Probing the dynamics of chiral SU (N) gauge theories via generalized anomalies

We study symmetries and dynamics of chiral SU(N) gauge theories with matter Weyl fermions in a two-index symmetric (ψ) or anti-symmetric tensor (χ) representation, together with N ± 4+ p fermions in the anti-fundamental (η) and p fermions in the fundamental (ξ) representations. They are known as the Bars-Yankielowicz (the former) and the generalized Georgi-Glashow models (the latter). The conventional ’t Hooft anomaly matching algorithm is known to allow a confining, chirally symmetric vacuum in all these models, with a simple set of massless baryonlike composite fermions describing the infrared physics. We analyzed recently one of these models (ψη model), by applying the ideas of generalized symmetries and the consequent, stronger constraints involving certain mixed anomalies, finding that the confining, chirally symmetric, vacuum is actually inconsistent. In the present paper this result is extended to a wider class of the Bars-Yankielowicz and the generalized Georgi-Glashow models. It is shown that for all these models with N and p both even, at least, the generalized anomaly matching requirement forbids the persistence of the full chiral symmetries in the infrared if the system confines. The most natural and consistent possibility is that some bifermion condensates form, breaking the color gauge symmetry dynamically, together with part of the global symmetry.


Introduction
A few steps have been taken recently [1,2] to go beyond the conventional 't Hooft anomaly matching analysis in understanding the infrared dynamics of chiral gauge theories. The standard anomaly matching constraints and other generally accepted ideas, are usually not sufficient to pinpoint what happens in the infrared, where the system gets strongly coupled and perturbation theory has a limited power in predicting the phase and global symmetry realization patterns.
The tools which allow these new results come from the idea of the generalized symmetries, of gauging some 1-form discrete center symmetries and studying the consequences of mixed-'t Hooft-anomaly-matching conditions [3]- [16]. Most concrete applications of these new techniques so far refer to vectorlike gauge theories, such as pure SU(N) Yang-Mills, or adjoint QCD, where there is an exact center symmetry ( N for SU(N) theories), or QCD where the color center symmetry can be combined with U(1) V to give a color-flavor locked 1-form center symmetry. In these, vectorlike, gauge theories, the results from the new approach can be corroborated by the extensive literature, based on some general theorems [17,18], on lattice simulations [19]- [22], on the effective Lagrangians [23]- [26], on 't Hooft anomaly analysis [27], on the powerful exact results in N = 2 supersymmetrie theories [28,29], or on some other theoretical ideas such as the space compactification combined with 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 in the context of realistic model building beyond the standard model, e.g., with composite fermions, with composite Higgs bosons, or with dynamical composite models for dark matter, and so on.
It was these considerations that recently motivated the present authors to apply some of the new concepts and techniques to chiral gauge theories, to see if new insights in the physics of these theories can be gained by doing so [1,2]. In particular, in [2], a simple class of SU(N) gauge theories with Weyl fermions ("ψη model") was studied. For even N the (nonanomalous) symmetry of the system is where U(1) ψη is the anomaly-free combination of U(1) ψ and U(1) η , and (Z 2 ) F is the fermion parity, ψ, η → −ψ, −η.
In spite of the presence of fermions in the fundamental representation of SU(N) the system turns out to possess an exact discrete N center (1-form) symmetry 1 , which can be "gauged". Remember that the unfamiliar-sounding expression of gauging a discrete symmetry means simply that field configurations related by it are identified and the redundancy eliminated. This implies redefinition of the path-integral sum over the gauge field configurations appropriately. By applying this to the 1-form N of an SU(N) gauge theory, one arrives at an SU (N ) N gauge system, with consequent 1 N fractional instanton numbers. 2 Concretely, this can be done by introducing the 2-form gauge fields B (2) c , B (1) c , NB (2) c = dB (1) c , (1.5) and coupling to them the SU(N) gauge fields a and U(1) ψη × (Z 2 ) F gauge fields, A and A (1) 2 , appropriately. As for the SU(N) gauge field a, this can be achieved by embedding it into a U(N) gauge field a as and requiring the whole system to be invariant under the 1-form gauge transformation, (1.7) As the N is a color-flavor locked symmetry, Eq. (1.4), the U(1) ψη and (Z 2 ) F gauge fields must also be transformed simultaneously: (1.8) The relation (1.5) indicates that one has now an SU (N ) N connection rather than SU(N). It implies that there are nontrivial 't Hooft fluxes carried by the gauge fields in a closed two-dimensionl subspace, Σ 2 . On topologically nontrivial four dimensional spacetime of Euclidean signature containing such subspaces one has then 3 1 8π 2 where n ∈ N . The fermion kinetic term with the background gauge field is obtained by the minimal 2 In [2] we have gauged also the 1-form center symmetry N +4 ⊂ SU (N + 4), but the conclusion of the work did not depend on it. Here and in the rest of the present work, only the "color-flavor locked" N center symmetry will be considered. 3 Throughout, a compact differential-form notation is used. For instance, a ≡ T c A c µ (x) dx µ ; F = da+a 2 ; F 2 ≡ F ∧ F = 1 2 F µν F ρσ dx µ dx ν dx ρ dx σ = 1 2 ǫ µνρσ F µν F ρσ d 4 x = F µνF µν d 4 x , and so on.
coupling procedure as 4 with the obvious notation. We compute the anomalies by applying the Stora-Zumino descent procedure starting with a 6D anomaly functional 5 (1.12) The rest of the procedure for computing the (Z 2 ) F anomaly is standard: (i) one first integrates to get the 5D boundary action containing A 2 (WZW action); (ii) the variations of the form δA 13) leads to, via the anomaly-in-flow, the seeked-for anomaly in the 4D theory. The result is the partition function changes sign, under ψ, η → −ψ, −η, that is, there is a (Z 2 ) F anomaly. As the ( 2 ) F − [ N ] 2 mixed anomaly is obviously absent in the IR, we conclude that the confining chirally symmetric vacuum, in which conventional 't Hooft anomalies are saturated in the infrared by massless composite "baryons" (antisymmetric in A ↔ B), is not the correct vacuum of the system. As shown in [2], the dynamical Higgs vacuum, characterized by bifermion condensates, is instead found to be fully consistent. Several subtle features of the calculation and in the interpretation of the results are 4 The N charges of A and A (1) 2 in (1.11) are determined by the way U (1) ψη and (Z 2 ) F together reproduce ψ → e 4πi/N ψ and η → e −2πi/N η, as the reader can easily check. See [2]. 5 In going from (1.11) to (1.12) term are arranged so that the expression inside each bracket be 1-form gauge invariant. discussed carefully in [2].
The purpose of the present work is to investigate if the result found in the ψη model extends naturally to a wider class of the so-called Bars-Yankielowicz and the generalized Georgi-Glashow models. The gauge group is taken to be SU(N), and the matter fermion content is (p is a natural number) for the former (let us call them {S, N, p} models), and for the latter ({A, N, p} models). We will find that for all N and p, both even, the system possesses a (Z 2 ) F symmetry, which is nonanomalous, i.e., respected by standard instantons. Also, these models all enjoy a "color-flavor locked" N center symmetry, in spite of the presence of fermions in the fundamental (or anti-fundamental) representation. It is thus possible to gauge this center symmetry and study if, by doing so, the (Z 2 ) F symmetry becomes anomalous, as happened in the {S, N, 0} model. The paper is organized as follows. In Sec. 2 we discuss the conventional 't Hooftanomaly-matching analysis in all these models. A good part of this section is a review of [34]- [40], but there are some new results, especially concerning the Higgs phase, which we need later. As the global symmetry group is relatively large, the fact that one can find a set of gauge-invariant composite fermions which satisfy all the anomaly-matching equations at all, assuming the system to confine, is quite remarkable. Also, in all these models we find an alternative phase, also consistent with the anomaly matching criterion, characterized by certain bifermion condensates breaking color dynamically (dynamical Higgs phase) 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) effects of the strong SU(N) gauge interactions are taken into account. In Sec. 3, the symmetry of these 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, ( 2 ) F − [ N ] 2 , in all models with N and p both even, whereas such an anomaly is absent in the infrared (IR) in a confining vacuum with full global symmetry -one of the candidate vacua allowed by the conventional anomaly matching argument. Consistency implies that these vacua cannot be 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.
2 Theories and possible phases 2.1 {S, N, p} models The first class of theories is the ψη model with additional p pairs of fundamental and antifundamental fermions. Namely, the model is an SU(N) gauge theory with Weyl fermions in the direct-sum representation The indices run as i, j = 1, . . . , N , A = 1, . . . , N + 4 + p , a = 1, . . . , p .
These theories (the Bars-Yankielowicz models) will be denoted as {S, N, p} below. The ψη model corresponds to {S, N, 0}. The first coefficient of the beta function is and p is limited by 9 2 N − 3 before asymptotic freedom (AF) is lost. In the limit N fixed and p → ∞ we recover ordinary QCD with p flavors, although this is outside the regime of AF. The classical symmetry group is (2.5) We discuss for the moment only 0-form symmetries, leaving a more detailed discussion of the symmetry group to Sec. 3. 6 Anomaly breaks the symmetry group (2.5) to where the anomaly-free combination of U(1) ψ and U(1) η is with α ∈ Ê, and the anomaly-free combination of U(1) ψ and U(1) ξ is with β ∈ Ê. The choice of the two unbroken U(1)'s is somehow arbritrary, for example with γ ∈ Ê could be chosen as a generator. In Table 1 we summarize the fields and how they transform under the symmetry group. There are also discrete unbroken symmetries of the three U(1)'s: ( N +2 ) ψ , ( N +4+p ) η and ( p ) ξ . The relation between these discrete symmetries and the continuous non-anomalos group U(1) ψη × U(1) ψξ will be discussed in Sec. 3.

{A, N, p} models
The second class of models we are interested are SU(N) gauge theories with Weyl fermions in the direct-sum representation The indices run as Here p will be assumed to be less than 9 2 N + 3 so as to maintain AF. The symmetry group is (2.14) Anomaly breaks this group to where the anomaly-free combination of U(1) χ and U(1) η is 16) and the anomaly-free combination of U(1) ψ and U(1) ξ is Another possible anomaly-free combination is U(1) ηξ : In Table 2 we summarize the fields and how they transform under the symmetry group. There are also discrete unbroken symmetries:

Confining phase with unbroken global symmetries
The standard 't Hooft anomaly matching conditions were found to allow a chirally symmetric, confining vacuum in the model first proposed in [35]. Let us assume that no condensates form, the system confines, and the flavor symmetry is unbroken.

{S, N, p} models
The candidate massless composite fermions for the {S, N, p} models are the left-handed gauge-invariant fields: the first is anti-symmetric in A ↔ B and the third is symmetric in a ↔ b; their charges are given in Table 3. Writing explicitly also the spin indices they are all transforming under the { 1 2 , 0} representation of the Lorentz group. Table 4 summarizes the anomaly matching checks, via comparison between Table 1 and Table 3.

{A, N, p} models
The candidate massless composite fermions for the {A, N, p} model are: the first symmetric in A ↔ B and the third anti-symmetric in a ↔ b. Writing the spin indices explicitly they are: All anomaly triangles are saturated by these candidate massless composite fermions, see Table 6 ( Table 5 vs Tab. 2).

Dynamical Higgs phase in the {S, N, p} models
The broken phase for the {S, N, 0}, ψη model has also been studied earlier [40,46]. The composite scalar ψη in the maximal attractive channel is in the fundamental of both the gauge group and the flavor group. All details can be found in the references.
Something interesting happens for p > 0. Now there is another channel, ξη, which is gauge invariant and charged under the flavor group. We thus have a competition between two possible symmetry breaking channels, ψη and ξη. We assume that both condensates occur in the following way: where Λ is the renormailization-invariant scale dynamically generated by the gauge interactions 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 strengths of the one-gluon exchange potential for the two channels are, respectively, So the ψη channel is slightly more attractive, but such a perturbative argument is not really significant and we assume here that both types of condensates are formed. The resulting pattern of symmetry breaking is At the end the color gauge symmetry is completely (dynamically) broken, leaving color- (2.27) Making the decomposition of the fields in the direct sum of representations in the subgroup one gets Table 7. The composite massless baryons are subset of those in (2.19): It is quite straightforward (and actually almost trivial) to verify -we leave it to the reader as an excercise -that the UV-IR anomaly matching continues to work, with the UV fermions in Table 7 and the IR fermions in Table 8.

Dynamical Higgs phase in the {A, N, p} models
In the {A, N, p} model there is a competition between two possible bifermion symmetry breaking channels χη and ξη. This time, the MAC criterion would favor the ξη condensates against χη. Indeed, the strength of the one-gluon exchange potential for the two channels are, respectively, Again, these perturbative estimates are not excessively significant, and we assume that both condensates occur as: The pattern of symmetry breaking is The color gauge symmetry is partially (dynamically) broken, leaving color-flavor diagonal global SU(N − 4) cfη 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 − 4 + p) η generated by: Making the decomposition of the fields in the direct sum of representations in the subgroup one arrives at Table 9.  The composite massless baryons are subset of those in (2.21): In the IR these fermions saturate all the anomalies of the unbroken chiral symmetry. This can be seen by an inspection of Table 10 and Table 9, with the help of the following observation.
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, forming massive composite mesons, ∼ χ 3 χ 3 , which also decouples. It is again neutral with respect to all of To summarize, SU(4) c is invisible (confines) in the IR, and only the unpaired part of the η 1 fermion ¯ remains massless, and its contribution to the anomalies is reproduced exactly by the composite fermions, (2.34).
Comment: The massive mesons {χ 2 η 3 }, {η 4 ξ 2 }, {χ 3 χ 3 } are not charged with respect to the flavor symmetries surviving in the infrared. It is tempting to regard them as a toymodel "dark matter", as contrasted to the fermions B AB which constitute the "ordinary, visible" sector.

Symmetries
In the conventional 't Hooft anomaly analysis discussed above only the algebra of the group matters. In this section the symmetry of the models will be examined with more care, by taking into account the global aspects of the color and flavor symmetry groups. Let us first consider the Bars-Yankielowicz ({S, N, p}) models.
For a {S, N, p} model, the classical symmetry group of our system is given by The color group is G c = SU(N) c , and its center acts non-trivially on the matter fields: (n ∈ ). The division by Z N in Eq. (3.1) is due to the fact that the numerator overlaps with the center of the gauge group (see Sec. 3.2 below). Another, equivalent way of writing the flavor part of the classical symmetry group is 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 instanton vertex 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 : (see Eq. (2.7)-Eq. (2.9)). Three different discrete sub-groups left unbroken are The question is: which is the correct anomaly-free sub-group? The anomaly affects only the U(1) part of the group so that the total symmetry group is broken as follows ξ are all part of the anomalyfree sub-group, but one must find the minimal description, in order to avoid the doublecounting. H is at the bottom of the following sequence of covering spaces: The first arrow can be understood as follows. U(1) ηξ can always be obtained by a combination of the other two continuous groups, by choosing (using conventions for α, β, γ as in Eq. (3.4)) Also, the fundamental element of (Z N +4+p ) η can be obtained by a combination of the fundamental of (Z N +2 ) ψ (k = 1 in Eq. (3.5)) with the U(1) ψη element . (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 holds, i.e., whether the discrete part of the group can be entirely expressed as a subgroup of the continuous U(1) groups. The requirement (3.11) is equivalent to where ∼ means the equality with possible additional terms of the form 2π× integer allowed.
It follows from the last two equations that 13) which inserted in the first gives that is, (2 + p)m + np = 1 + (N + 2)ℓ , m, n, ℓ ∈ . i.e., ( N +2 ) ψ is not entirely contained in U(1) ψη × U(1) ψξ ; only the even elements of ( N +2 ) ψ are: One can show however that for p, N both even where (Z 2 ) F is the fermion parity generated by In fact, admitting the presence of fermion parity the requirement (3.12) gets modified to and thus (2 + p)m + np = (N + 2)ℓ , m, n, ℓ ∈ . which always has a solution.
To summarize, when p and N are both even, one has i.e. it has two disconnected components. U(1) 1 and U(1) 2 are any two out of U(1) ψη , U(1) ψξ , and U(1) ηξ . If p and/or N is odd, instead, it has only one connected component.

N ⊂ H
We focus now on the center of the color SU(N) group, N . We first show that when N, p are both even, To prove this, ab absurdo, assume that U(1) ψη × U(1) ψξ does contains N : that is (Remember that the symbol "∼" here indicates equality modulo terms of the form 2πn, n ∈ .) We first eliminate α from the first two. As N, p are both even, multiply the first by N +2 2 and the second by N +4+p 2 (both integers) and add. We get We next prove that if at least one of N and p is odd, then 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, 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 when N and p are both even. This means that (cfr. Eq. (3.24)) where the anomaly acts on the U(1) part as  are the nonanomalous symmetry group of the system, but we need a minimum set without redundancy. For p = 0, the χη model, the result is:

Illustration
Let us illustrate the symmetry of our systems graphically, taking a few concrete models of the type, {S, N, p}.
It is convenient to introduce the following notation. We parameterize a generic U(1) ⊂ so that This U(1) winds gcd(t 1 , t 2 , t 3 )-times around the three-torus T 3 . In general, given a specific direction, we choose the "fundamental" generator for which gcd(t 1 , t 2 , t 3 ) = 1 so that periodicity in θ is exactly 2π. In this notations the three fundamental U(1)'s are generated by and the non-anomalus ones are generated by • For p = 0, the ψη model, this has been discussed in detail in [2] and the result is: • For p = 1, independentely on N, H has only one connected component. In Figure 1 we show the case N = 3. One possible way to parameterize H is Note that U(1) ψξ contains ( N +2 ) ψ and U(1) ηξ contains ( N +5 ) η , so together they contain the whole discrete lattice ( N +2 ) ψ × ( N +5 ) η . We can define the group U (1) as the one that contains Z N , and is the one generated by • For p = 2, N odd, H has only one connected component. In Figure 2 we show the graphs for the case N = 3. One possible way to parameterize H is 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 2 division in (3.50). The group U (1) that contains Z N is the one generated by (3.49). If we define U(1) generated by we can write • 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 We can define the group U(1) generated by (3.49) but this time it contains only ZN 2 . In general it is not possible to write Z N ⊂ U(1) ′ × (Z 2 ) F , both U(1)'s are necessary, for the {S, 3, 2} model.
although N = 4, p = 2 is an exception as we will see in the warmup example in Sec. 4.1.
The generalized (mixed) anomaly of the type ( 2 ) F − [ N ] 2 was studied in detail in [2] for the {S, N, 0} ("ψη") model. We have briefly reviewed the method and results found there at the end of Introduction. This study is extended below to a wider class of models discussed in Sec. 2 and Sec. 3. The global structure of the anomaly-free symmetry group revealed in 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 will set focus.

A warmup example {S, 4, 2}
We first consider a simplest, nontrivial model {S, 4, 2} and set up the calculation of the mixed anomalies, making a brief note on some general features of the gauging of the discrete 1-form N symmetry, on the idea of "( 2 ) F gauge field", and paying special attention to the way the fermions transform nontrivially under the 1-form N gauge transformation. The same procedure can then be easily extended to more general cases discussed later. Even though the fact that has been proven in general in Sec. 3.2, we need an explicit solution for this model, to fix the charges of the fermion fields under the 1-form N symmetry. From we see that a simple solution in this case is to take β = 0, and γ = + π 2 . It is easily seen that N is realized as a U ηξ (1) × ( 2 ) F transformation with 2 : ψ : e +iπ ; η : e −iπ ; ξ : e +3iπ . We introduce accordingly, • A: U(1) ηξ 1-form gauge field, • A 2 : (Z 2 ) F 1-form gauge field, •ã: U(N) c 1-form gauge field, • B (2) c : Z N 2-form gauge field.
The original SU(N) gauge field a is embedded in a U(N) gauge fieldã as As explained in [3], [4], one defines this way a globally well-defined SU(N)/ N connection. The imposition of the local, 1-form gauge invariance (4.6) below, eliminates the apparent increase of the degrees of freedom (in going from SU(N) to U(N)) on the one hand, and at the same time allows to "gauge away" the center N variation of Polyakov or Wilson loops e i a → e 2πi/N e i a , (4.5) on the other.
The 1-form gauge transformation acts on these fields as: As we are here dealing with a N which is a color-flavor locked symmetry the fermion fields also 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 "( 2 ) F gauge field", A 2 , which formally looks like an ordinary U(1) gauge field. Restoring momentarily the suffices for the differential forms, can be regarded as an invariant form of the ( 2 ) F gauge field, 2A is a 2π periodic scalar function (angle). It is an example of an "almost flat connection": it satisfies 2 dA  See [2] for more discussions. The fermion kinetic terms are: each of which is indeed invariant under (4.6) and (4.7). Note that the choice of the 2 charges, (1, −1, +3) for (ψ, η, ξ) fields (see Eq. (4.3)) is dictated by the requirement that the redundancy (4.1) involving the discrete symmetries 2 and N be formally expressed as an invariance under (4.6) with a continuous gauge function λ c = λ µ c (x) dx µ . The 1-form gauge invariant field tensors are, for the UV fermions ψ, η, ξ, By rearranging things so that each term in the bracket is manifestly invariant under (4.6) and (4.7), this can be rewritten as (4.12) In the confining vacuum with the full global symmetry, discussed in Sec. 2.3, the infrared degrees of freedom would be the (massless, by assumption) composite fermions B 1 , B 2 , B 3 , (2.19). Their kinetic terms are given by respectively. Though this formula appears to depend on B (2) c due to the way things have been arranged to make each term manifestly invariant, B c actually drops out completely, reflecting the fact that B 1 , B 2 , B 3 are all color SU(N) singlets: there are no gauge kinetic terms in their action. As a result, there would be no mixed anomalies in the IR due to the gauging of N 1-form symmetry.
Note that the same cannot be said of the formula Eq. (4.12) in the UV theory. Because, for instance, for the fundamental representation, the B c dependence of the expressions in Eq. (4.12) is not 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 invariance of our calculation step by step, the same final result can be obtained (as it should) by working with a not-term-by-term-manifestly-invariant expression Eq. (1.12). This is shown in Appendix A. As a bonus, the discussion there explains some interesting aspect of our results below.
The rest of the calculations follows that done in [2]. From Eq. (4.12) one finds the 6D anomaly functional in the UV theory 8 , Keeping only the relevant terms, the first line (ψ) gives the second line (η) gives  Following the usual procedure (e.g., Eq. (1.13), Eq. (1.14)) we find the mixed ( 2 ) F −[ N ] 2 anomaly in 4D: 8 Even though we follow here the Stora-Zumino descent procedure for calculating the anomalies, there is no problem obtaining the same resultsà la Fujikawa [52], staying in 4D: the idea of gauging the center N symmetry in itself has nothing to do with the introduction of the two extra dimensions. This was explicitly shown in [2] for the ψη model. Namely, the partition function suffers from a sign change under the fermion parity transformation. On the other hand, one would find no ( 2 ) F anomaly in the IR, if one would assume the chirally symmetric vacuum with the massless baryons B 1 , B 2 , B 3 of Sec. 2.3. The contradiction can be avoided by assuming that the system actually is in a dynamical Higgs phase such as the one discussed in Sec. 2.4.

General {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 warmup example, we verify anew for N, p both even, by solving the equations 9 : concretely. Indeed, it is sufficient to find one good solution. A possible solution is 10 which is a solution with the ( 2 ) F signs +, −, + for the ψ, η, ξ fields in Eq. (4.24), respectively. The above solution Eq. (4.25) can be simply rewritten as As in any anomaly calculation we couple the system to the appropriate background gauge fields, • A ψη : U(1) ψη 1-form gauge field, • A ψξ : U(1) ψξ 1-form gauge field, •ã: U(N) c 1-form gauge field, • B (2) c : Z N 2-form gauge field.
Under the 1-form gauge transformation the fields transform as where the charges follow from (4.24) and (4.26). The fermion kinetic terms are: It can be checked readily that each line is invariant under Eq. (4.27). In particular, the ( 2 ) F charges are fixed by this requirement. The 1-form gauge invariant field tensors are, for the UV fermions ψ, η, ξ, where appropriate factors of B (2) c are added and subtracted so that each term in the bracket is invariant under the 1-form gauge transformations (4.27). Of course, the final result does not depend on such a rewriting: see Appendix A.
The 6D anomaly functional is

3
(4.32) and the third line (ξ) gives: (4.33) Collecting terms, one finds that the coefficient of A somewhat curious feature of this result (and of Eq. (4.21)) is that only fermions in a higher representation contribute to the anomaly. The reason for this will become clear in an alternative derivation discussed in Appendix A.
Following the usual procedure one calculates the 4D mixed ( 2 ) F − [ N ] 2 anomaly. One finds an extra phase in the partition function associated with the fermion parity transformation in the presence of the N gauge fields, there is a ( 2 ) F − [ N ] 2 mixed anomaly in the theory. On the other hand, one finds no ( 2 ) F anomaly in the IR, if one assumes the symmetric vacuum 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 that the chirally symmetric vacuum described by the baryons B 1 , B 2 , B 3 cannot be realized dynamically.
We note again that such an inconsistency is avoided, assuming that the system is in the dynamical Higgs phase: the color-flavor locked 1-form symmetry is spontaneously broken. ("χη model"), has been studied, and the result of the analysis (unpublished) turns out to 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 2 ) F factor orthogonal to other continuous symmetry group. It gets anomalous under the 1-form gauging of a N center symmetry. This anomaly cannot be reproduced in the infrared, if the vacuum is assumed to be confining, and to keep the full global symmetries. Such a vacuum cannot be realized dynamically.
Below we study a more general class of {A, N, p} models, with p additional pairs of fermions in ⊕¯. We check first Call α and β the angles associated with U(1) χη and U(1) χξ , The condition (4.38) means that It turns out that any two of these imply the third: there is an arbitrariness to choose from multiple of solutions. A possible solution is which is a solution with the ( 2 ) F signs in Eq. (4.40), −π, +π, −π for the χ, η, ξ fields, respectively. Actually the solution Eq. (4.41) is, simply, The color-flavor locked N transformation, (4.40) and (4.42), together with the normalization of the 1-form gauge field λ c , fix the charges of the fermion fields in Eq. (4.44) below.
Under the 1-form gauge transformation The fermion kinetic terms are: (the charges follow from (4.42)) It is seen that each line is invariant under (4.43). In particular, the ( 2 ) F charges are fixed by this requirement. The 1-form gauge invariant field tensors are, for the UV fermions χ, η, ξ, (4.45) The 6D anomaly functional is Let us now extract the terms relevant to the ( 2 ) F − [ N ] 2 anomaly. From the χ contribution one has
In other words, we found a ( 2 ) F − [ N ] 2 mixed anomaly in the UV theory. On the other hand, one finds no ( 2 ) F anomaly in the IR, assuming the chirally symmetric vacuum with the massless baryons B 1 , B 2 , B 3 . This then cannot be the correct phase of the system.

Summary
In this work we have extended the study of mixed anomalies affecting a chiral discrete ( 2 ) F symmetry, found [2] in a simple chiral gauge theory (ψη model), to a wider class of models, 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 fermions as the global symmetry of these models G f can be written, for {S, N, p} models, for instance, as and similarly for {A, N, p} models, with a replacement, N + 4 + p → N − 4 + p. The division 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 a connected structure. It can be taken as where U(1) 1,2 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 satisfied with respect to G F , considering the mixed anomalies involving the 1-form discrete center symmetry Z N does not provide us with any new information about the candidate phase of the system. The UV-IR matching involving any new, mixed anomalies is a simple consequence of (i.e., included in) the conventional anomaly matching equations. This is similar 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, disconnected components, as On the other hand, such an obstruction could not occur in the chirally symmetric confining vacuum of Sec. 2.3, as the infrared fermions are all singlets of SU(N). Consistency requires 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 Higgs phase, Sec. 2.4 and Sec. 2.5), as U(1) χη , U(1) χξ and U(1) ηξ are broken spontaneously by the condensate, so is the color-flavor locked 1-form center N symmetry.
Note that the 0-form ( 2 ) 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 2π space rotation, a spontaneous breaking of ( 2 ) F would have meant the spontaneous breaking of the Lorentz invariance, which does not occur. In this respect, even though the mixed anomaly ( 2 ) F − [ N ] 2 found in [2] and confirmed here for an extended class of models, looks similar at first sight to the mixed anomaly CP − [ N ] 2 found recently [4] in the pure SU(N) Yang-Mills theory at θ = π, the way the mixed anomaly manifests itself in the infrared physics is different. In the latter case, the new anomaly is consistent with, or implies, the phenomenon of the double vacuum degeneracy and the consequent spontaneous CP breaking [49], which was known from the QCD Effective Lagrangian analysis [25,26] and also from soft supersymmetry breaking perturbation [50,51] of the exact Seiberg-Witten solutions [28,29] of pure N = 2 supersymmetric Yang-Mills theory.
In our case, the mixed anomaly ( 2 ) F − [ N ] 2 means instead that confinement and the full global chiral symmetries (no condensates) are incompatible: one or both must be abandoned. The dynamical Higgs phase discussed in Sec. 2.4, Sec. 2.5, seems to be fully consistent with this requirement.
To conclude, the analysis presented here confirms that the result found in [2] -that an extended symmetry consideration implies a dynamical Higgs phenomenon in a class of chiral gauge theories -is not an accidental one peculiar to the simplest models considered there, 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. For the purpose of finding the ( 2 ) F anomaly, we expand these, and integrate once to find the 5D WZW action proportional to A 2 . The variation of the form