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We study the third order pion-nucleon scattering amplitude obtained from heavy baryon chiral perturbation theory inside the Mandelstam triangle. We reconstruct the pion-nucleon amplitude in the unphysical region by use of dispersion relations and determine the pertinent low-energy constants by a fit to this amplitude. A detailed comparison with values obtained from phase shift analysis is given. Our analysis leads to a pion-nucleon σ-term of σ(0) = 40 MeV based on the Karlsruhe partial wave analysis. We have repeated the same procedure using the latest solution of the VPI group and find a much larger value for σ(0).

Experimental information on low energy πK scattering would shed light on the poorly known OZI suppressed sector of ChPT. I describe recent work aimed at generating such information based on available experimental data by setting up and then solving with appropriate boundary conditions a non linear system of equations of the Roy and Steiner type. First results of this analysis are presented.

We establish the framework for the comparison of πK scattering amplitudes from SU (3) chiral perturbation theory with suitable dispersive representations which result from the combination of certain fixed-t dispersion relations with dispersion relations on hyperbolic curves. This allows for predictions for some combinations of low energy constants appearing in higher order calculations of chiral perturbation theory. Using a simple parametrization for the lowest partial waves, first estimates for some combinations are presented.

A recently proposed set of sum rules, based on the pion-Kaon scattering amplitudes and their crossing-symmetric conjugates are analysed in detail. A key role is played by the $l=0$ $\pi\pi\to K\overline K$ amplitude which requires an extrapolation to be performed. It is shown how this is tightly constrained from analyticity, chiral counting and the available experimental data, and its stability is tested. A re-evaluation of the $O(p^4)$ chiral couplings $L_1$, $L_2$, $L_3$ is obtained, as well as a new evaluation of the large $N_c$ suppressed coupling $L_4$.Comment: 28 pages, 7 figures [v1: small corrections and improvements

A family of threshold parameters which probe the stability of chiral predictions is considered. The relevant criteria for the choice of threshold parameters are discussed. Sum rules for these quantities are derived from dispersion relations and evaluated from effective range formulae. Good agreement with two-loop chiral estimates for many of these quantities is found and interesting discrepancies are discussed.Keywords: Sum rules, ππ scattering, chiral perturbation theory PACS: 12.39.Fe, 13.75.Lb, 11.55.Fv, 25.80.Dj 1. Dispersion relations for ππ scattering amplitudes with two subtractions have been rigorously established in axiomatic field theory [1]. It is convenient to consider dispersion relations for s−channel amplitudes of definite iso-spin, T I s (s, t, u), I = 0, 1, 2, where s, t, u are the Mandelstam variables.Furthermore, each of the amplitudes may be written down in terms of a unique function as T 0 s (s, t, u) = 3A(s, t, u) + A(t, u, s) + A(u, s, t), T 1 s (s, t, u) = A(t, u, s) − A(u, s, t), T 2 s (s, t, u) = A(t, u, s) + A(u, s, t). Unitarity, analyticity and crossing symmetry have been used extensively to study this fundamental process of elementary particle physics. Introducing a partial wave expansion for these amplitudes via T I s (s, t, u) = 32πΣ(2l+1)f I l (s)P l ((t−u)/(s−4)), elastic unitarity implies above threshold and below the four-pion threshold that the partial wave amplitudes may be described in terms of the phase shifts δ I l (s) by f I l (s) = s/(s − 4) exp(iδ I l (s)) sin δ I l (s), where we have set the pion mass (m π = 139.6MeV) to unity. Note the threshold expansion this amplitude has now been computed to one-loop [7] and even two-loop order [9,10].) Furthermore this implies that the only non-vanishing threshold parameters are a 0 0 = 7/(32πF 2 π ), a 2 0 = −1/(16πF 2 π ),It is important to note the well known result that the set of functions:, where α and β are arbitrary real constants, verifies dispersion relations with two subtractions and vanishing absorptive parts. The Weinberg amplitude is a special case of this general linear amplitude, with α = β = 1. It may be noted that a generalized version of chiral perturbation theory that is motivated by considerations of a small quark condensate in the QCD vacuum allows α to vary over a range between unity and as much as three, and reorganizes the chiral power counting [11]. In this discussion we confine ourselves to the more predictive standard chiral perturbation theory with α = 1 for much of our discussion.Dispersion relations with two subtractions have been used to write down sum rules for (combinations) of some of these threshold parameters, for example the Wanders sum rules which maybe written down as:where σ I (ν) ≡ (4π/ ν(ν + 1)) · (2l + 1)Im f I l (ν) are the cross-sections. The Weinberg predictions for the quantities involved in these sum rules obey these relations identically with all the cross-sections set to zero, since the original amplitude obeys dispersion relations with vanishing absorptive parts. Furt...

A Roy equation analysis of the available ππ phase shift data is performed with the I = 0 S-wave scattering length a 0 0 in the range predicted by the one-loop standard chiral perturbation theory. A suitable dispersive framework is developed to extract the chiral coupling constantsl 1 ,l 2 and yieldsl 1 = −1.70±0.15 andl 2 ≈ 5.0. We remark on the implications of this determination to (combinations of) threshold parameter predictions of the three lowest partial waves.

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