No evidence for axions from Chandra observation of magnetic white dwarf

Ultralight axions with axion-photon couplings $g_{a\gamma\gamma} \sim {\rm few} \times 10^{-11}$ GeV$^{-1}$ may resolve a number of astrophysical anomalies, such as unexpected ~TeV transparency, anomalous stellar cooling, and X-ray excesses from nearby neutron stars. We show, however, that such axions are severely constrained by the non-observation of X-rays from the magnetic white dwarf (MWD) RE J0317-853 using ~40 ks of data acquired from a dedicated observation with the Chandra X-ray Observatory. Axions may be produced in the core of the MWD through electron bremsstrahlung and then convert to X-rays in the magnetosphere. The non-observation of X-rays constrains the axion-photon coupling to $g_{a\gamma\gamma} \lesssim 5.5 \times 10^{-13} \sqrt{C_{a\gamma\gamma}/C_{aee}}$ GeV$^{-1}$ at 95% confidence for axion masses $m_a \lesssim 5 \times 10^{-6}$ eV, with $C_{aee}$ and $C_{a\gamma\gamma}$ the dimensionless coupling constants to electrons and photons. Considering that $C_{aee}$ is generated from the renormalization group, our results robustly disfavor $g_{a\gamma\gamma} \gtrsim 4.4 \times 10^{-11}$ GeV$^{-1}$ even for models with no ultraviolet contribution to $C_{aee}$.

Axions are hypothetical ultralight pseudoscalar particles that couple through dimension-5 operators to the Standard Model.In particular the quantum chromodynamics (QCD) axion couples to QCD, which allows it to solve the strong-CP problem [1][2][3][4]; this coupling also generates a mass m QCD a ∼ Λ 2 QCD /f a for the particle, with f a the axion decay constant and Λ QCD the QCD confinement scale.In this work we probe axions with masses m a 10 −2 eV that do not couple to QCD (but see [5][6][7]) though they couple to electromagnetism and matter.Such ultralight axions, often referred to as axion-like particles, are especially motivated theoretically in the context of the String Axiverse [8][9][10][11][12][13].In the Axiverse it is natural to expect a large number N of ultralight axions, with m a m QCD a .One linear combination couples to QCD and receives a mass from QCD, becoming the QCD axion, while the rest of the N − 1 states remain ultralight and retain their non-QCD couplings to the Standard Model.It is well established that axions may be produced within stars including white dwarfs (WDs) (see e.g.[14][15][16]) and escape the stars due to their weak interaction strengths with matter.Recently it has been pointed out that such axions could produce X-ray signatures through axion-photon conversion in magnetic WD (MWD) magnetospheres [17] (see [18][19][20][21][22][23] for related discussions in neutron star (NS) magnetospheres).In this work we collect and analyze data from the MWD RE J0317-853 to look for evidence of this process.
The couplings of the axion a with mass m a to electromagnetism and electronic matter are described through the Lagrangian terms g aγγ aF µν F µν + g aee 2m e (∂ µ a)ēγ µ γ 5 e , (1) with F ( F ) the (dual) quantum electrodynamics (QED) field strength, e the electron field, and m e the electron mass.It is convenient to parameterize the cou- GeV −1 at 95% confidence for low ma from the non-observation of Xrays from the MWD RE J0317-853.We translate this result to constraints on gaγγ assuming: (i) a tree-level axion-electron coupling with Caee = Caγγ, and (ii) the loop-induced Caee ≈ 1.5 • 10 −4 Caγγ that represents a conservative W -phobic axion (the loop-induced Caee is generically larger).The expected 68% (95%) containment region for the power-constrained 95% upper limit is shaded in green (gold) for the Caee = Caγγ scenario.Previous constraints are shaded in grey [24].
As described in [17] axions may be produced within the cores of MWD stars through electron bremsstrahlung off of ions, using the g aee coupling, and converted to Xrays in the stellar magnetospheres with the g aγγ term in (1).Ref. [17] identified RE J0317-853 as being the most promising currently-known MWD because of a combination of (i) the close distance d = 29.38 ± 0.02 pc, as measured by Gaia [48], (ii) the large magnetic field B pole ∼ 500 MG, and (iii) the high core temperature T core ∼ 1.5 keV.The predicted axion-induced X-ray signal is expected to be roughly thermal at the core temperature, meaning that it should peak at a few keV where Chandra is the most sensitive currently-operating X-ray telescope.
We observed the MWD RE J0317-853 on 2020-12-18 using the Chandra ACIS-I instrument with no grating for a total of 37.42 ks (PI Safdi, observation ID 22326).After data reduction -see the Supplementary Material (SM) -we produce pixelated counts maps in four energy bins from 1 to 9 keV of width 2 keV each.Each square pixel in right ascension (RA) and declination (DEC) has physical length of ∼0.492 (note in the RA direction this is the width in RA × cos(Dec)).In Fig. 2 we show the binned counts over 1-9 keV in the vicinity of the MWD; note that in this region no pixel has more than one count.The figure is centered at the current location of the MWD, labeled 'Dec.2020 (calib.)':RA 0 ≈ 49 • 18 37. 77, DEC 0 ≈ −85 • 32 25.81.Fig. 2 also shows intermediate source locations determined during the astrometric calibration process (see the SM).The 68% energy containment radius at 1 keV (9 keV) is approximately 0. 5 (0. 6).The inset illustrates the expected template for emission associated with the MWD at 1 keV.No photon counts are observed near the MWD.The circle in Fig. 2 has radius 5 and is the extent of our region of interest (ROI); that is, we exclude pixels whose centers are beyond this radius in our analysis.
We analyze the pixelated data d = {n i,j }, with n i,j the number of counts in energy bin i and pixel j, in the context of the axion model, which is discussed more shortly, using the joint Poisson likelihood with M denoting the joint signal and background model, with model parameters θ = {A bkg , g aee g aγγ , m a }, and N pix the number of spatial pixels.The model predicts µ i,j (θ) counts in energy and spatial pixel i, j.The back- ground parameter vector A bkg consists of a single normalization parameter in each of the four energy bins that re-scales the background counts spatial template.For our background template, which we profile over, we use the exposure map, which is flat to less than 0.5% over our ROI.The signal model has the two parameters {g aee g aγγ , m a }, which predict the counts in each of the four energy bins.The signal template is centered on the MWD and accounts for the point spread function (PSF), as illustrated in the inset of Fig. 2. At a fixed m a we construct the profile likelihood for g aγγ g aee by maximizing the log-likelihood over A bkg at each g aγγ g aee .Our 95% upper limit on g aγγ g aee is constructed directly by Monte Carlo simulations of the signal and null hypotheses instead of relying on Wilks' theorem, since we are in the low-counts limit (see e.g.[49] for details).A priori we decided to power constrain [50] our limits to account for the possibility of under fluctuations, though this was not necessary in practice.
We also analyze the data using the Poisson likelihood in the individual energy bins to extract the spectrum dF/dE, which is illustrated in Fig. 3.In that figure we overlay the axion model prediction, which we now detail.For production via axion bremsstrahlung from electronion scattering [15,51], we broadly follow the formalism developed in [17], though we make improvements thanks to updated WD models and luminosity data from Gaia.Firstly, we improve our modeling of the density profile and composition of RE J0317-853 using MESA [52] version 12778.We simulate a WD of RE J0317-853's mass from stellar birth until it has cooled below RE J0317-853's observed luminosity.These simulations account for core electrostatic effects including ionic correlations and crystallization in the core that modify the profiles from that of a fully degenerate ideal electron gas, which were neglected in [17].We find RE J0317-853 has a predominantly oxygen-neon core because it completed carbonburning while ascending the asymptotic giant branch, typical for a WD of its mass undergoing single-star evolution.We take as our fiducial profiles those density and composition profiles from the model for which the luminosity matches the observed luminosity of RE J0317-853 (see Sec. IV of the SM for further details).
The second improvement we make is in estimating the core temperature of RE J0317-853.Ref. [17] estimated the core temperature from an empirical core temperature-luminosity relation using an assumed luminosity from [53].Ref. [53] used Hubble parallax and photometric data along with WD cooling sequences to estimate the luminosity of RE J0317-853.Here, we estimate the core temperature from WD cooling sequences [54] which predict Gaia DR2 band magnitudes.These cooling sequences are improved over those of [53] because they better account for ionic correlation effects than previous sequences, and our use of Gaia data rather than Hubble represents an improvement because of smaller uncertainties on the magnitudes, partly due to improved parallax measurements.In particular, we fit the models in [54] over cooling age and mass to the measured RE J0317-853 Gaia DR2 data [55].Although previous measurements indicated a mass for RE J0317-853 of 1.26 M , we find that the 1.22 M model provides the best fit to the data.In the context of that model, we find that the Gaia data prefers a core temperature T c = 1.388 ± 0.005 keV.Therefore we use this model and to be conservative assume a core temperature at the lower 1σ allowed value, T c = 1.383 keV, since the emissivity increases with increasing T c .
Axion emission from the stellar interior primarily results from the bremsstrahlung scattering e + (A, Z) → e + (A, Z) + a where an electron is incident on a nucleus with atomic number Z and mass number A. The electrons in a WD core are strongly degenerate with a temperature T p F that is much smaller than the Fermi momentum p F .In this regime, the axion emissivity spectrum is thermal and given by [15,51] which includes a sum over the species s of nuclei that are present in the plasma; Z s is the atomic number, A s is the mass number, ρ s is the mass density, and u 931.5 MeV is the atomic mass unit.The species-dependent, dimensionless factor F s accounts for medium effects, including screening of the electric field and interference between different scattering sites.For a strongly-coupled .The energy spectrum found from our analysis of the Chandra data from the MWD RE J0317-853.In each of the four energy bins the best-fit fluxes are consistent with zero (the 68% containment intervals are shown).We also illustrate the predicted axion-induced signal that would be seen from an axion with the indicated couplings and ma 10 −5 eV.
plasma [56] we use the empirical fitting functions provided by [57].Note that the axion luminosity is given by the integral of the emissivity over the WD core.
Our fiducial WD model leads to the predicted axion luminosity L a ≈ 8 • 10 −4 L (g aee /10 −13 ) 2 .Accounting for modeling uncertainties on RE J0317-853 we estimate the limit on g aγγ may be ∼10% stronger, as illustrated in SM Fig. S4.Axions may also be produced by the g aγγ coupling from electro-Primakoff production, which we compute in the SM, though as we show in SM Figs.S2 and S3 this process is subdominant compared to bremsstrahlung for RE J0317-853.
The axions then undergo conversion to X-rays in the MWD magnetic fields.The conversion probability p a→γ may be calculated numerically for arbitrary magnetic field configurations and axion masses m a by solving the axion-photon mixing equations in the presence of g aγγ , though it is important to incorporate the Euler-Heisenberg Lagrangian term which modifies the propagation of photons in strong magnetic fields and suppresses the mixing [19].The magnetic field of the MWD is found to vary over the rotation period between 200 MG and 800 MG [58]; we follow [17] and assume a dipole field of strength 200 MG, to be conservative.Note that at low axion masses and high B-field values the dependence of the conversion probability on magnetic field is mild: p a→γ ∝ B 2/5 [17].Using the offset dipole model from [58] increases the conversion probabilities by up to ∼50% [17] at low masses, which may increase the limit by ∼10% relative to our fiducial case.Numerically the conversion probabilities are O(10 −4 )× g aγγ /10 −11 GeV −1 2 for m a 10 −5 eV and drop off for higher masses.The GeV −1 assuming ma 10 −5 eV.For ma 10 −7 eV the leading constraint on gaγγ is from the CAST experiment [32] and HB star cooling [33], while for ma 10 −10 eV it is from X-ray observations of SSCs [25].The leading limit on gaee is from WD cooling [59], while the 68% containment region for explaining stellar cooling anomalies [16], along with the best-fit coupling, is also indicated and in tension with our null results.
distance is fixed at the central value measured by Gaia d = 29.38 pc [48] because the distance uncertainty only leads to a ∼0.1% uncertainty on the flux.In Fig. 3 we illustrate the energy-binned spectrum prediction from axion-induced emission from the MWD for m a 10 −5 eV and g aee g aγγ = 10 −25 GeV −1 .
We find no evidence for the axion model, with the bestfit coupling combination being zero for all masses.We thus set 95% one-sided upper limits on the coupling combination g aee g aγγ at fixed axion masses m a using the profile likelihood procedure.For low masses m a 10 −5 eV the limit is g aee g aγγ 1.3 × 10 −25 GeV −1 .This limit is around three orders of magnitude stronger than that set by the CAST experiment on this coupling combination [32].Our limit also severely constrains the low-mass axion explanation of stellar cooling anomalies [16], which prefer g aγγ g aee ∼ 2×10 −24 GeV −1 as illustrated in Fig. 4, where we show our low-mass limit in the g aγγ −g aee plane, along with current constraints.
It is instructive to translate our limit to one on g aγγ alone by assuming a relation between the dimensionless coupling constants C aee and C aγγ .Note that in the DFSZ QCD axion model there is a tree-level coupling between the axion and electron, such that C aee ∼ C aγγ , while in the KSVZ model no ordinary matter is charged under the Peccei-Quinn (PQ) symmetry and so C aee = 0 at tree level, though it is generated at one loop [60].The loop-induced value of C aγγ depends on the relative coupling of the axion to SU (2) L versus hypercharge [17,60,61] and the SM).To be conservative we assume in Fig. 1 the W -phobic axion scenario, where the axion only couples to U (1) Y (but see SM Fig. S2).We also show the limit on g aγγ for axion models with C aee = C aγγ , which is nearly two orders of magnitude stronger than the loopinduced limit.
Our results have strong implications for a number of astrophysical anomalies and planned laboratory experiments.For example, the WD cooling anomaly prefers g aee ∼ 1.6 × 10 −13 [16].In order for a low mass axion to explain this result and be compatible with our upper limit, one would need C aγγ 2.2C aee (g aγγ 8.1×10 −13 GeV −1 ), which would not be able to also explain the axion-photon coupling g aγγ ∼ 10 −11 GeV −1 suggested by the global fit to stellar cooling data [16] (see Fig. 4) or the TeV transparency anomalies, which prefer g aγγ 2 × 10 −11 GeV −1 for m a 10 −8 eV [62].Anomalous X-ray emission from nearby isolated Magnificent Seven NSs may be interpreted as low-mass (m a 10 −5 eV) axion production from nucleon bremsstrahlung in the NS cores and conversion to X-rays in the NS magnetospheres [22,63].The required coupling combination to explain the X-ray excesses is g aγγ g aN N 10 −21 GeV −1 , with g aN N = C aN N m N /f a the axion-nucleon coupling, with m N the nucleon mass and C aN N the dimensionless coupling.The non-observation of X-rays in this work from the MWD implies that if axions explain the Magnificent Seven excess they must be electro-phobic, with C aee 4 C aN N .Lastly, we note that our results are especially relevant for the upcoming ALPS II light-shiningthrough-walls experiment [64].The last stage of the experiment will have sensitivity to g aγγ 2 • 10 −11 GeV −1 for m a 10 −4 eV, meaning that much of the axion parameter space to be probed is constrained by the current analysis (see SM Fig. S2).
As evident in e.g.Fig. 2 with ∼40 ks of Chandra data we are able to perform a nearly zero-background search; an additional order of magnitude in exposure time would allow us to improve the sensitivity to g aγγ by a factor ∼1.5.The proposed Lynx X-ray Observatory [65] aims to improve the point source sensitivity by roughly two orders of magnitude compared to Chandra.A ∼400 ks observation with Lynx or a similar future telescope of RE J0317-853 (see SM Fig. S1) may be sensitive to axions with g aγγ ∼ 10 −13 GeV −1 for C aee ∼ C aγγ , which may probe photo-philic QCD axion models in addition to vast regions of uncharted parameter space for the hypothetical Axiverse.
search used resources from the National Energy Research Scientific Computing Center (NERSC) and the Lawrencium computational cluster provided by the IT Division at the Lawrence Berkeley National Laboratory, supported by the Director, Office of Science, and Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.Support for this work was provided by the National Aeronautics and Space Administration through Chandra Award Number GO0-21013X issued by the Chandra X-ray Center (CXC), which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060.The scientific results reported in this article are based to a significant degree on observations made by the Chandra Xray Observatory.This research has made use of software provided by the CXC in the application package CIAO.
Supplementary Material for: No evidence for axions from Chandra observation of magnetic white dwarf Christopher Dessert, Andrew J. Long, Benjamin R. Safdi This Supplementary Material (SM) is organized as follows.Sec.I provides Supplementary Figures that are referenced in the main Letter.Sec.II gives further information on our data reduction and calibration procedure.In Sec.III we review the renormalization group evolution of the axion-electron coupling to justify the values taken in the main text.In Sec.IV we describe our modeling procedure for the MWD in more detail.Sec.V presents our calculation of the Electro-Primakoff axion production rate.

I. SUPPLEMENTARY FIGURES
In this section we illustrate Figs.S1, S2, S3, and S4, which are cited and described in the main Letter.  1 but projecting future sensitivity from deeper observations of RE J0317-853.A factor of 10 increase in Chandra exposure time would lead to the projected expected 95% upper limits indicated, while in the future the Lynx X-ray observatory will allow for a significant increase in sensitivity.To generate the Lynx projections, we use the package SOXS to generate expected counts maps, exposure maps, and the Lynx PSF.We then run our Chandra pipeline with the Lynx files.As in Fig. 1 but showing the 95% upper limits from this work interpreted in the context of limits on gaγγ assuming loop-induced couplings to Caee for the W -phobic (Caee = 1.6 × 10 −4 Caγγ) and W -philic (Caee = 4.8 × 10 −4 Caγγ) UV completions.Models that couple to both SU (2)L and U (1)Y will generically have loop-induced couplings between these two extremes, assuming no fine-tuned cancellations (for example, models that couple in a way that preserve the Grand Unification group symmetry may have Caee ≈ 2.7×10 −4 Caγγ).Note that UV contributions to Caee may also exist.We compare these limits to the projected sensitivity from the ALPS-II experiment.We also show our limits only accounting for the electro-Primakoff process, which does not involve Caee -this process is seen to be subdominant compared to the bremsstrahlung process.  .As in Fig. 3 but comparing the (red) and electro-Primakoff (dashed blue) production rates, for the indicated couplings. .As in Fig. 1 but comparing the W -phobic loop-induced upper limit (red) for our fiducial stellar model to that for the alternate stellar model that differs in two ways: (i) the MWD mass in assumed to be higher at 1.29 M , and (ii) the temperature is taken at the upper value of the 1σ containment interval from fitting the stellar model to the Gaia luminosity data.The difference between these two limits gives an estimate for the magnitude of the astrophysical uncertainties, which are around 10%.

II. DATA REDUCTION AND CALIBRATION
The data from the 37.42 ks Chandra ACIS-I Timed Exposure observation of RE J0317-853 (PI Safdi, observation ID 22326) is reduced as follows.For the data reduction process, we use the Chandra Interactive Analysis of Observations (CIAO) [66] version 4.11.We reprocess the observation with the CIAO task chandra repro, which produces an events file filtered for flares and updated for the most recent calibration.We create counts and exposure images (units [cm 2 s]) with pixel sizes of 0. 492 with flux image.
We account for the astrometric uncertainty of Chandra, which is expected to be on the order of 0. 5 [67], through the following procedure: we (i) run the point source (PS) finding algorithm celldetect on the full Chandra image to find high-significance PSs ( 10σ significance), and then (ii) cross-correlate these sources with the Gaia early data release 3 (EDR3) catalog [48] evolved to the Dec. 2020 epoch.(Note that there are no already-known X-ray sources within the field of view to use as references.)Two of the high-significance sources have nearby matches with Gaia sources (Gaia source IDs 4613614905421384320 and 4613614974140862464).Although we were not able to verify the identity of these two sources from our observation, the Gaia sources both appear in the WISE catalog on active galactic nuclei [68], as J031629.01-852836.0 and J031821.59-852751.5 respectively.Both sources are localized by celldetect to within ∼0. 2. However, both Chandra sources are displaced from their Gaia matches by ∼0.6 in approximately the same direction (the offset is (0. 53, 0. 25) for one source and (0. 57, −0.05) for the other, in (RA cos(DEC), DEC)).We average these two offsets to determine our overall calibration and shift all RA, DEC values accordingly.The uncalibrated location is shown in Fig. 2. Note that we cannot exclude the possibility that the Chandra PSs are falsely matched with the Gaia sources, though this appears less likely given that the two position offsets are nearly the same.Additionally, using the uncalibrated source location produces nearly identical results to using the calibrated location, since the calibration error is relatively minor and there are no photons in the vicinity of either location.
In addition to the calibration, we also account for the proper motion of the WD.In particular, RE J0317-853 was observed by Gaia in the EDR3 with location RA ≈ 49 • 18 42.51, DEC ≈ −85 • 32 25.75 at the reference epoch of J2016.0 [48].We use the proper motion measurements from Gaia to infer the position in December 2020, which accounts for the small shift between Gaia 2016 and Dec. 2020 shown in Fig. 2.

III. LOOP-INDUCED AXION-ELECTRON COUPLING
In this section we review the loop-induced axion-electron coupling in order to justify the fiducial values taken in the main text for the W -phobic and W -philic axion with no ultraviolet (UV) axion-electron coupling.Recall that under the renormalization group and at energy scales µ > M Z , with M Z the mass of the Z-boson, where C µ e is the dimensionless axion-electron coupling at energy scale µ < Λ, with Λ the UV cutoff [17,60,61].The dimensionless axion couplings to weak isospin and hypercharge are denoted by C Λ W and C Λ B , respectively.Note that these couplings are topologically protected and do not evolve under the renormalization group.The weak isospin and hypercharge couplings constants are denoted by g and g , respectively.
It is common to integrate (S1) down to M Z and yet take g and g to be their low-energy values, at scales well below M Z .Below M Z the axion-electron coupling continues to evolve under the renormalization group equation and this contribution to C e at the scale µ = m e is also typically found by integrating (S2) and taking α EM to be the value at the scale m e .Here, we do not complete a full two-loop computation of C e but we try to be slightly more precise by accounting for the running of α EM , g, and g .To one-loop and within the Standard Model these couplings evolve as where g(M Z ) denotes the coupling at energy scale M Z , while g(Λ) is the coupling at the UV scale and similarly for g .At the Z-pole α EM (M Z ) ≈ 1/127 and sin 2 θ W ≈ 0.231, with θ W the Weinberg angle.Taking a benchmark value Λ = 10 9 GeV we then find Accounting for the running of α EM from M Z down to the electron mass we then find Note that the axion-photon coupling is defined by To be conservative, in our fiducial loop-induced model we consider a "W-phobic" axion and take C Λ W = 0 such that C aee ≈ 1.6 × 10 −4 C aγγ .We do note, though, with some amount of fine tuning the loop-induced contribution could be made smaller.For example, if then the two contributions to C Me e would roughly cancel each other.We do not consider this possibility further because it would require a conspiracy between the UV and IR contributions to the running.Note, also, that the relations in (S6) could be modified by the existence of beyond the Standard Model physics below the UV cutoff ∼10 9 GeV.

IV. MODELING RE J0317-853
In this section we detail our modeling of the interior of RE J0317-853.To compute the axion luminosity, we need to know the core temperature, the density profile, and the composition profiles.Note that we assume the core temperature is uniform throughout the interior due to the high thermal conductivity of the degenerate matter, while the density and composition can change throughout the interior.
We analyze WD cooling sequences [54] to infer the core temperature of RE J0317-853.These cooling sequences are improved over older ones in that they take ionic correlations into account, which are expected to be important for RE J0317-853 due to its high mass and low surface temperature.Included with the sequences are corresponding Gaia DR2 G, G BP , and G RP band absolute magnitudes as a function of cooling age.The sequences are available for WD masses of 1.10, 1.16, 1.22, and 1.29M .RE J0317-853's measured apparent magnitudes in the DR2 Gaia dataset [55] are G = 14.779 ± 0.005 G BP = 14.565 ± 0.017 G RP = 14.987 ± 0.012 (S7) where we have converted linear errors on flux to linear errors on magnitude.For reference, the G-band covers wavelengths between ∼300 and ∼1100 nm, G BP between ∼300 and ∼700 nm, and G RP between ∼600 and ∼1100 nm, although with wavelength-dependent efficiencies.Note we use EDR3 astrometric and distance data elsewhere in this work, but there do not yet exist cooling sequences incorporating EDR3 bands.We infer the core temperature T c of RE J0317-853 with a joint Gaussian likelihood over the three bands as a function of cooling age t for each WD mass available.We find that the 1.22M model provides the best fit to the data, as shown in the left panel of Fig. S5.Note that this is a lower mass for RE J0317-853 than previously inferred, but it is a conservative choice with respect to the 1.29M model, which is closer to previous mass estimates [53].
In the right panel of Fig. S5, we show the resulting likelihood profile as a function of T c for the best-fit 1.22M model.The ±1σ ages are extracted by solving for the age where ∆χ 2 increases by 1 on each side of the best-fit point.We find t = 0.369 ± 0.003 Gyr, corresponding to a core temperature T c = 1.388 ± 0.005 keV.We adopt the lower 1σ value of T c = 1.383 keV in our fiducial analysis to be conservative.We also show the axion luminosity, for which changes are minor over the range considered.
The 1.29 M model is disfavored in our analysis relative to the 1.22 M model at a level ∼5σ (the measured G BP and G RP are in tension with the model expectations).Therefore, when we determine the properties of RE J0317-853 in the context of the 1.29 M model, we broaden the likelihood profile so that at the best-fit point, ∆χ 2 /dof= 1.We find a lower cooling age of 0.301 ± 0.008 Gyr and a higher T c = 1.77 ± 0.02 keV by following the same procedure.SM Fig. S4 compares our limits computed using the fiducial model and the 1.29 M model, with T c at the upper end of the 1σ band; the differences are seen to be minor, indicating that our results are likely not significantly affected by astrophysical mismodeling.We run simulations with MESA from which we determine the density and composition profiles for RE J0317-853.MESA is a 1-dimensional modular stellar modeling code that outputs these profiles, along with others, as a function of time since stellar birth.We use the default parameters from the test suite inlist make o ne wd, but change the initial stellar mass to 11.1 (11.9)M , which produces a 1.22 (1.29) M WD.We evolve the star through the pre-WD stages and allow it to cool until its luminosity reaches 10 −3 L .
We then select the model for which the stellar luminosity matches the observed value and choose the profiles corresponding to this model, shown in Fig. S6, to be our fiducial density and composition profiles.We find that the core is predominantly oxygen and neon as expected for an isolated WD of its mass, and reaches densities ρ > 10 6 g/cm 3 , which means that the electron gas is strongly correlated.For ρ 10 7 g/cm 3 , the interior transitions to the lattice phase, which tends to reduce the axion emissivity.In the left panel of Fig. S7, we show the value of F as defined in (3) across the profile of the star for the four dominant ions in our WD model.The discontinuities in the profiles (except carbon) are due to the transition from the liquid phase to the lattice ion structure in the inner core of the WD.In general, F decreases with increasing density, although because the axion emissivity ε a ∼ ρF , the center of the star is still the most emissive.
Note that our choice of test suite is not the driving force behind why our WD is modeled as having an oxygenneon core-this is simply because, under the assumption of single-star evolution, the initial stellar mass of the WD progenitor is high enough so that the star depletes its core carbon on the asymptotic giant branch (this is the case for WDs with masses 1.1M [54,69]).If the star has evolved from a binary channel, then it may host a carbon-oxygen core instead.However, we consider this to be unlikely, as [53] finds that if RE J0317-853 has an effective temperature 40000 K, the single-star evolution is more likely.Indeed, our Gaia analysis prefers an effective temperature 25570±50 K.Note that although RE J0317-853 has a binary companion, they are too far apart to have interacted [53].
Given the core temperature, the density profile, and composition profiles, we have the tools to compute the axion luminosity of RE J0317-853 due to both axion bremsstrahlung and electro-Primakoff.We compute the axion emissivity at each radial slice in the MESA-generated profiles and integrate over the star to obtain the axion luminosity spectrum dL a /dω (in, e.g., ergs/s/keV) as for a stellar radius R. For axion bremsstrahlung, dε a /dω is computed using (3); for electro-Primakoff, (S29).Because of the geometric factors in the integrand in (S8) that suppress the contribution from the stellar core, the axion luminosity profile dL a /dr peaks around half the WD radius.
For our fiducial analysis, we model the magnetic field as a dipole field of strength 200 MG at the pole.To compute the axion-photon conversion probability p a→γ (ω), we follow the formalism developed in [17].The axion-induced photon flux dF γa /dω at Earth is then (S9)

V. ELECTRO-PRIMAKOFF AXION PRODUCTION
This section provides a derivation of the axion emissivity from the core of a WD from the electro-Primakoff production mechanism.Note that while the bremsstrahlung process dominates for our MWD, the electro-Primakoff process may be important for WDs with higher core temperatures, and this computation has not appeared elsewhere.

A. Cross section
Consider the scattering of an electron e and a nucleus N = (A, Z) that results in the emission of an axion a:         If the axion-photon coupling is dominant, then axion production is dominated by the electro-Primakoff channel.The leading-order Feynman graph is shown in Fig. S8, and corresponding matrix element is Note that the amplitude as ω = k 0 → 0, since 4-momentum conservation implies ρσαβ (p The spin-averaged, squared matrix element is given by |M| 2 = (g e g N ) −1 s |M| 2 where g e = g N = 2 counts the two spin states of the electron and the nucleus.
The differential cross section for axion emission is calculated from the squared matrix element as where the Lorentz-invariant flux factor is , and where the Lorentz-invariant phase space volume element is dΠ s (p) = d 3 p/(2π) 3 /2E s (p) for s = e, N, a.All 4-momenta are evaluated on shell with

B. Thermal-averaging
The thermal environment leads to Pauli-blocking and Bose-enhancement of the final-state particles.We take this into account by defining the thermally-suppressed/enhanced differential cross section where f e , f N , and f a are the phase space distribution functions for electrons, nuclei, and axions, respectively.The electrons are in equilibrium and their distribution function (in the rest frame of the plasma) is given by the Fermi-Dirac distribution where T e and µ e are the electrons' temperature and chemical potential.The nuclei are also in thermal equilibrium, and we could also write their distribution function as a Fermi-Dirac distribution.However, since their temperature is so low, T N m N , it turns out that the nuclei are effectively at rest v N ∼ T /m N 1.To a good approximation we can write the nuclei phase space distribution function (in the rest frame of the plasma) as where n N is the total number density of nuclei and g N = 2 counts the two spin states.This also lets us approximate 1 − f N ≈ 1 in (S13).Finally the axions are out of thermal equilibrium, and their distribution function satisfies and we can approximate 1 + f a ≈ 1 in (S13).

C. Axion emissivity
Using the differential cross section from (S13), we construct the thermally-suppressed/enhanced differential scattering rate density, which is where the Møller velocity is , where the thermally-weighted differential number density of incident particles is dn s (p) = g s d 3 p f s (p)/(2π) 3 for s = e, N , and where g e = g N = 2 counts the redundant internal degrees of freedom (spin).The differential axion emissivity (in the rest frame of the plasma) is where we multiply by the axion energy and sum over the spins of all the particles.Using the expression for dγ gives dε a = g e g N 32 where the factors of E a have cancelled, and all 4-momenta are on-shell.

D. Evaluating phase space integrals
To calculate the emissivity, we evaluate the phase space integrals as follows.First, we use the momentum-conserving Dirac delta function to evaluate the integral over the recoiling nucleus's momentum, which sets p 4 = p 1 + p 2 − p 3 − k.Next we write p 1 , p 3 , and k in polar coordinates, where i denotes the initial-state electron, f denotes the final-state electron, and ω = E a (k).We use the remaining Dirac delta function to evaluate the integral over E f , which gives Next we make use of the distribution functions in (S14).These let us approximate 1 − f N ≈ 1 and 1 + f a ≈ 1.Additionally, f N ∝ δ(p) and the p 2 integral sets p 2 = 0. Finally we note that the scattering is statistically isotropic, since the distributions of incident particles have no preferred direction.It suffices to suppose that Ω f and Ω a are measured with respect to Ω i , which is then treated as the orientation of the polar axis.Then the integral over Ω i reduces to the trivial integral over the polar axis (net rotation of the whole system), which just gives dΩ i = 4π, and dε a = g e n N 128π 5  To evaluate the squared matrix element, we approximate m a ≈ 0 implying ω ≈ |k|.We can also approximate the recoiling nucleus as non-relativistic, implying E 4 ≈ m N + p 2 4 /(2m N ), and here it is important to keep the sub-leading term in the energy expansion, since the would-be leading order contribution to the squared matrix element cancels.Then the squared matrix element reduces to (Zg aγγ e 2 ) 2 g e g N 32m 2 N ω 2 q 4  13 q 4 24 where we have dropped terms that are O(m 1 N ).Here we have also written p 1 • p 3 = p i p f c if and p 1 • k = p i ωc ia and p 3 • k = p f ωc f a .The momentum transfers are We have also used e 2 = 4πα EM and g aee 2 = 4πα aee and set g e = g N = 2.Note that our assumption E N (p 4 ) ≈ m N implies the simple relation E f ≈ E i − ω.
If the plasma is degenerate, T p F = E 2 F − m 2 e , then the thermal factor can be approximated as which contains the angular integrals.The momentum transfer factors have become q 2 13 ≈ q 2 24 ≈ −q 2 + O(ωp F ) where and we neglect the ω-suppressed terms.

E. Emissivity and luminosity
Now generalizing to a plasma with multiple species of ions, labeled by s, the emissivity spectrum is written as where we have used n s = ρ s /u and u ≈ 931.5 MeV is the atomic mass unit, and we have assumed that all species have a common temperature T s = T .Note that the emissivity spectrum, dε a /dω a , is almost a thermal spectrum, except that there's an additional factor of ω 2 , which follows from the momentum-dependent axion-photon coupling.
The integral over ω evaluates to 8π 6 T 6 /63, and the total emissivity is found to be Note that these relations hold for either relativistic or non-relativistic electrons; i.e., p F ≈ E F m e or p F E F ≈ m e .
In the derivation above, we have neglected medium effects, which are now taken into account following Ref.[15].Free electrons in the medium will screen the photon propagator, introducing an effective photon mass k 2 TF = 4α EM p F E F /π, which is the Thomas-Fermi screening scale.Additionally interference and correlation effects are captured by the static structure factor S ions (|q|).For a strongly-coupled plasma, such as the one in a WD core, the static structure factor has been calculated in Refs.[51,57], and the factor F is also evaluated for axion emission via electron-bremsstrahlung scattering.As a rough estimate, we simply carry over that estimate of F here, though future work using this result should calculate F more precisely.
The axion luminosity is evaluated by integrating L a = dV ε a over the volume of the WD star.To a good approximation, the core temperature T ≈ T c is approximately uniform throughout the star, due to the degenerate matter's high thermal conductivity.On the other hand, the Fermi momenta p F,s , medium factors F s , and mass fractions R s = ρ s /ρ tot have radial-dependent profiles.To provide a rough estimate, we neglect these effects and the volume integral gives dV ρ tot = M , which is the mass of the star.Then the axion luminosity is

14 E 1 Figure 3
Figure3.The energy spectrum found from our analysis of the Chandra data from the MWD RE J0317-853.In each of the four energy bins the best-fit fluxes are consistent with zero (the 68% containment intervals are shown).We also illustrate the predicted axion-induced signal that would be seen from an axion with the indicated couplings and ma 10 −5 eV.

Figure 4 .
Figure 4.The 95% one-sided limit on the axion-photon and axion-electron coupling from this work gaeegaγγ < 1.3×10 −25 GeV −1 assuming ma 10 −5 eV.For ma 10 −7 eV the leading constraint on gaγγ is from the CAST experiment[32] and HB star cooling[33], while for ma 10 −10 eV it is from X-ray observations of SSCs[25].The leading limit on gaee is from WD cooling[59], while the 68% containment region for explaining stellar cooling anomalies[16], along with the best-fit coupling, is also indicated and in tension with our null results.

Expected
Figure S1.As in Fig.1but projecting future sensitivity from deeper observations of RE J0317-853.A factor of 10 increase in Chandra exposure time would lead to the projected expected 95% upper limits indicated, while in the future the Lynx X-ray observatory will allow for a significant increase in sensitivity.To generate the Lynx projections, we use the package SOXS to generate expected counts maps, exposure maps, and the Lynx PSF.We then run our Chandra pipeline with the Lynx files.
Figure S2.As in Fig.1but showing the 95% upper limits from this work interpreted in the context of limits on gaγγ assuming loop-induced couplings to Caee for the W -phobic (Caee = 1.6 × 10 −4 Caγγ) and W -philic (Caee = 4.8 × 10 −4 Caγγ) UV completions.Models that couple to both SU (2)L and U (1)Y will generically have loop-induced couplings between these two extremes, assuming no fine-tuned cancellations (for example, models that couple in a way that preserve the Grand Unification group symmetry may have Caee ≈ 2.7×10 −4 Caγγ).Note that UV contributions to Caee may also exist.We compare these limits to the projected sensitivity from the ALPS-II experiment.We also show our limits only accounting for the electro-Primakoff process, which does not involve Caee -this process is seen to be subdominant compared to the bremsstrahlung process.

Figure S3
Figure S3.As in Fig.3but comparing the (red) and electro-Primakoff (dashed blue) production rates, for the indicated couplings.
Figure S4.As in Fig.1but comparing the W -phobic loop-induced upper limit (red) for our fiducial stellar model to that for the alternate stellar model that differs in two ways: (i) the MWD mass in assumed to be higher at 1.29 M , and (ii) the temperature is taken at the upper value of the 1σ containment interval from fitting the stellar model to the Gaia luminosity data.The difference between these two limits gives an estimate for the magnitude of the astrophysical uncertainties, which are around 10%.

Figure 8 ρ [g/cm 3 ]
Figure S5.A color-magnitude diagram with RE J0317-853's Gaia DR2 data shown with the black error bars.We show the curves predicted by the cooling simulation for three masses: 1.16, 1.22, and 1.29M .Note that MG refers to the absolute G-band magnitude, while the color BP − RP = GBP − GRP.(Right) The likelihood profile for the 1.22M model as a function of Tc.The best fit Tc is shown as the dashed vertical line, while the 1 and 2σ containment regions on Tc are shown as green and yellow bands, respectively.We also show, on the right y-axis, the axion luminosity (dashed red) as a function of Tc for gaee = 10 −13 .

Figure
Figure S7.(Left)The F -profile evaluated for the 1.22 M star, evaluated using the parametrization provided by[57], considered in our emissivity calculation.(Right) The sum in (3) evaluated for both mass models (1.22 M and 1.29 M ).

e(p 1
, s 1 ) + N (p 2 , s 2 ) → e(p 3 , s 3 ) + N (p 4 , s 4 ) + a(k) .(S10) e(p 1 , s 1 ) < l a t e x i t s h a 1 _ b a s e 6 4 = " q U A 9 w a p D u B v I r w 3 t C Y m n P J w L d 7 s = " > A A A C Z 3 i c l V D J S g N B E O 2 M e 9 w S B R G 8 t A Z B Q T R H Y 0 P H I l e / 6 Z V k m s n v w L Z q g v B r n R D P 8 X c 4 e E y x R d 4 + H P f n + D m / 1 q e F A 9 u j q s 1 G t 5 9 7 N o A 2 2 h H R S i Y 1 R H 5 + g S N R B F j + g Z v a D X w r u 3 7 K 1 5 6 5 9 W r 5 B n V t G 3 8 T Y / A O s B u Y s = < / l a t e x i t > e(p 3 , s 3 )< l a t e x i t s h a 1 _ b a s e 6 4 = " G T C W Y j Z v u h T i /A M h 7 v F E P C 2 U 5 k Q = " > A A A C Z 3 i c l V D J S g N B E O 2 M W 4 x b o i C C l 9 Y g K E h m x i h 6 E k E P X g Q F o 0 I S h p 5 O J T b p Z e j u U c O Q T / G q 3 + Q n + B d2 4 h z c L h Y U P N 5 C F S 9 O O D M 2 C N 4 K 3 s T k 1 P R M c b Y 0 N 7 + w u F S u L N 8 Y l W o K D a q 4 0 n c x M c C Z h I Z l l s N d o o G I m M N t 3 D 8 d 6 b c P o A 1 T 8 t o O E m g L 0 p O s y y i x j o r K F d j O W r H A y T C q 7 2 I T 1 X e i c j W o B e P B v 0 G Y g y r K 5 z K q F I 5 b H U V T A d J S T o x p h k F i 2 x n R l l E O w 1 I r N Z A Q 2 i c 9 a D o o i Q D T z s a / D / G W Y z q 4 q 7 R b a f G Y / Z r I i D B m I G L n F M T e m 5 / a i P x L a 6 a 2 e 9 T O m E x S C 5 J + H u q m H F u F R 0 X g D t N A L R 8 4 Q K h m 7 l d M 7 4 k m 1 L q 6 v l 3 x G 8 Z R P p E d D Y 9 c y Z 5 / p l U S q y f / g h n q i 0 F u N M P / x d w h 4 T I l 1 3 j 4 s 9 / f 4 G a v F t Z r B 1 f 7 1 Z O j v P s i W k e b a B u F 6 B C d o H N 0 i R q I o k f 0 j F 7 Q a + H d W / J W v b V P q 1 f I M y v o 2 3 g b H / L T u Y 8 = < / l a t e x i t > N (p 2 , s 2 ) < l a t e x i t s h a 1 _ b a s e 6 4 = " I x T W Y 4 O j k 4 a V e W W n H L n z 3 l h u n a M o n 9 0 r q r H L k 8 y o R / + L 2 U P c Z g q 2 c f 9 n v 7 9 B s 1 r x j y o n t 8 f l e i 3 r P g + 2 w S 7 Y B z 4 4 B X V w C W 5 A A 2 A w B C / g F b z l P p w 1 Z 9 P Z + r I 6 u S y z A a b G 2 f k E w N O 5 d g = = < / l a t e x i t > N (p 4 , s 4 ) < l a t e x i t s h a 1 _ b a s e 6 4 = " n M J b x D t l l Y N U v + b m G 0 P R y 8 s Z B + w

1 <
o n 9 0 r q r H L k 8 y o R / + L 2 U P c Z g q 2 c f 9 n v 7 9 B 8 6 j i H 1 d O b q v l e i 3 r P g + 2 w S 7 Y B z 4 4 B X V w C W 5 A A 2 A w B C / g F b z l P p w 1 Z 9 P Z + r I 6 u S y z A a b G 2 f k E y K W 5 e g = = < / l a t e x i t > p l a t e x i t s h a 1 _ b a s e 6 4 = " e v i / / S Y 0 x h / M m S 9 5 t F V m b m f H H w

2 <
6 d k m E g 3 9 w H q r H L x 4 l R T / 4 X s 4 u 4 z W R t 4 / 5 y v 6 u g f l b 2 z 8 u X j x e l y m n S f Q Y c g i N w A n x w B S r g H l R B D W D A w T v 4 A J + p b y f t 5 J z 8 3 O q k k s w + W B j n 4 A c r I b b 5 < / l a t e x i t > p l a t e x i t s h a 1 _ b a s e 6 4 = " P b / 1 9 r p 3 i p 8 B S M 2 O C a 3 C v L r l S P E = " > A A A C X X i c l V B N S w J B G B 6 3 L D M z r U O H L k M S d A h 3 1 4 o 6 h V C H L o F B f o S K z I 6 j D s 7 H M j N b y e K v 6 F o / r F N / p V H 3 U N q l F w Y e n g / e d 5 4 g Z F Q b z / t M O W v r 6 Y 3 N z F Z 2 O 7 e T 3 y 0 U 9 x p a R g q T

3 <
g 0 2 4 k 0 C R E e o y F p W y g Q J 7 o b z y + e w m P L 9 O F A K v u E g X P 2 Z y J G X O s J D 6 y T I z P S y 9 q M / E t r R 2 Z w 1 Y 2 p C C N D B F 4 s G k Q M G g l n 3 4 d 9 q g g 2 b G I B w o r a W y E e I Y W w s S X 9 2 u L W t a V c J P q K v D A p h u 6 t k m E g X 9 1 7 q r H L J 4 l R T / 8 X s 4 u 4 z W R t 4 / 5 y v 6 u g U S n 7 Z + W L h / N S 9 T T p P g M O w R E 4 A T 6 4 B F V w B 2 q g D j D g 4 A 2 8 g 4 / U l 5 N 2 c k 5 + Y X V S S W Y f / B r n 4 B s t E r b 6 < / l a t e x i t > p l a t e x i t s h a 1 _ b a s e 6 4 = " c O F

4 <p 1 p 3 <
g 0 2 4 k 0 C R E e o y F p W y g Q J 7 o b z y + e w m P L 9 O F A K v u E g X P 2 Z y J G X O s J D 6 y T I z P S y 9 q M / E t r R 2 Z w 1 Y 2 p C C N D B F 4 s G k Q M G g l n 3 4 d 9 q g g 2 b G I B w o r a W y E e I Y W w s S X 9 2 u L W t a V c J P q K v D A p h u 6 t k m E g X 9 1 7 q r H L J 4 l R T / 8 X s 4 u 4 z W R t 4 / 5 y v 6 u g c V b 2 K + W L h / N S 9 T T p P g M O w R E 4 A T 6 4 B F V w B 2 q g D j D g 4 A 2 8 g 4 / U l 5 N 2 c k 5 + Y X V S S W Y f / B r n 4 B s v A 7 b 7 < / l a t e x i t > p l a t e x i t s h a 1 _ b a s e 6 4 = " m M j a H n Y I r U m U S U O E b v L 8 B r / 7 j 2 M = " > A A A C X X i c l V B N S w J B G B 6 3 L D M z r U O H L k M S d A h 3 t 4 w 6 h V C H L o F B f o S K z I 6 j D s 7 H M j N b y e K v 6 F o / r F N / p V H 3 U N q l F w Y e n g / e d 5 4 g Z F Q b z / t M O W v r 6 Y 3 N z F Z 2 O 7 e T 3 y 0 U 9 x p a R g q TO p Z M q l a A N G F U k L q h h p F W q A j i A S P N Y H w z 0 5 v P R G k q x a O Z h K T L 0 V D Q A c X I W O o p 7 g Q c h t N e p V c o e W V v P n A V + A k o g W R q v W L q u t O X O O J E G M y Q 1 m 3 f C 0 0 3 R s p Q z Mg 0 2 4 k 0 C R E e o y F p W y g Q J 7 o b z y + e w m P L 9 O F A K v u E g X P 2 Z y J G X O s J D 6 y T I z P S y 9 q M / E t r R 2 Z w 1 Y 2 p C C N D B F 4 s G k Q M G g l n 3 4 d 9 q g g 2 b G I B w o r a W y E e I Y W w s S X 9 2 u L W t a V c J P q K v D A p h u 6 t k m E g X 9 1 7 q r H L J 4 l R T / 8 X s 4 u 4 z W R t 4 / 5 y v 6 u g c V b 2 z 8 s X D 5 V S 9 T T p P g M O w R E 4 A T 6 4 B F V w B 2 q g D j D g 4 A 2 8 g 4 / U l 5 N 2 c k 5 + Y X V S S W Y f / B r n 4 B s w 9 L b 8 < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " 4 m v u + Z E X J 6 + E c Q 2 2 / W j h w 2 P A + e w = " > A A A C a 3 i c l V D L S s N

p 2 p 4 <
a U i b T a T t 0 H m F m o p b Q j 3 G r X + R H + A 9 O 2 w i + N h 6 4 c D j 3 H s 7 l R D G j 2 v j + m + P O z M 7 N L + Q W 8 0 v L K 6 t r h e L 6 r Z a J w i T E k k l 1 H y F N G B U k N N Q w c h 8 r g n j E y F 0 0 O B / v 7 x 6 I 0 l S K G z O M S Y u j n q B d i p G x U r u w k TY j D u N R O 4 A H 8 J N X 2 4 W y X / E n g L 9 J k J E y y F B v F 5 2 z Z k f i h B N h M E N a N w I / N q 0 U K U M x I 6 N 8 M 9 E k R n i A e q R h q U C c 6 F Y 6 + X 8 E d 6 3 S g V 2 p 7 A g D J + p X R 4 q 4 1 k M e 2 U u O T F / / 3 I 3 F v 3 a N x H R P W y k V c W K I w N O g b s K g k X B c B u x Q R b B h Q 0 s Q V t T + C n E f K Y S Nr e x b i h d q K 3 l I d B R 5 Z F L 0 v A s l 4 0 g + e V d U Y 4 8 P s 0 M 9 + p / N B n H r y d v G g 5 / 9 / i a 3 h 5 W g W j m + P i r X 9 r P u c 2 A L 7 I A 9 E I A T U A O X o A 5 C g E E K n s E L e H X e 3 Z K 7 6 W 5 P T 1 0 n 8 5 T A N 7 i 7 H 9 h E u 1 Q = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " b / w r e w Z b 7 d W T w y x e A G e J G I q FO X c = " > A A A C a 3 i c l V D L S s N A F J 3 E V 6 2 v 1 u J G X Q w W w Y U 2 i Y i 6 k o I u 3 A g V r A p t K J P p t B 0 6 j z A z U U P o x 7 j V L / I j / A e n b Q R f G w 9 c O J x 7 D + d y o p h R b X z / z X F n Z u f m F w q L x a X l l d W 1 U n n 9 V s t E Y d L E k k l 1 H y F N G B W k a a h h 5 D 5 W B P G I k b t o e D 7 e 3 z 0 Q p a k U N y a N S c h R X 9 A e x c h Y q V P a y N o R h / G o c w g P 4 C c / 6 p S q f s 2 f A P 4 m Q U 6 q I E e j U 3 b O 2 l 2 J E 0 6 E w Q x p 3 Q r 8 2 I Q Z U o Z i R k b F d q J J j P A Q 9 U n L U o E 4 0 W E 2 + X 8 E d 6 3 S h T 2 p 7 A g D J + p X R 4 a 4 1 i m P 7 C V H Z q B / 7 s b i X 7 t W Y n q n Y U Z F n B g i 8 D S o l z B o J B y X A b t U E W x Y a g n C i t p f I R 4 g h b C x l X 1 L 8 Zr a S h 4 S X U U e m R R 9 7 0 L J O J J P 3 h X V 2 O N p f q h H / 7 P Z I G 4 9 R d t 4 8 L P f 3 + T 2 s B Y c 1 / z r o 2 p 9 P + + + A L b A D t g D A T g B d X A J G q A J M M j A M 3 g B r 8 6 7 W 3 E 3 3 e 3 p q e v k n g r 4 B n f 3 A 9 u U u 1 Q = < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " e b e + v b V P P p M E X c c 8 R o T + w m Z J + A I = " >A A A C X n i c l V B N S w J B G B 6 3 L 7 N M r U v Q Z U g C u 7 i 7 E e U p h D p 0 C Q z y g 1 R k d h x 1 c D 6 W m d l K F v 9 F 1 / p f 3 f o p j b q H 1 C 6 9 M P D w f P C + 8 w Q h o 9 p 4 3 l f K 2 d j c 2 t 5 J 7 2 b 2 9 r M H u X z h s K F l p D C p Y 8 m k a g V I E 0 Y F q R t q G G m F i i A e M N I M x r c z v f l C l K Z S P J l J S L o c D Q U d U I y M p Z 5 R K e 4E H I 6 n 5 7 1 8 0 S t 7 8 4 H r w E 9 A E S R T 6 x V S N 5 2 + x B E n w m C G t G 7 7 X m i 6 M V K G Y k a m m U 6 k S Y j w G A 1 J 2 0 K B O N H d e H 7 y F J 5 Z p g 8 H U t k n D J y z v x M x 4 l p P e G C d H J m R X t V m 5 F 9 a O z K D S j e m I o w M E X i x a B A x a C S c / R / 2 q S L Y s I k F C C t q b 4 V 4 h B T C x r a 0 t M W t a 0 u 5 S P QV e W V S D N 0 7 J c N A v r k P V G O X T x K j n v 4 v Z h d x m 8 n Y x v 3 V f t d B 4 6 L s X 5 W 9 x 8 t i t Z J 0 n w Y n 4 B S U g A + u Q R X c g x q o Aw w E e A c f 4 D P 1 7 W w 7 W S e 3 s D q p J H M E l s Y 5 / g G N W r c q < / l a t e x i t > a(k) < l a t e x i t s h a 1 _ b a s e 6 4 = " C Y 0 u o 9 y 5 U z M n C C q B j t q t q t k L 6 D g = " > A A A C W 3 i c l V D L S g M x F M 1 M f d S x a q u 4 c h M s g g v p z I i o K y n o w o 1 Q w T 6 g L S W T p m 1 o H Figure S8.The Feynman graph for axion production via the channel.
dE i (E i ) 1 − f e (E f ) p i p f k ω m N E N (p 4 ) |M| 2 .
+ m 2 e 2c if p i p f − 2c ia c f a p i p f − p 2 i s 2 ia − p 2 f s 2 f a + E i E f −2c if p i p f + 2c ia c f a p i p f + p 2 e (E i ) 1 − f e (E f ) S24)Then the integral over E i sets E i ≈ E F and E f ≈ E F − ω and gives dE i ≈ ω.This lets us writedε a = n N Z 2 α 2 EM α aγγ 2π 2