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J. Blümlein

Publications and source records attributed to J. Blümlein.

At least 73 records · Page 4Linked to original sources
arXiv2019

The unpolarized two-loop massive pure singlet Wilson coefficients for deep-inelastic scattering

We calculate the massive two--loop pure singlet Wilson coefficients for heavy quark production in the unpolarized case analytically in the whole kinematic region and derive the threshold and asymptotic expansions. We also recalculate the corresponding massless two--loop Wilson coefficients. The complete expressions contain iterated integrals with elliptic letters. The contributing alphabets enlarge the Kummer-Poincaré letters by a series of square-root valued letters. A new class of iterated integrals, the Kummer-elliptic integrals, are introduced. For the structure functions $F_2$ and $F_L$ we also derive improved asymptotic representations adding power corrections. Numerical results are presented.

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arXiv2019

The $O(α^2)$ Initial State QED Corrections to $e^+e^-$ Annihilation to a Neutral Vector Boson Revisited

We calculate the non-singlet, the pure singlet contribution, and their interference term, at $O(α^2)$ due to electron-pair initial state radiation to $e^+ e^-$ annihilation into a neutral vector boson in a direct analytic computation without any approximation. The correction is represented in terms of iterated incomplete elliptic integrals. Performing the limit $s \gg m_e^2$ we find discrepancies with the earlier results of Ref.~\cite{Berends:1987ab} and confirm results obtained in Ref.~\cite{Blumlein:2011mi} where the effective method of massive operator matrix elements has been used, which works for all but the power corrections in $m^2/s$. In this way, we also confirm the validity of the factorization of massive partons in the Drell-Yan process. We also add non-logarithmic terms at $O(α^2)$ which have not been considered in \cite{Berends:1987ab}. The corrections are of central importance for precision analyzes in $e^+e^-$ annihilation into $γ^*/Z^*$ at high luminosity.

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arXiv2018

Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable

We present an algorithm which allows to solve analytically linear systems of differential equations which factorize to first order. The solution is given in terms of iterated integrals over an alphabet where its structure is implied by the coefficient matrix of the differential equations. These systems appear in a large variety of higher order calculations in perturbative Quantum Field Theories. We apply this method to calculate the master integrals of the three--loop massive form factors for different currents, as an illustration, and present the results for the vector form factors in detail. Here the solution space emerging is given by the cyclotomic harmonic polylogarithms and their associated special constants. No special basis representation of the master integrals is needed. The algorithm can be applied as well to more general cases factorizing at first order, which are based on more general alphabets, iterated integrals and associated constants.

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arXiv2018

The asymptotic behavior of the heavy quark form factors at higher order

In the asymptotic limit $Q^2 \gg m^2$, the heavy quark form factors exhibit Sudakov behavior. We study the corresponding renormalization group equations of the heavy quark form factors which do not only govern the structure of infrared divergences, but also control the high energy logarithms. This enables us to obtain the complete logarithmic three--loop and partial four--loop contributions to the heavy quark form factors in perturbative Quantum Chromodynamics.

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arXiv2018

Numerical Implementation of Harmonic Polylogarithms to Weight w = 8

We present the FORTRAN-code HPOLY.f for the numerical calculation of harmonic polylogarithms up to w = 8 at an absolute accuracy of $\sim 4.9 \cdot 10^{-15}$ or better. Using algebraic and argument relations the numerical representation can be limited to the range $x \in [0, \sqrt{2}-1]$. We provide replacement files to map all harmonic polylogarithms to a basis and the usual range of arguments $x \in ]-\infty,+\infty[$ to the above interval analytically. We also briefly comment on a numerical implementation of real valued cyclotomic harmonic polylogarithms.

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arXiv2018

NLO PDFs from the ABMP16 fit

We perform a global fit of parton distribution functions (PDFs) together with the strong coupling constant $α_s$ and the quark masses $m_c$, $m_b$ and $m_t$ at next-to-leading order (NLO) in QCD. The analysis applies the $\overline{\mathrm{MS}}\, $ renormalization scheme for $α_s$ and all quark masses. It is performed in the fixed-flavor number scheme for $n_f=3, 4, 5$ and uses the same data as the previous fit of the ABMP16 PDF at next-to-next-to-leading order (NNLO). The new NLO PDFs complement the set of ABMP16 PDFs and are to be used consistently with NLO QCD predictions for hard scattering processes. At NLO we obtain the value $α_s^{(n_f=5)}(M_Z) = 0.1191 \pm 0.0011$ compared to $α_s^{(n_f=5)}(M_Z) = 0.1147 \pm 0.0008$ at NNLO.

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arXiv2018

Two-mass three-loop effects in deep-inelastic scattering

We report on recent results on the two-mass corrections for massive operator matrix elements at 2- and 3-loop orders in QCD. These corrections form the building blocks of the variable flavor number scheme. Due to the similar values of the charm and bottom quark masses the two-mass corrections form an important contribution.

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arXiv2018

The $ρ$ parameter at three loops and elliptic integrals

We describe the analytic calculation of the master integrals required to compute the two-mass three-loop corrections to the $ρ$ parameter. In particular, we present the calculation of the master integrals for which the corresponding differential equations do not factorize to first order. The homogeneous solutions to these differential equations are obtained in terms of hypergeometric functions at rational argument. These hypergeometric functions can further be mapped to complete elliptic integrals, and the inhomogeneous solutions are expressed in terms of a new class of integrals of combined iterative non-iterative nature.

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arXiv2018

Heavy Quark Form Factors at Three Loops in the Planar Limit

We compute the color-planar and complete light quark non-singlet contributions to the heavy quark form factors in the case of the axialvector, scalar and pseudoscalar currents at three loops in perturbative QCD. We evaluate the master integrals applying a new method based on differential equations for general bases, which is applicable for all first order factorizing systems. The analytic results are expressed in terms of harmonic polylogarithms and real-valued cyclotomic harmonic polylogarithms.

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arXiv2018

The Variable Flavor Number Scheme at Next-to-Leading Order

We present the matching relations of the variable flavor number scheme at next-to-leading order, which are of importance to define heavy quark partonic distributions for the use at high energy colliders such as Tevatron and the LHC. The consideration of the two-mass effects due to both charm and bottom quarks, having rather similar masses, are important. These effects have not been considered in previous investigations. Numerical results are presented for a wide range of scales. We also present the corresponding contributions to the structure function $F_2(x,Q^2)$.

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arXiv2018

The Two-mass Contribution to the Three-Loop Gluonic Operator Matrix Element $A_{gg,Q}^{(3)}$

We calculate the two-mass QCD contributions to the massive operator matrix element $A_{gg,Q}$ at $\mathcal{O} (α_s^3)$ 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.

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arXiv2017

The Heavy Quark Form Factors at Two Loops

We compute the two-loop QCD corrections to the heavy quark form factors in case of the vector, axial-vector, scalar and pseudo-scalar currents up to second order in the dimensional parameter $ε= (4-D)/2$. These terms are required in the renormalization of the higher order corrections to these form factors.

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arXiv2017

The massive 3-loop operator matrix elements with two masses and the generalized variable flavor number scheme

We report on our latest results in the calculation of the two--mass contributions to 3--loop operator matrix elements (OMEs). These OMEs are needed to compute the corresponding contributions to the deep-inealstic scattering structure functions and to generalize the variable flavor number scheme by including both charm and bottom quarks. We present the results for the non-singlet and $A_{gq,Q}$ OMEs, and compare the size of their contribution relative to the single mass case. Results for the gluonic OME $A_{gg,Q}$ are given in the physical case, going beyond those presented in a previous publication where scalar diagrams were computed. We also discuss our recently published two--mass contribution to the pure singlet OME, and present an alternative method of calculating the corresponding diagrams.

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arXiv2017

Iterative and Iterative-Noniterative Integral Solutions in 3-Loop Massive QCD Calculations

Various of the single scale quantities in massless and massive QCD up to 3-loop order can be expressed by iterative integrals over certain classes of alphabets, from the harmonic polylogarithms to root-valued alphabets. Examples are the anomalous dimensions to 3-loop order, the massless Wilson coefficients and also different massive operator matrix elements. Starting at 3-loop order, however, also other letters appear in the case of massive operator matrix elements, the so called iterative non-iterative integrals, which are related to solutions based on complete elliptic integrals or any other special function with an integral representation that is definite but not a Volterra-type integral. After outlining the formalism leading to iterative non-iterative integrals,we present examples for both of these cases with the 3-loop anomalous dimension $γ_{qg}^{(2)}$ and the structure of the principle solution in the iterative non-interative case of the 3-loop QCD corrections to the $ρ$-parameter.

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arXiv2017

Heavy Flavor Wilson Coefficients in Deep-Inelastic Scattering: Recent Results

We present recent analytic results for the 3-loop corrections to the massive operator matrix element $A_{Qg}^{(3)}$for further color factors. These results have been obtained using the method of arbitrarily large moments. We also give an overview on the results which were obtained solving all difference and differential equations for the corresponding master integrals that factorize at first order.

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arXiv2017

The two-mass contribution to the three-loop pure singlet operator matrix element

We present the two-mass QCD contributions to the pure singlet operator matrix element at three loop order in x-space. These terms are relevant for calculating the structure function $F_2(x,Q^2)$ at $O(α_s^3)$ as well as for the matching relations in the variable flavor number scheme and the heavy quark distribution functions at the same order. The result for the operator matrix element is given in terms of generalized iterated integrals that include square root letters in the alphabet, depending also on the mass ratio through the main argument. Numerical results are presented.

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arXiv2017

Heavy quark form factors at two loops in perturbative QCD

We present the results for heavy quark form factors at two-loop order in perturbative QCD for different currents, namely vector, axial-vector, scalar and pseudo-scalar currents, up to second order in the dimensional regularization parameter. We outline the necessary computational details, ultraviolet renormalization and corresponding universal infrared structure.

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