On amplitude zeros at threshold

Abstract

The occurrence of zeros of 2 to n amplitudes at threshold in scalar theories is studied. We find a differential equation for the scalar potential, which incorporates all known cases where the 2 to n amplitudes at threshold vanish for all sufficiently large $n$, in all space-time dimensions, $d\ge 1$. This equation is related to the reflectionless potentials of Quantum Mechanics and to integrable theories in 1+1 dimensions. As an application, we find that the sine-Gordon potential and its hyperbolic version, the sinh-Gordon potential, also have amplitude zeros at threshold, ${\cal A}(2\to n)=0$, for $n\ge 4$ and $d\ge 2$, independently of the mass and the coupling constant.Comment: 6 pages, Latex, CERN-TH.6762/9

Multi-parton cross sections at hadron colliders

“…to the full amplitude in any field theory, is possible. Another approach in this direction based on the so called ALPHA algorithm [21,22] or the Dyson-Schwinger recursion equations has been developed [23][24][25][26][27][28][29][30][31][32], where the multi-parton amplitude can be constructed without referring to individual Feynman diagrams. In the latter case, apart from the summation over colour in the colour flow basis, which will be briefly described in the next section, integration over a continuous set of colour variables (as well as flavour) was introduced.…”

“…We have three possibilities to obtain a gluon (A, B) described by the recursive relation Eq. (26). First, it can be obtained when a quark with colour assignment (C, 0) is merged with an antiquark of anticolour (0, D).…”

Nonperturbative Methods

“…Using the unitarity relation for the imaginary part of the loop graphs, one immediately concludes that this can only be if the tree amplitudes of the on-shell processes 2 → n are all zero at the threshold for n > 4 in the theory without SSB [83,87] and for n > 2 in the theory with SSB [84]. This can be traced to the special properties of the reflectionless potential −6/(cosh τ ) 2 in equation (30) and generalized [88,76] to other theories, where the problem of the 2 → n scattering is reduced to finding the Green's function in the reflectionless potential −N(N + 1)/(cosh τ ) 2 with integer N. The known additional cases are the following:…”

“…The recursion method can be extended to other theories [74] as well as to loop graphs [75,76] and to an analysis of higher loops [77]. However a more convenient method for further analysis is the one suggested by Brown [78] and is based on a functional technique.…”

Multiboson threshold production by fermion–antifermion pair and hidden symmetry

“…First, one can consider the "hard-soft" scattering, where two initial particles of opposite momenta scatter to produce an arbitrary number of final bosons, all at rest. It has been shown that, for a quite wide variety of processes, the relevant threshold amplitudes vanish provided the parameters of the lagrangian are related in a certain way [2] ÷ [9]. Typically, these relations take the form of "quantization conditions" and the integer number entering there gives an upper bound for the number of final bosons which can be produced.…”