3,399 research outputs found
Renormalization, isogenies and rational symmetries of differential equations
We give an example of infinite order rational transformation that leaves a
linear differential equation covariant. This example can be seen as a
non-trivial but still simple illustration of an exact representation of the
renormalization group.Comment: 36 page
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
Quantum discord for general two--qubit states: Analytical progress
We present a reliable algorithm to evaluate quantum discord for general
two--qubit states, amending and extending an approach recently put forward for
the subclass of X--states. A closed expression for the discord of arbitrary
states of two qubits cannot be obtained, as the optimization problem for the
conditional entropy requires the solution to a pair of transcendental equations
in the state parameters. We apply our algorithm to run a numerical comparison
between quantum discord and an alternative, computable measure of non-classical
correlations, namely the geometric discord. We identify the extremally
non-classically correlated two--qubit states according to the (normalized)
geometric discord, at fixed value of the conventional quantum discord. The
latter cannot exceed the square root of the former for systems of two qubits.Comment: 8 pages, 2 figure
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