204 research outputs found

### Integration by parts identities in integer numbers of dimensions. A criterion for decoupling systems of differential equations

Integration by parts identities (IBPs) can be used to express large numbers
of apparently different d-dimensional Feynman Integrals in terms of a small
subset of so-called master integrals (MIs). Using the IBPs one can moreover
show that the MIs fulfil linear systems of coupled differential equations in
the external invariants. With the increase in number of loops and external
legs, one is left in general with an increasing number of MIs and consequently
also with an increasing number of coupled differential equations, which can
turn out to be very difficult to solve. In this paper we show how studying the
IBPs in fixed integer numbers of dimension d=n with $n \in \mathbb{N}$ one can
extract the information useful to determine a new basis of MIs, whose
differential equations decouple as $d \to n$ and can therefore be more easily
solved as Laurent expansion in (d-n).Comment: 31 pages, minor typos corrected, references added, accepted for
publication in Nuclear Physics

### Schouten identities for Feynman graph amplitudes; the Master Integrals for the two-loop massive sunrise graph

A new class of identities for Feynman graph amplitudes, dubbed Schouten
identities, valid at fixed integer value of the dimension d is proposed. The
identities are then used in the case of the two loop sunrise graph with
arbitrary masses for recovering the second order differential equation for the
scalar amplitude in d=2 dimensions, as well as a chained sets of equations for
all the coefficients of the expansions in (d-2). The shift from $d\approx2$ to
$d\approx4$ dimensions is then discussed.Comment: 30 pages, 1 figure, minor typos in the text corrected, results
unchanged. Version accepted for publication on Nuclear Physics

### An Elliptic Generalization of Multiple Polylogarithms

We introduce a class of functions which constitutes an obvious elliptic
generalization of multiple polylogarithms. A subset of these functions appears
naturally in the \epsilon-expansion of the imaginary part of the two-loop
massive sunrise graph. Building upon the well known properties of multiple
polylogarithms, we associate a concept of weight to these functions and show
that this weight can be lowered by the action of a suitable differential
operator. We then show how properties and relations among these functions can
be studied bottom-up starting from lower weights.Comment: 27 pages plus three appendices, v2: references added, typos
corrected, accepted for publication on NP

### Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph

We consider the calculation of the master integrals of the three-loop massive
banana graph. In the case of equal internal masses, the graph is reduced to
three master integrals which satisfy an irreducible system of three coupled
linear differential equations. The solution of the system requires finding a $3
\times 3$ matrix of homogeneous solutions. We show how the maximal cut can be
used to determine all entries of this matrix in terms of products of elliptic
integrals of first and second kind of suitable arguments. All independent
solutions are found by performing the integration which defines the maximal cut
on different contours. Once the homogeneous solution is known, the
inhomogeneous solution can be obtained by use of Euler's variation of
constants.Comment: 39 pages, 3 figures; Fixed a typo in eq. (6.16

### Three-loop mixed QCD-electroweak corrections to Higgs boson gluon fusion

We compute the contribution of three-loop mixed QCD-electroweak corrections
($\alpha_S^2\alpha^2$) to the $gg \to H$ scattering amplitude. We employ the
method of differential equations to compute the relevant integrals and express
them in terms of Goncharov polylogarithms.Comment: 21 pages, associated ancillary files distributed with the paper or
available from external repository. Correct typos and reference

### Double-real contribution to the quark beam function at N$^{3}$LO QCD

We compute the master integrals required for the calculation of the
double-real emission contributions to the matching coefficients of 0-jettiness
beam functions at next-to-next-to-next-to-leading order in perturbative QCD. As
an application, we combine these integrals and derive the double-real gluon
emission contribution to the matching coefficient $I_{qq}(t,z)$ of the quark
beam function.Comment: 28 pages, 1 figure; updated ancillary file (accessible through url in
the section "Results"

### Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral

We introduce a class of iterated integrals that generalize multiple
polylogarithms to elliptic curves. These elliptic multiple polylogarithms are
closely related to similar functions defined in pure math- ematics and string
theory. We then focus on the equal-mass and non-equal-mass sunrise integrals,
and we develop a formalism that enables us to compute these Feynman integrals
in terms of our iterated integrals on elliptic curves. The key idea is to use
integration-by-parts identities to identify a set of integral kernels, whose
precise form is determined by the branch points of the integral in question.
These kernels allow us to express all iterated integrals on an elliptic curve
in terms of them. The flexibility of our approach leads us to expect that it
will be applicable to a large variety of integrals in high-energy physics.Comment: 22 page

### Triple-real contribution to the quark beam function in QCD at next-to-next-to-next-to-leading order

We compute the three-loop master integrals required for the calculation of
the triple-real contribution to the N$^3$LO quark beam function due to the
splitting of a quark into a virtual quark and three collinear gluons, $q \to
q^*+ggg$. This provides an important ingredient for the calculation of the
leading-color contribution to the quark beam function at N$^3$LO.Comment: 31 pages, 2 figures; published version, updated ancillary file
(accessible through url in the section "Results"

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