New constraint from supernova explosions on light particles beyond the Standard Model

We propose a new constraint on light (sub-GeV) particles beyond the Standard Model that can be produced inside the proto-neutron star core resulting from the core-collapse supernova explosion. It is derived by demanding that the energy carried by exotic particles being transferred to the progenitor stellar envelopes must not exceed the explosion energy of $\lesssim 2\cdot 10^{51}$~erg of observed supernovae. We show specifically that for the case of dark photon which kinetically mixes with the SM photon and decays predominantly to an $e^\pm$ pair, smaller mixing parameter of one order of magnitude below the well-established supernova cooling bound can be excluded. Furthermore, our bound fills the gap between the cooling bound and the region constrained by (non-)observation of $\gamma$-rays produced from supernovae for dark photon lighter than $\sim 20$~MeV.


I. INTRODUCTION
The Standard Model (SM) of particle physics has been the most successful theory that describes the fundamental properties and interactions between elementary particles. However, various hints from either the theoretical considerations or the cosmological and astrophysical observations point to the possibility that it is not a complete theory and new particles beyond the SM (bSM) that only couple to the SM sector very weakly may exist.
Among the imperative searches and constraints of bSM particles, one important criterion comes from the observation of electron antineutrinos (ν e ) associated with the seminal corecollapse supernova (CCSN) event, SN1987A. The observedν e burst duration of about 12 s, with individual energies up to 40 MeV, as well as the integrated total energy ∼ 5·10 52 erg [1][2][3][4][5][6][7][8], strongly supported the standard picture of neutrino cooling of the proto-neutron star (PNS): The total gravitational binding energy, E G ∼ 3 · 10 53 erg, released while forming a compact PNS with a mass M PNS ∼ 1.4 M and radius R PNS ∼ 10 km are roughly equipartitioned by all six flavours of (anti)neutrinos. Consequently, any bSM particles that can be produced inside the PNS and escape by taking away an energy comparable to E G would have shortened the observed timescale of theν e burst to be incompatible with the observation [9].
Other than affecting the PNS cooling, bSM particles produced inside the PNS may directly decay to photons, or indirectly produce the 511 keV lines via the pair-annihilation by first decaying into e ± , outside the surface of the progenitor stars, R * 10 14 cm. The (non-)observation of γ-rays associated with SN1987A, as well as the observed flux of 511 keV photons from the Milky Way have been used to put constraints on bSM particles that couple electromagnetically to the SM sector [33][34][35]. Such derived bounds mostly complement with those from the PNS cooling because for bSM particles to decay outside R * , the required coupling to SM sector is usually not large enough to affect the PNS cooling.
In this paper, we propose a new constraint that bridges those from the PNS cooling and the γ-ray (non-)observation. Our new constraint is based on a very basic fact: The known explosion energy of the CCSN of a progenitor star with 10 M < ∼ M * < ∼ 20 M is 1 B ≡ 10 51 erg [36,37]. Among which, most of the energy is carried by the kinetic energy of the expanding ejecta, with a mass of ∼ O(10) M and a velocity of ∼ 0.01 c, when we observed the emitted (quasi-)thermal photons at ≥ O(1) d after the core-bounce [38]. In the absence of bSM physics, the prevalent theory is that the neutrinos emitted from the PNS within ∼ 1 s after the core-bounce, can deposit a few percent of their energy to the stalled shockwave at ∼ O(10 2 ) km to revive it [39]. The shock then wipes out the outer stellar envelopes at a speed of ≤ 0.1 c, giving rise to the observed explosion.
However, if bSM particles produced from the PNS can transfer the energy that they carry into the stellar envelopes or the shocked material before leaving the progenitor star, they would serve as a new energy source contributing to the total explosion energy (see Fig. 1 for a schematic plot). As a result, if this energy deposition mediated by bSM particles exceeds the observed explosion energy, after subtracting the gravitational binding of the stellar envelopes, such bSM particle is then ruled out by CCSN observation.
Before working out a specific example, we first demonstrate analytically how this new bounds can improve the constraint derived from the PNS cooling. A well-known analytic criterion formulated by G. Raffelt of such states: For a novel cooling agent X that freestreams after production, its specific energy lossε is bounded by [9] where L ν ∼ E G /10 3 · 10 52 erg s −1 being the energy luminosity of all (anti)neutrinos andε X being evaluated at a typical core condition at ∼ 1 s post the core-bounce, with a temperature of 30 MeV and a density of 3 · 10 14 g cm −3 .
The upper bound of the observed explosion energy of CCSNe associated with progenitor stars with zero-age-main-sequence (ZAMS) mass between 10 and 20 M are mostly under E expl = 2 B (see e.g., the compilations in Refs. [36,37,40]), while the typical binding energy of the stellar envelopes is E b < ∼ 1 B (see later section for detail). Therefore, our proposed new constraint can be expressed by: where ∆t 10 s, and 0 < K ≤ 1 denotes the efficiency of energy transfer into the region between a radius R p , within which the particle X can be produced efficiently, and R * .
Comparing Eqs. (1) and (2), it is obvious that the new bound can exclude the bSM particle whose emissivity is ∼ two orders of magnitude less than the one constrained by the PNS cooling, for cases where K ∼ 1. For the rest of the paper, we consider a specific example of the dark photon that decays predominantly to an e ± pair.

II. NEW CONSTRAINT ON DARK PHOTON
We consider the minimal extension of the SM with a U (1) dark sector. The dark photon (A ) is the gauge boson of the broken U (1) symmetry which kinetically mixes with the hypercharge boson. When the dark photon mass is much smaller than the electroweak symmetry breaking scale, the mixing is effectively only with the photon (A). The effective Lagrangian for the photon-dark photon system is (see e.g. Ref. [41] for the transformation from the dark photon gauge eigenstates to the mass eigenstates) Here f is SM fermion with electric charge q f , and m A and are the mass and the kinetic mixing parameter of the dark photon in the physical basis, respectively. Strategies for dark photon search at colliders and fixed-target experiments, existing constraints on ( , m A ), as well as anticipated sensitivities of planned experiments, can be found in the reports [42,43].
The in-medium physical eigenstates are quite distinct from those in vacuum due to the presence of the photon polarisation tensor Π = Π R + iΠ I (see e.g. Ref. [44]) in the inverse propagator matrix of the photon-dark photon system. As a consequence, in hot or dense stars the collective effects of the stellar plasma can significantly change the dark photon production rate [45,46]. Refs. [24,25] found that plasma effects in the PNS qualitatively weaken the supernova cooling bound at dark photon masses below ∼ 10 MeV.
In this work we calculate the dark photon production rate following closely Refs. [24,25].
For dark photon weakly-coupled to the thermal bath, i.e. when 1, one can invoke the in-medium effective kinetic mixing parameter Dark photons are reabsorbed in the supernovae mainly by the decay process A → e + e − when it is kinematically allowed. As pointed out in Ref. [25], in the PNS core region dark photon decay is prevented due to the high electron chemical potential, unless m A is larger than twice the effective electron mass in the plasma [47]. In this work we are interested in the case when dark photon can escape the production region and decay freely in the stellar layers. The produced e ± then quickly interact with the medium and lose their kinetic energy of ∼ 10 − 100 MeV to the surrounding in a length scale much shorter than R * [48]. This effectively leads to an efficient transfer of the thermal energy from the PNS core region to the stellar envelope (K 1 in Eq. (2)).
For a given dark photon mass m A and kinetic coupling , the total energy carried by the dark photons to a distance R ≥ R p is calculated by m|L,T (r, ω) · Γ iBr|L,T (r, ω) + Γ sC|L,T (r, ω) , assuming that the nucleons and electrons are in local thermal equilibrium at temperature T (r). Here v = 1 − m 2 A /ω 2 , and Γ iBr and Γ sC are the photon absorption rates due to the inverse bremsstrahlung and the semi-Compton processes, respectively. For Γ iBr we adopt the soft-radiation approximation, and neglect many-body effects in the nuclear medium, as Ref. [23]. We calculate the optical depth for a dark photon produced at radius r with energy ω, which travels radially outward to R by and include a correction factor to relate τ (r) to τ radial out (r) as suggested by Ref. [24]. The dark photon absorption rate Γ abs receives contributions from the inverse bremsstrahlung processes, semi-Compton scattering, and decay to e ± pairs. We have checked that outside R p , Pauli-blocking can be ignored and one can use the decay rate in vacuum for Γ e + e − . The supernova cooling bound is determined by L A (R p ) ≤ L ν (cf. Eq. (1)) in the dark photon (m A , ) parameter space. Our new bound, Eq. (2), is by requiring that the energy deposited by the decay of A between R p and R * being smaller than the sum of the observed SN explosion energy and the total gravitational binding energy between these two radii: Here ∆E g (R) ≡ E g (R * ) − E g (R), with the gravitational binding energy inside radius R, and M enc (r) the total mass enclosed in the region inside r. We fix the emission duration ∆t = 10 s. Note that in Eq. (7) we have neglected the kinetic energy of the shocked material, as well as that of the stellar envelope, which contribute at most ∼ 10% of E expl .
The dark photon deposited energy E d (R p ) and the gravitational binding energy of the stellar envelop ∆E g (R p ) depend on the structure of the PNS and the mass of the stellar progenitor. We examine two cases using the radial profile of the mass density, temperature, electron fraction, and electron chemical potential obtained by SN simulations of progenitor stars with 10.8 M and 18 M masses [49], chosen at t = 1 s post the core-bounce. As those SN simulations do not contain the structure of the outer most hydrogen layer of the progenitor star, we extend the profile to R * using the pre-SN structure provided by Ref.
[50]. For both cases, we have used the same R p = 25 km (slightly larger than R ν ) so as to encompass all the dark photon resonant production sites.  from the (non-)observation of γ-rays (dotted green curve), taken from Ref. [35].

III. DISCUSSIONS
We have shown that the observed explosion energy of the CCSNe can be used to derive important constraint on light bSM particles that may be copiously produced from the PNS core. For dark photons that kinetically mixes with SM photons, we show that our new bound excludes a larger parameter space than that derived using the observed neutrino burst from SN1987a. Moreover, it overlaps with the region recently obtained using the (non-)observation of γ-rays produced by supernovae. Therefore, all three constraints together exclude a large range of parameter space that is not accessible by current terrestrial experiments or by cosmological observation.
Several uncertainties may affect the exact excluded region derived with the simple argument presented in this work for dark photons. For examples, improved description of the dark photon emission from the nuclear bremsstrahlung beyond the soft radiation approximation adopted here, as well as the time-dependence of the PNS structure and the stellar envelope profile, may introduce some minor corrections. Nevertheless, the main message of this paper remains solid: the observed explosion energy of core-collapse supernovae places improved constraint on bSM particles that are able to transfer energy efficiently from the PNS core to the stellar mantle.
On the other hand, a detailed SN light-curve modelling taking into account ejecta driven by bSM particles can potentially provide even better constraints. For instance, even if the bSM particles only unbind the outermost part of the stellar envelope with an energy smaller than E expl (see e.g., the case with = 10 −11 in Fig. 2), the standard neutrinodriven mechanism can still work to eject the entire inner layers. Depending on their relative velocity, those two ejecta may collide at times of days after the core-collapse and leads to very luminous events not compatible with observations. The hydrogen layer of the stellar envelope may also be driven off by the energy deposition from bSM particles with a speed much larger than typical SN ejecta velocity, resulting in an electromagnetic precursor prior to the main supernova peak lights, or reducing the line feature of hydrogen. All these aspects require more dedicated work beyond the scope of this paper and deserve further exploration.