A Relativistic Quantum Approach to Neutrino and Antineutrino Emissions via the Direct Urca Process in Strongly Magnetized Neutron-Star Matter

We study neutrino and antineutrino emission from the direct Urca process in neutron-star matter in the presence of strong magnetic fields. We calculate the neutrino emissivity of the direct Urca process, whereby a neutron converts to a proton, an electron and an antineutrino, or a proton-electron pair converts to a neutron-neutrino pair. We solve exact wave functions for protons and electrons in the states described with Landau levels. We find that the direct Urca process can satisfy the kinematic constraints even in density regions where this process could not normally occur in the absence of a magnetic field.

to lead to a strong reduction of the neutrino emissivity and affect the neutrino-antineutrino emission by the Cooper pair breaking and formation mechanism [7].
The DU process is one of the most feasible candidates to explain the rapid cooling of NSs [8,9]. One usually examines the rapid cooling associated with the DU process to compare temperatures and ages of isolated NSs. Recently another method has been realized by observing NSs in binary systems. Indeed, by analyzing the x-ray emission in MXB 1659-29, evidence of the direct Urca process has been found [10] Because of energy-momentum conservation and Fermi statistics for NS matter composed of protons, neutrons and electrons, the DU process occurs in a density region, where the proton density ρ p is larger than 1/8 of the neutron density ρ n (ρ p ≥ ρ n /8); this condition cannot be achieved in low density regions.
However, if the symmetry energy is proportional to the density, and its slope is sufficiently large, the condition for the DU process can be satisfied even at rather low density. Many other approaches have been proposed to satisfy this condition by considering other phases such as pion [11] or kaon condensation [12,13] and matter including hyperon degrees of freedom [6,14]. However, the latter case turns out to affect the masses of NSs, which is closely related to the hyperon puzzle in NSs [15].
On the other hand, magnetars, which are associated with a very strong magnetic field [16,17], have properties different from normal NSs. The strength of their magnetic field is about 10 14 − 10 15 G in the surface region, and can reach 10 17 G inside the star. Soft gamma repeaters (SGR) and anomalous X-ray pulsars (AXPs) correspond to magnetars [18]. Magnetars emit energetic photons. Furthermore, the surface temperature of magnetars is T ≈ 0.28 − 0.72 keV, which is larger than those of normal NSs T ≈ 0.01 − 0.15 keV at a similar age [19].
There are a number of theoretical works that show the influence of the magnetic field on the equation of state (EOS) of NS matter [20,21,22,23,24,25,26,27] and how the structure of NSs is altered by including magnetic fields [28,29] . We note however, that since our purpose here is to explore effects of the DU in NS matter and not the structure of NSs, we do not re-examine NS structure in this study. For the reader interested in magnetic effects on NS structure a list of works can be found in Ref. [28] (see also the "Lorene" website: https://lorene.obspm.fr/data/magnetNS.html).
A strong magnetic field can supply energy and momentum into the cooling process and changes the restriction caused by energy-momentum conservation by introducing transitions between Landau levels [30,31,32]. Indeed, the magnetic field influences the beta-decay process in NSs [24] and axion emission from magnetars [33]. Leinson [34] studied the effect of magnetic fields perturbatively and showed that a magnetic field increases the antineutrino emissivity. Thus, the associated strong magnetic fields may play a significant role in the cooling of magnetars.
In a previous paper [30] we have studied νν-pair emissions from the transition between two Landau levels in strongly magnetized NS matter. In that work, when the strength of the magnetic field is about B ∼ 10 15 G, emission energies of this process are larger than those of the MU process with zero magnetic field, B = 0. All of this suggests that the magnetic field may increase the emission of particles because the strong magnetic field can supply momentum into the emission processes.
Leinson and Pérez [35] calculated the neutrino emissivity from the DU process in strong magnetic fields, and showed that the emissivity becomes larger as the magnetic field increases. In that calculation the magnetic field was so strong that all charged particles populated the lowest Landau level. This means that one has not yet estimated realistic magnetic effects in NSs with realistic magnetic field strengths.
In the present paper, therefore, we apply our quantum theoretical approach to the DU process in strong magnetic fields and calculate it through the transition between Landau levels for electrons and protons. As noted above, only this quantum approach can exactly describe the momentum transfer from the magnetic field.
The low energy Lagrangian of the weak interaction between baryons and leptons is written as where ψ l 1.2 is the field of the lepton l 1,2 which denotes the electron or muon or neutrino, and ψ α is the field of the baryon α which denotes the proton or neutron, G F is the Fermi coupling constant, while c V and c A are the vector and axial-vector coupling constants which include the contribution from the Cabbibo angle [36].
In the single particle model, the l-neutrino emissivity of the DU process is written as with where the subscript l indicates the electron or muon, while ν indicates the ν e or ν µ , ψ a is the single particle wave-function, E a is the energy of the particle a(= n, p, e, µ, ν e , ν µ ), and α a denotes the quantum numbers such as momentum, spin, iso-spin and Landau-number. The quantity n a (E) is the Fermi-distribution function for the particle a, where T is the temperature, and µ a is the chemical potential of particle a.
We assume a uniform magnetic field along the z-direction, B = (0, 0, B), and take the electromagnetic vector potential A µ to be A = (0, 0, xB, 0) at the position r ≡ (x, y, z). The relativistic wave function ψ is obtained from the following Dirac equation: where κ is the anomalous magnetic moment (AMM), e is the elementary charge and ζ = ±1 or 0 is the sign of the particle charge. U s and U 0 are the scalar field and time component of the vector field, respectively. These are determined from relativistic mean-field (RMF) theory [37], although the AMM and the mean-fields are taken to be zero for charged leptons.
The single particle energy for a charged particle (ζ = ±1) is then written as with M * = M − U s , where n is the Landau number, p z is the z-component of momentum, and s = ±1 is the spin. Furthermore, the wave-function overlap at points r 1 and r 2 for charged particles such as the proton and electron are written as n,s,p z (r 2 ) = e i(p y y+p z z) R y R zF where Σ z = diag(1, −1, 1, −1), and R a (a = x, y, x) is the size of the system along the a-direction.
For neutral particles (ζ = 0) such as the neutron, the wave function is written as an eigenstate of momentum p ≡ (p T , p z ). So, the single particle energy is given by and the wave-function overlap becomes Substituting the above wave-functions into Eq. (2), we obtain with where R x = R y = R z is assumed, and f n is the n-th harmonic oscillator wave function. In addition, the neutron momentum is taken to be p n = (E n , 0, p nT , p nz ), where without loss of generality the neutron transverse momentum is assumed to be directed along the y-axis.
In cool NSs the temperature is less than 1 keV, and the emitted neutrino energy is of the order of the temperature e ν ∼ T , so that we can take the lowest order term of the temperature by using the following approximation with Therefore, the neutrino emissivity in the low temperature limit can be written as In addition, we assume chemical equilibrium for the system. Hence, the chemical potentials satisfy the following relations: At the low temperature limit, the chemical potentials are taken to be the Fermi energies at zero temperature, and the effective chemical potentials for baryons µ * in the presence of a magnetic field are written where s = ±1, n p is the Landau number of the proton, and k pz is the z-component of the proton Fermi momentum which depends on n p . In addition, k nT and k nz are the transverse component and z-component of the neutron Fermi momentum, whose values are not the same. That is the momentum-distribution breaks the spherical symmetry through the AMM.
For comparison we also show the neutrino emissivity from the DU process when B = 0 as where ∆U 0 = U 0 (p)−U 0 (n); this expression is equivalent to that of Ref. [38]. This emissivity is non-zero , the Fermi momentum of the particle a. This momentum matching condition determines the density region for the DU process when B = 0. In contrast, for B 0, the magnetic field carries momentum and breaks the momentum matching condition, and the DU process can occur in any density region.
In this work we discuss the relation between the DU emissivity and the density dependence of the symmetry energy as well as the effects of the magnetic field. For the calculation of nuclear matter, we use the following Lagrangian density in the relativistic mean-field approach: whereŨ[σ] is the self-energy term of the sigma meson which includes the non-linear effect for the scalar mean-field, and the other terms are written as the zero-range interaction between two nucleons. In addition, C IV s and C IV v are the coupling constants for the iso-vector interaction in the Lorentz scalar and vector channels, respectively [39]. Then, the scalar-field U s and the time component of the vector-field are given by In the above formalism we omit the magnetic field energy density B 2 /(8π) in the EOS. This term contributes to the pressure and affects the composition of NS matter significantly when B 10 18 G [27]. In the present calculation we adopt a smaller strength of the magnetic field, and this term does not contribute to the neutrino emissivity.
Parameter-Set  We use the parameter-set PM1 for symmetric matter as given in Ref. [40]. In addition, we give the three parameter-sets, SF1, SF2 and SF3, for the other parameters of the iso-vector parts, C IV s and C IV v , whose detailed values are shown in Table 1 . The symmetry energy is fixed to be e sym = 32 MeV at normal nuclear density though the parameter-sets give different density dependences of the symmetry energy. The SF1 includes only the Lorentz vector channel (C IV s = 0), the SF2 includes only the isovector Lorentz scalar channel (C IV v = 0), and the SF3 includes the negative value of the Lorentz vec-tor channel (C IV v < 0 < C IV s ). In addition, we list the values of the slope parameter L for these parameter-sets.
In Fig. 1 we show the density-dependence of the symmetry energy, We see that the three parametersets give quite different density dependences of the symmetry energy though all of them satisfy 32 MeV for the symmetry energy at normal nuclear density. This means that the parameter-sets, SF1, SF2 and SF3, represent different EOSs.
In actual calculations we substitute the mean-fields and the chemical potentials at B = 0 with these parameter-sets into Eq. (18) and calculate the neutrino emissivity. As mentioned above, surface temperatures of magnetars are T ≈ 0.28 − 0.72 keV [19]. Thus, we choose T = 0.50 keV for the present calculations. First, we restrict the matter to consist of only protons, neutrons and electrons. In Fig. 2 we show the density dependence of the neutrino emissivity in the DU process (a) and the density dependence of the proton fraction in NS matter, x p (b). In Fig. 2(a), the dashed line indicates the results for B = 0, which appear in the density region of x p ≥ 1/9; x p = 1/9 is denoted by the horizontal dashed line in the lower panel (b). For future reference, we define ρ DU as the critical density at which the proton fraction The calculations result in large fluctuations, which also appears the in νν-pair [30] and axion emission [31]. These fluctuations reflect the density of states in the xy-plane given by the Landau levels [32,41]. sivity agrees with that of the proton fraction, namely the effect of the magnetic field is related to the proton fraction and can appear when the proton fraction is close to 1/9.
In Fig. 3 we plot the results for SF1 on a linear scale. Here one can see that the spike of the neutrino emissivity is very narrow, and that its height is much larger than that with zero-magnetic field. To estimate the strength of the neutrino emission, we smooth the emissivity according to the following equation: We show this result for SF1 when ∆ρ = 0.01 fm −3 with a solid line. The smoothed value for the emissivity is still larger than that when B = 0, but the difference is not very drastic. Here, we add the following comment. In the study of pion [42,43], axion [31] and ννpair production [30] we showed a very large difference in the results between cases including the AMM of the protons and those not including it. In this calculation there is no significant difference between the cases with and without the AMM. Without a magnetic field the above particle production is forbidden by energy-momentum conservation. Thus, only the argument of the overlap function of Eq. (13) in the tail region contributes to the transition strength, and a small difference in input values makes a very large difference in the calculation results. In the DU process the transitions are not completely forbidden by the kinematics, and the value of the argument p nT / √ 2eB is not located in the tail region.
As a next step we study the DU process in NS matter including muons. In Fig. 4  For comparison, we also show the neutrino luminosities from the MU process [44] with the long-dashed lines. When B 10 15 G, the emissivities do not appear for ρ B < ρ DU , and the magnetic field does not play an important role for the DU process.
The emissivities from the NS matter with muons are larger than those without muons. Particularly, the emissivity for SF3 at B = 10 16 G is much larger in the matter with muons than that without muons though the contribution from the muon process is negligibly small.
As the strength of the magnetic field decreases, the amplitudes of the oscillations in the density dependence become smaller for both magnetic field strengths in NS matter. In the density region, ρ B < ρ DU , the neutrino emissivities are largely suppressed, and the DU process dominates in a narrower density region, as shown in the results by SF3, where the emissivities in the DU process are still much larger than that of the MU process.
Here, we observe the following facts: As the magnetic field strength decreases, the emission rate becomes larger for the νν-pairs [30] despite a smaller momentum transfer from the magnetic field.
The νν-pairs are produced via the transition of the proton and electron between different Landau levels; namely the initial and final particles are the same. As the magnetic field strength decreases, the energy interval between the initial and final states also becomes smaller. Hence, the emission strength becomes larger. In the DU process, however, the initial particle is a neutron which does not stay in the Landau level, and the energy interval is continuous, so that this effect does not appear.
In summary, we have used a relativistic quantum approach to study neutrino and anti-neutrino emission in the DU process from NS matter with strong magnetic fields, B = 10 15 − 10 17 G. We use three parameter-sets for the symmetric nuclear force to illustrate the relation between the proton fraction and the neutrino emissivity. If the proton fraction satisfies the DU condition x p ≥ 1/9, the neutrino emissivities are not much different from the case of B = 0, and the magnetic field does not significantly amplify the emission, though it causes very large fluctuations in the density dependence of the neutrino emissivity. In the usual forbidden region x p < 1/9, however, the magnetic field contributes to the emission and changes the kinematical condition. The effect is larger when the magnetic field strength is larger and the proton fraction is closer to 1/9.
If the magnetic field is rather weak B 10 15 G, the DU process does not appear when x p < 1/9, so that it does not occur in the surface region of NSs. However, neutrino emission from the Landau quantization may occur in the inner core of magnetars which are thought to have such strong magnetic fields. Those neutrinos are able to escape from the matter because they have very low energy, and their mean-free-paths are very long.
Magneto-hydrodynamic proto-NS simulations [45,46,47] have demonstrated that the magnetic field inside a NS can obtain a toroidal configuration. It has also been demonstrated [46] that the field strength of toroidal magnetic fields can be ∼100 times stronger than that of a poloidal magnetic field. Indeed, by analyzing the hard X-ray data and precession of pulsars. Makishima et al. [48] suggested evidence for the existence of a toroidal magnetic field whose strength is about one hundred times that of poloidal magnetic fields for cold NSs.
In the present calculation the neutrino emissivities vanish when x p 1/9 because we use the low temperature approximation whereby the Fermi distribution functions are approximated by a step function. As mentioned above, we exactly calculated the νν-pair production in strong magnetic fields [30] and showed that the neutrino emissivity is larger than that of the MU process in the case of zero magnetic field, even though the production rates described here are zero in the low temperature approximation.
This implies that there may be much larger neutrino emissivities in the DU process when x p 1/9; it may be possible to explicitly show that with an exact calculation. We defer exploring this point to a future work Our final remark is about the EOS as there are many different versions from many nuclear models. A salient feature is the density-dependence of the symmetry energy, which is related to the proton fraction in NS matter. One needs to explore using different EOSs [49,50] which we also defer to a future work. This