MnLargeSymbols’164 MnLargeSymbols’171 ††institutetext: Department of Physics,
University of California San Diego,
9500 Gilman Drive, La Jolla CA 92093-0319, USA

# A New Solution to the Callan Rubakov Effect

T. Daniel Brennan<EMAIL_ADDRESS>
###### Abstract

In this paper we study the scattering of massive fermions off of smooth,
spherically symmetric monopoles in $4d$ $SU(2)$ gauge theory. We propose a
complete physical picture of the monopole-fermion interaction which
encompasses all angular momentum modes. We show that as an in-going fermion
scatters off a monopole, it excites trapped $W$-bosons in the monopole core by
a version of the Witten effect so that the monopole can accrue charge and
transform into a dyon at parametrically low energies. The imparted electric
charge is then protected from decay by an emergent $\mathbb{Z}_{N}$
generalized global symmetry, creating a stable dyon. At sufficiently low
energies, the scattered fermion can be trapped by the dyon’s electrostatic
potential, forming a bound state, which can decay into spherically symmetric
fermion modes subject to the preserved ${\mathbb{Z}}_{N}$ global symmetry. We
propose that monopole-fermion scattering can be described in this way without
needing to add “new” states to the Hilbert space, thereby eliminating a long
standing confusion in the Callan Rubakov effect.

##  1 Introduction

Smooth monopoles are non-perturbative excitations of non-abelian gauge
theories that play an important role in both particle and formal high energy
theory. Since the discovery that monopoles occur naturally in grand unified
theories (GUT theories) tHooft:1974kcl ; Polyakov:1974ek , it has been
realized that there are many ways in which monopoles can lead to measurable
effects. One such way is known as the Callan Rubakov effect Callan:1982au ;
Callan:1982ah ; Callan:1982ac ; Rubakov:1982fp ; Rubakov:1981tf ;
Rubakov:1988aq ; Preskill:1984gd ; Yamagishi:1982wp ; Grossman:1983yf ;
Yamagishi:1983ua ; Yamagishi:1984zu ; Panagopoulos:1984ws ; Besson:1981ch ;
Dixon:1984xh ; Isler:1987xn ; Isler:1986ii ; Sen:1984qe ; Sen:1984kf ;
Balachandran:1983sw ; Nair:1983ps ; Nair:1983hc ; Smith:2019jnh ;
Smith:2020nuf ; Nelson:1983bu ; Affleck:1993np ; Hortacsu:1984gx ;
Hortacsu:1984bx .

The Callan Rubakov effect describes the interaction between (massless)
fermions and a smooth, spherically symmetric monopole in $4d$ gauge theory.
The physically interesting features of this interaction come from the fact
that the fermions probe the UV physics inside the monopole core. In the
Standard Model this is realized by fermions interacting with GUT monopoles,
which leads to baryon number violation such as through proton decay.

Recently the Callan Rubakov effect has been revisited with renewed interest
due to a long standing confusion of the monopole-fermion interaction
Kitano:2021pwt ; Csaki:2021ozp ; Hamada:2022eiv ; Brennan:2021ewu ;
vanBeest:2023dbu ; Csaki:2020inw ; Csaki:2022tvb ; Csaki:2022qtz . The feature
of interest is that an incoming particle appears to produce a collection of
“fractional” out-going particles. This effect goes all the way back to the
original analysis by Callan Callan:1982au  and has lead to a myriad of
attempts to explain the physics behind this computation.

Much of the original analysis is based on the constraints imposed by the
global symmetries that are preserved by the scattering process. As with all
physical processes, the scattering preserves gauge “symmetry.” Additionally,
the scattering process preserves angular momentum, which is especially simple
for the case of spherically symmetric monopoles as we consider here. These
constraints are also often supplemented with the expectation that the
asymptotic states should be matched by the free field quantization since the
energy density in the electromagnetic field falls off like $1/r^{4}$ so that
fields are well approximated by free fields far away from the magnetic
monopole.

There are roughly three different classes of explanations of fractional states
that arise in the Callan Rubakov effect:

  * •

The fractional states should be interpreted as a non-trivial density matrix
Callan:1983tm . This proposal has the problem that it can lead to gauge
charges that are only conserved up to “classical probability.”

  * •

There are asymptotic states in the monopole Hilbert space that differ from
that of free field quantization. These asymptotic states have been attributed
to different physical effects such as “vacuum polarization/soft fermionic
radiation” Polchinski:1984uw ; Brennan:2021ewu , “pancakes” Kitano:2021pwt ,
attachment of topological lines vanBeest:2023dbu , or simply that the in- and
out-going Hilbert spaces are in different super selection sectors parametrized
by a $U(1)$-valued parameter Hamada:2022eiv . These proposals claim that the
asymptotic structure of QED differs from the free field quantization in the
presence of a monopole.111 A point that contributes to the confusion of the
analysis of the second option is that all symmetries acting on the fermions
that are not violated by an Adler-Bell-Jackiw (ABJ) anomaly Adler:1969gk ;
Bell:1969ts  (which can be perturbatively activated in a monopole background)
are conserved and that all asymptotic states have integer charges with respect
to these symmetries. This makes it particularly tempting to conclude that the
fractional fermion states do in fact exist in nature.

  * •

The “fractional fermion states” are purified by higher angular momentum states
since the total angular momentum of a multi-particle state can be constructed
from single-particle states with higher angular momentum Csaki:2021ozp .
However, it is unclear what interaction would give rise to this entangling
effect given that the decaying dyonic mode is spherically symmetric.

###  1.1 Outline and Summary of Results

In this paper we will present an alternative proposal to describe the physics
of the Callan Rubakov effect. Here we will argue that there is an additional
physical mechanism that prevents the creation of asymptotic states with
fractional fermion number. This additional physical mechanism both preserves
all global symmetries and only requires asymptotic states that can be
described in terms of those from free field quantization.

To perform our analysis, we will take a slightly different approach and
consider _massive fermions_ interacting with spherically symmetric monopoles.
The reason is two-fold. First, massless charged fields are subtle and much of
our standard intuition from particle physics does not apply. For example,
classical massless charged particles do not have bound states in a background
electric charge. Perhaps more importantly, massless QED is not IR
finite222Here we differentiate between massive QED which is IR finite in the
sense that Fadeev-Kulish asymptotic scattering states Kulish:1970ut  can be
well defined for massive QED but are singular for massless QED Prabhu:2022zcr
.  and so it is possible that the confusion with understanding the Callan
Rubakov effect may come from unregulated IR divergences. The second reason we
consider massive fermions is that they are relevant for particle
phenomenology.

In this paper we will restrict our analysis to $SU(2)$ QCD with $N_{f}$
fundamental Weyl fermions with mass $m_{\psi}$. If we couple this theory to an
adjoint scalar field with potential, then RG flows along which the scalar
condenses above the confinement scale will lead to a $U(1)$ gauge theory with
charged fermions at energies below the non-abelian gauge boson mass (i.e.
$W$-boson mass): $E\ll m_{W}$. In this theory, there is a single spherically
symmetric, smooth monopole when we spontaneously break $SU(2)\to U(1)$ which
flows to the minimal monopole line in the IR $U(1)$ gauge theory. This
monopole contains a trapped $W$-boson inside of its core which in the low
energy effective theory gives rise to a charged $U(1)$-valued scalar field
(i.e. the phase mode of the trapped $W$-boson) on the monopole world volume
called the “dyon degree of freedom.”

We propose that the scattering of charged fermions off of the monopole occurs
as follows. An incident charged fermion will scatter off of the the monopole
by interacting with the trapped $W$-boson via the UV tree level interaction.
This excites the dyon degree of freedom and deposits charge on the monopole,
turning it into a dyon of charge 2. Naively, the dyon degree of freedom is
heavy and requires energy of the order of the GUT scale – $E_{GUT}\sim
m_{W}\gg m_{\psi}$ – to be excited. For this reason, it has often been assumed
that this charge must be radiated away.

However, our proposal is that the fermion can deposit charge on the monopole
at energies that are parametrically lower than $m_{W}$. Physically, this
occurs by the Witten effect. In the $1d$ quantum theory governing the dyon
degree of freedom, which can be modeled by a charged particle of mass $m_{W}$
on a ring, the scattering induces a non-trivial $\theta$-angle. The induced
$\theta$-angle leads to a shift of the energy levels so that the lowest energy
state sources a bulk electric field.

This reflects the fact that fermion scattering induces a shift in the $4d$
$\theta$-angle on the monopole. The shift in the $\theta$-angle is induced by
the phase mode of the fermions which acts as an axion field Peccei:1977ur ;
Peccei:1977hh ; Kim:1979if ; Shifman:1979if  that couples to the gauge field
according to the transformation properties of the associated fermion. In
fermion scattering, the axion winds, shifting the fermion vacuum which induces
a vacuum charge density of radius $R_{c}\sim 1/m_{\psi}$ that contributes
$E\sim m_{\psi}$ to the mass of the dyon instead of $E\sim m_{W}$. This is the
reason why the monopole can be transformed into a dyon by parametrically low
energies.

The electric charge that is deposited on the monopole is in fact protected
from decay by an emergent ${\mathbb{Z}}_{N_{f}}$ generalized global symmetry
which we will denote as ${\mathbb{Z}}_{N_{f}}^{(0,1)}$. This symmetry reflects
the fact that there is no way of distributing a non-zero charge amongst the
symmetrically coupled $N_{f}$ fermions unless the charge is a multiple of
$N_{f}$ without breaking the $SU(N_{f})$ global symmetry. This symmetry
prevents the creation of asymptotic states that have fractional fermion
number.

Since each scattering deposits 2-units of electric charge, our proposal leads
to charge accumulation on the monopole when we scatter less than
$\frac{{N_{f}}}{2}$ fermions off of the monopole. However, when we scatter
$\frac{{N_{f}}}{2}$ fermions off of the monopole, the
${\mathbb{Z}}_{N_{f}}^{(0,1)}$ symmetry does not protect the dyon from decay
and the charge is indeed radiated away. This reproduces the standard Callan
Rubakov effect for asymptotic states with integer fermion number. This can
also be seen by integrating out the dyon degree of freedom as in Fan:2021ntg
which leads to an ‘t Hooft vertex that is localized on the monopole world-
volume. This interaction relates $\frac{{N_{f}}}{2}$ in-coming fermion modes
to $\frac{{N_{f}}}{2}$ out-going fermion modes which can be activated by
scattering $\frac{{N_{f}}}{2}$-fermions. This argument is in the spirit of the
original analysis of Rubakov Rubakov:1982fp .

If the incident fermion does not have sufficient energy, the scattered fermion
can be trapped by the electrostatic potential of the dyon, forming a bound
state of radius $R_{0}$ which scales parametrically with the fermion mass as
$R_{0}\sim 1/{m_{\psi}}$ – similar to the hydrogen atom Zhang:1988ab ;
Ravendranadhan:1989vp ; Tang:1982fc .

We would like to make a couple important comments about our proposal:

  * •

Even though the higher angular momentum modes have no (plane wave)
normalizable zero energy states Brennan:2021ucy , the $j>0$ modes do not have
an energy gap. This means that indeed, there are higher angular momentum modes
in the low energy effective theory that may couple to the dyon degree of
freedom.

Furthermore, the higher angular momentum modes must couple to the dyon degree
of freedom. This can be seen from the fact that the independent modes in the
non-abelian monopole background have a net asymptotic charge flux. Since the
mode expansion of the fermions in the smooth monopole background are
constructed from positively and negatively charged IR modes, this means that
the two IR modes are related at the monopole core by the charged dyon degree
of freedom so that gauge symmetry is preserved.

The fact that the higher angular momentum modes couple to the dyon degree of
freedom should be anticipated from the results of Brennan:2021ewu  which
studies the Callan Rubakov proposal in general $SU(N)$ gauge theories with
fermions in generic representations and higher charge spherically symmetric
monopoles. When considering more general spherically symmetric monopoles and
fermion representations, the Callan Rubakov effect crucially relies on the
existence of higher orbital angular momentum modes (although not higher total
angular momentum modes) which experience an orbital angular momentum barrier.
It is usually argued that this angular momentum barrier prevents the modes
from reaching the monopole core, however the results of Brennan:2021ewu  show
that this would lead to a violation of gauge symmetry in the Callan Rubakov
effect.

The fact that the higher angular momentum modes should interact with the
monopole core can also be seen classically from the simple fact that a charged
point particle can reach the monopole core if it approaches along a magnetic
field line. In field theory, the analogous statement is that a localized wave
packet – which will generally be constructed out of a super position of higher
angular momentum modes – will similarly reach the monopole core if it
approaches the monopole along a magnetic field line.

  * •

We would like to further emphasize that our proposal preserves all conserved
symmetries and completely matches with the proposal of Callan and Rubakov for
scattering processes with asymptotic states that have integer fermion number.

In the case of the $SU(5)\to\frac{SU(3)\times SU(2)\times
U(1)}{{\mathbb{Z}}_{6}}$ GUT theory with a _single generation_ , the minimal
monopole can trap a single anti-proton:

$\displaystyle\begin{split}\bar{p}+M~{}\longrightarrow(D+2d+e^{c})~{},\end{split}$
(1)

at low enough scattering energies, but does not trap pairs of anti-protons:

$\displaystyle\begin{split}2\bar{p}+M~{}\longrightarrow~{}3e^{-}+p+M~{}.\end{split}$
(2)

These processes both lead to baryon and lepton number violation. Note that in
the first process, the dyon carries $U(1)_{B-L}$ charge due to the fact that
the generator of the $U(1)_{B-L}$ in the $SU(5)$ theory is given by

$\displaystyle\begin{split}Q_{B-L}=Q_{Y}+Q_{D}\end{split}$ (3)

where $Q_{Y}$ is the generator of hypercharge and $Q_{D}$ acts on the $SU(5)$
fermion multiplets with charges $Q_{D}[\mathbf{\bar{5}}]=-3$ and
$Q_{D}[\mathbf{10}]=1$. Since the monopole does not carry charge with respect
to $Q_{D}$, but does with respect to $Q_{Y}$, the dyon can carry charge with
respect to $Q_{B-L}$. When enough charge is accumulated, the $U(1)_{B-L}$
charge is radiated away, and thus $U(1)_{B-L}$ is conserved as in the original
Callan Rubakov effect.

  * •

For technical reasons, we will restrict to the limit where
$N_{f}<<\frac{1}{g^{2}}$ where $g$ is the gauge coupling which is
parametrically weak. Here, our assumptions allow us to ignore loop corrections
and prevents the screening of IR electric charges as discussed in
Aharony:2022ntz . Although we will comment on these effects, we will leave a
detailed analysis of these effects for future investigation.

The outline of the paper is as follows. In Section 2, we will begin by giving
some background on the spherically symmetric monopoles in $4d$ $SU(2)$ QCD
theory with $N_{f}$ Weyl fermions transforming in the fundamental
representation of the gauge group. There we will describe the mode expansion
of the fermion fields in the monopole background and demonstrate the lack of
an energy gap for the higher angular momentum modes. Then in Section 3 we will
study the symmetries of the theory in both the UV and IR and argue for our
proposal from analyzing the contraints from global symmetries. Then in Section
4 we will study the scattering of s-wave fermions off of the monopole which
can be described by an effective $2d$ theory. We will derive our new mechanism
and demonstrate how monopoles can capture fermions. Next, in Section 5 we will
generalize the $2d$ s-wave analysis to all angular momentum modes and describe
general monopole-fermion scattering. Finally, we conclude by discussing the
general application to Standard Model phenomenology and other future
directions.

##  2 Monopoles and Fermions in $SU(2)$ Yang-Mills

The setting for our paper is $4d$ $SU(2)$ gauge theory coupled to an adjoint
scalar field $\Phi$ and $N_{f}$ Weyl fermions $\psi$ that transform in the
fundamental representation. Here we include a potential for $\Phi$ so that it
condenses above the confinement scale. This spontaneously breaks the gauge
field $SU(2)\to U(1)$ and the theory flows to QED.

When the gauge symmetry is spontaneously broken $SU(2)\to U(1)$, the theory
has smooth dynamical monopoles of mass $m_{\rm mono}\sim m_{W}/g^{2}$ where
$m_{W}$ is the mass of the $W$-boson (i.e. massive gauge boson) and $g$ is the
gauge coupling. Here we will be focusing only on spherically symmetric
monopoles.

For $SU(2)$ gauge theory, there is a unique spherically symmetric monopole.
Spherically symmetric monopoles are those which are rotationally invariant up
to a gauge transformation. In other words, spherically symmetric monopoles are
invariant under the modified generator of angular momentum:

$\displaystyle\begin{split}\vec{J}=\vec{L}+\vec{S}+\vec{T}~{},\end{split}$ (4)

where $\vec{T}$ are the generators of an $SU(2)\subset G_{\rm gauge}$ and
${\vec{L}},{\vec{S}}$ are the standard generators of orbital- and spin-angular
momentum. The spherically symmetric monopole is a Yang-Mills-Higgs field
configuration, which in the Parasad-Sommerfield limit333This is the limit in
which the Higgs potential is set to zero while fixing the Higgs vev.
satisfies

$\displaystyle\begin{split}B_{\mu}=D_{\mu}\Phi~{}.\end{split}$ (5)

The explicit solution of this equation can be written as

$\displaystyle\begin{split}&A_{\rm mono}=T_{3}A_{\rm
Dirac}+\frac{i}{2}T_{+}\,M(r)\,e^{-i\phi}(d\theta-i\sin\theta
d\phi)-\frac{i}{2}T_{-}\,M(r)\,e^{i\phi}(d\theta+i\sin\theta d\phi)~{},\\\
&\Phi=T_{3}h(r)~{},\end{split}$ (6)

where

$\displaystyle\begin{split}M(r)=m_{W}r\,\,\textrm{csch}\,(m_{W}r)\quad,\quad
h(r)=m_{W}\coth(m_{W}r)-\frac{1}{r}~{},\end{split}$ (7)

and $A_{\rm Dirac}$ is the Dirac monopole connection.444Explicitly $A_{\rm
Dirac}=\frac{1}{2}(\sigma-\cos(\theta))d\phi$ where $\sigma=\pm 1$ on the
northern/southern hemisphere. For simplicity in this paper we will work in the
northern hemisphere coordinate patch. For simplicity, we will generally
restrict ourself to the Prasad-Sommerfield limit because 1.) the fermion-BPS
monopole interaction captures the physics of the general case and 2.) we have
explicit forms for the monopole connection in this limit.

Here we can see from the explicit form of $h(r)$ that although the Higgs vev
gives a mass to the $W_{\pm}$-bosons, the mass goes to zero inside the core of
the monopole. This localizes a charged degree of freedom to the monopole core
which is called the “dyon degree of freedom” that is described by a $1d$
charged periodic scalar field $\varphi(t)$. In fact, as discussed in
Brennan:2021ewu , all other non-abelian degrees of freedom of the gauge field
(that is all other degrees of freedom except the IR photon) acquire a mass of
order $m_{W}$ due to a combination of the profile of $M(r)$ and $h(r)$ in (7).

Explicitly, the dyon degree of freedom can be expressed in terms of the
(dynamical) phase of the monopole field configuration:

$\displaystyle\begin{split}A_{\rm mono}=T_{3}A_{\rm
Dirac}\pm\frac{i}{2}T_{\pm}\,M(r)\,e^{\pm i(\varphi-\phi)}(d\theta\mp
i\sin\theta d\phi)~{},\end{split}$ (8)

where the signs are all correlated and summed over. Because of the form of the
connection, we can interpret $\varphi(t)$ as the dynamical phase of a trapped
$W$-boson and the profile of $M(r)$ exponentially localizes the dyon degree of
freedom to the monopole core. Given this parametrization of $\varphi(t)$, it
is clear that that it transforms as a Goldstone mode under the bulk $U(1)$
gauge symmetry due to the fact that gauge transformations $e^{i\alpha T_{3}}$
shift $\varphi\to\varphi+2\alpha$.

Note that we could also infer the existence of the dyon degree of freedom from
consistency of the IR theory. The reason is that in the IR, the monopole line
is a defect operator which is defined by excising an infinitesimal 3-ball and
imposing boundary conditions for the magnetic field there. However, one cannot
impose simultaneous boundary conditions on both the electric and magnetic
field because they are not commuting operators. Therefore, when we impose
boundary conditions that fix the magnetic field, we must also allow for a
electric degree of freedom on the boundary; i.e. the dyon degree of freedom.
Said differently, magnetic boundary conditions are Dirichlet on the gauge
field $A$ which can only be made gauge invariant by introducing a charged
$U(1)$ scalar degree of freedom on the boundary.

If we plug in the background gauge configuration (8) into the action, the
dynamics of the dyon degree of freedom will be effectively described a $1d$
quantum mechanics of a charged particle on a ring:

$\displaystyle\begin{split}S_{1d}=\int\frac{1}{2m_{W}}(\dot{\varphi}-2a_{0})^{2}dt~{},\end{split}$
(9)

where $\varphi\sim\varphi+2\pi$ and $a_{0}=A_{0}\big{|}_{r=0}$. The associated
Hamiltonian is given in terms of the momentum operator
$p=\frac{1}{m_{W}}(\dot{\varphi}-2a_{0})$:

$\displaystyle\begin{split}H=\frac{m}{2}p^{2}+2a_{0}p~{}.\end{split}$ (10)

In the presence of a trivial electric field, the Hilbert space can be graded
by momentum eigen-states

$\displaystyle\begin{split}{\cal H}={\rm
span}\Big{\\{}|n\rangle~{},~{}n\in{\mathbb{Z}}~{}\Big{|}~{}\hat{p}|n\rangle=n|n\rangle\Big{\\}}~{},\end{split}$
(11)

which are quantized due to the periodicity of $\varphi$. The energy of these
states are given by

$\displaystyle\begin{split}\hat{H}|n\rangle=\frac{m_{W}}{2}n^{2}|n\rangle~{}.\end{split}$
(12)

The coupling between $a_{0}$ and $p$ indicates that the momentum eigen-states
source a bulk electric field: the state $|n\rangle$ sources a bulk electric
field corresponding to a point particle of charge $2n$.

It is important for our discussion later that this theory admits a topological
$\theta$-term:

$\displaystyle\begin{split}L=\frac{1}{2m_{W}}(\dot{\varphi}-2a_{0})^{2}+\frac{\theta}{2\pi}(\dot{\varphi}-2a_{0})~{}.\end{split}$
(13)

This deformation leads to a shift in the momentum operator
$\hat{p}=\frac{1}{m_{W}}(\dot{\varphi}-2a_{0})+\frac{\theta}{2\pi}$, without
changing the quantization of the states. However, the topological
$\theta$-term does shift the energy levels of the system from before

$\displaystyle\begin{split}E_{n}=\frac{m_{W}}{2}\left(n-\frac{\theta}{2\pi}\right)^{2}~{},\end{split}$
(14)

so that shifting $\theta\to\theta+2\pi$ also shifts the energy levels
$E_{n}\mapsto E_{n+1}$. Thus, if we have an initial $\theta=0$, then shifting
$\theta\to\theta+2\pi$ changes the ground state from the $|0\rangle$ state to
the $|1\rangle$ state which sources a bulk electric charge.

###  2.1 Fermions and the Dirac Monopole

Now we wish to study the behavior of the fermions in the presence of the
spherically symmetric $SU(2)$ monopole. In the IR, this is governed by the
interaction of the fermions with the Dirac monopole plus possible boundary
conditions/interactions with the dyon degree of freedom. To determine this
boundary interaction, we need the fact that the photon and $\varphi(t)$ are
the only low energy degrees of freedom of the full non-abelian gauge field.
This means that at low energies, the background monopole connection is rigid
(up to photon and $\varphi(t)$ fluctuations) and therefore in the IR, we
should expand the low energy fermions in terms of the eigen-spinors of the
Dirac operator with the fixed monopole background.555Said differently, there
is a 1-form magnetic symmetry that protects the monopole connection from being
screened. It is therefore appropriate to expand the fermion fields in terms of
eigen-spinors of the Dirac operator in this fixed background.  We can then
compute the interaction between the low energy fermions and the dyon degree of
freedom by matching the mode expansion of the fermions in the Dirac and non-
abelian monopole backgrounds.

In order to facilitate our analysis, we will first consider the IR fermion
modes. In the IR, $SU(2)$ QCD flows to QED with $N_{f}$ Weyl fermions each of
charge $+1$ and $-1$. Additionally, at long distances the smooth, spherically
symmetric non-abelian monopole flows to the minimal singular monopole which
has the connection

$\displaystyle\begin{split}A_{\rm
Dirac}=\frac{1}{2}(\sigma-\cos(\theta))d\phi~{},\end{split}$ (15)

and $\sigma=\pm 1$ on the northern/southern hemisphere. As before, we will
restrict our analysis to the northern hemisphere coordinate patch – similar
results will hold in the southern hemisphere which is related by a gauge
transformation $g=e^{-i\phi}$ along an equatorial patch.

It is now our goal to compute the spectrum of the Dirac operator in the
singular monopole background above. This monopole background is also
spherically symmetric with respect to the modified angular momentum generators
Wu:1976ge ; Wu:1977qk ; Brennan:2021ucy

$\displaystyle\begin{split}\vec{J}=\vec{L}+\vec{S}+\frac{1}{2}\hat{r}~{}.\end{split}$
(16)

See Brennan:2021ucy  to see how these angular momentum generators are related
to those of the full non-abelian monopole. Since this monopole is spherically
symmetric, we can decompose the eigenfunctions of the Dirac operator in terms
of angular momentum representations. We will label these representations by
quantum numbers $j,m$:

$\displaystyle\begin{split}J^{2}|j,m\rangle=j(j+1)|j,m\rangle\quad,\quad
J_{3}|j,m\rangle=m|j,m\rangle~{}.\end{split}$ (17)

Let us first find the spectrum of the massless Dirac operator. Here we will
consider a single fermion $\psi^{i}$ where $i=\pm$ denotes the gauge charge
$\pm 1$. Following from the analysis in Moore:2014jfa ; Brennan:2021ucy , the
Dirac equation with background gauge field (LABEL:Connection) can be
simplified by performing a frame rotation by

$\displaystyle\begin{split}\psi^{i}=U\hat{\psi}^{i}\quad,\quad
U=e^{\frac{i\phi}{2}\sigma^{0}\bar{\sigma}^{3}}e^{\frac{i\theta}{2}\sigma^{0}\bar{\sigma}^{2}}~{}.\end{split}$
(18)

The resulting Dirac equation is given by

$\displaystyle\begin{split}\Bigg{[}&-iE+\sigma^{3}\left(\partial_{r}+\frac{1}{r}\right)+\frac{{\cal
K}}{r}\Bigg{]}\hat{\psi}^{i}=0~{},\end{split}$ (19)

where

$\displaystyle\begin{split}{\cal
K}&=\sigma^{1}\left(\partial_{\theta}+\frac{1}{2}{\rm
cot}(\theta)\right)+\frac{\sigma^{2}}{\sin\theta}\Big{(}\left(\partial_{\phi}+i\frac{q}{2}\right)-i\cos(\theta)\frac{q}{2}\Big{)}~{},\end{split}$
(20)

and $q=\pm 1$ is the electric charge of the fermion. We can then expand
$\psi,\tilde{\psi}$ into angular momentum eigenstates as

$\displaystyle\begin{split}\hat{\psi}^{+}=\left(\begin{array}[]{c}f_{+}(r)\,e^{i\phi/2}{\cal
D}^{(j)}_{m,1}(\theta,\phi)\\\ g_{+}(r)\,e^{-i\phi/2}{\cal
D}^{(j)}_{m,0}(\theta,\phi)\end{array}\right)\quad,\quad\hat{\psi}^{-}=\left(\begin{array}[]{c}f_{-}(r)\,e^{i\phi/2}{\cal
D}^{(j)}_{m,0}(\theta,\phi)\\\ g_{-}(r)\,e^{-i\phi/2}{\cal
D}^{(j)}_{m,-1}(\theta,\phi)\end{array}\right)~{},\end{split}$ (21)

where ${\cal D}^{(j)}_{m,q}(\theta,\phi):={\cal
D}^{(j)}_{m,q}(\phi,\theta,\phi)$ are Wigner $D$-functions. Note that when
$j=0$, ${\cal D}^{(0)}_{m,\pm 1}$ is singular so $f_{+},g_{-}:=0$ and the
Dirac equation further simplifies.

Now, the Dirac equation reduces to a matrix ODE

$\displaystyle\begin{split}\left(\sigma^{3}\left(\partial_{r}+\frac{1}{r}\right)+\frac{1}{r}\left(\begin{array}[]{cc}0&-\lambda\\\
\lambda&0\end{array}\right)-iE\right)\Psi=0\end{split}$ (22)

for the spinors

$\displaystyle\begin{split}\Psi^{\pm}=\left(\begin{array}[]{c}f_{\pm}(r)\\\
g_{\pm}(r)\end{array}\right)\end{split}$ (23)

where $\lambda=\sqrt{j(j+1)}$. For $E=0$, the solutions take the simple form

$\displaystyle\begin{split}\Psi^{(j)}_{q}=a_{1}\frac{(m_{W}r)^{\lambda}}{r}\left(\begin{array}[]{c}1\\\
1\end{array}\right)+a_{2}\frac{(m_{W}r)^{-\lambda}}{r}\left(\begin{array}[]{c}1\\\
-1\end{array}\right)\quad,\quad\Psi^{(j=0)}_{q}=\frac{1}{r}\begin{cases}\left(\begin{array}[]{c}0\\\
1\end{array}\right)&q=1\\\ \left(\begin{array}[]{c}1\\\
0\end{array}\right)&q=-1\end{cases}\end{split}$ (24)

In order to assess the normalizability of these solutions, we need to use the
fact that we are in an IR effective theory with an effective boundary at some
finite boundary near $r\to 0$. Thus, the normalizability will be determined by
the asymptotic behavior of the solutions at $r\to\infty$ in addition to the
boundary conditions at $r\to 0$. For our current discussion, we will only
impose normalizabilty for $r\to\infty$ and leave the discussion of the
conditions at $r\to 0$ for the next section. 666Here the correct notion of the
normalizable is plane-wave normalizable. Thus, for $E=0$, we will discard the
$a_{1}$ solutions for $j>0$ since they are non-normalizable as $r\to\infty$.

For $E>0$, the solutions are formally given by solving the matrix ODE:

$\displaystyle\begin{split}\Psi=P\,e^{-\int{\cal
D}_{E}(r^{\prime})\,dr^{\prime}}\Psi_{0,{\rm IR}}\quad,\quad{\cal
D}_{E}(r)=\left(\begin{array}[]{cc}-iE+\frac{1}{r}&\frac{\lambda}{r}\\\
\frac{\lambda}{r}&iE+\frac{1}{r}\end{array}\right)~{}.\end{split}$ (25)

To see which of these solutions are normalizable, we again need to consider
the $r\to\infty$ limit. At $r\to\infty$, the Dirac equation can be written as

$\displaystyle\begin{split}\left(\sigma^{3}\left(\partial_{r}+\frac{1}{r}\right)-iE\right)\Psi=0~{},\end{split}$
(26)

which has plane wave solutions:

$\displaystyle\begin{split}\Psi\underset{r\to\infty}{\longrightarrow}\frac{e^{iE(t+\sigma^{3}r)}}{r}\Psi_{0,{\rm
IR}}~{}.\end{split}$ (27)

where $\Psi_{0,{\rm IR}}$ is a constant spinor. This means that all solutions
are asymptotically plane-wave normalizable for $E>0$.

This means that in addition to the $E=0$ modes, the normalizable, non-negative
energy solutions of the Dirac equation are those for which $E>0$ and
$\Psi_{0,{\rm IR}}$ is given by

$\displaystyle\begin{split}\Psi_{0,{\rm
IR}}^{(j>0)}=c_{1}\Psi_{0,q}^{(1)}+c_{2}\Psi_{0,q}^{(2)}=c_{1}\left(\begin{array}[]{c}1\\\
0\end{array}\right)+c_{2}\left(\begin{array}[]{c}0\\\
1\end{array}\right)\quad,\quad\Psi_{0,q}^{(j=0)}=\begin{cases}\left(\begin{array}[]{c}0\\\
1\end{array}\right)&q=1\\\ \left(\begin{array}[]{c}1\\\
0\end{array}\right)&q=-1\end{cases}\end{split}$ (28)

which have the asymptotic behavior:

$\displaystyle\begin{split}\Psi_{0,{\rm
IR}}^{(j>0)}=\frac{1}{r}\left(\begin{array}[]{c}c_{1}\,e^{iE(t+r)}\\\
c_{2}\,e^{iE(t-r)}\end{array}\right)\quad,\quad\Psi_{0,{\rm
IR}}^{(j=0)}=\begin{cases}\frac{1}{r}\left(\begin{array}[]{c}0\\\
e^{iE(t+r)}\end{array}\right)&q=1\\\
\frac{1}{r}\left(\begin{array}[]{c}e^{iE(t-r)}\\\
0\end{array}\right)&q=-1\end{cases}\end{split}$ (29)

In particular, this demonstrates that there is no energy gap between the $j=0$
(i.e. s-wave) modes and the higher angular momentum modes. Notice that the
$j=0$ modes are polarized so that the $q=1$ mode is in-going with positive
helicity and the $q=-1$ mode is out-going with negative helicity. This implies
that the boundary condition for the $j=0$ sector on the monopole will
necessarily relate the $q=1$ modes to the $q=-1$ modes.

Additionally, note that we can compute the net electric current flux from
these asymptotic states and we find that for a single Weyl fermion of charge
$q$:

$\displaystyle\begin{split}\oint_{S^{2}_{\infty}}j^{\mu}\,\hat{r}_{\mu}d^{2}\Omega=\oint_{S^{2}_{\infty}}q\,\bar{\psi}\bar{\sigma}^{\mu}\psi~{}\hat{r}_{\mu}\,d^{2}\Omega=c^{\dagger}c-q(c_{1}^{\dagger}c_{1}-c_{2}^{\dagger}c_{2})~{},\end{split}$
(30)

where $c,c^{\dagger}$ are the mode operators for the $j=0$ state.

We can now easily solve for the spectrum of the massive Dirac operator. If we
consider a pair of charged fermions $\psi,\tilde{\psi}$ and turn on a mass
term of the form

$\displaystyle\begin{split}{\cal L}_{\rm
mass}=m_{\psi}\psi\tilde{\psi}+c.c.~{},\end{split}$ (31)

where $m_{\psi}\in{\mathbb{R}}$, then the simplified Dirac equation becomes

$\displaystyle\begin{split}\left(\sigma^{3}\partial_{r}+\sigma^{3}{\cal
D}_{E}\right)\Psi=-im_{\psi}\bar{\tilde{\Psi}}~{},\\\
\left(\sigma^{3}\partial_{r}+\sigma^{3}{\cal
D}_{-E}\right)\bar{\tilde{\Psi}}=im_{\psi}\Psi~{}.\end{split}$ (32)

Here, the Weyl fermions effectively pair up to form Dirac fermions
$\Upsilon=(\Psi_{\alpha},\bar{\tilde{\Psi}}^{\dot{\alpha}})$. Since the mass
term is $r$-independent, it does not affect the $r\to 0$ limit – it only
affects the $r\to\infty$ limit. Now, the asymptotic Dirac equation is modified
to

$\displaystyle\begin{split}\left(\sigma^{3}\otimes\mathds{1}\left(\partial_{r}+\frac{1}{r}\right)-iE\sigma^{3}\otimes\sigma^{3}+m_{\psi}\mathds{1}\otimes\sigma^{2}\right)\left(\begin{array}[]{c}\Psi\\\
\bar{\tilde{\Psi}}\end{array}\right)=0~{}.\end{split}$ (33)

Again, the asymptotic solutions are plane waves of the form

$\displaystyle\begin{split}\Upsilon\underset{r\to\infty}{\longrightarrow}\frac{e^{i(Et+\omega\sigma^{3}r)}}{r}\Upsilon_{0,{\rm
IR}}~{},\end{split}$ (34)

where $\omega=\sqrt{E^{2}-m_{\psi}^{2}}$ and
$\Upsilon=(\Psi,\bar{\tilde{\Psi}})$. Due to the mass term, the solutions for
$\Upsilon_{0,{\rm IR}}$ are linear combinations of $\Psi_{0,{\rm
IR}},\tilde{\Psi}_{0,{\rm IR}}$:

$\displaystyle\begin{split}&\Upsilon_{0,{\rm
IR}}^{(j>0)}=c_{1}\Upsilon_{0,{\rm IR}}^{(1)}+c_{2}\Upsilon_{0,{\rm
IR}}^{(2)}+c_{3}\Upsilon_{0,{\rm IR}}^{(3)}+c_{4}\Upsilon_{0,{\rm
IR}}^{(4)}\\\ &=c_{1}\left(\begin{array}[]{c}-\sqrt{1+\gamma}\\\ 0\\\
\frac{m_{\psi}}{E}\frac{\gamma}{\sqrt{1+\gamma}}\\\
0\end{array}\right)+c_{2}\left(\begin{array}[]{c}\sqrt{1-\gamma}\\\ 0\\\
\frac{m_{\psi}}{E}\frac{\gamma}{\sqrt{1-\gamma}}\\\
0\end{array}\right)+c_{3}\left(\begin{array}[]{c}0\\\ \sqrt{1+\gamma}\\\ 0\\\
\frac{m_{\psi}}{E}\frac{\gamma}{\sqrt{1+\gamma}}\end{array}\right)+c_{4}\left(\begin{array}[]{c}0\\\
-\sqrt{1-\gamma}\\\ 0\\\
\frac{m_{\psi}}{E}\frac{\gamma}{\sqrt{1-\gamma}}\end{array}\right)\end{split}$
(35)

for $j>0$ and

$\displaystyle\begin{split}\Upsilon_{0,{\rm IR}}^{(j=0)}=b_{1}\Upsilon_{0,{\rm
IR}}^{(3)}+b_{2}\Upsilon_{0,{\rm IR}}^{(4)}=b_{1}\left(\begin{array}[]{c}0\\\
\sqrt{1+\gamma}\\\ 0\\\
\frac{m_{\psi}}{E}\frac{\gamma}{\sqrt{1+\gamma}}\end{array}\right)+b_{2}\left(\begin{array}[]{c}0\\\
-\sqrt{1-\gamma}\\\ 0\\\
\frac{m_{\psi}}{E}\frac{\gamma}{\sqrt{1-\gamma}}\end{array}\right)\end{split}$
(36)

for $j=0$ where $\gamma=\frac{1}{\sqrt{1-m_{\psi}^{2}/E^{2}}}$. Here the
asymptotic behavior is given by

$\displaystyle\begin{split}\Upsilon_{0,{\rm IR}}&=c_{1}e^{iE(t+\omega
r)}\Upsilon_{0,{\rm IR}}^{(1)}+c_{2}e^{iE(t-\omega r)}\Upsilon_{0,{\rm
IR}}^{(2)}+c_{3}e^{iE(t-\omega r)}\Upsilon_{0,{\rm
IR}}^{(3)}+c_{4}e^{iE(t+\omega r)}\Upsilon_{0,{\rm IR}}^{(4)}\\\
&+b_{1}e^{iE(t-\omega r)}\Upsilon_{0,{\rm IR}}^{(3)}+b_{2}e^{iE(t+\omega
r)}\Upsilon_{0,{\rm IR}}^{(4)}~{},\end{split}$ (37)

Using this mode expansion, we can compute the asymptotic charge flux:

$\displaystyle\begin{split}\oint_{S^{2}_{\infty}}j^{\mu},\hat{r}_{\mu}d^{2}\Omega&=\oint_{S^{2}_{\infty}}\left(\bar{\psi}\bar{\sigma}^{\mu}\psi+\tilde{\psi}\sigma^{\mu}\bar{\tilde{\psi}}\right)~{}\hat{r}_{\mu}\,d^{2}\Omega\\\
&=-c_{1}^{\dagger}c_{1}-c_{2}^{\dagger}c_{2}+c_{3}^{\dagger}c_{3}+c_{4}^{\dagger}c_{4}+b_{1}^{\dagger}b_{1}+b_{2}^{\dagger}b_{2}~{}.\end{split}$
(38)

For the case where $E=m_{\psi}$, the solutions have a similar asymptotic
structure as the case where $E,m_{\psi}=0$: there are always a pair of
solutions that have polynomial growth $r^{-1\pm\lambda}$ for
$\lambda=\sqrt{j(j+1)}$ at $r\to\infty$. There are additionally a pair of
solutions that grows like

$\displaystyle\begin{split}\Upsilon\sim\begin{cases}\frac{r^{\lambda}}{{\rm
log}(r)}&j>0\\\ 1&j=0\end{cases}\end{split}$ (39)

As we will see in the following section, the modes with this asymptotic
behavior will not arise in our analysis due to the boundary conditions on the
monopole and therefore we will not comment further on them.

###  2.2 Fermions and the Smooth Non-Abelian Monopole

We will now discuss the boundary conditions on the IR modes. Since the
dynamical dyon degree of freedom lives on the boundary, the fermion “boundary
conditions” will include a boundary interaction with $\varphi(t)$. We can
derive the interaction by computing the mode expansion of the fermions in the
fixed non-abelian monopole background (LABEL:Connection) with $\varphi(t)\to
0$. Using this solution, we can expand the asymptotic form of the non-abelian
fermion modes in terms of IR modes and then infer the coupling to $\varphi(t)$
by looking at the gauge orbit at $r\to 0$ since $\varphi(t)$ transforms as a
Goldstone under $U(1)$ gauge transformations. Equivalently, we can solve for
the mode expansion in the fixed monopole background with $\varphi(t)=0$
(LABEL:Connection), expand the solutions in terms of the IR modes, and then
plug the result into the action with the full $\varphi(t)$-dependent
connection (8).

Again, we can expand the fermion fields into representations of the angular
momentum operator (4). We will label these representations by the same quantum
numbers $j,m$. Here the angular momentum representations will have integer
total spin $j\in{\mathbb{Z}}_{\geq 0}$ because the fundamental Weyl fermions
transform in the spin-$\frac{1}{2}$ representation of the $SU(2)$ gauge group.

We are interested in finding the spectrum of the massive Dirac operator for
general total angular momentum $j$. Again, let us first find the spectrum of
the massless Dirac operator for a single fermion $\psi^{i}$ where $i=\pm$ is
the gauge index.

As before, we will simplify the Dirac equation with background gauge field
(LABEL:Connection) by performing a frame rotation by

$\displaystyle\begin{split}\psi^{i}=U(\theta,\phi)\hat{\psi}^{i}\quad,\quad
U(\theta,\phi)=e^{\frac{i\phi}{2}\sigma^{0}\bar{\sigma}^{3}}e^{\frac{i\theta}{2}\sigma^{0}\bar{\sigma}^{2}}~{}.\end{split}$
(40)

The resulting Dirac equation is given by

$\displaystyle\begin{split}\Bigg{[}&-iE+\sigma^{3}\left(\partial_{r}+\frac{1}{r}\right)+\frac{\hat{\cal
K}}{r}-\left(\sigma^{-}e^{-i\phi}M_{+}-\sigma^{+}e^{i\phi}M_{-}\right)\Bigg{]}\hat{\psi}^{i}=0~{},\end{split}$
(41)

where $\sigma^{\pm}=\frac{1}{2}(\sigma^{1}\pm i\sigma^{2})$,
$M_{\pm}=T_{\pm}\,m_{W}r\,\textrm{csch}\,(m_{W}r)$, and

$\displaystyle\begin{split}\hat{\cal
K}&=\sigma^{1}\left(\partial_{\theta}+\frac{1}{2}{\rm
cot}(\theta)\right)+\frac{\sigma^{2}}{\sin\theta}\Big{(}(\partial_{\phi}+iT_{3})-i\cos(\theta)T_{3}\Big{)}~{},\end{split}$
(42)

We can then expand $\psi,\tilde{\psi}$ into angular momentum eigenstates as

$\displaystyle\begin{split}\hat{\psi}^{+}=\left(\begin{array}[]{c}f_{+}(r)\,e^{i\phi/2}{\cal
D}^{(j)}_{m,1}(\theta,\phi)\\\ g_{+}(r)\,e^{-i\phi/2}{\cal
D}^{(j)}_{m,0}(\theta,\phi)\end{array}\right)\quad,\quad\hat{\psi}^{-}=\left(\begin{array}[]{c}f_{-}(r)\,e^{i\phi/2}{\cal
D}^{(j)}_{m,0}(\theta,\phi)\\\ g_{-}(r)\,e^{-i\phi/2}{\cal
D}^{(j)}_{m,-1}(\theta,\phi)\end{array}\right)~{},\end{split}$ (43)

where ${\cal D}^{(j)}_{m,q}(\theta,\phi):={\cal
D}^{(j)}_{m,q}(\phi,\theta,\phi)$ are Wigner $D$-functions. Again, when $j=0$,
${\cal D}^{(0)}_{m,\pm 1}$ is singular so $f_{+},g_{-}:=0$ and the Dirac
equation further simplifies.

Now, the Dirac equation reduces to a matrix ODE

$\displaystyle\begin{split}\left(\sigma^{3}\left(\partial_{r}+\frac{1}{r}\right)+\frac{1}{r}\left(\begin{array}[]{cc}0&-\lambda\\\
\lambda&0\end{array}\right)-iE+\frac{1}{r}\left(\sigma^{-}M_{+}-\sigma^{+}M_{-}\right)\right)\Psi=0\end{split}$
(44)

for the spinors

$\displaystyle\begin{split}\Psi=\left(\begin{array}[]{c}f_{+}(r)\\\
g_{+}(r)\\\ f_{-}(r)\\\ g_{-}(r)\end{array}\right)\end{split}$ (45)

where $\lambda=\sqrt{j(j+1)}$. Because the Dirac operator is everywhere
smooth, it means that the solutions of the Dirac operator are formally given
by solving the matrix ODE:

$\displaystyle\begin{split}\Psi=P\,e^{-\int{\cal
D}_{E}(r^{\prime})\,dr^{\prime}}\Psi_{0,{\rm UV}}\quad,\quad{\cal
D}_{E}(r)=\left(\begin{array}[]{cccc}-iE&\frac{\lambda}{r}&&\\\
\frac{\lambda}{r}&iE&\frac{1}{r}&\\\ &\frac{1}{r}&-iE&\frac{\lambda}{r}\\\
&&\frac{\lambda}{r}&iE\end{array}\right)+\frac{1}{r}\mathds{1}~{}.\end{split}$
(46)

These formal solutions may however be non-normalizable.777Here the correct
notion of the normalizable is plane-wave normalizable. The normalizability of
the solutions of this differential equation are now determined by the
asymptotic behavior of the solutions at both $r\to\infty$ and $r\to 0$.

For $E=0$, we can explicitly solve for the solutions. As shown in
Brennan:2021ucy , the spectrum of the angular momentum representations only
have no normalizable solutions for $j>0$. However, for $j=0$, the solution is
given explicitly by

$\displaystyle\begin{split}\Psi=\frac{\tanh(m_{W}r)}{r}\left(\begin{array}[]{c}0\\\
1\\\ -1\\\ 0\end{array}\right)~{}.\end{split}$ (47)

In the long distance limit, this solution is the combination of $E=0,$ $j=0$
modes of $\psi^{A}_{q}$ of opposite charges in the IR (24):

$\displaystyle\begin{split}\Psi\underset{r\to\infty}{\longrightarrow}\left(\begin{array}[]{c}\Psi^{(j=0)}_{+1}\\\
-\Psi^{(j=0)}_{-1}\end{array}\right)\end{split}$ (48)

This will imply that the boundary condition at the monopole, is given by
relating $\psi^{+}$ to $\psi^{-}$ via the dyon degree of freedom.

For $E>0$, the analysis is more involved. At $r\to\infty$, the Dirac equation
can be written as

$\displaystyle\begin{split}\left(\sigma^{3}\left(\partial_{r}+\frac{1}{r}\right)+\frac{1}{r}\left(\begin{array}[]{cc}0&-\lambda\\\
\lambda&0\end{array}\right)-iE\right)\Psi=0~{}.\end{split}$ (49)

The asymptotic behavior of the solutions match that of the IR:

$\displaystyle\begin{split}\Psi\underset{r\to\infty}{\longrightarrow}\frac{e^{iE(t-\sigma^{3}r)}}{r}\Psi_{0,{\rm
UV}}~{}.\end{split}$ (50)

where $\Psi_{0,{\rm UV}}$ is a constant spinor. All solutions are
asymptotically plane-wave normalizable for $E\neq 0$. Near $r\to 0$, the Dirac
operator can be written as

$\displaystyle\begin{split}\left(\sigma^{3}\left(\partial_{r}+\frac{1}{r}\right)+\frac{1}{r}\left(\begin{array}[]{cc}0&-\lambda\\\
\lambda&0\end{array}\right)+\frac{1}{r}\left(\sigma^{-}T_{+}-\sigma^{+}T_{-}\right)\right)\Psi=0~{}.\end{split}$
(51)

This operator has normalizable solutions which are given by

$\displaystyle\begin{split}\Psi\sim
c_{1}\frac{(m_{W}r)^{j+1}}{r}\left(\begin{array}[]{c}\sqrt{j}\\\
-\sqrt{j+1}\\\ \sqrt{j+1}\\\
-\sqrt{j}\end{array}\right)+c_{2}\frac{(m_{W}r)^{j}}{r}\left(\begin{array}[]{c}\sqrt{j+1}\\\
-\sqrt{j}\\\ -\sqrt{j}\\\ \sqrt{j+1}\end{array}\right)~{},\end{split}$ (52)

for $j>0$ and

$\displaystyle\begin{split}\Psi\sim\frac{1}{r}\left(\begin{array}[]{c}0\\\
1\\\ -1\\\ 0\end{array}\right)~{},\end{split}$ (53)

for $j=0$.

This means that the normalizable, positive energy solutions of the Dirac
equation are those for which $E>0$ and $\Psi_{0,{\rm UV}}$ is given by

$\displaystyle\begin{split}\Psi_{0,{\rm UV}}=c_{1}\Psi_{0,{\rm
UV}}^{(1)}+c_{2}\Psi_{0,{\rm
UV}}^{(2)}=c_{1}\left(\begin{array}[]{c}\sqrt{j}\\\ -\sqrt{j+1}\\\
\sqrt{j+1}\\\
-\sqrt{j}\end{array}\right)+c_{2}\left(\begin{array}[]{c}\sqrt{j+1}\\\
-\sqrt{j}\\\ -\sqrt{j}\\\ \sqrt{j+1}\end{array}\right)~{},\end{split}$ (54)

for $j>0$ and

$\displaystyle\begin{split}\Psi_{0,{\rm UV}}=\Psi_{0,{\rm
UV}}^{(0)}=\left(\begin{array}[]{c}0\\\ 1\\\ -1\\\
0\end{array}\right)\end{split}$ (55)

for $j=0$. In particular, this demonstrates that the interaction with the
monopole does not introduce an energy gap between the $j=0$ (i.e. s-wave)
modes and the higher angular momentum modes. In terms of the IR modes (28),
these can be matched by

$\displaystyle\begin{split}\Psi_{0,{\rm
UV}}^{(j>0)}=c_{1}\left(\begin{array}[]{c}\sqrt{j}\Psi^{(1)}_{0,+1}-\sqrt{j+1}\Psi^{(2)}_{0,+1}\\\
\sqrt{j+1}\Psi^{(1)}_{0,-1}-\sqrt{j}\Psi^{(2)}_{0,-1}\end{array}\right)+c_{2}\left(\begin{array}[]{c}\sqrt{j+1}\Psi^{(1)}_{0,+1}-\sqrt{j}\Psi^{(2)}_{0,+1}\\\
-\sqrt{j}\Psi^{(1)}_{0,-1}+\sqrt{j+1}\Psi^{(2)}_{0,-1}\end{array}\right)~{},\end{split}$
(56)

for $j>0$ and

$\displaystyle\begin{split}\Psi_{0,{\rm
UV}}=\left(\begin{array}[]{c}\Psi_{0,+1}^{(j=0)}\\\
-\Psi_{0,-1}^{(j=0)}\end{array}\right)\end{split}$ (57)

for $j=0$. It is important to note that these solutions all source an electric
charge flux at infinity:

$\displaystyle\begin{split}\oint_{S^{2}_{\infty}}j^{\mu}~{}\hat{r}_{\mu}d^{2}\Omega=\int_{S^{2}_{\infty}}\bar{\Psi}\bar{\sigma}^{\mu}\Psi\,\hat{r}_{\mu}\,d^{2}\Omega=2(c_{1}^{\dagger}c_{1}-c_{2}^{\dagger}c_{2}+c^{\dagger}c)~{},\end{split}$
(58)

for fixed $E$ where $c_{i}$ are as in (54) and $c^{\dagger},c$ are the
creation and annihilation operators for the $j=0$ modes.

We can now easily solve for the spectrum of the massive Dirac operator. If we
turn on a mass term of the form

$\displaystyle\begin{split}{\cal L}_{\rm
mass}=m_{\psi}\psi\tilde{\psi}+c.c.~{},\end{split}$ (59)

where $m_{\psi}\in{\mathbb{R}}$, then the simplified Dirac equation becomes

$\displaystyle\begin{split}\left(\sigma^{3}\partial_{r}+\sigma^{3}{\cal
D}_{E}\right)\Psi=-im_{\psi}\bar{\tilde{\Psi}}~{},\\\
\left(\sigma^{3}\partial_{r}+\sigma^{3}{\cal
D}_{-E}\right)\bar{\tilde{\Psi}}=im_{\psi}\Psi~{}.\end{split}$ (60)

Here, the fermions effectively pair up to form Dirac fermions
$\Upsilon=(\Psi_{\alpha},\bar{\tilde{\Psi}}^{\dot{\alpha}})$. Since the mass
term is $r$-independent, it does not affect the $r\to 0$ limit. However, it
does affect the $r\to\infty$ limit. Now, the asymptotic behavior is modified
to

$\displaystyle\begin{split}\Upsilon\sim\frac{1}{r}e^{\pm i\omega
r}\Upsilon_{0,{\rm UV}}~{},\end{split}$ (61)

for $E\neq m_{\psi}$ where $\omega=\sqrt{E^{2}-m_{\psi}^{2}}$. Here, the
constant Dirac fermion $\Upsilon_{0,{\rm UV}}$ takes the form

$\displaystyle\begin{split}\Upsilon_{0,{\rm
UV}}&=a_{1}\left(\begin{array}[]{c}\Psi_{0,{\rm UV}}^{(1)}\\\
-\frac{m_{\psi}}{E}\frac{\gamma}{1+\gamma}\bar{\Psi}_{0,{\rm
UV}}^{(1)}\end{array}\right)+a_{2}\left(\begin{array}[]{c}\Psi_{0,{\rm
UV}}^{(1)}\\\ \frac{m_{\psi}}{E}\frac{\gamma}{1-\gamma}\bar{\Psi}_{0,{\rm
UV}}^{(1)}\end{array}\right)\\\ &+a_{3}\left(\begin{array}[]{c}\Psi_{0,{\rm
UV}}^{(2)}\\\ -\frac{m_{\psi}}{E}\frac{\gamma}{1+\gamma}\bar{\Psi}_{0,{\rm
UV}}^{(2)}\end{array}\right)+a_{4}\left(\begin{array}[]{c}\Psi_{0,{\rm
UV}}^{(2)}\\\ \frac{m_{\psi}}{E}\frac{\gamma}{1-\gamma}\bar{\Psi}_{0,{\rm
UV}}^{(2)}\end{array}\right)\end{split}$ (62)

for $j>0$ and

$\displaystyle\begin{split}\Upsilon_{0,{\rm
UV}}=b_{1}\left(\begin{array}[]{c}\Psi_{0,{\rm UV}}^{(0)}\\\
-\frac{m_{\psi}}{E}\frac{\gamma}{1+\gamma}\bar{\Psi}_{0,{\rm
UV}}^{(0)}\end{array}\right)+b_{2}\left(\begin{array}[]{c}\Psi_{0,{\rm
UV}}^{(0)}\\\ \frac{m_{\psi}}{E}\frac{\gamma}{1-\gamma}\bar{\Psi}_{0,{\rm
UV}}^{(0)}\end{array}\right)~{},\end{split}$ (63)

for $j=0$, where again $\gamma=\frac{1}{\sqrt{1-m_{\psi}^{2}/E^{2}}}$ .

The asymptotic behavior of these solutions is given by

$\displaystyle\begin{split}\Upsilon&\sim\frac{e^{i(Et+\omega\sigma^{3}r)}}{r}\left\\{a_{1}\left(\begin{array}[]{c}\Psi_{0,{\rm
UV}}^{(1)}\\\ -\frac{m_{\psi}}{E}\frac{1}{1+\gamma}\bar{\Psi}_{0,{\rm
UV}}^{(1)}\end{array}\right)+a_{3}\left(\begin{array}[]{c}\Psi_{0,{\rm
UV}}^{(2)}\\\ -\frac{m_{\psi}}{E}\frac{1}{1+\gamma}\bar{\Psi}_{0,{\rm
UV}}^{(2)}\end{array}\right)\right\\}\\\
&+\frac{e^{i(Et-\omega\sigma^{3}r)}}{r}\left\\{a_{2}\left(\begin{array}[]{c}\Psi_{0,{\rm
UV}}^{(1)}\\\ \frac{m_{\psi}}{E}\frac{\gamma}{1-\gamma}\bar{\Psi}_{0,{\rm
UV}}^{(1)}\end{array}\right)+a_{4}\left(\begin{array}[]{c}\Psi_{0,{\rm
UV}}^{(2)}\\\ \frac{m_{\psi}}{E}\frac{\gamma}{1-\gamma}\bar{\Psi}_{0,{\rm
UV}}^{(2)}\end{array}\right)\right\\}~{},\end{split}$ (64)

for $j>0$ and

$\displaystyle\begin{split}\Upsilon&\sim\frac{e^{i(Et+\omega\sigma^{3}r)}}{r}b_{1}\left(\begin{array}[]{c}\Psi_{0,{\rm
UV}}^{(0)}\\\ -\frac{m_{\psi}}{E}\frac{\gamma}{1+\gamma}\bar{\Psi}_{0,{\rm
UV}}^{(0)}\end{array}\right)+\frac{e^{i(Et-\omega\sigma^{3}r)}}{r}b_{2}\left(\begin{array}[]{c}\Psi_{0,{\rm
UV}}^{(0)}\\\ \frac{m_{\psi}}{E}\frac{\gamma}{1-\gamma}\bar{\Psi}_{0,{\rm
UV}}^{(0)}\end{array}\right)~{},\end{split}$ (65)

for $j=0$. Again, these can be expanded in terms of the IR modes
(LABEL:massiveIRsol). For brevity, we will not give the full expressions.
However, it is clear by comparing the form of the $\Upsilon_{0,{\rm
IR}}^{(a)}$ and their associated charges in (LABEL:massiveIRsol) – (38) and
form of the $\Upsilon_{0,{\rm UV}}$ in (62), that these solutions relate an IR
mode with positive asymptotic charge flux to and IR mode with negative
asymptotic charge flux. However, these combinations are such that the total
asymptotic flux is given by

$\displaystyle\begin{split}\oint_{S^{2}_{\infty}}j^{\mu}~{}\hat{r}_{\mu}d^{2}\Omega=\oint_{S^{2}_{\infty}}\bar{\Upsilon}\gamma^{\mu}\Upsilon\,\hat{r}_{\mu}\,d^{2}\Omega=2(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}-a_{3}^{\dagger}a_{3}-a_{4}^{\dagger}a_{4})+2(b_{1}^{\dagger}b_{1}+b_{2}^{\dagger}b_{2})~{}.\end{split}$
(66)

For the case where $E=m_{\psi}$, the solutions have a similar asymptotic
structure as the case where $E,m_{\psi}=0$: there is a $j=0$ mode that has
$1/r$ behavior at $r\to\infty$ at energy $E=m_{\psi}$ but no normalizable
$j>0$ modes with energy $E=m_{\psi}$.

Additionally, one can show by explicitly checking, that solutions with
$|E|<|m_{\psi}|$ are not normalizable because the $\Upsilon_{0,{\rm UV}}$ that
lead to normalizable solutions in the $r\to 0$ limit turn on the exponential
growth at $r\to\infty$. Physically, this reflects the fact that turning on
masses induces an energy gap. Therefore, there is again a continuous spectrum
of states for massive fermions which has the same behavior for $E\geq
m_{\psi}$ as the spectrum of states for massless fermions at energies $E\geq
0$.

##  3 Generalized Symmetries and the Callan Rubakov Effect

Now let us discuss the symmetries of $4d$ $SU(2)$ gauge theory with $N_{f}$
fundamental fermions $\psi^{A}$ where $A=1,...,N_{f}$. We will take
$N_{f}=2n_{f}$ so that the theory has no gauge anomaly. Here we will use the
language of generalized global symmetries; for a review see Brennan:2023mmt ;
Schafer-Nameki:2023jdn ; Bhardwaj:2023kri ; Shao:2023gho .

Let us first consider the symmetries of the massless theory. This theory is
described by the action

$\displaystyle\begin{split}S=\int\frac{1}{2g^{2}}\,\textrm{Tr}\,[F\wedge\ast
F]+i\bar{\psi}\not{D}\psi+\,\textrm{Tr}\,|D\Phi|^{2}-V(\Phi)~{},\end{split}$
(67)

where again we choose a potential $V(\Phi)$ so that $\Phi$ condenses above the
confinement scale. This theory has a $SU(N_{f})^{(0)}$ 0-form global symmetry
that rotates the fermions which we will often refer to as the 0-form _flavor
symmetry_. Note that there is no independent $U(1)$ chiral symmetry because it
is broken to the center of $SU(N_{f})^{(0)}$ by an ABJ anomaly.

In the IR the theory flows to $U(1)$ gauge theory with $2N_{f}$ fermions which
we write as $\psi^{+A}=\psi^{A}$ an $\psi^{-A}=\tilde{\psi}^{A}$. Here we see
in the IR that the theory has a $SU(N_{f})^{(0)}\times SU(N_{f})^{(0)}$ global
flavor symmetry where each factor acts only on $\psi^{A}$ or
$\tilde{\psi}^{A}$ in the fundamental representation. Additionally, this
theory has a $U(1)^{(1)}$ 1-form magnetic symmetry which protects the monopole
lines from decay. 888Note that in this theory there is a mixed
$(SU(N_{f})^{(0)}\times SU({N_{f}})^{(0)})^{2}$-$U(1)$ mixed anomaly which
gives rise to a 2-group global symmetry involving the magnetic 1-form global
symmetry vanBeest:2023dbu . This 2-group global symmetry is broken by fermion
mass terms and therefore will not play a role in our analysis.

The theory also develops a ${\mathbb{Z}}_{N_{f}}$ generalized symmetry which
we denote as ${\mathbb{Z}}_{N_{f}}^{(0,1)}$. This follows from the fact that
the fermions transform faithfully under Brennan:2022tyl ; Delmastro:2022pfo ;
Hsin:2020nts ; Lee:2021crt

$\displaystyle\begin{split}G_{\rm ferm}=\frac{U(1)_{\rm gauge}\times
SU(N_{f})^{(0)}\times SU(N_{f})^{(0)}}{{\mathbb{Z}}_{N_{f}}}~{}.\end{split}$
(68)

We would like to elaborate on the nature of this generalized global
symmetry.999We would like to especially thank Konstantinos Roumpedakis and
Pierluigi Niro for related discussions.

A 1-form center symmetry typically arises in $G$ gauge theories when there is
matter that is uncharged under a subgroup $\Gamma$ of the center of the gauge
group: $\Gamma\subset Z(G)$. When $\Gamma$ is non-trivial, we can restrict the
path integral to only integrate over $G/\Gamma$ gauge fields with a fixed,
non-trivial discrete flux $w_{2}(\Gamma)\in H^{2}(BG/\Gamma,\Gamma)$ that
obstructs lifting the $G/\Gamma$ gauge field to a $G$ gauge field. The action
of the 1-form $\Gamma^{(1)}$-symmetry is implemented by a $\Gamma$-valued
large gauge transformation of the $G/\Gamma$ gauge field which corresponds to
a shift in the choice of integer lift of $w_{2}(\Gamma)$. This large gauge
transformation acts non-trivially on Wilson lines $W_{R}$ of representations
$R$ that transform non-trivially under $\Gamma\subset Z(G)$ because it is not
a large $G$ gauge transformation. This action multiplies $W_{R}(\gamma)\mapsto
W_{R}(\gamma)\,e^{in\oint_{\gamma}\Lambda_{1}}$ where $\Lambda_{1}$ is the
$\Gamma$-valued transformation parameter. This symmetry can be enacted by a
co-dimension 2 topological symmetry defect operator (SDO) which induces a
large gauge transformation on a Wilson line when the defect crosses the line.
Physically, the SDO operator can be viewed as measuring the electric flux
through the surface, or conversely as a topological flux tube that induces a
$\Gamma$-valued Aharanov-Bohm phase along any linking circle.

In our setting, the fields are invariant under a diagonal combination of
${\mathbb{Z}}_{N_{f}}\subset Z(U(1))$ and ${\mathbb{Z}}_{N_{f}}\subset
Z\Big{(}SU(N_{f})\times SU(N_{f})\Big{)}$. This means that the theory can be
described by $U(1)$ gauge theory with a $G_{f}/{\mathbb{Z}}_{N_{f}}$ global
symmetry where $G_{f}=SU(N_{f})\times SU(N_{f})$. This theory admits all
$U(1)$ Wilson lines and we are allowed to dress any line operator by
$G_{f}/{\mathbb{Z}}_{N_{f}}$ background Wilson lines – i.e. we are allowed to
add local counter terms which are dependent on the
$G_{f}/{\mathbb{Z}}_{N_{f}}$ background gauge connection.101010The reason that
we are only allowed to add local counter terms that are dependent on the
$G_{f}/{\mathbb{Z}}_{N_{f}}$ connection (and not a $G_{f}$ connection) is that
a counter term involving a $G_{f}$ connection would necessarily contain a
dynamical ${\mathbb{Z}}_{N_{f}}\subset U(1)_{\rm gauge}$ term and introduce an
additional interactions in the theory.

As in the case of standard 1-form center symmetries, we can restrict the path
integral to $U(1)/{\mathbb{Z}}_{N_{f}}$ gauge fields with fixed discrete flux
$w_{2}({\mathbb{Z}}_{N_{f}})$ if we additionally couple the theory to a
$G_{f}/{\mathbb{Z}}_{N_{f}}$ 0-form background gauge field with the same
discrete flux. The path integral is then invariant under simultaneous large
${\mathbb{Z}}_{N_{f}}$ gauge transformations of the
$U(1)/{\mathbb{Z}}_{N_{f}}$ and $G_{f}/{\mathbb{Z}}_{N_{f}}$ gauge fields (up
to possible anomalies):

$\displaystyle\begin{split}Z[w_{2}({\mathbb{Z}}_{N_{f}})+N_{f}\Lambda_{2}]=Z[w_{2}({\mathbb{Z}}_{N_{f}})]~{}.\end{split}$
(69)

The invariance of the path integral under this shift indicates a
${\mathbb{Z}}_{N_{f}}$ symmetry which appears similar to a 1-form global
symmetry due to the fact that the matter fields are invariant under the
diagonal ${\mathbb{Z}}_{N_{f}}\subset U(1)\times G_{f}$. Indeed, the
simultaneous ${\mathbb{Z}}_{N_{f}}$ large gauge transformations of the
$U(1)/{\mathbb{Z}}_{N_{f}}$ dynamical and $G_{f}/{\mathbb{Z}}_{N_{f}}$
background gauge fields acts on the Wilson lines as $W_{q}(\gamma)\mapsto
W_{q}(\gamma)\,e^{iq\int_{\gamma}\Lambda_{1}}$ where $\Lambda_{1}$ is the
large ${\mathbb{Z}}_{N_{f}}$ gauge transformation parameter.

However, there are several subtleties that arise in trying to understand in
what way the invariance of the path integral under the large
${\mathbb{Z}}_{N_{f}}$ gauge transformations is a symmetry. The first that
arises is how one would construct the topological operator that enacts the
symmetry since the gauge field for $G_{f}/{\mathbb{Z}}_{N_{f}}$ is not an
operator in our theory. A symmetry defect operator for this “symmetry” must
induce large ${\mathbb{Z}}_{N_{f}}$ transformations in
$G_{f}/{\mathbb{Z}}_{N_{f}}\times U(1)/{\mathbb{Z}}_{N_{f}}$. The shift of the
$U(1)/{\mathbb{Z}}_{N_{f}}$ field is enacted by the (broken) 1-form center
symmetry generator. Similarly, the shift operator for
$G_{f}/{\mathbb{Z}}_{N_{f}}$ is generated by a flux operator which can be
thought of as a flux tube in $G_{f}/{\mathbb{Z}}_{N_{f}}$ that generates a
holonomy that winds around a non-trivial element of
$\pi_{1}(G_{f}/{\mathbb{Z}}_{N_{f}})$. Since $G_{f}$ is simply connected, this
winding is indeed topologically non-trivial (i.e.
$\pi_{1}(G_{f}/{\mathbb{Z}}_{N_{f}})\neq 0$) and the existence of the flux
operator is guaranteed by the fact that we are allowed to turn on any
$G_{f}/{\mathbb{Z}}_{N_{f}}$ bundle up to the matching of discrete flux.
Similar constructions were considered in Hsin:2022heo .

If we take the product of these two non-topological operators, we construct a
topological operator ${\cal U}_{g}(\Sigma)$. This operator is topological
since it generates a non-trivial flux associated to the Poincaré dual of
$\Sigma$111111Here $[\Sigma]^{\vee}$ is the 2-form which is Poincaré dual to
$\Sigma\in H_{2}(M;{\mathbb{Z}})$. This is defined by the relation
$\oint_{\Sigma}[\Sigma]^{\vee}=1$.

$\displaystyle\begin{split}{\cal
U}_{g}(\Sigma)~{}\longmapsto~{}w_{2}({\mathbb{Z}}_{N_{f}})=n\cdot[\Sigma]^{\vee}\quad,\quad
g=e^{\frac{2\pi in}{{N_{f}}}}~{}.\end{split}$ (70)

Since a deformation of $\Sigma$ acts trivially on its topological class, we
see that a deformation of the surface $\Sigma$ will induce a gauge
transformation of $w_{2}({\mathbb{Z}}_{N_{f}})$:

$\displaystyle\begin{split}\delta_{\Sigma}{\cal
U}_{g}(\Sigma)~{}\longmapsto~{}\delta
w_{2}({\mathbb{Z}}_{N_{f}})=[\Sigma+\delta\Sigma]^{\vee}-[\Sigma]^{\vee}~{}={N_{f}}\Lambda_{2}=[0]\in
H^{2}(M;{\mathbb{Z}})~{}.\end{split}$ (71)

Since the action is invariant under large gauge transformations of
$w_{2}({\mathbb{Z}}_{N_{f}})$ along the diagonal embedding of
$U(1)/{\mathbb{Z}}_{N_{f}}\times G_{f}/{\mathbb{Z}}_{N_{f}}$, the associated
operator is indeed topological.

Figure 1: In this figure (left) we illustrate how ${\mathbb{Z}}_{N_{f}}$
symmetry defect operator (blue) must come with a choice of “invisible” 0-form
$G_{f}/{\mathbb{Z}}_{N_{f}}$ symmetry defect operators (black) in $3d$. Here,
$g_{1},g_{2},g_{3}\in G_{f}/{\mathbb{Z}}_{N_{f}}$ and
$g_{1}g_{2}g_{3}^{-1}\in{\mathbb{Z}}_{N_{f}}$ such that the
$G_{f}/{\mathbb{Z}}_{N_{f}}$ winding is implemented by the 0-form SDOs. On the
right, we give an alternative construction where the symmetry defect operator
(blue) wraps a $S^{1}$ is dressed by 0-form SDOs (green) on $D^{2}$.

This defect acts on Wilson lines by linking: pulling the defect through the
Wilson line operator induces a large ${\mathbb{Z}}_{N_{f}}$ gauge
transformation on the Wilson line similar to an ordinary 1-form global
symmetry. However, since the fermions have charge $\pm 1$ in our theory, the
Wilson lines can be cut by fermion operators. Despite this, the
${\mathbb{Z}}_{N_{f}}$ symmetry is preserved due to the fact that the end
points of a cut Wilson line transform under a projective
$G_{f}/{\mathbb{Z}}_{N_{f}}$ representation. Because of this, we must consider
the lift of the $G_{f}/{\mathbb{Z}}_{N_{f}}$ bundle to a $G_{f}$ bundle. In
general, this is not possible without introducing singularities which we can
interpret as an Aharanov-Bohm phase. However, if we consider locally the
deformation of the bundle due to the insertion of ${\cal U}_{g}(\Sigma)$, then
there is locally a lift when $\Sigma$ is contractible.121212The reason is that
if $\Sigma$ is contractible, the obstruction to lift the
$G_{f}/{\mathbb{Z}}_{N_{f}}$ bundle to a $G_{f}$ bundle is trivializable.
This lift requires making a choice of ${\mathbb{Z}}_{N_{f}}$ 0-form symmetry
defect operators that end on $\Sigma$ and implement the ${\mathbb{Z}}_{N_{f}}$
winding of the associated $G_{f}$ 0-form gauge field on circles that link
$\Sigma$. Such as in Figure 1. In other words, ${\cal U}_{g}$ should come with
a choice of dressing by ${\mathbb{Z}}_{N_{f}}\subset G_{f}$ symmetry defect
operators that end on ${\cal U}_{g}$ which, due to the identification of
${\mathbb{Z}}_{N_{f}}\subset U(1)\sim{\mathbb{Z}}_{N_{f}}\subset G_{f}$, do
not act on gauge invariant local operators.

Now, if we try to unlink any charged Wilson line from the symmetry defect
operator ${\cal U}_{q}$ by cutting with charged point operators, the end
points will see the ${\mathbb{Z}}_{N_{f}}\subset G_{f}$ 0-form symmetry defect
operators attached to ${\cal U}_{q}$. The action of these
${\mathbb{Z}}_{N_{f}}$ 0-form SDOs will implement the ${\mathbb{Z}}_{N_{f}}$
action on the cut line configuration see Figure 2 so that the
${\mathbb{Z}}_{N_{f}}$ charge is preserved. Additionally, the charge of the
fermions does not forbid them to dynamically screen the Wilson lines. However,
due to the representation of the fermions, this will lead to a line operator
that has $G_{f}/{\mathbb{Z}}_{N_{f}}$ symmetry fractionalization
Delmastro:2022pfo ; Brennan:2022tyl ; Barkeshli:2014cna ; Chen:2014wse  – i.e.
a $G_{f}/{\mathbb{Z}}_{N_{f}}$ world volume anomaly – and again will be a line
operator that is charged under ${\mathbb{Z}}_{N_{f}}$ symmetry.

Figure 2: In this figure we illustrate how a Wilson line in a theory that
couples to $\frac{G_{\rm gauge}\times G_{f}}{{\mathbb{Z}}_{N_{f}}}$-bundles
can be charged under the ${\mathbb{Z}}_{N_{f}}$ symmetry even though it can be
cut by a local operator. The un-cut line operator $(a.)$ is acted on $(b.)$ by
the $g$-symmetry defect operator by crossing the 1-form symmetry defect
(blue). The line operator can then be cut by inserting a pair of charged local
operators $(c.)$. The string can be un-linked from ${\cal U}_{q}$ by pulling
the end-points through the 0-form defect (green) which is the inverse of the
linking action $(d.)$. Here the symmetry acts as a phase in correlation
functions due to the fact that ${\cal U}_{q}$ implements a singular large
gauge transformation.

We can turn on a background gauge field for this symmetry (i.e. turn on
discrete flux associated to the quotient) by wrapping the symmetry defect
operator on a non-trivial 2-cycle $\Sigma$, which, upon choosing a consistent
set of $G_{f}/{\mathbb{Z}}_{N_{f}}$ 0-form SDOs, is equivalent to a choice of
$G_{f}/{\mathbb{Z}}_{N_{f}}$ bundle with discrete flux
$w_{2}({\mathbb{Z}}_{N_{f}})=[\Sigma]^{\vee}$ in addition to turning on a
correlated discrete flux in the gauge group. However, this symmetry is un-
gaugable without also gauging all of $G_{f}$ because summing over
$w_{2}({\mathbb{Z}}_{N_{f}})$ obstruction classes/discrete fluxes requires
making a choice of $G_{f}/{\mathbb{Z}}_{N_{f}}$ bundle for each flux sector.
For this reason, we will call such a symmetry a _non-gaugable symmetry_.

We anticipate that this construction of a topological operator by dressing a
non-topological co-dimension $p$ defect operator, that is associated to a
broken symmetry, by a non-genuine, non-topological operator that lives at the
junction of $q<p$-form symmetry defects is much more general. As such, we will
refer to such symmetries as $(q,p)$-form global symmetries and denote them as
$G^{(q,p)}$. In our setting, we are discussing a $(0,1)$-form symmetry, hence
the notation ${\mathbb{Z}}_{N_{f}}^{(0,1)}$. We plan to further explore such
symmetries in future investigations.

Note that the case of a (0,1)-form global symmetry differs from the case where
a theory has a 0-form global symmetry that is the product of simply connected
Lie groups modulo an abelian group $G^{(0)}=\frac{G_{1}\times G_{2}}{\Gamma}$.
Again, we can couple to $G^{(0)}$ background gauge fields with an obstruction
class $w_{2}(\Gamma)\in H^{2}(BG^{(0)};\Gamma)$, but now the large $\Gamma$
transformations do not act on any non-trivial operators (without symmetry
fractionalization). Therefore, this does not lead to an additional symmetry.

It is important to note that these IR symmetries are broken to the UV
symmetries at energy scales $E\sim m_{W}$ by the interaction with the massive
$W$-bosons. Importantly, the $W$-boson interaction breaks the
${\mathbb{Z}}_{N_{f}}^{(0,1)}$ (0,1)-form global symmetry because the large
${\mathbb{Z}}_{N_{f}}$ gauge transformation shifts
$\dot{\varphi}\mapsto\dot{\varphi}+2\Lambda_{1}$. This locally “breaks” the
${\mathbb{Z}}_{N_{f}}^{(0,1)}$ symmetry, reflecting the fact that the dyon
lines are charged under the ${\mathbb{Z}}_{N_{f}}^{(0,1)}$ due to their
electric charge.

Now let us study the symmetries of the theory with massive fermions. Turning
on a mass breaks the flavor symmetry to (a subgroup of) $SU(N_{f})^{(0)}\to
SO(N_{f})^{(0)}$. We will consider the case where the fermions have an
$SU(n_{f})^{(0)}\subset SO(N_{f})^{(0)}$ preserving mass

$\displaystyle\begin{split}{\cal L}_{\rm
mass}=m_{\psi}\epsilon_{ij}\psi^{ia}\tilde{\psi}^{j}_{a}+c.c.\end{split}$ (72)

$m_{\psi}\in{\mathbb{R}}$ and $i,j=1,2$ are $SU(2)$ indices and
$a=1,...,n_{f}$ are $SU(n_{f})^{(0)}$ indices where here we have artificially
divided the $N_{f}$ Weyl fermions into two sets of $n_{f}$:
$\psi^{ia},\tilde{\psi}^{i}_{a}$ that transform in the fundamental and anti-
fundamental representations of $SU(n_{f})^{(0)}$ respectively.131313Here we
take $m_{\psi}\in{\mathbb{R}}_{>0}$ so that we have a $CP$-preserving mass.

In the IR the theory flows to massive $U(1)$ gauge theory with $2N_{f}$
fermions of mass $m_{\psi}$. We can then write this in terms of $N_{f}$
fermions

$\displaystyle\begin{split}\psi^{A}=\left(\psi^{+a},\tilde{\psi}^{+}_{a}\right)\quad,\quad\tilde{\psi}_{A}=\left(\tilde{\psi}_{a}^{-},\psi^{-a}\right)\quad,\quad
A=1,...,N_{f}~{},\end{split}$ (73)

so that we can write the mass term as

$\displaystyle\begin{split}{\cal L}_{\rm
mass}=m_{\psi}\left(\psi^{+a}\tilde{\psi}^{-}_{a}+\tilde{\psi}^{+}_{a}\psi^{-a}\right)+c.c.=m_{\psi}\psi^{A}\tilde{\psi}_{A}+c.c.~{}.\end{split}$
(74)

Here we see in the IR that the theory has a $SU(N_{f})^{(0)}$ global symmetry
that acts on these $\psi^{A},\tilde{\psi}_{A}$. Here we have organized the
fermions based on their electric charge (as in the massless case) so that the
positively (negatively) charged fermions transform in the fundamental (anti-
fundamental) representation of $SU(N_{f})^{(0)}$. This means that the symmetry
group that acts faithfully is

$\displaystyle\begin{split}G_{\rm ferm}=\frac{U(1)_{\rm gauge}\times
SU(N_{f})^{(0)}}{{\mathbb{Z}}_{N_{f}}}~{}.\end{split}$ (75)

Again, the theory has a $U(1)^{(1)}$ 1-form magnetic symmetry and the quotient
implies an additional emergent ${\mathbb{Z}}_{N_{f}}^{(0,1)}$ (0,1)-form
global symmetry at energies below $E\sim m_{W}$.

###  3.1 Constraints on Monopole-Fermion Scattering from Symmetries

Now we will discuss the constraints that the conserved global symmetries
impose on monopole-fermion scattering. Here we consider scattering fermions at
energies $E\ll m_{W}$ so that the entire scattering process can be described
in the low energy effective QED theory. The standard description of the Callan
Rubakov effect is that an incident fermion will interact with the dyon degree
of freedom due to the polarization of the s-waves in the fermion mode
expansion (29). Then, by conservation of angular momentum and gauge symmetry,
fermion scattering in the $j=0$ sector leads to charge accumulation on the
monopole which is then radiated away due to the heavy mass of $\varphi(t)$.
For the moment, we will assume that charge accumulation is a general property
of monopole-fermion scattering and discuss what the generalized symmetries
imply.

Consider scattering a charged fermion off of a monopole. An incident fermion
will generically reach the monopole core. Besides examining the explicit
fermion mode expansion, this follows from classical physics because a point
particle can run into the monopole core if it approaches along the radial
magnetic field lines due to the lack of a Lorentz force. Inside the monopole
core, the scalar vev vanishes and $SU(2)$ gauge symmetry is restored. As the
fermion scatters, it interacts with the dyon degree of freedom via the $SU(2)$
gauge interaction with the $W$-boson. Momentarily forgetting about energetics,
this excites a $W$-boson and transforms the monopole into a dyon of charge 2.

Now, the fact that there is a (0,1)-form ${\mathbb{Z}}_{N_{f}}^{(0,1)}$ global
symmetry, which acts on electrically charged line operators, implies that
electric charge of the dyon cannot be radiated away without breaking the
0-form flavor symmetry. This explicitly forbids the creation of asymptotic
states with fractional fermion number and contradicts the original analysis of
the Callan Rubakov effect which did not make use of this additional symmetry.

If the dyon charge cannot dissipate, then the scattered fermion is propagating
in the presence of an electrically charged line operator. This changes the
relevant mode quantization and allows for a fermion bound state if the
scattered fermion does not have enough energy to overcome the electrostatic
potential.

In the case where we scatter $n_{f}=\frac{N_{f}}{2}$ fermions there is $N_{f}$
charge accumulated and the ${\mathbb{Z}}_{N_{f}}^{(0,1)}$ symmetry does not
protect the dyon degree of freedom from decaying. Here, the dyon will decay
into bulk, s-wave fermion modes by distributing the electric charge among the
$N_{f}$ out-going s-wave modes. Here, we can see this from the fact that
integrating out the dyon degree of freedom gives a local ‘t Hooft operator for
the minimal $SU(2)$ instanton Fan:2021ntg . In our theory, the fact that there
are $N_{f}$ flavors means that there is an $N_{f}$-fermion local operator
through which the dyon degree of freedom can decay.

We would like to emphasize that since we are scattering at energies $E\ll
m_{W}$, that the ${\mathbb{Z}}_{N_{f}}^{(0,1)}$ (0,1)-form global symmetry is
not violated. This simple analysis from symmetries suggests that monopole-
fermion scattering behaves qualitatively different than previous proposals.

To complete the analysis of monopole-fermion scattering from symmetry
constraints, we must now discuss how our proposal is consistent with
conservation of energy. Naively, the energy required to excite the monopole
into a dyon is on the order of the cutoff scale $E\sim m_{W}$. This can be
seen from a number of arguments, but the simplest is probably to just note
that the dyon is a monopole with an excited gauge boson which has mass
$m_{W}$. However, not all dyons are constructed in this way.

Let us recall the Witten effect Witten:1979ey . Here we can take $U(1)$
Maxwell theory with a $4d$ $\theta$-angle:

$\displaystyle\begin{split}S=\int\frac{1}{2g^{2}}F\wedge\ast
F+\frac{i\theta_{4d}}{8\pi^{2}}\int F\wedge F~{}.\end{split}$ (76)

As discussed in Witten:1979ey , a non-zero $\theta_{4d}$ induces an electric
charge on the monopole. Classically, this occurs because the $\theta$-term can
be written as

$\displaystyle\begin{split}\frac{i\theta_{4d}}{8\pi^{2}}\int F\wedge
F=\theta_{4d}\int\vec{E}\cdot\vec{B}~{}.\end{split}$ (77)

Since the magnetic monopole acts as a source for $B$, the interaction above
acts to source an electric charge $q_{e}=\frac{\theta_{4d}}{2\pi}q_{m}$. In
particular, shifting $\theta$ by $4\pi$ shifts the electric charge of the
minimal monopole by $+2$.

In the quantum theory, a $\theta$-angle can be generated by integrating out
heavy fermions that have a $CP$-violating mass term
$m_{\psi}=M_{\psi}e^{i\theta_{4d}}$ for $M_{\psi}>0$. Here, the electric
charge is generated by the fermion vacuum which can be computed by summing
over the contribution to the (renormalized) charge density from the continuum
of negative energy states in the Dirac sea. The vacuum charge density was
computed in Yamagishi:1982wp ; Grossman:1983yf  and results in the effective
charge density:

$\displaystyle\begin{split}\rho(r)=-\frac{M_{\psi}\,\sin(\theta_{4d})}{\pi}\int_{M_{\psi}}^{\infty}\frac{k\,e^{-2kr}dk}{\sqrt{k^{2}-M_{\psi}^{2}}(k+M_{\psi}\cos(\theta_{4d}))}~{}.\end{split}$
(78)

Here the integrand is dominated by $k$ near $M_{\psi}$ so that the charge
density is localized around the monopole with a radius of order $R_{c}\sim
1/|m_{\psi}|$. This contributes to the mass of the dyon on the order of
$|m_{\psi}|$ due to the energy in the electric field.

These energetics are also reflected in the quantum mechanics governing the
dyon degree of freedom. In the monopole background, we can spherically reduce:

$\displaystyle\begin{split}\frac{i\theta_{4d}}{2\pi}\int\frac{F\wedge
F}{4\pi}=\frac{i\theta_{4d}}{2\pi}\int
F_{tr}\,dtdr=\frac{i\theta_{4d}}{4\pi}\int(\dot{\varphi}-2a_{0})\,dt~{},\end{split}$
(79)

so that the $4d$ $\theta$-term induces a $1d$ $\theta$-term with
$\theta_{1d}=\frac{\theta_{4d}}{2}$ in the dyon quantum mechanics where in the
final step we inferred the coupling to $\varphi$ by imposing gauge invariance
on the boundary. As we have discussed above, shifting $\theta_{4d}$ from 0 to
$4\pi$ shifts $\theta_{1d}$ by $2\pi$ so that the ground state of the quantum
mechanics is given by the momentum eigenstate $|1\rangle$ which sources a bulk
electric charge field of charge 2. We would like to emphasize that these
effects are not simply correlated, but rather are manifestations of the same
physical effect as they both can be understood as coming from a shift in the
$4d$ fermionic vacuum which couples to the gauge field as $\theta_{4d}$.

Together the constraints from gauge symmetry and angular momentum and from
${\mathbb{Z}}_{N_{f}}^{(0,1)}$ symmetry imply that monopole scattering, at
least in the $j=0$ sector, leads to charge accumulation on the monopole which
cannot be radiated away unless the total charge is a multiple of $N_{f}$. This
suggests, due to the fact that the mass of the dyon degree of freedom is given
by $m_{W}$, that the charge accumulation must be possible at low energy such
as via the Witten effect. Indeed, as we will discuss in detail in the
following sections, the phase mode of the fermion acts like an axion field so
that propagating charged fermions dynamically rotate the effective
$\theta$-angle. For example, we will show that in s-wave scattering, where a
positively charged in-going fermion scatters into a negatively charged out-
going fermion, there is an effective $4\pi$ $\theta$-angle rotation on the
monopole that accounts for the conservation of gauge charge. One of the main
goals of the rest of the paper to demonstrate this dynamical Witten effect
explicitly by analyzing the fermion modes similarly to the original analysis
of Callan:1982ah .

We would also like to comment on what our proposal implies for the massless
limit. Due to the fact that the electric charge of the dyon degree of freedom
is generated by the fermion vacuum, the charge radius is parametrically given
by $R_{c}\sim 1/m_{\psi}$. In the massless limit, we see that the total charge
of the dyon remains fixed, but that the charge radius goes to infinity and the
charge density becomes infinitely dilute. In the strictly massless theory,
this is similar to a DC shift of the vacuum electric charge, but this only
strictly makes sense with an IR regulator. This is similar to the idea of
Intriligator:2013lca  which suggests that the Hilbert space has different
super-selection sectors corresponding to the plane-wave normalizable zero-
energy modes so that fermion scattering shifts the vacuum state along the
space of these super selection sectors. In the case at hand, these super-
selection sectors may come from to the $E=0$, $j=0$ fermion modes which are
non-normalizable but generate a constant charge density.141414 Since these
modes contribute a finite charge density, they will generate infinite total
charge in infinite volume which requires regulation/renormalization matching
with our discussion above.

##  4 Monopole-Fermion S-Wave Scattering

Now we would like to study the scattering of spherically symmetric (i.e.
$j=0$) fermion modes off of the spherically symmetric BPS monopole.

First, we would like to argue that there is a consistent truncation to the
theory of $j=0$ modes. As we have discussed, all non-abelian modes of the UV
gauge field except for the dyon degree of freedom are gapped out with an
effective mass of order $m_{W}$ Brennan:2021ewu . This means that the $j=0$
fermion modes do not interact with the higher angular momentum modes of the
$W$-boson in the monopole core. Since the dyon is a spherically symmetric
mode, this implies that the interaction between the dyon degree of freedom and
the fermions does not mix the fermionic angular momentum modes. Therefore an
in-coming $j=0$ fermion mode can only scatter into an out-going $j=0$ fermion
mode, while at most exciting the dyon due to conservation of angular
momentum.151515Note that the gauge field can mediate interactions between the
different angular momentum sectors, but these are perturbatively small and we
will ignore their effect in this paper. Additionally, it also implies that the
dyon can only decay into the $j=0$ modes. The reason is that we can think of
the excited dyon state as the trivial in-coming state which has vanishing
angular momentum and therefore the interaction only allows for radiation into
the $j=0$ modes. This fact is also obvious since the source term is
spherically symmetric and hence can only decay in a spherically symmetric
way.161616Here since we are interested in the monopole-fermion interaction,
for now we will also ignore the fluctuations of the higher angular momentum
modes of the dynamical $U(1)$ gauge field. We will comment on their effects in
the case of general scattering in Section 5 where the interaction will simply
be related to the effects of a propagating charged fermions in $4d$.  All
together, this implies that there is a consistent truncation to the $j=0$
sector of the theory.

Now, we would like to study in detail the scattering of a spherically
symmetric fermion off of the monopole. This scenario will encapsulate much of
the physics of and act as a warm up for the general fermion-monopole
scattering which we will discuss in the following section.

First we will derive the effective theory for massless fermions and then
consider the effect of turning on a mass. Because of our previous discussion,
the s-wave scattering can be studied in the truncated theory where we restrict
to only the $j=0$ modes of the fermions. As first discussed by Callan:1982au ;
Callan:1982ah ; Callan:1982ac , the truncated theory is equivalent to an
effective $2d$ theory of radially propagating fermion shells.

To see this, first note that the $j=0$ angular momentum modes of the massless
fermions can be decomposed as

$\displaystyle\begin{split}\psi^{A}=\frac{U(\theta,\phi)\,e^{-i\phi/2}}{r}\int
dk\left(\begin{array}[]{c}0\\\
e^{ik(t+r)}\chi_{k}^{A}\end{array}\right)~{},\quad\tilde{\psi}^{A}=\frac{U(\theta,\phi)\,e^{i\phi/2}}{r}\int
dk\left(\begin{array}[]{c}e^{ik(t-r)}\tilde{\chi}_{k}^{A}\\\
0\end{array}\right)~{},\end{split}$ (80)

where $A=1,...,N_{f}$ and $\chi_{k}^{A},\tilde{\chi}_{k}^{A}$ are the
Grassman-valued mode operators and $U(\theta,\phi)$ is the frame rotation as
described in the previous section in (40).

Because of this decomposition, we can reduce the $4d$ fermions to effective
$2d$ left- and right-moving fermions $\chi^{A},\tilde{\chi}^{A}$:

$\displaystyle\begin{split}\psi^{A}=\frac{U(\theta,\phi)\,e^{-i\phi/2}}{r}\left(\begin{array}[]{c}0\\\
\chi^{A}(t+r)\end{array}\right)\quad,\quad\tilde{\psi}^{A}=\frac{U(\theta,\phi)\,e^{i\phi/2}}{r}\left(\begin{array}[]{c}\tilde{\chi}^{A}(t-r)\\\
0\end{array}\right)~{},\end{split}$ (81)

as described in Callan:1982au ; Callan:1982ah ; Callan:1982ac . Plugging the
$j=0$ mode expansion into the $4d$ UV Lagrangian leads to effective $2d$
theory of fermions propagating on the $(r,t)$ half-plane that interact with
the dyon degree of freedom on the boundary at $r=0$ as Brennan:2021ewu

$\displaystyle\begin{split}S_{int}&=\int
d^{2}x\,\delta(x)e^{-i\varphi}\bar{\tilde{\chi}}_{A}\chi^{A}+c.c.~{}.\end{split}$
(82)

This boundary interaction imposes the boundary condition on the fermions

$\displaystyle\begin{split}\chi^{A}=e^{i\varphi}\tilde{\chi}^{A}\big{|}_{r=0}~{}.\end{split}$
(83)

Note that the boundary condition comes from the non-abelian interaction with
the $W$-boson and locall breaks $SU(N_{f})^{(0)}\times SU(N_{f})^{(0)}\to
SU(N_{f})^{(0)}$ and the ${\mathbb{Z}}_{N_{f}}^{(0,1)}$ (0,1)-form global
symmetry. This local “breaking” of the (0,1)-form symmetry is the statement
that the dyonic lines are charged under the 1-form global symmetry equivalent
to how charged local operators transform non-trivially under gauge
transformations of a 0-form symmetry.

Now by doing a field redefinition

$\displaystyle\begin{split}\chi\to
e^{i\varphi/2}\chi\quad,\quad\tilde{\chi}\to
e^{-i\varphi/2}\tilde{\chi}~{},\end{split}$ (84)

we can exchange this boundary condition/interaction for a bulk interaction
term as in Polchinski:1984uw :171717Here we drop the Heaviside step function
which is used as a regulator. Additionally, note that the spherical reduction
of the gauge kinetic term in the above action comes with a factor of $r^{2}$
due to measure for spherical coordinates. The spatially dependent factor
prevents the gauge theory from confining as is standard in $2d$ QED.

$\displaystyle\begin{split}S=&\int
d^{2}x\left(i\bar{\chi}\left(D_{+}+\frac{i}{2}\dot{\varphi}\right)\chi+i\bar{\tilde{\chi}}\left(D_{-}-\frac{i}{2}\dot{\varphi}\right)\tilde{\chi}+\frac{r^{2}}{2g^{2}}F_{tr}^{2}\right)\\\
&+\frac{1}{2m_{W}}\int dt(\dot{\varphi}-2a_{0})^{2}~{}.\end{split}$ (85)

After the field redefinition (84), the boundary conditions for the fermion
fields are given

$\displaystyle\begin{split}\chi=\tilde{\chi}\Big{|}_{r=0}~{},\end{split}$ (86)

which is still gauge invariant due to the fact that the field redefinition
involved the Goldstone field $\varphi$.

Now we can consider the effect of turning on a mass term. In the spherical
reduction, this becomes a $2d$ mass term

$\displaystyle\begin{split}{\cal
L}_{4d}=m_{\psi}\psi^{A}\tilde{\psi}_{A}+c.c.\quad\longrightarrow\quad{\cal
L}_{2d}=m_{\psi}\chi^{A}\tilde{\chi}_{A}+c.c.~{}.\end{split}$ (87)

However, in rewriting the mass term in a $SU(N_{f})^{(0)}$ symmetric way, we
also had to redefine our fields (73) so that the gauge interaction now becomes

$\displaystyle\begin{split}S_{int}&=\int
d^{2}x\,\delta(x)e^{-i\varphi}\epsilon_{AB}\bar{\tilde{\chi}}^{A}\chi^{B}+c.c.~{}.\end{split}$
(88)

where here $\epsilon_{AB}=\mathds{1}_{n_{f}}\otimes\sigma^{1}$. This
interaction leads to the boundary condition on the fermions

$\displaystyle\begin{split}\chi^{A}=\epsilon^{AB}e^{i\varphi}\tilde{\chi}_{B}~{}.\end{split}$
(89)

Again, this boundary condition breaks the 0-form global symmetry
$SU(N_{f})^{(0)}\to SU(n_{f})^{(0)}$ to its UV form and breaks
${\mathbb{Z}}_{N_{f}}^{(0,1)}$ (0,1)-form global symmetry and implies that the
dyonic lines are charged under the (0,1)-form global symmetry.

It will also be instructive to study the bosonized theory. Here we will
bosonize the fermions as181818As stressed in vanBeest:2023dbu , here we are
simply bosonizing the Cartan of the full $U(N_{f})$ current algebra generated
by the fermions.

$\displaystyle\begin{split}\bar{\chi}_{A}\chi^{A}=D_{+}H_{A}\quad,\quad\bar{\tilde{\chi}}^{A}\tilde{\chi}_{A}=D_{-}{\widetilde{H}}_{A}~{},\end{split}$
(90)

for fixed $A$ and $H_{A},{\widetilde{H}}_{A}$ are left-/right-moving chiral
bosons. The boundary conditions (90) are given in terms of the
$H_{A},{\widetilde{H}}_{A}$

$\displaystyle\begin{split}H_{A}={\widetilde{H}}_{A}\big{|}_{r=0}~{},\end{split}$
(91)

for the massless case and

$\displaystyle\begin{split}H_{A}=\epsilon_{AB}{\widetilde{H}}_{B}\big{|}_{r=0}~{},\end{split}$
(92)

for the massive case. As in the fermionic description, these boundary
conditions break $SU(N_{f})^{(0)}\times SU(N_{f})^{(0)}\to SU(N_{f})^{(0)}$
and $SU(N_{f})^{(0)}\to SU(n_{f})^{(0)}$ respectively but preserve gauge
invariance due to the field redefinition (84).

We can then write the action in terms of the complex scalar fields
$h_{A}=H_{A}+{\widetilde{H}}_{A}$ as

$\displaystyle\begin{split}S=\int d^{2}x\left(|\partial
h|^{2}-F_{tr}h+\frac{r^{2}}{2g^{2}}F_{tr}^{2}\right)+\int
dt\left(\frac{(\dot{\varphi}-2a_{0})^{2}}{2m_{W}}+\frac{(\dot{\varphi}-2a_{0})}{4\pi}h\right)\end{split}$
(93)

for the massless case and and as

$\displaystyle\begin{split}S=&\int d^{2}x\left(|\partial
h|^{2}-F_{tr}h-m_{\psi}^{2}\cos\left(\frac{h_{A}}{2}\right)+\frac{r^{2}}{2g^{2}}F_{tr}^{2}\right)\\\
&+\int
dt\left(\frac{(\dot{\varphi}-2a_{0})^{2}}{2m_{W}}+\frac{(\dot{\varphi}-2a_{0})}{4\pi}h\right)\end{split}$
(94)

for the massive case where $h=\sum_{A}h_{A}$. It will also be useful to
consider the decomposition of $h(t,r)$ into left- and right-handed components:
$h=\Theta(t-r)+\tilde{\Theta}(t+r)$.

As discussed in vanBeest:2023dbu ; Callan:1982au , there are quantum
corrections to the effective action. The corrections investigated there can be
interpreted as the 1-loop contribution to the gauge-mediated 4-fermion
interaction restricted to the $j=0$ fermion modes.191919Note that there is no
2-fermion mass term generated for massless fermions at 1-loop order by gauge
interactions due to symmetry. In the $2d$ effective action, this becomes a
4-fermion term which, upon bosonization, maps to a cosine-potential with
coupling strength that goes like $g^{2}/r^{2}$. However, this 1-loop
correction couples the $j=0$ modes to all higher angular momentum modes at the
same order in perturbation theory. This interaction requires further analysis
as we do not usually expect 4-fermion interactions to gap out fermions in
$4d$. Here, we will consider the weak coupling limit and ignore these
corrections, leaving a more complete analysis for future
investigation.202020Another reason why we may be able to safely ignore these
corrections is the following. As discussed in vanBeest:2023dbu ; Callan:1982au
, one of the corrections lead to a repulsive $1/r^{2}$ potential. In $2d$,
this is a short range potential – it corresponds to a $1/r^{4}$ potential in
$4d$. We expect that the physics with a short-range repulsive potential is
similar to the interacting resonant model (IRM) in $2d$ RLM1 ; RLM2 . The IRM
theory describes $2d$ fermions interacting with a defect with a delta function
repulsive potential. The IRM model is integrable and has been solved exactly –
the bulk fermion modes can tunnel through the short-range repulsive potential
to interact with the defect degrees of freedom. We expect that a similar
physical picture may occur here due to the short range nature of $1/r^{2}$
potentials in $2d$.

Now we would like to consider the ${\mathbb{Z}}_{N_{f}}^{(0,1)}$ (0,1)-form
global symmetry. As we discussed, the ${\mathbb{Z}}_{N_{f}}^{(0,1)}$ global
symmetry must act on $\varphi$ since
$\dot{\varphi}\mapsto\dot{\varphi}+2\Lambda_{1}$ under large gauge
transformations. In the field redefinition (84) we shift the bulk fields by
the dyon degree of freedom so that the fermions, which originally are
invariant under the diagonal ${\mathbb{Z}}_{N_{f}}$ subgroup of $U(1)_{\rm
gauge}$ and $SU(N_{f})\times SU(N_{f})$ or $SU(N_{f})$, now transform non-
trivially under ${\mathbb{Z}}_{N_{f}}^{(0,1)}$ transformations (96):

$\displaystyle\begin{split}dH_{A}\longmapsto dH_{A}+\Lambda_{1}\quad,\quad
d{\widetilde{H}}_{A}\longmapsto
d{\widetilde{H}}_{A}-\Lambda_{1}\quad,\quad\dot{\varphi}\longmapsto\dot{\varphi}+2\Lambda_{1}~{}.\end{split}$
(95)

To determine the intrinsic action on the fields, one must realize that in a
non-trivial ${\mathbb{Z}}_{N_{f}}^{(0,1)}$ background,
$H_{A},{\widetilde{H}}_{A},\varphi$ are sections of a $U(1)$ bundle. We can
think of them as defined in local patches where the
${\mathbb{Z}}_{N_{f}}^{(0,1)}$ background gives a rule for transformation
between the patches by a ${\mathbb{Z}}_{N_{f}}^{(0)}$ action:

$\displaystyle\begin{split}H_{A}\longmapsto H_{A}+\frac{2\pi
n}{N_{f}}\quad,\quad{\widetilde{H}}_{A}\longmapsto{\widetilde{H}}_{A}-\frac{2\pi\tilde{n}}{N_{f}}\quad,\quad\varphi\longmapsto\varphi+\frac{4\pi}{N_{f}}~{}.\end{split}$
(96)

Importantly, the fields $\Theta,\tilde{\Theta}$ are invariant and hence define
global sections because they have charge $N_{f}$ under the
${\mathbb{Z}}_{N_{f}}$ symmetry. This means that the coupling between
$\varphi(t)$ and $\Theta,\tilde{\Theta}$ is not invariant under under
${\mathbb{Z}}_{N_{f}}^{(0,1)}$ global symmetry which reflects the fact that
the dyons are charged with respect to this symmetry.

###  4.1 Dyon Decay

Now we will look more carefully at the process of dyon decay. We will now
explicitly demonstrate that the conserved (0,1)-form
${\mathbb{Z}}_{N_{f}}^{(0,1)}$ global symmetry implies that the dyon does not
decay unless it has vanishing charge mod${}_{N_{f}}$.

Let us again consider the quantum mechanics describing the dyon degree of
freedom. This is essentially given by the charged particle on the ring coupled
to an external magnetic flux:

$\displaystyle\begin{split}L=\frac{1}{2m_{W}}(\dot{\varphi}-2a_{0})^{2}+\frac{\theta}{2\pi}(\dot{\varphi}-2a_{0})~{},\end{split}$
(97)

where $\theta$ is a topological theta angle. As we have discussed in Section
2, this theory has a conserved momentum
$\hat{p}=\frac{1}{m_{W}}(\dot{\varphi}-2a_{0})+\frac{\theta}{2\pi}$ which
grades the Hilbert space:

$\displaystyle\begin{split}\hfil\displaystyle{\cal
H}=\displaystyle\Big{\\{}|n\rangle~{},~{}n\in{\mathbb{Z}}~{}\Big{|}~{}\hat{p}|n\rangle=n|n\rangle\Big{\\}}~{}.\end{split}$
(98)

In a trivial background electric field, each state $|n\rangle$ has an energy

$\displaystyle\begin{split}\hat{H}|n\rangle=\frac{m_{W}}{2}\left(n^{2}-\frac{\theta}{2\pi}\right)^{2}|n\rangle~{}.\end{split}$
(99)

and sources a non-trvial bulk $A_{0}$ corresponding to a point particle of
charge $n$.

In our setup, $\varphi(t)$ couples to a dynamical $\theta$ which is the
boundary value of a $2d$ periodic field

$\displaystyle\begin{split}\theta=\frac{h}{2}\big{|}_{r=0}=\frac{1}{2}\left(\Theta+\tilde{\Theta}\right)\big{|}_{r=0}~{}.\end{split}$
(100)

Note here that $h,\Theta,\tilde{\Theta}$ are all invariant under the
$SU(N_{f})^{(0)}\times SU(N_{f})^{(0)}$ or $SU(N_{f})^{(0)}$ flavor symmetries
as appropriate and that $\Theta,\tilde{\Theta}$ have gauge charges $\pm N_{f}$
respectively.

The coupling between $\varphi(t)$ and $\Theta+\tilde{\Theta}$ leads to an
instability of the quantum mechanical theory. This instability allows the
energy levels in (14) to decay to $|0\rangle$ by emitting bulk solitons. For
example, the boundary value of $\Theta+\tilde{\Theta}$ can jump by $\pi N_{f}$
by emitting a $\tilde{\Theta}$ (winding) kink solution. This shifts the energy
levels by $n_{f}$ units as can be seen from the energy equation (14).

One may then be tempted to conclude that the the dyon can decay by a single
energy level by a kink where $\tilde{\Theta}\to\tilde{\Theta}+4\pi$. However,
such a process will violate the global symmetries of the theory. In the
following, we will describe the dynamics in terms of chiral fields. Let us
parametrize the scalar degrees of freedom in terms of $\Theta,\tilde{\Theta}$
as above and: 212121Here there is a ${\mathbb{Z}}_{N_{f}}$ redundancy from the
fact that  $\displaystyle\begin{split}\Theta=\sum_{n}n\,\Theta_{n}~{}{\rm
mod}_{N_{f}}\quad,\quad\tilde{\Theta}=\sum_{n}n\,\tilde{\Theta}_{n}~{}{\rm
mod}_{N_{f}}~{}.\end{split}$ (101) This ${\mathbb{Z}}_{N_{f}}$ relation is
enforced by a ${\mathbb{Z}}_{N_{f}}$ gauge field under which both sides of the
above equation have charge 1. Note that there is not a unique charge
assignment of the $\Theta_{i},\tilde{\Theta}_{i}$ due to the individual
periodicity of the $\Theta_{i}$. The choice is equivalent to choosing a
representative of the generator of the center of $SU(N_{f})^{(0)}$. One can
make the simple choice that $\Theta_{1},\tilde{\Theta}_{1}$ have charge 1 and
all other $\Theta_{i},\tilde{\Theta}_{i}$ are uncharged.

$\displaystyle\begin{split}\Theta_{i}=H_{i}-H_{i+1}~{},\quad\tilde{\Theta}_{i}={\widetilde{H}}_{i}-{\widetilde{H}}_{i+1}~{},\quad
i=1,...,N_{f}-1~{}.\end{split}$ (102)

The periodicity of the new fields is given by

$\displaystyle\begin{split}\Theta\sim\Theta+2\pi
N_{f}\quad,\quad\Theta_{i}\sim\Theta_{i}+2\pi N_{f}~{},\end{split}$ (103)

and similarly for $\Theta,\tilde{\Theta}_{i}$ with boundary conditions:

$\displaystyle\begin{split}\Theta-\tilde{\Theta}\big{|}_{r=0}=0\quad,\quad\Theta_{A}-\tilde{\Theta}_{A}\big{|}_{r=0}=0~{}.\end{split}$
(104)

Now let us consider the charge 2 dyon state. This state is uncharged under the
flavor $SU(N_{f})^{(0)}$ or $SU(N_{f})^{(0)}\times SU(N_{f})^{(0)}$ global
symmetry, but carries charge under the ${\mathbb{Z}}_{N_{f}}^{(0,1)}$
(0,1)-form global symmetry. The dyon degree of freedom couples to the bulk
fields $\Theta,\tilde{\Theta}$ as well as possibly indirectly to
$\Theta_{i},\tilde{\Theta}_{i}$. The fields $\Theta_{i},\tilde{\Theta}_{i}$
are all charged under the global 0-form symmetry $SU(N_{f})^{(0)}$ or
$SU(N_{f})^{(0)}\times SU(N_{f})^{(0)}$ as appropriate while the fields
$\Theta,\tilde{\Theta}$ are uncharged. Additionally, following our previous
discussion we also see that all of the bulk fields
$\Theta_{i},\tilde{\Theta}_{i},\Theta,\tilde{\Theta}$ are uncharged under the
${\mathbb{Z}}_{N_{f}}^{(0,1)}$ global symmetry. This means that any dyon state
which has non-trivial charge under the ${\mathbb{Z}}_{N_{f}}^{(0,1)}$ global
symmetry cannot decay to any of the bulk $\Theta,\tilde{\Theta}$ modes.
Together, the conservation of the 0-form flavor symmetry and the (0,1)-form
symmetry forbids the decay of the dyon level except by multiples of $N_{f}$
via a $\tilde{\Theta}$-winding mode.

Another way to rephrase the fact that there is a preserved
${\mathbb{Z}}_{N_{f}}^{(0,1)}$ global symmetry, is the statement that the
fermion-dyon interaction term does not facilitate the radiation of bulk
fermion modes with charges less than $N_{f}$ due to the preserved
$SU(N_{f})^{(0)}\times SU(N_{f})^{(0)}$ or $SU(N_{f})^{(0)}$ symmetry. The
reason is that fermion modes can only carry integer charge, so there is no way
to divide up $Q\neq kN_{f}$ electric charge among $N_{f}$ fermions in an
$SU(N_{f})^{(0)}\times SU(N_{f})^{(0)}$ or $SU(N_{f})^{(0)}$ symmetric way.

As described in vanBeest:2023dbu , the out-going state described in the
standard Callan Rubakov effect corresponds to the $2\pi$ $\tilde{\Theta}$-kink
which is generated by $e^{i\frac{\tilde{\Theta}}{N_{f}}}$. However, this
operator is not well defined since it is not invariant under
${\widetilde{H}}_{A}\sim{\widetilde{H}}_{A}+2\pi$ which is a type of gauge
symmetry. The proposal of vanBeest:2023dbu  is that this issue can be cured by
attaching a line operator which compensates for the gauge non-invariance.
Physically, this is equivalent to taking the $N_{f}^{th}$-root of a well
defined operator $e^{i\tilde{\Theta}}=\prod_{A}\tilde{\chi}_{A}$ which
introduces a co-dimension 1 space time branch cut in all correlation
functions.

###  4.2 Scattering and Energetics

Now let us describe what happens when we scatter a single fermion mode off of
the monopole. Consider sending in a $j=0$ fermion state for $\psi^{A}$ with
fixed $A$. In the bosonized description, this is the same thing as sending in
a kink configuration for $H_{A}$ that winds by $2\pi$. This can be seen from
the action (93) – (94) where the coupling between
$h_{A}=H_{A}+{\widetilde{H}}_{A}$ and the gauge field is equivalent to the
coupling of the gauge field to the winding current of $h_{A}$.

Now from the boundary conditions on $H_{A}$ (91) – (92), we see that the
incoming kink reflects as a ${\widetilde{H}}_{A}$ kink. This shifts the
effective $1d$ $\theta$-angle by $2\pi$ and shifts the boundary state momentum
by 1 unit. This means that we have effectively excited the dyon degree of
freedom to the $n=1$ state so that the monopole has now become a dyon of
charge $2$:

$\displaystyle\begin{split}\psi^{A}+M~{}\longrightarrow\tilde{\psi}^{A}+D~{}.\end{split}$
(105)

Now we would like to explain the most important feature of this scattering:
how does an incident fermion impart gauge charge onto the monopole? First, we
will consider the case of massive fermions – we will then discuss the massless
limit $m_{\psi}\to 0$.

Let us address the fact that the monopole-fermion scattering can impart a
${\mathbb{Z}}_{N_{f}}^{(0,1)}$ global charge on the monopole. This fact may
appear strange because the fermion fields are invariant under the
${\mathbb{Z}}_{N_{f}}^{(0,1)}$ symmetry. However, the crucial point is that
physical scattering states are necessarily gauge invariant. These are
constructed by dressing the bare (i.e. gauge non-invariant) fermion scattering
state by gauge fields which allows them to transform non-trivially under the
${\mathbb{Z}}_{N_{f}}^{(0,1)}$ symmetry. These states can be thought of as
being created by fermion fields that are connected to Wilson lines and hence
can transform under the (0,1)-form symmetry because the operator that creates
the state is now no longer local. Said differently, asymptotic scattering
states are gauge invariant, but carry $SU(N_{f})^{(0)}$ (or
$SU(N_{f})^{(0)}\times SU({N_{f}})^{(0)}$) charge and therefore must transform
non-trivially under ${\mathbb{Z}}_{N_{f}}\subset SU({N_{f}})^{(0)}$ (or
${\mathbb{Z}}_{N_{f}}\subset SU(N_{f})^{(0)}\times SU(N_{f})^{(0)}$). This
means that the asymptotic scattering states can carry and impart
${\mathbb{Z}}_{N_{f}}^{(0,1)}$ (0,1)-form global charge.

Next we would like to discuss in detail how gauge charge is imparted to the
monopole at parametrically low energies. As we have mentioned the coupling
between the effective $2d$ fermions and the dyon degree of freedom can be
written as a dynamical $1d$ $\theta$-angle, the interaction can induce a shift
of the energy levels of the QM by a $1d$ Witten Effect. As we have discussed,
an in-coming fermion mode is described by an $H_{A}$-kink which winds by
$2\pi$ and the out-coming mode is a ${\widetilde{H}}_{A}$-kink which winds by
$2\pi$ so that the effective $1d$ $\theta$-angle in the dyon quantum mechanics
shifts by $2\pi$. This leads to a shift in the energy levels so that the
quantum mechanics ground state shifts from $|0\rangle\to|1\rangle$ and turns
the monopole into a dyon of charge 2.

In terms of the bulk $4d$ fields, the $H_{A},{\widetilde{H}}_{A}$ fields
describe the phase mode of the $\psi^{A},\tilde{\psi}^{A}$ $j=0$ modes. Thus,
a bosonic kink describes a propagating phase shift of
$\psi^{A},\tilde{\psi}^{A}$ by $2\pi$ which induces a $4d$ $\theta$-angle:
$\theta_{4d}=4\pi$. As we discussed in Section 3, this induces an electric
charge on the monopole by a shift in the fermion vacuum charge which is
distributed in a cloud around the monopole of radius $R_{c}\sim 1/m_{\psi}$.
This bulk electric field contributes to the mass of the dyon

$\displaystyle\begin{split}\Delta E_{\rm dyon}\sim m_{\psi}~{},\end{split}$
(106)

so that exciting the dyon degree of freedom/fermion vacuum via the shift in
$\theta$ only requires energy scales that are comparable to the energy
required for fermion scattering.

After scattering, the dyon charge sources an electric field that interacts
with the out-going fermion modes. Since the charge cloud and out-going
fermions have opposite charge so that the dyon can trap the out-going fermions
in its electrotatic potential. As discussed in Zhang:1988ab ;
Ravendranadhan:1989vp ; Tang:1982fc , the dyon has bound states that are
similar to that of the hydrogen atom with binding energy that scales
parametrically as

$\displaystyle\begin{split}E_{\rm binding}\sim m_{\psi}~{}.\end{split}$ (107)

This suggests that incident fermions will be captured by the excited dyon if
they are scattered at energy scales of $E_{\rm scatt}\lesssim 2m_{\psi}$
whereas for $E_{\rm scat}>>m_{\psi}$, fermions will have enough energy to
escape the dyon’s electrostatic potential.

In the case of massless fermions where $m_{\psi}\to 0$, the dyon can be
interpreted as a monopole with an infinitely disperse cloud of fermions
(carrying electric charge). Additionally, in this limit the binding energy of
the dyon goes to zero which means that any scattered fermion will impart a
charge by shifting the fermion vacuum and then will escape off to infinity. It
is possible that this may be realized in the renormalization of the infinite
vacuum contribution to the charge density.

We can think of the Hilbert space of our theory as belonging to a family of
Hilbert spaces that are indexed by this vacuum charge density. Such a family
of Hilbert spaces with varying vacuum charge density naturally arises in our
theory from considering the family of vacuum states that arise from quantizing
the zero energy mode of the $j=0$ fermion modes.222222Note that the zero
energy mode belongs to a continuum of scattering states and so it may seem
strange to separate this mode from the others in the continuum which can be
used to construct smooth states by Fourier transform smooth distributions.
However, since the zero energy mode is at the bottom of the continuum, all
smooth states will be constructed by the Fourier transform of a function
$F(k)$ with $\lim_{k\to 0}F^{(n)}(k)=0$ $\forall n\geq 0$ and hence in a sense
is distinct from the continuum scattering states.  As discussed in
Intriligator:2013lca , these modes are not usually quantized because they are
non-normalizable and hence lead to disconnected super selection sectors. We
believe these subtle issues are the intrinsic source of confusion about the
Callan Rubakov effect and deserve further investigation.

Now we would like to discuss the decay of the dyon degree of freedom. Recall
that the ${\mathbb{Z}}_{N_{f}}^{(0,1)}$ (0,1)-form symmetry only protects the
dyon from decaying mod${}_{N_{f}}$which allows for the emission of electric
charge in multiples of $N_{f}$. And indeed, we expect that this is
energetically favorable due to the fact that the dyon can decay through
angular momentum modes with energies that are arbitrarily close to $m_{\psi}$.
This should also be expected from the fact that integrating out the dyon
degree of freedom gives a local ‘t Hooft operator Fan:2021ntg :

$\displaystyle\begin{split}V_{\rm`t~{}Hooft}\sim
e^{i\frac{\Theta+\tilde{\Theta}}{2}}\delta(r)~{},\end{split}$ (108)

which has the effect of transforming an in-going state with $n_{f}$ fermions
to an out-going state with $n_{f}$ fermions with the opposite chirality.

From our previous analysis, we see that the the dyon quantum mechanics can
only emit gauge charge via $\tilde{\Theta}=\sum_{A}{\widetilde{H}}_{A}$
winding modes which are the combined winding mode of each
${\widetilde{H}}_{A}$. Upon returning to the fermionic description, this is
the statement that the dyon will decay by sending out a single quanta of each
$\tilde{\psi}_{A}$.

In the original setting of the Callan Rubakov effect, $SU(5)$ GUT theory with
a single generation, the interaction of Standard Model fermions with the
minimal spherically symmetric monopole (with respect to $U(1)_{Y}$
hypercharge) reduces to the interaction of $N_{f}=4$ Weyl doublets interacting
with the spherically symmetric $SU(2)$ monopole.232323Here we pick the
magnetic charge to be embedded as $T_{3}={\rm diag}(0,0,1,-1,0)$ as in
Callan:1983tm ; Callan:1982au ; Callan:1982ac ; Gardner:1984zd ;
Gardner:1983uu ; Brennan:2021ewu . Here there is an effective $SU(4)\times
SU(4)$ flavor symmetry in the effective $SU(2)\to U(1)_{Y}$ gauge theory which
plays the role of the global 0-form symmetry in our (and the standard Callan
Rubakov) analysis.242424Note that the $SU(3)\times SU(2)$ gauge symmetry does
not embed into the $SU(4)\times SU(4)\times U(1)_{\rm gauge}$ due to the fact
that the magnetic charge does not commute with the generators of $SU(3)\times
SU(2)$.  Our analysis shows that the process

$\displaystyle\begin{split}2\bar{p}+M~{}\longrightarrow~{}3e^{-}+p+M~{},\end{split}$
(109)

does indeed occur, but that scattering a single proton may be captured by the
monopole:

$\displaystyle\begin{split}\bar{p}+M~{}\longrightarrow~{}(e^{c}+2d+D)~{},\end{split}$
(110)

depending on energetics, where $D$ is the dyon and $(\,\underline{~{}~{}}+D)$
indicates a fermion-dyon bound state.

Now we would like to make a few comments about the application of our analysis
to scattering processes in the Standard Model:

  * •

As discussed in the introduction, both of these processes lead to baryon and
lepton number violation. In the second process, the dyon carries $U(1)_{B-L}$
charge due to the fact that the generator of the $U(1)_{B-L}$ in the $SU(5)$
theory is given by

$\displaystyle\begin{split}Q_{B-L}=Q_{Y}+Q_{D}\end{split}$ (111)

where $Q_{Y}$ is the generator of hypercharge and $Q_{D}$ acts on the $SU(5)$
fermion multiplets with charges $Q_{D}[\mathbf{\bar{5}}]=-3$ and
$Q_{D}[\mathbf{10}]=1$. Since the monopole does not carry charge with respect
to $Q_{D}$, but does with respect to $Q_{Y}$, the dyon can carry charge with
respect to $Q_{B-L}$. When enough charge is accumulated, the $U(1)_{B-L}$
charge is radiated away, and thus $U(1)_{B-L}$ is conserved as in the original
Callan Rubakov effect.

  * •

Here our analysis only strictly applies when the fermions all have an
identical mass. This would require that the Yukawa couplings be diagonal and
identical, and that the CKM matrix is trivial. It would be interesting to
study how our proposal is modified by weak symmetry breaking effects.

  * •

Here, it may seem strange that the down quarks carry color charge whereas the
anti-proton carries no color charge. The reason is that actually the
spherically symmetric monopole is also embedded non-trivially along the
$SU(3)$ part of the Standard Model gauge group and thus in a sense carries
color charge itself.

  * •

Note that as discussed in Aharony:2022ntz , Wilson line operators of large
electric charge $Q\sim\frac{1}{g^{2}}$ are screened in QED. Here we expect a
similar behavior and thus, there may be additional effects for the Callan
Rubakov effect when $N_{f}\gtrsim\frac{1}{g^{2}}$. For this reason, we only
claim our results hold for $N_{f}\ll\frac{1}{g^{2}}$.

In summary, we see that our analysis produces the original Callan Rubakov
effect for s-wave scattering for asymptotic states with integral fermion
number (i.e. that monopoles can catalyze proton decay/baryon number violation)
but modifies it so as to avoid the fractional fermion states.

##  5 General Monopole-Fermion Scattering

It is now somewhat straight forward to analyze the scattering for general
fermions beyond the $j=0$ mode. Due to the spherical symmetry of the monopole,
all higher angular momentum modes can be reduced to effective $2d$ fermions on
the half space. To write down the effective theory, we simply need to
understand the boundary conditions for the higher angular momentum modes. This
is dictated by the tree level interaction of the UV theory of the angular
momentum modes with the $W$-boson. As we have discussed, the fact that all
higher angular momentum modes of the non-abelian degrees of freedom are gapped
out with a mass of order $m_{W}$ implies that the dyon degree of freedom is
only allowed to decay into the $j=0$ modes as discussed in the previous
section.

The effective interaction is then determined by

$\displaystyle\begin{split}S_{int}=\int
e^{i\varphi}\,\bar{\psi}\,T_{+}(\gamma^{\theta}+i\gamma^{\phi})\psi\,M(r)\,d^{4}x+c.c.\end{split}$
(112)

Again, since the dyon degree of freedom is spherically symmetric, this
interaction does not mix the angular momentum states. Therefore, an
asymptotically in-going mode with angular momentum $|j,m\rangle$ will scatter
into an out-going state of the same $|j,m\rangle$.

The tree level interaction between the fermions and dyon degree of freedom is
determined explicitly by the mode expansion as in (62) – (64), which have
$j$-angular momentum spherical wave modes that propagate both in- and out-
ward. Using the explicit expressions in Section 2 we see that in the monopole
background, the independent, orthogonal modes generate an asymptotic current
flux at infinity as in (66):

$\displaystyle\begin{split}\Delta
Q=\int_{S^{2}_{\infty}}\bar{\Upsilon}\gamma^{\mu}\Upsilon\,\hat{r}_{\mu}\,d^{2}\Omega=2(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}-a_{3}^{\dagger}a_{3}-a_{4}^{\dagger}a_{4})~{}.\end{split}$
(113)

This above analysis captures the tree level interaction of the fermions with
the dyon degree of freedom at tree level. Thus, we find that if we send in a
quantum of $a_{1},a_{2}$ ($a_{3},a_{4}$), then we deposit $+2$ ($-2$) electric
charge and excite the dyon. Note that although the dyon degree of freedom
couples to asymptotic modes which have positive and negative (unlike the $j=0$
modes) asymptotic current, it cannot radiate away its charge through these
modes. The reason is that such a decay is prevented by conservation of angular
momentum – this forces the dyon to only decay through the $j=0$ modes.

Our analysis for the $j=0$ modes suggests that this deposits electric charge
by again inducing a Witten effect on the monopole. Here, the crucial
observation is that the phase mode of the fermions carries their gauge charge.
Separating out the phase mode from the other parts of the fermion generates an
axion field according to the transformation properties of the original fermion
field. This means that scattering a charged fermion – i.e. sending in gauge
current – contains a winding mode of this effective axion field which then
induces a shift in the $\theta$-angle, thereby depositing electric charge in a
cloud of radius $R_{c}\sim 1/m_{\psi}$.252525One could in principle truncate
to the $j$\- and 0- angular momentum modes which is described by a consistent
effective $2d$ theory. Due to the universality of the coupling between
$\varphi$ and the $2d$ fermions (112), the analysis should proceed almost
identically to that of Section 4 where the incoming $j\neq 0$ bosonic kinks
induces a shift in the effective $\theta_{1d}$, thereby converting the
monopole into a dyon. Again, these bosonic kinks describe the phase mode –
i.e. axionic mode – of the bulk $4d$ fermions which leads to the conservation
of electric charge in $4d$ via the dynamical Witten effect.

Again, the dyon can then either decay or not according to the conserved
${\mathbb{Z}}_{N_{f}}^{(0,1)}$ (0,1)-form global symmetry. As discussed in
detail in the previous section, the dyon can decay into the fermionic $j=0$
modes due to spherical symmetry.

This gives us the general picture of scattering a fermion off of a monopole.
Here, the fermions will excite the dyon via the tree-level UV interaction with
the (spherically symmetric) trapped $W$-boson. This can then decay (or not) to
the spherically symmetric modes fermion modes:

$\displaystyle\begin{split}\psi+M~{}&\longrightarrow~{}(\tilde{\psi}+D)~{},\\\
\psi^{n_{f}}+M~{}&\longrightarrow~{}\tilde{\psi}^{n_{f}}+\prod_{A}\bar{\tilde{\psi}}^{A}+M~{},\end{split}$
(114)

where the asymptotic out-going angular momentum modes for the second process
are given in terms of the normalizable states at $r\to 0$ as in (62).

This implies the general physical picture for monopole-fermion scattering.
When any fermion wave packet is sent into the monopole core, it will interact
with the dyon degree of freedom via the tree level UV coupling of the fermion
to the $W$-boson. Here the coupling to the $4d$ photon (and its higher angular
momentum modes) plays no special role – we simply have a propagating charged
fermion in QED. The incident fermion will scatter as the UV charge conjugate
field with respect to the UV $SU(2)$ gauge symmetry. The scattering will also
impart a charge onto the monopole via a shift in the fermion vacuum which
leads to a charge cloud of radius $R_{c}\sim 1/m_{\psi}$. This charge will
enforce charge conservation and may or may not trap the out-going fermion due
to the generated electrostatic potential.

Again, in the massless limit, this picture becomes much more subtle. As
before, the electric charge of the dyon is concentrated in a fermion cloud of
radius $R_{c}\sim 1/m_{\psi}$ around the monopole core due to the Witten
effect. As in the case of $j=0$ scattering, the limit $m_{\psi}\to 0$ sends
the radius of this cloud to infinity and sends the local charge density to
zero. Similarly, as discussed in Zhang:1988ab ; Ravendranadhan:1989vp ;
Tang:1982fc , the binding energy of the dyon-fermion system for finite angular
momentum modes scales as $E_{\rm bind}\sim m_{W}$ so that again in the limit
$m_{\psi}\to 0$ there is no possibility of a bound state and all scattered
fermions escape to infinity. This leads to the same conclusion as in the
scattering of $j=0$ fermion modes that scattering of charged fermions off of a
monopole is dictated by the tree level interaction and that the conservation
of gauge symmetry may possibly be attributed to shifting the super-selection
sector of the Hilbert space, although this requires regularization in order to
be made sensible.

##  6 Discussion: Implications for Standard Model and Future Directions

Here we have proposed a comprehensive picture of the monopole-fermion
interaction. We have discussed the interaction for massive particles at all
angular momentum and suggested how this leads to an interpretation of the
massless monopole-fermion interaction. Our discussion suggests that the
confusion surrounding the original analysis of the Callan Rubakov effect is
associated to an issue with IR regulation.

We expect that the new physical effect of fermion trapping by monopoles should
lead to a new class of phenomenological effects. Even though the Callan
Rubakov effect still occurs – i.e. that there are baryon number violating
interactions which are catalyzed by monopoles – we expect that this effect
leads to less efficient baryon number violation in cosmological scenarios than
originally proposed. The reason is that after a monopole participates in a
baryon violating process, it acquires an electric charge which favors un-doing
the process. This suggests that baryon number violating processes will be less
efficient than originally proposed and hence may lead to a less efficient
Langacker-Pi scenario which washes out early universe baryon number asymmetry
Langacker:1980kd ; Davis:1992ca .

Additionally, we expect interesting effects from the fact that the dyons
generated by the monopole-fermion interaction can carry $U(1)_{B-L}$ charge.
This means that interactions with the monopoles could generate a universe
which appears to have a net $U(1)_{B-L}$ charge by constructing a scenario in
which monopoles accrue $U(1)_{B-L}$ charge and then are inflated away.

This project also leads to many interesting future directions. Besides
generalizing to more complex spherically symmetric monopoles and higher
dimensional fermion representations, the logic of this paper can be
generalized (with some modifications) to describe the interaction of fermions
with non-spherically symmetric monopoles. There, the interaction between the
fermions and dyon degree of freedom will mix angular momentum modes according
to the angular momentum of the monopole itself. Similarly, it should also have
the ability to decay into different angular momentum modes. We believe that
the dynamical Witten effect is the key to understanding the more general
interaction.

We also believe it would be interesting to further study the details our
proposal such as the life time of the dyon after accumulating $N_{f}$ electric
charge as well as the detailed structure of fermion bound- and scattering-
states in the presence of the dyon.262626We believe that there will be an
interesting structure similar to that described in Iengo:2002eb .
Additionally, it would be interesting to study the effects of 1-loop quantum
corrections to the interaction and study the effects of small symmetry
breaking effects. These would be especially interesting to the application of
the Standard model due to the unequal masses in nature.

It is also interesting to ask how the Callan Rubakov effect is modified when
the fermions are given a mass by a GUT scale Yukawa interaction. In this case,
the monopole actually supports fermion zero-modes which may lead to a
different physical effect. Interestingly, we do not expect that this effect
occurs in supersymmetric Yang-Mills (SYM) theories without bare masses. The
reason is that in these theories, the effective mass of the fermions is given
by $m_{\psi}=m_{W}$ so that any fermion scattering occurs at high energies
outside of the $U(1)$ effective field theory. However, this effect may be
visible in SYM theories with bare masses $m_{0}$. In these theories, the
effective fermion masses are shifted $m_{\psi}=m_{W}\pm m_{0}$ so that by
taking large $m_{0}$, we can generate low effective mass fermions.

It would also be interesting to better understand the role that the higher
group and more general categorical symmetries play in the case of massless
fermions. Although we have proposed a physical interpretation for the
scattering process for massless fermions, it would still be interesting to
better understand the subtleties associated with this limit.

Another interesting application of this work is to the study of black hole
physics. Recently, Maldacena:2020skw  showed how magnetically black holes can
also lead to baryon number violation in the Standard Model due to electroweak
symmetry restoration near the black hole horizon. Additionally, in
Alford:1992ef , the authors showed that there is a Callan Rubakov-type
enhancement of s-wave scattering of fermions off of magnetically charged black
holes. In light of our proposal, we believe it is would be interesting to
revisit these ideas.

## Acknowledgements

The author would like to thank Seth Koren, Sungwoo Hong, Konstantinos
Roumpedakis, Pierluigi Niro, Andrea Antinucci, Giovanni Galati, Giovanni Rizi,
Ken Intriligator, Zohar Komargodski, Marieke Van Beest, Diego Delmastro, Matt
Reece, Liantao Wang, Colin Rylands, Gautam Satishchandran, David Tong, John
Terning, Jeffrey Harvey, Aneesh Manohar, Jorge Russo, Clay Cordova, and Thomas
Dumitrescu for helpful discussions and related collaborations. TDB would also
especially like to thank Seth Koren and Aneesh Manohar for comments on the
draft. TDB is supported by Simons Foundation award 568420 (Simons
Investigator) and award 888994 (The Simons Collaboration on Global Categorical
Symmetries).

## References

  * (1) G. ’t Hooft, “Magnetic Monopoles in Unified Gauge Theories,” Nucl. Phys. B 79 (1974) 276–284.
  * (2) A. M. Polyakov, “Particle Spectrum in the Quantum Field Theory,” JETP Lett. 20 (1974) 194–195.
  * (3) C. G. Callan, Jr., “Dyon-Fermion Dynamics,” Phys. Rev. D 26 (1982) 2058–2068.
  * (4) C. G. Callan, Jr., “Disappearing Dyons,” Phys. Rev. D 25 (1982) 2141.
  * (5) C. G. Callan, Jr., “Monopole Catalysis of Baryon Decay,” Nucl. Phys. B 212 (1983) 391–400.
  * (6) V. A. Rubakov, “Adler-Bell-Jackiw Anomaly and Fermion Number Breaking in the Presence of a Magnetic Monopole,” Nucl. Phys. B 203 (1982) 311–348.
  * (7) V. A. Rubakov, “MONOPOLE INDUCED BARYON NUMBER NONCONSERVATION,”.
  * (8) V. A. Rubakov, “Monopole Catalysis of Proton Decay,” Rept. Prog. Phys. 51 (1988) 189–241.
  * (9) J. Preskill, “MAGNETIC MONOPOLES,” Ann. Rev. Nucl. Part. Sci. 34 (1984) 461–530.
  * (10) H. Yamagishi, “THE FERMION MONOPOLE SYSTEM REEXAMINED,” Phys. Rev. D 27 (1983) 2383–2396.
  * (11) B. Grossman, “DOES A DYON LEAK?,” Phys. Rev. Lett. 50 (1983) 464.
  * (12) H. Yamagishi, “The Fermion - Monopole System Reexamined. 2.,” Phys. Rev. D 28 (1983) 977.
  * (13) H. Yamagishi, “FERMION - MONOPOLE SYSTEM REEXAMINED. 3.,” Phys. Rev. D 32 (1985) 1576.
  * (14) H. Panagopoulos and H. Yamagishi, “FERMION - MONOPOLE SYSTEM REEXAMINED. 4,” Phys. Rev. D 32 (1985) 2113.
  * (15) C. Besson, “FERMIONS AND MAGNETIC MONOPOLES,” other thesis, 10, 1981.
  * (16) L. J. Dixon, “Spherically Symmetric Monopoles and Dyons With Spin,” Nucl. Phys. B 248 (1984) 90–104.
  * (17) K. Isler, C. Schmid, and C. A. Trugenberger, “MONOPOLE CORE EXCITATIONS AND THE RUBAKOV-CALLAN EFFECT,” Nucl. Phys. B 294 (1987) 925.
  * (18) K. Isler, C. Schmid, and C. A. Trugenberger, “CHARGE EXCHANGE AND HELICITY FLIP SCATTERING IN THE FIELD OF A ’T HOOFT-POLYAKOV MONOPOLE,” Phys. Rev. D 34 (1986) 2508.
  * (19) A. Sen, “Role of Conservation Laws in the Callan-Rubakov Process with Arbitrary Number of Generation of Fermions,” Phys. Rev. Lett. 52 (1984) 1755.
  * (20) A. Sen, “Comparison of the Canonical Hamiltonian and the Hamiltonian of Callan and Rubakov for the Monopole Fermion System,” Phys. Rev. D 31 (1985) 433.
  * (21) A. P. Balachandran and J. Schechter, “The Missing Charge in the Rubakov Process,” Phys. Rev. Lett. 51 (1983) 1418.
  * (22) V. P. Nair, “Breakdown of Flavor Conservation in a Monopole Background,” Phys. Rev. D 28 (1983) 2673.
  * (23) V. P. Nair, “Catalysis of Flavor Violation by Monopoles,” Phys. Rev. Lett. 51 (1983) 631.
  * (24) P. B. Smith and D. Tong, “Boundary States for Chiral Symmetries in Two Dimensions,” JHEP 09 (2020) 018, arXiv:1912.01602 [hep-th].
  * (25) P. B. Smith and D. Tong, “What Symmetries are Preserved by a Fermion Boundary State?,” arXiv:2006.07369 [hep-th].
  * (26) P. C. Nelson and A. Manohar, “Global Color Is Not Always Defined,” Phys. Rev. Lett. 50 (1983) 943.
  * (27) I. Affleck and J. Sagi, “Monopole catalyzed baryon decay: A Boundary conformal field theory approach,” Nucl. Phys. B 417 (1994) 374–402, arXiv:hep-th/9311056.
  * (28) M. Hortacsu, J. Kalayci, and N. K. Pak, “A POSSIBLE PROBLEM WITH THE RUBAKOV-CALLAN CONDENSATE FORMATION FOR MASSIVE FERMIONS,” Phys. Lett. B 145 (1984) 411–415.
  * (29) M. Hortacsu and N. K. Pak, “IS ELECTRIC CHARGE CONSERVED IN MONOPOLE FERMION SCATTERING?,” J. Phys. G 10 (1984) 1477–1485.
  * (30) R. Kitano and R. Matsudo, “Missing final state puzzle in the monopole-fermion scattering,” arXiv:2103.13639 [hep-th].
  * (31) C. Csáki, Y. Shirman, O. Telem, and J. Terning, “Monopoles Entangle Fermions,” arXiv:2109.01145 [hep-th].
  * (32) Y. Hamada, T. Kitahara, and Y. Sato, “Monopole-fermion scattering and varying Fock space,” JHEP 11 (2022) 116, arXiv:2208.01052 [hep-th].
  * (33) T. D. Brennan, “Callan-Rubakov effect and higher charge monopoles,” JHEP 02 (2023) 159, arXiv:2109.11207 [hep-th].
  * (34) M. van Beest, P. Boyle Smith, D. Delmastro, Z. Komargodski, and D. Tong, “Monopoles, Scattering, and Generalized Symmetries,” arXiv:2306.07318 [hep-th].
  * (35) C. Csaki, S. Hong, Y. Shirman, O. Telem, J. Terning, and M. Waterbury, “Scattering amplitudes for monopoles: pairwise little group and pairwise helicity,” JHEP 08 (2021) 029, arXiv:2009.14213 [hep-th].
  * (36) C. Csáki, Z.-Y. Dong, O. Telem, J. Terning, and S. Yankielowicz, “Dressed vs. pairwise states, and the geometric phase of monopoles and charges,” JHEP 02 (2023) 211, arXiv:2209.03369 [hep-th].
  * (37) C. Csáki, Y. Shirman, O. Telem, and J. Terning, “Pairwise Multiparticle States and the Monopole Unitarity Puzzle,” Phys. Rev. Lett. 129 no. 18, (2022) 181601.
  * (38) C. g. Callan, Jr., “THE MONOPOLE CATALYSIS S MATRIX,” in Workshop on Problems in Unification and Supergravity.  1983\.
  * (39) J. Polchinski, “Monopole Catalysis: The Fermion Rotor System,” Nucl. Phys. B 242 (1984) 345–363.
  * (40) S. L. Adler, “Axial vector vertex in spinor electrodynamics,” Phys. Rev. 177 (1969) 2426–2438.
  * (41) J. S. Bell and R. Jackiw, “A PCAC puzzle: $\pi^{0}\to\gamma\gamma$ in the $\sigma$ model,” Nuovo Cim. A 60 (1969) 47–61.
  * (42) P. P. Kulish and L. D. Faddeev, “Asymptotic conditions and infrared divergences in quantum electrodynamics,” Theor. Math. Phys. 4 (1970) 745.
  * (43) K. Prabhu, G. Satishchandran, and R. M. Wald, “Infrared finite scattering theory in quantum field theory and quantum gravity,” Phys. Rev. D 106 no. 6, (2022) 066005, arXiv:2203.14334 [hep-th].
  * (44) R. Peccei and H. R. Quinn, “Constraints Imposed by CP Conservation in the Presence of Instantons,” Phys. Rev. D 16 (1977) 1791–1797.
  * (45) R. Peccei and H. R. Quinn, “CP Conservation in the Presence of Instantons,” Phys. Rev. Lett. 38 (1977) 1440–1443.
  * (46) J. E. Kim, “Weak Interaction Singlet and Strong CP Invariance,” Phys. Rev. Lett. 43 (1979) 103.
  * (47) M. A. Shifman, A. Vainshtein, and V. I. Zakharov, “Can Confinement Ensure Natural CP Invariance of Strong Interactions?,” Nucl. Phys. B 166 (1980) 493–506.
  * (48) J. Fan, K. Fraser, M. Reece, and J. Stout, “Axion Mass from Magnetic Monopole Loops,” Phys. Rev. Lett. 127 no. 13, (2021) 131602, arXiv:2105.09950 [hep-ph].
  * (49) J.-z. Zhang and Y.-c. Qi, “WAVE FUNCTIONS OF BOUND STATES OF A FERMION AND A DIRAC DYON AND MATRIX ELEMENTS IN AN EXTERNAL ELECTROMAGNETIC FIELD,” J. Math. Phys. 31 (1990) 1796–1799.
  * (50) M. Ravendranadhan and M. Sabir, “FERMION - DYON BOUND STATES AND FERMION NUMBER FRACTIONIZATION,” J. Phys. G 15 (1989) 741–749.
  * (51) J.-F. Tang, “FERMION BOUND STATES IN A DYON FIELD,” Phys. Rev. D 26 (1982) 510–514.
  * (52) T. D. Brennan, “Index-like Theorem for Massless Fermions in Spherically Symmetric Monopole Backgrounds,” arXiv:2106.13820 [hep-th].
  * (53) O. Aharony, G. Cuomo, Z. Komargodski, M. Mezei, and A. Raviv-Moshe, “Phases of Wilson Lines in Conformal Field Theories,” Phys. Rev. Lett. 130 no. 15, (2023) 151601, arXiv:2211.11775 [hep-th].
  * (54) T. T. Wu and C. N. Yang, “Dirac Monopole Without Strings: Monopole Harmonics,” Nucl. Phys. B 107 (1976) 365.
  * (55) T. T. Wu and C. N. Yang, “Some Properties of Monopole Harmonics,” Phys. Rev. D 16 (1977) 1018–1021.
  * (56) G. W. Moore, A. B. Royston, and D. Van den Bleeken, “Parameter counting for singular monopoles on $\mathbb{R}^{3}$,” JHEP 10 (2014) 142, arXiv:1404.5616 [hep-th].
  * (57) T. D. Brennan and S. Hong, “Introduction to Generalized Global Symmetries in QFT and Particle Physics,” arXiv:2306.00912 [hep-ph].
  * (58) S. Schafer-Nameki, “ICTP Lectures on (Non-)Invertible Generalized Symmetries,” arXiv:2305.18296 [hep-th].
  * (59) L. Bhardwaj, L. E. Bottini, L. Fraser-Taliente, L. Gladden, D. S. W. Gould, A. Platschorre, and H. Tillim, “Lectures on Generalized Symmetries,” arXiv:2307.07547 [hep-th].
  * (60) S.-H. Shao, “What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetry,” arXiv:2308.00747 [hep-th].
  * (61) T. D. Brennan, C. Cordova, and T. T. Dumitrescu, “Line Defect Quantum Numbers & Anomalies,” arXiv:2206.15401 [hep-th].