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One of the techniques used to compute Feynman integrals in dim-reg is the so-called "differential equations method". Exploiting integration-by-parts identities (linear functional relations between Feynman diagrams), one can set up differential equations for a basis of "master integrals". The DEs method is built upon the conjecture that it is always possible to put this differential equations on a convenient form, upon finding a correct change of basis of master integrals. Such form is called "canonical form" and looks like $$d \vec{f} = \epsilon d\tilde{A} \vec{f},$$ where $\vec{f}$ is the basis of integrals, $\epsilon$ comes from dim-reg in $D = 4-2 \epsilon$ dimensions and $\tilde{A}$ is a matrix of the form $$\tilde{A} = \sum_{i} q_i \log(x-x_i),$$ $x$ being the scale(s) of the problem and $x_i$ certain singular points in the kinematical space.

If one achieve to find the basis $\vec{f}$ where the above form of the DEs holds, then one can immediately integrate the equations in terms of Goncharov polylogs (GPL), which have a well studied algebraic structure and for which series expansions are known (If one is interested in numerical computations, e.g. for phenomenology).

In tipical cases, however, one is able to find a basis where the DE have the above form but with $$\tilde{A} = \sum_{i} q_i \log(a_i),$$ where $a_i$ are $algebraic$ functions of the scale(s) x, which tipically involve square roots. Note that if the $a_i$ were rationals, by partial fractioning one would be able to get back to the desired form whose integration in terms of GPL is immediate.

To address this problem, one could try to get rid of the roots by finding change of variables such that "rationalise" the roots: a beautiful and old problem in algebraic geometry (finding rational parametrizations of surfaces). However, often one does not find such rationalizations and therefore content himself in integrating the DE in terms of Chen Iterated integrals, finding $$ \vec{f}(x,\epsilon) = \mathbb{P} \exp\left(\epsilon \int_0^x \tilde{A}(t) dt\right) \vec{f}(0),$$ starting from some boundary condition $\vec{f}(0)$ which is usually easy to obtain from the definition of Feynman integrals as...integrals. $\mathbb{P}$ is the "path ordered" exponential.

My questions are: what are the advantages of this representation of Feynman integrals? What are the practical (pheno oriented) and conceptual gains over any other integral representation (loop integration, Feynman parametric representation...)?

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  • $\begingroup$ Are you following a reference for this? $\endgroup$ – Qmechanic Mar 24 at 14:55

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