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 $\Delta A_{Qg}^{(3)}$, 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.

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. . 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 Δ A_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(ε^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 ∈ ]0,∞[ using highly precise series expansions to obtain the imaginary part of the physical amplitude for x ∈ ]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.

The non-first-order-factorizable contributions1 to the unpolarized and polarized massive operator matrix elements to three-loop order, AQg(3) and ΔAQg(3), are calculated in the single-mass case. For the F12-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(ε5) 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∈]0,∞[ using highly precise series expansions to obtain the imaginary part of the physical amplitude for x∈]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.

The non-first-order-factorizable contributions1 to the unpolarized and polarized massive operator matrix elements to three-loop order, AQg(3) and ΔAQg(3), are calculated in the single-mass case. For the F12-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(ε5) 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∈]0,∞[ using highly precise series expansions to obtain the imaginary part of the physical amplitude for x∈]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.

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

A survey is given on the new 2- and 3-loop results for the heavy flavor contributions to deep-inelastic scattering in the unpolarized and the polarized case. We also discuss related new mathematical aspects applied in these calculations.

We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi,$ $\log(2)$ or zeta values. In order to perform these simplifications, we view the series as specializations of generating series. For these generating series, we derive integral representations in terms of root-valued iterated integrals or directly in terms of cyclotomic harmonic polylogarithms. Using substitutions, we express the root-valued iterated integrals as cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclotomic harmonic polylogarithms, we derive expressions in terms of several constants. The methods are implemented in the computer algebra package HarmonicSums.

We calculate the gluonic massive operator matrix elements in the unpolarized and polarized cases, A_gg,Q(x,μ^2) and Δ A_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.

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 multiscale 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.

We extend the (continuous) multivariate Almkvist-Zeilberger algorithm in order to apply it for instance to special Feynman integrals emerging in renormalizable Quantum field Theories. We will consider multidimensional integrals over hyperexponential integrands and try to find closed form representations in terms of nested sums and products or iterated integrals. In addition, if we fail to compute a closed form solution in full generality, we may succeed in computing the first coefficients of the Laurent series expansions of such integrals in terms of indefinite nested sums and products or iterated integrals. In this article we present the corresponding methods and algorithms. Our Mathematica package MultiIntegrate, can be considered as an enhanced implementation of the (continuous) multivariate Almkvist Zeilberger algorithm to compute recurrences or differential equations for hyperexponential integrands and integrals. Together with the summation package Sigma and the package HarmonicSums our package provides methods to compute closed form representations (or coefficients of the Laurent series expansions) of multidimensional integrals over hyperexponential integrands in terms of nested sums or iterated integrals.

A summary of the calculation of the color-planar and complete light quark contributions to the massive three-loop form factors is presented. Here a novel calculation method for the Feynman integrals is used, solving general uni-variate first order factorizable systems of differential equations. We also present predictions for the asymptotic structure of these form factors.

We introduce a general reduction strategy that enables one to search for solutions of parameterized linear difference equations in difference rings. Here we assume that the ring itself can be decomposed by a direct sum of integral domains (using idempotent elements) that enjoys certain technical features and that the coefficients of the difference equation are not degenerated. Using this mechanism we can reduce the problem to find solutions in a ring (with zero-divisors) to search solutions in several copies of integral domains. Utilizing existing solvers in this integral domain setting, we obtain a general solver where the components of the linear difference equations and the solutions can be taken from difference rings that are built e.g., by RΠΣ-extensions over ΠΣ-fields. This class of difference rings contains, e.g., nested sums and products, products over roots of unity and nested sums defined over such objects.

We describe the qFunctions Mathematica package for q-series and partition theory applications. This package includes both experimental and symbolic tools. The experimental set of elements includes guessers for q-difference equations and recurrences for given q-difference and fitting/finding explicit expressions for sequences of polynomials. This package can symbolically handle formal manipulations on q-differential, q-difference equations and recurrences, such as switching between these forms, finding the greatest common divisor of recurrences, and formal substitutions. The package qFunctions has implementations that automate the recurrence system creation of the weighted words approach as well as a scheme on cylindric partitions. We also discuss a slight extension of the classical method of weighed words and present some new sum-product identities.

We present the two-mass QCD contributions to the polarized pure singlet operator matrix element at three loop order in x-space. These terms are relevant for calculating the polarized structure function g1(x,Q2) at O(αs3) as well as for the matching relations in the variable flavor number scheme and the polarized heavy quark distribution functions at the same order. The result for the operator matrix element is given in terms of generalized iterated integrals. These integrals depend on the mass ratio through the main argument, and the alphabet includes square–root valued letters.

We give a proof of two identities involving binomial sums at infinity conjectured by Z-W Sun. In order to prove these identities, we use a recently presented method i.e. we view the series as specializations of generating series and derive integral representations. Using substitutions, we express these integral representations in terms of cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclotomic harmonic polylogarithms, we derive the results. These methods are implemented in the computer algebra package HarmonicSums.

We calculate the two-mass QCD contributions to the massive operator matrix element Agg,Q at O(αs3) in analytic form in Mellin N- and z-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 N-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 z-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.

We calculate the massive polarized three-loop pure singlet operator matrix element AQq(3),PS in the single mass case in the Larin scheme. This operator matrix element contributes to the massive polarized three-loop Wilson coefficient HQq(3),PS in deep-inelastic scattering and constitutes a three-loop transition matrix element in the variable flavor number scheme. We provide analytic results in Mellin N and in x space and study the behaviour of this operator matrix element in the region of small and large values of the Bjorken variable x.

Using the method of massive operator matrix elements, we calculate the subleading QED initial state radiative corrections to the process e+e−→γ⁎/Z⁎ for the first three logarithmic contributions from O(α3L3),O(α3L2),O(α3L) to O(α5L5),O(α5L4),O(α5L3) and compare their effects to the leading contribution O(α6L6) and one more subleading term O(α6L5). The calculation is performed in the limit of large center of mass energies squared s≫me2. These terms supplement the known corrections to O(α2), 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 w = 6 of argument z and (1−z) and in Mellin N space by generalized harmonic sums. Numerical results are presented on the position of the Z peak and corrections to the Z width, ΓZ. The corrections calculated result into a final theoretical accuracy for δMZ and δΓZ which is estimated to be of O(30keV) at an anticipated systematic accuracy at the FCC_ee of ∼100keV. This precision cannot be reached, however, by including only the corrections up to O(α3).

Higher order calculations in perturbative Quantum Field Theories often produce coupled linear systems of differential equations which factorize to first order. Here we present an algorithm to solve such systems in terms of iterated integrals over an alphabet the structure of which is implied by the coefficient matrix of the given system. We apply this method to calculate the master integrals in the color-planar and complete light quark contributions to the three-loop massive form factors.