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 $n \in \mathbb{N}$ one can extract the information useful to determine a new basis of MIs, whose differential equations decouple as $d \to n$ 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

Observation of Events with Isolated Charged Leptons and Large Missing Tra nsverse Momentum and of Events with Multi-Electrons at HERA

Striking events with isolated charged leptons, large missing transverse momentum and large transverse momentum of the hadronic final state (PTX) have been observed at the electron proton collider HERA. In the full HERA-I data sample corresponding to an integrated luminosity of about 130 invpb, the H1 experiment observes 10 events with isolated electrons or muons and with PTX >25 GeV. Only 2.9 pm 0.4 events are expected from Standard Model (SM) processes. Six of these events have PTX >40 GeV, while 1.1 pm 0.2 events are expected. The ZEUS experiment observes good agreement with the SM. However, in a preliminary search ZEUS has found two events with a similar event topology, but tau-leptons instead of electrons or muons in the final state . Only 0.12 pm 0.02 events are expected from SM processes. Moreover, six events with two or more electrons forming an invariant mass bigger than 100 GeV have been observed by the H1 experiment. Three events have two electrons and three events have three electrons, while only 0.25 events are expected in each case. The ZEUS measurement is in agreement with the SM expectation.Comment: talk given at 38th Recontres de Moriond Electroweak Interactions and Unified Theories, Les Arc (France) 200

A hierarchical Bayesian approach to record linkage and population size problems

We propose and illustrate a hierarchical Bayesian approach for matching statistical records observed on different occasions. We show how this model can be profitably adopted both in record linkage problems and in capture--recapture setups, where the size of a finite population is the real object of interest. There are at least two important differences between the proposed model-based approach and the current practice in record linkage. First, the statistical model is built up on the actually observed categorical variables and no reduction (to 0--1 comparisons) of the available information takes place. Second, the hierarchical structure of the model allows a two-way propagation of the uncertainty between the parameter estimation step and the matching procedure so that no plug-in estimates are used and the correct uncertainty is accounted for both in estimating the population size and in performing the record linkage. We illustrate and motivate our proposal through a real data example and simulations.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS447 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Schouten identities for Feynman graph amplitudes; the Master Integrals for the two-loop massive sunrise graph

A new class of identities for Feynman graph amplitudes, dubbed Schouten identities, valid at fixed integer value of the dimension d is proposed. The identities are then used in the case of the two loop sunrise graph with arbitrary masses for recovering the second order differential equation for the scalar amplitude in d=2 dimensions, as well as a chained sets of equations for all the coefficients of the expansions in (d-2). The shift from $d\approx2$ to $d\approx4$ dimensions is then discussed.Comment: 30 pages, 1 figure, minor typos in the text corrected, results unchanged. Version accepted for publication on Nuclear Physics

Flavour transitions of Dirac-Majorana neutrinos

From a phenomenological point of view, we study active-active and active-sterile flavour-changing (and flavour-conserving) oscillations of Dirac-Majorana neutrinos both in vacuum and in matter. The general expressions for the transition probabilities in vacuum are reported. We then investigate some interesting consequences following from particular simple forms of the neutrino mass matrices, and for the envisaged scenarios we discuss in detail neutrino propagation in matter. Special emphasis is given to the problem of occurrence of resonant enhancement of active-active and active-sterile neutrino oscillations in a medium. The peculiar novel features related to the Dirac-Majorana nature of neutrinos are particularly pointed out.Comment: latex 2e, 19 pages, 1 figure; to be published in The European Journal of Physics

An Elliptic Generalization of Multiple Polylogarithms

We introduce a class of functions which constitutes an obvious elliptic generalization of multiple polylogarithms. A subset of these functions appears naturally in the \epsilon-expansion of the imaginary part of the two-loop massive sunrise graph. Building upon the well known properties of multiple polylogarithms, we associate a concept of weight to these functions and show that this weight can be lowered by the action of a suitable differential operator. We then show how properties and relations among these functions can be studied bottom-up starting from lower weights.Comment: 27 pages plus three appendices, v2: references added, typos corrected, accepted for publication on NP

Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph

We consider the calculation of the master integrals of the three-loop massive banana graph. In the case of equal internal masses, the graph is reduced to three master integrals which satisfy an irreducible system of three coupled linear differential equations. The solution of the system requires finding a $3 \times 3$ matrix of homogeneous solutions. We show how the maximal cut can be used to determine all entries of this matrix in terms of products of elliptic integrals of first and second kind of suitable arguments. All independent solutions are found by performing the integration which defines the maximal cut on different contours. Once the homogeneous solution is known, the inhomogeneous solution can be obtained by use of Euler's variation of constants.Comment: 39 pages, 3 figures; Fixed a typo in eq. (6.16

Comparing parametric and semi-parametric approaches for bayesian cost-effectiveness analyses in health economics

We consider the problem of assessing new and existing technologies for their cost-effectiveness in the case where data on both costs and effects are available from a clinical trial, and we address it by means of the cost-effectiveness acceptability curve. The main difficulty in these analyses is that cost data usually exhibit highly skew and heavytailed distributions, so that it can be extremely difficult to produce realistic probabilistic models for the underlying population distribution, and in particular to model accurately the tail of the distribution, which is highly influential in estimating the population mean. Here, in order to integrate the uncertainty about the model into the analysis of cost data and into cost-effectiveness analyses, we consider an approach based on Bayesian model averaging: instead of choosing a single parametric model, we specify a set of plausible models for costs and estimate the mean cost with its posterior expectation, that can be obtained as a weighted mean of the posterior expectations under each model, with weights given by the posterior model probabilities. The results are compared with those obtained with a semi-parametric approach that does not require any assumption about the distribution of costs. 1 IntroductionHealthcare cost data, cost-effectiveness analysis, mixture models, Bayesian model averaging