Study of the all orders multiplicative renormalizability of a local confining quark action in the Landau gauge
aa r X i v : . [ h e p - t h ] N ov Study of the all orders multiplicative renormalizability ofa local confining quark action in the Landau gauge
M. A. L. Capri ∗ , D. Fiorentini † , S. P. Sorella ‡ , Departamento de F´ısica Te´orica, Instituto de F´ısica, UERJ - Universidade do Estado do Rio de Janeiro,Rua S˜ao Francisco Xavier 524, 20550-013 Maracan˜a, Rio de Janeiro, Brasil
Abstract
The inverse of the Faddeev-Popov operator plays a pivotal role within the Gribov-Zwanziger ap-proach to the quantization of Euclidean Yang-Mills theories in Landau gauge. Following a recentproposal [1], we show that the inverse of the Faddeev-Popov operator can be consistently coupledto quark fields. Such a coupling gives rise to a local action while reproducing the behaviour of thequark propagator observed in lattice numerical simulations in the non-perturbative infrared region.By using the algebraic renormalization framework, we prove that the aforementioned local action ismultiplicatively renormalizable to all orders.
Nowadays, the so called Gribov-Zwanziger framework [2, 3, 4, 5] is a powerful tool in order to studynon-perturbative aspects of gluon and quark confinement. Going a step beyond of the perturbativequantization method of Faddeev-Popov, Gribov called attention to the fact that the presence of zeromodes of the Faddeev-Popov operator, given by M ab = − ∂ µ ( δ ab ∂ µ − gf abc A cµ ) , (1)results in the existence of Gribov copies [2], i.e. equivalent gauge field configurations fulfilling thesame gauge-fixing condition . As a consequence, the Faddeev-Popov quantization procedure becomesill-defined, meaning that it is not possible to pick up a unique gauge field configuration for each gaugeorbit through a local and covariant gauge-fixing condition [8]. The existence of the Gribov copies is anon-perturbative phenomenon which has deep consequences on the infrared behaviour of confining Yang-Mills theories.To deal with the existence of equivalent gauge field configurations, Gribov was able to show that alarge number of copies could be eliminated by restricting the domain of integration in the functional inte-gral to a certain region Ω in field space [2]. This region is known as the Gribov region and is given by allfield configurations which fulfil the Landau gauge condition, ∂ µ A aµ = 0, and for which the Faddeev-Popovoperator M ab is strictly positive, namelyΩ = { A aµ ; ∂ µ A aµ = 0 ; M ab = − ( ∂ δ ab − gf abc A cµ ∂ µ ) > } . (2) ∗ [email protected] † diegofi[email protected] ‡ [email protected] For a pedagogical introduction to the Gribov problem, see [6, 7] • i) it is convex and bounded in all directions in field space. Its boundary ∂ Ω is the first Gribovhorizon, where the first non vanishing eigenvalue of M ab shows up. • ii) every gauge orbit intersects at least once the region Ω. This last property gives a well definedsupport to Gribov’s original proposal of cutting off the functional integral at the Gribov horizon.Later on, Zwanziger [3, 4, 5] showed that the restriction of the domain of integration to the region Ω isequivalent to adding to the original Faddeev-Popov action a nonlocal term H ( A ), called horizon function,given by H ( A ) = g Z d xd y f abc A bµ ( M − ) ad ( x, y ) f dec A eµ ( y ) , (3)with ( M − ) ab being the inverse of the Faddeev-Popov operator (1).Thus, for the partition function one writes [2, 3, 4, 5] Z = Z Ω D A δ ( ∂A ) det ( M ) e − S YM = Z D A δ ( ∂A ) det ( M ) e − ( S YM + γ H ( A ) − V γ N − = Z D A D c D ¯ c D b e − S GZ , (4)where S GZ is the Gribov-Zwanziger action S GZ = S FP + γ H ( A ) − V γ N − , (5)with S FP denoting the Faddeev-Popov action in Landau gauge S FP = Z d x (cid:18) F aµν F aµν + b a ∂ µ A aµ − ¯ c a M ab c b (cid:19) , (6)The field b a in expression (6) is the Lagrange multiplier enforcing the Landau gauge condition, ∂ µ A aµ = 0,while ( c a , ¯ c a ) are the Faddeev-Popov ghosts. Also, the massive parameter γ in eq.(5) is the so calledGribov parameter. It is not a free parameter, being determined in a self-consistent way through a gapequation, called the horizon condition, which reads ∂ E v ∂γ = 0 , (7)where E v ( γ ) is the vacuum energy defined by e − V E v = Z . (8)It is worth to point out that, even if the horizon term H ( A ), eq.(3), is non-local, the action (5) can be castin local form [3, 4, 5] by means of the introduction of a set of auxiliary fields (¯ ω abµ , ω abµ , ¯ ϕ abµ , ϕ abµ ), where( ¯ ϕ abµ , ϕ abµ ) are a pair of bosonic fields, while (¯ ω abµ , ω abµ ) are anti-commuting. For the local formulation ofthe theory, we have Z = Z D A D c D ¯ c D b D ¯ ω D ω D ¯ ϕ D ϕ e − S GZ , (9)where S GZ is now given by the local expression S GZ = S FP + S + S γ , (10)2here S and S γ read S = Z d x (cid:16) ¯ ϕ acµ ( ∂ ν D abν ) ϕ bcµ − ¯ ω acµ ( ∂ ν D abν ) ω bcµ − gf amb ( ∂ ν ¯ ω acµ )( D mpν c p ) ϕ bcµ (cid:17) , (11)and S γ = γ Z d x (cid:16) gf abc A aµ ( ϕ bcµ + ¯ ϕ bcµ ) (cid:17) − γ V ( N − . (12)The local action (10) exhibits the important property of being multiplicative renormalizable to all orders[3, 4, 5].Recently, a refinement of the Gribov-Zwanziger action has been worked out by the authors [12, 13, 14], bytaking into account the existence of dimension two condensates. The Refined Gribov-Zwanziger (RGZ)action reads [12, 13, 14] S RGZ = S GZ + Z d x (cid:18) m A aµ A aµ − µ (cid:16) ¯ ϕ abµ ϕ abµ − ¯ ω abµ ω abµ (cid:17)(cid:19) . (13)As the Gribov parameter γ , the massive parameters ( m , µ ) have a dynamical origin, being related tothe existence of the dimension two condensates h A aµ A aµ i and h ¯ ϕ abµ ϕ abµ − ¯ ω abµ ω abµ i , [12, 13, 14]. The gluonpropagator obtained from the RGZ action turns out to be suppressed in the infrared region, attaining anon-vanishing value at zero momentum, k = 0, i.e. h A aµ ( k ) A bν ( − k ) i = δ ab (cid:18) δ µν − k µ k ν k (cid:19) D ( k ) , (14) D ( k ) = k + µ k + ( µ + m ) k + 2 N g γ + µ m . (15)Also, the ghost propagator stemming from the Refined theory is not enhanced in the deep infrared G ab ( k ) = h ¯ c a ( k ) c b ( − k ) i (cid:12)(cid:12)(cid:12) k ∼ ∼ δ ab k . (16)The infrared behaviour of the gluon and ghost propagators obtained from the RGZ action turns outto be in very good agreement with the most recent numerical lattice simulations on large lattices[15, 16, 17, 18, 19]. Furthermore, from the numerical estimates [17] of the parameters ( m , µ , γ ) itturns out that the RGZ gluon propagator (14) displays complex poles and violates reflection positivity.This kind of propagator lacks the K¨all´en-Lehmann spectral representation and cannot be associated withthe propagation of physical particles. Rather, it indicates that, in the non-perturbative infrared region,gluons are not physical excitations of the spectrum of the theory, i.e. they are confined. Let us men-tioning here that the RGZ gluon propagator has been employed in analytic calculation of glueball states[20, 21], yielding results which compare well with the available numerical simulations as well as withother approaches, see [22] for an account on this topic.As illustrated above, the RGZ framework, turns out to capture important aspects of the gluon con-finement. Nevertheless, what can be said about the matter sector? A first answer to this question wasproposed recently in [1] where a new term, very similar to Zwanziger’s horizon function, was investigated.Such new term was inspired by recent lattice numerical simulations [23] of the correlation function Q abcdµν ( x − y ) = D R abµ ( x ) R cdν ( y ) E , (17)with R acµ ( x ) = Z d z ( M − ) ad ( x, z ) gf dec A eµ ( z ) , (18)3ore precisely, in [23], it has been shown that the Fourier transform of expression (17) is non-vanishingand behaves as k in the deep infrared, a result which is again in agreement with the RGZ framework, i.e. h ˜ R abµ ( k ) ˜ R cdν ( − k ) i (cid:12)(cid:12)(cid:12) k ∼ ∼ k . (19)As observed in [23], this behaviour can be understood by making use of the analysis of [24], i.e. of thecluster decomposition h ˜ R abµ ( k ) ˜ R cdν ( − k ) i ∼ g G ( k ) D ( k ) , (20)where D ( k ) and G ( k ) correspond to the gluon and ghost propagators, eqs.(15),(16). A non-enhancedghost propagator, i.e. G ( k ) (cid:12)(cid:12)(cid:12) k ∼ ∼ k , and an infrared finite gluon propagator, i.e. D (0) = 0, nicelyyield the behaviour of eq.(19).In [1], the idea that the quantity R abµ ( x ) and the correlation function Q abcdµν ( x − y ), eqs.(17), (18), couldbe generalized to the case of matter fields was exploited in details. The main argument developed in[1] can be summarized as follows. Let F i ( x ) be a generic matter field, i.e. a scalar or a spinor field, ina given representation of the gauge group SU ( N ), and let R ai F ( x ) and its complex conjugate ¯ R ai F ( x ) bedefined by R ai F ( x ) := g Z d z ( M − ) ab ( x, z )( T b ) ij F j ( z ) , ¯ R ai F ( x ) := g Z d z ( M − ) ab ( x, z )¯ F j ( z )( T b ) ji , (21)where ( T a ) ij are the generators of the representation and a = 1 , . . . , ( N − R ai F ( x ) isneeded if the generic field F i is complex. Notice how similar eq’s (18) and (21) are. Then, a non-trivialcorrelation function Q abij F ( x − y ) = D R ai F ( x ) ¯ R bj F ( y ) E , (22)can be obtained from a theory constructed by adding to the usual matter action a term similar to thehorizon function, namely S matter → S matter + M H matter , (23)where, in complete analogy with the horizon function (3), H matter is given by H matter = − g Z d xd y ¯ F i ( x )( T a ) ij ( M − ) ab ( x, y )( T b ) jk F k ( y ) . (24)Also, the mass parameter M plays a role akin to that of the Gribov parameter γ .The present work aims at pursuing the analysis started in [1]. In particular, we shall prove the mul-tiplicative renormalizability of the model constructed according to (23) when the matter field underconsideration is the quark field, i.e. , when F i ≡ ψ i . It is worth underlining that this approach providesa tree level quark propagator that can be written as (cid:10) ψ i ( p ) ¯ ψ j ( − p ) (cid:11) = − ip µ γ µ + A ( p ) p + A ( p ) δ ij , (25)where the quark mass function A ( p ) is given by A ( p ) = m ψ + g (cid:18) N − N (cid:19) M p + µ ψ . (26) In the case in which the field F i is a scalar field, the power of the mass parameter M appearing in expression (23) isfour, due to the fact that a scalar field has dimension one [1]. m ψ stands for the quark mass, while µ ψ is related to the condensation of a lower dimen-sional operator in the matter sector [1], a feature which shares great similarity with the lower dimensionalcondensates of the RGZ action, eq.(13).It is worth underlining here that a quark propagator of the kind of eqs.(25), (26) fits very well thelattice numerical data, see [25] and the discussion reported in [1]. In particular, one notices that thequark mass function does not vanish at zero momentum for vanishing quark mass m ψ = 0, signallinga dynamical breakdown of the chiral symmetry. As such, expressions (25),(26) capture nontrivial non-perturbative aspects of quark confinement.The present work is organized as follows. In Section 2 we construct the complete local action in thematter sector which takes into account the term (24). In Section 3 we establish the large set of Wardidentities fulfilled by the local action. In Section 4 we characterize the most general local invariantcounterterm by means of the algebraic renornalization procedure [26] and we establish the all ordermultiplicative renormalizability of the model. In Section 5 we collect our conclusions. In order to identify the local classical action implementing the framework discussed above, we startwith the Gribov-Zwanziger action supplemented with a non-local matter term, as described in eq.(24), i.e. S = S GZ + S matter + M H matter , (27)where S GZ is the Gribov-Zwanziger action, eq. (10), and S matter , H matter given, respectively, by S matter = Z d x h ¯ ψ iα ( γ µ ) αβ D ijµ ψ jβ − m ψ ¯ ψ iα ψ iα i , (28)and H matter = − g Z d x d y ¯ ψ iα ( x )( T a ) ij ( M − ) ab ( x, y )( T b ) jk ψ kα ( y ) . (29)Despite its non-locality, the term H matter can be cast in local by means of the introduction of a set oflocalizing field, in a way similar to the localization of the horizon function (3) [3, 4, 5]. Therefore, for thelocal version of the starting action, we write S = S GZ + S localmatter (30)where S localmatter = Z d x h ¯ ψ iα ( γ µ ) αβ D ijµ ψ jβ − m ψ ¯ ψ iα ψ iα i + Z d x h +¯ λ aiα ( − ∂ µ D abµ ) λ biα + ¯ η aiα ( − ∂ µ D abµ ) η biα − gf abc ( ∂ µ ¯ η aiα )( D bdµ c d ) λ ciα i + gM / Z d x h ¯ λ aiα ( T a ) ij ψ jα + ¯ ψ iα ( T a ) ij λ ajα i , (31)where (¯ λ ai , λ ai ) are anti-commutating spinor fields and (¯ η ai , η ai ) are commutating ones. It is easilychecked that integration over the auxiliary fields (¯ λ ai , λ ai , ¯ η ai , η ai ) gives back the non-local expression(29). 5s in the case of the Gribov-Zwanziger action [1, 3, 4, 5, 13, 27, 28, 29, 30, 31, 32], the local action S exhibits a soft breaking of the BRST symmetry, namely sS = γ ∆ γ + M / ∆ M , (32)with ∆ γ = Z d x h − gf abc ( D adµ c d )( ϕ bcµ + ¯ ϕ bcµ ) + gf abc A aµ ω bcµ i , (33)∆ M = Z d x h ig ( T a ) ij ( T b ) jk ¯ λ aiα c b ψ kα − ig ( T b ) ki ( T a ) ij ¯ ψ kα c b λ ajα − g ( T a ) ij ¯ ψ iα η ajα i , (34)and s denoting the nilpotent BRST transformations sA aµ = − D abµ c b , sc a = g f abc c b c c , s ¯ c a = b a , sb a = 0 ,s ¯ ω abµ = ¯ ϕ abµ , s ¯ ϕ abµ = 0 , sϕ abµ = ω abµ , sω abµ = 0 ,sψ iα = − ig ( T a ) ij c a ψ jα , s ¯ ψ iα = − ig ¯ ψ jα ( T a ) ji c a ,s ¯ η aiα = ¯ λ aiα , s ¯ λ aiα = 0 , sλ aiα = η aiα , sη aiα = 0 . (35)Notice that, being of dimensions less than four in the fields, the breaking terms ∆ γ , ∆ M are soft. Thiskind of breaking can be kept under control through the renormalization process. To that purpose, wefollow the strategy already employed in the case of the GZ and RGZ actions [3, 4, 5, 13, 27, 28, 29, 30, 31]and we embed the action S into a larger action exhibiting exact BRST invariance.Following [3, 4, 5, 13, 27, 28, 29, 30, 31], let us introduce two set of BRST quartet of external sources: s ¯ N abµν = ¯ M abµν , s ¯ M abµν = 0 , sM abµν = N abµν , sN abµν = 0 ; (36) s ¯ U ijαβ = ¯ V ijαβ , s ¯ V ijαβ = 0 , sV ijαβ = U ijαβ , sU ijαβ = 0 . (37)Therefore, we replace the action S by the BRST invariant action S inv = Z d x (cid:20) F aµν F aµν + ¯ ψ iα ( γ µ ) αβ D ijµ ψ jβ − m ψ ¯ ψ iα ψ iα (cid:21) + s Z d x (cid:16) ¯ c a ∂ µ A aµ + ¯ ω acν ∂ µ D abµ ϕ bcν − ¯ N acµν D abµ ϕ bcν − M acµν D abµ ¯ ω bcν − ¯ N abµν M abµν (cid:17) + s Z d x h ¯ η aiα ( − ∂ µ D abµ ) λ biα + ¯ U jkαβ ¯ ψ iα g ( T a ) ij λ akβ + V jkαβ ¯ η akβ g ( T a ) ij ψ jα + ζm ψ ¯ U ijαβ V ijαβ i = Z d x (cid:26) F aµν F aµν + ¯ ψ iα ( γ µ ) αβ D ijµ ψ jβ − m ψ ¯ ψ iα ψ iα + b a ∂ µ A aµ + ¯ c a ∂ µ D abµ c b + ¯ ϕ acν ∂ µ D abµ ϕ bcν − ¯ ω acν ∂ µ D abµ ω bcν − gf abc ( ∂ µ ¯ ω aeν )( D bdµ c d ) ϕ ceν − ¯ M acµν D abµ ϕ bcν + ¯ N acµν h D abµ ω bcν + gf abd ( D deµ c e ) ϕ bcν i − M acµν h D abµ ¯ ϕ bcν + gf abd ( D deµ c e )¯ ω bcν i − N acµν D abµ ¯ ω bcν − (cid:16) ¯ M abµν M abµν − ¯ N abµν N abµν (cid:17) +¯ λ aiα ( − ∂ µ D abµ ) λ biα + ¯ η aiα ( − ∂ µ D abµ ) η biα − gf abc ( ∂ µ ¯ η aiα )( D bdµ c d ) λ ciα + ¯ V jkαβ ¯ ψ iα g ( T a ) ij λ akβ + ¯ U jkαβ h ig ( T a ) ij ( T b ) li ¯ ψ lα c b λ akβ + ¯ ψ iα g ( T a ) ij η akβ i + V ikαβ h ¯ λ akβ g ( T a ) ij ψ jα − ig ¯ η akβ ( T a ) ij ( T b ) jl c b ψ lα i + U ikαβ ¯ η akβ g ( T a ) ij ψ jα + ζm ψ (cid:16) ¯ V ijαβ V ijαβ − ¯ U ijαβ U ijαβ (cid:17)(cid:27) , (38)6ith sS inv = 0 . (39)The last term in expression (38), ζm ψ (cid:16) ¯ V ijαβ V ijαβ − ¯ U ijαβ U ijαβ (cid:17) , is a vacuum term allowed by power-counting,while ζ is a dimensionless coefficient. The starting action S is recovered from the invariant action S inv when the external sources attain a particular value, usually called the physical value, i.e. S inv (cid:12)(cid:12)(cid:12) phys value = S (40)where M abµν (cid:12)(cid:12)(cid:12) phys = ¯ M abµν (cid:12)(cid:12)(cid:12) phys = γ δ ab δ µν , N abµν (cid:12)(cid:12)(cid:12) phys = ¯ N abµν (cid:12)(cid:12)(cid:12) phys = 0 ; (41) V ijαβ (cid:12)(cid:12)(cid:12) phys = ¯ V ijαβ (cid:12)(cid:12)(cid:12) phys = M / δ ij δ αβ , U ijαβ (cid:12)(cid:12)(cid:12) phys = ¯ U ijαβ (cid:12)(cid:12)(cid:12) phys = 0 . (42)As the action S is obtained as a particular case of the extended action S inv , renormalizability of S inv willimply that of S . We thus proceed by focussing on the action S inv .In order to discuss the renormalizability of S we notice that the BRST transformations of the gauge,ghost and matter fields, eqs.(35), are nonlinear. As such, we need to properly take into account thecorresponding composite operators, a task which is achieved by introducing external invariant sources(Ω aµ , L a , ¯ J iα , J iα ) coupled to the nonlinear BRST transformations, i.e. S ext = Z d x (cid:2) Ω aµ sA aµ + L a sc a + ¯ J iα sψ iα + ( s ¯ ψ iα ) J iα (cid:3) . (43)Therefore, for the complete starting classical action Σ we obtainΣ = S inv + S ext = Z d x (cid:26) F aµν F aµν + ¯ ψ iα ( γ µ ) αβ D ijµ ψ jβ − m ψ ¯ ψ iα ψ iα + b a ∂ µ A aµ + ¯ c a ∂ µ D abµ c b + ¯ ϕ acν ∂ µ D abµ ϕ bcν − ¯ ω acν ∂ µ D abµ ω bcν − gf abc ( ∂ µ ¯ ω aeν )( D bdµ c d ) ϕ ceν − ¯ M acµν D abµ ϕ bcν + ¯ N acµν h D abµ ω bcν + gf abd ( D deµ c e ) ϕ bcν i − M acµν h D abµ ¯ ϕ bcν + gf abd ( D deµ c e )¯ ω bcν i − N acµν D abµ ¯ ω bcν − (cid:16) ¯ M abµν M abµν − ¯ N abµν N abµν (cid:17) − ¯ λ aiα ∂ µ D abµ λ biα − ¯ η aiα ∂ µ D abµ η biα − gf abc ( ∂ µ ¯ η aiα )( D bdµ c d ) λ ciα + ¯ V jkαβ ¯ ψ iα g ( T a ) ij λ akβ + ¯ U jkαβ h ¯ ψ iα g ( T a ) ij η akβ + ig ( T a ) ij ( T b ) li ¯ ψ lα c b λ akβ i + V ikαβ h ¯ λ akβ g ( T a ) ij ψ jα − ig ¯ η akβ ( T a ) ij ( T b ) jl c b ψ lα i + U ikαβ ¯ η akβ g ( T a ) ij ψ jα + ζm ψ (cid:16) ¯ V ijαβ V ijαβ − ¯ U ijαβ U ijαβ (cid:17) − Ω aµ D abµ c b + g f abc L a c b c c − ¯ J iα ig ( T a ) ij c a ψ jα − ig ¯ ψ jα c a ( T a ) ji J iα (cid:27) . (44)Before proceeding with the analysis of the renormalizability, it turns out to be useful to introduce amulti-index notation. Following [3, 4, 5], we first introduce the multi-index I ≡ { a, µ } , I = 1 ... N − i.e. (cid:16) ϕ abµ , ¯ ϕ abµ , ω abµ , ¯ ω abµ (cid:17) ≡ (cid:0) ϕ aI , ¯ ϕ aI , ω aI , ¯ ω aI (cid:1) , (cid:16) M abµν , ¯ M abµν , N abµν , ¯ N abµν (cid:17) ≡ (cid:0) M aIµ , ¯ M aIµ , N aIµ , ¯ N aIµ (cid:1) . (45)7s pointed out in [3, 4, 5], the possibility of introducing the multi-index I ≡ { a, µ } relies on an exact U ( f ) symmetry, f = 4( N − i.e. Q abµν (Σ) = 0 , (46)where Q abµν ≡ Z d x (cid:18) ϕ caµ δδϕ cbν − ¯ ϕ cbν δδ ¯ ϕ caµ + ω caµ δδω cbν − ω cbν δδω caµ + M caσµ δδM cbσν − ¯ M cbσν δδ ¯ M caσµ + N caσµ δδN cbσν − ¯ N cbσν δδ ¯ N caσµ (cid:19) . (47)Also, the trace of the operator Q abµν defines the q f -charge, displayed in the tables below.Similarly, a second composite index ˆ I ≡ { i, α } can be introduced, (cid:0) λ aiα , ¯ λ aiα , η aiα , ¯ η aiα (cid:1) ≡ (cid:16) λ a ˆ I , ¯ λ a ˆ I , η a ˆ I , ¯ η a ˆ I (cid:17) , (cid:16) V ijαβ , ¯ V ijαβ , U ijαβ , ¯ U ijαβ (cid:17) ≡ (cid:16) V iα ˆ I , ¯ V iα ˆ I , U iα ˆ I , ¯ U iα ˆ I (cid:17) , (48)due to a second exact U ( ˆ f ) symmetry, ˆ f = 4 N , i.e. ˆ Q ijαβ (Σ) = 0 , (49)with ˆ Q ijαβ ≡ Z d x (cid:18) λ aiα δδλ ajβ − ¯ λ ajβ δδ ¯ λ aiα + η aiα δδη ajβ − ¯ η ajβ δδ ¯ η aiα + V kiγα δδV kjβγ − ¯ V kjγβ δδ ¯ V kiαγ + U kiγα δδU kjβγ − ¯ U kjγβ δδ ¯ U kiαγ (cid:19) . (50)The trace of the operator ˆ Q ijαβ defines the q ˆ f -charge, also displayed in the tables below. Making use ofthe multi-indices ( I, ˆ I ), for the complete action Σ we getΣ = S inv + S ext = Z d x (cid:26) ¯ ψ iα ( γ µ ) αβ D ijµ ψ jβ − m ψ ¯ ψ iα ψ iα + 14 F aµν F aµν + b a ∂ µ A aµ + ¯ c a ∂ µ D abµ c b + ¯ ϕ aI ∂ µ D abµ ϕ bI − ¯ ω aI ∂ µ D abµ ω bI − gf abc ( ∂ µ ¯ ω aI )( D bdµ c d ) ϕ cI − ¯ M aIµ D abµ ϕ bI + ¯ N aIµ h D abµ ω bI + gf abc ( D cdµ c d ) ϕ bI i − M aIµ h D abµ ¯ ϕ bI + gf abc ( D cdµ c d )¯ ω bI i − N aIµ D abµ ¯ ω bI − (cid:0) ¯ M aIµ M aIµ − ¯ N aIµ N aIµ (cid:1) − ¯ λ a ˆ I ∂ µ D abµ λ b ˆ I − ¯ η a ˆ I ∂ µ D abµ η b ˆ I − gf abc ( ∂ µ ¯ η a ˆ I )( D bdµ c d ) λ c ˆ I + ¯ V j ˆ Iα ¯ ψ iα g ( T a ) ij λ a ˆ I + ¯ U j ˆ Iα h ¯ ψ iα g ( T a ) ij η a ˆ I + ig ( T a ) ij ( T b ) ki ¯ ψ kα c b λ a ˆ I i + V i ˆ Iα h ¯ λ a ˆ I g ( T a ) ij ψ jα − ig ¯ η a ˆ I ( T a ) ij ( T b ) jk c b ψ kα i + U i ˆ Iα ¯ η a ˆ I g ( T a ) ij ψ jα + ζm ψ (cid:16) ¯ V iα ˆ I V iα ˆ I − ¯ U iα ˆ I U iα ˆ I (cid:17) − Ω aµ D abµ c b + g f abc L a c b c c − ¯ J iα ig ( T a ) ij c a ψ jα − ig ¯ ψ jα c a ( T a ) ji J iα (cid:27) . (51)Before ending this section, let us display the quantum numbers of all fields and external sources. Thenature of the fields/sources is denoted by “B” for bosonic fields/sources and by “F” for anti-commutingvariables. Also, the e -charge is the charge associated with a global U (1) gauge invariance.8 ields A aµ ψ iα ¯ ψ iα c a ¯ c a b a ϕ aI ¯ ϕ aI ω aI ¯ ω aI λ a ˆ I ¯ λ a ˆ I η a ˆ I ¯ η a ˆ I Dimension
Ghost number − − − q f -Charge − − q ˆ f -Charge − − e -Charge − − − Nature
B F F F F B B B F F F F B B
Sources Ω aµ L a J iα ¯ J iα M aIµ ¯ M aIµ N aIµ ¯ N aIµ V iα ˆ I ¯ V iα ˆ I U iα ˆ I ¯ U iα ˆ I Dimension
Ghost number − − − − − − q f -Charge − − q ˆ f -Charge − − e -Charge − Nature
F B B B B B F F B B F F
In this section we derive the large set of Ward identities fulfilled by the complete action (Σ). TheseWard identities will be the starting point for the analysis of the algebraic characterization of the mostgeneral invariant counterterm. It is easily checked that Σ obeys the following identities: • The Slavnov-Taylor identity: S (Σ) = 0 , (52)with S (Σ) ≡ Z d x (cid:18) δ Σ δ Ω aµ δ Σ δA aµ + δ Σ δL a δ Σ δc a + δ Σ δ ¯ J iα δ Σ δψ iα + δ Σ δ ¯ ψ iα δ Σ δJ iα + b a δ Σ δ ¯ c a + ¯ ϕ aI δ Σ δ ¯ ω aI + ω aI δ Σ δϕ aI + ¯ M aIµ δ Σ δ ¯ N aIµ + N aIµ δ Σ δM aIµ +¯ λ i ˆ I δ Σ δ ¯ η i ˆ I + η i ˆ I δ Σ δλ i ˆ I + ¯ V iα ˆ I δ Σ δ ¯ U iα ˆ I + U iα ˆ I δ Σ δV iα ˆ I (cid:19) (53)Let us also introduce, for further use, the so called linearized Slavnov-Taylor operator [26] B Σ ,defined as B Σ = Z d x (cid:18) δ Σ δ Ω aµ δδA aµ + δ Σ δA aµ δδ Ω aµ + δ Σ δL a δδc a + δ Σ δc a δδL a + δ Σ δ ¯ J iα δδψ iα + δ Σ δψ iα δδ ¯ J iα + δ Σ δ ¯ ψ iα δδJ iα + δ Σ δJ iα δδ ¯ ψ iα + b a δδ ¯ c a + ¯ ϕ aI δδ ¯ ω aI + ω aI δδϕ aI + ¯ M aIµ δδ ¯ N aIµ + N aIµ δδM aIµ + ¯ λ i ˆ I δδ ¯ η i ˆ I + η i ˆ I δδλ i ˆ I + ¯ V iα ˆ I δδ ¯ U iα ˆ I + U iα ˆ I δδV iα ˆ I (cid:19) . (54)The operator B Σ has the important property of being nilpotent B Σ B Σ = 0 . (55)9 The Landau gauge-fixing condition [26]: δ Σ δb a = ∂ µ A aµ . (56) • The anti-ghost equation [26]: δ Σ δ ¯ c a + ∂ µ δ Σ δ Ω aµ = 0 . (57) • The integrated ghost equation [26]: G a (Σ) = ∆ a , (58)where G a (Σ) ≡ Z d x δ Σ δc a + gf abc ¯ c b δ Σ δb c + ¯ ω bI δ Σ δ ¯ ϕ cI + ϕ bI δ Σ δω cI + ¯ N bIµ δ Σ δ ¯ M cIµ + M bIµ δ Σ δN cIµ +¯ η b ˆ I δ Σ δ ¯ λ c ˆ I + λ b ˆ I δ Σ δη c ˆ I (cid:19) + ig ( T a ) ij V i ˆ Iα δ Σ δU j ˆ Iα − ig ( T a ) ij ¯ U j ˆ Iα δ Σ δ ¯ V i ˆ Iα ! , and ∆ a = Z d x (cid:16) gf abc A cµ Ω bµ − gf abc L b c c − ¯ J iα ig ( T a ) ij ψ jα + ig ¯ ψ jα ( T a ) ji J iα (cid:17) . (59)Notice that the breaking term ∆ a is linear in the quantum fields. As such, it is a classical breaking,not affected by the quantum corrections [26]. • The exact U ( f ) symmetry (46), f = 4( N − Q IJ (Σ) = 0 , (60)with Q IJ (Σ) ≡ Z d x ϕ aI δ Σ δϕ aJ − ¯ ϕ aJ δ Σ δ ¯ ϕ aI + ω aI δ Σ δω aJ − ω aJ δ Σ δω aI + M aIµ δ Σ δM aJµ − ¯ M aJµ δ Σ δ ¯ M aIµ + N aIµ δ Σ δN aJµ − ¯ N aJµ δ Σ δ ¯ N aIµ ! . • The U ( ˆ f ) symmetry, ˆ f = 4 N : ˆ Q ˆ J ˆ I (Σ) = 0 , (61)where ˆ Q ˆ J ˆ I (Σ) ≡ Z d x λ a ˆ I δ Σ δλ a ˆ J − ¯ λ a ˆ J δ Σ δ ¯ λ a ˆ I + η a ˆ I δ Σ δη a ˆ J − ¯ η a ˆ J δ Σ δ ¯ η a ˆ I + V iα ˆ I δ Σ δV iα ˆ J − ¯ V iα ˆ J δ Σ δ ¯ V iα ˆ I + U iα ˆ I δ Σ δU iα ˆ J − ¯ U iα ˆ J δ Σ δ ¯ U iα ˆ I ! • The U (1) invariance: N e (Σ) = 0 , (62)10ith N e (Σ) ≡ Z d x ψ iα δ Σ δψ iα − ¯ ψ iα δ Σ δ ¯ ψ iα + J iα δ Σ δJ iα − ¯ J iα δ Σ δ ¯ J iα + λ a ˆ I δ Σ δλ a ˆ I − ¯ λ a ˆ I δ Σ δ ¯ λ a ˆ I + η a ˆ I δ Σ δη a ˆ I − ¯ η a ˆ I δ Σ δ ¯ η a ˆ I ! . This symmetry gives rise to a conserved charge, called e in the previous Tables. . • The ghost number Ward identity: N ghost (Σ) = 0 , (63) N ghost (Σ) ≡ Z d x c a δ Σ δc a − ¯ c a δ Σ δ ¯ c a + ω aI δ Σ δω aI − ¯ ω aI δ Σ δ ¯ ω aI + η a ˆ I δ Σ δη a ˆ I − ¯ η a ˆ I δ Σ δ ¯ η a ˆ I + N aIµ δ Σ δN aIµ − ¯ N aIµ δ Σ δ ¯ N aIµ + U i ˆ Iα δ Σ δU i ˆ Iα − ¯ U i ˆ Iα δ Σ δ ¯ U i ˆ Iα − Ω aµ δ Σ δ Ω aµ − L a δ Σ δL a − J iα δ Σ δJ iα − ¯ J iα δ Σ δ ¯ J iα ! . • The linearly broken Ward identities: δ Σ δ ¯ ϕ aI + ∂ µ δ Σ δ ¯ M aIµ = − gf abc ¯ M bIµ A cµ , (64) δ Σ δω aI + ∂ µ δ Σ δN aIµ − gf abc δ Σ δb a ¯ ω aI = − gf abc A cµ ¯ N bIµ , (65) δ Σ δ ¯ ω aI + ∂ µ δ Σ δ ¯ N aIµ − gf abc M bIµ δ Σ δ Ω aµ = gf abc A cµ ¯ N bIµ , (66) δ Σ δϕ aI + ∂ µ δ Σ δM aIµ − gf abc (cid:18) δ Σ δb c ¯ ϕ bI − δ Σ δc c ¯ ω bI + ¯ N bIµ δ Σ δ Ω cµ (cid:19) = − gf abc ¯ M bIµ A cµ , Z d x (cid:18) δ Σ δη a ˆ I + gf abc ¯ η a ˆ I δ Σ δb a (cid:19) = Z d x g ( T a ) ij ¯ U jα ˆ I ψ iα , (67) Z d x (cid:18) δ Σ δλ a ˆ I − g ( T a ) ij ¯ U j ˆ I δ Σ δJ iα + gf abc (cid:18) δ Σ δb c ¯ λ b ˆ I − δ Σ δ ¯ c b ¯ η c ˆ I (cid:19)(cid:19) = − Z d x g ( T a ) ij ¯ V jα ˆ I ¯ ψ iα . (68) Z d x δ Σ δ ¯ η a ˆ I − gV i ˆ Iα ( T a ) ij δ Σ δ ¯ J jα ! = Z d xgU i ˆ Iα ( T a ) ij ψ jα (69) Z d x δ Σ δ ¯ λ a ˆ I = Z d xgV i ˆ Iα ( T a ) ij ψ jα (70)11 the exact integrated Ward identities: Z d x (cid:18) c a δ Σ δω aI + ¯ ω aI δ Σ δ ¯ c a + ¯ N aI δ Σ δ Ω aµ (cid:19) = 0 , (71) Z d x (cid:18) c a δ Σ δϕ aI − ¯ ϕ aI δ Σ δ ¯ c a − ¯ M aI δ Σ δ Ω aµ + δ Σ δω aI δ Σ δL a (cid:19) = 0 , (72) Z d x (cid:18) c a δ Σ δη a ˆ I + ¯ η a ˆ I δ Σ δ ¯ c a − ¯ U i ˆ Iµ δ Σ δJ iα (cid:19) = 0 , (73) Z d x (cid:18) c a δ Σ δλ a ˆ I − ¯ λ a ˆ I δ Σ δ ¯ c a − δ Σ δη a ˆ I δ Σ δL a − ¯ V iα ˆ I δ Σ δJ iα (cid:19) = 0 . (74) In order to determine the most general invariant counterterm which can be freely added to each order,we follow the algebraic renormalization framework [26] and perturb the complete action Σ by adding anintegrated local polynomial in the fields and sources with dimension four and vanishing ghost number,Σ ct , and we require that the perturbed action, (Σ+ ε Σ ct ), where ε is an infinitesimal expansion parameter,obeys the same Ward identities fulfilled by Σ to the first order in the parameter ε , obtaining the followingconstraints: B Σ Σ ct = 0 , (75) δ Σ ct δb a = 0 , δ Σ ct δ ¯ c a + ∂ µ δ Σ ct δ Ω aµ = 0 , (76) G a Σ ct = 0 , (77) Q IJ Σ ct = 0 , ˆ Q ˆ J ˆ I Σ ct = 0 , (78) N e Σ ct = 0 , N ghost Σ ct = 0 , (79) δ Σ ct δ ¯ ϕ aI + ∂ µ δ Σ ct δ ¯ M aIµ = 0 , δ Σ ct δω aI + ∂ µ δ Σ ct δN aIµ − gf abc δ Σ ct δb a ¯ ω aI = 0 , (80) δ Σ ct δ ¯ ω aI + ∂ µ δ Σ ct δ ¯ N aIµ − gf abc M bIµ δ Σ ct δ Ω aµ = 0 , Z d x (cid:18) δ Σ ct δη a ˆ I + gf abc ¯ η a ˆ I δ Σ ct δb a (cid:19) = 0 , (81) δ Σ ct δϕ aI + ∂ µ δ Σ ct δM aIµ − gf abc (cid:18) δ Σ ct δb c ¯ ϕ bI − δ Σ ct δc c ¯ ω bI + ¯ N bIµ δ Σ ct δ Ω cµ (cid:19) = 0 , Z d x (cid:18) δ Σ ct δλ a ˆ I − g ( T a ) ij ¯ U j ˆ I δ Σ ct δJ iα + gf abc (cid:18) δ Σ ct δb c ¯ λ b ˆ I − δ Σ ct δ ¯ c b ¯ η c ˆ I (cid:19)(cid:19) = 0 , (82) Z d x δ Σ ct δ ¯ η a ˆ I − gV i ˆ Iα ( T a ) ij δ Σ ct δ ¯ J jα ! = 0 , Z d x δ Σ ct δ ¯ λ a ˆ I = 0 , (83)12 d x (cid:18) c a δ Σ ct δω aI + ¯ ω aI δ Σ ct δ ¯ c a + ¯ N aI δ Σ ct δ Ω aµ (cid:19) = 0 , (84) Z d x (cid:18) c a δ Σ ct δϕ aI − ¯ ϕ aI δ Σ ct δ ¯ c a − ¯ M aI δ Σ ct δ Ω aµ + δ Σ δω aI δ Σ ct δL a + δ Σ δL a δ Σ ct δω aI (cid:19) = 0 , (85) Z d x (cid:18) c a δ Σ ct δη a ˆ I + ¯ η a ˆ I δ Σ ct δ ¯ c a − ¯ U i ˆ Iµ δ Σ ct δJ iα (cid:19) = 0 , (86)and Z d x (cid:18) c a δ Σ ct δλ a ˆ I − ¯ λ a ˆ I δ Σ ct δ ¯ c a − δ Σ δη a ˆ I δ Σ ct δL a − δ Σ δL a δ Σ ct δη a ˆ I − ¯ V iα ˆ I δ Σ ct δJ iα (cid:19) = 0 . (87)The first condition, eq.(75), tells us that Σ ct belongs to the cohomolgy of the operator B Σ in the space ofthe local integrated polynomials in the fields and external sources of dimension bounded by four. Fromthe general results on the cohomolgy of Yang-Mills theories, see [26] and refs. therein, it follows that Σ ct can be parametrized as followsΣ ct = a S Y M + a m ψ Z d x ¯ ψ iα ψ iα + B Σ (cid:0) ∆ − (cid:1) (88)where ∆ − is an integrated polynomial of dimension three and ghost number −
1. The most generalexpression for ∆ − is given by∆ − = Z d x (cid:26) a Ω aµ A aµ + a L a c a + a ¯ J iα ψ iα + a ¯ ψ iα J iα + a gf abc ¯ N aIµ ϕ bI A cµ + a ¯ N aIµ ∂ µ ϕ aI + a gf abc M aIµ ¯ N aIµ + a gf abc M aIµ A cµ ¯ ω cI + a M aIµ ∂ µ ¯ ω bI + a A aµ ∂ µ ¯ c a + a f abc A aµ ∂ µ ϕ bI ¯ ω cI + a gf abc A aµ ϕ bI ∂ µ ¯ ω cI + a gf abc A aµ ∂ µ λ b ˆ I ¯ η c ˆ I + a gf abc A aµ λ b ˆ I ∂ µ ¯ η c ˆ I + a ∂ ϕ aI ¯ ω aI + a ∂ λ b ˆ I ¯ η c ˆ I + a gf abc ¯ c a c b ¯ c c + a ¯ c a b a + a f abc ¯ c a ϕ bI ¯ ϕ cI + a gf abc ¯ c a λ b ˆ I ¯ λ c ˆ I + a gf abc ¯ c a ω bI ¯ ω cI + a gf abc ¯ c a η b ˆ I ¯ η c ˆ I + a gf abc b a ϕ bI ¯ ω cI + a gf abc b a λ b ˆ I ¯ η c ˆ I + a ζm ψ V iα ˆ I ¯ U iα ˆ I + a m ψ ϕ aI ¯ ω aI + a m ψ λ a ˆ I ¯ η a ˆ I + a g ( T a ) ij ¯ ψ iα λ a ˆ I ¯ U jα ˆ I + a g ( T a ) ij ψ jα ¯ η a ˆ I V iα ˆ I + a ( γ µ ) αβ ( T a ) ij A aµ V iα ˆ I ¯ U jβ ˆ I + a ( γ µ ) αβ ( ∂ µ V iα ˆ I ) ¯ U iβ ˆ I + C abcd A aµ A bµ ϕ cI ¯ ω dI + C abcd A aµ A bµ λ c ˆ I ¯ η d ˆ I + C abcdeIJLM c a ϕ bI ϕ cJ ¯ ω dL ¯ ω eM + C abcde ˆ I ˆ J ˆ L ˆ M c a λ b ˆ I λ c ˆ J ¯ η d ˆ L ¯ η e ˆ M + C abcd ϕ aI ¯ ω bI λ c ˆ I ¯ λ d ˆ I + C abcd ϕ aI ¯ ϕ bI λ c ˆ I ¯ η d ˆ I + C abcdIJLM ϕ aI ¯ ω bI ω cL ¯ ω dM + C abcdIJLM ϕ aI ¯ ϕ bI ϕ cL ¯ ω dM + C abcd ˆ I ˆ J ˆ L ˆ M λ a ˆ I ¯ η b ˆ J η c ˆ L ¯ η d ˆ M + C abcd ˆ I ˆ J ˆ L ˆ M λ a ˆ I ¯ λ b ˆ J λ c ˆ L ¯ η d ˆ M (cid:27) . (89)After imposition of the remaining conditions, els.(76)-(87), and after a rather long algebra, it turns outthat the only non-vanishing coefficients are a = − a = a = a = a = a = a = a = a = − a = a = 0 , (90) a = a = a = a = 0 , a = 0 . (91) After application of the linearized BRST operator, the pure external source term a ( γ µ ) αβ ( ∂ µ V iα ˆ I ) ¯ U iβ ˆ I gives riseto a ( γ µ ) αβ (cid:16) ( ∂ µ U iα ˆ I ) ¯ U iβ ˆ I + ( ∂ µ V iα ˆ I ) ¯ V iβ ˆ I (cid:17) , which identically vanishes when the sources attain their physical values.Therefore, from now on we shall set a = 0. ct = Z d x (cid:26) a F aµν F aµν + a m ψ ¯ ψ iα ψ iα + a (cid:20) δS Y M δA aµ A aµ + ∂ µ ¯ c a ∂ µ c a + Ω aµ ∂ µ c a + gf abc (cid:16) ∂ µ c a ¯ N bIµ ϕ cI + ∂ µ c a M bIµ ¯ ω cI − ∂ µ c a ϕ bI ∂ µ ¯ ω cI − ∂ µ c a λ b ˆ I ∂ µ ¯ η c ˆ I (cid:17) +¯ ω aI ∂ ω aI − ¯ ϕ aI ∂ ϕ aI + η aI ∂ ¯ η aI − λ aI ∂ ¯ λ aI + ¯ M aIµ ∂ µ ϕ aI + ω aI ∂ µ ¯ N aIµ + N aIµ ∂ µ ¯ ω aI − ¯ ϕ aI ∂ µ M aIµ + ¯ M aIµ M aIµ − ¯ N aIµ N aIµ (cid:3) + a ζm ψ (cid:16) ¯ V iα ˆ I V iα ˆ I − ¯ U iα ˆ I U iα ˆ I (cid:17) + a h − ψ iα ( γ µ ) αβ D ijµ ψ jβ + 2 m ψ ¯ ψ iα ψ iα i(cid:27) (92)To complete the analysis of the algebraic renormalization of the model, we need to show that thecounterterm Σ ct can be reabsorbed into the starting action Σ through a redefinition of the fields { φ } , φ = ( A, c, ¯ c, b, ϕ, ¯ ϕ, ω, ¯ ω, ψ, ¯ ψ, λ, ¯ λ, η, ¯ η ), sources { S } , S = (Ω , L, J, ¯ J , ¯ N , ¯ M , N, M, ¯ U , ¯ V , U, V ), and pa-rameters τ , τ = ( g, ζ, m ψ ), namelyΣ( φ, S, τ ) + ε Σ ct ( φ, S, τ ) = Σ( φ , S , τ ) + O ( ε ) , (93)where ( φ , S , τ ) stand for the so-called bare fields, sources and parameters. φ = Z / φ φ , S = Z S S , τ = Z τ τ , (94)By direct inspection, for the renormalization factors we find Z / A = 1 + ε (cid:16) a a (cid:17) , Z g = 1 − εa , (95) Z / ψ = Z / ψ = 1 − εa , Z / m ψ = 1 − εa , (96) Z / b = Z − / A , Z / c = Z / c = Z − / A Z − / g , (97) Z / ϕ = Z / ϕ = Z − / A Z − / g , Z / ω = Z − g , (98) Z / ω = Z − / A , Z / λ = Z / λ = Z − / A Z − / g , (99) Z / η = Z − g , Z / η = Z − / A , (100) Z ¯ M = Z M = Z − / A Z − / g , Z ¯ N = Z − g , (101) Z N = Z − / A , Z ¯ V = Z V = Z − / ψ Z − / g Z / A , (102) Z ¯ U = Z − / ψ Z − g Z / A , Z U = Z − / ψ , (103) Z ζ = (1 + εa ) Z − m ψ Z ψ Z g Z − / A . (104) Z Ω = Z − / A Z − / c Z − g , Z L = Z / A , Z J = Z ¯ J = Z − g Z − / c Z − / ψ . (105)This ends the analysis of the all orders algebraic renormalization of the action Σ, and thus of the startingaction S , eq.(30). 14 Conclusion
In this work we have pursued the investigation started in [1], where the coupling between the inverseof the Faddeev-Popov operator and quark matter fields has been introduced through the operators R aiψ ( x ) := g Z d z ( M − ) ab ( x, z )( T b ) ij ψ j ( z ) , ¯ R aiψ ( x ) := g Z d z ( M − ) ab ( x, z ) ¯ ψ j ( z )( T b ) ji , (106)giving rise to a non-trivial correlation function [1] Q abijψ ( x − y ) = D R aiψ ( x ) ¯ R bjψ ( y ) E . (107)This correlation function can be directly studied in lattice numerical simulations, as done recently in thecase of gluons [23].As shown in [1], the correlation function (107) can be obtained from a theory constructed by addingto the usual matter action a term similar to Zwanzige’s horizon function [3, 4, 5], namely S matter → S matter + M H matter , (108)where, in complete analogy with the horizon function for the gluon sector [3, 4, 5], H matter is given by H matter = − g Z d xd y ¯ ψ i ( x )( T a ) ij ( M − ) ab ( x, y )( T b ) jk ψ k ( y ) . (109)with the mass parameter M playing a role similar to that of the Gribov parameter γ . Remarkably, theintroduction of such a non-local term gives rise to a quark propagator of the kind (cid:10) ψ i ( p ) ¯ ψ j ( − p ) (cid:11) = − ip µ γ µ + A ( p ) p + A ( p ) δ ij , (110)with the quark mass function A ( p ) given by A ( p ) = m ψ + g (cid:18) N − N (cid:19) M p + µ ψ , (111)This propagator fits very well the available lattice numerical data, see [25] and the discussion in [1].In the present paper we have been able to show that, despite its non-locality, expression (109) canbe cast in local form by means of a set of suitable localizing fields. Moreover, when cast in local form,the resulting action fulfils a large set of Ward identities which have enabled us to prove that the theoryis multiplicative renormalizable to all orders. This is a non-trivial result which supports the idea thatthe Faddeev-Popov operator couples in a universal way to both gauge and quark fields, as expressed byequations (18) and (106).This coupling allows us to reproduce from a local and renormalizable action the behaviour of the gluonand quark propagators observed in numerical simulations, reinforcing the belief that the inverse of theFaddeev-Popov operator plays an important role in the infrared dynamics of confining Yang-Mills theo-ries. 15 cknowledgments The Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq-Brazil), the Faperj,Funda¸c˜ao de Amparo `a Pesquisa do Estado do Rio de Janeiro, the Coordena¸c˜ao de Aperfei¸coamentode Pessoal de N´ıvel Superior (CAPES) are gratefully acknowledged.
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