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Johannes Blümlein

Publications and source records attributed to Johannes Blümlein.

At least 55 records · Page 3Linked to original sources
arXiv2009

Structural Relations of Harmonic Sums and Mellin Transforms up to Weight w = 5

We derive the structural relations between the Mellin transforms of weighted Nielsen integrals emerging in the calculation of massless or massive single--scale quantities in QED and QCD, such as anomalous dimensions and Wilson coefficients, and other hard scattering cross sections depending on a single scale. The set of all multiple harmonic sums up to weight five cover the sums needed in the calculation of the 3--loop anomalous dimensions. The relations extend the set resulting from the quasi-shuffle product between harmonic sums studied earlier. Unlike the shuffle relations, they depend on the value of the quantities considered. Up to weight {\sf w = 5}, 242 nested harmonic sums contribute. In the present physical applications it is sufficient to consider the sub-set of harmonic sums not containing an index $i = -1$, which consists out of 69 sums. The algebraic relations reduce this set to 30 sums. Due to the structural relations a final reduction of the number of harmonic sums to {15} basic functions is obtained. These functions can be represented in terms of factorial series, supplemented by harmonic sums which are algebraically reducible. Complete analytic representations are given for these {15} meromorphic functions in the complex plane deriving their asymptotic- and recursion relations. A general outline is presented on the way nested harmonic sums and multiple zeta values emerge in higher order calculations of zero- and single scale quantities.

hep-ph↗
arXiv2009

Mellin Moments of the {$O(α_s^3$)} Heavy Flavor Contributions to unpolarized Deep-Inelastic Scattering at $Q^2 \gg m^2$ and Anomalous Dimensions

We calculate the $O(α_s^3)$ heavy flavor contributions to the Wilson coefficients of the structure function $F_2(x,Q^2)$ and the massive operator matrix elements (OMEs) for the twist--2 operators of unpolarized deeply inelastic scattering in the region $Q^2 \gg m^2$. The massive Wilson coefficients are obtained as convolutions of massive OMEs and the known light flavor Wilson coefficients. We also compute the massive OMEs which are needed to evaluate heavy flavor parton distributions in the variable flavor number scheme (VFNS) to 3--loop order. All contributions to the Wilson coefficients and operator matrix elements but the genuine constant terms at $O(α_s^3)$ of the OMEs are derived in terms of quantities, which are known for general values in the Mellin variable $N$. For the operator matrix elements $A_{Qg}^{(3)}, A_{qg,Q}^{(3)}$ and $A_{gg,Q}^{(3)}$ the moments $N = 2$ to 10, for $A_{Qq}^{(3), \rm PS}$ to $N = 12$, and for $A_{qq,Q}^{(3), \rm NS}$, $A_{qq,Q}^{(3),\rm PS}$, $A_{gq,Q}^{(3)}$ to N=14 are computed. These terms contribute to the light flavor +-combinations. For the flavor non-singlet terms, we calculate as well the odd moments N=1 to 13, corresponding to the light flavor $-$-combinations. We also obtain the moments of the 3--loop anomalous dimensions, their color projections for the present processes respectively, in an independent calculation, which agree with the results given in the literature.

hep-ph↗
arXiv2009

Structural Relations of Harmonic Sums and Mellin Transforms at Weight w=6

We derive the structural relations between nested harmonic sums and the corresponding Mellin transforms of Nielsen integrals and harmonic polylogarithms at weight {\sf w = 6}. They emerge in the calculations of massless single--scale quantities in QED and QCD, such as anomalous dimensions and Wilson coefficients, to 3-- and 4--loop order. We consider the set of the multiple harmonic sums at weight six without index $\{-1\}$. This restriction is sufficient for all known physical cases. The structural relations supplement the algebraic relations, due to the shuffle product between harmonic sums, studied earlier. The original amount of 486 possible harmonic sums contributing at weight {\sf w = 6} reduces to 99 sums with no index $\{-1\}$. Algebraic and structural relations lead to a further reduction to 20 basic functions. These functions supplement the set of 15 basic functions up to weight {\sf w = 5} derived formerly. We line out an algorithm to obtain the analytic representation of the basic sums in the complex plane.

math-ph↗
arXiv2008

Higher Twist Contributions to Deep-Inelastic Structure Functions

We report on a recent extraction of the higher twist contributions to the deep inelastic structure functions $F_2^{ep,ed}(x,Q^2)$ in the large $x$ region. It is shown that the size of the extracted higher twist contributions is strongly correlated with the higher order corrections applied to the leading twist part. A gradual lowering of the higher twist contributions going from NLO to N$^4$LO is observed, where in the latter case only the leading large $x$ terms were considered.

hep-ph↗
arXiv2008

Higher Twist Contributions to the Structure Functions F_2^p(x,Q^2) and F_2^d(x,Q^2) at Large x and Higher Orders

The higher twist contributions to the deeply inelastic structure functions $F_2^{p}(x,Q^2)$ and $F_2^{d}(x,Q^2)$ for larger values of the Bjorken variable $x$ are extracted extrapolating the {twist--2} contributions measured in the large $W^2$ region to the region $4 \GeV^2 \leq W^2 \leq 12.5 \GeV^2$ applying target mass corrections. We compare the results for the NLO, NNLO and N$^3$LO analyzes and include also the large $x$ at N$^4$LO to the Wilson coefficients. A gradual lowering of the higher twist contributions going from NLO to N$^4$LO is observed, which stresses the importance of higher order corrections.

hep-ph↗
arXiv2008

Two--Loop Massive Operator Matrix Elements for Unpolarized Heavy Flavor Production to $O(ε)

We calculate the $O(α_s^2)$ massive operator matrix elements for the twist--2 operators, which contribute to the heavy flavor Wilson coefficients in unpolarized deeply inelastic scattering in the region $Q^2 \gg m^2$, up to the $O(ε)$ contributions. These terms contribute through the renormalization of the $O(α_s^3)$ heavy flavor Wilson coefficients of the structure function $F_2(x,Q^2)$. The calculation has been performed using light--cone expansion techniques without using the integration-by-parts method. We represent the individual Feynman diagrams by generalized hypergeometric structures, the $ε$--expansion of which leads to infinite sums depending on the Mellin variable $N$. These sums are finally expressed in terms of nested harmonic sums using the general summation techniques implemented in the {\tt Sigma} package.

hep-ph↗
arXiv2007

The Status of the Polarized Parton Densities

A survey is given on the present knowledge of the polarized parton distribution functions. We give an outlook for further developments desired both on the theoretical as well on the experimental side to complete the understanding of the spin--structure of nucleons in the future.

hep-ph↗
arXiv2007

Structural Relations between Harmonic Sums up to w=6

Multiply nested finite harmonic sums $S_{a_1 ... a_n}(N)$ occur in many single scale higher order calculations in Quantum Field Theory. We discuss their algebraic and structural relations to weight {\sf w=6}. As an example, we consider the application of these relations to the soft and virtual corrections for Bhabha-scattering to $O(α^2)$.

hep-ph↗
arXiv2007

Two-Loop Massive Operator Matrix Elements and Unpolarized Heavy Flavor Production at Asymptotic Values Q^2 >> m^2

We calculate the $O(α_s^2)$ massive operator matrix elements for the twist--2 operators, which contribute to the heavy flavor Wilson coefficients in unpolarized deeply inelastic scattering in the region $Q^2 \gg m^2$. The calculation has been performed using light--cone expansion techniques. We confirm an earlier result obtained in \cite{Buza:1995ie}. The calculation is carried out without using the integration-by-parts method and in Mellin space using harmonic sums, which lead to a significant compactification of the analytic results derived previously. The results allow to determine the heavy flavor Wilson coefficients for $F_2(x,Q^2)$ to $O(α_s^2)$ and for $F_L(x,Q^2)$ to $O(α_s^3)$ for all but the power suppressed terms $\propto (m^2/Q^2)^k, k \geq 1$.

hep-ph↗
arXiv2007

Universal Higher Order Singlet QED Corrections to Unpolarized Lepton Scattering

We calculate the universal flavor-singlet radiative QED corrections to unpolarized lepton scattering applicable to general differential scattering cross sections, involving charged fermions or photons in initial or final states. The radiators are derived to $O((α\ln(Q^2/m_f^2))^5)$ in analytic form. Numerical illustrations are given.

hep-ph↗
arXiv2006

Non--Singlet QCD Analysis of Deep Inelastic World Data at \boldmath $O(α_s^3)$

A non--singlet QCD analysis of the structure function $F_2(x,q^2)$ in LO, NLO, NNLO and N$^3$LO is performed based on the world data for charged lepton scattering. We determine the valence quark parton densities and present their parameterization and that of the correlated errors in a wide range of $x$ and $Q^2$. In the analysis we determined the QCD--scale $Λ_{\rm QCD, N_f = 4}^{\bar{\rm MS}} = 265 \pm 27 \MeV {\rm (NLO)}, 226 \pm 25 \MeV {\rm (NNLO)}, 234 \pm 26 \MeV {\rm (N^3LO)}$, with a remainder uncertainty of $\pm 2\MeV$ for the yet unknown 4--loop anomalous dimension, corresponding to $α_s(M_Z^2) = 0.1148 \pm 0.0019 {\rm NLO}, 0.1134 {\tiny{\begin{array}{c} +0.0019 -0.0021 \end{array}}} {\rm NNLO}, 0.1141 {\tiny{\begin{array}{c} +0.0020 -0.0022 \end{array}}} {\rm N^3LO}$. A comparison is performed to other determinations of the QCD scale and $α_s(M_Z^2)$ in deeply inelastic scattering. The higher twist contributions of $F_2^p(x,Q^2)$ and $F_2^d(x,Q^2)$ are extracted in the large $x$ region, subtracting the twist--2 contributions obtained in the NLO, NNLO and N$^3$LO analysis.

hep-ph↗
arXiv2006

Target mass and finite momentum transfer corrections to unpolarized and polarized diffractive scatterin

A quantum field theoretic treatment of inclusive deep--inelastic diffractive scattering is given. The process can be described in the general framework of non--forward scattering processes using the light--cone expansion in the generalized Bjorken region. Target mass and finite $t$ corrections of the diffractive hadronic tensor are derived at the level of the twist--2 contributions both for the unpolarized and the polarized case. They modify the expressions contributing in the limit $t, M^2 \to 0$ for larger values of $β$ or/and $t$ in the region of low $Q^2$. The different diffractive structure functions are expressed through integrals over the relative momentum of non--perturbative $t$--dependent 2--particle distribution functions. In the limit $t, M^2 \to 0$ these distribution functions are the diffractive parton distribution. Relations between the different diffractive structure functions are derived.

hep-ph↗
arXiv2005

Analytic Continuation of the Harmonic Sums for the 3--Loop Anomalous Dimensions

We present for numerical use the analytic continuations to complex arguments of those basic Mellin transforms, which build the harmonic sums contributing to the 3--loop anomalous dimensions. Eight new basic functions contribute in addition to the analytic continuations for the 2--loop massless Wilson coefficients calculated previously. The representations derived have a relative accuracy of better than $10^{-7}$ in the range $x ε[10^{-6},0.98]$.

hep-ph↗
arXiv2004

Universal Higher Order QED Corrections to Polarized Lepton Scattering

We calculate the universal radiative QED corrections to polarized lepton scattering for general scattering cross sections in analytic form. The flavor non--singlet and singlet radiation functions are calculated to $O((α{\rm L})^5)$. The resummation of the non--singlet and singlet contributions to the QED--anomalous dimensions $\propto(α\ln^2(x))^k$ is performed to all orders. Numerical results are presented for the individual radiation functions. Applications to polarized deeply inelastic lepton--nucleon scattering are given.

hep-ph↗
arXiv2004

Reduction of Multiple Harmonic Sums and Harmonic Polylogarithms

The alternating and non-alternating harmonic sums and other algebraic objects of the same equivalence class are connected by algebraic relations which are induced by the product of these quantities and which depend on their index calss rather than on their value. We show how to find a basis of the associated algebra. The length of the basis $l$ is found to be $\leq 1/d$, where $d$ is the depth of the sums considered and is given by the 2nd {\sc Witt} formula. It can be also determined counting the {\sc Lyndon} words of the respective index set. The relations derived can be used to simplify results of higher order calculations in QED and QCD.

hep-ph↗