quantum chromodynamics and number theory

E. Elizalde, S. Leseduarte and S. Zerbini, "Mellin transform techniques for zeta-function resummations"

"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 "

A. Petermann, "The so-called renormalization group method applied to the specific prime number logarithmic decrease"

"A so-called Renormalization Group (RG) analysis is performed in order to shed some light on why the density of prime numbers in N* decreases like the single power of the inverse naperian logarithm."

"...in this note, our aim is to look for the deep reason why the density of primes decreases with the single power of the natural logarithm. We hope that we have been able to shed some light on this fact: the breaking of a symmetry, namely that of scale invariance...is the very factor repsonsible for this specific decrease.

The coincidence of the results obtained is striking when compared to the formulas of the first non-trivial approximation of Quantum ChromoDynamics...But a main common feature emerges: in both cases the two fields are afflicted by the same broken symmetry, that of scale invariance."

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."

renormalisation, regularisation and number theory


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