305 research outputs found
Elliptic integral evaluations of Bessel moments
We record what is known about the closed forms for various Bessel function
moments arising in quantum field theory, condensed matter theory and other
parts of mathematical physics. More generally, we develop formulae for
integrals of products of six or fewer Bessel functions. In consequence, we are
able to discover and prove closed forms for with integers and , obtaining new results
for the even moments and . We also derive new closed forms
for the odd moments with and for with , relating the latter to Green functions on
hexagonal, diamond and cubic lattices. We conjecture the values of
, make substantial progress on the evaluation of ,
and and report more limited progress regarding
, and . In the process, we obtain 8
conjectural evaluations, each of which has been checked to 1200 decimal places.
One of these lies deep in 4- dimensional quantum field theory and two are
probably provable by delicate combinatorics. There remains a hard core of five
conjectures whose proofs would be most instructive, to mathematicians and
physicists alike.Comment: 51 pages, 1 Postscript figure, uses amsmath.sty, added reference
Random Aharonov-Bohm vortices and some exact families of integrals: Part III
As a sequel to [1] and [2], I present some recent progress on Bessel
integrals \int_0^{\infty}{\rmd u}\; uK_0(u)^{n}, \int_0^{\infty}{\rmd u}\;
u^{3}K_0(u)^{n}, ... where the power of the integration variable is odd and
where , the Bessel weight, is a positive integer. Some of these integrals
for weights n=3 and n=4 are known to be intimately related to the zeta numbers
zeta(2) and zeta(3). Starting from a Feynman diagram inspired representation in
terms of n dimensional multiple integrals on an infinite domain, one shows how
to partially integrate to n-2 dimensional multiple integrals on a finite
domain. In this process the Bessel integrals are shown to be periods.
Interestingly enough, these "reduced" multiple integrals can be considered in
parallel with some simple integral representations of zeta numbers. One also
generalizes the construction of [2] on a particular sum of double nested Bessel
integrals to a whole family of double nested integrals. Finally a strong PSLQ
numerical evidence is shown to support a surprisingly simple expression of
zeta(5) as a linear combination with rational coefficients of Bessel integrals
of weight n= 8.Comment: 13 pages. arXiv admin note: substantial text overlap with
arXiv:1209.103
Application of the lattice Green's function for calculating the resistance of an infinite networks of resistors
We calculate the resistance between two arbitrary grid points of several
infinite lattice structures of resistors by using lattice Green's functions.
The resistance for dimensional hypercubic, rectangular, triangular and
honeycomb lattices of resistors is discussed in detail. We give recurrence
formulas for the resistance between arbitrary lattice points of the square
lattice. For large separation between nodes we calculate the asymptotic form of
the resistance for a square lattice and the finite limiting value of the
resistance for a simple cubic lattice. We point out the relation between the
resistance of the lattice and the van Hove singularity of the tight-binding
Hamiltonian. Our Green's function method can be applied in a straightforward
manner to other types of lattice structures and can be useful didactically for
introducing many concepts used in condensed matter physics
A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
We define the rigidity of a Feynman integral to be the smallest dimension
over which it is non-polylogarithmic. We argue that massless Feynman integrals
in four dimensions have a rigidity bounded by 2(L-1) at L loops, and we show
that this bound may be saturated for integrals that we call marginal: those
with (L+1)D/2 propagators in (even) D dimensions. We show that marginal Feynman
integrals in D dimensions generically involve Calabi-Yau geometries, and we
give examples of finite four-dimensional Feynman integrals in massless
theory that saturate our predicted bound in rigidity at all loop orders.Comment: 5+2 pages, 11 figures, infinite zoo of Calabi-Yau manifolds. v2
reflects minor changes made for publication. This version is authoritativ
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