#
Strong anomaly and
phases of chiral gauge theories

Stefano Bolognesi(1,2), Kenichi Konishi(1,2), Andrea Luzio(3,2)
(1)Department of Physics “E. Fermi”, University of Pisa,
Largo 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

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###### Abstract

We present a simple argument which seems to favor, when applied to a large
class of strongly-coupled chiral gauge theories, a dynamical-Higgs-phase
scenario, characterized by certain bifermion condensates. Flavor symmetric
confining vacua described in the infrared by a set of baryonlike massless
composite fermions saturating the conventional ’t Hooft anomaly matching
equations, appear instead disfavored. Our basic criterion is that it should be
possible to write a strong-anomaly effective action, analogous to the one used
in QCD to describe the solution of the $U(1)_{A}$ problem in the low-energy
effective action, by using the low-energy degrees of freedom in the
hypothesized infrared theory. We also comment on some well-known ideas such as
the complementarity and the large $N$ planar dominance in the context of these
chiral gauge theories. Some striking analogies and contrasts between the
massless QCD and chiral gauge theories seem to emerge from this discussion.

###### Contents

  1. 1 Introduction
  2. 2 Nambu-Goldstone (NG) bosons and condensates
    1. 2.1 $N_{\rm f}$ \- flavored QCD
    2. 2.2 NonAbelian NG bosons in the $\psi\eta$ model
      1. 2.2.1 Colored $\psi\eta$ NG bosons
    3. 2.3 $U(1)$ NG boson(s) in the $\psi\eta$ model
    4. 2.4 $U(1)$ symmetries in the $\chi\eta$ model
  3. 3 Strong anomaly and effective Lagrangian
    1. 3.1 Strong anomaly, $U(1)$ problem and the $\theta$ dependence in QCD
      1. 3.1.1 Veneziano -Yankielowicz and Affleck-Dine-Seiberg superpotentials
    2. 3.2 Strong anomaly and effective action in the $\psi\eta$ model
    3. 3.3 Strong anomaly effective action in the generalized BY models
    4. 3.4 Strong anomaly in the $\chi\eta$ model
    5. 3.5 Strong anomaly in the generalized GG models
    6. 3.6 $SU(6)$ model with a single fermion in a self-adjoint representation
    7. 3.7 Adjoint QCD with $N_{\rm c}=N_{\rm f}=2$
  4. 4 Complementarity in the $\chi\eta$ model?
  5. 5 Large $N$ planar dominance and $U(1)$ NG bosons in chiral gauge theories
  6. 6 Multifield versus bi-fermion condensates
  7. 7 Summary and Discussion
  8. A Chirally symmetric phase in the $\psi\eta$ model
  9. B Color-flavor locked Higgs phase in the $\psi\eta$ model
  10. C Chirally symmetric phase in the $\chi\eta$ model
  11. D Color-flavor locked Higgs vacuum in the $\chi\eta$ model
  12. E Confining phase with unbroken global symmetries of the BY models
  13. F Dynamical Higgs phase in the BY models
  14. G Confining phase with unbroken global symmetries of the GG models
  15. H Dynamical Higgs phase in the generalized GG models

##  1 Introduction

Our comprehension of the dynamics of strongly-coupled chiral gauge theories is
still largely unsatisfactory, in spite of their potential role in describing
the physics of fundamental interactions beyond the standard model. The results
of many years of study of these theories, by using mixture of wisdom and
certain consistency conditions, the most significant among them being ’t
Hooft’s anomaly matching constraints [1], do restrict possible lists of
dynamical scenarios and symmetry realization patterns [2]-[17], but they are
usually far from being capable to determine the infrared physics of these
systems uniquely.

It is customary in these discussions of the dynamics and symmetry realization
in chiral gauge theories, to take into account the nonanomalous symmetries
only. For instance, the ’t Hooft anomaly matching requirements are normally
applied exclusively on nonanomalous global symmetries.

From the point of view of renormalization group, assuming that the color
interactions become strongly coupled towards the infrared, and that the low-
energy effective degrees of freedom are not the original gluons and quarks
(and similar color gauge bosons and matter fermions in general chiral
theories), the anomalies present in the underlying theory must be reproduced
for consistency either by composite massless fermions (’t Hooft) if the
symmetry remains unbroken, or, if spontaneously broken, by massless Nambu-
Goldstone (NG) bosons. In the latter case, the condition of “anomaly matching”
is nothing but the well-known procedure for calculating the amplitudes (such
as $\pi\to\gamma\gamma$, or much more generally, all anomalous amplitudes
included in Wess-Zumino-Witten effective action) containing the NG bosons.

A third type of the application of renormalization-group invariance of anomaly
concerns certain $U(1)$ symmetry(ies), which is (are) affected by the
topologically nontrivial gauge field configurations. This phenomenon goes
under the name of strong anomaly, and the “$U(1)_{A}$ problem” and its
solution in QCD [18, 19] is a renowned example in which the strong anomaly
plays the central role. Probably because this appeared in the context of some
characteristic aspects of QCD which is a vectorlike gauge theory, such as the
chiral symmetry breaking and the low-energy sigma models, a similar question
has not been discussed much in the context of strongly-coupled chiral gauge
theories, to the best of our knowledge, with an exception being [5].

The aim of the present note is to discuss the implication of strong anomaly in
strongly-coupled chiral gauge theories. Rather surprisingly, the requirement
that the assumed set of infrared effective degrees of freedom (massless NG
bosons and/or massless composite fermions à la ’t Hooft) should be able to
describe the strong-anomaly effective action in a way analogous to the famous
strong-anomaly QCD effective action [20]-[24] containing certain logarithmic
function, yields a rather solid indication about which type of the infrared
phase is more plausible than others.

The models we will discuss in some details are the so-called Bars-Yankielowicz
(BY) and generalized Georgi-Glasow (GG) models. In particular, our discussions
will be set up first by using two simplest classes of models. One is an
$SU(N)$ gauge theory with left-handed fermions in the reducible, complex
representation,

$\yng(2)\oplus(N+4)\,{\bar{{\yng(1)}}}\,$ (1.1)

that is,

$\displaystyle\psi^{\\{ij\\}}\,,\quad\eta_{i}^{B}\,,\qquad
i,j=1,2,\ldots,N\;,\quad B=1,2,\ldots,N+4\;,$ (1.2)

which is the simplest of the so-called Bars-Yankielowicz models [4]. This
model will be called “$\psi\eta$” model below, for short.111In some earlier
literature these fields were denoted by $S$ (the symmetric tensor $\psi$), $A$
(the antisymmetric tensor $\chi$), and ${\bar{F}}$ (the antifundamental,
$\eta$). The global symmetry group (actually the local property of the
symmetry group) is

$G_{\rm f}=SU(N+4)\times U(1)_{\psi\eta}\;,$ (1.3)

where $U(1)_{\psi\eta}$ indicates the anomaly-free combination of
$U(1)_{\psi}$ and $U(1)_{\eta}$, associated with the two types of matter Weyl
fermions of the theory. Another model we consider is an $SU(N)$ gauge theory
with fermions

$\yng(1,1)\oplus(N-4)\,{\bar{{\yng(1)}}}\,$ (1.4)

that is,

$\displaystyle\chi^{[ij]}\,,\quad\eta_{i}^{B}\,,\qquad
i,j=1,2,\ldots,N\;,\quad B=1,2,\ldots,N-4\;,$ (1.5)

(the simplest Georgi-Glashow model). We will refer to it as “$\chi\eta$” model
below. The global symmetry group of the $\chi\eta$ model is

$G_{\rm f}=SU(N-4)\times U(1)_{\chi\eta}\;.$ (1.6)

Our interest is to understand how these symmetries are realized in the
infrared.

More general Bars-Yankielowicz and Georgi-Glasow models, which are similar to
the above two models but with $p$ additional pairs of fermions in the
fundamental and antifundamental representations, will also be considered.

In all these models the conventional ’t Hooft anomaly matching discussion
apparently allows a confining phase, with no condensates and with full
unbroken global symmetry, with some simple set of massless composite fermions
saturating the anomaly matching equations. See Appendix A for the $\psi\eta$
model, Appendix C for the $\chi\eta$ model and Appendices E and G for more
general classes of BY and GG chiral gauge theories.

At the same time, the anomaly constraints are also consistent with a dynamical
Higgs phase, in which the color and (part of) the flavor symmetry are
dynamically broken by certain bifermion condensates, see Appendices B, D, F
and H (taken from [25]). These results are mostly known from the earlier work
[4]-[17] and partly completed by ourselves, but are recorded here to make
our discussion self-contained.

There is an aspect of the conventional ’t Hooft anomaly matching procedure in
the dynamical Higgs phase, which is perhaps not widely appreciated. The Higgs
phase of these chiral theories are, in general, described by massless NG
bosons as well as some massless fermions. These fermions saturate the
conventional ’t Hooft anomaly triangles with respect to the unbroken flavor
symmetries. The way they do is, however, quite remarkable, and in our view,
truly significant. As can be seen from Table 2, Table 4, and in similar Tables
8, 9, 12 and 13 for the generalized BY and GG models (see Appendices B, D, F,
H), the set of fermions remaining massless in UV and those in the IR are
identical  as regards their quantum numbers, charges, and multiplicities.
Therefore, the matching of anomalies (in the unbroken global symmetries) is
completely automatic, and is natural. No arithmetic equations need to be
solved to justify a solution which might look sometimes miraculous. The
significance of such a solution of ’t Hooft’s equations (the dynamical Higgs
phase) is its stablity 222As noted in [26, 25], in the study making use of
generalized anomalies one might try to make further gauging and study
associated anomalies in the low-energy effective theory. From the identities
of the sets of massless fermions in the UV and IR, it is seen that such an
extra gauging would not produce any new unmatched anomalies. .

As we will find out below, the consideration of the strong anomaly appears to
give us a rather clear indication that this second type of vacua - those in
dynamical Higgs phase, characterized by certain bifermion condensates breaking
color and part of the flavor symmetry - describe correctly the infrared
dynamics of these chiral gauge theories.

This paper is organized as follows. In order to present the discussion
systematically, we first review (for QCD) and work out for the $\psi\eta$ and
$\chi\eta$ theories, the chiral Ward-Takahashi identities and identify the
interpolating fields for all NG bosons, assuming the dynamical Higgs phase for
these models (Appendices B and D). This is done in Sec. 2

After this preparation, in Sec. 3 we write the strong-anomaly low-energy
effective action for the chiral gauge theories, $\psi\eta$, $\chi\eta$, as
well as all other BY and GG models, following the procedure used for QCD. This
discussion confirms the consistency of the dynamical Higgs phase of these
models, in agreement with the recent mixed-anomaly study in these chiral gauge
theories [26, 25], but it actually somewhat strengthens the conclusion 333 For
instance the argument based on strong anomaly does not depend on whether $N$
is even or odd, whereas the mixed-anomaly calculation of [26, 25] were done
for models with even $N$..

The basic observation is that there is no way of writing the strong-anomaly
effective action if confining flavor symmetric vacua are assumed, i.e., by
using only the massless baryons (Appendices A, C, E, and G). The fermion-zero-
mode (in the instanton background) counting would not work, in any of the
chiral theories considered.

We shall comment briefly on some other types of theories (other than BY and GG
models) as well, and discuss the implication of the strong anomaly on their
phases.

Related questions regarding apparent complementarity in (only) one of the
models (the $\chi\eta$ model) and a large $N$ argument concerning $U(1)$ NG
bosons in some chiral gauge theories, will be briefly commented upon in Sec. 4
and in Sec. 5, respectively.

In Sec. 7, our results are summarized. From the analyses of Sec. 2 $\sim$ Sec.
5 some striking analogies and at the same time contrasts, between vectorlike
theories and chiral gauge theories seem to emerge. Better stated, one perhaps
learns from these discussions a more precise meaning of what is similar or
what is dissimilar between the strong interaction dynamics of vectorlike and
chiral gauge theories beyond certain loose use of terminologies; as a result
one gets a somewhat clearer understanding of the dynamics of strongly-coupled
chiral gauge theories than before.

##  2 Nambu-Goldstone (NG) bosons and condensates

Consider any global continuous symmetry $G_{\rm f}$ and the associated
conserved current $J_{\mu}$ and charge $Q$, the field $\phi$ (elementary or
composite) which condenses and breaks $G_{\rm f}$, and the field
${\tilde{\phi}}$ which is transformed into $\phi$ by the $G_{\rm f}$
transformation:

$Q\equiv\int
d^{3}xJ_{0}\;,\qquad[Q,{\tilde{\phi}}]=\phi\;,\qquad\langle\phi\rangle\neq
0\;.$ (2.1)

Thus

$\displaystyle\lim_{q_{\mu}\to 0}iq^{\mu}\int d^{4}x\,e^{-iq\cdot x}\langle
0|T\\{J_{\mu}(x)\,{\tilde{\phi}}(0)\\}|0\rangle=\lim_{q_{\mu}\to 0}\int
d^{4}x\,e^{-iq\cdot x}\partial_{\mu}\langle
0|T\\{J_{\mu}(x)\,{\tilde{\phi}}(0)\\}|0\rangle=$ $\displaystyle=\int
d^{3}x\langle 0|[J_{0}(x),{\tilde{\phi}}(0)]|0\rangle=\langle
0|[Q,{\tilde{\phi}}(0)]|0\rangle=\langle 0|\phi(0)|0\rangle\neq 0\;.$ (2.2)

This Ward-Takahashi like identity implies that the two-point function

$\int d^{4}x\,e^{-iq\cdot x}\langle
0|T\\{J_{\mu}(x)\,{\tilde{\phi}}(0)\\}|0\rangle$ (2.3)

is singular at $q^{\mu}\to 0$. If the $G_{\rm f}$ symmetry is broken
spontaneously such a singularity is due to a massless scalar particle in the
spectrum -Nambu-Goldstone (NG) boson-, a “pion” below, symbolically, such that

$\langle
0|J_{\mu}(q)|\pi\rangle=iq_{\mu}F_{\pi}\;,\qquad\langle\pi|{\tilde{\phi}}|0\rangle\neq
0\;.$ (2.4)

Two point function (2.3) (times $q^{\mu}$) behaves as

$\lim_{q^{\mu}\to 0}\,q^{\mu}\cdot
q_{\mu}\,\frac{F_{\pi}\langle\pi|{\tilde{\phi}}|0\rangle}{q^{2}}\sim{\rm
const}\;.$ (2.5)

The constant $F_{\pi}$ represents the amplitude for the broken current to
produce the pion from the vacuum (the pion decay constant).

###  2.1 $N_{\rm f}$ \- flavored QCD

In the standard QCD with $N_{\rm f}$ light flavors, the quarks are

$\psi_{L}^{i},\,\psi_{R}^{i}\;,\qquad i=1,2,\ldots,N_{\rm f}\;.$ (2.6)

We take

$\phi={\bar{\psi}_{R}}\psi_{L}+{\rm{\rm
h.c.}}\;,\qquad{\tilde{\phi}}={\bar{\psi}_{R}}t^{b}\psi_{L}-{\rm{\rm
h.c.}}\;;\qquad
J^{5,a}_{\mu}=i{\bar{\psi}_{L}}{\bar{\sigma}}_{\mu}t^{a}\psi_{L}-(L\leftrightarrow
R)\;,$ (2.7)

where $t^{a}$ is the $SU(N_{\rm f})$ generators, $a=1,2,\ldots N_{\rm
f}^{2}-1$. It is believed, and confirmed by lattice simulations, that for
sufficiently small $N_{\rm f}$, the field $\phi$ condenses,

$\langle\phi\rangle=\langle{\bar{\psi}_{R}}\psi_{L}+{\rm{\rm
h.c.}}\rangle\sim-\Lambda^{3}\;,$ (2.8)

leaving $SU(N_{\rm f})_{V}\times U(1)_{V}$ unbroken. The axial $SU(N_{\rm
f})_{A}$ is spontaneously broken:

$\langle{[Q_{5}^{a},{\tilde{\phi}}^{b}]}\rangle=-\delta^{ab}\langle{\bar{\psi}}_{R}\psi_{L}+{\bar{\psi}}_{L}\psi_{R}\rangle=c\,\delta^{ab}\,\Lambda^{3}\neq
0\;,$ (2.9)

where $c$ is a constant of the order of unity. The axial $U(1)_{A}$ is also
spontaneously broken, but due to the strong anomaly the associated NG boson
gets mass (the $U(1)_{A}$ problem: see Sec. 3.1).

###  2.2 NonAbelian NG bosons in the $\psi\eta$ model

In the dynamical Higgs phase of the $\psi\eta$ model (Appendix B) the
nonanomalous symmetry is broken as

$SU(N)\times SU(N+4)\times
U(1)_{\psi\eta}\xrightarrow{\langle\psi\eta\rangle}SU(N)_{{}_{\rm cf}}\times
SU(4)\times U(1)^{\prime}\;.$ (2.10)

Let us first concentrate our attention to $8N$ NG bosons associated with the
$SU(N+4)$ breaking, leaving the discussion of the $U(1)$ NG boson and an
unbroken $U(1)^{\prime}$ symmetry to the next subsection. An $SU(N+4)$ current
is ($T^{a}$ is an $SU(N+4)$ generator)

$J^{a}_{\mu}=i\,{\bar{\eta}}^{j,m}{\bar{\sigma}}_{\mu}(T^{a})_{mn}\,\eta_{j}^{n}\;,\qquad
m,n=1,2,\ldots N+4$ (2.11)

and the charges are

$Q^{a}=\int d^{3}x\,J^{a}_{0}\;.$ (2.12)

In particular, consider the broken symmetry currents ($8N$ components):

$J^{a}_{\mu}=i\,{\bar{\eta}}^{j,m}{\bar{\sigma}}_{\mu}(T^{a})_{mn}\,\eta_{j}^{n}\;,$
(2.13)

with

$m=1,2,\ldots N\;,\quad n=N+1,\ldots N+4\;,\qquad{\rm or\,vice\,versa},$
(2.14)

with charges

$Q^{a}=\int d^{3}x\,J^{a}_{0}=\int
d^{3}x\,{\bar{\eta}}^{j,m}(T^{a})_{mn}\eta_{j}^{n}\,.$ (2.15)

The associated symmetry is broken by the condensates

$\langle\psi^{\\{ij\\}}\eta_{j}^{m}\rangle=\,c_{\psi\eta}\,\Lambda^{3}\delta^{im}\;,\qquad\quad\
i,m=1,2,\dots N\;,$ (2.16)

and there will be $8N$ NG bosons associated with the currents, $J^{a}_{\mu}$.
A natural choice for the pion interpolating field is a gauge-invariant
composite,

${\phi}^{\tilde{a}}=\left(\psi^{ik}\eta_{k}^{n}\right)^{*}(T^{\tilde{a}})_{nm}\,\left(\psi^{ij}\eta_{j}^{m}\right)\;.$
(2.17)

The relevant commutators are:

$\langle[Q^{a},{\phi}^{\tilde{a}}]\rangle=-[T^{a},T^{\tilde{a}}]_{mn}\langle\left(\psi^{ik}\eta_{k}^{m}\right)^{*}\,\left(\psi^{ij}\eta_{j}^{n}\right)\;\rangle\;.$
(2.18)

Let us consider various $SU(2)$ subalgebras ($\sigma^{i}$’s are the Pauli
matrices)

$T^{a}=\frac{\sigma^{1}}{2}\;\qquad
T^{\tilde{a}}=\frac{\sigma^{2}}{2}\;,\qquad T^{0}=\frac{\sigma^{3}}{2}$ (2.19)

living in the $2\times 2$ subspace $(m,n)$,

$m\in\\{1,2,\ldots N\\}\;,\quad n\in\\{N+1,\ldots N+4\\}\;,$ (2.20)

for the broken charge $Q^{a}$ (2.15) and the pion field ${\phi}^{\tilde{a}}$
(2.17), respectively. The commutator (2.18) now gives

$\langle[Q^{a},{\phi}^{\tilde{a}}]\rangle=-\frac{1}{2}\langle\left(\psi^{ik}\eta_{k}^{m}\right)^{*}\,\left(\psi^{ij}\eta_{j}^{m}\right)\rangle+\frac{1}{2}\langle\left(\psi^{ik}\eta_{k}^{n}\right)^{*}\,\left(\psi^{ij}\eta_{j}^{n}\right)\rangle\;$
(2.21)

(no summation over $m$ or $n$ here and below, in (2.24))444The summation over
the repeated color indices, $j$, $k$ are done, as usual.. Upon condensation of
color-flavor diagonal form (2.16), the above becomes

$\langle[Q^{a},{\phi^{\tilde{a}}}]\rangle=-\frac{1}{2}(c_{\psi\eta}\,\Lambda^{3})^{2}\neq
0\;,$ (2.22)

implying that $Q^{a}$ generates a massless NG bosons from the vacuum, whose
“wave function” is roughly

$\langle 0|{\phi}^{\tilde{a}}(x)|\pi^{a}(p)\rangle=f_{p}(x)\;.$ (2.23)

Note that, by inserting (2.16), the pion interpolating field (2.17) can also
be written in a simpler, gauge dependent form:

${\phi}^{\tilde{a}}\sim\left(\psi^{mk}\eta_{k}^{n}\right)^{*}(T^{\tilde{a}})_{nm}+(T^{\tilde{a}})_{mn}\,\left(\psi^{mj}\eta_{j}^{n}\right)=2\,{\Re}\,\\{(T^{\tilde{a}})_{mn}\left(\psi^{mj}\eta_{j}^{n}\right)\\}\;.$
(2.24)

Clearly, the pairing between the charge and the pion field can be
interchanged:

$T^{a}=\frac{\sigma^{2}}{2}\;\qquad
T^{\tilde{a}}=\frac{\sigma^{1}}{2}\;,\qquad T^{0}=\frac{\sigma^{3}}{2}$ (2.25)

(instead of (2.19)); therefore one finds $4N\times 2=8N$ NG bosons.

A simpler way to state the result is, as can be seen by inspection of (2.19),
(2.25), and (2.24), that the $8N$ NG bosons of non diagonal
$\frac{SU(N+4)}{SU(N)\times SU(4)}$ generators are just the real and imaginary
parts of $\psi^{mj}\eta_{j}^{n}$ ($m\leq N,\,n\geq N+1$).

####  2.2.1 Colored $\psi\eta$ NG bosons

The $N^{2}-1$ colored NG bosons

$\displaystyle{\tilde{\phi}}^{b}$ $\displaystyle\sim$
$\displaystyle\left(\psi^{mk}\eta_{k}^{n}\right)^{*}(T^{b})_{nm}+(T^{b})_{nm}\,\left(\psi^{nj}\eta_{j}^{m}\right)\;,$
(2.26) $\displaystyle=$
$\displaystyle\left(\psi^{nk}\eta_{k}^{m}\right)^{*}(T^{b})_{mn}+(T^{b})_{nm}\,\left(\psi^{nj}\eta_{j}^{m}\right)\;,$
$\displaystyle\sim$
$\displaystyle{\Re}\,\\{(T^{b})_{nm}\,\left(\psi^{nj}\eta_{j}^{m}\right)\\}\;,\qquad
n\leq N\;,\quad m\leq N\;,\quad T^{b}\in su(N)\;,$

generated by the condensates (2.16) are absorbed by the $SU(N)$ gauge bosons
by the Englert-Brout-Higgs mechanism, so do not appear as physical massless
particles.

###  2.3 $U(1)$ NG boson(s) in the $\psi\eta$ model

Assuming again that we are in the dynamical Higgs phase of the $\psi\eta$
model, let us examine the fate of the two nonanomalous $U(1)$ symmetries,
$U(1)_{\psi\eta}$ and a $U(1)_{\rm f}\subset SU(N+4)$. One combination is
broken spontaneously, giving rise to a physical NG boson, and the other
combination, $U(1)^{\prime}$, remains unbroken, as a manifest symmetry of the
low-energy theory.

For the $U(1)_{\psi\eta}$ symmetry, writing the matter fermions together as

$\left(\begin{array}[]{c}\psi\\\
{\eta}\end{array}\right)=\left(\begin{array}[]{c}\psi^{\\{ij\\}}\\\
\eta^{1}_{i}\\\ \vdots\\\ \eta^{N+4}_{i}\end{array}\right)\;,$ (2.27)

the $U(1)_{\psi\eta}$ current is

$J^{\mu}=i\big{(}{\bar{\psi}}\,\,\,{\bar{\eta}}\big{)}\,T_{\psi\eta}\,{\bar{\sigma}}^{\mu}\left(\begin{array}[]{c}\psi\\\
\eta\end{array}\right)\;,\qquad Q=\int d^{3}x\,J^{0}\;,$ (2.28)

where

$T_{\psi\eta}=\left(\begin{array}[]{cccc}(N+4)\,{\mathbf{1}}_{\frac{N(N+1)}{2}}&\\\
&-(N+2)\,{\mathbf{1}}_{(N+4)\cdot N}\\\ \end{array}\right)$ (2.29)

and the $U(1)_{\psi\eta}$ charge operator is

$Q_{\psi\eta}=\int
d^{3}x\,\Big{[}(N+4){\bar{\psi}}_{ij}\psi^{ij}-(N+2)\sum_{m}{\bar{\eta}}^{i}_{m}\eta_{i}^{m}\Big{]}\;,$
(2.30)

so

$\displaystyle[Q_{\psi\eta},\psi^{k\ell}\eta_{\ell}^{n}]=\left(-(N+4)+N+2\right)\psi^{k\ell}\eta_{\ell}^{n}=-2\psi^{k\ell}\eta_{\ell}^{n}\;,$
$\displaystyle[Q_{\psi\eta},{\bar{\psi}}_{k\ell}{\bar{\eta}}^{\ell}_{n}]=\left(N+4-(N+2)\right){\bar{\psi}}_{k\ell}{\bar{\eta}}^{\ell}_{n}=2{\bar{\psi}}_{k\ell}{\bar{\eta}}^{\ell}_{n}\;.$
(2.31)

The condensate is (2.16):

$\langle\psi^{ij}\eta_{j}^{m}\rangle=c_{\psi\eta}\,\delta^{im}\,\Lambda^{3}\;,\qquad
i,m\leq N\;,$ (2.32)

so

$\langle[Q_{\psi\eta},\psi^{k\ell}\eta_{\ell}^{n}]\rangle\neq 0\;,\qquad
k=n=1,2,\ldots,N\;.$ (2.33)

On the other hand, a diagonal $SU(N+4)$ generator (it acts only on $\eta$)
which mixes with $U(1)_{\psi\eta}$ is

$T_{\rm f}=\left(\begin{array}[]{ccc}\,{\mathbf{0}}_{\frac{N(N+1)}{2}}&&\\\ &\
4\,{\mathbf{1}}_{N\cdot N}&\\\ &&-N\,{\mathbf{1}}_{4\cdot
N}\end{array}\right)\;,$ (2.34)

with charge operator

$Q_{\rm f}=\int
d^{3}x\,\Big{[}4\sum_{m=1}^{N}{\bar{\eta}}^{i}_{m}\eta_{i}^{m}-N\sum_{m=N+1}^{N+4}{\bar{\eta}}^{i}_{m}\eta_{i}^{m}\Big{]}\;,$
(2.35) $[Q_{\rm f},\psi^{k\ell}\eta_{\ell}^{n}]=\begin{cases}\ \ \
4\,\psi^{k\ell}\eta_{\ell}^{n}&\text{for}\qquad n\leq N\;;\\\
-N\psi^{k\ell}\eta_{\ell}^{n}&\text{for}\qquad n>N\;.\end{cases}$ (2.36)

The same condensate (2.32) breaks also $Q_{\rm f}$, as

$\langle[Q_{\rm f},\psi^{k\ell}\eta_{\ell}^{n}]\rangle\neq 0\;,\qquad
k=n=1,2,\ldots,N\;.$ (2.37)

But the combination $U(1)^{\prime}$ generated by

$Q^{\prime}=2Q_{\psi\eta}+Q_{\rm f}$ (2.38)

remains unbroken:

$\langle[2Q_{\psi\eta}+Q_{\rm
f},\psi^{k\ell}\eta_{\ell}^{n}]\rangle=0\;\qquad(\forall k,\forall n)\,.$
(2.39)

Any other combination of $Q_{\rm f}$ and $Q_{\psi\eta}$ is spontaneously
broken, see Eqs. (2.31), (2.36), so there is one physical $U(1)$ NG boson in
this model. A natural choice for the interpolating field for this physical
$U(1)$ NG boson would be

$\sum_{n,j}^{N}\psi^{nj}\eta_{j}^{n}\
=\mathop{Tr}\nolimits(\psi\eta)\propto{\mathbf{1}}+\frac{i}{F_{\pi}^{(0)}}\,{\phi}_{0}+\ldots\;,$
(2.40)

where the field are appropriately normalized and $F_{\pi}^{(0)}$ is a constant
with a mass dimension. This is an analogue of
${\bar{\psi}_{R}}\psi_{L}={\bar{u}_{R}}u_{L}+{\bar{d}_{R}}d_{L}+\ldots$ in
QCD, and also analogous to the nonAbelian NG bosons, (2.24).

Unlike ${\bar{\psi}_{R}}\psi_{L}$ in QCD, (2.40) is not gauge-invariant.
However, it is not difficult to find a natural gauge-invariant form for the
interpolating field for the same NG boson: it could be written as

$\det U\;,\qquad U^{k\ell}=\psi^{kj}\eta_{j}^{\ell}\;,$ (2.41)

by using the first $N$ flavors, $\eta_{j}^{a}$, $a=1,2,\ldots,N$. Note that
this composite field is a singlet of the surviving symmetry of the $\psi\eta$
system, (2.10). This will play an important role in the discussion of the
strong anomaly below.

Expanding around the VEV, (2.32), (2.40) and (2.41) give the same physical
field,

$\sim{\rm
const}+\sum_{k,j}^{N}\left(\psi^{kj}\eta_{j}^{k}\right)^{\\{q\\}}\;,$ (2.42)

where $\left(\psi^{kj}\eta_{j}^{k}\right)^{\\{q\\}}$ indicates the fluctuation
part of the composite field, $\psi^{kj}\eta_{j}^{k}$.

So far, we considered two nonanomalous $U(1)$ symmetries, $U(1)_{\psi\eta}$
and a $U_{f}(1)\subset SU(N+4)$, and found that one combination remains a
manifest symmetry, call it $U(1)^{\prime}$, while the other (let us indicate
as $U(1)_{\rm NG}$) gets broken and generates an associated, physical massless
NG boson.

Actually, the system possesses one more, independent, $U(1)$ symmetry, though
anomalous. Any combination of $U(1)_{\psi}$ and $U(1)_{\eta}$ other than
$U(1)_{\psi\eta}$ is anomalous (let us call it $U(1)_{\rm an}$) hence cannot
be expressed as a linear combination of $U(1)^{\prime}$ and $U(1)_{\rm NG}$.

It is perhaps useful to compare the situation here with the massless QCD
vacuum, with quark condensate, (2.9). In the latter, there is one manifest
symmetry, $U(1)_{V}$, and an axial symmetry $U(1)_{A}$ which is anomalous, and
is also spontaneously broken (associated with a would-be NG boson which
becomes massive by the strong anomaly, as reviewed below, Sec. 3.1). In the
$\psi\eta$ model under consideration, there is one manifest symmetry,
$U(1)^{\prime}$, one spontaneously broken, nonanomalous, $U(1)_{NG}$, with an
associated physical massless NG boson, and an anomalous $U(1)_{\rm an}$
symmetry, independent of $U(1)^{\prime}$ and $U(1)_{\rm NG}$. There is thus
one more (counting also anomalous and/or spontaneously broken) $U(1)$ symmetry
in our model, as compared to the massless QCD.

It is tempting to regard the $U(1)_{\rm an}$ symmetry as an analogue of the
$U(1)_{A}$ symmetry in QCD. However, as it turns out, the nature of $U(1)_{\rm
an}$ (whether or not it is spontaneously broken), and related to that
question, the true form of the interpolating field for the physical massless
NG boson ((2.40), (2.41), or something else), can only be found after an
appropriate study of the strong anomaly effective action for the $\psi\eta$
system. This will be done in Sec. 3.2. Somewhat surprisingly, it will be found
that the dynamical Higgs phase of the $\psi\eta$ model is not exhaustively
characterized by the condensate of $\det U$. The proper form of the strong-
anomaly effective action leads to a second, bi-baryon type condensate besides
$\langle\det U\rangle$, see (3.26). This turns out to be the missing piece of
the puzzle. The true interpolating function for the physical massless
$U(1)_{\rm NG}$ boson will be found, not to be given by $\phi_{0}$ of (2.40),
but by $\phi$ in (3.33).

###  2.4 $U(1)$ symmetries in the $\chi\eta$ model

In the dynamical Higgs phase of the $\chi\eta$ model the symmetry breaking
proceeds as

$\displaystyle SU(N)\times SU(N-4)\times U(1)_{\chi\eta}$
$\displaystyle\xrightarrow{\langle\chi\eta\rangle}$ $\displaystyle
SU(N-4)_{\rm cf}\times SU(4)_{\rm c}\times U(1)^{\prime}$ (2.43)
$\displaystyle\longrightarrow$ $\displaystyle SU(N-4)_{\rm cf}\times
U(1)^{\prime}\;,$

where the residual color $SU(4)_{\rm c}$, unbroken by the condensates

$\langle\chi^{ij}\eta_{j}^{m}\rangle=c_{\chi\eta}\,\delta^{im}\,\Lambda^{3}\;,\qquad
i,m=1,2,\ldots,N-4\;,$ (2.44)

evolves further towards infrared, confines, develops another condensate,

$\langle\chi\chi\rangle\neq 0\;,$ (2.45)

and gives rise to a hidden (dark matter?) sector invisible to the massless
sector.

For the $U(1)_{\chi\eta}$ symmetry, writing the matter fermions together as

$\left(\begin{array}[]{c}\chi\\\
{\eta}\end{array}\right)=\left(\begin{array}[]{c}\chi^{[ij]}\\\
\eta^{1}_{i}\\\ \vdots\\\ \eta^{N-4}_{i}\end{array}\right)\;,$ (2.46)

the $U(1)_{\chi\eta}$ current is

$J^{\mu}=i\big{(}{\bar{\chi}}\,\,\,{\bar{\eta}}\big{)}\,T_{\chi\eta}\,{\bar{\sigma}}^{\mu}\left(\begin{array}[]{c}\chi\\\
\eta\end{array}\right)\;,\qquad Q=\int d^{3}x\,J^{0}\;,$ (2.47)

where

$T_{\chi\eta}=\left(\begin{array}[]{cccc}(N-4)\,{\mathbf{1}}_{\frac{N(N-1)}{2}}&\\\
&-(N-2)\,{\mathbf{1}}_{(N-4)\cdot N}\\\ \end{array}\right)$ (2.48)

and the charge operator is

$Q_{\chi\eta}=\int
d^{3}x\,\Big{[}(N-4){\bar{\chi}}_{ij}\chi^{ij}-(N-2)\sum_{m}{\bar{\eta}}^{i}_{m}\eta_{i}^{m}\Big{]}\;,$
(2.49)

so

$\displaystyle[Q_{\chi\eta},\chi^{k\ell}\eta_{\ell}^{n}]=\left(-(N-4)+N-2\right)\psi^{k\ell}\eta_{\ell}^{n}=2\,\chi^{k\ell}\eta_{\ell}^{n}\;,$
$\displaystyle[Q_{\chi\eta},{\bar{\chi}}_{k\ell}{\bar{\eta}}^{\ell}_{n}]=\left(N-4-(N-2)\right){\bar{\chi}}_{k\ell}{\bar{\eta}}^{\ell}_{n}=-2\,{\bar{\chi}}_{k\ell}\,{\bar{\eta}}^{\ell}_{n}\;.$
(2.50)

The condensate (2.44) leads to

$\langle[Q_{\chi\eta},\chi^{k\ell}\eta_{\ell}^{m}]\rangle\neq 0\;,\qquad
k=m=1,2,\ldots,N-4\;.$ (2.51)

On the other hand, a diagonal color $U(1)_{\rm c}\subset SU(N)$ generator
(which mixes with $U(1)_{\psi\eta}$) is

$T_{\rm c}=\left(\begin{array}[]{cc}4\,{\mathbf{1}}_{N-4}&\\\
&-(N-4){\mathbf{1}}_{4}\end{array}\right)\;,$ (2.52)

with charge operator $Q_{\rm c}$

$[Q_{\rm
c},\chi^{k\ell}\eta_{\ell}^{n}]=\begin{cases}4\,\chi^{k\ell}\eta_{\ell}^{n}&\text{for}\qquad
k\leq N-4\;;\\\ -(N-4)\,\chi^{k\ell}\eta_{\ell}^{n}&\text{for}\qquad
k>N-4\;.\end{cases}$ (2.53)

The same condensate (2.44) breaks both $U(1)_{\rm c}$ and $U(1)_{\chi\eta}$,
but the combination $U(1)^{\prime}$ generated by

$2Q_{\chi\eta}-Q_{\rm c}$ (2.54)

remains unbroken:

$[2Q_{\chi\eta}-Q_{\rm c},\chi^{k\ell}\eta_{\ell}^{n}]=0\;.$ (2.55)

Unlike in the $\psi\eta$ model, therefore, no physical massless NG bosons
appear in the $\chi\eta$ model, as the potential NG boson is eaten up by the
$U(1)_{\rm c}\subset SU(N)$ color gauge boson by the Englert-Brout-Higgs
mechanism.

As in the $\psi\eta$ model, there is actually another $U(1)$ symmetry (any
combination of $U(1)_{\chi}$ and $U(1)_{\eta}$, other than $U(1)_{\chi\eta}$).
This $U(1)$ is also spontaneously broken by the condensates, (2.44), but the
associated pseudo-NG boson gets mass by strong anomaly, as in the $U(1)_{A}$
NG boson of QCD. The fact that there are no physical, massless NG bosons in
the $\chi\eta$ model does not mean that this anomalous $U(1)$ symmetry is
unimportant for the discussion of the infrared dynamics, see the next section.

A closely related question is what is eventually the gauge-invariant form of
the condensate, (2.44). A natural choice for the $\chi\eta$ model is
($(\chi\eta)^{im}\equiv\chi^{ij}\eta_{j}^{m}$)

$\displaystyle U$ $\displaystyle=$ $\displaystyle\epsilon_{i_{1}i_{2}\ldots
i_{N}}\epsilon_{m_{1}m_{2}\ldots
m_{N-4}}(\chi\eta)^{i_{1}m_{1}}(\chi\eta)^{i_{2}m_{2}}\ldots(\chi\eta)^{i_{N-4}m_{N-4}}\chi^{i_{N-3}i_{N-2}}\chi^{i_{N-1}i_{N}}$
(2.56) $\displaystyle\sim$
$\displaystyle\epsilon\,(\chi\eta)^{N-4}\chi\chi\;.$

##  3 Strong anomaly and effective Lagrangian

In this section, we first review the well-known strong-anomaly effective
action in QCD and then, following the same steps, write down the analogous
effective action for chiral gauge theories.

###  3.1 Strong anomaly, $U(1)$ problem and the $\theta$ dependence in QCD

In the discussion of the dynamics of QCD the consideration of the anomalous
axial $U(1)_{A}$ symmetry has been quite important, in relation to the so-
called $U(1)$ problem and its solution [18, 19]. Even though the NG boson(s)
associated with the anomalous $U(1)_{A}$ ($\eta,\eta^{\prime}$) 555Here
$\eta,\eta^{\prime}$ are the singlet pseudoscalar mesons of the real world, as
in the Particle Data Booklet. The attentive reader will not confuse them with
the Weyl fermion in the chiral $\psi\eta$ or $\chi\eta$ models being studied
in the present work. get mass by the strong interaction dynamics ($m_{\eta}\gg
m_{\pi}$, $m_{\eta^{\prime}}\gg m_{K}$) the presence of the anomalous $U(1)$
symmetry has a deep implication on the spontaneous breaking of the
nonanomalous chiral symmetries,

$SU(N_{\rm f})_{L}\times SU(N_{\rm f})_{R}\to SU(N_{\rm f})_{V},$ (3.1)

which generates physical lightest NG bosons, the pions.

Such a logical connection is best seen in the effective Lagrangian approach
for QCD in the large $N$ limit. Generalizing the standard sigma model
Lagrangian to include the effect of strong anomaly, the authors of [20]-[24]
write

$L=L_{0}+{\hat{L}}\;,$ (3.2)

where $L_{0}$ is the standard sigma model effective Lagrangian

$L_{0}=\frac{F_{\pi}^{2}}{2}\mathop{Tr}\nolimits\,\partial_{\mu}U\partial^{\mu}U^{\dagger}+\mathop{Tr}\nolimits
M\,U+{\rm h.c.}+\ldots\;;\qquad U\equiv{\bar{\psi}}_{R}\psi_{L}\;,$ (3.3)

and ${\hat{L}}$ represents the strong anomaly

${\hat{L}}=\frac{i}{2}q(x)\,\log\det
U/U^{\dagger}+\frac{N}{a_{0}F_{\pi}^{2}}q^{2}(x)-\theta\,q(x)\;,$ (3.4)

$q(x)$ is the topological density

$q(x)=\frac{g^{2}}{32\pi^{2}}F_{\mu\nu}^{a}{\tilde{F}}^{a,\mu\nu}\;,$ (3.5)

$a_{0}$ is a constant of the order of unity, $F_{\pi}$ the pion decay
constant, and $\theta$ is the QCD vacuum parameter. The $U(1)_{A}$ anomaly
under

$\Delta S=2N_{\rm f}\alpha\int
d^{4}x\frac{g^{2}}{32\pi^{2}}F_{\mu\nu}^{a}{\tilde{F}}^{a,\mu\nu}\;,\qquad\psi_{L}\to
e^{i\alpha}\psi_{L}\;,\quad\psi_{R}\to e^{-i\alpha}\psi_{R}\;,$ (3.6)

is reproduced by the $\log\det U/U^{\dagger}$ term of the effective action.

Treating $q(x)$ as an auxiliary field, and integrating, one gets another form
of the anomaly term [20]-[24]

${\hat{L}}=-\frac{F_{\pi}^{2}\,a_{0}}{4N}\Big{(}\theta-\frac{i}{2}\log\det
U/U^{\dagger}\Big{)}^{2}\;.$ (3.7)

It has been noted that a multi-valued effective Lagrangian involving $\log\det
U/U^{\dagger}$ is only well defined because

$\langle U\rangle\propto{\mathbf{1}}\neq 0\;:$ (3.8)

the effective potential is defined as its expansion around its VEV 666We
indicated the $U(1)_{A}$ NG boson as $\eta$ here, as the real-world
pseudoscalar mesons $\eta$ or $\eta^{\prime}$. The attentive reader will not
confuse it with the $\eta$ field in the $\psi\eta$ model under consideration.
,

$U\propto
e^{i\tfrac{\pi^{a}t^{a}}{F_{\pi}}+i\tfrac{\eta\,t^{0}}{F_{\pi}^{(0)}}}={\mathbf{1}}+i\frac{\pi^{a}t^{a}}{F_{\pi}}+i\frac{\eta\,t^{0}}{F_{\pi}^{(0)}}+\ldots$
(3.9)

Inverting the logics, one might actually argue that the presence of such an
effective action reproducing the strong anomaly implies a nonvanishing
condensate, $\langle U\rangle=\langle\bar{\psi}_{R}\psi_{L}\rangle\neq 0$, and
hence indirectly the spontaneous breaking of nonanomalous chiral symmetry,
$SU(N_{\rm f})_{L}\times SU(N_{\rm f})_{R}\to SU(N_{\rm f})_{V}$, affecting
the low-energy physics.

####  3.1.1 Veneziano -Yankielowicz and Affleck-Dine-Seiberg superpotentials

In the context of ${\cal N}=1$ supersymmetric gauge theories, the strong-
anomaly effective action is expressed by the so-called Veneziano-Yankielowicz
(VY) and Affleck-Dine-Seiberg (AffDinSei) superpotentials [27]-[29]. They
correctly reproduce in the infrared effective theory the effects of
instantons, supersymmetric Ward-Takahashi identities, and the anomaly of [30,
31]. The VY and ADS superpotentials are crucial in determinig the infrared
dynamics and phases of the ${\cal N}=1$ supersymmetric gauge theories (see
[32] for a review).

###  3.2 Strong anomaly and effective action in the $\psi\eta$ model

Unlike ${\bar{\psi}_{R}}\psi_{L}$ in QCD,
${\tilde{\phi}}=\sum_{n,j}^{N}\psi^{nj}\eta_{j}^{n}$ is not gauge invariant.
This is not a problem if the system is assumed to be in dynamical Higgs phase
of the $\psi\eta$ model, in which the low-energy symmetry is

$SU(N)_{{}_{\rm cf}}\times SU(4)\times U(1)^{\prime}\;,$ (3.10)

(Appendix B). A gauge invariant form of the condensate, consistent with such a
symmetry is:

$\det U\;,\qquad U^{k\ell}\equiv\psi^{kj}\eta_{j}^{\ell}\;.$ (3.11)

In terms of this composite field, one may write the low-energy effective
Lagrangian describing the strong anomaly,

${\hat{L}}=\frac{i}{2}q(x)\log\det U/U^{\dagger}\;,\qquad
q(x)=\frac{g^{2}}{32\pi^{2}}F_{\mu\nu}^{a}{\tilde{F}}^{a,\mu\nu}\;,$ (3.12)

which looks very much in analogy with the strong-anomaly effective action of
QCD, (3.4).777$\theta$ parameter is absent in chiral gauge theories. The
multivalued, logarithmic potential is well defined, as we are in the dynamical
Higgs phase, $\langle U\rangle\propto{\mathbbm{1}}$. As in QCD, one may
actually reverse the logics and argue that the strong anomaly effective
action, which should be present in the low-energy effective theory for
faithfully representing all the symmetries of the UV theory, implies the
nonvanishing condensates, $\langle\det U\rangle\neq 0$, which means the global
symmetry breaking as in (3.10), i.e., the system is in Higgs phase.

Though (3.12) appears to be a natural choice in the broken phase, it has a
defect of not being invariant under the full symmetry of the underlying
theory,

$SU(N)\times SU(N+4)\times U(1)_{\psi\eta}\;.$ (3.13)

A correct low-energy effective action should be invariant under the full
symmetry, and must describe (at least, be consistent with) the breaking from
(3.13) to (3.10). This observation brings us back to the questions on the
nature of $U(1)$ NG bosons, raised at the end of Sec. 2.3.

In order to get to the right form of the strong-anomaly effective action, we
start from the beginning,

${\cal L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+{\cal L}^{\rm fermions}$ (3.14)
${\cal L}^{\rm
fermions}=-i\overline{\psi}{\bar{\sigma}}^{\mu}\left(\partial+\mathcal{R}_{\mathrm{S}}(a)\right)_{\mu}\psi\;-i\overline{\eta}{\bar{\sigma}}^{\mu}\left(\partial+\mathcal{R}_{\mathrm{F}^{*}}(a)\right))_{\mu}\eta\;$
(3.15)

($a$ is the $SU(N)$ gage field, and the matrix representations appropriate for
$\psi$ and $\eta$ fields are indicated in an obvious notation). Change the
variables by

${\cal L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+{\cal L}^{\rm
fermions}+\mathop{Tr}\nolimits[(\psi\eta)^{*}U]+{\rm
h.c.}+{B}\,(\psi\eta\eta)^{*}+{\rm h.c.}\;,$ (3.16)

where $U$ is the composite scalars of $N\times(N+4)$ color-flavor mixed matrix
form,

$\mathop{Tr}\nolimits[(\psi\eta)^{*}U]\equiv(\psi^{ij}\eta_{j}^{m})^{*}U^{im}\;$
(3.17)

and ${B}$ are the baryons $B\sim\psi\eta\eta$,

$B^{mn}=\psi^{ij}\eta_{i}^{m}\eta_{j}^{n}\;,$ (3.18)

antisymmetric in $m\leftrightarrow n$. Here we have anticipated the fact that
these baryonlike composite fields, present in the Higgs phase together with
the composite scalars $~{}\psi\eta$ (see Appendix B), are also needed to write
down the strong-anomaly effective action, see below.

Integrating over $\psi$ and $\eta$ one gets

${\cal L}^{\rm
eff}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\mathop{Tr}\nolimits({\cal
D}U)^{\dagger}{\cal
D}U-i\overline{B}\,{\bar{\sigma}}^{\mu}\partial_{\mu}{B}-V\;.$ (3.19)

The potential $V$ is assumed to be such that its minimum is of the form,
(2.16):

$\langle U^{im}\rangle=\,c_{\psi\eta}\,\Lambda^{3}\delta^{im}\;,\qquad\quad\
i,m=1,2,\dots N\;,$ (3.20)

and among other terms, it contains the strong anomaly term, ${\hat{L}}$,

$V=V^{(0)}+{\hat{L}}\;,$ (3.21)

which we choose as

${\hat{L}}={\rm const}\,\left[\log\left(\epsilon\,{B}{B}\det
U\right)-\log\left(\epsilon\,{B}{B}\det U\right)^{\dagger}\right]^{2}\;.$
(3.22)

The argument of the logarithm

$\epsilon\,{B}{B}\det
U\equiv\epsilon^{m_{1},m_{2},\ldots,m_{N+4}}\epsilon^{i_{1},i_{2},\ldots,i_{N}}{B}_{m_{N+1},m_{N+2}}{B}_{m_{N+3},m_{N+4}}U_{i_{1}m_{1}}U_{i_{2}m_{2}}\ldots
U_{i_{N}m_{N}}\;$ (3.23)

is invariant under the full (nonanomalous) symmetries,

$SU(N)_{\rm c}\times SU(N+4)\times U(1)_{\psi\eta}\;.$ (3.24)

It contains $N+2$ $\psi$’s and $N+4$ $\eta$’s, the correct numbers of the
fermion zeromodes in the instanton background: in other words, it corresponds
to a ’t Hooft’s instanton n-point function, e.g.,

$\langle\psi\eta\eta(x_{1})\psi\eta\eta(x_{2})\psi\eta(x_{3})\ldots\psi\eta(x_{N+2})\rangle\;.$
(3.25)

Up to now, we have assumed that the $\psi\eta$ system is in dynamical Higgs
phase, characterized by the bifermion $\psi\eta$ condensate, (3.20). In fact,
a crucial observation is that there is no way of saturating the fermion zero
modes, $N_{\psi}=N+2$; $N_{\eta}=N+4$, by using the baryon fields
(${B}\sim\psi\eta\eta$) only. This is a strong hint that such a confining
phase in which the only infrared degrees of freedom are the baryons (Appendix
A) cannot be the correct vacuum of the system.

The problem of multi-valuedness of the effective action is solved by assuming
that

$\langle\epsilon^{(4)}{B}{B}\rangle\neq 0\;,\qquad\langle\det U\rangle\neq
0\;,$ (3.26)

where

$\epsilon^{(4)}{B}{B}=\epsilon_{\ell_{1}\ell_{2}\ell_{3}\ell_{4}}{B}^{\ell_{1}\ell_{2}}{B}^{\ell_{1}\ell_{2}}\;,\qquad\ell_{i}=N+1,\ldots,N+4\;.$
(3.27)

As

$\langle\det U\rangle\propto{\mathbf{1}}_{N\times N}\;$ (3.28)

takes up all flavors up to $N$ (by using the full $SU(N+4)$ symmetry to orient
the symmetry breaking direction), ${B}{B}$ must be made of the four remaining
flavors, as in (3.27). These baryons were not considered in earlier studies
[10, 25], but are assumed to be massless here, and indicated as
$B^{[A_{2}B_{2}]}$ in Table 2 (Appendix B). Note that this appears to present
us with another puzzle. If these extra baryons were massive, how could such
composites appear in the low-energy effective action? Actually, there is an
elegant answer. The point is that these extra baryons do not have conventional
triangle anomalies with respect to the unbroken flavor symmetry,

$G_{\rm f}^{\prime}=SU(N)_{\rm cf}\times SU(4)\times U(1)^{\prime}$ (3.29)

(in particular, they have a vanishing $U(1)^{\prime}$ charge), as can be seen
in Table 2 (Appendix B). We assume therefore that the baryons, indicated as
$B^{[A,B]}$ in Table 2, are all massless, including those carrying the flavor
indices ($A,B=N+1,\ldots,N+4$). The conventional ’t Hooft criterion with
respect to $SU(N)_{\rm c}\times SU(4)\times U(1)^{\prime}$ did not require
them but does not exclude them either.

We are now ready to answer the questions raised at the end of Sec. 2.3. We
expand around the VEV,

$\displaystyle\det U=\langle\det
U\rangle+\ldots\propto{\mathbf{1}}+\frac{i}{F_{\pi}^{(0)}}\,\phi_{0}+\ldots\;;$
$\displaystyle\epsilon^{(4)}{B}{B}=\langle\epsilon^{(4)}{B}{B}\rangle+\ldots\propto{\mathbf{1}}+\frac{i}{F_{\pi}^{(1)}}\,\phi_{1}+\ldots\;,$
(3.30)

where we appropriately re-normalized the fields, $\phi_{0}$ is the fluctuation
of $\sum_{n}(\psi\eta)^{nn}$, (2.40), and $F_{\pi}^{(0)}$ and $F_{\pi}^{(1)}$
are some constants with dimension of mass.

Note that under the nonanomalous $U(1)$ symmetries discussed in Sec. 2.3,

$\langle[Q_{\psi\eta},\epsilon\,{B}{B}\det
U]\rangle=0\;,\quad\langle[Q_{f},\epsilon\,{B}{B}\det U]\rangle=0\;,$ (3.31)

as expected: the fluctuation of $\epsilon\,{B}{B}\det U$,

${\tilde{\phi}}\equiv
N_{\pi}\left[\frac{1}{F_{\pi}^{(0)}}\,\phi_{0}+\frac{1}{F_{\pi}^{(1)}}\,\phi_{1}\right]\;,\qquad
N_{\pi}=\frac{F_{\pi}^{(0)}F_{\pi}^{(1)}}{\sqrt{\big{(}F_{\pi}^{(0)}\big{)}^{2}+\big{(}F_{\pi}^{(1)}\big{)}^{2}}}\;,$
(3.32)

therefore does not represent the physical NG boson. Indeed, the strong-anomaly
effective action (3.22) gives a quadratic mass term for ${\tilde{\phi}}$.

Vice versa, an orthogonal combination

${\phi}\equiv
N_{\pi}\left[\frac{1}{F_{\pi}^{(1)}}\phi_{0}-\frac{1}{F_{\pi}^{(0)}}\phi_{1}\right]\;,$
(3.33)

does not get mass from (3.22). It is this field that represents (i.e., is the
interpolating field of) the physical $U(1)_{NG}$ NG boson of the $\psi\eta$
model, not the naïve expectation $\phi_{0}$, discussed in Sec. 2.3. The
presence of two condensates, $\langle\det U\rangle$ and
$\langle\epsilon^{(4)}{B}{B}\rangle,$ is thus the key to answer the questions
brought up at the end of Sec. 2.3. In particular, any anomalous combination of
$U_{\psi}(1)$ and $U_{\eta}(1)$ is spontaneously broken also, and $U(1)_{an}$
can indeed be regarded as a good analogue of the axial $U(1)_{A}$ symmetry in
QCD.

Let us check that everything fits together. We note that
$\langle\epsilon^{(4)}{B}{B}\rangle\neq 0\;$ but $\langle{B}\rangle=0$. On the
contrary, $\langle\det U\rangle\neq 0$ implies the condensates $\langle
U\rangle\propto{\mathbf{1}}_{N\times N}$, as $U\sim\psi\eta$ is a scalar
composite. Furthermore, the condensate $\langle\epsilon^{(4)}{B}{B}\rangle\neq
0\;$ is a singlet of the unbroken symmetry ${G_{\rm f}}^{\prime}$: it does not
modify the symmetry breaking pattern in the Higgs phase of the $\psi\eta$
model, determined by the color-flavor locked bifermion condensate,
$\langle\psi\eta\rangle\propto{\mathbf{1}}$, (3.10). Also, there are four-
baryon couplings from (3.22), and due to Nambu-Jona-Lasinio mechanism,
${B}\sim B^{[A_{2}B_{2}]}$ acquires mass: they disappear from the massless
spectrum, leaving the massless spectrum of the $\psi\eta$ model considered
earlier [10, 25]. It might appear that this effectively brings us back to the
simple-minded approach to the strong anomaly sketched at the beginning of this
section, but as we saw above, a more careful treatment was really needed.

The full understanding of the problem would require clarification of the
mechanism for the condensation of the baryon pair together with that of $\det
U$, (3.26). The absolute values of the condensates $BB$ and $\det
U=\det(\psi\eta)$ are flat directions of the potential (3.22). The instanton-
induced potential is thus insufficient to determine them in itself. Even
though the condensation of $\det U$ is analogous to the quark condensates in
QCD, and can be understood as due to the strong interactions among $\psi$’s
and $\eta$’s with the color gauge bosons, the condensation of the color-
singlet baryons is less obvious. Being the components of the baryon
$B\sim\psi\eta\eta$ charged under color, it is natural to expect residual
dipole-like interactions between the $B$’s. The flat directions of the
instanton potential (3.22) will be lifted, in some way, by quantum
corrections. Such an information is implicitly in the (unkown) potential $V$
in (3.19), but certainly a more in-depth study (as in the Coleman-Weinberg
effective action) is needed to understand the mechanism for the condensation
of $BB$, and the relation with the $\det U$ condensate.

###  3.3 Strong anomaly effective action in the generalized BY models

As the solution given above on the $\psi\eta$ model is remarkably subtle, one
might wonder whether a similar mechanism is at work in the so-called general
Bars-Yankielowicz model, an $SU(N)$ gauge theory with Weyl fermions

$\displaystyle\psi^{ij}\,,\quad\eta_{i}^{A}\,\,,\quad\xi^{i,a}$ (3.34)

in the direct-sum representation

$\yng(2)\oplus(N+4+p)\,{\bar{{\yng(1)}}}\;\oplus p\,{{{\yng(1)}}}\;.$ (3.35)

Without repeating the analysis we recall simply [25] that a chirally symmetric
confining vacuum, with massless baryons

${({B}_{1})}^{[AB]}=\psi^{ij}\eta_{i}^{A}\eta_{j}^{B}\;,\qquad{({B}_{2})}^{a}_{A}=\bar{\psi}_{ij}\bar{\eta}^{i}_{A}\xi^{j,a}\;,\qquad{({B}_{3})}_{\\{ab\\}}=\psi^{ij}{\bar{\xi}}_{i,a}{\bar{\xi}}_{j,b}\;,$
(3.36)

(the first is anti-symmetric in $A\leftrightarrow B$ and the third is
symmetric in $a\leftrightarrow b$), saturating all conventional ’t Hooft
anomaly triangles (see Appendix E) cannot be the correct vacuum of the system.
A $({\mathbbm{Z}}_{2})_{F}-[{\mathbbm{Z}}_{2}]^{2}$ mixed anomaly, present in
the UV theory, is absent in the IR.

A dynamical Higgs phase (see Appendix F) with condensate formation,

$\displaystyle\langle
U^{iB}\rangle=\langle\psi^{ij}\eta_{i}^{B}\rangle=\,c_{\psi\eta}\,\Lambda^{3}\delta^{jB}\neq
0\;,\qquad\ \ j,B=1,\dots,N\;,$ $\displaystyle\langle
V^{aA}\rangle=\langle\xi^{i,a}\eta_{i}^{A}\rangle=\,c_{\eta\xi}\,\Lambda^{3}\delta^{N+4+a,A}\neq
0\;,\quad a=1,\dots,p\;,\quad A=N+5,\dots,N+4+p\;,$ (3.37)

and with symmetry breaking

$\displaystyle SU(N)_{\mathrm{c}}\times SU(N+4+p)_{\eta}\times
SU(p)_{\xi}\times U(1)_{\psi\eta}\times U(1)_{\psi\xi}$
$\displaystyle\xrightarrow{\langle\xi\eta\rangle,\langle\psi\eta\rangle}SU(N)_{{\rm
cf}_{\eta}}\times SU(4)_{\eta}\times SU(p)_{\eta\xi}\times
U(1)_{\psi\eta}^{\prime}\times U(1)_{\psi\xi}^{\prime}\;$ (3.38)

turns out instead to be consistent [25].

A strong anomaly effective action for these theories can be constructed in a
way analogous to the $\psi\eta$ model. Instead of (3.23), one has now

$\displaystyle\epsilon\ {{B}_{1}}{{B}_{1}}\det U\det
V\equiv\epsilon^{m_{1},m_{2},\ldots,m_{N+4+p}}\epsilon^{i_{1},i_{2},\ldots,i_{N}}\epsilon^{k_{1},k_{2},\ldots,k_{p}}\times$
$\displaystyle\qquad\times\,{B}_{1}^{[m_{N+1},m_{N+2}]}{B}_{1}^{[m_{N+3},m_{N+4}]}U^{i_{1}m_{1}}U^{i_{2}m_{2}}\ldots
U^{i_{N}m_{N}}V^{m_{N+5}k_{1}}\ldots V^{m_{N+4+p}k_{p}}\;,$ (3.39)

where $B_{1}$ are the baryons $\sim\psi\eta\eta$ defined in (3.36). The rest
of the analysis closely follows that of the $\psi\eta$ model discussed in Sec.
3.2. We shall not pursue further the details of the analysis here, except for
noting that the strong anomaly effective action with such a logarithm, is
perfectly consistent with (implies?) the condensates, (3.37), together with
$\langle{B}_{1}{B}_{1}\rangle\neq 0$, where ${B}_{1}$ are the baryons defined
in (3.36), (F.6). Writing extensively,

$\displaystyle\langle{B}_{1}{B}_{1}\rangle=\langle\epsilon_{C_{1}C_{2}C_{3}C_{4}}(\psi^{ij}\eta_{i}^{C_{1}}\eta_{j}^{C_{2}})(\psi^{k\ell}\eta_{k}^{C_{3}}\eta_{\ell}^{C_{4}})\rangle\neq
0\;,$ $\displaystyle C_{1}\sim C_{4}=N+1,\dots,N+4\;:$ (3.40)

i.e., the system is in dynamical Higgs phase, described in Appendix F.

On the contrary, it is clearly not possible to write the strong-anomaly
effective action with logarithmic argument (3.39), in terms of massless
composite fermions (3.36).

###  3.4 Strong anomaly in the $\chi\eta$ model

Any combination of $U(1)_{\chi}$ and $U(1)_{\eta}$ other than
$U(1)_{\chi\eta}$ (see Table 4) suffers from the strong anomaly. It means that
the low-energy effective action should contain a term analogous to (3.4) for
QCD or (3.12) for the $\psi\eta$ model. The natural choice of a gauge-
invariant condensate (2.56) suggests an effective action of the form for the
$\chi\eta$ model:

$\frac{i}{2}q(x)\log(\chi\eta)^{N-4}\chi\chi+{\rm h.c.}\;,$ (3.41)

($q(x)$ is the topological density defined in (3.5)) where

$\displaystyle(\chi\eta)^{N-4}\chi\chi\equiv\epsilon_{i_{1}i_{2}\ldots
i_{N}}\epsilon_{m_{1}m_{2}\ldots
m_{N-4}}\,(\chi\eta)^{i_{1}m_{1}}(\chi\eta)^{i_{2}m_{2}}\ldots(\chi\eta)^{i_{N-4}m_{N-4}}\chi^{i_{N-3}i_{N-2}}\chi^{i_{N-1}i_{N}}\;.$
(3.42)

The argument of the logarithmic function taken here reflects the correct
number of the fermion zeromodes in the instanton background ($N_{\chi}=N-2$
and $N_{\eta}=N-4$); the epsilon tensors take care of the invariance under the
full (nonanomalous) symmetry of the $\chi\eta$ system,

$SU(N)_{\rm c}\times SU(N-4)\times U(1)_{\chi\eta}\;.$ (3.43)

This anomaly effective action agrees with the one proposed by Veneziano [5]
for the special case of $SU(5)$ $\chi\eta$ model, and generalizes it to all
$SU(N)$ $\chi\eta$ models.

An important observation we share with [5] is that this strong anomaly
effective action, which should be present in the low-energy theory to
reproduce correctly the (anomalous and nonanomalous) symmetries of the UV
theory, implies nonvanishing condensates,

$\langle\chi\eta\rangle\neq 0\;,\qquad\langle\chi\chi\rangle\neq 0\;,$ (3.44)

i.e., the dynamical Higgs phase, Appendix D.

Another important observation here is that there is no way of writing the
strong anomaly effective action (3.41) in terms of the “baryons”,
$B\sim\chi\eta\eta$, of the assumed confining, chirally symmetric phase
(Appendix C). No combination of the baryons can saturate the correct number of
the fermion zeromodes, cfr (3.42).

Even though, contrary to the $\psi\eta$ model, the $\chi\eta$ system has no
physical $U(1)$ NG boson (it is eaten by a color $SU(N)$ gauge boson), the
counting of the broken and unbroken $U(1)$ symmetries is basically similar in
the two models. Of the two nonanomalous symmetries ($U(1)_{\rm c}$ and
$U(1)_{\chi\eta}$), a combination remains a manifest symmetry, and the other
becomes the longitudinal part of the $T_{\rm c}$ gauge boson. Still another,
anomalous, $U(1)$ symmetry exists, any combination of $U(1)_{\chi}$ and
$U(1)_{\eta}$ other than $U(1)_{\chi\eta}$. This symmetry is also
spontaneously broken hence must be associated with a NG boson, though it will
get mass by the strong anomaly.

By expanding the composite $\chi\eta$ and $\chi\chi$ fields around their
VEV’s,

$\displaystyle(\det U)^{\prime}=\langle(\det
U)^{\prime}\rangle+\ldots\propto{\mathbf{1}}+\frac{i}{F_{\pi}^{(0)}}\,\phi_{0}^{\prime}+\ldots\;,$
$\displaystyle\chi\chi=\langle\chi\chi\rangle+\ldots\propto{\mathbf{1}}+\frac{i}{F_{\pi}^{(1)}}\,\phi_{1}^{\prime}+\ldots\;,$
(3.45)

where $(\det U)^{\prime}$ is defined in the $N-4$ dimensional color-flavor
mixed space, and

$\chi\chi=\epsilon_{i_{1}i_{2}i_{3}i_{4}}\chi^{i_{1}i_{2}}\chi^{i_{3}i_{4}}\;,\qquad
N-3\leq i_{j}\leq N\;.$ (3.46)

Now the strong-anomaly effective action (3.41) gives mass to

${\tilde{\phi}}^{\prime}\equiv
N_{\pi}\left[\frac{1}{F_{\pi}^{(0)}}\,\phi_{0}^{\prime}+\frac{1}{F_{\pi}^{(1)}}\,\phi_{1}^{\prime}\right]\;,\qquad
N_{\pi}=\frac{F_{\pi}^{(0)}F_{\pi}^{(1)}}{\sqrt{\big{(}F_{\pi}^{(0)}\big{)}^{2}+\big{(}F_{\pi}^{(1)}\big{)}^{2}}}\;,$
(3.47)

whereas an orthogonal combination

${\phi}^{\prime}\equiv
N_{\pi}\left[\frac{1}{F_{\pi}^{(1)}}\phi_{0}^{\prime}-\frac{1}{F_{\pi}^{(0)}}\phi_{1}^{\prime}\right]\;$
(3.48)

remains massless: it is this potential NG boson which is absorbed by the color
$T_{\rm c}$ gauge boson.

###  3.5 Strong anomaly in the generalized GG models

The structure of the strong anomaly action in the $\chi\eta$ turned out to be
markedly simpler than that in the $\psi\eta$ model. The argument of the
logarithm is made of composite scalar fields only. One might wonder if such a
simple description is valid also for more general Georgi Glashow models [25],
an $SU(N)$ gauge theory with Weyl fermions

$\displaystyle\chi^{[ij]}\,,\quad\eta_{i}^{A}\,\,,\quad\xi^{i,a}$ (3.49)

in the direct-sum representation

$\yng(1,1)\oplus(N-4+p)\,{\bar{{\yng(1)}}}\;\oplus p\,{{{\yng(1)}}}\;.$ (3.50)

It turns out that the construction (3.41) straightforwardly generalizes to

$\frac{i}{2}q(x)\log\epsilon\chi\chi\det\,(\chi\eta)\det(\xi\eta)+{\rm
h.c.}\;,$ (3.51)

where we used a shorthand notation

$\displaystyle\epsilon\chi\chi\det\,(\chi\eta)\det(\xi\eta)=\epsilon_{i_{1}i_{2}\ldots
i_{N}}\,\epsilon_{k_{1}k_{2}\ldots k_{p}}\,\epsilon_{m_{1}m_{2}\ldots
m_{N-4+p}}$ $\displaystyle\qquad\ \
\times\,(\chi\eta)^{i_{1}m_{1}}(\chi\eta)^{i_{2}m_{2}}\ldots(\chi\eta)^{i_{N-4}m_{N-4}}\chi^{i_{N-3}i_{N-2}}\chi^{i_{N-1}i_{N}}(\xi\eta)^{m_{N-3}k_{1}}\ldots(\xi\eta)^{m_{N-4+p}k_{p}}\;.$
(3.52)

Note that this is a singlet of the full symmetry group of the model. The
strong anomaly action (3.51) implies the condensates

$\displaystyle\,\langle\chi^{ij}\eta_{i}^{A}\rangle=c_{\chi\eta}\,\Lambda^{3}\delta^{jA}\neq
0\;,\qquad\qquad\ j=1,\dots,N-4\;,\quad A=1,\dots,N-4\;,$
$\displaystyle\langle\xi^{i,a}\eta_{i}^{B}\rangle=c_{\xi\eta}\,\Lambda^{3}\delta^{N-4+a,B}\neq
0\;,\qquad a=1,\dots,p\;,\quad B=N-3,\dots,N-4+p\;,$

and

$\langle\chi^{j_{1}j_{2}}\chi^{j_{3}j_{4}}\rangle=c_{\chi\chi}\,\epsilon^{j_{1}j_{2}j_{3}j_{4}}\Lambda^{3}\neq
0\;,\qquad j_{1},\ldots,j_{4}=N-3,\ldots,N\;.$ (3.54)

These are precisely the set of condensates expected to occur in the Higgs
phase of the GG models, Appendix H [25].

Vice versa, in the confining vacuum with unbroken global symmetry, Appendix G,
there is no way the baryons (G.2) saturate all the fermion zeromodes, as in
(3.52).

###  3.6 $SU(6)$ model with a single fermion in a self-adjoint representation

After the studies of BY and GG models above, it may be of some interest to see
whether our argument based on the strong anomaly may give some relevant
information in other types of models. Let us discuss just a few examples. The
first is an $SU(6)$ model with a single left-handed fermion in the
representation,

${\underline{20}}\,=\,\yng(1,1,1)\,,$ (3.55)

studied in [33, 34]. The gauging of the 1-form ${\mathbbm{Z}}_{3}^{C}$
symmetry gives rise to a new anomaly of the nonanomalous
${\mathbbm{Z}}_{6}^{\psi}\,$ symmetry,

${\mathbbm{Z}}_{6}^{\psi}\longrightarrow{\mathbbm{Z}}_{2}^{\psi}\;,$ (3.56)

leading to a three-fold vacuum degeneracy[33, 34]. The interpretation and the
details of such a breaking depends on which kind of condensates are formed in
the infrared. As (in this particular model), a scalar bifermion composite
cannot be a gauge singlet, the author of [33] suggested a four-fermion
condensate

$\langle\psi\psi\psi\psi\rangle\sim\Lambda^{6}\neq
0\;,\qquad\langle\psi\psi\rangle=0\;,$ (3.57)

whereas the authors of [34] proposed the gauge symmetry breaking condensate

$\langle\psi\psi\rangle\sim\Lambda^{3}\neq 0\;,$ (3.58)

with $\psi\psi$ in the adjoint representation of $SU(6)$. Both scenarios turn
out to be consistent with the discrete symmetry breaking, (3.56), and hence
agree on a three-fold vacuum degeneracy, but the infrared physics associated
with these assumptions are quite different, and one wonders which of the
assumptions describes the system in the infrared.

The strong anomaly effective action in this model has the form,

$\frac{i}{2}q(x)\log\psi\psi\psi\psi\psi\psi+{\rm h.c.}\;.$ (3.59)

Requiring that the argument of the log acquires a nonvanishing VEV, the
assumption of four-fermion condensate (3.57) appears to be somewhat unnatural,
whereas the bifermion condensates (3.58) looks perfectly consistent, with

$\langle\psi\psi\psi\psi\psi\psi\rangle\sim\langle\psi\psi\rangle^{i}_{j}\langle\psi\psi\rangle^{j}_{k}\langle\psi\psi\rangle^{k}_{i}\neq
0\;,$ (3.60)

where the color indices are briefly restored.

###  3.7 Adjoint QCD with $N_{\rm c}=N_{\rm f}=2$

Another interesting model is the adjoint QCD, widely studied in the
literature. Let us however consider a particular case, $N_{\rm c}=N_{\rm
f}=2$. The fermions are two Weyl fermions $\lambda^{i}$, $i=1,2$, both in the
adjoint of $SU(2)$. The conventional thinking assumed that a gauge invariant
bifermion condensate

$\langle\lambda\lambda\rangle\neq 0$ (3.61)

forms, breaking the flavor symmetry as $SU(2)_{\rm f}\to SO(2)_{\rm f}$,
leading to $2$ NG bosons, and reducing the discrete ${\mathbbm{Z}}_{8}$
symmetry to ${\mathbbm{Z}}_{2}$ resulting four degenerate vacua.

A special interest in this model was raised by the work by Anber and Poppitz
[35], which postulates that the system develops a condensates,

$\langle\lambda\lambda\lambda\lambda\rangle\neq
0\;,\qquad\langle\lambda\lambda\rangle=0\;,$ (3.62)

breaking the discrete symmetry as ${\mathbbm{Z}}_{8}$ symmetry to
${\mathbbm{Z}}_{4}$, hence predicting a doubly degenerate vacua. The flavor
$SU(2)_{\rm f}$ remains unbroken; massless baryons

$\sim\lambda\lambda\lambda$ (3.63)

(which is necessarily a doublet of $SU(2)_{\rm f}$) saturates Witten’s $SU(2)$
anomaly.

The generalized, mixed anomaly study in this model predicts [35, 34] the
discrete symmetry breaking ${\mathbbm{Z}}_{8}\to{\mathbbm{Z}}_{4}$: such a
result is consistent with both of the dynamical scenarios mentioned above,
which are markedly different from physics point of view.

Does our strong-anomaly argument tell anything significant? The analogue of
the strong anomaly effective action, such a (3.4, (3.41) and (3.59), is in
this case,

$\frac{i}{2}q(x)\log\lambda\lambda\ldots\lambda+{\rm h.c.}\;,$ (3.64)

with eight $\lambda$’s inside the argument of the logarithmic function. In
this model, in contrast to what we saw in the preceding model, Sec. 3.6, our
algorithm seems to be consistent with either of the dynamical possibilities,
(3.61), or (3.62).

It would be interesting if a lattice study could give a definitive answer.

As a side comment, in the case with $N_{\rm f}=1$, arbitrary $N_{\rm c}$, the
adjoint QCD becomes ${\cal N}=1$ supersymmetric Yang-Mills, and (3.64) with
$2N_{\rm f}N_{\rm c}=2N_{\rm c}$ $\lambda$’s, reduces precisely to the
Veneziano-Yankielowicz [27] effective potential. And in that case, the
assumption of the bifermion condensate, $\langle\lambda\lambda\rangle\neq 0$,
the breaking of the discrete symmetry ${\mathbbm{Z}}_{2N_{\rm
c}}\to{\mathbbm{Z}}_{2}$, and the resulting $N_{\rm c}$ fold degeneracy of the
vacua (equal to Witten’s index), are by now generally accepted as the correct
answer for these systems 888Lattice studies [36]-[38] also seem to confirm
this. We thank Stefano Piemonte for bringing these references to our
attention. .

##  4 Complementarity in the $\chi\eta$ model?

In the $\chi\eta$ model the symmetry of the massless sector in the dynamical
Higgs phase (2.43) (Appendix D) happens to coincide 999As noted in [25] this
occurs only for the $\chi\eta$ model. In all other generalized Bars-
Yankielowicz and Georgi-Glasow models, the dynamical Higgs phase has global
symmetry distinct from that in the confining, no-condensate phase, in spite of
the fact that one of the bifermion condensate is in the fundamental
representation of the gauge group. See Appendices A-H.  with that of the UV
theory hence with that of the symmetric, confining phase (Appendix C). This
fact might lead some, based on the so-called complementarity picture [40], to
think that in the $\chi\eta$ model these two phases are actually one and the
same.

The following discussion however shows that the coincidence of the global
symmetries in the two candidate vacua in the $\chi\eta$ model is an accidental
one. There are, indeed, clear indications that they are physically distinct
[10]. Of course it would be difficult to think that a phase in which

$\langle\chi\eta\rangle\propto\Lambda^{3}\neq
0\;,\qquad\langle\chi\chi\rangle\propto\Lambda^{\prime\,3}\neq 0$ (4.1)

(with no adjustable constants in front, and $\Lambda\sim\Lambda^{\prime}$ is
the dynamical mass scale generated by the strong $SU(N)$ gauge interactions.),
and another, without any condensates, can be the same phase. There are no
coupling constants which can be varied continuously so that the two possible
“phases” (one with (4.1), one with no condensates) are connected without phase
transition.

Another relevant issue could be the fact that in the dynamical Higgs phase
(Appendix D) there appear $(N-4)^{2}-1$ massive gauge bosons of degenerate
mass, by the Higgs mechanism together with the remnant color-flavor diagonal
$SU(N-4)$ global symmetry. This is a clear prediction which can be tested by
lattice simulations. Even though it is true that these massive bosons can be
re-expressed as gauge-invariant composites, there is no particular reason why
precisely these massive bosons and not others should appear at the mass scale
$\sim\Lambda$, if the system is in confining, symmetric phase of Appendix C.

Also, to understand the infrared dynamics, it is indispensable to take into
account of the effects of strong anomaly correctly. Its natural form is
(3.41), as discussed in the previous section, by taking into account the
fermion zeromodes in an instanton background. The solution of the effective
equation of motion would then lead to a massive NG boson, but as in QCD and in
the $\psi\eta$ model, the very solution implies nonvanishing vacuum
condensate, (4.1). It means that the system is in dynamical Higgs phase.

Crucially, as already noted at the end of Sec. 3.4, there is no way of writing
the strong-anomaly effective action in terms of the massless composite
“baryons” ($\sim\chi\eta\eta$): the zero-mode counting
($N_{\chi}=N-2>N_{\eta}=N-4$) cannot work. This is a clear sign that the
“confining phase” (Appndix C) can neither be equivalent to the Higgs phase
(Appendix D) nor is one of consistent vacuum phases of the $\chi\eta$ model.

Of course, one of the strongest arguments which discriminates the two
candidate vacua comes from the mixed anomalies and the consequent,
generalized-anomaly-matching analysis [26, 25] (see also [39]). The presence
of a mixed anomaly of the type
$({\mathbbm{Z}}_{2})_{F}-[{\mathbbm{Z}}_{N}]^{2}$ in the UV theory and the
absence of this anomaly in the “symmetric confining phase”, show that such a
phase cannot be the correct vacuum of the theory [26, 25]. As in the
$\psi\eta$ model, in the dynamical Higgs phase characterized by the bifermion
condensates (4.1), the color-flavor locked ${\mathbbm{Z}}_{N}$ symmetry is
spontaneousy broken by the condensates, as the $U(1)_{\chi\eta}$ symmetry used
to form ${\mathbbm{Z}}_{N}\subset
U(1)_{\chi\eta}\times({\mathbbm{Z}}_{2})_{F}$ is spontaneously broken.

To summarize: both the mixed-anomaly analysis [26, 25] and the consideration
of the strong-anomaly effective action Sec. 3.4, lead to the same conclusion
that one of the candidate phase (dynamical Higgs phase) is consistent whereas
the other (confining, flavor symmetric phase) is inconsistent, but not that
these two phases are inequivalent, perhaps separated by a phase transition of
some sort.

##  5 Large $N$ planar dominance and $U(1)$ NG bosons in chiral gauge theories

Large $N$ counting and planar Feynman diagram dominance are perturbative
concepts and are not adequate for describing condensates. Nevertheless, an
argument was presented some time ago [7] which states that in some chiral
gauge theories such as $\psi\eta$ or $\chi\eta$ models, a $U(1)$ symmetry
cannot be spontaneously broken, if the system is assumed to confine. Such a
statement, if taken without due care, might mislead the reader to conclude
that, e.g., in the $\psi\eta$ model, the only possible phase is the chirally
symmetric phase, with no condensates and with massless composite fermions
(Appendix A).101010Let us remember that the notation in [7] and in the present
work are simply related by $S=\psi$; ${\bar{F}}=\eta$.

To be scrupulous, the authors of [7] do not claim that a dynamical Higgs phase
(with gauge dependent bifermion condensate) is impossible in the $\psi\eta$
model, or in general chiral gauge theories. Indeed one of the two main
conclusions of [7] is that a dynamical Higgs phase is likely to be the correct
answer for some chiral gauge theories (e.g., the model in Sec. 5 of [7]).
Their claim concerning the $\psi\eta$ model (Sec. 3 of [7]) is that if the
$\psi\eta$ system is in an exact confinement phase, then a two point function
of a broken $U(1)$ current $\langle 0|T\\{J_{\mu}(x)J_{\nu}(0)\\}|0\rangle$
cannot have, in the leading $N$ approximation, an intermediate state having
the quantum numbers of the $U(1)$ NG boson, therefore a spontaneous $U(1)$
breaking is not possible.

Nevertheless, there seem to be a number of issues in this argument. First of
all, no definition of “exact confinement” is given in [7]. We noted already
that there is no global center symmetry which can be used with the Polyakov
loop, to set a criterion for discriminating confinement/Higgs phases. The
authors of [7] describe the “confinement phase” as a phase in which all
observable states are gauge invariant, but this is also problematic, as in a
local gauge theory gauge noninvariant (gauge-dependent) states can always be
re-expressed in a (though more complicated) gauge-invariant representation.
See (2.40) versus (2.41), for instance.

If the distinction (“exact confinement” or not) is whether or not a gauge-
dependent condensate forms, this cannot be used as a criterion either: as
shown in Sec. 2.3, the gauge-dependent $\psi\eta$ condensate can be re-
expressed as a gauge invariant multifermion condensate, (2.41).

If “exact confinement” means, in the context of the $\psi\eta$ model, that no
condensates $\langle\psi\eta\rangle\neq 0$ form, then no $U(1)$ symmetry is
broken spontaneously, hence no NG bosons would appear in the system anyway,
independently of the $\frac{1}{N}$ counting. The observation [7] that in the
large $N$ limit, assuming the planar diagram dominance, the intermediate
states are all made of pairs of outgoing and incoming $\psi$’s and $\eta$’s,
hence are all neutral with respect to the $U(1)$ charge, is not exactly
pertinent in this case.

Vice versa, if the condensate $\langle\psi\eta\rangle$ does occur, and thus
the system is in Higgs phase, then there is a $U(1)$ symmetry which is
spontaneously broken, and a physical NG boson appears (see Sec. 2.3 above).
This NG boson can appear as an intermediate state of the two-point function of
the associated currents, in the leading $N$ planar graph.

Indeed, for the large $N$ counting of Feynman diagrams in the $\psi\eta$
model, it is sufficient to remember [7] that neither $\psi$ nor $\eta$ loops
are suppressed by $\frac{1}{N}$: the standard $\frac{1}{N}$ suppression for
the fundamental-fermion ($\eta$) loop inside a planar graph is compensated by
a factor $\sim N$ from the flavor multiplicity. This leads to the observation
[7] that the diagrams with $B{\bar{B}}$, $B{\bar{B}}B{\bar{B}}$, etc.,
intermediate states 111111$B$ are the hypothetical massless composite fermions
(“baryons”) of (A.1), $B\sim\psi\eta\eta$.  are not suppressed by a
$\frac{1}{N}$ factor.

For exactly the same reason, neither are the planar graphs with the
intermediate states $\eta\psi\psi^{*}\eta^{*}$,
$\eta\psi\eta\psi(\psi\eta)^{*}(\psi\eta)^{*}$, and $\det U(\det U)^{*}$
121212The $\psi$ and $\eta$ loops (rings) in the graph must be ordered one
inside the other, alternatively, in order to keep the leading $\frac{1}{N}$
order. Arbitrary gluon exchanges can be added in a planar graph as in Fig. 1.
. A simplest such graph is shown in Fig. 1. Allowing for nonvanishing
condensates $\langle\psi\eta\rangle$ to occur in the diagram, one can see that
the $U(1)$ NG boson does appear as an intermediate state in the current two-
point function, Fig. 2. There is no difficulty showing the same by using the
gauge-invariant condensate and pion field, (2.41), even though it becomes more
cumbersome to draw a picture.

Recapitulating, a large N counting argument neither discriminates the possible
phases nor prohibits a $U(1)$ symmetry to be broken spontaneously.

Figure 1: A leading $N$ planar graph for the two-point function $\langle
0|T\\{J_{\mu}(x)J_{\nu}(0)\\}|0\rangle$ with an $\eta^{*}\psi^{*}\psi\eta$
intermediate state (Fig.1a, left). On the right figure (Fig. 1b) the color
(full) and flavor (dashed) lines are shown. The wavy lines inside the graph
are the gluons. Figure 2: The same graph as Fig. 1, with insertion of
$\langle\psi\eta\rangle$ condensates (shown with $\otimes$). The intermediate
state now corresponds to the $U(1)$ NG boson $\sim\psi\eta$ of Eq. (2.40).

##  6 Multifield versus bi-fermion condensates

One of the key elements of our argument based on the strong anomaly is the
statement that the multifield condensate such as the one in the $\chi\eta$
model,

$\langle(\chi\eta)^{N-4}\chi\chi\rangle\neq 0$ (6.1)

(see Eq. (3.42) for the exact meaning) implies the condensation of the
component bi-fermion scalars,

$\langle\chi\eta\rangle\neq 0\;,\qquad\langle\chi\chi\rangle\neq 0\;,$ (6.2)

where

$\langle\chi\eta\rangle\
=\langle\chi_{[ij]}{\eta}^{B\,j}\rangle\sim\,\Lambda^{3}\delta_{i}^{B}\;,\qquad
i,B=1,2,\ldots,N-4\;,$ (6.3)
$\langle\chi\chi\rangle=\langle\epsilon_{i_{1}i_{2}i_{3}i_{4}}\chi^{i_{1}i_{2}}\chi^{i_{3}i_{4}}\rangle\sim\Lambda^{3}\;,\qquad
N-3\leq i_{j}\leq N\;,$ (6.4)

in an appropriate gauge and with flavor orientation. A similar step is used in
all other models considered here. As this point is central in our discussion,
let us pose, before summarizing and concluding this paper, to make a few more
comments. We will use the $\chi\eta$ model (the simplest of the GG models) for
this purpose, for concreteness and to keep the notations simple.

We have not proven that (6.2) follows from (6.1). We do believe this is
correct, but our deduction is based on a collection of plausibility
considerations and consistency checks: dealing with a strongly-coupled gauge
theories, to mathematically prove this type of derivation is not always an
easy task.

Here are a few more considerations, supporting our deduction.

(a)

Quark condensates in QCD versus $\chi\eta$ condensate:

In QCD, the quark condensate

$\langle{\bar{\psi}}_{R}\psi_{L}\rangle\sim\Lambda^{3}\neq 0$ (6.5)

does not imply

$\langle{\bar{\psi}}_{R}\rangle\neq 0\;,\qquad\langle{\psi}_{L}\rangle\neq
0\;.$ (6.6)

However this is not because the quarks are not gauge invariant, but because
they are fermions.

But then as we recalled in Sec. 3.1, the multifield condensate

$\langle\det U\rangle=\langle\det{\bar{\psi}}_{R}\psi_{L}\rangle\neq 0$ (6.7)

following from the strong-anomaly effective action, does seem to imply the bi-
fermion scalar condensates, $\langle{\bar{\psi}}_{R}\psi_{L}\rangle\neq 0$. We
will further argue in Sec. 7 that the scalar composites
${\bar{\psi}}_{R}\psi_{L}$ in QCD and $\chi\eta$ in the GG model (and
similarly $\psi\eta$ in the BY model, etc.) should be regarded as analogous
objects.

We may further add that if the $SU(2)_{WS}\times U(1)_{Y}$ gauge interactions
are taken into account, the quark condensates
$\langle{\bar{\psi}}_{R}\psi_{L}\rangle\neq 0$ are no longer gauge-invariant
objects 131313As is well known, the quark condensate breaks the Weinberg-Salam
gauge symmetry dynamically, even though only by a tiny amount, insufficient to
explain in itself the observed electroweak gauge symmetry breaking. . In this
sense, too, there is no qualitative difference between the scalar composites
${\bar{\psi}}_{R}\psi_{L}$ in QCD and $\chi\eta$ in the GG model.

(b)

The standard-model Higgs doublet:

In the Weinberg-Salam model, the vacuum is at the gauge-invariant minimum of
the potential,

$\langle\sum_{i=1}^{2}\phi^{i\,*}\phi^{i}\rangle\neq 0\;.$ (6.8)

This is taken to mean a nonvanishing Higgs VEV, in an appropriate gauge,

$\langle\phi\rangle=\left(\begin{array}[]{c}v\\\ 0\end{array}\right)\;,\qquad
v\neq 0\;.$ (6.9)

Our deduction (6.1) $\leftrightarrow$ (6.2) is analogous to this, even though
here we have a dynamical, composite “Higgs” scalars. One can argue against the
use of this analogy, on the basis that the Weinberg-Salam model is a weakly
coupled theory, while here we have a strongly coupled one. We believe that
this difference (the former described by a potential, the latter not) is not
essential. What excludes the conclusion that the Weinberg-Salam model with
(6.8) is in confinement phase with

$\langle\sum_{1}^{2}\phi^{i\,*}\phi^{i}\rangle\neq
0\;,\qquad\langle\phi\rangle=0\;,$ (6.10)

is the mass spectrum ($W$, $Z$, $\gamma$, etc). The attempt to rewrite the
whole Weinberg-Salam model as a confining theory [41] fails to reproduce
exactly the mass spectra (though it does almost) of the standard Higgs phase
description.

Our observation here is similar: even though the global symmetry may look the
same in two “complementary” descriptions of the $\chi\eta$ model
141414Remember that the “complementarity” aspect is specific to the $\chi\eta$
model, not shared by any other GG or BY models., the exact mass spectra are
probably different, as we pointed out in Sec. 4.

(c)

Consistency:

A more indirect, consistency argument is the following. Let us assume the
multifield condensate (6.1) forms but with

$\langle\chi\eta\rangle=\langle\chi\chi\rangle=0\;,$ (6.11)

and with the full (nonanomalous) symmetry of the $\chi\eta$ system,

$SU(N)_{\rm c}\times SU(N-4)\times U(1)_{\chi\eta}\;$ (6.12)

intact. The low-energy system is then described, as the conventional ’t Hooft
anomaly argument suggests, by the set of massless baryons (C.1). But as there
is no way to describe the mutifermion condensate (6.1) in terms of these
massless fields, we conclude that the assumption (6.11) is inconsistent.

One may ask why the confining phase with baryons cannot have some composite
scalar $\sim(\chi\eta)^{N-4}\chi\chi$ entering strong anomaly effective
action. To answer this, we recall our criterion: the infrared degrees of
freedom of the assumed phase must be able to describe the strong anomaly
effectively at low energies.  Just as the NG boson fields do in the Higgs
phase of the GG models, and as they do in the confining phase in QCD. In the
BY models, the massless baryons and NG bosons together saturate the argument
of the logarithm, in the dynamical Higgs phase, as we discussed in Sec. 3.

If one adopts this criterion, then the confining, symmetric phase is
disfavored (inconsistent) in all of the GG and BY models.

##  7 Summary and Discussion

Consideration of the strong-anomalies, not always taken into account in the
study of dynamics of chiral gauge theories, turns out to provides us with a
powerful new argument supporting the conclusion of [26, 25]: the system is in
dynamical Higgs phase in the Bars-Yankielowicz and Georgi-Glashow models. The
essential requirement we make is that it should be possible to write a strong-
anomaly effective action by utilizing only the massless degrees of freedom,
assumed to be present in the infrared. This simple requirement is naturally
satisfied in all models considered, if assumed to be in the dynamical Higgs
vacuum with bifermion condensates. On the contrary, in the putative,
confinement vacua with massless composite fermions (baryons) as the only
infrared degrees of freedom, we found no way of writing the correct strong-
anomaly effective action. In other words, it seems that in these latter vacua,
the anomalous $U(1)$ symmetry(ies) cannot be properly realized in the
infrared.

The fact that the matter fermions in our models have larger multiplicities
than the quarks in QCD (for instance $\eta$ in the $\psi\eta$ model has the
flavor multiplicity, $N+4\sim N$ at large $N$), may mean that the mass of the
anomalous $U(1)$ NG bosons will turn out to be of the oder of $\Lambda$. Even
though we do write the strong-anomaly effective action in terms of the
massless degrees of freedom of the presumed infrared theory, the fact that its
solution (the minimum of the effective potential) is at $\mu\sim\Lambda$,
means that the answer cannot be regarded as reliable quantitatively. We are
however interested here only in a qualitative question of the validity of
confinement vs dynamical Higgs phase in the infrared, and we do believe that
our discussion of Sec. 3 makes reasonable sense.

The fact that both mixed anomalies [26, 25] and the strong-anomaly effective
action (this paper) imply dynamical Higgs phase in chiral BY and GG models \-
is actually not a pure coincidence. Both indeed arise by taking properly the
strong chiral $U(1)$ anomalies into account.

Certain analogies and contrasts between the strong-interaction dynamics of
vectorlike and chiral gauge theories seem to emerge from these discussions. As
the “confining, flavor symmetric” vacua have been shown to be disfavored, we
assume below that these chiral gauge theories are indeed in dynamical Higgs
phase.

And let us compare the physics of the $\psi\eta$, $\chi\eta$ and of more
general BY and GG models, with that of the standard QCD with $N_{\rm f}$ light
flavors of quarks and antiquarks. Both in QCD and in these chiral gauge
theories, the theory without the matter fermions is the same, pure $SU(N)$
Yang-Mills theory. We have here nothing new to add to the known and unknown
properties of the pure $SU(N)$ dynamics. Our interest here is the role the
light matter fermions play in determining the infrared dynamics, and how the
resulting phase(s) depend on the types of the matter fermions present.

In many senses, the bifermion condensates such as $U=\psi\eta$ in the
$\psi\eta$ model (and $U=\chi\eta$ and $\chi\chi$ condensates in the
$\chi\eta$ model), are a good analogue of the quark condensate
$U={\bar{\psi}}_{R}\psi_{L}$ in QCD. This “analogy” is based on the fact that
all of these composite scalar fields enter the strong-anomaly effective action
in the same way, as

${\hat{L}}=\frac{i}{2}q(x)\log\det U/U^{\dagger}\;,\qquad
q(x)=\frac{g^{2}}{32\pi^{2}}F_{\mu\nu}^{a}{\tilde{F}}^{a,\mu\nu}\;.$ (7.1)

(see Sec. 3 for more careful discussions). And in all cases this implies
condensation, $\langle U\rangle\propto{\mathbf{1}}$, although it’s meaning
depends on the system: the color-flavor locked Higgs phase in the chiral
models, and the chiral-symmetry-breaking confining vacuum in QCD.

Also, in a more general Bars-Yankielowicz models, there are two natural
bifermion condensation channels ($(\cdot)$ meaning an $SU(N)$ singlet):

$\displaystyle\psi\Big{(}\raisebox{-2.0pt}{\yng(2)}\Big{)}\,\eta\Big{(}\bar{\raisebox{-2.0pt}{\yng(1)}}\Big{)}\qquad\
$ $\displaystyle{\rm forming}\qquad\raisebox{-2.0pt}{\yng(1)}\;,$ (7.2)
$\displaystyle\qquad\xi\Big{(}{\raisebox{-2.0pt}{\yng(1)}}\Big{)}\,\eta\Big{(}\bar{\raisebox{-2.0pt}{\yng(1)}}\Big{)}\qquad\
$ $\displaystyle{\rm forming}\qquad(\cdot)\;:$

the gluon-exchange strengths in the two channels are, respectively,
proportional to
$-\frac{(N+2)(N-1)}{N}$ and $-\frac{N^{2}-1}{N}$ . The $\psi\eta$ channel is
slightly more attractive, but the strengths are identical in the large $N$
limit. Note that in our notation $\langle\xi\eta\rangle$ has the same quantum
numbers as $\langle{\bar{\psi}}_{R}\psi_{L}\rangle$ in QCD. Similarly for the
comparison between the condensates $\langle\chi\eta\rangle$ and
$\langle\xi\eta\rangle$ in the general Georgi-Glashow models. These
considerations, which are based on rather naïve MAC [2] like idea and
therefore are not very rigorous, nevertheless show that the quark condensates
in QCD and the bifermions condensates in the chiral gauge theories under
study, are really on a very “similar” footing.

Of course, the fact that the quark condensate
$\langle{\bar{\psi}}_{R}\psi_{L}\rangle$ is a color singlet, $SU(N_{\rm
f})_{L}\times SU(N_{\rm f})_{R}$ flavor matrix, whereas
$\langle\psi\eta\rangle$ is in a color-flavor bifundamental form, breaking the
color completely, makes a world of differences in other aspects of these
(vectorlike or chiral) systems. Most importantly, the existence of colored NG
bosons in the $\psi\eta$ (or in the $\chi\eta$) model, see Sec. 2.2.1, means
that these are coupled linearly to the color gauge bosons, and via Englert-
Brout-Higgs mechanism make them massive. These processes are absent in QCD,
all NG bosons being color singlets. It is in this sense that we talk about
confinement phase in QCD, in spite of the fact that the linear potential
between any two test particles can be always flattened by the pair production
of quarks from the vacuum, and Higgs phase in the chiral models.

Also, the mass spectra get arranged quite differently in QCD and in the chiral
models discussed here. Apart from the pattern of certain degenerate massive
vector bosons (see Sec. 4), the massless spectrum exhibits striking
differences. In all chiral gauge theories considered here, it contains in
general both a number of composite fermions (baryons) as well as some
composite scalars (pions), a feature not shared by massless QCD. In other
words, the way the chiral symmetries of the UV theory are realized in the IR,
is distinctly different.

Perhaps a closer analogy - from a formal point of view - between the
vectorlike theories and chiral theories, comes from the consideration of color
superconductivity in the high-density limit of QCD [42, 43]. In such a
situation the dynamics of QCD is believed to be such that the colored di-quark
condensates form,

$\langle\psi_{L}\psi_{L}\rangle\neq
0\;,\qquad\langle\psi_{R}\psi_{R}\rangle\neq 0\;.$ (7.3)

In particular, for $N_{\rm f}=3$, they are condensates of color-flavor
diagonal forms, somewhat similar to $\langle\psi\eta\rangle$ or
$\langle\chi\eta\rangle$ in the chiral theories discussed here, although the
details of the dynamics can obviously be quite different.

What lessons should one draw from these discussions? Clearly, there are both
similarities and differences between the dynamics of QCD and that of the
chiral gauge theories discussed here. The consideration of the strong anomaly
effective actions discussed here seems to lead us to clearer meaning of these
comparisons, and with that, to point to a better understanding of the dynamics
of strongly-coupled chiral gauge theories.

## Acknowledgments

We thank Gabriele Veneziano for bringing [5] to our attention and for a
comment, and David Tong for exchanges of views on some of the topics discussed
here. The work is supported by the INFN special research initiative grant,
“GAST” (Gauge and String Theories).

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##  Appendix A Chirally symmetric phase in the $\psi\eta$ model

An interesting possibility for the $\psi\eta$ model is that no condensates
form, the system confines and the flavor symmetry is unbroken [4]. The
candidate massless degrees of freedom in the IR are $\frac{(N+4)(N+3)}{2}$
“baryons”,

$B^{[AB]}=\psi^{\\{ij\\}}\eta_{i}^{A}\eta_{j}^{B}\;,\qquad
A,B=1,2,\ldots,N+4\;,$ (A.1)

antisymmetric in $A\leftrightarrow B$. All the $SU(N+4)\times U(1)$ anomalies
are saturated by $B^{[AB]}$ as can be seen by using the data in Table 1. The
discrete anomaly $(\mathbbm{Z}_{N+2})_{\psi}-[SU(N)]^{2}$ is also matched as
can be easily checked, and all other discrete anomalies are also matched as a
consequence.

| fields | $SU(N)_{\rm c}$ | $SU(N+4)$ | ${U(1)_{\psi\eta}}$
---|---|---|---|---
i UV | $\psi$ | ${\yng(2)}$ | $\frac{N(N+1)}{2}\cdot(\cdot)$ | $N+4$
| $\eta^{A}$ | $(N+4)\cdot{\bar{\yng(1)}}$ | $N\,\cdot\,{\yng(1)}$ | $-(N+2)$
IR | $B^{[AB]}$ | $\frac{(N+4)(N+3)}{2}\cdot(\cdot)$ | ${\yng(1,1)}$ | $-N$
Table 1: Chirally symmetric “confining” phase of the $\psi\eta$ model. As in
other Tables of the text, the multiplicity, charges and the representation are
shown for each set of fermions. $(\cdot)$ stands for a singlet representation.

##  Appendix B Color-flavor locked Higgs phase in the $\psi\eta$ model

Another possibility, in the $\psi\eta$ model, is that of a color-flavor locked
phase [10, 16], with

$\langle\psi^{\\{ij\\}}\eta_{i}^{B}\rangle=\,C\,\Lambda^{3}\delta^{jB}\;,\qquad
j,B=1,2,\dots N\;,$ (B.1)

in which the symmetry is reduced to

$\displaystyle SU(N)_{\rm cf}\times SU(4)_{\rm f}\times U(1)^{\prime}\,.$
(B.2)

A subset of the same baryons ($B^{[A_{1}B_{1}]}$ and $B^{[A_{1}B_{2}]}$ in the
notation of Table 2) saturate all of the triangles associated with the reduced
symmetry group, see Table 2. The massless degrees of freedom are
$\tfrac{N^{2}+7N}{2}$ massless baryons $B^{[A_{1}B]}$ and $8N+1$ NG bosons. To
reproduce correctly the strong anomaly in the IR, however, another condensate
$\langle{B}{B}\rangle\sim\langle\psi\eta\eta\psi\eta\eta\rangle$ and another
set of massless baryon $B^{[A_{2}B_{2}]}$ (see Table 2) are needed. This does
not alter neither the symmetry breaking pattern (B.2), nor the anomaly
matching. See Sec. 3.2 for more discussion.

| fields | $SU(N)_{\rm cf}$ | $SU(4)_{\rm f}$ | $U^{\prime}(1)$
---|---|---|---|---
i UV | $\psi$ | ${\yng(2)}$ | $\frac{N(N+1)}{2}\cdot(\cdot)$ | $1$
| $\eta^{A_{1}}$ | ${\bar{\yng(2)}}\oplus{\bar{\yng(1,1)}}$ | $N^{2}\,\cdot\,(\cdot)$ | $-1$
| $\eta^{A_{2}}$ | $4\cdot{\bar{\yng(1)}}$ | $N\,\cdot\,{\yng(1)}$ | $-\frac{1}{2}$
IR | $B^{[A_{1}B_{1}]}$ | ${\bar{\yng(1,1)}}$ | $\frac{N(N-1)}{2}\cdot(\cdot)$ | $-1$
| $B^{[A_{1}B_{2}]}$ | $4\cdot{\bar{\yng(1)}}$ | $N\,\cdot\,{\yng(1)}$ | $-\frac{1}{2}$
| $B^{[A_{2}B_{2}]}$ | $6\cdot(\cdot)$ | ${\yng(1,1)}$ | $0$
Table 2: Color-flavor locked phase in the $\psi\eta$ model. $A_{1}$ or $B_{1}$
stand for the first $N$ flavors ($A_{1},B_{1}=1,2,\ldots,N$), whereas $A_{2}$
or $B_{2}$ run over the rest of the flavor indices, $N+1,\ldots,N+4$. The set
of potentially massless baryons $B^{[A_{2}B_{2}]}$, which were not explicitly
taken into account in [10, 25], do not contribute to $SU(N)_{\rm cf}\times
SU(4)_{\rm f}\times U^{\prime}(1)$ anomalies.

##  Appendix C Chirally symmetric phase in the $\chi\eta$ model

Let us first examine the possibility that no condensates form, the system
confines and the flavor symmetry is unbroken [3]. The massless baryons are

$B^{\\{CD\\}}=\chi_{[ij]}\,{\eta}^{i\,C}{\eta}^{j\,D}\;,\qquad
C,D=1,2,\ldots(N-4)\;,$ (C.1)

symmetric in $C\leftrightarrow D$. The matching of the anomalies can be read
off Table 3.

| fields | $SU(N)_{\rm c}$ | $SU(N-4)$ | $U(1)_{\chi\eta}$
---|---|---|---|---
UV | $\chi$ | ${\bar{\yng(1,1)}}$ | $\frac{N(N-1)}{2}\cdot(\cdot)$ | $N-4$
| ${\eta}^{A}$ | $(N-4)\cdot{{\yng(1)}}$ | $N\,\cdot\,{\yng(1)}$ | $-(N-2)$
i IR | $B^{\\{AB\\}}$ | $\frac{(N-4)(N-3)}{2}\cdot(\cdot)$ | ${\yng(2)}$ | $-N$
Table 3: Confinement and unbroken symmetry in the $\chi\eta$ model

##  Appendix D Color-flavor locked Higgs vacuum in the $\chi\eta$ model

It was pointed out [10] that this system may instead develop a condensate of
the form

$\langle\chi_{[ij]}{\eta}^{B\,j}\rangle=C\,\Lambda^{3}\delta_{i}^{B}\;,\qquad
i,B=1,2,\ldots,N-4\;,$ (D.1)

namely,

${\bar{\yng(1,1)}}\otimes\yng(1)\to{\bar{\yng(1)}}\oplus\ldots\;.$ (D.2)

The symmetry is broken to

$\displaystyle SU(N-4)_{\rm cf}\times SU(4)_{\rm c}\times U(1)^{\prime}\,.$
(D.3)

The massless baryons (C.1) saturate all the anomalies associated with
$SU(N-4)_{\rm cf}\times U(1)^{\prime}$. There remains the $\chi_{i_{2}j_{2}}$
fermions which remain massless and strongly coupled to the $SU(4)_{\rm c}$. We
may assume that $SU(4)_{\rm c}$ confines, and the condensate

$\langle\chi\chi\rangle\neq 0\;,$ (D.4)

in

${\bar{\yng(1,1)}}\otimes{\bar{\yng(1,1)}}\,\to\,{\bar{\yng(1,1,1,1)}}\,\oplus\ldots\;,$
(D.5)

forms and $\chi_{i_{2}j_{2}}$ acquire dynamically mass. Assume that the
massless baryons are:

$B^{\\{AB\\}}=\chi_{[ij]}\,{\eta}^{i\,A}{\eta}^{j\,B}\;,\qquad
A,B=1,2,\ldots(N-4)\;,$ (D.6)

the saturation of all of the triangles associated can be seen in Table 4.

| fields | $SU(N-4)_{\rm cf}$ | $U(1)^{\prime}$ | $SU(4)_{\rm c}$
---|---|---|---|---
UV | $\chi_{i_{1}j_{1}}$ | ${\bar{\yng(1,1)}}$ | $N$ | $\frac{(N-4)(N-5)}{2}\cdot(\cdot)$
| $\chi_{i_{1}j_{2}}$ | $4\cdot{\bar{\yng(1)}}$ | $\frac{N}{2}$ | $(N-4)\cdot{\bar{\yng(1)}}$
| $\chi_{i_{2}j_{2}}$ | $\frac{4\cdot 3}{2}\cdot(\cdot)$ | $0$ | ${\bar{\yng(1,1)}}$
| ${\eta}^{i_{1},A}$ | $\yng(2)\oplus\yng(1,1)$ | $-N$ | $(N-4)^{2}\cdot(\cdot)$
| ${\eta}^{i_{2},A}$ | $4\,\cdot\,{\yng(1)}$ | $-\frac{N}{2}$ | $(N-4)\cdot\yng(1)$
i IR | $B^{\\{AB\\}}$ | ${\yng(2)}$ | $-N$ | $\frac{(N-4)(N-3)}{2}\cdot(\cdot)$
Table 4: Color-flavor locking in the $\chi\eta$ model. The color index $i_{1}$
or $j_{1}$ runs up to $N-4$ and the rest is indicated by $i_{2}$ or $j_{2}$.

The complementarity [40] apparently works here, in the sense that the massless
sector of the dynamical Higgs phase has the same $SU(N-4)\times U(1)$
symmetry. See Sec. 4 for more discussion.

##  Appendix E Confining phase with unbroken global symmetries of the BY
models

The matter fields of the BY model are:

| $SU(N)_{\rm c}$ | $SU(N+4+p)$ | $SU(p)$ | ${U}(1)_{\psi\eta}$ | ${U}(1)_{\psi\xi}$
---|---|---|---|---|---
$\psi$ | ${\yng(2)}$ | $\frac{N(N+1)}{2}\cdot(\cdot)$ | $\frac{N(N+1)}{2}\cdot(\cdot)$ | $N+4+p$ | $p$
$\eta$ | $(N+4+p)\cdot{\bar{\yng(1)}}$ | $N\cdot{\yng(1)}$ | $N(N+4+p)\,\cdot(\cdot)$ | $-(N+2)$ | $0$
$\xi$ | $p\cdot{{\yng(1)}}$ | $Np\,\cdot(\cdot)$ | $N\cdot{{\yng(1)}}$ | $0$ | $-(N+2)$
Table 5: The multiplicity, charges and the representation are shown for each
set of fermions in the BY model.

The candidate massless composite fermions for the Bars-Yankielowitz models are
the left-handed gauge-invariant fields:

${({B}_{1})}^{[AB]}=\psi^{ij}\eta_{i}^{A}\eta_{j}^{B}\;,\qquad{({B}_{2})}^{a}_{A}=\bar{\psi}_{ij}\bar{\eta}^{i}_{A}\xi^{j,a}\;,\qquad{({B}_{3})}_{\\{ab\\}}=\psi^{ij}{\bar{\xi}}_{i,a}{\bar{\xi}}_{j,b}\;,$
(E.1)

the first is anti-symmetric in $A\leftrightarrow B$ and the third is symmetric
in $a\leftrightarrow b$; their charges are given in Table 6. Writing
explicitly also the spin indices they are

$\displaystyle{({B}_{1})}^{[AB],\alpha}=\frac{1}{2}\epsilon_{\beta\gamma}\psi^{ij,\beta}\eta_{i}^{A,\gamma}\eta_{j}^{B,\alpha}+\frac{1}{2}\epsilon_{\beta\gamma}\psi^{ij,\beta}\eta_{i}^{A,\alpha}\eta_{j}^{B,\gamma}\;,$
$\displaystyle{({B}_{2})}^{a,\alpha}_{A}=\epsilon_{\dot{\alpha}\dot{\beta}}\bar{\psi}_{ij}^{\dot{\alpha}}\bar{\eta}^{i,\dot{\beta}}_{A}\xi^{j,a,\alpha}\;,\qquad{({B}_{3})}_{\\{ab\\}}^{\alpha}=\epsilon_{\dot{\beta}\dot{\gamma}}\psi^{ij,\alpha}{\bar{\xi}}_{i,a}^{\dot{\beta}}{\bar{\xi}}_{j,b}^{\dot{\gamma}}\;:$
(E.2)

all transforming under the $\\{\tfrac{1}{2},0\\}$ representation of the
Lorentz group.

| $SU(N)_{\rm c}$ | $SU(N+4+p)$ | $SU(p)$ | ${U}(1)_{\psi\eta}$ | ${U}(1)_{\psi\xi}$
---|---|---|---|---|---
${{B}_{1}}$ | $\frac{(N+4+p)(N+3+p)}{2}\cdot(\cdot)$ | ${\yng(1,1)}$ | $\frac{(N+4+p)(N+3+p)}{2}\cdot(\cdot)$ | $-N+p$ | $p$
${{B}_{2}}$ | $(N+4+p)p\cdot(\cdot)$ | $p\cdot\bar{\yng(1)}$ | $(N+4+p)\cdot{\yng(1)}$ | $-(p+2)$ | $-(N+p+2)$
${{B}_{3}}$ | $\frac{p(p+1)}{2}\cdot(\cdot)$ | $\frac{p(p+1)}{2}\cdot(\cdot)$ | $\bar{\yng(2)}$ | $N+4+p$ | $2N+4+p$
Table 6: Chirally symmetric phase of the BY model.

Table 7 summarizes the anomaly matching checks [25] via comparison between
Table 5 and Table 6.

| UV | IR
---|---|---
$SU(N+4+p)^{3}$ | $N$ | $N+p-p$
$SU(p)^{3}$ | $N$ | $N+4+p-(p+4)$
$SU(N+4+p)^{2}-{U}(1)_{\psi\eta}$ | $-N(N+2)$ | $-(N+2+p)(N-p)-p(p+2)$
$SU(N+4+p)^{2}-{U}(1)_{\psi\xi}$ | $0$ | $(N+2+p)p-p(N+p+2)$
$SU(p)^{2}-{U}(1)_{\psi\eta}$ | $0$ | $-(N+4+p)(p+2)+(p+2)(N+p+4)$
$SU(p)^{2}-{U}(1)_{\psi\xi}$ | $-N(N+2)$ | $-(N+4+p)(N+p+2)+(p+2)(2N+p+4)$
${U}(1)_{\psi\eta}^{3}$ | $\frac{N(N+1)}{2}(N+4+p)^{3}-N(N+4+p)(N+2)^{3}$ | $-\frac{(N+4+p)(N+3+p)}{2}(N-p)^{3}-(N+4+p)p(p+2)^{3}+$
|  | $+\frac{p(p+1)}{2}(N+4+p)^{3}$
${U}(1)_{\psi\xi}^{3}$ | $\frac{N(N+1)}{2}p^{3}-Np(N+2)^{3}$ | $\frac{(N+4+p)(N+3+p)}{2}p^{3}-(N+4+p)p(N+p+2)^{3}+$
|  | $\ +\frac{p(p+1)}{2}(2N+4+p)^{3}$
${\rm Grav}^{2}-{U}(1)_{\psi\eta}$ | $\frac{N(N+1)}{2}(N+4+p)-N(N+4+p)(N+2)$ | $-\frac{(N+4+p)(N+3+p)}{2}(N-p)-(N+4+p)p(p+2)+$
|  | $+\frac{p(p+1)}{2}(N+4+p)$
${\rm Grav}^{2}-{U}(1)_{\psi\xi}$ | $\frac{N(N+1)}{2}p-Np(N+2)$ | $\frac{(N+4+p)(N+3+p)}{2}p-(N+4+p)p(N+p+2)+$
|  | $\ +\frac{p(p+1)}{2}(2N+4+p)$
$SU(N+4+p)^{2}-({\mathbbm{Z}}_{N+2})_{\psi}$ | $0$ | $N+2+p-p=0\ {\rm mod}\ N+2$
$SU(p)^{2}-({\mathbbm{Z}}_{N+2})_{\psi}$ | $0$ | $-(N+4+p)+p+2=0\ {\rm mod}\ N+2$
${\rm Grav}^{2}-({\mathbbm{Z}}_{N+2})_{\psi}$ | $1$ | $1-1+1$
Table 7: Anomaly matching checks for the IR chiral symmetric phase of the BY
model. For $N$ odd, the last three equalities are consequences of other
equations.

##  Appendix F Dynamical Higgs phase in the BY models

The broken phase for the simplest of this class, the $\psi\eta$ model, has
also been extensively in the main text, see also [10, 16, 26, 25]. Something
interesting happens for models with $p>0$ additional pairs of fermions in the
fundamentals $(\eta,\xi)$. Now there is another channel, $\xi\eta$, which is
gauge invariant and charged under the flavor group. We thus have a competition
between two possible symmetry breaking channels, $\psi\eta$ and $\xi\eta$. We
assume that both condensates occur in the following way:

$\displaystyle\langle\psi^{ij}\eta_{i}^{B}\rangle=\,c_{\psi\eta}\,\Lambda^{3}\delta^{jB}\neq
0\;,\qquad j,B=1,\dots,N\;,$
$\displaystyle\langle\xi^{i,a}\eta_{i}^{A}\rangle=\,c_{\eta\xi}\,\Lambda^{3}\delta^{aA}\neq
0\;,\qquad a=1,\dots,N\;,\quad A=N+1,\dots,N+p\;,$ (F.1)

where $\Lambda$ is the renormailization-invariant scale dynamically generated
by the gauge interactions and $c_{\eta\xi},c_{\psi\eta}$ are coefficients both
of order one. According to the tumbling scenario [2], the first condensate to
occur is in the maximally attractive channel (MAC). The strengths of the one-
gluon exchange potential for the two channels

$\displaystyle\psi\Big{(}\raisebox{-2.0pt}{\yng(2)}\Big{)}\,\eta\Big{(}\bar{\raisebox{-2.0pt}{\yng(1)}}\Big{)}\qquad\
$ $\displaystyle{\rm forming}\qquad\raisebox{-2.0pt}{\yng(1)}\;,$ (F.2)
$\displaystyle\quad\
\xi\Big{(}{\raisebox{-2.0pt}{\yng(1)}}\Big{)}\,\eta\Big{(}\bar{\raisebox{-2.0pt}{\yng(1)}}\Big{)}\qquad\
$ $\displaystyle{\rm forming}\qquad(\cdot)\;,$

are, respectively,

$\displaystyle\frac{N^{2}-1}{2N}-\frac{(N+2)(N-1)}{N}-\frac{N^{2}-1}{2N}$
$\displaystyle=$ $\displaystyle-\frac{(N+2)(N-1)}{N}\;,$ $\displaystyle
0-2\frac{N^{2}-1}{2N}$ $\displaystyle=$ $\displaystyle-\frac{N^{2}-1}{N}\;.$
(F.3)

So the $\psi\eta$ 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

$\displaystyle SU(N)_{\mathrm{c}}\times SU(N+4+p)_{\eta}\times
SU(p)_{\xi}\times U(1)_{\psi\eta}\times U(1)_{\psi\xi}$
$\displaystyle\xrightarrow{\langle\xi\eta\rangle,\langle\psi\eta\rangle}SU(N)_{{\rm
cf}_{\eta}}\times SU(4)_{\eta}\times SU(p)_{\eta\xi}\times
U(1)_{\psi\eta}^{\prime}\times U(1)_{\psi\xi}^{\prime}\;.$ (F.4)

At the end the color gauge symmetry is completely (dynamically) broken,
leaving color-flavor diagonal $SU(N)_{{\rm cf}_{\eta}}$ symmetry.
$U(1)_{\psi\eta}^{\prime}$ and $U(1)_{\psi\xi}^{\prime}$ are combinations
respectively of $U(1)_{\psi\eta}$ and $U(1)_{\xi\eta}$ with the element of
$SU(N+4+p)_{\eta}$ generated by

$t_{SU(N+4+p)_{\eta}}=\left(\begin{array}[]{c|c|c}(-\alpha(p+2)-p\beta){\bf
1}_{N\times N}&&\\\ \hline\cr&\frac{\alpha(N-p)-\beta p}{2}{\bf 1}_{4\times
4}&\\\ \hline\cr&&(\alpha+\beta)(N+2){\bf 1}_{p\times p}\\\
\end{array}\right)\;.$ (F.5)

Making the decomposition of the fields in the direct sum of representations in
the subgroup one gets Table 8.

| $SU(N)_{{\rm cf}_{\eta}}$ | $SU(4)_{\eta}$ | $SU(p)_{\eta\xi}$ | $U(1)_{\psi\eta}^{\prime}$ | $U(1)_{\psi\xi}^{\prime}$
---|---|---|---|---|---
$\psi$ | ${\yng(2)}$ | $\frac{N(N+1)}{2}\cdot(\cdot)$ | $\frac{N(N+1)}{2}\cdot(\cdot)$ | $N+4+p$ | $p$
$\eta_{1}$ | ${\bar{\yng(2)}}\oplus{\bar{\yng(1,1)}}$ | $N^{2}\cdot(\cdot)$ | $N^{2}\cdot(\cdot)$ | $-(N+4+p)$ | $-p$
$\eta_{2}$ | $4\cdot{\bar{\yng(1)}}$ | $N\cdot{\yng(1)}$ | $4N\cdot(\cdot)$ | $-\frac{N+p+4}{2}$ | $-\frac{p}{2}$
$\eta_{3}$ | $p\cdot{\bar{\yng(1)}}$ | $Np\cdot(\cdot)$ | $N\cdot\bar{\yng(1)}$ | $0$ | $N+2$
$\xi$ | $p\cdot{{\yng(1)}}$ | $Np\cdot(\cdot)$ | $N\cdot{{\yng(1)}}$ | $0$ | $-(N+2)$
Table 8: UV fieds in the BY model, decomposed as a direct sum of the representations of the unbroken group of Eq. (F.4).  | $SU(N)_{{\rm cf}_{\eta}}$ | $SU(4)_{\eta}$ | $SU(p)_{\eta\xi}$ | $U(1)_{\psi\eta}^{\prime}$ | $U(1)_{\psi\xi}^{\prime}$
---|---|---|---|---|---
${B}_{1a}$ | ${\bar{\yng(1,1)}}$ | $\frac{N(N-1)}{2}\cdot(\cdot)$ | $\frac{N(N-1)}{2}\cdot(\cdot)$ | $-(N+4+p)$ | $-p$
${B}_{1b}$ | $4\cdot{\bar{\yng(1)}}$ | $N\cdot{\yng(1)}$ | $4N\cdot(\cdot)$ | $-\frac{N+p+4}{2}$ | $-\frac{p}{2}$
${B}_{1c}$ | $6\cdot(\cdot)$ | ${\yng(1,1)}$ | $6\cdot(\cdot)$ | $0$ | $0$
Table 9: IR massless fermions in the BY model in the Higgs phase.

The composite massless baryons are subset of those in (3.36):

$\displaystyle{B}_{1a}^{[AB]}=\psi^{ij}\eta_{i}^{A}\eta_{j}^{B}\;,\qquad{B}_{1b}^{[AC]}=\psi^{ij}\eta_{i}^{A}\eta_{j}^{C}\;,\qquad{B}_{1c}^{[CD]}=\psi^{ij}\eta_{i}^{C}\eta_{j}^{D}\;,$
$\displaystyle A,B=1,\dots,N\;,\quad C,D=N+1,\dots,N+4\;.$ (F.6)

It is quite straightforward (see the remark in Introduction) to verify that
the UV-IR anomaly matching continues to work, with the UV fermions in Table 8
and the IR fermions in Table 9.

As in the $\psi\eta$ model, the baryons indicated as $B_{1c}$ in Table 9 were
not considered in the earlier work on the BY models [10, 25], but assumed here
to be present and massless. These extra baryons do not contribute to the
triangle anomalies with respect to unbroken symmetry group, see Table 9,
therefore do not affect the anomaly matching argument. However, as in the
$\psi\eta$ model, the condensate
$\langle{B}_{1c}{B}_{1c}\rangle\sim\langle\psi\eta\eta\,\psi\eta\eta\rangle$
is needed in order to reproduce the strong anomaly in the IR correctly. This
however does not alter neither the symmetry breaking pattern (F.4), nor the
conventional anomaly matching. See Sec. 3.3 for more discussions.

##  Appendix G Confining phase with unbroken global symmetries of the GG
models

The candidate massless composite fermions for the Georgi-Glashow models are:

${({B}_{1})}^{\\{AB\\}}=\chi^{ij}\eta_{i}^{A}\eta_{j}^{B}\;,\qquad{({B}_{2})}^{a}_{A}=\bar{\chi}_{ij}\bar{\eta}^{i}_{A}\xi^{j,a}\;,\qquad{({B}_{3})}_{[ab]}=\chi^{ij}{\bar{\xi}}_{i,a}{\bar{\xi}}_{j,b}\;,$
(G.1)

the first symmetric in $A\leftrightarrow B$ and the third anti-symmetric in
$a\leftrightarrow b$. Writing the spin indices explicitly they are:

$\displaystyle{({B}_{1})}^{\\{AB\\},\alpha}=\frac{1}{2}\epsilon_{\beta\gamma}\,\chi^{ij,\beta}\eta_{i}^{A,\gamma}\eta_{j}^{B,\alpha}+\frac{1}{2}\epsilon_{\beta\gamma}\,\chi^{ij,\beta}\eta_{i}^{A,\alpha}\eta_{j}^{B,\gamma}\;,$
$\displaystyle{({B}_{2})}^{a,\alpha}_{A}=\epsilon_{\dot{\beta}\dot{\gamma}}\
\bar{\chi}_{ij}^{\dot{\beta}}\bar{\eta}^{i,\dot{\gamma}}_{A}\xi^{j,a,\alpha}\;,\qquad{({B}_{3})}_{[ab]}=\epsilon_{\dot{\beta}\dot{\gamma}}\
\chi^{ij}{\bar{\xi}}_{i,a}^{\dot{\beta}}{\bar{\xi}}_{j,b}^{\dot{\gamma}}\;.$
(G.2)

All anomaly triangles are saturated by these candidate massless composite
fermions, see Table 11.

| $SU(N)_{\rm c}$ | $SU(N-4+p)$ | $SU(p)$ | ${U}(1)_{\chi\eta}$ | ${U}(1)_{\chi\xi}$
---|---|---|---|---|---
${B}_{1}$ | $\frac{(N-4+p)(N-3+p)}{2}\cdot(\cdot)$ | ${\yng(2)}$ | $\frac{(N-4+p)(N-3+p)}{2}\cdot(\cdot)$ | $-N+p$ | $p$
${B}_{2}$ | $(N-4+p)p\cdot(\cdot)$ | $p\cdot\bar{\yng(1)}$ | $(N-4+p)\cdot{\yng(1)}$ | $-(p-2)$ | $-(N+p-2)$
${B}_{3}$ | $\frac{p(p-1)}{2}\cdot(\cdot)$ | $\frac{p(p-1)}{2}\cdot(\cdot)$ | $\bar{\yng(1,1)}$ | $N-4+p$ | $2N-4+p$
Table 10: IR massless fermions in the chirally symmetric phase of the GG model.  | UV | IR
---|---|---
$SU(N-4+p)^{3}$ | $N$ | $N+p-p$
$SU(p)^{3}$ | $N$ | $N-4+p-(p-4)$
$SU(N-4+p)^{2}-{U}(1)_{\chi\eta}$ | $-N(N-2)$ | $-(N-2+p)(N-p)-p(p-2)$
$SU(N-4+p)^{2}-{U}(1)_{\chi\xi}$ | $0$ | $(N-2+p)p-p(N+p-2)$
$SU(p)^{2}-{U}(1)_{\chi\eta}$ | $0$ | $-(N-4+p)(p-2)+(p-2)(N-4+p)$
$SU(p)^{2}-{U}(1)_{\chi\xi}$ | $-N(N-2)$ | $-(N-4+p)(N+p-2)+(p-2)(2N-4+p+0)$
${U}(1)_{\chi\eta}^{3}$ | $\frac{N(N-1)}{2}(N-4+p)^{3}-N(N-4+p)(N-2)^{3}$ | $-\frac{(N-4+p)(N-3+p)}{2}(N-p)^{3}-(N-4+p)p(p-2)^{3}+$
|  | $+\frac{p(p-1)}{2}(N-4+p)^{3}$
${U}(1)_{\chi\xi}^{3}$ | $\frac{N(N-1)}{2}p^{3}-Np(N-2)^{3}$ | $\frac{(N-4+p)(N-3+p)}{2}p^{3}-(N-4+p)p(N+p-2)^{3}+$
|  | $\ +\frac{p(p-1)}{2}(2N-4+p)^{3}$
${\rm Grav}^{2}-{U}(1)_{\chi\eta}$ | $\frac{N(N-1)}{2}(N-4+p)-N(N-4+p)(N-2)$ | $-\frac{(N-4+p)(N-3+p)}{2}(N-p)-(N-4+p)p(p-2)+$
|  | $+\frac{p(p-1)}{2}(N-4+p)$
${\rm Grav}^{2}-{U}(1)_{\chi\xi}$ | $\frac{N(N-1)}{2}p-Np(N-2)$ | $\frac{(N-4+p)(N-3+p)}{2}p-(N-4+p)p(N+p-2)+$
|  | $\ +\frac{p(p-1)}{2}(2N-4+p)$
$SU(N-4+p)^{2}-({\mathbbm{Z}}_{N-2})_{\chi}$ | $0$ | $N-2+p-p=0\ {\rm mod}\ N-2$
$SU(p)^{2}-({\mathbbm{Z}}_{N-2})_{\chi}$ | $0$ | $-(N-4+p)+p-2=0\ {\rm mod}\ N-2$
${\rm Grav}^{2}-({\mathbbm{Z}}_{N-2})_{\chi}$ | $1$ | $1-1+1$
Table 11: Anomaly matching checks for the IR chiral symmetric phase of the GG
model.

##  Appendix H Dynamical Higgs phase in the generalized GG models

In the generalized Georgi-Glashow models there is a competition between two
possible bifermion symmetry breaking channels $\chi\eta$ and $\xi\eta$. This
time, the MAC criterion would slightly favor the $\xi\eta$ condensates against
$\chi\eta$. Indeed, the strength of the one-gluon exchange potential for the
two channels

$\displaystyle\chi\left({\raisebox{-9.0pt}{\yng(1,1)}}\right)\,\,\eta\Big{(}\bar{\raisebox{-2.0pt}{\yng(1)}}\Big{)}\qquad\
$ $\displaystyle{\rm forming}\quad\raisebox{-2.0pt}{\yng(1)}\;,$ (H.1)
$\displaystyle\xi\Big{(}{\raisebox{-2.0pt}{\yng(1)}}\Big{)}\,\eta\Big{(}\bar{\raisebox{-2.0pt}{\yng(1)}}\Big{)}\qquad\
$ $\displaystyle{\rm forming}\quad(\cdot)\;,$

are, respectively,

$\displaystyle\frac{N^{2}-1}{2N}-\frac{(N-2)(N+1)}{N}-\frac{N^{2}-1}{2N}$
$\displaystyle=$ $\displaystyle-\frac{(N-2)(N+1)}{N}\;,$ $\displaystyle
0-2\frac{N^{2}-1}{2N}$ $\displaystyle=$ $\displaystyle-\frac{N^{2}-1}{N}\;.$
(H.2)

Again, these perturbative estimates are not excessively significant, and we
assume that both condensates occur as:

$\displaystyle\langle\chi^{ij}\eta_{i}^{A}\rangle=\,c_{\chi\eta}\,\Lambda^{3}\delta^{jA}\neq
0\;,\qquad j=1,\dots,N-4\;,\quad A=1,\dots,N-4\;,$
$\displaystyle\langle\xi^{i,a}\eta_{i}^{B}\rangle=\,c_{\eta\xi}\,\Lambda^{3}\delta^{aB}\neq
0\;,\qquad a=1,\dots,p\;,\quad B=N-4+1,\dots,N-4+p\;.$

The pattern of symmetry breaking is

$\displaystyle SU(N)_{\mathrm{c}}\times SU(N-4+p)_{\eta}\times
SU(p)_{\xi}\times U(1)_{\chi\eta}\times U(1)_{\chi\xi}$
$\displaystyle\xrightarrow{\langle\xi\eta\rangle,\langle\chi\eta\rangle}SU(4)_{{\rm
c}}\times SU(N-4)_{{\rm cf}_{\eta}}\times SU(p)_{\eta\xi}\times
U(1)_{\chi\eta}^{\prime}\times U(1)_{\chi\xi}^{\prime}\;.$ (H.4)

The color gauge symmetry is partially (dynamically) broken, leaving color-
flavor diagonal global $SU(N-4)_{{\rm cf}_{\eta}}$ symmetry and an
$SU(4)_{{\rm c}}$ gauge symmetry. $U(1)_{\chi\eta}^{\prime}$ and
$U(1)_{\chi\xi}^{\prime}$ are a combinations respectively of $U(1)_{\chi\eta}$
and $U(1)_{\chi\xi}$ with the elements of $SU(N)_{\mathrm{c}}$ and
$SU(N-4+p)_{\eta}$ generated by:

$\displaystyle\qquad t_{SU(N)_{\rm
c}}=\left(\begin{array}[]{c|c}2\frac{\alpha(N-4+p)+\beta p}{N-4}{\bf
1}_{(N-4)\times(N-4)}&\\\ \hline\cr&-\frac{\alpha(N-4+p)+\beta p}{2}{\bf
1}_{4\times 4}\\\ \end{array}\right)\;,$ (H.7) $\displaystyle
t_{SU(N-4+p)_{\eta}}=\left(\begin{array}[]{c|c}-\frac{p(\alpha+\beta)(N-2)}{N-4}{\bf
1}_{(N-4)\times(N-4)}&\\\ \hline\cr&(\alpha+\beta)(N-2){\bf 1}_{p\times p}\\\
\end{array}\right)\;.$ (H.10)

Making the decomposition of the fields in the direct sum of representations in
the subgroup one arrives at Table 12.

| $SU(N-4)_{{\rm cf}_{\eta}}$ | $SU(4)_{{\rm c}}$ | $SU(p)_{\eta\xi}$ | $U(1)_{\chi\eta}^{\prime}$ | $U(1)_{\chi\xi}^{\prime}$
---|---|---|---|---|---
$\chi_{1}$ | ${\yng(1,1)}$ | $\frac{(N-4)(N-5)}{2}\cdot(\cdot)$ | $\frac{(N-4)(N-5)}{2}\cdot(\cdot)$ | $\frac{(N-4+p)N}{(N-4)}$ | $p\frac{N}{N-4}$
$\chi_{2}$ | $4\cdot{\yng(1)}$ | $(N-4)\cdot{\yng(1)}$ | $4(N-4)\cdot(\cdot)$ | $\frac{(N-4+p)N}{2(N-4)}$ | $\frac{pN}{2(N-4)}$
$\chi_{3}$ | $6\cdot(\cdot)$ | ${\yng(1,1)}$ | $6\cdot(\cdot)$ | $0$ | $0$
$\eta_{1}$ | ${\bar{\yng(2)}}\oplus{\bar{\yng(1,1)}}$ | $(N-4)^{2}\cdot(\cdot)$ | $(N-4)^{2}\cdot(\cdot)$ | $-\frac{(N-4+p)N}{(N-4)}$ | $-\frac{pN}{N-4}$
$\eta_{2}$ | $p\cdot\bar{\yng(1)}$ | $p(N-4)\cdot(\cdot)$ | $(N-4)\cdot\bar{\yng(1)}$ | $-2-2\frac{p}{N-4}$ | $N-2-\frac{2p}{N-4}$
$\eta_{3}$ | $4\cdot{\bar{\yng(1)}}$ | $(N-4)\cdot{\bar{\yng(1)}}$ | $4(N-4)\cdot(\cdot)$ | $-\frac{(N-4+p)N}{2(N-4)}$ | $-\frac{pN}{2(N-4)}$
$\eta_{4}$ | $4p\cdot(\cdot)$ | $p\cdot{\bar{\yng(1)}}$ | $4\cdot\bar{\yng(1)}$ | $\frac{N-4+p}{2}$ | $N-2+\frac{p}{2}$
$\xi_{1}$ | $p\cdot{{\yng(1)}}$ | $p(N-4)\cdot(\cdot)$ | $(N-4)\cdot{{\yng(1)}}$ | $2+2\frac{p}{N-4}$ | $-(N-2)+\frac{2p}{N-4}$
$\xi_{2}$ | $4p\cdot(\cdot)$ | $p\cdot{{\yng(1)}}$ | $4\cdot{{\yng(1)}}$ | $-\frac{N-4+p}{2}$ | $-(N-2)-\frac{p}{2}$
Table 12: UV fieds in the GG model, decomposed as a direct sum of the representations of the unbroken group of Eq. (H.4).  | $SU(N-4)_{{\rm cf}_{\eta}}$ | $SU(p)_{\eta\xi}$ | $U(1)_{\chi\eta}^{\prime}$ | $U(1)_{\chi\xi}^{\prime}$
---|---|---|---|---
${B}$ | ${\bar{\yng(2)}}$ | $\frac{(N-4)(N-3)}{2}\ \cdot(\cdot)$ | $-\frac{(N-4+p)N}{(N-4)}$ | $-\frac{pN}{N-4}$
Table 13: IR fied in the GG model in the dynamical Higgs phase.

The composite massless baryons are subset of those in (G.1):

$\displaystyle{B}^{\\{AB\\}}=\chi^{ij}\eta_{i}^{A}\eta_{j}^{B}\;,\qquad
A,B=1,\dots,N-4\;.$ (H.11)

In the IR these fermions saturate all the anomalies of the unbroken chiral
symmetry. This can be seen by an inspection of Table 13 and Table 12, with the
help of the following observation.

In fact, there is a novel feature in the GG models, which is not shared by the
BY models. As seen in Table 13, there is an unbroken strong gauge symmetry
$SU(4)_{{\rm c}}$, with a set of fermions,

$\chi_{3}\;,\quad\chi_{2}\;,\quad\eta_{3}\;,\quad\eta_{4}\;,\quad\xi_{2}\;,$
(H.12)

charged with respect to it. However, the pairs $\\{\chi_{2}\;,\,\eta_{3}\\}$
and $\\{\eta_{4}\;,\,\xi_{2}\\}$ can form massive Dirac fermions and decouple.
These are vectorlike with respect to the surviving infrared symmetry, (H.4),
hence are irrelevant to the anomalies.151515Actually, with matter fermions
(H.12) $SU(4)_{{\rm c}}$ is asymptotically free only for $50-2N-2p>0$. If
$50-2N-2p<0$, $SU(4)_{{\rm c}}$ will remain weakly coupled in the infrared,
but the fact that the fermions (H.12) do not contribute to the anomalies with
respect to the remaining flavor symmetries (H.14) stays valid. On the other
hand, the fermion $\chi_{3}$ can condense

$\langle\chi_{3}\chi_{3}\rangle\ $ (H.13)

forming massive composite mesons, $\sim\chi_{3}\chi_{3}$, which also
decouples. It is again neutral with respect to all of

$SU(N-4)_{{\rm cf}_{\eta}}\times SU(p)_{\eta\xi}\times
U(1)_{\chi\eta}^{\prime}\times U(1)_{\chi\xi}^{\prime}\;.$ (H.14)

To summarize, $SU(4)_{\rm c}$ is invisible (confines) in the IR, and only the
unpaired part of the $\eta_{1}$ fermion $\big{(}{\bar{\yng(2)}}\big{)}$
remains massless, and its contribution to the anomalies is reproduced exactly
by the composite fermions, (H.11).

Comment: The massive mesons $\\{\chi_{2}\,\eta_{3}\\}$,
$\\{\eta_{4}\,\xi_{2}\\}$, $\\{\chi_{3}\,\chi_{3}\\}$ are not charged with
respect to the 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.