Spontaneous breaking of the BRST symmetry in presence of the Gribov horizon: renormalizability
M. A. L. Capri, D. Dudal, M. S. Guimaraes, I. F. Justo, L. F. Palhares, S. P. Sorella
aa r X i v : . [ h e p - t h ] J un Spontaneous breaking of the BRST symmetry in presence of theGribov horizon: renormalizability
M. A. L. Capri a ∗ , D. Dudal b † , M. S. Guimaraes a ‡ , I. F. Justo a § , L. F. Palhares a , c ¶ , S. P. Sorella a k∗∗ a 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 b Ghent University, Department of Physics and Astronomy, Krijgslaan 281-S9, 9000 Gent, Belgium c Institut f¨ur Theoretische Physik, Heidelberg University, Philosophenweg 16, 69120 Heidelberg, Germany
Abstract
An all orders algebraic proof of the multiplicative renormalizability of the novel formulation of the Gribov-Zwanziger action proposed in [1], and allowing for an exact but spontaneously broken BRST symmetry, isprovided.
In recent years much attention has been devoted to the study of the issue of the Gribov copies [2] and of itsrelevance for confinement in Yang-Mills theories . The existence of the Gribov copies is a general feature of thegauge fixing quantization procedure, being related to the impossibility of finding a local gauge condition whichpicks up only one gauge configuration for each gauge orbit [5]. As it has been shown by Gribov and Zwanziger[2, 6, 7], a partial resolution of the Gribov problem in the Landau gauge can be achieved by restricting the domainof integration in the functional Euclidean integral to the first Gribov horizon. Remarkably, this restriction hasresulted into a local and renormalizable action, known as the Gribov-Zwanziger action [6, 7].More recently, a Refined version of the Gribov-Zwanziger action has been worked out in [8, 9, 10], leading to atree level gluon propagator whose behavior in the infrared region is in very good agreement with the most recentlattice numerical simulations [11, 12, 13, 14, 15, 16]. This propagator displays complex poles in momentumspace. As such, it cannot describe the propagation of physical excitations. Rather, it is suited for a kind of effectivedescription of gluon confinement, see also [17]. In spite of the appearance of complex poles, the Refined-Gribov-Zwanziger gluon propagator has been successfully employed to investigate the correlation functions of gauge ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] k [email protected] ∗∗ Work supported by FAPERJ, Fundac¸ ˜ao de Amparo `a Pesquisa do Estado do Rio de Janeiro, under the program
Cientista do NossoEstado , E-26/101.578/2010. See refs.[3, 4] for a pedagogical introduction to the Gribov problem. J PC = ++ , ++ , − + , are in qualitative agreement with the available numerical dataon the spectrum of the glueballs [20]. Let us also mention that such type of gluon propagator has also been usedin previous studies in hadron physics [21, 22], see also [23, 24] for a recent attempt to generalize the Refined-Gribov-Zwanziger action by including quarks and associated chiral symmetry breaking. Recently, complex polepropagators were also considered in terms of semi-analytical approaches to the QCD phase diagram [25, 26],partially motivated by fits to finite temperature lattice gluon data [27, 28].Although the aforementioned results can be taken as evidence of the fact that the Refined Gribov-Zwanzigertheory can be effectively employed to investigate the physical spectrum of a confining Yang-Mills theory, thereare still many aspects of the theory which remain to be understood. Certainly, the systematic construction ofa set of composite operators whose correlation functions can be directly related to the physical spectrum of aconfining Yang-Mills theory is one of the most challenging aspects of the Gribov-Zwanziger framework for colorconfinement. At present, the characterization of the analyticity and of the unitarity properties of these correlationfunctions seems a highly cumbersome task, taking into account that explicit calculations have to be done byemploying a confining gluon propagator exhibiting complex poles.Amongst the various open aspects of the Gribov-Zwanziger framework, the issue of the BRST symmetry is asource of continuous investigations, see for example [6, 7, 9, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39] for anoverview of what has been already done on this topic. We expect that a better understanding of the role of theBRST symmetry in confining Yang-Mills theories would be of great relevance in order to face the characterizationof the physical spectrum.In a recent work [1], some of the authors have been able to obtain an equivalent formulation of the Gribov-Zwanziger action which displays an exact BRST symmetry which turns out to be spontaneously broken by therestriction of the domain of integration to the Gribov horizon. In particular, in [1], the BRST operator s retains theimportant property of being nilpotent, i.e. s =
0. This feature enabled us to make use of the powerful tool of thecohomology of s [41, 40] in order to prove that the set of colorless gauge invariant operators corresponding to thecohomology classes of s is closed under time evolution [1]. Moreover, it has also been shown that the Goldstonemode associated to the spontaneous breaking of s is completely decoupled.The aim of the present article is to fill a gap not addressed in the previous work [1], namely, the renormalizabilityto all orders of the spontaneous symmetry breaking formulation of the Gribov-Zwanziger theory in d =
4. As weshall see, the action obtained in [1] enjoys a large set of Ward identities which enables us to prove that it is, in fact,multiplicatively renormalizable to all orders.The paper is organized as follows. In Sect. 2 we provide a short summary of the BRST spontaneous symmetrybreaking formulation of the Gribov-Zwanziger action. In Sect. 3 we derive numerous Ward identities fulfilled bythe action in the novel formulation. In Sect. 4, the renormalizability to all orders of the model is established bymeans of the algebraic renormalization procedure [40]. 2
A novel formulation of the Gribov-Zwanziger action and the spontaneous break-ing of the BRST symmetry
Let us start by recalling the expression of the Gribov-Zwanziger action which enables us to restrict the Euclideanfunctional integral to the first Gribov horizon, namely S GZ = Z d x F aµ n F aµ n + Z d x (cid:16) ib a ¶ µ A aµ + c a ¶ µ D abµ c b (cid:17) + Z d x (cid:16) j acµ ¶ n D am n j mcµ − w acµ ¶ n D am n w mcµ − g (cid:0) ¶ n w acµ (cid:1) f abm ( D n c ) b j mcµ (cid:17) + Z d x (cid:16) − g g f abc A aµ ( j bcµ + j bcµ ) − (cid:0) N − (cid:1) g (cid:17) . (1)The field b a stands for the Lagrange multiplier implementing the Landau gauge condition, ¶ µ A aµ =
0, and c a , c a are the corresponding Faddeev-Popov ghosts. The fields j abµ , j abµ are a pair of bosonic fields, while w abµ , w abµ area pair of anticommuting fields. These fields are needed in order to implement the restriction to the first Gribovhorizon in a local way [6, 7, 9]. All fields belong to the adjoint representation of the gauge group SU ( N ) . Themassive parameter g , called the Gribov parameter, is not a free parameter, being determined in a self-consistentway through the gap equation ¶ E vac ¶g = , (2)where E vac stands for the vacuum energy of the theory [2, 6, 7], e − E vac = Z [ D F ] e − S GZ . (3)It turns out that the Gribov-Zwanziger action (1) does not exhibits an exact BRST invariance [6, 7, 9], which issoftly broken by the Gribov parameter g . Indeed, sS GZ = − g Z d x (cid:16) g f abc D akµ c k ( j bcµ + j bcµ ) + g f abc A aµ w bcµ (cid:17) , (4)where s stands for the nilpotent BRST operator sA aµ = − D abµ c b = − ( ¶ µ d ab + g f acb A cµ ) c b , sc a = g f acb c b c c , sc a = ib a , sb a = , s w abµ = j abµ , s j abµ = , s j abµ = w abµ , s w abµ = , s = . (5)In ref.[1] it was proposed to replace expression (1) by the following action S ′ GZ = Z d x F aµ n F aµ n + Z d x (cid:16) ib a ¶ µ A aµ + c a ¶ µ D abµ c b (cid:17) + Z d x (cid:16) j acµ ¶ n D am n j mcµ − w acµ ¶ n D am n w mcµ − g (cid:0) ¶ n w acµ (cid:1) f abm ( D n c ) b j mcµ (cid:17) + Z d x (cid:16) − G abµ n ¶ G abµ n + F abµ n ¶ F abµ n − G abµ n D akµ j kb n − g f ak ℓ G abµ n D ℓ pµ c p w kb n (cid:17) + Z d x (cid:18) − b G abµ n ¶ b G abµ n + b F abµ n ¶ b F abµ n + b F abµ n D akµ w kb n − b G abµ n D akµ j kb n + g f ak ℓ b F abµ n D ℓ pµ c p j kb n (cid:19) + Z d x (cid:18) H abµ n (cid:16) G abµ n − d µ n d ab g (cid:17) + b H abµ n (cid:18) b G abµ n − d µ n d ab g (cid:19) − G abµ n b G abµ n (cid:19) , (6)3here we have introduced two new BRST quartets [40] of fields consisting of F abµ n , F abµ n (commuting), G abµ n , G abµ n (anticommuting) and their hat-counterparts: s F abµ n = G abµ n , s G abµ n = , s b F abµ n = b G abµ n , s b G abµ n = , s G abµ n = F abµ n , s F abµ n = , s b G abµ n = b F abµ n , s b F abµ n = , (7)as well as the singlet fields H abµ n , b H abµ n s H abµ n = s b H abµ n = . (8)It is easily checked that the action (6) can be rewritten as S ′ GZ = Z d x F aµ n F aµ n + s Z d x (cid:0) c a ¶ µ A aµ + w acµ ¶ n D am n j mcµ (cid:1) + s Z d x (cid:18) − F abµ n ¶ G abµ n − G abµ n D akµ w kb n − b F abµ n ¶ b G abµ n − b F abµ n D akµ j kb n (cid:19) + Z d x (cid:18) H abµ n (cid:16) G abµ n − d µ n d ab g (cid:17) + b H abµ n (cid:18) b G abµ n − d µ n d ab g (cid:19) − G abµ n b G abµ n (cid:19) , (9)from which it can be established that S ′ GZ has an exact BRST invariance sS ′ GZ = , (10)whereby we have preserved the nilpotency of the BRST operator, s = S ′ GZ is equivalent to that of the original Gribov-Zwanziger action S GZ . Let us therefore point out that, using the algebraic exact equations of motion of the fields ( H , b H ) , G abµ n = b G abµ n = g d ab d µ n , (11)we immediately recover the g -dependent part of the Gribov-Zwanziger action, namely Z d x (cid:18) − G abµ n D akµ j kb n − b G abµ n D akµ j kb n − G abµ n ¶ G abµ n − b G abµ n ¶ b G abµ n − G abµ n b G abµ n + H abµ n (cid:16) G abµ n − d µ n d ab g (cid:17)(cid:19) H − , b H − EOM −→ Z d (cid:16) − g g f abc A aµ (cid:16) j bcµ + j bcµ (cid:17) − ( N − ) g (cid:17) . (12)Moreover, the integration over the fields F , F , b F and b F turns out to generate a unity in the partition function, ascan be seen by performing the following simultaneous changes of integration variables: b F abµ n → b F abµ n − ¶ (cid:16) D akµ w kb n + g f ak ℓ D ℓ pµ c p j kb n (cid:17) , w tb n → w tb n + (cid:2) ( ¶ D ) − (cid:3) tk (cid:16) g f ak ℓ G abµ n D ℓ pµ c p (cid:17) , F abµ n → F abµ n − ¶ (cid:16) D akµ w kb n + g f ak ℓ D ℓ pµ c p j kb n (cid:17) , (13)from which the equivalence between the two formulations, expressed by means of (1) and (6), follows.Let us proceed by showing how the spontaneous breaking of the BRST symmetry is realized in the new formulationof the Gribov-Zwanziger action. To that end, let us rewrite the action S ′ GZ by again making explicit use of the4quations of motion (11). Thus S ′ GZ = Z d x F aµ n F aµ n + Z d x s (cid:0) c a ¶ µ A aµ + w acµ ¶ n D am n j mcµ (cid:1) + Z d x (cid:18) F abµ n ¶ F abµ n − g s ( D akµ w kaµ ) + b F abµ n ¶ b F abµ n − g D akµ j kaµ + b F abµ n s ( D akµ j kb n ) − g d ( N − ) (cid:19) . (14)This expression turns out to be left invariant by the following nilpotent BRST transformations: sA aµ = − D abµ c b = − ( ¶ µ d ab + g f acb A cµ ) c b , sc a = g f acb c b c c , sc a = ib a , sb a = , s w abµ = j abµ , s j abµ = , s j abµ = w abµ , s w abµ = , s F abµ n = g d ab d µ n , s b F abµ n = g d ab d µ n , s G abµ n = F abµ n , s F abµ n = , s b G abµ n = b F abµ n , s b F abµ n = , (15)with sS ′ GZ = , s = . (16)Furthermore, from equations (15), it follows that the BRST operator suffers from spontaneous symmetry breaking.In fact h s F abµ n i = g d ab d µ n , h s b F abµ n i = g d ab d µ n . (17)Let us end this short summary by mentioning the important feature that the Goldstone mode associated to thespontaneous symmetry breaking of the BRST operator turns out to be completely decoupled from the theory, see[1] for the argument. The first step in order to prove the all orders renormalizability of the novel formulation is to establish the set ofWard identities obeyed by the action S ′ GZ , eq.(6). To that end, and following the algebraic renomalization procedure[40], we introduce a set of external sources l abi , r abi , K aµ , L a transforming as s l abi = r abi , s r abi = , sK aµ = sL a = , (18)and the complete BRST invariant action SS = Z d x (cid:26) F aµ n F aµ n + ib a ¶ µ A aµ + ¯ c a ¶ µ D abµ c b + ¯ f ai ¶ µ D abµ f bi − ¯ w ai ¶ µ D abµ w bi − g f abc ( ¶ µ ¯ w ai )( D bdµ c d ) f bi − G aµi ¶ G aµi + F aµi ¶ F aµi − G aµi D abµ ¯ f bi + g f abc G aµi ( D bdµ c d ) ¯ w ci − b G aµi ¶ b G aµi + b F aµi ¶ b F aµi + b F aµi D abµ w bi − b G aµi D abµ f bi − g f abc b F aµi ( D bdµ c d ) f ci + H aµi (cid:16) G aµi − d aµi g (cid:17) + b H aµi (cid:16) b G aµi − d aµi g (cid:17) − G aµi b G aµi − K aµ D abµ c b + g f abc L a c b c c + r abi b F aµi D bcµ c c − l abi b G aµi D bcµ c c (cid:27) , (19) s S = , (20)where, as done in the original work by Zwanziger [6, 7], we have introduced the multi-index notation i ≡ ( a , µ ) , i = , ..., f = ( N − ) , which turns out to be very useful in the discussion of the renormalizability. As pointed5ut in [6, 7], the possibility of introducing the multi-index i ≡ ( a , µ ) relies on the existence of a global symmetry U ( f ) . Thus, the term d aµi appearing in expression (19) stands for d aµi ≡ d ab d µ n . (21)As one can see from the expression S , the external sources K aµ , L a are introduced in order to properly define thecomposite operators D abµ c b and g f acb c b c c , corresponding to the nonlinear BRST transformations of the fields A aµ and c a , eqs.(5). Moreover, it turns out to be useful to also couple the BRST doublet of external fields l abi , r abi tothe composite operators b G aµi D bcµ c c and b F aµi D bcµ c c .We are now ready to derive an extensive set of Ward identities fulfilled by the action S . These are: • the Slavnov-Taylor identity S ( S ) ≡ Z d x (cid:26) dSd K aµ dSd A aµ + dSd L a dSd c a + ib a dSd ¯ c a + ¯ f ai dSd ¯ w ai + w ai dSdf ai + G aµi dSd F aµi + F aµi dSd G aµi + b G aµi dSd b F aµi + b F aµi dSd b G aµi + r abi dSdl abi (cid:27) = . (22) • the linearly broken Ward identity for the Gribov parameter g ¶S¶g = − Z d x (cid:16) H aaµµ + b H aaµµ (cid:17) . (23)As we shall see in the next section, this identity will be responsible for the nonrenormalizability properties of theGribov parameter g . Notice that the left hand side of (23) is linear in the quantum fields, i.e. it is a linear breaking.It is well established that this kind of breaking is not affected by quantum corrections, see [40]. • the gauge-fixing condition and the anti-ghost equation: dSd b a = ¶ µ A aµ , dSd ¯ c a + ¶ µ dSd K aµ = , (24) • the equations of motion of the auxiliary fields: dSd G aµi = − ¶ G aµi , dSd b G aµi = − ¶ b G aµi , (25) dSd F aµi = − ¶ F aµi , dSd F aµi = ¶ F aµi , dSd b F aµi = − ¶ b F aµi , (26) • the equations of motion of the Lagrange multipliers: dSd H aµi = G aµi − d aµi g , dSd b H aµi = b G aµi − d aµi g , (27)6 the equations of motion of the localizing fields: F ai ( S ) ≡ dSd ¯ f ai + ¶ µ dSd b G aµi − D abµ dSd H bµi − ¶ µ l abi dSd K bµ ! = − ¶ ¶ µ b G aµi + ¶ µ b H aµi + ¶ µ G aµi + g g f abc A cµ d bµi , (28) W ai ( S ) ≡ dSd ¯ w ai + ¶ µ dSd b F aµi − g f abc dSd H bµi + d bµi g ! dSd K cµ + ¶ µ r abi dSd K bµ ! = ¶ ¶ µ b F aµi , (29) F ai ( S ) ≡ dSdf ai + ¶ µ dSd G aµi + ig f abc ¯ f bi dSd b c − g f abc ¯ w bi dSd ¯ c c + g f abc dSdr bci − D abµ dSd b H bµi = − ¶ ¶ µ G aµi + ¶ µ H aµi − ¶ µ b G aµi − g g f abc d cµi A bµ . (30) • the Ward identities: U i ( S ) = Z d x (cid:26) c a dSdw ai + ¯ w ai dSd ¯ c a − d ab dSdr abi (cid:27) = , (31) V i ( S ) = Z d x (cid:26) − c a dSdf ai + ¯ f ai dSd ¯ c a + dSd L a dSdw ai + (cid:18) dSd b H aµi + g d aµi (cid:19) dSd K aµ (cid:27) = . (32) • the linearly broken U ( ( N − )) Ward identity Q i j ( S ) ≡ Z d x (cid:26) f ai dSdf aj − ¯ f aj dSd ¯ f ai + w ai dSdw aj − ¯ w aj dSd ¯ w ai + G aµi dSd G aµ j − G aµ j dSd G aµi − b G aµ j dSd b G aµi + b G aµi dSd b G aµ j − b F aµ j dSd b F aµi + b F aµi dSd b F aµ j − H aµ j dSd H aµi + b H aµi dSd b H aµ j + r abi dSdr abj + l abi dSdl abj (cid:27) = g Z d x (cid:16) d aµi H aµ j − d aµ j b H aµi (cid:17) . (33) • the exact integrated Ward identities: T ( ) i j ( S ) ≡ Z d x F aµi dSd G aµ j − G aµ j dSd F aµi ! = , T ( ) i j ( S ) ≡ Z d x b F aµi dSd G aµ j − G aµ j dSd b F aµi ! = , T ( ) i j ( S ) ≡ Z d x b F aµi dSd b G aµ j − b G aµ j dSd b F aµi ! = , T ( ) i j ( S ) ≡ Z d x F aµi dSd b G aµ j − b G aµ j dSd F aµi ! = , ( ) i j ( S ) ≡ Z d x F aµi dSd G aµ j + G aµ j dSd F aµi ! = , T ( ) i j ( S ) ≡ Z d x F aµi dSd F aµ j − F aµ j dSd F aµi ! = , T ( ) i j ( S ) ≡ ( d ik d jl − d jk d il ) Z d x G aµk dSd G aµl = , T ( ) i j ( S ) ≡ Z d x b G aµi dSd G aµ j − G aµ j dSd b G aµi ! = , T ( ) i j ( S ) ≡ ( d ik d jl − d jk d il ) Z d x b G aµk dSd b G aµl = , T ( ) i j ( S ) ≡ Z d x F aµi dSd F aµ j − F aµ j dSd F aµi ! = . (34) • the SL ( , R ) Ward identity D ( S ) ≡ Z d x (cid:18) c a dSd ¯ c a − i dSd b a dSd L a (cid:19) = . (35) • the linearly broken rigid SU ( N ) symmetry W a ( S ) = − g Z d x g f abc ( H bµi d cµi + b H bµi d cµi ) , (36)with W a ≡ g f abc Z d x ( (cid:229) y ∈ O y b dd y c + r bdi ddr cdi + r dbi ddr dci + l bdi ddl cdi + l dbi ddl dci ) (37)where O stands for O = (cid:8) A aµ , b a , ¯ c a , c a , f ai , ¯ f ai , w ai , ¯ w ai , . . . (cid:9) , (38) i.e. , the set O is the set of all fields and sources that have only one color index, where we have not taken intoaccount the color index hidden in the multi-index i = ( a , µ ) . • the equation of motion of the source l abi L abi ( S ) ≡ dSdl abi − dSd b H aµi + g d aµi dSd K bµ = . (39) • the Q f chargeWe can combine the operators Q i j and T ( ) i j , appearing in eqs.(33) and (34), respectively, and construct the follow-ing operator: Q Ti j = Q i j + T ( ) i j . (40)8he operator Q Ti j commutes with the BRST operator s [ s , Q Ti j ] = . (41)Then, the trace of Q Ti j defines a new charge: Q Tii ≡ Q f : = Z d x (cid:26) f ai ddf ai − ¯ f ai dd ¯ f ai + w ai ddw ai − ¯ w ai dd ¯ w ai + G aµi dd G aµi − G aµi dd G aµi − b G aµi dd b G aµi + b G aµi dd b G aµi − b F aµi dd b F aµi + b F aµi dd b F aµi − H aµi dd H aµi + b H aµi dd b H aµi + r abi ddr abi + l abi ddl abi + F aµi dd F aµi − F aµi dd F aµi (cid:27) , (42)where f ≡ ( N − ) . The Q f charge gives rise to a powerful linearly broken Ward identity when acting on S ,namely Q f ( S ) = g Z d x (cid:16) d aµi H aµi − d aµi b H aµi (cid:17) . (43)This is actually the Ward identity which enables us to make use of the multi-index i = ( a , µ ) . Having established the Ward identities obeyed by the action S , eqs.(22)-(43), we can proceed to show the renor-malizability to all orders of the model. Let us begin with the algebraic characterization of the most general localinvariant counterterm that is compatible with all Ward identities. In order to characterize the most general local invariant counterterm which can be freely added to all orders inperturbation theory, we follow the general setup of the algebraic renormalization [40] and perturb the startingaction S by adding an integrated local polynomial in the fields and sources, S count , with dimension bounded byfour and with vanishing ghost number. We thus demand that the perturbed action, S + h S count , (44)where h is an expansion parameter, fulfills, to the first order in h , the same set of Ward identities obeyed by S ,eqs.(22)-(43). This requirement gives rise to the following constraints for the counterterm S count : S S ( S count ) = , (45) dd ¯ c a + ¶ µ dd K aµ ! S count = , (46) dd b a S count = , ¶¶g S count = , dd G aµi S count = , dd b G aµi S count = , dd F aµi S count = , dd F aµi S count = , dd b F aµi S count = , dd H aµi S count = , dd b H aµi S count = , (47)9 ai ( S count ) = , W a , i S ( S count ) = , F ai ( S count ) = , U i ( S count ) = , V i S ( S count ) = , D S ( S count ) = , W a ( S count ) = , L ab , i S ( S count ) = , Q i j ( S count ) = , T ( n ) i j ( S count ) = , n = , . . . , , (48) Q f ( S count ) = . (49)Here, the operators with the subscript “ S ” represent the so called linearized operators corresponding to the Wardidentities which are nonlinear in S , see [40]. For example, S S is the linearized operator corresponding to theSlavnov-Taylor identity (22), namely S S = Z d x (cid:26) dSd K aµ dd A aµ + dSd A aµ dd K aµ + dSd L a dd c a + dSd c a dd L a + ib a dd ¯ c a + ¯ f ai dd ¯ w ai + w ai ddf ai + G aµi dd F aµi + F aµi dd G aµi + b G aµi dd b F aµi + b F aµi dd b G aµi + r abi ddl abi (cid:27) . (50)As the BRST operator, also S S is nilpotent, i.e. S S S S = . (51)The remaining linearized operators are given by: W a , i S = dd ¯ w ai + ¶ µ dd b F aµi − g f abc dSd H bµi + d bµi g ! dd K cµ + g f abc dSd K bµ dd H cµi + ¶ µ r abi dd K bµ ! , V i S = Z d x (cid:26) − c a ddf ai + ¯ f ai dd ¯ c a + dSd L a ddw ai + dSdw ai dd L a + (cid:18) dSd b H aµi + g d aµi (cid:19) dd K aµ + dSd K aµ dd b H aµi (cid:27) , D S = Z d x (cid:18) c a dd ¯ c a − i dSd b a dd L a − i dSd L a dd b a (cid:19) , L ab , i S = ddl abi − dSd b H aµi + g d aµi dd K bµ − dSd K bµ dd b H aµi . (52)Let us now turn to the characterization of the counterterm. The constraints (47) imply that S count is independentfrom the fields b , G , b G , F , b F , F , H , b H , as well as from the Gribov parameter g . Equation (46) means that S count depends on ¯ c and K only through the combination ( ¶ µ ¯ c a + K aµ ) . Moreover, from eq.(49) it follows that S count has zero Q f -charge. Finally, relying on well known properties of the cohomology of Yang-Mills theories [40],condition (45) allow us to construct the countertem in the form: S count = a S YM + S S D ( − ) , S YM = Z d x F aµ n F aµ n , (53)where a is a dimensionless coefficient and D ( − ) is an integrated polynomial in the fields and sources with dimen-sion four and ghost number −
1. Collecting all this information, and making use of Table 1 and of Table 2, one can10rite D ( − ) as D ( − ) = Z d x (cid:26) a ( ¶ µ ¯ c a + K aµ ) A aµ + a L a c a + a b F aµi ¶ µ f ai + a g f abc b F aµi A cµ f bi + a ¯ w ai ¶ f ai + a g f abc ( ¶ µ ¯ w ai ) A cµ f bi + a g f abc ¯ w ai A cµ ¶ µ f bi + a g f abc ¯ w ai A cµ G bµi + a ¯ w ai ¶ µ G aµi + a b F aµi G aµi + t abcd ¯ w ai f bi ¯ f cj f dj + t abcd ¯ w ai f bj ¯ f ci f dj + t abcd ¯ w ai f bi ¯ w cj w dj + t abcd ¯ w ai f bj ¯ w ci w dj + a abcd l abi b F cµi ¶ µ c d + a abcd l abi ( ¶ µ b F cµi ) c d + a abcd l abi b G cµi A dµ + a abcd r abi b F cµi A dµ + a abcd l abi ( ¶ µ ¯ w ci ) ¶ µ c d + a abcd l abi ( ¶ ¯ w ci ) c d + a abcd l abi ¯ w ci ¶ c d + a abcd ( ¶ l abi ) ¯ w ci c d + a abcd l abi ( ¶ µ ¯ f ci ) A dµ + a abcd l abi ¯ f ci ¶ µ A dµ + a abcd r abi ( ¶ µ ¯ w ci ) A dµ + a abcd r abi ¯ w ci ¶ µ A dµ + b abcde l abi b F cµi c d A eµ + b abcde l abi ( ¶ µ ¯ w ci ) c d A eµ + b abcde l abi ¯ w ci ( ¶ µ c d ) A eµ + b abcde l abi ¯ w ci c d ( ¶ µ A eµ ) + b abcde l abi ¯ f ci A dµ A eµ + b abcde r abi ¯ w ci A dµ A eµ + t abcde f l abi ¯ w ci c d A eµ A fµ + M abcde f g l abi l cdj ¯ f ei ¯ f fj c g + M abcde f g l abi l cdi ¯ f ej ¯ f fj c g (cid:27) . (54)In this expression, a i , i = , ...,
10, are dimensionless coefficients, while { t } , { a } , { b } , { t } , { M } stand for invarianttensors of the gauge group SU ( N ) . Following an observation already employed in previous works [8, 9, 10], itturns out that the coefficient a vanishes. This is due to the fact that, as the term L a c a is already of dimension 4,the coefficient a cannot depend on the Gribov parameter g , and it vanishes when g = h S S , dd b a i = − i (cid:16) dd ¯ c a + ¶ µ dd K aµ (cid:17) , n S S , dd F aµi o = dd G aµi , (cid:26) S S , dd b F aµi (cid:27) = dd b G aµi , (cid:26) S S , dd F aµi (cid:27) = , h S S , dd G aµi i = , (cid:20) S S , dd b G aµi (cid:21) = , (cid:2) S S , F ai (cid:3) = − W ai S , n S S , W ai S o = , [ S S , F ai ] = − g f abc L bc , i S , [ S S , U i ] = Z d x (cid:16) V i S + d ab L ab , i S (cid:17) , (cid:8) S S , V i S (cid:9) = , n S S , L ab , i S o = , n S S , T ( ) i j o = T ( ) i j , n S S , T ( ) i j o = T ( ) i j , n S S , T ( ) i j o = T ( ) i j , n S S , T ( ) i j o = − T ( ) ji , h S S , T ( ) i j i = , [ S S , D S ] = , [ S S , W a ] = , h S S , Q Ti j i = , (55)it follows that, after a lengthy analysis, only the coefficient a remains free. The a priori quite monstrous expressionfor D ( − ) , eq. (54), eventually thus reduces considerably to the following form D ( − ) = a Z d x h ( ¶ µ ¯ c a + K aµ ) A aµ + ( ¶ µ ¯ w ai + b F aµi ) D abµ f bi + G aµi D abµ ¯ w bi + b F aµi G aµi − l abi b F aµi D bcµ c c i . (56)Summarizing, the most general invariant counterterm S count compatible with all constraints (45)–(49) has twoindependent free coefficients, a , a , and is given by S count = a Z d x F aµ n F aµ n + S S D ( − ) . (57)11 b ¯ c c ¯ f f ¯ w w G G F F b G b G b F b F H b H dimension 1 2 2 0 1 1 1 1 2 0 2 0 2 0 2 0 2 2ghost − − − − Q f -charge 0 0 0 0 − − − − − − − K L r l dimension 3 4 1 1ghost − − − Q f -charge 0 0 1 1Table 2: Quantum numbers of the external sources Having characterized the most general invariant local counterterm S count , eq.(57), compatible with all Ward iden-tities (22)-(43), it remains to check if S count can be reabsorbed into the starting action S through a multiplicativerenormalization of the fields, sources and parameters of theory, namely S ( f , J ) + hS count ( f , J ) = S ( f , J ) + O ( h ) , (58)with f = Z / f f , J = Z J J , (59)where ( f , f ) is a shorthand notation for the bare and renormalized fields, while ( J , J ) stand for the bare andrenormalized sources and parameters. Making use of expression (57), for the renormalization factors { Z } oneobtains Z / A = + h (cid:16) a + a (cid:17) + O ( h ) , Z g = − h a + O ( h ) , (60) Z / c = Z / c = Z / f = Z / f = Z − / g Z − / A , Z / w = Z − g , Z / w = Z − / A , Z / G = Z / b G = Z g = Z − / g Z − / A , Z / G = Z / b G = Z / g Z / A , Z / b F = Z − g , Z / b F = Z g , Z / F = Z / F = , Z / H = Z / b H = Z / g Z / A , Z K = Z / g Z / A , Z L = Z g Z / A , Z r = Z / g Z / A , Z l = Z − g Z − / A . (61)12his finalizes the proof of the all orders algebraic renormalization of the novel formulation of the Gribov-Zwanzigertheory. Let us conclude by observing that the renormalization factor Z g of the Gribov parameter g is not an inde-pendent quantity, being expressed in terms of the renormalization factors of the gauge coupling constant g and ofthe gauge field A aµ , i.e. Z g = Z − / g Z − / A . This feature expresses the nonrenormalization properties of g , alreadyestablished in [6, 7, 8, 9, 10], here a simple consequence of the powerful Ward identity (23). In this work we have pursued the investigation of the novel formulation of the Gribov-Zwanziger action proposedin [1], which allows for an exact BRST invariance of the action implementing the restriction to the Gribov horizon.As shown in [1], the BRST symmetry turns out to be spontaneously broken, the breaking parameter being nothingbut the Gribov mass g . It is worth mentioning that in this reformulation the BRST operator s does keep itsnilpotency, i.e. s =
0, a crucial feature which enables us to employ the powerful results on the cohomology of s in order to construct the set of colorless local composite gauge invariant operators [40].In the present paper we have presented the all orders algebraic proof of the renormalizability of the new for-mulation. In particular, as one can see from eq. (60), only two independent renormalization factors are needed,namely Z A and Z g , a feature which is shared by the original formulation of the Gribov-Zwanziger action. This isan important check of the equivalence between the two formulations at the quantum level.Certainly, many aspects of the role of the BRST symmetry in the presence of the Gribov horizon remain to beunraveled. Though, we believe that the current formulation in which the BRST symmetry is spontaneously brokenmight be helpful in order to face the hard and still open problem of identifying a set of renormalizable compositeoperators whose correlation functions display the necessary analytical and unitarity properties allowing to makecontact with the physical spectrum of a confining Yang-Mills theory. The results of [1], together with those ofthe current follow-up paper already learn that we can introduce the subspace of renormalizable gauge invariantoperators which is furthermore preserved under time evolution, based on BRST cohomology tools. The furtherextraction of a physical subspace with the desired spectral properties is now subject to further investigation.Finally, although the proof of the renormalizability given here refers to the new Gribov-Zwanziger action in 4 d , itis worth to mention that it immediately generalizes to the case of the Refined Gribov-Zwanziger action [8, 9, 10],both in 4 d and in 3 d [43]. Acknowledgments
The Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq-Brazil), the Faperj, Fundac¸ ˜ao deAmparo `a Pesquisa do Estado do Rio de Janeiro, the SR2-UERJ, the Coordenac¸ ˜ao de Aperfeic¸oamento de Pessoalde N´ıvel Superior (CAPES) are gratefully acknowledged. D. D. is supported by the Research-Foundation Flanders.L. F. P. is supported by an Alexander von Humboldt Foundation fellowship.
References [1] D. Dudal and S. P. Sorella, Phys. Rev. D , 045005 (2012).[2] V. N. Gribov, Nucl. Phys. B , 1 (1978).[3] R. F. Sobreiro and S. P. Sorella, hep-th/0504095. 134] N. Vandersickel and D. Zwanziger, Phys. Rept. , 175 (2012).[5] I. M. Singer, Commun. Math. Phys. , 7 (1978).[6] D. Zwanziger, Nucl. Phys. B , 513 (1989).[7] D. Zwanziger, Nucl. Phys. B , 477 (1993).[8] D. Dudal, S. P. Sorella, N. Vandersickel and H. Verschelde, Phys. Rev. D , 071501 (2008).[9] D. Dudal, J. A. Gracey, S. P. Sorella, N. Vandersickel and H. Verschelde, Phys. Rev. D , 065047 (2008).[10] D. Dudal, S. P. Sorella and N. Vandersickel, Phys. Rev. D , 065039 (2011).[11] A. Cucchieri and T. Mendes, Phys. Rev. Lett. , 241601 (2008)[12] V. G. Bornyakov, V. K. Mitrjushkin and M. Muller-Preussker, Phys. Rev. D , 074504 (2009).[13] D. Dudal, O. Oliveira and N. Vandersickel, Phys. Rev. D , 074505 (2010).[14] A. Cucchieri, D. Dudal, T. Mendes and N. Vandersickel, Phys. Rev. D , 094513 (2012).[15] O. Oliveira and P. J. Silva, Phys. Rev. D , 114513 (2012).[16] D. Dudal, O. Oliveira and J. Rodriguez-Quintero, Phys. Rev. D , 105005 (2012).[17] M. Stingl, Z. Phys. A , 423 (1996).[18] L. Baulieu, D. Dudal, M. S. Guimaraes, M. Q. Huber, S. P. Sorella, N. Vandersickel and D. Zwanziger, Phys.Rev. D , 025021 (2010).[19] A. Windisch, M. Q. Huber and R. Alkofer, Phys. Rev. D , 065005 (2013).[20] D. Dudal, M. S. Guimaraes and S. P. Sorella, Phys. Rev. Lett. , 062003 (2011).[21] C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys. , 477 (1994).[22] M. Bhagwat, M. A. Pichowsky and P. C. Tandy, Phys. Rev. D , 054019 (2003).[23] L. Baulieu, M. A. L. Capri, A. J. Gomez, V. E. R. Lemes, R. F. Sobreiro and S. P. Sorella, Eur. Phys. J. C ,451 (2010).[24] D. Dudal, M. S. Guimaraes, L. F. Palhares and S. P. Sorella, arXiv:1303.7134 [hep-ph].[25] S. Benic, D. Blaschke and M. Buballa, Phys. Rev. D , 074002 (2012).[26] K. Fukushima and K. Kashiwa, arXiv:1206.0685 [hep-ph].[27] R. Aouane, V. G. Bornyakov, E. M. Ilgenfritz, V. K. Mitrjushkin, M. Muller-Preussker and A. Sternbeck,Phys. Rev. D , 034501 (2012).[28] A. Cucchieri and T. Mendes, PoS FACESQCD , 007 (2010).[29] N. Maggiore and M. Schaden, Phys. Rev. D , 6616 (1994).[30] L. Baulieu and S. P. Sorella, Phys. Lett. B , 481 (2009).[31] S. P. Sorella, Phys. Rev. D , 025013 (2009). 1432] D. Dudal and N. Vandersickel, Phys. Lett. B , 369 (2011).[33] D. Dudal, S. P. Sorella, N. Vandersickel and H. Verschelde, Phys. Rev. D , 121701 (2009).[34] M. A. L. Capri, A. J. Gomez, M. S. Guimaraes, V. E. R. Lemes, S. P. Sorella and D. G. Tedesco, Phys. Rev.D , 105019 (2010).[35] M. A. L. Capri, A. J. Gomez, M. S. Guimaraes, V. E. R. Lemes, S. P. Sorella and D. G. Tedesco, Phys. Rev.D , 105001 (2011).[36] L. von Smekal, D. Mehta, A. Sternbeck and A. G. Williams, PoS LAT , 382 (2007).[37] L. von Smekal, M. Ghiotti and A. G. Williams, Phys. Rev. D , 085016 (2008).[38] J. Serreau and M. Tissier, Phys. Lett. B , 97 (2012).[39] P. Lavrov, O. Lechtenfeld and A. Reshetnyak, JHEP , 043 (2011).[40] O. Piguet and S. P. Sorella, Lect. Notes Phys. M , 1 (1995).[41] G. Barnich, F. Brandt and M. Henneaux, Phys. Rept. , 439 (2000).[42] A. Blasi, O. Piguet and S. P. Sorella, Nucl. Phys. B , 154 (1991).[43] D. Dudal, J. A. Gracey, S. P. Sorella, N. Vandersickel and H. Verschelde, Phys. Rev. D78