# Quantum calculation of axion-photon transition in electromagnetodynamics for
cavity haloscope

Tong Li<EMAIL_ADDRESS>Rui-Jia Zhang<EMAIL_ADDRESS>School of Physics, Nankai University, Tianjin 300071, China

###### Abstract

The Witten effect implies the presence of electric charge of magnetic monople
and possible relationship between axion and dyon. The axion-dyon dynamics can
be reliably built based on the quantum electromagnetodynamics (QEMD) which was
developed by Schwinger and Zwanziger in 1960’s. A generic low-energy axion-
photon effective field theory can also be realized in the language of
“generalized symmetries” with higher-form symmetries and background gauge
fields. In this work, we implement the quantum calculation of axion-single
photon transition rate inside a homogeneous electromagnetic field in terms of
the new axion interaction Hamiltonian in QEMD. This quantum calculation can
clearly imply the enhancement of conversion rate through resonant cavity in
axion haloscope experiments. We also show the promising potentials on the
cavity search of new axion-photon couplings in QEMD.

##  I Introduction

It is well-known that the strong CP problem in quantum chromodynamics (QCD)
arises from the source of CP violation in QCD Lagrangian with $\theta
G^{a~{}\mu\nu}\tilde{G}^{a}_{\mu\nu}$. The Peccei-Quinn (PQ) mechanism solves
the strong CP problem by introducing a pseudo-Goldstone boson $a$ called axion
after the spontaneous breaking of a QCD anomalous $U(1)_{\rm PQ}$ global
symmetry Peccei and Quinn (1977a, b); Weinberg (1978); Wilczek (1978); Baluni
(1979); Crewther  _et al._ (1979); Kim (1979); Shifman  _et al._ (1980); Dine
_et al._ (1981); Zhitnitsky (1980); Baker  _et al._ (2006); Pendlebury  _et
al._ (2015). The chiral transformation of the quark fields with PQ charges
also leads to the anomaly under QED and the coupling
$g_{a\gamma\gamma}aF^{\mu\nu}\tilde{F}_{\mu\nu}$ between axion and
electromagnetic fields. In 1979, E. Witten showed that a CP violating term
$\theta F^{\mu\nu}\tilde{F}_{\mu\nu}$ with non-zero vacuum angle $\theta$
provides an electric charge $-\theta e/2\pi$ for magnetic monopoles Witten
(1979). This so-called Witten effect implies a close relationship between
axion and magnetic monopole due to the axion-photon coupling
$g_{a\gamma\gamma}a\vec{E}\cdot\vec{B}$.

This axion-dyon dynamics was first derived by W. Fischler et al. under the
classical electromagnetism Fischler and Preskill (1983) and was proposed to
solve cosmological problems in recent years Kawasaki  _et al._ (2016); Nomura
_et al._ (2016); Kawasaki  _et al._ (2018); Houston and Li (2017); Sato  _et
al._ (2018). In their works, however, the magnetic monopoles were treated as
quasi-classical external sources and the quantization of electromagnetism is
not complete. A reliable quantization in the presence of magnetic monopoles
was developed by J. S. Schwinger and D. Zwanziger in 1960’s and called quantum
electromagnetodynamics (QEMD) Schwinger (1966); Zwanziger (1968, 1971).
Recently, based on the QEMD framework, Ref. Sokolov and Ringwald (2022)
constructed a more generic axion-photon Lagrangian in the low-energy axion
effective field theory (EFT). Besides the Witten effect term, more anomalous
axion-photon interactions and couplings arise assuming the existence of heavy
PQ-charged fermions with electric and magnetic charges. This is in contrary to
the conventional axion EFT $g_{a\gamma\gamma}aF^{\mu\nu}\tilde{F}_{\mu\nu}$ in
the quantum electrodynamics (QED) framework. As a result of the above generic
axion-photon Lagrangian, the conventional axion Maxwell equations Sikivie
(1983) are further modified and some consequent new detection strategies of
axion are studied in recent years Li  _et al._ (2023a); Tobar  _et al._
(2022); McAllister  _et al._ (2022); Li  _et al._ (2023b).

The key property of QEMD is to extend the $U(1)_{\rm EM}$ gauge group in the
Standard Model (SM) by substituting two $U(1)$ gauge groups $U(1)_{\rm
E}\times U(1)_{\rm M}$ to introduce both electric and magnetic charges. Then,
two four-potentials $A^{\mu}$ and $B^{\mu}$ (instead of only one in QED) are
introduced corresponding to the two $U(1)$ gauge groups, respectively. A non-
trivial form of equal-time canonical commutation relations between them can be
built Zwanziger (1971). It guarantees the preservation of the right degrees of
freedom of physical photon.

The other property of QEMD is that it seemingly acts like a non-local quantum
field theory (QFT). To obtain the covariant Maxwell equations in the presence
of conserved electric current $j_{e}$ and magnetic current $j_{m}$, one needs
to introduce an arbitrary spacelike vector $n^{\mu}=(0,\vec{n})$. The
electromagnetic field strength tensor $F_{\mu\nu}$ and its dual tensor
$F_{\mu\nu}^{d}$ 111Below we define
$X^{d}_{\mu\nu}\equiv\tilde{X}_{\mu\nu}=\epsilon_{\mu\nu\alpha\beta}X^{\alpha\beta}/2$
as the Hodge dual of tensor $X_{\mu\nu}$. Also, $(Y\wedge Z)^{\mu\nu}\equiv
Y^{\mu}Z^{\nu}-Y^{\nu}Z^{\mu}$ for any four-vectors $Y$ and $Z$. are given in
this way

$\displaystyle F=\partial\wedge A-(n\cdot\partial)^{-1}(n\wedge
j_{m})^{d}\;,~{}~{}F^{d}=\partial\wedge B+(n\cdot\partial)^{-1}(n\wedge
j_{e})^{d}\;,$ (1)

where the integral operator $(n\cdot\partial)^{-1}$ satisfies
$n\cdot\partial(n\cdot\partial)^{-1}(\vec{x})=\delta(\vec{x})$. The second
terms on the right-hand side of Eq. (1) induce likely non-local property in
QEMD. This non-locality in QEMD can also be realized in the language of
“Generalized Global Symmetries” with higher-form group symmetries Kapustin and
Seiberg (2014); Seiberg (2010); Gaiotto  _et al._ (2015). We will show the
action of higher-form symmetry transformation as a topological operator on the
dynamical fields in a theory of two $U(1)$ one-form symmetries. One can prove
that the non-local part does not play any role in physical processes and the
Lorentz invariance is not violated Brandt  _et al._ (1978, 1979); Sokolov and
Ringwald (2023).

The QCD axion (see Ref. Di Luzio  _et al._ (2020) for a recent review) can
become as dark matter (DM) candidate through the misalignment mechanism
Preskill  _et al._ (1983); Dine and Fischler (1983). The conventional axion
haloscope experiments such as ADMX Du  _et al._ (2018); Bartram  _et al._
(2021) are built based on resonant cavity technique to search for
$\mathcal{O}(10)~{}\mu{\rm eV}$ axion DM. The cosmic axions resonantly convert
into a monochromatic photon with the enhancement from a high quality factor
$Q$ when the cavity resonant frequency is tuned to the axion mass $m_{a}$. The
mean number of thermal photons in the cavity at finite temperature $T$ is
given by

$\displaystyle n(\omega_{a},T)={1\over e^{\omega_{a}/k_{B}T}-1}\;,$ (2)

where $k_{B}$ is Boltzmann’s constant. When
$m_{a}\gtrsim\mathcal{O}(10)~{}\mu{\rm eV}$ and $T\approx 20~{}{\rm mK}$, the
occupation number of thermal photon is quite low and the cavity can be treated
as a single photon emitter. Although the usual electromagnetic power radiated
in cavity is calculated in classical field theory Sikivie (1983), the actual
description is quantum mechanical process of axion to photon conversion
Sikivie (2021). To describe this axion-single photon conversion $a\to\gamma$,
the calculation of transition rate should be performed at quantum level
Beutter  _et al._ (2019); Yang  _et al._ (2022). In this work, we implement
the quantum calculation of photon $|0\rangle\to|1\rangle$ transition rate
inside a homogeneous electromagnetic field in terms of the new axion
interaction Hamiltonian in QEMD. This quantum calculation can clearly imply
the enhancement of conversion rate through resonance which is not certain in
the classical picture. Our work will show the basic method for the generic
cavity search of the new axion-photon couplings in QEMD.

This paper is organized as follows. In Sec. II, we introduce the generic
axion-photon interactions. We will show the realizations of this theory in
both QEMD and generalized symmetry. In Sec. III, we perform a complete quantum
calculation of $a\to\gamma$ transition rate based on the new axion interaction
Hamiltonian in QEMD. The transition rates from different types of cavity mode
are obtained under external magnetic or electric background. We also show the
sensitivity of resonant cavity to axion-photon couplings in QEMD in Sec. IV.
Our conclusions are drawn in Sec. V.

##  II The realizations of generic axion-photon interactions in QEMD and
generalized symmetry

###  II.1 The generic axion-photon interactions in QEMD

In the QEMD theory, the photon is described by two four-potentials $A^{\mu}$
and $B^{\mu}$. Correspondingly, the gauge group of QEMD becomes $U(1)_{\rm
E}\times U(1)_{\rm M}$ which inherently introduces both electric and magnetic
charges. The equal-time canonical commutation relations between the two four-
potentials were obtained Zwanziger (1971)

$\displaystyle[A^{\mu}(t,\vec{x}),B^{\nu}(t,\vec{y})]$ $\displaystyle=$
$\displaystyle i\epsilon^{\mu\nu}_{~{}~{}\kappa
0}n^{\kappa}(n\cdot\partial)^{-1}(\vec{x}-\vec{y})\;,$ (3)
$\displaystyle~{}[A^{\mu}(t,\vec{x}),A^{\nu}(t,\vec{y})]$ $\displaystyle=$
$\displaystyle[B^{\mu}(t,\vec{x}),B^{\nu}(t,\vec{y})]=-i(g_{0}^{~{}\mu}n^{\nu}+g_{0}^{~{}\nu}n^{\mu})(n\cdot\partial)^{-1}(\vec{x}-\vec{y})\;.$
(4)

As a result of Eq. (1), the electromagnetic field strength tensors $F$ and
$F^{d}$ are then introduced in the way that

$\displaystyle n\cdot F=n\cdot(\partial\wedge A)\;,~{}~{}n\cdot
F^{d}=n\cdot(\partial\wedge B)\;,$ (5)

where $n^{\mu}=(0,\vec{n})$ is an arbitrary fixed spatial vector. Apparently,
the two four-potentials have opposite parities. In the absence of electric and
magnetic currents $j_{e},j_{m}$, one has a simplified form

$\displaystyle F=\partial\wedge A=-(\partial\wedge
B)^{d}\;,~{}~{}~{}F^{d}=\partial\wedge B=(\partial\wedge A)^{d}\;.$ (6)

The $n$-related terms produce the non-locality in this theory. This non-local
property can be realized by the two-particle irreducible representation in the
QFT theory with both electric charge $q$ and magnetic charge $g$. Each two-
particle state $(i,j)$ is characterized by the Dirac-Schwinger-Zwanziger (DSZ)
quantization condition

$\displaystyle q_{i}g_{j}-q_{j}g_{i}=2\pi N\;,~{}~{}~{}N\in\mathbb{Z}\;.$ (7)

Thus, the cluster decomposition principle is obviously violated by irreducible
two-particle state Sokolov and Ringwald (2023) and the Lorentz invariance is
seemingly violated in this QEMD theory. However, it was formally showed that
the observables of the QEMD are Lorentz invariant using the path-integral
approach Brandt  _et al._ (1978, 1979); Sokolov and Ringwald (2023). After all
the quantum corrections are properly accounted for, the dependence on the
spatial vector $n_{\mu}$ in the action $S$ factorizes into an integer linking
number $L_{n}$ multiplied by the combination of charges in the DSZ
quantization condition $q_{i}g_{j}-q_{j}g_{i}$. This $n$ dependent part is
then given by $2\pi N$ with $N$ being an integer. Since $S$ contributes to the
generating functional in the form of exponent $e^{iS}$, this Lorentz-violating
part does not play any role in physical processes.

The Lagrangian for the anomalous interactions between axion $a$ and photon in
QEMD is given by Sokolov and Ringwald (2022)

$\displaystyle\mathcal{L}$ $\displaystyle\supset$ $\displaystyle-{1\over
4}g_{aAA}~{}a~{}{\rm tr}[(\partial\wedge A)(\partial\wedge A)^{d}]-{1\over
4}g_{aBB}~{}a~{}{\rm tr}[(\partial\wedge B)(\partial\wedge B)^{d}]$ (8)
$\displaystyle-{1\over 2}g_{aAB}~{}a~{}{\rm tr}[(\partial\wedge
A)(\partial\wedge B)^{d}]\;.$

The first two dimension-five operators are CP-conserving axion interactions.
Their couplings $g_{aAA}$ and $g_{aBB}$ are governed by the $U(1)_{\rm
PQ}U(1)_{\rm E}^{2}$ and $U(1)_{\rm PQ}U(1)_{\rm M}^{2}$ anomalies,
respectively. As $A^{\mu}$ and $B^{\mu}$ have opposite parities, the third
operator is CP-violating one and its coupling $g_{aAB}$ is determined by the
$U(1)_{\rm PQ}U(1)_{\rm E}U(1)_{\rm M}$ anomaly. The inclusion of this term
accounts for the intrinsic CP violation in a dyon theory. It is analogous to
the interaction between electromagnetic field and a scalar $\phi$ with
positive parity $\phi F^{\mu\nu}F_{\mu\nu}$ Donohue  _et al._ (2021). In terms
of classical electromagnetic fields, the above axion-photon Lagrangian becomes
222We use symbol “H” rather than “B” to denote magnetic field in order not to
conflict with the four-potential $B^{\mu}$.

$\displaystyle\mathcal{L}$ $\displaystyle\supset$ $\displaystyle-{1\over
4}(g_{aAA}-g_{aBB})~{}a~{}F_{\mu\nu}F^{d~{}\mu\nu}+{1\over
2}g_{aAB}~{}a~{}F_{\mu\nu}F^{\mu\nu}$ (9) $\displaystyle=$
$\displaystyle(g_{aAA}-g_{aBB})~{}a~{}\vec{H}\cdot\vec{E}+g_{aAB}~{}a~{}(\vec{H}^{2}-\vec{E}^{2})\;.$

This is the form of interactions that we will use for the quantum calculation
of $a\to\gamma$ transition rate below. Taking care of the above anomalies, one
can calculate the coupling coefficients as

$\displaystyle g_{aAA}={Ee^{2}\over 4\pi^{2}v_{\rm
PQ}}\;,~{}~{}g_{aBB}={Mg_{0}^{2}\over 4\pi^{2}v_{\rm
PQ}}\;,~{}~{}g_{aAB}={Deg_{0}\over 4\pi^{2}v_{\rm PQ}}\;,$ (10)

where $e$ is the unit of electric charge, $g_{0}$ is the minimal magnetic
charge with $g_{0}=2\pi/e$ in the DSZ quantization condition, and $v_{\rm PQ}$
is the $U(1)_{\rm PQ}$ symmetry breaking scale. $E(M)$ is the electric
(magnetic) anomaly coefficient and $D$ is the mixed electric-magnetic CP-
violating anomaly coefficient. They can be computed by integrating out heavy
PQ-charged fermions with electric and magnetic charges. Ref. Sokolov and
Ringwald (2022) performed the calculation of the anomaly coefficients by
following Fujikawa’s path integral method Fujikawa (1979). As the DSZ
quantization condition indicates $g_{0}\gg e$, we have the scaling of the
axion-photon couplings as $g_{aBB}\gg|g_{aAB}|\gg g_{aAA}$.

###  II.2 Generalized symmetry realization

The QEMD theory describing monopole dynamics can also be realized in the
language of higher-form symmetries as a topological QFT (TQFT) Kapustin and
Seiberg (2014); Seiberg (2010); Gaiotto  _et al._ (2015). The above generic
axion couplings then naturally arise when an axion-Maxwell theory couples to a
TQFT Brennan  _et al._ (2023). Below we briefly show the spirit of generalized
symmetry and the realization of above generic axion couplings in TQFT.

We consider a general $p$-form symmetry in $d$ dimensions. The symmetry
transformation as an operator is associated with a co-dimension $p+1$ manifold
$M^{(d-p-1)}$

$\displaystyle U_{g}(M^{(d-p-1)})\;,$ (11)

where $g$ is an element of symmetry group $G$. The operators obey the
multiplication rule

$\displaystyle
U_{g}(M^{(d-p-1)})U_{g^{\prime}}(M^{(d-p-1)})=U_{g^{\prime\prime}}(M^{(d-p-1)})\;,$
(12)

where $g^{\prime\prime}=gg^{\prime}\in G$. The dependence of operator
$U_{g}(M^{(d-p-1)})$ on manifold $M^{(d-p-1)}$ is topological and remains
unchanged unless the deformation of $M^{(d-p-1)}$ crosses an operator $V$. The
topological operator $U_{g}(M^{(d-p-1)})$ acts on a $p$-dimensional operator
$V$ of manifold $C^{(p)}$ in the form of Gaiotto  _et al._ (2015)

$\displaystyle U_{g}(M^{(d-p-1)})V(C^{(p)})=g(V)^{\langle
M^{(d-p-1)},C^{(p)}\rangle}V(C^{(p)})U_{g}(M^{(d-p-1)})\;,$ (13)

where $g(V)$ is the representation of group element $g$ of $V$, and $\langle
M^{(d-p-1)},C^{(p)}\rangle$ is the linking number for manifolds $M^{(d-p-1)}$
and $C^{(p)}$. It is then natural to couple the system to flat background
gauge field of the higher-form symmetry. Taking a $d=4$ Maxwell theory for
illustration, one claims there are two one-form $U(1)$ symmetries, i.e.
$U(1)_{\rm E}\times U(1)_{\rm M}$. The symmetries are generated by the
integral of $d-1=3$-form currents

$\displaystyle U^{\rm E}(M^{(2)})$ $\displaystyle=$ $\displaystyle
e^{i\alpha\oint_{M^{(2)}}j_{e}}\;,~{}~{}g_{\rm E}=e^{i\alpha}\in U(1)_{\rm
E}\;,$ (14) $\displaystyle U^{\rm M}(M^{(2)})$ $\displaystyle=$ $\displaystyle
e^{i\beta\oint_{M^{(2)}}j_{m}}\;,~{}~{}g_{\rm M}=e^{i\beta}\in U(1)_{\rm
M}\;,$ (15)

where $g_{\rm E(M)}$ is the element of group $U(1)_{\rm E(M)}$, and the
electric and magnetic charges are given by $q(M^{(2)})=\oint_{M^{(2)}}j_{e}$
and $g(M^{(2)})=\oint_{M^{(2)}}j_{m}$, respectively. The representation then
generally becomes

$\displaystyle(g_{\rm E(M)})^{Q\langle M^{(2)},C^{(1)}\rangle}\;,$ (16)

where $Q$ denotes the conserved charge. These two operators act on the Wilson
loop operator and the ’t Hooft loop operator, respectively. Thus, we are able
to introduce two two-form background gauge fields of higher-form symmetries,
i.e. $A^{\mu}$ and $B^{\mu}$ as those in Zwanziger’s QEMD theory.

Inspired by this kind of higher-form symmetry realization, one can consider an
axion-Maxwell theory coupled to a $\mathbb{Z}_{n}$ TQFT as shown in Ref.
Brennan  _et al._ (2023). The gauge field $A^{\mu}$ is an one-form gauge field
and $F^{(2)}_{A}\equiv\partial\wedge A$ is its two-form field strength for the
$U(1)_{\rm EM}$ group in the SM. The action of the axion-Maxwell theory in
this sector becomes

$\displaystyle S_{0}={1\over 2g^{2}}\int F^{(2)}_{A}F^{(2)}_{A}-{iK_{A}\over
8\pi^{2}f_{a}}\int aF^{(2)}_{A}(F^{(2)}_{A})^{d}\;,$ (17)

where $f_{a}$ is the axion decay constant, and $K_{A}\in\mathbb{Z}$ is a
discrete coupling constant. This axion-Maxwell theory is considered to couple
to a $\mathbb{Z}_{n}$ TQFT. In this TQFT sector, the action of axion theory is

$\displaystyle S_{1}={in\over 2\pi}\int B^{(2)}(F^{(2)}_{B})^{d}-{iK_{B}\over
4\pi^{2}f_{a}}\int aF^{(2)}_{B}(F^{(2)}_{B})^{d}-{iK_{AB}\over
8\pi^{2}f_{a}}\int aF^{(2)}_{A}(F^{(2)}_{B})^{d}\;,$ (18)

where $F^{(2)}_{B}\equiv\partial\wedge B^{(1)}$ is the two-form field strength
of an one-form $\mathbb{Z}_{n}$ gauge field $B^{(1)}$ associated with another
$U(1)$ gauge group, and $B^{(2)}$ is a two-form gauge field associated with
one-form $\mathbb{Z}_{n}^{(1)}$ gauge symmetry. Then, the action of theory
with topological QFT couplings via axion-portal can be given by $S_{0}+S_{1}$
Brennan  _et al._ (2023). The second term in $S_{0}$ and the last two terms in
$S_{1}$ agree with the generic QEMD axion interactions shown in Eq. (8).

##  III Quantum calculation of axion-photon transition in QEMD

In this section we perform the quantum calculations of axion-photon transition
in QEMD under an external magnetic field or electric field as background.

###  III.1 Magnetic background

Suppose an external magnetic field $H_{0}$ along the z-direction, according to
Eq. (9), the axion and photon interaction can be written as

$\displaystyle\mathcal{L}_{a\gamma\gamma}=(g_{aAA}-g_{aBB})a\vec{E}\cdot\vec{H_{0}}+g_{aAB}a\vec{H}\cdot\vec{H_{0}}\;,$
(19)

where we have set external electric field to zero, i.e.,
$\vec{H_{0}}=\hat{z}H_{0}\neq 0$, $\vec{E_{0}}=0$. Due to the extremely light
mass and low velocity of the axion DM, its de Brogile wavelength is of the
order of $10^{3}/m_{a}$. It is much larger than the typical size of the cavity
in haloscope experiments $\sim 1/m_{a}$. Thus, the axion field inside the
cavity can be approximately viewed as spatially independent and be given in
the form of cosine oscillation: $a(\vec{x},t)\approx
a_{0}\cos{\omega_{a}t}=\frac{\sqrt{2\rho_{a}}}{m_{a}}\cos{\omega_{a}t}$. The
Hamiltonian for the above interaction can be written as follows

$\displaystyle
H_{I}=-\int{d^{3}x\mathcal{L}_{a\gamma\gamma}}=\frac{\sqrt{2\rho_{a}}}{m_{a}}H_{0}\cos{(\omega_{a}t)}\left[(g_{aBB}-g_{aAA})\int{d^{3}x\hat{z}\cdot\vec{E}}-g_{aAB}\int{d^{3}x\hat{z}\cdot\vec{H}}\right]\;.$
(20)

We find that the key difference of axion in QEMD and QED lies in the axion
induced electromagnetic fields. In QEMD, the axion induced fields $\vec{E}$
and $\vec{H}$ can occur simultaneously due to the presence of three couplings.
As a result, the interactions between the axion field $a$ and the
electromagnetic fields are divided into two parts: $g_{aBB}-g_{aAA}$ and
$g_{aAB}$, corresponding to $\vec{E}$ and $\vec{H}$ respectively. However, in
the traditional axion QED, only the $\hat{z}\cdot\vec{E}$ term can appear
given the substitution of $(g_{aBB}-g_{aAA})\to-g_{a\gamma\gamma}$.

Next, we will use a quantization approach to deal with the integrals
$\int{d^{3}x\hat{z}\cdot\vec{E}}$ and $\int{d^{3}x\hat{z}\cdot\vec{H}}$. In
QEMD, the magnetic field and electric field can be given by using the curl of
two vector potentials $\vec{A}$ and $\vec{B}$, respectively Zwanziger (1971)

$\displaystyle\vec{H}=\nabla\times\vec{A},~{}\vec{E}=-\nabla\times\vec{B}\;.$
(21)

We can expand the vector potentials $\vec{A}$ and $\vec{B}$ in terms of
creation and annihilation operators as well as mode functions
$\mathbf{u}_{k}(\mathbf{x})$

$\displaystyle\vec{A}(\mathbf{x},t)=\sum_{k}\frac{1}{\sqrt{2\omega_{k}V}}(a_{k}\mathbf{u}_{k}^{(A)}(\mathbb{x})e^{-i\omega_{k}t}+a_{k}^{\dagger}\mathbb{u}_{k}^{(A)*}(\mathbb{x})e^{i\omega_{k}t})\;,$
(22)
$\displaystyle\vec{B}(\mathbf{x},t)=\sum_{k}\frac{1}{\sqrt{2\omega_{k}V}}(a_{k}\mathbf{u}_{k}^{(B)}(\mathbb{x})e^{-i\omega_{k}t}+a_{k}^{\dagger}\mathbb{u}_{k}^{(B)*}(\mathbb{x})e^{i\omega_{k}t})\;,$
(23)

where $V$ is the volume of a cavity, and $\mathbf{u}_{k}(\mathbf{x})$ satisfy
the equations of motion with cavity-wall boundary conditions

$\displaystyle\frac{n\cdot\partial}{n^{2}}(n\cdot\partial
A^{\mu}-\partial^{\mu}n\cdot A-n^{\mu}\partial\cdot
A-\epsilon^{\mu}_{\nu\rho\sigma}n^{\nu}\partial^{\rho}B^{\sigma})=0\;,$ (24)
$\displaystyle\frac{n\cdot\partial}{n^{2}}(n\cdot\partial
B^{\mu}-\partial^{\mu}n\cdot B-n^{\mu}\partial\cdot
B-\epsilon^{\mu}_{\nu\rho\sigma}n^{\nu}\partial^{\rho}A^{\sigma})=0\;.$ (25)

The operators $a_{k}$ and $a_{k}^{\dagger}$ can annihilate and create physical
single-photon state even though two vector potentials are introduced to
describe photon. The above equations of motion are two first-order
differential equations. They constrain $A$ and $B$ together with the equation
corresponding to the gauge-fixing condition

$\displaystyle\partial^{2}n\cdot A+\partial^{2}n\cdot B=0\;.$ (26)

They reduce the four degrees of $A$ and $B$ to the two degrees of freedom for
a massless vector field. Furthermore, due to the equal-time commutation
relations in Eq. (3) and Eq. (4), the total degrees of freedom of photon can
be further reduced to two. Therefore, even if QEMD introduces two potentials
$A$ and $B$, the physical degrees of freedom of photon are preserved. Using
the relations given by Eq. (21), the electromagnetic fields can be obtained.
Although the exact forms of $\mathbf{u}_{k}^{(A,B)}$ are unknown, their curl
in a cavity can be given by

$\displaystyle\nabla\times\mathbf{u}_{k}^{(A)}=\omega_{k}\mathbf{u}_{k}^{H},~{}~{}~{}\nabla\times\mathbf{u}_{k}^{(B)}=\omega_{k}\mathbf{u}_{k}^{E}\;,$
(27)

where $\mathbf{u}_{k}^{H}$ and $\mathbf{u}_{k}^{E}$ are the actual
electromagnetic field modes inside the cavity with the normalization
$\frac{1}{V}\int{d^{3}x|\mathbf{u}_{k}^{E,H}|^{2}}=1$.

One can then calculate the $|0\rangle\to|1\rangle$ photon transition matrix
element as well as the transition probability inside the cavity with an
external magnetic field $\vec{H_{0}}$ Yang  _et al._ (2022). Up to the first
order, we have

$\displaystyle\begin{split}P&\approx\left|\langle
1|\int_{0}^{t}dtH_{I}|0\rangle\right|^{2}\\\
&\approx\frac{\rho_{a}}{m_{a}^{2}}H_{0}^{2}V\Bigg{[}(g_{aBB}-g_{aAA})^{2}\sum_{k}\omega_{k}C_{k}^{E}\\\
&+g_{aAB}^{2}\sum_{k}\omega_{k}C_{k}^{H}+2g_{aAB}(g_{aBB}-g_{aAA})\sum_{k}\omega_{k}C_{k}^{EH}\Bigg{]}\times\frac{\sin^{2}[(\omega_{k}-\omega_{a})t/2]}{4[(\omega_{k}-\omega_{a})/2]^{2}}\;,\end{split}$
(28)

where the relations in Eq. (27) are plugged into the above result, and
$C_{k}^{E,H},~{}C_{l}^{EH}$ are the form factors that characterize the
coupling strength of cavity mode $k$ to axions

$\displaystyle C_{k}^{E}=\frac{|\int
d^{3}x\hat{z}\cdot\mathbf{u}_{k}^{E}|^{2}}{V\int{d^{3}x|\mathbf{u}_{k}^{E}|^{2}}},~{}C_{k}^{H}=\frac{|\int
d^{3}x\hat{z}\cdot\mathbf{u}_{k}^{H}|^{2}}{V\int{d^{3}x|\mathbf{u}_{k}^{H}|^{2}}},~{}C_{k}^{EH}=\frac{\mathbf{Re}\left[\int
d^{3}x\hat{z}\cdot\mathbf{u}_{k}^{E}\int
d^{3}x\hat{z}\cdot\mathbf{u}_{k}^{H*}\right]}{V\sqrt{\int
d^{3}x|\mathbf{u}_{k}^{E}|^{2}\int d^{3}x|\mathbf{u}_{k}^{H}|^{2}}}\;.$ (29)

In practice, the transition emission process of a single photon is expected to
take a long time $t$. The time factor can thus approximately become

$\displaystyle\frac{\sin^{2}[(\omega_{k}-\omega_{a})t/2]}{4[(\omega_{k}-\omega_{a})/2]^{2}}\approx\pi
t\delta(\omega_{k}-\omega_{a})/2\;.$ (30)

Finally, the transition rate in the cavity can be obtained as

$\displaystyle
R=dP/dt=\frac{\pi}{2}\frac{\rho_{a}}{m_{a}^{2}}H_{0}^{2}VQ\left[(g_{aBB}-g_{aAA})^{2}C_{\omega_{a}}^{E}+g_{aAB}^{2}C_{\omega_{a}}^{H}+2g_{aAB}(g_{aBB}-g_{aAA})C_{\omega_{a}}^{EH}\right],$
(31)

where the discrete summation over the cavity modes $k$ is converted into
continuous integrals with
$\sum_{k}C_{k}^{E,H}\omega_{k}\delta(\omega_{k}-\omega_{a})\approx
QC_{\omega_{a}}^{E,H},~{}\sum_{k}C_{k}^{EH}\omega_{k}\delta(\omega_{k}-\omega_{a})\approx
QC_{\omega_{a}}^{EH}$. For a given cavity in a haloscope experiment, $Q$ is
assumed to be universal for any terms at the same moment $t$, we can thus
factor out $Q$ outside the parentheses. This result clearly shows the
enhancement of axion-photon transition by the cavity’s quality factor when
$\omega_{k}\approx\omega_{a}$. The modes existing in a cylindrical cavity
include TE modes (transverse electric modes with $E_{z}=0,~{}H_{z}\neq 0$) and
TM modes (transverse magnetic modes with $H_{z}=0,~{}E_{z}\neq 0$), and there
may be also some TEM modes ($E_{z}=0,~{}H_{z}=0$) embedded in. Based on
different types of cavity mode, the transition rate $R$ in Eq. (31) can then
be simplified as

$\displaystyle\begin{split}R_{\text{TE}}&=\frac{\pi}{2}\frac{\rho_{a}}{m_{a}^{2}}H_{0}^{2}VQg_{aAB}^{2}C_{\omega_{a}}^{H},~{}\text{with}~{}C^{E}=C^{EH}=0\;,\\\
R_{\text{TM}}&=\frac{\pi}{2}\frac{\rho_{a}}{m_{a}^{2}}H_{0}^{2}VQ(g_{aBB}-g_{aAA})^{2}C_{\omega_{a}}^{E},~{}\text{with}~{}C^{H}=C^{EH}=0\;.\end{split}$
(32)

This indicates that TEM modes have vanishing form factors for pure transverse
fields, while only TE and TM modes induce non-zero form factors with
electromagnetic field in longitudinal direction.

In a cavity, the distribution of the electromagnetic field is usually quite
different from that in vacuum. A cylindrical microwave resonant cavity can be
viewed as a circular waveguide of length $L$ with short-circuit at both ends.
Two movable bulk copper rods can be placed inside the cavity to achieve tuning
frequency Bartram  _et al._ (2021). The internal distribution of the
electromagnetic field must satisfy both the Helmholtz equation and the
corresponding boundary conditions, including those at the ends and on the
walls of the cavity. The Helmholtz equation is

$\displaystyle\nabla^{2}u(r,\phi,z)+k^{2}u(r,\phi,z)=0,~{}\vec{E}(r,\phi,z)~{}{\rm
or}~{}\vec{H}(r,\phi,z)=\hat{z}u(r,\phi,z)\;,$ (33)

where only the $z$-component of the modes couples to the axion field according
to the definition of the form factors in Eq. (29). Their solutions in
different modes satisfy the following conditions

$\displaystyle\text{TE
modes}~{}(E_{z}=0,~{}H_{z}=u(r,\phi,z)):\left\\{\begin{array}[]{ll}r,~{}~{}\dfrac{\partial
u(r,\phi,z)}{\partial r}\bigg{|}_{r=a}=0\\\
\phi,~{}~{}u(r,\phi,z)=u(r,\phi+2\pi m,z)\\\
z,~{}~{}u(r,\phi,z)\bigg{|}_{z=0,~{}L}=0\;,\end{array}\right.$ (37)
$\displaystyle\text{TM
modes}~{}(H_{z}=0,~{}E_{z}=u(r,\phi,z)):\left\\{\begin{array}[]{ll}r,~{}~{}u(r,\phi,z)\bigg{|}_{r=a}=0\\\
\phi,~{}~{}u(r,\phi,z)=u(r,\phi+2\pi m,z)\\\ z,~{}~{}\dfrac{\partial
u(r,\phi,z)}{\partial z}\bigg{|}_{z=0,~{}L}=0\;,\end{array}\right.$ (41)

where $a$ is the radius of circular cross section, $L$ is the length of cavity
along the $z$-axis, and $m=(0,\pm 1,\pm 2,\cdots)$ represents a series of
integers required by the periodic boundary conditions. The solutions of the
above differential equations yield a series of possible electromagnetic
resonant modes that can exist inside the cavity

$\displaystyle\begin{split}\text{TE}_{mnp}:~{}\hat{z}\cdot\mathbf{u}_{k}^{H}(r,\phi,z)=H_{z}(r,\phi,z)_{mnp}&=H_{mnp}J_{m}(k_{\rho}r)\left\\{\begin{array}[]{ll}\cos{m\phi}\\\
\sin{m\phi}\end{array}\right\\}\sin\left(\frac{p\pi z}{L}\right)\;,\\\
\text{TM}_{mnp}:~{}\hat{z}\cdot\mathbf{u}_{k}^{E}(r,\phi,z)=E_{z}(r,\phi,z)_{mnp}&=E_{mnp}J_{m}(k_{\rho}r)\left\\{\begin{array}[]{ll}\cos{m\phi}\\\
\sin{m\phi}\end{array}\right\\}\cos\left(\frac{p\pi
z}{L}\right)\;,\end{split}$ (42)

where $H_{mnp}$ and $E_{mnp}$ are dimensionless coefficients to ensure the
mode normalization.

From these solutions, it can be seen that the modes inside the cavity are
represented by three integers $m$, $n$ and $p$ under the fixed boundary
conditions. For TE modes, they are
$m=(0,1,2,\cdots),~{}n=(1,2,3,\cdots),~{}p=(1,2,3,\cdots)$, and for TM modes
$m=(0,1,2,\cdots),~{}n=(1,2,3,\cdots),~{}p=(0,1,2,\cdots)$. The eigenvalue of
the radial part of Helmholtz equation is $k_{\rho}=(x_{m,n}^{\prime}/a)$ for
TE modes or $k_{\rho}=(x_{m,n}/a)$ for TM modes with $x$ and $x^{\prime}$
being the $n$-th zero points of the Bessel function $J_{m}(x)$ and its first
derivative $J^{\prime}_{m}(x)$, respectively. In this way, the form factor can
be obtained by plugging the above solutions into their definitions in Eq. (29)
and integrating over the volume of the cavity. For the TM modes, only when
$m=p=0$, the integrals in the $z$ and $\phi$ directions are non-zero.
Therefore, we only consider the $\text{TM}_{010}$ mode. For the TE modes,
unlike TM modes, they satisfy the second boundary condition in $r$ direction
and the derivative of field is zero on the cavity wall. This leads to its
vanishing integral over the cavity wall, even if those in $z$ and $\phi$
directions are non-zero ($m=0,~{}p=1,3,5,\cdots$). Fig. 1 shows the radial
distributions of two modes, $\text{TM}_{010}$ and $\text{TE}_{011}$. For the
$\text{TM}_{010}$ mode, the amplitude of field strength $E_{010}$ decreases
from the maximum at the cavity center to zero on the cavity wall. The field
strength $H_{011}$ decreases to zero at the red circle and instead increases
outside the circle, resulting in the cancellation of the cavity response to
the axion.

Figure 1: Radial distributions of $\text{TM}_{010}$ (left) and
$\text{TE}_{011}$ (right), where the transverse cross-section of TE mode is
taken at $z=L/2$.

Thus, one can conclude that in a cylindrical cavity TE mode has no coupling
with axion, i.e $C^{H}=0$, and $R_{\text{TE}}$ in Eq. (32) is zero. It means
under an external magnetic field $H_{0}$, only the coupling
$(g_{aBB}-g_{aAA})\approx g_{aBB}$ can be measured through the TM mode. For
illustration, we show the transition rate $R_{\text{TM}}$ in terms of
practical units as

$\displaystyle R_{\text{TM}}\approx
3.63~{}\text{Hz}\left(\frac{\rho_{a}}{7.1\times
10^{-25}~{}\text{g}/\text{cm}^{3}}\right)\left(\frac{10^{-5}~{}\text{eV}}{m_{a}}\right)^{2}\left(\frac{H_{0}}{10~{}\text{mT}}\right)^{2}$
$\displaystyle\cdot\left(\frac{V}{0.001~{}\text{m}^{3}}\right)\left(\frac{Q}{10^{5}}\right)\left(\frac{g_{aBB}}{10^{-12}~{}\text{GeV}^{-1}}\right)^{2}\left(\frac{C^{E}_{\omega_{a}}}{1}\right)\;.$
(43)

Given this parameter setup, the axion cavity can be viewed as a device that
emits single photon with a slow rate. The resolution of existing linear
detectors is large enough to detect this signal. This approach is exactly the
same as that measuring the conventional axion coupling $g_{a\gamma\gamma}$ in
axion electrodynamics.

###  III.2 Electric background

When the external magnetic field is replaced by an external electric field
$E_{0}$, Eq. (19) can be rewritten as

$\displaystyle\mathcal{L}_{a\gamma\gamma}=(g_{aAA}-g_{aBB})a\vec{H}\cdot\vec{E_{0}}-g_{aAB}a\vec{E}\cdot\vec{E_{0}}\;.$
(44)

Similarly, the photon emission rates for TE and TM modes in the axion cavity
can be obtained as

$\displaystyle\begin{split}R_{\text{TE}}&=\frac{\pi}{2}\frac{\rho_{a}}{m_{a}^{2}}E_{0}^{2}VQ(g_{aBB}-g_{aAA})^{2}C_{\omega_{a}}^{H},~{}\text{with}~{}C^{E}=C^{EH}=0\;,\\\
R_{\text{TM}}&=\dfrac{\pi}{2}\dfrac{\rho_{a}}{m_{a}^{2}}E_{0}^{2}VQg_{aAB}^{2}C_{\omega_{a}}^{E},~{}\text{with}~{}C^{H}=C^{EH}=0\;.\end{split}$
(45)

We find that replacing $H_{0}$ with $E_{0}$ is associated with exchanging the
couplings constrained by TE and TM modes. Now, the sensitivity to $g_{aAB}$ is
still given by the transverse magnetic wave which couples to axion. Similar to
Eq. (43), the transition rate $R_{\text{TM}}$ under the external electric
field is given by

$\displaystyle R_{\text{TM}}=0.41~{}\text{Hz}\left(\frac{\rho_{a}}{7.1\times
10^{-25}~{}\text{g}/\text{cm}^{3}}\right)\left(\frac{10^{-5}~{}\text{eV}}{m_{a}}\right)^{2}\left(\frac{E_{0}}{10^{3}~{}\text{kV/m}}\right)^{2}$
$\displaystyle\cdot\left(\frac{V}{0.001~{}\text{m}^{3}}\right)\left(\frac{Q}{10^{5}}\right)\left(\frac{g_{aAB}}{10^{-12}~{}\text{GeV}^{-1}}\right)^{2}\left(\frac{C^{E}_{\omega_{a}}}{1}\right)\;.$
(46)

##  IV Sensitivity of resonant cavity to axion-photon couplings in QEMD

Based on the above arguments, we propose the following scheme for detecting
axion interactions in QEMD through the cavity haloscope. One can apply an
external magnetic field $H_{0}$ to measure $(g_{aBB}-g_{aAA})\approx g_{aBB}$
coupling or an external electric field $E_{0}$ to measure $g_{aAB}$ coupling.
The corresponding sensitivities are given by the experimental configuration
and the form factor $C^{E}_{\omega_{a}}$ in TM mode.

For the $\text{TM}_{0n0}$ mode, the form factor $C^{E}_{\omega_{a}}$ has a
simple analytic result: $C^{E}_{\omega_{a}}=C^{E}_{0n0}=4/(x_{0,n})^{2}$. Once
the mode is fixed, the value of the form factor can be determined and is
independent of $a$ and $L$. For $\text{TM}_{010}$, one gets $C^{E}_{010}=0.69$
Sikivie (2021). Note that the axion energy is determined by the eigenvalues
$k_{\rho}$ and $k_{z}$ of the Helmholtz equation in Eq. (41), that is,
$\omega_{a}=k=\sqrt{k_{\rho}^{2}+k_{z}^{2}}$ which simplifies to
$\omega_{a}=(x_{0,1})/a$ for the $\text{TM}_{010}$ mode with $x_{0,1}=2.4$. It
means that the axion mass corresponding to the experimentally measured axion
coupling is only related to the radius $a$ of the cavity. When $a$ is 5 cm,
one has $\omega_{a}\approx 9.5\times 10^{-6}$ eV. To determine the sensitivity
in other mass regions, tuning is required by changing the value of $a$. In
practical experiments such as ADMX Bartram  _et al._ (2021), two movable bulk
copper rods are placed inside the cavity to achieve tuning. Here, we do not
intend to explore the details of techniques for adjusting the resonant
frequency by changing the cavity structure, but only provide the cavity radius
$a$ required for the resonance condition when the axion mass is
$m_{a}=\omega_{a}$.

In an external magnetic field, the signal power is given by

$\displaystyle
P_{\text{signal}}=m_{a}R_{\text{TM}}=\frac{\pi}{2}\frac{\rho_{a}}{m_{a}}H_{0}^{2}VQg_{aBB}^{2}C^{E}_{010}\;,$
(47)

where the cavity volume $V$ is regarded as a function of $m_{a}$, i.e. $V=\pi
a^{2}L=\pi L(x_{0,1}/m_{a})^{2}$, to ensure that $m_{a}$ corresponding to each
sensitivity is always at the resonant point. Similarly, the signal power under
external electric field can be obtained by making a replacement
$g_{aBB}H_{0}\rightarrow{g_{aAB}E_{0}}$. A typical detection device of axion
cavity consists of a main cavity and an amplification chain. The main source
of noise comes from the cryogenic high electron mobility transistor (HEMT). It
contributes to an effective noise temperature $T_{\text{eff}}$ around a few
Kelvin, but it will be further enhanced to above 10 K due to the microwave
loss in channel as the microwave signal is emitted from cavity and then
received by HEMT Yang  _et al._ (2022). Currently, many axion cavity
experiments have applied Josephson Parametric Amplifiers (JPA) to enhance
detection sensitivity. For example, in ADMX, one can control the noise
temperature at the order of $\mathcal{O}(10^{2})$ mK Bartram  _et al._ (2021).
Therefore, the signal-to-noise ratio is given by

$\displaystyle\text{SNR}=\frac{P_{\text{signal}}}{k_{B}T_{\text{eff}}}\sqrt{\frac{t}{b}}\;,$
(48)

where $k_{B}$ is the Boltzmann constant, $t$ is the observation time and the
ratio of frequency and $Q$ factor $b=f/Q$ is the detector bandwidth. To
estimate the sensitivity of cavity experiment to $g_{aBB}$ or $g_{aAB}$, we
take $Q=10^{5}$ and limit the SNR to 5. The results of sensitivity bound are
shown in Fig. 2. Assuming an observation time of 7 days and a cavity length of
1 m, we obtain the corresponding bounds on couplings $g_{aBB}$ and $g_{aAB}$
in the external magnetic field $H_{0}=10$ T and electric field $E_{0}=10^{4}$
kV/m, respectively. For $g_{aBB}$, we assume that the temperature parameter
inside the cavity is taken to be $T_{\text{eff}}=0.5$ K, and for $g_{aAB}$ it
is $T_{\text{eff}}=0.1$ K. It should be noticed that $m_{a}$ cannot be
arbitrarily small or large since the smaller $m_{a}$ is, the larger the cavity
radius $a$ is required to satisfy the resonance condition, and vice versa for
large mass of axion. On the other hand, when $m_{a}$ is too small, the
transition rate $R$ increases sharply to GHz level and exceeds the detector
resolution. Thus, we only consider axions with the mass range of $10^{-6}\sim
10^{-4}$ eV. It turns out that the theoretical predictions of new axion
couplings can be probed in this mass range. The result of current cavity
experiments such as ADMX measuring the conventional axion coupling
$g_{a\gamma\gamma}$ can apply to confine the $g_{aBB}$ coupling. The existing
ADMX bound given $H_{0}=7.5$ T and $T_{\text{eff}}=0.5$ K has already excluded
a part of parameter space of $g_{aBB}$ coupling. To measure $g_{aAB}$, a
strong electric field $E_{0}$ and lower temperature are both required.

Figure 2: The expected sensitivity of $g_{aAB}$ (red solid line) and $g_{aBB}$
(black solid line) in the cavity haloscope experiment. The dashed lines
indicate the corresponding theoretical predictions (red for $g_{aAB}$, black
for $g_{aBB}$). Existing limits from ADMX (2021) Bartram  _et al._ (2021) and
ADMX SLIC Crisosto  _et al._ (2020) are also shown for comparison.

##  V Conclusion

Motivated by the Witten effect, the axion-dyon dynamics can be reliably built
based on the quantum electromagnetodynamics. We show a generic low-energy
axion-photon effective field theory can also be realized in the language of
“generalized symmetries” with higher-form symmetries and gauge fields. Two
$U(1)$ gauge groups and two four-potentials $A^{\mu}$ and $B^{\mu}$ are
introduced to describe electric charge, magnetic charge and photon in this
framework. As a result, three anomalous interactions between axion and photon
arise in contrary to the conventional axion-photon coupling
$g_{a\gamma\gamma}$.

In this work, we provide a complete quantum calculation of axion-single photon
transition rate inside a homogeneous electromagnetic field in terms of the new
axion interaction Hamiltonian in QEMD. This quantum calculation can clearly
imply the enhancement of conversion rate through resonant cavity in axion
haloscope experiments. Our work provides the basic method for the generic
cavity search of new axion-photon couplings in QEMD. We find that an external
magnetic field $H_{0}$ can be set to measure $(g_{aBB}-g_{aAA})\approx
g_{aBB}$ coupling or an external electric field $E_{0}$ to measure $g_{aAB}$
coupling. The corresponding sensitivity bounds are given by the cavity
experimental configuration and the form factor $C^{E}_{010}$ in TM mode for
the QEMD axion couplings.

###### Acknowledgements.

We thank Anton V. Sokolov, Andreas Ringwald, Yu Gao and Qiaoli Yang for useful
comment and discussion. T. L. is supported by the National Natural Science
Foundation of China (Grant No. 11975129, 12035008) and “the Fundamental
Research Funds for the Central Universities”, Nankai University (Grants No.
63196013).

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