热密强相互作用物质的量子涨落
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摘要
量子色动力学是标准模型的重要组成部分，它描写的是由胶子传递的强相互作用。不同于高能时的渐近自由，在中低能时强作用物质拥有十分丰富多彩的相结构。其中以颜色解禁闭相变和手征相变最为引人注目。由于颜色在中低能下是禁闭的，所以采用明显色单态的有效模型来进行手征相变的研究是比较方便的。与高能时的微扰论的计算不同，在中低能区量子色动力学是高度非微扰的，这就要求我们在技术上也必须采用与之相符的手段。本文中我们在无规相近似和泛函重整化群的框架下对有限温度密度的强作用体系的相变行为进行了讨论。同时也介绍了传统的超出平均场的自洽Hatree近似的方法。
在有限温度和同位旋化学势时，我们利用无规相近似的方法在NJL模型中讨论了平均场基础上的量子涨落。通过计算作为涨落的介子模式对夸克静态势的贡献，我们发现对应于同位旋对称性自发破缺的Goldstone模式控制了静态势的行为，并使得其所处的π超流体始终处于强耦合的状态。为了进一步自洽的考虑热力学涨落和量子涨落，我们采用泛函重整化群的方法在三味线性σ模型中同时讨论了UA(1)和手征对称性的温度行为。泛函重整化群是一种可以比较合理的考虑涨落的方法，利用它可以得到正确的相变临界行为。本文中在保证Nambu–Goldstone定理的前提下，我们发现手征和UA(1)对称性将随着温度的升高而得到恢复。同时介子谱的行为和对称性的恢复相吻合。
As an important part of the standard model, quantum chromodynamics describes thestrong interaction which is mediated by gluons. Diferent from the asymptotic freedomat a high-energy scale, the phase structure is very complicated at a low and intermediate-scale. Among these phases transitions the most interesting ones are deconfinement andchiral phase transition. For the color is confined in hadrons at low and intermediate-scale, efective models without color interaction are usually used to study the chiralphase transition. Meanwhile at this energy scale, because of the strong coupling con-stant，nonperturbative methods are required. In this thesis we study the phase structureof the strongly interacting matter in QCD efective models with the random phase ap-proximation and functional renormalization group. Besides, the self-consistent Hatreeapproximation is also introduced.
With the random phase approximation, the quantum fluctuations above mean fieldare constructed in the NJL model at finite temperature and isospin chemical potential.By calculating the meson contribution to the static quark potential, the Goldstone mode,which corresponds to the isospion symmetry spontaneous breaking, controls the poten-tial. In the pion superfluid quarks are always strongly coupled by the Goldstone mode. Inorder to consider the thermal and quantum fluctuations more self-consistently, we adoptthe functional renormalization group method to study the UA(1) and chiral symmetries atfinite temperature in the linear sigma model. In the framework of the functional renor-malization group, symmetries can be easily respected and the correct critical behavior atphase transitions can be obtained. In this thesis, by guaranteeing the Nambu–Goldstonetheorem, the UA(1) and chiral symmetries are both restored as temperature increases. Asa result, the obtained meson spectrum agrees well with the symmetry restoration.
在有限温度和同位旋化学势时，我们利用无规相近似的方法在NJL模型中讨论了平均场基础上的量子涨落。通过计算作为涨落的介子模式对夸克静态势的贡献，我们发现对应于同位旋对称性自发破缺的Goldstone模式控制了静态势的行为，并使得其所处的π超流体始终处于强耦合的状态。为了进一步自洽的考虑热力学涨落和量子涨落，我们采用泛函重整化群的方法在三味线性σ模型中同时讨论了UA(1)和手征对称性的温度行为。泛函重整化群是一种可以比较合理的考虑涨落的方法，利用它可以得到正确的相变临界行为。本文中在保证Nambu–Goldstone定理的前提下，我们发现手征和UA(1)对称性将随着温度的升高而得到恢复。同时介子谱的行为和对称性的恢复相吻合。
As an important part of the standard model, quantum chromodynamics describes thestrong interaction which is mediated by gluons. Diferent from the asymptotic freedomat a high-energy scale, the phase structure is very complicated at a low and intermediate-scale. Among these phases transitions the most interesting ones are deconfinement andchiral phase transition. For the color is confined in hadrons at low and intermediate-scale, efective models without color interaction are usually used to study the chiralphase transition. Meanwhile at this energy scale, because of the strong coupling con-stant，nonperturbative methods are required. In this thesis we study the phase structureof the strongly interacting matter in QCD efective models with the random phase ap-proximation and functional renormalization group. Besides, the self-consistent Hatreeapproximation is also introduced.
With the random phase approximation, the quantum fluctuations above mean fieldare constructed in the NJL model at finite temperature and isospin chemical potential.By calculating the meson contribution to the static quark potential, the Goldstone mode,which corresponds to the isospion symmetry spontaneous breaking, controls the poten-tial. In the pion superfluid quarks are always strongly coupled by the Goldstone mode. Inorder to consider the thermal and quantum fluctuations more self-consistently, we adoptthe functional renormalization group method to study the UA(1) and chiral symmetries atfinite temperature in the linear sigma model. In the framework of the functional renor-malization group, symmetries can be easily respected and the correct critical behavior atphase transitions can be obtained. In this thesis, by guaranteeing the Nambu–Goldstonetheorem, the UA(1) and chiral symmetries are both restored as temperature increases. Asa result, the obtained meson spectrum agrees well with the symmetry restoration.
引文
[1] Glashow S. Partial Symmetries of Weak Interactions. Nucl.Phys.,1961,22:579–588.
[2] Weinberg S. A Model of Leptons. Phys. Rev. Lett.,1967,19:1264–1266.
[3] Goldstone J, Salam A, Weinberg S. Broken Symmetries. Phys.Rev.,1962,127:965–970.
[4] Weinberg S. The Quantum theory of fields. Vol.1: Foundations.1995.
[5] Yang C N, Mills R L. Conservation of Isotopic Spin and Isotopic Gauge Invariance. Phys.Rev.,1954,96:191–195.
[6] Yukawa H. On the interaction of elementary particles. Proc.Phys.Math.Soc.Jap.,1935,17:48–57.
[7] Belloni M J. An Operator approach to Gauss’s law in Yang-Mills theory and quantum chro-modynamics.1997.
[8] Faddeev L, Popov V. Feynman Diagrams for the Yang-Mills Field. Phys.Lett.,1967, B25:29–30.
[9] Gribov V. Quantization of Nonabelian Gauge Theories. Nucl.Phys.,1978, B139:1.
[10] Singer I. Some Remarks on the Gribov Ambiguity. Commun.Math.Phys.,1978,60:7–12.
[11] Esposito G, Pelliccia D N, Zaccaria F. Gribov problem for gauge theories: A Pedagogicalintroduction. Int.J.Geom.Meth.Mod.Phys.,2004,1:423–441.
[12] Zwanziger D. NONPERTURBATIVE MODIFICATION OF THE FADDEEV-POPOV FOR-MULA AND BANISHMENT OF THE NAIVE VACUUM. Nucl.Phys.,1982, B209:336.
[13] Cutkosky R. HARMONIC FUNCTIONS AND MATRIX ELEMENTS FOR HYPERSPHER-ICAL QUANTUM FIELD MODELS. J.Math.Phys.,1984,25:939.
[14] Cutkosky R, Wang K. VACUUM AND EXCITED STATES OF COULOMB GAUGE SU(n)YANG-MILLS FIELDS. Phys.Rev.,1988, D37:3024.
[15] Cutkosky R. CHROMOELECTRIC AND CHROMOMAGNETIC EFFECTS IN THE CON-FINEMENT OF GLUONS. Czech.J.Phys.,1990,40:252–261.
[16] Vandersickel N, Zwanziger D. The Gribov problem and QCD dynamics. Phys.Rept.,2012,520:175–251.
[17]侯伯元，侯伯宇.物理学家用微分几何.科学出版社,2004.
[18] Atiyah M, Singer I. The index of elliptic operators on compact manifolds. Bull.Am.Math.Soc.,1969,69:422–433.
[19] Wilson K G. Confinement of Quarks. Phys.Rev.,1974, D10:2445–2459.
[20] Giusti L, Paciello M, Parrinello C, et al. Problems on lattice gauge fixing. Int.J.Mod.Phys.,2001, A16:3487–3534.
[21] Shibata A, Kondo K I, Kato S, et al. A New description of lattice Yang-Mils theory and non-Abelian monopoles as the quark confiner. PoS,2008, LATTICE2008:268.
[22] Shibata A, Kondo K I, Kato S, et al. Topological configurations of Yang-Mills field responsiblefor magnetic-monopole loops as quark confiner. PoS,2009, LAT2009:232.
[23] Nielsen H B, Ninomiya M. Absence of Neutrinos on a Lattice.1. Proof by Homotopy Theory.Nucl.Phys.,1981, B185:20.
[24] Nielsen H B, Ninomiya M. Absence of Neutrinos on a Lattice.2. Intuitive Topological Proof.Nucl.Phys.,1981, B193:173.
[25] Susskind L. Lattice Fermions. Phys.Rev.,1977, D16:3031–3039.
[26] Sharatchandra H, Thun H, Weisz P. Susskind Fermions on a Euclidean Lattice. Nucl.Phys.,1981, B192:205.
[27] Forcrand P, Philipsen O. Towards the QCD phase diagram. PoS,2006, LAT2006:130.
[28] Schmidt C. QCD thermodynamics at zero and non-zero density. PoS,2006, CPOD2006:002.
[29] Kaczmarek O, Karsch F, Zantow F, et al. Static quark anti-quark free energy and the runningcoupling at finite temperature. Phys.Rev.,2004, D70:074505.
[30] Shibata A, Kondo K I, Kato S, et al. Gauge-independent’Abelian’ dominance and magneticmonopole dominance in SU(3) Yang-Mills theory. PoS,2010, LATTICE2010:286.
[31] Abbott L. Introduction to the Background Field Method. Acta Phys.Polon.,1982, B13:33.
[32] Hooft G. Computation of the Quantum Efects Due to a Four-Dimensional Pseudoparticle.Phys.Rev.,1976, D14:3432–3450.
[33] Hooft G. Magnetic Monopoles in Unified Gauge Theories. Nucl.Phys.,1974, B79:276–284.
[34] Polyakov A M. Particle Spectrum in the Quantum Field Theory. JETP Lett.,1974,20:194–195.
[35] Cho Y. A Restricted Gauge Theory. Phys.Rev.,1980, D21:1080.
[36] Cho Y. EXTENDED GAUGE THEORY AND ITS MASS SPECTRUM. Phys.Rev.,1981,D23:2415.
[37] Faddeev L, Niemi A J. Decomposing the Yang-Mills field. Phys.Lett.,1999, B464:90–93.
[38] Faddeev L D, Niemi A J. Spin-Charge Separation, Conformal Covariance and the SU(2) Yang-Mills Theory. Nucl.Phys.,2007, B776:38–65.
[39] Shabanov S V. An Efective action for monopoles and knot solitons in Yang-Mills theory.Phys.Lett.,1999, B458:322–330.
[40] Shabanov S V. Yang-Mills theory as an Abelian theory without gauge fixing. Phys.Lett.,1999,B463:263–272.
[41] Kondo K I, Taira Y. NonAbelian Stokes theorem and quark confinement in SU(N) Yang-Millsgauge theory. Prog.Theor.Phys.,2000,104:1189–1265.
[42] Kondo K I. Wilson loop and magnetic monopole through a non-Abelian Stokes theorem.Phys.Rev.,2008, D77:085029.
[43] Wilson K G. The Renormalization Group: Critical Phenomena and the Kondo Problem.Rev.Mod.Phys.,1975,47:773.
[44] Hooft G, Veltman M. Regularization and Renormalization of Gauge Fields. Nucl.Phys.,1972,B44:189–213.
[45] Gross D, Wilczek F. Ultraviolet Behavior of Nonabelian Gauge Theories. Phys.Rev.Lett.,1973,30:1343–1346.
[46] Politzer H D. Reliable Perturbative Results for Strong Interactions? Phys.Rev.Lett.,1973,30:1346–1349.
[47] Bethke S. Determination of the QCD coupling αs. J.Phys.,2000, G26:R27.
[48] Nambu Y, Jona-Lasinio G. Dynamical Model of Elementary Particles Based on an Analogywith Superconductivity.1. Phys.Rev.,1961,122:345–358.
[49] Nambu Y, Jona-Lasinio G. DYNAMICAL MODEL OF ELEMENTARY PARTICLES BASEDON AN ANALOGY WITH SUPERCONDUCTIVITY. II. Phys.Rev.,1961,124:246–254.
[50] Volkov M. Efective chiral Lagrangians and the Nambu-Jona-Lasinio model. Phys.Part.Nucl.,1993,24:35–58.
[51] Barrois B C. Superconducting Quark Matter. Nucl.Phys.,1977, B129:390.
[52] Bailin D, Love A. SUPERFLUID QUARK MATTER. J.Phys.,1979, A12:L283.
[53] Son D. Superconductivity by long range color magnetic interaction in high density quarkmatter. Phys.Rev.,1999, D59:094019.
[54] Schaefer T, Wilczek F. Superconductivity from perturbative one gluon exchange in high densityquark matter. Phys.Rev.,1999, D60:114033.
[55] Migdal A B, Chernoutsan A, Mishustin I. PION CONDENSATION AND DYNAMICS OFNEUTRON STARS. Phys.Lett.,1979, B83:158–160.
[56] Son D, Stephanov M A. QCD at finite isospin density. Phys.Rev.Lett.,2001,86:592–595.
[57] Andersen J O. Pion and kaon condensation at finite temperature and density. Phys.Rev.,2007,D75:065011.
[58] Nambu Y. Quasiparticles and Gauge Invariance in the Theory of Superconductivity. Phys.Rev.,1960,117:648–663.
[59] Goldstone J. Field Theories with Superconductor Solutions. Nuovo Cim.,1961,19:154–164.
[60] Nair V. Quantum field theory: A modern perspective.2005.
[61] Watanabe H, Brauner T. Spontaneous breaking of continuous translational invariance.Phys.Rev.,2012, D85:085010.
[62] Nielsen H B, Chadha S. On How to Count Goldstone Bosons. Nucl.Phys.,1976, B105:445.
[63] Hooft G. On the Phase Transition Towards Permanent Quark Confinement. Nucl.Phys.,1978,B138:1.
[64] Weiss N. The Efective Potential for the Order Parameter of Gauge Theories at Finite Temper-ature. Phys.Rev.,1981, D24:475.
[65] Engelhardt M, Langfeld K, Reinhardt H, et al. Deconfinement in SU(2) Yang-Mills theory asa center vortex percolation transition. Phys.Rev.,2000, D61:054504.
[66] Shifman M A, Vainshtein A, Zakharov V I. QCD and Resonance Physics. Sum Rules. Nu-cl.Phys.,1979, B147:385–447.
[67] Aoki Y, Borsanyi S, Durr S, et al. The QCD transition temperature: results with physicalmasses in the continuum limit II. JHEP,2009,0906:088.
[68] Borsanyi S, et al. Is there still any Tc mystery in lattice QCD? Results with physical masses inthe continuum limit III. JHEP,2010,1009:073.
[69] Gasiorowicz S, Gefen D. Efective Lagrangians and field algebras with chiral symmetry.Rev.Mod.Phys.,1969,41:531–573.
[70] Huang M. Color superconductivity at moderate baryon density. Int.J.Mod.Phys.,2005,E14:675.
[71] He L, Jin M, Zhuang P. Pion superfluidity and meson properties at finite isospin density.Phys.Rev.,2005, D71:116001.
[72] Cornwall J M, Jackiw R, Tomboulis E. Efective Action for Composite Operators. Phys.Rev.,1974, D10:2428–2445.
[73] Larsen A. SYMMETRY RESTORATION IN THE LINEAR SIGMA MODEL AT FINITETEMPERATURE. Z.Phys.,1986, C33:291.
[74] Roh H S, Matsui T. Chiral phase transition at finite temperature in the linear sigma model.Eur.Phys.J.,1998, A1:205–220.
[75] Polchinski J. Renormalization and Efective Lagrangians. Nucl.Phys.,1984, B231:269–295.
[76] Wetterich C. Exact evolution equation for the efective potential. Phys.Lett.,1993, B301:90–94.
[77] Ellwanger U, Wetterich C. Evolution equations for the quark-meson transition. Nucl.Phys.,1994, B423:137–170.
[78] Berges J, Jungnickel D, Wetterich C. Two flavor chiral phase transition from nonperturbativeflow equations. Phys.Rev.,1999, D59:034010.
[79] Litim D F. Optimized renormalization group flows. Phys.Rev.,2001, D64:105007.
[80] Tetradis N, Wetterich C. The high temperature phase transition for phi**4theories. Nucl.Phys.,1993, B398:659–696.
[81] Tetradis N, Wetterich C. High temperature phase transitions without infrared divergences.Int.J.Mod.Phys.,1994, A9:4029–4062.
[82] Cherny A Y, Dorokhov A, Han N S, et al. Bound States in Gauge Theories as the PoincareGroup Representations. Phys.Atom.Nucl.,2013,76:382–396.
[83] Reuter M. Renormalization of the topological charge in Yang-Mills theory. Mod.Phys.Lett.,1997, A12:2777–2802.
[84] Reuter M, Wetterich C. Gluon condensation in nonperturbative flow equations. Phys.Rev.,1997, D56:7893–7916.
[85] Gies H, Wetterich C. Universality of spontaneous chiral symmetry breaking in gauge theories.Phys.Rev.,2004, D69:025001.
[86] Braun J, Gies H, Pawlowski J M. Quark Confinement from Color Confinement. Phys.Lett.,2010, B684:262–267.
[87] Braun J. The QCD Phase Boundary from Quark-Gluon Dynamics. Eur.Phys.J.,2009,C64:459–482.
[88] Kondo K I. Toward a first-principle derivation of confinement and chiral-symmetry-breakingcrossover transitions in QCD. Phys.Rev.,2010, D82:065024.
[89] Braun J. Fermion Interactions and Universal Behavior in Strongly Interacting Theories.J.Phys.,2012, G39:033001.
[90] Brouzakis N, Tetradis N. The Universal equation of state near the critical point of QCD.Nucl.Phys.,2004, A742:144–164.
[91] Kamikado K, Strodthof N, Smekal L, et al. Fluctuations in the quark-meson model for QCDwith isospin chemical potential. Phys.Lett.,2013, B718:1044–1053.
[92] Fukushima K, Kamikado K, Klein B. Second-order and Fluctuation-induced First-order PhaseTransitions with Functional Renormalization Group Equations. Phys.Rev.,2011, D83:116005.
[93] Strodthof N, Schaefer B J, Smekal L. Quark-meson-diquark model for two-color QCD.Phys.Rev.,2012, D85:074007.
[94] Higashijima K, Itou E. Three-dimensional nonlinear sigma models in the Wilsonian renormal-ization method. Prog.Theor.Phys.,2003,110:563–578.
[95] Flore R, Wipf A, Zanusso O. Functional renormalization group of the non-linear sigma modeland the O(N) universality class. Phys.Rev.,2012, D87:065019.
[96] Krahl H, Wetterich C. Functional renormalization group for d-wave superconductivity in Hub-bard type models. Phys.Lett.,2007, A367:263–267.
[97] Krippa B. Exact renormalisation group flow for ultracold Fermi gases in unitary limit. J.Phys.,2009, A42:465002.
[98] Friederich S, Krahl H C, Wetterich C. Functional renormalization for spontaneous symmetrybreaking in the Hubbard model. Phys.Rev.,2011, B83:115125.
[99] Metzner W, Salmhofer M, Honerkamp C, et al. Functional renormalization group approach tocorrelated fermion systems. Rev.Mod.Phys.,2012,84:299.
[100] Berges J, Mesterhazy D. Introduction to the nonequilibrium functional renormalization group.Nucl.Phys.Proc.Suppl.,2012,228:37–60.
[101] Klevansky S. The Nambu-Jona-Lasinio model of quantum chromodynamics. Rev.Mod.Phys.,1992,64:649–708.
[102] Jaeckel J, Wetterich C. Flow equations without mean field ambiguity. Phys.Rev.,2003,D68:025020.
[103] Meisinger P N, Ogilvie M C. Coupling the deconfining and chiral transitions. Nu-cl.Phys.Proc.Suppl.,1996,47:519–522.
[104] Meisinger P N, Ogilvie M C. Chiral symmetry restoration and Z(N) symmetry. Phys.Lett.,1996, B379:163–168.
[105] Fukushima K. Chiral efective model with the Polyakov loop. Phys.Lett.,2004, B591:277–284.
[106] Ratti C, Thaler M A, Weise W. Phases of QCD: Lattice thermodynamics and a field theoreticalmodel. Phys.Rev.,2006, D73:014019.
[107] Ratti C, Roessner S, Thaler M, et al. Thermodynamics of the PNJL model. Eur.Phys.J.,2007,C49:213–217.
[108] Stokic B, Friman B, Redlich K. The Functional Renormalization Group and O(4) scaling.Eur.Phys.J.,2010, C67:425–438.
[109] Schaefer B J, Pirner H J. Renormalization group flow and equation of state of quarks andmesons. Nucl.Phys.,1999, A660:439–474.
[110] Lenaghan J T, Rischke D H, Schafner-Bielich J. Chiral symmetry restoration at nonzerotemperature in the SU(3)(r) x SU(3)(l) linear sigma model. Phys.Rev.,2000, D62:085008.
[111] Papp G, Schaefer B, Pirner H, et al. On the convergence of the expansion of renormalizationgroup flow equation. Phys.Rev.,2000, D61:096002.
[112] Blaizot J P, Ipp A, Mendez-Galain R, et al. Perturbation theory and non-perturbative renormal-ization flow in scalar field theory at finite temperature. Nucl.Phys.,2007, A784:376–406.
[113] Bohr O, Schaefer B, Wambach J. Renormalization group flow equations and the phase transi-tion in O(N) models. Int.J.Mod.Phys.,2001, A16:3823–3852.
[114] Adams J A, Berges J, Bornholdt S, et al. Solving nonperturbative flow equations.Mod.Phys.Lett.,1995, A10:2367–2380.
[115] Berges J, Jungnickel D U, Wetterich C. The Chiral phase transition at high baryon densityfrom nonperturbative flow equations. Eur.Phys.J.,2000, C13:323–329.
[116] Jiang Y, Ren K, Xia T, et al. Meson Screening Mass in a Strongly Coupled Pion Superfluid.Eur.Phys.J.,2011, C71:1822.
[117] Fujikawa K. Comment on Chiral and Conformal Anomalies. Phys.Rev.Lett.,1980,44:1733.
[118] Witten E. Current Algebra Theorems for the U(1) Goldstone Boson. Nucl.Phys.,1979,B156:269.
[119] Veneziano G. U(1) Without Instantons. Nucl.Phys.,1979, B159:213–224.
[120] Bornyakov V, Ilgenfritz E M, Martemyanov B, et al. Topology across the finite tempera-ture transition studied by overimproved cooling in gluodynamics and QCD. arXiv,2013,1304.0935.
[121] Jungnickel D, Wetterich C. Efective action for the chiral quark-meson model. Phys.Rev.,1996, D53:5142–5175.
[122] Jiang Y, Zhuang P. Functional Renormalization for Chiral and UA(1) Symmetries at FiniteTemperature. Phys.Rev.,2012, D86:105016.
[123] Benic S, Horvatic D, Kekez D, et al. eta’ Multiplicity and the Witten-Veneziano relation atfinite temperature. Phys.Rev.,2011, D84:016006.
[124] Wantz O, Shellard E. The Topological susceptibility from grand canonical simulations in theinteracting instanton liquid model: Chiral phase transition and axion mass. Nucl.Phys.,2010,B829:110–160.
[125] Weinberg S. The quantum theory of fields. Vol.2: Modern applications.1996.
[2] Weinberg S. A Model of Leptons. Phys. Rev. Lett.,1967,19:1264–1266.
[3] Goldstone J, Salam A, Weinberg S. Broken Symmetries. Phys.Rev.,1962,127:965–970.
[4] Weinberg S. The Quantum theory of fields. Vol.1: Foundations.1995.
[5] Yang C N, Mills R L. Conservation of Isotopic Spin and Isotopic Gauge Invariance. Phys.Rev.,1954,96:191–195.
[6] Yukawa H. On the interaction of elementary particles. Proc.Phys.Math.Soc.Jap.,1935,17:48–57.
[7] Belloni M J. An Operator approach to Gauss’s law in Yang-Mills theory and quantum chro-modynamics.1997.
[8] Faddeev L, Popov V. Feynman Diagrams for the Yang-Mills Field. Phys.Lett.,1967, B25:29–30.
[9] Gribov V. Quantization of Nonabelian Gauge Theories. Nucl.Phys.,1978, B139:1.
[10] Singer I. Some Remarks on the Gribov Ambiguity. Commun.Math.Phys.,1978,60:7–12.
[11] Esposito G, Pelliccia D N, Zaccaria F. Gribov problem for gauge theories: A Pedagogicalintroduction. Int.J.Geom.Meth.Mod.Phys.,2004,1:423–441.
[12] Zwanziger D. NONPERTURBATIVE MODIFICATION OF THE FADDEEV-POPOV FOR-MULA AND BANISHMENT OF THE NAIVE VACUUM. Nucl.Phys.,1982, B209:336.
[13] Cutkosky R. HARMONIC FUNCTIONS AND MATRIX ELEMENTS FOR HYPERSPHER-ICAL QUANTUM FIELD MODELS. J.Math.Phys.,1984,25:939.
[14] Cutkosky R, Wang K. VACUUM AND EXCITED STATES OF COULOMB GAUGE SU(n)YANG-MILLS FIELDS. Phys.Rev.,1988, D37:3024.
[15] Cutkosky R. CHROMOELECTRIC AND CHROMOMAGNETIC EFFECTS IN THE CON-FINEMENT OF GLUONS. Czech.J.Phys.,1990,40:252–261.
[16] Vandersickel N, Zwanziger D. The Gribov problem and QCD dynamics. Phys.Rept.,2012,520:175–251.
[17]侯伯元，侯伯宇.物理学家用微分几何.科学出版社,2004.
[18] Atiyah M, Singer I. The index of elliptic operators on compact manifolds. Bull.Am.Math.Soc.,1969,69:422–433.
[19] Wilson K G. Confinement of Quarks. Phys.Rev.,1974, D10:2445–2459.
[20] Giusti L, Paciello M, Parrinello C, et al. Problems on lattice gauge fixing. Int.J.Mod.Phys.,2001, A16:3487–3534.
[21] Shibata A, Kondo K I, Kato S, et al. A New description of lattice Yang-Mils theory and non-Abelian monopoles as the quark confiner. PoS,2008, LATTICE2008:268.
[22] Shibata A, Kondo K I, Kato S, et al. Topological configurations of Yang-Mills field responsiblefor magnetic-monopole loops as quark confiner. PoS,2009, LAT2009:232.
[23] Nielsen H B, Ninomiya M. Absence of Neutrinos on a Lattice.1. Proof by Homotopy Theory.Nucl.Phys.,1981, B185:20.
[24] Nielsen H B, Ninomiya M. Absence of Neutrinos on a Lattice.2. Intuitive Topological Proof.Nucl.Phys.,1981, B193:173.
[25] Susskind L. Lattice Fermions. Phys.Rev.,1977, D16:3031–3039.
[26] Sharatchandra H, Thun H, Weisz P. Susskind Fermions on a Euclidean Lattice. Nucl.Phys.,1981, B192:205.
[27] Forcrand P, Philipsen O. Towards the QCD phase diagram. PoS,2006, LAT2006:130.
[28] Schmidt C. QCD thermodynamics at zero and non-zero density. PoS,2006, CPOD2006:002.
[29] Kaczmarek O, Karsch F, Zantow F, et al. Static quark anti-quark free energy and the runningcoupling at finite temperature. Phys.Rev.,2004, D70:074505.
[30] Shibata A, Kondo K I, Kato S, et al. Gauge-independent’Abelian’ dominance and magneticmonopole dominance in SU(3) Yang-Mills theory. PoS,2010, LATTICE2010:286.
[31] Abbott L. Introduction to the Background Field Method. Acta Phys.Polon.,1982, B13:33.
[32] Hooft G. Computation of the Quantum Efects Due to a Four-Dimensional Pseudoparticle.Phys.Rev.,1976, D14:3432–3450.
[33] Hooft G. Magnetic Monopoles in Unified Gauge Theories. Nucl.Phys.,1974, B79:276–284.
[34] Polyakov A M. Particle Spectrum in the Quantum Field Theory. JETP Lett.,1974,20:194–195.
[35] Cho Y. A Restricted Gauge Theory. Phys.Rev.,1980, D21:1080.
[36] Cho Y. EXTENDED GAUGE THEORY AND ITS MASS SPECTRUM. Phys.Rev.,1981,D23:2415.
[37] Faddeev L, Niemi A J. Decomposing the Yang-Mills field. Phys.Lett.,1999, B464:90–93.
[38] Faddeev L D, Niemi A J. Spin-Charge Separation, Conformal Covariance and the SU(2) Yang-Mills Theory. Nucl.Phys.,2007, B776:38–65.
[39] Shabanov S V. An Efective action for monopoles and knot solitons in Yang-Mills theory.Phys.Lett.,1999, B458:322–330.
[40] Shabanov S V. Yang-Mills theory as an Abelian theory without gauge fixing. Phys.Lett.,1999,B463:263–272.
[41] Kondo K I, Taira Y. NonAbelian Stokes theorem and quark confinement in SU(N) Yang-Millsgauge theory. Prog.Theor.Phys.,2000,104:1189–1265.
[42] Kondo K I. Wilson loop and magnetic monopole through a non-Abelian Stokes theorem.Phys.Rev.,2008, D77:085029.
[43] Wilson K G. The Renormalization Group: Critical Phenomena and the Kondo Problem.Rev.Mod.Phys.,1975,47:773.
[44] Hooft G, Veltman M. Regularization and Renormalization of Gauge Fields. Nucl.Phys.,1972,B44:189–213.
[45] Gross D, Wilczek F. Ultraviolet Behavior of Nonabelian Gauge Theories. Phys.Rev.Lett.,1973,30:1343–1346.
[46] Politzer H D. Reliable Perturbative Results for Strong Interactions? Phys.Rev.Lett.,1973,30:1346–1349.
[47] Bethke S. Determination of the QCD coupling αs. J.Phys.,2000, G26:R27.
[48] Nambu Y, Jona-Lasinio G. Dynamical Model of Elementary Particles Based on an Analogywith Superconductivity.1. Phys.Rev.,1961,122:345–358.
[49] Nambu Y, Jona-Lasinio G. DYNAMICAL MODEL OF ELEMENTARY PARTICLES BASEDON AN ANALOGY WITH SUPERCONDUCTIVITY. II. Phys.Rev.,1961,124:246–254.
[50] Volkov M. Efective chiral Lagrangians and the Nambu-Jona-Lasinio model. Phys.Part.Nucl.,1993,24:35–58.
[51] Barrois B C. Superconducting Quark Matter. Nucl.Phys.,1977, B129:390.
[52] Bailin D, Love A. SUPERFLUID QUARK MATTER. J.Phys.,1979, A12:L283.
[53] Son D. Superconductivity by long range color magnetic interaction in high density quarkmatter. Phys.Rev.,1999, D59:094019.
[54] Schaefer T, Wilczek F. Superconductivity from perturbative one gluon exchange in high densityquark matter. Phys.Rev.,1999, D60:114033.
[55] Migdal A B, Chernoutsan A, Mishustin I. PION CONDENSATION AND DYNAMICS OFNEUTRON STARS. Phys.Lett.,1979, B83:158–160.
[56] Son D, Stephanov M A. QCD at finite isospin density. Phys.Rev.Lett.,2001,86:592–595.
[57] Andersen J O. Pion and kaon condensation at finite temperature and density. Phys.Rev.,2007,D75:065011.
[58] Nambu Y. Quasiparticles and Gauge Invariance in the Theory of Superconductivity. Phys.Rev.,1960,117:648–663.
[59] Goldstone J. Field Theories with Superconductor Solutions. Nuovo Cim.,1961,19:154–164.
[60] Nair V. Quantum field theory: A modern perspective.2005.
[61] Watanabe H, Brauner T. Spontaneous breaking of continuous translational invariance.Phys.Rev.,2012, D85:085010.
[62] Nielsen H B, Chadha S. On How to Count Goldstone Bosons. Nucl.Phys.,1976, B105:445.
[63] Hooft G. On the Phase Transition Towards Permanent Quark Confinement. Nucl.Phys.,1978,B138:1.
[64] Weiss N. The Efective Potential for the Order Parameter of Gauge Theories at Finite Temper-ature. Phys.Rev.,1981, D24:475.
[65] Engelhardt M, Langfeld K, Reinhardt H, et al. Deconfinement in SU(2) Yang-Mills theory asa center vortex percolation transition. Phys.Rev.,2000, D61:054504.
[66] Shifman M A, Vainshtein A, Zakharov V I. QCD and Resonance Physics. Sum Rules. Nu-cl.Phys.,1979, B147:385–447.
[67] Aoki Y, Borsanyi S, Durr S, et al. The QCD transition temperature: results with physicalmasses in the continuum limit II. JHEP,2009,0906:088.
[68] Borsanyi S, et al. Is there still any Tc mystery in lattice QCD? Results with physical masses inthe continuum limit III. JHEP,2010,1009:073.
[69] Gasiorowicz S, Gefen D. Efective Lagrangians and field algebras with chiral symmetry.Rev.Mod.Phys.,1969,41:531–573.
[70] Huang M. Color superconductivity at moderate baryon density. Int.J.Mod.Phys.,2005,E14:675.
[71] He L, Jin M, Zhuang P. Pion superfluidity and meson properties at finite isospin density.Phys.Rev.,2005, D71:116001.
[72] Cornwall J M, Jackiw R, Tomboulis E. Efective Action for Composite Operators. Phys.Rev.,1974, D10:2428–2445.
[73] Larsen A. SYMMETRY RESTORATION IN THE LINEAR SIGMA MODEL AT FINITETEMPERATURE. Z.Phys.,1986, C33:291.
[74] Roh H S, Matsui T. Chiral phase transition at finite temperature in the linear sigma model.Eur.Phys.J.,1998, A1:205–220.
[75] Polchinski J. Renormalization and Efective Lagrangians. Nucl.Phys.,1984, B231:269–295.
[76] Wetterich C. Exact evolution equation for the efective potential. Phys.Lett.,1993, B301:90–94.
[77] Ellwanger U, Wetterich C. Evolution equations for the quark-meson transition. Nucl.Phys.,1994, B423:137–170.
[78] Berges J, Jungnickel D, Wetterich C. Two flavor chiral phase transition from nonperturbativeflow equations. Phys.Rev.,1999, D59:034010.
[79] Litim D F. Optimized renormalization group flows. Phys.Rev.,2001, D64:105007.
[80] Tetradis N, Wetterich C. The high temperature phase transition for phi**4theories. Nucl.Phys.,1993, B398:659–696.
[81] Tetradis N, Wetterich C. High temperature phase transitions without infrared divergences.Int.J.Mod.Phys.,1994, A9:4029–4062.
[82] Cherny A Y, Dorokhov A, Han N S, et al. Bound States in Gauge Theories as the PoincareGroup Representations. Phys.Atom.Nucl.,2013,76:382–396.
[83] Reuter M. Renormalization of the topological charge in Yang-Mills theory. Mod.Phys.Lett.,1997, A12:2777–2802.
[84] Reuter M, Wetterich C. Gluon condensation in nonperturbative flow equations. Phys.Rev.,1997, D56:7893–7916.
[85] Gies H, Wetterich C. Universality of spontaneous chiral symmetry breaking in gauge theories.Phys.Rev.,2004, D69:025001.
[86] Braun J, Gies H, Pawlowski J M. Quark Confinement from Color Confinement. Phys.Lett.,2010, B684:262–267.
[87] Braun J. The QCD Phase Boundary from Quark-Gluon Dynamics. Eur.Phys.J.,2009,C64:459–482.
[88] Kondo K I. Toward a first-principle derivation of confinement and chiral-symmetry-breakingcrossover transitions in QCD. Phys.Rev.,2010, D82:065024.
[89] Braun J. Fermion Interactions and Universal Behavior in Strongly Interacting Theories.J.Phys.,2012, G39:033001.
[90] Brouzakis N, Tetradis N. The Universal equation of state near the critical point of QCD.Nucl.Phys.,2004, A742:144–164.
[91] Kamikado K, Strodthof N, Smekal L, et al. Fluctuations in the quark-meson model for QCDwith isospin chemical potential. Phys.Lett.,2013, B718:1044–1053.
[92] Fukushima K, Kamikado K, Klein B. Second-order and Fluctuation-induced First-order PhaseTransitions with Functional Renormalization Group Equations. Phys.Rev.,2011, D83:116005.
[93] Strodthof N, Schaefer B J, Smekal L. Quark-meson-diquark model for two-color QCD.Phys.Rev.,2012, D85:074007.
[94] Higashijima K, Itou E. Three-dimensional nonlinear sigma models in the Wilsonian renormal-ization method. Prog.Theor.Phys.,2003,110:563–578.
[95] Flore R, Wipf A, Zanusso O. Functional renormalization group of the non-linear sigma modeland the O(N) universality class. Phys.Rev.,2012, D87:065019.
[96] Krahl H, Wetterich C. Functional renormalization group for d-wave superconductivity in Hub-bard type models. Phys.Lett.,2007, A367:263–267.
[97] Krippa B. Exact renormalisation group flow for ultracold Fermi gases in unitary limit. J.Phys.,2009, A42:465002.
[98] Friederich S, Krahl H C, Wetterich C. Functional renormalization for spontaneous symmetrybreaking in the Hubbard model. Phys.Rev.,2011, B83:115125.
[99] Metzner W, Salmhofer M, Honerkamp C, et al. Functional renormalization group approach tocorrelated fermion systems. Rev.Mod.Phys.,2012,84:299.
[100] Berges J, Mesterhazy D. Introduction to the nonequilibrium functional renormalization group.Nucl.Phys.Proc.Suppl.,2012,228:37–60.
[101] Klevansky S. The Nambu-Jona-Lasinio model of quantum chromodynamics. Rev.Mod.Phys.,1992,64:649–708.
[102] Jaeckel J, Wetterich C. Flow equations without mean field ambiguity. Phys.Rev.,2003,D68:025020.
[103] Meisinger P N, Ogilvie M C. Coupling the deconfining and chiral transitions. Nu-cl.Phys.Proc.Suppl.,1996,47:519–522.
[104] Meisinger P N, Ogilvie M C. Chiral symmetry restoration and Z(N) symmetry. Phys.Lett.,1996, B379:163–168.
[105] Fukushima K. Chiral efective model with the Polyakov loop. Phys.Lett.,2004, B591:277–284.
[106] Ratti C, Thaler M A, Weise W. Phases of QCD: Lattice thermodynamics and a field theoreticalmodel. Phys.Rev.,2006, D73:014019.
[107] Ratti C, Roessner S, Thaler M, et al. Thermodynamics of the PNJL model. Eur.Phys.J.,2007,C49:213–217.
[108] Stokic B, Friman B, Redlich K. The Functional Renormalization Group and O(4) scaling.Eur.Phys.J.,2010, C67:425–438.
[109] Schaefer B J, Pirner H J. Renormalization group flow and equation of state of quarks andmesons. Nucl.Phys.,1999, A660:439–474.
[110] Lenaghan J T, Rischke D H, Schafner-Bielich J. Chiral symmetry restoration at nonzerotemperature in the SU(3)(r) x SU(3)(l) linear sigma model. Phys.Rev.,2000, D62:085008.
[111] Papp G, Schaefer B, Pirner H, et al. On the convergence of the expansion of renormalizationgroup flow equation. Phys.Rev.,2000, D61:096002.
[112] Blaizot J P, Ipp A, Mendez-Galain R, et al. Perturbation theory and non-perturbative renormal-ization flow in scalar field theory at finite temperature. Nucl.Phys.,2007, A784:376–406.
[113] Bohr O, Schaefer B, Wambach J. Renormalization group flow equations and the phase transi-tion in O(N) models. Int.J.Mod.Phys.,2001, A16:3823–3852.
[114] Adams J A, Berges J, Bornholdt S, et al. Solving nonperturbative flow equations.Mod.Phys.Lett.,1995, A10:2367–2380.
[115] Berges J, Jungnickel D U, Wetterich C. The Chiral phase transition at high baryon densityfrom nonperturbative flow equations. Eur.Phys.J.,2000, C13:323–329.
[116] Jiang Y, Ren K, Xia T, et al. Meson Screening Mass in a Strongly Coupled Pion Superfluid.Eur.Phys.J.,2011, C71:1822.
[117] Fujikawa K. Comment on Chiral and Conformal Anomalies. Phys.Rev.Lett.,1980,44:1733.
[118] Witten E. Current Algebra Theorems for the U(1) Goldstone Boson. Nucl.Phys.,1979,B156:269.
[119] Veneziano G. U(1) Without Instantons. Nucl.Phys.,1979, B159:213–224.
[120] Bornyakov V, Ilgenfritz E M, Martemyanov B, et al. Topology across the finite tempera-ture transition studied by overimproved cooling in gluodynamics and QCD. arXiv,2013,1304.0935.
[121] Jungnickel D, Wetterich C. Efective action for the chiral quark-meson model. Phys.Rev.,1996, D53:5142–5175.
[122] Jiang Y, Zhuang P. Functional Renormalization for Chiral and UA(1) Symmetries at FiniteTemperature. Phys.Rev.,2012, D86:105016.
[123] Benic S, Horvatic D, Kekez D, et al. eta’ Multiplicity and the Witten-Veneziano relation atfinite temperature. Phys.Rev.,2011, D84:016006.
[124] Wantz O, Shellard E. The Topological susceptibility from grand canonical simulations in theinteracting instanton liquid model: Chiral phase transition and axion mass. Nucl.Phys.,2010,B829:110–160.
[125] Weinberg S. The quantum theory of fields. Vol.2: Modern applications.1996.