# The Phases of Non-supersymmetric Gauge Theories: the $SO(N_{c})$ Case Study

Csaba Csáki<EMAIL_ADDRESS>Department of Physics, LEPP, Cornell University,
Ithaca, NY 14853, USA  Andrew Gomes<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 investigate the IR phases of non-supersymmetric (non-SUSY) $SO(N_{c})$
gauge theories with $N_{F}$ fermions in the vector representation obtained by
perturbing the SUSY theory with anomaly mediated SUSY breaking (AMSB). We find
that of the wide variety of phases appearing in the SUSY theory only two
survive: for $N_{F}<\frac{3}{2}(N_{c}-2)$ the theory confines, breaking the
$SU(N_{F})$ global symmetry to $SO(N_{F})$, while for
$\frac{3}{2}(N_{c}-2)<N_{F}<3(N_{c}-2)$ the theory flows to a
(super)-conformal fixed point. The abelian Coulomb and free magnetic phases do
not survive and collapse to the confining phase. We also investigate the
behavior of loop operators in order to provide a clear distinction between the
confining and screened phases. With the choice of $Spin(N_{c})$ for the global
structure of the gauge group, we find that the electric Wilson loop indeed
obeys an area law, providing one of the first demonstrations of true
confinement with chiral symmetry breaking in a non-SUSY theory. We identify
monopole condensation as the dynamics underlying confinement. These monopoles
arise naturally for $N_{F}=N_{c}-2$. The case with smaller number of flavors
can be obtained by integrating out flavors, and we confirm numerically that
the monopole condensate persists in the presence of AMSB and mass
perturbations.

##  I Introduction

Understanding the phases of strongly coupled gauge theories is one of the most
important outstanding goals of particle physics. A great deal of progress was
made on this problem in the mid-90’s, led by Seiberg Affleck:1983mk ;
Seiberg:1994bz ; Seiberg:1994pq , when the vacuum structures of a large
variety of supersymmetric (SUSY) gauge theories were established, including
${\cal N}=1$ SUSY QCD, ${\cal N}=2$ Seiberg-Witten theories, and also various
interesting chiral gauge theories. A wide variety of gauge theory phases were
encountered in this process: the confining phase, the screened/Higgs phase
(which are indistinguishable due to complementarity), a free magnetic phase
with an IR free dual gauge group, true conformal field theories (CFTs) with a
non-trivial interacting fixed point dubbed the non-abelian Coulomb phase, and
finally the abelian Coulomb phase, which in the SUSY case usually includes
points in the moduli space with massless monopoles Seiberg:1994rs . These
phases form an intricate web which can be nicely connected by integrating out
flavors or Higgsing the theories. One particularly interesting example is the
case of $SO(N_{c})$ gauge theories with $N_{f}$ chiral superfields in the
vector $N_{c}$ representation: as $N_{f}$ is varied between $1$ and $3N_{c}-2$
all of the above phases actually appear, making $SO(N_{c})$ an ideal test lab
for studying the phases of gauge theories.

SUSY gauge theories are very interesting in their own right, though one would
obviously like to ask what phases appear for the non-SUSY versions of these
theories. This question was investigated in a series of papers in the late
90’s Evans:1995rv ; Evans:1995ia ; Aharony:1995zh ; Alvarez-Gaume:1996qoj ;
Alvarez-Gaume:1996vlf ; Evans:1996hi ; Konishi:1996iz ; Alvarez-Gaume:1997bzm
; Alvarez-Gaume:1997wnu ; Evans:1997dz ; Cheng:1998xg ; Martin:1998yr ;
Luty:1999qc , by perturbing SUSY gauge theories with soft supersymmetry
breaking terms (see also the more recent Cordova:2018acb ). Recently, a simple
and efficient method for perturbing SUSY dualities via anomaly mediated SUSY
breaking (AMSB) was recently proposed in Murayama:2021xfj . The main advantage
of this method is that a single spurion provides both the positive scalar mass
squared and the gaugino mass, thereby preparing exactly the UV theory of
interest. By the UV insensitive nature of AMSB the determination of the low-
energy phase of the theory can be reliably performed within the dual low-
energy effective theory. Indeed many interesting results have been obtained
this way including the phases and global symmetry breaking pattern of SUSY QCD
Murayama:2021xfj  as well as in the simplest non-supersymmetric chiral gauge
theories with an antisymmetric Csaki:2021aqv  or symmetric Csaki:2021xhi
fermion. These results allowed the reexamination of old conjectured phases of
non-SUSY chiral gauge theories based on the tumbling/most attractive channel
(MAC) framework Raby:1979my ; Dimopoulos:1980hn . It was found that all cases
needed some form of modification, and in some cases it was not the condensate
in the MAC that was actually generated. The theories with larger number of
flavors included more chiral symmetry breaking and also examples of
superconformal fixed points from non-SUSY UV theories.

However all of the theories explored so far with AMSB contained some particles
in the fundamental representation, which led to screening and a perimeter law
for all of their Wilson loops. Consequently, their chiral symmetry breaking
took place in a screened/Higgs phase, rather than a genuine confining phase.
This will no longer be the case for gauge theories with Lie algebra
$\mathfrak{so}(N_{c})$ (more specifically $Spin(N_{c})$) and matter in the
vector representation, where the Wilson loop in the spinor can not be
screened. This makes the appearance of an area law possible – which is the
strict definition of a confining phase.

In this paper we study the low-energy dynamics of the $SO(N_{c})$ gauge
theories with $N_{F}$ Weyl fermions in the vector representation, obtained by
perturbing the corresponding ${\cal N}=1$ $SO(N_{c})$ theory via AMSB. We will
also include a discussion of the global properties of these theories, which is
necessary to distinguish the truly confining theories with an area law for the
Wilson loop from those with a perimeter law. Thus, as will be explained later,
we will have to distinguish between $Spin(N_{c})$ and the $SO(N_{c})_{\pm}$
groups. The $N_{F}=N_{c}-2$ case will be particularly interesting: the SUSY
theory in this case corresponds to a pure Coulomb branch, where a pair of
monopoles becomes massless at one point, together with a full multiplet of
dyons becoming massless at the origin. We will show that the AMSB perturbation
of this theory leads to the condensation of monopoles, along with (partial)
breaking of the chiral global symmetries at the true ground state. Such
monopole (or dyon) condensation was conjectured by Mandelstam Mandelstam1976
and ’t Hooft tHooft:1981bkw  long ago to be the dynamics leading to
confinement via the dual Meissner effect. This has indeed been found to be the
case by Seiberg and Witten when perturbing pure ${\cal N}=2$ to pure ${\cal
N}=1$ supersymmetric Yang–Mills (SYM) Seiberg:1994rs , and also by
Aharony:1995zh ; Alvarez-Gaume:1996vlf ; Konishi:1996iz ; Evans:1995rv  who
studied the small non-SUSY perturbations of the Seiberg-Witten theory. Our
results provide one of the first examples of (true) confinement with chiral
symmetry breaking in a non-SUSY gauge theory.

By considering mass deformations to the $N_{F}=N_{c}-2$ case, we will be able
to show that all of the cases with $N_{F}<\frac{3}{2}(N_{c}-2)$ exhibit the
same behavior as for $N_{F}=N_{c}-2$, i.e., electric confinement via monopole
condensation, together with chiral symmetry breaking. In contrast, for
$\frac{3}{2}(N_{c}-2)<N_{F}\leq 3(N_{c}-2)$ the non-SUSY UV theory flows as
expected to a (super)-conformal theory in the IR. The case with
$N_{F}=\frac{3}{2}(N_{c}-2)$ is marginal, and will be discussed in future
work. In other words, with the AMSB perturbation the many phases present in
the SUSY case collapse down to just two: the confining phase, and the
conformal phase. The Coulomb and free magnetic phases do not survive the
breaking of supersymmetry.

The paper is organized as follows. First we present a short summary of the
moduli space and symmetries of ${\cal N}=1$ SUSY $SO(N_{c})$ theories with
$N_{F}$ vectors. The quantum vacuum structure of the entire SUSY series has
been worked out in a beautiful paper by Intriligator and Seiberg in
Intriligator:1995id , which will be the basis of our analysis for the AMSB
perturbations. We begin our discussion of the various cases with the most
novel $N_{F}=N_{c}-2$ case, which is the only one known example so far giving
rise to monopole condensation with chiral symmetry breaking in a non-SUSY
theory. We then show that whenever “non-trivial” (spinorial) Wilson lines
exist in the theory with $N_{F}=N_{c}-2$, they exhibit an area law, signaling
true electric confinement. By considering mass deformations to the
$N_{F}=N_{c}-2$ case, we demonstrate this fact, as well as chiral symmetry
breaking, for all $SO(N_{c})$ theories with $1<N_{F}\leq N_{c}-2$. While there
are several special cases to examine following the analysis of
Intriligator:1995id , in the end the global minimum of the theory for
$1<N_{F}\leq N_{c}-2$ is always the one with the $SU(N_{F})\rightarrow
SO(N_{F})$ chiral symmetry breaking pattern. We have also summarized this
important result in a companion letter Csaki:2021jax .

We then continue on to consider the cases with a larger number of flavors. We
find results similar to the case of SUSY QCD. For
$N_{c}-2<N_{F}<\frac{3}{2}(N_{c}-2)$ we again find a global minimum where the
chiral symmetry is broken to $SO(N_{F})$. In this case monopole condensation
does not directly appear in the description, but non-trivial Wilson loops
still exhibit an area law. To see this, note that at a generic point in the
moduli space of the dual theory, the dual quarks are integrated out, leaving
us with pure $SO(N_{F}-N_{c}+4)$ SYM. Consequently, whenever the dual gauge
group allows for dyonic (’t Hooft) loops, these exhibit an area (perimeter)
law. By the duality of Aharony:2013 , whenever the original theory allows for
non-trivial Wilson (’t Hooft) loops, they exhibit an area (perimeter) law.

Finally, for $\frac{3}{2}(N_{c}-2)<N_{F}\leq 3(N_{c}-2)$ we find that the
theory flows to a superconformal fixed point, even though the UV theory
explicitly breaks supersymmetry. For $3(N_{c}-2)<N_{F}$, the theory with AMSB
has tachyonic squarks and hence no ground state, and is not continuously
connected to the non-supersymmetric $SO(N_{c})$ theory.

##  II $SO(N_{c})$ with $N_{F}$ Fundamentals

We consider a supersymmetric gauge theory with gauge group $SO(N_{c})$,
$N_{F}$ vectors, and no tree-level superpotential. This theory has been
thoroughly studied in e.g., Intriligator:1995id ; Intriligator:1995au . The
anomaly-free global symmetry of the theory is $\left(SU(N_{F})\times
U(1)_{R}\times\mathbb{Z}_{2N_{F}}\times\mathbb{Z}_{2}/\mathbb{Z}_{N_{F}}\right)$,
where the $\mathbb{Z}_{2}$ is charge conjugation 111For $N_{F}=3$ the discrete
part of the global symmetry is enhanced from $Z_{2N_{F}}$ to $Z_{4N_{F}}$.
Under the continuous part of the global symmetry, the matter fields transform
as
$Q(\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}})_{\frac{N_{F}-N_{c}+2}{N_{F}}}$.

For $N_{F}<N_{c}$, the $D$-flat directions of the theory are given, up to
gauge and global transformations, by

$\displaystyle
Q=\left(\begin{array}[]{c|c}\begin{array}[]{ccc}\varphi_{1}&&\\\ &\ddots&\\\
&&\varphi_{N_{F}}\end{array}&0\end{array}\right)\,,$ (5)

For $N_{F}<N_{c}$ $D$-flat directions are conveniently parameterized by the
$\frac{1}{2}N_{F}(N_{F}+1)$ gauge invariant “meson” operators
$M^{ij}=Q^{i}Q^{j}$. Along the $D$-flat directions, they are given by

$\displaystyle
M=\text{diag}\left(\varphi^{2}_{1},\ldots,\varphi^{2}_{N_{f}}\right)\,.$ (6)

For $N_{F}\geq N_{c}$, the $D$-flat direction is given by

$\displaystyle Q=\left(\begin{array}[]{c}\begin{array}[]{ccc}\varphi_{1}&&\\\
&\ddots&\\\ &&\varphi_{N_{c}}\end{array}\\\ \hline\cr 0\end{array}\right)\,$
(12)

It is conveniently parameterized in terms of meson operators, as well as the
baryon operators $B^{[i_{1},\ldots i_{N_{c}}]}=Q^{[i_{1}}\ldots
Q^{i_{N_{c}}]}$, where the $i$ are flavor indices and we take the gauge
singlet out of the tensor product of $N_{c}$ fundamentals of $SO(N_{c})$. The
gauge invariant operators are given, up to global transformations, by

$\displaystyle
M=\left(\begin{array}[]{c|c}\text{diag}\left(\varphi^{2}_{1},\ldots,\varphi^{2}_{N_{c}}\right)&0\\\
\hline\cr 0&0_{{N_{F}-N_{c}}\times{N_{F}-N_{c}}}\end{array}\right)$ (15)
$\displaystyle B^{1,\ldots{N_{c}}}=\varphi_{1}\ldots\varphi_{N_{c}}\,.$ (16)

The IR behavior of the theory strongly depends on the relative magnitudes of
$N_{c}$ and $N_{F}$ and is summarized in Table 1. Below we will show that
adding AMSB to the theory leads to chiral symmetry breaking for all
$1<N_{F}<\frac{3}{2}(N_{c}-2)$. Furthermore, in this range the theory
confines; below we give an exact meaning to this statement in terms of the
loop operators of the theory. We assume throughout that $N_{c}>3$, and leave
the $N_{c}=3$ to future work.

Range SUSY +AMSB $N_{F}=1$ run-away confinement $1<N_{F}<N_{c}-4$ run-away
confinement$+\chi$SB $N_{F}=N_{c}-4$ 2 branches confinement$+\chi$SB
$N_{F}=N_{c}-3$ 2 branches confinement$+\chi$SB $N_{F}=N_{c}-2$ Coulomb
confinement$+\chi$SB $N_{F}=N_{c}-1$ $\begin{array}[]{c}\text{free
magnetic}\\\ \text{2 branches}\end{array}$ confinement$+\chi$SB $N_{F}=N_{c}$
$\begin{array}[]{c}\text{free magnetic}\\\ \text{2 branches}\end{array}$
confinement$+\chi$SB $N_{c}+1\leq N_{F}<\frac{3}{2}(N_{c}-2)$ free magnetic
confinement $+\chi$SB $\,\frac{3}{2}(N_{c}-2)<N_{F}\leq 3(N_{c}-2)\,$ CFT CFT
$3(N_{c}-2)<N_{F}$ IR free run-away

Table 1: Summary of IR Behavior of $SO(N_{c})$ theories with $N_{F}$
fundamentals with AMSB. $\chi$SB stands for chiral symmetry breaking. For
$N_{F}=N_{c}-1$ and $N_{c}$, two branches appear along the flat direction of
the maximum rank of the meson $M^{ij}$, yet the AMSB chooses one over the
other, resulting in the $\chi$SB.

##  III Phases of Gauge Theories

One often hears the word “confinement” describing the situation in which
colored degrees of freedom are bound into color-singlet states, even if a
linear potential is lacking due to screening from quark-antiquark production.
We will be more careful with the word and, following Aharony:2013 , only use
it in the narrow context of a particular Wilson/’t Hooft/dyonic loop operator
222By a loop operator we mean a closed line around which we take a test
charge, monopole or dyon in some representation allowed by the gauge group.
exhibiting an area law, in which case we will say that the given loop operator
confines. We are especially interested in the confinement of non-trivial loop
operators - the ones which transform non-trivially under the center of the
gauge group. Note that in some of the literature, e.g., Aharony:2013 , these
closed loops are referred to as line operators - we will use the more
conventional name “loop operators” to stress their gauge invariance.

The allowed non-trivial loop operators in the theory depend on the choice of
the global properties of the gauge group – for example, with the Lie algebra
$\mathfrak{so}(N_{c})$ the gauge group can be $Spin(N_{c}),\,SO(N_{c})$, and
so on. Depending on the choice of gauge group, the allowed non-trivial loop
operators can be Wilson, ’t Hooft, or dyonic loops. Whatever the choice may
be, these loops exhibit either a perimeter or an area law, depending on the
local physics, which is in and of itself insensitive to the global properties
of the gauge group.

One possible choice of the gauge group is $Spin(N_{c})$, which is the
universal cover of all Lie groups that share the Lie algebra
$\mathfrak{so}(N_{c})$. In this case the non-trivial loop operators are Wilson
loops, while others are forbidden by Dirac quantization. An area law for non-
trivial Wilson loops indicates the confinement of the electric degrees of
freedom associated with it. Other choices for the global structure are
obtained by modding out the $Spin(N_{c})$ by subgroups of its center, which is
$\mathbb{Z}_{2}$ for odd $N_{c}$ and larger for even $N_{c}$ Aharony:2013 .
Here we only consider $SO(N_{c})=Spin(N_{c})/\mathbb{Z}_{2}$. In this case the
non-trivial loop operators are either purely magnetic ’t Hooft loops, a choice
denoted by $SO(N_{c})_{+}$, or dyonic loops, a choice denoted by
$SO(N_{c})_{-}$. In each case, other non-trivial loop operators are forbidden
by Dirac quantization. Below, when we speak of “the loop” in a particular
theory, we will be referring to the single non-trivial loop that the theory
admits, whether electric, magnetic, or dyonic. Additionally, when we are only
concerned with the local physics we will simply refer to the gauge group as
$SO(N_{c})$.

While $Spin(N_{c})$ and $SO(N_{c})_{\pm}$ exist on equal footing as possible
gauge groups, we will be particularly interested in $Spin(N_{c})$, as it can
provide what eludes us for $SU(N)$ gauge theories with fundamental matter: an
order parameter for electric confinement. Whereas the fundamental Wilson loop
of $SU(N)$ (which is its own universal cover) can be screened by the
fundamental matter, the spinorial Wilson loop of $Spin(N_{c})$ cannot be
screened by the vectorial quarks.

The main objective of our study is to determine the phases of supersymmetric
$SO(N_{c})$ gauge theories when perturbed by AMSB. We will be able to
determine the local behavior of the theory – be it chiral symmetry breaking,
monopole condensation, etc. As a result, we will be able to establish the
behavior of the allowed non-trivial loop operators in the theory, and whether
they exhibit an area or a perimeter law. Our final results about the phase
structure (along with the corresponding SUSY phases) are summarized in Tab. 1.
We can see that in the non-SUSY theory the only surviving phases are those of
confinement (with chiral symmetry breaking), or a conformal phase. The abelian
Coulomb and the free magnetic phases do not survive the AMSB perturbation, and
they all collapse to the generic confining+$\chi$SB phase.

##  IV Confinement with Chiral Symmetry Breaking for
$1<N_{F}<\frac{3}{2}(N_{c}-2)$

Next we present a detailed analysis of the vacuum structure of $SO(N_{c})$
with $N_{F}$ flavors in the presence of AMSB. For $N_{F}=N_{c}-2$ we show
monopole condensation with chiral symmetry breaking – leading to the
confinement of non-trivial Wilson lines for $Spin(N_{c})$333Alternatively, for
$SO(N_{c})_{+}$ the non-trivial ’t Hooft line exhibit a perimeter law, while
for $SO(N_{c})_{-}$, the non-trivial dyonic line exhibits an area law.. For
$N_{F}<N_{c}-2$ we can still explicitly see chiral symmetry breaking, while
monopole condensation and thus electric confinement is established by adding
mass deformations to the $N_{F}=N_{c}-2$ case. Finally, theories with
$N_{c}-2<N_{F}<\frac{3}{2}(N_{c}-2)$ exhibit chiral symmetry breaking, while
confinement is demonstrated by finding a “hidden” monopole condensate in the
dual theory.

###  IV.1 $N_{F}=N_{c}-2$

In this case the supersymmetric theory is in an abelian Coulomb phase
Intriligator:1995id . Since the $M^{ij}$ are not charged under $U(1)_{R}$,
there is no superpotential even at the quantum level, and hence the theory has
a quantum moduli space. On this moduli space, the gauge symmetry is higgsed to
a $SO(2)\simeq U(1)$, namely, the theory is on the Coulomb branch. On the
moduli space, the gauge coupling
$\tau=\frac{\theta}{2\pi}+\frac{i8\pi}{g^{2}}$ is given only as a function of
the $SU(N_{F})$ invariant $U\equiv\text{det}M$. It is singular at two points
$U=0$ ($U=U_{1}\equiv 16\Lambda^{2N_{F}}$), where the dyons $q^{\pm}_{i}$
(monopoles $E^{\pm}$) of the $U(1)$ gauge symmetry become massless. In the
original paper Intriligator:1995id , the authors chose to label the particles
condensing at $U=0$ and $U=U_{1}$ as monopoles and dyons, respectively. Our
opposite labeling leads to line behaviors consistent with those in
Aharony:2013  for all confining theories, and is also consistent with the
finding in Witten:2000nv .

Around the singular point $U=0$ the relevant light degrees of freedom are the
dyons $q^{\pm}_{i}$ with magnetic charge $\pm 1$, which transform under the UV
global symmetry $SU(N_{F})\times U(1)_{R}$ as
$q^{\pm}_{i}(\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}})_{1}$.
These have a dynamically generated superpotential about $U=0$ of

$\displaystyle W_{\text{dyon}}\,$ $\displaystyle=$
$\displaystyle\,\frac{1}{\mu}\,f(t)\,M^{ij}q^{+}_{i}q^{-}_{j}\,,$ (17)

where $\mu$ is an effective mass scale, $t=U\Lambda^{4-2N_{c}}$, and $f(t)$ is
a holomorphic function in the neighborhood of $t=0$, normalized so that
$f(0)=1$. Expanding $f$ to higher orders in $t$ introduces tree level AMSB,
but it is highly suppressed by powers of the meson VEV over $\Lambda$ and
results in no qualitative changes. Exactly at $U=0$, ’t Hooft anomaly matching
is saturated by $q^{\pm}_{i},\,M^{ij}$, and the photinos
$\mathcal{W}_{\alpha}\sim W_{\alpha}Q^{N_{c}-2}$ Intriligator:1995id , whose
charges are given in Table 2.

$SO(N_{c})$ $SU(N_{F})$ $U(1)_{R}$ $Q^{i}$ $0$ $\lambda$ $\bf 1$ $1$ $M_{ij}$
$\bf 1$ $0$ $q^{\pm}_{i}$ $-$
$\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$
$1$ $\lambda_{\text{mag}}$ $-$ $\bf 1$ $1$

Table 2: DOF of the $SO(N_{c})$ theory with $N_{F}=N_{c}-2$ near $M=0$.
$\lambda$ are the $SO(N_{c})$ gauginos, while $\lambda_{\text{mag}}$ are the
photinos of the unbroken (magnetic) $U(1)$ in the IR. For the supersymmetric
theory at the origin, the full global symmetry is unbroken. With AMSB there is
a local minimum, where the global symmetry is broken to $SU(N_{F}-2)$.

Using the formulae for loop level AMSB Eq. (84), we can explore the local
minima around the origin of moduli space. The IR free nature of the $U(1)$
gauge theory gives a tachyonic contribution to the dyon masses. However, the
dyons also receive a positive mass-squared contribution from the Yukawa-like
coupling to the meson field Eq. (17). The co-dependence of the Yukawa and
gauge beta functions results in a flow to a fixed ratio between the two
couplings. This ratio is such that the mass-squared due to loop AMSB is
positive for both the meson and the dyons. Thus, the loop-level AMSB trilinear
term in combination with the tree-level quartic potential gives a local
minimum a distance $\mathcal{O}(\frac{m}{16\pi^{2}})$ from the origin. To
understand the symmetry breaking pattern at this minimum, we must examine the
form of the tree-level potential in terms of the $SU(N_{F})$ matrix $M$ and
vectors $q^{\pm}$,

$\displaystyle V~{}$ $\displaystyle=$ $\displaystyle~{}\frac{1}{2}(q^{+}\cdot
q^{+*})(q^{-}\cdot q^{-*})$ (18)
$\displaystyle+|Mq^{+}|^{2}+|Mq^{-}|^{2}+V_{\text{AMSB}}$

The dot product term is due to the symmetric nature of the meson matrix (i.e.,
$M^{ij}$ couples to $q_{i}^{+}q_{j}^{-}$ and $q_{j}^{+}q_{i}^{-}$). This term
encourages the $q^{\pm}$ VEVs to point in different directions in flavor
space. We find a minimum along the direction,

$\displaystyle q^{+}=\left(\begin{array}[]{c}1\\\ 0\\\ \hline\cr 0\\\
\vdots\\\ 0\end{array}\right)\alpha,\quad q^{-}=\left(\begin{array}[]{c}0\\\
1\\\ \hline\cr 0\\\ \vdots\\\ 0\end{array}\right)\alpha,$ (29) $\displaystyle
M\propto\left(\begin{array}[]{cc|ccc}0&1&0&\cdots&0\\\ 1&0&0&\cdots&0\\\
\hline\cr 0&0&0&\cdots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&\cdots&0\end{array}\right),$ (35)

Breaking the global flavor symmetry to $SU(N_{F}-2)$. The vacuum energy of
this minimum is $V=-\mathcal{O}(\frac{m}{16\pi^{2}})^{4}$.

In the vicinity of the singular point $U=U_{1}$, on the other hand, the light
degrees of freedom are the monopoles $E^{\pm}\sim q^{\pm}_{i}Q^{i}$, whose
magnetic charges are $\pm 1$. These transform under the $SU(N_{F})\times
U(1)_{R}$ global symmetry of the UV theory as $E^{\pm}(\bf 1)_{1}$. Since
$\text{det}M\equiv U\neq 0$, the global $SU(N_{F})\times U(1)_{R}$ is
spontaneously broken to $SO(N_{F})\times U(1)_{R}$. In the neighborhood of
$U=U_{1}$, the theory generates a dynamical superpotential

$\displaystyle W_{\text{mon}}\,$ $\displaystyle=$
$\displaystyle\,\tilde{f}\left(\frac{U-U_{1}}{\Lambda^{2N_{F}}}\right)\,E^{+}\,E^{-}\,.$
(36)

Here $\tilde{f}(t)=t+\dotsb$ is holomorphic near $t=0$. For all practical
purposes, only the leading order in $\tilde{f}$ matters for the stabilization
of the minimum. Using canonically normalized fields we have

$\displaystyle W_{\text{mon}}\,$ $\displaystyle=$
$\displaystyle\,\Lambda\left(\frac{\tilde{U}}{\Lambda^{N_{F}}}-16\right)\,\tilde{E}^{+}\,\tilde{E}^{-}\,,$
(37)

where $\tilde{E}^{\pm}=E^{\pm}/\sqrt{\Lambda}$,
$\tilde{U}=\text{det}\tilde{M}$, and $\tilde{M}=M/\Lambda$ is the canonically
normalized meson. Exactly at $\tilde{U}=\tilde{U}_{1}\equiv
16\Lambda^{N_{F}}$, ’t Hooft anomaly matching is saturated by
$E^{\pm},\,\tilde{M}^{ij}$, and the photinos $\mathcal{W}_{\alpha}\sim
W_{\alpha}Q^{N_{c}-2}$, whose charges are given in Table 3.

Explicitly,

$U(1)_{R}\,{\rm gravity}^{2}$ and $U(1)^{3}_{R}:$

$\displaystyle(-1)N_{F}N_{c}+(1)\frac{N_{c}(N_{c}-1)}{2}=(1)+(-1)\frac{N_{F}(N_{F}+1)}{2},$
(38)

$U(1)_{R}\,SO(N_{F})^{2}$:

$\displaystyle(-1)(1)N_{c}$ $\displaystyle=(-1)(N_{F}+2).$ (39)

$SO(N_{c})$ $SU(N_{F})$ $U(1)_{R}$ $U(1)_{\text{mag}}$ $SO(N_{F})$ $Q^{i}$ $0$
$-$ $\lambda$ $\bf 1$ $1$ $-$ $\bf 1$ $M_{ij}$ $\bf 1$ $0$ $-$ $\bf
1+\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}$
$E^{\pm}$ $-$ $\bf 1$ $1$ $\pm 1$ $\bf 1$ $\lambda_{\text{mag}}$ $-$ $\bf 1$
$1$ $0$ $\bf 1$

Table 3: DOF of the $SO(N_{c})$ theory with $N_{F}=N_{c}-2$ near
$M^{ij}\sim\delta^{ij},U=U_{1}$. The unbroken global symmetry near
$M^{ij}\sim\delta^{ij},U=U_{1}$ is $SO(N_{F})\times U(1)_{R}$, with $U(1)_{R}$
explicitly broken by AMSB.

Contrary to the point $\tilde{U}=0$, here adding AMSB generates a tree-level
contribution to the scalar potential from (83). This results in the global
minimum at $\tilde{U}=\tilde{U}_{1}$. In particular, the scalar potential
along $\tilde{M}^{ij}=\tilde{M}\delta^{ij}$ is given locally as

$\displaystyle
V_{\tilde{U}\sim\tilde{U}_{1}}\,=\,\Lambda^{2}\left|{\left(\frac{\tilde{M}}{\Lambda}\right)}^{N_{F}}-16\right|^{2}\left(|\tilde{E}^{+}|^{2}+|\tilde{E}^{-}|^{2}\right)$
$\displaystyle+\frac{1}{kN_{F}}\left|N_{F}{\left(\frac{\tilde{M}}{\Lambda}\right)}^{N_{F}-1}\right|^{2}|\tilde{E}^{+}\tilde{E}^{-}|^{2}\,+\,V_{\text{AMSB}}\,.$
(40)

Note the $(kN_{F})^{-1}$ factor in the second line, which comes from the
Kähler term $kN_{F}\tilde{M}^{\dagger}\tilde{M}$ for $\tilde{M}$, where $k$ is
an unknown $\mathcal{O}(1)$ normalization factor. The tree-level AMSB
contribution is given by (83), i.e.,

$\displaystyle
V_{\text{AMSB}}=m\Lambda\left[16+(N_{F}-1){\left(\frac{\tilde{M}}{\Lambda}\right)}^{N_{F}}\right]\,\tilde{E}^{+}\tilde{E}^{-}+\text{c.c.}$

This potential has a minimum at

$\displaystyle\tilde{M}\,=\,16^{\frac{1}{N_{F}}}\Lambda~{}~{}~{},~{}~{}~{}|\tilde{E}^{+}||\tilde{E}^{-}|=16^{\frac{2}{N_{F}}-1}\,km\Lambda$
$\displaystyle~{}~{}V_{\text{min}}\,=\,-16^{\frac{2}{N_{F}}}N_{F}\,km^{2}\Lambda^{2}\,.$
(42)

Since $\tilde{M}^{ij}=\tilde{M}\delta^{ij}$ in this minimum, the global
symmetry is broken to $SO(N_{f})$, and there are no ’t Hooft anomalies to
match. Because it is generated by a tree-level contribution from AMSB, it is
lower than the local minimum near the origin $U\approx 0$, and so it is the
global minimum of the theory. A similar phenomenon of AMSB leading to a global
minimum generated by a tree-level contribution and a local minimum generated
at loop-level was seen in Csaki:2021aqv . It is easy to see that there are no
minima with $M^{ij}\neq 0,\,M^{ij}\not{\propto}\delta^{ij}$.

Notably, the minimum (IV.1) involves the condensation of monopoles $E^{\pm}$,
which in turn leads to confinement tHooft:1981bkw ; Cardy:1981qy ; Cardy1982 ,
in the sense that non-trivial Wilson lines get an area law. Famously, monopole
condensation has also been seen in the breaking of $\mathcal{N}=2$ Seiberg-
Witten theory to $\mathcal{N}=1$ via a tree level superpotential for the
matter field Seiberg:1994rs . In Evans:1996hi ; Konishi:1996iz , it was shown
in a non-supersymmetric theory by introducing soft SUSY breaking on top of the
breaking to $\mathcal{N}=1$. Here, on the other hand, monopole condensation
and SUSY breaking emerge together as a result of AMSB. Furthermore, since the
global $SU(N_{F})$ symmetry is spontaneously broken to $SO(N_{F})$, this is a
demonstration of confinement with chiral symmetry breaking in a non-
supersymmetric setting.

We can also connect the chiral symmetry breaking observed here to the familiar
one due to fermion bilinears. To see this, note that the UV theory of quarks
has no superpotential and their F-components vanish. Therefore the only
contribution to the $F$-component of the meson superfield comes from fermion
bilinears:

$\displaystyle\langle\psi_{i}^{*}\psi_{j}^{*}\rangle=F^{*}_{M_{ij}}=16\Lambda^{2}M^{-1}_{ij}E^{+}E^{-}\propto\delta_{ij}km\Lambda^{2}\neq
0.$ (43)

In other words, our analysis demonstrates the condensation of fermion
bilinears in a non-supersymmetric theory, in addition to the monopole
condensate.

###  IV.2 $N_{F}<N_{c}-4$

At a generic point in the moduli space, the gauge group $SO(N_{c})$ is higgsed
down to $SO(N_{c}-N_{F})$ pure SYM, whose gaugino condensation induces an
Affleck–Dine–Seiberg (ADS) superpotential given by Affleck:1983vc

$\displaystyle W_{\text{ADS}}$
$\displaystyle=\frac{N_{c}-N_{F}-2}{2}\omega^{k}\left(\frac{16\Lambda^{3N_{c}-N_{F}-6}}{\text{det}M}\right)^{\frac{1}{N_{c}-N_{F}-2}}\,,$
(44)

where $\Lambda$ is the strong scale of the theory and $\omega=e^{2\pi
i/(N_{c}-N_{F}-2)}$ with $k=0,1,\dots,N_{c}-N_{F}-3$.

The Kähler potential of $M$ is singular at the origin and writing
$M^{ij}=\varphi^{2}\delta^{ij}$, we identify $\varphi$ as the canonical DOF.
Turning on AMSB stabilizes the runaway behavior of the superpotential via the
tree-level scalar potential

$V_{\text{AMSB}}=-m\Lambda^{3}\,\frac{3N_{c}-N_{F}-6}{2}\left(\frac{16\Lambda^{2N_{F}}}{\varphi^{2N_{F}}}\right)^{\frac{1}{N_{c}-N_{F}-2}}+\text{c.c.}\,,$
(45)

which together with the scalar potential derived from the superpotential (44)
gives a minimum

$\displaystyle\varphi$ $\displaystyle=$
$\displaystyle~{}~{}2^{\frac{2}{N_{c}-2}}\,{\left(f_{N_{F}}\frac{\Lambda}{m}\right)}^{\frac{N_{c}-N_{F}-2}{2(N_{c}-2)}}~{}\Lambda$
$\displaystyle V_{\text{min}}$ $\displaystyle=$
$\displaystyle-2^{\frac{4}{N_{c}-2}}\frac{N_{c}-2}{f^{2}_{N_{F}}}{\left(f_{N_{F}}\frac{\Lambda}{m}\right)}^{\frac{N_{c}-N_{F}-2}{N_{c}-2}}~{}m^{2}\Lambda^{2}\,.$

with $f_{N_{F}}=\frac{N_{c}+N_{F}-2}{3N_{c}-N_{F}-6}$. We see that the minimum
is at $\varphi\gg\Lambda$, which justifies a weakly coupled analysis in an
asymptotically free theory. Since $\tilde{M}^{ij}\propto\delta^{ij}$, in this
minimum the global symmetry is broken to $SO(N_{F})$. There are no minima with
$\tilde{M}^{ij}\neq 0,\,\tilde{M}^{ij}\not{\propto}\delta^{ij}$. Since the
$U(1)_{R}$ symmetry was explicitly broken by AMSB, there are no ’t Hooft
anomaly matching conditions to check in this scenario.

The non-trace components of $\tilde{M}^{ij}$ are split into massless
Nambu–Goldstone bosons (NGBs), massive fermions, and massive scalar partners
of the NGBs, where masses are $\mathcal{O}(m)$. The NGBs form the chiral
Lagrangian on the $SU(N_{F})/SO(N_{F})$ coset space. Once the massive fermions
are integrated out, the one-loop diagram DHoker:1984izu  produces the
Wess–Zumino–Witten (WZW) term Wess:1971yu ; Witten:1983tw  because
$\pi_{5}(SU(N_{F})/SO(N_{F}))={\mathbb{Z}}$ ($N_{F}\geq 3)$. For $N_{F}=2$,
there is no WZW term. To summarize, we establish that the $1<N_{F}<N_{c}-4$
case with AMSB has a global minimum in which the chiral symmetry is broken to
$SO(N_{F})$, similar to the $N_{F}=N_{c}-2$ case.

The case $N_{F}=1$ is an exception because the meson has only one component.
There is no exact flavor symmetry, no massless NGB, and the theory is gapped.

###  IV.3 $N_{F}=N_{c}-4$

In this case the gauge symmetry is higgsed on the moduli space to $SO(4)\simeq
SU(2)_{L}\times SU(2)_{R}$. The theory has two distinct branches corresponding
to the gaugino condensates in $SU(2)_{L,R}$ having the same, or opposite
signs. On the first branch with aligned condensates, the superpotential is of
the same form as (44), while on the second branch, it vanishes. The second
branch contains the point $M=0$, at which there is confinement without chiral
symmetry breaking. In Csaki:1997aw , it was shown that on this branch there is
also a VEV for the exotic baryon $S=W_{\alpha}W^{\alpha}Q^{N_{F}}$, which
breaks the discrete global symmetry $\mathbb{Z}_{2F}$ down to
$\mathbb{Z}_{F}$.

With AMSB, the theory on the first branch develops a minimum identical to
(IV.2), breaking the global symmetry down to $SO(N_{F})$. This is the global
minimum of the theory. As for the second branch, the identically zero
superpotential means we need to consider a more general version of Eq. (83)
that accounts for a non-canonical Kähler potential. We find for a general $W$
and $K$

$\displaystyle V_{\rm tree}=$
$\displaystyle\partial_{i}Wg^{ij^{*}}\partial_{j}^{*}W^{*}+m^{*}m\left(\partial_{i}Kg^{ij^{*}}\partial_{j}^{*}K-K\right)$
$\displaystyle+m\left(\partial_{i}Wg^{ij^{*}}\partial_{j}^{*}K-3W\right)+c.c.,$
(47)

where $g^{ij}$ is the inverse of the Kähler metric
$g_{ij}=\partial_{i}\bar{\partial}_{j}K$. With a vanishing superpotential,
interactions originate only from the Kähler potential. Higher order terms in
the Kähler potential will give rise to irrelevant interactions, and as the
theory is IR free, we expect the effects of AMSB to be highly suppressed by
the dynamical scale. To estimate these effects, consider the leading
corrections to the canonical Kähler potential for $M$,

$\displaystyle K~{}$ $\displaystyle=$
$\displaystyle~{}\mathrm{Tr}{M^{\dagger}M}+\frac{a}{\Lambda^{2}}\left(\mathrm{Tr}{M^{\dagger}M}\right)^{2}$
(48)
$\displaystyle+\frac{b}{\Lambda^{2}}\mathrm{Tr}{M^{\dagger}MM^{\dagger}M}\,,$

where $a$, $b$ are order one numbers. Note that cubic terms are forbidden
because $M$ is in the symmetric of $SU(N_{F})$.

In this case, only the second term in Eq. (IV.3) will contribute, and at
leading order gives,

$\displaystyle V$ $\displaystyle\sim\pm\frac{m^{2}}{\Lambda^{2}}|M|^{4}$ (49)

The potential in this theory arises exclusively from AMSB. Clearly the power
series expansion makes sense only up to $M\sim\Lambda$, and the maximum
contribution of the higher dimension terms to the potential is
$O(m^{2}\Lambda^{2})$. Note that the minimum we obtained in Eq. (IV.2) (and is
also relevant for the branch with nonzero superpotential) is parametrically
enhanced by $(\Lambda/m)^{2/(N_{c}-2)}$. Therefore the branch with $W=0$ does
not yield the global minimum, which instead arises from the branch with the
ADS-type superpotential.

###  IV.4 $N_{F}=N_{c}-3$

Here the gauge symmetry on the moduli space is higgsed down to $SO(3)$. As in
Intriligator:1995id , it is useful to first turn on the VEVs of $N_{F}-1$ of
the fundamentals, in which case the theory is higgsed to $SU(2)_{L}\times
SU(2)_{R}$. Then, the VEV of the last fundamental higgses $SU(2)_{L}\times
SU(2)_{R}$ to the diagonal $SO(3)$. The superpotential for this theory is
dynamically generated by a combination of gaugino condensation in the
unhiggsed $SO(3)$ and instantons in the broken $SU(2)_{L}\times
SU(2)_{R}/SO(3)$. The theory again has two branches: on the first, the gaugino
contribution is aligned with the instanton contributions, and the
superpotential is of the form (44). On the second branch, the contributions
cancel out, and the ADS-type superpotential vanishes. However, it was shown
via a mass deformation to the $N_{F}=N_{c}-4$ case that the second branch has
a dynamically generated superpotential:

$\displaystyle W_{\text{dyn}}\,$ $\displaystyle=$
$\displaystyle\,\frac{1}{2\mu}\,f(t)\,M^{ij}q_{i}q_{j}\,,$ (50)

where
$q_{i}=\left(W_{\alpha}W^{\alpha}Q^{N_{F}-1}\right)_{i}/\Lambda^{N_{F}+1}$ is
the exotic baryon of the theory. Here $\mu$ is an effective mass scale,
$t=\frac{1}{\Lambda^{2N_{F}+4}}\text{det}M\,M^{ij}q_{i}q_{j}$, and $f(t)$ is a
holomorphic function in the neighborhood of $t=0$, normalized so that
$f(0)=1$. We can canonically normalize the superpotential yielding

$\displaystyle W_{\text{dyn}}\,$ $\displaystyle=$
$\displaystyle\,\frac{1}{2}f(t)\,\tilde{M}^{ij}q_{i}q_{j}\,,$ (51)

where $\tilde{M}$ is the canonically normalized meson.

As usual, adding AMSB to the theory generates for the first branch the minimum
(IV.2) and breaks the global symmetry down to $SO(N_{F})$. This is again the
global minimum of the theory. On the second branch, the tree-level AMSB
contribution vanishes at $\mathcal{O}(t^{0})$, while the loop-level
contribution (84) generates a minimum in the neighborhood of
$\tilde{M}^{ij}=0$, with

$\displaystyle V\,$
$\displaystyle\approx\,-\left(\frac{m}{16\pi^{2}}\right)^{4}\,,$ (52)

Again the $\mathcal{O}(t)$ corrections give a sub-leading contribution. Around
the origin loop-level AMSB is again the dominant perturbation, which will
clearly not be the global minimum of the theory, since its (negative) height
is loop suppressed.

###  IV.5 $N_{c}-1\leq N_{F}<\frac{3}{2}(N_{c}-2)$

In this case the correct IR description of the theory is in terms of its IR
free Seiberg dual $SO(N_{F}-N_{c}+4)$ with $N_{F}$ fundamentals $q_{i}$ and
$\frac{1}{2}N_{F}(N_{F}+1)$ singlets $M^{ij}$ in the
$q_{i}(\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}})_{\frac{N_{c}-2}{N_{F}}}$
and
$M^{ij}(\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-0.4pt\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}})_{\frac{2(N_{F}-N_{c}+2)}{N_{F}}}$
representations of the global $SU(N_{F})\times U(1)_{R}$, respectively. First
we will focus on the case when $N_{F}\geq N_{c}+1$, leaving the
$N_{F}=N_{c}-1$ and $N_{F}=N_{c}$ special cases to the end of the section. For
$N_{F}\geq N_{c}+1$, the dual theory has a superpotential

$\displaystyle W_{\text{dual}}\,$ $\displaystyle=$
$\displaystyle\,\frac{1}{2\mu}\,M^{ij}q_{i}q_{j}\,.$ (53)

The scales of the original and dual theories are related by

$2^{8}\Lambda^{3(N_{c}-2)-N_{F}}\tilde{\Lambda}^{2N_{F}-3(N_{c}-2)}~{}=~{}(-1)^{N_{f}-N_{c}}\mu^{N_{F}}\,.$
(54)

For later convenience, we switch to canonically normalized fields $\tilde{M}$,

$\displaystyle W_{\text{dual}}\,$ $\displaystyle=$
$\displaystyle\,\frac{1}{2}\,\tilde{M}^{ij}q_{i}q_{j}\,,$ (55)

with $\tilde{M}=M/\mu$. When we turn on AMSB, the situation is similar to the
one encountered in Csaki:2021aqv  and to the $N_{F}=N_{c}-2$ case in the
present work. Near $M=0$ there is a local minimum generated by the loop level
AMSB contribution. The tree-level contribution vanishes as usual because the
superpotential (55) is marginal. Again we expect only a local minimum with
$V=-\mathcal{O}(\frac{m}{16\pi^{2}})^{4}$ along the direction,

$\displaystyle q\propto\left(\begin{array}[]{ccc}1&\cdots&0\\\
\vdots&\ddots&\vdots\\\ 0&\cdots&1\\\ \hline\cr 0&\cdots&0\\\
\vdots&\ddots&\vdots\\\ 0&\cdots&0\end{array}\right),$ $\displaystyle\quad
M\propto\left(\begin{array}[]{ccc|ccc}1&\cdots&0&0&\cdots&0\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\ 0&\cdots&1&0&\cdots&0\\\
\hline\cr 0&\cdots&0&0&\cdots&0\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
0&\cdots&0&0&\cdots&0\end{array}\right)\,,$ (68)

where $q$ is a $N_{F}\times(N_{F}-N_{c}+4)$ matrix.

At nonzero $M$ with $\text{rank}(M)=N_{F}$ on the moduli space for $N_{F}\geq
N_{c}+1$, the dual quarks $q_{i}$ are integrated out, and we are left with
pure $SO(N_{F}-N_{c}+4)$ SYM, with a scale

$\displaystyle\tilde{\Lambda}_{L}^{3N_{F}-3(N_{c}-2)}\,$ $\displaystyle=$
$\displaystyle\,\text{det}(\tilde{M})~{}\tilde{\Lambda}^{2N_{F}-3(N_{c}-2)}$
(69)

Gaugino condensation in the dual theory now generates a dynamical
superpotential

$\displaystyle
W_{\lambda\lambda}\,=\,2^{-\frac{N_{F}-(N_{c}-2)-4}{N_{F}-(N_{c}-2)}}\epsilon_{N_{F}-(N_{c}-2)}\,\,\tilde{\Lambda}^{3}_{L}$
(70)

With AMSB this superpotential leads to the tree-level SUSY breaking scalar
potential

$\displaystyle V_{\text{AMSB}}=$
$\displaystyle-2m\tilde{\Lambda}^{3}\,\frac{\frac{3}{2}(N_{c}-2)-N_{F}}{N_{F}-(N_{c}-2)}\times$
$\displaystyle\left(\frac{16^{\frac{1}{N_{F}}}\text{det}(\tilde{M})}{\tilde{\Lambda}^{N_{F}}}\right)^{\frac{1}{N_{F}-(N_{c}-2)}}+c.c.$

For $N_{F}>N_{c}$, this potential, together with the usual scalar potential
from the superpotential, is minimized at

$\displaystyle\tilde{M}^{ij}$ $\displaystyle\sim$ $\displaystyle
4^{\frac{N_{F}-(N_{c}-2)-2}{2(N_{c}-2)-N_{F}}}\left(f\frac{m}{\Lambda}\right)^{\frac{N_{F}-(N_{c}-2)}{2(N_{c}-2)-N_{F}}}\,\Lambda\,\delta^{ij}$
$\displaystyle V_{\text{min}}$ $\displaystyle\sim$
$\displaystyle-4^{\frac{N_{F}-4}{2((N_{c}-2)-N_{F}}}\frac{2(N_{c}-2)-N_{F}}{[N_{F}-(N_{c}-2)]^{2}}\times$
(72)
$\displaystyle\left(f\frac{m}{\Lambda}\right)^{\frac{2(N_{c}-2)}{2(N_{c}-2)-N_{F}}}\,\Lambda^{4}\,,$

where
$f=\frac{1}{N_{c}-2}\left[N_{F}-(N_{c}-2)\right]\left[\frac{3}{2}(N_{c}-2)-N_{F}\right]$.
Noting that $N_{c}-2<N_{F}\leq\frac{3}{2}(N_{c}-2)$ from (IV.5), this minimum
satisfies $\tilde{\Lambda}_{L}\ll\tilde{M}\ll\tilde{\Lambda}$, below the
Landau pole of the UV magnetic theory and above the scale of gaugino
condensation in the IR pure SYM. This justifies our weakly coupled analysis.
Again the global symmetry at the minimum is broken down to $SO(N_{F})$.

We now comment on the $N_{F}=N_{c}-1$ and $N_{F}=N_{c}$ cases. For
$N_{F}=N_{c}-1$, the dual gauge group is $SO(3)$. The superpotential of the
dual now includes a contribution from instantons:

$\displaystyle W_{\text{dual},N_{c}-1}\,$ $\displaystyle=$
$\displaystyle\,\frac{1}{2\mu}\,M^{ij}q_{i}q_{j}\,-\,\frac{\text{det}M}{64\tilde{\Lambda}^{2N_{c}-5}}\,.$
(73)

The extra contribution can be seen by deforming the dual for $N_{F}=N_{c}$ by
a mass term for the last flavor, which leads to instantons in the broken
$SU(2)_{L}\times SU(2)_{R}/SO(3)$ in the magnetic theory Csaki:1998vv . Here
the scale matching is given by

$2^{14}\,\left(\Lambda^{2N_{c}-5}\,\tilde{\Lambda}^{4-N_{c}}\right)^{2}~{}=~{}\left(\mu^{N_{c}-1}\right)^{2}\,.$
(74)

Interestingly, this relation looks like the square of the usual relation (54)
- for more details, see the original Intriligator:1995id . When $M$ has full
rank on the moduli space, the $q_{i}$ become massive and the gaugino
condensation in the IR $SO(3)$ SYM generates a superpotential

$\displaystyle W_{\lambda\lambda,\text{dual},N_{c}-1}\,$ $\displaystyle=$
$\displaystyle\,\epsilon_{2}\,\frac{\text{det}M}{64\Lambda^{2N_{c}-5}}\,.$
(75)

Note that this superpotential has the same magnitude as the instanton
contribution in (73), and so the theory again has two branches: one in which
the two contributions add, leading to an AMSB minimum of the form (IV.5), and
another with vanishing superpotential. On the second branch there is no
superpotential, and we can repeat the arguments concerning the second branch
in the case of $N_{F}=N_{c}-4$. Again, any minimum produced by AMSB and a non-
canonical Kähler is parametrically suppressed in comparison with the first
branch.

For $N_{F}=N_{c}$, the magnetic gauge group is $SU(2)_{L}\times SU(2)_{R}$.
There is no instanton contribution to the tree-level superpotential, but when
$M$ is given full rank, there are again two branches: one with aligned gaugino
condensates in the IR pure $SU(2)_{L}\times SU(2)_{R}$ SYM, and one with
opposite sign condensates and vanishing superpotential. As usual, with AMSB
the first branch leads to a global minimum of the form (IV.5), while any
minimum in the second branch is again subdominant.

Finally, we note that for $N_{c}<6$ we have $\frac{3}{2}(N_{c}-2)<N_{c}$, and
so the theories with $N_{F}=N_{c}=4,5$ have to be considered separately.
Indeed, for $N_{F}=N_{c}=4,5$ the magnetic theory is no longer IR free, but
rather has an IR fixed point.

###  IV.6 Monopole Condensation for $N_{F}<N_{c}-2$
via Mass Deformations

When discussing the theory with $N_{F}=N_{c}-2$, the non-supersymmetric vacuum
of the theory explicitly involved monopole condensation. In the next section,
this will enable us to determine the behavior of the loop operators in the
theory, and in particular establish confinement of non-trivial Wilson loops
for $Spin(N_{c})$. In this section we wish to make contact between the
$N_{F}=N_{c}-2$ case and the cases with fewer flavors, by treating the latter
as the $N_{F}=N_{c}-2$ deformed by a supersymmetric mass $\mu$, with
$\mu\gg\Lambda$.

We begin by considering the $N_{F}=N_{c}-2$ theory in the supersymmetric
limit, with just one mass term for the last flavor,

$\displaystyle W$ $\displaystyle=$
$\displaystyle\Lambda\left(\frac{\det\tilde{M}}{\Lambda^{N_{F}}}-16\right)\,\tilde{E}^{+}\,\tilde{E}^{-}+\frac{1}{2}\mu\Lambda\tilde{M}_{N_{F}N_{F}}$
(76)

The equation of motion for $\tilde{M}_{N_{F}N_{F}}$ gives

$\displaystyle\tilde{E}^{+}\tilde{E}^{-}\,$ $\displaystyle=$
$\displaystyle\,-\frac{1}{2}\frac{\mu\Lambda^{N_{F}}}{\det\tilde{M}^{\prime}}\,,$
(77)

where $\tilde{M}^{\prime}$ is the matrix of the remaining mesons. As already
demonstrated, tree level AMSB corrections to the ADS superpotential stabilize
the runaway, and the finite VEV of $M^{\prime}$ ensures that a non-vanishing
monopole condensate persists. In Fig. 1 we show this explicitly by studying
the minimum of the mass-deformed theory (76) in the presence of AMSB with
$m<\mu$. Since we are ultimately interested in the infinite $\mu$ limit, this
does not interfere with our extrapolation to the non-SUSY limit with large
$m$. As can be seen in the plot, the VEV of the first $N_{c}-3$ flavors
interpolates between the minimum (IV.1) for $\mu=0$, and the ADS+AMSB minimum
(IV.2) with $N_{f}=N_{c}-3$ and $\Lambda\rightarrow\Lambda_{N_{F}=N_{c}-3}$ in
the large $\mu$ limit. We can see that the monopole condensate persists in the
large $\mu$ limit.

To correctly reproduce the ADS+AMSB minimum, we had to interpolate the Kähler
potential between the neighborhood of $\text{det}\tilde{M}\sim\tilde{U}_{1}$,
where it is canonical in $\tilde{M}$, to large $\text{det}\tilde{M}$, where
the Kähler potential is canonical in $\varphi\sim\sqrt{\tilde{M}\Lambda}$.
More specifically, we used the following interpolating Kähler potential in the
numerical study:

$\displaystyle
K_{\text{interp.}}~{}=~{}\Lambda^{2}\,\sqrt{1+\frac{\tilde{M}\tilde{M}^{\dagger}}{\Lambda^{2}}}\,.$
(78)

For $\mu<m$, the UV theory has a runaway at $E^{+}E^{-}=0$ and
$M_{i}\rightarrow\infty$. This is a consequence of the mass term in (76) in
the presence of AMSB. In this regime we follow the local minimum which goes
over to the global minimum for $\mu>m$. Importantly, the condensation of
monopoles in the large $\mu$ limit is independent of this subtlety.

Figure 1: The supersymmetry breaking minimum for the theory with AMSB and
$N_{F}=N_{c}-2$, with the extra mass term $\frac{1}{2}\mu M^{N_{F}N_{F}}$. $E$
is the VEV of the monopoles $\tilde{E}^{\pm}$, while $M_{i}$ is the VEV of the
first $N_{c}-3$ flavors. The labeling on the different curves indicates
different values of $m/\Lambda$. For $\mu=0$, the curves are at the $N_{c}-2$
minimum (IV.1) . As $\mu$ grows, the VEV of $M_{i}$ initially decreases, but
then starts increasing as $\mu$ passes $m$. For large
$\mu/\Lambda\rightarrow\infty$, the vacuum of the theory goes over to Eq.
(IV.2) with $N_{F}=N_{c}-3$, while the monopole condensate persists. The
relation (77) is shown in the dashed line for $\mu=50\Lambda$. We chose
$N_{c}=13$ for this plot.

Similarly, we can give a mass term to any number of flavors in the
$N_{F}=N_{c}-2$ theory and show that monopole condensation persists. In Fig. 2
we present the case where all of the flavors get the same mass term, resulting
in a pure SYM theory with monopole condensation. In the small $\mu$ limit, the
minimum is given by the $N_{F}=N_{c}-2$ vacuum (IV.1). The $\mu\gg m$ case can
be fully understood in the supersymmetric limit – the monopoles get a VEV

$\displaystyle\tilde{E}^{+}\tilde{E}^{-}\,$ $\displaystyle=$
$\displaystyle\,-\frac{\mu\Lambda}{2}{\left(\frac{\Lambda}{\tilde{M}}\right)}^{N_{F}-1}\,,$
(79)

where $\tilde{M}$ is the common VEV of all of the flavors. This generates an
ADS superpotential for the $\tilde{M}$, which is balanced by the $\mu$ term
and leads to an overall minimum at

$\displaystyle\tilde{M}\,$ $\displaystyle=$
$\displaystyle\,16^{\frac{1}{N_{F}}}\Lambda\,,$
$\displaystyle\tilde{E}^{+}\tilde{E}^{-}\,$ $\displaystyle=$
$\displaystyle\,-2^{\frac{4}{N_{F}}-5}\,\mu\Lambda\,.$ (80)

Notably, the VEV of $\tilde{M}$ in this case is equal to the pure
$N_{F}=N_{c}-2$ case. This vacuum is the one depicted by the dashed line in
Fig. 2.

Figure 2: Location of the minimum in the theory with AMSB and $N_{F}=N_{c}-2$,
deformed by a universal mass term $\frac{1}{2}\mu M^{ii}$ for all flavors. $E$
is the VEV of the condensed monopoles, while $M_{i}$ is the common VEV for all
of the flavors. The different curves are labeled by the value of $m/\Lambda$.
The curves start at the $N_{c}-2$ minimum (IV.1) for $\mu=0$. As
$\mu/\Lambda\rightarrow\infty$, the theory goes over to pure SYM, while the
VEV of the monopoles $E$ is given by (IV.6), represented by the dashed line.
We have again chosen set $N_{c}=11$ for this plot.

Though we don’t show this explicitly, the same conclusion persists for any
number of flavors that are integrated out from the $N_{F}=N_{c}-2$ theory, and
so we explicitly see monopole condensation for the entire range $0\leq
N_{F}\leq N_{c}-2$.

###  IV.7 Loop Operators and Confinement

The observed monopole condensation for $N_{F}\leq N_{c}-2$ implies the
confinement of electric and dyonic loop operators and a perimeter law for
magnetic ’t Hooft loops. This is in agreement with arguments made in
Aharony:2013 . For $N_{F}<N_{c}-4$, the VEV of the meson field Higgses the
gauge group to pure YM with more than four colors, in which Wilson and dyonic
loops confine while ’t Hooft loops do not.

With $N_{c}-4$ flavors, the situation is a bit more subtle because the
unbroken $SO(4)\simeq SU(2)_{L}\times SU(2)_{R}$ gauge theory forms a gaugino
condensate for each $SU(2)$ factor. The branch where the gaugino condensates
are aligned is connected by a mass deformation to the $N_{c}-2$ case, and so
by the argument of the previous section it involves monopole condensation.
Consequently, magnetic loops acquire a perimeter law while dyonic loops
acquire an area law. The other branch with anti-aligned condensates is related
to the first one by a shift $\theta_{1}\rightarrow\theta_{1}+2\pi$
Aharony:2013 , and so by the Witten effect, on this branch it is the dyons
that condense. On this branch, dyonic loops acquires a perimeter law while
magnetic loops acquires an area law. In both cases the electric Wilson loop is
confined.

In the case of $N_{c}-3$ flavors, the unbroken gauge group is again special
because $SO(3)_{+}$ is in fact related to $SO(3)_{-}$ by a shift of the vacuum
angle $\theta\rightarrow\theta+2\pi$ Aharony:2013 . Such a shift permutes the
magnetic and dyonic loops, however the shift also exchanges the two
orientations of the gaugino condensate, thus exchanging the ADS and exotic
baryon branches. Again the global minimum exhibits the same loop operator
behavior as of all other values of $N_{F}\leq N_{c}-2$. We are free to
interpret this case as either monopoles condensing in $SO(3)_{-}$ or dyons
condensing in $SO(3)_{+}$.

For $N_{c}-1\leq N_{F}\leq\frac{3}{2}(N_{c}-2)$, the global minimum of the
dual theory finds the meson with non-vanishing VEV. Integrating out the dual
quarks leaves behind pure YM, for which we already demonstrated monopole
condensation in the previous section. Thus, the electric and dyonic loops of
the dual theory confine. Matching the behavior of the single non-trivial loop
in each of the dual theories, the correspondence of gauge groups is
Aharony:2013 ; Gaiotto:2014kfa ,

$\displaystyle Spin(N_{c})\longleftrightarrow SO(N_{F}-N_{c}+4)_{-}$
$\displaystyle SO(N_{c})_{+}\longleftrightarrow SO(N_{F}-N_{c}+4)_{+}$ (81)
$\displaystyle SO(N_{c})_{-}\longleftrightarrow Spin(N_{F}-N_{c}+4)$

Put concisely, the duality exchanges the electric and dyonic loops. The
duality (IV.7) therefore implies the same loop operator behavior as in the
$N_{F}\leq N_{c}-2$ case.

As first noted in Intriligator:1995id  and elaborated in Aharony:2013 , the
cases $N_{F}=N_{c}-1,\,N_{c}$ are special in the sense that there are extra
dual descriptions for the same theory. Let us first focus on the
$N_{F}=N_{c}-1$ case, and take the original theory to be $Spin(N_{c})$. In
that case the dual is $SO(3)_{-}$ with $N_{f}=N_{c}-1$ flavors and a
superpotential (73). But we know that this theory is equivalent to $SO(3)_{+}$
with $\theta$ shifted by $2\pi$, which results in an exchange of the ADS and
the exotic baryon branch. Dualizing back, we find another IR description of
the theory in terms of $SO(N_{c})_{+}$ and a superpotential $W_{\text{2nd
dual}}\,=\,-\,\frac{\text{det}M}{32\Lambda^{2N_{c}-5}}$. By the arguments
above, at the AMSB minimum the dyonic loop of the $SO(3)_{-}$ description and
the Wilson loop of the $Spin(N_{c})$ description confine, while the ’t Hooft
loop of the $SO(N_{c})_{+}$ description has a perimeter law. A similar logic
applies if we choose the original theory to be $SO(N_{c})_{-}$, in which case
the first dual is $Spin(3)$ and the second dual is again $SO(N_{c})_{-}$. Here
both the Wilson loop of $Spin(3)$ and the dyonic loop of $SO(N_{c})_{-}$
confine. For $N_{F}=N_{c}$ there are again two dual descriptions of the
original theory, albeit with a different superpotential for the second dual.
Since the loop behavior is identical to the $N_{F}=N_{c}-1$ case, we do not
repeat the analysis here.

In summary, theories with $N_{F}\leq\frac{3}{2}(N_{c}-2)$ and AMSB all
experience monopole condensation and the same behavior for their loop
operators. In particular, the non-trivial Wilson loop has an area law,
signaling electric confinement.

##  V Non-Confining Phases

For the supersymmetric theories with $\frac{3}{2}(N_{c}-2)<N_{F}\leq
3(N_{c}-2)$, there is an IR fixed point, while for $3(N_{c}-2)<N_{F}$ the
theories are IR free. Below we explore the behavior of the theory in these
ranges when we add AMSB.

###  V.1 $\frac{3}{2}(N_{c}-2)<N_{F}\leq 3(N_{c}-2)$

In this regime the theory has an IR fixed point. The IR dynamics is described
by either the electric or the magnetic theory. At the fixed point, the
electric and magnetic degrees of freedom pick up an anomalous dimension
$\gamma_{i}=3R_{i}-2$, where $R_{i}$ is the $R$-charge. Since there is no
additional anomaly-free global $U(1)$ symmetry, the $R$-symmetry in this case
is uniquely defined. The anomalous dimensions are then:

$\displaystyle\gamma_{Q}~{}$ $\displaystyle=$
$\displaystyle~{}\frac{N_{F}-3N_{c}+6}{N_{F}}\,,$ $\displaystyle\gamma_{q}~{}$
$\displaystyle=$ $\displaystyle~{}\frac{3N_{c}-2N_{F}-12}{N_{F}}\,,$
$\displaystyle\gamma_{M}~{}$ $\displaystyle=$
$\displaystyle~{}\frac{4N_{F}-6N_{c}+12}{N_{F}}\,.$ (82)

It’s easy to see that these anomalous dimensions are consistent with the
vanishing of the NSVZ beta function. With AMSB, the theory becomes
supersymmetric again at the IR fixed point, as in Murayama:2021xfj ;
Csaki:2021aqv . This is reminiscent of the IR restoration of supersymmetry
presented in Lanzagorta:1995ai ; Sundrum:2009gv . At intermediate scales, the
gaugino mass is power-suppressed, and approaches zero quickly in the IR
Murayama:2021xfj ; Csaki:2021aqv .

###  V.2 $3(N_{c}-2)<N_{F}$

Here the theory is in the free-electric phase, and so there is no
superpotential. With AMSB, the squarks get a negative soft mass from (84) and
become tachyonic. The theory then has no ground sate. The theory is then not
continuously connected to non-SUSY $SO(N_{c})$ with $3(N_{c}-2)<N_{F}$
fundamentals.

##  VI Conclusions

We have examined the low-energy phase structure of $SO(N_{c})$ gauge theories
with $N_{F}$ Weyl fermions in the vector representation, obtained by
perturbing the SUSY version of this theory via AMSB. We found that the
intricate phase structure of the SUSY theory does not survive the non-SUSY
perturbation. Instead the phase structure is very simple: for small number of
flavors $N_{F}<\frac{3}{2}(N_{c}-2)$ the theory is confining with chiral
symmetry breaking, while for $\frac{3}{2}(N_{c}-2)<N_{F}<3(N_{c}-2)$ it flows
to a (super)-conformal fixed point. This suggests that the free magnetic and
abelian Coulomb phases are rather special to supersymmetry, and are lifted as
soon as SUSY is broken. We have also paid special attention to the loop
operators of the theory that can be used as proper order parameters. In the
most interesting case of $Spin(N_{c})$ (in which case the electric Wilson loop
in the spinor can not be screened by the dynamical matter fields) we indeed
find an area law behavior corresponding to true confinement for all
$N_{F}<\frac{3}{2}(N_{c}-2)$, while the $SU(N_{F})$ global symmetry is broken
to $SO(N_{F})$. The dynamics leading to confinement is monopole condensation.
This is most clearly seen for the $N_{F}=N_{c}-2$ special case, where massless
monopoles (and dyons) indeed appear at special points in the moduli space.
With AMSB, we indeed find a non-vanishing monopole condensate in accordance
with the original conjecture by Mandelstam and ’t Hooft. By considering mass
deformations to the $N_{F}=N_{c}-2$ case, we have numerically verified that
the monopole condensate persists for all $N_{F}\leq N_{c}-2$. For the
$N_{c}-2<N_{F}<\frac{3}{2}(N_{c}-2)$, the AMSB vacuum is obtained when the
quarks of the dual theory are integrated out and the dual theory becomes pure
YM, for which we already established monopole condensation as the special case
$N_{F}=0$.

##  VII Anomaly Mediation

Anomaly mediation of supersymmetry breaking (AMSB) is parameterized by a
single number $m$ that explicitly breaks supersymmetry in two different ways.
One is the tree-level contribution based on the superpotential

$\displaystyle{V}_{\rm tree}$ $\displaystyle=m\left(\varphi_{i}\frac{\partial
W}{\partial\varphi_{i}}-3W\right)+c.c.$ (83)

The other is the loop-level supersymmetry breaking effects in tri-linear
couplings, scalar masses, and gaugino masses,

$\displaystyle A_{ijk}(\mu)$
$\displaystyle=-\frac{1}{2}(\gamma_{i}+\gamma_{j}+\gamma_{k})(\mu)\,m,$ (84)
$\displaystyle m_{i}^{2}(\mu)$
$\displaystyle=-\frac{1}{4}\dot{\gamma}_{i}(\mu)\,m^{2},$ (85) $\displaystyle
m_{\lambda}(\mu)$ $\displaystyle=-\frac{\beta(g^{2})}{2g^{2}}(\mu)\,m.$ (86)

Here, $\gamma_{i}=\mu\frac{d}{d\mu}\ln Z_{i}(\mu)$,
$\dot{\gamma}=\mu\frac{d}{d\mu}\gamma_{i}$, and
$\beta(g^{2})=\mu\frac{d}{d\mu}g^{2}$. When the gauge theory is asymptotically
free, $m_{i}^{2}>0$ which stabilizes the theory against run-away behaviors.
Note that Eqs. (83,84) also break the $U(1)_{R}$ symmetry explicitly and hence
we do not need to study its anomaly matching conditions.

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