P H Y S I C A L R E V I E W D V O L U M E 1 2 , N U M B E R 2 1 5 J U L Y 1 9 7 5

Asymptotic freedom of Yang-Mills fields in the Coulomb gauge

Ahmed Ali* and Jeremy Bernsteint Cenrro de Investigacion del 1P.V. Aparrado Postal 14-740. MCxico 14, D.F.

and Stevens Institute of Technology, Hoboken, New Jersey 07030

Arnulfo Zepeda : Cenrro de Investigacion del IPN, Aparrado Postal 14-740, Mexico 14 , D.F.

(Received 10 March 1975)

We show that the slope of P ( g ) at the orig~n is negative in the Coulomb gauge and agrees with the result obtained In covariant gauges. We also make a detailed comparison between the results obtained from the Bjorken-Johnson-Low limit and perturbation theory for the Yang-Mills propagator.

I. INTRODUCTION

In a recent paper' two of u s (A. A , and J. B.) studied the Bjorken-Johnson-Low (BTL) l imit of the self-coupled Yang-Mills vector-meson prop- agator. The equations of motion for the equal-time commutator [by(%, 0), bj (O)] w e r e solved to o rder gt in the Coulomb gauge, the solution being

where

a and b a r e internal indices, and i and j label the spatial components in Minkowski space. This yields the expression for the BJL limit fo r the t ransverse propagator in momentum space

which t r a c e s itself back to the fact that in the Yang-Mills theory the f ie lds bi and bi a r e not canonically conjugate. In Eq. (3) the value of the constant on the right-hand s ide i s immaterial s ince it can be absorbed in the t e r m ln(A/p) by a redefinition of A.

It i s c lea r that it would be very interesting to compare this resu l t with what would be obtained by doing s traight Feynman-Faddeev-Popov per - turbation theory for the propagator to the s a m e order and then passing to the

yo- m, 6' fixed (4)

l imit . It i s well known2 that fo r a l l field theories so f a r examined these two methods do not a g r e e unless in the diagrammatic method one imposes the prescr ipt ion of-first passing to the l imit and then performing the integrations, a prescr ipt ion that i s totally unjustifiable mathematically. A s we show in the body of the paper, th i s i s a l so what happens in the m a s s l e s s Yang-Mills theory which w a s not considered in the previous work.' If we make this l a s t prescr ipt ion we reproduce Eq. (3) exactly. If we simply c a r r y out the cal- culation for the propagator fo r finite 4, we a r e able to extract the renormalization constant 2, in the Coulomb gauge which tu rns out to be3

where p i s an a r b i t r a r y 3-momentum subtraction point. We have used Slavnov-Taylor-Ward4 iden- t i t ies to compute 2, in Coulomb gauge, which tu rns out to be

F r o m these two expressions we can compute /3 in the Coulomb gauge,5 where

and

We find

in p r e c i s e agreement with the s tandard resul t for the covariant gauges6 which means that the theory i s a l so asymptotically f r e e in the Coulomb gauge. Kallosh and Tyutin7 have given a proof that /3 i s gauge-independent to a l l o r d e r s in go in the co- variant gauges. In fact, they show that g i s gauge-

504 A H M E D A L I , J E R E M Y B E R N S T E I N , A N D A R N U L F O Z E P E D A - 12

independent fo r the covariant gauges which i s ex- ceedingly reasonable physically; g i s , a f te r a l l , the observable coupling constant. We conjecture that their argument can be extended t o all renor - malizable gauges but have not c a r r i e d out a proof. This would mean that a search for the z e r o s of 0 , which i s connected to the possible f ini teness of the Yang-Mills theory, ' i s a gauge-independent enter- p r i se . In the body of the paper the detai ls of our work a r e presented.

11. PERTURBATION THEORY CALCULATION

Up to second o r d e r in go the Yang-Mills prop- agator i s of the f o r m

qq ( q ) = ~ ( 0 ) " ~ r j ( 4 ) + ~ ! ; ) " ~ ( 9 ) $4 ( q ) ~ , ( ; ) ~ * ( q ) , where the zeroth-order t e r m i s

and the amputated second-order two-point func- tion nib(q) i s the sum of the graphs in Fig. 1 and is of the f o r m

nibr(9) = O a b n b l ( q ) C z ( G ) / 2 , ( 1 1 )

FIG. 1. (a), (b) Vector-meson contributions, (c) ghost contribution to the vector-meson propagator.

where C , ( G ) i s the quadratic Cas imir operator f o r the adjoint representat ion of the internal group G . F o r concreteness w e will consider SU(2), fo r which C , ( G ) = 2 . The tensor n, , (q) has to be of the f o r m

nkl(4) = 6 , 1 n ( 4 ) + q k 41 N ( q ) . ( 1 2 )

Thus

4; ( 9 ) = D I ; ) " ~ ( ~ ) d ( q ) , ( 1 3 )

where

d ( q ) = 1 + i n ( q ) / q Z . ( 1 4 )

With the rules8 given in Fig. 2 we obtain

+ ( q - 2k)k A,,@ - q ) ( k + q ) , + ( q - 2k)l Ak,(k) &,(k - q ) ( k + q ) ,

+ ( Q - 2k)k (9 - 2 k ) 1 4 , ( k ) 4, ( k - q ) + ( k -- q - k ) ]

+ [ ~ k l ( k ) ( q + k ) ; &(k - 4 ) + ( k - 4 - k ) ) +[(P - 2k)k ( q - 2 k ) , h o ( k ) h 0 ( q - k ) ]

2kk(k - 9 ) I + [ z 2 ( G - G ) 2 ] + [ - 2 i A k l ( k ) - 2 ~ b k 1 ( & o ( k ) - ~ i i ( k ) ) ] ' \ ' ( 1 5 )

where

represen ts the Coulomb interaction and

In Eq. (15) we have separated by square b racke ts five different types of contributions: (a) the con- tribution of vector mesons to the bubble diagram; (b) the contribution of single Coulomb interaction to the bubble diagram; (c) the contribution of dou- ble Coulomb interaction to the bubble diagram; (d) the ghost contribution; and (e) the leaf graph.

The contributions of the Coulomb interaction in the leaf, the ghost, the double Coulomb interac- tion in the bubble, and the par t of the single Cou- lomb interaction in the bubble diagram that i s pro-

portional to k; sum to ze ro . The r e s t of the con- tribution to the leaf graph can be showng by dimen- sional regularization to be ze ro .

We can now d iscuss the q o - * l imit. If we take the l imit inside the integral, we obtain

which can be integrated immediately over ko to give

This reproduces exactly the resu l t of the BJL l imit . Using dimensional regularization Eq. ( 1 9 ) can be wri t ten a s

A S Y M P T O T I C F R E E D O M O F Y A N G - M I L L S F I E L D S I N T H E . . . 505

vector propagator and Coulomb interaction

k a ......,...... b i sob /E2 ghost propagator

a, P xu (€cob Eecd + Eead ecd E )

+g/", g ,,p (flab cede + ceac E edb

c9 P d , 0- +gpY gp '7 ( EeaC gebd + cead ee b C ) ]

FIG. 2. The Feynman rules for a pure Yang-Mills theory (all momenta flow into the vertices). The Bjorken-Drell metric is used.

n ( q ) - - i d q Z l n T + c o n s t . ( ,:I ) (19') i t can be easily shown to contain two powers of

3 a 2 O q, in the numerator . Performing the integration

If we do not interchange limit and integration, we in (20) with the help of dimensional regularization,

can immediately s e e that by the previous procedure taking then the limit, and adding the Eq.

we have missed a t e r m ar i s ing f r o m t h e contribu- (18) we obtain

tion of the vector mesons to the bubble diagram: ,2 i (q ) - - i 5 q:(ln + const

- 4 8 / d 4 k k , k, 4.W) 4.(k - s) . (274

(20) 1

The r e s t can be neglected in the l imit 9,- s ince +is qa( ln I~ : -? I~ " c o n ) (21)

506 A H M E D A L I , J E R E M Y B E R N S T E I N , A N D A R N U L F O Z E P E D A

in disagreement with the resu l t of the B J L limit. I t should be evident that, in general , if pe r tu r -

bation theory produces an expression that, in the limit qo- i s of the fo rm A ln(A/q,) + B l n ( ~ / / G / ) , then the B J L limit reproduces the second t e r m but does not detect the f i r s t one. The t e r m A ln(A/q,) wil lappear whenever dispersion relat ions need subtractions. This is the reason why in QED the B J L limit gives for the photon propagator only q-' instead of the resul t q-'[ 1 - ( ~ 1 / 3 n ) l n ( A ~ q ~ - ~ ) ] obtained f r o m perturbation theory.

F o r finite go the application of dimensional reg- ularization gives

where A i s a constant and f (x)- const fo r x - m.

Defining Z3 by -. d(qo, 151 , A ) =z3dc(qo, 1 9 1 , li), (23)

where y i s the 3-momenta subtraction point, we obtain f rom (22)

This concludes the discussionof 2,. To compute L, we use the Taylor-Ward identity that re la tes the renormalization constants 2: and 2: of the ghost field to those of the Yang-Mills fields

Z ; / Z ~ =Z3/Zl. (25)

Up to second order in go, Z y i s trivially given by

s ince the ghost-ghost-vector-meson t r iangle graph is finite in the Coulomb gauge. The second- o rder contribution to 2: i s obtained s traight- forwardly f r o m the graph of Fig. 3, and the r e - su l t i s

Thus

Equations (24) and (28) lead to the value of P given

FIG. 3. Second-order contribution to the ghost propaga- tor.

in Eq. (9) and hence the gauge invariance of 6 to this o rder . A general proof to a l l o r d e r s in a l l renormalizable gauges would be welcome.

111. CONCLUSIONS

The work of the previous section was c a r r i e d out only to the lowest nontrivial o r d e r in per tu r - bation theory. In this concluding sect ion we dis- cuss whether m o r e general nonperturbative con- clusions can be drawn. In a previous note' two of us (A. A. and J. B.) derived, by solving the Callan- Symanzik equation in the Coulomb gauge f o r d , , the following expression which i s , presumably, exact:

where A A = - l i '

F r o m our computation of ,3 we have

if g i s in the domain of attraction of the origin, while to o r d e r g2 one finds

Hence if we allow A to approach infinity in Eq. (29) we find

s ince

a s 7 becomes large. The factor

12 - A S Y M P T O T I C F R E E D O M O F Y A N G - M I L L S F I E L D S I N T H E . . . 507

simply approaches a finite number as T goes to infinity.

The conclusion f r o m this, we believe, i s that 2, i s infinite if g i s in the domain of a t t ract ion of the origin. We feel that this means that the theory i s not consistent f o r such values of g. As we s t r e s s e d in our previous work, the theory could b e ultraviolet-finite and consistent if 9 has a t least two m o r e zeros . At the third z e r o we might have, just to take an example,

?=- I c I (?-g),

i.e., a s imple z e r o with a negative slope. Thus

j j = g + e - I ~ ! r .

(At the second z e r o the slope would have to be positive.) We might then expand y , i .e. ,

and if

Y (g) = 0

we would then have for large 7

This s e t of conditions, it turns out, a l so renders 2, finite in the Coulomb gauge f o r coupling con- s tants in the domain of attraction of the third ze ro . We do not know if the Yang-Mills theory sat isf ies these conditions.

ACKNOWLEDGMENT

This work was s ta r ted a t the Stevens Institute of Technology, Hoboken, and was completed a t the Centro de Investigacibn del IPN, Mexico. One of us (A. Z . ) would like to thank the Stevens Institute of Technology f o r the hospitality extended to him. Two of us (A. A. and J. B.) would like to thank Prof . M. Zaidi and the faculty of Centro d e Investigacibn del IPN for the hospitality extended to them a t the Centro. We a r e thankful to Profes - s o r M. A. B . BCg for severa l discussions on the question of renormalizability of Yang-Mills theor- i es .

*Work partially supported by NSF under Grant No. GP-36777.

?Work partially supported by NSF United States-Latin American Scientific Exchange Program.

$Work partially supported by CONACfl (Mexico) under contract No. 540.

'Ahmed Ali and Jeremy Bernstein, Phys. Rev. D 2, 336 (1975). Equation (23) in this paper contains a sign mis- take which is corrected in Eq. (3) of the present paper.

'see, for example, R. Jackiw and G. Preparata, Phys. Rev. 2, 1748 (l969); A. Zepeda, Phys. Rev. D 4, 1072 (1971).

3 ~ e always restrict ourselves to the propagation of the transverse field components j , i.e., the physical de- grees of freedom.

4A. Slavnov, Teor. Mat. Fiz. g, 153 (1972) [Theor. Math. Phys. E, 99 (1972)l; J. C. Taylor, Nucl. Phys. B33, 436 (1971); B. W. Lee and J. Zinn-Justin, Phys. -

Ref. D 5 , 3121 (1972); 5, 3137 (1972); 8, 4654(E) (1973); 5 , 3155 (1972).

5 ~ . Karowski and S. Meyer [Phys. Lett. g, 79 (1974)l have independently computed p in the Coulomb gauge with results that agree with ours. No details are given in their note, so we cannot compare the calculational details.

'see, for example, D. J. Gross and F. Wilczek, Phys. Rev. D 8 , 3683 Q973), and H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973).

'R. E. Kallosh and I. V . Tyutin, Yad. Fiz. 20, 1247 (1974). One of the authors (J.B.) would like to thank R. Jackiw for calling his attention to this paper.

'E. S. Fradkin and I. V. Tyutin, Phys. Rev. D 2 , 2841 (1970).

9 ~ . M. Capper and G. Leibrandt, J. Math. Phys. 15, 86 (1974).