Higher threshold parameters in ππ scattering

Abstract

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...

Cross sections for inelastic meson-meson scattering via quark-antiquark annihilation

“…Including one-and two-loop corrections in chiral perturbation theory and imposing unitarity, the resonances appear and the experimental data are reproduced. This has been shown in approaches based on the results of chiral perturbation theory, for example, the master formula approach [38], the Padé method [39], the large-N f expansion [40], the N/D method [41], the inverse amplitude method [42], the K-matrix method [43], the current algebra unitarization [44], the Roy equations [45], the coupled-channel Lippmann-Schwinger equations [46], the Bethe-Salpeter approach [47], and the approaches based on effective meson Lagrangians [48][49][50][51][52][53][54][55].…”

Comparison of $\pi K$ scattering in SU(3) chiral perturbation theory and dispersion relations

“…The masses of the pion, kaon, and eta, the latter entering the calculation only through the loop-functions, are set to M = 139.56 MeV, m = 497.67 MeV, and m η = 547.30 MeV, respectively. The decay constants are F π = 92.4 MeV, F K = 1.22F π (we take the well-established analysis for the ratio F K /F π in the present work; new analyses are now available [27], and these will be incorporated at the time the fresh Steiner-Roy equation fits to the data are ready [28]). Furthermore, the renormalization scale µ is set to m ρ = 769.30 MeV.…”

“…Due to the importance of these equation in such an analysis, the Steiner-Roy equations in the S-and P -wave approximation for the S-and P -waves are given here explicitly. (It may be noted that one can proceed to analyze πN scattering [20] in an analogous manner. )…”

Isospin breaking corrections to low-energy π-Kscattering

“…In addition to the experimental data a large-N c arguments were used to settle the marginal relevance of some operators (those entering together with L 4 and L 6 ). The use of π − K data in the T 3/2 channel can disentangle (in principle) the value of L r 4 , due to its product with M 2 K which enhances its sensitivity to the role of m s [28].…”