1. Pions in Neutron Stars Evgeni E. Kolomeitsev
(Matej Bel University, Banska Bystrica, Slovakia
LTP, JINR, Dubna, Russia )

2. Hideki Yukawa 1935 proposed a quantum of strong interactions
1940 professor in Kyoto University
1949 Nobel Prize in Physics
1953 the first chairman of YITP

3. Pion in neutron star, to be or not to be?
1.1 Pionization. Chiral shield against it
1.2 P-wave pion condensation. Short-range correlations
2. Neutron star cooling
2.1 “Modified Urca” reactions (dynamical pions)
2.2 Medium “Modified Urca” – a mechanism to understand NS cooling
3. R-modes and viscosity of NS matter
3.1 R(Rossby)-mode instability. Age-P-dot diagram of pulsars.
3.2 Shear viscosity. In-medium pions for bulk viscosity.
3.3 Layered neutron star

4. Nucleon-nucleon interaction1 2 r p1 p2
several scales are involved
non-relativistic description
vector mesons: mw,r~800 MeV , r~0.24 fm
correlated 2¼ exchange: m~ MeV r~0.3—1fm
1-pion exchange: m¼=140 MeV r~1.4 fm
relativistic description
Equilibrium density of an atomic nucleus n0=0.16 fm-3
inter-nuclear distance (n0)-1/3=1.8 fm +

5. QCD with light quarks

6. Pions – Goldstone bosons of chiral symmetry breaking
gradient couplings
expansion in small pion momenta
and masses
chiral gap
(½+)
(½-)
parity partners

7. PIONIZATION OF THE NEUTRON STAR MATTER

8. Equation of state of nuclear matter
symmetry energy
There is a correlation among parameters: J, L, Ksym

9. If we assume some model for the density dependence of the symmetry energy
Analysis of 36 RMF models gives
[Dong, et al PRC85, (2012)]
More details and other relations in
Tews, Lattimer et al, ApJ848, 105(2017)

10. Simple model for NS matter
n+p+e+m matter
Lightest negatively charged bosons: p-
minimum at k=0
Weak reactions start
Pionization of neutron star matter

11. Chiral symmetry for pion-nucleon interaction
isospin even and odd amplitudes
Chiral symmetry: for forward scattering amplitude:
Weinberg Tomazawa theorem
repulsive in neutron reach matter

12. chiral shield against pionization
29 MeV<J<38 MeV
Detailed analysis of the possibility of the s-wave pion condensation in
Onishi, Jido et al, PRC 80, (2009)

13. P-WAVE PION CONDENSATION

15. Baym
Migdal
Scalapino
1974 Tbilisi

16. Tensor forces in NN interaction
Resummed, enhanced pion exchange
nucleon arrangement

17. Alternating-layer-spin configurations
Ryozo Tamagaki
Tatsuyuki Takatsuka
1972’-73’
p0 condensate
tensor force contribution to the energy

18. THE FERMI-LIQUID THEORY
PION CONDENSATION
WITHIN
THE FERMI-LIQUID THEORY

19. Fermi liquid theory
system of strongly interacting fermions (no pairing)
single-particle excitation mechanism
quasiparticle
particle-hole interaction n n’
on Fermi surface
short-range
long-range
particle-hole propagator
pole parts
Landau parameters

20. density of states at the Fermi surface
number of fermion types
neutron matter:
(1 parameter in each channel)
nuclear matter:
(3 parameters in each channel)
In matter of arbitrary isospin composition these parameters are independent.
Fermi-liquid renormalization is different for these parameters.
small isospin disballance
(2 parameters in each channel)
In nuclear physics one uses also the normalization on the nuclear Fermi surface
constant, independent of density
Density dependence? Residual momentum dependence ?

21. There are relations between some Landau parameters and bulk properties of the system
effective mass
compressibility
symmetry energy
In general Landau parameter are to be fitted to empirical information (nucleus properties)

22. Solutions for zeroth harmonics
keep only zeroth harmonics
Lindhard function

23. Lindhard function Imaginary part
Results of expansions depends on the expansion order:
Temperature corrections

24. zero-sound modes “diffuson” unstable mode Scalar modes I. II.
Landau damping I.
zero-sound modes
II.
“diffuson”
stable mode
unstable mode
III.

25. Fermi-liquid with pions
pion with residual (irreducible in NN-1 and  N-1) s-wave  N interaction and  scattering``
Part of the interaction involving  isobar is analogously constructed:

26. Spin channel pure neutron matter similar solutions as
Consider now the spin channel
include the pion exchange explicitly
similar solutions as
in scalar channel
s-wave and D parts
of polarization operator
pion nucleon coupling constant
If there is a “diffuson” solution
Can change the sign at
some momentum and density!
Instability: pion condensation

27. Critical density of pion condensation
Without short-range correlations (g’) the critical density of the pion condensation
would be 0.3 n0 !

28. Spectrum of diffusive pion mode
We search for a diffusive solution
roton-like spectrum
effective pion gap

29. Nucleon-nucleon interaction in dense matter
based on a separation of long and short scales
full pion propagator:
enhancement of the
amplitude
dressed vertex:
suppression
Similar to Debye screening in plasma
Landau-Migdal parameters of short-range interaction are extracted from atomic nuclei
Poles yield zero-sound modes in scalar and spin channels
known phenomena in Fermi liquid
provided short-scale interaction
can be reduced to the local one
[Migdal et al., Phys. Rep. 192 (1990) 179]

30. pion propagator for pion gap reconstruction of pion spectrum
on top of the pion condensate
LM parameters increase with density
saturation of pion softenning
no pion condensate
amplitude of the pion condensate
[Migdal, Rev.Mod. Phys. 50 (1978), Migdal, Sapershtein,Troitsky, Voskresensky Phys. Rept. 192 (1990)]

31. Green’s functions only
Vertex renormalization in FL
Coupling of an external field to a particle
pole parts of
Green’s functions only
“bare” (FL renormalized) vertex
Effects:
Reduce couplings. “A shield” against pion condensation
Produce sound modes contributing to response functions
Enhance reactions in some channels

32. NEUTRON STAR COOLING

33. Pulsar age Pulsar rotation period/frequency changes with time:
e.m. wave
emission
grav. wave
emission
angle between rotation a
and magnetic axes
e neutron star eccentricity
initial period
current period
spin-down age:
braking index:
kinematic age:
3) historical events
1) age of the associated SNR
Crab : 1054 AD
Cassiopeia A: 1680 AD
Tycho’s SN: 1572 AD
2) pulsar speed and position w.r. to the
geometric center of the associated SNR

34. Cooling scenario [neutrino production]
Given:
EoS
Cooling scenario [neutrino production]
Mass of NS
Cooling curve

35. Neutron Star Cooling Data
3 groups:
>103 in emissivity
slow cooling
intermediate cooling
rapid cooling
All groups can be described within one cooling scenario if there is a strong dependence of neutrino luminocity on the NS mass strong density-dependence of emissivity
[Voskresensky, Senatorov 1986]

36. Neutrino emission reactions
neutron star is transparent for neutrino
CV – heat capacity, L - luminosity
emissivity
each leg on a Fermi surface / T
neutrino phase space ´ neutrino energy

37. standard exotic modified Urca (MU)
pair formation-breaking process (PFB)
exotic
Direct Urca (DU)
Direct Urca on pion (PU)

38. EoS should produce a large DU threshold in NS matter !
DU process schould be „exotics“ (if DU starts it is dificult to stop it)
[Blaschke, Grigorian, Voskresensky A&A 424 (2004) 979]
EoS should produce a large DU threshold in NS matter !
[EEK, Voskresensky NPA759 (2005) 373]

39. medium bremsstrahlung reactions
enhancement factors w.r.t. MU emissivity
medium MU reactions
medium bremsstrahlung reactions
vertex correlation function

40. Neutron star cooling
[Blaschke, Grigorian, Voskresensky PRC88 (2013)065805]

41. R-MODES IN NEUTRON STARS

42. Age-period diagram for pulsars
The ATNF pulsar catalogue
recycled
pulsars
stability of
old pulsars?
Young pulsars periods
are much larger
than the Kepler limit.

43. Rossby waves on Earth: in oceans and atmosphere
The returning force for these wave is the Coriolis force
Carl-Gustaf Rossby

44. R-mode instability of rotating neutron star and viscosity
In a superdense system like a neutron star the Rossby waves are sources of
gravitational radiation.
1998 it was shown by Andersson, Friedman and Morsnik showed that this
radiation leads to an increase of the amplitude of the mode.
so there is an instability
Large R-modes can either destroy the star
or the star stop rotating
Viscosity of the dense nuclear matter can damp r-modes and save the rotating star

45. R-modes in rotating neutron star
Consider a neutron star with a radius R rotating with a rotation frequency W
and a perturbation in the form
oscillation amplitude
oscillation frequency
(dominant modes
for l=m=2)
for a<<1
amplitude changes
viscous damping
[Andersson ApJ 502 (98) 708;
Friedman, Morsink, ApJ 502 (98) 714]
gravitational radiation
drives r-mode unstable
[Lindblom, Owen, Morsink, PRL80 (98) 4843]
If the r-modes are undamped, the star would lose its angular moment on the time scale of tG, because of an enhanced emission of gravitation waves.

46. R-mode is unstable if the time >0
R-modes stability
R-mode is unstable if the time
>0
gravitational time scale
damping rate due to the shear viscosity
damping rate due to the bulk viscosity
[Lindblom, Owen, Morsnik PRL 80 (98) 4843
Owen, Lindblom, Cutler, Schutz, Vecchio, Andersen PRD58 (98) ]
profile averages

47. Viscosity Dissipation after Dissipation when there
Euler equation:
shear viscosity
bulk viscosity
Dissipation after
uniform volume change
Dissipation when there
is a velocity gradient
Maxwell (1860) kinetic theory calculations
Units:
Nuclear matter:
(naïve estimate)

48. Lepton shear viscosity
Lepton shear viscosity = electron + muon contribution
low T, Fermi liquid results
Lepton collision time tl is determine by lepton-lepton and lepton-proton collisions
1979
Flowers and Itoh:
Important role of the phonon modification (plasmon exchange)
for QCD plasma [Heiselberg, Pethick Phys. Rev. D 48 (1993) 2916]
[Shternin, Yakovlev, Phys. Rev. D 78 (2008) ]
leading terms for small T
muon contributions are important

49. Lepton shear viscosity vs. neutron star mass
Effects of proton pairing on
lepton shear viscosity
[Shternin, Yakovlev, Phys. Rev. D 78 (2008) ]

50. Nucleon shear viscosity
Fermi liquid
result
[Shternin, Yakovlev, Phys. Rev. D 78 (2008) ]
effective NN
cross section
FOPE:
MOPE:
modification factor
[Bacca et al, PRC80]

51. Neutrino shear viscosity
With the temperature increase the neutrino mean free path decreases and for sufficiently high temperatures neutrinos become trapped inside the neutron star interior.
Neutrino mean free path is determined by inverse MMU and PU processes
weak T dependence
contributes only in regions where neutrinos are trapped
opacity radius is determined as

52. Lepton shear viscosity

53. nucleon relaxation time;
Bulk viscosity
collisional
[Sykes, Brooker, Ann. Phys. 56(1970) 1]
F0 is the zeroth harmonics of the dimensionless scalar Landau-Migdal parameter, F0~ 1
nucleon relaxation time;
small contribution

54. Energy dissipation of the mode: is neutrino emissivity
Bulk viscosity
Energy dissipation of the mode:
is neutrino emissivity
Energy of the mode decreases if the pressure depends on an order parameter, which variation is delayed with respect to the variation of the density
[Mandelstam, Leontovich, ZhETF 7(1937)438]
soft mode
[Sawyer PRD39, Haensel, Levenfish, Yakovlev A&A357, A&A372]
order parameter is
lepton concentration
relaxation time
average over the perturbation period

55. [calculated with FOPE] or medium MU (MMU) [calculated with MOPE];
for
sum over various processes changing dXl
direct Urca (DU);
modified Urca (MU)
[calculated with FOPE] or
medium MU (MMU)
[calculated with MOPE];
pion condensate Urca (PU)

56. Shear and bulk viscosities. Results
MMU+DU

57. R-mode stability window
shear
bulk

58. Young pulsars
1. star is born hot and rapidly rotating (point A)
2. cooling time (heat transport!) >> spin-down time
for max. r-mode amplitude
3. star moves along line AB because of r-modes 4. line BC, cooling and magnetic breaking
Minimum point B must be above 62 Hz (PSRJ )

59. Minimum of the stability window

60. LMXB recycled pulsars
Rotation of LMXB pulsars cannot be explained. Shear viscosity is too small.
Alternative mechanisms
differential drift in magnetic field
[Rezzolla, Lamb, Shapiro]
-weak reactions with hyperons +hyperon pairing
[Jones; Nayyar Owen]
core-crust coupling [Bildsten, Ushomirsky, Levin]
saturation of r-mode amplitude at small values [Arras, Bondaresku, Wasserman]
non-linear decay of r-modes
[Kastaun]
coupling to more stable modes
[Gusakov, Chugunov, Kantor]
-vortex flux-tube interactions
[Haskell, Glampedakis, Andersson]

61. Soft bosonic modes in rapidly rotating systems
dispersion laws of the low-lying bosonic excitations
zero-sounds in normal fermionic system
AB, Carlson-Goldman
in fermionic superfluid
pion and kaon
p-wave condensates
What happens if the medium flows with a velocity v>vc?
[Pitaevskii ’84; Voskresenky 93;Baym, Pethick 12]
There appears a Bose condensate of excitations,
which will carry a part of the momentum and
the fluid will move with smaller velocity.
differential rotation of the star

62. Outer part of the core does not rotates. Crust rotates
Minimal rc is determined by the size of the proton paring zone

63. Pion in neutron star, to be or not to be?
Conclusions
Pion in neutron star, to be or not to be?
pionization? NO!
p-wave pion condensation. Yes!? But
(g’ short-range correlations?, mN*)
2. Neutron star cooling
medium “Modified Urca” – a mechanism to understand NS cooling
3. R-modes and viscosity of NS matter
moderate increase nucleon shear viscosity for small NS masses
partial trapping of neutrinos in the star core at T~2x109 – 1010 K
large neutrino shear viscosity for T>2x109
strong increase of the bulk viscosity

64. STAR MODEL

65. Neutron star model HDD EoS n,p,e,m, matter
[Blaschke, Grigorian, Voskresensky,
PRC 88 (13) ]
based on Heiselberg-Hjorth-Jensen (HHJ) analytical parameterization of
the microscopic Akmal-Pandharipande-Ravenhall (APR) EoS with A18+ dv+U IX* forces
n,p,e,m, matter
binding energy per nucleon:
scaling function:
The scaling function makes EoS harder for n>5n0, but the EoS remains causal.

66. HDD EoS
Friedman-Pandharipande-Skyrme EoS
mass and radius
proton and electron concentration
thresholds of Direct Urca
reactions
thresholds of Modified Urca
reactions on protons
density profiles
similar to the Tolman V solution

67. 1S0 neutron and proton pairing
parameterizations from
[Kaminker, Yakovlev, Gnedin, A&A383(2002)1076] motivated by the calculations of Wambach, Ainsworth, Pines (1993) and Schulze, Cugnon, Lejeune, Baldo, Lombardo (1996) including in-medium modifications of the NN interaction
BCS calculations with vacuum interaction, Vlow-k
[Hebeler, Schwenk, Friman, PLB 648 (2007) 176]
3P2 neutron pairing
[Schwenk, Friman, PR92 (2004) (2004)] triplet paring is supperessed
by medium-induced spin-orbit interaction.
[Ding, Witte, Dickhoff, Dussan, Rios, Polls, arXiv: ]
long-range density and spin fluctuations preclude triplet pairing
We will ignore triplet pairing

68. Profiles of critical temperatures for the singlet neutron and proton pairing