@**techreport**{RISC6629,author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. Goedicke and A. von Manteuffel and C. Schneider and K. Schoenwald},

title = {{The Unpolarized and Polarized Single-Mass Three-Loop Heavy Flavor Operator Matrix Elements $A_{gg, Q}$ and $Delta A_{gg, Q}$}},

language = {english},

abstract = {We calculate the gluonic massive operator matrix elements in the unpolarized and polarized
cases, $A_{gg,Q}(x,mu^2)$ and $Delta A_{gg,Q}(x,mu^2)$, at three--loop order for a single
mass. These quantities contribute to the matching of the gluon distribution in the variable
flavor number scheme. The polarized operator matrix element is calculated in the Larin scheme.
These operator matrix elements contain finite binomial and inverse binomial sums in Mellin
$N$--space and iterated integrals over square root--valued alphabets in momentum fraction
$x$--space. We derive the necessary analytic relations for the analytic continuation of these
quantities from the even or odd Mellin moments into the complex plane, present analytic expressions
in momentum fraction $x$--space and derive numerical results. The present results complete the
gluon transition matrix elements both of the single-- and double--mass variable flavor number
scheme to three--loop order.
},

number = {22-15},

year = {2022},

month = {November},

note = {arXiv:2211.05462 [hep-ph]},

keywords = {Feynman integrals, linear difference equations, linear differential equations, binomial sums, harmonic sums, iterative integrals, computer algebra},

length = {48},

license = {CC BY 4.0 International},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Altenberger Straße 69, 4040 Linz, Austria},

issn = {2791-4267 (online)}

}