# More Exact Results on Chiral Gauge Theories: the Case of the Symmetric
Tensor

Csaba Csáki<EMAIL_ADDRESS>Department of Physics, LEPP, Cornell University,
Ithaca, NY 14853, USA  Hitoshi Murayama<EMAIL_ADDRESS><EMAIL_ADDRESS>Hamamatsu Professor Department of Physics,
University of California, Berkeley, CA 94720, USA  Kavli Institute for the
Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa
277-8583, Japan  Ernest Orlando Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA  Ofri Telem<EMAIL_ADDRESS>Department of Physics,
University of California, Berkeley, CA 94720, USA  Ernest Orlando Lawrence
Berkeley National Laboratory, Berkeley, CA 94720, USA

###### Abstract

We study dynanics of $SU(N-4)$ gauge theories with fermions in rank-2
symmetric tensor and $N$ anti-fundamental representations, by perturbing
supersymmetric theories with anomaly-mediated supersymmetry breaking. We find
the $SU(N)\times U(1)$ global symmetry is dynamically broken to $SO(N)$ for
$N\geq 17$, a different result from conjectures in the literature. For $N<17$,
theories flow to infrared fixed points.

##  I Introduction

Understanding the dynamics of strongly coupled chiral gauge theories remains a
difficult challenge, as we are lacking effective tools to study them. Even
progress on lattice simulations has been slow, since they are hampered by the
notorious doubling problem. So far the best we can do is to resort to educated
guesses, based on some general guiding principles. The most well-known such
framework is “tumbling” Raby:1979my ; Dimopoulos:1980hn , where one tries to
find the most attractive channel (MAC) among the various fermions, and
postulate condensates that will successively break the gauge symmetry till one
arrives at a non-chiral QCD-like theory. Tumbling has indeed been used to
propose a plausible vacuum structure for the simplest chiral gauge theories:
$SU(N)$ with a Weyl fermion in the antisymmetric (or in the symmetric)
representation of the gauge group, as well as anti-fundamentals to cancel the
gauge anomalies. For both examples the proposal contains a symmetry breaking
pattern that satisfies non-trivial ’t Hooft anomaly matching conditions, hence
appears to be passing some very non-trivial checks. Recently, these proposals
have undergone further scrutiny in Bolognesi:2020mpe ; Bolognesi:2021yni , by
applying new discrete anomaly matching conditions Csaki:1997aw  involving the
center symmetry $Z_{n}$ of the gauge group, in the spirit of Gaiotto:2017yup ;
Tanizaki:2018wtg ; Bolognesi:2019fej . We do not elaborate more on these
generalized consistency conditions, since they seem to automatically hold for
our proposal of the IR dynamics, due to its continuous connection to the
supersymmetric theory.

The study of the supersymmetric (SUSY) versions of these theories opens up a
new method for finding candidate vacua for chiral gauge theories. SUSY -
thanks in most part due to holomorphy - allows for a much greater control of
the IR dynamics, and together with anomaly matching often enables us to fully
pin down the vacuum structure of the theory. The obvious challenge then is to
be able to deduce results for the non-SUSY theory by perturbing the SUSY
results in a controlled manner. While several attempts along these lines were
initiated in the 90’s Aharony:1995zh ; Cheng:1998xg ; ArkaniHamed:1998wc ,
recently a UV-insensitive method for perturbing SUSY dualities based on
anomaly mediation Randall:1998uk ; Giudice:1998xp  has been proposed in
Murayama:2021xfj . Applying the AMSB method to chiral gauge theories (wherever
exact results for the supersymmetric limit is known) will always produce a
candidate vacuum structure that will automatically satisfy all consistency
conditions, since it will be continuously connected to the vacuum of the SUSY
theory without the AMSB perturbations, yielding a well-defined procedure for
generating a candidate vacuum solution to the non-SUSY chiral gauge theories.
Of course this does not automatically imply that we have found the correct
ground state of the theory, since one can not rule out the possibility of a
phase transition as the SUSY breaking mass terms are raised above the strong
coupling scale. Nevertheless we find it significant that at least a plausible
conjecture can be formulated this way, which eventually will be compared to
the results from the lattice simulations. It may even be the case that
holomorphy prohibits a phase transition Csaki:2021xhi .

Recently we have used this method to examine the simplest chiral gauge
theories based on $SU(N)$ with a rank-2 antisymmetric representation for
$N\geq 5$ and have identified the vacuum structure of the resulting non-SUSY
chiral theory Csaki:2021xhi . We have found that the global symmetry breaking
pattern is in fact different than initially conjectured based on tumbling: for
odd $N$ we found that the global $SU(N-4)\times U(1)$ symmetry is broken to
$Sp(N-5)\times U(1)$, while for the even case to $Sp(N-4)$. While this
symmetry breaking pattern did not agree with the original predictions from
tumbling, we have explained that assuming additional condensates in the second
most attractive channel will fully resolve the discrepancy.

In this paper we extend our previous work to examine the other examples of
non-SUSY chiral gauge theories for which a simple prediction for the vacuum
structure based on tumbling exists: the case of $SU(N-4)$ with a fermion $S$
in the rank-2 symmetric representation and $N$ anti-fundamentals $\bar{F}$ to
cancel the gauge anomalies. There exist two proposals for the symmetry
breaking pattern of this model. The tumbling prediction for this case would be
a MAC leading to a condensate of the symmetric and $N-4$ anti-fundamentals,
leading to color-flavor locking with an unbroken $SU(N-4)\times SU(4)\times
U(1)$ global symmetry. The anomalies are matched by a composite fermion
$S\bar{F}_{i}\bar{F}_{j}$ antisymmetric in the $i,j$ flavor indices. Another
interesting option is that the entire group confines without breaking any of
the global symmetries via condensates, since the same fermion composite
actually matches the ’t Hooft anomalies of the entire $SU(N)\times U(1)$
global symmetry.

Similar to the case of the anti-symmetric tensor, we will show that the AMSB
method results in a prediction different from either of these two scenarios.
The details of the analysis for the symmetric case turn out to be quite
different from that of the antisymmetric, since here we have to make use of
the Seiberg dual found by Pouliot and Strassler Pouliot:1995sk  in terms of a
magnetic $Spin(8)$ group. We find that for $N\geq 17$ the remaining global
symmetry is only $SO(N-4)$, and no massless composite fermion is needed in the
absence of ’t Hooft anomalies. On the other hand for $N<17$ the theory flows
to a conformal fixed point in the IR. At the fixed point the supersymmetry
breaking terms all vanish, and one is left with a genuine superconformal
theory.

The paper is organized as follows. We first briefly review the anomaly
mediation of supersymmetry breaking (AMSB), and the SUSY limit of the
$SU(N-4)$ gauge theories with a symmetric tensor and $N$ anti-fundamentals.
Then we combine them to find consistent vacua of the non-SUSY theories that
can be extrapolated to decouple supersymmetry. We show that we can understand
the symmetry breaking pattern à la tumbling, even though we need to rely on
fermion bilinear condensates that are not in the MAC. The case of infrared
fixed points are discussed in the end.

##  II Anomaly Mediation

In scenarios with anomaly-mediated supersymmetry breaking (AMSB),
supersymmetry is broken in a sequestered sector, and is mediated to the
visible sector via the superconformal anomaly. The magnitude of the breaking
is given by a single number $m$, which enters both at tree and at loop level.
The tree-level contribution to the scalar potential is derived from the
superpotential,

$\displaystyle{\cal L}_{\rm tree}$
$\displaystyle=m\left(\phi_{i}\frac{\partial
W}{\partial\phi_{i}}-3W\right)+c.c.$ (1)

In addition, there is loop-level supersymmetry breaking, which generates in
tri-linear couplings, scalar masses, and gaugino masses Pomarol:1999ie ,

$\displaystyle A_{ijk}(\mu)$
$\displaystyle=-\frac{1}{2}(\gamma_{i}+\gamma_{j}+\gamma_{k})(\mu)m,$ (2)
$\displaystyle m_{i}^{2}(\mu)$
$\displaystyle=-\frac{1}{4}\dot{\gamma}_{i}(\mu)m^{2},$ (3) $\displaystyle
m_{\lambda}(\mu)$ $\displaystyle=-\frac{\beta(g^{2})}{2g^{2}}(\mu)m.$ (4)

Here, $\gamma_{i}=\mu\frac{d}{d\mu}\ln Z_{i}(\mu)$,
$\dot{\gamma}=\mu\frac{d}{d\mu}\gamma_{i}$, and
$\beta(g^{2})=\mu\frac{d}{d\mu}g^{2}$.

##  III $SU(N-4)$ with $S$ and $(N)\,\bar{F}_{i}$

First we present a summary of the duality explored in Pouliot:1995sk , which
will be the basis of our explorations. We consider a supersymmetric $SU(N-4)$
gauge theory with a rank-2 symmetric tensor $S$ and antifundamentals
$\bar{F}_{i}$ ($i=1,\cdots,N$). For $N\leq 16$, it has an interacting IR fixed
point, while for $N\geq 17$, it is in a free magnetic phase. The theory has a
global $SU(N)\times U(1)\times U(1)_{R}$ symmetry, under which the charges of
the matter fields are shown in Table 2.

The $SU(N-4)$ theory (henceforth the ‘electric’ theory) has a magnetic dual,
which is a non-chiral $Spin(8)$ gauge theory (the double cover of $SO(8)$)
with $N$ vectors $q^{i}$, a spinor $p$ and the $Spin(8)$ singlets
$M_{ij}=S\bar{F}_{i}\bar{F}_{j}$ and $U={\rm det}S$. The magnetic theory has a
tree-level superpotential:

$\tilde{W}_{\text{tree}}~{}=~{}\frac{1}{\mu^{2}_{1}}\,M_{ij}q^{i}q^{j}\,+\,\frac{1}{\mu^{N-5}_{2}}\,Upp\,,$
(5)

where the scales $\mu_{1,2}$ are related to the electric (magnetic) strong
scales $\Lambda\,(\tilde{\Lambda})$ by

$(\Lambda^{2N-11})^{2}\tilde{\Lambda}^{17-N}~{}=~{}\mu^{2N}_{1}\mu^{N-5}_{2}\,.$
(6)

The ’t Hooft anomaly matching conditions are satisfied and imply that the
fields $M$ and $U$ have regular Kähler potential at the origin. For later
convenience, we switch to canonically normalized fields $\tilde{M}$ and
$\tilde{U}$ and introduce the Yukawa couplings $y_{M,U}$,

$\tilde{W}_{\text{tree}}~{}=~{}y_{M}\,\tilde{M}qq\,+\,y_{U}\,\tilde{U}pp\,.$
(7)

$SU(N-4)$ $SU(N)$ $U(1)$ $U(1)_{R}$ $S$ $\bf 1$ $-2N$ $\frac{12}{(N+1)(N-4)}$
$\bar{F}_{i}$
$\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$
$2N-4$ $\frac{6(N-5)}{(N+1)(N-4)}$ $M_{ij}=S\bar{F}_{i}\bar{F}_{j}$ $\bf 1$
$2N-8$ $\frac{12}{N+1}$ $U=\text{det}S$ $\bf 1$ $\bf 1$ $2N(4-N)$
$\frac{12}{N+1}$

Table 1: Particle content of the electric $SU(N-4)$ theory. We omit the
baryons $B,\,B_{n}$ since they are not dynamical and don’t play a role in our
analysis.

$Spin(8)$ $SU(N)$ $U(1)$ $U(1)_{R}$ $SO(N)$ $q^{i}$ $\bf 8_{v}$
$\bar{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$
$4-N$ $\frac{N-5}{N+1}$ $p$ $\bf 8_{s}$ $\bf 1$ $N(N-4)$ $\frac{N-5}{N+1}$
$\bf 1$ $M_{ij}$ $\bf 1$ $2N-8$ $\frac{12}{N+1}$ $\bf
1+\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}$
$U$ $\bf 1$ $\bf 1$ $2N(4-N)$ $\frac{12}{N+1}$ $\bf 1$

Table 2: Particle content of the magnetic $Spin(8)$ theory. We omit the
baryons $b,\,b_{n}$ since they are not dynamical don’t play a role in our
analysis. Note that we use the same name for $M_{ij},\,U$ as in the electric
theory, due to their indentical representations.

The duality between the $SU(N-4)$ and $Spin(8)$ also maps the composite
operators of the theory: $S\bar{F}_{i}\bar{F}_{j}\leftrightarrow M_{ij}$ and
$\text{det}\,S\leftrightarrow U$. There are also gauge invariant baryonic
operators on both the electric and the magnetic side, but they do not play a
role in the dynamics below and so we do not discuss them further in this
paper.

##  IV Adding AMSB

Perturbing the duality via the AMSB mechanism will result on the electric side
in positive scalar masses as well as gaugino masses, leaving in the IR the
non-supersymmetric chiral gauge theory of interest. We will then have to
identify the effect of the AMSB on the magnetic $Spin(8)$ theory and find the
global minimum of its supersymmetry breaking potential. We first focus on the
case $N\geq 17$, in which the theory is in the free magnetic phase. A naive
local minimum is obtained by directly adding the AMSB to the tree level
potential (7). In this case the tree-level AMSB contribution from (1)
vanishes, and so the supersymmetry breaking is generated by the loop level
$A$-terms (2) and soft masses (3,4), leading to a local minimum at

$\displaystyle V\,$
$\displaystyle\approx\,-\left(\frac{\lambda^{2}}{16\pi^{2}}\right)^{4}\,m^{4}\,,$
(8)

where $\lambda$ is some $\mathcal{O}(1)$ combination the gauge and
superpotential couplings $g,\,y_{M},$ and $y_{U}$. While this is indeed a
local minimum of the potential, we will now show, this is not the global
minimum of the theory.

To find the global minimum, we first consider the magnetic theory on the
moduli space by turning on $\left\langle M\right\rangle$ of rank $N$, as well
as $\left\langle U\right\rangle$. As we shall see, the global minimum end up
being at small values of these VEVs, justifying our weakly coupled analysis in
the (initially) IR free magnetic theory. Once $q_{i},p$ are integrated out,
the magnetic theory becomes asymptotically free again: indeed the IR theory is
a pure $Spin(8)$ SYM, with a scale

$\displaystyle\tilde{\Lambda}_{L}^{18}\,=\,\frac{\text{det}\tilde{M}\,\tilde{U}}{\tilde{\Lambda}^{N-17}}\,.$
(9)

It develops a gaugino condensate with a dynamically generated superpotential
Intriligator:1995id 111Here and below, we absorb scheme-dependent numerical
constants Finnell:1995dr  into the definition of the scale
$\tilde{\Lambda}_{L}$ which does not affect any of the discussions below.

$W_{\text{dyn}}\,=\,e^{i\frac{\pi
k}{3}}(\tilde{\Lambda}^{18}_{L})^{1/6}\,=\,e^{i\frac{\pi
k}{3}}\left(\frac{\text{det}\tilde{M}\,\tilde{U}}{\tilde{\Lambda}^{N-17}}\right)^{1/6}\,,$
(10)

where $k=0,\ldots,5$ denoting six different vacua, originating from taking the
sixth root in the gaugino condensate. Taking this dynamical superpotential
into account, and using (1), we find the tree-level AMSB contribution

$\displaystyle{\cal L}_{\rm tree}$
$\displaystyle=m\frac{N-17}{6}\,W_{\text{dyn}}+c.c.$ (11)

The loop-level AMSB terms (2,3,4) are negligible with respect to this tree-
level contribution. The scalar potential from the superpotential (10) and the
tree-level AMSB contribution (11) has a supersymmetry breaking minimum at

$\displaystyle\tilde{M}_{ij}\,$
$\displaystyle\approx\,\delta_{ij}m\left(\frac{\tilde{\Lambda}}{m}\right)^{\frac{N-17}{N-11}},\quad\tilde{U}\,\approx\,m\left(\frac{\tilde{\Lambda}}{m}\right)^{\frac{N-17}{N-11}},$
$\displaystyle V\,$
$\displaystyle\approx\,-m^{4}\,{\left(\frac{\tilde{\Lambda}}{m}\right)}^{\frac{2(N-17)}{N-11}}\,.$
(12)

Indeed, for $N>17$ this minimum is deeper than the one in (8). We will come
back to the marginal case $N=17$ shortly. On the complex plane of $z=({\rm
det}\tilde{M}\ \tilde{U})$, six branches are connected through branch cuts
from one Riemann sheet to another. The minimum of the potential finds itself
on the sheet where $mW_{\rm dyn}$ in Eq. (11) can be real and negative. We
recall that the magnetic theory is IR free for $N\geq 17$ and in this case the
VEVs are both much smaller than the Landau pole and much larger than the scale
$\Lambda_{L}$ of the gaugino condensate. For this reason the minimum eq. (IV)
is at weak coupling.

At the minimum the global $SU(N)\times U(1)$ symmetry is broken to $SO(N)$,
(the $U(1)_{R}$ is explicitly broken by the AMSB) and all fermions become
massive. There are $\frac{1}{2}N(N+1)$ Nambu-Goldstone bosons corresponding to
the breaking, and none of them are eaten since the $Spin(8)$ gauge symmetry
remains unbroken. Since $SO(N)$ is anomaly free, there are no non-trivial ’t
Hooft anomaly matching conditions. In the IR theory, the $M_{ij}$ and $U$ all
play the role of Goldstone superfields, each one containing exactly one of the
$\frac{1}{2}N(N+1)$ Nambu-Goldstone boson for the $SU(N)\times U(1)/SO(N)$
breaking, except for one combination of $U$ and ${\rm Tr}M$ which is the
direction of the potential and is not a NGB.

Note that the VEV $M_{ij}$ has the full rank $N$ in the magnetic theory, while
in the electric theory their maximal rank is $N-4$. This is completely
consistent with the duality: in the supersymmetric limit, the rank condition
is enforced on $M_{ij}$ dynamically Intriligator:1995au , because the
superpotential Eq. (10) requires $\tilde{M}^{-1}{\rm det}\tilde{M}\
\tilde{U}={\rm det}\tilde{M}=0$. Therefore, along $F$-flat directions
corresponding to potential SUSY preserving minima ${\rm
rank}\,\tilde{M}_{ij}<N$. Directions with maximal rank satisfying ${\rm
det}\tilde{M}\ \tilde{U}\neq 0$ (as we assumed in deriving Eq. (10))
necessarily correspond to SUSY breaking vacua. In our case AMSB indeed
stabilizes the minimum (IV) along the direction where
$\langle\tilde{M}_{ij}\rangle$ has maximal rank, and away from the classically
expected rank condition. This situation is similar to the way the full-rank
meson field is found in the ISS model when $N_{c}<N_{f}$ in the electric
theory Intriligator:2006dd , with the mass perturbation stabilizing the
minimum away from the classical rank condition.

The marginal case $N=17$ is a little subtle. In this case the one-loop beta
function of the magnetic gauge coupling vanishes, though the two-loop beta
function does not and is IR free. Consequently, the theory is still in the
free magnetic phase. While in this case the minima (IV) and (8) have the same
dependence on $m$, the minimum (8) is much shallower, since it depends on the
slow 2-loop running of the gauge coupling, and quasi-IR-fixed point behavior
of the Yukawa couplings that tracks the gauge coupling. Then (IV) remains the
global minimum for $N=17$. We also verified that the two-loop mass-squared for
$\tilde{M}$ and $\tilde{U}$ is smaller than the square of the one-loop
$A$-term, and the origin is unstable. Therefore we find
$\tilde{M}\sim\tilde{U}\sim\frac{\lambda^{2}}{16\pi^{2}}m\neq 0$, and the
symmetry breaking pattern is the same as the rest of $N\geq 17$ cases.

So far the results obtained are exact. As in previous analyses with the AMSB
Murayama:2021xfj ; Csaki:2021xhi , we now take the limit $m\rightarrow\infty$
to extrapolate to the non-supersymmetric theory. Though there might be a phase
transition on the way, our analysis yields a plausible conjecture for the IR
behavior of non-supersymmetric $SU(N-4)$ with a symmetric $S$ and $N$ anti-
fundamentals. As a continuous limit of a self consistent supersymmetric
analysis, our method is guaranteed to fulfill all ’t Hooft matching
conditions, including generalized ones Gaiotto:2017yup ; Tanizaki:2018wtg ;
Bolognesi:2019fej ; Bolognesi:2020mpe ; Bolognesi:2021yni . Note also that in
a complete AMSB model our SUSY breaking spurion $m$ (the $F$-component of the
compensator) would originate from the constant term in the superpotential
leading to a relation $m=\frac{1}{M_{\rm Pl}^{2}}W_{0}$, where $W_{0}$ is the
superpotential in the SUSY breaking sector. If there is a phase transition
while increasing the amount of SUSY breaking it would happen for
$|m|\approx|\Lambda|$, corresponding to the condition $|\Lambda M_{\rm
Pl}^{2}|^{2}\approx|W_{0}|^{2}$. Since both $\Lambda$ and $W_{0}$ are chiral
superfields (or products thereof), any relation among them defining the phase
boundaries should be given by a holomorphic expression. Consequently, the
phase boundary should be even dimensional: either isolated points or the
entire complex plane. Since the latter is implausible it would have to be
isolated points. However those can not corresponds to the relation $|\Lambda
M_{\rm Pl}^{2}|^{2}\approx|W_{0}|^{2}$ which would imply the phase boundary to
be a circle on the complex $W_{0}$ plane. This suggests that such a phase
boundary should not exist. It would be interesting to see if this argument can
be made more rigorous.

Irrep $C_{2}$ $\frac{n^{2}-1}{2n}$ $\frac{(n+1)(n-2)}{n}$
$\frac{(n-1)(n+2)}{n}$ $\frac{2(n^{2}-4)}{n}$ $\frac{2(n^{2}+n-4)}{n}$

Irrep |  |  ⋮
---|---|---
$C_{2}$ | $\,\,\frac{2(n+4)(n-1)}{n}\,\,$ | $\,\,\frac{n^{2}-6n^{2}-7n+80}{2n}\,\,$

Table 3: Quadratic Casimirs for $SU(n)$. The Casimirs are the same for irreps
and their conjugates. For the last one, there are $n-1$ boxes vertically for
the first column.

Constituents Channel $\,C_{2}(\text{channel})-C_{2}(1)-C_{2}(2)\,$
$S,\,\bar{F}$ $-\frac{(n-1)(n+2)}{n}$ $S,\,S$ $-\frac{2(n+2)}{n}$
$\bar{F},\,\bar{F}$ $-\frac{n+1}{n}$ $S,\,S$ $-\frac{4}{n}$
$\bar{F},\,\bar{F}$ $\frac{n-1}{n}$ $S,\,S$ $\frac{4(n-1)}{n}$ $S,\,\bar{F}$ ⋮
$\frac{n^{2}-9n^{2}-9n+85}{2n}$

Table 4: $SU(n)$ Channels, ordered by most to least attractive. Note that in
our case $n=N-4$.

Irrep $\,\bf 1\,$ $C_{2}$ $0$ $\,\frac{(n-1)}{2}\,$ $n-2$ $n$ $\,2n-2\,$

Irrep |  |  |  |
---|---|---|---|---
$C_{2}$ | $\,\frac{3(n+1)}{2}\,$ | $2n+4$ | $\frac{3(n-1)}{2}$ | $2n$

Table 5: Quadratic Casimirs for $SO(n)$.

Constituents Channel $\,C_{2}(\text{channel})-C_{2}(1)-C_{2}(2)\,$ $S,\,S$
$\bf 1$ $-2n$ $S,\,S$ $-n-2$ $S,\,\bar{F}$ $-n$ $S,\,S$ $-n$
$\bar{F},\,\bar{F}$ $\bf 1$ $1-n$ $S,\,S$ $-2$ $\bar{F},\,\bar{F}$ $-1$
$S,\,\bar{F}$ $-1$ $S,\,S$ $0$ $\bar{F},\,\bar{F}$ $1$ $S,\,\bar{F}$ $2$
$S,\,S$ $4$

Table 6: $SO(n)$ Channels, ordered by most to least attractive. Note that in
our case $n=N-4$.

##  V Tumbling Interpretation

Here we would like to interpret our SUSY+AMSB analysis of $SU(N-4)$ with a
symmetric $S$ and $N$ $\bar{F}$ in the heuristic tumbling approach Raby:1979my
; Dimopoulos:1980hn . The unbroken global $SO(N)$ symmetry in the IR hints at
a symmetric $\bar{F}_{\\{i,}\bar{F}_{j\\}}$ condensate which breaks the gauge
group $SU(N-4)\rightarrow SO(N-4)$ and the global symmetry $SU(N)\rightarrow
SO(N)$. However, a symmetric condensate is not an attractive channel for the
$SU(N-4)$ gauge symmetry. The solution to this conundrum is the simultaneous
condensation of two channels. The first is

$\displaystyle\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-13.4pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}:\qquad
S_{ab}S_{cd}-S_{ad}S_{cb}\,\propto\,\delta_{ab}\delta_{cd}-\delta_{ad}\delta_{cb}\,.$
(13)

This condensate is attractive in $SU(N-4)$. It breaks the $U(1)$ global
symmetry, and higgses the gauge symmetry down to $SO(N-4)$. Under the reduced
gauge symmetry, the theory is now vector-like and confines. The symmetric
$\bar{F}_{\\{i,}\bar{F}_{j\\}}$ is now a color singlet. It is attractive, and
condenses as

$\displaystyle\delta_{ab}\bar{F}^{a}_{i}\bar{F}^{b}_{j}\propto\delta_{ij}$
(14)

breaking the global $SU(N)$ symmetry down to $SO(N)$. The candidate Nambu-
Goldstone bosons for the $SU(N)/SO(N)$ coset are
$\bar{F}_{\\{i,}\bar{F}_{j\\}}$, while the Goldstones for the two broken
$U(1)$s can be taken to be ${(\bar{F}^{a}_{i})}^{2}$ and $S^{2}_{ab}$. By
examining tables 4-6, we see that the condensates described above are
attractive, but they are certainly not the most attractive channel (MAC). Our
suggestion for the IR dynamics of the theory is then different from the one
suggested by tumbling, or the other proposed phase with fully unbroken global
symmetries and no condensates.

##  VI Non Abelian Coulomb Phase

For $N<16$, the theory has an IR fixed point Pouliot:1995sk , in which both
the electric and magnetic descriptions of the theory are equally useful. In
the supersymmetric limit the anomalous dimensions $\gamma_{i}$ are scale
independent and so $\dot{\gamma}_{i}=0$ and the AMSB soft masses vanish at the
IR fixed point. More specifically $\gamma_{i}=3R_{i}-2$ where $R_{i}$ is the
$R$-charge. However, in the presence of non-anomalous $U(1)$ symmetries, the
definition of the $R$-symmetry is ambiguous. The combination that appears in
the super-conformal algebra is fixed by $a$-maximalization Intriligator:2003jj
, and the resultant $a$ is the Euler trace anomaly coefficient that always
decreases with the renormalization-group flow Komargodski:2011vj  analogous to
the Zamolodchikov’s $c$-theorem Zamolodchikov:1986gt . The Euler trace anomaly
coefficient $a$ is defined by Anselmi:1997am

$\displaystyle a$ $\displaystyle=\frac{3}{32}\left(3{\rm Tr}R^{3}-{\rm
Tr}R\right),$ (15)

where the trace sums over all fermions in the theory. Obviously the definition
is common between the electric and magnetic theories as long as the ’t Hooft
anomaly matching conditions are satisfied. Using the $U(1)$ charge $Q$ in
Table 2, we maximize $a$ using the combination $R^{\prime}=R+tQ$, and find the
(local) maximum at

$\displaystyle
t=\frac{9N^{2}-24N+39-(N+1)\sqrt{73N^{2}-362N+433}}{6(N-4)(N-1)(N+1)}\ .$ (16)

$t$ is very small, $t\approx 0.0245$ for $N=6$ and even smaller for larger
$N$. We can see that $t=0$ for $N=17$, implying that the dimension of $M$ and
$U$ become 1 at that point, signaling the beginning of the free magnetic
phase, where $M$ and $U$ are indeed free fields. With the anomalous dimensions
$\gamma_{i}=3R^{\prime}_{i}-2$, the anomaly-free condition for
$U(1)^{\prime}_{R}$ guarantees that the NSVZ beta function vanishes
Anselmi:1997am ; deGouvea:1998ft .

Supersymmetry breaking quickly vanishes as the theory approaches the IR fixed
point. For example, we can work out the gaugino mass by expanding the beta
function to the first order around the coupling $g_{*}$ at the fixed point,

$\displaystyle\beta(g^{2})\,$
$\displaystyle=\,\beta^{\prime}_{*}\left(g^{2}-g^{2}_{*}\right)\,+O\left(g^{2}-g^{2}_{*}\right)^{2},$
(17)

and so in the vicinity of the IR fixed point ($\beta^{\prime}_{*}>0$),

$\displaystyle g^{2}(\mu)\,$
$\displaystyle=\,g^{2}_{*}\,+\,\left[g^{2}(\mu^{\prime})-g^{2}_{*}\right]{\left(\frac{\mu}{\mu^{\prime}}\right)}^{\beta^{\prime}_{*}}\,,$
(18)

for energy scales $\mu>\mu^{\prime}>\Lambda_{*}$ where $\Lambda_{*}$ is the
energy scale of the IR fixed point. Eq. (4) then gives the gaugino mass

$\displaystyle m_{\lambda}(\mu)$
$\displaystyle=-m\,\frac{\beta^{\prime}_{*}}{2}\frac{g^{2}(\mu^{\prime})-g^{2}_{*}}{g^{2}_{*}+\left(g^{2}(\mu^{\prime})-g^{2}_{*}\right){\left(\frac{\mu}{\mu^{\prime}}\right)}^{\beta^{\prime}_{*}}}{\left(\frac{\mu}{\mu^{\prime}}\right)}^{\beta^{\prime}_{*}}$
$\displaystyle\approx
m\,\frac{\beta^{\prime}_{*}}{2}\left[1-\frac{g^{2}(\mu^{\prime})}{g^{2}_{*}}\right]{\left(\frac{\mu}{\mu^{\prime}}\right)}^{\beta^{\prime}_{*}}\,,$
(19)

where in the last step we assumed $\mu\ll\mu^{\prime}$ and neglected the
power-suppressed second term in the denominator. The gaugino mass as expected
is power-law suppressed, and tends quickly to zero as we approach the IR.

##  VII Conclusions

We have identified the IR phase of the non-SUSY chiral $SU(N-4)$ gauge theory
with a fermion in the symmetric representation, as well as $N$ anti-
fundamentals, obtained via perturbing the relevant SUSY duality with the AMSB.
For $N\geq 17$ the theory is confining with the $SU(N)$ global symmetry broken
to $SO(N)$ and no massless composite fermions are needed to match anomalies.
For $N<17$ the theory flows to a (super-)conformal fixed point, providing
another interesting example of a non-SUSY theory flowing to a SUSY fixed point
in the IR.

While the results are obtained in the limit of small SUSY breaking
$m\ll\Lambda$, they do provide a plausible candidate vacuum structure for
these theories even when $m\gg\Lambda$, satisfying all possible consistency
conditions by construction. The resulting vacuum structure obtained with this
method differs significantly from either of the two conjectured phases of the
non-SUSY theory, and can be given an interesting interpretation in the
tumbling framework via two condensates, neither of which would correspond to
the MAC (and one of them becomes attractive only in the presence of the first
condensate). Whether this is indeed the correct phase of the non-SUSY theory,
or if a phase transition occurs at $m\sim\Lambda$ will have to be eventually
verified by dedicated lattice simulations.

##  VIII Acknowledgments

###### Acknowledgements.

We thank Ken Intriligator for careful reading of the manuscript and useful
comments. CC is supported in part by the NSF grant PHY-2014071 as well as the
US-Israeli BSF grant 2016153. OT and HM were supported in part by the DOE
under grant DE-AC02-05CH11231. HM was also supported in part by the NSF grant
PHY-1915314, by the JSPS Grant-in-Aid for Scientific Research JP20K03942, MEXT
Grant-in-Aid for Transformative Research Areas (A) JP20H05850, JP20A203, by
WPI, MEXT, Japan, and Hamamatsu Photonics, K.K.

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