Properties of the free energy density using the principle ofmaximum conformality

Shi Bu; Xing-Gang Wu; Jian-Ming Shen; Jun Zeng

doi:10.1088/1674-1137/42/8/083105

Chinese Physics C> 2018, Vol. 42> Issue(8) : 083105 DOI: 10.1088/1674-1137/42/8/083105

Properties of the free energy density using the principle ofmaximum conformality

1.
Department of Physics, Chongqing University, Chongqing 401331, China

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Abstract：
We present a detailed study on the properties of the free energy density at high temperature by applying the principle of maximum conformality (PMC) scale-setting method within effective field theory. The PMC utilizes the renormalization group equation recursively to identify the occurrence and pattern of the non-conformal {β_i}-terms, and determines the optimal renormalization scale at each order. Our analysis shows that a more accurate free energy density up to g_s⁵-order level without renormalization scale dependence can be achieved by applying the PMC. We also observe that by using a smaller factorization scale around the effective parameter m_E, the PMC prediction is consistent with the lattice QCD prediction derived at low temperature.
- perturbative QCD ,
- free energy density ,
- renormalization ,
- QCD high-order corrections

References

[1]	K. Kajantie, M. Laine, K. Rummukainen, and Y. Schroder, Phys. Rev. Lett., 86:10 (2001)
[2]	E. Braaten and A. Nieto, Phys. Rev. D, 53:3421 (1996)
[3]	G. Boyd, J. Engels, F. Karsch, E. Laermann, C. Legeland, M. Lutgemeier, and B. Petersson, Nucl. Phys. B, 469:419 (1996)
[4]	M. Okamoto et al (CP-PACS Collaboration), Phys. Rev. D, 60:094510 (1999)
[5]	C. Bernard et al (MILC Collaboration), Phys. Rev. D, 71:034504 (2005)
[6]	Y. Aoki, Z. Fodor, S. D. Katz, and K. K. Szabo, JHEP, 0601:089 (2006)
[7]	C. Bernard et al, Phys. Rev. D, 75:094505 (2007)
[8]	M. Cheng et al, Phys. Rev. D, 77:01451 (2008)
[9]	G. Endrodi, Z. Fodor, S. D. Katz, and K. K. Szabo, PoS LATTICE, 2007:228 (2007)
[10]	F. Di Renzo, M. Laine, Y. Schroder, and C. Torrero, JHEP, 0809:061 (2008)
[11]	A. Hietanen, K. Kajantie, M. Laine, K. Rummukainen, and Y. Schroder, Phys. Rev. D, 79:045018 (2009)
[12]	A. Bazavov et al, Phys. Rev. D, 80:014504 (2009)
[13]	S. Borsanyi, G. Endrodi, Z. Fodor, A. Jakovac, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szabo, JHEP, 1011:077 (2010)
[14]	S. BorsSnyi, G. Endrodi, Z. Fodor, S. D. Katz, and K. K. Szab_, PoS Lattice, 2010:171 (2014)
[15]	S. Borsanyi, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szabo, JHEP, 1208:053 (2012)
[16]	S. Borsanyi, G. Endrodi, Z. Fodor, S. D. Katz, and K. K. Szabo, JHEP, 1207:056 (2012)
[17]	S. Borsanyi et al, JHEP, 1208:126 (2012)
[18]	A. Bazavov et al (HotQCD Collaboration), Phys. Rev. D, 86:034509 (2012)
[19]	A. Bazavov et al, Phys. Rev. Lett., 109:192302 (2012)
[20]	O. Philipsen, Prog. Part. Nucl. Phys., 70:55 (2013)
[21]	S. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szabo, Phys. Rev. Lett., 111:062005 (2013)
[22]	A. Bazavov et al, Phys. Rev. Lett., 111:082301 (2013)
[23]	U. Gursoy, Acta Phys. Polon. B, 47:2509 (2016)
[24]	E. V. Shuryak, Sov. Phys. JETP, 47:212 (1978)
[25]	J. I. Kapusta, Nucl. Phys. B, 148:461 (1979)
[26]	T. Toimela, Phys. Lett., 124B:407 (1983)
[27]	P. B. Arnold and C. X. Zhai, Phys. Rev. D, 50:7603 (1994)
[28]	P. B. Arnold and C. X. Zhai, Phys. Rev. D, 51:1906 (1995)
[29]	C. X. Zhai and B. M. Kastening, Phys. Rev. D, 52:7232 (1995)
[30]	E. Braaten and A. Nieto, Phys. Rev. Lett., 76:1417 (1996)
[31]	K. Kajantie, M. Laine, K. Rummukainen, and Y. Schroder, Phys. Rev. D, 67:105008 (2003)
[32]	A. D. Linde, Rept. Prog. Phys., 42:389 (1979)
[33]	A. D. Linde, Phys. Lett., 96B:289 (1980)
[34]	D. J. Gross, R. D. Pisarski, and L. G. Yaffe, Rev. Mod. Phys., 53:43 (1981)
[35]	T. Matsubara, Prog. Theor. Phys., 14:351 (1955)
[36]	W. Celmaster and R. J. Gonsalves, Phys. Rev. D, 20:1420 (1979)
[37]	L. F. Abbott, Phys. Rev. Lett., 44:1569 (1980)
[38]	A. J. Buras, Rev. Mod. Phys., 52:199 (1980)
[39]	G. Grunberg, Phys. Lett., 95B:70 (1980)
[40]	P. M. Stevenson, Phys. Rev. D, 23:2916 (1981)
[41]	S. J. Brodsky, G. P. Lepage, and P. B. Mackenzie, Phys. Rev. D, 28:228 (1983)
[42]	X. G. Wu, S. J. Brodsky, and M. Mojaza, Prog. Part. Nucl. Phys., 72:44 (2013)
[43]	X. G. Wu, Y. Ma, S. Q. Wang, H. B. Fu, H. H. Ma, S. J. Brodsky, and M. Mojaza, Rept. Prog. Phys., 78 (2015) 126201.
[44]	M. Gell-Mann and F. E. Low, Phys. Rev., 95:1300 (1954)
[45]	M. Beneke, Phys. Rept., 317:1 (1999)
[46]	E. Gardi and G. Grunberg, Phys. Lett. B, 517:215 (2001)
[47]	F. Karsch, A. Patkos, and P. Petreczky, Phys. Lett. B, 401:69 (1997)
[48]	S. Chiku and T. Hatsuda, Phys. Rev. D, 58:076001 (1998)
[49]	J. O. Andersen, E. Braaten, and M. Strickland, Phys. Rev. D, 61:074016 (2000)
[50]	J. O. Andersen, E. Braaten, and M. Strickland, Phys. Rev. Lett., 83:2139 (1999)
[51]	J. O. Andersen, E. Braaten, and M. Strickland, Phys. Rev. D, 63:105008 (2001)
[52]	J. O. Andersen, L. E. Leganger, M. Strickland, and N. Su, Phys. Lett. B, 696:468 (2011)
[53]	J. O. Andersen, L. E. Leganger, M. Strickland, and N. Su, JHEP, 1108:053 (2011)
[54]	N. Haque, M. G. Mustafa, and M. Strickland, Phys. Rev. D, 87:no. 10, 105007 (2013)
[55]	N. Haque, A. Bandyopadhyay, J. O. Andersen, M. G. Mustafa, M. Strickland, and N. Su, JHEP, 1405:027 (2014)
[56]	M. Strickland, J. O. Andersen, A. Bandyopadhyay, N. Haque, M. G. Mustafa, and N. Su, Nucl. Phys. A, 931:841 (2014)
[57]	N. Haque, PoS ICPAQGP, 2015:057 (2017)
[58]	B. M. Kastening, Phys. Rev. D, 56:8107 (1997)
[59]	T. Hatsuda, Phys. Rev. D, 56:8111 (1997)
[60]	G. Cvetic and R. Kogerler, Phys. Rev. D, 66:105009 (2002)
[61]	G. Cvetic and R. Kogerler, Phys. Rev. D, 70:114016 (2004)
[62]	T. Hatsuda and T. Kunihiro, Phys. Rept., 247:221 (1994)
[63]	K. Fukushima, Phys. Rev. D, 68:045004 (2003)
[64]	K. Fukushima, Phys. Lett. B, 591:277 (2004)
[65]	C. Ratti, M. A. Thaler, and W. Weise, Phys. Rev. D, 73:014019 (2006)
[66]	S. Mukherjee, M. G. Mustafa, and R. Ray, Phys. Rev. D, 75:094015 (2007)
[67]	A. Bhattacharyya, P. Deb, S. K. Ghosh, and R. Ray, Phys. Rev. D, 82:014021 (2010)
[68]	A. Bhattacharyya, P. Deb, A. Lahiri, and R. Ray, Phys. Rev. D, 83:014011 (2011)
[69]	M. Bluhm and B. Kampfer, Phys. Rev. D, 77:034004 (2008)
[70]	V. M. Bannur, JHEP, 0709:046 (2007)
[71]	V. M. Bannur, Phys. Rev. C, 78:045206 (2008)
[72]	F. G. Gardim and F. M. Steffens, Nucl. Phys. A, 825:222 (2009)
[73]	B. J. Schaefer, M. Wagner, and J. Wambach, PoS CPOD, 2009:017 (2009)
[74]	B. J. Schaefer, M. Wagner, and J. Wambach, Phys. Rev. D, 81:074013 (2010)
[75]	V. Skokov, B. Friman, and K. Redlich, Phys. Rev. C, 83:054904 (2011)
[76]	S. J. Brodsky and X. G. Wu, Phys. Rev. D, 85:034038 (2012)
[77]	S. J. Brodsky and X. G. Wu, Phys. Rev. Lett., 109:042002 (2012)
[78]	M. Mojaza, S. J. Brodsky, and X. G. Wu, Phys. Rev. Lett., 110:192001 (2013)
[79]	S. J. Brodsky, M. Mojaza, and X. G. Wu, Phys. Rev. D, 89:014027 (2014)
[80]	X. G. Wu, J. M. Shen, B. L. Du, and S. J. Brodsky, arXiv:1802.09154[hep-ph].
[81]	X. G. Wu, S. Q. Wang, and S. J. Brodsky, Front. Phys., 11:111201 (2016)
[82]	P. H. Ginsparg, Nucl. Phys. B, 170:388 (1980)
[83]	T. Appelquist and R. D. Pisarski, Phys. Rev. D, 23:2305 (1981)
[84]	S. Nadkarni, Phys. Rev. D, 27:917 (1983)
[85]	A. Bazavov, N. Brambilla, X. Garcia i Tormo, P. Petreczky, J. Soto, and A. Vairo, Phys. Rev. D, 907:074038 (2014)
[86]	S. Q. Wang, X. G. Wu, S. J. Brodsky, and M. Mojaza, Phys. Rev. D, 94:053003 (2016)
[87]	S. Q. Wang, X. G. Wu, X. C. Zheng, G. Chen, and J. M. Shen, J. Phys. G, 41:075010 (2014)

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Shi Bu, Xing-Gang Wu, Jian-Ming Shen and Jun Zeng. Properties of the free energy density using the principle ofmaximum conformality[J]. Chinese Physics C, 2018, 42(8): 083105. doi: 10.1088/1674-1137/42/8/083105

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沈阳化工大学材料科学与工程学院沈阳 110142

Properties of the free energy density using the principle ofmaximum conformality

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