Casimir scaling and renormalization of Polyakov loops in large-N gauge theories

Journal of High Energy Physics

Volumn 2012, Issue 5, 2012, Pages

  Mykkänen, Anne ^a   Panero, Marco ^a,b   Rummukainen, Kari ^a  

^a UNIVERSITY OF HELSINKI   (Finland)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

't Hooft and Polyakov loops; 1 N expansion; Lattice gauge field theories; Quark gluon plasma; Wilson

Indexed keywords

EID: 84861908547     PISSN: 11266708     EISSN: 10298479     Source Type: Journal    
DOI: 10.1007/JHEP05(2012)069     Document Type: Article

References (173)

1. N. Cabibbo and G. Parisi, Exponential hadronic spectrum and quark liberation, Phys. Lett. B 59 (1975) 67.
2. J.C. Collins and M. Perry, Superdense matter: neutrons or asymptotically free quarks?, Phys. Rev. Lett. 34 (1975) 1353.
3. E.V. Shuryak, Quark-gluon plasma and hadronic production of leptons, photons and psions, Phys. Lett. B 78 (1978) 150
4. [Sov. J. Nucl. Phys. 28 (1978) 408]
5. [Yad. Fiz. 28 (1978) 796].
6. U.W. Heinz and M. Jacob, Evidence for a new state of matter: An Assessment of the results from the CERN lead beam program, nucl-th/0002042.
7. M. Gyulassy and L. McLerran, New forms of QCD matter discovered at RHIC, Nucl. Phys. A 750 (2005) 30 [nucl-th/0405013].
8. PHENIX collaboration, K. Adcox et al., Formation of dense partonic matter in relativistic nucleus-nucleus collisions at RHIC: experimental evaluation by the PHENIX collaboration, Nucl. Phys. A 757 (2005) 184 [nucl-ex/0410003].
9. BRAHMS collaboration, I. Arsene et al., Quark gluon plasma and color glass condensate at RHIC? The perspective from the BRAHMS experiment, Nucl. Phys. A 757 (2005) 1 [nucl-ex/0410020].
10. B. Back et al., The PHOBOS perspective on discoveries at RHIC, Nucl. Phys. A 757 (2005) 28 [nucl-ex/0410022].
11. STAR collaboration, J. Adams et al., Experimental and theoretical challenges in the search for the quark gluon plasma: the STAR collaboration's critical assessment of the evidence from RHIC collisions, Nucl. Phys. A 757 (2005) 102 [nucl-ex/0501009].
12. T.K. Nayak, Heavy ions: results from the Large Hadron Collider, arXiv:1201.4264.
13. A.D. Linde, Infrared problem in thermodynamics of the Yang-Mills gas, Phys. Lett. B 96 (1980) 289.
14. D.J. Gross, R.D. Pisarski and L.G. Yaffe, QCD and instantons at finite temperature, Rev. Mod. Phys. 53 (1981) 43.
15. J.-P. Blaizot, Quantum fields at finite temperature 'from tera to nano Kelvin', arXiv:1108.3482.
16. P.B. Arnold and C.-X. Zhai, The three loop free energy for pure gauge QCD, Phys. Rev. D 50 (1994) 7603 [hep-ph/9408276].
17. P.B. Arnold and C.-X. Zhai, The three loop free energy for high temperature QED and QCD with fermions, Phys. Rev. D 51 (1995) 1906 [hep-ph/9410360].
18. 5, Phys. Rev. D 52 (1995) 7232 [hep-ph/9507380].
19. J.O. Andersen, E. Braaten and M. Strickland, Hard thermal loop resummation of the free energy of a hot gluon plasma, Phys. Rev. Lett. 83 (1999) 2139 [hep-ph/9902327].
20. J.-P. Blaizot, E. Iancu and A. Rebhan, Approximately selfconsistent resummations for the thermodynamics of the quark gluon plasma. 1. Entropy and density, Phys. Rev. D 63 (2001) 065003 [hep-ph/0005003].
21. 6ln(1/g), Phys. Rev. D 67 (2003) 105008 [hep-ph/0211321].
22. A. Hietanen, K. Kajantie, M. Laine, K. Rummukainen and Y. Schroder, Three-dimensional physics and the pressure of hot QCD, Phys. Rev. D 79 (2009) 045018 [arXiv: 0811.4664] INSPIRE].
23. J.O. Andersen, M. Strickland and N. Su, Gluon thermodynamics at intermediate coupling, Phys. Rev. Lett. 104 (2010) 122003 [arXiv: 0911.0676].
24. J.O. Andersen, L.E. Leganger, M. Strickland and N. Su, NNLO hard-thermal-loop thermodynamics for QCD, Phys. Lett. B 696 (2011) 468 [arXiv: 1009.4644].
25. C. DeTar and U. Heller, QCD thermodynamics from the lattice, Eur. Phys. J. A 41 (2009) 405 [arXiv:0905.2949].
26. M. Laine, Finite-temperature QCD, PoS(LAT2009)006 [arXiv: 0910. 5168].
27. K. Kanaya, Finite temperature QCD on the lattice - Status 2010, PoS(LATTICE 2010)012 [arXiv: 1012.4247].
28. C. DeTar, QCD thermodynamics on the lattice: recent results, arXiv: 1101.0208.
29. L. Levkova, QCD at finite temperature and density, PoS(LATTICE 2011)012.
30. G. 't Hooft, On the phase transition towards permanent quark confinement, Nucl. Phys. B 138 (1978) 1.
31. A.M. Polyakov, Thermal properties of gauge fields and quark liberation, Phys. Lett. B 72 (1978) 477.
32. L.D. McLerran and B. Svetitsky, Quark liberation at high temperature: a Monte Carlo study of SU(2) gauge theory, Phys. Rev. D 24 (1981) 450.
33. J. Kuti, J. Polonyi and K. Szlachanyi, Monte Carlo study of SU(2) gauge theory at finite temperature, Phys. Lett. B 98 (1981) 199.
34. T. Matsui and H. Satz, J/ψ suppression by quark-gluon plasma formation, Phys. Lett. B 178 (1986) 416.
35. S. Digal, P. Petreczky and H. Satz, String breaking and quarkonium dissociation at finite temperatures, Phys. Lett. B 514 (2001) 57 [hep-ph/0105234].
36. A. Mocsy and P. Petreczky, Quarkonia correlators above deconfinement, Phys. Rev. D 73 (2006) 074007 [hep-ph/0512156].
37. L. Kluberg and H. Satz, Color deconfinement and charmonium production in nuclear collisions, arXiv: 0901. 3831.
38. O. Jahn and O. Philipsen, The Polyakov loop and its relation to static quark potentials and free energies, Phys. Rev. D 70 (2004) 074504 [hep-lat/0407042].
39. O. Philipsen, Static potentials for quarkonia at finite temperatures, Nucl. Phys. A 820 (2009) 33C-40C [arXiv:0810.4685].
40. J. Ghiglieri, Effective field theories of QCD for heavy quarkonia at finite temperature, arXiv: 1201. 2920.
41. V. Dotsenko and S. Vergeles, Renormalizability of phase factors in the nonabelian gauge theory, Nucl. Phys. B 169 (1980) 527.
42. J. Ambjørn, P. Olesen and C. Peterson, Stochastic confinement and dimensional reduction. 1. Four-dimensional SU(2) lattice gauge theory, Nucl. Phys. B 240 (1984) 189.
43. L. Del Debbio, M. Faber, J. Greensite and Š. Olejník, Casimir scaling versus Abelian dominance in QCD string formation, Phys. Rev. D 53 (1996) 5891 [hep-lat/9510028].
44. M. Faber, J. Greensite and Š. Olejník, Casimir scaling from center vortices: towards an understanding of the adjoint string tension, Phys. Rev. D 57 (1998) 2603 [hep-lat/9710039].
45. G.S. Bali, Casimir scaling of SU(3) static potentials, Phys. Rev. D 62 (2000) 114503 [hep-lat/0006022].
46. C. Piccioni, Casimir scaling in SU(2) lattice gauge theory, Phys. Rev. D 73 (2006) 114509 [hep-lat/0503021].
47. F. Bursa and M. Teper, Casimir scaling of domain wall tensions in the deconfined phase of D = 3 + 1 SU(N) gauge theories, JHEP 08 (2005) 060 [hep-lat/0505025].
48. 2 and SU(N) gauge theories, Phys. Rev. D 75 (2007) 034501 [hep-lat/0609050].
49. M. Döring, K. Hübner, O. Kaczmarek and F. Karsch, Color screening and quark-quark interactions in finite temperature QCD, Phys. Rev. D 75 (2007) 054504 [hep-lat/0702009].
50. P. Bicudo, M. Cardoso and O. Oliveira, Study of the gluon-quark-antiquark static potential in SU(3) lattice QCD, Phys. Rev. D 77 (2008) 091504 [arXiv:0704.2156].
51. B. Bringoltz and M. Teper, Closed k-strings in SU(N) gauge theories: 2 + 1 dimensions, Phys. Lett. B 663 (2008) 429 [arXiv:0802.1490].
52. 2 lattice gauge theory, Phys. Rev. D 78 (2008) 074501 [arXiv:0807.1390].
53. 2 gluodynamics, Phys. Rev. D 83 (2011) 016001 [arXiv:1006.2305].
54. V.I. Shevchenko and Y.A. Simonov, Casimir scaling as a test of QCD vacuum, Phys. Rev. Lett. 85 (2000) 1811 [hep-ph/0001299].
55. V.I. Shevchenko and Y.A. Simonov, On Casimir scaling in QCD, hep-ph/0104135.
56. Y.A. Simonov, Gluelumps and confinement in QCD, arXiv:1003.3608.
57. Y. Burnier, M. Laine and M. Vepsäläinen, Dimensionally regularized Polyakov loop correlators in hot QCD, JHEP 01 (2010) 054 [arXiv:0911.3480].
58. N. Brambilla, J. Ghiglieri, P. Petreczky and A. Vairo, The Polyakov loop and correlator of Polyakov loops at next-to-next-to-leading order, Phys. Rev. D 82 (2010) 074019 [arXiv:1007.5172].
59. O. Kaczmarek, F. Karsch, P. Petreczky and F. Zantow, Heavy quark anti-quark free energy and the renormalized Polyakov loop, Phys. Lett. B 543 (2002) 41 [hep-lat/0207002].
60. A. Dumitru, Y. Hatta, J. Lenaghan, K. Orginos and R.D. Pisarski, Deconfining phase transition as a matrix model of renormalized Polyakov loops, Phys. Rev. D 70 (2004) 034511 [hep-th/0311223].
61. S. Gupta, K. Hübner and O. Kaczmarek, Renormalized Polyakov loops in many representations, Phys. Rev. D 77 (2008) 034503 [arXiv:0711.2251].
62. S. Digal, S. Fortunato and P. Petreczky, Heavy quark free energies and screening in SU(2) gauge theory, Phys. Rev. D 68 (2003) 034008 [hep-lat/0304017].
63. K. Hübner and C. Pica, Renormalized Polyakov loops in various representations in finite temperature SU(2) gauge theory, PoS(LATTICE 2008)197 [arXiv:0809.3933].
64. R.V. Gavai, Renormalized Polyakov loop in the fixed scale approach, Phys. Lett. B 691 (2010) 146 [arXiv:1001.4977].
65. G. 't Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461.
66. E. Witten, Baryons in the 1/n Expansion, Nucl. Phys. B 160 (1979) 57.
67. L.G. Yaffe, Large-N limits as classical mechanics, Rev. Mod. Phys. 54 (1982) 407.
68. A.V. Manohar, Large-N QCD, hep-ph/9802419.
69. c baryons, Ann. Rev. Nucl. Part. Sci. 48 (1998) 81 [hep-ph/9803349].
70. Y. Makeenko, Large-N gauge theories, hep-th/0001047.
71. c, Nucl. Phys. A 796 (2007) 83 [arXiv:0706.2191].
72. S. Lottini and G. Torrieri, A percolation transition in Yang-Mills matter at finite number of colours, Phys. Rev. Lett. 107 (2011) 152301 [arXiv:1103.4824].
73. J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1133
74. [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200].
75. S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109].
76. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150].
77. E. Witten, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131].
78. D.T. Son and A.O. Starinets, Viscosity, black holes and quantum field theory, Ann. Rev. Nucl. Part. Sci. 57 (2007) 95 [arXiv:0704.0240].
79. D. Mateos, String theory and quantum chromodynamics, Class. Quant. Grav. 24 (2007) S713 [arXiv:0709.1523].
80. S.S. Gubser and A. Karch, From gauge-string duality to strong interactions: a pedestrian's guide, Ann. Rev. Nucl. Part. Sci. 59 (2009) 145 [arXiv:0901.0935].
81. J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal and U.A. Wiedemann, Gauge/string duality, hot QCD and heavy ion collisions, arXiv:1101.0618.
82. U. Gürsoy, E. Kiritsis, L. Mazzanti, G. Michalogiorgakis and F. Nitti, Improved holographic QCD, Lect. Notes Phys. 828 (2011) 79 [arXiv:1006.5461].
83. M.J. Teper, Glueball masses and other physical properties of SU(N) gauge theories in D = (3 + 1): a review of lattice results for theorists, hep-th/9812187.
84. B. Lucini and M. Teper, Confining strings in SU(N) gauge theories, Phys. Rev. D 64 (2001) 105019 [hep-lat/0107007].
85. B. Lucini and M. Teper, SU(N) gauge theories in four-dimensions: exploring the approach to N = ∞, JHEP 06 (2001) 050 [hep-lat/0103027].
86. B. Lucini, M. Teper and U. Wenger, The deconfinement transition in SU(N) gauge theories, Phys. Lett. B 545 (2002) 197 [hep-lat/0206029].
87. B. Lucini, M. Teper and U. Wenger, The high temperature phase transition in SU(N) gauge theories, JHEP 01 (2004) 061 [hep-lat/0307017].
88. c, Nucl. Phys. B 715 (2005) 461 [hep-lat/0401028].
89. B. Lucini, M. Teper and U. Wenger, Properties of the deconfining phase transition in SU(N) gauge theories, JHEP 02 (2005) 033 [hep-lat/0502003].
90. B. Lucini, M. Teper and U. Wenger, Glueballs and k-strings in SU(N) gauge theories: calculations with improved operators, JHEP 06 (2004) 012 [hep-lat/0404008].
91. H. Meyer and M. Teper, Confinement and the effective string theory in SU(N → ∞): a lattice study, JHEP 12 (2004) 031 [hep-lat/0411039].
92. B. Bringoltz and M. Teper, The pressure of the SU(N) lattice gauge theory at large-N, Phys. Lett. B 628 (2005) 113 [hep-lat/0506034].
93. B. Bringoltz and M. Teper, In search of a Hagedorn transition in SU(N) lattice gauge theories at large-N, Phys. Rev. D 73 (2006) 014517 [hep-lat/0508021].
94. L. Del Debbio, B. Lucini, A. Patella and C. Pica, Quenched mesonic spectrum at large-N, JHEP 03 (2008) 062 [arXiv: 0712.3036].
95. c from lattice QCD, JHEP 09 (2008) 110 [arXiv: 0806. 2278].
96. M. Panero, Thermodynamics of the QCD plasma and the large-N limit, Phys. Rev. Lett. 103 (2009) 232001 [arXiv: 0907.3719].
97. M. Panero, Thermodynamics of the strongly interacting gluon plasma in the large-N limit, PoS(LAT2009) 172 [arXiv:0912.2448].
98. c) pure gauge theory, Phys. Rev. D 80 (2009) 114504 [arXiv: 0909. 5591].
99. c plasma, Phys. Rev. D 82 (2010) 114505 [arXiv: 1006.0938].
100. M.J. Teper, SU(N) gauge theories in (2 + 1)-dimensions, Phys. Rev. D 59 (1999) 014512 [hep-lat/9804008].
101. R.W. Johnson and M.J. Teper, String models of glueballs and the spectrum of SU(N) gauge theories in (2 + 1)-dimensions, Phys. Rev. D 66 (2002) 036006 [hep-ph/0012287] INSPIRE].
102. H.B. Meyer and M.J. Teper, Glueball Regge trajectories in (2 + 1)-dimensional gauge theories, Nucl. Phys. B 668 (2003) 111 [hep-lat/0306019].
103. A. Athenodorou, B. Bringoltz and M. Teper, The closed string spectrum of SU(N) gauge theories in 2 + 1 dimensions, Phys. Lett. B 656 (2007) 132 [arXiv: 0709.0693].
104. M. Caselle, L. Castagnini, A. Feo, F. Gliozzi and M. Panero, Thermodynamics of SU(N) Yang-Mills theories in 2 + 1 dimensions I - The confining phase, JHEP 06 (2011) 142 [arXiv: 1105. 0359].
105. M. Caselle et al., Thermodynamics of SU(N) Yang-Mills theories in 2 + 1 dimensions II. The deconfined phase, arXiv: 1111.0580.
106. c) gauge theories using the gauge-string duality, Phys. Rev. D 81 (2010) 045011 [arXiv:0910.1261].
107. O. Andreev, Renormalized Polyakov loop in the deconfined phase of SU(N) gauge theory and gauge/string duality, Phys. Rev. Lett. 102 (2009) 212001 [arXiv:0903.4375].
108. E. Megías, H. Pirner and K. Veschgini, QCD thermodynamics using five-dimensional gravity, Phys. Rev. D 83 (2011) 056003 [arXiv:1009.2953].
109. O. Andreev, Some thermodynamic aspects of pure glue, fuzzy bags and gauge/string duality, Phys. Rev. D 76 (2007) 087702 [arXiv:0706.3120].
110. E. Megías, E. Ruiz Arriola and L. Salcedo, Dimension two condensates and the Polyakov loop above the deconfinement phase transition, JHEP 01 (2006) 073 [hep-ph/0505215].
111. E. Megías, E. Ruiz Arriola and L. Salcedo, Correlations between perturbation theory and power corrections in QCD at zero and finite temperature, Phys. Rev. D 81 (2010) 096009 [arXiv:0912.0499].
112. F. Xu and M. Huang, Electric and magnetic screenings of gluons in a model with dimension-2 gluon condensate, arXiv:1111.5152.
113. R.D. Pisarski, Why the quark gluon plasma isn't a plasma, hep-ph/0101168.
114. A. Dumitru and R.D. Pisarski, Explosive collisions at RHIC?, Nucl. Phys. A 698 (2002) 444 [hep-ph/0102020].
115. O. Scavenius, A. Dumitru and A. Jackson, Explosive decomposition in ultrarelativistic heavy ion collision, Phys. Rev. Lett. 87 (2001) 182302 [hep-ph/0103219].
116. A. Dumitru and R.D. Pisarski, Degrees of freedom and the deconfining phase transition, Phys. Lett. B 525 (2002) 95 [hep-ph/0106176].
117. c, Phys. Rev. D 66 (2002) 096003 [hep-ph/0204223].
118. P.N. Meisinger, T.R. Miller and M.C. Ogilvie, Phenomenological equations of state for the quark gluon plasma, Phys. Rev. D 65 (2002) 034009 [hep-ph/0108009].
119. A. Dumitru, J. Lenaghan and R.D. Pisarski, Deconfinement in matrix models about the Gross-Witten point, Phys. Rev. D 71 (2005) 074004 [hep-ph/0410294].
120. A. Dumitru, Y. Guo, Y. Hidaka, C.P.K. Altes and R.D. Pisarski, How wide is the transition to deconfinement?, Phys. Rev. D 83 (2011) 034022 [arXiv:1011.3820].
121. Á. Mócsy, F. Sannino and K. Tuominen, Confinement versus chiral symmetry, Phys. Rev. Lett. 92 (2004) 182302 [hep-ph/0308135].
122. D. Gross and E. Witten, Possible third order phase transition in the large-N lattice gauge theory, Phys. Rev. D 21 (1980) 446.
123. S. Wadia, A study of U(n) lattice gauge theory in two-dimensions, EFI-79-44-CHICAGO (1979).
124. D.J. Gross and W. Taylor, Two-dimensional QCD is a string theory, Nucl. Phys. B 400 (1993) 181 [hep-th/9301068].
125. D.J. Gross and W. Taylor, Twists and Wilson loops in the string theory of two-dimensional QCD, Nucl. Phys. B 403 (1993) 395 [hep-th/9303046].
126. S. Samuel, On the large-N phase transition in the heat kernel lattice gauge theory, Nucl. Phys. B 205 (1982) 221.
127. K. Rummukainen, QCD-like technicolor on the lattice, AIP Conf. Proc. 1343 (2011) 51 [arXiv:1101.5875].
128. L. Del Debbio, The conformal window on the lattice, arXiv:1102.4066.
129. A. Mykkänen, M. Panero and K. Rummukainen, Renormalization of Polyakov loops in different representations and the large-N limit, PoS(LATTICE 2011)211 [arXiv:1110.3146].
130. F. Gliozzi, The Stefan-Boltzmann law in a small box and the pressure deficit in hot SU(N) lattice gauge theor, J. Phys. A 40 (2007) F375 [hep-lat/0701020].
131. M. Panero, Geometric effects in lattice QCD thermodynamics, PoS(LATTICE 2008)175 [arXiv:0808.1672].
132. K.G. Wilson, Confinement of quarks, Phys. Rev. D 10 (1974) 2445.
133. P. Weisz, Continuum limit improved lattice action for pure Yang-Mills theory. 1, Nucl. Phys. B 212 (1983) 1.
134. P. Weisz and R. Wohlert, Continuum limit improved lattice action for pure Yang-Mills theory. 2, Nucl. Phys. B 236 (1984) 397 [Erratum ibid. B 247 (1984) 544].
135. B. Beinlich, F. Karsch, E. Laermann and A. Peikert, String tension and thermodynamics with tree level and tadpole improved actions, Eur. Phys. J. C 6 (1999) 133 [hep-lat/9707023].
136. S.L. Adler, An overrelaxation method for the Monte Carlo evaluation of the partition function for multiquadratic actions, Phys. Rev. D 23 (1981) 2901.
137. F.R. Brown and T.J. Woch, Overrelaxed heat bath and Metropolis algorithms for accelerating pure gauge Monte Carlo calculations, Phys. Rev. Lett. 58 (1987) 2394.
138. M. Creutz, Monte Carlo study of quantized SU(2) gauge theory, Phys. Rev. D 21 (1980) 2308.
139. A. Kennedy and B. Pendleton, Improved heat bath method for Monte Carlo calculations in lattice gauge theories, Phys. Lett. B 156 (1985) 393.
140. N. Cabibbo and E. Marinari, A new method for updating SU(N) matrices in computer simulations of gauge theories, Phys. Lett. B 119 (1982) 387.
141. M. Lüscher, Symmetry breaking aspects of the roughening transition in gauge theories, Nucl. Phys. B 180 (1981) 317.
142. M. Lüscher, K. Symanzik and P. Weisz, Anomalies of the free loop wave equation in the WKB approximation, Nucl. Phys. B 173 (1980) 365.
143. J. Polchinski and A. Strominger, Effective string theory, Phys. Rev. Lett. 67 (1991) 1681.
144. M. Caselle, M. Panero and S. Piemonte, in preparation.
145. H. Weyl, The classical groups. Their invariants and representations, Princeton University Press, Princeton, U.S.A. (1997).
146. C. Itzykson and M. Nauenberg, Unitary groups: representation and decompositions, Rev. Mod. Phys. 38 (1966) 95.
147. U.M. Heller and F. Karsch, One loop perturbative calculation of Wilson loops on finite lattices, Nucl. Phys. B 251 (1985) 254.
148. M. Laine and Y. Schröder, Two-loop QCD gauge coupling at high temperatures, JHEP 03 (2005) 067 [hep-ph/0503061].
149. M. Laine and M. Vepsäläinen, On the smallest screening masses in hot QCD, JHEP 09 (2009) 023 [arXiv:0906.4450].
150. O. Aharony and E. Karzbrun, On the effective action of confining strings, JHEP 06 (2009) 012 [arXiv:0903.1927].
151. F. Gliozzi, Dirac-Born-Infeld action from spontaneous breakdown of Lorentz symmetry in brane-world scenarios, Phys. Rev. D 84 (2011) 027702 [arXiv:1103.5377].
152. O. Aharony and M. Dodelson, Effective string theory and nonlinear Lorentz invariance, JHEP 02 (2012) 008 [arXiv:1111.5758].
153. T. Goto, Relativistic quantum mechanics of one-dimensional mechanical continuum and subsidiary condition of dual resonance model, Prog. Theor. Phys. 46 (1971) 1560.
154. Y. Nambu, Strings, monopoles and gauge fields, Phys. Rev. D 10 (1974) 4262.
155. M. Teper, Large-N and confining flux tubes as strings - A view from the lattice, Acta Phys. Polon. B 40 (2009) 3249 [arXiv:0912.3339].
156. K.J. Juge, J. Kuti and C. Morningstar, Fine structure of the QCD string spectrum, Phys. Rev. Lett. 90 (2003) 161601 [hep-lat/0207004].
157. M. Caselle, M. Hasenbusch and M. Panero, String effects in the 3D gauge Ising model, JHEP 01 (2003) 057 [hep-lat/0211012].
158. M. Panero, A numerical study of confinement in compact QED, JHEP 05 (2005) 066 [hep-lat/0503024].
159. M. Panero, A numerical study of a confined QQ̄ system in compact U(1) lattice gauge theory in 4D, Nucl. Phys. Proc. Suppl. 140 (2005) 665 [hep-lat/0408002].
160. F. Gliozzi, S. Lottini, M. Panero and A. Rago, Random percolation as a gauge theory, Nucl. Phys. B 719 (2005) 255 [cond-mat/0502339].
161. P. Giudice, F. Gliozzi and S. Lottini, The confining string beyond the free-string approximation in the gauge dual of percolation, JHEP 03 (2009) 104 [arXiv:0901.0748].
162. Y. Hidaka and R.D. Pisarski, Zero point energy of renormalized Wilson loops, Phys. Rev. D 80 (2009) 074504 [arXiv:0907.4609].
163. E. Megías, E. Ruiz Arriola and L. Salcedo, Trace anomaly, thermal power corrections and dimension two condensates in the deconfined phase, Phys. Rev. D 80 (2009) 056005 [arXiv:0903.1060].
164. 2 corrections and gauge/string duality, Phys. Rev. D 73 (2006) 107901 [hep-th/0603170].
165. R.D. Pisarski, Fuzzy bags and Wilson lines, Prog. Theor. Phys. Suppl. 168 (2007) 276 [hep-ph/0612191].
166. T. Umeda et al., Fixed scale approach to equation of state in lattice QCD, Phys. Rev. D 79 (2009) 051501 [arXiv:0809.2842].
167. A. Armoni and M. Shifman, On k string tensions and domain walls in N = 1 gluodynamics, Nucl. Phys. B 664 (2003) 233 [hep-th/0304127].
168. A. Armoni and M. Shifman, Remarks on stable and quasistable k strings at large-N, Nucl. Phys. B 671 (2003) 67 [hep-th/0307020].
169. J. Greensite, B. Lucini and A. Patella, k-string tensions and the 1/N expansion, Phys. Rev. D 83 (2011) 125019 [arXiv:1101.5344].
170. C.P. Korthals Altes and H.B. Meyer, Hot QCD, k-strings and the adjoint monopole gas model, hep-ph/0509018.
171. H.B. Meyer, Transport properties of the quark-gluon plasma: a lattice QCD perspective, Eur. Phys. J. A 47 (2011) 86 [arXiv:1104.3708].
172. F. Iachello, Lie algebras and applications, Lectures Notes in Physics volume 708, Springer U.S.A. (2006).
173. A.M. Perelomov and V.S. Popov, Casimir operators for U(n) and SU(n), Sov. J. Nucl. Phys. 3 (1966) 676.