1 Introduction
###### Abstract

The center-stabilized multiflavor QCD* theories formulated on exhibit both Abelian and non-Abelian confinement as a function of the radius, similar to the Seiberg–Witten theory as a function of the mass deformation parameter. For sufficiently small number of flavors and small , we show occurence of a mass gap in gauge fluctuations, and linear confinement. This is a regime of confinement without continuous chiral symmetry breaking (SB). Unlike one-flavor theories where there is no phase transition in , the multiflavor theories possess a single phase transition associated with breaking of the continuous S. We conjecture that the scale of the SB is parametrically tied up with the scale of Abelian to non-Abelian confinement transition.

SLAC-PUB-13516, FTPI-MINN-09/05, UMN-TH-2735/09

Multiflavor QCD on : Studying Transition

[1mm] From Abelian to Non-Abelian Confinement

M. Shifman and M. Ünsal

William I. Fine Theoretical Physics Institute,

University of Minnesota, Minneapolis, MN 55455

SLAC, Stanford University, Menlo Park, CA 94025

Physics Department, Stanford University, Stanford, CA 94305

## 1 Introduction

In supersymmetric Yang–Mills theories slightly deformed to by a mass term for chiral superfield linear confinement is a result of the dual Meissner effect [1]. In the limit of small amenable to analytic studies [1] confinement is Abelian (for a definition of Abelian vs. non-Abelian confinement see e.g. [2]). Many theorists believe that in passing to large , (i.e. ), a smooth transition to non-Abelian confinement – pertinent to pure supersymmetric Yang-Mills – takes place in the Seiberg–Witten model. This is sometimes referred to as the same universality class hypothesis. In non-supersymmetric theories a construction serving the same purpose – studying the transition from Abelian to non-Abelian confinement by tuning an adjustable parameter – was engineered in [3] (see also [4]). There we considered SU Yang–Mills theories on treating the radius of the compact dimension as a free parameter. At small (i.e. ) we introduced a double-trace deformation stabilizing the vacuum of the theory at a center-symmetric point. With this stabilization, the Polyakov mechanism [5] guarantees linear Abelian confinement both, in pure Yang–Mills theories and in those with one massless quark in various representations of the SU gauge group [3]. A discrete chiral symmetry (S) inherent to two-index representations is spontaneously broken. Both effects, linear confinement and SB were caused by topological excitations in the vacuum (monpole-instantons, bions, etc.) which are under complete control at small . No obvious phase transitions in passing from the weak coupling Abelian regime at small- to the strong coupling decompactification/non-Abelian confinement regime at was detected. The trace of the Polyakov line remains vanishing in both regimes. Thus, the same universality class hypothesis is not necessarily tied up with supersymmetry.

In this paper we turn to SU gauge theories with several flavors, within the same theoretical framework.111Neither the number of colors nor the number of flavors are asumed to be large. A crucial distinction with the previously considered cases is the presence of continuous chiral symmetries. At small , at weak coupling, the continuous chiral symmetries remain unbroken, while Abelian confinement of the Polyakov type sets in, much in the same way as in pure Yang–Mills or single-flavor theories. Linear confinement coexists with the unbroken chiral symmetry in the quark sector.222There is no contradiction with the Casher argument [6] since the latter does not apply in 2+1 dimensions.

In the strong coupling decompactification limit one expects non-Abelian confinement and spontaneous SB. We study the dynamical origin and other details of the SB phenomenon within our theoretical framework. We observe a chiral phase transition in passing from small to large in the multiflavor case. We conjecture that the scale of the SB is tied with the passage from Abelian to non-Abelian confinement, and is of the order . This surprising suppressed scale would be a natural scale of SB were the center symmetry stable all the way down to arbitrarily small .

## 2 Theoretical framework

The general design is as follows. We consider SU Yang–Mills theories with flavors where . Each flavor is described by the Dirac fermion field in the complex representation

 R={F,S,AS,BF} (1)

where F stands for fundamental, AS/S/BF stand for two index antisymmetric, symmetric and bifundamental representations. We assume to be sufficiently small so that asymptotic freedom is preserved and the theory at hand is below the lower boundary of the conformal window. For simplicity we will focus on . The action for multiflavor QCD-like theories on takes the form

 S=∫R3×S11g2[12TrF2MN+i¯Ψa⧸\kern-3.0ptDΨa] (2)

where is the flavor index and is the covariant derivative acting in the representation . For QCD(BF) the gauge group is SUSU, and the gauge part of the action (2) must be replaced by

 F2MN→F21,MN+F22,MN. (3)

On a small cylinder , one can deform the original theory by adding a double-trace operator where

 P[U(x)]=2π2L4[N2]∑n=1dn|TrUn(x)|2, (4)

are numerical parameters of order one, and denotes the integer part of the argument in the brackets. The deformed action is

 S∗=S+∫R3×S1P[U(x)]. (5)

For judiciously chosen , the center symmetry remains unbroken in the vacuum while – due to weak gauge coupling and center symmetric holonomy – the gauge symmetry SU spontaneously breaks,

 SU(N)→U(1)N−1. (6)

The eigenvalues of the Polyakov line in the vacuum have a regular pattern

 uk=e2πikN,k=0,1,...,N−1 (7)

depicted in Fig. 1. diagonal gauge bosons – photons – remain perturbatively massless, while off-diagonal gauge bosons acquire masses . For what follows it will be convenient to introduce

 ⟨Az⟩ = 1Ldiag{−2π[N/2]N,−2π([N/2]−1)N,....,2π[N/2]N}, (8)

a matrix whose main diagonal is proportional to . If is odd, one of the eigenvalues is zero, and there is a fermionic mode which is massless.

On fermions we will impose a boundary condition with a U-twist

 Ψa(\boldmathx,x4+L)=e2πiωΨa(% \boldmathx,x4),g24D≪ω≪1. (9)

which is equivalent to turning on an overall U(1) Wilson line for the background holonomy. The U(1)-shifted holonomy generates three-dimensional real mass terms for the fermion fields which does not break any of the chiral symmetries (10) inherent to the multiflavor theories on . This is unlike the complex four-dimensional mass which would explicitly violate S. The U-twist plays the role of an infrared regulator in loops with (otherwise massless) fermions.

The chiral symmetry group of the action (2) is

 SU(Nf)L×SU(Nf)R×U(1)V×Z2hNfZNf×ZNf×Z2 . (10)

The factors in the denominator eliminate double-counting of the symmetries. is the number of fermion zero modes in the background of the Belavin–Polyakov–Schwarz–Tyupkin (BPST) instanton [7] on . For the fermionic representations of interest

 2h={2,2N+4,2N−4,2N}forR={F,S,AS,BF}. (11)

The action of (10) on the Weyl fermions is

 SU(Nf)L:λa→(Uλ)a,¯ψa→¯ψa, SU(Nf)R:λa→λa,¯ψa→(V¯ψ)a, U(1)V:λa→eiδλa,¯ψa→eiδ¯ψa, Z2hNf:λa→e2πik2hNfλa,¯ψa→e−2πik2hNf¯ψa. (12)

Note that the four-dimensional Weyl fermion reduces to the Dirac fermion in the long-distance three-dimensional gauge theory. Thus, the relation between the four-dimensional Dirac fermion and the three-dimensional Dirac fermion obtained upon reduction is

 Ψ=(λ¯ψ). (13)

At small the eigenvalues of the Polyakov line weakly fluctuate near their vacuum values depicted in Fig. 1. It is only the sum of the eigenvalues that vanishes in the vacuum. At large each eigenvalue is expected to wildly fluctuate and average to zero. The same universality class hypothesis (say, for pure Yang–Mills) is that the passage from one regime to another is smooth. In principle, the smoothness, as opposed to a phase transition on the way from small to large values of , can be tested on lattices.

## 3 Qcd∗ with two flavors

For definiteness, let us start from the case of fundamental fermions. Relevant introductory material and notation can be found in [3]. There are distinct U(1)’s in this model, corresponding to distinct electric charges. Each component ( and ) will be characterized by a set of charges, which we will denote by ,

 \boldmathqΨi=g\boldmathHii≡g([H1]ii,[H2]ii…,[HN−1]ii),i=1,...,N, (14)

where is the set of Cartan generators. If is odd, fermion components remain massless (two ’s and two ’s). More exactly, these modes are nearly massless, with mass . Other components become massive and can be integrated out. If is even, to keep fermions in the low-energy limit, we will have to tune . The fermions that survive in the low-energy limit are charged and therefore appear in loops.

The infrared dynamics can be described as compact QED with light fermionic matter. Due to gauge symmetry breaking (6) via a compact scalar (the holonomy), there are types of elementary instanton-monopoles. These topological excitations are uniquely labeled by their magnetic charges valued in the affine root system . The operators corresponding to the topological excitations are expressed in terms of the dual variables for the photons by using Abelian duality. Summing over the instanton-monopole contributions, the non-perturbative low-energy effective Lagrangian takes the form

 SQCD(F)∗=∫R3[14g23\boldmathF2+1g23i¯Ψa[γμ(∂μ+i\boldmathqΨ\boldmathAμ)+γ42πiωL]Ψa +e−S0(~μei\boldmathα1\boldmathσdeta,b=1,2{λaψb}+μ∑\boldmathαj∈(Δ0aff−\boldmathα1)ei\boldmathαj\boldmathσ+H.c.)+...], (15)

where and are dimensionful coefficients of the monopole operators and the ellipses stand for higher order terms in the topological expansion and ignored massive modes. The linearly independent instanton-monopole operators render all dual photons massive, with masses proportional to . This switches on the Polyakov linear confinement.333There are distinct strings. In principle, they can break due to the fermion pair creation, but the breaking is exponentially suppressed.

Let us now discuss the fermion sector of the low-energy theory (15). To establish the vacuum structure we note that at distances larger than , the photons are gapped and are pinned at the bottom of the instanton-monopole induced potential. Consequently, to find the symmetry of the vacuum we explore the long-distance Lagrangian

 SNJL=∫R3{1g23i¯Ψa[γμ∂μ+γ42πiωL]Ψa+e−S0(~μdeta,b=1,2{λaψb}+H.c.)}. (16)

In essence, it describes a three-dimensional Nambu–Jona-Lasinio (NJL) model with chiral symmetry at the Lagrangian level. It is known that at arbitrarily weak coupling (in the units of the cut-off scale), the chiral symmetry of the NJL model remains unbroken. This is the phase of confinement without SB, with massless (or light) fermions in the spectrum whose masslessness is protected by unbroken S.

As the coupling increases and approaches unity in the domain , the chiral symmetry (10) is expected to spontaneously break down to the diagonal vector subgroup, SU. This breaking must result in massless Nambu–Goldstone (NG) bosons. The SB phase transition occurs at the boundary of the region of validity of the low-energy theory (15).

Now let us extend the above discussion to fermions in the two-index representations (still keeping ).

QCD(AS/S/BF)*: In QCD* theories with two-index fermions in the representations we also observe two different phases as is the case with QCD(F)*. These are

 L

In the latter phase we have confinement with (continuous) SB while in the former confinement without SB. At any radius, the discrete chiral symmetry pattern is [3], probed by a determinantal, continuos S-singlet order parameter, . Thus, at small , these theories have isolated vacua and at large , isolated coset spaces. Below, we highlight the differences in the analyses of two- and one-index representation fermions. We take QCD(BF)* as our main example due to its simplicity. QCD(AS/S) analysis is analogous, up to minor differences that can be filled in by using Ref. [3].

In two-flavor QCD(BF)*, the zero modes of the BPST instanton (which can be viewed as instanton-monopoles) split into groups of four zero modes each [8]. (This is the reason why the instanton-monopoles must play a more prominent role on than the four-dimensional BPST instanton.) Thus, unlike QCD(F)*, the instanton-monopoles appearing at the order do not cause confinement, but may induce SB. The magnetic bions which appear at the order lead to confinement, and mass gap for the gauge field fluctuations. The leading monopole and bion induced nonperturbative effects are

 LQCD(BF)∗nonpert = ∑αi∈Δ0aff[~μe−S0(e+i\boldmathαi\boldmathσ1+e+i\boldmathαi\boldmathσ2)(deta,bλaiψbi+deta,bλai+1ψbi+1) + μe−2S0[c1(ei(\boldmathαi−\boldmathαi−1)\boldmathσ1+ei(%\boldmath$α$i−\boldmathαi−1)\boldmathσ2) + c2(2ei(\boldmathαi\boldmathσ1−\boldmathαi\boldmathσ2)+ei(\boldmathαi\boldmathσ1−\boldmathαi−1\boldmathσ2)+ei(\boldmathαi\boldmathσ1−\boldmathαi+1% \boldmathσ2))]+H.c.

Consequently, the gauge structure of the theory undergoes a two-stage breaking:444In the second stage of the gauge structure reduction, the gauge group is not Higgsed, but, rather, the dual photons acquire gauge invariant masses.

 SU(N)×SU(N)\lx@stackrelHiggsing⟶[U(1)]N−1×[U(1)]N−1\lx@stackrelnonperturbative⟶ZN. (19)

The discrete gauge group (DGG) appearing at the final stage is another important difference between various QCD theories. In general, the gauge symmetry breaking pattern pertinent to QCD theories can be described as

 G\lx@stackrelHiggsing⟶Ab(G)\lx@stackrelnonperturbative⟶DGG(R) (20)

where is the Abelian gauge structure, as in the Seiberg–Witten theory, and DGG is the discrete gauge group which survives in the infrared. is equal to where is determined by the representation of massless fermions. For even

while for odd , , respectively. Note that DGG is a subgroup of the center group and can be obtained as the quotient of the center group by equivalences imposed by massless matter. In QCD, the charges which are non-neutral under DGG= are confined. This leads to the second difference between QCD(F)* and QCD where . In the first problem, strictly speking, the strings can break. In the latter case, the area law behavior of large Wilson loops (typically due to magnetic bions) is exact. This guarantees that charges nonvanishing under DGG are confined. Nonetheless, the gauge fluctuations are gapped in both cases.

As stated above (see (3)), the QCD theories possess two phases of confinement, with and without SB. Returning to QCD(BF)*, at distances larger than , the photons are gapped, and the vacuum structure is determined by the fermion action

 S=∫R3N∑i=1[i¯¯¯¯Ψai[γμ∂μ+γ42πiωL]Ψa,i+2~μe−S0(deta,bλaiψbi+h.c)]. (22)

This is again, an NJL-type model, with the same consequences as those discussed around (3). For , we have massless or light fermions in the spectrum – no complex Dirac mass is generated. For , the chiral symmetry is broken via the bilinear , inducing a complex four-dimensional Dirac mass for the fermion. This phase possesses massless NG-bosons due to SB.

## 4 Abelian to non-Abelian confinement

In the small- regime, the mechanism of confinement is Abelian, by virtue of in the gauge structure chain (20).555In fact, this is true for all the analytically controlled mechanisms of confinement known so far, including the Seiberg–Witten theory [1] or Polyakov’s mechanism [5]. The reason for the applicability of the semiclassical analysis is the appearance of an structure at some length scale. At , one looses the separation of scales between the lightest -bosons and the heaviest nonperturbatively gapped photons, so that the long distance theory based on looses its validity. This is also the scale at which one expects the long-distance NJL-model to induce the SB. We believe that, the scale of the passage from the Abelian to non-Abelian confinement and that of the chiral phase transition are parametrically tied up, and the bilinear chiral order parameter probes both.

This suggests that, in multiflavor QCD-like gauge theories, confinement without SB is a property of the Abelian confinement, whereas, continuous SB is associated with non-Abelian confinement.

One other surprising aspect of this chiral transition is its scale. In this work we dealt with small , small theories. However, if we let to be arbitrarily large, we would observe that the scale of the chiral transition is a sliding (or suppressed) scale as a function of . We found, by either an order of magnitude estimate based on the NJL Lagrangian, or by employing more powerful large- volume independence theorem of the center symmetric theories, that . This also means that in the limit, the region of Abelian confinement shrinks to zero, in compliance with the volume independence. (See section 5 of Ref. [4] and references therein.) The emergence of such -suppressed physical scales in QCD-like theories is rather surprising by itself, and is outside the reach of perturbation theory and non-perturbative holographic (supergravity) constructions. It is testable by numerical lattice simulations.

## Acknowledgments

We are grateful to E. Poppitz for useful discussions. M.S. is supported in part by DOE Grant DE-FG02-94ER-40823. The work of M.Ü. is supported by the U.S. Department of Energy Grant DE-AC02-76SF00515.

## References

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• [2] M. Shifman and M. Ünsal, Confinement in Yang–Mills: Elements of a Big Picture, arXiv:0810.3861 [hep-th].
• [3] M. Shifman and M. Ünsal, Phys. Rev. D 78, 065004 (2008) [arXiv:0802.1232 [hep-th]].
• [4] M. Shifman and M. Ünsal, On Yang-Mills Theories with Chiral Matter at Strong Coupling, arXiv:0808.2485 [hep-th].
• [5] A. M. Polyakov, Nucl. Phys. B 120, 429 (1977).
• [6] A. Casher, Phys. Lett. 83B, 395 (1979).
• [7] A. A. Belavin, A. M. Polyakov, A. S. Schwarz and Yu. S. Tyupkin, Phys. Lett. B 59, 85 (1975).
• [8] E. Poppitz and M. Ünsal, Index theorem for topological excitations on and Chern-Simons theory, arXiv:0812.2085 [hep-th].
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