Role of magnetic interactions in neutron stars

In this work, we present a calculation of the non-Fermi liquid correction to the specific heat of magnetized degenerate quark matter present at the core of the neutron star. The role of non-Fermi liquid corrections to the neutrino emissivity has been calculated beyond leading order. We extend our result to the evaluation of the pulsar kick velocity and cooling of the star due to such anomalous corrections and present a comparison with the simple Fermi liquid case.


Advantages of quark matter
• The DURCA process cannot occur because it is not kinetically possible at such temperature of interest.
• The MURCA requires a bystander particle. The neutrino emission rate is found to be insignificant.

Quark dispersion relation
• Interactions within the medium severely modify the self-energy of the quarks. For quasiparticles with momenta close to the Fermi momentum pF, the one-loop self-energy is dominated by the soft gluon exchanges.

Quark self energy
• The analytical expression for the one-loop quark self energy (for T ∼ |E − μ| ≪ g μ ≪ μ) exhibits a logarithmic singularity close to the Fermi Surface.
• Low temperature expansion of the on-shell fermion self energy for the ultrarelativistic case is given as: Not over yet..
• HDL (Hard Dense Loop) resummation for gluon propagator required because higher order diagrams can contribute to lower order in coupling constant which is missing in bare p-QCD; resummation done by means of

Mean free path of the degenerate neutrinos •
The absorption process: • The scattering process: Mean free path of the nondegenerate neutrinos • The absorption process: • The scattering process:

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The relation between neutrino emissivity and neutrino mean free path is obtained as:

Specific heat capacity of degenerate quark matter
• The specific heat of normal(non-color superconducting) degenerate quark matter shows NFL behavior at low temperature.
• Thus, at low temperatures, the resulting deviation of the specific heat from its FL behavior is significant in case of normal quark matter and thus of potential relevance for the cooling rates of NS with a quark matter component. Think..

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In this work, we have calculated the MFP of degenerate and non-degenerate neutrinos both for the scattering and absorption processes.

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We then find the expression for neutrino emissivity for nondegenerate neutrinos with NLO corrections. It is seen that both MFP and emissivity contain terms at the higher order which involve fractional powers in (T /µ).

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We have found that there is a decrease in the MFP due to NLO corrections.

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We reconfirm that the leading order correction to the quantities like MFP or emissivity are significant compared to the Fermi liquid results.

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The NLO corrections, which we derive here, have however been found to be numerically close to the LO results.

Conclusions
• In this work, we have derived the expressions of the pulsar kick velocity including the NFL corrections to the specific heat of the degenerate quark matter core.

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The contributions from the electron polarization (χ) for different cases has also been taken into account to calculate the velocities.

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We have included the effect of the external magnetic field into the specific heat of the degenerate quark matter for the calculation of the pulsar kick velocity. The calculation of the specific heat of the degenerate quark matter in magnetic field for the NFL LO and NLO are new.

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We have found that the NFL LO contributions are significant while calculating the radius-temperature relationship as seen from the graphs presented for the case of the neutron star with moderate and high magnetic field. The anomalous corrections introduced to the pulsar kick velocity due to the NFL (LO) behavior increases appreciably the kick velocity for a particular value of radius and temperature. However, for all the cases, no appreciable change in the R-T relationship has been observed for the NLO correction with respect to the LO case.

Dynamical screening
• The longitudinal and transverse HDL propagators are given as: • For q 0 → 0 longitudinal photons acquire an effective mass m D 2 =2m 2 which screens IR singularities.
• For q 0 → 0 transverse (or magnetic) interactions are NOT screened; only dynamical screening.
• Retaining the leading term for (q 0 /q → 0) we obtain: • Frequency dependent screening with a frequency dependent cutoff.
• This cut-off is able to screen IR singularities so that finite results are obtained.