Neutrino emissivities in quark matter

Neutrino emissivities in quark matter Jens Berdermann (University Rostock) David Blaschke (University Wrocław) WORKSHOP III OF THE VI,,DENSE HADRONIC MATTER & QCD PHASE TRANSITION 16.10.2006 Rathen Neutrino emissivities in quark matter p.1/26

Content Introduction Urca process (Iwamoto) Limitations Nambu-Gorkov formalism Results Summary/Outlook Neutrino emissivities in quark matter p.2/26

URCA-Process [Iwamoto Ann.Phys. 141(1982) 1] d W A p 2 p 4 p V 1 p 3 d u + e + ν e θ 13 p 3 p 1 u d ν e e u ε = 6 4Y Z i=1 d 3 p i (2π) 3 2E i E 2 W fi n(p 1 )[1 n(p 3 )][1 n(p 4 )] W fi = (2π) 4 δ (4) (p 1 p 2 p 3 p 4 ) M 2 M 2 1 X M fi 2 = 64 G 2 cos 2 θ c (p 1 p 2 ) (p 3 p 4 ) 2 θ 14 p 4 e θ 3 4 σ d,σ u,σ e = 64 G 2 cos 2 θ c E 1 E 2 E 3 E 4 (1 cos θ 34 ) µ i = p i F [1 + (2/3π)α s], i = u, d µ i p i F [1 + 0.5(m i/p i F )2 ], i = u, d, e p d F pu F pe F 1 2 pe F θ2 14, θ 14 θ 34 p d F pu F pe F p e F (2/3π)α s, α s = g 2 /4 2!! p d F pu F pe F 1 2 2 pe 4 pd F m d F p e pu F F p e F p d F! m u p u F! 2! 3 m 2 e 5 p e F Neutrino emissivities in quark matter p.3/26

Emissivities (perturbative) ε α s (457/630) G 2 cos 2 θ c α s p d F pu F pe F T 6 ε m (457π/1680) G 2 cos 2 θ c m 2 d f pu F T 6, f 1 (m u /m d ) 2 (p d F /pu F ) (m e/m d ) 2 (p d F /pe F ) 10 33 m u = 5 MeV, m d = 10 MeV ε [erg / m 3 s] 10 32 10 31 10 30 10 29 DU (α s ) DU (mass) 10 28 10 27 0 20 40 60 80 100 T [kev] Neutrino emissivities in quark matter p.4/26

QCD phase diagram 120 NQ 100 g2sc gcfl 80 gusc T [MeV] 60 2SC CFL 40 20 M s =200 MeV 175 150 0 300 350 400 450 500 550 µ [MeV] Figure F.Sandin [Blaschke et. al. Phys.Rev D 72,065020 (2005)] Neutrino emissivities in quark matter p.5/26

Quark mass and diquark gap 2SC Ω (µb,µ Q,µ 8,T ) = φ2 u + φ 2 d 8 G 1 + 2 4 G 2 1 π 2 Z 0 dp p 2 (T ln " 1 + exp E d,b (p) + 0.5 µ Q T " + T ln 1 + exp E+ d,b (p) 0.5 µ!# " Q +T ln 1 + exp E u,b (p) 0.5 µ! Q T T " + T ln 1 + exp E+ u,b (p) + 0.5 µ!# " Q +2T ln 1 + exp ξ d,r (p) + 0.5 µ! Q T T " + 2T ln 1 + exp ξ+ d,r (p) 0.5 µ!# "! Q +2T ln 1 + exp ξ u,r(p) 0.5 µ Q T T "!# 2 3 + 2T ln 1 + exp ξ+ u,r(p) + 0.5 µ Q + X 4E f (p) + X ) ξ j f,r T (p) 5 f=u,d j=+,!# q ξ f,r = [E f,r (p)]2 + g(p) 2 2, f = u, d E f,r (p) = E f (p) 1 µ B + 1 «3 2 µ Q 1 3 µ 8, E f,b (p) = E f (p) 1 3 q E f (p) = p 2 + Mf 2(p), M f (p) = m f + g(p)φ f µ B + 1 «2 µ Q ± 2 3 µ 8, Neutrino emissivities in quark matter p.6/26

Quark mass and diquark gap 2SC (sym. matter) Λ = 0.6023 GeV, G1 Λ 2 = 2.319, m u,d = 5.5MeV Ω φ f = 0 Ω φ f = Ω = 0 400 400 300 T= 0 MeV M u, M d 300 T= 0 MeV η= 0.75 M u, M d, M [MeV] 200, M [MeV] 200 100 100 200 300 400 500 µ q [MeV] 200 300 400 500 µ q [MeV] Neutrino emissivities in quark matter p.7/26

Quark mass and diquark gap 2SC (β-equil.,charge cons.) 400 400 300 T= 0 MeV η= 0.75 M u, M d M u M d 300 T= 0 MeV η= 1 M u, M d M u M d ud ud, M [MeV] 200, M [MeV] 200 100 100 300 350 400 450 500 µ q [MeV] 300 350 400 450 500 µ q [MeV] Neutrino emissivities in quark matter p.8/26

Quark mass and diquark gap (β-equil.,charge cons.) 400 400 300 T= 50 MeV η= 0.75 M u, M d M u M d 300 T= 50 MeV η= 1 M u, M d M u M d ud ud, M [MeV] 200, M [MeV] 200 100 100 300 350 400 450 500 µ q [MeV] 300 350 400 450 500 µ q [MeV] Neutrino emissivities in quark matter p.9/26

Kinetic equation i µ x Tr[γ µg < ν (X, q 2)] = Tr[G > ν (X, q 2)Σ < ν (X, q 2) Σ > ν (X, q 2)G < ν (X, q 2)], X = (t, x) W ν + d u W + + ν e + Σ < ν (t, q 2 ) = G2 F 2 Σ > ν (t, q 2 ) = G2 F 2 Z d 4 q 1 (2π 4 ) γµ (1 γ 5 )(γ α q 1,α + µ e γ 0 )γ ν (1 γ 5 ) Π > µν(q 1 q 2 ) π q 1 f e (t, q 1 )δ(q 0 1 + µ e q 1 ), Z d 4 q 1 (2π 4 ) γµ (1 γ 5 )(γ α q 1,α + µ e γ 0 )γ ν (1 γ 5 ) Π < µν(q 1 q 2 ) π q 1 [1 f e (t, q 1 )]δ(q 0 1 + µ e q 1 ig < ν (t, q 2 ) = (γ β q 2,β + µ ν γ 0 ) π {f ν (t, q 2 )δ(p 0 2 + µ ν q 2 ) q 2 [1 f ν (t, q 2 )]δ(q2 0 + µ ν + q 2 )} ig > ν (t, q 2 ) = (γ β q 2,β + µ ν γ 0 ) π {[1 f ν (t, q 2 )]δ(q2 0 + µ ν q 2 ) q 2 f ν (t, q 2 )δ(q2 0 + µ ν + q 2 )} Neutrino emissivities in quark matter p.10/26

Kinetic equation t f ν(t, q 2 ) = i G2 F 16 Z d 3 q 1 (2π) 3 q 1 q 2 Lµν (q 1, q 2 ){[1 f ν (t, q 2 )]f e (t, q 1 )Π > µν(q) f ν (t, q 2 )[1 f e (t, q 1 )]Π < µν(q)} Z = G2 F d 3 q 1 8 (2π) 3 q 1 q 2 Lµν (q 1, q 2 ) n F ( q 1 µ e )n B ( q 2 + µ e q 1 )ImΠ R µν(q Π > (q) = 2i[1 + n B (q 0 )]ImΠ R (q) Π < (q) = 2i n B (q 0 )ImΠ R (q) n B (ω) 1/(e ω/t 1) (Bose) n F (ω) 1/(e ω/t + 1) (Fermi) L µν (q 1, q 2 ) = Tr[γ µ (1 γ 5 ) q 1 γ ν (1 γ 5 ) q 2 ] = 8[q µ 1 qν 2 g µν (q 1 q 2 ) + q ν 1 q µ 2 iɛµανβ q 1α q 2β ] Neutrino emissivities in quark matter p.11/26

Hadronic loop u u <ud> W W W d d W W W <du> Π µν (q) = i 2 Z d 4 p (2π) 4 Tr Z,D [Γ Z µ S p Γ Z ν S p+q ], u, d p, p + q ˆp, ˆk Γ Z i = 0 @ Γ i 0 0 Γ + i 1 A Γ ± i = γ i (1±g A γ 5 ) i = µ, ν ; S j = 0 @ G+ j F + j F j G j 1 A j = p, p+q. S 1 S = 1 ( [S 0 + ] 1 + [S0 ] 1 ) ( A C B D ) = ( 1 0 0 1 ) Neutrino emissivities in quark matter p.12/26

Nambu-Gorkov propagators I : [S + 0 ] 1 A + C = 1 II : [S + 0 ] 1 B + D = 0 III : + A + [S 0 ] 1 C = 0 IV : + B + [S 0 ] 1 D = 1 A + = [(S0 ) 1 Σ + ] 1 = G + B = S + 0 G = F C = S 0 + G + = F + D = [(S 0 ) 1 Σ ] 1 = G S ± 0 (p 0, p) = γ 0 Λ p p 0 Ep + γ 0 Λ + p p 0 + E p ± h 1 S ± 0 (p 0, p)i = γ 0 (p 0 ± µ) γp m = γ 0 (p 0 Ep )Λ + p + γ 0 (p 0 + E p ± )Λ p E p = E p µ (particle/hole), E ± p = E p ± µ (antiparticle/antihole), E p = p p 2 + m 2 = i ε ik ɛ αβb γ 5, + = γ 0 ( ) γ 0 (diquark condensat) Σ ± = S 0 ± Neutrino emissivities in quark matter p.13/26

Nambu-Gorkov propagators G ± = [(S ± 0 ) 1 S 0 ± ] 1 = p 0 + E p p 2 0 (ξ p ) 2 γ 0 Λ p + F ± = S 0 ± G ± = ± p 2 0 (ξ± p ) 2 Λ+ p + ± p 2 0 (ξ p ) 2 Λ p Pole p0 = ±ξ p und p 0 = ξ + p mit (ξ ± p ) 2 = (E ± p ) 2 + 2 p 0 E ± p p 2 0 (ξ± p ) 2 γ 0 Λ + p (Quasiparticle/Quasihole- and Quasiantiparticle/Quasiantihole excitation energies) Energy projectors Λ ± p = 1 2 (1 ± γ 0 S + p ) Λ ± p = 1 2 (1 ± γ 0 S p ) 9 = ; S± p = γ ˆp ± ˆm, ˆp = p E p ˆm = m E p Neutrino emissivities in quark matter p.14/26

Polarisation tensor Π µν (q) = i T 2 X Z n d 3 p (2π) 3 Tr Z,D [Γ Z µ S p Γ Z ν S p+q ] Tr D [Γ µ G + p Γ ν G + p+q + Γ+ µ G p Γ + ν G p+q + Γ µ F p Γ + ν F + p+q + Γ+ µ F + p Γ ν F p+q ] = (p 0 + E p )(p 0 + q 0 + E k ) [p 2 0 (ξ p ) 2 ][(p 0 + q 0 ) 2 (ξ k )2 ] {T + µν(ˆp, ˆk) + g 2 A e T + µν(ˆp, ˆk) g A [ f W + µν(ˆp, ˆk) + W + µν(ˆp, ˆk)]} + (p 0 E p )(p 0 + q 0 E k ) [p 2 0 (ξ p ) 2 ][(p 0 + q 0 ) 2 (ξ k )2 ] {T µν(ˆp, ˆk) + g 2 A e T µν(ˆp, ˆk) + g A [ f W µν(ˆp, ˆk) + W µν(ˆp, ˆk)]} 2 [p 2 0 (ξ p ) 2 ][(p 0 + q 0 ) 2 (ξ k )2 ] {[T µν (ˆp, ˆk) + T + µν (ˆp, ˆk)] + g 2 A [e T µν (ˆp, ˆk) + e T + µν (ˆp, ˆk)] g A [ f W + µν(ˆp, ˆk) + W + µν(ˆp, ˆk) f W µν(ˆp, ˆk) W µν(ˆp, ˆk)]} p0 = i(2n + 1)πT, q 0 = i2mπt fermionic and bosonic Matsubara frequencies T e ± µν (ˆp, ˆk) = Tr[γ 0 γ µ Λ e± p γ 0 γ ν Λ ± k ],f W µν(ˆp, ± ˆk) = Tr[γ 0 γ µ Λ e± p γ 0 γ ν Λ ± k γ 5], T µν(ˆp, ± ˆk) = Tr[γ 0 γ µ Λ ± p γ 0 γ ν Λ ± k ], W± µν(ˆp, ˆk) = Tr[γ 0 γ µ Λ ± p γ 0 γ ν Λ ± k γ 5] (hadronic tensors) Neutrino emissivities in quark matter p.15/26

Polarisation tensor Π µν (q 0, q) = i 2 Z d 3 p (2π) 3 A+ (E p, E k ){T µν + (ˆp, ˆk) + T e µν + (ˆp, ˆk) [ W f µν + (ˆp, ˆk) + W µν + (ˆp, ˆk)] + A (E p, E k ){Tµν (ˆp, ˆk) + T e µν (ˆp, ˆk) + [ W f µν (ˆp, ˆk) + Wµν (ˆp, ˆk)]} 2 B(E p, E k ){T µν(ˆp, ˆk) + T + µν(ˆp, ˆk) + e T µν(ˆp, ˆk) + e T + µν(ˆp, ˆk) [ f W + µν(ˆp, ˆk) + W + µν(ˆp, ˆk) f W µν(ˆp, ˆk) W µν(ˆp, ˆk)]} A ± (E p, E k ) = 1 2ξp 2ξ k B(E p, E k ) = 1 2ξp 2ξ k X s 1 s 2 =± X s 1 s 2 =± (ξp + s 1 Ep )(ξ k + s 2E k ) n F (±s 1 ξp )n F ( s 2 ξ k ) q 0 ± s 1 ξp s 2 ξ k n B (±s 1 ξp s 2 ξ k ) 1 n F (s 1 ξp )n F ( s 2 ξ k ) q 0 + s 1 ξp s 2 ξ k n B (s 1 ξp s 2 ξ k ) Neutrino emissivities in quark matter p.16/26

Neutrino emissivities ε ν t Z d 3 q 2 (2π) 3 q 2 [f ν (t, q 2 ) + f ν (t, q 2 )] = 2 t Z d 3 q 2 (2π) 3 p F,ν f ν (t, q 2 ) t f ν(t, q 2 ) = G2 F 8 Z d 3 q 1 (2π) 3 p F,e p F,ν L µν (q 1, q 2 ) n F (p F,e µ e )n B (p F,ν +µ e p F,e )ImΠ R µν(q) ε ν = π 8 G2 F cos2 θ c Z d 3 q 2 (2π) 3 p F,ν n B (p F,ν + µ e p F,e ) " X s 1 s 2 =± Z d 3 q 1 (2π) 3 p F,e p F,ν 2B s 1 p Bs 2 k Lµν (q 1, q 2 ) H (n) µν (ˆp, ˆk) 2 2ξ p 2ξ k δ(q 0 + s 1 ξ p s 2 ξ k ) n F (s 1 ξ p )n F ( s 2 ξ k ) n B (s 1 ξ p s 2 ξ k ). Z d 3 p (2π) 3 n F (p F,e µ e ) L µν (q 1, q 2 ) H (a) µν (ˆp, ˆk) L µν (q 1, q 2 )H (n) µν (ˆp, ˆk) = 64q 0 1q 0 2(1 ˆq 1 ˆp)(1 ˆq 2 ˆk) Neutrino emissivities in quark matter p.17/26

Emissivities (Quark mass effect) 10 34 10 33 10 32 DU (α s ) DU (mass, b =0 MeV) η = 1 µ B = 1.2 GeV, α s, r,g = 0 MeV µ B = 1.2 GeV, M u/d = 74/20 MeV, r,g = 0 MeV µ B = 1.2 GeV,M u/d = 32/18 MeV, r,g = 205 MeV µ B = 1.3 GeV,M u/d = 22/14 MeV, r,g = 211 MeV µ B = 1.4 GeV,M u/d = 17/11 MeV, r,g = 214 MeV ε [erg / m 3 s] 10 31 10 30 10 29 10 28 10 27 0 20 40 60 80 100 T [kev] Neutrino emissivities in quark matter p.18/26

Computation scheme T Start, µ B Computation scheme Ω grand canonical thermodynamic potential (EOS) Ω = Ω φ f = 0 Μ f, µ B, µ Q, µ 8 µ f,c, µ e c=r,g,b f=u,d microscopic calculation of kinetic parameters,cv ε = P T Ω Ω µ T µ P = Ω T Next cooling eq. ε ν,λ / L, t, T transport code, R m cooling curves pulsar kicks GRB Tolman TOV Oppenheimer Volkoff equation (star structure) Neutrino emissivities in quark matter p.19/26

Cooling t = T Z f T i X i,j C i v(t ) L j (T ) dt ; i = quark, e, γ, gluon, ν; j = ν, γ µ B =1200 MeV, η=1 µ B =1400 MeV, η=1 6,4 6,2 Crab α s α s, Gap mass mass, Gap Model IV 6,4 6,2 Crab α s α s, Gap mass mass, Gap Model IV log 10 (T s [K]) 6 5,8 5,6 3C58 Vela RX J1856 log 10 (T s [K]) 6 5,8 5,6 3C58 Vela RX J1856 5,4 5,4 5,2 5,2 5 2 3 4 5 6 log 10 (t[yr]) 5 2 3 4 5 6 log 10 (t[yr]) Neutrino emissivities in quark matter p.20/26

Conclusion Iwamoto Nambu-Gorkov formalism Neutrino emissivities in quark matter p.21/26

Conclusion Iwamoto Nambu-Gorkov formalism (non-)perturbative regime (effective chiral quark model) Neutrino emissivities in quark matter p.21/26

Conclusion Iwamoto Nambu-Gorkov formalism (non-)perturbative regime (effective chiral quark model) Mass effect due to chiral phase transition Neutrino emissivities in quark matter p.21/26

Conclusion Iwamoto Nambu-Gorkov formalism (non-)perturbative regime (effective chiral quark model) Mass effect due to chiral phase transition Color-superconductivity (diquark gap) Neutrino emissivities in quark matter p.21/26

Conclusion Iwamoto Nambu-Gorkov formalism (non-)perturbative regime (effective chiral quark model) Mass effect due to chiral phase transition Color-superconductivity (diquark gap) Influence to the cooling of quark stars Neutrino emissivities in quark matter p.21/26

Outlook Cooling (color-superconductivity, regime chiral symmetry breaking) Neutrino emissivities in quark matter p.22/26

Outlook Cooling (color-superconductivity, regime chiral symmetry breaking) Influence of collective fields (vector mean fields sym/asym chem. pot. shift) Neutrino emissivities in quark matter p.22/26

Outlook Cooling (color-superconductivity, regime chiral symmetry breaking) Influence of collective fields (vector mean fields sym/asym chem. pot. shift) High temperatures (chiral/diquark gap change, neutrino trapping, mean free path) Neutrino emissivities in quark matter p.22/26

Outlook Cooling (color-superconductivity, regime chiral symmetry breaking) Influence of collective fields (vector mean fields sym/asym chem. pot. shift) High temperatures (chiral/diquark gap change, neutrino trapping, mean free path) Cooling with transport (diffusion equation) Neutrino emissivities in quark matter p.22/26

Outlook Cooling (color-superconductivity, regime chiral symmetry breaking) Influence of collective fields (vector mean fields sym/asym chem. pot. shift) High temperatures (chiral/diquark gap change, neutrino trapping, mean free path) Cooling with transport (diffusion equation) Phenomenology (Pulsar Kicks,Gamma-Ray-Bursts,Supernova) Neutrino emissivities in quark matter p.22/26

leptonic tensors L 00 (q 1, q 2 ) = 8(q1q 0 2 0 + q 1 q 2 ) L 0i (q 1, q 2 ) = 8[q1 0 qi 2 + qi 1 q0 2 iɛijk q 1j q 2k ] L i0 (q 1, q 2 ) = 8[q1q 0 2 i + q1q i 2 0 + iɛ ijk q 1j q 2k ] L ij (q 1, q 2 ) = 8[δ ij (q1q 0 2 0 q 1 q 2 ) + q1q i j 2 + qj 1 qi 2 iɛ ijkl q 1k q 2l ] T ± 00 (ˆp, ˆk) = 1 + ˆp ˆk + ˆm u ˆm d T ± 0i (ˆp, ˆk) = ±(ˆp i + ˆk i ) T ± i0 (ˆp, ˆk) = ±(ˆp i + ˆk i ) T ± ij (ˆp, ˆk) = δ ij (1 ˆp ˆk ˆm u ˆm d ) + ˆp iˆkj + ˆk i ˆp j et ± 00 (ˆp, ˆk) = 1 + ˆp ˆk ˆm u ˆm d et ± 0i (ˆp, ˆk) = ±(ˆp i + ˆk i ) et ± i0 (ˆp, ˆk) = ±(ˆp i + ˆk i ) et ± ij (ˆp, ˆk) = δ ij (1 ˆp ˆk + ˆm u ˆm d ) + ˆp iˆkj + ˆk i ˆp j W ± 00 (ˆp, ˆk) = 0 W ± 0i (ˆp, ˆk) = iɛ ijk ˆp j ˆkk W ± i0 (ˆp, ˆk) = +iɛ ijk ˆp j ˆkk W ± ij (ˆp, ˆk) = iɛ ijk (ˆp k ˆk k ) + iɛ ijkl ˆp kˆk fw ± 00 (ˆp, ˆk) = 0 fw ± 0i (ˆp, ˆk) = iɛ ijk ˆp j ˆkk fw ± i0 (ˆp, ˆk) = +iɛ ijk ˆp j ˆkk fw ± ij (ˆp, ˆk) = iɛ ijk (ˆp k ˆk k ) + iɛ ijkl ˆp kˆk Neutrino emissivities in quark matter p.23/26

Projection operator Main- and transformation properties Λ ± p Λ ± p = Λ ± p Λ ± p Λ p = 0 Λ + p + Λ p = 1 and γ 0 Λ ± p γ 0 = Λ p γ 5 Λ ± p γ 5 = Λ ± p Neutrino emissivities in quark matter p.24/26

Polarisation tensor Π µν (q 0, q) = i 2 A ± 1 (E p, E k ) = 2ξp 2ξ k 1 B(E p, E k ) = 2ξp 2ξ k Z d 3 p (2π) 3 A+ (E p, E k ){T + µν(ˆp, ˆk) + e T + µν(ˆp, ˆk) [ f W + µν(ˆp, ˆk) + W + µν(ˆp, ˆk)]} + A (E p, E k ){T µν(ˆp, ˆk) + e T µν(ˆp, ˆk) + [ f W µν(ˆp, ˆk) + W µν(ˆp, ˆk)]} 2 B(E p, E k ){T µν(ˆp, ˆk) + T + µν(ˆp, ˆk) + e T µν(ˆp, ˆk) + e T + µν(ˆp, ˆk) [ f W + µν(ˆp, ˆk) + W + µν(ˆp, ˆk) f W µν(ˆp, ˆk) W µν(ˆp, ˆk)]} X s 1 s 2 =± X s 1 s 2 =± ImΠ µν (q 0, q) = π 2 cos2 θ c Z d 3 p (2π) 3 (ξ p + s 1 E p )(ξ k + s 2E k ) q 0 ± s 1 ξ p 1 q 0 + s 1 ξ p s 2 ξ k s 2 ξ k n F (±s 1 ξ p )n F ( s 2 ξ k ) n B (±s 1 ξ p s 2 ξ k ) n F (s 1 ξ p )n F ( s 2 ξ k ) n B (s 1 ξ p s 2 ξ k ) 2 A (E p, E k )H µν (n) 2 B (E p, E k )H µν (a)!, A (E p, E k ) = X s 1 s 2 =± B 1 (E p, E k ) = 2ξp 2ξ k ξ p X + s 1 E p 2ξ p s 1 s 2 =±! ξ k + s 2E k 2ξ k! δ(q 0 + s 1 ξ p s 2 ξ k ) n F (s 1 ξ p )n F ( n B (s 1 ξ p s 2 δ(q 0 + s 1 ξ p s 2 ξ k ) n F (s 1 ξ p )n F ( s 2 ξ k ) n B (s 1 ξ p s 2 ξ k ) Neutrino emissivities in quark matter p.25/26

derivatives Ω φ u = Ω φ d = φ u 1 4 G 1 π 2 + E u,r ξ u,r + E d,r ξ d,r φ d 1 4 G 1 π 2 Ω = 2 G 2 1 π 2 + + 1 ξ + u,r 1 ξ + d,r Z 0 dp p 2 g(p)m u(p) E u (p) ˆ1 2 nf (ξ u,r 0.5µ Q) + E+ u,r Z 0 dp p 2 g(p)m d(p) E d (p) ( h i 1 n F (E + u,b + 0.5µ Q) n F (E u,b 0.5µ Q) ξ + u,r h i 1 2 n F (ξ d,r + 0.5µ Q) + E+ d,r Z 0 dp p 2 g(p) 2 ( ˆ1 2 nf (ξ + u,r + 0.5µ Q) ) ( h i 1 n F (E d,b + 0.5µ Q) n F (E + d,b 0.5µ Q) 1 ξ u,r ˆ1 2 nf (ξ + u,r + 0.5µ Q ) + 1 h i ) 1 2 n F (ξ + d,r 0.5µ Q) ξ + d,r h i ) 1 2 n F (ξ + d,r 0.5µ Q) ˆ1 2 nf (ξ u,r 0.5µ Q ) ξ d,r h i 1 2 n F (ξ d,r + 0.5µ Q) Neutrino emissivities in quark matter p.26/26