Probing the dynamics of chiral SU(N) gauge theories via generalized anomalies
aa r X i v : . [ h e p - t h ] J a n Probing the dynamics ofchiral
S U ( N ) gauge theoriesvia generalized anomalies Stefano Bolognesi (1 , , Kenichi Konishi (1 , , Andrea Luzio (3 , (1) Department of Physics “E. Fermi”, University of PisaLargo Pontecorvo, 3, Ed. C, 56127 Pisa, Italy (2)
INFN, Sezione di Pisa, Largo Pontecorvo, 3, Ed. C, 56127 Pisa, Italy (3)
Scuola Normale Superiore, Piazza dei Cavalieri, 7 56127 Pisa, Italy [email protected], [email protected], [email protected]
Abstract
We study symmetries and dynamics of chiral SU ( N ) gauge theories with matterWeyl fermions in a two-index symmetric ( ψ ) or anti-symmetric tensor ( χ ) representa-tion, together with N ± p fermions in the anti-fundamental ( η ) and p fermions inthe fundamental ( ξ ) representations. They are known as the Bars-Yankielowicz (theformer) and the generalized Georgi-Glashow models (the latter). The conventional ’tHooft anomaly matching algorithm is known to allow a confining, chirally symmet-ric vacuum in all these models, with a simple set of massless baryonlike compositefermions describing the infrared physics.We analyzed recently one of these models ( ψη model), by applying the ideas ofgeneralized symmetries and the consequent, stronger constraints involving certainmixed anomalies, finding that the confining, chirally symmetric, vacuum is actuallyinconsistent.In the present paper this result is extended to a wider class of the Bars-Yankielowiczand the generalized Georgi-Glashow models. It is shown that for all these modelswith N and p both even, at least, the generalized anomaly matching requirementforbids the persistence of the full chiral symmetries in the infrared if the systemconfines. The most natural and consistent possibility is that some bifermion conden-sates form, breaking the color gauge symmetry dynamically, together with part ofthe global symmetry. ontents {S , N, p } models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 {A , N, p } models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Confining phase with unbroken global symmetries . . . . . . . . . . . . . . 92.3.1 {S , N, p } models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 {A , N, p } models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Dynamical Higgs phase in the {S , N, p } models . . . . . . . . . . . . . . . 112.5 Dynamical Higgs phase in the {A , N, p } models . . . . . . . . . . . . . . . 13 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Z N ⊂ H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 {A , N, p } models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 ( Z ) F − [ Z N ] anomaly 24 {S , , } . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 General {S , N, p } models with generic N and p even . . . . . . . . . . . . 294.3 {A , N, p } models with N and p even . . . . . . . . . . . . . . . . . . . . . 32 ( Z ) F − [ Z N ] anomaly: an alternative derivation 41 A few steps have been taken recently [1, 2] to go beyond the conventional ’t Hooft anomalymatching analysis in understanding the infrared dynamics of chiral gauge theories. Thestandard anomaly matching constraints and other generally accepted ideas, are usually notsufficient to pinpoint what happens in the infrared, where the system gets strongly coupledand perturbation theory has a limited power in predicting the phase and global symmetryrealization patterns.The tools which allow these new results come from the idea of the generalized symme-tries, of gauging some 1-form discrete center symmetries and studying the consequences ofmixed-’t Hooft-anomaly-matching conditions [3]-[16]. Most concrete applications of thesenew techniques so far refer to vectorlike gauge theories, such as pure SU ( N ) Yang-Mills,2r adjoint QCD, where there is an exact center symmetry ( Z N for SU ( N ) theories), orQCD where the color center symmetry can be combined with U (1) V to give a color-flavorlocked 1-form center symmetry. In these, vectorlike, gauge theories, the results from thenew approach can be corroborated by the extensive literature, based on some general the-orems [17, 18], on lattice simulations [19]-[22], on the effective Lagrangians [23]-[26], on ’tHooft anomaly analysis [27], on the powerful exact results in N = 2 supersymmetrie theo-ries [28,29], or on some other theoretical ideas such as the space compactification combinedwith semi-classical analyses [30]-[33].Most of these theoretical tools are however unavailable for the study of strongly-coupled chiral gauge theories, except for some general wisdom, the large- N approximation, and the’t Hooft anomaly considerations. Together, they offer significant, but not very stringent,information on the infrared dynamics, phases, and symmetry realization (see [34]- [47]).Such a situation is doubtlessly limiting our capability of utilizing chiral gauge theories inthe context of realistic model building beyond the standard model, e.g., with compositefermions, with composite Higgs bosons, or with dynamical composite models for darkmatter, and so on.It was these considerations that recently motivated the present authors to apply someof the new concepts and techniques to chiral gauge theories, to see if new insights in thephysics of these theories can be gained by doing so [1, 2]. In particular, in [2], a simpleclass of SU ( N ) gauge theories with Weyl fermions ψ ij , η Ai , i, j = 1 , . . . , N , A = 1 , . . . , N + 4 , (1.1)in the direct-sum representation ⊕ ( N + 4) ¯ , (1.2)(“ ψη model”) was studied. For even N the (nonanomalous) symmetry of the system is SU ( N ) × G f , G f = SU ( N + 4) × U (1) ψη × ( Z ) F Z N × Z N +4 , (1.3)where U (1) ψη is the anomaly-free combination of U (1) ψ and U (1) η , and ( Z ) F is the fermionparity, ψ, η → − ψ, − η .In spite of the presence of fermions in the fundamental representation of SU ( N ) thesystem turns out to possess an exact discrete Z N center (1-form) symmetry , Z N ⊂ SU ( N ) × { U (1) ψη × ( Z ) F } , (1.4)which can be “gauged”. Remember that the unfamiliar-sounding expression of gauging a Let us recall that a 1-form symmetry acts on extended operators such as closed Wilson or Polyakovloops, but not on a local operator as in conventional (0-form) symmetries. Z N of an SU ( N )gauge theory, one arrives at an SU ( N ) Z N gauge system, with consequent N fractional instantonnumbers. Concretely, this can be done by introducing the 2-form gauge fields (cid:0) B (2)c , B (1)c (cid:1) , N B (2)c = dB (1)c , (1.5)and coupling to them the SU ( N ) gauge fields a and U (1) ψη × ( Z ) F gauge fields, A and A (1)2 , appropriately. As for the SU ( N ) gauge field a , this can be achieved by embedding itinto a U ( N ) gauge field e a as e a = a + 1 N B (1)c , (1.6)and requiring the whole system to be invariant under the 1-form gauge transformation, B (2)c → B (2)c + d λ c , B (1)c → B (1)c + N λ c , e a → e a + λ c . (1.7)As the Z N is a color-flavor locked symmetry, Eq. (1.4), the U (1) ψη and ( Z ) F gauge fieldsmust also be transformed simultaneously: A → A − λ c , A (1)2 → A (1)2 + N λ c . (1.8)The relation (1.5) indicates that one has now an SU ( N ) Z N connection rather than SU ( N ). Itimplies that there are nontrivial ’t Hooft fluxes carried by the gauge fields12 π Z Σ B (2)c = n N , n ∈ Z N , (1.9)in a closed two-dimensionl subspace, Σ . On topologically nontrivial four dimensionalspacetime of Euclidean signature containing such subspaces one has then π Z Σ ( B (2)c ) = nN , (1.10)where n ∈ Z N .The fermion kinetic term with the background gauge field is obtained by the minimal In [2] we have gauged also the 1-form center symmetry Z N +4 ⊂ SU ( N + 4), but the conclusion of thework did not depend on it. Here and in the rest of the present work, only the “color-flavor locked” Z N center symmetry will be considered. Throughout, a compact differential-form notation is used. For instance, a ≡ T c A cµ ( x ) dx µ ; F = da + a ; F ≡ F ∧ F = F µν F ρσ dx µ dx ν dx ρ dx σ = ǫ µνρσ F µν F ρσ d x = F µν ˜ F µν d x , and so on. ψγ µ (cid:16) ∂ + R S ( e a ) + N + 42 A + A (cid:17) µ P L ψ + ηγ µ (cid:16) ∂ + R F ∗ ( e a ) − N + 22 A − A (cid:17) µ P L η , (1.11)with the obvious notation. We compute the anomalies by applying the Stora-Zuminodescent procedure starting with a 6 D anomaly functional T = 124 π tr R S (cid:20) R S (cid:0) F (˜ a ) − B (2)c (cid:1) + N + 42 (cid:16) d A + B (2)c (cid:17) + (cid:16) d A (1)2 − N B (2)c (cid:17)(cid:21) , T = 124 π tr R ∗ F (cid:20) − (cid:0) F ( e a ) − B (2)c (cid:1) − N + 22 (cid:16) d A + B (2)c (cid:17) − (cid:16) d A (1)2 − N B (2)c (cid:17)(cid:21) . (1.12)The rest of the procedure for computing the ( Z ) F anomaly is standard: (i) one firstintegrates to get the 5 D boundary action containing A (1)2 (WZW action); (ii) the variationsof the form δA (1)2 = 12 ∂δA (0)2 , δA (0)2 = ± π , (1.13)leads to, via the anomaly-in-flow, the seeked-for anomaly in the 4 D theory. The result is δS = − N π Z Σ ( B (2)c ) δA (0)2 = − N × Z N ( ± π ) = ± π × Z : (1.14)the partition function changes sign, under ψ, η → − ψ, − η , that is, there is a ( Z ) F anomaly.As the ( Z ) F − [ Z N ] mixed anomaly is obviously absent in the IR, we conclude thatthe confining chirally symmetric vacuum, in which conventional ’t Hooft anomalies aresaturated in the infrared by massless composite “baryons” B AB = ψ ij η Ai η Bj , A, B = 1 , . . . , N + 4 , (1.15)(antisymmetric in A ↔ B ), is not the correct vacuum of the system. As shown in [2], thedynamical Higgs vacuum, characterized by bifermion condensates, h ψ ij η Bi i = c Λ δ jB = 0 , j, B = 1 , . . . N , c ∼ O (1) (1.16)is instead found to be fully consistent.Several subtle features of the calculation and in the interpretation of the results are The Z N charges of A and A (1)2 in (1.11) are determined by the way U (1) ψη and ( Z ) F togetherreproduce ψ → e πi/N ψ and η → e − πi/N η , as the reader can easily check. See [2]. In going from (1.11) to (1.12) term are arranged so that the expression inside each bracket be 1-formgauge invariant. ψη modelextends naturally to a wider class of the so-called Bars-Yankielowicz and the generalizedGeorgi-Glashow models. The gauge group is taken to be SU ( N ), and the matter fermioncontent is ( p is a natural number) ⊕ ( N + 4 + p ) ¯ ⊕ p (1.17)for the former (let us call them {S , N, p } models), and ⊕ ( N − p ) ¯ ⊕ p (1.18)for the latter ( {A , N, p } models). We will find that for all N and p , both even, the systempossesses a ( Z ) F symmetry, which is nonanomalous, i.e., respected by standard instantons.Also, these models all enjoy a “color-flavor locked” Z N center symmetry, in spite of thepresence of fermions in the fundamental (or anti-fundamental) representation. It is thuspossible to gauge this center symmetry and study if, by doing so, the ( Z ) F symmetrybecomes anomalous, as happened in the {S , N, } model.The paper is organized as follows. In Sec. 2 we discuss the conventional ’t Hooft-anomaly-matching analysis in all these models. A good part of this section is a reviewof [34]- [40], but there are some new results, especially concerning the Higgs phase, whichwe need later. As the global symmetry group is relatively large, the fact that one can finda set of gauge-invariant composite fermions which satisfy all the anomaly-matching equa-tions at all, assuming the system to confine, is quite remarkable. Also, in all these modelswe find an alternative phase, also consistent with the anomaly matching criterion, char-acterized by certain bifermion condensates breaking color dynamically (dynamical Higgsphase) accompanied by a partial breaking of the global symmetry.In the conventional ’t Hooft anomaly matching equations, only the perturbative (local)aspect of the flavor symmetry group matters, though nonperturbative (instanton) effectsof the strong SU ( N ) gauge interactions are taken into account. In Sec. 3, the symmetry ofthese models is re-analyzed more carefully, taking into account the global properties (e.g.,the connectedness).In Sec. 4.2 we calculate and find a mixed anomaly of the type, ( Z ) F − [ Z N ] , in allmodels with N and p both even, whereas such an anomaly is absent in the infrared (IR) ina confining vacuum with full global symmetry - one of the candidate vacua allowed by theconventional anomaly matching argument. Consistency implies that these vacua cannotbe realized dynamically in the infrared, in all {S , N, p } and {A , N, p } models, with N and p are both even.We summarize and discuss our results in Sec. 5.6 Theories and possible phases {S , N, p } models The first class of theories is the ψη model with additional p pairs of fundamental and anti-fundamental fermions. Namely, the model is an SU ( N ) gauge theory with Weyl fermions ψ ij , η Ai , ξ i,a (2.1)in the direct-sum representation ⊕ ( N + 4 + p ) ¯ ⊕ p . (2.2)The indices run as i, j = 1 , . . . , N , A = 1 , . . . , N + 4 + p , a = 1 , . . . , p . (2.3)These theories (the Bars-Yankielowicz models) will be denoted as {S , N, p } below. The ψη model corresponds to {S , N, } . The first coefficient of the beta function is b = 11 N − ( N + 2) − ( N + 4 + 2 p ) = 9 N − − p , (2.4)and p is limited by N − N fixedand p → ∞ we recover ordinary QCD with p flavors, although this is outside the regimeof AF. The classical symmetry group is SU ( N ) c × U (1) ψ × U ( N + 4 + p ) η × U ( p ) ξ . (2.5)We discuss for the moment only 0-form symmetries, leaving a more detailed discussion ofthe symmetry group to Sec. 3. Anomaly breaks the symmetry group (2.5) to p = 0 : SU ( N ) c × SU ( N + 4) η × U (1) ψη ,p = 1 : SU ( N ) c × SU ( N + 5) η × U (1) ψη × U (1) ψξ ,p > SU ( N ) c × SU ( N + 4 + p ) η × SU ( p ) ξ × U (1) ψη × U (1) ψξ , (2.6)where the anomaly-free combination of U (1) ψ and U (1) η is U (1) ψη : ψ → e i( N +4+ p ) α ψ , η → e − i( N +2) α η , (2.7) To be precise, (2.5) is a covering space of the “real” symmetry group. As the conventional ’t Hooftanalysis depends only on the algebra of the symmetry group, this is for the moment sufficient. α ∈ R , and the anomaly-free combination of U (1) ψ and U (1) ξ is U (1) ψξ : ψ → e i pβ ψ , ξ → e − i( N +2) β ξ , (2.8)with β ∈ R . The choice of the two unbroken U (1)’s is somehow arbritrary, for examplealso U (1) ηξ U (1) ηξ : η → e i pγ η , ξ → e − i( N +4+ p ) γ ξ , (2.9)with γ ∈ R could be chosen as a generator. In Table 1 we summarize the fields and howthey transform under the symmetry group. There are also discrete unbroken symmetries SU ( N ) c SU ( N + 4 + p ) SU ( p ) U (1) ψη U (1) ψξ ψ N ( N +1)2 · ( · ) N ( N +1)2 · ( · ) N + 4 + p pη ( N + 4 + p ) · ¯ N · N ( N + 4 + p ) · ( · ) − ( N + 2) 0 ξ p · N p · ( · ) N · − ( N + 2) Table 1:
The multiplicity, charges and the representation are shown for each set of fermions in the {S , N, p } model. ( · ) stands for a singlet representation. of the three U (1)’s: ( Z N +2 ) ψ , ( Z N +4+ p ) η and ( Z p ) ξ . The relation between these discretesymmetries and the continuous non-anomalos group U (1) ψη × U (1) ψξ will be discussed inSec. 3. {A , N, p } models The second class of models we are interested are SU ( N ) gauge theories with Weyl fermions χ ij , η Ai , ξ i,a (2.10)in the direct-sum representation ⊕ ( N − p ) ¯ ⊕ p . (2.11)The indices run as i, j = 1 , . . . , N , A = 1 , . . . , N − p , a = 1 , . . . , p . (2.12)These (the generalized Georgi-Glashow) models will be indicated as {A , N, p } . The firstcoefficient of the beta function is b = 11 N − ( N − − ( N − p ) = 9 N + 6 − p . (2.13)8ere p will be assumed to be less than N + 3 so as to maintain AF. The symmetry groupis SU ( N ) c × U (1) χ × U ( N − p ) η × U ( p ) ξ . (2.14)Anomaly breaks this group to p = 0 : SU ( N ) c × SU ( N − η × U (1) χη ,p = 1 : SU ( N ) c × SU ( N − η × U (1) χη × U (1) χξ ,p > SU ( N ) c × SU ( N − p ) η × SU ( p ) ξ × U (1) χη × U (1) χξ , (2.15)where the anomaly-free combination of U (1) χ and U (1) η is U (1) χη : χ → e i( N − p ) α χ , η → e − i( N − α η , (2.16)and the anomaly-free combination of U (1) ψ and U (1) ξ is U (1) χξ : χ → e i pβ χ , ξ → e − i( N − β ξ . (2.17)Another possible anomaly-free combination is U (1) ηξ : U (1) ηξ : η → e i pγ η , ξ → e − i( N − p ) γ ξ . (2.18)In Table 2 we summarize the fields and how they transform under the symmetry group.There are also discrete unbroken symmetries: ( Z N − ) ψ , ( Z N − p ) η and ( Z p ) ξ . SU ( N ) c SU ( N − p ) SU ( p ) U (1) χη U (1) χξ χ N ( N − · ( · ) N ( N − · ( · ) N − p pη ( N − p ) · ¯ N · N ( N − p ) · ( · ) − ( N −
2) 0 ξ p · N p · ( · ) N · − ( N − Table 2:
The multiplicity, charges and the representation are shown for each set of fermions in the {A , N, p } model. The standard ’t Hooft anomaly matching conditions were found to allow a chirally symmet-ric, confining vacuum in the model first proposed in [35]. Let us assume that no condensatesform, the system confines, and the flavor symmetry is unbroken.9 .3.1 {S , N, p } models The candidate massless composite fermions for the {S , N, p } models are the left-handedgauge-invariant fields:( B ) [ AB ] = ψ ij η Ai η Bj , ( B ) aA = ¯ ψ ij ¯ η iA ξ j,a , ( B ) { ab } = ψ ij ¯ ξ i,a ¯ ξ j,b , (2.19)the first is anti-symmetric in A ↔ B and the third is symmetric in a ↔ b ; their chargesare given in Table 3. Writing explicitly also the spin indices they are( B ) AB,α = ǫ βγ ψ ij,β η A,γi η B,αj + ǫ βγ ψ ij,β η A,αi η B,γj , ( B ) a,αA = ǫ ˙ α ˙ β ¯ ψ ˙ αij ¯ η i, ˙ βA ξ j,a,α , ( B ) αab = ǫ ˙ β ˙ γ ψ ij,α ¯ ξ ˙ βi,a ¯ ξ ˙ γj,b : (2.20)all transforming under the { , } representation of the Lorentz group. Table 4 summarizes SU ( N ) c SU ( N + 4 + p ) SU ( p ) U (1) ψη U (1) ψξ B N +4+ p )( N +3+ p )2 · ( · ) ( N +4+ p )( N +3+ p )2 · ( · ) − N + p p B ( N + 4 + p ) p · ( · ) p · ¯ ( N + 4 + p ) · − ( p + 2) − ( N + p + 2) B p ( p +1)2 · ( · ) p ( p +1)2 · ( · ) ¯ N + 4 + p N + 4 + p Table 3:
Chirally symmetric phase of the {S , N, p } model. the anomaly matching checks, via comparison between Table 1 and Table 3. UV IR SU ( N + 4 + p ) N N + p − pSU ( p ) N N + 4 + p − ( p + 4) SU ( N + 4 + p ) − U (1) ψη − N ( N + 2) − ( N + 2 + p )( N − p ) − p ( p + 2) SU ( N + 4 + p ) − U (1) ψξ N + 2 + p ) p − p ( N + p + 2) SU ( p ) − U (1) ψη − ( N + 4 + p )( p + 2) + ( p + 2)( N + p + 4) SU ( p ) − U (1) ψξ − N ( N + 2) − ( N + 4 + p )( N + p + 2) + ( p + 2)(2 N + p + 4) U (1) ψη N ( N +1)2 ( N + 4 + p ) − N ( N + 4 + p )( N + 2) − ( N +4+ p )( N +3+ p )2 ( N − p ) − ( N + 4 + p ) p ( p + 2) ++ p ( p +1)2 ( N + 4 + p ) U (1) ψξ N ( N +1)2 p − Np ( N + 2) N +4+ p )( N +3+ p )2 p − ( N + 4 + p ) p ( N + p + 2) ++ p ( p +1)2 (2 N + 4 + p ) Grav − U (1) ψη N ( N +1)2 ( N + 4 + p ) − N ( N + 4 + p )( N + 2) − ( N +4+ p )( N +3+ p )2 ( N − p ) − ( N + 4 + p ) p ( p + 2)++ p ( p +1)2 ( N + 4 + p )Grav − U (1) ψξ N ( N +1)2 p − Np ( N + 2) ( N +4+ p )( N +3+ p )2 p − ( N + 4 + p ) p ( N + p + 2)++ p ( p +1)2 (2 N + 4 + p ) SU ( N + 4 + p ) − ( Z N +2 ) ψ N + 2 + p − p = 0 mod N + 2 SU ( p ) − ( Z N +2 ) ψ − ( N + 4 + p ) + p + 2 = 0 mod N + 2Grav − ( Z N +2 ) ψ − Table 4:
Anomaly matching checks for the IR chiral symmetric phase of the {S , N, p } model. For N odd, the last three equalities are consequences of other equations. .3.2 {A , N, p } models The candidate massless composite fermions for the {A , N, p } model are:( B ) { AB } = χ ij η Ai η Bj , ( B ) aA = ¯ χ ij ¯ η iA ξ j,a , ( B ) [ ab ] = χ ij ¯ ξ i,a ¯ ξ j,b , (2.21)the first symmetric in A ↔ B and the third anti-symmetric in a ↔ b . Writing the spinindices explicitly they are:( B ) AB,α = ǫ βγ χ ij,β η A,γi η B,αj + ǫ βγ χ ij,β η A,αi η B,γj , ( B ) a,αA = ǫ ˙ β ˙ γ ¯ χ ˙ βij ¯ η i, ˙ γA ξ j,a,α , ( B ) ab = ǫ ˙ β ˙ γ χ ij ¯ ξ ˙ βi,a ¯ ξ ˙ γj,b . (2.22)All anomaly triangles are saturated by these candidate massless composite fermions, seeTable 6 (Table 5 vs Tab. 2). SU ( N ) c SU ( N − p ) SU ( p ) U (1) χη U (1) χξ B N − p )( N − p )2 · ( · ) ( N − p )( N − p )2 · ( · ) − N + p p B ( N − p ) p · ( · ) p · ¯ ( N − p ) · − ( p − − ( N + p − B p ( p − · ( · ) p ( p − · ( · ) ¯ N − p N − p Table 5:
IR massless fermions in the chirally symmetric phase of the {A , N, p } model. UV IR SU ( N − p ) N N + p − pSU ( p ) N N − p − ( p − SU ( N − p ) − U (1) χη − N ( N − − ( N − p )( N − p ) − p ( p − SU ( N − p ) − U (1) χξ N − p ) p − p ( N + p − SU ( p ) − U (1) χη − ( N − p )( p −
2) + ( p − N − p ) SU ( p ) − U (1) χξ − N ( N − − ( N − p )( N + p −
2) + ( p − N − p + 0) U (1) χη N ( N − ( N − p ) − N ( N − p )( N − − ( N − p )( N − p )2 ( N − p ) − ( N − p ) p ( p − ++ p ( p − ( N − p ) U (1) χξ N ( N − p − Np ( N − N − p )( N − p )2 p − ( N − p ) p ( N + p − ++ p ( p − (2 N − p ) Grav − U (1) χη N ( N − ( N − p ) − N ( N − p )( N − − ( N − p )( N − p )2 ( N − p ) − ( N − p ) p ( p − p ( p − ( N − p )Grav − U (1) χξ N ( N − p − Np ( N − ( N − p )( N − p )2 p − ( N − p ) p ( N + p − p ( p − (2 N − p ) SU ( N − p ) − ( Z N − ) χ N − p − p = 0 mod N − SU ( p ) − ( Z N − ) χ − ( N − p ) + p − N − − ( Z N − ) χ − Table 6:
Anomaly matching checks for the IR chiral symmetric phase of the {A , N, p } model. {S , N, p } models The broken phase for the {S , N, } , ψη model has also been studied earlier [40, 46]. Thecomposite scalar ψη in the maximal attractive channel is in the fundamental of both thegauge group and the flavor group. All details can be found in the references.11omething interesting happens for p >
0. Now there is another channel, ξη , which isgauge invariant and charged under the flavor group. We thus have a competition betweentwo possible symmetry breaking channels, ψη and ξη . We assume that both condensatesoccur in the following way: h ψ ij η Bi i = c ψη Λ δ jB = 0 , j, B = 1 , . . . , N , h ξ i,a η Ai i = c ηξ Λ δ aA = 0 , a = 1 , . . . , N , A = N + 1 , . . . , N + p , (2.23)where Λ is the renormailization-invariant scale dynamically generated by the gauge interac-tions and c ηξ , c ψη are coefficients both of order one. According to the tumbling scenario [34],the first condensate to occur is in the maximally attractive channel (MAC). The strengthsof the one-gluon exchange potential for the two channels ψ (cid:16) (cid:17) η (cid:16) ¯ (cid:17) forming ,ξ (cid:16) (cid:17) η (cid:16) ¯ (cid:17) forming ( · ) , (2.24)are, respectively, N − N − ( N + 2)( N − N − N − N = − ( N + 2)( N − N , − N − N = − N − N . (2.25)So the ψη channel is slightly more attractive, but such a perturbative argument is notreally significant and we assume here that both types of condensates are formed.The resulting pattern of symmetry breaking is SU ( N ) c × SU ( N + 4 + p ) η × SU ( p ) ξ × U (1) ψη × U (1) ψξ h ξη i , h ψη i −−−−−→ SU ( N ) cf η × SU (4) η × SU ( p ) ηξ × U (1) ′ ψη × U (1) ′ ψξ . (2.26)At the end the color gauge symmetry is completely (dynamically) broken, leaving color-flavor diagonal SU ( N ) cf η symmetry. U (1) ′ ψη and U (1) ′ ψξ are combinations respectively of U (1) ψη (2.7) and U (1) ξη (2.8) with the element of SU ( N + 4 + p ) η generated by t SU ( N +4+ p ) η = ( − α ( p + 2) − pβ ) N × N α ( N + p ) − βp × ( α + β )( N + 2) p × p . (2.27)Making the decomposition of the fields in the direct sum of representations in the subgroupone gets Table 7. The composite massless baryons are subset of those in (2.19):12 U ( N ) cf η SU (4) η SU ( p ) ηξ U (1) ′ ψη U (1) ′ ψξ ψ N ( N +1)2 · ( · ) N ( N +1)2 · ( · ) N + 4 + p pη ¯ ⊕ ¯ N · ( · ) N · ( · ) − ( N + 4 + p ) − pη · ¯ N · N · ( · ) − N − p +42 − p η p · ¯ N p · ( · ) N · ¯ 0 N + 2 ξ p · N p · ( · ) N · − ( N + 2) Table 7:
UV fieds in the {S , N, p } model, decomposed as a direct sum of the representations of theunbroken group of Eq. (2.26). SU ( N ) cf η SU (4) η SU ( p ) ηξ U (1) ′ ψη U (1) ′ ψξ B ¯ N ( N − · ( · ) N ( N − · ( · ) − ( N + 4 + p ) − p B · ¯ N · N · ( · ) − N − p +42 − p Table 8:
IR fieds in the {S , N, p } model, the massless subset of the baryons in Tab. 3 in the Higgs phase. B [ AB ]1 = ψ ij η Ai η Bj , B [ AC ]2 = ψ ij η Ai η Cj ,A, B = 1 , . . . , N , C = N + 1 , . . . , N + 4 . (2.28)It is quite straightforward (and actually almost trivial) to verify - we leave it to the reader asan excercise - that the UV-IR anomaly matching continues to work, with the UV fermionsin Table 7 and the IR fermions in Table 8. {A , N, p } models In the {A , N, p } model there is a competition between two possible bifermion symmetrybreaking channels χη and ξη . This time, the MAC criterion would favor the ξη condensatesagainst χη . Indeed, the strength of the one-gluon exchange potential for the two channels χ (cid:18) (cid:19) η (cid:16) ¯ (cid:17) forming ,ξ (cid:16) (cid:17) η (cid:16) ¯ (cid:17) forming ( · ) , (2.29)are, respectively, N − N − ( N − N + 1) N − N − N = − ( N − N + 1) N , − N − N = − N − N . (2.30)13gain, these perturbative estimates are not excessively significant, and we assume thatboth condensates occur as: h χ ij η Ai i = c χη Λ δ jA = 0 , j = 1 , . . . , N − , A = 1 , . . . , N − , h ξ i,a η Bi i = c ηξ Λ δ aB = 0 , a = 1 , . . . , p , B = N − , . . . , N − p . (2.31)The pattern of symmetry breaking is SU ( N ) c × SU ( N − p ) η × SU ( p ) ξ × U (1) χη × U (1) χξ h ξη i , h χη i −−−−−→ SU (4) c × SU ( N − cf η × SU ( p ) ηξ × U (1) ′ χη × U (1) ′ χξ . (2.32)The color gauge symmetry is partially (dynamically) broken, leaving color-flavor diag-onal global SU ( N − cf η symmetry and an SU (4) c gauge symmetry. U (1) ′ χη and U (1) ′ χξ are a combinations respectively of U (1) χη (2.16) and U (1) χξ (2.17) with the elements of SU ( N ) c and SU ( N − p ) η generated by: t SU ( N ) c = α ( N − p )+ βpN − ( N − × ( N − − α ( N − p )+ βp × ! ,t SU ( N − p ) η = − p ( α + β )( N − N − ( N − × ( N − ( α + β )( N − p × p ! . (2.33)Making the decomposition of the fields in the direct sum of representations in the subgroupone arrives at Table 9. SU ( N − cf η SU (4) c SU ( p ) ηξ U (1) ′ χη U (1) ′ χξ χ N − N − · ( · ) ( N − N − · ( · ) ( N − p ) N ( N − p NN − χ · ( N − · N − · ( · ) ( N − p ) N N − pN N − χ · ( · ) 6 · ( · ) 0 0 η ¯ ⊕ ¯ ( N − · ( · ) ( N − · ( · ) − ( N − p ) N ( N − − pNN − η p · ¯ p ( N − · ( · ) ( N − · ¯ − − pN − N − − pN − η · ¯ ( N − · ¯ 4( N − · ( · ) − ( N − p ) N N − − pN N − η p · ( · ) p · ¯ 4 · ¯ N − p N − p ξ p · p ( N − · ( · ) ( N − · pN − − ( N −
2) + pN − ξ p · ( · ) p · · − N − p − ( N − − p Table 9:
UV fieds in the {A , N, p } model, decomposed as a direct sum of the representations of theunbroken group of Eq. (2.32). U ( N − cf η SU ( p ) ηξ U (1) ′ χη U (1) ′ χξ B ¯ ( N − N − · ( · ) − ( N − p ) N ( N − − pNN − Table 10:
IR fied in the {A , N, p } model in the dynamical Higgs phase. The composite massless baryons are subset of those in (2.21): B { AB } = χ ij η Ai η Bj , A, B = 1 , . . . , N − . (2.34)In the IR these fermions saturate all the anomalies of the unbroken chiral symmetry. Thiscan be seen by an inspection of Table 10 and Table 9, with the help of the followingobservation.In fact, there is a novel feature in the {A , N, p } models, which is not shared by the {S , N, p } models. As seen in Table 10, there is an unbroken strong gauge symmetry SU (4) c ,with a set of fermions, χ , χ , η , η , ξ , (2.35)charged with respect to it. However, the pairs { χ , η } and { η , ξ } can form massiveDirac fermions and decouple. These are vectorlike with respect to the surviving infraredsymmetry, (2.32), hence are irrelevant to the anomalies. On the other hand, the fermion χ can condense h χ χ i (2.36)forming massive composite mesons, ∼ χ χ , which also decouples. It is again neutral withrespect to all of SU ( N − cf η × SU ( p ) ηξ × U (1) ′ χη × U (1) ′ χξ . (2.37)To summarize, SU (4) c is invisible (confines) in the IR, and only the unpaired part of the η fermion (cid:0) ¯ (cid:1) remains massless, and its contribution to the anomalies is reproducedexactly by the composite fermions, (2.34). Comment:
The massive mesons { χ η } , { η ξ } , { χ χ } are not charged with respect tothe flavor symmetries surviving in the infrared. It is tempting to regard them as a toy-model “dark matter”, as contrasted to the fermions B AB which constitute the “ordinary,visible” sector. Actually, with matter fermions (2.35) SU (4) c is asymptotically free only for 50 − N − p >
0. If50 − N − p < SU (4) c will remain weakly coupled in the infrared, but the fact that the fermions (2.35)do not contribute to the anomalies with respect to the remaining flavor symmetries (2.37) stays valid. Symmetries
In the conventional ’t Hooft anomaly analysis discussed above only the algebra of the groupmatters. In this section the symmetry of the models will be examined with more care, bytaking into account the global aspects of the color and flavor symmetry groups. Let usfirst consider the Bars-Yankielowicz ( {S , N, p } ) models.For a {S , N, p } model, the classical symmetry group of our system is given by G class = G c × G f = SU ( N ) c × U (1) ψ × U ( N + 4 + p ) η × U ( p ) ξ Z N . (3.1)The color group is G c = SU ( N ) c , and its center acts non-trivially on the matter fields: Z N : ψ → e π i nN ψ , η → e − π i nN η , ξ → e π i nN ξ , (3.2)( n ∈ Z ). The division by Z N in Eq. (3.1) is due to the fact that the numerator overlapswith the center of the gauge group (see Sec. 3.2 below). Another, equivalent way of writingthe flavor part of the classical symmetry group is G f = SU ( N + 4 + p ) × SU ( p ) × U (1) ψ × U (1) η × U (1) ξ Z N × Z N +4+ p × Z p . (3.3)Quantum mechanically one must consider the effects of the anomalies and SU ( N )instantons which reduce the flavor group down to its anomaly-free subgroup. The instantonvertex explicitly breaks the three independent U (1) rotations for ψ , η and ξ down to two U (1)’s, to be chosen among U (1) ψη , U (1) ψξ , and U (1) ξη : U (1) ψη : ψ → e i( N +4+ p ) α ψ , η → e − i( N +2) α η ,U (1) ψξ : ψ → e i pβ ψ , ξ → e − i( N +2) β ξ ,U (1) ηξ : η → e i pγ η , ξ → e − i( N +4+ p ) γ ξ (3.4)(see Eq. (2.7)-Eq. (2.9)). Three different discrete sub-groups left unbroken are( Z N +2 ) ψ : ψ → e π i kN +2 ψ , ( Z N +4+ p ) η : η → e π i kN +4+ p η , ( Z p ) ξ : ξ → e π i kp ξ . (3.5)The question is: which is the correct anomaly-free sub-group? The anomaly affects onlythe U (1) part of the group U (1) ψ × U (1) η × U (1) ξ anomaly −−−−→ H (3.6)16o that the total symmetry group is broken as follows G f anomaly −−−−→ SU ( N + 4 + p ) × SU ( p ) × H Z N × Z N +4+ p × Z p . (3.7) H Clearly, U (1) ψη , U (1) ψξ , U (1) ηξ , ( Z N +2 ) ψ , ( Z N +4+ p ) η , ( Z p ) ξ are all part of the anomaly-free sub-group, but one must find the minimal description, in order to avoid the double-counting. H is at the bottom of the following sequence of covering spaces: U (1) ψη × U (1) ψξ × U (1) ηξ × ( Z N +2 ) ψ × ( Z N +4+ p ) η × ( Z p ) ξ ↓ U (1) ψη × U (1) ψξ × ( Z N +2 ) ψ ↓H (3.8)The first arrow can be understood as follows. U (1) ηξ can always be obtained by a combi-nation of the other two continuous groups, by choosing (using conventions for α , β , γ asin Eq. (3.4)) α = − pγN + 2 , β = ( N + 4 + p ) γN + 2 . (3.9)Also, the fundamental element of ( Z N +4+ p ) η can be obtained by a combination of thefundamental of ( Z N +2 ) ψ ( k = 1 in Eq. (3.5)) with the U (1) ψη element α = − N + 4 + p )( N + 2) . (3.10)Similarly ( Z p ) ξ can always be expressed as part of U (1) ψξ × ( Z N +2 ) ψ .The question now (the second arrow) is whether( Z N +2 ) ψ ⊂ U (1) ψη × U (1) ψξ (3.11)holds, i.e., whether the discrete part of the group can be entirely expressed as a subgroupof the continuous U (1) groups. The requirement (3.11) is equivalent to( N + 4 + p ) α + p β ∼ πN + 2 , − ( N + 2) α ∼ , − ( N + 2) β ∼ , (3.12)where ∼ means the equality with possible additional terms of the form 2 π × integer allowed.17t follows from the last two equations that α = 2 πmN + 2 , β = 2 πnN + 2 , m, n ∈ Z , (3.13)which inserted in the first gives2 πm ( N + 4 + p ) N + 2 + 2 πnpN + 2 ∼ πN + 2 , (3.14)that is, (2 + p ) m + np = 1 + ( N + 2) ℓ , m, n, ℓ ∈ Z . (3.15)If one (or both) of N and p is odd, Eq. (3.15) has solutions. That is Eq. (3.11) is valid,and H has only one component connected to the identity. This also means that, in thecontext of the conventional anomaly matching discussion, the anomaly matching require-ment involving ( Z N +2 ) ψ , ( Z N +4+ p ) η , or ( Z p ) ξ is automatically satisfied when the trianglescontaining U (1) ψη × U (1) ψξ × SU ( N + p + 4) × SU ( p ) are UV-IR matched.Vice versa, if p and N are both even there are no solutions of Eq. (3.15): i.e., ( Z N +2 ) ψ is not entirely contained in U (1) ψη × U (1) ψξ ; only the even elements of ( Z N +2 ) ψ are: (cid:0) Z N +22 (cid:1) ψ ⊂ U (1) ψη × U (1) ψξ . (3.16)One can show however that for p , N both even( Z N +2 ) ψ ⊂ U (1) ψη × U (1) ψξ × ( Z ) F , (3.17)where ( Z ) F is the fermion parity generated by ψ → − ψ , η → − η , ξ → − ξ . (3.18)In fact, admitting the presence of fermion parity the requirement (3.12) gets modified to( N + 4 + p ) α + p β ∼ πN + 2 + π , − ( N + 2) α ∼ π , − ( N + 2) β ∼ π , (3.19)and thus (2 + p ) m + np = ( N + 2) ℓ , m, n, ℓ ∈ Z . (3.20)which always has a solution.To summarize, when p and N are both even, one has H = U (1) × U (1) × ( Z ) F , (3.21)18.e. it has two disconnected components. U (1) and U (1) are any two out of U (1) ψη , U (1) ψξ , and U (1) ηξ . If p and/or N is odd, instead, H = U (1) × U (1) : (3.22)it has only one connected component. Z N ⊂ H We focus now on the center of the color SU ( N ) group, Z N . We first show that when N , p are both even, Z N U (1) ψη × U (1) ψξ . (3.23)To prove this, ab absurdo , assume that U (1) ψη × U (1) ψξ does contains Z N : that is( N + 4 + p ) α + p β ∼ πN , − ( N + 2) α ∼ − πN , − ( N + 2) β ∼ πN . (3.24)(Remember that the symbol “ ∼ ” here indicates equality modulo terms of the form 2 πn , n ∈ Z .) We first eliminate α from the first two. As N , p are both even, multiply the firstby N +22 and the second by N +4+ p (both integers) and add. We get p N + 2) β ∼ πN N + 22 − πN N + 4 + p ∼ π − πpN . (3.25)On the other hand multiplying the third of Eq. (3.24) by p (also an integer) gives p N + 2) β ∼ − πpN . (3.26)Eq. (3.25) and Eq. (3.26) contradict each other. Q.E.D.
We next prove that if at least one of N and p is odd, then Z N ⊂ U (1) ψη × U (1) ψξ , (3.27)that is, Eq. (3.24) has solutions. To prove this, we repeat the procedure above, noting thatthere may be now extra terms on the right hand side. As a result, Eq. (3.25) is replacedby p N + 2) β = π − πp N + 2 πm N + 22 + 2 πn N + 4 + p , (3.28)19hile Eq. (3.26) is replaced by p N + 2) ˜ β = − πp N + 2 πℓ · p , (3.29) m, n, ℓ ∈ Z . (3.30)Now when one or both of N and p is odd, it is always possible to find appropriate integers m, n, ℓ such that the right hand sides of Eq. (3.28) and Eq. (3.29) are equal, that is, π + 2 πm N + 22 + 2 πn N + 4 + p ∼ πℓ · p . (3.31)When both N and p are even, exceptionally, this equality does not hold for any choice of m, n, ℓ , as has been already noted.Finally, we prove that Z N ⊂ U (1) ψη × U (1) ψξ × ( Z ) F , (3.32)when N and p are both even. This means that (cfr. Eq. (3.24))( N + 4 + p ) α + p β ∼ πN + π , − ( N + 2) α ∼ − πN + π , − ( N + 2) β ∼ πN + π . (3.33)Let us repeat the procedure Eq. (3.24)-Eq. (3.26), by keeping the extra terms coming from π on the right hand sides. Eq. (3.25) is replaced by p N + 2) β ∼ π − πpN + p + 22 π , (3.34)whereas Eq. (3.26) is modified to p N + 2) β ∼ − πpN + p π . (3.35)The right hand sides of Eq. (3.34) and Eq. (3.35) now agree.To sum up, we have shown that Z N ⊂ H (3.36)for any choice of N and p , for the {S , N, p } models.20 .3 {A , N, p } models So far, our analysis concentrated on the {S , N, p } models for definiteness. For the {A , N, p } models, the result is very similar. The symmetry group is G f = SU ( N − p ) × SU ( p ) × U (1) χ × U (1) η × U (1) ξ Z N × Z N − p × Z p , (3.37)where the anomaly acts on the U (1) part as U (1) χ × U (1) η × U (1) ξ anomaly −−−−→ H . (3.38)Clearly all U (1) χη , U (1) χξ , U (1) ηξ defined in Eq. (2.16)-Eq. (2.18) together with the discretegroups( Z N − ) χ : χ → e π i kN − χ , ( Z N − p ) η : η → e π i kN − p η , ( Z p ) ξ : ξ → e π i kp ξ . (3.39)are the nonanomalous symmetry group of the system, but we need a minimum set withoutredundancy. For p = 0, the χη model, the result is: N odd : H = U (1) χη ,N even : H = U (1) χη × ( Z ) F . (3.40)For greater p , as for the {S , N, p } model, H is:gcd( N, p,
2) = 1 : H = U (1) × U (1) , gcd( N, p,
2) = 2 : H = U (1) × U (1) × ( Z ) F . (3.41)where U (1) , are any two out of U (1) χη , U (1) χξ and U (1) ηξ . Again, Z N ⊂ H (3.42)for any choice of N and p . The proof for the {A , N, p } models is entirely analogous to theone given for {S , N, p } and is omitted. Let us illustrate the symmetry of our systems graphically, taking a few concrete models ofthe type, {S , N, p } .It is convenient to introduce the following notation. We parameterize a generic U (1) ⊂ = U (1) ψ × U (1) η × U (1) ξ with a triplet of integer numbers t = t t t ∈ Z , (3.43)so that U (1) : ψηξ → e it θ ψe it θ ηe it θ ξ , ≤ θ < π . (3.44)This U (1) winds gcd( t , t , t )-times around the three-torus T . In general, given a specificdirection, we choose the “fundamental” generator for which gcd( t , t , t ) = 1 so thatperiodicity in θ is exactly 2 π . In this notations the three fundamental U (1)’s are generatedby t U (1) ψ = , t U (1) η = , t U (1) ξ , (3.45)and the non-anomalus ones are generated by t U (1) ψη = N +4+ p gcd( N +4+ p,N +2) − N +2gcd( N +4+ p,N +2) , t U (1) ψξ = p gcd( p,N +2) − N +2gcd( p,N +2) , t U (1) ηξ = p gcd( N +4+ p,p ) − N +4+ p gcd( N +4+ p,p ) . (3.46)We give now specific examples for p = 0 , , • For p = 0, the ψη model, this has been discussed in detail in [2] and the result is: N odd : H = U (1) ψη ,N even : H = U (1) ψη × ( Z ) F . (3.47) • For p = 1, independentely on N , H has only one connected component. In Figure 1we show the case N = 3. One possible way to parameterize H is H = U (1) ψξ × U (1) ηξ . (3.48)Note that U (1) ψξ contains ( Z N +2 ) ψ and U (1) ηξ contains ( Z N +5 ) η , so together theycontain the whole discrete lattice ( Z N +2 ) ψ × ( Z N +5 ) η . We can define the group e U (1)as the one that contains Z N , and is the one generated by t e U (1) = 2 t U (1) ψξ − t U (1) ηξ . (3.49)22 ( Z Z ηU (1) ηξU (1) ψξ U (1) ξU (1) ψ U (1) η U (1) ψ U ( ) ψ η U (1) η : e U ( ) U (1) ηU (1) ξ U ( ) η ξ U (1) ξ U ( ) ψ ξ U (1) ψ Figure 1:
The three-torus U (1) ψ × U (1) η × U (1) ξ broken to U (1) ψξ × U (1) ηξ for the {S , , } model. • For p = 2, N odd, H has only one connected component. In Figure 2 we show thegraphs for the case N = 3. One possible way to parameterize H is H = U (1) ψξ × U (1) ηξ Z (3.50)Note that U (1) ψξ contains ( Z N +2 ) ψ × ( Z ) ξ and U (1) ηξ contains ( Z N +6 ) η × ( Z ) ξ so U (1) ψξ × U (1) ηξ contains Z N +2) × Z N +6) which twice redundant with respectto ( Z N +2 ) ψ × ( Z N +6 ) η × ( Z ) ξ . We can also see this in the following way. U (1) ψξ contains a non-trivial element of U (1) ηξ . If we take the element of U (1) ψξ with β = π we obtain ψ → ψ , η → η , ξ → − ξ (3.51)which is exactly the element of U (1) ηξ with γ = π . This is the reason for the Z division in (3.50). The group e U (1) that contains Z N is the one generated by (3.49).If we define b U (1) generated by t b U (1) = − t U (1) ψξ + 12 t U (1) ηξ , (3.52)we can write H = U (1) ψξ × b U (1) = U (1) ηξ × b U (1) . (3.53) • For p = 2, N even, H has two components. In Figure 3, we illustrate the case N = 4, p = 2. One possible way to parameterize H is H = U (1) ψξ × U (1) ηξ × ( Z ) F . (3.54)We can define the group e U (1) generated by (3.49) but this time it contains only Z N .In general it is not possible to write Z N ⊂ U (1) ′ × ( Z ) F , both U (1)’s are necessary,23 U (1) U (1) η U ( ) ψ η U (1) ψU (1) ηξU (1) ψξ U (1) ξU (1) ψ U (1) η b U (1) :( Z ηU (1) ηU (1) ξ U ( ) η ξ :( Z U (1) ψ U ( ) ψ ξ U (1) ξ Figure 2:
The three-torus U (1) ψ × U (1) η × U (1) ξ broken to U (1) ψξ × U (1) ηξ Z for the {S , , } model. although N = 4, p = 2 is an exception as we will see in the warmup example inSec. 4.1. U (1) ηξ e U (1) U (1) ψξ :( Z ) η ⊂ ( Z ) ηU (1) ξU (1) ψ U (1) η U (1) ψ U (1) ξ U ( ) ψ ξ :( Z ) c ⊂ ( Z ) c U (1) η U (1) ξ U ( ) η ξ U (1) η U ( ) ψ η U (1) ψ Figure 3:
The three-torus U (1) ψ × U (1) η × U (1) ξ broken to U (1) ψξ × U (1) ηξ × ( Z ) F for the {S , , } model. ( Z ) F − [ Z N ] anomaly The generalized (mixed) anomaly of the type ( Z ) F − [ Z N ] was studied in detail in [2] forthe {S , N, } (“ ψη ”) model. We have briefly reviewed the method and results found there atthe end of Introduction. This study is extended below to a wider class of models discussedin Sec. 2 and Sec. 3. The global structure of the anomaly-free symmetry group revealedin Sec. 3 teaches us that the most interesting class of models for the present purpose are {S , N, p } and {A , N, p } models with N and p both even, on which our analysis below willset focus. {S , , } We first consider a simplest, nontrivial model {S , , } and set up the calculation of themixed anomalies, making a brief note on some general features of the gauging of the discrete1-form Z N symmetry, on the idea of “( Z ) F gauge field”, and paying special attention to24he way the fermions transform nontrivially under the 1-form Z N gauge transformation.The same procedure can then be easily extended to more general cases discussed later.Even though the fact that Z N ⊂ U (1) ψξ × U (1) ηξ × ( Z ) F (4.1)has been proven in general in Sec. 3.2, we need an explicit solution for this model, to fixthe charges of the fermion fields under the 1-form Z N symmetry. From U (1) ψξ : ψ : e iβ ; ξ : e − i N +22 β = e − iβ ; U (1) ηξ : η : e iγ ; ξ : e − i N +62 γ = e − iγ ; Z N : ψ : e πi/N = e πi ; η : e − πi/N = e − iπ/ ; ξ : e πi/N = e iπ/ ; Z : ψ : e ± iπ ; η : e ± iπ ; ξ : e ± iπ , (4.2)we see that a simple solution in this case is to take β = 0, and γ = + π . It is easily seenthat Z N is realized as a U ηξ (1) × ( Z ) F transformation with Z : ψ : e + iπ ; η : e − iπ ; ξ : e +3 iπ . (4.3)We introduce accordingly, • A : U (1) ηξ • A : ( Z ) F • ˜ a : U ( N ) c • B (2)c : Z N SU ( N ) gauge field a is embedded in a U ( N ) gauge field ˜ a as e a = a + 1 N B (1)c , N B (2)c = dB (1)c . (4.4)As explained in [3], [4], one defines this way a globally well-defined SU ( N ) / Z N connection.The imposition of the local, 1-form gauge invariance (4.6) below, eliminates the apparentincrease of the degrees of freedom (in going from SU ( N ) to U ( N )) on the one hand, andat the same time allows to “gauge away” the center Z N variation of Polyakov or Wilsonloops e i H a → e πi/N e i H a , (4.5)on the other. 25he 1-form gauge transformation acts on these fields as: B (2)c → B (2)c + d λ c , B (1)c → B (1)c + N λ c , ˜ a → ˜ a + λ c , F (˜ a ) → F (˜ a ) + dλ c ; , (4.6) A → A − λ c ,A → A + N λ c = A + 2 λ c . (4.7)As we are here dealing with a Z N which is a color-flavor locked symmetry the fermion fieldsalso transform as well, appropriately. Their charges above follow from Eq. (4.2), Eq. (4.3).It is perhaps not useless, before proceeding, to remind ourselves of the meaning of a“( Z ) F gauge field”, A , which formally looks like an ordinary U (1) gauge field. Restoringmomentarily the suffices for the differential forms,2 A (1)2 − B (1)c = dA (0)2 (4.8)can be regarded as an invariant form of the ( Z ) F gauge field, 2 A (1)2 = dA (0)2 , where A (0)2 is a 2 π periodic scalar function (angle). It is an example of an “almost flat connection”:it satisfies 2 d A (1)2 − N B (2)c = 0 locally. However it cannot be set to zero everywhere, asa non vanishing flux through a closed two-dimensional surface may be present, allowing anontrivial ( Z ) F holonomy H A (1)2 = 2 πm/ , m ∈ Z , along a noncontractible closed loop.A kind of partial gauge fixing would allow us to work with the gauge field B (1)c and gaugefunction λ c , satisfying always I B (1)c = 2 πn , I λ c = 2 πℓN , ( n ∈ Z , ℓ ∈ Z ) . (4.9)See [2] for more discussions.The fermion kinetic terms are: ψγ µ (cid:0) ∂ + R S ( e a ) − A (cid:1) µ P L ψ + ηγ µ (cid:0) ∂ + R F ∗ ( e a ) + A + A (cid:1) µ P L η + ξγ µ (cid:0) ∂ + R F ( e a ) − A − A (cid:1) µ P L ξ , (4.10)each of which is indeed invariant under (4.6) and (4.7). Note that the choice of the Z charges, (1 , − , +3) for ( ψ, η, ξ ) fields (see Eq. (4.3)) is dictated by the requirement thatthe redundancy (4.1) involving the discrete symmetries Z and Z N be formally expressedas an invariance under (4.6) with a continuous gauge function λ c = λ µc ( x ) dx µ . The 1-form26auge invariant field tensors are, for the UV fermions ψ , η , ξ , R S F (˜ a ) − d A , R ∗ F F (˜ a ) + d A + d A , R F F (˜ a ) − A − A . (4.11)By rearranging things so that each term in the bracket is manifestly invariant under (4.6)and (4.7), this can be rewritten as R S (cid:0) F (˜ a ) − B (2)c (cid:1) − (cid:0) d A − B (2)c (cid:1) , R ∗ F (cid:0) F (˜ a ) − B (2)c (cid:1) + (d A + B (2)c ) + (cid:0) d A − B (2)c (cid:1) , R F (cid:0) F (˜ a ) − B (2)c (cid:1) − A + B (2)c ) − (cid:0) d A − B (2)c (cid:1) . (4.12)In the confining vacuum with the full global symmetry, discussed in Sec. 2.3, the infrareddegrees of freedom would be the (massless, by assumption) composite fermions B , B , B ,(2.19). Their kinetic terms are given by B γ µ (cid:0) ∂ + 2 A + A (cid:1) µ P L B + B γ µ (cid:0) ∂ − A − A (cid:1) µ P L B + B γ µ (cid:0) ∂ + 10 A + 5 A (cid:1) µ P L B . (4.13)The corresponding invariant tensors are2 (d A + B (2)c ) + (cid:2) d A − B (2)c (cid:3) , − A + B (2)c ) − (cid:2) d A − B (2)c (cid:3) ,
10 (d A + B (2)c ) + 5 (cid:2) d A − B (2)c (cid:3) , (4.14)respectively. Though this formula appears to depend on B (2)c due to the way things havebeen arranged to make each term manifestly invariant, B (2)c actually drops out completely,reflecting the fact that B , B , B are all color SU ( N ) singlets: there are no gauge kineticterms in their action. As a result, there would be no mixed anomalies in the IR due to thegauging of Z N F (˜ a ) = N B (2)c , (4.15)for the fundamental representation, the B (2)c dependence of the expressions in Eq. (4.12) isnot exhausted by the explicit B (2)c factors. Even though we shall use the formula Eq. (4.12)for the calculation of the mixed anomalies below, for manifest 1-form gauge invarianceof our calculation step by step, the same final result can be obtained (as it should) byworking with a not-term-by-term-manifestly-invariant expression Eq. (1.12). This is shown27n Appendix A. As a bonus, the discussion there explains some interesting aspect of ourresults below.The rest of the calculations follows that done in [2]. From Eq. (4.12) one finds the 6 D anomaly functional in the UV theory ,124 π tr R S h(cid:8)(cid:0) F (˜ a ) − B (2)c (cid:1) − (cid:0) d A − B (2)c (cid:1)(cid:9) i +124 π tr R ∗ F h(cid:8)(cid:0) F (˜ a ) − B (2)c (cid:1) + (d A + B (2)c ) + (cid:0) d A − B (2)c (cid:1)(cid:9) i +124 π tr R F h(cid:8)(cid:0) F (˜ a ) − B (2)c (cid:1) − A + B (2)c ) − (cid:0) d A − B (2)c (cid:1)(cid:9) i . (4.16)Keeping only the relevant terms, the first line ( ψ ) gives124 π (cid:20) − N + 2)tr (cid:0) F (˜ a ) − B (2)c (cid:1) (cid:16) d A − B (2)c (cid:17) − N ( N + 1)2 (cid:16) d A − B (2)c (cid:17) (cid:21) , (4.17)the second line ( η ) gives124 π (cid:20) N + 6)tr (cid:0) F (˜ a ) − B (2)c (cid:1) (cid:16) d A − N B (2)c (cid:17) + N ( N + 6) (cid:16) d A − B (2)c + . . . (cid:17) (cid:21) , (4.18)the third line ( ξ ) gives:124 π (cid:20) − · · (cid:0) F (˜ a ) − B (2)c (cid:1) (cid:16) d A − N B (2)c (cid:17) + 2 N (cid:16) − A + B (2)c + . . . (cid:17) (cid:21) . (4.19)Collecting the relevant terms, one finds that the coefficient of18 π ( B (2)c ) d A (4.20)is equal to N ( N + 2) − N ( N + 1)2 4 + ( N + 6)( − N ) + N ( N + 6) + 2 N N ( − − N = − . (4.21)Following the usual procedure (e.g., Eq. (1.13), Eq. (1.14)) we find the mixed ( Z ) F − [ Z N ] anomaly in 4 D : − N π Z Σ ( B (2)c ) δA (0)2 = − N × Z N ( ± π ) = ± π × Z . (4.22) Even though we follow here the Stora-Zumino descent procedure for calculating the anomalies, thereis no problem obtaining the same results `a la Fujikawa [52], staying in 4 D : the idea of gauging the center Z N symmetry in itself has nothing to do with the introduction of the two extra dimensions. This wasexplicitly shown in [2] for the ψη model. Z ) F anomaly in the IR, if one wouldassume the chirally symmetric vacuum with the massless baryons B , B , B of Sec. 2.3.The contradiction can be avoided by assuming that the system actually is in a dynamicalHiggs phase such as the one discussed in Sec. 2.4. {S , N, p } models with generic N and p even Let us now discuss {S , N, p } systems with general N , p , both even. As in the warmupexample, we verify anew Z N ⊂ U (1) ψη × U (1) ψξ × Z (4.23)for N , p both even, by solving the equations : N + 4 + p α + p β = 4 πN ± π , − N + 22 α = − πN ± π , − N + 22 β = 2 πN ± π , (4.24)concretely. Indeed, it is sufficient to find one good solution. A possible solution is α = 4 πN ( N + 2) + 2 πN + 2 , β = − πN ( N + 2) − πN + 2 , (4.25)which is a solution with the ( Z ) F signs + , − , + for the ψ, η, ξ fields in Eq. (4.24), respec-tively. The above solution Eq. (4.25) can be simply rewritten as α = 2 πN , β = − πN . (4.26)As in any anomaly calculation we couple the system to the appropriate background gaugefields, • A ψη : U (1) ψη • A ψξ : U (1) ψξ • A : ( Z ) F • ˜ a : U ( N ) c The charges here are taken half of those in (3.4). They would really have be chosen as in Eq. (3.46) inorder to ensure that the angles α and β take the canonical range of 2 π , but the following derivation of themixed anomaly is not affected by the different choices of the normalization of the charges and the angles. This time we first solved the second and third of Eq. (4.24), inserted the solutions to the first, checkingthat it is indeed satisfied, with appropriate signs for the ( Z ) F terms. B (2)c : Z N B (2)c → B (2)c + d λ c , B (1)c → B (1)c + N λ c , ˜ a → ˜ a + λ c , ˜ F (˜ a ) → ˜ F (˜ a ) + dλ c ,A ψη → A ψη − λ c ,A ψξ → A ψξ + λ c ,A → A + N λ c , (4.27)where the charges follow from (4.24) and (4.26). The fermion kinetic terms are: ψγ µ (cid:16) ∂ + R S ( e a ) + N + 4 + p A ψη + p A ψξ + A (cid:17) µ P L ψ + ηγ µ (cid:16) ∂ + R F ∗ ( e a ) − N + 22 A ψη − A (cid:17) µ P L η + ξγ µ (cid:16) ∂ + R F ( e a ) − N + 22 A ψξ + A (cid:17) µ P L ξ . (4.28)It can be checked readily that each line is invariant under Eq. (4.27). In particular, the( Z ) F charges are fixed by this requirement.The 1-form gauge invariant field tensors are, for the UV fermions ψ , η , ξ , T = R S (cid:0) F (˜ a ) − B (2)c (cid:1) + N + 4 + p A ψη + B (2)c ) + p A ψξ − B (2)c ) + (cid:16) d A − N B (2)c (cid:17) , T = R ∗ F (cid:0) F (˜ a ) − B (2)c (cid:1) − N + 22 (d A ψη + B (2)c ) − (cid:16) d A − N B (2)c (cid:17) , T = R F (cid:0) F (˜ a ) − B (2)c (cid:1) − N + 22 (d A ψξ − B (2)c ) + (cid:16) d A − N B (2)c (cid:17) , (4.29)where appropriate factors of B (2)c are added and subtracted so that each term in the bracketis invariant under the 1-form gauge transformations (4.27). Of course, the final result doesnot depend on such a rewriting: see Appendix A.The 6 D anomaly functional is124 π Z tr R S ( T ) + 124 π Z tr R ∗ F ( T ) + 124 π Z tr R F ( T ) . (4.30)Let us now extract the terms relevant to the ( Z ) F − [ Z N ] anomaly. From the ψ contri-bution one has124 π (cid:20) N + 2)tr (cid:0) F (˜ a ) − B (2)c (cid:1) (cid:16) d A − N B (2)c (cid:17) + N ( N + 1)2 (cid:16) d A + 2 B (2)c + . . . (cid:17) (cid:21) , (4.31)30 gives124 π (cid:20) N + 4 + p )tr (cid:0) F (˜ a ) − B (2)c (cid:1) (cid:16) d A − N B (2)c (cid:17) − N ( N + 4 + p ) (cid:16) d A + B (2)c + . . . (cid:17) (cid:21) (4.32)and the third line ( ξ ) gives:124 π (cid:20) p tr (cid:0) F (˜ a ) − B (2)c (cid:1) (cid:16) d A − N B (2)c (cid:17) + pN (cid:16) d A + B (2)c + . . . (cid:17) (cid:21) . (4.33)Collecting terms, one finds that the coefficient of18 π ( B (2)c ) d A (4.34)is equal to − N ( N + 2) + N ( N + 1)2 · − N ( N + 4 + p ) + N ( N + 4 + p ) − N p + pN = N . (4.35)A somewhat curious feature of this result (and of Eq. (4.21)) is that only fermions in ahigher representation contribute to the anomaly. The reason for this will become clear inan alternative derivation discussed in Appendix A.Following the usual procedure one calculates the 4 D mixed ( Z ) F − [ Z N ] anomaly.One finds an extra phase in the partition function associated with the fermion paritytransformation in the presence of the Z N gauge fields, N π Z Σ ( B (2)c ) δA (0)2 = N × Z N ( ± π ) = ± π × Z : (4.36)there is a ( Z ) F − [ Z N ] mixed anomaly in the theory.On the other hand, one finds no ( Z ) F anomaly in the IR, if one assumes the symmetricvacuum of Sec. 2.3. This can be seen, as in the warmup example of the previous section,by simply noting that all infrared degrees of freedom are color-singlet. We conclude thatthe chirally symmetric vacuum described by the baryons B , B , B cannot be realizeddynamically.We note again that such an inconsistency is avoided, assuming that the system is in thedynamical Higgs phase: the color-flavor locked 1-form symmetry is spontaneously broken.31 .3 {A , N, p } models with N and p even The simplest of this class of models, {A , N, } , with matter fermions ⊕ ( N −
4) ¯ , (4.37)(“ χη model”), has been studied, and the result of the analysis (unpublished) turns outto be similar to that in the ψη model of [2], reviewed in Introduction. For even N the(nonanomalous) symmetry of the system contains a nonanomalous ( Z ) F factor orthogonalto other continuous symmetry group. It gets anomalous under the 1-form gauging of a Z N center symmetry. This anomaly cannot be reproduced in the infrared, if the vacuum isassumed to be confining, and to keep the full global symmetries. Such a vacuum cannotbe realized dynamically.Below we study a more general class of {A , N, p } models, with p additional pairs offermions in ⊕ ¯ . We check first Z N ⊂ U (1) χη × U (1) χξ × Z . (4.38)Call α and β the angles associated with U (1) χη and U (1) χξ , U (1) χη : χ → e i N − p α χ , η → e − i N − α η ,U (1) χξ : χ → e i p β χ , ξ → e − i N − β ξ . (4.39)The condition (4.38) means that N − p α + p β = 4 πN ± π , − N − α = − πN ± π , − N − β = 2 πN ± π . (4.40)It turns out that any two of these imply the third: there is an arbitrariness to choose frommultiple of solutions. A possible solution is α = 4 πN ( N − − πN − , β = − πN ( N −
2) + 2 πN − , (4.41)which is a solution with the ( Z ) F signs in Eq. (4.40), − π, + π, − π for the χ, η, ξ fields,respectively. Actually the solution Eq. (4.41) is, simply, α = − πN , β = 2 πN . (4.42)32he color-flavor locked Z N transformation, (4.40) and (4.42), together with the normal-ization of the 1-form gauge field λ c , fix the charges of the fermion fields in Eq. (4.44)below.We introduce the background gauge fields • A χη : U (1) χη • A χξ : U (1) χξ • A : ( Z ) F • ˜ a : U ( N ) c • B (2)c : Z N B (2)c → B (2)c + d λ c , B (1)c → B (1)c + N λ c , ˜ a → ˜ a + λ c , ˜ F (˜ a ) → ˜ F (˜ a ) + dλ c ,A χη → A χη + λ c ,A χξ → A χξ − λ c ,A → A + N λ c . (4.43)The fermion kinetic terms are: (the charges follow from (4.42)) χγ µ (cid:16) ∂ + R A ( e a ) + N − p A χη + p A χξ − A (cid:17) µ P L χ + ηγ µ (cid:16) ∂ + R F ∗ ( e a ) − N − A χη + A (cid:17) µ P L η + ξγ µ (cid:16) ∂ + R F ( e a ) − N − A χξ − A (cid:17) µ P L ξ . (4.44)It is seen that each line is invariant under (4.43). In particular, the ( Z ) F charges are fixedby this requirement.The 1-form gauge invariant field tensors are, for the UV fermions χ , η , ξ , T = R A (cid:0) F (˜ a ) − B (2)c (cid:1) + N − p A χη − B (2)c ) + p A χξ + B (2)c ) − (cid:16) d A − N B (2)c (cid:17) , T = R F ∗ (cid:0) F (˜ a ) − B (2)c (cid:1) − N −
22 (d A χη − B (2)c ) + (cid:16) d A − N B (2)c (cid:17) , T = R F (cid:0) F (˜ a ) − B (2)c (cid:1) − N −
22 (d A χξ + B (2)c ) − (cid:16) d A − N B (2)c (cid:17) . (4.45)33he 6 D anomaly functional is124 π Z tr R A ( T ) + 124 π Z tr R F ∗ ( T ) + 124 π Z tr R F ( T ) . (4.46)Let us now extract the terms relevant to the ( Z ) F − [ Z N ] anomaly. From the χ contri-bution one has124 π (cid:20) − N − (cid:0) F (˜ a ) − B (2)c (cid:1) (cid:16) d A − N B (2)c (cid:17) − N ( N − (cid:16) d A − B (2)c + . . . (cid:17) (cid:21) , (4.47) η gives124 π (cid:20) N − p )tr (cid:0) F (˜ a ) − B (2)c (cid:1) (cid:16) d A − N B (2)c (cid:17) + N ( N − p ) (cid:16) d A − B (2)c + . . . (cid:17) (cid:21) (4.48)and the third line ( ξ ) gives:124 π (cid:20) − p tr (cid:0) F (˜ a ) − B (2)c (cid:1) (cid:16) d A − N B (2)c (cid:17) − pN (cid:16) d A − B (2)c + . . . (cid:17) (cid:21) . (4.49)Collecting terms, one finds that the coefficient of18 π ( B (2)c ) d A (4.50)is equal to N ( N − − N ( N − · N − p )( − N ) + N ( N − p ) + N p − pN = − N . (4.51)Following the usual procedure one calculates the 4 D mixed ( Z ) F − [ Z N ] anomaly, − N π Z Σ ( B (2)c ) δA (0)2 = N × Z N ( ± π ) = ± π × Z . (4.52)That is, the partition function changes sign under the fermion parity, χ, η, ξ → − χ, − η, − ξ .In other words, we found a ( Z ) F − [ Z N ] mixed anomaly in the UV theory.On the other hand, one finds no ( Z ) F anomaly in the IR, assuming the chirally sym-metric vacuum with the massless baryons B , B , B . This then cannot be the correctphase of the system. 34 Summary
In this work we have extended the study of mixed anomalies affecting a chiral discrete( Z ) F symmetry, found [2] in a simple chiral gauge theory ( ψη model), to a wider class ofmodels, the general Bars-Yankielowicz and the generalized Georgi-Glashow models.Writing the effects of instantons on the three U (1)’s associated with the three fermionsas U (1) ψ × U (1) η × U (1) ξ anomaly −−−−→ H , (5.1)the global symmetry of these models G f can be written, for {S , N, p } models, for instance,as G f anomaly −−−−→ SU ( N + 4 + p ) × SU ( p ) × H Z N × Z N +4+ p × Z p (5.2)and similarly for {A , N, p } models, with a replacement, N + 4 + p → N − p . Thedivision by various centers has been explained in Sec. 3.In both classes of the models, if one of N and p (or both) is odd, H , hence G f , has aconnected structure. It can be taken as H = U (1) × U (1) , (5.3)where U (1) , are arbitrary two of the nonanomalous combinations, U (1) ψη , U (1) ψξ , and U (1) ξη . It follows that, once the conventional anomaly matching equations are all satisfiedwith respect to G F , considering the mixed anomalies involving the 1-form discrete centersymmetry Z N does not provide us with any new information about the candidate phaseof the system. The UV-IR matching involving any new, mixed anomalies is a simpleconsequence of (i.e., included in) the conventional anomaly matching equations. This issimilar to what was found in [2] for odd N ψη models.For this reason, the main part of our analysis here has been focused on the models with N and p , both even. In all cases of this type, the global symmetry G f has two, disconnectedcomponents, as H = U (1) × U (1) × ( Z ) F . (5.4)( Z ) F is nonanomalous, as all other factors in G F , but the fact that it is nonanomaloushinges upon the integer instanton numbers18 π (cid:18)Z Σ tr F (cid:19) ∈ Z (5.5)and is not a simple result of an algebraic cancellation of the contributions from differentfermions, as is the case for the continuous, nonanomalous symmetries U (1) ψξ × U (1) ηξ .This can be checked by inspecting Eqs. (4.13), (4.28) and (4.44). For instance, in the35armup example of the {S , , } model, the effect of the chiral transformations, ψ → e − iπ ψ , η → e iπ η , ξ → e − iπ ξ , (5.6)(see Eq. (4.13)) is the extra phase in the partition function {− ( N + 2) + ( N + 6) − · } π (cid:18)Z Σ tr F (cid:19) · π = − π Z : (5.7)which is indeed irrelevant, but only because the instanton numbers are quantized to inte-gers. The nonanomalous ( Z ) F symmetry has thus a different status as compared to other,continuous nonanomalous symmetries such as U (1) χη , U (1) χξ and U (1) ηξ .But this means that, once all fields are coupled to the 1-form center Z N gauge fields (cid:0) B (2)c , B (1)c (cid:1) N B (2)c = dB (1)c , (5.8)and fractional ’t Hooft fluxes are allowed, a mixed ( Z ) F − [ Z N ] anomaly may arise. Inother words, there may be an obstruction against gauging the 1-form center Z N symmetryand 0-form ( Z ) F symmetry simultaneously.Our calculations show that such an obstruction (a generalized ’t Hooft anomaly) isindeed present.On the other hand, such an obstruction could not occur in the chirally symmetricconfining vacuum of Sec. 2.3, as the infrared fermions are all singlets of SU ( N ). Consistencyrequires that either the assumption of confinement or that of unbroken global symmetry(no condensates), or both, must be abandoned.There is no inconsistency in the other, possible vacua in the infrared (dynamical Higgsphase, Sec. 2.4 and Sec. 2.5), as U (1) χη , U (1) χξ and U (1) ηξ are broken spontaneously bythe condensate, so is the color-flavor locked 1-form center Z N symmetry.Note that the 0-form ( Z ) F symmetry itself does not need to be, and indeed is not,spontaneously broken, since all bifermion condensates are invariant under ψ, η, ξ → − ψ, − η, − ξ . (5.9)In fact, as this fermion parity coincides with an angle 2 π space rotation, a spontaneousbreaking of ( Z ) F would have meant the spontaneous breaking of the Lorentz invariance,which does not occur.In this respect, even though the mixed anomaly ( Z ) F − [ Z N ] found in [2] and con-firmed here for an extended class of models, looks similar at first sight to the mixed anomaly CP − [ Z N ] found recently [4] in the pure SU ( N ) Yang-Mills theory at θ = π , the way themixed anomaly manifests itself in the infrared physics is different. In the latter case, thenew anomaly is consistent with, or implies, the phenomenon of the double vacuum degen-eracy and the consequent spontaneous CP breaking [49], which was known from the QCD36ffective Lagrangian analysis [25, 26] and also from soft supersymmetry breaking pertur-bation [50, 51] of the exact Seiberg-Witten solutions [28, 29] of pure N = 2 supersymmetricYang-Mills theory.In our case, the mixed anomaly ( Z ) F − [ Z N ] means instead that confinement andthe full global chiral symmetries (no condensates) are incompatible: one or both must beabandoned. The dynamical Higgs phase discussed in Sec. 2.4, Sec. 2.5, seems to be fullyconsistent with this requirement.To conclude, the analysis presented here confirms that the result found in [2] - thatan extended symmetry consideration implies a dynamical Higgs phenomenon in a class ofchiral gauge theories - is not an accidental one peculiar to the simplest models consideredthere, but holds true in a much larger class of theories. Such a result should, in our view,be regarded as a general property of strongly-coupled chiral gauge theories. Acknowledgments
This work is supported by the INFN special research project grant “GAST” (Gauge andString Theories).
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A The mixed ( Z ) F − [ Z N ] anomaly: an alternativederivation In this Appendix, we show that our results on the mixed anomaly ( Z ) F − [ Z N ] foundin Sec. 4 do not depend on the rearrangement of the fermion tensors to term-by-termmanifestly invariant form, as done in Eq. (4.12), Eq. (4.14), Eq. (4.29), and Eq. (4.45).For concreteness, let us first take the warmup example of Sec. 4.1. The 6 D anomalyfunctional is, from (1.12), 124 π tr R S (cid:2) { F (˜ a ) − d A } (cid:3) +124 π tr R ∗ F (cid:2) { F (˜ a ) + d A + d A } (cid:3) +124 π tr R F (cid:2) { F (˜ a ) − A − A } (cid:3) . (A.1)For the purpose of finding the ( Z ) F anomaly, we expand these, and integrate once to findthe 5 D WZW action proportional to A . The variation of the form δA = 12 ∂ δA (0)2 , δA (0)2 = ± π , (A.2)then leads to an anomalous surface term - the anomaly in 4 D theory - given by the phase18 π Z Σ P δA (0)2 , δA (0)2 ± π (A.3)where P = − tr R S (cid:2) F (˜ a ) (cid:3) + ( N + p + 4) tr R ∗ F (cid:2) F (˜ a ) (cid:3) − p tr R F (cid:2) F (˜ a ) (cid:3) , (A.4)41 N = 4, p = 2), where the trace taken in a representation R is indicated by tr R . Nowtr R h(cid:0) F (˜ a ) (cid:1) i = tr R h(cid:0) F (˜ a ) − B (2) c + B (2) c (cid:1) i = tr (cid:2) R R (cid:0) F (˜ a ) − B (2) c (cid:1) + N ( R ) B (2) c d ( R ) (cid:3) = tr h R R (cid:0) F (˜ a ) − B (2) c ) + N ( R ) (cid:0) B (2) c (cid:1) d ( R ) i , (A.5)where R R indicates the matrix form appropriate for the representation R , N ( R ) its N -ality,and use was made of the fact thattr R (cid:0) F (˜ a ) − B (2) c (cid:1) = 0 , (A.6)valid for an SU ( N ) element in any representation. d ( R ) stands for the d ( R ) × d ( R ) unitmatrix, where d ( R ) is the dimension of the representation R . Calculating the above, onefinds tr R h(cid:0) F (˜ a ) (cid:1) i = D ( R ) tr F h(cid:0) F (˜ a ) − B (2) c (cid:1) i + d ( R ) N ( R ) (cid:0) B (2) c (cid:1) == D ( R ) tr F [ F (˜ a )] + (cid:2) − D ( R ) · N + d ( R ) N ( R ) (cid:3) (cid:0) B (2) c (cid:1) , (A.7)where D ( R ) is twice the Dynkin index T R ,tr (cid:0) t aR t bR (cid:1) = T R δ ab , (A.8)normalized as T R = 12 , D ( R ) = 1 , R = or ¯ . (A.9)Now 18 π Z Σ tr F (cid:2) F (˜ a ) (cid:3) ∈ Z , (A.10)and the first term in Eq. (A.7) corresponds to the conventional instanton contribution tothe ( Z ) F anomaly, which is known to be absent (for instance, see Eq. (5.7)) .Thus the new, mixed ( Z ) F − Z N anomaly is given by the second term of Eq. (A.7), X fermions ( d ( R ) N ( R ) − N · D ( R )) 18 π Z Σ (cid:0) B (2) c (cid:1) = X fermions ( d ( R ) N ( R ) − N · D ( R )) Z N . (A.11) The combination 18 π Z Σ { tr ˜ F − tr ˜ F ∧ tr ˜ F } is the second Chern number of U ( N ) and is an integer. The second term of the above is also an integer. {S , , } model of Sec. 4.1, Eq. (A.1), one gets ( ± π times)18 π Z Σ (cid:26) − (cid:18) · N ( N + 1)2 − N ( N + 2) (cid:19) + 10 ( N − N ) − N − N ) (cid:27) (cid:0) B (2) c (cid:1) = − N π Z (cid:0) B (2) c (cid:1) , (A.12)which is indeed the result found in Sec. 4.1.Note that for R = F (the fundamental) or R = F ∗ (antifundamental), d ( R ) = N , N ( R ) = D ( R ) = 1, therefore d ( R ) N ( R ) − N · D ( R )) = 0 , ( R = or ¯ ) : (A.13)these fermions do not contribute to the ( Z ) F − [ Z N ] mixed anomaly. And this explainsa somewhat curious feature in the results observed earlier in Eq. (4.21), Eq. (4.22) andEq. (4.51).The formula (A.11) is valid for a fermion in a generic representation, so it can beapplied at once to the general {S , N, p } and {A , N, p } models, yielding an extra phase inthe partition function under the fermion parity,∆ S = ± πN (cid:0) d ( S ) · N ( S ) − N · D ( S ) (cid:1) = ± πN (cid:18) N ( N + 1)2 · − N ( N + 2) (cid:19) = ± π , (A.14)for the {S , N, p } model, and∆ S = ± πN (cid:0) d ( A ) · N ( A ) − N · D ( A ) (cid:1) = ± πN (cid:18) N ( N − · − N ( N − (cid:19) = ± π , (A.15)for the {A , N, p }}