1. Neutrino emissivity of the dense superfluid nuclear matter
and problems of neutron star cooling
Evgeni Kolomeitsev
Matej Bel University, Banska Bystrica
We want to learn about properties of microscopic excitations in dense matter
How to study response function of the NS?
How to look inside the NS?

2. important for supernova
At temperatures smaller than the opacity temperature (Topac~1-few MeV)
mean free path of neutrinos and antineutrinos is larger than the neutron star radius
white body radiation problem
After >105 yr –black body radiation of photons
At temperatures T>Topac
neutrino transport problem
important for supernova

3. Cooling scenario [neutrino production]
Given:
EoS
Cooling scenario [neutrino production]
Uncertainties: Data extraction; Tin-Tsurf relation
Mass of NS
Cooling curve
Uncertainty: spin-down age vs. kinematic age

4. neutron star is transparent for neutrino
CV - specific heat, L - luminosity
emissivity
each leg on a Fermi surface / T
neutrino phase space ´ neutrino energy

5. ~T6 ~T8 Cooling: role of crust and interior?
most important are reactions in the interior
(The baryon density is
where n0 is the nuclear saturation density)
one-nucleon reactions:
direct URCA (DU)
~T6
modified URCA (MU)
two-nucleon reactions:
~T8
nucleon bremsstrahlung (NB)
URCA=Gamow’s acronym for “Un-Recordable Coolant Agent”

6. volume neutrino radiation
DU:
neutrino cooling
MU:
DATA Tn
photon cooling

7. How to calculate reaction rates in medium?
Let a lepton pair (l1,l2) be coupled to a boson (B) or to a fermion pair (F1,F2)
Lepton production rate in medium consisting of the bosons and fermions is given by
Feynman diagrams
summation over the phase space of initial particle (occupation factors)
Introduce a coupling among boson and fermions:
Background Blue 124

8. consisting of fermions F1 the boson propagator is
resummed diagrams
polarization operator
Spectrum of excitations with the quantum numbers of bosons B
two characteristic mass gaps

9. To calculate the lepton production rates we cannot use Feynman diagrams with in-medium (dressed propagators).
It can lead to double counting!! +
…. additional complications due to vertex corrections

10. In general case one should deal with
Perturbative diagrams are irrelevant for calculation of in-medium processes.
In general case one should deal with
closed diagrams in terms of dressed Green's functions
[Voskresensky, Senatorov, Sov. Nucl. Phys. 45 (1987); Knoll, Voskresensky, Ann. Phys. 249 (1996)]
one-nucleon reactions
two-nucleon reactions
for superfluid systems: [Kolomeitsev, Voskresensky, Phys. Atom. Nucl.

11. weak current susceptibility vector (V) and axial (A) currents
non-relativistic limit
relativistic corrections can be large !

12. In medium there exists a single-particle excitation mechanism
Green’s function of interacting nucleon possesses a pole
pole residue
q.p. energy
q.p. width
small for T<<F
complicated background part

13. T<<eF Only particles on the Fermi surface take part in reactions.
particle-hole interaction:
particle-particle interaction:
Interaction in this two channels can be essentially different !
spin zero
spin one
expansion in Legendre polynomials
Landau-Migdal constants:
empirical info., calculated from
NN potential

14. Graphically, the resummation is straightforward and yields:
full pion propagator
Poles yield zero-sound modes in scalar and spin channels
dressed vertex

15. Pion modes in nuclear medium
A.B. Migdal, G.E. Brown
dressed vertices
quasi-particle modes
“pion gap”
n=1n0
pion propagator has a complex pole
when
instability
pion condensation
[A.B.Migdal et al, Phys. Rept. 192 (1990) 179]

16. |*|~ amplitude of the condensate
reconstruction of pion spectrum
on top of the pion condensate
pion gap for n<ncPU
no pion condensate
LM parameters increase with density
saturation of pion softenning
no pion condensate
1st-order phase transition
|*|~ amplitude of the condensate
[Blaschke, Grigorian, Voskresensky, A&A 424, 979–992 (2004)]

17. emissivity: larger smaller Very strong density dependence
[Voskresensky, Senatorov, Sov. Nucl. Phys. 45 (1987)]

18. 1S0 proton pairing 1S0 neutron pairing 3P2 neutron pairing
[Schwenk Friman]

19. Ground state Excited state unpaired fermions paired fermions pair
breaking
“exciton” D
pairing gap
excitation spectrum
emission spectrum

20. minimal
standard
exotic

21. Urbana-Argonne A18+d v +UIX* based EoS.
Blaschke, Grigorian, Voskresensky, A&A 424, 979 (2004).
Grigorian,Voskresensky, A&A 444, 913 (2005).
Urbana-Argonne A18+d v +UIX* based EoS.
Medium effects included in all processes.
3P2 gaps from Schwenk & Friman model.
Passed log N -log S (population synthesis) control [S.Popov et al., astro-ph/ ]

22. particle
hole
normal Green’s function
particle  hole
hole particle
anomalous Green’s function
Number of excitations is not conserved !
amplitude of 2 particle annihilation
amplitude of 2 particle creation
Fs, Gs and Ds are connected by the Gorkov’s and gap equations
S wave pairing

23. bare vertices:
ext. field create 2 particles
dressed vertices
ext. field create 2 holes
Without vertex corrections the current conservation is violated !

24. Larkin-Migdal equations
bare vertices
dressed vertices
Larkin-Migdal equations
For T=0 and S-pairing written by Larkin, Migdal 1963 [Sov.Phys.JETP 17, 1146].
For finite T and S-pairing re-derived by Leggett (no Gw terms!).
[Phys.Rev. 140, A1869 (1965), Phys.Rev. 147, 119 (1966)].
Applied to weak interactions in [EEK, Voskresensky, Phys.Rev.C 77, (2008)].
Equivalence of Leggett’s and Larkin-Migdal’s approaches for finite T
[EEK, Voskresensky, Phys.Rev.C 81, (2010)].
General structure for arbitrary pairing and non-equlibrium systems
[EEK, Voskresensky, Phys. Atom. Nucl. (2010)].

25. Main contribution is due to the axial current.
moderate suppression
strong suppression
Leinson,Perez (2006), Kolomeitsev,Voskresensky (2008)
Kolomeitsev,Voskresensky (2008)
with free vertices
Main contribution is due to the axial current.
Suppression is of the order ~0.1

26. Consider axial weak currents (dominate the emissivity)
Gamov-Teller transitions in nuclei
particle-particle interaction:
s-wave paring:
spin zero channel
next possible harmonics , which is expected to be much smaller!
spin one channel
next possible harmonics , which is expected to be much smaller!
particle-hole interaction:
does not contribute to the axial channel
drop here for simplicity 
corrections
neglect
density states at Fermi surface

27. C<0 exciton modes for w<2D
C>gT(0) diffusive modes for w>2D
[Kolomeitsev, Voskresensky, PRC 84, (2011)]

28. We need neutrino emissivity for description of neutron star cooling
Closed diagram technique is the convenient scheme to calculate reaction rates in medium. (No double counting!)
Pion modes are softened in dense nuclear matter.
enhancement of a nucleon-nucleon interaction enhanced emissivity
Pair formation and breaking reactions are important neutrino source in superfluid
nuclear matter .
In the case of pairing, vertex corrections are important:
conservation of vector current
new excitation modes