19,645 research outputs found
Dimensional recurrence relations: an easy way to evaluate higher orders of expansion in
Applications of a method recently suggested by one of the authors (R.L.) are
presented. This method is based on the use of dimensional recurrence relations
and analytic properties of Feynman integrals as functions of the parameter of
dimensional regularization, . The method was used to obtain analytical
expressions for two missing constants in the -expansion of the most
complicated master integrals contributing to the three-loop massless quark and
gluon form factors and thereby present the form factors in a completely
analytic form. To illustrate its power we present, at transcendentality weight
seven, the next order of the -expansion of one of the corresponding
most complicated master integrals. As a further application, we present three
previously unknown terms of the expansion in of the three-loop
non-planar massless propagator diagram. Only multiple values at integer
points are present in our result.Comment: Talk given at the International Workshop `Loops and Legs in Quantum
Field Theory' (April 25--30, 2010, W\"orlitz, Germany)
Evaluating `elliptic' master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points
This is a sequel of our previous paper where we described an algorithm to
find a solution of differential equations for master integrals in the form of
an -expansion series with numerical coefficients. The algorithm is
based on using generalized power series expansions near singular points of the
differential system, solving difference equations for the corresponding
coefficients in these expansions and using matching to connect series
expansions at two neighboring points. Here we use our algorithm and the
corresponding code for our example of four-loop generalized sunset diagrams
with three massive and two massless propagators, in order to obtain new
analytical results. We analytically evaluate the master integrals at threshold,
, in an expansion in up to . With the help of
our code, we obtain numerical results for the threshold master integrals in an
-expansion with the accuracy of 6000 digits and then use the PSLQ
algorithm to arrive at analytical values. Our basis of constants is build from
bases of multiple polylogarithm values at sixth roots of unity.Comment: Discussion extende
On Thermodynamics of Rational Maps. II: Non-Recurrent Maps
The pressure function p(t) of a non-recurrent map is real analytic on some interval (0,t_*) with t_* strictly greater than the dimension of the Julia set. The proof is an adaptation of the well known tower techniques to the complex dynamics situation. In general, p(t) need not be analytic on the whole positive axis
Four-loop quark form factor with quartic fundamental colour factor
We analytically compute the four-loop QCD corrections for the colour
structure to the massless non-singlet quark form factor. The
computation involves non-trivial non-planar integral families which have master
integrals in the top sector. We compute the master integrals by introducing a
second mass scale and solving differential equations with respect to the ratio
of the two scales. We present details of our calculational procedure.
Analytical results for the cusp and collinear anomalous dimensions, and the
finite part of the form factor are presented. We also provide analytic results
for all master integrals expanded up to weight eight.Comment: 16 pages, 2 figure
Three-loop massive form factors: complete light-fermion and large- corrections for vector, axial-vector, scalar and pseudo-scalar currents
We compute the three-loop QCD corrections to the massive quark form factors
with external vector, axial-vector, scalar and pseudo-scalar currents. All
corrections with closed loops of massless fermions are included. The
non-fermionic part is computed in the large- limit, where only planar
Feynman diagrams contribute.Comment: 33 page
The Four-Loop Dressing Phase of N=4 SYM
We compute the dilatation generator in the su(2) sector of planar N=4 super
Yang-Mills theory at four-loops. We use the known world-sheet scattering matrix
to constrain the structure of the generator. The remaining few coefficients can
be computed directly from Feynman diagrams. This allows us to confirm previous
conjectures for the leading contribution to the dressing phase which is
proportional to zeta(3).Comment: 19 pages, v2: referenced adde
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