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We evaluate the .rrg scattering amplitude at next-to-leading order in chiral perturbation theory. We discuss the conventional approach of the pseudoscalar Goldstone bosons ( a , K , g ) only and an extension including explicit resonance fields. The contributions of these resonances saturate the low-energy constants at order E 4 . We present predictions for the scattering lengths and phase shifts of the low partial waves.Green's functions of quark currents can be analyzed at low energies in the framework of chiral perturbation theory (CHPT) as dictated by chiral symmetry and its spontaneous breakdown. It is most economical to make use of an effective Lagrangian involving the pseudoscalar Goldstone bosons [ I ] eventually accompanied by explicit resonance degrees of freedom [2,3]. In the first version (conventional CHPT), one evaluates physical observables for a given loop order which is equivalent to a systematic expansion in external momenta and quark masses. In most cases, the one-loop approximation suffices (i.e., taking terms up to and including order E 4 ) . Whenever resonances can contribute to a process under consideration, the one-loop approximation encounters a barrier. This naturally leads one to include the low-lying resonance multiplets. Chiral symmetry dictates the form of the interaction vertices between the Goldstone pseudoscalars and the resonances at leading order ( E 2 ) . At order E 4 , it can be demonstrated that the momentum-independent parts of the resonance propagators generate polynomial terms whose coefficients (low-energy constants) are very close to the empirically determined values of these constants for the conventional approach (working at the resonance scale, p = M , ) [2]. Taking into account the full resonance propagators, one includes some terms of order E6 and higher. In Ref.[4] it has been shown that with the inclusion of the resonances the chiral predictions for n n [5-71 and n K [8] scattering are in fact improved. In the following, we will be concerned with the elastic n-7 scattering amplitude in chiral SU(3 I L X SU(3 ), . We will consider the conventional C H P T at next-to-leading order as well as the enlarged effective Lagrangian with explicit resonance degrees of freedom. In the absence of any phase-shift analysis in the n 7 system, we will predict the scattering lengths and phase shifts of the low partial waves in what follows. Even if it will take a long time before accurate phases might be available, we believe that our results are useful for the model builders.In the presence of external sources of scalar, pseudoscalar, vector, and axial-vector type (s,p, u, a ), the QCD generating functional at next-to-leading order takes the form Here, U =exp[iQ/Fo] embodies the eight pseudoscalar Goldstone bosons with Fo the pion decay constant in the chiral limit. The subscripts 2 and 4, respectively, refer to the leading (tree-ievel) and next-to-leading (one-loop) order. It is common (but not necessary) to split the oneloop contribution into terms of tadpole type ( Z j ) , the ...

I review recent developments in chiral perturbation theory (CHPT) which is the effective field theory of the standard model below the chiral symmetry breaking scale. The effective chiral Lagrangian formulated in terms of the pseudoscalar Goldstone bosons (π, K, η) is briefly discussed. It is shown how one can gain insight into the ratios of the light quark masses and to what extent these statements are modelindependent. A few selected topics concerning the dynamics and interactions of the Goldstone bosons are considered. These are ππ and πK scattering, some non-leptonic kaon decays and the problem of strong pionic final state interactions. CHPT also allows to make precise statements about the temperature dependence of QCD Green functions and the finite size effects related to the propagation of the (almost) massless pseudoscalar mesons. A central topic is the inclusion of matter fields, baryon CHPT. The relativistic and the heavy fermion formulation of coupling the baryons to the Goldstone fields are discussed. As applications, photo-nucleon processes, the πN Σ-term and non-leptonic hyperon decays are presented. Implications of the spontaneously broken chiral symmetry on the nuclear forces and meson exchange currents are also described. Finally, the use of effective field theory methods in the strongly coupled Higgs sector and in the calculation of oblique electroweak corrections is touched upon.

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