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

Abstract
HTML
Reference
Related

PDF

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 _s ⁵-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)

[1]	Xu-Dong Huang,Xing-Gang Wu,Xu-Chang Zheng,Jiang Yan,Zhi-Fei Wu,Hong-Hao Ma. Precise determination of the top-quark on-shell mass ${\boldsymbol M_t}$ via its scale- invariant perturbative relation to the top-quark ${\overline{\bf MS}}$ mass ${{\overline {\boldsymbol m}}_{\boldsymbol t}({\overline {\boldsymbol m}}_{\boldsymbol t})}$ . Chinese Physics C, 2024, 48(5): 053113.doi:10.1088/1674-1137/ad2dbf
[2]	Zhi-Qing Zhang,Yan-Chao Zhao,Zhi-Lin Guan,Zhi-Jie Sun,Zi-Yu Zhang,Ke-Yi He. Quasi-two-body ${{\boldsymbol { B_{\bf (s)}\to} {\boldsymbol K^*\gamma\to} {\boldsymbol K\pi\gamma}}}$ decays in the perturbative QCD approach. Chinese Physics C, 2022, 46(12): 123105.doi:10.1088/1674-1137/ac89d1
[3]	Jun Zeng,Xing-Gang Wu,Xu-Chang Zheng,Jian-Ming Shen. Gauge dependence of the perturbative QCD predictions under the momentum-space subtraction scheme. Chinese Physics C, 2020, 44(11): 113102.doi:10.1088/1674-1137/abae4e
[4]	Yueling Yang,Yupei Guo,Junfeng Sun,Na Wang,Qin Chang,Gongru Lu.B_u→ψMdecays andS-Dwave mixing effects. Chinese Physics C, 2018, 42(11): 113102.doi:10.1088/1674-1137/42/11/113102
[5]	Pazilet Obul,Sayipjamal Dulat,Tie-Jiun Hou,Ablikim Tursun,Nijat Yalkun. Next-to-leading order QCD and electroweak corrections to Higgs-strahlung processes at the LHC. Chinese Physics C, 2018, 42(9): 093105.doi:10.1088/1674-1137/42/9/093105
[6]	Qing-Feng Sun,An-Min Wang. Next-to-leading order QCD corrections to the decay of Higgs to vector meson and Z boson. Chinese Physics C, 2018, 42(3): 033105.doi:10.1088/1674-1137/42/3/033105
[7]	Wei Sun,Andrei Alexandru,Ying Chen,Terrence Draper,Zhaofeng Liu,Yi-Bo Yang. Anatomy of the ρ resonance from lattice QCD at the physical poin. Chinese Physics C, 2018, 42(6): 063102.doi:10.1088/1674-1137/42/6/063102
[8]	Han Guo,Zi-Ping Zhang. A method for measuring D^*electromagnetic form factors in e⁺e^-annihilation. Chinese Physics C, 2017, 41(9): 093001.doi:10.1088/1674-1137/41/9/093001
[9]	Ai-Jun Ma,Ya Li,Wen-Fei Wang,Zhen-Jun Xiao.S-wave resonance contributions to B_(s)⁰→η_c(2S)π⁺π^-in the perturbative QCD factorization approach. Chinese Physics C, 2017, 41(8): 083105.doi:10.1088/1674-1137/41/8/083105
[10]	Chao Zhou,Mao Song,Gang Li,Ya-Jin Zhou,Jian-You Guo. Next-to-leading order QCD corrections to Higgs boson decay to quarkonium plus a photon. Chinese Physics C, 2016, 40(12): 123105.doi:10.1088/1674-1137/40/12/123105
[11]	WANG Rui,CHEN Ying,GONG Ming,LIU Chuan,LIU Yu-Bin,LIU Zhao-Feng,MA Jian-Ping,MENG Xiang-Fei,ZHANG Jian-Bo. Phase transition in finite density and temperature lattice QCD. Chinese Physics C, 2015, 39(6): 063103.doi:10.1088/1674-1137/39/6/063103
[12]	ZHANG Zhi-Qing,FU Jian-Hua. Analysis of B→a₁(1260)(b₁(1235))K^*decays in the perturbative QCD approach. Chinese Physics C, 2015, 39(3): 033102.doi:10.1088/1674-1137/39/3/033102
[13]	ZOU Zhi-Tian,ZHOU Rui,LÜ Cai-Dian. Pure annihilation type decays B⁰→D_s^-K₂^*+and B_s→Da₂in perturbative QCD approach. Chinese Physics C, 2013, 37(1): 013103.doi:10.1088/1674-1137/37/1/013103
[14]	ZOU Zhi-Tian,LI Cheng,LÜ Cai-Dian. Pure annihilation type D→PP(V) decays in the perturbative QCD approach. Chinese Physics C, 2013, 37(9): 093101.doi:10.1088/1674-1137/37/9/093101
[15]	WANG Wen-Fei,FAN Ying-Ying,XIAO Zhen-Jun. Semileptonic decays B_c→(η_c,J/Ψ)lvin the perturbative QCD approach. Chinese Physics C, 2013, 37(9): 093102.doi:10.1088/1674-1137/37/9/093102
[16]	JIN Dan,YANG Ya-Dong. Space- and time-like kaon electromagnetic form factors in perturbative QCD. Chinese Physics C, 2012, 36(10): 941-946.doi:10.1088/1674-1137/36/10/004
[17]	JIN Dan,YANG Ya-Dong. Radiative decays J/ψ → η^(')γ in perturbative QCD. Chinese Physics C, 2011, 35(8): 703-707.doi:10.1088/1674-1137/35/8/001
[18]	LIU Xin,ZHANG Zhi-Qing,XIAO Zhen-Jun. B→(J/ψ,η_c) K decays in the perturbative QCD approach. Chinese Physics C, 2010, 34(7): 937-943.doi:10.1088/1674-1137/34/7/002
[19]	LI Jing-Wu,YOU Fu-Yi. Perturbative QCD Study of B_(s)→φρ Decays. Chinese Physics C, 2007, 31(6): 523-531.
[20]	WANG Hong-Min,SUN Xian-Jing,HOU Zhao-Yu. Next-to-Leading Order QCD Corrections to Energy Loss Effect in Drell-Yan Process. Chinese Physics C, 2006, 30(7): 628-632.

Access

Get Citation

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

RIS(for EndNote,Reference Manager,ProCite)

BibTex

Txt

Milestone

Received: 2018-04-08

Fund

Supported by Natural Science Foundation of China (11625520)

Article Metric

Article Views(3560)
PDF Downloads(18)
Cited by(0)

Policy on re-use

To reuse of Open Access content published by CPC, for content published under the terms of the Creative Commons Attribution 3.0 license (“CC CY”), the users don’t need to request permission to copy, distribute and display the final published version of the article and to create derivative works, subject to appropriate attribution.

通讯作者:陈斌, bchen63@163.com

1.
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

Abstract：

References

Access

Article Metrics

Metrics

通讯作者:陈斌, bchen63@163.com

Email This Article

Properties of the free energy density using the principle ofmaximum conformality

HTML

目录