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Radoje Belu]evi~
Neutral Kaons With 67 Figures
Springer
Dr. Radoje Belu~evi~ High Energy Accelerator Research OrganizationKEK Department of Physics 1-10ho, Tsukuba-shi 3o5-o8ol Ibaraki-ken, Japan Email:
[email protected] Physics and Astronomy Classification Scheme (PACS): 14.4o.A, o3.65.-w, 13.2o.Eb, 13.25.Es, ll.3o.Er, 12.15.Ff, 12.15.Ji
ISSN oo81-3869 ISBN 3-54o-65645-6 Springer-Verlag Berlin Heidelberg New York Library of Congress Cataloging-in-Publication Data applied for. Die Deutsche Bibliothek - CIP Einheitsaufnahme Belu~evi~, Radoje: Neutral kaons/Radoje Belugevi~. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 1999 (Springer tracts in modern physics; Vol. 153) ISBN 3-54o-65645-6 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1999 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data conversion by EDV-Beratung E Herweg, Hirschberg Cover design: design &production GmbH, Heidelberg SPIN: lO7O9232
56/3144- 5 4 3 21 o - Printed on acid-free paper
Dedicated to Jack S t e i n b e r g e r
Foreword
Science Museum, London. Science & Society Picture Library
Among the five thousand stereoscopic photographs of cosmic ray showers obtained by George Rochester and Clifford Butler at Manchester University, using a cloud chamber placed in a magnetic field, there was a picture containig "forked tracks of a very striking character". In the lower right-hand side of the picture, just below a 3-cm lead plate mounted across the centre of the chamber, they observed, on 15th October 1946, a pair of tracks forming a two-pronged fork (an inverted V) with the apex in the gas (see the reprinted image). The direction of the magnetic field was such that a positively charged particle moving downward is deflected in an anticlockwise direction. They determined that the particle corresponding to the upper track had positive charge and a momentum of 340 + 100 MeV/c; the lower particle had negative charge and a momentum of 350 + 150 MeV/c. The ionization and curvature
VIII
Foreword
of the tracks showed that they were due to particles much less massive than the proton. If the tracks were associated with a collision process, one would have expected several hundred times as m a n y of these interactions in the lead plate as in the gas. Since very few events similar to this were observed in the plate, they argued that the fork "must be due to some type of spontaneous process for which the probability depends on the distance travelled and not on the amount of m a t t e r traversed". This conclusion is supported by the following argument: if the fork were due to a deflection of a backscattered charged particle by a nucleus, the m o m e n t u m transfer would be so large as to produce a visible recoiling nucleus at the apex. Based on their past experience, the electron pair production by a highenergy photon in the Coulomb field of the nucleus was excluded because the two tracks would have to be much closer together if they were an electronpositron pair. They also excluded the possibility of this picture representing the decay of a charged pion or muon coming up from below the chamber, since in that case conservation of energy and m o m e n t u m would require the incident particle to have a minimum mass of 1280me (me is the electron
mass).
Rochester and Butler therefore concluded that this had to be a photographic image of the decay of a new type of uncharged elementary particle into two lighter charged particles. For the case where the incident particle decays into two particles of equal mass, they determined the mass of the parent particle to be 870 + 200 M e V / c 2, for an assumed secondary particle mass of 200me.
Preface
Enormous progress has been made in the field of high-energy, or elementary particle, physics over the past three decades. The existence of a subnuclear world of quarks and leptons, whose dynamics can be described by quantum field theories possesing local gauge symmetry (gauge theories), has been firmly established. The cosmological and astrophysical implications of experimental results and theoretical ideas from particle physics have become essential to our understanding of the formation of the universe. For example, a tiny violation of CP symmetry, which has been observed so far only in the K ° system, is believed to have played an important role in the early stages of cosmic evolution. The main purpose of this book is to convey the unique beauty of a quantum-mechanical system that contains so many of the aspects of modern physics. Inevitably, this imposes considerable constraints on the content and nature of the presentation. In outlining the basic formalism necessary to describe the K ° system and its time evolution in both vacuum and matter, effort was made to keep the presentation as clear as possible and to justify the main steps in the derivations. To highlight their quantum-mechanical origin, extraordinary properties of neutral kaons are illustrated through analogous experiments with polarized light and atomic beams. A formal theory of the discrete symmetry operations C (charge conjugation), P (parity transformation) and T (time reversal) is presented. These subtle concepts are discussed in the context of parity violation, time reversal asymmetry and CP noninvariance in kaon decays. In order to emphasize the complementary roles of theory and measurement, a number of "classic" experiments with neutral K mesons are described and some major current projects and proposals are reviewed. A detailed and pedagogical discussion of the K ° physics within the framework of gauge theories of the electroweak interactions is also provided. Athough this book was written primarily for graduate students and researchers in high-energy physics, I have endeavored to make its content accessible to curious undergraduates and physicists not specializing in the field.
Acknowledgements I would like to thank Bruce Winstein and Italo Mannelli for valuable comments regarding the experiments E731 at Fermilab and NA31 at CERN.
X
Preface
I have benefitted from discussions with Robert Sachs about the K ° phenomenology in the presence of T and C P T violation, and with Kaoru Hagiwara, Makoto Kobayashi, Yasuhiro Okada and Yasuhiro Shimizu concerning K°-/~ ° mixing and rare kaon decays in the Standard Model. Helpful comments and suggestions by Volker Hepp, Martin Wunsch and Sher Alam are appreciated. I am particularly indebted to Asish Satpathy and Bruce Winstein for their interest, help and advice. For permission to reprint various plots and drawings I am grateful to Bill Carithers, Val Fitch, Erwin Gabathuler, Jack Ritchie, Jack Steinberger and Bruce Winstein. I wish to express my special gratitude to Hans KSlsch, Victoria Wicks and the production team at Springer for their help in preparing the manuscript for publication. Support from Prof. Sakue Yamada, Head of the Institute for Particle and Nuclear Studies at KEK, and the Japanese Ministry of Education, Science and Culture (Monbusho) is gratefully acknowledged. Tsukuba-shi Februar~ 1999
R. Belu~evid
Contents
1.
.
Introduction .............................................. 1.1 K ° a n d / ~ 0 as E i g e n s t a t e s of S t r a n g e n e s s . . . . . . . . . . . . . . . . . . 1.2 C P E i g e n s t a t e s of N e u t r a l K a o n s : K ° a n d K ° . . . . . . . . . . . . . 1.3 D u a l i t y of N e u t r a l K a o n s : ( K ° , / ~ 0 ) vs. ( g °, K °) . . . . . . . . . . . 1.4 T h e E i n s t e i n - P o d o l s k y R o s e n P a r a d o x in t h e K ° S y s t e m . . . . 1.5 S t r a n g e n e s s Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 T h e K 10 - K 20 M a s s Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 6 8 9 10 13
of Neutral Kaons in Matter .................. The K ° Regeneration ................................... Coherent Regeneration Amplitude ........................ K 01 - K 1o Interference a n d t h e Sign of A m k . . . . . . . . . . . . . . . . . .
17 17 20 28
Propagation
2.1 2.2 2.3
CP Violation
3.
3.1 3.2 3.3 3.4 3.5
in K ° Decays ............................... Discovery of C P V i o l a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P h e n o m e n o l o g i c a l I m p l i c a t i o n s of K ° --+ 27r . . . . . . . . . . . . . . . . Unitarity, C P T Invariance and T Violation ................ Isospin A n a l y s i s of K°,L -~ 21r . . . . . . . . . . . . . . . . . . . . . . . . . . . K L0- K s0 I n t e r f e r e n c e as E v i d e n c e for CP V i o l a t i o n . . . . . . . . . . .
1
33 33 35 39 43 49
4.
Interference in Semileptonic and Pionic Decay Modes .... 4.1 S e m i l e p t o n i c D e c a y s of N e u t r a l K a o n s . . . . . . . . . . . . . . . . . . . . 4.2 K ° - K ° Interference in 7r+Tr- a n d giTr=F~ D e c a y s . . . . . . . . . . . . 4.3 K ° - K ° I n t e r f e r e n c e W i t h o u t R e g e n e r a t o r . . . . . . . . . . . . . . . . . .
57 57 62 65
5.
Precision Measurements of 0oo, ¢+- and e'/e ............ 5.1 T h e E x p e r i m e n t NA31 a t C E R N . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 T h e E x p e r i m e n t E 7 3 1 / E 7 7 3 a t F e r m i l a b . . . . . . . . . . . . . . . . . . 5.3 C o m p a r i s o n of NA31 a n d E731 E x p e r i m e n t a l Techniques . . . .
71 72 75 80
Neutral Kaons in Proton-Antiproton Annihilations ....... 6.1 T h e C P L E A R E x p e r i m e n t a t C E R N . . . . . . . . . . . . . . . . . . . . . . 6.2 Is CP V i o l a t i o n C o m p e n s a t e d by T i m e - R e v e r s a l A s y m m e t r y ?
81 81 87
.
XII
Contents
7.
Neutral Kaons in Electron-Positron Annihilations 7.1 T h e D A C N E P r o j e c t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.
Neutral Kaons in Fixed-Target Experiments .............. 8.1 T h e E x p e r i m e n t s K T e V a n d NA48 . . . . . . . . . . . . . . . . . . . . . . .
.
The 9.1 9.2 9.3
9.4
K ° System in the Standard Model ................... C a l c u l a t i o n of A m k a n d % . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B°-/~ ° M i x i n g a n d C o n s t r a i n t s on C K M P a r a m e t e r s . . . . . . . . Rare K a o n Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 KL° --+ 7r°v~ a n d K + -+ 7r+v~ . . . . . . . . . . . . . . . . . . . . . . 9.3.2 K ° - + p + # - a n d K ° - + e + e - . . . . . . . . . . . . . . . . . . . . . . Direct CP V i o l a t i o n ( # ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendices
A B C D E F
........
89 89 93 93 99 99 109 116 117 125 139
...................................................
145 145 Forward S c a t t e r i n g A m p l i t u d e a n d the O p t i c a l T h e o r e m . . . . 147 W a t s o n ' s T h e o r e m a n d the Decay A m p l i t u d e s K ° , / ~ o __+ 27r. 150 T i m e Reversal a n d CPT Violation . . . . . . . . . . . . . . . . . . . . . . . . 153 T r a n s f o r m a t i o n P r o p e r t i e s of Dirac Fields U n d e r C, P a n d T 156 The Vacuum Insertion Approximation . . . . . . . . . . . . . . . . . . . . . 169
CP P r o p e r t i e s of K --+ 27r a n d K --+ 37r . . . . . . . . . . . . . . . . . . . .
References ....................................................
171
Name Index
175
..................................................
Subject Index
................................................
Subject Index (Decays) .......................................
177 183
1. I n t r o d u c t i o n
"This is one of the greatest achievements of theoretical physics. It is not based on an elegant mathematical hocus-pocus such as the general theory of relativity yet the predictions are just as important as, say, the prediction of positrons."
Richard Feynman, The Theory of Fundamental Processes The neutral K meson (neutral kaon), K ~ and its antiparticle, /7/~ form a remarkable quantum-mechanical two-state system that has played an important role in the history of elementary particle physics. Indeed, ever since the discovery of K ~ half a century ago, neutral kaons have been a rich source of unique and fascinating phenomena associated with their production, decay and propagation in both vacuum and matter. What makes the K ~ system so special is that K ~ and /7/~ which have the same charge, mass, spin and parity, but different strangeness quantum number, S, cannot always be distinguished from one another. Whereas in strangeness-conserving strong interactions K ~ (S = +1) a n d / ~ 0 (S = - 1 ) are as distinct as the neutron and antineutron, this distinction is erased in strangeness-violating weak interactions, thus allowing K ~ ++/7/o transitions. 1 As a consequence, an initially pure IK ~> or I/( ~> state will gradually evolve into a state of mixed strangeness, ]K~
> a(t)]K ~ + b(t)l/~~
in accordance with the principle of superposition of amplitudes in quantum mechanics. This strangeness oscillation effect has a nice optical analogy: right-circularly polarized light rapidly acquires a large left-circularly polarized component while passing through a crystal that absorbs predominantly x-polarized light. The K ~ and /~0 mesons are two unconnected, degenerate (mao = m~o) states in the absence of the weak interaction. As is well known from quantum mechanics, the mixing of two degenerate levels in vacuum must result in level splitting (this splitting shows up in the hydrogen molecular ion and in the inversion spectrum of ammonia). The application of ordinary perturbation theory to the K~ ~ system produces the following result: the weak interaction, /t/w, slightly shifts the value of the kaon mass, mko, and splits the degenerate levels by a tiny amount:
Amk --
Iml - m21 -- ,
where I~> is the unperturbed state. We can consequently apply ordinary degenerate perturbation theory to the K ~ system. S{nce neutral lmons decay through a number of channels, our Hilbert space should, in principle, be expanded to include all possible transitions. However, we keep the analysis simple by restricting ourselves to the two-dimensional Hilbert space spanned by [K ~ and 1/~o>, in which case the effect of decays is incorporated into an effective hamiltonian
He/= Hw+ Z
/:/w In> (n I/:/w
+
(1.29)
n ^ eft" H~ is determined by the virtual transitions to all intermediate states n outside the two-particle subspace. Now,
[f/sq-em -~-/:/ewff]I~> ~-[mko ~-m'] Ik~>
(1.30)
14
1. Introduction
where a/:/~fr [ g ~ + b/:/ef t/f ~ = m '
[alK ~ + bl/;;~
(1.31)
is a small p e r t u r b a t i o n due to the weak interaction. Taking the inner p r o d u c t of (1.31) with (K~ and then (/ ml, as shown in Sect. 2.3.
1.6 The K1-K2 o o Mass Difference
J
m+ z~m
"~"
m- txm
f.1
m'~
Hw
J
K~
Hs + Hem
15
The KO_/~ o mixing results in level splitting
Fig. 1 .4.
and (KOl
Ae~ H7 IK~ =
[ = (/7(o I /2/w I K~ = 0, thus confirming t h a t first-order weak interactions obey the empirical rule IASI ] _ a + b []KO ) _ i/~0)] + a - b []KO ) + ]/~o)] 1 (a + b)lK ~ + 1 (a - b)lK~
(2.1)
Since a r b, we conclude that after all the K ~ mesons in the initial b e a m have decayed away, some can be regenerated by passing the pure K ~ b e a m through matter. Figure 2.1 contains a schematic drawing of such an experiment. Like the K ~ ~ oscillation, the K ~ regeneration phenomenon is a direct consequence of the principle of superposition of amplitudes in quantum mechanics. The analogous experiment with polarized light is shown in Fig. 2.2. This analogy follows from the wave aspect of q u a n t u m theory. There is another close analogy to the K ~ regeneration which beautifully illustrates the concept of quantization. It is based on the Stern-Gerlach atomic
18
2. Propagation of Neutral Kaons in Matter
~_Kt + K2 o
Negative pion [-] beam
K2
o
Proton target
O
0
K2 + K1
Regenerator
Fig. 2.1. Schematic drawing of a K ~ regeneration experiment
K0absorbed Re! enerator Polarized
f~Rotator
~r~
K0
K2
K~I decays
Rotator
o
Ko
K01 decays
K20
Fig. 2.2. An experiment with polarized light analogous to K ~ regeneration beam experiment, i.e., on the fact that it is impossible to quantize the spin components of the beam along two orthogonal axis simultaneously. As shown in Fig. 2.3, an unpolarized atomic beam of spin 1/2 propagating in the z direction enters an inhomogeneous magnetic field that points along the y axis (Hy). This causes the beam to split equally into two components, one deflected upward and the other downward. The two components correspond to the atoms quantized by the field in the spin eigenstates ay = + 1 / 2 and O'y ~ - 1 / 2 , respectively. If the "lower" component is then sent through an inhomogeneous magnetic field pointing in the direction perpendicular to the y axis (the x direction), it will again split into two equal components, one deflected in the x direction (a= = + 1 / 2 ) and the other in the opposite
unpolarized r spin 1/2 / a t o ~
r~ absorbed
. protontarget.~f ~ ~ 1 7 6 I //~/ .~,~
/.x
"y
If' ~ Kl~decays f ~
~,oabsorbed
K~~,,
Hy
regenerator,,
Fig. 2.3. Analogy between the Stern-Cerlach and K ~ regeneration experiments
2.1 The K ~ Regeneration
]9
direction (ax = - 1 / 2 ) . We now repeat the above two steps starting with the ax -- - 1 / 2 component, and assume that after each beam-splitting the a~,y = + 1 / 2 component is absorbed in matter. The analogy between this and the K1~ regeneration experiment is as follows. Each time the atomic beam passes through Hy corresponds to the selection of strangeness eigenstates K ~ a n d / ~ 0 via strong interactions in the proton target or the regenerator. The beam splitting in /Ix is analogous to the selection of CP eigenstates K ~ and K ~ through weak decays. Note that once the beam passes through an inhomogeneous magnetic field in, say, the y direction, all preexisting information about quantization along the x axis is lost (the axis singled out in space is defined by the magnetic field). This means that it is impossible to quantize the spin components of the beam along two orthogonal axes; in other words, the spin operators &x and ~y do not commute. Similarly, the operators S and C/~ also do not commute; hence K ~ states are eigenstates of either S or C/~, but not both. The first experimental confirmation of the regeneration phenomenon, and o 20 mass difference, was that by O. Picalso the first measurement of the K 1-K cioni and his collaborators [9]. To produce K ~ mesons, they passed negative pions through a liquid-hydrogen target (see (1.1)), and then allowed the K1~ component of the K ~ beam to decay away. About 200 K1~ mesons were regenerated in a 30-inch propane bubble chamber fitted with lead and iron plates (see Fig. 2.4), from a beam of approximately 105 K~) particles. A particularly elegant way of measuring Amk is based on the variable gap method. Refer to Fig. 2.5. A K ~ beam passes through two thin slabs of material separated by distance g, which can be varied. Let us assume for the sake of simplicity that t h e / ~ 0 component is totally absorbed in each of the two slabs. After exiting from the first slab (t = 0), the beam is pure K ~
Incident Kz beam
lllllll -W.,~~
30"
J I
,,^~, K,meson !/
", ,~+
I \/ i /
T 19"
Fig. 2.4. A regenerated K ~ decays in the experiment by R. Good et al. [9]
20
2. Propagation of Neutral Kaons in Matter ~-
g
Absorber
Regenerator
Fig. 2.5. A sketch of the variable gap method Just before it enters the second slab (regenerator), the beam is described by (1.19), where tp = t 9 = t / 7 = g/e'~ = g / [ ( p k / E k ) ( E k / m k ) ] = g ( m k / p k ) is the proper time to traverse g. Immediately after the regenerator (t~ >> tslab), where t h e / ~ 0 component is totally absorbed, the K ~ amplitude reads (see (1.20)) AKo(t~) = (K~ I ~P(tg)) = ~1( K ~ with r
I ~(t~)) = ~ 1
[e~l + e ~2]
(2.2)
given by (1.21). The corresponding decay intensity is (F1 >>/"2)
I(g) _= AK o~ ~-4-1(0) [1 + e -rltg + 2e -rltg/2 cos(Amktg)] ,
(2.3)
where I(0) = I < K ~
2 =
1
(2.4)
The mass difference Amk can be obtained by measuring the K1~ -+ 7r+Trdecay intensity after the regenerator as a function of g. This is, of course, a very simplified description of the method (see [10] regarding the original variable gap experiment).
2.2 Coherent
Regeneration
Amplitude
We have seen that regeneration occurs because K ~ and/~0 mesons interact differently with matter. Let us rewrite (2.1) as ]k~after)
_
_
a + b []KO ) + t~lK10)] 2
'
t) ~
a-b a+b'
(2.5)
where 0, the regeneration parameter, is a complex number that can be related to the physical properties of the regenerator, as will be shown in what follows. The regenerated K ~ mesons will coincide in direction and momentum with the incident K ~ beam whenever the forward scattering amplitudes for
2.2 Coherent Regeneration Amplitude
21
K ~ and /~0 are different. This situation is called coherent regeneration, or regeneration by transmission, and is a result of the constructive contribution from all scatterers in the target. The coherence is preserved in a region that is typically contained within a micro-radian. As in the case of forward scattering of neutrons, light, X-rays, etc., the forward scattering of neutral kaons is always coherent, and its interference with the incoming beam gives rise to the refractive index of the scattering material (see Appendix B): 2rN n = 1 + - - ~ - f ( w , 0),
(2.6)
where N is the number of scatterers per unit volume, k the wave number of the incident particles and f(w, 0) the scattering amplitude in the forward direction (0 = 0). The forward amplitude is related to the total cross-section by the optical theorem 47r O'tot = T ]m ~(w, 0).
(2.7)
A particle traversing a slab of thickness z picks up an extra phase, which is proportional to the refractive index n: = k(n - 1)z.
(2.8)
Not only are the K ~ a n d / ~ 0 mesons absorbed differently in matter, but their elastic scattering amplitudes also differ, just as those for K + p --4 K + p and K - p --4 K - p do. Since K ~ and /2(0 have different total cross-sections, they must also have different indices of refraction; hence they acquire unequal phase shifts while propagating through matter. To see what this implies, we first write down the time development of K ~ a n d / ~ 0 states in vacuum (see (1.19)): = ~1 [ IK~
Ig~
=
1
er
-~-[K0) er 2] ,
[igO ) er 1 _
[KO) er
'
(2.9)
where r
=
-
(i- 1,2 + rl,2/2) t,
(2.10)
and tp = t / 7 = t x / 1 - v 2 is the proper time: z = vt = t p V / V ~ - v 2. The two-state SchrSdinger equation describing the system (2.9) reads 9 1
-
-
dtp
- -
~m - i6F/2
rh
iF~2 ] ~Pvac,
(2.tl)
9 The K ~ system is described by two coupled differential equations because of the K 0 ++ /~0 mixing. If neutral kaons interacted only through strong and electromagnetic interactions, both of which conserve S, there would be no transitions between the K ~ and/~o states, and each state would be separately described by a SchrSdinger equation. In fact, (2.11) is not a "real" SchrSdinger equation.
22
2. Propagation of Neutral Kaons in Matter
where ~va~ -- (IK~
IR~
~
(2.12)
)
and
_ ml + m2 /~ _ I"1 + ['2 ~m -- ml -- m2 5F -- F I _ _ -F2. - (2.13) 2 ' 2 ' 2 ' 2 While traversing a distance z in matter, the K~ extra phase: IK~
~ IK~
ir
IK~
~ meson acquires an
--+ I/7/~
(2.14)
i•,
where = k(n
27cN v
-
1)z- k~tpf(0), (2.15) 27rN v
r = k ( f i - 1)z - k lx/]_:~_ v2 tpf(0). It is straightforward to show that (2.11) now becomes . d~matt 1 - -
dtp
( ~ t - i/~/2 - k ~ l ( 0 ) =
5m-iSF/2
5m - i ~ r / 2
rh - i/~/2
27rgv
-
k~_~f(0) ]
k~mat t.
(2.16)
The matrix equation (2.16) may be expressed as 'matt a9d ~dt-----~
__
(M - iF)
2~rNv
~matt
k 1~/~_ v 2
(~(:)
0 ) k~matt
7(0)
(2.17)
or
i dk~matt -~p - ( M ' - i F ' )
(2.18)
~matt,
with
ir
ir
,2,9,
and -
2~Nv
(2.20)
kx/1 - v 2"
The time development of the K ~ system in vacuum and in matter is therefore described by the SchrSdinger equations 9d~vac 1 - H~wc dtp
and
dk~mat t
i--
dtp
'
- - H ~[tmatt ,
(2.21)
2.2 Coherent Regeneration Amplitude
23
respectively, with H_--M-iF,
H'__M'-iF',
(2.22)
where M and F are 2 x 2 matrices. 1~ M is called the mass matrix and F the decay matri3:, both are hermitian (M~k = M k j , F~k = Fkj) because they represent observable quantities. However, H and H' are not hermitian, otherwise K ~ and/~0 would not decay. To understand this statement note that the decay process is the equivalent of an absorption, and that an absorbing medium can be described in terms of a complex index of refraction, the imaginary part of which is associated with attenuation (see Appendix B). We may now proceed with our main task, which is to obtain the intensity of the regenerated K ~ component at z. To this end we express ~ m a t t in (2.16) in terms of the C P eigenstates K ~ and K ~ and write the result down as a system of two coupled differential equations (IK~ = K~ i d
:
(,h
-
i/~/2
-
~f)
+ (Sin - i 6 r / 2 )
i ~dp [K~
-
K o] =
K2o]
+
[K ~ -
K~
(2.23)
(sin -i~r/2) [K ~ + K ~ + (~h - i / ~ / 2 - ~ ) [K1~ - K ~
Adding and subtracting equations (2.23) yields .dP1 2 dtp
0)
( m l - iF1/2 m2-iF2/2 0
(
f+
P1,2 - z
~'~-f--i)~t12'(2.24)
with P1,2 = ~ KO ] .
(2.25)
We see that d'Pl,2/dtp has two components: the first one describes the propagation of free particles, and the second one the scattering with nuclei inside the regenerator. The change of P1,2 with respect to z as the beam passes through matter may thus be expressed as
d 12 _ [d 12]
dz
L-h-~z'-Jvac+L dz
J~c,
(2.26)
The second term in (2.26) is readily obtained from (2.24) by setting dz = v2:
v d t = v'ydtp = v d t p / V ~ -
lo Since we have restricted ourselves to a two-dimensional Hilbert space, H and H' are effective hamiltonians: H = Hstrong + Helect...... gnetic +/-/weak. From (2.22) it follows that M = (H + Ht)/2 and F = i(H - Ht).
24
2. Propagation of Neutral Kaons in Matter
- ~ z ].uc ~
k
\f-~
f+
'
Regarding the first term, recall that an unstable particle propagating in vacuum may be described by a plane wave
~P(z) = eikz-zl2Ao(o), where we used Therefore,
(2.28)
Ftp/2 = z/2/37v = z/2A (A - 137T is the decay length).
--~-z'-J vac =
0
ik2 -
1/2A2 ~1,2.
Expressions (2.27) and (2.29) form a system of two coupled differential equations:
dK~ [ikl _ l/2Al + ilrN ] dz = --~1 (f+~) K~
iTcN
(f-~)K~
- CK~ + :DK~
(2.30)
d K ~ _ [ ik2 - 1/2A2 + i~rN (f + ~) 1 n ~ + brN (~ - ~) K ~ dz ~ - . 4 K o + t~K ~
The coefficients A, B, C and :D in (2.30) represent the rates of change of the K ~ and K ~ amplitudes due to propagation, decay and (coherent) scattering. The coupled differential equations (2.30) can be formally solved for the z dependence of the amplitudes. However, the solution is simplified, and underlying physics made more transparent, by noting that K2~ >> K ~ and/"1 >> F2. We can thus neglect the contribution of K ~ to the change in K ~ and write the second equation in (2.30) as
~ ik2-1/2A2+~(f+~)
dz.
(2.31)
This yields lnKO(z)
inN
A2_.~zik2z + ~2 (~ +9) z + In K~
(2.32)
i.e.
KO(z) = eia~z e - N f .... /2 KO(0),
(2.33)
where we defined 11 11 Writing r
0) ---- drtot(K O) = [(Ttot(K 0) + atot ( / ~ ~ ---- 47r[Im I + Im~]/2k = -4~_i[iImJ + iIm~]/2k we could identify ftot with O'tot provided iImf = f and iIm I = I, i.e. the forward scattering amplitudes were purely imaginary (the beam was only attenuated). In general this is not the case because of refraction.
2.2 Coherent Regeneration Amplitude 47ri ~+~ k 2 Inserting (2.33) into the first equation (2.30) gives
(2.34)
ftot ~
dKl~ ~zz
[ -- i k l - 1 / 2 A l + iTrN ~- ~ - 1 (~
-
25
brN ] ~-1 ( f + ~ ) K~
~) eik2z e-gft~
g~
(2.35)
dz
Fig. 2.6. Schematic drawing of a solid regenerator of thickness l A K ~ travels as a K ~ with a wave number k2 until it is regenerated, and after that as a K ~ with a wave number kj. Since Amk is so tiny, kl ~ k2. Note also that Avac ~ Amatt. The first term on the right-hand side of (2.35) describes the behavior of K ~ mesons, created before z, in the interval z, z+dz. The second term gives the amplitude of K ~ particles created between z and z + dz (see Fig. 2.6). To see how this amplitude is modified at the end of the regenerator due to propagation, decay and coherent scattering, we integrate the first term in the interval l - z, where l is the thickness of the regenerator, with the result (2.36)
(~(l -- Z) = e ikl (l-z) e-(l-z)/2A~ e - N f t o t ( l - z ) / 2 .
Since we are dealing with coherent scattering, the amplitudes of K1~ mesons created in all dz intervals have to be added. This gives the following K ~ amplitude at the end of the regenerator: /q0q) = --
cq - z)dV(z) brN
X
(f-f)/q~176
f
e ikl(/-z) e -(l-z)/2A~ e -NIt~
f
e i(kl-k2)(l-z) e -(l-z)/2A1 d z
e ik2z e -N/t~
dz
brN
•
-
-
i~N kl ( ~ - ~ ) K 2 (o/ ) fo z e(l-z)[i(k*-k~)-l/2A1]dz
(2.37)
26
2. Propagation of Neutral Kaons in Matter
with g~
=- e -Nft~
e ik2/g~
= 0).
(2.38)
The simple integration above results in KO(l ) _
iNA1Alfr [1 --e -(-ihm+l/2)~] K ~ - i h m + 1/2
-
ocK~
(2.39)
where fr = ( ~ - 7 ) / 2 ,
A1 - 27r/kl,
~ =-l/A1,
(2.40)
and
6m-- (m2
ml)c2
h/'Q
-- (kl - k2) A1
(2.41)
is the K ~ ~ mass difference expressed in units of the K1~ mass width h/T1. Expression (2.41) requires a few words of explanation. As discussed in Appendix B, for coherent scattering the regenerator as a whole absorbs the difference in m o m e n t u m Pl - P~ = h(kt - k2) = h A k and recoils with the m o m e n t u m - h A k , taking away from the meson the energy (hAk)2/2Mreg. Since Mreg is very large, this energy is negligible compared with m l - m2. Hence E2 = El, which we express as (h, c ~ 1) h
-
:
-
c 2
i.e. kt - k2 m2 - m l . . . . kh 2, me 2 where k - (kl + ka)/2 and m - (ml + m 2 ) / 2 can be taken as the m o m e n t u m and mass of either kaon. Expression (2.41) follows from the above result if we note that k (like [) is determined in the rest-frame of the scatterers: k = p/h = flE/h = fl~m/h. Equation (2.39) defines the coherent regeneration amplitude Oc. The intensity of K ~ mesons emerging from the regenerator is thus given by
IK~
-
(NA1AI)252 + 1/4 Ifrl2 [1 + e -(
--
loci 2
tKO(1).
-- 2e -~/2 COS(~m~)] IK~
-ms o ' (2.42)
Note that expression (2.42) has the same oscillatory form as that for the K~ ~ intensity (see (1.23), (1.24)). This is to be expected since in both cases oscillations arise from the interference of the two eigenstates of the hamiltonian, which have the same masses and lifetimes in the two cases. The above results are summarized in Fig. 2.7. The magnitude of 6m can be determined by measuring 0c as a flmction of regenerator thickness. This method is relatively simple and does not require a knowledge of ft. Using an iron regenerator, a spark chamber experiment
2.2 Coherent Regeneration Amplitude
27
IK2> + .~1 KO 1>
I K2>
Fig. 2.7. K ~ beam composition before and immediately after a regenerator
[11] obtained 5 m = 0.82 • 0.12 by plotting the K ~ intensity as a function of the iron thickness I (see Fig. 2.8). Since IKo(/) ~ e -N/t~
(2.43)
----e - l / t L
they measured the nuclear mean path, #, in a separate attenuation experiment.
I 900
I
I
I
I
8,1.5
9 9
800
9
RUN I RUN ~ RUN m
700 =0.8
600 0
m
500
Z
400
~=o
300
ZOO
100
o~
i
t
I
I
0
5
I0
15
20
THICKNESS OF IRON, r
Fig. 2.8. The magnitude of Amk measured by [11]
25
|
28
2. Propagation of Neutral Kaons in Matter
2 . 3 K x -0K
10 I n t e r f e r e n c e
and the
Sign of Amk
We next describe a regeneration experiment which determined the magnitude, as well as the sign, of the K ~ ~ mass difference A m k . The measurement was based on the observation of the interference between the regenerated (reg) and originally produced (orig) K ~ mesons, which proves unequivocally that the states Kl~ and K~ are quantum-mechanically identical [12]. Consider a pure K ~ beam impinging on a regenerator of thickness I placed at a distance d from the production point. Both l and d can be so adjusted that the two interfering waves, g ~(orig) and K ~(reg), have comparable amplitudes, thus maximizing the interference effect. At the exit of the regenerator
~after :/C~
+ K:l~
(2.44)
Since {K~ = ~
1
[tK ~ § IK~
(2.45)
> K:~ -- 0) = K:~ -- 0),
the above two amplitudes read (see (2.36), (2.39) and (2.43)) K:~
= e (ik~-l/2A~)(d+O e -1/2"/C~
-- 0)
K:~
- - t [1 - e [i(kl-k2)-l/2A1]l] e ik:(d+l) e -l/2t~ ~ ~
(2.46)
and = O)
: teikl(d+l) [e-i(kl-k2)(d+l) _ e-i(kl-k~)d-l/2A1] x e -t/2" 1C~
= O)
:e-ihmd/Al r, [e-ih~l/A1 _e-l/2A1] • e ikl(d+l) e -I/2~ K;~ = 0),
(2.47)
where iNA1AI~r - i h m + 1/2"
(2.48)
Hence,
)i~after ~ [e-(d+l)/2A1 q-e-ihmd/Axt(e-ihml/A1 --e-l/2A1)] • e ikm(d+l)-l/2tz K:~ -- 0).
(2.49)
Defining A(I) ~-r [e -ihm//A1 _ e-//2A,] -- {A(/)I e i arg A(l)
(2.50)
means the (normalized) intensity of K ~ mesons right after the regenerator can be expressed as
2.3 K1-K1 0 o Interference and the Sign of Am~
I~afterl2
I~o(d + z)
29
- e -(d+t)/A' -4- IA(Z)I 2 + 2 IA(Z)I
• e -(a+l)/2nl cos [arg A(1) -
5md/A1].
(2.51)
T h e last term in (2.51) describes the K o1 - K 1o interference. T h e difference in phase between/C~ and/C~ depends on the proper time elapsed between p r o d u c t i o n and regeneration because of Amk. By altering d, this phase difference can be changed to maximize the destructive interference between the two waves. Tile K ~ intensity will exibit a pronounced m i n i m u m when cos [argA(/) - 5md/A1] = - 1
i.e.
a r g A ( / ) - ~ m d / a l = 180 ~
(2.52)
Since the m a x i m u m interference occurs when the amplitudes of the two waves are a b o u t equal, l can be p r o p o r t i o n a t e l y reduced as d is increased to keep the ratio of intensities ]/C~ ]]CO(orig)l 2
IA(I)I2 e-(d+l)/AI
(2.53)
close to unity. In the experiment [12], a K + b e a m of 900 M e V / c at Berkeley was used to p r o d u c e K ~ mesons via K + n --+ K ~ in a copper target (see Fig. 2.9). S p a r k - c h a m b e r pictures of K ~ -4 7r+Tr- events were taken behind an iron regenerator. B y measuring the K ~ intensity as a function of (d + l)/A1 one can deduce b o t h the m a g n i t u d e and sign of A m k , provided the m a g n i t u d e and phase of [r arc known. T h e latter were determined by the authors in a separate scattering experiment with charged kaons on iron nuclei. 12 T h e phase of A(1) is the sum of the four nonzero phases in 13
A(1) = i N )~l Alfr e - i h m l / A1 -- r - i h m + 1/2
(2.54)
Based on reference [12], we obtain 14 arg(i) =
arg [e i~r/2] L J
=
71"/2 =
90 ~
arg([r) --
~ ~ 50~
~
12 The imaginary part of ~r can be determined from the K i cross-sections on nucleons: a ( K + n) = a ( K o p), a ( K - p ) = a(fi[ 0 n), etc. The real part of [ is obtained from the interference between Coulomb and nuclear scattering ([ is usually assumed to be purely imaginary). To calculate [r for iron nuclei, optical-model fits to nucleon data are used. 13 The phase of a complex function f ( z ) = fl(z)f2(z)f3(z) ... is the algebraic sum of the phases of its individual factors: arg f ( z ) ----arg f l ( z ) + arg f2(z) + arg f3 (z) + . . . . Recall also that arg[fl (z)/f2 (z)] = arg fl(z) - arg f2 (z). 14 The average momentum of the K ~ beam was approximately 760 MeV/c, which corresponds to A1 = 3.98 cm (l/A1 = 1.9).
30
2. Propagation of Neutral Kaons in Matter
Coil
Magnet
Coil
_
_
~
~2???ers.Y_./_~ L__
~'Cutargets ~ Foil sparkchamber K+ Beam . z,.,X,~,~I ~,~. r gapsof89 9so.eV/cc.:ll,, ,ml]l~,^ ~ , lO~'/pulse
Smal"'l " spark chambers
]1
~ I~, A
u
9
./
- Cz
IJ
////, Magnet
~"//~/
Coil
L .~
Coil
Fig. 2.9. The experimental set-up of W. Mehlhop et al. [12]
arg [1/2 -- iSm]-1 ---=- arg [1/2 - i6m] = arcsin
26m
+45 , 6m = +0.5, =
arg [e-i~m//A1 - e
- - 4 5 0 , ~m = - - 0 . 5 ,
-U2A'] = arctan [
1 ~ T~=,/2,~,j
sin(-Sml/A1)
Lcos(-~
_ ~ - 7 6 ~ (~m -~ + 0 . 5 , - - [ + 7 6 ~ 5m = - 0 . 5 .
Hence, arg A(l) = - 6 0 ~ =]=31~ for ~rn = "4-0.5 (see Fig. 2.10). The minimum is given by (2.52), which yields d/A1 = 3.1(6m = +0.5) and 7.3(6m = --0.5). The K1~ --+ 7r+Tr- decay intensity measured as a function of (d + l)/A1 = d/A1 + 1.9 is shown in Fig. 2.11. This distribution has a minimum at 5.5 (in units of ~h), corresponding to d/A1 = 5 . 5 - 1.9 = 3.6, which clearly favor8 the value of 3.1 predicted for 6m = +0.5. We thus conclude that K ~ is heavier than K ~ If one assumes a value for 6m, the experiment can provide the magnitude and phase of ~r. Excellent agreement was found between I~rl and Or obtained this way for 6m = 0.46 and the corresponding values from the scattering experiment with charged kaons mentioned above.
2.3 K ~
~ Interference and the Sign of A m k
. ~0 for 8 < 0
for
S>0
A for S>0
F i g . 2.10. Phase relation between Kl~ and g ~ [12]
10CO--
-
10
! ~.5
,t 3.5
5,5
I, 75 Ks LIFETIHES
I cl5
! II,S
f 135
F i g . 2.11. The rate of 7r+rr - decays as a function of distance measured by W. Mehlhop et al. [12]
31
3. C P V i o l a t i o n in K ~ D e c a y s
"But then in 1964 these same particles, in effect, dropped the other shoe."
Val Fitch, Nobel Prize lecture (1980) We have assumed up to now that the combined operation of charge conjugation and parity transformation, C P , which turns a particle state into an antiparticle state, is conserved in weak interactions. Considering that parity violation is such a large effect (as we mentioned earlier, all neutrinos are left-handed and all antineutrinos are right-handed), and that both P and C are not conserved in weak interactions (applying C to a left-handed neutrino changes it into a left-handed antineutrino), one may wonder if this assumption is justified. It turns out that it is not, as we will now explain. ^
^
3.1 Discovery of C P Violation In 1964, J. Christenson, J. Cronin, V. Fitch and R. Turlay 15 detected one 27r event among 500 or so common decays of the long-lived neutral kaon clear evidence of C P violation. Subsequent studies of semileptonic decays of neutral kaons have confirmed this finding (see Sect. 4.1). Unlike parity violation, which is maximal in weak interactions, C P is violated only infinitesimally (at a rate of about 10-3). Moreover, C P violation has been observed so far only in the K ~ system. While the nonconservation of parity was readily incorporated into the theory of weak interactions, primarily because the neutrino is adequately described by the Dirac equation for massless particles (the W e y l equation), a "natural" way to accomodate C P violation has yet to be found. Invariance under C P implies a particle antiparticle symmetry in nature. As it happens, there is practically no animatter in the universe. From measurements of galactic masses and nucleosynthesis calculations, and from the temperature of the microwave background radiation, the ratio of baryon to photon densities at the present time is found to be nb -- ~tb / ] o b s e r v e d
- -
-
-
n~
- -
10-7/cm 3 ~ 10_9. 400/cm 3
Since the universe is electrically neutral (n~ - ne+ = rip), there is also an excess of electrons over positrons of about 10-7/cm 3. On the other hand the 15 A Princeton University group.
3. CP Violation in K ~ Decays
34
theoretical prediction, based on the assumption of baryon number conservation and initial symmetry between matter and antimatter 16, is ~lth ~ 10-1s As shown in Sect. 4.1, the long-lived neutral kaon decays more often (about 3 • 10 -3 times) into a positron than into an electron. If the K2~ meson were a pure CP eigenstate, and if CP were strictly conserved, the two decay modes (4.5), which transform into one another under CP, would be equally probable. The nonconservation of CP therefore permits particle-antiparticle "discrimination" and thus may be responsible for the observed asymmetry between matter and antimatter in the universe. The charge asymmetry in the decay of neutral kaons not only distinguishes between matter and antimatter, but also provides an unambigous definition of positive charge: it is the electric charge carried by the lepton preferentially emitted in the decay of K ~ In essence, CP violation implies that the laws of nature do make an arbitrary distinction between left and right and between particles and antiparticles. So far, CP violation has been observed via the CP forbidden decays K ~ -+ Ir+Tr- [13a], 7r%r~ [135] and 7r+Ir-7 [13c], and in the form of charge asymmetry in the semileptonic decays of K ~ [26, 27]. In the celebrated CP-violation experiment of J. Christenson et al. [13a], a beryllium target was placed in the circulating 30 GeV proton beam of the Brookhaven A.G. Synchrotron. Neutral beams of approximately 1 GeV/c, emitted at 30 ~ to the proton direction, pass through two collimators and a sweeping magnet before entering a plastic "bag" filled with helium gas at atmospheric pressure, placed 17 m from the target. At this point the K ~ component has decayed away leaving a pure K ~ beam. Pairs of charged particles originating from the (cross-hatched) area inside the helium bag (see Fig. 3.1) are analyzed by two spectrometers consisting of bending magnets and spark chambers, triggered on a coincidence between water Cerenkov and scintillator counters. The helium bag serves to minimize secondary interactions in the decay region ("cheap vacuum"). Spark-chamber photographs were measured on machines equiped with digitized angular encoders, The rare 27r decays are distinguished from the c o m m o n semileptonic and 37r decays on the basis of their invariant mass ]7 and the direction 0 of their resultant m o m e n t u m vector relative to the incident b e a m (0 ~ 0 for K ~ --+ ~r+~r-). The apparatus was calibrated for 21r events by measuring K ~ -+ ~r+~r- decays produced by coherent regeneration in a tungsten regenerator successively placed at intervals of 28 c m along the sensitive decay region. Since the regenerated K ~ mesons have the same m o m e n t u m and direction as the K ~ beam, their decays simulate the CP-violating K ~ -+ Ir+Tr- decay. For these measurements a thin anticoincidence counter was placed immediately 16 In the context of the cosmological Big Bang model. 17
Tytlr+,r
-
=
/ytk 0
=
i=1
--
(Y]~L1p~)2]
1/2
3.2 Phenomenological Implications of K ~ -+ 27r Water Scintillator ~ e ~ o v
Plan view
.
.
.
35
>parkcham/r
.
57 ft to < internal target
Scintillat ,r -'..I Water Cerenkov
Fig. 3.1. The experimental arrangement of J. Christenson et al. [13a]
behind the regenerator to ensure that K ~ mesons decay downstream from it, thereby eliminating neutron-induced background events. Taking into account the relative detection efficiency for two- and threebody decays, they retained 45 + 9 events from the helium gas that appeared indentical with those from the coherent regeneration in tungsten in both mass and angular distribution. The total corrected sample of K ~ decays was 22700. They concluded that K ~ decays to two pions with a branching ratio of T~ - K ~ --+ 7r+Tr- - (2.0 + 0.4) • 10 -3. K ~ --+ all
3.2 Phenomenological
Implications
(3.1)
o f K ~ --+ 27r
Let us assume - - and this is generally believed to be the case - - t h a t the weak hamiltonian is not invariant under CP. As a consequence, its eigenstates are not CP eigenstates, but linear superpositions of CP-odd and CP-even components. Since CP violation is so small, we expect the particle eigenstates to differ only slightly from the CP eigenstates K ~ and K ~ We therefore write the short- and long-lived eigenstates of the weak hamiltonian, /:/w, as Ig~> - IK~ + el IK~ (3.2) IK~ ~ IK~ -4- e21K~ i.e. IK~ = ICP = +1) + (10 -3) x ICP : - 1 ) , (3.3)
IK~ = ICP = - 1 ) + (10 -3) x ICP : +1>,
36
3. CP Violation in K ~ Decays
where el,2 is generally a complex number. In the above expressions we neglected the normalization factors l / v / 1 + lel2, which are second order in c. The new states K ~ and K ~ are clearly not CP eigenstates. Moreover, they are not even orthogonah
( K~ I K~ = el + ~ .
(3.4)
This lack of orthogonality is to be expected since K~ and K ~ have the same decay modes. From (1.8), (1.10) and (3.2) it follows (neglecting terms proportional to c1~2) that
1
IK~ = ~
1
IK~ = ~
[(1 + el)lK~ + (1 - el)l/-
(1 + c1)lK~
.
If we assume that weak interactions are invariant under the combined operation of charge conjugation, C, parity transformation,/5, and time reversal, 7~, then CPT invariance means ci = e2 -- e,
(3.7)
where e is a measure of nature's deviation from perfect CP invariance. The above assumption is in accordance with the CPT theorem, which states that any quantum theory that is based on relativistic invariance and locality is automatically invariant under ~/5~. An important consequence of this theorem is that a particle and its antiparticle must have the same mass, decay lifetime and magnetic moment. To prove (3.7), consider a transition 2[(Pi, si) --+ .T(pi, si) , where 2[ stands for one or more particles in the initial state and $v for particles in the final state; p and s are the corresponding momenta and spins, respectively. This transition is described by the matrix element
.A.fi = (.T" I A [ Z}.
(3.8)
The operation of C flips the signs of internal charges, such as the electric charge, baryon number, etc., but spins and momenta are not affected; /5 reverses 3-momenta; under T, initial and final states are interchanged and spins and momenta are reversed. The effect of the combined C/ST operation is thus
I A I z>
.
(3.10)
For spinless kaons this implies ( C P T invariance)
(KO i/:/[ KO) = (/~o i/:/i ko).
(3.11)
If we define
~--~--' ~-=--&~' ~--~-e2
1 + et
1
~--v~'
-
•1
-
1 + e2
1 -
(3.12)
expressions (3.5) and (3.6) can be written as (IK~) = g~, etc.) K~=aK
~
~
g~
~
~
(3.13) K ~ = S g ~ + Z K o,
~:~ = .~KO _ ~ K ~
By using (2.9) and (2.10), the time development of the K ~ state is given by g ~ = ,~K~ e ~S + Z K ~ e *L = ~ ( , ~ g ~
~ er + ~ ('~g ~ - , ~ k ~ e %
i.e. K ~ = (6ae Cs + fl'),eeL) g ~ + (6fie r - flSe r
k ~
(3.14)
Similarly, /~o = (3~aeCs _ a,),eCL) g o + (~fleCs + a6eCL)/~o.
(3.15)
From (3.11) and (3.14) and (3.15) it follows that 6a e Cs + air eel = 7fl e Cs + a6 e r
---+ 6oe = ~/fl.
Therefore, 1 - e2
l+e2
1 e1 l+el -
--
-
-
) ~1
z~2
~s
By assuming C P T invariance, expressions (3.5) and (3.6) become
IK~) = f(e) [(] + e)lK~ + (1 - e)lR~ IK ~ =/(,)
(3.16)
[(1 + e)lK ~ - (1 - e)IR~
and
IK~
= f(e) [(1 -
e)lK~>
+ (1 - ~)IK~
Ig~
= f(e) [(1 +
e)lK~)
- (1 + e)lK~
(3.17)
where we included the normalization factor 1
f(e) - V/2( 1 +
lel2).
(3.18)
38
3. C P Violation in K ~ Decays CP
Found in nature
Not found in nature
/ Fig. 3.2. K ~ is a superposion of K ~ and /~o, with a slightly larger amplitude for K ~ This violates CP symmetry
/ / /
T h e C P n o n s y m m e t r i c state IK ~ is illustrated in Fig. 3.2. Expression (3.4) now reads (K~ ] K~ -
e + e* 2Re c _ ( K o ] KO), C P T i n v a r i a n c e . (3.19) 1 + ]~l2 - --------5 1 + [el
To relate a possible C P T - v i o l a t i n g mass difference mko - mko to the measurable q u a n t i t y AmL,S, we consider (3.14), (3.15) in a small time interval At. In this case e--(ims,L+Fs,L/2)t =
e_i~elS,Lt t~O 1 -- iA/Is,LAt.
(3.20)
Hence, K ~ -+ K ~ - i A t [(Sa.MIs +/33~ML) K ~ +/35 ( M s - J~L) NO] ,
(3.21) /~o _+ KO _ iAt [Ta ( M s - .A/[L) K 0 -f- ("//3MS A- C~6ML) R 0] , where we used (Sa+/37) K~
~
~
etc.
We can express (3.21) as . d~vac
1
dtp
(3.22)
( 1 (.A/IS-q-./~L) q- (MS --.~L)~ 1(.s
h4L) ( 1 _ 2~)
1 (MS--.A4L)(1 q-2~ ) 89(A4S +.A4L)_ ( M S _ M L ) g ] k~vac,
i.e.
9dl/fvac 1 dtp
{/J~ 11J~ 12 "~ -- \.A,~21
M22 ] ~vac
(3.23)
with _ _el _- e2 2 '
~ = -s --~- e2 2
(3.24)
Now, M22 - Mll
= (ML
- Ms)
(e, - e2)
= [(m~ - ms) + i (Fs - FL)/2]
(mL -- m s ) (el -- e2) (1 + i)
(e, -- e2)
3.3 Unitarity, CPT Invariance and T Violation
39
since ZlrnL,S ~ Fs/2 (see (1.43)) and Fs >> FL. Therefore, Re (M22 - M u ) -= mko - m~o = (mL -- ms) (cl -- e2) _< 3.5 • 10 -9 eV, which sets the upper limit on a possible CPT-violating K~ ence. Experimentally [32, 36], mko
-
m~o
mko
(3.25)
~ mass differ-
< 1.3 x 10 -18.
(3.26)
The above results can be summarized in the following way. According to (3.24), 2~=el+e2,
2~=el-e2
>el=~+~,
*
e2 ~
~*
-- ~-*,
which we use to express (3.4) as (3.27)
{KL~ 1 7 6
If CPT is conserved (and consequently T is violated), then according to (3.7) and (3.11) J~ll
=-M22, (K ~ I K ~ = 2 R e e (el =c2), C P T invariance. (3.28)
If, on the other hand, T is conserved (which means that CPT is violated), then ~ ( e 1 = - c 2 ) , T invariance.
A/ll2 =A/I21, (K ~ t K ~
(3.29)
because time reversal interchanges indices of the initial and final states. Ignoring the small CP violation, we find that m L -- m s
= -- (.hd12 + .hd~2 ) = - 2 R e
M12.
(3.30)
Recall that the mass and decay matrices are both hermitian: M21 = M{2,
3.3 U n i t a r i t y ,
F21 =/'1"2.
CPT
(3.31)
Invariance
and
T Violation
The observed CP violation in the neutral kaon decay implies that either T is violated but CPT is conserved, or CPT is violated but T is conserved, or both T and CPT are violated. To test these possibilities we will rely on the unitarity relations, first derived by J. Bell and J. Steinberger [14a]. These relations are direct consequences of probability conservation, as shown in what follows. The time development of the (K ~ K ~ system is given by -- ~s e-i'MStlK~ + as
e-i'MLtlK~
(3.32)
40
3. CP Violation in K ~ Decays
with A4S,L as defined in (3.20). The states IK~ due to CP violation. We can thus write
IK~ are not orthogonal
1~->, e t c The I1, 0) state is not symmetric under particle interchange because it changes sign for 7r+ ~ ~r-. This leaves us with 12, 0) and 10, 0) as the only allowed isospin states. To express the experimentally observed pion states 17r+Tr- ) and 17r%r~ in terms of the allowed isospin states 12, 0) and }0, 0), note that 17r+Tr- ) is the symmetric combination of the two states 17r+Tr~-) and I~r~-~r+): I~+~r- ) = ~
1
[Ir+rr2) + ]~2~+)].
(3.54)
Inverting the two remaining expressions (3.53) gives
I~+~-) =
d
,Tr~176 = ~ , 2 ,
12, o) +
d 1
0) - ~ , 0 ,
Io, o), (3.55) 0).
In the isotopic spin space, the K ~ --+ 27r decay is specified by the following four transition amplitudes (Trr I = 0 I f/w I K~
+
Qc]g~
= 0)>.
Here we set the KL~ amplitude to unity, in which case the amplitude of K~ is just the regeneration amplitude Qc = I~odeir
(3.88)
At a later time t, I+(t)) = e-iMLt]KL0(t = 0)) +
Oce-iMStlK~
= 0)).
(3.89)
The ~P -+ 7r%r- amplitude reads A~+=-
I#w
= (7r+1r I +(t)) = Atcso+~+ ~_ [/]+_e -i2vtLt § ~Oce-i'A4st]
(3.90)
and the corresponding decay rate (per K ~ meson) is
I.+.- =
Fs,+- {l~/+-12e -rLt + le~12e -r~' + 21,1+_ll0cle-(rs+rL)~/= cos [5.# -- (r
- r
(3.91)
where t is the proper time of decay. The value of 17/+_12 was obtained from the rate of K ~ -+ 7r+Tr- decays without regenerator; 21 that of ]0~]2 was determined by measuring the ~ + u decay rate immediately behind a dense regenerator (~c >> ~7+-), in which case the interference is small (for this measurement a 7.6 cm solid piece of berillium was employed). ~1 The result was in excellent agreement with that of Christenson et al. [13a].
3.5
o so Interference as KL-K
Evidence for
CP Violation
51
The maximum interference occurs when the KL~ and K ~ amplitudes are about equal. For a berillium regenerator this corresponds to a density of roughly 0.1 g/cm 3, or N ~ 7 • 1021 nuclei/cm 3. To attain low density, they used berillium plates 0.5 mm thick, separated by 1 cm; the whole assembly was 1 m long. Note that granularity effects in this case are negligible, since the element spacing is small compared with the K ~ decay length As = (pk/mk)/Fs (1.3GeV/0.5GeV) • 10-1~ ~ 2.6 • 10 - l ~ • 3 • 101~ ~ 7.5cm and the oscillation length lo~c ~ 12As ~ 90 cm. This arrangement is, therefore, equivalent to a uniform distribution of the same amount of berillium over lo~c. Since the length of the diffuse regenerator is considerably larger than As, the coherent regeneration amplitude is independent of ~ ~ I/As at distances sufficiently far from the face of the regenerator:
Qc l>>As iNoAsAsfr -iSm + 1/2 --: Co-
(3.92)
T h a t is to say, at large l the number of regenerated K ~ mesons is equal to the number of those which decay (AKo = c o n s t ) . The interference experiments are always performed over a time scale t 0, as demonstrated by W. Mehlhop et al. [12]). Since we will describe shortly an experiment that measured the phase difference
54
3. C P Violation in K ~ Decays
9
L
ID
w Z v
r~
~
o
9 a s ~ . O i I N . L N 3 A ~J
..o
hill
J J l
(2 ~$
R c - O i l SJ, N 3 A 3
-
//
IN
:Das,~.O~ I S J . N 3 A 3
Fig. 3.7. Event rates measured by C. AlffSteinberger et al. [18]
~ K S~ Interference as Evidence for 3.5 K L-
CP Violation
55
I'(
0'"
-0,
-0" -b
-1.{
0
~
/
)[
~ 'lP(Z IO"~)
0
,
Z
3
4
,
.
,
.
1. ( s NO'l~
F i g . 3.8. Interference terms measured by C. Alff-Steinberger et al. Bott-Bodenhausen et al. (right)
(left) and M.
r = r - r a n d , concurrently, t h e r e g e n e r a t i o n p h a s e r m u c h m o r e precisely t h a n t h e C E R N g r o u p s did, we do n o t p r e s e n t t h e values of t h e i r fitted parameters.
4. Interference in Semileptonic and Pionic Decay Modes
4.1 Semileptonic Decays of N e u t r a l Kaons In our discussion of the K ~ system so far we have associated the (K ~ doublet with strong interactions and the (K ~ K ~ doublet with weak decays. This classification is, in fact, not entirely correct. The reason is that the semileptonic weak decays of neutral kaons (see [21]), Neutral kaon --+ e+rrmve,
#+Trmv~,
(4.1)
have also been observed. These final states are clearly not CP eigenstates, for they transform into one another under CP. Consequently, they cannot be described in terms of the CP eigenstates K ~ and K~ instead they are decay modes of the strangeness eigenstates K ~ and/~o. The semileptonic decays of strange particles obey the selection rule AStrangeness = ACharge
(AS = AQ)
(4.2)
first postulated by R. Feynman and M. Gell-Mann in 1958. As an example of this rule, the decay ~- -+n+e-
(4.3)
+~
is observed (branching ratio ~ 10-3), whereas 57+ --~ n + e + + ve
(4.4)
is not (branching ratio < 5 • 10-6). In the case of neutral kaons, the AS = AQ rule implies K ~ -~ e%r-v~,
R ~ ~ e-Tr+~.
(4.5)
A test of (4.2) is shown in Fig. 4.1, where the measured distribution of K ~ ~ e+Tr-ve and /7/0 __+ e-rr+#e events from an initially pure K ~ state is plotted as a function of the K ~ decay time [22]. The result is in good agreement with the strangeness oscillation plot of Fig. 1.3, thus confirming the AS = AQ rule (to about 2%). A related measurement is that of J. Steinberger and his collaborators [23], who obtained the time dependence of the charge asymmetry: N ( K ~ ~ e+r-v~) - N ( f ( ~ --+ e-Tr+p~) N + - NA ( t ) = N ( K o --+ e+~r_ue) ~ N - - ~ - - ~ e-zc+o~) =-- N + ~ N-
(4.6)
58
4. Interference in Semileptonic and Pionic Decay Modes
1.0
9 N+ o N"
--x,O 0.8
0.6
0.4
0.2
I
.2
t
.4
I .6
I
I T
"
98"10"9s
eigentime
Fig. 4.1. A test of the AS A Q rule usin K ~ -+ e+Tc-uc and K w --+ e-~+~c decays [22])
(see Fig. 4.2). From expressions for the K ~ a n d / ~ 0 probabilities (1.23) and (1.24), it follows 23 that .A(t) = 2e-(Fs+FL)t/2 COS((~mt) = 2COS((~mt) e-FS t + e--FL t e - A F t ~- e+AFt
(4.7)
for a pure K ~ at t = 0 ( A F -- (Fs - FL)/2 and t is the proper time). Again, the measurement is in good agreement with (4.7) in the strangeness oscillation region. Note the apparent decay rate a s y m m e t r y between the K ~ --+ e+~r-L,~ a n d / ~ 0 ~ e - ~ r + ~ decays for large values of the K ~ decay time in Fig. 4.2. Assuming the validity of the A S = A Q rule, this a s y m m e t r y represents the CP-violating effect mentioned in Sect. 3.1. 23 For a beam of neutral kaons, the probabilities can be replaced by the number densities of particles and antiparticles in the beam.
4.1 Semileptonic Decays of Neutral Kaons
59
0.075 7" 0.05 0.025
z+
§
0
~-§
.
I
i J
"
I
-0.025
'
'
j
*~'
~
-
'
:'
2"
' r (lO-l%er 0 10* 'Decaytime
-0.05 -0.075 Fig. 4.2. Charge asymmetry as a function of the Kl3 decay time [23] In order to study the semileptonic decays Ks ~ K 0 --+ ~4-71"T/2 in more detail, we define the following transition amplitudes f = (t+~r-v [/:/w [K~
A S = AQ,
g - (e+~-~ I//w I K~
AS = - A Q .
(4.8)
Assuming CPT invariance, we have f* ---- (e-~+~ I ~qw I K~
as = aq, AS = -AQ.
g* - (e-~+~ I/tw I K~
(4.9)
The operation of ~/5 transforms particles into antiparticles. If CP is conserved, both f and g are real. The amplitudes g and g* violate the AS = AQ rule. This violation is small (g/f > 1 yields 2Re e = (K~ ] K~>. To keep the effects of K ~ ~ interference as low as possible, they selected events according to t~3 > 12.75 • 10-1~ and t#3 > 14.75 • 10 - l ~ s. Based on a total of 34 million Kr and 15 million K~3 events, their measurement yielded (see [24]) Re r = (1.67 + 0.08) x 10 -3.
(4.26)
Using (3.80) Re c =
17/+- I v/1 + (2AmL,s/Fs) 2
(4.27)
and the measured values of ZimL,S, ]U+-[ and Fs (see Sect. 4.3), they computed Ir/+-I = (1.66 J: 0.03) • 10 -3, v/1 + (2n,nL,s/rs) 2
(4.28)
in good agreement with the above result.
4.2
o o Interference K~-K~
in ~-+~-- a n d s
Decays
We now describe a high-precision K ~ ~ interference experiment which measured the phase difference r -r using the pionic decay modes K ~ --+ 7r+Tr- and, concurrently, the regeneration phase r from the timedependent charge asymmetry in K~ --+ e+zrT=v~ and #+Tr~=v~ decays [25]. In this experiment, a KL~ beam of 4 to 10 GeV/c momentum from the Brookhaven AGS traversed an 81-cm-long block of carbon. Multiwire proportional chambers were used to measure time distributions of 7r+Tr- and semileptolliC decays behind the regenerator.
K L0 - K s0 Interference in 7r+~r- and g•
4.2
Decays
63
The semileptonic decay rates F+ and F_ behind a regenerator are obtained from (3.89) by repeating the steps which led to (4.13), with the result F+(0c) c( +2Re e(1 - I x [
2) {Iocl2e -rSt + e -rL+ }
+ I1 + xl21ocl2e -rSt + IX - x[2e - r L t
+ 2 {2Re e • (1 - I x 1 2 ) }
loci
-- 410olin x e - r t sin(6mt + r
x e - r t COS0mt + r
(4.29)
In the interference region, e -rLt ~ 1 and e - r s t r
-41 ~
[Smt+(OS + r 1 6 2 1 6 2 1 6 2 1 6 2
~-87~
(4.37)
~ 46 ~
4 . 3 K ~o - K so I n t e r f e r e n c e
Without
Regenerator
The CP violating phase r can be measured independently of the regeneration phase Ce by observing K L0- K s0 interference in vacuum close to the K ~ 1 6 3~ production point. This so-called vacuum interference method requires the mass difference A~mL,S to be known very accurately: a 1% error in z~mL,s corresponds to an uncertainty of 3 ~ in 0+_. Here we describe an experiment by J. Steinberger and his collaborators at the CERN Proton Synchrotron, 24 who used results from two high precision measurements of Ama,s in the same detector [23, 28] to obtain r They were the first to employ multiwire proportional chambers (MWPC), invented by G. Charpak. MWPCs can handle event rates that are hundreds of times higher than those possible with spark chambers. The detector, described below, was also used in the charge asymmetry measurement discussed earlier. 24 A CERN-Heidelberg collaboration.
66
4. Interference in Semileptonic and Pionic Decay Modes
The apparatus is sketched in Fig. 4.6. The neutral kaons, produced by 24 GeV/c protons hitting a platinum target, were selected by a collimator at an angle of roughly 75 mrad. Protons with such momenta produce at small angles about three times as many K~ a s / ( ~ The kaon momenta were in the range 3-15 GeV/c. The collimator was followed by a 9-m-long decay volume filled with helium ("cheap vacuum"). A 6-m-long threshold Cerenkov counter, containing hydrogen gas at atmospheric pressure, was used to identify electrons. Muons were identified by two counters behind a concrete absorber at the far end of the detector. The decay region, extending 2.2 m to 11.6 m after the target, permitted detection in the proper time interval 3.5 • ]0 -10 S < tp < 30 z 10 -10 S. The momenta of charged decay products of neutral kaons were measured in a spectrometer consisiting of four MWPCs and a bending magnet. A total of 109 events was registered, with an average rate of about 1000 events per machine cycle. To select K~ --+ 7c+7r- events only inward bending pairs of charged particles were retained. They also required that: (a) there must be no signal in the Cerenkov counter and no coincidence between the two muon counters; and (b) the momenta of both particles must lie in the interval 1.5 GeV/c to 8.5 GeV/c, i.e., above the minimum momentum to traverse the muon absorber (1.45 GeV/c) and below the threshold for pion detection in the Cerenkov counter (8.4 GeV/c). To derive the 7r~rdecay distribution from the interfering K ~ and K~ states, let us assume that at t = 0 a pure K~ ~ beam is produced. At a later time t (see (4.10)) [~P(t)) = ~ 1 {(1 =7 e)e -iA4st lKs) 0 :h (1 ~
e)e-i~LtlK~
(4.38)
where the upper (lower) signs refer to a K ~ (/~0). Note that the phase between KL~ and K ~ at t = 0 is 0 ~ (180 ~ if the original state is a K ~ (/~0). The decay amplitudes read 1 A~r = AKo._+Tr~-~ e -irnst X {(1 ~ e ) e -rst/2 + (1Te)lrll e-i(~mt-r and the corresponding decay rates (we omit the factor Fs,+_/2, i.e.,
(4.39)
Fs,oo/2)
F,r. o( (1 T 2Re e) • {[r/12e-FLt + e - r s t
-t-2[rile -(rs+f'L)t/2 COS(Smt--r
(4.40)
The interference term changes sign when K ~ is replaced by/~o, resulting in different ~rTr decay distributions for the two states, as shown in Fig. 4.7. This illustrates nicely the violation of CP symmetry. The distribution (4.40) is practically identical to that behind a regenerator (see (3.91)), with the regeneration amplitude 0c = 1.
4.3 K Lo- K so Interference Without Regenerator
67
rr IJJ
_k
LJ~ rr laJ
r W ~D O re
a: r
ot tLJ r
Z
i r
t~J r~
~w~N
~7
tlJ m
{2.
Fig. 4.6. The apparatus used by J. Steinberger and his collaborators at CERN
68
4. Interference in Semileptonic and Pionic Decay Modes 10-3,
10 - 4 -
"
K 0 "-* 7rTr
r~
10-s
~o..~
~..
:. :
: : 9
10 -6
0
i
i
5
l0
'"
o
i
i
i
I5
20
25
30
['st
Fig. 4.7. Difference in ;rrr decay distributions between initially pure K ~ and /2/o beams, illustrating the violation of CP symmetry
T h e magnitude of rj+_ was obtained by measuring the ;r+rr - decay rate at Fst >> 1, FLt 15 x 10 -1~ T h e phase of 77+_ was measured in the interference region (5 x 10-1~ < t < 15 x 10 - l ~ s) by isolating the cosine-term. The results were [29, 30] r
= 45-9~ • 1-6~
I~+-I = (2.3 4- 0.035) x 10 -3.
(4.41)
To obtain r they used AmL,S = (0.5338 4-0.00215) x 101~ -1, the combined value of the two measurements of AmL,S in the same detector referenced earlier (one measurement was based on the variable gap regeneration method, and the other one on charge a s y m m e t r y in semileptonic K ~ decays). In fact, their analysis was slightly more complicated than this simplified description because the neutral kaon b e a m is an incoherent mixture of K ~ a n d / ~ 0 particles, as explained in Sect. 4.1. The rrTr decay intensity is therefore a linear combination of the two distributions (4.40):
x
{ I.12e-rL ~+ e - r s t + 2V(p)lule-(r~-cL)~/2cos(Star -
r } (4.42)
4.3 KL-K o so Interference Without Regenerator
69
(see (4.21)). Expression (4.42) was fitted to the ~ + ~ - data to obt&in Fs, I~+-], r S(p) &nd S(p), assuming that AmL,s and FL are known (see Fig. 4.8). The fit yielded Fs = (1.119 • 0.006) x 101~ s - ] .
A'~ ~ ~'~"
(a)
event
(4.43)
rate
10 5
-- ~ ~
I O"
With interference interfelnce
10 3
10 2
~
+I
L
l
L
(b)
Ai
g I
o u
5
.L 10 ro 10" io sec
j 15
.L__
Fig. 4.8. (a) The measured K ~ -+ 7r+Tr- event rate and (b) the extracted interference term [29]
5. P r e c i s i o n and #/e
Measurements
of r
r
The decay modes KL~ --+ 7r~ ~ are considerably more difficult to investigate experimentally than their charge counterparts K~ --+ 7r+Tr- . The reason is that neutral pions cannot be observed directly: a 7r~ decays within 10-16s into two photons. Instead, one has to detect and measure electromagnetic "showers" associated with the reaction shown in Fig. 5.1. The difficulty lies in measuring accurately the direction and energy of the final state photons. Y
K~
_~ " ' ~
Y
F i g . 5 . 1 . K ~ --4 lr%r~ --4 47
An additional complication arises from the fact that the C P conserving decay K ~ --+ 37r~ --+ 67 is two hundred times more frequent than the CP violating decay K ~ --+ 27r~ --4 47. Since 37r~ decays can simulate 47 events if two photons are not detected, one has to rely on distinct kinematical features of the 27r~ decay mode to eliminate this background. As the photon energy increases, its measurement precision improves. The two experiments described below used intense fluxes of neutral kaons with energies of about 100 GeV. High beam intensities are essential to achieve adequate statistical accuracy, especially in the K ~ -+ 27r~ channel. Both experiments measured the CP-violating parameter st/s, which can be related to the K~ -+ 7r~ ~ 7r+Tr- decay intensities in the following way. Using
170o 12 - rL,oo
rs,oo '
17+-I2 - /'L,+-rs,+_
(5.1)
(see (3.46)) we form the double ratio of decay intensities
]~7oo ]2 _ F ( K o _+ 7r%O)/F(KO __+ ~-o~-o) I~+_l 2 r ( K o ~ ~ + ~ - ) / V ( K O ~ ~§
(5.2)
72
5. Precision Measurements of r
r
and
~'/~
From expressions (3.72) and (3.73) it follows that ~?oo ~ c ( 1 - 2 ~ )
,
r/+_ ~ e ( 1 + ~ ) .
(5.3)
Hence,
] and Re
(@)
1[ ~ ~ 1
(5.6)
[ rio012 ] 1~+_12j .
By observing all four K L,S ~ ~ 7r%r~ 7r+~r- decay modes simultaneously, or at least two at a time, beam intensities and detection efficieneies cancel in the double ratio (5.2), thus minimizing systematic uncertainties in the measurement of e'/c.
5.1 The
Experiment
HA31
at CERN
We first describe an experiment by J. Steinberger and his coworkers at the CERN Super Proton Synchrotron (SPS). 25 Intense beams of KL~ and K ~ mesons with energies around 100 GeV were produced alternately by 450 GeV protons (1011 and 107 protons per pulse, respectively) at two different targets (see Fig. 5.2). The ~r%r~ and 7r+Tr- decay modes were detected concurrently, however. The K ~ data were taken with the corresponding target displaced in steps of 1.2 m, which resulted in uniform K ~ and K ~ decay distributions over a 48-m-long decay region, despite the short Ks~ decay length (6 m on the average). The decay region was evacuated and the space between two tracking wire chambers, set 25 m apart, was filled with helium. Photons from 7r~ decays were measured in a liquid-argon/lead calorimeter which was also used, together with an iron/scintillator calorimeter, to measure the energy of charged pions. There was no magnetic spectrometer. The KL~ --~ ~r+Tr- decays were reconstructed from hits in the two wire chambers, and the K ~ ~ 7r%r~ decays from the measured positions and energies of the photons. The energy spectra of accepted 7r%r~ and 7r+Tr- events are shown in Fig. 5.3. After corrections for various systematic uncertainties, the analysis, based on their 1986 data, yielded Re ( - ~ ) = ( 3 . 3 + 1 . 1 ) x 1 0 -3, 25 The HA31 Collaboration.
(5.7)
5.1 The Experiment NA31 at CERN
73
anficount er -ring!;
V 'q
] .....
Ilffll
beam dump
/
i , , .... ~176 ,
| k
'
II
I' M
co,,ma*0~s \
~-----~1
tllll
lifl-....
veto counters
I Ill~" h,d~o. calorimeter phofon calorimeter
wire chamber 1
wire chamber 2
Fig. 5.2. The NA31 experimental set-up at CERN which was interpreted as the first evidence of direct CP violation in the KL~ -+ 2~r decay [31@ Using data collected in 1988 and 1989, they reported Re (e'/g) = (2.0 + 0.7) • 10 -3 [315]. The NA31 collaboration also determined the phases of the CP-violating parameters U00 and 77+_ fi'om the time dependence of the 27r decay rates by using (4.42). F ~ is most sensitive to r in the region where K ~ and K ~ decay rates are about equal ( ~ 12K ~ lifetimes). The original beam layout was therefore modified to obtain the maximum acceptance in the interference region. The phases r and 0o0 were determined from the ratio of decay distributions for two different target positions (one near to the detector, ]Cnear, and one far from the detector, ]Cf~r), which renders the acceptance correction negligible. The maximum sensitivity is obtained when the interference patterns from the two targets are displaced by ~/2; at 100 G e V / c this corresponds to a distance of about 15 m. The measured decay rates from the combined /Cnear and Kfar data are shown in Fig. 5.4. The interference term was extracted by subtracting the fitted lifetime distribution without interference from the data. The phases r and r were obtained in the following way. They took the ratio of events observed from the two targets (see Fig. 5.5). A simultaneous fit to (~Cnear/K:far)00 and (K~near/]~far) +_ was made using bins of 5 GeV/c momentum and 0.5~'s (rs was computed for each event from the mid-point between the two targets). The phases r and r and the dilution factor :D(p) were varied in the fit, while TS, TL, I~+--I, I~lool and AmL,s were fixed. Taking into account various systematic uncertainties, they obtained [32] r r
= (46.9 ! 2.2) ~ = (47.1 4- 2.8) ~
(5.8) )r
- r
= (0.2 4- 2.9) ~
74
5. Precision Measurements of r
r
and r
K~--> 7T+n .~, 105
K~---> ~"TT ~
K~'--> ~*TT-
10 4
K~--'>~o~o 103
102
60
80
120
100
160
160
180
F i g . 5.3. Energy spectra of ~r~ ~ and Tc+Tr- decays a n d the corresponding event statistics [31a]
200 GEV
KAON ENERGY
K~--> .ri.Orf ~
K ~ --~ ~ * ~ . . . .
1.6 ~
:i~',"
K~--> n'+/r-
1.2
1 t
/tt t
E 107
106 -1.6~
6
8
10
12
6
14 16 18 Ks Lifetimes
8
10
12
14 16 18 Ks Lifetimes
105
oD m
104
i 5
7.5
10
t2.5
15
17~
20 2z5 25 K s Lifetime~
5
,
,
i
7.5
,
,
i
10
,
,
i , ,
12.~
i
15
,
,
]
,,
17.$
i
,
-1 i
,
L
20 22.5 2~ K, Lifetimes
F i g . 5.4. Acceptance-corrected lifetime distributions. Insets show the difference between a fit without the interference term and the data, averaged over energy [32]
5.2 The Experiment E731/E773 at Fermilab
75
KO > ~+~14 __ 12
_
,
IO
With int*r~=rence _
9
~thout ;hterference
12
P = I O0 GeV
~o
__
With interference
_ _
W~thout i n t e r f e r e n c e 9
P=
IOOGeV.
@ 6
2
$
7.5
10
12.5
I$
t7.5
2
22.$
25
5
7.5
I0
t2.$
15
17.$
20
22.5
25
Ks L i f e t i m e s
Ks L i f e t i m e s
Fig. 5.5. Ratio of decay distributions for two target positions [32]
5.2 The Experiment E731/E773 at Fermilab The second experiment was performed at Fermilab by B. Winstein and his collaborators, 26 who used 800 GeV protons incident on a berillium target to produce two parallel kaon beams, one pure K ~ and one with coherently regenerated K ~ mesons (see Fig. 5.6). In this experiment IPr ~ 10 • I~l, and so the 2~r-decays from the regenerator b e a m were mostly K s~ mesons. This way they obtained K ~ and K ~ beams with almost identical m o m e n t u m and spatial distributions. The regenerator alternated between the beams once
Photon Veto..~ .........
PhotonVeto
Muon Veto Lead Glass "...
/i~.,,~:i ....................li
i~
' iiil.ii.1, I' :i I ......i.
~ch~,s-?-":~!i::il I.......i...........j........... I "" = 2,omI Tngger Planes
Magnet
, 10 m
Fig. 5.6. Detector layout of the experiment E731 at Fermilab every minute, thus essentially eliminating any small difference in b e a m intensity or detector acceptance for decays from the two beams. However, since TL >~> TS, the detector acceptance as a function of decay vertex must be 26 The E731/E773 Collaborations.
76
5. Precision Measurements of r
o.•'"•,S'•~,,
10 3 --
KS
r
and c'/e
I OATA ,' MONTECARLO
-
-
~, ~t
0
10 2
0 tn
r
~
Z tJJ Q
bJ
10
tt4 t j ,: I 120
I , 130
VERTEX ( m )
rl 110
150
Fig. 5.7.7r~ ~ decay positions
[331
precisely known. This acceptance was determined by using a highly detailed Monte Carlo simulation which relied on K~3 and 37r~ decays. To minimize systematic uncertainties, K ~ and K~ decays to 7r~ ~ and 7r+:r - final states were detected simultaneously. A drift chamber spectrometer was employed to determine the 7r+Tr- momenta, mass and decay vertex. The energies and positions of the four photons from the :r%r~ decays were measured with a lead-glass calorimeter. The KL~ --+ ~r%r~ decay position and the 7r%r~ effective mass were obtained from the best pairing of photons into two pions (see Figs. 5.7 and 5.8 [33]). Semileptonic events were removed from the 7r+:r - sample using the ratio of shower energy to track momentum (for Ke3 decays) and a muon "hodoscope" (for Ku3 decays). The 37r~ backround to the 7r~ ~ data was estimated by Monte Carlo calculations (Fig. 5.8). After background subtraction, the full E731 data set contained (3.27 • 1057r+~r- , 4.1 x 1057r%r~ vacuum events and (1.06 • 10%r+Tr- , 8.0 • 105:r~ ~ regenerator events. The data were collected in 1987 and 1988 at the Fermilab "Tevatron" accelerator. To obtain Re (s'/g), the ratio of vacuum to regenerator events was fitted in momentum (p) and decay position (z) bins by using the following expression for the event rate downstream of the regenerator:
dpdzdN o( ~c(p)e_Z(Fs/2_iz~mL.s)//3. w ~- T/e_ZFL/2/3.yc 2 ,
(5.9)
5.2 The Experiment E731/E773 at Fermilab I
i
i
i
77
i
10 3 9 3~T" BkCKGROUNO
f OkTX
>
[0 2
O4
W
tO o 9 1 4 99
w.
1
420
[
I
]
I
I
440
460
480
500
520
HAS$ ( HeY
540
Fig. 5.8. 7r~ ~ effective mass
]
where c is the speed of light in vacuum, -), = Ek/rnk, 77+_ = s(1 + s'/e) and ~1oo = e(1 - 2el/e). In the fit they used (a) their own values of AmL,S and Fs, (b) the world average of Ir/+_l for H , (c) r = arctan(2AmL,s/Fs), (d) r = (43 + 6) ~ and (e) the empirical power-law parametrization of the regeneration amplitude
~0c(p) (2( p--C~e--i(2--c~)Tr/2
(5.10)
(see [34] regarding the above parametrization). Fits were first done for each decay mode separately, setting s / = 0, to extract a and Pc. The 7c+7r- and 7c~ ~ results were found to be mutually consistent, which points to a small value of el/s. A grand fit was next made to both modes simultaneously for the value of Re (s'/s), allowing the regeneration parameters to vary. They obtained [35] Re(~)=(0.74•215
-3,
(5.11)
a value not significantly different from zero. The full E731 data set was also used to measure the neutral kaon parameters ArnL,s, TS, r and Ar =-- r - r To extract AmL,S and ~-s they fixed r
~ r
~ arctan \
~
= 43.7 ~
(5.12)
78
5. Precision Measurements of r ....
I ....
r
I ....
i ....
I ....
I ....
L ....
I'''
....
i ....
I ....
I ....
I ....
I ....
i'''
ill,
,,It
,,,,
,,,i
,,,,
JlJJ
,ll
and e'/e
1.2 ~
0.8
o~
0.4
"
0
-04 -0.8 ~
-1.2
[-. eo
M
10"~ 10 4
10: i,i,
2 3 Time a (21 0 "l&s) Proper
,
'
I
'
I
'
I
'
I
""
Fig. 5.9a,b. Distributions in proper time for lr+lr - decays. The lines are the best fit results described in [35]
, I
'
1.2
[-..
0.8
O
0.4
~ ~
0
r/
-0,4
N -0.s *'* -1.2 l
(b)
10.1 O
r
f~'b,,
t 0 "J 10.4
0
2
4
6
8
I0
Proper Time (•176
12
Fig. 5.10a, b. Distributions in proper time for 2~r~ decays. The lines are the best fit results described in [35]
and simultaneously varied rs, ArnL,S, a and Oc(P = 70 G e V / c ) in expressions (5.9) and (5.10)). T h e extracted interference and exponential terms, together with the superposed best fits, are shown in Figs. 5.9a,b and 5.10a,b [35]. Combining the values for ~ m L , s and TS obtained from the two decay modes, they found
AmL,S =
(0.5286 + 0.0028) x 101~ -1,
'rs = (0.8929 -t- 0.0016) x 10 -1~ s.
(5.13)
5.2 The Experiment E731/E773 at Fermilab
79
The AmL,S result was lower than the existing world average by about two standard deviations. Based on the reported dependences upon AmL,S, they corrected the best previous measurements of r by using their value of Z~mL,S. The corrected values were found to be in excellent agreement with each other and with (5.12), as expected from CPT symmetry. To extract Ar a simultaneous fit to the charged and neutral data was made, allowing r Ar and s//g to vary, with the result 27 r
- r
= ( - 1 . 6 + 1.2) ~
(5.14)
A similar fit with AmL,S floating yielded r
= (42.2 =t= 1.4) ~
(5.15)
in agreement with (5.12), which is based on the world-average values for AmL,S and Fs. The apparatus of experiment E773, which took d a t a in 1991, was essentially the same as that of experiment E731, the main difference being that K ~ mesons this time impinged on two different regenerators, one placed 117m and the other one 128 m from the target. For this run a new "active" regenerator made of plastic scintillator was used, thereby reducing inelastic regeneration by a factor of 10 (kaons scattered inelastically may be assigned to the wrong beam). Downstream of its regenerator each b e a m is a coherent superposition of K ~ and K s~ mesons. The 27r decay rate is given by (5.9). The phases CQc - r and A r _ r -- 0 + - were extracted from the measured decay rates into b o t h neutral and charged pions [36] (see Fig. 5.11). From the fits, performed simultaneously to both regenerators, they found 102
102
~+~"
l i_.......... .... , IllJlllJ
t,lJ~]ntlllll[lll
120
130
140
Z decay (m)
150
120
130
IIII
140
I JI
150
Z decay (m)
Fig. 5.11. Measured rates for decay into 7r+Tr- and ~%r ~ The predictions from the fits with (solid line) and without (dotted line) the interference term are also shown [36]
27 Since they used their own values for Ts and AmL,S, derived assuming (5.12), they could not report 4)+- in the same fit.
80
5. Precision Measurements of r r
= (43.53 =t=0.97) ~
As with the cos [tAmL,s -- (r for 7r~ ~ events.
r
- r
r
and c'/~ = (0.62 =t= 1.03) ~
(5.16)
E731 data, the interference terms were fitted by -- Cec)] for 7r+~r- and cos [tAmL,s -- (r -- r + Ar
5.3 Comparison of NA31 and E731 Experimental Techniques We conclude this chapter with a few brief comments regarding the experiments NA31 and E731/E773 (see also [37]). The presence of a regenerator in E731 leads to the quantum-mechanical interference between the K ~ --+ 27r and K ~ --+ 27r amplitudes, the measurement of which can provide independent confirmation of an cl/c signal. The NA31 experiment had to be concerned with possible shifts in the overall detection efficiency, since the K ~ -+ 27r and K ~ --~ 27r decays were collected at different times under different rate conditions. The K ~ decay distribution was not uniform in E731, resulting in large relative acceptance corrections. Because of the shorter decay region, the residual background from 37r~ decays in the K ~ --+ 27r~ sample in E731 is considerably smaller than in NA31. The energy and position resolutions of the NA31 electromagnetic calorimeter are superior to those of E731. However, the plane resolution of its tracking chambers is much worse. As a consequence, the background in the K ~ -+ 7c+7r- sample is significantly smaller in E731 (NA31 was forced to discard about 40% of its ~r+Tr- d a t a in order to keep the eTrv background low). Concerning backgrounds, it should be noted that a 1% shift in the double ratio (5.2) corresponds to 1.6 • 10 -3 in e//e. The largest backgrounds in E731 and NA31 were at a few-percent level. In the Fermilab experiment the regenerator b e a m flux was significantly reduced by a 66 cm carbon absorber. Consequently, lack of statistics prevented t h e m from extracting r from the time-dependent charge a s y m m e t r y in semileptonic decays. The presence of D(p) in (4.42) is a fundamental deficiency in this class of experiments, as a source of uncertainty in the NA31 data analysis.
6. N e u t r a l K a o n s in P r o t o n - A n t i p r o t o n A n n i h i l a t i o n s
Proton antiproton annihilations were first used as a source of neutral kaons in early 1960s (see, e.g., Armenteros et al. [38]). In a typical experiment of this kind, J. Steinberger and his collaborators produced K ~ a n d / ~ 0 mesons in equal, but relatively small, numbers in the annihilation of low-energy antiprotons in a liquid-hydrogen chamber at Brookhaven: ~ + p -+/~~
+ pions,
D+p--+K~
(6.1)
Due to strangeness conservation in strong interactions, the K ~ (/~o) is "tagged" by the charge sign of the accompanying kaon. 2s Figure 6.1 shows the time distribution of the semileptonic decays K~ ~ --+ g• measured by F~anzini et al. [38].
6.1 The
CPLEAR
Experiment at
CERN
High-precision studies of CP violation based on this idea began a quarter of a century later at CERN, following the construction of the Low Energy Antiproton Ring (LEAR) and a dedicated detector. The C P L E A R experiment produces intense fluxes of tagged K ~ and /7/0 mesons by stopping low-energy antiprotons (200 MeV/c, 106 antiprotons/second) from LEAR in a low-density hydrogen target:
(PP)rest -+ /~~
(PP)rest --+ K ~
(6.2)
The branching ratio for each of the above two processes is about 0.2%, which means that K ~ and K ~ mesons are produced in equal numbers. However, the tagging efficiencies for K ~ and /7/0 are not identical because of different cross-sections for interactions of K + and K - mesons in the detector material. Note also that K ~ and/7/o undergo coherent regeneration in the detector, which must be taken into account. Tagged K ~ a n d / ~ 0 beams offer the possibility of observing directly K ~ K ~ interference. The K ~ 1 7 6 decay rate to any final state f reads (see (4.39), (4.40)) 28 Reactions (6.1) have also been used to test charge conjugation invariance in strong interactions.
82
6. Neutral Kaons in Proton Antiproton Annihilations |
t
I
I
I
!
i
I
I
I w
36109 EVENTS 32
Z8- 1 m 24Z
--
> 20tlJ
o e,-
167
W
--
~E I 2 Z
84-
q
o 2
4
6
8 I0 12 t x I0 "l~ sec
14
16
18
20
Fig. 6.1. The result from Franzini et al. [38] on the time-distribution of the leptonic decays of an equal mixture of K ~ and/s The solid curve is the prediction of the AS = AIQ rule
FKO R o o f = ~ ( 1 T 2 R e
e)FKo__+f{l~fl2e -rLt + e - r s t
+ 21wle-(rs+rL)t/2 eos(Smt -- e l ) } , where
Ft.~0.8 0
0.5 0.4
0.2 0.1 0
-0.1
2
4-
6
8
10
12
14
16
18
20
Fig. 6.5. The asymmetry .Azure(t) as a function of the decay time (in units of Ts). The solid line represents the result of the fit [40]
86
6. Neutral Kaons in Proton-Antiproton Annihilations Their most recent results on A m L s and Re x (full statistics) are [41a] AmL,s = (0.5295 + 0.00202) x 101~ s -1, (6.11) Re x = [-1.8 + 4.1(stat) + 4.5(syst)] • 10 -a.
As explained in Appendix A, the K ~ meson may decay into the kinematicssupressed and CP-allowed final state 7r+Tr-Tr~ with L = 1 = 1 and CP = +1, or into the kinematics-favored and CP-forbidden state :r+Tr-zr~ with L = l = 0 and CP = - 1 . This results in a Dalitz plot distribution which is symmetric with respect to the 7r+ and :r- for the CP-violating amplitude and antisymmetric for the CP-conserving amplitude. Thus by integrating the decay amplitude over the entire phase space of the K ~ -+ 7r+Tr-Tr~ decay, the CP-allowed contribution can be eliminated. From (4.38), and defining T/+- 0 ~
AKg-~+~- ~~
the decay amplitude for K ~
A+-o =
(6.12)
AKo-+Tr+;r- ~o
AKo~r+Tr-Tro
~-+Tr-~ ~ can
~ -+
be expressed as
1 e_imLt
x {(1Te)e-FLt/2+(l:Fe)rl+_oe(ia=-vs/2)t},
(6.13)
where the upper (lower) signs refer to K ~ (KO). The corresponding decay rates read F + - o c( (1 T 2Re e){e -rLt +
I +_olZe-r
-t-e-(rs+rL)t/2 [,~__oe-iamt + ,+_o ei~''t] }.
(6.14)
The time-dependent decay rate asymmetry, which is a direct measure of the KL-K o so interference, is given by
A+-o(t) =
r+_o(t) F+-o(t)
-
r+_o(t)
+ V+-o(t)
2Re e - 2e -(rs-FL)t/2 x
{Re ~7+-o cos(Smt) -- [m U+-o sin(e~mt)},
(6.15)
where F and P are the KO and K ~ decay rates, respectively. A recent result on the CP-violating p a r a m e t e r ~+-0, based on the full statistics of CPLEAR, is [42a] Re 7/+-o = [ - 2 + 7(stat)+4(syst)] • 10 -3,
(6.16)
I m ~/+-o = [ - 2 + 9(stat)+_2(syst)] x 10 -3. Additional information about their measurement of ~?+-o can be found elsewhere [42b].
6.2 Is CP Violation Compensated by Time-Reversal Asymmetry?
87
The Fermilab experiment E621 has also published a result on Im ~+-o by fixing Re rl+-o = Re c and assuming C P T invariance [43]: Im rl+_o = [-15 + 30] x 10 -3.
(6.17)
6.2 Is C P Violation Compensated by Time-Reversal Asymmetry? If C P T is conserved, the observed CP violation demonstrates the failure of time-reversal invariance. A direct test of T asymmetry in the K ~ system was suggested by Aharony and Kabir in 1970 [44]. As shown in Appendix D, the operation of time reversal gives the identity
(K0 i e-ira i/?0 > = (f;o i e-~HTt i K0 >
(6,1S)
where /2/T is the time-reverse of the hamiltonian /2/. If /2/T ---- /2/, the two amplitudes are equal. A nonzero value of the ratio of transition intensities F(I~O _+ K o) _ F ( K o _+ [(o) A T ( t ) -- r ( K o -~ g o ) + F ( K o ~ fi2o),
(6.19)
where
F(K ~ ~ K ~ - (KO e_i/:/t [KO ) 2 (6.20) r ( g o _+ [(o) == (KO e_i/:/t ] Ko ) 2,
would thus imply /2/T r /2/, 1.e., a violation of time-reversal invariance. The first evidence for time-reversal noninvariance has been reported by the C P L E A R collaboration based on semileptonic decays of tagged neutral kaons [41b].They extracted A T ( t ) = -F+(t) - F_(t) m 4Re e + 2Ira sin((~mt) (6.21) F+ (t) + F_ (t) X c o s h ( A F t ) - cos((~mt)
from the measured decay rates F + ( t ) - F [/~~ = 0) ~ g+Tr-u] and F_(t) F [K~ = 0) --9 f-lr+O] (see Fig. 6.6). Expression (6.21), which was derived assuming C P T invariance, follows readily from (4.13), with A F -- ( F s - F L ) / 2 (see also Appendix D). The C P L E A R measurement yielded AT(t) = [6.6 + 1.3(star) + 1.0(syst)] x 10 -3,
(6.22)
which should be compared with 4Re e ~ 6.6 x 10 -a (see (6.21), (4.26), and (4.28)), assuming Im x = 0.
88
6. Neutral Kaons in P r o t o n - A n t i p r o t o n Annihilations
t.. 0.04 0.05 0.02
._~ .
0.01
-0.01 -0.02
2
4
6
8
10 12 14 16 18 20 Neutral-kaon decay time ["rs]
F i g . 6.6. The asymmetry .AT, indicating a violation of T invariance in semileptonic decays of neutral kaons. The solid line represents the fitted average (.AT(t)) [41b]
7. N e u t r a l K a o n s in E l e c t r o n - P o s i t r o n
7.1 The
DA(I)NE
Annihilations
Project
The DA(I)NE project at Frascati (Italy) will study the process e+e - -+ virtual photon --+ ~5 -+ K~
~
jPC _- 1 - - ,
(7.1)
by producing at rest about 5000 r mesons per second at the centre-of-mass energy x/~ -- 1020 MeV. As explained in Sect. 1.4, the neutral kaon pair in (7.1) is in a pure C = - 1 quantum state: 1
{[K~176
- IK~176
= --~ { I K ~ 1 7 6
[K~176
[4~> = ~
1
} (7.2)
0 S0 - K s0K L. 0 With an i.e., the final state is either K ~ ~ - / ( ~ 1 7 6 or KLK expected collider luminosity of 5 • 1032 c m -2 s -1, about 10 l~ coherently produced K~ pairs per year will provide a particularly beautiful method of quantum interferometry. To study the time evolution of the state (7.2), we denote by f l ( t l , +z) and f2(t2,-z) any two final states in the neutral-kaon decay, and define the amplitude ratios
~ _= 0 and A t < 0 distributions. T h e interference p a t t e r n s for different combinations of final states shown in Fig. 7.1 can be used to extract Re (e'/e), I m (el/e), z~mL,S, [T/TrTrI, r etc., as discussed in [46b, c]. T h e a s y m m e t r y in K ~ --+ g+lrT=v decays provides tests o f T and C P T invariance. 1.2 3.5
b
3 2.5
0.8
'~ f '2 '~
2
0.6
1.5
0.4 1 0.2
o .,s
\
\
\
0.5
:iO . ?
" ~ .... ; 0 i s
0 -30
'-i0
:i0
~,'oio-)o
-15-10 -$
0
$
10 15 20 25 30
~l=(tFt2)l~ s
Fig. 7.1. Calculated interference patterns for the following final states in two-kaon decays: (a) fl : wTTr-, f2 = w~176 (b) fl = l+Tv-v, f2 : /-Tr+v; (c) 11 = 2~, f2 = lzcv
Measuring el/r at DAgPNE requires very accurate reconstruction of t + _ and too. T h e high-statistics d a t a from K ~ --+ g• 3~r decays will be used to m a p detector acceptance and reconstruction efficiency. A b o u t 107 7r+Tr-Tr~ 0 decays are needed for a statistical error of 10 -4 on Re (r the CP-violating K ~ --+ 27r decays being the modes with the limiting statistics. In this context, the K ~ and K ~ decay lengths from ~5 --+ K ~ K ~ are As = 7/3c7"s = 0.592 cm and AL = 7~CTL = 343 cm, respectively. A b o u t one quarter of K ~ mesons are expected to decay within the tracking volume of the K L O E detector at DA(I)NE.
92
7. Neutral Kaons in Electron Positron Annihilations
The DAONE project is, undoubtedly, very versatile. It provides a novel method of q u a n t u m interferometry and precision tests of the discrete symmetries C, P and T not readily achieved in other experiments. The neutral kaons in the reaction (7.1) change their identity continually and in a completely correlated way. This can be used to test q u a n t u m mechanics by studying correlations of the Einstein-Podolsky-Rosen type (see Sect. 1.4 and [47, 48]). Note also t h a t a 4~ factory is characterized by low background, since about a third of the 4~-decay final states are neutral-kaon pairs ( K + K - : K ~ ~ : 0~r = 0.49 : 0.34 : 0.13).
8. Neutral Kaons in Fixed-Target Experiments
8.1 The
Experiments
KTeV
and
NA48
Presently there are two major fixed-target experiments with neutral kaons: NA48 at C E R N and KTeV at Fermilab. The elegance and sophistication of these experiments reflect years of experience with K ~ beams, especially that gained with their predecessors, NA31 and E731, respectively. The main aim of each of the two groups is to determine Re (e//e) with a precision of ~ 10 -4. In order to achieve this goal it is necessary to collect roghly 4 million KL~ --+ 27r~ decays, an increase of about an order of magnitude compared with NA31 or E731. The KTeV apparatus is shown in Fig. 8.1. The experiment retains the basic features of E731: it uses two beams, which are side by side and identical in shape, and records all four decay modes simultaneously. To reduce inelastic regeneration, an "active" regenerator made of plastic scintillator is employed. The most significant improvement with respect to E731 is the use of an Muon
Analysis Magnet Photon Veto Detectors i
i .................. Ku
beams
[email protected] ...........t,I............ F::::II IH 1 1.............. lq.. Vacuum DecayRegion
IT
V
,
.......... : ..... acuum Window l l ~ ; ~ : T .
Drift ;5' .
.
li
~'~"' ;' ' ) ....... -.- ] ,
| IllIf i
]~egenerat~ [' I I /il~~ilTr,, i
Muon
Filter Veto i" ii
.
.
g
/
Hadron Veto with Lead Wall I
120
I
I
140
I
I
I
160 Distance from Target (m)
Fig. 8.1. A schematic drawing of the KTeV detector
180
i
94
8. Neutral Kaons in Fixed-Target Experiments
~t
E Regenerator beam data Prediction without interference > -,
40-50 GeV kaon energy
lO
10 4 .....~
10 3
125
130
135
140
145
150 155 Distance from target (m)
Fig. 8.2. Interference in K ~ -+ 7r+~r- decays measured by KTeV undoped CsI electromagnetic calorimeter. This should result in a much better energy resolution and thus more efficient background rejection in the 2~r~ decay channel. KTeV expects to achieve an error of 0.5 ~ in the determination of r and r and also of r using semileptonic decays. Fig. 8.2 shows their preliminary result on kaon interference. The NA48 detector is sketched in Fig. 8.3. An important element of the new design is the concurrent presence of (almost) collinear K~ and K ~ beams in the detector, produced by protons hitting two different targets. A channeling crystal is used to simultaneously attenuate and deflect protons emerging from the KL~ target, which are then sent toward the K ~ target located close to the detector (see Fig. 8.4 [49]). The protons producing the K~ component are tagged in order to distinguish between the K~ and KL~ beams. A major improvement compared with NA31 is a liquid-krypton calorimeter with superior energy and position resolution. A magnetic spectrometer has been added for charged decay modes.
8.1 The Experiments KTeV and NA48 lavetocounters, hadroncalorimeter
"x
li filter (iron) triggerhodoscope anticounter ~
~
~
homogeneousLKr calorimeter
magnet
/
/
~
ch r am s be/
beampipe Fig. 8.3. The NA48detector at CERN Detectors 1012ppp "~-
KS t a r g e ~ ....~
Z
~
~
target ~ j~" tagging ~ . 107.ppp/ counter Channeling crystal ~ Fig. 8.4. Simultaneous K~ and K~ beams in NA48
"
95
96
8. Neutral Kaons in Fixed-Target Experiments
Supplement:
Kaon
Beams
As a supplement to the preceding section, we will describe (a) the production of K mesons by a high-energy proton beam striking a stationary target, and (b) the techniques of electrostatic and RF separation of K + beams. Our discussion is partly based on the lecture notes by B. Winstein [104]. Other sources of K-mesons are discussed in Sects. 6.1 (proton antiproton annihilations) and 7.1 (e+e - collisions). An experiment that uses K + "beams" at rest is described in Sect. 9.3.1. The production of charged kaons by incident protons on a Be target can be expressed as a yield per incident proton per GeV/c kaon momentum per steradian of the solid angle ~2. An approximate parametrization of the yield at angle 0 is given by d2a [( mb r] P(~I - ~-~)a (1 + 5e-20x) dpdf2 G e ~ / c ) s t o( j -=pf(x, Pt)
(S8.1)
with x = P/Pbeam and Pt = Op. Using d3p = p2dpd~, we can write d d2a p d ~ -p2d3a~pp
d3cr'~_ ,~,p ( E--~p] -= p O'inv,
(88.2)
where "inv" denotes Lorentz invariance. The k~on yield per incident proton therefore reads d2N P O'inv Yield-= d p d ~ - ~rinel '
($8.3)
where ainel is the inelastic cross-section. From ($8.1) we see that the kaon production is most copious in the forward direction (Pt = 0). The most significant difference in the production of K + and K - mesons is in the x dependence: K + (x: (1 - x) 3,
K-
(x (1 -- X) 6.
(S8.4)
By simple quark-counting, one can show that the relative production rates of charged and neutral kaons are given by K ~ ,-~ ( K + + K - ) / 2 ,
[(o ~ K - ,
($8.5)
i.e., K o _/~o K 0 +/7/o
K + - KK + + 3K-"
(s8.6)
The asymmetry in the K ~ and/~o production spectra is due to an interplay between associated production and baryon number conservation. Expressions ($8.5) and (88.6) are corroborated by a measurement of the "dilution factor" KO(p) _ [gO(p) D(p) - KO(p ) T fiO(p ) at the CERN SPS (the NA31 collaboration [32]; see Sect. 5.1).
($8.7)
8.1 The Experiments KTeV and NA48
97
Therefore, a proton b e a m produces a mixture of K ~ a n d / ~ o mesons that depends upon energy and angle. At high energies (large x), the K ~ production dominates. As was mentioned in Sect. 4.1, if K ~ and /~0 mesons were produced with equal spectra (D(p) = 0), the interference t e r m in (4.42) could not be observed in a target experiment! Sufficiently far from the production target, where only the KL~ component can survive, the initial composition of the kaon b e a m becomes irrelevant. In the forward direction (0 = 0~ the neutral b e a m is dominated by photons and neutrons. Photons, which originate mainly from rr~ decays, are easily removed with a lead converter near the production target. It is sufficient to have a converter t h a t is about 10 radiation lengths thick ( ~ 6cm), since the probability of photon non-conversion is e - ( 7 / 9 ) x 1 0 ~ 4 • 10 - 4 . The charged products of the resulting electromagnetic shower are swept away by a magnetic field. The neutrons are produced with the invariant cross-section 25 mb Oinv(n ) ~ (GeM/c)2 e -5pt,
($8.8)
which does not depend on x = P/Pbeam" The neutron flux is m a n y times the kaon flux at 0 = 0 ~ and thus presents a much more serious problem t h a n the photon beam. Fortunately, the kaon to neutron ratio can be enhanced very efficiently by targeting away from 0 = 0% It follows from the above differential distributions t h a t an enhancement factor of over 100 can be obtained with less than a factor of 2 loss in the kaon flux at 0 ~ 5 m r (see [104]). In the design of a K ~ beam, one also has to consider: (a) the "soft" component of neutral particles in and around the b e a m ("beam halo"), (b) the forward hadronic "jet" and (c) the b e a m of noninteracting particles. The first component can be reduced considerably by collimators that subtend little solid angle compared with ~2beam- To dispose of the other two components, a "beam dump" is used t h a t is thick enough to completely absorb the hadronic shower, and is also sufficiently close to the target to reduce the muon flux originating from ~ and K decays. As an example, we outline the main features of the KTeV beamline. The kaons are produced by 800 GeV protons striking a BeO target. The proton b e a m is delivered every minute in spills lasting 23 s. About half of the 3 • 1012 protons per spill interact in the target, while the other half are absorbed in a b e a m dump. Neutral kaons produced in the horizontal plane at an angle of 4.8 mrad relative to the proton direction point toward the detector. Two identical sets of collimators produce two secondary neutral beams at • mrad in the horizontal plane. The size of the beams is about 8 cm • 8 cm at 180 m from the target. A large sweeping magnet downstream of the b e a m d u m p removes muons from both the primary target and the dump. The photon flux is reduced using a 7.6-cm-long lead absorber t h a t transmits 55% of kaons and neutrons.
98
8. Neutral Kaons in Fixed-Target Experiments
To further enhance the kaon to neutron ratio, a 52-cm-long Be absorber is used. To see how this enhancement works, note that the neutron and kaon total cross-sections have the following atomic number dependences: O'tot(/t ) ~,~ 49mb • A ~
5rtot(g )
~
24rob • A ~
A > 7.
($8.9)
The cross-section ratio is thus greatest for small A. The number of interaction lengths, Xint, in the Be absorber (A = 9, L -- 52 cm) is Xint ~- (Tt~ ( - ~ )
L -~ { 0.97, 1"74'
kaons, neutrons,
($8.10)
where No ~ 6 • 1023 is Avogadro's number and ~) = 1.85 g/cm 3 is the density of Be. The respective transmission coefficients are e -Xint -- 0.18 (neutrons) and 0.38 (kaons), resulting in an enhancement factor of about 2. As mentioned in Sect. 9.3, the purity of K + beams at Brookhaven (experiment E787) has been considerably improved through electrostatic (DC) separation. This method of particle separation works in the following way. An almost parallel beam enters a separator which has a vertical electric field and a horizontal magnetic field. The separator is tuned in such a way that the action of the magnetic field cancels the action of the electric field for the kaons, whereas the pions are deflected. At a vertical focus downstream of the seperator, the pion beam is displaced from the axis and then stopped. The separation is given by Separation (in rad) .-~
,
($8.11)
P where, in the case of E787, E = 60 K V / c m and L = 220 cm (p is the beam momentum). The method of RF separation employs, in general, two cavities and a system of quadrupole magnets between them. The first RF cavity imposes a transverse momentum kick on the beam of a few MeV/c per meter of cavity length. The second cavity, located downstream of the first one, is so tuned that the Tr+ mesons arrive with the same phase that they had in the first cavity. Since the 7r+ momenta have been reversed in the quadrupoles, the positive pions end up with no net kick. If the RF frequency and the distance between the cavities are such that the ~+s and protons are 360 ~ out of phase at the second cavity, the protons will also receive no net kick. The K + mesons, on the other hand, are 90 ~ out of phase with respect to the 7r+s, and thus get a net transverse kick corresponding to ~ 1 mrad. A beam plug downstram of the second cavity stops the pions and protons. RF-separated K + beams can be used, for example, to search for the rare decay K + --+ 7r+vO (see Sect. 9.3). Alternatively, they can be focused on a target to produce K ~ mesons by charge exchange.
9. T h e K ~ S y s t e m in t h e S t a n d a r d M o d e l
9.1 C a l c u l a t i o n
of Am k and ek
So far our description of the K ~ system has been restricted to its quantummechanical properties and phenomenological implications of the observed CP violation. In this section we consider K~ ~ mixing within the framework of gauge theories of the electroweak and strong interactions (the Standard Model (SM)). We will outline the calculation of the off-diagonal elements of the K~176 mass matrix that generate the tiny K~ mass difference and the CP-violating parameter e (see (3.30) and (3.82)). This is the forerunner of all the calculations that are forbidden in the lowest order of the electroweak theory. The smallness of the K ~ ~ mass difference (Amk = 3.5 • 10-6 eV) enables the quantum-mechanical interference effects between the K~ and K ~ components of a K ~ beam to spread out over macroscopic distances (see Sect. 1.5). Phenomenologically, it played an important role in the establishment of the charm hypothesis, as explained in what follows. There is overwhelming experimental evidence that strangeness-changing neutral currents are heavily suppressed. For example, F ( K ~ -+ #+#-) ~ 10_8, r ( K ~ ~ all)
F ( K • --+ ~• F ( K • --4 all)
.~ 10_10.
(9.1)
The suppression of K ~ --+ # + # - was very surprising because the analogous charged decay K + --+ #+v~ has a branching ratio of 63% (see Fig. 9.1, where the mesons are represented, in accordance with SM, as bound states of quark-antiquark pairs). To explain this puzzle, Glashow, Iliopoulos and Maiani (GIM) incorporated the charm quark into the hadronic weak current originally proposed by Cabibbo. The weak charged current in the GIM model reads [50] j a CC -
(UC)L?~
S'
(9.2)
'
L
where
=
{' cosOr sin 0r "~ (d) k,-sinOr cosOr s L'
r
s')= V r
s),
(9.3)
100
9. The K ~ System in the Standard Model
Zo 9 s9 $ U
K+
r
g+
,
sinOr
) o
KL
(b)
(a) e+
w+{
I
sinOr
V:o
K+ U
11
(c) Fig. 9.1. Allowed decays K + -+ #+v, (a) and K + --+ rc~ first-order contribution to K ~ --~ # + # - (b)
(c); forbidden
g is the SU(2) weak coupling, y~ are Dirac matrices and 0c is the Cabibbo angle. The subscript L denotes "left-handed" spinors: qL = ~1 (1 --~5)q. The quarks have charges Q~,c,t = + 2 / 3 and Qd,s,b = --1/3. Since the flavor quantum numbers are not conserved in weak interactions, the weak eigenstates d~ and s ~ are mixtures of the mass eigenstates d and 8. 29 The mixing matrix V, which also couples the up (u) as well as the charm (e) quark to a linear combination of the down (d) and strange (s) quarks, is unitary: vW
= v-~v
(9.4)
= 11
that is to say, the inverse of V is its hermitian conjugate (ll is the unit matrix). The unitarity of V ensures the absence of strangeness-changing neutral currents. Indeed, ~0r
= CV t O r e
= ~OV t Yr = r162
(9.5)
for any arbitrary operator 0 . From (9.5) we conclude that the neutral component of the hadronic weak current does not contain terms that mix quark flavours. Changesof flavor alwaysinvolve change of charge (hence A S -- AQ). This explains why the amplitude in Fig. 9.1b is zero. 29 It is sufficient to mix ("rotate") either
(ds) or (uc).
9.1 Calculation of Am k and ek cosO,=
d
W +
[ l i i ui i i i i l l sinOc W-slnOc
g co~O~
101
,
~'-
W+
W-
g+
Fig. 9.2. Second-order contributions to K ~ --~ #+#-
The decay K ~ -+ # + # - can also proceed through the diagrams in Fig. 9.2. The amplitudes corresponding to these diagrams are proportional to cos 0c sin 0c and - sin 0c cos 0c, respectively. The second amplitude cancels most of the first. If the u and c quarks had the same mass, the cancellation would be exact (see (9.13)). The foregoing discussion demonstrates the importance of the Cabibbo angle 0c, which not only suppresses the weak transitions that are proportional to 0c ~ 13~ but also removes strangeness-changing neutral currents via the charm quark. The origin of 0c, however, is not explained in the Standard Model. It is instructive to calculate the amplitude that corresponds to the lowestorder diagrams contributing to Amk (see Fig. 9.3a,b). The lowest order for which the transition sd ++ ,~d is possible is fourth order in the weak hamiltonian/:/w: the two W bosons in Fig. 9.3a,b are emitted and reabsorbed, so that a total of four weak "vertices" are involved. Of the two sets of diagrams in Fig. 9.3a,b it is sufficient to compute the first one: it can be readily verified that, in the limit of vanishing external momenta, the amplitude corresponding to the second set is identical. We ignore, for the moment, the contribution of the top (t) quark, which is justified when computing A m k (but not e), as explained later on. Neglecting the external quark momenta, 3~ the electroweak Feynman rules applied to the box diagrams in Fig. 9.3a yield, in the so-called t'Hooft-Feynman gauge, 31
30 In the K ~ rest frame, their components are of the order of mk and thus can be neglected compared with the W boson and heavy quark masses. 31 The presence of unphysical scalars in this gauge can be ignored when the top quark is not included, since their coupling to fermions (f) is proportional to
m//m~
~ I.
102
9. The K ~ System in the Standard Model Vid i i i
d
.~--S m
w+:' l k 1 W"
K~
o t i
v.~, ,.~.~
v~
(a) v~ d
K~
~
w +
v~ ~s
. . . . . . . . .
qi
v.~
~
7 mcS >> mu2 [rn~ = (80.3+0.05) GeV/c 2, rnc = (1.3+0.3) GeV/cS]. Therefore, i M --
--ig4
2
28 2m
2
mc -- mu sin20c cos20c
• [~'7~'~.TQLUdO~'ye'~ ~/~Lvd ].
(9.13)
The factor (m~ - m u s) / m ~ s represents the typical GIM suppression mentioned above (it contains m~ 2 because the loop integration is cut off by m~). A single box diagram would yield A4 o( g4mj,2. The Dirac algebra in (9.13) can be simplified by virtue of
7~'~,'~e = ( g~,~,g~ + g~,~,g,~Q - g:~g~,o + ie~o'~5 ) 7 ~, .~e~.,,/~ = ( ga~'g~,~ + ge.g,Z~ _ ge~g,~ + iEQ~-~/5 ) ~/a (Co1~3 = + 1, e ~ 7 7v7 (1 -
(9.14)
= - 1 ). It is then straightforward to show that |
-
= 4
"(1 - 75) | 7 (1 - 75),
(9.15)
where the symbol | separates the matrices from two different fermion lines. Expressing the SU(2) weak coupling g in terms of the Fermi coupling constant GF (g2/8m 2 = GF/V~), it follows (mc2 >> m 2) that M - - G2 m 2 sinS0r c0s20r [ ~ 7 ~ ( 1 - "/5)Ud$)s'~a(1 8~s
-
-
~5)Vd ]
(9.16)
The Feynman amplitude (9.16) describes the AS = 2 transitions sd ++ Sd, with quark-antiquark pairs in the intermediate states. What we are actually concerned with are the hadronic transitions K ~ ++/~o mediated by genuine physical states. We thus express the above amplitude (obtained in the limit
104
9. The K ~ System in the Standard Model
of vanishing external quark momenta) as the matrix element of an equivalent four-fermion operator between the K ~ and/22o states (normalized to unity in a volume V):
M12- (K~176
(9.17)
2mk 2
_ - a ~ m_z~ sin20r cos20r 16~ 2 2rnk
~"(1
- ~)d~.(1
- ~ ) d IK~
In deriving this expression the spinors in (9.16) were replaced by the field operators ~s,d -- s, d. Also, a factor of 1/2 was included to compensate for the fact that /2/eft contains four terms which contribute to sd ++ sd, two corresponding to Fig. 9.3a and two to Fig. 9.3b, as can be readily verified by writing the fields in terms of creation and annihilation operators. This brings us to the most obscure part in the calculation of M12, the evaluation of the hadronic matrix element. Following the early calculations of Area based on the box diagrams in Fig. 9.3a,b [51a], it is customary to insert the vacuum intermediate state in the middle of the four-fermion operator, which amounts to neglecting strong-interaction effects. Since the renormalized operator /?/eft cannot really be treated as a product of two factors, the whole procedure is dubious, to say the least. These uncertainties are embodied in the parameter
~ : _ < o I ~-yo(1 - 75)d
IK~
(9.18)
s'Ta(1 - 75)d gTa(1 - ~5)d I K ~
Using the definition of the
(OIg~,75dlK~
K2t decay constant fk 32,
=- fkqo,, f~xp ~ 160MeV,
(9.19)
we obtain, in the K ~ rest frame, (/;;~ ~ ( 1
- 7 5 ) d I 0)(01 g~,,~(1- 75)d IK ~ --
~(fkmk) 2.
(9.20)
The presence of the additional factor of 8/3 is explained in Appendix .F (see also [98]). Therefore,
Amk = - 2 Re M12 = ~G2 f~ BKmkm2c sin 2 0c cos 20c,
(9.21)
which is in good agreement with the measured value of Amk, provided BK 1 and m~ ~ 1.5 G e V / c 2. Historically, a correct upper limit was set on the charm mass in this way [51b] before chacmonium (the bound state of a charmanticharm pair) was observed. In the light of what we said earlier, however, this must be viewed as a fortuitous coincidence. 32 In the matrix element of a vector-axial vector current between the vacuum and a pseudoscalar state only the axial current contributes (see (9.105)).
9.1 Calculation of Am k and ek
105
Since the K~ ~ mass splitting is caused by weak interactions, one would expect Amk to be comparable to the K ~ decay width, which is indeed the c a s e (see (1.41)). The quarks inside the K mesons are "glued" together by the strong force, resulting in gluonic corrections to the box diagrams of Fig. 9.3a,b. It is beyond the scope of this book to discuss strong-interaction effects on A m k (see [54] for an extensive review of the subject), except to mention that among various nonperturbative calculations of the denominator in (9.18), the lattice gauge theory yields the most accurate value: BK = 0.8 • 0.2. We also note that the real part of 71//12 is dominated by low momenta (k < m~) in the loop diagram. In this region the effect of "virtual" low-energy transitions K ~ --+ ~r~, 7r, ~, 6, 77' --~/~0 is important, yet difficult to estimate reliably. We now turn to the evalulation of the CP-violating parameter e based on the box diagrams in Fig. 9.3a,b. According to (3.82), e - - - v /Amk ~ Im M12.
(9.22)
With its two weak isospin doublets (ud') and (cs'), the GIM model cannot account for an imaginary part of the matrix element M12 (see below). This motivated Kobayashi and Maskawa (KM) to introduce, in 1973, a third quark doublet. Their proposal is not a trivial extension of the four-quark model because it allows for the existence of CT' violation within SM. The KM model [52] was proposed before the discovery of even the charm quark (see Chap. 1). In this context, recall that a third quark generation is not required to explain K~ ~ mixing. The GIM model can be extended to include the two additional quarks b (bottom, or beauty) and t (top) by defining, in analogy with (9.2) and (9.3),
jcc=
g, - ~ ( gv~ t ) / 3 ~~V ( ~ ) b \
vt
=v-l'
(9.23)
/L
where V is the unitary, 3 x 3 Cabibbo-Kobayashi-Maskawa (CKM) matrix. The matrix elements of V can be expressed in terms of a certain number of independent parameters. A unitary n x n matrix for n quark generations is characterized by no = (n - 1)n/2 rotation angles and n~ = (n - 1)(n 2)/2 physical phases. For n = 2 we have n5 = 0 and no = 1, the only parameter being the Cabibbo angle 0r For three generations no = 3 and n~ = 1, i.e., in addition to three mixing angles there is also a nonvanishing phase 5. Therefore, the CIM matrix is real and has only one parameter, whereas the CKM matrix is complex and contains four independent parameters. There is a number of ways (three dozen, in fact) to express the elements of V in terms of three rotation angles and one phase. The parametrization suggested by Wolfenstein [53] is particularly convenient because it emphasizes the observed strong hierarchy of the CKM matrix elements:
106
9. The K ~ System in the Standard Model
V-
(
Yu~Yus Yub) Ycd Vcs Ycb Ytd Vts Vtb 1 - A2/2
A
AA3re -i~
-A(1 + A2A4re i~) 1 - A2/2 - A2A6re i~ AA3(1 - re i~)
-AA~(1 + A2re i~)
AA2
) ,
(9.24)
1
where A and r are of the order of unity and A - sin 0c = 0.22 • 0.002. The notation for V~j on the left in (9.24) may seem peculiar: the matrix element Vud, for example, does not refer to the mixing of u and d quarks (they cannot mix because of charge conservation), but rather to that of the d' and d states (see (9.3)). Recall, however, that V may also be viewed as containing the transition amplitudes for weak processes, in which case Vii represents the relative strength of the transition i ++ j. The rows and columns of V must satisfy E j ]Viii 2 = E i [Viii 2 = 1. From (9.24) we infer that (a) the quarks of one generation are coupled to those of the successive generations with decreasing strength: Vub > me. This justifies our earlier claim that the top quark can be ignored as far as Amk is concerned. Regarding the imaginary part of M12, note that the contributions from the three terms in (9.32), which are of the same order, are suppressed by the common factor A~)~6 sinS. This "explains" why the CP-violating parameter e is so small, and shows explicitely that e can be attributed to the presence of a complex phase factor in the CKM matrix. From (9.22), (9.32) and (9.33) it follows that
G~f~ mk BKA2A6~? 6v/-2 7r2Amk X {m2c [ln(mt2/rn2c)- 1]-f-mt2A2A4(1- co)}.
s ~ eir
(9.34)
The result of a more detailed calculation of e, which includes stronginteraction (QCD) corrections to the lowest-order electroweak amplitude, can be expressed as [54] (the original calculation is due to Inami and Lim [55]) 2
2
2
GFfk m k m w BK ] e ] - 12v~Tr2Amk x {~XcIm~ 2 + 2VctE(xc, xt)Im~c~t + VtE(xt)Im~ 2 }
= C~I3KA2)~6rl {[rktE(xc, xt) - VcXc] + A2A%?tE(xt)( 1 - co)}- (9.35) Here xi = m 2i / m ~2, the coefficients rh are perturbative QCD corrections ( ~ = 1.38, ~?t = 0.57, ~?~t= 0.47) and 2
2
2
GFI~ rnkmw -- 3.8 x 104 Cr =- 6v/~ rr2 Amk
(GF = 1.17 x 10 -5 G e V - 2 ) .
(9.36)
The functions E(xt) and E(xr obtained after the loop integration in (9.31), depend weakly on the top quark mass: 34 34 When next-to-leading-order QCD corrections are included, the "current-quark" mass Iilt(mt) rn;[ 1 - (4/3)a~(mt)/Tr] should be used (rn~" is the pole mass measured in collider experiments). =
9.2 B~
~ Mixing and Constraints on CKM Parameters
[4 - llxt_+_xt2 E ( x , ) = x, L 4(1 - xt) 2
E(xc, xt)=xr
[
109
3xt2 in xt ] 20:x-~t)3J
3x, 4(1-x,)
3x2!nxt 4(1-x,)2]
(9.37)
] "
Since ImM12 is dominated by large loop momenta (k > me), the lowenergy mesonic transitions K ~ --~ 7rzr,~, 7/, 0, ~# --+ /~0, while significant for Amk, do not affect e. As explained in [54], 7/~ depends strongly on the QCD parameter A, a number that is not fixed by the theory (one can think of A as the energy at which quasi-free quarks and gluons bind themselves together to become hadrons). Luckily, ~kxc is smaller than either of the other two terms in (9.35).
9.2 B~
~ Mixing and Constraints on CKM Parameters
The computed value of e depends on the CKM parameters A, ~/and Q (A is known to a high precision). From (9.24),
A- IVcb[ 2,
?]2 = ;1 GVub
(9.38)
Combining measurements of inclusive semileptonic B decays to charmed mesons with those of the exclusive decays B ~ --+ ( D * + ) D + g - G it is possible to deduce [56] [V~bl = 0.038 4- 0.002 --+ A = 0.79 4- 0.04.
(9.39)
The ratio [Vub/Vcb[can be obtained from semileptonic decays of B mesons, produced on the T(4S) resonance, by measuring the lepton energy spectrum above the endpoint allowed for the predominant B --~ Dgu transitions. There are large theoretical uncertainties in the predicted lepton spectrum used to extract Vub. We quote [56]
Vub
~r
= 0.08 4- 0.015 --+ r b _----V/~ + ~/2 = 0.36 + 0.07.
(9.40)
The magnitude of Vtd places an additional constraint on ~oand ~1. [Vtd] can be determined from "virtual" processes involving t ~ d transitions. The only available process is B~ ~ mixing, identified by the presence of "same-sign" leptons in semileptonic decays of B~ ~ pairs: e+e - -+ T(4S) --+ B~ ~ --~ ~+~+ + anything.
(9.41)
This reaction is possible if one of the neutral B-mesons can transform to its antiparticle before decaying. Otherwise, the flavor-specific transitions B ~ --+ ~ - ~ X and/~o __+ g+ PX would result in opposite-sign dilepton events in (9.41). It does not require much imagination, nor effort, to extend our description of K~ ~ mixing to the B ~ system. The Feynman diagrams responsible for
110
9. The K ~ System in the Standard Model Vtd
d
~
Bo
vg
t
= b
l I I
W +,J
d
~o
W-
v~
w*
v,;
vd
w-
~d
"-
Bo
I I
V~ (a)
(b)
Fig. 9.4a,b. Box diagrams for B~
~ mixing
B ~ ~ oscillation are shown in Fig. 9.43,b. Whereas the top quark contribution to kaon mixing is suppressed by [YtdYt*l2 ~/~10 in the case of neutral n mesons 35 .2 Am(B~)t c<mt2 I~dVtbJ 2
m2A2A 6 [(1 - Q)2 +~]2],
* 2
(9.42)
m t 2 A 2 A4.
2 4 , the diagrams with t quarks Since Am(B~ (x m2A 6 and Am(B~ o( m~A in b o t h internal fermion lines dominate the B~ ~ mass difference. The expressions for Am(B~,s) follow directly from (9.31) and (9.37):
Am(BO=d,s) = -~2[ C~ m BfBBB]d,~?B 2 x
f
G2 -
67r2
IV~qV~;12m~[ 1
d4k k2--~:~-~t2) 2 2
2
3k4 1
4(k2~n2)2 ] 9 2
mw[mBfBBBld,~lVtqVtbl r]BE(Xt),
(9.43)
where msd = 5 . 2 8 G e V / c 2, roSs = 5.38GeV/& and the QCD correction factor rlB = 0.55 + 0.01. The parameter BB, analogous to •K, is related to the probability of (rib) or (sb) quarks forming a B ~ meson. The product f2BBe contains all the uncertainties associated with the hadronic matrix element. Prom lattice QCD calculations, fBe BV/~d = (200+40) MeV (see [54]). Solving (9.43) for IVtd~;I yields
IV~d%l ~ IV~dl= (8.3 • 1.85)
(9.44)
x 10 - 3 ,
where we used the world-average value [57]
Am(B~) = (3.0 • 0.13) x 10 -4 eV -+ Am(B~ Amk -- 85.8 • 3.6.
(9.45)
Since 1 Vtd
V~b = V/(o -- 1)2 + ~/2,
35 In contrast to the K ~ system, there are two neutral B mesons: the the B~
(9.46)
B~
and
9.2 B~
~ Mixing and Constraints on CKM Parameters
111
it follows that ~t -= V/(Y - 1) 2 + r/2 = 0.99 • 0.22.
(9.47)
The constraints on COand r/ given by (9.40) and (9.47) have the form of rings centred at (0, 0) and (1, 0) in the (co,rl) plane, with radii % and t~, respectively. Another constraint on these parameters comes from CPviolating decays of KL~ mesons, as given by (9.35). The allowed values of 0 and rI are shown in Fig. 9.5.
0.6 0.5 0.4
~-o.3 0.2 0.1 0 -0.5
-0.4
-0.2
0
0.2
0.4
0.6
P Fig. 9.5. Allowed region in the ~
plane
As mentioned earlier, the CKM matrix can be parametrized in many different ways. We will now demonstrate, however, that there exists a quantity that "measures" the amount of CP violation in a parametrizationindependent manner, and in doing so will provide a simple geometrical interpretation of the unitarity equation (vvt)ij = 5~j. Expressions (9.27) lead to six relations that can be represented as triangles in the complex (CO,r/) plane. Consider, for example, the unitarity relation vudv
Since
(9.48)
+ v sv * + Y bS; -- 0.
V,,d ,-~ Vtb ~ 1, Vt*, ,~ - Vcb and V=s = ~, we can write (9.48) as vub
_
= 0.
(9.49)
+Y-7
Equation (9.48) requires the sum of three complex quantities to vanish. Geometrically, it defines a triangle in the (& q) plane, with sides
-~1 ~VubI = V / ~ + r l 2 = % '
xllVtd~cb : k//(aO-- 1)2-}-r/2 -----~t t '
1.(9.50)
112
9. The K ~ System in the Standard Model
The six triangles that represent the unitarity relations (9.27) have very different shapes, yet they all contain the same area. To show this we multiply each term in (9.48) by the phase factor V~dVtd/lV~dVtd I =-- a/tal, which leaves the shape and the area of the triangle intact, with the result la I + a VusVt____~* ~ + a VubVt*b _ O. lal lal
(9.51)
From (9.51) we infer that 1 Area(triangle) = ~ Im Vu*dYtdYt*sVus
:
- -
1 -~ Im Vu*dYtdVt*bVub .
(9.52)
Multiplying (9.48) by V~sVts/IV~sVts I leads to yet another expression for the same area: 1 Area(triangle) = ~ Im Vu*sYtsYt*bVub . (9.53) Repeating this analysis for other pairs of rows or columns, we obtain the following result [59] 2 • Area(each triangle) = Im V*jV~kV~kVzj =-- J.
(9.54)
In the Wolfenstein parametrization, J
= A2/~ 6
sin 5.
(9.55)
All C'P-violating observables within the Standard Model are proportinal to this quantity (see, for example, (9.34)). Note that each of the subscripts in (9.54) appears twice, once with V and once with V*. The quantity J is thus invariant under redefinitions of the phases of quark fields: qL --+ eiCqqL" In other words, unlike the mixing matrix itself, J is parametrization independent. We conclude this section with a brief review of some aspects of B~ ~ mixing that are relevant to our discussion of the CKM parameters ~ and 6. Ignoring C P violation, the time evolution of "flavor eigenstates" (K ~ or (B 0'/~0) is given by (D.19): IM~
= f+(t)lM~
+ f-(t)lM~
IM~
=/+(t)l ~~
+ f_(t)lM~
(9.56)
The functions f+(t) and f _ ( t ) , defined in (D.20), can also be written as f +(t) = e-imte -Ft/2
COS
[(Am/2 - iAF/4)t] , (9.57)
f _ ( t ) --~ ie-imte -Ft/2 sin [(Am/2 - i n r / 4 ) t ] , where F = (F1 +/'2)/2 is the average width of M ~ and M ~ m = (ml +m2)/2, Am = m2 - rnl and A F = F2 - F1 (we define 1 and 2 such that A m > 0). The probabilities of finding IM ~ or I-~/~ at time t are (see also (1.23) and (1.24))
9.2 B~
~ Mixing and Constraints on CKM Parameters
113
p ( M o _). M o) = p(]~,/o _> AT/o) = if+(Oi =
e-Pt 2
[ c o s h ( A F / 2 ) t + cos
p ( M o _.)./~/-o) = p(gT/-o _> M o) = -
Amt] (9.58)
[f_(t)]2
e-Ft 2 [cosh(AF/2)t-cosAmt].
The above expressions describe how an initially pure ]M ~ or ]1~/o) state evolves with time into a state of mixed flavor. As it decays, the system oscillates between M ~ and _~/0 with frequency A m . This deviation from a simple exponential time evolution is an unambiguous sign of mixing. For oscillations to be detected, the system must not decay away too fast. The magnitudes of Am and A F relative to F are therefore crucial parameters: Am F
lifetime mixing time"
(9.59)
Although (9.56) (9.58) apply equally well to both ( K ~ and (B ~ there are significant differences in the behavior of the two systems. Because of the light l~on mass, the dominant decay mode is K ~ --+ 7r~ (the CP-odd kaon state decays into the phase-space suppressed 3~ mode); hence Fs >> FL. In contrast, B ~ and B ~ have a number of important decay modes. However, the channels that are common to both B ~ a n d / ~ (and are thus responsible for a width difference AF) have branching ratios _< 10 -3, leading to A F / F < 0.01 for B ~ and < 0.2 for/~o. We can therefore neglect A F / F in the case of B~ ~ mixing. On the other hand, A m ( B ~ >> Amk (see (9.45) and the text below). Experimentally [57, 58], xk :
Z~mk --
-- 0.473 + 0.0018,
Fs
n.~(B ~
Xd -- ~
xs=
nm(B ~ -
-
Fs
-- 0.728 • 0.025,
(9.60)
> 10.5.
The parameter x expresses the oscillation frequency in terms of the average lifetime. Using (9.58), we obtain the following ratio of time-integrated probabilities: f o P( B~ -+/}~ dt r -= f o P( B~ -+ B~ dt (rim) 2 + (nv/2) 2 2r2 + (n.~)2 - ( n r / 2 ) ~
x2
- -
2 + x~'
0 < r < 1.
Another useful parameter is the oscillation probability:
(9.61)
114
9. The K ~ System in the Standard Model
fo~ P(B ~ -->/3~ X =- fo~ p(Bo --> BO)d t + J o P( B~ ~ / 3 ~ r 1 -
0 < X< -5"
l+r'
(9.62)
The measurement of B~ ~ mixing requires the flavour quantum number, B, of the neutral meson to be identified at both its production and decay. Since B (like S) is conserved in strong and electromagnetic interactions, B mesons are produced in pairs. The flavour of B ~ (/~o) can be traced by observing semileptonic decays B ~ -+ g - v X and/~0 _~ g+~X. One thus expects
B0 mi~ B0 d_~ ~+~X.
(9.63)
Experimentally, the amount of mixing is determined through
Tg = N(BB) + N(JBB) = U(g-g-) + N(g+g +) N(B[~) + N(BB) N(g+g -) '
(9.64)
where N(BB) is the number of B B final states in a sample of events from a process where a B/3 pair is initially produced, etc. Note that N(BB) and N(BB) are experimentally indistinguishable. When a B/3 pair is produced incoherently, 36 which occurs at energies well above the bb threshold, the time evolution of one meson is independent of the other. In this case, 7~-
2X(1-X) (1 - X) 2 + X 2
_
2____f__r incoherent production, r 2'
(9.65)
1+
since P ( B B ) = P(/3/3) = P(Ba oscillates) x P(Bb remains the same) = X ( 1 X), P(B/3) = (1 - ~()2 (neither oscillates) and P(/3B) = ?(2 (both oscillate). At L E P and hadron colliders, where both the B ~ and B ~ are produced, one measures the sum of their mixing probabilities, weighted by the corresponding production fractions: ;~ = fdXd + fsXs. The situation is quite different on the Y(4S) resonance, or at the B/3* threshold, where the two mesons are produced coherently (i.e., they form a quantum-mechanical state of definite orbital angular momentum, g, and parity). The Y(4S) resonance is a P wave bb bound state with C = - 1 and P = - 1 t h a t lies just above the Ba[~a threshold (its mass is less than 2 m(B~ The Y(4S) state decays strongly into B+B - or B~ ~ (see (9.41)). Since it is produced via a "virtual" photon, the B~ ~ pair is in a pure C = - 1 q u a n t u m state (see Sects. 1.4 and 7.1): 1
IB~ ~ = ~
{IB~176
-IB~176
(9.66)
The two B mesons are strongly correlated: at no time can the original B~ ~ system evolve into two identical states in the Y(4S) rest frame. As 36 That is, the angular momentum and parity of the pair are different for each event, i.e., the final state is a superposition of many angular-momentum states.
9.2 B~
~ Mixing and Constraints on CKM Parameters
115
explained in Sect. 7.1, if the mesons were to decay at the same time to the same final state, there would be two identical J = 0 bosonic systems in an overall P wave. But this would violate the rule that two identical spinless bosons cannot be in an antisymmetric spatial state. The B ~ and/~o propagate coherently until one of them decays. Only then will the state of the second particle be uniquely defined: it will have the flavor quantum number opposite to that of the first B meson. Suppose that the two decays occur at times tl and t2. Using expressions (9.56), which describe the time evolution of flavor eigenstates IB~ and IB~ we obtain (with f~_ ~ f + ( t l ) , etc.) 1 2 _ fl_f2_) ]BOBO) ]BOBO(t)) ~ (f~_f2_ _ f~_f2) iBOBO I + (f~_fr
1 2 +(f_f__ I 2 IB~176 + if_f+ 1 2 /;f+)
(9.67)
From (9.67) we see that P ( B ~ ~ = p(/~o/~o) and P ( B ~ ~ = P(/~~176 hence,
Tr = f o I f + ( t l ) f - ( t 2 ) - f - ( t l ) f + ( t 2 ) l 2 dtldt2 _ Af
(9.68)
f o If+(tl)f+(t2) - f - ( t l ) f - ( t 2 ) ] 2dtldt2 - :D" Writing
f•
= l e - i m t e - C t / 2 [eiZ~mt/2e z~Ft/4 • e-iZ~mt/2e -z~Ft/4]
(9.69)
2
it follows that .M"
T)
=
=
dtldt2e -rt
/j
e z~Fz~t/2 + e -zlcz~t/2 T 2 R e e - i A m A t
dtldt2 [e-&tle -Fit2 -t- e-lht~e -F2t~
:~22 R e e - F t l eiAmtl e - F t ~ e -iAmt2 ]
2 -
FIF2
:F
2 : r2
_
(
2Re
1
1
F - iAm F + iAm 2
r/2)2
+
(9.70)
Therefore 7~=r-
1 -XX '
coherent production (g odd).
(9.71)
The T(4S) resonance decays to B~ ~ or B + B - , and so the observed number of N(t~+/-) events has to be corrected for leptons coming from charged B mesons, a procedure that is not entirely unambiguous. The first evidence for B/~ mixing was provided in 1987 by the UA1 experiment a7 at the CERN pp collider [60]. Soon thereafter the ARGUS collaboration at the DORIS e+e - storage ring observed large B d0- B- 0d mixing 37 In 1983, the UA1 collaboration, led by C. Rubbia, discovered the intermediate vector bosons W J: and Z ~
116
9. The K ~ System in the Standard Model
(r -- 0.21 4- 0.08) among B mesons produced in T(4S) decays [61]. Their result strongly suggested that the top quark was much heavier than expected. Until recently, all measurements of B / ) mixing were time integrated. These studies are insensitive to x when mixing is maximal because x --+ ec as 0 -0 X ~ 0.5 (see (9.60)-(9.62)). To measure Bs-B s transitions one thus needs to determine the time evolution of B ~ mesons, which is not an easy task given their rapid oscillation rate. The oscillation period gives a direct measurement of the mass difference between the CP eigenstates B1~ and B ~ provided the proper time of the Bmeson decay, tp, is known with sufficient accuracy: tp = L/~'~ = L(m/p), where L is the measured decay length; m and p are are the mass and momentum of the meson, respectively. The typical experimental resolution of L E P experiments is 2.5 ps in tp. The value of A m is found from the fraction of "mixed" or "unmixed" events as a function of tp by using (9.58). Based on data collected between 1991 and 1994, the DELPHI collaboration at LEP has reported [58]
Am(B ~ > 6.5ps -1 (4.3 x 10-3eV)
at 95% CL
(9.72)
corresponding to xs > 10.5, where xs = Am(B~ ~ and ~-(B ~ = (1.61 40.1) ps. As we mentioned earlier, the measurement of IVtd[ suffers from large theoretical uncertainties associated with fB V/BBB.This uncertainty can be reduced by measuring
Am(B ~ m.~ Vts 2 mB ~ ~s2 Am(B ~ - robe Vtd ~ - mBd A2 [ ( Q - 1 ) 2+r/21 '
(9.73)
where is is the ratio of hadronic matrix elements for the B ~ and B ~ [54]: fB~ ~
_ 1.16 4- 0.05.
(9.74)
Unfortunately, it is much more difficult to determine Arn(B ~ than Arn(B~) because (a) the fraction of B ~ mesons produced in b decays is considerably smaller than that of B ~ particles, and (b) the large value of A m ( B ~ (experimentally, Am(B~ ~ > 14) leads to rapid oscillations that complicate the measurement.
9.3 Rare Kaon Decays Over the past forty years, studies of rare meson decays have contributed significantly to our present understanding of weak interactions. As explained in Chap. 1, the observation of both K -+ 27r and K --~ 37c decays led to the discovery of parity violation. Parity is maximally violated in weak interactions
9.3 Rare Kaon Decays
117
in the sense t h a t all neutrinos are left-handed and all antineutrinos are righthanded. This motivated Feynman, Gell-Mann and others to formulate the weak interaction in terms of vector-axial vector (V-A) currents. The helicity suppressed decay 3s ~r --~ cue provided crucial support for the "V-A theory" ,39 which has been very successfifl in explaining most of the low-energy weakinteraction phenomena. The observed violation of C P s y m m e t r y in K ~ decays (at a rate of about 10 -3) may be a fundamental property of nature, with important implications for the early evolution of the universe. A deeper insight into C P violation is expected to be gained from precision measurements of theoretically "clean" rare kaon decays, such as K ~ --e 7r~ The suppression of the KL~ -+ # + p decay (at the level of 10-s), discussed in Sect. 9.1, suggested the existence of the charm quark, and thus played an important role in the development of the Standard Model (the GIM mechanism). In the same section we described how the sensitivity of K ~ ~ mixing to energies higher than the kaon mass scale was used to predict the mass of the charm quark. Similarly, rare kaon decays t h a t are dominated by one-loop Feynman diagrams with top quark exchange can yield valuable measurements of the C K M matrix elements Vtd and Vts. Since the branching ratio for t -+ d + W is very small, it is difficult to determine the coupling Vtd directly from t decays. Rare kaon decays are an important source of information on higher-order effects in electroweak interactions, and can therefore serve as a probe into physics beyond the Standard Model. Experiments at the highest-energy particle colliders, and those studying the rarest of K-meson decays at low energies, are pursuing different aspects of the same physics. In what follows we will concentrate on those processes that are theoretically best understood. Our exposition is meant to be pedagogical, rather than comprehensive, in order to highlight the underlying physics. We will not discuss decays t h a t violate lepton number conservation.
9.3.1 K ~ --+ 7r~
and K + -+ n'+v0
Within the Standard Model, these transitions are loop-induced semileptonic decays of the type s -+ d + g + g. They are entirely due to second-order weak processes determined by Z~ and W-box diagrams: since photons do not couple to neutrinos, there is no electromagnetic contribution. 3s Since the pion has spin zero and, according to the V-A law, the neutrino is left-handed, the lepton in 7r -+ gv~ must have negative helicity (~ = -1). The probability for a lepton of velocity v to be left-handed is P()~ = - 1 ) = 1 v/c. This probability is much smaller for the light electron than for the muon ( m u / m ~ ~ 200). The electronic decay mode is even more suppressed in the K -+ gu~ case because the electron is more relativistic than in the pion decay. The phase space can do little to improve its odds against the muonic decay mode. 39 In fact, the suppression is proportional to ( m ~ / m u ) z for any arbitrary mixing of V and A couplings: ,.7~e = ftt",/~,(Cv + CA",/5)yv --+ F(71" --4' g//g) (X 4(C~ + C~)m~.
118
9. The K ~ System in the Standard Model
Both decays are theoretically "clean" because the hadronic transition amplitudes are matrix elements of quark currents between mesonic states, which can be extracted from the leading semileptonic decays by using isospin symmetry. The process K ~ --4 7r~ offers the most transparent window into the origin of CP violation proposed so far. It proceeds almost entirely through direct CP violation [62], and is completely dominated by "short-distance" loop diagrams with top quark exchange. Although this decay has a miniscule branching ratio (about 10 -11 ) and is experimentally very challenging, its measurement, which is complementary to those planned in the B ~ system, is feasible and certainly worth the effort. The main features of the decay K --4 ~rup, summarized above, can be discerned from the relatively simple calculation of the box diagram in Fig. 9.6a. Neglecting the charged lepton mass and external quark momenta, the corresponding amplitude reads (the contribution of unphysical scalars in the t'Hooft-Feynman gauge can be ignored because their coupling to leptons is proportional to me/m~ 3,
(9.79)
and
VudV&=
- vc~E~
- ~dE~
(9.8o)
(see (9.28)), it is straightforward to show that V/dV/s I
.A/[ -- (47r)2m~ E
~-~ :~-)2-
i=-c,t
) (9.81)
• [~sT~LUd] [~tv%Lvv]
with xi =- m i2/ r n w2, xu ,-~ 0, x~ ~ 2.6 • 10 -4 and xt ~-- 5. As for (9.17), we can express this amplitude as the matrix element of an equivalent operator between the states IK) and [Tr):
i:c,t
x (Tr I $ ' ~ d
[ K)[P~/~(1
- ~5)u],
(9.82)
where [] stands for "box diagram", GF ~ CK =- V ~ 27rsin20w ' -
"TD(xi) =
Xi(X i -- ln xi -- 1) ( x i - 1) 2
(9.83)
120
9. The K ~ System in the Standard Model
and 0w is the weak mixing, or Weinberg, angle: e = gsin0w, a - e2/47r. Since Ir and K have the same parity, only the vector current contributes to The amplitudes @l.i7~d]K) are much simpler objects than the matrix element of the four-quark operator in (9.17). In the limit of exact isospin symmetry, which is a very good approximation, @+l$7~d[K +) = v/2 @~176 ). Moreover, the matrix element of the weak current Sy~d between K + and ~+ is related by isospin to the known matrix element of the operator gT~u between K + and 7r~ @+l$7"d]K +) = v/2@~ I ~7"u ] K + ) .
(9.84)
The operator $7~u is measured in the decay K + --~ ~~ for this transition is given by (see Fig. 9.1c)
A( K+ --+ 7~~
The amplitude
= ~22V~( :r176I s7 ~u I K+) [~Ta(1 - 75)e] 9
(9.85)
Neglecting the positron mass, the branching ration for K + -+ 7c+vi per neutrino flavor reads (the decays K + --+ ~r and K + -+ 7:~ have essentially the same phase space)
B(K+--+zc+u~')=B(K+--+Tc~ = B ( K + ~ 7r~
27r sin 2a 0w 7)/-~2 Vus]2 (x/2) 2
a2 [7)[2 sin40w A2 "
(9.86)
The complex coefficient 7) depends on the charm and top quark masses: 7) = E
V/dV~; ~-(x,).
(9.87)
i=c,t
To show that the decay K ~ -+ 7r~ is CP violating, consider the behavior of the corresponding interaction lagrangian under CP: _- qSKLO,q5 o~7~( 1 _ 75)V c P [_4~KL] OU [--4i~o][-- PTt'(1 -- 75)V] Using (3.2), (1.8) and (9.82), we can write [62]
.A(K ~ ~ T:Ov~,) = ~ A(K~ --+ 7c~
+ A ( K ~ -+ 7r~
(9.88)
where 4~
40 Note that (Tr~ (see Appendix E).
"~ = (~~
~ = (.~176
9.3 Rare Kaon Decays
A(K o ~ .o~)=
121
1 [~4(KO --~ 7rOmp) + ~4(k o -+ 7ro~) ] = Re A ( K + -+ 7r+~,~),
A(Ko ~ ~o.v) :
(9.89)
1 [A(K o -~ ~ o ~ ) _ A ( k o _~ ~o.v)]
= iIm A ( K + --+ 7r+v~,). Summed over three neutrino flavors, the branching ratios for the indirect and direct CP-violating contributions are, respectively, B ( K L0 --> 71"0VV)indirect ~,~ 3 I~l2 ~KL • 2.8 • 10 -~
TK+
• [br(Xc) + A2A4(1
-
Lo).~(xt)]
2
(9.90) U(KL0 --~ 71"0/-'P)direct ~ 3 TKL • 2.8 • 10 -6 [A2A4~.T(xt)] 2 TK +
for B ( K + -+ 7r~ = 0.0482, sin 2 0w = 0.23 and c~(mw) = 1/128. The small value of e renders the contribution from indirect C P violation (and hence from the charm quark) insignificant. Therefore, B ( K ~ --4 T:~ ') ~ B ( K ~ ~ 7r0up)direct = 8 • 10 -11 [~$'(Xt)] 2
(9.91)
based on TKL = 4.18~-K+ and A = 0.8. To complete the calculation of B ( K -~ 7r~D), we consider the remaining diagrams in Fig. 9.6. The presence of unphysical scalars in Fig. 9.6c cannot be ignored because of the large top quark mass. The result of a somewhat lengthy calculation yields Az(K
V~dV~* $'z(X~) (~')V-A,
~ Try,F,) = r
(9.92)
i:c,t
where jZz(xi ) -_ xi [x~ + xi(31nxi - 7) + 21nxi + 6] 8(xi - 1) 2
(9.93)
Combining this result with (9.83), it follows that xi [ 3xi - 6 2 + x~ ] 9~ ( x i ) = .,~z(Xi) + 9rD(xi) = ~- [(~-i :-1~ 2 lnxi + xi - l J"
(9.94)
We thus finally obtain, for ~] = 0.36 and mt = 180 GeV/c 2, B ( K ~ -+ ~r~
,,~ 2.8 • 10 -11.
(9.95)
Isospin-violating quark mass effects and electroweak radiative corrections reduce this branching ratio by 5.6% [63]. Next-to-leading-order QCD effects are known to within +1% [54]. The overall theoretical ambiguity in the calculation of B ( K ~ --+ 7r~ is below 2%. This uncertainty does not include
122
9. The K ~ System in the Standard Model
the error on the CKM p a r a m e t e r ~, as given by the (correlated) constraints (9.40) and (9.47). The detection of K ~ --+ 7r~ presents a formidable challenge. The experimental signature of this decay is a single unbalanced 7r~ which makes background rejection very difficult. The direction of photons from the decay 7r~ --+ 2"y can be determined through their conversion to e+e - pairs. In general, the most important backgrounds are K ~ --+ 2"y (B ~ 5 • 10-4), K ~ --+ 27r~ (/~ .~ 10-3), neutron interactions at residual gas atoms in the decay region t h a t produce 7r~ A --+ n~r~ decays, etc. The K ~ --+ 2"~ decay, for example, can be discriminated by using both the transverse m o m e n t u m balance of the two-gamma system and the position of the detected photons with respect to the b e a m axis. Alternatively, the final state can be defined by selecting those events in which the ~r~ undergoes the Dalitz decay 7r~ ~ e + e - % In this case it is possible to reconstruct the vertex of the decay and hence the invariant mass of the Ir~ Another advantage over the 2"y final state is t h a t a relatively wide b e a m can be used. However, this method has the disadvantage that (a) the 7r~ --~ e + e - ' y decay has a small branching ratio (about 1%) and (b) the final state in the radiative decay K ~ --+ 7r+eT'yp looks like e + e - 7 + "nothing" if the 7r+ is misidentified as e +. All a t t e m p t s to detect K ~ --+ ~r~ thus far have relied on the Dalitz decay mode. The best published limit to date is B ( K ~ --+ 7r~ < 5.8 z 10 -5 (90% CL) from Fermilab experiment E731/799 [64]. There are several proposals to measure B ( K ~ -+ ~r~ The KTeV experiment, described in Sect. 8.1, is expected to reach a sensitivity of 10 - s by identifying ~r~ through the Dalitz decay. The K A M I collaboration [65] has proposed to use the Main Injector at Fermilab as a source of very highintensity and high-energy neutral kaons, and to detect 7r~ -+ 27 decays in the pure CsI crystals of the KTeV apparatus. They aim at a sensitivity of better than 10 -12. An experiment at the K E K laboratory [66] intends to employ an array of CeF3 crystals to measure the energy and position of the two g a m m a s from the ~r~ --+ "y'~ decay, and a lead/scintillator barrel calorimeter to select two-photon events. The experiment E926 at Brookhaven [67] would exploit high b e a m intensities of the AGS proton synchrotron, which will be able to provide, by the year 2000, over 1014 protons/pulse. The Brookhaven group proposes to obtain low-momentum kaons ((Pk) =700 MeV/e) from a microbunched proton beam. This would allow them to determine the m o m e n t u m of the decaying KL~ using time of flight measurements. The expression for 13(K + -+ 7r+v~) with three quark and lepton families was originally derived by T. Inami and C. Lim [55]:
9.3 Rare Kaon Decays B(I( + ~ ~ + ~ ) =
123
a 2 ( m w ) B ( K + --+ 7rOe+u~)
2zr2 sin 4 0w
x }]
E~=~,~v,~v~;7(x~,y~)
(9.96)
(cf. (9.86) and (9.87)), with J:(x,y)-
lay xy 16 x - y
(y-4
+~--
~
\~]
7
x nx[x x4 x
+8
3x
\~L-~_l] + 1 +
(
16(x--l)
----------7 (x 1)
]
3) l+y_l
"
(9.97)
In the above two equations, xi - m i2/ m w2, i = c, t (quarks) and Yt =- m e2/ m w2, = e, tt, T (leptons). For y --+ 0, (9.97) reduces to (9.94). Experimentally, xt ~ 5, xc, y~- ..~ 10 -4 (mr ~ 1.78GeV/c2), y~ -~ 10 -6 and yr ~ 10 -11. We can thus write lnxr - 1] . % ( x t , y t ) ~ : F ( x t ) , .%(x~,yt),~xc. x~lnz--~- y--elnye F - (9.98) t Ye - x~ 4
[
and B ( K + --~ it+uP) ~-. 2.8 x 10-6{3 [A2 A4r] 2F(xt)] 2
+ ~
f(~c,y~) + A~a4(1 - o)f(~t)
, (9.99)
#.=e,,u,Tk
where, to a very good approximation, .T(xe, y~) ~ .T(Xc, y,) -~ ~'(xc, 0). In contrast to the CP-violating decay K ~ --+ 7r~ the charm and top quark contributions to K + --+ 7r+u0 are of comparable size: the smallness of ~-(x~, Ye) in comparison with .T(xt) is compensated by the strong CKM suppression of the t contribution. Isospin-violating quark-mass effects and electroweak radiative corrections (which do not affect the short-distance structure of K -+ zruO) result in a decrease of the branching ratio by 10% [63]. Possible long-distance contributions are estimated to be negligibly small [68]. Short-distance QCD effects represent the most important class of radiative corrections to this process. The inclusion of next-to-leading-order QCD corrections yields [69, 54] 0.88 x 10 - l ~ = (0l g%75dl K~ =- fkqa.
(9.108)
Therefore,
A ( K ~ ~ #p) = - CK~-~ 6(x~)( 0 IV~dV~s(ST~d) i=c,t + Vi~V~ (dT~ 75 s ) I g ~ (#7~75#)
=
-
x/2Cgae E VidVis* 6( X i)(O ]gTa75u]K+>(fi'7'~75#) i=c,t aGFfkq~ Re E V~dVi;6( i)(#7 75#), 27r sin20w i=c,t
(9.109)
al Since K ~ is (mainly) CP-odd, only the CP-odd combination of axial currents gT~v5d + dV~75s contributes to the transition K ~ -4 vacuum.
9.3 Rare Kaon Decays
where we used IK~ ~ [IK~ -/
-
($9.1)
(c.f. (9.85)). Measurements of the ~~ angular distribution in the decay K + ~ 7r~ reveal that the scalar and tensor contributions to Jh~ are very small [80] (see Fig. 9.11). Taking also into account that 7r and K have the same parity, we infer that Jh~ contains only the vector current. The hadronic amplitude must be formed from the available 4-vectors. It is convenient to write
(~ I
aft I K )
: f+(q2)kZ + f_(q2)qZ,
($9.2)
where
k - k K + k~,
q2 = (k K _ k~)2 = (p€ + pe)2.
($9.3)
By virtue of the Dirac equation, 5(p~)(~(1 - ?5)v(pe) = me~(p~)(1 + ?5)v(pe). The amplitude ($9.1) can thus be written as
GF
.54 : ~
{f+kZ~(p,),'/Z(1-,.fh)v(pe) + f_mefi(p~)(1 +?5)v(pe) }.
(S9.4)
Summing over the spin states of the leptons, we obtain _~_
Tr
2^
^ ^
2
2^
^
+ f + f - m e p ~ k ( p e + me) + f _ f + m e p , ( p e + me)k
= 4ag{f
[2(kp.)(kpe) - k2(p~,pe)]
+ f~-2m2e(P~,Pe) + 2 R e ( f + f _ ) m ~ ( k p ~ ) } .
($9.5)
134
9. The K ~ System in the Standard Model
200 Z
100
-1
-0.5
0 r
0.5
l
Ot
Fig. 9.11. Cosine of the angle between 7r~ and L~ in the dilepton CM system f<JrK,a decay. Predictions for vector (V), scalar (S) and tensor (T) couplings are plotted against data [80]
To derive ($9.5) we used
which reflects the fact that there are ~a[y three tadepeedeat 4 - m a m e n ~ (s is the aamp[ete~y ~n~isyn]raet~iv L~vi-Civita ~e~so;: ~ = +I, In the ka~n rest frame, m k = E~ + E F + E V and p~ 4- p . -~ k , = 0 t ~ e d ~n this~ the 4-vector products in ($9.5) can be expressed in terms ~f the center-~f-mass (CM) energies of the lepton and pion as Iellows k 2 = m 2 + m,~2 + 2mkErr ,
p~.p~
=
{m~ + . ~ - 2
m~- 2m~E~}/2,
($9.7)
+ m~ - 2 . ~ ( 2 E ~ + E ~ ) } / 9 , k.p~ = {2m~(2E~ + E~) - m~ - m~ In the CM system, the decay rate is given by d r = I~1--~ d~3, 2m k where d ~ 3 is the differential element of the three-particle phase space
1 f Ca--(27r) 5 J
dak~ d3pe d3p, (~(3)(krr -[- Pt + Pv) 2E~ 2Ee 2 E ,
(S9.8)
9.3 Rare Kaon Decays x
_
135
5(ink - E . - E~ - ~:.)
1 f k~ dlk.l d n . m ~dlpl d ~ 8(27r) 5 J E~ Ee x
5(m k - E~ - Ee - E . ) Eu
(S9.9)
Since pd]p I = E d E and d~2 = d(cos0)dr this becomes
27:
[ d~2~dE~dEe Ik~]lP~ d(cos0~)
~3 -- 8(27F)5 j
E,,
x 5(m k - E~ - Ee - E~).
(S9.10)
To simplify ($9.10), note that
E~ = p2 -_ k~ + p~ + 21k=flps cos0=~,
($9.11)
which yields
E, dE, = Ik=llp~d(cos0~),
(S9.12)
after differentiation with respect to 0.~, while keeping Ik.I and IP~ fixed. Substituting ($9.12) in ($9.10) and integrating out the (f-function results in d~ 3 = ~
1
dE~dE,~.
($9.13)
When ($9.5) and ($9.7) are substituted in the expression for the decay rate, it reads
G2mk ~ 2
d2F
dEedE~ --
8~3 l f + [ P 2 - ( m
k_E~_2E~) 2
m2
+ ~@2(SmkE~ + 6mkE~ + m 2 -~,,%
+#_ m~
_
3m2)]
[m2k + m ~ - m ~ - 2rnkE~]
($9.14) If the charged lepton is a positron, we can neglect the terms proportional to rn~, in which case d2F-
G2Fmk 8~r3 f~ {p~ -- (mk -- E~ - 2Er 2 } dE~dE~.
($9.15)
To determine the kinematic limits of phase-space integration, consider p~ = _ ( p . + kTr) ~
Hence,
p2e = k ,2 + p . 2 + 2lk~l]p~ cos0,,.
(S9.16)
136
9. The K ~ System in the Standard Model ax Pe m min
2
=
([p~[-4-[k.[) 2
(Ev • [kTr[)2 ($9.17)
= (ink - E . - E~ + Ik.I) 2,
i.e. E~
-m~
= (ink - E= • Ik.I) ~ + E~ - 2(ink - E . • Ik.I)E~.
m a x or m i n
Therefore, (E~) mien x : (mk - E , • Ik.I) 2 + m~
($9.18)
2 ( m k - E= • Ik=l)
and we find that the positron energy at a fixed E , varies within the range m k - E , - Ik,I < 2E~ ~ mk - E . + Ik.I.
($9.19)
Analogously, the pion energy at a given E~ is constrained by max
(E.)mi
n __
(mk - E~ • [p~) 2 + m .
2
(S9.20)
2(m~ - E~ + IP~) If the positron mass is neglected, 2
(ink - 2E~) 2 + m 2 < E , < m~ + m , 2(ink - 2Ee) 2m k
(S9.21)
Expressions ($9.18) and ($9.20) define the contour of a Dalitz plot. Assuming that f+ (q2) ~ constant, the distribution ($9.15) can be readily integrated over d e e from E m i n t o E m a x , with the result dF-
G~mk f~ (E~ 127r3
-m2)3/2 d E , .
($9.22)
A measurement of the pion energy spectrum gives the q2 dependence of f+. Using ($9.21) and rewriting expression ($9.15) in a slightly different form, we have dF dEC-
G2Fmk 8~ 3 f2 • /E ~m~ { ( ~
- 2E~)[2E.
- (.~
- 2E~)] - ~n~} d E . .
(S9.23)
A simple integration yields the positron energy spectrum:
dF G2mk E2(W~ - E~)2 dE--~ - -------527~ f2 m k - 2E~ '
($9.24)
where We is the maximum energy of the positron in K~3 decay (see ($9.29) and ($9.30)). The neutrino spectrum in the rest-frame of the kaon can be constructed from the measured momentum distributions of the pion and electron in the laboratory frame:
9.3 Rare Kaon Decays 2 =
-
pe-k~)Lab
2(E~E~ -
137
-
(S9.25)
2m k
This expression follows from
(kK -
2 = (k~ + P~)aab 2 9 P-)CM
($9.26)
To find the m a x i m u m energy of a particle in a t h r e e - b o d y decay M -+ m I + m 2 + m 3 , let M12 be the invariant mass of particles 1 and 2. In the rest s y s t e m of M , M122 = (El + E2) 2 - (Pl -4- p2) 2 = ( M - E3) 2 - (p3) 2.
(S9.27)
Since P3 = E32 - m32, we have M22 = M 2 - 2 M E 3 + m 2
> E3 =
M r + m2 - M~2
($9.28)
2M
E3 has the m a x i m u m value for M A in = rn 1 + m : . Therefore, W3 = g ~ nax --
M 2 + m~ - ( m 1 + m2) 2 2M
($9.29)
In our case one of the particles is a massless neutrino. We thus obtain the following kinematic limits 2
2
m,~ < E . < m~ + m,~ - m~ _
_
2m k
me < Ee < m~ + m~ - r n , '
_
_
($9.30)
2m k
T h e Ke3 decay rate is therefore given by F-
G2mk
127r3
2 fmk(1Ta)/2 3 / f~_ [ E,~tl-
.,m,~
,
m2
2
dE,~ ,
~2
2 2k. Setting 1 - m ~2 / E ~ 2 = where a =- mTr/rn obtain
F-
,3/2 X2
($9.31)
and integrating by parts, we
5
GFmk
7687r 3 f 2 {(1 -- a)3(1 + a) -- 6a(1 -- a)(1 + a) -- 12a 2 l n a } .($9.32)
If only the leading orders in a are kept, this simplifies to F-
G F2 m ks
(
7681r a f 2
1-
Sm~'~ m2 j .
($9.33)
Of special interest is the m u o n p o l a r i z a t i o n in the decay K + -+ 7r~ S u m m i n g over the spin states of the neutrino (see the text preceding expressions (9.133) and (9.134) in Sect. 9.3), we obtain
IMI 2 = - ~ Tr[(f+k~
+ f _ q ~ ) ( f ~ _ k ~' + if_q)')
Sv
x/5.3,Z(1 - "/5) (15u - m , ) ( 1 + 75g,)'),x(1 - 75)] = G.__}_~T r [ ( f + k Z + f _ q Z ) ( f ~ _ k ~ + f_* qa)(1 - ~'s) 2 x ~,'),~(~, + rnu~u)7)~].
($9.34)
138
9. The K ~ System in the Standard Model
To derive ($9.34) we used pt*. st* = 0 (see (9.134)). A straightforward evaluation of the traces yields
E JMI = Sv
1
IMI 2 + 2c , t* {
- k2(v -st*)]
+ f2_ [2(q.p~)(q.st*) - q2(p~-su) ] +2Re ( f + f * ) [(k.p~)(q.st*) + ( q . p ~ ) ( k . s u ) -
(k.q)(p,.st*)]
or } +2Im ( f + f *_) e , X ~ kZa)~_a ._ = V,%
($9.35)
with ]~]2 given by ($9.5). In the kaon rest frame,
q.st* = p~.st* = E , Pu'St* (PdSt*)(Pu'P~) rot* P~'Su - mt*(Et* + mt*) ' ($9.36)
k.st*
(2k g
--
pt* - p , ) . s u
=
2m k PdSt* - -
_
p
.st,
mt*
(st* is the spin unit vector of the muon in its rest frame) and el3Xar
kt3a,~na.qr "~ "* r v - l *
=
al~Aar(2kK
--
a r pt* - p.)~(pt* + p~) .X p.st*
= 2 gl3.~ar kJ3nAna'qr i kst* = 2m k (pt*• p,)" s u. = 2mkeioyk pill2,
($9.37)
Each term in ($9.35), with the exception of the last one, contains components of the muon polarization in the decay plane. The last term gives rise to a polarization component normal to that plane. Since this term changes sign under time reversal, its presence would violate T invariance. Indeed, the time-reversed amplitude of ($9.1) and ($9.2) GF
.h/[ T) - ~ [f~_(q2)k ~ + f._(q2)qz] ~(V.)~,Z(1 _ %)v(Pe)
(S9.38)
also leads to ($9.35), except that in this case the last term is ~ Im ( f ~ f _ ) = - I m ( f + f _ * ) . If the decay is caused by an interaction that is T invariant, Im (f+f*_) = 0 (f+ and f_ have the same phase). To obtain ($9.38) we used Table E.2 in Appendix E and the fact that the operation of time reversal implies charge conjugation. We now define ( ~ = ~m-t * [ 1 and rewrite
(S9.35)
f~+] = ~mt* - [1-~(q2)],
9~=2Rec~, 3 ~ - 2 I m ~ ,
(S9.39)
as
= 8 G F f + [A + B . s ~ ] ,
($9.40)
Sv
where A = [2(kK.p,) - fftmt*] (kK.P~) -- (m2k -- a2)(pdp~),
($9.41)
9.4 Direct CP Violation (e')
139
B = [2m,(kK.pu ) - fJt(p,.pu)]k ~ - [rn,(rn~ + a 2 ) _ N(kK.pt,)]p~ ar + ~:Je ~ , k ~ q, x pus,.
(S9.42)
It can be readily verified, using ($9.36) and ($9.37), that in the kaon rest frame
B'%:{a(()k~+[b(~)+a(()(?
~-:p-~ + r n k - - E ~ ) ] P , \ L', + rn,
+ 3 rnk(k,~x p~) }. s,.
($9.43)
In the above expression,
mtt
l
($9.44)
b(() = m : [ 2 E , -
~R ( W . - E~)]. mtt (W,~ is the maximum energy of the pion; see ($9.29) and ($9.30)). From ($9.43) it follows immediately that the muon polarization vector is given by [81] ~ ' ( ~ ) - I~t(~)l'
($9.45)
[
.,4 = a(~)k,~ + b(~) + a(~) \ E, + mr, mk(k~ x p,). Since the pion and K meson are spinless and the neutrino has a definite helicity, the muon is completely polarized. The direction of its polarization is fixed by the kinematics of the decay. The nonvanishing mass of the muon gives rise to a polarization component along the pion momentum. As we explained earlier, if the decay is T invariant, ( is real and the polarization vector lies entirely in the decay plane.
9.4 Direct C P Violation (e') The parameter e', defined by (3.75), determines the amount of direct CP violation in AS : 1 transitions: e'-- x/~ A 0ReA2 Re 1 [IIA2ReA2
Re~00ImA~
ei(Tr/2+52_5o )
(9.148)
Any CP-violating observable must involve an interference of two amplitudes. In the case of e/, the interfering amplitudes are
140
9. The K ~ System in the Standard Model
(rTr, I = al/:/w]K ~ = A~e i~~
(~ = 0, 2,
(9.149)
where [Trlr,I = 0, 2) are the I = 0 and I = 2 isospin states of the 27r system. Remarkably, the phase factor in (9.148) is very close to 7r//4, i.e., the phase of c ~ is almost the same as that of ~. Using (3.74), (3.81) and (3.82), we can express the parameter a, which measures the interference between K ~ --+ ~Tr and K ~ -+/7/0 _+ 7rTr, as follows: e ~ ~22 eir
[ ImM12 ImAo] [2ReM12 + ReAoJ "
(9.150)
The expressions for e ~ and c are independent of phase convention. To show this we redefine the phases of the strangeness eigenstates K ~ and K~ ]K ~ -+ e-i;~lK~ Ig~ ~ e~lk~ For Iml 0 are supressed by kinematics. We can thus write JP. = L (-1)L+~ -- 0 - , 1+, 2 - , . . . .
(A.5)
From (A.2) and (A.5) we infer that J3P,~ = 0 - , 2 - , 4 - , . . . .
(A.6)
Measurements of the charged 37r decay modes favor Jk = 0. This result is corroborated by the absence of the decay K • --+ 7r+ 4- 7, which is forbidden if the K + spin is zero. Furthermore, the muon polarization in the decay
146
Appendices
K + --+ #+ + v~ is the same as in rr + --+ #• + vt,, thus indicating t h a t Jk = J~ = 0. Therefore, g P ~ = 0 +, JP~ = 0 - .
(A.7)
T h e kaon parity can be determined from the reaction K - + 4He--+ 4H A 4- rr0 ( K - capture from an a t o m i c S state), where the hypernucleus 4H A contains a b o u n d A h y p e r o n in place of a neutron. Measurements of 4H 4 weak decay modes suggest t h a t J = t? = 0 on b o t h sides of the reaction. Hence j R __ 0 - , if the parity of the A-particle is defined to be positive, like t h a t of the nucleon. Application of the charge conjugation o p e r a t o r 0 interchanges the rr + and 7r- in the decay K ~ --+ rr+r~ - . In this case the operation is equivalent to space inversion, 0g%+,~- = Pg%+~- = (-1)e~P~+.-,
(A.8)
and thus 0/SkV=+rr - = (-1)2g~P~+~ - = + ~P~+~r-"
(A.9)
For the 7r~ ~ final state 0/5g%o~o =/3~V~o,o = (-1)~,P.o.o = +~P.o,~o
(A.10)
because two identical bosons must be in an overall s y m m e t r i c state. If we also take into account t h a t the kaon is spinless, conservation of angular m o m e n t u m requires ~ = 0 in the 27r state. Hence, 0~2~ = / 3 ~ 2 , = + ~P2,,
(i.ll)
Regarding the decay K ~ --+ 3 r ~ we have shown t h a t for a system of three pions, at least two of which are identical, P = - 1 : 0/3g'3~o = - g'3~o.
(A.12)
As for the charged 3rr decay mode, for K ~ --+ rt+rr-rr ~ we take the relative orbital angular m o m e n t u m in the r~+r~- state to be s and of the rr ~ relative to the lr+lr - centre of mass to be L. Now,
OPO~+~-.o = 015 {~+~- (s = (--1)L+I~P~+~_~o.
= + ~+~- P ~ o ( n ) (A.13)
We see that, in contrast to the 7r+Tr- , 7r~ ~ and 39r~ final states, the r~+rr-rr ~ system does not have a well-defined C P eigenvalue: C P = ( - 1 ) L+I. T h e kaon is spinless, and so the total angular m o m e n t u m in the 37r-system must be zero, which means t h a t L must be m a t c h e d by g = L. If L = 0 or an even integer, then C P = - 1 ; if L is odd, C P = +1. As explained above, states with ~ > 0 are supressed by kinematics. Therefore, the decay K ~ -+ rr+rr-rc ~ is expected to be d o m i n a t e d by the CP-allowed decay K ~ --+ rr+rc-rr ~ with L = t? = 0 and C P = - 1 . T h e K ~ m a y decay into the kinematics-supressed
B Forward Scattering Amplitude and the Optical Theorem
147
and CP-allowed final state with L = g = 1 and C P = +1, or into the kinematics-favored and CP-forbidden state with L = g = 0 and C P -- - 1 . The charge conjugation operator C changes the sign of all charges, including the charge of the sources and thus of the electromagnetic fields they produce: Au(x)
~ - A t ( x ),
under C.
(i.14)
If we consider an electromagnetic field to be a collection of photons, then the electromagnetic potential A u ( x ) can be taken to represent the photon wavefunction. From (A.14) we infer that the photon has odd charge-conjugation parity C)l')') = -17},
(A.15)
i.e., 1"),}is an eigenstate of 0 with eigenvalue C = -1. Since C is multiplicative, 0In'),) = Ch,1)l-Y2)... [')'n} ~---(--1)nln3'}.
(A.16)
The rr~ decays predominantly to two photons. Hence, C)l~r~ = + Ire~
0/51rr ~ = -Irl~
(A.17)
B Forward Scattering Amplitude and the Optical Theorem Suppose in the z (see Fig. outward
an incident beam of particles, represented by a plane wave traveling direction, ~i = eikZ, impinges normally on a thin slab of material B.1). The scattered beam at large r is a spherical wave propagating from each of the scatterers: eikr ~Psc = ~(0) , (n.1) r
dx a X
z
\
p
Fig. B.1. A plane wave impinges on a thin slab of material
148
Appendices
where ~(0) is the scattering amplitude. Note that in (B.1) there is no dependence on azimuthal angle r because the z component of the angular mom e n t u m is zero. We have also ignored the factor e -iEt, since for coherent scattering the slab as a whole takes up the recoil, and therefore very little energy is exchanged in the process: E1 - E2 -
P~a~
2Mslab
= le-(p, s)). Under charge conjugation, the roles of creation and annihilation operators are interchanged: b~(p) ~ bt~(p) and ds(p) ++ d~(p). This symmetry operation flips the signs of internal charges, such as the electric charge, baryon number etc., but spins and momenta are not affected. For example, C turns a left-handed neutrino into a left-handed antineutrino, a state which does not exist. The operation of charge conjugation, therefore, does not transform a particle into its antiparticle; this can be accomplished through the combined CPT operation. Like parity conservation, charge conjugation invariance too is violated in weak interactions. Direct evidence for this violation is provided, for example, by the fact that the positive and negative electrons in the decay #+ --+ ed=~P have opposite longitudinal polarization. This effect was first observed in 1957 by measuring the circular polarization of bremsstrahlung photons emitted by e + and e- in the muon decay (the total transmission cross-section for photons propagating through magnetized iron depends on their helicity).
162
Appendices
Having defined C conjugation for free Dirac fields, we will now examine the effect of this symmetry transformation on the interaction parts of the lagrangian, ~int. For two Dirac fields r and r interacting with a vector field V~, ~int = g (~1")'/~r
+ w2! Wl /~, )
(E.29)
where g is a real coupling constant. The second term is the hermitian conjugate of the first one, thus ensuring the hermiticity of the lagrangian. It should be remembered that every quantized field theory that obeys commutation or anticommutation rules must be properly symmetrized or antisymmetrized. Thus all the bilinear forms of the Dirac field must be antisymmetrized. We will encounter shortly an important consequence of this rule. The adjoint field transforms as
: r
: __eTa-l,
(E.30)
where we used (E.20) and (E.21). Hence,
--~ --~2 ^/ttr
(E.31)
(the superscript "T" can be omitted because ~ 7 u r is a number, i.e., a oneby-one "matrix"). The origin of the minus sign in (E.31) is both subtle and important; it is related to the connection between spin and statistics. Since the fermion fields are antisymmetric, a minus sign must be introduced when we move one spinor past another. For the electron current this implies
jU(x) =- - e ~,.,/ur ~ _ jU(x).
(E.32)
The above result was anticipated: C conjugation flips the signs of all charges, including that associated with the current operator. We see that the method of second quantization is indeed essential for a self-consistent formulation of charge conjugation invariance. From (E.31) it follows that the lagrangian (E.29) is invariant under charge conjugation provided Vu -+ - V t in which case this symmetry transformation merely turns each term in the lagrangian into its hermitian conjugate. The electromagnetic potential Au(x ) and the current jU(x) are related through a simple differential operator (see the beginning of this appendix). Hence they must have the same transformation properties under C, which means that the combination jUA u is invariant under C conjugation. If we associate a photon with the field A , ( x ) , then C~ = - 1 . As shown in Appendix A, for a state with n photons, C = ( - 1 ) % In the decay of positronium, C invariance implies that the 1S0 singlet state decays to two photons and the 3S0 triplet state to three photons.
E Transformation Properties of Dirac Fields Under C, P and T
163
We conclude our discussion of charge conjugation by showing that ~r satisfies the anticommutation rules (E.9a). In terms of field components, a
~
Lo'
where we dropped the superscript T since it pertains to the complete field operator, not to its components. Now,
= (7~
(~(3)(x - y),
(E.33)
and similarly {r
x), r
y)} = {--c r
r
_- 0.
(E.34)
T i m e Reversal Turning next to the time-reversal transformation (t, x) ~ ( - t , x), we will demonstrate that the free Dirae field does not possess a unique time direction, i.e., that it is invariant under this symmetry operation. As explained in Appendix D, the time-reversal operator T is antilinearunitary (or antiuuitary): = unitary transformation (U) • complex conjugation (/4).
(E.35)
Since complex conjugation is involved in time reflections, the Dirac equation for the time-reversed state reads (cf. (E.18)) {-iT~
+i(--TJ)T~xj
-- m}-~T(--t,x)=0.
(E.36)
From (E.1) and (E.36 it follows that x) -
(E.37)
x)
and ~-17o~ = 7~ = -~-17o~ '
for the Dirac equation to remain invariant under time reversal. Clearly, the operators T and C are of similar nature. In the Dirac-Pauli representation, = i7275 = 717370
) ~b = -- ~ - 1 = _ 5hi, ~2 = _ 1
(E.39)
and
~%~'T(__t,x) = .,f173,.~0 [r where
T = 7 17 3~ 9 ( - t , x ) ,
(E.40)
164
Appendices
"Y5~ i70717273 =
(0:) ]1
/End/
"
The adjoint of the time-reversed spinor is given by
-~t(t,~) : [~/%~Z(--t,X)]t'),0 : [~'(~3'(--t,x)70)Z]'~ 0 = [r
x)70] * T*3~0 : ~pT(--t, x ) T -1
since ~t = ~b-1 and T-17~ = ~/o~-1. Therefore,
~(t,x) ~ -~t(t,x) = cT(--t,x)T -1.
(E.42)
Using (E.40) and (E.42), it can be readily shown that the quantization conditions remain invariant under time reversal: =
(~/l~/3)aa(~O~/3"yl)o~3{~)~(--t
,
X), r
= (71~3)~. (~0).~(70)~(70~37~)~,
y)}
~(3)(~ _ y)
= (../0)c~~ (~(3)(X -- y).
(E.43)
Similarly, {r
x), r
y)} = {r
(t, x), r
y)} = 0.
(E.44)
In analogy with our treatment of parity and charge conjugation, we seek an antiunitary transformation in Hilbert space that transforms ~b(x) into Ct(x). A clue is provided by expression (D.12) from Appendix D:
(BIOIA)
= (At 10t I Ut),
(E.45)
where
Ot -- ~b0tT -1
(E.46)
and 0 is a linear operator. Note that, due to complex conjugation, the time reversal transformation exchanges "bra" and "ket" vectors. This represents the exchange of the initial and final states in an interaction. In view of (E.46), we postulate the existence of an antiunitary operator V ( T ) -- U ( T ) K that satisfies
V (T)Ot (t, x) V - I (T) = ~Pt(t, x) =- T~T(--t, x).
(E.47)
Expressed in terms of field components, equation (E.47) reads V(T)r
(t, ~ ) V -1 (T) = (71737~ z [r ( - t , x)7~
w
= (713~3)a,O~(-t, x),
(E.48a)
U ( T ) ~ ( t , x ) U -1 (T) = (713~3)~p~(-t, x).
(E.4Sb)
i.e.,
E Transformation Properties of Dirac Fields Under C, P and T
165
By Fourier-transforming this expression into momentum space, we obtain
U(T)r
(21r)3/2
+ U(T)bs (p)U-l(T)Us (p)e -ipx]
=-i (,.)','d'" + bs(p)')'l"y3U*(p)e-i(Et+p':)].
(E.49)
Now, (E.50)
7 ~/ u,(p) =io "2 tr*. Xs and -Xs=-l/2
io'2Xs=+l/2 =
Xs=A_I/2 (E.51)
i 2 tr*
{-a'(--P))G=-I/2
O" (
,, O. (__p)Xs=+l/2
"P)~s=•
Equation (E.50) can thus be written as
s) =
', -s),
(E.52a)
with p' = ( E , - p ) . Similarly
3'l"y3v*(p, S) ---- (--1)s+l/2v(ff,--S).
(E.52b)
When (E.52a) is substituted in (E.49) and p changed to - p in the second integral, we find that
U(T)bt(p, s)U-I(T) = ( - 1 ) s-1/2 bt(-p,-s), (E.53)
V(T)di(p, s)U-I(T) = ( - 1 ) s-l/2 dt(-p, -s),
where the first relation (E.53) is the hermitian conjugate of Ub(p, s)U -1 = ( - 1 ) s-l/2 b(-p, -s). The phase factor in (E.53) implies that the result of two time reversal transformations performed on the Dirac field is the original field multiplied by a minus sign. Indeed, from (E.48a) it follows that
V2(T)r
x)V-2(T) = U(T) [-~173@(-t, x)]* U-I(T) = (ffl")'3)(~1~3)~b(t, x) ---- - r
and from (E.53),
x),
(E.54)
166
Appendices
U2(T)bt(p, s)U-2(T) = ( - 1 ) s - l / 2 V ( T ) b t ( - p , - s ) U - I ( T ) = - b t (p, s).
(E.55)
Assuming that U(T)I O) = 10), expressions (E.53) yield
U(T) l e•
-- (-1)s-1/21e+(-p,-s)).
(E.56)
This symmetry transformation, therefore, reverses the momentum and spin of an electron (positron) with respect to the original orientation along the z axis. This is to be expected, since p is the time derivative of x and the spin transforms as angular momentum (x x p). According to (E.45), the expectation values of the observables () and Or, which are constructed from the field operators r and Ct, respectively, satisfy
(A I O I A ) = ( At l Ot l At),
(E.57)
where
IAt)=- V(T)IA),
vtv=
v v t = l.
(E.58)
With the existence of an antiunitary operator that transforms r into Ct previously established, expression (E.57) demonstrates time-reversal invariance for a free Dirac field. Transformation P r o p e r t i e s o f Dirac Bilinears By virtue of Lorentz invariance, the quark and lepton spinors appear in bilinear forms in the lagrangians of quantum field theories. The transformation properties of the Dirac bilinears r 1 6 2(scalar), ~'Y5r (pseudoscalar), r162 (vector), ~'~'Y5r (axial vector) and r162 (tensor) under the discrete symmetry operations CP, T and CPT are given in Table E.2. Table E.2.
CP T
(t, ~) (t, -x) (--t, x)
r162
~_~~ Cb
~Jb~Ja -~a~Jb
--~Jb75~Ja --~bTiz~Ja --~b~fDTS~Ja --~JbO'l~r -~a75~)b ~aT~Jb ~aTi~75~Jb --~aO'tzv~Jb
~o~"~
~o~"~r
~_~cr'~r
The lagrangian of a local field theory must be hermitian 46 and behave either as a scalar or a pseudoscalar under Lorentz transformations. Based on this one can show, referring to Table E.2, that CPT is a good symmetry. For example, a term in the lagrangian s that includes only scalars and/or pseudoscalars transforms under CPT as 46 Hermiticity of the lagrangian ensures probability conservation (unitarity condition). A "local" lagrangian is composed only of terms containing products of fields at the same space-time point.
E Transformation Properties of Dirac Fields Under C, P and T
s
x) = g (-~aCb) (ir162
~
s
167 (E.59)
where we used the fact that T implies charge conjugation: c number --+ (c number)*. For ~ to be hermitian, it must also contain a term ~ . The sum s + s is evidently CPT invariant. The same holds true for a combination of vector and/or axial vector fields, e.g.,
s
x) = g VU(t, x)Au(t, x) + h.c. CPT s
--x),
(E.60)
where h.c. denotes hermitian conjugate. Since tensors transform as products of vectors and/or axial vectors, we conclude that
[CPT] s
[CPT] -1 = ~ ( - x ) .
(E.61)
Ignoring irrelevant phases, transformation relations (E.14), (E.26) and (E.53) amount to
bt(p, s) P bt(-p, s) c dt(_p, s) T dt(p ' -s).
(E.62)
The combined CPT operation converts particles to antiparticles and exchanges kets and bras. Under this symmetry transformation, the momentum of the particle is unchanged because both space and time are reflected, but the sign of the spin state is reversed. We will round off our discussion of the discrete symmetry transformations C, P and T by deriving the entries in Table E.2. Consider first the transformation properties of the Dirac bilinears CaF~Cb under CP:
r
r~ r
= V(C) r U - ' ( C ) ~U(C) r U-~(C) = ~ ( t , - x ) F~Cpr
-x),
(E.63)
where i - ~ (V~')y - 7"7 ~)
F~ = ll, 75, 7 ~, V~V5, r
(E.64)
( II is the unit 4 • 4 matrix ) and F~cp -- "~~(C-1FiC)TT~
(E.65)
Expressions (E.63) and (E.65) were derived by using
v(c) r
-l(C) = ~0 v ( c ) r = - C~ ~ ~ ( t ,
V(C) ~ f ( x ) V -1 (C)
-1(6)
-~)
=
g(c)-~a(t, - x ) V
-_
[v(c)r
(E.66)
-1(C)~ '0 =
CdT ( ,t- ~ ) 7
~
168
Appendices
To evaluate Ficp, note that &-iT5&
= 75 = ( 7 5 ) T
c-l,-)fl'*TvC = C - 1 7 t t C C - 1 7 v C ~-- (7/z)T(Tv) T -------(TttTv) T,
(E.67)
C-1,-),tt75 ~ = (---,//z)Tc-175 ~ _ (--7tt)T(7t) T = (7#7t) T,
based on (E.20). It is then straightforward to show that F cp = ]1,
-75,
-7~,
-7~75,
(E.68)
-a~v,
where we set 7~ ~ = 7g. Under the time-reversal transformation,
g,~Fi ~,~ = V(T) -~a(x)V-l(T) Fi* V(T) ~ ( x ) V - I ( T ) = ~a(--t, x) Fit r X),
(E.69)
Fit = (7173)t Fi,7,7 3
(n.70)
--t
where and (see (E.48a)) a7
V(T) r
= 7173r
V(T) r
= V(T) ~ / ) ~ ' ( x ) V - 1 7 = [V(T)r =
x),
(E.71) 0
t 70
~(-t, .) (7173)*
From (E.70) we obtain ~t = 11,75, 7,, 7~75, -(Y~.
(E.72)
Finally, under the combined CPT transformation, ~.cpt Fi CbPt ~_ V ( T ) ~cp V-I(T)/~/* V ( T ) r cp V - I ( T ) = ~--~b(--x)., FcPt Ca(-x),
(E.73)
where ~cvt ~ (7570 Fi,7075)W : 7570 Fit7o75.
(E.74)
To derive (E.73) and (E.74), we used V ( T ) ~ P V - I ( T ) -- 7173~/)bP(--t, x ) ----')'173 (-- C70 ~bb) = 7%~(-x)
V(T)-~:pv-I(T)
:
(E.75) (~/):P)t (?'73)t 70 = (__ ~70~-~ra) t (_ 71737 ~
= _ r
a7 Note that v r
-1 --- u c T u -1 -- [Ur
~
t = [V~bV-1] t.
F The Vacuum Insertion Approximation
169
Upon substituting (7") t : "),07"70
(E.76)
in expression (E.74), it yields
~iicpt ~-~ ]]-,
--75,
--7",
--7"75,
(E.77)
attu"
In general, the transformation properties of a free Dirac field under the discrete symmetry operations P, C and T can be expressed as U(P)r
%r162 -x), = ~; ~(t,-x)7 ~ = ~cS~(t, x), =
U(P)-~(x)U-~(P) u(c)r u(c)~(x)U-'(c) V(T)r V(T)-~(x)V-I(T)
=
-- ?~c ~2T(x) ~ - 1 '
,1,~r = ~, r
=
,--
(E.78)
~), ^
x)'P,
where Op, ~c and 7/t are arbitrary phases. One can easily show that these phases do not enter into the transformation laws for the Dirac bilinears.
F The
Vacuum
Insertion
Approximation
If we define O , =- 7u (1 -75), the matrix element of the four-fermion operator in (9.17) reads
M =- (K~ g O u d g O " d l K ~ = o~joke