49,306 research outputs found
Proof of a modular relation between 1-, 2- and 3-loop Feynman diagrams on a torus
The coefficients of the higher-derivative terms in the low energy expansion
of genus-one graviton Type II superstring scattering amplitudes are determined
by integrating sums of non-holomorphic modular functions over the complex
structure modulus of a torus. In the case of the four-graviton amplitude, each
of these modular functions is a multiple sum associated with a Feynman diagram
for a free massless scalar field on the torus. The lines in each diagram join
pairs of vertex insertion points and the number of lines defines its weight
, which corresponds to its order in the low energy expansion. Previous
results concerning the low energy expansion of the genus-one four-graviton
amplitude led to a number of conjectured relations between modular functions of
a given , but different numbers of loops . In this paper we shall
prove the simplest of these conjectured relations, namely the one that arises
at weight and expresses the three-loop modular function in terms of
modular functions with one and two loops. As a byproduct, we prove three
intriguing new holomorphic modular identities.Comment: 38 pages, 9 figures, in version 2: Appendix D added and corrections
made in section
Nonlinear, nonbinary cyclic group codes
New cyclic group codes of length 2(exp m) - 1 over (m - j)-bit symbols are introduced. These codes can be systematically encoded and decoded algebraically. The code rates are very close to Reed-Solomon (RS) codes and are much better than Bose-Chaudhuri-Hocquenghem (BCH) codes (a former alternative). The binary (m - j)-tuples are identified with a subgroup of the binary m-tuples which represents the field GF(2 exp m). Encoding is systematic and involves a two-stage procedure consisting of the usual linear feedback register (using the division or check polynomial) and a small table lookup. For low rates, a second shift-register encoding operation may be invoked. Decoding uses the RS error-correcting procedures for the m-tuple codes for m = 4, 5, and 6
Generic Construction of Efficient Matrix Product Operators
Matrix Product Operators (MPOs) are at the heart of the second-generation
Density Matrix Renormalisation Group (DMRG) algorithm formulated in Matrix
Product State language. We first summarise the widely known facts on MPO
arithmetic and representations of single-site operators. Second, we introduce
three compression methods (Rescaled SVD, Deparallelisation and Delinearisation)
for MPOs and show that it is possible to construct efficient representations of
arbitrary operators using MPO arithmetic and compression. As examples, we
construct powers of a short-ranged spin-chain Hamiltonian, a complicated
Hamiltonian of a two-dimensional system and, as proof of principle, the
long-range four-body Hamiltonian from quantum chemistry.Comment: 13 pages, 10 figure
Non-planar double-box, massive and massless pentabox Feynman integrals in negative dimensional approach
Negative dimensional integration method (NDIM) is a technique which can be
applied, with success, in usual covariant gauge calculations. We consider three
two-loop diagrams: the scalar massless non-planar double-box with six
propagators and the scalar pentabox in two cases, where six virtual particles
have the same mass and in the case where all of them are massless. Our results
are given in terms hypergeometric functions of Mandelstam variables and for
arbitrary exponents of propagators and dimension as well.Comment: Latex, 12 pages, 2 figures, uses axodraw (included
Modular Graph Functions
In earlier work we studied features of non-holomorphic modular functions
associated with Feynman graphs for a conformal scalar field theory on a
two-dimensional torus with zero external momenta at all vertices. Such
functions, which we will refer to as modular graph functions, arise, for
example, in the low energy expansion of genus-one Type II superstring
amplitudes. We here introduce a class of single-valued elliptic multiple
polylogarithms, which are defined as elliptic functions associated with Feynman
graphs with vanishing external momenta at all but two vertices. These functions
depend on a coordinate, , on the elliptic curve and reduce to modular
graph functions when is set equal to . We demonstrate that these
single-valued elliptic multiple polylogarithms are linear combinations of
multiple polylogarithms, and that modular graph functions are sums of
single-valued elliptic multiple polylogarithms evaluated at the identity of the
elliptic curve, in both cases with rational coefficients. This insight suggests
the many interrelations between modular graph functions (a few of which were
established in earlier papers) may be obtained as a consequence of identities
involving multiple polylogarithms, and explains an earlier observation that the
coefficients of the Laurent polynomial at the cusp are given by rational
numbers times single-valued multiple zeta values.Comment: 42 pages, significant clarifications added in section 5, minor typos
corrected, and references added in version
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