Vector gauge boson radiation from neutron star binaries in a gauged Lμ − Lτ scenario

gauged Lμ − Lτ scenario Tanmay Kumar Poddar, Subhendra Mohanty, and Soumya Jana Theoretical Physics Division, Physical Research Laboratory, Ahmedabad 380009, India Abstract The orbital period of the binary neutron star systems decrease due to mainly quadrupole gravitational radiation which agrees with the observation to within one percent. Other types of radiation such as ultralight scalar or pseudoscalar radiation, massive vector boson radiation also contribute to the decay of orbital period within the permissible limit. Due to large chemical potential and degenerate electrons, the neutron stars have a large muon charge. We obtain the energy loss for the radiation of massive vector proca field from neutron star binaries. We also derive the energy loss for the U(1)Lμ−Lτ gauge boson radiation from the binaries. For the radiation of vector boson the mass is restricted by Mz′ < Ω where Ω ' 10−19eV are the orbital frequencies of PSR B1913+16 and PSR J0737-3039 neutron star binaries. Using the orbital period decay formula, we put constraints on the coupling constant of the gauge boson in anomaly free Lμ−Lτ gauged theory for the HulseTaylor (PSR B1913+16) binary pulsar and the Double Pulsar (PSR J0737-3039). For vector gauge boson muon coupling we find the coupling constant for Hulse-Taylor binary is g < 2.207 × 10−18 and for PSR J0737-3039 the coupling is g < 2.17 × 10−19. We also obtain the exclusion plot for the massive vector proca field from the two binaries.


I. INTRODUCTION
The decrease in the orbital period with time of Hulse-Taylor binary pulsar (PSR B1913+16) provided the first indirect evidence of gravitational waves [1][2][3]. Although the decay of the orbital period is due to mainly the quadrupole gravitational radiation [4], radiation of other massless or ultralight scalar or pseudoscalar particles [5,6] can contribute about one percent of the observed decay of the orbital period [7]. In this paper we calculate the orbital energy loss due to radiation of proca vector boson and massive vector gauge boson of L µ − L τ [8][9][10][11] anomaly free gauge theory.
The standard model (SM) of particle physics is a SU (3) c × SU (2) L × U (1) Y gauge theory and it remains invariant under four global symmetries corresponding to the lepton numbers of the three lepton families and the baryon number. These are not the gauge symmetries but one can construct three combinations in an anomaly free way and they can be gauged in the standard SU (2) × U (1) group. These gauge symmetries are L e − L µ , L e − L τ and L µ − L τ . The L e − L τ and L e − L µ long range forces from the electrons can be probed in neutrino oscillation experiments [12][13][14]. The L µ − L τ gauge force is not generated in a macroscopic body like the earth and the Sun and it can not be probed in the neutrino oscillation experiments. In this paper we point out that neutron star can have large charge of muons and therefore the neutron star binaries can radiate ultralight L µ − L τ vector gauge bosons.
Besides neutrons there are electrons, protons and muons in lower fraction inside the neutron star. There are around 10 55 number of muons compared to about 10 57 number of neutrons [15][16][17][18] for a typical old neutron star. The main uncertainties in our following calculation are the chemical potential and muon content in NS which should be at most a factor of two [18][19][20].The chemical potential of relativistic degenerate electron is, Where m e is the mass of the electron, k f is the Fermi momentum, ρ is the nuclear matter density and Y e is the electron fraction. From the charge neutrality of the neutron star Y p = Y e + Y µ and Y n + Y p = 1. Above the nuclear matter density when µ e exceeds the mass of muon (∼ 105M eV , non-relativistic) electrons can convert into muons at the edge of the Fermi sphere. So e − → µ − + ν e +ν µ , p + µ − → n + ν µ and n → p + µ − +ν µ may be energetically favourable. Hence both muons and electrons can stay in neutron star and stabilize through beta equilibrium. Thus the β stability condition becomes, Where Y µ is the muon fraction inside the neutron star [21]. Muon decay (µ − → e − +ν e + ν µ ) inside the neutron star is prohibited by Fermi statistics. The Fermi energy of the electron (relativistic) is roughly 380M eV whereas the Fermi energy of the muon (non relativistic) is roughly 30M eV . Hence the muon decay cannot take place as the energy levels of the electron are all filled upto the Fermi surface and the final state electron is Fermi blocked.
For massive vector gauge boson radiation from the neutron star binaries, the orbital frequency of the binary orbit should be greater than the mass of the particle which restricts binaries. In this paper we say that ultra light vector L µ − L τ gauge bosons can radiate from the neutron star binaries which can reduce the orbital period.
The paper is organized as follows. In section II, we derive the energy loss due to proca vector field radiation. In section III, we derive the energy loss due to massive vector gauge boson radiation. In section IV we put constraints on the gauge couplings in L µ − L τ gauge for vector gauge boson radiation from two compact neutron star binaries (PSR B1913+16: Hulse Taylor binary pulsar [1][2][3] and PSR J0737-3039:double pulsars [22]). We also plot the allowed regions for the coupling and mass of the proca vector boson in the mass range M Z = 10 −20 eV to 10 −34 eV for those two compact binaries. In section V we summarize and discuss our results.
In this paper we have used the natural system of units: = c = 1, and G = 1/M 2 pl .

II. ENERGY LOSS DUE TO RADIATION OF MASSIVE PROCA VECTOR FIELD COUPLING WITH MUONS
If there is a mismatch between the observed period loss of the binary system and the prediction due to gravitational quadrupole radiation, then other particles that may also be radiated from the binaries suggesting new physics. Neutron stars have large number of muon charge (N ≈ 10 55 ) and Z massive proca vector boson can be emitted from the neutron star in addition to the gravitational radiation to satisfy the observed orbital period decay. A NS of typical size 10km can be treated as a point source because the Compton wavelength of radiation (λ = 10 12 m) is much larger than its dimension. We will treat the radiation of massive Z vector boson from the neutron star classically. The classical current of muons J µ in the NS is determined from the Kepler orbits and assuming the interaction vertex as gZ µ J µ where g is the coupling constant. So the rate of massive Z boson radiation is given by, where J µ (k ) is the Fourier transform of J µ (x) and λ µ (k) is the polarization vector of massive vector boson. The polarization sum is given as, Therefore the emission rate is, Now the momentum four vector of the Z boson is k µ = (ω, − k), k i = | k|n i and k j = | k|n j .
The third term in the first bracket will not contribute anything because, Therefore the rate of energy loss due to massive Z boson radiation is Now the current density for the binary stars denoted by a = 1, 2 may be written as, where Q a is the total charge of the neutron star due to muons and x a (t) denotes the Kepler orbit of the binaries. u µ a = (1,ẋ a ,ẏ a , 0) is the non relativistic four velocity in the x-y plane of the neutron star. A Kepler orbit in the x-y plane can be written in the parametric form as, where e is the eccentricity, a is the semi major axis of the elliptic orbit, and Ω = G[ m 1 +m 2 is the fundamental frequency. The angular velocity is not constant in an eccentric orbit and this means that the Fourier expansion must sum over the harmonics nΩ of the fundamental.
The Fourier transform of Eq. (8) for the spatial part with ω = nΩ is, We expand the e ik .x = 1 + ik .x + ... and retained the leading order term as k .x ∼ Ωa 1 for binary star orbits. Hence Eq. (10) becomes In the c.o.m coordinates we have is the reduced mass of the compact binary system. Hence we rewrite the current density as The Fourier transform of the velocity in the Kepler orbit can be evaluated as follows: where T = 2π/Ω and from Eq. (9) we have used the fact thatẋdt = −a sin ξdξ. Similarly we writeẏ From Eq. (9) we use the fact thatẏdt = a √ 1 − e 2 cos ξdξ and we obtaiṅ 2π 0 e in(ξ−e sin ξ) cos ξdξ Using the identity of the Bessel function in Eqs. (13) and (15), we obtain the velocities in Fourier space aṡ where the prime over the Bessel function denotes derivative with respect to the argument.
Hence we have Similarly Hence From Eq. (8) we have Going to the c.o.m frame the integral results in where x(ω) = aJ n (ne)/n and y(ω) = ia √ 1 − e 2 J n (ne)/ne are the fourier transform of the orbital coordinates. The first term in Eq. (22) does not contribute due to the delta function δ(ω). Therefore considering the second term as the leading order contribution we obtain where we have used < k 2 x >=< k 2 y >= k 2 /3 and ω = nΩ. Using Eqs. (20) and (23) in Eq. (7) we obtain the rate of energy loss Since the mass of the proca vector boson is very small compare to the angular frequency of the orbit (n 0 /n 1), we can expand the sum in Eq. (24) in O(M 6 Z ). Thus we write Eq. (24) as where We define and use these to rewrite Eq. (24) as This is the energy loss due to radiation of proca vector massive boson.

III. ENERGY LOSS DUE TO RADIATION OF MASSIVE L µ −L τ GAUGE BOSON
If the Z boson is a gauge field then from gauge invariance k µ J µ = 0 and, consequently, the second term in the polarization sum Eq. (4) will not contribute to the energy loss formula.
Using the same procedure we obtain the rate of energy loss Where K 2 (n, e) is defined earlier Eq. (31). Since K 2 (n, e) n 0 =0 ≥ K 2 (n, e) n 0 =0 the massless limit gives the stronger bound on the energy loss. This is the energy loss due to massive vector gauge boson radiation which has the similar form as given in [23].
Since the mass of the gauge boson is very small compare to the angular frequency of the orbit (n 0 /n 1), we can expand the sum in Eq. (33) in O(M 6 Z ). Therefore we write Eq. (33) as where, f 2 (e) and f 3 (e) are defined earlier in (Eq. 28) and (Eq. 29).
The rate of change of the orbital period due to energy loss is where dE GW dt is the rate of energy loss due to quadrupole formula for the gravitational radiation and is given by [4] dE GW dt = 32 5 GΩ 6 M 2 a 4 (1 − e 2 ) −7/2 1 + 73 24 e 2 + 37 96 In the massless limit of the vector gauge boson i.e; M Z = 0 implies n 0 = 0 the rate of energy loss from Eq. (35) becomes If the orbit is circular then the angular velocity is a constant over the orbital period and the Fourier expansion of the orbit contains only one term for ω = Ω. In an eccentric orbit the angular velocity is not constant and that means the Fourier expansion must sum over the harmonics nΩ of the fundamental.
In the following we will put constraints on the mass of the vector gauge boson and on the L µ − L τ coupling constant from the decay of orbital period of two binary neutron star From the fifth force constraint the ratio of the fifth force to the gravitational force should be less than unity which implies g 2 s N 2 4πGm 1 m 2 < 1.
This gives the upper bound on g as g < 4.998 × 10 −17 for HT binary.

B. PSR J0737-3039
The double binary star system (PSR J0737-3039A/PSR J0737-3039B) [22] consists of two neutron stars and both of them are pulsars emitting electromagnetic waves in the radio wavelength range. This compact binary system has an average orbital period P b = 2.4h. From fifth force constraint we can write, This gives the upper bound on g as g < 4.584 × 10 −17 for PSR J0737-3039. to mass ratio is different for two neutron stars. We have shown the exclusion plots of g vs M Z for the radiation of massive vector proca field from the two neutron star binaries.
Due to the fact of K 2 (n, e) n 0 =0 ≥ K 2 (n, e) n 0 =0 the massless limit gives the stronger bound for the radiation of massive vector gauge boson radiation. The main uncertainty of the