Goldstone bosons in the CFL phase

Goldstone bosons in the CFL phase Verena Werth 1 Michael Buballa 1 Micaela Oertel 2 1 Institut für Kernphysik, Technische Universität Darmstadt 2 Observatoire de Paris-Meudon Dense Hadronic Matter and QCD Phase Transitions Workshop III Rathen, October 15, 2006

Outline 1 Motivation 2 Formalism Model Method 3 Results Equal quark masses Dependence on the strange quark mass Calculation of f π 4 Summary & Outlook

Phase diagram of neutral dense quark matter [Rüster, V.W., Buballa, Shovkovy, Rischke Phys. Rev. D 72 (2005)] first order phase transition second order phase transition normal quark matter: ud = us = ds = 0 2SC phase: ud 0, us = ds = 0 usc phase: ud, us 0, ds = 0 CFL phase: ud, us, ds 0

Phase diagram of neutral dense quark matter [Rüster, V.W., Buballa, Shovkovy, Rischke Phys. Rev. D 72 (2005)] first order phase transition second order phase transition normal quark matter: ud = us = ds = 0 2SC phase: ud 0, us = ds = 0 usc phase: ud, us 0, ds = 0 CFL phase: ud, us, ds 0

Why are we interested in Goldstone bosons? chiral symmetry breaking Goldstone bosons lightest excitations: mesons effective theories predict low meson masses and meson condensation

Why are we interested in Goldstone bosons? chiral symmetry breaking Goldstone bosons lightest excitations: mesons effective theories predict low meson masses and meson condensation influence of the pseudoscalar meson condensates studied by Warringa [hep-ph/0606063]

CFL phase Diquark condensates diquark condensates: q T Ôq Ô antisymmetric scalar diquark condensates ud = 2 q T Cγ 5 τ 2 λ 2 q us = 5 q T Cγ 5 τ 5 λ 5 q ds = 7 q T Cγ 5 τ 7 λ 7 q τ A : flavor space λ A : color space

CFL phase Diquark condensates diquark condensates: q T Ôq Ô antisymmetric scalar diquark condensates ud = 2 q T Cγ 5 τ 2 λ 2 q us = 5 q T Cγ 5 τ 5 λ 5 q ds = 7 q T Cγ 5 τ 7 λ 7 q color-flavor locking Condensates invariant under q e iθa τa λt a 2 q τ A : flavor space λ A : color space

CFL phase Diquark condensates diquark condensates: q T Ôq Ô antisymmetric scalar diquark condensates ud = 2 q T Cγ 5 τ 2 λ 2 q us = 5 q T Cγ 5 τ 5 λ 5 q ds = 7 q T Cγ 5 τ 7 λ 7 q τ A : flavor space λ A : color space color-flavor locking Condensates invariant under q e iθa τa λt a 2 q chiral symmetry breaking A not invariant under q e iθa τa 2 γ 5 q Goldstone bosons

Nambu Jona-Lasinio model Lagrangian L eff = q(i / ˆm)q + H [( qiγ5 τ A λ A C q T ) ( q T Ciγ 5 τ A λ A q ) A=2,5,7 A =2,5,7 + ( qτ A λ A C q T ) ( q T Cτ A λ A q )]

Nambu Jona-Lasinio model Lagrangian L eff = q(i / ˆm)q + H [( qiγ5 τ A λ A C q T ) ( q T Ciγ 5 τ A λ A q ) A=2,5,7 A =2,5,7 + ( qτ A λ A C q T ) ( q T Cτ A λ A q )] work in Nambu-Gorkov space ψ = 1 ( ) q 1 2 C q T, ψ = 2 ( q, q T C )

Nambu Jona-Lasinio model Lagrangian L eff = q(i / ˆm)q + H [( qiγ5 τ A λ A C q T ) ( q T Ciγ 5 τ A λ A q ) A=2,5,7 A =2,5,7 + ( qτ A λ A C q T ) ( q T Cτ A λ A q )] four vertices in Nambu-Gorkov space ( ) Γ ll 0 0 s = iγ 5 τ A λ A 0 ( ) Γ ll 0 0 ps =, Γ ur ps = τ A λ A 0, Γ ur s = ( ) 0 iγ5 τ A λ A 0 0 ( ) 0 τa λ A 0 0

Ingredients inverse propagator in Nambu-Gorkov space p/ + ˆµγ 0 ˆm A γ 5 τ A λ A S 1 = A=2,5,7 Aγ 5 τ A λ A p/ ˆµγ 0 ˆm A=2,5,7 scalar diquark condensates ud = 2 q T Cγ 5 τ 2 λ 2 q us = 5 q T Cγ 5 τ 5 λ 5 q ds = 7 q T Cγ 5 τ 7 λ 7 q

Ingredients inverse propagator in Nambu-Gorkov space p/ + ˆµγ 0 ˆm A γ 5 τ A λ A S 1 = A=2,5,7 Aγ 5 τ A λ A p/ ˆµγ 0 ˆm A=2,5,7 scalar diquark condensates ud = 2 q T Cγ 5 τ 2 λ 2 q us = 5 q T Cγ 5 τ 5 λ 5 q ds = 7 q T Cγ 5 τ 7 λ 7 q A and µ 8 derived from self-consistent solution of the CFL gap equation + neutrality conditions

Bethe-Salpeter equation Bethe-Salpeter equation = + + +... = + i ˆT = i ˆV + i ˆV ( iĵ)i ˆT

Bethe-Salpeter equation Bethe-Salpeter equation = + + +... = + i ˆT = i ˆV + i ˆV ( iĵ)i ˆT ˆT = Γ i T ij Γ j ˆV = Γ i V ij Γ j Γ i ĴΓ j = J ij

Bethe-Salpeter equation Bethe-Salpeter equation = + + +... = + i ˆT = i ˆV + i ˆV ( iĵ)i ˆT T -matrix ˆT = Γ i T ij Γ j Γ i ĴΓ j = J ij ˆV = Γ i V ij Γ j T = V + V JT = T = ( 1 V J) 1 V

Polarization function Γ j p + q Γ k ij Γ j Γ k (q) = d 4 p (2π) 4 1 2 Tr[Γ j is(p + q)γ kis(p)] p

Polarization function Γ j p + q Γ k ij Γ j Γ k (q) = d 4 p 1 (2π) 4 2 Tr[Γ j is(p + q)γ kis(p)] p applying Matsubara formalism and restriction to q = 0 q=0 ij Γ j Γ (q) = i k 2 T n d 3 p (2π) 3 1 2 Tr[Γ j is(iω n + iω m, p)γ k is(iω n, p)]

Polarization function Γ j p + q Γ k ij Γ j Γ k (q) = d 4 p 1 (2π) 4 2 Tr[Γ j is(p + q)γ kis(p)] p applying Matsubara formalism and restriction to q = 0 q=0 ij Γ j Γ (q) = i k 2 T n d 3 p 1 (2π) 3 2 Tr[Γ j is(iω n + iω m, p)γ k is(iω n, p)] only a few vertex-combinations non-vanishing J can be transformed into block-diagonal structure

Structure of T choose the basis in which J is block-diagonal V is diagonal T is block-diagonal problem can be decomposed into smaller blocks: six 2x2 blocks (correspond to π +, π, K +, K, K 0, and K 0 ) one 6x6 block (corresponds to π 0, η, and η )

Mesons ˆT couples to external meson current Γ M

Mesons ˆT couples to external meson current = we have to calculate which diquark vertices contribute to the loop Γ M Γ j Γ M

Mesons ˆT couples to external meson current = we have to calculate which diquark vertices contribute to the loop Γ M Γ j Γ M meson vertices Γ π ± = i «τ1 ± iτ 2 0 2 γ5 0 τ 1 iτ 2 Γ K ± = i «τ4 ± iτ 5 0 2 γ5 0 τ 4 iτ 5 Γ ( ) = i «τ6 ± iτ 7 0 K 0 2 γ5 0 τ 6 iτ 7

Mesons ˆT couples to external meson current = we have to calculate which diquark vertices contribute to the loop Γ M Γ j Γ M meson vertices Γ π ± = i «τ1 ± iτ 2 0 2 γ5 0 τ 1 iτ 2 Γ K ± = i «τ4 ± iτ 5 0 2 γ5 0 τ 4 iτ 5 Γ ( ) = i «τ6 ± iτ 7 0 K 0 2 γ5 0 τ 6 iτ 7 non-vanishing combinations Γ π + Γ ll 75, Γ ur 57 Γ π Γ ll 57, Γ ur 75 Γ K + Γ ll 72, Γ ur 27 Γ K Γ ll 27, Γ ur 72 Γ K 0 Γ ll 52, Γ ur 25 Γ K0 Γ ll 25, Γ ur 52

Meson mass for equal quark masses π ±, K ±, K 0, K 0 12 10 m meson [MeV] 8 6 4 2 0 0 5 10 15 20 25 30 m q [MeV]

Meson mass for equal quark masses π ±, K ±, K 0, K 0 m meson [MeV] 12 10 8 6 4 2 0 0 5 10 15 20 25 30 m q [MeV] prediction from EFT: m M = am q r 8A a = f 2 π A = 3 2 4π 2 fπ 2 21 8 ln 2 µ 2 = 18 2π 2 [Son, Stephanov, Phys. Rev. D 61 (2000)]

Meson mass for equal quark masses π ±, K ±, K 0, K 0 m meson [MeV] 12 10 8 6 4 2 calculated from EFT 0 0 5 10 15 20 25 30 m q [MeV] 77.26 MeV : ) a fit 0.38 a EFT 0.58 prediction from EFT: m M = am q r 8A a = f 2 π A = 3 2 4π 2 fπ 2 21 8 ln 2 µ 2 = 18 2π 2 [Son, Stephanov, Phys. Rev. D 61 (2000)] fit 35 % smaller

Meson mass for equal quark masses π ±, K ±, K 0, K 0 m meson [MeV] 12 10 8 6 4 2 calculated from EFT 0 0 5 10 15 20 25 30 m q [MeV] 51.62 MeV : ) a fit 0.29 a EFT 0.39 prediction from EFT: m M = am q r 8A a = f 2 π A = 3 2 4π 2 fπ 2 21 8 ln 2 µ 2 = 18 2π 2 [Son, Stephanov, Phys. Rev. D 61 (2000)] fit 28 % smaller

Meson mass for equal quark masses π ±, K ±, K 0, K 0 m meson [MeV] 12 10 8 6 4 2 calculated from EFT 0 0 5 10 15 20 25 30 m q [MeV] 36.31 MeV : ) a fit 0.21 a EFT 0.27 prediction from EFT: m M = am q r 8A a = f 2 π A = 3 2 4π 2 fπ 2 21 8 ln 2 µ 2 = 18 2π 2 [Son, Stephanov, Phys. Rev. D 61 (2000)] fit 23 % smaller

Meson mass for equal quark masses π ±, K ±, K 0, K 0 m meson [MeV] 12 10 8 6 4 2 calculated from EFT 0 0 5 10 15 20 25 30 m q [MeV] 21.74 MeV : ) a fit 0.14 a EFT 0.16 prediction from EFT: m M = am q r 8A a = f 2 π A = 3 2 4π 2 fπ 2 21 8 ln 2 µ 2 = 18 2π 2 [Son, Stephanov, Phys. Rev. D 61 (2000)] fit 14 % smaller

Meson mass for equal quark masses π ±, K ±, K 0, K 0 m meson [MeV] 12 10 8 6 4 2 calculated from EFT 0 0 5 10 15 20 25 30 m q [MeV] 9.53 MeV : ) a fit 0.071 a EFT 0.071 prediction from EFT: m M = am q r 8A a = f 2 π A = 3 2 4π 2 fπ 2 21 8 ln 2 µ 2 = 18 2π 2 [Son, Stephanov, Phys. Rev. D 61 (2000)] deviation < 1%

Dependence on the coupling comparison with EFT 1 0.95 a calc / a EFT 0.9 0.85 0.8 0.75 0.7 0.65 0 10 20 30 40 50 60 70 80 [MeV] for weaker interaction meson masses approach the EFT result

Dependence on the strange quark mass m u = m d = 30 MeV q 0 (pole) [MeV] 50 40 30 20 10 0-10 -20-30 -40-50 π ± K -, K 0 K +, K 0 40 60 80 100 120 140 160 m s [MeV]

Dependence on the strange quark mass m u = m d = 30 MeV prediction from EFT q 0 (pole) [MeV] 50 40 30 20 10 0-10 -20-30 -40-50 π ± K -, K 0 K +, K 0 40 60 80 100 120 140 160 m s [MeV] [Bedaque, Schaefer, Nucl.Phys. A697 (2002)] q meson(pole) = µ meson + m meson q π ± (pole) = m2 d m2 u 2µ s 4A + fπ 2 m s(m u + m d ) q K ± (pole) = m2 s m 2 u 2µ s 4A + fπ 2 m d (m u + m s) q K 0, K 0(pole) = m2 s m 2 d 2µ s 4A + fπ 2 m u(m d + m s)

Dependence on the strange quark mass m u = m d = 30 MeV prediction from EFT q 0 (pole) [MeV] 50 40 30 20 10 0-10 -20-30 -40-50 π ± K -, K 0 K +, K 0 40 60 80 100 120 140 160 m s [MeV] [Bedaque, Schaefer, Nucl.Phys. A697 (2002)] q meson(pole) = µ meson + m meson q π ± (pole) = m2 d m2 u 2µ s 4A + fπ 2 m s(m u + m d ) q K ± (pole) = m2 s m 2 u 2µ s 4A + fπ 2 m d (m u + m s) r 4A f 2 π a fit 2 q K 0, K 0(pole) = m2 s m 2 d 2µ s 4A + fπ 2 m u(m d + m s)

Kaon propagator m u = m d = 30 MeV inverse K propagator for m s = 30 MeV 1000 comparison with a free boson propagator with chemical potential µ K and mass m K t K - -1 [MeV 2 ] 0-1000 -2000-3000 -4000-40 -30-20 -10 0 10 20 30 40 q 0 [MeV] t 1 K = (q0 + µ K )2 m 2 K g 2 µ K = m2 s m 2 u 2µ r afit m K = m d (m u + m s) 2

Kaon propagator m u = m d = 30 MeV inverse K propagator for m s = 30 MeV 1000 comparison with a free boson propagator with chemical potential µ K and mass m K t K - -1 [MeV 2 ] 0-1000 -2000-3000 -4000 calculated fitted -40-30 -20-10 0 10 20 30 40 q 0 [MeV] t 1 K = (q0 + µ K )2 m 2 K g 2 µ K = m2 s m 2 u 2µ r afit m K = m d (m u + m s) 2

Kaon propagator m u = m d = 30 MeV inverse K propagator for m s = 60 MeV 1000 comparison with a free boson propagator with chemical potential µ K and mass m K t K - -1 [MeV 2 ] 0-1000 -2000-3000 -4000 calculated fitted -40-30 -20-10 0 10 20 30 40 q 0 [MeV] t 1 K = (q0 + µ K )2 m 2 K g 2 µ K = m2 s m 2 u 2µ r afit m K = m d (m u + m s) 2

Kaon propagator m u = m d = 30 MeV inverse K propagator for m s = 120 MeV 1000 comparison with a free boson propagator with chemical potential µ K and mass m K t K - -1 [MeV 2 ] 0-1000 -2000-3000 -4000 calculated fitted -40-30 -20-10 0 10 20 30 40 q 0 [MeV] t 1 K = (q0 + µ K )2 m 2 K g 2 µ K = m2 s m 2 u 2µ r afit m K = m d (m u + m s) 2

Meson propagators inverse π propagators inverse K propagators t π - -1 [MeV 2 ] 1000 500 0-500 -1000-1500 -2000 m s =30 MeV m s =60 MeV -2500 m s =120 MeV -40-30 -20-10 0 10 20 30 40 q 0 t K - -1 [MeV 2 ] 500 0-500 -1000-1500 -2000-2500 -3000-3500 -4000-4500 m s =30 MeV m s =60 MeV m s =120 MeV -40-30 -20-10 0 10 20 30 40 q 0

Pion decay constant A µ π A 0 π = i ( ) + 2 τ1 iτ 2 γ0 γ 2 0 5 0 τ 1 + iτ 2 calculate loop at q 0 = q 0 (pole) d 4 p 1 [ ] f π q 0 = (2π) 4 2 Tr A 0 π S(p + q)(g + 1 Γ ll 75 + g 2 Γ ur 57)S(p)

Pion decay constant A µ π A 0 π = i ( ) + 2 τ1 iτ 2 γ0 γ 2 0 5 0 τ 1 + iτ 2 calculate loop at q 0 = q 0 (pole) d 4 p 1 [ ] f π q 0 = (2π) 4 2 Tr A 0 π S(p + q)(g + 1 Γ ll 75 + g 2 Γ ur 57)S(p) for µ = 500 MeV, T = 0, and m u = m d = m s = 30 MeV we find f π 95.1 MeV comparison with effective theory: 21 8 ln 2 µ f π = 104.3 MeV 36 π

Pion decay constant A µ π A 0 π = i ( ) + 2 τ1 iτ 2 γ0 γ 2 0 5 0 τ 1 + iτ 2 calculate loop at q 0 = q 0 (pole) d 4 p 1 [ ] f π q 0 = (2π) 4 2 Tr A 0 π S(p + q)(g + 1 Γ ll 75 + g 2 Γ ur 57)S(p) for µ = 500 MeV, T = 0, and m u = m d = m s = 30 MeV we find f π 95.1 MeV 9 % smaller than in EFT comparison with effective theory: 21 8 ln 2 µ f π = 104.3 MeV 36 π

Dependence on the coupling comparison with EFT f π [MeV] 105 104 103 102 101 100 99 98 97 96 95 0 10 20 30 40 50 60 70 80 [MeV] for weaker interaction f π approaches the EFT result

Summary & Outlook Summary description of mesons with diquark loops good agreement with low-energy effective theory

Summary & Outlook Summary description of mesons with diquark loops good agreement with low-energy effective theory Outlook wider range of chemical potentials finite temperature include additional interactions quark-antiquark interactions mesons in the new CFL+meson groundstate