View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Direct URCA process in neutron stars with strong magnetic fields. L. B. Leinson Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation RAS, 142092 Troitsk, Moscow Region, Russia A. P´erez Departamento de F´ısica Te´orica, Universidad de Valencia 46100 Burjassot (Valencia) Spain We calculate the emissivity for the direct URCA process in strongly magnetized, degenerate matter in neutron stars, under β-equilibrium. We show that, if magnetic field is large enough for protons and electrons to be confined to the ground Landau levels, the field-free threshold condition on proton concentration no longer holds, and direct URCA reactions are open for an arbitrary proton concentration. For densities around the saturation point and a magnetic field larger than B ' 7 × 1017G, this process leads to fast neutron star cooling. Magnetic fields as large as 1018 G, and even more intense, have been recently discussed [1], [2], [3] as possibly existing inside neutron stars and some other compact astrophysical objects. Effects of strong magnetic fields on fundamental weak interactions are of great importance for neutron star evolution. It is already shown [4], [5] that, induced by an intense magnetic field, anisotropic neutrino emission from a supernova can impart a very large recoil momentum to newly-born neuron stars. Superstrong magnetic fields are also capable to shift β-equilibrium composition in the neutron star interior [6]. In this Letter we study the emissivity of direct URCA reactions B p + e− → n + ν (1) B n → p + e− + ν (2) in strongly magnetized neutron stars under β-equilibrium. We consider degenerate nonrelativistic nuclear matter and ultrarelativistic degenerate electrons to be totally transparent for neutrinos and antineutrinos. We also assume that at zero temperature the neutron star is β-stable, but at non-zero temperature T EFi (i=n, p, e) reactions (1) and − (2) proceed near the Fermi energies EFi of participating particles. In the field-free case, for reactions p + e → n + ν ; n → p + e− + ν could occur , momentum conservation near the Fermi surfaces requires the following inequality among Fermi momenta of participating particles: (p) (e) (n) pF + kF ≥ qF (3) In conjunction with charge neutrality condition np = ne, the above inequality leads to the well-known threshold for proton concentration Yp ≡ np/ (np + nn) ≥ 1/9, where np and nn are the number densities of protons and neutrons, respectively. In other words, in the field free case, direct URCA reactions are strongly suppressed by Pauli blocking in the neutron-rich nuclear matter (assuming matter is formed only by protons, neutrons and electrons). √ In magnetized media, transversal momenta of charged particles are defined only within an accuracy ∆p⊥ ∼ eB. √ (n) We show in this Letter that strong magnetic fields eB ∼ qF break down condition (3) so that direct URCA process is allowed in the degenerate n-p-e system under β-equilibrium for an arbitrary proton concentration. We examine reactions (1) and (2) in a constant magnetic field large enough so that 1 k2 B> F (4) 2e √In this case the Fermi energy of degenerate electrons is smaller than the energy of the first excited Landau level m2 +2eB . This means that, at zero temperature, all electrons occupy only the ground Landau band and they are polarized in the direction opposite to the magnetic field . The number density of electrons reads 1Here e is the absolute value of electron charge. For an electron having a Fermi momentum of 50 Mev one obtains B>2×1017 G. We set ~ = c = kB =1. 1 eB kF eB n = dk = k (5) e 4π2 3 2π2 F Z−kF where kF is the Fermi momentum of degenerate electrons. Following charge neutrality np = ne, one obtains the proton Fermi momentum pF to be equal to kF . Therefore, if inequality (4) is satisfied, then degenerate protons are also confined to the ground Landau state and are totally polarized in the direction paralell to the external magnetic field B =(0,0,B). The emissivity for reactions (1) and (2) can be computed using standard charged current β-decay theory. The matrix element for the V-A interaction reads G 3 µ Mσ = √ d rΨnσ (r) γµ (1 − gAγ5)Ψp (r)Ψν (r)γ (1 − γ5)Ψe (r)(6) 2 Z where Ψν (r)andΨe represent the neutrino and electron fields , respectively, while Ψp and Ψn stand for the proton and neutron; G = GF cos θC with θC being the Cabibbo angle, and gA ' 1.261 is the axial-vector coupling constant. By using the asymmetric (Landau) gauge A =(0,Bx,0) , the four-dimensional polarized wave functions can be expressed in terms of stationary states in the normalization volume V = L1L2L3. The electron wave function, corresponding to the ground Landau band with energy 2 2 ε0 = m + k3 (7) q reads 1 Ψe (r)=√ √ exp(−iε0t)exp[i(k3z+k2y)] ϕ0(ξ)ue (8) 2ε0 L2L3 with √ k ξ = eB x + 2 (9) eB and 0 1 (ε0 + m) ue = √ (10) ε0 + m 0 k − 3 k2,k3 are electron momenta in the y and z directions, respectively. The function ϕ0(ξ) is the eigenfunction of the one-dimensional harmonic oscillator, normalized with respect to x: √ 1/2 eB ϕ (x)= √ exp(−eBx2/2) (11) 0 π In order to take into account strong interactions in nuclear matter, we consider nonrelativistic protons, with an ∗ effective mass Mp , moving in the self-consistent uniform nuclear potential Up. Then the proton wave function reads 1 Ψp (r)=√ exp(−iE0t)exp[i(p3z+p2y)] ϕ0 (ζ) uP (12) L2L3 where p2,p3 are proton momenta in the y and z directions, respectively, and √ p2 ζ = eB x − (13) eB Polarized protons are given by the following four-spinor W u = + (14) p 0 † with W+ being a nonrelativistic spin-up bispinor normalized by the condition W+W+ = 1. By introducing the anomalous magnetic moment of the proton, its energy reads [7]: 2 2 p3 E0 ' Mp + εp + Up εp = (15) 2Mp Here f f e M = M ∗ (g 1) B (16) p p − ∗ p − 2Mp with the proton’s Lande factor gp =2.79 f ∗ In a similar way, we consider nonrelativistic neutrons, of effective mass Mn, moving in the self-consistent uniform nuclear potential Un. One can describe neutron polarization by the four-spinors unσ corresponding to two different spin polarizations σ = ± with respect to the magnetic field W u = σ (17) nσ 0 † where Wσ are nonrelativistic bispinors normalized by the condition Wσ Wσ = 1. Then the neutron wave function reads 1 Ψnσ (r)=√ exp(−iEqσt)exp[i(q3z+q2y+q1x)] unσ (18) L2L3L3 Here the energy of a neutron with spin polarization σ is q2 E = M∗ + ε + U ε = σµ B (19) qσ n qσ n qσ ∗ − n 2Mn which incorporates neutron interaction with the external magnetic field due to its anomalous magnetic moment e µn = gn (20) 2Mp with gn = −1.91 and Mp the proton free mass. According to Eq.(19), neutrons polarized along the magnetic field (σ = +), and neutrons polarized in the opposite direction ( σ = −), have different momenta qF ± at the Fermi surface F εq± = εn .Takingintoaccount q2 εF = F± µ B (21) n ∗ ± n 2Mn one finds 2 2 ∗ qF − = qF + +4MnµnB (22) Explicit evaluation of the matrix elements in (6) yields 2 2 2 2 2 1q1 +(p2 +k2) |M+| =2G 1+gA (ε0 +k3)(ω+κ3)exp − (23) 2 eB ! 2 2 1q +(p2 +k2) |M |2 =8G2g2 (ε +k )(ω−κ )exp − 1 (24) − A 0 3 3 2 eB ! We have neglected here the neutrino momentum κ = ω ∼ T , which is small as compared to neutron momenta qF ±. The neutrino emissivity reads : eBL1/2 eBL1/2 3 3 1 L2dk2 L3dk3 L2dp2 L3dp3 Vd q Vd κ ν = (25) V 2L2L2 2π 2π 2π 2π (2π)3 (2π)3 2 3 Z−eBL1/2 Z Z−eBL1/2 Z Z Z 2 ω |M+| δ (Eq+ + ω − E0 − ε0)(1−fn+)fefp × 2π 2 2ω2ε0 +|M | δ(E +ω−E −ε )(1−f )f f − q− 0 0 n− e p ×2πδ (q3 + κ3 − p3 − k3)2πδ (q2 + κ2 − p2 − k2) 3 Here (1 − fnσ) fefp are statistical factors corresponding to Fermi-Dirac distributions of participating particles with ∗ F F F chemical potentials ψn ' Mn + Un + εn , ψe ' εe and ψp ' Mp + εp + Up. Fermi energy of neutrons is defined in (21). For electrons and protons they are given by : f 2 F 2 2 F pF εe = m + kF εp = (26) 2M q p With this definitions one has f 1 fn± = F (27) exp [(εq± − εn ) /T ]+1 1 fe = F (28) exp [(ε0 − εe ) /T ]+1 1 f = (29) p F exp εp − εp /T +1 where particle energies are defined in (7), (15), (19) and ω =|κ|. Since the integrand in (25) depends on p2 and k2 only through the combination p2 + k2, we can introduce a new variable k2 + p2 → p2 and perform integration over k2. We can also neglect the small neutrino momentum in the momentum conservation δ-functions. Thus, we obtain 2 2 2 G eB 3 3 (ε0 +k3) q1 + p2 ν = 7 dk3dp2dp3d qd κδ(q3 −p3 −k3)δ(q2 −p2) exp − (30) (2π) ε0 2eB Z 1 2 ∗ 2 1+gA (ω+κ3)δ Mn −Mp +Un −Up +εq+ +ω−εp −ε0 (1 − fn+) fefp × 2 ∗ +2g (ω− κ3) δ M − Mp + Un − Up + εq− + ω − εp − ε0 (1− fn−) fefp A n f The integrals can be performed in a straightforwardf way by using the following identity ∗ fe (ε0) fp (εp)(1−fn(εn±)) δ Mn − Mp + Un − Up + εq± + ω − εp − ε0 (31) ∞ ∞ ∞ = dε3 dε2 dε1fe (ε1) fp (ε2f)(1−fn (ε3)) Z−∞ Z−∞ Z−∞ ∗ ×δ Mn − Mp + Un − Up + ε3 + ω − ε2 − ε1 δ (ε3 − εn±) δ (ε2 − εp) δ (ε1 − ε0) F F F Then, due to Pauli blocking, contributionf to the integrals peaks at ε1 ≈ εe , ε2 ≈ εp and ε3 ≈ εn .