# Probing Light Mediators in the Radiative Emission of Neutrino Pair

Shao-Feng Ge<EMAIL_ADDRESS>Tsung-Dao Lee Institute & School of Physics and
Astronomy, Shanghai Jiao Tong University, China  Key Laboratory for Particle
Astrophysics and Cosmology (MOE) & Shanghai Key Laboratory for Particle
Physics and Cosmology, Shanghai Jiao Tong University, Shanghai 200240, China
Pedro Pasquini<EMAIL_ADDRESS>Tsung-Dao Lee Institute & School of
Physics and Astronomy, Shanghai Jiao Tong University, China  Key Laboratory
for Particle Astrophysics and Cosmology (MOE) & Shanghai Key Laboratory for
Particle Physics and Cosmology, Shanghai Jiao Tong University, Shanghai
200240, China

###### Abstract

We propose a new possibility of using the coherently enhanced neutrino pair
emission to probe light-mediator interactions between electron and neutrinos.
With typical momentum transfer at the atomic $\mathcal{O}(1$ eV) scale, this
process is extremely sensitive for the mediator mass range
$\mathcal{O}(10^{-3}\sim 10^{4}$) eV. The sensitivity on the product of
couplings with electron ($g^{e}$ or $y^{e}$) and neutrinos ($g^{\nu}$ or
$y^{\nu}$) can touch down to $|y^{e}y^{\nu}|<10^{-9}\sim 10^{-19}$ for a
scalar mediator and $|g^{e}g^{\nu}|<10^{-15}\sim 10^{-26}$ for a vector one,
with orders of improvement from the existing constraints.

##  I Introduction

Since the reactor mixing angle $\theta_{r}$ ($\equiv\theta_{13}$) was measured
by the Daya Bay DayaBay  and RENO RENO  experiments in 2012, the neutrino
study has entered a precision era deSalas:2020pgw ; Esteban:2020cvm ;
Capozzi:2021fjo . With the next-generation neutrino oscillation experiments,
the neutrino mixing angles and the mass squared differences would be precisely
measured. For example, the solar mixing angle $\theta_{s}$
($\equiv\theta_{12}$) and $\Delta m^{2}_{a}$ ($\equiv\Delta m^{2}_{31}$) at
JUNO Li:2016txk ; CaoNuTurn2012 ; Ge:2012wj ; Ge:2015bfa ; JUNO:2015zny as
well as the atmospheric mixing angle $\theta_{a}$ ($\equiv\theta_{23}$) at
atmospheric Ge:2013zua ; Ge:2013ffa ; Chatterjee:2013qus ; Choubey:2013xqa ;
Ghosh:2015ena and accelerator Agarwalla:2013ju ; Ghosh:2015ena ;
Minakata:2013eoa ; Ballett:2016daj ; Coloma:2014kca ; Choubey:2013xqa ;
Chatterjee:2013qus ; Das:2016bwe ; Singh:2017vvh experiments. The only
exception is the Dirac CP phase and the neutrino interactions. While the CP
measurement is also approaching discovery threshold with T2K T2K and NO$\nu$A
NOvA , the neutrino interaction is still much less constrained Farzan:2017xzy
; Proceedings:2019qno . Without precision measurement on the neutrino
interactions, especially those possiblility beyond the Standard Model (SM),
one cannot conclude that the nonzero neutrino mass is the only new physics in
the neutrino sector. Most importantly, without new interactions beyond the SM
(BSM), there is even no explanation of the nonzero neutrino mass. It is
natural to expect neutrinos to participate BSM interactions and urgent to find
sensitive probes.

The BSM neutrino interactions can appear as non-standard interactions (NSI)
Farzan:2017xzy ; Proceedings:2019qno and effective neutrino masses Ge:2018uhz
; Venzor:2020ova . In addition, new interactions can also lead to neutrino
beamstrahlung Berryman:2018ogk , neutrino trident production
Altmannshofer:2014pba ; Magill:2016hgc ; Ge:2017poy ; Magill:2017mps ;
Ballett:2018uuc ; Shimomura:2020tmg , modified spectrum from leptonic meson
decays Barger:1981vd ; Lessa:2007up , and neutrino decay Berezhiani:1991vk ;
Acker:1992eh ; Dias:2005jm ; Coloma:2017zpg ; deGouvea:2019qre . The presence
of new light particles have impact in the early Universe by generating extra
degrees of freedom Babu:2019iml ; Chigusa:2020bgq or delaying the neutrino
freeze-out Bell:2005dr ; Cyr-Racine:2013jua . Despite these ways of probing
neutrino interactions, most of them involve large momentum transfer and hence
cannot probe light mediators much below $\mathcal{O}(1$ MeV). The only
exception is the NSI from the coherent scattering that involves zero momentum
transfer by definition and hence can probe very light mediators
Smirnov:2019cae ; Babu:2019iml . However, NSI as matter effect is only
sensitive to the ratio between coupling and mass as a whole, being unable to
provide more information about the mediator.

On the other hand, astrophysics and cosmology can give very good constraints.
For example, big bang nuclesynthesis (BBN) may provide some constraints from
the degree of freedom of relativistic species in early universe Babu:2019iml ;
Chigusa:2020bgq in addition to the effective neutrino masses Venzor:2020ova .
And the most stringent bound comes from cooling of astrophysical sources
Hardy:2016kme ; Heurtier:2016otg which are model dependent DeRocco:2020xdt .
Nevertheless, most of these can only indirectly constrain the coupling with
either electron or neutrino, not both of them. This highlights the importance
of Earth-based experiments to test the possible BSM interactions that
neutrinos can experience. Jaeckel:2006xm .

The dark matter (DM) direct detection experiments can also probe neutrinos and
their interaction with electron or nuclei. The recent observations of electron
recoil at Xenon detectors Aprile:2020tmw ; PandaX-II:2020udv ; PandaX-
II:2021nsg have prompted several possible explanations with light mediator
between neutrino and electron, including scalar Boehm:2020ltd ; Khan:2020vaf ;
Khan:2020csx ; Gao:2020wfr ; GPS and vector Boehm:2020ltd ; Khan:2020vaf ;
Khan:2020csx ; Bally:2020yid ; AristizabalSierra:2020edu ; Lindner:2020kko ;
Karmakar:2020rbi ; Amaral:2020tga ; Chala:2020pbn ; Ibe:2020dly ; GPS bosons,
eletromagnetic interactions Khan:2020vaf ; Khan:2020csx ; Amaral:2020tga ;
Miranda:2020kwy ; Babu:2020ivd ; Chala:2020pbn ; Brdar:2020quo ;
AristizabalSierra:2020zod with massless photon, and sterile neutrino
Ge:2020jfn ; Shoemaker:2020kji ; Chen:2021uuw ; Shakeri:2020wvk . Due to the
intrinsic energy threshold $\sim\mathcal{O}(1$ keV) of DM experiments, the
typical momentum transfer is around $\mathcal{O}(10$ keV) for electron recoil
and $\mathcal{O}(1$ MeV) for nuclei recoil. Although the situation can be
improved for electron recoil when compared with the usual neutrino
experiments, the improvement is not significant enough.

In this work we present a new possibility of using the radiative emission of
neutrino pair (RENP) Yoshimura:2006nd ; Fukumi:2012rn ; Hiraki:2018jwu ;
Tashiro:2019ghs to probe the BSM interactions mediated by a light mediator.
The $\mathcal{O}($eV) energy scale of atomic transitions are perfect for
investigating the existence and properties of light mediators. We start by
briefly summarizing the SM calculation of the neutrino pair emission process
in Sec. II for both Dirac and Majorana neutrinos. Then Sec. III considers the
modifications to the spectral function of the photon emission in the presence
of light vector/scalar mediators with mass in the range of
$\mathcal{O}(10^{-3}\sim 10^{4}$) eV. The smallness of the mediator mass
implies a significant modification and hence unprecedented sensitivity as
shown in Sec. IV. Finally, we summarize the most prominent results and
features in Sec. V.

##  II Neutrino Pair Emission

The radiative emission of neutrino pair (RENP) Yoshimura:2006nd ;
Yoshimura:2011ri ; Fukumi:2012rn ; Hiraki:2018jwu ; Tashiro:2019ghs is a
coherently induced superradiance Yoshimura:2012tm of a neutrino pair and a
photon when an excited atomic state $|e\rangle$ transits to the ground state
$|g\rangle$, $|e\rangle\rightarrow|g\rangle+\gamma+\nu\bar{\nu}$. In the SM,
the emission of neutrino pair is mediated by the heavy $W/Z$ gauge bosons, as
illustrated in Fig. 1. Since the gauge boson masses are much larger than the
typical atomic energy,
$m_{W}=80.4\,\mbox{GeV},m_{Z}=91.2\,\mbox{GeV}\gg\mathcal{O}(1\sim
10)\,\mbox{eV}$, a weak process at low energy is highly suppressed by the
Fermi constant, $G_{\rm F}\approx 1.67\times 10^{-5}$ GeV-2. Fortunately, the
RENP can be enhanced by two quantum mechanical effects Yoshimura:2012tm .
First, the bosonic nature of photon allows stimulated photon emission enhanced
by a factor of the photon number density $n_{\gamma}$. Second, the coherent
behaviour of atoms can further boost the decay rate by a factor of $n_{a}^{2}$
where $n_{a}$ is the number of atoms behaving coherently Dicke:1954zz . For
macroscopically coherent material, the enhancement can be large enough to
allow practical measurement. Since the emitted photon spectrum can be
precisely measured with electromagnetic instead of weak interactions, the RENP
is especially sensitive to neutrino properties Yoshimura:2009wq ; Dinh:2012qb
, including the neutrino mass Song:2015xaa ; Zhang:2016lqp , the unitarity
test of the neutrino mixing matrix Huang:2019phr , the Dirac/Majorana nature
of neutrinos Dinh:2012qb ; Fukumi:2012rn , the leptonic CP phases Dinh:2012qb
; Yoshimura:2015yja , the cosmic neutrino background Takahashi:2007ec ;
Yoshimura:2014hfa , and light sterile neutrinos Dinh:2014toa . In the current
section, we summarize the key elements of RENP in the SM for both Dirac and
Majorana neutrinos.

###  II.1 The SM Interactions for Atomic Transitions

The RENP process involves both electromagnetic and weak interactions,

$H=H_{0}+D_{\gamma}+H_{W}.$ (1)

The zeroth order $H_{0}$ describes the atomic energy levels of electrons in an
atom, namely, $H_{0}|a\rangle=E_{a}|a\rangle$, for the excited ($a=e$),
virtual ($a=v$), and ground ($a=g$) states, respectively. For comparison,
$D_{\gamma}$ represents the electric dipole interaction while $H_{W}$ is the
weak counterpart.

The excited state $|e\rangle$ should be meta-stable with $E_{v}>E_{e}>E_{g}$
such that the transition between $|e\rangle$ and $|g\rangle$ cannot happen by
emitting a single photon. Otherwise, the excited state $|e\rangle$ would decay
much faster ($\sim 1$ ns Fukumi:2012rn ) to the ground state directly via
$|e\rangle\rightarrow|g\rangle+\gamma$ than the RENP. This can be achieved by
selecting the atom with appropriate $|e\rangle$ and $|g\rangle$. For example,
the Xe (Yb) element has angular momentum $J_{e}=2$ and $J_{g}=0$ ($J_{e}=0$
and $J_{g}=0$) Song:2015xaa for the excited and ground states, respectively.
With emission of a single photon that carries only $J_{\gamma}=1$, the angular
momentum can not conserve for the $J_{e}\rightarrow J_{g}$ transition. This
significantly reduces the background that cannot be observationally
distinguished from the RENP signal
$|e\rangle\rightarrow|g\rangle+\gamma+\nu\bar{\nu}$.

With the direct transition from $|e\rangle$ to $|g\rangle$ forbidden, the
neutrino pair emission is at most a second order process through a virtual
atomic state $|v\rangle$ at higher energy,
$|e\rangle\rightarrow|v\rangle\rightarrow|g\rangle$. The whole transition is
then a combined transition of the E1$\times$M1 type. The current experimental
configuration Huang:2019phr ; Hiraki:2018jwu allows an M1 transition for
$|e\rangle\rightarrow|v\rangle+\nu\bar{\nu}$ with neutrino pair emission and
an E1 transition for $|v\rangle\rightarrow|g\rangle+\gamma$ with photon
emission. Since $E_{v}>E_{e}$, the first step $|e\rangle\rightarrow|v\rangle$
cannot happen separately. The two transitions have to happen simultaneously.

Figure 1: Feynman diagrams for the radiative emission of neutrino pair with
the SM heavy gauge bosons ($Z/W$) and light mediators ($Z^{\prime}/\phi$).

For the electromagnetic part, the dominant contribution comes from an electric
dipole transition (E1) induced by $D_{\gamma}$. More explicitly, the electric
dipole is linearly related to the laser electric field ${\bf E}({\bf x})$ with
trigger frequency $\omega$,

$\displaystyle\langle g|D_{\gamma}|v\rangle\equiv\mathcal{M}_{D}e^{-i\omega
t+{\bf k}\cdot{\bf x}},\quad\mathcal{M}_{D}\equiv-{\bf d}_{gv}\cdot{\bf
E}_{0},$ (2)

where $\omega$ and ${\bf k}$ are the photon energy and momentum, respectively.
The 3-vector ${\bf d}_{gv}$ sandwiched between the virtual and ground states
is the dipole operator for the $v\rightarrow g$ transition and ${\bf E}_{0}$
is the stimulating electric field.

The $H_{W}$ term contains the SM weak interactions between electron and
neutrinos,

$\displaystyle\mathcal{H}_{W}\equiv\sqrt{2}G_{F}\left(v_{ij}J^{\mu}_{V}-a_{ij}J^{\mu}_{A}\right)\bar{\nu}_{iL}\gamma_{\mu}\nu_{jL}.$
(3)

The electron current $J_{e}^{\mu}\equiv v_{ij}J^{\mu}_{V}-a_{ij}J_{A}^{\mu}$
is a linear combination of the vector current $J^{\mu}_{V}\equiv\langle
v|\overline{e}\gamma^{\mu}e|e\rangle$ and the axial current
$J^{\mu}_{A}\equiv\langle v|\overline{e}\gamma^{\mu}\gamma_{5}e|e\rangle$. The
coefficients

$\displaystyle a_{ij}\equiv U_{ei}U^{*}_{ej}-\frac{\delta_{ij}}{2},~{}{\rm
and}~{}v_{ij}\equiv a_{ij}+2s^{2}_{w}\delta_{ij},$ (4)

contain the PMNS matrix $U$ of neutrino mixing which is a function of mixing
angles and leptonic CP phases Zyla:2020zbs . From the weak Hamiltonian (3), we
can obtain the matrix element for the $|e\rangle\rightarrow|v\rangle$
transition,

$\displaystyle\langle v|H_{W}|e\rangle$ $\displaystyle\equiv$
$\displaystyle\mathcal{M}_{W}^{(D)}e^{-i(p_{\nu}+p_{\bar{\nu}})\cdot x}$ (5a)
$\displaystyle\mathcal{M}_{W}^{(D)}$ $\displaystyle\equiv$
$\displaystyle\sqrt{2}G_{\rm
F}(J_{e}^{\mu})_{ij}\bar{u}(p_{\nu_{i}})\gamma_{\mu}P_{L}v(p_{\bar{\nu}_{j}}),$
(5b)

for Dirac neutrinos. The M1 transition is mainly contributed by the axial
current that has only spatial components and is proportional to the electron
spin in the nonrelativistic limit, $J_{A}^{\mu}\approx(0,\langle v|2{\bf
S}|e\rangle)$ Song:2015xaa . For comparison, the electric dipole ${\bf
d}_{gv}$ is parity odd (E1) while the spin operator ${\bf S}$ is parity even
(M1).

###  II.2 Dirac vs Majorana Neutrino Matrix Elements

The matrix element $\mathcal{M}^{(D)}_{W}$ (5b) for Dirac neutrinos has only
one term, with $\bar{u}(p_{\nu_{i}})$ for neutrino and $v(p_{\bar{\nu}_{j}})$
for anti-neutrino in the final state. In comparison, Majorana neutrinos can
contribute an extra term since there is no distinction between neutrino and
anti-neutrino now. In the formalism of second quantization, the Majorana
neutrino field, $\nu\sim au+a^{\dagger}v$, has only one set of annihilation
($a$) and creation ($a^{\dagger}$) operators. Both $\bar{\nu}$ and $\nu$
fields in (3) can create/annihilate neutrino and anti-neutrino final states
Dinh:2012qb ,

$\displaystyle\mathcal{M}_{W}^{(M)}=\sqrt{2}G_{F}J_{A}^{\mu}\Bigl{[}$
$\displaystyle
a_{ij}\bar{u}(p_{\nu_{i}})\gamma_{\mu}P_{L}v(p_{\overline{\nu}_{j}})$
$\displaystyle-$ $\displaystyle
a_{ij}^{*}\bar{u}(p_{\overline{\nu}_{j}})\gamma_{\mu}P_{L}v(p_{\nu_{i}})\Bigr{]}.$
(6)

The minus sign comes from the commutation property of fermion fields and we
have used the property $a_{ji}=a_{ij}^{*}$ to write the coefficients all in
terms of $a_{ij}$. Using the spinor relations $\overline{v}=-u^{T}C$ and
$u=-C\overline{v}^{T}$ and
$C^{-1}\left(\gamma_{\mu}P_{L}\right)^{T}C=-\gamma_{\mu}P_{R}$, where $C$ is
the charge conjugation operator, the squared matrix elements become,

$\displaystyle|\mathcal{M}_{W}^{(D)}|^{2}$ $\displaystyle=$ $\displaystyle
2G_{F}^{2}J_{A}^{\mu}J_{A}^{\nu*}|a_{ij}|^{2}$ (7a) $\displaystyle\times$
$\displaystyle\bar{u}(p_{\nu_{i}})\gamma_{\mu}P_{L}v(p_{\overline{\nu}_{j}})\bar{v}(p_{\overline{\nu}_{j}})\gamma_{\nu}P_{L}u(p_{\nu_{i}}),$
$\displaystyle|\mathcal{M}_{W}^{(M)}|^{2}$ $\displaystyle=$
$\displaystyle|\mathcal{M}_{W}^{(D)}|^{2}-2G_{F}^{2}{\rm
Re}\Bigl{[}a_{ij}^{2}J_{A}^{\mu}J_{A}^{\nu*}$ (7b)
$\displaystyle\bar{u}(p_{\nu_{i}})\gamma_{\mu}P_{L}v(p_{\overline{\nu}_{j}})\bar{v}(p_{\overline{\nu}_{j}})\gamma_{\nu}P_{R}u(p_{\nu_{i}})\Bigr{]}.\quad$

We have also included a factor of $1/2$ in $|\mathcal{M}_{W}^{(M)}|^{2}$ to
avoid double counting in the phase space integration. The appearance of both
chirality projection operators $P_{L}$ and $P_{R}$ will result in chirality
flip and hence the extra term is proportional to a product of both neutrino
masses $m_{i}m_{j}$. Since the final-state momentum of neutrinos from atomic
transition are typically $\mathcal{O}(1$eV), this chirality-flipping term is
no longer suppressed. This is the reason why RENP can be sensitive to the
neutrino masses and hence its Dirac/Majorana nature.

The squared electroweak matrix element after averaging over spins and all the
possible atomic magnetic numbers is,

$\displaystyle\overline{|\mathcal{M}_{W}^{ij}|^{2}}$ $\displaystyle=$
$\displaystyle\frac{8}{3}(2J_{v}+1)|a_{ij}|^{2}G_{\rm F}^{2}C_{ev}$
$\displaystyle\times$ $\displaystyle\Bigl{\\{}2{\bf p}_{\nu_{i}}\cdot{\bf
p}_{\bar{\nu}_{j}}+3p_{\nu_{i}}\cdot p_{\bar{\nu}_{j}}-3\delta_{M}{\rm
Re}[a_{ij}^{2}]m_{i}m_{j}\Bigr{\\}}.$

The prefactor $\delta_{M}=1$ for Majorana neutrinos and $\delta_{M}=0$ for
Dirac ones. To obtain (II.2), the electron spins have also been summed over
Dinh:2012qb ,

$\frac{1}{(2J_{e}+1)}\sum_{m_{e}m_{v}}\langle v|S^{i}|e\rangle\langle
e|S^{j}|v\rangle=(2J_{v}+1)\frac{C_{ev}}{3}\delta^{ij},$ (9)

where $J_{e}$ and $J_{v}$ are the total spins of the excited ($|e\rangle$) and
the virual ($|v\rangle$) states, respectively. The constant $C_{ev}$ is a
factor that depends on the atom. For Yb we have $(2J_{v}+1)C_{ev}=2$
Song:2015xaa .

###  II.3 Emission Rate and Spectrum

The photon emission rate of the reaction is Dinh:2012qb ; Song:2015xaa ;
Zhang:2016lqp ,

$\displaystyle
d\Gamma_{ij}=\frac{\Gamma_{0}}{(E_{vg}-\omega)^{2}\omega}\frac{\overline{|\mathcal{M}^{ij}_{W}|^{2}}}{8G_{F}^{2}C_{ev}(2J_{v}+1)}dE_{\nu},$
(10)

with the reference decay width $\Gamma_{0}$ Song:2015xaa ; Zhang:2016lqp ,

$\displaystyle\Gamma_{0}$ $\displaystyle\equiv$
$\displaystyle(2J_{v}+1)\frac{n_{a}^{2}C_{ev}G_{\rm F}^{2}|{\bf
d}_{gv}\cdot{\bf E}_{0}|^{2}}{\pi}$ (11a) $\displaystyle\approx$
$\displaystyle 0.002{\rm s}^{-1}\left(\frac{V}{10^{2}{\rm
cm}^{3}}\right)\left(\frac{n_{a}^{2}n_{\gamma}}{10^{21}{\rm
cm}^{-3}}\right)\eta(t).$ (11b)

For illustration, we have used Yb. The decay width scales linearly with the
target volume $V$ and the fraction $\eta$ of atoms that behave coherently. An
$\eta\approx 1$ is claimed to be achievable BoyeroGarcia:2015dye and we set
$\eta=1$ for simplicity.

The photon kinematics of the RENP process is fixed by the two back-to-back
trigger laser beams whose frequencies Song:2015xaa ; Zhang:2016lqp
($\omega_{1}$ and $\omega_{2}$) sum up to $\omega_{1}+\omega_{2}=E_{e}-E_{g}$.
The stimulated emission imposes a fixed emitted laser frequency
$\omega=\omega_{1}<\omega_{2}$ in the same direction of the $\omega_{1}$ laser
beam and also enforces the energy/momentum conservation,

$\displaystyle E_{e}-E_{g}=\omega+E_{\nu}+E_{\overline{\nu}}\quad{\rm
and}\quad{\bf k}=-{\bf p}_{\nu}-{\bf p}_{\overline{\nu}}\,.$ (12)

The invariant mass of the neutrino pair is then related to the photon energy,
$m^{2}_{\nu\bar{\nu}}=E^{2}_{eg}-2E_{eg}\omega\geq(m_{i}+m_{j})^{2}$, which
imposes a frequency upper limit on the emitted photon,

$\displaystyle\omega_{ij}^{\rm
max}\equiv\frac{E_{e}-E_{g}}{2}-\frac{1}{2}\frac{(m_{i}+m_{j})^{2}}{(E_{e}-E_{g})}.$
(13)

Only for $\omega<\omega_{ij}^{\rm max}$, the RENP process can be triggered.

With the emitted photon frequency $\omega$ fixed, the 3-body final state only
needs 2-body phase space integration which can reduce to a single integration
(10) over the neutrino energy $E_{\nu}$. The integration limit is a function
of the photon frequency $\omega$,

$\bar{E}-\frac{\omega\Delta_{ij}(\omega)}{2}\leq
E_{\nu}\leq\bar{E}+\frac{\omega\Delta_{ij}(\omega)}{2},$ (14)

where,

$\displaystyle\Delta_{ij}(\omega)$ $\displaystyle\equiv$
$\displaystyle\frac{\sqrt{E_{eg}(E_{eg}-2\omega)-(m_{i}+m_{j})^{2}}}{E_{eg}(E_{eg}-2\omega)}$
(15a) $\displaystyle\times$
$\displaystyle\sqrt{E_{eg}(E_{eg}-2\omega)-(m_{i}-m_{j})^{2}},$
$\displaystyle\bar{E}\equiv\frac{[E_{eg}(E_{eg}-2\omega)-\Delta
m^{2}_{ij}](E_{eg}-\omega)}{2E_{eg}(E_{eg}-2\omega)}.$ (15b)

After integration over the kinematic range in (12) and sum over all the
neutrino mass eigenstates, the total decay width becomes
$\Gamma\equiv\Gamma_{0}\mathcal{I}(\omega)$ Dinh:2012qb . The spectral
function $\mathcal{I}(\omega)$ is defined as,

$\displaystyle\mathcal{I}(\omega)\equiv\sum_{ij}$
$\displaystyle\frac{\Delta_{ij}(\omega)}{(E_{vg}-\omega)^{2}}\Theta(\omega-\omega_{ij}^{\rm
max})$ (16) $\displaystyle\left[|a_{ij}|^{2}I_{ij}^{(D)}-\delta_{M}{\rm
Re}[a_{ij}^{2}]I_{ij}^{(M)}\right],$

with a Heaviside theta function $\Theta$ to impose frequency requirement and,

$\displaystyle I_{ij}^{(D)}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{3}\left\\{E_{eg}(E_{eg}-2\omega)-\frac{1}{2}(m_{i}^{2}+m_{j}^{2})\right.$
$\displaystyle+$
$\displaystyle\left.\frac{\omega^{2}}{2}\left[1-\frac{1}{3}\Delta_{ij}(\omega)^{2}\right]-\frac{(E_{eg}-\omega)^{2}(\Delta
m_{ij}^{2})^{2}}{2E_{eg}^{2}(E_{eg}-2\omega)^{2}}\right\\},$ $\displaystyle
I_{ij}^{(M)}$ $\displaystyle\equiv$ $\displaystyle m_{i}m_{j}\,.$ (17b)

There are two candidate elements that allow RENP process: Xenon (Xe) and
Ytterbium (Yb) Dinh:2012qb . In this work we focus on Yb that can reach
$\mathcal{O}(20)$ events per day for a target volume of $100$ cm3 and atomic
density of $10^{21}$ cm-3 Zhang:2016lqp . Although these parameters are not
achievable yet, there have been some advances recently Hiraki:2018jwu .

The black line of Fig. 2 illustrates the SM spectral function (16) as a
function of the trigger laser frequency $\omega$ for the normal neutrino mass
ordering (NO) and lightest neutrino mass $m_{1}=0.01$ eV with Dirac neutrino.
We can see several kinks that correspond to the 6 kinematic upper limits in
(13). The two most prominent kinks are $\omega_{33}\approx 1.0693$ eV and
$\omega_{13}\approx 1.0710$ eV. The curve drops to zero at the highest upper
limit $\omega^{\rm max}_{11}\approx 1.0716$ eV.

##  III Light Mediators

For the SM interactions discussed above, the $W/Z$ gauge bosons essentially
contribute as contact operators, $1/(q^{2}-m^{2})\approx 1/m^{2}$ with
$m^{2}\gg q^{2}$. For atomic transitions, the momentum transfer is typically
eV scale, $q^{2}\sim\mathcal{O}(\mbox{eV}^{2})$, in contrast to
$m^{2}_{W},m^{2}_{Z}\sim\mathcal{O}(10^{21}\mbox{eV}^{2})$. The hierarchy
between the cut-off scale and the physical energy is as large as 21 orders at
the amplitude level and more than 40 orders for the decay width. If the
mediator mass is lowered down to the atomic scale, the decay width can be
significantly enhanced. In other words, the RENP is very sensitive to probing
the coupling of light mediators with electron and neutrinos.

Generally speaking, a light mediator between electron and neutrinos can be
either scalar or vector boson. In principle, both charged and neutral
mediators are allowed. However, a charged light mediator with mass at eV scale
can be easily detected and hence has probably been excluded already. We only
consider neutral light mediators for illustration.

A scalar boson can have both scalar and pseudo-scalar couplings with fermion
while a vector boson can mediate both vector and axial-vector interactions.
However, the $|e\rangle\rightarrow|v\rangle+\nu\bar{\nu}$ transition only
allows M1 transition for the coupling with electron. Not all interactions can
contribute. In Tab. 1 we summarize all the possible Lorentz structures for new
interactions with electron in the non-relativistic limit. Only the pseudo-
scalar and axial-vector interactions are M1 type. Nevertheless, the neutrino
part can still have all possible interaction types including the vector and
scalar ones.

Figure 2:  The spectral function $\mathcal{I}(\omega)$ of Yb as a function of the trigger laser frequency $\omega$ for the normal ordering (NO) and lightest neutrino mass $m_{1}=0.01$ eV in the SM (black) as well as in the presence of a light mediator with mass $m_{\phi,Z^{\prime}}=20$ keV (colorful lines). For comparison, we show four different scenarios of the BSM couplings: $g^{e}g^{\nu}_{L,11}=8\times 10^{-16}$ (red dashed) $g^{e}g^{\nu}_{R,11}=1.5\times 10^{-15}$ (green dotter) $y^{e}y^{\nu}_{S,11}=6\times 10^{-9}$ (brown dotted) $y^{e}y^{\nu}_{P,11}=6\times 10^{-9}$ (blue dot-dashed). Notice that the scalar and pseudo-scalar contributions almost overlap with each other. Coupling | $\mathcal{L}_{\rm new}$ | Non-Relativistic Transition | Type
---|---|---|---
scalar | $y^{e}_{S}\bar{e}e$ | $\langle f|i\rangle$ | E1
pseudo-scalar | $iy^{e}_{P}\bar{e}\gamma_{5}e$ | $\frac{{\bf q}}{2m_{e}}\cdot\langle f|\boldsymbol{\sigma}|i\rangle$ | M1
vector | $g^{e}_{V}\bar{e}\gamma^{\mu}e$ |  $(\langle f|i\rangle,\frac{{\bf q}}{2m_{e}}\cdot\langle f|\boldsymbol{\sigma}\boldsymbol{\sigma}|i\rangle$) | E1
axial-vector | $g^{e}_{A}\bar{e}\gamma^{\mu}\gamma_{\mu}e$ | $(\frac{{\bf q}}{2m_{e}}\cdot\langle f|\boldsymbol{\sigma}|i\rangle,\langle f|\boldsymbol{\sigma}|i\rangle)$ | M1
Table 1: The electron atomic transition currents from an initial state
$|i\rangle$ to a final state $|f\rangle$ for non-relativistic electrons and
momentum transfer ${\bf q}$ Weinberg:1995 .

Light mediators with masses below $\mathcal{O}(10)$ keV that couple to
neutrinos can appear in many models. An incomplete list includes $U(1)$
extensions of the SM gauge group Gelmini:1980re ; Chikashige:1980qk ;
Machado:2010ui ; Machado:2013oza ; Fayet:2016nyc ; Heeck:2010pg , fifth force
Wise:2018rnb ; Dror:2020fbh ; Bustamante:2018mzu , $U(1)_{R}$ symmetry
connected to a hidden right-handed neutrino sector Alikhanov:2021dhb ;
Xu:2020qek , neutrino mass models Bonilla:2016zef ; Shakya:2018qzg ;
Chauhan:2020mgv , the neutrinophilic two-Higgs doublet models Machado:2015sha
; Lindner:2020kko , the neutrinophilic axion-like particles Mohapatra:1982tc ;
Arason:1990sg ; Huang:2018cwo ; Baek:2019wdn ; Baek:2020ovw , neutrino non-
standard interaction with light mediators Smirnov:2019cae ; Coloma:2020gfv .
Also, light mediators can be a connection between neutrinos and ultra-light
dark matter Berlin:2016woy ; Brdar:2017kbt ; Farzan:2019yvo ; Cline:2019seo ;
Ge:2019tdi ; Choi:2019zxy ; Dev:2020kgz ; Alonso-Alvarez:2021pgy and inflation
Joshipura:2020ibd . Some discussions on light mediator and neutrino
interactions can also be found in He:1998ng ; Ding:2020yen . The rich
phenomenology of light mediators in the leptonic sector can be found in
Babu:2019iml . It is well motivated to find possible ways of probing light and
ultra-light mediators using low enough momentum transfer as we elaborate
below.

###  III.1 Vector Mediator

Since the vector coupling with electron does not contribute to the RENP
process, the relevant new interactions with a vector mediator $Z^{\prime}$
are,

$\displaystyle\mathcal{L}_{V}=g^{e}\bar{e}\gamma^{\mu}\gamma_{5}eZ^{\prime}_{\mu}+\bar{\nu}_{i}\gamma^{\mu}(g^{\nu}_{L,ij}P_{L}+g^{\nu}_{R,ij}P_{R})\nu_{j}Z^{\prime}_{\mu}.\qquad$
(18)

Extra terms can arise as correction to the scattering matrix elements (5b) and
(6), $\mathcal{M}_{W}^{ij}\rightarrow\mathcal{M}_{\rm
tot}^{ij}\equiv\mathcal{M}_{W}^{ij}+\mathcal{M}_{\rm New}^{ij}$, where
$\mathcal{M}_{\rm New}^{ij}$ is the new contribution,

$\displaystyle\mathcal{M}_{\rm tot}^{ij}=$ $\displaystyle\sqrt{2}G_{\rm
F}J^{\mu}_{A}\Bigl{[}\bar{u}(p_{\nu_{i}})\gamma_{\mu}\left(a^{L}_{ij}P_{L}+a^{R}_{ij}P_{R}\right)v(p_{\bar{\nu}_{j}})$
(19)
$\displaystyle-\delta_{M}\bar{u}(p_{\bar{\nu}_{j}})\gamma_{\mu}\left(a^{L}_{ji}P_{L}+a^{R}_{ji}P_{R}\right)v(p_{\nu_{i}})\Bigr{]}.$

Although only the left-handed neutrino current is involved in the SM, the new
interactions allow the right-handed component to also participate,

$\displaystyle a^{L}_{ij}$ $\displaystyle\equiv$ $\displaystyle
a_{ij}+\frac{g^{e}g^{\nu}_{L,ij}}{\sqrt{2}G_{F}(q^{2}-m^{2}_{Z^{\prime}})}\delta_{ij},$
(20a) $\displaystyle a^{R}_{ij}$ $\displaystyle\equiv$
$\displaystyle\frac{g^{e}g^{\nu}_{R,ij}}{\sqrt{2}G_{F}(q^{2}-m^{2}_{Z^{\prime}})}\delta_{ij}.$
(20b)

The first term $a_{ij}$ is the SM contribution and those terms involving
$1/(q^{2}-m^{2}_{Z^{\prime}})$ come from the $Z^{\prime}$ propagator. Since an
overall Fermi constant $G_{F}$ has been extracted in (19), the new terms also
contain a $G_{F}$ in the denominator although the weak scale is not
necessarily relevant here.

The total matrix element squared $\overline{|\mathcal{M}^{ij}_{\rm tot}|^{2}}$
after averaging over all spin and atomic state is,

$\displaystyle\frac{8}{3}(2J_{v}+1)C_{ev}G^{2}_{F}$ (21)
$\displaystyle\times\left[\Bigl{(}|a^{L}_{ij}|^{2}+|a^{R}_{ij}|^{2}-2\delta_{M}{\rm
Re}[a^{L}_{ij}a^{R}_{ij}]\Bigr{)}(2{\bf p}_{i}\cdot{\bf p}_{j}+3p_{i}\cdot
p_{j})\right.$ $\displaystyle\left.\hskip 11.38109pt+3m_{i}m_{j}\left(2{\rm
Re}[a^{L*}_{ij}a^{R}_{ij}]-\delta_{M}{\rm
Re}[(a^{L}_{ij})^{2}+(a^{R}_{ij})^{2}]\right)\right].$

It is interesting to observe that the first line in the bracket has the same
structure as (II.2) with the overall factor substitution
$|a_{ij}|^{2}\rightarrow|a^{L}_{ij}|^{2}+|a^{R}_{ij}|^{2}-2\delta_{M}{\rm
Re}[a^{L}_{ij}a^{R}_{ij}]$. The last term in the bracket comes from the
interference of the right- and left-handed components and is proportional to
the neutrino masses. It is also similar to the Majorana contribution in the SM
with the replacement, ${\rm Re}[a_{ij}^{2}]\rightarrow 2{\rm
Re}[a^{L}_{ij}a^{R}_{ij}]-\delta_{M}{\rm
Re}[(a^{L*}_{ij})^{2}+(a^{R}_{ij})^{2}]$.

Correspondingly, the spectral function $\mathcal{I}(\omega)$ in (16) receives
the following correction,

$\displaystyle\mathcal{I}_{Z^{\prime}}$ $\displaystyle\equiv$
$\displaystyle\sum_{ij}\frac{\Delta_{ij}(\omega)}{(E_{vg}-\omega)^{2}}\Theta(\omega-\omega_{ij}^{\rm
max})$ (22) $\displaystyle\times$
$\displaystyle\left[\Bigl{(}|a^{L}_{ij}|^{2}+|a^{R}_{ij}|^{2}-2\delta_{M}{\rm
Re}[a^{L}_{ij}a^{R}_{ij}]\Bigr{)}I_{ij}^{(D)}\right.$
$\displaystyle+\left.\Bigl{(}\delta_{M}{\rm
Re}\left[(a^{L}_{ij})^{2}+(a^{R}_{ij})^{2}\right]-2{\rm
Re}\left[a^{L*}_{ij}a^{R}_{ij}\right]\Bigr{)}I_{ij}^{(M)}\right].\qquad$

Fig. 2 shows the changes in the spectral function $\mathcal{I}$ for a light
mediator mass of $m_{Z^{\prime}}=20$ keV and two different coupling
combinations (1) $g^{e}g^{\nu}_{L,11}=8\times 10^{-16}$ (red dashed) and (2)
$g^{e}g^{\nu}_{R,11}=1.5\times 10^{-15}$ (dotted green). As we can see, the
$g_{L}^{\nu}$ coupling can be smaller than $g_{R}^{\nu}$ and still produce a
comparable modification of the spectral function. This is because of the
interference between the SM coupling $a_{ij}$ and the new couplings as squared
terms of $a^{L}_{ij}$. This interference with the large SM coupling $a_{ij}$
only applies to $g^{e}g^{L}_{ij}$ but not $g^{e}g^{R}_{ij}$. In addition,
there is an interference term between $a_{ij}^{L}$ and $a_{ij}^{R}$ due to the
chirality flip. For the Dirac case it is proportional to the neutrino masses
and hence is a small correction. In comparison, the interference term for
Majorana neutrinos is much larger with its coefficient being momentum instead
of the tiny neutrino masses.

###  III.2 Scalar Mediator

As explained above, only pseudo-scalar interaction on the electron side can
contribute the required M1 transition while the neutrino side is general,

$\displaystyle\mathcal{L}_{S}$ $\displaystyle\equiv$ $\displaystyle
iy^{e}_{P}\bar{e}\gamma_{5}e\phi+\bar{\nu}_{i}(y^{\nu}_{S,ij}+i\gamma_{5}y^{\nu}_{P,ij})\nu_{j}\phi+h.c.,\quad$
(23)

with both scalar ($y^{\nu}_{S,ij}$) and pseudo-scalar ($y^{\nu}_{P,ij}$)
interactions. The corresponding new matrix element is,

$\displaystyle\mathcal{M}^{ij}_{S}$ $\displaystyle=$
$\displaystyle\frac{y^{e}J_{P}\bar{u}(p_{\nu_{i}})(y^{\nu}_{S,ij}+i\gamma_{5}y^{\nu}_{P,ij})v(p_{\overline{\nu}_{j}})}{E_{eg}(E_{eg}-2\omega)-m^{2}_{\phi}}$
(24) $\displaystyle-$
$\displaystyle\delta_{M}\frac{y^{e}J_{P}\bar{u}(p_{\overline{\nu}_{j}})(y^{\nu}_{S,ji}-i\gamma_{5}y^{\nu}_{P,ji})v(p_{\nu_{i}})}{E_{eg}(E_{eg}-2\omega)-m^{2}_{\phi}},\qquad$

where the second line arises only for Majorana neutrinos.

The total averaged matrix element squared,

$\displaystyle\overline{|\mathcal{M}_{\rm
tot}^{ij}|^{2}}=\overline{|\mathcal{M}^{ij}_{W}|^{2}}+\overline{|\mathcal{M}^{ij}_{S}|^{2}}+2{\rm
Re}\left(\overline{\mathcal{M}^{ij}_{W}\mathcal{M}^{ij*}_{S}}\right),\qquad$
(25)

receives corrections $\overline{|\mathcal{M}^{ij}_{S}|^{2}}$ from the scalar
mediator and the $W,Z/\phi$ interference term. Note that the matrix element
$\mathcal{M}^{ij}_{W}$ contributed by the weak interactions contains an axial-
vector current $J^{\nu}_{A}$ of electron while the scalar one
$\mathcal{M}^{ij}_{S}$ by the light scalar mediator contains a pseudo-scalar
counterpart $J_{P}\equiv\frac{1}{2m_{e}}{\bf q}\cdot\langle
e|\boldsymbol{\sigma}|v\rangle$ as summarized in Tab. 1. Then, the square of
the new matrix element contains a term $|J_{A}|^{2}$ and the interference one
$J_{A}^{\mu}J^{*}_{P}$. Using the spin relation (9) we can obtain,

$\displaystyle\frac{1}{(2J_{e}+1)}\sum_{m_{e}m_{v}}|J_{P}|^{2}$
$\displaystyle=$
$\displaystyle\frac{C_{ev}}{3}(2J_{v}+1)\frac{\omega^{2}}{m_{e}^{2}}\,,$ (26a)
$\displaystyle\frac{1}{(2J_{e}+1)}\sum_{m_{e}m_{v}}J_{A}^{\mu}J_{P}^{*}$
$\displaystyle=$ $\displaystyle\frac{2C_{ev}}{3}(2J_{v}+1)\left(0,\frac{\bf
q}{m_{e}}\right).\qquad$ (26b)

Since the emitted photon energy or momentum transfer is much smaller than the
electron mass, $\omega,{\bf q}\ll m_{e}$, the pseudo-scalar atomic current
introduces a suppression factor $\omega^{2}/m_{e}^{2}\sim 10^{-11}$ for the
new non-interference term and $\omega/m_{e}\sim 10^{-5}$ for the interference
one, as indicated by (26). So the interference term would dominate. Putting
things together, the averaged matrix element squared (II.2) becomes,

$\displaystyle\overline{|\mathcal{M}^{ij}_{S}|^{2}}$ $\displaystyle=$
$\displaystyle\frac{2(2J_{v}+1)|y^{e}|^{2}C_{ev}}{[E_{eg}(E_{eg}-2\omega)-m^{2}_{\phi}]^{2}}\frac{\omega^{2}}{3m_{e}^{2}}$
$\displaystyle\times$ $\displaystyle\left\\{|y^{\nu}_{S,ij}|^{2}(p_{i}\cdot
p_{j}-m_{i}m_{j})\right.$
$\displaystyle+\left.(1-\delta_{M})|y^{\nu}_{P,ij}|^{2}(p_{i}\cdot
p_{j}+m_{i}m_{j})\right\\},$
$\displaystyle\overline{\mathcal{M}^{ij}_{W}\mathcal{M}^{ij*}_{S}}$
$\displaystyle=$
$\displaystyle\frac{2(2J_{v}+1)\sqrt{2}G_{F}y^{e}a_{ij}C_{ev}}{3m_{e}[E_{eg}(E_{eg}-2\omega)-m^{2}_{\phi}]}$
$\displaystyle\times\left\\{y_{S,ij}^{\nu}\left[m_{i}({\bf p}_{j}\cdot{\bf
q})-m_{j}({\bf p}_{i}\cdot{\bf q})\right]\right.$
$\displaystyle\left.i(1-\delta_{M})y_{P,ij}^{\nu}\left[m_{i}({\bf
p}_{j}\cdot{\bf q})-m_{j}({\bf p}_{i}\cdot{\bf q})\right]\right\\},$

where the pseudo-scalar part disappears for the Majorana case.

Combining everything into the differential cross-section in (10) and integrate
over $E_{\nu}$, we can calculate the change in the SM spectral function due to
the BSM scalar and pseudo-scalar interactions,

$\displaystyle\mathcal{I}_{\phi}=\sum_{ij}\frac{\Delta_{ij}(\omega)}{(E_{vg}-\omega)^{2}}\Theta(\omega-\omega_{ij}^{\rm
max})\left[I_{ij}^{\rm SM}(\omega)+\delta I_{ij}(\omega)\right],\qquad$ (28)

where the correction term $\delta I_{ij}(\omega)$ is,

$\displaystyle\delta I_{ij}$ $\displaystyle=$
$\displaystyle\frac{|y^{e}|^{2}\omega^{2}}{m_{e}^{2}G_{\rm
F}^{2}}\frac{\left[|y^{\nu}_{S,ij}|^{2}+(1-\delta_{M})|y^{\nu}_{P,ij}|^{2}\right]E_{eg}(E_{eg}-2\omega)-|y^{\nu}_{S,ij}|^{2}(m_{i}+m_{j})^{2}-(1-\delta_{M})|y^{\nu}_{P,ij}|^{2}(m_{i}-m_{j})^{2}}{24[E_{eg}(E_{eg}-2\omega)-m^{2}_{\phi}]^{2}}$
$\displaystyle+$
$\displaystyle\frac{y^{e}\omega^{2}}{6\sqrt{2}G_{F}}\left\\{\frac{{\rm
Re}\left[a_{ij}y^{\nu}_{S,ij}\right](m_{i}-m_{j})\left[1-\frac{(m_{i}+m_{j})^{2}}{E_{eg}(E_{eg}-2\omega)}\right]}{m_{e}[E_{eg}(E_{eg}-2\omega)-m^{2}_{\phi}]}-(1-\delta_{M})\frac{{\rm
Im}\left[a_{ij}y^{\nu}_{P,ij}\right](m_{i}+m_{j})\left[1-\frac{(m_{i}-m_{j})^{2}}{E_{eg}(E_{eg}-2\omega)}\right]}{m_{e}[E_{eg}(E_{eg}-2\omega)-m^{2}_{\phi}]}\right\\}.$

In Fig. 2 we show the changes in the spectral function $\mathcal{I}$ due to a
light scalar/pseudo-scalar mediator with mass $m_{\phi}=20$ keV and couplings
(1) $y^{e}y^{\nu}_{S,11}=6\times 10^{-9}$ (blue dot-dashed) or (2)
$y^{e}y^{\nu}_{P,11}=6\times 10^{-9}$ (brown dotted). Both the scalar and
pseudo-scalar contributions have almost the same shape for Dirac neutrinos as
shown in Fig. 2 since the only difference is a sign in front of $m_{j}$ shown
in (III.2) and the mass term is suppressed when compared with the leading
energy term. For Majorana neutrinos, only the scalar couplings can contribute
a correction.

##  IV Sensitivity and Physics Reach

To estimate the sensitivity of new physics parameter, we follow the
experimental setup proposed in Song:2015xaa and take a typical experiment
configuration with $V=100$ cm3 as well as $n_{a}=n_{\gamma}=10^{21}$cm-3. In
addition, the experiment will run for equal time $T$ at three different
frequencies $\omega_{i}=1.0688,1.0699,1.0711$ eV. Although other works
Zhang:2016lqp ; Huang:2019phr propose different frequency locations, our
results presented here should not change much since the effect is not
sensitive to the threshold locations but rather event rate. The QED background
$|e\rangle\rightarrow|g\rangle+n\gamma$ with $n\geq 3$ photons in the final
state can be large Yoshimura:2015fna . Fortunately, this background can be
removed using the photonic crystals wave guides Tanaka:2016wir that forbid the
emission of photons with a wavelength $\lambda\leq\pi/L$, where $L$ is the
size of the wave guide. Recent advances show that the background can be
significantly reduced to achieve background free environment Tanaka:2019blr .

The sensitivity is evaluated with the $\chi^{2}$ function below,

$\displaystyle\chi^{2}\equiv 2\sum_{\omega_{i}}\Bigl{\\{}$ $\displaystyle
N^{\rm true}(\omega_{i})-N^{\rm test}(\omega_{i})$ $\displaystyle-$
$\displaystyle N^{\rm true}(\omega_{i})\log\left[N^{\rm
true}(\omega_{i})/N^{\rm test}(\omega_{i})\right]\Bigr{\\}},\quad$ (30)

in terms of the total numbers $N^{\rm true}(\omega_{i})$ and $N^{\rm
test}(\omega_{i})$ of the true and test event samples, respectively,

$\displaystyle N(\omega)\equiv
0.002\left(\frac{T}{s}\right)\left(\frac{V}{100~{}{\rm
cm}^{3}}\right)\left(\frac{n_{a}\mbox{ or }n_{\gamma}}{10^{21}{\rm
cm}^{-3}}\right)^{3}\mathcal{I}(\omega).\qquad$ (31)

For a test experiment, we expect $\mathcal{O}(20)$ events with an exposure
time $T=2.3$ days for the SM interactions.

Figure 3: Left: Sensitivity on the combinations
$\left|g^{e}g^{\nu}_{L,ij}\right|$ and $\left|g^{e}g^{\nu}_{R,ij}\right|$ as a
function of the vector mediator mass $m_{Z^{\prime}}$ for Dirac and Majorana
neutrinos. Right: Sensitivity on the combinations $|y^{e}y^{\nu}_{S,ij}|$, and
$|y^{e}y^{\nu}_{P,ij}|$ as a function of the scalar mediator mass $m_{\phi}$.
In these plots we assume the normal neutrino mass ordering (NO) with the
lightest mass $m_{1}=0.01$ eV and an exposure time of $T=2.3$ days at a Yb-
based experiment.

The sensitivity curves shown in Fig. 3 are obtained for $\Delta\chi^{2}=5.99$,
corresponding to 95% C.L. The region above these curves will be excluded if no
signal observed. We can see that the sensitivity on the coupling constants
decreases with increasing mediator mass for both vector and scalar cases. The
sensitivity can reach $\left|g^{e}g^{\nu}_{L,ij}\right|$,
$\left|g^{e}g^{\nu}_{R,ij}\right|\lesssim 10^{-15}\sim 10^{-26}$
($|y^{e}y^{\nu}_{S,ij}|$, $|y^{e}y^{\nu}_{P,ij}|\lesssim 10^{-9}\sim
10^{-19}$) for the vector (scalar) mediator. In both cases, we take one
diagonal neutrino coupling in the mass basis, $g_{11},y_{11}\neq 0$, at a
time. Notice that there is no $y^{\nu}_{P}$ curve for Majorana neutrinos since
the pseudo-scalar term vanishes in this situation, as noted in (III.2).

The behaviour of the curve can be understood as follows. For large mass,
$m_{Z^{\prime},\phi}$ dominates over $E_{eg}(E_{eg}-2\omega)$ and the curve
decreases with $m_{Z^{\prime},\phi}^{2}$ as a line with slope $2$ in the log-
log scale. For very small mass, the situation is the opposite and
$E_{eg}(E_{eg}-2\omega)$ dominates over $m_{Z^{\prime},\phi}$. As result, the
event rate becomes insensitive to the mediator mass and the sensitivity curve
flattens. In between, the propagator has a pole at
$\omega=\frac{E_{eg}}{2}-\frac{m_{Z^{\prime},\phi}^{2}}{2E_{eg}}$ and hence
the diverging sensitivity. Since our benchmark experiment has only three
frequency points, $\omega=1.0688,1.0699,1.0711$ eV, the divergence happens at
three masses $m_{Z^{\prime},\phi}=0.11,0.088,0.053$ eV, respectively. However,
these dips in the sensitivity curves are not physical. The mediator decay into
a pair of neutrinos would lead to a nonzero decay width
$\Gamma_{Z^{\prime},\phi}$ in the mediator propagator and hence modulate the
sensitivity behavior. But it is not straightforward to implement in Fig. 3
since the decay width only relies on the neutrino coupling $g^{\nu}$ or
$y^{\nu}$. For a fixed value of the vertical axis $g^{e}g^{\nu}$
($y^{e}y^{\nu}$), there are too many choices for $g^{\nu}$ ($y^{\nu}$). So
there is no unique way of implementing the mediator decay width in Fig. 3. For
simplicity, the dips are kept but we shall keep in mind that the sensitivity
is not that dramatic at these points.

Fig. 3 also shows that there is no difference in the sensitivity of
$g^{e}g_{L,11}^{\nu}$ between Dirac and Majorana neutrinos. This happens
because the extra contribution due to the Majorana property, $I^{(M)}_{ij}$ of
(17), is small. Taking $m_{1}=0.01$ eV, $I^{(M)}_{ij}/I^{(D)}_{ij}\sim
6m_{i}m_{j}/\omega^{2}\approx 10^{-3}$ which is negligible. When only
$g_{L}^{\nu}$ coupling is non-zero, the change to the
$\mathcal{I}_{Z^{\prime}}$ in (22) that is proportional to $I^{(M)}$ is tiny.
For $g_{R}^{\nu}\neq 0$, the situation is different. There is a mixing term
contribution for Majorana neutrinos, $-2{\rm
Re}[a^{L}_{ij}a^{R}_{ij}]I^{(D)}_{ij}$ in (22) that can be large.

For comparison, also show several other constraints in Fig. 3. The vector
mediator case has three bounds: The BBN bound for a $Z^{\prime}$ was obtained
from the relativistic degrees of freedom $N_{\rm eff}$ in the early Universe
Ibe:2020dly . Since BBN happens at $\mathcal{O}(\mbox{MeV})$, the light
mediator mass in the considered range $[10^{-3},10^{4}]$ eV is negligibly
small. Consequently, the BBN bound is insensitive to the $Z^{\prime}$ mass and
flat. The red giant (RG) and horizontal branch (HB) bounds from stellar
cooling Hardy:2016kme are based on a specific $B-L$ model with the coupling to
leptons. The dotted curves for the RG and HB constraints with $U(1)_{B-L}$ in
the left panel of Fig. 3 are then obtained from the bounds on $g_{B-L}$ and
the model property that $g^{e}=g^{\nu}=g_{B-L}$. Finally, the NSI bound
(green) is obtained from the global fit of neutrino experiments Coloma:2020gfv
. Since the NSI effect is due to the coherent scattering of neutrinos in
matter with zero momentum transfer, it is only sensitive to the ratio between
coupling and mediator mass, $g^{e}g^{\nu}/m^{2}_{Z^{\prime}}$, reflected as a
straight line with slope 2 in the plot.

Most bounds for the scalar mediator case are for the coupling with neutrino or
electron separately. The bound on the pseudo-scalar coupling with the electron
neutrino $\nu_{e}$, $|y^{\nu}_{ee}|<1.5\times 10^{-3}$, is from pion  Figure
4:  Sensitivity on the coupling combinations $\left|g^{e}g^{\nu}\right|$ for
vector mediator (left) and $\left|y^{e}y^{\nu}\right|$ for the scalar one
(right) as a function of the lightest neutrino mass $m_{1}$ with the normal
mass ordering. For both cases, a vanishing mediator mass
$m_{Z^{\prime}},m_{\phi}=0$ eV is adopted. To make comparison, we have
considered both Dirac and Majorana neutrinos. These sensitivities are obtained
with an exposure of $T=2.3$ days at a Yb-based experiment. The gray region
shows the combined constraint from the current cosmological data Zyla:2020zbs
and neutrino oscillation measurements deSalas:2020pgw . decay Pasquini:2015fjv
; Berryman:2018ogk ; deGouvea:2019qaz ; Dror:2020fbh . For the coupling with
electron, the bound comes from the electron anomalous magnetic moment
$(g-2)_{e}$ measurement Mohr:2015ccw which is the most stringent experiment on
Earth. Astrophysical sources like RG and HB can also provide constraints from
the star cooling process Knapen:2017xzo ; DeRocco:2020xdt . If these bounds
follow a half-normal distribution, the product distribution follows the Bessel
function of the first kind and the 95% upper bound is a product of individual
ones, $(y^{e}y^{\nu})_{95\%}\approx
0.57\times(y^{e})_{95\%}\times(y^{\nu})_{95\%}$, to combine the separate
bounds on electron and neutrino couplings.

We also include a BBN bound Venzor:2020ova due to the effective neutrino mass
via matter effect Ge:2018uhz in the early Universe. This is the only bound
implementing the electron and neutrino couplings simultaneously. We also
include a more general bound arising from the production of relativistic
degrees of freedom. The result in DeRocco:2020xdt was obtained for the
electron coupling only, $y^{e}<10^{-9}$. In order to obtain the bound for the
combination $y^{e}y^{\nu}$, we combine with the pion decay data as before,
resulting in $\left(y^{e}y^{\nu}\right)_{95\%}<10^{-12}$. This bound starts to
dominate the BBN constraint at $m_{\phi}>0.4$ eV and is of same size as the
lower region of the SN bound.

With coupling to electron, a light mediator, such as axion Huang:2019rmc and
dark photon Bhoonah:2019eyo , can also be directly produced in this coherently
stimulated process, $|e\rangle\rightarrow|g\rangle+\gamma+a/\gamma^{\prime}$.
Experimentally, there is no distinction whether the final state is a neutrino
pair, axion, or dark photon. Although it can also provide a sensitive probe,
this direct production mode can only test the coupling with electron but not
the one with neutrinos. In addition, the emitted axion or dark photon cannot
be heavier than the emitted energy $E_{eg}$. For comparison, the neutrino pair
emission with light mediator as an intermediate particle can simultaneously
probe the couplings to both electron and neutrino without limitation on the
mediator mass.

Fig. 4 shows the dependence of the coupling sensitivities on the lightest
neutrino mass $m_{1}$. To avoid complication from the mediator mass as shown
in Fig. 3, we focus on the massless mediator limit where the sensitivity is a
constant and essentially independent of $m_{Z^{\prime}}/m_{\phi}$. The normal
ordering is adopted for illustration with the corresponding lightest mass
$m_{1}$ varies from 0 eV to 0.05 eV. The neutrino pair emission is more
sensitive with vanishing $m_{1}$. Increasing $m_{1}$ to 0.05 eV introduces
roughly a factor of 4 reduction in the sensitivity. This happens because the
allowed phase-space shrinks with increasing neutrino mass and consequently
smaller event rates. Again, there is no difference in the sensitivity for
$g^{e}g_{L,11}^{\nu}$ between Dirac (black line) and Majorana (dotted red
line) neutrinos as shown in the left panel of Fig. 4. This happens because
contribution $I_{ij}^{(M)}$ purely due to the Majorana property only
contributes $10^{-3}$ of $\mathcal{I}_{Z^{\prime}}$ in (22). For comparison,
the gray region shows the 95% C.L. upper bound on the lightest neutrino mass
$m_{1}$ from cosmology Zyla:2020zbs and oscillation measurements
deSalas:2020pgw .

##  V Conclusion

The sensitivity to light mediators is typically limited by the involved
momentum transfer. The various existing constraints on the coupling constant
can no longer improve once the light mediator mass decreases to become
comparable with momentum transfer. This is most transparent in the BBN, RG,
and HB constraints in Fig. 3. The only exception is the NSI bound with
neutrino coherent scattering but its sensitivity is limited to the neutrino
oscillation experiment precision which is just entering percentage era and it
cannot disentangle coupling from the mediator mass. In this paper, we point
out the promising future of the neutrino pair emission process as a very
sensitive probe of light vector and scalar mediators around the
$\mathcal{O}$(eV) scale. It can enhance the sensitivity by 3 orders more than
the NSI above $\mathcal{O}(0.1$eV) for the vector mediator case and the BBN
bound for the scalar case. The best upper limit can be as low as
$|y^{e}y^{\nu}|<10^{-9}\sim 10^{-19}$ for a scalar mediator and
$|g^{e}g^{\nu}|<10^{-15}\sim 10^{-26}$ for a vector one, respectively. The
RENP process can not only become an ideal place for testing the neutrino
nature and parameters but also provide a sensitive probe of the associated new
physics with neutrinos.

## Acknowledgements

This work is supported by the Double First Class start-up fund (WF220442604),
the Shanghai Pujiang Program (20PJ1407800), National Natural Science
Foundation of China (No. 12090064), and Chinese Academy of Sciences Center for
Excellence in Particle Physics (CCEPP). The authors are grateful to Xiao-Dong
Ma and Jin Sun for reading the draft and providing useful comments.

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