1,648 research outputs found
Analytic Results for Massless Three-Loop Form Factors
We evaluate, exactly in d, the master integrals contributing to massless
three-loop QCD form factors. The calculation is based on a combination of a
method recently suggested by one of the authors (R.L.) with other techniques:
sector decomposition implemented in FIESTA, the method of Mellin--Barnes
representation, and the PSLQ algorithm. Using our results for the master
integrals we obtain analytical expressions for two missing constants in the
ep-expansion of the two most complicated master integrals and present the form
factors in a completely analytic form.Comment: minor revisions, to appear in JHE
The Dimensional Recurrence and Analyticity Method for Multicomponent Master Integrals: Using Unitarity Cuts to Construct Homogeneous Solutions
We consider the application of the DRA method to the case of several master
integrals in a given sector. We establish a connection between the homogeneous
part of dimensional recurrence and maximal unitarity cuts of the corresponding
integrals: a maximally cut master integral appears to be a solution of the
homogeneous part of the dimensional recurrence relation. This observation
allows us to make a necessary step of the DRA method, the construction of the
general solution of the homogeneous equation, which, in this case, is a coupled
system of difference equations.Comment: 17 pages, 2 figure
Foundation and generalization of the expansion by regions
The "expansion by regions" is a method of asymptotic expansion developed by
Beneke and Smirnov in 1997. It expands the integrand according to the scaling
prescriptions of a set of regions and integrates all expanded terms over the
whole integration domain. This method has been applied successfully to many
complicated loop integrals, but a general proof for its correctness has still
been missing. This paper shows how the expansion by regions manages to
reproduce the exact result correctly in an expanded form and clarifies the
conditions on the choice and completeness of the considered regions. A
generalized expression for the full result is presented that involves
additional overlap contributions. These extra pieces normally yield scaleless
integrals which are consistently set to zero, but they may be needed depending
on the choice of the regularization scheme. While the main proofs and formulae
are presented in a general and concise form, a large portion of the paper is
filled with simple, pedagogical one-loop examples which illustrate the
peculiarities of the expansion by regions, explain its application and show how
to evaluate contributions within this method.Comment: 84 pages; v2: comment on scaleless integrals added to conclusions,
version published in JHE
An Analytic Result for the Two-Loop Hexagon Wilson Loop in N = 4 SYM
In the planar N=4 supersymmetric Yang-Mills theory, the conformal symmetry
constrains multi-loop n-edged Wilson loops to be basically given in terms of
the one-loop n-edged Wilson loop, augmented, for n greater than 6, by a
function of conformally invariant cross ratios. We identify a class of
kinematics for which the Wilson loop exhibits exact Regge factorisation and
which leave invariant the analytic form of the multi-loop n-edged Wilson loop.
In those kinematics, the analytic result for the Wilson loop is the same as in
general kinematics, although the computation is remarkably simplified with
respect to general kinematics. Using the simplest of those kinematics, we have
performed the first analytic computation of the two-loop six-edged Wilson loop
in general kinematics.Comment: 17 pages. Extended discussion on how the QMRK limit is taken. Version
accepted by JHEP. A text file containing the Mathematica code with the
analytic expression for the 6-point remainder function is include
A New Algorithm For The Generation Of Unitarity-Compatible Integration By Parts Relations
Many multi-loop calculations make use of integration by parts relations to
reduce the large number of complicated Feynman integrals that arise in such
calculations to a simpler basis of master integrals. Recently, Gluza, Kajda,
and Kosower argued that the reduction to master integrals is complicated by the
presence of integrals with doubled propagator denominators in the integration
by parts relations and they introduced a novel reduction procedure which
eliminates all such integrals from the start. Their approach has the advantage
that it automatically produces integral bases which mesh well with generalized
unitarity. The heart of their procedure is an algorithm which utilizes the
weighty machinery of computational commutative algebra to produce complete sets
of unitarity-compatible integration by parts relations. In this paper, we
propose a conceptually simpler algorithm for the generation of complete sets of
unitarity-compatible integration by parts relations based on recent results in
the mathematical literature. A striking feature of our algorithm is that it can
be described entirely in terms of straightforward linear algebra.Comment: 20 pages; My apologies to Krzysztof Kajda for misspelling his name in
v1; in v3: the labeling of the variables in (4.5) and eqs. (4.20) and (4.21)
was adjusted to match the notation used in the rest of Section 4. I thank
York Schroeder for pointing out the notational inconsistenc
The Form Factors and Quantum Equation of Motion in the sine-Gordon Model
Using the methods of the 'form factor program' exact expressions of all
matrix elements are obtained for several operators of the quantum sine-Gordon
model alias the massive Thirring model. A general formula is presented which
provides form factors in terms of an integral representation. In particular
charge-less operators as for example the current of the topological charge, the
energy momentum tensor and all higher currents are considered. In the breather
sector it is found the quantum sine-Gordon field equation holds with an exact
relation between the 'bare' mass and the normalized mass. Also a relation for
the trace of the energy momentum is obtained. All results are compared with
Feynman graph expansion and full agreement is found.Comment: TCI-LaTeX, 21 pages with 2 figur
Hepta-Cuts of Two-Loop Scattering Amplitudes
We present a method for the computation of hepta-cuts of two loop scattering
amplitudes. Four dimensional unitarity cuts are used to factorise the integrand
onto the product of six tree-level amplitudes evaluated at complex momentum
values. Using Gram matrix constraints we derive a general parameterisation of
the integrand which can be computed using polynomial fitting techniques. The
resulting expression is further reduced to master integrals using conventional
integration by parts methods. We consider both planar and non-planar topologies
for 2 to 2 scattering processes and apply the method to compute hepta-cut
contributions to gluon-gluon scattering in Yang-Mills theory with adjoint
fermions and scalars.Comment: 37 pages, 6 figures. version 2 : minor updates, published versio
An Integrand Reconstruction Method for Three-Loop Amplitudes
We consider the maximal cut of a three-loop four point function with massless
kinematics. By applying Groebner bases and primary decomposition we develop a
method which extracts all ten propagator master integral coefficients for an
arbitrary triple-box configuration via generalized unitarity cuts. As an
example we present analytic results for the three loop triple-box contribution
to gluon-gluon scattering in Yang-Mills with adjoint fermions and scalars in
terms of three master integrals.Comment: 15 pages, 1 figur
Application of the DRA method to the calculation of the four-loop QED-type tadpoles
We apply the DRA method to the calculation of the four-loop `QED-type'
tadpoles. For arbitrary space-time dimensionality D the results have the form
of multiple convergent sums. We use these results to obtain the
epsilon-expansion of the integrals around D=3 and D=4.Comment: References added, some typos corrected. Results unchange
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