Novel Astrophysical Probes of Light Millicharged Fermions through Schwinger Pair Production

The extreme properties of neutron stars provide unique opportunities to put constraints on new particles and interactions. In this paper, we point out a few interesting ideas that place constraints on light millicharged fermions, with masses below around an eV, from neutron star astrophysics. The model independent bounds are obtained leveraging the fact that light millicharged fermions may be pair produced copiously via non-perturbative processes in the extreme electromagnetic environments of a neutron star, like a Magnetar. Magnetar energetics, magnetic field evolution and spin-down rates may all be influenced to various degrees by the presence of these millicharged particles.


I. INTRODUCTION
The Standard Model of particle physics has been an incredibly successful theory whose predictions have been tested to an immaculate degree. Nevertheless, it is considered incomplete and one of the foremost hints in this direction is the presence of dark matter in the universe.
An interesting possibility for a dark matter component are milli-charged particles (mCPs) -particles carrying fractional electric charges [1][2][3]. They arise naturally in a large class of Standard Model extensions [4][5][6][7][8][9] and have been subjects of intense investigations in the context of observational anomalies in the recent past [10][11][12]. They are also intriguing from the viewpoint of charge quantisation. Recently, mCPs in the mass range of a few GeV have also garnered attention due to the anomalous 21-cm absorption profile observed by the EDGES collaboration [13], and its possible theoretical interpretation in terms of mCPs [14]. All these reasons make mCPs of much current interest [15,16].
In this paper, we consider the effects of light mCPs (m mCP 1 eV) on neutron stars, via their nonperturbative production. They key idea is that the light mCP states could be Schwinger pair produced in the neutron star regions with large electric fields and distort the energetics of the over all system. During the completion of this work, two studies appeared placing interesting constraints on magnetic monopoles [17,18], through their Schwinger pair production in neutron stars. Few of their arguments are in the same spirit as ours, but as we shall see the case for electrically charged mCPs and pertinent astrophysical considerations are very different from monopole scenario. It will be demonstrated that the bounds thereby obtained are relatively model independent and stronger than previous stellar cooling bounds, the latter of which may be evaded in certain models. mCP SPP was first considered in accelerator * mrunal.korwar@students.iiserpune.ac.in † thalapillil@iiserpune.ac.in cavities, obtaining projected bounds of 10 −7 [19]. Using the constraints from neutron star energetics and related ideas, we shall demonstrate that robust, model independent bounds as strong as 10 −12 may be obtained.
In Sec. II we briefly review the theoretical underpinnings behind mCPs and survey constraints in the lowmass region, along with caveats to these. Then, in Sec. III we outline features of neutron stars and magnetars relevant for the study. In Sec. IV we argue how non-perturbative production of mCPs in the electromagnetic environment of neutron stars could very generically affect their energetics, magnetic field evolution, and spindown rates, thereby placing strong non-trivial bounds on mCPs. We summarise and conclude in Sec. V.

II. MILLICHARGED PARTICLES AND CONSTRAINTS
mCPs may be incorporated into Standard Model extensions directly, or more naturally through kinetic mixing with a singlet dark sector (denoted by D). In latter scenarios, there may be new gauge groups in the dark sector whose gauge fields mix with the gauge fields of the Standard Model through kinetic mixing [4], i.e. gauge kinetic terms in the Lagrangian are off-diagonal. Fermion mCPs are particularly attractive since chiral symmetry may render their masses small in a natural way. We are interested in this low-mass region and will assume the mCPs to be fermions.
In the simplest case with a single U (1) D gauge field A D α , that is massless, the Lagrangian density is The last term in Eq. (1) is a kinetic mixing term [4] between U (1) D and Standard Model hypercharge U (1) Y . ξ is generated at some high scale via loop-diagrams involving heavy particles, charged under both U (1) Y and U (1) D . The field redefinition A D α → A D α − ξB α makes the gauge kinetic term canonical and eliminates the mixing term. This now leads to an effective coupling of the χ D fermions to B α with effective charge ξe D , which could be fractional and very small [4]. After electroweak symmetry breaking, these χ D therefore couple to the U (1) QED photon with a small, fractional electromagnetic charge of magnitude ξe D cos θ W (θ W is the electroweak mixing angle). This in units of electron charge (e) is The mCP parameter space (m χ , ) is tightly constrained by various laboratory, cosmological and astrophysical bounds [15,16,[20][21][22][23], which are illustrated in Fig. 1 for the relevant parameter space region. The most stringent limits are from red giant and white dwarf stellar cooling considerations. If mCPs are present, they could be produced in the stellar plasma and take away significant energy, altering conventional stellar evolution histories. One seemingly requires [15,16,[20][21][22][23] to be viable, based on these stellar cooling arguments. These astrophysical limits [20][21][22][23] nevertheless have some model dependence and may be evaded in various cases [24][25][26][27]. Many of these models have more than one U (1) D gauge group, and associated gauge field, with the feature that the effective mCP charge (say q mCP (k 2 ) for momentum transfer k) in plasma differs significantly from that in vacuum [24] Here, ω 2 P is the plasma frequency. This charge screening in plasma renders the stellar cooling bounds impotent, but these models have to possibly contend with some fine-tuning as well [24]. In general, viable mCP couplings all the way upto ∼ 10 −7 or larger [24][25][26][27] may therefore be possible. We will generically refer to fermion mCPs, in any model, as χ D .

III. NEUTRON STARS AND MAGNETARS
Neutron stars (NS) are supernovae collapse end products of very massive stars [28,29]. Isolated neutron stars, that are not part of a binary system, may be categorised into radio pulsars and X-ray pulsars [30,31]. Radio pulsars are thought to be rotationally powered (i.e. rotational energy losses power their electromagnetic emissions). The second category consists loosely of two groups -soft-gamma repeaters and anomalous X-ray pulsars [31] and exhibit both persistent emissions as well as short-lived burst activities. This latter category may be accommodated in the so-called Magnetar model [32][33][34]. In a Magnetar (MG), the persistent luminosities and burst activities are thought to be powered by the dissipation and decay of super-strong magnetic fields [32][33][34].
NS are compact, rotating objects with large magnetic fields in general. Radio pulsars already are thought to have magnetic fields typically approaching 10 11 − 10 13 G, while Magnetars are thought to have even larger fields in a range 10 14 − 10 15 G or higher. The NS rotation and large magnetic fields lead to the generation of large external electric fields [30,35]. The generated lorentz forces greatly exceed the gravitational force on the surface and lead to extraction of particles from the NS surface. The extracted particles form a co-rotating envelope around the NS called the NS magnetosphere. Once this plasma forms, a force-free condition occurs -the distribution of charges in the magnetospheric plasma shorts-out the induced electric field, E GJ +( Ω NS × r)× B NS = 0. These ideas constitute the basic Goldreich-Julian model [36] and describes the salient principles behind NS electrodynamics. The various regions for a typical NS are illustrated in Fig. 2.
Interestingly, the force-free state is not maintained in all magnetospheric regions though, and many models generically predict the existence of 'vacuum gap' regions where the plasma density is very low or vanishing [30,35]. In these regions, the Goldreich-Julian model co-rotating condition and force-free criteria break down and electric fields are non-vanishing. This is a crucial observation for the arguments we put forward.
We will be interested specifically in the polar gap vacuum regions (see Fig. 2 magnitude [30,35] Here, B NS is the polar magnetic field on the NS surface, R NS is the NS radius and θ is the star-centred polar angle. This electric field is parallel to the polar magnetic field. Taking representative MG parameter values (denoted 'M') -rotation period τ M = 10 s, radius R M = 10 Km, and B M = 10 15 G, one gets in the MG case The polar gap radius is approximately given by R pol. 150 m(τ NS /s) − 1 2 [35], where τ NS is the NS rotation period. Specialising to MGs, with τ M = 10 s, one obtains R pol. 50 m. The polar gap height and characteristic slot-gap widths are determined by the pair-formation front [37][38][39]. The typical pair formation front height and slot-gap width for an MG may be taken to be ∼ 10 m [39]. With these dimensions and assuming | E M | is significant in the slot-gap at least all the way upto a height O(2R NS ), we may estimate a relevant polar gap volume (V pol. ).
The rates based on Eq. (7), for B M = 10 15 G and induced | EM| = 10 14 Vm −1 , are significant and many orders of magnitude larger than e + e − -SPP rates for these field values (see Fig. 3). This has the potential to affect MG energetics. This is the central idea of the paper. We will work in = c = 1 units. In MGs, as we commented earlier, the persistent luminosities are powered by superstrong magnetic field decays [32][33][34]. These magnetically sourced radiation losses therefore have to be included in any consideration of MG energetics. Now, the rate of energy loss from the MG, averaged over a lifetime T M , should be bound approximately by (8) d 2 E rad. /dt dV represents the rate of energy loss, per unit volume, due to radiation losses. d 2 E χχ SPP /dt dV similarly quantifies energy losses, per unit volume, due to potential mCP SPP - Here, E M and E M are the average electric field values over the respective distance ranges. The first term in Eq. (9) is the energy extracted per unit volume per unit time from the E M field for SPP. l 0 is the inter-mCP distance at the instant of SPP. The second term in Eq. (9) is the subsequent work that may be done by E M in accelerating one of the χ D particles out by a distance l − l 0 . In (m χ , ) regions where inter-mCP dark Coulombic attraction (F Coul. , no mCPs accelerate out of the MG. The pairs would instead annihilate soon after SPP. Thus, the second term in Eq. (9) gives no contribution in these regions and the only energy extracted from the electromagnetic field is to initiate SPP. Rate computations at strong coupling [45][46][47] suggest that mCP SPP with the dark Coulombic interaction included would give a correction ∼ exp[e 2 D /4] to the exponent in Eq. (7), and only further enhance rates. In other regions where F E > F Coul. D , energy is extracted from the electromagnetic field both for SPP and to subsequently accelerate mCPs out of the MG. The precise details of mCP evolution subsequent to SPP is less important as long as no significant energy is deposited back into the magnetic field. This is true to good approximation -mCP SPP is a dissipative process.
In Eq. (8), dV denotes integration over a relevant volume for each term. An estimate of the volume over which mCP SPP may be significant is given by V pol. , as discussed earlier. For the electromagnetic energy stored in the MG, we assume that most of it is within a distance ∼ R NS of the magnetosphere.
As an estimate for the radiation loss component, we take the average of the persistent quiescent X-ray emissions, for all currently known MG candidates, from the MG catalog , the only energy extracted from the electromagnetic field is to achieve SPP. This situation is tantamount to putting l = l 0 in Eq. (9). The inter-mCP distance l 0 at the instant of SPP is given by This is valid for both strong coupling and large fields by energy conservation and symmetry. With these consid- In the (m χ , ) parameter space of interest, l 0 10 m and particle separations are always within the polar gap at the time of SPP. This, along with the relevant Compton wavelengths, support the validity of Eq. (7) and the neglect of any field inhomogeneities to leading order [42,47].
Note that these limits only depend on the fact that fermion mCPs have an effective coupling with the U (1) QED photon. Any model dependent charge screening mechanism in plasma [24,26,27], that makes stellarcooling bounds weak, is also irrelevant in the the vacuum regions.
To augment the central idea from energetics, let us now regard the related effects from mCP SPP on MG magnetic field decay and spin-down rates. Ohmic and Hall drift contributions [49,50] are conventionally responsible for field attenuations in the NS interior. The details of these contributions and how magnetic field configurations evolve in a NS are not fully know, and are topics of intense study (see for instance, [51,52] and references therein). We may nevertheless try to capture some salient features by incorporating appropriate timescales and terms relevant in the conventional evolution of Magnetar magnetic fields (see for instance Eqs. (16)- (20) in [53]). The new addition now is that potentially mCP SPP may also non-perturbatively contribute to field decays in the MG. We assume that the final source of energy in the Magnetar is electromagnetic and that since SPP leads to the non-perturbative dissipation of electromagnetic fields directly, that the energy extracted is dominantly from this source. With this reasonable ansatz, we equate the energy loss due to mCP SPP to a corresonding change in the electromagnetic energy stored in the dipole field (also see Eqs. (15)-(17) of [18] for the case of millicharged monopoles) d dt One may then phenomenologically model the overall magnetic field evolution, in the toy model of the MG system, as B M (0) is the initial magnetic field, which we take as 10 15 G. Ω M (t) is the MG angular velocity. V pol. as before is the relevant polar gap volume. For the Ohmic and Hall drift time constants, we take τ ohm = 10 6 yrs and τ hall = 10 4 yrs following typical values from literature [49,53]. Realistically, the time constants are complex functions of temperature and density, but the above values have been found to phenomenologically capture relevant behavior [53]. Moreover, an equation of the above form, without the SPP term, is known to semi-quantitaively reproduce [53] results from more detailed magneto-thermal simulations [53][54][55]. Note though that the basic principle is largely independent of modelling and depends only on the fact that there is a potentially new nonperturbative dissipative contribution from mCP SPP. For typical MG parameters, the coefficients in Eq. (15) are given by ρ ohm = 2.1×10 −38 GeV, ρ hall = 2.1×10 −36 GeV, ρ SPP = 3.1 × 10 32 3 GeV −2 ,ρ SPP = 1.24 × 10 −19 GeV and ρ SPP = 4.1 × 10 −37 −1 (m χ /1 eV) 2 GeV 3 .

mCP-SPP Suppressed
One reasonable supposition could be that for viable (m χ , ) values, the field decays due to mCP SPP should not overwhelm the conventional B M (t) evolution in the MG. From the viewpoint of Eq. (15) a criteria could be This gives a constraint (18) The bounds obtained from Eqs. (8) and (16) are comparable, which make sense -in the conventional scenario, without mCP SPP, the Ohmic and Hall terms lead to B M dissipation, which subsequently power persistent emissions. Thus, our arguments based on MG energetics are related to those based on overall B M (t) evolution in the system. The complete exclusion regions based on these arguments from Eqs. (8) and (16) are shown in Fig. 4. We have assumed for the coupling strength, e D ∼ e conservatively.
For a relationΩ NS (t) = −λ(t) Ω b NS (t), the true braking-index (b true ) is given by If λ(t) is a constant, one obtains b true = b. For instance, a rotating, constant magnetic dipole has b true = 3. In general, b true is time dependent as seen from Eq. (19). Now, assuming a predominantly dipolar magnetic field in the NS exterior [56][57][58], the spin-down due to magnetic-braking torque (from radiation reaction) is given by [59] Here, I NS is the NS moment-of-inertia and α is the angle between the NS rotation and magnetic axes. Without loss of generality we take α = π/4 and neglect the small time dependence that I NS may have. Specialising to a MG and approximating it to a spinning rigid sphere with I M = 2 5 M M R 2 M , we geṫ Ω M (t) = − 5 24 We solve the coupled differential equations, Eqs. (15) and (21), for B M (t) and Ω M (t) over a time-scale [1.0, 1 × 10 5 ] yrs. Based on the solution we may calculate b true (t) for various (m χ , ) values. Define the deviation of b true (t) from the pure magnetic dipole braking index as ∆b true (t) ≡ b true (t) − 3. The ratio of this quantity, without (∆bS PP true ) to that with (∆b SPP true ) mCP SPP, is shown in Fig. 5. The curves are for parametric values = {10 −11 , 10 −12 , 10 −13 } and m χ = 10 −2 eV. For these values F E > F Coul. D . For large values, the ratio ∆bS PP true /∆b SPP true differs significantly from unity and also shows appreciable time evolution.

V. SUMMARY
Neutron stars may provide novel constraints on exotic particles and interactions. In this paper we point out a few new ideas to place limits on electrically charged, light, fermion mCPs by considering their non-perturbative production in neutron stars. Depending on the mCP parameters, the neutron star energetics, magnetic field evolu-tion and spin-down rates may all be significantly modified. These effects provide a new method whereby light fermion mCPs, below around 1 eV, may be constrained in a relatively model independent way. It would be interesting to investigate the effects further by incorporating a more realistic modelling of the magneto-thermal evolution of the MG system in the presence of mCPs, as well as incorporating any effects of temperature on Schwinger pair-production when E B [63] (in the pure electric or magnetic field case see for instance [18,[60][61][62]). Another intriguing avenue to consider is what effects transient electromagnetic fields in the NS magnetosphere may have.