Lorentz Invariant Baryon CHPT

Using the example of the elastic $\pi N$-amplitude, we discuss the low energy expansion of QCD amplitudes in the sector with baryon number one. We show that the chiral expansion of these amplitudes breaks down in certain regions of phase space and present a framework which leads to a coherent description throughout the low energy region, while keeping Lorentz and chiral invariance manifest at every stage of the calculation. We explain how to construct a representation of the pion nucleon scattering amplitude in terms of functions of a single variable, which is valid to $O(q^4)$ and properly accounts for the $\pi\pi$- and $\pi N$-cuts required by unitarity.Comment: Latex, 12 pages. Plenary talk given at "Chiral Dynamics 2000: Theory and Experiment", Newport News, USA, 17-22 July 200

Low energy analysis of pi N --> pi N

We derive a representation for the pion nucleon scattering amplitude that is valid to the fourth order of the chiral expansion. To obtain the correct analytic structure of the singularities in the low energy region, we have performed the calculation in a relativistic framework (infrared regularization). The result can be written in terms of functions of a single variable. We study the corresponding dispersion relations and discuss the problems encountered in the straightforward nonrelativistic expansion of the infrared singularities. As an application, we evaluate the corrections to the Goldberger-Treiman relation and to the low energy theorem that relates the value of the amplitude at the Cheng-Dashen point to the \sigma-term. While chiral symmetry does govern the behaviour of the amplitude in the vicinity of this point, the representation for the scattering amplitude is not accurate enough to use it for an extrapolation of the experimental data to the subthreshold region. We propose to perform this extrapolation on the basis of a set of integral equations that interrelate the lowest partial waves and are analogous to the Roy equations for \pi\pi scattering.Comment: 97 pages (LaTeX), 16 figures. Two references added, correction in table one. Published versio

Enhanced non-perturbative effects through the collinear anomaly

We show that non-perturbative effects are logarithmically enhanced for transverse-momentum-dependent observables such as q_T-spectra of electroweak bosons in hadronic collisions and jet broadening at e^+e^- colliders. This enhancement arises from the collinear anomaly, a mechanism characteristic for transverse observables, which induces logarithmic dependence on the hard scale in the product of the soft and collinear matrix elements. Our analysis is based on an operator product expansion and provides, for the first time, a systematic, model-independent way to study non-perturbative effects for this class of observables. For the case of jet broadening, we relate the leading correction to the non-perturbative shift of the thrust distribution.Comment: 5 pages, 2 figures. v2: Minor changes, references added. Journal versio

Mathematic & mathematics education: searching for common ground, edited by M. Fried and T. Dreyfus, New York, Springer, 2014, 402 pp., £90, ISBN 978-94-007-7472-8

This is an Accepted Manuscript of an article published by Taylor & Francis in Research in Mathematics Education on 22nd Aug 2014, available online: http://dx.doi.org/10.1080/14794802.2014.93735

Isospin violation in the vector form factors of the nucleon

A quantitative understanding of isospin violation is an increasingly important ingredient for the extraction of the nucleon's strange vector form factors from experimental data. We calculate the isospin violating electric and magnetic form factors in chiral perturbation theory to leading and next-to-leading order respectively, and we extract the low-energy constants from resonance saturation. Uncertainties are dominated largely by limitations in the current knowledge of some vector meson couplings. The resulting bounds on isospin violation are sufficiently precise to be of value to on-going experimental studies of the strange form factors.Comment: 13 pages, 8 figures, uses RevTe

On the Structure of Infrared Singularities of Gauge-Theory Amplitudes

A closed formula is obtained for the infrared singularities of dimensionally regularized, massless gauge-theory scattering amplitudes with an arbitrary number of legs and loops. It follows from an all-order conjecture for the anomalous-dimension matrix of n-jet operators in soft-collinear effective theory. We show that the form of this anomalous dimension is severely constrained by soft-collinear factorization, non-abelian exponentiation, and the behavior of amplitudes in collinear limits. Using a diagrammatic analysis, we demonstrate that these constraints imply that to three-loop order the anomalous dimension involves only two-parton correlations, with the possible exception of a single color structure multiplying a function of conformal cross ratios depending on the momenta of four external partons, which would have to vanish in all two-particle collinear limits. We argue that such a function does not appear at three-loop order, and that the same is true in higher orders. Our formula predicts Casimir scaling of the cusp anomalous dimension to all orders in perturbation theory, and we explicitly check that the constraints exclude the appearance of higher Casimir invariants at four loops. Using known results for the quark and gluon form factors, we derive the three-loop coefficients of the 1/epsilon^n pole terms (with n=1,...,6) for an arbitrary n-parton scattering amplitude in massless QCD. This generalizes Catani's two-loop formula proposed in 1998.Comment: 46 pages, 9 figures; v2: improved treatment of collinear limits, references added; v3: improved discussion of non-abelian exponentiation, references updated; v4: typo in eq. (17) fixed, references updated; v5: additional term in (17

Sudakov Resummation for Subleading SCET Currents and Heavy-to-Light Form Factors

The hard-scattering contributions to heavy-to-light form factors at large recoil are studied systematically in soft-collinear effective theory (SCET). Large logarithms arising from multiple energy scales are resummed by matching QCD onto SCET in two stages via an intermediate effective theory. Anomalous dimensions in the intermediate theory are computed, and their form is shown to be constrained by conformal symmetry. Renormalization-group evolution equations are solved to give a complete leading-order analysis of the hard-scattering contributions, in which all single and double logarithms are resummed. In two cases, spin-symmetry relations for the soft-overlap contributions to form factors are shown not to be broken at any order in perturbation theory by hard-scattering corrections. One-loop matching calculations in the two effective theories are performed in sample cases, for which the relative importance of renormalization-group evolution and matching corrections is investigated. The asymptotic behavior of Sudakov logarithms appearing in the coefficient functions of the soft-overlap and hard-scattering contributions to form factors is analyzed.Comment: 50 pages, 10 figures; minor corrections, version to appear in JHE

Information on the structure of the a1 from tau decay

The decay $\tau\to \pi\pi\pi\nu$ is analysed using different methods to account for the resonance structure, which is usually ascribed to the a1. One scenario is based on the recently developed techniques to generate axial-vector resonances dynamically, whereas in a second calculation the a1 is introduced as an explicit resonance. We investigate the influence of different assumptions on the result. In the molecule scenario the spectral function is described surprisingly well by adjusting only one free parameter. This result can be systematically improved by adding higher order corrections to the iterated Weinberg-Tomozawa interaction. Treating the a1 as an explicit resonance on the other hand leads to peculiar properties

Direct photon production with effective field theory

The production of hard photons in hadronic collisions is studied using Soft-Collinear Effective Theory (SCET). This is the first application of SCET to a physical, observable cross section involving energetic partons in more than two directions. A factorization formula is derived which involves a non-trivial interplay of the angular dependence in the hard and soft functions, both quark and gluon jet functions, and multiple partonic channels. The relevant hard, jet and soft functions are computed to one loop and their anomalous dimensions are determined to three loops. The final resummed inclusive direct photon distribution is valid to next-to-next-to-leading logarithmic order (NNLL), one order beyond previous work. The result is improved by including non-logarithmic terms and photon isolation cuts through matching, and compared to Tevatron data and to fixed order results at the Tevatron and the LHC. The resummed cross section has a significantly smaller theoretical uncertainty than the next-to-leading fixed-order result, particularly at high transverse momentum.Comment: 42 pages, 9 figures; v2: references added, minor changes; v3: typos; v4: typos, corrections in (16), (47), (72