A pathology of the Landau gauge in euclidean Yang-Mills theories

Volume 78B, number 1 PHYSICS LETTERS 11 September 1978

A PATHOLOGY OF THE LANDAU GAUGE IN EUCLIDEAN Y A N G - M I L L S THEORIES

Kiyomi ITABASHI Department of Physics, Tohoku University, Sendai 980, Japan

Received 4 April 1978

It is argued that the Landau gauge in euclidean Yang-Mills theories exhibits the following pathological features similar to those of the Coulomb gauge: (1) As far as asymptotically 0(3) symmetric potentials are concerned, the potential for the field configuration with I ql > 1 is necessarily singular, where q is the topological quantum number of the configuration. (2) If furthermore the boundary condition rA ~ 0 as r ~ ~ is imposed, then even the Iql = 1 configuration necessitates singularities of the potential.

Recent observations by Gribov [1 ] have st imulated much at tent ion for the pathological features of the Coulomb gauge in non-abelian Yang-Mil ls theories [2,3]. Particularly in connection with pseudoparticle phenomena [ 4 - 6 ] the euclidean Yang-Mil ls theories are investigated. For example, it was shown that [3] the Coulomb gauge necessitates discontinuities of the potentials Au(x ) in order to accomodate a non-vanishing topological quantum number q, as far as the boundary condit ion

lim rA(x) = 0, A = (A 1 , A 2 , A 3 ) , (1) r - +

is imposed, where (also used hereafter)

r = l x l , x = ( x l , x 2 , x 3 ) , x 2 = r 2 + x 2 . (2)

Then, very naturally the question arises: how about the Landau gauge? Because of the similarity between the Landau gauge condit ion in euclidean space - t ime and the ordinary Coulomb gauge condition, it has already been suggested in ref. [1] that also the former gauge may have ambiguities analogous to those of the latter. The purpose of the present paper is to clarify a further parallelism between the pathological features in these two gauges. Our main conclusions are I and II given after eq. (27).

Throughout the present paper, we deal with SU(2) Yang-Mil ls theory in four-dimensional euclidean space- t ime , and adopt the usual matr ix representation

of the gauge po ten t i a lA u and the field strength tensor Guy:

A u -~ ieAa~o a, Guy 1. a a = = - : l e G ~ v o , (3)

Guy = OuA v - OvA ~ + [Au,Av] ,

where e is the coupling constant, the o's are the Pauli matrices, and gt, v ( 1 - 4 ) and a ( 1 - 3 ) are respectively the space - t ime and SU(2) indices.

As a preliminary step, let us first consider the 0 (3 ) symmetric pure gauge potential

A(O)(x) =g-lOug, g= exp[ict(r, x4 )~x / r ]. (4)

Then, the Landau gauge condit ion, OuA(O) --- 0, be- comes

02~ + 02ct -I 2 Oct 1 sin(2ct) = 0. (5)

0x2 0r 2 r Or r 2

Alternatively a can be regarded as a function o f r/x 4 and x 4/)t:

a = c t ( u , o ) , u=r /x4 , V=Xa/)t , (6)

where k is a scale parameter which necessarily enters into eq. (6) for dimensional reasons. In terms of u and o, eq. (5) is transformed as

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(1 + u 2) ~j2a +~2(1 - - u + u 2 ~ g g u au 2

_ ~ 2 ~ + v 2 32°~ 2 u v ~ = O.

3v 2

1 - - - - sin(2a)

u2 (7)

Here, of course, we are not concerned with the trivial v-dependence of a, such as produced simply by the replacement x 4 ~ x 4 + X in a v-independent a. It is trivial, since the ambiguity of potentials related to the freedom in the choice of the origin x u = 0 is inherent in sourceless theories. Apart from such a trivial v-dependence, we do not know whether eq. (7) has any solution that depends on v in a non-trivial way. If it has a non-trivially v-dependent solution a ( r / x 4 , x4/X), then we have a family of a continuously infinite number of non-trivial solutions, each of them being speci- fied by a particular value of X. If this were actually the case, it would mean that the Landau gauge is extreme- ly ambiguous, even for the vacuum! It seems very un- likely, though unfortunately we could not prove the

nonexistence of (non-trivially) v-dependent solutions to eq. (7) which satisfy the physical boundary conditions.

In any case, to make the problem more tractable and to go further, let us look for v-independent solutions to eq. (7). Namely, hereafter a is regarded as a function o f u alone, and our working equation (7) be- comes the ordinary second-order differential equation:

d 2 ~ + 2 . d~ ~2 ( l + u 2) du 2 u ( l + u 2 ) ~ - _ s i n ( 2 ~ ) = O . (8)

By the transformation of variable

u = 1/sinh t, (9)

eq. (8) can be cast to the form of a "pendulum equa- t ion"

ii - (tanh t)& -- s in(2a) = 0, (10)

where the dot denotes derivative with respect to the new variable t.

The correspondence between (r, x4) and t is ob- vious:

< ix4 < ot (x4 > o (x4 r = 0 I \ r = ~ I \ r =o° I \ r = 0 I (11)

t = - o ~ ~ t = - - 0 -+ t = + O ~ t=+oo

As is well known, the form (4) requires that c~ = m n

at r = 0 (X 4 4: 0), where m is an arbitrary integer. Fur- thermore, since

& = - i u l x/1 + u ~ d a / d u , (12)

we have the following boundary conditions in terms of t:

ct = 0, & = 0 or infinitesimal at t = - ~ , (13)

a = ran, & = 0 or infinitesimal at t = +~ .

Here we have arbitrarily chosen a = 0 at t = - ~ , and furthermore hereafter we assume & <t: 0 at t = - ~ : If some a satisfies eq. (10), then do also a + mrr and - a . Therefore it is sufficient to consider only the solution with ~ = 0 and & < 0 at t = - ~ .

The form o feq . (10) resembles the corresponding pendulum equation ~ + & - sin (2a) = 0 appearing in the Coulomb gauge problem [1,7] except that in the present case the "fr ic t ion" term is - ( t a n h t )& It is really a ( t-dependent friction for t < 0; but for t 3> 0 it acts as a ( t-dependent) "anti-fr ict ion", i.e., it accelerates the motion of c~. The potential is - s i n 2 ~ , which is the same as in the case of the Coulomb gauge. Thus

(d /d t ) [~& 2 - sin2o~] = (tanh t)& 2. (14)

The right-hand side o feq . (14) is of the same sign as t. Therefore, unless a is identically zero, the " to ta l energy,, I • 2 c~ -- sin2a of the pendulum is certainly negative for - ~ < t ~< 0. In other words,

0 < a < ~ r for - ~ , < t ~< 0, (15a)

or otherwise

= 0 for all t. (15b)

Now let us enumerate several immediate conse- quences of eqs. (10 ) - (15 ) .

(a) If, following ref. [3] , one requires eq. (1), i.e.,

lira r A = O so that s i n a ( t = 0 ) = 0 , (16) / '--+co

then owing to eqs. (15) the vacuum is uniquely A u = 0

for all t. This situation is the same as in the Coulomb

gauge [31. (b) We already know a non-trivial solution to eq.

(8): it is the phase function ~ of the famous pseudoparticle solution by Belavin et al. [4]

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a =/3, /3 = - t a n - l u . (17)

In terms of the variable t, the "mot ion" o f this solution is as follows:

3 = 0 , / J ~ 0 , at t = - o o ,

/3 = 7r/2, ~ = 1, at t = -T-0, (18)

/3=~r, / J ~ 0 , at t=+oo.

It does not oscillate, i.e.,/3 > 0 all through - ~ < t < oo. In terms of this/3, the pseudoparticle potential A (B) by Belavin et al. is given by

: ( r ) A(B)(x) x2 + X2 g-l~)ug, g = exp i/3 ~x , (19)

where ), is an arbitrary scale parameter. This A (B) is regular everywhere, satisfies the Landau gauge condition OuAu = 0, and it has nothing to do with the "vacuum" so long as condition (16) is imposed on the latter. In order to accomodate this pseudoparticle to condition (16), one needs a transformation out of the Landau gauge.

(c) There cannot exist a solution to eq. (10) such that la(t = +oo) _ a(t = -oo)l > 7r. The proof goes as follows: Suppose that there exists a solution such that it starts from a = 0 at t = _oo and passes a = 7r at some time to, that is, a = 7r and & > 0 at t = t 0. This t o should be positive because of eqs. (15), and according-

1o 2 ly the total "energy" : a - sin2a of this pendulum in creases all through t ~> t O (see eq. (14)). On the other hand this ~ "energy" is already positive at t = t 0. Ob- viously such an a cannot meet the boundary condition (13) which requires that the above "energy" should be zero at t = +oo. q.e.d.

Let us now proceed to the discussion of a general field Guy and its potential A u. The topological quantum number q of the field Gup is defined by

a fTr(au, ,~, .w ) d4x, q - 167r 2

1 "Guy =:euvxoGxo, e1234 = 1.

If the potential A u is regular everywhere (in the sense that [0x, 0o] A u = 0 for any ~, P , t0 , then eq..(20) can be rewritten as a surface integral over the three-dimensional boundary surface S 3 of the whole four-dimensional euclidean space-t ime:

X4

T

• r

Fig. 1.

1 1 q = 47r 2 3~ " eafs~ ~ f Tr[AoA.rA ~ ] d3oa. (21)

$3(X2__~ ~) /

We are considering, of course, a field of finite action so that at large x 2 the potential A u should approach sufficiently rapidly to the pure gauge type, A u _~ g- l~ug as x 2 -~ large. Eq. (21) together with this asymptotic form ofA u clarifies the topological mean- ing of q.

In general, the potential Au(x ) may involve some scale parameter ~, and also be 0(3) non-symmetric. However, in the present paper, we limit our considera- tions to potentials that, for x 2 >> ),2, become 0(3) symmetric and h-independent. In other words, the asymptotic form ofA u considered here is given by

(0) of eq. (4), with the phase function c~ depending A u only on u = r/x 4. It means that all the previous preliminary discussions and particularly the statements (a) and (c) are applicable in the asymptotic region.

With such an asymptotic form of A u, it is easy to evaluate the surface integral (21). As usual, we consider a (four-dimensional) cylindrical region depicted in fig. 1, whose temporal extension is I - T , + T] and spa- tial radius is R. Then, we have

q = ¢+ - ¢ - + eL, (22)

where ¢+ and q~_ are respectively the contributions from the top and bot tom surface of the cylinder,

4~_+ = - ( l / n ) l i m [ot (r ,x 4 = +-T) T , R ~ o *

1 . r=R - -~sm 2a(r,x 4 = +T)] r=0 '

(23)

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and q~L is the contribution from the lateral surface,

CL= +(1/n) lim [ a ( r = R , x 4 ) T,R--, ~ (24)

i . --[ x4=+T -- 5" sm 2a(r = R , x4 ) 1 x4 =- T "

Note that these ~b's are gauge-dependent, but the re- suiting q is gauge-independent. Furthermore, although the ¢'s depend also on the order and the way of taking the limit T, R -+ ~ , the resulting q does not. Substitut- ing eqs. (23) and (24) into eq. (22) and taking into ac- count the boundary condition sin a = 0 at r = 0, we ob- tain

q = (1/if) [0~(r = 0, x 4 -+ +co) _ ct(r = 0 , x 4 --> _oo)]. (2s)

This holds irrespective of the choice of gauge. In terms of the variable t defined by eq. (9) (see also eq. (11)),

q = (1/lr)[a(t : +~) - a ( t : - ~ ) ] . (26)

Therefore from statement (c) it follows that

tql ~< 1, (27)

if the potential A u ( x ) is subjected to the Landau gauge condition, O u A u = O.

In the course of the above estimation of q, the potential has been assumed to be regular everywhere. Therefore our final conclusion is as follows:

I. If the asymptotic form of the potential is restricted to be 0(3) symmetric and X-independent, then, in the Landau gauge, the potential for the field configuration with [ql > 1 is necessarily singular. This statement is independent of condit ion (16) ( l imr~ = rA = 0). If we impose this condit ion, then, owing to eqs. (15) or statement (a), we arrive at the stronger conclusion:

II. If furthermore condition (16) (limr__, ~ rA = O) is imposed, then, also in the Landau gauge, the vacuum is uniquely A u = 0 (at least for sufficiently large x 2) and even the field configuration with Iql = 1 necessitates some singularities of the potential. In other words, in so far as condit ion (16) is imposed and we still want to confine ourselves to the non-singu-

lar and asymptotical ly 0 (3) symmetric potentials, we have to step out of the Landau gauge in order to accomodate field configurations having non-vanishing q and realize the vacuum tunneling process.

In the present paper, we have restricted our consid- erations to potentials that are asymptotically 0 (3 ) symmetric and ?vindependent. Examination of a more general class of potentials is a remaining problem. Nevertheless, comparing our conclusions with those of ref. [3] , we see the very similarity between the Landau gauge and the Coulomb gauge; this is completely in ac- cordance with a naive expectation. In any case these two gauges seem to be peculiar and very different from the other gauges, say, the temporal gauge. We feel that it will be too restrictive to impose a differential type of gauge condit ion (such as OuAu = 0) throughout whole space- t ime (e.g., for all times through the vacuum tunneling process).

References

[1] V.N. Gribov, Lecture at the 12th Winter School of the Leningrad Nuclear Physics Institute (1977) ed. H.D.I. Abarbanel, SLAC-TRANS-176.

[2] C.M. Bender, T. Eguchi and H. Pagels, Rockefeller Univ. preprint (1977); D. Amati and A. Rouet, Phys. Lett. 73B (1978) 39.

[3] S. Sciuto, Phys. Lett. 71B (1977) 129; M. Ademollo, E. Napolitano and S. Sciuto, CERN preprint TH.2412 (1977); R. Jackiw, I. Muzinich and C. Rebbi, BNL preprint BNL- 2336 (1977).

[4] A.M. Polyakov, Phys. Lett. 59B (1975) 82; A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Yu.S. Tyupkin, Phys. Lett. 59B (1975) 85.

[5] G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8; R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976) 172; A.M. Polyakov, Nucl. Phys. B120 (1977) 429; C.G. Callan, R.F. Dashen and D.J. Gross, Phys. Lett. 63B (1976) 334; 66B (1977) 375; R. Jackiw, Rev. Mod. Phys. 49 (1977) 681.

[6] E. Witten, Phys. Rev. Lett. 38 (1977) 121; R. Jackiw, C. Nohl and C. Rebbi, Phys. Rev. D15 (1977) 1642; J.J. Giambiagi and K.D. Rothe, Nucl. Phys. B129 (1977) 111.

[7] S. Wadia and T. Yoneya, Phys. Lett. 66B (1977) 341.

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A pathology of the Landau gauge in euclidean Yang-Mills theories