We calculate the non-singlet, the pure singlet contribution, and their interference term, at O(α2) 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≫me2 we find discrepancies with the earlier results of Ref. 1 and confirm results obtained in Ref. 2 where the effective method of massive operator matrix elements has been used, which works for all but the power corrections in m2/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(α2) which have not been considered in 1. The corrections are of central importance for precision analyses in e+e− annihilation into γ⁎/Z⁎ at high luminosity.
We calculate the two-mass QCD contributions to the massive operator matrix element Agg,Q at O(αs3) 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.
Hypergeometric structures in Feynman integrals Blümlein, J.; Saragnese, M.; Schneider, C.
Annals of mathematics and artificial intelligence,
10/2023, Letnik:
91, Številka:
5
Journal Article
Recenzirano
Odprti dostop
For the precision calculations in perturbative Quantum Chromodynamics (QCD) gigantic expressions (several GB in size) in terms of highly complicated divergent multi-loop Feynman integrals have to be ...calculated analytically to compact expressions in terms of special functions and constants. In this article we derive new symbolic tools to gain large-scale computer understanding in QCD. Here we exploit the fact that hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package Sigma in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code HypSeries transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code solvePartialLDE is designed. Generalized hypergeometric functions, Appell-, Kampé de Fériet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton–type functions are considered. We illustrate the algorithms by examples.
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 F2(x,Q2) at ...O(αs3) 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.
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 F2(x,Q2).
Heavy quark form factors at two loops Ablinger, J.; Behring, A.; Blümlein, J. ...
Physical review. D,
05/2018, Letnik:
97, Številka:
9
Journal Article
Recenzirano
Odprti dostop
We compute the two-loop QCD corrections to the heavy quark form factors in the case of the vector, axial-vector, scalar and pseudoscalar 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.
Starting at 3-loop order, the massive Wilson coefficients for deep-inelastic scattering and the massive operator matrix elements describing the variable flavor number scheme receive contributions of ...Feynman diagrams carrying quark lines with two different masses. In the case of the charm and bottom quarks, the usual decoupling of one heavy mass at a time no longer holds, since the ratio of the respective masses, η=mc2/mb2∼1/10, is not small enough. Therefore, the usual variable flavor number scheme (VFNS) has to be generalized. The renormalization procedure in the two-mass case is different from the single mass case derived in 1. We present the moments N=2,4 and 6 for all contributing operator matrix elements, expanding in the ratio η. We calculate the analytic results for general values of the Mellin variable N in the flavor non-singlet case, as well as for transversity and the matrix element Agq(3). We also calculate the two-mass scalar integrals of all topologies contributing to the gluonic operator matrix element Agg. As it turns out, the expansion in η is usually inapplicable for general values of N. We therefore derive the result for general values of the mass ratio. From the single pole terms we derive, now in a two-mass calculation, the corresponding contributions to the 3-loop anomalous dimensions. We introduce a new general class of iterated integrals and study their relations and present special values. The corresponding functions are implemented in computer-algebraic form.
The post-Newtonian and post-Minkowskian solutions for the motion of binary mass systems in gravity can be derived in terms of momentum expansions within effective field theory approaches. In the ...post-Minkowskian approach the expansion is performed in the ratio GN/r, retaining all velocity terms completely, while in the post-Newtonian approach only those velocity terms are accounted for which are of the same order as the potential terms due to the virial theorem. We show that it is possible to obtain the complete post-Minkowskian expressions completely algorithmically, under most general purely mathematical conditions from a finite number of velocity terms and illustrate this up to the third post-Minkowskian order given in 1 and compare to expressions obtained in the effective one body formalism.
We calculate the massive flavor non-singlet Wilson coefficient for the heavy flavor contributions to the structure function F2(x,Q2) in the asymptotic region Q2≫m2 and the associated operator matrix ...element Aqq,Q(3),NS(N) to 3-loop order in Quantum Chromodynamics at general values of the Mellin variable N. This matrix element is associated with the vector current and axial vector current for the even and the odd moments N, respectively. We also calculate the corresponding operator matrix elements for transversity, compute the contributions to the 3-loop anomalous dimensions to O(NF) and compare to results in the literature. The 3-loop matching of the flavor non-singlet distribution in the variable flavor number scheme is derived. All results can be expressed in terms of nested harmonic sums in N space and harmonic polylogarithms in x-space. Numerical results are presented for the non-singlet charm quark contribution to F2(x,Q2).
The QED initial state corrections are calculated to the forward–backward asymmetry for e+e−→γ⁎/Z0⁎ in the leading logarithmic approximation to O(α6L6), with L=ln(s/me2), extending the known ...corrections up to O(α2L2) in analytic form. We use the method of massive on-shell operator matrix elements and present the radiators both in Mellin-N and momentum fraction z-space. Numerical results are presented for various energies around the Z-peak by also including energy cuts. These corrections are of relevance for the precision measurements at the FCC_ee.
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