The Off-Shell Nucleon-Nucleon Amplitude: Why it is Unmeasurable in Nucleon-Nucleon Bremsstrahlung

Nucleon-nucleon bremsstrahlung has long been considered a way of getting information about the off-shell nucleon-nucleon amplitude which would allow one to distinguish among nucleon-nucleon potentials based on their off-shell properties. There have been many calculations and many experiments devoted to this aim. We show here, in contrast to this standard view, that such off-shell amplitudes are not measurable as a matter of principle. This follows formally from the invariance of the S-matrix under transformations of the fields. This result is discussed here and illustrated via two simple models, one applying to spin zero, and one to spin one half, processes. The latter model is very closely related to phenomenological models which have been used to study off-shell effects at electromagnetic vertices.Comment: 6 pages, Latex, uses FBSsuppl.cls - Invited plenary talk at the Asia Pacific Conference on Few Body Problems in Physics, Noda/Kashiwa, Japan, August, 1999 - To be published in Few Body Systems Supp

Baryon chiral perturbation theory with virtual photons and leptons

We construct the general pion-nucleon SU(2) Lagrangian including both virtual photons and leptons for relativistic baryon chiral perturbation theory up to fourth order. We include the light leptons as explicit dynamical degrees of freedom by introducing new building blocks which represent these leptons.Comment: 11 page

Proportionate vs disproportionate distribution of wealth of two individuals in a tempered Paretian ensemble

We study the distribution P(\omega) of the random variable \omega = x_1/(x_1 + x_2), where x_1 and x_2 are the wealths of two individuals selected at random from the same tempered Paretian ensemble characterized by the distribution \Psi(x) \sim \phi(x)/x^{1 + \alpha}, where \alpha > 0 is the Pareto index and $\phi(x)$ is the cut-off function. We consider two forms of \phi(x): a bounded function \phi(x) = 1 for L \leq x \leq H, and zero otherwise, and a smooth exponential function \phi(x) = \exp(-L/x - x/H). In both cases \Psi(x) has moments of arbitrary order. We show that, for \alpha > 1, P(\omega) always has a unimodal form and is peaked at \omega = 1/2, so that most probably x_1 \approx x_2. For 0 < \alpha < 1 we observe a more complicated behavior which depends on the value of \delta = L/H. In particular, for \delta < \delta_c - a certain threshold value - P(\omega) has a three-modal (for a bounded \phi(x)) and a bimodal M-shape (for an exponential \phi(x)) form which signifies that in such ensembles the wealths x_1 and x_2 are disproportionately different.Comment: 9 pages, 8 figures, to appear in Physica