Stellar cooling bounds on new light particles: plasma mixing effects

Strong constraints on the coupling of new light particles to the Standard Model (SM) arise from their production in the hot cores of stars, and the effects of this on stellar cooling. For new light particles which have an effective in-medium mixing with the photon, plasma effects can result in parametrically different production rates to those obtained from a naive calculation. Taking these previously-neglected contributions into account, we make updated estimates for the stellar cooling bounds on light scalars and vectors with a variety of SM couplings. In particular, we improve the bounds on light (m<~ keV) scalars coupling to electrons or nucleons by up to 3 orders of magnitude in the coupling squared, significantly revise the supernova cooling bounds on dark photon couplings, and qualitatively change the mass dependence of stellar bounds on new vectors. Scalars with mass<~ 2 keV that couple through the Higgs portal are constrained to mixing angle<~ 3*10^-10, which gives the dominant bound for scalar masses above ~ 0.2 eV.


Introduction
Many Beyond-Standard-Model (BSM) physics scenarios include new light, weakly-coupled particles. If a new particle has mass 100 MeV, then it will be produced in the hot cores of stars/supernovae, and will contribute to energy transport in the star. The non-observation of anomalous energy transport, in various different types of star, can then place strong constraints on the coupling of the particle to the Standard Model (SM) particles that make scenarios in which our calculations may be useful, among them early-universe cosmology.
The main aim of this paper is to illustrate how plasma mixing effects may have important consequences for the thermal production of new light particles, not to calculate precise and robust constraints on such particles. Consequently, we do not do the detailed stellar modelling that would be required to derive properly reliable bounds, instead taking representative stellar models and approximate analytic versions of observational bounds from the literature. However, this suffices to demonstrate the parametric changes that our new physical effects bring about, and shows that they would have important effects in a detailed analysis.
The structure of this paper is as follows: In Section 2 we introduce the machinery of thermal field theory, and obtain the proper expressions for production rates of new light BSM states. Following this, in Section 3 we apply these results to stellar cooling, and update the constraints on BSM couplings from observations. Finally, in Section 4, we discuss our results, and review directions for future work.

Thermal production rates and plasma mixing
The conceptually-simplest way to handle medium effects in a thermalised plasma, such as that found in a stellar core, is to use the apparatus of thermal field theory (TFT) -for a comprehensive overview, see e.g. [10]. In this framework, weakly-coupled excitations in a medium correspond to poles in the thermal propagators of fields, with the decay and production rates of these excitations being related to the imaginary parts of the pole locations. If we have a state X that couples weakly to the species making up a thermalised bath, then writing the thermal propagator in terms of the free propagator and the 'thermal selfenergy', D X = 1 D F X −Π X , we are interested in the imaginary part of Π X (see Appendix A.1 for a short review of propagators in the real-time thermal field theory formalism). This paper's central point is that, if X mixes with the bath fields, either through terms in the Lagrangian, or through medium-induced effects, then Π X is not just given by the sum of one-particle-irreducible (1PI) TFT diagrams. Instead, diagrams with intermediate bath species propagators must also be included. This is illustrated, in the case where X mixes with a single bath species A, by Figure 1. At the level of a leading-order 'kinetic theory' calculation, the extra terms correspond to including X production through its mediuminduced mixing with A. This point may seem trivial, but we will see that many calculations in the literature correspond to ignoring mixing effects, which can lead to parametrically incorrect results.
To see the effects of mixing more explicitly, we can consider the species-non-diagonal terms. Supposing again that X mixes with a single bath species A (in the models we consider, A will generally be the SM photon), the in-medium propagation eigenstates will be given by null eigenvectors of the 2 × 2 matrix where K = (ω, k) is the 4-momentum, and we have taken m A = 0. Here, the self-energies Π AX etc. do correspond to the sum of 1PI diagrams (evaluated at real energy-momentum).
If the couplings of X to bath states are O(g), where g is small, then since Π AX = O(g) and Π XX = O(g 2 ), for a given k there exist null eigenvectors when where ω c ≡ ω + iω i is the complex frequency. The second expression corresponds to the weakly-coupled state, and we can see how the self-energy corresponds to the sum of the terms from Figure 1. These expressions make sense if g is small compared to all other parameters. In particular, if Π AA −m 2 becomes very small ('on-resonance'), they may cease to apply, as discussed in Section 2.2. The canonically-normalised propagation eigenstates are, in the (A, X) basis, where is the wavefunction renormalisation factor for A, with the derivative being evaluated onmass-shell. The renormalisation factor for the mostly-X eigenstate is 1 + O(g 2 ).
The mostly-X eigenstate has at least g-suppressed interactions with the thermal bath. So, if the bath is of finite size (such as a star) and g is small enough, it will free-stream out of the bath, rather than coming to thermal equilibrium. In this case, we are interested in the production rate of X by the bath. If the order-g couplings of X to the bath involve production/absorption of one X, rather than scattering of X, then the production rate can be related, via detailed balance, to the damping rate given by the imaginary part of the frequency. Since the production and absorption rates satisfy Γ prod = e −ω/T Γ abs , and the overall damping rate is Γ = Γ abs −Γ prod (for a bosonic excitation), we have Γ prod = f B (ω)Γ, where f B (E) ≡ (e E/T − 1) −1 is the bosonic thermal occupation number. The production rate per volume for the (assumed bosonic) mostly-X state is therefore given by The essential point of this paper is that, for production of new weakly-interacting particles from a SM bath, many previous analyses correspond to only considering the Π XX term, whereas in fact the other term can sometimes be very important. 1 One straightforward  Figure 1. Illustration of the 1PI contributions to the thermal field theory self-energy of a particle X, versus the 'mixing' contributions involving intermediate bath species propagators, where we take X to mix with a single bath species A. The double-lined A propagator indicates the full (resummed) TFT propagator. The third row shows some of the tree-level contributions to the imaginary part of the self-energy, with the 'mixing' diagrams corresponding to X production involving its mediuminduced mixing with A (indicated by the star). Many calculations of thermal particle production in the literature only consider the class of diagrams show in the 1PI column, ignoring the 'mixing' contributions.
example where this cannot be ignored is the case of low-mass dark photon production, as has been noted in a number of papers [6][7][8]. We will see that this also occurs in other cases of phenomenological interest.

SM plasma oscillations
Suppose that there exists an as-yet-undiscovered light bosonic state, which couples weakly to the SM (a new fermionic state, if it is not detectably SM-charged, can only mix with the SM neutrinos, and such a mixing would not be important in most scenarios we consider). If this state does not carry SM quantum numbers, then, in the TFT treatment of a lowenergy (T Λ QCD ) SM plasma, it can only mix with the photon field. In this section, we review the photon field's TFT behaviour (more comprehensive reviews can be found in [1,11]).
We can write the in-medium photon self-energy, in an isotropic medium, as where ± are the usual orthogonal polarisation vectors [1], and L µ = 1 √ K 2 (−|k|, ωˆ k) is the longitudinal polarisation vector (working in the rest frame of the medium). We will write Π = Π r + iΠ i for the real and imaginary parts of the self-energy. For a non-relativistic, dilute plasma, the real parts are [1] Π T,r = ω 2 p 1 + (2.8) to first order in the small average electron velocities, where with T the temperature of the medium and n e the electron number density. Figure 2 plots the corresponding dispersion relations. The essential physics is that the usual transverse modes are simply modified by the presence of a small effective plasma mass, whereas the longitudinal modes oscillate at an almost-fixed frequency set by the same plasma mass, propagating only through thermal diffusion effects. This unusual dispersion relation for the longitudinal modes has the very important consequence that, for a light new particle of mass m < ω p , there is always a k where the dispersion relations of the new particle and the longitudinal photon cross. This means that the denominator in equation 2.6 can become small, allowing resonant production of the new state, as pointed out in the case of a dark photon by [12]. Resonant production from mixing with transverse photons is also possible, but only for m in a narrow range around ω p .
The situation changes somewhat for relativistic or degenerate plasmas. As we increase the typical electron velocity, the propagation speed of the longitudinal mode increases, moving the point where the longitudinal dispersion relation crosses the lightcone to higher k. In the ultra-relativistic limit, both dispersion relations are always above the lightcone. However, the ratio of the cross-over point to the plasma frequency increases only logarithmically with the typical electron energy, so even for the highly relativistic plasma in a supernova core, cross-over still occurs at ω only a few times the temperature. Thus, there is still the possibility of resonant production for very light, weakly-coupled states through mixing with the longitudinal mode. Details are given in Appendix A.3.
One point to note is, well below the light-cone, the photon self-energy will have more complicated behaviour, and the longitudinal propagator will generally have extra poles corresponding to ion acoustic oscillations etc. (c.f. [13]). However, we are interested in the emission of weakly-coupled massive particles, so in the behaviour above the light-cone -in particular, resonant production can only occur for the longitudinal and transverse photon oscillations.
For notational convenience, we define m T (K) 2 ≡ Π T,r (K), (K 2 /ω 2 )ω L (K) 2 ≡ Π L,r ; the longitudinal definition factors out the K 2 /ω 2 dependence that appears automatically from the polarisation contractions. From the previous section, the width for transverse modes is Γ T (K) −Π T,i /ω in the dilute non-relativistic limit (in fact, as per [11], this is always true to good accuracy), and for longitudinal modes we write −Π L,i ≡ (K 2 /ω 2 )ωσ L (K), where this equation defines σ L (K). In the non-relativistic case,

Dark photons
Dark photons provide the simplest example of a new state mixing with SM photons. The physics in this case has been described in previous literature [6,7], but we go over it here to show the similarity with other cases to follow, and since we will extend the dark photon constraints to include SN cooling and stellar trapping.
Suppose we have a new vector A , the 'dark photon', which couples to the SM EM current J, Note that, after a field redefinition, this is equivalent to kinetic mixing L ⊃ − 1 We will assume that there are no additional BSM states at low energies, so that the mass m is a Stueckelberg mass (this is natural even with very small m, since J is a conserved current), rather than coming from a low-scale Higgs mechanism. (2.11) In [6,7], this result was obtained by going to the active/sterile basis, in which the sterile state couples to the SM only via its mass mixing with the active state. As per Section 2, we then have no Π XX 1PI contribution, but just the (Π AX ) 2 /(Π AA − m 2 ) term. The end result is, of course, independent of the basis chosen, as illustrated by our choice of the (vacuum) mass basis. If the SM photon oscillations are weakly damped, Π i Π r (as is generally the case for the environments of interest here), we can split the production rate into four different contributions, with parametrically different properties: • Continuum transverse production: the production of transverse weakly-coupled states has rate (2.12) For |m 2 T − m 2 | ωΓ T , i.e. off-resonance, this is For m m T , this is simply the 'naive' production rate 2 Γ T,prod , while for m m T , it is suppressed by m 4 /m 4 T . In an active/sterile basis, this suppression comes from the propagator for the heavy active state [6].
If we consider production at a given frequency ω, then for a spatially-varying medium, there is some region for which the resonance condition holds. The size of this region will be set by ∼ ωΓ T /(∂ x m 2 T ), so the overall power emitted will be independent of Γ T , in the small-width approximation. If we assume a spherically-symmetric medium, with m T a function of ω, k, r, we have [12] where the integral is over ω such that we match the resonance condition at some r.
• Continuum longitudinal production: the production of longitudinal weakly-coupled states has rate (2.17) Off-resonance, this is As expected, the production rate is always suppressed by m 2 (since σ L has already had the leading m 2 dependence factored out). Since the EM current is conserved, longitudinal emission is always suppressed by m 2 , in vacuum and in medium [6].
• Resonant longitudinal production: for |ω 2 − ω 2 L | < ωσ L , 3 As noted in Section 2.1, this differs from the resonant transverse case in that, if m < ω L , there is always a k such that the resonance condition holds. Therefore, we can find the emissivity at a given position, at leading order in the small-σ L approximation, via As expected, at large masses m m T , ω L we are in the kinetic theory regime for longitudinal and transverse emission. At masses small compared to the plasma frequencies, longitudinal production is dominant, being suppressed by m 2 . Since, as discussed in Appendix A.3, the effects of increasing temperature and chemical potential once T, µ m e are almost degenerate, the right-hand plot is qualitatively similar to the shape of the dispersion relations in e.g. a SN core.

B − L vector
As well as the electromagnetic current, there is another non-anomalous (assuming Dirac neutrinos) conserved current in the SM corresponding to B − L. A light vector coupling weakly to this current, where J B is the SM B − L current, is a popular candidate for new vector portal physics, e.g. [14]. Since B − L does not have any mixed anomalies with the SM gauge groups, there are no order-g 2 production processes that are enhanced by powers of E 2 /m 2 , where E is the energy of the process. This means that a B − L vector avoids the tight constraints, coming from high-energy experiments, that apply to vectors coupling to non-conserved currents. 4 In this paper, we will suppose that the only light fermions coupling to X are those of the SM (the phenomenology we study is not significantly affected by whether neutrinos are Dirac or Majorana), and that X has a Stueckelberg mass. In a medium consisting entirely of protons and electrons, the tree-level behaviour of a B − L vector is exactly the same as for a dark photon. We therefore expect differences in thermal production of a light B − L vector vs a dark photon to be driven by the neutron content of the medium.
For a dilute, non-relativistic plasma, the dominant contribution to the real parts of the X and A self-energies is from the electrons, and the imaginary parts are also dominated by emission from electrons (either electron-ion bremsstrahlung, or Compton scattering). So, we can write where α and β are suppressed by some function of the small parameter m e /m i . Then, where Π ≡ Π AA is the photon self-energy, and we have ignored higher order terms in α, β.
As expected, the only part that is not α, β suppressed is the dark photon rate m 4 Π i , which will dominate at high masses, when the m 4 factor does not suppress it. In particular, we have the same resonant production contributions as for a dark photon, which were not visible in the 'kinetic theory' rate corresponding to Π XX The part of the production that is unsuppressed as m → 0 depends on α i − 2β i + O(α 2 , β 2 ) (for transverse emission; longitudinal emission is always suppressed by m 2 ). The α i and β i contribution from electron-ion bremsstrahlung are O(m e /m n ) (arising from interference between emission from electrons and from ions), but these cancel at leading order, . This becomes clear if we go to a basisÃ = A + (g/e)X, X = X − (g/e)A, whereX only couples to neutrons -then, andX production in electron-ion bremsstrahlung is suppressed by m 2 e /m 2 n , sinceX can only be emitted from the ion leg.
Compton production has no interference between emission from electrons and ions, and has α i , β i = O(m 2 e /m 2 n ) at leading order. The lowest-order in m e /m n contribution to m-independent production therefore comes from bremsstrahlung between different ion species, which is suppressed by (m e /m n ) 3/2 . At masses well below the plasma frequencies, this contribution will dominate the production, while at masses comparable or above, the dark-photon-like contribution is largest. 5 The production of B − L bosons in relativistic and/or degenerate plasmas is more complicated, and we leave it to future work. 5 As discussed in Appendix A.2, for sufficiently low-temperature plasmas, there is the additional complication that ion-ion bremsstrahlung is further suppressed by the ratio vi/α, where vi ∼ T /mi is the typical velocity of an ion, if this ratio is 1. This occurs since the Coulomb interaction between ions is repulsive, and for low velocities, prevents them from getting within a de Broglie wavelength of each other, so limiting the accelerations they feel in the collision. For example, in the solar core, the velocity of protons is ∼ 10 −3 α. In this case, the ion-ion bremsstrahlung contribution is suppressed by √ T m 3/2 e /(m 2 n α). Since me/mn ∼ 0.02, there is the potential that m 2 e /m 2 n contributions may be numerically important in such circumstances. For sufficiently dense plasmas, plasma screening will also be important, since the relevant ion separations during collisions is large due to Coulomb repulsion.

φf f scalar
A scalar with renormalisable couplings to SM fermions, in the low-energy theory, will also couple coherently to SM plasma oscillations. This action can arise, for example, from mixing with the SM Higgs. The low-energy SM does not contain any scalar states, so vacuum φ-SM mixing is not possible. However, in a plasma, φ can mix with the in-medium 'longitudinal photon' mode.
For non-relativistic (relative to f ) momentum transfers,f f f γ 0 f , so the contribution to the mixing self-energy from f will be where Q f is the EM charge of f . Thus, on mass-shell, there is a ∼ k/m enhancement over the longitudinal vector self-energy. As we will see below, this will translate into a different mass dependence of the overall production rate. The damping rate for φ is, from equation 2.6, given by Considering a φēe coupling as an example, this gives (2.30) For ω p < T , this has a continuum contribution which is approximately the naive kinetic theory rate. If m < ω p , then there is also a resonant contribution, with emission at ω ω p at a rate dṄ prod dV 1 4π Note that, unlike the case for a vector, this resonant contribution is not suppressed at small m. This resonant contribution may be larger or smaller than the continuum one, depending on the properties of the plasma. If m ω p < T , then using ω 2 p n e e 2 /m e , resonant emission gives while Compton and bremsstrahlung continuum emission give (from Appendix A.2) We see that resonant production occurs at the same order in α as Compton production, but for a dense plasma where n e > T 3 , the resonant rate is larger. Bremsstrahlung production is suppressed by α compared to resonant production, but is enhanced by the inverse electron velocity. Since electron velocities in stellar cores can be almost relativistic, the resonant contribution can dominate, as we will see in Section 3.4. For relativistic / degenerate electrons, the calculations are slightly more complicated (see Appendix A.4 for details). However, the overall picture of resonant production being unsuppressed at small m, and being potentially larger than the continuum contribution, remains.

Coupling to nucleons
In the low-energy SM, we can consider a scalar coupling to nucleons, φnn and φpp. For simplicity, we will take the n and p couplings to be equal here (this is approximately true for e.g. a Higgs portal scalar) -it is simple to extend to unequal couplings.
The real part of the mixing self-energy is, for non-relativistic nucleons, where Ω eB is suppressed by O(m 2 e /m 2 n ) relative to the plasma frequency. For a single species of ion, Ω 2 eB (g/e)(m e /m n )ω 2 p . The damping rate is again given by equation 2.29, which simplifies to At ω ω L , the second term is suppressed by Ω 2 eB /ω 2 , so production is dominated by −Π φφ i , as expected. We also have a resonant contribution to the production rate, given by Both the resonant and non-resonant contributions will, in the case of equal couplings to protons and neutrons, be suppressed by m 2 e /m 2 n relative to the φēe rates. As per the B − L case, different ions have couplings almost proportional to their mass, resulting in a suppression of ion-ion bremsstrahlung, which would otherwise contribute at (m e /m n ) 3/2 . The emissivity relative to Compton and bremsstrahlung is parametrically the same as in the previous section.

Bounds on weakly-coupled bosons
We now evaluate some of the parametrically-new production rates discussed in the previous section for the physical plasmas inside stellar cores, and use these to estimate bounds on the masses and couplings of new bosons. We emphasise that our calculations should not be viewed as precise constraints, which would generally require a detailed numerical study of the observational data and stellar models. In addition, we do not aim to be comprehensive -we choose a number of new physics scenarios to illustrate our points, leaving many possibilities for future work.
Following [1], there are a number of scenarios in which we are particularly sensitive to novel energy losses from stars: • The Sun. The Sun is not an exceptional star, but it is better-measured than others -in particular, it is the only star whose neutrino emission . With modern solar models, novel forms of energy loss can be constrained down to around 10% of the measured solar luminosity [7,15] (global modelling using specific energy loss profiles can improve this limit to the few percent level [16]). This gives a limit on the average energy loss per unit mass of new 0.2 erg g −1 sec −1 . In the solar core, which is roughly half hydrogen and helium by mass, (3.1) • Red Giant (RG) cores just before helium ignition. Red Giant cores before the onset of helium fusion are basically small, hot white dwarfs at the centre of the enormouslylarger stellar atmosphere, whose electrons are mostly degenerate. Since the core is inert (not undergoing fusion), the main energy transfer process is energy loss by neutrino emission, ν ∼ 4 erg g −1 sec −1 . Novel energy loss processes can be constrained down to around this magnitude, new 10 erg g −1 sec −1 , since a more efficient loss process would delay the onset of helium ignition, in disagreement with observations that match stellar models (e.g. [17]). The conditions in the core just before helium ignition are approximately T core ∼ 10 keV , ρ core ∼ 10 6 g cm −3 , µ e ∼ 1.3m e , ω p ∼ 20 keV . (3.2) • Horizontal branch (HB) star cores during helium-burning. During their heliumburning phase, the energy released by fusion puffs up the core of the star, lowering its density and making it non-degenerate. The power from helium fusion is If there are additional energy-loss processes, these cause the core to contract, heating it up, enhancing the rate of the helium fusion, and so shortening the helium-burning lifetime of the star. This lifetime is measured to around the ∼ 10% level, and agrees with standard stellar models, so novel energy losses are constrained to be new 10 erg g −1 sec −1 constraints (e.g. [18]). The core properties are approximately, • Core-collapse supernovae (SN). In the first few seconds of a core-collapse supernova, the core is basically a very hot proto-neutron star, with degenerate electrons and almost-degenerate nucleons. It is sufficiently hot and dense to be optically thick even to neutrinos. The dominant energy loss mechanisms is via the outwards diffusion of neutrinos, which occurs over a ∼ 10 s timescale -the resulting neutrino burst was measured for SN1987A [1], with properties that agreed with SN model predictions. In order for a new energy-loss mechanism not to have disrupted the SN1987A neutrino burst, the averaged energy loss rate would have to have been less than that from the neutrinos, new 10 19 erg g −1 sec −1 [1].
(3.4) As we will see, for particle emission processes that do not depend on high powers of the temperature or density, stellar limits are generally more constraining than SN limits for masses 50 keV. However, SN limits apply up to masses ∼ 100 MeV.
It should be emphasised that there is still significant uncertainty regarding the behaviour of core-collapse supernovae [19], with conflicting models in the literature of even the very basic energy source involved [20]. As a result, SN constraints on new light particles should be viewed as order-of-magnitude estimates at best.
These have the desirable properties of fairly inefficient SM energy-loss mechanisms (either neutrino emission, or surface photon emission), as well as reasonable temperature and high density compared to the aforementioned stellar core situations. However, the emission of new particles in scattering process is generally highly suppressed by Pauli-blocking. As a result, WDs can only impose similar constraints to those from stars in some cases, while NSs do not seem to be competitive [1]. For resonant emission, the situation is ever worse: since these highly degenerate systems have plasma frequencies m T , ω L T , emission at resonant frequencies is heavily Boltzmann-suppressed, so is never significant.

Emission and trapping
The production calculations in Section 2 looked at the production rate per unit volume for a weakly-coupled in-medium mode, assuming a uniform, time-independent thermal medium. For stellar cooling bounds, we are interested in the energy flux that escapes the core -that is, in the energy transport between regions with different medium properties. However, as long as medium properties vary slowly enough, then a weakly-coupled mode emitted from one region will propagate adiabatically into the weakly-coupled mode for the new region (if the weakly-coupled and strongly-coupled modes go through level-crossing during the propagation, there will be resonant conversion, which needs to be taken into account).
Consequently, as long as absorption (and resonant conversion) during the course of the propagation is small, almost all of the energy emitted in weakly-coupled excitations at the centre is transported to weakly-coupled excitations further out. When the whole star is optically thin to weakly-coupled excitations, energy is lost to infinity, and the total energy loss rate is obtained by integrating over the per-volume production rate.
If weakly-coupled excitations are not able to free-stream through the whole star, then they contribute to energy transport within the star, and let the outer layers of the star lose energy to infinity (once the optical depth becomes low). Since our models of stellar structure fit observations fairly well, there is generally no room for a whole additional degree of freedom to contribute to radiative heat transport, unless that degree of freedom is heavy enough that its Boltzmann-suppressed abundance is too small to contribute significantly [1]. We will see in Sections 3.3 and 3.4 how this opens up some extra parameter space at high masses and couplings, for stellar bounds on new particles. As discussed in Section 3.2.1, numerical simulations of supernovae also seem to indicate that energy transport by particles with mean free path significantly larger than that of neutrinos is not compatible with SN1987A observations, allowing similar energy-transport bounds to be placed.

Dark photon production in supernovae
The uncertainty surrounding the behaviour of core-collapse supernovae is an obstacle to using detailed explosion models to constrain new physics. As mentioned in the Introduction, our focus in this paper is on highlighting the physical consequences of plasma effects, rather than attempting comprehensive and robust stellar modelling. Accordingly, we adopt a single representative supernova model (from [21], which is similar to others found in the literature, see e.g. the review [22]), and impose the simple 'Raffelt bound' approximation on the energy loss from new particles. That is, we demand that the average energy loss per unit mass from the core via new particles is 10 19 erg g −1 sec −1 [1] -in a variety of simulations, this is within a factor ∼ 10 of the critical energy loss rate at which the neutrino signal would be detectably modified [23]. Thus, the constraints we derive should not be considered as robust parameter space exclusions. However, as we will see, plasma mixing can lead to orders-of-magnitude differences in the production and absorption rates, compared to a kinetic theory calculation [4,5,9]. Such effects will therefore be very important to include in any eventual robust study, and our calculations indicate how such exclusion bounds will vary with dark photon mass.
In the supernova environment, the real part of the photon self-energy is dominantly generated by electrons 6 , the form of which is discussed in Section 2.1 and Appendix A.3. The imaginary parts of the photon self energy gets contributions from various processes, and the large rate of nucleon-nucleon interactions means that neutron-proton bremsstrahlung is dominant. Following [5], we can approximate the bremsstrahlung production rate by relating it to proton-neutron scattering data, using the soft approximation for photon emission. This is only strictly correct in the limit that the emitted energy is small compared to the kinetic energy of the collision, but from [5], we expect that the overall error from this approximation will be comparable to the other uncertainties in our calculation. Further details are given in Appendix A.6. For some supernova models, the SM photon oscillations may become broad resonances in the supernova core, in which case the split of dark photon production into continuum and resonant contributions loses its applicability.
In Figure 3 we plot the bounds obtained for our SN model after including the mixing effects discussed in Section 2.2 -the bounds that would be obtained neglecting such effects are also shown. 7 At masses below the typical plasma frequency, transverse continuum emission has the expected ∼ 1/m 2 scaling. Meanwhile, longitudinal continuum and resonant emission both scale as ∼ 1/m. At masses around the plasma frequency, resonant transverse emission is possible, and leads to a stronger constraint, while at higher masses continuum emission of longitudinal and transverse modes is close to the kinetic theory prediction.
For dark photon masses below approximately 0.1 MeV, bounds from stellar cooling become dominant, also shown in Figure 3. Production in these environments is less suppressed since the plasma frequency is lower, and they are better modelled and measured than supernovae. For masses below 2m e there are also strong constraints from cosmology -a small dark photon abundance is produced in the early universe, which then decays to three photons, which would be visible in the observed gamma ray background [8] (or, for sufficiently early decays, through effects on the CMB or BBN). Such constraints could be relaxed by allowing the dark photon to decay to light hidden sector states, though the viability of this possibility is model dependent. There are also cosmological constraints for m > 2m e mixings [26,27], but these constrain significantly smaller mixings than the supernova cooling limits.
In addition to affecting the neutrino signal through energy loss from the SN core, new particles that decay to SM could have other experimentally-visible signatures in SN. [9] considers some of these signatures for a dark photon, including gamma radiation from the injection of e + e − pairs when dark photons decay slightly outside the star. However, they use the kinetic theory calculation for dark photon production, so their constraints would need re-calculating with the correct production and absorption rates. We leave this to future work.

Trapping and decay
If is large enough, then for the SN core there is some (frequency-dependent) radius within which dark photon radiation gets reprocessed into nearly black-body radiation. 8 The core will then lose energy, at a given frequency, approximately like a black-body sphere of that radius (with surface temperature the medium temperature at that radius), along with 7 Our results are compatible with [25], which considers a range of SN profiles, finding roughly an order of magnitude variation in the bounds obtained. Our SN profile is close to those denoted "Fischer" in [25], and leads to similar constraints. 8 Along a given 'line of sight' out of the core, the occupation number f of a weakly-coupled state propagating in that direction will evolve as v df /dx = −f Γ abs + (1 + f )Γ prod = Γ prod − f Γ damp where v is the velocity of the weakly-coupled state. So, for intervals over which the 'damping depth' τ d ≡ dx Γ damp /v is large, the occupation number converges to the black-body value fB = Γ prod /Γ damp , while for intervals over which τ d 1, we simply integrate the production rate along the line. For frequencies ω T , we have Γ damp Γ abs , so the damping depth is approximately the optical depth. For ω T , Γ damp (ω/T )Γ abs , so the damping depth is significantly smaller than the optical depth. The limit on that would be obtained from a kinetic theory calculation is shown in dotted black. Assuming purely SM decays of the dark photon, the parameter space above the shaded region is not excluded by the energy loss constraint we impose because of absorption and/or decays to electron-positron pairs (at masses 2 MeV) (although energy transport via the dark photon may mean that the neutrino signal is still observably shortened -see Section 3.2.1). Under the same assumption, beam dump experiments exclude the region shaded in blue [28]. The orange shaded region is excluded by stellar cooling bounds [6,7] (where we have also incorporated the effects of trapping inside the star, as per Section 3.1), while in models without hidden sector decays masses below ∼ MeV will be constrained by cosmological observations (green, [8] -the small gap at large mixing will very probably be closed by beam dump experiments such as [29]). It should be stressed that models of core-collapse supernovae are very uncertain, and limits such as those derived here, which assume a particular SN model [21], should be treated as order-of-magnitude estimates at best. volume emission from the region outside. If these black-body radii are large enough, and correspond to small enough medium temperatures, then the overall energy loss rate via the dark photon will become small compared to the neutrino rate.
We make the assumption that, to significantly affect the neutrino signals, there needs to be energy loss greater than the Raffelt bound from the part of the core at temperature 1.5 MeV [5]. As per Section 3.1, it is likely that provided the dark photon mean free path is greater than of neutrinos in the core, the increased efficiency of energy transport would observably shorten the neutrino signal [21,30,31]. Imposing this condition would allow dark photon coupling roughly an order of magnitude stronger than those constrained by the energy loss bound to be ruled out; however, detailed simulations would be required to obtain quantiative bounds. Since such strong couplings are already ruled out by beam dump experiments, we only show the energy loss bounds in Figure 3.
For the supernova profile we use, the T = 1.5 MeV boundary is at a radius of approximately 14.5 km, by which point the electron chemical potential has dropped significantly, to ∼ MeV. This corresponds to the typical region from which neutrinos can free stream out of the supernova. Note however, it might be that energy has to escape to much larger distances, before the dark photons suppress neutrino productions significantly [25].
Up to such radii, continuum absorption of dark photons is dominated by inverse protonneutron bremsstrahlung, expressions for which are given in Appendix A.6 (at much larger radii, inverse Compton scattering off electrons may become important). The ω −3 dependence of the bremsstrahlung absorption rate means that low-frequency dark photons are trapped more efficiently than higher-frequency ones, and so the frequency dependence of the effective black-body radius is important. For dark photon masses 100 keV, continuum absorption of transverse dark photons is more efficient than longitudinal continuum absorption, because of the small plasma frequency in the outer part of the supernova core. For small enough dark photon masses, transverse absorption will be suppressed by the expected m 4 /ω 4 p factor, so transverse photons will escape most easily, but such small masses are constrained by stellar cooling observations. Longitudinal dark photons can be resonantly absorbed if there is a shell where the supernova conditions are such that ω 2 − ω 2 L < ωσ L is satisfied, and if present this shell is typically optically thick for at the trapping bound. Meanwhile, resonant reabsorption of transverse modes occurs for dark photon masses around the typical plasma frequencies.
If the dark photons decay to electrons and positrons before escaping the neutrino emission region, this will also prevent efficient energy loss (although as for absorption, the neutrino signal may still be observably shortened). In the centre of the supernova, decay to electrons is blocked by the high chemical potential, unless the mass of the dark photon is above twice the effective electron mass, m eff 15 MeV. However, towards the outside, the chemical potential drops sharply, and the effective electron mass is close to its vacuum value. In Appendix A.5 we give explicit formula for the decay rate.
The effect on the allowed values of is plotted in Figure 3. For dark photon masses m 2m e , decays dominate the absorption, although this is sensitive to the details of the supernova profile. If the dark photon can decay to lighter hidden sector states, which interact sufficiently weakly with the supernova medium that they can escape, decays and absorption would be reduced and larger values of constrained. This could also relax limits from beam dump experiments, and, as mentioned, alter cosmological constraints. The interplay of these effects would be interesting to study in the future.

B − L vector bounds
From Section 2.3, B-L production in a dilute, non-relativistic plasma should be wellapproximated by the dark photon production rate, plus a mass-independent production rate that is suppressed by at least (m e /m n ) 3/2 . As discussed in footnote 5, the proton velocity in the Sun is small enough that the extra ∼ v/α suppression of ion-ion bremsstrahlung makes it numerically comparable to the (m e /m n ) 2 contribution from electron-ion bremsstrahlung.  [32][33][34][35][36]), stellar cooling (see text) and cosmological observations (green, [8]). The solar bounds (red) take into account the mass-independent part of the production rate discussed in Section 2.3, shown by the solid line -the dotted line shows the constraints that would arise if the coupling to neutrons were neglected. As explained in the text, the value of the mass-independent contribution to the production shown here is an order-of-magnitude estimate. The HB star bounds (orange) do not take into account the coupling to neutrons, so are shown dotted at low masses, where the neutron contribution will be important. The dashed red line indicates the bounds that would be obtained from a naive 'kinetic theory' calculation, which ignored in-medium mixing effects.
The plasma screening length inside the solar core is also small enough that it may affect the ion-ion bremsstrahlung calculation. We have not treated all of these contributions precisely, so our numerical value for the mass-independent production rate should be treated as an order-of-magnitude estimate. For the Sun, where the core is roughly half hydrogen and half 4 He by mass, we obtain the bounds shown in in Figure 4. There will also be low-mass bounds from HB and RG stars -however, since these are unlikely to rule out significant areas of parameter space not already constrained by force bounds, we restrict ourselves to the solar calculation as an illustration. Figure 4 also shows the trapping bounds for emission from the Sun and from HB stars, as discussed in Section 3.1 -these constraints translate directly to the dark photon case, in the high-mass region.
Comparing to the existing literature, stellar bounds on massive B − L vectors have been considered in [2,37,38]. [2] performs the naive kinetic theory calculation, ignoring plasma mixing effects. [37,38] treat the production of on-shell vectors as being the same as for dark photons, which ignores the mass-independent production rate from non-hydrogen nuclei calculated in Section 2.3. [37,38] also claim that there is a strong bound from the enhanced emission of SM neutrinos. However, the B − L mediated emission of neutrinos has rate ∼ g 4 B−L , which is extremely small, while the rates in the aforementioned papers appear to be ∼ g 2 B−L . Figure 4 also shows the bounds on a light B − L vector from fifth-force tests [32][33][34][35][36], which are more constraining than stellar cooling bounds for masses eV. It also indicates the cosmological bounds arising from late-time three-photon decays of a B − L population, produced in the early-universe plasma. These can be derived directly from the dark photon bounds [8], since the dominant contribution to early-universe production happens around electron-positron freeze-out, when only the electron coupling is important. The cosmological bounds depend on the B − L vector not having hidden sector decays. Another possible probe of B − L vectors or dark photons would be the direct detection of weakly-coupled states emitted from the Sun, in dark matter experiments on Earth [39,40].
As is the case for dark photons, there will also be supernova cooling bounds on B − L vectors of mass a few hundred MeV. Since neutrons play an important role in a supernova core, we expect the constraints to be rather different from those in the dark photon case. We leave an analysis of these to future work.
The importance of plasma mixing effects does not just apply to B − L vectors or dark photons, but to any new vector coupling to the SM. For example, the bounds on a new vector coupling to baryon number derived in [3,5] did not consider mixing effects, so would need to recomputed with those taken into account. Again, we would expect kinetictheory-like emission at energies above the plasma frequencies, some enhancement from resonant emission at masses around the plasma frequencies, and then some suppression due to interference effects at lower masses.

φf f scalar bounds
As discussed in Section 2.4, the production rate for a scalar that couples coherently to SM plasmons has a resonant contribution for m < ω p , and a continuum contribution that is roughly similar to the naive kinetic theory value. The resonant contribution is lower-order in α than the continuum bremsstrahlung rate, and is expected to be larger for plasmas with large electron velocities.

Coupling to electrons
Using the non-relativistic resonant production rate from equation 2.31, and taking a sample HB star model from [41], imposing the rough constraint 10 erg g −1 sec −1 gives for m keV, where α φēe ≡ g 2 φēe /(4π). For comparison, the kinetic theory production rate would give α φēe 1.5 × 10 −29 (in agreement with the estimate from [1]). Since the continuum production is dominated by 4 He bremsstrahlung, we expect the ratio of resonant to transverse power per volume to be given by equation 2.34, Since our HB model has T /10 8 K 0.5 averaged over the core, this gives the correct ratio of ∼ 20 between the continuum and resonant constraints. HB-cont Figure 5. Stellar cooling constraints on a scalar φ with a φēe coupling (left), or a φN N coupling to nucleons (right), where we assume that φ couples equally to protons and nucleons. Both the continuum and resonant limits continue to very small φ masses. Continuum production of φ in HB stars gives approximately the same energy loss as a naive kinetic theory calculation. There will also be continuum production from RG cores, but this is more complicated to calculate due to electron degeneracy, so we leave it to future work (we expect it to be sub-dominant to resonant production at small masses). At high masses and couplings, the contribution of φ to energy transport in the star becomes small enough that it is not constrained by observations (Section 3.1), as shown the in left-hand plot (the trapping constraints occur at larger α than shown in the right-hand plot). In a full model, there will also be other constraints on these parameter spaces, including force bounds and cosmological observations (for example, Figure 6 shows the constraints on a Higgs portal scalar model).
For a RG core, where the electrons are degenerate, we can use the formulae from Appendix A.4 to calculate the resonant production rate. Using a sample RG core model from the MESA package [42], and imposing 10 erg g −1 sec −1 , we obtain for m 10 keV. Calculating the continuum production rate in this case is more difficult, due to the electron degeneracy, but we expect resonant production to dominate at small masses. To summarise, for small scalar masses, we have the constraints  The corresponding mass-dependent constraints are shown in Figure 5.
A calculation similar to that performed in Section 3.2 could be done to find SN bounds on the φēe coupling. We are not aware of any existing calculation in the literature, and we leave such a treatment to future work.

Coupling to nuclei
Performing the analogous calculations for a scalar coupling to nucleons, assuming that we couple equally to protons and neutrons, we obtain  for small scalar masses (note that our continuum production bound is slightly tighter than that given in [1], since only Compton scattering is considered there, and electron-ion bremsstrahlung production is dominant for low-mass scalars). We again obtain a ratio of ∼ 20 between resonant and continuum production for HB stars, and a significantly stronger bound from RGs. Figure 5 shows the mass-dependent bounds.

Higgs portal scalar
The |H| 2 operator is the unique renormalisable portal through which a neutral scalar can be coupled to the SM. In the low-energy SM, its effects can be parameterised by a mixing angle sin θ, with the scalar φ coupling to fermions as (m f /v) sin θφf f , where v 246 GeV is the electroweak vev. This gives a coupling to electrons of g φēe = (m e /v) sin θ = 2 × 10 −6 sin θ, and a coupling to nucleons of g φN N 8 × 10 −4 sin θ [43] (this is approximately the same for neutrons and protons, since most of the coupling does not come from the valence quarks). Since g φēe /g φN N 2 × 10 −3 > m e /m n 5 × 10 −4 -as expected, since most of the nucleon mass comes from QCD, not from the Higgs VEV -the φēe coupling dominates the φ production rate in stellar cores, resulting in the stellar cooling bounds shown in Figure 6. An interesting point is that the enhanced stellar cooling constraints are able to probe potentially natural parameter space for a Higgs portal scalar, as indicated in Figure 6.
Comparing these to other constraints, the stellar cooling bounds are more stringent than fifth-force constraints at masses 0.2 eV. Having a UV complete model also enables us to consistently compare the stellar bounds to cosmological constraints (assuming that the universe was reheated above the electroweak scale, and that there was no cosmologicallyimportant new physics below the EW scale). [44] estimates the early-universe φ production, which is dominated by temperatures around the electroweak scale, and then constrains the later energy injection from φ → γγ decays. This gives the bounds shown in Figure 6, indicating that the stellar bounds are probably the most constraining up to masses ∼ 10 keV. We leave an improved calculation of early-universe φ production, incorporating plasma effects, to future work. It would also be interesting to calculate the minimal bounds, assuming only that the reheating temperature is MeV.

Discussion
In this work, we have demonstrated how, in the medium of a SM thermal bath, plasma effects can introduce an effective in-medium mixing between hypothetical new, weakly-  [32][33][34][35][36] are in blue, stellar cooling bounds (see text) in red and orange, and cosmological constraints in green. The latter arise from a freeze-in relic abundance decaying around the present day, contributing to the observed photon background [44] (we show these bounds as extending up to couplings at which φ would decay at BBN time, though it would require further work to verify whether the constraints are valid there). Since most of the nucleon mass comes from QCD, rather than from the Higgs VEV, the ratio of nucleon to electron φ couplings is smaller than the nucleon-electron mass ratio. The electron coupling therefore dominates the φ-SM mixing, and thus the resonant φ production, as well as dominating the continuum production. The stellar cooling bounds are therefore the appropriate rescaling of the φēe bounds show in the left-hand panel of Figure 5, and likewise continue to very small φ masses. The grey line shows the approximate fine-tuning bound sin θ m φ /v EW [45], demonstrating that our new stellar constraints are able to probe natural parameter space.
coupled bosons and SM photons (if the new bosons couple coherently to SM plasma oscillations). Such mixing affects the production of the new bosons by allowing resonant production from SM photons, and also by allowing cancellation between new boson and photon emission amplitudes. This can make a parametric difference to production rates from plasmas where chemical potentials are important, such as stellar cores. Dark photons provide the simplest example of these effects, which were treated correctly in some, though not all, of the previous literature. In Section 3.2, we estimate the production of dark photons in core-collapse supernovae, previous calculations of which ignored mixing effects [4,5,9], and use this to place new constraints in the mass-mixing parameter space. Note that there are very probably extra constraints one could place from arguments beyond simple energy-loss-from-core [9], but we leave those to future work. We also update the stellar-cooling constraints to include trapping effects, for heavy and more-strongly-coupled dark photons.
Extending these results to other types of new vectors, we illustrate the effects in the case of a weakly-coupled B − L vector. In Section 3.3, we estimate B − L stellar cooling bounds, correcting previous literature [2,37,38]. Weakly-coupled scalars can have an in-medium mixing with longitudinal photon oscillations. Resonant production of such scalars, in particular from the cores of RG stars, places significantly stronger constraints (Section 3.4, Figure 6) than previous calculations which ignored mixing effects. Other BSM particle candidates which could result in similar effects include vectors coupling to other SM current (though the non-conservation of these currents generally leads to strong constraints from high-energy experiments), other forms of scalar coupling, and higher-spin new particles (e.g. KK gravitons, as considered in [23]). We leave the calculation of such constraints to future work.
Since the low-energy SM preserves parity, axion-like particles do not mix with SM plasma excitations, up to effects suppressed by weak-scale masses. However, if there is a parity-violating background, such as a strong magnetic field, then plasma mixing effects may become important. Nevertheless, even for a supernova, which may have magnetic fields of order 10 15 Gauss [46], rough estimates indicate that this effect will be sub-leading. The temperature in a SN core is also hot enough that mixing with pions will have a small effect, but the pion mass is large enough that this will also be sub-dominant to other processes.
Up until the recombination era (i.e. for temperatures eV), the universe was a hot, basically homogeneous plasma, so our calculations of the production rate for light, weakly coupled particles from this plasma will apply. Such production can also, with some assumptions, be used to place constraints on the properties of light particles, and as per stellar cooling calculations, plasma effects may make an important difference. For example, the cosmological bounds on a light B − L vector, as shown in Figure 4, are from the late-time decay of a small relic B − L abundance produced in the early universe, and at the masses indicated, this abundance is dominated by resonant production around electron-positron freeze-out [8]. However, the early universe differs from stellar cores in a number of wayschemical potentials were probably very small, and variation was in time instead of space -so the new particle candidates for which plasma effects make an important difference to observable quantities are not necessarily the same as those for stellar cooling bounds. We leave calculations of plasma effects on such early universe physics to future work. 9 'Medium effects' of the kind discussed in this paper are sometimes important in laboratory searches for BSM particles, particularly at very low masses (e.g. [47,48]). If we are interested in particle masses well above the plasma frequency of bulk materials, ∼ eV, then bulk medium effects will be negligible. For particle production in processes involving nuclei, the effective 'plasma frequency' of the nuclear 'medium', which we expect to be ∼ 10 MeV, may be significant. For example, the beam dump constraints on dark photons shown in Figure 3 come from the production of dark photons in high-energy collisions between electrons/protons and target nuclei. However, the very small size of nuclei introduces important differences with respect to the stellar production case; in particular, nuclei are optically thin to SM photons as well as to dark photons. Thus, the previous arguments about only needing to consider the production of the weakly-coupled in-medium state will not hold. Further work would be required to calculate the nuclear medium effects on particle production in experiments such as beam dumps.
An important point is that the mixing effects giving rise to resonant production etc are better viewed as 'plasma' effects, rather than 'thermal' ones. In particular, although we have used the apparatus of thermal field theory to calculate them, they will also occur out of thermal equilibrium. In Appendix A.7, we show how modified production rates can be derived by considering classical plasma oscillations in the fluid approximation, including the presence of a new weakly-coupled field in addition to electromagnetism. This serves as an independent check of our thermal field theory calculations, as well as indicating how to extend them to other situations.
Future observations of stars, and of the early universe, will improve our sensitivity to anomalous energy losses, and if deviations from Standard Model predictions are found, may provide hints of new, light BSM particles. In either case, correct calculations of BSM production rates are vital in translating such observations into knowledge about new physics.

Note added
In the final stages of preparing this manuscript, the paper [25] which also studies plasma effects on supernova dark photon bounds, was posted on the arXiv.

A Appendices
A.1 Thermal field theory -real time formalism In the real-time formalism of thermal field theory [10], for each physical field φ 1 we introduce a 'ghost' field φ 2 , with the in-medium propagator being a matrix that mixes the 1, 2 fields. This doubling of degrees allows the formalism to keep track of the analytic structure of the contour along which the time integral is evaluated. We can write the full in-medium propagator D ab (with a, b ∈ {1, 2}) as where D F is the free-field propagator in the thermal bath, and Π ab is the self-energy. From [10], the propagator can be diagonalised in the {1, 2} basis by the matrix for a single species of mass M , where Π(K) is the 'real-time' self-energy.
A.2 Damping rates in a dilute non-relativistic plasma For a vector coupling to electrons with coupling g, the damping rate from Thomson scattering is [7], writing α g = g 2 /4π, for ω > ω p , and zero for smaller ω (since ω p is the minimum energy carried by a SM photon in the plasma). The damping rate from electron-ion bremsstrahlung is to lowest order in m e /m i , whereḡ is the thermally-averaged Gaunt factor [49]. In the Born approximation,ḡ = √ 3 π e ω/(2T ) K 0 (ω/(2T )) , (A.6) (this approximation is valid for ω T , butḡ is order-1 throughout). In the simplest case of emitting a massless vector, integrating over ω gives a power per unit volume of The leading-order expressions for σ L are the same. For a vector coupling to nucleons, we also have ion-ion bremsstrahlung. The situation here is slightly more complicated, since the ions interact through a repulsive Coulomb potential, rather than an attractive one as for electron-ion collisions. In the Born approximation, the damping rates are as for the attractive case, if we substitute (X 1 /m 1 −X 2 /m 2 ) 2 for 1/m 2 e , and the reduced mass µ for the rest of the m e , where m 1 and m 2 are the masses of the ions, and X 1 , X 2 their charges under the new vector.
If the ion thermal velocities are low enough -in particular, if Z 1 Z 2 e 2 /v i 1, where v i ∼ T /m i is a typical ion thermal velocity -then the ion wavefunctions are significantly distorted during the collision, and the Born approximation ceases to hold. In this regime, the emission can be treated as approximately classical. [50] gives the classical dipole radiation spectrum from collisions with a given initial approach velocity v 0 ; for high frequencies, this is suppressed by then the above expressions are We see that the dispersion relations both have ω = ω p at k = 0, and that the frequency at which the longitudinal dispersion relation crosses the lightcone goes to infinity as the electron velocity goes to one. However, the crossing frequency increases rather slowly. For example, in a SN core, with T 30 MeV and µ e 350 MeV, we have v * 1−2×10 −6 . On the light-cone, this gives 1 + H(v 2 s ) 18.7, and since ω p 20 MeV, the crossing frequency is (1 + H(v 2 * ))ω p 85 MeV. In general, To evaluate resonant longitudinal production rates (e.g. 2.21), we need the expression Since δ > m 2 /ω 2 , this factor is always O(1). (A.29)

A.5 Decay to electrons and positrons
In Appendix A.3, we noted that for a photon excitation of small invariant mass, the inmedium mass of the electron prevents the decay γ → e + e − . However, if the new weakly coupled state has mass m > 2m e , wherem e is the effective electron mass, the weaklycoupled state can decay into an electron and a positron, and we need the corresponding imaginary parts of Π XX , Π XA and Π AA to evaluate the absorption rate from this decay. The simplest case, and the only one which is relevant to the calculations in this paper, is that of a heavy dark photon, where we only need to calculate Π AA i at large invariant mass. From [51], the effective electron mass (at k = 0) is For example, in a SN core with T = 30 MeV, µ = 350 MeV, we havem e 12 MeV. For an ultra-relativistic electron, the dispersion relation has ω 2 − k 2 ranging from m 2 eff at k = 0 to 2m 2 eff as k → ∞. The decay rate for longitudinal excitations is (ignoring the longitudinal residue factor, and in the rest frame of the plasma) where f − (E) = 1/(e (E−µ)/T + 1) is the electron occupation number, giving Pauli blocking, g(E) is an order-1 function summarising the in-medium residue factor for the electron/positron, and E ± are the maximum and minimum decay energies in the plasma rest frame. Ignoring the frequency variation in the electron effective mass, E ± = (1/2) (ω ± β cm k), where β cm ≡ 1 − 4m 2 e /m 2 . For transverse excitations (again ignoring the transverse residue factor), which is the correct decay rate in vacuum.

A.6 Dark photon production and absorption in supernovae
Here we give cross sections for dark photon production and absorption in the soft approximation, following [5] but adapting to include thermal mixing effects (which requires separating the longitudinal and transverse cases). Assuming the energy of an emitted photon is small compared to the energy of the proton-neutron interaction, the cross section for the emission of a photon can be related to that for elastic neutron-proton scattering by where J is the dipole current J = P 1 P 1 .K − P 2 P 2 .K , (A. 37) with K the four-momentum of the emitted photon, and P 1 and P 2 the initial and final four-momentum of the proton. As well as a dipole contribution from neutron-proton interactions there will also be quadrupole pieces from proton-proton interactions, however these are smaller.
For the longitudinal polarisation we have where the notation is the same as in Section 2.1, with ω L given by equation (A.21). As discussed in Section 2.2, for resonant emission the value of the imaginary part of the photon self energy cancels out when computing the overall energy emitted. The absorption cross sections are obtained analogously, and for the dark photon the inverse mean free path is is a thermally averaged cross section, and numerical values as a function of temperature are again given in [5]. 10

A.7 Classical equations of motion
To illustrate how our 'mixing effects' arise outside the thermal field theory formalism, we can look at the classical EoM for plasma oscillations in the fluid approximation, with the presence of an additional weakly-coupled field (as per e.g. [47]). For example, considering the case of longitudinal oscillations in the presence of a φēe scalar, the electric field E and scalar field φ are sourced by (assuming that ions have Z = 1) where we have included a 'frictional drag' term from electron-ion collisions, parameterised by an 'effective collision frequency' ν (we have ignored magnetic fields, since the velocities are non-relativistic).
To find the equation of motion for small oscillations, we linearise about the equilibrium values of quantities, writing n e = n 0 (1 + δ e ), n i = n 0 (1 + δ i ), φ = φ 0 + δφ, and and v e , v i , E as small quantities themselves. Taking the limit where the ion mass is much heavier than the electron mass, we can set v i , δ i = 0 throughout. Writing the linearised quantities as e −i(ωt−kx) (i.e. taking propagation to be in the x direction), and solving for v e , δ e in terms of E, δφ, we obtain the equations For g = 0, we obtain the separate equations (A.52) 10 Our expressions match those of [25] and differ from those of [5] by numerical factors.
Analogous calculations can be done for the new particle candidates discussed in this work, replicating the results of the thermal field theory calculations as required.