1 ngb and their parameters gradient expansion: parameters of the ngb’s masses of the ngb’s ...
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NGB and their parametersNGB and their parameters
Gradient expansion: parameters of the NGB’s
Masses of the NGB’s
The role of the chemical potential for scalar fields: BE condensation
Dispersion relations for the gluons
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Hierarchies of effective Hierarchies of effective lagrangianslagrangians
Integrating out Integrating out heavy degrees of heavy degrees of freedom we have freedom we have
two scales. The gap two scales. The gap and a cutoff, and a cutoff, above which we above which we integrate out. integrate out.
Therefore: Therefore: two two different effective different effective theories, theories, LLHDETHDET and and
LLGoldsGolds
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Gradient Gradient expansion: NGB’s expansion: NGB’s
parametersparameters
AB ABA† BD *
AB AB
iV DdvL (L R)
iV D4
Recall from HDET that in the CFL Recall from HDET that in the CFL phasephase
9A
i A iA 1
1( )
2
and in the basis
AB A AB A
A 1, ,8A
A 9 29
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AA† AD
A
iVdvL (L R)
iV4
AABAB 2
A A
VS
V V V
Propagator
Coupling to the Coupling to the U(1)U(1) NGB: NGB:i / f i / fU e , V e
f , f
i( ) i( )L L R R
i i
e , e
U e U, V e V
†L RUV , UV Invariant Invariant
couplingscouplings
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†2A† AA
D 2A
iV UdvL
4 U iV
Consider now the case of the U(1)B NGB. The invariant Lagrangian is:
At the lowest order in 2
2A† A
A 2
2
2i 20
f fdvL
4 2i 20
f f
generates 3-linear and 4-linear couplings
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Generating functional:†i A( )†Z[ ] D D e
21 A A
0 0 12
0 1
2i 2A( ) S
f f
0 1 0 1,
1 0 1 0
A10
A
VS
V
1Tr[log A( )]1/ 2 2Z[ ] (det[A( )]) e
eff
iS [ ] Tr[log A( )]
2
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21
0 12
n2n 1 21
0 12n 1
2i 2iTr[log A( )] iTr log S 1 S S
f f
( 1) 2i 2iTr logS i iS i iS i
n f f
At the lowest order:
eff 0 0
2
12
i iS(y,x)2i (x) iS(x, y)2i (y)S dxdyTr i i
4 f f
i iS(x,x)2 (x)dxTr i
2 f
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Feynman rules
For each fermionic internal line
AABAB AB 2
A A
ViiS i S(p)
V V V
For each vertex a term iLiLintint
For each internal momentum not For each internal momentum not constrained by momentum conservation:constrained by momentum conservation:
2 22
04 3
4d d d
(2 ) 4
Factor 2x(-1) from Fermi statistics and spin. A factor 1/2 from replica trick.
A statistical factor when needed.
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+2 2
Aeff I II 3 2
A
2 2 22 A
A A A
1 dviL iL (p) iL (p)
2 4 f
V ( p) V V ( p) V 2 2d
D ( p)D ( ) D ( )
2A AD ( ) V V i
I IIL (0) L (0) 0 Goldstone theorem:
Expanding in p/
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2
eff 2 2
9 1 dvL (x) (V ) (x)(V ) (x)
f 2 4
1 0 0 0
10 0 0
3dvV V 1
4 0 0 03
10 0 0
3
2 22 2
eff 02 2
22 2
2
1 9L (x) v
2 f
1 9v , f
3
CFL
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For the V NGB same result in CFL, whereas in 2SC
22 2
2
1 4v , f
3
2SC
With an analogous calculation:
2 8a 2 2 a 2
eff 02 2a 1
(21 8log 2) 1L ( ) v | |
36 F 2
22 2 2
T 2
1 (21 8log 2)v , F F
3 36
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Dispersion relation for the NGB’s
1E | p |
3
Different way of computing:
a b ab 0 10 | J | iF p , p p , p
3
Current conservation:
2 21p p E | p | 0
3
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Masses of the Masses of the NGB’sNGB’s
QCD mass term: L RM h.c.
2 LZM M
† T 4i T(Y X) e
1 † 2masses
† †
L c det[M]Tr[M h.c. c ' det( )Tr[(M ) ]
c" Tr[M ]Tr[M ]
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Calculation of the coefficients from QCD
Mass insertion in QCD
Effective 4-fermi
Contribution to the vacuum energy
2
2
3c , c ' 0, c" 0
2
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Consider:
a aQCD 0 L R R L
1L (iD ) M M G G
4
Solving for as in HDET,L
,L 0 ,L 0 ,R
1i D M
2
† † 2D ,L ,L ,L ,L
†
†1L (iV D) ( D )
2
L
MM
R, M M
like chemical potential
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Consider fermions at finite density:
0L gi i
as a gauge field A0
Invariant under: i ( t )e , (t)
Define: † †L R
1 1X MM , X M M
2 2
Invariance under:
,L ,L ,R ,R
† †L L 0
† †R R 0
L(t) , R(t) ,
X L(t)X L (t) iL(t) L (t),
X R(t)X R (t) iR(t) R (t)
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The same symmetry should hold at the level of the effective theory for the CFL phase (NGB’s), implying that
T T† †
0 0 0
MM M Mi i
2 2
The generic term in the derivative expansion of the NGB effective lagrangian has the form mn p† 2
2 2 q †r0NGB 2
iMM / 2 ML F
F
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mn p† 22 2 q †r0
NGB 2
iMM / 2 ML F
F
Compare the two contribution to quark masses:
kinetic term4 4
2 22 2 2 2
m 1 mF
F
mass insertion2 2 2
2 22 2 2
m 1 mF
F F F
Same order of magnitude for since
m F
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The role of the chemical The role of the chemical potential for scalar fields: potential for scalar fields:
Bose-Einstein condensationBose-Einstein condensation
A conserved current may be coupled to the a gauge field.
Chemical potential is coupled to a conserved charge.
The chemical potential must enter as the fourth component of a gauge field.
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Complex scalar field:
2† † 2 † †0 0
2† 2 † †0 0
2 † †
L i i m
im
negative mass term
breaks C
Mass spectrum:
2 2 20
2 2 2
p (m ) 2 Qp 0 (Q 1)
(E Q) m | p |
P,Pm m For < mm
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At = m = m, second order phase transition. Formation of a condensate obtained from:
22 2 † †V m
2 2
† mm
2
† 2 2V2 m
Charg
e densit
yGround state = Bose-Einstein condensate
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2 22v m
, v2
i (x ) / v1
(x) v h(x) e2
2 22 0
1 1L h h v h 2 h
2 2
Mass spectrum2 2
2
p 2 v 2i Edet 0
2i E p
At zero momentum
2 2 2 2M M 2 v 4 0
2
2 2 2
M 0
M 6 2m
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At small momentum
2 2
NGB 2 2
2 22 2 2
massive 2 2
mE | p |
3 m
9 mE 6 2m | p |
6 m
2 22NGB 2 2
m
m 33
1v
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Back to CFL. From the structure P,Pm m
0 0
2 2d u
u d s2
2 2s u
u s d2K
2 2s d
d s u2K ,K
m m 2cm (m m )m ,
2 F
m m 2cm (m m )m ,
2 F
m m 2cm (m m )m
2 F
First term from “chemical potential” like kinetic term, the second from mass insertions
†MM
2
2
3c
2
22
2
(21 8log 2)F
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For large values of ms:
0 0
u d s2
2 2s s
s d s u2 2K K ,K
2cm (m m )m ,
F
m 2c m 2cm m m , m m m
2 F 2 F
and the masses of K+ and K0 are pushed down. For the critical value
1/ 322 23 3
s u,d u,d2 2crit
s crit
12m m 3.03 m ,
F
m 40 110 MeV
masses vanish
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For larger values of ms these modes become unstable. Signal of condensation. Look for a kaon condensate of the type:
4i 24 4e 1 (cos 1) i sin
(In the CFL vacuum, = 1) and substitute inside the effective lagrangian
22 2s s
2
1 m 2cm mV( ) F sin (1 cos )
2 2 F
negative contribution from the “chemical potential”
positive contribution from mass insertion
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Defining
2
20s seff K 2
m 2cm m, m
2 F
2 2 2 0 2eff K
1V( ) F sin (m ) (1 cos )
2
with solution 20
K 0eff K2
eff
mcos , m
and hypercharge density
40K2
Y eff 4eff eff
mVn F 1
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Mass terms break original SU(3)c+L+R to
SU(2)IxU(1)Y. Kaon condensation breaks
this to U(1)
3 8
1 1Q , [Q, ] 0
2 3
I Y
0 0 0SU(2) U(1)
( , ) (K , K ) (K , K ) ( )
breaking through the doublet as in the SM
Only 2 NGB’s from K0, K+ instead of expected 3 (see Chada & Nielsen 1976)
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Chada and Nielsen theorem: The number of NGB’s depends on their dispersion relationI. If E is linear in k, one NGB for any
broken symmetry
II. If E is quadratic in k, one NGB for any two broken generators
In relativistic case always of type I, in the non-relativistic case both
possibilities arise, for instance in the ferromagnet there is a NGB of type II, whereas for the antiferromagnet there
are to NGB’s of type I
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Dispersion relations for the Dispersion relations for the gluonsgluons
The bare Meissner mass
The heavy field contribution comes from the term
2
† †h h h h
D DDP
2 iV D 2 iV D
1P g V V V V
2
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Notice that the first quantized hamiltonian is:
2
2 20 0
gH p gA eA | p | gA gv A | A | (v A)
2 | p |
Since the zero momentum propagator is the density one gets
3 22 2
f 3 2|p|
d p 1 (p A)g 2 N Tr A
(2 ) 2 | p | | p |
spin
2 2a a 2 a a
f BM2a a
2 22BM f 2
g 1 1N A A m A A ,
6 2 2
gm N
6
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Gluons self-energy
Vertices froma
aigA J
Consider first 2SC for the unbroken gluons:
00 2ab ab
kl kl,self klab ab ab
2 2 2 2 2
2
kl kl kl 20ab ab ab 02 2 2
0
2
2 2
2 2
2 2
k2 2
2a 20 k
b ab
(p) | p | ,
(p) (p) (p)
g p g1 p ,
3 6 3
(p)
g
18
g
18
p pg
18
2BMfrom m
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Bare Meissner mass cancels out the constant contribution from the s.e.
All the components of the vacuum polarization have the same wave function renormalization
a a b a a a a a aa ab i i i i i i
2 2
2 2
1 1 1 kL F F A A E E B B E E
4 2 2 2
gk
18
Dielectric constant = k+1, and magnetic permeability =1 1
vg
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Broken gluons
a (0) - ij(0)
1-3 0 0
4-7 3mg2/2 mg
2/2
8 3mg2 mg
2/3
2 22g 2
gm
3
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But physical masses depend on the wave function renormalization
2 2
2
g
Rest mass defined as the energy at zero momentum:
R
R
m 2 , a 4,5,6,7
gm , a 8
The expansion in p/ cannot be trusted, but numerically
Rm 0.9 , a 4,5,6,7
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In the CFL case one finds:2 2
2 2 2D 2
2 2 22 DM 2
gm (21 8log 2) g F
36
g 11 2 1 mm log 2
36 27 2 3
Recall that from the effective lagrangian we got:
2 2 2 2 2 2 2D T T M S Tm g F , m v g F
from bare Meissner mass
implying and fixing all the parameters.
S T 1
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We find:2 2
DR 1 2 2
1
R
m g 16m , 7 log 2
216 33
m 1.70
Numerically Rm 1.36
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LOFF phaseLOFF phase
Different quark masses
LOFF phase
Phonons
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Different quark massesDifferent quark masses
We have seen that for one massless flavors and a massive one (ms), the condensate may be disrupted for 2
sm
2
The radii of the Fermi spheres are:
1 2
22 2 s
F s F
mp m , p
2
As if the two quarks had different chemical potential (ms
2/2)
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Simulate the problem with two massless quarks with different chemical potentials:
u d
u d u d
,
,2 2
Can be described by an interaction hamiltonian
†I 3H
Lot of attention in normal SC.
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LOFF:LOFF: ferromagnetic alloy with ferromagnetic alloy with paramagnetic impuritiesparamagnetic impurities. .
The impurities produce a constant exchangeThe impurities produce a constant exchange fieldfield acting upon the electron spins giving rise acting upon the electron spins giving rise to anto an effective difference in the chemical effective difference in the chemical potentials of the opposite spins.potentials of the opposite spins.
Very difficult experimentally but claims of Very difficult experimentally but claims of observations in heavy fermion observations in heavy fermion superconductorssuperconductors ( (Gloos & al 1993Gloos & al 1993) ) and in quasi-and in quasi-two dimensional layered organic two dimensional layered organic superconductorssuperconductors ( (Nam & al. 1999, Manalo & Klein 2000Nam & al. 1999, Manalo & Klein 2000))
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HI changes the inverse propagator
10 *
3
3
VS
V
and the gap equation (for spin up and down fermions):
2 2
2 2 2 20
dv dig
4 (2 ) ( )
This has two solutions:
2 20 0 0a) b, ): : 2
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Grand potential:
0
0
22
2 00
0( 0)
H dg| |
g g g
d
2
02
0
d 2 dg
g
Also:
20 0( ) (0)
2
Favored solution
0
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44
Also: 2 20 0( ) 2
4
First order transition to the normal state at
01
2
For constant Ginzburg-Landau expanding up to
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LOFF phaseLOFF phase
In 1964 Larkin, Ovchinnikov and Fulde, Ferrel, argued the possibility that close to the first order-line a new phase could take place.
According LOFF possible condensation with non zero total
momentum of the pair1p k q
2p k q
xqi2e)x()x(
xqi2
mm
mec)x()x(More generallyMore generally
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q2pp 21
|q|
|q|/q
fixed variationallyfixed variationally
chosen chosen spontaneouslyspontaneously
)qk(E)p(E
qvF
Gap equation: ),p(
nn1
)2(
pd
2
g1 du
3
3
du nn
Non zero total
momentum
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1e
1n
T/)),p((d,u
ForFor T T 00
))()(1(),p(
1
)2(
pd
2
g1
3
3
||blocking blocking regionregion
The blocking region reduces the gap:The blocking region reduces the gap:
BCSLOFF
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Possibility of a crystalline structure (Larkin
& Ovchinnikov 1964, Bowers & Rajagopal 2002)
xqi2
2.1|q|iq
i
i
e)x()x(
The qi’s define the crystal pointing at its vertices.
The LOFF phase has been studied via a Ginzburg-Landau expansion of the
grand potential
see latersee later
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49
642
32
(for regular crystalline structures all the q are equal)
The coefficients can be determined microscopically for the different structures (Bowers and Rajagopal (2002)Bowers and Rajagopal (2002) ))
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Gap equationGap equation
Propagator expansion
Insert in the gap equation
General strategy
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We get the equation
053
Which is the same as
0
with
3
5
The first coefficient has universal structure, independent on the
crystal. From its analysis one draws the
following results
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22normalLOFF )(44.
)2(4
2BCS
2normalBCS
)(15.1 2LOFF
2/BCS1 BCS2 754.0
Small window. Opens up in QCD?
(Leibovich, Rajagopal & Shuster 2001; Giannakis, Liu
& Ren 2002)
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Single plane wave
Critical line from
0q
,0
Along the critical line
)2.1q,0Tat( 2
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Preferred structure:
face-centered
cube
Bowers and Bowers and Rajagopal Rajagopal
(2002)(2002)
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In the LOFF phase translations and rotations are broken
phonons
Phonon field through the phase of the condensate (R.C., Gatto, Mannarelli & Nardulli 2002R.C., Gatto, Mannarelli & Nardulli 2002):
)x(ixqi2 ee)x()x(
xq2)x(
introducing
xq2)x()x(f
1
PhononPhononss
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2
22||2
2
2
222
phonon zv
yxv
2
1L
Coupling phonons to fermions (quasi-particles) trough the gap term
CeC)x( T)x(iT
It is possible to evaluate the parameters of LLphononphonon (R.C., Gatto, Mannarelli & R.C., Gatto, Mannarelli &
Nardulli 2002Nardulli 2002)
153.0|q|
12
1v
2
2
694.0
|q|v
2
2||
++
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Cubic structureCubic structure
i
)i(i
i
iik
;3,2,1i
)x(i
;3,2,1i
x|q|i28
1k
xqi2 eee)x(
i)i( x|q|2)x(
i)i()i( x|q|2)x()x(
f
1
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0)x(
4)x(
4)x(
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Coupling phonons to fermions (quasi-particles) trough the gap term
i
)i(i
;3,2,1i
T)x(iT CeC)x(
(i)(i)(x) (x) transforms under the group O Ohh of the cube. . Its e.v. ~ xi breaks O(3)xO O(3)xOhh ~ ~ OOhh
diagdiag
2(i)(i) 2
phononi 1,2,3 i 1,2,3
2(i) (i) ( j)i i j
i 1,2,3 i j 1,2,3
1 aL | |
2 t 2
bc
2
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we get for the coefficients
12
1a 0b
1
|q|3
12
1c
2
One can evaluate the effective lagrangian for the gluons in tha
anisotropic medium. For the cube one finds
Isotropic propagationIsotropic propagation
This because the second order invariant for the cube and for the rotation group
are the same!
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Compact stellar objectsCompact stellar objects
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Compact stellar objectsCompact stellar objects
High density core of a compact star, a good lab for testing QCD at high density.
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Some features of a compact starFor simplicity consider a gas of free massless fermions.
3
f p p3
4
f 2
d p2N V ( V) ( )
2N
1) 2(
Grand potential:
Density:3
f 2VN
3
Eq. of state:f
4 / 34
2P VN
12P K
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For a non-relativistic fermion:3/ 2 3/ 2
5/ 2 3/ 2f f2 2
5/ 3
8 m 4 mP VN , VN
15 32
P K
2
More generally assumed P K
For high densities inverse beta decay becomes importante p n
At the equilibrium
e p n
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1/ 3 1/ 3 1/ 3e p n e p n
From charge neutrality e p
p
n
1
8
Neutron star
Radius of a neutron star (Landau 1932)
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N fermions in a box of volume V. Number density
3
Nn
R
Position uncertainty 1/ 31/ 3
Rn
N
Uncertainty principle1/ 3 1/ 3
/ 3F
1F
Np n
R
NE c
R
Gravitational energy per baryon
2B
G
GNmE
R
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1/ 3 2B
G F
cN GNmE E E
R R
E > 0 otherwise not bounded. This condition gives
3/ 2
571/
m xB
3 2B
a 2
cN 2 10
Gm
cN GNm0 N
R R
Maximum mass
maxM 1.4M Chandrasekhar limit
max max BM N n 1.5M
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1/ 2
2 1/ 3F max 2
B
c c cE mc N
R R Gm
1/ 28
e52
nB
c 5 10 cm (m m )R
3 10 cm (m m )mc Gm
Typical neutron star density
15 3 15 3Nnm
1.3ferminm
m10 g / cm , 0.15 10 g / cm5 6
V
min
16 3max
R
M2.5 10 gr / cm
V Density (f or a neutron star)
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Neutron stars are a good laboratory to test hadronic matter at high density and
zero temperature
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In neutron stars CS can be studied at T = 0 (TTnsns~10~105 5 KK)
)K10MeV1(
100)MeV(201010T
10
BCS76
BCS
ns
Orders of magnitude from a crude model: 3 free quarks
0M,0MM sdu
Consider the LOFF state. From BCSBCS14 (MeV) 70
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s,d,ui
iqeqees,d,ui
ii NNQNNN
0Qe
2
s,d,ui
iF2B )p(
3
1
3
1
Weak equilibrium:
2s
2s
sFes
ddFed
uuFeu
Mp,3
1
p,3
1
p,3
2
Electrical neutrality:
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n.m.n.m. is the saturation nuclear density ~ .15x10~ .15x1015 15 g/cmg/cm
At the core of the neutron star B B ~ 10~ 101515 g/cm g/cm
65.m.n
B Choosing ~ 400 ~ 400
MeVMeVMs = 200 pF = 25
Ms = 300 pF = 50Right
ballpark (14 - 70
MeV)
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)10Ω/Ω( 6
Glitches: discontinuity in the period of the pulsars Standard explanation: metallic crust
+ neutron superfluide inside
LOFF region inside the star providing the crystalline structure + superfluid CFL phase
dipole emission
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In the superfluid phase there are vortices pinned to the crust. When the star slows down the vortices do not participate in the motion until an instability is produced. Then there is a release of angular momentum to the crust seen as a jump in the rotational frequency.
The presence of the LOFF phase might avoid the main objection against the existence of strange stars (made of u,d,s quarks in equal ratios) since they cannot have a crust.
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ConclusionsConclusions
SC almost 100 years old, but still actual
Important technological applications
Source of inspiration for other physical theories (SM as an example)
Deep implications in QCD at very high density: very rich phase structure
Possible applications for compact stellar objects
Unvaluable theoretical laboratory