We present Extended Double BRST on the lattice and extend the Neuberger
problem to include the ghost/anti-ghost symmetric formulation of the non-linear
covariant Curci-Ferrari (CF) gauges. We then show how a CF mass regulates the
0/0 indeterminate form of physical observables, as observed by Neuberger, and
discuss the gauge-parameter and mass dependence of the model.Comment: Prepared for 7th Conference on Quark Confinement and the Hadron
  Spectrum, Ponta Delgada, Azores, Portugal, 2-7 Sep 2006. 3p

Ghiotti, M.

von Smekal, L.

Williams, A. G.

English

arXiv

M. Ghiotti

L. von Smekal

A. G. Williams

Crossref

PUBLISHED VERSION  
 
 
Ghiotti, Marco; von Smekal, Lorenz Johann Maria; Williams, Anthony Gordon.  
Extended double lattice BRST Curci-Ferrari mass and the neuberger problem, Quark Confinement 
and the Hadron Spectrum VII: 7th Conference on Quark Confinement and the Hadron Spectrum - 
QCHS7. 2-7 September, 2006 / J. Emilio, F. T. Ribeiro, N. Brambilla, A. Vairo, K. Maung, and G. M. 
Prosperi (eds.): pp.180-182.  
 © 2007 American Institute of Physics. This article may be downloaded for personal use only. Any other use 
requires prior permission of the author and the American Institute of Physics. 
 
The following article appeared in Quark Confinement and the Hadron Spectrum VII: 7th Conference on Quark 
Confinement and the Hadron Spectrum - QCHS7. 2-7 September, 2006 / J. Emilio, F. T. Ribeiro, N. Brambilla, 
A. Vairo, K. Maung, and G. M. Prosperi (eds.): pp.180-182 and may be found at 
http://link.aip.org/link/?APCPCS/892/180/1 
  
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
http://hdl.handle.net/2440/43852 
 
 
 
 
PERMISSIONS 
http://www.aip.org/pubservs/web_posting_guidelines.html 
The American Institute of Physics (AIP) grants to the author(s) of papers submitted to or 
published in one of the AIP journals or AIP Conference Proceedings the right to post and 
update the article on the Internet with the following specifications. 
On the authors' and employers' webpages:  
 There are no format restrictions; files prepared and/or formatted by AIP or its vendors 
(e.g., the PDF, PostScript, or HTML article files published in the online journals and 
proceedings) may be used for this purpose. If a fee is charged for any use, AIP 
permission must be obtained.  
 An appropriate copyright notice must be included along with the full citation for the 
published paper and a Web link to AIP's official online version of the abstract.  
 
31st March 2011 
 
 
 
date ‘rights url’ accessed / permission obtained: (overwrite text) 
Extended Double Lattice BRST, Curci-Ferrari 
Mass and the Neuberger Problem 
M. Ghiotti, L. von Smekal and A. G. Williams 
Centre for the Subatomic Structure of Matter, University of Adelaide, SA 5005, Australia 
Abstract. We present Extended Double BRST on the lattice and extend the Neuberger problem to 
include the ghost/anti-ghost symmetric formulation of the non-linear covariant Curci-Ferrari (CF) 
gauges. We then show how a CF mass regulates the 0/0 indeterminate form of physical observables, 
as observed by Neuberger, and discuss the gauge parameter and mass dependence of the model. 
Keywords: Lattice BRST, TFT, Neuberger Problem, Curci-Ferrari Mass 
PACS: ll.15.Ha, 11.30.Ly, 11.30.Pb 
INTRODUCTION 
In the covariant continuum formulation of gauge theories one has to deal with the redun-
dant degrees of freedom due to gauge invariance. Within the language of local quantum 
field theory, the machinery for this is based on the Becchi-Rouet-Stora-Tyutin (BRST) 
symmetry which can be considered the quantum version of local gauge invariance. Be-
yond perturbation theory one faces the famous Gribov ambiguity: the existence copies of 
gauge-configurations that satisfy the Lorentz condition (or any other local gauge fixing 
condition) but are related by gauge transformations, and are thus physically equivalent. 
As a result, the usual definitions of a BRST charge fail to be globally valid. A rigorous 
non-perturbative framework is provided by lattice gauge theory. Its strength and beauty 
derives from the fact that gauge-fixing is not required. However, in order to arrive at 
a non-perturbative definition of non-Abelian gauge theories in the continuum, from a 
lattice formulation, we need to be able to perform the continuum limit in a formally wa-
tertight way. The same ambiguity then shows in another form when attempting to fix a 
gauge via BRST formulations on the lattice. There it is known as the Neuberger prob-
lem which asserts that the expectation value of any gauge invariant (and thus physical) 
observable in a lattice BRST formulation will always be of the indefinite form 0/0 [1]. 
In this talk we present the ghost/anti-ghost symmetric Curci-Ferrari gauges with 
double BRST on the lattice. We show how Neuberger's argument can be extended 
to include these non-linear covariant gauges, and how the indeterminate form 0/0 of 
expectation values is regulated by CF mass term [2] thereby decontracting the double 
BRST algebra to its extended version. Finally, we discuss how the gauge-parameter | 
dependence of the model can be compensated by adjusting the CF mass with | . 
In pure SU{N) lattice gauge theory, the gauge transformation of link Utj is defined 
as Uf- = giUjjg'-. BRST and anti-BRST transformations s and s in the topological 
setting do not act on the link variables U directly, but on the gauge transformations 
gi like infinitesimal left translations in the gauge group with real ghost and anti-ghost 
CP892, Quark Confinement and the Hadron Spectrum VII 
edited by J. E. F. T. Ribeiro 
© 2007 American Institute of Physics 978-0-7354-0396-3/07/$23.00 
180 
Downloaded 01 Apr 2011 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions
Grassmann fields c", c" as parameters, sgt = Cjgj and sgj = Cjgj, where c, = c"X
a and 
Cj = c"Xa. For the normalization of the anti-Hermitian generators Xa in the fundamental 
representation we use XxXaXh = —8ab/2. The action of the topological lattice model for 
gauge fixing a la Faddeev-Popov with double BRST invariance can then be written as 
5CF = isi(v[US]+ig 2trc,Ci) = £ (tf?i?[t/*] - ^M?[U8,c] + \{b
a
1f + | ( c " x ct)
2) , 
where V[Ug] = —S;Sy~;trf/^ = — 2XXjURetrU^ is the gauge fixing functional of 
covariant gauges which here assumes the role of a Morse potential on a gauge orbit. 
F"[US] = 0 is the gauge-fixing condition andMf[U8, c] = ^j-Mffc1!- defines the Faddeev-
Popov operator of the ghost/anti-ghost symmetric gauges. Note the occurrence of quartic 
ghost self-interactions °= (c, x c,)2 = (/"6cc*c^)2 which make the Neuberger problem 
somewhat less obvious in these gauges. Details will be presented elsewhere. 
REGULARISATION OF 0/0 AND % INDEPENDENCE 
Following Neuberger, we introduce an auxiliary parameter / upfront the Morse potential, 
to write the Euclidean partition function used as the gauge-fixing device, with double 
BRST, 
ZCT(0 = Jd\g,b,c,c] exp{ - i s ^ t V p S ] + i% YpciC^ } (1) 
which is independent of {£/} and t,. Because a derivative w.r.t. / produces a BRST-exact 
operator in the integrand, it is in fact independent oft also, i.e., Zl(t) = 0. For / = 0 on 
the other hand we find that ZCF(0) = 0; and this is the reason for the indeterminate form 
of 0/0 for all observables first derived for the standard linear covariant gauges in [1]. 
The fact that this conclusion holds also in the ghost/anti-ghost symmetric formulation 
with its quartic self-interactions directly relates to the topological interpretation [3] of 
the Neuberger zero: ZCF can be viewed as the partition function of a Witten-type TQFT 
which computes the Euler character % of the gauge group. On the lattice the gauge group 
is a direct product of SU(N)'s per site, and ZCF = x{SU(N)
#sites) = x{SU(N))#sites = 
Q#sites p o r ^ = o the action decouples from the link-field configuration and ZCF(0), albeit 
computing the same topological invariant, has no effect in terms of fixing a gauge. In 
the present formulation, ZCF(0) factorises into independent Grassmann integrations per 
site of the quartic term containing the curvature of SU(N), each of which computes the 
vanishing Euler character of SU(N) via the Gauss-Bonnet theorem [4]. 
As proposed in [5], this zero can be regularised, however, by introducing a Curci-
Ferrari mass m, such that the gauge-fixing action SC¥ is replaced by 
SmCF{t) = i{ss-im
2) (tVu[g}+i^tic,c^. (2) 
The corresponding partition function ZmCF(t) no-longer vanishes at / = 0, and this part 
in Neuberger's disastrous conclusion is thus avoided. We have explicitly calculated 
ZmCF(0), which is polynomial in i;m
4, for 577(2) and 577(3). The original zero is obtained 
181 
Downloaded 01 Apr 2011 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions
for m2 —> 0 which corresponds to a Wigner-Ionu contraction of the so-called extended 
double BRST superalgebra. While a non-vanishing m2 thereby breaks the nilpotency of 
BRST and anti-BRST transformations, which is known to result in a loss of unitarity, it 
also serves to regulate the 0/0 indeterminate form of expectation values in lattice BRST 
formulations, and to obtain finite results for m2 —> 0 via l'Hospital's rule. 
For gauge fixing we need to have t ^ 0. The partition function ZmCF(/) of the mas-
sive CF model is no-longer /-independent because s and s are no-longer nilpotent and 
the simple argument above fails, i.e., ZrJ(t) ^ 0 for m
2 ^ 0. However, the existence 
of 3 independent parameters /, | and m2 is an illusion. A change in / can always be 
compensated by changing the gauge parameter | and m2. In fact, simple scaling argu-
ments and explicit calculations show that ZmCF only depends on 2 combinations of the 
3, we can parametrise ZmCF = / ( /
2 / | , | / w 4 ) = f(x2,mA), where we defined x2 = t2/| 
and fhA = \mA. Our explicit calculations for / = 0 yield /(0,/w4). Independence of / 
then comes together with gauge parameter | independence. To achieve this, we allow 
in2 = in2 (x) so that ZmCF = f(x, m
A (x)). This means that we adjust the CF mass in2 with x 
such that our x = 0 results remain unchanged. In particular, we must have 
d_ =td_ ti$:JL\z = 0 (3) 
dx Voix dx amL 1 
which can be used to determine the derivative of m2(x). This is always possible. The 
crucial question at this point is whether it can be done independent of the link con-
figuration {£/}. As our explicit calculations are restricted to x = 0 we have explicitly 
verified that m2l(0) is finite and independent of {£/}. While this is merely necessary, but 
not sufficient, it demonstrates that we can get away from x = 0, at least infinitesimally. 
This is of qualitative importance as a non-zero value of x = tl \J%, no matter how small, 
corresponds to a large but finite | at / = 1 and thus eliminates the gauge freedom. 
CONCLUSIONS 
The massive Curci-Ferrari model with extended double BRST symmetry can be formu-
lated on the lattice without the 0/0 problem. The parameter m2 is not interpreted as a 
physical mass but to meaningfully define a limit m2 —> 0 in the spirit of l'Hospital's rule. 
At finite m2 the topological nature of the gauge-fixing partition function seems lost. It 
is possible, however, to tune the CF mass with the gauge parameter | so that the limit 
m2 —> 0 can be defined along a certain trajectory in parameter space independent of | . 
An interesting open question might then be the topological interpretation of the model 
within the extended double BRST superalgebra framework. 
REFERENCES 
1. H. Neuberger, Phys. Lett. B175, 69 (1986); ibid B183, 337 (1987). 
2. G. Curci, and R. Ferrari, Phys. Lett. B63, 91 (1976); Nuovo dm. A35, 1 (1976). 
3. B. Sharpe, J. Math. Phys. 25 (1984) 3324; L. Baulieu and M. Schaden, Int. J. Mod Phys. A13, 985 
(1998); M. Schaden, Phys. Rev. D59, 014508 (1999). 
4. D. Birmingham, M. Blau, M. Rakowski, and G. Thompson, Phys. Kept. 209, 129 (1991). 
5. A. C. Kalloniatis, L. von Smekal, and A. G. Williams, Phys. Lett. B609, 424 (2005). 
182 
Downloaded 01 Apr 2011 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions


Extended Double Lattice BRST, Curci-Ferrari Mass and the Neuberger Problem

We present Extended Double BRST on the lattice and extend the Neuberger problem to include the ghost/anti-ghost symmetric formulation of the non-linear covariant Curci-Ferrari (CF) gauges. We then show how a CF mass regulates the 0/0 indeterminate form of physical observables, as observed by Neuberger, and discuss the gauge-parameter and mass dependence of the model.M. Ghiotti, L. von Smekal, A.G. William

Von Smekal, L.

Williams, A.

Adelaide Research &amp; Scholarship

PUBLISHED VERSION    Ghiotti, Marco; von Smekal, Lorenz Johann Maria; Williams, Anthony Gordon.  Extended double lattice BRST Curci-Ferrari mass and the neuberger problem, Quark Confinement and the Hadron Spectrum VII: 7th Conference on Quark Confinement and the Hadron Spectrum - QCHS7. 2-7 September, 2006 / J. Emilio, F. T. Ribeiro, N. Brambilla, A. Vairo, K. Maung, and G. M. Prosperi (eds.): pp.180-182.   © 2007 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.  The following article appeared in Quark Confinement and the Hadron Spectrum VII: 7th Conference on Quark Confinement and the Hadron Spectrum - QCHS7. 2-7 September, 2006 / J. Emilio, F. T. Ribeiro, N. Brambilla, A. Vairo, K. Maung, and G. M. Prosperi (eds.): pp.180-182 and may be found at http://link.aip.org/link/?APCPCS/892/180/1                    http://hdl.handle.net/2440/43852     PERMISSIONS http://www.aip.org/pubservs/web_posting_guidelines.html The American Institute of Physics (AIP) grants to the author(s) of papers submitted to or published in one of the AIP journals or AIP Conference Proceedings the right to post and update the article on the Internet with the following specifications. On the authors' and employers' webpages:   There are no format restrictions; files prepared and/or formatted by AIP or its vendors (e.g., the PDF, PostScript, or HTML article files published in the online journals and proceedings) may be used for this purpose. If a fee is charged for any use, AIP permission must be obtained.   An appropriate copyright notice must be included along with the full citation for the published paper and a Web link to AIP's official online version of the abstract.   31st March 2011    date ‘rights url’ accessed / permission obtained: (overwrite text) Extended Double Lattice BRST, Curci-Ferrari Mass and the Neuberger Problem M. Ghiotti, L. von Smekal and A. G. Williams Centre for the Subatomic Structure of Matter, University of Adelaide, SA 5005, Australia Abstract. We present Extended Double BRST on the lattice and extend the Neuberger problem to include the ghost/anti-ghost symmetric formulation of the non-linear covariant Curci-Ferrari (CF) gauges. We then show how a CF mass regulates the 0/0 indeterminate form of physical observables, as observed by Neuberger, and discuss the gauge parameter and mass dependence of the model. Keywords: Lattice BRST, TFT, Neuberger Problem, Curci-Ferrari Mass PACS: ll.15.Ha, 11.30.Ly, 11.30.Pb INTRODUCTION In the covariant continuum formulation of gauge theories one has to deal with the redun-dant degrees of freedom due to gauge invariance. Within the language of local quantum field theory, the machinery for this is based on the Becchi-Rouet-Stora-Tyutin (BRST) symmetry which can be considered the quantum version of local gauge invariance. Be-yond perturbation theory one faces the famous Gribov ambiguity: the existence copies of gauge-configurations that satisfy the Lorentz condition (or any other local gauge fixing condition) but are related by gauge transformations, and are thus physically equivalent. As a result, the usual definitions of a BRST charge fail to be globally valid. A rigorous non-perturbative framework is provided by lattice gauge theory. Its strength and beauty derives from the fact that gauge-fixing is not required. However, in order to arrive at a non-perturbative definition of non-Abelian gauge theories in the continuum, from a lattice formulation, we need to be able to perform the continuum limit in a formally wa-tertight way. The same ambiguity then shows in another form when attempting to fix a gauge via BRST formulations on the lattice. There it is known as the Neuberger prob-lem which asserts that the expectation value of any gauge invariant (and thus physical) observable in a lattice BRST formulation will always be of the indefinite form 0/0 [1]. In this talk we present the ghost/anti-ghost symmetric Curci-Ferrari gauges with double BRST on the lattice. We show how Neuberger's argument can be extended to include these non-linear covariant gauges, and how the indeterminate form 0/0 of expectation values is regulated by CF mass term [2] thereby decontracting the double BRST algebra to its extended version. Finally, we discuss how the gauge-parameter | dependence of the model can be compensated by adjusting the CF mass with | . In pure SU{N) lattice gauge theory, the gauge transformation of link Utj is defined as Uf- = giUjjg'-. BRST and anti-BRST transformations s and s in the topological setting do not act on the link variables U directly, but on the gauge transformations gi like infinitesimal left translations in the gauge group with real ghost and anti-ghost CP892, Quark Confinement and the Hadron Spectrum VII edited by J. E. F. T. Ribeiro © 2007 American Institute of Physics 978-0-7354-0396-3/07/$23.00 180 Downloaded 01 Apr 2011 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissionsGrassmann fields c", c" as parameters, sgt = Cjgj and sgj = Cjgj, where c, = c"Xa and Cj = c"Xa. For the normalization of the anti-Hermitian generators Xa in the fundamental representation we use XxXaXh = —8ab/2. The action of the topological lattice model for gauge fixing a la Faddeev-Popov with double BRST invariance can then be written as 5CF = isi(v[US]+ig 2trc,Ci) = £ (tf?i?[t/*] - ^M?[U8,c] + \{ba1f + | ( c " x ct)2) , where V[Ug] = —S;Sy~;trf/^ = — 2XXjURetrU^ is the gauge fixing functional of covariant gauges which here assumes the role of a Morse potential on a gauge orbit. F"[US] = 0 is the gauge-fixing condition andMf[U8, c] = ^j-Mffc1!- defines the Faddeev-Popov operator of the ghost/anti-ghost symmetric gauges. Note the occurrence of quartic ghost self-interactions °= (c, x c,)2 = (/"6cc*c^)2 which make the Neuberger problem somewhat less obvious in these gauges. Details will be presented elsewhere. REGULARISATION OF 0/0 AND % INDEPENDENCE Following Neuberger, we introduce an auxiliary parameter / upfront the Morse potential, to write the Euclidean partition function used as the gauge-fixing device, with double BRST, ZCT(0 = Jd\g,b,c,c] exp{ - i s ^ t V p S ] + i% YpciC^ } (1) which is independent of {£/} and t,. Because a derivative w.r.t. / produces a BRST-exact operator in the integrand, it is in fact independent oft also, i.e., Zl(t) = 0. For / = 0 on the other hand we find that ZCF(0) = 0; and this is the reason for the indeterminate form of 0/0 for all observables first derived for the standard linear covariant gauges in [1]. The fact that this conclusion holds also in the ghost/anti-ghost symmetric formulation with its quartic self-interactions directly relates to the topological interpretation [3] of the Neuberger zero: ZCF can be viewed as the partition function of a Witten-type TQFT which computes the Euler character % of the gauge group. On the lattice the gauge group is a direct product of SU(N)'s per site, and ZCF = x{SU(N)#sites) = x{SU(N))#sites = Q#sites p o r ^ = o the action decouples from the link-field configuration and ZCF(0), albeit computing the same topological invariant, has no effect in terms of fixing a gauge. In the present formulation, ZCF(0) factorises into independent Grassmann integrations per site of the quartic term containing the curvature of SU(N), each of which computes the vanishing Euler character of SU(N) via the Gauss-Bonnet theorem [4]. As proposed in [5], this zero can be regularised, however, by introducing a Curci-Ferrari mass m, such that the gauge-fixing action SC¥ is replaced by SmCF{t) = i{ss-im2) (tVu[g}+i^tic,c^. (2) The corresponding partition function ZmCF(t) no-longer vanishes at / = 0, and this part in Neuberger's disastrous conclusion is thus avoided. We have explicitly calculated ZmCF(0), which is polynomial in i;m4, for 577(2) and 577(3). The original zero is obtained 181 Downloaded 01 Apr 2011 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissionsfor m2 —> 0 which corresponds to a Wigner-Ionu contraction of the so-called extended double BRST superalgebra. While a non-vanishing m2 thereby breaks the nilpotency of BRST and anti-BRST transformations, which is known to result in a loss of unitarity, it also serves to regulate the 0/0 indeterminate form of expectation values in lattice BRST formulations, and to obtain finite results for m2 —> 0 via l'Hospital's rule. For gauge fixing we need to have t ^ 0. The partition function ZmCF(/) of the mas-sive CF model is no-longer /-independent because s and s are no-longer nilpotent and the simple argument above fails, i.e., ZrJ(t) ^ 0 for m2 ^ 0. However, the existence of 3 independent parameters /, | and m2 is an illusion. A change in / can always be compensated by changing the gauge parameter | and m2. In fact, simple scaling argu-ments and explicit calculations show that ZmCF only depends on 2 combinations of the 3, we can parametrise ZmCF = / ( / 2 / | , | / w 4 ) = f(x2,mA), where we defined x2 = t2/| and fhA = \mA. Our explicit calculations for / = 0 yield /(0,/w4). Independence of / then comes together with gauge parameter | independence. To achieve this, we allow in2 = in2 (x) so that ZmCF = f(x, mA (x)). This means that we adjust the CF mass in2 with x such that our x = 0 results remain unchanged. In particular, we must have d_ =td_ ti$:JL\z = 0 (3) dx Voix dx amL 1 which can be used to determine the derivative of m2(x). This is always possible. The crucial question at this point is whether it can be done independent of the link con-figuration {£/}. As our explicit calculations are restricted to x = 0 we have explicitly verified that m2l(0) is finite and independent of {£/}. While this is merely necessary, but not sufficient, it demonstrates that we can get away from x = 0, at least infinitesimally. This is of qualitative importance as a non-zero value of x = tl \J%, no matter how small, corresponds to a large but finite | at / = 1 and thus eliminates the gauge freedom. CONCLUSIONS The massive Curci-Ferrari model with extended double BRST symmetry can be formu-lated on the lattice without the 0/0 problem. The parameter m2 is not interpreted as a physical mass but to meaningfully define a limit m2 —> 0 in the spirit of l'Hospital's rule. At finite m2 the topological nature of the gauge-fixing partition function seems lost. It is possible, however, to tune the CF mass with the gauge parameter | so that the limit m2 —> 0 can be defined along a certain trajectory in parameter space independent of | . An interesting open question might then be the topological interpretation of the model within the extended double BRST superalgebra framework. REFERENCES 1. H. Neuberger, Phys. Lett. B175, 69 (1986); ibid B183, 337 (1987). 2. G. Curci, and R. Ferrari, Phys. Lett. B63, 91 (1976); Nuovo dm. A35, 1 (1976). 3. B. Sharpe, J. Math. Phys. 25 (1984) 3324; L. Baulieu and M. Schaden, Int. J. Mod Phys. A13, 985 (1998); M. Schaden, Phys. Rev. D59, 014508 (1999). 4. D. Birmingham, M. Blau, M. Rakowski, and G. Thompson, Phys. Kept. 209, 129 (1991). 5. A. C. Kalloniatis, L. von Smekal, and A. G. Williams, Phys. Lett. B609, 424 (2005). 182 Downloaded 01 Apr 2011 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions

Extended double lattice BRST Curci-Ferrari mass and the neuberger problem

PUBLISHED VERSION  
 
 
Ghiotti, Marco; von Smekal, Lorenz Johann Maria; Williams, Anthony Gordon.  
Extended double lattice BRST Curci-Ferrari mass and the neuberger problem, Quark Confinement 
and the Hadron Spectrum VII: 7th Conference on Quark Confinement and the Hadron Spectrum - 
QCHS7. 2-7 September, 2006 / J. Emilio, F. T. Ribeiro, N. Brambilla, A. Vairo, K. Maung, and G. M. 
Prosperi (eds.): pp.180-182.  
 © 2007 American Institute of Physics. This article may be downloaded for personal use only. Any other use 
requires prior permission of the author and the American Institute of Physics. 
 
The following article appeared in Quark Confinement and the Hadron Spectrum VII: 7th Conference on Quark 
Confinement and the Hadron Spectrum - QCHS7. 2-7 September, 2006 / J. Emilio, F. T. Ribeiro, N. Brambilla, 
A. Vairo, K. Maung, and G. M. Prosperi (eds.): pp.180-182 and may be found at 
http://link.aip.org/link/?APCPCS/892/180/1 
  
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
http://hdl.handle.net/2440/43852 
 
 
 
 
PERMISSIONS 
http://www.aip.org/pubservs/web_posting_guidelines.html 
The American Institute of Physics (AIP) grants to the author(s) of papers submitted to or 
published in one of the AIP journals or AIP Conference Proceedings the right to post and 
update the article on the Internet with the following specifications. 
On the authors' and employers' webpages:  
 There are no format restrictions; files prepared and/or formatted by AIP or its vendors 
(e.g., the PDF, PostScript, or HTML article files published in the online journals and 
proceedings) may be used for this purpose. If a fee is charged for any use, AIP 
permission must be obtained.  
 An appropriate copyright notice must be included along with the full citation for the 
published paper and a Web link to AIP's official online version of the abstract.  
 
31st March 2011 
 
 
 
date ‘rights url’ accessed / permission obtained: (overwrite text) 
Extended Double Lattice BRST, Curci-Ferrari 
Mass and the Neuberger Problem 
M. Ghiotti, L. von Smekal and A. G. Williams 
Centre for the Subatomic Structure of Matter, University of Adelaide, SA 5005, Australia 
Abstract. We present Extended Double BRST on the lattice and extend the Neuberger problem to 
include the ghost/anti-ghost symmetric formulation of the non-linear covariant Curci-Ferrari (CF) 
gauges. We then show how a CF mass regulates the 0/0 indeterminate form of physical observables, 
as observed by Neuberger, and discuss the gauge parameter and mass dependence of the model. 
Keywords: Lattice BRST, TFT, Neuberger Problem, Curci-Ferrari Mass 
PACS: ll.15.Ha, 11.30.Ly, 11.30.Pb 
INTRODUCTION 
In the covariant continuum formulation of gauge theories one has to deal with the redun-
dant degrees of freedom due to gauge invariance. Within the language of local quantum 
field theory, the machinery for this is based on the Becchi-Rouet-Stora-Tyutin (BRST) 
symmetry which can be considered the quantum version of local gauge invariance. Be-
yond perturbation theory one faces the famous Gribov ambiguity: the existence copies of 
gauge-configurations that satisfy the Lorentz condition (or any other local gauge fixing 
condition) but are related by gauge transformations, and are thus physically equivalent. 
As a result, the usual definitions of a BRST charge fail to be globally valid. A rigorous 
non-perturbative framework is provided by lattice gauge theory. Its strength and beauty 
derives from the fact that gauge-fixing is not required. However, in order to arrive at 
a non-perturbative definition of non-Abelian gauge theories in the continuum, from a 
lattice formulation, we need to be able to perform the continuum limit in a formally wa-
tertight way. The same ambiguity then shows in another form when attempting to fix a 
gauge via BRST formulations on the lattice. There it is known as the Neuberger prob-
lem which asserts that the expectation value of any gauge invariant (and thus physical) 
observable in a lattice BRST formulation will always be of the indefinite form 0/0 [1]. 
In this talk we present the ghost/anti-ghost symmetric Curci-Ferrari gauges with 
double BRST on the lattice. We show how Neuberger's argument can be extended 
to include these non-linear covariant gauges, and how the indeterminate form 0/0 of 
expectation values is regulated by CF mass term [2] thereby decontracting the double 
BRST algebra to its extended version. Finally, we discuss how the gauge-parameter | 
dependence of the model can be compensated by adjusting the CF mass with | . 
In pure SU{N) lattice gauge theory, the gauge transformation of link Utj is defined 
as Uf- = giUjjg'-. BRST and anti-BRST transformations s and s in the topological 
setting do not act on the link variables U directly, but on the gauge transformations 
gi like infinitesimal left translations in the gauge group with real ghost and anti-ghost 
CP892, Quark Confinement and the Hadron Spectrum VII 
edited by J. E. F. T. Ribeiro 
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Grassmann fields c", c" as parameters, sgt = Cjgj and sgj = Cjgj, where c, = c"Xa and 
Cj = c"Xa. For the normalization of the anti-Hermitian generators Xa in the fundamental 
representation we use XxXaXh = —8ab/2. The action of the topological lattice model for 
gauge fixing a la Faddeev-Popov with double BRST invariance can then be written as 
5CF = isi(v[US]+ig 2trc,Ci) = £ (tf?i?[t/*] - ^M?[U8,c] + \{ba1f + | ( c " x ct)2) , 
where V[Ug] = —S;Sy~;trf/^ = — 2XXjURetrU^ is the gauge fixing functional of 
covariant gauges which here assumes the role of a Morse potential on a gauge orbit. 
F"[US] = 0 is the gauge-fixing condition andMf[U8, c] = ^j-Mffc1!- defines the Faddeev-
Popov operator of the ghost/anti-ghost symmetric gauges. Note the occurrence of quartic 
ghost self-interactions °= (c, x c,)2 = (/"6cc*c^)2 which make the Neuberger problem 
somewhat less obvious in these gauges. Details will be presented elsewhere. 
REGULARISATION OF 0/0 AND % INDEPENDENCE 
Following Neuberger, we introduce an auxiliary parameter / upfront the Morse potential, 
to write the Euclidean partition function used as the gauge-fixing device, with double 
BRST, 
ZCT(0 = Jd\g,b,c,c] exp{ - i s ^ t V p S ] + i% YpciC^ } (1) 
which is independent of {£/} and t,. Because a derivative w.r.t. / produces a BRST-exact 
operator in the integrand, it is in fact independent oft also, i.e., Zl(t) = 0. For / = 0 on 
the other hand we find that ZCF(0) = 0; and this is the reason for the indeterminate form 
of 0/0 for all observables first derived for the standard linear covariant gauges in [1]. 
The fact that this conclusion holds also in the ghost/anti-ghost symmetric formulation 
with its quartic self-interactions directly relates to the topological interpretation [3] of 
the Neuberger zero: ZCF can be viewed as the partition function of a Witten-type TQFT 
which computes the Euler character % of the gauge group. On the lattice the gauge group 
is a direct product of SU(N)'s per site, and ZCF = x{SU(N)#sites) = x{SU(N))#sites = 
Q#sites p o r ^ = o the action decouples from the link-field configuration and ZCF(0), albeit 
computing the same topological invariant, has no effect in terms of fixing a gauge. In 
the present formulation, ZCF(0) factorises into independent Grassmann integrations per 
site of the quartic term containing the curvature of SU(N), each of which computes the 
vanishing Euler character of SU(N) via the Gauss-Bonnet theorem [4]. 
As proposed in [5], this zero can be regularised, however, by introducing a Curci-
Ferrari mass m, such that the gauge-fixing action SC¥ is replaced by 
SmCF{t) = i{ss-im2) (tVu[g}+i^tic,c^. (2) 
The corresponding partition function ZmCF(t) no-longer vanishes at / = 0, and this part 
in Neuberger's disastrous conclusion is thus avoided. We have explicitly calculated 
ZmCF(0), which is polynomial in i;m4, for 577(2) and 577(3). The original zero is obtained 
181 
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for m2 —> 0 which corresponds to a Wigner-Ionu contraction of the so-called extended 
double BRST superalgebra. While a non-vanishing m2 thereby breaks the nilpotency of 
BRST and anti-BRST transformations, which is known to result in a loss of unitarity, it 
also serves to regulate the 0/0 indeterminate form of expectation values in lattice BRST 
formulations, and to obtain finite results for m2 —> 0 via l'Hospital's rule. 
For gauge fixing we need to have t ^ 0. The partition function ZmCF(/) of the mas-
sive CF model is no-longer /-independent because s and s are no-longer nilpotent and 
the simple argument above fails, i.e., ZrJ(t) ^ 0 for m2 ^ 0. However, the existence 
of 3 independent parameters /, | and m2 is an illusion. A change in / can always be 
compensated by changing the gauge parameter | and m2. In fact, simple scaling argu-
ments and explicit calculations show that ZmCF only depends on 2 combinations of the 
3, we can parametrise ZmCF = / ( / 2 / | , | / w 4 ) = f(x2,mA), where we defined x2 = t2/| 
and fhA = \mA. Our explicit calculations for / = 0 yield /(0,/w4). Independence of / 
then comes together with gauge parameter | independence. To achieve this, we allow 
in2 = in2 (x) so that ZmCF = f(x, mA (x)). This means that we adjust the CF mass in2 with x 
such that our x = 0 results remain unchanged. In particular, we must have 
d_
 =td_ ti$:JL\z = 0 (3) 
dx Voix dx amL 1 
which can be used to determine the derivative of m2(x). This is always possible. The 
crucial question at this point is whether it can be done independent of the link con-
figuration {£/}. As our explicit calculations are restricted to x = 0 we have explicitly 
verified that m2l(0) is finite and independent of {£/}. While this is merely necessary, but 
not sufficient, it demonstrates that we can get away from x = 0, at least infinitesimally. 
This is of qualitative importance as a non-zero value of x = tl \J%, no matter how small, 
corresponds to a large but finite | at / = 1 and thus eliminates the gauge freedom. 
CONCLUSIONS 
The massive Curci-Ferrari model with extended double BRST symmetry can be formu-
lated on the lattice without the 0/0 problem. The parameter m2 is not interpreted as a 
physical mass but to meaningfully define a limit m2 —> 0 in the spirit of l'Hospital's rule. 
At finite m2 the topological nature of the gauge-fixing partition function seems lost. It 
is possible, however, to tune the CF mass with the gauge parameter | so that the limit 
m
2
 —> 0 can be defined along a certain trajectory in parameter space independent of | . 
An interesting open question might then be the topological interpretation of the model 
within the extended double BRST superalgebra framework. 
REFERENCES 
1. H. Neuberger, Phys. Lett. B175, 69 (1986); ibid B183, 337 (1987). 
2. G. Curci, and R. Ferrari, Phys. Lett. B63, 91 (1976); Nuovo dm. A35, 1 (1976). 
3. B. Sharpe, J. Math. Phys. 25 (1984) 3324; L. Baulieu and M. Schaden, Int. J. Mod Phys. A13, 985 
(1998); M. Schaden, Phys. Rev. D59, 014508 (1999). 
4. D. Birmingham, M. Blau, M. Rakowski, and G. Thompson, Phys. Kept. 209, 129 (1991). 
5. A. C. Kalloniatis, L. von Smekal, and A. G. Williams, Phys. Lett. B609, 424 (2005). 
182 
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Extended Double Lattice BRST, Curci-Ferrari Mass and the Neuberger Problem

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Extended Double Lattice BRST, Curci-Ferrari Mass and the Neuberger Problem

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