Local analyticity properties of the n particle scattering amplitude

The connected part F/sub /c(p) of the scattering amplitude (p/sub 1 /...p/sub /r mod S-1 mod p/sub r+1/,..., p/sub n/) defined on the mass shell p/sub i//sup 2/=m/sub i//sup 2/ and deduced from a local field theory involving only (stable) particles with strictly positive masses can be represented in a suitable neighbourhood of any physical point p as a finite sum f/sub /c(p)= Sigma /sub 1//sup N/F/sub i/(p) of partial amplitudes', each F/sub i/(k) analytic in a certain domain F /sub i/ of the complex mass shell k/sub i//sup 2/=m/sub i//sup 2/. The mentioned real neighbourhood lies on the boundary of each F/sub i/. The above decomposition may fail to hold only at points p where any two incoming or any two outgoing four-momenta become parallel (thresholds). The number N as well as the shape of the domains F/sub i / depend on the number n and on the real neighbourhood considered. For a generic configuration p the intersection of the domains F/sub i/ is empty. When this does not happen, F/sub i/(p) is the boundary value of a single analytic function. This is illustrated on the case of the five point function, where it is shown that when D identical to det(p /sub /rp/sub s/)>m/sub i//sup 2/m/sup 2//sub 2/m/sub 3//sup 2/, D being the Gram determinant of the scalar products of the three outgoing momenta p/sub 1/, p/sub 2/, p/sub 3/, the scattering amplitude is the boundary value of a single analytic function. It is also indicated on the same example how these local results may be improved; one finds in the equal mass case m/sub /r=m that the five- point scattering amplitude is the boundary value of a single analytic function whenever M>4, 8m, M being the total centre-of-mass energy of the three outgoing momenta. (29 refs).