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Eur. Phys. J. C (2010) 66: 451–464DOI 10.1140/epjc/s10052-010-1250-5
Regular Article - Theoretical Physics
Renormalizability of a quark–gluon modelwith soft BRST breaking in the infrared region
L. Baulieu1,2,a, M.A.L. Capri3,b, A.J. Gómez3,c, V.E.R. Lemes3,d, R.F. Sobreiro4,e, S.P. Sorella3,f
1Theoretical division CERN, 1211, Genève 23, Switzerland2LPTHE, CNRS and Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France3UERJ—Universidade do Estado do Rio de Janeiro, Instituto de Física, Departamento de Física Teórica, Rua São Francisco Xavier 524,20550-013, Maracanã, Rio de Janeiro, Brazil
4UFF—Universidade Federal Fluminense, Instituto de Física, Avenida Gal. Milton Tavares de Souza s/n, 24210-346, Niterói, Brazil
Received: 27 February 2009 / Revised: 23 September 2009 / Published online: 11 February 2010© Springer-Verlag / Società Italiana di Fisica 2010
Abstract We prove the renormalizability of a quark–gluonmodel with soft breaking of the BRST symmetry, which ac-counts for the modification of the large distance behavior ofthe quark and gluon correlation functions. The proof is validto all orders of perturbation theory, by making use of softlybroken Ward identities.
1 Introduction
The Gribov–Zwanziger framework [1–3] consists in restrict-ing the domain of integration in the Feynman path integralwithin the Gribov horizon. It has motivated extended stud-ies of non-perturbative tools for investigating the infraredbehavior of the gluon and ghost correlation functions, see[4] for the use of modified Schwinger–Dyson equations forQCD, [5–8] for recent analytical results and [9–11] for nu-merical data obtained through lattice simulations.
Zwanziger has been able to show that the restriction ofthe path integral within the Gribov horizon for the gluon canbe achieved by adding to the Faddeev–Popov action in theLandau gauge a local action, depending on new fields, withwell-defined interactions with the gluons and their Faddeev–Popov ghosts in the Landau gauge [2, 3]. However, this lo-cal action violates the BRST symmetry by a soft term, sothat BRST symmetry is only enforced in the scaling limit.
a e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]
The yet unorthodox point of view that the BRST symme-try can be broken in the IR region of QCD was heuristi-cally anticipated by Fujikawa [12]. It is not in contradic-tion with any given physical principle, since it is by nomeans necessary that the QCD microscopic theory pos-sesses a unitary sector for its partons, namely the quarks andgluons, to warrantee unitarity properties among the sectorof bound states that constitute its spectrum. The only re-quirement is that the modified theory remains renormaliz-able and that the BRST symmetry is recovered in the ul-traviolet region, in order to suitably describe the asymp-totic properties in terms of almost deconfined partons, aspredicted by asymptotic freedom and short distance expan-sion. From a physical point of view, it is gratifying for thequark–gluon model introduced in [13] that the modificationof the usual Feynman propagators into a Gribov-type prop-agators eliminates from the beginning all partons from thespectrum, since their modified propagators have no poles onthe real axis, a property that anticipates quite well the con-finement.
What justified our previous work [13] is that the genuinegeometrical approach of Zwanziger leaves aside the quarks,which do not participate in the Gribov phenomenon, whilethe idea of a parton model suggests that quarks and gluonsshould be treated on the same footing, and quark propagatorsshould have an analogous behavior as the gluon one in theinfra-red domain. An idea for getting such modified quarkpropagators was thus needed, which goes beyond the Gribovquestion. A generalization of the work of Zwanziger wasalso needed [6–8] in order to achieve a different behavior forthe shape of the genuine Gribov–gluon propagator D(q2)
that vanishes at q2 = 0, for accommodating recent latticesimulations that seem to indicate that the gluon propagatorgoes to a non-vanishing constant at very small q2 [9–11].
452 Eur. Phys. J. C (2010) 66: 451–464
To provide such an improved, local and renormalizable,quantum field theory that gives the wanted modifications inthe infra-red region, both for quarks and gluons, we pro-posed in [13] the following picture. Given a theory of par-tons (e.g. quark and gluons in four dimensions, or a scalarfields in dimensions such that the ultraviolet divergences arerenormalizable or super-renormalizable), it can always becoupled to a topological field theory made of new fieldsarranged as BRST trivial doublets, in such a way that thepartons are already confined at the tree level by their mix-ing with the unphysical fields of the topological field the-ory. We found that such a mixing can be generally allowedby a soft breaking of the BRST symmetry. Here, confine-ment is meant in a very simple way. The propagators ofall fields have only poles at complex positions.1 This im-plies that the theory has no vacuum for the partons and allobservables are made of composite operators, defined bysolving the cohomology of the BRST operator. These com-posite operators can be renormalized in the standard way,with expectation values related to the parameters of the softbreaking mechanism. This idea was inspired by the alge-braic characterization of the local terms that Zwanziger in-troduced to complete the Faddeev–Popov action for the glu-ons. Part of the Zwanziger action can be recognized as atopological action, which involves bosons and fermions thattransform under the BRST symmetry as a system of twoBRST trivial doublets, and it is BRST-exact. The remainingpart of Zwanziger’s action breaks the BRST symmetry in asoft way, and yields a Gribov propagator for the transversegluon.
The addition of the new fields arranged as BRST-exactdoublets eventually allows for the introduction of massiveparameters that can be used to modify the infrared behav-ior of the theory. As will be discussed later on in Sect. 3,in order to have positivity, the observables are identifiedby the cohomology of the BRST operator with the addi-tion of soft local terms, which are uniquely determined bythe softly broken Slavnov–Taylor identities that ensure therenormalizability of the theory. The necessity of breakingthe BRST symmetry can be easily understood within thisframework. If the added action were BRST-exact, nothingwould be changed for the predictions of the original par-ton theory. In fact, in the absence of the breaking, the inte-gration over the new fields would only consist in multiply-ing the partition function by one. So, if the new field de-pendence is through BRST-invariant terms (and thus BRST-exact terms, because they transform as BRST-exact dou-blets), there is no way to improve the infra-red behavior ofthe amplitudes. One can notice that the possibility of adding
1More precisely, the propagators display violation of reflection posi-tivity, a feature which invalidates the interpretation of partons as exci-tations of the physical spectrum of the theory.
such a BRST-exact terms can be related to the formal invari-ance of the path integral under arbitrary changes of variablesfor the parton fields. By doing such changes of variablesand playing with Lagrange multipliers and determinants for-mula, one can indeed recover the class of BRST-exact ac-tion that are bilinear in the new fields that have the oppo-site statistics to the partons.2 To obtain the wanted modifica-tion of the infra-red behavior of parton correlators, one mustgo further, and consider the possibility of an explicit softbreaking of the BRST symmetry. This can modify the par-ton propagators in the low energy domain, without changingtheir ultraviolet behavior, provided the theory is renormaliz-able.
In the case of the improved quark–gluon theory, the prop-erties of the breaking and the respective consequences forthe theory can be summarized as follows.
– The breaking term is soft, meaning that its dimensionis smaller than the space–time dimension. As a conse-quence, the breaking can be neglected in the ultraviolet re-gion, where one recovers the notion of exact BRST sym-metry.
– The soft breaking of the BRST symmetry is introduced ina way compatible with the renormalizability. More pre-cisely, it is associated to a quadratic term in the fields ob-tained by demanding that the additional fields couple lin-early to the original fields of the theory, so that the result-ing propagators turn out to be modified in the infrared re-gion. As a consequence, a given correlation function candisplay a different behavior, when going from the deep ul-traviolet to the infrared region in momentum space. Fur-thermore, we point out that the renormalizability of themodel, encoded in a set of softly broken Slavnov–Tayloridentities, is of great relevance for the construction of therenormalized physical composite operators and of theircorrelation functions. As a consequence of the soft BRSTbreaking, the renormalized version of a gauge invariantcomposite operator often requires a suitable mixing witha set of operators which, besides quantities which areBRST exact, contains also BRST non-invariant soft oper-ators displaying an explicit dependence from the soft pa-rameters. This is the case, for example, of the compositeoperator F 2(x) = Fa
μν(x)F aμν(x) whose renormalization
has been recently worked out in [17], within the frame-work of the Gribov–Zwanziger theory. We are checkingthis interesting property of quantum field theory in a sep-arate work using a toy model built up with scalar fieldsonly. We underline here that the aforementioned mixingmight play a pivotal role in the cancellation mechanism
2In string theory, this idea has been already used by adding so-called“topological packages” to any given 2d-string action, which allows oneto show in this way possible relationships between the different stringmodels [14–16].
Eur. Phys. J. C (2010) 66: 451–464 453
of the unphysical cuts present in the correlation functionsof gauge invariant operators like, e.g. 〈F 2(x)F 2(y)〉. Al-though being beyond the aim of the present work, a wholesection will be devoted to the discussion of this importantissue, see Sect. 3.
– The soft BRST symmetry breaking is meant to be explicitbreaking, i.e. it is not a spontaneous symmetry breaking,which would give rise to Goldstone massless fields, andthus to a quite different framework, as it will be explainedelsewhere.
– All massive parameters should be related to the uniquescale of the theory, namely ΛQCD, by requiring that themassive soft parameters satisfy suitably gap-type equa-tions which allow to determine them in a self-consistentway. This is the case, for example, of the massive Gribovparameter γ which is fixed by a gap equation [2, 3, 5–7],see Sect. 4.
Following the procedure described above, in [13] a modelaccounting for a soft BRST symmetry breaking givingrise to a modification of the long distance behavior of thequark propagator was established. In fact, the model pre-dicts
⟨ψ(k)ψ(−k)
⟩ = iγμkμ + A(k)
k2 + A2(k), (1)
instead of the standard Dirac propagator for the quarksiγμkμ
k2 . In this equation
A(k) = 2M21M2
k2 + m2, (2)
is a function depending on the soft breaking mass parame-ters M1, M2 and m. From expression (2) one sees that thefunction A(k) vanishes in the deep ultraviolet region wherethe usual perturbative behavior for the propagator is recov-ered. It is worth mentioning that expression (2) provides agood fit for the dynamical mass generation for quarks in theinfrared region in the Landau gauge, as reported by latticenumerical simulations of the quark propagator [18, 19]. Werecall that the function (2) is analogous to the one appear-ing in the gluon propagator within the Gribov–Zwanzigerframework [2, 3]. In fact, as discussed in [13], the Gribov–Zwanziger action can be recovered through the introductionof a soft breaking, related to the appearance of the Gri-bov parameter γ . This parameter is needed to implementthe restriction to the Gribov horizon, which turns out to beat the origin of the soft breaking of the BRST symmetry[6, 7]. In particular, for the tree level gluon propagator, onefinds
⟨Aa
μ(k)Abν(−k)
⟩ = δab
(δμν − kμkν
k2
)1
k2 + M2(k), (3)
with
M2(k) = γ 4
k2 + μ2, (4)
where the second mass parameter μ accounts for the non-trivial dynamics of the auxiliary fields needed to localizeZwanziger’s horizon function, see [6, 7].
It is of course striking how similar are the modified gluonand quark propagators, in the sense that the introduction ofthe massive parameters by the above mechanism has elimi-nated the infra-red problem for both type of fields. Withinthe context of plain perturbation theory, these parameterscan be fined-tuned, and even be used as plain infra-red reg-ulators that are controlled by softly broken Ward identi-ties. However, in a confining theory like QCD, they mustbe related to physical observables, and non-perturbativelycomputed as a function of the basic parameter of the the-ory, ΛQCD. The clear advantage of our approach is that themodified theory is no more submitted to the Gribov ambi-guity and the parton (quark and gluons) propagators haveno poles at real values. The model excludes their appear-ance in the spectrum from the beginning, and its pertur-bative expansion can be possibly compared to the predic-tion of non-perturbative approach such as the lattice formu-lation in the Landau gauge. The method solves in a quitesimple conceptual way the necessity of introducing mas-sive parameters, not only for the Gribov ambiguities butalso for defining the scale of the chiral symmetry break-ing.
The goal of this paper is to prove an elementary, but nec-essary property of this model: it has to be multiplicativelyrenormalizable, to ensure that the breaking of the BRSTsymmetry remains soft, at any given finite order of per-turbation theory, and that no further parameters than thosewhich modify the infra-red behavior of propagators can ap-pear. We will do this by employing the algebraic renor-malization [21]. Notice that such a proof can be general-ized to other cases, where one may wish to introduce aninfra-red cut-off for other theories with non-trivial gaugeinvariance and control its effect by softly broken Wardidentities, within the context of locality. This could beof relevance for the study of non-exactly solvable super-renormalizable theories, like 3-dimensional gauge theories,where one must control the way the infra-red regularizationis compatible with the BRST symmetry, or even for gravity,when it is computed at a finite order of perturbation the-ory.
The paper is organized as follows. In Sect. 2 we providea short overview of the model and of its soft BRST sym-metry breaking. In Sect. 3 we provide a discussion aboutthe possible physical meaning of the model. Sections 4, 5are devoted to the algebraic proof of the renormalizabil-ity. In Sect. 6 we consider the inclusion of the Gribov–Zwanziger action and we discuss the renormalizability of
454 Eur. Phys. J. C (2010) 66: 451–464
the resulting model. Finally, in Sect. 7 we collect our con-clusions.
2 The quark model and its BRST soft breaking term
We start with the Yang–Mills action quantized in the Landaugauge,
Sinv =∫
d4x
(1
4Fa
μνFaμν + ψ i
α(γμ)αβDijμ ψ
jβ + iba∂μAa
μ
+ ca∂μDabμ cb
), (5)
where
Dijμ = ∂μδij − ig
(T a
)ijAa
μ, (6)
is the covariant derivative in the fundamental representationof the SU(N) gauge group, with generators (T a)ij , and
Dabμ = ∂μδab − gf abcAc
μ, (7)
is the covariant derivative in the adjoint representation. Thefirst set of small Latin indices {i, j, . . .} ∈ {1, . . . ,N} willbe used to denote the fundamental representation, whilethe second set {a, b, . . . , h} ∈ {1, . . . ,N2 − 1} will be em-ployed for the adjoint representation. The set of Greek in-dices {α,β, γ, δ} stand for spinor indices. The remainingGreek indices will denote space–time indices.
The action (5) is invariant under the BRST transforma-tions, namely
sAaμ = −Dab
μ cb,
sψiα = −igca
(T a
)ijψj
α ,
sψ iα = −igψj
αca(T a
)ji,
(8)sca = 1
2gf abccbcc,
sca = iba,
sba = 0.
Following [13], a model for the dynamical quark mass gen-eration can be constructed by introducing two BRST dou-blets of spinor fields (ξ i, θ i) and (ηi, λi), transforming as
sξ iα = θi
α, sθ iα = 0,
(9)sηi
α = λiα, sλi
α = 0.
The propagation of these fields is described by the followingBRST-exact action:
Sξλ = s
∫d4x
(−ηiα∂2ξ i
α + ξ iα∂2ηi
α + m2(ηiαξ i
α − ξ iαηi
α
))
=∫
d4x(−λi
α∂2ξ iα − ξ i
α∂2λiα − ηi
α∂2θiα + θ i
α∂2ηiα
+ m2(λiαξ i
α + ξ iαλi
α + ηiαθ i
α − θ iαηi
α
)), (10)
where m is a mass parameter. Further, we introduce the cou-pling of the spinors (ξ i, θ i) and (ηi, λi) with the matter fieldψi
α ,
SM =∫
d4x(M2
1
(ξ iαψi
α + ψ iαξ i
α
) − M2(λi
αψiα + ψ i
αλiα
)).
(11)
Evidently, the action (10) is BRST-invariant. This is not thecase of SM , which will give rise to a soft breaking of theBRST symmetry, parameterized by the two soft mass para-meters M1, M2. In fact,
sSM = Δ, (12)
where
Δ =∫
d4x(M2
1
(θ iαψi
α − ψ iαθ i
α
)
+ igM21ca
(ξ iα
(T a
)ijψj
α − ψ iα
(T a
)ijξ jα
)
+ igM2ca(λi
α
(T a
)ijψj
α − ψ iα
(T a
)ijλj
α
)). (13)
Notice that Δ is a soft breaking, i.e. it is of dimension lessthan four.
3 A discussion about the possible physical meaningof the model
Before facing the issue of the renormalizability of the model,it is worth to discuss the physical content of the quark modeldescribed by the action
S = Sinv + Sξλ + SM. (14)
To that purpose, this section will be divided in two parts.In the first one we discuss the role played by the auxiliaryspinor fields (ξ i, θ i) and (ηi, λi), and we clarify the contextin which our model might be relevant. In the second partwe address the issue of the physical content of the model.Relying on an explicit example borrowed from the Gribov–Zwanziger model, we shall outline a possible way to char-acterize the physical operators of the theory.
Eur. Phys. J. C (2010) 66: 451–464 455
3.1 The role of the auxiliary spinor fields
From expression (10) one sees that the fields (ξ i, θ i) and(ηi, λi) have been introduced in such a way that they couplelinearly with the matter fields ψ , ψ . Moreover, as is apparentfrom the presence of the Laplacian ∂2 in (10), these fieldsare not coupled to the gauge fields Aa
μ. As a consequence,the fields (ξ i, θ i), (ηi, λi) can be integrated out, yielding anon-local action for ψ and ψ , namely
Sψ =∫
d4x
(ψ i
α(γμ)αβDijμ ψ
jβ
− 2M21M2ψ
iα
(1
∂2 − m2
)ψi
α
). (15)
Expression (15) allows us to capture the physical meaningof the model which we are proposing as well as to elucidatethe physical situation for which it might be relevant. As isapparent from (15), we are introducing a running mass termfor the quark field ψ , encoded in the non-local operator
∫d4x−2M2
1M2ψiα
(1
∂2 − m2
)ψi
α =∫
d4x ψiα A
(∂2)ψi
α,
(16)
where the running mass is given by
A(∂2) = 2M2
1M2
−∂2 + m2. (17)
From expression (17) one sees that the running mass A(∂2)
vanishes in the high energy region, where one recovers thestandard massless quark action. However, A(∂2) entails adeep modification of the quark propagator in the infrared,due to the presence of the soft parameters M1, M2,m. Inparticular, from expression (1), it turns out that the matterpropagator 〈ψ(k)ψ(−k)〉 exhibits complex poles, which in-validate the usual particle interpretation. In other words, dueto the presence of the running mass A(∂2), quarks cannot beconsidered anymore as belonging to the physical spectrumof the theory in the infrared region. Evidently, what we havein mind is to propose a possible mechanism accounting forquark confinement.
Nevertheless, willing to treat the action (15) within a con-sistent quantum field theory framework, the non-local char-acter of the running mass A(∂2) has to be faced. The onlyway out at our disposal to treat non-local actions is thatof localizing the non-local term through the introduction ofa suitable set of auxiliary fields. This is precisely the roleplayed by the spinor fields (ξ i, θ i) and (ηi, λi). These fieldshave to be seen as a useful device in order to localize thenon-local operator A(∂2), so that the model can be treatedwithin a local quantum field theory framework, which en-ables us to discuss its renormalizability. At the end, after
having proven that the model is renormalizable, these auxil-iary fields have to be integrated out again, so that we end upwith the renormalized version of the running mass A(∂2).Let us anticipate the output of our forthcoming analysis byremarking that the action (14) turns out to be multiplica-tively renormalizable. This means that no new parameterswill show up and that the most general counterterm has pre-cisely the form of the action S, up to multiplicative renor-malizations. This will enable us to integrate out the auxil-iary fields at the end. We point out that this would be not bethe case if the auxiliary fields (ξ i, θ i), (ηi, λi) would havebeen coupled to the gauge fields Aa
μ through the introductionof covariant derivatives, e.g. λi
α∂2ξ iα → λi
αD2ξ iα . The pres-
ence of the covariant derivatives would introduce new in-teractions which would require quartic counterterms in thespinor fields (ξ i, θ i) and (ηi, λi). As a consequence, thesefields could not be integrated out at the end of the renor-malization process. It is thus important to provide a detailedproof of the renormalizability of the action S.
3.2 A few remarks on the physical content of the theory
Having clarified the role of the auxiliary spinor fields, weshall outline here a possible framework in order to extractuseful information on the physical content of the theory. Inparticular, we shall point out that the existence of the softbreaking of the BRST symmetry might have important con-sequences on the construction of the physical operators. Inorder to illustrate the main reasoning, we shall rely on an ex-plicit example taken from the Gribov–Zwanziger theory, see[2]. Let us start by giving a look at the gluon propagator ofthe Gribov–Zwanziger model, which can be obtained fromthe expression (3) by setting μ = 0, namely
⟨Aa
μ(k)Abν(−k)
⟩ = δab
(δμν − kμkν
k2
)k2
k4 + γ 4. (18)
As is apparent, expression (18) displays complex poles atk2 = ±iγ 2, which invalidate the interpretation of gluons asexcitations of the physical spectrum. In other words, in theinfrared, gluons cannot be considered as part of the phys-ical spectrum: they are confined. The physical excitationsof the theory would correspond to colorless bound states ofgluons, i.e. glueballs. We expect thus that information aboutthe physical spectrum could be obtained by looking at thecorrelation function of colorless gauge invariant operatorsas, for example:
G(p) =∫
d4x e−ipx⟨F 2(x)F 2(0)
⟩, (19)
where F 2(x) = Faμν(x)F a
μν(x). This correlation function isuseful in order to investigate the properties of the scalar spinzero glueball state. Within the Gribov–Zwanziger theory, it
456 Eur. Phys. J. C (2010) 66: 451–464
has to be evaluated with the Feynman rules obtained by em-ploying the confining gluon propagator (18). The explicitfirst order evaluation of expression (19) can be found in [2],and can be summarized as follows:
G(p) = Gphys(p) + Gunphys(p). (20)
The unphysical part, Gunphys(p), displays cuts beginning atthe unphysical values p2 = ±4iγ 2, whereas the physicalpart, Gphys(p), has a cut beginning at the physical thresh-old p2 = −2γ 2. Moreover, the spectral function of Gphys(p)
turns out to be positive, so that it obeys the Kallen–Lehmannrepresentation [2]. As such, Gphys(p) is an acceptable cor-relation function for physical glueball excitations. What isinteresting in expression (20) is that a physical cut hasemerged in the correlation function of gauge invariant quan-tities, even if it has been evaluated with a gluon propaga-tor exhibiting only unphysical complex poles. This is pre-cisely what one expects from a confining theory. Gluons arenot physical excitations, but the glueball correlation functiondisplays a physical singularity.
Of course, we have still to face the very difficult presenceof the unphysical part Gunphys(p). Here, we can argue thatthe presence of the soft BRST breaking might be a welcomefeature. In fact, we can figure out that, in the presence ofthe Gribov horizon, the correlation functions of the physi-cal operators should receive contributions from the horizon,which could be accounted for by the breaking term, throughthe presence of the auxiliary fields. This reasoning can bemade on a quantitative basis by recalling that in ordinaryYang–Mills theories the construction of the physical opera-tors is obtained through the cohomology of the BRST oper-ator [21]. In the present case, however, the BRST symmetryis softly broken. Nevertheless, it turns out that it is possibleto write down suitable softly broken Slavnov–Taylor iden-tities which ensure that the theory remains renormalizable.Moreover, as shown in the recent work [17], these identi-ties can be employed to determine how the renormalizedversion of a physical operator is explicitly affected by thepresence of the horizon. Taking as an example the compos-ite operator F 2(x), it turns out that its renormalized version,Rglueb(x), mixes with s-exact terms as well as with non-invariant BRST soft terms proportional to the Gribov para-meter γ 2 [17], namely
F 2(x) → Rglueb(x), (21)
with Rglueb(x) given by
Rglueb(x) = F 2(x) + sO(x) + γ 2 Pγ (x), (22)
where Pγ (x) stands for a non-invariant BRST soft term ofdimension two, built up with the auxiliary fields (ϕab
μ , ϕabμ )
present in the Gribov–Zwanziger theory [17], see Sect. 6,
Pγ = Dμϕaaμ + Dμϕaa
μ . (23)
From expression (22) it becomes apparent that the physicaloperators get modified by the presence of the Gribov hori-zon, encoded in the explicit γ -dependent term Pγ . As a con-sequence, for the glueball correlation function, we would get
⟨Rglueb(x)Rglueb(y)
⟩
= ⟨F 2(x)F 2(y)
⟩
+ ⟨s(F 2(x)O(y) + F 2(y)O(x)
)⟩
+ γ 2⟨F 2(x)Pγ (y) + F 2(y)Pγ (x)⟩
+ γ 2⟨(sO(x))
Pγ (y) + (sO(y)
)Pγ (x)
⟩
+ ⟨(sO(x)
)(sO(y)
)⟩
+ γ 4⟨Pγ (x)Pγ (y)⟩. (24)
In the absence of the Gribov horizon, i.e. when γ = 0,the inclusion of the BRST exact term sO(x) is completelyirrelevant, due to the fact the BRST symmetry is unbro-ken. As a consequence, the vacuum expectation value of s-exact quantities vanishes, so that the correlator 〈Rglueb(x)
Rglueb(y)〉 reduces to 〈F 2(x)F 2(y)〉. However, in the pres-ence of the Gribov horizon, i.e. for γ �= 0, BRST sym-metry is softly broken and the vacuum expectation valueof s-exact quantities no longer vanishes. We see thus thatboth operators O(x) and Pγ contribute to the correlationfunction 〈Rglueb(x)Rglueb(y)〉. Although the form of thenon-invariant operator Pγ turns out to be uniquely fixedby the softly broken Slavnov–Taylor identities [17], a cer-tain freedom is left out in the characterization of the ex-act operator O(x). This feature is a consequence of thefact that the BRST operator remains nilpotent. We couldthus search for a suitable term O(x) which would en-able us to cancel, order by order, the unphysical contribu-tions contained in Gunphys(p). Thus, the correlation function〈Rglueb(x)Rglueb(y)〉 would display only physical poles,corresponding to colorless gluon bound states.
So far, although the case of Yang–Mills theories has notyet been faced, a simpler toy model has been constructedwhere the disappearance of unphysical cuts by this mecha-nism seems to work on the examples that we have been ableto evaluate, at the one and two loop level, using expectationvalues analogous to 〈F 2(x)F 2(y)〉 and 〈F 2(x) F 2(y)F 2(z)〉[20]. Certainly, it is a point worth to be investigated. If true,this would mean that the Gribov horizon combines in a nicefashion with the soft breaking of the BRST operator in sucha way that the unphysical cuts present in the correlationfunctions of physical operators could be removed order byorder. The presence of the auxiliary fields would thus be aquite welcome feature, as they would give us a way to con-struct the exact BRST term O(x) needed to account for theunphysical cuts.
Eur. Phys. J. C (2010) 66: 451–464 457
4 Introducing sources for controlling the soft symmetrybreaking
In order to prove that the model described by the action S,(14), is multiplicatively renormalizable, we follow the pro-cedure outlined by Zwanziger in the study of the Gribovhorizon in the Landau gauge [2, 3]. It amounts to embeddingthe action (14) in a larger model, S → S0, displaying exactBRST invariance. This is achieved by treating the breakingterm Δ as a composite operator, which is introduced into thetheory through a suitable set of external sources. The origi-nal action S is thus recovered from the extended action S0 bydemanding that the sources acquire a particular value, whichwe shall refer to as the physical value. The renormalizabil-ity of S follows thus by proving the renormalizability of theextended action S0.
In order to introduce the extended invariant action S0, wemake use of the following set of sources (J,H), (J , H ),(K,G), (K, G) and (N,P ), assembled in BRST doublets,3
i.e.
sJijαβ = H
ijαβ, sH
ijαβ = 0,
sJijαβ = H
ijαβ, sH
ijαβ = 0,
sKijαβ = G
ijαβ, sG
ijαβ = 0, (25)
sKijαβ = G
ijαβ, sG
ijαβ = 0,
sN = P, sP = 0.
The invariant action that accounts for the extra spinor fieldsand for the breaking term is thus defined as
SJK = s
∫d4x
(−ηiα∂2ξ i
α + ξ iα∂2ηi
α + P(ηi
αξ iα − ξ i
αηiα
)
+ σNP + Jijαβ ξ i
αψjβ + J
ijαβψ
jβξ i
α
+ Kijαβλi
αψjβ + K
ijαβψ
jβλi
α
), (26)
which, explicitly, reads
SJK =∫
d4x(−λi
α∂2ξ iα − ξ i
α∂2λiα − ηi
α∂2θiα + θ i
α∂2ηiα
+ P(λi
αξ iα + ξ i
αλiα + ηi
αθ iα − θ i
αηiα
) + σP 2
+ Hijαβ ξ i
αψjβ + H
ijαβψ
jβξ i
α + Gijαβλi
αψjβ + G
ijαβψ
jβλi
α
− Jijαβ
(θ iαψ
jβ − ξ i
αigca(T a
)jkψk
β
)
− Jijαβ
(igψk
βca(T a
)kjξ iα − ψ
jβθ i
α
)
+ Kijαβλi
αigca(T a
)jkψk
β − Kijαβigψk
βca(T a
)kjλi
α
),
(27)
3For future purposes, we recall that all quantum numbers of fields andsources are displayed in Tables 1 to 6, including the charge Q4N thatwill be defined through expression (32).
Table 1 Quantum numbers of the ordinary Yang–Mills fields
Fields A b c c ψ ψ
UV dim. 1 2 0 2 3/2 3/2
Ghost no 0 0 1 −1 0 0
Q4N -charge 0 0 0 0 0 0
Spinor no 0 0 0 0 1 −1
Statistics 0 0 1 −1 1 −1
Table 2 Quantum numbers of the extra fields
Fields ξ ξ λ λ
UV dim. 1/2 1/2 3/2 3/2
Ghost no 0 0 0 0
Q4N -charge 1 −1 1 −1
Spinor no 0 0 0 0
Statistics 1 −1 1 −1
Table 3 Quantum numbers of more extra fields
Fields θ θ η η
UV dim. 1/2 1/2 3/2 3/2
Ghost no 1 1 −1 −1
Q4N -charge 1 −1 1 −1
Spinor no 0 0 0 0
Statistics 2 0 0 −2
Table 4 Quantum numbers of the ordinary Yang–Mills sources
Sources Ω L Y Y
UV dim. 3 4 5/2 5/2
Ghost no −1 −2 −1 −1
Q4N -charge 0 0 0 0
Spinor no 0 0 1 −1
Statistics −1 −2 0 −2
Table 5 Quantum numbers of the extra sources with vanishing physi-cal values
Sources J J K K N
UV dim. 2 2 1 1 2
Ghost no −1 −1 −1 −1 −1
Q4N -charge 1 −1 1 −1 0
Spinor no 1 −1 1 −1 0
Statistics 1 −3 1 −3 −1
whose BRST invariance is manifest. The parameter σ , inexpression (27), is a dimensionless parameter needed for
458 Eur. Phys. J. C (2010) 66: 451–464
Table 6 Quantum numbers of the extra sources with non-vanishingphysical values
Sources H H G G P
UV dim. 2 2 1 1 2
Ghost no 0 0 0 0 0
Q4N -charge 1 −1 1 −1 0
Spinor no 1 −1 1 −1 0
Statistics 2 −2 2 −2 0
renormalization purposes. For the so-called physical valuesof the external sources, we have
Jijαβ
∣∣phys = J
ijαβ
∣∣phys = 0,
Kijαβ
∣∣phys = K
ijαβ
∣∣phys = 0,
Hijαβ
∣∣phys = H
ijαβ
∣∣phys = M2
1 δij δαβ,(28)
Gijαβ
∣∣phys = G
ijαβ
∣∣phys = −M2δ
ij δαβ,
P∣∣phys = m2,
N∣∣phys = 0.
The BRST-invariant extended action S0 is thus defined as
S0 = Sinv + Sξλ + SJK + Sext. (29)
It is easily checked that the starting action S, (14), is recov-ered from the extended action S0 when taking the physicalvalues of the sources, (28), namely
S0|phys = S +∫
d4xσm4. (30)
It is worth noticing that the action S0 possesses an additionalU(4N) global symmetry, provided by
QijαβS0 = 0, (31)
where
Qijαβ
=∫
d4x
(ξ iα
δ
δξjβ
− ξ iα
δ
δξjβ
+ λjβ
δ
δλiα
− λjβ
δ
δλiα
+ θiα
δ
δθjβ
− θ iα
δ
δθjβ
+ ηjβ
δ
δηiα
− ηjβ
δ
δηiα
+ Jjkβσ
δ
δJ ikασ
− Jjkβσ
δ
δJ ikασ
+ Kikασ
δ
δKjkβσ
− Kikασ
δ
δKjkβσ
+ Gikασ
δ
δGjkβσ
− Gikασ
δ
δGjkβσ
+ Hjkβσ
δ
δH ikασ
− H ikασ
δ
δHjkβσ
). (32)
The trace of the operator Qijαβ , i.e. Qαα
ii = Q4N , defines anew conserved charge, and thus an additional quantum num-ber for the fields and sources, allowing for the introductionof a multi-index I = (i, α), with I ∈ {1, . . . ,4N}, for thefields φ = (ξ, η,λ, θ) and sources Γ = (K,J,H,G). Ac-cordingly, we shall set
φiα = φI ,
(33)Γ
ijαβ = Γ
Ijβ .
From now on, we shall make use of the multi-index notation.
5 Algebraic proof of the renormalizability
Let us face now the issue of the renormalizability of the ex-tended action S0, a task which we shall undertake by makinguse of the algebraic renormalization [21]. Let us start by es-tablishing the set of Ward identities fulfilled by the actionS0. To that purpose, we add external sources, La , Ωa
μ, Y iα
and Y iα , coupled to the non-linear BRST variations of the
fields, namely
Sext = s
∫d4x
(Ωa
μAaμ + Laca + Y i
αψiα + ψ i
αY iα
)
=∫
d4x
(−Ωa
μDμca + 1
2gf abcLacbcc
− igY iαca
(T a
)ijψj
α − igψjαca
(T a
)jiY i
α
), (34)
with
sΩaμ = sLa = sY = sY = 0. (35)
The term (34) allows one to convert the BRST symme-try into the corresponding Slavnov–Taylor identity. Thus, toprove the renormalizability of the model we shall considerthe more general action
Σ = S0 + Sext. (36)
5.1 Ward identities
The complete action (36) fulfills a rich set of Ward Identities,namely:
– The Slavnov–Taylor identity
S(Σ) = 0, (37)
where
S(Σ)
=∫
d4x
(δΣ
δΩaμ
δΣ
δAaμ
+ δΣ
δY iα
δΣ
δψiα
− δΣ
δY iα
δΣ
δψiα
Eur. Phys. J. C (2010) 66: 451–464 459
+ δΣ
δca
δΣ
δLa+ iba δΣ
δca+ θ I δΣ
δξ I+ λI δΣ
δηI+ θI δΣ
δξI
+ λI δΣ
δηI+ HIj
α
δΣ
δJIjα
+ GIjα
δΣ
δKIjα
+ H Ijα
δΣ
δJIjα
+ GIjα
δΣ
δKIjα
+ PδΣ
δN
). (38)
– The gauge condition:
δΣ
δba= i∂μAa
μ. (39)
– The antighost equation:
δΣ
δca+ ∂μ
δΣ
δΩaμ
= 0. (40)
– The ghost equation:
GaΣ = Δacl, (41)
with
Ga =∫
d4x
(δ
δca− if abccb δ
δbc
− ig(T a
)jk(
J Ijα
δ
δHIkα
+ KIjα
δ
δGIkα
)
− ig(T a
)kj(
J Ijα
δ
δH Ikα
+ KIjα
δ
δGIkα
)), (42)
and
Δacl =
∫d4x
(gf abcΩb
μAcμ + gf abcLbcc
+ igY iα
(T a
)ijψj
α − igψiα
(T a
)ijY j
α
). (43)
– The classical rigid invariance
RIJ Σ = 0, (44)
where
RIJ =∫
d4x
(λI δ
δξ J− λJ δ
δξI+ H Ik
α
δ
δGJkα
− HJkα
δ
δGIkα
+ J Jkα
δ
δKIkα
+ J Ikα
δ
δKJkα
). (45)
– The equations of motion of the doublet fields:
δΣ
δηI= −∂2θI + PθI ,
δΣ
δθI= ∂2ηI − PηI − J Ij
α ψjα ,
(46)δΣ
δλI+ KIj
α
δΣ
δYjα
= −∂2ξI + PξI + GIjα ψj
α ,
δΣ
δξ I+ J Ij
α
δΣ
δYjα
= −∂2λI + PλI + HIjα ψ
ja .
– The Ward identity for the source N :
δΣ
δN= 0. (47)
5.2 The invariant counterterm
In order to characterize the most general invariant coun-terterm which can be freely added to all orders in pertur-bation theory [21], we perturb the classical action Σ byadding an integrated local polynomial Σcount of dimensionbounded by four, and with vanishing ghost number as wellas Q4N -charge. We thus demand that the perturbed action,(Σ + εΣcount), where ε is an expansion parameter, fulfills,to the first order in ε, the same Ward identities as fulfilled bythe classical action Σ , i.e. (37)–(47). This requirement givesrise to the following constraints for the counterterm Σcount:
BΣΣcount = 0, (48)
δ
δbaΣcount = 0, (49)
(δ
δca+ ∂μ
δ
δΩaμ
)Σcount = 0, (50)
GaΣcount = 0, (51)
RIJ Σcount = 0, (52)
δΣcount
δN= 0, (53)
δΣcount
δηI= 0, (54)
δΣcount
δθ I= 0, (55)
δΣcount
δλI+ KIj
α
δΣcount
δYjα
= 0, (56)
δΣcount
δξ I+ J Ij
α
δΣcount
δYjα
= 0, (57)
where the operator BΣ in (48) stands for the nilpotent lin-earized Slavnov–Taylor operator,
BΣ =∫
d4x
(δΣ
δΩaμ
δ
δAaμ
+ δΣ
δAaμ
δ
δΩaμ
− δΣ
δψiα
δ
δY iα
− δΣ
δY iα
δ
δψiα
+ δΣ
δY iα
δ
δψiα
+ δΣ
δψiα
δ
δY iα
+ δΣ
δca
δ
δLa
+ δΣ
δLa
δ
δca+ iba δ
δca+ θ I δ
δξ I+ λI δ
δηI+ θI δ
δξ I
+ λI δ
δηI+ HIj
α
δ
δJIjα
+ GIjα
δ
δKIjα
+ H Ijα
δ
δJIjα
+ GIjα
δ
δKIjα
+ Pδ
δN
). (58)
460 Eur. Phys. J. C (2010) 66: 451–464
The first constraint, (48), identifies the invariant countertermas the solution of the cohomology of the operator BΣ in thespace of the integrated local field polynomials of dimensionfour. From the general results on the cohomology of Yang–Mills theories [21], it follows that Σcount can be written as
Σcount = a0
4
∫d4xFa
μνFaμν + BΣΔ(−1), (59)
where Δ(−1) is a local integrated polynomial in all fields andsources, with dimension four, ghost number minus one andvanishing Q4N -charge, namely
Δ(−1) =∫
d4x(a1A
aμΩa
μ + a2∂μcaAaμ + a3c
aLa + a4caba
+ a5g
2f abccacbcc + a6λ
IKIjα ψj
α + a7ψjαKIj
α λI
+ a8ηIGIj
α ψjα + a9ψ
jαGIj
α ηI + a10θIKIj
α Y jα
+ a11Yjα KIj
α θI + a12ξIGIj
α Y jα + a13Y
jα GIj
α ξ I
+ a14ξI J Ij
α ψjα + a15ψ
jα J Ij
α ξ I + a16ψiαY i
α
+ a17Yiαψi
α + a18HIjα J Ij
α + a19JIjα HIj
α
+ a20cacaKIj
α GIjα + a21c
acaN + a22AaμAa
μN
+ a23AaμAa
μGIjα KIj
α + a24GIjα ∂2KIj
α
+ a25GIjα GIj
α N
+ a27ηI ∂2ξI + a28ξ
I ∂2ηI + a29AaμAa
μηI ξ I
+ a30AaμAa
μξ I ηI + a31cacaηI ξ I + a32c
aca ξ I ηI
+ a33P ηI ξ I + a34P ξ I ηI + a35cacaKIj
α GIjα
+ a36GIjα ∂2KIj
α + a37PN + a38 ξ I λIN
+ a39λI ξ IN + a40θ
I ηIN + a41ηI θIN
+ a42AaμAa
μGIjα KIj
α
). (60)
The coefficients {a0, a1, a2, . . . , a42} in expressions (59)and (60) stand for arbitrary constants parameters.
After a straightforward analysis using the conditions(48)–(57) it turns out that the only non-vanishing coeffi-cients are: a0, a1, a2, a6, a7, a14, a15, a16, a17, a37, withthe following relations between them
a2 = a1, (61)
and
a16 = a15 = a7,(62)
−a17 = a14 = a6.
Then, after redefining
a6 + a7 �→ a2,(63)
a37 �→ σa3,
for the form of the final allowed counterterm one finds
Σcount =∫
d4x
((a0 + 4a1
4
)Fa
μνFaμν − a1∂μcaΩa
μ
− a1∂μca∂μca − a1igψiα(γμ)αβT aijψ
jβAa
μ
− a2ψiαDij
μ (γμ)αβψjβ + a3σP 2
), (64)
which corresponds to the usual Yang–Mills counterterm inthe Landau gauge with the addition of an energy vacuumterm, σP 2, related to the mass m.
5.3 Stability
It remains now to discuss the stability of the model, i.e. tocheck that the counterterm Σcount can be reabsorbed in theclassical action Σ by means of a multiplicative redefinitionof the coupling constant g, of the parameter σ , and of thefields and sources [21], namely
Σ(g,σ,φ,Φ) + εΣcount = Σ(g0, σ0, φ0,Φ0) + O(ε2),
(65)
where φ stands for all fields and Φ for the sources,
φ ∈ {A,ψ, c, c, b, ξ, ξ , λ, λ, θ, θ , η, η},(66)
Φ ∈ {Ω,L,Y, Y , J, J ,K, K,G, G,H, H ,N,P }.Thus, by defining
φ0 = Z1/2φ φ,
Φ0 = ZΦΦ,(67)
g0 = Zgg,
σ0 = Zσ σ,
we obtain
Z1/2A = 1 + ε
2(a0 + 2a1),
Zg = 1 − εa0
2,
(68)Z
1/2ψ = Z
1/2ψ
= 1 + ε
2a2,
Zσ = 1 + εa3.
Expressions (68) constitute the independent renormalizationfactors. All the remaining factors can be expressed in termsof the renormalization factors appearing in (68). In fact, forthe Lagrange multiplier and Faddeev–Popov ghost fields wehave
Zb = Z−1/2A ,
(69)Z
1/2c = Z
1/2c = Z
−1/2g Z
−1/4A ,
Eur. Phys. J. C (2010) 66: 451–464 461
while the renormalization of the external BRST sources arefound
ZΩ = Z−1/2g Z
−1/4A ,
ZL = Z1/2A , (70)
ZY = ZY = Z−1/2g Z
1/4A Z
−1/2ψ .
As expected from (64), the renormalization properties of theusual Yang–Mills sector are preserved. For the doublet fieldswe obtain
Z1/2ξ = Z
1/2ξ = Z
1/2λ = Z
1/2λ = 1,
(71)Z
1/2θ = Z
1/2θ
= Z−1/2η = Z
−1/2η = Z
1/2g Z
−1/4A .
Finally, for the remaining sources we have
ZH = ZH = ZG = ZG = Z−1/2ψ ,
ZJ = ZJ = ZK = ZK = Z−1/2g Z
1/4A Z
−1/2ψ , (72)
ZP = 1.
This ends the proof of the multiplicative renormalizabil-ity of the model proposed in this article. For completeness,let us give the expression of the bare action written in termsof the renormalized fields and parameters:
Σ =∫
d4x
(1
2ZA
(∂μAa
ν − ∂νAaμ
)∂μAa
ν
+ ZgZ3/2A gf amn∂μAa
νAmμAn
ν
+ 1
4Z2
gZ2Ag2f abcf amnAb
μAcνA
mμAn
ν
+ Zψψiα(γμ)αβ∂μψi
β
− iZgZψZ1/2A gψi
α(γμ)αβAaμ
(T a
)ijψ
jβ + iba∂μAa
μ
+ igY iαca
(T a
)ijψj
α − igψjαca
(T a
)jiY i
α
+ (∂μca + Ωa
μ
)(Z−1
g Z−1/2A ∂μca − gf abcAc
μcb)
+ 1
2gLaf abccbcc − λI ∂2ξI − ξ I ∂2λI
− ηI ∂2θI + θ I ∂2ηI
+ mλI ξI + mξIλI + mηI θI − mθI ηI + M21 ξ i
αψiα
+ M21 ψ i
αξ iα − M2λ
iαψi
α − M2ψiαλi
α + Zσ σm2)
. (73)
From expression (73) one can obtain the renormalized ver-sion of (15), namely
Srenψ =
∫d4x
(Zψψi
α(γμ)αβ∂μψiβ
− iZgZψZ1/2A gψi
α(γμ)αβAaμ(T a)ijψ
jβ
− 2ZψM21 M2ψ
iα
(1
∂2 − m2
)ψi
α
). (74)
6 Inclusion of the Gribov–Zwanziger term
6.1 A brief overview of the Gribov–Zwanziger action andof its soft BRST breaking term
The Gribov–Zwanziger framework [1–3] enables one to takeinto account the existence of the Gribov copies, which affectthe Landau gauge.4 This is done by restricting the domainof integration in the Feynman path integral to the so-calledGribov region Ω , defined as the set of fields fulfilling theLandau gauge condition and for which the Faddeev–Popovoperator, Mab = −∂μDab
μ (A), is strictly positive,
Ω = {Aa
μ, ∂μAaμ = 0, Mab > 0
}. (75)
As shown in [2, 3], the implementation of the restriction tothe region Ω is done by adding to the starting action a non-local term, known as the horizon function, namely
SGZ = −g2γ 4∫
d4x f abcAbμ
[(∂ · D)−1]ad
f decAeμ. (76)
The parameter γ has the dimension of a mass and is knownas the Gribov parameter. It is not a free parameter, beingdetermined in a self-consistent way through the gap equation
δΓ
δγ 2= 0, (77)
where Γ stands for the effective action evaluated in the pres-ence of the horizon function (76). Despite of its non-localcharacter, the term SGZ can be cast in local form by intro-ducing a suitable set of auxiliary fields (ϕab
μ , ϕabμ ,ωab
μ , ωabμ ),
e−SGZ =∫
DϕDϕ DωDω e−SLocalGZ , (78)
SLocalGZ =
∫d4x
(−ϕacμ ∂νD
abν ϕbc
μ + ωacμ ∂νD
abν ωbc
μ
+ (∂νωacμ )gf abdϕbc
μ Ddeν ce
)
+ gγ 2∫
d4x
(f abc(ϕab
μ − ϕabμ )Ac
μ
− 4
g(N2 − 1)γ 2
). (79)
4See [22] for an introduction to the subject of the Gribov copies.
462 Eur. Phys. J. C (2010) 66: 451–464
Here, (ϕabμ , ϕab
μ ) form a pair of complex commuting fields,while (ωab
μ , ωabμ ) form a pair of complex anti-commuting
fields. These fields are assembled in BRST doublets
sϕabμ = ωab
μ , sωabμ = 0,
(80)sωab
μ = ϕabμ , sϕab
μ = 0,
and, as pointed out in [5–7, 13], the local action (79) givesrise to a soft breaking of the BRST symmetry, due to thepresence of the Gribov parameter γ . In fact, it turns out thatexpression (79) can be written as
SLocalGZ = Sϕω
+ gγ 2∫
d4x
(f abc
(ϕab
μ − ϕabμ
)Ac
μ
− 4
g
(N2 − 1
)γ 2
), (81)
with
Sϕω = −s
∫d4x ωac
μ ∂νDabν ϕbc
μ
=∫
d4x(−ϕac
μ ∂νDabν ϕbc
μ + ωacμ ∂νD
abν ωbc
μ
+ (∂νω
acμ
)gf abdϕbc
μ Ddeν ce
), (82)
so that
sSLocalGZ = γ 2Δγ ,
Δγ =∫
d4x(gf abcωab
μ Acμ − gf abc
(ϕab
μ − ϕabμ
)Dcd
μ cd).
(83)
In order to keep control of the soft BRST breaking term,we proceed as before and introduce a set of external sources(Uab
μν, Uabμν,V
abμν , V ab
μν ) transforming as
sV abμν = Uab
μν, sUabμν = 0,
(84)sUab
μν = V abμν , sV ab
μν = 0,
and whose physical values are defined by
V abμν
∣∣phys= −V ab
μν
∣∣phys= −γ 2δabδμν,
(85)Uab
μν
∣∣phys= U ab
μν
∣∣phys= 0.
Thus, we can replace the breaking term in (81) by the fol-lowing BRST-invariant source term:
Ssource = s
∫d4x
(U ab
μνDacμ ϕcb
ν + V abμν Dac
μ ωcbν − U ab
μνVabμν
)
=∫
d4x(V ab
μν Dacμ ϕcb
ν
− U abμν
[Dac
μ ωcbν + gf acd(Dde
μ ce)ϕcbν
]
+ UabμνD
acμ ωcb
ν
+ V abμν
[Dac
μ ϕcbν + gf acd(Dde
μ ce)ωcbν
]
− (Uab
μνUabμν − V ab
μν V abμν
)). (86)
Notice that the source term Ssource gives back the originalBRST soft breaking term when the sources attain their phys-ical values (85). In fact, after a little algebra, one finds
Ssource|phys = gγ 2∫
d4x
(f abc
(ϕab
μ − ϕabμ
)Ac
μ
− 4
g
(N2 − 1
)γ 2
). (87)
We are now ready to discuss the inclusion of the Gribov–Zwanziger term (79) into our starting action Σ , (36). To thatpurpose we consider the more general action
Σtot = Σ + Sϕω + Sμ + Ssource, (88)
where
Sμ = μ2s
∫d4x ωab
μ ϕabμ = μ2
∫d4x
(ϕab
μ ϕabμ − ωab
μ ωabμ
).
(89)
As discussed in [6, 7], the term Sμ takes into account thenon-trivial dynamics of the auxiliary localizing fields (ϕab
μ ,
ϕabμ ). The introduction of the BRST-invariant term Sμ fol-
lows from the observation that the dimension two con-densate 〈ϕab
μ ϕabμ − ωab
μ ωabμ 〉 has a non-zero value for non-
vanishing Gribov parameter γ , namely
⟨ϕab
μ ϕabμ − ωab
μ ωabμ
⟩ = 3(N2 − 1)
64π21/2gN1/2γ 2. (90)
The existence of this condensate is taken into accountthrough the mass parameter μ which, in a way similar tothe Gribov parameter γ , is determined by a variational prin-ciple, see [6, 7].
6.2 Renormalizability of the quark-gluon modelin the presence of the Gribov–Zwanziger term
In order to discuss the renormalizability of expression (88),one uses the Ward identities that have already been estab-lished in [3, 5–7, 23]. Moreover, it exhibits the followingsymmetry:
ΘabμνΣtot = 0, (91)
where the operator Θabμν is given by
Eur. Phys. J. C (2010) 66: 451–464 463
Θabμν
=∫
d4x
(ϕac
μ
δ
δϕbcν
− ϕbcν
δ
δϕacμ
+ ωacμ
δ
δωbcν
− ωbcν
δ
δωacμ
+ V acμσ
δ
δV bcνσ
− V acμσ
δ
δV bcνσ
+ Uacμσ
δ
δUbcνσ
− U acμσ
δ
δUbcνσ
).
(92)
The Ward identity (91) expresses the invariance of the ac-tion (88) under a global U(4(N2 − 1)) transformation. Thissymmetry works exactly as the global U(4N) symmetry as-sociated with the spinor sector of the theory through the op-erator (32). These global symmetries ensure in fact that nomixing terms between the two set of BRST doublet fields,i.e. (ϕab
μ ,ϕabμ , ωab
μ ,ωabμ ) and (ξ i
α, θ iα, ηi
α, λiα), arise in the
allowed counterterm. All Ward identities of the Gribov–Zwanziger action remain valid in the present case. Thisis also the case of the identities (37)–(47). Of course, theSlavnov–Taylor identity (37) is supplemented by suitableextra terms accounting for the BRST new doublets of thegluon sector, (80) and (84),
S(Σ) → S(Σtot) +∫
d4x
(ωab
μ
δΣtot
δϕabμ
+ ϕabμ
δΣtot
δωabμ
+ Uabμν
δΣtot
δV abμν
+ V abμν
δΣtot
δUabμν
). (93)
Also, the ghost equation (41) needs a little modificationwhich, due to the presence of the Gribov–Zwanziger term,generalizes to
Ga → Ga + gf abc
∫d4x
(ϕbd
μ
δ
δωcdμ
+ ωbdμ
δ
δϕcdμ
+ V dbμν
δ
δUdcμν
+ U dbμν
δ
δV dcμν
), (94)
while the classical breaking term (43) remains unmodified.The previous algebraic analysis can be now repeated for themore general action (88). The final output is that the ac-tion (88) remains renormalizable to all orders.
7 Conclusion
In this work we have considered a model that accountsfor a modification of the infrared behavior of quark andgluon propagators in Yang–Mills theories. This is achievedthrough the introduction of suitable mass parameters, whichgive rise to a soft breaking of the BRST symmetry, as out-lined in [13].
Being soft, the breaking term can be neglected in the ul-traviolet region, where the standard massless quark propaga-tor is recovered as well as the notion of exact BRST invari-ance. Moreover, in the infrared region the quark propagator
turns out to be deeply modified, as shown by expression (2).The physical reasoning behind the introduction of the softBRST breaking and of the ensuing modification of the prop-agator relies on quark confinement and on the breaking ofthe chiral symmetry, both occurring in the non-perturbativeinfrared region. It is worth remarking that the quark propa-gator (2) is in fact in qualitative agreement with the fittingformulas employed in the numerical studies of the quarktwo-point function through lattice simulations in the Landaugauge [18, 19].
The main result of the present article is the analysis ofthe renormalizability of the model, which we have shownto hold at any given finite orders of perturbation theory, bymaking use of the algebraic renormalization [21]. The inclu-sion of the Gribov–Zwanziger term, which enables us to im-plement the restriction to the Gribov region Ω, has also beentaken into account. Despite the presence of the soft BRSTbreaking term, the renormalizability of the model is guaran-teed by the large set of Ward identities, (37)–(47), which canbe established.
We expect that the mechanism of introducing non-pertur-bative infrared effects through the soft breaking of the BRSTsymmetry [13] applies as well to other kinds of models, in-cluding supersymmetric and topological field theories.
Acknowledgements The Conselho Nacional de DesenvolvimentoCientífico e Tecnológico (CNPq-Brazil), the Faperj, Fundação de Am-paro à Pesquisa do Estado do Rio de Janeiro, the Latin American Cen-ter for Physics (CLAF) the SR2-UERJ and the Coordenação de Aper-feiçoamento de Pessoal de Nível Superior (CAPES) are gratefully ac-knowledged for financial support. This work has been partially sup-ported by the contract ANR (CNRS-USAR), 05-BLAN-0079-01.
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