Shining Light on the Mass Scale and Nature of Neutrinos with $e\gamma \to e\nu\overline{\nu}$

The discovery of neutrino oscillations invites many fundamental physics questions that have yet to be answered. Two of these questions are simple, easy to state, and essential: What are the values of the neutrino masses? Are neutrinos Majorana fermions? The reason we don't know the answer to those questions is that it is difficult to measure neutrino properties outside of the ultrarelativistic regime. We discuss the physics of $e\gamma\to e\nu\bar{\nu}$ near threshold, where one has access to nonrelativistic neutrinos and only nonrelativistic neutrinos. Near threshold, $e\gamma\to e\nu\bar{\nu}$ is a rich phenomenon and its cross section is sensitive to the individual values of the neutrino masses and the nature of the neutrinos. We show that if one could scan the threshold region, it would be simple to identify the mass of the lightest neutrino, the neutrino mass ordering, and whether the neutrinos are Majorana fermions. In practice, however, event rates are tiny and backgrounds are huge; the observation of $e\gamma\to e\nu\bar{\nu}$ in the sub-eV regime appears to be utterly inaccessible in the laboratory. Our results, nonetheless, effectively illustrate the discriminatory power of nonrelativistic neutrino observables.

The discovery of neutrino oscillations invites many fundamental physics questions that have yet to be answered. Two of these questions are simple, easy to state, and essential: What are the values of the neutrino masses? Are neutrinos Majorana fermions? The reason we don't know the answer to those questions is that it is difficult to measure neutrino properties outside of the ultrarelativistic regime. We discuss the physics of eγ → eνν near threshold, where one has access to nonrelativistic neutrinos and only nonrelativistic neutrinos. Near threshold, eγ → eνν is a rich phenomenon and its cross section is sensitive to the individual values of the neutrino masses and the nature of the neutrinos. We show that if one could scan the threshold region, it would be simple to identify the mass of the lightest neutrino, the neutrino mass ordering, and whether the neutrinos are Majorana fermions. In practice, however, event rates are tiny and backgrounds are huge; the observation of eγ → eνν in the sub-eV regime appears to be utterly inaccessible in the laboratory. Our results, nonetheless, effectively illustrate the discriminatory power of nonrelativistic neutrino observables.

I. INTRODUCTION
The discovery of neutrino oscillations reveals that neutrinos do not behave as prescribed by the Standard Model (SM). The unambiguous feature of neutrino physics that has been unearthed from these oscillations is that at least two of the three known neutrinos have nonzero masses. Furthermore, these masses are orders of magnitude less than those of all other known fermions. In the last two decades, there has been great theoretical interest in trying to understand the mechanism by which neutrinos acquire these tiny masses. While numerous candidates have been proposed, none has emerged as the single most compelling explanation. Part of the difficulty is that, even two decades into this enterprise, there is still insufficient information on the neutrino masses themselves and the nature of the neutrinos is unknown.
The magnitudes of the differences of the squares of the neutrino masses -∆m 2 21 and ∆m 2 31 -have been measured at the 3% level [1] by oscillation experiments, but the sign of the latter -the neutrino mass hierarchy -remains undetermined. * There are physical observations sensitive to the overall scale of the neutrino masses, for instance, the shape of the endpoint of the spectrum for electrons produced in tritium beta decay depends on a particular combination of neutrino masses [2][3][4][5], and nonzero neutrino masses contribute to the observed anisotropies of the cosmic microwave background [6]. While the upper bounds derived from these phenomena are nontrivial and point to neutrino masses below the eV scale, a precise determination of the values of the neutrino masses is still lacking.
It is also unknown whether or not the neutrino and its antiparticle are distinct physical objects, i.e., whether neutrinos are Dirac or Majorana fermions. The answer to this question has profound impact on the mechanism responsible for nonzero neutrino masses. If neutrinos are Majorana fermions, there are additional avenues for pursuing the absolute neutrino masses. In particular, in a large class of models, the rates of lepton-number-violating processes will be governed by elements of the Majorana neutrino mass matrix. The most famous such process is neutrinoless double beta decay (0νββ), (A, Z) → (A, Z +2)+2e − [7,8]. Other possible lepton-number-violating processes include muon-to-positron conversion in nuclei [9][10][11] as well as rare lepton [9,12,13] and meson [9,13] decays, but the bounds on these are significantly weaker than those derived from 0νββ. Upper bounds on neutrino masses derived from 0νββ are competitive with the bounds from β-decay and cosmology but are moot if neutrinos are Dirac fermions.
In this paper, we discuss another process in which neutrino masses can be important: e − γ → e − νν, which we refer to as stimulated νν emission † . Specifically, we consider this process when a sub-eV photon impinges on an electron at rest; we probe the threshold region, in which the final state neutrinos can be nonrelativistic. This is a SM process that, as far as we can tell, has not been previously discussed in the literature, and with good reason: its cross section is orders of magnitude below what could have been detected in any laboratory experiment to date. However, motivated by the landmark detection of coherent elastic neutrino-nucleus scattering at the COHERENT experiment [14] more than four decades after its prediction [15], we ask what we would learn about neutrino properties if stimulated νν emission were measured in a terrestrial setting and discuss the challenges associated with performing such a measurement. Additionally, we investigate the extent to which physics beyond the Standard Model can affect this process; since stimulated νν emission occurs with weak-interaction strength, it may be possible for stronger-thanweak physics to manifest itself in a nontrivial way.
The rest of this paper is organized as follows. In Sec. II, we study stimulated νν emission in the Standard Model, for both Dirac and Majorana neutrinos, as well as either a normal or inverted hierarchy. In Sec. III, we consider event rates and the size of SM backgrounds to any potential search for this process. In Sec. IV, we assess the impact of several new-physics scenarios on this process, focusing on (a) the introduction of a fourth, sterile neutrino, (b) a large magnetic or electric dipole moment induced by heavy new physics, and (c) the effects of a light (∼MeV-scale) dark photon of gauged B − L. In Sec. V, we offer some concluding thoughts.

II. STIMULATED νν EMISSION
In the Standard Model, the process e − γ → e − ν i ν j exists. If the electron is at rest, then the threshold energy of the incoming photon is where m i is the mass of ν i , m j is the mass of ν j , and m e is the mass of the electron. Regardless of the nature of the neutrinos (Majorana or Dirac fermions) or the ordering of their masses, this threshold behavior, at which only the lightest mass eigenstate can be produced, is a clear signal of the overall neutrino mass scale. As the photon energy increases, additional combinations of mass eigenstates become kinematically accessible, until enough energy exists to produce all states. When the photon energy is well above this final threshold, the distinction between mass-eigenstate and flavor-eigenstate final states becomes negligible. Since the energies we will consider are far below the scale of electroweak physics, amplitudes will be calculated using the effective four-point νν interaction obtained by integrating out the W and Z bosons; see, for instance, Ref. [16] for details. The Lagrangian can be written 2) † While we will be interested in this process for both Dirac-and Majorana-type final state neutrinos, we will use Dirac-type language and refer to the process as eγ → eνν.
where L CC and L NC are the relevant charged-and neutral-current pieces of the Lagrangian, respectively, G F is the Fermi constant, s W is the sine of the Weinberg angle and α, β = e, µ, τ are flavor indices. This Lagrangian can be rewritten in terms of the neutrino mass eigenstates by identifying where U is the unitary leptonic mixing matrix and i = 1, 2, 3 is a mass eigenstate index. This identity and a Fierz transformation allow us to recast the interactions as where we introduce the vector and axial couplings Since the only charged leptons considered in this work are electrons, we will make the simplification g ij V,A ≡ g eeij V,A . The following diagrams are relevant to the evaluation of the amplitude: In Appendix A, we give analytic forms for the squared matrix element for our process of interest, assuming the neutrinos are either Dirac or Majorana fermions, in terms of the four-momenta of the incoming and outgoing states. In the case i = j, off-diagonal final states, the vector and axial couplings are equal. The |M| 2 simplify significantly, then, all being proportional to 2|g ij V,A | 2 = 2|U ei | 2 |U ej | 2 . Using experimental information on the magnitudes of the elements of the leptonic mixing matrix [1], the largest contribution to |M| 2 comes from the channel {i, j} = {1, 2} (or {2, 1}). Ignoring effects due to available phase space, the {i, j} = {1, 3} contribution is ∼ 8% of that from {i, j} = {1, 2}, and the {i, j} = {2, 3} channel is roughly 3 − 4% of that from {i, j} = {1, 2}.
For diagonal final states, i = j, determining which channel dominates is less straightforward. There is no interference between the vector and axial-vector contributions for same-mass final-state neutrinos. Additionally, for E γ m e , axial contributions dominate over vector ones. The i = j final state that dominates for energies of interest will be the one with the largest |g ii A | -From Eq. II.5, we have |g 11 A | = 0.18, |g 22 A | = 0.20, and |g 33 A | = 0.48, so we expect i, j = 3, 3 to dominate. For completeness, the vector couplings are |g 11 V | = 0.62, |g 22 V | = 0.25, and |g 33 V | = 0.03. The cross section for stimulated νν emission is insensitive to the value of the CP -violating phase δ CP . When i = j, both the charged-current and neutral-current terms in g ii V and g ii A are real and independent of δ CP . When i = j, the neutral-current term vanishes and g ij V and g ij A are identical making their overall phase irrelevant. The analogous process involving muons or taus would, however, depend on δ CP , as changing δ CP for fixed values of the mixing angles changes the magnitudes of U µ1 , U µ2 , U τ 1 and U τ 2 . This is a consequence of the parameterization of the leptonic mixing matrix: fixing the mixing angles is sufficient to fix the magnitudes of the elements of its first row, whereas the other rows require additional input in the form of δ CP . Moreover, none of the U ei depend on θ 23 in this parameterization; because one cannot determine if θ 23 is nonzero from these elements, and because δ CP is unphysical if any of the mixing angles vanish, the scattering cross section cannot depend on the value of δ CP . We emphasize that this dependence  on the value of the CP -violating phase does not indicate that CP symmetry is violated. Changing its value changes the magnitudes of g ij V and g ij A , but it does not introduce a relative phase between them that could yield different cross sections for scattering involving positrons instead of electrons. ‡

A. Normal vs. Inverted Mass Hierarchies
The arguments made above regarding which final states dominate the overall cross section well above threshold are independent of the values of the neutrino masses m 1,2,3 , and are a consequence purely of the leptonic mixing matrix. Because the elements of the mixing matrix are well known, we use their central values in our calculations, and explore how other properties of the neutrino mass spectrum influence this process.
The values of the neutrino mass splittings ∆m 2 ij ≡ m 2 i − m 2 j are also well constrained. What remains to be known, however, is the overall mass scale of the neutrinos. This is typically parameterized in terms of the lightest neutrino mass, which we designate m 0 . We do not label it specifically m 1 , given the possibility that the third mass eigenstate is the lightest. This latter scenario is known as the inverted mass hierarchy.
In the two mass hierarchies, normal (NH) and inverted (IH), for the same lightest neutrino mass m 0 , different combinations of final state neutrinos become kinematically accessible for different photon energies. For instance, in the IH, the final state {i, j} = {3, 3}, which is the dominant diagonal contribution, is the first to become accessible, while it requires a significantly higher energy in the NH. The same is true for the final state {i, j} = {1, 2} becoming available at lower energy for the NH than the IH. ‡ At higher order in the weak interactions, the cross sections for stimulated neutrino emission from positrons and electrons could be different and hence violate CP invariance. We do not explore this issue here. Extending to different values of m 0 , the ratio between the NH and IH cross sections as a function of E γ and m 0 is depicted in Fig. 3. Again, the region of interest for the largest deviations between these two cross sections is between the energies at which the {i, j} = {1, 2} final state becomes accessible for the two hierarchies.
In summary, if one were to measure the threshold effect of eγ → eνν, then the lightest neutrino mass could be determined. A more detailed measurement of the excitation curve would reveal the mass hierarchy, because there are some qualitative features that distinguish between the two hierarchies near threshold. In

B. Dirac vs. Majorana Fermions
The cross section for stimulated neutrino emission is different when one considers the final state neutrinos to be Dirac or Majorana fermions. This difference is proportional to ∼ (m i m j /s) cos (2∆α ij ), where ∆α ij ≡ α j − α i is the difference between the Majorana phases associated to the mass eigenstates ν i and ν j and s 2m e E γ is the center-of-mass energy squared. As expected, the distinction between Majorana and Dirac neutrinos goes to zero as the photon energy increases. Therefore, there is an opportunity to discriminate between the two hypotheses near threshold. For i = j, because Dirac neutrinos have more available chirality states, the cross section is larger for Dirac neutrinos than for Majorana neutrinos. For i = j, the dominant cross section depends on ∆α ij . We have computed the ratio between cross sections for Dirac and Majorana neutrinos close to threshold. For more details, see Appendix A 2. The contribution of kinematic effects (e.g., thresholds) does not depend on the nature of the neutrino, § so all differences are properly captured by the squared matrix elements. If i = j, then ∆α ij = 0, and (II.7) The value of this ratio depends strongly on the magnitude of U ei . We estimate the matrix-element ratios § Several factors of 1/2 and 2 arise when comparing the Dirac and Majorana final state cross sections. These conspire in such a away that the total cross section well above threshold is the same for Majorana and Dirac neutrinos. For i = j, the Majorana final state cross section consists of two diagrams, however a factor of 1/2 arises in the phase space integration because the final state particles are identical. For i = j, we compare the νiνj Majorana final state to the νiνj + νjνi Dirac final states, both of which have two distinct diagrams. (II.8) Since the neutrino masses are small relative to that of the electron, and given that these ratios do not depend on the elements of the mixing matrix, they vary between 2/3 and 2, depending on ∆α ij .

III. RATES AND BACKGROUNDS
In light of the cross section calculations of the previous section, we estimate the rate for stimulated νν emission in an unrealistically optimistic experimental setup. We imagine a configuration in which a 1 eV (1240 nm, near-infrared) laser with a power of ∼ 2 W [17] is directed at a target of electrons at rest with density n e ∼ 10 23 cm −3 and length of 1 m. With the results of Sec. II, such an experiment could expect one signal event every ∼ 10 20 years. This cross section grows as E 4 γ for E γ < m e , so the event rate improves rapidly for energies significantly above threshold. However, moving high enough above threshold to achieve an appreciable rate limits the usefulness of making this measurement, given our interest in probing the nonrelativistic regime.
Because of the extremely low energies involved, the photon-electron collisions under study benefit from the absence of hadronic activity in the final state. Moreover, since higher-order weak corrections will be strongly suppressed relative to the leading-order contribution, estimating the backgrounds is an exercise in pure QED. The dominant background for stimulated νν production is Thomson scattering, e − γ → e − γ, the total cross section for which, in the limit E γ m e with unpolarized initial particles, is well known to be [18] Because this is a 2 → 2 process, one could exploit the correlations between the energies of the final state electron and photon and their directions to identify these background events and separate them from the signal. However, compare this to the signal cross section calculated in the previous section for photon energies E γ = 1 eV, σ ∼ O 10 −47 b. This casts into sharp relief the infeasibility of detecting stimulated νν emission in a terrestrial experiment: for instance, one would need to ensure that no more than one in O(10 47 − 10 48 ) Thomson photons escapes detection in order to effectively identify this background, an objective that is, for all intents and purposes, impossible.
To compound the matter, it is insufficient to consider single Thomson scattering as a background; one must also consider multiple Thomson scattering. * Multiple Thomson scattering has been previously discussed in, for instance, Refs. [19][20][21]. One would naively expect that the cross section for n-photon emission would scale as σ n ∼ α n−1 σ Thomson ; while this is true in the high-energy regime (E γ m e ), up to logarithmic corrections [22,23], in the nonrelativistic regime the cross section has the form [21,24] where C n is a numerical coefficient of ∼ O(0.1−10). The factor (E γ /m e ) 2n is a consequence of the necessity for a cutoff in photon energy. The rate of emission of multiple photons is formally divergent unless an infrared (IR) regulator is introduced; in an experiment, the cross section is regulated by the finite photon energy detection threshold. Introducing this IR regulator means that one is not integrating over all of phase space in calculating the cross section, and an additional suppression appears relative to the naive expectation.
On one hand, this extra suppression means that the backgrounds are not as daunting a challenge as one might originally suspect; one does not have to contend with O(20)-photon backgrounds in order to study stimulated νν emission near threshold. On the other, we find that, for E γ ∼ O 10 −2 − 1 eV, the cross section for n = 4, 5 (depending on the choice of IR cutoff) still utterly swamps the signal cross section. Moreover, while one can, in principle, calculate angular and energy correlations between the final state particles to attempt to characterize and reduce these backgrounds, (1) the computational power required for a detailed study quickly becomes prohibitive (though not impossible), and (2) practical constraints exist on how well these backgrounds can be measured.
The bottom line is that a terrestrial experiment will almost certainly never be able to measure stimulated νν emission at threshold. While we still regard this process as interesting, we move forward with the mentality that this process is one of purely theoretical relevance.

IV. NEW PHYSICS CONTRIBUTIONS
We consider how the presence of new physics beyond neutrino masses and lepton mixing would modify the results discussed in Sec. II. We focus on three new-physics scenarios: the existence of a sterile neutrino, nonstandard neutrino electromagnetic properties, and the coupling of the SM neutrino to a dark photon.

A. Sterile Neutrinos
Here, we consider the existence of a sterile neutrino species associated to a fourth neutrino mass eigenstate with mass around or below the eV scale. This scenario is qualitatively similar to the three-neutrino case outlined in the previous section, but mixing with the fourth neutrino induces important changes in the νν interaction in Eq. (II.4). Here, we simply quote the new values of the vector and axial coupling constants (Eq. (II.5)); for details, see Appendix B: where U si is the ith element of the fourth row of the 4 × 4 extension of the leptonic mixing matrix. The inclusion of a sterile neutrino has two effects on these couplings. First, changes to the U αi change * We are referring to a single event in which an incident photon is split into multiple final state photons, not to a single photon Thomson scattering multiple times.  the charged-and neutral-current contributions to g ij V,A in a way that may either increase or decrease the magnitudes of the couplings. Second, the neutral-current contribution is no longer strictly diagonal in the mass basis, stemming from the absence of a coupling between the Z boson and the sterile neutrino; see Appendix B for more details.
We present three scenarios for the existence of sterile neutrinos with oscillation parameters given in Table I. The first two cases are consistent with current bounds on the existence of a fourth neutrino presented in Ref. [25] for either a light (∆m 2 41 = 1.0 × 10 −5 eV 2 ) or heavy (∆m 2 41 = 1.3 eV 2 ) fourth neutrino; the latter of these is consistent with the global best-fit point from ν e /ν e disappearance data in Ref. [25]. The third case is already excluded at high confidence by existing oscillation data, but we use it to demonstrate important features of the cross section and to check the consistency of our calculations. In the rest of this work, we assume that the neutrino masses follow a normal hierarchy (i.e., ∆m 2 31 > 0) and that the sign of ∆m 2 41 is positive. The cross section for Case 1 is depicted in Fig. 6 (left); we show the breakdown of the total cross section into components for only this case, in the interest of conserving space. In Fig. 6 (right), we show how the total cross sections for all three cases in Table I compare to the three-neutrino cross section. In the high energy (E γ > 10 eV) limit, Cases 1 and 3 are ∼ 10% below the νSM (SM plus nonzero neutrino masses of the Dirac or Majorana-type) -the relatively large mixing with the sterile neutrino results in a reduction of the charged-current contribution to the cross section -while Case 2 is essentially consistent with the three-neutrino cross section. For lower energies, however, the differences can be quite sizable; the maximum excursion is ∼ 20% for Cases 1 and 3, where the active-sterile mixing angles are relatively large, while this value is ∼ 5% for Case 2. The convergence of Cases 1 and 3 at high energy indicates that our calculation is consistent: for E γ 10 eV, the neutrino masses are irrelevant, so the overall cross section cannot distinguish between scenarios with different neutrino masses but the same mixing matrix. While these ∼ O(10%) effects are not insignificant, we remind the reader that these effects are subdominant to other effects considered related to the physics of neutrinos -the nature of the neutrinos, the mass hierarchy, etc. -and could only be meaningfully observed after these other characteristics have been pinned down elsewhere.
Extending the mixing matrix introduces new sources of CP violation, characterized by two new CPviolating phases. The placement of these new phases depends on the parameterization; we follow the conventions employed in Ref. [26], where the three new phases, {η 1 , η 2 , η 3 }, are the (negative of the) phases of U e3 , U e4 and U µ4 , respectively. The presence of the U si U * sj contribution to the neural-current term in g ij V,A spoils the independence of the stimulated νν emission cross section on these CP -violating phases that we encountered in Sec. II. For i = j, these phases change the neutral-current contribution; for i = j, g ij V and g ij A are no longer equal and so may have a physically relevant relative phase. To study the impact of these CP -violating phases on the cross sections, we compare the cross sections for Case 1 and Case 2 with all CP -violating phases set to zero against the same with each phase, in turn, set to ±π/2. For Case 1, in which some of the mixing angles are sizable, varying the CP -violating phases  FIG. 6: Left: Cross sections for individual final states (colored) and total cross section for the process e + γ → e + ν i + ν j , assuming a fourth neutrino with parameters given for Case 1 in Table I. Here, we assume neutrinos are Dirac particles with a normal mass hierarchy. Right: Ratio between cross sections assuming four neutrinos exist to the cross section assuming three neutrinos exist given parameters for Case 1 (blue), Case 2 (magenta), or Case 3 (green) from Table I. can have a significant impact on the cross section. The largest deviation from the CP -conserving scenario is ∼ 12% but is typically around ∼ 2 − 3% in the threshold region. This effect is more modest for Case 2; due to the smallness of the mixing angles, the changes to the cross section are only as large as ∼ 2% and are mostly around the sub-percent level. Additionally, we have investigated the differences between η i = +π/2 and η i = −π/2 for each of the η i and find that the total cross section depends only on the magnitude of the phase. Despite the dependence of the cross section on these CP -violating phases, there is no CP violation present in stimulated νν emission (at least at tree level).

B. Neutrino Magnetic and Electric Dipole Moments
If new physics exists but is heavier than available energies, then its associated degrees of freedom cannot be directly produced in experiments. However, it will still indirectly affect physical processes via modifications of leading-order couplings, or by introducing couplings that would not otherwise exist. This is typically described using the machinery of effective field theory [27,28]. In fact, this is precisely how we formulated our discussion in Sec. II; the energies we consider are far below the electroweak gauge boson masses, but the effects of these are included via the Fermi operator. In this subsection, we will concern ourselves, in a model-independent way, with how heavy new physics may induce nonstandard electromagnetic properties for neutrinos. We can only hope to provide a cursory overview the subject; for in-depth discussions, see Refs. [29][30][31].
The most general photon-neutrino-neutrino ν i (k i ) → ν j (k j ) + γ(q) vertex can be written as The functions f ij X (q 2 ), X = Q, M, E, A are, respectively, the charge, magnetic dipole, electric dipole and anapole form factors which, in the limit q 2 → 0, become the neutrino charge q ij ν , * magnetic dipole moment (MDM) µ ij ν , electric dipole moment (EDM) ε ij ν , and anapole moment a ij ν . If neutrinos are Dirac fermions, then the f ij X are hermitian matrices in flavor space; if they are Majorana fermions, then the f ij X are antisymmetric. We assume that, since the momenta are small relative to most energy scales in the process of interest, any dependence of the form factors on q 2 is negligible, allowing us to replace them with their respective moments. The neutrino charge and anapole moments are severely constrained by existing data. A collection of bounds on q ν are tabulated in Table IV of Ref. [29], but the authors of that reference derive q ν = (−0.6 ± 3.2) × 10 −21 e from the (non)neutrality of matter in Ref. [32]. We find that neutrino charges of this order of magnitude make a negligible contribution to the stimulated νν emission cross section. Bounds on the neutrino anapole moments are derived from bounds on the neutrino charge radius, r 2 ν ; for exactly neutral neutrinos, we have a ν = − r 2 ν /6. Current limits on the neutrino charge radius are shown in Table V of Ref. [29]; the strongest limits give a ν O 10 −32 cm 2 , an order of magnitude above their SM predictions [33,34]. We have verified that anapole moments of this size are thoroughly subdominant to the SM contribution to the cross section of interest. We will therefore disregard neutrino charge and anapole moments and instead focus on MDMs and EDMs.
Some experimental limits on neutrino MDMs are tabulated in Table III of Ref. [29]; more recent limits can be found in, for instance, Refs. [35,36]. The strongest bounds from terrestrial experiments are µ ν O(10 −11 )µ B , where µ B (= 2.96 × 10 −7 eV −1 ) is the Bohr magneton [37]; astrophysical limits can be as strong as µ ν O(10 −12 )µ B , but we will take µ ν = 10 −11 µ B as our benchmark in this analysis. Moreover, every neutrino ever detected has been ultrarelativistic; in this regime, the vertex in Eq. (IV.2) can be taken to be σ µν q ν (µ ν − iε ν ), so experiments have only been able to constrain the quantity |µ ν − iε ν |. As such, current bounds on |µ ij ν | also apply to |ε ij ν |. Refs. [38][39][40] have calculated the neutrino MDMs and EDMs by minimally extending the SM with right-handed neutrinos -making neutrinos Dirac particles -and find where the + applies for µ ij ν and the − applies for ε ij ν . Numerically, this expression becomes for the diagonal elements, whereas for the off-diagonal elements, we find (i = j) (IV.6) * We rely on context to distinguish between the neutrino electric charge and νth component of the four-vector q. Here, we also display the total cross section discussed in Section II, increased by a factor of 10 12 for comparison. Here we assume that neutrinos are Dirac fermions and the mass hierarchy is normal.
Experimental bounds on µ ij ν and ε ij ν are many orders of magnitude above the predicted values from these scenarios, leaving plenty of opportunity for this new physics to be discovered in, for instance, stimulated νν emission.
Generic theoretical arguments for the expected sizes of Majorana neutrino MDMs and EDMs have also been employed in the literature [40] (see also the references in Sec. IV B of Ref. [29]), but are more model dependent and we will not discuss them here explicitly. We will remark, however, that MDMs and EDMs can be large in several classes of models -potentially many orders of magnitude larger than the predictions discussed above -and constitute precisely the kind of new physics that motivates these searches. Figure 7 depicts eγ → eν i ν j cross sections for either a nonzero Dirac MDM (blue lines) or a nonzero Dirac EDM (red lines) for the ν 1 ν 1 final state (solid lines) and the ν 1 ν 2 + ν 2 ν 1 final state (dashed lines). In both cases, we have chosen µ ij ν = ij ν = 10 −11 µ B for all i, j and have only kept diagrams that are first order in the neutrino MDM/EDM, to wit, The SM contribution discussed is also shown in black, multiplied by 10 12 for comparison. There are a couple of factors that govern the relative sizes of weak and MDM/EDM cross sections. The first is that the MDM/EDM considered here is significantly larger than the predictions of Eq. (IV.3). One would expect that a MDM/EDM of SM strength should appear as a one-loop weak correction to the tree-level νSM cross section and is hence completely negligible even close to νν threshold, as one can infer from Fig. 7. The second factor is that since neutrinos are light, the cross section benefits from a nearly-soft, nearly-collinear enhancement from the virtual photon propagator; this overcomes the energy dependence of the vertex, resulting in a significant enhancement of the differential cross section. There is also an obvious difference in scaling of the cross section with energy -σ ∼ E 2 γ here, compared with σ ∼ E 4 γ , as before -meaning the magnetic moment contribution would dominate the cross section for energies below At high energies, the MDM-and EDM-induced cross sections in Fig. 7 converge. In the ultrarelativistic limit, the different-chirality final states decouple and the cross section only depends on |µ ij ν ±iε ij ν |, making it impossible to disentangle the effects of a nonzero MDM and a nonzero EDM. However, a difference between nonzero MDMs and nonzero EDMs emerges at low energies: the γ 5 that accompanies ε ij ν (see Eq. (IV.2)) causes the different-chirality final states to destructively interfere with one another, whereas the implicit 1 that accompanies µ ij ν allows these contributions to interfere constructively. If one could detect this process when the final state neutrinos are not ultrarelativistic, then one should be able to disentangle the effects of a nonzero MDM and nonzero EDM by measuring the shape of the cross section in this regime.
Dirac and Majorana dipole moments are qualitatively similar but with two important distinctions. (1) The Majorana dipole moment matrices must be antisymmetric, i.e., only transition moments exist, whereas the Dirac moment matrices are only constrained to be Hermitian. The absence of diagonal final states would be a strong (though not necessarily ironclad) indication of the Majorana nature of neutrinos. (2) For Majorana magnetic dipole moments, we must make the replacement |µ ij ν | 2 → 4|µ ij ν | 2 , and similarly for electric dipole moments, in the calculations above. As with Majorana masses, this change in dependency arises from interference between the ν i ν j and ν j ν i final states, but unlike the former scenario, this interference does not vanish at high energies.
In this subsection, we discuss the consequences of one particular model of new physics: the gauging of baryon-number-minus-lepton-number symmetry, U (1) B−L . While this particular model faces strong constraints, we use it to represent how similar (though more complicated) models may result in cross sections that can rival the SM prediction in magnitude.
We introduce a new gauge boson X, which we refer to as the dark photon, that couples to the SM fermions proportionally to their B − L charge. The relevant part of the new-physics Lagrangian is where g X is the new gauge coupling and M X is the dark photon mass. We have only shown the couplings of X to leptons, though it must also couple to quarks, and we ignore potential kinetic mixing between the X and the SM photon. In order for B − L to be free of anomalies, one must introduce a right-handed neutrino. Consequently, we assume neutrinos to be Dirac particles. Again ignoring contributions from the weak interactions, the amplitude for γ + e − → e − + ν + ν can be calculated from the following diagrams: As is the case of the SM neutral-current interactions, the final state neutrino and antineutrino must be in identical mass eigenstates. Limits on the U (1) B−L dark photon mass and coupling can be found in, for instance, Refs. [41,42]. Null searches for a fifth, long-range force constrain the dark photon mass to be O(100 eV); for sub-eV incident photon energies, the dark photon will always be off-shell, and we can treat the new interaction in the framework of effective field theory, similar to what we did previously for the weak interactions. We incorporate the effects of this dark photon at low energies by modifying the vector coupling g ij V : where G X is the Fermi-like coupling of gauged U (1) B−L . There is a sliver of allowed parameter space in Refs. [41,42] that allows for a U (1) B−L dark photon with g X = 10 −6 and M X = 1 MeV, corresponding to G X /G F 0.1. As such, a dark photon in this region of parameter space would only contribute subdominantly to the weak interactions.
There are available regions of parameter space, particularly at higher M X , where the coupling can be quite large: for instance, the point g X = 10 −4 , M X = 1 GeV is allowed by current experiments. However, the increase in the coupling constant in the numerator of G X is never enough to outstrip the growth in its denominator, so this high-mass region cannot yield larger cross sections than those of the previous paragraph. For lower values of M X , however, there exists a region of parameter space -g X ∼ O(10 −7 ) and M X ∼ O(1 keV) -where such a dark photon may be safe from solar cooling constraints; see Refs. [41,42] for details. In this region of parameter space, not only do we have G X /G F ∼ 10 3 , resulting in new physics dominating over the SM, but if the energy of the incident photon were high enough (E γ M X ), then the effective-operator approach would break down and there would be an additional enhancement to the cross section from the production of on-shell X. While it is intriguing that this process may be sensitive to such a strong, new force, † our interest in this work is around the threshold region, so we do not consider this scenario further. We conclude this section by noting that the contribution of the dark photon of U (1) B−L to the neutrino MDM and EDM is highly suppressed, and does not lead to enhanced contributions of the type discussed in the previous subsection.

V. DISCUSSION AND CONCLUSIONS
The discovery of neutrino oscillations reveals that neutrinos have nonzero masses and leptons mix. It also invites many fundamental physics questions that have the potential to qualitatively change our understanding of particle physics. Two of these questions are simple, easy to state, and essential: what are, even roughly, the values of the neutrino masses -we only have information on the neutrino masssquared differences -and are neutrinos Majorana fermions? The reason we don't know the answers to those questions is that neutrino masses are very small compared to the typical laboratory neutrino energies in experiments.
We discussed a process that involves nonrelativistic neutrinos: eγ → eνν. When the neutrinos are nonrelativistic, eγ → eνν is a rich phenomenon and the cross section is sensitive to the individual values of the neutrino masses and the Dirac or Majorana nature of the neutrinos. If one could scan the threshold region, then it would be simple to identify the mass of the lightest neutrino, the neutrino mass ordering, and whether the neutrinos are Dirac or Majorana fermions. As we remarked on several occasions -see, for example, Figs. 2  There are a few other processes that provide access to nonrelativistic neutrinos. The 2 → 4 scattering process e − e ± → − ± νν ( , = e, µ, τ ) could also be studied at low energies and the outgoing charged leptons would contain enough information to measure the properties of the neutrinos, similar to what we discussed here. Nuclear β-decay also includes nonrelativistic neutrinos close to the end point of the electron energy spectrum. In principle, details around the endpoint of the spectrum contain information about the individual neutrino masses (see Ref. [43] and references therein). Nuclear β-decay energy spectra, however, cannot be used to distinguish Majorana from Dirac neutrinos, as there is only one neutrino in the final state. At threshold, eγ → eνν is different in the sense that all neutrinos involved are nonrelativistic and the neutrinos are pair-produced; it is sensitive to both the neutral and charged currents and there is a manifest distinction between Majorana and Dirac fermions. The cosmic neutrino background also serves as an intense, rich source of cold neutrinos that are mostly nonrelativistic. While they are yet to be observed, future precision measurements could access information about the neutrino masses and the nature of the neutrino [44].
Low-energy eγ → eνν and the processes summarized above necessarily suffer from tiny cross sections because neutrino masses are very small and the weak interactions are aptly named at low energies. In Sec. III, we discussed event rates and backgrounds to eγ → eνν. In particular, multiple-photon backgrounds utterly overwhelm the νν signal, rendering a measurement of the latter virtually impossible in a terrestrial context. Nonetheless, eγ → eνν processes should occur at nonnegligible rates in astrophysical systems. For instance, the flux of keV-scale (and below) neutrinos produced by several thermal production mechanisms in the Sun -including stimulated νν emission -has been calculated in Ref. [45]. That work indicates that the thermal neutrino flux would dominate the flux of neutrinos produced by proton-proton fusion at ∼ keV energies (see Fig. 18 therein). The spectrum of the solar neutrinos at sub-eV energies contains many of the features discussed in this work. A detailed calculation of the role of threshold effects in νν production in astrophysical sources is beyond the scope of this work.
where the vector and axial coupling constants are defined in Eq. (II.5).

Appendix B: Four-Neutrino Formalism
We revisit the four-point νν interaction Eq. (II.4) in the context of nonzero mixing with a sterile neutrino. The Lagrangian of Eq. (II.2) remains unchanged, but we emphasize that the implicit sum over α, β is only over the active flavor eigenstates, i.e., e, µ, τ . Using Eq. (II.3) to replace the flavor eigenstates in the charged-current Lagrangian in favor of the mass eigenstates proceeds similarly to the three-neutrino case, the only difference being that the indices of U αi can take the values α = e, µ, τ, s and i = 1, 2, 3, 4.
In the neutral-current interaction, however, there is an additional subtlety: since the Lagrangian only sums over the active neutrino flavors, the Lagrangian will not end up diagonal after rotating to the neutrino mass basis. To wit, the neutrino neutral-current term becomes α=e, µ, τ (ν α γ µ P L ν α ) = i, j α=e, µ, τ U αi U * αj (ν j γ µ P L ν i ) = i, j δ ij − U si U * sj (ν j γ µ P L ν i ) ; (B.1) the result of this is to modify the vector and axial couplings g ij V,A ≡ g eeij V,A , precisely as stated in Eq. (IV.1).