arXiv subjects

Sebastian Klein

Publications and source records attributed to Sebastian Klein.

30 records · Page 2Linked to original sources

arXiv2009

Heavy Flavor Contributions to DIS Structure Functions at $O(α_s^3)$

We calculate moments of the $O(α_s^3)$ heavy flavor contributions to the Wilson coefficients of the structure function $F_2(x,Q^2)$ in the region $Q^2\gg m^2$. The massive Wilson coefficients are obtained as convolutions of massive operator matrix elements (OMEs) and the known light flavor Wilson coefficients. The calculation of moments of the massive OMEs involves a first independent recalculation of moments of the fermionic contributions to all 3--loop anomalous dimensions of the unpolarized twist--2 local composite operators stemming from the light--cone expansion. The expressions for the massive OMEs are now known for general values of the Mellin variable $N$ apart of one coefficient in the constant term, for which only a series of moments is computed.

hep-ph↗

arXiv2009

Mellin moments of heavy flavor contributions to F_2(x,Q^2) at NNLO

This thesis is concerned with the calculation of fixed moments of the O(a_s^3) heavy flavor contributions to the Wilson coefficients of the structure function F_2(x,Q^2) in the limit Q^2 >> m^2, neglecting power corrections. The massive Wilson coefficients in the asymptotic region are given as convolutions of massive operator matrix elements (OMEs) and the known light flavor Wilson coefficients. The former derive from the twist--2 operators emerging in the light--cone--expansion and are calculated at the 3--loop level for fixed moments. We also compute the massive OMEs which are needed to evaluate heavy flavor parton distributions in the variable flavor number scheme to the same order. All contributions to the Wilson coefficients and OMEs but the genuine constant terms at O(a_s^3) of the OMEs are derived in terms of quantities, which are known for general values in the Mellin variable N. For the OMEs A_{Qg}^(3), A_{qg,Q}^(3) and A_{gg,Q}^(3) the moments N = 2 to 10, for A_{Qq}^{(3), PS} to N = 12, and for A_{qq,Q}^{(3), NS}, A_{qq,Q}^{(3), 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 moments of the terms ~ T_F of the 3-loop anomalous dimensions in an independent calculation, which agree with results given in the literature. The mathematical structure of the occurring momentum integrals and of the final results in terms of harmonic sums is discussed. We study applications of the same techniques to the polarized and transversity case at the NLO and NNLO level as well.

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

On the Funk transform on compact symmetric spaces

We prove that a function on an irreducible compact symmetric space M, which is not a sphere, is determined by its integrals over the shortest closed geodesics in M. We also prove a support theorem for the Funk transform on rank one symmetric spaces which are not spheres.

math.DG↗

arXiv2008

Totally geodesic submanifolds in Riemannian symmetric spaces

In the first part of this expository article, the most important constructions and classification results concerning totally geodesic submanifolds in Riemannian symmetric spaces are summarized. In the second part, I describe the results of my classification of the totally geodesic submanifolds in the Riemannian symmetric spaces of rank 2.

math.DG↗

arXiv2008

Totally geodesic submanifolds of the exceptional Riemannian symmetric spaces of rank 2

The present article is the final part of a series on the classification of the totally geodesic submanifolds of the irreducible Riemannian symmetric spaces of rank 2. After this problem has been solved for the 2-Grassmannians in my previous papers cited in the present paper as [K1] and [K2], and for the space SU(3)/SO(3) in [K3], Section 6, we now solve the classification for the remaining irreducible Riemannian symmetric spaces of rank 2 and compact type: SU(6)/Sp(3), SO(10)/U(5), E6/(U(1)*Spin(10)), E6/F4, G2/SO(4), SU(3), Sp(2) and G2. Similarly as for the spaces already investigated in the earlier papers, it turns out that for many of the spaces investigated here, the earlier classification of the maximal totally geodesic submanifolds of Riemannian symmetric spaces by Chen and Nagano ([CN], Paragraph 9) is incomplete. In particular, in the spaces Sp(2), G2/SO(4) and G2, there exist maximal totally geodesic submanifolds, isometric to 2- or 3-dimensional spheres, which have a "skew" position in the ambient space in the sense that their geodesic diameter is strictly larger than the geodesic diameter of the ambient space. They are all missing from [CN].

math.DG↗

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↗

arXiv2008

Reconstructing the geometric structure of a Riemannian symmetric space from its Satake diagram

The local geometry of a Riemannian symmetric space is described completely by the Riemannian metric and the Riemannian curvature tensor of the space. In the present article I describe how to compute these tensors for any Riemannian symmetric space from the Satake diagram, in a way that is suited for the use with computer algebra systems. As an example application, the totally geodesic submanifolds of the Riemannian symmetric space SU(3)/SO(3) are classified. The submission also contains an example implementation of the algorithms and formulas of the paper as a package for Maple 10, the technical documentation for this implementation, and a worksheet carrying out the computations for the space SU(3)/SO(3) used in the proof of Proposition 6.1 of the paper.

math.DG↗

arXiv2007

Totally geodesic submanifolds of the complex and the quaternionic 2-Grassmannians

In this article, I classify the totally geodesic submanifolds in the complex 2-Grassmannians and in the quaternionic 2-Grassmannians. It turns out that for both of these spaces, the earlier classification of maximal totally geodesic submanifolds in Riemannian symmetric spaces of rank 2, published by Chen and Nagano (B.-Y. Chen, T. Nagano, "Totally geodesic submanifolds of symmetric spaces, II", Duke Math. J. 45 (1978), 405--425) is incomplete. For example, G_2(H^n) with n >= 7 contains totally geodesic submanifolds isometric to a HP^2, its metric scaled such that the minimal sectional curvature is 1/5; they are maximal in G_2(H^7). Also G_2(C^n) with n >= 6 contains totally geodesic submanifolds which are isometric to a CP^2 contained in the HP^2 mentioned above; they are maximal in G_2(C^6). Neither submanifolds are mentioned in the cited paper by Chen and Nagano.

math.DG↗

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↗

arXiv2006

Totally geodesic submanifolds of the complex quadric

In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Riemannian symmetric spaces. In this way a classification of the totally geodesic submanifolds in the complex quadric $Q^m := \SO(m+2)/(\SO(2) \times \SO(m))$ is obtained. It turns out that the earlier classification of totally geodesic submanifolds of $Q^m$ by Chen and Nagano is incomplete: in particular a type of submanifolds which are isometric to 2-spheres of radius $\tfrac{1}{2}\sqrt{10}$, and which are neither complex nor totally real in $Q^m$, is missing.

math.DG↗