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,
which increases with energy, is a clear signature f o r j e t s .
2.2
Inclusive Distributions
We s t a r t with a b r i e f description of the formalism f o r i n c l u s i v e production /33,34/ +
-
e e § h+x The hadron h is characterized by i t s mass m and i t s four momentum p = (E,~). The cross section f o r h production (when summing over the p o l a r i z a t i o n of h,e +, =
and e ) is described by two structure f u n c t i o n s , corresponding to the l o n g i t u d i n a l and transverse p o l a r i z a t i o n components of the v i r t u a l photon d2o ~ h I h 92 dXR~ ~ ~XR(-F1+@XRF2Sln 6)
(2.4)
with x R = 2 E / ~ . The structure functions Fh1 and Fh 2 depend on E,s, and on the type of h. At high energies, where m is small, one gets the scaling l i m i t for p a r t i c l e production via intermediate spin I / 2 partons: F~(E,s,m) § F~(x R) =~XRF2(XR) I h
(2.5)
sdo/dx R is then expected to scale as /33,34/ sd~/dx R § (1-xR)n/x R
(2.6)
The term I / x accounts f o r the r i s e of the mean m u l t i p l i c i t y with s. In the l i m i t E >>m, x R can be replaced by x = 2 1 ~ I / ~ .
Experimental r e s u l t s f o r d~/dx are shown
in Figs. 2.6 and 2.7 f o r the energy range 3.0 ~ /s ~ 7.4 and 5.0 ~ v~ ~ 31.6 GeV, respectively. In the low-energy region data from the SLAC-LBL /35/ detector has been chosen as a representative sample, the high-energy r e s u l t s come from the
I
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I
I
~
I
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~r
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Ec.m.=4.8 GeV
>~ Od~~Offs
::k
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I
I
x 5 GeV DASP 07.4 GeV SLAC-LBL e(3 GeV TASSO
=3.0 GeV
~17GeV TASSO m27.4-31.6 G.:'V
o~.!~'
Io' E- o
k
i
l,i ~ ~ T xx IFr t .
v
v
TASSO P r elimJnoty~
,p' .o
::L ,o ~
FO"I
0"10
I
0.2
1
I
0,4
o.6
o.
l
l.o
x : 2p/Ec.m. Fi9.2.6. Inclusive cross section sdo/dx vs s. Data from the SLAC-LBL detector /44/
I
0.2
I
x=
I
o.4
[
I
0.6
i
t
0.8
2p/Earn.
Fig.2.7. Inclusive cross section sdo/dx from DASP / 3 6 / , SLAC-LBL / 3 5 / , an~ TASSO /24/. The curves ( I / x ) ( 1 - x ) ~ and ( I / x ) ( 1 - x ) 3 shown f o r comparison are normalized at x = 0.2
TASSO detector at PETRA /24/. With the exceptions of the threshold region x ~ O(m//s) and eventually except the very lowest energy, a l l spectra scale in x w i t h i n the accuracy of the measurements. For f u r t h e r studies and to enable a comparison with the j e t s observed in hadronhadron c o l l i s i o n s , i t is convenient to use quantities r e f e r r i n g to the j e t a x i s , l i k e the r a p i d i t y y = ( I / 2 ) I n [ ( E + p , , ) / ( E - p , , ) ] , or the transverse momentum p . These q u a n t i t i e s are, of course, somewhat sensitive to the way the j e t axis is defined, the influence being strongest f o r very slow ~ I~I ~ 0 (300 MeV)] and very fast (x§
particles.
Figure 2.8 shows the r a p i d i t y d i s t r i b u t i o n per event, ( I / ~ ) ( d ~ / d y ) as measured by the SLAC-LBL /22/ and TASSO /24/ detectors, with respect to the s p h e r i c i t y (SLAC-LBL) and t h r u s t axis (TASSO). For the SLAC-LBL data, a fast p a r t i c l e (x > 0.3) in one of the j e t s is required in order to allow a r e l i a b l e determination of a j e t axis already at energies as low as 5 GeV. The density shown is twice the density observed in the hemisphere opposite to the fast p a r t i c l e . As shown in / 2 2 / , t h i s method introduces nearly no bias.
I0 The r a p i d i t y d i s t r i b u t i o n s show the f o l l o w i n g features: - the length of the r a p i d i t y i n t e r v a l populated increases as In(s) -
-
at high energies, the r a p i d i t y d i s t r i b u t i o n develops a plateau around y = 0 the height of the plateau seems to increase s l i g h t l y with increasing cms energy. For comparison, the r a p i d i t y d i s t r i b u t i o n observed for mesons produced in proton-
proton i n t e r a c t i o n s at ~ = 31GeV /37/ is included in Fig.2.8. I t is in surp r i s i n g l y good agreement with the high-energy TASSO data, taking i n t o account that in proton-proton i n t e r a c t i o n s a sizeable f r a c t i o n of energy is carried o f f by leading nucleons, so that as f a r as meson production is concerned, the protonproton data at ~
= 31 GeV should be compared with e+e- a n n i h i l a t i o n s at ~ 1 5 - 2 0
Ge~. There are indications that the height of the plateau is s l i g h t l y larger f o r the e+e- data. Furthermore, the d i s t r i b u t i o n s seem to develop a hole at y = O. The second e f f e c t may be due to biases introduced by the d e f i n i t i o n of the j e t axis. Figure 2.9 shows the d i s t r i b u t i o n of transverse momenta at ~ ~ 13-17 and ~
~ 7 /22/,
~ 30 GeV /24/. As a reference, the p~ d i s t r i b u t i o n of pions
produced in proton-proton c o l l i s i o n at ~
~ 14-20 GeV is shown / 3 8 / . Taking into
account that at the lowest energy transverse momenta above I GeV/c are damped due to energy and momentum conservation, a l l data in the range v~s ~ 7 GeV to ~
~ 20
GeV agree. Again there is no severe difference between hadrons produced in protonproton and in e l e c t r o n - p o s i t i o n c o l l i s i o n s . However, the p~ d i s t r i b u t i o n at ~ 30 GeV develops a strong t a i l
towards higher p 's, s t a r t i n g at p~ ~ 0.5 GeV2.
There is no correspondent feature observed in proton-proton i n t e r a c t i o n s ; in the p~ range studied here the slope of the p~ d i s t r i b u t i o n is nearly independent of f o r v~s > 15 GeV. We shall return to these phenomena in Sect.2.3. The dependence of the mean transverse momentum on x,, is shown in Fig.2.10 for two energies. Note that the shape of t h i s curve is sensitive to the way the j e t axis is defined, systematic changes at low and high x,, may be as large as 20%-30%. Like the corresponding proton-proton data, the d i s t r i b u t i o n s show a pronounced seagull e f f e c t at x,, ~ O.
2.3
Scaling V i o l a t i o n s at High Energies
In Sect.2.2 we found that in the energy range 5 < ~
< 20 GeV the i n c l u s i v e par-
t i c l e d i s t r i b u t i o n s in j e t s scale in x R and have a f i x e d , l i m i t e d p~ with respect to the j e t a x i s , the deviations from universal d i s t r i b u t i o n s at low x R and at large p~ being induced by phase space effects. At ~
~ 30 GeV, however, the p~ d i s t r i -
bution changes q u a l i t a t i v e l y , i t develops a t a i l towards larger p~, which must have
11 101~
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Fi9.2.9
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o.4 o'.G o.8 X,
Lo
-"
PP->iT+ § X
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o.2
o.,~ o.~ o:B ~.~ X,
Fig.2.8. Rapidity distributions for charged particles assuming m = m~, measured at v~ = 4.8, 7.4 /22/ and at 13,17,27.6-31.6 GeV /24/. The f u l l l i n e shows the d i s t r i bution of secondaries in i n e l a s t i c pp interactions at v~ = 31GeV /37/ Fig.2.9. Distribution of transverse momentawith respect to the sphericity axis /22,24/. The SLAC-LBL and the proton-proton data /38/ are normalized a r b i t r a r i l y Fig.2.10. Seagull plot
vs x,, from charged tracks /23/, compared to data from proton-proton interactions ~ = 6.8 and 14 GeV /38-40/
12
a dynamical o r i g i n (Fig.2.9). In the following we shall investigate t h i s e f f e c t in more d e t a i l . For the discussion, we w i l l mainly use results from TASSO /24/; simil a r results have been reported from the PLUTO /41/, MARK-J /42/ and JADE /43/ collaborations. The widening of the p~ d i s t r i b u t i o n can have the following origins: the production of a new quark f l a v o r ; the p~ d i s t r i b u t i o n for quark fragmentation is energy dependent, the average p~ grows as s increases; as in hadron-hadron interactions /6-11/, single hard scattering processes dominate the cross section at large p~ and s. Candidates for such processes are, e . g . , the emission of hard gluon bremsstrahlung under large angles by the primary quark /44,45/ or the production of meson resonances with large transverse momenta by constituent interchange mechanisms /46/. The f i r s t p o s s i b i l i t y can be immediately ruled out by other data /24/. The other two p o s s i b i l i t i e s can be discriminated by examining the phase space structure of the events: an increase of the fragmentation-p~ with energy s t i l l
yields p a r t i c l e
d i s t r i b u t i o n s which are symmetric in azimuth with respect to the j e t axis. Hard parton processes produce a t h i r d j e t , e i t h e r by gluon fragmentation in quark-gluon models or by the decay of the excited meson in the ClM model /46/. In most cases t h i s j e t w i l l be more or less aligned with one of the i n i t i a l
j e t s , and the cross
section of the cylinder w i l l be deformed into an e l l i p s o i d ; sometimes, a planar t h r e e - j e t structure w i l l be recognizable. In search for planar events, the TASSO collaboration assigned an event plane in such a way as to minimize the sum of the momentum components squared out of this plane /47/. Obviously, the j e t axis as defined via the minimum sphericity is contained in that plane. Figure 2.11 shows the mean transverse momentum squared in the event plane, IN' and normal to the event plane, oUT'- per event at ~ from 13 to 17 GeV and ~ at ~ a r o u n d 30 GeV /24/. By d e f i n i t i o n of the event plane, IN is larger than oUT" The comparison of the Iow-o and high-energy data proves that t h i s bias is not able to explain the t a i l at IN in the 30 GeV data. In f a c t , models with a fragmentation symmetrically in azimuth are not able to describe t h i s t a i l consistently. This observation is very much in favor of the l a s t hypothesis. In f a c t , events s i t t i n g in the t a i l of the IN d i s t r i b u t i o n show a t h r e e - j e t signature. Fig.2.12 gives a " t y p i c a l " example. The planar events were analyzed as t h r e e - j e t events. The d i s t r i b u t i o n of the transverse momentum with respect to the axis was measured for each j e t and was compared to the inclusive p~ d i s t r i b u t i o n of two-jet events at ~ =
12 GeV (Fig.2.13).
Obviously the events which lead to large
values when treated as two-jet events have the canonical
value of -0.3 GeV when analyzed as t h r e e - j e t events.
13 ~
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m,,
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0.2
0.2
0.4
0.6
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1.0
1.2
1J.
< p~> (OeVlc)2
F.!gt2.11
Fig.2.12
I
10
o
i
2-Jets
12GeV
t30OeV
Fig.2.11. Mean transverse momentum squared in and perpendicular to the event plane per event. The f u l l and dotted lines are predictions of j e t models using particle distributions symetrically in azimuth /24/ Fig.2.12o A three-jet event projected into the event plane /24/ Observed transverse momentum distribution for noncollinear planar events at ~ 3 0 GeV, when analyzed as three j e t events, compared to the inclusive p~ distribution of events at / s ~ 12 GeV, analyzed as two-jet events /24/
0.1
0.01 0
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Fi 9 . 2.
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14
Does the hypothesis of a hard-bremsstrahlung process explain the sudden onset of the e f f e c t when going from s = 17 to s ~ 30? The p r o b a b i l i t y f o r emission of a hard f i e l d quantum is given by /48/ w ~ const. -~S - In(s/N 2)
(2.7)
where a s is the strong coupling constant and N is a cut o f f parameter of the order I GeV. Rather slowly does w vary with energy. Two f u r t h e r circumstances are relevant however. F i r s t , a t h i r d j e t emitted at large angles can be recognized as soon as the mean l o n g i t u d i n a l momentum of fragments is larger than
. Since is roughly proportional to the j e t energy, t h i s c r i t e r i o n introduces a much f a s t e r v a r i a t i o n in the f r a c t i o n of events which can be i d e n t i f i e d as m u l t i j e t events. Secondly, there are two competing mechanisms governing the energy loss of a f a s t quark: hard bremsstrahlung and soft hadronization (although we shall see in Chap.5 that a s t r i c t d i s t i n c t i o n of the two mechanisms is not j u s t i f i e d ) .
Their r e l a t i v e importance de-
pends on the r a t i o of the corresponding time scales. As we shall note in Chap.4, the fragmentation time of the j e t is of the order
(2.8)
~frag- ~ TO 2
with m0 being a t y p i c a l hadronic mass scale. On the other hand, the p a r t i a l l i f e time f o r emission of a quantum carrying a f r a c t i o n z of the quarks momentum at an angle e is given by /48/
~
brems,
!
(2.9)
v~ z(1_z)sin28/2
From (2.8) and (2.9) follows that the r e l a t i v e importance of hard processes increases proportional to s. This explains the fast t r a n s i t i o n from the hadronisationdominated regime to the region where hard i n t e r a c t i o n s are obvious. A unique i d e n t i f i c a t i o n of the hard process as gluon bremsstrahlung is d i f f i c u l t at the present q u a l i t y of data; however, the main features of the data are cons i s t e n t with t h i s i n t e r p r e t a t i o n /49,50/. A more detailed discussion on hard processes in j e t fragmentation w i l l f o l l o w in Chap.5.
3. Jets in Longitudinal Phase Space Models
As we have seen in the last chapter, the physics of jets at small energies, up to v~s ~ 10-20 GeV, is dominated by phase space effects due to the nonzero transverse momenta and particle masses. To arrive at a quantitative description of the l i m i tations imposed by pure four-momentum conservation and to separate j e t dynamics from kinematics, i t is useful f i r s t to study a simple phase space model for j e t production. This section is organized as follows. We shall present the uncorrelated j e t model for e+e" annihilations, discuss the physical meaning of the parameters involved, and give the asymptotic behavior of the model. Next, scaling variables and the approach to the scaling l i m i t w i l l be discussed. In the remainder of the chapter, possible generalizations of the model are presented, e.g., the inclusion of resonance production and the use of more sophisticated matrix elements.
3.1
The Uncorrelated Jet Model (UJM)
In the UJM the f u l l y exclusive decay probability of a virtual photon of the four momentum Q = (~s-,0) into N particles is assumed to factorize as /51,52/
i:I Ti
f(p~'pi)
64(Zpi_Q)
(3. I)
neglecting spins and photon polarization. The four momentum of the i th secondary is Pi = (Ei'-Pi)' which for simplicity w i l l be assumed to be a chargeless pion. To arrive at final state jets, the invariant momentum space element d3_pi/Ei /53/ is weighted with a matrix element f depending on the longitudinal and transverse momenta with respect to the j e t axis defined by a unit vector p,, : ~-~; p : l~x~l
(3.2)
16 The ~-function in (3.1) accounts for energy-momentum conservation. As the n - p a r t i c l e matrix element factorizes into n independent p r o b a b i l i t i e s , the model contains no dynamical correlations. In the following we shall adopt the simplest choice for f , generating a transverse-momentum cutoff f(p,,,p~) ~ exp(-~p~)
(3.3)
Such models have been discussed by many authors; we shall follow the presentation of /54/.
In terms of the momentum space volume (the grand p a r t i t i o n function) N
~(~,Q) = Z NT. < ~N N=2
(3.4)
the inclusive s i n g l e - p a r t i c l e spectrum in the photon rest frame, normalized to the t o t a l cross section o, is given by (E/o)(d3o/dp 3) = Kexp(-kp~)~(~,Q-p)/~(~,Q)
(3.5)
with the t r i v a l sum rules (x R = 2E/V~)
I fdXRdp2 d3c~ I fdXRdp2XR d3o dXRdp2 = ;-~ dXRdP-~ ---- = 2
(3.6)
The physical significance of the parameters K and ~ is evident: ~-I determines the transverse j e t width and K characterizes the m u l t i p l i c i t y d i s t r i b u t i o n , an increase in K giving a higher weight to larger p a r t i c l e numbers. Asymptotically, one obtains
E d3o I I d2o ..... = Kexp(-~p~) dp3 o ~ dydp2 S § y fixed
(3.7)
Consequently, the p a r t i c l e density per unit of r a p i d i t y is 1 do
(3.8)
K'~ S ->co
y fixed leading to a mean m u l t i p l i c i t y
17 = ~ l n ( S ) S§
(3.9)
The asymptotic behavior of the inclusive x R d i s t r i b u t i o n
is given by (3.1) and
(3.5) i f we note that Q(~,Q-p) is a function of the particles transverse mass m and of the longitudinal missing mass ML ^ 2 I/2 / m ~ I/2 = v~V-XR§ ~ - / ML = [(vrs-E) 2 - (e~) ]
(3.10)
with the asymptotic l i m i t
M~~-2 ~(~,Q-p) ~ in(ML/m ) ~+O(In-1(ML/m~ )]
(3.11)
m y> s v ~ cosh ~-I
(3.18)
resulting in a r a p i d i t y plateau of length OEln(S)] with a shoulder width of roughly 2-3 units in y (Fig.3.1). One further feature of inclusive spectra in the UJM is worth noting. Equation (3.7) suggests a f a c t o r i z a t i o n of E d3o/dp 3 in terms of y and p . Due to the factor E in the i n v a r i a n t momentum space volume, t h i s does not hold i f one replaces y by other longitudinal variables as x or m,, = (m2+p~)I / 2 . This is c l e a r l y demonstrated in Fig.3.2, where
is plotted vs m,,. §
-
This "seagull e f f e c t " is observed in e e annihilations /24/ as well as in hadronhadron interactions /40/ and is often interpreted as a d i r e c t evidence for a parton
19
structure of the produced particles /56-58/. One should, however, be aware that part of the effect may be simple j e t kinematics.
/~
~--3
3
(GeV)
"13 b
-o
2
--~
~ ~
.3
e'~e
b
r
1 J 8
6
J /. y-InS
t
2
/
I ~ I 0 -2
F i g . 3 . 1 . Asymptotic r a p i d i t y d i s t r i b u t i o n s in the UJM (<m > = 0.35)
3.2
//
I
1
2
3 m. (GeV)
F i ~ . 3 . 2 . Mean transverse momentum o f secondaries as a f u n c t i o n o f the l o n g i t u d i n a l mass f o r Z = 6.2 GeV- I in the UJM
The Approach to Scaling
At typical SPEAR- or DORIS-energies, where the bulk of e+e- data was produced, masses and transverse momenta are not a priori negligible. The approach to scaling /54/ w i l l therefore be discussed in more detail. Obviously, the scaling l i m i t of do/dxR is reached latest for xR close to 0 and I. At x R ~ <m >/v~, scaling is violated because of threshold effects; in the v i c i nity of xR ~ I (for In -1[(S/4m2)(1-xR~ I )
the unobserved system has a low missing
mass ML (3.10) and hence ~(~,Q-p) (3.5) is not asymptotic. Qualitatively, scaling in do/dxR w i l l be approached from below at low xR and from above at high xR. To study the approach to scaling quantitatively, i t is more convenient to calculate ~(~,Q) by numerical integration /54,59,50/. The results for (I/a)(XRdO/dxR) are compared in Fig.3.3 for v?s = 3.0, 3.8, 4.9 and 20 GeV with the asymptotic l i m i t , using the "standard" choice o f parameters ~ ~ 6 GeV- I ,
o r
~ 0.33 GeV, and
= 3 / 5 4 / . Although t h e r e seems t o be an e a r l y s c a l i n g a l r e a d y a t v~ = 3-5 GeV f o r x R > 0.2, the c a l c u l a t i o n shows t h a t one i s s t i l l
f a r from the s c a l i n g l i m i t ,
which is reached w i t h i n O (10%) a t V~ = 20 GeV f o r 0.05 < x R < 0.8.
2O 6
/. o
~ - 13 \
,/ \ \ \ \ ~ - ~ :
VS = 3,0 GeV 38 6ev
X
I 0.2
0,4
0.6
0.8
XR= 2E/V~-
Fig.3.3. The i n c l u s i v e s i n g l e - p a r t i c l e spectra in the UJM f o r ~ = 6.2 GeV- I , = 3 /54/
1.0
; T t III
5
10
I
I
20 V"s (5eV)
I
I Y Is,f...
50
co
Fig.3.4. Normalized p a r t i c l e densit i e s at y = 0 in the UJM. ( ~ ) d i r e c t production of ~ , ~ = 3 . ( - - - ) production via p decay, K = 1.5 /61/ P
Figure 3.4 shows the height of the r a p i d i t y plateau at y = 0 as a function of v~ /61/. At energies of /s = 3-10 GeV, the p a r t i c l e density is ~30% below the asympt o t i c value of ( 3 . 8 ) ; even at the highest energies available at present accelerators (ISR) the UJM predicts the plateau height not to be constant. Another nonasymptotic e f f e c t , namely the influence of the production of heavy p a r t i c l e s l i k e K or p, becomes evident i f one t r i e s to compare the UJM model with + -
data from e e
storage rings.
Of course, the UJM is not able to give a r e a l i s t i c description of the charge, spin or isospin structure of e+e - f i n a l states. However, with appropriate constants K and ~ i t should describe the gross features of i n c l u s i v e spectra averaged over p a r t i c l e types, quantum numbers, etc. In Fig.3.5a, the UJM prediction for ~ = 3 and = 6.2 GeV-I is shown together with i n c l u s i v e ~• spectra (s/6)(do/dx R) in e+e- ann i h i l a t i o n s at v~ = 4.46-4.90 GeV /36,62/. The UJM values have been evaluated f o r v~ = 4.8 GeV using numerical techniques. There is a strong disagreement between the predicted and the measured x R slope; furthermore, as we have seen in Chap.2, the high-x R pion cross sections scale over the whole DORIS/SPEAR and PETRA energy range, in contrast to the model, In p r i n c i p l e , the f i r s t
point could be cured simply by increasing ~ (3.8). This,
however, leads to m u l t i p l i c i t i e s
incompatible with experiments (3.9).
21 e*e'~ '
A
'
'
Tc-*X '
I
'
'
e+e - ~ ,
, - - - ,
,
,
K*-X
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. . . . . . . . .
h*X i
,
,
, ,
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DASP ~ " = .',.46- 4 90 6#r
DATA UJFI
GcV
027.4
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u)
I
GcV
I
MPP
i
i
i
i
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i
x~: 2 E~I',F~
J
,
,
,
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,
,
,
i
i
i
Q5
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i
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i
i
os
xR:2ExlJs"
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0.1
i
xR : 2 E I ' , / ~
Fi9.3.5
I
r
O. 2
0.4
0.6
0.8
X : PIPBEA M
Fig.3.6.
Fig.3.5a-c. Comparison of UJM predictions ( ) with data from e+e- storage rings /36,62/. (a) Inclusive ~• spectrum; (b) i n c l u s i v e K• spectrum; (c) UJM compared to the sum ( . . . . . ) of the 7 • (a), K= (b) and p,~ spectra, a f t e r subtraction of charm c o n t r i b u t i o n s . The normalization is a r b i t r a r y +
-
Fig.3.6. Comparison of the asymptotic l i m i t of the UJM with data from e e a n n i h i l a t i o n s at ~ ~ 17 /24/. The UJM parameters are ~ = 3, X = 6.2 GeV- I , as in Fig.3.5. The normalization is a r b i t r a r y
There is a more natural explanation possible. F i r s t , at V~ = 4.8 GeV charm production contributes a sizeable f r a c t i o n to the pion cross section below x = 0.2, /36/ which has to be subtracted. Second, in the UJM a l l p a r t i c l e s were assumed to be pions, whereas in practice, also kaons and protons are produced. At moderate v~, the rest masses of these p a r t i c l e s eat up a good f r a c t i o n of the available energy, r e s u l t i n g in a steeper decrease of pion spectra at high x. A better;(however not f u l l y l e g i t i m a t e ) way of comparison is to add up the pion (Fig.3.5a), kaon (Fig.3.5b), and proton spectra (Fig.3.5c).
I f one f u r t h e r notes that systematic uncertainties
between d i f f e r e n t experiments /35,36,62/ are of the order 2 at high x R, the agreement is quite impressive. The early scaling of the experimental pion x R spectra can thus be interpreted as a cancellation of two nonscaling effects: the production of "heavy" p a r t i c l e s and the nonasymptotic ~(~,Q). At higher energies, v~ ~ 17 GeV, the UJM is in rather good agreement with data +
from e e
-
annihilations (Fig.3.6).
22 3.3
The UJM and Short Range Correlations
The UJM discussed in the previous chapters describes the production of i d e n t i c a l p a r t i c l e s without any short range c o r r e l a t i o n . There i s , however, strong evidence from experiments studying p a r t i c l e production in hadronic i n t e r a c t i o n s /63/ that the f a c t o r i z a t i o n property of the t r a n s i t i o n matrix N
fN(~1,~2 . . . . . PN,S) = ~ f ( P i , S ) i=I
(3.19)
holds only in the l i m i t of large i n v a r i a n t masses (pi+Pj) 2 >> I GeV2 whereas low-mass p a r t i c l e systems show strong c o r r e l a t i o n s . Part of these correlations are explained by the c l u s t e r model / 6 4 / , which states that the p r i m a r i l y produced e n t i t i e s are not the f i n a l state p a r t i c l e s , but instead are excited states which in turn decay i s o t r o p i c a l l y i n t o stable p a r t i c l e s . The most natural way to enclose short range correlations in the UJM is to assume that the intermediate states, called c l u s t e r s , are emitted independently. In a second step, charge, energy and momentum conservation in the c l u s t e r decay provide low-mass c o r r e l a t i o n s . Experiments suggest that the bulk of pions is produced via intermediate vector mesons /65/. As an example, l e t us take the p meson as a representative " c l u s t e r " . Figure 3.7 shows UJM r e s u l t s f o r emission of p mesons decaying i n t o two pions compared to d i r e c t production in the scaling l i m i t S§
The p meson density is chosen
such as to give the same plateau height as f o r d i r e c t emission. Figure 3.7 demonstrates that the i n c l u s i o n of c l u s t e r decays does not a l t e r very much the basic UJM p r e d i c t i o n s , as f a r as i n c l u s i v e spectra are concerned. However, the introduction of the new ( c l u s t e r mass) scale f u r t h e r delays the approach to the scaling l i m i t /61,66/. A second source of correlations is related to the quantum nature of secondaries /67/. As f a r as the emission of i d e n t i c a l p a r t i c l e s is concerned, the i n v a r i a n t momentum space representation of the t r a n s i t i o n matrix (3.1) is not f u l l y adequate / 6 8 / , since these f i n a l state p a r t i c l e s obey Boltzmann s t a t i s t i c s .
In r e a l i t y , pion
production f o r example is governed by Bose-Einstein (BE) s t a t i s t i c s ,
leading to an
" a t t r a c t i o n " of l i k e - s i g n pions, since two pions " l i k e " to be in the same quantum state. The range in momentum space of t h i s type of dynamical c o r r e l a t i o n can be calculated by noting that two p a r t i c l e s are in the same quantum state i f
23 1o 5
2
~
--
direct n
- [I from?
),
.1 ~5
0
t
I
.2
.4
X
I
I
.6
.8
1.0
Fig.3.7. x d i s t r i b u t i o n s in the UJM for d i r e c t , and resonance production of pions. The matrix elements are chosen to give identical heights of the r a p i d i t y plateau
I(~I-E2)IR ~ 9 ~ I , or I~i-P2 [ ~ 200 MeV R is the uncertainty of the point of emission, or in other words, the radius of the p a r t i c l e source, which is expected to be of the order I fm ~ (200 MeV)- I . An a l t e r n a t i v e derivation of this result has been given by COCCONI /69/ and by KOPYLEV and PODGORETZKI /70/ using higher-order interference effects in analogy to optics. Such a correlation modifies the usual phase space weights in such a way as to favor events which have high p a r t i c l e numbers in identical states. At large rapid i t i e s or transverse momenta, this tendency is balanced by the f i n i t e amount of energy available; therefore, the enhancement of inclusive densities is strongest at low p~ and y /71/. Models of this type have been used to explain the s l i g h t rise in do/dy at y = 0 in the ISR energy range /72/. Nevertheless, one should keep in mind that conventional phase space models also predict a nonasymptotic behavior at those energies. Anyhow, these results should be taken q u a l i t a t i v e l y rather than q u a n t i t a t i v e l y . In r e a l i t y , a l o t of physically d i f f e r e n t particles are produced, which of course show no BE a t t r a c t i o n . Furthermore, the model assumes that a l l particles are produced in the same volume R4 in space-time because otherwise the particles could be labeled by t h e i r production coordinates (within the l i m i t s imposed by the uncertainty r e l a t i o n ) and hence were not i d e n t i c a l . We shall see l a t e r (Chap.4,5), that in quark j e t s , for example, a BE interference is nearly impossible because of this l a s t argument.
24 3.4
Summary
The uncorrelated j e t model for independent emission of particles enables us to study the main implications of four-momentum conservation in j e t production. The explicit addition of short-range correlations is possible via resonance or cluster production. Irrespective of details of the model, and nearly independent of the matrix elements involved, the asymptotic scaling l i m i t is f a i r l y well described by
XR do = 2~(1_XR)~-} The approach to scaling is rather slow, even at PETRA or ISR energies the scaling l i m i t is not f u l l y reached.
4. Jets and Parton Models
In the l a s t decade, the quark-parton model / I - 4 , 7 3 / has proven to be one of the most useful and i n t u i t i v e concepts in high-energy physics. In the quark-parton model, hadrons are composed of partons. The binding forces of partons are governed by time scales of the order T ~ I/m with m being a typical mass of a few hundred MeV. In a moving hadron, the time scale is d i l a t e d by the Lorentz factor y. Then we have two main classes of hadron interactions in the quark-parton model. I f the i n t e r a c t i o n time TI of hadrons is large compared to yT, a l l partons p a r t i c i p a t e coherently in the i n t e r a c t i o n . I f ~I is small compared to u
i n d i v i d u a l partons may
i n t e r a c t . In the asymptotic l i m i t T I § 0 the hadron seems to be composed of free, on-shell quarks. The consistency of the model requires the partons to carry f r a c t i o n a l charges, and a new quantum number, color /17,74,75/. Since no f r a c t i o n a l charged objects have been observed up to now, there must be a nonasymptotic mechanism confining partons to integer-charged states with the observed hadron quantum numbers. These states turn out to be color singlets. Meanwhile, i t became customary to turn these arguments upside down and to consider color confinement as the primary mechanism, and the nonexistence of f r a c t i o n a l l y charged p a r t i c l e s as a secondary consequence. Consider now a process where a large four momentum is transferred to one patton out of a color s i n g l e t system. Obviously, due to the large i n v a r i a n t mass of the parton f i n a l state, color cannot be confined to a single stable hadron anymore. One of the basic assumptions of the quark-parton model is that in such a case the confinement of color leads to a bunch of p a r t i c l e s moving e s s e n t i a l l y along the direction of the scattered parton and that t h i s process happens with the p r o b a b i l i t y I . In t h i s chapter, we shall discuss the dynamics of j e t formation in more d e t a i l , +
s t a r t i n g with the simplest type of j e t s : quark jets produced in e e
-
annihilations.
26 4.1
Jets from Quark Confinement
Confinement is most e a s i l y v i s u a l i z e d in terms of a QCD-supported bag model /76/. QCD - Quantum Chromodynamics - /17/ is the most promising candidate for a theory of strong i n t e r a c t i o n s . I t is a nonabelian gauge theory in which the interactions between colored quarks are mediated by 8 colored gauge bosons, the gluons. The 3 color charges are generators of an unbroken SU(3) c symmetry. QCD is an asymptotic a l l y free theory, i t s dimensionless coupling constant mS/~ tends to zero as the energy increases, allowing a perturbative expansion. At low energies, Q2 ~ 0 (I GeV2), the coupling constant diverges. This low energy behavior of QCD is hoped (but not yet proven) to explain confinement: recent f i e l d t h e o r e t i c a l i n v e s t i g a t i o n s i n d i cate that the QCD ground state is a n o n t r i v i a l two phase vacuum /77/. In the normal phase, outside hadrons, color f i e l d s cannot propagate. This e f f e c t is known as "chromodynamic Meissner e f f e c t " in analogy to the propagation of magnetic f i e l d s in a superconductor. There is f u r t h e r an abnormal phase which may contain quarks and gluons as quasi-free p a r t i c l e s . Since surface and volume energy is required to create the abnormal phase, the abnormal regions form l i t t l e
bubbles, or "bags"
w i t h i n the normal phase. The dynamics of quark and gluon f i e l d s inside a bag is governed l o c a l l y by the f i e l d equations of QCD. I t can be shown that such a bag, embedded in the physical vacuum, is stable in i t s time e v o l u t i o n ; the volume energy B (or vacuum pressure) and surface energy S are balanced by the pressure exerted by the gluon f i e l d s which are reflected at the phase boundary /76/. B and S can be arranged to give t y p i c a l bag sizes of I cubic fermi. Such a bag containing a quark and a antiquark or three quarks can be i d e n t i f i e d as a meson or as a baryon, resp e c t i v e l y . A c t u a l l y the spectrum of bag e x c i t a t i o n modes reproduces the observed
[Gev-'] ~R
, [Gev-'] gr 1
~ tube
2
3
4
The shape of one quadrant of a typical bag /76/ for different quark-antiparations. The arrow indicates the radius of an ideal cylindrical vortex
27 hadronic mass spectrum f a i r l y well /76/. What happens, i f a large amount of momentum is transferred to a quark inside a bag? As the quark moves away, the bag changes i t s shape from a sphere to a cylinder (Fig.4~
and the kinetic energy of the quark
is converted into surface and volume energy of the bag. When the bag reaches a c r i t i c a l length, i t is energetically more favorable to create a new quark-antiquark pair in between the i n i t i a l
quarks. These new quarks screen the i n i t i t a l
color
f i e l d s and the bag breaks up into two new, spherical bags, each containing one initial
and one new quark. One of these bags is a slowly moving ground state bag,
while the other, which contains the fast quark, starts to become c y l i n d r i c a l , repeating the process u n t i l the whole i n i t i a l energy is converted into a series of bags moving along the direction of the i n i t i a l momentum transfer. The transverse momentum with respect to the j e t axis of these bags is determined by the transverse bag size.
~ 0(I fermi - I ) ~ 0(200 MeV)
4.2
Space-Time Development of Quark Jets
The confinement mechanism proposed above, via successive polarization and de-excitation of the vacuum has been investigated already in 1962 by SCHWINGER/78/. He showed that in certain gauge theories of charged fermion f i e l d s , the only asympt o t i c a l l y stable particles are massive neutral bosons. In such theories, e l e c t r i c charge is a confined quantum number, in analogy to the (yet unproven) color confinement in QCD. This phenomenon occurs in two-dimensional (space-time) QED. I t may also happen in four-dimensional gauge theories i f the coupling constant exceeds a certain c r i t i c a l value /78/. Examples of f i e l d theories exhibiting the Schwinger phenomenon have been discussed by CASHER et al. /79/. In the following, we shall use two-dimensional QED /78,79/ to give a quantitative description of the process of j e t formation. Of course, we are aware of the two essential l i m i t a t i o n s of this picture, the use of only one space dimension and the absence of photon s e l f couplings in QED. On the other hand, i f we keep the a t t i t u d e of the QCD bag model discussed in the previous section, the nonabelian nature of QCD is mainly responsible for the chromodynamic Meissner effect which in turn reduces the problem to a one-dimensional one, the dynamics of a color string. We shall proceed as follows. F i r s t , j e t development is discussed using the formalism of two-dimensional QED. Next, we t r y to transform these results into an in-
28 t u i t i v e physical picture. F i n a l l y , a simple graph technique is presented, which further i l l u s t r a t e s the mechanisms of j e t development and enables quantitative predictions. I+I dimensional (space-time) QED is defined by the Lagrangian density
-" l y ~ ~ - LF L = @ 4 NvFNv-g~YN@A
(4.1)
This model is exactly solvable. I t has already been mentioned that in this model there are no asymptotic fermion states; the only stable p a r t i c l e is a boson of mass m m = g/~
(4.2)
The equation of motion in the presence of external current sources is given by (~+m2)j~(x) = -m2j~xt(X)
(4.3)
Let us consider the simplest case, the production of a q-q pair by a v i r t u a l photon. The external currents in the photon rest system (which correspond to color currents in QCD) are defined by
Joext = g6(z-t)+g6(z+t) J l e x t =-gS(z-t)+g$(z+t)
(4.4)
(In our notation, x is a two-vector with the components x 0 = z, x I = t ) . I t is now convenient to express the current in terms of a dipole density r j~ = ~
r
(4.5)
in analogy to the well known equations 2r , J =-~--~ 2r p =-~-~
(4.6)
The external dipole density is then
@ext : -ge(t+z)8(t-z) and the induced polarization density s a t i s f i e s
(4.7)
29
(D+m2)@ = gm20(t+z)O(t-z)
(4.8)
The resulting dipole density is constant on hyperbolas in space-time, vanishing near the l i g h t cone and approaching a constant for distances Ix[ >m-I from the o r i gin. As the hyperbolas approach the l i g h t cone, the regions containing the polarization charge are confined to a length of the order (tm2) - I . The polarization charge combines with the outgoing charge to form a neutral boson as soon as t h e i r distance becomes of the order m-I in t h e i r common rest frame. In the cms frame this happens a f t e r a time t
C
t c ~ FQQ2/m2
(4.9)
The existence of the induced dipole density is equivalent to the creation of chargeanticharge bound states. The momentum d i s t r i b u t i o n of these bosons can be calculated from the f i e l d @. One obtains f i n a l l y 2 I E~-~=d~~d-yl do _ (2~) 2 1 [fd2x exp(ipx)g@ext 12 =--~g = I
(4.10)
The time evolution of the main quantities @, do/dy and p is summarized in Fig.4.2.
(~
P
C
/
J
\
I Z
ZlYl
t = t3 t = t2 t = t1 t=O
Fig.4.2a-c. Development of charge confinement in the Schwinger model. (a) Lines of constant polarization density in space-time. Particles are produced in the shaded regions. (b) Spatial d i s t r i b u t i o n of the polarization density at d i f f e r e n t times ( O > t I > t 2 > t 3 ) . This is equivalent to the rapidity d i s t r i b u t i o n of the emitted particles. (c) Density of charge at d i f f e r e n t times. (The i n i t i a l quarks are represented by delta functions.)
How can we interpret these results? Particle production happens in the cloud of polarization charge, which is induced by the primary charge, i . e . , color sources. Particle creation starts at low r a p i d i t i e s .
As time goes on, the i n i t i a l
quark feeds
30 energy into accelerating the polarization cloud u n t i l the cloud overtakes the leading quark and neutralizes i t s charge. One should note that the motion of the polarization cloud proceeds through the creation of new quark pairs in front of the cloud and through the recombination of quark pairs into neutral bosons at t h e i r end; i t need not be a unique quark in the cloud which is accelerated. The process of charge confinement (or "quark fragmentation") turns out to be characterized by a rather long time scale (4.9). Furthermore, the cascade extends over large distances in space, as compared to typical p a r t i c l e sizes (Fig.4.2); the extension of the cascade is proportional to the quark energy. The maximum distance between the quark charge and the screening polarization charge, however, stays f i n i t e at a l l times (Fig.4.2c). There is one difference between QCD and QED model worth noting. In QED, the charge of the leading quark is constant during the whole process. In the nonabelian QCD, the colors of quark and polarization cloud change due to gluon exchange, the quark color being opposite to the cloud color in the average. The whole system -
quark, cloud, and gluons between them -
is a color singlet.
The p r o p o r t i o n a l i t y between quark momentum and confinement time is important since i t ensures that at high energies the outgoing fermions remain free s u f f i c i e n t l y long to j u s t i f y calculations based on the naive parton model. This model has f i r s t are produced f i r s t
been advocated by BJORKEN /80/. Since the slowest hadrons
in time, i t is known as an "inside-outside" cascade.
Of course i t is questionable to what extent d e t a i l s of the model should be taken seriously and whether i t can be generalized to four dimensions. I t has been shown that i f the Schwinger phenomenon occurs in the four-dimensional f i e l d theory, the model gives a r a p i d i t y plateau and an appropriate p~ cutoff /79/. The problem is~ however, to prove that the Schwinger mechanism works, e . g . , in QCD. Nevertheless, CASHER et al. /79/ succeeded to prove that the inside-outside evolution of the cascade is a feature common to a l l fragmentation models describing color confinement. Their argument is quite simple. Imagine a model where the cascades s t a r t at the two i n i t i a l
quarks (e.g., via a multiperipheral chain of low momentum transfers).
To neutralize t h e i r residual charges, the two cascades have to j o i n somewhere in themiddle. Since the fragmentation chain represents a random walk,in transverse momentum space and due to the long time scales involved, the p r o b a b i l i t y that the ends of the two cascades overlap in space-time goes to zero as Q2 goes to i n f i n i t y . Quarks of energies >> I GeV would not be confined in such a model. Recently, ANDERSSONet al. have presented a semiclassical model for quark-jet fragmentation /81/, which incorporates a l l features of the Schwinger model and gives a reasonable description of the j e t s observed in deep i n e l a s t i c lepton scattering
31 /83/. In their model, quarks are treated as classical particles, which move on well-defined trajectories in two-dimensional space-time. I t is easily seen that the classical treatment is j u s t i f i e d as far as longitudinal motions along the j e t axis are concerned /84,85/. The relative momentum smearing due to the uncertainty relation dy ~ dp ~ ~ . p - Rp '
R ~ quark bag radius
(4.11)
is small i f fast quarks are considered, e . g . , r a p i d i t y and position in space of a quark are well defined simultaneously. Furthermore, one may speculate that i n t e r ference effects are suppressed since the various p a r t i c l e production points w i l l turn out to be well separated in space-time. The use of only one space dimension is based on the assumption of a "QCD-Meissner e f f e c t " ; more precisely i t means that no transverse degrees of freedom of the elongated quark bags ( " f l u x tubes") are excited /86/. Since there is only one space dimension, the force acting between two oppositely charged (or colored) partons is independent of t h e i r distance and is a t t r a c t i v e . In the zero-mass l i m i t , the world lines of the two valence quarks inside a meson are described by Fig.4.3a: The quarks sta;rt to separate, losing momentum at a rate dp/dt = •
(4.12)
After a certain time, the motion is reversed, the quarks move together, start to separate again and so on. Using the uncertainty relation to relate the mean quark separation and momentum, one obtains for the mass of the meson ground state /81/ m ~ gl~ 2 The constant g is related to the Regge slope m'via g = 2/m' /85/, or can be taken from linear potential charmonium models /82/, yielding meson masses of the order of a few hundred MeV. In a moving meson, the period of oscillation becomes timedilated (Fig.4.3b). In this model, j e t formation is visualized in diagrams like Fig.4.4. Two i n i t i a l quarks move in opposite directions. As soon as a certain energy is stored in the color f i e l d (shaded area), the f i e l d breaks up somewhere and produces a quark pair. The break up can be visualized as a classical tunneling effect /85/: somewhere in the color f l u x tube a virtual quark-antiquark pair is generated by a fluctuation.
32 a
~
Fig.4.3a~b. World l i n e s of quarks inside a meson (a) meson at r e s t ; (b) moving meson
•
z
•
~
Fig.4.4. Space-time development of a quarkantiquark cascade
A
The pair moves in a potential 2 E = -2ET+~-~d
(4.13)
22 where ET is the transverse energy ET = /Cp +m ) of each of the new quarks, and d is the separation between quark and antiquark. This second term arises since the new quarks screen the external color f i e l d in between them. The pair is assumed to be created with zero net momentum since a system with negative t o t a l energy cannot propagate with real momentum components / 8 5 / . I f the quarks succeed to tunnel from d = 0 to d = 8~ET/g2, t h e i r energy becomes p o s i t i v e and they m a t e r i a l i z e as two on-shell quarks. In the WKB approximation the p r o b a b i l i t y per u n i t time per u n i t volume to produce on-shell pairs with transverse momentum PT is given by /85,86/
33
g2 dp~
--
8~4
/
(4.14)
As soon as kinematically allowed, new pairs are produced in the region between the i n i t i a l quark and a quark of the f i r s t pair, etc. Quarks and antiquarks produced at different points C and D in space-time may join to form a meson E. This is possible, i f at the time where quark and antiquark meet in space, the q~ invariant mass equals the meson mass, or equivalently
(Zc-ZD)2-(tc-tD)2 = m2(4--~)2 g
(4.15)
Equation 4.15 determines the minimum space-time distance of the production points of quarks appearing in f i n a l state mesons. As a consequence of (4.15), p a r t i c l e s are emitted from a hyperbola-like spacetime region as in the Schwinger model. Two remarks for the sake of completeness w i l l be made: In p r i n c i p l e i t is possible that a quark pair has a large invariant mass m >> I GeV. This would s p l i t the event into four separate jets. This processes w i l l be rare; i f the coupling constant is large enough, quark pairs are created and recombined as soon as possible. -
Figure 4.4, as well as (4.12-14) are s l i g h t l y oversimplified. Actually quarks are produced in three colors, only one of which screens the external color f i e l d . However, i t has been shown that the production of the "screening" color is favored and one may argue that the production of other colors leads to a configuration which f i n a l l y discharges the color f i e l d via baryon production /85/.
On the average, the slowest particles are produced f i r s t . There is, however, no s t r i c t mapping between the rapidity and the time of emission of a particle. Only the average production time increases with rapidity. Such statements on rapidities and times are not Lorentz invariant. Usually they w i l l be invalidated by a change of the reference frame. ANDERSSONet al. /81/ have investigated this point in detail. Their conclusion is that these statements are true in any reference frame, proving the self consistency of the model. Define now DbC(z) as the density of mesons with the valence quark flavor b~ in a the fragmentation region of quark a. z is the fractional momentum of the meson. Masses and transverse momenta are neglected for the moment. Averaging over a l l possible diagrams like Fig.4.4, the model yields a set of coupled integral equations for the D's
34
~
=
+
I dz'
bc z fdDd (~T)
(4.16)
f
denotes the p r o b a b i l i t y f o r a color f i e l d to create a pair of quark flavors cc. c The break up is assumed to happen with equal p r o b a b i l i t y anywhere between the quarks generating the f i e l d , otherwise f depends on z or z ' . Color and spin degrees of freedom are i m p l i c i t l y summed over, with the r e s t r i c t i o n that bE is a color s i n g l e t . S i m p l i f i e d to the case of one f l a v o r ,
(4.16) reads
I D(z) : 1+f--~,dz: D(~,) z Equation 4.13 is e a s i l y solved to give D(z) = ~I or ~1- }d~ = I
(4.17)
Equation 4.17 holds only for production of quarks with n e g l i g i b l e constituent masses and transverse momenta. The i n c l u s i o n of these effects changes the behavior of D(z) from D(z)+1 to D ( z ) § z§ z+l The reason is e a s i l y seen. For nonzero masses, quark and antiquark of a pair mater i a l i z e at a distance d ~ ET (4.13). However, a break up cannot happen with equal p r o b a b i l i t y anywhere along a f l u x tube. A minimum average distance d/2 between the center of a pair and the end of a f l u x tube is required, y i e l d i n g a reduced number of large-momentum secondaries. One must also keep in mind that the mesons described by (4.16,17) are not the observed stable hadrons. According to the spin weights, 3/4 of them are vector mesons decaying into a number of pions. This results in a smearing of the r a p i d i t y d i s t r i b u t i o n , i t f u r t h e r increases the height of the plateau. Adjusting the f ' s (4.16) to give a suppression of heavy-quark production, as predicted on the basis of (4.14) /178/
fup = fdown = 0.4; fstrange = 0.2; fcharm = fbottom = "'" = 0
(4.18)
(4.16) can be solved numerically, taking into account resonance decays, etc. The r e s u l t s are in rough agreement with the data (see, e . g . , F i g . 4 . 5 ) , although recent experiments indicate that the amount of SU(3) breaking (4.18) is s t i l l what underestimated /162,179/.
some-
35 i
=
i
~
I
(~ 2 (ep~eltaX) .~it"
~ l/2(e*e---'l~"X)
2 1 N 0..: -:1-:
,-,:1-o,0.; 0.!
--.-- Anderssonet cI.L
I
-- Field and Feynman --- Sehga[
0.05
[ ",,X,~, ~ I
'
7 0.2
I 0.~
~
o!6
'
o'.8
'
1.0
Fi~.4.5. Predictions of various j e t models /86,87,90/ compared to quark fragmentation functions measured in deep i n e l a s t i c lepton-nucleon reactions /91/
z
4.3
An Algorithm f o r Simulation of Quark Jets
To provide a standard j e t model representing the state of the a r t , FEYNMANand FIELD /87/ have proposed a s i m i l a r j e t model based on parton phenomenology. The model uses a recursive generation p r i n c i p l e o r i g i n a l l y introduced by KRZYWICKI and PETERSSON /88/ and by FINKELSTEIN and PECCEI /89/. The j e t model is intended to serve as a reference for design and comparison of experiments. I t generates exclusive m u l t i p a r t i c l e f i n a l states including resonance production, transverse momenta and f i nite-energy effects. Furthermore, i t allows detailed predictions on two- and morep a r t i c l e correlations and on t h e i r quantum-number dependence. The main difference to the model discussed above is that the Feynman-Field (FF) model involves four free parameters adjusted to f i t
the data: I ) one a r b i t r a r y function u l t i m a t e l y de-
termines the momentum d i s t r i b u t i o n of the hadrons, 2) the degree that SU(3) is broken in the formation of new quark-antiquark pairs, 3) the spin of the primary mesons and 4) the mean transverse momentum given to t h i s mesons. In the o r i g i n a l version of the model / 8 7 / , the production of baryons and of heavy-quark f l a v o r s , l i k e charm or beauty is neglected. The ansatz is based on the idea that a quark of type "a" coming out at some momentum PO creates a color f i e l d in which new quark pairs are produced. Quark "a" then combines with an antiquark, say "6", from the new pair "bb" to form a meson "ab" leaving the remaining quark "b" to combine with a next quark "E" and so on. The primary meson "ab" may be d i r e c t l y observed as a
36
stable meson, or i t may be an unstable resonance which subsequently decays. A hierachy of primary mesons is formed in which "ab" is f i r s t
in rank, "b~" is second in
rank, "cd" is t h i r d in rank, etc, as shown in Fig.4.6 (here and in a l l f u r t h e r diagrams we have adopted the convention to plot antiquarks as quarks moving backwards along t h e i r world l i n e s ) . The chain decay ansatz assumes that i f the primary meson of rank I carries away a momentum ~ I ' the f u r t h e r cascade s t a r t s with a quark "b" with the momentum Pl = PO-~I and the remaining hadrons are d i s t r i b u t e d in the same way as i f they came from a j e t originated by a quark of type "b" with momentum PI' I t is assumed that the momenta are high enough, so that a l l d i s t r i b u t i o n s scale.
FINAL STATE
RESONANCE DECAY
FORMATION OF PRIMARY MESONS
QUARK PAIR PRODUCTION
INITIAL QUARK
Fi9.4.6. Production of a cascade via successive q~ creation and recombination
Given these assumptions and defining f ( z ) by f ( z ) d z = p r o b a b i l i t y that the rank I quarks momentum, with ~f(z)dz = I ,
I primary meson carries a f r a c t i o n z of the i n i t i a l
the l o n g i t u d i n a l structure of the quark j e t is uniquely determined. For t6e single p a r t i c l e density of primary mesons (regardless of i t s rank) I do D(z) : ~ dz
with
Z
-
Pmeson P0
we have the i n t e g r a l equation 1 dz' ~,. D(z) : f ( z ) + f ---~-i-rtl-Z )D(~-r) i
Z
(4.19)
37 (assuming t h a t ZpO >> m~). Generalized to many-quark f l a v o r s ,
oTC/z :
aDc
czl+ d
(4.19) reads
z TT
[ ( i n the n o t a t i o n of ( 4 . 1 6 ) ] . The f i r s t
term is the p r o b a b i l i t y f o r the meson to have rank I , the second term
arises a sum over a l l higher ranks w i t h the rank I meson being at z'
Feynman and
Field assume t h a t f b ( z ) (the p r o b a b i l i t y t h a t a rank I meson "ab" is formed at z) can be f a c t o r i z e d as f o l l o w s fb(z) = f~.f(z)
(4.20)
where I ff(z)dz = I 0
(4.21)
and ~ f [u = I b
(4.22)
The breaking of the SU(N) f l a v o r symmetry is put in the f ~
i d e a l l y one has in
SU(N) o I fb = Guided by experiments, Feynman and Field use fo = fo = 0.4; fo = 0.2; fo = fo up down strange charm bottom = " ' " = 0
(4.24)
in agreement w i t h (4.18). In contrast to the model by Andersson e t a l . , t h e mesons is kept as a free parameter. The best f i t p r o b a b i l i t y f o r spin 0 mesons p r o b a b i l i t y f o r spin I mesons probability
for
spin
= 0.5 P = a = 0.5 v
spin s t r u c t u r e of the primary to the data is quoted f o r
= a
~ 2 mesons = a T = 0
(4.25)
38
Of course, the smaller f r a c t i o n of vector mesons requires a change of f ( z ) compared to / 9 1 / . Feynman and Field use f ( z ) = (1-a)+3a(1-z) 2 with a ~ 0.77
(4.26)
As in (4.14), i t is assumed that the new quark pairs have no net transverse momentum. Transverse momenta ~
and -p~ are assigned to the quark or antiquark of a p a i r ,
with a d i s t r i b u t i o n 1 do ~ exp~p~/(2p~)] o dp2
(4.27)
The transverse momentum of a primary meson is given by the vector sum of the transverse momenta of i t s quarks. To reproduce the observed mean transverse momentum of the f i n a l state pions, P+O ~ 350 MeV is required. The mean transverse momentum of primary mesons is of the order of 440 MeV. The recursive scheme of j e t generation is now evident: I)
choose the momentum f r a c t i o n of the rank I meson from (4.26)
II)
generate a quark pair according to (4.24,27). The rank I meson is made of the
III)
decide on s p i n - p a r i t y of the meson (4.25). Use the known pseudoscalar and vec-
old quark and the new antiquark t o r - m i x i n g angles to determine the precise type of the meson. I f necessary, l e t i t decay. IV)
I f there is s t i l l
enough energy l e f t over, repeat the procedure s t a r t i n g with
the residual quark. At t h i s stage, a few remarks concerning the philosophy of the model have to be made. Regarding the recursive p r i n c i p l e , one is tempted to consider the Feynman-Field j e t as an " o u t s i d e - i n s i d e " cascade, where p a r t i c l e s of rank I are produced f i r s t
at high
y in the overall cmso The authors themselves considered t h i s point as one of the maj o r drawbacks of t h e i r model / 8 7 / . A comparison of the integral equations (4.19) with the equations describing the inside-outside cascade of ANDERSSONet a l . (4.16) however proves that both point of views lead to very s i m i l a r mathematical structures, the (4.16) being as well compatible with an inside-outside development of the j e t .
The recursive p r i n c i p l e proposed by Feynman and Field thus should not be
considered as a physical model, but merely as a mnemonic s i m p l i f y i n g the bookkeeping of momenta, quantum numbers, etc.
39
A second remark is more technical. The FF model describes the fragmentation of a single quark. Consequently, one single antiquark with i t s charge, color and (small) momentum has to be thrown away at the end of the cascade a f t e r n mesons have been produced. In r e a l i t y , these q u a n t i t i e s have to be neutralized by the corresponding ones in the opposite j e t . This unbalance of conserved quantities is not a s p e c i f i c f a i l u r e of the FF technique to generate j e t s , instead i t is a r e f l e c t i o n of the fact that i t is meaningless to t a l k about a single j e t . The only systems of physical relevance are color s i n g l e t s , l i k e q~ or qqq states. A d i v i s i o n of such systems into single jets is an approximation possible for fast p a r t i c l e s , i t is however completely meaningless and a r b i t r a r y for slow fragmentation products.
4.4
Properties of Quark Jets
In t h i s section we shall discuss the expected properties of quark j e t s , and compare them to the experiment. The FF model w i l l be used as the typical parton-jet reference. The j e t energy should be chosen such that mass effects are n e g l i g i b l e , on the other hand i t should be low enough so that the QCD corrections to be discussed in the next section w i l l not dominate. These conditions are f u l f i l l e d
in the lower
PETRA energy range. Consider the fragmentation of u,d, and s quarks (the decay of charmed quarks w i l l be discussed l a t e r ) . As "stable" f i n a l state p a r t i c l e s we have u,d,s
§
+, O, - ; K+,KO,K-, KO',
In p r i n c i p l e , there are 3•
T
(4.28)
fragmentation functions (plus the corresponding ones
for antiquarks). I h Dq(Z) - ~
q
dOh dz
(4.29)
Parton models allow to reduce the number of independent D's d r a s t i c a l l y . E s s e n t i a l l y , there are two main fragmentation mechanisms, usually known as favored and unfavored fragmentation /90/. A fragmentation is favored i f the fragment contains the i n i t i a l quark. In the language of Feynman and Field i t may then be the ( f a s t ) meson of rank I. In the unfavored case, where the meson does not contain the i n i t i a l
quark, i t has
to be of rank 2 or higher, implying a smaller mean momentum. Since the f l a v o r of the quarks generated in the f i r s t
break up of the color f l u x tube should not depend on
the nature of the primary quark, a l l unfavored D's should be equal, except f o r a
40 constant suppression factor ~ for strange secondaries [according to ( 4 . 1 4 ~ . Favored fragmentation functions are obtained by adding an universal d i s t r i b u t i o n DO of rank I mesons to the corresponding unfavored fragmentation function Dunf , e . g . , Dr DKu = Dunf; u = ~Dunf +
Dr
u
+
= Dunf+Do; D
= ~(Dunf+Do)
Of course, these expressions are only approximations because of the presence of a second mechanism f o r "unfavored" fragmentation: the decay of vector mesons, e . g . , 0
+
-
u § p +anvthina § ~ ~ +anvthina which competes with the d i r e c t suppressed channel. A l l t h i s is more e a s i l y demonstrated in Table 4.1, where the f r a c t i o n of momentum carried by the various decay products is given, and by Fig.4o7, where the favored and unfavored fragmentation functions of a u-quark into ~'s and K's are shown as predicted by the FF model.
ioo
\.
-....\
\
~
u
\-'O.I
W> 4GeV
012 -
4
Fig.4.8, u-quark fragmentation functions measured in neutrino nucleus interactions /93/ compared to predictions of the FF model and of a phase space model (LPS) Fig.4.9a,b. Distribution in z for the fastest and the second fastest charged hadrons, compared with FF and LPS in (a) ~N and (b) ON i n t e r actions /94/
0 I~ --0'12-+0"13 1 ;; ............
0 -0,2
-T .........
.
1
~
0.007 0,02 0.05 0,14 0,37' z : a p, i~,
-
Fig.4.10. Distribution of the charge dQ/d In(z) plotted vs In(z) for u- and dquark fragmentation /95/. The curve is from Feynman and Field
43
vation. The m u l t i p l i c i t y
has been constrained to the observed one. Obviously, much
of the difference observed between the "favored" and the "unfavored" d i s t r i b u t i o n of p o s i t i v e and negative hadrons is accounted f o r by simple charge conservation. Even the increase of the difference with z is reproduced. Events with a p a r t i c l e close to z ~ I have a low missing mass and therefore a low m u l t i p l i c i t y ;
effects due to
charge conservation are more important than in average events. A more d i f f e r e n t i a l q u a n t i t y , the d i s t r i b u t i o n of the fastest and second fastest charged hadron, is shown in Fig.4.9 /94/. The FF model as well as the naive phase space model agree with data. F i n a l l y , Fig.4.10 i l l u s t r a t e s the net charge density in u- and d-quark jets as a function of the r a p i d i t y Yz = I n ( z ) /95/. The integrated charge content of the jets is u = 0.55•
and d = -0.12•
for u- and d-
quarks, respectively. The FF model has the tendency to reproduce the observed asymmetry between and . To summarize so f a r , the FF model seems to be in reasonable agreement with data from deep i n e l a s t i c reactions even in d e t a i l s ; however, the comparison with the l o n g i t u d i n a l phase space model proves that most of t h i s agreement is required by conservation laws, once the m u l t i p l i c i t y d i s t r i b u t i o n is c o r r e c t l y adjusted. Let us now study in more d e t a i l the quantum-number structure of quark j e t s . We have seen in Fig.4.10 that quark jets tend to r e t a i n the charge of the quark at high r a p i d i t i e s . This is the object of a suggestion by FEYNMAN/ 4 / . He conjectured that the t o t a l quantum numbers of a l l the fragments in t h i s region averaged over events should be equal to the ( f r a c t i o n a l ) quantum numbers of the parent quark. A necessary, but not s u f f i c i e n t condition f o r retention of quantum numbers is that the rank I mesons stay at large r a p i d i t i e s in the j e t . This is e a s i l y demonstrated. Assume we cut the j e t somewhere, e . g . , at z 0 = 0.01 and sum up the quantum numbers of a l l p a r t i c l e s with higher z. In nearly a l l cases the i n i t i a l
quark is contained
among these mesons. Yet, since we s t a r t with a quark and count an integer number of mesons, there is always one quark pair whose quark is below z 0 and whose antiquark is above. I f z0 is low enough, t h i s quark pair w i l l stem from the sea. The sum A of quantum numbers f o r z > z 0 is then given by
= A +
q
(4.30)
s
Aq describes the quantum numbers of the i n i t i a l
quark (or antiquark) and are
the mean quantum numbers of an antiquark (or quark) from the sea. Exact quantumnumber retention happens only i f = O. Let us consider a few examples: a SU(2) S
sea consisting of u , u , d , d ; a SU(3) symmetric sea; and the FF quark sea with SU(3) broken due to the higher s-quark mass. We get for S
44
Sea
Charge
SU(2)
Strangeness
Baryon number
-I/6
0
-I/3
0
0
I/3
-I/3
0
-I/15
I/5
-I/3
0
SU(3) SU(3)FF
3 rd Component o f isospin
(4.31)
In none of these models are a l l quantum numbers retained. For the FF sea, we expect approximate r e t e n t i o n of charge and I z. The mean quantum numbers of t h e i r j e t s are
Inital quark
Mean j e t charge
Mean Jet strangeness
Mean I Z
u
0.60
0.20
0.50
d
-0.40
0.20
0.50
s
-0.40
0.80
0.00
(4.32)
o.04Fa 9 , . . . . , . . . . , . . . . , . . . . , . . . . .
oooF . . . . . .
-o.o2 -O.04F .... 9 0.04 o ....
=
, ......... ,I .... 2 ....
-~176 _O,04J ~ . . . z . .... .1 .... ,~ ....
o.o4'
Y_
1
_z. . . . . . . . . . . . . . ,~ .... ,4 .... ~ ....
1 6
Y
1
............. .3 .... ,4 .... ,5 ....
:l
c
0.02
# ~ , ' : ~'T~::1
0.00 -0.02 -0.04
1
0
' 1
' 2
Yz 3
4
5
F i g . 4 . 1 1 a - c . D i s t r i b u t i o n o f (a) charge Q, t o t a l Q = 0.6, (b) t h i r d component o f i s o s p i n I 3 , t o t a l 13 = 0.5, and (c) strangeness S, t o t a l S = 0.2, along the Yz[Yz = - I n ( z ) ] a x i s f o r a u-quark j e t
45 These numbers are confirmed by the e x p l i c i t Monte Carlo simulations of j e t s /87/, the results for quantum-number density in quark jets are shown in Fig.4.11. The most remarkable consequence of these considerations is that even i f a j e t starts from a nonstrange quark, i t f i n a l l y acquires a net strangeness, which has to be compensated by the opposite antiquark j e t . Furthermore, in any j e t mechanism leading to a r a p i d i t y plateau (that means the decay function f ( z ) (4.16) is f i n i t e for z § O) only the high-y region of the j e t is influenced by the parents quantum numbers, and the plateau e s s e n t i a l l y stays neutral.
4.5
Summary
Based on the quark-parton model, the space-time evolution of a quark j e t can be visualized as an "inside-outside" cascade. Independent of the reference frame chosen, slow fragments are produced e a r l i e s t . The production of leading fragments, or equivalently the f i n a l neutralization of the primary color charge occurs a f t e r a time T proportional to the quark momentum Q. A quantitative description of t h i s process can be given using an analogy to two-dimensional QED, or a semiclassical quark model. The recursive scheme for j e t generation proposed by Feynman and Field is consistent with these models; i t produces the measured inclusive p a r t i c l e dist r i b u t i o n s in quark j e t s , and the data on correlations, f a i r l y well. One should note, however, that much of this agreement results simply from constraints due to four-momentum and charge conservation and can be obtained as well in an UJM (Chap. 3), once the mean m u l t i p l i c i t y is correctly chosen.
5. Parton Jets and QCD
The quark-parton model was o r i g i n a l l y developed to provide a useful and simple desc r i p t i o n of the physics of deep i n e l a s t i c phenomena / I - 4 , 9 6 , 9 7 / . The modern foundation for the parton model is the gauge theory of strong interactions based on the color degree of freedom: QCD. In QCD scaling of the deep i n e l a s t i c structure functions, which describe the quark d i s t r i b u t i o n within the nucleon, is predicted to be broken by logarithms of q2 = _Q2. Physically, these scaling v i o l a t i o n s are due to the emission or absorption of gluons during the hard-scattering process. Thus although the naive parton model, s t r i c t l y speaking, f a i l s , i t is possible to rephrase the parton language by assigning a well specified Q2 dependence to the parton densities /98/ G(x) ~ G(x,Q 2)
(5.!)
Already through the principle of analytic continuation of the scattering amplitude a Q2 dependence of the structure functions, describing the d i s t r i b u t i o n of partons in a hadron, also automatically induces a Q2 dependence of the fragmentation functions, which parameterize the d i s t r i b u t i o n of hadrons in a parton /99,100/. The dynamical mechanisms leading to this scale breaking are sketched in Fig.5.1. +
"Before" the quarks emitted, e . g . , in e e
-
annihilations, reach the surface of the
confining bag, they radiate gluons which may in turn convert into new quark-antiquark pairs. The bulk of low-momentum gluons is reflected at the bag surface and compensates the "vacuum pressure". Gluons and quarks in the high-momentum t a i l of the radiation however deform the surface of the bag and create a number of incoherent f i n a l state j e t s . The "parton shower" inside the bag is described by QCD, whereas the f i n a l confining stage has to be described by phenomenological models l i k e the FF j e t . Here we shall follow the h i s t o r i c a l approach and f i r s t discuss the mechanisms of scale breaking for the example of structure functions. F i n a l l y , we generalize the results to the case of fragmentation functions.
47
!NT
5.1
Fig.5.1. The perturbative QCD scale breaking in parton j e t s through gluon emission and conversion. Full and wavy lines refer to quarks and gluons, respectively
Scale Breaking in QCD
The formal and t h e o r e t i c a l l y rigorous d e r i v a t i o n of scale breaking in QCD is f o r mulated in the language of renormalization group equations for the c o e f f i c i e n t functions of local operators in the light-cone expansion for the product of two currents /I01/. A more i n t u i t i v e model for scaling v i o l a t i o n s in scale i n v a r i a n t theories has been given by KOGUT and SUSSKIND /44/ and we shall f o l l o w t h e i r lines of argument a t i o n /44,102,103/. We assume that matter organizes i t s e l f into "clusters" /104/. For example, molecules are made of atoms which are made of nuclei which are made of nucleons and so on. Each c l u s t e r is characterized by a certain size and time scale. The r e l a t i o n between these scales appears to be accidental. However, as smaller and smaller scales are probed in high-energy physics some kind of regular i t i e s may emerge; certain f i e l d theories suggest that the connections between adjacent scales may become universal. Consider time and length scales of ordinary hadrons, 10-13 cm, and denote those as N = 0 clusters which in turn are composite of N = I clusters (partons). Renormalization group investigations suggest that the dynamics of clusters of type N which form a c l u s t e r of type N-I can be described by a Hamiltonian HN without explicitly
refering to smaller clusters (N+I, for example). In an i n f i n i t e momentum
frame, HN depends on the f r a c t i o n a l momenta x of clusters of the type N and on t h e i r transverse momenta. The i n t u i t i v e picture suggested by WILSON /105/ and POLYAKOV / I 0 6 / was that at large N the coupling constant describing parton-parton interactions becomes constant and thus the r a t i o of t y p i c a l scales RN+I/RN = ~ approaches
48 a constant at large N. In that case, the transformation connecting the Hamiltonians HN+1 and HN is also independent of N. Imagine now an experiment, which probes hadrons with p a r t i c l e s of large four momentum q. Assume the q is spacelike. Then a reference frame can be chosen such that q consists only of a three vector, q = (0,9). Such a p r o j e c t i l e w i l l be able to resolve structures of the order of i t s wavelength ~ = Q-I = (_q2)-I/2. As the "target" hadron moves with roughly the speed of l i g h t , the time scales probed are of the order Q-I as w e l l . The parton d i s t r i b u t i o n which is relevant for the desc r i p t i o n of the process is that of clusters with RN ~ ~, since clusters of the type N+I cannot be resolved by the probe, whereas clusters of type N-I no longer appear p o i n t l i k e . This means ~ Ro6N
(5.2)
or, with ~ ~ Q-I : N ~ -In(RoQ)/In(6). To calculate the x d i s t r i b u t i o n of clusters of type N w i t h i n the o r i g i n a l hadron I dON GN(X) - o h dx
(5.3)
we introduce the function hN+1,N(Z) which gives the p r o b a b i l i t y per u n i t z of f i n ding a c l u s t e r of type N+I and l o n g i t u d i n a l momentum zx' in a c l u s t e r of type N and l o n g i t u d i n a l momentum x ' .
Invariance under l o n g i t u d i n a l boosts requires that
h depends only on z and not on the absolute momenta of the clusters. We neglect transverse momenta for the moment being. The d i s t r i b u t i o n of type N+I clusters is given by the convolution
I dx' GN(X ' )
GN+I(X ) = f ~ x
(x
hN+I,N~ )
(5.4)
We shall now loosen the assumption of the existence of discrete scales; real f i e l d theories may not contain such abrupt t r a n s i t i o n s between scales. A better descript i o n is given using a smooth connection between d i f f e r e n t N and Q. With the replacements
GN(X)
+ G(x,t)
hN+I,N(Z)
+ h(z,t)
3G(x,t) GN+I(X)-GN(X) + 21n(I/~) ~t
(5.5)
49 and t = In(Q2/Q~)
(5.6)
we obtain the following equation for G(x,t)
G(x,t) = } dx' ~t x' [G(x',tlh(
21n(6-I)
,tl-G(x
X !
',t)6(I-~-)]
(5.7)
x or with the redefinition p(z,t) = [h(z,t)-6(1- 89
-I)
(5.8)
(5.7) reads simply ~G(x,t) } dx' x , ~t 7 p(~T,t)G(x ,t)
(5.9)
X
Equation (5.9) describes the number density of partons in an object which is probed at a momentum transfer t = In(Q2/Q~). ~ It can be generalized to the case were the probe is only sensitive to the distribution of partons carrying a set of quantum numbers i
~Gi(x't)
~ 7dx' -ZI
~t
x
pi~(~,t)Gi(x,,t) j
J
(5.10)
Pij is the transition matrix between the various species of partons. To solve (5.9) i t is convenient to rewrite i t in terms of moments I
G(~,t) = f d x x ~ - I G ( x , t ) 0
(5.11)
and I
p(~,t) = f d x x~ - I p ( x , t ) 0
(5.12)
Equation (5.9) reads now
@G(~,t) @t = p(~,t)G(~,t) resp. for (5.10)
(5.13)
50 ~G.(~,t) 1 ~t - ~ Pij(a,t)Gj(~,t) J
(5.14)
The structure function G(x,t) can be obtained from i t s moments via the inverse Melf i n transform or by approximative numerical methods /107/. Let us now t r y to concatenate this picture and QCD as a description of parton dynamics. We i d e n t i f y a subnuclear "cluster" as a quark-parton, which, when probed at a higher q 2 looks l i k e one o f f - s h e l l quark surrounded by a cloud of gluons and quark-antiquark pairs. The coupling between quarks and gluons tends to zero as Q2 increases; QCD is an aymptotically free theory /74,75/: as(Q2 ) ~
1 b!n(q2/A 2)
(5.15)
with b = 33-2_~f 12~
(5.16)
f is the number of active flavors. For further calculations we write the t r a n s i t i o n p r o b a b i l i t y p as a product of the e f f e c t i v e coupling constant ~s(Q2)/2~ and a term depending only on the momentum ratios
as(t) p(x,t) = 2~ p(x)
(5.17)
Equation (5.13) 8G(a,t) _ as(t) ~t T p(a)G(a,t) is now e a s i l y solved: 2 2 ~p(a)/2~b [a(0)IP(a)/2~b = G (a) |In(Q /A )I G(~,t) : G 0 ( a ) [ ~ j
(5.18)
Thus QCD leads to logarithmic v i o l a t i o n s of scaling. In real QCD, we have not simply one parton-structure function G but instead the structure functions qi and g of 2f quark flavors (i = u~u,d,a . . . . ) and the gluon. Summing over a l l color states, we get from (5.10):
51 ~qi(x,t) = ~-~ x x'
3t
Pij(~-)qj (x (5.19)
ag(x,t)
~t
= ~-~
~--
x
. pgj(~)qj(x
[j
These are the well known ALTARELLI-PARISI master equations /98/, I t must be pointed out here that the master equations describe the t dependence of the parton d i s t r i bution function in momentum space. They do not refer to the time development of parton densities during the scattering process, The p functions can be evaluated using standard graph techniques /98/ (Fig.5.2). In the leading log approximation all terms containing powers in ~sln(Q 2) are summed up; the direct terms in Pii and pgg are then given through momentum conservation.
.
.
.
.
12 2
Fig.5.2. F i r s t - and second-order contributions to the p functions
-[ l+xz]
4 l+x 2 +26('1-x)~ Pij : 311-x ]+ i j ij 3 1+(1-x) 2 Pgi - 4 x (5.20) Pig = 89 x2+(1-x)2] pgg = 6 [ _ ~ +
I I 6(x-I )I (~T~-)+x(1-x)-~
-~f~(x-1
)
+
In terms of moments, (5.19) yields a system of coupled differential
equations which
can be p a r t i a l l y diagonalized by choosing suitable combinations of the parton den-
52
s i t i e s : the flavor sing!et and flavor nonsinglet (valence) components qS(x,t) and qnS(x,t)
q
ns
(x,t) = Z qi(x't)-Z qi(x't) u,d,s..o G,d,s... (5.21)
qS(x, t)
= Z
qi(x,t)+Z
qi ( x , t )
u,d,s Define now G as the vector of the t h
moments of the parton densities
G(~,t) = lqnS(~,t),qS(~,tl,g(~,tli
(5.22)
The master equations read then
-~G(~,t)
aS = ~A(a)G(a,t)
(5.23)
with the matrix A(~)
[
Pqq(~)
0
A(~) = I
0
Aqq(~)
|l
0
Agq(~)
0
1 (5.24)
2fAqg(~) Agg(~)J
The coefficients A etc. are combinations of the moments of the p functions. The qq t dependence of the nonsinglet component is obvious, i t is given by (5.18). Equation (5.23) has been shown to describe the measured nucleon-structure functions, once higher-order corrections and bound-state effects are taken into account /108/. We are now prepared to discuss the QCD scale breaking of fragmentation functions /109/. Following again the ideas of KOGUT, SUSSKIND /44/ and POLYAKOV/106/, we +
-
assume that in a deep i n e l a s t i c reaction or in e e annihilation,
a parton of type
N ~ -In(RoQ)/In(6) is emitted. This parton "decays" now in a series of N-I type partons which decay into N-2 type partons, etc. Consequently, the distance of the partons four momentum from the mass shell increases with N /106/.
53
In close analogy to (5.4-10) we get I dx'
x
GN_I(X) = ~ ~7--GN(X)kN-I,N(xT)
(5.25)
~G(x,t) _ } dx' r(~,t)G(x',t)
(5.26)
or
~t
x ~7-
Since we are s t i l l
dealing with quark and gluon interactions governed by QCD, the
r-functions in (5.26) are identical to the p-functions in (5.19). With the d e f i n i tions of (5.21,22) we arrive at - ~-~G(~,t)
= ~ aSA ( ~ ) G ( ~ , t )
For s i m p l i c i t y ,
(5.27)
we write the moments of the parton-fragmentation functions D as a
matrix as well, the element D~(~,t) describing the decay of the j t h parton component (nonsinglet, singlet, gluon) into the hadron i ( ~ + , ~ ~
The moments of
the final state hadron density are then ~(~) : D(~,t)G(~,t)
(5.28)
There are now two ways to describe the fragmentation of a quark at tq, as sketched in Fig.5.3. The fragmentation function D(~,tq) gives a direct mapping of the parton d i s t r i b u t i o n at t q
PARTON
HADRONIC
REPRESENTATION
FIHAL STATES
OF TWE J E T
cs.ol olx,t,l 0
o. I
Fig.5.3. The fragmentation of a quark at tq can be considered as a mapping of the point G(x,tq) = [6(1-x),6(1-x),O] onto a hadronic final state o h . This can be done in two steps: G(x,tq) is f i r s t mapped onto G(X,tl); next G(x,t I) is mapped onto ~h
54 G(x,tq) : ~(1-x),~(1-X),0] or
G(m,tq) = (1,1,0)
(5.29)
onto the hadronic f i n a l state. On the other hand, G(m,tq) can be described in terms of a superposition of partons at t I , 0 ~ t I ~ tq. The r e l a t i o n between G(m,tq) and G(m,t I) is given by (5.27). The hadronic f i n a l state can now be evaluated by applying D(m,tl) on G(m,tl). Of course the f i n a l state d i s t r i b u t i o n function has to be independent of t I . So we obtain
at I
[D(m,t 1)G(m,t I)] = 0
or, with (5.27) aS ~-~ D(~,t) = T~ D(~,t)A(~)
(5.3o)
To solve (5.30) one needs a boundary condition at a reference Q2. We could use the FF j e t model at Q2 ~ 100 GeV where QCD effects should be negligible. The explicit
calculation of the scale breaking in parton jets is however complicated by
the fact that a physical quark or gluon is never a pure f l a v o r nonsinglet. Through the s i n g l e t term in the quark density G(~,tq) quark and gluon terms become mixed up. Thus to find the quark fragmentation function, one has to know as well the gluon fragmentation functions at the reference Q2. To a r r i v e at a more quantitat i v e understanding of the process, we shall consider the parton densities G(x,0) 2 2 within a j e t i n i t i a t e d by a quark at Q2 >> Q0~ Q0 is chosen as low as possible (of course, the v a l i d i t y of the perturbative QCD expansion has s t i l l
to be guaranteed).
That means, we observe the "parton shower" i n i t i a t e d by a primary quark at a point j u s t before the nonperturbative conversion into hadrons starts. De GRAND /110/ has shown that in the high Q2 l i m i t the d i s t r i b u t i o n qnS (favored nonsinglet component), unf (unfavored component, e . g . , antiquarks) and g (gluons) from (5.27) may be
q
approximated by
55
qqns (x ,0) x§
(l-x) 16~/3-I ~>0.05
q~nf(x,O) ~ (l_x) 16r
(5.31)
x§ gq(x,O)
~ (l-x) 16~/3 x§
~>0.05
The subscript q refers to the nature of the parent parton, ~ is defined by
Obviously, as Q2 goes to i n f i n i t y , a l l parton (and hadron) d e n s i t i e s concentrate a t x = 0. The r a t i o of the various components stays constant. Compared to the valence quark, gluons and sea quarks are suppressed by f a c t o r s ( l - x ) and ( l - x ) 2 r e s p e c t i v e l y . Thus the leading p a r t i c l e e f f e c t , which means the dominance a t high x of p a r t i c l e s containing the o r i g i n a l quark is not destroyed by the QCD c o r r e c t i o n s . The f i n a l hadron spectrum is given by the convolution of (5.31) with the fragment a t i o n function Dh(x,0). Therefore, the steepening of the leading hadron d i s t r i butions occurs a t the same r a t e as in q~S. Comparing t y p i c a l DORIS and PETRA ener/
g i e s , we get
x dO)Q2 ~ (I_x)0.5 x (d_~)Q2 (d--~ ~ 1000 GeV2 ~ ~ 10 GeV2 x§
5.2
(5.33)
Preconfinement
I f one compares the QCD results on parton fragmentation (5.30) with the results of naive confinement models (Chap.4), a paradox seems to emerge. In QCD, the decay 2 of a parton at Q2 into many partons of lower QO is entirely governed by the color charge of the fragmenting parton. The pictorial flux tube connecting the two color sources plays no role at a l l , a second source is not even needed! Furthermore, the QCD " f i n a l " state at t = 0 contains a number of color t r i p l e t and octet sources, which are assumed to decay independently according to fragmentation functions D(x,O), whereas naively one expects a system of color f l u x tubes joining these sources.
56
The origin of these problems is the following. The confinement and QCD picture apply to different stages of the process and even to different kinematical regions. The naive confinement picture mainly addresses the question how do nearly on-shell partons transform into hadrons to form a rapidity plateau; whereas QCD describes the evolution of fast partons, which are far off shell within the j e t . Nevertheless, the question remains how do the two different approaches, the confinement model at Q2 ~ Q~, ~ and the QCD picture at high Q2 join smoothly. Recently, KONISHI et al. /111/ have proposed a generalization of (5.30) to describe the t-dependence of n-parton cross section in jets. They describe the parton fragmentation as a branching process /112/; an exact derivation of their method has been given by KIRSCHNER/113/ using the leading log approximation. In close analogy to the model of KOGUTand SUSSKIND /44/, i t is shown that in the QCD evolution of a j e t , partons of size QK decay ("branch") into two partons of size QK-I" The process may be visualized as real decay in spacetime /114/. The lifetime ?K associated to the propagator of a patton in the Kth generation is /114/ (5.34)
?K = XKQ/Q~
in agreement with the naive expectation from the uncertainty relation, r K = y/AE, with y = XKQ/QK and AE = QK" From (5.34) we get the total time for j e t development
r ~ ~
Q----2 K-I 2
K=I QK
= Q~exp [In(Q2/Q~)] I/2 2
-
(5.35)
QO
The factor 2K-I in (5.35) arises since the Kth generation contains 2K-! partons. This result is in qualitative agreement with that obtained from the naive fragmentation model of Sect.4.1. The j e t development, by branching of a parton of mass QK into two partons close in mass instead of decay by on-shell bremsstrahlung, is a specific feature of the parton formfactor in QCD. One gets 2 Q K + I exp I_2 ~ n(Q~/Q~)] I/21
(5.36)
One of the most important advantages of this scheme is that i t allows to calculate the distribution of color sources in the final state. AMATI and VENEZIANO /115/ calculated the distribution of color among the n final partons of mass QO' with the astonishing result that already perturbative QCD provides a "preconfinement" of
57 color. They discovered that these "final state" partons are grouped into colorless clusters in a number s u f f i c i e n t to "exhaust" the final state, but s t i l l possessing 2 a f i n i t e average mass of the order QO" The result is peculiar of QCD, in particular of its nonabelian nature. Let me b r i e f l y sketch t h e i r derivation.
In the axial gauge
and at the leading log level, the flow of color during the degradation of a high Q2 2 quark into partons at QO is determined by planar diagrams, which means that color flow lines must not cross. Nonplanar diagrams are suppressed by I/Ncolo r factors. This is visualized by Fig.5.4. Note that a quark corresponds to a single color l i n e , whereas a gluon can be displayed as a color t r i p l e t - a n t i t r i p l e t
state corres-
ponding to two color lines. Obviously, i t is always possible to group the " f i n a l state" partons into nonoverlapping, colorless clusters. Partons inside each cluster are nearby in the branching-tree structure of the j e t . Consequently, the clusters have f i n i t e masses, as demonstrated in Fig.5.5.
QUARK GLUON ".
ALLOWED PROCESSES ., .~--J" _ ~
q"
q~ g
g-
q'~
g_
g§
FORBIDnEN
>
(~S/~)In(~)In(2~).
q
q
HARD
GLUON
SOFT OLUON
Fig.5.9. Diagrams contributing to the transverse width of QCD quark jets in f i r s t order of aS /123/
Extensions of this formula to a wider range of parameters and calculations in higher orders have been published by several authors /125-127/. The main results of (5.41), however, remain unchanged: neglecting the effects due to confinement, QCD jets are well collimated in angle. At / s = 7 GeV one obtains 70% of all events having more than 80% of their energy within a double cone of half angle 130. At increasing energy and constant fractions, this angle decreases as E- I / 4 . Thus, the conjecture that jets look narrower at higher energies is supported by QCD. The rate of shrinking of the opening angle is, however, much less than expected for fixed p,. jets. At f i n i t e energies, (5.41) does not describe the final state hadrons. There, the nonperturbative smearing of transverse momenta has to be included. The interplay of the two components is demonstrated in Fig.5.10, which shows results of Monte Carlo calculations taken from /124/. In Fig.5.10 the half angle of a double cone containing 90% of the energy in a fraction f of all events is plotted as a function of Q2. Dotted and dashed lines denote the contributions from the nonperturvative and perturbative components alone, respectively. One recognizes that below Q2 ~ 1000 GeV2, QCD effects are small; above Q2 ~ 104_105 GeV2, one has nearly pure QCD jets.
5.5
Gluon Jets
Up to now, we have dealt with the simplest type of jets, e.g., quark-antiquark jets resulting from the confinement of a color t r i p l e t - a n t i t r i p l e t system. However, color confinement is, of course, not restricted to color t r i p l e t s ; any attempt to separate nonsinglet color systems w i l l produce jets. In QCD the second basic colored state besides the t r i p l e t is an octet, the gluon.
64 0.01
,......C......{] . / f f .o.3 ,",',,"
F a
o 4s
J"
o O,I
t ~
~ ..
,,
f=O,7
.," .." .. 6 GeV
W > 4 GeV
t
0.1
t
ill,
0.1
4
i
tm:
0.01
0.01
0.01
b i
I
i i I
I
i
i
I i I
i
i
n
IO
i
IO0 q2 GeV 2
0.1
iI
I
I
r
I
,I
i
i
!
I0
q 2, GeV z
Fig.6.15a,b. Nonsinglet moments Dns(n,Q 2). = D+(n,Q2)-D-(n,Q 2) vs Q2 for m = 2-7 /164/. (a) Averaged over all W. The curves show f i t of the QCD formula (5.33) to the data points above Q2 = I GeV2; (b) for W > 4 GeV
86
10
LnM~
2~0
~
3'0 --~ [GeV]
~
n
-I
-2
-3
.... -
-
QCD nonper
7 turbative
FIB.6.16. Q2 dependence of the logarithms of moments from nonperturbative quark fragmentation with respect to the scaling variable x F compared to perturbative QCD p r e d i c t i o n calculated with AZ = 0.5 GeVz. Horizontal l i n e s indicate the scaling l i m i t s of the nonperturbative moments /165/
Clean "beams" of one quark species can be produced in the f o l l o w i n g reactions: a) neutrinoproduotion
~N §
I- + u + x
~N § l + + d + x f o r x > 0.I b) eleotroproduction
Ip § 1 + u + x for x § I
where 1 refers to an electron or muon. The cuts in x are necessary to suppress i n t e r actions with sea quarks in a). The l i m i t x § I in b) makes use of the f a c t that f o r x § I a proton consists of pure u-quarks, in addition to the f a c t that the u-quark cross section is enhanced by a f a c t o r 4 compared to d-quarks because of i t s charge. The main experimental l i m i t a t i o n s of the study of quantum number retention comes again from the low masses of the hadronic f i n a l state which induce a considerable s p i l l o v e r between the target and the current fragmentation region. This is evident from Fig.6.17. The t o t a l negative charge in the d-quark fragmentation region in iN reactions increases with W, since then the contamination due to p o s i t i v e fragments of the target nucleon becomes less important /166/. The real mean charge of quark fragments was derived from Fig.6.17 by extrapolation l i n e a r in W-I to W = ~ (Fig. 6.18). The r e s u l t , = - ( 0 . 4 6 •
is in good agreement with the expectation
87 15' ~ N e - H 2
++++
T i
-~176I -0.1 5 I
15' ONe - H 2 i I
t
^
=-(OJ4• 772
v ~5tel
Events-
~
#
'-r r
-oo5 +-_4__F-+- w=4-6 ~v
I,U ~" : -0.10
0.4
< A Q ~ 9m:-(.46+,O81-
\
I-
=-(024 ~0.05)
831 Events
-0.15
0r
O2
~ -OIX ~W ~ Z -015
i
W>SGeV \\
0.01
r
5+++
W=6-15GeV =
- (O,54•
,4o4 E,;.ts I a 3 RAPIDITY'--->
!
\
0 u.
I\
I-MJ Z J
]
o
Fig.6.17. D i s t r i b u t i o n of charge per event in the cms r a p i d i t y f o r hadrons going forward in the cms in events with x > 0.1 and Q2 > I GeV2 f o r three i n t e r v a l s of W. From 15' ~Ne-H2 /166/
' 0,5 OA I/W(GeV) L-~
Fi9.6.18. Total net charge per event of hadrons going forward in the cms vs I/W. From 15' 5Ne-H2 /166/
Table 6.1. Measured net charges in the u,d fragmentation region Zxperiment
Ref.
Selection c r i t e r i a and cuts
(in u n i t s of GeV, GeV2 e t c . ) - 0.46-+ 0.08
15' ~H2Ne
/166/
W§
x >O.I,Q 2 > I , x F > 0
3EBC vH2
/164/
W§
x >0.I,Q2> I , x F> 0
0.59-+0.10
~EBC vH2
/164/
W§
x >O.I,Q 2 > I , ZB > 0
0.52 -+0.08
3EBC ~,vNeH 2
/168/
W>4, x>O.I,Q 2 > I ,
~EBC v, ~NeH2
/168/
W>~, x >O.I,Q 2 > I , ZB > 0
)ECO ep
/161/
W> 2.5, x§
ZB>O
I , xF > 0
0.55 -+ 0.06
- 0.12-+ 0.08
-0.3
"+0.1
0.48 -+ 0.05
based on the FF j e t model. The experimental information on charge retention in quark jets is summarized in Table 6.1. In the high energy l i m i t , a l l data are in q u a l i t a t i v e agreement with the expect a t i o n of approximate charge retention. Q u a n t i t a t i v e l y , the data tend to confirm an SU(3) breaking f o r sea quarks as used in the FF model.
88 6.3
Jet Universality
At moderate energies the j e t physics is dominated by the parton fragmentation regions investigated in the last section. At asymptotic energies, however, the d i r e c t quark or spectator fragments populate only a r e l a t i v e l y small region of the whole r a p i d i t y range; the dominant contribution to p a r t i c l e production comes from the plateau region. The idea of j e t universality t r i e s to relate the structure of the r a p i d i t y plateaus observed in the various deep inelastic reactions. The idea of an "universal" plateau, whose properties are asymptotically independent of target and current in current-induced reactions was f i r s t KEN and KOGUT /169/ based on correspondence principles.
suggested by BJOR-
In an universal plateau the
density of each p a r t i c l e species (I/~)(d2~/dydp2) is independent of the final state mass and the reaction considered. Consequently, one predicts an universal behavior of m u l t i p l i c i t i e s . = noln(W2)
(6.13)
Taking into account our present ideas on parton fragmentation, this model has to be modified s l i g h t l y .
In two-dimensional quark-gluon bremsstrahlung models, as discussed
in Chaps.4 and 5, the mean m u l t i p l i c i t y
per unit of r a p i d i t y grows with the square
of the partons color charge. This observation led BRODSKYand GUNION /15/ to the following universality principle:
the properties of the hadronic plateau are unique-
ly determined by the type of color charges separated, independent of the flavor content of the color sources. In QCD, the two basic sources of color are t r i p l e t s ,
such as quarks or diquarks,
and octets, such as gluons or quark-antiquark pairs. In current-nucleon interactions at high Q2, one should observe a t r i p l e t of the length Y3x3 ~ In(Q2) and on octet
plateau of the length Y8•
plateau
~ In(I/x-l)
(Fig.6.2c), thus, yielding the asymptotic m u l t i p l i c i t y = n3x~In(Q2)+n8x~In(I/x-1) x§ Q2§
(6.14)
More generally, and taking into account scaling v i o l a t i o n s , we get
= N3• x§ Q2§
r-
2 2 3x~)+N8x~(Y8x~,Q 8x~)
(6.1s)
89 with the asymptotic prediction from QCD 4_ N . N3x3 = 9 8x8
Unifying models based on dual unitarization schemes /171,172/ and on dual topological unitarization /173,174/ arrive at similar conclusions /139/. +
-
Referring to a comparison of hadronic events in e e annihilations and in currentnucleon reactions, the predictions of (6.14,15) are obvious: same height of the rapidity plateau, as measured at YCMS = 0 (e+e-) and at
-
YBREIT = 0 (IN), respectively; - for valence quark scattering, the total m u l t i p l i c i t i e s observed in the two reactions should be identical for same W, up to a small additive constant from a possible difference in quark and diquark fragmentation; - in IN reactions the total m u l t i p l i c i t y at fixed W increases with decreasing x, because then the relative contribution of the color octet part grows. Figure 6.19 shows dq/dy from vp reactions /164/ for 8 ~ W~ 16 GeV, compared to §
-
e e annihilations at W = vrs = 13 GeV /24/. The height of the charged-particle plateau, (dn/dy) ~ 2, is consistent. In Fig.6.20 the total m u l t i p l i c i t y measured in vp +
=
reactions /160,164/ is compared with data from e e annihilations (see Chap.2). Again the values agree surprisingly well, except at very low energies where the phase space available to fragments is significantly reduced in vp interactions due to the existence of a final state nucleon. Figure 6.21 demonstrates the dependence of on Q2 (or, equivalently, x) at fixed Wfor vp interactions. In contrast to the expectation based on (6.14,15), shows no significant dependence on Q2. This feature is in basic agreement with other experiments studying deep inelastic lepton-nucleon interactions /175-177/. When interpreting Figs.6.19-21 in terms of j e t universality, one should note that at these energies the m u l t i p l i c i t y is far from being dominated by the plateau region. Below W = 5 GeV, the m u l t i p l i c i t y is more or less fixed once the transverse momentum smearing of the produced particles is given and is nearly independent on the choice 2 of matrix elements. The invariance of with 2 Q w i l l be p a r t i a l l y due to this fact. I t may further indicate that at these Q either reactions with a coherent spectator system dominate and no octets are separated or that the typical m u l t i p l i cities of t r i p l e t and octet plateaus are similar. Clearly, the present data are not sufficient to decide whether j e t universality holds exactly; nevertheless, since the data are consistent with universality, we shall keep i t as a working hypothesis.
90 i
I
i
i
I
BEBCvp~H*-+X,
I
W=B...16
i 7 GeV
6
2
o o 9
v p Ref. 150 v p Ref, 164 v p Ref. 1,57
/
5
D__N
DY
k 1
3
o*
2 I
0
9
-3
-2
-1
0
1
2
3
1
0
u
I 3
I ,c.,
I 5
I 6
f 7
I B
I 9 10
W (SeV)
Fig.6.20
Fig.6.19
Fig.6.19. Rapidity distributions in the cms of secondaries in ~p reactions at W = 8-16 GeV /164/, and in e+e- annihilations at v;s = 13 GeV /24/
W = 8 -10 GeV
Fig.6.20. Mean charged hadronic multiplicity measured in ~p reactions and in e+e- annihilations
Fig.6.21. in~ions
"W=4-66eV
A
Mean number of charged hadrons produced in vN as a function of Qz for fixed W /160/
V W = 3-46ev
.
.
4
8
.
.
.
tl .
16
.
.
.
32
.
.
.
64
Qz(Gev=)
Fig.6.21
6.4
Spectator Fragmentation
Present data seem to be consistent with the hypothesis of approximate factorization and jet universality. In a certain sense lepton-nucleon reactions therefore don't +
-
give qualitatively new information on quark jets as compared to e e annihilations. Lepton-nucleon reactions, however, offer the unique possibility to study the fragmentation of compound quark states, such as diquarks. I t will be shown that the measurement of their fragmentation functions reveals information on the wave function and dynamics of multiparton states. Furthermore, once the fragmentation properties of multiquark systems are known, e.g., from vN reactions, the mechanisms of other deep inelastic phenomena like muon pair production in hadron-hadron inter-
91 actions can be reconstructed by studying the quark contents of the spectators l e f t over after the hard process. A spectator system can be described by its flavor content, i t s color state, i t s mass, and by the degree of coherence of i t s wave function. Table 6.2 gives a l i s t of possible spectators in charged current neutrino-nucleon interactions.
Table 6.2. Spectator systems in v, ~N interactions. Coherent subsystems of the spectator are put in brackets. Only u and d sea quarks are taken into account Reaction Flavor v-valence quark (Fig.6.1) v-valence quark (Fig.6.1)
v-primordial sea (Ffg.6.3a) ~-primordial sea (Fig.6.3a) v-pointlike sea (Fig.6.3a)
~ - p o i n t l i k e sea (Fig.6.3b)
Spectator Color
Mass
(uu) (ud)
] ]
O(mn) D(mn)
(uuda) or (uudu)
~
m (71x-l)
(uudu) or (uudd)
3 3
mn(I/x-1)
(uud)u
(uud)a or
(B) 3 (8) 3
mn(1/x-1)
(uud)u or (uud)d
(8) 3 (8) 3
mn(I/x-1)
3
n
The spectator is thus a much more complex object than a single active quark. To reduce the number of variables, we shall assume the following simplifications based on our knowledge on quark jets. First, the studies of QCD jets presented in Chap.5 have shown that the fragmentation function of a system consisting of incoherent components can be represented as the incoherent sum of their fragmentation functions, at least as far as fast (x >> 0) fragments are concerned. Second, we shall assume that factorization holds here as well. That means the distribution of fast fragments depends only on the flavor contents of the spectator, whereas i t s color contents influences the plateau region. We are now ready to define the spectator fragmentation function bh(z) which refers
to the decay of a coherent parton system. We use a reference frame where the
i n i t i a l nucleon has a large momentum PO" Since the maximummomentum available to fragments is P"MAX ~ PO( l - x ) ' the natural scaling variable is z = p,,/[po(1-x)]. In
the cms, we have z ~ -x F, with x F defined in the usual way as 2p,,/W.
92
Do we expect scaling to hold for spectator fragmentation? The picture of asympt o t i c freedom, together with the diagrams of Fig.6.1 suggest that the spectator completely ignores the hard process. The energy used to build up the preconfinement plateau is radiated by the active quark; the color charge of the spectator participates only in the f i n a l confining step at fixed Q~. Does this imply that the spectator fragmentation function is independent of Q2? In f a c t , a Q2 dependence of spectator fragmentation arises in a rather i n d i r e c t , process dependent way as can be seen from Fig.6.6b. The incoming proton dissociates into the active quark at x' and the spectator at ( 1 - x ' ) . The active quark radiates an energy fraction ( x ' - x ) and is then h i t by the hard probe. Therefore, the maximum momentum f r a c t i o n of spectator fragments is ( 1 , x ' ) and not ( l - x ) as naively expected. Since the amount of radiation ( x ' - x ) increases with increasing Q2, the spect a t o r fragmentation function D(z) shrinks with increasing Q2 and the p a r t i c l e dens i t y at low z increases. Q u a l i t a t i v e l y these effects of scale breaking equal those observed in quark fragmentation; q u a n t i t a t i v e l y they d i f f e r and depend on x as well as on Q2. This difference of the perturbative j e t structures in e+e- and in IN reactions has been emphasized by various authors /181-184/. In r e a l i t y , however, these differences are small compared to nonperturbative e f f e c t s , at least at present values of Q2 and W. This is evident from Fig.6.22 where +
i
the d i s t r i b u t i o n of sphericity is compared for e e events and for IN reactions in two bins of W. The d i s t r i b u t i o n s agree within the error bars. Therefore, we feel that in the following discussion scale breaking effects can be neglected without i n t r o ducing much bias. Let us now consider the fragmentation of the coherent part of the spectator. Because i t s decay is governed by s o f t , c o l l e c t i v e processes, one may ask i f the perturbative quark-parton model or, more generally, the concept of individual quarks may be applied. Consider the decay of the spectator (mass ms ) into a parton b and a core of mass mx. Let z be the fractional momentum of b and p~ i t s transverse momentum. From simple kinematics we get 2 2 2 P~+Zmx Pb = - - ~ +
2 Zms
(6.16)
I f the core z contains valence constituents of the spectator, i t s mass should be of the order O(GeV) /189-191/. That means, however, that for z § I the active parton is f a r o f f s h e l l . A fragmentation which contains this parton thus actually tests the parton structure at large Q2 and the application of the perturbative parton concept appears to be j u s t i f i e d /192/. To conclude: although spectator fragmenta-
93 t i o n is b a s i c a l l y a soft process, the quantum numbers of fast fragments should ref l e c t the constituent structure of the spectator. Unfortunately, experimental data at s u f f i c i e n t l y high Q2 and W suited to test these ideas are rare. Figure 6.23 shows the x F d i s t r i b u t i o n of p o s i t i v e and negat i v e fragments observed in ~p reactions in the 15 f t bubble chamber /160/ f o r events with W > 4 GeV and x > 0.05, and with typical Q2 of the order of 2-10 GeV2. Similar results have been reported by BEBC /164/. In Figo6.23 target fragments populate the region x F §
For comparison, the d i s t r i b u t i o n s observed in the proton fragmen+
t a t i o n region in i n e l a s t i c ~ p reactions at v~s = 5.6 GeV /185/ are included as dotted l i n e s .
3.0 2.0
I
I
k/!
~.
i
I | ....
I
I (o) e +e-
1.0
-Ib
(b).
9 "~p, vp
4.0
2.0
o
0.2
o,4
S'
o,6
0.8
Fig.6.22. (a) Sphericity d i s t r i b u t i o n for 5p reactions with W = 6.6 GeV and for e+eannihilation at W = 7 GeV; (b) sphericity d i s t r i b u t i o n for ~p plus 5p reactions at W = 10,5 GeV and for e+e- annihilation at W : 13 GeV. From 15' ~H2 /167/
The s i m i l a r i t y of the pion spectra obtained in the two reactions is s t r i k i n g . Furthermore, in each case proton production dominates f o r x F § - I . The absolute proton y i e l d , however, is a factor of 5 less in the deep i n e l a s t i c reaction f o r x F § - I . A dominant production of neutral baryons can be excluded. Baryon number conservation requires the proton d i s t r i b u t i o n to peak at lower x F, in contrast to +
the ~ p events. The f u l l
lines in Fig.6.23 are the r e s u l t of a phase space model (UJM). The mo-
del uses constant matrix elements to describe pion production. The matrix element for proton production has been adjusted to f i t
the experimental proton spectrum.
94
,
,
,
,
,
,
lO
-.r-,-l--=-l-.--
~ ,u- h+ X I DATA (A =I.D. PROTONS) - - UI4C
',- 5
Up
i
1 .~" 1.o
EO
?
I,'p -> ~z-h-X 10 ~{ DATA - - UA~C
10
'
5
"2
!i,\
ii
B)
~ 3 GeV, is j u s t at the threshold where i t is possible to recognize a j e t structure on a s t a t i s t i c a l
t16 basis averaged over many events. The f o l l o w i n g results r e l y on a comparison with Monte Carlo models, taking i n t o account the detector acceptance, the r e l i a b i l i t y of the track recognition, and r a d i a t i v e corrections. Let me f i r s t
discuss r e s u l t s from DASP I I /225,247/. Since the DASP detector
measures d i r e c t i o n s of p a r t i c l e s , but no momenta, four "pseudo" topological v a r i ables have been used: Pseudosphericity
= (3/2)min <sin28>
Pseudospherocity
= [(4/~)min ] 2
Pseudothrust
= max
Pseudoacoplanarity = 4(min ) 2 where 0 is the angle of each track with respect to the preferred axis or, for pseudoacoplanarity, with respect to the preferred plane. < > means the average over p a r t i c l e s of one event. Figure 6.41 shows the mean pseudosphericity as a function of v~s. Obviously, events from T decays are less j e t l i k e .
Table 6.4 summarizes the
r e l a t i v e changes of these variables on and o f f the T resonance compared to a Monte Carlo c a l c u l a t i o n . The model, based on the FF algorithm, assumes gluons are produced according to (6.39) and decay exactly l i k e quark j e t s .
Table 6.4. Change o n / o f f resonance
Exp. /247/
Model /247/
18.8 • 4.0
23.0
9.1 • 1.8
9.8
9.5 • 2.0
14.0
- 2.8 • 0.6
- 3.1
<Multiplicity>
8.4 • 2.0
7.0
The agreement is f a i r l y good, X2 = 1.4/NDF. T decays are d e f i n i t i v e l y d i f f e r e n t 2 from normal t w o - j e t events, a f i t in terms of two-jet models gives • = 22/NDF. The DASP I I group pointed out that i f the T decays into three gluons, the decay properties of these gluons are i d e n t i c a l to those of quarks. A change of the fragmentation function from
Dg(Z) = Dq(Z) to Dg(Z) ~ (1-Z)Dq(Z), as predicted by asymp-
t o t i c QCD, completely spoils the agreement. S i m i l a r l y , a s i g n i f i c a n t change in the mean transverse momentum with respect to the j e t axis can be excluded.
117
Recent and very detailed studies of the hadr6n distribution from T decays by the PLUTO /248/ and DHHM/246/ groups strongly support this picture of a three-jet decay of the T. This is demonstrated by a three-jet analysis using the t r i p l i c i t y method /249/ as performed by the PLUTO group. The t r i p l i c i t y method assigns three j e t axes to each event by grouping the detected particles into three disjunct groups in a way as to maximize the sum of the momenta of the jets. Let x I ~ x2 ~ x3 be the normalized j e t momenta and eI ~ 02 ~ 03 the angles between the jets. Figure 6.44 shows the experimental distributions of thrust, x I , x3, 01 , and 03 for direct T decays and for two-jet events from the continuum /248/. Included are Monte Carlo model predictions for two-jet, three-jet and phase-space like events. Consider f i r s t the two-jet data off resonance. An attempt to assign three j e t axes to a two-jet event w i l l lead to one j e t axis coinciding with one of the jets, and the two other axes pointed in opposite directions. Consequently, x I is peaked close to I , and 83 is close to 1800. The distribution in x3 should be rather f l a t , since the momentum of one original j e t is shared among two reconstructed jets, and the angle between these jets, 81 should be small. This is exactly what is observed. For direct T decays, 03 decreases and 01 increases, proving that i t is reasonable to assign three distinct axes. Data are in good agreement with the three-gluon decay model (which assumes that gluon jets behave like quark j e t s ) ; a description by pure phase space seems to be ruled out. I t has been pointed out that the spin of the three decay partons can be determined from the angular distribution of the axis of the fastest j e t with respect to the beam axis /232/. In the l i m i t x I § I , the d i s t r i b u t i o n of the angle 0 between the event axis and the beam d i r e c t i o n is given by
d~/dcosO ~ I + a cos2e
(6.46)
with a ~
I f o r spin I partons
a ~ -I f o r spin 0 partons.
Averaged over a l l values of x I , one predicts a ~ 0.39 /232/ for spin I partons and a < 0 for spin 0 partons. The distribution of the sphericity axis of direct T decays is shown in Fig.6.45 as measured by the PLUTO and DHHMdetectors. Taking into account that the DHHMdetector does not allow a precise measurement of the momenta of fast charged hadrons and thus has the tendency to smear the distribution towards isotropy, the spin I assignment seems to be s l i g h t l y favored.
118 ,,
,
i,,
, , i , , , , i ~
, . , i , ,
Fi9.6.44. Experimental distributions of thrust ?, reconstructed gluon energies Xl,X 3 and reconstructed angles 01,03 between gluons compared to various models. From PLUTO /292/
i ,
~,y direct --3-g[uon ~, off resonance -- pl~se space ,-. - - - Feynman,\. ~, Field
0,5
06
0.7
08
09T 1.0
!,' x -,o
,Z
,,,," ,y:.o
',
x '~
0.5
O.l
O.B
-
2[
0.8 xl 1.0
o.~s
x]
0.7
~ "k-'~'~'~ ~- F
I
O'
. . . . . fiB" E)~
120'
~2o'
~so'
~so"
0
o]
5
3-15
3-14 3-15
0 5-4
1 5-4
I-4
0-4
0-3
2 5-8
0-4
2-8 0-5
I-5
,
P.[GeV/c]
Trigger
~
~
110~
50~ ~ 900
60~
50~
60~ ~
90~ 900
45 ~
90~
90~ 90 ~
90~
90~
20~ ~
530-90 ~
45~ ~
90o
90~176
90~ 450-90~
90 ~
Ocms
I
1274/ /275,276/
/272,273/
1271/
/270/
/267/ /268/ /269/
/266/
/264/ /265/
/261-263/
/7,258,259/ /260/
/257/
/256/
/255/
/254/
/10,252/ /8,253/
/9/
Some ref.
03
127 where ql,q2,q, q' denote parton momenta. In analogy to the treatment of lepton-nucleon interactions we assume that the cross section d3~/dydy'dp~ for the production of two partons q and q' factorizes as dO(hlh2§
:
Z d~(h1§247247 ql,q2
')
(7.5)
The f i r s t two terms are the well known hadron structure functions do(h§
: Gq(x)dx/x
(7.6)
For the cross section we write
d~
dj2 dt
q2+qq' )
(~,~)ds
(7.7)
^
~
2
Using (dXl/Xl)(dx2/x2)dt = dydy dp~ (7,7) f i n a l l y yields (for s i m p l i c i t y , we omit the sum over ql,q2 ),
do
ql(xl)G (x2 )
dydy' dp2 =Ghl
(7.8)
d-~
Writing o as .
.
d!(s,t) d~
.
.
=~ s s
f
=
f(~)
(7.9)
we obtain the scaling law do : F ' dydy'dp~ ms (y'y 'x~)p~2n
(with x = 2p~/V~)
(7.10)
where the angular dependence of do/dt and the structure functions are summarized in F, and the term p~2n reflects the ~ dependence of do/dt at fixed angle 0 [note that scale breaking effects are neglected in (7.10)!] Using dimensional arguments, n is given by
os > n which yields n = 2 for theories with a dimensionless coupling constant.
(7.11)
128 The cross sections (7.9) and (7.10) refer to the production of large p~ partons h or j e t s . The single-particle cross section ~ is given by a convolution of ~ over the parton fragmentation function D(z) d3~h
; d3~ , ~ ,
3
=
(7
2)
In order to enable simple analytical calculations, we parameterize ~ locally as E(d3~/dp 3) ~ Ap[ k and choose a rather general ansatz for D(z) D~(z) = (B/z)(1-z)m+L+Ta(1-z)
(7.13)
The ~ term takes into account the possible emergence of stable hadrons from certain subprocesses. From (7.12) and (7.13) we obtain for the ratio of inclusive single particle and j e t cross sections /277-279/
,
3
h .3 o__~_o (Edp3 )/(Edp3 )
= 2B
m!(k-3) l L " * ~: (k+m-2)! ( k - l )
+T
(7.14)
Within our approximation the single-particle spectrum has the same slope in p~ as the distribution of jets, however, the absolute yield is much smaller. For example, by choosing the standard quark fragmentation function with m ~ 2-3, L "~
: -6--=
P*
j - z- ~ L -~p -~
(C)
D(z)
f d~zr d3---~{ ~ ) D ( z ) ~ h z 2 ~ dp3
Bmt(k-4)!
(k+m-3)'t §
Bmt(k-3)!
+ ~L +
+
T] T]
(7.15)
and oh drops. Experiments triggered on a single hadron at large p~ thus select a special type of parent j e t s , those consisting essentially of one very fast fragment. This effect is known as the trigger bias /278,279/. Experiments triggered by jets of large p: are not subjected to this bias. However, a similar bias is introduced i f the solid angle covered by the j e t detector is comparable to the j e t size. Then the trigger condition enhances narrow, well collimated jets /273/.
129 Another important consequence of the trigger bias is that for a mixture of d i f ferent parent partons the species with the f l a t t e s t fragmentation function is favored by the single particle trigger condition. Of course, only the j e t containing the trigger particle is affected by the t r i g ger bias. The recoiling parton in the opposite azimuthal hemisphere decays unbiased within the limits imposed by energy-momentum conservation. The fragmentation of this second "away" j e t is usually described in terms of a variable x E referring to the h trigger momentum p~ h'/ h x E = p~ p~
(7.16)
where h' is a hadron in the away j e t . Neglecting transverse momenta in the parton h' fragmentation, x E is related to the "correct" scaling variable z referring to the momentum of the away j e t z
h'
h = XEZ
(7.17)
(Note that x E may exceed I . ) The particle density (I/~)(do/dx E) is obtained from h I do =
~ X~E
f zd--zEd-~(~--~)D(z)D' (XEZ) dp h
(7.18)
f~dz E d3~(zB_~)D(Z)dp
D(z) and D'(z) are fragmentation functions of the towards and the away jets, resprectively. Two points should be kept in mind. Since do/dx E depends both on D(z) and on the inclusive j e t cross section -d3~/dp3, scaling of the fragmentation functions does not necessarily imply scaling of do/dx E in x E, Furthermore, the dependence of do/dx E on the trigger-side fragmentation function D(z) may induce correlations between the towards and the away j e t . Compare, for example, events where +
scattered u- or d-quarks create ~- and K- trigger particles. Since DK f a l l s u,d steeper in z than Du,d, the mean momentum of the towards j e t is larger for the Ktrigger, and do/dx E is f l a t t e r . Care is needed not to confuse such kinematical effects with dynamical correlations due to the scattering mechanism i t s e l f . To complete this discussion let me quote the relations concerning the two spectator jets at low p . The energy available to the spectators is reduced when compared to the total cms energy, The sum of the energies or momenta of the spectator jets is
130 E' = r p' =
(cosh y+cosh y ' )
(7.19)
-m (sinh y+sinh X')
Hence, the usual Feynman variable 2p,,/r
is no longer suitable to describe the frag-
mentation of spectators. The most natural choice is to use a reduced energy Vs' for the system of the two spectator jets r
= (E,2_p,2) !/2
The appropriate scaling variable is then the reduced longitudinal momentum of a secondary in the spectator system /259/ x' = 2p,~//~'
7.1.1
(7.20)
The QCD Approach
In a hard scattering picture, the favored candidates for the active partons are quarks and gluons, t h e i r interactions governed by QCD. The basic ingredients of the model are shown in Figs.7.3-5. The main subprocesses are quark-quark, quarkgluon, and gluon-gluon scattering with the cross sections /280,281/ given by A
d_s
d{
(qlq2§
f(O) 6 6 , ~s2(Q2)7 ql q q2q
')
(7.21)
tg f (~)) ' 2
0o
3PO o
610 o
gO lO
I
120~
I
150~
IBO~
Fig.7.3. Angular dependence of partonparton cross sections in f i r s t order of QCD /280,281/
131 SCALE BREAKING A:O.4 GeV/c (a} Electroproduction Structure Funchon of Proton 0.5 ,; . . . . . . . . . . . . . , .... , .... ~, ...... Q2= 4
,.~
_._
0.4 .~,~ v W 2 ( x , Q 2) "i.'..'.~ 0.3 ~ ~ ' ~ . .
Q2 :
(b) Gluon Distribution in Proton
tO
___ Q2 : 50 _=_ Q2 = 5 0 0 ~ FFI RESULTS
,,,i
....
i ....
x g ( x , Q 2)
i , , V,a , , , Q2= 4
......
4 ~
0.2
_._
Q2=I 0
-_-
QZ=50
_ = _ Q2 = 5 0 0
2
OH
0.0
Fi
5pl
"', ",
0.0
0.2
0.4
0.6
-"
0.8
1.0
X
0.0
0.2
0.4
-0.6
0.8'
1.0
X
Fig.7.4. Typical parameterization of quark and gluon structure functions of the proton for various Q2 /282/
zD(z,Q2) VERSUS z lOll: . . . .
, ....
, ....
J: (o) u"O'Tr ~ f
, .... , ....
A:O.4 ........ Q2 =10
ioO~
-.-~
,t,,,,
....
t
--~=5oo ~ .
"
I
~ ' ~ ,,~,,,:...-> ~ ,,~.. 0.15
(7.28)
Compared to large p ~" t r i g g e r s , which in the ClM are almost e n t i r e l y due to qM§
the away j e t f o r a K- t r i g g e r should be d i f f e r e n t , and i t s contents
of K+ should be enhanced in contrast to QCD where no strong f l a v o r c o r r e l a t i o n exists between the two jets at large p~.
Ill)
I f one of the c o l l i d i n g hadrons contains a valence antiquark, the q~§
sub-
process w i l l considerably enhance the cross section at large x , as compared to QCD p r e d i c t i o n s .
7.2
General Characteristics of High p~ Events
Scattering of quark or gluon partons was shown to account for single-particle yields at large p~, once higher-order and higher twist effects are taken into account. In this section we shall discuss further evidence for a basic two-body scattering process, using two sets of data which are insensitive to details of parton fragmentation. The mechanisms of fragmentation and the properties of jets in large p~ events will be discussed in Sects.7.3,4.
7.2.1
Particle and BeamRatios
Ratios of single particle or jet cross sections at large x for different particle species, beam, and target types essentially test the ratio of structure functions of the interacting partons. They have the great advantage that both theoretical and experimental uncertainties tend to cancel. The ratio of jet to single-particle cross sections (Fig.7.10) tests the idea that scattered partons carry color and hence must fragment. This would yield a tremendous ratio for ~Jet(p~)/oh(p~) of the order of 103 at large x /282/ (7.14). The measured ratio is in good agreement with QCD predictions. For pure CIM processes which directly produce color-singlet mesons, the jet cross section is lowered by about one order
139
"'~m
. E260 JET DCP ('rr~l."rr-)
q,'~" '., ~
Io5
',20 "-,-f o ",,-~-, "' ",
"",, J.
,o 3 pp ~++X p p "-->l~m +
",, . o o
,,,, \ \
lO
mb
I0i
eL.
-,.
"",,~
,,I,,,,I
....
2
0
I,~,
I ....
i ....
4 P• GeV/c
I
I,~
6
9
I
I
.2
3
I
I
I
.~,
.5
.6
XT
Fig.7.11
I
i
i
i
~l I
8
0
I.O
I
I
I
I
I
}
: 23.4 GeV o O,m 90 p- p collisions
m L)
~ = 23.4 GeV 0,.= 90*
0 D-
p- p collisions
n
I
I
I
I
I
b
0
0
I
t//
700
Fig.7.10 1.0
X
I
.01
1.0 2.0 30 40 50 60 70 P- (GeV/c)
I
I
I
I
I
I
1.0 20 30 4.0 5.0 6,0 P- (GeV/c)
Fig.7.12
Fi9.7.10. Jet cross section at /~ = 19.4 GeV compared with the s i n g l e - p a r t i c l e cross section /271/. The f u l l l i n e s r e f e r to a QCD c a l c u l a t i o n f o r f i x e d j e t energy and f i x e d j e t momentum, r e s p e c t i v e l y Fig.7.11. P a r t i c l e r a t i o pp§ at Ocms ~ 900 from /268/. (-) vrs = 19 GeV; (m) /~ = 23 GeV; (e) ~ = 27 GeV and /353/; (v) ~ = 19 GeV compared to QCD /282/ and CIM /287/ p r e d i c t i o n s F i q . 7 . 1 2 a . b . P a r t i c l e r a t i o s pp§ v~s = 23.4 GeV (o): / 2 5 3 / , (o) /268/
and p p §
at 8 ~ 90 o vs p~ f o r cms
140 of magnitude /302,304/ as compared to QCD. In the subprocess expansion (7.32) combining QCD and CIM graphs, the relative abundance of processes can be adjusted such that the major contributions to the j e t cross section come from QCD graphs, whereas CIM terms dominate single-particle production, favored by the trigger bias. Within present experimental errors /271,273,307/ such a combination is indistinguishable from pure QCD. Figures 7.11,12 show ~+/~-, k+/~ + and k-/~- ratios at large p~ for proton-proton collisions. In almost any hard scattering model referring to quark partons, ~+,k + and ~- mesons at large x
or x contain an u and d valence quark, respectively from
one of the incident hadrons. Consequently, the ~+/~- ratio at large x
reflects the
ratio I / ( I - x ) of u- and d-quark structure functions, while k+/~ + should be constant for x §
and k /~
will drop with increasing p~. Data show all these features, pro-
ving that the standard valence quarks are involved in large x The same argument shows that the beam ratio pp§
+ X/~p§
particle production. + X f a l l s with in-
creasing x , since the x distribution of valence quarks is f l a t t e r in a pion, as compared to a proton, simply because momentum is shared by only two valence quarks. Again data agrees with the QCD ideas (Fig.7.13). The CIM predictions, where the process qM§
is considerably enhanced due to the incident meson, f a l l s below the da-
ta. Beam ratios also offer a simple way to decide whether the underlying process is a scattering (t channel) or fusion (s channel) mechansim. The ratio p p § p~§
should be about I in the f i r s t case and small compared to unity else.
The conclusion from Fig.7.14 is evident.
R ( p/-rr ) 2,5 . . . . I . . . . .t. . .E26~I~"'IjET . . . . I . . . . I " ' q o E 2 6 0 SLNG.PART, 2,0
9 E 5 9 5 JET
'3
OL~llplli~il,~,,l 0 2 JET
. . . . I, 4
iLll,,~l,,,, 6
P.L (GeV/c)
Fi9.7.13. Beam ratio pp+T O+X/~p§ ~ and pp+Jet+X/~p§ /271,272/ compared to QCD /306/ and CIM /287/ predictions. The shaded area gives the QCD prediction for jets (upper boundary) and single particles (lower boundary). The CIM prediction refers to single particles
141 F i n a l l y , one can t r y to study the angular dependence of the basic scattering process by considering the p a r t i c l e r a t i o ~-/~+ in ~-p c o l l i s i o n s as a function of the cms angle. I f do/do is peaked forward, one expects that ~-/~+, measured at f o r ward angles in the pion hemisphere, is large and increases with p~ since the incoming pion contains two negative, but no positive valence quarks. Figure 7.15 compares very recent experimental results with QCD predictions for cms angles of 90o , 77o , and 60o . The expected tendency is not observed, except perhaps for the two points at highest p~ and Ocms ~ 60o . A possible explanation for the discrepancy may be that the gluon contribution is underestimated since the gluon fragmentation function is chosen too steep. In that case the expected behavior shows up only at larger p~ or smaller Ocms. However, i t is obvious that these data, i f confirmed by future experiments, pose a serious problem for QCD.
2.0
BEAM
R A T I O p/15
i
i
i
i
i
5.00
, , i/60o
1.5
F-F~/, F-F/~/N,, ~/77~
1.0 RATIO
QCD
, ~ I ~
05 14'';
pj.(GeV/c)
Fig.7.14. Beam ratio pp§247 and pp§176+ X/p~§176+ X vs p~ at v~ = 19 GeV /270,271/ compared to QCD /306/ and CIM /287/predictions
7.2.2
:/?
.;.,.'
9 de* o ~o
0.01 0,0
9o~
PT(geV/c) i
i
]
!
40
t
i
i
80
Fig.7.15. Particle ratio ~-p§ + X/~-p§ + X as a function of p~ for cms angles of 600 (A), 77 (1), and 900 (o), at = 19.4 GeV, compared to QCD predictions /269/
Structure of Large p~ Events
The inspection of p a r t i c l e and beam ratios supports the idea that in events with a secondary at large x , a valence quark is struck out of one of the incident hadrons. What remains to be proven is that this happens by a two-body process, i . e . , that a large p~ event contains four jets of p a r t i c l e s , two at large p~ and two low p. j e t s made of spectator fragments.
142
RAPIDITY
~U'P RAPIDITY
TRIGGER
AWAY
3 "= PTRIG": 4.5 GeV/c 1"~R (y,~p) for associated particles with PT > 0.5 GeV/c
Fig.7.16o Ratio of p a r t i c l e densities in large p~ events and in normal i n e l a s t i c events as a function of y and @. From BFS /262/
Figure 7.16 shows the r a t i o of p a r t i c l e densities observed in pp c o l l i s i o n s with a large p~ p a r t i c l e at y t ~ 0 to those observed for normal i n e l a s t i c events /262/ as a function of the r a p i d i t y y and the azimuthal angle @. The observed structure is not too f a r from what one expects in a hard scattering model. Near the t r i g g e r hadron, the p a r t i c l e density is enhanced, i n d i c a t i n g the existence of a "towards" j e t . The increase in density opposite in azimuth to the t r i g g e r is commonly i n t e r preted as due to the away j e t . Since the momenta of the c o l l i d i n g partons vary, part i c l e s from the away j e t are smeared out over a wide range in y, when averaged over many events. Contributions from the two jets at large p~ are more c l e a r l y seen in Fig.7.17. In the central r a p i d i t y region, the p a r t i c l e density peaks at @ = @trigger = 0 and at @ = ~. With increasing p~ of the secondaries, the f l a t background of spectator fragments diminishes, and the jets are better collimated in angle /264/.
143 0.3 < PT < 1.
a)
b)
0.3< pT< I. 1,0
.4
a..~n
dn
d,
d"~ .2
0
0
-~/P-
0
=~12
"~/2
"~"
3"rr/2
l. < P T < 2 .
I. < P T < 2 , .2
f~ -.12
0
2. < p T
3-3Gev/c I O- ]
eee 9
ee Fully inclusive
9
distribution
r
§ §
i 0 -2
+
+
++++
10-3
t 10-4
J
i
i
,
I I
+
,
i
i
2
I
~
t
t
i
3
I 4
% G~
Fig.7.20. Density of particles produced opposite in @to the trigger particle as a function of the transverse momentum of secondaries compared to the inclusive cross section. From CCRS/255/
Pt
(a)
5L R
(b)
o 9 150 MeV/c
9 Some-side
~~--~--{
LC
~'--L_~___
0,5
R ,,o
05
Pt > 8 0 0 MeV/c
Some - s ide
I 0,3
Pt
1.5
!.S
0.~
(c)
o Opposite - side
9 > 0 0 0 MeV/c
I 0,6
I 0.9
0
0.3
0.6
1',-'7ol
0,9
0
i 0.3
l 0.6
! 0.9
I 1.2
L 1.5
i",-"Jl
Fig.7.21a-c. Pair correlation functions for rapidities of towards and away secondaries. (a) Between the trigger n~ and charged particles on the same side, (b) between the trigger 7~ and charged particles with p~ >800 MeV/c, (c) between charged particles in the away region. From CS /265/
146
0
5
I0
15
2 0 x l O "a p o i r s / e v e n t / ( A y : l )
t
CORRELATION C(y~,yg)
-4
O
-2
2
Yl
4
Fig.7.22. Contour p l o t of c o r r e l a t i o n between away p a r t i c l e s with p~ > 800 MeV/c for a mean t r i g g e r p~ o f ~ 2.5 GeV/c. From CCHK /258/
AWAY-SIDE PTSPECTRUM dN 14oo d~
1200
EVENTS
3.0 < ET > 10.0 GeV
Fl
PER 0.2
1095 events
~J [7 IllF11
160-
Ll
PT(TRIGGER)
I000
120-
BOO 600
20O 0 O'
AWAY-SIDE= I (A) ALL EVENTS- - -
,
L~ B FIDUCIALCUT- -
I
FJ
80-
400
rj j
1
40r]
I
I
I.. i~
o i
I
I
I
~)
I
I 80"
Fig.7.23. D i s t r i b u t i o n of azimuthal angle between two large p~ ~ ~ f o r 8 ~ E ~ 10 GeV/c. From ABCY /266/
o PT (AWAY) (GeV/c)
Fig.7.24 ~ Transverse momentum detected in the away side calorimeter when t r i g gering on a j e t with 3.95 < p~ < 4.2 GeV/c. From E 395 /272/
the UJM (Chap.3), the production p r o b a b i l i t y depends only on EL and not on the angle between ~~
Figure 7.23 proves that even at f i x e d EL the d i s t r i b u t i o n peaks at
A~ ~ 0~ and at A@ ~ 180~
147
Finally, is there exactly one away j e t in each event, and does i t compensate the whole p~ of the trigger particle? Figure 7.24 shows the momentum distribution of the away j e t as obtained from a calorimeter experiment /272,273/. The discrepancy between the trigger p~ and the mean momentum of the away j e t cannot be explained by the limited acceptance of the calorimeter /307/. A natural explanation is given by assuming a parton transverse momentum of ~I GeV/c. In this case, the transverse motion of the active partons is aligned along the trigger direction, with the recoil being taken by the spectators. This interpretation is supported by the investgation of spectator fragmentation /7,261/ (Sect.7.4). In Fig.7.25 the frequency of reconstructed away jets with p~ > 1.5 GeV/c is plotted as a function of the trigger p . As a reference, two curves from a Monte Carlo simulation of the j e t reconstruction procedure are shown, one for zero parton transverse momentum, and one assuming a momentum difference between trigger and away j e t of ~0.8 GeV/c due to parton k . The away j e t was chosen to resemble +
-
those observed in e e annihilations. The result for nonzero parton k supports the idea that an away j e t is Dresent in each large p~ event.
p~
:,. 1.5GeV/c
0.5 /
II
/'
I
'1%1
< I
3 GeV, the onset of scaling being evident from Fig.7.28. In the scaling region, x E spectra from d i f f e r e n t experiments and f o r charged and neutral secondaries agree remarkably well (Fig.7.29) and are described by ( I / o ) ( d o / d x E) ~ exp(-3.TXE)/X E. The scaling v i o l a t i o n s at low pt can be understood as a consequence of a nonzero patton transverse momentum and p a r t l y as a contamination by spectator fragments / 7 / . Since these effects are n e g l i g i b l e at s u f f i c i e n t l y large p#, i t seems j u s t i f i e d to use (7.18) to predict x E spectra f o r the known quark fragmentation f u n c t i o n . For
149 10
~ 1.0
{
pTt>3
"
P'r (trig.) 9 2,0-2,2
GeV/c ....
9 2,2-2.6
i
9 2.6 -3.2 9 3.2 - 4 , 0
.... W x
w
i!
b
E "0
'.,.,>. . \i
0.5
10-I
~ "~,~\'~.
",~ \'~k"..
"v,'-.',:.l~:," 0.6
0.8
1,0
lO-2[_
I
I
I
I
I
0
0.2
0.4
0.6
0.8
10
1,2
XE
XE
Fi9.7.26. Distribution in x F of away secondaries, (I/o)(do/dx E) ~or four intervals of transverse momentum of the trigger. From CCHK /7/
|
I
I
I'OKlyI 5 GeV/c, [] p~ > 7 GeV/c, from CCOR /264/), compared to data from vN reactions /312/ (A) and e+e- annihilations /22/ ( ). Distributions are normalized to unity in the interval 0.2 < z < 0.8
0
I
2
i
I
4
I
I
6 PTGev/c
Fig.7.34. Slope parameter b of an exponential f i t to the j e t fragmentation function, D(z) = exp(-bz) for 0.2 ~ z ~ 0.8, compared to data from e+e- annihilations /22/. From BFS /263
gluon jets in the trigger with a f a i r l y soft fragmentation function. The dependence of D(z) on the p~ of the trigger is interpreted as a scale-breaking effect. I t should be pointed out, however, that results on inclusive distributions in a t r i g ger j e t rely heavily on details of the Monte Carlo used to simulate the experiment (in contrast to data on ratios of j e t cross sections where the major uncertainties tend to cancel), and that some problems, like the inclusion of parton transverse momentum, are not yet solved in a satisfying manner. Figure 7.36 compares the mean charged m u l t i p l i c i t y of jets in large p~ events with data from e+e- annihilations. Although the overall agreement is not bad, especially the BSF data /263/, for low v~ the data show an increase of the mean mult i p l i c i t y by s l i g h t l y more than one unit. The excess can be traced to an increase of particle density by a factor 2 for z ~ 0 to 0.2, and the authors claim that the effect cannot be accomodated within a model with "standard" quark jets. I t could be regarded as a sign for gluon jets. On the other hand, z ~ 0.1 corresponds to particle momenta of about 0.5 GeV/c in the cms of the two jets of large p~, and factorization is not expected to hold for such fragments. In the s p i r i t of the QRM
154 io 2
--i-YTTi-+l~,,l,,~l,, %
9
~
3 < p~et< 4
l~ 4 < p•Jet < 5 9 5 < pJet i- 02- 0.3
~s v
C 0L 0
i
Fig.7.39
I
95 xE
I0
ct
"2
J
,
i
5
,
p~
i,=l
10 vT
pq
IPOUT
20
Fig.7.40
Fig.7.38. Angular distribution of j e t fragments around the reconstructed axis of a j e t at large p~ for d i f f e r e n t cuts in the transverse momentum of the secondaries. Since the jets are detected at 900 in the cms, these cuts in p~ correspond to cuts in the momentum fraction parallel to the j e t axis. From CCOR/264/ Fig.7.39. Mean transverse momentum with respect to the j e t axis. (a) As a function of xE. From CCOR/264/. (b) As a function of the cms energy of the two large p~ jets. Only hadrons at large z are taken into account. From BFS /263/ and CS /265/. As a comparison, points from IN interactions /313/ and e+e- annihilations /22,41/ are added Fig.7.40. Definition of p~u~. All momenta are projected into the transverse momentum plane, pt, ph, pq andUp~' are the momenta of the trigger particle, of a hadron in the awayj e t , and of the two scattered partons, respectively
157
in the accuracy of the measurements the values do not differ significantly from + -
those obtained in e e annihilations (Fig.7.39b). QCD effects can be investigated by studying the quantity k , which denotes the amount of parton transverse momentum in the nucleon required to describe the effective imbalance of the transverse momenta of the two active partons after the scattering. The Q2 or s dependenceof k can be measured directly in two ways. Either the component of k in the scattering plane is determined by measuring the magnitude of the transverse momentum imbalance of the two jets at large p~ which are detected in two opposite calorimeters, or one studies the component of k out of the scattering plane, which gives rise to a noncollinearity of the jet transverse momenta. This is usually done by investigating the momentum component Pout out of the scattering plane defined by the trigger momentum and the collision axis (Fig.7.40). The mean square of Pout is given by = ~ 1 2 + x2 I 2 ) E (~+
(7.31)
where is the mean p~ squared of j e t fragments. The t r i g g e r hadron has been assumed to have z ~ I . The factor I / 2 enters since only one component of transA
verse momentum is used. QCD effects should manifest themselves in a rise of with Q2 or s at f i x e d x E. Whereas f o r t r i g g e r transverse momenta between 2 and 4 GeV/c no v a r i a t i o n of was found / 7 / , more recent experiments /264,266/ report a strong increase of with the t r i g g e r p= f o r t r i g g e r momenta between 3 and 11GeV/c (Fig. 7,41)o Figure 7.42a summarizes the information of
as derived from and
from measurements of the momentum balance. Obviously, rises with the t r i g g e r p , In QCD, is expected to be proportional to a mass scale, such as ~ , a function of dimensionless variables such as x
times
= 2p=/v~ (note that x2 = ~/s) and
~s /320/, i . e . ,
u
,
1.2
+
0
0.8
a v
e PTt>7 GeVlc
04
z~pTt>5 i
0
G2
J
I x PTt :> ~
0.4
0.6 xE
0.8
I 1.0
Fig.7.41. Mean value for IPo tI as a function ~ r different ranges o~ trigger p~. From CCOR/264/
158 1.B - a 1.6 1.L~
+
1.2
/ ~ and x~ = 2p~/vrs. Small i n s e r t : same p l o t a f t e r s u b t r a c t i o n of a constant primord i a l k~ of 600 MeV/c from
V~ : 27 GeV V'~ : 52 GeV
A, .2 .0 0
I
I
2
4
z~
CCOR
g~ : 63',GeV
9
CS
V'~: 635eV
9
ABZS
g~" :
I 6
I B
63 5eV I
10
12
Pz (GeV/c}
.12
_
~0
9
ISR
A
WPLF v~ :
V'~ :
O
WPLF V'~ =
a
WPLF V~ :
9..63 Gev 16 nev
NO CORRECTION FOR PRIMORDIALk.L
19 GeV
t
Z< kj.>"08 .L:- T .06
cou~ciEoFo~
+.
.0k .02
0
o
b 0
= ~
I
I
0.1
0.2
f ( x ,ms) ~ ~
I
0.3 x~ : 2 p ~ l ~ "
=,
Q~
I
I
0.4
03
f(x )
(7.32)
Due to the slow v a r i a t i o n o f a s, the dependence of f on a s can be neglected in a first
approximation. Figure 7.42b shows ~
= 2/vrs as a f u n c t i o n of x . Where-
as data points a t vrs = 16, 19, and 27 GeV scale p e r f e c t l y , they disagree w i t h values obtained at the ISR f o r ~
~ 50-60 GeV.
Part o f the disagreement can be traced back to the f a c t t h a t e s p e c i a l l y in the WPLF c a l o r i m e t e r experiment, i t
is not c l e a r what the measured r e a l l y means
159
since i t is not known i f a gluon radiated by a scattered quark is caught by the calorimeter. Nevertheless, there is a way in which the two sets of data can be made consistent. Rememberthat besides the perturbative component of k described by (7.32), there exists a nonperturbative k inherent in the hadrons wave function. From the study of Drell-Yang processes, we know the mean primordial k is about 500-600 MeV/c /314/. In fact, after subtracting quadratically a primordial k of 600 MeV/c, all data approximately scale (Fig.7.42b). To summarize so far: the fragmentation function D(z) obtained for fast (p >> m) fragments of the scattered partons in large p~ events agree with the properties of +
-
quark jets measured in e e annihilations, within the experimental accuracy. Also, the transverse size of jets is similar and mainly determined by nonperturbative effects. A large asymmetry between the fragmentation of the towards and away jets, as expected in the CIM model, is excluded. On the other hand, a dominant contribution from gluons jets with a fragmentation function D~(z) = (1-z)D~(z) seems to be incompatible with the data. QCD effects are visible as an increase of k with Q2 and are qualitatively compatible with predictions. For definite quantitative conclusions, however, the systematic uncertainties of the experiments seem to be s t i l l too large.
7.3.3
Quantum-NumberCorrelations
The next step in the investigation of jets in large p~ events is to study correlations between quantum numbers of particles emitted within one j e t , or in different jets. Correlations in normal hadronic interactions are governed by the principle of "short-range order" which states that flavor quantum numbers of secondaries tend to be conserved locally /315,316/. The situation differs drastically for large p~ events. In the hard scattering process, both confined quantum numbers (such as fractional charge or color) and nonconfined quantum numbers (such as strangeness) propagate over large distances in phase space and give rise to long range quantum number correlations. In the remainder of Sect.7.3 we discuss correlations between the trigger flavor and hadrons in the opposite j e t of large p . In contrast to the constituent-interchange model, QCD predicts that the flavors of the towards and the away parton emitted in proton-proton collisions (and, to a large extent, in proton-nucleus collisions) are nearly uncorrelated since in QCD subprocesses no flavors are exchanged. However, the argument does not hold for interactions where beam and target particles d i f f e r . For example, in ~-p interactions a positive towards parton always stems from the incident proton, and consequently, the away parton is a ne-
160 gative pion constituent. Whereas for a negative trigger parton the mean charge of away partons will be close to zero. Let us now study correlations in proton-proton collisions. Consider f i r s t the influence of the trigger charge. Figure 7.43 shows the net charge density (I/o)(da+/dy-do-/dy) for events with a positive and negative large p~ particle at yt = 0 /262/. o+ and o- are the cross sections for production of additional pos i t i v e and negative particles, respectively. I t seems that the charge of the t r i g ger particle is balanced by other particles close to i t in rapidity. A precise measure for the correlation is given by the difference of the two curves in Fig.7.43, the so called "associated charge-density balance" ~q(y,yt) /316/. Naively speaking, Aq(y,y t) is the answer to the question "which particles in a high p~ event know about the trigger charge?" or "where do the valence constituents in the trigger particle come from?". Figure 7.44 displays Aq(y,y t) for events with a large p~ hadron at y ~ -0.9 /317/, The peak in Aq follows the trigger rapidity; the whole distribution looks quite similar compared to the compensation of the charge of a low p~ "trigger" particle in normal events /316/. Figure 7.45 shows ~q as a function of the azimuthal angle r and the difference y_yt for two event configurations, with the two jets of large p~ being in the same rapidity hemisphere ("back-antiback") and with the two jets in opposite hemispheres ("back-to
0.50 dO dy
{
~
/
/
%- trigger inclusive
0.25
"~o.////,.o~" / Tr, + trigger
3
-0.25
4-
-
Fig.7.43. Average charge density dQ/dy for events with a large p~ ~ or ~ trigger at y = 0, compared to the charge density in normal inelastic events. From BFS /262/
161
I ASSOC,ATEDBALCAHNAcREGEDENSITY h h (I} 2 GeVA: ~ PT' PT >
1,0
=
normal inelastic
--
I
I
Z_>,
>. 0,5 o- 0.5-I, provide a reasonable measure for the flavor of the away parton, and that only for p ' s of the secondaries well above I GeV the background due to spectator fragments is small~ Figure 7.47 summarizes information from three experiments on the ratio of the mean number per event of fast positive and negative hadrons in the away j e t for pp and pN collisions. The two experiments with trigger p~ greater than 3 GeV/c agree that the flavor composition of the away j e t does not depend on whether the trigger particle is a ~+, ~- , K+ or p and agree with QCD predictions. The large h+/h- ratio obtained by E 260 for ~ triggers is probably due to a contamination from spectator fragments /271/. As far as those triggers are concerned which have no valence quarks in common with the incident hadrons, such as K- or p, the experimental situation is controversial. For example, for K', BFS claims significant deviations from the "standard" behavior, whereas E 494 sees no effect. More recent data from E 494 is shown in Fig.7.48. The correlation function R for two hadrons emitted with large and opposite transverse momenta is defined as R = Wij/WiWj, where Wij is the prob a b i l i t y to find a flavor combination i , j among the two particle triggers and
164 EXP
TRIGGERp~ (BeY/c)
BFS E260 E~%
XE
pp 3 - z~.5 pp 1.5-L,.0 pBe 3- L~
RATIO PLOTTED
->.L5 h§ ->.A.O h*/h-> 35 ~"/TC
t I
i
" I
i
J
r~+ TC k+
I
m
i,,,
k-
p
p 'k+p
i
Fi9.7.47. Ratio of densities of p o s i t i v e and negative hadrons in the away j e t f o r d i f f e r e n t t r i g ger p a r t i c l e s obtained by the BFS /262,263/, E 260 /271/ and E 494 /274/ groups in proton-proton and proton-nucleus c o l l i s i o n s
i
k-p
TRIBGER TYPE
2.5 2.O
3.9<m'< 5.9 GeV
9 K t - Corrected
1.5
t
+
LO
+ ,t
.5
t
It
+
I
T r ' " r r ' ' n - K " K~ K* K- K - K - P P P ~ /) p l('p p "n'*'n'- "rr-'n"'rr-K" "n'* "rr-K" "rr''rr-K" "rr" "n- K" K-K- P
5' K-
P P
2.5 2.0
5.9<m
'
0 2 GeV/c
" ~ < 0.2
....
MtN. BIAS MIN BIAS WITH TRACK IN l y l < 0 . 5 , PT2OeVlc
((~PTR,G } =2.TGeVIc)
L-~ooo
L__
o.1
-Jz
~176176176
L-- I
;":
t
"....
0.01 0.2
r
I
f
0.3
0.4
0.5
0.6
0.7
0.8
"I
,.__,
0.9
1.0
1.t
12
IXI
Fi9.7.50. Distribution of positive spectator fragments in Feynman x for events with a large p~ particle. Full and dashed lines refer to the corresponding distributions in inelastic and in nondiffractive proton-proton interactions. From BFS /261/
168
POSITIVE PARTICLES
"-~
"::0.2
~( : PYI(PBEAM- 2 "PTRIG) O POS. TRIGGERS 9 NEG. TRIGGERS
0.4,
0,5
0
-
neg. fragments I
+
. . . . . . .+. ., 1.0
0,5
--
a
neg.fragments ]
0.5
TRIGGER
I -5
o
A
1
I
I
-4
-2
0
TRIGGER
I
I
I
0
I
5
Y
I
I
2
4
5
Fig.7.56
~+
Fiq.7.55. Ratio of the d i s t r i b u t i o n s of spectator fragments f o r a trigger [(I/a)(do/dy)~+ t r i g g e r ~ P(WI~+)] and for a ~- t r i g g e r [ p ( y l ~ - ~ as a function of r a p i d i t y . The region y > 0 is populated by fragments from the accompanying spectator. From CCHK /259/ Fig.7.56. As Fig.7.63, but for proton and 7+ t r i g g e r . From CCHK /259/
171
7.4.2
Quantum-Number Correlations
As already discussed, the quark-parton model predicts strong correlations between trigger and spectator quantum numbers. Figure 7.55 shows the ratio of rapidity distributions of positive secondary particles for a ~+ trigger to those with a ~- t r i g ger and similarly for negative secondaries. The trigger particle at yt ~ 2 has a mean p~ of 2~5 GeV/c. Similarly, the corresponding ratios for p/~+ triggers and for K-/~- triggers are given in Figs.7.56,58 (note that the "K- triggers" contain certain , -30%, contamination of large p~ antiprotons and that the "p trigger" includes some K+, ~20% /259/. A clear correlation is seen between the nature of the trigger particle and charged particles in the accompanying spectator j e t ; for y > 2,
+
where spectator fragments dominate, the ratios deviate markedly from unity.
The size of the effect is as expected: the accompanying spectator in ~ triggered events, which is a ud-quark system, contains less positive fragments than the uuspectator in ~ triggers. The different quark composition of the accompanying spectator is once more demonstrated in Fig.7.57, where the p~+ mass spectra is shown
RATIO OF SECONDARY PARTICLE DISTRIBUTIONS FOR K" AND "n'- TRIGGER
50-
I
]~ TRIGGER
trigger:
I 1,5
i
1.0
.
.
.
.
i
1.5
.
.
.
.
i
.
.
.
~ e (O',360") I
I
I
I pOS" fragments I
i ~"
.
'~+ TRIGGER
1.5
I
2.0 GeV
20
1.0
~ [t:-, U ) 20 ~ Ii frogments : gr E(0,25,0,8)
~(t)~ o', y(tl~he
2.0
~ neg. fi'agments ]
1,5 i
Fi9.7.57. p~+ invariant mass distributions in the accompanying spectator of events triggered on a forward ~+ or ~meson at large p . From ACCDHW/318/ Fig.7.58. As Fig.7.63, but for K- and ~7- triggers. From CCHK/259/
1.0
0,5
-
r -5
A
TRIGGER
I
I 0
t
I 5
172 f o r f a s t fragments. In ~- triggered events (uu-spectator) a clear A++ (uuu) signal is v i s i b l e , which disappears completely for ~+ triggers (ud-spectator).
7.4.3
Spectator Fragmentation Functions
For a forward ~+ and ~- t r i g g e r p a r t i c l e , the quark composition of the accompanying spectator are assumed to be ud and uu, respectively. Using dimensional counting rules or the quark recombination model, the scaling d i s t r i b u t i o n of fast fragments can be predicted and compared with experiment. Vice versa, one can t r y to i d e n t i f y the scattering mechanism by inspecting the quark composition of the spectator, an application which is p a r t i c u l a r l y interesting for the proton and K- triggers. As a scaling variable, we use x' as defined by (7.20) and make the following approximations for y, y' and m : y
=y t
(trigger rapidity), (experimentally: _O J I I i ! J i =
1
I
FC+
- - - x Du (x'). . . . (1-x')
QRM
I0 r c
I
I
0.1
0,2
I
(i-x')
I
I
0,5
I
I t I
I0
Fi9.7.59. Distribution in x' of positive fragments in the spectator accompanying large p~ pions compared to favored quark fragmentation functions D~+, the counting U rule prediction (1-x'), and the prediction of the QRM, for a diquark-quark recombination. From CCHK/259/
174 Figure 7.59 shows the d i s t r i b u t i o n of positive fragments in the spectator accompanying a pion t r i g g e r . The DCR prediction f i t s the data reasonably w e l l , whereas the QRM d i s t r i b u t i o n drops a l i t t l e
too f a s t at large x, an e f f e c t which had been
observed already in IN interactions (Fig.6.30). As a less model dependent compari7§
son, the quark fragmentation function Du
is shown, assuming that the fragmentation
of a quark into a meson w i l l be s i m i l a r to the fragmentation of a diquark into a baryon.
RATIO
NEGATIVE SPECTATOR FRAGMENt" ACCOMPANYING
37- +
OF
SPECTATOR
FRAGMENT
DISTRIBUTIONS FOR " r r - AND -rr + T R I G G E R
TRIGGER
i
trigger: I Y(l)"l'8
r ---
50
-
-
y_>6 ~
'
i
I
i
I fragments: PT E(0.25,0.8)
~(ILIBOa,Yit}*"I'8J (~('(50~I~0")~176176
I SE(50~lSO~)or(23 O" 310")
9I @ i'Lm~176 I I
trigger:
fragments:QrE(0 25,0 81 I I r
- - - ( 1 - x ' ) 3. . . . (1-x') i Q R M
(l_x') ~
I
QRM INCL.-
-~ #
I
i
i
i
I I i t
pos. fragments ] 2,0
2O
1,0 k
~
IO
/P' J
-
c~ o
O,5
Ioeg,,aome~
7"
/ll.r-f~//
"x
./
0,2
O,l
O,
I I
I-
0.2.
I
,
0,5
I
I I
1,0
(I-x') Fi9.7.60~ Spectrum of negative part i c l e s in the spectator accompanying a 7+ t r i g g e r compared to model predictions and to the f u l l y inclusive 7- spectrum. From CCHK /259/
/
./
OJ
/
/
/1
I 0.2
I I 0.5
I II I.O
(l-x')
Fiq.7.61~ Ratios of d i s t r i b u t i o n of spect a t o r fragments in events with a 7- and a 7+ t r i g g e r . The QRM prediction for negative fragments refers to pions; for positive fragments, pions were assumed to dominate at small x' and protons at large x ' . From CCHK /259/
175 §
The spectrum of negative particles in the spectator accompanying a ~ trigger is given in Fig.7.60. The shape of the spectrum coincides with the f u l l y inclusive pion distribution in proton-proton collisions /319/ and is reproduced by both QRM and DCR models. To demonstrate the influence of the trigger charge the ratios of x' distributions for secondaries in the spectator j e t accompanying a ~- trigger and a ~+ trigger are shown in Fig.7.61 for both positive and negative secondaries. As expected, there are less fast negative fragments in ~- triggered events than for ~+ triggers, proving that in the f i r s t case the valence d-quark has been struck. The model calculations included in Fig.7.61 are straightforward as far as negative secondaries are concerned; the QRMmodel describes data well even at low x ' . The prediction based on "standard" DCR seems to be too steep at low x ' , but below x' ~ 0.3-0.4 the model is not expected to hold anyway, The QRM values for positive particles include three components. At low x ' , the ratio is dominated by ~+ production. At large x' the rat i o is determined by the proton yields from i)
ud § p
+ X for trigger
ii)
uu § p
+ X for
trigger
§
~
i i i ) uu § A++ + X for trigger ~§ § p~ Based on the d i r e c t term i ) and i i ) alone, one expects the r a t i o plotted in Fig.7.61 to increase with increasing x' since in the average the uu-system has a larger momentum than the ud-spectator. Contributions from i i i ) ,
however, increase the r a t i o
to values larger than I in agreement with experiment. Note that the prediction is not sensitive to the assumption that a l l baryons are produced in t h e i r ground states. The corresponding ratios of particles produced in association with a K- and ~t r i g g e r are shown in Fig.7.62. In the K- events, positive spectator fragments are enhanced and the density of negative particles is reduced. This r e s u l t is compat i b l e with the prediction of standard DCR for scattering of strange quarks, or with p o i n t l i k e DCR for strange quark or gluon scattering, taking into account that the r a t i o for negative fragments is probably s t i l l
compatible with ~const f o r x § I .
I t i s , however, hard to accommodate these observations in the framework of a recombination model. Since the K- contains no d-quarks, i t seems impossible to imagine any scattering mechanism which picks up the valence d-quark with the same efficiency than for ~- triggers. Without such a mechanism, the r a t i o shown in Fig. 7.62 is predicted to increase with x ' , for negative secondaries!
176 RATIO OF A CC O M P A NYIN G FRAGMENT K-AND trigger y(t)
SPECTATOR
DISTRIBUTION
FOR
"n'-TRIGGER
fragments : PT ~'(0.25,0.8)
~ e 15o'j so~
I 8
i
(z 3o', slo ~}
y_>o i ,
,
, ii
I
\i
---- ( I - x ' )
E" #
2,0
- - Q~m
i.O
iX
--- 0,5 Q"
i 02L
///
O'IJ 9neg'~fmgments opos.
0,1
0,2
0.5 ( l-x' )
i.O
Fig.7.62.
As Fig.7.61 but for K and
triggers
Another example where new information can be derived from the analysis of spect a t o r fragmentation is the production of forward large p~ baryons in proton-proton c o l l i s i o n s . There are strong indications that high p~ protons r e s u l t from the scattering of a quasi-bound diquark system out of one of the incident protons. A det a i l e d discussion of t h i s point is found in /259/. Let me summarize so f a r : using the dimensional counting rules, the shape of part i c l e spectra in the accompanying spectator can be described. The quark recombination model using a Kuti-Weisskopf matrix element f i t s the d i s t r i b u t i o n of spectator mesons in events with pion triggers quite w e l l , but f a i l s for "exotic" large p~ part i c l e s l i k e K-. There are two possible reasons for this f a i l u r e . a) The Kuti-Weisskopf model for the proton wave function makes a somewhat a r t i f i c i a l d i s t i n c t i o n between valence quark and sea quark matrix elements. I t may happen that although the model accounts f o r correlations between valence quarks, i t underestimates the correlation between a valence quark and a fast sea quark. b) In the QRM the fragments r e f l e c t the instantaneous d i s t r i b u t i o n s of quarks at the moment of the scattering. There, the spectator is f a r from i t s equilibrium. After scattering a quark at x ~ 0.3, 70% of the spectator momentum turns out to be carried by gluons. Assuming that this system regains i t s equilibrium before the fragmentation s t a r t s , the q u a l i t a t i v e agreement with experiments can be improved.
177
7.4.4
Particle Correlations in Spectator Jets
The short-range correlations observed between hadrons emitted in normal inelastic hadron reactions are usually described in terms of a cluster model /64,316,321/, assuming that in a f i r s t stage of the interaction, hadronic clusters with masses of about I-1.5 GeV are produced. There is some evidence that a "cluster" is a synonym for a sum over the known vector and tensor meson resonances /322/. The mean decay m u l t i p l i c i t y and the width of the correlation induced by cluster decay are characteristic and energy independent features of low p~ interactions. To investigate such short-range rapidity correlations among spectator fragments, the perturbation by particles from the two jets at large p~ has to be minimized. To achieve this goal the CCHKgroup /317/ studied events where both the towards and the away j e t are in the same rapidity hemisphere at rapidities y < -0.7. The j e t rapidities are taken as the rapidity of the trigger particle and as the rapidity of the fragment with largest p~ in the away j e t region. Correlations were studied among spectator fragments at positive rapidities. Figure 7.63 shows the two particle density correlation
C ' ( y l l y 2) : p(yiIY2)-p(yl ) for Y2 ~ 2 and Y2 ~ 4. The correlation function C ' ( y l l y 2) describes the change in particle density at Yl for events which have a particle at Y2 as compared to the inclusive rapidity distribution. C' is closely related to the correlation function C defined by (4.47) /316/. From Fig.7.63 a strong short-range correlation is evident. Both size and width of the correlation essentially agree with the corresponding distributions obtained in ordinary nondiffractive proton-proton collisions at the same cms energy /316/. In Fig.7.64, the charge balance Aq(yl,Y2 ) is plotted as a function of the rapidity YI" Aq(YI'Y21 (Fig.7.44) measures where in rapidity the charge of a particle selected at Y2 is balanced. Charge is seen to be conserved locally in phase space, like in normal nondiffractive hadron reactions /316/. The s i m i l a r i t y of correlations in inelastic events at low p~ and in spectator fragmentation at large p~ suggest that the same mechanism of fragmentation or confinement occurs, supporting the hypothesis of j e t universality in i t s most general sense. Moreover, the observed correlations are in qualitative agreement with the preconfinement concept in QCD which states that during the evolution of the rapidity plateau color singlet clusters with masses of the order of I GeV are created. From parton diagrams describing j e t formation via successive branching (Fig.5.6), one learns that these states tend to be flavor neutral. Assuming that such clusters
178 CHARGE DENSITY BALANCE ASSOCIATED WITH SPECTATOR FRAGMENT ~, high P-r -normal inelastic
,
'
! .O
--
0,5
/
i
O,*7t '-9--xsL:-; - - , ' -4
-2
0 Yl
2
~
4
,~T~]-J-!T" T - ~ - , '' -q
-2
0 Yt
2
4
i9.7.63_. Two-particle density correlation C'(y I y2 ) for spectator fragments in #arge p~ events for two r a p i d i t i e s Y2 = 2 and Y2 = 4. The f u l l lines show the corresponding d i s t r i b u t i o n s in normal i n e l a s t i c events. From CCHK /317/
TWO-PARTICLE ~, high i
"PT T
I
CORRELATION ASSOCIATED WITH SPECTATOR FRAGMENT - normal inek~slic I
i
0.2 0,t
-O'Ifllll~
-0.2 -0.:5
0 >~ v ~o -O.1
~
-O,2 _I -,
I
-2
I
I
i
O
2
4
Yl
~
-4
,
-2
0
2
4
Yl
Fi9.7.64. Associated charge-density balance A q ( y l l y 2) for spectator fragments as a function of Yl f o r Y2 = 2 and Y2 = 4. Full lines show the corresponding d i s t r i b u tions obtained in normal i n e l a s t i c events. From CCHK /317/
decay without a large reshuffling of quark l i n e s , one immediately predicts both size and width of the observed correlations and explains the observed local conservation of charge.
179
7.5
Summary
The distribution in phase space of secondaries in hadron-hadron interactions with large p~ particles is consistent with the four-jet structure characterizing a basic two-body parton-parton scattering. The main features of both inclusive spectra and particle ratios at large p~ are described in the QCD model where quark-quark, quark-gluon and gluon-gluon subprocesses contribute. Especially at medium p~ ~ 2-5 GeV/c, scale-breaking effects and corrections due to parton k have to be taken into account. Significant contributions from constituent interchange processes, like qq § MR, or qM § q'M' are excluded. The properties of the jets at large p~ agree with those of quark jets observed +
-
in e e annihilations, except for a possible increase of the plateau height. There is no clear evidence for the presence of gluon jets with fragmentation functions markedly different from quark fragmentation. However, based on the experience gained from the study of T decays, such a difference is not expected to show up at j e t momenta below 5 to 10 GeV. In agreement with QCD predictions, the quantum numbers of partons at large p~ are essentially uncorrelated, proving that no flavors are exchanged in the subprocess. Only for production of "exotic" particles which cannot result from favored fragmentation of a scattered valence quark, certain correlations are observed. However, the experimental situation is s t i l l ambiguous. A further test of the QCD model was obtained by studying spectator fragmentation. +
I t is shown that in proton-proton reactions with a forward large p~ ~ and ~ , an u- and d-quark, respectively, is scattered out of the proton moving in the same longitudinal direction as the pion, proving further that the underlying partonparton cross section is strongly peaked forward: A comparison of p~ distributions and two particle correlations in the spectator region with those obtained in normal inelastic events proves that similar mechanisms of fragmentation act in both cases.
8. Hadron-Hadron Interactions at Low p .
Motivated by the success of the quark-parton model in the description of deep ine l a s t i c processes, one is tempted to apply this picture to normal i n e l a s t i c hadron interactions as well. Many of the general aspects of low p~ hadron interactions have been referenced already in Chaps.2 and 7. - The secondaries form jets around the c o l l i s i o n axis. - Similar to the jets observed in lepton induced reactions, these jets consist of a fragmentation region carrying the quantum numbers of the incident p a r t i c l e and of a plateau region. The retention of quantum numbers l i k e charge is i l l u s t r a t e d in Fig.8.1, where p a r t i c l e density and charge density d i s t r i b u t i o n s per event are shown as a function of the r a p i d i t y y, for pp reactions at vrs = 52 GeV /316/. - The s dependence of p a r t i c l e spectra is described by the concept of l i m i t i n g fragmentation /2,323/. Inclusive spectra in the fragmentation region scale in Feynman x, respectively in x E = 2 E / ~ . With increasing s, the length of the plateau increases, but i t s other characteristics e s s e n t i a l l y stay unchanged (Fig.8o2). = Inclusive fragmentation spectra factorize in a sense that at high energies, beam and target fragmentation depend only on the type of beam and target p a r t i c l e . This is visualized in Fig.8.3. -
Transverse momenta with respect to the c o l l i s i o n axis are l i m i t e d . Mean momenta are of the same size as the nonperturbative component of p~ in reactions at large momentum transfers.
All this is precisely what is expected in parton models where fragmentation spectra are related to the d i s t r i b u t i o n of valence quarks in the incident particles /13-15/. In the following section, we shall concentrate on the main characteristic of parton processes, the x d i s t r i b u t i o n of fragments.
181
PARTICLE l
AND CHARGE DENSITY I
I
I
9 p(Y) =p+ (y)+ p-(y ) o q (y):p+ ( y ) - O-(y )
I
IECC model
3
/
\
0 -4
-2
0
2
4
Y
8.1
Fig.8.1. Particle and charge density in h i g h - m u l t i p l i c i t y proton-proton reactions at ~ = 52 GeV as a function of the rapidity y. From CCHK/316/
Longitudinal Fragmentation Spectra
To apply models l i k e DCR or QRM to reactions at low p~, one has to identify the basic interaction which gives rise to the fragmentation. In deep inelastic scattering the process i n i t i a t i n g the fragmentation is well known: a large momentum transfer leads to a separation of color carriers in space. The existence of a rapidity plateau connecting the fragmentation regions of low p~ events indicates that in the i n i t i a l interaction "something" is exchanged between the incident hadrons which gives rise to a long-range force. The most natural candidate for this "something" is color..Within the general concept of QCD there are two mechanisms which could provide a color exchange. - color-octet, vector-gluon exchange (Fig.8.4b) /325-327/, - c o l o r - t r i p l e t quark exchange, respectively quark-antiquark annihilation (Fig.8.4a) /4,15,73/. The gluon exchange mechanism was the starting point for the Pomeron model of LOW /325/ and NUSSINOV /326/ and has the advantage of automatically generating a constant high-energy cross section. The quark exchange or annihilation mechanism is the QCD equivalent of Feynmans wee-parton exchange /4/. The quark exchange model yields total cross sections rising as In(S) for seaquark-sea-quark interactions and constant cross sections for valence-quark-sea-quark
182 I
!
'L = 0.4
GeVlc
(
I
{
I
(
,
j
I
(
p 9 p--K;..
..~l~
P. P ~
,,
o.
~--*
,/ / /
p.p~K% t
~
i
le
.
|
e
If'
,e
.
U
f J/
3 E
&
&
Ct
~1~ 0.1
UJ
eO~ /0 / / / / /
~0
BC.
-e
C.H.L M
BS.
9 11 9
II ~
SS
O B
23 GeV 3I 45
O
---
,%
6.8
/ O0
/
84
/
I I I I I
t o
J o~
/
( 1.o
~ , 1.s
} 20
]
I
)
I
2.5
3.0
3.5
4.0
YLAB = YMAX - y
Fig.8,2. Inclusive distributions of particles produced in proton-proton interactions at v~ = 6.8 to 53 GeV plotted as a function of YMAX-Y ~ -In(x) at fixed p . /324/
183 1,0
,
os
,
r
,
,
,
,
,
LO
,
"n'*P~Tr~~
I
i
I
i
i
i
i
i
i
pp__.rr•
-
0.81-
-~ 0,6
0'6 f
"O
-Ib'-
0.4
0.4
II
0'2t
II
m:>,, 0.2
Q. 0.0'
-5.0 -3.0 -I.0
y*
1.0
0.0 '
-5.0 -3.0 -I.0
3.0 5.0
1.0
5,0
5.0
y* 4.0
i
i
|
E
!
i
i 1 I
3.01L ,~Q" 2.0
~
..§
| .I
'T"
~
R~
3.(
+"
'-l"
'
'
'
4-
3.0 ~
RK
-C 2.0 't 0.0 ~
,
,
-4.0 -2.0
,
i
0.0 y.
,
,
,~
2.0 4.0
Q"
0.0 -4.0
'
'
-2.0
'
0.0
yO
'
z
2.0 4.0
Fi9.8.3. (a) Particle density per event (I/o)(do/dy) as a function of the cms rapidity y for ~+p § 7+ , 7- (shaded) and pp § ~+, 7- at ~ ~ 14 GeV. (b) Ratios of the positive and negative pion densities for ~+p § 7+X/pp + ~+X (R~) and for K+p § ~+X/ pp § ~+X (R~). Particle densities in the target fragmentation regions (y < O) are independent of the type of the beam p a r t i c l e /328/
reactions /193/. Processes such as valence-quark-valence-quark fusion (e.g., in 7N reactions) give a negligible contribution to the total high-energy cross section since the valence wave function of beam and target barely overlap. A p o s s i b i l i t y to distinguish between the quark and gluon exchange mechanisms has been proposed by BRODSKYand GUNION /193/. I t is based on the ansatz that p a r t i c l e spectra at large x are determined by those quark diagrams containing the minimum number of quark lines. According to the DCR, for example, each additional spectator line damps the x distributions of fragments by at least ( l - x ) (see Sect.6.6). Consider
184 d ~tlTd,
\
u u
/
u u o -
>
~
\
u
9
u~u / /
/" ~
0.9 r e s u l t from tripple-Regge contributions /197/
Figure 8.6c shows the predictions of " p o i n t l i k e " counting rules for the favored and unfavored fragmentation function for quark j e t s . The predictions ( l - x ) and ( l - x ) 2 are s l i g h t l y too f l a t in x compared to the measured spectra for 0.3 ~ x ~ 0.8 which is the same range of x as used in the discussion of spectator fragments. An explanation for this tendency may be given by scale-breaking effects: in the evolution of a quark j e t , the formation of f i n a l state hadrons w i l l occur at 2 Q2. some scale QO ~ I GeV2 A2 >> QO
(8.5)
2 where A is the QCD scale parameter. In the fragmentation from Q2 to QO partons are generated which increase the number of spectator f i e l d s as compared to the minimum number [see (6.36)]: zD(z) ~ ( l - z ) neff(Q2)-1
Recently quantification of these ideas within the framework of the QRM has been t r i e d /335/. QCD (5.27) is used to describe how a quark at Q2 fragments into a
190 2 "valence quark" and a number of "sea" partons of size QO" This evolution is usually characterized by the parameter ~ [see (5.32 9
- InL
_
J
2=0
2=0 ~=1.0
0.01
,
t?
0.5
1.0
Fig.8.8. Single-parton (quark, antiquark, and gluon) inclusive d i s t r i b u t i o n in a quark j e t for d i f f e r e n t values of ~ ~ In[~s(Q~)/~(Q) ] . The curves are calculated using an approximation ~o the exact QCD matrix elements [see (5.23~ /336/
Figure 8.8 shows how the parton density in a quark j e t changes from ~(1-x) at = 0 to a function decreasing with increasing x for large ~ /336/. Final state mesons are formed by recombination according to (6.!7) dz q
dzq
where G(Zq,Z~,O~), is the two-parton density within the j e t . To deal with gluons, one may use the prescription /335/: "at QO 2 all gluons convert into quark-antiquark pairs". Practical applications of this scheme suffer from the fact that the shape of 2 depends c r u c i a l l y on the unknown r a t i o Qo/A and that for the region G(Zq,Zq,Q O) of i n t e r e s t QO ~ I GeV, the perturbative expansion starts to become unreliable. In /335/ Qo/A ~ 1.1 is used as extracted from deep inelastic IN scattering (in a model-dependent way), and an " e f f e c t i v e " A is chosen to account for higher-order correctibns. Although this procedure is somewhat questionable, the results (Fig.8.9) look encouraging, especially i f vector meson production is taken into account.
191 I0.
DASP O = 5.2 GeV {
\\ i.
I.
rm 0.1
0.01
I 0.2
I 0.4
MPP(SPEAR) Q = 4.8 GeV
% I
I
0.6 Z
I
0.8
Fig.8.9. Absolute QRM predictions for quark frag~ n functions with (.... ) and without (- - -) vector meson contributions /335/
9. Summary
The existence of j e t l i k e structures is shown to be a general characteristic of hadronic f i n a l states in reactions of elementary p a r t i c l e s at high energies. The quarkparton model suggests that these j e t s r e s u l t from the fragmentation of colored partons. Although the partons produced in various reactions may d i f f e r in t h e i r quark contents, in t h e i r color state, and in the momentum transfer Q2 at which they are probed, the resulting jets are expected to e x h i b i t many common features. In the present work, properties of j e t s observed in +
-
- e e annihilations into hadrons, -
-
-
hadronic decays of the T meson, deep i n e l a s t i c lepton-nucleon scattering, hadron-hadron reactions with large p~ p a r t i c l e s , and
- normal i n e l a s t i c hadronic reactions are compared, and phenemenological approaches to describe t h e i r properties are discussed. In the parton model', the space-time evolution of a j e t may be visualized as an "inside-outside" cascade: the color charge of the active parton induces a p o l a r i zation cloud which follows the parton and f i n a l l y neutralizes i t s color. I t is shown that phenomenological models for j e t fragmentation, l i k e the FeynmanField algorithm, can be derived from this concept. The model of "inside-outside" cascades is further supported by QCD calculations. Here, parton fragmentation is treated as a successive branching into partons of smaller mass, y i e l d i n g the same space-time structure of jets as the very naive parton model. According to these concepts, a j e t should consist of a fragmentation region and a plateau region. Parton quantum numbers are e s s e n t i a l l y retained in the fragmentation region. Jets in d i f f e r e n t environments and at d i f f e r e n t energies are related via scaling fragmentation functions, environmental independence of the fragmentat i o n , and by the u n i v e r s a l i t y of the r a p i d i t y plateau.
193 +
-
Data on quark jets observed in e e annihilations and in lepton-nucleon interactions are consistent with these ideas. Small violations of the idealized concepts can be understood quantitatively as due to phase-space effects at low energies and QCD bremsstrahlung corrections at high energies. In contradiction to QCD predictions, the investigation of gluon jets in T decays shows no evidence for a softening of fragmentation functions as compared to quark j e t s , and average m u l t i p l i c i t i e s are very similar for both types of jets. However, parton fragmentation at these energies is governed by phase-space effects rather than by asymptotical parton concepts. In hadron-hadron interactions involving large momentum transfers, pairs of jets with opposite large transverse momenta are observed. Both longitudinal and transverse properties of these jets are essentially consistent with those of "standard" quark jets. This fact, as well as measurements of inclusive particle ratios at large p~, favor QCD models where quarks or gluons are scattered, in contrast to constituent interchange mechanisms. The absence of correlations between the charge of a pion at large p~ and the quantum numbers of the opposite recoiling parton, proves further that parton scattering occurs through the exchange of flavor singl e t f i e l d s , such as gluons. In such interactions, where a quark is scattered out of a hadron, or more gener a l , where a current of large Q2 interacts with a single parton, a colored spectator system is l e f t over and fragments. In the present work, special emphasis is put on the investigation of spectator fragments, since only here can the relation between the quark contents of a composed colored system and i t s fragmentation functions be studied systematically. Fragmentation spectra of diquark spectator systems produced in vN interactions are shown to be compatible with predictions based on counting rules (DCR) or the quark recombination model (QRM). Since the shape of fragmentation spectra characterizes the quark contents of the spectator, this provides a new method to determine the type of constituent subprocess occurring, i . e . , in events with large p~ particles. This method, having been applied to proton-proton interactions, has demonstrated that forward high-p~ pions are fragments of valence quarks scattered at small angles. In an analogous way fragmentation models can be used to describe hadron production at low p~ in normal inelastic hadronic interactions, once the mechanism i n i t i a t i n g the fragmentation is known. The absence of correlations between fast pions emitted in opposite rapidity hemispheres points towards a primary gluon exchange as compared to flavor exchange mechanisms.
194 Fragmentation spectra of incident hadrons into mesons and baryons are essentia l l y consistent with the quark recombination picture and with a " p o i n t l i k e " version of counting rules. There are indications that a combination of the perturbative evolution of parton densities in j e t s , as given by QCD, and of a recombination mechanism provides an universal tool to understand parton fragmentation. The d i s t r i b u t i o n of low "mass" partons within a j e t of large Q2 can be derived using QCD, resp. is given by the primordial parton d i s t r i b u t i o n in reactions at moderate momentum transfers. The recombination p r i n c i p l e t e l l s how these quarks convert into hadrons.
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201 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330
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Subject Index
Accompanying spectator 166 ms 26, 50, 68
inside-outside 30
-
ALTARELLI-PARISl equations 51 Annihilation
Cascade Charge
i n t o gluons 108
Approach to scaling 19
-
-
Associated charge density 160 Away j e t 129, 162 -
charge 161
balance 159, 177 correlations in high p~ events 143, 157, 177
- d i s t r i b u t i o n in quark jets 41, 43, 87 -
retention, see retention
- charge r a t i o 164
Charm production in j e t s 60
- c o r r e l a t i o n with t r i g g e r 161
Charge r a t i o
Away spectator 166 Azimuthal d i s t r i b u t i o n of p a r t i c l e s in j e t s 156
-
away j e t 164
- spectator j e t s 93, 100, 172 - t r i g g e r 139, 141 Chromodynamic Meissner e f f e c t 26
Back-to-back j e t configuration 162
CIM, see Constituent Interchange Model
Back-antiback j e t configuration 162
Cluster model 22
Bag model 26
Cluster, c o l o r - s i n g l e t 57
Beam matios
Cms frame, in lepton-nucleon scattering 75
- high p~ events 138
Coherent spectator states 74
- low p~ reactions 183
Color 26
Boltzmann s t a t i s t i c s 22
-
Bose-Einstein correlations 22
- f l u x tubes 27, 31, 55, 109, 122
Bose-Einstein s t a t i s t i c s 22
-
center frame 110 octet exchange 181
Bottom quarks 108
- octet, fragmentation 63, 110
Branching 56
- octet j e t s 63, 110
B r e i t frame 75
- screening 30
Breaking of co]or strings 32, 34
- separation in QCD jets 56 -
singlets 57
- strings 32, 34
204 - s t r i n g s , tension I~0
- s p h e r i c i t y d i s t r i b u t i o n s 92
- s t r u c t u r e , high p~ events 121 +
valence quark s c a t t e r i n g 73, 99
=
- structure, e e
a n n i h i l a t i o n s 27, 110
Dimensional Counting Rules 102
- s t r u c t u r e , T decays 110 - triplet
exchange 181
fragmentation functions 106, 173 - helicity
f a c t o r s 106
Compensation of t r i g g e r charge 160
- low p~ i n t e r a c t i o n s 183
Constituent Interchange Model
- p o i n t l i k e coupling 107, 173
+
-
- e e
a n n i h i l a t i o n s 12
scaling v i o l a t i o n s 106
-
- high p~ reactions 136
- spin e f f e c t s 106
Conservation of quantum numbers 159
- s t r u c t u r e functions 105 Dipole density 28
Confinement 25, 55 time scales 29, 56
-
two-dimensional QED 27
-
Diquark 72, 90 fragmentation 90, 100, 172
-
Correlation +
-
- f u n c t i o n 177, 184
e e
- high p~ events 143, 159, 177
- r a p i d i t y plateau 9
quantum numbers 157, 177
-
- short range 22, 177
annihilations 4
scaling 8
-
scaling v i o l a t i o n s 10, 67
-
- spectator fragmentation region 177
- t h r e e - j e t events 12
- Uncorrelated Jet Model 15
- t o t a l cross-section 4
Counting r u l e s , see "Dimensional Counting Rules" Cross-Section, see also " i n c l u s i v e " +
- e e
-
annihilation 4
transverse momenta 7
-
Early s c a l i n g 21 Enhanced quark sea 97 Environmental independence 76
- parton-parton s c a t t e r i n g 127
Event plane 12, 68
Current fragmentation region 73, 86
Evolution equations 48
-
D(z) 33, 39, 53
parton d e n s i t i e s 48, 5 1 , 5 5 , 66,132, 189
-
DCR, see Dimensional Counting Rules
Exchange
Decays, T 108
- c o l o r octets 181 color triplets
Deep i n e l a s t i c s c a t t e r i n g 46, 71, 121 - c o l o r s t r u c t u r e 73 -
f i n a l states 73
- j e t development 72
181
quarks 181
-
- wee quarks 181 Experiments, high p~ 123
- proton production 94, 99 -
-
-
quark-parton model 76, 79 reference frames 75 sea-quark s c a t t e r i n g 73, 99
F a c t o r i z a t i o n 76, 79, 122, 127, 180 -
-
experimental evidence 77 hadron-hadron reactions 180
205 -
Hard scattering expansion 137
v i o l a t i o n 80
Hard-gluon radiation
Favored fragmentation 39 Feynman-Field model 35
-
deep i n e l a s t i c lepton-nucleon reactions
Flavor c o r r e l a t i o n s 159, 174
79 +
-
Fock states 98
- e e annihilations 10
Fragmentation
- high p~ 155, 169
- current 73, 86
Heavy-quark production 59
diquark 90, 100, 172
H e l i c i t y factors in DCR 106
Fragmentation function 33, 39, 53, 82
Higher twists 80
behavior at large x and Q2 54
- high p~ reactions 137
- coler octet 63, 110 -
High p~ events
Dimensional Counting Rules 106,173
-
- favored 39 _
-
charge correlations 160
- exotic trigger particles 165
Q2 dependence 46, 52, 79
-
scale breaking 47, 66, 132
-
kinematics 125 momentum balance 157
- spectator 90, 100, 172
- multiplicity
- towards/away j e t in high p~ events
-
148, 153
153
quantum number correlations 159 spectator jets 129, 166
- unfavored 39
- t w o - p a r t i c l e correlations 143, 146,
Frequency of j e t s , in high p= events
160, 177
147
- t r i g g e r bias 128 - x E d i s t r i b u t i o n 129, 148
Gluon
- z of t r i g g e r 128
- bremsstrahlung 14, 46, 62, 79, 135
High p~ reactions
- conversion into qu~rks 97, 190
-
- r a d i a t i o n , high p~ events 135,
-
155, 168
-
- spin 68, 117
-
beam r a t i o s 138 i n c l u s i v e cross sections 128 natural mechanisms 137
- p a r t i c l e r a t i o s 138
structure function 50, 132
_ @3 theory 137
T decays 108
- QCD 130
Gluon jets 63, 110 - asymptotic behavior 64, 66
-
quark-parton model 125
High p~ experiments 123
i n c l u s i v e spectra 66 - m u l t i p l i c i t y 66, 112 -
scaling v i o l a t i o n 66
Inclusive cross sections +
-
- e e annihilations 8 - jets 124, 127, 140
Hadron-hadron interactions 180 - f a c t o r i z a t i o n 180
high p~ reactions 128, 133, 136 Inclusive spectra 8
206 ÷
=
- e e annihilations 8
K
gluon fragments 66
-
- quark fragments 39 Regge contributions 185
-
production in high p~ events 138, 165, 176
Kaon production in T decays 119 Kinematics, high p~ events 125
- spectator fragments 90, 100, 172
KINOSHITA-LEE-NAUENBERG theorem 62
- uncorrelated j e t model 17
KUTI-WEISSKOPF model 98
T decays 115
-
Inside-outside cascade 30, 38
Large p~, see high p~
Integral equations
Leading particle effect 55
- QCD jets 51
Lepton-nucleon scattering 46, 71
- quark fragmentation 34, 36
Lepton-nucleon quasielastic scattering 75
Isospin distribution in quark jets 43
Limiting fragmentation 180 Limited transverse momenta 7, 16, 61, 155, 168, 180
Jet cross section 124, 127, 140 Jet development
Local conservation of quantum numbers 159
- deep i n e l a s t i c reactions 72
Long range correlations 159
- high p~ reactions 121
Lorentz invariance, j e t formation 33
- integral equations 34, 36
Lorentz i n v a r i a n t phase space 15
- Lorentz invariance 33
LOW-NUSSINOV model 181
-
parton model 25, 71, 79, 125
Low p~ phenomena 180 low p~ interactions
- QCD 26, 46
Dimensional Counting Rules 183
- two-dimensional QED 27
-
- T decays 109
-
Jetiness 5
LUND model 31
Quark Recombination Model 185
Jet leader 161 - charge 161
Mandelstam variables 125
Jets 4, see also "fragmentation",
Matrix elements
" i n c l u s i v e spectra", "gluon j e t s " , "spectator jets",~Uncorrelated ÷
Uncorrelated Jet Model 17 Master equations 51
Jet Model" Jets in e e
KUTI-WEISSKOPF model 98
-
annihilations 4
- l i m i t e d p~ 7, 16, 61, 82, 155, 168
Mass of color singlets in jets 57 Meissner effect 26
- v i o l a t i o n of scaling 12, 19, 47, 66, 132 Meson wave function 95 Jets in hadron-hadron reactions 180
Moments 48, 85
Jets, space-time development 27, 32, 56
- nonsinglet 52
Jet u n i v e r s a l i t y 88
- singlet 52
Jet t r i g g e r 124
Momentum balance, in high p~ events 157 Monte-Carlo simulation of patton cascades 59
207
Multiplicity
Planar diagrams 57
- of color singlet clusters in jets
"Pointlike" Dimensional Counting Rules 107, 173
58 -
deep i n e l a s t i c scattering 88 +
Polarization charge 28
-
- e e annihilations 4, 59
Pomeron model of LOW, NUSSINOV 181
- gluon jets 65, 112
POUT' d e f i n i t i o n
- high p~ events 153
Preconfinement 55, 72
- quark jets 4, 59
- high p~ reactions 135
-
-
157
T decays 111, 114
Primordial transverse momentum 135, 157
x and Q2 dependence 89
Primary mesons 36 Proton production
Natural mechanisms 137
-
Nonsinglet moments 52. 85
- hadron-hadron reactions 182
deep i n e l a s t i c scattering 94 high p~ reactions 171, 175, 176
-
Off-shell partons 52, 79
Probabilistic quark model 98 P r o b a b i l i t y , for gluon bremsstrahlung 14
Particles, high p~ 121
Pseudosphericity 116 2 12 P-IN
Particle r a t i o s , high p~ 138 Patton -
-
P~ OUT 12
cascade 30. 54 cascade, Monte-Carlo simulation 59
- density in jets 46, 55, 66, 189 -
Q2 dependence
dynamical transverse momentum 157
- structure functions 46, 79
- jets 46 -
-
fragmentation functions 46, 52, 79
-
- formfactor 56
QCD, see Quantum Chromo Dynamics
model 25, 76, 125
Q
model, time scales of interactions 25
E
D
two-dimensional 27
,
QRM, see Quark Recombination Model Quantum Chromo Dynamics 26, 46, 50, 130
- -parton cross section 127
- correction of order (~S) 80
- -parton scattering 125
- corrections, quark-patton model 79
-
-
+
-
primordial transverse momentum 157
- e e annihilations 67
shower 30, 54
- high p~ reactions 130
- t r a j e c t o r i e s , classical approximation 31 transverse momentum 134, 147, 157 Phase space 15
-
subprocesses 130
Quantum number correlations 159
scaling violations 78, 84 @3 theory, high p. reactions
- j e t s , threshold effects 60 - j e t s , transverse width 61
- trigger/spectator 171 137
Quantum numbers, local conservation 159
208 Quantum number s t r u c t u r e , quark j e t s 43 Quantum number r e t e n t i o n 43, 84, 180
R a p i d i t y plateau 9, 16, 34 gluon j e t s 65
-
- r i s e 20, 23
Quark - c r e a t i o n , in j e t s 31
Uncorrelated Jet Model 16
-
- exchange 181
- quark j e t s 189
- fragmentation 4, 26, 39
- spectator fragmentation 175
fragmentation f u n c t i o n 39, 82 - fragmentation region 74 - fusion 181
Recombination - f u n c t i o n 95 model 96
-
- hole 74
- multiquark 95
- j e t s 4, 26, 39
- spectator fragmentation 100, 172
- jets
distribution
of charge 43
R e c o i l , parton p~ 168
- jets
distribution
of isospin 43
Reference frames, deep i n e l a s t i c scat-
- jets
distribution
of strangeness 43
jets
-
- jets -
-
extension in space-time 30 multiplicity
Regge slope 31
4, 59, 88, 153
jets
Quark Recombination Model 189
Resonance production
jets
s i m u l a t i o n 35, 59
- quark j e t s 34, 37
-
e t s , vector meson production 37 model 4, 25, 76, 125
-
t e r i n g 75 Regge c o n t r i b u t i o n s 185
- model, QCD c o r r e c t i o n s 79 - model, deep i n e l a s t i c reactions 79 - model, p r o b a b i l i s t i c 98 - -quark s c a t t e r i n g 121
- Uncorrelated Jet Model 22 - spectator fragmentation 171 Retention charge 43, 87, 180
-
quantum numbers 43, 84, 180
-
p meson production 22
- r e s t frame 75 -
-
sea 97
Scale breaking 47
sea enhancement 97
- fragmentation functions 52, 66, 79,
Quark Recombination Model 94 - low p~ hadron i n t e r a c t i o n s 185 Q u a s i e l a s t i c s c a t t e r i n g 75
82, 132, 154 - high p~ reactions 131, 154 phase space e f f e c t s 84
-
s t r u c t u r e functions 51, 131 R4
Scale invariance 47
Radial s c a l i n g 186
Scaling 8, 19, 77, 180
Rapidity 9
- approach to s c a l i n g 19
Rapidity distribution
- i n c l u s i v e spectra 8
+
- e e
-
a n n i h i l a t i o n s 10
hadron-hadron i n t e r a c t i o n s 182 - proton-proton reactions 10
-
-
-
r a d i a l 186 Uncorrelated Jet Model 19 v a r i a b l e s 75, 77
209 - variables, spectator fragmentation
-
91, 130, 168, 172
-
Scaling violations 12, 47, 50, 78, 86 -
-
-
deep i n e l a s t i c scattering 84
Spectator, quantum number correlations 171
dimensional counting rules 106
Sphericity 5, 92 -
deep i n e l a s t i c scattering 92 +
-
phase space effects 78
- e e annihilations 7
Quantum Chromo Dynamics 50, 131
- T decays 117
- Uncorrelated Jet Model 20
Spin, Dimensional Counting Rules 106
Scaling in x E 129, 148
Spin, gluon 68, 117
Scattering
S p l i t t i n g functions 51
- quark-quark 121
STERMAN-WEINBERG jets 62
- parton-parton 125
Strange-meson production in jets 34, 37, 61
Schwinger model 27, 64
Strangeness d i s t r i b u t i o n in quark jets 43
Schwinger phenomenon 30
Strong coupling constant 26, 50, 68
Seagull e f f e c t 10, 18, 156
Structure functions 8, 46, 127
Secondary mesons 36
-
Short-range correlations 177 -
transverse momentum 168
- two-particle correlations 177
- fragmentation functions 52 -
time scales 106, 176
-
Uncorrelated Jet Model 22
-
Simulation of quark j e t s 35, 59
_
Singlet moments 52
-
Dimensional Counting Rules 105 gluon 132 KUTI-WEISSKOPF model 98 Q2 dependence 46, 79 scale breaking 51, 131
Single p a r t i c l e t r i g g e r 124
Structure of high p~ events 141
Soft gluon radiation, high p~
SU(3) breaking 37
events 168 Space-time development - parton jets 27, 32
jets 34 - quark creation 31 -
sea quarks 87
- QCD j e t s 56 Spectator 72, 90
Target mass corrections 80
- diquark 72, 100, 172
Tension, color strings 110
Spectator fragmentation 74, 90, 166
Tests, QCD 67
- azimuthal d i s t r i b u t i o n in high p~
Three-jet structure
events 168 -
-
-
+
-
- e e annihilations 12
charge balance 177
- T decays 116
Dimensional Counting Rules 106, 172
Threshold effects in jets 61
- high p~ events 172 -
Target fragmentation region 74, 86
- coherence 74
Quark Recombination Model 97, 175 scaling variables 91, 130, 167, 172
Thrust 7 Time scales confinement 29, 56
210 parton i n t e r a c t i o n s 25 - spectator fragmentation 106, 176
-
-
decays, kaon production 119 decays, matrix-element 109
Towards j e t 142, 151
- decays, s p h e r i c i t y a x i s 117
Transverse momentum
- decays, t h r u s t 117
- c u t - o f f 7, 16, 61, 155, 168, 180
- decays, t h r e e - j e t s t r u c t u r e 116
- partons in j e t s 61 - quark j e t s 38
Vector meson production, quark j e t s 37
- secondaries in high p~ j e t s 155
V i o l a t i o n of
spectator fragments 168
-
Uncorrelated Jet Model 16
-
-
-
f a c t o r i z a t i o n 80 s c a l i n g 12, 47, 50, 78, 86
- s c a l i n g , Uncorrelated Jet Model 20
T r i p l i c i t y 117 Trigger bias 128 - j e t trigger 128 Trigger j e t 151
Wee parton 181 -
Tunnel effect, in color f l u x tubes 31
exchange 181
Width of QCD j e t s 62
Two-dimensional QED 27 Two-particle correlations 18, 22, 159, 177, 184
xE 129
- hadron-hadron i n t e r a c t i o n s 184 high p~ events 143 Uncorrelated Jet Model 18, 22
-
x d i s t r i b u t i o n , gluons in T decays 109
T w o - p a r t i c l e density 177
xE distribution 148 xE scaling 129, 148 -
approach to scaling 148
xj 161 xR 16
UJM, see Uncorrelated Jet Model Uncorrelated Jet Model 15, 94 +
- e e
-
a n n i h i l a t i o n s 20
r a p i d i t y plateau 16
-
resonance production 22
-
- short-range c o r r e l a t i o n s 22 - t w o - p a r t i c l e c o r r e l a t i o n s 18 Unfavored fragmentation 39 Universality - j e t s 88 - multiplicity -
88
r a p i d i t y plateau 88
T meson 108 -
a n n i h i l a t i o n i n t o gluons 108
- decays, i n c l u s i v e spectra 115
< z >, high p~ t r i g g e r p a r t i c l e
128
Classified Index Springer Tracts in Modern Physics, Volumes 36-90 This cumulative index is based upon the Physics and Astronomy Classification Scheme (PACS) developed by the American Institute of Physics
General 04 Relativity and Gravitation
Heintzmann, H., Mittelstaedt, P. : Physikalische Gesetze in beschleunigten Bezugssystemen (Vol. 47) Stewart, J., Walker, M.: Black Holes: the Outside Story (Vol. 69) 05 Statistical Physics
Agarwal, G. S.: Quantum Statistical Theories of Spontaneous Emission and their Relation to Other Approaches (Vol. 70) Graham, 1~: Statistical Theory of Instabilities in Stationary Nonequilibrium Systems with Applications to Lasers and Nonlinear Optics (Vol. 66) Haake, F: Statistical Treatment of Open Systems by Generalized Master Equations (Vol. 66)
07 Specific Instrumentation
Godwin, R, P.: Synchrotron Radiation as a Light Source (Vol. 51)
The Physics of Elementary Particles and Fields 11 General Theory of Fields and Particles
Brandt, R. A.: Physics on the Light Cone (Vol. 57) Dahmen, H. D.: Local Saturation of Commutator Matrix Elements (Vol. 62) Ferrara, S., Gatto, R., Grillo, A. F.: Conformal Algebra in Space-Time and Operator Product Expansion (Vol. 67) JaclOw, R.: Canonical Light-Cone Commutators and Their Applications (Vol. 62) Kundt, W..: Canonical Quantization of Gauge Invariant Field Theories (Vol. 40) Riihl, W..:Application of Harmonic Analysis to Inelastic Electron-Proton Scattering (Vol. 57) Symanzik, K.: Small-Distance Behaviour in Field Theory (Vol. 57) Zimmermann, W.: Problems in Vector Meson Theories (Vol. 50)
11.30 Symmetry and Conservation Laws
Barut, A. O.: Dynamical Groups and their Currents. A Model for Strong Interactions (Vol. 50)
Ekstein, H.: Rigorous Symmetrics of Elementary Particles (Vol. 37) Gourdin, M.: Unitary Symmetry (Vol. 36) Lopuszahski, J. T.: Physical Symmetrics in the Framework of Quantum Field Theory (Vol. 52) PaulL W.: Continuous Groups in Quantum Mechanics (Vol. 37) Racah, G.: Group Theory and Spectroscopy (Vol. 37) Riihl, W.: Application of Harmonic Analysis to Inelastic Electron-Proton Scattering (Vol. 57) Wess, J.: Conformal Invariance and the Energy-Momentum Tensor (Vol. 60) Wess, J.:Realisations of a Compact, Connected, Semisimpie Lie Group (Vol. 50)
11.40 Currents and Their Properties
Furlan, G, Paver, N., VerzegnassL C.: Low Energy Theorems and Photo- and Electroproduction Near Threshold by Current Algebra (Vol. 62) Gatto, R.: Cabibbo Angle and SU2 x SU2 Breaking (Vol. 53) Genz, H.: Local Prop erties of o-Terms: A Review (VoL 61) Kleinert, 1t..:Baryon Current Solving SU (3) Charge-Current Algebra (Vol. 49) Leutwyler, H.: Current Algebra and Lightlike Charges (Vol. 50) Mendes, R. K, Ne'eman, I(.: Representations of the Local Current Algebra. A Constructional Approach
(Vol.60)
Miiller, K E: Introduction to the Lagrangian Method (Vol. 50) Pietschmann, H.: Introduction to the Method of Current Algebra (Vol. 50) Pilkuhn, t1.: Coupling Constants from PCAC (Vol. 55) Pilkuhn, H.: S-Matrix Formulation of Current Algebra (Vol. 50) Rennet, B.: Current Algebra and Weak Interactions (Vol. 52) Rennet, B.: On the Problem of the Sigma Terms in MesonBaryon Scattering. Comments on Recent Literature (Vol. 61) Soloviev, L. D.: Symmetries and Current Algebras for Electromagnetic Interactions (Vol. 46) Stech, B.: Nonleptonic Decays and Mass Differences of Hadrons (Vol. 50) Stichel, P. Current Algebra in the Framework of General Quantum Field Theory (Vol. 50) Stichel, P.: Current Algebra and Renormalizable Field Theories (Vol. 50) Stichel, P.: Introduction to Current Algebra (Vol. 50) VerzegnassL C.: Low Energy Photo- and Electroproduction, Multipole Analysis by Current Algebra Commutators (Vol. 59) Weinstein, M.: Chiral Symmetry. An Approach to the Study of the Strong Interactions (Vol. 60)
12 Specific Theories and Interaction Models
Amaldi, E., Fubinl, S. P., Furlan, G.: Electroproduction at Low Energy and Hadron Form Factors (Vol. 83) Hofman, W.: Jets of Hadrons (Vol. 90) Wiik, B. H., Wolf, G., Electron-Positron Interactions (Vot. 86)
I2.20 Quantum Eleetrodynamics
Kiillkn, G.: Radiative Corrections in Elementary Particle Physics (Vol. 46) Olsen, H. A.: Applications of Quantum Electrodynamics (Vol. 44)
van Hove, L.: Theory of Strong Interactions of Elementary Particles in the GeV Region (Vol. 39) Huang, K.: Deep Inelastic Hadronic Scattering in DualResonance Model (Vol. 62) Landshoff, P. V.: Duality in Deep Inelastic Electroproduction (Vol. 62) Michael, C.: Regge Residues (Voi. 55) Oehme, R.: Complex Angular Momentum (Vol. 57) Oehme, R.: Duality and Regge Theory (Vol. 57) Oehme, R.:Rising Cross-Sections (Vol, 61) Rubinstein, H. R.: Duality for Real and Virtual Photons (Vol. 62) Rubinstein, H. R.: Physical N-Pion Functions (Vol. 57) Satz, H.: An Introduction to Dual Resonance Models in Multiparticle Physics (Vol. 57) Schrempp-Otto, B., Schrempp, E: Are Regge Cuts Still Worthwhile? (Vol. 61) Squires, E. J..: Regge-Pole Phenomenology (VoL 57)
12.30 Weak Interactions
Barut, A. O.: On the S-Matrix Theory of Weak Interactions (VoL 53) Dosch, 11. G.: The Decays of the K0-K0 System (Vol. 52) Gasiomwicz, S.: A Survey of the Weak Interactions (Vol. 52) 8atto, R.: Cabibbo Angle and SU2 x SU2 Breaking (Vol. 53) yon Gehlen, G.: Weak Interactions at High Energies (Vol. 53) Kabir, P. If..: Questions Raised by CP-Nonconservation (Vol. 52) Kummer, W.: Relations for Semileptonic Weak Interactions Involving Photons (Voi. 52) Mi;tller, V..F.: Semiteptonic Decays (Vol. 52) Paul, E.: Status of Interference Exp eriments with Neutral Kaons (Vol. 79) Pietschmann, H.: Weak Interactions at Small Distances (Vol. 52) Primako~£ 1t.: Weak Interactions in Nuclear Physics (Vol. 53) Rennet, B.: Current Algebra and Weak Interactions (Vol. 52) Riazuddin: Radiative Corrections to Weak Decays Involving Leptons (Vol. 52) Rothleitner, Z :Radiative Corrections to WeakInteractions (Vol. 52) Segre, G.: Unconventional Models of Weak Interactions (Vol. 52) Stech, B.: Non Leptonic Decays (Vol. 52)
12.40 Strong Interactions, Regge Polee Theory, Dual Models
Ademollo, M.: Current Amplitudes in Dual Resonance Models (Vol. 59) Chung-I Tan: High Energy Inclusive Processes (VoL 60) Collins, P. D. B.: How Important Are ReggeCuts? (Vol.60) Collins, P. D. B., Gault, E D.: The Eikonal Model for Regge Cuts in Pion-Nucleon Scattering (Vol. 63) Collins, P. D. B., Squires, E. J.: Regge Poles in Particle Physics (Vol. 45) Contogouris, A. P.: Certain Problems of Two-Body Reactions with Spin (Vol. 57) Contogouris, A. P.: Regge Analysis and Dual Absorptive Model (Vol. 63) Dietz, K.: Dual Quark Models (Vol. 60)
13.40 Electromagnetic Properties of Hadrons
Buchanan, C. D., Collard, H., Crannell, C., Frosch, R., Griffy, T. A., Hofstadter, R., Hughes, E. B., N61deke, G.K. Oakes, R. J., Van Oostrum, K. J., Rand, R. E., Suelzle, L., Yearian, M. R., Clark, B., Herman, R., Ravenhall, D. G.: Recent High Energy Electron Investigations at Stanford University (Vol. 39) Gatto, R.: Theoretical Aspects of Colliding Beam Experiments (Vol. 39) Gourdin, M.: Vector Mesons in Electromagnetic Interactions (Vol. 55) Huang, K: Duality and the Pion Electromagnetic Form Factor (VoL 62) Wilson, R.: Review of Nucleon Form Factors (Vol. 39)
13.60 Photon and Lepton Interactions with Hadrons
Brinkmann, P.: Polarization ofRecoil Nucleons from Single Non Photoproduction. Experimental Methods and Results (Vol. 61) Donnaehie, A.: Exotic Electromagnetic Currents (Vol. 63) Drees, J.: Deep Inelastic Electron-Nucleon Scattering (Vol. 60) Drell, S. D.: Special Models and Predictions for Photoproduction above 1 GeV (Vol. 39) Fischer, H.: Experimental Data on Photoproduction of Pseudoscalar Mesons at Intermediate Energies (Vol. 59) Fo~, L : Meson Photoproduction on Nuclei (Vol. 59) Froyland, J.: High Energy Photoproduction of Pseudoscalar Mesons (Vol. 63) Furlan, G., Paver, N., Verzegnassi, C.: Low Energy Theorems and Photo- and Electroproduction Near Threshold by Current Algebra (VoL 62) yon Gehlen, G.: Pion Electroproduction in the Low-Energy Region (Vol. 59) Heinloth, K.: Experiments on Eleetroproduction in High Energy Physics (Vol. 65) H6hler, G.: Special Models and Predictions for Pion Photoproduction (Low Energies) (Vol. 39) von Holtey, G.: Pion Photoproduction on Nucleons in the First Resonance Region (Vol. 59) Landshoff P. V.:Duality in Deep Inelastic Electroproduction (Vol. 62)
Llewellyn Smith, C H.: Patton Models of Inelastic Lepton Scattering (Vol. 62) Liicke, D., S6ding, P.: Multipole Pion Photoproduction in the s-Channel Resonance Region (Vol. 59) Osborne, L. S.: Photoproduction of Mesons in the GeV Range (Vol. 39) Pfeil, W., Sehwela, D.: Coupling Parameters of Pseudoscalar Meson Photoproduction on Nucleons (Vol. 55) Renard, E M..'Q -~ Mixing (Vol. 63) Rittenberg, E: Scaling in Deep Inelastic Scattering with Fixed Final States (Vol. 62) Llewellyn Smith, C. H.: Parton Models of Inelastic Lepton Scattering (Vol. 62) Liike, D., S6ding, P.: Multipole Pion Photoproduction in the s-Channel Resonance Region (Vol. 59) Rollnik, H., Stichel, P.: Compton Scattering (Vol. 79) Rubinstein, ILL R.: Duality for Real and Virtual Photons (Vol. 62) Ri~hl, W.: Application of Harmonic Analysis to Inelastic Electron-Proton Scattering (Vol. 57) Schildknecht, D.: Vector Meson Dominance, Photo- and Electroproduction from Nucleons (Vol. 63) Schilling, K.: Some Aspects of Vector Meson Photoproduction on Protons (Vol. 63) Schwela, D.: Pion Photoproduction in the Region of the A (1230) Resonance (Vol. 59) Wolf, G.: Photoproduction of Vector Mesons (Vol. 57)
Nuclear Physics 21 Nuclear Structure
Arenh6vel, H., Weber, H. J.:Nuclear Isobar Configurations (Vot. 65) Cannata, E, Oberall, H.: Giant Resonance Phenomena in Intermediate-Energy Nuclear Reactions (Vol. 89) Racah, G.: Group Theory and Spectroscopy (Vol. 37) Singer, P.: Emission of Particles Following Muon Cap ture in Intermediate and Heavy Nuclei (Vol. 71) Oberall, H.: Study of Nuclear Structure by Muon Capture (Vol. 71) Wildermuth, K., McClure, W..:Cluster Representations of Nuclei (Vol. 41) 21.10 Nuclear Moments
Donner, W., Siiflmann, G.: Paramagnetische Felder am Kernort (Vol. 37) Zu Putlitz, G.: Determination of Nuclear Moments with Optical Double Resonance (Vol. 37) Schmid, D.: Nuclear Magnetic Double Resonance - Principles and Applications in Solid State Physics (Vol. 68)
21.30 Nuclear Forces and 13.75 Hadron-Indnced Reactions
21.40 Few Nucleon Systems
AtkSnson, D.: Some Consequences of Unitary and Crossing. Existence and Asymptotic Theorems (Vol. 57) Basdevant, J. L.,: nn Theories (Vol. 61) DeSwart, J. J., Nagels, M. M., Rijken, T. A., Verhoeven, P.A. : Hyperon-Nucleon Interaction (Vol. 60) Ebel, G., Julius, D., Kramer, G., Martin, B. R., Miillensiefen, A., Oades, G., Pilkuhn, H., Pig~t, J., Roos, M., Schierholz, G., Schmidt, W..,Steiner, F., DeSwart, J. J.: Compilating of Coupling Constants and Low-Energy Parameters (Vol. 55 ) Gustafson, G., Hamilton, J.: The Dynamics of Some rt-N Resonances (Vol. 57) Hamilton, J.: New Methods in the Analysis ofnN Scattering (Vol. 57) Kramer, G. :Nucleon-Nucleon Interactions below 1 GeV/c (Vol. ss) Liehtenberg, D. B.: Meson and Baryon Spectroscopy (Vol. 36) Martin, A. D.: The AKN Coupling and Extrapolation below the KN Threshold (Vol. 55) Martin, B. R.: Kaon-Nucleon Interactions below 1 GE V/c (Vol. 55) Morgan, D., Pi~t, J.: Low Energy Pion-Pion Scattering (Vol. 55) Oades, G. C.: Coulomb Corrections in the Analysis ofnN Experimental Scattering Data (Vol. 55) Pig~t, J.: Analytic Extrapolations and the Determination of Pion-Pion Shifts (Vol. 55) Wanders, G.: Analyticity, Unitary and Crossing-Symetry Constraints for Pion-Pion Partial Wave Amplitudes (Vol. 57) Zinn-Justin, J.: Course on Pad6 Approximants (Vol. 57)
DeSwart, J. J., Nagels, M. M., Rijken, T.A., Yerhoeven, P.A.: Hyperon-Nucleon Interactions (Vol. 60) Kramer, G.: Nucleon-Nucleon Interactions below 1 GeV/c (Vol. 55) Levinger, J. S.: The Two and Three Body Problem (Vol. 71 ) 23 Weak Interactions Gasiorowicz, S.: A Survey of the Weak Interaction (Vol. 52) Primakoff H.: Weak Interactions in Nuclear Physics (Vol. 53) 25.30 Lepton-Induced Reactions and Sca~ering
Theiflen, H.: Spectroscopy of Light Nuclei by Low Energy .. (70 MeV) Inelastic Electron Scattering (Vol. 65) Uberall, H.: Electron Scattering, Photoexcitation and Nuclear Models (Vol. 49) 28.20 Neutron Physics
Koester, L.: Neutron Scattering Lengths and Fundamental Neutron Interactions (Vol. 80) Springer, T.: Quasi-Elastic Scattering of Neutrons for the Investigation of Diffusive Motions in Solids and Liquids (Vol. 64) Steyerl, A.: Very Low Energy Neutrons (Vol. 80) 29 Experimental Methods
Panofsky, W. K H.: Experimental Techniques (Vol. 39) Strauch, K.: The Use of Bubble Chambers and Spark Chambers at Electron Accelerators (Vol. 39)
Atomic and Molecular Physics
Fluids, P l a s m a s
31 Electronic Structure of Atoms and Molecules, Theory
51 Kinetics and Transport Theory of Fluids; Physical Properties of Gases
Donner, W., Siiflmann, G.: Paramagnetische Felder am Kernort (Vol. 37)
Geiger, W.., Hornberger, H., Schramm K.-H.: Zustand der Materie unter sehr hohen Drficken und Temperaturen (Vol. 46) Hess, S.: Depolarisierte Rayleigh-Streuung und Str6mungsdoppelbrechung in Gasen (Vol. 54)
32 Atomic Spectra and Interactions with Photons Racah, G.: Group Theory and Spectroscopy (Vol.37) Zu Putlitz, G.: Determination and Nuclear Moments with Optical Double Resonance (Vol. 37)
34 Atomic and Molecular Collision Processes and Interactions Dettmann, K~: High Energy Treatment of Atomic Collisions (Vol. 58) Langbein, D.: Theory of Van der Waals Attraction (Vol. 72) Seiwert, T.: Unelastische St6Be zwischen angeregten und unangeregten Atomen (Vol. 47)
C l a s s i c a l F i e l d s of P h e n o m e n o l o g y 41.70 Particles in Electromagnetic Fields
C o n d e n s e d Matter, M e c h a n i c a l and T h e r m a l Properties 61 Structure of Liquids and Solids Behringer, J.: Factor Group Analysis Revisited and Unified (Vol. 68) Dederichs, P. H., Zeller, R.: Dynamical Properties of Point Defects in Metals (Vol. 87) Laemann, R.: Die Gleichgewichtsform yon Kristallen und die Keimbildungsarbeit bei der Kristallisation (Vol. 44) Langbein, D.: Theory of Van der Waals Attraction (Vol. 72) Leibfried, G., Breuer, N.: Point Defects in Metals I: Introduction to the Theory (Vol. 81) Sehroeder, K.: Theory of Diffusion Controlled Reactions of Point Defects in Metals (Vol. 87) Springer, T.: Quasi-elastic Scattering of Neutrons for the Investigation of Diffusive Motions in Solids and Liquids (Vol. 64) Steeb, S.: Evaluation of Atomic Distribution in Liquid Metals and Alloys by Means of X-Ray. Neutron and Electron Diffraction (Vol. 47)
Olson, C L.: Collective Ion Acceleration with Linear Electron Beams (Vol. 84) Schumacher, U.:Collective Ion Acceleration with Electron Rings (Vol.84)
62 Mechanical and Acoustical Properties and
41.80 Particle Optics
Ludwig, W.: Recent Developments in Lattice Theory (Vol. 43) Schramm, K.-H.: Dynamisches Verhalten yon Metallen unter StoBwellenbelastung (Vol. 53)
Hawkes, P. IV..:Quadrupole Optics (Vol. 42)
42 Optics
42.50 Quantum Optics AgarwaL G. S.: Quantum Statistical Theories of Spontaneous Emission and their Relation to Other Approaches (Vol. 70) Graham, R.: Statistical Theory of Instabilities in Stationary Nonequilibrium Systems with Applications to Lasers and Nonlinear Optics (Vol. 66) Haake, F: Statistical Treatment of Open Systems by Generalized Master Equations (Vol. 66) Schwabl, E, Thirring, W..:Quantum Theory of Laser Radiation (Vol. 36)
42.72 Optical Sources Godwin, R. P.: Synchrotron Radiation as a Light Source (Vol. 51)
63 Lattice Dynamics
C o n d e n s e d Matter: E l e c t r o n i c Structure, Electrical, M a g n e t i c and O p t i c a l P r o p e r t i e s 71 Electron States and
72 Electronic Transport in Condensed Matter Bauer, G.: Determination of Electron Temperatures and of Hot-Electron Distribution Functions in Semiconductors (Vol. 74) Bennemann, K.H.: A New Self-consistent Treatment of Electrons in Crystals (Vot. 38) Daniels, J.: v. Festenberg, C., Raether, H., Zeppenfeld, K.: Optical Constants of Solids by Electron Spectroscopy (Vol. 54) Dornhaus, R., Nimtz, G.: The Properties and Applications of Hgl_xCdxTe Alloy Systems (Vol. 78) Feitknecht, J.: Silicon Carbide as a Semiconductor (Vol. 58)
Grosse, P.: Die Festk6rpereigenschaften von Tellur (Vol. 48) Raether, H.: Solid State Excitations by Electrons (Vol. 38) Schmid, D.: Nuclear Magnetic Double Resonance Principles and Applications in Solid State Physics (Vol. 68) Schnackenberg, J.: Electron-Phonon Interaction and Boltzmann Equation in Narrow Band Semiconductors (Vol. 51)
Pick, H.: Struktur von St6rstellen in Alkalihalogenidkristallen (Vol. 38) Raether, H.: Solid State Excitations by Electrons (Vol. 38) Richter, W. Resonant Raman Scattering in Semiconductors (vol. 78)
Related Areas of Science and Technology 73 Properties of Surfaces and Thin Films H61zl, J., Schulte, FK,: Work Function of Metals (vol. 85) Miiller, K~:How Much Can Auger Electrons Tell Us about Solid Surfaces? (vol. 77) Raether, tL: Excitation of Plasmons and Interband Transitions by Electrons (Vol. 88) Wagner, H.: Physical and Chemical Prop erties of Stepped Surfaces (Vol. 85) Wissmann, P.: The Electrical Resistivity of Pure and Gas Covered Metal Films (VoL 77)
74 Superconductivity Liiders, G., Usadel, K-D.: The Method of the Correlation Function in Superconductivity Theory (Vol. 56) Ullmaier, H.: Irreversible Properties of Type II Superconductors (Vol. 76)
85.70 Magnetic Devices Lehner, G.: Uber die Grenzen der Erzeugung sehr hoher Magnetfelder (Vol. 47)
Geophysics, Astronomy, and Astrophysics 95 Theoretical Astrophysics Kundt, W.: Recent Progress in Cosmology (Isotropy of 3 deg Background Radiation and Occurrence of SpaceTime Singularities) (Vol.47) Kundt, W.: Survey of Cosmology (Vol. 58)
97 Stars Bdrner, G.: On the Properties of Matter in Neutron Stars (Vol. 69)
75 Magnetic Properties Fischer, K : Magnetic Impurities in Metals: the s - d Exchange Model (Vol. 54) Schmid, D.: Nuclear Magnetic Double Resonance - Principles and Applications in Solid State Physics (Vol. 68) Stiemtadt. K.: Der Magnetische Barkhauseneffekt (Vol. 40)
78 Optical Properties Bh'uerle, D.: Vib rational Spectra of E1ectron and Hydrogen Centers in Ionic Crystals (Vol. 68) Bendow, B.: Polariton Theory of Resonance Raman Scattering in Solids (Vol. 82) Borstel, G., Falge, H.J., Otto, A. :Surface and BulkPhononPolariton Observed by Attenuated Total Reflection (Vol. 74) Claus, R., Merten, L., Brandmiiller, J.: Light Scattering by Phonon-Polaritons (Vol. 75) Daniels, J., v. Festenberg, C., Raether, H., Zeppenfeld, K.: Optical Constants of Solids by Electron Spectroscopy (Vol. 54) Excitons at High Density, Haken, H., Nilatine, S., (Volume Ed!tors). Contributors: Bagaev, E S., Biellmann, J., Bivas, A., Goll, J., Grosmann, M., Grun, J. B., Haken, H., Hanamura, E., Levy, R., Mahr, t-L, Nikitine, S., Novikov, B. V..,Rashba, E. L, Rice, T. M., Rogachev, A.A., Schenzle, A., Shaklee, K. L..(Vol. 73) Godwin, R. P.: Synchrotron Radiation as a Light Source (Vol. 51) Lengeler, B. : de Haas-van Alphen Studies of the Electronic Structure of the Noble Metals and Their Dilute Alloys (vol. 82)
