P H Y S I C A L R E V I E W D V O L U M E 1 8 , N U M B E R 2 1 5 J U L Y 1 9 7 8

Ambiguities of the natural gauge in Yang-Mills theories

G. Lazarides Department of Physics, University of Ioannina, Ioannina, Greece

(Received 22 March 1978)

We study the ambiguities of the natural gauge condition for the Euclidean SU(2) Yang-Mills theory in four dimensions. Then, we show that, in the stationary-phase approximation, these ambiguities do not affect the contribution of the sector with Pontryagin index q = 1 to the correlation functions of gauge-invariant operators. They affect only the higher-order corrections to this contribution.

I . INTRODUCTION

Recently, i t has been observed by Gribovl that, f o r non-Abelian gauge theories , the t ransversal i ty condition 8,A = 0 (i = 1 , 2 , 3 ) fai ls to uniquely specify the gauge. This gauge-fixing degeneracy, which means that one can find a whole family of gauge-equivalent t ransverse fields, provides u s with a possible m e c h a n i ~ m " ~ f o r color con- finement in quantum chromodynamics. More specifically, i t has been shown2 that the Coulomb propagator becomes very s ingular f o r vanishing momentum, and consequently the potential en- e rgy between two charges increases a t l a rge dis tances. Notice that the gauge independence of these resu l t s i s s t i l l a n open question.2s3

In this paper , we will study the ambiguities of the natural gauge condition4 for the Euclidean S U ( ~ ) Yang-Mills theory in four space-t ime dimensions. This gauge condition has been used4 to calculate the contribution of the s e c t o r with Pontryagin index5 q = 1 to the correlat ion functions.

The work is organized in the following way. In Sec. 11, we study the ambiguities of the natural gauge f o r a par t icular c lass of gauge fields in- cluding the pseudoparticle solution.' We find that there ex i s t s a continuous multiplicity of fields that a r e connected by finite gauge t ransformations to the or iginal field and sat isfy the natural gauge condition. In addition, we prove that the fields belonging to this continuous multiplicity exhibit the phenomenon of level accumulation2 which in the Coulomb gauge leads to color confinement. In Sec. 111, we show that, in the stationary-phase approximation, the ambiguities of the natural gauge do not affect the contribution of the sec tor with Pontryagin index q = 1 to the correlat ion functions of gauge-invariant opera tors . These ambiguities affect only the higher-order cor - rect ions to this contribution.

11. AMBIGUITIES OF THE NATURAL GAUGE

The pseudoparticle solution to the S U ( ~ ) Yang- Mills theory in Euclidean four-dimensional space-

t ime i s given by596

where the as's a r e the Paul i mat r ices , x = ( X ~ X , ) ~ / ~ , and g = (x, - i azxi)x- ' . This solution, which has Pontryagin index q = 1, can a l s o be wri t ten a s 7

where the 7]lV's a r e defined in Ref. 7 and sat isfy the identities

rl;,=-rl,n,=i ~uv; lp?7Xnpt

~ ~ v ~ ~ h = 6 a b 6 v h f ' a b c ~ ~ h ,

and (2.3)

9 ~ v 1 1 % = 6 , , ~ v , - ~ , , 6 v , + ' . v ~ .

Let u s now consider the field configuration

A,=f ( x ) A i l , (2.4)

where f ( x ) i s a regu la r function of x = ( x , ~ , ) " ~ and f (x) - 1 a s x - .o. A, has Pontryagin index q = 1 and sat isf ies the natural gauge condition4

V ~ ~ ' A ~ = ~ , A , + [ A :',A,] = O . (2.5)

We will s e a r c h f o r gauge t ransformations

U = U 4 + i u i U i , U,U,=l ( p = 1 , 2 , 3 , 4 ) (2.6)

which t ransform the field A , to A:,

A , - A ~ = U - l A u U + U - l a , U , (2.7)

s o that A a l s o sat isf ies the natural gauge con- dition, i.e., V,A~'A' ,=O. This means that we will look for solutions of the equation

V , A ( U V ~ ~ ' U - ~ ) = O . (2.8)

Restr ic t ing ourselves to the case where U depends only on x = ( x , ~ , ) ' / ~ and using Eqs. (2.3), Eq. (2.8) reduces to

18 - A M B I G U I T I E S O F T H E N A T U R A L G A U G E I N Y A N G - M I L L S . . . 39 1

where = aha,. Then, using the ansatz

~ ( x ) = exp[i a ( x ) t i oil cos a ( x ) + i t i ai s i n a ( x )

[x = (x,x,)"~] , (2.10)

where the ti's a r e constants and t i t i = 1, Eqs. (2.9) give

Changing variables f r o m x to s = lnx , we obtain

which i s the equation of a pendulum''218 with fr ic t ion in a variable gravitational field g ( s ) = 4 f ( s ) ( l + e-2S)-2. 2 a ( s ) is the angle of the pendulum measured f rom the position of unstable equilibrium in a positive field g ( s ) .

This equation has two tr ivial solutions,

a ( s ) = O (modn) (2.13)

and

a ( s ) = Q n ( m o d n ) . (2.14)

F o r s --a the gravitational field g ( s ) tends to z e r o exponentially, and Eq. (2.12) becomes

Requiring that a ( s ) remain regu la r a t s =-a ( x = 0), we conclude that the only solution of Eq. (2.15) is a ( s ) = a = const. Therefore ,

a ( s ) - a (modn), a s s - - m , (2.16)

where 0 c a <n. In the spec ia l cases where a = 0 and a = $ a , we obtain the t r ivial solutions (2.13) and (2.14), respectively. In any o ther c a s e we obtain a nontrivial solution. F o r s -m, Eq. (2.12) reduces to the equation of a pendulum with friction in a constant gravitational field g(m) = 4,',2*a

At s = m, the pendulum eventually stops a t the top [a(m)=O (modn)] o r a t the bottom [a(m)=;n (modn)]. Then, linearizing Eq. (2.171, we obtain

+ n + c r ( a ) e - " c o s [ w s + $ ( a ) ] ( m o d n ) , s- P

(2.19)

where c , c', and g5 a r e constants depending on

a and where w2 = 7. The initial condition a t s = -m, i.e., the value of a, together with the f o r m of the functiong(s)"(-.o c s ~ m ) determine which of the two asymptotic beh&viors (2.18) and (2.19) will be realized.

Equation (2.12) implies that

= -2 ( ) - s i n s ) . (2.20)

In the r e s t of this paper we will r e s t r i c t ourselves to the case where g ( s ) is a monotonically in- creasing function of s. Then, f o r a #O (modn), the above relation gives dE(s)/ds < 0. Thus,

On the other hand, the asymptotic behavior (2.18) implies that E ( s = m) = 0, which contradicts in- equality (2.21). Therefore , in this case, the asymptotic behavior (2.19) definitely holds f o r a l l the solutions of Eq. (2.12) except the t r ivial one exhibited in Eq. (2.13). Notice that in the important c a s e where A,=A C,'(f = 1), the gravi- tational field g ( s ) = 4(1 + e-2S)-2 increases mono- tonically with s.

The special solutions of Eq. (2.8) which we have found, depend on three continuous p a r a m e t e r s a (O<a<n) , cp(O~cp <2a), and 8 ( 0 6 8 ST) , where cp and 8 a r e the polar angles in the space spanned by the th ree vectors ia, (i = 1 , 2 , 3 ) . Therefore, Eq. (2.7) gives a th ree-parameter family of fields A: which a r e gauge equivalent to the field A, = f ( x ) A ',' and sat isfy the natural gauge condition

An infinitesimal variat ion of any one of the p a r a m e t e r s a , cp and 8 leads to a n infinitesimal gauge t ransformation of A:, FA: = V ,A'#. Then, Eq. (2.22) gives

This means that the opera tor VfC' v,A', which appears in the Faddeev-Popov determinant,4$" has three independent z e r o modes, $,, $,, and $,, corresponding to the three p a r a m e t e r s a , cp , and 9.

T o b e m o r e specific, le t us now choose t i = F i , (cp=O,$=+n). Then Eq. (2.7) becomes

392 G . L A Z A R I D E S 18 -

The asymptotic behavior of^;'' i s given by

where the plus sign corresponds to a = 1 and the minus sign to a = 2 and a =3. Thus, A: "O(x-I) a s x - m, i.e., it has the same asymptotic behavior a s A ,.

The three zero modes of the operator ViC1 v:', which a r e mutually orthogonal, can be written a s follows:

It i s easily seen from Eq. (2.19) that these zero modes a r e not normalizable. Thus, they belong to the continuous spectrum of the operator Vfcl v$" and we should not worry about them4' lo

For $ = $4, and $ = the equation v F1 v f ' $ = 0 reduces to Eq. (2.111, which definitely holds. For $ = $,, it reduces to the SchrGdinger equation

Notice that this equation can also be obtained by differentiating Eq. (2.11) with respect to a. For x-m, the potential in Eq. (2.27) becomes negative and behaves a s -8Y2. Thus,

$;(x) rv c1xm1 COS(W lnx +c , ) , (2.28) x- -

where c , and c, a r e constants depending on a and where w2 = 7. Equation (2.28) shows that the zero mode #(x) has an infinite number of zeros. Therefore, the Hamiltonian operator H in Eq. (2.27) has an infinite se t of negative eigenvalues with an accumulation point a t A s imi lar level accumulation phenomenon in the Coulomb gauge provides us with a possible mechanism for color confinement.'

111. THE CORRELATION FUNCTIONS

The contribution of the sector with Pontryagin index q = 1 to the partition function is given by

Here, F$,=a,A,-a,A,+[A,,A,]. The simplest way to eliminate the gauge, t rans-

lational and dilatational freedom i s to use the method of Faddeev and Popov.' Thus, we define the quantities A(A) and @(A)

and

~ a ( j . ( A ) [ ( x , - R , ) ~ - 2 * ~ ] d ~ x ) ,

(3.4)

where g E S U ( ~ ) and A ;=g-'A,g +g-'a,g. It is easily seen that

s ( A ~ ) =s(A), A ( A ~ ) =A(A) , $ ( A ~ ) =#(A 1,

vg E ~ ~ ( 2 1 (3.5)

and

In addition, we assume that the integration mea- su re (dA) is also gauge, translationally, and dilatationally invariant. Now, we insert the left- hand side of Eq. (3.4) into the integrand of Eq. (3.1) and perform the transformation

Then, we insert the left-hand side of Eq. (3.3) into the integrand of the resulting expression and perform the gauge transformationA,-A:-' Finally, we obtain

where we integrate over a l l field configurations A , for which q(A,) = 1 and where

18 .- A M B I G U I T I E S O F T H E N A T U R A L G A U G E I N Y A N G - M I L L S . . . 393

The integrand of this expression depends on A through the rescaled ~ u t o f f . ~

Let us now ca l lA 2' ( ~ E I ) the field configurations which a r e gauge equivalent to A C,' and satisfy the natural gauge condition v f C 1 ~ 2 ) = 0. To calculate the integral in Eq. (3.8) in the stationary-phase (semiclassical) approximation, we must calculate it, in this approximation, around each A y ) and then sum the results.

For A, in the neighborhood of A Li', the quantities A(A) and @ ( A ) can be replaced by A(A '") = A(AC1) and @(A "') = @(A"), respectively. These can be calculated using Eqs. (3.3), (3.41, and the rela- tions4

and (3.9)

The result i s

and (3.10)-

where SC1 =S(A 'I). A , can be expanded around A Li' a s follows4:

=A;) + a, , (3.11)

(i) *(i) where V f a, F :ji), and x x F ,, a r e the gauge, translational, and dilational zero modes of [6'S(A )/6A,6Ay],,,tr,, respectively. a: (i' a r e the nonzero eigenmodes and satisfy the equation

All the eigenmodes in Eq. (3.11) a r e mutually orthogonal. Moreover, the nonzero modes a r e normalized a s follows:

The orthogonality of to the gauge modes v:'"'a implies that

Let u s now write

A ; ~ ) = u - ~ A ; ~ u + U - I a, u . (3.15)

Then, it is easily seen that

F ~ ( i ) = U - l F ~ C I U x LI ,, a n d x , ~ : y ) = ~ - ~ x , F C ~ .

(3.16)

Moreover, Eq. (3.12) implies that

an.(i)=~-la,~l~ and f i m ( i ) = ~ n . c l = f i n u - . (3.17)

The following relations a r e also obvious [see Eqs, (3.16)]:

and

d e t ( v ~ ( i ) , A ( ~ ) ) = d e t ( V , ~ ~ ' v f ~ ~ ) . (3.18)

From Eq. (3.11) we obtain

E ( A ) = E ( A " ' ) + ~ T ~ ( F , A ~ ~ ) v $ ' ~ ' ~ , ) + . * * , (3.19)

which gives4

/ f (A)x ,d4x=-2 r ,N t +.

and (3.20)

The functional measure in Eq. (3.8) can be written a s

and the action S(A) can be expanded around A L i ) a s follo\vs:

Using Eqs. (3.81, (3.10), (3.111, (3.1'71, (3.181, (3.201, (3.211, and (3.22) we can now write the 2''' in the stationary-phase approximation,

394 G . L A Z A R I D E S 18 -

- dX z'l'l (1') 2 - 5 ( g 2 ~ c i ) 5 ~ ~ 1 h ~ t 2 e x P ( - ~ c') ,/ o 1 d 4 ~ ( ~ f i . ) - 1 / 2 d e t 1 " ( v f c ' n v$') de t -~(vfc l v , c i ) ) A ( A ~ ~ )

Comparing this formula with the known4 expression Equation (3.11) can be wri t ten a s f o r Z'", we conclude that the existence of gauge- A ( i )

A ~ = A ; ~ ) + ~ , + v , a , fixing degeneracies does not affect the value of (3.25)

2''' in the semic lass ica l (stationary-phase) ap- where v ,A '~ ' $,= 0. Then the integratibn measure proximation. in Eq. (3.8) becomes

Let u s ca l l G"' the contribution of the s e c t o r with q = 1 to the correlat ion function (dA),,, = (d 6,)(da) d e t ' 1 2 ( ~ f c 1 v,A"') . (3.26)

G = ( o I T (o,(A , ( x , ) ) . ~ 0, (A,(%,))) 10) , Performing the integration over ( d a ) we obtain

where the 0, (A i (x i ) ) (i = 1, . . . , n) a r e gauge- I ( d n ) a ( ~ ~ ~ ~ A , ) . ~ * invariant operators . This can be calculated by inserting the product O,(A ,(x,)).. On(An(xn)) into the integrand of Eq. (3.1). Repeating the previous manipulations, we end up with Eq. (3.8), where the product

Thus, a f t e r this integration, Eq. (3.25) becomes O , ( ~ A , , on(; A,, (v)) *,=A ;i)+;,-

IL

is inser ted into the integrand. T o calculate the The quantity A(A) i s then given by semic lass ica l contribution of the neighborhood of A Li ' to this correlat ion function, we must r e - Aml(A) = x d e t - ' ( V , ~ ~ ~ V: ) , (3.29) place A, in the product (3.24) by A Li' . The gauge A'

invariance of the opera tors O,(i = 1,. . . ,n) implies where A, i s given by Eq. (3.28) and the s u m is that the product (3.24) does not depend on (i) and defined over al lAL that a r e gauge equivalent to the previous argument goes through. Therefore,

A, and sat isfy the natural gauge condition. There- we conclude that the ambiguities of the natural fo re , 2"' can finally b e written a s gauge do not affect the value of G'" in the s ta - tionary-phase approximation. Z(''= Z 1 (d&,) d e t l " ( ~ f l ~ : ' i ) d e r l ( ~ " c i VA"'

On the contrary, higher-order correct ions to ( i ) LL L L )

~ ( 1 ) and G'" a r e in general affected by the ex- X A(A ).. . / , = - ( v ~ c i v ~ ( i ) ) - ~ v ~ c i ~ l , , istence of these ambiguities. We will now d iscuss

these higher-order correct ions v e r y briefly. (3.30)

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