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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, in neutron stars, under β-equilibrium. We show that, if magnetic field is large enough for protons and 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 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 emission from a 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 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 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 2 2 2 1q1 +(p2 +k2) |M−| =8G gA(ε0+k3)(ω−κ3)exp − (24) 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 . By this reason, one F F F can substitute δ εn − εn± δ εp − εp δ εe − ε0 instead of δ (ε3 − εn±) δ (ε2 − εp) δ (ε1 − ε0). Considering electrons as being ultrarelativistic, we obtain that the contribution from (ε0 + k3) is not small only when k3 ≥ 0. We finally arrive to   

∗ 457π 2 6 MpMn ν =2 G eBT × (32) 40320 pF f q2 − 4p2 q2 [ 1+g2 Θ q2 −4p2 exp − F + F +4 1+g2 Θ q2 exp − F + A F+ F 2eB A F+ 2eB       q2 − 4p2  q2 +2g2 Θ q2 − 4p2 exp − F − F +8g2Θ q2 exp − F − ] A F − F 2eB A F− 2eB       Where the charge neutrality condition kF = pF was used. Considering the energy loss rate due to reaction (2) we would arrive to the same expression for emissivity as (25) but with different Pauli blocking factors. Under the β-equilibrium assumption (39), one has the following relationship :

fe (ε0) fp (εp)(1−fn(εn±)) = (1 − fe (ε0)) (1 − fp (εp)) fn (εn±) (33) Therefore, the neutron decay process (2) gives the same energy loss rate as process (1), although giving final antineu- trinos. In this way, the total luminosity per unit volume of the direct URCA process is twice that of Eq.(25). Factor

4 2 in (32) takes this into account. Since both electrons and protons are at the ground Landau level, the total third component momentum conservation reads either q3 = kF + pF , when proton and electron go in the same direction, along the magnetic field, or q3 = kF − pF when the initial particles move in opposite directions . Since kF = pF and 2 2 2 2 2 2 2 2 qF ± >q3, four possibilities are open : qF + > (2pF ) , qF − > (2pF ) , qF + > 0andqF− >0. This is the origin of the four terms in the latter equation, which are multiplied by their corresponding step functions, where Θ (x) gives +1 if the argument exceeds 0, and is 0 otherwise. As we will show later, opening or closing of these channels will give rise to jumps on the emissivity. We can rewrite Eq.(32) as follows :

∗ 27 6 Mp Mn B ν =1.04 × 10 T9 × pF Mp B0 f q2 − 4p2 q2 [ 1+g2 Θ q2 −4p2 exp − F + F +4 1+g2 Θ q2 exp − F + A F+ F 2eB A F+ 2eB       q2 − 4p2  q2 +2g2 Θ q2 − 4p2 exp − F − F +8g2Θ q2 exp − F − ] (34) A F − F 2eB A F− 2eB       3 9 2 where the emissivity is given in units of (ergs/cm s), T9 is the temperature in units of 10 K and B0 = Mp /e ' 1.5 × 1020G. In order to estimate numerically the emissivity of the direct URCA process in a strongly magnetized neutron star, we consider nuclear matter and electrons in β-equilibrium within a nonrelativistic Hartree approach in the linear σ-ω-ρ model, as it was made (relativistically) in [8] but with taking into account anomalous magnetic moments of proton and neutron in the energies for these particles. Then the interaction energies for protons and neutrons are given by

g 2 1 g 2 U = ω n + ρ ρ (35) p m b 4 m 3  ω   ρ 

2 2 gω 1 gρ U = n − ρ (36) n m b 4 m 3  ω   ρ  where nb = np + nn is the total number density of and ρ3 = np − nn with (see Eq. (5))

2 Mp B np = ne = 2 pF (37) 2π B0 and 1 n = q3 + q3 (38) n 6π2 F + F −

 2 2 The parameters for the coupling constants and mesons masses are taken from [9] to be gσ (M/mσ) = 357.47, 2 2 2 2 gω (M/mω) = 273.87, and gρ (M/mρ) = 97. In this model, mass M = 939MeV is the same for proton and neutron. To make our estimates for the emissivity, we solved for variables qF + and pF the β-equilibrium condition

ψp + ψe = ψn (39) with the help of equations (35-38) and (22) for a given density nb. In order to be fully consistent with incorporating magnetic moments of proton and neutron, one would also like to find the effective mass M ∗ from this modified model. However, this would add some (perhaps unessential) calculation problems. Our purpose in this Letter is to point out some qualitative new results and motivate more realistic computations. For this reason , we have chosen for our estimates the values of the effective mass that are given by [8]. More precisely, we have taken ∗ −3 ∗ M =0.6M for nb equal to nuclear saturation density, which is n0 =0.15fm in this model, and M =0.3M for nb =2n0. In Fig. 1 we have plotted the emissivity of the direct URCA process, calculated from (34) as a function of the magnetic field intensity (in units of B/B0) for two values of the baryon density : nb = n0 and nb =2n0.Wehave chosen, as a representative value for the temperature, T =109K. As one can see from this figure, if the magnetic field is larger than B ' 7 × 1017G ,theννemissivity from the neutron star is many orders of magnitude larger than

5 that in the standard cooling scenario. Another feature which is readily observed is that, as magnetic field increases, 2 2 jumps on the emissivity can appear. The first jump in the curve nb = n0 is due to qF − − 4pF becoming negative 2 when B/B0 & 0.014. The second, much smaller one, at B/B0 ∼ 0.026 , corresponds to qF − becoming zero. For the 2 2 curve nb =2n0 we observe a different situation : qF − − 4pF is always negative, but an abrupt increase appears due 2 2 to the fact that qF + − 4pF is first negative and, as magnetic field increases, it becomes positive at B/B0 ∼ 0.03. In this Letter we have considered the direct URCA process in degenerate nuclear matter under β-equilibrium in a strong magnetic field. These reactions are strongly suppressed by Pauli blocking in the field-free case, but we showed that, in a magnetic field larger than B ' 7 × 1017G , direct URCA reactions are open for an arbitrary proton concentration, and will lead to fast neutron star cooling. We restricted ourselves to these values of the magnetic field because in this case all protons, as well as electrons, occupy only the ground Landau bands at zero temperature. However, as one can see from Fig. 1, the direct URCA process can substantially contribute to neutron star cooling, even if the magnetic field is smaller than the value considered above. A more detailed study of this problem requires some additional complicated calculations. Also, we have considered matter densities not much larger than saturation densities. For these densities, can be treated as non-relativistic particles. However, if one intends to go to larger densities, relativistic effects should be taken into account. This will be the subject of a future work.

ACKNOWLEDGMENTS

This work has been partially supported by Spanish DGICYT Grant PB94-0973 and CICYT AEN96-1718. L.B. L. would like to thank for partial support of this work by RFFR Grant 97-02-16501.

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24 0.01 0.02 0.03 0.04 FIG. 1. Neutrino emissivity of the direct URCA process as a function of the magnetic field intensity B (B in units of the 9 proton critical field B0), for two values of the baryon density and for a temperature T =10 K.

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