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QCD_at_Work 2003 International Workshop on Quantum
Chromodynamics Theory and Experiment Conversano
(Bari, Italy) June 14-18  2003
Inhomogeneous color superconductivity
Roberto Casalbuoni
Department of Physics and INFN Florence

CERN TH Division - Geneva
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Summary

• Introduction
• Effective theory of CS
• Gap equation
• The inhomogeneous phase (LOFF) phase diagram and
  crystalline structure
• Phonons
• LOFF phase in compact stellar objects
• Outlook

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Introduction

• mu, md, ms ltlt m CFL phase

• mu, md ltlt m ltlt ms 2SC phase

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In this situation strange quark decouples. But
what happens in the intermediate region of m? The
interesting region is for (see later)
m ms2/D
Possible new inhomogeneous phase of QCD
LOFF phase
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Effective theory of Color Superconductivity
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Relevant scales in CS
Fermi momentum defined by
The cutoff is of order wD in superconductivity
and gt LQCD in QCD
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Hierarchies of effective lagrangians
LQCD
Microscopic description
p pF gtgt d
pF d
Quasi-particles (dressed fermions as electrons in
metals). Decoupling of antiparticles (Hong 2000)
LHDET
d gtgt p pF gtgt D
D ltlt d ltlt pF
pF D

Decoupling of gapped quasi-particles. Only light
modes as Goldstones, etc. (R.C. Gatto Hong,
Rho Zahed 1999)
LGold
D
p pF ltlt D
pF
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Physics near the Fermi surface
Relevant terms in the effective description
(see Polchinski, TASI 1992, also
Hong 2000 Beane, Bedaque Savage 2000, also
R.C., Gatto Nardulli 2001)
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SM gives rise di-fermion condensation producing a
Majorana mass term. Work in the
Nambu-Gorkov basis
Near the Fermi surface
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Dispersion relation
At fixed vF only energy and momentum along vF are
relevant
v1
v2
Infinite copies of 2-d physics
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Gap equation
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For T T 0
At weak coupling
density of states
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With G fixed by cSB at T 0, requiring
Mconst 400 MeV

and for typical values of m 400 500 MeV one
gets
Evaluation from QCD first principles at
asymptotic m (Son 1999)
Notice the behavior exp(-c/g) and not exp(-c/g2)
as one would expect from four-fermi interaction
For m 400 MeV one finds again
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The inhomogeneous phase (LOFF)

• In many different situations the would be
  pairing fermions belong to Fermi surfaces with
  different radii
• Quarks with different masses
• Requiring electrical neutrality and/or weak
  equilibrium

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Consider 2 fermions with m1 M, m2 0 at the
same chemical potential m. The Fermi momenta are
To form a BCS condensate one needs common momenta
of the pair pFcomm
Grand potential at T 0 for a single fermion
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Pairing energy
Pairing possible if
The problem may be simulated using massless
fermions with different chemical potentials
(Alford, Bowers Rajagopal 2000)
Analogous problem studied by Larkin
Ovchinnikov, Fulde Ferrel 1964. Proposal of a
new way of pairing. LOFF phase
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• LOFF ferromagnetic alloy with paramagnetic
  impurities.
• The impurities produce a constant exchange field
  acting upon the electron spins giving rise to an
  effective difference in the chemical potentials
  of the opposite spins.
• Very difficult experimentally but claims of
  observations in heavy fermion superconductors
  (Gloos al 1993) and in quasi-two dimensional
  layered organic superconductors (Nam al. 1999,
  Manalo Klein 2000)

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or paramagnetic impurities (dm H) give rise to
an energy additive term
Gap equation
Solution as for BCS D DBCS, up to (for T 0)
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According LOFF, close to first order line,
possible condensation with non zero total
momentum
More generally
fixed variationally
chosen spontaneously
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Simple plane wave energy shift
Gap equation
For T T 0
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The blocking region reduces the gap
Possibility of a crystalline structure (Larkin
Ovchinnikov 1964, Bowers Rajagopal 2002)
see later
The qis define the crystal pointing at its
vertices.
The LOFF phase is studied via a Ginzburg-Landau
expansion of the grand potential
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(for regular crystalline structures all the Dq
are equal)
The coefficients can be determined
microscopically for the different structures
(Bowers and Rajagopal (2002))
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• Gap equation

• Propagator expansion

• Insert in the gap equation

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We get the equation
Which is the same as
with
The first coefficient has universal structure,
independent on the crystal. From its analysis one
draws the following results
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Small window. Opens up in QCD? (Leibovich,
Rajagopal Shuster 2001 Giannakis, Liu Ren
2002)
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Results of Leibovich, Rajagopal Shuster (2001)
m(MeV) dm2//DBCS (dm2 - dm1)/DBCS
LOFF 0.754 0.047
400 1.24 0.53
1000 3.63 2.92
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Single plane wave
Critical line from
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General analysis (Bowers and Rajagopal (2002))
Preferred structure face-centered cube
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Phonons
In the LOFF phase translations and rotations are
broken
phonons
Phonon field through the phase of the condensate
(R.C., Gatto, Mannarelli Nardulli 2002)
Introduce
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Coupling phonons to fermions (quasi-particles)
trough the gap term
It is possible to evaluate the parameters of
Lphonon (R.C., Gatto, Mannarelli Nardulli 2002)
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Cubic structure
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Using the symmetry group of the cube one gets
Coupling phonons to fermions (quasi-particles)
trough the gap term
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we get for the coefficients
One can also evaluate the effective lagrangian
for the gluons in the anisotropic medium. For the
cube one finds
Isotropic propagation
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LOFF phase in CSO
Why the interest in the LOFF phase in QCD?
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In neutron stars CS can be studied at T 0
For LOFF state from dpF 0.75 DBCS
Orders of magnitude from a crude model 3 free
quarks
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• rn.m.is the saturation nuclear density
  .15x1015 g/cm3
• At the core of the neutron star rB 1015 g/cm3

Choosing m 400 MeV
Ms 200 dpF 25
Ms 300 dpF 50
Right ballpark (14 - 70 MeV)
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Glitches discontinuity in the period of the
pulsars.

• Standard explanations require metallic crust
  superfluide inside (neutrons)
• LOFF region inside the star might provide a
  crystalline structure superfluid CFL phase
• New possibilities for strange stars

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Outlook

• Theoretical problems Is the cube the optimal
  structure at T0? Which is the size of the LOFF
  window?
• Phenomenological problems Better discussion of
  the glitches (treatment of the vortex lines)
• New possibilities Recent achieving of
  degenerate ultracold Fermi gases opens up new
  fascinating possibilities of reaching the onset
  of Cooper pairing of hyperfine doublets.
  Possibility of observing the LOFF crystal?