Abstract
We consider the analytic calculation of a two-loop non-planar three-point function which contributes to the two-loop amplitudes for toverline{t} production and γγ production in gluon fusion through a massive top-quark loop. All subtopology integrals can be written in terms of multiple polylogarithms over an irrational alphabet and we employ a new method for the integration of the differential equations which does not rely on the rationalization of the latter. The top topology integrals, instead, in spite of the absence of a massive three-particle cut, cannot be evaluated in terms of multiple polylogarithms and require the introduction of integrals over complete elliptic integrals and polylogarithms. We provide one-fold integral representations for the solutions and continue them analytically to all relevant regions of the phase space in terms of real functions, extracting all imaginary parts explicitly. The numerical evaluation of our expressions becomes straightforward in this way.
Highlights
Been known for a long time, starting at the two-loop order the expansion of Feynman integrals can involve new mathematical structures which lie beyond the realm of multiple polylogarithms
Once the homogeneous solution is known, an integral representation for the inhomogeneous solution is provided by Euler’s variation of constants. Even if usually such integrals cannot be expressed in terms of known special functions, from a practical point of view we are interested in obtaining results that allow for fast and reliable numerical evaluations over the physical phase space. It was shown in [34] that the study of the imaginary part and of the corresponding dispersion relations of Feynman integrals within the differential equations framework can facilitate obtaining compact one-fold integral representations for the two-loop massive Sunrise and the Kite integral
Feynman integrals which evaluate to classes of functions outside the realm of multiple polylogarithms constitute the bottleneck for many multiloop calculations relevant for LHC phenomenology
Summary
As a result we find 11 different master integrals: 9 for the subtopologies and 2 for the top topology. There it was shown for various examples that the possibility of decoupling the differential equations in the limit d → 4 (and of writing the result in terms of multiple polylogarithms) is signaled by a degeneracy of the integration by parts identities in the limit of even numbers of dimensions, d → 2 n with n ∈ N. M9) for the master integrals of the subtopologies, which fulfil canonical differential equations. The two structures, mix up once considering the differential equations for the two master integrals of the top sector, m10 and m11. With the basis given in (2.3), the differential equations for the master integrals of the top topology read d m10 = B(x) m10 + D(x) m10 + N10( ; x) dx m11.
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