Unique Probe of Neutrino Electromagnetic Moments with Radiative Pair Emission

The neutrino magnetic and electric moments are zero at tree level but can arise in radiative corrections. Any deviation from the Standard Model prediction would provide another indication of neutrino-related new physics in addition to the neutrino oscillation and masses. Especially, Dirac and Majorana neutrinos have quite different structures in their electromagnetic moments. Nevertheless, the recoil measurements and astrophysical stellar cooling can only constrain combinations of neutrino magnetic and electric moments with the limitation of not seeing their detailed structures. We propose using the atomic radiative emission of neutrino pair to serve as a unique probe of the neutrino electromagnetic moments with the advantage of not just separating the magnetic and electric moments but also identifying their individual elements. Both searching strategy and projected sensitivities are illustrated in this letter.

Introduction -In the Standard Model (SM) of particle physics, there is no tree-level coupling between neutrino (ν) and photon (A) [1]. However, the neutrino electromagnetic interactions are expected to arise from radiative corrections [2], The two terms account for the magnetic (µ ν ≡ f M (0)) and electric ( ν ≡ f E (0)) dipole moments at vanishing momentum transfer, q 2 = 0, respectively.
Observing a non-zero diagonal element (µ ν ) ii or ( ν ) ii is then a direct evidence of Dirac neutrinos. In addition, a nonzero diagonal ( ν ) ii also indicates CP violation.
Generally speaking, the explicit form of µ ν is modeldepended and its size is many orders smaller than the Bohr magneton µ B = e/2m e [4,5,[9][10][11][12]. Interestingly, if the neutrino mass and magnetic moment arise from the same effective operator, the magnetic moment for Majorana neutrino is typically 5 orders larger than the Dirac one [13,14]. Currently, the experimental sensitivity is already around the threshold for discovering the Majorana neutrino magnetic moments.
The neutrino electromagnetic moments can be tested in various ways. Typically, the neutrino scattering cross section with electron peaks in the low momentum transfer region to provide a sizable signal in the electron recoil. Both solar and reactor neutrinos can be used for such recoil measurement. The best sensitivity comes from the reactor experiment GEMMA, µ eff α < 2.9 × 10 −11 µ B [15], and the solar experiment Borexino, µ eff α < 2.8 × 10 −11 µ B [16], both at 90% C.L. However, the recoil measurement probes not just the magnetic moment µ ν but also the electric one ν as a combination [17,18], where U αk is the neutrino mixing matrix element [1]. The sensitivity of scattering experiments only applies to the combination µ eff α but not the individual electromagnetic moments due to possible cancellation among them. In other words, there is no unique probe of the neutrino magnetic or electric moment.
Although having multiple experimental ways, the current probe of neutrino electromagnetic moments has intrinsic limitations. In addition to the fact that the aforementioned measurements cannot distinguish the magnetic moment from the electric counterpart, the presence of the mixing matrix leads to blind spots in the allowed parameter space [36]. Moreover, existing measurements can not truly probe the magnetic moment at zero momentum transfer but instead have an O(keV) threshold. It is desirable to find new ways of exploring the neutrino electromagnetic properties.
In this letter, we present a novel way to probe the neutrino magnetic and electric moments by using the proposed radiative emission of neutrino pair (RENP) [37][38][39]. Although the RENP transition is yet to be observed, the coherent superradiance has been demonstrated with two-photon emission from hydrogen molecules [40]. Further discussions on experimental realization and background suppression can be found in [41][42][43].
The RENP process with O(eV) momentum transfer is a perfect place for probing light mediator interactions [44]. With massless photon being the mediator, the neutrino electromagnetic interactions fall exactly into this category. It allows the possibility of scanning the detailed structure of neutrino magnetic and electric moments in the mass eigenstate basis as a unique probe.
Electromagnetic Emission of Neutrino Pair -The radiative emission of a neutrino pair is an atomic transition from an excited state |e to the ground state |g . With the direct transition |e → |g + γ being forbidden, the emission arises at the second order in perturbation theory. The atom first goes from an excited state |e to a virtual state |v and then falls to the ground state |g , |e → |v +νν → |g + γ +νν.
This spontaneous process is very slow, but can be greatly enhanced by superradiance using a trigger laser beam [37,45]. The total Hamiltonian describing the reaction contains three parts, The zeroth-order Hamiltonian, H 0 , accounts for the electron state, H 0 |a = E a |a where a = v, e, or g. With energies E v > E e > E g , the two-step process |e → |v → |g renders |e meta-stable. By proper selection of |e and |g , the whole transition is of M1×E1 type with one electric (E1) and one magnetic (M1) dipole transitions. The photon is emitted from the second step, |v → |g + γ, by the E1-type electric dipole term D γ [46]. The corresponding amplitude is, where ω and k are the photon energy and momentum, respectively. The matrix element M D is a product of the dipole operator d gv for the atomic transition |v → |g and the photon electric field E 0 . On the other hand, the neutrino pair emission |e → |v +ν j ν i during the first step is of M1 type dictated The atomic transition diagram for the RENP process. In a typical experimental configuration, the emission of a neutrino pair is emitted first from |e → |v +νν while the photon emission occurs for |v → |g + γ.
by the weak Hamiltonian H W . In the SM, the leading contribution comes from the electroweak (EW) charged and neutral currents, where the prefactor a ij ≡ U ei U * ej − δ ij /2 is a function of the neutrino mixing matrix elements U ei . Although both vector and axial-vector currents are present, only the axial part of the electron current contributes since the transition is of the M1 type [46].
Non-zero neutrino magnetic and electric moments in Eq. (1) can also contribute to the M1 type transition as depicted in Fig. 1. For Dirac neutrinos, the amplitude for the magnetic one is v|H The momentum transfer is defined as and similarly for the electric moment case with the vertex replacement (µ ν ) ij σ βµ → ( ν ) ij σ βµ γ 5 . The minus sign comes from the anti-commutation property of fermion fields and the factor of 1/2 from the Lagrangian interaction normalization of Majorana neutrinos. Using = M E . For non-relativistic atomic states, only the spatial components of the atomic currents v|eγ µ γ 5 e|e in Eq. (7) and v|eσ µν e|e in Eq. (9) contribute significantly. They are proportional to the atomic spin operator S [47], v|eγγ 5 e|e = 2S ve and v|eσ ij e|e = −2 ijk S k ve . The summation over the electron spins m e and m v follows the identity [48], The other factors J a are the total spin of the excited (a = e) and virtual (a = v) states. For Yb and Xe, (2J v + 1)C ve = 2 [46]. In addition to the SM contribution |M W | 2 [38,44,46], the neutrino magnetic/electric moment first contributes an spin averaged term, where θ is the angle between the photon and the neutrino momentum. Between the magnetic and electric moments, the mass eigenvalue m j flips a sign which comes from the γ 5 matrix in the second term of Eq. (1). It is also possible to have interference between the SM and electromagnetic moment contributions, withĒ ≡ (E eg − ω)[q 2 − ∆m 2 ji ]/2q 2 . However, the interference between magnetic and electric moment contributions is zero after integrating over E νi .
The differential emission rate [46,48,49] is, with regulates the total number of decays. In practice only a fraction η of the total volume V can be enhanced by n 2 a n γ where n a and n γ are the atomic and photon number densities.
The momentum conservation fixes the value of the integration range to beĒ − ω∆ ij /2 ≤ E νi ≤Ē + ω∆ ij /2 where the relative energy width is ∆ ij ≡ cannot survive the neutrino energy integration. So the total decay rate contains only three parts Γ = Γ W +Γ M,E where Γ W is the SM contribution and Γ M (Γ E ) the magnetic (electric) moment one.
Integrating Eq. (14) over the neutrino energy E νi renders the decay rate for the magnetic (electric) moment For each pair of neutrinos ν i and ν j , the emitted photon energy has an upper limit ω max ij [38,48,50], ω max Fig. 2. Apart from E eg , these frequency thresholds ω max ij are functions of the neutrino absolute masses m i and m j . Smaller mass leads to higher ω max ij . The final-state photon has the same frequency ω as the trigger laser. Tuning the trigger laser frequency allows detailed scan of the I(ω) function. Especially, scanning the thresholds at ω max ij can be used to determine the neutrino mass hierarchy and the absolute masses [46,48,49,51]. This requires tuning the trigger laser with a precision of 10 −5 eV [48].
Searching Strategy and Sensitivity Estimation -Interestingly, the threshold scan can also be used to separate the magnetic and electric moments. As shown in the left panel of Fig. 2, the contribution of (µ ν ) ij to the total spectral function I ≡ I W + |(µ ν ) ij | 2 I M is non-zero only if ω < ω max ij . In other words, the frequency region ω < ω max 33 receives contribution from all elements (µ ν ) ij while the region ω max 33 < ω < ω max 23 cannot be affected by (µ ν ) 33 . Two independent measurements below and above ω max 33 can identify a nonzero (µ ν ) 33 . Similarly, two independent measurements in the regions of (ω max 33 , ω max 23 ) and (ω max 23 , ω max 22 ) can identify (µ ν ) 23 . Carrying out this procedure recursively, all the six (µ ν ) ij elements can be identified. The process is equivalent for ν . For m 1 = 0.01 eV, we take six trigger laser frequencies ω i=1···6 = (1.069,  1.07, 1.0708, 1.0712, 1.0716, 1.07164) eV.
In addition, the sign flip m j → −m j in Eq. (16) allows separating the magnetic moment contribution from the electric one. The difference ∆I = I M −I E ∝ 12m i m j /q 2 is relatively significant and can even reach 100% near the threshold. With two measurements, one near and another away from threshold, it is possible to distinguish the magnetic and electric moments. As a conservative estimation, we assign a universal extra frequency ω 0 = 1.068 eV below ω max 33 to resolve the ambiguity. We try to estimate the sensitivity on the neutrino electromagnetic moments by taking the conservative setup with n γ = n a = 10 21 cm −3 [46,52]. The number of photon events for the trigger laser frequency ω i is T day V 100 cm 3 n a or n γ 10 21 cm −3 3 I(ω). (17) For an exposure of T = 10 days and a volume of V = 100 cm 3 that are equally assigned for all the seven frequencies ω i , we expect the SM background events to be N i=0···6 ≈ (120, 107, 87, 82, 79, 39, 2.5), respectively, with a total of 512 events. Fig. 3 shows the 90% C.L. sensitivity curves evaluated in Poisson statistics [44] versus the expected RENP event number that can be translated for future experiments with different configurations as required design targets. The sensitivity can reach (µ ν ) ij < (1.5 ∼ 3.5) × 10 −11 µ B (( ν ) ij < (2 ∼ 9) × 10 −11 µ B ) for a conservative number of 500 events and further touch (0.8 ∼ 2) × 10 −12 µ B ((1.1 ∼ 5.5)×10 −12 µ B ) for 5000 events. Among the magnetic moment elements, (µ ν ) 11 has the best sensitivity while the worst case is (µ ν ) 33 due to two reasons. First, the (µ ν ) 33 curve is probed at only a single frequency, ω 1 = 1.0688 eV while (µ ν ) 11 contributes at all the 6 frequencies. So the event statistic is the smallest for (µ ν ) 33 and the largest for (µ ν ) 11 . Secondly, the SM background is also larger for lower frequency which makes it harder to probe (µ ν ) 33 in comparison with other parameters. All electric dipole moments have relatively worse sensitivity than their magnetic counterparts. This is because the electric dipole contribution is suppressed by the sign flip of m j . For larger mass, the suppression is larger. The extreme case happens for the heaviest neutrino with mass m 3 where the sensitivity of ( ν ) 33 is around 2.75 times worse than (µ ν ) 33 . The expected 90% C.L. RENP sensitivity on the neutrino magnetic moment (µν )ij (blue lines) and electric one ( ν )ij (red lines) as a function of the expected RENP event number. For comparison, the green curves show the Borexino sensitivities [16] while the dotted lines are the bounds from stellar cooling of White Dwarf (black) [20] and Red Giant (red) [19]. The gray region is the 90% C. L. neutrino magnetic moment explanation for the Xenon1T excess [24].
For comparison, we also show the sensitivity of the Borexino experiment [16] (green). The result is translated into neutrino magnetic moments in the mass basis by the collaboration using the constraint µ eff α < 2.8 × 10 −11 µ B , taking ( ν ) ij = 0 and only one non-zero (µ ν ) ij at a time. Although (µ ν ) 33 still has the worst sensitivity, the best one occurs for (µ ν ) 12 < 2.7×10 −11 µ B instead of (µ ν ) 11 . The RENP experiment can exceed this limit for all components of µ ν with 1300 events. The gray band shows the neutrino magnetic moment explanation to the Xenon1T anomaly, µ eff ν ∈ (0.9 ∼ 3.5) × 10 −11 µ B [24]. Our proposed RENP setup can probe this region with 500 events.
For comparison, the astrophysical constraints have even smaller numbers at 90% C.L. with µ ν < 2.9 × 10 −12 µ B (yellow-dotted) for white dwarfs [22] and µ ν < 2.2 × 10 −12 µ B (red-dotted) for red giants [19]. Although we show these two sensitivities in Fig. 3 for comparison, one needs to keep in mind that there are various uncertainties for astrophysical measurements.
Conclusions -With O(eV) momentum transfer, the RENP process is sensitive to light mediator including the massless photon. This feature provides a sensitive probe of the neutrino electromagnetic moments. The sensitivity can reach (1.5 ∼ 3.5)×10 −11 µ B for the magnetic moment and (2 ∼ 9) × 10 −11 µ B for the electric one with 500 events. Further reduction by a factor of 2 is possible with 5000 events. The six components of µ ν or ν in the mass basis appear in different frequency regions which allows frequency scan to identify each component step-wisely. Once measured, the different dependence on the trigger laser frequency allows separation of the magnetic and electric moments. All these features make the RENP a unique probe of the neutrino eletromagnetic moments and the fundamental new physics behind them.