Inverse amplitude method and chiral zeros

Abstract

The inverse amplitude method has previously been successfully applied to ππ scattering in order to extend the range of applicability of chiral perturbation theory. However, in order to take the chiral zeros into account systematically, the previous derivation of the inverse amplitude method has to be modified. It is shown how this can be done to both one and two loops in the chiral expansion. In the physical region, the inclusion of these chiral zeros has very little significance, whereas they become essential in the sub-threshold region. Finally, the crossing properties of the inverse amplitude method are investigated in the sub-threshold region. PACS number(s): 13.75. Lb, 11.30.Rd, 11.55.Fv, 11.80.Et Chiral perturbation theory (ChPT) [1,2] has become a very successful methodology for low-energy hadron physics. Within this methodology one obtains a systematic expansion in powers of external momenta and light quark masses, or equivalently in the number of loops. However, unitarity is only satisfied perturbatively in the chiral expansion, which gives a severe restriction on the applicability of ChPT. Therefore, in order to extend the validity of ChPT, several methods have been proposed to combine exact unitarity and the chiral expansion. One such method is the inverse amplitude method (IAM), which has been analyzed to both one-loop [3] and two-loop [4] order in the chiral expansion. This has shown that the IAM is indeed a successful and systematic method to extend the range of applicability of ChPT.However, the IAM has to be somewhat modified when there are zeros in the amplitudes [5]. This is the case for ππ scattering where chiral dynamics demand that the S waves have zeros below threshold, whereas P and higher partial waves have fixed kinematical zeros at threshold. The previous derivation of the IAM [3,4] has been based upon the assumption that these zeros occur at the same energy as for the lowest order ChPT result. This assumption is in fact true for the P and higher partial waves, whereas the same is not necessarily the case for the S waves. Thus, the derivation of the IAM should be somewhat modified in the case of the S waves in order to systematically account for the occurrence of the chiral zeros. In this brief report it is shown how this is possible to both one and two-loop order in the chiral expansion. Since the main interest will be in the sub-threshold behavior, the following derivation of this generalized IAM will be restricted to the elastic approximation.The elastic unitarity relation for the ππ partial waves t is given bywhere s is the square of the c.m. energy, I and l are isospin and angular momentum indices, and σ is the phase-space factor. In ChPT the elastic ππ partial waves are now known to two loops in the chiral expansion [6,7]. These partial waves are given by the following expansionwhere t (0) is the lowest order result, t (1) is the one-loop correction, and t (2) the additional two-loop correction. They satisfy the elastic unitarity relation (1) perturbativelySince perturbative un...

Inverse amplitude method in ππ scattering in chiral perturbation theory to two loops

“…Moreover, if the crossing violation definition of Refs. [30,34] is evaluated we see from Table 5 that these sum rules are better fulfilled. The effect of uncertainties on the violations has been overlooked in previous works [30,31].…”

“…[37]). In those methods, unitarity is restored in the different partial wave amplitudes, while crossing is violated [30,34,40]. In the complex energy plane this corresponds to exactly take into account the unitarity right hand elastic cut ( s > 4m 2 π ) but to approximate the left hand cut ( s < 0 ).…”

“…[34] a generalized IAM to two loops has been proposed. If s 0 is the chiral Adler zero to lowest order t IJ (s 0 ) = 0 then the following expression for the inverse amplitude is suggested [34] …”

“…[40] and Refs. [30,34] respectively since they give an idea of how large are these violations in percentage terms.…”

“…(28) the generalized IAM of Ref. [34] reduces to the standard IAM method of Eq. (20) and also unitarity is exactly fulfilled.…”