Neutron star cooling with lepton-flavor-violating axions

The cores of dense stars are a powerful laboratory for studying feebly-coupled particles such as axions. Some of the strongest constraints on axionlike particles and their couplings to ordinary matter derive from considerations of stellar axion emission. In this work we study the radiation of axionlike particles from degenerate neutron star matter via a lepton-flavor-violating (LFV) coupling that leads to muon-electron conversion when an axion is emitted. We calculate the axion emission rate per unit volume (emissivity) and by comparing with the rate of neutrino emission, we infer upper limits on the LFV coupling that are at the level of $|g_{ae\mu}| \lesssim 10^{-6}$. For the hotter environment of a supernova, such as SN 1987A, the axion emission rate is enhanced and the limit is stronger, at the level of $|g_{ae\mu}| \lesssim 10^{-11}$, competitive with laboratory limits. Interestingly, our derivation of the axion emissivity reveals that axion emission via the LFV coupling is suppressed relative to the familiar lepton-flavor-preserving channels by a factor of $T^2 E_{F,e}^2 / (m_\mu^2 - m_e^2)^2 \sim T^2/m_\mu^2$, which is responsible for the relatively weaker limits.

Due to their weak interactions with SM particles, detecting axions in terrestrial experiments is exceptionally challenging.Therefore, it is highly motivated to search for evidence of axions in astrophysical systems where their feeble couplings are partially compensated by high temperatures and densities [20].For instance, probing axion emission from the white dwarf luminosity function [21][22][23][24] places a stringent limit on the axion-electron coupling at the level of g aee ≲ 10 −13 .Additionally, the axion's interaction with nucleons is probed by neutron star (NS) cooling [25][26][27] and supernova neutrino emission [28][29][30][31][32][33], which imply tight upper limits at the level of g aN N ≲ 10 −10 .
As an extension of the SM, there is no strong reason for the ultraviolet theory of axions to respect the lepton flavor symmetry, an accidental one of the SM broken by tiny neutrino masses.The axions whose ultraviolet theory is responsible for the breaking of the flavor symmetry are known as flavons or familons [34][35][36], which can also explain the strong CP problem if they have a coupling to gluons [37,38].Even if the underlying theory preserves lepton flavor, lepton-flavor-violating (LFV) effects can arise from radiative corrections [39][40][41][42].It has been shown that LFV interactions can explain the anomalies related to the muon and electron magnetic moments [43] and account for the production of dark matter through thermal freeze-in [44].Tests of lepton flavor conservation thus provide important information about new physics.
Laboratory tests of lepton-flavor violation serve as an indirect probe of the axion's LFV interactions.Notably, charged lepton flavor violation would lead to rare lepton decays [45].If the axion were heavier than the muon, an effective field theory approach could be used to study decays such as µ → eγ, µ → 3e and µ − e conversion, being the best process to detect LFV in the eµ sector. 1 For lighter axions, µ → ea could be the dominating channel and the current limit on Br(µ → ea) is of order 10 −6 [50] or 10 −5 [51] depending on the axion mass and chirality of the interaction.The limit will be improved in the future experiments MEG II [52,53] and Mu3e [54] by up to two orders of magnitude [55].
In this work, we aim to establish an astrophysical limit on the axion's LFV interactions based on NS cooling arguments, as a complement to current lab limits.The basic idea is illustrated in figure 1; if axions are produced in NS cores, they must not carry energy out of the star more efficiently than standard neutrino-mediated cooling channels [20].In a NS core, unlike nondegenerate stars or even white dwarf stars, the particle densities are so high that the electron Fermi energy exceeds the muon mass, and an appreciable population of muons is present [56].As such, NSs provide a unique opportunity to probe the axion's LFV coupling with muons and electrons.
Axions with LFV couplings.-Weconsider a LFV coupling among the electron, muon, and axion, which is expressed as where Ψ e (x) is the electron field, Ψ µ (x) is the muon field, a(x) is the axion field, m e ≈ 0.511 MeV is the electron mass, m µ ≈ 106 MeV is the muon mass, and g aeµ is the axion's LFV coupling.The coupling may also be written in terms of the axion decay constant f a as g aeµ = C aeµ (m e + m µ )/(2f a ).This interaction can naturally arise, e.g., in the models of the LFV QCD axion, the LFV axiflavon, the leptonic familon and the majoron (see [55] and references therein).Past studies of charged lepton flavor violation, from both terrestrial experiments and cosmological / astrophysical observations, furnish constraints on the axion LFV coupling g aeµ , which we summarize here.
The LFV interaction opens an exotic decay channel for the muon µ → ea, as long as the axion mass is not too large m a < m µ − m e .The branching ratio is predicted to be [57] Br(µ → ea) ≈ Γ(µ → ea) Γ(µ → eνν) = 7.0 × 10 15 g2 aeµ . ( Initial searches for the two-body muon decay were performed by Derenzo using a magnetic spectrometer, resulting in an upper limit on the branching ratio of 2 × 10 −4 for the mass range 98.1-103.5 MeV [58].Jodidio et al. constrained the branching ratio for a massless familon to be < 2.6 × 10 −6 , which was later extended to massive particles up to ∼ 10 MeV [55].Bryman & Clifford analyzed data of muon and tauon decays obtained from NaI(Tl) and magnetic spectrometers, concluding an upper limit of 3 × 10 −4 for masses less than 104 MeV [59].Bilger et al. studied muon decay in the mass range 103-105 MeV using a high purity germanium detector and established a limit of 5.7 × 10 −4 [60], while the PIENU collaboration improved the limit in the mass range 87.0-95.1 MeV [61].The TWIST experiment performed a broader search for masses up to ∼ 80 MeV by accommodating nonzero anisotropies, resulting in an upper limit of 2.1 × 10 −5 for massless axions [51].These constraints on Br(µ → ea) translate into upper limits on the LFV coupling g aeµ , and we summarize the current status in Tab.I. Apart from terrestrial experiments, cosmological and astrophysical observations also constrain the axion's LFV interaction.If this interaction were too strong, relativis-tic axions would be produced thermally in the early universe; however, the presence of a dark radiation in the universe is incompatible with observations of the cosmic microwave background anisotropies.Constraints on dark radiation are typically expressed in terms of a parameter N eff called the effective number of neutrino species.A recent study of flavor-violating axions in the early universe finds that current observational limits on N eff require the LFV coupling to obey |2f a /C aeµ | > 2.5 × 10 8 GeV [62].Astrophysical probes of the axion's LFV interaction have not been extensively explored.Calibbi et al. considered the bound on Br(µ → ea) from SN 1987A associated with the cooling of the proto-NS [55].Assuming that the dominant energy loss channel is free muon decay µ → ea, they derive an upper limit on the branching ratio at the level of 4 × 10 −3 .We find that a stronger constraint is obtained from the 2-to-3 scattering channels, such as µp → epa, and we discuss this result further below.
Axion emission via LFV couplings.-Theemission of axions from NS matter via the LFV interaction can proceed through various channels.One might expect the dominant channel to be the decay of free muons µ → ea; however, since the electrons in NS matter are degenerate, this channel is Pauli blocked, and its rate is suppressed in comparison with scattering channels.Since NS matter consists of degenerate electrons, muons, protons, and neutrons, various scattering channels are available.We denote these collectively as where a lepton l = e, µ is converted to another l ′ = µ, e with the spectator particle f = p, e, µ.We consider channels in which the neutron star's muon is present in the initial state, and channels in which muons are created thanks to the large electron Fermi momentum.The scattering is mediated by the electromagnetic interaction (photon exchange), and channels involving neutrons are neglected.Assuming that all particles are degenerate, scattering predominantly happens for particles at the Fermi surface.These processes are kinematically allowed a The PIENU collaboration obtained upper limits on the branching ratio from 10 −4 to 10 −5 for the considered mass range.
TABLE I.A summary of constraints on the axion's LFV coupling in the e-µ sector, where stronger constraints are presented at the bottom.See the main text for more detailed descriptions.For the NS cooling limit, we calculate the axion emissivity via l + f → l ′ + f + a and compare with the neutrino emissivity via Murca channels.For the SN 1987A limit, we compare with the upper bound on energy loss rate.
implying the existence of a threshold momentum of the spectator particle Here we have introduced the Fermi momentum p F,i of the particle species i.
The quantities of interest are the axion emissivities ε (lf ) a , which corresponds to the energy released in axions per unit volume per unit time through the channel lf → l ′ f a.We assign (E 1 , p 1 ) and (E ′ 1 , p ′ 1 ) for the initial and final four-momenta of the converting leptons l and l ′ , (E 2 , p 2 ) and (E ′ 2 , p ′ 2 ) for the spectator f , and (E ′ 3 , p ′ 3 ) for the axion.Then the axion emissivity is calculated as where S is the symmetry factor accounting for identical initial and final state particles, M (lf ) is the Lorentz invariant matrix element, f i and f ′ i are the Fermi-Dirac distribution functions, the factor (1 − f ′ i ) takes into account the Pauli blocking due to particle degeneracy, and dp ≡ d 3 p/[(2π) 3 2E] is the Lorentz-invariant differential phase space element.We do not include a factor of (1 + f ′ 3 ), since f ′ 3 ≪ 1 and there is no Bose enhancement of axion production since NSs are essentially transparent to axions for the currently allowed parameter space.
Calculating the emissivity (5) requires evaluating the 15 momentum integrals along with the 4 constraints from energy and momentum conservation.We evaluate all but 2 of these integrals analytically using the Fermi surface approximation, and we calculate the last 2 integrals using numerical techniques.The Fermi surface approximation assumes that the integrals are dominated by momenta near the Fermi surface |p| ≈ p F ; smaller and larger momenta do not contribute because of Pauli blocking or Boltzmann suppression.See the Supplemental Material for details of the derivation.We find the axion emissivity of the lf → l ′ f a channel to be where α ≈ 1/137 is the electromagnetic fine-structure constant, E F,i is the Fermi energy, β F,i ≡ p F,i /E F,i is the Fermi velocity, T is the plasma temperature, and F (lf ) is a factor depending on both the specific process and the Fermi velocity of the scattering particles.To derive (6), we have assumed that the axion mass is small compared to the NS temperature m a ≪ T , muons and electrons are in the beta equilibrium (i.e., E F,e ≈ E F,µ ), electrons are ultra relativistic but muons are not (i.e., p F,µ ≲ m µ ), and T ≪ m 2 µ /E F,e .The temperature dependence of the axion emissivity ( 6) is especially interesting and important for understanding the limits from neutron star cooling.For comparison, note that axion bremstrahlung via leptonflavor-preserving (LFP) interactions (such as ep → epa or µp → µpa) goes as ε a ∝ T 6 .In other words, the LFV interaction leads to an emissivity that's suppressed by an additional factor of T 2 E 2 F,e /(m 2 µ −m 2 e ) 2 ∼ T 2 /m 2 µ , which is of order (100 keV/100 MeV) 2 ∼ 10 −6 for T ∼ 10 9 K.A detailed discussion appears in the Supplemental Material, but the essential idea can be understood as follows.The phase-space integrals over momenta can be converted to energy integrals, and each integral for degenerate leptons and protons is restricted to the Fermi surface of thickness ∼ T , giving a factor of T 4 .The phase-space integral of axions (i.e., d 3 p ′ 3 /E ′ 3 ) gives a factor of T 2 .The axions are emitted thermally and have for the LFV process l + f → l ′ + f + a, given by equation ( 6), as a function of the muon Fermi velocity βF,µ.The top axis, in a nonlinear scale, represents the corresponding mass density of a NS assuming the npeµ matter.Here we take gaeµ = 10 −11 and T = 10 9 K, and more generally ε an energy ∼ T .The energy conservation delta function gives T −1 .The squared matrix element has a temperature dependence T 2 .Putting all these together, we see that the emissivity is proportional to T 8 .In comparison, the squared matrix element for the LFP interactions has no temperature dependence since one power of T from the coupling vertex is canceled by T −1 from the lepton propagator.
We numerically evaluate the axion emissivities (6) and present these results in figure 2 for the six channels lf → l ′ f a, where the effective mass of protons is taken to be 0.8m p (see [67] and references therein). 3The emissivities are equal for the channels ef → µf a and µf → ef a, so the plot only shows three curves corresponding to different spectator particles f = p, e, µ.The channels with a spectator proton (f = p) have the largest emissivity across the range of muon Fermi momenta shown here; this is a consequence of the enhanced matrix element and the larger available phase space for these scatterings.For the channels with a spectator muon (f = µ), the emissivity drops to zero below β F,µ ≈ 0.34; this corresponds to a violation of the kinematic threshold in (4).For all channels, the emissivity decreases with decreasing muon Fermi velocity due to the reduced kinematically allowed phase space.On the other hand, for larger muon Fermi velocity, the channels with spectator electrons and muons coincide, since both particles can be regarded as massless.For the top axis in figure 1, we show the corresponding mass density of a NS assuming the npeµ model; see the Supplemental Material for more details. 3Thanks to the electric charge neutrality and the beta equilibrium condition E F,e ≈ E F,µ , the emissivity can be fully determined once the effective proton mass and β F,µ are given.
The total axion emissivity is obtained by summing over the six channels.For this estimate we set β F,µ = 0.84.We find the axion emissivity via LFV interactions to be where T 9 ≡ T /(10 9 K) and 10 9 K ≈ 86.2 keV.
Implications for NS cooling.-Inlow-mass NSs, slow cooling could occur via neutrino emission by the modified Urca (Murca) processes nn → npeν, npe → nnν or slightly less efficient processes such as the nucleon bremsstrahlung [68,69].At the density ρ = 6ρ 0 , where ρ 0 = 2.5 × 10 14 g cm −3 is the nuclear saturation density [70], and with the effective nucleon mass taken to be 0.8m N [67], the emissivity of the Murca process is given by ε ν = 4.4 × 10 21 T 8 9 erg cm −3 s −1 [71].Comparing this rate with (7), one finds that the axion emission from LFV couplings dominates the neutrino emission unless which is indeed the case based on the existing constraints.
In heavier NSs, the LFV emission of axions tends to have a less significant impact.This is because fast neutrino emission could occur via the direct Urca processes [72].
In the presence of superfluidity, the formation of Cooper pairs can dominate over the Murca process [73,74], further diminishing the role of LFV axion emission.Axions are predominantly produced in NSs through the nucleon bremsstrahlung process nn → nna.At the same core conditions, its emissivity is given by ε (nn) a ≃ 2.8 × 10 38 g 2 ann T 6 9 erg cm −3 s −1 [75,76].The nucleon bremsstrahlung process dominate the LFV processes if The current best constraint on the axion-neutron coupling is |g ann | ≲ 2.8 × 10 −10 [26].Therefore, it is unlikely for the LFV couplings to play a significant role in NSs with an age ≳ 1 yr, where the temperature has cooled to 10 9 K [77].These limits on the axion's LFV coupling are relatively weak, and this is a consequence of the ε LFV a ∝ T 8 scaling, which is suppressed compared to LFP channels by a factor of (T /m µ ) 2 , which is tiny in old NSs.However, in the proto-NS that forms just after a supernova, this ratio can be order one, which suggests that stronger limits can be obtained by considering the effect of axion emission on supernova rather than neutron stars.Since our analysis has focused on neutron star environments, adapting our results to the more complex proto-NS system requires some extrapolation.We estimate the axion emissivity from a supernova by extrapolating (7) to high temperatures.By imposing the bound on the energy loss of SN 1987A, ε a /ρ ≲ 10 19 erg g −1 s −1 [20], one finds that at a typical core condition ρ ∼ 8 × 10 14 g cm −3 , which is to be evaluated at T ∼ (30 − 60) MeV.This constraint is more stringent than that obtained from considering µ → ea in a supernova and is comparable to the current best terrestrial limit.One should note that at typical core conditions of a proto-NS, nucleons and muons are at the borderline between degeneracy and nondegeneracy, and we expect a similar constraint if a nondegenerate emission rate is used. 4iscussion.-Inthis letter, we study the astrophysical signatures of an axionlike particle's LFV coupling with muons and electrons.We focus on axion emission from NS cores, where the electron Fermi energy is large enough to maintain a high abundance of muons.Our limits on the LFV coupling g aeµ derive from comparing the axion emission rate with the energy loss rate due to neutrino emission, since excessively strong axion emission would conflict with the observations of old NSs and SN 1987A.
Further research is needed to assess the impact of axion's LFV interactions on the entire cooling history of the star, including a careful treatment of equations of state and nuclear interactions.Stronger nuclear interactions would result in higher number densities of protons and muons at the same mass density, thereby enhancing the rate of the LFV interactions.Such an analysis is particularly motivated for axion emission from proto-NSs formed after type-II supernovae, where the transition from nondegenerate to degenerate matter and the creation of the muon population could impact axion emissivities.Our work highlights the importance of assessing both the free muon decay channel µ → ea as well as scattering channels lf → l ′ f a in such studies. where T , and z ≡ E ′ 3 /T .The approximation symbols arise from extending the limits of integration to infinity.The second equality is derived using the technique in [81].For n = 2, we obtain For the angular integral, we first integrate d 3 p ′ 2 with the momentum delta function and dp 1 , dp 2 , dp ′ 1 with the Fermi surface delta function.It is convenient to align all angles with respect to p 1 , so d 2 Ω 1 simply gives 4π.The angular integral A becomes where c ij denotes the cosine of the angle between p i and p j , u φ ≡ φ

D. Axion emissivity
In summary, the axion emissivity is given by The dc 13 ′ integral can be evaluated analytically.We calculate the other integrals using numerical techniques and present the result for F (lf ) in figure S2.In the left panel we vary the muon Fermi velocity β F,µ = p F,µ /E F,µ .From the right panel we see that F (lp) is not sensitive to β F,p if protons are nonrelativistic, i.e., β F,p ≲ 0.5, which is expected in NSs.Therefore, we use the values of F (lf ) shown in the left panel to calculate the emissivity shown in the main text.

E. Different temperature dependence from LFV and LFP interactions
In the main text we contrast the temperature dependence of the axion emissivity for LFV and LFP interactions.The LFP interaction leads to axion emission via channels such as l + f → l + f + a with an emissivity that scales as ε a ∝ T 6 (similar for nn → nna [20]).By considering the LFV interaction here, we find that channels such as l + f → l ′ + f + a lead to an emissivity ε a ∝ T 8 instead.This different scaling may be understood by inspecting the form of the matrix element.Consider the Feynman diagram in the left panel of figure S1.The fermion propagator and the axion vertex contribute factors of in the (−, +, +, +) metric signature and neglecting the axion mass E ′ 3 = |p ′ 3 |.The axion energy E ′ 3 in the numerator arises from the derivative nature of the axion interaction.The temperature dependence enters via the typical axion energy, E ′ 3 ∼ T .For LFP channels such as µp → µpa, we have m ′ 1 = m 1 , the E ′ 3 ∼ T factor in the numerator is canceled by the factor in the denominator, and consequently the squared matrix element is insensitive to the temperature.On the other hand, for the LFV channels, the m ′ 2 1 −m 2 1 term dominates in the denominator.Consequently, the LFV axion emissivity is suppressed relative to the LFP calculation by a factor of order T 2 E 2 F,e /(m 2 µ −m 2 e ) 2 ∼ T 2 /m 2 µ ∼ 7×10 −7 T 2 9 .

60 70 80 FIG
FIG.S2.The factor F (lf ) as a function of the Fermi velocity of muons (left) and protons (right).Here we have set βF,p = 0.3 and βF,µ = 0.8 for the left and right panels respectively for the f = p processes.