Noncovariant gauges : quantization of Yang-Mills and Chern-Simons theory in axial-type gauges

7 = (rcosff) ,

0 < r < o o ,

2

, (4.27)

we o b t a i n

h = - V -

T/2

*/2

00

/ dr r

5

/

<# cos 0 s i n

3

c o s 4> s i n fi

0 2

2

x (r sm 9)-

or, d e f i n i n g s i n f l = x a n d s i n

u l +

2

1

A

ir~ e- ^"'>'

d> =

,

y,

1 h = ~~T(3-u)

1

f dx x<-*» {

2

A{x,y;u )

2

2

(4.28)

f dy [Aix.y-f))"-*

(4.29)

2

= x y(\ - y)p + g ( l - r ) + * L J i > 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 <•

; ; yy <.* * p

* - - 2 ( - i - z)h =

(2

)2 2

r(± -

*>r(i+q

— ioa

r(3 -

+ t ) r ( z - p r ( - z ) r ( < j - 2 + 1 - 2z)r(a> r(i-l-l-;)F(2w-4-2z)

X

\ V ( p «) (p n) J \ np 2

2

2

-2-0

2 S <

(4.42)

The presence of the gamma function f ( i — z) implies that the new contour .2 C2 now lies to the right of z = — ^, but to the /e_ff of the origin (since

Nonco variant Gnu get

52

Rez < 0 from Eq. (4.37)), as depicted in Fig. 4.5. Collapsing C to the right and taking a -* 0, we obtain from the s-integral in Eq. (4.42), 2

2

+i

fdz

T{z - t)T{± - z)T(u -2 + t-

J

2z)T(-z)

r(i +1 - z)T(2w - 4 - 2z)

(

n

2

V

\ ( P • »)V

— ioo

*

r(i+*jn>-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
r(3 - w + ()r(w - 2 - f) = ir/sin(ir(w - 2 - ()) , to go from Eq. (4.44) to Eq. (4.45). The final step consists of collapsing the contour clockwise, thereby picking up another minus sign. The contour lies to the left of the origin, but arbitrarily close to i t . The only contributions to the integral (4.45) come from the poles at t — 0 and f = w — 2. Hence,

53

Overview of tWonco variant (Gauges

Im z

\

c

2

' Rez

-1

0

-1/2

F i g . 4.5. Relation (4.41) permits us to shift the contour C% to the right of the line 2 = —^. 2

lim K p-0

-2*"(P )' 2 W

•pTiRcs (( = 0) + 2xt"Res (t = w - 2)] ,

1

2

f2y+ (p )"\ n

8

2

r r > - 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,

As the name suggests, discretized light-cone quantization alludes to the discretization of space-time: one defines the theory under discussion in a box of finite length L and expands the fields in a complete set of plane waves satisfying periodic (or anti-periodic) boundary conditions. Imposition of periodic boundary conditions for a single particle then leads to discretized light-cone momenta k of the f o r m +

57

k*=2*n/'L,

n = 1,2,... ,K ,

(4.49)

and to discretized light-cone energies k~ :

K(F)

= I&,

and

k-{B)=-^,

(4.50)

K being the upper bound for the single-par tide momentum. Of course, the above procedure breaks Lorentz invariance, but the latter is supposed to be recovered in the continuum limit as the box length L approaches zero: L —• 0. Note that in the definition of fc+ in Eq. (4.49), n starts at n = 1, and not at t i = 0. One may show, for instance for the Schwinger model, that the range n = l , . . . ,K, yields the correct particle spectrum.

Overview of Noncovariant

Gauges

55

In the discretized case, application of the light-cone gauge condition n -A = 0, n = 0, likewise leads to spurious factors of the form (p- n ) , so that the choice of prescription for (p • n ) becomes a matter of the utmost importance, just as for gauge theories. However, unlike traditional theories in the path-integral formalism, where a consistent prescription for (p • n ) is known to exist (cf. Eqs. (4.18)), no satisfactory prescription is currently available in the discretized Hamiltonian approach. An additional concern are zero gauge modes which have been analyzed by several authors including McCartor, Heinzl, Krusche and Werner " and P a u l i . For instance, in the presence of spontaneous symmetry breaking, the accompanying zero modes are being viewed as physical modes, whereas in the absence of spontaneous symmetry breaking the zero mode sector is assumed to be spurious. 2

- 1

- 1

- 1

58-60

61

63

64

The method of discretized light-cone quantization has been applied in Q E D i i t o the massless and massive Schwinger model, to d> theory and Q C D , as well as to the bound-state problem of positronium and heavy quarkonia in 3 + 1 dimensions. * The technique holds promise for the treatment of strongly interacting fields, but whether it is destined for stardom will depend on its adaptability to 3+1-dimensional non-Abelian theories. For the latest developments in this direction and related work on the light-cone Tamm-Dancoff a p p r o x i m a t i o n the interested reader may wish to consult the recent l i t e r a t u r e . 85

4

66

+

67

6

6 9

70-75

76-78

References 1. 2. 3. 4. 5. 6. 7.

8. 9. 10. 11.

R. P. Feynman, Phys. Rev. 76, 749 (1949). D. M. Capper and G. Leibbrandt, Phys. Rev. D25, 1002 (1982). D. M. Capper and G. Leibbrandt, Phys. Rev. D25, 1009 (1982). A. Bassetto, G. Nardelli and R. Soldati, Yang-Mills Theories in Algebraic Non-Covariant Gauges (World Scientific, Singapore, 1991). D. J. Pritchard and W. J. Stirling, Nucl. Phys. B165, 237 (1980). T. R. Taylor, Phys. Lett. B93, 437 (1980). K. Konishi, in Perturbative Quantum Chromodynamics, 1981 AIP Conf. Proc. No. 74, Particles and Fields Subseries No. 24, eds. D. W. Duke and J. F. Owens, AIP, New York, p. 265. J, Kalinowski, K. Konishi and T. R. Taylor, Nucl. Phys. B181, 221 (1981). J. Kalinowski, K. Konishi, P. N . Scharbach and T. R. Taylor, Nucl. Phys. B181, 253 (1981). G. Curci, W. Furmanski and R. Petronzio, Nuc. Phys. B175, 27 (1980). S. Mandelstam, Nucl. Phys. B213, 149 (1983).

56

Noncovariant Gauges

12. L. Brink, O. Lindgren and B. E. W. Nilsson, Nucl. Phys. B212, 401 (1983); Phys. Lett. B123, 323 (1983). 13. M . B. Green and J. H. Schwarz, Phys. Lett. B148, 117 (1984). 14. P. J . Doust and J . C. Taylor, Phys. Lett. B197, 232 (1987). 15. J. C. Taylor, in Physical and Nonstandard Gauges, Lecture Notes i n Physics 361, eds. P. Gaigg, W. Kummer and M. Schweda (Springer Verlag, Berlin, Heidelberg, 1990) p. 137. 16. A. Bassetto, M . Dalbosco and R. Soldati, Phys. Rev. D36, 3138 (1987). 17. A. Bassetto, G. Nardelli and R. Soldati, Mod. Phys. Lett. A3, 1663 (1988). 18. A. Bassetto, G. Nardelti and R. Soldati, Proc. of the XVIIInt. Con/, on Group Theoretical Methods in Physics- Montreal (Canada) 1988, (World Scientific, Singapore, 1989). 19. A. Bassetto, G. NardclH and R. Soldati, Yang-Mills Theories in Algebraic Non-Covariant Gauges (World Scientific, Singapore, 1991). 20. A. Andrasi, G. Leibbrandt and S.-L. Nyeo, Nucl. Phys. B276, 445 (1986). 21. G. Leibbrandt and S.-L. Nyeo, Nucl. Phys. B276, 459 (1986). 22. B. Cheng and E.-C. Tsai, Phys. Rev. Lett. 57, 511 (1986). 23. H. Cheng and E.-C. Tsai, Phys. Rev. D36, 3196 (1987). 24. J. Fienkel, Phys. Rev. D13, 2325 (1976). 25. G. Leibbrandt, Rev. Mod. Phys. 59, 1067 (1987). 26. 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 lines 27. T. Matsuki, Phys. Rev. D19, 2879 (1979). 28. G. 't Hooft and M . Veltman, Nucl. Phys. B44, 189 (1972). 29. N . N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (3rd ed., John Wiley, New York, 1980). 30. J. Schwinger, Phys. Rev. 130, 402 (1963). 31. Y.-P. Yao, J. Math. Phys. 5, 1319 (1964). 32. J. Frenkel and J. C. Taylor, Nucl. Phys. B109, 439 (1976). 33. W. Kummer, Acta Phys. Austriaca 41, 315 (1975). 34. W. Konetschny, Phys. Rev. D28, 354 (1983). 35. D. M . Capper and G. Leibbrandt, Phys. Rev. D25, 1002 (1982). 36. D. M. Capper and G. Leibbrandt, Pfiyj. fieti. D25, 1009 (1982). 37. S. Caracciolo, G. Curci and P. Menotti, Phys. Lett. 11113, 311 (1982). 38. P. V. Landshoff, Phys. Lett. B169, 69 (1986). 39. H. Cheng and E.-C. Tsai, Phys. Rev. D34, 3858 (1986). 40. H. Cheng and E.-C. Tsai, Phys. Rev. D36, 3196 (1987). 41. 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; X X I International Conference on High-Energy Physios, Paris, 1982, eds. P. Petiau and M . Porneuf, Les Editions de Physique, Paris, p. 331. 42. G. Leibbrandt, On the Light-Cone Gauge, Univ. of Cambridge, Cambridge, DAM TP seminar (1982).

Overview of Noncovariant

Ganges

57

43. G. Leibbrandt, The light-cone gauge in Yang-Milts theory, Univ. of Cambridge, Cambridge Report No. DAMTP 83/10, 1983, unpublished. 44. G. Leibbrandt, Phys. Rev. D29, 1699 (1984). 45. G. Leibbrandt, seminar at the Summer Theoretical Physics Institute in Quantum Field Theory. Univ. of Western Ontario, London, July 28-August 10, 1985. 46. G. Leibbrandt, A General Prescription for Three Prominent Non-Covariant Gauges, CERN preprint, Report No. TH-4910/87 (1987). 47. G. Leibbrandt, Nucl. Phys. B310, 405 (1988). 48. P. Gaigg, M. Kreuzer, O. Piguet and M. Schweda, /. AfarA. Phys. 28, 2781 (1987). 49. P. Galgg and M. Kreuzer, Phys. Lett. B205, 530 (1988). 50. H. Buffel, P. V. Landshoff and J . C. Taylor, Phys. Lett. B217, 147 (1989); A. Bassetto, I. A. Korchemskaya, G. P. Korehemsky and G. Nardelli, Nucl. Phys. B408, 62 (1993). 51. A. Andrasi and J. C. Taylor, Nucl. Phys. B375, 341 (1992); Nucl. Phys. B414, 856E (1994). 52. S.-L. Nyeo, Z. Phys. C52, 685 (1991). 53. A. C. Kalloniatis, Quantization and Renormalization in the Homogeneous Axial Gauge, Ph.D thesis, Univ. of Adelaide (June 1992). 54. D. M. Capper and G. Leibbrandt, unpublished (Queen Mary College, London, 1982). 55. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, New York, London, 4th ed. 1965). 56. H.-C. Pauli and S. J. Brodsky, Phys. Rev. D32, 1993 (1985). 57. H.-C. Pauli and S. J. Brodsky, Phys. Rev. D32, 2001 (1985). 58. G. McCartor, Z. Phys. C41, 271 (1988). 59. G. McCartor, Light-Cone Gauge Schwinger Model, Southern Methodist Univ., Report No. SMUTH/91-02, 1991 (unpublished). 60. G. McCartor and D. G. Robertson, Z. Phys. C53, 679 (1992). 61. Th. Heinzl, St. Krusche and E. Werner, Regensburg Report No. TPR 90-44 (unpublished). 62. Th. Heinzl, St. Krusche and E . Werner, Phys. Lett. B256, 55 (1991); B272, 54 (1991). 63. Th. Heinzl, St. Krusche and E . Werner, Phys. Lett. B275, 410 (1992). 64. H.-C. Pauli, On the Gauge Mode in Gauge Field Theory, Heidelberg Report No. MPIH-V21-91, 1991 (unpublished). 65. T. Eller, H.-C. Pauli and S. J. Brodsky, Phys. Rev. D35, 1493 (1987). 66. A. Haruidranath and J . P. Vary, Phys. Rev. D36, 1141 (1987). 67. K. Hornbostel, S. J. Brodsky and H.-C. Pauli, Phys. Rev. D41, 3814 (1990). 68. A. C. Tang, S. J . Brodsky and H.-C. Pauli, Phys. Rev. D44, 1842 (1991). 69. M. Krautgartner, H.-C. Pauli and F. Wolz, Phys. Rev. D45, 3755 (1992). 70. I. Tamm, J. Phys. (U.S.S.R.) 9, 449 (1945). 71. S. M. Dancoff, Phys. Rev. 78, 382 (1950).

58

Noncovariant

Gauges

72. R. J. Perry, A. Harindranath and K. G. Wilson, Phys. Rev. Lett. 65, 2959 (1990). 73. D. Mustaki, S. Pinsky, J. Shigemitsu and K. G. Wilson, Phys. Rev. D43, 3411 (1991). 74. R. J. Perry and A. Harindranath, Phys. Rev. D43, 4051 (1991). 75. D. Mustaki and S. Pinsky, Phys. Rev. D45, 3775 (1992). 76. S. J. Brodsky and H.-C. Pauli, in Recent Aspects of Quantum Fields, H. Mitter and H. Gausterer, eds., Lecture Notes in Physics, Vol. 396 (Springer-Verlag, Berlin, Heidelberg, New York, 1991). 77. S. D. Glazek and R. J. Perry, Phys. Rev. D45, 3740 (1992). 78. K. Hornbostel, Phys. Rev. D45, 3781 (1992).

CHAPTER 5 GAUGES OF T H E A X I A L

KIND

This Chapter is devoted to the family of axial gauges, consisting of the pure axial gauge ( n < 0), the temporal gauge ( n > 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),

"
=(\nj.\,n),

ng <

u

« ) * = (N.-n). (»? )*= ([iii|,-ii), « «

m

p

2

o,

(5.34b)

(5.34c)

r=(i«ui. )* = (no,nj.,+i|nx|),

with the abbreviations: lc = hght-cone gauge, ax — axial gauge, pi = planar gauge and temp = temporal gauge. The general prescription (5.34) has tremendous advantages: it facilitates comparison between individual gauges, it ensures that Feynman integrals need only be computed once, and it simplifies the analysis of basic concepts such as locality, unitarity and renormalization. But what about the practical side of implementing prescription (5.34), specifically Eqs. (5.21), (5.29) and (5.32)? In view of the different components of n,,, Eq. (5.34b), computations are bound to be awkward, unless the process of integration can somehow be streamlined. To achieve this streamlining, we replace the original set {9^,1,1} by a new set of variables {Qp,N }. For instance, in the case of the temporal gauge in Euclidean space, the correspondence between the two sets is given by u

q± = Qx, OL = NX,

73 = Q<, 94 = Qs> n = -1N4, n = N, 3

A

A

(5.35)

70

Noncovariant

Ganges

thereby enabling us to rewrite Eq. (5.29) as 1

temp

- ( Q N

= lim Eucl

f—«

( Q N ^

+ . + G

W

1

a

^ + ^ J '

>

0

_

{

m

*

Precisely the same form can be deduced from Table 4 for the axial and light- cone gauge, so that our streamlined prescription for all four gauges reads (Euclidean space): ,

IQaNa

Q -N +

A

.. = (-1) lim

A =1,2,3,...;

2

/i >0.

(5.37)

We shall illustrate formula (5.37) in the next section for A = 1 and A = 2.

T a b l e 4. Correspondence between {q^, n^} and {Qp,

n

lu'i p\

gauge

94: "4

Temporal gauge

Pure axial gauge

0*;-iW| = i | n x |

Q*;As

Qr.N

Qi-.-iNt. = - f t a i l

3

Np}.

Light-cone gauge

Q*;-iN = -i\n\ t

5 . 3 . Calculations at One Loop 5 . 3 . 1 . Two-propagator

integral

As our first illustration of formula (5.37) we consider the following twopropagator integral in Euclidean space,

y-Euci

(q-pWq

ny '

(5.38)

which possesses a spurious pole of order two (A = 2 in Eq. (5.37)). Application of prescription (5.37), together with the Schwinger representation (3.12) for propagators, yields

Ganges of the Axial Kind

71 2

1

(

1 ) 2

dQ (Q • N + iQiN*) i™/(Q-P)2[(Q.N) -rO" N + 2

2

a

4

2 U

2

j '

2uj

dQ = d Q = d^q, OO

CO d

= &§j

(5.39)

a

a

aP

J 0

0

3

pe- '- "

iQ Nt) e

4

4

o

y y /3 -° -^ |y 2iN J y oo

oo

pa

= Km

do

0

+

2

J dQ dQ (Q • N +

dj9

e

0

dQ Q - N e ~

4

a

E

dQ

- j y j y d Q e - y dQ Q e~ B

2

4

E <

4

2

at) ( Q • N ) e

Q e

_ B

y

<(Q e 4

_ f t

4

J ,

(5.41)

where

2

2

E = a Q — 2cQ • P + 0(Q • N ) , E

i

2

= (a +

2w

d Q

2 u

1

= d Q d ' - Q = dQ dQ, 4

4

2

0N )Q -2cQ P . i

A

All Q-integrations range from —oo to +00, unless otherwise indicated. Substituting the appropriate integrals from Appendix C . l into Eq. (5.41) and calling N = a , N\ = p , we can express the integral as 2

2

2

f = 7

where the divergent portion J

and the finite term 1™ by

d l v

d i v

fin

+ J ,

is given by

(5.42)

Noncovariant

72

1*" = x" lim / *-*Jo

da

(

a

t

M

=

^ 2

(

dpp Jo

f(P N ) * [ (5,1)

D

Ganges

3

2ipP-NP _ p'PjH

3 - «

4

+

^

(3,3)

^

+

/2

E

- D

(1,5)J

)

+

^

'

,

( 5 4

2

b

)

(5.42c)

2 kl2

with (j, k) = (a + pa y (a + Pp ) . It is possible to integrate out both 7 ' and 7 . However, since the pole term is clearly of greater interest to us, we shall ignore 1*" for the time being and compute 7 instead. (For details on see Appendix A in Ref. 12.) Rescaling P,p—> 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

-

<

5 4 4 )

or _ (.r'-^K^-u.)

/ ^ r - ' a - o

2

- "

(5.45) Perhaps the most reliable way of handling the final integration is to compute the (-integral both for general u and in closed form. The approach may or may not be successful depending on the functional structure of f'(&!*)• If 77 is sufficiently complicated as in Eq. (5.44), it is best to expand the integrand in Eq. (5.45) in an infinite series about itr= 2 :

73

Gaugei of the Axial Kind d,v

/ (p,w) a

3

_ (o-

-p )n"T(2-u) lim 2p*

1+

X <

(ff'-^ji-rp-w) 2p<

3

BO ''

(
2

x

1 +

i -W)*---]}-

B Bi

(5.46)

where we have set (2 — iLi)r(2 — u) — T(3 — w) in the curly bracket { . . . } . Expansion of T(2 — u>) about w = 2 yields: 2

2

a

T + * ( l ) - * ' ( l ) + 0((2- ) ),

1

r(2-w) = ( 2 - w ) - + * ( l ) +

W

(5.47) with *(w) = (d/dw)logr(w), so that (5-46) becomes div

/ (p,w) (
u 1

2-w)- +*(l) + i ( 2 - ) w

o-p(a + p) .

^

2

2

2

+ 4 ( l ) - * ' ( l ) + 0 ( ( 2 - ) ) + second term in Eq. (5.46), w

fff - p)x" • (l) + _ ( 2 - ) + cp(a + p)(2 - w) ap(a + p) w

+ $ 2 ( 1 ) _ * ' ( ! ) + 0((2-w) ) + second term in Eq. (5.46). 2

(5.48) This is probably as good a time as any to review the meaning of »p, F„, a and p in the unified-gauge formalism. We recall from Eqs. (5.23) and

74

Noncovariant Gauges

(5.24) that these symbols were originally defined in the light-cone gauge (= 1c) as follows: ic

r>£ = (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

=
.

12

Similar definitions hold in the pure axial gauge (= a) : a

»*, = (n,n ),

a

a

£ < ) * = ( n , - m ) =- (tr F ,p F^

4

,

(5.50)

with cr" = |n|, p" = \n±\, so that F* = (W\Ft)

= (£ i)-, t

a

n-F°

= o- +p°

,

and in the temporal gauge (= f) :

4 = (nx.tUjns), 2

2

( n ) ' = (nx.n4-.-n3) 4

1

=

( O * I V J S )

,

(5.51)

2

where
Equations (5.49) to (5.51) suggest several relations which are common to all three gauges, namely: 2

2


2

=n,


2

(F„) = 0 and F = i . 4

+

Returning to Eq. (5.48) and taking the limit as u —> 2 , we obtain

+

^ r j a j

3

( i - b o ^

n ^ " J•

(5.52)

Gauges oj the Axial Kind

75

where

(5.54)

o

2 /

,Eucl

=

=

d

i

v

2

1

( 2 ^ o y ^(p-?) ]- . Accordingly, the divergent part of I in Eq. (5.38) reads ^

/

(

p

-

A

-

^

r

1

^

( 5

-

5 5 )

and is valid for all four gauges. In the light-cone gauge, for instance, the r.h.s. of (5.55) reduces to d i v

J

/ 7

Wt 2

(p-9) (?

wl"=

0

5

'

56

< ' >

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

«,-
OO

pp

= - H m y da dp d-te- *-°»*

= (J,/ ). 4

r

V

V

H

J dQ(Q, Q )(Q • N + ig A )e" " ' , 4

4

4

(5-59)

76

Noncovariant

Gauge!

At this point the reader might wish to consult Table 4 which gives the connection between the sets {q,rt} and {Q,N}.

The components J and J A

read 01

0

J = - l i m J da dp dye- "- ^

j dQ Q ( Q • N +

A

4

iQ N )e A

A

(5.59a)

o oo

J = - l i m j da d0 dye' "-'"' 131

1

J dQ Q ( Q • N + A

iQ N )e A

v-v

(

A

(5.59b)

o where the functions V and V in the exponentials are given by: A

V = A Ql A

- 20Q P , A

4

V = ( + 0)Q

A = y + 0 + aN

A

A

-20Q-P

2

7

2

,

+ o(Q • N ) . 2

(5.59c)

Let us first compute the scalar integral J . A

(a) The integral J

A

The computation of J i n Eq. (5,59a) is similar i n procedure to that of / in Eq. (5.38), differing merely in the degree of complexity. Performing the momentum integrations and then reseating the parameter a, a —• a/N , where N = t r , JVf = p , we find that A

2

2

2

z

CO C

f—OJ

3 2

A'

Here Ai = a + 0 + y, A = y + 0 + ap /cr , 2

2

and

0

D(a,0, ,n) T

=

0P -

2 P 2

2

0 + y [

« ( P - N ) 31
X

0 P\ 2

au

An

a

2

The parameter integrals are most readily attacked in spherical polar coordinates ( r , 9,4>) '• 2

a = (rcos0) ,

0 = (rsinflcos^) , 2

y = (rsin(?sm»

2

,

(5.62)

Ganges of the Axial Kind

77

so that T/2

oo

Jdad0dj

T/2

= J dB j

o

co

d<j> j dr (Jacobian),

o

o

o 5

3

with Jacobian determinant (Jacobian) = - 8 r sin 9 cos 9 sin &> cos <j>. Hence J4 in Eq. (5.60) assumes the form w/2

.

5

8sr" ..

/ dB ( s i n t f ) - ^ c o s f l

1 2

f'

„

.

,

, f°° ,

.

.

0 x lp

4

A

4

P - N s i n 9 cos * + £

1

+ '/^WgcosV j ^ M j

(

^

j_ (

5

g

3

)

where

2

2

D{9,4>]u) = sin 0 cos ^

2

2

, p cos 9 M = sin 0 + r . 2

The reintegration is trivial, 00

2 J dr

3 r

T

-^ - '

D

e

= T(2 - w ) D " "

2

,

(5.64a)

= T(3 - w J D " -

3

,

(5.64b)

0 00

2 y dr *-a-6-''* P

B

+

and leads, for w —• 2 , to a natural separation of J into divergent and convergent components: A

Noncovariant

78

Gauges

*/2ir/2 w

2ipw T(2-u) J a

,.

=

f

2-w

f dB d sin 8 cos 0 sin d> cos <j>

i-o/ J o o ir/2 i/2

+

9

47rT(3-tj) 1

tr-

2

5

df>
lim «-

0

0

(5.65a)

=

d

<M

J

+

4

(5.65b)

-

v

7 ' and J j " are proportional to T(2 - w) and T(3 - « ) , respectively. The ^4 term turns out to be finite for u —* 0 and w —• 2 , and will not be examined further. As for the divergent part of Eq. (5.65a), expansion of +

{airii£)2-u

a

b

o

u

t

t

h

e

p

o

i

n

t

_

u

2

g

i

v

e

s

.

TT/2IT/2 tt

2 i / M r T ( 2 - w ) ,. lim

f '

/ d$ di^ sin 8 cos $ sin 0 cos

M

o o 2

(2-w),

/sin f)\

2

(2-w) /,

2

sin 0\"

+ ...

ir/2»/2

2i.p7r"T(2 - w) f

f d8 d sin 8 cos 8 sin d> 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
2

v

or finally, JV*jfV,

(5.66)

where

J

<

- -

n

F

- F "

4

+

- ^ -

,w/2 >*/S y

F

4

rf9

rf

2

j sin f cos 6 sin 0 cos (4 ,

/sin S i

0

o

4 —

(5.67) Combining the results from Eqs. (5.65b) and (5.66), wefindthat the scalar integral is given by JA = / 4

d i v

+

4

N

,

i.e. J = -L_F 4

+ (j^-rJt").

i

(5-68)

(b) The integral J

The computation of J in Eq. (5.59b), CO pp

J dQ Q(Q • N + i Q / V ) e - - ' ' ,

m

v

J(p,w) = - l i m y da d$ tjye- *- '

4

v

4

(5.69) follows the method for 7 in part (a). Thus, performing the momentum integrations, rescaling a —> a/a , and using the spherical coordinates defined in Eq. (5.62), we obtain 4

2

Noncevariani

80

Gauges

x/7x/2 3

J(p,w) = —y lim /

dQ d® sin 8cos0sintpcosd>

o o U(sinfl) - " 7 2

x J dr r 5

2 w

2

e-

r l D

J-*,-,-* .

| .

(5.70)

Integrating next over r with the help of Eqs. (5.64), we find that J splits into a divergent part, J , and afinitecomponent, J : d l v

f i n

J =J

d i v

+J

f i n

,

(5.71)

where x/2x/2 div

J (p,w) = ^ " ^ N l i m

/

/ d0 dtp sin0cos6sin<j> cos6

o o 2-w

and fin

J (p, ) w

ir/2n/2 4 T T T ( 3 - u)

..

7

0

3

1 2

0 3

x

2

lim J J d0d

2

2

, N(P - N ) cos 0 cos
v

r

fin

ipP

A

(5.72)

J (p,w) is of no immediate interest to us. Repeating the steps between Eqs. (5.65) and (5.67) for the divergent portion J , we get d i v

G a » j e i of the Axial

d i

J

>,

= 2) = — F

W

+j

Kind

f i n

81

,

F = fi/er ,

(5.73)

so that J(p

= 2) = - ^ - F + a fl • r

l W

n n

+J

f l n

),

(5.74)

where r

J

=

-F

-F +

F

•• f —— urn. /

f / dO d sin 8 ros# sin A cos A

0

x fix jfin

=

4 T * T ( 3 - w)

K

lim y

0

fl

/ d$ d>(sin tf) " " cos sin

2

3

a o Af""

3

\

The expression for the integral 7 combining Eqs.

V cos 0 cos A +

2

2

P P - N cos ^ - N (^-^

( 5 . 6 8 ) and

M

tr

2

/

M

in Eq. (5.58) is finally obtained by

(5.74):

I„=(J,/ ) , 4

= (V,F*)~

+ ( j f + 4", j

f l n

+J

f l n

) ;

henc •

^

^ Li

(5.75)

with (5.76)

5 . 3 . 3 . Gluon self-energy

in a uniform

gauge

To illustrate the effectiveness of the unified-gauge formalism we shall outline the computation of the gluon self-energy to one-loop order (Fig. 5.3). We shall work specifically with the Yang-Mills Lagrangian density

Nonco variant Gauges

82

L

2

= -\(F^) -^(n-A")

VM

2

(5.77)

*-»0,

l

2

where n • A" = 0 is the axial-type gauge condition, with n = 0 or n' ^ 0, and assume that the components of the constant vector n defined in Eq. (5.34b), are all different from zero, i.e. n,, = ("o,"i,n ,ns) in Minkowski space. Moreover, it is convenient to adopt the "working prescription" (5.37) for the spurious poles of (q • n)~ . v

2

x

F i g . 5 . 3 . Yang-Mills self-energy to one-loop order.

Applying the Feynmann rules (5.3) and (5.6), together with the appropriate integrals from Appendix D, we obtain the following expression for the infinite part of the gluon self-energy: 12

—--unified

ab

where c A {p)= uv

acd

=

bcd

f

f ,

and

22 / y(p J -P P,) + n V ( p V - W , ) j

d?"a

!

2

F

u

q

2

i

q

_

p

)

2

i

q

.

n

)

2

•

(5.78a) P • " ( p ^ +P F )-PV

U

p• n The C^-term can be split into two parts,

Ffaf**

j

+ <Wto)

(5.78b)

Givgtt

oj ike Axial

Kind

83

with eg? = j f ^ y ? ' * *

+

2

P

- 2 • F f o A + P„F„) P

2

F

n - FppP* ~ p n - F +p-

p-n

p p-F.

F ( p „ n „ + p n„)

g>lv

¥

nn u

pn

2

n p.F

+(P«)

A

p n • Fg

2

4

- p n • Fp^p„ - p ( F n „ + F„n„)

ou

(1

3

2

+ p p • F(p n II

v

3

+ p n„)

+ p p - n(p^F„ + p^F^) - 2p • np • Fp^P*

v

2

+

n p-F p•nn • F

2

- 2p F F + (1

v

p-n

u

p•n

4

-I

- F ( p F „ + p„F„)

P

2

p F^F„ - p p • F (

(P-n)

P f l

F

+ p F ^ } + (p

v

t

J

F) p„p,

(5.78c) (n Fg

2

c< >-_^!_/-?l "* " ( n - F ) I 4<rp

+ nF)

a

e

n=

a

2

2

9*&{PpPv - P 9w) + 4p Pp9 a

+

uv

- 2(p p„Q j„ 0

J

n ^ B ^ t +P ,"i/fffl i} -PaP/j(npPi* + C

/

+p p 90,,) a

u

n p^)j y

p-n I 2

+ ^

p

«

2

^

fi

j ,

I P ffajiffpV ~ P (PoPuSpV ~ PcPv90 ) + VaPpPy-P* U

(5.78d) The expression for n u " '

f i e d

n

a

s

a

n u m r j e

r of remarkable features:

84

Noncova-na.nl Gauges 3

—

(i) It contains the correct gauge-in dependent factor (ll/3)(p ffU»/ PuPv) and is seen to possess only simple poles. (ii) Il)l" conserves n , is nonlocal in the external momentum p„, and respects the Ward identity derived in Eq. (5.103) and Eq. (5.108), ,fied

u

(iii) Equation (5.78) is valid in all axial-type gauges. For instance, to extract the gluon self-energy in the hght-cone gauge, we simply set n = 0 in Eq. (5.78) and replace F -# Fl =
1 f 12 H(p, n, O = - c r ( 2 - u)l y (

9

c

a

l c

ai

-

2 P

2 P

9 ^ - P.P.)

(nx+«i-«;)+Vp'

(5.80) Expressions similar to Eq. (5.80) may be deduced from Eq. (5.78) in the temporal and axial gauges with the assistance of Eqs. (5.50) and (5.51), and by recalling the appropriate definitions of n„ and n* from Eqs. (5.34b) and (5.34c). 5 . 4 . Ward

Identities

5.4.1. Ward identity in the lighi-cone gauge

The Ward identity for massless Yang-Mills theory in the light-cone gauge can be derived from the generating functional for complete Green functions Z[J$

= N

J^IDMDWe

D(A) = lHll[dAl{x),

1

'/^^"''-*"

A 1

1+J

' > ' **t*™»SVl

dz = d^z.

(

(5.81)

as a ft

Since ghost fields are known to decouple in any Feynman diagram, whether the ghost Unes are open or closed, the ghost termii n''Dj|*w* in Eq. (5.81) 16

a

Gauge! of the Axial Kind

85

may be omitted from Eq. (5.81) in the derivation of the Ward identity. I t suffices, therefore, to work with the generating functional * W 3 = N j D{A)Z[J%, Z[J«] = expij

(5.82) a

dz U^F^f

- ^ ( n • A)

2

+ J" • A"

,

which must be invariant under the gauge transformation for the field J4J, a

6A (x) =

+ gf^A^w^x)

a

;

(5.83)

c

ui (x) is usually called the gauge function and N is defined by Z[0] = 1 = N j D(A)Z[Q] .

(5.84)

2

Since D(A) and ( - 1 / 4 ) ( F ™ „ ) are both invariant under (5.83), so that 2

6D(A) = 0 = 6(F£„) ,

variation of Z leads to

m = Q = W I D(A)Z6 j dz [-l(FZ ) v

0 = iN

I D(A)Z J dz J-^n

0 = iJV J D(A)Z J dz |-I

V

+

2

+ J* -A'

0

A n"-|-y'"'J

n

a

J "^

/ ")jr ^K( )] a

^

• A)

a

a

n

a

- ^{n

• A n" + J

0 = iN j D(A)Z j dz^(-^n-A n"+

+ ^

2

S^d^w^z)

lj

2

.

(5.85)

The right-hand side of Eq. (5.85) is not yet in manageable form, since the first term is proportional to <9*tu (s) rather than to w (z). Integrating by parts, however, and assuming that the surface term vanishes at the boundary, we find that c

c

86

Noncovariant Gauges j

A

= -J

a n

*

dzw"{z)

a

+

a

J ^d*w (z)

f-ln

(5.86)

• d'n • A" +

The expression a

lc

i

c

~ -A n"9r A {z)v {z) n

u

J

in Eq. (5.85) vanishes due to the anti-symmetry of f , reduces to

so that Eq. (5.85)

abc

IN j

c

0 = i j dzw (z)

D(A)Z

(5.87) c

or, since w (z) is arbitrary, to 0 = TV jD(A)Z[J£]

a

a

t

al

±n-d*n-A (x)-dZJ > (z)

h

:

+ gf *J >'(x)A< (z)

.

a

(5.88) The next step in the derivation of the Ward identity consists of differentiating Eq. (5.88) functionally with respect to the external current J%,(y), and then equating J" to zero. Abbreviating the square bracket in the integrand of Eq. (5.87) by

a

H [J)

H'[J],

a

= i » . a - n . A (x)

- d^J""^)

+

gf'^J^WAXz)

(5.89)

J=0

(5.90)

we obtain from Eq. (5.88)

0

here

Gauges of the Axial Kind

i ^ j j r = iZ[J) j dz&*%A*Hz)6(*

87

- »).

= iZ[J] j dzA""(z)S(z - y),

« * t > - I f ) , a % - y) ,

^

|

= ^ ( z - y) ,

pa

-
(5.91a)

and £ rra r

Tl = -6* 6(z - y){»6» a

c

+ r 6»6( 9

c

- y)A (x) .

x

)1

(5.91b)

Equation (5.90) now reads 0 = hfj

D(A)Z[J]

fl

W

x [iAZ(y) f i n -fl'n• A"{x) - 6^ "(x) + , / - J » " ( i ) A ^ ( * ) )

x

(y)n - d*n • A"{x) - d J(x - y)S" + gf^Slx = xW y-A" a D(v4)Z[0] a

-

c

y)A (x) a

(5.92) Finally, exploiting the definition of N in Eq. (5.84), we obtain the following Ward identity in coordinate space: z

a

- d n • A (x)Al(y)

x

a

c

- 8 J{x - y)6? + r 6(x

- y)Al(x) = 0. (5.93)

9

To derive the corresponding expression in momentum space, we first take vacuum expectation values of the time-ordered products in Eq. (5.93), x

-r."n • 0*(O|T ^ ( * ) ^ ( j / ) | 0 ) - 8 J(x - y)6"> a

a

e

+ gf > {0\T6(x

- y)Al(x)\Q) = 0 ,

(5.94)

SS

JVoncovariant

Gauges

then use the Fourier representations 2

SO -y)

= f » - * ' J dqe*^

d'J(x -y)

,

(5-95a)

y d«e'"«<—»>«„ ,

=

(5.95b)

together w i t h : (0\T Al(x)A' (y)\Q)

= j dqe^'-^D^q)

a

n

6~(0|T Al(x)A' (y)\0)

a

= i J dqe'^'-^q

a

,

• nD / (q) a

(5.95c)

,

(5.95d)

and 2

O)" "1 o 9 B * « ^ ^ (

(0|T ff(* - y)A%(x)\0) =

S

) .

(5.95e)

T denotes the time-ordering operator, D° (q) is the gluon propagator, and Ba(q) the Fourier transform of the vacuum expectation value of the time-ordered product of 6(x - y)A„(x). W i t h the help of formulas (5.95), the momentum-space version of Eq. (5.94) reads r

Since B£(g) corresponds to a massless tadpole which vanishes i n dimensional regularization, and i f D°fi,(q) = (2w)- 6 ''D (q), Eq. (5.96) simplifies to the form 2u

a

uu

n

" n"D (q) a tta

+ iq = Q,

(5.97)

a

i.e. = -iq"{D-%

.

u

(5.98)

The easiest way of verifying the Ward identity (5.98) is to express i t first in terms of H B K I * - P Yang-Mills self-energy. Expanding according to (see Fig. 5.4), n e o n e

| 0 O

n

aa

= G

( M

Are,

(j„° + n"o«)

,

(5.99)

Gauges of the Axial

89

Kind

n.

F i g . 5.4. One-loop expansion of

In E q . (5.99). - 1

-1

and multiplying (5.99) from the left and right by ( G ) „ and ( £ > ) „ 0 , respectively, we obtain T

(erVfeii =

fr,

+ If* w

{^"{D^u

l

> 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. <

sca!ar

>

+ 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<(q a

- p)

XO-P.9.P-1). +

or, as u —* 2 ,

: ^ - ^ ^ ( G ) ^ * j y ( p V - P , . ) + f^(***S + P

+

2

P

a

n

~ . [2p nuf» - p • n ^ p , , + n^p-)] - — ^ - r ( n , . I + n * ' ^ ) } . p• nn • n n •n ) v

(6.6)

100

Noncovariant Gauge>

where we used Eq. (5.9) for the bare gluon propagator and Eq. (5.3) for the three-gluon vertex in the hght-cone gauge. Adding Eqs. (6.4)-(6.6) we obtain the following expression for the total gluon self-energy:

n =

(total)

16ir», +

P

1 ^ ^ " - + ^ 2

- .[2p n n.„ - p • n(n p p•nn • n M

6.2.2. Contribution.*

v

p

+ n )] uPv

from additional

^-z("nK n-n

+ n„n* ) \ ) (6.7)

diagrams

The complete calculation of the N = 4 supersymmetric Yang-Mills selfenergy involves three additional sets of diagrams summarized below. (a) The fermion self-energies: There are three contributions from fermion self-energy diagrams. The fermion-gluon self-energy ^ (s )i i u o n

Y?

(Bluon) =

f

\

(6.8)

has been evaluated in Ref. 29, while the fermion-scalar self-energy (scalar), £

(scalar) =

*

>

and the fermion-pseudoscalar self-energy

(pseudoscalar) =

^

/

\

^

Yj

( .9a) 6

(pseudoscalar),

*\

^

(6.9b)

Application

W

'

t h

of the Li§ht-Co*e

Gauge to SxpeTiymmetrj

v-^/

101

/

^""^ (scalar) = (pseudoscalar) , (6.10) were computed by Bassetto and Dalbosco. Addition of Eqs. (6.8)-(6.9) yields 17

17

y!

< >+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 «/-(«) A<(x) 0

S

SS

s

+ <Sw (i) ^ " ( x ) + ^ " ( x ) «w*(«) a

2

o- = 0. (7.10)

The solution of the renormalization Eq. (7.9) invokes several basic concepts, such as the application of a quantum action principle and the existence of an insertion D. Quantum action principles have their origin in the action principles of Schwinger , Lowenstein and L a m . ' For details of this elegant program, we refer the reader to Ref. 19. 14,15

16

17

18

Renormalization

in the Pretence

of Nonlocal

Term*

109

Here we merely recall that the insertion D represents a polynomial of normal products (for a definition of normal product, see Sec. 2 of Ref. 19) in thefieldsand their derivatives, and that the solution of Eq. (7.9) is given by -D = Y + " and J* must appear in the combination (J'+npQ") EE L * , so that the counterterm will likewise depend only on 2

ll

B

19

The functional Y and X should possess the proper mass dimension and ghost number N . If the ghost number of the Lagrangian density L is taken to be zero, i.e. N h[L] = 0, we may assign the following values to the fields: sn

g

11

NghiAft = 0,

N [u'] tk

a

= -1,

= 1,

N [Q ] Qh

a

N {K )

= 2,

gh

= 1,

(7.12a)

in which case N, [D] = Q, H

N, [
N, [X]=l. h

(7.12b)

The next task is to obtain the appropriate renormalization constants by: (a) making an appropriate ansatz for X (Sec. 7.2.2); (b) determining the divergent constants aj (Sec. 7.3); (c) obtaining the complete expression for the counterterm action AS (Sec. 7.4, Eq. (7.22)); (d) eliminating all nonlocal terms in AS to get a simplified version for AS (Sec. 7.4, Eq. (7.23)); and, finally, (e) deducing the one-loop renormalization constants for massless YangMills theory in the light-cone gauge (Eq. (7.29)).

110

Noncovariant

7 . 2 . 2 . The functional

Gaugct

X

According to Eq. (7.11), we can write the counterterm action to one loop as AS = Y +
Y =

dx = d^x

jdx\-\a,(F'^

X = X .i

,

+ Xnonlocah

loca

(7.14a)

where * i o « i = / dx{a A" Ll 2

^nonlocal —

j

+ a^A^Ll

u

a

,

t

i

dx{a [n*,d {n\d\) A

+ a u'K'}

(7.14b)

a

n ,A ]n L l

v

ll

ll

1

+ ^gr^dArixdxr^A^nrA^n^)- ^^}

; (7.14c)

a,-, i = 1 , . . . ,6 are divergent constants to be determined. The nonlocal expressions in Eq. (7.14c) are chosen so as to match the nonlocal terms in the self-energy and vertex functions. Notice also that the nilpotent operator a in Eq. (7.13) effectively depends on L ^ , i.e. 6 \6A%(x) t) 6L*(x)

+

6S

SS

6

r

J

^

6S )

^

6 R

+

SLl(x) SA^(x)

+

1 /

6w^)

jKHx)

, •

7

<-

14d

>

The detailed structure of cX is given in Eq. (7.21). 7.3. Determination of Divergent Constants In order to determine the divergent constants dj, we simply compare the theoretical expression (7.13) with the calculated answer obtained from oneloop computations. This tedious, albeit sobering, exercise requires the

Renormalization

in the Presence

of Nonlocal

Terms

111

evaluation of: the three-gluon vertex TffiAp, 0, - p ) , the gluon self-energy Ily^(P)' ghost contributions (cf. Figs. 5 in Ref. 21). a

n

a

t

n

e

1/2 x

p-q

1/2 x 0) F i g . 7.1. Yang-Mills self-energy diagrams. (aj One-loop self-energy diagram; (b) massless tadpole diagram. A l l lines correspond to Yang-Mills fields.

(a) The gluon self-energy Ylnpip)

+ p•nn

m

Fig. 7.1

•n

2

1p - -^( >- P n

n

n

1 n

n

+ P ' )f

i

u

a

the nonlocal term being proportional to 4p p • n*(p • nn •

(7.15) 1

n')~ n n . li

fi

Nona/variant

112

Ganger

+ 1/2X

p,a,|i

-q-p.c,(

• -q-p.c,p W q.b.v

1/2 x

+ l/2x -q-p,c,p

P,a,|x

-q-p,c,p

P,a.|i

(?) Fig. 7.2. Three-gluon "swordfish" diagrams.

vertex

diagrams, (a) Triangle diagram; (b),

(b) The three-gluon vertex Tff (p,§,-p)

and (d) are

in Fig. 7.2

p

1

(c),

(22

—[3p,(tt,|fl£ + 16p • n" n n'p n

p

4p • 71", . —— (n„6

uu

p

+

n n*) - P,i(n>,> + n„n*) - pp(n n*„ + n„n* )] p

8p • n* n • n'(p • n)-

u

5 1

*

* (7.16)

Renormalization

in the Pretence

of Nonlocal

(<=)

Termt

113

(d)

F i g . 7 . 3 . Ghost related diagrams, (a) J-ui ghost diagram, with dashed line* representing ghost-fields; (b) J-A-w vertex diagram; (c) J-A-w vertex diagram; (d) K-w vertex diagram.

where K =

CYM = N

(4T)- Cy (2- j)- ,

2 ff

2

1

M

t

for the group

Equation (7.16) contains two nonlocal terms: 16p -n*(p • nn • and -op • n'(n • n*(p • n) )~ p n n n . 2

l

2

ll

v

p

SU(N). 1

n*)~ p n, n , v

i

p

Noncovariant Gauges

114

(c) The four ghost diagrams in Fig. 7.3 Although the contributions from these diagrams vanish identically, they nevertheless place constraints on several of the coefficients a,-. Explicitly, 11

2K

a = a = 0, 2

a

5

= - a = a« = -

3

4

(7.17)

n • n"

The value a = 0, for instance, is obtained by comparing the covariant part of the vertex r j ' ^ p , 0, - p ) with the Feynman rules for the A -terms, while the value as = 0 is due to the vanishing of the ghost diagrams in Fig. 7.3. The values for 0 3 and a , on the other hand, are obtained by comparing Yip,, with the Feynman rules for the j4 -terms. 2

3

4

3

7 . 4 . Determination of Renormalization Constants In this subsection we derive the renormalization constants for Green functions in the light-cone gauge. The presentation follows the elegant treatment in Nyeo and effectively starts from the counterterm action, A S = Y + o-X, in Eq. (7.13). 22

,

2

We begin by casting S = / dxL' and Y = f
IS

BA'Jx)

SL'Jx)

6W(x)

a

6K (x)

2Y = j d x ^ A l W - J ^ + ^K'ix) SA'Jx)

d a

:

6K (x)_

- 9

- a,g

as dg

ds_ dg

(7.18)

(7.19)

respectively. In order to evaluate trX, we operate with tr from Eq. (7.14d) on the various components of X in Eqs. (7.14b) and (7.14c). Recalling that "a = 1 5 = 0, we obtain the intermediate expressions 6X 6K°(x)

= 0,

(7.20a)

SX = 0, Sw (x)

(7.20b)

a

J

Rtnormalization

in the Preaence of Nonlocal

Terms

(7.20c) 6X

cotcr„*a / „

a \-l„

i*'W_.. a.\—i_

r*.

(7.20d) where the minus sign in the last term of Eq. (7.20c) arises from integration by parts. Substitution of Eqs. (7.20c) and (7.20d) into o~X yields:

/

SS dx ^{a n' Alnp 6Aa

z

l

+

v

a [n d {n d )- n A ]n i

g

a

T

T

e

v

fl

SS

/ abc

+ + a6ff/

1

1

a gf nldAnrdr)- [^Ai(n^)- n L }n s

o t c

p

1

p

u

l

K^(n 3 )- n ,A:](n a )- n,i> } . T

r

t

A

A

u

(7-21) Combining Eqs. (7.19) and (7.21), we get:

Woneowan'onl Gauges

116

2AS = 2Y + 2cX

c

I

1

-2a r» (n ,9 )- {K9 (n 3 )- n , 4;]r Ann^ 6 9

)

( )

f f

T

r

a

+ j dxjj^{2 n L (x)n' a3

v

r

1

J

l A

+

ll

ie

1

la^n'Mnrdrr^LKx)]^ 1

+ 2a r n;c3 ,(n 9 )" K<("A^)" "^^]n . 6j?

(

l

T

T

1

/

1

+ 2a 9r >lM»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<

+ (n • F )

4

n F

p' n p- F -2p F,F„-rpF(p p n

(8.3)

2

3

where K and the null vector F„ are denned in Eqs. (7.16) and (5.50), respectively.

q-k F i g . 8.1. Pincer diagram for the one-loop contribution to Wavy lines correspond to Yang-Mills fields.

bc

E {p) ¥

in the planar gauge.

The result in Eq. (8.3) looks certainly more intimidating than in the light-cone gauge: there now appear extra nonlocal terms proportional to either n or n . The challenge is to find suitable BRS-invariant connterterms that will match precisely all divergent local and nonlocal expressions in Eq. (8.3), a task further complicated by the non-trans vers ality of [pi**"*' 2

4

Connterterma

in Ike Planar

123

Gauge

+ « > ^ 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<]n D?F%,

n •

A']n D?F*„

a

- a [ F • 8(n • 0 )

- 1

3

Gauges

u

H

l

- ovalF • $fn • d)~ F

• A']F D?F*

1 + (A5)

a

gho

.t •

J

(8.11)

It is worth noting that the individual terms in (AS) ho«t contain only single g

A'B*

(AS)

g h o s t

ab

= J dx!-

aigf 'A

abc

a

b

a

- af F

b

• An • Lu

ig

c

abc

• Lw c

- a gf n alc

- asgf F

b

• A"F •

3

• A"F -

e

Lu b

Lu

e

-aegr^F-din-dr'ADLlu' aic

• 3{n • d^n

bc

• 6(n • d)~ F

- a gf [F 7

a

l

- &9 f [F abc

aic

a

l

e

Lu

• A*)F • i V

• d(n • ty^F • A ]F • L*hi'

10

L

e

a

- a gf [F a i

a

• d(n • d)~ n

9

h

• A ]n •

l

- a gf [F

-

a

• A \n • L w

• D w" - a F • L"JI • D w ab

ab

b

3

- a n • L"F • A

- a F • L"F • D*w* - a [F • d(n • 8)' 6

_ 1

ab

b

a

1

ai

- a [F

9

+ ±«

10

• d(n • d^'F

a

• L ]F •

c

l l j 7

b

s

- a [ F • 3(n • S ) " / • £ > • D J llU

ab

- a [F • d(n • fi)-*n • L ]F • D w

7

1

b

LflD^u*

6

- a [F • d(n • a ) n • L']n - D u

- a 'D?L\,

1

at

Dw

r* ff°uV} .

(8.12)

To determine the at,i = 0 , . . . ,11 for the two-point function, we extract from Eq. (8.11) all terms proportional to A \,. .]A and Fourier-transform them to momentum space (9 —* ip ). The counterterms in momentum space corresponding to the various terms in AS are listed below: u

M

U

u

6

al

b

Du

Coimttrlermi in (At Planar Gauge Term in A S

127

Counterterm in momentum space

- a ^ D ^ F * , <>2[n • piPuhn + p n „ ) - 2 p n n , ] 2

u

-o3Ti •

b

b

A"F D* F ll

o [ F - pfp^riL, +

u

3

(i

pvtip) — p ( n F „ + n „ F | ] 2

H

2

o [ p - n ( p ^ F + p F„) -p (n F„ 4

-

o

F - A'F^DfF^

s

t

u

t

M

)J

1

pS )

(p„p,, -

6

+ n F )]

+ p ^ ) - 2» F F„]

u

2a ^

H

M

a

as[p • F(p^F

-^[F-8[n.a)-'4;iDX

t

llv

p-n -

n

7

[ F . 3 ( n . a ) - ' n . ^ > ^

- 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

+ <« " l)*}

(9.6)

and then defines n„ = (1,0,0,0) to deduce the photon propagator in the Coulomb gauge (a —* 0):

Noncovariant

Ganges

9ft9v T } ' i ) ( , n 1 . +gt- '<.)1

—»

,

(9.7) with 2

3

A>o(? ) = i / q ,

(9.8)

where g = q — g , and q n = q •») - goWn = -9o"o- The propagator (9.7) gives rise, as expected, to noncovariant-gauge integrals of the type 2

2

2

/

4

2

2

M

2

t» q [g -r-2g.p+A/ ] ' 2

which are characterized by the factor 1/q . We shall see in Sec. 9.4 that absence of the g^-component in the denominator of 1/q leads to severe difficulties in non-Abelian theories beyond the two-loop level. Heckathorn's one-loop Q E D computations are obviously immune to this problem, since all required integrals can be calculated consistently with dimensional regularization. 2

19

Our next example surveys two papers by Sapirstein on the determination of binding corrections in the Coulomb gauge. His first article describes a novel approach for calculating the self-energy contribution to the Lamb shift, while the second one deals with one-loop corrections to the ground-state hyperfine splitting in hydrogenic atoms. Sapirstein remarks that the electromagnetic mass shift of the electron could have been obtained in a covariant gauge, of course, but with considerably more effort. 13,14

14

Detailed Coulomb-gauge computations may also be found in Adkins who derived one-loop expressions for the self-energy function and self-mass of the electron, and for the renormalization constants Z\, Zo.. In another paper Adkins used a compact, albeit un-integrated, form for the vertex function to deduce the one-loop vertex correction to the decay rate of p arap osi t roniu m . 11

l a

9.4. Recent Developments In 1986 Cheng and Tsai re-examined the operator-ordering problem of Schwinger and Christ and Lee from a different perspective. Working with an interpolating Feynman-Coulomb gauge and realizing that Gauss' 4

5

20-23

133

Tkt Coulomb Gauge

law is not an operator relation, Cheng and Tsai confirmed the presence of the anomalous Vi, t^-terms of Christ and Lee. They argued that the Vi, ^-contributions are indispensable for achieving gauge invariance and equivalent to divergent energy-integrals of the form 5

23

E= f f ° go - J 2r J 2 T ( p g - | p | i + i O ( « g - | q P + « ) " d p a

dq

P

o

/A

I

N

^

-.

"

J

These integrals emerge for the first time in two-loop diagrams and cannot be regulated by the continuous dimension method. The approach of Cheng and Tsai was soon generalized by Doust who stressed the importance of replacing the Lagrangian phase-space formulation by a Hamiltonian formalism. Doust distinguished between two types of energy integrals: linearly divergent expressions such as 19

and two-loop integrals of the form **-] F

f Po f ffoo Po 2*} 2n(p -\p\

go

d

2

2

2

+ h)( -\ \ q

q

2

, *

Q

+ i )C

1

, *

9

J

Whereas the i?i-integrals may, with a phase-space Hamiltonian, be cancelled systematically to all orders of perturbation theory, no such luck prevails in the case of E . The latter first appears at two loops and is generated precisely by the ope rat or-ordering related Vi, V^-terms of Christ and Lee. Doust and Taylor have demonstrated that the ^-integrals may in fact be eliminated to two loops, provided the integrals over the three-momenta converge. They achieve cancellation by inserting quark loops in all possible ways into any ghostless one-loop diagram that contains energy divergences of type Ex, Unfortunately, this clever procedure cannot be extended to three loops and higher loops. The major difficulty appears related to the fact that the Coulomb gauge generates ultraviolet divergences at 2w = 4, as well as divergent energy integrals such as E\, E , and there is currently no prescription capable of handling both types of divergences in a satisfactory manner. One obvious implication of this embarrassing situation is that Green functions in the Coulomb gauge have not been shown to be renormalizable. Clearly, much more effort, coupled with some novel, perhaps even radical concepts, is required to place this gauge on a rigorous mathematical footing. Further discussions on the Coulomb gauge can be 19

19

2

24

2

24

134

Noncovariant

Gauge*

found in the papers by Frenkel and Taylor, and Halpern.

25

Burnel,

26

Nyeo,

27

and Chan

28

References 1. J . C. Taylor, in Physical and Nonstandard Gauges, eds. P. Gaigg, W. Kummer and M. Schweda, Lecture Notes in Physics, Vol. 361 (Springer-Verlag, Berlin, 1990) p. 137. 2. K. Johnson, Ann. Phys. (NY.) 10, 536 (1960). 3. C. R. Bagen, Phys. Rev. 130, 813 (1963). 4. J . Schwinger, Phys. Rev. 127, 324 (1962). 5. N. B. Christ and T. D. Lee, Phys. Rev. D22, 939 (1980). 6. R. N. Mohapatra, Phys. Rev. D4, 378 (1971). 7. V. N. Gribov, Materials for the XII Winter School of the Leningrad Nuclear Research Institute (1977), Vol. 1, p. 147. 8. V. N. Gribov, Nucl. Phys. B139, 1 (1978). 9. S. Mandelstam, Lecture at the American Physical Society Meeting, Washington, D.C. (1977), unpublished. 10. I. M. Singer, Comm. Math. Phys. 60, 7 (1978). 11. R. Jaclriw, I. J. Muzinich and C. Rebbi, Phys. Rev. D17, 1576 (1978). 12. I. J. Muzinich and F. E. Paige, Phys. Rev. D21, 1151 (1980). 13. J. R. Sapirstein, Phys. Rev. Lett. 47, 1723 (1981). 14. J. R. Sapirstein, Phys. Rev. Lett. 51, 985 (1983) . 15. R. Utiyama and J. Sakamoto, Prog. Theor. Phys. 55, 1631 (1976). 16. D. Heckathorn, jVuci. Phys. B156, 328 (1979). 17. G. S. Adkins, Phys. Rev. D27, 1814 (1983). 18. G. S. Adkins, Phys. Rev. D34, 2489 (1986). 19. P. J . Doust, Ann. Phys. (NY.) 177, 169 (1987). 20. H. Cheng and Er-Cheng Tsai, Phys. Rev. Lett. 57, 511 (1986). 21. H. Cheng and Er-Cheng Tsai, Phys. Lett. B176, 130 (1986). 22. H. Cheng and Er-Cheng Tsai, Phys. Rev. D34, 3858 (1986). 23. H. Cheng and Er-Cheng Tsai, Phys. Rev. D36, 3196 (1987). 24. P. J. Doust and J. C. Taylor, Phys. Lett. B197, 232 (1987). 25. J. Frenkel and J. C. Taylor, Nucl. Phys. B109, 439 (1976). 26. A. Burnel, Phys. Rev. D32, 450 (1985). 27. S.-L. Nyeo, Phys. Rev. D36, 2512 (1987). 28. H. S. Chan and M. B. Halpern, Phys. Rev. D33, 540 (1986).

C H A P T E R 10 CHERN-SIMONS THEORY 10.1. Background Most of the discussion in the preceding Chapters dealt with noncovariant gauges in four-dimensional Yang-Mills theory. In this Chapter we shall apply the light-cone gauge to another popular gauge model, namely ChernSimons theory in three dimensions. The transition from four to three dimensions in not necessarily a trivial one. There is ample evidence in the literature to suggest that changing the dimensionality of space-time can alter the underlying theory or the nature of its solutions. Here are two examples. First, let us recall the Einstein-Hilbert Lagrangian density for pure gravity, 1

£ = i(-ff) 'V*iW,

9 = det ffr ,

(Id)

rG 2

where g"" is the metric tensor, fl the Ricci tensor, tc = 32wG, G being the Newtonian constant, and g is defined by g^^g"" = 6°. Replacing jf"* by the tensor density g " (weight = +1), uv

1

uu

u

?*ft/V=*,

(10-2)

we may express L in the more convenient form 1

=

W

~ - j j ^ h ^

~

1

Au

2 0 j v - ) 9 ,i> 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
51

,

(10.11)

to the D-dimensional version of Eq. (10.7); m is called the regulator mass. The D-dimensional definition of the e^-tensor is given in Eq. (10.17). The final action 5 ( D ) , m

S {D) m

= S(D) +

SY {D) M

D

= J dx je*" (

-^-{n 2

• A'f

a

^

M

he

; +

~r A' A A^j a

a

p

+ Qn • D " V - J - f - F"""! , J 4

m

a -

0, (10.12)

leads to a proper inverse of the kinetic term and permits regularization of the UV-divergent Feynman integrals by analytic continuation to complex values of D. Although S (D) breaks the D-dimensional Lorentz symmetry 50(1, D- 1), i t still possesses 50(1,2) ® 5 0 ( D - 3) symmetry. Moreover, 5 ( D ) remains invariant under infinitesimal BRS transformations, m

m

Chern-Simont

sA° = D»*w*

Theory

141

a

su> = —n - A" ,

a

sw = - | / * H r V ,

(10.13a) 62

s being the traditional D-dimensional BRS transformation. However, S {D) is no longer invariant under the following additional BRS-type symmetry of the action in Eq. (10.7): m

58

s^Q" = - A°,

Spu" = 0 ,

a

a

s b"=d Q , a

a

b = n-A

u

»„Al =-t^n'u*

,

.

(10.13b)

We are not aware of any regularization method that simultaneously preserves the symmetries in Eqs. (10.13a) and (10.13b). Regularization methods which explicitly preserve (10.13b), but break BRS invariance, have been studied in Ref. 58. To derive the Green functions G°»;; ° ; ( P i , . . . ,p„) of SU(N) ChernSimons theory, we apply the following double limit to the Green functions G£}"//J*(pi,... , P n ; , D) which arise from the action in Eq.(10.12): m

Cr*£g(Pi

Pr.) = Jim^ H m G ^ ; ; . ; ; ( p i , . . . , p „ ; m , D ) . 3

Although the existence of this limit has not been demonstrated in the case at hand, it may nevertheless be inferred from the U V finiteness of our model. We shall presently see that the double limit exists in several important instances. The D-dimensional action in Eq. (10.12) leads to a complicated propagator for the gauge field. However, since we are only interested in the D —* 3 limit of our dimensionally regularized Feynman diagrams, it suffices to work with the effective D-dimensional propagator shown below. Use of such a simplified propagator requires, of course, a careful study of the complete propagator in the limit as D —• 3 . In the sequel we shall employ the following Feynman rules: 63

gauge propagator.

"/in

(,)=

r

ab 2

q —m

2

9u« + —

[—im n°Cp

av

+ q^n,, + q^n ) u

; (10.14a)

Noncovariant

142

Ganges

ghost propagator. ai

G"\q) = -iS /q

•n ;

(10.14b)

three-gauge vertex: V;^(p,q,r) = -9f

+ ^((s -

aic

r

+ ( - p)"ff"f + &~

r

)»°»e

9ha ») a

; (10.14c)

four-gauge vertex:

*

*

m ace

bd

+ f f '(g^g

ade

- g^g.c)

afi

bt

+ f r (g^g

- g^a.p)}

vP

;

(10.14d) ghost-gauge-ghost vertex: V?

C

aic

=-igf n

a

.

(10.14e) -1

Notice the appearance of the spurious factor ( 5 ' r t ) in the gauge and ghost propagators, and the dependence in Eqs. (10.14a,c,d) on the regulator mass m. These Feynman rules are supplemented by a number of technical rules, namely: (a) Application of dimensional regularization in complex D-dimensional space-time, thereby guaranteeing BRS invariance of the regularized theory. (b) Use of prescription (5.14) for the spurious poles: _ L - lim -, q•n f—o q • nq • n" + if

e> 0 .

(10.15)

(c) Reduction of the integrated expressions to manageable form by means of Martin's identities, such as 58

-L-n*t

aav

= ~

5

(

' »'

f

+

~

5 T n

<

'(!i''"'"»" (">)]

• (10.16)

Ckcrn-Simoti)

Theory

143

(d) For treatment of the £,,,,„-tensor, adoption of the 't Hooft-Veltman recipe in the context of dimensional regularization, that was later systematized by Breitenlohner and Maison, and Collins. The completely antisymmetric epsilon tensor satisfies 64

65

66

Suji-i S M J I ' I 9iiii/2 9nsvi 9>llV3 9 list's

fffiiva 9II1KJ

(10.17)

4 4 , 6 1 , 5 1

where g is the three-dimensional Minkowski m e t r i c . (e) Computation of the relevant contribution coming from Feynman integrals by means of the limiting procedure uv

51

(10.18) which is illustrated in the next section by two examples. 10.3. Massive C h e r n - S i m o n s I n t e g r a l s The computation of massive Chern-Simons integrals differs from conventional calculations in four-dimensional Yang-Mills theory essentially by the limiting procedure (10.18). In the examples to be discussed below, the integrals are functions of the regulator mass m and are initially defined over D dimensions. E x a m p l e 1. Consider D

dq

D

q"

dq

which is UV-divergent by power counting.

We begin by rescaling the

momentum variable q^ —> mq^,

=

m

° ~

2

1 * 7 - 2 - ^ 3

J

(« -l)ff-n

and then make the ansatz

/

2

(9 - i ) g

n

•

(

1 0

-

2 0

>

144

Noncovariant

Gaugti

The coefficient A is actually zero on dimensional grounds, since A would have to be proportional to ( n ) which is ill defined in the light-cone gauge; therefore, 3

- 1

S^T^W-n

( 1

=

°

2 2

»

Multiplying Eq. (10.22) by n so that a

1

B = (ft . n - ) " J dqtf - I ) '

1

,

and integrating by dimensional regularization, we obtain in the limit D —• 3 the finite value 2

B = - —/, n •n

/ = I/(8T) .

(10.23)

Hence,

E x a m p l e 2. The UV-divergent integral q" dq-r-z TTT ; — , ( g m ) (o-p)-n / 3

2

1

p* = external momentum,

(10.25) '

v

is similar to J " in Eq. (10.19), but its evaluation requires more care. Setting (? P)u = Qu, and repealing —» mQ , so that _

p

J » -

M

D - 2

fQ

W+P")

d

Q

J

((Q

=,-P" P

+ P)*-W»'

we next employ the identity p)2_, (Q + P)

=

m

1

(

1

0

2

6

)

51,61

Q l - l ^

(Q + p ) 2 - l )

'

(

1 0 2 7

>

to obtain D

J* ~ m "

3

Q

fdC

V

J

V

"

(\

2

(Q -l)Q

(JR^±fl\

n ^

(Q + p )

3

- l

(10.28)

Chern-Simons

145

Theory

The first integral on the right-hand side of Eq. (10.28) has already been evaluated in Example 1;

/ V-°i)g j

=

» ^ " ^

( 1 0

2 9 )

<

i o 3 o

the fourth one is zero since it is odd in Q",

h f f - V ,

=

i

'

2

»

61,67

while the integrals proportional to p vanish in the limit of large m. Operating with l i m l i m on the remaining two integrals and applying once m-foo D —

'J

more identity (10.27), we get Jn = l i m l i m /** = l i m l i m m—-oo D—3

x

D

{ -

-

m-.co

2

2

^ -

- 2 ^ - V p , / ^

m

(

D-.3

D

g

2

^ / -

(

Q

- i )

2

g . n (1031)

2

The first integral in Eq. (10.31) vanishes by symmetric integration (we replace Q"Q by 3^ Q V ) * i.e. V

V

2

The second integral is UV-finite,

but proportional to 1/m as D —• 3, so that lim

D

lim m -

m-000-3

4

/ JO . = lim (-—I-n'A J (Q - 1) Q • n n-»\mn'n' 7

2

/

= 0. (10.34)

Therefore, J " has the same finite value as J": / J

•! (q - m ){q - p) • n 2

2

.

=

n • »*

(10.35)

Noncovariant

116

Ganges

We stress again that the limit m —• oo should only be taken after the contributions from all diagrams of a given order have been summed. This completes our illustration of the integration procedure (10.18). All integrals listed in Appendix E were computed by the same method. 10.4. T h e V a c u u m Polarization Tensor

To familiarize ourselves with the oddities of perturbative Chern-Simons theory in the light-cone gauge, we shall summarize the computation of the vacuum polarization tensor rj°* to one loop. We shall see that the answer for is surprisingly simple, and that all major technical aspects, such as prescription (10.15), dimensional regularization and the idiosyncrasies of light-cone gauge physics go through just as in Yang-Mills theory. The diagrams contributing to the vacuum polarization tensor are depicted in Fig. 10.1. The first diagram, Fig. 10.1(a), yields 61

2

al

= Cg6

I -£-Xy ,

v

(

„ 0 , 5, -q - p)^ Sl

( l l

1

Vlll

+ P)V

UVlU3

{-p -q,q

+ p) ,

>

*

(10.36)

2cab L - a = cg 6 8lp e v

1

6ip • H* —n n•n

a i i v

I,

I = */(8*) ,

(10.37)

where A ^ g ) and ^ ( p , f , r ) are listed in Eqs. (10.14a) and (10.14c), respectively, and f 'f = c 6 \ c„ is the Casimir operator of SU{N) in the adjoint representation. The tadpole diagram in Fig. 10.1(b) gives a

ad

kic

ab

v

n

( 2 , a l

llpr

( )= P

W

f ^ L v ^ ^ M J (2TT)

=

2

cgS v

at

D

W4m M T W W, '

—7 K ^ + l a O (10.38)

Chern-Simona

147

Theory

1/2 x

0) F i g . 10.1. One-loop contributions to the vacuum polarization tensor, diagram, (b) Tadpole diagram. T h e wavy lines represent gauge fields.

(a) Self-energy

which has just the correct magnitude and sign to cancel the corresponding expression in Eq. (10.37), i.e. prior to letting m —* -f-oo. Accordingly, we see that n i , ' " ' " is indeed independent of m and UV finite: 1

1

_.(!o[)ai

, _.(l)at>

=

, _.(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 —<• k + c„ sign (£), where it = 4 J T / I J , is identical to the shift obtained with a covariant gauge. The following discussion is based on reference. 2

61

Introducing source fields J£, K" of mass dimension 1, 1 and ghost number —1, —2, respectively, we may write the one-loop effective action as r( 4) = r J

where r

0

= s, c

+ J dx 3

0

+ r _ 1

[j^u)"

l o o p

-

:!

(10.40)

^/^K^V

(10.41)

+ o(n ),

and r i _ i p satisfies the BRS equation 0 0

o-ri_ , (A,u,Q;J,K)

(10.42)

= 0;

lr ofr

IT is the nilpotent BRS operator (cf. E q . (7.10)) £r

0

+

STp S

SR* £w° +' Su" 6K-

2

rr = 0 . (10.43)

The solution of the cohomology problem (10.42) reads fl-loop =

cS , c

+

cX

,

(10.44)

where X is an arbitrary, nonlocal integrated functional of the fields A", W", w", J£, K", with mass dimension 2 and ghost number - 1 ; the unknown

Chera-Simona

Theory

149

constant c preceding S , can be determined by evaluating the one-loop three-point function r " ' ^ (Sec. 10.6). Notice that the gauge-noninvariant contribution to Ti-ioop comes entirely from oX in Eq. (10.44), while S , is, of course, gauge invariant. It turns out that X has the structure c

c

61

X = ja^xp^^ a

with J

+ XiA^.Q-lK)

,

(10.45)

and u>° 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,

« <-i(9)K. «« (3,P3>-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 <

B

J

( P 2 " «)

im

— [ - i m r t ' E , , , ^ , + ( p - g ) , , ^ , -t- ( p - q),,^,] 2

C(p2

g)

T7i )

,

2

(

1

0

6

6

)

Noncovariant

154

Ganges

with similar definitions for A ^ ( ? ) and A j ^ f g + p ); the vertices V have already been defined (cf. Eq. (10.61)). Notice that the basic integral associated with the amplitude in (10.65) contains the maximum number of three spurious factors: u t

3

apy

1 2

* (9 -

2

™ )((P2 -

2

l)

2

3

~ u ) ( ( g + p ) - rn*)q • n(p - s) • n(q + ps) • n' (10.67) 3

2

tL

Accordingly, rJ,Vw is characterized by double nonlocalities of the type — a n d — , while the remaining terms in Eq. (10.64) contain only single poles of the form ,3 = 1, 2,3; the pj's are external momenta. This analysis completes our application of the light-cone gauge to perturbative Chern-Simons theory. For additional details, the reader may wish to consult. 63

References 1. D. M. Capper, G. Leibbrandt and M. Ramon Medrano, Phys. Rev. D8, 4320 (1973). 2. J. N. Goldberg, Phys. Rev. I l l , 315 (1958). 3. G. L. Lamb, Jr., Phys. Lett. A25, 181 (1967). 4. A. Barone, F. Esposito and C. J. Magee, Riv. Nuovo Cimento 1, 227 (1971). 5. A. C. Scott, F . Y . F. Chu and D. W. McLaughlin, Proc. IEEE 61, 1443 (1973). 6. S. Coleman, Phys. Rev. D l l , 2088 (1975). 7. G. Leibbrandt, Phys. Rev. B15, 3353 (1977). 8. G. Leibbrandt, J. Math. Phys. 19, 960 (1978). 9. A. V. Backlund, Math. Ann. I X , 297 (1876). 10. A. V. Backlund, Math. Ann. X V I I , 285 (1880). 11. A. V. Backlund, Math. Ann. X I X , 387 (1882). 12. L. P. Eisenhart, Differential Geometry of Curves and Surfaces (Dover, New York, 1960). 13. G. Leibbrandt, Phys. Rev. Lett. 41, 435 (1978). 14. G. Leibbrandt, R. Morf and S.-S. Wang, /. Math. Phys. 21, 1613 (1980). 15. S.-S. Chern and J. Simons, ^nrt. Math. 99, 48 (1974). 16. A. S. Schwarz, Lett. Math. Phys. 2, 247 (1978). 17. A. S. Schwarz, Comm. Math. Phys. tJ7, 1 (1979). 18. D. B. Ray and I. M. Singer, Adv. Math. 7, 145 (1971).

Chern-Simons

Theory

155

19. V. F. R. Jones, Ann. Math. 126, 335 (1987). 20. P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickoiish, K. MUlett and A. Ocneanu, Bull. Am. Math. Soc. 12, 239 (1985). 21. E . Witten, Comm. Math. Phys. 121, 351 (1989). 22. G. Moore and N. Seiberg, Phys. Lett. B220, 422 (1989). 23. M. Bos and V. P. Nair, Phys. Lett. B223, 61 (1989). 24. G. V. Dunne, R. Jackiw and C. A. Trugenberger, Ann. Phys. (N.Y.) 194, 197 (1989). 25. S. Elitzur, G. Moore, A. Schwimmer and N. Seiberg, Nucl. Phys. B326, 108 (1989) . 26. T. P. KiUingback, Phys. Lett. B219, 448 (1989). 27. S. Axelrod, S. Delia Pietra and E . Witten, /. Diff. Geom. 33, 787 (1991). 28. J. M. F. Labastida and A. V. Ramallo, Phys. Lett. B227, 92 (1989); Phys. Lett. B228, 214 (1989). 29. J. FrShlich and C. King, Comm. Math. Phys. 126, 167 (1989). 30. P. Cotta-Ramusino, E . Guadagnini, M. Martellini and M. Mintchev, Nucl. Phys. B330, 557 (1990). 31. E. Guadagnini, M. Martellini and M. Mintchev, Nucl. Phys. B336, 581 (1990) . 32. E . Guadagnini, Int. J. Mod. Phys. A7, 877 (1992). 33. K. Gawedzki and A. Kupiainen, Comm. Math. Phys. 135, 531 (1991). 34. R. Jackiw and S. Templeton, Phys. Rev. D23, 2291 (1981). 35. J. F. Schonfeld, Nucl. Phys. B185, 157 (1981). 36. S. Deser, R. Jackiw and S. Templeton, Ann. Phys. (N.Y.) 140, 372 (1982). 37. R.D. Pisarski and S. Rao, Phys. Rev. D32, 2081 (1985). 38. J. D. Lykken, J. Sonnenschein and N. Weiss, Int. J. Mod. Phys. A6, 5155 (1991) . 39. D. S. Freed and R. E. Gompf, Phys. Rev. Lett. 66, 1255 (1991). 40. M. F, Atiyah, The Geometry and Physics of Knots (Cambridge Univ. Press, Cambridge, 1990). 41. E. Guadagnini, M. Martellini and M. Mintchev, Phys. Lett. B227, 111 (1989). 42. L. Alvarez-Gaume, J. M. F. Labastida and A. V. Ramallo, Nucl. Phys. B334, 103 (1990). 43. M. Asorey and F. Falceto, Phys. Lett. B241, 31 (1990). 44. C. P. Martin, Phys. Lett. B241, 513 (1990). 45. W. Chen, G. W. Semenoff and Yong-Shi Wu, Mod. Phys. Lett. A5, 1833 (1990). 46. D. Birmingham, R. Kantowski and M. Rakowski, Phys. Lett. B251, 121 (1990). 47. A. Brandhuber, M. Langer, O. Piguet and S. P. Sorella, Phys. Lett. B300, 92 (1993). 48. G. P. Korchemsky, Mod. Phys. Lett. A6, 727 (1991). 49. D. Bar-Natan, Perturbative Aspects of Chern-Simons Topological Quantum Field Theory, Princeton Univ. Ph.D Thesis, 1990.

156

Noncovariant Gauges

50. D. Daniel and N. Dorey, Phys. Lett. B246, 82 (1990). 51. G. Giavarini, C. P. Martin and F. Ruiz Rub, Nucl. Phys. B381, 222 (1992). 52. E . Guadagnini, M. Martellini and M. Mintchev, Phys. Lett. B228, 489 (1989) . 53. D. Bar-Natan, Perturbative Chern-Simons Theory Princeton University Preprint, May 1990. 54. M. Alvarez and J. M. F. Labastida, Analysis of observables in Chern-Simons perturbative theory, US-FT-10/91 preprint. 55. A. Bias and R. Collina, Nucl. Phys. B345, 472 (1990). 56. F. Delduc, C. Lucchesi, O. Piguet and S. P. Sorella, Nucl. Phys. B346, 313 (1990) . 57. S. Emery and O. Piguet, Helv. Phys. Acta 64, 1256 (1991). 58. C. P. Martin, Phys. Lett. B263, 69 (1991). 59. D. Birmingham, M. Blau, M. Rakowski and G. Thompson, Phys. Rep. 209, 129 (1991). 60. P. A. M. Dirac, Lectures on Quantum Mechanics (Yeshiva Univ., New York, 1964). 61. G. Leibbrandt and C. P. Martin, JVucf. Phys. B377, 593 (1992). 62. C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980). 63. G. Leibbrandt and C. P. Martin, Nucl. Phys. B416, 351 (1994). 64. G. 't Hooft and M. Veltman, Nucl. Phys. B44, 189 (1972). 65. P. Breitenlohner and D. Maison, Comm. Math. Phys. 52, 11 (1977). 66. J. C. Collins, Renormalization (Cambridge Univ. Press, Cambridge, 1984). 67. Y . Hahn and W. Zimmermann, Comm. Math. Phys. 10, 330 (1968). 68. G. Leibbrandt and S.-L. Nyeo, Nucl. Phys. B276, 459 (1986).

APPENDIX A COVARIANT-GAUGE FEYNMAN INTEGRALS The following list of 2w-dimensional integrals is divided into two categories: the integrals in the first category (Sees. A . l and A.3) hold for massive particles (m ^ 0), whereas those in the second category (Sec. A.2) are valid for integrals associated with massless fields (m = 0). 1. Massive integrals Formulas (A1)-(A6) below are taken from Appendix A in Ref. 1. I n transferring them we have, for the sake of consistency, replaced the complex variable n by 2u and divided each integral by (2ir) . These integrals hold for m ^ 0, and y arbitrary. 2w

2

/ J ( 2 j r ) ( o + 2k • q + m ) 2w

f J (2x) »(q 2

2

2

. a

r

' (4ir)"(m 2

'

<»-") T(a)

d^qg, _ i + 2k q + m )° (47r)"(m - k ) -*

2

2

d** q

2

2

Tjc-u) T(a)

a

f

A

l

1

K

'

1 (

'

'

A 2 }

2

9

2

2

/ ( 2{2w) * »(q

+ 2k • q + m?Y

"

r

(

^ . - . | . ' * r ( . - . - . H . ' - *

u

f (P q q q„ J ( 2 T ) * - ( j + 2k • q + m )

i

u

a

2

x |r(c - w)k k u

v

a

2

2

a

(4jr)"(m - k ) ~

+ T(a - 1 - u - ) i ^ ( m u

157

2

w

- k) j 2

_J_ T(a) ,

(A4)

Noncovariant

158

Gaugci

«P"g g^qx

f 2

2

3

J (2jr) <"(fl + 2* • q + ml)" x f

1

>

(45r)"(m -

' I »

w)kpk kx - T{a - 1 - w) u

x

+ «rA**)(™ - * ) J , a

+

2u

2

2

2

(A5)

2

J (2TT) '(3 + 2fc • g + m )«

(4jr)"'(m -

T(a)

2

x (-k ){T(a

2

2

- « ) * + T(a - 1 - w)(w + l)(m - it )} .

M

(A6) 2. Massless Integrals The following massless integrals arise in the treatment of Yang-Mills theory and quantum gravity. There are no factors of i present, since all integrals are defined over Euclidean space. 2

1 (4ir)T(2u - 2)

d^q 2

/

{2^q (q

- p)

2

2

l)(p r-

x T(2 - w)T( - lJFfw U

J

( 2 ^

2

(

9

2

- P )

3

2

2

=

=

(2n)^q ( -p) q

P

" '

*»"

,

(A7)

'

2

h

=h

2

+

{

P

P

""

/4

{

'

A

A

9

8

)

)

2u

d g q^ q q v

2

/

j

y

2

(2n) "q (q- y

=Vr>P»Pih

P

f

+ E^I

6

,

(A10)

d^t1 gggHErgg . ,/i r , rr T (2ff) D (g - p) ~ PpPvPlP" ! + We70 o + Huvjah , d

2u

2

r

2

1

J

where = ^ p + tf^p,, -I- 6-,„p„ , t

7

(A12)

Appendix A

159

Gpy-fB = buvP-tPo + $f-tP(LP<7 +tivgPpP-i+ 6 yPl>Po + tpo-Pi/Py + G-fuPpP (A13) u

Huvyo- = $pv$-fa + t> i/,o'v + fW
(A14)

y

The integrals I 2 , . . . ,1c, have the following simple structure in terms of the basic integral Iy in Eq. (A7): h =

3

'

J

J2 l i ,

(A15)

= 74(2w 7 ^ -- 71) 1 'i.

-» = 2(2u n / o "-

z

1r; ) i •

i) . = ±( f±+ A 4(2w - 1)

h

(A16) (

,

h

A 1 7

>

(Ai8)

-P 2 1) i > = 8(2w , » - n 2

If

v

7

(

a

A l f

0

4(4w - 1) 2

2

-tV + 1)P 8(4w2 - 1)

ft = 16(4^2 i « , 2 - 1) H i • 7

(A22)

3. T h e Divergent P a r t s of Some Other Massive Covariant-Gauge F e y n m a n Integrals The integrals below occur in the computation of the quark self-energy and the quark-quark-gluon vertex functions and are defined over Euclidean space. The symbol f has the following meaning here: 3

d i v

/ Hence,

d i v

= divergent part of /

f

q

=

.

160

Noncovariant Gavget

^

fo

/

(

-

t

^

"

V

-

^

]

= ^

+

' '

)

I

f

e

(

'

(>

A

2

4

)

( A 2 5 )

/( -,)w-»rl -"-'*' t

3) [(9 + P ) - ™ ] 2

2

2

= s ( 2 t t „ - 4 , l > „ - t p > + aiVft) / ( i

# 1

d i v

2

2

+ ^fv[3m -(p+i) ]

b

1 J

I * (A26)

div

6

/

div

1

2

(A27)

9 ( * - g ) [ ( 9 + p) -m>] 3

2

2

= A12-1 L W * * - P*) + W * * - P„) + 6»,(*r

3

~ P*)] r "

,

(A29)

rf * 2

9 {*-«) [(*-p-*) -m ] 2

2

2

2

= T12-1^ W Z ^ + P ^ H W 2 * „ + p.0 + M 2 j f c + p ) ] J * * , (i

Hv

/

/J

(A30)

d^qpqiq, * (*-«) [(9 + p) -m ] 3

j( = P^T^,

2

2

2

7o are Dirac matrices,

(A31)

Appendix A

j

iiv

2

2

161

2

2

? (*-
(A32)

,^

d i v

Iiv

=

+ * • PT„ +

+ *h + Jfeu I*" •

(A33)

References 1. G. 't Hooft and M. Veltman, Nucl. Phys. B44, 189 (1972). 2. D. M. Capper, G. Leibbrandt and M. Ramon Medrano, Phys. Rev. D8, 4320 (1973). 3. G. Leibbrandt and S.-L. Nyeo, Phys. Rev. D39, 1752 (1989).

This page is intentionally left blank

APPENDIX B MASSLESS A X I A L - T Y P E INTEGRALS IN T H E PV-PRESCRIPTION We list the divergent parts of some massless one-loop integrals in the axial-planar gauge (cf. Chapters 4 and 5). Here n ^ 0, d^q = dq, and / ' is defined by (see either Eq. (3.18) or Eq. (A7)): 2

d

v

d i v

f

2

2

= divergent part of / dq[q (q - p ) ] " 2

1

,

5r /(2 — w),

Euclidean space,

J V / ( 2 — w),

Minkowski space.

2

The integrals listed below have been collected from Ref. 1: dq

2p • n

2

(4~P) 9

n

dq q? (q - p) q • n

I

_2pn

2

f

dq

dq q (,-p)

fafr

2

2 9

n

-2pnp

2

{{p-n)\

-2pnp

2

/ 2 ( « + l)(p • n )

3n

2

V

2p - n p

2

!

P «

/ ^ ( p r . ) p n 2

163

2

3

:

3

2

4(p-n) _ _

2

-3

J

7° u = 3

164

Noncovariant Ganges

dq

-

2

C?-P) (9

n)

2

dq q

2 jdiv 2

~ n

P

(9 -p)Hi-

n)

2

n)

2

~ n

2

P

\»

dq q q u

v

2

(9 -p) (q 2

=

2(p-n) / . »4

2

n 2 . - Tp—nyP^" + —(Pp '+P* *) n

(V

n

4

\

RFIV

~ ^"M"*J ^

.

2

/ dqg J (q-p) (q 2

n)

2

^(^-'L'*-¥(^-')^ f

dq

/"

^ U S

/"

- _L

n

/div

p•n /

dq

J ^-p)H

n

/•

=

2n^ ^

1

+ —(PM"> + P ^ ) - ^ » v J i

^9

7 ( 9 - p ) ( l - i ) V n = finite, a

2

2

7 (g-p) (g-*) 3-«"«

2

2

"

'

\ ,

A ppendix B

d o

f

2

(q - p) (q - k) q • n

/"

J

tutu

2

J

^ 2

(q-p) (q-mq

_ (u(p

«'

n)

\

+ k)-n\ n )

d i v

2

_ 2(p + ,fc)n 2

u

=

2

Reference 1. D. M. Capper and G. Leibbrandt, Phys. Rev. D25, 1009 (1982).

This page is intentionally left blank

APPENDIX C LIGHT-CONE GAUGE INTEGRALS IN T H E ^ - P R E S C R I P T I O N This appendix contains a partial list of massless and massive one-loop integrals in the light-cone gauge n*-prescription. (See particularly Chapters 4 and 5.) 1. Gaussian Integrals (a) Gaussian integrals in one dimension:

2

Vc, = Aq\-2Bq ,

E = B /A,

A

A,B are arbitrary coefficients,

0

1

7T '

2

1 2

A' — oo +

0O

J

V

dq q e- ° A

A

3/2

A

— OO +oo 3

2A /

2

5

A /

2

— oo

I

3

v

dq
5

2A /2

-oo

167

.

S 7

!

A /2

.

E o

166

.V'";r.ivariflTi(

Gauges

(b) Gaussian integrals in (2ui — 1) dimensions:

a

2

V s q - 2 / ) q . p + o(q-D) , T

a,fi,-y are arbitrary coefficients,

L x

,

on

2

h - ' —

yd

20 +

q (q n)q e f

1

3cm

2

t (

2

( , p

—

= 2

, /? /

2a(p-n)

A

2

2

2

2

a n (p-n) \l

+

%

E

) r

•

^ .

2a(p n )

2

2

2

3

« n (p • n ) \ l

E

2. One-loop massless Feynman integrals in 2u> space All integrals in this section and the next have been derived by applying the light-cone prescription, Eq. (4.18b). The variable / is denned in Appendix B, and d^q = dg. d i v

Appendix C

(a) Two propagators: f

— (g - p ) g 2

J

f

n

dq q q v

p-n*

v

(-p

• np n* n - n"

= rT^ I,

(g-p)V"

/

nn*

2pn " ^r7<*" '

.

n

p-n*,

.

+ 2(pn)

2

- ^ 7 ^ 7 ( P n f + Pf",.) + P^P>' + ( u

2p • np- n * , +

/•

dg g

7 (g - P ) 9

=

dq

./ ( 9 - p ) ( 9 - » )

"""-

+

n • n*

2

+

j d i v

/

2

2p n* (« " ' )

/-2p 2

"\

u

• np • n' \ ~ n n*

.' .)2 /• n

2(p - n * )

_ 3p-np-n'\

" ' "* \

_

qfl 2

2

,

J

2

^

n

= finite,

2


P

"


2 T.- /

2

2

(

(n-n>)»

* *

2 d i v

2

170

Ifancovariant Ganges

(b) Three propagators:

J

rit-vrq-n

f dq q J q (q-p) q

u

2

2

_ 1 . -div n - n n' " '

- P • nn' til - p • n'fanl u

/

9 ( -p^ ) («n) 2

3

+ n^)]

div

/

,

finite,

2

9

dq q

u

I

2

finite, 2

2

q {q-p) {q-n) f

dq q^q

I

q U-pY{q-n?

„*\-2 *„*

v

rdiv

M

2

=

•

•^

•

+ 6

"^'"Z »>

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) ~ ™ ][(
2

_

2

2

2

'

2

q [{q-p) -m }[(q-k) -m ]q-n

/"

dg q^qy [{q-p) -m ][{n-k) -m )q-n 2

J

2

2n • n- ( ( p + * ) X + (P ^ d i v

"' K

2

2

+

^ + " X ) + (P + ^ ) ' " ' ^ ) /

2

where 7 = ijr (2/e), 2w = 4 - e, and the expressions that are known exactly.

d i v

+ ^io,

's, j = 1,2,... , 10, are finite

2

4. Special Integrals (n = 0) The following integrals arise in the computation of two-loop massless Feynman integrals in the light-cone gauge: 2

174

No nc ov avian I Ganges

2

I —

dq(q ) p ) ( , • n) 3

2

2w-4

^

x

v p

2

2(1 - » ) ( ! - ^ ) p - » p »* j

+

(b) 2

fdq(q

u

+

tg-nq-n-) -

1

0 2u-4

x (1 + uvtntr*

(vp* + 2 ( l - t > ) ( 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
3

2

Appendix C References 1. G. Leibbrandt and S.-L. Nyeo, Phys. Lett. B140, 417 (1984). 2. G. Leibbrandt and S.-L. Nyeo, J. Math. Phys. 27, 627 (1986).

175

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APPENDIX D UNIFIED-GAUGE INTEGRALS IN T H EG E N E R A L I Z E D n*-PRESCRIPTION In this appendix we give a partial list of unified-gauge integrals in the generalized n*-prescription, Eq. (5.34). Only the divergent parts of the corresponding integrals are given. The null vector and the associated scalars
J

J

d i v

2u

2

2

= divergent part of J d q{q {q - p) ]

£j

:

Minkowski space,

= I, in Eq. (5.54):

Euclidean space.

1. Massless One-Loop Integrals in the Unified-Gauge Formalism The majority of the integrals listed below occurs in the computation of the Yang-Mills self-energy and quark-quark-gluon vertex functions in Q C D (see also Ref. 3). Only the divergent parts of the integrals are given. 1

(a) Simple spurious poles (q • n) -

2

p) q • n

2P -P .div n-F

d^g g„ ^ 2p • F p) q n n-F 2

l

"Pp n-F " 177

p-F 2n-F

2

"

+

np • F 1 2(n-F) J 2

d i v

'

2

178

Noncovariant

w

2

(P qq p) g - n

div

r i d

2pF n-F

2

d^g

div 2

/

a

, P

3p-np -F , n (p • / • ) * ! n-F + (n-F) ,

~

rf

2

= 0,

2

J 9 (g - p ) g d i v

Gauges

»

fa

- J _

7<"v

F

d2ui

div

/

q q^q„ q (q - p) q • n 2

2

1 2(n-F) -pF(n F IA

p-Fn-

2

-FSfu, + n - Ffp^F* + p^F^) - p • n F ^ F ,

+ nF)

v

ll

rdiv

+

v

n-F

(b) Double spurious poles (q • n) 2

d "g

2

:

n

2

rdiv

1 d V

2

I (q-p) (q

d i V

/

'

( g -

n)

^ n )

P

2

2

<rp(n-F)

2

=

F^j, —
div/ 2

!

9 (g - P) (g • » ) div

div

/

y

2

2

g (g-p) (g.n) 2

d * g g^g* g (g-p) (gn)

Note that

2

3

2

2

2

(

( - V

P

I

+

'

2^

/ d l V

+

n-F

( ~F)2^' n

—n rdiv <rpp (n • F ) 2

2

= 0, 1 (F^F,+F^)I (n-F) 2

d l v

'

.

i

A ppendir D lim

d^q q q„ - p) ] - (g • « )

/

finite value,

u

2

m

2

u

2

d^q a

—i+J

a

«>[(«-p) ] -(«-n)»

finite value,

whereas

/

2

d "q 2

9 (?'«)

2

= 0.

2. Massive One-Loop Integrals in the Unified-Gauge Formalism (a) Simple spurious poles (q • n) - l .

a

2

[(9+p) -"» ]3«

n-F

w

d? qq» 2

2

[{l + p ) - f" ]° • « 2p • F n-F

— F

p-n n-F

"

p-F 2nF

d^q q q„ q [(q-p-h) -m ]q-n u

^

2

l

2

2

- (p + fc) • nFpF,, - ( + t ) • F ( F n „ + n^F,,) P

+ JiL p fc).FF (

+

( 1

rdiv F , | r" 1

u

2

*

n p-F 2(n-F) 2 '

C

180

No BCD variant Gauge* div

=

2

2

2

2

q) [{q-p-k) -m ]qn

(

1

( 2 f c +

f)a{

r i

- (2k+p)

p )

F

'

n

+

w

A-

(

n

2

2

2

q

F

/

r

k

{

-Fn-Fj

+

< 2

*

+

P )

F

" "

2

+

nFlk-pr+fp-Fl-k-np-Fr

I *,

P

t)>-ra»]«-

gtfa [(9 + P ) - ™ ][(l + k) - m }q 2

2

div/

^

/ I(9

+

P

) ^ ] (

2

2

(b) Double spurious poles (q-n)

^

'

f J-W

Ffi + P nf] + k F Fr-^-y}

2

W

F

q

/ [(« + p)3-m»]Kfl +

d i v

^

d^qpqkqj q (k- ) [(p+k- ) -m }q-n

-k-F\p

d l V

P

+

v

1

-

l 4

+

*

)

f J

1V

F [ { 2

u

f

d i v

d

""

• nF„F - (2fc + p) • F(F n

n•r

=

F

'

- 2 , , i"" 6

-

n

)

2

~ ° '

_L „ n n-F =

/ „ . f H v

=

=

n~F

™ *

+

(n-Jy^l

2

[(q + p) -m ]{(

q

tP <, +

2

2

2

k) -m ]( -n) q

+

^

'

u

2

!-» .

F

:

,„

g

n

'

1

Appendix D

1V

2

!

2

181

a

2

J [(« + p ) ~ ™ ][(? + * ) ~ ™ ](« • " ) " d % J q

Hiv / 7 g [( +P) 2

9

2

g

1

fa " IP p - m*](« - " ) " O - F ) ^ 2

2

2

2

J (k + q)H(q + + pP) ) -" m ] ( g • n ) 2

d^qk

div

<rp(„ • F )

qi

a

, P \ Tdi« + *W '

'

2

-1

2

3

[(« + p ) - " • ' ] [ ( « + * ) - m l ( « - n ) »

-2 [(p + i ) - F f (n.F) ?

div/

2

'

+ (p + i ) ' ' £

( 1

) W

]

r*,

d^q q* [(« + p ) - m » ] [ ( + l ) * - m a ] ( - n ) * 3

a

2

5

2

= 7 ^ T 5 T j { 4 [ ( p - F ) + (k - F) + (p • F)(Jr - F )

+ W

+ P V + i W f c ] "

2

2

"V+* -2m n

/div

References 1. G- Leibbrandt, /Vuci. Phys. B310, 405 (1988). 2. G. Leibbrandt and S.-L. Nyeo, Phys. Rev. D39, 1752 (1989). 3. G. Leibbrandt and K. A. Richardson, Phys. Rev. D46 , 2578 (1992).

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APPENDIX E CHERN-SIMONS I N T E G R A L S The following noncovariant-gauge Feynman integrals occur in the computation of the vacuum polarization tensor in Eq. (10.39). D is a complex number and m the regulator mass. 2

h=rn Jda

(

q

_

2

m

2

)

(

^

i

p

)

_

2

m

2

)

q

n

=0,

(9 - ™ ) ( ( g + p) — m )q - n 2

2

2

2

(3 + p) h = / dq— • - T T — = 0 • ™ ) { ( ? + p) — m )q - n (I 2

2

2

h = jd

2

2

9f (g -m )((g + p ) - m ) g - n 2

2

2

2

f ,

T 5

~

m

J

ffl9

3

I = m jdq 6

_ i , n ~ n n* "'

( 2-m )((g+p) -m )g 2

2

2

j _ J_ 8ir '

71

0

I _f_ -—,«* ( o 2 - 2 ) ( ( + p ) 2 - 2 ) g . n ~ 2n n" m

2

m

i_ i" = m j dq77 — . — — — •' g - m )((g + p) - m )q • n 7

2

J / , h = mjdq / .

2

2

(g+p) 2 _

7

2

{ q 2

_

m 2 ) { ( q

+

p ) 2

g

m

2

2

)

s

fl

/ - ^-^p • 3

183

7

. R

,

184

Noncovariant

h

l

m

=

a

/*(9»-m»)((j+rt*-« ).-»

= ^ . .yi 2

Gauget

n

n

n

n

n

• *(ffflfP • * + P» 'p +

n

- n njjj • n* - n* njp • r» - n,jn*p • n*], u

T _

m

1 2

J( „ ?

7 f .

j

I

2

(g2-m2)(( -rp) -m=) . n n (g + p) g„ 3mJ 9

g

n

2

s

7

m

2e

d

a

a

*(I -™ )((9 + p ) - » ' ) • • » " "

"'

/i4 = /
p

2

1

hi

5

_

9

=J

2

2

2

2

g

' _ „ _ = 0, (g2-m2)(( )2_ 2),.„ a +

p

m

1_ • r —-. ((3 + P ) - "> )g - n

ht = j i i , ,

2

2

2

(g + p) ?,. m )((g + p ) - m ) , n

/is = m

2

2

3m / - n ^ ^ +

2

, +

2

p /

.

1 2 n ^ ^ -

J

6(n n - )

2 ( 2 p

P

pn'/ T V

3

n

n

' " " ' '" 2

2

+

P

' ' P

< '

2

n , ) 2 n

"*

2

m )((g+p) -m )gn 2

3m / , n n* * 5J

&

2

p / , 5 n'7 12n n- " ' 3n-n"

" 6(n n-)2

P

(2p

2

"P "'nj + (p - n') n„)

Appendix E

dq

mlp • n*

1 2

2

2

2

(g -m )((g + p ) ~ m ) g - n

2n

'

2

dq

(g + p) mlp • n* ( o - m ) ( ( g + p ) - m ) g ri ~ 2n r»* '

iq

g -Zrnlp • n* {g - m )((o + p) - m )g - n ~ 2n n* '

a

3

a

3

2

3

a

a

2

a

!

9 (9+P) (g -m )((g + p ) - m ) g n 3

3

3

3

-3m/p » ' 2n-n* '

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APPENDIX F T H E G R A V I T Y T E N S O R S T*„ , i<M

1

We list the 14 independent tensors that appear in the text in connection with the graviton propagator, Eq. (5.13). The tensors T*^ , i = 1,... , 14, are formed from n^p^S^, and satisfy ^ = Tf = I*„ = T^ : p a

2

U

f V

i<rp

a

i l v

'Fp.e.pa — ^p.l/^pa i Tt«W =

+ &poP»Pv) .

{P )~ ^P."PPP'> 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 (
Q QCD see quantum chromodynamics QED see quantum electrodynamics quadratic form 1 quantization of Maxwell-Dirac held in the temporal gauge 3 quantum action principles 108 quantum chromodynamics 7, 40 ladder diagrams, or planar diagrams, in 40 non-planar diagrams in 40 quantum electrodynamics 3, 12 Lagrangian density for 12 quantum held theory 1 quantum gravity 7, 12, 43 massless integrals in 158 independent tensor in 187, Appendix F quantum Hamiltonian in the Coulomb gauge 130 quantum mechanics 2, 3 quantum theory see quantum mechanics

207

hdcx quark-quark-gluon vertex and unified-gauge integrals R radiative corrections in Chern-Simons theory 150 radiation held 12 radiation gauge 3, 5, 129 see also Coulomb gauge Ray-Singer torsion 136 redundant gauge degrees 23 see also unwanted gauge degrees re-gauging see umeichen regularization in Chern-Simons theory 141 regulator mass in Chern-Simons theory 140, 142, 143 relativistic invariance preservation of 27 renormalization 40, 69, 116 in the light-cone gauge 106 n on-multiplicative 91 of Yang-Mills theory 105, 119 renormalization equation in the light-cone gauge 107, 108 planar gauge 123 renormalization constants 106, 109, 114 in the Coulomb gauge 132 for Green functions 118 Ricci tensor 135 S S-matrix elements 102 scalar boson field 54 scalar-fermion loop 101 scalar ghost 43 scalar ghost field 42, 43 scalar ghost propagator 62 scalar-gluon self-energy 101 scale factor 2 scale invariance 1-2, 6 scale transformation 3 Schwinger's exponential representation Schwinger model 54

31, 35, 47, 70

177

208

Index

in Q E D 55 Schwinger parameters 75 Schwinger representation see Schwinger's exponential representation sine-Gordon equation in 1 + 1 dimensions 136 in 2 + 1 dimensions 136 Slav no v-Taylor identity 107 sources external BRS sources 106 in the generating functional for connected Green functions space-time dimensionality of complex 31 discretization of 54 Minkowski 28, 30 space-like axial gauge see axial gauge spherical polar coordinates 48, 76, 79 spinor held 12 charge and mass of 12 spontaneous symmetry breaking and discretized light-cone quantization 55 and zero modes 55 spurious factors 37, 102, 154 see also spurious poles spurious poles in Chern-Simons theory 142 in Yang-Mills theory 63, 70, 75, 82, 96 Streckeneichung 2 strong interactions 7 structure constants 11, 106, 138 subsidiary condition and the Gauss law 19 superconductivity 137 superficial degree of divergence 96 superfield 96, 102 supergraph 96 and finiteness 96 supergravity 7 superstring theory 7, 40 superstrings 12 supersymmetry 7, 40 "swordfish" diagrams/graphs 1 + 1

in Chern-Simons theory

151, 152

107

Index in Yang-Mills theory 112 symmetries integration in Chern-Simons integrals 145 T tadpole integrals 32, 33, 102 massive, in Chern-Simons theory 146 massless 32, 33, 43, 88, 102, 111 Tamm-DancofF approximation 55 temporal gauge 3, 5, 7, 8, 38, 43, 59, 69, 105, 137 definition of 67 prescription for 67 tensor density g " 135 weight of 135 tensor method 32, 33 for covariant-gauge Feynman integrals 28 noncovariant-gauge Feynman integrals 39 test function integration over 28 theory of electromagnetism 1 theory of relativity see general theory of relativity theory of superstrings see superstring theory 't Hooft gauges 4 't Hooft-Veltman gauge 6 time-ordering operator 88 time-ordered product 87, 88 vacuum expectation value of 87, 88 three-gauge vertex in Chern-Simons theory 142 three-gluon vertex and ghost diagrams 113, 114 and renormalization 110 three-gluon vertex in Yang-Mills theory 60 three-gluon vertex diagrams "'s word fish" diagrams/graphs 112 triangle diagram 112 three-point function in Chern-Simons theory 151 in Yang-Mills theory 111 topological gauge theory 137, 138 invariant 136 1

1

209

210

Index

mass term 137 trace 43 transformations gauge 2, 12 infinitesimal 12, 20, 21 scale 3 transverse Landau gauge see Landau gauge transverse mode 40 see also mode two-loop integrals, in Yang-Mills theory

Sec. 3.4

U U(l) gauge symmetry 12, 14 ultraviolet divergence and naive power counting 34 ultraviolet finiteness of the N = 4 supersymmetric Yang-Milis model two-point function 96 Chern-Simons model 141 ultraviolet infinities 140 ultraviolet regulator 138 umeichen 2 unification 7 unified-gauge formalism 73, 81, 179 unified-gauge integrals 177, Appendix D uniform prescription, or unifying prescription for axiai-type gauges 69, 128 unitarity 69 unitary gauge 4, 8 unitary transformation 12 unphysical degrees of freedom 24, 38 and Lagrangian density 24 unphysical poles 40 see also unphysical singularities unphysical singularities 28, 30 unwanted gauge degrees 19 V Vi, Vi terms 130, 133 vacuum expectation value -amed 136 *f : .e= field 4 ;

T:

40, 63, 95

211

lad ci

of lime-ordered product 87, 88 vacuum polarization tensor in Chern-Simons theory nonlocal structure of 149 vector potential 19 volume factor 20 W bosons 7 Ward identity diagrammatic representation of 91, 92 for gluon self-energy 84 in the axial/temp oral gauge Sec. 5.4.2 in Chern-Simons theory 147 in the light-cone gauge 84, 87, 89, Sec. 5.4.1 in the planar gauge 123, Sec. 5.4.3 and alpha-prescription 46 wave function bound-state 54 renormalization in Chern-Simons theory 150 Wess-Zumino gauge 6 Wess-Zumino model 7 Weyl gauge 5 see also temporal gauge Weyl-ordering 130 Wick rotation 30, 39, 68 for axial-type gauges 30, 39 for covariant gauges 27, 31 in the axial gauge 68 in the light-cone gauge 63, 65 in Minkowski space 45 in the temporal gauge 67 to Euclidean space 35, 45, 65 and principal-value prescription 45 and two-loop integrals 35 Wilson loop 46 and Chern-Simons theory 136, 137 and principal-value prescription 45 world geometry, Weyl's 2, 3 Y Yang-Mills field 12, 37, 97 Yang-Mills self-energy 88 and unified-gauge integrals

'77, Appendix D

139, 146, 148, 183

Index

212

see also gluon self-energy 81, 82 Yang-Mills theory 12, 40, 41, 46, 105, 130, 136, 137, 143, 146, 158 finiteness of sup ersym me tries 95 in the light-cone gauge 63, 84, 106, 119 in the planar gauge 121 in the temporal gauge 41 supersymmetric 40 Yang-Mils Ward identity in the light-cone gauge 84-89 in the planar gauge 123 Z a

Z boson 7 zero gauge modes

55