Ginzburg-Landau approach to the three flavor LOFF phase of QCD

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1 Ginzburg-Landau approach to the three flavor LOFF phase of QCD R. Casalbuoni Dipartimento di Fisica, Università di Firenze, I Firenze, Italia and I.N.F.N., Sezione di Firenze, I Firenze, Italia R. Gatto Départ. de Physique Théorique, Université de Genève, CH-1211 Genève 4, Suisse N. Ippolito, G. Nardulli, and M. Ruggieri Università di Bari, I Bari, Italia and I.N.F.N., Sezione di Bari, I Bari, Italia (Dated: July 16, 2005) We explore, using a Ginzburg-Landau expansion of the free energy, the Larkin-Ovchinnikov-Fulde- Ferrell (LOFF) phase of QCD with three flavors, using the NJL four-fermion coupling to mimic gluon interactions. We find that, above the point where the QCD homogeneous superconductive phases should give way to the normal phase, Cooper condensation of the pairs u s and d s is still possible, but in the form of the inhomogeneous LOFF pairing. PACS numbers: Aw, Lg BARI-TH 515/05 I. INTRODUCTION At high quark density and small temperatures Quantum-Chromo-Dynamics (QCD) predicts Cooper pairing of quarks due to the existence of an attractive quark interaction in the color antisymmetric channel, see [1 3] and for reviews [4, 5]. At extreme densities the energetically favored phase is the Color-Flavor-Locking (CFL) phase, characterized by a spin 0 diquark condensate antisymmetric in both color and flavor [6]; at intermediate densities the situation is much more involved, because one cannot neglect the strange quark mass and the differences δµ in the quark chemical potentials induced by β equilibrium. Several ground states have been considered in the literature, from the 2SC phase [2], to the gapless phases g2sc [7] and gcfl [8]. The gapless phases are instable as shown by imaginary gluon Meissner masses (for g2sc see [9], for gcfl see [10] and [11]). This seems to be connected to the existence of gapless modes in these phases [12]. An instability is present also in the 2SC phase [9]. Though this phase has no gapless mode, imaginary gluon masses are present when the gap and δµ satisfy the condition / 2 δµ. Another superconductive state discussed in the literature is the Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) [13] phase. The relevance of this phase is based on the possibility that, for appropriate values of δµ, it can be advantageous for quarks to form pairs with non-vanishing total momentum: p 1 + p 2 = 2q 0, see [14, 15] and for a review [16]. As far as instability is concerned, the authors in [17] have shown that, with two flavors, the instability of 2SC implies that the LOFF phase is energetically favored. Moreover, in the LOFF phase with two flavors the gluon Meissner masses are real [18]. Thus far only the case of two quark species for the LOFF phase has been studied. This is not justified in QCD. At intermediate densities all the three quarks: u, d and s should be considered. The three flavor problem is however much more involved and difficult to work out. We present here a first attempt to study the three flavor LOFF phase of QCD. Our approach is based on a Ginzburg-Landau (GL) expansion of the free energy. Differently from the CFL phase, where quark matter is in β equilibrium while being also electrically and color neutral, here we should impose these conditions. We shall consider in the sequel only β-equilibrated and electrically neutral quark matter, while assuming that the color-chemical potentials vanish. This is an approximation we discuss below. II. THE GAP EQUATION To get the gap equation in the Ginzburg Landau approximation, we start with the Lagrangian density for three flavor ungapped quarks: L = ψ ( ) iα i D/ αβ ij M αβ ij + µ αβ ij γ 0 ψ βj. (1) M αβ ij = δ αβ diag(0, 0, M s ) is the mass matrix and D αβ ij = δ αβ δ ij + iga a Ta αβ δ ij ; µ ij αβ is a diagonal color-flavor matrix depending in general on µ (the average quark chemical potential), µ e (the electron chemical potential), and µ 3, µ 8,

2 related to color [8]. We do not require color neutrality and we work in the approximation µ 3 = µ 8 = 0, which might be justified by the results of ref. [8] for the gcfl phase that show that the values of µ 3 and µ 8 are in general small. Therefore in this paper µ αβ ij = (µδ ij µ e Q ij )δ αβ = µ i δ ij δ αβ (2) where Q is the quark electric-charge matrix. We treat the strange quark mass at the leading order in the 1/µ expansion; this corresponds to a shift in the chemical potential of the s quark: µ s µ s M s2. This is the same approximation used in Refs. [8, 10] for the study 2µ of the gcfl phase. Therefore: µ u = µ 2 3 µ e, µ d = µ µ e, µ s = µ µ e M s2 2µ. (3) Another approximation we employ is the High Density Effective Theory (HDET), see [19 21] and, for a review, [5]. Here one decomposes the quark momentum into a large component, proportional to µ, and a residual small component: p = µn + l; n is a unit vector and l is the small residual momentum. Moreover one introduces n-dependent fields ψ n and Ψ n by the Fourier decomposition dn ψ(x) = 4π ei µn x (ψ n (x) + Ψ n (x)) ; (4) ψ n and Ψ n correspond to positive and negative energy solutions of the Dirac equation. Substituting the expression (4) in the Eq. (1) one gets at the leading order in 1/µ dn ( ) L = 4π ψ n,iα iv D αβ ij + µ i δ αβ δ ij ψ n,βj, (5) where V µ = (1, n), Ṽ µ = (1, n) and µ i = µ i µ. It is convenient to change the basis for the spinor fields by defining ψ A = (ψ ur, ψ dg, ψ bs, ψ dr, ψ ug, ψ sr, ψ ub, ψ sg, ψ db ). This change of basis can be performed by unitary matrices F A, whose explicit expression can be found in Ref. [10]. To the Lagrangian in Eq. (5) we add a Nambu-Jona Lasinio four fermion coupling treated in the mean field approximation. The gap term in the resulting Lagrangian is conveniently treated by introducing the Nambu-Gorkov field χ A = 1 ( ) ψn 2 C ψ n (6) A so that the Lagrangian reads L = 1 2 A,B dn 4π ( ) de dξ (E ξ + µa ) δ AB AB (r) (2π)2 χ A AB (r) (E + ξ µ χ A) δ B (7) AB where E is the energy, ξ l n is the component of the residual momentum along n and satisfies: ξ < δ, with δ an ultraviolet cutoff. Moreover ( µ) A = ( µ u, µ d, µ s, µ d, µ u, µ s, µ u, µ s, µ d ). We assume the pairing ansatz with < ψ iα C γ 5 ψ βj >= 3 I (r) ɛ αβi ɛ iji (8) I=1 I (r) = I exp (2iq I r). (9) In other words, for each inhomogeneous pairing we assume a Fulde-Ferrell ansatz; 2q I represents the momentum of the Cooper pair. The gap matrix AB in (7) can be expressed in terms of the three independent functions 1 (r) 2 (r) 3 (r) describing respectively d s, u s and u d pairing. The explicit expression of AB can be found in [8, 10]. 2

3 3 To write down the gap equation it is useful to introduce the following components of the free quark propagator The quark propagator is the matrix [ S 11 0 ]AB δ AB, E ξ + µ A whose components satisfy the Gorkov equations S AB = [ S 22 0 ]AB δ AB. (10) E + ξ µ A ( ) S11 S 12 S 21 S 22 AB S 11 = S S 11 0 (r)s 21, S 21 = S 22 0 (r)s 11. (12) S 21 is the anomalous propagator involved in the gap equation. The wave vectors q I should be derived by minimizing the free energy. We will fix the norms q I by a minimization procedure. As to their directions, we will limit the analysis to four structures, choosing among them the one with the smallest value of the energy. The first structure has all q I along the positive z axis; Structure 2, 3, 4 have, respectively, q 1, q 2, q 3 along the positive z axis (the remaining two momenta along the negative z axis). This is obviously a limitation. It is justified by our final results, that show the existence of a range of values of the strange quark mass where the LOFF phase, even with these limitations, is favored in comparison with other QCD phases. The gap equation in the HDET formalism can be written as follows [5] AB(r) = i 3G V µ Ṽ ν 9 C,D=1 d n h AaCh DbB 4π (11) d3 l de (2π)3 2π S 21(E, l) CD g µν δ ab, (13) where S 21 is given in Eq. (12); in the above equation h DbB is a Clebsch-Gordan coefficient. It is expressed by the formula h DbB = Tr[F D T bf B ] in terms of the unitary matrices F A used to write the quark fields as in (6), i.e. in the basis A = 1,, 9. G is the Nambu-Jona Lasinio coupling constant, of dimension mass 2. In what follows, we shall rule out G introducing the value of the CFL gap parameter 0 as a measure of the strength of the interaction (see Eq. (18)). III. THE GINZBURG-LANDAU EXPANSION Performing the Ginzburg-Landau expansion of the propagator we get S 21 = S 22 0 S S 22 0 S 11 0 S 22 0 S O( 5) (14) I = Π I I + J J IJ I J 2 + O( 5), I = 1, 2, 3. (15) Let us comment on the functions Π I and J IJ appearing in this expansion. Π I are defined as follows: Π 1 = Π(q 1, δµ ds ), Π 2 = Π(q 2, δµ us ), Π 3 = Π(q 3, δµ ud ), with and δµ ud µ d µ u 2 = µ e 2, δµ us µ s µ u = µ e 2 2 M s2 4µ, δµ ds µ s µ d = M s2 2 4µ. (16) Π(q, δµ) = 1 + 2Gµ2 π2 ( 1 δµ 2q log q + δµ q δµ 1 ) 2 log 4(q2 δµ2) 0 2. (17) We note that Π is analogous to the function determining the behavior of the free energy in the GL approximation of the LOFF phase with two flavors. We have introduced the parameter 0 to get rid of the ultraviolet cutoff δ. It is defined by { 0 2δ exp π2 }. (18) 2 Gµ2

4 0 is equal to the CFL gap for M s = 0 and µ e = 0 in the weak coupling limit, with no sextet condensation. As for J IJ, we have, for the diagonal components: J 11 J 1 J(q 1, δµ ds ), J 22 J 2 J(q 2, δµ us ), J 33 J 3 J(q 3, δµ ud ), with The off-diagonal term J 12 is J 12 = 3 Gµ2 dn 2 π2 4π J(q, δµ) = Gµ2 2π2 1 q2 δµ2. (19) 1 (2q 1 n + µ s µ d ) (2q 2 n + µ s µ u ) ; (20) J 13 is obtained from J 12 in (20) by the exchange q 2 q 3 and µ s µ d ; J 23 is obtained from J 12 by the exchange q 1 q 3 and µ s µ u. 4 IV. THE FREE ENERGY Let us now consider the free energy Ω. It is obtained by integrating the gap equation. The result is 3 Ω = Ω n + α I 2 I2 + β I 4 I4 + β IJ 4 I2 J 2 + O( 6) (21) I=1 J I with Ω n = 3 12π2 (µ u4 + µ d 4 + µ s 4) µ e4 12π2 where the chemical potentials for quarks are defined in Eq. (3) and the coefficients are defined by (22) α I = 2 (1 Π I) G Electric neutrality is obtained by imposing the variational condition, β I = 2 J I G, β IJ = 2 J IJ G. (23) Ω µ e = 0, (24) which, together with the gap equations, gives, for each value of the strange quark mass, the electron chemical potential µ e and the gap parameters I. Moreover one should determine q I by searching for the energetically favored solution. This is a complex task as it would require the simultaneous solution of the previous equations (24) and (15) together with: 0 = Ω α I = I + I q I q I 3 J=1 J 2 β IJ q I, I = 1, 2, 3. (25) Moreover one should look for the most energetically favored orientations of the three vectors q I in space. A complete analysis is postponed to a future paper; as discussed above we have limited the analysis to the four structures characterized by all vectors q I parallel or antiparallel to the same axis. Even with this limitation we are able to prove that there exists a window of values of M s where the LOFF phase is favored in comparison with other phases of QCD, as will be seen below. As to the norms q I, since we work in the GL approximation, we can neglect the O( 2) terms in (25). As a consequence we simply get α I = 0, which, being identical to the condition for two flavors, gives the q I result q I = δµ I [13, 14]. V. DISCUSSION OF THE RESULTS Our results are summarized in Figs In Fig. 1 we give Ω LOF F Ω norm (in unit 10 6 MeV 4 ) as a function of M 2 s /µ (in MeV) for the four structures considered above. We see that, except for M s small, the most favored

5 structure is the number 4 (solid line), i.e. the one characterized by q 3 along the positive z axis and q 1, q 2 in the opposite direction. For M s small the favored LOFF phase would be characterized by the alignment labelled as 3, but in this region LOFF pairing is not favored in comparison to the CFL phase, as will be seen below. We also note that for M s sufficiently large the solutions 3 and 4 coincide. The results in this figure and in the subsequent ones are obtained for µ = 500 MeV (for µ = 400 MeV the results are qualitatively similar). The value of the CFL gap is fixed at 0 = 25 MeV. 5 Energy Difference [10 6 MeV 4 ] M 2 s/µ [MeV] FIG. 1: Free energy differences Ω LOF F Ω norm in units of 10 6 MeV 4 plotted versus M 2 s /µ (in MeV) for the four structures considered in the text. The results are obtained for µ = 500 MeV and 0 = 25 MeV. The dot-dashed curve is the result for structure 1; the long-dashed curve refers to the structure 2; the short-dashed curve to the structure 3 and the solid curve to the structure 4. In Fig. 2 we give the gaps I / 0 as functions of M 2 s /µ (in MeV). The discontinuity at M 2 s /µ 30 MeV corresponds to the change from the structure 3 to the structure 4. However, as discussed below, for these values of M 2 s /µ the favored phase is the CFL state, not the LOFF state. We also notice that, for M 2 s /µ larger than 30 MeV, 1 vanishes. For this reason above this threshold we have either u s or u d pairing; q 1 plays no role, which is why the structures 3 and 4, that would differ only for the role of q 1, have the same free energy. Ι / M 2 s/µ [MeV] FIG. 2: Gaps I / 0 as functions of M 2 s /µ (in MeV). Dashed line: 1 ; dot-dashed line: 2 ; solid line, 3. The discontinuity at M 2 s /µ 30 MeV corresponds to the change from the the structure 3 to the structure 4 (see text). In Fig. 3 we present results for the electron chemical potential µ e. This result shows that µ e vanishes for M 2 s /µ < 70

6 MeV. This is an indication that, below this threshold, quark matter is in β equilibrium and electrically neutral without electrons, a peculiar feature of the CFL phase. 6 µ e [MeV] M 2 /µ [MeV] s FIG. 3: The electron chemical potential µ e as a function of M 2 s /µ. Units are MeV. VI. COMPARISON AMONG DIFFERENT PHASES Finally in Fig. 4 we present comparison of different phases of QCD. In order to comment this figure, let us start assuming that all the other phases are stable, meaning that in some way it is possible to cure the instability due to the imaginary gluon masses. In this case, following the graph for decreasing values of Ms 2 /µ, we see that at about Ms 2 /µ = 150 MeV the LOFF phase has a free energy lower than the normal one. This is a first order transitions as it can be seen from Fig. 2, since the gap 3 is not zero at the transition. Then the LOFF state is energetically favored till the point where it meets the gcfl line at about Ms 2 /µ = 119 MeV. Again we have a first order transition from the LOFF phase to gcfl since all the gaps are different in the two phases (for the gcfl case, see [8]). Then the system stays in the gcfl phase up to about Ms 2 /µ = 48 MeV where it turns into the CFL phase via a second order transition (see [8]). However, if the gapless phases are unstable, then they should not be considered, and the LOFF phase is the stable phase from Ms 2 /µ = 150 MeV down to about Ms 2 /µ = 95 MeV where the LOFF line meets the 2SC line. Notice, that the 2SC phase line is the one obtained in Ref. [22]. This is defined by considering three quarks with a massive strange quark but with only u and d quarks paired and requiring electrical neutrality. This has the consequence that the electron chemical potential µ e is equal to Ms 2 /2µ and then the transition point between the g2sc and 2SC phases occurs at Ms 2 /µ 4 2SC 113MeV, since in [22] 2SC has been assumed 31 MeV at M s = 0. Therefore the 2SC phase here defined is slightly different from the canonical one, where the difference in the chemical potential of the two quarks does not depend on M s. Given this difference, if we extrapolate the results of Ref. [9] we expect the 2SC phase to have gluons with imaginary mass in the interval which goes from the transition point of g2sc and 2SC at Ms 2 /µ = 113 MeV down to 113/ 2 80 MeV. Therefore in this case the LOFF phase is the stable one from Ms 2 /µ 150 down to Ms 2 /µ 85 MeV where it goes to the CFL phase with a first order transition (this can be seen by comparing our gaps with the CF L 23 MeV at this value of Ms 2 /µ). We should also add that at the moment it is still unknown if the LOFF phase with three flavors suffers of chromomagnetic instabilities. This problem is left to future investigations. VII. CONCLUSION AND OUTLOOK In conclusions, we have explored in the framework of the Ginzburg-Landau expansion the LOFF phase of QCD in the very involved case of three flavors, using the NJL four-fermion coupling to mimic the gluon interactions. We have

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