are obviously desirable in order to have control over errors in approximations, especially
when the convergence of expansion in powers of mass ratio works slowly.
On the theoretical side, unravelling the mathematical structure of Feynman integrals
will be important to handle the complexity of their calculation and may help us to obtain
a better understanding of the perturbative quantum field theory. The study of the math-
ematical properties of Feynman integrals has attracted increasing attention both by the
physics and the mathematics communities. Significant progresses were achieved in under-
standing the analytical computations of multi-loop Feynman integrals in the last years.
One of the powerful methods to evaluate the master integrals analytically is the method
of differential equations [15–19]. Along with the recent years’ development [20–24], this
method is becoming more and more powerful. It is pointed out in ref. [20] that for multi-
loop Feynman integrals calculations, a suitable basis of master integrals can be chosen, so
that the corresponding differential equations are greatly simplified, and their iterative so-
lutions become straightforward in terms of dimensional regularization parameter =
4−D
2
.
The choice of canonical basis will also simplify the determination of boundary conditions
considerably. Following this proposal, substantive analytical computations of various phe-
nomenology processes have been completed [25–36].
The two-loop master integrals of QED electron form factors have been calculated
in [37], for on shell electrons of finite squared mass and arbitrary momentum transfer. The
calculations of those master integrals have been refined by a suitable choice of basis [23].
The analytical results of two-loop master integrals for form factors of heavy fermion to
massless fermion have been obtained in [6, 38–40]. All the master integrals for these
two processes can be expressed in terms of Harmonic polylogarithms. However, for two-
loop vertex integrals with two different type of massive fermions, the master integrals
will contain one more scale, to the best of our knowledge, the integrals have not been
calculated analytically in the literature. Furthermore, understanding the structure of loop
integrals more generally is an interesting and important challenge. In this work, employing
the method of differential equations, along with a proper choice of canonical basis, we
calculate all the master integrals for two-loop heavy-to-light form factors of two massive
fermions, the results are expressed in terms of Goncharov polylogarithms.
The outline of the paper is as follows. In section 2, the kinematics and notations
are introduced for the processes we concern. We also present the generic form of the dif-
ferential equations with respect to the kinematics variables in terms of the derivatives of
the external momentum. In section 3, the Goncharov polylogarithms as well as Harmonic
polylogarithms are introduced. In section 4, the canonical basis are explicitly presented, fol-
lowed by the discussion of their solutions. In sections 5, the determination of the boundary
conditions, as well as the analytical continuation are explained. Discussions and conclu-
sions are made in section 6. In appendix A, we present all the rational matrices of the
system of differential equations in canonical form. All the analytical results up to weight
four from our computation as well as the rational matrices in electronic form are collected
as supplementary material accompanying the published version of this paper.
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