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

2Introduction  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 mass m <    Two  most interesting cases: N f = 2, 3.  Due to asymptotic freedom quarks are almost free at high density and we expect difermion condensation in the color channel 3 *.

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

4 Only possible pairings LL and RR Favorite state for N f = 3, CFL (color-flavor locking) (Alford, Rajagopal & Wilczek 1999) Symmetry breaking pattern For  m u, m d, m s

5 Why CFL?

6 What happens going down with  ? If  << m s, we get 3 colors and 2 flavors (2SC) However, if  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.)

7 Difficulties with lattice calculations  Define euclidean variables:  Dirac operator with chemical potential  At   Eigenvalues of D(0) pure immaginary  If eigenvector of D(0), eigenvector with eigenvalue -  If eigenvector of D(0), eigenvector with eigenvalue -

8 For  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  and close to the critical line (Fodor and Katz)

9 Hierarchies of effective lagrangians Integrating out heavy degrees of freedom we have two scales. The gap  and a cutoff,  above which we integrate out. Therefore: two different effective theories, L HDET and L Golds

10  L HDET is 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.  L Golds describes the low energy modes ( E<<  ), as Goldstone bosons, ungapped fermions and holes and massless gauge fields, depending on the breaking scheme.

11 Effective theory at the Fermi surface (HDET) Starting point: L QCD at asymptotic  QCD,

12 Introduce the projectors: and decomposing:  States   close to the FS  States   decouple for large 

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

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

15 Momentum integration for the light fields v1v1v1v1 v2v2v2v2 The Fourier decomposition becomes For any fixed v, 2-dim theory

16 In order to decouple the states corresponding to substituting inside L QCD and using Momenta from the Fermi sphere

17 Using: One gets easily the result. Proof:

18 Eqs. of motion:

19 At the leading order in  At the same order: Propagator:

20 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:

21 Decomposing one gets When integrating out the heavy fields

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

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

24 We define and neglect L int. Therefore

25 Nambu-Gor’kov basis

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

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

28 In the 2SC case 4 pairing fermions

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

30 Gap equation in QCD

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

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

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

34 implying in general In NJL case with cutoff 800 MeV, constituent mass 400 MeV and  400 MeV

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

36  Breaking of U(1) B makes the CFL phase superfluid  The CFL condensate is not gauge invariant, but consider Notice:  is gauge invariant and breaks the global part of G QCD. Also break

37 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:

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

39 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

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

41 In-medium electric charge The condensate breaks U(1) em but leaves invariant a combination of Q and T 8. CFL vacuum: Define: leaves invariant the ground state

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

43 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

44 In the 2SC case, new Q and B are conserved 1/20 -1/

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

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

47 Simple model: condensation energy Condensation if:

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

49 Effective lagrangians  Effective lagrangian for the CFL phase  Effective lagrangian for the 2SC phase

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

51 Since The number of NG fields is 8 of these fields give mass to the gluons. There are only 10 physical NG bosons corresponding to the breaking of the global symmetry (we consider also the NGB associated to U(1) A )

52 Better use fields belonging to SU(3). Define and transforming as The breaking of the global symmetry can be described by the gauge invariant order parameters

53 , d X, d Y are 10 fields describing the physical NG bosons. Also , d X, d Y are 10 fields describing the physical NG bosons. Also shows that  T transforms as the usual chiral field. Consider the currents:

54 Most general lagrangian up to two derivatives invariant under G, the space rotation group O(3) and Parity (R.C. & Gatto 1999)

55 Using SU(3) c gauge invariance we can choose: Expanding at the second order in the fields Gluons acquire Debye and Meissner masses (not the rest masses, see later)

56 Low energy theory supposed to be valid at energies << gap. Since we will see that gluons have masses of order  they can be decoupled

57 ~  -lagrangian we get the gauge invariant result: