Low-energy ππ and πK scatterings revisited in three-flavour resummed chiral perturbation theory

Abstract

Chiral symmetry breaking may exhibit significantly different patterns in two chiral limits: N f = 2 massless flavours (m u = m d = 0, m s physical) and N f = 3 massless flavours (m u = m d = m s = 0). Such a difference may arise due to vacuum fluctuations of ss pairs related to the violation of the Zweig rule in the scalar sector, and could yield a numerical competition between contributions counted as leading and next-to-leading order in the chiral expansions of observables. We recall and extend Resummed Chiral Perturbation Theory (ReχPT), a framework that we introduced previously to deal with such instabilities: it requires a more careful definition of the relevant observables and their one-loop chiral expansions. We analyse the amplitudes for low-energy ππ and πK scatterings within ReχPT, which we match in subthreshold regions with dispersive representations obtained from the solutions of Roy and Roy-Steiner equations. Using a frequentist approach, we constrain the quark mass ratio as well as the quark condensate and the pseudoscalar decay constant in the N f = 3 chiral limit. The results mildly favour significant contributions of vacuum fluctuations suppressing the N f = 3 quark condensate compared to its N f = 2 counterpart.

Chiral extrapolation of the strangeness changing scalar Kπ form factor

“…Indeed, if large values for L r 4 and L r 6 as expected from a large violation of the OZI rule were confirmed in the future then the use of standard ChPT as done here would not really be appropriate. A way of solving the problem could be for example to work within resummed ChPT [72,73]. A study along this line is in progress [74].…”

Topological susceptibility on the lattice and the three-flavour quark condensate

“…In refs. [16][17][18][43][44][45][46][47], we have argued that the pattern of N f = 3 chiral symmetry breaking could be affected significantly by vacuum fluctuations of ss pairs, leading to the suppression of the quark condensate, the enhancement of L 6 (and L 4 ) and finally a numerical competition between LO and NLO contributions in N f = 3 chiral expansions. Such a problem would occur for eq.…”

“…A scenario where such features could occur was discussed in a series of articles [16][17][18][43][44][45][46][47]: significant vacuum fluctuations of ss pairs, hinted at by the violation of the Zweig rule in the scalar sector, could lead to a paramagnetic suppression of LO chiral order parameters (quark condensate and pseudoscalar decay constant). In the N f = 3 chiral expansions of observables such as decay constants, masses, form factors.…”

“…(3.8) shows that in the physical case, the topological susceptibility has no sensitivity on the three-flavour condensate X(3). Indeed, up to 1/r-suppressed corrections (which are small since r ≥ 15 [3,7,46]), eq. (3.7) combines the (LO) three-flavour condensate and the (NLO) low-energy constant L 6 in the same way as F 2 π M 2 π , or equivalently the two-flavour condensate X(2):…”

“…Indeed, in refs. [16][17][18][43][44][45][46][47], we have considered the decay constants and masses as they could be obtained from the NLO expression of A µ A ν where the propagators are explicitly expressed in terms of the physical masses, considering either the correlator at 0 (for F 2 π ) or its residue at the pseudoscalar pole (for F 2 π M 2 π ). Thanks to our diagrammatical analysis of the two-point correlator χ in χPT, we can easily identify the contribution from the propagation of the η meson in eq.…”