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Dimensionally regularized Polyakov loop correlators in hot QCD

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Abstract

A popular observable in finite-temperature lattice QCD is the so-called singlet quark-antiquark free energy, conventionally defined in Coulomb gauge. In an effort to interpret the existing numerical data on this observable, we compute it at order \( \mathcal{O}\left( {\alpha_s^2} \right) \) in continuum, and analyze the result at various distance scales. At short distances (r ≪ 1/πT) the behaviour matches that of the gauge-independent zero-temperature potential; on the other hand at large distances (r ≫ 1/πT) the singlet free energy appears to have a gauge-fixing related power-law tail. At infinite distance the result again becomes physical in the sense that it goes over to a gauge-independent disconnected contribution, the square of the expectation value of the trace of the Polyakov loop; we recompute this quantity at \( \mathcal{O}\left( {\alpha_s^2} \right) \), finding for pure SU(N c ) a different non-logarithmic term than in previous literature, and adding for full QCD the quark contribution. We also discuss the value of the singlet free energy in a general covariant gauge, as well as the behaviour of the cyclic Wilson loop that is obtained if the singlet free energy is made gauge-independent by inserting straight spacelike Wilson lines into the observable. Comparisons with lattice data are carried out where possible.

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References

  1. T. Matsui and H. Satz, J/ψ suppression by quark-gluon plasma formation, Phys. Lett. B 178 (1986) 416 [SPIRES].

    ADS  Google Scholar 

  2. M. Laine, O. Philipsen, P. Romatschke and M. Tassler, Real-time static potential in hot QCD, JHEP 03 (2007) 054 [hep-ph/0611300] [SPIRES].

    Article  ADS  Google Scholar 

  3. A. Beraudo, J.P. Blaizot and C. Ratti, Real and imaginary-time \( Q\bar Q \) correlators in a thermal medium, Nucl. Phys. A 806 (2008) 312 [arXiv:0712.4394] [SPIRES].

    ADS  Google Scholar 

  4. M.A. Escobedo and J. Soto, Non-relativistic bound states at finite temperature: the hydrogen atom, Phys. Rev. A 78 (2008) 032520 [arXiv:0804.0691] [SPIRES].

    ADS  Google Scholar 

  5. N. Brambilla, J. Ghiglieri, A. Vairo and P. Petreczky, Static quark-antiquark pairs at finite temperature, Phys. Rev. D 78 (2008) 014017 [arXiv:0804.0993] [SPIRES].

    ADS  Google Scholar 

  6. A. Rothkopf, T. Hatsuda and S. Sasaki, Proper heavy-quark potential from a spectral decomposition of the thermal Wilson loop, arXiv:0910.2321 [SPIRES].

  7. O. Kaczmarek, F. Karsch, E. Laermann and M. Lütgemeier, Heavy quark potentials in quenched QCD at high temperature, Phys. Rev. D 62 (2000) 034021 [hep-lat/9908010] [SPIRES].

    ADS  Google Scholar 

  8. 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] [SPIRES].

    ADS  Google Scholar 

  9. P. Petreczky and K. Petrov, Free energy of a static quark anti-quark pair and the renormalized Polyakov loop in three flavor QCD, Phys. Rev. D 70 (2004) 054503 [hep-lat/0405009] [SPIRES].

    ADS  Google Scholar 

  10. O. Kaczmarek and F. Zantow, Static quark anti-quark interactions in zero and finite temperature QCD. I: Heavy quark free energies, running coupling and quarkonium binding, Phys. Rev. D 71 (2005) 114510 [hep-lat/0503017] [SPIRES].

    ADS  Google Scholar 

  11. WHOT-QCD collaboration, Y. Maezawa et al., Heavy-quark free energy, Debye mass and spatial string tension at finite temperature in two flavor lattice QCD with Wilson quark action, Phys. Rev. D 75 (2007) 074501 [hep-lat/0702004] [SPIRES].

    ADS  Google Scholar 

  12. S. Gupta, K. Hübner and O. Kaczmarek, Renormalized Polyakov loops in many representations, Phys. Rev. D 77 (2008) 034503 [arXiv:0711.2251] [SPIRES].

    ADS  Google Scholar 

  13. A. Bazavov, P. Petreczky and A. Velytsky, Static quark anti-quark pair in SU(2) gauge theory, Phys. Rev. D 78 (2008) 114026 [arXiv:0809.2062] [SPIRES].

    ADS  Google Scholar 

  14. F. Zantow, O. Kaczmarek, F. Karsch and P. Petreczky, Short distance physics with heavy quark potentials, Nucl. Phys. (Proc. Suppl.) 106 (2002) 519 [hep-lat/0110103] [SPIRES].

    Article  ADS  Google Scholar 

  15. WHOT-QCD collaboration, Y. Maezawa et al., Free energies of heavy quarks in full-QCD lattice simulations with Wilson-type quark action, Nucl. Phys. A 830 (2009) 247c [arXiv:0907.4203] [SPIRES].

    Google Scholar 

  16. 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 [SPIRES].

    ADS  Google Scholar 

  17. A.D. Linde, Infrared problem in thermodynamics of the Yang-Mills gas, Phys. Lett. B 96 (1980) 289 [SPIRES].

    ADS  Google Scholar 

  18. D.J. Gross, R.D. Pisarski and L.G. Yaffe, QCD and instantons at finite temperature, Rev. Mod. Phys. 53 (1981) 43 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  19. P.H. Ginsparg, First order and second order phase transitions in gauge theories at finite temperature, Nucl. Phys. B 170 (1980) 388 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  20. T. Appelquist and R.D. Pisarski, High-temperature Yang-Mills theories and three-dimensional Quantum Chromodynamics, Phys. Rev. D 23 (1981) 2305 [SPIRES].

    ADS  Google Scholar 

  21. E. Braaten and A. Nieto, Free energy of QCD at high temperature, Phys. Rev. D 53 (1996) 3421 [hep-ph/9510408] [SPIRES].

    ADS  Google Scholar 

  22. E. Gava and R. Jengo, Perturbative evaluation of the thermal Wilson loop, Phys. Lett. B 105 (1981) 285 [SPIRES].

    ADS  Google Scholar 

  23. M. Laine and Y. Schröder, Two-loop QCD gauge coupling at high temperatures, JHEP 03 (2005) 067 [hep-ph/0503061] [SPIRES].

    Article  ADS  Google Scholar 

  24. E. Braaten and A. Nieto, Asymptotic behavior of the correlator for Polyakov loops, Phys. Rev. Lett. 74 (1995) 3530 [hep-ph/9410218] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  25. S. Nadkarni, Non-Abelian Debye screening: the color-averaged potential, Phys. Rev. D 33 (1986) 3738 [SPIRES].

    ADS  Google Scholar 

  26. A.K. Rebhan, The non-Abelian Debye mass at next-to-leading order, Phys. Rev. D 48 (1993) R3967 [hep-ph/9308232] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  27. E. Braaten and A. Nieto, Next-to-leading order Debye mass for the quark-gluon plasma, Phys. Rev. Lett. 73 (1994) 2402 [hep-ph/9408273] [SPIRES].

    Article  ADS  Google Scholar 

  28. A.M. Polyakov, Gauge fields as rings of glue, Nucl. Phys. B 164 (1980) 171 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  29. V.S. Dotsenko and S.N. Vergeles, Renormalizability of phase factors in the non-Abelian gauge theory, Nucl. Phys. B 169 (1980) 527 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  30. J.-L. Gervais and A. Neveu, The slope of the leading Regge trajectory in Quantum Chromodynamics, Nucl. Phys. B 163 (1980) 189 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  31. I.Ya. Arefeva, Quantum contour field equations, Phys. Lett. B 93 (1980) 347 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  32. N. Brambilla, J. Ghiglieri, P. Petreczky and A. Vairo, The Polyakov loop and correlator of Polyakov loops at next-to-leading order, to appear [TUM-EFT 2/09].

  33. K. Kajantie, M. Laine, K. Rummukainen and M.E. Shaposhnikov, 3d SU(N) + adjoint Higgs theory and finite-temperature QCD, Nucl. Phys. B 503 (1997) 357 [hep-ph/9704416] [SPIRES].

    Article  ADS  Google Scholar 

  34. M. Laine and M. Vepsäläinen, On the smallest screening mass in hot QCD, JHEP 09 (2009) 023 [arXiv:0906.4450] [SPIRES].

    Article  Google Scholar 

  35. M. Cheng et al., The spatial string tension and dimensional reduction in QCD, Phys. Rev. D 78 (2008) 034506 [arXiv:0806.3264] [SPIRES].

    ADS  Google Scholar 

  36. J. Alanen, K. Kajantie and V. Suur-Uski, Spatial string tension of finite temperature QCD matter in gauge/gravity duality, Phys. Rev. D 80 (2009) 075017 [arXiv:0905.2032] [SPIRES].

    Google Scholar 

  37. E. Megías, E. Ruiz Arriola and L.L. Salcedo, Dimension two condensates and the Polyakov loop above the deconfinement phase transition, JHEP 01 (2006) 073 [hep-ph/0505215] [SPIRES].

    Article  ADS  Google Scholar 

  38. 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] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  39. M. Laine and M. Vepsäläinen, Mesonic correlation lengths in high-temperature QCD, JHEP 02 (2004) 004 [hep-ph/0311268] [SPIRES].

    Article  ADS  Google Scholar 

  40. M. Vepsäläinen, Mesonic screening masses at high temperature and finite density, JHEP 03 (2007) 022 [hep-ph/0701250] [SPIRES].

    Article  ADS  Google Scholar 

  41. S. Nadkarni, Non-Abelian Debye screening. II. The singlet potential, Phys. Rev. D 34 (1986) 3904 [SPIRES].

    ADS  Google Scholar 

  42. 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] [SPIRES].

    ADS  Google Scholar 

  43. Y. Burnier, M. Laine and M. Vepsäläinen, Heavy quark medium polarization at next-to-leading order, JHEP 02 (2009) 008 [arXiv:0812.2105] [SPIRES].

    Article  ADS  Google Scholar 

  44. W. Fischler, Quark-anti-quark potential in QCD, Nucl. Phys. B 129 (1977) 157 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  45. Y. Schröder, The static potential in QCD to two loops, Phys. Lett. B 447 (1999) 321 [hep-ph/9812205] [SPIRES].

    ADS  Google Scholar 

  46. P. Arnold and L.G. Yaffe, The non-Abelian Debye screening length beyond leading order, Phys. Rev. D 52 (1995) 7208 [hep-ph/9508280] [SPIRES].

    ADS  Google Scholar 

  47. M. Laine and O. Philipsen, The non-perturbative QCD Debye mass from a Wilson line operator, Phys. Lett. B 459 (1999) 259 [hep-lat/9905004] [SPIRES].

    Article  ADS  Google Scholar 

  48. N. Iqbal and H.B. Meyer, Spatial correlators in strongly coupled plasmas, JHEP 11 (2009) 029 [arXiv:0909.0582] [SPIRES].

    Article  Google Scholar 

  49. S. Aoyama, The renormalization of the string operator in QCD, Nucl. Phys. B 194 (1982) 513 [SPIRES].

    Article  ADS  Google Scholar 

  50. O. Kaczmarek, O. Philipsen and M. Tassler, private communication.

  51. J.L. Albacete, Y.V. Kovchegov and A. Taliotis, Heavy quark potential at finite temperature in AdS/CFT revisited, Phys. Rev. D 78 (2008) 115007 [arXiv:0807.4747] [SPIRES].

    ADS  Google Scholar 

  52. Y. Burnier, M. Laine and M. Vepsäläinen, Heavy quarkonium in any channel in resummed hot QCD, JHEP 01 (2008) 043 [arXiv:0711.1743] [SPIRES].

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Faculty of Physics, University of Bielefeld, D-33501, Bielefeld, Germany

    Y. Burnier & M. Laine

  2. Department of Physics, University of Helsinki, P.O.Box 64, FI-00014, Helsinki, Finland

    M. Vepsäläinen

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  1. Y. Burnier
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  2. M. Laine
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  3. M. Vepsäläinen
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Correspondence to Y. Burnier.

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Burnier, Y., Laine, M. & Vepsäläinen, M. Dimensionally regularized Polyakov loop correlators in hot QCD. J. High Energ. Phys. 2010, 54 (2010). https://doi.org/10.1007/JHEP01(2010)054

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