Abstract
We derive analytic results for scalar massless bosonic vacuum sum-integrals at two loops. Building upon a recent factorization proof of massive two-loop vacuum integrals, we are able to solve the corresponding Matsubara sums and map the result onto one-loop structures, thereby proving factorization also in the sum-integral setting. Analytic results are provided for generic integer-valued propagator- and numerator-powers of the class of sum-integrals under consideration, allowing to eliminate them from any perturbative expansion, dramatically simplifying the evaluation of some observables encountered e.g. in hot QCD.
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References
F. Gross et al., 50 Years of Quantum Chromodynamics, Eur. Phys. J. C 83 (2023) 1125 [arXiv:2212.11107] [INSPIRE].
J.I. Kapusta and C. Gale, Finite-Temperature Field Theory, Cambridge University Press (2023).
M.L. Bellac, Thermal Field Theory, Cambridge University Press (2011).
M. Laine and A. Vuorinen, Basics of Thermal Field Theory, Springer (2016) [https://doi.org/10.1007/978-3-319-31933-9] [INSPIRE].
J. Ghiglieri, A. Kurkela, M. Strickland and A. Vuorinen, Perturbative Thermal QCD: Formalism and Applications, Phys. Rept. 880 (2020) 1 [arXiv:2002.10188] [INSPIRE].
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] [INSPIRE].
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] [INSPIRE].
G. Boyd et al., Thermodynamics of SU(3) lattice gauge theory, Nucl. Phys. B 469 (1996) 419 [hep-lat/9602007] [INSPIRE].
S. Borsanyi et al., Precision SU(3) lattice thermodynamics for a large temperature range, JHEP 07 (2012) 056 [arXiv:1204.6184] [INSPIRE].
L. Giusti and M. Pepe, Equation of state of the SU(3) Yang-Mills theory: A precise determination from a moving frame, Phys. Lett. B 769 (2017) 385 [arXiv:1612.00265] [INSPIRE].
E. Braaten and R.D. Pisarski, Resummation and gauge invariance of the gluon damping rate in hot QCD, Phys. Rev. Lett. 64 (1990) 1338 [INSPIRE].
E. Braaten and R.D. Pisarski, Soft amplitudes in hot gauge theories: a general analysis, Nucl. Phys. B 337 (1990) 569 [INSPIRE].
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] [INSPIRE].
N. Haque et al., Three-loop HTLpt thermodynamics at finite temperature and chemical potential, JHEP 05 (2014) 027 [arXiv:1402.6907] [INSPIRE].
J.O. Andersen, N. Haque, M.G. Mustafa and M. Strickland, Three-loop hard-thermal-loop perturbation theory thermodynamics at finite temperature and finite baryonic and isospin chemical potential, Phys. Rev. D 93 (2016) 054045 [arXiv:1511.04660] [INSPIRE].
F. Karsch, A. Patkos and P. Petreczky, Screened perturbation theory, Phys. Lett. B 401 (1997) 69 [hep-ph/9702376] [INSPIRE].
P.H. Ginsparg, First order and second order phase transitions in gauge theories at finite temperature, Nucl. Phys. B 170 (1980) 388 [INSPIRE].
T. Appelquist and R.D. Pisarski, High-temperature yang-mills theories and three-dimensional quantum chromodynamics, Phys. Rev. D 23 (1981) 2305 [INSPIRE].
K. Kajantie, M. Laine, K. Rummukainen and M.E. Shaposhnikov, Generic rules for high temperature dimensional reduction and their application to the standard model, Nucl. Phys. B 458 (1996) 90 [hep-ph/9508379] [INSPIRE].
E. Braaten and A. Nieto, Free energy of QCD at high temperature, Phys. Rev. D 53 (1996) 3421 [hep-ph/9510408] [INSPIRE].
K. Kajantie, M. Laine, K. Rummukainen and Y. Schröder, The pressure of hot QCD up to g6 ln(1/g), Phys. Rev. D 67 (2003) 105008 [hep-ph/0211321] [INSPIRE].
A. Vuorinen, The pressure of QCD at finite temperatures and chemical potentials, Phys. Rev. D 68 (2003) 054017 [hep-ph/0305183] [INSPIRE].
T. Gorda, R. Paatelainen, S. Säppi and K. Seppänen, Equation of state of cold quark matter to \( O\left({\alpha}_s^3\ln {\alpha}_s\right) \), Phys. Rev. Lett. 131 (2023) 181902 [arXiv:2307.08734] [INSPIRE].
F. Di Renzo et al., The Leading non-perturbative coefficient in the weak-coupling expansion of hot QCD pressure, JHEP 07 (2006) 026 [hep-ph/0605042] [INSPIRE].
M. Laine and Y. Schröder, Quark mass thresholds in QCD thermodynamics, Phys. Rev. D 73 (2006) 085009 [hep-ph/0603048] [INSPIRE].
Y. Schröder, Weak-coupling expansion of the hot QCD pressure, PoS JHW2005 (2006) 029 [hep-ph/0605057] [INSPIRE].
P. Navarrete and Y. Schröder, Tackling the infamous g6 term of the QCD pressure, PoS LL2022 (2022) 014 [arXiv:2207.10151] [INSPIRE].
J. Österman, P. Schicho and A. Vuorinen, Integrating by parts at finite density, JHEP 08 (2023) 212 [arXiv:2304.05427] [INSPIRE].
K.G. Chetyrkin and F.V. Tkachov, Integration by parts: The algorithm to calculate β-functions in 4 loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
S. Laporta, High-precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
M. Nishimura and Y. Schröder, IBP methods at finite temperature, JHEP 09 (2012) 051 [arXiv:1207.4042] [INSPIRE].
A.I. Davydychev and J.B. Tausk, Two loop selfenergy diagrams with different masses and the momentum expansion, Nucl. Phys. B 397 (1993) 123 [INSPIRE].
A.I. Davydychev and Y. Schröder, Recursion-free solution for two-loop vacuum integrals with “collinear” masses, JHEP 12 (2022) 047 [arXiv:2210.10593] [INSPIRE].
A.A. Vladimirov, Method for computing renormalization group functions in dimensional renormalization scheme, Theor. Math. Phys. 43 (1980) 417 [INSPIRE].
O.V. Tarasov, Generalized recurrence relations for two loop propagator integrals with arbitrary masses, Nucl. Phys. B 502 (1997) 455 [hep-ph/9703319] [INSPIRE].
Y. Schröder, Loops for hot QCD, Nucl. Phys. B Proc. Suppl. 183 (2008) 296 [arXiv:0807.0500] [INSPIRE].
I. Ghisoiu and Y. Schröder, A new three-loop sum-integral of mass dimension two, JHEP 09 (2012) 016 [arXiv:1207.6214] [INSPIRE].
I. Ghisoiu and Y. Schröder, A new method for taming tensor sum-integrals, JHEP 11 (2012) 010 [arXiv:1208.0284] [INSPIRE].
M. Laine, P. Schicho and Y. Schröder, A QCD Debye mass in a broad temperature range, Phys. Rev. D 101 (2020) 023532 [arXiv:1911.09123] [INSPIRE].
M. Laine, M. Meyer and G. Nardini, Thermal phase transition with full 2-loop effective potential, Nucl. Phys. B 920 (2017) 565 [arXiv:1702.07479] [INSPIRE].
Acknowledgments
We wish to thank I. Kondrashuk for discussions at early stages of this work. The work of A.D. was partially supported by CONICYT PCI/MEC 80180071. P.N. has been supported by an ANID grant Magíster Nacional No. 22211544, and by the Academy of Finland grants No. 347499, 353772 and 1322507. Y.S. acknowledges support from ANID under FONDECYT projects No. 1191073 and 1231056.
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Davydychev, A.I., Navarrete, P. & Schröder, Y. Factorizing two-loop vacuum sum-integrals. J. High Energ. Phys. 2024, 104 (2024). https://doi.org/10.1007/JHEP02(2024)104
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DOI: https://doi.org/10.1007/JHEP02(2024)104