quantum chromodynamics and number theory

"Making use of inverse Mellin transform techniques for analytical continuation, an elegant proof and an extension of the zeta function regularization theorem is obtained...As an application of the method, the summation of the series which appear in the analytic computation (for different ranges of temperature) of the partition function of the string - basic in order to ascertain if QCD is some limit of a string theory - is performed."

E. Elizalde, "Effective Lagrangian for ordinary quarks in a background field", Nuclear Physics B 243 (1984) 398-410

[abstract:] "The explicit form of the effective QCD Lagrangian for ordinary, massive quarks in the presence of a constant, color-magnetic background field is given. Its calculation involves a non-trivial computation of the derivative of the generalized Riemann zeta function. The confining properties of the resulting Lagrangian, to one-loop order, are exhibited. They turn out to be very similar to those of the more simplified Lagrangians (without quarks) used by Adler and Piran in their attempt at a proof of quark confinement "

S. Abreu, M. Becchetti, C. Duhr and M.A. Ozcelik, "Two-loop master integrals for pseudo-scalar quarkonium and leptonium production and decay", Journal of High Energy Physics 09 (2022) 194

[summary:] The master integrals needed for two-loop pseudoscalar quarkonium and leptonium processes are evaluated analytically and to high numerical precision. Their function space goes beyond ordinary multiple polylogarithms to elliptic multiple polylogarithms and iterated integrals of Eisenstein series.

T. Gehrmann, P. Jakubčík, C.C. Mella, N. Syrrakos and L. Tancredi, "Two-loop helicity amplitudes for H+jet production to higher orders in the dimensional regulator", Journal of High Energy Physics 04 (2023) 016

[summary:] One- and two-loop helicity amplitudes for Higgs-plus-jet production are computed to higher powers of the dimensional regulator. A canonical basis turns the differential equations for the master integrals into expressions in two-variable multiple polylogarithms through transcendental weight six – a modern example of graded special-function structure arising inside a precision collider calculation.

L.R. Surguladze and M.A. Samuel, "On the renormalization group ambiguity of perturbative QCD for R(s) in e⁺e^- annihilation and \(R_{\tau}\) in \(\tau\)-decay", Physics Letters B 309 (1993) 157-162

[abstract:] "The \(O(\alpha_{s}^{3})\) perturbative QCD result for R(S) in e⁺e^- annihilation is given with explicit dependence on the scale parameter. We apply the three known approaches for resolving the scheme-scale ambiguity and we fix the scale for which all of the criteria tested are satisfied. We find the four-loop R(s) within the new scheme with flavor independent perturbative coefficients. . .

We find a remarkable cancellation of the Riemann zeta-functions at the 3-loop level. The theoretical uncertainty of the QCD effect in R(s) is estimated at 4%. The results of the analysis of \(R_{\tau}\) in \(\tau\)-decay are presented."

E. de Rafael, "Large-Nc QCD, harmonic sums and the Riemann zeros" (Invited talk at the Montpellier QCD 2010 Conference)

[abstract:] "It is shown that in large-Nc QCD, two-point functions of local operators become harmonic sums. We comment on the properties which follow from this fact. This has led us to an aside observation concerning the zeros of the Riemann zeta-function seen from the point of view of dispersion relations in quantum theory. "

E. de Rafael, "Large Nc QCD and Harmonic Sums" (preprint 11/2011, based on a talk at "Raymond Stora's 80th Birthday Party", LAPP, July 11th 2011)

[abstract:] "In the large-Nc limit of QCD, two-point functions of local operators become harmonic sums. I review some properties which follow from this fact and which are relevant for phenomenological applications. This has led us to consider a class of analytic number theory functions as toy models of large-Nc QCD which I also discuss."