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Charged kaon condensation in high density quark matter

We show that at asymptotically high densities the ``color-flavor-locked + neutral kaon condensate'' phase of QCD develops a {\it charged} kaon condensate through the Coleman-Weinberg mechanism. At densities achievable in neutron stars a charged kaon condensate forms only for some (natural) values of the low energy constants describing the low-lying excitations of the ground state.

It was realized a long time ago [1,2,3] that the attraction between two quarks close to the Fermi surface in high density strongly interacting matter leads to the formation of Cooper pairs of quarks and the spontaneous breaking of color symmetry. This phenomenon was more recently studied using Nambu-Jona-Lasinio models and renormalization group methods [4,5,6,7], instanton models [8] and, at asymptotically high densities where a weak coupling expansion is valid, perturbative QCD [9,10,11]. The main lessons learned in these studies were that i) the gap can be large, up to 100 MeV, ii) in the case of three-flavors of quarks with the same mass the ground state is the so-called color-flavor-locked (CFL) state , and chiral symmetry remains broken at arbitrarily high densities and iii) the low-lying excitations carry the quantum numbers of the pseudo-scalar octet familiar from the zero density case (plus two other scalars related to the spontaneous breaking of baryon number and axial charge) and iv) a form of electromagnetism survives: a combination of the photon and the eight gluon is not "Higgsed" and remains massless in the CFL phase. The equation of state is left nearly unchanged by the pairing. A number of electromagnetic and transport properties of quark matter are, however, sensitively dependent on what phase the system finds itself and what the low energy excitation are. This dependence provides a unique opportunity to study quark matter in the interior of neutron stars (or rule out its existence).
The fact that the low-lying excitation of the ground state are very similar to the low-lying excitations of the vacuum (pions, kaons,...) allows us to studied small perturbations around this ground state with the techniques of chiral perturbation theory. This brings about two advantages. At asymptotic high densities where perturbative QCD is valid it organizes perturbative calculations that would be very complicated otherwise. More importantly, it provides a method to systematically expand around the CFL phase in inverse powers of the density and/or the gap, which is particularly useful if the use of perturbative QCD is not legitimate. One of the important perturbations around the CFL phase is the presence of realistic quark masses. In the case of free quarks it is easy to determine the response of the system to these masses. As the mass of the strange quark increases towards its realistic value its density decreases in such a way that its Fermi energy equals the Fermi energy of the up quark (plus the Fermi energy of the electrons) and the weak decay s → u + e − + ν e becomes forbidden. Charge neutrality is guaranteed by the presence of electrons (we assume that the neutrinos leave the system). The interacting case may be qualitatively different. If the interactions are such that quarks of different flavors are paired, the change of flavor caused by flavor changing decays would result in two unpaired quarks, what is not energetically favorable. As a result, the system is rigid against small enough flavor asymmetries.
That is what happens in two-flavor QCD, where up and down quarks (of two of the three colors) are paired. An asymmetry in mass or chemical potential between the flavors causes little change in the ground state if they are small enough, the small change coming entirely from the unpaired quarks of the third color [12]. One might think that the same effect occurs in the three-flavor, CFL phase, since there all quarks are paired with quarks of different flavors. The CFL phase has, however, a another way of responding to mass asymmetries that costs little energy but is not available in the free or two-flavor system : it can condense mesons carrying strangeness, that are particularly light [13]. This was demonstrated in [13] on very general grounds and, in the case of weak coupling, through explicit computations of the response function to mass asymmetries.
For realistic values of the quark masses and densities it was found that the K 0 is the meson that condenses and this ground state will be referred to from now on as the "CF L + K 0 " phase. For more general asymmetries, including asymmetries on chemical potential present before weak equilibrium is achieved and/or neutrinos leave the system, a rich phase diagrams results, with kaonic, pionic, neutral and charged condensates forming with different values of the parameters [19]. The pattern of symmetry breaking caused by the K 0 condensation (in the isospin limit) is the same one found in the standard electroweak model SU I (2) × U Y (1) → U Q (1) (I =isospin,Y =hypercharge and Q = modified electric charge). Due to the lack of Lorentz symmetry, only two, and not three, Goldstone bosons are generated, one neutral and another charged [20,21]. There are stable, superconducting topological vortices [22,23] and almost stable non-topological domains walls in the "CF L + K 0 " phase [24]. In the CFL phase there is an equal number of quarks of the three flavors, and the system achieves electrical neutrality in the absence of any electrons, making it a perfect insulator [25]. The presence of the K 0 condensate, being neutral, does not change this situation. A charged kaon condensate however would change quark matter from a perfect insulator to a (electrical) superconductor. It is a generic feature of charged massless scalars that the strong long wavelength fluctuations of the gauge field lead to condensation of the scalar field (Coleman-Weinberg mechanism ([26])). In the CFL+K 0 phase there is one almost massless charged scalar field. Its mass comes from isospin breaking contributions coming from the quark mass difference and electromagnetic mass effects. In this paper we consider the competition between the isospin breaking mass terms and the fluctuations of the electromagnetic field in order to determine the fate of the charged kaons and of the possibility of a (electromagnetic) superconducting phase in quark matter.
In the absence of quark masses, the symmetry breaking pattern generated by diquark condensation in the CFL phase is [6] The electromagnetic U Q (1) is a subgroup of the chiral group SU L (3) × SU R (3) and the there is a surviving local U Q (1) "electromagnetism" in SU c+L+R (3) that is a combination of the photon and one of the gluons. The axial U A (1) is only an approximate symmetry of high density QCD due to the instanton suppression in the medium. This symmetry breaking pattern implies the existence of two singlet Goldstone bosons associated with the broken baryon number and axial symmetry, and an octet of pseudoscalars. The singlets will not play a role in our analysis and will be dropped from now on. At low (excitation) energies below the gap, QCD is equivalent to the most general theory of an octet of pseudoscalars and photons with the same symmetries of QCD. This theory has been extensively analyzed [27,28,29,30,31] recently. The leading terms of its lagrangian are where µ is the baryon chemical potential, ∆ is the gap, was computed in [27] where it was found that ǫ = 1 + 8 9π αµ 2 ∆ 2 . The magnetic permeability was argued to be unchanged from the vacuum value because the diquark condensate carries no spin.
Finally, the value of the gap is estimated to be around 50 − 100 MeV in phenomenological models at µ ≃ 500MeV and is given in perturbation theory by ∆ = 512(2/N F ) 5 /2π 4 /g 5 e For values of ∆ satisfying the minimum of the potential is found at [13] with It is convenient to use a modified Landau gauge fixing procedure, that is, we add a term The one-loop effective potential is then given by the sum of diagrams shown in Fig.(1): where we work in D = 4+η dimensions and λ is an arbitrary renormalization scale. The ultraviolet divergences are absorbed in the terms proportional to b and c in Eq. (1), which suggests that the natural values for these (renormalized) constants at the cutoff scale λ ≃ 2∆ are whereb andc are numbers of order one 1 . In addition we expect on physical grounds thatb < 0, what guarantees a positive electromagnetic contribution to the mass square for the mesons (the photon loop contribution vanishes in the Landau gauge). This estimate of the coefficient b agrees with the ones in [17,18] (where no attempt was made to count factors of 4π) and [19] (where no attempt was made to count factors of ǫ or v). Other electromagnetic terms not renormalized at one loop order are assumed to be suppressed. Also, the contribution from meson loops is proportional 2 and is strongly suppressed. Finally, the effective potential including the one photon loop correction becomes Whether the electromagnetic corrections computed above can modify the position of the minimum of the potential depends on the hierarchy assumed for the scales µ, ∆, etc.
It is probably instructive to compare the present situation with the simpler one of scalar QED.
The presence of a finite mass term will not destroy this minimum if, at φ =φ, it is smaller than the other terms. This condition translates into m 2 e − 1 2 + λ α 2 < αM 2 . In our case, we have mass terms of the order α∆ 2 , (m d − m u )m s ∆ 2 /µ 2 , and M 2 ∼ ∆ 2 . The role of the self-interaction is played by terms coming from the electromagnetic mass term (λ ∼ α∆ 2 /µ 2 ), the quark mass term (λ ∼ (m d − m u )m s ∆ 2 /µ 4 ) and the electromagnetic interaction term (λ ∼ α 2 ). Assuming Asymptotic limit: To analyze the simultaneous condensation of neutral and charged kaons we use the parameterization Due to charge and hypercharge invariance the potential does not depend on the phases θ 1 and θ 2 .
Isospin breaking effects create a dependence on x. The effective potential (8) becomes with χ = 2 sin 2 x(cos 2 x + (1 + cos( φ √ 2f )) sin 2 x) and we approximate cos( √ 2φ f ) ≃ 0. Let us now consider the limit µ → ∞. For values of µ such that (but still satisfying the condition for K 0 condensation in Eq. (2), that is violated only around 10 6 GeV) the first term in Eq. (11) is smaller than the second one and can be disregarded. Numerically, condition (12) is satisfied for µ > 3GeV. Let us momentarily put aside the second the term in Eq. (11) (electromagnetic mass). Minimizing in relation to x we find a solution At this value of x, the electromagnetic mass term can be disregarded compared to the one we For many, but not all, natural values ofb andc this condition is satisfied and the solution in Eq. (13) can be trusted. Unfortunately, the asymptotic values of these parameters in the limit µ → ∞ are not known (the computation ofc involves the calculation of four loops diagrams) and we cannot determine whether (14). In Fig.(2) we show, as an example, the effective potential for A better idea of the likelihood of charged kaon condensation at these densities can be perhaps obtained through the use of QCD models to estimate the unknown low energy constants in the density range inaccessible to perturbation theory.
We have considered the possibility of charged kaon condensation and (electromagnetic) superconductivity at high dense quark matter. At asymptotically high densities, where perturbative QCD applies and the question can be decided on first principles, a complicated computation of some low energy constants are necessary to settle the issue. We find however that for most natural values of these constants charged kaon condensation indeed occurs. At lower densities the situation is the opposite. For most reasonable values of the low energy constants the quark mass effects overwhelm the electromagnetic effects and there is no K + condensation.