We show that the pseudogap of the quark density of states is formed in hot
quark matter as a precursory phenomenon of the color superconductivity on the
basis of a low-energy effective theory. We clarify that the decaying process of
quarks near Fermi surface to a hole and the diquark soft mode (qq)_{soft} is
responsible for the formation of the pseudogap. Our result suggests that the
pseudogap is a universal phenomenon in strong coupling superconductors.Comment: Introduction is largely rewritten and minor changes are made in other
  parts of the text. Some referenes with comments are added. Numerical errors
  in the figures are corrected. To appear in Phys. Rev. 

D. J. Thouless

M. Iwasaki

M. Kitazawa

T. Koide

T. Kunihiro

Y. Nemoto

Y. Yanase

English

arXiv

Crossref

Pseudogap of color superconductivity in heated quark matter

Kitazawa, M

Koide, T

Kunihiro, T

Nemoto, Y

Kyoto University Research Information Repository

Title Pseudogap of color superconductivity in heated quark matterAuthor(s)Kitazawa, M; Koide, T; Kunihiro, T; Nemoto, YCitationPHYSICAL REVIEW D (2004), 70(5)Issue Date2004-09URL http://hdl.handle.net/2433/50052RightCopyright 2004 American Physical SocietyType Journal ArticleTextversionpublisherKyoto UniversityPseudogap of color superconductivity in heated quark matterM. Kitazawa,1 T. Koide,2 T. Kunihiro,2 and Y. Nemoto31Department of Physics, Kyoto University, Kyoto 606-8502, Japan2Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan3RIKEN BNL Research Center, BNL, Upton, New York 11973, USA(Received 3 September 2003; published 14 September 2004)We show that the pseudogap of the quark density of states is formed in hot quark matter as aprecursory phenomenon of the color superconductivity on the basis of a low-energy effective theory. Weclarify that the decaying process of quarks near Fermi surface to a hole and the diquark soft modeqqsoft is responsible for the formation of the pseudogap. Our result suggests that the pseudogap is auniversal phenomenon in strong coupling superconductors.DOI: 10.1103/PhysRevD.70.056003 PACS numbers: 25.75.Nq, 11.15.Ex, 12.38.AwIt is an intriguing problem to examine how rich is thephase structure of the high-density () QCD matter atvanishing or moderate temperature (T). It is now believedthat quark matter at an extremely high density undergoesa Cooper instability leading to the color superconductiv-ity (CS) [1–4]: As might be ensured by the asymptotic-free nature of QCD, the quark matter at extremely highdensities is usually treated as a Fermi liquid, and themean-field (MF) theoretical approach a` la BCS theoryis employed; see, for reviews on recent exciting develop-ment, [5].Various complications are, however, expected on thenature of quark matter at intermediate baryon densities orchemical potential  where the strong coupling nature ofQCD may show up and invalidate the MF approximation[2]: The strong coupling may make the so-calledGinzburg region so wide that precursory fluctuations ofthe quark-pair field can have a prominent strength andmay give rise to physically significant effects above thecritical temperature (Tc) [6]. In this paper we will showthat the fluctuation effects of the CS can be so significantthat such quark matter may share some basic propertieswith the cuprates of the high-Tc superconductivity(HTSC), having a non-Fermi liquid nature above Tc.One of the typical non-Fermi liquid properties of thecuprates is the existence of the pseudogap, i.e., an anoma-lous depression of density of states (DOS) N! as afunction of the Fermion energy ! around the Fermisurface above Tc. Although the mechanism of the pseu-dogap associated with the HTSC is still controversial,precursory fluctuations of the pair field and the quasi-two-dimensionality of the system seem to be basic ingre-dients to realize the pseudogap [7,8]. In this work, weshall show that the pseudogap of the quark density ofstates exists as a precursory phenomenon of the CS in aconsiderable range of T above Tc even in the three-dimensional system. We calculate the quark self-energyincorporating the preforming pair field of the CS for thefirst time in the T-matrix approximation (T-approximation[7,9,10]), where the amplitude fluctuations are assumed tobe dominant over the phase fluctuations of the pair field[8,11]. Our result shows that the pseudogap may appearalso in a relativistic system solely owing to the amplitudefluctuations, and suggests that the pseudogap phenomenonis universal in strong coupling superconductors, irrespec-tive of the dimensionality as suggested from the study ofnuclear matter [9]. Our results should also provide aninsight into the physics of proto-neutron stars andheavy-ion collisions as well, where the  is relativelylow and the effect of finite T plays an important role.To describe a system at relatively low T and , it isappropriate to adopt a low-energy effective theory ofQCD [3,12,13]. Here we employ a simplified version ofthe instanton-induced interaction in the two-flavor caseknown as the Nambu-Jona-Lasinio model [14,15] withthe scalar-diquark interaction in the chiral limit,L   i@6  GCXA  i52A C  Ci52A GS   2    i5 ~ 2; (1)where Cx  C  Tx, withC  i20 being the chargeconjugation operator. Here, 2 and A mean the antisym-metric flavor SU(2) and color SU(3) matrices, respec-tively. The coupling GS and the three-dimensionalmomentum cutoff   650 MeV are determined so asto reproduce the pion decay constant f  93 MeV andthe chiral condensate h   i  250 MeV3 in the chirallimit [13]. We choose GC so as to reproduce the phasestructure calculated using the instanton-induced interac-tion [12], i.e., GC  3:11 GeV2 [13]. We neglect thegluon degrees of freedom, especially their fluctuation,which is known to make the CS phase transition firstorder in the weak coupling region [2,16]; notice that theCS is a type I in this regime. However, as is emphasized inRef. [17], nothing definite is known on the characteristicsof the CS in the intermediate density region. In this work,simply assuming that the fluctuations of the pair fielddominates that of the gluon field as is the case for type IIcolor superconductors, we examine the effects of thePHYSICAL REVIEW D, VOLUME 70, 0560030556-2821=2004=70(5)=056003(5)$22.50 70 056003-1  2004 The American Physical Societyprecursory fluctuations of the diquark-pair field on thequark sector in the T-approximation.The DOSN! is calculated from the spectral functionAk; ! of a single quark, which is defined through thespectral representation of the retarded Green function ofthe quark field:GRk; ! Zd!0Ak; !0!!0  i ; (2)and accordingly, Ak; !  1=  ImGRk; ! 1=  GR  0GRy0=2i. From the rotational and par-ity invariances, the spectral function has the followingmatrix structure: Ak; !  0k; !0  vk; !k^  sk; !, where k^  k=jkj and  (  0; v; s) stillhave color and flavor indices. Since the quark number isgiven by N  R d3xh  0 i, the DOS is solely given by0k; !,N!  4Z d3k23 Trc;f0k; !; (3)with Trc;f denoting the trace over color and flavor indices.The GR in Eq. (2) is given by the analytic continuationof the imaginary-time (Matsubara) Green function G,which obeys the following Dyson-Schwinger equation:G k; !n  G0k; !nf1 ~k; !nGk; !ng; (4)where G0k; !n and ~k; !n denote the free Greenfunction and the self-energy in the imaginary time, re-spectively. In the normal phase, G0 is reduced toG0k; !n  i!n 0  k  1 with the Mat-subara frequency !n  2n 1T for fermions.As was shown in [6], the fluctuating diquark-pair fielddevelops a collective mode (the soft mode of the CS) at Tabove but in the vicinity of Tc, in accordance with theThouless criterion [18]. Our point in this work is thatthe soft mode in turn contributes to the self-energy of thequark field, thereby it can modify the DOS so much togive rise to a pseudogap. The quark self-energy ~ owingto the soft mode may be obtained by the infinite series ofthe ring diagrams shown in Fig. 1:~k; !n  TXn1Z d3k123~k k1; !n !n1G0k1; !n1; (5)~k; n  8GC1GCQk; n1; (6)with the lowest particle-particle correlation functionQk; n [6] and n  2nT being the Matsubara fre-quency for bosons. Notice that the thin quark lines inFig. 1 are the free Green function, so we have taken theso-called non-self-consistent approximation, on whichwe shall make a comment later.Inserting Eqs. (5) and (6) into Eq. (4) and performingthe analytic continuation to the upper half of the complexenergy plane, we obtain the retarded Green function,GRk; !  G10 k; ! i Rk; !1, withRp; p0 Z d3q23Z d!2ImRp q; !! p0  Eq  iEq0  q  2EqtanhEq 2T coth!2TEq ! Eq; (7)where Eq  jqj, Rk; !  ~k; !nji!n!i, andRk; !  ~k; njin!i. The matrix structureof the self-energy is the same as the spectral fun-ction, Rk; !  0k; !0 vk; !k^  sk; !  0  , where   10  k^=2 denotes the projection operators onto the posi-tive and negative energy states, and accordingly  0  v represents the self-energies of the particles andantiparticles, respectively; notice that sk; !  0 in thechirally restored phase in the chiral limit. Then it can beshown that 0 is expressed as a sum of the positive- andnegative-energy parts: 0k; !  1=2 PIm=fRk; !2  Im2g, where Rk; !  ! jkj  Rek; !.A remark is in order here: The Thouless criterionmentioned above tells us that the denominator of R, 1GCQ0; 0, vanishes at T  Tc because of the self-consistency condition for the diquark condensate at T Tc [6], where Qk; !  Qk; njin!i.For the numerical calculation, we employ the followingcutoff scheme [14,19]: First we notice that the imaginarypart of Q is free from ultraviolet divergences. Thereforewe first evaluate the imaginary part exactly without in-troducing a cutoff, and then use the dispersion relationwith the imaginary part just obtained to calculate the realpart introducing a three-dimensional cutoff . Theimaginary part is nicely found to have the followingcompact form:FIG. 1. The Feynman diagrams representing the quark Green function. The thin lines represent the free propagator G0, while thebold ones represent the full propagator G.M. KITAZAWA, T. KOIDE, T. KUNIHIRO, AND Y. NEMOTO PHYSICAL REVIEW D 70 056003056003-2ImQk; !  NfNc  1T ! 22  k22k!j! 2j  k lncosh! k=4Tcosh! k=4T!j! 2j  k ln 1 e!k=2T1 e!k=2T: (8)We remark that each term has the respective interpreta-tion in terms of the kinetic processes [6]. We emphasizethat the compact expression of ImQk; ! above ex-tremely improves the efficiency of the numerical calcu-lations. Nevertheless, the calculations still involvemultiple integrations for obtaining R and hence 0.Since 0k; ! for!> is well approximated solelyby the positive-energy part, we first see the characteristicproperties of . Figure 2 shows a typical behavior of thereal and imaginary parts of  with k  kF at  400 MeV and the reduced temperature "  T Tc=Tc  0:01; we remark that Tc  40:04 MeV in thepresent case [6]. From the figure, one can see that Reshows a rapid increase around the Fermi energy !  0.The quark dispersion relation !  !k therefore alsoshows a similar behavior around the Fermi surface;!kis the solution of Rk; !  0. Hence the density ofstates proportional to @!=@k1 becomes smallernear the Fermi surface, which suggests the existence ofa pseudogap, provided that the imaginary part Im isneglected, which will be discussed shortly. One can alsosee that !k  kF ’ k since Re at !  0 isvanishingly small, which will be found important forthe pseudogap formation around the Fermi surface.A numerical calculation shows that as the momentum kis varied from kF, the peak of jImj at !  0 seen inFig. 2 moves along !  k [20]. This means thatthe quasiparticles with this energy are dumped modes.Figure 1 tells us that Im describes a decay process of aquark to a hole and a diquark, q ! h qq, where thehole is on shell with a free dispersion relation !h  jkhj. The essential point for the pseudogap formation isthat the above process is enhanced when the diquark (qq)makes a collective mode, which we have emphasized isthe case; the diquark soft mode qqsoft has a prominentstrength at vanishingly small energy !s and momentumks near Tc. Owing to the energy-momentum conserva-tion, the energy momentum of the decaying particle!p;kp should satisfy !p !h  !s ’ 0 and kp kh  ks ’ 0. It means that when the decaying particlehas almost the same energy as the free quark,jImk; !j has the largest value.The spectral function 0k; ! is shown in Fig. 3, at thesame  and " as those in Fig. 2. One can see two familiesof peaks around !  !k  k and !  k,which correspond to the quasiparticle peaks of the quarksand antiquarks, respectively. A notable point is that thequasiparticle peak has a clear depression around !  0,i.e., the Fermi energy. The mechanism for the depressionis easily understood in terms of the characteristic prop-erties of the self-energy mentioned above: In fact,RkF;! ’  kF ’ 0 and jImk; !j becomes largewhen !  k. Thus 0k ’ kF;! ’ 0 ’1=2ImkF;! ’ 0, which is found to besuppressed.Integrating 0, one obtains the DOSN!: Fig. 4 showsthe DOS at   400 MeV and various values of thereduced temperature " together with that of the freequark system, N0!. As anticipated, one can see a re-markable depression of N!, i.e., the pseudogap, aroundthe Fermi energy !  0; N!=N0!j!0 ’ 0:64 at " 0:01. One sees that the smaller ", the more remarkable therate of depression. The clear pseudogap structure surviveseven at "  0:05. One may thus conclude that there is apseudogap region within the normal phase above Tc up to-30-20-10 0 10-200 -100  0  100  200Σ -(k=k F,ω) [MeV]ω [MeV]Im Σ-Re Σ-FIG. 2. The self-energy  with k  kF at   400 MeVand "  T  Tc=Tc  0:01. One observes a peak in Im at!  0 and a rapid increase of Re at the same !.-400-200 0 200ω 100 200 300 400 500k 00.010.020.03ρ0(ω,k)1000-100 300 400 500 0 0.005 0.01 0.015FIG. 3. The spectral function 0 at   400 MeV and " 0:01. The upper figure is an enlargement of the one near theFermi surface. The peaks at !  k and !  kcorrespond to the quark and antiquark quasiparticles, respec-tively. Notice that there is a depression around !  0, which isresponsible for the pseudogap formation.PSEUDOGAP OF COLOR SUPERCONDUCTIVITY IN . . . PHYSICAL REVIEW D 70 056003056003-3T  1:05–1:1Tc at   400 MeV, for instance. Anumerical calculation shows that " dependence of thewidth of the pseudogap region hardly changes for320<< 500 MeV.We notice that the pseudogap region obtained in thepresent work is more than 1 order of magnitude wider inthe unit of " than in the nuclear matter [9] where the clearpseudogap is seen up to "  0:0025. This is just a reflec-tion of the strong coupling nature of the QCD at inter-mediate density region. Our result obtained for a three-dimensional system tells us that a considerable pseudogapcan be formed without the help of the low dimensionalityas in the HTSC and that the pseudogap phenomena ingeneral may be universal in any strong couplingsuperconductivity.In this paper we have found that the pseudogap can beformed as a precursory phenomenon of the CS in a ratherwide region of T above Tc. This may imply that quarkmatter shares some basic properties with the cuprates ofthe HTSC. It should be noted that our work is the firstcalculation to show the formation of the pseudogap in therelativistic framework.In the present work, we have employed the non-self-consistent T-approximation, while the self-consistent ap-proximation apparently seems better. However, it may notbe the case [22]; higher-order terms with the vertexcorrections, which are usually discarded in the self-consistent approximation, cancel with each other, whichmeans that the lowest-order calculation such as ours ismore reliable than the self-consistent one [22].As a future problem, one should consider how to ob-serve the pseudogap in heavy-ion collisions or proto-neutron stars. It is also interesting to see what wouldhappen if the phase transition of the CS is strong firstorder.We thank K. Iida and D. Rischke for their valuablecomments and for drawing our attention to [17,21],respectively. T. Kunihiro is supported by Grant-in-Aidefor Scientific Research by Monbu-Kagaku-sho(No. 14540263).[1] B. C. Barrois, Nucl. Phys. B129, 390 (1977).[2] D. Bailin and A. Love, Phys. Rep. 107, 325 (1984).[3] R. Rapp, T. Schafer, E.V. Shuryak, and M. Velkovsky,Phys. Rev. Lett. 81, 53 (1998); M. G. Alford, K.Rajagopal, and F. Wilczek, Nucl. Phys. B537, 443 (1999).[4] M. Iwasaki and T. Iwado, Prog. Theor. Phys. 94, 1073(1995); D. T. Son, Phys. Rev. D 59, 094019 (1999); R. D.Pisarski and D. H. Rischke, Phys. Rev. Lett. 83, 37(1999); T. Schafer and F. Wilczek, Phys. Rev. D 60,114033 (1999).[5] K. Rajagopal and F. Wilczek, hep-ph/0011333; M. G.Alford, Annu. Rev. Nucl. Part. Sci. 51, 131 (2001);R. Casalbuoni and G. Nardulli, Rev. Mod. Phys. 76,263 (2004); J. A. Bowers, hep-ph/0305301.[6] M. Kitazawa, T. Koide, T. Kunihiro, and Y. Nemoto,Phys. Rev. D 65, 091504(R) (2002).[7] Y. Yanase and K. Yamada, J. Phys. Soc. Jpn. 70, 1659(2001).[8] See, for example, V. M. Loktev, R. M. Quick, and S. G.Sharapov, Phys. Rep. 349, 1 (2001).[9] A. Schnell, G. Ropke, and P. Schuck, Phys. Rev. Lett. 83,1926 (1999).[10] L. P. Kadanoff and G. Baym, Quantum StatisticalMechanics (Benjamin, New York, 1962).[11] A possible pseudogap formation in association with thechiral transition due to the phase fluctuation was dis-cussed earlier by others: E. Babaev, Int. J. Mod. Phys. A16, 1175 (2001), in which the possible relevance of thepseudogap to the CS is also mentioned; see alsoK. Zarembo, JETP Lett. 75, 59 (2002).[12] J. Berges and K. Rajagopal, Nucl. Phys. B538, 215(1999).[13] T. M. Schwarz, S. P. Klevansky, and G. Papp, Phys. Rev. C60, 055205 (1999).[14] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345(1961); Phys. Rev. 124, 246 (1961).[15] S. P. Klevansky, Rev. Mod. Phys. 64, 649 (1992);T. Hatsuda and T. Kunihiro, Phys. Rep. 247, 221 (1994).[16] T. Matsuura, K. Iida, T. Hatsuda, and G. Baym, Phys. Rev.D 69, 074012 (2004).[17] R. D. Pisarski, Phys. Rev. C 62, 035202 (2000); see alsoS. Mo, J. Hove, and A. Sudbø, Phys. Rev. B 65, 104501(2002).[18] D. J. Thouless, Ann. Phys. (N.Y.) 10, 553 (1960). 0 5 10 15 20-400 -300 -200 -100  0  100N(ω)/104ωε=0.20ε=0.05ε=0.02ε=0.01freeFIG. 4. Density of states at   400 MeV and various " T  TC=TC. The dotted line shows that of the free quarks. Aclear pseudogap structure is seen, which survives up to " 0:05.M. KITAZAWA, T. KOIDE, T. KUNIHIRO, AND Y. NEMOTO PHYSICAL REVIEW D 70 056003056003-4[19] S. Klimt, M. Lutz, U. Vogl, and W. Weise, Nucl. Phys.A516, 429 (1990); U. Vogl, M. Lutz, S. Klimt, andW. Weise, Nucl. Phys. A516, 469 (1990).[20] The peak structure of the jImj around !  0 is atypical non-Fermi liquid behavior, although the wavefunction renormalization Z of the quasiparticle atthe Fermi surface is found to be Z  1dRek; !=d!1kkF;!0  0:7, at   400 MeV and"  0:01, which suggets that the non-Fermi liquid pic-ture is not complete; see, for other possible origins of thenon-Fermi liquid behavior of quark matter, [21].[21] D. Boyanovsky and H. J. deVega, Phys. Rev. D 63, 034016(2001).[22] S. Fujimoto, J. Phys. Soc. Jpn. 71, 1230 (2002).PSEUDOGAP OF COLOR SUPERCONDUCTIVITY IN . . . PHYSICAL REVIEW D 70 056003056003-5

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Pseudogap of Color Superconductivity in Heated Quark Matter

Pseudogap of Color Superconductivity in Heated Quark Matter

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