Color Superconductivity Giuseppe Nardulli Physics Department and INFN Bari Varenna, June 29, 2006

Collaboration Bari - Firenze- Geneva R. Casalbuoni, M. Ciminale, R. Gatto, N. Ippolito, M. Mannarelli, G. Nardulli, M. Ruggieri Review papers F. Wilczek, K. Rajagopal, hep-ph/0011333 G. N. Riv. N. Cim.(2002) hep-ph/0202037 R. Casalbuoni and G.N. Rev. Mod. Phys, (2004) hep-ph/0305069

Nuclear matter and QCD At small nuclear density and low temperature the relevant degrees of freedom of nuclear matter are the hadrons: nucleons, hyperons, mesons. At high density and/or high temperature the relevant dof are quarks and gluons. The corresponding theory is Quantum ChromoDynamics QCD is a gauge theory based on the local gauge group SU(3), with quarks belonging to the fundamental and gluons to the adjoint representation QCD enjoys the property of Asymptotic Freedom

Accelerators (RHIC,LHC, GSI) and compact stars

Continuous emission of e.m. radiation and slowing down Structure of a Pulsar Continuous emission of e.m. radiation and slowing down

Phenomenology of pulsar: glitches Sudden increases of pulsar rotational frequency Vela Psr 0833-45

Further phenomenology Effects on EOS Cooling by neutrino emission Neutrinos from supernovae Shear viscosity and other transport properties

Nuclear matter at T=0, large μ Quarks have S=1/2. They are in three colors (r,g,b) and, if light, in three flavors (u,d,s). When two quarks scatter (3 x 3 ) they are either in a sextet (color symmetric) or in antitriplet (color antisymmetric) state Gluon exchange in the antitriplet is attractive

Free fermion gas and BCS mechanism Per f(E) E

Free energy Adding a particle at the Fermi surface Extracting a particle, i.e. forming a hole Free energy unchanged

SC more efficient in QCD For an arbitrary attractive interaction it is energetically convenient to form particle pairs (Cooper pairs) In metals SC realized only in special conditions, when the attractive interactions between electrons, due to phonon exchange beats the repulsive e.m. forces QCD:attractive channels with SC more efficient in QCD

QCD phases at T=0, large μ As a consequence of the attractive gluonic interaction in the color antisymmetric channel and the Cooper’s theorem: Formation of Cooper pairs of quarks (diquarks) and color superconductivity The condensate is colored, not white as in the quark-antiquark channel, relevant at zero density

Cooper pairs a) Normal preferred state: Ppair=0

b) Preferred total spin Spair=0 Two particle w.f. ψ(r) = χ Σk gk eikr r = r1 – r2 The spin wf χ with S=0 is antisymmetric in spin therefore the orbital part has behavior cos kr (symmetric in r), while S=1 has behavior sin kr; coupling disfavored at small r

Color Superconductivity at extreme densities QCD with 3 flavors: attractive interaction in color antisymmetric (antitriplet) channel For S=0 state Pauli principle implies antisymmetry in flavor Asymptotically (very large densities):Color Flavor Locking (CFL) < 0 | ψi α ψj β |0> = Δ εijγ ε αβγ Valid for μ >> mq

CFL (color flavor locking) phase Symmetries of the CFL (color flavor locking) phase < 0 | ψ ψ |0>  0 broken spontaneously. Also U(1) global symmetry (baryonic number) broken. 8 gluons acquire mass (Higgs-Anderson). 8+1 Goldstone bosons similar to π and K of chiral symmetry breaking

CS at intermediate densities Analysis complicated because the three light quarks are different: strange quark has mass comparable with μ for μ of the order of 500 MeV, while up, down are massless These are the densities expected in the core of neutron stars where CS most likely takes place In stars quark matter must be neutral and in beta equilibriun due to d -> u  e and u e -> d 

Neutrality and β equilibrium Non interacting quarks μd,s = μu + μe If the strange quark is massless this equation has solution Nu = Nd = Ns , Ne = 0; quark matter electrically neutral with no electrons

Strange quark mass

Choosing the true vacuum: Mass effects, neutrality, beta equilibrium: The result is that there are different gaps for quarks of different flavor. Gapless phases: uniform (gCFL) or not uniform (LOFF). Three gaps < 0 | ψi α ψj β |0> = γΔγ εijγ ε αβγ Choosing the true vacuum: The true ground state has the minimum value of the free energy (grand potential)

Free energy Gapless CFL Gap Parameters M.Alford, P.Jotwani, C. Kouvaris, J. Kundu, K.Rajagopal Gapless CFL

The importance of being gapless Gapless modes: relevant for astrophysical effects (phenomenology of compact stars) µ= 500 MeV Δ= 25 MeV M2s/2µ= 80 MeV

Meissner masses in the gCFL phase Solid line a=1,2; Gluons 3, 8: same behavior; 4,5,6,7: no instability R. Casalbuoni, M. Mannarelli, G. N., M. Ruggieri, R. Gatto Imaginary masses: instability Symptom that it is not the true vacuum

LOFF= Larkin-Ovchinnikov& Loff Phase If δμ0 it can be energetically favourable to have states where the Cooper pair has total momentum 2q 0 LOFF= Larkin-Ovchinnikov& Fulde-Ferrel (1964); R. Casalbuoni, G.N. Rev. Mod. Phys.(2004) Δ ( r) = < 0 | ψ( r)ψ( r)|0> = Δ exp[i(p1 +p2 )r]

Two flavors Condensate BCS (independent of δμ) Condensate LOFF Ω(BCS) - Ω (normal) Ω(LOFF)-Ω (normal)

Symmetry breaking for LOFF phase with 3 flavors T(3) x O(3) Oh Cube group

LOFF phase: three flavors Needed for realistic calculations R.Gatto,R.Casalbuoni,N.Ippolito,GN, M.Ruggieri : Ginzburg Landau expansion of the gap equation and free energy and M.Mannarelli, K. Rajagopal, R. Sharma: one plane wave K. Rajagopal, R. Sharma: Cristallography < 0 | ψi α ψj β |0> = γΔγ εijγ ε αβγ q exp iqr

Ginzburg-Landau expansion of the gap equation Δ (small)

Gap parameters: one plane wave K( r) = K e2i qk r For each inhomogeneous pairing a Fulde-Ferrell ansatz; 2 qk represents the momentum of the Cooper pair. This is the simplest ansatz, other structures should be examined Three independent functions 1( r), 2( r), 3( r), describing respectively d-s, u-s and u-d pairing.

Expected: μe =m2s/4μ Δ1 = 0; Δ2 = Δ3 Near the transition point du,us couplings, equal strength, no ds pairing Expected: μe =m2s/4μ Δ1 = 0; Δ2 = Δ3

q2 q3 parallel favored by phase space

=n+ 1/4I=13 (2I I2+ II4 + JIIJI2J2) +O(6) Ginzburg-Landau expansion =n+ 1/4I=13 (2I I2+ II4 + JIIJI2J2) +O(6) n = -3/122(u4+d4+s4) -1/ 122 e4 -k  = 0 minimum in k -e = 0 electrical neutrality Approximation: 3'8' 0 (confirmed by M.Ciminale et al., in preparation: Valid near the transition point, since this is the result for the normal phase) The numerical results in this regime confirm 1=0; 2  3 R. Casalbuoni, R.Gatto,, N.Ippolito,GN, M.Ruggieri

Comparison among different CSC phases Ginzburg Landau Approximation Calculation at leading order in μ R.Gatto, R.Casalbuoni, N.Ippolito, G.Nardulli, M.Ruggieri M.Mannarelli, K. Rajagopal, R. Sharma

Free energy expansion for more plane waves Ω= αΔ2+βΔ4+γΔ6 +δΔ8… The favored structures for both 2 and 3 flavors have cubic symmetry (Bowers, Rajagopal, Sharma).

Chromomagnetic instability In gCFL : imaginary gluon Meissner masses Studied in LOFF model with 2 flavors (Giannakis&Ren, Fukushima, Gorbar, Hashimoto&Miransky). LOFF favored in comparison to gapless uniform conductive states. At least in the GL region (small gap) no chromomagnetic instability For 3 flavors: recent results (Ciminale,Gatto,GN,Ruggieri)

Feynman Diagrams at O(Δ4)

Gluon Meissner masses:no instability Transverse mass Longitudinal mass M.Ciminale, R.Gatto,GN, M.Ruggieri

Compact stars and CS Model of neutron star with quark core Laboratory for the study ogf color superconductivity Model of neutron star with quark core

Specific heat and neutrino emissivity Cv  T for electrons and gapless or free quarks; Cv exponentially suppressed for gapped quarks E=neutrino emissivity E T6 for gapless or free quarks; E T8 for nuclear matter (URCA [beta decay] processes forbidden, modified URCA allowed [ n+X->p+X+e+ν]

Cooling of a compact star L dt = heath lost in time dt -V Cv dT =change in the energy of star Taking into account quark and nuclear matter for neutrino emission as well as photon emission, the rate of change in T: dT = -- Vnm Enm + Vqm Eqm + Lγ dt Vnm Cnm + Vqm Cqm V=volumes of nm (nuclear) or qm (quark)

Cooling of neutron star (preliminary)

Conclusions Color superconductivity is a theoretical prediction based on the theory of hadronic interactions, QCD Best laboratory to study it: future dedicated experiments with heavy ions or the core of compact stars Compact star lab: remote, but experimental signatures can be found and should allow to find the correct QCD vacuum at high, but not very high nuclear density