We show that a modification of the BRST lattice quantization allows to
circumvent an old paradox, formulated by Neuberger, related to lattice Gribov
copies and non-perturbative BRST invariance. In the continuum limit the usual
BRST formulation is recovered.Comment: 7 pages, LaTe

Faddeev

Gribov

M. Testa

Neuberger

Singer

Wilson

Physics Letters B

English

arXiv

TESTA, Massimo

Archivio della ricerca- Università di Roma La Sapienza

Lattice gauge fixing, Gribov copies and BRST symmetry

We show that a modification of the BRST lattice quantization allows to circumvent an old paradox, formulated by Neuberger, related to lattice Gribov copies and non-perturbative BRST invariance. In the continuum limit the usual BRST formulation is recovered.We show that a modification of the BRST lattice quantization allows to circumvent an old paradox, formulated by Neuberger, related to lattice Gribov copies and non-perturbative BRST invariance. In the continuum limit the usual BRST formulation is recovered

Testa, M.

CERN Document Server

LATTICE GAUGE FIXING, GRIBOV COPIESANDBRST SYMMETRYM. Testa1;21 Theory Division, CERN, 1211 Geneva 23, Switzerland?.2 Dipartimento di Fisica, Universita di Roma \La Sapienza"Sezione INFN di RomaP.le A. Moro 2, 00185 Roma, Italy??.AbstractWe show that a modication of the BRST lattice quantization allows to circumventan old paradox, formulated by Neuberger, related to lattice Gribov copies and non-perturbative BRST invariance. In the continuum limit the usual BRST formulation isrecovered.? Address until August 31st, 1998.?? Permanent address.1 IntroductionBesides being a landmark in our understanding of non-perturbative hadron dynamics,the lattice regularization of gauge theories[1] is an important laboratory where formaleld theoretical ideas can be cross-checked for their validity or possible flaws may beidentied.One of these ideas is the gauge xing. In fact we are in the fortunate positionthat lattice regularization does not require any gauge-xing, so that the steps towardsa gauge-xed version of the theory are completely non formal. In particular a nonformal meaning could be attached to the Faddeev-Popov procedure[2], so essential formany applications. The Faddeev-Popov procedure has afterwards been replaced by themore formal apparatus of the so-called BRST symmetry[3]. Despite these importantdevelopments, some problems are still open. It is in fact known[4, 5] that compactgauge xing is aected by the presence of Gribov copies and one is therefore faced tothe problem of dealing, in the functional integral, with several gauge equivalent copiesof the same conguration. An appealing solution[6], naturally suggested by the BRSTformulation, is that the various Gribov copies, weighted by the Faddeev-Popov deter-minant, contribute to the functional integral with alternating signs. This opens theway to a cancellation in which, nally, only the contribution from a single copy shouldsurvive. It was shown by Neuberger[7], on very general grounds, that a cancellationtakes place indeed, with the disastrous result of leaving physical observables in theembarrassing indeterminate form 00 . The situation becomes even more confused when,as it happens for chiral gauge theories, a gauge invariant discretization is missing.One possible scheme for their quantization, known as the Rome approach[8], requiresgauge-xing and BRST quantization as fundamental ingredients. While the validityof this approach can be checked within perturbation theory, most of the objectionsraised against it[9] concern the lack of control of the gauge xing procedure at thenon-perturbative level, although Neuberger’s argument is not directly relevant in thiscase.In this paper we will show how the paradox presented in ref.[7] can be avoided,through a slight modication of the BRST scheme, in the case of theories admitting agauge invariant discretization.In section 2 we will review the Neuberger’s argument. In section 3 we will clarifyits Gribov-like nature through a very simplied one dimensional integral. In section 4we will discuss the modication required in order to obtain a consistent BRST quan-tization.2 The paradoxFollowing ref.[7] we consider a discretized gauge-invariant theory in a nite volume, sothat the functional integral reduces to a nite-dimensional one. The expectation valueof any gauge invariant operator O(U) is given by:hOi =1ZZDUe−S(U)O(U) (1)Z ZDUe−S(U)1where DU denotes the group-invariant integration measure over links and S(U) is thegauge invariant euclidean action.The process of gauge-xing requires, rst of all, the choice of a gauge condition, i.e.a gauge non invariant function, f(U; x), and the introduction of auxiliary degrees offreedom: the grassmannian ghost and anti-ghost variables, c and c, and the Lagrangemultipliers , to be integrated over the whole real axis.After gauge-xing, the expectation value in eq.(1), is replaced by:hOi =1Z 0ZD e−S(U)e−2R2 exp(Zfc) O(U) (2)Z 0 ZD e−S(U)e−2R2 exp(Zfc)whereD  DU d dc dc (3)and  denotes the nilpotent (2 = 0) BRST transformation which acts on the links Uas an innitesimal gauge transformation with parameter c and:c = ic =12cc (4) = 0In eq.(2) both the measure and the integrand are BRST invariant. The expectationvalue hOi computed through eq.(2) is formally equal to the one computed througheq.(1).However let us examine the quantity:FO(t) ZD e−S(U)e−2R2 exp(t Zfc) O(U) (5)which coincides with the numerator in eq.(2), when the real parameter t is equal to 1.We have:dFO(t)dt=ZD [Zfc] e−S(U)e−2R2 exp(t Zfc) O(U) = (6)=ZD Zfc e−S(U)e−2R2 exp(t Zfc) O(U)= 0because the integral of a total BRST variation vanishes identically. On the other handwe also have:FO(0) =ZD e−S(U)e−2R2 O(U) = 0 (7)according to Berezin rules of grassmannian integration, because there are no ghost oranti-ghost variables in the integrand of eq.(7). Eqs.(6) and (7) imply:FO(1) = 0 (8)Similar considerations show that:Z 0 = 0 (9)thus reducing eq.(2) to an indeterminate form 00 .We stress that these manipulations are completely justied by the nite dimension-ality of the regularized functional integral.23 The nature of the paradoxIn order to visualize where the problem comes from, we consider a toy (abelian) modelwith only one degree of freedom, one "link" variable U , which we choose to parametrizethrough its phase, as:U = eiaA (10)In eq.(11), A is a variable with values ranging between a and a is a parameter,reminiscent of the lattice spacing, in more realistic situations, whose limit a ! 0 willbe used to connect the periodic, compact case to the non compact one. The integralwe consider is the trivial one:N Z a−adA (11)"Gauge xing" proceeds exactly as in section 2: we choose a periodic function f(A)and consider:N 0 =Z a−adAZ +1−1dZdc dc e−22 exp([cf(A)]) (12)where  is the BRST variation (2 = 0), dened by:A = c (13)c = ic = 0 = 0c = 0 is of course peculiar to the present abelian situation.Eqs.(12) and (13) give:N 0 =Z a−adAZ +1−1dZdc dc e−22eif(A)e−cf0(A)c = (14)=Z a−adAZ +1−1d e−22eif(A)f 0(A) ==r2Z a−adf(A) e−f(A)22 = 0for a periodic f(A).Why must we choose a periodic f(A)?Because we want the expectation value of a BRST variation to vanish identically,as a consequence of BRST invariance.Let us choose, for instance:Γ  [cF (A)] = iF (A)− cF 0(A)c (15)A simple computation gives:hΓi Z a−adAZ +1−1dZdc dc e−22eif(A)e−cf0(A)c Γ = (16)=r2Z a−adAddA(F (A) e−f(A)22 )3Eq.(16) shows that hΓi is zero, only if both f(A) and F (A) are periodic functions.Choosing  = 0 in eq.(14), we get:N 0 = 2Z a−adA f 0(A) (f(A)) (17)which discloses the Gribov-like nature of the zero obtained in eq.(14): eq.(17) tellsus that N 0 gets contributions only from the zeros of f(A). Each zero contributesalternatively a factor 1 and the periodicity of f(A) implies an even number of zeros.4 The solutionThere is, however, a way out which exploits the compact nature of the variable A. Thebasic idea is simple: it amounts to change the BRST formulation, so that in eq.(17)the periodic -function appears. As we will show in a moment, this allows the choiceof a function f(A) periodic up to a shift, i.e.:f(A+2a) = f(A) +2ka(18)with an integer k, still leaving a periodic integrand in eq.(17). In this way the numberof zeroes is odd and the Gribov problem is overcome.To be more precise, let us start recalling some properties of the periodic -function,P (x). P (x) is dened as[10]:P (x) 1Xn=−1(x − n2a) (19)and may also be represented as:P (x) =a21Xn=−1einax (20)The modication I propose, is to allow the variable  in eq.(12) to assume discretevalues only:n  an (21)with integer n, and replace eq.(12) by:N 00 = a+1Xn=−1Z a−adAZdc dc e−2n2 exp([cf(A)]) (22)The BRST transformations have the same form as those dened in eq.(13), only with replaced by n. The same steps leading to eq.(14) give now:N 00 = a+1Xn=−1Z a−adA e−2n2einf(A)f 0(A) (23)We can now choose a function f(A) obeying the condition stated in eq.(18): whilef(A) itself cannot be expressed globally in terms of the link U , dened in eq.(10), both4einf(A) and f 0(A) can, being strictly periodic functions of A. We can easily check thatthe consequences expected from BRST invariance are satised by the new denition,eq.(23). In particular the expectation value of Γ, dened in eq.(15), still vanishes fora periodic F (A). However, in general, N 00 6= 0: the argument of ref.[7], reviewedin section 2, fails to apply in this case because, while einf(A) is a periodic function,eitnf(A) is not periodic, except for integer values of t.It is worth noticing that, in the "continuum limit", a! 0, eq.(23) converges to theusual BRST expression:lima!0N 00 =Z +1−1dAZ +1−1d e−22eif(A)f 0(A)) (24)thus recovering a continuous valued  variable.The above arguments seem to translate directly into the much more complicatedcase of lattice gauge theories where we have a set of Lagrangian multipliers, (x),for each space-time point x. The basic move is the discretization of the range of theLagrange multipliers, n(x), which allows the introduction of "twisting" gauge xings,as exemplied in eq.(18). We stress again that this amounts to a non conventionalchoice of the gauge condition, f(A;x), which should not be expressible in terms of thelink variables, U(x) = eiaA(x), while eiRd4xn(x)f(A;x) should.In realistic cases, of course, it is not easy to identify gauge-xing conditions apt toavoid the vanishing results implied by eqs.(8) and (9): it is, however, most encouragingthat, in presence of discretized n’s, eqs.(8) and (9) cannot be shown to be valid anymore.5 ConclusionsWe have proposed a possible way out to an old paradox concerning non-perturbativeBRST symmetry in lattice regularized gauge theories. The solution consists in chang-ing the BRST formulation by allowing the Lagrange multipliers, which enforce thegauge condition, to range over discrete values only. This modication gives more free-dom in the choice of the gauge condition, allowing, in simple examples, to circumventthe paradox1. Even more convincing is the fact that, within this modied BRSTformulation, the steps leading to the paradox cannot be implemented.In the continuum limit, the modied BRST formulation naturally converges towardsthe conventional continuum one, in the sense that the discretized Lagrange multipliersbecome continuous variables as a! 0.It is hard to think that the modied BRST will have any impact on actual numericalsimulations, because its implementation seems rather involved. To this we must addthat Neuberger’s argument, although relevant to gauge-xings used in practice, wasalways ignored by the practitioners in the eld.Are there any relevant consequences for the Rome approach? I don’t think so. Inthe Rome approach one starts from a discretized theory which is not gauge invariantand tries to merge it, as a ! 0, into a continuum gauge xed, gauge invariant (i.e.1The possibility of a multivalued gauge-xing condition is hinted in the concluding remarks of ref.[7], butit is not pursued any further.5satisfying the continuum BRST identities) theory. As stressed several times, within theRome approach one does not deal with regularized, exactly gauge-invariant systems.AcknowledgementsI thank the CERN Theory Division for the kind hospitality.I also thank G. C. Rossi for discussions.References[1] K. Wilson, Phys. Rev. D14 (1974) 2455[2] L. D. Faddeev, V. N. Popov, Phys. Lett. B25 (1967) 29[3] C. Becchi, A. Rouet, R. Stora, Phys. Lett. B52 (1974) 344;I. V. Tyupkin Gauge invariance in eld theory and statistical physics in operatorialformulation, preprint of Lebedev Physics Institute n.39 (1975)[4] V. N. Gribov, Nucl. Phys. B139 (1978) 1[5] I. M. Singer, Comm. Math. Phys. 60 (1978) 7[6] P. Hirschfeld, Nucl. Phys. B157 (1979) 37;B. Sharpe, J. Math. Phys. 25 (1984) 3324[7] H. Neuberger, Phys. Lett. B183 (1987) 337[8] A. Borrelli, L. Maiani, G. C. Rossi, R. Sisto, M. Testa, Nucl. Phys. B333 (1990)335;A. Borrelli, L. Maiani, G. C. Rossi, R. Sisto, M. Testa, Phys. Lett. B221 (1989)360;L. Maiani, G. C. Rossi, M. Testa, Phys. Lett. B261 (1991) 479;L. Maiani, G. C. Rossi, M. Testa, Phys. Lett. B292 (1992) 397;M. Testa, The Rome Approach to Chirality, invited talk at APCTP - ICTP JointInternational Conference (AIJIC 97) on Recent Developments in NonperturbativeQuantum Field Theory, Seoul, Korea, 26-30 May 1997. hep-lat/9707007[9] H. Neuberger, Remarks on Lattice BRST Invariance, hep-lat/9801029[10] M. J. Lighthill, Introduction to Fourier analysis and generalised functions, Cam-bridge Univ. Press 19586

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