1. 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

2. 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. 141 1 (1982)

3. 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 10-20 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 ??????????????

4. 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 D70 051501 (2004) Casalbuoni, et al., PL B605 362 (2005) Fukushima, PR D72 074002 (2005) AlforD and Wang, J. Phys. G31 719 (2005) BUT THERE IS SOMETHING THAT MAY ENLIGHT THE WAY Ciminale, et al., PL B636 317 (2006) T=0

5. 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/0211046 1 32 n 0 = 0.16 fm -1 M = 1.4 M O.

6. 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

7. 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

8. 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)

9. 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

10. 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.

11. 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/0605316]. 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.

12. 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/0607341, whose results underlie the present talk

13. 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/0411560] P 1 bu P 2 bu

14. 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

15. 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 B627 89 (2005) [2] Rajagopal et al., hep-ph/0603076 [3] Ciminale, Gatto et al., PL B636 317 (2006) (1) Larkin and Ovchinnikov; Fulde and Ferrell (1964)

16. 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 B627 89 (2005) where  0 is the CFL gap. Rajagopal et al., hep-ph/0605316 (1) (2)

17. 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/0606242

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

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

20. 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.

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

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

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

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