69,395 research outputs found
Feynman Diagrams and Differential Equations
We review in a pedagogical way the method of differential equations for the
evaluation of D-dimensionally regulated Feynman integrals. After dealing with
the general features of the technique, we discuss its application in the
context of one- and two-loop corrections to the photon propagator in QED, by
computing the Vacuum Polarization tensor exactly in D. Finally, we treat two
cases of less trivial differential equations, respectively associated to a
two-loop three-point, and a four-loop two-point integral. These two examples
are the playgrounds for showing more technical aspects about: Laurent expansion
of the differential equations in D (around D=4); the choice of the boundary
conditions; and the link among differential and difference equations for
Feynman integrals.Comment: invited review article from Int. J. Mod. Phys.
Boosting the Maxwell double layer potential using a right spin factor
We construct new spin singular integral equations for solving scattering
problems for Maxwell's equations, both against perfect conductors and in media
with piecewise constant permittivity, permeability and conductivity, improving
and extending earlier formulations by the author. These differ in a fundamental
way from classical integral equations, which use double layer potential
operators, and have the advantage of having a better condition number, in
particular in Fredholm sense and on Lipschitz regular interfaces, and do not
suffer from spurious resonances. The construction of the integral equations
builds on the observation that the double layer potential factorises into a
boundary value problem and an ansatz. We modify the ansatz, inspired by a
non-selfadjoint local elliptic boundary condition for Dirac equations
Differential Equations for Two-Loop Four-Point Functions
At variance with fully inclusive quantities, which have been computed already
at the two- or three-loop level, most exclusive observables are still known
only at one-loop, as further progress was hampered so far by the greater
computational problems encountered in the study of multi-leg amplitudes beyond
one loop. We show in this paper how the use of tools already employed in
inclusive calculations can be suitably extended to the computation of loop
integrals appearing in the virtual corrections to exclusive observables, namely
two-loop four-point functions with massless propagators and up to one off-shell
leg. We find that multi-leg integrals, in addition to integration-by-parts
identities, obey also identities resulting from Lorentz-invariance. The
combined set of these identities can be used to reduce the large number of
integrals appearing in an actual calculation to a small number of master
integrals. We then write down explicitly the differential equations in the
external invariants fulfilled by these master integrals, and point out that the
equations can be used as an efficient method of evaluating the master integrals
themselves. We outline strategies for the solution of the differential
equations, and demonstrate the application of the method on several examples.Comment: 26 pages, LaTeX; some explanatory comments added; several typos
correcte
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 one can
extract the information useful to determine a new basis of MIs, whose
differential equations decouple as 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
High-precision calculation of multi-loop Feynman integrals by difference equations
We describe a new method of calculation of generic multi-loop master
integrals based on the numerical solution of systems of difference equations in
one variable. We show algorithms for the construction of the systems using
integration-by-parts identities and methods of solutions by means of expansions
in factorial series and Laplace's transformation. We also describe new
algorithms for the identification of master integrals and the reduction of
generic Feynman integrals to master integrals, and procedures for generating
and solving systems of differential equations in masses and momenta for master
integrals. We apply our method to the calculation of the master integrals of
massive vacuum and self-energy diagrams up to three loops and of massive vertex
and box diagrams up to two loops. Implementation in a computer program of our
approach is described. Important features of the implementation are: the
ability to deal with hundreds of master integrals and the ability to obtain
very high precision results expanded at will in the number of dimensions.Comment: 55 pages, 5 figures, LaTe
Dynamical gravities
It is offered that modified gravities can be considered as
nonperturbative quantum effects arising from Einstein gravity. It is assumed
that nonperturbative quantum effects gives rise to the fact that the connection
becomes incompatible with the metric, the metric factors and the square of the
connection in Einstein - Hilbert Lagrangian have nonperturbative additions. In
the simplest approximation both additions can be considered as functions of one
scalar field. The scalar field can be excluded from the Lagrangian obtaining
gravity. The essence of quantum correction to the affine connection as a
torsion is discussed.Comment: discussion on quantum corrections is adde
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