2,664 research outputs found
Evaluating single-scale and/or non-planar diagrams by differential equations
We apply a recently suggested new strategy to solve differential equations
for Feynman integrals. We develop this method further by analyzing asymptotic
expansions of the integrals. We argue that this allows the systematic
application of the differential equations to single-scale Feynman integrals.
Moreover, the information about singular limits significantly simplifies
finding boundary constants for the differential equations. To illustrate these
points we consider two families of three-loop integrals. The first are
form-factor integrals with two external legs on the light cone. We introduce
one more scale by taking one more leg off-shell, . We analytically
solve the differential equations for the master integrals in a Laurent
expansion in dimensional regularization with . Then we show
how to obtain analytic results for the corresponding one-scale integrals in an
algebraic way. An essential ingredient of our method is to match solutions of
the differential equations in the limit of small to our results at
and to identify various terms in these solutions according to
expansion by regions. The second family consists of four-point non-planar
integrals with all four legs on the light cone. We evaluate, by differential
equations, all the master integrals for the so-called graph consisting of
four external vertices which are connected with each other by six lines. We
show how the boundary constants can be fixed with the help of the knowledge of
the singular limits. We present results in terms of harmonic polylogarithms for
the corresponding seven master integrals with six propagators in a Laurent
expansion in up to weight six.Comment: 27 pages, 2 figure
A planar four-loop form factor and cusp anomalous dimension in QCD
We compute the fermionic contribution to the photon-quark form factor to
four-loop order in QCD in the planar limit in analytic form. From the divergent
part of the latter the cusp and collinear anomalous dimensions are extracted.
Results are also presented for the finite contribution. We briefly describe our
method to compute all planar master integrals at four-loop order.Comment: 19 pages, 3 figures, v2: typo in (2.3) fixed and coefficients in
(2.6) corrected; references added and correcte
Two-loop planar master integrals for the production of off-shell vector bosons in hadron collisions
We describe the calculation of all planar master integrals that are needed
for the computation of NNLO QCD corrections to the production of two off-shell
vector bosons in hadron collisions. The most complicated representatives of
integrals in this class are the two-loop four-point functions where two
external lines are on the light-cone and two other external lines have
different invariant masses. We compute these and other relevant integrals
analytically using differential equations in external kinematic variables and
express our results in terms of Goncharov polylogarithms. The case of two equal
off-shellnesses, recently considered in Ref. [1], appears as a particular case
of our general solution.Comment: 28 pages, many figures; ancillary files included with arXiv
submissio
Magic identities for conformal four-point integrals
We propose an iterative procedure for constructing classes of off-shell
four-point conformal integrals which are identical. The proof of the identity
is based on the conformal properties of a subintegral common for the whole
class. The simplest example are the so-called `triple scalar box' and `tennis
court' integrals. In this case we also give an independent proof using the
method of Mellin--Barnes representation which can be applied in a similar way
for general off-shell Feynman integrals.Comment: 13 pages, 12 figures. New proof included with neater discussion of
contact terms. Typo correcte
Iteration of Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory at Three Loops and Beyond
We compute the leading-color (planar) three-loop four-point amplitude of N=4
supersymmetric Yang-Mills theory in 4 - 2 epsilon dimensions, as a Laurent
expansion about epsilon = 0 including the finite terms. The amplitude was
constructed previously via the unitarity method, in terms of two Feynman loop
integrals, one of which has been evaluated already. Here we use the
Mellin-Barnes integration technique to evaluate the Laurent expansion of the
second integral. Strikingly, the amplitude is expressible, through the finite
terms, in terms of the corresponding one- and two-loop amplitudes, which
provides strong evidence for a previous conjecture that higher-loop planar N =
4 amplitudes have an iterative structure. The infrared singularities of the
amplitude agree with the predictions of Sterman and Tejeda-Yeomans based on
resummation. Based on the four-point result and the exponentiation of infrared
singularities, we give an exponentiated ansatz for the maximally
helicity-violating n-point amplitudes to all loop orders. The 1/epsilon^2 pole
in the four-point amplitude determines the soft, or cusp, anomalous dimension
at three loops in N = 4 supersymmetric Yang-Mills theory. The result confirms a
prediction by Kotikov, Lipatov, Onishchenko and Velizhanin, which utilizes the
leading-twist anomalous dimensions in QCD computed by Moch, Vermaseren and
Vogt. Following similar logic, we are able to predict a term in the three-loop
quark and gluon form factors in QCD.Comment: 54 pages, 7 figures. v2: Added references, a few additional words
about large spin limit of anomalous dimensions. v3: Expanded Sect. IV.A on
multiloop ansatz; remark that form-factor prediction is now confirmed by
other work; minor typos correcte
Fermionic corrections to quark and gluon form factors in four-loop QCD
We analytically compute all four-loop QCD corrections to the photon-quark and Higgs-gluon form factors involving a closed massless fermion loop. Our calculation of nonplanar vertex integrals confirms a previous conjecture for the analytical form of the nonfermionic contributions to the collinear anomalous dimensions of quarks and gluons
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