Abstract
We compute the non-singlet nh terms to the massive three loop vector-, axialvector-, scalar- and pseudoscalar form factors in a direct analytic calculation using the method of large moments. This method has the advantage, that the master integrals have to be dealt with only in their moment representation, allowing to also consider quantities which obey differential equations, which are not first order factorizable (elliptic and higher), already at this level. To obtain all the associated recursions, up to 8000 moments had to be calculated. A new technique has been applied to solve the associated differential equation systems. Here the decoupling is performed such, that only minimal depth ε–expansions had to be performed for non–first-order factorizing systems, minimizing the calculation of initial values. The pole terms in the dimensional parameter ε can be completely predicted using renormalization group methods, as confirmed by the present results. A series of contributions at O(ε0) have first order factorizable representations. For a smaller number of color–zeta projections this is not the case. All first order factorizing terms can be represented by harmonic polylogarithms. We also obtain analytic results for the non–first-order factorizing terms by Taylor series in a variable x, for which we have calculated at least 2000 expansion coefficients, in an approximation. Based on this representation the form factors can be given in the Euclidean region and in the region q2≈0. Numerical results are presented.
Highlights
The knowledge of the massive three–loop form factor is essential ingredient to the calculation for a series of massive processes at e+e− and hadron colliders, determined by vector, axialvector, scalar and pseudoscalar currents
The differential equations given by the integration by parts (IBP) relations [16,17,18,19,20,21,22,23] are transformed into recursions, through which a large number of moments for the master integrals and the form factors are generated using the package SolveCoupledSystems [15]
If λ ≥ 5, the uncoupling step failed by space-time resources or produced a not digestible output: the degrees of the polynomials bi,j (x, ε) in (5.5) were very high yielding linear recurrences with orders close to 1000. For these more complicated systems λ ≥ 5 we developed another variant of our large moment method, that is implemented in our package SolveCoupledSystem.m
Summary
The knowledge of the massive three–loop form factor is essential ingredient to the calculation for a series of massive processes at e+e− and hadron colliders, determined by vector, axialvector, scalar and pseudoscalar currents. We compute the non-singlet nh contributions of the massive three–loop form factor for vector, axialvector, scalar and pseudoscalar currents. The differential equations given by the integration by parts (IBP) relations [16,17,18,19,20,21,22,23] are transformed into recursions, through which a large number of moments for the master integrals and the form factors are generated using the package SolveCoupledSystems [15]. In the expansion of the form factors to master integrals usually higher order terms in ε = 2 − D/2 are contributing These are containing, elliptic and more involved contributions. In the appendix we present a series of deeper ε–expansions for some integrals defining the initial conditions
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