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I
2
I f w e were to set u
=
,
{
0 at this early stage,
2
( 1
2
2
- x )x y
.
"
(4.30)
the x-integration
would
b e c o m e u n d e f i n e d a n d w e w o u l d get t h e w r o n g v a l u e for J i , a n d h e n c e for / .
Accordingly, great care needs to be exercised w h e n e x p a n d i n g the
y - i n t e g r a l for s m a l l v a l u e s of
2
u.
Overview
of Noncovariant
Ganges
49
We proceed by first rewriting A in Eq. (4.30) as X = y ( l - y ) V [ l + j(l +«)]. *(l-*')( n)* I
P
* ~
- »)»V
(1
2
'
1
" * V ( p • n) '
J
'
so that / i becomes l h = - ^ r ( 3 -
w
) (
2 P
3
r -
l
| j dx dy « - v - * ( i - » r 0
-
3
0
a
x [ l + j ( l + o)]"- ,
ifc(3-w)>0.
(4.32)
The next challenge is to find a suitable integral representation for the factor [1 + §{% + a ) ] " . From integral tables [e.g., Ref. 55] we know that (l + z ) = F(-n,0;0;-z), 0 arbitrary, and - 3
n
+ico F { a
'^-
z )
J
-r(c,)T(m*i
ftTTTj
'
( 4
-
3 3 )
— 100
where |arg(j)| < it; the path of integration is chosen such that the poles of the functions r ( o 4- £) and T(0 +1) lie to the left of the path of integration, while the poles of the function r(—t) lie to the right of i t . Application of formula (4.33) to [1 + g(l + a)]"- , 55
3
P+.d
+
->r-
3
°°dt r ( 3 - u + *)r(-ob(i + aft r ( 3 - u)
=
— ICQ
[«8(fftl + « ) ) ! < » .
(4-34)
transforms I\ into the form
^ r ! / * r
1+
(
3 -
U
+
J
1
,r(-o[^ J|,
1 rW11
sibf//'' '' -'' ' 0
0
-'
1
w-3-t
(4.35)
50
Noneovaria.nl
Gavgct
It remains to expand 2
t 1 +
2
2
x y (p-
V n)
r
I f dzT(z - t)T(-z) = -L f T/(-f) 2xi J
2
in which case +
2
-2*"(
P
2
3 '°°
)
jjdzdtY{Z-u
(2JTI
+ t)T(z -
t)T(-
with
= j
d
x
2
x
- -
2 i
, _ r ( - i - « ) r ( ' + i) = 2r{i + ( - z ) '
7
(i-x y
i Y = J
3+
2
dyy»- '- '(l-yr--
o T{w-2
+ t-2z)T(w-2-t) f f > - 4 - 2z)
Hence
JJdzdtr(o-u,
+ t)T(z -
r ( - i - ) r ( t + i ) r ( ^ - 2 + t - 2*)r(w z
r(i-i-i-s)r(2w-4~2z) 2
the only dependence on u being of the form
2
[u )'.
t)T(-z)
-2-0
Overview of Noncovariavit
Gauges
51
The results (4.38) impose various restrictions on t and z, such as Ke(z-f-l/2) < 0 and Re(t + l) > 0 from Eq. (4.38a), and also Re(u-2~t) > 0 and Re(w-2+t-2z) > 0 from Eq. (4.38b). The condition f t e ( z + l / 2 ) < 0, for instance, tells us that the contour Ci in Fig. 4.4 must lie to the left of the point z = —1/2.
•
Rez 0
-1/2
-1
F i g . 4.4. Original position, of contour C\-
2
2
To compute K = ( 1 + 2/ d/dii )I differentiate the ( / i ) - t e r m , 1
2
in Eq. (4.24), it suffices to
1
!
2
8
(l + 2 ^ t W ) ( M r = - 2 ( - i -«)fji )* , and then to combine ( - 1 - z) with _ ( - I - z)T{-\2
(4.40)
- z) in Eq.(4.39): z) = - 2 r ( ! - *)
(4.41)
Thus, + IOO
2w-3
f r(i-l-l-;)F(2w-4-2z)
X
\ V ( p «) (p n) J \ np 2
2
2
-2-0
2 S
-4)
(
•
4
'
4
3
)
Of course, there are other poles i n the complex z plane, but they are of no consequence since ft —* 0 in Eq. (4.42). Accordingly, K reduces to 2
+ioo z
i ™
A
2x^VT2
3
f dt r ( 3 - ^ + i ) r ( t + l ) r ( - t )
2
~
2xin
J
T(i + r)r(2w - 4)
—ioo
xr( -2-or(i)r( -2 + w
w
2
2ir"(p )"~
+
3
+
2 r V
2
r J
i)(^^y
dt x- T(±)^(w-2
,
+ t)a
(4.44)
,
r(| + t ) r ( 2 w - 4) sin(Tt) sin » ( « - 2 - t ) '
— im
(4.45) where we used
r(-«)r(i + 1) =
-*/sm(*t),
a =
(p-n)V - 2) r(2w - 4) sinir(w - 2) a
r(4)n-i)"" 11 r(« - §)r(« - l ) s m i r ( w - 2 ) J /
w - 3 +
' (4.46)
The divergent part of the integral / in Eq.(4.21) is, therefore, given by d i v
/ "57
\i =
d i v
(
Um
K
)
= -Tl
1
•
4
47
(- )
2
I f f i is equated to zero prematurely, for instance in Eq. (4.30), an incorrect value for I is obtained. 4.7. D i s c r e t i z e d L i g h t - C o n e Q u a n t i z a t i o n 56,57
During the mid 1980's, Pauli and B r o d s k y initiated a novel quantization scheme within the Hamiltonian formalism called discretized light-cone quantization. Its purpose was to handle strongly interacting fields and, especially, bound-state problems. Since the scheme exploited the notion of light-cones, the appropriate coordinates for this technique in 1 + 1 dimen-
54
Noncovariant Gauges l
sions were the light-cone variables {x+,x } , where x+ = (x° + x )f\/2 is defined as the light-cone time and xT = (x° - x )/^ as the hght-cone 1
position; the metric tensor reads g'"' =
^
J J , ft,H = +,—•
The
technique of discretized light-cone quantization was originally developed in 1 + 1 dimensions in the context of the interacting boson-fermion system with Lagrangian density 56
- ^ # 7 * * - (m
F
+ A^)** ,
(4.48)
where * is a fermion field with bare mass mp,
0), the light-cone gauge (n = 0) and the planar gauge ( n < 0). The first three gauges are defined by the constraint 2
2
2
2
n-A"(x) = G,
L
1
n
x
a
--(2c,)- (n-A )
2
,
a -
0,
(5.1)
while the planar gauge is characterized by
a
n-A (x)
a
= B (x),
L
fi)t
= - ± - A let n
a
( ^ ] n - A \ \ /
a = 1 , (5.2)
n
a
where B (x) is an arbitrary function of x. Moreover, we shall endeavour to keep the vector as general as possible, i.e. we shall refrain from assigning specific values to the components of = (no, n). The collective treatment of these four gauges is motivated, at least in part, by the discovery of a general prescription for (q • n ) (see Eq. (5,34)). This unifying prescription allows us to streamline computations and avoid duplication of effort. A brief historical account of these gauges can be found in Ref. 1. We begin this Chapter with a review of the Feynman rules in Yang-Mills theory and a detailed discussion of the uniform prescription for {qn)-K - 1
5.1. F e y n m a n Rules The Yang-Mills Lagrangian density (4.5) yields the following axial-gauge Feynman rules.
1,2
59
Noncovariant
60
5.1.1.
Ganges
Vertices
Three-gluon vertex (Fig. 5.1(a)): V;#(p,3,r)
F i g . 5 . 1 ( a ) . Three-gluon vertex.
Four-gluon vertex (Fig. 5.1(b)): d
W;l\ { ,s,r) p
PA
a
a
= - f ( 2 » ) * - i - ( p + , + r + #) f
Ghost-ghost-gluon vertex (Fig.5.1(c)): ^ 5.1.2.
flare
f l t e
( p , ft. 5) =
gluon
+ P " 9)-
propagators 2
In the general axial gauge, n ^ 0 , a ^ 0 (Fig. 5.1(d)):
2
In the pure axial gauge, n < 0, a = 0
Noncovariant
Gauje«
a F i g . 5 . 1 ( d ) . Gauge boson propagator
G"t(9,a = 0) =
-i6
ab
3uc (2ir) »{q +if) . O O . 2
3
!
(5.7)
2
In the temporal gauge, n > 0, a = 0 : 3
= 0) =
2w
2
(2jr) (g +ie) e > 0.
9n" -
1
" + Qti9r
2
(9«) J' (5.8)
q• a
2
In the light-cone gauge, n = 0, a = 0 : ah
-iS
q^n + q T\y. v
f » " ( < , + ie) Sm 2
2
v
,
e > 0.
(5.9)
qn
2
In the general planar gauge, n ^ 0, a ^ 0 : -i6 2
+ q„n^ , (1 - a)n (. q — — q n {q - rc)
ab
2
(27r) "(g + it)
ff
qil
u
2
(5.10)
t > 0. 2
In the planar gauge, n ^ 0, a = 1 : G£(«,«=l) =
n
9*i
(2T) -( 2
2 9
+ W)
+ 9w (i
9 "
,
e > 0.
(5.11)
The scalar ghost propagator reads (Fig. 5.1(e)): a 6
G ( ) =
(5.12)
9
For the sake of completeness we also list here the bare graviton propagator in the pure axial gauge: ' 3 4
&xp, (q,a pa
= 0) =
2 i 2 j r )
J
f q 2
+
Wlg.ro
ie)
~ lie,?*).
e>0,
(5.13)
Gaagti
of the Axial
63
Kind
q
F i g . 5 . 1 ( e ) . Scalar ghost propagator.
where
Ipv,tv
=
d^Kd dpxd x, UK
d^ = S^,, — ——q,,n
a
v
.
Compared with the Yang-Miils case, where only single and double poles of (q - n) occur, there now appear spurious poles of order three and four in (q • n). This list completes our summary of axial-type Feynman rules. 5.2. U n i f o r m P r e s c r i p t i o n f o r Axial—Type Gauges - 1
A meaningful prescription for (q • n ) was developed between 1982-1988 in two stages. First, Mandelstam and, independently, Leibbrandt derived the proper prescription in the light-cone gauge. Then, between 1985 and 1988, L e i b b r a n d t and researchers from V i e n n a generalized the light-cone prescription for (q • n ) to include the axial gauge, the temporal gauge and the planar gauge. 5,6
7-9
10-12
13,14
- 1
5.2.1. Prescription
for the light-cone
gauge
6
_ 1
Early in 1982, Mandelstam proposed a light-cone prescription for (q • n ) that differed radically from the principal-value prescription (4.14) and used it to demonstrate the ultraviolet finiteness of N = 4 supersymrnetric YangMills theory, Later that year, Leibbrandt independently discovered the following equivalent prescription and implemented i t in the framework of dimensional regularization: 6
7-9
' - = lim — —, q•n f—o q - nq • n + if 9
c > 0,
(5.14)
where = ( n , n ) , n'^ — ( n , - n ) are vectors in Minkowski space with rt = ( n * ) = 0, n* being the dual vector introduced in Eq. (4.18b). To motivate prescription (5.14) we shall first look at its structure in Minkowski and Euclidean space and address the question of Wick rotation. 0
a
2
0
61
Noncovariant Gango
Minkowski space
(i) To begin with we recall that in covariant-gauge propagators, the denominators are semi-definite forms of the intermediate momenta, as in (q + if)' , or in (jf - m ) = (4 + m)/(g - m ), for instance. The poles lie, therefore, in the second and fourth quadrants of the complex qa plane (Fig. 5.2). 2
_ 1
1
2
2
-1
(ii) The second remark concerns the constant vector in (g-n) . Since the constraint n — 0 implies n = ±|n|, the value of (g • n ) is ambiguous, because there are now two values for tip : nj, ' = (+|n|,n) and n^ ' = (—|n|,n). Lest we somehow remove this ambiguity in (g • n ) prior to computation, the final Feynman integrals will either be wrong or internally inconsistent. Accordingly, we utilize both signs in n = ± | n | by defining the two light-like vectors n)P and nj, ', 2
- 1
0
1
2
- 1
2
0
1
nt > = n„ = (|n| n), )
-nf> = » ' = ( | n | - B ) , 1
1
2
n = 0,
(5.15a)
2
(n') = 0,
(5.15b) 1
and then replacing (q • n)" by g • n'(q • nq • n* + ie)" to arrive at formula (5.14).
Gauges of the Axial
Kind
65
Euclidean space To motivate prescription (5-14) in Euclidean space we observe that the condition n = 0 = n\ + | n | implies that no, = ± i | n | , so that 2
2
1 qn
1 9 n + q-n 4
1 ±ig |n| + q • n
4
(5.16a)
4
the choice n = — i | n | leads to 4
1 q n
_
1 q
il -
!(/.?
|n[
(5.16b)
2
The ambiguity in n = 0 now manifests itself through the complex factor i in the denominator of Eq. (5.16b). To remove this ambiguity and, at the same time, ensure the positive semi- defini ten ess of the denominator, we simply rationalize the denominator of (5.16b): 1 q-n
q-n + tftH ( q • n ) + q\a? 2
q-n" q • nq • n* '
or, finally, qn
Eud
= lim„- J " " ; , , c—o q • nq • n' •+ p?
(5.17)
where n,, = ( n , n ) = (—t|n|,n), n* = (i'|n|,n), and where a small real part / i has been added to the denominator to ensure its positive definiteness. 4
2
Wick rotation in the light-cone gauge A crucial test of the Minkowski-spare prescription (5.14) is whether or not it can be Wick-rotated to Euclidean space to yield expression (5.17). To answer this question we use again r>„ = ( n , n ) = ( | n | , n ) , 0
and
n* = ( n o , - n ) = ( | n | , - n ) ,
with q n = q \n\ - q n , a
q n' = o ! n | + q n , 9
which leads to the form
(5.18)
Nancov&riant Gadget
66
q-n
1
= lim
Mink
f - o \q
•n'
• nq
|
= umf , *
n
+tej +
'
q
; "
(
• )•
>
0
-
5
w
< - >
To make the transition from Minkowski to Euclidean space, we simply define 9o = '94.
q =
no = irt4,
n = n ,
1.
(5.20)
2
and replace the te-term by a /i -term, so that Eq. (5.19) becomes 1 1c = lim q • n Eud c—0
-(iq \n\ 4
\ 9 |
n
2
+ q • n)
+ ( l
n
)
=
2
+
2
t* .
,) ,
^>0.
(5.21)
Prescriptions (5.21) and (5.17) are seen to be identical, except for an overall negative sign. This extra sign makes perfect sense, because Eq. (5.21) originated in Minkowski space. We shall adopt Eq. (5.21) as our prescription in Euclidean space. Prescriptions (5.19) and (5.21) are equivalent to Mandelstam's version for (q-n)- in the light-cone gauge. In terms of the n*-vector, Mandelstam's original prescription can be cast into the form 1
5,6
1
'=
q•n
Mink
= lim
e—o
1 —
-,
q • n + it sign q n'
e> 0.
(5.22)
Before extending the above prescription to the axial and temporal gauges, we shall introduce the following algebraic simplification. We shall on occasion "replace" the dual vector n* by its normalized version F^ = ( F , F ) . Defining 12
4
(5.23a)
n; = ( n , - n ) s ( f f P , p ^ ) , 4
we see that where c = |n| ,
F = —, tr F
4
= -n
4
where p = \ - n | = |n|, 4
n = —i|n.|. 4
(5.23b)
Gauges of the Axial Kind
67
Hence, < a W ( r , ^
S
| l f t
,
(5.24)
2
where f ' is a null vector: (F^) = 0. a
5.2.2. Pre s c r i p t t o n for
axial
and
temporal
gauges
Since the prescription (5.19), = lim g • n iMink
2
- J ,
t > 0,
f—o \ g - ng - n* + i £ /
gives satisfactory results in the hght-cone gauge and, moreover, avoids the problems of the PV technique, the prescription was extended to include both the temporal gauge and the axial gauge. Below we shall illustrate the generalization of (5.19) in the case of the temporal gauge. The temporal gauge is defined by n A" = 0 with n > 0, i.e. n > n , where n = ( " o , n j . , rts) and n i = (n n ) . To ensure n > 0, no ^ 0, we may either choose n such that n , > n > 0, or such that njj > n — 0. For simplicity we select n = 0 — n ^ + w ., keeping | n x | ^ 0. The constraint n = 0 then implies n$ = ± i | n j . | . Choosing the minus sign, we get 12
2
2
2
2
A
it
2
2
2
2
2
2
2
2
n,, = ( n , n j _ , - i j n i | ) ,
n = -i|n±| ,
0
3
(5.25a)
and j • n = qono — q_L • n j . — 9 3 ^ 3 = 9ono — q± • n ± + i g 3 | i j . | .
(5.25b)
Next we introduce, in analogy with the light-cone gauge, the dual vector n'p which is the complex conjugate of n^ in Eq. (5.25a): n* = ( n , n , ! | n i | ) , 0
(5.26a)
±
so that g - n* = q n a
- q j . • n j . - igsln-il •
a
The Minkowski-space prescription for (q n ) reads 1
q-n
t e m P
Mink
= lim ^
_ 1
in the temporal gauge then
n
*° ° — 1-L • "J. — iflalnxi
f—o L(gnn - qj. • n j . ) a
= lim
(5.26b)
q
n
-,
f—0 q - nq •rt"+ te
2
+ g^n ^ + i f J 2
£> 0 .
(5.27)
Noncovariant Gtntfci
Mimicking the procedure in the light-cone gauge, one performs a Wick rotation to Euclidean space, 9o = »94,
r»o = »«4.
q = q;
(5.28)
ri — n
such that l e m P
q n Eucl
= lim [ ~(94"4 + qj. n± +'93l"J-l) *i-o [(o; n + qj. • nj.) + qla\ + f* \ ' 2
4
a
^ > 0 . (5.29)
2
4
A similar formula may be established for the pure axial gauge, defined by n • A" = 0, n < 0, with n„ = (n ,n) and n = n - n|. The choice n , = 0, ^ 0, now implies n = ±]nj.| (we shall take no = +|njj), and guarantees n < 0, provided n ^ 0. In Minkowski space, the components of n and n* read, respectively, 3
2
n
0
2
0
2
3
u
= (|nj.l,n).
and
(5.30a)
n* = (|nj.|,-n),
leading to the scalar products q • n = q \a± | - q • n,
(5.30b)
q • n* = q \nx j + q • n .
0
Q
Therefore,
=bm[ r
Lr
i
n
+ w , l f
y .1. l
( >0 (5.31)
2
• n lM.uk
f-o [ql,n\ - (q • n ) + it j = Bm( * ' * . ) , e—o \q • nq • n* + te J
which possesses the same structure as Eq. (5.19) in the light-cone gauge. The corresponding expression in Euclidean space may again be deduced with the help of a Wick rotation (q = t"g4,q = q;n = i n , n = n), and reads 0
0
I ax q-n
Eucl
- ( q n + ig |n±|) (q n ) + B M + n J ' 4
= lim
2
»i—0
2
4
u >0
(5.32)
n = -ijnj.|,
(5.33a)
n ^ 0.
(5.33b)
2
fhere n„ = (n ,n) = (-i|ni],n), 4
= ( - n , n ) = (thai.|,n), 4
4
3
Gavgr.r of the Anal
Kind
It is clear from Eqs. (5.29) and (5.32) that the temporal-gauge and axial-gauge prescriptions are identical in form to the light-cone gauge prescription (5.21). Since the same can be said of the planar gauge, we conclude that the spurious poles of (q • n)~ can be treated by a single, uniform prescription: x
12,15
\q • n)
A
t—a \q • nq n' +te J
t > 0,
X = 1,2,3,... . (5.34a)
The components of n and n* possess the following structure (Minkowski space): v
2
< = (Kn), n$
r. = o , 0
2
=(|nx|,n),
" P/p , and then defining d
v
fin
dlw
2
DO
a = X(l-(),
P = K,
L
CO
jdajdp=
oo
jdt\jdXX,
(5.43)
we obtain
/
d
i
v
= $ H i
* *
( 1
"
e ) 1
~" [ ( T ^ o ^ -
J^W^l
0
where m m
- a -o
(pi+X - % ) ) 7 ' +
B
=
1
- ^
2
-
£ = (n,n ),
( < ) " = ( n , - « ) = (W,p F}f)
4
k
,
4
(5.49)
iQ
where cr = |n|, p = |n|, so that lc
C
J $ = ( F , F ) = tt, i ) ; 4
n -F
l c
lc
=
ty\
2
whereas in the temporal gauges, n = n\ > 0, we obtain dq (p-q) {qnf\
|
2
lem
P
n\l
2
" | n Kft + n ) ±
2
\ + (n
2
+ n\yl )
L
2
'
(5.57) 5.3.2. Three-propagator
integral
As our second example we consider the vector integral Eucl 2 f 2
j, ' %
•
^ B * ,
(5.58)
2
q (p-q) qn
/
- 1
which has a simple spurious pole. Applying prescription (5.37) to (g • n ) and introducing the Schwinger parameters 7, 0 and a for Q , {P — Q) and [(Q • N ) + Q ^ + /l ], respectively, we get 2
2
!
2
2
Q2(P - Q ) [(Q - N ) + Q ^ 2
- = ' ^ y
2
a
2
+ ,2 j •
2
«,- cos tj> MV2
0
0 ir/2ir/2
2 i / n r r ( 3 - w) ,.
f
u
+
— v
2
— i s * /
j 0
[,
0 2
(2-w),
(sin .
2!
D
+
In the limit u —*2 ,
2
f dB dd> sin 8 cos 8 sin <j> cos A , Mm
l
o
fe\n 8
H ~ c T x
Gauges oj ihe Axial Kind a
r(2--'» n n •• rF
"*'~
79
a
o~
*/2xf2 */2x/2
f
2
f d$ do sin 8 cos 9 sin A cos A ,
*l™oJ J
/ sin 8
Mm F
n-F(2-w)
:
F + ^ - V 4 + - V lim(...) , n-F - I f ) , a % - y) ,
^
|
= ^ ( z - y) ,
pa
- w )
,
i.e.
(5.100) Gav is the bare gluon propagator.
2
q
+ i£
q-n
+
c >0, (9 (5.101)
l
and (G )/tu its inverse, (G '
a / 0) = i (Vs^
-
+ ^ n * ^
•
(5.102)
Equations (5.100) and (5.102) enable us to deduce from Eq. (5.98) the following Ward identity in the light-cone gauge; .q_n a —iq n •»( i = —-—n^+tq Iq
1
- q q^ + u
^ J .
or, finally, (5.103) So the self-energy is transverse in the light-cone gauge, at least to one loop. Notice that, as a rule, Ward identities are insensitive to the type of prescription used in computing JJ , apart from having to respect the
Noncovariant
90
transversality tttXi of Sec.
Ganges
We shall return to this important point at the end
5.4.3.
5 . 4 . 2 . Ward identity in the axial/temporal
gauge
Since the gauge-fixing term for the axial and temporal gauges is identical to that in the light-cone gauge, i.e. L
a
=
R x
-~(n-A )\
la
the corresponding Ward identities have the same form as in Eq. namely —n*D (q)
+ iq
M¥
= 0;
¥
(5.97),
(5.104)
D {q) denotes the gluon propagator in the axial/temporal gauge. Repeating the procedure between Eqs. ( 5 . 9 8 ) and ( 5 . 1 0 3 ) , we obtain uu
"JJ
^vu
?
where now G v a
a
n
a
=
'9 • " _ , j,/n-ii + * { ^ % a
_!13%
8
,
U
(5.105)
given by
1
{G~ )
MU
——
9»» q + k r " 2
r q q,
q n
u
2
(q • n)
e > 0,
(5.106)
and
(G )^^,* ^ -1
0) =
2
i (q g
- q q„ + ^ « * « * J
Mlf
Substitution of Eq. ( 5 . 1 0 7 ) into Eq. axial/temporal gauge:
p
(5.105)
n
•
(5.107)
gives the Ward identity in the
ut .temp («) = 0 ,
(5.108)
which is the same as in the light-cone gauge. Of course, it remains to be shown that the computed self-energy n ™ , ' " > Eq. ( 5 . 7 8 ) , does indeed respect formula ( 5 . 1 0 8 ) . mp
5 . 4 . 3 . Ward identity in the planar 17
gauge
Although the planar gauge is just a variant of the axial gauge, the unusual structure of its gauge-fixing term
Gauges of the Axial
91
Kind
= 0
F i g . 5.5. Diagrammatic representation of the Ward identity in the axial, temporal and light-cone gauge (cf. Eqs. (5.103) and (5.108)). The double bar denotes amputation of right leg.
3 L
1
n x
3
a
= -(2a)- nA"-^n-A ,
a = 1,
(5.109)
generates the relatively complicated Ward identity (see Fig. 5.7) e,anar
q"S^l[
(,
q Q
bc
f 0) = -Lgf^Et (q,a)
bc
.
(5.110)
>e
shown in
E denotes the amputated one-loop contribution to Wl (q,a), the pincer diagram of Fig. 5.6, ¥
Wfiq.a)
!
bc
= Gy (q,a)Et (q,c,)
,
(5.111)
and GjJ* is the bare gluon propagator in the planar gauge,
G
»"
{ q
'
a
9
f > ) * - ( , » + « ) K ""
q-n—)
'
m
€
>
°(5.112)
The nontransversality of Yl^u Eq. (5.110) has profound implications for the re normalization program: i t implies that Yang-Mills theory is no longer multiplicatively renormalizable in the planar gauge. This section completes our analysis of the Ward identities (5.103), (5.108) and (5.110). They will turn out to play a central role in the successful application of the axial-type gauges. Before turning to practical matters, however, we should like to comment on the effectiveness of Ward identities in finding meaningful pole prescriptions. We recall that the structure of Ward identities depends only on the gauge, not on the type of prescription used to compute the integrals in n « « - Accordingly, Ward identities cannot be invoked to test the relative merits of two competing prescriptions. In I8_2D
92
Nonet/variant Gauges k
q-k 8
Fig. 5.6. Pincer diagram for the one-loop contribution to EJ* in the planar gauge. Wavy lines correspond to Yang-Mills fields.
= 0
Fig. 5.7. Diagrammatic representation of the planar-gauge Ward identity Eq. (5.111). short, Ward identities provide a necessary, but not sufficient, test of pole prescriptions. References 1. G. Leibbrandt, Rev. Mod. Phys. 59, 1067 (1987). 2. C . Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980). 3. T. Matsuki, Phys. Rev. D19, 2879 (1979). 4. D.M. Capper and G. Leibbrandt, Phys. Rev. D25, 1009 (1982). 5. S. Mandelstam, Light-cone superspace and the vanishing of the beta-function for the N = 4 model, University of California, Berkley, Report No. UCBPTH-82/10; XXI International Conference on High-Energy Physics, Paris, 1982, eds. P. Petiau and M. Porneuf, Les Editions de Physique, Paris, p. 331. 6. S. Mandelstam, Nucl. Phys. B213, 149 (1983). 7. G. Leibbrandt, On the Light-Cone Gauge, Univ. of Cambridge, Cambridge, DAM TP seminar (1982). 8. G. Leibbrandt, The light-cone gauge in Yang-Mills theory, Univ. of Cambridge, Cambridge Report No. DAMTP 83/10, 1983, unpublished.
Gauges of ike Axial
Kind
93
9. G. Leibbrandt, Phys. Rev. D29, 1699 (1984). 10. G. Leibbrandt, seminar at the Summer Theoretical Physics Institute in Quantum Field Theory, Univ. of Western Ontario, London, July 28-August 10, 1985. 11. G. Leibbrandt, A General Prescription for Three Prominent Non-Covariant Gauges, CERN preprint, Report No. TH-4910/87 (1987). 12. G. Leibbrandt, Nucl. Phys. B310, 405 (1988). 13. P. Gaigg, M. Kreuzer, O. Piguet and M. Schweda, J. Math. Phys. 28, 2781 (1987). 14. P. Gaigg and M. Kreuzer, Phys. Lett. B205, 530 (1988). 15. G. Leibbrandt, Nucl. Phys. B337, 87 (1990). 16. J. C. Taylor, Private Communication (1986). The author is grateful to Professor J. C. Taylor for providing him with this analysis in terms of open and closed ghost Unes. 17. Yu. L. Dokshitzer, D. I. Dyakonov and S. I. Troyan, Phys. Rep. 58, 269 (1980). 18. D. M. Capper and G. Leibbrandt, Phys. Lett. B104, 158 (1981). 19. A. 1. Mil'shtein and V. S. Fadin, Yad. Fiz. 34, 1403 (1981); Sov. J. Nucl. Phys. 34, 779 (1981). 20. A. Andrasi and J. C. Taylor, Nucl. Phys. B192, 283 (1981).
CHAPTER 6 APPLICATION OF T H E LIGHT-CONE G A U G E TO S U P E R S Y M M E T R Y 6.1. Introduction Ever since the advent of supersymmetry, theorists have been intrigued by the possibility that certain non-Abelian models might be ultraviolet finite. Of special interest in the early 1980's was the supersymmetric N = 4 Yang-Mills model in four dimensions, conjectured already in 1977 by Geil-Mann and Schwarz to be ultraviolet convergent. The finiteness problem was tackled by two distinct techniques: the Lorentz-covariant method and the noncovariant Hght-cone gauge technique. Using the Lorentz-covariant approach, several groups succeeded in demonstrating finiteness of the JV = 4 supersymmetric Yang-Mills model to three loops. Various N — 1 theories were also shown to be finite to three loops. The first proof of all-order finiteness of the N — 4 model was given by Sohnius and West and was subsequently made rigorous by Piguet and his co-workers. A class of N = 2 models, consisting of JV* = 2 Yang-Mills coupled to N = 2 matter, was likewise proven to be finite to all orders of perturbation theory," with an explicit calculation to two loops given in Ref. 9. Another interesting revelation was the fact that addition of certain soft terms, such as mass terms or interaction terms, which break some or all of the supersymmetries, did not spoil the finiteness arguments. For example, Parkes and West demonstrated that the addition of N = 1 supersymmetric mass terms to the N = 4 supersymmetric Yang-Mills theory did not affect the UV properties of the theory. For further details, the curious reader may wish to consult the original articles or any number of books or review articles on the subject (see for instance Ref. 13). 1-4
5 6
7
10-12
10
For noncovariant gauges the success rate was equally impressive. Exploiting the reductive powers of the Hght-cone gauge, Brink, Lindgren 95
96
Noncovariant 14
Gauges
15
and Nilsson, as well as Mandelstam, managed to prove that the N = 4 supersymmetric Yang-Mills model was ultraviolet convergent to all orders of perturbation theory. In their proof the above authors concentrated on the three-point functions and higher-point functions and accentuated the transverse components of the fields. Their proof was later completed in two stages. Ultraviolet convergence of the two-point functions was demonstrated in 1987 by Taylor and L e e , whereas Bassetto and Dalbosco proved finiteness for the nontranverse components of the Lagrangian density [cf. E q . (6.1)]. Both analyses were carried out in the light-cone gauge. In Sec. 6.2 we shall briefly examine the N = 4 model by using component fields, but first let us say a few words about the superfield approach. (For a review see Howe and Stelle. ) 16
17
4
This formalism exploits the elegant method of superfields and supergraphs and is contingent upon successful implementation of the light-cone gauge condition n^A^x) = 0, n = 0. Specifically, one has to express the Lagrangian density for the N = 4 model in terms of a complex, scalar tight-cone gauge superfield. The advantages of supergraphs over ordinary Feynman graphs have been extolled in numerous research papers and books since the mid 1970's (see, for instance, Salam and Strathdee, Ferrara ei a/., and Gates ei a". ) An exceptionally potent property concerns the degree of divergence of a supergraph. It was shown some time ago in a covariant-gauge formalism that the superficial degree of divergence of an n-point supergraph was actually z e r o . 2
18
19
20
21,20
15
Even more surprising, however, was the discovery by Mandelstam and by Brink, Lindgren and Nilsson that judicious integration by parts reduces the superficial degree of divergence from zero to minus one, provided a physical gauge is employed such as the light-cone gauge. In other words, all supergraphs turn out to be finite. Of course, a nontrivial component in this entire discussion on finiteness is the use of a consistent prescription for the spurious singularities of (p • n ) , such as Eq. (5.14). 14
22,23
- 1
6.2. C o m p o n e n t - F i e l d F o r m a l i s m The one-loop finiteness of the N = 4 model may also be illustrated by using the method of component fields. The Lagrangian density for this theory can be written a s 24,25
26
Application
of the Light-Cone
2 -~H » a
xir"
H
al)
Gauge to SjtpeTigmmetry
xH , yl
fi v=
97
0,1,2,3,
y
(6.1) F„„ = d A u
v
— d„A„ + gA,, x A„,
D„ =fl„+ gA x , v
where A is a Yang-Mills field, a scalar field, and a,0 = 1,2,3,4 are SU(4) indices. The chiral ferrnion field 4 „ has the component form u
26
*
(6.2)
1/A
Q
=2
Gauge indices have been omitted, and allfieldsare in the adjoint representation of the gauge group. Moreover, C is the charge conjugation matrix, g the coupling constant and the superscript T denotes the "transpose''. We shall now summarize the divergent parts of the various two-point Green functions in the light-cone gauge. 27,17
6.2.1. Total gluon self-energy
f]^ (total)
The total gluon self-energy in the light-cone gauge consists of four components: 27
nr. (
t o t a i
>=rC
( f e r m i o n ) +
nr.
+ TT°* (pseudoscalar) + TT"' (gluon).
(6.3)
(a) Gluon-fermion loop (wavy hues denote gluon lines, solid lines denote fermion lines):
98
Nonco variant
Gauge q
q-p
dq
x div
—i 2 ai o V e Sf C 2 (G)5 (pV where e = 2 - w, f***f*'* = C (G)6
ab
- p„) ,
w-
P(i
2+ , (6.4)
and /t is a mass scale,
2
(b) Gluon-scalar loop (large broken lines are scalar lines):
JJ** (scalar) = | x
\
j
= -|ffV - c (G)C-« ») 3 w
4
u
2
X
^
/ ( 2 , )
2
V ( ^ p )
2
(
2
g
'
P
)
'
,
(
2
g
-
p
)
-
Application
Ij£
of the Light-Cone
Gauge to Superayramelry
2
al
2
(scalar) = ^L- C (G)S >(p g^ 32» e g a
2
- p p„} . H
99
(6.5a)
(c) Gluon-pseudoscalar loop (broken dotted lines denote pseudoscalar lines):
TT j j
a i
1 (pseudoscalar) = - x
n
ot
(6.5b)
(scalar) .
(d) Pure-gluon loop:.28
ab
•n
1
q • P
= ^ div |
- ^ ^ ( p ,
ff
- p)G? (-q)Gi+EV)+y! (« S
I U
N
° )=
4 s a t
+
u A
A
•
(MI)
where a
yt = i» Ca(G)«"*/(32»M.
and f**f*+*
ah
C (G)6 ,
E
t = 2 - w.
2
(b) *TAe scalar self-energies: There are two expressions here, the scalar-gluon self-energy J2 (gluon), H
E («
l u o n
s
)=
l
1
and the pseudoscalar gluon self-energy £
(e
| , , o n
)"
p
l
(*•")
(gluon),
j^Z^
(
6
1
3
)
(c) Finally, we have contributions from the scalar-fermion loop f ] , (fermion),
]J
s
(fermion) =
(6.14)
and the pseudoscalar-fermion loop [~[
pa
Y[
(fermion) =
(fermion),
(6.15)
102
Nonet/variant
Gauges luon
f r o m
E o
s
6
12
It turns out that the sum (gluon) + D » ( g ) - ' ( - ) and (6.13) is equal, but opposite in sign, to the sum fT (fermion) + r j (fermion) from Eqs. (6.14) and (6.15), i.e. P
s
z
s
+ £
P
S
+ n
s
+ n
P
p 4
. = ° .
This skeleton summary completes our discussion of the component-field formalism in the light-cone gauge, with Eqs. (6.6), (6.7), (6.11) and (6.16) containing the most important information. According to Bassetto and Dalbosco the above one-loop answers can be generalized to Green functions of arbitrary order. This result confirms the conclusions from superfields, namely that the N = 4 supersymmetric Yang-Mills theory is ultraviolet finite to any order of perturbation theory. 17
Three other points are worth mentioning. First, the pure gluon selfenergy (6.6) was derived with the gauge-breaking term - (2et) (n • A ) , and by applying the light-cone prescription (5.14) to all spurious factors of the form (q • n) * , /? = 1,2,3, Second, the total gluon self-energy J"]*' (total), Eq. (6.7), contains only gauge-dependent terms and is transverse, in agreement with the Ward identity -1
-
0
2
3
p"n^(
t o t a I
) =° •
(6.17) +
Although fj** (total) appears to diverge as u —* 2 (i.e. e —» 0), the expression is actually harmless since the gauge dependent terms vanish when computed between physical 5-matrix elements. Finally, all momentum integrals were computed by the technique of dimensional regularization, and all massless tadpole integrals, such as 17
were equated to zero. The literature on the application of the light-cone gauge to ordinary Yang-Mills theory and to supersymmetry is fairly extensive. The interested reader may wish to consult, for instance, the articles by Capper, Dulwich and Litvak, Capper and Jones, Capper, Jones and Packman, Amati and Veneziano, Lee and Milgram " Dalbosco, Nyeo, and Smith. Additional references may be found in Refs. 42, 43. 30
31,32
34
40,41
33
35
37
38
39
Application
of the Light-Cone
Gauge to SupcTsymmetry
103
References 1. L. V. Avdeev, O. V. Tarasov and A. A. Vladimirov, Phys. Lett. 96B, 94 (I960). 2. M. T. Grisaru, M. Rocek and W. Siegel, Phys. Rev. Lett. 45, 1063 (1980). 3. W. E . Caswell and D. Zanon, Phys. Lett. B100, 152 (1981). 4. P. S. Howe and K. S. Stelle, J/nf. J. Mod. Phys. 4A, 1871 (1989). 5. A. J. Parkes and P. C. West, Phys. Lett. B138, 99 (1984). 6. A. J. Parkes and P. C. West, Nucl. Phys. B256, 340 (1985). 7. M. F. Sohnius and P. C. West, Phys. Lett. B10O, 245 (1981). 8. P. S. Howe, K. S. Stelle and P. C. West, Phys. Lett. B124, 55 (1983). 9. P. S. Howe and P. C. West, Nucl. Phys. B242, 364 (1986). 10. A. J. Parkes and P. C. West, Phys. Lett. B122, 365 (1983). 11. A. J. Parkes and P. C. West, Nucl. Phys. B222, 269 (1983). 12. A. J . Parkes and P. C. West, Phys. Lett. B127, 353 (1983). 13. P.C. West, Introduction to Supersymmetry and Supergravity (World Scientific Publishing, Singapore, 1986). 14. L. Brink, O. Lindgren and B. E. W. Nilsson, Nucl. Phys. B212, 401 (1983); Phys. Lett. B123, 323 (1983). 15. S. Mandelstam, Nucl. Phys. B213, 149 (1983). 16. J. C. Taylor and H. C. Lee, Phys. Lett. B185, 363 (1987). 17. A. Bassetto and M. Dalbosco, Mod. Phys. Lett. A3, 65 (1988). 18. A. Salam and J. Strathdee, Nucl. Phys. B76, 477 (1974). 19. S. Ferrara, J . Wess and B. Zumino, Phys. Lett. 51B, 239 (1974). 20. S. J. Gates, Jr., M. T. Grisaru, M. Rocek and W. Siegel, Superspace (Benjamin/Cummings, Reading, MA, 1983). 21. D. M. Capper and G. Leibbrandt, Nucl. Phys. B85, 492 (1975). 22. A. Salam and J. Strathdee, Nucl. Phys. B80, 499 (1974). 23. D. M. Capper and G. Leibbrandt, Nucl. Phys. B85, 503 (1975). 24. F. Gliozzi, D. Olive and J. Scherk, Phys. Lett. B65, 282 (1976). 25. F. Gliozzi, D. Olive and J. Scherk, Nucl. Phys. B122, 253 (1977). 26. M. A. Namazie, A. Salam and J. Strathdee, Phys. Rev. D28, 1481 (1983). 27. G. Leibbrandt and T. Matsuki, Phys. Rev. D31, 934 (1985). 28. G. Leibbrandt, Phys. Rev. D29, 1699 (1984). 29. G. Leibbrandt and S.-L. Nyeo, Phys. Lett. B140, 417 (1984). 30. D. M. Capper, J. J . Dulwich and M. J. Litvak, Nucl Phys. B241, 463 (1984). 31. D. M. Capper and D. R. T. Jones, Phys. Rev. D31, 3295 (1985). 32. D. M. Capper and D. R. T. Jones, Nucl. Phys. B252, 718 (1985). 33. D. M. Capper, D. R. T. Jones and M. N. Packman, Nucl. Phys. B263, 173 (1986). 34. D. Amati and G. Veneziano, Phys. Lett. B157, 32 (1985). 35. H. C. Lee and M. S. Milgram, Phys. Rev. Lett. 55, 2122 (1985). 36. H. C. Lee and M. S. Milgram, Z. Phys. C28, 579 (1985). 37. H. C. Lee and M. S. Milgram, Nucl. Phys. B268, 543 (1986).
104
38. 39. 40. 41. 42. 43.
Noncovariant
Gauges
M. Dalbosco, Phys. Lett. B163, 181 (1985). S.-L. Nyeo, Nucl. Phys. B273, 195 (1986). A. Smith, Nucl. Phys. B261, 285 (1985). A. Smith, Nucl. Phys. B267, 277 (1986). G. Leibbrandt, Rev. Mod. Phys. 59, 1067 (1987). A. Bassetto, G. Nardelli and R. Soldati, Yang-Mills Theories in Algebraic Non-Covariant Gauges (World Scientific, Singapore, 1991).
CHAPTER 7 R E N O R M A LIZ ATIO N I N T H E P R E S E N C E OF NONLOCAL T E R M S 7.1. Introduction
One of the by-products of the unified-gauge prescription (5.34) is the appearance of nonlocal expressions in certain loop integrations. As we have seen in Sees. 4.2 and 5.3.3, these nonlocal terms arise whenever the Feynman integrand contains two or more noncovariant factors such as *"* 1 • n(g - p) • n
qn(q
e t c
.
-p)n(q-k)n
p,,, k,, being external momenta. Application of the separation formula 1 q n(q-p)
= n
7 ? p • n \{q - p) • n
*)> q n j
P-»*0, - 1
is then seen to yield nonlocal terms proportional to (p • n ) , and the question is how or to what extent these nonlocal terms are likely to affect the renormalization program. As in the case of covariant gauges, the answer to this question depends largely on the gauge used. Concerning the renormalization of Yang-Mills theory, there has been considerable success in the light-cone gauge, but only limited progress for the temporal and axial gauges. For this reason we shall concentrate on the light-cone gauge, emphasizing in particular the construction of nonlocal, but BRS-invariant, counterterms to one-loop order. A different approach, advocated by Bassetto, Soldati and their coworkers goes beyond the one-loop level and is described in detail in Ref. t. Of course, there have been numerous other advances over the years in this field, beginning with the extension of the BRS formalism for covariant gauges by Kluberg-Stern and Zuber, ' and Piguet and Sibold, 2 3
105
4
106
tfoneovariant
Gauge*
and the subsequent generalization of this extension to noncovariant gauges by Gaigg, Piguet, Rebhan and Schweda. Equally important have been the contributions by Bagan and Martin, " Hiiffel, LandshotTand Taylor, and by Gaigg, Kreuzer and Pollak on the re normalization in axial-type gauges. 5
6
8
9
10
7.2. R e normalization in the Light-Cone Gauge
In this section we examine the renormalization of pure Yang-Mills theory in the light-cone gauge. Working to one loop we shall first derive the counterterm action, then the appropriate renormalization constants for Green functions. The discussion is complicated by the presence of divergent nonlocal terms. 7.2.1. £17.5 transformations
and the ^normalization
equation
The BRS-invariant Yang-Mills Lagrangian density in the light-cone gauge n"A°(x) = 0, n = 0, reads (no distinction is made between upper and lower indices): 2
2
L' = L - -^-{n • A") ,
o —• 0,
a = gauge parameter,
(7.1a)
11
where
F% = d„Al - M J + af^A^K ah
ahc
,
c
= 6 d +gf A ; ll
(7.1b)
ll
hc
g is the gauge coupling constant, f are the group structure constants; w°, u> denote ghost fields, and J', K are external BRS sources. The action S = Jd*xL' is invariant under the following Becchi-Rouet-Stora (BRS) transformation: a
a
12-13
6AI = \D?u>
»* = A being an anticommuting constant.
h
,
.
(7.2)
Rcnormalization
in the Presence
of Nonlocal Terms
107
The first priority is to obtain the renormalization equation for the divergent part of the one-loop generating functional for one-particleirreducible (1PI) vertices, and then to solve this equation for the counterterms. The derivation of the renormalization equation involves the following basic steps: 0
r
a
(a) Introduction of external sources j R , € , € " f ° the fields A^,w",Q , respectively, and construction of the generating functional Z for the complete Green functions:
a
a
= j . DA Dw°Dw exp^i
j cPxL' + i j a S f i ^ . + f V + f l * * * j j
M
a
a
a
~exp{iW\j ^ ,^;J^K }}
,
(7.3)
where W is the generating functional for connected Green functions. (b) Definition of a new generating functional T for lPI-Green functions in terms of the Legendre transform of W with respect to the sources
a
a
r[Al,u ,Q°;JZ,K ] =
Wtii,t',t';JZ,K°)
- (
rf*ifj-(iM-Kr)+r(rK(*)
+ «'(*)r(x)] •
7
(7-4)
(c) Derivation of a set of dual relations from Eqs. (7.3) and (7.4) such as:
Sj-(x)
a
6t (x)
" =
SA%{x) T ^ = ?(*), 6u'(x)
' "
etc.
(7.5)
(d) Replacement of T by the modified effective action T, i
2
f = r+±Jd x(n>-Al)
leading to the Slavnov-Taylor identity
,
(7.6)
Noncovariant
108
J
sr
6T
Gaugct
sr
sr
+
(7.7a)
= 0
Eq. (7.7a) is constrained by the ghost equation a
S
"*«;(*)
t
te-(«)
(7.7b)
= 0.
Finally, (e) Expansion of T in powers of fi = 1, 1
s
f = fW + f< >+f< > + . . . ,
(7.8)
where (divergent = div)
The one-loop divergent contribution fjj-; = —Z) then satisfies ffte renormalization equation
(7.9)
-o-D = 0 ,
where o~ denotes the nilpotent BRS operator 6S S 6Afr)6JZ( )
+
x
SS
12-13
6S S «/-(«) AlM»rd )- n Al](n dx)- n L Ti } 6
T
v
x
p
p
a
.
(7.22)
Traditional renormalization demands that the counterterm AS be of the same functional form as S. Yet, comparing Eq. (7.22) with the original action in Eq. (7.18) we see that AS differs appreciably in structure from S : not only does AS contain nonlocal expressions, but it also has terms with five A's, such as the fourth term in Eq. (7.22) which is proportional to (—2a ). By contrast, S contains only local terms and at most four A's, so that AS cannot be absorbed into S in its present form. Does this mean that massless Yang-Mills theory is unrenormalizable in the light-cone gauge? Certainly not, as can be seen by exploiting two specific properties of the light-cone gauge. 6
The argument goes as follows. According to Eq. (7.22), every" nonlocal term is proportional either to n^A^x) or n„L°(x), where L {x) = Jp{z) + u"n . Consider first the terms proportional to n„Lp(x). It was shown in Ref. 21 that the vertices connected with n • L" lead to vanishing ghost diagrams so that the last three expressions in Eq. (7.22) drop out. The nonlocal terms containing n„A°(x) also vanish but for a different reason: this time it is the expectation valves like p
u
{0|r[n,AJ(z)At(y)]|0) that go to zero (a -* 0) by virtue of the gauge constraint n A^(x) = 0. In summary, all nonlocal expressions in Eq. (7.22) drop out so that the u
Renormalization
117
in He Presence of Nonlocal Term*
counterterm action AS reduces to 3S_
2AS =
'6A°(x)
-ai9
6K'(x)
(7.23) Since the functional structures of AS and S now coincide, we may add Eqs. (7.23) and (7.18) to get 2(5 + AS) = (j„„ + a , „ + 2 f l n > „ ) l 9 (
/dxA°{x)
3
SS SAXx)
a
(ff„v + 2 a n „ n ; )
fdxL {x)
3
v
Ui{x) SS
6K'(x)
- ( l + oi)ff dg
(7.24)
' 5S
+ 5
55
+ (7.25) where 2
*it =
9i>i/{^ +
a
i ) + 2030*0,, =
= g„v + ^3^"'
2l = 1 + f»l •
u
zig^v + 2a3«*n , (i
.
(7.26)
The renormalized quantities "P are then related to the bare quantities by
118
jVoncov*riant Gauges
— Ziii/A^x)
a
u W(x)
= «'(*), =
a
K ^(x) 9
z\'*K*{x) -1/2
m
= *i
(7.27)
9;
2 and z „, the renormalization constants for Green functions, are defined respectively as: M t
u
22
12
l
Zp» = z\
[fa - (1 - {zi)~ )n^n' /n v
-ri*] ,
Iw- = 9^ - (1 - (*i) ' K n j n • "* , = 1 - a n • n' ,
(7.28)
3
or, utilizing Eq. (7.17), as 1/2
2«
h = 1 + 2K,
Zi = 1 + y K .
(7.29)
In conclusion we state the counterterm in the light-cone gauge:
• counterterm
=
n,D F ¥
• flt'Wy^^^
,
1,20,23
"
26
(7-30)
where Dj, and F'"' denote, respectively, the covariant derivative and field strength tensor (with gauge indices omitted). The nonlocal operator (n • D)" may be represented formally as an infinite series in the nonlocal operator (n • 3 ) : 1
- 1
1
1
n •D
n d
1
+ •n-d
n • A,
n-d
+ ...
(7.31)
119
Re norm a liza tio n in the Presence of Nonlocal Terms
Notice that in the expression for the counterterm in Eq. (7.30), the gauge field A only appears implicitly through the symbols D and F"". The above analysis completes our discussion of the renormalization of Yang-Mills theory in the light-cone gauge to one loop. We emphasize once again the dual role played by nonlocal quantities. Although nonlocal terms do not contribute to Green functions, they do generate factors with external n^'s and also contribute to higher-order vertex functions. Fortunately, however, nonlocal terms do not generate higher-order gauge independent quantities. 2S
U
V
22
References 1. A. Bassetto, G. Nardelli and R, Soldati, Yang-Mills Theories in Algebraic Non-Covariant Gauges (World Scientific, Singapore, 1991). 2. H. Kluberg-Stern and J.-B. Zuber, Phys. Rev. D l 2 , 467 (1975). 3. H. Kluberg-Stern and J.-B. Zuber, Phys. Rev. D l 2 , 482 (1975). 4. O. Piguet and K. Sibold, Nucl. Phys. B248, 301 (1984). 5. P. Gaigg, O. Piguet, A. Rebhan and M. Schweda, Phys. Lett. B17S, 53 (1986). 6. E. Bagan and C. P. Martin, Phys. Lett. B223, 187 (1989). 7. E. Bagan and C. P. Martin, Int. J. Mod. Phys. A5, 867 (1990). 8. E. Bagan and C. P. Martin, Nucl. Phys. B341, 419 (1990). 9. H. Huffel, P. V. Landshoff and J. C. Taylor, Phys. Lett. B217, 147 (1989). 10. P. Gaigg, M. Kreuzer and G. Pollak, Phys. Rev. D38, 2559 (1988). 11. G. Leibbrandt and S.-L. Nyeo, Z. Phys. C30, 501 (1986). 12. C. Becchi, A. Rouet and R. Stora, Phys. Lett. 52B, 344 (1974); Comm. Math. Phys. 42, 127 (1975). 13. C. Becchi, A. Rouet and R. Stora, Ann. Phys. (N.Y) 98, 287 (1976). 14. J. Schwinger, Phys. Rev. 82, 914 (1951). 15. J. Schwinger, Phys. Rev. 91, 713 (1953). 16. J. H. Lowenstein, Comm. Math. Phys. 24, 1 (1971). 17. Y. M. P. Lam, Phys. Rev. D6, 2145 (1972). 18. Y. M. P. Lam, Phys. Rev. D7, 2943 (1973). 19. O. Piguet and A. Rouet, Phys. Rep. C76, 1 (1981). 20. G. Leibbrandt and S.-L. Nyeo, Nucl. Phys. B276, 459 (1986). 21. A. Andrasi G. Leibbrandt and S.-L. Nyeo, Nucl. Phys. B276, 445 (1986). 22. S.-L. Nyeo, Phys. Rev. D34, 3842 (1986). 23. A. Andrasi and J. C. Taylor, Nucl. Phys. B302, 123 (1988). 24. M. Dalbosco, Phys. Lett. B163, 181 (1985). 25. A. Bassetto, M. Dalbosco and R. Soldati, Phys. Rev. D36, 3138 (1987). 26. H. Skarke and P. Gaigg, Phys. Rev. D38, 3205 (1988).
CHAPTER 8 COUNTERTERMS IN T H EPLANAR
GAUGE
8.1. Introduction In the preceding section we renormalized the Yang-Mills action to one loop in the powerful light-cone gauge. Renormalization was achieved in the Becchi-Rouet-Stora (BRS) formalism and despite the appearance of nonlocal terms in the self-energy and vertex functions. The purpose of this section is to demonstrate the treatment of nonlocal terms in the unifying prescription Eq. (5.34) for the fashionable planar gauge. The latter differs from the light-cone gauge on two important counts, (i) In the planar gauge, the self-energy is no longer transverse and (ii) the gauge-breaking part of the Lagrangian density is more complicated than in the light-cone gauge. Yang-Mills theory in the planar gauge had originally been analyzed with the principal-value (PV) prescription and found to be non-mvltiphcaltvely renormalizable. 1-3
Here we work with the unified-gauge prescription, Eq. (5.34), and use the BRS-invariant Lagrangian density a
2
2
V = L + — n • A id /n )n
• A",
a = -1 ,
(8.1)
where a is the gauge parameter and L is defined in Eq. (7.1b). The propagator is given by Eq. (5.10) but with (1 + o ) n in the third term, and the three-gluon vertex by Eq. (5.3). Application of these Feynman rules and of the general prescription (cf. Eq. (5,34)) 2
-L
q•n
= lim( s—
" n' .) , o \q • nq • n* + i f /
c>0,
leads to the following answer for the self-energy to one loop: 121
(8,2) 4,5
122
n
Noncovariant
Gangci
p 22
1 • col
2
(n • F }
P
2
n (n • F )
P 2
n
^ v ^ ^
- 0
Fig. 8.2. Yang-Mills Ward identity in the planar gauge. 1
The non-vanishing of p* n^'™" may be traced back to an additional Feynman diagram in the Ward identity, called a pincer diagram (see Figs. 8.1, 8.2). Explicitly, 3
^lC>=^
^ ^-
e W
N
(8-4)
tc
tc
where i?° (p) is the amputated one-loop contribution to tv"/ (p), e
a
abe
W/* (p) = G^ ( )E ( ) P
,
P
(8.5a)
namely 4 tabc gn*f
P-F 2
2
2(n • F ) (2T) -
n • Fp n
2
2 F„ + P pF 2
P
n-Fpn
p- n I,
n„ — 3p • Fp
2
u
I = H T(2-u)
.
(8.5b)
8.2. Counterterm Action
We must now match both the local and nonlocal divergent terms in ffif*"" with a finite number of BRS-invariant counterterms. Proceeding in the spirit of Chapter 7, we write the solution of the renormalization equation CTD = 0 as (cf. Eq. (7.11)) -D
= AS = Y + rrX ,
(8.6)
where D EE f|JJ is the one-loop divergent part of the effective action f in Eq. (7.6), A S denotes the counterterm action and c = 0. As before, the gauge-invariant functional Y, 2
124
JVoncDvon'on* Gauges
Y = - | j dxa^f
,
M
may only depend on local expressions, so that any nonhcaliiy must reside in the functional X; the constant a has to be determined from explicit calculations. The trick is to make a judicious ansatz for X, so that the local and nonlocal components of trX are equal in magnitude, but of opposite sign to the corresponding terms in n?*""> 1 - ( ' ) ' appropriate choice for X turns out to be: 0
E
8
3
T
h
e
4
X = -Ylocal + ^nonlocal i
(8-8)
where X ^
a
a
= J dx[a,A
a
• L" + a n - A n • L" + a n • A F • L 2
a
a
+ a F - A n • L" + a F • A"F • L + anu'K'] A
X„
o n l o c a l
d)- A* )Ll
e
Q
u
_ 1
a
a
1
a
n • A ]n • L + a [F • 8(n • 3 ) " rf -A ]n • L"
7
s
l
+ ag[F • d(n • d)~ n + a [F
(8.8a)
l
= / dx[a [F • d(n •
l0
j
5
+ a [F - d(n • )
a
3
a
• d(n • )
_
1
a
• A ]F • L
a
a
F • A"}F • L ] ;
(8.8b)
a
LJJ = Jp+n w , and F,, is the noncovariant null vector defined in Eq. (5.50). The ghost fields ( w , w ° ) and external BRS sources {J ,K ) satisfy the relations tl
u
a
K , / Y
6
]
=
O,
a
t
[ j ; , / c ] = o.
Using Eqs. (7.14d), (8.1) and (8.8), together with S = / dzL, we can compute the eight factors needed in cX, . SS SX
Thus,
SS SX
SS SX
SS
6X1
(8.9)
Countertermt
in tlie Planar Gauge
125
SS
6S SASS Su
i
a
r
9
i
e
V ; (8.10a)
W
similarly, SX Su
SX
a
aK n
a
SX 777 = aiAl + a n • A n a
2
a
+ a n • A"F„ + a*F • A'n„ + a F • A F^
v
3
%
5
l
+ a F • 3(n • d)- A*
+ a F • 9(n . 0 ) - n • A*n„
6
7
l
a
l
+ a F - d{n - d)~ F • A n „ + a F • d{n • d)~ n • A"F 8
-+a F 10
7— -
a i
L
a
9
l
•d(n d)- F-A"F
l
2
a
+ a f • d(n • d)- L
7
l
• L'n,,
a
l
a
• L F„ + a F • d(n • d)~ F • L n„
8
l0
4
+ a F • 3(n • d^n
u
+ a F • d(n • d)- n + aF
a
+ a F - L n „ + a n • L"F„ + a$F • L F„
a
6
,
ll
a
+ a n • L'n 2
li
9
l
a
• d(n • d)~ F • L F„
.
(8.10b)
Substituting Eqs. (8.10) into Eq. (8.9) and then adding aX to the counterterm AS (terms of order hD are omitted), we obtain
- n n • A'FHD?#*,
- aF
3
A
l
h
b
- a [F • d(n • d)- Al]D: F „ 6
a
a
b
A n D „ F^ u
- aF • 5
- a [F • d(n • dy'n 7
a
A F D?F*„ u
•A ^ i U }
126
Nonce-variant l
- a [F • d(n • d)~ F
•
A ^
- a [ F - 9(-i • 9 } - " F • 8
F
^
a
2
a-, —— [p • n ( p n „ + p „ n ^ ) - 2 p n „ n J p-n M
A ]n„DfF
b u
ag ——\p -n(PiiF + p„F„) u
P-n
J
- p ( n F „ + n ,F ,)] M
- a [ F • S(n • 9)-*n • A*lF»Dtr*% 9
as
l
(
[p • F(p Tt!, + p „ n „ ) u
p- n 2
-p (n^F -a
1 0
[ F - 8(n • B )
_ 1
F • ^ J F ^ D ^ ^
+ n ,F )]
t
1
fi
2
"10 — [p • F ( ^ F „ + p„F^) - 2p F „ F „ ] . p-n P
(8.13) Although (A5) host does not contribute terms quadratic in — (AS) host contains only single A's—ghosts are necessary nevertheless for a consistent determination of some of the divergent constants. By matching the eight expressions in YJffp**' Eq. (8.3), with the corresponding terms on the RHS of Eq. (8.11), and using ( A S ) h for consistency, we obtain the following unique values for a, : g
g
g
o s t
4
ax = a = 2
03
= an = 0 ,
(8.14a)
Noncovariant
128
Ganges
as well as a =jK, 0
"4 =
ai0 =
-O7 =
K
2
=
2
1
ff (4 r)- C (2- )1
yjlf
W
,
~K , n •r
K
{8
-(>r7r -
-
14b)
This completes our treatment of the planar-gauge counterterms in the presence of nonlocal terms. The derivation was achieved in the framework of the BRS-formal ism, where ghosts play an essential role, and with the aid of a unifying prescription for (q • n ) , Eq. (5.34). The general expression for the nonlocal counterterm(s) will again contain the by now familiar factor (n • D ) , reminiscent of our discussion on the light-cone gauge in Sec. 7.4. In principle, an infinite number of n-point functions is required to fix infinitely many nonlocal terms which, however, may be summed as 1/n • D7 _ 1
- 1
References 1. A. L MiTshtein and V. S. Fadin, Yad. Fiz. 34, 1403 (1981); Sov. J. Nucl. Phys. 34, 779 (1981). 2. A. Andrasi and J. C. Taylor, iVucf. Phys. B192, 283 (1981). 3. D. M. Capper and G. Leibbrandt, Phys. Lett. B104, 158 (1981). 4. G. Leibbrandt and S.-L. Nyeo, Mod. Phys. Lett. A3, 1085 (1988). 5. G. Nardelli and R. Soldati, Phys. Lett. B206, 495 (1988). 6. S.-L. Nyeo, unpublished lecture notes (Univ. of Guelph, 1988). 7. S.-L. Nyeo, Private communication, 1992.
CHAPTER 9 THE COULOMB
GAUGE
9.1. Introduction During the past dozen years much effort has been devoted to solving one of field theory's truly annoying problems: how to quantize non-Abelian gauge theories in the ghost-free Coulomb gauge in a mathematically rigorous fashion. Our present goal is to draw the reader's attention to typical quantization problems in the Coulomb gauge, as well as highlight some recent technical advances. The Coulomb gauge, defined by V A
= 0 ,
(9.1)
is a physical, or ghost-free, gauge which first appeared on the scene in the 1930's and has since been amazingly effective in Abelian computations. However, its success rate in non-Abelian models such as Yang-Mills theory, where V • A" = 0,
a = internal symmetry label,
(9.2)
is far from impressive, and there is no denying that the Coulomb gauge (also called radiation gauge) continues to be plagued by serious difficulties. For instance, there exist no consistent rules for quantizing and renormalizing non-Abelian theories in that gauge. 1
9.2. E a r l y Treatments The purpose of this subsection and the next is to review some of the more noteworthy developments in the treatment of the radiation gauge. The latter has proven most useful in quantizing Abelian models such as Maxwell's theory, as reflected by the large number of practical applications. By contrast, only a small fraction of the papers examines the thornier issues 2,3
129
Nonce-variant Gauges
130
of this baffling gauge. One of the earliest critiques of the Coulomb gauge is due to Schwinger who discussed a relati vis tic ally invariant formulation of a non-Abelian vector field coupled to a spin-1/2 Fermi field. Schwinger managed to show that the associated quantum Hamiltonian differs from the classical Hamiltonian by an instantaneous Coulomb interaction term, later called Vi by Christ and Lee. In 1971, Mohapatra, working in the context of canonical quantization, succeeded in deriving covariant Feynman rules for a massless Yang-Mills field in the physical radiation gauge. He verified that the noncovariant terms, generated by the gauge condition (9.2), drop out to all orders in g for tree diagrams, and to order g for one-loop diagrams. Moreover, he emphasized that the so-called Vi-term of Schwinger and of Christ and Lee is essential if quantization in the radiation gauge is to be consistent with Lorentz invariance. Towards the end of the decade the Coulomb gauge came under further scrutiny by Grihov, Singer and Mandelstam in the context of "Gribov copies", and by Jackiw, Muzinkh and Rebbi. The latter authors analyzed the behaviour of the gauge for large Yang-Millsfieldsand noted that it remained ambiguous even after imposition of an additional constraint, 4
5
6
2
4
7-10
5
11
fl
lim(rA ) = 0 .
(9.3)
r—ctj
Yet, despite some glaring deficiencies, the Coulomb gauge has proven superior to covariant gauges in at least one significant respect, namely in the treatment of static problems in both QED and QCD. Muzinich and Paige, for instance, used the radiation gauge to justify the Okubo-ZweigIizuka rule dealing with the decay of very heavy quark-antiquark states, while Sapirstein employed it to gain a sharper understanding of the ground-state hyperfine splitting in hydrogenic atoms. Further progress was achieved by Christ and Lee in the framework of Yang-Mills theory. They employed Weyl-ordering to deduce the correct operator ordering for the associated Hamiltonian density, and then converted this canonical system to path-integral Lagrangian form. The proper Weylordered Hamiltonian in the Coulomb gauge now led to a Lagrangian density containing additional, nonlocal interactions. Christ and Lee labeled these new interaction terms (Vi + V ), and stressed their significance in attaining the appropriate Feynman rules. While the Vi -contribution had already been scrutinized by Schwinger, the expression for V% was definitely new. We should mention that the operator-ordering problem in the radiation 12
13,14
5
2
4
5
13!
The Coulomb Gauge 5
15
gauge is also discussed in a paper by Utiyama and Sakamoto, but their approach differs somewhat from Christ and Lee's. 9.3. One-Loop Applications in Q E D The role of the Coulomb gauge in quantum electrodynamics is far less problematic than in non-Abelian models, as underscored by a host of applications to static problems. For instance, in the case of bound-state problems, separation of the binding interactions from the perturbing interactions is more easily achieved in the Coulomb gauge than in any of the covariant gauges. Below we shall illustrate some of the more appealing characteristics of the radiation gauge by referring to the work of A d k i n s , Sapirstein and Heckathorn. 16
17,18
13,14
18
Motivated by the absence of an explicit construction of a renormalized theory of Q E D in the Coulomb gauge, Heckathorn re-examined the issue in 1979, evaluating all noncovariant-gauge Feynman integrals by the technique of dimensional regularization. We shall use Heckathorn's notation to pinpoint the troublesome spurious singularities and to highlight similarities between the Coulomb gauge and axial-type gauges. Heckathorn considered the traditional Q E D Lagrangian density 16
L = *(x)(t? - m ) * 0 ) - ±F (x)F'"'{x)
+ e*(xy,"*(x)A (x)
liV
F„{x)
= d„A (x)
- dyA^x),
u
u
? = fPfy ,
, (9.4)
together with the following gauge-fixing part, i
2
f i x
= - — [d^A^x) + rj^A^x)}
.
(9.5)
The vector l}p (which is not defined in Heckathorn's article) is reminiscent of the noncovariant vector n in the definition of the axial gauge constraint n-A = 0. From Eqs. (9.4) and (9.5) and keeping a ^ 0, Heckathorn derives the bare propagator M
D^{q,a^G) r
g
2 _
r
9f9v + i • ){q.f )'> + gov?) .
—i U
fM"
q' + (q-n)
3
+
q " [?
2
We
+ s , ; uo
(10.3) D is the dimension of complex space-time, while the comma in g^" ,p denotes covariant differentiation. In four dimensions, the Lagrangian density (10.3) reduces to Goldberg's version and is clearly free of poles, whereas in two dimensions Eq. (10,3) possesses a simple pole. By lowering the 2
135
Noncovariant
136
Gaugei
dimensionality from four to two, we seem to have altered the character of the theory in a nontrivial way. Our second illustration is taken from the theory of nonlinear secondorder partial differential equations which can be notoriously difficult to solve. Consider, for instance, the ubiquitous sine-Gordon e q u a t i o n both in 1 + 1 dimensions, 3-8
(£-^)*^=^*^*>'
(io
-
4)
and in 2 + 1 dimensions,
Here * is a massless scalar field, x, y are spatial coordinates and t is the time variable (ft = c = 1). I t is common knowledge that there exists a Backlund transformation that leads to exact solutions of Eq. (10.4). But in 2 + 1 dimensions, no genuinely three-dimensional Backlund system is available as yet and, hence, neither are exact multi-soliton s o l u t i o n s . I n this case, the increase in dimensionality from two to three has effectively prevented us from finding meaningful solutions of the sine- Gordon equation (10.5). In summary, a change in the number of dimensions should not be taken lightly. 9-12
13,14
But let us return to the task at hand, namely the analysis of perturbative Chern-Simons theory in the light-cone gauge. We shall find that the tools and methodology developed in perturbative four-dimensional YangMills theory work equally well for the topological SU(N) Chern-Simons model on IR3. A hint of the topological content of the Chern-Simons model on a given manifold comes from the fact that its classical action is the integral over the manifold of the Chern-Simons three-form, the latter having been introduced by S.-S. Chern and J. Simons in 1974 in a paper entitled "Characteristic forms and geometric invariants". Abelian Chern-Simons theory was proposed by A. S. Schwarz to give a Feynman path-integral definition of the topological invariant of oriented three-dimensional manifolds known as the Ray-Singer torsion of the manifold. 15
16,17
18
In vide an and its loops.
1989, Witten introduced non-Abelian Chern-Simons theory to prointrinsically three-dimensional definition of the Jones polynomial generalizations as framed vacuum expectation values of Wilson W i t t e n also showed that the model was exactly soluble and 19
20
21
Chem-Simoni
Theory
137
could be used to give a three-dimensional explanation of two-dimensional conformal field theories. This seminal work was subsequently studied by many authors " who quantized theories on a manifold of the type E®R, by using the Hamiltonian formalism in a non-perturbative setting; here E is a compact two-dimensional Riemann manifold. In this context the temporal gauge seemed a good starting point. A non-perturbative quantization of Chern-Simons theory on an arbitrary oriented three-dimensional manifold without boundary was carried out by the authors of Ref. 28. Frohlich and King, on the other hand, set up a non-perturbative quantization framework of Chern-Simons theory in the light-cone gauge. Further work on the connection between the Chern-Simons model and link invariants was carried out by Cotta-Ramusino, Guadagnini, Martellini and Mintchev For a rigorous study of the quantum states of SU(2) Chern-Simons theory on E ® R, E being a genus-zero compact Riemann surface without boundary, the reader should consult Ref. 33. 22
27
29
3 0 - 3 2
Present-day interest in the Chern-Simons model presumably stems from the fact that this topological gauge theory is both UV- and IR-finite and possesses amazing connections with both two-dimensional conformal field theory and knot theory. But there are other reasons for its popularity: for instance, there is the fact that the Chern-Simons action provides a topological mass term for Yang-Mills theories, and the remarkable phenomenon of fractional spin and statistics that occurs in three-dimensional models with matter coupled to gauge fields. Such models with matter fields might help explain the behaviour of some of the degrees of freedom which are involved in high-temperature superconductivity (see Ref, 38 and references therein). 34-37
Since many of the exact non-perturbative results of Chern-Simons theory were derived from path integrals which are known to be mathematically ill defined, a perturbative derivation of some of these properties is highly desirable, if not essential. Of course, the ultimate goal is to obtain a series expansion in the observables of the theory (e.g. Wilson loops and the partition function) and thereby arrive at a perturbative definition of the Jones polynomial and of Witten's invariant of the manifold. 39
4D
There exist numerous articles on perturbative SU(N) Chern-Simons theory. Most of these deal with the computation of the effective action, and only very few analyze the lower-order terms of the perturbative expansion of the Wilson loop. Analysis of these lower-order terms has led 41-51
31,52-54
Noncovariant Gauges
138
to new relationships among the coefficients of the Jones polynomial and its generalization. While the majority of researchers preferred to employ a covariant gauge such as the Landau gauge, only a tiny fraction of the authors considered the perturbative Chern-Simons model in the context of noncovariant gauges. Emery and Piguet, for example, examined the relationship between SU(N) Chern-Simons theory and two-dimensional SU(N) current algebra. Loop calculations in an axial-type gauge appear to have been first carried out by Martin. The latter evaluated the complete perturbative effective action for a particular class of UV regulators. The fact that this result has as yet not been duplicated in a Lorentz covariant gauge is clear proof of the power of axial-type gauges. For an excellent review on topological gauge theories the reader is referred to. 52
55,56
57
58
59
10.2. Action and F e y n m a n Rules in the L i g h t - C o n e Gauge The classical SU(N) Chern-Simons action reads
)
(10.6)
where A£ is the gauge field over ffi with Minkowski metric, g is the dimension less coupling constant and / are the real, totally antisymmetric structure constants of SU(N). The metric independence of Eq. (10.6) implies, at least formally, that we are dealing here with a topological gauge field theory. In the light-cone gauge, defined by n • A = 0, n = 0, the Chern-Simons action assumes the form (we drop Stg on the integral sign) 3
o t c
a
2
(10.7) where L = ^
(±Ald,Al
!/ «**
+
A"
a -* 0
with aic
D'* = 6"% + gf A%
;
A'
AC
)
Chem-Simona
Theory
139
a
w , u" are ghost, anti-ghost fields, respectively, and a denotes the gauge parameter; u,p,v ... are Lorentz indices, and a,b,c... SU(N) gauge indices. The presence of Lf\„ and L hosi implies that the action 5 is no longer metric independent. Before proceeding with the calculation of the vacuum polarization tensor, we observe that the light-cone condition n • A" = 0, n — 0, somehow neutralizes the interaction term in L , Eq. (10.7). To see this consider the three-dimensional vectors x^ = (z°, r , x ) = (x ,%~ ,x ), t
g
2
1
2
+
i f = (y°,y\y )
1
= {y ,y~,y l
2
+
and A„ = M o , i , , ^ ) =
l
{A+,A-M
where ±
x
I EI'/\/2.
2
= (x°±x )/V2, +
= 2(x x~ +
x • y = x y~ a
transverse,
2
- x ) , T
+ x~y
+
1
1
- xy
+
- x y~
A = (A ±A )l^ together with the null vector n* , ±
T=
T
+
+ x~ y
- 2x y T
T
,
A =AilJi..
2
T
1
1
2
+
n" = ( n ° , n , n ) = (1,0,1),
1
i.e. n" = ( n , n~, n ) = (^2,0, 0) . (10.8)
Since n M „ = A_ = 0 ,
(10.9)
the interaction term in Eq. (10.6) vanishes, because (we take n,p,v ahc
a
h
gf €>""'A A Al ll
= o/
(l
a t c
+
4
e -M;A _^ = 0 .
=
(10.10)
The fact that the Chern-Simons action collapses to a Gaussian action might, therefore, seem to "explain" the remarkable simplifications induced by the light-cone gauge (10.9), But whatever the reason, or reasons, for these simplifications, it is essential to treat the interaction term as being nonzero, at least initially, since premature implementation of the gauge condition is apt to yield ambiguous results. 60
Our immediate task is to obtain from Eq. (10.7) a set of consistent Feynman rules. Unfortunately, this is not possible since the corresponding Feynman diagrams are generally not UV-convergent. Indeed, the twoand three-point functions develop linear and logarithmic divergences, respectively, when the loop momenta approach infinity simultaneously. The
Noncovariant
140
Gauges
appearance of TJV infinities in quantum field theory requires introduction of an intermediate regularization scheme prior to renormalization of the theory. The traditional choice of regularization is dimensional regularization, since i t preserves at least formally the structure of the action, BO that the regularized Green functions have the same appearance as the MIIregularized ones. However, due to the presence of the (Wr*-tensor, we cannot apply dimensional regularization to Eq. (10.7), i f we want to preserve gauge invariance explicitly and if the D-dimensional counterpart of tW is to satisfy a set of algebraically consistent equations. The crux of the problem is that the D-dimensional version of Eq. (10.7) does not have an inveriible kinetic term, so that perturbation theory does not exist (the D-dimensional ^•"•-tensor is defined in Eq. (10.17)). For a detailed discussion of this issue the interested reader may wish to c o n s u l t . ' To circumvent the invertibility problem of the kinetic term and still preserve BRS invariance explicitly, we shall adopt the procedure outlined in Refs. 44, 61, 51: we shall simply add a D-dimensional Yang-Mills term 5 Y M 44,61
5
V M
= ~
f
D
d
=
, _.(2)ai
ab
ic H v9
n
P "
n•n
n
J
(10.39)
It is worth noting that the term proportional to p ° e , matches the result in the covariant Landau gauge, d^A^x) = 0 . The presence of the nonlocal term proportional to p • n"/p • n, on the other hand, is a firm reminder that we are working in a noncovariant gauge. The nonlocality is, in fact, necessary if Hin/* ' transverse and obey the Ward identity: a j l l
42
s
t o
r
e
m
a
i
n
Noncovariant
148
Gauges
In conclusion, calculation of the vacuum polarization tensor \ \ ° " in Eq. (10.39) confirms (a) the finiteness of the three-dimensional ChernSimons model; (b) the validity of the light-cone prescription (5.14), or (10.15); and (c) the preservation of gauge invariance of our hybrid regularization, consisting of dimensional regularization and a Yang-Mills action proportional to m~ Jd x(F*„) . u
l
3
s
10.5. Treatment of Nonlocal T e r m s The persistence of nonlocal terms in the light-cone gauge makes it advisable to examine the vacuum polarization tensor in the context of BRS-theory. Specifically, one would like to know whether the nonlocal expression, proportional to p • n*/p • n in Eq. (10.39), can he matched unambiguously by counterterms using BRS-techniques, and whether the shift of the ChernSimons parameter k, k —° appearing necessarily in the combination pl = J^ + n Q li
a
,
(10.46a)
and $*{A) having mass dimension unity. Since the functional X obeys ffX = Q,
(10.46b)
Eq. (10.44) reduces to ri-taop = cS , + j JtjjttyiA)
.
c
(10.47)
We are now faced with the task of finding an appropriate ansatz for that will match, in particular, the nonlocal structure of the vacuum polarization tensor, Eq. (10.39). In the light-cone gauge, the proper choice for is 61
n,
n-D
at
ab
c
= d^n - 8 + gf 'n
•A ,
(10.48) - 1
the nonlocality being clearly displayed by the inverse factors (n • D " * ) . Comparison of the quadratic parts in Eqs. (10.48) and (10.39) yields the unique values 1
4 = Z-W so that
,
c = -i-c„ 2
2 f f
,
(10.49)
becomes
Finally, substituting Eq. (10.47) into Eq. (10.40), we arrive at the one-loop effective action,
150
Noncovariant
V(A)=(^-y (A)
+
ci
Gangtt
y*«^«2(A) + 0(ft»),
(10.51)
where k is the bare Chern-Simons parameter. Much has been made in recent years of the shift in k. Working perturbatively in a covariant gauge, researchers succeeded in showing that the one-loop radiative corrections manifested themselves merely as a shift in fc, and that the renormalized effective action was just the tree level action, with an appropriate renormalization of the coefficient k. Precisely the same conclusion can be drawn in the physical light-cone gauge by performing the following three transformations on r(A), Eq. (10.51): (a) a finite non-multiplicative wave function renormalization, Al^A'f
= Al+*°
,
(10.52)
which reduces Eq. (10.51) to the form T(A',t)
(10.53)
=
(b) a rescaling transformation, A* - r A«™) = gA'Z = ^/brJkA''
,
(10.54)
leading, for general it, to (10.55) and, finally, (c) a finite "coupling constant" renormalization, cen
k -Htfc< >= k + c„ sign (*) .
(10.56)
The one-loop effective action in the light-cone gauge assumes, therefore, the simple form J.(ren) n
r(A(™ >,
= l—SdAW)
,
(10.57)
Chem-Simona
151
Theory
which is identical to that derived in a covariant gauge, but with the following proviso: i f the renormalization procedure is gauge invariant, as in our case, the shift in it is nonzero and finite, but with a gauge-nontnuarionf regularization, the shift is actually z e r o . 42,43,44,51
41,45
10.6. T h e T h r e e - P o i n t
Function
The computation of the general Chern-Simons vertex function r j j £ , ( p i , p 2 , P3) in the light-cone gauge is patterned after the calculation of the vacuum polarization tensor H ^ t ( p ) Sees. 10.2-10.5. In particular, the same Feynman rules, regularization and means of evaluating massive UV-divergent integrals are employed. A knowledge of r°*^, is needed to find the unknown coefficient c i n Eq. (10.44) and, on a grandioser scale, to complete the renormalization program of the Chern-Simons model in the presence of nonlocal terms. However, in view of the intricacy and length of the calculation, we have decided to omit here all details in favour of some general remarks. m
63
F i g . 10.2. Three-gauge vertex diagrams in Chem-Simons theory, (a) Triangle diagram; (b), (c) and (d) denote "swordfish" diagrams.
152
Noncovariant Gauge* The contributions to the one-loop vertex function rgJJ, arise from 1
the gauge-vertex I ^ S f i * , depicted in Fig. 10.2(a), and from the three "swordfish" graphs in Figs. 10.2(d), (b) and (c), represented collectively
e
rJ,y:* (pi,ft,»)+ri2^(Pi,pa.ft) •
r^(pi.P2,P3) =
UO-58)
The computation of the Chern-Simons vertex rj,™*' is much lengthier than that of rJiv£* (pi,p2,P3) and requires, in addition, a much higher level of technical sophistication than is needed for the corresponding fourdimensional Yang-Mills vert ex. With its gauge indices and gauge factors omitted, the D-dimensional version of r j u L reads: e
68
f i l l (Pi, Pa, Pa) 7^^^ x
A
M a M i
( p i , P a - J, J+Ps)A,
« -8-P3)A , 1
a
I
t l
where, for example, A „ (10.14c)): A
V l f l I
1 ( i a
and V „
M3/il
i ( J i
: l ( 1 1
( p 2 - q)V„ , (q )Vt 3
- P2,p , - 9 ) 2
( -|-p3) ,
(10.59)
g
are given by (cf. Eqs. (10.14a) and
( p - g) 2
" (w -
g) E(p 3
2
s
- g) - m»] { "
( P 2
x [ - i m n ' t r , ^ + (p -
"* + (p2 - t )
2
^ (p2-5)n
V l
n
P l
]| , (10.60)
and V^
a / 1
,(Pi.P2-g,«-r-P3) =
-i^tum
+ —Kipa - Pi -ffJ^SWa
+ (P3 - P2 +
WnHntUx + (Pi - P3 -
9W,I*I,] ;
(10.61) n is the usual noncovariant vector appearing in the light-cone gauge condition (10.9). M
Chcnt-Simoni
Theory
153
To facilitate the calculation of i t is convenient to divide each of the gauge propagators in Eq. (10.59) into light-cone (L) and nonlight-cone (N) components, i.e. A (q) op
so that
N
= A (q)
+ (q • ny'A^iq)
g
,
(10.62)
assumes the structure -
r(l)LLL
, T-(1)LNL
. fr(l)NLL
, / (l)IfVjV r
, (l)NNL
, p(l)ttW\
, (l)NLN\
r
, (l)NNN
r
.
r
(10.63) the superscript combinations {LLL}, {NLL}, etc. just label the three propagators in the triangle vertex. For example, T }^ refers to the subdiagram with propagators A ^ ^ , A f ; ^ , and &as0 - Fortunately, only 1
LL
P
3
half of the terms in Eq.(10.63) contribute to r j / i L - By invoking general power-counting arguments, we may prove the vanishing of the last four terms in Eq. (10.63) for large values of the regulator mass. Hence, 63
r
UVLLL
(l)
, (1)NLL
, r ( 1 ) t W L , Ul)LLN
T
t
The trickiest component is T j})^
LL
MO 641
with its three light-cone propaga-
tors: £
r#" (pi>P2,P3)
* V„ (q tVVa
x A£
3 / i ]
- P2,P2,-9)A^„ (9)V , l
u
l W W 3
( ( ( , p 3 , -q - pa)
( g + Pa)[g • n(p2 - q) • n(q + p ) • n ] "
1
(10.65)
3
is given by
K +
+ n • n*(p„n"n* + Pt>R*n* + p,,n£n') - 2p • n'(n n" n' + n„n*n' + n n " n ' ) p
u
p
p
aiv
-2p-nn»t] I f / "77
dq
^
2
J ? (?-p)V 2
=
dq q gu a
.
n
« (g-p)V"
It
fimte
,
finite,
finite.
The remaining integrals in this section and the next have been obtained with the help of the decomposition formulas:
171
Appendix C
(q-p)nq-n
P • n \(q - p) • n
(q - p) • n(q • n )
/ J 9° 2
j
2
dq n(q-p)
2
2
(p • n) (g - p) • n
u
p • n(q • n)
d
v
( p - n \ + 2p-n«;) J ' , (n • n*) p • n
2
dqq qi> q q-n(q-p)-n /
2
(p-n) g-n
- 2 p «n • n'p • n
n
dq q q q-n(q-p)-n
J
q n
2
u
=
2
(n •n-)"p n
[ n
P
P
'"* '" '
" ^
dq
_
J 9 ( ° - P) • "(s • n ) 2
/ /
dq q„ q (q - p) • n(g • n ) 2
dg g^g, q (q-p)-n(q-n) 2
2
2 n
n
- 2{p • n ) ( n > ; ) ] Z
d i v
m
rdiv
n • n"(p • n (pn*n (n • n*) (p • n )
2
2
_ 2
-2p-n
_
n
' *> * " 2
- 2p • np • n * ( n ^ ; + f
F
2
M
+ 2pT n;)/
d
t
p • w* { n - n'p • np • n'6 (n • n") (p • n)*
uv
3
- 2[p • np • n*(n^n; + n„»*)
dg
_
2
2
P) (q - P) - "(g • « )
/ /
dg (9-P) (7-P)-"(9-«) 2
dq 2
g
qii
2
n
P • *
jdiv
n - n'(p • n )
2
p-nt>-n* ,-2n.n*p^) / («-n-) (p-n)
2
2
u 2
{9 - P) (? " P) " "(9 ' n)
n(
2
-p-n' ~ (n - n'^p
.nf
[ 2 (
P
" '
- n • n'p • n'fpj.n,, +
P
" "
pn) v
a
u
s
d
+ §(p-n*) n,n,-2(p- ) ;n;] 7 " . n
n
d , v
,
Noncovariant Gangei
172
(c) Four progagators:
/
2 q
dq (q-p) q-n(q-p)-Ti 2
dq q? 2
2
J q (q-p) q
n{q-p)
=
finite,
=
finite,
n
dq q^q
u
p ) V "(3 - p ) "
2
q (q-
/ / /
dq = q (q - p) (q - p) • n(q • n) 2
2
2
2
2
dq q
u
=
q (q - p) (q - p) • *(q • n)
finite,
finite,
2
dq q q q (q-p) (q-p)-n{q-n) M
2
w
2
2
+
= (n-^Hp-n)^"" •
+
•
3. Massive light-cone gauge integrals in 2u space 2
In the following one-loop integrals, m is a mass, n = 0, and dP^q = dq. d q
f = j \(q — p) — m ]q • n 3
2
2
f dq J [(q-p) -m ]q
p
n
diV
' ' I n • n*
+ F
t
qil
2
2
n
2
1 m - \ n - ^
n
2p.np.n(n n*) *» 3
" dq
I
2
2
q [(q-p)
2
- ™ ]l
n
+
2p • n* ITrF^
(p-n') ~ J^rT) ^) 2
2
\
d i 1
+
F 2
'
1
Appendix C
f
173
dq q q„ u
J
2
2
2
2
2
9 [(? - p) - m ]q • n
2
2
2
J ? [(5 " P) ~ ™ }[(q - k) - m )q • n f dSJh J 9 [(? " P ) " m ][(q - k) - m*]q 2
2
j
2
dq q^qy 2
J
2
2
2
F
_ 2
2
q [(° ~ P) ~ ™ ][( ) ( l - t ,
^r,g
+
) p
-n -n-\ P
3
where t is a parameter and n\ = n ;
/ =
dq q ? [(?-p) ]'(?-«) 2
2
u
i(-7r)"r( T +1 - u ^ K T{c)n • n* r(o-)n J l
o i
w
2i(-ir) r(cr
+ 2 - w)p • » p - n "
a 1
ui — l rjuj—o—2 dr dy xy^H I»(n-n*)2 o 2i(-jr) T(r/-r-2-w)(p-n*) where H = (1 - y)p + 2zyp • np • n*/n • n*, and 2
1
2
1
Ttv.pa = ( P • " T ' t V P p ' * + 6^p n a
Tpv.po*
= (P )~ ( HPPPPV 2
+ o^p^n, + * ,p n^) ,
p
p(
+ &p,*PvPp + GvpPp-Po + t>v
1 6
(2p • n) ' [ ( ^ p i . + 5 ,p^)n + {6 V(
0
+
uaPu
6^ p )n a
+ {^p^u + SvpTi^pa + (S^riv + S n )p ] va
T
l»,e°
T
"10
=
2
v
li
u
p
,
p
2
iP )~ P 'P-'Pi>P'' . l
_
-„ 4
„,-i.
uv,pc = ( P P • ") 2
("(iPvPpPo + KvPpPpPv + n p p p p
li
u
+
a
UBP^PP)
,
2
= (2p n ) ^ P ^ n p n o + PpPai^nf) , T
2
^ ^ ^ =P [- (P' n ) " ]f -l ^ ^ " - +Pv v){Pp e ll,P° 4
4
- 1
2
n
1
n
+P*n ) P
,
1
ttv.po= ( P ' nn )" (p».«^' ,.n d - p ^ n ^ t . + P p i n n / + P„n n n ) D
14
pi,pt>
T
_
/ _ J2 \ - 22 .
= (n )~ n n .n,,n fl
1
0
(1
ff
i
.
References 1. T . Matsuki, Phys. Rev. D19, 2879 (1979). 2. D. M. Capper and G. Leibbrandt, Phys. Rev. D25, 1009 (1982).
187
fl
v
p
,
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INDEX A Abelian field 11, 12 Abelian gauge 6 Abelian gauge theory 11, 12 canonical quantization of 17 Abelian gauge transformation 12 Abelian subgroup 11 action 22 action principles 108 adjoint representation 97, 146 a-pr ascription properties of 46 and gluon seif-energy 46 anomalies cancellation of 40 and supeistring theory 40 anti-com muting field see ghost field antisymmetry of structure constants 11, 42, 86 auxiliary equation see constraint equation axial gauge or pure axial gauge or homogeneous axial gauge axial gauge 5, 37, 38, 43, 59, 105, Chapter 5 prescription for 67, Sec. 5.2.2 see also axial-type gauges "axial" condition 37 axial-type gauges 8, 38, 59, 131, Chapter 5 Feynman rules for 59 uniform prescription for 63, Sec. 5.2 see also axial gauge light-cone gauge planar gauge temporal gauge axial-type integrals 189
5, 7, 8
Index
190
in the P V prescription 163, Sec. 4.6, Appendix B in the n^-prescription 70-81 axial vector n,, 5, 38, 46 axial vector n* 32, 46 B BRS see Becchi-Rouet-St or a background held 4 background field gauge 4 bare gluon propagator in the genera! axial gauge 60 in the general planar gauge 62 in the light-cone gauge 62 for general gauge parameter 89 in the planar gauge 62, 91 in the pure axial gauge 61 in the temporal gauge 62 inverse of 89 Backlund transformation in 1 + 1 dimensions 136 see also sine-Gordon equation Becchi-Rouet-Stora identities 42 transformations 106 bosons W ,Z° 7 bound-state problems 53, 131 bound-state wave function 54 BRS equation 148 BRS formalism for covariant gauges 105 BRS operator tr 108, 110, 114, 124, 148 BRS sources 109 external 106, 124 BRS transformations and Chern-Simons theory 141 BRS transformations and noncovariant gauges 41 ±
C canonical coordinates 14 complete set o( 15 for Maxwell's theory 15 for Yang-Mills theory 17 canonical momenta 14 for Maxwell's theory 14 for Yang-Mills theory 17, 18
Index
canonical quantization 12, 14, 17 Casimir operator 146 Cauchy residue theorem 28 causal behaviour of integrand 28 causal prescription see Feynman's ie-prescription see n'-prescription charge density 13 charge conjugation matrix 97 Chern-Simons three-form 136 massive integrals 143, 183, Appendix E vertex function 151 see also three-point function Chern-Simons action 137 Feynman rules for 139 Chern-Simons parameter k 148, 150 shift in 150, 151 renormalization of 150 Chern-Simons theory/model 40, 135, Chapter 10 Abelian 136 action for 138 counterterms in 148 D-dimensional action of 140 finiteness of 148 Green functions of 141 manifold and 136 non-Abelian 136 nonlocality in 149 non-perturbative 137 one-loop effective action in 148 perturbative SU(N) 137, 146, 154 quantization of 137 regularization of 141, 151 renormalization of 140, 151 and conformal field theories 137 and dimensional regularization 140 and fractional spin and statistics 137 and knot theory 137 and link invariants 137 and n'-prescription 46, 142 and nonlocal terms 147 and perturbative effective action 138 in the light-cone gauge 46, 137, 138
191
192
Men
in the temporal gauge 137 chiral fermion field component form of 97 cohomology problem 148 colour electric field 17 colour magnetic field 17 commutation relations 16 equal-time 14, IS incompatibility with the Gauss equation 16 for group generators 11 components of n n * 46, 69 conformal field theories connection with Chern-Simons theory 137 conjugate momenta 14 see also canonical momenta constraint equation 15, 16, 20 continuous dimension method 133 see also dimensional regularization contour gauges 6 coordinate gauge 6 see also Fock-Schwinger gauge cosmological constant 2 Coulomb condition 19 see also Coulomb gauge Coulomb gauge 3, 5, 8, 37, 129, Chapter 9 application to non-Abelian models 4, 40 beyond the two-loop level 132 definition of 129 in quantum electrodynamics 131 lack of consistent prescription for 133 photon propagator in the 131 quantization in the 129 and binding corrections 132 and canonical quantization 130 and dimensional regularization 132 and hyperfine splitting 130 and spurious singularities 131 and static problems 130 Coulomb interaction term 130 counterterm 116, 119, 125 BRS-invariant 105, 122 equation for 107 counterterm in the light-cone gauge 118 expressions for counterterms in the planar gauge Pl
121, 127, Chapter 8
193
Index nonlocal 105, 128 counterterm action 106, 109, 110, 114 in the light-cone gauge 109, 114 in the planar gauge 123, Sec. 8.2 counterterms in momentum space, planar gauge 127 coupling constant 42, 97, 138 in boson- fermion system 54 coupling constant renormalization in Chern-Simons theory covariance of canonical quantization 17 covariant derivative 18, 20, 118 covariant gauges 4, 8, 27 quantization procedures for 27 relativistic invariance of 27 see also background-held gauge de Donder gauge Fermi gauge Feynman gauge generalized Lorentz gauge Landau gauge Lorentz gauge 't Hooft gauge transverse Landau gauge {see Landau gauge) unitary gauge covariant-gauge Feynman integrals 27, 28, 157, 159 causal behaviour of integrand of 28 generalized version of 29 Lorentz invariance of 28 power counting for 27 principal techniques for 30 properties of 27 table of massive integrals of table of massless integrals of current algebra 138 current density 13 curvature scalar 2
150
157, 159, Appendix A . l , A.3 158, Appendix A.2
D d' Alembertian operator 14 de Donder gauge 8 decomposition formula of splitting formula
40, 170, 171
194
Index
decoupling or ghost field 41 and closed ghost Unes 42 and open ghost lines 42 degree of divergence of a supergraph 96 superficial 96 degrees of freedom 19, 38 physical 40 distance gauging see Streckeneichung 5 (0)-terms and dimensional regularization 32 delta functional 23 differential form 1 dimensional regularization 32, 131, 132, 144 and f*(0)-terms 32 and hght-cone gauge 63 and massless tadpoles 43, 88 dimension of group 11, 20 Dirac gauge 6 discretized Hamiltonian approach 55 light-cone energy 54 light-cone momentum 54 light-cone quantization 53 divergent constants 109, 110, 114 and ghosts 127 double-pole prescription 44, 47 see also principle-value prescription dual vector a*, 46, 63, 66, 67 normalized version of 66 dual tensor 13 dynamical equations 16 4
E effective action 107, 123, 150 renormalized 150 in Chern-Simons theory in the hght-cone gauge Eichamt 2 Eichgewicht 2 Eichinvarianz 1-3 Eichnormierung 2
149, 150
Index Eichung 2 Eichverhaltnis 2 electric field 13 electricity 3 electromagnetic current conservation of 15 electromagnetic potentials 1 electromagnetic waves 3 electromagnetism 1 electroweak model/theory 7 energy integrals in the Coulomb gauge 133 epsilon tensor 13, 140 in three dimensions 138, 143 in four dimensions 13 and dimensional regularization 143 equal-lime commutation relations in Maxwell's theory 14 in Yang-Mills theory 18 expectation value and renormalization 116 exponential representation see Schwinger's exponential representation external current sec external source external source/field/current 23, 42, 86, 106, 148 Euclidean-space integral 47 Euclidean space 27 transition from Minkowski space to 30 integrals in 31 and n„-prescription 65 F fy, definition 66 in the axial gauge 74 in the light-cone gauge 66, 74 in the temporal gauge 74 Faddeev-Popov determinant in the Feynman gauge 21,22 in the light-cone gauge 21, 22 exponentiation of 43 Faddeev-Popov ghost 27 decoupling of 38
195
196
Index
Faddeev-Popov matrix 20 in the Feynman gauge 21 in the light-cone gauge 22 Faddeev-Popov term 42 Fermi gauge 4 fermion-gluon self-energy 100 fermion-pseudoscalar self-energy 100 fermion-scalar self-energy 100 Fermi statistics 24 and ghost particles 24, 42 Feynman diagram 29 "swordfish", in Yang-Mills theory 112 "s wordfish",in Chern-Simons theory 151, 152 triangle, in Yang-Mills theory 112 triangle, in Chern-Simons theory 151 Feynman gauge 4, 21, 22, 34 Feynman's ic-prescription 27, 28, 30, 32, 39 and Wick rotation 30 Feynman integrals covariant-gauge 28, 30, 157, 159 no ncovariant-gauge 39, 131, 183 properties of noncovariant-gauge 39 rules for noncovariant-gauge 39 and tensor method 39 Feynman rules for axial-type gauges in Yang-Mills theory four-gluon vertex 60, 61 gauge boson propagator 60, 62 ghost-ghost gluon vertex 60, 61 three-gluon vertex 60 Feynman's trick 31 see also Feynman's ie-prescription field strength tensor 12, 13, 109, 118 dual of 13 in Yang-Mills theory 17 flow gauges 6 Fock's gradient transformation 3 Fock-Schwinger gauge 6 four-gauge vertex in Chern-Simons theory 142 four-gluon vertex in Yang-Mills theory 60, 61 four-vector potential 3 in QED 13 fractional spin and statistics 137 functional 123
Sec. 5.1
Index
nonlocal integrated differentiation 86
148
G gamma function 49 gauge physical 35, 38-11 see also Eichung see also covariant gauges, and noncovariant gauges gauges of the axial type see axial-type gauges gauge boson propagator 61, 62 gauge choice 23 and local functional 15 see also gauge condition gauge condition 20, 22 implementation of 23, 139 gauge constraint 12, 17, 22, 116 implementation of 22 see also gauge condition gauge coupling constant 106 gauge degrees of freedom 23 gauge density see Eichgewicht gauge-equivalent orbits 23 integration over 23 gauge-equivalent fields 20, 23 integration over 20, 23 gauge factor or gauge ratio see Eichverhaltnis gauge field 11 gauge function 3, 20, 85 infinitesimal 12 gauge group 11, 42 Abelian 11 adjoint representation of 97 global 12 local 12 gauging 2, 7 see also Eichung gauge invariance 1, 3, 6, 11, 12, 15, 133 modern version of 3 new principle of 3
197
198
Index
and model building 6 gauge normalization see Eichnoimieiung gauge office see Eichamt gauge parameter 4, 6 gauge principle see principle of gauge invariance gauge propagator in Chern-Simons theory 141 in Yang-Mills theory 61, 62 gauge symmetry 1, 3, 7, 12, 24 as a new symmetry principle 7 gauge theory 7 Abelian 11, 12, 17 non-Abelian 7, 17 topological 136, 137 gauge transformation 12, 14 Abelian 12 for Al 85 non-Abelian 17 time-independent 19 Gauss equation in Maxwell's theory 16 in Yang-Mills theory 18 Gauss law 16, 17, 19, 132, 133 in Maxwell's theory 16 in Yang-Mills theory 17 Gauss operator in Maxwell's theory 16 in Yang-Mills theory 19 Gaussian integrals 167, Appendix C . l Gaussian weight function 23 general relativity see general theory of relativity general theory of relativity 1 general prescription for axial-type gauges see unifying prescription, or uniform prescription generalized Gaussian integral 31 generalized Lorentz gauge 4 generating functional 22-24, 43 for complete Green functions 84, 107 for connected Green functions 23, 107 for Green functions 23, 24
199
for one-particle-irreducible vertices 107 normalization factor of 22 generators 11, 12 of gauge group 11 geometry see world geometry ghost ghost-gauge-ghost vertex in Chern-Simons theory 142 ghost-gauge-gluon vertex in Yang-Mills theory 60 diagrams 40, 113, 114 equation 108 loops 41 number 109 oriented vector 43 scalar 43 vertex 42 see also ghost field, or ghost particle ghost-free gauge 38 ghost field or ghost particle 84, 106, 124 decoupling of 40, 41 see also ghost ghost lines closed 42, 43, 84 open 42, 43, 84 ghost propagator, scalar in Chern-Simons theory 142 in Yang-Mills theory 62 gluon-fermion loop 97, 98 gluon loop, pure 99 gluon-pseudoscalar loop 99 gluon-scalar loop 98 gluon propagator see bare gluon propagator gluon self-energy infinite part of 82 in the light-cone gauge 84 in a uniform gauge, or unifying gauge Sec. 5.3.3 nonlocalily of 84 nontransversality of 91 transversality of 89, 90 and axial-gauge condition 82 and Ward identity 84 gradient transformation 2 see also Fock's gradient transformation
200
Him
graviton propagator in the axial gauge, using the principal-value prescription tensors appearing in 187 gravity 3 Green (unctions 24, 102, 133 Gribov ambiguities 27 Gribov copies 130 group fundamental representation of 11 generic element of 20 structure constants of 11, 106 of unitary transformations 12 see also Lie group group space 23 integration in 23 H Hamiltonian function 14 in Maxwell's theory 14, 17 in Yang-Mills theory 18 Hamiltonian density and operator ordering 130 Hamiltonian formalism 12, 15, 53, 133 Hamiltonian quantization see canonical quantization Hei sen berg- Pauli gauge see temporal gauge, or Weyl gauge Hilbert space 19 homogeneous axial gauge, or pure axial gauge 5 see axial gauge hypergeometric function integral representation for 49 hyperfine splitting in hydrogenic atoms 132 hypersurface 15, 20 I identities in Chern-Simons theory inhomogeneous axial gauge 5 insertion 108, 109 integral divergent part of 27 massless tadpole 32
142
62, 63
201
Index
relativistic invariance of 32 regular part of 32 two-loop 34, 35 invariance under gradient transformation
2-3
J Jacobian determinant 20, 77 matrix 20, 21 Jones polynomial and Chern-Simons theory
137
K Kaluza-Klein theory knot theory 137
7
L ladder diagram/graph 40, 41 Lagrangian density Einstein-Hilbert 135 external-source part of 24 gauge-fixing part of 24, 38 gauge-fixing part in the Coulomb gauge 131 gauge-fixing part in the Feynman gauge 21 gauge-fixing part in a uniform gauge 82 gauge-fixing part in the light-cone gauge 84, 85 gauge-invariant part of 24 ghost part of 24 kinetic part of 15 Lagrangian density for boson-fermion system 54 electromagnetic field 14 N = 4 supersymmetric Yang-Mills model 96, 97 QED in the Coulomb gauge 131 Yang-Mills theory 17, 81 Lamb shift 132 Landau gauge 4, 8 Laplace's equation 2 Laurent expansion 32 leading logarithmic approximation (LLA) in the light-cone gauge 40 Legendre transform 107 Levi-Civita tensor
202
Index
see epsilon tensor Lie algebra 11, 20 Lie group 11 Abelian 11 compact 11, 12, 22 identity of 12 semi-simple 11 simple 11, 22 light-cone coordinate, position, time, variables: in 1 + 1 dimensions energies 54 momenta 54 light-cone gauge 4, 5, 8, 22, 32, 37, 70, 74 applications of 46 definition of 37 integrals 167, Appendix C prescription for Sec. 5.2.1 prescription in Chern-Simons theory 148 Yang-Mills Ward identity in the 89 and Chern-Simons theory 46 and component-field formalism Sec. 6.2 and e-p collisions 40 and ordinary Yang-Mills theory 46 and superstring theory 46 and supersymmetry 40 and Wick rotation 63 and Wilson loop 46 light-front gauge 7 linear form 1 link invariant 137 see Chern-Simons theory /mod el local functional 15, 20 locality 69 Lorentz condition 19 Lorentz constraint 15 see also Lorentz gauge Lorentz covariance and light-cone gauge 38 Lorentz gauge 4 Lorentz symmetry and Chern-Simons theory 140
54
Index
M magnetic field see magnetic induction magnetic induction 13 mass dimension 109, 148, 149 mass scale 33 mass spectrum 54 massless tadpole integral 32 see also tadpole integral massive Chern-Simons integrals 143, 183, Appendix E Maflstab-Invarianz 1, 2, 3 material waves 3 matter 3 Maxwell's equations 12-13, 16 Maxwell's theory 1, 11, 16 measure 23 metric tensor 1, 135 in discretized light-cone quantization 54 three-dimensional 143 Minkowski space 27 and n„-prescription 63 and transition to Euclidean space 30 and Wick rotation 27 mode physical 38 nonpropagating 38 transverse 40 multi-loop integral 32 multi-soliton solutions 136 N n , fixed four-vector 5, 22 definition 37 dual vector 32, 39 definition 46, 64 n^-prescription 45, 47, 177 definition of 45 for axial gauge 46 for light-cone gauge 45 for planar gauge 46 for temporal gauge 46 in Euclidean space 65 in Minkowski space 64 u
203
Index
204
and non-Abelian theories 46 and Chern-Simons theory 142 naive power counting 27, 39 Newton's gravitational constant 135 non-Abelian gauge held 17 noncovariant gauges 3, 5, 7, 27 and lack of Lorentz covariance 39 and physical degrees of freedom 40 nonlinear gauge conditions 6 nonlocal operators (n • 9) *,(»» • D)~ 118 nonlocal terms/quantities 40, 41, 105, 111, 116, 119, 128 in Chern-Simons theory 148 in the light-cone gauge 105, Chapter 7 in the planar gauge 121, Chapter 8 in the self-energy function 84, 110, 111 in the vertex function 110, 112 in Yang-Mills theory 84 and counterterm action 109, 110 and spurious factors 37 noii locality 154 in noncovariant gauges 41 see also nonlocal terms non-planar diagram 40 nonpropagating mode 38 normalization constant/factor in the generating functional 21, 23, 43 normalized version of n' 66 see also F^ nontransversality of gluon self-energy 91 implication for renormalization program 91 null vector n„ 5, 22, 37, 139 null vector F„ 66, 67, 122, 124, 177 —
l
O Okubo-Zweig-Iizuka rule 130 operator ordering in the Coulomb gauge 130, 132, 133 overlapping divergences 34 P PV prescription see principal-value prescription
205
Index
parapositronium in the Coulomb gauge 132 particle spectrum 54 phase factor 2,3 phase transformation 3 photon field 12 physical gauge 35, 38, 40, 96 see also ghost-free gauge see also noncovariant gauges physical mode 38 and discretized light-cone quantization 55 physical states and the Gauss operator 16, 17 pincer diagram 91, 92, 123 planar diagram see ladder diagram planar gauge 5, 8, 38, 59 counterterms 121, 122 gauge parameter 121 Lagrangian density 121 nonlocal expressions 122 nontransversality 122 pincer diagram 122 prescription .69 self-energy 121 Yang-Mills Ward identity 123 and principal-value prescription 121 and nonlocal terms 123 and uniform prescription, or unifying prescription Poincare gauge 6 poles double 63 pinching 45 simple 35, 63 spurious 63 unphysical 40 positronium and heavy quarkonia 55 power counting 27, 34, 39 see also naive power counting prescription a- 46 causal 39, 45 Feynman's te- 39 principal-value 34, 41, 45, 63 unifying or uniform 39
121
Index
206
prescription foe covariant gauges 27 light-cone gauge 45, 46, 63 (
