Clemens G. Raab

Bio: Clemens G. Raab is an academic researcher from Johannes Kepler University of Linz. The author has contributed to research in topics: Deep inelastic scattering & Feynman diagram. The author has an hindex of 10, co-authored 31 publications receiving 479 citations.

Iterated binomial sums and their associated iterated integrals

Abstract: We consider finite iterated generalized harmonic sums weighted by the binomial 2kk in numerators and denominators. A large class of these functions emerges in the calculation of massive Feynman diagrams with local operator insertions starting at 3-loop order in the coupling constant and extends the classes of the nested harmonic, generalized harmonic, and cyclotomic sums. The binomially weighted sums are associated by the Mellin transform to iterated integrals over square-root valued alphabets. The values of the sums for N → ∞ and the iterated integrals at x = 1 lead to new constants, extending the set of special numbers given by the multiple zeta values, the cyclotomic zeta values and special constants which emerge in the limit N → ∞ of generalized harmonic sums. We develop algorithms to obtain the Mellin representations of these sums in a systematic way. They are of importance for the derivation of the asymptotic expansion of these sums and their analytic continuation to N∈C. The associated convolution ...

Iterated elliptic and hypergeometric integrals for Feynman diagrams

Abstract: We calculate 3-loop master integrals for heavy quark correlators and the 3-loop quantum chromodynamics corrections to the ρ-parameter. They obey non-factorizing differential equations of second order with more than three singularities, which cannot be factorized in Mellin-N space either. The solution of the homogeneous equations is possible in terms of 2F1 Gaus hypergeometric functions at rational argument. In some cases, integrals of this type can be mapped to complete elliptic integrals at rational argument. This class of functions appears to be the next one arising in the calculation of more complicated Feynman integrals following the harmonic polylogarithms, generalized polylogarithms, cyclotomic harmonic polylogarithms, square-root valued iterated integrals, and combinations thereof, which appear in simpler cases. The inhomogeneous solution of the corresponding differential equations can be given in terms of iterative integrals, where the new innermost letter itself is not an iterative integral. A new class of iterative integrals is introduced containing letters in which (multiple) definite integrals appear as factors. For the elliptic case, we also derive the solution in terms of integrals over modular functions and also modular forms, using q-product and series representations implied by Jacobi’s ϑi functions and Dedekind’s η-function. The corresponding representations can be traced back to polynomials out of Lambert–Eisenstein series, having representations also as elliptic polylogarithms, a q-factorial 1/ηk(τ), logarithms, and polylogarithms of q and their q-integrals. Due to the specific form of the physical variable x(q) for different processes, different representations do usually appear. Numerical results are also presented.

Iterated Binomial Sums and their Associated Iterated Integrals

Abstract: We consider finite iterated generalized harmonic sums weighted by the binomial $\binom{2k}{k}$ in numerators and denominators. A large class of these functions emerges in the calculation of massive Feynman diagrams with local operator insertions starting at 3-loop order in the coupling constant and extends the classes of the nested harmonic, generalized harmonic and cyclotomic sums. The binomially weighted sums are associated by the Mellin transform to iterated integrals over square-root valued alphabets. The values of the sums for $N \rightarrow \infty$ and the iterated integrals at $x=1$ lead to new constants, extending the set of special numbers given by the multiple zeta values, the cyclotomic zeta values and special constants which emerge in the limit $N \rightarrow \infty$ of generalized harmonic sums. We develop algorithms to obtain the Mellin representations of these sums in a systematic way. They are of importance for the derivation of the asymptotic expansion of these sums and their analytic continuation to $N \in \mathbb{C}$. The associated convolution relations are derived for real parameters and can therefore be used in a wider context, as e.g. for multi-scale processes. We also derive algorithms to transform iterated integrals over root-valued alphabets into binomial sums. Using generating functions we study a few aspects of infinite (inverse) binomial sums.

Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams

Abstract: We calculate 3-loop master integrals for heavy quark correlators and the 3-loop QCD corrections to the $\rho$-parameter. They obey non-factorizing differential equations of second order with more than three singularities, which cannot be factorized in Mellin-$N$ space either. The solution of the homogeneous equations is possible in terms of convergent close integer power series as $_2F_1$ Gaus hypergeometric functions at rational argument. In some cases, integrals of this type can be mapped to complete elliptic integrals at rational argument. This class of functions appears to be the next one arising in the calculation of more complicated Feynman integrals following the harmonic polylogarithms, generalized polylogarithms, cyclotomic harmonic polylogarithms, square-root valued iterated integrals, and combinations thereof, which appear in simpler cases. The inhomogeneous solution of the corresponding differential equations can be given in terms of iterative integrals, where the new innermost letter itself is not an iterative integral. A new class of iterative integrals is introduced containing letters in which (multiple) definite integrals appear as factors. For the elliptic case, we also derive the solution in terms of integrals over modular functions and also modular forms, using $q$-product and series representations implied by Jacobi's $\vartheta_i$ functions and Dedekind's $\eta$-function. The corresponding representations can be traced back to polynomials out of Lambert--Eisenstein series, having representations also as elliptic polylogarithms, a $q$-factorial $1/\eta^k(\tau)$, logarithms and polylogarithms of $q$ and their $q$-integrals. Due to the specific form of the physical variable $x(q)$ for different processes, different representations do usually appear. Numerical results are also presented.

Event Generation with Sherpa 2.2

Abstract: Sherpa is a general-purpose Monte Carlo event generator for the simulation of particle collisions in high-energy collider experiments. We summarize essential features and improvements of the Sherpa 2.2 release series, which is heavily used for event generation in the analysis and interpretation of LHC Run 1 and Run 2 data. We highlight a decade of developments towards ever higher precision in the simulation of particle-collision events.

The ABM parton distributions tuned to LHC data

Abstract: We present a global fit of parton distributions at next-to-next-to-leading order (NNLO) in QCD. The fit is based on the world data for deep-inelastic scattering, fixed-target data for the Drell-Yan process and includes, for the first time, data from the Large Hadron Collider (LHC) for the Drell-Yan process and the hadroproduction of top-quark pairs. The analysis applies the fixed-flavor number scheme for ${n}_{f}=3$, 4, 5, uses the $\overline{\mathrm{MS}}$ scheme for the strong coupling ${\ensuremath{\alpha}}_{s}$ and the heavy-quark masses and keeps full account of the correlations among all nonperturbative parameters. At NNLO this returns the values of ${\ensuremath{\alpha}}_{s}({M}_{Z})=0.1132\ifmmode\pm\else\textpm\fi{}0.0011$ and ${m}_{t}(\text{pole})=171.2\ifmmode\pm\else\textpm\fi{}2.4\text{ }\text{ }\mathrm{GeV}$ for the top-quark pole mass. The fit results are used to compute benchmark cross sections for the Higgs production at the LHC to NNLO accuracy. We compare our results to those obtained by other groups and show that differences can be linked to different theoretical descriptions of the underlying physical processes.

Discrete math = 離散数学

Abstract: What is discrete math? • The real numbers are continuous in the senses that: * between any two real numbers there is a real number • The integers do not share this property. In this sense the integers are lumpy, or " discrete " So discrete math is the study of mathematical objects that are discrete. " It's all the math that counts " Some discrete mathematical concepts: • Integers: Between two integers there is not another integer. • Propositions: Either true or false, there are no 1/2 truths (in math) • Sets: An item is either in a set or not in a set, never partly in and partly out. • Relations: A pair of items are related or not. • Networks (graphs): Between two terminals of a network connection there are no terminals. Propositional Logic Propositions are statements that are either true or false. Principles: Substituting an equivalent statement. Replacing a logic variable in a tautology. Defn algebraic proof

arXiv : Higgs Boson Production at Hadron Colliders at N$^3$LO in QCD

Abstract: We present the Higgs boson production cross section at Hadron colliders in the gluon fusion production mode through N3LO in perturbative QCD. Specifically, we work in an effective theory where the top quark is assumed to be infinitely heavy and all other quarks are considered to be massless. Our result is the first exact formula for a partonic hadron collider cross section at N3LO in perturbative QCD. Furthermore, our result is an analytic computation of a hadron collider cross section involving elliptic integrals. We derive numerical predictions for the Higgs boson cross section at the LHC. Previously this result was approximated by an expansion of the cross section around the production threshold of the Higgs boson and we compare our findings. Finally, we study the impact of our new result on the state of the art prediction for the Higgs boson cross section at the LHC.