The ${\Bbb Z}_2$ anomaly in some chiral gauge theories

We revisit the simplest Bars-Yankielowicz (BY) model (the $\psi\eta$ model), starting from a model with an additional Dirac pair of fermions in the fundamental representation, together with a complex color-singlet scalar $\phi$ coupled to them through a Yukawa interaction. This model possesses a color-flavor-locked 1-form ${\Bbb Z}_N$ symmetry, due to the intersection of the color $SU(N)$ and two nonanomalous $U(1)$ groups. In the bulk, the model reduces to the $\psi\eta$ model studied earlier when $\phi$ acquires a nonzero vacuum expectation value and the extra fermions pair up, get massive and decouple (thus we will call our extended theory as the ``X-ray model"), while it provides a regularization of the $\Bbb Z_2$ fluxes needed to study the $\Bbb Z_2$ anomaly. The anomalies involving the 1-form ${\Bbb Z}_N$ symmetry reduce, for $N$ even, exactly to the mixed ${\Bbb Z}_2$ anomaly found earlier in the $\psi\eta$ model. The present work is a first significant step to clarify the meaning of the mixed ${\Bbb Z}_2-[{\Bbb Z}_N^{(1)}]^2$ anomaly found in the $\psi\eta$ and in other BY and Georgi-Glashow type $SU(N)$ models with even $N$.


Introduction
The dynamics of two wide classes of chiral SU(N) gauge theories -the so-called Bars-Yankielowicz (BY) and generalized Georgi-Glashow (GG) models [1]- [6] -has been reexamined recently [7][8][9], in the light of a gauged color-flavor locked N 1-form symmetry 1  and of the stronger forms of 't Hooft anomaly matching constraints following from that.In particular, certain mixed anomalies involving a 2 symmetry were found to imply, in a class of theories with even N, 2 that chirally symmetric confining vacua in these models, where the global symmetries in the infrared are saturated by the hypothetical massless composite fermions were inconsistent.These massless "baryons" reproduce the conventional 't Hooft anomalies but do not match the mixed 2 − [ N ] 2 anomaly.Dynamical Higgs vacua, characterized by color-flavor locked bifermion condensates, are 1 From now on, whenever there might be confusion, we will indicate a 1-form symmetry with the apex notation, e.g. the N 1-form symmetry as (1) N . 2 More precisely, with even N and with an even number p of Dirac pairs of fermions in the fundamental representation [7][8][9].We call this class of models Type I in this note; others will be referred to as Type II.
instead found to be compatible with the indications coming from the tighter consistency conditions involving the 2 anomaly [7][8][9].An independent argument [10], following from the requirement that the so-called strong anomalies be reproduced correctly in an effective low-energy action in terms of the assumed set of infrared degrees of freedom, provides a solid support for the dynamical Higgs scenario.
The arguments based on the mixed 2 − [ N ] 2 anomalies have been put in question in [11].The problem boils down to the singular nature of the external " 2 gauge field" A 2 , introduced in [7][8][9] to construct the color-flavor 1-form N symmetry which is due to the intersection3 SU(N) ∩ { 2 × U(1) ψη }.The 2 gauge field needs to wind along a closed loop L, to parametrize the holonomy and to give the color-flavor-locked 1-form N symmetry. 4Such a field contains necessarily a singularity (i.e., a singular 2 vortex) [7] somewhere inside the closed 2D space Σ 2 bounded by L. The authors of [11] show that, by choosing instead a (regular, hence legitimate) " 2 gauge field" A 2 such that (cfr.(1.1)) the flux carried by the N gauge field B (2) c becomes twice those used in [7], and accordingly the anomalies found there would disappear.However, (1.3) means that such a background 2 gauge field corresponds to the trivial holonomy i.e., no transformation (an identity element of 2 ).
To grasp correctly the main issue it is indeed necessary to distinguish the concepts of the global 1-form N symmetry from the gauged version of it.The former, a color-flavor locked N symmetry, is a generalization of the familiar center symmetry of pure SU(N) Yang-Mills theory.This symmetry certainly exists in the ψη and other models studied in [7][8][9], but in itself it does not lead to any consistency condition.It is another story if one tries to gauge this 1-form N symmetry, by introducing the N gauge field B (2) c with a proper N flux (cfr.(1.4)) [12][13][14] Such a gauging may encounter a topological obstruction (a 't Hooft anomaly).If it does, then there are new, nontrivial UV-IR matching conditions [15]- [36].This is indeed what was found in [7][8][9].The question is whether the anomalies and their consequences discussed there are to be trusted, in view of the fact that the argument made use of a singular external (non-dynamical) A 2 gauge field, (1.1).The present work aims to clarify the sense of the anomalies found in [7][8][9].We start with the simplest BY model ("ψη" model) with an extra pair of fermions (q, q) in the fundamental representation, which acts as a sort of regulator field.When a gauge-invariant, complex scalar field coupled to them through a Yukawa potential term gets a nonvanishing vacuum expectation value (VEV), v, the fermions q, q get mass and decouple,5 below ∼ v. Namely, this extended model (which we call the X-ray model) reduces, below the decoupling mass scale v, to the previously considered ψη model. 6his work is organized as follows.In Sec. 2 we introduce the extended model and discuss its symmetries.Before taking into account the scalar VEV, the model is of type II: conventional 't Hooft anomaly matching discussion allows a chirally symmetric, confining vacuum as well as a dynamical Higgs phase characterized by certain bifermion condensates.The model reduces to the previously studied ψη model at mass scales below the scalar VEV, v, where the extra fermions pair in a Dirac fermion, get massive and decouple.Sec. 3 is dedicated to the gauging of the color-flavor locked 1-form N symmetry and to the calculation of the consequent mixed anomalies.The generalized anomaly found in the X-ray model, which is free from the subtleties related to the singular A 2 field [7], reduces to the 2 − [ N ] 2 anomaly [7], precisely for even N (i.e.type I) theories.In Sec. 4 we discuss a few subtle issues related to the decoupling of the fermions q, q.The summary and conclusion are in Sec. 5.

The model and the color-flavor-locked 1-form N (center) symmetry
We consider the ψη model, in which a Dirac pair of fermions in the fundamental representation of SU(N) c , q and q, are added.In other words, we start with a generalized Bars-Yankielowicz model, with Weyl fermions7 in the direct-sum representation The global symmetry of the model is where U(1) ψη and U(1) ψξ are two anomaly-free combinations of the chiral U(1) symmetries associated with the fermions, ψ, η and χ.
We shall rename the fields as η N +5 = q and ξ = q below, so that the matter content is We furthermore add a color-singlet complex scalar φ coupled to the (q, q) pair as, ∆L = g Y φ q q + h.c. .
The Yukawa coupling (2.5) breaks the global symmetry as where the charges are given in Table 1.
The Yukawa coupling breaks explicitly part of the global symmetry of the original model, (2.1), (2.2).The implications of the conventional 't Hooft anomaly-matching conditions [37], with respect to the unbroken global symmetry, therefore remain the same as in the original generalized Bars-Yankielowicz model, (2.1),(2.2).The model is of Type II: 't Hooft anomaly matching allows both dynamical Higgs phase (with bifermion condensates) and confining, chirally symmetric phase (with no condensate formation).See App. A.
We assume that the potential for the φ field is such that it acquires a nonvanishing VEV, φ = v ≫ Λ ψη . (2.7) The fields and charges of the X-ray model with respect to the nonanomalous symmetries.The last symmetry, Ũ (1), is not linearly independent, but it is particularly useful to define it for our discussion.
The system at mass scales reduces exactly to the ψη model, studied in [7]- [9], as the fermions q and q get mass and decouple.The global U(1) V and Ũ(1) symmetries remain unbroken, they reduce respectively to the identity ½ and to U(1) ψη when the fermions q and q decouple.The U(1) 0 symmetry is broken as where 2 acts as ψ → −ψ , η → −η . (2.10) We refer to this model as the X-ray theory.
Clearly, besides the ψη model, the breaking of U(1) 0 introduces also a massless NGB.However, the NGB cannot couple to the ψη degrees of freedom through relevant or marginal operators:8 in the limit Λ ≪ v, the NGB sector decouples.
As U(1) 0 and Ũ (1) symmetries are free of (strong) anomalies, one may introduce external regular gauge fields, A 0 and Ã, respectively.

Color-flavor locked 1-form N symmetry
As the idea of color-flavor locked N 1-form symmetry is central below, let us briefly review it.Let us consider an SU(N) gauge theory with a set of the massless matter Weyl fermions {ψ k }.In general, the color N symmetry is broken by the fermions (unless the fermions present are all in the adjoint representation of SU(N)).However the situation changes if some global, nonanomalous U(1) symmetries, U(1) i , i = 1, 2, . .., are present, such that when U(1) i are gauged (in the usual sense, by the introduction of external gauge fields A µ i ), the color N ⊂ SU(N) and the U(1) i transformations can compensate each other for the fermions.This allows to define a global color-flavor locked N generator on Wilsons loops that stretch along the non-contractible loop L is ) i : where a ≡ a A µ t A dx µ is the SU(N) gauge field, A i is the U(1) i gauge field, and the integers p i defines an embedding of N ֒→ U(1) i .
As, locally, (2.11) can be realized as a gauge transformation, it can fail to be a symmetry only if it ruins the periodicity9 of the fermion fields.To check it, one should compute the action of (2.11) on the ψ k Wilson loop, i.e.
(here ψ k transforms under SU(N) in the irrep R k with N-arity N k , and has charge the fermions periodicity conditions are preserved and (2.11) defines a new color-flavor locked N 1-form symmetry.
As the ordinary N center transformation, such a color-flavor combined N center symmetry is still just a global 1-form symmetry.
A more powerful idea is to introduce the gauging of this 1-form symmetry and studying possible topological obstructions in doing so (generalized 't Hooft's anomalies) [15]- [36].As in the case of conventional gauging of 0-form symmetries, the idea of gauging is that of identifying the field configurations connected by the given symmetry transformations, and of eliminating the double counting in the sum over field configurations.However, as one is now dealing with a 1-form symmetry, the associated gauge transformations are parametrized by a 1-form Abelian gauge function10 λ = λ µ (x)dx µ , see (3.9) below.
3 Gauging 1-form N symmetry: mixed anomalies We consider now the gauging of the 1-form N symmetry in the X-ray model, that arises because the subgroup (see Table 2) acts trivially on any field of the theory. 11In other words, the symmetry group that acts faithfully on the fundamental fields is so to get all the 't Hooft anomalies of the theory we should consider a gauge connection of (3.3) rather than by the simple product principal bundle To gauge (3.4) it is enough to introduce the U(1) gauge connections C and C 0 in addition to the dynamical color gauge SU(N) field, a.However, by doing so, one obtains only a subset of all the possible gauge connections allowed by the gauging of (3.3): gauging (3.3) one can allow C, C 0 and a not to be proper gauge connection, individually, e.g. one can allow fractional Dirac quantization for C and C 0 .A very convenient way to describe a generic gauge connection for (3.3) is by introducing a pair of fields [15] where B c is a well-defined 12 U(1) gauge connection, and B c is a 2-form gauge field that satisfies NB (2)  c = dB (1)  c . (3.6) 11 Also, as it is possible to gauge the 1-form N symmetry together with U (1) ψη , U (1) V and U (1) 0 .Here we choose to proceed with gauging N lying in the intersection (3.1).
12 With well-defined U (1) connection we mean that they satisfy the usual Dirac quantization condition.
Then we embed a, C and C 0 into where a is a U(N) connection, and A 0 and Ã are well-defined U(1) connections 13 .Doing so, the N 1-form symmetry of the original group is embedded in a continuous 1-form symmetry parameterized by the U(1) gauge connection λ c , which cancel any local degrees of freedom introduced by B c .Local physics is not affected by these global issues, so the fermionic Lagrangian (locally) still reads However, as the faithful symmetry group is (3.3), we can express this Lagrangian in terms 13 In this definition, there is an ambiguity, as we could have set c instead.The construction would be equivalent, but, to describe the same background, we would need to add some integer flux for A 0 .The same sign ambiguity is present also for the ψη model.We will comment on the consequences of this sign choice on anomalies in footnote 16.
of well-defined geometrical entities (well-defined gauge connection) as which is explicitly invariant under the 1-form symmetry (3.9).The effective field-strength tensors acting on the fermions are accordingly: Note that by turning off the 1-form gauge fields B (2) c = 0 , one goes back to the standard SU(N) × Ũ(1) × U(1) 0 gauge theory.
The anomalies are compactly expressed by a six-dimensional (6D) anomaly functional [38,39] Expanding the 6D anomaly functional (3.13), one finds by making use of the known formulas for the traces of quadratic and cubic forms in different representations.Note that the terms proportional to tr c ( Fc − B c ) 3 and tr c ( Fc − B c ) 2 in (3.14) cancel completely as they should.Thus the anomalies are expressed by the last four lines of (3.14) only: Below we are going to extract the mixed anomalies, involving the U(1) 0 or Ũ (1) gauge fields, A 0 , Ã, together with the 1-form N gauge field, (B c ).To compute such anomalies explicitly it is useful to take as our spacetime manifold the 4-torus, T 4 = T 2 1 ×T 2 2 , and We recall again that if (B c , B c ) is set to zero, the UV anomalies simply express the conventional 't Hooft anomaly triangles involving the U(1) 0 × Ũ (1) background fields, and by construction those are matched by the assumed set of the massless baryons of a candidate IR theory such as the one discussed in Appendix B. What we shall exhibit below is only the new, stronger anomalies introduced by the gauging of the 1-form N symmetry.As will be discussed below (Sec.3.3) the consequence of these is that the confining, symmetric vacuum with just one massless baryon and no other nontrivial sectors, is not consistent.

Ã − B (2) c 2 anomaly
To calculate the anomaly in Ũ (1) caused by the introduction of the 1-form N gauge fields, let us briefly recall the procedure for calculating the anomalies in 4D theory according to the Stora-Zumino descent procedure [38][39][40], starting from the 6D anomaly functional, (3.15), in our case. 14One collects the terms of the form, B (3.17) yields, by anomaly inflow, the anomalous variation in the (boundary) 4D theory The Ũ(1) symmetry is broken (i.e., gets anomalous) by the generalized 1-form gauging of the N .

3.2
An analogous calculation leads to the U(1) 0 anomaly due to the 1-form gauging of the N symmetry, This appears to imply that the U(1) 0 symmetry is also broken by the 1-form gauging of the N symmetry.However, the scalar VEV φ = v breaks spontaneously the U(1) 0 symmetry to 2 .It means that, in contrast to (3.19), (3.20), the variation (3.21) cannot be used to examine the generalized UV-IR anomaly matching check.For that purpose, we can use only the nonanomalous 15 and unbroken symmetry operation, i.e., variations corresponding to a nontrivial 2 transformation δα 0 = ±π.Taking into account the nontrivial 't Hooft flux (3.7, 3.16)), and the crucial coefficient of the anomaly, K 0 = N 2 (N + 3), it is seen that the partition function changes sign for even 16 N.We reproduce exactly the 2 anomaly found in [7].

Chirally symmetric vacuum versus dynamical Higgs phase
Now what is the implication of the mixed anomalies found in the X-ray model, (3.19), (3.22) to the physics in the infrared, that is, the phase of the ψη model?We consider here two particularly interesting dynamical possibilities, a confining, chirally symmetric vacuum and a dynamical Higgs phase, which are both known to be compatible with the conventional 't Hooft anomaly-matching constraints.
If we assume that the infrared system was confining, chirally symmetric one, with no bifermion condensates forming, then the conventional 't Hooft anomalies would be matched by a low-energy theory consisting just of a single color-singlet massless composite fermion, the baryon B 11 ∼ ψηη (see Appendix B).Knowing its quantum numbers, we can construct the infrared anomaly functional, following the same procedure used at the beginning of this section.The answer is the expression (B.3), which does not contain the 1-form gauge field B (2) c : it reproduce neither of the mixed anomalies, (3.19) or (3.22).We must conclude that such a vacuum, with just B 11 ∼ ψηη and nothing else, cannot represent the correct IR physics of the ψη model, as the X-ray model reduces to it in the infrared.
On the other hand, the dynamical Higgs phase (analyzed in Appendix C) is characterized by bifermion condensates (3.23) 15 In the sense of the standard strong anomaly. 16By taking the equivalent definition of C 0 in footnote 13, one obtains K 0 = − 1 2 N 2 (N + 2)(N + 3), which signals an 2 anomaly only for N = 0 mod 4. Exactly the same happens in the ψη model.One might wonder how is it possible that the two constructions lead to different anomalies.However, the puzzle is only apparent, as the system has also a A 0 (dA 0 ) 2 anomaly, and, if one takes also it into account, the anomalous phase under a F 2 transformation depends only on the background and not on the sign convention chosen.Moreover, the choice of the convention is totally irrelevant to discuss the 't Hooft anomaly matching with the confining phase, as the [ 2 ] 3 anomaly is matched for every N .
Under this assumption, both U(1) 0 and Ũ(1) are broken by the condensate so, if one requires the condensate (3.23) to be everywhere non-vanishing, then, as it is charged under U(1) 0 and Ũ (1), one cannot allow any non-vanishing B (2) c fields.If, on the other way around, one imposes a non-vanishing B (2) c field, then ψ ij η B i cannot condense everywhere, and, similarly with φ form the X-ray to the UV, there must be vortices where the condensate (3.23) vanishes.We leave a more in-depth description of the matching in this case for subsequent work, but, disregarding the details, the matching must work as one can arrive at the same phase perturbatively, by substituting the composite operator ψ ij η B i by a fundamental scalar field with the same quantum number of it and a suitable potential.
This can be understood as a consistent way in which the infrared dynamics reflects the impossibility (an anomaly), (3.19), of gauging the color-flavor locked 1-form symmetry, (3.5), found in the UV theory. 17 Reduction to the ψη model, 2 vortex and the fermion zeromodes In order to make the argument of the present work water-tight, let us discuss here a subtle question associated with the reduction of the X-ray theory to the ψη model in the infrared.The basic statement is that nonvanishing VEV φ gives mass to the extra Dirac pair of fermions, q, q, and that the system indeed reduces in the infrared to the ψη model (the simplest BY model), studied in [7][8][9].The point is that the generalized, mixed anomalies (3.19) and (3.22), occur in the background of the external Ũ(1) and U(1) 0 gauge fields with fluxes, (3.7).In the case of the Ũ(1) gauge field Ã this does not present a problem.On the other hand, U(1) 0 is spontaneously broken to a 2 by the φ VEV, see Table 1.This means that the relevant background fields (A 0 , φ) correspond to a (regular) 2 vortex configuration.Again this does not present any issue in itself: there is nothing wrong in considering such a particular (and convenient) background and asking if the gauging of the color-flavor-locked 1-form N symmetry encounters a topological obstruction (a 't Hooft anomaly).This is what is studied in Sec.3.1, Sec.3.2 and Sec.3.3.
A (possible) problem is that q, q fields are massive everywhere and decouple from the system, except along the vortex core, where φ = 0 and m q,q = 0.As is well known, such a system develops a chiral two-dimensional q, q zero-mode, traveling along the vortex core with light velocity.They will produce an anomaly in the Ũ (1) gauge symmetry in the 2D vortex worldsheet, as discussed, e.g., by Callan and Harvey [41].To make the parallelism with the problem discussed in [41] complete, let us for the moment forget about the contribution of the fermions ψ and η in Table 1.It will be taken care of later.
In a 4D system considered in [41], a Dirac fermion Ψ, with an electric charge, is coupled to a complex scalar field Φ via a Yukawa interaction, and Φ is assumed to get a nonvanishing VEV, Φ = v = 0.The axial U(1) A is spontaneously broken by the condensate, whereas the vector (electromagnetic) symmetry U(1) em remains exact.Such a system can develop a solitonic vortex, Now the zero-mode for Ψ which develops on the string (vortex core) turns out to have a chiral nature in the vortex worldsheet (x 0 , x 1 ).As Ψ is charged, such a massless fermion causes a 2D chiral anomaly where A µ and J µ are the U(1) em gauge field and its covariant current.As U(1) em is supposed to be an exact conserved symmetry of the system, this appears to present a paradox.
The solution to this puzzle [41] is the following.As the system suffers from the ABJ anomaly for the axial U(1) A symmetry (U(1) A −[U(1) em ] 2 triangle), the spontaneous breaking of the U(1) A (1) means that the low-energy (µ ≪ v) 4D effective action has an axion-like (or better, π 0 − 2γ like) term, where π(x) is the pion field, Φ(x) = v e iπ(x)/v .( Now, in the presence of the soliton vortex, the pion field π(x) is ill-defined as one goes around the vortex string, see (4.2).As a result, the U(1) em variation δA µ = ∂ µ ω in L π 0 γγ turns out to be nonvanishing.The nontrivial vorticity in π(x) ∼ θ(x) indeed gives rise [41] to ("the anomaly-inflow") δL π 0 γγ in the vortex worldsheet (x 0 , x 1 ), which precisely cancels the 2D chiral anomaly (4.3) generated by the fermion zeromode.The Callan-Harvey argument exactly applies to our model, upon identifying (see Table 1), as long as the effects of the other fermions ψ and η are not considered.
In our model q i − qi form, in the bulk, a Dirac fermion fundamental of SU(N) c , meaning that also the 2D world-sheet fermion is fundamental under SU(N) c .Because of that the same mechanism (a local 2D anomaly, canceled by a bulk inflow) happens also for SU(N) c , without any significant difference.
More interestingly, the fact that the world-sheet fermions are coupled with the bulk gauge field means that, as we continue to follow the RG-flow and approach µ ∼ Λ, something should happen.In this work, we do not prescribe in detail what happens: we assume that what remains of the vortices in IR does not contribute to the 't Hooft anomaly matching of the anomalies found above. 18 As was recalled at the end of Sec.3.2, the 't Hooft fluxes (3.7), (3.22) mean that one is working in a bi-torus, T 1 × T 2 spacetime.The associated fractional flux A 0 (3.7) hence the 2 vortex, must accordingly be considered both in T 1 and in T 2 .The Callan-Harvey solution of an apparent puzzle associated with the vortex (a point on T 1 ) and the fermion zeromodes propagating in the vortex worldsheet T 2 , has been adapted to our problem as explained above.Exactly the same argument eliminates any issue concerning the second vortex punctuating T 2 and the chiral fermion zero-mode generating an anomaly in T 1 .The details will appear elsewhere.
As a final remark, we note that the questions (the fermion zeromodes traveling along the vortex core, etc.) discussed here concern perturbative, infinitesimal Ũ (1) variations of the system.Regarding the 2 − (1) N 2 discussed in subsection 3.2, apparently, the analysis might be more involved, and the 2D chiral fermions might, in principle, contribute to this anomaly.However, this is not the case: by explicit calculation both in the X-ray model (as shown in subsection 3.2) and in the ψη model (as shown in Ref. [36]) we have found a nontrivial 2 − (1) N 2 anomaly, thus, being them 2 anomalies, they must agree, and the overall contribution of the vortex physics must vanish.

Discussion and Summary
All Bars-Yankielowicz (BY) and generalized Georgi-Glashow (GG) models [1]- [6] possess a nonanomalous fermion parity symmetry ( 2 ) F 19 where i labels the fermions present in the model.In the standard quantization, the instanton analysis tells us that (5.1) is a nonanomalous symmetry of the quantum theory. 18If we lift this hypothesis some other interesting possibilities might arise.We will discuss them in a future work. 19( 2 ) F is equivalent to a subgroup of the proper Lorentz group.The point is whether or not in the non-trivial 2-form gauge background, B c , the symmetry is broken by a ('t Hooft) anomaly.
However, in some cases with even N (models of type I20 ), this statement holds because its anomaly is given by with i b i = even integer = 0 , (5.3) is the standard integer instanton number.It is essential to realize that the ( 2 ) F anomaly is absent because the sum of the anomaly coefficients i b i is a nonzero even number, not because it vanishes.
For the ψη model, The group G is doubly-connected (Π 0 ( G) = 2 ) [8].This always happens in models of type I. Instead, in type II models, where ( 2 ) F is a subset of a continuous G.
In general, in a type I theory, the gauging of the 1-form N symmetry leads to the ( 2 ) F anomaly, given by a master formula [9] 21 ∆S (Mixed anomaly) = (±π) (5.5) The calculation gives The aim of the present work was to cure the defect of the original analysis [7], i.e., the use of a singular ( 2 ) F gauge field.In a theory with a regulator Dirac pair of fields q, q (the X-ray theory), the singular 2 vortex background needed in [7] is replaced by a regular 2 vortex, without affecting the crucial holonomy, (1.1).The 1-form N symmetry lies now in the intersection between SU(N) and two nonanomalous U(1) symmetries, (3.1).In other words, the model is described by a well-defined principal bundle, (3.3).The generalized cocycle condition is met exactly as in [25].
In the X-ray theory the new anomalies are of the type, Ã − B ) and its UV-IR mismatch occur both for even and odd N (of the SU(N) color group).Therefore the statement in the X-ray model is somewhat stronger than in the ψη model. 22As for the U(1) 0 − (1) N 2 anomaly, (3.21), U(1) 0 is spontaneously broken by the scalar VEV, therefore only the variations 2 ⊂ U(1) 0 can be used in the UV-IR anomaly matching algorithm.For N even, the anomaly found here reduces to the 2 anomaly found in [7].
U(1) ψη × U(1) ψξ (see Table 3) can be matched by gauge-invariant (candidate) massless composite fermions, the first is anti-symmetric in A ↔ B; their charges are listed in Table 4.The anomaly  matching can be verified straightforwardly via a comparison between Table 3 and Table 4 (see [8] for explicit checks).Note that this model is an extended BY model with p = 1 (one additional Dirac pair of fermions in the fundamental representation): it is a Type II model.The 2 is not a genuine independent symmetry.The gauging of a color-flavor locked N symmetry by introducing (B c ) gauge fields does not lead to any new constraints as compared with the conventional 't Hooft anomaly matching.
The situation is the same when a scalar field φ is introduced with the Yukawa coupling to the (q, q) pair, but without taking into account the scalar VEV and the consequent decoupling of (q, q).The Yukawa term simply reduces the symmetry as of which three of the U(1) symmetries are independent.The decomposition of the UV fermions as a sum of the irreducible representations of the reduced symmetry group is given in Table 1 in the main text.The decomposition of the "massless baryons" (Table 4) in the direct sum of the irreps of G ′ F is in Table 5.Since this model (with or without the Yukawa coupling, but without the scalar VEV, v) is of type II, the massless baryons in Table 4 or Table 5 reproduce all the conventional with respect to SU(N + 4) × Ũ (1), but not with respect to U(1) 0 , which is however broken to 2 .Possible operators that mimic the mechanism that gives mass to these baryons are the Yukawa couplings with the scalar

C The dynamical Higgs phase
It was noted in [6][7][8][9] that in all the BY and GG models another possible phase is a dynamical (color-flavor-locked) Higgs vacuum, in which the color SU(N) is completely broken and the global symmetry is partially realized in the Nambu-Goldstone mode.In the X-ray model considered in this work, (2.1) -(2.5), the proposed bifermionic condensates, (3.23), together with the scalar condensate φ , break the global symmetries as where U(1) ′ ψη is generated by an appropriate linear combination of the SU(N +4) generator, 4 1 N −N 1 4 and that of U(1) ψη .The fermions in the UV are decomposed into the sum of irreducible representations of the unbroken group, in Table 6.The baryons which remain massless among those in Table 4 are listed in Table 7.
Table 6: UV fields in the model, Table 3, are decomposed as a direct sum of the representations of the unbroken group G ′ F of (C.1).
Finally, we note that both U(1) 0 and Ũ (1) of the X ray model, (2.1) -(2.7), and hence the color-flavor locked N ⊂ SU(N) c × ( Ũ(1) × U(1) 0 ) itself, are spontaneously broken by the bifermion condensates, (3.23).It follows that the mixed anomalies found in the X-ray  model in Sec. 3, are perfectly consistent with the physics of the dynamical Higgs phase, in contrast to the case of the confining, chirally symmetric phase discussed in Appendix.B. Now, unlike the somewhat mysterious matching equations in the hypothetical confining phase (as those fully exposed in [8]), the conventional, 't Hooft anomaly matching constraints with respect to the unbroken group G ′ F in the dynamical Higgs phase are trivially satisfied, as can be seen by inspection of Table 6 and Table 7. 24

φ B 12 B
21 + h.c. and φ * B 22 B 31 + h.c.(B.4) Giving mass to B 12 , B 21 , B 22 , B 31 in this way does not leave just B 11 in the IR, but also fermion zero modes localized on vortices where φ = 0.This confining symmetric theory with B 11 in the bulk plus degrees of freedom localized on vortices will require further investigations in the future.
The partition function changes sign under ( 2 ) F , in the ψη model with N even, and in all other type I models: the mixed ( 2 ) F − [ N ] 2 anomaly.As the candidate massless baryons do not support this generalized anomaly (see(B.3)in the simplest, ψη model), such a confining vacuum cannot represent a correct phase in type I models.

Table 3 :
The multiplicity, charges, and representation are shown for each set of fermions in the BY model, (2.1) -(2.4).(•) stands for a singlet representation.