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Asymmetric neutrino production in strongly magnetizedproto-neutron starsTomoyuki Maruyama,1,2 Myung-Ki Cheoun,2,3 Jun Hidaka,4 Toshitaka Kajino,2,5 Takami Kuroda,6Grant J. Mathews,7 Chung-Yeol Ryu,3 Tomoya Takiwaki,2 and Nobutoshi Yasutake21College of Bioresource Sciences, Nihon University, Fujisawa 252-0880, Japan2National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan3Department of Physics, Soongsil University, Seoul 156-743, Korea4Meisei University, Hino, Tokyo 191-8506, Japan5Department of Astronomy, Graduate School of Science, University of Tokyo,Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan6Department of Physics, University of Basel, CH-4056 Basel, Switzerland7Center of Astrophysics, Department of Physics, University of Notre Dame,Notre Dame, Indiana 46556, USA(Received 28 May 2014; published 23 September 2014)We calculate the neutrino production cross-section through the direct URCA process in proto-neutronstar matter in the presence of a strong magnetic field. We assume isoentropic conditions and introduce anew equation of state parameter set in the relativistic mean-field approach that can allow a neutron star massup to 2.1M as required from observations. We find that the production process increases the flux ofemitted neutrinos along the direction parallel to the magnetic field and decreases the flux in the oppositedirection. This means that the neutrino flux asymmetry due to the neutrino absorption and scatteringprocesses in a magnetic field becomes larger by the inclusion of the neutrino production process.DOI: 10.1103/PhysRevD.90.067302 PACS numbers: 25.30.-c, 26.60.-c, 24.10.JvThemagnetic field in neutron stars plays an important rolein the interpretation of many observed phenomena. Indeed,strongly magnetized neutron stars (dubbed magnetars[13]) hold the key to understanding the asymmetry insupernova (SN) remnants, and the still unresolved mecha-nism of nonspherical SN explosions. Such strong magneticfields are also closely related to the unknown origin of thekick velocity [4,5] that proto-neutron stars (PNSs) receive atbirth. Although several post-collapse instabilities have beenstudied as a possible source to trigger a nonsphericalexplosion leading eventually to a pulsar kick, there remainuncertainties in the global initial asymmetric perturbationsand the numerical simulations [6,7].In strongly magnetized PNSs, asymmetric neutrinoemission emerges from parity violation in the weakinteraction [8,9] and/or an asymmetric distribution of themagnetic field [10]. Kusenko, Segre and Vilenkin [11]criticized this conclusion through calculation of onlyneutrino-neutron collisions. However they neglected thePauli-blocking effect, and their proof is only applicable inthe very low-density region. In addition, they do not takeinto account the neutrino absorption, which makes a largecontribution to the asymmetry [12]. Recent theoreticalcalculations [1214] have suggested that even a 1%asymmetry in the neutrino emission out of a total neutrinoluminosity 1053 ergs might be enough to explain theobserved pulsar kick velocities.In our previous work [12,15], we calculated neutrinoscattering and absorption cross sections in hot, densemagnetized neutron-star matter including hyperons in arelativistic mean field (RMF) theory [16]. We evaluated boththe associated pulsar kick velocities [15] and the spindeceleration [17] for PNSs. The magnetic field was shownto enhance the scattering cross-section for final neutrinomomenta in a direction parallel to the magnetic field and toreduce the absorption cross-section for initial neutrinomomentum along the same direction. When the neutrinomomentum is anti-parallel to the magnetic field, the oppositeeffect occurs. For a magnetic field strength of B 2 1017 G and densities in excess of nuclear matterB 1 50, the enhancement in the scattering cross-sec-tion was calculated to be about 1% [15], while the reductionin the neutrino absorption was 24%. This enhancement andreduction were conjectured to increases the neutrinomomen-tum flux emitted along the north magnetic pole, whiledecreasing the flux along the south pole when the magneticfield has a poloidal distribution. By exploiting a one-dimen-sional Boltzmann equation in the attenuation approximationand including only neutrino absorption, we estimated that thepulsar kick velocity is about 520 km=s for a star with baryonmassMNS 1.68 M, B 2 1017 G, T 20 MeV, andET 3 1053 erg. It was suggested in the previous worksthat the neutrino asymmetries produced in the deep interior ofthe star at densities well above the saturation density arealmost washed out. We confirmed [15] that the neutrinoswhich propagate toward the lower densities near the surfaceof the star eventually determine the final neutrino asymmetry.In those calculations, however, we did not consider theneutrino production process through the direct URCA (DU)and modified URCA (MU) processes. A strong magneticPHYSICAL REVIEW D 90, 067302 (2014)1550-7998=2014=90(6)=067302(5) 067302-1 2014 American Physical Societyhttp://dx.doi.org/10.1103/PhysRevD.90.067302http://dx.doi.org/10.1103/PhysRevD.90.067302http://dx.doi.org/10.1103/PhysRevD.90.067302http://dx.doi.org/10.1103/PhysRevD.90.067302field may lead to an angular-dependence of the neutrinoproduction in the URCA process because of the spinpolarization of electrons and positrons in matter [18,19].It has also been reported [20,21] that the Landau levels dueto a magnetic field can cause an asymmetry in the neutrinoemission which causes a pulsar kick velocity. Furthermore,an angular dependence of the neutrino production causedby a magnetic field has even been reported [22,23] to occurin a pion condensation phase or in a quark-matter color-super conducting phase [24].Therefore, the neutrino production process in the pres-ence of a magnetic field may also lead to asymmetricneutrino emission from PNSs. In this report, we take thisproduction process into account in our model by calculat-ing the cross-sections using the PM1L1 parameter set [25]with an isothermal neutron-star model. In this parameter setwe cannot reproduce the observed neutron star mass of1.97 M for PSR J1614-2230 [26] and 2.01 M for PSRJ0348 0432 [27] when including particles. In thiswork, therefore, we improve the RMF parameter set toallow a more massive neutron star. We then study theneutrino absorption and production though the DU processusing the RMF approach in an isoentropic neutron star asshown below.We start from the RMF Lagrangian comprised ofnucleons, fields, sigma and omega meson fields, andthe iso-scalar and Lorentz vector interaction betweennucleons. We parametrize the nucleon mean-fields toreproduce the consensus nuclear-matter properties, i.e. abinding energy of 16 MeV, a nucleon effective mass ofM=M 0.65, and a incompressibility coefficient of K 250 MeV in symmetric nuclear matter at a saturationdensity of 0 0.17 fm3. In the analysis of heavy-ionexperiments [28] an EOS with M=M 0.65 and K 200 400 MeV simultaneously reproduces the results ofthe transverse-flow and sub-threshold K-productionexperiments.In order to stiffen the EOS when including the lambda() particles, we introduce an additional - interactionterm with a Lagrangian written asL h2s2m2sfg2 h2v2m2vfgfg; 1where hs and hv are the scalar and vector couplingsbetween the two s, respectively.The scalar and vector meson masses are taken to bems 550 MeV and mv 783 MeV, as in previous calculations[12,15,17]. The and couplings are taken to be2=3 that of the nucleon, i.e. g; 2=3g; by takingaccount of the quark degrees of freedom. For the -interaction we use hs 0.3467g and hv 0.5g. Our newEOS including particles can reproduce massive neutronstars up to 2.1 M. Various theoretical attempts to repro-duce such a heavy neutron star by including hyperons arediscussed in literature [29].Figure 1 shows total energies per baryon ET=A, temper-ature profiles, and number fractions for various constituentparticles in an isoentropic system with entropy per baryonS=A 1 or 2. The proton fraction is xp 0.3 in all densityregions. When one includess in the system, they appear ata density B 20 and the number fraction x increaseswith increasing density. In these isoentropical models,the proton fraction slightly decreases even when the sappear, while in an isothermal model xp decreases morerapidly.Using the above EOS we calculated the neutrinoabsorption and production cross-sections. In this workwe assume a uniform dipole magnetic field along thez-direction, i.e. B Bz. Since even for an astronomicallystrong magnetic field the associated energy scale is stillmuch weaker than the strong interactions,ffiffiffiffiffiffieBp a,where a is the chemical potential of the particle a, wecan treat the magnetic field perturbatively. Hence, weignore the contribution from the convection current andonly consider the spin-interaction [12,15].We set B 1017 G as a representative maximum fieldstrength inside a neutron star. This value correspondsto NB 0.32 MeV which satisfies jbBj N . Theinitial momentum here is taken to be the chemicalpotential jkij .We calculated the absorption (e e) neutrino cross-sections perturbatively, and separated the cross section intothe two parts: A 0A A, where 0A is independent ofB, and A is proportional to B. Related weak couplingsare taken from Ref. [30].0100200300400500ET / A (MeV)p,np, n, (a) S/A = 1020406080T (MeV) (b)0 1 2 3 4 50.00.20.40.6B / 0x p,(c)(d) S/A = 2(e)0 1 2 3 4 5B / 0(h)FIG. 1 (color online). Upper panels (a) and (d): Densitydependence of the total energy per baryon ET=A of neutron-starmatter at a fixed lepton fraction YL 0.4 and for entropiesS=A 1 (a) and 2 (d). Middle panels (b) and (e): Temperatureprofiles for entropies S=A 1 (b) and 2 (e). Solid and dashedlines represent results with and without particles, respectively,in the EOS. Lower panels (c) and (f): Number fractions of protonsxp and of particles x for entropies S=A 1 (c) and 2 (f). Solidand dashed lines stand for xp with and without particles,respectively. Dot-dashed lines show the x fraction.BRIEF REPORTS PHYSICAL REVIEW D 90, 067302 (2014)067302-2In Fig. 2(ad), we show the magnetic part of theabsorption cross-section as a function of the initial neutrinoangle for entropy S=A 1 without (a) and with (b) particles in the EOS, and for S=A 2 without (c) and with(d) particles.At B 0, the magnetic field suppresses the absorptioncross-sections in a direction parallel to the magnetic field Bby about 8.3% for an entropy of S=A 1 and by about 4%for S=A 2. Hence, the magnetic field increases the emittedneutrino flux in the direction of the north magnetic pole anddecreases the flux along the south magnetic pole. Thesuppression of A for S=A 1 at B 0 turns out to bemuch larger than in an isothermal model with T 20 MeV.This is because the temperature at this density in theisoentropic model is only about T 7 MeV. At lowertemperature the magnetic contribution becomes larger.However, at higher densities and temperatures the suppres-sion is comparable in the two models.As discussed in Ref. [15], the normal parts of the cross-sections, 0, decrease as the temperature and the densitybecome lower. In contrast, the magnetic parts, , increaseas the temperature becomes lower. Also, as the densitydecreases the magnetic part decreases more slowly than thenormal part. This is because is approximately propor-tional to the fractional area of the distorted Fermi surfacecaused by the magnetic field. Because of these two effects,the relative strength A=0A becomes significantly largerwhen the density and entropy are small.Now considering the neutrino production process, wedefine the integrated cross-section for neutrino productionas follows,prk Zd3ki23 neeikid3dk3prk; ki; 2where eiki is the single particle energy of electrons withmomentum ki. The cross-section and the electron momen-tum distribution function in the presence of a magnetic fieldare separated in a perturbative way into the two partsd3prdk3 d30prdk3 d3prdk3;neeiki nejkij neki: 3The first terms are independent of the B field, and thesecond terms are proportional to B. Then, the neutrinophase-space distribution for the DU process in the presenceof a magnetic field also separates into the two partsprk 0prk prkZd3k23 n0e k d30prdk3Zd3k23n0e k d3prdk3 nekd30prdk3:4Detailed expressions for ei and ne are given inRef. [17]. As shown in Ref. [17], we can obtain thecross-section for e Bi Bf e by exchanging thelepton chemical potentials for the neutrinos and electrons inthe cross-section for e Bi Bf e.Figure 3 shows the magnetic part of the integratedproduction cross-section pr normalized to 0pr as afunction of f. Calculations were made for densities inthe range 0 B 50 as indicated. Final neutrinoenergies were taken to be equal to the neutrino chemicalpotential, jkj .At B 0 with entropy S=A 1, the magnetic part isenhanced by about 10% for 0 and suppressed byabout 6% for 180. This is true for both systems withand without particles. Hence, the magnetic-field givesrise to an about 8% asymmetry in the production process.As the density increases, the magnetic contributionbecomes smaller, particularly in the system with par-ticles. For S=A 2, the asymmetry is about 6% at 0and 4% for 180 at N 0, so that the asymmetry isslightly smaller than for S=A 1. At higher density theasymmetry also becomes smaller, particularly whenS=A 2.In any condition the neutrino production becomes largerin a direction parallel to the magnetic field B, and smaller inthe opposite direction. The net result is that the magneticfield increases the momentum flux of neutrinos emittedalong the north magnetic polar direction while decreasingthe flux in the south polar direction. This magnetic effect on0.050.000.05 A / A0(a) S/A = 1p, n(c) S/A = 2p, n1.0 0.5 0.0 0.50.050.000.05 cos i A / A0(b) S/A = 1p, n, 1.0 0.5 0.0 0.5 cos iB /0 = 1B /0 = 3B /0 = 5(d) S/A = 2p, n, FIG. 2 (color online). Ratio of the magnetic part of theabsorption cross-section A to the cross-section without amagnetic-field 0A. Lines are drawn for matter without s (a)and with s (b) at T 20 MeV. Solid, dot-dashed and dashedlines represent the results at B 0, 30 and 50, respectively.Neutrino incident energies are taken to be equal to the neutrinochemical potentials.BRIEF REPORTS PHYSICAL REVIEW D 90, 067302 (2014)067302-3the production process turns out to be of the same sign andmagnitude as the absorption process. Hence, the totalasymmetry induced by the magnetic field from bothprocesses should be about twice that from absorptionalone [15].In summary, we have calculated the magnetic contribu-tion to the neutrino production through the direct URCAprocess and the absorption during transport. We haveutilized an isoentropic model for the proto-neutron starand employed RMF theory (with and without particles)for the EOS and to compute the production cross-section.The asymmetry in the absorption becomes larger at B 0 than that in our previous calculation based upon anisothermal model. Furthermore, the asymmetry in theproduction cross-section is found to be also enhanced bythe magnetic-field with the same magnitude and sign as inthe absorption process.Since the scattering process also enhances the neutrinoasymmetry [15] in addition to the direct URCA process, wecan conclude that the magnetic-field effect causes asym-metric neutrino emission from a PNS through the combi-nation of the production process as well as absorption andscattering. Therefore, the neutrino emission asymmetryfrom the neutrino sphere should be significantly larger thanpreviously estimated. 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