183 research outputs found
Numerical Implementation of Harmonic Polylogarithms to Weight w = 8
We present the FORTRAN-code HPOLY.f for the numerical calculation of harmonic
polylogarithms up to w = 8 at an absolute accuracy of
or better. Using algebraic and argument relations the numerical representation
can be limited to the range . We provide replacement
files to map all harmonic polylogarithms to a basis and the usual range of
arguments to the above interval analytically. We also
briefly comment on a numerical implementation of real valued cyclotomic
harmonic polylogarithms.Comment: 19 pages LATEX, 3 Figures, ancillary dat
Subleading Logarithmic QED Initial State Corrections to to
Using the method of massive operator matrix elements, we calculate the
subleading QED initial state radiative corrections to the process for the first three logarithmic contributions from
to and compare their effects to the leading
contribution and one more subleading term .
The calculation is performed in the limit of large center of mass energies
squared . These terms supplement the known corrections to
, which were completed recently. Given the high precision at
future colliders operating at very large luminosity, these corrections are
important for concise theoretical predictions. The present calculation needs
the calculation of one more two--loop massive operator matrix element in QED.
The radiators are obtained as solutions of the associated Callen--Symanzik
equations in the massive case. The radiators can be expressed in terms of
harmonic polylogarithms to weight {\sf w = 6} of argument and and
in Mellin space by generalized harmonic sums. Numerical results are
presented on the position of the peak and corrections to the width,
. The corrections calculated result into a final theoretical accuracy
for and which is estimated to be of O(30 keV) at
an anticipated systematic accuracy at the FCC\_ee of \sim 100 keV. This
precision cannot be reached, however, by including only the corrections up to
.Comment: 58 pages, 3 Figure
The two-mass contribution to the three-loop pure singlet operator matrix element
We present the two-mass QCD contributions to the pure singlet operator matrix
element at three loop order in x-space. These terms are relevant for
calculating the structure function at as well as
for the matching relations in the variable flavor number scheme and the heavy
quark distribution functions at the same order. The result for the operator
matrix element is given in terms of generalized iterated integrals that include
square root letters in the alphabet, depending also on the mass ratio through
the main argument. Numerical results are presented.Comment: 28 papges Latex, 3 figure
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
Iterated integrals over letters induced by quadratic forms
An automated treatment of iterated integrals based on letters induced by
real-valued quadratic forms and Kummer--Poincar\'e letters is presented. These
quantities emerge in analytic single and multi--scale Feynman diagram
calculations. To compactify representations, one wishes to apply general
properties of these quantities in computer-algebraic implementations. We
provide the reduction to basis representations, expansions, analytic
continuation and numerical evaluation of these quantities.Comment: 14 pages LATEX, 1 anc. fil
Heavy Flavor Wilson Coefficients in Deep-Inelastic Scattering: Recent Results
We present recent analytic results for the 3-loop corrections to the massive
operator matrix element for further color factors. These results
have been obtained using the method of arbitrarily large moments. We also give
an overview on the results which were obtained solving all difference and
differential equations for the corresponding master integrals that factorize at
first order.Comment: 11 pages Latex, To appear in the Proceedings of: QCDEV2017, JLAB,
Newport News, VA, USA, May 22-26, 2017; Po
The 3-Loop Pure Singlet Heavy Flavor Contributions to the Structure Function and the Anomalous Dimension
The pure singlet asymptotic heavy flavor corrections to 3-loop order for the
deep-inelastic scattering structure function and the corresponding
transition matrix element in the variable flavor number
scheme are computed. In Mellin- space these inclusive quantities depend on
generalized harmonic sums. We also recalculate the complete 3-loop pure singlet
anomalous dimension for the first time. Numerical results for the Wilson
coefficients, the operator matrix element and the contribution to the structure
function are presented.Comment: 85 pages Latex, 14 Figures, 2 style file
Large scale analytic calculations in quantum field theories
We present a survey on the mathematical structure of zero- and single scale
quantities and the associated calculation methods and function spaces in higher
order perturbative calculations in relativistic renormalizable quantum field
theories.Comment: 25 pages Latex, 1 style fil
Calculating Three Loop Ladder and V-Topologies for Massive Operator Matrix Elements by Computer Algebra
Three loop ladder and -topology diagrams contributing to the massive
operator matrix element are calculated. The corresponding objects can
all be expressed in terms of nested sums and recurrences depending on the
Mellin variable and the dimensional parameter . Given these
representations, the desired Laurent series expansions in can be
obtained with the help of our computer algebra toolbox. Here we rely on
generalized hypergeometric functions and Mellin-Barnes representations, on
difference ring algorithms for symbolic summation, on an optimized version of
the multivariate Almkvist-Zeilberger algorithm for symbolic integration, and on
new methods to calculate Laurent series solutions of coupled systems of
differential equations. The solutions can be computed for general coefficient
matrices directly for any basis also performing the expansion in the
dimensional parameter in case it is expressible in terms of indefinite nested
product-sum expressions. This structural result is based on new results of our
difference ring theory. In the cases discussed we deal with iterative sum- and
integral-solutions over general alphabets. The final results are expressed in
terms of special sums, forming quasi-shuffle algebras, such as nested harmonic
sums, generalized harmonic sums, and nested binomially weighted (cyclotomic)
sums. Analytic continuations to complex values of are possible through the
recursion relations obeyed by these quantities and their analytic asymptotic
expansions. The latter lead to a host of new constants beyond the multiple zeta
values, the infinite generalized harmonic and cyclotomic sums in the case of
-topologies.Comment: 110 pages Latex, 4 Figure
The massive 3-loop operator matrix elements with two masses and the generalized variable flavor number scheme
We report on our latest results in the calculation of the two--mass
contributions to 3--loop operator matrix elements (OMEs). These OMEs are needed
to compute the corresponding contributions to the deep-inealstic scattering
structure functions and to generalize the variable flavor number scheme by
including both charm and bottom quarks. We present the results for the
non-singlet and OMEs, and compare the size of their contribution
relative to the single mass case. Results for the gluonic OME are
given in the physical case, going beyond those presented in a previous
publication where scalar diagrams were computed. We also discuss our recently
published two--mass contribution to the pure singlet OME, and present an
alternative method of calculating the corresponding diagrams.Comment: 20 pages Latex, 5 Figures, different style file
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