\nu \bar{\nu} -Pair Synchrotron Emission in Neutron-Star Matter based on a Relativistic Quantum Approach

We study the neutrino anti-neutrino pair synchrotron emission from electrons and protons in a relativistic quantum approach. This process occurs only under a strong magnetic field, and it is considered to be one of effective processes for neutron star cooling. In this work we calculate the luminosity of the neutrino anti-neutrino pairs emitted from neutron-star-matter with a magnetic field of about 10^{15} G. We find that the energy loss is much larger than that of the modified Urca process. The neutrino anti-neutrino pair emission processes in strong magnetic fields is expected to contribute significantly to the cooling of the magnetars.

Neutrino antineutrino (νν)-pair emission is also an important cooling processes in the surface region of NSs. Pairs can be emitted by synchrotron radiation in a strong magnetic field [6,7,8,9] and by bremsstrahlung through two particle collisions [10,11].
Landstreet [6] studied this process for B ∼ 10 14 G and applied it to the cooling of white dwarfs.
In that study the magnetic field is very low, and the discontinuity due to the Landau levels was ignored when calculating the νν-pair luminosity. It was concluded that this process is insignificant van Dalen et al. [9] calculated the νν-pair emission in a strong magnetic field of B ≥ 10 16 G. In such strong magnetic fields and low temperatures, T ≤ 1 MeV, energy intervals between two states with different Landau numbers are much larger than the temperature. Hence, they treated only the spin-flip transition between states with the same Landau number.
In Ref. [12,13], we introduced Landau levels in our framework and calculated pion production though proton synchrotron radiation strong magnetic fields. In that work we showed that quantum calculations gave much larger production rates than semi-classical calculations.
In Ref. [14] we calculated the axion production in the same way, and found that the transition between two states with different Landau numbers gives significant contributions even if the temperature is low, T ≤ 1 keV, when the strength of the magnetic field is large, B = 10 15 G. In this case the energy interval between the two states is much larger than the temperature.
In the present paper, then, we apply our quantum theoretical approach to νν-pair synchrotron production in strong magnetic fields and calculate this through the transition between different Landau levels for electrons and protons. Only this quantum approach can exactly describe the momentum transfer from the magnetic field.
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 anormalous magnetic moment (AMM), e is the elementary charge and ζ = ±1 is the sign of the particle charge. U s and U 0 are the scalar field and time components of the vector field, respectively.
In our model charged particles are protons and electrons. The mean-fields are taken to be zero for electrons, while for protons they are given by relativistic mean-field (RMF) theory [16]. The single particle energy is then written as with M * = M − U s , where n is the Landau number, p z is a z-component of momentum, and s = ±1 is the spin. The vector-field U 0 plays the role of shifting the single particle energy and does not contribute to the result of the calculation. Hence, we can omit the vector field in what follows.
The weak interaction part of the Lagrangian density is written as where ψ ν is the neutrino field, ψ α is the field of the particle α, where α indicates the proton or electron, while G F , c V and c A are the coupling constants for the weak interaction [15].
By using the above wave function and interaction, we obtain the differential decay width of the protons and electrons into νν-pairs. where Here, γ 2 and γ 3 denote the second and third Dirac gamma matrices, respectively, Σ z ≡ diag(1, −1, 1, −1), and h n (x) is the harmonic wave function with the quantum number n.
In actual calculations, we use the parameter-sets of Ref. [17] for the equation of state (EOS) of neutron-star matter, which we take to be comprised of neutrons, protons and electrons. In this work we take the temperature to be very low, T ≪ 1 MeV, and use the mean-fields obtained at zero temperature.
In Fig. 1   First, we note that in the moderate temperature region, T 3−5 keV, the luminosities change rapidly while they vary slowly in the higher temperature region. This qualitative behavior is very similar to the axion luminosities in Ref. [14].
The energy of the νν-pair for a charged particle transition is obtained as where The initial and final states are near the Fermi surface in the low temperature region, and |q z | ≪ √ eB, so that the energy interval of the dominant transition is given by with The luminosities are proportional to the Fermi distribution of the initial state and the Pauli-blocking F , the strength is concentrated in the narrow energy region between E * F − T and E * F + T for both the E i and the E f . When T ∆E ≈ eB/E * F , however, neither the initial nor the final states reside in the region, Then, the luminosities rapidly decrease at low temperature as the temperature becomes smaller as can be seen in Fig. 1. For example, when B = 10 15 G, we find that eB/E * F = 9.4 keV at ρ B = ρ 0 for protons. Indeed, the change of the νν-pair luminosities becomes more abrupt for T eB/E * F . The energy step is much larger for protons than electrons because the proton mass is much larger than the electron mass, and the emission from protons becomes the dominant source of νν-pairs.  behavior has also been seen in the pion production [12]. When the temperature is very low, however, the positive additional energy makes the energy interval ∆E larger than the temperature, and it suppresses the luminosity. The roles of the two contributions reverse at temperatures above the inflections in Fig. 1.
In Fig. 3, we show the density dependence of the luminosities at a temperature of T = 0.5, 0.7 and 1 keV. The solid and dashed lines represent the contributions from the protons and electrons, respectively.
For comparison, we give theν-luminosities from the modified Urca (MU) process. In addition, we plot the proton contribution without the AMM and the axion luminosity [14] at T = 0.7 keV on the right panel (b), where the strength of the luminosity is taken to be 10 −2 of that in Ref. [14].
The calculation results include fluctuations. The density dependence of the factor f ( does not smoothly vary for strong magnetic fields and very low temperatures because the energy intervals between the initial and final states are larger than the temperature as discussed above. However, these fluctuations are much smaller than that of the axion luminosity because the invariant mass of the νν-pair is not fixed while the axion mass is approximately zero.
The proton contributions are dominant at least when ρ B < 3ρ 0 . The energy intervals for electrons are much larger than those for protons because the electron mass is much smaller than the proton mass.
When the AMM does not exist, the electron contributions rapidly increase while the proton contribution gradually decreases.
At ρ B = 0 and B = 10 15 G, eB/m e ≈ 11.6 MeV for electrons, and eB/M ≈ 6.3 keV for protons. For electrons the energy interval is too large, and the transition probability is negligibly small around zero density, while the energy interval for protons is also large but much smaller than that for electrons, and the proton contribution gives a finite value. The solid and dashed lines represent the νν-pair emission luminosity from protons with and without the AMM, respectively.
The thin line indicates the axion luminosity.
As the density becomes larger, the Fermi energy of the electrons rapidly increases in the low density region, and the electron contribution increases. In contrast, the effective Fermi energy E * F for protons gradually decreases with increasing the density because of the density dependence of the effective mass M * , and thta fact that the contribution from protons without the AMM gradually decreases. We should note that E * F increases in the higher density region, and that the proton contributions to the νν-luminosities must increase at higher densities.
When the AMM is included, we see that there are peaks in the proton contribution around ρ B ≈ As the magnetic field strength increases, the momentum transfer from the magnetic field becomes larger, and the energy interval between the initial and final states is also larger. The former effect enhances the emission rate, but the latter effect suppresses it. In the region of the magnetic field for magnetars, the latter effect is larger. Indeed, the axion production is largest around B = 10 14 G [14].
Thus, the νν-pair emission process has a much larger effect than that of the MU process in strong magnetic fields. We can conclude that the νν-pair emission process is dominant in the low density region, ρ B ρ 0 , for a cooling process of magnetars whose magnetic field strength is 10 14 − 10 15 G. In the high density region, ρ B 3ρ 0 , the direct Urca process must appear, and its contribution is much larger than that of the νν-pair emission.
In summary, we have studied the νν-pair emission from neutron-star matter with a strong magnetic field, B ≈ 10 15 G, in a relativistic quantum approach. We calculated the νν-pair luminosities due to the transitions of protons and electrons between different Landau levels. In such strong magnetic fields the quantum calculation is necessary because the energies of νν-pairs are much larger than the temperature.
In the semi-classical calculations energies of the νν-pairs are assumed to be almost zero, and the momentum transfer from the magnetic field cannot be taken into account exactly. This would cause the neutrino energy spectra to shift to lower energies in the semi-classical calculation, resulting in a much smaller total luminosity than that of the quantum calculations.
In actual magnetars the magnetic field is weaker than 10 15 G in the low density region, so that in low density region the νν-pair luminosity is expected to be much larger than that of neutrinos due to the MU process. Therefore, the present results suggest that one needs to introduce the νν-pair emission process when calculating the cooling rate of magnetars.
We expect that the cooling rate would increase due to the νν-pair emission process. On the other hand, additional energy made by transitions between Landau levels could contribute to high energy part of the thermal spectra of neutrinos and anti-neutrinos, which may heat the ambient gas surrounding magnetars through absorption. Thus, the νν-pair production process may contribute to both the heating and cooling of magnetars whose surface temperature is larger than that of normal neutron stars. More careful calculations of neutrino transport including these processes are highly desirable to quntity this speculation.