Neutrino mean free paths in spin-polarized neutron Fermi liquids.

# Neutrino mean free paths in spin-polarized neutron Fermi liquids.

M. Ángeles Pérez-García Department of Fundamental Physics and IUFFyM,
University of Salamanca, Plaza de la Merced s/n 37008 Salamanca
July 19, 2019
###### Abstract

Neutrino mean free paths in magnetized neutron matter are calculated using the Hartree-Fock approximation with effective Skyrme and Gogny forces in the framework of the Landau Fermi Liquid Theory. It is shown that describing nuclear interaction with Skyrme forces and for magnetic field strengths , the neutrino mean free paths stay almost unchanged at intermediate densities but they largely increase at high densities when they are compared to the field-free case results. However the description with Gogny forces differs from the previous and mean free paths stay almonst unchanged or decrease at densities . This different behaviour can be explained due to the combination of common mild variation of the Landau parameters with both types of forces and the values of the nucleon effective mass and induced magnetization of matter under presence of a strong magnetic field as described with the two parametrizations of the nuclear interaction.

###### pacs:
21.30.Fe,21.65.Cd,26.60.-c.

## I Introduction

The study of the behaviour of hadronic matter in the density-temperature, , diagram allows to have a deeper understanding of matter under extreme conditions. In this context, the high density, low temperature limit can be addressed for a fermion system using the Landau Fermi Liquid Theory (FLT) bookFL (). From a theoretical point of view the properties of this type of normal quantum systems can be studied calculating the interaction matrix element of quasiparticle (qp) excitations close to the Fermi surface. The inclusion of an additional component in the problem, a magnetic field, , allows further testing the properties of magnetized Fermi Liquids. The role of magnetic fields in bulk properties and equation of state has been partially analyzed in the past for nuclear matter latt () chak () and quark matter quark1 () quark2 (). Due to the tiny value of the neutron magnetic moment ( MeV pdb () and in order to provide a sizable magnetization, huge magnetic fields are needed.

The only scenarios where we have indication of such intense fields are, first, from estimates of the background magnetic fields created in heavy-ion collisions like those at RHIC rhic () and, second, in a subgroup of pulsars called magnetars. For these astrophysical objects surface magnetic field strengths are of the order G thom (); lazzati (). Recent numerical simulations sim () of formation of proto-neutron stars show that the field configuration plays a significant role in the dynamics of the core if the initial magnetic field is large enough. In particular, in the rapid cooling of the newly formed neutron-rich object neutrino transport is an important ingredient cooling (). However, some of these simulations lack from accurate and consistent neutrino transport, missing the impact of magnetic fields in the microphysics input that affects the dynamics of the collapsing dense objects.

In most of the existing calculations of nuclear matter (either symmetric, pure neutron or beta-equilibrated) the effect due to the presence of strong magnetic fields and the consistently induced spin polarization are discarded in a first approximation. Either relativistic prakash (); chakra () or effective approaches vida1 () have been used to obtain some insight into the equation of state (EOS) or some structure properties latt () in presence of magnetic fields. These include a possible transition to a ferromagnetic state, although simulations using realistic potentials seem to prevent it ferro (). In general, a non-vanishing magnetization in a low temperature nuclear plasma angprc () produces a resolution of some degenerated observables as obtained in the context of the FLT ang2 (); ang3 ().

## Ii Formalism

In this work we are interested in the response of a spin-polarized pure neutron plasma to a weak neutrino probe. It can be seen prakash () that for the density range , where the quark deconfinement is not expected to take place, and for magnetic field strengths of maximum strength G, allowed in principle by the scalar virial theorem, the neutral system is mostly neutrons. The maximum magnetic field strength we will consider is (as measured in units of the electron critical field with G) and the neutron fraction is prakash (). So the neutral plasma is mostly neutrons but leptons and additional baryons are also present in a tiny fraction that should be considered for full application in an astrophysical scenario where -equilibrium holds.

We are interested in exploring the effect of a strong magnetic field and the spin polarization of a pure neutron plasma through the structure functions, which provide information on density and spin density in-medium correlations. The homogeneous system under study is under the presence of an internal magnetic field, populated by species with paricle density , where is the spin -projection. is the spin excess and is the total particle density. For given thermodynamical conditions is obtained by minimizing the Helmholtz free energy per particle, , where is the energy per particle. Note that parallel (antiparallel) aligned magnetic moments (spins) are energetically favoured. We have considered an effective approach to describe the nuclear interaction using zero-range Skyrme forces vautherin () with two of the most widely used parametrizations given by the Lyon group SLy4 and SLy7 chabanat1 (); chabanat2 () and finite range Gogny with D1P d1p () and D1S d1s () forces. All of them provide good values for binding of nuclei and also for neutron matter EOS.

In the context of the FLT the properties of non-magnetized systems at low temperature have been evaluated plbbackman () by calculating the qp matrix element around the Fermi surface where the only dependence is on fermionic densities and the qp scattering angle, , involved. In the usual formalism, for the non-magnetized case the qp matrix element is written as a multipolar expansion in Legendre polinomials,

 Vph=∞∑l=0[fl+glσ1.σ2]Pl(cosθ), (1)

and are the so-called Landau parameters of multipolarity . In the more general case where any two possible spin orientations are taken into account, the polarized qp matrix elements notes () ang2 () are a crucial ingredient to compute the response functions to a a weakly interacting neutrino probe that excites a collective mode under the presence of a magnetic field . The Lindhard function in the system, , satisfies the Bethe-Salpeter equation and can be written in the dipolar () case in the random phase approximation (RPA) as a coupled system,

 χ(σ,σ′) = χ(σ)0δ(σ,σ′)+χ(σ)0∑σ′′=+,−f(σσ′′)0χ(σ′′σ′) (2) +γ(σ)1∑σ′′=+,−f(σ,σ′′)1Γ(σ′′,σ′),
 Γ(σ,σ′) = γ(σ)1δ(σ,σ′)+γ(σ)1∑σ′′=+,−f(σσ′′)0χ(σ′′σ′) (3) +γ(σ)2∑σ′′=+,−f(σ,σ′′)1Γ(σ′′σ′),

with the auxiliar definitions, and . Notice that the qp propagators have been given in annals () and the expressions for the coefficients can be written notes () in the Landau limit as and where . The qp effective mass in a magnetized system depends on the polarized dipolar coefficients bookFL (),

 m∗σ/m=1+13N0σ[f(σ,σ)1+(k2F,−σk2F,σ)f(σ,−σ)1] (4)

where is the quasiparticle level density at each polarized Fermi surface with momentum .

The generalized parameters are obtained by derivating the Helmhotz free energy with respect to the polarized density component, ang2 (), setting momenta on the polarized Fermi surfaces and expanding the resulting expression as a series in Legendre polinomials of multipolarity . These generalized parameters fullfill the following relations recovering the usual ones in FLT in the limit ang2 (),

 fl=f(σ,σ)l+f(σ,−σ)l2, (5)
 gl=f(σ,σ)l−f(σ,−σ)l2. (6)

With the generalized paramters and using the expressions in Eq. (2) the corresponding Lindhard function for the isovector () response of the plasma can be written as,

 χ(S=0)=χ(++)+χ(−−)+χ(+−)+χ(−+), (7)

and for the vector-axial () response as,

 χ(S=1)=χ(++)+χ(−−)−χ(+−)−χ(−+). (8)

Then the previous expression of the Lindhard function in RPA approximation rpa () include in-medium correlations at zero temperature. From them, one can obtain the structure functions given by,

 SS=0,1(ω,q)=−1πImχS=0,1(ω,q). (9)

The structure function allows to calculate the non-relativistic differential cross section of neutrinos scattering off matter via neutral currents from peth ()

 1VdσdΩdω=G2F8π3E′2[C2V(1+cosθ)S0(ω,q)+C2A(3−cosθ)S1(ω,q)] (10)

where is the incoming (outgoing) neutrino energy and is the neutrino incoming (outgoing) three-momentum. The transferred energy is and the transferred three-momentum is . The neutral current vector and axial vector charges are and where pdb (). is the Fermi coupling constant. Once the response has been evaluated it is straightforward to evaluate the neutrino mean free paths in the medium , .

## Iii Results

In this section we include the effect of in-medium correlations in the neutron magnetized system as obtained in the Hartree-Fock approximation in the presence of a strong magnetic field. In Fig. 1 the ratio of effective neutron mass as compared to the free value at saturation density, , is shown as a function of the logarithm of the magnetic field strength for the Skyrme SLy7 (a) and Gogny D1P (b) parametrizations. For each model upper (lower) curves correspond to spin up (down) polarized particles. With Skyrme description the intense field affects more both effective nucleon mass absolute and relative (up-down polarized components) values than with Gogny. The impact of density effects on the mean free path can be seen in Fig. 2. We plot the ratio of change of neutrino mean free paths in the RPA dipolar approximation for a fixed value of magnetic field strength, G, with respect to the field-free case, , as a function of the density (in units of nuclear saturation density, ). We consider Skyrme SLy7 (solid line), SLy4 (long dashed line) Gogny D1P (short dashed line) and D1S (dotted line) parametrizations and set as a typical value of neutrino incoming energy MeV. While Gogny forces show almost unchanged or very mild reduction of mean free paths at densities in the range , the Skyrme forces show a high density dramatic increase with respect to the field-free case. Note that all standard Skyrme forces predict the onset of a ferromagnetic transition in the range , and in our selection of interactions for the study cases it is near . However this feature is not present in the Gogny forces that prevent ferromagnetic transitions. At densities , effects due to the energetic contribution of the magnetic perturbation, introduced by the neutron magnetic moment, are small at the selected field ( G) with respect to changes in other single particle properties like effective masses. For even lower densities (i.e. ) it can be seen (see Fig. 7 in angprc ()) that the term produces a relevant contribution to the magnetization. For application in astrophysical scenarios and at low densities one should consider that the effect of a non-zero proton fraction determines the appearance of pasta phases pasta () where electromagnetic and nuclear interactions are frustrated and clustering of matter arises. As density grows, at fixed values of , the spin polarization decreases forming a plateau at intermediate densities before the possible appearance of a phase transition in the system.

In Fig. 3 we plot the ratio of change of neutrino mean free paths in the RPA dipolar approximation computed for a generic value of magnetic field strength with respect to the field-free case as a function of the logarithm (base 10) of the magnetic field strength, . We set a value of density and use SLy7 (solid line), SLy4 (long dashed line), D1P (short dashed line) and D1S (dotted line) parametrizations. For fields below G there is almost no change in the ratio but for larger strengths there is a decrease (increase) as computed with Gogny (Skyrme) forces. For this high density case the change can be as computed with the SLy7 parametrization while the Gogny D1P predicts a relative change . Note that the main contribution to the mean free paths comes from the fact that, as shown in Fig. 1, the Skyrme parametrization predicts a larger change in the absolute and relative values of the two effective masses of the spin polarized components. The Landau parameters and the energetic contribution of the magnetic perturbation ang2 () show a minor contribution to the structure functions, that in turn determine the mean free paths. It is worth mentioning that the Lindhard function,, has a rich structure in () that has been studied in ang3 (), however, the smallnes of the magnetic perturbation is washed out in the response of the system by the influence of the magnetization angprc () and density effects in the neutron effective mass. As we can see from Fig. 3, this result shows not only quantitative but also qualitative differences in the neutrino transparency of magnetized neutron matter.

## Iv Conclusions

We have investigated for the first time in the context of the Landau Theory of normal Fermi Liquids, the effect of a strong magnetic field on the variation of the neutrino mean free path in a partially magnetized pure neutron system within the framework of the non-relativistic Hartree-Fock approximation comparing Skyrme and Gogny forces. We find that for fields up to the maximum strength studied in this work, G, Skyrme forces show at high density an enhancement of neutrino transparency of the system, while Gogny forces predict a small decrease. These results can be explanined due to the fact that for the density, and field range considered in this work the variation of Landau parameters is a minor contribution compared to the effective mass and magnetization.

Acknowledgments
We acknowledge discussions with J. Navarro and A. Polls. This work has been partially funded by the Spanish Ministry of Science and Education under projects FIS2006-05319, FIS2009-07238 and Junta de Castilla y Leon Excellence program GR234. We also acknowledge support by CompStar, a research networking programme of the European Science Foundation.

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