Search

Hadron production in lepton-proton scattering (semi-inclusive deep inelastic scattering, SIDIS) probes the structure of hadrons at a higher level of detail than fully inclusive processes. A wealth of SIDIS data is available especially from fixed-target experiments. Here we review our calculation for the NNLO corrections to the full set of polarized and unpolarized SIDIS coefficient functions and present some selected analytical expressions. Our results enable for the first time a fully consistent treatment of hadron fragmentation processes in polarized and unpolarized DIS at NNLO and provide the basis for studies of hadron structure, hadron fragmentation and identified particle cross sections at colliders., Comment: Contribution to: Loops and Legs in Quantum Field Theory (LL2024)

The ComPair balloon instrument is a prototype of the All-sky Medium Energy Gamma-ray Observatory (AMEGO) mission concept. AMEGO aims to bridge the spectral gap in sensitivity that currently exists from $\sim$100 keV to $\sim$100 MeV by being sensitive to both Compton and pair-production events. This is made possible through the use of four subsystems working together to reconstruct events: a double-sided silicon strip detector (DSSD) Tracker, a virtual Frisch grid cadmium zinc telluride (CZT) Low Energy Calorimeter, a ceasium iodide (CsI) High Energy Calorimeter, and an anti-coincidence detector (ACD) to reject charged particle backgrounds. Composed of 10 layers of DSSDs, ComPair's Tracker is designed to measure the position of photons that Compton scatter in the silicon, as well as reconstruct the tracks of electrons and positrons from pair-production as they propagate through the detector. By using these positions, as well as the absorbed energies in the Tracker and 2 Calorimeters, the energy and direction of the incident photon can be determined. This proceeding will present the development, testing, and calibration of the ComPair DSSD Tracker and early results from its balloon flight in August 2023., Comment: SPIE Astronomical Telescopes + Instrumentation Conference

In these proceedings, we consider the factorizable contributions of the two-loop electroweak corrections to Higgs boson pair production at the Large Hadron Collider. We discuss the classification of factorizable diagrams and their renormalization. Compact analytic results for the renormalized form factors are presented in a computer-readable form., Comment: Contribution to proceedings of the Loops and Legs in Quantum Field Theory conference (LL2024), April 14-19, 2024, Wittenberg, Germany

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, Comment: 12 pages

We compute next-to-next-to-leading order QCD corrections to the partonic processes $b\to c \bar{u} d$ and $b\to c \bar{c} s$, which constitute the dominant decay channels in standard model predictions for $B$-meson lifetimes within the heavy quark expansion. We consider the contribution from the four-quark operators $O_1$ and $O_2$ in the $\Delta B =1$ effective Hamiltonian. The decay rates are obtained from the imaginary parts of four-loop propagator-type diagrams. We compute the corresponding master integrals using the "expand and match" approach which provides semi-analytic results for the physical charm and bottom quark masses. We show that the dependence of the decay rate on the renormalization scale is significantly reduced after including the next-to-next-to-leading order corrections. Furthermore, we compute next-to-next-to-leading order corrections to the Cabibbo-Kobayashi-Maskawa-suppressed decay channels $b \to u \bar c s$ and $b \to u \bar u d$., Comment: v2: ref. added; numerical values in CKM-suppressed contribution and fig.10 updated; conclusions unchanged

We compute three-loop corrections of $\mathcal{O}(\alpha_{s}^3)$ to form factors with one massive and one massless quark coupling to an external vector, axialvector, scalar, pseudoscalar, or tensor current. We obtain analytic results for the color-planar contributions, for the contributions of light-quark loops, and the contributions with two heavy-quark loops. For the computation of the remaining master integrals we use the "expand and match" approach which leads to semi-analytic results for the form factors. We implement our results in a {\tt Mathematica} and a {\tt Fortran} code which allows for fast and precise numerical evaluations in the physically relevant phase space. The form factors are used to compute the hard matching coefficients in Soft-Collinear Effective Theory for all currents. The tensor coefficients at light-like momentum transfer are used to extract the hard function in $\bar B \to X_s \gamma$ to three loops., Comment: 51 pages

We compute three-loop corrections to the process $gg\to HH$ originating from one-particle reducible diagrams. This requires the computation of two-loop corrections to the gluon-gluon-Higgs vertex with an off-shell gluon. We describe in detail our approach to obtain semi-analytic results for the vertex form factors and present results for the two form factors contributing to Higgs boson pair production., Comment: 19 pages, 6 figures

The ComPair gamma-ray telescope is a technology demonstrator for a future gamma-ray telescope called the All-sky Medium Energy Gamma-ray Observatory (AMEGO). The instrument is composed of four subsystems, a double-sided silicon strip detector, a virtual Frisch grid CdZnTe calorimeter, a CsI:Tl based calorimeter, and an anti-coincidence detector (ACD). The CsI calorimeter's goal is to measure the position and energy deposited from high-energy events. To demonstrate the technological readiness, the calorimeter has flown onboard a NASA scientific balloon as part of the GRAPE-ComPair mission and accumulated around 3 hours of float time at an altitude of 40 km. During the flight, the CsI calorimeter observed background radiation, Regener-Pfotzer Maximum, and several gamma-ray activation lines originating from aluminum.

Semi-inclusive hadron production in longitudinally polarized deep-inelastic lepton-nucleon scattering is a powerful tool for resolving the quark flavor decomposition of the proton's spin structure. We present the full next-to-next-to-leading order (NNLO) QCD corrections to the coefficient functions of polarized semi-inclusive deep-inelastic scattering (SIDIS) in analytical form, enabling the use of SIDIS measurements in precision studies of the proton spin structure. The numerical impact of these corrections is illustrated by a comparison with data of polarized single-inclusive hadron spectra from the DESY HERMES and CERN COMPASS experiments., Comment: revised version, ancillary files cleaned up, all figures redone with BDSSV24 polarized PDFs

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 computation of master integrals from their differential equations requires boundary values to be supplied by an independent method. These boundary values are often desired at singular kinematical points. We demonstrate how the auxiliary mass flow technique can be extended to compute the expansion coefficients of master integrals in a singular limit in an analytical manner, thereby providing these boundary conditions. To illustrate the application of the method, we re-compute the phase space integrals relevant to initial-final antenna functions at NNLO, now including higher-order terms in their $\epsilon$-expansion in view of their application in third-order QCD corrections., Comment: 28 pages, 4 figures, v2-published version

The unpolarized and polarized massive operator matrix elements $A_{Qg}^{(3)}$ and $\Delta A_{Qg}^{(3)}$ 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-$\zeta$ 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-Poincar\'e 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-Poincar\'e 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., Comment: 58 pages, 4 Figures

The B -> X_s gamma branching ratio is currently measured with around 5% accuracy. Further improvement is expected from Belle II. To match such a precision on the theoretical side, evaluation of O(alpha_s^2) corrections to the partonic decay b -> X_s^part gamma are necessary, which includes the b -> s gamma, b -> s g gamma, b -> s g g gamma, b -> s qbar q gamma decay channels. Here, we evaluate the unrenormalized contribution to b -> s gamma that stems from the interference of the photonic dipole operator Q_7 and the current-current operators Q_1 and Q_2. Our results, obtained in the cut propagator approach at the 4-loop level, agree with those found in parallel by Fael et al. who have applied the amplitude approach at the 3-loop level. Partial results for the same quantities recently determined by Greub et al. agree with our findings, too., Comment: published version, 9 pages, 1 figure

We compute three-loop vertex corrections to $b\to s\gamma$ induced by current-current operators. The results are presented as expansions in $m_c/m_b$ with numerical coefficients which allow to cover all relevant values for the heavy quark masses in different renormalization schemes. Moreover we provide for the first time analytic results for the next-to-leading order contribution. Our results present an important building block to the next-to-next-to-leading order interference contributions of the current-current operators $Q_1$ and $Q_2$ with the electric dipole operator $Q_7$., Comment: 22 pages;v2: minor modifications,agrees with published version

The Compton Pair (ComPair) telescope is a prototype that aims to develop the necessary technologies for future medium energy gamma-ray missions and to design, build, and test the prototype in a gamma-ray beam and balloon flight. The ComPair team has built an instrument that consists of 4 detector subsystems: a double-sided silicon strip detector Tracker, a novel high-resolution virtual Frisch-grid cadmium zinc telluride Calorimeter, and a high-energy hodoscopic cesium iodide Calorimeter, all of which are surrounded by a plastic scintillator anti-coincidence detector. These subsystems together detect and characterize photons via Compton scattering and pair production, enable a veto of cosmic rays, and are a proof-of-concept for a space telescope with the same architecture. A future medium-energy gamma-ray mission enabled through ComPair will address many questions posed in the Astro2020 Decadal survey in both the New Messengers and New Physics and the Cosmic Ecosystems themes. In this contribution, we will give an overview of the ComPair project and steps forward to the balloon flight., Comment: Proceedings for the 38th International Cosmic Ray Conference (ICRC2023)

We compute two-loop electroweak corrections to Higgs boson pair and Higgs plus jet production, taking into account all sectors of the Standard Model. All diagrams with virtual top quarks are computed in an expansion for large top quark mass up to order $1/m_t^8$ or more. We present analytic results for the form factors and discuss the convergence properties. For the process $gg\to gH$ we also consider QCD corrections in the large-$m_t$ limit., Comment: Minor modifications; matches published version

We compute next-to-next-to-leading order corrections to the semileptonic decay rate of $B$ mesons for arbitrary values of the final-state quark mass. For the contribution with one massive quark in the final state, we extend the literature result and obtain analytic expressions in terms of iterated integrals. For the complete contribution, which also includes contributions with three massive quarks in the final state, we present a semi-analytic method, which leads to a precise approximation formula for the decay rate. Our results agree with the expansions available for $b\to c\ell \bar{\nu}_\ell$ and $b\to u\ell \bar{\nu}_\ell$ in the literature. The main emphasize of this paper is on the technical aspects of the calculation which are useful for a wider range of applications., Comment: 25 pages, 8 figures

We consider light-fermion three-loop corrections to $gg\to HH$ using forward scattering kinematics in the limit of a vanishing Higgs boson mass, which covers a large part of the physical phase space. We compute the form factors and discuss the technical challenges. The approach outlined in this letter can be used to obtain the full virtual corrections to $gg\to HH$ at next-to-next-to-leading order., Comment: 12 pages, v2: fig. 4 updated

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., Comment: Proc RADCOR 2023, 7 pages, 1 figure

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., Comment: 43 pages Latex

We consider two-loop corrections to $2\to 2$ scattering processes with massive particles in the final state and massive particles in the loop. We discuss the combination of analytic expansions in the high-energy limit and for small Mandelstam variable~$t$. For the example of double Higgs boson production we show that the whole phase space can be covered and time-consuming numerical integrations can be avoided., Comment: 25 pages

We compute three-loop corrections to the singlet form factors for massive quarks using a semi-analytic method which provides precise results over the whole kinematic range. Particular emphasis is put on the anomaly contribution originating from an external axial-vector current. We also discuss in detail the contribution for a pseudoscalar current and verify the chiral Ward identity to three-loop order. Explicit results are presented for the low- and high-energy regions and the expansions around threshold., Comment: 38 pages

We calculate the quarkonic $O(\alpha_s^2)$ massive operator matrix elements $\Delta A_{Qg}(N), \Delta A_{Qq}^{\rm PS}(N)$ and $\Delta A_{qq,Q}^{\rm NS}(N)$ 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 $O(\alpha_s^2)$ 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 $O(\varepsilon)$., Comment: 58 pages Latex, 12 Figures

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., Comment: 48 p LATEX, 4 figures

There is a growing interest in the science uniquely enabled by observations in the MeV range, particularly in light of multi-messenger astrophysics. The Compton Pair (ComPair) telescope, a prototype of the AMEGO Probe-class concept, consists of four subsystems that together detect and characterize gamma rays in the MeV regime. A double-sided strip silicon Tracker gives a precise measure of the first Compton scatter interaction and tracks pair-conversion products. A novel cadmium zinc telluride (CZT) detector with excellent position and energy resolution beneath the Tracker detects the Compton-scattered photons. A thick cesium iodide (CsI) calorimeter contains the high-energy Compton and pair events. The instrument is surrounded by a plastic anti-coincidence (ACD) detector to veto the cosmic-ray background. In this work, we will give an overview of the science motivation and a description of the prototype development and performance.

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., Comment: 159 pages Latex

We report on the calculation of the three-loop polarized and unpolarized flavor non-singlet and the polarized singlet anomalous dimensions using massless off-shell operator matrix elements in a gauge-variant framework. We also reconsider the unpolarized two-loop singlet anomalous dimensions and correct errors in the foregoing literature., Comment: 12 pages Latex, Contributeion to the Proceedings of LL2022

We report on our recent calculation of massive quark form factors using a semi-numerical approach based on series expansions of the master integrals around singular and regular kinematic points and numerical matching. The methods allows to cover the whole kinematic range of negative and positive values of the virtuality $s$ with at least seven significant digits accuracy., Comment: 9 pages, 3 figures, contribution to the proceedings of Loops and Legs in Quantum Field Theory (LL2022), Ettal, Germany

We analytically compute two-loop Yukawa corrections to Higgs boson pair production in the high-energy limit. Such corrections are generated by an exchange of a Higgs boson between the virtual top quark lines. We propose two approaches to obtain expansions of the massive two-loop box integrals and show that precise results are obtained for transverse momenta of the Higgs bosons above about 150 GeV. We discuss in detail the computation of all 140 master integrals and present analytic results., Comment: 39 pages

We consider Quantum Chromodynamics with external vector, axial-vector, scalar and pseudo-scalar currents and compute three-loop corrections to the corresponding vertex function taking into account massive quarks. We consider all non-singlet contributions as well as those singlet contributions where the external current couples to a massive quark loop. We apply a semi-numerical method which is based on expansions around singular and regular kinematical points. They are matched at intermediate values of the squared partonic center-of-mass energy $s$ which allows to cover the whole kinematic range for negative and positive values of $s$. Our method permits a systematic increase of the precision by varying the expansion depth and the choice of the intermediate matching points. In our current set-up we have at least seven significant digits for the finite contribution of all form factors. We present our results as a combination of series expansions and interpolation functions which allows for a straightforward use in practical applications., Comment: 73 pages, 15 figures

We study the impact of third-order QCD corrections for several kinematic moments of the inclusive semileptonic $B$ decays, to first order in the $1/m_b$ expansion. We consider the first four moments of the charged-lepton energy $E_\ell$ spectrum, the total leptonic invariant mass $q^2$ and the hadronic invariant mass $M_X^2$. No experimental cuts are applied. Our analytic results are obtained via an asymptotic expansion around the limit $m_b \simeq m_c$. After converting the scheme for the bottom mass to the kinetic scheme we compare the size of higher QCD corrections to the contributions from $1/m_b^2$ and $1/m_b^3$ power corrections and to the relative uncertainties., Comment: 36 pages, 2 figures

For the determination of the Cabbibo-Kobayashi-Maskawa matrix element $|V_{cb}|$ from inclusive data a precise knowledge of the semileptonic $b \to c$ decay rate is necessary. Since this observable has a bad convergence behavior when the heavy quark masses are expressed in the on-shell or $\overline{\text{MS}}$ scheme the latest determinations have been obtained in the so called kinetic mass scheme. The relation between the different schemes needs to be known to high precision as well. In this proceedings we present our recent calculations which push the precision of both ingredients to $\mathcal{O}(\alpha_s^3)$. The results can be used to improve the inclusive determination of $|V_{cb}|$., Comment: Contribution to the 11th International Workshop on the CKM Unitarity Triangle (CKM2021)

We compute the matching coefficients between QCD and non-relativistic QCD for external vector, axial-vector, scalar and pseudo-scalar currents up to three-loop order. We concentrate on the non-singlet contributions and present precise numerical results with an accuracy of about ten digits. For the vector current the results from arXiv:1401.3004 are confirmed, increasing the accuracy by several orders of magnitude., Comment: 9 pages

The precise knowledge of the QED initial state corrections is of instrumental importance in studying high luminosity measurements in $e^+e^-$ annihilation at facilities like LEP, the International Linear Collider ILC, CLIC, a Giga-$Z$ facility, and the planned FCC\_ee. Logarithmic corrections of up to $O(\alpha^6 L^5)$ are necessary with various subleading terms taken into account. This applies to both the inclusive measurement of processes like $e^+ e^- \rightarrow \gamma^*/Z^*$ and also the forward-backward asymmetry. As has been shown recently, techniques from massive QCD, such as the computation of massive on-shell operator matrix elements, can be used for these calculations. We give an introduction to this topic and present both the calculation methods and the numerical corrections having been reached so far., Comment: 24 pages LATEX, 5 Figures

We compute the three-loop non-singlet corrections to the photon-quark form factors taking into account the full dependence on the virtuality of the photon and the quark mass. We combine the method of differential equations in an effective way with expansions around regular and singular points. This allows us to obtain results for the form factors with an accuracy of about eight to twelve digits in the whole kinematic range., Comment: 7 pages

We calculate the unpolarized and polarized two--loop massless off--shell operator matrix elements in QCD to $O(\varepsilon)$ in the dimensional parameter in an automated way. Here we use the method of arbitrary high Mellin moments and difference ring theory, based on integration-by-parts relations. This method also constitutes one way to compute the QCD anomalous dimensions. The presented higher order contributions to these operator matrix elements occur as building blocks in the corresponding higher order calculations up to four--loop order. All contributing quantities can be expressed in terms of harmonic sums in Mellin--$N$ space or by harmonic polylogarithms in $z$--space. We also perform comparisons to the literature., Comment: 101 pages Latex

Future high luminosity polarized deep--inelastic scattering experiments will improve both the knowledge of the spin sub--structure of the nucleons and contribute further to the precision determination of the strong coupling constant, as well as, reveal currently yet unknown higher twist contributions in the polarized sector. For all these tasks to be performed, it is necessary to know the QCD leading twist scaling violations of the measured structure functions. Here an important ingredient consists in the polarized singlet anomalous dimensions and splitting functions in QCD. We recalculate these quantities to three--loop order in the M--scheme by using the traditional method of space--like off--shell massless operator matrix elements, being a gauge--dependent framework. Here one obtains the anomalous dimensions without referring to gravitational currents, needed when calculating them using the forward Compton amplitude. We also calculate the non--singlet splitting function $\Delta P_{\rm qq}^{(2), \rm s, NS}$ and compare the final results to the literature, also including predictions for the region of small values of Bjorken $x$., Comment: 28 pages Latex, 1 figure

We present our recent calculation of the third order corrections to the semileptonic $b \to c$ and the muon decays. The calculation has been performed in an expansion around the limit $m_c \sim m_b$, but shows decent convergence even for $m_c=0$ from which the contribution to the muon decay can be extracted. For the semileptonic $b \to c$ decay we find large perturbative corrections in the on-shell scheme which can be significantly reduced by changing to the kinetic scheme for the bottom quark mass. These results are important input for the inclusive determination of $|V_{cb}|$ and the Fermi coupling constant $G_F$., Comment: 9 pages, 3 figures. Contribution to the 15th International Symposium on Radiative Corrections (RADCOR) and the XIX Workshop on Radiative Corrections for the LHC and Future Colliders (LoopFest)

We discuss a practical approach to compute master integrals entering physical quantities which depend on one parameter. As an example we consider four-loop QCD corrections to the relation between a heavy quark mass defined in the $\overline{\rm MS}$ and on-shell scheme in the presence of a second heavy quark., Comment: 7 pages, Contribution to the 15th International Symposium on Radiative Corrections (RADCOR) and the XIX Workshop on Radiative Corrections for the LHC and Future Colliders (LoopFest)

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., Comment: 14 pages Latex, 4 Figures, Contribution to the Proceedings of DIS 2021 and RADCOR 2021