The one-loop polarisation tensor and the propagation of ``in-medium'' photons
of colour superconductors in the 2SC and CFL phase is discussed. For a study of
thermal corrections to the low energy effective theory in the 2SC phase, a
classical transport theory for fermionic quasiparticles is invoked.Comment: 5 pages, talk given at the International Conference on "Statistical
  QCD", Bielefeld, August 26-30, 200

Litim, Daniel F.

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Daniel Litim (4461961)

Sussex Research Online

Transport theory and low energy properties of colour superconductors

D.F. Litim

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The one-loop polarisation tensor and the propagation of ``in-medium'' photons of colour superconductors in the 2SC and CFL phase is discussed. For a study of thermal corrections to the low energy effective theory in the 2SC phase, a classical transport theory for fermionic quasiparticles is invoked.The one-loop polarisation tensor and the propagation of ``in-medium'' photons of colour superconductors in the 2SC and CFL phase is discussed. For a study of thermal corrections to the low energy effective theory in the 2SC phase, a classical transport theory for fermionic quasiparticles is invoked

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1Transport theory and low energy properties of colour superconductorsDaniel F. LitimTheory Group, CERN, CH { 1211 Geneva 23, Switzerland.CERN-TH-2001-315The one-loop polarisation tensor and the propagation of \in-medium" photons of coloursuperconductors in the 2SC and CFL phase is discussed. For a study of thermal correctionsto the low energy eective theory in the 2SC phase, a classical transport theory forfermionic quasiparticles is invoked.1. IntroductionQuantum chromodynamics (QCD) under extreme conditions displays a very rich phasestructure. At low temperature and high baryonic density, quarks form Cooper pairs dueto attractive interactions amongst them. The diquark condensate modies the groundstate of QCD and leads to the phenomenon of colour superconductivity [1]. This phase istypically characterised by the Anderson-Higgs mechanism and an energy gap associated tothe fermionic quasiparticles. It is expected that a colour superconducting state of matteris realized in compact stars.For two light quark flavours the diquark condensate breaks the SUc(3) group down toSUc(2) (2SC phase) [1]. Five gluons acquire a mass, while three gluons remain masslessand exhibit connement. Furthermore, quarks of one colour remain ungapped, as they donot participate in the condensation. A global axial UA(1) is broken at asymptotically largedensities, leading to a pseudo Nambu-Goldstone mode, similar to the η0 meson. For threelight quark flavours the condensates lock the colour and flavour symmetry (CFL phase)[1]. Colour, chiral and baryon number symmetry are spontaneously broken. All gluonsbecome massive and all quarks are gapped. The light degrees of freedom are the Nambu-Goldstone bosons { eight associated to the breaking of chiral symmetry, one associated tothe breaking of baryon number symmetry, and, at asymptotically large densities, an extrapseudo Nambu-Goldstone boson associated to the breaking of UA(1) { which dominatethe long distance physics of the superconductor.In this contribution, we discuss results obtained in collaboration with C. Manuel in [2,3].First, we include electromagnetic interactions and study the one-loop photon polarisationtensor in colour superconductors [2]. Second, we discuss a semi-classical transport theoryfor quark quasiparticles in the 2SC phase [3]. This is a convenient starting point forboth the construction of low energy eective theories or the computation of transportcoecients of dense quark matter.22. Photons in colour superconductorsA diquark condensate breaks, apart from the colour and flavour symmetry, also the U(1)symmetry of electromagnetism [1]. However, the linear combination A˜ of the standardphoton and the eighth gluon remains massless,A˜ = − sin θ G8 + cos θ A , G˜8 = cos θ G8 + sin θ A , e˜ = e cos θ . (1)The eld A˜ plays the role of the \in-medium" photon in the superconductor, with thein-medium gauge coupling e˜. Hence, the usual U(1) charge generator Q is replaced bythe in-medium generator Q˜ [1]. Notice also that the rotation angle θ diers for the 2SCand CFL phases,cos θCFL = g (g2 + 43e2)−1/2 , cos θ2SC = g (g2 + 13e2)−1/2 . (2)In the CFL phase, the charged particles are the Goldstone bosons pi, K, four gluons, andfour quarks, all with Q˜-chargee˜. The condensates are Q˜-neutral. The spectrum containsthe same number of positively and negatively charged particles. In consequence, quarkmatter in the CFL phase is Q˜-neutral. This holds also in the presence of K0 condensates.CFL matter is also Q-neutral, even for a non-vanishing strange quark mass or chemicalpotential [4]. No further charge carriers are needed to make the system electrically neutral.In the 2SC phase, four gluons and four quarks have half integer charges 12 e˜, while theup quark which is not participating in the condensate has the Q˜-charge e˜. Additionalheavy strange quarks have the Q˜-charge −12 e˜. Again, the condensates are electricallyneutral. Q˜-neutrality of the full system requires additional negative charges, either as abackground of heavy strange quarks or electrons [1].The photon self energy, to one-loop order, is given by all one-loop diagrams with twoexternal photon lines and internal loops of charged particles. The nite parts of heavygluon loops are suppressed in the infrared, as are those from the light pions and kaons.Hence, photon polarisation eects are entirely dominated in the infrared limit by thequark loops [2]. A simple physical picture has emerged from the explicit computation ofthe photon polarisation tensor [2]. For intermediate photon momenta   p0, p  µ,the polarisation tensor due to gapped quarks reduces to its well-known hard-dense-loopcounterpart in the normal phase. This is easily understood. The photon wavelength islarger than the size of the Cooper pair. Hence, the photons scatter o the gapped quarksjust as they would do in the phase without condensation. They are Landau damped,static magnetic elds remain unscreened while static electric elds are eectively Debyescreened with a mass of the order  e˜µ, due to all charged quarks. In the 2SC phase,an additional contribution to the Debye mass arises for non-vanishing electron chemicalpotential. For photons in this momentum regime, and apart from numerical dierences forthe value of the Debye mass and the couplings, the 2SC and CFL phases look qualitativelythe same.The situation changes signicantly for photons with low momenta p0, p  . Here,their Compton wave length is too large to resolve the condensates. The condensatesare electrically neutral, but they have an electrical dipole moment and hence modifythe dielectric constant ˜ of the medium. Condensates with vanishing spin and angularmomentum (J = 0) have no magnetic moment. Therefore, the magnetic permeability3remains unchanged. To leading order in p0, p  , one nds for the longitudinal andtransversal components of the polarisation tensor gap arising from the gapped quarks [2]gapL (p0,p) = −κ˜ p2 , gapT (p0,p) = −κ˜ p20 , κ˜ =c18pi2e˜2µ22. (3)The coecient c counts the charges squared of all gapped quarks participating in thecondensation: c2SC = 1 and cCFL = 4. The dielectric constant of the medium is˜ = 1 +c18pi2e˜2µ22> 1 . (4)For realistic values of the gauge coupling, the gap and the quark chemical potential, κ˜ isof order one. It becomes very large for asymptotic large densities and xed e˜.In the CFL phase, the low momentum limit of the polarisation tensor is solely given by(3). Due to the absence of ungapped quarks, electric elds are no longer Debye screenedin the infrared limit. Since all Q˜-charged hadronic excitations in the CFL phase acquirea gap for non-vanishing quark masses, the photon cannot scatter if its energy is belowthe energy of the lightest charged mode: CFL matter is a transparent insulator [1]. Thepolarisation of the medium implies that photons propagate with a velocity smaller thanthe velocity of light in vacuum. The refraction index of CFL matter with the normalphase is n˜ = ˜1/2 > 1.In the 2SC phase, the low momentum limit of the polarisation tensor is given by (3)plus an additional contribution from the charged ungapped quark and, possibly, electrons.In the static limit, they combine into L = −(m˜2 + κ˜ p2) with a Debye mass due to theungapped quark and the electrons. Therefore, photons still scatter from the chargedfermions, and Debye screening, though now with a dierent Debye mass, persists even forlow momenta: 2SC matter is an opaque conductor.To conclude, we have seen that the propagation of photons in a colour superconductingphase is very dierent from the normal phase, and also dierent for the dierent coloursuperconducting phases. This result is an important input for a quantitative study oftransport coecients like the electrical or thermal conductivity of the medium. Althoughour results are based only on a one-loop computation, the corrections to the dielectricconstant are small for realistic values of the parameters. Therefore, one may expect thathigher loop orders do not change the qualitative picture discussed here.3. Transport in colour superconductorsNext, we study thermal corrections to the low energy physics of a two-flavour coloursuperconductor [3]. For T = 0, an infrared eective theory has been given recently forthe massless SU(2)c gauge elds [5]. The gauge eld dynamics diers from the vacuumtheory, because the condensates, although neutral with respect to the unbroken SU(2),polarise the medium [6]. This is fully analogous to what has been found above for thephoton interactions. For momenta q  , the eective theory isST=0eff [A] =∫d4x(2Ea Ea − 12Ba Ba), (5)where Eai  F a0i and Bai  12ijkF ajk are the SU(2) electric and magnetic elds. Here, = 1 + g2µ2/(18pi22) is the colour dielectric susceptibility [5]. The colour magnetic4susceptibility remains unchanged. This theory is conning with a highly reduced scale ofconnement 0QCD   exp (−2p2pi11µg∆). Due to asymptotic freedom, it is expected thatperturbative computations are reliable for energy scales larger than 0QCD.At non-vanishing temperature, thermal excitations modify the low energy physics. Thecondensate melts at the critical temperature Tc  0.570, where 0 is the gap at vanishingtemperature. We restrict the discussion to temperatures within 0QCD  T < Tc. In thisregime, the main contribution stems from the thermal excitations of the gapped quarks.Those of the massless gauge elds are of the order g2T 2 and subleading for suciently largeµ, and the gapless quarks and the η meson do not couple to the SU(2) gauge elds. Weintroduce a transport equation for the quark quasiparticles, which are described in termsof an on-shell one-particle phase space density f(x,p, Q), xµ = (t,x). The distributionfunction depends on time, the phase space variables position x, momentum p, and onSU(2) colour charges Qa, with the colour index a = 1, 2 and 3. The quasiparticles carrySU(2) colour charges simply because the constituents of the condensate do. Using naturalunits kB = h = c = 1, the on-shell condition for massless quarks mq = 0 relates the energyof the quasiparticle excitation to the chemical potential and the gap as p0  p, withp =√(p− µ)2 + 2(T ) , vp  ∂p∂p=jp− µj√(p− µ)2 + 2(T )p^ . (6)Hence, the quasiparticle velocity vp depends on both the chemical potential and the gap.In the presence of the gap, their propagation is suppressed, jvpj < 1. The one-particledistribution function obeys the transport equation[Dt + vp D− gQa (Ea + vp Ba) ∂∂p]f = C[f ] . (7)Here, we have introduced the short-hand notation Dµf  [∂µ − gabcQcAµb ∂Qa ]f for thecovariant derivative acting on f . The rst two terms on the left-hand side of (7) combine toa covariant drift term vµp Dµ, where vµp = (1,vp) and Dµ = (Dt,D). The terms proportionalto the colour electric and magnetic elds provide a force term. The right-hand side of (7)contains an unspecied collision term C[f ]. Eq. (7) is very similar to transport equationsused earlier for hot or dense QCD in the normal phase [7,8]. The main dierence isthe non-trivial dispersion relation (6), which leads to modied Wong equations in thepresent case. The thermal quasiparticles carry a SU(2) charge, and provide a SU(2)colour current. We introduce the colour densityJa(x,p) = g∫dQQaf(x,p, Q) , (8)where an implicit sum over species or helicity indices on f is understood. The colour mea-sure obeys∫dQQa = 0 and∫dQQaQb = C2δab (C2 =12 for quarks in the fundamental) [8].The induced colour current of the medium follows from (8) as Jµa (x) =∫ d3p(2pi)3vµp Ja(x,p).It is covariantly conserved.As an application of (7), we study the collisionless dynamics C[f ] = 0 close to thermalequilibrium and to leading order in the gauge coupling. Consider the distribution functionf = f eq + gf (1) where f eq = (exp p/T + 1)−1 is the fermionic equilibrium distribution5function and gf (1) a slight deviation from equilibrium. Expanding the transport equation(7) in g, and taking the two helicities per quasiparticle into account, we nd[Dt + vp D] J(x,p) = g2Nf vp  E(x) dfeqdp. (9)This equation can be solved for J(x,p) and gives the full colour current as a functionalof the gauge elds only, J(x) = J [A](x) [3]. It denes the "hard superconducting loop"eective action ΓHSL[A] through J [A] = −δΓHSL[A]/δA, in full analogy to the derivationof the hard thermal loop eective theory within classical transport theory [8]. Hence, thelow energy eective theory for modes with q   in the 2SC phase at nite temperatureis STeff [A] = ST=0eff [A] + ΓHSL[A] to leading order in g.Consider the thermal polarisation tensor which follows from J [A] [3]. It describesLandau damping and Debye screening of the SU(2) gauge elds in the 2SC phase. Forlow momenta, it agrees with the ndings of [6], based on a eld-theoretical derivation.In contrast to the normal phase, the dispersion relations for longitudinal and transversegluons are modied. For example, the Debye mass is m2D = (2/pi2T )1/2g2µ2 exp−/Tin the limit of large density and low temperature. In the general case, the Debye masscan only be computed numerically. An approximate expression for the polarisation tensorcould be given in terms of an eective mean squared velocity v for the quasiparticles,similar to the approximation used in [9] for the normal phase.In conclusion, we have discussed a very simple classical transport equation for quarkquasiparticles in the 2SC phase. To leading order in g, it leads to the same result as afull eld theoretical computation. Due to its simplicity, the present approach seems tobe a good starting point for the computation of transport coecients in the 2SC phase,like the colour conductivity. Here, the techniques of classical transport theory can beexploited [7,8,10]. It would also be interesting to provide a similar approach for the CFLphase, where the low energy degrees of freedom are substantially dierent.Acknowledgements: I thank C. Manuel for collaboration, and the organisers for a stim-ulating conference and nancial support. This work has been supported by the EuropeanCommunity through the Marie-Curie fellowship HPMF-CT-1999-00404.REFERENCES1. See K. Rajagopal and F. Wilczek, hep-ph/0011333 for a review and relevant literature.2. D. F. Litim and C. Manuel, Phys. Rev. D 64 (2001) 094013 [hep-ph/0105165].3. D. F. Litim and C. Manuel, Phys. Rev. Lett. 87 (2001) 052002 [hep-ph/0103092].4. K. Rajagopal and F. Wilczek, Phys. Rev. Lett. 86 (2001) 3492 [hep-ph/0012039].5. D. H. Rischke, D. T. Son and M. A. Stephanov, Phys. Rev. Lett. 87 (2001) 062001.6. D. H. Rischke, Phys. Rev. D62 (2000) 054017 [nucl-th/0003063].7. U. W. Heinz, Phys. Rev. Lett. 51 (1983) 351; Annals Phys. 161 (1985) 48.8. D. F. Litim and C. Manuel, hep-ph/0110104.9. E. Braaten and D. Segel, Phys. Rev. D 48 (1993) 1478 [hep-ph/9302213].10. D. F. Litim and C. Manuel, Phys. Rev. Lett. 82 (1999) 4981 [hep-ph/9902430]; Nucl.Phys. B 562 (1999) 237 [hep-ph/9906210].

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Transport theory and low energy properties of colour superconductors

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