28 research outputs found
Constant terms in threshold resummation and the quark form factor
We verify to order alpha_s^4 two previously conjectured relations, valid in
four dimensions, between constant terms in threshold resummation (for Deep
Inelastic Scattering and the Drell-Yan process) and the second logarithmic
derivative of the massless quark form factor. The same relations are checked to
all orders in the large beta_0 limit; as a byproduct a dispersive
representation of the form factor is obtained. These relations allow to compute
in a symmetrical way the three-loop resummation coefficients B_3 and D_3 in
terms of the three-loop contributions to the virtual diagonal splitting
function and to the quark form factor, confirming results obtained in the
literature.Comment: 39 pages, no figure; version 2: same content, but improved
presentation, with a new section devoted to the variety of resummation
procedures; version 3: journal version, where a remark about the all orders
validity of the conjecture in the DIS case is reporte
Target annihilation by diffusing particles in inhomogeneous geometries
The survival probability of immobile targets, annihilated by a population of
random walkers on inhomogeneous discrete structures, such as disordered solids,
glasses, fractals, polymer networks and gels, is analytically investigated. It
is shown that, while it cannot in general be related to the number of distinct
visited points, as in the case of homogeneous lattices, in the case of bounded
coordination numbers its asymptotic behaviour at large times can still be
expressed in terms of the spectral dimension , and its exact
analytical expression is given. The results show that the asymptotic survival
probability is site independent on recurrent structures (),
while on transient structures () it can strongly depend on the
target position, and such a dependence is explicitly calculated.Comment: To appear in Physical Review E - Rapid Communication
The projective translation equation and unramified 2-dimensional flows with rational vector fields
Let X=(x,y). Previously we have found all rational solutions of the
2-dimensional projective translation equation, or PrTE,
(1-z)f(X)=f(f(Xz)(1-z)/z); here f(X)=(u(x,y),v(x,y)) is a pair of two (real or
complex) functions. Solutions of this functional equation are called projective
flows. A vector field of a rational flow is a pair of 2-homogenic rational
functions. On the other hand, only special pairs of 2-homogenic rational
functions give rise to rational flows. In this paper we are interested in all
non-singular (satisfying the boundary condition) and unramified (without
branching points, i.e. single-valued functions in C^2\{union of curves})
projective flows whose vector field is still rational. We prove that, up to
conjugation with 1-homogenic birational plane transformation, these are of 6
types: 1) the identity flow; 2) one flow for each non-negative integer N -
these flows are rational of level N; 3) the level 1 exponential flow, which is
also conjugate to the level 1 tangent flow; 4) the level 3 flow expressable in
terms of Dixonian (equianharmonic) elliptic functions; 5) the level 4 flow
expressable in terms of lemniscatic elliptic functions; 6) the level 6 flow
expressable in terms of Dixonian elliptic functions again. This reveals another
aspect of the PrTE: in the latter four cases this equation is equivalent and
provides a uniform framework to addition formulas for exponential, tangent, or
special elliptic functions (also addition formulas for polynomials and the
logarithm, though the latter appears only in branched flows). Moreover, the
PrTE turns out to have a connection with Polya-Eggenberger urn models. Another
purpose of this study is expository, and we provide the list of open problems
and directions in the theory of PrTE; for example, we define the notion of
quasi-rational projective flows which includes curves of arbitrary genus.Comment: 34 pages, 2 figure
Large Nc QCD and Harmonic Sums
In the Large-Nc limit of QCD, two--point functions of local operators become
Harmonic Sums. I review some properties which follow from this fact and which
are relevant for phenomenological applications. This has led us to consider a
class of Analytic Number Theory Functions as toy models of Large-Nc QCD which I
also discuss.Comment: Based on my talk at "Raymond Stora's 80th Birthday Party", LAPP, July
11th 201
Harmonic Sums and Mellin Transforms up to two-loop Order
A systematic study is performed on the finite harmonic sums up to level four.
These sums form the general basis for the Mellin transforms of all individual
functions of the momentum fraction emerging in the quantities of
massless QED and QCD up to two--loop order, as the unpolarized and polarized
splitting functions, coefficient functions, and hard scattering cross sections
for space and time-like momentum transfer. The finite harmonic sums are
calculated explicitly in the linear representation. Algebraic relations
connecting these sums are derived to obtain representations based on a reduced
set of basic functions. The Mellin transforms of all the corresponding Nielsen
functions are calculated.Comment: 44 pages Latex, contract number adde