1,363 research outputs found
Harmonic sums, Mellin transforms and Integrals
This paper describes algorithms to deal with nested symbolic sums over
combinations of harmonic series, binomial coefficients and denominators. In
addition it treats Mellin transforms and the inverse Mellin transformation for
functions that are encountered in Feynman diagram calculations. Together with
results for the values of the higher harmonic series at infinity the presented
algorithms can be used for the symbolic evaluation of whole classes of
integrals that were thus far intractable. Also many of the sums that had to be
evaluated seem to involve new results. Most of the algorithms have been
programmed in the language of FORM. The resulting set of procedures is called
SUMMER.Comment: 31 pages LaTeX, for programs, see http://norma.nikhef.nl/~t68/summe
The seven-gluon amplitude in multi-Regge kinematics beyond leading logarithmic accuracy
We present an all-loop dispersion integral, well-defined to arbitrary
logarithmic accuracy, describing the multi-Regge limit of the 2->5 amplitude in
planar N=4 super Yang-Mills theory. It follows from factorization, dual
conformal symmetry and consistency with soft limits, and specifically holds in
the region where the energies of all produced particles have been analytically
continued. After promoting the known symbol of the 2-loop N-particle MHV
amplitude in this region to a function, we specialize to N=7, and extract from
it the next-to-leading order (NLO) correction to the BFKL central emission
vertex, namely the building block of the dispersion integral that had not yet
appeared in the well-studied six-gluon case. As an application of our results,
we explicitly compute the seven-gluon amplitude at next-to-leading logarithmic
accuracy through 5 loops for the MHV case, and through 3 and 4 loops for the
two independent NMHV helicity configurations, respectively.Comment: 56 pages, 4 figures, 1 table; v2: minor corrections and
clarifications, matches published versio
The Two-mass Contribution to the Three-Loop Gluonic Operator Matrix Element
We calculate the two-mass QCD contributions to the massive operator matrix
element at in analytic form in Mellin
- and -space, maintaining the complete dependence on the heavy quark mass
ratio. These terms are important ingredients for the matching relations of the
variable flavor number scheme in the presence of two heavy quark flavors, such
as charm and bottom. In Mellin -space the result is given in the form of
nested harmonic, generalized harmonic, cyclotomic and binomial sums, with
arguments depending on the mass ratio. The Mellin inversion of these quantities
to -space gives rise to generalized iterated integrals with square root
valued letters in the alphabet, depending on the mass ratio as well. Numerical
results are presented.Comment: 99 pages LATEX, 2 Figure
Algorithms to Evaluate Multiple Sums for Loop Computations
We present algorithms to evaluate two types of multiple sums, which appear in
higher-order loop computations. We consider expansions of a generalized
hypergeometric-type sums, \sum_{n_1,...,n_N} [Gamma(a1.n+c1) Gamma(a2.n}+c2)
... Gamma(aM.n+cM)] / [Gamma(b1.n+d1) Gamma(b2.n+d2) ... Gamma(bM.n+dM)]
x1^n1...xN^nN with , etc., in a small parameter
epsilon around rational values of ci,di's. Type I sum corresponds to the case
where, in the limit epsilon -> 0, the summand reduces to a rational function of
nj's times x1^n1...xN^nN; ci,di's can depend on an external integer index. Type
II sum is a double sum (N=2), where ci,di's are half-integers or integers as
epsilon -> 0 and xi=1; we consider some specific cases where at most six Gamma
functions remain in the limit epsilon -> 0. The algorithms enable evaluations
of arbitrary expansion coefficients in epsilon in terms of Z-sums and multiple
polylogarithms (generalized multiple zeta values). We also present applications
of these algorithms. In particular, Type I sums can be used to generate a new
class of relations among generalized multiple zeta values. We provide a
Mathematica package, in which these algorithms are implemented.Comment: 30 pages, 2 figures; address of Mathematica package in Sec.6; version
to appear in J.Math.Phy
Regularized inner products of meromorphic modular forms and higher Green's functions
In this paper we study generalizations of quadratic form Poincar\'e series,
which naturally occur as outputs of theta lifts. Integrating against them
yields evaluations of higher Green's functions. For this we require a new
regularized inner product, which is of independent interest
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