aa r X i v : . [ a s t r o - ph ] J un Hyperon bulk viscosity in strong magnetic fields
Monika Sinha and Debades Bandyopadhyay
Theory Division and Centre for Astroparticle Physics,Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata-700064, India
Abstract
We study the bulk viscosity of neutron star matter including Λ hyperons in the presence ofquantizing magnetic fields. Relaxation time and bulk viscosity due to both the non-leptonic weakprocess involving Λ hyperons and direct Urca processes are calculated here. In the presence ofa strong magnetic field of 10 G, the hyperon bulk viscosity coefficient is reduced whereas bulkviscosity coefficients due to direct Urca processes are enhanced compared with their field free caseswhen many Landau levels are populated by protons, electrons and muons.
PACS numbers: 97.60.Jd, 26.60.-c, 04.40.Dg . INTRODUCTION R-mode instability plays an important role in regulating spins of newly born neutronstars as well as old and accreting neutron stars in low mass x-ray binaries [1]. Gravitationalradiation drives the r-mode unstable due to Chandrasekhar-Friedman-Schutz mechanism[2, 3, 4, 5, 6, 7, 8, 9, 10]. R-mode instability could be a promising source of gravitationalradiation. It would be possible to probe neutron star interior if it is detected by gravitywave detectors.Like gravitational radiation, electromagnetic radiation also drives the r-mode unstablethrough Chandrasekhar-Friedman-Schutz mechanism. There exists a class of neutron starscalled magnetars [11] with strong surface magnetic fields 10 − G as predicted byobservations on soft gamma-ray repeaters and anomalous x-ray pulsars [12, 13]. The effectsof magnetic fields on the spin evolution and r-modes in protomagnetars were investigated bydifferent groups [14, 15, 16]. On the one hand, it was shown that the growth of the r-modedue to electromagnetic and Alfv´en wave emission for strong magnetic field and slow rotationcould compete with that of gravitational radiation [15]. On the other hand, it was arguedthat the distortion of magnetic fields in neutron stars due to r-modes might damp the modewhen the field is ∼ G or more [14, 16].The evolution of r-modes proceeds through three steps [17]. In the first phase, themode amplitude grows exponentially with time. In the next stage, the mode saturatesdue to nonlinear effects. In this case viscosity becomes important. Finally, viscous forcesdominate over gravitational radiation driven instability and damp the r-mode. This showsthat viscosity plays an important role on the evolution of r-mode. Bulk and shear viscositieswere extensively investigated in connection with the damping of the r-mode instability [1,18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. In particular, itwas shown that the hyperon bulk viscosity might effectively damp the r-mode instability[25]. However all these calculations of viscosity were performed in the absence of magneticfields. The only calculation of bulk viscosity due to Urca process in magnetised neutronstar matter was presented in Ref.[37]. This motivates us to investigate bulk viscosity due tonon-leptonic process involving hyperons in the presence of strong magnetic fields. It is tobe noted that the magnetic field in neutron star interior might be higher by several ordersof magnitude than the surface magnetic field [38]. Further it was shown that neutron stars2ould sustain strong interior magnetic field ∼ G [39, 40].The paper is organised in the following way. In Section II we describe hyperon matter instrong magnetic fields. We calculate bulk viscosity due to the non-leptonic process involvingΛ hyperons and due to leptonic processes in Section III. We discuss results in Section IVand a summary is given in Section V.
II. HYPERON MATTER IN MAGNETIC FIELD
We describe β equilibrated and charge neutral neutron star matter made of neutrons,protons, Λ hyperons, electrons and muons within a relativistic mean field approach [41, 42].The baryon-baryon interaction is mediated by σ , ω and ρ mesons. In the absence of magneticfield, the baryon-baryon interaction is given by the Lagrangian density [43, 44] L B = X B = n,p, Λ ¯ ψ B ( iγ µ ∂ µ − m B + g σB σ − g ωB γ µ ω µ − g ρB γ µ t B · ρ µ ) ψ B + 12 (cid:0) ∂ µ σ∂ µ σ − m σ σ (cid:1) − U ( σ ) − ω µν ω µν + 12 m ω ω µ ω µ − ρ µν · ρ µν + 12 m ρ ρ µ · ρ µ . (1)The scalar self interaction term [43, 44, 45] is, U ( σ ) = 13 g m N ( g σN σ ) + 14 g ( g σN σ ) , (2)and ω µν = ∂ ν ω µ − ∂ µ ω ν , (3) ρ µν = ∂ ν ρ µ − ∂ µ ρ ν . (4)In mean field approximation, the effective mass of baryons B is m ∗ B = m B − g σB σ , (5)where σ is given by its ground state expectation value σ = 1 m σ X B g σB n BS − ∂U∂σ ! . (6)3he scalar density is given by n BS = 2(2 π ) Z k FB m ∗ B q k B + m ∗ B d k B . (7)The chemical potential for baryons B is µ B = q k F B + m ∗ B + ω g ωB + ρ g ρB I B , (8)where I B is the isospin projection and ω = 1 m ω X B g ωB n B , (9) ρ = 1 m ρ X B g ρB I B n B . (10)The total baryon number density is n b = P B n B .Now we consider the effects of strong magnetic fields on hyperon matter. The motionof charged particles in a magnetic field is Landau quantized in the plane perpendicular tothe direction of the field. We solve Dirac equations for charged particles using the gaugecorresponding to the constant magnetic field B m along the z axis as A = 0, ~A = (0 , xB m , ψ α = (cid:16) √ b ν ν ! √ π (cid:17) / p L y L z e − ξ / e − i ( ǫt − k y y − k z z ) U α,ν ( k, x ) , (11)with ξ = √ b (cid:16) x − k y qB m (cid:17) and b = qB m .The positive energy spinors, U ν ( k, x ), [47, 48, 49, 50] are given by U ↑ ,ν ( k, x ) = q ǫ ′ + m ∗ p H ν ( ξ )0 p z ǫ ′ + m ∗ p H ν ( ξ ) −√ νbǫ ′ + m ∗ p H ν +1 ( ξ ) , (12)and U ↓ ,ν ( k, x ) = q ǫ ′ + m ∗ p H ν ( ξ ) −√ νbǫ ′ + m ∗ p H ν − ( ξ ) − p z ǫ ′ + m ∗ p H ν ( ξ ) , (13)4here ǫ ′ = p p z + m ∗ p + 2 νqB m .The proton number density n p and scalar density n pS are given by [46] n p = qB m π ν max X ν =0 g ν k p ( ν ) , (14) n pS = qB m π m ∗ p ν max X ν =0 g ν ln k p ( ν ) + µ ∗ p q ( m ∗ p + 2 νqB m ) , (15)where µ ∗ B = q k F B + m ∗ B and k p ( ν ) = q k F p − νqB m . Maximum number of Landaulevels populated is denoted by ν max and the Landau level degeneracy g ν is 1 for ν = 0 and2 for ν >
0. Similarly, we treat noninteracting electrons and muons in constant magneticfields.The total energy density of neutron star matter is ε = 12 m σ σ + U ( σ ) + 12 m ω ω + 12 m ρ ρ + X B = n, Λ π (cid:18) k F B µ ∗ B − k F B m ∗ B µ ∗ B − m ∗ B ln k F B + µ ∗ B m ∗ B (cid:19) + qB m (2 π ) ν max X ν =0 g ν k p ( ν ) µ ∗ p + ( m ∗ p + 2 νqB m ) ln k p ( ν ) + µ ∗ p q ( m ∗ p + 2 νqB m ) + qB m (2 π ) X l = e,µ ν max X ν =0 k l ( ν ) µ l + ( m l + 2 νqB m ) ln k l ( ν ) + µ l p ( m l + 2 νqB m ) ! + B m π . (16)Similarly the total pressure of the system is given by P = − m σ σ − U ( σ ) + 12 m ω ω + 12 m ρ ρ + 13 X B = n, Λ J B + 12 π Z k FB k dk ( k + m ∗ B ) / + qB m (2 π ) ν max X ν =0 k p ( ν ) µ ∗ p − ( m ∗ p + 2 νqB m ) ln k p ( ν ) + µ ∗ p q ( m ∗ p + 2 νqB m ) + qB m (2 π ) X l = e,µ ν max X ν =0 ( k l ( ν ) µ l − ( m l + 2 νqB m ) ln k l ( ν ) + µ l p ( m l + 2 νqB m ) ) + B m π , (17)where k l ( ν ) = q k F l − νqB m . The relation between pressure and energy density definesthe equation of state (EoS). 5 II. BULK VISCOSITY
The macroscopic compression (or expansion) of a fluid element leads to departure fromchemical equilibrium. Non-equilibrium processes cause dissipation of energy which is theorigin of bulk viscosity in neutron stars. Weak interaction processes bring the system backto equilibrium. In this calculation, we consider the non-leptonic reaction n + p −→ p + Λ , (18)as well as direct Urca (dUrca) processes which are represented by n −→ p + l − + ¯ ν l , (19)where l stands for e or µ . When the chemical equilibrium is achieved, chemical potentialsinvolved in above reactions satisfy µ n − µ Λ = 0 and µ n − µ p − µ l = 0 respectively. In this casethe forward and reverse reaction rates, Γ f and Γ r are same. The departure from chemicalequilibrium due to macroscopic perturbation gives rise to the difference between forward andreverse reaction rates, Γ = Γ f − Γ r = 0. For a rotating neutron star, the r-mode oscillationprovides the macroscopic perturbation which drives the system out of chemical equilibrium.The real part of bulk viscosity coefficient can be written as [51] ζ = − n b τ ωτ ) (cid:18) ∂P∂n n (cid:19) d ¯ x n dn b , (20)where ¯ x i = n i /n b is the equilibrium fraction of i -th species, ω is the angular velocity of ( l, m )r-mode and τ is the microscopic relaxation time. For a neutron star rotating with angularvelocity Ω, the angular velocity ( ω ) of ( l, m ) r-mode is given by ω = 2 ml ( l + 1) Ω . (21)We are interested in l = m = 2 r-mode in this calculation. The relaxation time is given by1 τ = Γ δµ δµn b δx n (22)where δµ refers to the chemical imbalance. Here Γ is the total reaction rate.The partial derivative of pressure with respect to neutron number density can be evaluatedfrom the EoS under consideration as ∂P∂n n = k F n µ ∗ n − g σN m σ m ∗ n µ ∗ n D X B n B g σB m σ m ∗ B µ ∗ B + g ωN ω + g ρN I n ρ , (23)6 = 1 + X B (cid:18) g σB m σ (cid:19) ∂n BS ∂m ∗ B + 1 m σ ∂ U∂σ . (24)The total derivative dx n /dn b can be evaluated numerically.Now, we calculate relaxation times for above mentioned processes in presence of magneticfield B m using the EoS as described in section II. A. Non-leptonic process
Here we consider the non-leptonic process given by Eq. (18). In this case, only protonsare affected by magnetic fields. The reaction rate is given byΓ = Z V d k n (2 π ) Z L z dk p iz π Z bLx − bLx L y dk p iy π Z L z dk p fz π Z bLx − bLx L y dk p fy π Z V d k Λ (2 π ) W fi × F ( ǫ n , ǫ p i , ǫ p f , ǫ Λ ) , (25) k p iz and k p fz being the z component of momenta of initial and final protons respectively and k n and k Λ denote momenta of neutrons and Λ hyperons. The Pauli blocking factor is givenby F ( ǫ n , ǫ p i , ǫ p f , ǫ Λ ) = f ( ǫ n ) f ( ǫ p i ) { − f ( ǫ p f ) }{ − f ( ǫ Λ ) } − f ( ǫ Λ ) f ( ǫ p f ) { − f ( ǫ p i ) }{ − f ( ǫ n ) } , (26)with the Fermi distribution function at temperature Tf ( ǫ i ) = 11 + e ǫi − µkT . (27)The matrix element W fi is given by W fi = 1 V ( L y L z ) (2 π ) ǫ n ǫ p i ǫ p f ǫ Λ |M| e − Q δ ( ǫ ) δ ( k y ) δ ( k z ) , (28)where Q = ( k nx − k Λ x ) + ( k p iy − k p fy ) b and δ ( k ) ≡ δ ( k n + k p i − k p f − k Λ ) . (29)7he invariant amplitude squared for the process is |M| = 4 G F sin θ c (cid:2) m ∗ n m ∗ p m ∗ Λ (1 − g np )(1 − g p Λ ) − m ∗ n m ∗ p ( k p i · k Λ )(1 − g np )(1 + g p Λ ) − m ∗ p m ∗ Λ ( k n · k p f )(1 + g np )(1 − g p Λ )+ ( k n · k p i )( k p f · k Λ ) { (1 + g np )(1 + g p Λ ) + 4 g np g p Λ } +( k n · k Λ )( k p i · k p f ) { (1 + g np )(1 + g p Λ ) − g np g p Λ } (cid:3) . (30)In calculating the matrix element given by Eq. (28) we use the solutions of Dirac equationfor protons in magnetic field given by Eqs. (12) and (13). We also assume that the magneticfield is so strong that only zeroth Landau level is populated. Now we integrate over k p iy and k p fy using δ ( k y ) and obtainΓ = L y L z (2 π ) V bL x Z d k n Z dk p iz Z dk p fz Z d k Λ (cid:18) |M| ǫ n ǫ p i ǫ p f ǫ Λ (cid:19) δ ( k y ) × e − [( k nx − k Λ x ) +( k ny − k Λ y ) ] / b F ( ǫ n , ǫ p i , ǫ p f , ǫ Λ ) δ ( ǫ n + ǫ p i − ǫ p f − ǫ Λ ) δ ( k z ) . (31)Here the subscript δ ( k y ) denotes that this condition has been imposed on the quan-tity within the parenthesis. Next we perform the integration over k n and k Λ and write d k = k dk d (cos θ ) dφ . The delta function of z-components of momenta implies k nz + k p iz = k p fz + k Λ z . Here we note that when protons occupy only the zeroth Landaulevel, they have momenta along the direction of magnetic field i.e. in z direction. Hence wehave k pz = k F p . Then depending upon whether the initial and final protons are moving inthe same or opposite direction we have k Λ z − k nz = 0 or k Λ z − k nz = 2 k F p . Next we performthe angle integrations using δ ( k z ) and change variable k to ǫ to getΓ = b (2 π ) Z dǫ n dǫ pi dǫ pf dǫ Λ k F Λ k F p k F p (cid:0) ( |M| ) θ int (cid:1) δ ( k y ) ,δ ( k z ) × h Θ { ( k F n − k F Λ ) } e − [( k Fn − k F Λ ) ] / b + Θ { ( k F n − k F Λ ) − k F p } e − [( k Fn − k F Λ ) − k Fp ] / b i × F ( ǫ n , ǫ p i , ǫ p f , ǫ Λ ) δ ( ǫ n + ǫ p i − ǫ p f − ǫ Λ ) . (32)Here the subscript θ int denotes the angle integrated value. As particles reside near theirFermi surfaces in a degenerate matter we replace momenta and energies under integrationby their respective values at their Fermi surfaces.8he matrix element squared is rewritten as, (cid:0) ( |M| ) θ int (cid:1) δ ( k y ) ,δ ( k z ) = 4 G F sin θ c (cid:2) m ∗ n m ∗ p m ∗ Λ (1 − g np )(1 − g p Λ ) − m ∗ n m ∗ p µ p µ Λ (1 − g np )(1 + g p Λ ) − m ∗ p m ∗ Λ µ n µ p (1 + g np )(1 − g p Λ )+ µ n µ p µ Λ { (1 + g np )(1 + g p Λ ) + 4 g np g p Λ } + µ n µ p µ Λ − k F p µ p ! { (1 + g np )(1 + g p Λ ) − g np g p Λ . (33)As δµ << kT , the energy integration of Eq. (32) can be written as [51] Z dǫ n dǫ p i dǫ p f dǫ Λ F ( ǫ n , ǫ p i , ǫ p f , ǫ Λ ) δ ( ǫ n + ǫ p i − ǫ p f − ǫ Λ ) = ( kT ) π δµ. (34)Finally we getΓ = 1384 π qB m k F Λ k F p (cid:0) ( |M| ) θ int (cid:1) δ ( k y ) ,δ ( k z ) h Θ { ( k F n − k F Λ ) } e − [( k Fn − k F Λ ) ] / b +Θ { ( k F n − k F Λ ) − k F p } e − [( k Fn − k F Λ ) − k Fp ] / b i ( kT ) δµ. (35)The expression of the reaction rate for a zero magnetic field is given by[51]Γ = 1192 π h|M| i k F Λ ( kT ) δµ, (36)where the angle averaged matrix element squared is same as given by [51].Now the quantity δµ/δx n in Eq. (22) is to be evaluated under the condition of totalbaryon number conservation [51] δn n + δn Λ = 0 , (37)which leads to δµδx n = α nn − α n Λ − α Λ n + α ΛΛ , with α ij = ∂µ i ∂n j . (38)Further we have α ij = π k F i µ ∗ i δ ij − m ∗ i µ ∗ i (cid:16) g σi m σ (cid:17) (cid:16) g σj m σ (cid:17) m ∗ j µ ∗ j D + 1 m ω g ωi g ωj + 1 m ρ g ρi I i g ρj I j . (39)Here D is the same as given by Eq. (24). Next we evaluate the relaxation time of thenon-leptonic reaction at a given baryon density using Eq. (22) along with Eqs. (35), (38)and (39).As soon as we know the relaxation time, we can calculate the bulk viscosity coefficient ζ due to the non-leptonic interaction at a given baryon density from Eq. (20).9 . Leptonic processes Here we consider dUrca processes involving nucleons, electrons and muons in a magneticfield. The forward reaction rate for this process is then given by [47, 48, 49]Γ f = Z V d k n (2 π ) Z V d k ν (2 π ) Z L z dk zp π Z bLx − bLx L y dk yp π Z L z dk zl π Z bLx − bLx L y dk yl π W fi × F ( ǫ n , ǫ p , ǫ l ) . (40)Here F ( ǫ n , ǫ p , ǫ l ) is given by F ( ǫ n , ǫ p , ǫ l ) = f ( ǫ n ) { − f ( ǫ p ) }{ − f ( ǫ l ) } . (41)Using the solutions of Dirac equations for protons and electrons in magnetic field, we obtainthe matrix element W fi = (2 π ) V ( L y L z ) |M| δ ( ǫ ) δ ( k y ) δ ( k z ) . (42)Firstly we treat the case following the prescription of Baiko and Yakovlev [48] whenprotons and electrons populate large numbers of Landau levels. In this case, we have X s n ,s p |M| = 2 G F cos θ c (1 + 3 G A ) F , (43)where F is Laguerre functions for both protons and electrons [48]. The forward reactionrate is given by, Γ f = 32 πG F cos θ c m ∗ n m ∗ p µ l (2 π ) R qcB Z dǫ ν ǫ ν J ( ǫ ν ) , (44)where R qcB = 2 Z Z − d cos θ p d cos θ l K F p K F l b F N p ,N l ( u )Θ( k F n − | k F p cos θ p + k F l cos θ l | ) , (45)and J ( ǫ ν ) = Z dǫ n dǫ p dǫ l F ( ǫ n , ǫ p , ǫ l ) δ ( ǫ n − ǫ p − ǫ l − ǫ ν ) , = ( kT ) π + ( ǫ ν /kT ) e ǫ ν /kT . (46)As there is chemical imbalance due to the perturbation, the reverse reaction rate (Γ r ) differsfrom the forward reaction rate and the net reaction rate is given by [26, 48]Γ l = 32 πG F cos θ c m ∗ n m ∗ p µ l (2 π ) R qcB Z dǫ ν ǫ ν { J ( ǫ ν − δµ ) − J ( ǫ ν + δµ ) } . (47)10ne important aspect of dUrca process is the opening of this channel in the forbiddenregime K F n > K F p + K F l which was otherwise closed in field free case [48]. The dUrcaprocess also operates in the allowed domain K F p + K F l > K F n in the presence of a magneticfield. We adopt fitting formulas for R qcB in both domains as given by Ref.[48].Next we focus on the case when both protons and electrons populate zeroth Landau levels[47, 48, 49]. In this case we write the matrix element as W fi = (2 π ) V ( L y L z ) 116 ǫ n ǫ ν ǫ p ǫ e |M| e − Q δ ( ǫ ) δ ( k y ) δ ( k z ) , (48) Q = ( k nx − k νx ) + ( k py + k ly ) b . (49)In a magnetic field neutrons will be polarized because of their anomalous magnetic mo-ments. Hence for two different spin states of neutrons, matrix elements should be evaluatedseparately. The invariant amplitude squared is then |M| = |M + | + |M − | , where |M ± | = G F X s { ¯ V νs ( k ν )(1 + γ ) γ ν U l − ( k l ) }{ ¯ U n ± ( k n )(1 − g np γ ) γ ν U p + ( k p ) } (50) ×{ ¯ U p + ( k p ) γ µ (1 + g np γ ) U n ± ( k n ) }{ ¯ U l − ( k e ) γ µ (1 − γ ) V νs ( k ν ) } , (51)and ± signs denote the up and down spins respectively. The spinors for non-relativisticneutrons are given by U n ± = p ǫ n + m ∗ n χ ± , (52)where χ + = and χ − = . (53)For non-relativistic protons in the zeroth Landau level, the spinor has the same form asgiven by Eq. (52). For spin down relativistic leptons in the zeroth Landau level, the spinoris given by U l − = √ ǫ l + m l − p lz ǫ l + m l (54)11or spin up and down neutrons, invariant amplitudes squared are |M + | = 8 G F cos θ c m ∗ n m ∗ p (1 + g np ) ( ǫ l + p l )( ǫ ν + p νz ) , (55)and |M − | = 32 G F cos θ c m ∗ n m ∗ p g np ( ǫ l + p l )( ǫ ν − p νz ) . (56)Following the same procedure as described in subsection III A and neglecting the neutrinomomenta in momentum conserving delta functions, the final expression of forward reactionrate Γ f is given byΓ f = b (2 π ) m ∗ n m ∗ p µ l k F p k F l h(cid:0) |M + | d (cid:1) δ ( k y ) ,δ ( k z ) + (cid:0) |M − | d (cid:1) δ ( k y ) ,δ ( k z ) i × h Θ { k F n − ( k F p − k F l ) } e − [ k Fn − ( k Fp − k Fl ) ] / b +Θ { k F n − ( k F p + k F l ) } e − [ k Fn − ( k Fp + k Fl ) ] / b i × Z dǫ ν ǫ ν Z dǫ n dǫ p dǫ l F ( ǫ n , ǫ p , ǫ l ) δ ( ǫ n − ǫ p − ǫ l − ǫ ν ) , (57)where (cid:0) |M + | d (cid:1) δ ( k y ) ,δ ( k z ) = 8 G F cos θ c (1 + g np ) (cid:18) p l ǫ l (cid:19) (cid:18) p νz ǫ ν (cid:19) . (58)Similarly we have, (cid:0) |M − | d (cid:1) δ ( k y ) ,δ ( k z ) = 32 G F cos θ c g np (cid:18) p l ǫ l (cid:19) (cid:18) − p νz ǫ ν (cid:19) . (59)It is to be noted that z-component of neutrino momentum is smaller than its energy. WeobtainΓ f = b (2 π ) m ∗ n m ∗ p µ l k F p k F l h(cid:0) |M + | d (cid:1) δ ( k y ) ,δ ( k z ) + (cid:0) |M − | d (cid:1) δ ( k y ) ,δ ( k z ) i × h Θ { k F n − ( k F p − k F l ) } e − [ k Fn − ( k Fp − k Fl ) ] / b +Θ { k F n − ( k F p + k F l ) } e − [ k Fn − ( k Fp + k Fl ) ] / b i × Z dǫ ν ǫ ν J ( ǫ ν ) . (60)Now if the reverse reaction rate is Γ r and there is slight departure from chemical equilibrium δµ , then the net reaction rate is [26],Γ l = Γ r − Γ f = b (2 π ) m ∗ n m ∗ p µ l k F p k F l h(cid:0) |M + | d (cid:1) δ ( k y ) ,δ ( k z ) + (cid:0) |M − | d (cid:1) δ ( k y ) ,δ ( k z ) i × h Θ { k F n − ( k F p − k F l ) } e − [ k Fn − ( k Fp − k Fl ) ] / b +Θ { k F n − ( k F p + k F l ) } e − [ k Fn − ( k Fp + k Fl ) ] / b i × Z dǫ ν ǫ ν { J ( ǫ ν − δµ ) − J ( ǫ ν + δµ ) } . (61)12sing the following result from Ref. [26] Z dǫ ν ǫ ν { J ( ǫ ν − δµ ) − J ( ǫ ν + δµ ) } = 17( πkT ) δµ, (62)we getΓ l = 17 qB m π m ∗ n m ∗ p µ l k F p k F l G F cos θ c (cid:18) p l ǫ l (cid:19) (cid:20)
14 (1 + g np ) + g np (cid:21) × h Θ { k F n − ( k F p − k F l ) } e − [ k Fn − ( k Fp − k Fl ) ] / b +Θ { k F n − ( k F p + k F l ) } e − [ k Fn − ( k Fp + k Fl ) ] / b i × ( kT ) δµ. (63)The zero magnetic field result is given byΓ l ( B m = 0) = 17240 π m ∗ n m ∗ p µ l ( |M| d ) θ int ( kT ) δµ, (64)where( |M| d ) θ int = G F cos θ c (cid:26) (1 + g np ) (cid:18) − k F n m ∗ n (cid:19) + (1 − g np ) (cid:18) − k F p m ∗ p (cid:19) − (1 − g np ) (cid:27) . (65) IV. RESULTS AND DISCUSSION
Nucleon-meson coupling constants of the model are obtained by reproducing the prop-erties of nuclear matter such as binding energy
E/B = − . M eV , saturation density n = 0 . f m − , asymmetry energy coefficient a asy = 32 . M eV and incompressibility K = 240 M eV and taken from Ref [52]. The coupling strength of Λ hyperons with ω mesons is determined from SU(6) symmetry of the quark model [53, 54, 55]. The couplingstrength of Λ hyperons to σ mesons is determined from the potential depth of Λ hyperonsin normal nuclear matter U Λ = − g σ Λ σ + g ω Λ ω . (66)We take the potential depth U Λ = − M eV as obtained from the analysis of Λ hypernuclei[54, 56].We adopt a profile of magnetic field given by [57], B ( n b /n ) = B s + B c (cid:18) − e − β “ nbn ” γ (cid:19) . (67)We consider different values for central field B c = 10 and 10 G whereas surface fieldstrength is taken as B s = 10 G in this calculation. We chose β = 0 .
01 and γ = 3. The13agnetic field strength depends on baryon density in the above parameterization. Furtherthe magnetic field at each density point is constant and uniform. The effects of anomalousmagnetic moments of nucleons and contributions of the magnetic field to energy densityand pressure are negligible because magnetic fields considered in this calculation are not toostrong.Numbers of Landau levels populated by electrons and protons, are sensitive to the mag-netic field strength and baryon density. As the field strength increases, the population ofLandau levels decreases. In a weak magnetic field, when many Landau levels are populated,we treat charged particles unaffected by the magnetic field. Further the effects of magneticfields are most pronounced when only zeroth Landau levels are populated. Protons, elec-trons and muons populate zeroth Landau levels if central field strength B c ∼ G. Figure1 shows fractions of various particle species with normalised baryon density. We find largenumbers of Landau levels of charged particles even when the magnetic field reaches its cen-tral value 10 G. Populations of charged particles are enhanced in the magnetic field dueto Landau quantization than those of field free case (not shown in the figure). It is notedin Fig. 1 that the threshold density of Λ hyperons is shifted to 1.7 n from its zero magneticfield value of 2.6 n because of phase space modifications of charged particles in a magneticfield.The variation of pressure with energy density in the presence of a magnetic field withcentral field strength B c = 10 G (solid curve) is shown in Fig. 2. The dashed curve denotesthe EoS without a magnetic field. The EoS in the presence of the magnetic field becomesstiffer when charged particles are Landau quantised. Here magnetic field contributions tothe energy density and pressure are insignificant.Now we compute the relaxation time for both non-leptonic and leptonic reactions as givenby Eq. (22). To calculate the matrix element we take g np = − .
27 and g p Λ = − .
72 [51],and the Cabibbo angle ( θ c ) is given by sin θ c = 0 . B c = 10 Gover entire density range considered in our calculation. For the non-leptonic process, whenprotons populate large number of Landau levels, we use the field free expression of Γ asgiven by Eq. (36). For leptonic reactions we use the expression as given by Eq. (47) whenleptons and protons populate finite numbers of Landau levels. Chemical potentials andFermi momenta of constituent particles are obtained from the EoS. The partial derivative14f chemical potentials with respect to baryon density can be calculated from the EoS. Usingthese inputs, we can compute relaxation times for both reactions as a function of baryondensity at a particular temperature. Figure 3 displays relaxation time ( τ ) of the non-leptonicprocess involving Λ hyperons in a magnetic field having its central value B c = 10 G and atdifferent temperatures as a function of normalised baryon number density. Here τ decreaseswith increasing baryon density. Further the relaxation time in a magnetic field increaseswith decreasing temperature as was earlier noted in the field free case [23].Relaxation times for dUrca reactions involving electrons and muons in a magnetic fieldwith B c = 10 G and at different temperatures are plotted in Figs. 4 and 5 respectively.For leptonic processes, relaxation times are affected by the magnetic field. For the fieldfree case, the dUrca process sets in at 1.4 n . In the magnetic field, relaxation times due todUrca reactions drop sharply from large values in the forbidden domain K F n > K F p + K F e .This is attributed to the behaviour of R qcB which we discuss in details in connection withbulk viscosity due to dUrca processes below. The forbidden domain joins with the alloweddomain K F p + K F e > K F n at a point from which relaxation times increase with baryondensity. Like the non-leptonic case, relaxation times for dUrca processes also increase withdecreasing temperature.Now we focus on the calculation of bulk viscosity due to the non-leptonic and leptonicprocesses. As soon as we know relaxation times of non-leptonic and leptonic reactions,we compute bulk viscosity coefficients for the respective processes from Eq. (20). In thiscalculation we consider l = m = 2 r-mode and hence ω = 2 / s − . In the temperature regime considered here, we have always ωτ << ζ also depends on temperature. The bulk viscosity coefficient for the non-leptonic processin a magnetic field with B c = 10 G (dashed curve) and in the absence of a magnetic field(solid curve) are exhibited as a function of normalised baryon number density in Fig. 6at different temperatures. The non-leptonic reaction involves protons that populate manyLandau levels in the magnetic field with B c = 10 G. In this case, we adopt the field free15xpression of the reaction rate as given by Eq. (36) for the calculation of relaxation timeand hyperon bulk viscosity coefficient in Eq. (20). Therefore, the effects of magnetic fieldenter into hyperon bulk viscosity coefficient through the EoS which is modified by Landauquantization of charged particles. In Fig. 6, we find hyperon bulk viscosity in the magneticfield is suppressed compared with the field free case.We display bulk viscosity coefficient for the dUrca process in a magnetic field with B c =10 G and at a temperature T = 10 K as a function of normalised baryon density in Fig. 7.In this case electrons and protons populate many Landau levels. The dotted line representsthe dUrca contribution in the forbidden domain K F n > K F p + K F e . In this regime, reactionkinetics are characterised by two parameters x = K Fn − ( K Fp + K Fe ) K Fp N − / Fp and y = N / F p , where N F p is the number of proton Landau levels. The dUrca reaction in the forbidden domain is anefficient process as long as x ≤
10. This corresponds to baryon density ≤ n . The largeenhancement of bulk viscosity coefficient in this domain is attributed to the behaviour of R qcB [48]. It was noted R qcB = 1 / x = 0 and it becomes very small when x >
10 [48].At x = 0, the forbidden domain merges with the allowed domain K F p + K F e > K F n of thedUrca process. The dUrca bulk viscosity in the allowed domain is shown by the dash-dottedline. The result of zero field is shown by the solid line. The bulk coefficient increases withmagnetic field in the allowed domain at higher baryon densities.Figure 8 and Figure 9 show bulk viscosity coefficients for dUrca processes involving elec-trons and muons in the presence of the magnetic field with central value B c = 10 G and atdifferent temperatures as a function of normalised baryon density. In both cases contribu-tions to bulk viscosity coefficients due to dUrca processes come from the forbidden as well asallowed domains. As discussed above, the forbidden domain merges with the allowed domainat x = 0. For temperatures T = 10 and 10 K, bulk viscosity coefficients due to dUrcaprocesses increase with baryon density. However the bulk viscosity for T = 10 K initiallydecreases and later increases with baryon density. This behaviour can be understood in thefollowing way. For dUrca processes at 10 K, we have ωτ <
1. On the other hand, we find ωτ > K and 10 K. Consequently bulk viscosity coefficientshave a T dependence when ωτ > T − dependence when ωτ <
1. Thisinversion of temperature dependence of dUrca bulk viscosity coefficients is not found in thecase of hyperon bulk viscosity.Finally, we point out what happens in case of superstrong fields. We find that charged16articles populate zeroth Landau levels when B c ∼ . Populations of charged particles areenhanced because of strong modification of their phase spaces. Further the EoS is modifieddue to magnetic field contributions to the energy density and pressure. The strong magneticfield enhances the hyperon bulk viscosity compared with the field free case. Similarly we notesignificant modification in bulk viscosity coefficients due to dUrca processes when leptonsand protons are in zeroth Landau levels. However, there is no observational evidence forsuperstrong field ∼ G in neutron star’s interior so far.
V. SUMMARY
We have investigated bulk viscosity of non-leptonic process involving Λ hyperons anddUrca processes in the presence of strong magnetic fields. In this calculation we considermagnetic fields with different central values B c = 10 and 10 G. The equation of statehas been constructed using the relativistic field theoretical model. Many Landau levels ofcharged particles are populated for above values of central field. For a particular tempera-ture, the hyperon bulk viscosity coefficient is reduced compared with that of the zero fieldcase. Further it is noted that the hyperon bulk viscosity decreases with increasing tem-perature as was earlier reported for the field free case. Bulk viscosity coefficients due todUrca processes in a magnetic field have contributions from the forbidden as well as alloweddomains. The bulk viscosity coefficients in magnetic fields having central values B c = 10 and 10 G are enhanced in the allowed domain at higher baryon densities than those offield free cases. We find an inversion of the temperature dependence of dUrca bulk viscositycoefficients at 10 K. We briefly discuss the effects of a superstrong magnetic field ∼ G on hyperon and dUrca bulk viscosities when zeroth Landau levels of charged particles arepopulated. However, such a superstrong magnetic field may not be a possibility in neutronstars.In this calculation, we adopt the field free hyperon bulk viscosity relation when protonspopulate large number of Landau levels. This may be an approximate treatment of the actualcase. However the exact treatment of the effects of a magnetic field on the non-leptonic bulkviscosity would be worth studying when protons populate many Landau levels. Further theinvestigation of bulk viscosity in magnetic fields has important implications for the r-modes17n magnetars. This will be reported in a future publication. [1] M. Nayyar and B.J. Owen, Phys. Rev. D , 084001 (2006).[2] S. Chandrasekhar, Phys. Rev. Lett. , 611 (1970).[3] J. L. Friedman and B. F. Schutz, Astrophys. J. , 937 (1978);J. L. Friedman and B. F. Schutz, Astrophys. J. , 281 (1978);J. L. Friedman Commun. Math. Phys. , 247 (1978).[4] N. Andersson and K.D. Kokkotas, Int. J. Mod. Phys. D10 , 381 (2001).[5] N. Andersson, Class. Quant. Grav. , R105 (2003).[6] N. Andersson, Astrophys J. , 708 (1998).[7] J. L. Friedman and S. M. Morsink, Astrophys. J. , 714 (1998).[8] L. Lindblom, B.J. Owen and S. M. Morsink, Phys. Rev. Lett. , 4843 (1998).[9] N. Andersson, K. D. Kokkotas and B. F. Schutz, Astrophys. J. , 846 (1999).[10] N. Stergioulas, Liv. Rev. Rel. , 3 (2003).[11] C. Thompson and R. C. Duncan, Astorphys. J. L115, (1999).[13] G. Vasisht and E. V. Gotthelf, Astorphys. J.
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