arXiv subjects

J. Blümlein

Publications and source records attributed to J. Blümlein.

At least 19 recordsLinked to original sources

arXiv2026

The $q$-extension of iterated integrals and nested sums in quantum field theory

Analytic calculations of zero- and single-scale quantities in perturbative quantum field theory result into special numbers and functions, the first of which have been revealed during the last decades. These are generalizations of the polylogarithm in form of Kummer-Poincaré iterative integrals over special alphabets and extensions thereof.With growing order in the coupling constant, the polylogarithms, Nielsen integrals, the iterated integrals over linear denominator terms, cyclotomic letters, letters induced by quadratic forms, square-root valued letters, and more general functions contribute. For the nested sums we consider nested harmonic sums, generalized harmonic sums, nested sums implied by quadratic forms, cyclotomic harmonic sums, and nested sums containing central binomials. We construct the $q$-extensions of these special functions and of the nested sums, which are associated to them by the series expansion at $x=0$, and their Mellin transform in the $q$-free case. These functions are expected to play a role in perturbative calculations in the case of $q$-deformed commutation relations. For the simpler function spaces closed form solutions are presented. For more involved alphabets we present the algorithmic steps leading to the $q$-extension for the individual cases. We also derive the determining differential and difference equations of these higher transcendental functions. The $q$-extended special functions are quite different form the corresponding $μ$-extended functions.

math-ph↗

arXiv2026

The variable flavor number scheme to three-loop order

We describe the variable flavor number scheme to three-loop order, which modifies the massless parton densities by single- and two-mass effects and introduces heavy-quark parton distribution functions for charm and bottom. A renormalization group analysis shows the validity of this picture at large scales $Q^2$, where it resembles the non-power-suppressed heavy-flavor corrections completely. We also provide numerical implementations of a series of charged and neutral current Wilson coefficients.

hep-ph↗

arXiv2026

The $μ$-extension of iterated integrals and nested sums

The analytic integration of single-scale Feynman integrals emerging in perturbative calculations in quantum field theories can be performed within special classes of functions, which appear as consecutive generalizations of the polylogarithm in form of Kummer-Poincaré iterative integrals over special alphabets and extensions thereof. These are the polylogarithms, Nielsen integrals, the iterated integrals over linear denominator terms, cyclotomic letters, letters induced by quadratic forms, and square-root valued letters. These integrals are solutions of first-order factorizing differential equations. They are related to specific nested sums via the Mellin transform and their expansions around $x=0$. We construct the $μ$-extensions of these iterated integrals and the associated nested sums. We present closed form solutions or provide algorithms in the case of more involved cases to derive the respective $μ$-extensions and study the algebras of the $μ$-extended function spaces. Except for the case of square-root valued alphabets, the $μ$-extension maps into the same function space polynomially in $μ$. This is also the case for the associated nested sums. For square-root valued alphabets or sums containing central binomials, the $μ$-extension leads to higher transcendental functions. In all other cases the $μ$-extension preserves the Hopf algebra structure implied by the (quasi)shuffle product, by supplementing $μ$ to the ground field.

hep-th↗

arXiv2026

The complete three-loop unpolarized and polarized massive operator matrix elements and asymptotic Wilson coefficients

We report on the three-loop unpolarized and polarized massive operator matrix elements, with single- and two-mass corrections, and the associated deep-inelastic massive Wilson coefficients in the region $Q^2 \gg m_Q^2$, the calculation of which has been completed recently. We also provide fast and precise numerical representations of the massless Wilson coefficients, splitting functions to tree-loop order, and target-mass corrections in $x$-space well suited for QCD-fitting codes.

hep-ph↗

arXiv2026

The three-loop single-mass heavy-flavor corrections to the structure functions $F_2(x,Q^2)$ and $g_1(x,Q^2)$

We present quantitative results on the single-mass heavy-flavor contributions in the region of large virtualities $Q^2$ up to three-loop order to the unpolarized structure function $F_2(x,Q^2)$ and the polarized structure function $g_1(x,Q^2)$ for the first time. These results are relevant for precision QCD analyses of the World deep-inelastic data and the data taken at future colliders, such as the Electron--Ion Collider, since the scaling violations due to massless and massive Wilson coefficients are significantly different. In order to measure the strong coupling constant $α_s(M_Z^2)$ and the twist-2 parton distribution functions consistently at highest precision, the next-to-next-to-leading order corrections have to be taken into account. Furthermore, the complete three-loop corrections will allow to reduce the present theory error of the charm mass $m_c$ as measured form deep-inelastic data. We provide a fast and precise public numerical code for the unpolarized and polarized massive Wilson coefficients in the asymptotic region.

hep-ph↗

arXiv2025

The heavy quark-antiquark asymmetry in the variable flavor number scheme

The twist-2 heavy-quark and antiquark distributions, as defined in the variable flavor number scheme, turn out to be different due to QCD corrections from three-loop onward. This is caused by terms containing the color factor $d_{abc} d^{abc}$ in the heavy-flavor massive pure-singlet operator matrix elements (OMEs) $A^{\rm PS, s, (3)}_{Qq}$ for odd moments in the unpolarized case and for $ΔA^{\rm PS, s, (3)}_{Qq}$ for even moments in the polarized case. The dependence on the factorization scale of the OMEs is ruled by the anomalous dimensions $γ^{\rm NS, s, (2)}_{qq}$ and $Δγ^{\rm NS, s, (2)}_{qq}$. The polarized calculations are performed in the Larin scheme. We compute the corresponding three-loop heavy-flavor distributions $(Δ) f_Q(x,Q^2) - (Δ) f_{\overline{Q}}(x,Q^2)$. Compared to the sum of the heavy-quark and antiquark parton distributions, their difference is small, however, non-vanishing.

hep-ph↗

arXiv2025

The two-mass contributions to the three-loop massive operator matrix elements $\tilde{A}_{Qg}^{(3)}$ and $Δ\tilde{A}_{Qg}^{(3)}$

We calculate the two-mass three-loop contributions to the unpolarized and polarized massive operator matrix elements $\tilde{A}_{Qg}^{(3)}$ and $Δ\tilde{A}_{Qg}^{(3)}$ in $x$-space for a general mass ratio by using a semi-analytic approach. We also compute Mellin moments up to $N = 2000 (3000)$ by an independent method, to which we compare the results in $x$-space. In the polarized case, we work in the Larin scheme. We present numerical results. The two-mass contributions amount to about $50 \%$ of the full \textcolor{blue}{$O(T_F^2)$} and \textcolor{blue}{$O(T_F^3)$} terms contributing to the operator matrix elements. The present result completes the calculation of all unpolarized and polarized massive three-loop operator matrix elements.

hep-ph↗

arXiv2025

The Single-Mass Variable Flavor Number Scheme at Three-Loop Order

The matching relations in the unpolarized and polarized variable flavor number scheme at three-loop order are presented in the single-mass case. They describe the process of massive quarks becoming light at large virtualities $Q^2$. In this framework, heavy-quark parton distributions can be defined. Numerical results are presented on the matching relations in the case of the single-mass variable flavor number scheme for the light parton, charm and bottom quark distributions. These relations are process independent. In the polarized case we generally work in the Larin scheme. To two-loop order we present the polarized massive OMEs also in the $\overline{\rm MS}$ scheme. Fast numerical codes for the single-mass massive operator matrix elements are provided.

hep-ph↗

arXiv2024

The three-loop single-mass heavy flavor corrections to deep-inelastic scattering

We report on the status of the calculation of the massive Wilson coefficients and operator matrix elements for deep-inelastic scatterung to three-loop order. We discuss both the unpolarized and the polarized case, for which all the single-mass and nearly all two-mass contributions have been calculated. Numerical results on the structure function $F_2(x,Q^2)$ are presented. In the polarized case, we work in the Larin scheme and refer to parton distribution functions in this scheme. Furthermore, results on the three-loop variable flavor number scheme are presented

hep-ph↗

arXiv2024

Next-to-Next-to-Leading Order Evolution of Polarized Parton Densities in the Larin Scheme

In many calculations involving polarized twist-2 parton densities to higher order in the strong coupling constant one uses the Larin scheme to describe chiral effects in dimensional regularization. Upon forming observables, the scheme dependence cancels. Still one needs a corresponding regularization scheme to compute the contributing building blocks, like massless and massive Wilson coefficients, as well as the massive 3-loop operator matrix elements used in the variable flavor number scheme. These are matched to the evolved parton distribution functions in the Larin scheme. Starting with suitable input distributions we provide the solution of scale evolution of the different polarized parton distribution functions in Bjorken $x$ space for a wide range of virtualities $Q^2$ in the Larin scheme, at next-to-leading, and to next-to-next-to-leading order for the first time. We also illustrate the deviation between the parton distributions in the Larin and $\overline{\rm MS}$ schemes numerically.

hep-ph↗

arXiv2024

The non-first-order-factorizable contributions to the three-loop single-mass operator matrix elements $A_{Qg}^{(3)}$ and $ΔA_{Qg}^{(3)}$

The non-first-order-factorizable contributions (The terms 'first-order-factorizable contributions' and 'non-first-order-factorizable contributions' have been introduced and discussed in Refs. \cite{Behring:2023rlq,Ablinger:2023ahe}. They describe the factorization behaviour of the difference- or differential equations for a subset of master integrals of a given problem.) to the unpolarized and polarized massive operator matrix elements to three-loop order, $A_{Qg}^{(3)}$ and $ΔA_{Qg}^{(3)}$, are calculated in the single-mass case. For the $_2F_1$-related master integrals of the problem, we use a semi-analytic method based on series expansions and utilize the first-order differential equations for the master integrals which does not need a special basis of the master integrals. Due to the singularity structure of this basis a part of the integrals has to be computed to $O(\varepsilon^5)$ in the dimensional parameter. The solutions have to be matched at a series of thresholds and pseudo-thresholds in the region of the Bjorken variable $x \in ]0,\infty[$ using highly precise series expansions to obtain the imaginary part of the physical amplitude for $x \in ]0,1]$ at a high relative accuracy. We compare the present results both with previous analytic results, the results for fixed Mellin moments, and a prediction in the small-$x$ region. We also derive expansions in the region of small and large values of $x$. With this paper, all three-loop single-mass unpolarized and polarized operator matrix elements are calculated.

hep-ph↗

arXiv2023

The first-order factorizable contributions to the three-loop massive operator matrix elements $A_{Qg}^{(3)}$ and $ΔA_{Qg}^{(3)}$

The unpolarized and polarized massive operator matrix elements $A_{Qg}^{(3)}$ and $ΔA_{Qg}^{(3)}$ contain first-order factorizable and non-first-order factorizable contributions in the determining difference or differential equations of their master integrals. We compute their first-order factorizable contributions in the single heavy mass case for all contributing Feynman diagrams. Moreover, we present the complete color-$ζ$ factors for the cases in which also non-first-order factorizable contributions emerge in the master integrals, but cancel in the final result as found by using the method of arbitrary high Mellin moments. Individual contributions depend also on generalized harmonic sums and on nested finite binomial and inverse binomial sums in Mellin $N$-space, and correspondingly, on Kummer-Poincaré and square-root valued alphabets in Bjorken-$x$ space. We present a complete discussion of the possibilities of solving the present problem in $N$-space analytically and we also discuss the limitations in the present case to analytically continue the given $N$-space expressions to $N \in \mathbb{C}$ by strict methods. The representation through generating functions allows a well synchronized representation of the first-order factorizable results over a 17-letter alphabet. We finally obtain representations in terms of iterated integrals over the corresponding alphabet in $x$-space, also containing up to weight {\sf w = 5} special constants, which can be rationalized to Kummer-Poincaré iterated integrals at special arguments. The analytic $x$-space representation requires separate analyses for the intervals $x \in [0,1/4], [1/4,1/2], [1/2,1]$ and $x > 1$. We also derive the small and large $x$ limits of the first-order factorizable contributions.

hep-ph↗

arXiv2023

Recent 3-Loop Heavy Flavor Corrections to Deep-Inelastic Scattering

We report on recent progress in calculating the three loop QCD corrections of the heavy flavor contributions in deep--inelastic scattering and the massive operator matrix elements of the variable flavor number scheme. Notably we deal with the operator matrix elements $A_{gg,Q}^{(3)}$ and $A_{Qg}^{(3)}$ and technical steps to their calculation. In particular, a new method to obtain the inverse Mellin transform without computing the corresponding $N$--space expressions is discussed.

hep-ph↗

arXiv2023

The inverse Mellin transform via analytic continuation

We present a method to calculate the $x$--space expressions of massless or massive operator matrix elements in QCD and QED containing local composite operator insertions, depending on the discrete Mellin index $N$, directly, without computing the Mellin--space expressions in explicit form analytically. Here $N$ belongs either to the even or odd positive integers. The method is based on the resummation of the operators into effective propagators and relies on an analytic continuation between two continuous variables. We apply it to iterated integrals as well as to the more general case of iterated non--iterative integrals, generalizing the former ones. The $x$--space expressions are needed to derive the small--$x$ behaviour of the respective quantities, which usually cannot be accessed in $N$--space. We illustrate the method for different (iterated) alphabets, including non--iterative $_2F_1$ and elliptic structures, as examples. These structures occur in different massless and massive three--loop calculations. Likewise the method applies even to the analytic closed form solutions of more general cases of differential equations which do not factorize into first--order factors.

hep-ph↗

arXiv2022

$O(α_s^2$) Polarized Heavy Flavor Corrections}to Deep-Inelastic Scattering at $Q^2 \gg m^2$

We calculate the quarkonic $O(α_s^2)$ massive operator matrix elements $ΔA_{Qg}(N), ΔA_{Qq}^{\rm PS}(N)$ and $ΔA_{qq,Q}^{\rm NS}(N)$ for the twist--2 operators and the associated heavy flavor Wilson coefficients in polarized deeply inelastic scattering in the region $Q^2 \gg m^2$ to $O(\varepsilon)$ in the case of the inclusive heavy flavor contributions. The evaluation is performed in Mellin space, without applying the integration-by-parts method. The result is given in terms of harmonic sums. This leads to a significant compactification of the operator matrix elements and massive Wilson coefficients in the region $Q^2 \gg m^2$ derived previously in \cite{BUZA2}, which we partly confirm, and also partly correct. The results allow to determine the heavy flavor Wilson coefficients for $g_1(x,Q^2)$ to $O(α_s^2)$ for all but the power suppressed terms $\propto (m^2/Q^2)^k, k \geq 1$. The results in momentum fraction $z$-space are also presented. We also discuss the small $x$ effects in the polarized case. Numerical results are presented. We also compute the gluonic matching coefficients in the two--mass variable flavor number scheme to $O(\varepsilon)$.

hep-ph↗

arXiv2022

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

We calculate the gluonic massive operator matrix elements in the unpolarized and polarized cases, $A_{gg,Q}(x,μ^2)$ and $ΔA_{gg,Q}(x,μ^2)$, 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.

hep-ph↗

arXiv2022

The massless three-loop Wilson coefficients for the deep-inelastic structure functions $F_2, F_L, xF_3$ and $g_1$

We calculate the massless unpolarized Wilson coefficients for deeply inelastic scattering for the structure functions $F_2(x,Q^2), F_L(x,Q^2), x F_3(x,Q^2)$ in the $\overline{\sf MS}$ scheme and the polarized Wilson coefficients of the structure function $g_1(x,Q^2)$ in the Larin scheme up to three--loop order in QCD in a fully automated way based on the method of arbitrary high Mellin moments. We work in the Larin scheme in the case of contributing axial--vector couplings or polarized nucleons. For the unpolarized structure functions we compare to results given in the literature. The polarized three--loop Wilson coefficients are calculated for the first time. As a by--product we also obtain the quarkonic three--loop anomalous dimensions from the $O(1/\varepsilon)$ terms of the unrenormalized forward Compton amplitude. Expansions for small and large values of the Bjorken variable $x$ are provided.

hep-ph↗

arXiv2022

Gravity in binary systems at the fifth and sixth post-Newtonian order

Binary sources of gravitational waves in the early inspiral phase are accurately described by a post-Newtonian expansion in small velocity and weak interaction. We compute the conservative dynamics to fifth and partial sixth order using a non-relativistic effective field theory. We give predictions for central observables and determine the required coefficients for the construction of an Effective One-Body Hamiltonian, extending the applicability of our results to the late inspiral and merger phases.

gr-qc↗