Color Superconductivity: CFL and 2SC phases

Color Superconductivity: CFL and 2SC phases • Introduction • Hierarchies of effective lagrangians • Effective theory at the Fermi surface (HDET) • Symmetries of the superconductive phases

Introduction • Ideas about CS back in 1975 (Collins & Perry-1975, Barrois-1977, Frautschi-1978). • Only in 1998 (Alford, Rajagopal & Wilczek; Rapp, Schafer, Schuryak & Velkovsky) a real progress. • The phase structure of QCD at high-density depends on the number of flavors with massm <m. • Twomost interesting cases:Nf = 2, 3. • Due to asymptotic freedom quarks are almost free at high density and we expect difermion condensation in the color channel 3*.

Consider the possible pairings at very high density a, b color; i, j flavor; a,b spin • Antisymmetry in spin (a,b) for better use of the Fermi surface • Antisymmetry in color (a, b) for attraction • Antisymmetry in flavor (i,j) for Pauli principle

Only possible pairings LL and RR For m >> mu, md, ms Favorite state for Nf = 3, CFL(color-flavor locking) (Alford, Rajagopal & Wilczek 1999) Symmetry breaking pattern

Why CFL?

What happens going down with m? Ifm << ms, we get 3 colors and 2 flavors (2SC) However, if m is in the intermediate region we face a situation with fermions having different Fermi surfaces (see later). Then other phases could be important (LOFF, etc.)

Difficulties with lattice calculations • Define euclidean variables: • Dirac operator with chemical potential • At m = 0 • Eigenvalues of D(0) pure immaginary • If eigenvector of D(0), eigenvector with eigenvalue - l

For m not zero the argument does not apply and one cannot use the sampling method for evaluating the determinant. However for isospin chemical potential and two degenerate flavors one can still prove the positivity. For finite baryon density no lattice calculation available except for small m and close to the critical line (Fodor and Katz)

Hierarchies of effective lagrangians Integrating out heavy degrees of freedom we have two scales. The gap Dand a cutoff, d above which we integrate out. Therefore: two different effective theories, LHDET and LGolds

LHDETis the effective theory of the fermions close to the Fermi surface. It corresponds to the Polchinski description. The condensation is taken into account by the introduction of a mean field corresponding to a Majorana mass. The d.o.f. are quasi-particles, holes and gauge fields. This holds for energies up to the cutoff. • LGoldsdescribesthe low energy modes (E<< D), as Goldstone bosons, ungapped fermions and holes and massless gauge fields, depending on the breaking scheme.

Effective theory at the Fermi surface (HDET) Starting point: LQCD at asymptotic m >> LQCD,

Introduce the projectors: and decomposing: • States y+ close to the FS • States y- decouple for large m

Field-theoretical version: Choosing 4 d.of. Separation of light and heavy d.o.f.

Separation of light and heavy d.o.f. heavy light light heavy

Momentum integration for the light fields The Fourier decomposition becomes v1 v2 For any fixed v, 2-dim theory

In order to decouple the states corresponding to Momenta from the Fermi sphere substituting inside LQCD and using

Proof: Using: One gets easily the result.

Eqs. of motion:

At the leading order in m: At the same order: Propagator:

Integrating out the heavy d.o.f. For the heavy d.o.f. we can formally repeat the same steps leading to: Eliminating the E- fields one would get the non-local lagrangian:

Decomposing one gets When integrating out the heavy fields

Contribute only if some gluons are hard, but suppressed by asymptotic freedom This contribution from LhD gives the bare Meissner mass

HDET in the condensed phase Assume (A, B collective indices) due to the attractive interaction: Decompose

We define and neglect Lint . Therefore

Nambu-Gor’kov basis

From the definition: one derives the gap equation (e.g. via functional formalism)

Four-fermi interaction one-gluon exchange inspired Fierz using:

In the 2SC case 4 pairing fermions

G determined at T = 0. M, constituent mass ~ 400 MeV with For Similar values for CFL.

Gap equation in QCD

Hard-loop approximation For small momenta magnetic gluons are unscreened and dominate giving a further logarithmic divergence electric magnetic

Results: from the double log To be trusted only for but, if extrapolated at 400-500 MeV, gives values for the gap similar to the ones found using a 4-fermi interaction. However condensation arises at asymptotic values of m.

Symmetries of superconducting phases Consider again the 3 flavors, u,d,s and the group theoretical structure of the two difermion condensate:

implying in general In NJL case with cutoff 800 MeV, constituent mass 400 MeV and m = 400MeV

Original symmetry: anomalous broken to anomalous # Goldstones massive 8 give mass to the gluons and 8+1 are true massless Goldstone bosons

Notice: • Breaking of U(1)B makes the CFL phasesuperfluid • The CFL condensate is not gauge invariant, but consider S is gauge invariant and breaks the global part of GQCD. Also break

is broken by the anomaly but induced by a 6-fermion operator irrelevant at the Fermi surface. Its contribution is parametrically small and we expect a very light NGB (massless at infinite chemical potential) Spectrum of the CFL phase Choose the basis:

Inverting: for any 3x3 matrix g We get quasi-fermions

gluons Expected NG coupling constant but wave function renormalization effects important (see later) NG bosons Acquire mass through quark masses except for the one related to the breaking of U(1)B

NG boson masses quadratic in mqsince the approximate invariance and quark mass term: Notice: anomaly breaks through instantons, producing a chiral condensate (6 fermions -> 2), but of order (LQCD/m)8

In-medium electric charge The condensate breaks U(1)em but leaves invariant a combination of Q and T8. CFL vacuum: Define: leaves invariant the ground state

Integers as in the Han-Nambu model rotated fields new interaction:

The rotated “photon” remains massless, whereas the rotated gluons acquires a mass through the Meissner effect A piece of CFL material for massless quarks would respond to an em field only through NGB: “bosonic metal” For quarks with equal masses, no light modes: “transparent insulator” For different masses one needs non zero density of electrons or a kaon condensate leading to massless excitations

In the 2SC case, new Q and B are conserved

Spectrum of the 2SC phase Remember No Goldstone bosons (equal gap) Light modes:

2+1 flavors It could happen Phase transition expected The radius of the Fermi sphere decrases with the mass

Simple model: condensation energy Condensation if:

Notice that the transition must be first order because for being in the pairing phase (Minimal value of the gap to get condensation)

Effective lagrangians • Effective lagrangian for the CFL phase • Effective lagrangian for the 2SC phase

Effective lagrangian for the CFL phase NG fields as the phases of the condensates in the representation Quarks and X, Y transform as

Color Superconductivity: CFL and 2SC phases