A new approach to the epsilon expansion of generalized hypergeometric functions

Assumed that the parameters of a generalized hypergeometric function depend linearly on a small variable $\varepsilon$, the successive derivatives of the function with respect to that small variable are evaluated at $\varepsilon=0$ to obtain the coefficients of the $\varepsilon$-expansion of the function. The procedure, quite naive, benefits from simple explicit expressions of the derivatives, to any order, of the Pochhammer and reciprocal Pochhammer symbols with respect to their argument. The algorithm may be used algebraically, irrespective of the values of the parameters. It reproduces the exact results obtained by other authors in cases of especially simple parameters. Implemented numerically, the procedure improves considerably the numerical expansions given by other methods.Comment: Some formulae adde

Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin-Barnes Representation

Using a method mixing Mellin-Barnes representation and Borel resummation we show how to obtain hyperasymptotic expansions from the (divergent) formal power series which follow from the perturbative evaluation of arbitrary "$N$-point" functions for the simple case of zero-dimensional $\phi^4$ field theory. This hyperasymptotic improvement appears from an iterative procedure, based on inverse factorial expansions, and gives birth to interwoven non-perturbative partial sums whose coefficients are related to the perturbative ones by an interesting resurgence phenomenon. It is a non-perturbative improvement in the sense that, for some optimal truncations of the partial sums, the remainder at a given hyperasymptotic level is exponentially suppressed compared to the remainder at the preceding hyperasymptotic level. The Mellin-Barnes representation allows our results to be automatically valid for a wide range of the phase of the complex coupling constant, including Stokes lines. A numerical analysis is performed to emphasize the improved accuracy that this method allows to reach compared to the usual perturbative approach, and the importance of hyperasymptotic optimal truncation schemes.Comment: v2: one reference added, one paragraph added in the conclusions, small changes in the text, corrected typos; v3: published versio

Asymptotic expansions of Feynman diagrams and the Mellin-Barnes representation

In this talk, we describe part of our recent work \cite{FGdeR05} (see also \cite{F05,G05}) that gives new results in the context of asymptotic expansions of Feynman diagrams using the Mellin-Barnes representation.Comment: Talk given at the $12^{th}$ High-Energy Physics International Conference on Quantum Chromodynamics, 4-8 July (2005), Montpellier (France

The Froissart--Martin Bound for ππ\pi\pi Scattering in QCD

The Froissart--Martin bound for total $\pi\pi$ scattering cross sections is reconsidered in the light of QCD properties such as spontaneous chiral symmetry breaking and the counting rules for a large number of colours \Nc.Comment: Mispints corrected. Version published in the Phys. Rev.

Assuming Regge trajectories in holographic QCD: from OPE to Chiral Perturbation Theory

The soft wall model in holographic QCD has Regge trajectories but wrong operator product expansion (OPE) for the two-point vectorial QCD Green function. We modify the dilaton potential to comply OPE. We study also the axial two-point function using the same modified dilaton field and an additional scalar field to address chiral symmetry breaking. OPE is recovered adding a boundary term and low energy chiral parameters, $F_\pi$ and $L_{10}$, are well described analytically by the model in terms of Regge spacing and QCD condensates. The model nicely supports and extends previous theoretical analyses advocating Digamma function to study QCD two-point functions in different momentum regions.Comment: Major changes to improve the presentation of the paper but main results unchanged. Added appendix on Regge progressio

Standard Model and New Physics contributions to KLK_L and KSK_S into four leptons

We study the $K_L$ and $K_S$ decays into four leptons ($e\bar{e}e\bar{e}, \, \mu\bar{\mu}\mu\bar{\mu},\, e\bar{e}\mu\bar{\mu}$) where we use a form factor motivated by vector meson dominance, and show the dependence of the branching ratios and spectra from the slopes. A precise determination of short distance contribution to $K_L\to \mu\mu$ is affected by our ignorance on the sign of the amplitude $\mathcal{A}(K_L\to \gamma\gamma)$ but we show a possibility to measure the sign of this amplitude by studying $K_L$ and $K_S$ decays in four leptons. We also investigate the effect of New Physics contributions for these decays.Comment: Improvements of the text and references adde

On the amplitudes for the CP-conserving K±(KS)→π±(π0)ℓ+ℓ−K^\pm(K_S)\to\pi^\pm(\pi^0)\ell^+\ell^- rare decay modes

The amplitudes for the rare decay modes $K^\pm\to\pi^\pm\ell^+\ell^-$ and $K_S\to\pi^0\ell^+\ell^-$ are studied with the aim of obtaining predictions for them, such as to enable the possibility to search for violations of lepton-flavour universality in the kaon sector. The issue is first addressed from the perspective of the low-energy expansion, and a two-loop representation of the corresponding form factors is constructed, leaving as unknown quantities their values and slopes at vanishing momentum transfer. In a second step a phenomenological determination of the latter is proposed. It consists of the contribution of the resonant two-pion state in the $P$ wave, and of the leading short-distance contribution determined by the operator-product expansion. The interpolation between the two energy regimes is described by an infinite tower of zero-width resonances matching the QCD short-distance behaviour. Finally, perspectives for future improvements in the theoretical understanding of these amplitudes are discussed.Comment: 49 pages, 11 figures, matches the published versio

Muon Anomaly from Lepton Vacuum Polarization and The Mellin--Barnes Representation

We evaluate, analytically, a specific class of eighth--order and tenth--order QED contributions to the anomalous magnetic moment of the muon. They are generated by Feynman diagrams involving lowest order vacuum polarization insertions of leptons $l=e,\mu$, and $\tau$. The results are given in the form of analytic expansions in terms of the mass ratios $m_e/m_\mu$ and $m_\mu/m_\tau$. We compute as many terms as required by the error induced by the present experimental uncertainty on the lepton masses. We show how the Mellin--Barnes integral representation of Feynman parametric integrals allows for an easy analytic evaluation of as many terms as wanted in these expansions and how its underlying algebraic structure generalizes the standard renormalization group properties. We also discuss the generalization of this technique to the case where two independent mass ratios appear. Comparison with previous numerical and analytic evaluations made in the literature, whenever pertinent, are also made.Comment: v2, minor changes in the introduction, typos corrected, two references added; to appear in Phys. Rev.