On threshold amplitudes II: Amplitudes nullification via classical symmetries

The Ward identities for amplitudes at the tree level are derived from symmetries of the corresponding classical dynamical systems. The result are aplied to some 2 → n amplitudes supported by the Lódź University grant No. 269. supported by KBN grant No. 5 P03B 060 21 1 I Introducion Threshold amplitudes provide one of the rare examples of problems in quantum field theory which are both interesting and tractable, to some extend at least, by analytical methods. They were subject of numerous studies ( see [1] for a review). An interesting aspect of the problem is amplitude nullification which exhibit some models [2]. In particular, Libanov et al. [3] have shown that tree amplitudes vanish on the threshold in the model with 0(2)-symmetry broken softly by the mass term. Moreover, they gave a beautiful argument in favour of the idea that nullification is related to the symmetries of the reduced classical mechanical system [4](see also [5]). The line of reasoning presented in Ref. [4] can be used to construct other theories exhibiting amplitude nullification [6]. Moreover, the conclusions of [4] were confirmed by more traditional approach based on Ward identities following from the symmetry [7]. In the present note we derive the Ward identities for tree-level threshold amplitudes by functional techniques. We assume that the reduced hamiltonian system posses some symmetry and compute the corresponding Ward identities at the tree level. These Ward identities are shown to imply amplitude nullification. The paper is organized as follows. In Sec. II we remind the description of canonical symmetries on lagrangian level. Then, in Sec. III, the Ward identities for tree-level amplitudes are computed assuming the existence of symmetry for the reduced dynamical system obtained by neglecting space-dependence of the initial system. In Sec.IV these results are applied to specific 2 → n processes. Finally, Sec.V is devoted to some conclusions. II Canonical symmetries on lagrangian level Generically, the symmetries enforcing amplitudes nullification are canonical rather than point transformations. However, it is more convenient to do the pertubation theory on the lagrangian level. Therefore, we shall first remind how the canonical symmetries show up in the lagrangian formalism. As a first step consider a generalized lagrangian formalism; the generalization consists in the assumption that the second time derivatives are also admitted in the lagrangian function L = L(q, q̇, q̈) (1) The corresponding equations of motion read ∂L ∂qi − d dt ( ∂L ∂q̇i ) + d dt2 ( ∂L ∂q̈i ) = 0 (2) The class of point transformations allowed is now wider: new coordinates can depend also on old velocities. The transformation (q, t) → (q, t) is a symmetry, if L(q, dq dt′ , dq dt′2 ) dt dt = L(q, dq dt , dq dt2 ) + df(q, dq dt , t) dt ; (3)