An Astrophysical Application of Crystalline Color Superconductivity Roberto Anglani Physics Department - U Bari Istituto Nazionale di Fisica Nucleare, Italy SM&FT 2006 XIII workshop on Statistical Mechanics and non perturbative Field Theory

Bari, SMFT 20.IX.06Anglani (U Bari)2/12 Direct and Modified URCA processes Neutrino emission due to direct URCA process is the most efficient cooling mechanism for a neutron star in the early stage of its lifetime. In stars made of nuclear matter only modified URCA processes can take place [1] because the direct processes n → p + e + and e + p → n + are not kinematically allowed. If hadronic density in the core of neutron stars is sufficiently large, deconfined quark matter could be found. Iwamoto [2] has shown that in quark matter direct URCA process, d → u + e + and e + u → d + are kinematically allowed, consequently this enhances drammatically the emissivity and the cooling of the star [1] Shapiro and Teukolski, White Dwarfs, Black Holes and Neutron Stars. J.Wiley (New York) [2] Iwamoto, Ann. Phys (1982)

Bari, SMFT 20.IX.06Anglani (U Bari)3/12 Color Superconductivity in the CS core Aged compact stars T < 100 KeV T CS is of order of MeV:. Asymptotical densities: Color-Flavor- Locked phase is favored. But direct URCA processes are strongly suppressed in CFL phase because thermally excited quasiquarks are exponentially rare. Relevant density for compact stars: not asymptotic! Matter in the core could be in one of the possible Color Superconductive phases effects due to the strange quark mass m s must be included. β – equilibrium Color neutrality Electrical neutrality a mismatch between Fermi momenta of different quarks depending on the in-medium value of ms. GROUND STATE ??????????????

Bari, SMFT 20.IX.06Anglani (U Bari)4/12 The Great Below of gapless phases μ Asymtptotia Temple Great below of GAPLESS phases CHROMOMAGNETC INSTABILITY DANGER Huang and Shovkovy, PR D (2004) Casalbuoni, et al., PL B (2005) Fukushima, PR D (2005) AlforD and Wang, J. Phys. G (2005) BUT THERE IS SOMETHING THAT MAY ENLIGHT THE WAY Ciminale, et al., PL B (2006) T=0

Bari, SMFT 20.IX.06Anglani (U Bari)5/12 Simplified models of toy stars 5 km 10 km 5 km Normal quark matter n ~ 9 n 0 LOFF matter n ~ 9 n 0 Noninteracting nuclear matter 12 km - n ~ 1.5 n 0 Noninteracting nuclear matter n ~ 1.5 n 0 Alford and Reddy nucl-th/ n 0 = 0.16 fm -1 M = 1.4 M O.

Bari, SMFT 20.IX.06Anglani (U Bari)6/12 Dispersion laws for ( r d – g u) and ( r s – b u) 1.LOFF phase is gapless 2.Dispersion laws around gapless modes could be considered as linear

Bari, SMFT 20.IX.06Anglani (U Bari)7/12 “The importance of being gapless” The contribution of gapped modes are exponentially suppressed since we work in the regime T<<  <<  Each gapless mode contributes to specific heat by a factor ~ T

Bari, SMFT 20.IX.06Anglani (U Bari)8/12 Neutrino Emissivity We consider the following  – decay process for color  = r, g, b. Neutrino emissivity = the energy loss by  -neutrino emission per volume unit per time unit. Electron capture process Thermal distributionsBogoliubov coefficients Transition rate Neutrino Energy (1) (2)

Bari, SMFT 20.IX.06Anglani (U Bari)9/12 Cooling laws NUCLEAR matter [Shapiro]LOFFmatter UNPAIRED Q. matter [Iwamoto] -Luminosity -Luminosity ~ T 8 ~ T 6 Specific Heat ~ T  -Luminosity ~ T 2.2 (1) t < ~1 Myr main mechanism is neutrino emission t > ~1Myr main mechanism is photon emission

Bari, SMFT 20.IX.06Anglani (U Bari)10/12Results A star with LOFF matter core cools faster than a star made by nuclear matter only. REM.: Similarity between LOFF and unpaired quark matter follows from linearity of gapless dispersion laws : ε~T 6 c V ~T. Normal quark matter curve: only for comparison between different models.

Bari, SMFT 20.IX.06Anglani (U Bari)11/12Conclusions 1.We have shown that due to existence of gapless mode in the LOFF phase, a compact star with a quark LOFF core cools faster than a star made by ordinary nuclear matter only. 2.These results must be considered preliminary. The simple LOFF ansatz should be substituted by the favored more complex crystalline structure [Rajagopal and Sharma, hep-ph/ ]. 3.In this case (2.) identification of the quasiparticle dispersion laws is a very complicated task but probable future work. For this reason it is also difficult to attempt a comparison with present observational data.

Bari, SMFT 20.IX.06Anglani (U Bari)12/12Acknowledgments In these matters the only certainty is that nothing is certain. PLINY THE ELDER Roman scholar and scientist (23 AD - 79 AD) Thanks to M. Ruggieri, G. Nardulli and M. Mannarelli for the fruitful collaboration which has yielded the work hep-ph/ , whose results underlie the present talk

Bari, SMFT 20.IX.06Anglani (U Bari)13/12 A look at the HOT BOTTLE L  ~ T 2.2 c V ~  0.5 T 0.5 c V ~ T L  ~ T 2.2 Alford et al. [astro-ph/ ] P 1 bu P 2 bu

Bari, SMFT 20.IX.06Anglani (U Bari)14/12 LOFF3 Dispersion laws Every quasiquark is a mixing of coloured quarks, weighted by Bogolioubov – Valatin coefficients. “Coloured” components of quasiparticles can be easily found in the sectors of Gap Lagrangean in an appropriate color-flavor basis. Sector 123Sector 45 Sector 67 Sector 89 ruru gdgd bsbs RdRd gugu rsrs bubu gsgs bdbd det S –1 = 0 Dispersion laws Ref. prof. Buballa

Bari, SMFT 20.IX.06Anglani (U Bari)15/12 Larkin-Ovchinnikov-Fulde-Ferrel state of art The simplified ansatz crystal structure is i, j = 1, 2, 3 flavor indices; ,  = 1, 2, 3 color indices; 2q I represents the momentum of Cooper pair and  1,  2,  3  describe respectively d – s, u – s, u – d pairings. LOFF phase has been found energetically favored [1,2] with respect to the gCFL and the unpaired phases in a certain range of values of the mismatch between Fermi surfaces. [Ref. Ippolito’s talk and Buballa’s lecture]. This phase results chromomagnetically stable [3] [1] Casalbuoni, Gatto et al., PL B (2005) [2] Rajagopal et al., hep-ph/ [3] Ciminale, Gatto et al., PL B (2006) (1) Larkin and Ovchinnikov; Fulde and Ferrell (1964)

Bari, SMFT 20.IX.06Anglani (U Bari)16/12 Neutral LOFF quark matter - 1 The GL approximation is reliable in a region close to the second order phase transition point where the crystal structure is characterized by 1.Three light quarks u, d, s, in a color and electrically neutral state 2.Quark interactions are described employing a Nambu-Jona Lasinio model in a mean field approximation 3.We employ a Ginzburg-Landau expansion [1] Requiring color and electric neutrality, the energetically favored phase results in  1 = 0;  2 =  3 =  < 0.3  0 [1] q 2 =q 3 =q = m 2 s /(8  z q ); z q ~ 0.83 [1] [1] Casalbuoni et al., PL B (2005) where  0 is the CFL gap. Rajagopal et al., hep-ph/ (1) (2)

Bari, SMFT 20.IX.06Anglani (U Bari)17/12 Neutral LOFF quark matter - 2  0 = 25 MeV Finally, for our numerical estimates we use To the leading order approximation in  /  one obtains  3 =  8 = 0 and  e =m s 2 /4  [1]  = 500 MeV The LOFF phase is energetically favored with respect to gCFL and normal phase in the range of chemical potential mismatch of y = m s 2 /  [130,150] MeV y = 140 MeV (2) (3) (4) (1) (5) [1] Casalbuoni, Gatto, Nardulli et al., hep-ph/

Bari, SMFT 20.IX.06Anglani (U Bari)18/12 Dispersion laws for ( r u – g d – b s)

Bari, SMFT 20.IX.06Anglani (U Bari)19/12 Appendix A: Emissivity

Bari, SMFT 20.IX.06Anglani (U Bari)20/12 Appendix B: Specific Heat μ = 500 MeV; m s = (μ 140) 1/2 MeV;  1 = 0;  2 =  3 =  ~ 6 MeV.

Bari, SMFT 20.IX.06Anglani (U Bari)21/12 Appendix C: Dispersion laws

Bari, SMFT 20.IX.06Anglani (U Bari)22/12 Appendix D: Dispersion laws 3X3

Bari, SMFT 20.IX.06Anglani (U Bari)23/12 Appendix E: Cooling laws

Bari, SMFT 20.IX.06Anglani (U Bari)24/12 Appendix F: Redifinition of gapless modes