Latest results. Pairing fermions with different momenta Neutrality and beta equilibrium Chromomagnetic instability Three flavors and the LOFF phase

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1 Latest results Pairing fermions with different momenta Neutrality and beta equilibrium Chromomagnetic instability Three flavors and the LOFF phase 1 1

2 What do we know about the ground state of the color superconducting phase of QCD? At asymptotic densities and T = 0, the ground state of QCD is the CFL phase (highly symmetric diquark condensate) Understanding the interior of CSO s Study of the QCD phase diagram at T~ 0 and moderate density (phenomenological handle?) Real question: does this type of phase persist at relevant densities (~5-6 r 0 )?

3 Pairing fermions with different Fermi momenta M s not zero Neutrality with respect to em and color Weak equilibrium no free energy cost in neutral singlet,! (Amore et al. 003) All these effects make Fermi momenta of different fermions unequal causing problems to the BCS pairing mechanism 3

4 Weak equilibrium makes chemical potentials of quarks of different charges unequal: d ueν μ -μ = μ d u e From this: μ = μ + Q μ i i Q and μ = -μ e Q N.B. m e is not a free parameter: neutrality requires: V Q = - = 0 μ e 4

5 Neutrality and β Non interacting quarks equilibrium μ = μ +μ d,s u e If the strange quark is massless this equation has solution N u = N d = N s, N e = 0; quark matter electrically neutral with no electrons 5

6 By taking into account M s d u u s e F F F F s μ p - p p - p M / 4μ d s μ F F e p - p Fermi surfaces for neutral and color singlet unpaired quark matter at the equilibrium and M s not zero. In the normal phase m 3 = m 8 = 0. 6

7 As long as dm is small no effects on BCS pairing, but when increased the BCS pairing is lost and two possibilities arise: The system goes back to the normal phase Other phases can be formed Notice that there are also color neutrality conditions V μ V = T = 0, = T = μ 3 8 7

8 The point dm = D is special. In the presence of a mismatch new features are present. The spectrum of quasiparticles is D E(p) = δμ ± (p -μ) + Δ E blocking region dm = dm = D dm > D p For dm < D, the gaps are D - dm and D + dm For dm = D, an unpairing (blocking) region opens up and gapless modes are present (relevant in astrophysical applications) gapless modes Energy cost for pairing Energy gained in pairing E(p) = 0 p = μ ± δμ - Δ begins to unpair δμ > Δ 8

9 The case of 3 flavors gcfl (Alford, Kouvaris & Rajagopal, 005) 0 ψ ψ 0 = Δ + Δ + Δ α β αβ1 αβ αβ3 al bl 1 ab1 ab 3 ab3 Different phases are characterized by different values for the gaps. For instance (but many other possibilities exist) CFL : 1 3 gsc : 0, gcfl: 3 1 9

10 Q ru gd bs rd gu rs bu gs bd ru gd bs rd gu rs bu gs bd D D 1 3 Gaps in gcfl : ds - pairing : us - pairing D : ud -pairing 10

11 Strange quark mass effects: Shift of the chemical potential for the strange quarks: M s μαs μαs - μ Color and electric neutrality in CFL requires Ms μ 8 = -, μ 3 = μ e = 0 μ The transition CFL to gcfl starts with the unpairing of the pair ds with (close to the transition) δμ ds s M = μ 11

12 It follows: M s μ D Energy cost for pairing Energy gained in pairing begins to unpair M s >Δ μ Calculations within a NJL model (modelled on onegluon exchange): Write the free energy: V(μ,μ,μ,μ,Δ ) 3 8 e i Solve: Neutrality Gap equations V V V = = = 0 μ μ μ e 3 8 V = 0 D 1 i

13 CFL # gcfl nd order transition at M s /m ~ D, when the pairing ds starts breaking Gap Parameters [MeV] ~ D D 3 D D 1 Energy Difference [10 6 MeV 4 ] CFL unpaired SC ~ D gcfl M /m [MeV] s ~ D gsc (Alford, Kouvaris & Rajagopal, 005) (D 0 = 5 MeV, m = 500 MeV) M S /m[mev] 13

14 gcfl has gapless quasiparticles, and there are gluon imaginary masses (RC et al. 004, Fukushima 005). m (M ) M m (0) M s , M s m M m (M ) m (0) Instability present also in gsc (Huang & Shovkovy 004; Alford & Wang 005) M s ,5 6, M s m 14

15 How to solve the chromomagnetic instability Gluon condensation. Assuming artificially <A m 3 > or <A m 8 > not zero (of order 10 MeV) this can be done (RC et al. 004). In gsc the chromomagnetic instability can be cured by a chromo-magnetic condensate (Gorbar, Hashimoto, Miransky, 005 & 006; Kiriyama, Rischke, Shovkovy, 006). Rotational symmetry is broken and this makes a connection with the inhomogeneous LOFF phase (see later). At the moment no extension to the three flavor case. 15

16 CFL-K 0 phase. When the stress is not too large (high density) the CFL pattern might be modified by a flavor rotation of the condensate equivalent to a condensate of K 0 mesons (Bedaque, Schafer 00). This occurs for m s > m 1/3 D /3. Also in this phase gapless modes are present and the gluonic instability arises (Kryjevski, Schafer 005, Kryjevski, Yamada 005). With a space dependent condensate a current can be generated which resolves the instability. Again some relations with the LOFF phase. No extension to the three flavor case. 16

17 Single flavor pairing. If the stress is too big single flavor pairing could occur but the gap is generally too small. It could be important at low m before the nuclear phase (see for instance Alford 006) Secondary pairing. The gapless modes could pair forming a secondary gap, but the gap is far too small (Huang, Shovkovy, 003; Hong 005; Alford, Wang, 005) Mixed phases of nuclear and quark matter (Alford, Rajagopal, Reddy, Wilczek, 001) as well as mixed phases between different CS phases, have been found either unstable or energetically disfavored (Neumann, Buballa, Oertel, 00; Alford, Kouvaris, Rajagopal, 004). 17

18 Chromomagnetic instability of gsc makes the crystalline phase (LOFF) with two flavors energetically favored (Giannakis & Ren 004), also there are no chromomagnetic instability although it has gapless modes (Giannakis & Ren 005). 18

19 Results about LOFF with three flavors Recent study of LOFF with 3 flavors within the following simplifying hypothesis (RC, Gatto, Ippolito, Nardulli & Ruggieri, 005) Study within the Landau-Ginzburg approximation. Only electrical neutrality imposed (chemical potentials m 3 and m 8 taken equal to zero). M s treated as in gcfl. Pairing similar to gcfl with inhomogeneity in terms of simple plane waves, as for the simplest LOFF phase. 3 I al bl I abi I I I = 1 = D ( x), D ( x) = D e I iq x 19

20 A further simplifications is to assume only the following geometrical configurations for the vectors q I, I=1,,3 (a more general angular dependence will be considered in future work) The free energy, in the GL expansion, has the form 3 I I 4 IJ 6 - normal = D I + D I + DID J + O( D ) I = 1 4 I J normal = - ( mu + md + ms )- me 1 1 with coefficients I, I and IJ calculable from an effective NJL four-fermi interaction simulating one-gluon exchange 0

21 1 1 M s D 0 = D BCS, mu = m - me, md = m + me, ms = m + me m I 4m dmi qi + dmi 1 4( qi -dm I = - 1 log log - - qi qi -dmi D0 =- 1 ( q, dm ) = I I I m I I 3m dn dm 4 ( q n + m - m )( q n + m - m ) 1 s d s u Others by the exchange : 1 3, 1 13, m s s q m m m d u 1

22 We require: = = = D q m I I e 0 At the lowest order in D I I = 0 = 0 q q I I since I depends only on q I and dm i we get the same result as in the simplest LOFF case: q = 1. I dm I In the GL approximation we expect to be pretty close to the normal phase, therefore we will assume m 3 = m 8 = 0. At the same order we expect D = D 3 (equal mismatch) and D 1 = 0 (ds mismatch is twice the ud and us).

23 Once assumed D 1 = 0, only two configurations for q and q 3, parallel or antiparallel. The antiparallel is disfavored due to the lack of configurations space for the up fermions. 3

24 (we have assumed the same parameters as in Alford et al. in gcfl, D 0 = 5 MeV, m = 500 MeV) D D 1 3 : ds - pairing : us - pairing D : ud -pairing D = 0, D = D 1 3 4

25 Comparison with other phases LOFF phase takes over gcfl at about 18 MeV and goes over to the normal phase at about 150 MeV (RC, Gatto, Ippolito, Nardulli, Ruggieri, 005) Confirmed by an exact solution of the gap equation (Mannarelli, Rajagopal, Sharma, 006) 5

26 No chromo-magnetic instability in the LOFF phase with three flavors (Ciminale, Gatto, Nardulli, Ruggieri, 006) Transverse masses Longitudinal masses M = M = M = M M = M 6 7 6

27 Extension to a crystalline structure (Rajagopal, Sharma 006), always within the simplifying assumption D 1 = 0 and D = D 3 ud D exp( iq r), us D exp( iq r) a a 3 3 a a The sum over the index a goes up to 8 q ia. Assuming also D = D 3 the favored structures (always in the GL approximation up to D 6 ) among 11 structures analyzed are CubeX Cube45z 7

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29 Conclusions Various phases are competing, many of them having gapless modes. However, when such modes are present a chromomagnetic instability arises. Also the LOFF phase is gapless but the gluon instability does not seem to appear. Recent studies of the LOFF phase with three flavors seem to suggest that this should be the favored phase after CFL, although this study is very much simplified and more careful investigations should be performed. The problem of the QCD phases at moderate densities and low temperature is still open. 9