Influence of Regge poles on rainbow angular scattering for state-to-state chemical reactions using Heisenberg’s S matrix programme

Two powerful theories for state-to-state chemical reactions are brought together for the first time. The first theory incorporates Regge pole positions and residues into the partial wave (PW) scattering (S) matrix. The second theory is a ‘weak’ version of Heisenberg’s Scattering Matrix Programme (wHSMP). It uses four general physical principles to suggest simple parameterised forms for the S matrix. The wHSMP is particularly useful for understanding generic structures in differential cross sections (DCSs). Our initial S matrix parameterisation has no Regge poles, but it exhibits a rainbow in the DCS using a Legendre PW series. We then introduce Regge poles for three examples into the S matrix and investigate their influence on the rainbow scattering. We find that inclusion of Regge poles ‘pushes’ the rainbow angle to larger values. We also employ Nearside-Farside (NF) PW and Local Angular Momentum PW theories, including up to three resummations. The recently introduced ‘CoroGlo’ test is used to distinguish between glory and corona scattering in the forward direction. We apply full and NF asymptotic (semiclassical) rainbow theories: the uniform and transitional Airy approximations for the farside scattering. We prove that structure in the no-pole and with-pole DCSs are examples of primary and supernumerary rainbows. GRAPHICAL ABSTRACT


Introduction
A key quantity in the theory of state-to-state chemical reactions is the scattering matrix (S matrix) or scattering operator (S operator) because, in principle, they allow a complete characterisation of a reaction. For example, the S matrix permits collisional observables to be calculated, such as differential cross sections (DCSs), which contain detailed information on the mechanism and dynamics of a reaction. The interpretation of DCSs, which often exhibit complicated interference patterns, is therefore an important problem [1][2][3][4][5][6].
The computation of DCSs from first principles for known initial states to known final states usually follows numerical errors. (c) In the semiclassical (or asymptotic, h → 0) limit, the S matrix elements obtained in a largescale computation often consist of long list of complex numbers, which, in more complicated cases, can be difficult (or impossible) to understand and interpret. (d) A new scattering calculation is required for a new potential energy surface and a new reaction. For additional discussion, see ref. [7]. Clearly, it is desirable to investigate alternative and complementary approaches for the computation of dynamical phenomena, in particular for the calculation of DCSs.
We have recently begun exploring a simple, yet powerful, alternative approach based on a 'weak' version of Heisenberg's Scattering Matrix Programme (wHSMP) [7][8][9][10][11]. Our approach can be summarised as follows initial states → S matrix → final states (R2) The reaction scheme R2 does not employ a potential energy surface -see ref. [7] for a full discussion of this point. Rather it is based on Heisenberg's fundamental insight that the S matrix contains, in principle, all the information needed to calculate dynamical phenomena [12][13][14][15]. Heisenberg hoped to use general physical principles, such as unitarity, causality and analyticity to determine the S matrix. This has not yet proved possible, so in our approach, we use instead four general physical principles relevant to state-to-state chemical reactions to suggest simple, yet realistic, parametrised forms for the S matrix. We denote our five previous papers in refs. [7,8,9,10,11] by H1, H2, H3, H4, H5 respectively. We have earlier explored the following topics: • In H1, we described HSMP and introduced wHSMP, applying it to the forward glory scattering of the H + D 2 → HD + D reaction. Historical remarks on HSMP can also be found in H1. • In H2, we obtained a surprising result: piecewise S matrix elements using simple linear, quadratic, stepfunction and top-hat parameterisations can reproduce the small-angle scattering of the H + D 2 → HD + D reaction. • In H3, we studied a DCS for the F + H 2 → FH + H reaction using the techniques in H1 and H2. We also applied a variant of wHSMP in which the parameterised S matrix is modified using simple Gaussiantype functions suggested by results from scheme R1. We called this variant, 'hybrid', and denoted it, hHSMP ≡ hwHSMP. • In H4, we proved the existence of pronounced Airy rainbows, supernumerary rainbows and interference effects in product DCSs arising from attractive (or farside) scattering. We kept the modulus of the S matrix fixed and systematically varied its phase.
• H5 is similar to H4, except that we kept the phase of the S matrix fixed and systematically varied its modulus. Thus, H4 and H5 together reported a systematic investigation of pronounced rainbows for a class of reactive systems.
It is also important to note that H1, H2, H3 are fundamentally different from H4, H5. In H1, H2, H3, a quadratic polynomial phase in J was used for the parameterised S matrix, whereas in H4, H5, a cubic polynomial phase was employed. Here J is the total angular momentum quantum number.
It is evident from the above, that wHSMP is a practical and powerful tool, which can complement calculations from the 'The Royal Road of Reaction Dynamics' [7][8][9][10][11]. In particular, the wHSMP can be used to quickly calculate DCSs from partial wave series (PWS). Then by making changes to the S matrix, we can rapidly investigate generic structures in DCSs for state-to-state chemical reactions, such as rainbows, glories and interference (Fraunhofer) effects. This is done by exploring different parameterisations for the modulus and phase of the S matrix [7][8][9][10][11]. A useful aim is to find simple parameterisations, which are capable of describing key features of the reaction dynamics. In addition, we can also determine the values of the parameters by fitting experimental DCSs, or the results of computer simulations. Finally, we note that the parameterised S matrix elements can be refined by adding information obtained from potential energy surface(s) computations, until eventually we arrive at the S matrix from The Royal Road.
The purpose of this paper is to incorporate, for the first time, Regge poles into wHSMP. In particular, we examine how Regge poles affect rainbow scattering. More generally, our aim is to gain a better understanding of the rôle Regge pole positions and residues play in understanding interference structures in reactive DCSs.
The theory presented in this paper is for the following generic state-to-state chemical reaction: where v i , j i , m i and v f , j f , m f are vibrational, rotational and helicity quantum numbers for the initial and final states, respectively. It is assumed that the reaction occurs at a fixed total energy, or equivalently a fixed initial translational wavenumber. Then the PWS for the scattering amplitude can be expanded in a basis set of Legendre polynomials. There are also many approximate theories for chemical reactions that use a Legendre PWS. Thus, the theory and results in this paper should have a wide utility. This paper is organised as follows. Section 2 presents the general physical principles used in wHSMP for the design of the S matrix. Sections 3 and 4 describe the parameterised forms that we employ in our PWS computations for the pre-exponential and phase components of the S matrix respectively. The theoretical techniques we use are presented in Section 5, which all use a PWS. We summarise in Section 6 the asymptotic (semiclassical ≡ SC) techniques we use to prove the existence of rainbows; in particular, we use uniform and transitional Airy SC approximations. In Section 7, we numerically sum the Legendre PWS for the scattering amplitude to calculate DCSs and Local Angular Momenta (LAMs) for the reference case of a smooth-step parameterisation for the S matrix. This example does not contain any explicit Regge poles in the PWS. In Section 8, we investigate the influence of Regge poles of the formã n /(J − J n ) in the PWS. Hereã n is the partial residue and J n is the position of the nth Regge pole. In Section 9, we present and discuss our DCS results when Regge poles are included for three examples. An interesting finding is that a Regge pole(s) 'pushes' the rainbow angle to larger angles in the DCS compared to the non-pole DCS. In both Sections 7 and 9, we apply the recently introduced 'CoroGlo' test [30,31], which lets us distinguish in a simple way between corona and glory scattering in a reactive DCS at small angles. We also apply Nearside-Farside (NF) theory [27,32,33] and LAM theory [34,35] together with resummations of the PWS [34][35][36][37][38][39][40]. Conclusions are in Section 10. The Appendix discusses a many-pole example in more detail.

Weak version of Heisenberg's S matrix programme
In this section, we describe the assumptions of wHSMP [7]. It is based on four general physical principles relevant to chemical reactions. Our design strategy for the construction of the S matrix elements is based on them. Note that no potential energy surface is used. As usual [7][8][9][10][11], we employ a modified S matrix element, denoted S J . The four assumptions of wHSMP are: (1) The forces responsible for chemical reactions are short ranged, of the order of 10 −10 m. This impliesS J → 0 as J → ∞. In practice, there is a maximum value of J, denoted, J max , beyond which partial waves make a negligible numerical contribution to the PWS. N.b., this assumption excludes reactions that are asymptotically Coulombic, for which the PWS is divergent. (2) Conservation of probability holds. This implies the S matrix is unitary with 0 ≤ |S J | ≤ 1. (3) Under semiclassical (SC) or asymptotic (h → 0) conditions, namely, J max >> 1, we can continue the set {S J } to a smoothly varying function,S(J), with simple properties. In our applications below,S(J) is an analytic function, i.e. one of class C ω in the notation used for the continuity and differentiability of functions [8]. (4) In the classical limit, we require a head-on collision to correspond to backward (or rebound) scattering of the products.

Notes:
• In Assumption 3, by 'analytic' we include functions with poles and branch points, as is usual in S matrix theory [7]. • Assumption 3 was used in a weaker form in H2 and H3, whereS(J) was allowed to be (i) a piecewise continuous function (of class C 0 ), with simple properties for the pieces, or (ii) a piecewise-discontinuous function (of class C −1 ), again with simple properties for the pieces. There is thus considerable flexibility in the way Assumption 3 is applied to different problems. • In Assumption 4 and elsewhere, we use the reactive scattering angle, θ R , defined as the angle between the outgoing diatomic molecule and the incoming atom in the centre-of-mass reference frame.
Using the above physical assumptions for wHSMP, we follow H1-H5 and parametriseS(J) for real J in the form where the subscript 'X' = 'step', 'pole', or 's/p ≡ step/pole', is an optional label added sometimes for clarity, and which are defined below, N is a real normalisation constant, andφ X (J) is a real phase. For the preexponential factor,s X (J), there are two possibilities: (1) When there are no explicit Regge poles,s(J) is positive or zero, withs(J) → 0 for J → ∞. We also have, argS(J) =φ(J). (2) When there are explicit Regge poles present of the type,ã n /(J − J n ), where J n is the nth Regge pole position with partial residueã n , thens(J) is complex valued, withs(J) → 0 for J → ∞. We also have, argS(J) = args(J) +φ(J).

Parameterisation ofs(J)
For the two possibilities mentioned in Section 2, we construct 'smooth step' and 'pole sum' parameterisations for s(J), as follows:

Smooth step
We use a simple smooth-step parameterisation, previously employed in H1 and H4, modelled on the H + D 2 reaction, namelỹ where J cut is the 'cut off' value of J and d cut acts as a 'diffuseness' parameter in J space. Both J cut and d cut are positive, although not necessarily integers.
In passing, we note that in the CAM plane,s step (J), has an infinite number poles lying on a line in the first and fourth quadrants of the complex J plane [7,41]. However this property is not required, since we only use, s step (J), for J = 0, 1, 2, . . . and its continuation to real values of J. The continuation is automatically supplied by Equation (2). We also sometimes write Equation (1) more explicitly asS ( 3 ) Figure 1 shows graphically, for the standard values of the parameters in Section 7, the properties that we derive analytically in this section, and the next one, for the smooth-step parameterisation.

Pole sum
We first define a 'pole sum' [21,42,43] bỹ where n max is a positive integer, and J n andã n are complex numbers. When J is allowed to take complex values in the CAM plane, J n andã n correspond to the position and and J 3 = J 3 (θ R ) for the F scattering. Also shown is J r , which is located at the minimum of the˜ step (J)/deg curve, wherẽ step (J = J r ) = −θ r R (pink arrow and dashed lines) together with J g 1 and J g 2 , which satisfy˜ step (J g 1 or J g 2 ) = 0. The three branches of the deflection function are labelled 1 (N, red) and 2, 3 (F, both blue). (d) (2J + 1)|S step (J) | versus J. Also marked are, J g 1 , J r , J g 2 , with dashed lines and arrows, and the N and F angular zones. [See colour online]. partial residue of the nth Regge pole respectively. For this case, we sometimes label Equation (1) in the form The form in Equation (4) is suggested by the Mittag-Leffler theorem for the expansion of a meromorphic function in terms of its poles.
In passing, we note that the full residue ofS pole (J) at the pole, J = J n , is, by definition, although we do not need this result in the present paper.

Smooth-step + pole parameterisation
We also define the sum of the smooth-step and pole parameterisations, denoted,s s/p (J) ≡s step/pole (J). We havẽ In Equation (6), θ damp (J) is a damping factor needed to ensure convergence of the PWS whens pole (J) is present. We use where J damp and d damp are positive parameters. Note that θ damp (J) is not needed whens pole (J) is absent. Finally, we sometimes writẽ Sinces s/p (J) in Equation (8)

Parameterisation ofφ(J)
The equations presented below are for the smoothstep parameterisation, for which argS step (J) =φ step (J). When Regge poles are present, we have, argS s/p (J) = args s/p (J) +φ s/p (J). Then the equations needed are modifications of the smooth-step case -this is discussed in Section 9.
It was shown in H4 and H5, that to describe rainbow scattering,φ step (J) has to be a cubic (or higher) polynomial in J. We writẽ where a 0 , a 1 , a 2 , a 3 are the four real phase parameters with a 3 = 0, and not to be confused with the partial residue, a n , in Equation (4). An important quantity in the asymptotic (h → 0) or SC theory of rainbows is the derivative ofφ step (J), which is called the quantum defection function (QDF) and writ-ten˜ step (J) = dφ step (J)/dJ [44]. For the smooth-step parameterisation, we have from Equation (9) Next, we must choose the phase parameters inφ step (J). We proceed as follows: • The PWS DCSs and SC DCSs are independent of the value of a 0 , so we usually choose a 0 = 0. • Equation (10) shows that,˜ step (J = 0) = a 1 , i.e. in a classical picture, a head-on collision determines a 1 . In accordance with Assumption 4 of wHSMP in Section 2, we usually choose, a 1 = π . • The parameters a 2 and a 3 determine the properties of the rainbow. In practice, it is more convenient to use as parameters, the rainbow angle, θ r R , and the corresponding rainbow angular momentum variable, J r , shown in Figure 1(c) for the smooth-step parameterisation. We next discuss this point in more detail.
The relation between {a 2 , a 3 } and {θ r R , J r } has been worked out in H4 using the properties,˜ step (J r ) = 0 and˜ step (J r ) = −θ r R . Here the prime indicates differentiation. The results for the smooth-step parameterisation are and Equations (11) and (12) let us define in a convenient way the coefficients of the cubic phase in terms of θ r R and J r , (also knowing, a 0 and a 1 ). Since θ r R , J r and a 1 are always positive in our applications, we see that a 2 is always negative, whereas a 3 is always positive.
To summarise, in the {a 0 , a 1 , θ r R , J r } representation, we have for the cubic phasẽ Another useful representation [10], makes an exact Taylor expansion ofφ step (J) in Equation (13) about the point J = J r . We find together with Equations (16) and (17) let us confirm that˜ step (J r ) = −θ r R and˜ step (J r ) = 0, respectively. From Equation (16), we can obtain explicit formulae for the stationary points Figure 1(c). Solving,˜ step (J(θ R )) = +θ R , for the N scattering gives whilst solving,˜ step (J(θ R )) = −θ R , for the F scattering results in and

Partial wave theory
Here we summarise the partial wave theory that we require and establish our notations. In particular, we outline the 'CoroGlo' test, nearside-farside (NF) and local angular momentum (LAM) theory, and resummations of the PWS.

Partial wave series
Since m i = m f = 0 in reaction R3, the PWS for the scattering amplitude can be expanded, as usual, in a basis set of Legendre polynomials where P J (•) is a Legendre polynomial of degree J, θ R is the reactive scattering angle and k is the incident translational wavenumber. In practice, the upper limit of infinity in the PWS is replaced by a finite value, J max , assuming that all partial waves with J > J max are negligible. The DCS is then given by In our applications, the PWS of Equation (21) contains about 100 numerically significant terms making its physical interpretation difficult or impossible. We also have the estimate, J max ≈ kR, where R is the reaction radius.
Our angular distributions in Sections 7 and 9 show that the full DCSs computed from Equations (21) and (22) exhibit a forward peak accompanied by oscillatory structures. To help understand these structures, we first analyse the small-angle scattering using the newly introduced 'CoroGlo' test [30,31]. For the scattering at larger angles, we make a NF decomposition of the scattering amplitude. These topics are outlined in the next two sections.

CoroGlo test
The recently introduced 'CoroGlo' test [30,31] lets us distinguish at small-angles in a DCS, whether the peak at θ R = 0 • arises from a corona or from a glory. Now both a corona and a glory give rise to a peak in the DCS(θ R ) at θ R = 0 • , accompanied by oscillations at larger angles displaying maxima. A corona in molecular collisions is usually modelled by Fraunhofer scattering from a hardsphere [30], whereas a glory arises from N, F interference, as illustrated in Figures 2 and 3. The CoroGlo test lets us distinguish, in the asymptotic limit, between these two possibilities from the ratio of the DCS(θ R ) at θ R = 0 • to the value of the DCS at its adjacent maximum. We have: [30] corona diffraction ratio (CDR) ≈ 57.1 glory diffraction ratio (GDR) ≈ 6.2

Notes:
• The values of 57.1 and 6.2, above, are the diffraction ratios for a corona and glory respectively. When applied to DCSs for chemical reactions, different values can occur and may indicate the presence of other mechanisms in the angular scattering. For example, in  reference [30] a ratio of 2.6 was found for a state-tostate transition in the H + HD → H 2 + D reaction, which was shown to arise from the additional presence of rainbow scattering at small angles. • The CoroGlo test does not prove the presence of e.g. glory scattering. To do this, we must construct the asymptotic (SC) limit of the PWS (21). See Section 6. • The CoroGlo test does not usually apply to elastic or inelastic DCSs, because there is often a contribution to the small-angle scattering from the long-range attractive interaction potential, which is usually absent in the reactive case.

Nearside-farside decomposition
We exactly decompose the full scattering amplitude into the sum of two contributing terms, the N and F subamplitudes [27,32,33].
where (25) and Q J (•) is a Legendre function of the second kind. Similar to Equation (22), the corresponding N and F DCSs are defined by Using the asymptotic properties of the P J (•) and Q J (•) in the limit J sin θ R >> 1, we obtain, e.g. refs. [27,33].
which have a standard travelling N,F angular wave interpretation.

Local angular momentum nearside-farside decomposition
A local angular momentum (LAM) analysis can also be used to provide information on the total angular momentum variable that contributes to the scattering at an angle θ R , under semiclassical conditions [34,35]. It is defined by The same idea can also be applied to the N and F subamplitudes in Equation (23). The corresponding N, F LAMs are defined by [34,35] Note that the args in Equations (27) and (28) are not necessarily principal values in order that the derivatives be well defined.

Resummation of the partial wave series
It is known that a resummation [34][35][36][37][38][39][40] of the PWS (21) can markedly improve the physical effectiveness of the N,F decomposition (23). Although we have investigated resummation orders of r = 0, 1, 2, and 3 in this paper, we present below the working equations for r = 1 and for r = 0 [no resummation, i.e. Equation (21)]. This is because resummation cleans the N,F DCSs of unphysical oscillations, and we find the biggest effect occurs on going from r = 0 to r = 1. Further resummations, r = 1 to r = 2 and r = 2 to r = 3 have a smaller cleaning effect. A detailed account of resummation has been given by Totenhofer et al. [40] for a Legendre PWS.

Introduction
The asymptotic (semiclassical ≡ SC) theory of rainbow scattering for reactive systems possessing deflection functions similar to Figure 1(c) is well established [3,16]. So in this section, we just present the working formulae for the asymptotic N and F subamplitudes, respectively. Note that we use superscript (∓) to label the N and F subamplitudes, respectively, to avoid confusion with the PWS subamplitudes, f (24) and (33) respectively. The full SC scattering amplitude is therefore The corresponding full SC DCS is (37) and the N and F SC DCSs are given by The SC theory uses the continuation of {S J }, with J = 0, 1, 2, . . . , to real values of J, which we denote, as usual, byS(J), i.e. the S matrix elements are now considered to be a continuous function of the total angular momentum variable. An important role in the asymptotic analysis is played by the quantum deflection function,˜ X (J), for the smooth-step parameterisation from Equations (9) and (10): As previously, the arg in Equation (38) is not necessarily the principal value in order that the derivative be well defined. Figure 1(c) shows a graph of˜ step (J)/deg versus J for the standard values of the S matrix parameters used in Section 7 for rainbow scattering for the smooth-step parameterisation. We note the following properties in Figure 1(c), which are used below in the SC analysis: • The rainbow angular momentum variable, J r , occurs where˜ (J) has a minimum in the F scattering. We write,˜ (J r ) = −θ r R , where θ r R is the rainbow angle.
• For˜ (J) = −θ R and θ R > θ r R , there are no real roots in the F scattering.
• The real roots are ordered, which define the three branches, 1, 2, 3, which are marked on the˜ (J) versus J plot. • For˜ (J) = −θ r R , J 2 and J 3 coalesce to J r . • Two glory angular momenta, J g 1 and J g 2 , are also marked in Figure 1(c); they satisfy˜ (J g i ) = 0 for i = 1, 2. They are not used in the asymptotic analysis in this paper. [They do, however, provide an alternative way of defining the branches of˜ (J)]. • There is a 4th branch in the˜ (J) plot for J ≥ J g 2 . It also plays no role because (2J + 1)|S(J)| ≈ 0 when J ≥ J g 2 as can be seen in Figure 1(d).
In the following, we adopt the following notations where J s = J r , J 1 , J 2 , J 3 and similarly for the second derivative,˜ (J s ).

Definitions
Before describing the uniform and transitional SC approximations, we first define the following phases and 'classical-like' DCSs, as they are needed in the expressions below. There are four SC N and F phases associated with J 1 ,J 2 , J 3 and J r , which are given by [10,28] There are also three 'classical-like' N and F DCSs defined by [10,28] Since,˜ (J r ) = 0, there is no analogous expression to Equations (45)-(47) for the case, J = J r .

Asymptotic (SC) rainbow analysis of the farside scattering
For the farside scattering, the stationary phase condition is˜ In Figure 1(c), there are two real roots of Equation (48) for a given value of θ R , provided θ R < θ r R , which is the bright side of the rainbow. At θ R = θ r R , the two real roots coalesce. The uniform Airy approximation for the bright side of the F rainbow subamplitude is [10,28], where and The prime on the Airy function, Ai (x), means dAi(x)/dx, and in Equation (50), the positive branch is taken so that, ς(θ R ) ≥ 0. In a systematic notation [10,28], the uniform Airy approximation is denoted, SC/F/uAiry, or uAiry for short.
On the dark side of the rainbow, where θ R > θ r R , Equation (48) has two complex-valued roots, which are more awkward to handle. To avoid this problem, we assume a quadratic approximation for˜ (J) about J = J r , i.e.˜ where which results in the transitional Airy approximation for the F subamplitude [10,28].
Equation (52) only depends on the properties ofS(J) at, J = J r . It can be used on both the bright and dark sides of the rainbow, as well as at, θ R = θ r R , where the uniform Airy approximation (49) is numerically indeterminate. In a systematic notation [10,28], the transitional Airy approximation is denoted, SC/F/tAiry, or tAiry for short. Equation (51) is exact for our S matrix smooth-step parameterisation. This does not mean that the uAiry and tAiry approximations become equivalent, because the tAiry formula contains additional approximations-see ref. [46] for details.

Asymptotic (SC) analysis of the nearside scattering
For the nearside scattering, the stationary phase condition is˜ (J) = +θ R In this case,˜ (J) has a similar behaviour to that for glory scattering. We can therefore use the primitive semiclassical approximation for the N subamplitude previously introduced for glory scattering. It is given by [29,44,47] where β 1 (θ R ) and σ 1 (θ R ) are given by Equations (41) and (45) respectively. In a systematic notation, Equation (53) is denoted SC/N/PSA or PSA for short, where PSA = Primitive Semiclassical Approximation [10,27,28].

Asymptotic (SC) analysis of the full differential cross section
The full asymptotic DCS is obtained by combining the results in Sections 6.3 and 6.4. We have where f (−) PSA (θ R ) given by Equation (53) and f (+) xAiry (θ R ) is either Equation (49) for x = u, or Equation (52) for x = t.

Asymptotic (SC) analysis of the local angular momentum
Analogous to the LAM definitions in Section 5.4, a LAM analysis can also be applied to the SC scattering amplitudes. We have for the full SC scattering amplitude For the F scattering we get For the tAiry approximation, the Ai(•) term dominates in Equation (52), which gives us the simple result which is a constant independent of θ R . For the N scattering, we obtain another simple result Note that the arguments of all the scattering amplitudes in Equations (55)- (58) are not necessarily principal values.

DCS and LAM results for the smooth-step parameterisation
In this section, we describe and discuss our results for the smooth-step parameterisation,S step (J), defined by Equations (2), (3) and (13). The seven parameters used are θ r R = 2π/9 rad = 40 • , J r = 60, d cut = 3, J cut = 70 as well as N = 0.03, a 0 = 0, a 1 = π . We call them the standard parameters or standard values because they act as reference values and do not change. The corresponding values for a 2 and a 3 in Equation (9)  Some properties ofS step (J) are displayed in Figure 1, which was used to illustrate our earlier analysis in Section 3.1. We note the following for Figure 1: • Figure 1(a,b), showing |S step (J)| versus J and argS step (J)/rad versus J respectively, behave as expected, with the argS step (J)/rad curve a cubic polynomial in J. • Figure 1(c) displays˜ step (J) versus J and is a polynomial of degree two in J. Three branches of the˜ step (J) curve are denoted 1, 2, 3. We have also labelled Figure 1(c), and to a lesser extent Figure 1(b,d), with the quantities needed for the asymptotic rainbow analysis of section 6. Note the rainbow angular momentum variable and rainbow angle, are related by˜ step (J r = 60) = −θ r R (= −40 • ). In passing, we observe there are also two glories, labelled J g 1 (≈ 34.4) and J g 2 (≈ 85.6), where˜ step (J g 1 or J g 2 ) = 0 • . • Figure 1 • In passing, we also note that Figure 1(d) shows that the contribution to the DCS from the glory at J = J g 2 (≈ 85.6) is negligible. Figure 2(a) shows the logarithm of the full dimensionless DCS (dDCS), defined as k 2 σ (θ R ), for the angular range, 0 • ≤ θ R ≤ 180 • . A more detailed plot of the rainbow region is displayed in Figure 2 We note the following: • In Figure 2(a,b), the full dDCS shows the characteristic shape of a primary rainbow for θ R ≈ 35.6 • . It is not clear whether there is a supernumerary rainbow, or not, close to θ R ≈ 20.8 • in the full dDCS. • In Figure 2(a), we plot N and F dDCSs for r = 0, 1, 2, 3. The largest cleaning effect is seen to occur at large angles in the F dDCS on going from r = 0 to r = 1.
Thus from now on, we only show N,F results for r = 1. • There is a change in mechanism from F-dominated to N-dominated at θ R ≈ 50 • , as θ R increases from small angles to larger angles. • The CoroGlo test of Section 5.2 gives for the smallangle diffraction ratio a value of 6.0, which is close to the forward glory ratio of 6.2. We do not carry out an asymptotic glory analysis in this paper, since this has been done for related systems in refs. [28][29][30]44,[47][48][49][50][51]. • The LAM plots in Figure 2(c,d) are complementary and consistent with the dDCSs plots. In particular, the change in mechanism at θ R ≈ 50 • is clearly visible. Also there are oscillations in the F (r = 1) LAM curve for θ R 55 • showing the existence of primary and supernumerary rainbows. • For θ R 75 • , the full LAM changes to a monotonic increase, where it overlaps with the N (r = 1) LAM (except for angles close to the backward direction). This behaviour for θ R 75 • is similar to the N LAM for the collision of two hard spheres [44,47]. Note that the oscillations in the F (r = 1) LAM for θ R 70 • are unphysical [44,47]. This is similar to the corresponding oscillations in the F (r = 0, 1, 2, 3) dDCS in Figure 2(a) at larger angles.
In Figure 3, we compare dDCSs and LAMs for the asymptotic (SC) analysis of Section 6 with the PWS results. We note the following: • The uAiry results for the bright side of the rainbow have been calculated up to θ R = 39.9 • , with the tAiry results evaluated at larger angles (recall, θ r R = 40 • ). • There is good agreement between the full PWS dDCS and LAM with the full SC dDCS and LAM respectively. Note that the full SC theory includes a contribution from the N scattering, as given by Equation (53) and denoted SC/N/PSA. • Comparing the SC N and F curves in Figure 3 with the corresponding PWS N and F curves in Figure 2, shows good agreement.

Properties of Regge poles relevant to reactive scattering
In this section, we derive and discuss some properties of s pole (J) which are important for understanding the contribution of Regge poles to DCSs in Section 9. We begin with the simplest case of the pole sum (4) with one term (n = 0).

Single Regge pole
If we write,J 0 = x 0 + y 0 i, where x 0 and y 0 are both real positive quantities, and for a partial residue of unity, then we haves  Figure 4(a) shows an Argand plot ofs pole (J), in which the arrow indicates the direction of increasing J. We see thats pole (J) rotates clockwise, starting in the second quadrant and moves into the first quadrant as J increases from 0 to 120 in steps of unity. Figure 4(b) shows the corresponding plot of args pole (J) versus J. We see that args pole (J) decreases, corresponding to the clockwise rotation ofs pole (J) in Figure 4(a). The most rapid change in args pole (J) occurs at J = x 0 (= 60). For y 0 → 0, the decrease in args pole (J) approaches π. Figure 4(c) shows a plot of |s pole (J) | versus J; the maximum occurs at J = x 0 (= 60).
We next study the contribution of the Regge pole to˜ pole (J). The graph of args pole (J) in Figure 4(b) is always monotonically decreasing, which means that pole (J) = d args pole (J)/d J is always negative, as illustrated in Figure 4(d). This panel illustrates two important properties: (1) A single pole contributes to the farside scattering.
(2)˜ pole (J) possesses a minimum, which corresponds to a rainbow. It is located at J = J r (pole) and occurs at˜ pole (J r ) = −θ r R (pole).
In more detail, from Equation (59), we have Differentiation then gives a Lorentzian-type function, namelỹ which has a minimum value of −1/y 0 ≡ −1/ImJ 0 at J = x 0 ≡ ReJ 0 (= 60). For y 0 = 10, the minimum is at −0.10 rad ≈ −5.7 • . Next we examine how the above results are modified upon using a complex-valued residue,ã 0 =r 0 exp(iβ 0 ) in place of unity. We havẽ Figure 5 shows graphs analogous to Figure 4, except in Equation (62) we use a partial residue ofr 0 = 10 and β 0 = 5π/18 for which,ã 0 = 6.428 + 7.660i. It can be seen that the effects of makingã 0 = 1 are These and other properties are readily proved using the following result for the m complex numbers, z 1 z 2 · · · z m , namely, the modulus of the product equals the product of the moduli, and the argument of the product equals the sum of the arguments. The results described above illustrate a difficulty we face when using a single Regge pole to describe rainbow scattering. The negative of the rainbow angle, −θ r R (pole), occurs at a small negative angle in the plot of˜ pole (J) versus J -see Figure 4(d). For J 0 = 60 + 10i, we found in Figures 4(d) and 5(d) that˜ pole (J) has a minimum at −0.1 rad(≈ −5.7 • ). Now, since the minimum occurs at −1/y 0 ≡−1/ImJ 0 , one way to decrease the minimum is to use a smaller value for ImJ 0 . A plot of˜ pole (J) versus J for J 0 = 60 + 2i is shown in Figure 6. Comparing Figure 6 with Figures 4(d) and 5(d), we see that the minimum is now at −0.5 rad (≈ −28.6 • ), a sizable change. However, the full width at half depth of the 'valley' has been reduced from ≈ 20 in Figures 4(d) and 5(d) to only 5 partial waves in Figure 6, namely J = 58, 59, 60, 61, 62. This illustrates that a single Regge pole becomes less semiclassical in its contribution to the PWS as ImJ 0 → 0. We recall that a pronounced rainbow requires a contribution from many partial waves close to J = J r (= 60) if a significant rainbow is to be observed in the DCS.

Many Regge poles
When many Regge poles contribute to the scattering, Equations together with To proceed further, we need information about the distribution of poles in the first quadrant of the CAM plane, together with their partial residues. Appendix considers the case where a string of poles lie on a straight line parallel to the Im J axis. We also show how the inclusion of several Regge poles can be used to make the pole-term contribution more significant, and in particular how to increase the number of partial waves affected bys pole (J) in the PWS.

Introduction
In this section, we present and analyse dDCSs based on the parameterisation of Equation (8). We show results for three examples: for a single pole (two examples) in the summand of Equation (4), and for a many-pole example having 6 poles. We do not discuss results already analysed in Section 7 for the smooth-step parameterisation, if the same (or similar) discussion also applies to the Regge pole case.
We want to employ a representation forS s/p (J) based on Equations (8) and (13). However, there is now a notational ambiguity. In Equation (13), J r and θ r R are defined by the equations,˜ step (J r ) = 0 and step (J r ) = −θ r R respectively. In the present case we have We again want J r and θ r R to be defined by properties of the full deflection function, namely,˜ s/p (J r ) = 0 and s/p (J r ) = −θ r R . This implies we have to change the notation for the phase in Equation (13). We do this by adding a prime as a label to J r and θ r R , which become J r and θ r R respectively.
To summarise, we use the following Regge pole parameterisation for the S matrix in the three examples of this sectioñ with a 0 = 0 a 1 = π a 2 = −(π + θ r R )/J r a 3 = (π + θ r R )/(3 J 2 r ) In many cases, we find, J r ≈ J r , and, θ r R ≈ θ r R , and if the pole sum in Equation (67) is absent, we have, J r ≡ J r , and, θ r R ≡ θ r R and can also replace θ damp (J) by unity. We can derive a useful limiting case from Equation (67), whens pole (J) consists of a single term (n = 0), and which dominates the scattering for J ≈ J r . i.e. the terms step (J), is neglected. We then have from Equation (66), and using Equations (68) and (69), together with the change in notation of s/p → pole [sincẽ s step (J) is absent], the result We know from Equation (61) that the minimum of the first term is −1/ImJ 0 for J = ReJ 0 = J r . Inserting J = J r in the remaining terms gives −θ r R , so the overall result is −θ r R − 1/ImJ 0 , or equivalently, the rainbow angle is moved to θ r R = θ r R + 1/ImJ 0 . In this special case, the importance of the Regge pole contribution depends on the values of θ r R and 1/ImJ 0 . If θ r R >> 1/ImJ 0 then the scattering is mainly due to the phase,φ pole (J). Whereas, if θ r R ≈ 1/ImJ 0 , then the scattering has contributions from both the Regge pole andφ pole (J).
Our starting point for the three examples is the smooth-step parameterisation of Section 7. We know from the discussion given previously for Figures 2 and 3, that this case exhibits a rainbow in its dDCS, when there is no Regge-pole sum present. By adding a pole term(s), we can examine how a Regge pole(s) affects the rainbow angular scattering.
For the three examples, we use the same seven standard parameters already employed for the smoothstep parameterisation, namely, θ r R = 2π/9 rad = 40 • , J r = 60, J cut = 70, d cut = 3, as well as N = 0.03, a 0 = 0, a 1 = π together with J damp = 75, d damp = 5. We do not display the LAMs because the information they contain is consistent with the dDCS results.

Single Regge pole at J 0 = 60 + 10i
We first consider a simple example where there is a single Regge pole in the representation (67). For the partial residue, we use,ã 0 = 20 exp[iφ s/p (J = 60)] ≈ −18.794 − 6.840i. This value has been chosen so that the argument of the pole term merges smoothly withφ s/p (J), when the contribution from the pole term becomes significant. Our choice also ensures that the pole term makes a large contribution for J ≈ ReJ 0 . We note the following in Figures 7 and 8: • Figure 7(a) shows graphs of, |S s/p (J) |, |S step (J) | (i.e. the non-pole term) and |S pole (J) |, all versus J. It can be seen that the pole term dominates for, J ≈ ReJ 0 . In Figure 7(b) we also plot˜ step (J), which we see is mostly similar to˜ s/p (J). In particular, Figure 7   range, 0 • ≤ θ R ≤ 90 • . Note the dDCS usingS step (J) is taken from Figures 2(a) and 3(a). We see that the primary rainbow in our Regge pole representation has clearly shifted towards larger angles. From our discussion above, we know, that if the pole term dominates at J ≈ ReJ 0 , then θ r R changes by, ≈ (1/ImJ 0 ) rad, which in this case is ≈ 6 • . This is consistent with Figure 8(a) where the rainbow angle is, θ r R = 44.46 • , which is 4.46 • greater than, θ r R = 40.00 • . • In Figure 8(b), we show the N,F PWS (r = 1) analysis of the full PWS dDCS for the angular range, 0 • ≤ θ R ≤ 180 • . We see that the F dDCS possesses two supernumerary rainbows with peaks at θ R ≈ 13.3 • and θ R ≈ 23.6 • , as well as the primary rainbow with a peak at θ R ≈ 38.8 • . If we compare with the F PWS (r = 1) dDCS in Figure 2(b), we see that the presence of the Regge pole has pushed the primary and first supernumerary rainbow peaks to larger angles by about 4 • . Due to this pushing effect, a vaguely hintedat shoulder/oscillation in the F PWS (r = 1) dDCS in Figure 2(b) at θ R ≈ 10 • can now be identified as an incipient supernumerary rainbow, which moves to θ R ≈ 13.3 • , when the Regge pole is present. • We present our SC angular distributions in Figure 8(c) for the angular range, 0 • ≤ θ R ≤ 90 • . It can be seen that the full SC dDCS is in very good agreement with the full PWS dDCS in Figure 8(c). Note in Figure 8(d) that the F uAiry dDCS also possesses a primary rainbow and two supernumerary rainbows at similar angles to the F PWS (r = 1) dDCS. • We have also applied the CoroGlo test of Section 5.2.
It gives for the small-angle diffraction ratio a value of 5.3, which is not close to the forward glory ratio of 6.2. This simple test tells us that inclusion of the Regge pole term contributes to the forward angle scattering. This can also be inferred from Figure 7(a) where |S pole (J) | is significant on branches 1 and 2, close to the forward glory angular momentum value of J g 1 ≈ 34.

Single Regge pole at J 0 = 60 + 2i
Our second example has ImJ 0 = 2. The partial residue is chosen to be, 5 exp[iφ s/p (J = 48)] ≈ 1.277 − 4.834i, in order to produce a smooth argS s/p (J) curve, and to make the pole term significant for J ≈ ReJ 0 . We note the following in Figure 9 [properties ofS s/p (J),S step (J) andS pole (J)] and Figure 10 (dDCSs): • In Figure 9(a), we plot graphs of, |S s/p (J) |, |S step (J) | and |S pole (J) |, all versus J, where we see that the pole term again dominates for J ≈ ReJ 0 . • Figure 9(b) shows graphs of˜ s/p (J) and˜ step (J). Our discussion above suggests that the change in the rainbow angle will be approximately, 1/ImJ 0 = 0.5 rad ≈ 28.6 • , providedS pole (J) dominates over the nonpole contribution. We observe that θ r R ≈ 60.7 • . This means, θ r R = 40.0 • (= θ r R ) + 20.7 • , so the above estimate for the shift in the rainbow angle is useful.
We also see that the presence of the pole term only changes˜ s/p (J) compared to˜ step (J) for a small range of J values around J = 60, where a 'minitrench' or 'teat' is created. In particular, Figure 9(b) shows that˜ s/p (J) has a minimum value of −60.7 • (thus, θ r R = +60.7 • ) at J r = 60.04. In this case, we note that J r = 60.04 and J r = 60.00 are very close in value. Figure 9(c) plots (2J + 1)|S s/p (J) | versus J. Figure 9(b,c) together shows that the F SC subamplitude corresponds to 34 J 86.
• Logarithmic plots of the full and N,F dDCSs are displayed in Figure 10(a-d), all for the angular range, 0 • ≤ θ R ≤ 180 • . In Figure 10(a), we again make a comparison between the full PWS dDCS usingS s/p (J) (black solid curve) with the full PWS dDCS using S step (J) (black dashed curve). As discussed above, the primary rainbow angle, θ r R , has shifted towards a larger angle by about 21 • , when the pole term is included.
An important observation compared to the smooth-step result is that the position of the supernumerary rainbow has not changed much due to the addition of the Regge pole. In fact, the two curves in Figure 10(a) are almost indistinguishable for θ R 30 • . Note also that the oscillatory region in the dDCS extends to θ R = 180 • compared with θ R ≈ 90 • in Figure 8(a). • In Figure 10(b), we show the N,F PWS (r = 1) analysis of the full PWS dDCS. We see that the F dDCS possesses a supernumerary rainbow with a peak at θ R ≈ 20.7 • , as well as the primary rainbow with a peak at θ R ≈ 38.7 • . If we compare with the F PWS (r = 1) dDCS in Figure 2(b), we see that the Regge pole has pushed the primary rainbow peak to larger angles by about 4 • . • We present our SC angular distributions in Figure 10(c,d). It can be seen that the full SC dDCS agrees closely (except for θ R ≈ 40 • ) with the full PWS dDCS up to θ R ≈ 135 • . It then tends toward the N SC dDCS, and the full SC oscillations become damped. • The disagreement between the full SC and full PWS dDCSs at θ R ≈ 40 • can be understood by examining Figure 9(b). For˜ s/p (J) ≈ −40 • , there are two flat shoulders on either side of the mini-trench. This means the slopes, |˜ (J 2 (θ R )) | and˜ (J 3 (θ R )) are small; as a result σ 2 (θ R ) and σ 3 (θ R ) will become large in the uAiry approximation of Equation (49), which will then itself be too large. It may be possible to overcome this limitation by using uniform asymptotic cuspoid techniques beyond the uAiry approximation, i.e. the cusp, swallowtail, butterfly, etc. approximations [52,53]. However this is beyond the scope of the present paper. • We observe in Figure 10(d) that the F uAiry dDCS possesses a primary rainbow and a supernumerary rainbow at similar angles to the F PWS (r = 1) dDCS. Finally, we note that the F tAiry dDCS becomes up to approximately 10 times smaller than the F PWS (r = 1) dDCS for 135 • θ R 180 • . This is the reason for the less structured full SC dDCS in this angular range [in contrast, note that the N SC dDCS and N PWS (r = 1) dDCS are in good agreement].
• We have applied the CoroGlo test of Section 5.2. It gives for the small-angle diffraction ratio a value of 6.0, which is close to a forward glory ratio of 6.2. This tells us that inclusion of the Regge pole term contributes little to the forward angle scattering. This can also be inferred from Figure 9(a,b), where |S pole (J) | is relatively small on branches 1 and 2 close to the forward glory angular momentum value of, J g 1 ≈ 35.

Many Regge poles
Finally, in this section, we discuss our results for a parameterisation containing 6 regge poles, labelled, n = 0(1)5.
We use the standard parameters as given above, but we also need to choose values for {J n } and {ã n }. For the Regge pole positions, we use, J n = 60 + (10 + 2n)i n = 0, 1, 2, 3, 4, 5 Note that the {J n } are equally spaced on a straight line parallel to the ImJ axis in the CAM plane. This case is discussed further in Appendix. The partial residues are calculated using the following equation: where C n max n = n max !/[n!(n max − n)!] is the binomial coefficient; the corresponding numerical values for the {ã n } are reported in Table 1. Equation (71) can be broken into three components. The factor 5 × 10 3 is chosen so that the Regge pole contribution dominates for J ≈ ReJ n . The factor, (−1) n C 5 n , is used to maximise the pole contribution to˜ s/p (J), which we have discussed in Section 8.2 and the Appendix. The remaining factor, exp[iφ s/p (J = 40)], is used to ensure the smoothness of argS s/p (J).
We next consider Figures 11 and 12 and note the following: • Figure 11(a-c) shows plots of |S(J) |,˜ (J), (2J + 1) |S(J) | versus J respectively, with different choices for the subscripts, when affixed to these quantities. Note that the two terms (step and pole) contributing to |S s/p (J) | interfere destructively around J ≈  60(= J r ). In Figure 11(b), notice that˜ s/p (J) has a minimum value of −87.0 • at J r = 61.16. If we compare Figure 11(b) with Figure 9(b), we see that the minitrench in the many-pole case is deeper and wider. That is, the pole term now affects˜ s/p (J) over a wider range of J (from ≈ 50 to ≈ 90) than previously. These observations mean there are more partial waves contributing to the rainbow scattering, which favours a pronounced primary rainbow and a supernumerary rainbow(s) in the dDCS. • Logarithmic plots of the full and N,F dDCSs are displayed in Figure 12(a-d), all for the angular range, 0 • ≤ θ R ≤ 180 • . We observe pronounced primary and supernumerary rainbows in the full PWS dDCS plots.
In more detail, Figure 12(a) shows full PWS dDCSs constructed for the S matrix parameterisation with (black solid curve), and without (black dashed curve), Regge poles. It can be seen once more that inclusion of the poles pushes the primary rainbow to larger angles. In particular, θ r R has increased from 40.0 • in the nonpole dDCS to θ r R = 87.0 • in the pole-included dDCS. In contrast the peak of the first supernumerary rainbow in the pole-included full PWS dDCS at θ R ≈ 37 • is similar to the position of the peak of the primary rainbow in the non-pole full PWS dDCS.
We show the N,F (r = 1) analysis of the full PWS dDCS in Figure 12(b). The F dDCS has broad oscillations for θ R 90 • , which correspond to the primary plus first and second supernumerary rainbows.
Our SC results are plotted in Figure 12(c,d). It is clear that the uAiry approximation reproduces the overall shape of the full PWS dDCS, in particular the primary plus the first and second supernumerary rainbows, although the tAiry approximation causes the NF interference oscillations to be overestimated for θ R 100 • .
The reason for these discrepancies, compared to Figure 8(c) for example, is that the shape of the minitrench in Figure 11(b), together with its flat shoulders, means that the uAiry and tAiry approximations are less accurate.
• We have applied the CoroGlo test of Section 5.2. We find for the small-angle diffraction ratio a value of 6.0, which is close to a forward glory ratio of 6.2. This tells us that inclusion of the Regge pole terms contribute little to the forward angle scattering. This can be seen in Figure 12(a), where the with-poles and withoutpoles PWS dDCSs in Figure 12(a) are very similar for θ R 40 • . In addition, this result can be inferred in Figure 11(a,b), where |S pole (J) | is relatively small on branches 1 and 2 close to the forward glory angular momentum value of, J g 1 ≈ 34.

Conclusions
We have incorporated, for the first time, Regge poles and residues into wHSMP for a Legendre PWS. In particular, we used pole terms of the formã n /(J − J n ) and then investigated their effect on the properties of the S matrix, dDCSs and LAMs. We showed results for three examples: for a single pole (two examples) and for a many-pole example having 6 poles. Our focus was on understanding the behaviour of primary and supernumerary rainbows in the product angular distributions. We began with a simple S matrix parameterisation with no Regge poles, but which exhibits a rainbow in the dDCS. We then introduced Regge poles for the three examples into the S matrix and investigated their influence on the rainbow scattering. We find that inclusion of Regge poles 'pushes' the rainbow angle to larger values. It is clear there is much more research that should be carried out on this topic of Regge poles in PWS. Our results are also relevant to other recent work using CAM theory and rainbows [54][55][56][57][58][59][60][61][62][63][64][65][66].
We also analysed the scattering using Nearside-Farside (NF) PWS theory for the dDCSs and NF PWS LAM theory, including up to three resummations of the PWS. We also applied the recently introduced 'CoroGlo' test, which lets us distinguish between glory and corona scattering close to the forward direction. In addition, we applied full and NF asymptotic (SC) rainbow theories to the PWS -in particular, the uniform Airy and transitional Airy approximations for the farside scattering. This let us prove that structure in the no-pole and with-pole DCSs are indeed examples of primary and supernumerary rainbows, as well as the presence of other interference effects.

Disclosure statement
No potential conflict of interest was reported by the author(s).