### Abstract

Three new contributions to the complex angular momentum (CAM) theory of differential cross sections (DCSs) for chemical reactions are reported. They exploit recent advances in the Padé reconstruction of a scattering (S) matrix in a region surrounding the axis, where *J* is the total angular momentum quantum variable, starting from the discrete values, *J* = 0, 1, 2, …. In particular, use is made of Padé continuations obtained by Sokolovski, Castillo, and Tully [Chem. Phys. Lett.313, 225 (Year: 1999)10.1016/S0009-2614(99)01016-7] for the S matrix of the benchmark F + H_{2}(*v* _{i} = 0, *j* _{i} = 0, *m* _{i} = 0) → FH(*v* _{f} = 3, *j* _{f} = 3, *m* _{f} = 0) + H reaction. Here *v* _{i}, *j* _{i}, *m* _{i} and *v* _{f}, *j* _{f}, *m* _{f} are the initial and final vibrational, rotational, and helicity quantum numbers, respectively. The three contributions are: (1) A new exact decomposition of the partial wave (PW) S matrix is introduced, which is called the QP decomposition. The P part contains information on the Regge poles. The Q part is then constructed exactly by subtracting a rapidly oscillating phase and the PW P matrix from the input PW S matrix. After a simple modification, it is found that the corresponding scattering subamplitudes provide insight into the angular-scattering dynamics using simple partial wave series (PWS) computations. It is shown that the leading *n* = 0 Regge pole contributes to the small-angle scattering in the centre-of-mass frame. (2) The Q matrix part of the QP decomposition has simpler properties than the input S matrix. This fact is exploited to deduce a parametrized (analytic) formula for the PW S matrix in which all terms have a direct physical interpretation. This is a long sort-after goal in reaction dynamics, and in particular for the state-to-state F + H_{2} reaction. (3) The first definitive test is reported for the accuracy of a uniform semiclassical (asymptotic) CAM theory for a DCS based on the Watson transformation. The parametrized S matrix obtained in contribution (2) is used in both the PW and semiclassical parts of the calculation. Powerful uniform asymptotic approximations are employed for the background integral; they allow for the proximity of a Regge pole and a saddle point. The CAM DCS agrees well with the PWS DCS, across the whole angular range, except close to the forward and backward directions, where, as expected, the CAM theory becomes non-uniform. At small angles, θ_{R} ≲ 40°, the PWS DCS can be reproduced using a nearside semiclassical subamplitude, which allows for a pole being close to a saddle point, plus the farside surface wave of the *n* = 0 pole sub-subamplitude, with the oscillations in the DCS arising from nearside-farside interference. This proves that the *n* = 0 Regge resonance pole contributes to the small-angle scattering.

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