Neutrino emission from triplet pairing of neutrons in neutron stars
aa r X i v : . [ a s t r o - ph . S R ] F e b Neutrino emission from triplet pairing of neutrons in neutronstars
L. B. Leinson
Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation RAS,142190 Troitsk, Moscow Region, Russia
Abstract
Neutrino emission due to the pair breaking and formation processes in the bulk triplet superfluidin neutron stars is investigated with taking into account of anomalous weak interactions. Weconsider the problem in the BCS approximation discarding Fermi-liquid effects. In this approach wederive self-consistent equations for anomalous vector and axial-vector vertices of weak interactionstaking into account the P − F mixing. Further we simplify the problem and consider thepure P pairing with m j = 0, as is adopted in the minimal cooling paradigm. As was expectedbecause of current conservation we have obtained a large suppression of the neutrino emissivity inthe vector channel. More exactly, the neutrino emission through the vector channel vanishes in thenonrelativistic limit V F = 0. The axial channel is also found to be moderately suppressed. Thetotal neutrino emissivity is suppressed by a factor of 1 . × − relative to original estimates usingbare weak vertices. . INTRODUCTION Thermal excitations in superfluid baryon matter of neutron stars, in the form of brokenCooper pairs, can recombine into the condensate by emitting neutrino pairs via neutral weakcurrents [1]. It is generally accepted that, for temperatures near the associated superfluidcritical temperatures, emission from pair breaking and formation (PBF) processes dominatesthe neutrino emissivities in many cases. Recently [2], it has been found however that theexisting theory of PBF processes based on the bare weak vertices violates conservationof vector weak current. Correct evaluations including anomalous interactions has shownthe neutrino emission by a nonrelativistic singlet superfluid is substantially suppressed.Consistent estimates of the inhibition factor can be found in Refs. [2]-[4]. The suppressionof neutrino emissivity from the S PBF processes was studied also in Refs. [5]-[8], althoughthese are controversial (see discussion in Refs. [3], [4]).Quenching of the neutrino emission found in the case of S pairing leads to higher tem-peratures that can be reached in the crust of an accreting neutron star. This allows to explainthe observed data of superbursts triggering [9], [10] which was in dramatic discrepancy withthe previous theory of the crust cooling. Numerical simulations of the neutron star coolingin the minimal scenario [11] have shown that the suppression of the PBF processes in thecrust of a neutron star has a significant effect at early times ( t < P and F channels [16], [17]. In Sec. IV, we present the general expressionfor the emissivity of the neutron PBF processes formulated in terms of the imaginary part ofthe current-current correlator. The widely used expression for the neutrino emissivity causedby the triplet pairing of neutrons was obtained in Ref. [12] with the aid of the Fermi goldenrule. Therefore before proceeding to the self-consistent calculation of the neutrino energylosses, in Sec. V, we reproduce this formula using the calculation technique developed in ourpaper so that an apposite comparison with Ref. [12] can be made. In Sec. VI, we considerthe anomalous vertices and the self-consistent superfluid response both in the vector andaxial channels. Here we focus on the P pairing with m j = 0, as is adopted in the minimalcooling paradigm [11]. Finally, in Sec. VII, we evaluate the self-consistent neutrino energylosses from the PBF processes in the triplet neutron superfluid. Section VIII contains ashort summary of our findings and the conclusion.In this work we use the standard model of weak interactions, the system of units ~ = c = 1 , and the Boltzmann constant k B = 1. II. PRELIMINARY NOTES AND NOTATIONA. The order parameter and Green functions.
The order parameter, ˆ D ≡ D αβ , arising due to triplet pairing of quasiparticles, representsa 2 × α, β = ↑ , ↓ ). The spin-orbit interaction amongquasiparticles is known to dominate in the nucleon matter of a high density. Therefore it isconventional to represent the triplet order parameter of the system ˆ D = P lm j ∆ lm j Φ ( jlm j ) αβ
3s a superposition of standard spin-angle functions of the total angular momentum ( j, m j ),Φ ( jlm j ) αβ ( n ) ≡ X m s + m l = m j (cid:18)
12 12 αβ | sm s (cid:19) ( slm s m l | jm j ) Y l,m l ( n ) . (1)For our calculations it will be more convenient to use vector notation which involves aset of mutually orthogonal complex vectors b lm j ( n ) defined as b lm j ( n ) = −
12 Tr (cid:16) ˆ g ˆ σ ˆΦ jlm j (cid:17) , (2)where ˆ σ = (ˆ σ , ˆ σ , ˆ σ ) are Pauli spin matrices, and ˆ g = i ˆ σ . We will use the normalizationcondition Z d n π b ∗ l ′ m ′ j b lm j = δ ll ′ δ m j m ′ j . (3)If the most attractive channel of interactions is assumed in the states with s = 1 , j =2 , l = j ± D ( n ) = X lm j ∆ lm j (cid:0) ˆ σ b lm j (cid:1) ˆ g . (4)We are mostly interested in the values of quasiparticle momenta p near the Fermi surface p ≃ p F , where the partial gap amplitudes, ∆ lm j ( p ) ≃ ∆ lm j ( p F ) , are almost constants, andthe angular dependence of the order parameter is represented by the unit vector n = p /p which defines the polar angles ( θ, ϕ ) on the Fermi surface.The ground state (4) occurring in neutron matter has a relatively simple structure (uni-tary triplet) [16], [17]: X lm j ∆ lm j b lm j ( n ) = ∆ ¯b ( n ) , (5)where ∆ is a complex constant (on the Fermi surface), and ¯b ( n ) is a real vector which wenormalize by the condition Z d n π ¯ b ( n ) = 1 . (6)Thus the triplet order parameter can be written asˆ D ( n ) = ∆ ¯bˆ σ ˆ g . (7)We will use the adopted graphical notation for the ordinary and anomalous propagators,as shown in Fig. 1. 4 = G = F = (2) F = (1) , , ,
FIG. 1: Diagrams depicting the ordinary and anomalous propagators of a quasiparticle.
The analytic form of the propagators can be found in the standard way [18], [19] , usingthe general form (7) of the gap matrix. Since the matter is assumed in thermal equilibriumat some temperature, we employ the Matsubara calculation technique. Thenˆ G ( p m , p ) = aG ( p m , p ) δ αβ , ˆ G − ( p m , p ) = aG − ( p m , p ) δ αβ , ˆ F (1) ( p m , p ) = aF ( p m , p ) ¯bˆ σ ˆ g , ˆ F (2) ( p m , p ) = aF ( p m , p ) ˆ g ˆ σ ¯b , (8)where a ≃ p m ≡ iπ (2 m + 1) T with m = 0 , ± , ± ... is the Matsubara’s fermion frequency, and the scalar Green’s functions areof the form G ( p m , p ) = − ip m − ε p p m + E p , G − ( p m , p ) = ip m − ε p p m + E p ,F ( p m , p ) = − ∆ p m + E p . (9)Here ε p = p M ∗ − p F M ∗ ≃ p F M ∗ ( p − p F ) , (10)with M ∗ = p F /V F being the effective mass of a quasiparticle. The quasiparticle energy isgiven by E p ≡ r ε p + 12 Tr ˆ D ( n ) ˆ D † ( n ) = q ε p + ∆ ¯ b , (11)where the (temperature-dependent) energy gap, ∆¯ b ( n ), is anisotropic. Here the fact is usedthat, in the absence of external fields, the gap amplitude ∆ is real.Green functions of a quasiparticle (8) involve the renormalization factor a ≃ ω, q , T (see e.g. [19]). The final outcomes are independent of this factor thereforewe will drop the renormalization factor in order to shorten the equations by assuming thatall the necessary physical values are properly renormalized.The following notation will be used below. We designate as L X,X ( ω, q ; p ) the analyticalcontinuation onto the upper-half plane of complex variable ω of the following Matsubara5ums: L XX ′ (cid:16) ω n , p + q p − q (cid:17) = T X m X (cid:16) p m + ω n , p + q (cid:17) X ′ (cid:16) p m , p − q (cid:17) , (12)where X, X ′ ∈ G, F, G − , and ω n = 2 iπT n with n = 0 , ± , ± ... .It is convenient to divide the integration over the momentum space into integration overthe solid angle and over the energy according to Z d p (2 π ) ... = ρ Z d n π Z ∞−∞ dε p ... (13)and operate with integrals over the quasiparticle energy: I XX ′ ( ω, n , q ; T ) ≡ Z ∞−∞ dε p L XX ′ (cid:16) ω, p + q , p − q (cid:17) . (14)These are functions of ω , q and the direction of the quasiparticle momentum p = p n . Hereand below ρ = p F M ∗ /π is the density of states near the Fermi surface.The loop integrals (14) possess the following properties which can be verified by a straight-forward calculation: I G − G = I GG − , I GF = −I F G , I G − F = −I F G − , (15) I G − F + I F G = ω ∆ I F F , (16) I G − F − I F G = − qv ∆ I F F . (17)For arbitrary ω, q , T one can obtain also I GG − + ¯ b I F F = A + ω − ( qv ) I F F , (18)where v is a vector with the magnitude of the Fermi velocity V F and the direction of n , and A ( n ) ≡ (cid:2) I G − G ( n ) + ¯ b ( n ) I F F ( n ) (cid:3) ω =0 , q =0 . (19)In the case of triplet superfluid the key role in the response theory belongs to the loopintegrals I F F and (cid:0) I GG ± ¯ b I F F (cid:1) . For further usage we indicate the properties of thisefunctions in the case of ω > q →
0. A straightforward calculation yields I F F ( ω, q = 0) = − Z ∞ dεE ω + i − E tanh E T , (20)and (cid:0) I GG + ¯ b I F F (cid:1) q → = 0 , (21)6 I GG − ¯ b I F F (cid:1) q → = − b I F F ( ω, ) . (22)The imaginary part of I F F arises from the poles of the integrand in Eq. (20) at ω = ± E :Im I F F ( ω > , q = 0) = Θ (cid:0) ω − b ∆ (cid:1) π ∆ ω p ω − b ∆ tanh ω T . (23)where Θ ( x ) is Heaviside step function. B. Gap equation
The block of the interaction diagrams irreducible in the channel of two quasiparticles,Γ αβ,γδ , is usually generated by the expansion over spin-angle functions (1). Using the vectornotation, the most attractive channel of pairing interactions with j = 2 can be written as ρ Γ αβ,γδ ( p , p ′ ) = − X l ′ lm j V ll ′ ( p, p ′ ) (cid:0) b lm j ( n ) ˆ σ ˆ g (cid:1) αβ (cid:16) ˆ g ˆ σ b ∗ l ′ m j ( n ′ ) (cid:17) γδ , (24)where V ll ′ ( p, p ′ ) are the corresponding interaction amplitudes, and | l − l ′ | ≤ lm j is of theform∆ lm j ( p ) = − X l ′ ρ Z dp ′ p ′ V ll ′ ( p, p ′ ) ∆ ( p ′ ) Z d n ′ π b ∗ l ′ m j ( n ′ ) ¯b ( n ′ ) T X m p m + E p ′ . (25)where ¯b ( n ) = 1∆ X lm j ∆ lm j b lm j ( n ) , (26)as defined in Eq. (5). These equations can be reduced to the standard form [17] with theaid of the identity T X m p m + E p ′ ≡ E ( p ′ ) tanh E ( p ′ )2 T , (27)and the relation 12 Tr (cid:16) ˆΦ jlm j ˆΦ ∗ jl ′ m j (cid:17) = b lm j ( n ) · b ∗ l ′ m j ( n ) . (28)We are interested in the processes occuring in a vicinity of the Fermi surface. Thereforewe now recast the gap equation to the more convenient form. We notice that1 p m + E p ≡ G ( p m , p ) G − ( p m , p ) + ¯ b F ( p m , p ) F ( p m , p ) , (29)7.e. Eq. (25) can be written as∆ lm j ( p ) = − ρ X l ′ Z dp ′ p ′ V ll ′ ( p, p ′ ) ∆ ( p ′ ) Z d n ′ π b ∗ l ′ m j ( n ′ ) ¯b ( n ′ ) × T X m (cid:2) G ( p m , p ′ ) G − ( p m , p ′ ) + ¯ b F ( p m , p ′ ) F ( p m , p ′ ) (cid:3) . (30)To get rid of the integration over the regions far from the Fermi surface we renormalizethe interaction as suggested in Ref. [15]: we define V ( r ) ll ′ ( p, p ′ ; T ) = V ll ′ ( p, p ′ ) − V ll ′ ( p, p ′ ) (cid:0) GG − (cid:1) n V ( r ) ll ′ ( p, p ′ ; T ) , (31)where the loop ( GG − ) n is evaluated in the normal (nonsuperfluid) state. In terms of V ( r ) ll ′ the gap equation becomes∆ lm j ( p ) = − ρ X l ′ Z dp ′ p ′ V ( r ) ll ′ ( p, p ′ ) ∆ ( p ′ ) Z d n ′ π b ∗ l ′ m j ( n ′ ) ¯b ( n ′ ) × T X m (cid:2) GG − − (cid:0) GG − (cid:1) n + ¯ b F F (cid:3) p m , p ′ . (32)and we may everywhere substitute V ( r ) ll ′ for V ll ′ provided that at the same time we understandby GG − element the subtracted quantity GG − − ( GG − ) n [( GG − ) n is to be evaluated for ω = 0 , q = 0 in all cases]. From now we will do this and drop the superscript r on V ( r ) ll ′ .Since the function GG − + ¯ b F F decreases rapidly along with a distance from the Fermisurface, we may replace Eq. (32) with∆ lm j = − ρ X l ′ V ll ′ ∆ Z d n π b ∗ l ′ m j ( n ) ¯b ( n ) 12 Z dpp T X m (cid:2) GG − + ¯ b F F (cid:3) p m , p , (33)assuming that in the narrow vicinity of the Fermi surface the smooth functions∆ lm j ( p ) , V ll ′ ( p, p ′ ) , ∆ ( p ′ ) may be replaced with constants: ∆ ( p ) ≃ ∆ ( p F ) ≡ ∆, ect..The function (19) is now to be understood as A ( n ) → (cid:2) I G − G − I ( G − G ) n + ¯ b I F F (cid:3) ω =0 , q =0 , (34)and the gap equations (33) become:∆ lm j = − ∆ X l ′ V ll ′ Z d n π b ∗ l ′ m j ( n ) ¯b ( n ) A ( n ) . (35)The function (34) can be found explicitly after performing the Matsubara’s summation: A ( n ) = 14 Z ∞−∞ dε √ ε + ∆ ¯ b tanh √ ε + ∆ ¯ b T − ε tanh ε T ! . (36)8 II. EFFECTIVE VERTICES AND THE CORRELATION FUNCTIONS
The field interaction with a superfluid should be described with the aid of four effectivethree-point vertices shown in Fig. 2.
FIG. 2: Diagrams of the ordinary and anomalous vertices for the quasiparticle interacting with theexternal field shown by the dash line.
There are two ordinary effective vertices corresponding to creation of a particle and ahole by the field that differ by direction of fermion lines. We denote these 2 × τ ( n ; ω, q ) ≡ τ αβ ( n ; ω, q ) and ˆ τ − ( n ; ω, q ) ≡ τ βα ( − n ; ω, q ), respectively. The anomalousvertices correspond to creation of two particles or two holes. We denote these matrices asˆ T (1) ( n ; ω, q ) and ˆ T (2) ( n ; ω, q ), respectively.Given by the sum of the ladder-type diagrams [14], the anomalous vertices are to satisfythe Dyson’s equations symbolically depicted by the graphs in Fig. 3. = + + + = + + + FIG. 3: Dyson’s equations for the anomalous vertices. The shaded rectangle represents the pairinginteraction.
Analytically the equations reduce to the following (we omit for brevity the dependence9f functions on ω and q) : T (1) αβ ( n ) = X lm j (cid:0) ˆ σ b lm j ( n )ˆ g (cid:1) αβ X l ′ V ll ′ × Z d n ′ π Tr h I GG − ˆ g (cid:16) ˆ σ b ∗ l ′ m j (cid:17) ˆ T (1) − I F F (cid:16) ˆ σ b ∗ l ′ m j (cid:17) (cid:0) ˆ σ ¯b (cid:1) ˆ g ˆ T (2) (cid:0) ˆ σ ¯b (cid:1) −I GF (cid:0) ˆ σ ¯b (cid:1) (cid:16) ˆ σ b ∗ l ′ m j (cid:17) ˆ τ + I F G − (cid:16) ˆ σ b ∗ l ′ m j (cid:17) (cid:0) ˆ σ ¯b (cid:1) (cid:0) ˆ g ˆ τ − ˆ g (cid:1)i n ′ , (37) T (2) αβ ( n ) = X lm j (cid:16) ˆ g ˆ σ b ∗ lm j ( n ) (cid:17) αβ X l ′ V ll ′ × Z d n ′ π Tr h I G − G (cid:0) ˆ σ b l ′ m j (cid:1) ˆ g ˆ T (2) − I F F (cid:0) ˆ σ b l ′ m j (cid:1) (cid:0) ˆ σ ¯b (cid:1) ˆ T (1) ˆ g (cid:0) ˆ σ ¯b (cid:1) + I G − F (cid:0) ˆ σ b l ′ m j (cid:1) ˆ g ˆ τ − ˆ g (cid:0) ˆ σ ¯b (cid:1) − I F G (cid:0) ˆ σ b l ′ m j (cid:1) (cid:0) ˆ σ ¯b (cid:1) ˆ τ (cid:3) n ′ . (38)To obtain these equations we used the identity ˆ g ˆ g = − ˆ1 and a cyclic permutation of thematrices under the trace signs.In general, the ordinary effective vertex is to be also found by ideal summation of theladder diagrams incorporating residual particle-hole interactions. Unfortunately, the Landauparameters for these interactions in asymmetric nuclear matter are unknown therefore wesimply neglect the particle-hole interactions and consider the pair correlation function in theBCS approximation. Thus, if the 2 × ˆ ξ ( n ,k ) is some vertex of a freeparticle, the ordinary vertices of a quasiparticle and a hole in the BCS approximation areto be taken as: ˆ τ ( n ,k ) = ˆ ξ ( n ,k ) , ˆ τ − ( n ,k ) = ˆ ξ T ( − n ,k ) . (39)Discarding the particle-hole interactions, we nevertheless assume that the ”bare” verticesare properly renormalized [14] in order to get rid of the integration over regions far fromthe Fermi surface, ε p ≫ ∆ . As mentioned above, we omit the renormalization factoreverywhere. G' = + ++
FIG. 4: Correction to the ordinary propagator of a quasiparticle in external field.
Variation of the Green function of a quasiparticle under the action of external field U ,ˆ G ′ = δ ˜ GδU , (40)10s given by the graphs [19] shown in Fig. 4, and can be written analytically as G ′ = GG ˆ τ + F F (cid:0) ˆ σ ¯b (cid:1) ˆ g ˆ τ − ˆ g (cid:0) ˆ σ ¯b (cid:1) + GF ˆ T (1) ˆ g (cid:0) ˆ σ ¯b (cid:1) + F G (cid:0) ˆ σ ¯b (cid:1) ˆ g ˆ T (2) , (41)where GG ≡ G ( p m + ω n , p + q / G ( p m , p − q / τ ( ω n , q ) = T X m Z d p π Tr (cid:16) ˆ τ ˆ G ′ (cid:17) . (42) IV. GENERAL APPROACH TO NEUTRINO ENERGY LOSSES
The PBF processes are kinematically allowed thanks to the existence of a superfluidenergy gap, which admits the quasiparticle transitions with time-like momentum transfer k = ( ω, q ), as required by the final neutrino pair: k = k + k . We consider the standardmodel of weak interactions. After integration over the phase space of escaping neutrinosand antineutrinos the total energy which is emitted into neutrino pairs per unit volume andtime is given by the following formula (see details, e.g., in Ref. [20]): ǫ = − G F N ν π Z ∞ dω Z d q ω Θ ( ω − q )exp (cid:0) ωT (cid:1) − µν weak ( ω, q ) (cid:0) k µ k ν − k g µν (cid:1) , (43)where N ν = 3 is the number of neutrino flavors; G F is the Fermi coupling constant; andΘ ( x ) is the Heaviside step-function. Π µν weak is the retarded weak polarization tensor of themedium.In general, the weak polarization tensor of the medium is a sum of the vector-vector, axial-axial, and mixed terms. The mixed axial-vector polarization has to be an antisymmetrictensor, and its contraction in Eq. (43) with the symmetric tensor k µ k ν − k g µν vanishes.Thus only the pure-vector and pure-axial polarizations should be taken into account. Wethen obtain Im Π µν weak ≃ C Im Π µν V + C Im Π µν A , where C V and C A are vector and axial-vectorweak coupling constants of a neutron, respectively. V. PRESENT STATUS OF THE PROBLEM
The widely used expression for the neutrino emissivity caused by the triplet pairing ofneutrons was obtained in Ref. [12] with the aid of the Fermi ”golden” rule. Therefore before11roceeding to the self-consistent calculation of the neutrino energy losses, it is instructiveto reproduce this formula using the calculation technique developed in our paper. We willprove the result of Ref. [12] can be obtained from our equations (43) and (42) if to removethe field interactions through anomalous vertices [second line in Eq. (41)]. We will label thecorresponding results with tilde.The authors of Ref. [12] state that the weak current of nonrelativistic neutrons is causedmostly by the temporal component of the vector current, ˆ J = Ψ + ˆ1Ψ, and by the spacecomponents of the axial-vector current, ˆ J i = Ψ + ˆ σ i Ψ. Consequently to reproduce theirresult we need to evaluate the temporal component of the polarization tensor in the vectorchannel and the spatial part of the axial polarization. Omitting the anomalous contributionsfor the temporal component of the vector polarization we have to substitute forˆ τ = ˆ τ − → ˆ1 , ˆ T (1 , → , (44)where ˆ1 is a unit 2 × ( ω, q ) = 4 ρ Z d n π (cid:0) I GG − ¯ b I F F (cid:1) . (45)In obtaining this expression we used Eqs. (12), (14) and the identity (cid:0) ˆ σ ¯b (cid:1) (cid:0) ˆ σ ¯b (cid:1) = ¯ b .Only small transferred momenta, q < ω ∼ T , contribute into the neutrino energy losses.Since the transferred momentum comes in the polarization function in a combination qV F ≪ ω, ∆ (Fermi velocity V F is small in a nonrelativistic system), to the lowest accuracy, we mayevaluate the polarization tensor in the limit q = 0. (In the same approximation the aboveauthors evaluate the matrix elements of a quasiparticle transition.) Then using Eqs. (22)and (23) we findIm ˜Π ( ω > , q = 0) = − πρ Z d n π ¯ b ∆ Θ (cid:0) ω − b ∆ (cid:1) ω p ω − b ∆ tanh ω T . (46)Polarization tensor in the axial channel can be evaluated in the same way. In this case,omitting the anomalous contributions we have to takeˆ τ ( n ,k ) → ˆ σ i , ˆ τ − ( n ,k ) → ˆ σ Ti , ˆ T (1 , → . (47)Then we find after some algebraic manipulations:˜Π ij A ( ω, q ) = 4 ρ Z d n π (cid:18) (cid:0) I GG − ¯ b I F F (cid:1) δ ij + I F F ¯ b i ¯ b j (cid:19) (48)12n obtaining this we used the identities ˆ g ˆ σ T ˆ g = ˆ σ , and ˆ σ (cid:0) ˆ σ ¯b (cid:1) = 2 ¯b − (cid:0) ˆ σ ¯b (cid:1) ˆ σ . With the aid of Eqs. (22) and (23) we find:Im ˜Π ij A ( ω > , q = 0) = − πρ Z d n π (cid:18) δ ij − ¯ b i ¯ b j ¯ b (cid:19) ¯ b ∆ Θ (cid:0) ω − b ∆ (cid:1) ω p ω − b ∆ tanh ω T (49)Inserting the imaginary part of the polarization tensor into Eq. (43) we calculate the con-traction of Im ˜Π µν weak with the symmetric tensor k µ k ν − k g µν to obtainIm ˜Π µν weak (cid:0) k µ k ν − k g µν (cid:1) = − πρ Z d n π ¯ b ∆ Θ (cid:0) ω − b ∆ (cid:1) ω p ω − b ∆ tanh ω T × (cid:2) C (cid:0) q k + q ⊥ (cid:1) + C (cid:0) (cid:0) ω − q k (cid:1) − q ⊥ (cid:1)(cid:3) , (50)where q k and q ⊥ are defined as q k = 1¯ b (cid:0) q¯b (cid:1) , q ⊥ = q − q k . (51)After a little algebra we obtain the neutrino emissivity in the form:˜ ǫ = G F N ν π p F M ∗ Z d n π ∆ n Z ∞ n dω ω (cid:0) ω T (cid:1) p ω − n (cid:0) C V + 2 C A (cid:1) , (52)where ∆ n ≡ ∆ ¯ b ( n ).With the aid of the change ω = 2 T p x + ∆ n /T one can recast this expression to theform obtained in Ref. [12]:˜ ǫ = ǫ Y KL ≡ G F N ν π p F M ∗ (cid:0) C V + 2 C A (cid:1) T Z d n π ∆ n T Z ∞ dx z (1 + exp z ) , (53)where z = p x + ∆ n /T .Apparently the contribution of the vector channel in this expression is a subject of incon-sistency, since conservation of the vector current in weak interactions requires ω Π ( ω, q ) = q i Π i ( ω, q ) , and thus one should expect Π ( ω > , q = 0) = 0 for the correct result insteadof Eq. (46). This however was not proved explicitly for the case of triplet pairing. We nowfocus on this calculation. 13 I. ANOMALOUS CONTRIBUTIONSA. Vector channel
The self-consistent longitudinal polarization function Π ( ω > , q ) incorporates theanomalous contributions. At finite transferred space momentum the problem of determin-ing the vertex corrections is much complicated. Typically massless Goldstone modes thatarise due to symmetry breaking play a crucial role in conservation of the vector current. Inthe anisotropic P phase rotational symmetry is broken and three Goldstone modes arise(termed angulons in Ref. [21]). However, since we are interested in the specific case of q = 0the temporal component of the anomalous vertex ˆ T µ ( µ = 0 , , ,
3) can be retrieved fromthe Ward identity which requires [19], [3]: ω ˆ T (1 , ( n ; ω, q ) − q ˆT (1 , ( n ; ω, q ) = ± D ( n ) . (54)From this identity we immediately findˆ T (1)0 ( n ; ω, q = ) = 2∆ ω ¯bˆ σ ˆ g , (55)and ˆ T (2)0 ( n ; ω, q = ) = − ω ˆ g ¯bˆ σ . (56)In the BCS approximation, the ordinary scalar vertices are to be taken, as given by Eq.(44). Inserting the above vertices into Eqs. (41), (42) we obtain after a little algebra:Π ( ω, q = 0) = 4 ρ Z d n π (cid:18) (cid:0) I GG − ¯ b I F F (cid:1) + 2∆ ω ¯ b I F G (cid:19) q =0 . (57)Using Eqs. (16), (17) yielding I F G = ω + qv I F F , (58)we finally find Π ( ω, q = 0) = 2 C V ρ Z d n π (cid:0) I GG + ¯ b I F F (cid:1) q =0 . (59)Comparing this with Eq. (21) we obtain Π ( ω, q = 0) = 0, as is required by the currentconservation condition. We found that the neutrino emissivity through the vector channelvanishes in the limit q = 0. This proves explicitly that the neutrino emissivity via the vectorchannel, as obtained in Eq. (53), is a subject of inconsistency.14 . Axial channel We now focus on the axial channel of the weak polarization. The order parameter in thetriplet superfluid varies under the action of axial-vector external field. Therefore the self-consistent axial polarization tensor also must incorporate anomalous contributions. Thenfrom Eqs. (41), (42) we obtain after simple algebraic manipulations:Π ij A ( ω ) = 4 ρ Z d n π (cid:20) (cid:0) I GG − ¯ b I F F (cid:1) δ ij + ¯ b I F F ¯ b i ¯ b j ¯ b − ω I F F
14 Tr (cid:16) ˆ σ i ˆ T (1) j ˆ g (cid:0) ˆ σ ¯b (cid:1) − ˆ σ i (cid:0) ˆ σ ¯b (cid:1) ˆ g ˆ T (2) j (cid:17)(cid:21) , (60)As in above, we focus on the case q = 0 and omit for brevity the dependence on n and ω .The anomalous axial-vector vertices ˆ T (1 , j ( j = 1 , ,
3) are to be found from Eqs. (37), (38),where the ordinary vertices are given by Eq. (47).Up to this point we have not discussed the n dependence of b lm j ( n ). This makes Eq.(59) valid in the case of tensor forces resulting in the P − F mixing, because the generalform of Eqs. (37), (38) for the anomalous vertices takes into account not only spin-orbitinteractions but the tensor interactions in the channel of two quasiparticles. Now we simplifythe problem according to approximation adopted in simulations of neutron star cooling [11]and consider the case of paring in the P channel, when l = 1, and V ll ′ = δ ll ′ V , and thevectors b m j ( n ) are given by b = r
12 ( − n , − n , n ) , b = − b ∗− = − r
34 ( n , in , n + in ) , b = b ∗− = r
34 ( n + in , in − n , , (61)where n = sin θ cos ϕ, n = sin θ sin ϕ, n = cos θ . From now on we will drop the subscript l = 1 by assuming b m j ≡ b ,m j , ∆ m ≡ ∆ ,m j , etc.We will focus on the p-wave condensation into the state P with m j = 0 which isconventionally considered as the preferable one in the bulk matter of neutron stars. In thiscase, Eq. (5) implies ¯b ( n ) = b ( n ) , ∆ = ∆ (62)15nd the gap equation (35) reads 1 = − V Z d n π ¯ b ( n ) A ( n ) . (63)From Eqs. (37) and (38) we obtain the vertex equations of the following form ( i = 1 , , T (1) i ( n ) = V X m j ˆ σ b m j ( n )ˆ g Z d n ′ π h I GG − Tr (cid:16) ˆ g (cid:16) ˆ σ b ∗ m j (cid:17) ˆ T (1) i (cid:17) − I F F Tr (cid:16)(cid:16) ˆ σ b ∗ m j (cid:17) (cid:0) ˆ σ ¯b (cid:1) ˆ g ˆ T (2) i (cid:0) ˆ σ ¯b (cid:1)(cid:17) − ω ∆ I F F i (cid:16) b ∗ m j × ¯b (cid:17) i i n ′ , (64)ˆ T (2) i ( n ) = V X m j ˆ g ˆ σ b ∗ m j ( n ) Z d n ′ π h I G − G Tr (cid:16)(cid:0) ˆ σ b m j (cid:1) ˆ g ˆ T (2) i (cid:17) − I F F Tr (cid:16)(cid:0) ˆ σ b m j (cid:1) (cid:0) ˆ σ ¯b (cid:1) ˆ T (1) i ˆ g (cid:0) ˆ σ ¯b (cid:1)(cid:17) − ω ∆ I F F i (cid:0) b m j × ¯b (cid:1) i i n ′ . (65)In obtaining the last line in these equations we used ˆ σ (cid:0) ˆ σ ¯b (cid:1) = 2 ¯b − (cid:0) ˆ σ ¯b (cid:1) ˆ σ along withTr (cid:16)(cid:16) ˆ σ b ∗ m j (cid:17) (cid:0) ˆ σ ¯b (cid:1) ˆ σ (cid:17) = 2 i (cid:16) b ∗ m j × ¯b (cid:17) , and Eqs. (15), (16).Inspection of the equations reveals that the anomalous axial-vector vertices can be foundin the following form ˆT (1) ( n , ω ) = X m j B (1) m j ( ω ) (cid:0) ˆ σ b m j (cid:1) ˆ g , (66) ˆT (2) ( n , ω ) = X m j B (2) m j ( ω ) ˆ g (cid:16) ˆ σ b ∗ m j (cid:17) . (67)These general expressions can be simplified due to the fact that the function I F F ( n ; ω ) givenby Eq. (20) is axial - symmetric, and the last (free) term, in Eqs. (64) and (65), can beaveraged over the azimuth angle to give Z dϕ π (cid:0) b ∗ × ¯b (cid:1) = Z dϕ π (cid:0) b ∗ × ¯b (cid:1) = Z dϕ π (cid:0) b ∗− × ¯b (cid:1) = 0 , (68)and i Z dϕ π (cid:0) b ∗ × ¯b (cid:1) = − e √
64 ¯ b , i Z dϕ π (cid:0) b ∗− × ¯b (cid:1) = − e ∗ √
64 ¯ b , (69)where e = (1 , − i,
0) is a constant complex vector in spin space. The following relations canbe also verified by a straightforward calculation: Z dϕ π b ∗ m j b m ′ j = δ m j m ′ j b ∗ m j b m j , (70)16 dϕ π (cid:16) ¯bb ∗ m j (cid:17) (cid:16) ¯bb m ′ j (cid:17) = δ m j m ′ j (cid:16) ¯bb ∗ m j (cid:17) (cid:0) ¯bb m j (cid:1) . (71)Relations (68) and (69) allow to conclude that B (1 , = B (1 , ± = 0, and ˆT (1) ( n ) = (cid:16) B (1)1 ( ˆ σ b ) + B (1) − ( ˆ σ b − ) (cid:17) ˆ g , ˆT (2) ( n ) = ˆ g (cid:16) B (2)1 ( ˆ σ b ∗ ) + B (2) − (cid:0) ˆ σ b ∗− (cid:1)(cid:17) . Inserting these expressions into Eqs. (64) and (65), taking the traces and using the orthog-onality relations (3) along with relations (70), (71), and¯ b ≡ b , b ∗ b = b ∗− b − , (72) (cid:0) ¯bb ∗ (cid:1) (cid:0) ¯bb (cid:1) = (cid:0) ¯bb ∗− (cid:1) (cid:0) ¯bb − (cid:1) , (73)we obtain the equations: B (1) ± = − V Z d n π h I GG − B (1) ± ( b b ∗ ) −I F F B (2) ∓ (cid:0) ( b ∗ b ) ¯ b − (cid:0) b ∗ ¯b (cid:1) (cid:0) ¯bb (cid:1)(cid:1) − ω ∆ I F F e √
64 ¯ b , (74)and B (2) ± = − V Z d n ′ π h I G − G B (2) ± ( b b ∗ ) −I F F B (1) ∓ (cid:0) ( b b ∗ ) ¯ b − (cid:0) b ¯b (cid:1) (cid:0) ¯bb ∗ (cid:1)(cid:1) + ω ∆ I F F e ∗ √
64 ¯ b . (75)Solution to Eqs. (74), (75) can be found in the form B (2)1 = − B (1) − , B (2) − = − B (1)1 , (76)where B = e f ( ω ) , B − = e ∗ f ( ω ) , (77)and the function f ( ω ) satisfies the equation f = − V Z d n π " (cid:0) I GG − + ¯ b I F F (cid:1) ( b ∗ b ) f − I F F (cid:0) b ∗ ¯b (cid:1) (cid:0) ¯bb (cid:1) f − ω ∆ I F F √
64 ¯ b . (78)Using Eq. (18) we can rewrite this as f = − V Z d n π " (cid:18) A + ω I F F (cid:19) ( b ∗ b ) f − I F F (cid:0) b ∗ ¯b (cid:1) (cid:0) ¯bb (cid:1) f − ω ∆ I F F √
64 ¯ b . (79)17t this point it is convenient to recast the left side of this equation according to Eq. (63): f = − V f Z d n π ¯ b ( n ) A ( n ) . (80)In this way we obtain the equation f Z d n π (cid:20)(cid:0) b ∗ b − ¯ b (cid:1) A + 2 (cid:18) ω ( b ∗ b ) − (cid:0) b ∗ ¯b (cid:1) (cid:0) ¯bb (cid:1)(cid:19) I F F (cid:21) = r ω Z d n π ¯ b I F F . (81)Since the function I F F ( n ; ω ) is axial - symmetric and¯ b = 12 (cid:0) n (cid:1) , b ∗ b = 34 (cid:0) n (cid:1) , (82) (cid:0) ¯bb ∗ (cid:1) (cid:0) ¯bb (cid:1) = 38 n (cid:0) − n (cid:1) , (83)Eq. (81) can be integrated over the azimuth angle, yielding the following solution f ( ω, q = 0) = 1 χ ( ω, q = 0) r ω Z dn (cid:0) n (cid:1) I F F ( n , ω, T ) . (84)where χ ( ω, q = 0) ≡ Z dn (cid:20) (cid:0) − n (cid:1) A ( n , T )+ 34 (cid:18) ω (cid:0) n (cid:1) − n (cid:0) − n (cid:1)(cid:19) I F F ( n , ω, T ) (cid:21) , (85)and the functions A ( n , T ) and I F F ( n , ω, T ) are given by Eqs. (36) and (20).Explicit evaluation of Eq. (84) for arbitrary values of ω and T appears to require nu-merical computation. However, we can get a clear idea of the behavior of this functionusing the angle-averaged energy gap in the quasiparticle energy, (cid:10) ∆ ¯ b (cid:11) ≡ ∆ . (Replacingangle-dependent quantities in the gap equation with their angular average has been foundto be a good approximation [22].) In this approximation the functions I F F ( ω, T ) and A ( T ),in Eqs. (84) and (85), can be moved beyond the integrals. Using also the fact that A Z dn (cid:0) − n (cid:1) = 0 (86)we find f = r
32 ∆ ωω − ∆ / f ( ω ) is real-valued and is indepen-dent of the temperature.Poles of the vertex function correspond to collective eigen-modes of the system. Therefore,the pole at ω = ∆ / ω > b ( θ ) ≥ √
2∆ and, to obtain a simple analytic approximation, weomit a small term ∆ / ˆT (1) ( n ) = r
32 ∆ ω ( e ( ˆ σ b ) + e ∗ ( ˆ σ b − )) ˆ g , (88) ˆT (2) ( n ) = r
32 ∆ ω ˆ g ( e ( ˆ σ b ) + e ∗ ( ˆ σ b − )) , (89)Having obtained this simple result we can evaluate the axial polarization function. In-serting (88) and (89) into Eq. (60) givesΠ ij A ( ω ) = 4 ρ Z d n π (cid:20) (cid:0) I GG − ¯ b I F F (cid:1) δ ij + ¯ b I F F ¯ b i ¯ b j ¯ b + ( δ ij − δ i δ j ) 34¯ b I F F (cid:21) q =0 . (90)The first line in Eq. (90) can be evaluated with the aid of Eq. (22). We find:Π ij A = − ρ Z d n π (cid:18) δ ij − ¯ b i ¯ b j ¯ b −
34 ( δ ij − δ i δ j ) (cid:19) ¯ b I F F ( ω, q = 0) . (91)Using Eq. (23) we obtain the imaginary part of axial polarization:Im Π ij A ( ω > , q = 0)= − πρ Z d n π (cid:18) δ ij − ¯ b i ¯ b j ¯ b −
34 ( δ ij − δ i δ j ) (cid:19) ¯ b ∆ Θ (cid:0) ω − b ∆ (cid:1) ω p ω − b ∆ tanh ω T (92)19 II. SELF-CONSISTENT NEUTRINO ENERGY LOSSES
As we have obtained Im Π µν V ( ω > , q = 0) = 0, using Eqs. (23) and (91) we findIm Π µν weak = − δ µi δ νj C πρ Z d n π (cid:18) δ ij − ¯ b i ¯ b j ¯ b −
34 ( δ ij − δ i δ j ) (cid:19) × ¯ b ∆ Θ (cid:0) ω − b ∆ (cid:1) ω p ω − b ∆ tanh ω T . (93)Contraction of this tensor with ( k µ k ν − k g µν ) gives:Im Π µν weak (cid:0) k µ k ν − k g µν (cid:1) = − C (cid:0) (cid:0) ω − q k (cid:1) − q ⊥ (cid:1) πρ Z d n π ¯ b ∆ Θ (cid:0) ω − b ∆ (cid:1) ω p ω − b ∆ tanh ω T , (94)where q k = 1¯ b (cid:0) q¯b (cid:1) , q ⊥ = q − q k . (95)The rest of the calculation is already performed in Sec. VII. The neutrino energy lossescan be written immediately after inspection of Eqs. (50) and (94). From this comparison itis clear that in order to obtain the correct neutrino energy losses, it is necessary to replacethe factor ( C + 2 C ) with (1 / C in Eq. (53). In this way we obtain ǫ ≃ π G F C N ν p F M ∗ T Z d n π ∆ n T Z ∞ dx z (1 + exp z ) , (96)where ∆ n ≡ ∆ ¯ b ( n ) = ∆ (1 + 3 cos θ ), and z = p x + ∆ n /T . Comparison of thisexpression with Eq. (53) shows that the neutrino energy losses caused by the P pairing inneutron matter are suppressed by the factor12 C ( C + 2 C ) ≃ .
19 (97)with respect to that predicted in Ref. [12].For a practical usage we reduce Eq. (96) to the traditional form ǫ ≃ . × (cid:18) M ∗ M (cid:19) (cid:16) p F M c (cid:17) T N ν C F t ergcm s , (98)where M and M ∗ are the effective and bare nucleon masses, respectively; c is speed of light,and F t = Z d n π ∆ n T Z ∞ dx z (1 + exp z ) . (99)20otice the gap amplitude ∆ ( T ) defined above is √ ( T ) used in Ref. [12] , where the same anisotropic gap ∆ n is written in the form ∆ n =∆ (1 + 3 cos θ ). However, the function F t , defined in Eq. (99), is independent of theparticular choice of the gap amplitude, therefore the analytic fit (B) suggested in Eq. (34)of Ref. [12], is valid and can be used in practical computations. VIII. SUMMARY AND CONCLUSION
In this paper we have performed a self-consistent calculation of the neutrino energy lossesdue to the pair breaking and formation processes in the triplet-correlated neutron matterwhich is generally expected to exist in the neutron star interior. Since the existing theory ofanomalous weak interactions in the fermion superfluid is well developed only for the case of S pairing we have generalized the corresponding equations for the triplet pairing includingthe case when the attractive tensor coupling is operative.Exact solution of the vertex equations is much complicated because of anisotropy of thetriplet order parameter. Fortunately only small values of the transferred space momenta aresignificant for the considered processes in the nonrelativistic approximation. Therefore theweak vertices as well as the polarization functions can be evaluated in the limit q = 0.Before proceeding to the self-consistent calculation we reproduced the neutrino energylosses as obtained in Ref. [12], using the calculation technique developed in our paper. Wehave shown that the result of Ref. [12] can be obtained in the BCS approximation from ourequations (43) and (42) if to remove the field interactions through anomalous vertices.The exact solution we found for the vector part of the weak polarization,Π ( ω > , q = 0) = 0, is consistent with the current conservation condition. This generalresult, which is obtained including the tensor couplings and the Fermi-liquid interactions,means that the neutrino emissivity in the vector channel, as obtained in Ref. [12], is asubject of inconsistency.The self-consistent consideration of the axial weak polarization is more complicated.In this case, inclusion of the tensor forces and the Fermi-liquid effects requires numericalcomputations even in the limit of q = 0. Therefore to obtain a simple analytic result we haveconsidered the P pairing in the state with m j = 0 which is conventionally considered as thepreferable one in the minimal cooling scenario of neutron stars. We have also neglected the21esidual particle-hole interactions since the Landau parameters are unknown for the neutronmatter at high density.Finally we used the self-consistent polarization functions for evaluation of the neu-trino energy losses due to PBF processes in the P neutron superfluid with m j = 0.The obtained self-consistent neutrino emissivity, is given by Eq. (96). This expressionneeds to be compared to the emissivity (53) originally derived in Ref. [12], ignoring theanomalous weak interactions. One can see the neutrino emissivity is strongly suppresseddue to the collective effects we have considered in this paper. The suppression factor is(1 / C / ( C + 2 C ) ≃ . P pairing occurs in the core, which contains more than 90% ofthe neutron star volume, the found quenching of the neutrino energy losses from the PBFprocesses can affect the minimal cooling paradigm. [1] E. Flowers, M. Ruderman, P. Sutherland, Astrophys. J. , 541 (1976).[2] L. B. Leinson and A. P´erez, Phys. Lett. B , 114 (2006).[3] L.B. Leinson, Phys. Rev. C , 015502 (2008).[4] L.B. Leinson, Phys. Rev. C , 045502 (2009).[5] J. Kundu and S. Reddy, Phys. Rev. C , 055803 (2004).[6] A. Sedrakian, H. M¨uther, and P. Schuck, Phys. Rev. C 76, 055805 (2007).[7] E. E. Kolomeitsev, D. N. Voskresensky, Phys. Rev. C , 065808 (2008).[8] A. W. Steiner, S. Reddy, Phys. Rev. C , 015802 (2009).[9] A. Cumming, J. Macbeth, J. J. M. I. Zand & D. Page, Astrophys. J. , 429 (2006).[10] S. Gupta, E. F. Brown, H. Schatz, P. Moller, and K.-L. Kratz, Astrophys. J. , 1118 (2007).[11] D. Page, J. M. Lattimer, M. Prakash, A. W. Steiner, Astrophys. J. , 1131 (2009).[12] D. G. Yakovlev, A. D. Kaminker, & K. P. Levenfish, A&A , 650 (1999).[13] L. B. Leinson and A. P´erez, e-Print: astro-ph/0606653[14] A. I. Larkin and A. B. Migdal, Sov. Phys. JETP , 1146 (1963).[15] A. J. Leggett, Phys. Rev. , 1869 (1965); A. J. Leggett, Phys. Rev. , 119 (1966).[16] R. Tamagaki, Prog. Theor. Phys. , 905 (1970).[17] T. Takatsuka, Prog. Theor. Phys. , 1517 (1972).
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