Probing light mediators in the radiative emission of neutrino pair

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.

1 Introduction

Since the reactor mixing angle $\theta _r$ ($\equiv \theta _{13}$) was measured by the Daya Bay [1] and RENO [2] experiments in 2012, the neutrino study has entered a precision era [3,4,5]. 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 [6,7,8,9,10] as well as the atmospheric mixing angle $\theta _a$ ($\equiv \theta _{23}$) at atmospheric [11,12,13,14,15] and accelerator [13,14,15,16,17,18,19,20,21] experiments. The only exception is the Dirac CP phase and the neutrino interactions. While the CP measurement is also approaching discovery threshold with T2K [22, 23] and NO$\nu$A [25], the neutrino interaction is still much less constrained [26, 27]. 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) [26, 27] and effective neutrino masses [28, 29]. In addition, new interactions can also lead to neutrino beamstrahlung [30], neutrino trident production [31,32,33,34,35,36], modified spectrum from leptonic meson decays [37, 38], and neutrino decay [39,40,41,42,43]. The presence of new light particles have impact in the early Universe by generating extra degrees of freedom [44, 45] or delaying the neutrino freeze-out [46, 47]. 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 [44, 48]. 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 [44, 45] in addition to the effective neutrino masses [29]. And the most stringent bound comes from cooling of astrophysical sources [49, 50] which are model dependent [51]. 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. [52].

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 [53,54,55] have prompted several possible explanations with light mediator between neutrino and electron, including scalar [56,57,58,59,60] and vector [56,57,58, 60,61,62,63,64,65,66,67] bosons, eletromagnetic interactions [57, 58, 65, 66, 68,69,70,71] with massless photon, and sterile neutrino [72,73,74,75]. 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) [76,77,78,79] 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 2 for both Dirac and Majorana neutrinos. Then 3 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 4. Finally, we summarize the most prominent results and features in 5.

2 Neutrino pair emission

The radiative emission of neutrino pair (RENP) [76,77,78,79,80] is a coherently induced superradiance [81] 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, $m_W = 80.4\,\text{ GeV }, m_Z = 91.2\,\text{ GeV } \gg \mathcal O(1 \sim 10)\,\text{ eV }$, a weak process at low energy is highly suppressed by the Fermi constant, $G_\mathrm{F}\approx 1.67\times 10^{-5}$ GeV$^{-2}$. Fortunately, the RENP can be enhanced by two quantum mechanical effects [81]. 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 [82]. 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 [83, 84], including the neutrino mass [85, 86], the unitarity test of the neutrino mixing matrix [87], the Dirac/Majorana nature of neutrinos [77, 84], the leptonic CP phases [84, 88], the cosmic neutrino background [89, 90], and light sterile neutrinos [91]. In the current section, we summarize the key elements of RENP in the SM for both Dirac and Majorana neutrinos.

2.1 The SM interactions for atomic transitions

The RENP process involves both electromagnetic and weak interactions,

$$\begin{aligned} H = H_0 + D_\gamma + H_W. \end{aligned}$$

(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 [77]) 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 $J_g = 0$) [85] 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 [78, 87] 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.

Fig. 1

Feynman diagrams for the radiative emission of neutrino pair with the SM heavy gauge bosons (Z/W) and light mediators ($Z'/\phi$)

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 $\mathbf{E}(\mathbf{x})$ with trigger frequency $\omega$,

$$\begin{aligned} \langle g|D_\gamma | v \rangle \equiv \mathcal M_D e^{-i\omega t + \mathbf{k}\cdot \mathbf{x}}, \quad \mathcal M_D \equiv - \mathbf{d}_{gv}\cdot \mathbf{E}_0, \end{aligned}$$

(2)

where $\omega$ and $\mathbf{k}$ are the photon energy and momentum, respectively. The 3-vector $\mathbf{d}_{gv}$ sandwiched between the virtual and ground states is the dipole operator for the $v\rightarrow g$ transition and $\mathbf{E}_0$ is the stimulating electric field.

The $H_W$ term contains the SM weak interactions between electron and neutrinos,

$$\begin{aligned} \mathcal H_W \equiv \sqrt{2} G_F \left( v_{ij} J^\mu _V - a_{ij} J^\mu _A \right) \bar{\nu }_{i L} \gamma _\mu \nu _{j L}. \end{aligned}$$

(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

$$\begin{aligned} a_{ij} \equiv U_{ei} U^*_{ej} - \frac{\delta _{ij}}{2}, ~ \mathrm{and} ~ v_{ij} \equiv a_{ij} + 2 s^2_w \delta _{ij}, \end{aligned}$$

(4)

contain the PMNS matrix U of neutrino mixing which is a function of mixing angles and leptonic CP phases [92]. From the weak Hamiltonian (3), we can obtain the matrix element for the $|e\rangle \rightarrow |v\rangle$ transition,

$$\begin{aligned} \langle v | H_W|e\rangle&\equiv \mathcal M_W^{(D)} e^{-i(p_\nu +p_{\bar{\nu }})\cdot x} \end{aligned}$$

(5a)

$$\begin{aligned} \mathcal M_W^{(D)}&\equiv \sqrt{2}G_\mathrm{F} (J_e^\mu )_{ij} \bar{u}(p_{\nu _i})\gamma _\mu P_L v(p_{\bar{\nu }_j}), \end{aligned}$$

(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, $J_A^\mu \approx (0,\langle v| 2 \mathbf{S} |e\rangle )$ [85]. For comparison, the electric dipole $\mathbf{d}_{gv}$ is parity odd (E1) while the spin operator $\mathbf{S}$ is parity even (M1).

2.2 Dirac vs majorana neutrino matrix elements

The matrix element $\mathcal M^{(D)}_W$ (5a5b) 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, $\nu \sim a u + a^\dagger v$, 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 [84],

$$\begin{aligned} \mathcal M_W^{(M)}&= \sqrt{2} G_F J_A^\mu \Bigl [ a_{ij} \bar{u} (p_{\nu _i}) \gamma _\mu P_L v (p_{\overline{\nu }_j}) \nonumber \\&\quad - a_{ij}^* \bar{u}(p_{\overline{\nu }_j}) \gamma _\mu P_L v (p_{\nu _i}) \Bigr ]. \end{aligned}$$

(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,

$$\begin{aligned} |\mathcal M_W^{(D)}|^2&= 2 G_F^2 J_A^\mu J_A^{\nu *} |a_{ij}|^2 \nonumber \\&\quad \times \bar{u}(p_{\nu _i}) \gamma _\mu P_L v (p_{\overline{\nu }_j}) \bar{v} (p_{\overline{\nu }_j}) \gamma _\nu P_L u(p_{\nu _i}), \end{aligned}$$

(7a)

$$\begin{aligned} |\mathcal M_W^{(M)}|^2&= |\mathcal M_W^{(D)}|^2 - 2 G_F^2 \mathrm{Re} \Bigl [ a_{ij}^2 J_A^\mu J_A^{\nu *} \nonumber \\&\quad \times \bar{u}(p_{\nu _i}) \gamma _\mu P_L v(p_{\overline{\nu }_j}) \bar{v}(p_{\overline{\nu }_j}) \gamma _\nu P_R u(p_{\nu _i}) \Bigr ]. \quad \end{aligned}$$

(7b)

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,

$$\begin{aligned} \overline{|\mathcal M_W^{ij}|^2}&= \frac{8}{3}(2 J_v + 1) |a_{ij}|^2 G_\mathrm{F}^2 C_{ev} \nonumber \\&\quad \times \Bigl \{ 2 \mathbf{p}_{\nu _i}\cdot \mathbf{p}_{\bar{\nu }_j} + 3 p_{\nu _i} \cdot p_{\bar{\nu }_j} - 3 \delta _M \mathrm{Re}[a_{ij}^2] m_i m_j \Bigr \}.\nonumber \\ \end{aligned}$$

(8)

The prefactor $\delta _M = 1$ for Majorana neutrinos and $\delta _M = 0$ for Dirac ones. To obtain (8), the electron spins have also been summed over [84],

$$\begin{aligned} \frac{1}{(2 J_e + 1)}\sum _{m_e m_v} \langle v|S^i |e\rangle \langle e| S^j | v\rangle = (2 J_v + 1) \frac{C_{ev}}{3} \delta ^{ij}, \end{aligned}$$

(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 $(2 J_v + 1) C_{ev} = 2$ [85].

2.3 Emission rate and spectrum

The photon emission rate of the reaction is [84,85,86],

$$\begin{aligned} d\Gamma _{ij} = \frac{\Gamma _0}{(E_{vg} - \omega )^2 \omega } \frac{\overline{|\mathcal {M}^{ij}_W|^2}}{8 G_F^2 C_{ev} (2J_v + 1)} dE_\nu , \end{aligned}$$

(10)

with the reference decay width $\Gamma _0$ [85, 86],

$$\begin{aligned} \Gamma _0\equiv & {} (2J_v + 1)\frac{n_a^2 C_{ev} G_\mathrm{F}^2 |\mathbf{d}_{gv}\cdot \mathbf{E}_ 0|^2}{\pi } \end{aligned}$$

(11a)

$$\begin{aligned}\approx & {} 0.002\mathrm{s}^{-1} \left( \frac{V}{10^2 \mathrm{cm}^3}\right) \left( \frac{n_a^2 n_\gamma }{10^{21}\mathrm{cm}^{-3}}\right) \eta (t). \end{aligned}$$

(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 [93] 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 [85, 86] ($\omega _1$ 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,

$$\begin{aligned} E_e - E_g = \omega + E_\nu + E_{\overline{\nu }} \quad \mathrm{and} \quad \mathbf{k} = - \mathbf{p}_\nu - \mathbf{p}_{\overline{\nu }}\,. \end{aligned}$$

(12)

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

$$\begin{aligned} \omega _{ij}^\mathrm{max} \equiv \frac{E_e - E_g}{2} - \frac{1}{2}\frac{(m_i + m_j)^2}{(E_e - E_g)}. \end{aligned}$$

(13)

Only for $\omega < \omega _{ij}^\mathrm{max}$, 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 $\omega$,

$$\begin{aligned} \bar{E} - \frac{\omega \Delta _{ij}(\omega )}{2} \le E_\nu \le \bar{E} + \frac{\omega \Delta _{ij}(\omega )}{2}, \end{aligned}$$

(14)

where,

$$\begin{aligned} \Delta _{ij}(\omega )\equiv & {} \frac{ \sqrt{E_{eg}(E_{eg}-2 \omega )-(m_i+m_j)^2} }{E_{eg} (E_{eg}-2 \omega )} \nonumber \\&\quad \times \sqrt{E_{eg}(E_{eg}-2 \omega )-(m_i-m_j)^2}, \end{aligned}$$

(15a)

$$\begin{aligned} \bar{E}\equiv & {} \frac{[E_{eg}(E_{eg}-2\omega ) - \Delta m^2_{ij}](E_{eg}-\omega )}{2E_{eg}(E_{eg}-2\omega )}. \end{aligned}$$

(15b)

After integration over the kinematic range in (12) and sum over all the neutrino mass eigenstates, the total decay width becomes $\Gamma \equiv \Gamma _0 \mathcal I(\omega )$ [84]. The spectral function $\mathcal I(\omega )$ is defined as,

$$\begin{aligned} \mathcal I(\omega )\equiv & {} \sum _{ij} \frac{\Delta _{ij}(\omega ) }{(E_{vg} - \omega )^2} \Theta (\omega - \omega _{ij}^\mathrm{max}) \nonumber \\&\quad \times \left[ |a_{ij}|^2 I_{ij}^{(D)} - \delta _M \mathrm{Re}[a_{ij}^2] I_{ij}^{(M)} \right] , \end{aligned}$$

(16)

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

$$\begin{aligned} I_{ij}^{(D)}\equiv & {} \frac{1}{3} \left\{ E_{eg} (E_{eg}-2\omega ) - \frac{1}{2}(m_i^2+m_j^2) \right. \end{aligned}$$

(17a)

$$\begin{aligned}&\quad + \left. \frac{\omega ^2}{2} \left[ 1-\frac{1}{3}\Delta _{ij}(\omega )^2\right] - \frac{(E_{eg}-\omega )^2(\Delta m_{ij}^2)^2}{2E_{eg}^2(E_{eg}-2 \omega )^2} \right\} , \nonumber \\ I_{ij}^{(M)}\equiv & {} m_i m_j\,. \end{aligned}$$

(17b)

There are two candidate elements that allow RENP process: Xenon (Xe) and Ytterbium (Yb) [84]. In this work we focus on Yb that can reach $\mathcal O(20)$ events per day for a target volume of 100 cm$^{3}$ and atomic density of $10^{21}$ cm$^{-3}$ [86]. Although these parameters are not achievable yet, there have been some advances recently [78].

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 $\omega ^\mathrm{max}_{11} \approx 1.0716$ eV.

3 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, $q^2 \sim \mathcal O(\text{ eV}^2)$, in contrast to $m^2_W, m^2_Z \sim \mathcal O(10^{21} \text{ eV}^2)$. 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 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.

Fig. 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 $m_{\phi , Z'} = 20$ 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

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 $\mathbf{q}$ [94]

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 [95,96,97,98,99,100], fifth force [101,102,103], $U(1)_R$ symmetry connected to a hidden right-handed neutrino sector [104, 105], neutrino mass models [106,107,108], the neutrinophilic two-Higgs doublet models [63, 109], the neutrinophilic axion-like particles [110,111,112,113,114], neutrino non-standard interaction with light mediators [48, 115]. Also, light mediators can be a connection between neutrinos and ultra-light dark matter [116,117,118,119,120,121,122,123] and inflation [124]. Some discussions on light mediator and neutrino interactions can also be found in [125, 126]. The rich phenomenology of light mediators in the leptonic sector can be found in [44]. It is well motivated to find possible ways of probing light and ultra-light mediators using low enough momentum transfer as we elaborate below.

3.1 Vector mediator

Since the vector coupling with electron does not contribute to the RENP process, the relevant new interactions with a vector mediator $Z'$ are,

$$\begin{aligned} \mathcal L_V = g^e \bar{e} \gamma ^\mu \gamma _5 e Z'_\mu + \bar{\nu }_i \gamma ^\mu (g^\nu _{L,ij} P_L + g^\nu _{R,ij} P_R) \nu _j Z'_\mu . \qquad \end{aligned}$$

(18)

Extra terms can arise as correction to the scattering matrix elements (5a5b) and (6), $\mathcal M_W^{ij} \rightarrow \mathcal M_\mathrm{tot}^{ij} \equiv \mathcal M_W^{ij} + \mathcal M_\mathrm{New}^{ij}$, where $\mathcal M_\mathrm{New}^{ij}$ is the new contribution,

$$\begin{aligned} \mathcal M_\mathrm{tot}^{ij} =&\sqrt{2} G_\mathrm{F} J^\mu _A \Bigl [ \bar{u}(p_{\nu _i}) \gamma _\mu \left( a^L_{ij} P_L + a^R_{ij} P_R \right) v (p_{\bar{\nu }_j}) \nonumber \\&- \delta _M \bar{u}(p_{\bar{\nu }_j}) \gamma _\mu \left( a^L_{ji} P_L + a^R_{ji} P_R \right) v (p_{\nu _i}) \Bigr ]. \end{aligned}$$

(19)

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

$$\begin{aligned} a^L_{ij}\equiv & {} a_{ij} + \frac{g^e g^\nu _{L,ij}}{\sqrt{2} G_F (q^2 - m^2_{Z'})} \delta _{ij}, \end{aligned}$$

(20a)

$$\begin{aligned} a^R_{ij}\equiv & {} \frac{g^e g^\nu _{R,ij}}{\sqrt{2} G_F (q^2 - m^2_{Z'})} \delta _{ij}. \end{aligned}$$

(20b)

The first term $a_{ij}$ is the SM contribution and those terms involving $1/(q^2 - m^2_{Z'})$ come from the $Z'$ 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 $\overline{|\mathcal M^{ij}_\mathrm{tot}|^2}$ after averaging over all spin and atomic state is,

$$\begin{aligned}&\frac{8}{3} (2 J_v + 1) C_{ev} G^2_F \nonumber \\&\quad \times \left[ \Bigl (|a^L_{ij}|^2 + |a^R_{ij}|^2 - 2 \delta _M \mathrm{Re}[a^L_{ij}a^R_{ij}] \Bigr ) (2 \mathbf{p}_i \cdot \mathbf{p}_j + 3 p_i \cdot p_j) \right. \nonumber \\&\quad \left. + 3 m_i m_j \left( 2 \mathrm{Re}[a^{L *}_{ij} a^R_{ij}] - \delta _M \mathrm{Re}[(a^L_{ij})^2 + (a^R_{ij})^2] \right) \right] . \end{aligned}$$

(21)

It is interesting to observe that the first line in the bracket has the same structure as (8) with the overall factor substitution $|a_{ij}|^2 \rightarrow |a^L_{ij}|^2 + |a^R_{ij}|^2 - 2 \delta _M \mathrm{Re} [ a^L_{ij} a^R_{ij}]$. 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, $\mathrm{Re}[a_{ij}^2] \rightarrow 2 \mathrm{Re}[a^L_{ij}a^R_{ij}] - \delta _M \mathrm{Re} [(a^{L *}_{ij})^2 + (a^R_{ij})^2]$.

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

$$\begin{aligned} \mathcal I_{Z'}&\equiv \sum _{ij} \frac{\Delta _{ij}(\omega ) }{(E_{vg} - \omega )^2} \Theta (\omega - \omega _{ij}^\mathrm{max}) \nonumber \\&\quad \times \left[ \Bigl ( |a^L_{ij}|^2 + |a^R_{ij}|^2 - 2 \delta _M \mathrm{Re} [a^L_{ij} a^R_{ij}] \Bigr ) I_{ij}^{(D)} \right. \nonumber \\&\quad + \left. \Bigl ( \delta _M \mathrm{Re} \left[ (a^L_{ij})^2 + (a^R_{ij})^2 \right] {-} 2\mathrm{Re} \left[ a^{L*}_{ij}a^R_{ij} \right] \Bigr ) I_{ij}^{(M)} \right] . \nonumber \\ \end{aligned}$$

(22)

Figure 2 shows the changes in the spectral function $\mathcal I$ for a light mediator mass of $m_{Z'} = 20$ 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.

3.2 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,

$$\begin{aligned} \mathcal L_S\equiv & {} i y^e_P \bar{e} \gamma _5 e \phi + \bar{\nu }_i (y^\nu _{S,ij} + i \gamma _5 y^\nu _{P,ij}) \nu _j \phi + h.c., \quad \end{aligned}$$

(23)

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

$$\begin{aligned} \mathcal M^{ij}_S&= \frac{ y^e J_P \bar{u}(p_{\nu _i}) (y^\nu _{S,ij} + i \gamma _5 y^\nu _{P,ij}) v(p_{\overline{\nu }_j}) }{E_{eg} (E_{eg} - 2 \omega ) - m^2_\phi } \nonumber \\&\quad - \delta _M \frac{ y^e J_P \bar{u}(p_{\overline{\nu }_j}) (y^\nu _{S,ji} - i \gamma _5 y^\nu _{P,ji}) v(p_{\nu _i}) }{E_{eg} (E_{eg} - 2 \omega ) - m^2_\phi }, \qquad \end{aligned}$$

(24)

where the second line arises only for Majorana neutrinos.

The total averaged matrix element squared,

$$\begin{aligned} \overline{|\mathcal M_{\mathrm{tot}}^{ij}|^2} = \overline{|\mathcal M^{ij}_W|^2} + \overline{|\mathcal M^{ij}_S|^2} + 2 {\mathrm{Re}} \left( \overline{ \mathcal M^{ij}_W \mathcal M^{ij *}_S} \right) , \qquad \end{aligned}$$

(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 $J_P \equiv \frac{1}{2 m_e} \mathbf{q} \cdot \langle e | \varvec{\sigma }| v \rangle$ as summarized in 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,

$$\begin{aligned} \frac{1}{(2 J_e + 1)} \sum _{m_e m_v} |J_P|^2= & {} \frac{C_{ev}}{3} (2 J_v + 1) \frac{\omega ^2}{m_e^2} \,, \end{aligned}$$

(26a)

$$\begin{aligned} \frac{1}{(2 J_e + 1)} \sum _{m_e m_v} J_A^\mu J_P^*= & {} \frac{2 C_{ev}}{3} (2 J_v + 1) \left( 0, \frac{q}{m_e} \right) . \end{aligned}$$

(26b)

Since the emitted photon energy or momentum transfer is much smaller than the electron mass, $\omega , \mathbf{q} \ll m_e$, 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 (). So the interference term would dominate. Putting things together, the averaged matrix element squared (8) becomes,

$$\begin{aligned} \overline{|\mathcal {M}^{ij}_S|^2}= & {} \frac{2 (2 J_v + 1)|y^e|^2 C_{ev}}{[E_{eg} (E_{eg} - 2 \omega ) - m^2_\phi ]^2} \frac{\omega ^2}{3 m_e^2}\nonumber \\&\times \left\{ |y^\nu _{S,ij}|^2 (p_i \cdot p_j - m_i m_j) \right. \nonumber \\&+ \left. (1 - \delta _M) |y^\nu _{P,ij}|^2 (p_i \cdot p_j + m_i m_j) \right\} , \end{aligned}$$

(27a)

$$\begin{aligned} \overline{\mathcal M^{ij}_W \mathcal M^{ij *}_S}= & {} \frac{ 2(2 J_v + 1)\sqrt{2} G_F y^e a_{ij} C_{ev} }{3 m_e[E_{eg} (E_{eg} - 2 \omega ) - m^2_\phi ]} \nonumber \\&\times \left\{ y_{S,ij}^\nu \left[ m_i (\mathbf{p}_j \cdot \mathbf{q}) - m_j (\mathbf{p}_i \cdot \mathbf{q}) \right] \right. \nonumber \\&\left. i (1 - \delta _M) y_{P,ij}^\nu \left[ m_i (\mathbf{p}_j \cdot \mathbf{q}) - m_j (\mathbf{p}_i \cdot \mathbf{q}) \right] \right\} ,\nonumber \\ \end{aligned}$$

(27b)

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,

$$\begin{aligned} \mathcal I_\phi = \sum _{ij} \frac{\Delta _{ij}(\omega ) }{(E_{vg} - \omega )^2} \Theta (\omega - \omega _{ij}^\mathrm{max}) \left[ I_{ij}^\mathrm{SM}(\omega ) + \delta I_{ij}(\omega ) \right] , \nonumber \\ \end{aligned}$$

(28)

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

$$\begin{aligned} \delta I_{ij}&= \frac{|y^e|^2 \omega ^2 }{m_e^2 G_\mathrm{F}^2} \frac{ \left[ |y^\nu _{S,ij}|^2 + (1 - \delta _M)|y^\nu _{P,ij}|^2 \right] E_{eg} (E_{eg} - 2 \omega ) - |y^\nu _{S,ij}|^2 (m_i + m_j)^2 - (1 - \delta _M) |y^\nu _{P,ij}|^2 (m_i - m_j)^2 }{24[E_{eg}(E_{eg}-2\omega ) - m^2_\phi ]^2} \nonumber \\&\quad + \frac{y^e\omega ^2}{6\sqrt{2} G_F} \left\{ \frac{ \mathrm{Re}\left[ a_{ij} y^\nu _{S,ij}\right] (m_i - m_j) \left[ 1 - \frac{(m_i + m_j)^2}{E_{eg}(E_{eg} - 2\omega )} \right] }{m_e [E_{eg} (E_{eg} - 2 \omega ) - m^2_\phi ]}- (1 - \delta _M) \frac{ \mathrm{Im}\left[ a_{ij} y^\nu _{P,ij}\right] (m_i + m_j) \left[ 1 - \frac{(m_i - m_j)^2}{E_{eg}(E_{eg} - 2\omega )} \right] }{m_e [E_{eg} (E_{eg} - 2 \omega ) - m^2_\phi ]} \right\} .\nonumber \\ \end{aligned}$$

(29)

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 (29) and the mass term is suppressed when compared with the leading energy term. For Majorana neutrinos, only the scalar couplings can contribute a correction.

4 Sensitivity and physics reach

To estimate the sensitivity of new physics parameter, we follow the experimental setup proposed in [85] and take a typical experiment configuration with $V = 100$ cm$^3$ 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 [86, 87] 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 $n \ge 3$ photons in the final state can be large [127]. Fortunately, this background can be removed using the photonic crystals wave guides [128] that forbid the emission of photons with a wavelength $\lambda \le \pi /L$, where L is the size of the wave guide. Recent advances show that the background can be significantly reduced to achieve background free environment [129].

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

$$\begin{aligned} \chi ^2&\equiv 2 \sum _{\omega _i} \Bigl \{ N^\mathrm{true}(\omega _i) - N^\mathrm{test}(\omega _i) \nonumber \\&\quad - N^\mathrm{true}(\omega _i) \log \left[ N^\mathrm{true}(\omega _i)/N^\mathrm{test}(\omega _i) \right] \Bigr \}, \quad \end{aligned}$$

(30)

in terms of the total numbers $N^\mathrm{true}(\omega _i)$ and $N^\mathrm{test}( \omega _i)$ of the true and test event samples, respectively,

$$\begin{aligned} N (\omega ) {\equiv } 0.002 \left( \frac{T}{s}\right) \left( \frac{V}{100~\mathrm{cm}^3}\right) \left( \frac{n_a \text{ or } n_\gamma }{10^{21} \mathrm{cm}^{-3}}\right) ^3 \mathcal I (\omega ).\nonumber \\ \end{aligned}$$

(31)

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

Fig. 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 $m_{Z'}$ 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

Fig. 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 $m_{Z'}, m_\phi = 0$ 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 [92] and neutrino oscillation measurements [3]

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, $g_{11}, y_{11} \ne 0$, 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 (29).

The behaviour of the curve can be understood as follows. For large mass, $m_{Z',\phi }$ dominates over $E_{eg}(E_{eg}-2\omega )$ and the curve decreases with $m_{Z',\phi }^2$ 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 $m_{Z',\phi }$. As result, the event rate becomes insensitive to the mediator mass and the sensitivity curve flattens. In between, the propagator has a pole at $\omega = \frac{E_{eg}}{2}- \frac{m_{Z',\phi }^2}{2E_{eg}}$ 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 $m_{Z',\phi } = 0.11, 0.088, 0.053$ 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 $\Gamma _{Z', \phi }$ 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.

Figure 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 (), 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 $\mathcal I_{Z'}$ in (22) that is proportional to $I^{(M)}$ is tiny. For $g_{R}^\nu \ne 0$, the situation is different. There is a mixing term contribution for Majorana neutrinos, $-2\mathrm{Re}[a^L_{ij} a^R_{ij}] I^{(D)}_{ij}$ 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 $Z'$ was obtained from the relativistic degrees of freedom $N_\mathrm{eff}$ in the early Universe [67]. Since BBN happens at $\mathcal O(\text{ MeV})$, the light mediator mass in the considered range $[10^{-3}, 10^4]$ eV is negligibly small. Consequently, the BBN bound is insensitive to the $Z'$ mass and flat. The red giant (RG) and horizontal branch (HB) bounds from stellar cooling [49] 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 [115]. 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, $g^e g^\nu / m^2_{Z'}$, 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 decay [30, 102, 130, 131]. For the coupling with electron, the bound comes from the electron anomalous magnetic moment $(g-2)_e$ measurement [132] which is the most stringent experiment on Earth. Astrophysical sources like RG and HB can also provide constraints from the star cooling process [51, 133]. 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 [29] due to the effective neutrino mass via matter effect [28] 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 [51] was obtained for the electron coupling only, $y^e < 10^{-9}$. In order to obtain the bound for the combination $y^ey^\nu$, 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 [134] and dark photon [135], can also be directly produced in this coherently stimulated process, $|e\rangle \rightarrow |g\rangle + \gamma + a/\gamma '$. 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.

Figure 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 $m_{Z'}/m_\phi$. 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 $\mathcal I_{Z'}$ in (22). For comparison, the gray region shows the 95% C.L. upper bound on the lightest neutrino mass $m_1$ from cosmology [92] and oscillation measurements [3].

5 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.