Some rigorous analyticity properties of the four-point function in momentum space

Analyticity properties and asymptotic behavior of scattering amplitude in higher dimensional theories

“…F (s, t) is an analytic function of the two Mandelstam variables, s and t, in a neighborhood ofs in an interval below the threshold, 4m 2 − ρ <s < 4m 2 and also in some neighborhood of t = 0, |t| < R(s). This statement hold due to the work of Bros, Epstein and Glaser [39,45].…”

“…This is defined by: |t| < R and s outside the cut s thr + λ = 4m 2 + λ, λ > 0. The importance of this result is recognized if we recall BEG theorem [39]. It was shown that in neighborhood of any point s 0 , t 0 , −T < t 0 ≤ 0, s 0 outside the cuts, there is analyticity in s and t in a region…”

“…Therefore, the amplitude is analytic in the domain: |t| < R ⊗ cut s-plane. Martin proved the theorem for s below the threshold from the BEG [39,45] results. The analytic continuation is applicable for 0 < s, s thr = 4m 2 .…”

“…Moreover, the four point amplitude depends only on the Lorentz invariant variables s and t. Thus if we obtain the BEG proof of the edge-of-thewedge theorem in this special choice of (momentum) frame, then it will be valid in any Lorentz frame. However, when BEG argument is adopted in this frame, we are to implement the analytic continuation in this four dimensional subspace from one domain to another domain following the arguments used for D = 4 theories [39,41]. Let us consider the simple case of n-point amplitude where 4 < n < D. Here we are unable to restrict the D-momenta of n-particles to a four dimensional subspace to apply BEG theorem (proved for D = 4 theories).…”

“…We have not rigorously proved existence of such a domain of analyticity for D-dimensional theories. This analyticity property was derived by [39] in the LSZ formulation, with micro causality. We have shown that there is analyticity in t in the Lehmann ellipse for D-dimensional theories and polynomial bounded (in s) in such higher dimensional theories.…”