Consistency of a low energy CHPT theorem with unitarity and perturbative QCD

Chiral perturbation theory (CHPT) provides a systematic low energy expansion of the QCD Green functions in the sector of light mesons. Calculations up to two loops were done recently for the scattering amplitudes and the electroweak form factors. The fundamental properties of causality and unitarity are important ingredients in CHPT. We shall consider a dispersive method suitable especially for form factors, which uses as input an information about the modulus along the unitarity cut, leading to constraints on the values inside the analyticity domain. The technique was for the first time combined with the operator product expansion (OPE) of a QCD polarization function. Recently, this formalism, supplemented with the predictions of the heavy quark effective theory (HQET), was successfully applied to the form factors describing the weak semileptonic decays of the heavy mesons. The predictive power of the technique was considerably increased by the simultaneous treatment of several form factors with different unitarity thresholds, connected among them near zero recoil by HQET. In the present paper we apply a similar technique for testing the consistency of a low energy CHPT theorem with unitarity and perturbative QCD. We consider the function Δ(t) depending on π, K+ and K0 form factors and on fKπ weak form factor to describe the semileptonic decay K →πlν. Sirlin's theorem requires the relations Δ'(0) = 0, Δ''(0) = 0,... which must hold up to symmetry breaking corrections. In CHPT the renormalization constants cancel in the combination of the form factors entering the function Δ(t). A recent O(p6) calculation in full chiral SU(3) x SU(3) perturbation theory gave the value: Δ'(0) = (0.0033 ± 0.0005) fm2. In order to test the CHPT theorem we consider the vacuum polarization function Π(q2) = Πelm(q2) + Πw(q2) due to the electromagnetic and weak currents of the light meson form factors. This function satisfies a dispersion relation correlating its absorptive part with Π(-Q2) in the Euclidean region Q2>0, where it is given by renormalization group improved perturbative QCD, with nonperturbative corrections included by means of OPE. On the other hand, the absorptive part ImΠ(t) is expressed by unitarity as a sum of moduli squared of form factors. By optimally exploiting this input one can derive constraints on the form factors at interior points or provide tests of the low energy relations. A nontrivial complication is the unphysical cut of the kaon form factors, which can be eliminated using information about the phase and a generalization of the Omnes technique. We tested the consistency of Sirlin's theorem using functional optimization techniques. Denoting by sk the derivatives of the function Δ(t) at t = 0, we defined a suitable quantity μ02, depending on sk and the physical input described above. The values of sk for which μ02 ≤ 1 are consistent with perturbative QCD and unitarity, while μ02 >1 is a signal of inconsistency. As an illustration, we evaluated the quantity μ02 using the value s1 = 0.0033 fm2 obtained by Post and Schilcher and several choices for the higher derivatives s2 and s3. A few results are given, showing in particular that Sirlin's theorem is verified within the present formalism. (author)