Renormalizability of the Center-Vortex Free Sector of Yang-Mills Theory

PHYSICAL REVIEW D 101, 085007 (2020)

Renormalizability of the center-vortex free sector of Yang-Mills theory

† ‡ D. Fiorentini ,* D. R. Junior, L. E. Oxman , and R. F. Sobreiro § UFF—Universidade Federal Fluminense, Instituto de Física, Campus da Praia Vermelha, Avenida Litorânea s/n, 24210-346 Niterói, RJ, Brasil.

(Received 11 February 2020; accepted 30 March 2020; published 17 April 2020)

In this work, we analyze a recent proposal to detect SUðNÞ continuum Yang-Mills sectors labeled by center vortices, inspired by Laplacian-type center gauges in the lattice. Initially, after the introduction of appropriate external sources, we obtain a rich set of sector-dependent Ward identities, which can be used to control the form of the divergences. Next, we show the all-order multiplicative renormalizability of the center-vortex free sector. These are important steps towards the establishment of a first-principles, well-defined, and calculable Yang-Mills ensemble.

DOI: 10.1103/PhysRevD.101.085007

I. INTRODUCTION was obtained in Euclidean spacetime [12,13], which provides a calculational tool similar to the one used in As is well known, the Fadeev-Popov procedure to the perturbative regime. Beyond the linear covariant quantize Yang-Mills (YM) theories [1], so successful in gauges, many efforts were also devoted to the maximal making contact with experiments at high energies, cannot Abelian gauges; see Ref. [14] and references therein. BRST be extended to the infrared regime [2,3]. In covariant invariance is an important feature to have predictive power gauges, this was established by Singer’s theorem [4]: for (renormalizability), as well as to show the independence of any gauge fixing, there are orbits with more than one gauge observables on gauge-fixing parameters. Another interest- field satisfying the proposed condition. In geometrical ing feature of the GZ approach is that the gluon Green’s language, the YM principal bundle has a topological functions get drastically modified in the infrared, pointing obstruction, which makes it impossible to find a global to a destabilization of the perturbative regime. The pertur- section f in the complete gauge field configuration bative pole that would correspond to an asymptotic mass- space fAμg. less particle is replaced by an infrared suppressed behavior. A way to cope with this problem was extensively studied However, as the Green’s functions are not gauge invariant in the last few decades, mostly in the Landau and linear objects, it is not clear how to define gluon confinement. A covariant gauges. This is the (refined) Gribov-Zwanziger related discussion regards the presence of complex poles in (GZ) approach [5–10], where the configurations to be path the gluon Green’s functions, which led to the search for integrated are restricted so as to avoid infinitesimal copies. correlators of gauge invariant operators only displaying In this respect, it is worth mentioning that this region is not physical poles [15]. In addition to the GZ treatment, free from copies, since it still contains those related with alternative analytical approaches have been used to deter- finite gauge transformations. In fact, the existence of finite mine correlation functions displaying complex poles; see Gribov copies inside the Gribov region is discussed in for instance Refs. [16–20]. Ref. [11] and references therein, where it is argued about A deeper issue is how to get closer to the confinement of the relation between the Gribov region and the fundamental quark probes in pure YM theories, whose order parameter modular region, the latter being free of copies. Along this is the Wilson loop. This phenomenon is associated with not line, a Becchi-Rouet-Stora-Tyutin (BRST) invariant action only a linearly rising potential but also the formation of a flux tube (see Refs. [21,22] and references therein) with transverse collective modes [23,24]. This is a physical *[email protected] † [email protected] object on its own, which is well beyond the language of ‡ ’ [email protected] Green s functions and the usual Feynman diagrams. Then, §[email protected] a fundamental question is if there is a first-principles formulation of YM theories which allows one to envisage Published by the American Physical Society under the terms of a connection with a confining flux tube. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to In this regard, since the very beginning of gauge theories, the author(s) and the published article’s title, journal citation, we were used to picking a gauge fixing and then perform- and DOI. Funded by SCOAP3. ing the calculations for this choice. When confronted with

2470-0010=2020=101(8)=085007(11) 085007-1 Published by the American Physical Society FIORENTINI, JUNIOR, OXMAN, and SOBREIRO PHYS. REV. D 101, 085007 (2020)

Singer’s theorem, this naturally led to a restriction on fAμg Configurations with regular S½A are in the same sector, to eliminate the associated Gribov copies. However, we which may be labeled by S0 ≡ I. This sector is expected to could try to resolve the obstruction possed by this theorem be the relevant one for describing weakly interacting in a different way. While Singer’s theorem states that it is particles in the UV regime. On the other hand, in the region impossible to construct a global section f, it does not rule around a distribution of closed surfaces, smooth variables Aμ out the possibility of covering fAμg with local regions Vα, may involve large gauge transformations with multivalued each one having its own well-defined local section fα, angles in their formulation. Accordingly, S½A and the for Aμ ∈ Vα. associated choice of S0 is characterized by a distribution In Ref. [25], an approach motivated by lattice center of center vortices and correlated monopoles with non- gauges was proposed in the continuum. There, a tuple of Abelian degrees of freedom (d.o.f.) [26]. Then, from this perspective, the Singer no-go theorem appears as the auxiliary adjoint fields ψ I ∈ suðNÞ, with I being a flavor index, were initially introduced by means of an identity, thus foundational basis for the YM theory to give place to a keeping the pure YM dynamics unchanged. The identity was YM ensemble. Concerning the possible relation with quark constructed using equations of motion that correlate Aμ with confinement, center vortices have been considered as relevant degrees to describe the infrared properties of YM ψ I. Next, a polar decomposition in terms of a “modulus” theory [27–29]. In the lattice, at asymptotic distances, they tuple qI and phase S was applied to the fields ψ I: account for a Wilson loop area law with N-ality [30–40]. −1 ψ I ¼ SqIS ;S∈ SUðNÞð1Þ More recently, phenomenological ensembles containing the possible defects that characterize S0 led to effective (for the definition of modulus, see Sec. II). As this was done models that can accommodate the properties of the confining U covariantly, a gauge transformed field Aμ is mapped to flux tube [26]. The aim of this work is to advance towards the establish- S½AU¼US½A: ð2Þ ment of a Yang-Mills ensemble from first principles. In this

respect, the renormalizability of each sector ZðS0Þ is Although Aμ is smooth, S½A generally contains defects, important to having well-defined calculable partial con- which cannot be eliminated by gauge transformations. This tributions. In Ref. [25], we showed that each VðS0Þ has its makes it impossible to define a global reference phase. own BRST transformation. Although the algebraic struc- Instead, we have to split fAμg into sectors VðS0Þ, formed by ture does not change from sector to sector, the regularity those Aμ that can be gauge transformed to some Aμ with conditions of the ghosts do change. This is needed in S½A¼S0. The condition for copies in the orbit of Aμ is order for the regularity conditions at the defects of S0 to be U ψ S½A ¼S0,whichduetoEq.(2) implies U ≡ I. Therefore, BRST invariant. That is, the field modes to expand I are on each sector, there are no copies. Indeed, the sectors VðS0Þ inherited by the ghosts. Here, we were able to prove the all- provide a partition of the configuration space: order multiplicative renormalizability of the center-vortex free sector, which we expect to be essentialy perturbative. A ∪ V V ∩ V 0 ∅ ≠ 0 f μg¼ S0 ðS0Þ; ðS0Þ ðS0Þ¼ ; if S0 S0; In other sectors, most of the Ward identities remain valid, but the ghost equation should be modified by sector- ð3Þ dependent terms. Additionaly, new counterterms arise located at the center-vortex guiding centers. The associated where the labels S0 are representatives of the classes obtained from the equivalence relation, difficulties for establishing the renormalizability of these sectors will be dealt with in a future contribution. S0 ∼ S iff S0 ¼ US; with regular U: ð4Þ II. THE YM QUANTIZATION V In particular, the different ðS0Þ cannot contain physically ON THE V(S0) SECTORS equivalent gauge fields. Therefore, the partition function and As proposed in Ref. [25], the correlation between the the average of an observable O over fAμg become a sum gauge field Aμ and the phase SðAÞ, used to fix the gauge on over partial contributions: each sector, can be done by means of the solutions to the X X Z S0 equations of motion: Z ¼ Z ; hOi ¼ hOi ð Þ ; ð5Þ YM ðS0Þ YM ðS0Þ Z ZYM S0 S0 δS 1 aux 0 abψ b acψ c Z a ¼ ;Saux ¼ Dμ I Dμ I þ Vaux ; δψ x 2 − I A SYM ZðS0Þ ¼ ½D μe ; ab ab acb c V Dμ ≡ δ ∂μ þ gf Aμ: ð7Þ ðS0ÞZ 1 − A SYM We initially choose the auxiliary potential V ψ to be the hOiðS0Þ ¼ ½D μOe : ð6Þ auxð Þ Z V ðS0Þ ðS0Þ most general one constructed in terms of antisymmetric

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−1 structure constants, and containing up to quartic terms. ψ I ¼ SqIS ; ½uI;qI¼0: ð10Þ Then, besides color symmetry, we take I ¼ 1; …;N2 − 1 and impose adjoint flavor symmetry: At the quantum level, to filter the configurations μ2 κ Aμ ∈ VðS0Þ, an identity was constructed from a Dirac ψ ψ aψ a abc ψ aψ bψ c Vauxð Þ¼ 2 I I þ 3 f fIJK I J K delta functional over the scalar field equations of motion, λ together with flavored ghosts cI [25]. Then, ψ I was γabcd ψ aψ bψ c ψ d þ 4 IJKL I J K L; ð8Þ restricted to have a polar decomposition with a local phase of the form S ¼ US0, where U is regular and S0 contains where γ is general a color and flavor invariant tensor, i.e., center-vortex and correlated monopole defects. In particular, by using a reference tuple uI in the vacua manifold, we aa0 bb0 cc0 dd0 γa0b0c0d0 γabcd → R R R R IJKL ¼ IJKL; require the asymptotic condition qI uI, plus appropriate II0 JJ0 KK0 LL0 abcd abcd regularity conditions at the defects of S0 (see Sec. III A). R R R R γ 0 0 0 0 γ ;R∈ Ad SU N ; 9 I J K L ¼ IJKL ð ð ÞÞ ð Þ Next, the gauge fixing in this sector was implemented by U changing variables to Aμ, Aμ ¼ Aμ . Thus, in the gauge- composed of combinations of antisymmetric structure A constants and Kronecker deltas, while μ and κ are mass fixed expressions, we can write Aμ in the place of μ, and parameters. In order for the procedure to be well defined, a potential with minima displaying SUðNÞ → ZðNÞ is essen- −1 ζ ≡ S0q S ð11Þ tial. This pattern, which can be easily accommodated by I I 0 N2 − 1 flavors [41], produces a strong correlation between A A μ and the local phase S½ . As in Ref. [25], a tuple qI ¼ in the place of ψ I. The presence of S0 occurs because, when −1 S ψ IS is called the modulus of ψ I,ifS ∈ SUðNÞ performing a shift of group variables within the Fadeev- 2 minimizes ðqI − uIÞ , where uI ∈ suðNÞ is a reference Popov procedure, it is impossible to eliminate S0 with a tuple of linearly independent vectors. Then, if we perform regular gauge transformation. The pure modulus condition η ζ 0 an infinitesimal rotation of qI with generator X, we get can also be written as ½ I; I¼ , using the classical η ≡ −1 ðqI − uI; ½qI;XÞ ¼ 0 for every X ∈ suðNÞ. That is, the (background) field I S0uIS0 . polar decomposition and the modulus condition become The full Yang-Mills action in this gauge is then given by

Z Σ ab ¯b ac c ab b acζc κ abc aζbζc − 2¯aζb c ¼ SYM þ fðDμ cI ÞDμ cI þðDμ bI ÞDμ I þ fIJKf ðbI J K cI KcJÞ x λγabcd aζbζc ζd 3¯a bζc ζd μ2 ¯a a aζb abc aηbζc ecd eba ¯aηbζc d abc ¯aηb c þ IJKLðbI J K L þ cI cJ K LÞþ ðcI cI þ bI I Þþif b I I þ f f c I I c þ if c I cI g; ð12Þ

Z Z 1 a 2 a a 2 S ¼ ð∂μAν Þ − ∂μAν ∂νAμ − ab ¯b acζc ¯a μ ζ κ IJK abcζbζc YM 2 S ¼ SYM s ½Dμ cI Dμ I þ cI ð I þ f f J KÞ x x abc a b c c þ gf AμAν ð∂μAν − ∂νAμÞ þ γabcd λc¯aζbζc ζd þ ifabcc¯aηbζc: ð15Þ IJKL I J K L I I 2 g abc dec a b d e þ 2 f f AμAν AμAν : ð13Þ A. Some remarks about the BRST invariance The action (12) is invariant under the following BRST Let us focus on the flavor sector, which implements the transformations [25]: correlation between the gauge field and the auxiliary adjoint scalar fields: i i a ab b − abc b c Z sAμ ¼ Dμ c ;sc¼ 2 f c c ; g ˜ ab ¯b ac c μ2 ¯a a aζa Sf ¼ SYM þ ½Dμ cI Dμ cI þ ðcI cI þ bI I Þ sc¯a ¼ −ba;sba ¼ 0; x κ abc aζbζc − 2¯aζb c ζa abcζb c a ¯a − abc ¯b c − a þ fIJKf ðbI J K cI KcJÞ s I ¼ if I c þ cI ;scI ¼ if cI c bI ; λγabcd aζbζc ζd 3¯a bζc ζd ab b acζc abc b c a − abc b c þ IJKLðbI J K L þ cI cJ K LÞþDμ bI Dμ I ; sbI ¼ if bI c ;scI ¼ if cI c : ð14Þ ð16Þ Conveniently, the gauge fixing terms can be written as a BRST-exact term, so that the action [Eq. (12)] is equivalent to which can be written as

085007-3 FIORENTINI, JUNIOR, OXMAN, and SOBREIRO PHYS. REV. D 101, 085007 (2020) Z Z δ δ ˜ a SH ab b ac c a VH S S − s c¯ S − s Dμ c¯ Dμ ζ c¯ ; 17 f ¼ YM I δψ a ¼ YM I I þ δψ a ð Þ x I ψ¼ζI x I ψ¼ζI with δV H μ2ζa κfIJKfabcζbζc λγabcd ζbζc ζd : 18 δψ a ¼ I þ J K þ IJKL J K L ð Þ I ψ¼ζI

As discussed in Refs. [42,43], in order to control the gauge-parameter independence of invariant correlation functions, it is convenient to extend the action of the BRST operator s on the gauge-fixinig parameters as

sμ2 ¼ U2;sU2 ¼ 0; sκ ¼ K;sK ¼ 0; sλ ¼ Λ;sΛ ¼ 0; ð19Þ where ðU2; K; ΛÞ are constant Grassmann parameters with ghost number 1 and mass dimensions 2, 1, and 0, respectively. Thus, since physical quantities must belong to the (nontrivial) cohomology of the s operator, all the gauge-fixing parameters will be on the trivial sector of the cohomology of this operator, in accordance with the BRST doublet theorem [43]. The complete classical action Sf is given by Z ˜ − 2 ¯aζa K IJK abc ¯aζbζc Λγabcd ¯aζbζc ζd Sf ¼ Sf ðU cI I þ f f cI J K þ IJKLcI J K LÞ; ð20Þ x which is invariant under the extended transformations in Eq. (19). The full action in the flavor sector is the following: Z Σ ab ¯b ac c ab b acζc μ2 ¯a a aζb κ abc aζbζc − 2¯aζb c I-sec ¼ SYM þ fðDμ cI ÞDμ cI þðDμ bI ÞDμ I þ ðcI cI þ bI I Þþ fIJKf ðbI J K cI KcJÞ x λγabcd aζbζc ζd 3¯a bζc ζd − 2 ¯aζa − K IJK abc ¯aζbζc − Λγabcd ¯aζbζdζe þ IJKLðbI J K L þ cI cJ K LÞ U cI I f f cI J K IJKLcI J I Jg: ð21Þ

III. THE COMPLETE ACTION AND ITS SYMMETRIES Owing to the nonlinearity of the BRST transformations [Eq. (14)], the renormalization of some composite operators is necessary. With this purpose, we need to couple a source to each nonlinear variation of the fields, which can be done in a BRST-exact manner by imposing the s variation of all these sources to be zero, i.e., Z Σð1Þ a a ¯ a a a ζa ¯ a a a ¯a a a sources ¼ ½KμðsAμÞþC ðsc ÞþQI ðs I ÞþLI ðscI ÞþLI ðscI ÞþBI ðsbI Þ Zx i 1 a ab b − ¯ a abc b c a abcζb c a − abc ¯ a b c − a abc ¯b c a abc a b c ¼ KμDμ c iC f c c þ QI ðif I c þ cI Þ if LI cI c þ LI ðif cI c þ bI Þþif BI bI c ; x g 2 ð22Þ

a ¯ a a ¯ a a a 0 ab ab with sðKμ; C ;QI ; LI ;LI ;BI Þ¼ . An additional pair of external sources ðMI ;NI Þ, to form a convenient composite operator, will also be necessary: Z Z 2 δΣ Σð Þ ¼ s Mabc¯bζb ¼ Nabc¯aζb − Mabbaζb − Mabc¯a : ð23Þ sources I I I I I I I δ b x x QI

These sources will be used to restore the Ward identities associated with the ghost and antighost equations, a key step towards the algebraic proof of renormalizability. After the introduction of the external sources, for the complete extended classical action Σ, one has

085007-4 RENORMALIZABILITY OF THE CENTER-VORTEX FREE … PHYS. REV. D 101, 085007 (2020) Z Σ ab ¯b ac c ab b acζc κ abc aζbζc − 2¯aζb c ¼ SYM þ ðDμ cI ÞDμ cI þðDμ bI ÞDμ I þ fIJKf ðbI J K cI KcJÞ x μ2 ¯a a aζb λγabcd aζbζc ζd 3¯a bζc ζd þ ðcI cI þ bI I Þþ IJKLðbI J K L þ cI cJ K LÞþ − 2 ¯aζa − K IJK abc ¯aζbζc − Λ abc cde ¯aζbζdζe abc aηbζc ¯aηb c U cI I f f cI J K f f cI J I J þ if ðb I I þ c I cI Þ 1 ecd eba a b c d i a ab b ¯ a abc b c a abc b c a abc ¯ a b c þ f f c¯ η ζ c þ KμðDμ c Þ − iC f c c þ Q ðif ζ c þ c Þ − if L c c þ I I g 2 I I I I I δΣ − Laðifabcc¯bcc þ baÞþifabcBabbcc þ Nabc¯aζb − Mabbaζb − Mabc¯a ; ð24Þ I I I I I I I I I I δ b QI

which is invariant under the full set of BRST transformations transformations. That is, for these roots we must impose α 0 sqI ¼ on the vortex worldsheet. A way to impose these a i ab b i abc b c conditions is to add to the action the following term: sAμ ¼ Dμ c ;sc¼ − f c c ; g 2 Z Z sc¯a ¼ −ba;sba ¼ 0; σ σ δ − ¯ σ σ λαζα ξα ζα Sb:c: ¼ d 1d 2 ðx xð 1; 2ÞÞð I I þ I s I Þ ζa abcζb c a ¯a − abc ¯b c − a Zx s I ¼ if I c þ cI ;scI ¼ if cI c bI ; λαζα ξα ζα sb ¼ ifabcbbcc;sca ¼ −ifabccbcc; ¼ Jð I I þ I s I Þ: ð28Þ I I I I x sμ2 ¼ U2;sU2 ¼ 0; ξα λα λα 0 The fields I , I are Lagrange multipliers satisfying s I ¼ sκ ¼ K;sK ¼ 0; ξα −λα ¯ σ σ and s I ¼ I , and xð 1; 2Þ is a parametrization of the sλ ¼ Λ;sΛ ¼ 0; vortex worldsheet. To account for a general sector, we should consider a general source JðxÞ localized on the sMab ¼ Nab;sNab ¼ 0; I I I various magnetic defects. UsingR these BRST transforma- ¯ a a a ¯ a a a 0 ξαζα sC ¼ sKμ ¼ sLI ¼ sLI ¼ sQI ¼ sBI ¼ : ð25Þ tions, we can write Sb:c: ¼ x sðJ I I Þ, and the full action in this sector turns out to be

ΣS0 Σ A. Regularity conditions in sectors labeled ¼ þ Sb:c: ¼ sðsomethingÞ; ð29Þ by magnetic defects As an example, let us consider a sector labeled by a where Σ is defined in Eq. (24). This is an important center vortex, In this case, the singular gauge transforma- construction, since the quantum action principle [44–48] iχ2Nω T tion S0 may be given by S0 ¼ e p p, with ω being a can be applied to the action in this form. weight of the fundamental representation (see Ref. [25] and 1 … − 1 references therein). The generators Tp, p ¼ ; ;N , B. Ward identities belong to the Cartan sector of suðNÞ, while χ is multi- In the center-vortex free sector, the action and BRST valued when we go around the vortex worldsheet. The color a transformations are those of Eqs. (24) and (25), respec- components of the field qI are defined by qI ¼ qI Ta, 2 tively, with ζ and η replaced by q and u , respectively. a ¼ 1…N − 1. The Lie basis consists of N − 1 Cartan I I I I † We now display the rich set of Ward identities enjoyed by elements Tq, and a pair of elements Eα, E−α ¼ Eα for this action: α su ζ each positive root of ðnÞ. To compute I in Eq. (11), (1) The Slavnov-Taylor identity: we can use Z δΣ δΣ δΣ δΣ δΣ δΣ −1 SðΣÞ¼ þ þ S0TqS0 ¼ Tq; ð26Þ δ a a δ ¯ a δ a δ a δ¯a x Kμ Aμ LI cI LI cI δΣ δΣ δΣ δΣ δΣ δΣ −1 2 ω α χ 2 ω α χ þ þ þ þ S0EαS0 ¼ cosð N · Þ Eα þ sinð N · Þ Eα: ð27Þ δQa δqa δBa δb δC¯ a δca I I I I δΣ δΣ δΣ Therefore, to ensure regularity, the components qα, such − ba þ Nab þ U2 I δc¯a I δMab δμ2 that α · β ≠ 0, must vanish on the vortex worldsheet. I δΣ δΣ However, as argued in Ref. [25], we must make sure þ K þ Λ ¼ 0: ð30Þ that this regularity condition is invariant under BRST δκ δλ

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In view of the algebraic characterization of the counterterm, we introduce the so-called linearized Slavnov-Taylor operator BΣ [43], defined as Z δΣ δ δΣ δ δΣ δ δΣ δ δΣ δ δΣ δ B Σ ¼ δ a a þ δ a δ a þ δ ¯ a δ a þ δ a δ ¯ a þ δ a δ¯a þ δ¯a δ a x Kμ Aμ Aμ Kμ LI cI cI LI LI cI cI LI δΣ δ δΣ δ δΣ δ δΣ δ δΣ δ δΣ δ þ þ þ þ þ þ þ δQa δqa δqa δQa δBa δb δba δBa δC¯ a δca δca δC¯ a I I I I I I I I δ δ δ δ δ − ba þ Nab þ U2 þ K þ Λ ; ð31Þ δ¯a I δ ab δμ2 δκ δλ c MI

2 which is nilpotent, BΣ ¼ 0. (6) Global flavor symmetry: (2) Gauge fixing condition: QΣ ¼ 0; ð39Þ δΣ ¼ ifabcubqc − Mabqb: ð32Þ δba I I I I where we have defined the flavor charge operator

(3) The antighost equation: δ δ δ δ δ Q ≡ qa − ba − c¯a ca − ua I δ a I δ a I δ¯a þ I δ a I δ a qI bI cI cI uI G¯ aΣ ab b ¼ NI qI ; ð33Þ δ δ δ − Qa Ba La I δ a þ I δ a þ I δ a þ QI BI LI with the antighost operator given by δ δ δ δ δ − L¯ a − κ − 2λ − K − 2Λ I δL¯ a δκ δλ δK δΛ ¯ δ δ δ I Ga ¼ þ Mab − ifabcub : ð34Þ δ δ δ¯a I δ b I δ c ab ab c QI QI − N − M : ð40Þ I δ ab I δ ab NI MI (4) The ghost equation: This symmetry can be used to define a new con- served quantum number in the auxiliary flavor GaΣ abc ¯ b c b c ¯ b c b ¯c b c ¼ if ðC c þ QI qI þ LI cI þ LI cI þ BI bI Þ sector, the Q charge. Thus, this symmetry forbids i ab b combinations of composite fields with nonvanishing þ Dμ Kμ; ð35Þ g Q charge. The corresponding values of this charge, for each field, source and parameter, is assigned in a with the ghost operator given by similar way to the ghost numbers [43]. (7) Exact rigid symmetry: δ δ Ga ¼ − fabcfcmnun : ð36Þ RΣ ¯ a a a ¯a − a a δ a I δ mb ¼ LI cI þ LI cI qI QI ; ð41Þ c NI

(5) Ghost number equation: where δ δ δ δ N Σ 0 R ¼ c¯a þ qa − ifabcua − Ba gh ¼ ; ð37Þ I δ a I δ a I δ bc I δ ¯ a bI cI NI LI Z δ δ δ δ δ δ δ δ þ L¯ a − κ − 2λ − Mab : ð42Þ N d4x ca − c¯a ca − c¯a I δQa δK δΛ I δNab gh ¼ I δ a I δ¯a þ δ a δ¯a I I cI cI c c δ δ δ þ U2 þ K þ Λ þ δU2 δK δΛ Notice that the right-hand sides of the broken Ward — δ ¯ δ ¯ δ δ identities that is, the gauge fixing condition, the antighost − Ka − 2Ca − 2La − Qa — δKa δC¯ a I δL¯ a I δQa equation, and the ghost equation are linear in the quantum I I δ δ fields. This is compatible with the quantum action princi- − Ba þ Nab : ð38Þ ple; i.e., they remain classical in the perturbative expansion I δ a I δ ab BI NI [43]. For further use, the quantum numbers of all fields,

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TABLE I. The field quantum numbers. G¯ a Σ abζb G¯ a G¯ a S0 ¼ NI I ; S0 ¼ : ð45Þ Fields Ab c c¯ q u ccb¯ I I I I I (4) Ghost number equation: Dimension 1 1 1 1 1 1 2 0 2 Z Ghost number 0 0 1 −1 00−1 10 δ N S0 ΣS0 0; N S0 ≡ N − d4x ξa : 46 Q charge 0 −1 1 −1 1 −1 000 gh ¼ gh gh I δξa ð Þ Nature B B F F B B F F B I (5) Global flavor symmetry:

TABLE II. The quantum numbers of external sources. δ δ QS0 Σ 0; QS0 ≡ Q − ξa − λa : 47 ¼ I δξa I δλa ð Þ ¯ ¯ I I Sources CKμ LI LI QI BI NI MI Dimension 4 3 3 3 3 3 1 1 (6) Exact rigid symmetry: Ghost number −2 −1 −2 0 −1 −1 10 Q −1 −1 −1 −1 charge 0 0 1 1 RS0 ΣS0 ¼ L¯ aca þ Lac¯a − ζaQa; Nature B F B B F F F B I I I I I I δ RS0 ≡ R ξa : 48 þ I δλa ð Þ I sources, and parameters are displayed in Tables I, II, and III (where B stands for bosonic and F for fermionic statistics). IV. RENORMALIZABILITY OF THE In a general sector, the action (29) satisfies most of these CENTER-VORTEX FREE SECTOR Ward identities, up to minor modifications. More precisely, Σ the following Ward identities are satisfied: In order to prove that the action [cf. Eq. (24)]is (1) The Slavnov-Taylor identity: multiplicatively renormalizable in the center-vortex free sector, we follow the algebraic renormalization setup [43]. Z δΣS0 δΣS0 δΣS0 δΣS0 δΣS0 δΣS0 By means of the Ward identities previously derived, we SS0 ΣS0 characterize the most general invariant local counterterm ð Þ¼ δ a a þ δ ¯ a δ a þ δ a δ¯a x Kμ Aμ LI cI LI cI Σc:t:., which can be freely added to the starting action Σ. δΣS0 δΣS0 δΣS0 δΣS0 According to the quantum action principle [44–48], Σc:t:.is þ δ a δζa þ δ a δ an integrated local polynomial in the fields and external QI I BI bI sources of dimension bounded by four, and it has the same δΣS0 δΣS0 δΣS0 δΣS0 Σ þ − ba þ Nab quantum numbers as the starting action . Further, we δ ¯ a δ a δ¯a I δ ab c:t: C c c MI require that the perturbed action Σ þ ϵΣ . satisfy the same 1 δΣS0 δΣS0 δΣS0 Ward identities and constraints of Σ [43], to the first order U2 K Λ þ δμ2 þ δκ þ δλ in the perturbation parameter ϵ. In this manner, we obtain the following set of constraints: δΣS0 − λa ¼ 0: ð43Þ I δξa c:t: ¯ a c:t: a c:t: I BΣΣ ¼ 0; G Σ ¼ 0; G Σ ¼ 0; N Σc:t: ¼ 0; QΣc:t: ¼ 0; RΣc:t: ¼ 0: ð49Þ (2) Gauge fixing condition: gh The first one means that Σc:t:. belongs to the cohomology of δΣS0 the nilpotent linearized operator BΣ, in the space of ¼ ifabcubζc − Mabζb: ð44Þ δba I I I I integrated local polynomials in the fields, sources, and parameters bounded by dimension four. From the general (3) The antighost equation. Note that the antighost results on the BRST cohomolgy of Yang-Mills theories, it operator is the same as that of the center-vortex follows that Σc:t:. can be decomposed as free sector: c:t: ð−1Þ Σ ¼ Δ þ BΣΔ ; ð50Þ

TABLE III. The quantum numbers of parameters. −1 where Δ denotes a four-dimensional integrated quantity Parameters μ2 κλU2 K Λ in the fields, sources, and parameters with ghost number −1 Dimension 2 1 0 2 1 0 Ghost number 0 0 0 1 1 1 1The algebraic renormalization technique [43] is a recursive Q charge 0 −1 −2 0 −1 −2 method. Hence, to show the renormalizability of a theory at first Nature B B B F F F order means that the proof is valid to all orders in perturbation theory.

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ð−1Þ “ ” and vanishing flavor charge. The term BΣΔ in the where the label 0 indicates bare quantities. By conven- equation above corresponds to the trivial solution—i.e., to tion, we choose the renormalization factors as the exact part of the cohomology of the BRST operator. On the other hand, the quantity Δ identifies the nontrivial ϵ F 1=2F 1 F solution, namely the cohomology of BΣ, meaning that 0 ¼ ZF ¼ þ 2 zF ; ˜ ˜ Δ ≠ BΣΔ, for some local integrated polynomial Δ. In order J J 1 ϵ J for the action to be multiplicatively renormalizable, the 0 ¼ ZJ ¼ð þ zJ Þ : ð52Þ counterterm must be reabsorbed in the classical action by a F multiplicative renormalization of the fields ( ), sources, The coefficients fzF ;zJ g are linear combinations of the free and parameters (J ): parameters in the counterterm, fa0;bigi¼1;… (see below). By direct inspection, and with the help of Tables I, II,andIII, Σ F J ϵ2 Σ F J ϵΣc:t: F J 0½ 0; 0þOð Þ¼ ½ ; þ ½ ; ; one can find that the most general counterterm with ¯ ¯ Q F ¼fAμ;qI; cI;cI;bI;b;c; cg; vanishing charge is of the form ¯ ¯ 2 2 J ¼fKμ;LI; LI;QI;BI;NI;MI; C; g; μ ; κ; λ;U ; K; Λg; c:t: ð−1Þ Σ ¼ a0S þ BΣΔ ; ð53Þ ð51Þ YM

Z Δð−1Þ ¯ a a a a ¯ a a abc ¯ a b c abc a bc a a a a ¼ ½b1C c þ b2KμAμ þ b3LI cI þ b4f LI qI c þ b5 BI MI þ b6LI cI þ b7QI qI x a a a a abc ∂ a ¯b c abcd a b ¯c d ¯a∂2 a ∂ a ¯a þ b8BI uI þ b9BI bI þ b10f ð μAμÞcI qI þ b11 AμAμcI qI þ b12cI qI þ b13ð μAμÞc abcd a ¯b c d abc a ¯b c abcd a ¯b ¯c d abc a ¯b ¯c þ b14;IJKLbI cJqKqL þ b15f bI c qI þ b16;IJKLcI cJcKqL þ b17f cI cI c abcd ¯a b c d IJK abcκ¯a b c abcd λ¯a b c d þ b18;IJKLcI uJqKqL þ b19f f cI qJqK þ b20;IJKL cI qJqKqL abcd ¯a b ¯c d abc a b ¯c abc ¯a b c μ2 ¯a a þ b21 cI qI c c þ b22f qI uI c þ b23f cI qI b þ b24 cI qI abc ¯a b c abc ¯a ¯b c abcde ¯a ¯b c d e ¯a a þ b25f c AμAμ þ b26f c c c þ b27;IJKLcI cJqKqLc þ b28c b abcde ab ¯c d e abcd ab ¯c c þ b29;IJKLMI cJqKqL þ b30 MI c qI : ð54Þ

Terms containing the parameters of the model are forbidden in Δ due to their BRST doublet structure in Eq. (19). Also, as in usual Yang-Mills theories gauged in a linear covariant way ala` Faddeev-Popov, the linear and quadratic terms in Aμ mixed with other fields vanish because of the BRST invariance. Any other possibility results in a combination of flavored fields with mass dimension four and vanishing ghost number and Q charge, all of them being BRST-exact forms. Thus, we conclude that the nontrivial cohomology of the present model is the usual cohomology corresponding to the Yang-Mills theories, namely

Δ ¼ a0SYM: ð55Þ

After a long but straightforward computation, the constraints imposed by the Ward identities [Eq. (49)] are the following:

b1 ¼ … ¼ b9 ¼ b13 ¼ b14 ¼ b16 ¼ b18 ¼ b22 ¼ b25 ¼ … ¼ b30 ¼ 0; cban mcn mba mbn cma b21 ¼ ib23ðf f þ f f Þ;

b15 ¼ b23 ¼ −b17 ¼ b24;

b10 ¼ gb12; abcd 2 acα αbd b11 ¼ −g f f b12; cban mcn mba mbn cma b21 ¼ ib23ðf f þ f f Þ; cbae ca be b30 ¼ −δ δ b7; mna mbcd mba nmcd mca nbmd mda nbcm 0 f b20;IJKL þ f b20;IJKL þ f b20;IJKL þ f b20;IJKL ¼ : ð56Þ

The most general counterterm consistent with all the Ward identities is therefore

085007-8 RENORMALIZABILITY OF THE CENTER-VORTEX FREE … PHYS. REV. D 101, 085007 (2020) Z c:t: a0 a 2 a0 a a a0 abc a b c a0 2 abc cde a b d e Σ ¼ ½ ð∂μAν Þ − ∂νAμ∂μAν þ gf AμAν ∂μAν þ g f f AμAν AμAν x 2 2 2 4 ∂ ¯a∂ a abc ¯a∂ b c abc∂ ¯a b c 2 abe cde a ¯b c d þ b12ð μcI μcI þ gf cI μcI Aμ þ gf μcI AμcI þ g f f AμcI AμcI ∂ a∂ a abc a∂ b c abc∂ a b c 2 abe cde a b c d þ μbI μqI þ gf bI μqI Aμ þ gf μbI AμqI þ g f f AμbI AμqI Þ IJK abc K¯a b c − κ a b c − 2κ¯a b c þ b19f f ð cI qJqK bI qJqK cI cJqKÞ abcd Λ¯a b c d − λ a b c d − 3λ¯a b c d þ b20;IJKLð cI qJqKqL bI qJqKqL cI cJqKqLÞ 2 ¯a a − μ2 a a − μ2 ¯a a þ b24ðU cI qI bI qI cI cI Þ: ð57Þ

Finally, it remains to check if this term can be reabsorbed analyzed, leading to the detection of percolating center through a multiplicative redefinition of the fields, sources, vortices and monopoles as dominant degrees in the infrared. coupling constant, and parameters of the starting action, On the other side, far from the YM fundaments, topological according to Eq. (51). Indeed, there is no contribution that solutions to effective Yang-Mills-Higgs (YMH) models is not already present in the original action [Eq. (24)]. By successfully reproduced asymptotic properties of the con- direct inspection, we obtain the following renormalization fining flux tube [49–51] (see also Refs. [41,52] and factors: references therein). Some connections have been established between these approaches. Center vortices lie on the a0 common boundary of the fundamental modular and zA ¼ a0;zg ¼ − ; 2 Gribov regions [53]. Recently, in Ref. [26],aphenomeno- 0 0 zqI ¼ ;zcI ¼ ; logical ensemble of percolating center-vortex worldsurfaces was generated by emergent gauge fields, which are the z¯ ¼ 2b12;z¼ 2b12; cI bI Goldstone modes in a condensate of center-vortex loops. zK ¼ −b12 − b19;zκ ¼ −b12 − b19; The inclusion of monopoles with non-Abelian d.o.f.’son center vortices was effectively represented by adjoint Higgs zΛ ¼ −b12 − b20;zλ ¼ −b12 − b20; fields. These elements were then related to effective YMH 2 − − 2 − − zU ¼ b12 b25;zμ ¼ b12 b25; models that can accommodate confining flux tubes with N-ality. z ¼ 0;z¯ ¼ 0; c c Then, we may envisage a long road that, starting from 0 0 zb ¼ ;zC¯ ¼ ; YM first principles, leads to an ensemble, and then from the σ ensemble to the confining flux tube. Regarding the possible z − ;z−b12; K ¼ 2 L ¼ transition from YM theory to a YM ensemble, we believe 0 − that a controlled initial step was made in Ref. [25], where a zL¯ ¼ ;zB ¼ b12; continuum version of the lattice Laplacian-type center 0 0 zQ ¼ ;zN ¼ ; gauges was proposed. In this formulation, the theory is defined on infinitely many sectors that give a partition of zM ¼ 0: ð58Þ the whole configuration space. Each sector has its own With these relations we end the proof of the algebraic BRST transformation and invariance, being labeled by a renormalizability in the center-vortex free sector of the distribution of center vortices and correlated monopoles gauge-fixing proposed in Ref. [25]. Particularly, for the with non-Abelian d.o.f.’s. These are precisely the above- renormalization of the gluon field and the coupling con- mentioned elements needed to make contact with effective −1 YMH models and confining flux tubes. stant, we obtained the relation ZA ¼ Zg . Moreover, the In this work, another step to settle the foundations of a Faddeev-Popov ghosts and flavored pair fqI;cIg do not YM ensemble was given. Initially, we pointed to the renormalize, Zc ¼ Zc¯ ¼ Zq ¼ Zc ¼ 1. I I existence of a set of nonintegrated renormalizable Ward identities that control the dependence on the (adjoint) V. CONCLUSIONS flavored auxiliary fields and ghosts. External sources were In the last few decades, various approaches aimed at also added to restore the Faddeev-Popov ghost equation understanding confinement have been extensively explored. and originate a new type of quantum number. Next, we They were based on numerous theoretical ideas, proved the renormalizability to all orders in the center- Monte Carlo simulations, phenomenological, and effective vortex free sector, by exploring its rich set of Ward models. The Gribov-Zwanzinger scenario is among those identities. In contrast to the Landau gauge, the ghost closer to the first principles of SUðNÞ YM theory. In the equation is not integrated, so it is more powerful. It implies lattice, SUðNÞ (Monte Carlo) configurations have also been that the counterterm cannot depend on ca, and that the

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Faddeev-Popov ghosts do not renormalize Zc ¼ Zc¯ ¼ 1, also pointed out that other extensions could effectively which is a very strong nonrenormalization theorem. implement the physically interesting thick center-vortex Another consequence is that the renormalization factors case. The generalized procedure could be applied in such for the gluon field and the coupling constant are not a context, as the field modes in center-vortex sectors −1 independent, ZA ¼ Zg , since the ghost equation eliminates satisfy nontrivial regularity conditions on the correspond- δΣ ing worldsheets, which is one of the properties of the the counterterm A δA. A similar property was obtained in the Abelian sector of the maximal Abelian gauge [54]. thick objects. In particular, we could approach properties These advances encourage further studies about the such as the stiffness and tension of center vortices, thus sectors labeled by magnetic topological degrees. Here, inferring the main properties of the Yang-Mills ensemble. we showed that not only the BRST transformations but In this manner, we could make contact with lattice-based also most Ward identities maintain the same algebraic phenomenological proposals used to describe quark structure. The only exception is the ghost equation, confinement in terms of percolating center vortices with which should be modified by sector-dependent terms. positive stiffness. Additionally, the counterterms would also contain divergences located at the center-vortex guiding centers. In a ACKNOWLEDGMENTS future work, we will investigate renormalizability in these sectors,whichwouldbeusefultocharacterizeanensem- The Conselho Nacional de Desenvolvimento Científico e ble from first principles. In this regard, quantum fluctua- Tecnológico (CNPq), the Coordenação de Aperfeiçoamento tions around configurations of thin vortices were studied de Pessoal de Nível Superior (CAPES), and the Fundação de in Ref. [55], by considering one of the possible self- Amparoa ` Pesquisa do Estado do Rio de Janeiro (FAPERJ) adjoint extensions to treat them. In that reference, it was are acknowledged for their financial support.

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