An Astrophysical Application of Crystalline Color Superconductivity

An Astrophysical Application of An Astrophysical Application of Crystalline Color Crystalline Color

SuperconductivitySuperconductivity

Roberto AnglaniRoberto AnglaniPhysics DepartmentPhysics Department - U- U BariBari

Istituto Nazionale di Fisica Nucleare, ItalyIstituto Nazionale di Fisica Nucleare, Italy

SM&FT 2006XIII workshop on Statistical Mechanics and non

perturbative Field Theory

Bari, SMFT 20.IX.06 Anglani (U Bari) 2/12

Direct and Modified Direct and Modified URCA processesURCA 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)


Color Superconductivity in the CS coreColor Superconductivity in the CS core

Aged compact stars T < 100 KeVTCS 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 ms

must be included.

β – equilibrium

Color neutrality

Electrical neutralitya mismatch between Fermi

momenta of different quarks depending on the in-medium

value of ms.GROUND

STATE ??????????????


The Great Below of gapless phasesThe Great Below of gapless phases

μAsymtptotia Temple

Great below of GAPLESS phases

CHROMOMAGNETC INSTABILITY DANGERHuang 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 WAYCiminale, et al., PL B636 317 (2006)

T=0


Simplified models of toy starsSimplified models of toy stars

5 km

10 km 10 km

5 km

Normal quark matter n ~ 9 n0

LOFF matter n ~ 9 n0

Noninteracting nuclear matter

12 km - n ~ 1.5 n0

Noninteracting nuclear matter

n ~ 1.5 n0

Alford and Reddy nucl-th/0211046

1

3 2

n0 = 0.16 fm-

1

M = 1.4 MO.


Dispersion laws forDispersion laws for ( (rrd –d – ggu) andu) and ( (rrs s – – bbu) u)

1. LOFF phase is gapless

2. Dispersion laws around gapless modes could be considered as linear


““The importance of being gapless”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


Neutrino EmissivityNeutrino EmissivityWe 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 distributions Bogoliubov coefficients

Transition rate

Neutrino Energy

(1)

(2)


Cooling lawsCooling laws

NUCLEARNUCLEAR mattermatter[Shapiro][Shapiro]

LOFFLOFF mattermatter

UNPAIREDUNPAIRED Q. Q. mattermatter

[Iwamoto][Iwamoto]

--LuminosityLuminosity ~ T~ T88 ~ T~ T66 ~ T~ T66

Specific HeatSpecific Heat ~ ~ TT ~ ~ TT ~ ~ TT

--LuminosityLuminosity ~ T~ T2.22.2 ~ T~ T2.22.2 ~ T~ T2.22.2

(1)

t < t < ~~11 MMyryrmain mechanism is neutrino

emission

t > t > ~~1M1Myr yr main mechanism is photon

emission


ResultsResults

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~T6 cV ~T. Normal quark matter curve: only for comparison between different models.


ConclusionsConclusions

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.


AcknowledgmentsAcknowledgments

In these matters the only certainty is that nothing is certain.

PLINY THE ELDERRoman 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


A look at the HOT BOTTLE

L ~ T2.2

cV ~ 0.5T0.5 cV ~ TL ~ T2.2

Alford et al.[astro-ph/0411560]

P1bu P2

bu


LOFF3 Dispersion lawsEvery 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 123

Sector 45

Sector 67

Sector 89

ru g

d

bs Rd gu rs bu gs bd

det S –1 = 0 Dispersion lawsDispersion lawsRef. prof. Buballa


Larkin-Ovchinnikov-Fulde-Ferrel state of artThe simplified ansatz crystal structure is

i, j = 1, 2, 3 flavor indices; , = 1, 2, 3 color indices; 2qI represents the momentum of Cooper pair and 1,2,3describe 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)


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 = < 0.30 [1]

q2=q3=q = m2s/(8 zq); zq ~ 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)


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=ms

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 = ms2/[130,150]

MeV

y = 140 MeV

(2)

(3)

(4)

(1)

(5)

[1] Casalbuoni, Gatto, Nardulli et al.,

hep-ph/0606242


Dispersion laws forDispersion laws for ( (rruu –– ggd –d – bbs)s)


Appendix A: Emissivity


Appendix B: Specific Heat

μ = 500 MeV; ms = (μ 140)1/2 MeV; 1 = 0; 2 = 3 = ~ 6 MeV.


Appendix C: Dispersion laws


Appendix D: Dispersion laws 3X3


Appendix E: Cooling laws


Appendix F: Redifinition of gapless modes

An Astrophysical Application of Crystalline Color Superconductivity

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