e_^_#1#1— ^#2#2— 11institutetext: Department of Physics & Astronomy, McMaster
University, 1280 Main Street West, Hamilton ON, Canada.  22institutetext:
Perimeter Institute for Theoretical Physics, 31 Caroline Street North,
Waterloo ON, Canada.  33institutetext: School of Theoretical Physics, Dublin
Institute for Advanced Studies,10 Burlington Road, Dublin, Ireland

# On the EFT of Dyon-Monopole Catalysis

S. Bogojević 1,2,3 and C.P. Burgess

###### Abstract

Monopole-fermion (and dyon-fermion) interactions provide a famous example
where scattering from a compact object gives a cross section much larger than
the object’s geometrical size. This underlies the phenomenon of monopole
catalysis of baryon-number violation because the reaction rate is much larger
in the presence of a monopole than in its absence. It is sometimes claimed to
violate the otherwise generic requirement that short distance physics
decouples from long-distance observables – a property that underpins the
general utility of effective field theory (EFT) methods. Decoupling in this
context is most simply expressed using point-particle effective field theories
(PPEFTs) designed to capture systematically how small but massive objects
influence their surroundings when probed only on length scales large compared
to their size. These have been tested in precision calculations of how nuclear
properties affect atomic energy levels for both ordinary and pionic atoms. We
adapt the PPEFT formalism to describe low-energy $S$-wave dyon-fermion
scattering with a view to understanding whether large catalysis cross sections
violate decoupling (and show why they do not). We also explore the related but
separate issue of the long-distance complications associated with polarizing
the fermion vacuum exterior to a dyon and show in some circumstances how PPEFT
methods can simplify calculations of low-energy fermion-dyon scattering in
their presence. We propose an effective Hamiltonian governing how dyon
excitations respond to fermion scattering in terms of a time-dependent vacuum
angle and outline open questions remaining in its microscopic derivation.

##  1 Introduction

Magnetic monopoles and their electrically charged cousins – dyons – are a gift
that keeps on giving. First came the observation that they could exist and, if
they did, how their presence provides an instance where nature explores
nontrivial topology Dirac . Next came the realization that they not only could
exist but are actually predicted to exist within the semiclassical limit of
some classes of nonabelian gauge theories tHooft:1974kcl ; Polyakov:1974ek .
Then dyonic systems provided a first glimpse at the rich structure of
weak/strong coupling duality in relativistic dynamics Montonen:1977sn ;
Goddard:1976qe ; Duality ; Schwarz:1993vs . To this was added the remarkable
phenomenon of monopole catalysis wherein monopole-mediated reactions can
sometimes proceed with strong-interaction cross sections unsuppressed by the
small monopole size Rubakov:1981rg ; Callan:1982ah . (For a review see
Preskill:1984gd ; Rubakov:1988aq ; Schnir .)

Monopole catalysis is the focus of this paper because it is sometimes held up
as a rare example where decoupling fails. For the present purposes
‘decoupling’ is the statement that short-distance physics contributes only in
a limited number of ways to long-distance physics, and its validity in nature
is one of the reasons why science is so successful; nature comes to us with
many scales but decoupling allows us to understand each scale on its own terms
without needing to understand everything at once.

Decoupling is also a property of the mathematics – quantum field theory – we
use to describe nature and the formalism that organizes calculations to
exploit this property as efficiently as possible has come to be known as
effective field theory (EFT) Weinberg:1968de ; Wilson:1969zs ; Weinberg:1978kz
(see EFTBook  for a review). Our goal here is to understand how monopole
catalysis works within an EFT framework and thereby to understand how
decoupling can be consistent with the large interaction cross sections
predicted by monopole dynamics. Besides being a conceptually interesting
question in its own right our hope is to develop tools that can eventually be
used to compute low-energy monopole/dyon scattering more efficiently.

To this end we adapt to dyons the formalism of point-particle effective
theories111PPEFTs are equivalent to EFTs like NRQED NRQED  or HQET HQET  but
differ from them in that they treat the massive heavy things as first-
quantized objects rather than second-quantized fields when describing their
interactions with lighter relativistic degrees of freedom. (PPEFTs) that
describe systematically the interactions of individual heavy but small objects
with their surroundings.222For simplicity we do not discuss here multiparticle
scattering processes for which the complications of Csaki:2021ozp ;
Csaki:2022tvb  might be relevant or those involving e.g. the twist operators
described in vanBeest:2023dbu ; vanBeest:2023mbs . The PPEFT formalism is
equivalent to more graphical first-quantized point-particle EFT applications –
such as Goldberger:2004jt  for instance – but as used here does not as heavily
rely on momentum-space Feynman rules. It instead relates the effective world-
line lagrangian for the small source to a set of near-source boundary
conditions satisfied by the relativistic fields with which the source
interacts. It does so by generalizing to arbitrary interactions the standard
Gauss’ Law technique that draws a small Gaussian pillbox around the compact
source and relates the total charge within the pillbox to the electric flux
through the pillbox’s surface (which can be regarded as the value of
$r^{2}\partial A_{0}/\partial r$ near the source) PPEFT ; PPEFT2 ; PPEFT3 .

Relating the near-source boundary condition to the source action in this way
allows a systematic expansion of the source’s influence at low energies that
includes but extends the usual multipole expansion to include nonlinear
interactions. PPEFT methods have been tested extensively in applications for
which answers are known by other methods, by using it to describe the
influence of finite nuclear size (and various nuclear moments) for atomic
energy levels Burgess:2020ndx ; Zalavari:2020tez , absorption by hot wires in
atom traps Plestid:2018qbf  and to describe the gravitational back-reaction of
codimension-two branes in various dimensions Burgess:2008yx ; Bayntun:2009im .
Applying these same techniques to dyons allows a comparison between how
$S$-wave scattering with no angular momentum barrier differs for a spinless
particle (a pion, say) interacting with a nucleus and a spin-half particle
interacting with a dyon. This comparison shows why scattering cross sections
in one case are set by the size of the nucleus but in the other can be much
larger.

The key difference is kinematic. For spinless particles interacting with a
nucleus there are two types of mode functions for each angular-momentum
quantum number $l$. One typically varies as $r^{l}$ for small $r$ and the
other varies as $r^{-l-1}$. Because one solution grows and the other shrinks
as $r\to 0$ the existence of two solutions defines a physical scale $R_{*}$:
when the specific solution exterior to a known source is decomposed in terms
of these two modes $R_{*}$ is the distance at which the two modes are
comparable in size. $R_{*}$ turns out to be the scale that determines the size
of scattering cross sections, since the $S$-wave cross section turns out to be
of order $\sigma_{s}\sim\pi R_{*}^{2}$.

For instance, for nuclear-strength interactions – such as is relevant for
pionic atoms – $R_{*}$ is comparable to the nuclear radius $R$. By contrast,
for electromagnetic scattering $R_{*}\sim\alpha R$ is much smaller (where
$\alpha=e^{2}/4\pi$ is the fine-structure constant). What is unusual about
fermion-dyon scattering is there is only a single $S$-wave mode, and so
decomposing a specific solution in terms of this mode does not introduce a
scale similar to $R_{*}$. Instead scattering turns out to be scale invariant
and predicts $\sigma_{s}\sim 2\pi/k^{2}$ (where $k$ is the relativistic
fermion energy). This is the EFT translation of the standard understanding
Rubakov:1981rg ; Callan:1982ah  of why dyon-fermion $S$-wave scattering does
not scale with the dyon size.

Although the above is a good start, the full story is of course more
complicated because the fermion-dyon interaction polarizes the fermion ground
state Goldhaber:1977xw ; Ezawa:1983vi ; CallanSMatrix ; Yamagishi:1982wp ;
Grossman:1983yf ; Kazama:1983rt ; Isler:1987xn . Within the PPEFT approach
this can be interpreted as a further renormalization of the effective
couplings as the radius of the underlying Gaussian pillbox grows to become
larger than the Compton wavelength of any light fermion species. For the
particular dyonic system we study here this has been argued to simply turn off
the underlying charge-changing dyon-fermion interactions in a calculable way
Polchinski ; converting them into helicity-flipping interactions. We argue
that the basic PPEFT scattering rates apply equally well in this case,
provided one works with the appropriate long-wavelength effective couplings
that contain these interaction effects.

We treat the fermion-dyon scattering within the Born-Oppenheimer approximation
with the idea that the fermions represent the fast degrees of freedom and the
dyonic response provides the slow degrees of freedom. Within this framework
the PPEFT formalism also allows us to discuss how the dyon responds to fermion
scattering and in principle allows estimates for the size of effects such as
vacuum angle evolution due to fermion scattering. We show how such evolution
can occur although we have been unable to find a regime where the dynamics
envisaged in Brennan:2021ewu  takes place in detail. Although we derive some
expressions that show how the dyon response to fermion scattering is
consistent with the flow of conserved quantities, we have not been able to
explicitly derive this effective dyon Hamiltonian by integrating out the
fermions.

Our results are presented as follows. §2 reviews the basic construction of the
dyon solutions we explore in detail and §3 sets up the PPEFT description for
the scattering of relativistic fermions with the dyon. It turns out that the
number of effective operators in the simplest effective description is larger
than the number of observable parameters and we argue why this difference can
be understood in terms of redundancy of PPEFT interactions. (A similar thing
happens for electromagnetic interactions with nuclei in atoms, where it
implies the various nuclear moments do not contribute independently to shifts
in atomic energy levels to fixed order in an expansion in powers of $Z\alpha$
Burgess:2020ndx ; Zalavari:2020tez .) §4 then explores the RG evolution of the
boundary conditions as the radius of the Gaussian pillbox is varied and shows
in more detail why the kinematics of $S$-wave dyon-fermion scattering leads to
scale-invariant cross sections even though the same does not happen for small
objects without magnetic charge (such as nuclei). This section also discusses
how the fermion condensation effects described in Polchinski  can be
incorporated within this framework. Finally §6 uses the PPEFT effective
couplings to compute fermion polarization and fermion-dyon scattering within
the Born-Oppenheimer approximation, after first warming up by computing
similar results perturbatively in the fermion-dyon interactions in §5. Our
conclusions are briefly summarized in §7.

##  2 The Julia-Zee dyon

To start we work within a simple model that supports simple dyon solutions:
the Georgi-Glashow style $SU(2)$ gauge theory spontaneously broken down to
$U(1)$ using an adjoint Higgs field Georgi:1972cj . To this we couple a
complex doublet of massless fermions333Because this corresponds to two
pseudoreal doublets, it is the minimal anomaly-free fermion content
Witten:1982fp . so that we can study fermion-dyon interactions in the dyon’s
low-energy regime (but where the fermions are nonetheless ultra-relativistic).

This model is described by the following action,444The distinction between
symmetric differentiation
$\overset{\leftrightarrow}{D}_{\mu}=\overset{\rightarrow}{D}_{\mu}-\overset{\leftarrow}{D}_{\mu}$
and the usual one-sided quantity in the Dirac action is a total derivative,
but much of the later discussion hinges on being careful with total
derivatives and boundary terms.

${S}=-\int\mathrm{d}^{4}x\left[\,\frac{1}{4}F^{a}_{\mu\nu}\,F^{\mu\nu}_{a}+\frac{1}{2}D_{\mu}\Phi^{a}\,D^{\mu}\Phi_{a}+\frac{1}{2}\overline{\psi}\,\gamma^{\mu}\overset{\leftrightarrow}{D}_{\mu}\psi-\frac{\mu^{2}}{2}\,\Phi^{a}\,\Phi_{a}+\frac{\lambda}{4}\,\left(\Phi^{a}\,\Phi_{a}\right)^{2}\right]\,,$
(1)

with covariant derivatives

$D_{\mu}\psi=\left(\partial_{\mu}-\frac{i}{2}\,eA^{a}_{\mu}\tau_{a}\right)\psi\quad\hbox{and}\quad(D_{\mu}\Phi)_{a}=\partial_{\mu}\Phi_{a}+\epsilon_{abc}\,eA^{b}_{\mu}\,\Phi^{c}\,,$
(2)

and nonabelian field strength
$F^{a}_{\mu\nu}=\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A^{a}_{\mu}+e\,\epsilon^{abc}A_{b\mu}\,A_{c\nu}$.
The gauge generators acting on the fermion’s doublet gauge indices are given
in terms of the usual Pauli matrices by $T_{a}=\frac{1}{2}\tau_{a}$ and so
satisfy the $SU(2)$ commutation relation $[T_{a},T_{b}]=i\epsilon_{abc}T_{c}$.
With respect to the gauge field gauging rotations in the $T_{3}$ direction
(say) we see that the members of the fermion doublet have charge
$\pm\frac{1}{2}\,e$ and the two gauge bosons spanned by $A^{1}_{\mu}$ and
$A^{2}_{\mu}$ have charge $\pm e$. The components of the adjoint Higgs
multiplet $\Phi^{a}$ similarly carry charges zero and $\pm e$ for this
symmetry.

The model’s parameters $\lambda$, $e$ and $\mu^{2}$ are all real and positive,
with dimensionless couplings $\lambda$ and $e$ chosen small enough to justify
the standard semiclassical analysis. In the semiclassical regime the choice
$\mu^{2}>0$ ensures the vacuum expectation value
$w^{a}:=\langle\Phi^{a}\rangle$ satisfies
$w^{2}:=w^{a}w_{a}=\mu^{2}/\lambda\neq 0$ so that the gauge symmetry is
spontaneously broken down to $U(1)$. The two charged gauge bosons then ‘eat’
the two charged fields in $\Phi^{a}$ and by doing so acquire nonzero mass
$m_{g}\simeq ew=\beta\mu$, where $\beta^{2}:=e^{2}/\lambda$. The remaining
spin-one particle is massless and so can be regarded as the ‘photon’ that
gauges the unbroken $U(1)$ symmetry. The uneaten physical scalar is also
neutral under the unbroken gauge symmetry and has mass
$m_{s}\simeq\sqrt{2\lambda}\,w=\sqrt{2}\,\mu$.

###  2.1 Classical dyon

There are a variety of soliton solutions to the field equations of this model.
Of these the so-called Julia-Zee dyon tHooft:1974kcl ; Julia:1975ff ;
Wu:1975es  is of most interest for the present purposes. This has the form

$e{{\cal A}}^{a}_{i}={\epsilon^{a}}_{ib}\,\hat{r}^{b}\,\left[\frac{1-{\cal
K}(r)}{r}\right]\,,\quad e{{\cal A}}^{a}_{0}=\hat{r}^{a}\,\frac{{\cal
J}(r)}{r}\quad\hbox{and}\quad e\varphi^{a}=\hat{r}^{a}\,\frac{{\cal
H}(r)}{r}\,,$ (3)

with vanishing $\psi$. Here $\hat{\bm{r}}$ denotes the unit vector in the
radial direction and the dimensionless functions ${\cal K}(r)$, ${\cal H}(r)$
and ${\cal J}(r)$ depend on $r$ only through the combination $\mu r$. They
satisfy second-order coupled ordinary differential equations given explicitly
in Julia:1975ff  that depend on $e$ and $\lambda$ only through the ratio
$e^{2}/\lambda=\beta^{2}$.

The integration constants are chosen to ensure boundedness at the origin — and
so ${\cal K}\to 1$ and ${\cal J},{\cal H}\to 0$ as $r\to 0$. They also have a
finite-energy falloff to the vacuum solution as $r\to\infty$, implying the
asymptotic behaviour

${\cal K}(r)\to 0\,,\quad{\cal J}(r)\to e(Q-vr)\quad\hbox{and}\quad{\cal
H}(r)\to hr\quad\hbox{as }\,r\to\infty\,,$ (4)

where $h=ew=\beta\mu$ and only one of $Q$ or $v$ is an independent parameter
because one combination of $Q$ and $v$ is a calculable function of $\beta$ and
$\mu$. The approach to these asymptotic forms is exponential in $\mu r$, and
this sets the ‘size’ of the solution. For instance the function ${\cal K}(r)$
behaves for large $r$ as ${\cal K}(r)\sim e^{-ar}$ with
$a=\sqrt{h^{2}-(ev)^{2}}$, which shows the solutions damp exponentially
provided $h>ev$.

In the special case where $\lambda\rightarrow 0$ and $e\to 0$ with
$\beta^{2}=e^{2}/\lambda$ and $\mu$ fixed the functions ${\cal K}(r),{\cal
J}(r)$ and ${\cal H}(r)$ can be solved in closed form BPS solution . In this
Prasad-Sommerfield limit the solution that is regular at the origin and for
which $\varphi^{a}\varphi_{a}$ approaches a constant555When $\lambda\to 0$ it
is no longer necessary to require ${\cal H}/r$ to approach the specific
constant $h$ as $r\to\infty$. at infinity is given explicitly by

${\cal K}(r)=\frac{cr}{\sinh(cr)},\quad{\cal
J}(r)=\sinh\varpi\,\Bigl{[}1-cr\coth(cr)\Bigr{]}\quad\text{and}\quad{\cal
H}(r)=\cosh\varpi\,\Bigl{[}cr\coth(cr)-1\Bigr{]},$ (5)

with $\varpi$ an arbitrary real constant and the field equations require
$c=\beta\mu$. Comparing with the large-$r$ limit of ${\cal J}(r)$ shows that
$v$ and $Q$ are given in terms of these constants by $ev=c\sinh\varpi$ and
$eQ=\sinh\varpi$ and so $Q=v/c$.

At any particular spacetime point the nonzero fields break the $SU(2)$ gauge
invariance down to a $U(1)$ subgroup, so just like for the vacuum the dyon
preserves a $U(1)$ symmetry and as a result one of the gauge modes – the
photon – remains precisely masses. But for the dyon the particular embedding
of the unbroken $U(1)$ within $SU(2)$ varies from place to place, as can be
seen by the mixing of gauge and spatial indices in (3). This makes it
convenient to change gauge to a form for which the massless gauge mode
corresponds to the same gauge direction everywhere in spacetime.

If the asymptotic gauge field is chosen to point along the $T_{3}$ direction
in $SU(2)$ space, the required gauge transformation is

$U({\bm{r}})=\frac{1}{\sqrt{2}}\left[\sqrt{1-\hat{{\bm{z}}}\cdot\hat{{\bm{r}}}}+\frac{i{\vec{\tau}}\cdot(\hat{{\bm{r}}}\times\hat{{\bm{z}}})}{\sqrt{1-\hat{{\bm{z}}}\cdot\hat{{\bm{r}}}}}\right],$
(6)

where we use arrows to denote vectors in gauge space and bold-face for vectors
in physical space and $\hat{{\bm{z}}}$ denotes the unit vector in the ‘3’ or
$z$ direction. This is a singular gauge transformation inasmuch as it
introduces a previously non-existent singularity into the asymptotic vector
potential if it is performed everywhere. Because the singularity is a gauge
artefact we can remove it by following the procedure outlined in Wu:1975es
where we define instead gauge-related but nonsingular configurations on each
of two hemispheres that enclose the dyon.

That is, defining the two overlapping regions, $R_{+}$ and $R_{-}$, by

$R_{+}:\left\\{0\leq\theta<\frac{\pi}{2}+\delta,\,r>0,\,0\leq\phi<2\pi\right\\}\quad\hbox{and}\quad
R_{-}:\left\\{\frac{\pi}{2}-\delta<\theta\leq\pi,\,r>0,\,0\leq\phi<2\pi\right\\},$
(7)

where $(\theta,\phi)$ are the usual spherical-polar angular coordinates,
$0<\delta\leq\frac{\pi}{2}$ is otherwise arbitrary and we define the gauge
field in region $R_{-}$ using the gauge function $U({\bm{r}})$ while the gauge
field in $R_{+}$ is additionally transformed using the gauge function
$V({\bm{r}})=e^{i\phi\,\tau_{3}}$. Defined in this way, the Julia-Zee dyon
field configuration is given by

$e\mathcal{A}^{a\pm}_{i}(\vec{r})=\frac{1}{r}\left[\frac{\pm
1-\cos\theta}{\sin\theta}\hat{\phi}_{i}\,\delta^{a}_{3}-{\cal
K}(r)\,\zeta^{a}_{i\pm}\right],\quad
e\mathcal{A}^{a}_{0}(\vec{r})=-\frac{{\cal
J}(r)}{r}\,\delta^{a}_{3}\quad\hbox{and}\quad e\Phi^{a}(\vec{r})=-\frac{{\cal
H}(r)}{r}\,\delta^{a}_{3}.$ (8)

where

${\bm{\zeta}}_{i\pm}:=\frac{1}{2}\,\zeta^{a}_{i\pm}\tau_{a}=\frac{1}{2}\left(i\hat{\theta}_{i}-\hat{\phi}_{i}\right)e^{\pm
i\phi}\tau_{+}-\frac{1}{2}\left(i\hat{\theta}_{i}+\hat{\phi}_{i}\right)e^{\mp
i\phi}\tau_{-}$ (9)

arises as the gauge transform of $\frac{1}{2}\hat{\bm{r}}\times{\vec{\tau}}$.
Here the subscripts $\pm$ refer to the regions $R_{\pm}$ and $\tau_{\pm}$ are
defined as usual by $\tau_{\pm}:=\frac{1}{2}(\tau_{1}\pm i\tau_{2})$.

The asymptotic large-$r$ form of the solution in this gauge makes its
electromagnetic properties explicit,

$e{\mathcal{A}}^{a\pm}_{i}\to\frac{\pm
1-\cos\theta}{r\sin\theta}\hat{\phi}_{i}\,\delta^{a}_{3}\qquad\hbox{and}\quad
e\mathcal{A}^{a}_{0}\to\left(ev-\frac{eQ}{r}\right)\delta^{a}_{3}\qquad\hbox{as}\quad
r\rightarrow\infty\,,$ (10)

since using this in Gauss’ law

$q_{{\scriptscriptstyle E}}=\frac{1}{4\pi}\int_{0}^{4\pi}{\bf E}\cdot\hat{{\bf
r}}\,{\rm d}^{2}S=Q\quad\hbox{and}\quad q_{{\scriptscriptstyle
M}}=\frac{1}{4\pi}\int_{0}^{4\pi}{\bf B}\cdot\hat{{\bf r}}\,{\rm
d}^{2}S=\frac{1}{e},$ (11)

reveals it to be a dyon that carries magnetic charge $q_{\scriptscriptstyle
M}=1/e$ and electric charge $q_{\scriptscriptstyle E}=Q$. Notice the magnetic
charge is consistent with the Dirac quantization condition that requires
$2q_{\scriptscriptstyle M}q$ to be an integer for any electric charge $q$
given that the particle with the smallest electric charge is the fermion, for
which $q_{\psi}=\pm\frac{1}{2}e$. The parameter $v$ is also seen to be the
electrostatic potential difference between the origin and infinity, which in
the general case is not independent of $Q$, with

$ev=\mu\;{\cal Z}\left(\beta,eQ\right)\,,$ (12)

for a calculable order-unity function ${\cal Z}$. (For instance in the Prasad-
Sommerfield limit (5) we have ${\cal Z}=eQ\beta$.) The monopole limit $Q\to 0$
corresponds to taking $v\to 0$ so ${\cal Z}(\beta,0)=0$.

The classical dyon mass is similarly given by evaluating the energy of the
classical solution:

$M\simeq\frac{4\pi\beta\mu}{e^{2}}\;F\left(\beta,v/\mu\right)\sim{\cal
O}\left(\frac{m_{g}}{\alpha}\right)\,,$ (13)

where $m_{g}\simeq\beta\mu=ew$ is the gauge boson mass, $\alpha=e^{2}/4\pi$ is
the fine-structure constant and $F$ is an explicitly calculable function that
is order unity when $e^{2}\sim\lambda$ and $v\sim\mu$. The mass is related to
the dyon size $R\sim m_{g}^{-1}$ by $MR\sim 4\pi/e^{2}\gg 1$ in the
semiclassical regime and so its Compton wavelength is much smaller than its
radius.

###  2.2 Semiclassical quantization

The interactions of a dyon with other degrees of freedom are found by
performing a semiclassical expansion of the quantum fields around the
classical dyonic background, with

$A_{\mu}^{a}(x)=\mathcal{A}_{\mu}^{a}(x)+\widehat{A}_{\mu}^{a}(x)\quad\hbox{and}\quad\Phi^{a}(x)={\varphi}^{a}(x)+\widehat{\Phi}^{a}(x),$
(14)

where the fields $\widehat{A}^{a}_{\mu}(x)$ and $\widehat{\Phi}^{a}(x)$ join
the fermion field ${\psi}(x)$ as quantum operators.

####  2.2.1 Collective coordinates

A special role is played by those fluctuations that are Goldstone directions
corresponding to symmetry transformations that act nontrivially on the
background dyon, since these cost little energy and so are important for
understanding a dyon’s low-energy response to external probes.

The counting of these zero modes is subtle for dyons SubtleCounting , but for
the case of the Julia-Zee dyon the physical collective coordinates are the
dyon’s center-of-mass position, ${\bm{y}}(t)$, and a charged collective
coordinate, ${\mathfrak{a}}(t)$. These can be regarded as the special cases
where $\widehat{A}_{\mu}^{a}$ and $\widehat{\Phi}^{a}$ are obtained by
transforming the dyon by a time-dependent spatial translation or a time-
dependent gauge rotation in the unbroken $U(1)$ gauge symmetry direction
(generated by $\tau_{3}$ in the gauge (8)),

$\delta{\cal A}_{\mu}^{a}(x)=y^{i}\partial_{i}{\cal
A}^{a}_{\mu}+\partial_{\mu}y^{i}{\cal
A}^{a}_{i}+\frac{1}{e}\,\delta^{a}_{3}\,\partial_{\mu}{\mathfrak{a}}-{\mathfrak{a}}\,\epsilon^{a3b}{\cal
A}^{b}_{\mu}\quad\hbox{and}\quad\delta\varphi^{a}(x)=y^{i}\partial_{i}\varphi^{a}-{\mathfrak{a}}\,\epsilon^{a3b}\varphi^{b},$
(15)

and the dependence on the background fields ensures these are localized near
the dyon’s location.

The semiclassical expansion proceeds by using (14) in the action (1) and
Taylor expanding in the fluctuation fields. The classical contribution to the
action (with $\widehat{A}_{\mu}^{a}=\widehat{\Phi}^{a}=\psi=0$) is order
$4\pi/e^{2}$ in the same way that eq. (13) implies that $M/m_{g}$ is order
$4\pi/e^{2}$. The leading dependence on fluctuations is quadratic and
describes free quantum fields evolving within the dyonic background. Because
the quadratic term is independent of $e$ the masses of the quanta destroyed by
the fluctuation fields are suppressed compared to the dyon mass by
$e^{2}/4\pi$. It is because successive terms in this expansion are suppressed
by still more powers of $e^{2}/4\pi$ that semiclassical methods are under
control in the weakly coupled regime.

In particular, the kinetic term for the collective coordinate ${\mathfrak{a}}$
is found by evaluating the Maxwell kinetic term using the ansatz

$e\mathcal{A}^{\pm}_{i}=\frac{\pm
1-\cos\theta}{2r\sin\theta}\hat{\phi}_{i}\tau_{3}-\frac{{\cal
K}(r)}{2r}\Big{[}(i\hat{\theta}_{i}-\hat{\phi}_{i})e^{i{\mathfrak{a}}(t)}e^{\pm
i\phi}\tau_{+}-(i\hat{\theta}_{i}+\hat{\phi}_{i})e^{-i{\mathfrak{a}}(t)}e^{\mp
i\phi}\tau_{-}\Big{]}$ (16)

for which the field strength contains

${\cal
F}_{0i}\ni\partial_{0}\mathcal{A}^{\pm}_{i}=-\frac{i\dot{{\mathfrak{a}}}\,{\cal
K}(r)}{2er}\Big{[}(i\hat{\theta}_{i}-\hat{\phi}_{i})e^{i{\mathfrak{a}}}e^{\pm
i\phi}\tau_{+}+(i\hat{\theta}_{i}+\hat{\phi}_{i})e^{-i{\mathfrak{a}}}e^{\mp
i\phi}\tau_{-}\Big{]}\,,$ (17)

and so

$-\text{Tr}({\cal F}_{0i}{\cal
F}^{0i})=\dot{{\mathfrak{a}}}^{2}(-i)^{2}\left[\frac{{\cal
K}(r)}{er}\right]^{2}(i\hat{\theta}_{i}-\hat{\phi}_{i})\cdot(i\hat{\theta}_{i}+\hat{\phi}_{i})\,\frac{\text{Tr}(\tau_{+}\tau_{-})}{2}=\left[\frac{{\cal
K}(r)}{er}\right]^{2}\dot{{\mathfrak{a}}}^{2}\,.$ (18)

This is localized near the position of the dyon because ${\cal K}(r)$ falls to
zero for large $r$. Writing this as $\frac{1}{2}\,{\cal
I}\dot{\mathfrak{a}}^{2}\,\delta^{3}(x)$ implies the constant ${\cal I}$ is

$\mathcal{I}=\frac{8\pi}{e^{2}}\int_{0}^{\infty}{\rm d}r\;{\cal
K}^{2}(r)\sim\frac{1}{\alpha m_{g}}\,,$ (19)

and so is parametrically large compared to the dyon size $R\sim m_{g}^{-1}$.

In the absence of fermion interactions the kinetic lagrangian for
${\mathfrak{a}}$ is $\frac{1}{2}\,{\cal I}\,\dot{\mathfrak{a}}^{2}$ and
because ${\mathfrak{a}}$ is a periodic variable it behaves like a quantum
rigid rotor, whose canonical momentum ${\mathfrak{p}}$ is both conserved and
quantized. Because ${\mathfrak{a}}$ shifts under electromagnetic gauge
transforms, with $\delta{\mathfrak{a}}=\omega$ when $\delta
A_{\mu}^{3}=\frac{1}{e}\partial_{\mu}\omega$, the conserved rotor momentum is
proportional to the rotor’s contribution to the dyon-localized electric
charge666We return to a more precise statement of rotor quantization including
the effects of a vacuum angle in §6.2 below.

${\cal Q}_{\scriptscriptstyle D}=-e\,{\mathfrak{p}}\,,$ (20)

and for ${\mathfrak{a}}$ identified with ${\mathfrak{a}}+2\pi$ the canonical
rotor momentum takes integer values ${\mathfrak{p}}|n\rangle=n|n\rangle$ (and
so ${\cal Q}_{\scriptscriptstyle D}=-en$). Unit steps in the rotor levels
differ in charge by $\delta{\cal Q}_{\scriptscriptstyle D}=\pm e$ and differ
in energy by

$\delta E_{n}=E_{n+1}-E_{n}=\frac{2n+1}{{\cal
I}}\sim\frac{e^{2}m_{g}}{4\pi}\ll m_{g}\,.$ (21)

These estimates show that dyonic excitations belong in the effective theory
for energies below $m_{g}$ precisely because the characteristic loop-counting
factor $e^{2}/4\pi$ suppresses $\delta E$ relative to $m_{g}$. But because the
steps in rotor energy are proportional to $e^{2}/4\pi$ changes to rotor
energies due to transitions can be negligible at a given order in the
semiclassical expansion, so care must be taken about the ordering of the
limits $E/m_{g}\to 0$ and $e^{2}/4\pi\to 0$ when working at low energies
within the semiclassical limit.

####  2.2.2 Fermion $S$-wave modes

Our main focus in this paper is how the bulk fermion interacts with the dyon,
so we now focus on the leading dependence of the action on the fermions, which
is given by

$S_{\psi}=-\int\mathrm{d}^{4}x\,\overline{\bm{\psi}}\,\gamma^{\mu}\left[\partial_{\mu}-\frac{i}{2}\,e\mathcal{A}^{a}_{\mu}\tau_{a}\right]\bm{\psi}.$
(22)

Far from the dyon core it is straightforward to find the fermionic energy
eigenstates predicted by this action. Near-source fermion-dyon interactions
are trickier to deal with since in principle they depend in detail on the
functions ${\cal K}(r)$ and ${\cal J}(r)$. The next section shows how EFT
reasoning allows the small size of the dyon to be exploited to circumvent this
issue for fermion states with $E\ll\mu$. Before doing so we first identify the
eigenmodes far from the dyon.

At low energies the partial waves that dominate in dyon scattering are those
that minimize the centrifugal barrier that must be penetrated in order to
reach the dyon core. What makes monopole and dyon interactions special is the
way that their magnetic charge adds a new contribution to the angular momentum
of an incident charged particle, and how this changes how the angular momentum
barrier depends on the orbital and total angular momenta Rubakov:1981rg ;
Callan:1982ah . For spinless charged particles the new contribution implies
there is always a centrifugal barrier, even with zero orbital angular
momentum. Unusually, it is only for spin-half charged particles that the
absence of a barrier is possible, and this only occurs for a very specific
type of $S$-wave state (on which we now focus).

In the present instance the conserved angular momentum $\vec{J}$ of an
external isodoublet doublet fermion moving in the presence of the Julia-Zee
dyon is

$\vec{J}=\vec{L}+\vec{S}+\vec{T},$ (23)

where $\vec{L}$ and $\vec{S}$ are the usual orbital and spin angular momentum
while $\vec{T}$ is the gauge isospin of the particle state. This last term is
the new one and it arises because the dyon background breaks the freedom to
perform independent gauge and physical rotations but leaves unbroken the
diagonal subgroup where gauge and physical rotations are peformed in unison.
As a consequence the radial magnetic field sourced by the dyon’s magnetic
charge contributes to a charged particle’s angular momentum.

For the isodoublet fermions of interest here, $\vec{T}^{2}$ and $\vec{S}^{2}$
both have eigenvalue $\frac{3}{4}$ as appropriate for a spin-half contribution
to the total angular momentum. The usual rules for combining angular momenta
show that the combination $\vec{S}+\vec{T}$ can either carry spin zero or spin
one. The state with no angular momentum barrier is the one where
$\vec{S}+\vec{T}$ combine with orbital angular momentum $\vec{L}$ to give zero
total angular momentum for $\vec{J}$. Writing the eigenvalues of
$\vec{L}^{2}=\ell(\ell+1)$ and $\vec{J}^{2}=j(j+1)$ the rules for combining
angular momenta show that $j=0$ can be obtained by combining the spin and
magnetic angular momentum with either of the two orbital angular momenta
$\ell=0,1$.

The explicit form for this type of $S$-wave fermion mode for the Julia-Zee
dyon was found for the solution (3) by Jackiw and Rebbi Jackiw and Rebbi
s-wave , and writing the spin index $i$ and isospin index $a$ as a matrix
$M_{ia}$ for each chirality of field the $S$-wave configuration has the
structure $M(x)=s(r,t)+ip(r,t)\,\hat{{\bm{r}}}\cdot\vec{\tau}$ and so
$\psi_{\scriptscriptstyle L}$ and $\psi_{\scriptscriptstyle R}$ each involve
two independent functions. Once transformed to the gauge where the solution
has the form (8) the result becomes

$\bm{\psi}(x)=\frac{1}{r}\begin{bmatrix}\psi_{+}(x)\\\
\psi_{-}(x)\end{bmatrix}\quad\hbox{where}\quad\psi_{\pm}(x)=\begin{pmatrix}f_{\pm}(r,t)\,\eta_{\pm}(\theta,\phi)\\\\[2.84526pt]
g_{\pm}(r,t)\,\eta_{\pm}(\theta,\phi)\end{pmatrix}\,,$ (24)

where square brackets denote gauge isodoublets and round brackets denote
4-component Dirac spinors in a basis for which $\gamma_{5}=\hbox{diag}(I,-I)$.
The 2-component Weyl spinors $\eta_{\pm}$ are defined to satisfy
$\sigma^{r}\eta_{\pm}=\pm\eta_{\pm}$ (see Appendix A for our Dirac matrix
conventions) and so are given explicitly by

$\eta_{+}(\theta,\phi)=\frac{1}{\sqrt{4\pi}}\left(\begin{array}[]{c}\cos{\frac{\theta}{2}}\,e^{-i\phi}\\\
\sin{\frac{\theta}{2}}\end{array}\right)\quad\hbox{and}\quad\eta_{-}(\theta,\phi)=\frac{1}{\sqrt{4\pi}}\left(\begin{array}[]{c}-\sin{\frac{\theta}{2}}\\\
\cos{\frac{\theta}{2}}\,e^{i\phi}\end{array}\right)$ (25)

in the region $R_{-}$ and by
$\eta^{\prime}_{\pm}(\theta,\phi)=\eta_{\pm}(\theta,\phi)\,e^{\pm i\phi}$ in
the region $R_{+}$.

At large distances from the dyon core only the far-field electromagnetic parts
of the monopole fields matter and so the $S$-wave sector Dirac equation
simplifies to

$\left(-\partial_{t}+\gamma^{0}\gamma^{r}\partial_{r}+\frac{i}{2}\,e\mathcal{A}^{3}_{0}\,\tau_{3}\right)(r\bm{\psi})=\begin{bmatrix}\left(-\partial_{t}+\partial_{r}+\frac{i}{2}\,e\mathcal{A}^{3}_{0}\right){f_{+}\eta_{+}}\\\
\left(-\partial_{t}-\partial_{r}+\frac{i}{2}\,e\mathcal{A}^{3}_{0}\right)g_{+}\eta_{+}\\\
\left(-\partial_{t}-\partial_{r}-\frac{i}{2}\,e\mathcal{A}^{3}_{0}\right)f_{-}\eta_{-}\\\
\left(-\partial_{t}+\partial_{r}-\frac{i}{2}\,e\mathcal{A}^{3}_{0}\right)g_{-}\eta_{-}\end{bmatrix}=0$
(26)

which uses $\gamma^{0}\gamma^{r}=\hbox{diag}(\sigma^{r},-\sigma^{r})$ (see
Appendix A) and $\sigma^{r}\eta_{\pm}=\pm\eta_{\pm}$. In principle we can
simultaneously diagonalize $\gamma_{5}$, $\gamma^{0}\gamma^{r}$ and
$\tau_{3}$, each of which has eigenvalues $\pm 1$, leading to eight possible
unique sets of quantum numbers,
$\\{{\mathfrak{c}},{\mathfrak{h}},{\mathfrak{s}}\\}$, where we denote the
eigenvalues of $\gamma_{5}$, $\gamma^{0}\gamma^{r}$ and $\tau_{3}$
respectively by ${\mathfrak{c}}$ (chirality), ${\mathfrak{h}}$ (helicity) and
${\mathfrak{s}}$ (for $\tau_{3})$. Of these, the $S$-wave condition (24) says
that the eigenvalues of $\tau_{3}$ and
$\gamma_{5}\gamma^{0}\gamma^{r}=\hbox{diag}(\sigma^{r},\sigma^{r})$ are the
same and so ${\mathfrak{s}}={\mathfrak{c}}{\mathfrak{h}}$. Also eq. (26) shows
that the direction of motion (radially ingoing or radially outgoing) is
correlated with the eigenvalue of $\gamma^{0}\gamma^{r}$ with
${\mathfrak{h}}=+1$ corresponding to infalling modes and ${\mathfrak{h}}=-1$
pairing with outgoing modes.

For an $S$-wave fermion passage through the origin inevitably brings a change
of radial fermion direction and this must change the sign of ${\mathfrak{h}}$.
Because ${\mathfrak{h}}={\mathfrak{c}}{\mathfrak{s}}$ we see why $S$-wave
scattering famously must involve a change in chirality (${\mathfrak{c}}$) or
in electric charge (${\mathfrak{s}}$). If it should be true that the
microscopic properties of the dyon preserve chirality – as is indeed the case
e.g. when these interactions are modeled by solving the Dirac equation in the
presence of a fixed dyon background Goldhaber:1977xw ; Blaer:1981ps ;
Marciano:1983md ; Ezawa:1983vi  – then ${\mathfrak{c}}$ cannot change and so
only charge changing processes ${\mathfrak{s}}^{\prime}\neq{\mathfrak{s}}$ are
possible. Famously, the rotor-fermion interactions can significantly modify
the fermionic vacuum which can in turn allow more complicated behaviour
Rubakov:1981rg ; Callan:1982ah ; CallanSMatrix ; Yamagishi:1982wp ;
Grossman:1983yf ; Kazama:1983rt ; Polchinski ; Csaki:2021ozp ; Csaki:2022tvb ;
vanBeest:2023dbu ; vanBeest:2023mbs  (more about which later). For this reason
we do not assume below that it is ${\mathfrak{s}}$ that must change during
$S$-wave fermion-dyon scattering.

We label the four independent modes by their quantum numbers ${\mathfrak{s}}$
and ${\mathfrak{c}}$ (from which the $S$-wave condition implies
${\mathfrak{h}}={\mathfrak{c}}{\mathfrak{s}}$), and then integrate to find the
mode functions with frequency $\omega$ once ${\cal A}_{0}^{3}$ is evaluated
using (8) specialized to the asymptotic form (10). This gives the following
explicit basis of far-field positive-frequency solutions
$u_{{\mathfrak{s}}{\mathfrak{c}}\omega}(x)$,

$\begin{bmatrix}u_{+{\mathfrak{c}}\omega}(x)\\\
0\end{bmatrix}\,,\;\begin{bmatrix}0\\\
u_{-{\mathfrak{c}}\omega}(x)\end{bmatrix}\quad\hbox{where}\quad
u_{{\mathfrak{s}}{\mathfrak{c}}\omega}(x):=\frac{1}{r}\,\xi_{\mathfrak{c}}\otimes\eta_{\mathfrak{s}}(\theta,\phi)\,e^{-i\omega
t}e^{-i({\mathfrak{s}}\omega+\frac{1}{2}ev)\,{\mathfrak{c}}\,r}\left(\frac{r}{r_{0}}\right)^{i{\mathfrak{c}}\,eQ/2},$
(27)

with $\sigma_{3}\xi_{\mathfrak{c}}={\mathfrak{c}}\,\xi_{\mathfrak{c}}$ and we
write $\gamma_{5}=\sigma_{3}\otimes I$. The length scale $r_{0}$ is arbitrary
and is introduced on dimensional grounds due to the singular Coulomb phase.
Here $\omega$ is bounded from below, but the asymptotic voltage $v$ implies
the floor is $\omega\geq-\frac{1}{2}{\mathfrak{s}}ev$. The oscillatory factors
in this expression can be equivalently written
$e^{-i(k-\frac{1}{2}{\mathfrak{s}}ev)t}e^{-i{\mathfrak{s}}{\mathfrak{c}}kr}$
for $k\geq 0$. The negative frequency counterparts similarly are

$\begin{bmatrix}v_{+{\mathfrak{c}}\omega}(x)\\\
0\end{bmatrix}\,,\;\begin{bmatrix}0\\\
v_{-{\mathfrak{c}}\omega}(x)\end{bmatrix}\quad\hbox{where}\quad
v_{{\mathfrak{s}}{\mathfrak{c}}\omega}(x):=\frac{1}{r}\,\xi_{\mathfrak{c}}\otimes\eta_{\mathfrak{s}}(\theta,\phi)\,e^{i\omega
t}e^{i({\mathfrak{s}}\omega-\frac{1}{2}ev)\,{\mathfrak{c}}\,r}\left(\frac{r}{r_{0}}\right)^{i{\mathfrak{c}}\,eQ/2}\,.$
(28)

Frequency $\omega$ is again bounded from below, but in this case the floor is
$\omega\geq+\frac{1}{2}{\mathfrak{s}}ev$ and the oscillatory factors can be
written
$e^{i(k+\frac{1}{2}{\mathfrak{s}}ev)t}e^{i{\mathfrak{s}}{\mathfrak{c}}kr}$ for
$k\geq 0$.

Keeping in mind that particle probability flux points in the opposite
direction to the momentum for negative-frequency states (see for example
Hansen:1980nc ), the sign of the radial direction of motion for these modes is
given by $-{\mathfrak{s}}{\mathfrak{c}}$ for particles and by
$+{\mathfrak{s}}{\mathfrak{c}}$ for antiparticles. These modes are normalized
so that
$i{\overline{u}}_{{\mathfrak{s}}{\mathfrak{c}}\omega}\gamma^{0}u_{{\mathfrak{s}}{\mathfrak{c}}\omega}=i{\overline{v}}_{{\mathfrak{s}}{\mathfrak{c}}\omega}\gamma^{0}v_{{\mathfrak{s}}{\mathfrak{c}}\omega}=(4\pi
r^{2})^{-1}$ and as a result
$i{\overline{u}}_{{\mathfrak{s}}{\mathfrak{c}}\omega}\gamma^{r}u_{{\mathfrak{s}}{\mathfrak{c}}\omega}=i{\overline{v}}_{{\mathfrak{s}}{\mathfrak{c}}\omega}\gamma^{r}v_{{\mathfrak{s}}{\mathfrak{c}}\omega}=-{\mathfrak{s}}{\mathfrak{c}}/(4\pi
r^{2})=-{\mathfrak{h}}/(4\pi r^{2})$.

####  2.2.3 2D formulation

As has been often noted elsewhere purely $S$-wave dynamics can be usefully
rewritten as a Dirac equation in 1+1 dimensions Rubakov:1981rg ; Callan:1982ah
, and we pause here to establish our conventions. In this 1+1 dimensional
formulation the only spatial direction corresponds to the radial direction in
the 3+1 dimensional formulation, and so runs only along the half-line
corresponding to positive $r$, with the dyon interior providing a more
complicated background solution only within a distance of order $\mu^{-1}$
about $r=0$.

Within the full theory fields are required to be nonsingular at the origin
(deep within the dyon) and this can be converted in the 2D formulation into a
boundary condition for fields at $r=0$ designed so that reflection at $r=0$
describes the effect in the 4D theory of passing through the origin.
Alternatively we can instead describe $S$-wave scattering within the 2D
picture by extending the spatial direction to cover the entire real line and
think of, say, $x<0$ as describing incoming waves and $x>0$ describing
outgoing waves.

In either formulation 4D fermions are described by independent 2D spinors,
$\chi$, for each fermion flavour and chirality in the following way777This is
to be compared with the 4D expression (24). These definitions ensure that
$\psi_{{\mathfrak{s}}}$ and $\chi_{{\mathfrak{s}}}$ transform in the same way
under the remaining Lorentz transformation in 2D i.e. radial boosts. These are
generated by $\gamma^{0}\gamma^{r}$ in 4D and
$\Gamma^{0}\Gamma^{1}=\Gamma_{c}$ in 2D, and since
${\mathfrak{h}}=-{\mathfrak{c}}$ for negatively charged fermions, the order of
$f_{-}(r,t)$ and $g_{-}(r,t)$ is switched when going to 2D.

$\chi_{+}(r,t)=\begin{pmatrix}f_{+}(r,t)\\\
g_{+}(r,t)\end{pmatrix}\quad\hbox{and}\quad\chi_{-}(r,t)=\begin{pmatrix}g_{-}(r,t)\\\
f_{-}(r,t)\end{pmatrix}=i\Gamma^{0}\begin{pmatrix}f_{-}(r,t)\\\
g_{-}(r,t)\end{pmatrix},$ (29)

with the subscript $\pm$ indicating both the isospin and the $U(1)$ charge of
each spinor (see Appendix A for our 2D Dirac-matrix conventions). Comparing to
(26) shows that their equations of motion can then be compactly written as a
2D Dirac equation in the background potential $\mathcal{A}^{3}_{0}(r)$ of the
form:

$\left(-\partial_{t}+\Gamma^{0}\Gamma^{1}\partial_{1}+\frac{i{\mathfrak{s}}}{2}\,e\mathcal{A}^{3}_{0}\right)\chi_{\mathfrak{s}}=0\,,$
(30)

such as would follow from the bulk action

$S_{2}=-\frac{1}{2}\sum_{{\mathfrak{s}}=\pm}\int{\rm
d}^{2}x\;{\overline{\chi}}_{\mathfrak{s}}\,\Gamma^{\alpha}\overset{\leftrightarrow}{D}_{\alpha}\chi_{\mathfrak{s}}\,,$
(31)

where ${\overline{\chi}}:=i\chi^{\dagger}\Gamma^{0}$ and
$D_{\alpha}\chi_{\mathfrak{s}}=(\partial_{\alpha}-\tfrac{1}{2}ie{\mathfrak{s}}{\cal
A}_{\alpha}^{3})\chi_{\mathfrak{s}}$.

Notice that (24) shows that the eigenvalue of the 2D chirality
$\Gamma_{c}:=\Gamma^{0}\Gamma^{1}=\sigma_{3}$ in this representation is
${\mathfrak{c}}{\mathfrak{s}}$ where ${\mathfrak{c}}$ is the eigenvalue of 4D
chirality and so is the same as ${\mathfrak{h}}$ and therefore should
correlate with the direction of motion. This correlation in the 2D language is
a direct consequence of the Dirac equation because
$\Gamma^{0}(\Gamma^{\alpha}\partial_{\alpha})=-\partial_{t}+\Gamma_{c}\partial_{1}$.

Dimensional reduction of the $S$-wave modes strips away their angular content,
but otherwise leaves them much as described in (27) and (28) above. For
instance, writing the doublet built from $\chi_{+}$ and $\chi_{-}$ by
$\bm{\chi}$ (and continuing to denote the isodoublets by square brackets and
spinor doublets with round brackets), the positive-frequency basis of
solutions ${\mathfrak{u}}_{{\mathfrak{s}}{\mathfrak{c}}\omega}(x)$
corresponding to (27) are

$\begin{bmatrix}{\mathfrak{u}}_{+{\mathfrak{c}}\omega}(x)\\\
0\end{bmatrix}\,,\;\begin{bmatrix}0\\\
{\mathfrak{u}}_{-{\mathfrak{c}}\omega}(x)\end{bmatrix}\quad\hbox{where}\quad{\mathfrak{u}}_{{\mathfrak{s}}{\mathfrak{c}}\omega}(x):=\xi_{{\mathfrak{s}}{\mathfrak{c}}}\,e^{-i\omega
t}e^{-i({\mathfrak{s}}\omega+\frac{1}{2}ev)\,{\mathfrak{c}}\,r}\left(\frac{r}{r_{0}}\right)^{i{\mathfrak{c}}\,eQ/2},$
(32)

with
$\Gamma_{c}\,\xi_{{\mathfrak{s}}{\mathfrak{c}}}=\sigma_{3}\,\xi_{{\mathfrak{s}}{\mathfrak{c}}}={\mathfrak{s}}{\mathfrak{c}}\,\xi_{{\mathfrak{s}}{\mathfrak{c}}}$.
The negative frequency counterparts similarly are

$\begin{bmatrix}{\mathfrak{v}}_{+{\mathfrak{c}}\omega}(x)\\\
0\end{bmatrix}\,,\;\begin{bmatrix}0\\\
{\mathfrak{v}}_{-{\mathfrak{c}}\omega}(x)\end{bmatrix}\quad\hbox{where}\quad{\mathfrak{v}}_{{\mathfrak{s}}{\mathfrak{c}}\omega}(x):=\xi_{{\mathfrak{s}}{\mathfrak{c}}}\,e^{i\omega
t}e^{i({\mathfrak{s}}\omega-\frac{1}{2}ev)\,{\mathfrak{c}}\,r}\left(\frac{r}{r_{0}}\right)^{i{\mathfrak{c}}\,eQ/2}\,.$
(33)

The frequency $\omega$ is (as above) bounded from below, with floors
$\omega\geq-\frac{1}{2}{\mathfrak{s}}ev$ and
$\omega\geq+\frac{1}{2}{\mathfrak{s}}ev$ for particle and antiparticle
respectively. These modes are normalized so that
$i{\overline{{\mathfrak{u}}}}_{{\mathfrak{s}}{\mathfrak{c}}\omega}\Gamma^{0}{\mathfrak{u}}_{{\mathfrak{s}}{\mathfrak{c}}\omega}=i{\overline{{\mathfrak{v}}}}_{{\mathfrak{s}}{\mathfrak{c}}\omega}\Gamma^{0}{\mathfrak{v}}_{{\mathfrak{s}}{\mathfrak{c}}\omega}=1$
and so the 2D and 4D fluxes are related by
$i{\overline{{\mathfrak{u}}}}_{{\mathfrak{s}}{\mathfrak{c}}\omega}\Gamma^{r}{\mathfrak{u}}_{{\mathfrak{s}}{\mathfrak{c}}\omega}=4\pi
r^{2}(i{\overline{u}}_{{\mathfrak{s}}{\mathfrak{c}}\omega}\gamma^{r}u_{{\mathfrak{s}}{\mathfrak{c}}\omega})$
and similarly for ${\mathfrak{v}}$ and $v$.

##  3 Dyonic EFT

The far-field modes given in (27) and (28) suffice to fully describe the state
of both an incident and departing fermion (or antifermion) when it is far from
the dyon, but scattering calculations relate the size of the departing wave to
the initially incident one and in principle this requires solving the full
Dirac equation obtained from (22) including the dyon’s interior structure. It
is only once this structure is included that the fermionic modes can be
required to be nonsingular at the origin, providing the information that
ultimately links the incoming and outgoing modes.

But solving the full Dirac equation including the dyon interior is difficult
(see however Goldhaber:1977xw ; Blaer:1981ps ; Marciano:1983md ; Ezawa:1983vi
) and also likely overkill, at least for incident fermion energy $E$ small
compared with the inverse dyon size $\mu$. In this regime the dyon is so small
that its physical implications should be capturable by a choice of boundary
condition near the origin. But this is where EFT techniques can usefully be
applied and to this end we apply here the PPEFT formalism PPEFT ; PPEFT2 ;
PPEFT3  — a framework explicitly designed for the purpose of constructively
deriving such boundary conditions starting from the low-energy effective
action for the compact object (for a review see e.g. EFTBook ). This section
applies this formalism to determine the form of the boundary conditions
required for the Julia-Zee dyon (and its more complicated counterparts), at
leading order in the small ratio $E/\mu$.

###  3.1 $S$-wave PPEFT

The strategy, as always for EFTs, starts by replacing the full dyon with an
effective description that captures its low-energy interactions. The required
EFT is defined along the dyonic world-line and describes the interactions
between low-energy bulk fields and the low-energy dyonic collective
coordinates. This begins with an enumeration of all possible lowest-dimension
interactions allowed by the assumed symmetries and particle content.

The dyon-localized fields to be included in the dyonic effective action
include the dyonic collective coordinates like the centre-of-mass position,
$x^{\mu}=y^{\mu}(s)$, of the dyon’s world-line $W$ and the charge-excitation
field ${\mathfrak{a}}(s)$, where $s$ is an arbitrary parameter along the
world-line. The ‘bulk’ fields888Here ‘bulk’ fields mean fields that are
defined everywhere in spacetime and not only along the dyon’s world-line. to
be included are those with masses much smaller than $\mu$: the fermion doublet
${\bm{\psi}}(x)$ and the massless bulk electromagnetic field
$\widehat{A}_{\mu}(x)=\widehat{A}^{\,3}_{\mu}(x)$, and (in principle) the
spacetime metric $g_{\mu\nu}(x)$.

The symmetries to be imposed are ($i$) world-line reparameterizations of $s$;
($ii$) Poincaré invariance in spacetime (or general coordinate invariance if a
general metric $g_{\mu\nu}$ is included); ($iii$) invariance under
electromagnetic gauge transformations; and ($iv$) any low-energy flavour
and/or discrete symmetries. Spacetime symmetries are built in by constructing
the action using the pull-backs of spacetime tensors to the world-
line,999Interactions involving the normals to the world-line can also be
constructed but do not play any role in what follows.

$\psi(s)=\psi[y(s)]\,,\quad\widehat{A}(s)=\dot{y}^{\mu}(s)\widehat{A}_{\mu}[y(s)]\quad\hbox{and}\quad\gamma(s)=\dot{y}^{\mu}(s)\dot{y}^{\nu}(s)g_{\mu\nu}[y(s)]\,,$
(34)

where over-dots denote differentiation with respect to $s$. One builds from
these a reparameterization invariant action in the standard way.

For these fields the electromagnetic gauge transformations are

$\delta{\bm{\psi}}(s)=\frac{i}{2}\,\omega(s)\,\tau_{3}{\bm{\psi}}(s)\,,\quad\delta\widehat{A}(s)=\frac{1}{e}\,\dot{\omega}(s)\quad\hbox{and}\quad\delta{\mathfrak{a}}(s)=\omega(s)\,,$
(35)

where $\omega(s)$ is related to the spacetime dependent $SU(2)$ gauge
transformation parameters $\omega^{a}(x)$ by $\omega(s)=\omega^{3}[y(s)]$.
This symmetry requires derivatives of ${\mathfrak{a}}$ to appear within a
covariant derivative

$D{\mathfrak{a}}:=\dot{\mathfrak{a}}-e\widehat{A}\,.$ (36)

####  3.1.1 Dyon world-line effective action

For these variables the lowest-dimension interactions involving the rotor
field ${\mathfrak{a}}$ and the dyon displacement $y^{\mu}$ located at the
dyonic position become

$S_{\rm dyon}=\int_{\scriptscriptstyle W}{\rm
d}s\left\\{\sqrt{-\gamma}\left[-M-\frac{{\cal
I}}{2\gamma}(D{\mathfrak{a}})^{2}-\frac{1}{2}\,{\overline{\bm{\psi}}}\,{\mathfrak{C}}({\mathfrak{a}})\,{\bm{\psi}}+\cdots\right]+\frac{\vartheta}{2\pi}D{\mathfrak{a}}\right\\}$
(37)

where $M$ is the classical dyon mass while ${\cal I}$ is the ‘rotor’
coefficient (19) and the ellipses include a variety of other higher-dimension
terms whose contributions to physics should be suppressed at low energies and
so whose detailed form is not required in what follows.101010Notice we only
include here the couplings of the fluctuation fields and not also the dyon
interactions giving rise to the background fields themselves. This is why no
term like $Q\widehat{A}$ appears describing the dyon’s classical charge, and
why no term is required expressing the dyon’s magnetic charge MagCharge ;
MagCharge1 . The same could also have been done for the metric if we’d
expanded about the dyon’s gravitational back-reaction, but we do not do so.
The most general fermion bilinear consistent with the field content, gauge and
spacetime symmetries is

$\displaystyle{\mathfrak{C}}({\mathfrak{a}})$ $\displaystyle:=$
$\displaystyle\hat{\mathfrak{c}}^{s}_{1}+i\,\hat{\mathfrak{c}}^{ps}_{1}\gamma_{5}+i\,\hat{\mathfrak{c}}^{v}_{1}\gamma_{\mu}\dot{y}^{\mu}+i\,\hat{\mathfrak{c}}^{pv}_{1}\gamma_{5}\gamma_{\mu}\dot{y}^{\mu}+\Bigl{(}\hat{\mathfrak{c}}^{s}_{3}+i\,\hat{\mathfrak{c}}^{ps}_{3}\gamma_{5}+i\,\hat{\mathfrak{c}}^{v}_{3}\gamma_{\mu}\dot{y}^{\mu}+i\,\hat{\mathfrak{c}}^{pv}_{3}\gamma_{5}\gamma_{\mu}\dot{y}^{\mu}\Bigr{)}\tau_{3}$
$\displaystyle\qquad\qquad+\Bigl{(}\hat{\mathfrak{c}}^{s}_{+}+i\,\hat{\mathfrak{c}}^{ps}_{+}\gamma_{5}+i\,\hat{\mathfrak{c}}^{v}_{+}\gamma_{\mu}\dot{y}^{\mu}+i\,\hat{\mathfrak{c}}^{pv}_{+}\gamma_{5}\gamma_{\mu}\dot{y}^{\mu}\Bigr{)}e^{i{\mathfrak{a}}}\tau_{+}$
$\displaystyle\qquad\qquad\qquad\qquad+\Bigl{(}\hat{\mathfrak{c}}^{s}_{-}+i\,\hat{\mathfrak{c}}^{ps}_{-}\gamma_{5}+i\,\hat{\mathfrak{c}}^{v}_{-}\gamma_{\mu}\dot{y}^{\mu}+i\,\hat{\mathfrak{c}}^{pv}_{-}\gamma_{5}\gamma_{\mu}\dot{y}^{\mu}\Bigr{)}e^{-i{\mathfrak{a}}}\tau_{-}\,,$

where (as before) $\tau_{\pm}=\frac{1}{2}(\tau_{1}\pm i\tau_{2})$.

This action is meant to capture the low-energy interactions of the underlying
dyon and the quantities $M$, ${\mathfrak{C}}$ and so on are found by matching
to the properties of the dyon’s microscopic description. In particular, this
relates the parameter $\vartheta$ to the vacuum angle appearing in the bulk
electromagnetic theta-term

${\cal L}_{\vartheta}=\frac{\vartheta e^{2}}{4\pi^{2}}\,{\bf E}\cdot{\bf
B}\,,$ (39)

since both ultimately descended from the underlying theta-term for the
microscopic nonabelian $SU(2)$ gauge interactions. (Equivalently, this
connection can also be established using invariance of the system under large
gauge transformations.)

To interpret (37) it is convenient to specialize to a dyon that is
perturbatively close to being at rest so
$\dot{y}^{\mu}(s)=\delta^{\mu}_{0}+\delta\dot{y}^{\mu}(s)$, choose the
Minkowski metric and choose the parameter $s=t$ to be time in the background
dyon’s rest frame. In this case
$-\gamma=-\eta_{\mu\nu}\dot{y}^{\mu}\dot{y}^{\nu}=1-\dot{\bm{y}}\cdot\dot{\bm{y}}+\cdots$
(where ${\bm{y}}$ is the spatial part of $\delta y^{\mu}$) and (37) becomes

$S_{\rm dyon}\simeq\int_{\scriptscriptstyle W}{\rm
d}t\left[\frac{M}{2}\dot{\bm{y}}\cdot\dot{\bm{y}}+\frac{{\cal
I}}{2}\Bigl{(}\dot{\mathfrak{a}}-e\widehat{A}_{0}\Bigr{)}^{2}-\frac{1}{2}\,{\overline{\bm{\psi}}}\,{\mathfrak{C}}({\mathfrak{a}})\,{\bm{\psi}}+\frac{\vartheta}{2\pi}\Bigl{(}\dot{\mathfrak{a}}-e\widehat{A}_{0}\Bigr{)}+\cdots\right]$
(40)

where to leading order – i.e. neglecting dyon recoil effects – the integral is
evaluated along the dyon’s world-line, which we choose to be located at
${\bm{r}}={\bm{0}}$. In the same approximation the matrix
${\mathfrak{C}}({\mathfrak{a}})$ controlling the fermion-dyon couplings
becomes

$\displaystyle{\mathfrak{C}}({\mathfrak{a}})$ $\displaystyle:=$
$\displaystyle\hat{\mathfrak{c}}^{s}_{1}+i\,\hat{\mathfrak{c}}^{ps}_{1}\gamma_{5}-i\,\hat{\mathfrak{c}}^{v}_{1}\gamma^{0}-i\,\hat{\mathfrak{c}}^{pv}_{1}\gamma_{5}\gamma^{0}+\Bigl{(}\hat{\mathfrak{c}}^{s}_{3}+i\,\hat{\mathfrak{c}}^{ps}_{3}\gamma_{5}-i\,\hat{\mathfrak{c}}^{v}_{3}\gamma^{0}-i\,\hat{\mathfrak{c}}^{pv}_{3}\gamma_{5}\gamma^{0}\Bigr{)}\tau_{3}$
$\displaystyle\quad+\Bigl{(}\hat{\mathfrak{c}}^{s}_{+}+i\,\hat{\mathfrak{c}}^{ps}_{+}\gamma_{5}-i\,\hat{\mathfrak{c}}^{v}_{+}\gamma^{0}-i\,\hat{\mathfrak{c}}^{pv}_{+}\gamma_{5}\gamma^{0}\Bigr{)}e^{i{\mathfrak{a}}}\tau_{+}+\Bigl{(}\hat{\mathfrak{c}}^{s}_{-}+i\,\hat{\mathfrak{c}}^{ps}_{-}\gamma_{5}-i\,\hat{\mathfrak{c}}^{v}_{-}\gamma^{0}-i\,\hat{\mathfrak{c}}^{pv}_{-}\gamma_{5}\gamma^{0}\Bigr{)}e^{-i{\mathfrak{a}}}\tau_{-}\,,$

which drops ${\bm{\gamma}}\cdot\dot{\bm{y}}$ dyon-recoil terms.

On dimensional grounds all sixteen of the effective couplings111111Since the
size of ${\mathfrak{c}}$ might naturally be expected to be set by the dyon
size $R\sim\mu^{-1}$ this is usually where the suppression by the monopole
scale would naively enter into observable scattering. Part of the purpose of
this exercise is to understand why this suppression does not actually arise in
fermion-dyon scattering, and how general a phenomenon this is.
$\hat{\mathfrak{c}}$ have dimensions (length)2. All of these effective fermion
interactions conserve electric charge, and this dictates the
${\mathfrak{a}}$-dependence of ${\mathfrak{C}}({\mathfrak{a}})$. If treated
perturbatively these interactions describe fermion scattering from the dyon,
possibly associated with ${\mathfrak{a}}$ excitation. The perturbative dyon
response to fermion scattering can be seen because the canonical momentum for
${\mathfrak{a}}$ is also its conserved charge,

${\mathfrak{p}}:=\frac{\delta S_{\rm dyon}}{\delta\dot{\mathfrak{a}}}={\cal
I}D{\mathfrak{a}}+\frac{\vartheta}{2\pi}=-\frac{{\cal Q}_{\scriptscriptstyle
D}}{e}\,,$ (42)

where – see e.g. eq. (44) – ${\cal Q}_{\scriptscriptstyle D}$ is the
contribution of dyonic excitations to the fluctuations’ electric charge:
${\cal Q}={\cal Q}_{\scriptscriptstyle D}+{\cal Q}_{\scriptscriptstyle F}$
(where ${\cal Q}_{\scriptscriptstyle F}$ is the electric charge carried by the
fermions). The canonical commutation relation
$[{\mathfrak{p}}(t)\,,{\mathfrak{a}}(t)]=-i$ therefore implies the quantities
$e^{\pm i{\mathfrak{a}}}$ act as raising/lowering operators for ${\cal
Q}_{\scriptscriptstyle D}$, since

$\Bigl{[}{\cal Q}_{\scriptscriptstyle D}/e\,,e^{\pm
i{\mathfrak{a}}}\Bigr{]}=\mp e^{\pm i{\mathfrak{a}}}\,.$

Interactions proportional to $e^{\pm i{\mathfrak{a}}}$ therefore describe
transitions that raise or lower the dyon charge by $e$, as required by charge
conservation for the reactions involving $\tau_{\pm}$ in (40) (which change
the fermion charge from $-\frac{1}{2}e$ to $+\frac{1}{2}e$ or vice versa).

Because these fermion-${\mathfrak{a}}$ couplings are not suppressed by powers
of small couplings like $e$ they are not intrinsically negligible even at
leading loop order, which does not justify expanding the exponentials in
powers of ${\mathfrak{a}}$. Furthermore the energy exchanged by exciting or
de-exciting ${\mathfrak{a}}$ is order ${\cal I}^{-1}$ which (21) reveals is
suppressed by $\alpha=e^{2}/(4\pi)$ compared to the characteristic scale $\mu$
and so has little intrinsic cost in the semiclassical limit. Such transitions
can be important to fermion scattering, though there is an order-of-limits
issue when working both at low fermion energies, $E\ll\mu$, and at leading
nontrivial order in the loop expansion, $\alpha=e^{2}/(4\pi)\ll 1$, because it
matters in practice whether or not the fermion energy $E$ is larger or smaller
than the dyon excitation scale $\alpha\mu$. As argued below, a natural way to
handle this fermion-dyon dynamics in the effective theory – at least in the
regime $\alpha\mu\ll E\ll\mu$ – is through the Born-Oppenheimer approximation
BornOppenheimer  (in which the fermions play the role of the ‘fast’ degrees of
freedom while the large size of ${\cal I}$ makes the dyonic excitations
‘slow’).

The interactions in (3.1.1) can also be classified by how they transform under
global ‘flavour’ transformations acting on the fermion field. In the present
instance the limited field content restricts this to two types of such
symmetries: an axial symmetry for which $\delta_{\scriptscriptstyle
A}{\bm{\psi}}=i\omega_{\scriptscriptstyle A}\,\gamma_{5}\,{\bm{\psi}}$ and a
fermion-number ‘baryon’ symmetry for which $\delta_{\scriptscriptstyle
B}{\bm{\psi}}=i\omega_{\scriptscriptstyle B}\,{\bm{\psi}}$ (notice the
difference between this and the gauge transformation (35)). We denote the
corresponding conserved charges by ${\cal Q}_{\scriptscriptstyle A}$ and
${\cal Q}_{\scriptscriptstyle B}$ to distinguish them from the fermionic
contribution to the gauge charge ${\cal Q}_{\scriptscriptstyle F}$. The
Noether currents for these two symmetries are $J^{\mu}_{\scriptscriptstyle
B}=i{\overline{{\bm{\psi}}}}\gamma^{\mu}{\bm{\psi}}$ and
$J^{\mu}_{\scriptscriptstyle
A}=i{\overline{{\bm{\psi}}}}\gamma^{\mu}\gamma_{5}{\bm{\psi}}$ and satisfy

$\partial_{\mu}J^{\mu}_{\scriptscriptstyle
B}=0\quad\hbox{and}\quad\partial_{\mu}J^{\mu}_{\scriptscriptstyle
A}=\frac{e^{2}}{32\pi^{2}}\,\epsilon^{\mu\nu\lambda\rho}F^{a}_{\mu\nu}F^{a}_{\lambda\rho}=-\frac{e^{2}}{4\pi^{2}}\,{\bf
E}^{a}\cdot{\bf B}^{a}\,,$ (43)

which shows that the axial symmetry is anomalous. Table 1 identifies which of
the various effective interactions of (3.1.1) preserves each of these two
flavour symmetries.

| $\Delta{\cal Q}_{\scriptscriptstyle F}=\Delta{\cal Q}_{\scriptscriptstyle A}=0$ |  $\Delta{\cal Q}_{\scriptscriptstyle F}=0$, $\Delta{\cal Q}_{\scriptscriptstyle A}\neq 0$ |  $\Delta{\cal Q}_{\scriptscriptstyle A}=0$ and $\Delta{\cal Q}_{\scriptscriptstyle F}\neq 0$ |  $\Delta{\cal Q}_{\scriptscriptstyle A}\neq 0$ and $\Delta{\cal Q}_{\scriptscriptstyle F}\neq 0$
---|---|---|---|---
coupling | $\hat{\mathfrak{c}}^{v}_{1},\hat{\mathfrak{c}}^{v}_{3},\hat{\mathfrak{c}}^{pv}_{1},\hat{\mathfrak{c}}^{pv}_{3}$ | $\hat{\mathfrak{c}}^{s}_{1},\hat{\mathfrak{c}}^{s}_{3},\hat{\mathfrak{c}}^{ps}_{1},\hat{\mathfrak{c}}^{ps}_{3}$ | $\hat{\mathfrak{c}}^{v}_{\pm},\hat{\mathfrak{c}}^{pv}_{\pm}$ | $\hat{\mathfrak{c}}^{s}_{\pm},\hat{\mathfrak{c}}^{ps}_{\pm}$
Table 1: Classification of interaction terms in the worldline action according
to whether they conserve the fermionic charges ${\cal Q}_{\scriptscriptstyle
F}$ (electric charge) and ${\cal Q}_{\scriptscriptstyle A}$ (axial charge) as
defined in the main text. All interactions conserve ${\cal
Q}_{\scriptscriptstyle B}$ (fermion number). Total electric charge (including
the contribution from ${\mathfrak{a}}$) is also conserved by all interactions.

####  3.1.2 Induced boundary conditions

In practice we wish to excise the dyon from the external world and replace it
with a gaussian ‘pillbox’, ${\cal P}_{\epsilon}$, whose radius is chosen much
larger than the dyon’s, $\epsilon\gg R\sim m_{g}^{-1}$, but also much smaller
than the distances of interest for the bulk fields used for low-energy probes
of the dyon. The idea is to replace the dyon with a set of near-dyon boundary
conditions on the surface $r=\epsilon$ of this pillbox, whose detailed form is
chosen to reproduce the interactions implied by (37). It is these boundary
conditions that communicate dyonic physics to the external world from which
the dyon is excised.

The boundary conditions can be obtained by a variety of means PPEFT ; PPEFT2 ;
PPEFT3 , such as by integrating the field equations with the dyon-localized
interaction written proportional to $\delta^{3}(x)$ and then regularizing the
appearance of divergent bulk fields, like $\widehat{A}(0)$, with its value on
the surface of the pillbox, $\widehat{A}(\epsilon)$. Applied to (40) this
leads to a familar expression: Gauss’ law itself, in the form

$\oint{\rm
d}^{2}\Omega\,\Bigl{(}r^{2}\partial_{r}\widehat{A}_{0}\Bigr{)}_{r=\epsilon}=4\pi\Bigl{(}r^{2}\partial_{r}\widehat{A}_{0}\Bigr{)}_{r=\epsilon}=\left(\frac{\delta
S_{\rm
dyon}}{\delta\widehat{A}_{0}}\right)_{r=\epsilon}=-e{\mathfrak{p}}=-e\left({\cal
I}D{\mathfrak{a}}+\frac{\vartheta}{2\pi}\right)\,.$ (44)

Comparing this to the bulk far-field Coulomb solution
$\widehat{A}_{0}\simeq-{\cal Q}_{\scriptscriptstyle D}/(4\pi r)$ with
integration constant ${\cal Q}_{\scriptscriptstyle D}$ given by the dyonic
fluctuation’s charge, this boundary condition fixes ${\cal
Q}_{\scriptscriptstyle D}=-e{\mathfrak{p}}$ as used above (and in passing
shows how the dyon acquires a $\vartheta$-dependent charge through the Witten
effect Witten:1979ey ).

We wish to make the same argument for the bulk fermion field. The same steps
PPEFT3  lead to the boundary condition

$\frac{1}{2}\oint{\rm
d}^{2}\Omega\,\Bigl{(}r^{2}\gamma^{r}{\bm{\psi}}\Bigr{)}_{r=\epsilon}=\left(\frac{\delta
S_{\rm
dyon}}{\delta{\overline{{\bm{\psi}}}}}\right)_{r=\epsilon}=-\frac{1}{2}\,\Bigl{[}{\mathfrak{C}}({\mathfrak{a}})\,{\bm{\psi}}\Bigr{]}_{r=\epsilon}\,,$
(45)

once projected onto the $S$ wave state. This can formally be recast as the
non-differential condition

$\displaystyle 0=\Bigl{\\{}\Bigl{[}\gamma^{r}+\Bigl{(}\hat{{\cal
C}}^{s}_{1}+i\,\hat{{\cal C}}^{ps}_{1}\gamma_{5}-i\,\hat{{\cal
C}}^{v}_{1}\gamma^{0}-i\,\hat{{\cal
C}}^{pv}_{1}\gamma_{5}\gamma^{0}\Bigr{)}+\Bigl{(}\hat{{\cal
C}}^{s}_{3}+i\,\hat{{\cal C}}^{ps}_{3}\gamma_{5}-i\,\hat{{\cal
C}}^{v}_{3}\gamma^{0}-i\,\hat{{\cal
C}}^{pv}_{3}\gamma_{5}\gamma^{0}\Bigr{)}\tau_{3}$ (46)
$\displaystyle\quad+\Bigl{(}\hat{{\cal C}}^{s}_{+}+i\,\hat{{\cal
C}}^{ps}_{+}\gamma_{5}-i\,\hat{{\cal C}}^{v}_{+}\gamma^{0}-i\,\hat{{\cal
C}}^{pv}_{+}\gamma_{5}\gamma^{0}\Bigr{)}e^{i{\mathfrak{a}}}\tau_{+}+\Bigl{(}\hat{{\cal
C}}^{s}_{-}+i\,\hat{{\cal C}}^{ps}_{-}\gamma_{5}-i\,\hat{{\cal
C}}^{v}_{-}\gamma^{0}-i\,\hat{{\cal
C}}^{pv}_{-}\gamma_{5}\gamma^{0}\Bigr{)}e^{-i{\mathfrak{a}}}\tau_{-}\Bigr{]}{\bm{\psi}}\Bigr{\\}}_{r=\epsilon}$

where the dimensionless coefficients $\hat{{\cal C}}^{\scriptscriptstyle
A}_{\scriptscriptstyle I}$ are related to the coefficients
$\hat{\mathfrak{c}}^{\scriptscriptstyle A}_{\scriptscriptstyle I}$ by

$\hat{{\cal C}}^{\scriptscriptstyle A}_{\scriptscriptstyle
I}=\frac{\hat{\mathfrak{c}}^{\scriptscriptstyle A}_{\scriptscriptstyle
I}}{4\pi\epsilon^{2}}\,,$ (47)

for all $A=s,ps,v,pv$ and $I=1,3,+,-$.

This expression is only ‘formal’ because the projection onto the $S$-wave
while regulating by displacing the field to $r=\epsilon$ is more subtle
because of the different angular dependence imposed on $\psi_{+}$ relative to
$\psi_{-}$ by the presence of the magnetic monopole background configuration.
These issues are summarized in Appendix C and the result is most conveniently
expressed in terms of the two-dimensional fields $\chi$ whose bulk dynamics is
described by (31). In this 2D formulation the fermionic terms in the dyon’s
point-particle world line action given by (40) and (3.1.1) can be written

$S_{\rm
dyon2}=-\frac{1}{2}\sum_{{\mathfrak{s}},{\mathfrak{s}}^{\prime}=\pm}\int{\rm
d}t\,\overline{\chi}_{{\mathfrak{s}}}\Bigl{(}\mathcal{C}^{s}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}+i\mathcal{C}^{ps}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}\Gamma_{c}+i\mathcal{C}^{v}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}\Gamma_{\alpha}\dot{y}^{\alpha}+i\mathcal{C}^{pv}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}\Gamma^{\alpha}\epsilon_{\alpha\beta}\dot{y}^{\beta}\Bigr{)}\,\chi_{{\mathfrak{s}}^{\prime}}\;e^{\frac{i}{2}({\mathfrak{s}}-{\mathfrak{s}}^{\prime})\,{\mathfrak{a}}}\,,$
(48)

where the fields are evaluated at $r=\epsilon$. Here (as above)
$\Gamma_{c}:=\Gamma^{0}\Gamma^{1}$ has eigenvalues ${\mathfrak{h}}=\pm 1$ (and
is diagonal in the basis used here) while the 2D Dirac matrices satisfy
$\Gamma_{c}\Gamma^{\alpha}=\epsilon^{\alpha\beta}\Gamma_{\beta}$ where our
Levi-Civita convention chooses $\epsilon^{01}=+1$. The coefficients ${\cal
C}^{\scriptscriptstyle A}_{ij}$ are dimensionless in the same way that the
$\hat{{\cal C}}^{\scriptscriptstyle A}_{\scriptscriptstyle I}$ are, and must
satisfy ${\cal C}^{{\scriptscriptstyle
A}*}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}={\cal C}^{\scriptscriptstyle
A}_{{\mathfrak{s}}^{\prime}{\mathfrak{s}}}$ if the action $S_{\rm dyon2}$ is
real (in which case they contain 16 independent real parameters).

Combining this ‘boundary’ action with the ‘bulk’ action (31) leads to the
following near-dyon boundary condition for the 2D fermions near a static dyon
(for which $\dot{y}^{\alpha}\simeq\delta^{\alpha}_{0}$)

$\sum_{\mathfrak{s}^{\prime}}\Bigl{(}\delta_{\mathfrak{s}\mathfrak{s}^{\prime}}\Gamma^{1}+\mathcal{C}^{s}_{\mathfrak{s}\mathfrak{s^{\prime}}}+i\mathcal{C}^{ps}_{\mathfrak{s}\mathfrak{s^{\prime}}}\Gamma_{c}-i\mathcal{C}^{v}_{\mathfrak{s}\mathfrak{s^{\prime}}}\Gamma^{0}+i\mathcal{C}^{pv}_{\mathfrak{s}\mathfrak{s^{\prime}}}\Gamma^{1}\Bigr{)}\,e^{\frac{i}{2}(\mathfrak{s}-\mathfrak{s}^{\prime}){\mathfrak{a}}(t)}{\chi}_{\mathfrak{s}^{\prime}}(\epsilon,t)=0,$
(49)

for all $t$. Equivalently

$\Bigl{[}\Gamma^{1}+O_{\mathcal{B}}({\mathfrak{a}})\Bigr{]}\left[\begin{matrix}\chi_{+}\\\
\chi_{-}\end{matrix}\right]_{r=\epsilon}=0,$ (50)

where

$\displaystyle O_{\mathcal{B}}({\mathfrak{a}})$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\mathcal{C}^{s}_{++}&\mathcal{C}^{s}_{+-}e^{i{\mathfrak{a}}}\\\
\mathcal{C}^{s}_{-+}e^{-i{\mathfrak{a}}}&\mathcal{C}^{s}_{--}\end{pmatrix}+i\begin{pmatrix}\mathcal{C}^{ps}_{++}&\mathcal{C}^{ps}_{+-}e^{i{\mathfrak{a}}}\\\
\mathcal{C}^{ps}_{-+}e^{-i{\mathfrak{a}}}&\mathcal{C}^{ps}_{--}\end{pmatrix}\Gamma_{c}$
(51)
$\displaystyle\qquad\qquad\qquad-i\begin{pmatrix}\mathcal{C}^{v}_{++}&\mathcal{C}^{v}_{+-}e^{i{\mathfrak{a}}}\\\
\mathcal{C}^{v}_{-+}e^{-i{\mathfrak{a}}}&\mathcal{C}^{v}_{--}\end{pmatrix}\Gamma^{0}+i\begin{pmatrix}\mathcal{C}^{pv}_{++}&\mathcal{C}^{pv}_{+-}e^{i{\mathfrak{a}}}\\\
\mathcal{C}^{pv}_{-+}e^{-i{\mathfrak{a}}}&\mathcal{C}^{pv}_{--}\end{pmatrix}\Gamma^{1}\,.$

Some intuition about the physical meaning of this boundary condition can be
obtained by considering what it implies for the flux of fermionic currents
through the surface at $r=\epsilon$. Consider, for example, a fermion current
of the form $j^{\alpha}=i\overline{\bm{\chi}}\Gamma^{\alpha}M\bm{\chi}$ where
$M$ is some matrix in spin and isospin space. When the action (31) is real the
boundary condition allows $\Gamma^{1}{\bm{\chi}}(\epsilon,t)$ to be replaced
by terms involving the effective couplings ${\cal C}^{\scriptscriptstyle
A}_{ij}$ and this can be used to learn something about the flux
$j^{1}(\epsilon,t)$. As applied to the currents for fermion number,
$j^{\alpha}_{\scriptscriptstyle
B}=i\overline{\bm{\chi}}\Gamma^{\alpha}\bm{\chi}$, electric charge,
$j^{\alpha}_{{\scriptscriptstyle
F}}=\tfrac{1}{2}ie\overline{\bm{\chi}}\Gamma^{\alpha}\,\tau_{3}\bm{\chi}$ and
axial symmetry121212$j^{\alpha}_{\scriptscriptstyle A}$ is chosen in the 2D
theory to match the $4$D axial current $J^{\mu}_{\scriptscriptstyle
A}=i{\overline{{\bm{\psi}}}}\gamma^{\mu}\gamma_{5}{\bm{\psi}}$ for $S$-wave
states, up to factors of $4\pi r^{2}$. $j^{\alpha}_{\scriptscriptstyle
A}=i\overline{\bm{\chi}}\Gamma^{\alpha}\Gamma_{c}\tau_{3}\bm{\chi}$ (for real
boundary action) the boundary conditions imply131313Divergences associated
with evaluating these fermion bilinears can be regulated in a way that
preserves this relation, as we show in appendix G. $j^{r}_{\scriptscriptstyle
B}(\epsilon,t)=0$,

$j^{r}_{\scriptscriptstyle
F}(\epsilon,t)=-\tfrac{1}{4}ie\,\overline{\bm{\chi}}(\epsilon,t)\Bigl{[}\tau_{3},O_{\mathcal{B}}({\mathfrak{a}})\Bigr{]}\bm{\chi}(\epsilon,t)\quad\hbox{and}\quad
j^{r}_{\scriptscriptstyle
A}(\epsilon,t)=\tfrac{1}{2}i\,\overline{\bm{\chi}}(\epsilon,t)\Bigl{\\{}\Gamma_{c}\tau_{3},O_{\mathcal{B}}({\mathfrak{a}})\Bigr{\\}}\bm{\chi}(\epsilon,t)\,,$
(52)

where $O_{\cal B}$ is defined by (51). These show that none of the dyon-
fermion interactions transfer fermion number to or from the dyon while the
terms in the boundary action involving $\tau_{\pm}$ contribute nonzero flux of
electrical current at $r=\epsilon$, as required by charge conservation when
exciting or de-exciting the dyonic field ${\mathfrak{a}}$.

Several things about the boundary condition (50) are noteworthy (for more
details and a discussion of how things look for a non-dyonic source see
Appendix B and PPEFT3 ).

  * •

Linearity: This boundary condition is linear in ${\bm{\psi}}$ (or
${\bm{\chi}}$) because the action (40) is quadratic in ${\bm{\psi}}$ (or (48)
is quadratic in ${\bm{\chi}}$). Furthermore, the action is quadratic because
this is the lowest-dimension interaction consistent with the field content and
symmetries and it is the lowest-dimension interactions that dominate at low
energies within an effective theory. It is through arguments like this that
PPEFTs explain the ubiquity of linear boundary conditions for many systems as
being a generic consequence of the low-energy limit.

  * •

Algebraic: This boundary condition does not involve derivatives – unlike (44),
for instance – because the fermion bulk action is linear in derivatives of the
fields. The first term in (46) comes from integrating by parts in the bulk
action, and when the same exercise is performed for bosonic fields one instead
finds first-derivative Robin-style boundary conditions because the bulk field
equations are second order in derivatives. We discuss some novel implications
of non-derivative boundary conditions at some length below.

####  3.1.3 Implications for mode functions

We next work out how these boundary conditions influence the bulk fermion mode
functions. This is complicated by the fact that the boundary condition (46)
explicitly involves the collective coordinate ${\mathfrak{a}}$, and this is
important because it shows how the dynamics of ${\mathfrak{a}}$ feeds back
onto the fermionic field as it interacts with the dyon. As emphasized in
Polchinski  it is the nonperturbative nature of this fermion-dyon interaction
that drives much of the unusual dynamics of the dyonic effective theory,
generating effects that are correlated between the ${\bm{\psi}}$ and
${\mathfrak{a}}$ sectors. In practice this complicates things by making the
fermion boundary condition (46) field-dependent. For the remainder of this
section we regard the field ${\mathfrak{a}}$ to be a specified classical
function while we explore the implications of (50) for the fermion field. §3.3
returns to justify how to interpret the resulting ${\mathfrak{a}}$-dependence
of our results using the Born-Oppenheimer approximation.

Eqs. (27) and (28) provide the general solution for the ${\bm{\psi}}$ mode
functions and so form a complete basis of solutions to the bulk field
equations. Energy eigenstates in the presence of the dyon are found by asking
these also to satisfy the near-dyon boundary condition (50), since this
captures the implications of the dyon for its surroundings. Subtleties to do
with the angle-dependence of the $S$-wave mode functions can be avoided if the
boundary conditions are phrased in terms of the two-dimensional fields
${\bm{\chi}}$. For instance, if a general positive-frequency solution is given
as a linear combination of the solutions (32), with coefficients
$\bm{\kappa}_{{\mathfrak{s}}{\mathfrak{c}}}$, then a near-source boundary
condition of the form (50) can be written

${{\cal
B}}({\mathfrak{a}})\begin{bmatrix}\bm{\kappa}_{++}\,{\mathfrak{u}}_{++\omega}(\epsilon,t)+\bm{\kappa}_{+-}\,{\mathfrak{u}}_{+-\omega}(\epsilon,t)\\\\[2.84526pt]
\bm{\kappa}_{-+}\,{\mathfrak{u}}_{-+\omega}(\epsilon,t)+\bm{\kappa}_{--}\,{\mathfrak{u}}_{--\omega}(\epsilon,t)\end{bmatrix}=0$
(53)

where ${\cal B}({\mathfrak{a}})\coloneqq\Gamma^{0}[\Gamma^{1}+O_{{\cal
B}}({\mathfrak{a}})]$ and so

$\mathcal{B}({{\mathfrak{a}}})=\begin{pmatrix}1+i(\mathcal{C}^{pv}_{++}+\mathcal{C}^{v}_{++})&-(\mathcal{C}^{ps}_{++}+i\mathcal{C}^{s}_{++})&i(\mathcal{C}^{pv}_{+-}+\mathcal{C}^{v}_{+-})e^{i{\mathfrak{a}}}&-(\mathcal{C}^{ps}_{+-}+i\mathcal{C}^{s}_{+-})e^{i{\mathfrak{a}}}\\\\[5.69054pt]
\mathcal{C}^{ps}_{++}-i\mathcal{C}^{s}_{++}&-1-i(\mathcal{C}^{pv}_{++}-\mathcal{C}^{v}_{++})&(\mathcal{C}^{ps}_{+-}-i\mathcal{C}^{s}_{+-})e^{i{\mathfrak{a}}}&-i(\mathcal{C}^{pv}_{+-}-\mathcal{C}^{v}_{+-})e^{i{\mathfrak{a}}}\\\\[5.69054pt]
i(\mathcal{C}^{pv}_{-+}+\mathcal{C}^{v}_{-+})e^{-i{\mathfrak{a}}}&-(\mathcal{C}^{ps}_{-+}+i\mathcal{C}^{s}_{-+})e^{-i{\mathfrak{a}}}&1+i(\mathcal{C}^{pv}_{--}+\mathcal{C}^{v}_{--})&-(\mathcal{C}^{ps}_{--}+i\mathcal{C}^{s}_{--})\\\\[5.69054pt]
(\mathcal{C}^{ps}_{-+}-i\mathcal{C}^{s}_{-+})e^{-i{\mathfrak{a}}}&-i(\mathcal{C}^{pv}_{-+}-\mathcal{C}^{v}_{-+})e^{-i{\mathfrak{a}}}&\mathcal{C}^{ps}_{--}-i\mathcal{C}^{s}_{--}&-1-i(\mathcal{C}^{pv}_{--}-\mathcal{C}^{v}_{--})\end{pmatrix}\,.$
(54)

Notice that when the ${\cal C}^{\scriptscriptstyle A}_{ij}$ couplings are
hermitian – i.e. when the dyon-fermion action (48) is real – the matrix ${\cal
B}({\mathfrak{a}})$ satisfies the useful identity

$\mathcal{B}^{\dagger}=2\begin{pmatrix}\Gamma_{c}&0\\\
0&\Gamma_{c}\end{pmatrix}-\mathcal{B}$ (55)

for all ${\mathfrak{a}}$, where $\Gamma_{c}=\sigma_{3}$ is the diagonal Pauli
matrix. The most general matrix satisfying this condition has 16 real
parameters, corresponding to the 16 real effective couplings contained in the
${\cal C}^{\scriptscriptstyle A}_{ij}$.

The physical implications of this boundary condition depend sensitively on the
rank of the matrix $\mathcal{B}$. For instance, if all couplings
$\mathcal{C}^{\scriptscriptstyle A}_{ij}$ were independent of one another
$\mathcal{B}$ would generically have rank 4, in which case the only solution
to (53) is ${\bm{\chi}}(\epsilon,t)=0$. If $\mathcal{B}$ instead has rank
$4-n$ for $n=0,1,2,3$ then there would be $n$ linearly independent nonzero
values for ${\bm{\chi}}(\epsilon,t)$ allowed by (53).

Now comes a key observation: when the boundary action $S_{\rm dyon2}$ is real
– and so the constants ${\cal C}^{\scriptscriptstyle A}_{ij}$ are hermitian –
then the rank of $\mathcal{B}$ cannot be smaller than two. This is because
both of the following quantities are ${\mathfrak{a}}$-independent and always
nonzero:

$\displaystyle\left|\begin{array}[]{c c}\mathcal{B}_{22}&\mathcal{B}_{24}\\\
\mathcal{B}_{42}&\mathcal{B}_{44}\end{array}\right|$ $\displaystyle=$
$\displaystyle
1+i(\mathcal{C}^{pv}_{--}-\mathcal{C}^{v}_{--}+\mathcal{C}^{pv}_{++}-\mathcal{C}^{v}_{++})-(\mathcal{C}^{pv}_{++}-\mathcal{C}^{v}_{++})(\mathcal{C}^{pv}_{--}-\mathcal{C}^{v}_{--})+\left|\mathcal{C}^{pv}_{+-}-\mathcal{C}^{v}_{+-}\right|^{2},$
(58) $\displaystyle\left|\begin{array}[]{c
c}\mathcal{B}_{11}&\mathcal{B}_{13}\\\
\mathcal{B}_{31}&\mathcal{B}_{33}\end{array}\right|$ $\displaystyle=$
$\displaystyle
1+i(\mathcal{C}^{pv}_{--}+\mathcal{C}^{v}_{--}+\mathcal{C}^{pv}_{++}+\mathcal{C}^{v}_{++})-(\mathcal{C}^{pv}_{++}+\mathcal{C}^{v}_{++})(\mathcal{C}^{pv}_{--}+\mathcal{C}^{v}_{--})+\left|\mathcal{C}^{pv}_{+-}+\mathcal{C}^{v}_{+-}\right|^{2}.$
(61)

To see why these cannot vanish, consider the first case. Notice that its
imaginary part can only vanish if
$\mathcal{C}^{pv}_{--}-\mathcal{C}^{v}_{--}=-(\mathcal{C}^{pv}_{++}-\mathcal{C}^{v}_{++})$,
but if this is true then the real part becomes
$1+(\mathcal{C}^{pv}_{++}-\mathcal{C}^{v}_{++})^{2}+\left|\mathcal{C}^{pv}_{+-}-\mathcal{C}^{v}_{+-}\right|^{2}\geq
1$. A similar argument goes through also for the second case.

As a consequence it is always possible to solve for two of the $\bm{\kappa}$’s
in terms of the other two. For instance, solving for $\bm{\kappa}_{++}$ and
$\bm{\kappa}_{--}$ gives

$\bm{\kappa}_{++}=-\bm{\kappa}_{+-}e^{2i(\omega+\frac{1}{2}ev)\epsilon}\,\left(\frac{\epsilon}{r_{0}}\right)^{-ieQ}\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{12}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{32}&\hat{\mathcal{B}}_{33}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{33}\end{array}\right|}-\bm{\kappa}_{-+}e^{2i\omega\epsilon}e^{i{\mathfrak{a}}}\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{14}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{34}&\hat{\mathcal{B}}_{33}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{33}\end{array}\right|},$ (63)

and

$\bm{\kappa}_{--}=-\bm{\kappa}_{+-}e^{2i\omega\epsilon}e^{-i{\mathfrak{a}}}\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{12}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{32}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{33}\end{array}\right|}-\bm{\kappa}_{-+}e^{2i(\omega-\frac{1}{2}ev)\epsilon}\left(\frac{\epsilon}{r_{0}}\right)^{ieQ}\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{14}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{34}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{33}\end{array}\right|}\,,$ (64)

which defines $\hat{\mathcal{B}}_{ij}:=\mathcal{B}_{ij}({\mathfrak{a}}=0)$ so
that all ${\mathfrak{a}}$-dependence is explicit. In later sections it is
sometimes useful instead to solve for $\bm{\kappa}_{+-}$ and
$\bm{\kappa}_{-+}$, which instead gives

$\bm{\kappa}_{+-}=-\bm{\kappa}_{++}e^{-2i(\omega+\frac{1}{2}ev)\epsilon}\,\left(\frac{\epsilon}{r_{0}}\right)^{ieQ}\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{21}&\hat{\mathcal{B}}_{24}\\\
\hat{\mathcal{B}}_{41}&\hat{\mathcal{B}}_{44}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{22}&\hat{\mathcal{B}}_{24}\\\
\hat{\mathcal{B}}_{42}&\hat{\mathcal{B}}_{44}\end{array}\right|}-\bm{\kappa}_{--}e^{-2i\omega\epsilon}e^{i{\mathfrak{a}}}\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{23}&\hat{\mathcal{B}}_{24}\\\
\hat{\mathcal{B}}_{43}&\hat{\mathcal{B}}_{44}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{22}&\hat{\mathcal{B}}_{24}\\\
\hat{\mathcal{B}}_{42}&\hat{\mathcal{B}}_{44}\end{array}\right|},$ (65)

and

$\bm{\kappa}_{-+}=-\bm{\kappa}_{++}e^{-2i\omega\epsilon}e^{-i{\mathfrak{a}}}\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{22}&\hat{\mathcal{B}}_{21}\\\
\hat{\mathcal{B}}_{42}&\hat{\mathcal{B}}_{41}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{22}&\hat{\mathcal{B}}_{24}\\\
\hat{\mathcal{B}}_{42}&\hat{\mathcal{B}}_{44}\end{array}\right|}-\bm{\kappa}_{--}e^{-2i(\omega-\frac{1}{2}{ev})\epsilon}\left(\frac{\epsilon}{r_{0}}\right)^{-ieQ}\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{22}&\hat{\mathcal{B}}_{23}\\\
\hat{\mathcal{B}}_{42}&\hat{\mathcal{B}}_{43}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{22}&\hat{\mathcal{B}}_{24}\\\
\hat{\mathcal{B}}_{42}&\hat{\mathcal{B}}_{44}\end{array}\right|}.$ (66)

If $\mathcal{B}$ is rank two then this is all that can be learned because the
other two equations in (53) are not independent. When $\mathcal{B}$ has rank
two the boundary condition has precisely enough information to determine ‘out’
states from the ‘in’ states (with no extra constraints) as is required for
scattering problems, so we henceforth assume the effective couplings
$\mathcal{C}^{\scriptscriptstyle A}_{ij}$ satisfy the conditions required to
ensure rank$(\mathcal{B})=2$. This should automatically be the case when the
microscopic physics of the source allows out states to be inferred from
arbitrary in states – such as for fermion scattering in the classical dyon
background of the full nonabelian theory – since the effective couplings
obtained by matching must give a consistent description.141414We have examples
of effective couplings for which rank$({\cal B})=3$, but defer an exploration
of their microscopic physical significance to future work. As is shown in
Appendix D the requirement that ${\cal B}$ be rank two imposes a total of 8
conditions on its coefficients (four of which amount to unitarity conditions)
and so removes half of the 16 real parameters that could have been encoded in
${\cal B}$ for general choices of hermitian ${\cal C}^{\scriptscriptstyle
A}_{ij}$’s.

###  3.2 Scattering states

We next construct the explicit single-particle scattering states appropriate
for fermion dyon scattering, assuming the rank of the matrix ${\cal B}$ is
two. This allows us to identify which combinations of the effective couplings
actually appear in scattering processes.

To this end we construct a basis of energy eigenmodes that either correspond
to a single type of particle moving towards the dyon ($in$ state) or a single
type of particle moving away from the dyon ($out$ state). These correspond to
choosing specific couplings $\bm{\kappa}_{{\mathfrak{s}}{\mathfrak{c}}}$ to
vanish, as described in detail below. Because the direction of motion of an
$S$-wave state correlates with the value of the quantum number
${\mathfrak{h}}={\mathfrak{s}}{\mathfrak{c}}$, it suffices to label in and out
states using just ${\mathfrak{s}}$ and momentum $k$ (or frequency $\omega$),
leading to positive (negative) frequency modes ${\mathfrak{u}}^{\rm
in}_{{\mathfrak{s}},k}$ and ${\mathfrak{v}}^{\rm in}_{{\mathfrak{s}},k}$ or
${\mathfrak{u}}^{\rm out}_{{\mathfrak{s}},k}$ and ${\mathfrak{v}}^{\rm
out}_{{\mathfrak{s}},k}$.

####  3.2.1 In modes

We define the positive-frequency in-states to be those modes for which there
is only one component with incoming momentum (heading towards the dyon).
Labelling these by the sign of the incoming particle’s electric charge
${\mathfrak{s}}$, they are given explicitly by

${{{\mathfrak{u}}}}^{\rm
in}_{+,k}=\begin{bmatrix}[l]\begin{pmatrix}e^{-ikr}\\\
e^{ik(r-2\epsilon)}\left(\frac{\epsilon}{r}\right)^{ieQ}\mathcal{T}^{++}_{{\rm
in}}\end{pmatrix}\\\\[11.38109pt] \begin{pmatrix}0\\\
e^{i(k-ev)r}e^{-i(2k-ev)\epsilon}\,e^{-i{\mathfrak{a}}}{\cal T}^{-+}_{{\rm
in}}\end{pmatrix}\end{bmatrix}{e^{-i(k-\frac{ev}{2})t}\left(\frac{r}{r_{0}}\right)^{ieQ/2}}$
(67)

and

${{{\mathfrak{u}}}}^{\rm in}_{-,k}=\begin{bmatrix}[l]\begin{pmatrix}0\\\
e^{i(k+ev)r}e^{-i(2k+ev)\epsilon}\,e^{i{\mathfrak{a}}}{\cal T}^{+-}_{{\rm
in}}\end{pmatrix}\\\\[11.38109pt] \begin{pmatrix}e^{-ikr}\\\
e^{ik(r-2\epsilon)}\,\left(\frac{r}{\epsilon}\right)^{ieQ}\,\mathcal{T}^{--}_{{\rm
in}}\end{pmatrix}\end{bmatrix}{e^{-i(k+\frac{ev}{2})t}\left(\frac{r}{r_{0}}\right)^{-ieQ/2}\,}\,.$
(68)

where $k=\omega+\frac{1}{2}{\mathfrak{s}}ev\geq 0$. (These can be obtained
from (65) and (66) by choosing $\bm{\kappa}_{++}=1,\bm{\kappa}_{--}=0$ for
${\mathfrak{u}}^{{\rm in}}_{+,k}$ and $\bm{\kappa}_{--}=1,\bm{\kappa}_{++}=0$
for ${\mathfrak{u}}^{{\rm in}}_{-,k}$.) The negative-frequency in-modes are
similarly defined by

${{{\mathfrak{v}}}}^{\rm in}_{+,k}=\begin{bmatrix}[l]\begin{pmatrix}e^{ikr}\\\
e^{-ik(r-2\epsilon)}\left(\frac{\epsilon}{r}\right)^{ieQ}\mathcal{T}^{++}_{{\rm
in}}\end{pmatrix}\\\\[11.38109pt] \begin{pmatrix}0\\\
e^{-i(k+ev)r}e^{i(2k+ev)\epsilon}\,e^{-i{\mathfrak{a}}}{\cal T}^{-+}_{{\rm
in}}\end{pmatrix}\end{bmatrix}{e^{i(k+\frac{ev}{2})t}\left(\frac{r}{r_{0}}\right)^{ieQ/2}}$
(69)

and

${{{\mathfrak{v}}}}^{\rm in}_{-,k}=\begin{bmatrix}[l]\begin{pmatrix}0\\\
e^{-i(k-ev)r}e^{i(2k-ev)\epsilon}\,e^{i{\mathfrak{a}}}{\cal T}^{+-}_{{\rm
in}}\end{pmatrix}\\\\[11.38109pt] \begin{pmatrix}e^{ikr}\\\
e^{-ik(r-2\epsilon)}\,\left(\frac{r}{\epsilon}\right)^{ieQ}\mathcal{T}^{--}_{{\rm
in}}\end{pmatrix}\end{bmatrix}{e^{i(k-\frac{ev}{2})t}\left(\frac{r}{r_{0}}\right)^{-ieQ/2}},$
(70)

where $k=\omega-\frac{1}{2}{\mathfrak{s}}ev\geq 0$. In both of these
expressions $r_{0}$ is an arbitrary length that contributes only to the
overall Coulomb phase.

The coefficients $\mathcal{T}^{++}_{{\rm in}}$, $\mathcal{T}^{+-}_{{\rm in}}$,
$\mathcal{T}^{-+}_{{\rm in}}$ and $\mathcal{T}^{--}_{{\rm in}}$ appearing in
these expressions are ${\mathfrak{a}}$-independent quantities given as
explicit functions of the fermion-dyon couplings by

$\mathcal{T}^{++}_{\rm
in}=-\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{21}&\hat{\mathcal{B}}_{24}\\\
\hat{\mathcal{B}}_{41}&\hat{\mathcal{B}}_{44}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{22}&\hat{\mathcal{B}}_{24}\\\
\hat{\mathcal{B}}_{42}&\hat{\mathcal{B}}_{44}\end{array}\right|},\;{\cal
T}^{+-}_{\rm
in}=-\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{23}&\hat{\mathcal{B}}_{24}\\\
\hat{\mathcal{B}}_{43}&\hat{\mathcal{B}}_{44}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{22}&\hat{\mathcal{B}}_{24}\\\
\hat{\mathcal{B}}_{42}&\hat{\mathcal{B}}_{44}\end{array}\right|},\;{\cal
T}^{-+}_{\rm
in}=-\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{22}&\hat{\mathcal{B}}_{21}\\\
\hat{\mathcal{B}}_{42}&\hat{\mathcal{B}}_{41}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{22}&\hat{\mathcal{B}}_{24}\\\
\hat{\mathcal{B}}_{42}&\hat{\mathcal{B}}_{44}\end{array}\right|},\;\mathcal{T}^{--}_{\rm
in}=-\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{22}&\hat{\mathcal{B}}_{23}\\\
\hat{\mathcal{B}}_{42}&\hat{\mathcal{B}}_{43}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{22}&\hat{\mathcal{B}}_{24}\\\
\hat{\mathcal{B}}_{42}&\hat{\mathcal{B}}_{44}\end{array}\right|}\,.$ (71)

Written directly in terms of the boundary couplings, these become

$\displaystyle\mathcal{T}^{++}_{{\rm in}}$ $\displaystyle=$
$\displaystyle\frac{(\mathcal{C}^{s}_{++}+i\mathcal{C}^{ps}_{++})(i-\mathcal{C}^{pv}_{--}+\mathcal{C}^{v}_{--})+(\mathcal{C}^{s}_{-+}+i\mathcal{C}^{ps}_{-+})(\mathcal{C}^{pv}_{+-}-\mathcal{C}^{v}_{+-})}{-|\mathcal{C}^{pv}_{+-}-\mathcal{C}^{v}_{+-}|^{2}+(-i+\mathcal{C}^{pv}_{++}-\mathcal{C}^{v}_{++})(\mathcal{C}^{pv}_{--}-\mathcal{C}^{v}_{--}-i)},$
$\displaystyle\mathcal{T}^{--}_{{\rm in}}$ $\displaystyle=$
$\displaystyle\frac{(\mathcal{C}^{s}_{--}+i\mathcal{C}^{ps}_{--})(i-\mathcal{C}^{pv}_{++}+\mathcal{C}^{v}_{++})+(\mathcal{C}^{s}_{+-}+i\mathcal{C}^{ps}_{+-})(\mathcal{C}^{pv}_{-+}-\mathcal{C}^{v}_{-+})}{-|\mathcal{C}^{pv}_{+-}-\mathcal{C}^{v}_{+-}|^{2}+(-i+\mathcal{C}^{pv}_{++}-\mathcal{C}^{v}_{++})(\mathcal{C}^{pv}_{--}-\mathcal{C}^{v}_{--}-i)},$
(72) $\displaystyle{\cal T}^{-+}_{{\rm in}}$ $\displaystyle=$
$\displaystyle\frac{(\mathcal{C}^{s}_{-+}+i\mathcal{C}^{ps}_{-+})(i-\mathcal{C}^{pv}_{++}+\mathcal{C}^{v}_{++})+(\mathcal{C}^{s}_{++}+i\mathcal{C}^{ps}_{++})(\mathcal{C}^{pv}_{-+}-\mathcal{C}^{v}_{-+})}{-|\mathcal{C}^{pv}_{+-}-\mathcal{C}^{v}_{+-}|^{2}+(-i+\mathcal{C}^{pv}_{++}-\mathcal{C}^{v}_{++})(\mathcal{C}^{pv}_{--}-\mathcal{C}^{v}_{--}-i)},$
$\displaystyle{\cal T}^{+-}_{{\rm in}}$ $\displaystyle=$
$\displaystyle\frac{(\mathcal{C}^{s}_{+-}+i\mathcal{C}^{ps}_{+-})(i-\mathcal{C}^{pv}_{--}+\mathcal{C}^{v}_{--})+(\mathcal{C}^{s}_{--}+i\mathcal{C}^{ps}_{--})(\mathcal{C}^{pv}_{+-}-\mathcal{C}^{v}_{+-})}{-|\mathcal{C}^{pv}_{+-}-\mathcal{C}^{v}_{+-}|^{2}+(-i+\mathcal{C}^{pv}_{++}-\mathcal{C}^{v}_{++})(\mathcal{C}^{pv}_{--}-\mathcal{C}^{v}_{--}-i)}.$

When the boundary action $S_{\rm dyon2}$ is real (and so the ${\cal
C}^{\scriptscriptstyle A}_{ij}$ are hermitian) these satisfy the following
unitarity conditions

$|\mathcal{T}^{+-}_{{\rm in}}|^{2}=|\mathcal{T}^{-+}_{{\rm
in}}|^{2}=1-|\mathcal{T}^{++}_{{\rm in}}|^{2}=1-|\mathcal{T}^{--}_{{\rm
in}}|^{2}\,,$ (73)

as well as

$\mathcal{T}^{--}_{{\rm in}}{\cal T}^{-+\,*}_{{\rm in}}+{\cal T}^{+-}_{{\rm
in}}\mathcal{T}^{++\,*}_{{\rm in}}=0,$ (74)

as identities (see Appendix D for a derivation). These relations imply that
the in amplitudes only carry 4 real parameters’ worth of information; they can
always be written

$\mathcal{T}^{++}_{{\rm
in}}=\rho\,e^{i\theta_{++}}\quad\text{and}\quad\mathcal{T}^{--}_{{\rm
in}}=\rho\,e^{i\theta_{--}},$ (75)

and

${\mathcal{T}}^{+-}_{{\rm
in}}=\sqrt{1-\rho^{2}}\,e^{i\theta_{+-}}\quad\text{and}\quad{\mathcal{T}}^{-+}_{{\rm
in}}=-\sqrt{1-\rho^{2}}\,e^{i(\theta_{++}+\theta_{--}-\theta_{+-})},$ (76)

where $\rho$, $\theta_{++}$, $\theta_{+-}$ and $\theta_{--}$ are the four
independent real parameters.

####  3.2.2 Out modes

A set of outgoing modes can be constructed in precisely the same way, with the
positive frequency modes given by

${{{\mathfrak{u}}}}^{{\rm
out}}_{+,k}=\begin{bmatrix}[l]\begin{pmatrix}e^{-ik(r-2\epsilon)}\left(\frac{r}{\epsilon}\right)^{ieQ}\mathcal{T}^{++}_{{\rm
out}}\\\ e^{ikr}\end{pmatrix}\\\\[11.38109pt]
\begin{pmatrix}e^{-i(k-ev)r}e^{i(2k-ev)\epsilon}\,e^{-i{\mathfrak{a}}}{\cal
T}^{-+}_{{\rm out}}\\\
0\end{pmatrix}\end{bmatrix}{e^{-i(k-\frac{ev}{2})t}\left(\frac{r}{r_{0}}\right)^{-ieQ/2}},$
(77)

and

${{{\mathfrak{u}}}}^{{\rm
out}}_{-,k}=\begin{bmatrix}[l]\begin{pmatrix}e^{-i(k+ev)r}e^{i(2k+ev)\epsilon}\,e^{i{\mathfrak{a}}}{\cal
T}^{+-}_{{\rm out}}\\\ 0\end{pmatrix}\\\\[11.38109pt]
\begin{pmatrix}e^{-ik(r-2\epsilon)}\left(\frac{\epsilon}{r}\right)^{ieQ}\,\mathcal{T}^{--}_{{\rm
out}}\\\
e^{ikr}\end{pmatrix}\end{bmatrix}{e^{-i(k+\frac{ev}{2})t}\left(\frac{r}{r_{0}}\right)^{ieQ/2}},$
(78)

with $k=\omega+\frac{1}{2}{\mathfrak{s}}ev\geq 0$ (these can be obtained from
(63) and (64) by choosing $\bm{\kappa}_{+-}=1$, $\bm{\kappa}_{-+}=0$ for
${\mathfrak{u}}^{{\rm out}}_{+,k}$ and $\bm{\kappa}_{-+}=1,\bm{\kappa}_{+-}=0$
for ${\mathfrak{u}}^{{\rm out}}_{-,k}$). The negative-frequency out-modes are

${{{\mathfrak{v}}}}^{{\rm
out}}_{+,k}=\begin{bmatrix}[l]\begin{pmatrix}e^{ik(r-2\epsilon)}\left(\frac{r}{\epsilon}\right)^{ieQ}\,\mathcal{T}^{++}_{{\rm
out}}\\\ e^{-ikr}\end{pmatrix}\\\\[11.38109pt]
\begin{pmatrix}e^{i(k+ev)r}e^{-i(2k+ev)\epsilon}\,e^{-i{\mathfrak{a}}}{\cal
T}^{-+}_{{\rm out}}\\\
0\end{pmatrix}\end{bmatrix}{e^{i(k+\frac{ev}{2})t}\left(\frac{r}{r_{0}}\right)^{-ieQ/2}},$
(79)

and

${{{\mathfrak{v}}}}^{{\rm
out}}_{-,k}=\begin{bmatrix}[l]\begin{pmatrix}e^{i(k-ev)r}e^{-i(2k-ev)\epsilon}\,e^{i{\mathfrak{a}}}{\cal
T}^{+-}_{{\rm out}}\\\ 0\end{pmatrix}\\\\[11.38109pt]
\begin{pmatrix}e^{ik(r-2\epsilon)}\left(\frac{\epsilon}{r}\right)^{ieQ}\mathcal{T}^{--}_{{\rm
out}}\\\ e^{-ikr}\\\
\end{pmatrix}\end{bmatrix}{e^{i(k-\frac{ev}{2})t}\,\left(\frac{r}{r_{0}}\right)^{ieQ/2}},$
(80)

for which $k=\omega-\frac{1}{2}{\mathfrak{s}}ev\geq 0$.

The ${\mathfrak{a}}$-independent coefficients $\mathcal{T}^{++}_{{\rm
out}},\mathcal{T}^{+-}_{{\rm out}},\mathcal{T}^{-+}_{{\rm
out}},\mathcal{T}^{--}_{{\rm out}}$ appearing in these expressions are defined
in terms of the dyon-fermion couplings by

$\mathcal{T}^{++}_{{\rm
out}}=-\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{12}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{32}&\hat{\mathcal{B}}_{33}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{33}\end{array}\right|},\;{\cal
T}^{-+}_{{\rm
out}}=-\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{12}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{32}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{33}\end{array}\right|},\;{\cal
T}^{+-}_{{\rm
out}}=-\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{14}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{34}&\hat{\mathcal{B}}_{33}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{33}\end{array}\right|},\;\mathcal{T}^{--}_{{\rm
out}}=-\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{14}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{34}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{33}\end{array}\right|},$ (81)

and so

$\displaystyle\mathcal{T}^{++}_{{\rm out}}$ $\displaystyle=$
$\displaystyle\frac{i(\mathcal{C}^{ps}_{++}+i\mathcal{C}^{s}_{++})(-i+\mathcal{C}^{pv}_{--}+\mathcal{C}^{v}_{--})+(-i\mathcal{C}^{ps}_{-+}+\mathcal{C}^{s}_{-+})(\mathcal{C}^{pv}_{+-}+\mathcal{C}^{v}_{+-})}{|\mathcal{C}^{pv}_{+-}+\mathcal{C}^{v}_{+-}|^{2}-(-i+\mathcal{C}^{pv}_{--}+\mathcal{C}^{v}_{--})(\mathcal{C}^{pv}_{++}+\mathcal{C}^{v}_{++}-i)},$
$\displaystyle\mathcal{T}^{--}_{{\rm out}}$ $\displaystyle=$
$\displaystyle\frac{i(\mathcal{C}^{ps}_{--}+i\mathcal{C}^{s}_{--})(-i+\mathcal{C}^{pv}_{++}+\mathcal{C}^{v}_{++})+(-i\mathcal{C}^{ps}_{+-}+\mathcal{C}^{s}_{+-})(\mathcal{C}^{pv}_{-+}+\mathcal{C}^{v}_{-+})}{|\mathcal{C}^{pv}_{+-}+\mathcal{C}^{v}_{+-}|^{2}-(-i+\mathcal{C}^{pv}_{--}+\mathcal{C}^{v}_{--})(\mathcal{C}^{pv}_{++}+\mathcal{C}^{v}_{++}-i)},$
(82) $\displaystyle{\cal T}^{-+}_{{\rm out}}$ $\displaystyle=$
$\displaystyle\frac{i(\mathcal{C}^{ps}_{-+}+i\mathcal{C}^{s}_{-+})(-i+\mathcal{C}^{pv}_{++}+\mathcal{C}^{v}_{++})+(-i\mathcal{C}^{ps}_{++}+\mathcal{C}^{s}_{++})(\mathcal{C}^{pv}_{-+}+\mathcal{C}^{v}_{-+})}{|\mathcal{C}^{pv}_{+-}+\mathcal{C}^{v}_{+-}|^{2}-(-i+\mathcal{C}^{pv}_{--}+\mathcal{C}^{v}_{--})(\mathcal{C}^{pv}_{++}+\mathcal{C}^{v}_{++}-i)},$
$\displaystyle{\cal T}^{+-}_{{\rm out}}$ $\displaystyle=$
$\displaystyle\frac{i(\mathcal{C}^{ps}_{+-}+i\mathcal{C}^{s}_{+-})(-i+\mathcal{C}^{pv}_{--}+\mathcal{C}^{v}_{--})+(-i\mathcal{C}^{ps}_{--}+\mathcal{C}^{s}_{--})(\mathcal{C}^{pv}_{+-}+\mathcal{C}^{v}_{+-})}{|\mathcal{C}^{pv}_{+-}+\mathcal{C}^{v}_{+-}|^{2}-(-i+\mathcal{C}^{pv}_{--}+\mathcal{C}^{v}_{--})(\mathcal{C}^{pv}_{++}+\mathcal{C}^{v}_{++}-i)}.$

When the boundary action $S_{\rm dyon2}$ is real (so the ${\cal
C}^{\scriptscriptstyle A}_{ij}$’s are hermitian) these satisfy

$|\mathcal{T}^{+-}_{{\rm out}}|^{2}=|\mathcal{T}^{-+}_{{\rm
out}}|^{2}=1-|\mathcal{T}^{++}_{{\rm out}}|^{2}=1-|\mathcal{T}^{--}_{{\rm
out}}|^{2}\,.$ (83)

and

$\mathcal{T}^{--}_{{\rm out}}{\cal T}^{-+\,*}_{{\rm out}}+{\cal T}^{+-}_{{\rm
out}}\mathcal{T}^{++\,*}_{{\rm out}}=0,$ (84)

in the same way as found earlier for ${\cal T}_{{\rm in}}$. There is of course
a good reason why ${\cal T}_{{\rm out}}$ and ${\cal T}_{{\rm in}}$ satisfy
similar conditions: they are not independent of one another. Since both the in
and out bases are complete, the $\mathcal{T}_{{\rm out}}$ amplitudes can be
expressed in terms of the $\mathcal{T}_{{\rm in}}$ amplitudes. This leads to
the relations (see Appendix D)

$\mathcal{T}^{++}_{{\rm out}}=(\mathcal{T}^{++}_{{\rm
in}})^{*},\quad\mathcal{T}^{--}_{{\rm out}}=(\mathcal{T}^{--}_{{\rm
in}})^{*},\quad\mathcal{T}^{+-}_{{\rm out}}=(\mathcal{T}^{-+}_{{\rm
in}})^{*}\quad\text{and}\quad\mathcal{T}^{-+}_{{\rm
out}}=(\mathcal{T}^{+-}_{{\rm in}})^{*}\,,$ (85)

and so the out amplitudes also depend only on the four parameters given in
(75) and (76).

The implications of the ${\cal T}_{{\rm in}}$ and ${\cal T}_{{\rm out}}$
amplitudes for fermion-dyon scattering problems are explored in some detail in
§5 and §6 below and suffice it to say that they carry all of the information
about the dyon that is relevant to describing transitions amongst the $S$-wave
fermion modes described above, at least at leading order at low energies. See
§4 for a discussion of the values predicted for these quantities by specific
microscopic choices for the underlying dyon solution.

But the above discussion introduces a minor puzzle: why are there only four
free real parameters within, say, the ${\cal T}_{{\rm in}}$’s when there are
16 possible real parameters in the hermitian effective couplings ${\cal
C}_{ij}^{\scriptscriptstyle A}$ appearing in $S_{\rm dyon2}$ in e.g. eqs. (40)
and (3.1.1)? We argue in §3.4 that all but four of the effective couplings
${\cal C}_{ij}^{\scriptscriptstyle A}$ are redundant (in the precise EFT sense
EFTBook ), but before doing so the next section first develops a required
tool. Along the way it also resolves a dangling technical issue: how to handle
systematically the ${\mathfrak{a}}$-dependence that is embedded in boundary
conditions like (50) or (53).

###  3.3 Dyonic response

Up to this point we have treated the matrix ${\cal B}$ as if it were specified
completely once the effective couplings ${\cal C}^{\scriptscriptstyle A}_{ij}$
are, but the appearance of the field ${\mathfrak{a}}$ within ${\cal B}$ makes
this not quite true. Having ${\cal B}$ be a function of ${\mathfrak{a}}$
complicates its use – as in (53) – to find fermionic mode functions. This
section provides two complementary ways to handle this field-dependence:
perturbation theory and the Born-Oppenheimer approximation.

####  3.3.1 Fermion-dyon perturbation theory

The first approach starts with the observation that not every term in the
matrix ${\cal B}$ depends on ${\mathfrak{a}}$. If the coefficients ${\cal
C}^{\scriptscriptstyle A}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}$ with
${\mathfrak{s}}\neq{\mathfrak{s}}^{\prime}$ were for some reason much smaller
than the others then they could be ignored when formulating the near-dyon
fermion boundary conditions with the effects of ${\mathfrak{a}}$-dependent
terms then included perturbatively.

For instance, suppose $S_{\rm dyon2}$ of (48) could be written $S_{\rm
dyon2}^{(0)}+S_{\rm dyon2}^{\rm int}$ with151515We choose the unperturbed
boundary terms here fairly arbitrarily and all that is important for our
arguments is that the dominant piece be ${\mathfrak{a}}$-independent.

$S_{\rm dyon2}^{(0)}:=-\frac{1}{2}\sum_{{\mathfrak{s}}=\pm}\int{\rm
d}t\,i\overline{\chi}_{{\mathfrak{s}}}\Gamma_{c}\,\chi_{{\mathfrak{s}}}\,,$
(86)

and

$S_{\rm dyon2}^{\rm
int}=-\frac{1}{2}\sum_{{\mathfrak{s}},{\mathfrak{s}}^{\prime}=\pm}\int{\rm
d}t\,\overline{\chi}_{{\mathfrak{s}}}\Bigl{(}\delta\mathcal{C}^{s}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}+i\delta\mathcal{C}^{ps}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}\Gamma_{c}-i\delta\mathcal{C}^{v}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}\Gamma^{0}+i\delta\mathcal{C}^{pv}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}\Gamma^{1}\Bigr{)}\,\chi_{{\mathfrak{s}}^{\prime}}\;e^{\frac{i}{2}({\mathfrak{s}}-{\mathfrak{s}}^{\prime})\,{\mathfrak{a}}}\,,$
(87)

where the coefficients $\delta{\cal C}^{\scriptscriptstyle A}_{ij}$ are
regarded as being perturbatively small161616Although, as noted above, the
${\cal C}^{\scriptscriptstyle A}_{ij}$’s are not generically suppressed by the
loop-counting parameter $\alpha=e^{2}/(4\pi)$ it is possible that in some
circumstances some of them are suppressed by another small quantity. and we
again specialize to the case of an approximately static dyon (for which
$\dot{y}^{\alpha}\simeq\delta^{\alpha}_{0}$). In this case only $S_{\rm
dyon2}^{(0)}$ need play a role in determining the boundary conditions of the
free modes appearing in the field expansions within the interaction picture.

With this choice the boundary condition matrix of (51) becomes
$O_{\mathcal{B}}^{(0)}=iI\otimes\Gamma_{c}$ and so the matrix appearing in
(54) becomes

$\mathcal{B}^{(0)}=\Gamma^{0}\left[\Gamma^{1}+O_{\cal
B}^{(0)}\right]=\begin{pmatrix}1&-1&0&0\\\\[5.69054pt] 1&-1&0&0\\\\[5.69054pt]
0&0&1&-1\\\\[5.69054pt] 0&0&1&-1\end{pmatrix}\,.$ (88)

which clearly has rank two. The boundary condition satisfied by the free mode
functions at $r=\epsilon$ therefore is

$\Bigl{(}\Gamma^{1}+i\Gamma_{c}\Bigr{)}\bm{\chi}(\epsilon,t)=0\,.$ (89)

The continuum normalized free in and out mode functions that satisfy (89) then
are

${{\mathfrak{u}}}^{{\rm
in}0}_{+,k}=\begin{bmatrix}[l]\begin{pmatrix}e^{-ikr}\\\
e^{ik(r-2\epsilon)}\left(\frac{\epsilon}{r}\right)^{ieQ}\end{pmatrix}\\\\[14.22636pt]
\hskip 14.22636pt\begin{pmatrix}\hskip 14.22636pt0\hskip 14.22636pt\\\ \hskip
14.22636pt0\hskip
14.22636pt\end{pmatrix}\end{bmatrix}{e^{-i(k-\frac{ev}{2})t}\left(\frac{r}{r_{0}}\right)^{ieQ/2}},\quad{{\mathfrak{u}}}^{{\rm
in}0}_{-,k}=\begin{bmatrix}[l]\hskip 14.22636pt\begin{pmatrix}\hskip
14.22636pt0\hskip 14.22636pt\\\ \hskip 14.22636pt0\hskip
14.22636pt\end{pmatrix}\\\\[14.22636pt] \begin{pmatrix}e^{-ikr}\\\
e^{ik(r-2\epsilon)}\,\left(\frac{r}{\epsilon}\right)^{ieQ}\end{pmatrix}\end{bmatrix}{e^{-i(k+\frac{ev}{2})t}\left(\frac{r}{r_{0}}\right)^{-ieQ/2}\,},$
(90) ${{\mathfrak{v}}}^{{\rm
in}0}_{+,k}=\begin{bmatrix}[l]\begin{pmatrix}e^{ikr}\\\
e^{-ik(r-2\epsilon)}\left(\frac{\epsilon}{r}\right)^{ieQ}\end{pmatrix}\\\\[14.22636pt]
\hskip 14.22636pt\begin{pmatrix}\hskip 14.22636pt0\hskip 14.22636pt\\\ \hskip
14.22636pt0\hskip
14.22636pt\end{pmatrix}\end{bmatrix}{e^{i(k+\frac{ev}{2})t}\left(\frac{r}{r_{0}}\right)^{ieQ/2}},\quad{{\mathfrak{v}}}^{{\rm
in}0}_{-,k}=\begin{bmatrix}[l]\hskip 14.22636pt\begin{pmatrix}\hskip
14.22636pt0\hskip 14.22636pt\\\ \hskip 14.22636pt0\hskip
14.22636pt\end{pmatrix}\\\\[14.22636pt] \begin{pmatrix}e^{ikr}\\\
e^{-ik(r-2\epsilon)}\,\left(\frac{r}{\epsilon}\right)^{ieQ}\end{pmatrix}\end{bmatrix}{e^{i(k-\frac{ev}{2})t}\left(\frac{r}{r_{0}}\right)^{-ieQ/2}},$
(91)

(where $k\geq 0$) and

${{\mathfrak{u}}}^{{\rm
out}0}_{+,k}=\begin{bmatrix}[l]\begin{pmatrix}e^{-ik(r-2\epsilon)}\left(\frac{r}{\epsilon}\right)^{ieQ}\\\
e^{ikr}\end{pmatrix}\\\\[14.22636pt] \hskip 14.22636pt\begin{pmatrix}\hskip
14.22636pt0\hskip 14.22636pt\\\ \hskip 14.22636pt0\hskip
14.22636pt\end{pmatrix}\end{bmatrix}{e^{-i(k-\frac{ev}{2})t}\left(\frac{r}{r_{0}}\right)^{-ieQ/2}},\quad{{\mathfrak{u}}}^{{\rm
out}0}_{-,k}=\begin{bmatrix}[l]\hskip 14.22636pt\begin{pmatrix}\hskip
14.22636pt0\hskip 14.22636pt\\\ \hskip 14.22636pt0\hskip
14.22636pt\end{pmatrix}\\\\[14.22636pt]
\begin{pmatrix}e^{-ik(r-2\epsilon)}\left(\frac{\epsilon}{r}\right)^{ieQ}\\\
e^{ikr}\end{pmatrix}\end{bmatrix}{e^{-i(k+\frac{ev}{2})t}\left(\frac{r}{r_{0}}\right)^{ieQ/2}},$
(92) ${{\mathfrak{v}}}^{{\rm
out}0}_{+,k}=\begin{bmatrix}[l]\begin{pmatrix}e^{ik(r-2\epsilon)}\left(\frac{r}{\epsilon}\right)^{ieQ}\\\
e^{-ikr}\end{pmatrix}\\\\[14.22636pt] \hskip 14.22636pt\begin{pmatrix}\hskip
14.22636pt0\hskip 14.22636pt\\\ \hskip 14.22636pt0\hskip
14.22636pt\end{pmatrix}\end{bmatrix}{e^{i(k+\frac{ev}{2})t}\left(\frac{r}{r_{0}}\right)^{-ieQ/2}},\quad{{\mathfrak{v}}}^{{\rm
out}0}_{-,k}=\begin{bmatrix}[l]\hskip 14.22636pt\begin{pmatrix}\hskip
14.22636pt0\hskip 14.22636pt\\\ \hskip 14.22636pt0\hskip
14.22636pt\end{pmatrix}\\\\[14.22636pt]
\begin{pmatrix}e^{ik(r-2\epsilon)}\left(\frac{\epsilon}{r}\right)^{ieQ}\\\
e^{-ikr}\\\
\end{pmatrix}\end{bmatrix}{e^{i(k-\frac{ev}{2})t}\,\left(\frac{r}{r_{0}}\right)^{ieQ/2}},$
(93)

and so the unperturbed modes satisfy ${\cal T}_{{\rm in}}^{++}={\cal T}_{\rm
in}^{--}=1$ and ${\cal T}_{{\rm in}}^{+-}={\cal T}_{{\rm in}}^{-+}=0$. These
modes are normalized so that

$\int^{\infty}_{\epsilon}\mathrm{d}r\,({{\mathfrak{u}}}^{{\rm
in}0}_{\mathfrak{s},k})^{\dagger}\,{{\mathfrak{u}}}^{{\rm
in}0}_{\mathfrak{s}^{\prime},k^{\prime}}=\int^{\infty}_{\epsilon}\mathrm{d}r\,({{\mathfrak{v}}}^{{\rm
in}0}_{\mathfrak{s},k})^{\dagger}\,{{\mathfrak{v}}}^{{\rm
in}0}_{\mathfrak{s}^{\prime},k^{\prime}}=2\pi\delta(k-k^{\prime})\,\delta_{\mathfrak{s}\mathfrak{s}^{\prime}},$
(94)

and

$\int^{\infty}_{\epsilon}\mathrm{d}r\,({{\mathfrak{u}}}^{{\rm
in}0}_{\mathfrak{s},k})^{\dagger}\,{{\mathfrak{v}}}^{{\rm
in}0}_{\mathfrak{s}^{\prime},k^{\prime}}=2\pi\delta(k+k^{\prime})\,\delta_{\mathfrak{s}\mathfrak{s^{\prime}}},$
(95)

and similarly for the out modes.

From here on perturbation theory proceeds in the standard way by moving to the
interaction picture and expanding the fermion field operator using these mode
functions,

$\displaystyle\bm{\chi}(x)$ $\displaystyle=$
$\displaystyle\sum_{{\mathfrak{s}}=\pm}\int^{\infty}_{0}\frac{\mathrm{d}k}{\sqrt{2\pi}}\Bigl{[}{\mathfrak{u}}^{{\rm
in}0}_{{\mathfrak{s}},k}(x)\,{a}^{{\rm
in}}_{{\mathfrak{s}},k}+{\mathfrak{v}}^{{\rm
in}0}_{{\mathfrak{s}},k}(x)\,(\overline{a}^{{\rm
in}}_{{\mathfrak{s}},k})^{\star}\Bigr{]}$ (96) $\displaystyle=$
$\displaystyle\sum_{{\mathfrak{s}}=\pm}\int^{\infty}_{0}\frac{\mathrm{d}k}{\sqrt{2\pi}}\Bigl{[}{\mathfrak{u}}^{{\rm
out}0}_{{\mathfrak{s}},k}(x)\,{a}^{{\rm
out}}_{{\mathfrak{s}},k}+{\mathfrak{v}}^{{\rm
out}0}_{{\mathfrak{s}},k}(x)\,(\overline{a}^{{\rm
out}}_{{\mathfrak{s}},k})^{\star}\Bigr{]},$

where the creation/annihilation operators satisfy the usual algebra
$\\{a^{{\rm in}}_{\mathfrak{s},k},\,(a^{{\rm
in}}_{\mathfrak{s}^{\prime},k^{\prime}})^{\star}\\}=\delta(k-k^{\prime})\,\delta_{\mathfrak{s}\mathfrak{s}^{\prime}}$
and $\\{\overline{a}^{{\rm in}}_{\mathfrak{s},k},\,(\overline{a}^{{\rm
in}}_{\mathfrak{s}^{\prime},k^{\prime}})^{\star}\\}=\delta(k-k^{\prime})\,\delta_{\mathfrak{s}\mathfrak{s}^{\prime}}$
appropriate for fermions (and similarly for the out operators). Because both
the in and out modes are complete, each can be expanded in terms of the other,
allowing the in and out operators to be related by

$a^{{\rm
out}}_{{\mathfrak{s}},k}=e^{-2ik\epsilon}\left(\frac{\epsilon}{r_{0}}\right)^{i{\mathfrak{s}}eQ}a^{{\rm
in}}_{{\mathfrak{s}},k}\quad\text{and}\quad({\overline{a}}^{{\rm
out}}_{{\mathfrak{s}},k})^{\star}=e^{2ik\epsilon}\left(\frac{\epsilon}{r_{0}}\right)^{i{\mathfrak{s}}eQ}\,({\overline{a}}^{{\rm
in}}_{{\mathfrak{s}},k})^{\star},$ (97)

and their adjoints. The above Bogoliubov relations depend on the arbitrary
scale $r_{0}$. Physical predictions should not depend on $r_{0}$ and in §5 –
which uses the perturbative framework set up here – we rephase the $out$
operators to absorb the $r_{0}$ dependence.

The perturbative formalism is useful for some kinds of questions – such as the
discussion of redundancy in §3.4 – and is used in §5 to compute some
scattering processes (which can be compared with calculations performed in §6
using the alternative approach we describe next). But it is not guaranteed
that specific microscopic realizations of the dyon must yield effective
couplings for which such a perturbative approach is a good approximation. This
is true in particular for massless fermion scattering in the classical
nonabelian dyon background because in this case conservation of chirality
implies the vanishing of all of the ${\mathfrak{a}}$-independent boundary
couplings ${\cal C}^{\scriptscriptstyle
A}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}$ with
${\mathfrak{s}}^{\prime}={\mathfrak{s}}$.

####  3.3.2 Born-Oppenheimer evolution of ${\mathfrak{a}}$

A more ambitious approach to computing the fermion-${\mathfrak{a}}$
interactions takes advantage of the fact that the mass ${\cal I}\sim(\alpha
m_{g})^{-1}$ appearing in the ${\mathfrak{a}}$ kinetic term is much larger
than the generic dyon scale $m_{g}^{-1}$ in the semiclassical limit (for which
$\alpha=e^{2}/4\pi\ll 1$). This means that the energy associated with
${\mathfrak{a}}$ excitation is order ${\cal I}^{-1}\sim\alpha m_{g}$ and so is
negligible – even within the low energy theory – for fermion momenta in the
range $\alpha m_{g}\ll k\ll m_{g}$.

Furthermore, the equations of motion say that
$\dot{\mathfrak{a}}\propto{\mathfrak{p}}/{\cal I}$ and so the timescale for
${\mathfrak{a}}$ to respond to order-unity changes in ${\mathfrak{p}}$ are
order $\tau\sim{\cal I}$. By contrast the timescale for a relativistic
fermionic wave packet of size $L$ to interact with a much smaller dyon is
order $L$ (which cannot be smaller than $1/k$ for fermions with momentum $k$).
So the rotor response is slow compared to the fermion interaction time for
fermion momenta in the range $\alpha m_{g}\ll L^{-1}\mathrel{\raise
1.29167pt\hbox{$<$\kern-8.50006pt\lower 4.30554pt\hbox{$\sim$}}}k$.

In the regime $\alpha m_{g}\ll k$ the ‘rotor’ field response to fermions is
slow and costs little energy in much the same way that the position of heavy
atomic nuclei respond to much lighter and faster electrons in everyday solids.
This electron/nucleus analogy suggests using the Born-Oppenheimer
approximation BornOppenheimer  to describe the interactions of relativistic
fermions with the slower dyonic excitation ${\mathfrak{a}}$. In this
approximation one first solves for the motion of the fast degrees of freedom
(the light fermions) with the slow degrees of freedom (the nuclear positions
${\bf R}$ or the field ${\mathfrak{a}}$) imagined to be fixed classical
variables. The idea is that the slow variables behave classically because they
do not have time to respond at all when hit by the fast ones. Once the fast
evolution is computed one calculates an effective Hamiltonian describing the
dynamics of the slow degrees of freedom obtained by averaging over the fast
degrees of freedom. This Hamiltonian is then used to solve for how the slower
variables evolve in response to its interactions with the much faster system.

As applied to dyon-fermion interactions the first part of the Born-Oppenheimer
procedure asks for a solution to fermion evolution with the field
${\mathfrak{a}}$ regarded as a fixed classical background. This is precisely
what is done in §3.2 above when finding the fermion scattering states in the
presence of an ${\mathfrak{a}}$-dependent boundary condition like (53). These
are used in §6 to calculate scattering rates also for fixed dyon
configurations. We return to the issue of how to determine the dyon reponse in
§6.2 below, but first close this section by gathering together several loose
ends and conceptual issues to do with using the PPEFT framework in the
monopole/dyon setting.

###  3.4 Redundant interactions

In this section we return to the puzzle alluded to at the end of §3.2 above:
why are the physical effects of the 16 real parameters in the couplings ${\cal
C}^{\scriptscriptstyle A}_{ij}$ in the action $S_{\rm dyon2}$ all encoded in
amplitudes ${\cal T}_{{\rm in}}$ (or ${\cal T}_{{\rm out}}$) that involve only
4 parameters. Why are there not more scattering options available given the
number of possible effective interactions? Part of the reason for this
reduction is the rank-2 condition we assume for ${\cal B}({\mathfrak{a}})$,
since – as shown in Appendix D.2 – this imposes 8 real conditions on the
components of ${\cal B}({\mathfrak{a}})$ (and so also on the ${\cal
C}^{\scriptscriptstyle A}_{ij}$’s). But the fact that the remaining 8
independent effective couplings only produce a 4-parameter number of
scattering outcomes strongly suggests that some of the remaining effective
couplings are actually redundant (in the precise EFT sense reviewed for
example in EFTBook ).

One way in which an effective operator can be seen to be redundant is if it
can be removed using a field redefinition. When working in perturbation theory
there is a very simple diagnostic for when such a field redefinition exists:
one asks whether the operator vanishes when it is evaluated at the solution to
the leading order field equations. In the present instance the fermion field
equation at the position $r=\epsilon$ is actually just the boundary condition
itself, and this suggests a redundancy test that can be performed, at least if
the dyon-fermion interactions are treated perturbatively (as they are in
§3.3.1 when perturbing the dyon effective couplings around the action
$S^{(0)}_{\rm dyon2}$).

When perturbing in this way we first ask what conditions the $\delta{\cal
C}^{\scriptscriptstyle A}_{ij}$’s must satisfy to ensure that the perturbed
boundary condition remains rank two. Perturbing the rank-$2$ conditions like
(D.2) of the appendix to linear order in the $\delta{\cal C}$s shows that
${\cal B}$ remains rank two only if the coefficients $\delta{\cal
C}^{pv}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}$ and $\delta{\cal
C}^{ps}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}$ all vanish. For the remaining
couplings the redundancy test then asks how many of the effective couplings in
(87) survive when simplified using the lowest-order fermion boundary condition
$(\Gamma^{1}+i\Gamma_{c}){\bm{\chi}}(\epsilon)=0$ that follows from the
unperturbed dyon-fermion action (86). Notice that because
$\Gamma_{c}=\Gamma^{0}\Gamma^{1}$ this is equivalent to the condition
$i\Gamma^{0}{\bm{\chi}}(\epsilon)={\bm{\chi}}(\epsilon)$ and because
$\Gamma^{0}$ is antihermitian this implies
${\bm{\chi}}^{\dagger}(i\Gamma^{0})={\bm{\chi}}^{\dagger}$ when evaluated at
$r=\epsilon$, and so also $\overline{\bm{\chi}}=\bm{\chi}^{\dagger}$ there.
These conditions allow us to rewrite

$\overline{\chi}_{\mathfrak{s}}(\epsilon)\chi_{{\mathfrak{s}}^{\prime}}(\epsilon)=\chi^{\dagger}_{\mathfrak{s}}(\epsilon)i\Gamma^{0}\chi_{\mathfrak{s}^{\prime}}(\epsilon)=\chi^{\dagger}_{\mathfrak{s}}(\epsilon)\chi_{\mathfrak{s}^{\prime}}(\epsilon)=\overline{\chi}_{\mathfrak{s}}(\epsilon)i\Gamma^{0}\chi_{\mathfrak{s}^{\prime}}(\epsilon),$
(98)

where the first equality uses the definition
${\overline{{\bm{\chi}}}}:=i{\bm{\chi}}^{\dagger}\Gamma^{0}$. This shows that
within perturbative calculations the couplings
$\delta\mathcal{C}^{s}_{\mathfrak{s}\mathfrak{s}^{\prime}}{\overline{\chi}}_{\mathfrak{s}}\chi_{{\mathfrak{s}}^{\prime}}-i\delta{\cal
C}^{v}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}{\overline{\chi}}_{\mathfrak{s}}\Gamma^{0}\chi_{{\mathfrak{s}}^{\prime}}$
appearing in expression (87) for $S_{\rm dyon2}^{\rm int}$ should only
contribute to physical predictions through the combination
$\delta\mathcal{C}^{s}_{\mathfrak{s}\mathfrak{s}^{\prime}}-\delta{\cal
C}^{v}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}$.

This is consistent with the more general expressions like (3.2.1) for the
amplitudes ${\cal T}_{\rm in}$ and ${\cal T}_{\rm out}$ within the Born-
Oppenheimer framework, since for rank-2 perturbations about the unperturbed
couplings ${\cal C}^{ps}_{++}={\cal C}^{ps}_{--}=1$ these give

$\displaystyle\mathcal{T}^{++}_{{\rm in}}\simeq 1+i(\delta{\cal
C}^{v}_{++}-\delta{\cal C}^{s}_{++}),\quad\mathcal{T}^{--}_{{\rm in}}\simeq
1+i(\delta{\cal C}^{v}_{--}-\delta{\cal C}^{s}_{--}),$ $\displaystyle{\cal
T}^{+-}_{{\rm in}}\simeq i(\delta{\cal C}^{v}_{+-}-\delta{\cal
C}^{s}_{+-}),\quad\hbox{and}\quad{\cal T}^{-+}_{{\rm in}}\simeq i(\delta{\cal
C}^{v}_{-+}-\delta{\cal C}^{s}_{-+}),$ (99)

showing at linear order that $\delta{\cal
C}^{s}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}$ and $\delta{\cal
C}^{v}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}$ really do only appear in the
combination $\delta{\cal
C}^{s}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}-\delta{\cal
C}^{v}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}$.

These arguments suggest that within the perturbative framework one combination
of $\delta{\cal C}^{s}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}$ and
$\delta{\cal C}^{v}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}$ is redundant in
the sense that it can be removed by performing a field redefinition at the
position of the dyon worldline. Indeed, writing

${\overline{{\bm{\chi}}}}{\cal
C}^{s}{\bm{\chi}}-i{\overline{{\bm{\chi}}}}{\cal
C}^{v}\Gamma^{0}{\bm{\chi}}=\tfrac{1}{2}{\overline{{\bm{\chi}}}}({\cal
C}^{s}+{\cal
C}^{v})(1-i\Gamma^{0}){\bm{\chi}}+\tfrac{1}{2}{\overline{{\bm{\chi}}}}({\cal
C}^{s}-{\cal C}^{v})(1+i\Gamma^{0}){\bm{\chi}}\,,$ (100)

we seek a redefinition that removes the first term. But under a small
variation ${\bm{\chi}}\to{\bm{\chi}}+\delta{\bm{\chi}}$ at $r=\epsilon$ the
variation of the bulk action and the unperturbed boundary action (86) becomes

$\delta
S(r=\epsilon)=-\tfrac{1}{2}{\overline{{\bm{\chi}}}}\Bigl{(}-\Gamma^{1}+i\Gamma_{c}\Bigr{)}\delta{\bm{\chi}}-\tfrac{1}{2}\delta{\overline{{\bm{\chi}}}}\Bigl{(}\Gamma^{1}+i\Gamma_{c}\Bigr{)}{\bm{\chi}}\,,$
(101)

which for $\delta{\bm{\chi}}=iA\,\Gamma_{c}{\bm{\chi}}$ (with hermitian $A$ in
flavour space) gives

$\delta
S(r=\epsilon)=-\tfrac{1}{2}i\,{\overline{{\bm{\chi}}}}A\Bigl{[}(-\Gamma^{1}+i\Gamma_{c})\Gamma_{c}+\Gamma_{c}(\Gamma^{1}+i\Gamma_{c})\Bigr{]}{\bm{\chi}}=\,{\overline{{\bm{\chi}}}}A\Bigl{(}1-i\Gamma^{0}\Bigr{)}{\bm{\chi}}\,,$
(102)

showing that $A$ can be chosen to remove the ${\cal C}^{s}+{\cal C}^{v}$ term
in (100), as claimed.

A very similar reduction in the number of independent couplings also happens
for more prosaic applications of the PPEFT framework to fermions. When used to
describe the influence of a non-pointlike nucleus on electronic energy levels
within an atom there turn out to be more effective interactions describing
various types of nuclear effective couplings on the nuclear worldline than
there are independent nuclear contributions to electronic energy levels. In
that case the redundancy of many of the apparent nuclear moments at low
energies ensures that nuclear uncertainties enter into atomic calculations in
fewer ways than one would naively expect Burgess:2020ndx ; Zalavari:2020tez

##  4 RG methods and catalysis

We now turn to the issue of $\epsilon$-dependence. The mode functions and
scattering amplitudes found above depend in detail on the boundary conditions
imposed at the surface $\partial P$ of the gaussian pillbox that surrounds
each source. But how can it be that physical quantities depend on an arbitrary
radius $r=\epsilon$ that defines the boundary of this pillbox?

This question is precisely what PPEFT methods are good for: they show how
effective couplings like the ${\cal C}^{\scriptscriptstyle A}_{ij}$’s
appearing in the dyon’s effective action must depend implicitly on (i.e. run
with) $\epsilon$ in order to ensure that physical predictions are
$\epsilon$-independent. A side benefit of this discussion is that it shows how
divergences that arise (even at the classical level) in the values of the
fields as $\epsilon\to 0$ get renormalized into the effective couplings of the
dyon action, along the lines described in Goldberger:2001tn . In this language
the physical scales of the UV physics – such as $\mu$ in the action (1) – are
described within the EFT as RG-invariant scales associated with this running.

In this section we set up how this works for $S$-wave scattering from the dyon
and in particular ask why this running makes scattering for monopoles so
different from scattering from other small massive objects like nuclei in
atoms (for which the corresponding issues are described in PPEFT3  and briefly
summarized in Appendix B). We discuss in turn two ways the running of
effective couplings of fermions to dyons differ from their couplings to more
run-of-the-mill compact objects: ($i$) dimensional scaling changes associated
with the kinematics of the fermion $S$-wave (which lies at the root of why
dyon-fermion scattering is insensitive to dyon size), and ($ii$) the large-
scale effects to do with the nontrivial fermionic interacting vacuum (what is
sometimes called the fermion ‘condensate’).

###  4.1 Scaling for compact objects

The most important difference between fermion-dyon and fermion-nucleus
scattering is the size of the interaction rates to which one is led. For
fermion-nucleus scattering the RG invariant scale that sets the size of
scattering cross sections depends on the nucleus’ small radius.171717More
precisely, the RG scales expressing the implications of the nuclear strong
interactions of pionic atoms, say, are set by the nuclear radius while the
nucleus’ electromagnetic effects relevant for electron-nuclear interactions
are suppressed relative to this by powers of the fine-structure constant
Burgess:2020ndx ; Zalavari:2020tez . But for fermion-dyon scattering the
$S$-wave cross sections are not suppressed by the small dyonic size $R\sim
m_{g}^{-1}$ even at energies $E\ll m_{g}$. Although the mechanism for this
(the special kinematics of $S$-wave scattering) has been well-understood for
quite some time Rubakov:1981rg ; Callan:1982ah  our discussion here embeds
this understanding into the wider PPEFT framework and shows how it can be
understood using the same standard EFT reasoning that also applies to nuclei.

### Scaling for nuclei

For nuclei the bulk fermion fields satisfy the Dirac equation away from the
nucleus (in the ‘bulk’)

$(\hbox to0.0pt{\hbox to8.55695pt{\hfil/\hfil}\hss}D+m)\psi=0\,,$ (103)

subject to nucleus-dependent boundary conditions at the surface $r=\epsilon$
of some small gaussian pillbox described by an effective action similar to the
one described above. The main difference from the dyon case is that the
absence of a magnetic monopole implies the total spin $j$ is half-integral
rather than integral (see Appendix B).

For instance, for $r>\epsilon$ the parity-even solutions are

$\psi=\left(\begin{array}[]{c}\psi_{\scriptscriptstyle L}\\\
\psi_{\scriptscriptstyle
R}\end{array}\right)=\left(\begin{array}[]{c}f(r)\,U^{+}(\theta,\phi)+ig(r)\,U^{-}(\theta,\phi)\\\
f(r)\,U^{+}(\theta,\phi)-ig(r)\,U^{-}(\theta,\phi)\end{array}\right)\,,$ (104)

and a similar expression exists for parity-odd ones. Here $U^{\pm}$ are
appropriate spinor harmonics and the radial functions $f(r)$ and $g(r)$ with
mode frequency $\omega$ solve the radial equations

$f^{\prime}=\left(m+\omega\right)g\quad\hbox{and}\quad
g^{\prime}+\frac{2g}{r}=\left(m-\omega\right)f\,,$ (105)

where primes denote differentiation with respect to $r$. The general solution
to these coupled first-order equations is a linear combination of two linearly
independent solutions

$f(r)=C_{1}f_{1}(r)+C_{2}f_{2}(r)\qquad\hbox{and}\qquad
g(r)=C_{1}g_{1}(r)+C_{2}g_{2}(r)\,,$ (106)

where $C_{1}$ and $C_{2}$ are the two expected integration constants.

Physical predictions (e.g. for atomic energy levels or scattering cross
sections) can be expressed in terms of $C_{2}/C_{1}$ and this ratio is in turn
determined in terms of nuclear properties through the near-nucleus boundary
conditions, which give the ratio of the functions $f$ and $g$ at $r=\epsilon$
in terms of the effective couplings $\hat{\mathfrak{c}}_{i}$ for fermion
bilinears of the form $\int_{\scriptscriptstyle W}{\rm
d}t\;\hat{\mathfrak{c}}_{i}\,{\overline{\psi}}M_{i}\psi$ in the nucleus’
world-line action (where $M_{i}$ are a basis of Dirac matrices). More
precisely, the boundary condition at the surface of a gaussian pillbox
centered on the nucleus is – compare with eqs. (40) and (3.1.1):

$\hat{\cal C}=\left(\frac{g}{f}\right)_{r=\epsilon}\,.$ (107)

Here $\hat{\cal C}$ is a linear combination of dimensionless quantities
$\hat{\cal C}_{i}=\hat{\mathfrak{c}}_{i}/(4\pi\epsilon^{2})$ – compare with
(47) (see PPEFT3  for details). Eq. (107) provides the solution to how the
properties of the source influence the bulk solutions for $\psi$ in the
source’s vicinity: the couplings $\hat{\cal C}_{i}$ determine
$(g/f)_{r=\epsilon}$ through the near-source boundary condition and this in
turn fixes the ratios of integration constants $C_{2}/C_{1}$ that control the
size of observable phenomena.

But it is still a potential puzzle why physical predictions can depend on the
relatively arbitrary radius, $r=\epsilon$, of the Gaussian pillbox, which is
just a scale that regularizes the boundary conditions that need not be
directly related to the underlying physical scales such as the physical size
of the actual compact object within the pillbox (which we denote by $R$). The
precise value of $\epsilon$ must therefore drop out of predictions for
observables. In detail, this happens because any explicit
$\epsilon$-dependence arising in calculations of an observable cancels an
implicit $\epsilon$-dependence buried within the ‘bare’ quantities $\hat{\cal
C}_{i}$. Physical predictions remain $\epsilon$-independent if quantities like
$\hat{\cal C}_{i}(\epsilon)$ are chosen to be $\epsilon$-dependent in a way
that ensures that ratios like $C_{2}/C_{1}$ remain fixed as $\epsilon$ is
varied.

This gives us another way to interpret eq. (107): rather than reading it as
fixing $f/g$ at $r=\epsilon$ given fixed values of the $\hat{\cal C}_{i}$ we
can instead read it as telling us how the $\hat{\cal C}_{i}(\epsilon)$ must
depend on $\epsilon$ in order to ensure that $C_{2}/C_{1}$ remains
$\epsilon$-independent. Within this rereading physical observables depend only
on the trajectory $(\epsilon,\hat{\cal C}_{i}(\epsilon))$ rather than
depending on $\epsilon$ and $\hat{\cal C}_{i}(\epsilon)$ separately. Changes
of $\epsilon$ with $C_{2}/C_{1}$ fixed (and so also with physical observables
fixed) can be regarded as defining a renormalization-group (RG) flow of the
$\hat{\cal C}_{i}(\epsilon)$’s, and physical observables must be invariant
with respect to this flow.

It turns out that this kind of RG flow defines a natural RG-invariant length
scale, and it is this length scale that both appears in physical predictions
(such as for scattering cross sections) and is determined in terms of physical
scales like $R$ when matching effective couplings to the full UV theory of the
source. To see how this scale arises in terms of radial mode functions like
$f(r)$ and $g(r)$ it is convenient to choose $\epsilon$ such that
$R\ll\epsilon\ll a$ where $a$ is a characteristic length scale of the bulk
theory far from the source (for instance, for nuclei in atoms we might have
$R$ of order the nuclear size and $a$ of order the Bohr radius). Radial mode
functions are often well-approximated by power laws in this regime, with

$f(r)=C_{1}\left(\frac{r}{a}\right)^{{\mathfrak{z}}-1}+C_{2}\left(\frac{r}{a}\right)^{-{\mathfrak{z}}-1}\qquad\hbox{and}\qquad
g(r)=\widetilde{C}_{1}\left(\frac{r}{a}\right)^{{\mathfrak{z}}-1}+\widetilde{C}_{2}\left(\frac{r}{a}\right)^{-{\mathfrak{z}}-1}\,,$
(108)

for some power181818Within atoms, for instance, the relevant power is
${\mathfrak{z}}=[(j+\tfrac{1}{2})^{2}-(Z\alpha)^{2}]^{1/2}$. ${\mathfrak{z}}$
with $\widetilde{C}_{i}\propto C_{i}$ in a way that depends on the relative
small-$r$ asymptotic behaviour of $f_{i}(r)$ and $g_{i}(r)$. One of these
solutions dominates for sufficiently small $r$ while the other wins when $r$
is sufficiently big. The precise crossover radius $R_{\star}$ between these
two regimes depends only on the value of the constant $C_{2}/C_{1}$, and so
provides a convenient RG-invariant characterization of the coupling evolution,
and one typically finds e.g. scattering cross sections with bulk fields of
size $\sigma\sim\pi R_{\star}^{2}$ for scattering of long-wavelength modes
($kR_{\star}\ll 1$) despite couplings like $\hat{\cal C}_{i}$ being
dimensionless – see PPEFT3 .

####  $S$-wave scaling for dyons and catalysis

With the above story in mind we can now use the same EFT language to see why
$S$-wave scattering from dyons is so different from scattering from other
compact objects. The key issue is not whether couplings like $\hat{\cal
C}_{i}$ are dimensionless or not (they are dimensionless for both nuclei and
dyons). The key issue is the size to be expected for RG-invariant scales like
$R_{\star}$.

The main issue is the difference between eqs. (105) for nuclei and (26) for
dyons. For nuclei (105) has two linearly independent solutions and so admits
two different power-law type asymptotic forms like (108) the transition
between which defines the RG-invariant scale $R_{\star}$. But for dyons the
$S$-wave condition removes one of these solutions leaving just a single first-
order equation (108) for each choice of external quantum numbers. This means
there is never a transition between two asymptotic power-law regimes; there is
only one power-law for each type of mode. As a result there is no RG-invariant
scale $R_{\star}$ on which measureable things like scattering cross sections
can depend.

For $S$-wave scattering from dyons the situation is similar to what would have
happened for nucleons if for some reason we were required to choose $C_{2}=0$.
In this case using the asymptotic form (108) in (107) would imply that
$\hat{\cal C}$ is $\epsilon$-independent. The $\epsilon$-independence of
physical quantities for $S$-wave dyons similarly requires quantities like
${\cal T}_{\rm in}$ or ${\cal T}_{\rm out}$ to be $\epsilon$-independent (as