we see below explicitly), and this asks all of the dimensionless rank-two
couplings ${\cal C}_{i}$ to themselves directly be $\epsilon$-independent.
This makes $S$-wave dyon scattering from massless fermions scale invariant and
so its size is mainly set by the projection of any incoming wave onto the
$S$-wave state, leading to cross sections that vary as
$\sigma\propto\pi/k^{2}$ when $kR\ll 1$ rather than $\sigma\propto\pi R^{2}$
(as we verify explicitly below). This is true for any $S$-wave process
regardless of whether or not the reaction in question violates a flavour
symmetry (like baryon number).

Arguments like these relying on the uniqueness of the $S$-wave kinematics are
standard ones Rubakov:1981rg ; Callan:1982ah  for explaining the large size of
catalysis cross sections. What the above arguments do is provide them with an
EFT veneer that shows why they do not undermine the usual notions of
decoupling (once these are carefully formulated).

### Matching

To this point the effective couplings ${\cal C}_{i}$ and amplitudes ${\cal
T}_{\rm in}$ have been treated as arbitrary parameters. But they really should
be regarded as being functions of microscopic parameters in any specific
theory and so take definite values once a given microscopic dyon construction
is chosen. Physical predictions for things like fermion-dyon scattering cross
sections within specific microscopic theories are then obtained by
substituting the appropriate values for the ${\cal C}_{i}$’s (or ${\cal
T}_{\rm in}$) into the general expressions for e.g. scattering cross sections
given in §5 and §6 below.

For simple calculations of semiclassical scattering of massless fermions
moving in a classical dyon field chirality is conserved by the Dirac equation.
As discussed below eq. (26) the change of direction of radial motion required
by $S$-wave scattering implies ${\mathfrak{h}}={\mathfrak{c}}{\mathfrak{s}}$
must change sign during scattering and so conservation of ${\mathfrak{c}}$
implies ${\mathfrak{s}}$ must change. This means that fermion charge is always
exchanged with the dyon and so ${\cal T}_{\rm in}^{++}={\cal T}_{\rm
in}^{--}=0$ for this type of semiclassical scattering. This then implies –
c.f. the unitarity conditions (73) – ${{\cal T}}^{-+}_{{\rm
in}}=e^{-i\delta},{{\cal T}}^{+-}_{{\rm in}}=e^{i\delta^{\prime}}$ for some
phases $\delta,\delta^{\prime}$. This is the situation that applies to the
majority of microscopic fermion scattering calculations (where typically
$\delta^{\prime}=\delta$) performed in the presence of a classical dyon
Goldhaber:1977xw ; Blaer:1981ps ; Marciano:1983md ; Ezawa:1983vi .

Alternatively Kazama et.al. Kazama:1976fm  compute scattering processes where
the fermion moving within the dyonic background has an anomalous magnetic
moment. In this case chirality is not conserved and chirality-changing
processes dominate, corresponding to the case $\mathcal{T}^{++}_{{\rm
in}}=\mathcal{T}^{--}_{{\rm in}}=i\,\text{sign}(\kappa)$ (where $\kappa$ is a
parameter of their model) and $\mathcal{T}^{+-}_{{\rm
in}}=\mathcal{T}^{-+}_{{\rm in}}=0$. In both chirality-preserving and
chirality-breaking cases our expressions for cross sections and currents found
in later sections agree with theirs once restricted to these choices.

###  4.2 Interaction effects

In practice the motion of a free fermion within a fixed dyonic background does
not provide a good approximation to fermion-dyon scattering. The free-fermion-
moving-in-a-fixed-background approximation breaks down because charge-changing
fermion interactions with the rotor field ${\mathfrak{a}}$ described by the
amplitudes ${\cal T}_{\rm in}^{+-}$ and ${\cal T}_{\rm in}^{-+}$ significantly
distort the ground state within the fermionic sector (more about this in §6.1
below) and this distortion cannot be neglected Rubakov:1981rg ; Callan:1982ah
; CallanSMatrix ; Yamagishi:1982wp ; Grossman:1983yf ; Kazama:1983rt ;
Polchinski . The radial extent of the fermionic vacuum polarization can extend
outside the dyon to distances of order the fermion Compton wavelength and so
can be much larger than the underlying classical dyon solution itself.

The back-reaction of this kind of dynamics can appreciably alter the RG flow
of couplings like the ${\cal C}_{i}$’s once $\epsilon$ is large enough to
include a significant component of fermionic polarization within the gaussian
pillbox. In this case it is the new values for ${\cal C}_{i}(\epsilon)$ that
are relevant when computing quantities like ${\cal T}_{\rm in}$ and the above
conclusion that ${\cal T}_{\rm in}$’s are $\epsilon$-independent changes. The
results of this type of evolution are studied in Rubakov:1981rg ;
Callan:1982ah ; CallanSMatrix ; Polchinski , and for general models the full
interpretation of the resulting physics remains incomplete (see for example
vanBeest:2023dbu ; vanBeest:2023mbs  for recent discussions). In the specific
model considered here, however, the upshot is fairly simple: the Coulomb
energies associated with the fermionic vacuum distortions turn out to convert
the UV semiclassical prediction ${\cal T}_{\rm in}^{++}={\cal T}_{\rm
in}^{--}=0$ for $\epsilon$ of order the monopole scale into the new prediction
$\mathcal{T}^{+-}_{{\rm in}}=\mathcal{T}^{-+}_{{\rm in}}=0$ for $\epsilon$
large enough to include the fermionic vacuum distortions Polchinski . This
conversion is intuitive inasmuch as the underlying distortion of the fermionic
vacuum is driven by nonzero ${\cal T}_{\rm in}^{+-}$ and ${\cal T}_{\rm
in}^{-+}$ (as we see in §6 below)

More generally the coefficients appropriate to any other particular
microscopic dyon construction can in principle be obtained in a similar way by
matching the EFT to the microscopic theory of interest, once this is known.
When doing this matching we typically choose $\epsilon$ such that
$R\ll\epsilon\ll a$. For the simplest applications $R\sim m_{g}^{-1}$ is of
order the dyon size, but for applications including fermion condensation in
the dyon field $R$ is instead of order the fermion’s Compton wavelength $R\sim
m_{\psi}^{-1}$. In either case taking $R\ll\epsilon\ll a$ remains justified
provided the low-energy focus is on sufficiently large $a$.

##  5 Perturbative calculations

We next turn to calculating some simple dyon-fermion processes as functions of
the general effective couplings described above. We do so first in this
section treating the fermion-dyon interactions perturbatively, and then in the
next section repeat the process within the Born-Oppenheimer approximation
(which works to all orders in the fermion-dyon interactions when the rotor
field ${\mathfrak{a}}$ is treated classically).

We start by computing several reaction rates treating the dyon-fermion
interactions perturbatively, within the framework described in §3.3.1 above.
This choice is just made for convenience to explore some perturbative
consquences of the fermion-dyon interactions, and might not be the dominant
reaction for particular microscopic descriptions of the underlying dyon. We
return to the more broadly applicable general case without this perturbative
approximation in §6 below.

The dyon-localized part of the hamiltonian governing dyon-fermion interactions
obtained from the lagrangian (40) (and including the electrostatic background
and fluctuation field) is

$H_{\rm dyon}=e{\mathfrak{p}}\,\widehat{A}_{0}(r=\epsilon)+\frac{1}{2{\cal
I}}\left({\mathfrak{p}}-\frac{\vartheta}{2\pi}\right)^{2}+\frac{1}{2}\Bigl{[}{\overline{\bm{\psi}}}\,{\mathfrak{C}}({\mathfrak{a}})\,{\bm{\psi}}\Bigr{]}_{r=\epsilon}$
(109)

where ${\mathfrak{p}}$ is the canonical momentum for ${\mathfrak{a}}$ given in
(42). In this section we drop the Coulomb fluctuation $\widehat{A}_{0}$ and
perturb the fermion boundary action about a simple
${\mathfrak{a}}$-independent boundary action along the lines described in
§3.3.1. Specializing to the $S$-wave and switching to 2D fields, we split the
bulk and boundary-localized hamiltonian into unperturbed and perturbed parts,
$H_{0}+H_{\rm int}$. The unperturbed hamiltonian is191919We follow standard
practice here and keep the rotor kinetic term despite it being order
$(e^{2}/4\pi)\mu$ in magnitude and so nominally being a higher-loop size.

$\displaystyle H_{0}=\frac{1}{2{\cal
I}}\left({\mathfrak{p}}-\frac{\vartheta}{2\pi}\right)^{2}+\frac{i}{2}\sum_{{\mathfrak{s}}=\pm}\left({\overline{{\chi}}}_{{\mathfrak{s}}}\Gamma_{c}\chi_{{\mathfrak{s}}}\right)_{r=\epsilon}+\frac{1}{2}\sum_{{\mathfrak{s}}=\pm}\int^{\infty}_{\epsilon}{\rm
d}r\,{\overline{{\chi}}}_{{\mathfrak{s}}}\left[\Gamma^{1}\overset{\leftrightarrow}{\partial}_{1}-i{\mathfrak{s}}\,\Gamma^{0}\left(ev-\frac{eQ}{r}\right)\right]\chi_{{\mathfrak{s}}},$
(110)

in which the first term describes the free rotor dynamics and the second term
gives the boundary conditions at $r=\epsilon$ satisfied by the fermions, whose
bulk dynamics in the presence of the background dyon charge is given by the
last term.

The unperturbed boundary term
$\frac{i}{2}\left({\overline{\bm{\chi}}}\Gamma_{c}\bm{\chi}\right)_{r=\epsilon}$
is chosen such that ($i$) it does not involve the rotor field
${\mathfrak{a}}$; ($ii$) the corresponding boundary matrix ${\cal B}^{(0)}$
has rank $2$ and ($iii$) the resulting modes describe fermion reflection from
the dyon with no phase change at $r=\epsilon$: that is, ${\cal T}^{++}={\cal
T}^{--}=1$ to zeroeth order in perturbation theory. This choice implies the
interaction picture field $\bm{\chi}$ satisfies the boundary condition (89)
and can be expanded in terms of either the modes ${\mathfrak{u}}^{\rm
in0}_{{\mathfrak{s}},k},{\mathfrak{v}}^{\rm in0}_{{\mathfrak{s}},k}$ or
${\mathfrak{u}}^{\rm out0}_{{\mathfrak{s}},k},{\mathfrak{v}}^{\rm
out0}_{{\mathfrak{s}},k}$, given in equations (90)-(93).

The interaction hamiltonian in the interaction picture is then given by

$H_{\rm
int}=\frac{1}{2}\sum_{{\mathfrak{s}},{\mathfrak{s}}^{\prime}=\pm}\overline{\chi}_{{\mathfrak{s}}}\Bigl{(}\delta\mathcal{C}^{s}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}+i\delta\mathcal{C}^{ps}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}\Gamma_{c}-i\delta\mathcal{C}^{v}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}\Gamma^{0}+i\delta\mathcal{C}^{pv}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}\Gamma^{1}\Bigr{)}\,\chi_{{\mathfrak{s}}^{\prime}}\;e^{\frac{i}{2}({\mathfrak{s}}-{\mathfrak{s}}^{\prime})\,{\mathfrak{a}}_{\scriptscriptstyle
I}}\,,$ (111)

where the fermion fields are evaluated at $r=\epsilon$ and
${\mathfrak{a}}_{\scriptscriptstyle I}$ is the interaction picture rotor field

${\mathfrak{a}}_{\scriptscriptstyle
I}(t):=\;e^{\frac{i}{2\mathcal{I}}\Pi^{2}t}{\mathfrak{a}}\,e^{-\frac{i}{2\mathcal{I}}\Pi^{2}t}\,,$
(112)

with $\Pi:={\mathfrak{p}}-(\vartheta/2\pi)$. As discussed in §3.4 this
perturbed problem also has rank two only when $\delta{\cal
C}^{pv}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}=\delta{\cal
C}^{ps}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}=0$, which we henceforth assume
(though this is not crucial for this perturbative discussion). In terms of
creation and annihilation operators eq. (111) can be written in the normal-
ordered form

$\displaystyle H_{\rm int}$ $\displaystyle=$ $\displaystyle
E_{0}+\sum_{\mathfrak{s},\mathfrak{s}^{\prime}=\pm}\int^{\infty}_{0}\frac{\mathrm{d}k\,\mathrm{d}k^{\prime}}{2\pi}\,(\delta\mathcal{C}^{s}_{\mathfrak{s}\mathfrak{s}^{\prime}}-\delta\mathcal{C}^{v}_{\mathfrak{s}\mathfrak{s}^{\prime}})\Big{[}({a}^{{\rm
in}}_{\mathfrak{s},k})^{\star}{a}^{{\rm
in}}_{\mathfrak{s}^{\prime},k^{\prime}}e^{i(k-k^{\prime})t}+({a}^{{\rm
in}}_{\mathfrak{s},k})^{\star}(\overline{{a}}^{{\rm
in}}_{\mathfrak{s}^{\prime},k^{\prime}})^{\star}e^{i(k+k^{\prime})t}$ (113)
$\displaystyle\qquad\qquad+\overline{{a}}^{{\rm
in}}_{\mathfrak{s},k}{{a}}^{{\rm
in}}_{\mathfrak{s}^{\prime},k^{\prime}}e^{-i(k+k^{\prime})t}-(\overline{{a}}^{{\rm
in}}_{\mathfrak{s}^{\prime},k^{\prime}})^{\star}\overline{{a}}^{{\rm
in}}_{\mathfrak{s},k}e^{-i(k-k^{\prime})t}\Big{]}\;e^{\frac{i}{2\mathcal{I}}\Pi^{2}t}e^{\frac{i}{2}(\mathfrak{s}-\mathfrak{s}^{\prime})({\mathfrak{a}}-evt)}e^{-\frac{i}{2\mathcal{I}}\Pi^{2}t}$

where we absorb an $r_{0}$-dependent phase using an appropriate constant shift
of ${\mathfrak{a}}$. We also drop factors of $e^{i(k\pm k^{\prime})\epsilon}$
because our EFT framework requires we choose $\epsilon$ to be smaller than the
characteristic bulk length scales of interest, as described in §4. In
particular this requires us to restrict our attention to the regime where
$k\epsilon$, $k^{\prime}\epsilon$ and $ev\epsilon$ are all much smaller than
unity.

In (113) $E_{0}$ denotes the vacuum expectation value of the interacting
hamiltonian $H_{\rm int}$, given by

$E_{0}:=\bra{0}H_{\rm
int}\ket{0}=\frac{eQ}{4\pi\epsilon}(\delta\mathcal{C}^{s}_{--}-\delta\mathcal{C}^{v}_{--}-\delta\mathcal{C}^{s}_{++}+\delta\mathcal{C}^{v}_{++}),$
(114)

as shown in Appendix G. Although this diverges as $\epsilon\to 0$ it can be
absorbed into the counterterm describing the mass of the dyon. Notice these
expressions depend on the boundary couplings only through the combination
$\delta{\cal C}^{s}-\delta{\cal C}^{v}$, as argued must be the case in §3.4.

The remainder of this section uses the above setup to calculate some
scattering observables within this perturbative framework.

###  5.1 Processes involving dyon charge eigenstates

We start by recording the amplitudes for single-fermion scattering processes
assuming the dyon is chosen to be in a charge – i.e. momentum – eigenstate at
both the initial and final times and working to first order in perturbation
theory. By focusing here on single-particle scattering we avoid the
complications of multiparticle state-definitions in a monopole background
described in Csaki:2021ozp ; Csaki:2022tvb  and the twist operators described
in vanBeest:2023dbu ; vanBeest:2023mbs .

####  5.1.1 Pair production

Among the charge-changing processes mediated at leading order by (113) is the
production or absorption of particle-antiparticle pairs carrying net charge.
The amplitude obtained at leading order from (113) for the production of a
pair of positive charge – i.e. for $\ket{0}\ket{{\Pi}}\rightarrow({a}^{{\rm
out}}_{+,k})^{\star}(\overline{a}^{{\rm
out}}_{-,k^{\prime}})^{\star}\ket{0}\ket{{\Pi}^{\prime}}$ – is

$\displaystyle{\cal
A}\Bigl{[}\Pi\to\Pi^{\prime}+f_{+}(k)+\bar{f}_{-}(k^{\prime})\Bigr{]}=-\delta_{{\Pi}^{\prime},{\Pi}+1}\delta\left(\omega_{+,k}+{\overline{\omega}}_{-,k^{\prime}}+\frac{\Pi^{\prime
2}-\Pi^{2}}{2\mathcal{I}}\right)i\left(\delta\mathcal{C}^{s}_{+-}-\delta\mathcal{C}^{v}_{+-}\right),$
(115)

where $\omega_{{\mathfrak{s}},k}=k-\frac{1}{2}{\mathfrak{s}}ev$ is the energy
of a particle with charge ${\mathfrak{s}}e/2$ and momentum $k$ and
$\overline{\omega}_{{\mathfrak{s}},k}=k+\frac{1}{2}{\mathfrak{s}}ev$ is the
energy of an antiparticle with charge $-{\mathfrak{s}}{e}/2$ and momentum $k$.

The amplitude for producing a negatively charged pair – i.e. for
$\ket{0}\ket{{\Pi}}\rightarrow({a}^{{\rm
out}}_{-,k})^{\star}(\overline{a}^{{\rm
out}}_{+,k^{\prime}})^{\star}\ket{0}\ket{{\Pi}^{\prime}}$ – similarly is

$\displaystyle{\cal
A}\Bigl{[}\Pi\to\Pi^{\prime}+f_{-}(k)+\bar{f}_{+}(k^{\prime})\Bigr{]}=-\delta_{{\Pi}^{\prime},{\Pi}-1}\delta\left(\omega_{-,k}+{\overline{\omega}}_{+,k^{\prime}}+\frac{\Pi^{\prime
2}-\Pi^{2}}{2\mathcal{I}}\right)i\left(\delta\mathcal{C}^{s}_{-+}-\delta\mathcal{C}^{v}_{-+}\right).$
(116)

These show that the rotor level can only change by one unit, as required for
it to absorb or emit the charge lost or gained by the fermions. The fermion
similarly gains or loses the energy required by this transition. These
processes cause rotor states with nonzero $\Pi$ to decay towards $\Pi=0$ by
emitting fermion pairs. It is energetically possible to emit positively
charged pairs when $k+k^{\prime}=ev-\frac{1+2\Pi}{2{\cal I}}\geq 0$ and so is
only possible when $\Pi\leq{\cal I}ev-\frac{1}{2}$. It is similarly possible
to emit negatively charged pairs if $k+k^{\prime}=-ev-\frac{1-2\Pi}{2{\cal
I}}\geq 0$ and this is only possible when $\Pi\geq{\cal I}ev+\frac{1}{2}$. No
pair production occurs for electrically neutral particle-antiparticle pairs
since energy conservation implies this can only happen if $k=k^{\prime}=0$.

Writing ${\cal A}={\cal M}\,\delta(E_{f}-E_{i})$ and using these amplitudes in
Fermi’s golden rule gives the following differential decay rate

$\mathrm{d}\Gamma=2\pi|{\cal M}|^{2}\,\delta(E_{f}-E_{i})\,\frac{{\rm
d}k}{2\pi}\,\frac{{\rm d}k^{\prime}}{2\pi}\,.$ (117)

Using this and performing the final-state momentum integrals, the integrated
rate for emitting positively charged fermions is nonzero for initial dyon
momenta satisfying $\Pi<{\cal I}ev-\frac{1}{2}\sim(4\pi/e)(v/\mu)$, and is
given by

$\Gamma\Bigl{[}\Pi\to(\Pi+1)+f_{+}(k)+\bar{f}_{-}(k^{\prime})\Bigr{]}=\frac{1}{2\pi}\left(ev-\frac{2\Pi+1}{2{\cal
I}}\right)|\delta{\cal C}^{s}_{+-}-\delta{\cal C}^{v}_{+-}|^{2}\,.$ (118)

The integrated rate for emitting negatively charged pairs is similarly nonzero
when $\Pi>{\cal I}ev+\frac{1}{2}$ with

$\Gamma\Bigl{[}\Pi\to(\Pi-1)+f_{-}(k)+\bar{f}_{+}(k^{\prime})\Bigr{]}=\frac{1}{2\pi}\left(-ev+\frac{2\Pi-1}{2{\cal
I}}\right)|\delta{\cal C}^{s}_{-+}-\delta{\cal C}^{v}_{-+}|^{2}\,.$ (119)

####  5.1.2 Scattering cross sections

The Hamiltonian (113) also describes scattering processes. The amplitude for
the charge-changing process $({a}^{{\rm
in}}_{+,k})^{\star}\ket{0}\ket{{\Pi}}\rightarrow({a}^{{\rm
out}}_{-,k^{\prime}})^{\star}\ket{0}\ket{{\Pi}^{\prime}}$ is given by

${\cal
A}\Bigl{[}f_{+}(k)+\Pi\to\Pi^{\prime}+f_{-}(k^{\prime})\Bigr{]}=-\delta_{{\Pi}^{\prime},{\Pi}-1}\delta\left(\omega_{-,k^{\prime}}-\omega_{+,k}+\frac{{\Pi^{\prime}}^{2}-\Pi^{2}}{2\mathcal{I}}\right)\,i\left(\delta\mathcal{C}^{s}_{-+}-\delta\mathcal{C}^{v}_{-+}\right),$
(120)

where (as before) $\omega_{{\mathfrak{s}},k}=k-{\mathfrak{s}}\frac{ev}{2}$.
Similarly the amplitude for $({a}^{{\rm
in}}_{-,k})^{\star}\ket{0}\ket{{\Pi}}\rightarrow({a}^{{\rm
out}}_{+,k^{\prime}})^{\star}\ket{0}\ket{{\Pi}^{\prime}}$ is

${\cal
A}\Bigl{[}f_{-}(k)+\Pi\to\Pi^{\prime}+f_{+}(k^{\prime})\Bigr{]}=-\delta_{{\Pi}^{\prime},{\Pi}+1}\delta\left(\omega_{+,k^{\prime}}-\omega_{-,k}+\frac{{\Pi^{\prime}}^{2}-\Pi^{2}}{2\mathcal{I}}\right)\,i\left(\delta\mathcal{C}^{s}_{+-}-\delta\mathcal{C}^{v}_{+-}\right).$
(121)

The analogous amplitudes for charge-changing antiparticle transitions can be
inferred from those above by crossing symmetry. These reactions can proceed so
long as the initial fermion energy satisfies $\omega_{i}\geq\frac{1}{2{\cal
I}}({\mathfrak{s}}^{\prime}2\Pi+1)-{\mathfrak{s}}^{\prime}\frac{ev}{2}$, where
${\mathfrak{s}}^{\prime}$ is the charge of the final particle, since
$\omega_{f}\geq-{\mathfrak{s}}^{\prime}\frac{ev}{2}$ for single particle
scattering.

The effective interactions in (113) also describe scattering processes that do
not exchange charge or energy with the dyon (and so necessarily flip the $4$D
chirality). The amplitude for an incoming positively charged particle to
scatter to an outgoing particle of the same charge is

${\cal
A}\Bigl{[}f_{+}(k)+\Pi\to\Pi^{\prime}+f_{+}(k^{\prime})\Bigr{]}=-\delta_{{\Pi}^{\prime},{\Pi}}\,\delta\left(\omega_{+,k^{\prime}}-\omega_{+,k}\right)i\Bigl{[}\,\int^{\infty}_{-\infty}{\rm
d}t\,\bra{0}H_{\rm
int}\ket{0}+\left(\delta\mathcal{C}^{s}_{++}-\delta\mathcal{C}^{v}_{++}\right)\Bigr{]},$
(122)

where we omit the zeroth order term in the perturbative expansion of the
$S$-matrix. The above expression includes a contribution from the vacuum
survival amplitude, given in terms of the vacuum expectation value of $H_{\rm
int}$. Although we formally include this in amplitudes such as (122) and (123)
below, we are primarily interested in inclusive scattering processes in which
the number of pairs produced by the dyon is unmeasured, as explained in more
detail in §6. In that case, the vacuum survival amplitude, which describes a
process in which no pairs are produced, drops out of physical predictions such
as cross section results. The corresponding amplitude for a negatively charged
incoming particle is

${\cal
A}\Bigl{[}f_{-}(k)+\Pi\to\Pi^{\prime}+f_{-}(k^{\prime})\Bigr{]}=-\delta_{{\Pi}^{\prime},{\Pi}}\delta\left(\omega_{-,k^{\prime}}-\omega_{-,k}\right)i\Bigl{[}\,\int^{\infty}_{-\infty}{\rm
d}t\,\bra{0}H_{\rm
int}\ket{0}+\left(\delta\mathcal{C}^{s}_{--}-\delta\mathcal{C}^{v}_{--}\right)\Bigr{]}.$
(123)

The amplitudes for transitions between antiparticles of the same charge can be
calculated similarly.

Fermion-dyon scattering reactions are most usefully described in terms of 4D
cross sections rather than 2D scattering rates, so we pause to make the
connection to these explicit. Because incoming initial states in 4D are plane
waves far from the dyon, they are not prepared in the $S$-wave. Their
scattering rates are therefore the product of the 2D $S$-wave scattering rate
times the probability, $p_{s}$, of finding the incoming plane-wave in the
$S$-wave. The 4D cross section then is obtained by dividing by the incoming 4D
particle flux ${\mathfrak{F}}_{i}$.

Combining these factors leads to the following factorized expression for the
4D single-particle $S$-wave differential cross section ${\rm d}\sigma_{s}$
computed using the above amplitudes with ${\cal A}={\cal
M}\,\delta(E_{f}-E_{i})$:

${\rm
d}\sigma_{s}\Bigl{[}f_{{\mathfrak{s}}{\mathfrak{c}}}(k)+\Pi\to\Pi^{\prime}+f_{{\mathfrak{s}}^{\prime}{\mathfrak{c}}^{\prime}}(k^{\prime})\Bigr{]}=\frac{p_{s}}{{\mathfrak{F}}_{i}}\;{\rm
d}\Gamma_{2}=\frac{\pi}{k^{2}}\,\delta_{{\mathfrak{s}},{\mathfrak{c}}}\delta_{{\mathfrak{s}}^{\prime},-{\mathfrak{c}}^{\prime}}\,|{\cal
M}|^{2}\,\delta(E_{f}-E_{i})\,{\rm d}k^{\prime}\,,$ (124)

where the factors $p_{s}$ and ${\mathfrak{F}}_{i}$ are computed explicitly in
Appendix E, where we also show how the $2$D rates are calculated. The factor
$\delta_{{\mathfrak{s}},{\mathfrak{c}}}$ ensures the result is nonzero only
when ${\mathfrak{s}}={\mathfrak{c}}$ corresponding to the observation made
below eq. (28) that ${\mathfrak{s}}{\mathfrak{c}}=+1$ for any incoming
$S$-wave fermion. Similarly
${\mathfrak{s}}^{\prime}{\mathfrak{c}}^{\prime}=-1$ for any outgoing
fermion.202020We restore the chirality label ${\mathfrak{c}}^{\prime}$ on the
final $S$-wave state to make all the quantum numbers explicit in ${\rm
d}\sigma_{s}$ Notice the proportionality to $1/k^{2}$ ensures the cross
section scales with energy as does the unitarity bound (so long as the fermion
mass is negligible).

For instance, using the amplitudes for fermion scattering given above we find
the charge-changing cross section to be

${\rm
d}\sigma_{s}\Bigl{[}f_{++}(k)+\Pi\rightarrow\Pi^{\prime}+f_{-+}(k^{\prime})\Bigr{]}=\frac{\pi}{k^{2}}\,\delta_{\Pi^{\prime},\Pi-1}\delta\left(k^{\prime}-k+ev+\frac{1-2\Pi}{2{\cal
I}}\right)|\delta{\cal C}^{s}_{-+}-\delta{\cal C}^{v}_{-+}|^{2}\,{{\rm
d}k^{\prime}}\,.$ (125)

Integrating over the final fermion momentum and marginalizing over the
unmeasured final dyon momentum then gives the total charge-changing cross
section

$\sigma_{s}\Bigl{[}f_{++}(k)+\Pi\rightarrow(\Pi-1)+f_{-+}(k^{\prime})\Bigr{]}=\Theta\left(k-ev+\frac{2\Pi-1}{2{\cal
I}}\right)\,\frac{\pi}{k^{2}}\,|\,\delta{\cal C}^{s}_{-+}-\delta{\cal
C}^{v}_{-+}|^{2}\,.$ (126)

In the same way for the charge-changing
$f_{-\,-}(k)+\Pi\rightarrow(\Pi+1)+f_{+-}(k^{\prime})$ transition, we get

$\displaystyle\sigma_{s}\Bigl{[}f_{-\,-}(k)+\Pi\rightarrow(\Pi+1)+f_{+-}(k^{\prime})\Bigr{]}=\Theta\left(k+ev-\frac{2\Pi+1}{2{\cal
I}}\right)\frac{\pi}{k^{2}}|\delta{\cal C}^{s}_{+-}-\delta{\cal
C}^{v}_{+-}|^{2}.$ (127)

Similar expressions can be found for the cross section for processes where the
$4$D chirality of the fermions changes but not their charge (such as the
amplitudes (122) and (123)) but we do not provide them explicitly here because
they depend more sensitively on our initial choice of unperturbed boundary
conditions. Expressions for these processes are instead derived in §6 below
using the Born-Oppenheimer approximation.

###  5.2 Transitions between dyon field eigenstates

For the purposes of comparing with Born-Oppenheimer results in §6 it is worth
computing the same processes as above but this time choosing the initial and
final rotor states to be eigenstates of ${\mathfrak{a}}$ rather than
${\mathfrak{p}}$ or $\Pi$. Strictly speaking, the interaction picture operator

${\mathfrak{a}}_{\scriptscriptstyle I}(t)={\mathfrak{a}}+\frac{\Pi\,t}{{\cal
I}}\,,$ (128)

is not conserved and so its eigenstates need not agree at different times.
However because ${\cal I}^{-1}\sim\alpha\mu\ll\mu$ in the semiclassical limit,
it is a good approximation to neglect the time-dependence of ${\mathfrak{a}}$
and so amplitudes for transitions between ${\mathfrak{a}}$ eigenstates
simplify considerably, because ${\mathfrak{a}}$ is approximately conserved.

####  5.2.1 Pair production

For instance, for static ${\mathfrak{a}}$ the amplitude to produce a particle-
antiparticle pair of positive charge is found by taking the matrix element of
(111), leading to

${\cal
A}\Bigl{[}{\mathfrak{a}}\to{\mathfrak{a}}^{\prime}+f_{+}(k)+\bar{f}_{-}(k^{\prime})\Bigr{]}\simeq-\delta_{{{\mathfrak{a}}}^{\prime}{{\mathfrak{a}}}}\delta\left(\omega_{+,k}+{\overline{\omega}}_{-,k^{\prime}}\right)e^{i{\mathfrak{a}}}\,i\left(\delta\mathcal{C}^{s}_{+-}-\delta\mathcal{C}^{v}_{+-}\right).$
(129)

The corresponding amplitude for producing a negatively charged pair vanishes
in this approximation because the energy conservation condition now implies
$k+k^{\prime}+ev=0$, which is never satisfied for $k,k^{\prime}\geq 0$,
$ev>0$. As we shall see, the above amplitudes exactly match the ones listed in
§6 below, once these are evaluated in the perturbative regime.

The resulting total rate for producing fermion pairs (marginalized over their
unmeasured quantum numbers and the dyon final state) then is

$\Gamma\Bigl{[}{\mathfrak{a}}\to{\mathfrak{a}}^{\prime}+f_{+}(k)+\bar{f}_{-}(k^{\prime})\Bigr{]}\simeq\frac{ev}{2\pi}\Bigl{|}\delta{\cal
C}^{s}_{+-}-\delta{\cal C}^{v}_{+-}\Bigr{|}^{2}.$ (130)

####  5.2.2 Scattering

The $S$-wave amplitudes for charge-changing fermion scattering in the same
static-${\mathfrak{a}}$ limit are

${\cal
A}\Bigl{[}f_{+}(k)+{\mathfrak{a}}\to{\mathfrak{a}}^{\prime}+f_{-}(k^{\prime})\Bigr{]}=-\delta_{{\mathfrak{a}}^{\prime}{\mathfrak{a}}}\delta\left(\omega_{-,k^{\prime}}-\omega_{+,k}\right)e^{-i{\mathfrak{a}}}\,i\left(\delta\mathcal{C}^{s}_{-+}-\delta\mathcal{C}^{v}_{-+}\right),$
(131)

and

${\cal
A}\Bigl{[}f_{-}(k)+{\mathfrak{a}}\to{\mathfrak{a}}^{\prime}+f_{+}(k^{\prime})\Bigr{]}=-\delta_{{{\mathfrak{a}}}^{\prime}{\mathfrak{a}}}\delta\left(\omega_{+,k^{\prime}}-\omega_{-,k}\right)e^{i{\mathfrak{a}}}\,i\left(\delta\mathcal{C}^{s}_{+-}-\delta\mathcal{C}^{v}_{+-}\right).$
(132)

The corresponding total total cross section for charge-exchange processes in
which the dyon remains in an ${\mathfrak{a}}$ eigenstate then is

$\sigma_{s}\Bigl{[}f_{{\mathfrak{s}}{\mathfrak{s}}}(k)+{\mathfrak{a}}\to{\mathfrak{a}}+f_{-{\mathfrak{s}}{\mathfrak{s}}}(k^{\prime})\Bigr{]}=\Theta\left(k-{\mathfrak{s}}ev\right)\,\frac{\pi}{k^{2}}\,\Bigl{|}\delta{\cal
C}^{s}_{-{\mathfrak{s}}{\mathfrak{s}}}-\delta{\cal
C}^{v}_{-{\mathfrak{s}}{\mathfrak{s}}}\Bigr{|}^{2}\,.$ (133)

Notice that both pair production and scattering involve only a fairly simple
condition on $k$ (if any) as opposed to the fairly complicated restrictions on
$\Pi$ that arose when the dyon was prepared in a charge eigenstate. This
relative simplicity arises because the ${\mathfrak{a}}$ eigenstate always has
an overlap with $\Pi$ eigenstates for which the processes are energetically
allowed. The ${\mathfrak{a}}$-eigenstate expressions are easier to compare
with the Born-Oppenheimer, and agree within their common domain of validity
with the cross sections found in §6.

##  6 Born-Oppenheimer approximation

We next compute the pair-production and scattering implied by the full mode
functions described within the Born-Oppenheimer approximation of §3.2, for
which the fermions are quantized with the bosonic field ${\mathfrak{a}}$
initially treated as a classical field. We start in §6.1 by examining how the
fast degrees of freedom (the fermions) evolve in the presence of a static
classical rotor field ${\mathfrak{a}}$, then continue in §6.2 with some
observations about the rotor’s response.

For these purposes recall that dyon interactions with $S$-wave fermions are
described by the hamiltonian $H=H_{\rm dyon}+H_{\rm bulk}$ that is the sum of
the dyon-localized terms of (109), reproduced for convenience here,212121We
follow standard practice here and keep the rotor kinetic term despite it being
order $(e^{2}/4\pi)m_{g}$ in magnitude and so nominally being higher-loop in
size. It can make sense to do so to the extent that all of the rotor’s
responses arise at this same order (and are included).

$H_{\rm dyon}=\frac{1}{2{\cal
I}}\left[{\mathfrak{p}}(t)-\frac{\vartheta}{2\pi}\right]^{2}+e{\mathfrak{p}}\,\widehat{A}_{0}(r=\epsilon,t)+\frac{1}{2}\Bigl{[}{\overline{\bm{\chi}}}\,O_{\cal
B}({\mathfrak{a}})\,{\bm{\chi}}\Bigr{]}_{r=\epsilon,t}$ (134)

where $O_{\cal B}({\mathfrak{a}})$ is given in terms of the couplings ${\cal
C}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}^{\scriptscriptstyle A}$ by (51) and
the bulk 2D hamiltonian is

$\displaystyle H_{\rm bulk}=H_{\scriptscriptstyle
C}+\frac{1}{2}\sum_{{\mathfrak{s}}=\pm}\int^{\infty}_{\epsilon}{\rm
d}r\,{\overline{{\chi}}}_{{\mathfrak{s}}}\left[\Gamma^{1}\overset{\leftrightarrow}{\partial}_{1}-i{\mathfrak{s}}\,\Gamma^{0}\left(ev-\frac{eQ}{r}+e\widehat{A}_{0}\right)\right]\chi_{{\mathfrak{s}}}\,,$
(135)

with $H_{\scriptscriptstyle C}$ denoting the part of the Maxwell action
depending on the Coulomb field $\widehat{A}_{0}$. We follow previous work and
perturb in the Coulomb interactions involving $\widehat{A}_{0}$, whose
contributions to the energy are suppressed by powers of $e^{2}$. We differ
from the warm-up calculations of §5 by not splitting $O_{\cal
B}({\mathfrak{a}})$ into an unperturbed and perturbed piece; instead treating
the interaction with ${\mathfrak{a}}$ using Born-Oppenheimer methods (as
motivated in §3.3.2). In practice this means that we treat ${\mathfrak{a}}$ as
a fixed classical field when determining the fermionic response and then
return to ask how this slower rotor field evolves in response to interactions
with the faster fermions.

###  6.1 Fermion evolution

We start by calculating the Bogoliubov coefficients that relate $in$ and $out$
modes, neglecting the Coulomb back-reaction of the distortions of the fermion
ground state. Subsequent subsections then consider some of the implications of
this fermionic distortion such as to pair-production rates and scattering
cross sections.

####  6.1.1 Bogoliubov relations

The starting point expands the $S$-wave fermion field in terms of the $in$ and
$out$ bases for the fermion modes in the presence of a dyon background
described in §3.2:

$\displaystyle\bm{\chi}(x)$ $\displaystyle=$
$\displaystyle\sum_{{\mathfrak{s}}=\pm}\int^{\infty}_{0}\frac{{\rm
d}k}{\sqrt{2\pi}}\,\Bigl{[}{\mathfrak{u}}^{\rm
in}_{{\mathfrak{s}},k}(x)\,a^{\rm in}_{{\mathfrak{s}},k}+{\mathfrak{v}}^{\rm
in}_{{\mathfrak{s}},k}(x)\,({\overline{a}}^{\rm
in}_{{\mathfrak{s}},k})^{\star}\Bigr{]}$ (136) $\displaystyle=$
$\displaystyle\sum_{{\mathfrak{s}}=\pm}\int^{\infty}_{0}\frac{{\rm
d}k}{\sqrt{2\pi}}\,\Bigl{[}{\mathfrak{u}}^{\rm
out}_{{\mathfrak{s}},k}(x)\,a^{\rm out}_{{\mathfrak{s}},k}+{\mathfrak{v}}^{\rm
out}_{{\mathfrak{s}},k}(x)\,({\overline{a}}^{\rm
out}_{{\mathfrak{s}},k})^{\star}\Bigr{]}$

with mode functions defined in eqs. (67) through (70) and (77) through (80)
and particle and antiparticle creation and annihilation operators satisfying

$\displaystyle\Bigl{\\{}{a}^{\rm in}_{{\mathfrak{s}},k},({a}^{\rm
in}_{{\mathfrak{s}}^{\prime},k^{\prime}})^{\star}\Bigr{\\}}$ $\displaystyle=$
$\displaystyle\Bigl{\\{}{\overline{{a}}}^{\rm
in}_{{\mathfrak{s}},k},({\overline{{a}}}^{\rm
in}_{{\mathfrak{s}}^{\prime},k^{\prime}})^{\star}\Bigr{\\}}=\delta(k-k^{\prime})\,\delta_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}},$
$\displaystyle\hbox{and}\quad\Bigl{\\{}{a}^{\rm
out}_{{\mathfrak{s}},k},({a}^{\rm
out}_{{\mathfrak{s}}^{\prime},k^{\prime}})^{\star}\Bigr{\\}}$ $\displaystyle=$
$\displaystyle\Bigl{\\{}{\overline{{a}}}^{\rm
out}_{{\mathfrak{s}},k},({\overline{{a}}}^{\rm
out}_{{\mathfrak{s}}^{\prime},k^{\prime}})^{\star}\Bigr{\\}}=\delta(k-k^{\prime})\,\delta_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}},$
(137)

with all other anticommutators vanishing.

Since both the $in$ and $out$ basis are complete, each can be expanded in
terms of the other. For each $k\geq 0$ we have

$\displaystyle{\mathfrak{u}}^{\rm in}_{-,k}(x)$ $\displaystyle=$
$\displaystyle\mathcal{T}^{--}_{{\rm
in}}e^{-2ik\epsilon}\left(\frac{\epsilon}{r_{0}}\right)^{-ieQ}\,{\mathfrak{u}}^{\rm
out}_{-,k}(x)+{\cal T}^{+-}_{{\rm
in}}e^{-i(2k+ev)\epsilon}\,e^{i{\mathfrak{a}}}\,{\mathfrak{u}}^{\rm
out}_{+,k+ev}(x)$ $\displaystyle{\mathfrak{v}}^{\rm in}_{+,k}(x)$
$\displaystyle=$ $\displaystyle\mathcal{T}^{++}_{{\rm
in}}e^{2ik\epsilon}\left(\frac{\epsilon}{r_{0}}\right)^{ieQ}\,{\mathfrak{v}}^{\rm
out}_{+,k}(x)+{\cal T}^{-+}_{{\rm
in}}e^{i(2k+ev)\epsilon}\,e^{-i{\mathfrak{a}}}\,{\mathfrak{v}}^{\rm
out}_{-,k+ev}(x)$ $\displaystyle{\mathfrak{u}}^{\rm in}_{+,k}(x)$
$\displaystyle=$ $\displaystyle\mathcal{T}^{++}_{{\rm
in}}e^{-2ik\epsilon}\left(\frac{\epsilon}{r_{0}}\right)^{ieQ}\,{\mathfrak{u}}^{\rm
out}_{+,k}(x)$ $\displaystyle\qquad+{\cal T}^{-+}_{{\rm
in}}e^{-i(2k-ev)\epsilon}\,e^{-i{\mathfrak{a}}}\Bigl{[}\Theta\left(k-ev\right){\mathfrak{u}}^{\rm
out}_{-,k-ev}(x)+\Theta\left(-k+ev\right){\mathfrak{v}}^{\rm
out}_{-,-k+ev}(x)\Bigr{]}$ $\displaystyle{\mathfrak{v}}^{\rm in}_{-,k}(x)$
$\displaystyle=$ $\displaystyle\mathcal{T}^{--}_{{\rm
in}}e^{2ik\epsilon}\left(\frac{\epsilon}{r_{0}}\right)^{-ieQ}\,{\mathfrak{v}}^{\rm
out}_{-,k}(x)$ $\displaystyle\qquad+{\cal T}^{+-}_{{\rm
in}}e^{i(2k-ev)\epsilon}\,e^{i{\mathfrak{a}}}\Bigl{[}\Theta\left(k-ev\right){\mathfrak{v}}^{\rm
out}_{+,k-ev}(x)+\Theta\left(-k+ev\right){\mathfrak{u}}^{\rm
out}_{+,-k+ev}(x)\Bigr{]},$

and similarly

$\displaystyle{\mathfrak{u}}^{\rm out}_{-,k}(x)$ $\displaystyle=$
$\displaystyle\mathcal{T}^{--}_{{\rm
out}}e^{2ik\epsilon}\left(\frac{\epsilon}{r_{0}}\right)^{ieQ}\,{\mathfrak{u}}^{\rm
in}_{-,k}(x)+{\cal T}^{+-}_{{\rm
out}}e^{i(2k+ev)\epsilon}\,e^{i{\mathfrak{a}}}\,{\mathfrak{u}}^{\rm
in}_{+,k+ev}(x)$ $\displaystyle{\mathfrak{v}}^{\rm out}_{+,k}(x)$
$\displaystyle=$ $\displaystyle\mathcal{T}^{++}_{{\rm
out}}e^{-2ik\epsilon}\left(\frac{\epsilon}{r_{0}}\right)^{-ieQ}\,{\mathfrak{v}}^{\rm
in}_{+,k}(x)+{\cal T}^{-+}_{{\rm
out}}e^{-i(2k+ev)\epsilon}\,e^{-i{\mathfrak{a}}}\,{\mathfrak{v}}^{\rm
in}_{-,k+ev}(x)$ $\displaystyle{\mathfrak{u}}^{\rm out}_{+,k}(x)$
$\displaystyle=$ $\displaystyle\mathcal{T}^{++}_{{\rm
out}}e^{2ik\epsilon}\left(\frac{\epsilon}{r_{0}}\right)^{-ieQ}\,{\mathfrak{u}}^{\rm
in}_{+,k}(x)$ $\displaystyle\qquad+{\cal T}^{-+}_{{\rm
out}}e^{i(2k-ev)\epsilon}\,e^{-i{\mathfrak{a}}}\Bigl{[}\Theta\left(k-ev\right){\mathfrak{u}}^{\rm
in}_{-,k-ev}(x)+\Theta\left(-k+ev\right){\mathfrak{v}}^{\rm
in}_{-,-k+ev}(x)\Bigr{]}$ $\displaystyle{\mathfrak{v}}^{\rm out}_{-,k}(x)$
$\displaystyle=$ $\displaystyle\mathcal{T}^{--}_{{\rm
out}}e^{-2ik\epsilon}\left(\frac{\epsilon}{r_{0}}\right)^{ieQ}\,{\mathfrak{v}}^{\rm
in}_{-,k}(x)$ $\displaystyle\qquad+{\cal T}^{+-}_{{\rm
out}}e^{-i(2k-ev)\epsilon}\,e^{i{\mathfrak{a}}}\Bigl{[}\Theta\left(k-ev\right){\mathfrak{v}}^{\rm
in}_{+,k-ev}(x)+\Theta\left(-k+ev\right){\mathfrak{u}}^{\rm
in}_{+,-k+ev}(x)\Bigr{]}.$

These lead to the following Bogoliubov relations between the $in$ and $out$
operators for each $k\geq 0$ (see appendix E for derivation)

$\displaystyle a^{\rm out}_{-,k}=\mathcal{T}^{--}_{{\rm in}}\;a^{\rm
in}_{-,k}+{\cal T}^{-+}_{{\rm in}}\,e^{-i{\mathfrak{a}}}\;a^{\rm
in}_{+,k+ev}\,,\qquad({\overline{a}}^{\rm
out}_{+,k})^{\star}=\mathcal{T}^{++}_{{\rm in}}\;({\overline{a}}^{\rm
in}_{+,k})^{\star}+{\cal T}^{+-}_{{\rm
in}}\,e^{i{\mathfrak{a}}}\;({\overline{a}}^{\rm in}_{-,k+ev})^{\star}$
$\displaystyle\qquad a^{\rm out}_{+,k}=\mathcal{T}^{++}_{{\rm in}}a^{\rm
in}_{+,k}+{\cal T}^{+-}_{{\rm
in}}\,e^{i{\mathfrak{a}}}\Bigl{[}\Theta\left(k-ev\right)a^{\rm
in}_{-,k-ev}+\Theta\left(-k+ev\right)({\overline{a}}^{\rm
in}_{-,-k+ev})^{\star}\Bigr{]}$ (140) $\displaystyle\quad({\overline{a}}^{\rm
out}_{-,k})^{\star}=\mathcal{T}^{--}_{{\rm in}}({\overline{a}}^{\rm
in}_{-,k})^{\star}+{\cal T}^{-+}_{{\rm
in}}\,e^{-i{\mathfrak{a}}}\Bigl{[}\Theta\left(k-ev\right)({\overline{a}}^{\rm
in}_{+,k-ev})^{\star}+\Theta\left(-k+ev\right)a^{\rm in}_{+,-k+ev}\Bigr{]},$

as well as their inverses

$\displaystyle a^{\rm in}_{-,k}=\mathcal{T}^{--}_{{\rm out}}\;a^{\rm
out}_{-,k}+{\cal T}^{-+}_{{\rm out}}\,e^{-i{\mathfrak{a}}}\;a^{\rm
out}_{+,k+ev}\,,\qquad({\overline{a}}^{\rm
in}_{+,k})^{\star}=\mathcal{T}^{++}_{{\rm out}}\;({\overline{a}}^{\rm
out}_{+,k})^{\star}+{\cal T}^{+-}_{{\rm
out}}\,e^{i{\mathfrak{a}}}\;({\overline{a}}^{\rm out}_{-,k+ev})^{\star}$
$\displaystyle\qquad a^{\rm in}_{+,k}=\mathcal{T}^{++}_{{\rm out}}a^{\rm
out}_{+,k}+{\cal T}^{+-}_{{\rm
out}}\,e^{i{\mathfrak{a}}}\Bigl{[}\Theta\left(k-ev\right)a^{\rm
out}_{-,k-ev}+\Theta\left(-k+ev\right)({\overline{a}}^{\rm
out}_{-,-k+ev})^{\star}\Bigr{]}$ (141) $\displaystyle({\overline{a}}^{\rm
in}_{-,k})^{\star}=\mathcal{T}^{--}_{{\rm out}}({\overline{a}}^{\rm
out}_{-,k})^{\star}+{\cal T}^{-+}_{{\rm
out}}\,e^{-i{\mathfrak{a}}}\Bigl{[}\Theta\left(k-ev\right)({\overline{a}}^{\rm
out}_{+,k-ev})^{\star}+\Theta\left(-k+ev\right)a^{\rm out}_{+,-k+ev}\Bigr{]}.$

These are consistent with the anticommutation relations (6.1.1) by virtue of
the unitarity identities (73) and (74) – and their counterparts (83) and (84)
– satsified by the ${\cal T}$’s. As in §5, we drop powers of $k\epsilon$,
$k^{\prime}\epsilon$ and $ev\epsilon$ and shift ${\mathfrak{a}}$ and rephase
the out-state creation and-annihilation operators to remove an
$r_{0}$-dependent phase.

####  6.1.2 Pair production

The mixing of creation and annihilation operators in the Bogoliubov
transformations (6.1.1) and (6.1.1) shows that the system is unstable to pair
production if ${\cal T}_{{\rm in}/{\rm out}}^{+-}$ is nonzero.

To compute the pair-production rate define the $in$ and $out$ vacua
$\ket{0_{{\rm in}}}$ and $\ket{0_{{\rm out}}}$ as usual:

$\bm{a}^{\rm in}_{{\mathfrak{s}},k}\ket{0_{{\rm in}}}={\overline{\bm{a}}}^{\rm
in}_{{\mathfrak{s}},k}\ket{0_{{\rm in}}}=0\quad\hbox{and}\quad\bm{a}^{\rm
out}_{{\mathfrak{s}},k}\ket{0_{{\rm out}}}={\overline{\bm{a}}}^{\rm
out}_{{\mathfrak{s}},k}\ket{0_{{\rm out}}}=0\,.$ (142)

We switch here for convenience to discretely normalized momentum states, for
which we denote the creation and annihilation operators using bold-faced
fonts, as in ($\bm{a},{\overline{\bm{a}}}$). (See appendix E for relation
between discrete and continuum normalized states.)

Writing the total vacuum as a tensor product over momenta,
$\ket{0}=\underset{k>0}{\prod}\ket{0^{\,k}}$, and using the above Bogoliubov
transformations implies for each $k\geq 0$ we have

$\ket{{0}^{\,{k}}_{{\rm
in}}}=\left\\{\Theta\left({k}-ev\right)+\Theta\left(-{k}+ev\right)\left[\mathcal{T}^{--}_{{\rm
in}}+{\cal T}^{+-}_{{\rm in}}e^{i{\mathfrak{a}}}(\bm{a}^{{\rm
out}}_{+,{k}})^{\star}(\bm{\overline{a}}^{{\rm
out}}_{-,{-k+ev}})^{\star}\right]\right\\}\ket{{0}^{\,{k}}_{{\rm out}}}$ (143)

and

$\ket{{0}^{\,{k}}_{{\rm
out}}}=\left\\{\Theta\left({k}-ev\right)+\Theta\left(-{k}+ev\right)\left[\mathcal{T}^{--\,*}_{{\rm
in}}+{\cal T}^{-+\,*}_{{\rm in}}e^{i{\mathfrak{a}}}(\bm{a}^{{\rm
in}}_{+,{k}})^{\star}(\bm{\overline{a}}^{{\rm
in}}_{-,{-k+ev}})^{\star}\right]\right\\}\ket{{0}^{\,{k}}_{{\rm in}}}\,,$
(144)

where as usual $\Theta(x)$ denotes the Heaviside step function. See appendix F
for derivation of the above expressions.

If ${\mathcal{T}}^{+-}=0$ then (73) shows that ${{\cal T}}^{-+}$ also vanishes
and $|{\cal T}^{--}|=|{\cal T}^{++}|=1$. In this case (143) and (144) imply
the $in$ and $out$ vacua are the same state. But when ${\mathcal{T}}^{+-}\neq
0$ for some momenta the $in$ vacuum contains occupied $out$ particles. This
shows that when charge-changing effective dyon-fermion interactions are
present the dyon spontaneously emits a fermion with charge $+\frac{1}{2}e$ and
the antiparticle of the charge $-\frac{1}{2}e$ state, ensuring the net charge
emission of $\frac{1}{2}e+\frac{1}{2}e=e$.

Total electric charge is conserved because this emission is accompanied by a
transition between rotor levels for ${\mathfrak{a}}$ that removes one unit
$+e$ of charge from the dyon, as can be seen222222This is made very explicit
in §5 for those who need convincing. from the ${\mathfrak{a}}$-dependence of
(145). The sign of the charge removed is dictated by the overall sign of the
dyon charge $Q$ which we’ve chosen to be positive: $Q>0$. These pairs are
produced through the Schwinger effect Schwinger2D ; Blaer:1981ps  by the
external voltage $v$ of the dyon between $r\to\infty$ and $r=0$, and act to
discharge the dyon’s net charge.

The energetics of the process is somewhat obscured within the Born-Oppenheimer
approximation because the rotor is treated as a classical field, which if
regarded as an eigenstate of ${\mathfrak{a}}$ is not an energy eigenstate.
When calculated perturbatively in §5 we do find that fermion emission extracts
an energy given by the spacing between rotor steps, which is of order ${\cal
I}^{-1}\sim\alpha m_{g}$. But in a semiclassical approximation this energy
transfer is the same size (relative to $m_{g}$) as loop corrections, which to
this order we neglect, and this is why (145) allows pair-production for the
entire momentum range $k\in(0,ev)$. Implicit in this treatment is the
assumption that the fermion energies of interest are much higher than the
rotor gap; one manifestation of the noncommuting order of limits discussed
below eq. (21).

The amplitude for this emission (for fixed ${\mathfrak{a}}$) in a specific
momentum mode is

$\bra{0^{\,k}_{{\rm out}}}\,\bm{{\overline{a}}}^{\rm
out}_{-,-k+ev}\,\bm{a}^{\rm out}_{+,k}\ket{0^{\,k}_{\rm
in}}=\Theta(-k+ev)\,{\cal T}^{+-}_{{\rm in}}\,e^{i{\mathfrak{a}}}\,,$ (145)

which has squared modulus ${\cal P}_{k}=|{\cal T}^{+-}_{\rm in}|^{2}=|{\cal
T}^{+-}_{\rm out}|^{2}$ when $0<k<ev$. The vacuum-survival amplitude for this
specific mode is similarly

$\braket{0^{\,k}_{{\rm out}}}{{0}^{\,{k}}_{{\rm
in}}}=\Theta\left({k}-ev\right)+\Theta\left(-{k}+ev\right)\mathcal{T}^{--}_{{\rm
in}}\,,$ (146)

which squares to unity when $k>ev$ but has squared modulus
$|\mathcal{T}^{--}_{{\rm in}}|^{2}=1-|\mathcal{T}^{+-}_{{\rm in}}|^{2}=1-{\cal
P}_{k}$ when $0<k<ev$. The likelihood of producing zero or one pairs sums to
unity because for fermions these are the only possible options.

The exclusive probability for the full vacuum to produce exactly one pair in a
specific mode $k\in(0,ev)$ is given by combining the above result for all $k$,
giving

${P}_{\text{pair}}(k)=|\bra{0_{{\rm out}}}\bm{{\overline{a}}}^{\rm
out}_{-,-k+ev}\,\bm{a}^{\rm out}_{+,k}\ket{0_{{\rm in}}}|^{2}={\cal
P}_{k}\overset{q<ev}{\prod_{q\neq k}}(1-{\cal P}_{q})=|{\cal T}^{+-}_{\rm
in}(k)|^{2}\overset{q<ev}{\prod_{q\neq k}}|\mathcal{T}^{--}_{{\rm
in}}(q)|^{2}.$ (147)

The total probability factorizes because the likelihood of pair-production in
each mode is independent of what happens for the other modes. More useful is
the inclusive probability for pair production of a specific mode with the
other modes unmeasured (and so marginalized over). The above expressions show
this is given by

$p_{\text{pair}}(k)={\cal P}_{k}=|{\cal T}^{+-}_{{\rm in}}|^{2}\quad\hbox{and
so}\quad p_{\rm no\,pair}(k)=1-{\cal P}_{k}=|\mathcal{T}^{--}_{{\rm
in}}|^{2}\,.$ (148)

We can calculate the average number of particles produced by making use of the
Bogoliubov relations in the form

$\displaystyle\bra{0_{{\rm in}}}N_{\rm out}\ket{0_{{\rm in}}}$
$\displaystyle=$ $\displaystyle\sum^{\infty}_{k=0}\bra{0_{{\rm
in}}}(\bm{a}^{{\rm out}}_{+k})^{\star}\,\bm{a}^{{\rm out}}_{+k}\ket{0_{{\rm
in}}}$ (149) $\displaystyle=$ $\displaystyle\sum^{ev}_{k=0}|{\cal
T}^{+-}_{{\rm in}}|^{2}\bra{0_{{\rm in}}}\overline{\bm{a}}^{{\rm
in}}_{-,-k+ev}(\overline{\bm{a}}^{{\rm in}}_{-,-k+ev})^{\star}\ket{0_{{\rm
in}}}=\frac{ev}{\Delta k}|{\cal T}^{+-}_{{\rm in}}|^{2},$

where $\Delta k=\pi/L$ is the spacing between momentum states (when these are
discretely normalized).232323We discretize momenta by putting the system in a
box $-L<r<L$ with near-dyon boundary conditions imposed at $r=\epsilon\approx
0$. In these conventions the relevant length of the system is $2L$, and the
density of states is $2\pi/(2L)=\pi/L$ The number of particles per unit length
is obtained by dividing by $2L$, and this has a sensible continuum limit as
$L\to\infty$, with ${\rm d}\langle N\rangle/{\rm d}x=ev|{\cal T}_{\rm
in}^{+-}|^{2}/(2\pi)$. Since each produced particle moves to larger $r$ at the
speed of light this means that these particles emerge at infinity with a rate
$\Gamma_{\infty}=ev|{\cal T}_{\rm in}^{+-}|^{2}/(2\pi)$. A similar counting
also applies to the number of produced antiparticles. Because each pair
contains one positively charged particle and one positively charged
antiparticle the number of produced pairs is also

$\Gamma_{\rm pair}=\frac{ev}{2\pi}|{\cal T}_{\rm in}^{+-}|^{2}\,.$ (150)

This result is independently computed below using the expectation values for
the fermionic currents.

Eq. (150) can also be compared with (130) (when restricted to the perturbative
domain). To this end we must expand the coefficients ${\cal T}_{\rm in}$
perturbatively to the same order in the $\delta{\cal C}$s used in §5. To
linear order the ${\cal T}_{{\rm in}}$ amplitudes are given by

${\cal T}^{{\mathfrak{s}}{\mathfrak{s}}}_{{\rm in}}=1+i(\delta{\cal
C}^{v}_{{\mathfrak{s}}{\mathfrak{s}}}-\delta{\cal
C}^{s}_{{\mathfrak{s}}{\mathfrak{s}}})\quad\text{and}\quad{\cal T}^{+-}_{{\rm
in}}=-{\cal T}^{-+\,*}_{{\rm in}}=i(\delta{\cal C}^{v}_{+-}-\delta{\cal
C}^{s}_{+-}),$ (151)

once the rank-2 conditions $\delta{\cal
C}^{ps}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}=\delta{\cal
C}^{pv}_{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}=0$ are used. Using these in
(150) then agrees with (130).

####  6.1.3 Vacuum currents

An alternative characterization of dyonic pair production that lends itself to
the continuum limit is the integrated contribution of the produced pairs to
current flow in the fermion sector. To display these currents we evaluate the
expectation value of the various conserved currents in the $in$ vacuum. Since
these expectation values in general diverge we regulate them by point-
splitting the two fermion fields involved by a distance $\varepsilon$, writing

$\overline{\bm{\chi}}(r,t)M\bm{\chi}(r,t)\to\overline{\bm{\chi}}(r+\varepsilon/2,t)M\bm{\chi}(r-\varepsilon/2,t)\,,$
(152)

with $\varepsilon\to 0$ taken at the end, after renormalizing. We quote here
expressions for the regularized currents – see appendix G for details of the
matrix-element calculations.

The regularized components of the fermion number current242424We do not go
through the exercise of renormalizing the fermion number current here, since
we only use $\bra{0_{{\rm in}}}{j}^{1}_{\scriptscriptstyle B}(x)\ket{0_{{\rm
in}}}$ to evaluate the conservation equations for
$j^{\alpha}_{\scriptscriptstyle F},j^{\alpha}_{\scriptscriptstyle A}$.
$j_{\scriptscriptstyle
B}^{\alpha}=i{\overline{{\bm{\chi}}}}\Gamma^{\alpha}{\bm{\chi}}$ are given by

$\bra{0_{{\rm in}}}{j}^{0}_{\scriptscriptstyle B}(x)\ket{0_{{\rm
in}}}=\bra{0_{{\rm out}}}{j}^{0}_{\scriptscriptstyle B}(x)\ket{0_{{\rm
out}}}=0\quad\hbox{and}\quad\bra{0_{{\rm in}}}{j}^{1}_{\scriptscriptstyle
B}(x)\ket{0_{{\rm in}}}=\bra{0_{{\rm out}}}{j}^{1}_{\scriptscriptstyle
B}(x)\ket{0_{{\rm out}}}=\frac{2i}{\pi\varepsilon},$ (153)

while those of the fermionic electromagnetic current
$j^{\alpha}_{\scriptscriptstyle
F}=\frac{1}{2}ie{\overline{{\bm{\chi}}}}\Gamma^{\alpha}\tau_{3}{\bm{\chi}}$
are

$\displaystyle\bra{0_{{\rm in}}}{j}^{0}_{\scriptscriptstyle
F}(r,t)\ket{0_{{\rm in}}}$ $\displaystyle=$ $\displaystyle\bra{0_{{\rm
out}}}{j}^{0}_{\scriptscriptstyle F}(r,t)\ket{0_{{\rm
out}}}=\frac{e^{2}v}{2\pi}|\mathcal{T}^{+-}_{{\rm in}}|^{2}-\frac{e^{2}Q}{2\pi
r}$ $\displaystyle\hbox{and}\quad\bra{0_{{\rm in}}}{j}^{1}_{\scriptscriptstyle
F}(r,t)\ket{0_{{\rm in}}}$ $\displaystyle=$ $\displaystyle-\bra{0_{{\rm
out}}}{j}^{1}_{\scriptscriptstyle F}(r,t)\ket{0_{{\rm
out}}}=\frac{e^{2}v}{2\pi}|\mathcal{T}^{+-}_{{\rm in}}|^{2}\,.$ (154)

We define the axial current by $j^{\alpha}_{\scriptscriptstyle
A}=i{\overline{{\bm{\chi}}}}\Gamma^{\alpha}\Gamma_{\scriptscriptstyle
A}{\bm{\chi}}$ with $\Gamma_{\scriptscriptstyle
A}\coloneqq\Gamma_{c}\,\tau_{3}$ rather than $\Gamma_{c}$ because this agrees
with the 4D axial current
$i{\overline{{\bm{\psi}}}}\gamma^{\mu}\gamma_{5}{\bm{\psi}}$ for $S$-wave
states (up to the usual 2D normalization factor of $4\pi r^{2}$). Its vacuum
matrix elements are

$\displaystyle\bra{0_{{\rm in}}}{j}^{0}_{\scriptscriptstyle
A}(r,t)\ket{0_{{\rm in}}}$ $\displaystyle=$ $\displaystyle-\bra{0_{{\rm
out}}}{j}^{0}_{\scriptscriptstyle A}(r,t)\ket{0_{{\rm
out}}}=-\frac{ev}{\pi}|\mathcal{T}^{+-}_{{\rm in}}|^{2}$
$\displaystyle\hbox{and}\quad\bra{0_{{\rm in}}}{j}^{1}_{\scriptscriptstyle
A}(r,t)\ket{0_{{\rm in}}}$ $\displaystyle=$ $\displaystyle\bra{0_{{\rm
out}}}{j}^{1}_{\scriptscriptstyle A}(r,t)\ket{0_{{\rm
out}}}=-\frac{ev}{\pi}|\mathcal{T}^{+-}_{{\rm in}}|^{2}+\frac{eQ}{\pi r}\,.$
(155)

Recalling that the 2D modes are normalized so that the 2D flux $j^{1}$ gives
the integrated radial flux $4\pi r^{2}J^{r}$ for the corresponding 4D $S$-wave
current, we see that at spatial infinity there is a nonzero flux of both
electric and axial charge:

$\bra{0_{{\rm in}}}{j}^{1}_{\scriptscriptstyle F}(\infty,t)\ket{0_{{\rm
in}}}=e{\mathfrak{F}}\quad\hbox{and}\quad\bra{0_{{\rm
in}}}{j}^{1}_{\scriptscriptstyle A}(\infty,t)\ket{0_{{\rm
in}}}=-2{\mathfrak{F}}\quad\hbox{where}\quad{\mathfrak{F}}=\frac{ev}{2\pi}|{\cal
T}_{\rm in}^{+-}|^{2}\,.$ (156)

This has a simple interpretation as a flux of pair-produced particles, since
each such pair carries electric charge $e$ and axial charge252525See appendix
E for discussion of asymptotic charges of $in,out$ states. $-2$ (and no net
fermion number), with integrated particle flux (or rate with which particle
pairs appear at infinity) given by $\Gamma_{\rm pair}={\mathfrak{F}}$, in
agreement with (150). Our result for the flux of the axial current is
consistent with Blaer:1981ps , when the ${\cal T}$ amplitudes are chosen to
match theirs262626Note that the axial current in Blaer:1981ps  has a relative
minus sign compared to our definition. i.e. when $|{\cal T}^{+-}|=1$.

These expressions are also consistent with (anomalous) current conservation.
As shown in appendix G, the above currents satisfy

$\displaystyle\partial_{\alpha}j^{\alpha}_{\scriptscriptstyle B}(x)$
$\displaystyle=$
$\displaystyle\frac{eQ}{2r^{2}}\underset{\varepsilon\rightarrow
0}{\lim}\Bigl{[}\varepsilon\,\overline{\bm{\chi}}(r+\varepsilon/2,t)\Gamma^{0}\,\tau_{3}\bm{\chi}(r-\varepsilon/2,t)\Bigr{]}=-\frac{iQ}{r^{2}}\underset{\varepsilon\rightarrow
0}{\lim}\Bigl{[}\varepsilon j^{0}_{\scriptscriptstyle F}(x)\Bigr{]}=0,$
$\displaystyle\partial_{\alpha}j^{\alpha}_{\scriptscriptstyle F}(x)$
$\displaystyle=$
$\displaystyle\frac{e^{2}Q}{4r^{2}}\underset{\varepsilon\rightarrow
0}{\lim}\Bigl{[}\varepsilon\,\overline{\bm{\chi}}(r+\varepsilon/2,t)\Gamma^{0}\bm{\chi}(r-\varepsilon/2,t)\Bigr{]}=-\frac{ie^{2}Q}{4r^{2}}\underset{\varepsilon\rightarrow
0}{\lim}\Bigl{[}\varepsilon j^{0}_{\scriptscriptstyle B}(x)\Bigr{]}=0,$
$\displaystyle\partial_{\alpha}j^{\alpha}_{\scriptscriptstyle A}(x)$
$\displaystyle=$
$\displaystyle-\frac{eQ}{2r^{2}}\underset{\varepsilon\rightarrow
0}{\lim}\Bigl{[}\varepsilon\,\overline{\bm{\chi}}(r+\varepsilon/2,t)\Gamma^{1}\bm{\chi}(r-\varepsilon/2,t)\Bigr{]}=\frac{ieQ}{2r^{2}}\underset{\varepsilon\rightarrow
0}{\lim}\Bigl{[}\varepsilon j^{1}_{\scriptscriptstyle
B}(x)\Bigr{]}=-\frac{eQ}{\pi r^{2}}\,.$ (157)

These agree with the standard 2D anomaly expressions, which in the present
instance tell us that $j^{\alpha}_{\scriptscriptstyle B}$ and
$j^{\alpha}_{\scriptscriptstyle F}$ are anomaly free and give

$\partial_{\alpha}j^{\alpha}_{\scriptscriptstyle
A}=\left[\frac{e}{2}-\left(-\frac{e}{2}\right)\right]\frac{1}{2\pi}\epsilon^{\alpha\beta}{\cal
F}_{\alpha\beta}=\frac{e}{\pi}\Bigl{[}\partial_{0}\mathcal{A}_{r}-\partial_{r}\mathcal{A}_{0}\Bigr{]}=\frac{1}{\pi}\partial_{r}\left(\frac{eQ}{r}-ev\right)=-\frac{eQ}{\pi
r^{2}},$ (158)

when evaluated with the background dyonic Coulomb field.

It is also noteworthy that (6.1.3) implies that particle production
significantly polarizes the fermionic ground state, inducing a charge density
with the opposite sign to the dyon charge that (for massless fermions) falls
off only as a power law as one moves away from the dyon. As has been remarked
elsewhere Preskill:1984gd ; Ezawa:1983vi ; CallanSMatrix ; Yamagishi:1982wp ;
Grossman:1983yf ; Kazama:1983rt , such charging of the fermion ground state
puffs up the dyon into a much bigger object than was the underlying classical
dyon configuration.

####  6.1.4 Dyon-fermion scattering

The Bogoliubov relations in the 2D EFT also allow us to calculate the cross
section for any $S$-wave fermion-dyon scattering process, by evaluating
amplitudes of the form $\langle A_{\rm out}|B_{\rm in}\rangle$. The Bogoliubov
transformation provides a succinct listing of the options for $\bra{A_{\rm
out}}$ that give nonzero amplitudes given a single-particle $\ket{B_{\rm in}}$
that can be seen by expanding the incoming state $(a^{{\rm
in}}_{\mathfrak{s},k})^{\star}$ in terms of $out$ operators – see (6.1.1). The
presence of pair-production means that any such process can be accompanied by
some number of spontaneously produced pairs.

For instance the $4$D chirality-changing amplitude for $f_{-}(k)\to
f_{-}(k^{\prime})$ accompanied by the emission of $n$ pairs is given by

$\displaystyle\bra{0_{{\rm out}}}\overline{a}^{{\rm
out}}_{-,-q_{n}+ev}{a}^{{\rm out}}_{+,q_{n}}...\overline{a}^{{\rm
out}}_{-,-q_{1}+ev}{a}^{{\rm out}}_{+,q_{1}}\,{a}^{{\rm
out}}_{-,k^{\prime}}({a}^{{\rm in}}_{-,k})^{\star}\ket{0_{{\rm in}}}$ (159)
$\displaystyle\qquad=\delta({k-k^{\prime}})\mathcal{T}^{--}_{{\rm
in}}\bra{0_{{\rm out}}}\overline{a}^{{\rm out}}_{-,-q_{n}+ev}{a}^{{\rm
out}}_{+,q_{n}}...\overline{a}^{{\rm out}}_{-,-q_{1}+ev}{a}^{{\rm
out}}_{+,q_{1}}\ket{0_{{\rm in}}},$

where $n=0,1,2,\cdots$. The amplitude for charge-changing processes like
$f_{\pm}(k)\to f_{\mp}(k^{\prime})$ (accompanied by $n$ spontaneously produced
pairs) is similarly

$\displaystyle\bra{0_{{\rm out}}}\overline{a}^{{\rm
out}}_{-,-q_{n}+ev}{a}^{{\rm out}}_{+,q_{n}}...\overline{a}^{{\rm
out}}_{-,-q_{1}+ev}{a}^{{\rm out}}_{+,q_{1}}\,{a}^{{\rm
out}}_{-,k^{\prime}}({a}^{{\rm in}}_{+,k})^{\star}\ket{0_{{\rm in}}}$ (160)
$\displaystyle\qquad=\delta(k^{\prime}+ev-k)e^{-i{\mathfrak{a}}}{\cal
T}^{-+}_{{\rm in}}\bra{0_{{\rm out}}}\overline{a}^{{\rm
out}}_{-,-q_{n}+ev}{a}^{{\rm out}}_{+,q_{n}}...\overline{a}^{{\rm
out}}_{-,-q_{1}+ev}{a}^{{\rm out}}_{+,q_{1}}\ket{0_{{\rm in}}},$

and

$\displaystyle\bra{0_{{\rm out}}}\overline{a}^{{\rm
out}}_{-,-q_{n}+ev}{a}^{{\rm out}}_{+,q_{n}}...\overline{a}^{{\rm
out}}_{-,-q_{1}+ev}{a}^{{\rm out}}_{+,q_{1}}\,{a}^{{\rm
out}}_{+,k^{\prime}}({a}^{{\rm in}}_{-,k})^{\star}\ket{0_{{\rm in}}}$ (161)
$\displaystyle\qquad=\delta({k^{\prime}-ev-k})e^{i{\mathfrak{a}}}{\cal
T}^{+-}_{{\rm in}}\bra{0_{{\rm out}}}\overline{a}^{{\rm
out}}_{-,-q_{n}+ev}{a}^{{\rm out}}_{+,q_{n}}...\overline{a}^{{\rm
out}}_{-,-q_{1}+ev}{a}^{{\rm out}}_{+,q_{1}}\ket{0_{{\rm in}}}\,.$

In particular the different voltages seen by the two charge states imply the
reaction $f_{+}(k)\to f_{-}(k^{\prime})$ (plus pair production) vanishes
unless $k>ev$ for want of fermion final states with the required energy.

The remaining reaction obtained with an initial incoming positively charged
fermion is slightly more complicated. On one hand the $4$D chirality-flipping
process $f_{+}(k)\to f_{+}(k^{\prime})$ (plus the production of $n$ pairs)
proceeds much as above. However when $k<ev$ the part of the Bogoliubov
transformation relating $({a}^{{\rm in}}_{+,k})^{\star}$ to
${\overline{a}}^{{\rm out}}_{-,-k+ev}$ also contributes to give an amplitude
for spontaneously emitting $n+1$ pairs from the vacuum (with one of the pairs
corresponding to a particle and antiparticle of momentum $k^{\prime}$ and
$-k+ev$, respectively), leading to:

$\displaystyle\bra{0_{{\rm out}}}\overline{a}^{{\rm
out}}_{-,-q_{n}+ev}{a}^{{\rm out}}_{+,q_{n}}...\overline{a}^{{\rm
out}}_{-,-q_{1}+ev}{a}^{{\rm out}}_{+,q_{1}}\,{a}^{{\rm
out}}_{+,k^{\prime}}({a}^{{\rm in}}_{+,k})^{\star}\ket{0_{{\rm in}}}$
$\displaystyle\qquad=\delta(k-k^{\prime})\mathcal{T}^{++}_{{\rm
in}}\bra{0_{{\rm out}}}\overline{a}^{{\rm out}}_{-,-q_{n}+ev}{a}^{{\rm
out}}_{+,q_{n}}...\overline{a}^{{\rm out}}_{-,-q_{1}+ev}{a}^{{\rm
out}}_{+,q_{1}}\ket{0_{{\rm in}}}$ (162)
$\displaystyle\qquad\quad+\Theta\left(-k+ev\right)e^{-i{\mathfrak{a}}}{\cal
T}^{-+}_{{\rm in}}\bra{0_{{\rm out}}}\overline{a}^{{\rm
out}}_{-,-q_{n}+ev}{a}^{{\rm out}}_{+,q_{n}}...\overline{a}^{{\rm
out}}_{-,-q_{1}+ev}{a}^{{\rm out}}_{+,q_{1}}{a}^{{\rm
out}}_{+,k^{\prime}}{\overline{a}}^{{\rm out}}_{-,-k+ev}\ket{0_{{\rm in}}}.$

In equations (159)-(6.1.4), the momenta $k^{\prime},q_{1},\cdots q_{n}$ are
all distinct and satisfy $ev>q_{1},\cdots q_{n}>0$ as well as
$k,k^{\prime}>0$. Similar formulae can be derived for the amplitudes with a
single antiparticle in the initial state.

The next step is to evaluate the pair production amplitudes appearing in
(159)-(6.1.4) by using the $out$ particle content of the $in$ vacuum, as in
appendix F. When counting pairs it is more convenient to switch to a discrete
normalization for momentum eigenstates, as we now do, in which case the
maximum number of $out$ pairs in the $in$ vacuum, $N$, is a finite but large
number272727$N$ is defined as the maximum value of $\bra{0_{{\rm in}}}N_{\rm
out}\ket{0_{{\rm in}}}={ev}/{\Delta k}\,|{\cal T}^{+-}_{{\rm in}}|^{2}$ and so
is given by $N={ev}/{\Delta k}$. In the continuum limit, $N$ goes to infinity
as the spacing $\Delta k$ between states vanishes (see Appendix E for
details).. The discrete normalization analogue of the amplitude for emitting
$n$ pairs appearing in (159) to (161) and the first line of the right-hand
side of (6.1.4) evaluates to

$\bm{{\cal A}}^{n}_{\rm pair}=\bra{0_{{\rm out}}}\overline{\bm{a}}^{{\rm
out}}_{-,-q_{n}+ev}\bm{a}^{{\rm out}}_{+,q_{n}}...\overline{\bm{a}}^{{\rm
out}}_{-,-q_{1}+ev}{\bm{a}}^{{\rm out}}_{+,q_{1}}\ket{0_{{\rm
in}}}=\left({\cal T}_{{\rm in}}^{--}\right)^{N-n}\left({\cal T}_{{\rm
in}}^{+-}\right)^{n}e^{in{\mathfrak{a}}},$ (163)

for $n\leq N$, and vanishes otherwise.282828We denote transition amplitudes
between discretely normalized states by $\bm{{\cal A}}$ and their continuum
normalization counterparts by ${\cal A}$. For $k<ev$ the amplitude for
emitting $n+1$ pairs encountered in the second line of (6.1.4) similarly
corresponds to

$\displaystyle\bm{{\cal A}}^{n+1}_{\rm pair}(k,k^{\prime})$ $\displaystyle=$
$\displaystyle\bra{0_{{\rm out}}}\overline{\bm{a}}^{{\rm
out}}_{-,-q_{n}+ev}\bm{a}^{{\rm out}}_{+,q_{n}}...\overline{\bm{a}}^{{\rm
out}}_{-,-q_{1}+ev}\bm{a}^{{\rm out}}_{+,q_{1}}\bm{a}^{{\rm
out}}_{+,k^{\prime}}{\overline{\bm{a}}}^{{\rm out}}_{-,-k+ev}\ket{0_{{\rm
in}}}$ (164) $\displaystyle=$ $\displaystyle-\delta_{kk^{\prime}}\left({\cal
T}_{{\rm in}}^{--}\right)^{N-(n+1)}\left({\cal T}_{{\rm
in}}^{+-}\right)^{n+1}e^{i(n+1){\mathfrak{a}}},$

when $n<N$ and vanishes otherwise. The two lines on the right-hand side of
(6.1.4) then combine to become

$\displaystyle\bra{0_{{\rm out}}}\overline{\bm{a}}^{{\rm
out}}_{-,-q_{n}+ev}\bm{a}^{{\rm out}}_{+,q_{n}}...\overline{\bm{a}}^{{\rm
out}}_{-,-q_{1}+ev}\bm{a}^{{\rm out}}_{+,q_{1}}\,\bm{a}^{{\rm
out}}_{+,k^{\prime}}(\bm{a}^{{\rm in}}_{+,k})^{\star}\ket{0_{{\rm in}}}$
$\displaystyle\qquad=\delta_{kk^{\prime}}\Bigl{[}\mathcal{T}^{++}_{{\rm
in}}\mathcal{T}^{--}_{{\rm in}}-{\cal T}^{-+}_{{\rm in}}{\cal T}^{+-}_{{\rm
in}}\Bigr{]}\left({\cal T}_{{\rm in}}^{--}\right)^{N-(n+1)}\left({\cal
T}_{{\rm in}}^{+-}\right)^{n}e^{in{\mathfrak{a}}},$ (165)

for $k<ev$ and $n<N$, when evaluated using discretely normalized states. The
corresponding continuum normalization amplitudes have a very similar form.

Notice that the amplitudes (159)-(161) factorize into a product of a single-
particle transition amplitude, ${\cal A}_{sc}$, from a particle with quantum
numbers $\mathfrak{s},k$ to one with quantum numbers
$\mathfrak{s}^{\prime},k^{\prime}$, times a product of pair-production (or
vacuum survival) amplitudes for all the modes. The amplitude (6.1.4)
factorizes in the same way for initial momenta $k>ev$, while for $k<ev$ the
product over pair production (or vacuum survival) amplitudes runs over all but
the $k$-th mode292929For $k<ev$, the amplitude (6.1.4) factorizes differently
than (159)-(161) because it describes two processes in which the number of
produced pairs is not the same. We define ${\cal A}_{sc}$ for this process as
in (169) so that the single-particle amplitude captures the relevant
contribution of (6.1.4) to inclusive observables. . The amplitudes factorize
in this way because scattering and pair production for different modes are
statistically independent. Inspection of the above formulae shows that the
single-particle transition amplitudes
$\mathcal{A}_{sc}[f_{{\mathfrak{s}}}(k)\to
f_{\mathfrak{s}^{\prime}}(k^{\prime})]$ are given by

$\mathcal{A}_{sc}[f_{-}(k)\to f_{-}(k^{\prime})]=\mathcal{T}^{--}_{{\rm
in}}\;\delta({k-k^{\prime}}),$ (166) $\mathcal{A}_{sc}[f_{+}(k)\to
f_{-}(k^{\prime})]=e^{-i{\mathfrak{a}}}{\cal T}^{-+}_{{\rm
in}}\;\delta(k^{\prime}+ev-k),$ (167) $\mathcal{A}_{sc}[f_{-}(k)\to
f_{+}(k^{\prime})]=e^{i{\mathfrak{a}}}{\cal T}^{+-}_{{\rm
in}}\;\delta({k^{\prime}-ev-k})\,,$ (168)

and

$\mathcal{A}_{sc}[f_{+}(k)\to
f_{+}(k^{\prime})]=\Bigl{[}\Theta(k-ev)\mathcal{T}^{++}_{{\rm
in}}+\Theta\left(-k+ev\right)\left[{\cal T}^{++}_{{\rm in}}{\cal T}^{--}_{{\rm
in}}-{\cal T}^{+-}_{{\rm in}}{\cal T}^{-+}_{{\rm
in}}\right]\Bigr{]}\,\delta(k-k^{\prime})\,.$ (169)

Notice that the unitarity constraints on the ${\cal T}_{{\rm in}}$ amplitudes
given in (73) and (74) imply that the expression ${\cal T}^{++}_{{\rm
in}}{\cal T}^{--}_{{\rm in}}-{\cal T}^{+-}_{{\rm in}}{\cal T}^{-+}_{{\rm in}}$
simplifies to a phase.

Low-energy scattering rates can now be computed much as in §5 by projecting
any incoming plane wave onto the $S$-wave. 4D cross sections can then be
computed by dividing by the appropriate incident particle flux. For instance,
factoring out the energy-conserving delta function from amplitudes (166)-(168)
as ${\cal A}_{sc}={\cal M}\,\delta(E_{f}-E_{i})$, the $4$D cross section for
scattering with no pair production is

${\rm d}\sigma^{s}_{\rm
exclusive}\Bigl{[}f_{{\mathfrak{s}}{\mathfrak{c}}}(k)\to
f_{{\mathfrak{s}}^{\prime}{\mathfrak{c}}^{\prime}}(k^{\prime})\Bigr{]}=\frac{p_{s}}{{\mathfrak{F}}_{i}}\;{\rm
d}\Gamma_{\rm
exclusive}=\frac{\pi}{k^{2}}\,\delta_{{\mathfrak{s}},{\mathfrak{c}}}\delta_{{\mathfrak{s}}^{\prime},-{\mathfrak{c}}^{\prime}}\,|{\cal
M}|^{2}\;|\braket{0_{{\rm out}}}{0_{{\rm in}}}|^{2}\,\delta(E_{f}-E_{i})\,{\rm
d}k^{\prime}\,,$ (170)

where the probability for the plane wave to be found in an $S$-wave, $p_{s}$
and the initial particle flux ${\mathfrak{F}}_{i}$ are defined as in §5 and
are calculated in Appendix E along with the 2D exclusive differential
interaction rate ${\rm d}\Gamma_{\rm exclusive}$. Similarly, the cross section
for scattering with no additional pair production for the final amplitude
(169) is

$\displaystyle{\rm d}\sigma^{s}_{\rm exclusive}\Bigl{[}f_{++}(k)\to
f_{+-}(k^{\prime})\Bigr{]}$ $\displaystyle=$
$\displaystyle\frac{\pi}{k^{2}}\,|{\cal M}|^{2}\;|\braket{0_{{\rm
out}}}{0_{{\rm in}}}|^{2}\,\delta(E_{f}-E_{i})\,{\rm d}k^{\prime}$ (171)
$\displaystyle\qquad\qquad\quad\times\Bigl{[}\Theta(k-ev)+\Theta(-k+ev)\left|{\cal
T}^{--}_{{\rm in}}\right|^{-2}\Bigr{]}\,,$

where ${\cal M}$ is defined through ${\cal A}_{sc}={\cal
M}\,\delta(E_{f}-E_{i})$ and the factor of $\left|{\cal T}^{--}_{{\rm
in}}\right|^{-2}$ is cancelled by similar factors in the overlap
$|\braket{0_{{\rm out}}}{0_{{\rm in}}}|^{2}$, making the cross section finite
even in the ${\cal T}^{--}_{{\rm in}}\to 0$ limit.

Of more practical interest are inclusive cross sections for which the number
of associated pair productions is unmeasured and so marginalized over.
Appendix F – see the discussion below eq. (330) – explicitly performs the
marginalization over the number of produced pairs and shows that the resulting
cross section can be expressed purely in terms of the single-particle
scattering amplitudes ${\cal A}_{sc}$, given in (166)-(169). The ability to do
so is a consequence of unitarity, and is also the reason the parameters $n$
and $N$ drop out of our final results. The inclusive 4D cross section becomes

${\rm d}\sigma_{s}\Bigl{[}f_{{\mathfrak{s}}{\mathfrak{c}}}(k)\to
f_{{\mathfrak{s}}^{\prime}{\mathfrak{c}}^{\prime}}(k^{\prime})\Bigr{]}=\frac{p_{s}}{{\mathfrak{F}}_{i}}\;{\rm
d}\Gamma_{\rm
inclusive}=\frac{\pi}{k^{2}}\,\delta_{{\mathfrak{s}},{\mathfrak{c}}}\delta_{{\mathfrak{s}}^{\prime},-{\mathfrak{c}}^{\prime}}\,|{\cal
M}|^{2}\,\delta(E_{f}-E_{i})\,{\rm d}k^{\prime}\,\,,$ (172)

where the 2D inclusive differential interaction rate ${\rm d}\Gamma_{\rm
inclusive}$ is defined in Appendix E.

Combining results and integrating over the final momentum $k^{\prime}$ leads
to the total inclusive $S$-wave cross sections. For charge-exchange processes
with $\mathfrak{s}=\mathfrak{c}$ we have

$\sigma_{s}[f_{\pm}\to
f_{\mp}]=\frac{\pi}{k^{2}}\Bigl{|}\mathcal{T}^{+-}_{{\rm
in}}\Bigr{|}^{2}\Theta(k-{\mathfrak{s}}ev).$ (173)

Similarly for processes in which the $4$D chirality changes, but the charge
doesn’t the cross section when $\mathfrak{s}=\mathfrak{c}=-$ is

$\sigma_{s}[f_{-}\to f_{-}]=\frac{\pi}{k^{2}}\Bigl{|}\mathcal{T}^{--}_{{\rm
in}}\Bigr{|}^{2},$ (174)

and

$\sigma_{s}[f_{+}\to
f_{+}]=\frac{\pi}{k^{2}}\Big{[}\Theta(k-ev)\Bigl{|}\mathcal{T}^{++}_{{\rm
in}}\Bigr{|}^{2}+\Theta\left(-k+ev\right)\Big{]},$ (175)

when $\mathfrak{s}=\mathfrak{c}=+$. These agree with the corresponding
perturbative expressions when their domains of validity overlap, and are
consistent with the cross section results given in Kazama:1976fm , when we
restrict to their choices: $|{\cal T}^{++}_{{\rm in}}|=|{\cal T}^{--}_{{\rm
in}}|=1$, ${\cal T}^{+-}_{{\rm in}}={\cal T}^{-+}_{{\rm in}}=0$.

These cross sections display catalysis inasmuch as they are independent of the
scale $R\sim m_{g}^{-1}$ of the underlying classical dyon Preskill:1984gd ;
Rubakov:1988aq ; Schnir . They also do not depend directly on the dyon
magnetic or electric charge (though there is a large Coulomb phase that drops
out of the cross section that does see the dyon’s electric charge). Their size
scales like the unitarity bound $\sigma\propto 1/k^{2}$ whose origin comes
purely from the projection of the incoming plane wave onto the $S$-wave that
dominates at low energies. Finally the rates are directly controlled by the
size of the effective couplings hidden within the $\mathcal{T}_{{\rm in}}$
amplitudes, rather than through the more microscopic scales associated with
the RG-invariants that would normally be needed once couplings are
renormalized to remove the spurious $\epsilon$-dependence. In the present
instance – and just for the special kinematics of the monopole $S$-wave – the
existence of only a single solution to the radial equation implies that the
dimensionless magnitudes $|{\cal T}_{\rm in}|$ themselves are already RG-
invariant.

###  6.2 Effective dyon dynamics

Having described fermion behaviour in the approximation where the ‘slow’
degree of freedom ${\mathfrak{a}}$ is a fixed background we here return to the
question of how ${\mathfrak{a}}$ responds to fermion scattering on longer
time-scales. The leading dynamics of ${\mathfrak{a}}$ is governed by the
hamiltonian (134)

$H_{\rm dyon}=\frac{1}{2{\cal
I}}\left[{\mathfrak{p}}(t)-\frac{\vartheta}{2\pi}\right]^{2}+\frac{1}{2}\Bigl{[}{\overline{\bm{\chi}}}\,O_{\cal
B}({\mathfrak{a}})\,{\bm{\chi}}\Bigr{]}_{r=\epsilon,t}$ (176)

where ${\mathfrak{p}}$ is the conjugate momentum for ${\mathfrak{a}}$ given in
(42), repeated here for convenience:

${\mathfrak{p}}:=\frac{\vartheta}{2\pi}+{\cal I}\dot{{\mathfrak{a}}}\,.$ (177)

####  6.2.1 Charge conservation and rotor evolution

In the Heisenberg picture the momentum ${\mathfrak{p}}$ satisfies the equation
of motion

$e\,\dot{{\mathfrak{p}}}(t)=\frac{i}{2}\left({\overline{\bm{\chi}}}\,\left[O_{{\cal
B}}({\mathfrak{a}}(t)),\frac{e}{2}\tau_{3}\right]\,\bm{\chi}\right)_{r=\epsilon}=j^{1}_{\scriptscriptstyle
F}(\epsilon,t)\,,$ (178)

where $O_{\cal B}({\mathfrak{a}})$ is given in (51) and the last equality
equating the result to the fermionic electromagnetic current flux is obtained
by using the boundary condition (50) c.f. eq. (52):

$\left[\Gamma^{1}+O_{{\cal
B}}({\mathfrak{a}}(t))\right]\bm{\chi}(\epsilon,t)=0.$ (179)

Eq. (178) expresses conservation of total electric charge in the sense that it
equates the change in the dyon charge $-e{\mathfrak{p}}(t)$ to the radial flux
of fermion electric charge $j^{1}_{\scriptscriptstyle F}(\epsilon,t)$
evaluated near the dyon.

Equation (178) integrates to give

${\mathfrak{p}}(t)={\mathfrak{p}}(t_{0})+\frac{1}{e}\int^{t}_{t_{0}}{\rm
d}t^{\prime}j^{1}_{\scriptscriptstyle F}(\epsilon,t^{\prime}),$ (180)

and this result can be used in (177) to evolve ${\mathfrak{a}}$, leading to

$\displaystyle{\mathfrak{a}}(t)$ $\displaystyle=$
$\displaystyle{\mathfrak{a}}(t_{0})+\frac{1}{{\cal I}}\int^{t}_{t_{0}}{\rm
d}t^{\prime}\left[{\mathfrak{p}}(t_{0})-\frac{\vartheta}{2\pi}+\frac{1}{e}\int^{t^{\prime}}_{t_{0}}{\rm
d}t^{{}^{\prime\prime}}j^{1}_{\scriptscriptstyle
F}(\epsilon,t^{\prime\prime})\right]$ (181) $\displaystyle=$
$\displaystyle{\mathfrak{a}}(t_{0})+\frac{t-t_{0}}{{\cal
I}}\left[{\mathfrak{p}}(t_{0})-\frac{\vartheta}{2\pi}\right]+\frac{1}{e{\cal
I}}\int^{t}_{t_{0}}{\rm d}t^{\prime}\int^{t^{\prime}}_{t_{0}}{\rm
d}t^{{}^{\prime\prime}}j^{1}_{\scriptscriptstyle
F}(\epsilon,t^{\prime\prime}).$

These expressions show how the rotor operator acquires a component that acts
within the fermionic part of the Hilbert space for times $t>t_{0}$.

Now comes the main point. We wish to use the above expressions to determine
how the slow variable ${\mathfrak{a}}$ evolves in response to its interactions
with the relativistic fermion field. If we use how nuclei are handled within
the Born-Oppenheimer approximation applied to atoms, the answer seems simple.
For atoms one first computes the electronic state $|\Psi({\bf R})\rangle$ as a
function of fixed classical nuclear positions, ${\bf R}$, and then determines
the nuclear positions by minimizing the nuclear energy $V({\bf
R})=\langle\Psi({\bf R})|H|\Psi({\bf R})\rangle$ given these atomic states. In
the present instance the first step corresponds to computing the fermion state
$\ket{\psi_{i};{\mathfrak{a}}}$ as a function of a classical initial value for
${\mathfrak{a}}$. Then we compute an interaction Hamiltonian that captures the
correct dynamics within the fermionic state $\ket{\psi_{i};{\mathfrak{a}}}$
and use it to find how the field ${\mathfrak{a}}$ evolves.

Suppose we assume that the rotor and fermion sectors are initially unrelated
to one another at $t=t_{0}$, at which point
${\mathfrak{a}}(t_{0})={\mathfrak{a}}_{0}$. The above reasoning suggests these
slow-moving rotor degrees of freedom see only an average over the fast
variables and so (178) is approximately given by

$\dot{{\mathfrak{p}}}_{\rm
eff}(t)\simeq\frac{1}{e}\bra{{\mathfrak{a}}_{0};\psi_{i}}j^{1}_{\scriptscriptstyle
F}(\epsilon,t)\ket{\psi_{i};{\mathfrak{a}}_{0}}\,,$ (182)

which integrates to

${\mathfrak{p}}_{\rm
eff}(t)={\mathfrak{p}}(t_{0})+\frac{1}{e}\int^{t}_{t_{0}}{\rm
d}t^{\prime}\bra{{\mathfrak{a}}_{0};\psi_{i}}j^{1}_{\scriptscriptstyle
F}(\epsilon,t^{\prime})\ket{\psi_{i};{\mathfrak{a}}_{0}},$ (183)

as well as

${\mathfrak{a}}_{\rm eff}(t)={\mathfrak{a}}_{0}+\frac{t-t_{0}}{{\cal
I}}\left[{\mathfrak{p}}(t_{0})-\frac{\vartheta}{2\pi}\right]+\frac{1}{e{\cal
I}}\int^{t}_{t_{0}}{\rm d}t^{\prime}\int^{t^{\prime}}_{t_{0}}{\rm
d}t^{{}^{\prime\prime}}\bra{{\mathfrak{a}}_{0};\psi_{i}}j^{1}_{\scriptscriptstyle
F}(\epsilon,t^{\prime\prime})\ket{\psi_{i};{\mathfrak{a}}_{0}}.$ (184)

For instance, for fermions initially prepared in the vacuum state
$\ket{\psi_{i};{\mathfrak{a}}_{0}}=\ket{0_{{\rm in}}}$ eq. (6.1.3) gives a
time-independent current expectation value

$\dot{{\mathfrak{p}}}_{\rm eff}(t)\simeq\frac{1}{e}\bra{0_{\rm
in}}j^{1}_{\scriptscriptstyle F}(\epsilon,t)\ket{0_{\rm
in}}=\frac{ev}{2\pi}|{\cal T}^{+-}_{{\rm in}}|^{2}\,,$ (185)

and so the time-evolution of ${\mathfrak{a}}$ and its conjugate momentum
${\mathfrak{p}}$ are approximately given by

${\mathfrak{p}}_{\rm
eff}(t)={\mathfrak{p}}(t_{0})+\frac{ev(t-t_{0})}{2\pi}|{\cal T}^{+-}_{{\rm
in}}|^{2},$ (186)

and

${\mathfrak{a}}_{\rm eff}(t)={\mathfrak{a}}(t_{0})+\frac{t-t_{0}}{{\cal
I}}\left[{\mathfrak{p}}(t_{0})-\frac{\vartheta}{2\pi}\right]+\frac{ev(t-t_{0})^{2}}{4\pi{\cal
I}}|{\cal T}^{+-}_{{\rm in}}|^{2}\simeq{\mathfrak{a}}_{0}\,,$ (187)

where the final approximate equality drops $\alpha$-supressed terms involving
${\cal I}^{-1}\sim\alpha m_{g}$ (which also assumes $t-t_{0}$ is not too
large).

####  6.2.2 Effective rotor hamiltonian

One can ask: is there an effective rotor hamiltonian whose equations of motion
have the same form as eqs. (177) and (178)? Strictly speaking, such a
hamiltonian need not exist because the rotor is an open quantum system once
the fermion degrees of freedom are ignored. For such systems an effective
hamiltonian only exists to the extent that there is a mean-field description
for which fluctuations in the ignored degrees of freedom – the ‘environment’ –
can be neglected relative to the mean evolution (for a review of these issues,
including a more precise statement of the mean-field criterion – see EFTBook
).

But even if an effective rotor hamiltonian exists, its description is likely
not as simple as generating a potential $V({\mathfrak{a}})$ for
${\mathfrak{a}}$, which would be the analogy expected based on the Born-
Oppenheimer description of the energetics of nuclear positions within an atom.
In particular any such hamiltonian must produce nonzero $\dot{\mathfrak{p}}$
that is independent of ${\mathfrak{a}}$. But while it is true that Hamilton’s
equation $\dot{\mathfrak{p}}=-\partial H/\partial{\mathfrak{a}}$ seems to
imply that nonzero $\dot{\mathfrak{p}}$ requires $H$ to depend on
${\mathfrak{a}}$ – such as by adding a potential $V({\mathfrak{a}})$ to $H$ –
there is also no choice for a potential satisfying
$V({\mathfrak{a}}+2\pi)=V({\mathfrak{a}})$ that can produce a nonzero
$\dot{\mathfrak{p}}$ that is independent of ${\mathfrak{a}}$.

To explore what a successful choice for a rotor Hamiltonian would look like
consider as a starting point the rotor dynamics implied by the Lagrangian of
(40), repeated here (including the coupling to $\widehat{A}_{0}$)

$L_{\rm dyon}\ni\frac{\vartheta_{\rm
eff}(t)}{2\pi}\Bigl{(}\dot{\mathfrak{a}}-e\widehat{A}_{0}\Bigr{)}+\frac{{\cal
I}}{2}\Bigl{(}\dot{\mathfrak{a}}-e\widehat{A}_{0}\Bigr{)}^{2}+\cdots\,,$ (188)

which writes the terms in order of dominance at low-energies. For later
purposes we temporarily entertain the possibility that $\vartheta_{\rm
eff}=\vartheta_{\rm eff}(t)$ is a specified function of time.

Consider first keeping just the leading term,

$S_{\rm rotor}=\int_{\scriptscriptstyle W}{\rm d}t\;\frac{\vartheta_{\rm
eff}(t)}{2\pi}\Bigl{(}\dot{\mathfrak{a}}-e\widehat{A}_{0}\Bigr{)}\,,$ (189)

in which case the canonical momentum and Hamiltonian are

${\mathfrak{p}}:=\frac{\delta S_{\rm
rotor}}{\delta\dot{\mathfrak{a}}}=\frac{\vartheta_{\rm
eff}(t)}{2\pi}\quad\hbox{and}\quad H={\mathfrak{p}}\,\dot{\mathfrak{a}}-L_{\rm
rotor}=\frac{e\vartheta_{\rm eff}(t)}{2\pi}\,\widehat{A}_{0}\,,$ (190)

where the momentum equation can be regarded as a constraint. The Hamiltonian
reveals the sole energy associated with this interaction to be the Coulomb
energy due to the dyon acquiring an additional induced charge
$-e\vartheta_{\rm eff}/2\pi$ (as expected from the Witten effect Witten:1979ey
). The evolution equation for ${\mathfrak{p}}$ is then

$\dot{\mathfrak{p}}=\frac{\dot{\vartheta}_{\rm eff}}{2\pi}.$ (191)

Interestingly, this equation does agree with (182) provided we identify

$\vartheta_{\rm eff}(t)\simeq\vartheta_{0}+\frac{2\pi}{e}\int_{t_{0}}^{t}{\rm
d}t^{\prime}\;\bra{{\mathfrak{a}}_{0};\psi_{i}}j^{1}_{\scriptscriptstyle
F}(\epsilon,t^{\prime})\ket{\psi_{i};{\mathfrak{a}}_{0}}\,.$ (192)

In the special case where $\ket{\psi_{i};{\mathfrak{a}}_{0}}=\ket{0_{\rm in}}$
this becomes

$\vartheta_{\rm eff}(t)\simeq\vartheta_{0}+ev|{\cal T}^{+-}_{{\rm
in}}|^{2}(t-t_{0})\,.$ (193)

The idea that fermion scattering might cause vacuum angle evolution was
earlier discussed in Brennan:2021ewu .

Extending the above to include the subdominant kinetic term for
${\mathfrak{a}}$ appearing in (188) – and again entertaining the possibility
that $\vartheta_{\rm eff}=\vartheta_{\rm eff}(t)$ is a function of time –
instead leads to the canonical momentum and Hamiltonian

${\mathfrak{p}}:=\frac{\delta
S}{\delta\dot{\mathfrak{a}}}=\frac{\vartheta_{\rm eff}(t)}{2\pi}+{\cal
I}(\dot{\mathfrak{a}}-e\widehat{A}_{0})\quad\hbox{and}\quad
H={\mathfrak{p}}\,\dot{\mathfrak{a}}-L=e{\mathfrak{p}}\,\widehat{A}_{0}+\frac{1}{2{\cal
I}}\left({\mathfrak{p}}-\frac{\vartheta_{\rm eff}}{2\pi}\right)^{2}\,,$ (194)

which shows how the previous momentum constraint
${\mathfrak{p}}=\vartheta_{\rm eff}/(2\pi)$ emerges as the momentum choice
that minimizes the energy, for fixed $e\widehat{A}_{0}$. To the extent that
the rotor evolution minimizes its energy one expects

$\dot{\mathfrak{p}}\simeq\frac{\dot{\vartheta}_{\rm
eff}}{2\pi}\quad\hbox{and}\quad\dot{\mathfrak{a}}=\frac{1}{{\cal
I}}\left({\mathfrak{p}}-\frac{\vartheta_{\rm eff}}{2\pi}\right)\simeq 0\,,$
(195)

which agree with eqs. (177) and (185) in the limit that ${\cal I}^{-1}$ can be
neglected (so $\dot{\mathfrak{a}}\simeq 0$), when $\vartheta_{\rm eff}(t)$ is
given by (193). Although the rotor charge ${\mathfrak{p}}$ evolves as it
follows $\vartheta_{\rm eff}$ the minimized value of the energy remains
unchanged.

The above picture apparently relies on ${\mathfrak{a}}$ and ${\mathfrak{p}}$
approximately behaving as slow classical variables so that ${\mathfrak{p}}$
can evolve continuously with time as $\vartheta_{\rm eff}$ does. This is
indeed a good approximation for the fermion energies $\alpha m_{g}\ll E\ll
m_{g}$ for which the Born-Oppenheimer approximation applies. In a fuller
quantum treatment the initial value ${\mathfrak{a}}_{0}$ appearing in (184)
should be regarded as an operator satisfying
$[{\mathfrak{a}}_{0}\,,{\mathfrak{p}}_{0}]=i$, and commutes with the fermion
degrees of freedom evaluated at $t_{0}$. The requirement that
${\mathfrak{a}}_{0}$ also be a periodic variable with
${\mathfrak{a}}_{0}\sim{\mathfrak{a}}_{0}+2\pi$ implies that its canonical
momentum ${\mathfrak{p}}_{0}$ is quantized with eigenvalues $\lambda_{n}=n$
where $n$ is an arbitrary integer. Evaluating the hamiltonian of (194) as a
function of ${\mathfrak{p}}_{\rm eff}(t)$ given in (186) with $\vartheta_{\rm
eff}(t)$ given by (193) then shows – in the absence of $\widehat{A}_{0}$ –
that

$H=\frac{1}{2{\cal I}}\left[{\mathfrak{p}}_{\rm eff}(t)-\frac{ev}{2\pi}|{\cal
T}^{+-}_{{\rm
in}}|^{2}(t-t_{0})-\frac{\vartheta_{0}}{2\pi}\right]^{2}=\frac{1}{2{\cal
I}}\left({\mathfrak{p}}_{0}-\frac{\vartheta_{0}}{2\pi}\right)^{2}$ (196)

and so has energy eigenvalues

$E_{n}=\frac{1}{2{\cal I}}\left(n-\frac{\vartheta_{0}}{2\pi}\right)^{2}\,,$
(197)

that are independent of time and quantized with step size of order ${\cal
I}^{-1}\sim\alpha m_{g}$ even as ${\mathfrak{p}}_{\rm eff}(t)$ varies
continuously.

###  6.3 Vacuum-angle evolution

The upshot of the previous section is that – somewhat surprisingly – an
effective Hamiltonian can exist that captures the rotor’s evolution equations
(177) and (178) if tracing out the fermion were to produce an effective
contribution to the effective lagrangian of the form $\Delta
L=\frac{1}{2\pi}\vartheta_{\rm eff}(t)D{\mathfrak{a}}$ with $\vartheta_{\rm
eff}(t)$ satisfying the matching condition (192). How might such an effective
interaction actually be generated when explicitly integrating out the bulk
fermion?303030This issue is also discussed in Brennan:2021ewu , though in a
way that invokes a bulk coupling of ${\mathfrak{a}}$ to fermions.

An effective lagrangian of the form (188) with a time-dependent
$\vartheta_{\rm eff}$ might arise if the fermion-dyon interaction involves
operators that contain one extra derivative relative to those in (40), such as

$\Delta L={\cal O}({\overline{{\bm{\chi}}}},{\bm{\chi}})\,D{\mathfrak{a}}\,,$
(198)

where ${\cal O}$ is an operator built from the bulk fermion field and
$D{\mathfrak{a}}=\dot{\mathfrak{a}}-e\widehat{A}_{0}$ as before. Taking the
expectation value of this in the $in$ vacuum shows that $\langle{\cal
O}\rangle$ plays the role of $\vartheta_{\rm eff}/(2\pi)$ and this can be
time-dependent (and calculable) if $\langle{\cal O}\rangle$ is. In principle
the operator we seek should satisfy

$\partial_{t}{\cal O}\simeq\frac{1}{e}j^{1}_{\scriptscriptstyle
F}(\epsilon,t^{\prime})=-\frac{1}{2}j^{0}_{\scriptscriptstyle
A}(\epsilon,t^{\prime})\,.$ (199)

in order to ensure that (192) is true.

This last condition suggests a guess for what the operator ${\cal O}$ should
be. In 1+1 dimensions our two Dirac fermions $\chi_{\mathfrak{s}}$ can be
bosonized into two real scalars $\phi_{\mathfrak{s}}$, where the map between
bosons and fermions implies – see also (52)

$j^{\alpha}_{\scriptscriptstyle
A}(\epsilon,t)\propto\partial^{\alpha}\phi_{+}-\partial^{\alpha}\phi_{-}\,.$
(200)

Comparing this to (199) suggests that the operator ${\cal O}$ we seek has a
simple expression in terms of the bosonized field: ${\cal
O}\propto\phi_{+}-\phi_{-}$. If so then (198) represents a dyon-localized
kinetic mixing between $\phi_{\mathfrak{s}}$ and ${\mathfrak{a}}$.

We have not yet found a convincing derivation of why $\Delta L$ of the form
(198) is generated once the fermions are integrated out, or why it arises with
the right coefficients. But the above discussion reinforces earlier work
Callan:1982ah ; CallanSMatrix ; Preskill:1984gd ; Brennan:2021ewu  that
suggests that the bosonized formulation of the scattered fermions might be
more useful for understanding how the dyonic excitation ${\mathfrak{a}}$
evolves.

##  7 Conclusions

This paper identifies an EFT description of fermion-dyon interactions and uses
this to compute several simple reaction rates in the presence of a dyon. This
is done by adapting a framework – point-particle effective field theories
(PPEFTs) – that are designed to describe how small compact sources interact
with surrounding relativistic ‘bulk’ fields.

As typical for an EFT approach the starting point is an effective action – in
this case a world-line action that describes how bulk fields couple to one
another and to dyon-localized degrees of freedom (such as collective
coordinates). Such an action is organized as usual as a succession of
operators of higher and higher dimension, with the lowest-dimension operators
dominating at low energies. For the dyon case the important terms in this
action governing the interactions with light fermions are given in (37) and
are not that remarkable since they resemble the kinds of terms that also arise
in more prosaic examples (like those describing nuclei within atoms PPEFT3 ).

The main novelty in PPEFT methods is an algorithm that maps this effective
action onto a set of boundary conditions for the bulk fields as they approach
the source. These boundary conditions depend explicitly on the couplings
appearing in the effective action and this is ultimately how the news about
the nature of the source reaches the observables of low-energy bulk physics.
For fermions interacting with dyons this leads to the boundary conditions (45)
or (46), which are also not that different from the kinds of things that
appear for fermions interacting with other compact sources.

The novelty of dyons and monopoles enters once one projects the bulk fields
onto the $S$-wave mode that dominates the interactions at low energies since
the dyon’s magnetic charge gives this mode a distinctive angular dependence
that is not shared by $S$-waves for sources without magnetic charges. The
projection of the low-energy physics onto this $S$-wave allows the full 4D
bulk physics to be described by an effective 2D bulk action that couples to a
boundary situated outside of the underlying dyon field at $r=\epsilon$, with
the effects of dyonic structure communicated through the boundary conditions
at $r=\epsilon$ following from the PPEFT action. For dyons the $S$-wave
projection become scale invariant in a way that excludes the emergence of a
microscopic length scale $R_{*}$ on which low-energy observables can depend,
leading to potentially enormous interaction cross sections that are not
suppressed at all by powers of the physical size of the underlying source
(unlike for nuclei in atoms).

We show how the resulting dynamics of the bulk fermion coupled to the dyon’s
‘rotor’ mode captures the well-known fermion-dyon physics, but in a way that
is very generally characterized by only a few parameters – e.g. the ${\cal
T}_{\rm in}^{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}$ of (3.2.1) – whose values
can be obtained by matching with microscopic physics for specific dyon
configurations. Our general expressions for e.g. scattering as a function of
the ${\cal T}_{\rm in}^{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}$ reduce to
those of the literature once these matched values are used. These expressions
also capture how scattering includes the nonperturbative effects of fermion-
rotor interactions as studied in Polchinski , which can be regarded as
providing a more complicated matching prescription to the microscopic physics
that changes the values found for the ${\cal T}_{\rm
in}^{{\mathfrak{s}}{\mathfrak{s}}^{\prime}}$ but not the expressions for how
observables depend on these parameters.

We also explore how the fermion scattering causes the dyonic excitions to
evolve and identify an effective hamiltonian that captures the dynamics
required by the model’s conservation laws at low energy. Although we do not
yet have a microscopic derivation of this Hamiltonian we explore several
preliminary options.

## Acknowledgements

We thank Peter Hayman, Markus Rummel and Laszlo Zalavari for helpful
conversations. This research was partially supported by funds from the Natural
Sciences and Engineering Research Council (NSERC) of Canada. Research at the
Perimeter Institute is supported in part by the Government of Canada through
NSERC and by the Province of Ontario through MRI.

##  Appendix A Gamma-matrix conventions

This appendix summarizes our Dirac matrix conventions in both four and two
dimensions.

###  A.1 4D Dirac matrices

We follow Weinberg’s spinor and metric conventions, so our metric has
signature $(-,+,+,+)$ and a Weyl representation for the gamma matrices in four
dimensions is given by

$\gamma^{0}=-i\,\left(\begin{array}[]{ccc}0&&I\\\
I&&0\end{array}\right),\quad\gamma^{j}=\left(\begin{array}[]{ccc}0&&-i\sigma_{j}\\\
i\sigma_{j}&&0\end{array}\right),$ (201)

where $I$ is the $2$ by $2$ unit matrix and $\sigma_{j}$ are Pauli matrices
(acting in spin space, as opposed to the Pauli matrices $\tau_{a}$ acting on
gauge doublet indices). These satisfy the Clifford algebra
$\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}=2\eta^{\mu\nu}$ where
$\eta^{\mu\nu}$ is the inverse Minkowski metric, and the representation is
chosen to diagonalize

$\gamma^{5}=-i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=\left(\begin{array}[]{ccc}I&&0\\\
0&&-I\end{array}\right)\,.$ (202)

Dirac conjugation in these conventions is given by
$\overline{\psi}=i\psi^{\dagger}\gamma^{0}$. In these conventions the left-
and right-handed chirality projectors are $\gamma_{\scriptscriptstyle
L}:=\frac{1}{2}(1+\gamma_{5})$ and $\gamma_{\scriptscriptstyle
R}:=\frac{1}{2}(1-\gamma_{5})$.

#### Spherical coordinates

The gamma matrices adapted to spherical coordinates are given in terms of the
unit coordinate vectors

$\displaystyle\hat{\bm{r}}:=\hat{r}_{i}\,{\bf e}_{i}$ $\displaystyle=$
$\displaystyle\sin\theta\cos{\phi}\,{\bf e}_{x}+\sin\theta\sin\phi\,{\bf
e}_{y}+\cos\theta\,{\bf e}_{z},$
$\displaystyle\hat{\bm{\theta}}:=\hat{\theta}_{i}\,{\bf e}_{i}$
$\displaystyle=$ $\displaystyle\cos\theta\cos{\phi}\,{\bf
e}_{x}+\cos\theta\sin\phi\,{\bf e}_{y}-\sin\theta\,{\bf e}_{z},$
$\displaystyle\hat{\bm{\phi}}:=\hat{\phi}_{i}\,{\bf e}_{i}$ $\displaystyle=$
$\displaystyle-\sin{\phi}\,{\bf e}_{x}+\cos\phi\,{\bf e}_{y}.$ (203)

by $\gamma^{r}:=\gamma^{i}\,\hat{r}_{i}$,
$\gamma^{\theta}:=\gamma^{i}\,\hat{\theta}_{i}$ and
$\gamma^{\phi}:=\gamma^{i}\,\hat{\phi}_{i}/\sin\theta$, and so

$\gamma^{r}=\left(\begin{array}[]{ccc}0&&-i\sigma^{r}\\\
i\sigma^{r}&&0\end{array}\right)\,,\quad\gamma^{\theta}=\left(\begin{array}[]{ccc}0&&-i\sigma^{\theta}\\\
i\sigma^{\theta}&&0\end{array}\right)\quad\hbox{and}\quad\gamma^{\phi}=\frac{1}{\sin\theta}\,\left(\begin{array}[]{ccc}0&&-i\sigma^{\phi}\\\
i\sigma^{\phi}&&0\end{array}\right).$ (204)

where the Pauli matrices in spherical coordinates are defined by

$\sigma^{r}=\left(\begin{array}[]{cc}\cos\theta&e^{-i\phi}\sin\theta\\\
e^{i\phi}\sin\theta&-\cos\theta\end{array}\right),\;\sigma^{\theta}=\left(\begin{array}[]{cc}-\sin\theta&e^{-i\phi}\cos\theta\\\
e^{i\phi}\cos\theta&\sin\theta\end{array}\right)\;\hbox{and}\;\;\sigma^{\phi}=\left(\begin{array}[]{cc}0&-ie^{-i\phi}\\\
ie^{i\phi}&0\end{array}\right).$ (205)

###  A.2 2D Dirac matrices

In $D=1+1$ spacetime dimensions we label coordinates with
$x^{\alpha}=\\{t,r\\}$ for $\alpha=0,1$ and use the following representation
of gamma matrices

$\Gamma^{0}=-i\sigma_{1},\quad\Gamma^{1}=\sigma_{2}\,.$ (206)

These satisfy the algebra
$\\{\Gamma^{\alpha},\Gamma^{\beta}\\}=2\eta^{\alpha\beta}$, where
$\eta^{\alpha\beta}=\hbox{diag}(-1,1)$ is the inverse Minkowski metric in
$1+1$ dimensions. The chiral matrix in 1+1 dimensions we then define to be

$\Gamma_{c}:=\Gamma^{0}\Gamma^{1}=\sigma_{3}\,,$ (207)

which has eigenvalues $\pm 1$ (and is diagonal in the basis used here). Notice
that these definitions imply
$\Gamma_{c}\Gamma^{\alpha}=\epsilon^{\alpha\beta}\Gamma_{\beta}$ where our
Levi-Civita convention chooses $\epsilon^{01}=+1$. Dirac conjugation is again
given by $\overline{\chi}=i\chi^{\dagger}\Gamma^{0}$ and in these conventions
the left- and right-handed helicity projectors are
$\Gamma_{+}:=\frac{1}{2}(1+\Gamma_{c})$ and
$\Gamma_{-}:=\frac{1}{2}(1-\Gamma_{c})$.

##  Appendix B PPEFTs and relativistic fermions

This Appendix summarizes the results of PPEFT3  describing how PPEFTs couple
to relativistic fermions. Formally we wish to couple a Dirac fermion to a
point-like source located at the origin

$S=-\int{\rm d}^{4}x\Bigl{[}\overline{\psi}(\hbox to0.0pt{\hbox
to8.55695pt{\hfil/\hfil}\hss}D+m)\psi+\overline{\psi}N\psi\,\delta^{3}(x)\Bigr{]}\,,$
(208)

where $N$ is a Dirac matrix. For definiteness in this Appendix we follow
PPEFT3  and consider the rotational and parity invariant case
$N=\hat{\mathfrak{c}}_{s}+i\hat{\mathfrak{c}}_{v}\gamma^{0}$ where
$\hat{\mathfrak{c}}_{s}$ and $\hat{\mathfrak{c}}_{v}$ are coupling constants
(but more general possibilities are entertained in the main text). The $\psi$
equation of motion including the coupling to the source is

$(\hbox to0.0pt{\hbox
to8.55695pt{\hfil/\hfil}\hss}D+m)\psi+N\psi\,\delta^{3}(x)=0\,.$ (209)

Formally we’d like to trade the delta function for a near-source boundary
condition, and following usual practice this would be obtained by integrating
(209) over a small Gaussian pillbox, $P$, of radius $\epsilon$ centred on the
source. This gives, in the limit $\epsilon\to 0$ of vanishingly small pillbox,
the result

$\int_{\partial P}{\rm d}^{2}x\;n_{\mu}\gamma^{\mu}\psi=\int{\rm
d}^{2}\Omega\,\epsilon^{2}\;\gamma^{r}\,\psi\simeq-N\psi(0)\,,$ (210)

where the solid-angle measure is ${\rm d}^{2}\Omega=\sin\theta\,{\rm
d}\theta{\rm d}\phi$ and $n_{\mu}$ is an outward-pointing unit normal to the
pillbox so in polar coordinates $n_{\mu}{\rm d}x^{\mu}={\rm d}r$ and $\psi(0)$
denotes the value of the field at the position of the source. The final
approximate equality drops the $m\psi$ term, as is appropriate for a
sufficiently small pillbox since this vanishes as $\epsilon\to 0$ provided
$\psi$ is sufficiently smooth near the origin. Our conventions on gamma-
matrices in polar coordinates are given in Appendix A.

###  B.1 Boundary action

The problem with the formal argument is that bulk fields like $\psi$ are
typically not smooth as $r\to 0$, which both complicates the neglect of the
$m\psi$ term when integrating (209) over the pillbox and makes $\psi(0)$
undefined. The PPEFT way for dealing with both of these issues is essentially
to regulate the source action by replacing it by a boundary action on the
boundary of the pillbox $\partial P$ at $r=\epsilon$. For configurations that
are spherically symmetric313131Non-spherically symmetric configurations can
also be handled by decomposing into spherical harmonics and treating each
harmonic separately on $\partial P$. very near the source this is particularly
simple to do by replacing the world-line action by its value integrated over
$\partial P$:

$\int_{r=0}{\rm
d}t\;{\overline{\psi}}N\psi\to\frac{1}{4\pi\epsilon^{2}}\int_{\partial P}{\rm
d}^{2}\Omega\,{\rm d}t\;\epsilon^{2}\,{\overline{\psi}}N\psi\,.$ (211)

This procedure makes the replacement
$N\psi(0)\to(4\pi\epsilon^{2})^{-1}\int{\rm
d}^{2}\Omega\,\epsilon^{2}N(\epsilon)\psi(\epsilon)$ on the right-hand side of
(210), leading (in the limit where $\epsilon$ is much smaller than all other
scales of interest) to the regulated boundary condition

$\int_{r=\epsilon}{\rm
d}^{2}\Omega\;\left[\gamma^{r}+\frac{N}{4\pi\epsilon^{2}}\right]\psi=\int_{r=\epsilon}{\rm
d}^{2}\Omega\;\left[\gamma^{r}+\frac{1}{4\pi\epsilon^{2}}\Bigl{(}\hat{\mathfrak{c}}_{s}+i\hat{\mathfrak{c}}_{v}\gamma^{0}\Bigr{)}\right]\psi=0\,.$
(212)

Notice this boundary condition is trivially satisfied if we’d tried to make
the same derivation using a pillbox that does not contain the source position.
This is because in this case the $N\psi$ term is no longer present and so the
boundary condition states $\int{\rm d}^{2}\Omega\,\gamma^{r}\psi=0$. Since
$\psi$ varies very slowly in a small enough region not containing the source,
it can be taken to be approximately constant across the pillbox and so the
integral over all directions of $\gamma^{r}$ gives zero trivially without
restricting $\psi$.

Returning to the case where the pillbox does enclose the source, the boundary
condition (212) can be written as $\int{\rm
d}^{2}\Omega\;B_{\epsilon}\,\psi(\epsilon)=0$ where

$B_{\epsilon}:=\gamma^{r}+\frac{N}{4\pi\epsilon^{2}}=\gamma^{r}+\hat{\cal
C}_{s}+i\hat{\cal C}_{v}\gamma^{0}=\left(\begin{array}[]{cc}\hat{\cal
C}_{s}&\hat{\cal C}_{v}-i\sigma^{r}\\\ {}\hat{\cal
C}_{v}+i\sigma^{r}&\hat{\cal C}_{s}\end{array}\right)\,,$ (213)

where $\hat{\cal C}_{s}=\hat{\mathfrak{c}}_{s}/(4\pi\epsilon^{2})$ and
$\hat{\cal C}_{v}=\hat{\mathfrak{c}}_{v}/(4\pi\epsilon^{2})$ are now
dimensionless effective couplings. The subscript $\epsilon$ on $B_{\epsilon}$
is meant to emphasize that the constants $\hat{\cal C}_{a}$ (and in general
also the original couplings $\hat{\mathfrak{c}}_{i}$ themselves) also must
carry an implicit $\epsilon$-dependence if physical quantities are to remain
unchanged as $\epsilon$ is varied (more about which below). In terms of the
left- and right-handed parts of $\psi$ the boundary condition becomes

$-\hat{\cal C}_{s}\int_{\epsilon}{\rm
d}^{2}\Omega\,\psi^{\pm}_{\scriptscriptstyle L}=\int_{\epsilon}{\rm
d}^{2}\Omega\,\Bigl{(}\hat{\cal
C}_{v}-i\sigma^{r}\Bigr{)}\psi^{\pm}_{\scriptscriptstyle
R}\quad\hbox{and}\quad-\int_{\epsilon}{\rm d}^{2}\Omega\,\Bigl{(}\hat{\cal
C}_{v}+i\sigma^{r}\Bigr{)}\,\psi^{\pm}_{\scriptscriptstyle L}=\hat{\cal
C}_{s}\int_{\epsilon}{\rm d}^{2}\Omega\,\psi^{\pm}_{\scriptscriptstyle R}\,.$
(214)

To see what these boundary conditions imply, imagine solving the bulk equation
$(\hbox to0.0pt{\hbox to8.55695pt{\hfil/\hfil}\hss}D+m)\psi=0$ for
$r>\epsilon$ and decomposing the result into rotation and parity eigenstates.
The parity-even solutions are

$\psi^{+}=\left(\begin{array}[]{c}\psi^{+}_{\scriptscriptstyle L}\\\
\psi^{+}_{\scriptscriptstyle
R}\end{array}\right)=\left(\begin{array}[]{c}f_{+}(r)\,U^{+}(\theta,\phi)+ig_{+}(r)\,U^{-}(\theta,\phi)\\\
f_{+}(r)\,U^{+}(\theta,\phi)-ig_{+}(r)\,U^{-}(\theta,\phi)\end{array}\right)\,,$
(215)

while the parity-odd ones are

$\psi^{-}=\left(\begin{array}[]{c}\psi^{-}_{\scriptscriptstyle L}\\\
\psi^{-}_{\scriptscriptstyle
R}\end{array}\right)=\left(\begin{array}[]{c}f_{-}(r)\,U^{-}(\theta,\phi)+ig_{-}(r)\,U^{+}(\theta,\phi)\\\
f_{-}(r)\,U^{-}(\theta,\phi)-ig_{-}(r)\,U^{+}(\theta,\phi)\end{array}\right)\,,$
(216)

where $U^{\pm}$ are the spinor harmonics that combine the particle’s spin-half
with orbital angular momenta $\ell=j\mp\frac{1}{2}$ to give total angular
momentum $j=\frac{1}{2},\,\frac{3}{2},\cdots$. The radial functions
$f_{\pm}(r)$ and $g_{\pm}(r)$ with mode frequency $\omega$ solve the radial
equations

$f_{+}^{\prime}=\left(m+\omega\right)g_{+}\quad\hbox{and}\quad
g_{+}^{\prime}+\frac{2g_{+}}{r}=\left(m-\omega\right)f_{+}\,,$ (217)

together with

$g_{-}^{\prime}=\left(m-\omega\right)f_{-}\quad\hbox{and}\quad
f_{-}^{\prime}+\frac{2f_{-}}{r}=\left(m+\omega\right)g_{-}\,,$ (218)

where primes denote differentiation with respect to $r$. The boundary
conditions (214) fix the ratio of the functions $f$ and $g$ at $r=\epsilon$
once the angular integrations are performed, giving

$\hat{\cal C}_{s}+\hat{\cal
C}_{v}=\left(\frac{g_{+}}{f_{+}}\right)_{r=\epsilon}\qquad\hbox{and}\qquad\hat{\cal
C}_{s}-\hat{\cal C}_{v}=\left(\frac{f_{-}}{g_{-}}\right)_{r=\epsilon}\,.$
(219)

###  B.2 RG evolution

Eq. (219) provides the solution to how the properties of the source influence
the bulk solutions for $\psi$ in the source’s vicinity. Given the general
solution,

$f_{\pm}(r)=C^{\pm}_{1}f_{1\pm}(r)+C^{\pm}_{2}f_{2\pm}(r)\qquad\hbox{and}\qquad
g_{\pm}(r)=C^{\pm}_{1}g_{1\pm}(r)+C^{\pm}_{2}g_{2\pm}(r)\,,$ (220)

to the radial part of the Dirac field equation we see that (219) show that the
couplings $\hat{\cal C}_{s}$ and $\hat{\cal C}_{v}$ determine the ratios of
integration constants $C^{+}_{2}/C^{+}_{1}$ and $C^{-}_{2}/C^{-}_{1}$ that
specify $(g_{\pm}/f_{\pm})_{r=\epsilon}$. Energy levels for states of either
parity and scattering amplitudes are then determined by the values of
$C^{\pm}_{2}/C^{\pm}_{1}$.

But it is still a potential puzzle why physical predictions can depend on the
radius, $r=\epsilon$, of the Gaussian pillbox which is not a physical scale
(arising just as a way to regularize the boundary conditions). The precise
value of $\epsilon$ must therefore drop out of predictions for observables
(unlike the physical size, $R$, of the underlying source, say). In detail,
this happens because any explicit $\epsilon$-dependence arising in a
calculation of an observable cancels an implicit $\epsilon$-dependence buried
within the ‘bare’ quantities $\hat{\mathfrak{c}}_{s}$ and
$\hat{\mathfrak{c}}_{v}$. Physical predictions remain $\epsilon$-independent
if $\hat{\mathfrak{c}}_{s}(\epsilon)$ and $\hat{\mathfrak{c}}_{v}(\epsilon)$
are chosen to ensure the ratios $C^{\pm}_{2}/C^{\pm}_{1}$ are held fixed as
$\epsilon$ is varied.

This gives us another way to interpret eq. (219). Rather than reading (219) as
fixing $f_{\pm}/g_{\pm}$ at a specific radius given known values of
$\epsilon$, $\hat{\mathfrak{c}}_{s}$ and $\hat{\mathfrak{c}}_{v}$ we can
instead read the equations

$\hat{\mathfrak{c}}_{s}(\epsilon)=\left[\frac{g_{+}(\epsilon)}{f_{+}(\epsilon)}+\frac{f_{-}(\epsilon)}{g_{-}(\epsilon)}\right]2\pi\epsilon^{2}\quad\hbox{and}\quad\hat{\mathfrak{c}}_{v}(\epsilon)=\left[\frac{g_{+}(\epsilon)}{f_{+}(\epsilon)}-\frac{f_{-}(\epsilon)}{g_{-}(\epsilon)}\right]2\pi\epsilon^{2}\,,$
(221)

as telling us how $\hat{\mathfrak{c}}_{s}(\epsilon)$ and
$\hat{\mathfrak{c}}_{v}(\epsilon)$ must depend on $\epsilon$ in order to
ensure that $C^{\pm}_{2}/C^{\pm}_{1}$ remains $\epsilon$-independent. Since we
choose $\epsilon$ much smaller than the typical scale of the external problem
(such as the Bohr radius, for applications to atoms), it suffices to use the
leading small-$r$ form of the solutions $f_{\pm}$ and $g_{\pm}$ when using
(221). In this regime solutions are usually well described by power laws, with
(220) reducing to

$f_{\pm}(r)=C^{\pm}_{1}\left(\frac{r}{a}\right)^{{\mathfrak{z}}-1}+C^{\pm}_{2}\left(\frac{r}{a}\right)^{-{\mathfrak{z}}-1}\qquad\hbox{and}\qquad
g_{\pm}(r)=\widetilde{C}^{\pm}_{1}\left(\frac{r}{a}\right)^{{\mathfrak{z}}-1}+\widetilde{C}^{\pm}_{2}\left(\frac{r}{a}\right)^{-{\mathfrak{z}}-1}\,,$
(222)

with $\widetilde{C}^{\pm}_{i}\propto C^{\pm}_{i}$ in a way that depends on the
relative small-$r$ asymptotic behaviour of $f_{i}(r)$ and $g_{i}(r)$. For such
solutions the choice of $C^{\pm}_{2}/C^{\pm}_{1}$ controls the precise radius
at which one of these solutions dominates the other one, and as a result the
RG evolution of the couplings implied by (221) in this regime describes the
cross-over between these two types of evolution.

##  Appendix C Regularized codimension-1 boundary action

This Appendix addresses the question of how to regularize the fermion-dyon
interactions defined on the dyon world-line, which appear in equations such as
(40).

As argued in the main text, the lowest-dimension interactions between the
fermion doublet ${\bm{\psi}}(x)$ and the dyon collective coordinates
$y^{\mu}(t),{\mathfrak{a}}(t)$ are given by

$\displaystyle S^{\rm int}_{\rm dyon}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\int{\rm
d}t\,{\overline{\bm{\psi}}}\,{\mathfrak{C}}({\mathfrak{a}})\,\bm{\psi}$
$\displaystyle=$ $\displaystyle-\frac{1}{2}\int{\rm
d}t\,{\overline{\bm{\psi}}}\Bigl{[}\hat{\mathfrak{c}}^{s}_{1}+i\,\hat{\mathfrak{c}}^{ps}_{1}\gamma_{5}-i\,\hat{\mathfrak{c}}^{v}_{1}\gamma^{0}-i\,\hat{\mathfrak{c}}^{pv}_{1}\gamma_{5}\gamma^{0}+\Bigl{(}\hat{\mathfrak{c}}^{s}_{3}+i\,\hat{\mathfrak{c}}^{ps}_{3}\gamma_{5}-i\,\hat{\mathfrak{c}}^{v}_{3}\gamma^{0}-i\,\hat{\mathfrak{c}}^{pv}_{3}\gamma_{5}\gamma^{0}\Bigr{)}\tau_{3}$
$\displaystyle\quad+\Bigl{(}\hat{\mathfrak{c}}^{s}_{+}+i\,\hat{\mathfrak{c}}^{ps}_{+}\gamma_{5}-i\,\hat{\mathfrak{c}}^{v}_{+}\gamma^{0}-i\,\hat{\mathfrak{c}}^{pv}_{+}\gamma_{5}\gamma^{0}\Bigr{)}e^{i{\mathfrak{a}}}\tau_{+}+\Bigl{(}\hat{\mathfrak{c}}^{s}_{-}+i\,\hat{\mathfrak{c}}^{ps}_{-}\gamma_{5}-i\,\hat{\mathfrak{c}}^{v}_{-}\gamma^{0}-i\,\hat{\mathfrak{c}}^{pv}_{-}\gamma_{5}\gamma^{0}\Bigr{)}e^{-i{\mathfrak{a}}}\tau_{-}\Bigr{]}\,\bm{\psi}$

where ${\bm{\psi}}$ is evaluated at $\bm{r}=0$ and we specialize to the dyon
rest frame and neglect dyon recoil effects so that
$\dot{y}^{\mu}\approx\delta^{\mu}_{0}$. We can regulate the operators
appearing in $S^{\rm int}_{\rm dyon}$ by replacing them with appropriate
interaction terms, defined on the boundary of a Gaussian pillbox at
$r=\epsilon$. Appendix B shows that for fermion field configurations which are
spherically symmetric near the source, the regularization procedure amounts to
replacing fermion bilinears such as ${\overline{\bm{\psi}}}M{\bm{\psi}}$,
where $M$ is a matrix in spin and isospin space, with their average over the
pillbox boundary $({4\pi\epsilon^{2}})^{-1}\int{\rm
d}^{2}\Omega\,\epsilon^{2}\,{\overline{\bm{\psi}}}(\epsilon){M}{\bm{\psi}}(\epsilon)$.
This is a valid prescription for operators in (C) describing
$\psi_{+},\psi_{-}$ self-interactions (mediated by
$\hat{\mathfrak{c}}_{1},\hat{\mathfrak{c}}_{3}$). For operators that couple
$\psi_{+}$ and $\psi_{-}$ we must use another prescription however, since the
angular dependence of the two components of the doublet is different, even
when restricting to the same partial wave. As a result, regularized fermion
bilinears such as
${\overline{\bm{\psi}}}(\epsilon)\,\hat{\mathfrak{c}}^{s}_{+}\tau_{+}{\bm{\psi}}(\epsilon)$
are generally not rotation-invariant and so can vanish after integration over
the boundary of the pillbox. This can be seen explicitly by specializing to
$S$-wave states for which we get e.g.:

$\frac{1}{4\pi\epsilon^{2}}\int{\rm
d}^{2}\Omega\,\epsilon^{2}\,{\overline{\bm{\psi}}}(\epsilon)\,\hat{\mathfrak{c}}^{s}_{+}\tau_{+}{\bm{\psi}}(\epsilon)=\frac{f^{*}_{+}(\epsilon,t)\,g_{-}(\epsilon,t)+g^{*}_{+}(\epsilon,t)\,f_{-}(\epsilon,t)}{4\pi\epsilon^{2}}\hat{\mathfrak{c}}^{s}_{+}\int{{\rm
d}^{2}\Omega}\,\eta^{\dagger}_{+}\,\eta_{-}=0,$ (224)

in the gauge where the Julia-Zee solution has the form (8) and $S$-wave
fermions are given by (24). That the form of angular momentum eigenstates
depends on their electric charge can be traced back to the expression for the
angular momentum operator $\vec{J}$, which includes a contribution from the
gauge isospin $\vec{T}$, as in (23).

To couple the two components of the doublet at the $r=\epsilon$ boundary, it
suffices to introduce additional matrices $P_{\pm}$, which turn the angular
dependence of $\psi_{+}$ into that of $\psi_{-}$ and vice versa. For $S$-wave
states and in the ‘abelian’ gauge of (8), these matrices act in spin space and
are defined through:

$P_{+}\,\frac{1}{r}\begin{pmatrix}f_{+}(r,t)\,\eta_{+}\\\
g_{+}(r,t)\,\eta_{+}\end{pmatrix}=\frac{1}{r}\begin{pmatrix}f_{+}(r,t)\,\eta_{-}\\\
g_{+}(r,t)\,\eta_{-}\end{pmatrix},\quad\text{and}\quad
P_{-}\,\frac{1}{r}\begin{pmatrix}f_{-}(r,t)\,\eta_{-}\\\
g_{-}(r,t)\,\eta_{-}\end{pmatrix}=\frac{1}{r}\begin{pmatrix}f_{-}(r,t)\,\eta_{+}\\\
g_{-}(r,t)\,\eta_{+}\end{pmatrix}.$ (225)

Since these equations do not uniqely determine $P_{\pm}$, we further choose:

$P_{-}\,\frac{1}{r}\begin{pmatrix}f_{+}(r,t)\,\eta_{+}\\\
g_{+}(r,t)\,\eta_{+}\end{pmatrix}=\begin{pmatrix}0\\\
0\end{pmatrix},\quad\text{and}\quad
P_{+}\,\frac{1}{r}\begin{pmatrix}f_{-}(r,t)\,\eta_{-}\\\
g_{-}(r,t)\,\eta_{-}\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix},$ (226)

which leads us to the following expressions for $P_{\pm}$ in the $R_{-}$ patch

$P_{+}=-\frac{ie^{i\phi}}{2}\begin{pmatrix}i\sigma_{\theta}+\sigma_{\phi}&&0\\\
0&&i\sigma_{\theta}+\sigma_{\phi}\end{pmatrix},\quad\text{and}\quad
P_{-}=-\frac{ie^{-i\phi}}{2}\begin{pmatrix}i\sigma_{\theta}-\sigma_{\phi}&&0\\\
0&&i\sigma_{\theta}-\sigma_{\phi}\end{pmatrix},$ (227)

as well as $P^{\prime}_{\pm}\coloneqq e^{\mp 2i\phi}P_{\pm}$ in the $R_{+}$
patch. It follows that for $S$-wave states, the world-line interactions in (C)
can be regulated using the following boundary action:

$\displaystyle I^{\rm int}_{\rm dyon}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\underset{r=\epsilon}{\int}{\rm d}t\,{\rm
d}^{2}\Omega\,\epsilon^{2}\,{\overline{\bm{\psi}}}\Bigl{[}\hat{{\cal
C}}^{s}_{1}+i\,\hat{{\cal C}}^{ps}_{1}\gamma_{5}-i\,\hat{{\cal
C}}^{v}_{1}\gamma^{0}-i\,\hat{{\cal
C}}^{pv}_{1}\gamma_{5}\gamma^{0}+\Bigl{(}\hat{{\cal C}}^{s}_{3}+i\,\hat{{\cal
C}}^{ps}_{3}\gamma_{5}-i\,\hat{{\cal C}}^{v}_{3}\gamma^{0}-i\,\hat{{\cal
C}}^{pv}_{3}\gamma_{5}\gamma^{0}\Bigr{)}\tau_{3}$
$\displaystyle\quad+\Bigl{(}\hat{{\cal C}}^{s}_{+}+i\,\hat{{\cal
C}}^{ps}_{+}\gamma_{5}-i\,\hat{{\cal C}}^{v}_{+}\gamma^{0}-i\,\hat{{\cal
C}}^{pv}_{+}\gamma_{5}\gamma^{0}\Bigr{)}e^{i{\mathfrak{a}}}P_{-}\,\tau_{+}+\Bigl{(}\hat{{\cal
C}}^{s}_{-}+i\,\hat{{\cal C}}^{ps}_{-}\gamma_{5}-i\,\hat{{\cal
C}}^{v}_{-}\gamma^{0}-i\,\hat{{\cal
C}}^{pv}_{-}\gamma_{5}\gamma^{0}\Bigr{)}e^{-i{\mathfrak{a}}}P_{+}\,\tau_{-}\Bigr{]}\,\bm{\psi}$

where we introduce the boundary couplings $\hat{{\cal
C}}^{A}_{I}=\frac{1}{4\pi\epsilon^{2}}\,\hat{\mathfrak{c}}^{A}_{I}$. The
boundary condition satisfied by $S$-wave fermions is then given by

$\displaystyle 0=\Bigl{\\{}\Bigl{[}\gamma^{r}+\Bigl{(}\hat{{\cal
C}}^{s}_{1}+i\,\hat{{\cal C}}^{ps}_{1}\gamma_{5}-i\,\hat{{\cal
C}}^{v}_{1}\gamma^{0}-i\,\hat{{\cal
C}}^{pv}_{1}\gamma_{5}\gamma^{0}\Bigr{)}+\Bigl{(}\hat{{\cal
C}}^{s}_{3}+i\,\hat{{\cal C}}^{ps}_{3}\gamma_{5}-i\,\hat{{\cal
C}}^{v}_{3}\gamma^{0}-i\,\hat{{\cal
C}}^{pv}_{3}\gamma_{5}\gamma^{0}\Bigr{)}\tau_{3}$ (229)
$\displaystyle\quad+\Bigl{(}\hat{{\cal C}}^{s}_{+}+i\,\hat{{\cal
C}}^{ps}_{+}\gamma_{5}-i\,\hat{{\cal C}}^{v}_{+}\gamma^{0}-i\,\hat{{\cal
C}}^{pv}_{+}\gamma_{5}\gamma^{0}\Bigr{)}e^{i{\mathfrak{a}}}P_{-}\,\tau_{+}+\Bigl{(}\hat{{\cal
C}}^{s}_{-}+i\,\hat{{\cal C}}^{ps}_{-}\gamma_{5}-i\,\hat{{\cal
C}}^{v}_{-}\gamma^{0}-i\,\hat{{\cal
C}}^{pv}_{-}\gamma_{5}\gamma^{0}\Bigr{)}e^{-i{\mathfrak{a}}}P_{+}\,\tau_{-}\Bigr{]}{\bm{\psi}}\Bigr{\\}}_{r=\epsilon},$

instead of (46) and is equivalent to the $2$D boundary condition (49) when the
$4$D and $2$D boundary couplings are related as follows:

${\cal C}^{s}_{{\mathfrak{s}}{\mathfrak{s}}}=\hat{{\cal
C}}^{s}_{1}+{\mathfrak{s}}\hat{{\cal C}}^{s}_{3},\quad{\cal
C}^{ps}_{{\mathfrak{s}}{\mathfrak{s}}}={\mathfrak{s}}\hat{{\cal
C}}^{ps}_{1}+\hat{{\cal C}}^{ps}_{3},\quad{\cal
C}^{v}_{{\mathfrak{s}}{\mathfrak{s}}}=\hat{{\cal
C}}^{v}_{1}+{\mathfrak{s}}\hat{{\cal C}}^{v}_{3},\quad{\cal
C}^{pv}_{{\mathfrak{s}}{\mathfrak{s}}}=-({\mathfrak{s}}\hat{{\cal
C}}^{pv}_{1}+\hat{{\cal C}}^{pv}_{3})$ (230)

as well as

${\cal C}^{s}_{+-}=-\hat{{\cal C}}^{v}_{+},\quad{\cal C}^{ps}_{+-}=i\hat{{\cal
C}}^{pv}_{+},\quad{\cal C}^{v}_{+-}=-\hat{{\cal C}}^{s}_{+},\quad{\cal
C}^{pv}_{+-}=i\hat{{\cal C}}^{ps}_{+}$ (231)

and

${\cal C}^{s}_{-+}=-\hat{{\cal C}}^{v}_{-},\quad{\cal
C}^{ps}_{-+}=-i\hat{{\cal C}}^{pv}_{-},\quad{\cal C}^{v}_{-+}=-\hat{{\cal
C}}^{s}_{-},\quad{\cal C}^{pv}_{-+}=-i\hat{{\cal C}}^{ps}_{-}.$ (232)

Notice that there is a mismatch between $2$D and $4$D chirality, which
explains why some (pseudo)scalar and (pseudo)vector $4$D couplings correspond
to (pseudo)vector and (pseudo)scalar $2$D couplings respectively, as well as
the presence of additional minus signs in the matching of pseudoscalar and
pseudovector couplings.

The $S$-wave boundary action could have equivalently been formulated in the
original spherical gauge of (3), in which we get:

$\displaystyle I^{\rm int}_{\rm dyon}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\underset{r=\epsilon}{\int}{\rm d}t\,{\rm
d}^{2}\Omega\,\epsilon^{2}\,{\overline{\bm{\psi}}}\Bigl{[}\hat{{\cal
C}}^{s}_{1}+i\,\hat{{\cal C}}^{ps}_{1}\gamma_{5}-i\,\hat{{\cal
C}}^{v}_{1}\gamma^{0}-i\,\hat{{\cal
C}}^{pv}_{1}\gamma_{5}\gamma^{0}+\Bigl{(}\hat{{\cal C}}^{s}_{3}+i\,\hat{{\cal
C}}^{ps}_{3}\gamma_{5}-i\,\hat{{\cal C}}^{v}_{3}\gamma^{0}-i\,\hat{{\cal
C}}^{pv}_{3}\gamma_{5}\gamma^{0}\Bigr{)}(-\tau_{r})$
$\displaystyle-\Bigl{(}\hat{{\cal C}}^{s}_{+}+i\,\hat{{\cal
C}}^{ps}_{+}\gamma_{5}-i\,\hat{{\cal C}}^{v}_{+}\gamma^{0}-i\,\hat{{\cal
C}}^{pv}_{+}\gamma_{5}\gamma^{0}\Bigr{)}e^{i{\mathfrak{a}}}P_{-}\,\tau^{-r}_{+}-\Bigl{(}\hat{{\cal
C}}^{s}_{-}+i\,\hat{{\cal C}}^{ps}_{-}\gamma_{5}-i\,\hat{{\cal
C}}^{v}_{-}\gamma^{0}-i\,\hat{{\cal
C}}^{pv}_{-}\gamma_{5}\gamma^{0}\Bigr{)}e^{-i{\mathfrak{a}}}P_{+}\,\tau^{-r}_{-}\Bigr{]}\,\bm{\psi},$

where $P_{\pm}$ are given by (227) and
$\tau^{-r}_{\pm}\coloneqq\frac{1}{2}e^{\pm i\phi}\left(\tau_{\theta}\mp
i\tau_{\phi}\right)$ act as raising and lowering operators on eigenstates of
$-\tau_{r}$, that is

$\tau^{-r}_{+}\eta_{+}(\theta,\phi)=\eta_{-}(\theta,\phi),\quad\tau^{-r}_{-}\eta_{-}(\theta,\phi)=\eta_{+}(\theta,\phi),\quad\text{and}\quad\tau^{-r}_{-}\eta_{+}(\theta,\phi)=\tau^{-r}_{+}\eta_{-}(\theta,\phi)=0,$
(234)

where $\eta_{\pm}(\theta,\phi)$ are vectors in isospin space given by (25),
which satisfy $(-\tau_{r})\eta_{\pm}(\theta,\phi)=\mp\eta_{\pm}(\theta,\phi)$.

##  Appendix D Properties of the amplitudes $\mathcal{T}_{{\rm
in}},\mathcal{T}_{{\rm out}}$

This appendix derives and summarizes several useful properties of the
amplitudes $\mathcal{T}_{{\rm in}}$ and $\mathcal{T}_{{\rm out}}$ that appear
in the construction of the $in$ and $out$ states. We assume when doing so that
the dyon-fermion action is real (and so the couplings ${\cal
C}^{\scriptscriptstyle A}_{ij}$ are hermitian).

###  D.1 Definition of the amplitudes

The main text shows that the $\mathcal{T}_{{\rm in}},\mathcal{T}_{{\rm out}}$
amplitudes are given in terms of the boundary matrix $\hat{\mathcal{B}}$ by

$\mathcal{T}^{++}_{{\rm
in}}=-\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{21}&\hat{\mathcal{B}}_{24}\\\
\hat{\mathcal{B}}_{41}&\hat{\mathcal{B}}_{44}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{22}&\hat{\mathcal{B}}_{24}\\\
\hat{\mathcal{B}}_{42}&\hat{\mathcal{B}}_{44}\end{array}\right|},{\cal
T}^{+-}_{{\rm
in}}=-\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{23}&\hat{\mathcal{B}}_{24}\\\
\hat{\mathcal{B}}_{43}&\hat{\mathcal{B}}_{44}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{22}&\hat{\mathcal{B}}_{24}\\\
\hat{\mathcal{B}}_{42}&\hat{\mathcal{B}}_{44}\end{array}\right|},{\cal
T}^{-+}_{{\rm
in}}=-\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{22}&\hat{\mathcal{B}}_{21}\\\
\hat{\mathcal{B}}_{42}&\hat{\mathcal{B}}_{41}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{22}&\hat{\mathcal{B}}_{24}\\\
\hat{\mathcal{B}}_{42}&\hat{\mathcal{B}}_{44}\end{array}\right|},\mathcal{T}^{--}_{{\rm
in}}=-\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{22}&\hat{\mathcal{B}}_{23}\\\
\hat{\mathcal{B}}_{42}&\hat{\mathcal{B}}_{43}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{22}&\hat{\mathcal{B}}_{24}\\\
\hat{\mathcal{B}}_{42}&\hat{\mathcal{B}}_{44}\end{array}\right|},$ (235)

and

$\mathcal{T}^{++}_{{\rm
out}}=-\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{12}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{32}&\hat{\mathcal{B}}_{33}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{33}\end{array}\right|},{\cal
T}^{-+}_{{\rm
out}}=-\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{12}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{32}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{33}\end{array}\right|},{\cal
T}^{+-}_{{\rm
out}}=-\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{14}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{34}&\hat{\mathcal{B}}_{33}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{33}\end{array}\right|},\mathcal{T}^{--}_{{\rm
out}}=-\frac{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{14}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{34}\end{array}\right|}{\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{33}\end{array}\right|}.$ (236)

Written directly in terms of the boundary couplings, these become

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

and

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

###  D.2 Rank-2 conditions

The above definitions assume the denominators do not vanish. This is satisfied
when the ${\cal C}^{\scriptscriptstyle A}_{ij}$ are all hermitian, as shown in
equation (58) and the text directly after it.

#### Conditions for rank-two boundary conditions

We next ask what is required to ensure that $\text{rank}(\mathcal{B})=2$. When
this is true one pair of linearly independent columns of $\mathcal{B}$ can be
written as a linear combination of the other two linearly independent columns.
When the ${\cal C}^{\scriptscriptstyle A}_{ij}$ are hermitian we’ve seen the
linearly independent pairs consist of the first and third columns of
$\mathcal{B}$ and the second and fourth columns of ${\cal B}$. Consequently
there exist nonzero coefficients $A_{1}$, $A_{3}$, $B_{1}$ and $B_{3}$ such
that

$\displaystyle\begin{pmatrix}\mathcal{B}_{11}&\mathcal{B}_{21}&\mathcal{B}_{31}&\mathcal{B}_{41}\end{pmatrix}$
$\displaystyle=$ $\displaystyle
A_{1}\begin{pmatrix}\mathcal{B}_{12}&\mathcal{B}_{22}&\mathcal{B}_{32}&\mathcal{B}_{42}\end{pmatrix}+B_{1}\begin{pmatrix}\mathcal{B}_{14}&\mathcal{B}_{24}&\mathcal{B}_{34}&\mathcal{B}_{44}\end{pmatrix},$
$\displaystyle\begin{pmatrix}\mathcal{B}_{13}&\mathcal{B}_{23}&\mathcal{B}_{33}&\mathcal{B}_{43}\end{pmatrix}$
$\displaystyle=$ $\displaystyle
A_{3}\begin{pmatrix}\mathcal{B}_{12}&\mathcal{B}_{22}&\mathcal{B}_{32}&\mathcal{B}_{42}\end{pmatrix}+B_{3}\begin{pmatrix}\mathcal{B}_{14}&\mathcal{B}_{24}&\mathcal{B}_{34}&\mathcal{B}_{44}\end{pmatrix},$
(239)

The required coefficients are given by

$A_{1}=\frac{\left|\begin{array}[]{c c}\mathcal{B}_{21}&\mathcal{B}_{24}\\\
\mathcal{B}_{41}&\mathcal{B}_{44}\end{array}\right|}{\left|\begin{array}[]{c
c}\mathcal{B}_{22}&\mathcal{B}_{24}\\\
\mathcal{B}_{42}&\mathcal{B}_{44}\end{array}\right|},\quad
B_{1}=\frac{\left|\begin{array}[]{c c}\mathcal{B}_{22}&\mathcal{B}_{21}\\\
\mathcal{B}_{42}&\mathcal{B}_{41}\end{array}\right|}{\left|\begin{array}[]{c
c}\mathcal{B}_{22}&\mathcal{B}_{24}\\\
\mathcal{B}_{42}&\mathcal{B}_{44}\end{array}\right|},\quad
A_{3}=\frac{\left|\begin{array}[]{c c}\mathcal{B}_{23}&\mathcal{B}_{24}\\\
\mathcal{B}_{43}&\mathcal{B}_{44}\end{array}\right|}{\left|\begin{array}[]{c
c}\mathcal{B}_{22}&\mathcal{B}_{24}\\\
\mathcal{B}_{42}&\mathcal{B}_{44}\end{array}\right|},\quad
B_{3}=\frac{\left|\begin{array}[]{c c}\mathcal{B}_{22}&\mathcal{B}_{23}\\\
\mathcal{B}_{42}&\mathcal{B}_{43}\end{array}\right|}{\left|\begin{array}[]{c
c}\mathcal{B}_{22}&\mathcal{B}_{24}\\\
\mathcal{B}_{42}&\mathcal{B}_{44}\end{array}\right|}.$ (240)

Comparing these solutions to (235) implies

$A_{1}=-\mathcal{T}^{++}_{{\rm in}},\quad B_{1}=-{\cal T}^{-+}_{{\rm
in}}e^{-i{\mathfrak{a}}},\quad A_{3}=-{\cal T}^{+-}_{{\rm
in}}e^{i{\mathfrak{a}}},\quad B_{3}=-\mathcal{T}^{--}_{{\rm in}}\,,$ (241)

and using these to trade $A_{1,3},B_{1,3}$ for the ${\cal T}_{{\rm in}}$’s in
equation (D.2) gives four tautologies as well as the following four complex
conditions that can be regarded as conditions required if the matrix ${\cal
B}$ is to have rank 2:

$\displaystyle\mathcal{E}_{1}$ $\displaystyle:=$
$\displaystyle\hat{\mathcal{B}}_{13}+{\cal T}^{+-}_{{\rm
in}}\hat{\mathcal{B}}_{12}+\mathcal{T}^{--}_{{\rm
in}}\hat{\mathcal{B}}_{14}=0,$ $\displaystyle\mathcal{E}_{2}$
$\displaystyle:=$ $\displaystyle\hat{\mathcal{B}}_{11}+\mathcal{T}^{++}_{{\rm
in}}\hat{\mathcal{B}}_{12}+{\cal T}^{-+}_{{\rm in}}\hat{\mathcal{B}}_{14}=0,$
(242) $\displaystyle\mathcal{E}_{3}$ $\displaystyle:=$
$\displaystyle\hat{\mathcal{B}}_{33}+{\cal T}^{+-}_{{\rm
in}}\hat{\mathcal{B}}_{32}+\mathcal{T}^{--}_{{\rm
in}}\hat{\mathcal{B}}_{34}=0,$ $\displaystyle\mathcal{E}_{4}$
$\displaystyle:=$ $\displaystyle\hat{\mathcal{B}}_{31}+\mathcal{T}^{++}_{{\rm
in}}\hat{\mathcal{B}}_{32}+{\cal T}^{-+}_{{\rm in}}\hat{\mathcal{B}}_{34}=0.$

These four complex equations amount to eight real conditions. Four of these
state

$\Re(\mathcal{E}_{2})=0,\quad\Re(\mathcal{E}_{3})=0,\quad\text{and}\quad\mathcal{E}_{1}+\mathcal{E}^{*}_{4}=0,$
(243)

and, once expressions (235) are used, can be written purely in terms of the
$\mathcal{T}_{{\rm in}}$ which must therefore satisfy

$\displaystyle\Re(\mathcal{E}_{2})$ $\displaystyle=$
$\displaystyle-(|\mathcal{T}^{++}_{{\rm in}}|^{2}+|{\cal T}^{-+}_{{\rm
in}}|^{2}-1)=0,$ $\displaystyle\Re(\mathcal{E}_{3})$ $\displaystyle=$
$\displaystyle-(|\mathcal{T}^{--}_{{\rm in}}|^{2}+|{\cal T}^{+-}_{{\rm
in}}|^{2}-1)=0,$ (244) $\displaystyle\mathcal{E}_{1}+\mathcal{E}^{*}_{4}$
$\displaystyle=$ $\displaystyle-2(\mathcal{T}^{--}_{{\rm in}}{\cal
T}^{-+\,*}_{{\rm in}}+{\cal T}^{+-}_{{\rm in}}\mathcal{T}^{++\,*}_{{\rm
in}})=0,$

as used in the main text. In deriving the above expressions we use the
identity (55), which follows when the boundary coupling constants are
hermitian. The relations (D.2) simply express the unitarity of the $S$ matrix
since the first two of these equations impose that the fermion number density
of the $in$ modes is conserved during scattering processes, while the third
imposes that the off-diagonal matrix elements of $S^{\dagger}S$ vanish.

The remaining four rank $(\mathcal{B})=2$ conditions within (D.2) are

$\mathcal{E}_{1}=0,\quad\Im(\mathcal{E}_{2})=0,\quad\text{and}\quad\Im(\mathcal{E}_{3})=0.$
(245)

These depend on more than just the ${\cal T}_{{\rm in}}$ amplitudes but can be
expressed in terms of both ${\cal T}_{{\rm in}}$ and ${\cal T}_{{\rm out}}$.
To see why the rank-two conditions (D.2) can be written entirely in terms of
$\mathcal{T}_{{\rm in}},\mathcal{T}_{{\rm out}}$ take the following linear
combinations

$\displaystyle-\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{33}\end{array}\right|^{-1}\left(\hat{\mathcal{B}}_{33}\,\mathcal{E}_{1}-\hat{\mathcal{B}}_{13}\,\mathcal{E}_{3}\right)$
$\displaystyle=$ $\displaystyle\mathcal{T}^{--}_{{\rm in}}{\cal T}^{+-}_{{\rm
out}}+{\cal T}^{+-}_{{\rm in}}\mathcal{T}^{++}_{{\rm out}}=0,$ (248)
$\displaystyle\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{33}\end{array}\right|^{-1}\left(\hat{\mathcal{B}}_{31}\,\mathcal{E}_{1}-\hat{\mathcal{B}}_{11}\,\mathcal{E}_{3}\right)$
$\displaystyle=$ $\displaystyle\mathcal{T}^{--}_{{\rm
in}}\mathcal{T}^{--}_{{\rm out}}+{\cal T}^{+-}_{{\rm in}}{\cal T}^{-+}_{{\rm
out}}-1=0,$ (251)
$\displaystyle\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{33}\end{array}\right|^{-1}\left(\hat{\mathcal{B}}_{31}\,\mathcal{E}_{2}-\hat{\mathcal{B}}_{11}\,\mathcal{E}_{4}\right)$
$\displaystyle=$ $\displaystyle{\cal T}^{-+}_{{\rm in}}\mathcal{T}^{--}_{{\rm
out}}+\mathcal{T}^{++}_{{\rm in}}{\cal T}^{-+}_{{\rm out}}=0,$ (254)
$\displaystyle-\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{33}\end{array}\right|^{-1}\left(\hat{\mathcal{B}}_{33}\,\mathcal{E}_{2}-\hat{\mathcal{B}}_{13}\,\mathcal{E}_{4}\right)$
$\displaystyle=$ $\displaystyle{\cal T}^{-+}_{{\rm in}}{\cal T}^{+-}_{{\rm
out}}+\mathcal{T}^{++}_{{\rm in}}\mathcal{T}^{++}_{{\rm out}}-1=0.$ (257)

where we again use
$\left|\begin{array}[]{ll}\hat{\mathcal{B}}_{11}&\hat{\mathcal{B}}_{13}\\\
\hat{\mathcal{B}}_{31}&\hat{\mathcal{B}}_{33}\end{array}\right|\neq 0$. These
can be solved to give the $\mathcal{T}_{{\rm out}}$ amplitudes in terms of
$\mathcal{T}_{{\rm in}}$ amplitudes and vice versa, giving

$\displaystyle{\cal T}^{+-}_{{\rm out}}=\frac{{\cal T}^{+-}_{{\rm
in}}}{\mathcal{T}^{-+}_{{\rm in}}\mathcal{T}^{+-}_{{\rm
in}}-\mathcal{T}^{--}_{{\rm in}}\mathcal{T}^{++}_{{\rm
in}}},\qquad\,\mathcal{T}^{--}_{{\rm out}}=-\frac{\mathcal{T}^{++}_{{\rm
in}}}{\mathcal{T}^{-+}_{{\rm in}}\mathcal{T}^{+-}_{{\rm
in}}-\mathcal{T}^{--}_{{\rm in}}\mathcal{T}^{++}_{{\rm in}}}$
$\displaystyle\mathcal{T}^{++}_{{\rm out}}=-\frac{\mathcal{T}^{--}_{{\rm
in}}}{\mathcal{T}^{-+}_{{\rm in}}\mathcal{T}^{+-}_{{\rm
in}}-\mathcal{T}^{--}_{{\rm in}}\mathcal{T}^{++}_{{\rm in}}},\qquad\,{\cal
T}^{-+}_{{\rm out}}=\frac{{\cal T}^{-+}_{{\rm in}}}{\mathcal{T}^{-+}_{{\rm
in}}\mathcal{T}^{+-}_{{\rm in}}-\mathcal{T}^{--}_{{\rm
in}}\mathcal{T}^{++}_{{\rm in}}}$ (258)

and

$\displaystyle{\cal T}^{+-}_{{\rm in}}=\frac{{\cal T}^{+-}_{{\rm
out}}}{\mathcal{T}^{-+}_{{\rm out}}\mathcal{T}^{+-}_{{\rm
out}}-\mathcal{T}^{--}_{{\rm out}}\mathcal{T}^{++}_{{\rm
out}}},\qquad\,\mathcal{T}^{--}_{{\rm in}}=-\frac{\mathcal{T}^{++}_{{\rm
out}}}{\mathcal{T}^{-+}_{{\rm out}}\mathcal{T}^{+-}_{{\rm
out}}-\mathcal{T}^{--}_{{\rm out}}\mathcal{T}^{++}_{{\rm out}}}$ (259)
$\displaystyle\mathcal{T}^{++}_{{\rm in}}=-\frac{\mathcal{T}^{--}_{{\rm
out}}}{\mathcal{T}^{-+}_{{\rm out}}\mathcal{T}^{+-}_{{\rm
out}}-\mathcal{T}^{--}_{{\rm out}}\mathcal{T}^{++}_{{\rm out}}},\qquad\,{\cal
T}^{-+}_{{\rm in}}=\frac{{\cal T}^{-+}_{{\rm out}}}{\mathcal{T}^{-+}_{{\rm
out}}\mathcal{T}^{+-}_{{\rm out}}-\mathcal{T}^{--}_{{\rm
out}}\mathcal{T}^{++}_{{\rm out}}}.$

#### Modulus and phase of $\mathcal{T}_{{\rm in}}$ and ${\cal T}_{{\rm out}}$

The constraint conditions (D.2) imply that the four complex ${\cal T}_{{\rm
in}}$ amplitudes satisfy four real constraints and so actually only contain