Threshold amplitudes: Amplitude nullification via classical symmetries

Abstract

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 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.

Multiboson threshold production by fermion–antifermion pair and hidden symmetry

“…The arguments presented in Ref. [11] were subsequently supported by diagrammatic [12] and functional [13] proofs of the relevant Ward identities expressing the underlying symmetry at the tree level. Moreover, they allowed to construct other models with vanishing soft-soft amplitudes [14].…”

“…One can pose the question whether the nullification of hard-soft amplitudes can be also explained as a consequence of some symmetry. Recently, the present authors proposed a partial solution to this problem [13] (see also [15]). Consider the 2 → n hard-soft process; we assume that both inital and final particles are scalars and , furthermore, they are of different kind.…”

“…The vanishing of the resulting soft-soft amplitude is then the result of integrability of reduced system (Garnier system for the example considered in Ref. [13]). In particular, the N = 1 case of the model (1) can be treated in this way.…”

“…We want to calculate the amplitude for ff → nρ process with all final ρ-particles at rest. This can be done diagrammatically [19] or by calculating the relevant fermionic propagator in external scalar field [6] or, else, by Feynman wave function method [2], [16], [15]. We adopt here the last approach; accordingly, we first solve the field equation for ρ (assuming ψ = 0 =ψ) under the condition…”

“…In order to find 2 → n soft-soft amplitude for the theory described by the lagrangian (13) we proceed in a standard way [2], [16], [15]. First, neglecting the space-dependence in L ef f we arrive at the reduced fermionic-bosonic system of finite degrees of freedom given by…”