Regge pole description of scattering of scalar and electromagnetic waves by a Schwarzschild black hole

Regge pole description of scattering of scalar and electromagnetic waves
by a Schwarzschild black hole

Antoine Folacci folacci@univ-corse.fr Equipe Physique Théorique, SPE, UMR 6134 du CNRS et de l’Université de Corse,
Université de Corse, Faculté des Sciences, BP 52, F-20250 Corte, France
   Mohamed Ould El Hadj med.ouldelhadj@gmail.com Equipe Physique Théorique, SPE, UMR 6134 du CNRS et de l’Université de Corse,
Université de Corse, Faculté des Sciences, BP 52, F-20250 Corte, France
Consortium for Fundamental Physics, School of Mathematics and Statistics, University of Sheffield,
Hicks Building, Hounsfield Road, Sheffield S3 7RH, United Kingdom
July 20, 2019
Abstract

We revisit the problem of plane monochromatic waves impinging upon a Schwarzschild black hole from complex angular momentum techniques. We focus more particularly on the differential scattering cross sections associated with scalar and electromagnetic waves. We provide an exact representation of these cross sections by replacing the discrete sum over integer values of the angular momentum which defines their partial wave expansions by a background integral in the complex angular momentum plane plus a sum over the Regge poles of the -matrix. We show that, surprisingly, the background integral is numerically negligible and, as a consequence, that the cross sections can be reconstructed, for arbitrary frequencies and scattering angles, in terms of Regge poles with a very good agreement. We show in particular that, for large values of the scattering angle, a small number of Regge poles permits us to describe the black-hole glory and that, by increasing the number of Regge poles, we can reconstruct very efficiently the differential scattering cross sections for small and intermediate scattering angles and therefore describe the orbiting oscillations. In fact, the sum over Regge poles allows us to extract by resummation the physical information encoded in the partial wave expansion defining a differential scattering cross section and, moreover, to overcome the difficulties linked to its lack of convergence due to the long-range nature of the fields propagating on the black hole.

I Introduction

Studies concerning the scattering of waves by black holes (BHs) are mainly based on partial wave expansions (see, e.g., Ref. Futterman et al. (2012)). This is due to the high degree of symmetry of the BH spacetimes usually considered and physically or astrophysically interesting. For example, the Schwarzschild BH is a static spherically symmetric solution of the vacuum Einstein’s equations while the Kerr BH is a stationary axisymmetric solution of these equations with, as a consequence, the separability of wave equations on these gravitational backgrounds (see, e.g., Ref. Frolov and Novikov (1998)). Even if the approach based on partial wave expansions is natural and very effective in the context of scattering of waves by BHs, it presents some flaws. Due to the long-range nature of the fields propagating on a BH, some partial wave expansions encountered are formally divergent (see, e.g., Ref. Futterman et al. (2012)) and, moreover, it is in general rather difficult to interpret physically the results described in terms of partial wave expansions. These problems can be overcome by using complex angular momentum (CAM) techniques (analytic continuation of partial wave expansions in the CAM plane, effective resummations involving the poles of the -matrix in the CAM plane, i.e., the so-called Regge poles, and the associated residues, semiclassical interpretations of Regge pole expansions, etc.). Such techniques, which proved to be very helpful in quantum mechanics (see, e.g., Refs. de Alfaro and Regge (1965); Newton (1982)), in electromagnetism and optics (see, e.g., Refs. Watson (1918); Sommerfeld (1949); Newton (1982); Nussenzveig (1992); Grandy (2000)), in acoustics and seismology (see, e.g., Refs. Überall (1992); Aki and Richards (2002)) and in high energy physics (see, e.g., Refs. Gribov (2003); Collins (1977); Barone and Predazzi (2002); Donnachie et al. (2005)) to describe and analyze resonant scattering are now used in the context of BH physics (see, e.g., Refs. Andersson and Thylwe (1994); Andersson (1994); Decanini et al. (2003); Glampedakis and Andersson (2003); Decanini and Folacci (2010, 2009); Dolan and Ottewill (2009); Decanini et al. (2010, 2011a, 2011b, 2011c); Dolan et al. (2012); Macedo et al. (2013); Nambu and Noda (2016); Folacci and Ould El Hadj (2018); Benone et al. (2018)).

In this article we revisit the problem of plane monochromatic waves impinging upon a Schwarzschild BH from CAM techniques. More precisely, we focus on the differential scattering cross sections associated with scalar and electromagnetic waves. It should be recalled that the partial wave expansions of these cross sections have been obtained a long time ago by Matzner Matzner (1968) for the scalar field and by Mashhoon Mashhoon (1973, 1974) and Fabbri Fabbri (1975) for the electromagnetic field and that many additional works have since been done which have theoretically and numerically completed these first investigations (see, e.g., Refs. Sanchez (1976, 1978); Matzner et al. (1985); Anninos et al. (1992); Andersson and Thylwe (1994); Andersson (1994, 1995); Glampedakis and Andersson (2001); Dolan (2008a); Crispino et al. (2009) for some articles directly relevant to our own study). Here, we construct an exact representation of these differential scattering cross sections by replacing the discrete sum over integer values of the angular momentum which defines their partial wave expansions by a background integral in the CAM plane plus a sum over the Regge poles of the -matrix. Surprisingly, we find that the background integral is numerically negligible and, as a consequence, that the differential scattering cross sections can be described in terms of Regge poles with a very good agreement for arbitrary frequencies and scattering angles. In fact, the sum over Regge poles allows us to extract by resummation the physical information encoded in the partial wave expansion defining a differential scattering cross section and, moreover, to overcome the difficulties linked to its lack of convergence due to the long-range nature of the fields propagating on the BH. We show in particular that, for large values of the scattering angle, i.e., in the backward direction, a small number of Regge poles permits us to describe the BH glory (see Refs. Matzner et al. (1985) and Andersson (1994) for semiclassical interpretations) and that, by increasing the number of Regge poles, we can reconstruct very efficiently the differential scattering cross sections for small and intermediate scattering angles and therefore describe the “orbiting” oscillations (see Refs. Anninos et al. (1992) and Andersson (1994) for semiclassical interpretations).

It is important to point out that our work extends (and corrects) the studies of Andersson and Thylwe Andersson and Thylwe (1994); Andersson (1994) where we can find the first application of CAM techniques in BH physics. In Ref. Andersson and Thylwe (1994), Andersson and Thylwe have considered the scattering of scalar waves by a Schwarzschild BH from a theoretical point of view and adapted the CAM formalism to this problem. They have established some properties of the Regge poles and of the -matrix in the CAM plane. In Ref. Andersson (1994), Andersson has used this formalism to interpret semiclassically the BH glory and the orbiting oscillations. He has, in particular, considered “surface waves” propagating close to the unstable circular photon (graviton) orbit at , i.e., near the so-called photon sphere, and associated them with the Regge poles. It is interesting to note that we have developed this point of view in a series of papers establishing that the complex frequencies of the weakly damped quasinormal modes (QNMs) are Breit-Wigner-type resonances generated by the surface waves previously mentioned. We have then been able to construct semiclassically the spectrum of the QNM complex frequencies from the Regge trajectories, i.e., from the curves traced out in the CAM plane by the Regge poles as a function of the frequency Decanini et al. (2003); Decanini and Folacci (2010); Folacci and Ould El Hadj (2018) establishing on a “rigorous” basis the physically intuitive interpretation of the Schwarzschild BH QNMs suggested, as early as 1972, by Goebel Goebel (1972) (see Refs. Decanini and Folacci (2009); Dolan and Ottewill (2009); Decanini et al. (2010, 2011a) for the extension of these results to other BHs and to massive fields). Moreover, from the Regge trajectories and the residues of the greybody factors, we have described analytically the high-energy absorption cross section for a wide class of BHs endowed with a photon sphere and explained its oscillations in terms of the geometrical characteristics (orbital period and Lyapunov exponent) of the null unstable geodesics lying on the photon sphere Decanini et al. (2011b, c, a). All these results highlight the interpretive power of CAM techniques in BH physics.

In this article, we consider not only the case of the scalar field but, in addition, we treat the important case of the electromagnetic field which is of fundamental interest with the emergence of multimessenger astronomy as well as with the possibility to produce experimentally, in a near future, the first BH image or, more precisely, to “photograph” with the Event Horizon Telescope the shadow of the supermassive BH at the Galactic Center Falcke et al. (2000); Castelvecchi (2017). Moreover, we correct the results numerically obtained by Andersson in Ref. Andersson (1994) for the residues associated to the Regge poles (see also Ref. Glampedakis and Andersson (2003)) and we show that the background integral in the CAM plane is numerically negligible. It is important to recall that, in Refs. Andersson and Thylwe (1994); Andersson (1994), Andersson and Thylwe tried to extract some physical information of this background integral and that, in electromagnetism and optics Watson (1918); Sommerfeld (1949); Newton (1982); Nussenzveig (1992); Grandy (2000) as well as in acoustics and seismology Überall (1992); Aki and Richards (2002)), it can be associated semiclassically with incident and reflected rays. In the context of scattering by BHs (or, more precisely, if we focus on differential scattering cross sections), it plays only a minor role.

Our paper is organized as follows. In Sec. II, by means of the Sommerfeld-Watson transform Watson (1918); Sommerfeld (1949); Newton (1982) and Cauchy’s theorem, we construct exact CAM representations of the differential scattering cross sections for plane scalar and electromagnetic waves impinging upon a Schwarzschild BH from their partial wave expansions. These CAM representations are split into a background integral in the CAM plane and a sum over the Regge poles of the -matrix involving the associated residues. In Sec. III, we obtain numerically the Regge poles of the -matrix and the associated residues and we show that the background integral can be numerically neglected. This permits us to reconstruct, for various frequencies of the impinging waves, the differential scattering cross sections of the BH from the Regge pole sum with a very good agreement. In the Conclusion, we summarize our main results and briefly consider possible extensions of our work. In an Appendix, we discuss the numerical evaluation of the background integrals. Due to the long-range nature of the fields propagating on a Schwarzschild BH, these integrals in the CAM plane (as the partial wave expansions) suffer of a lack of convergence. We overcome this problem, i.e., we accelerate their convergence, by extending to integrals the iterative method developed in Ref. Yennie et al. (1954) for partial wave expansions.

Throughout this article, we adopt units such that . We furthermore consider that the exterior of the Schwarzschild BH is defined by the line element where and is the mass of the BH while , , and are the usual Schwarzschild coordinates. We finally assume a time dependence for the plane monochromatic waves considered.

Ii Differential scattering cross sections for scalar and electromagnetic waves, their CAM representations and their Regge pole approximations

In this section, we recall the partial wave expansions of the differential scattering cross sections for plane monochromatic scalar and electromagnetic waves impinging upon a Schwarzschild BH and we construct exact CAM representations of these cross sections by means of the Sommerfeld-Watson transform Watson (1918); Sommerfeld (1949); Newton (1982) and Cauchy’s theorem. These CAM representations are split into a background integral in the CAM plane and a sum over the Regge poles of the -matrix involving the associated residues.

ii.1 Partial wave expansions of differential scattering cross sections

We recall that, for the scalar field, the differential scattering cross section is given by Matzner (1968)

(1)

where

(2)

denotes the scattering amplitude and that, for the electromagnetic field, the differential scattering cross section can be written in the form Mashhoon (1973, 1974) (see also Refs. Fabbri (1975); Crispino et al. (2009))

(3)

where the scattering amplitude is given by

(4)

with

(5)

and

(6a)
(6b)

The expression (4)-(6) takes into account the two polarizations of the electromagnetic field. In Eqs. (2) and (5), the functions are the Legendre polynomials Abramowitz and Stegun (1965). We also recall that the -matrix elements appearing in Eqs. (2) and (5) can be defined from the modes solutions of the homogenous Regge-Wheeler equation

(7)

(here denotes the tortoise coordinate) where

(8)

(here corresponds to the scalar field and to the electromagnetic field) which have a purely ingoing behavior at the event horizon (i.e., for )

(9a)
and, at spatial infinity (i.e., for ), an asymptotic behavior of the form
(9b)

In this last equation, the coefficients and are complex amplitudes and we have

(10)

ii.2 CAM representation of the scattering amplitude for scalar waves

By means of the Sommerfeld-Watson transformation Watson (1918); Sommerfeld (1949); Newton (1982) which permits us to write

(11)

for a function without any singularities on the real axis, we can replace in Eq. (2) the discrete sum over the ordinary angular momentum by a contour integral in the complex plane (i.e., in the complex plane with ). By noting that , we obtain

(12)

In Eqs. (11) and (II.2), the integration contour encircles counterclockwise the positive real axis of the complex plane, i.e., we take with . We can recover (2) from (II.2) by using Cauchy’s residue theorem and by noting that the poles of the integrand in (II.2) that are enclosed into are the zeros of , i.e., the semi-integers with . It should be recalled that, in Eq. (II.2), the Legendre function of first kind denotes the analytic extension of the Legendre polynomials . It is defined in terms of hypergeometric functions by Abramowitz and Stegun (1965)

(13)

In Eq. (II.2), denotes “the” analytic extension of . It is given by [see Eq. (10)]

(14)

where the complex amplitudes and are defined from the analytic extensions of the modes , i.e., from the function solution of the problem (7)-(9) where we now replace by . With the deformation of the contour in mind, it is important to note the symmetry property

(15)

of the -matrix which can be easily obtained from its definition (see also Ref. Andersson and Thylwe (1994)). It is also important to note that the poles of in the complex plane (i.e., the Regge poles) lie in the first and third quadrants of the CAM plane symmetrically distributed with respect to the origin of this plane. The poles lying in the first quadrant can be defined as the zeros with of the coefficient [see Eq. (14)]. They therefore satisfy

(16)

We now deform the contour in Eq. (II.2) in order to collect, by using Cauchy’s residue theorem, the contributions of the Regge poles lying in the first quadrant of the CAM plane (for more details, see, e.g., Ref. Newton (1982)). This must be done very carefully and, in particular, we must deal with the contributions coming from the quarter circles at infinity with great caution. We first note that

(17)

vanishes rapidly for and , that

(18)

vanishes rapidly for and and that

(19)

diverging for and , it is more convenient to replace it by

(20)

where the first term vanishes rapidly for and . In Eq. (II.2) we have introduced the Legendre function of the second kind which satisfies Abramowitz and Stegun (1965)

(21)

Then, by using (15) and the relation Abramowitz and Stegun (1965)

(22)

we finally obtain

(23)

where

(24a)
with
(24b)
and
(24c)

is a background integral contribution [here we have with and we can note that the path of integration defining the background integral (24a) is a continuous one running down first the positive imaginary axis and then running along , i.e., slightly below the positive real axis] and where

(25)

is a sum over the Regge poles lying in the first quadrant of the CAM plane. In Eq. (II.2) we have introduced the residue of the matrix at the pole defined by

(26)

Of course, Eqs. (23), (24) and (II.2) provide an exact representation of the scattering amplitude for the scalar field equivalent to the initial partial wave expansion (2). From this CAM representation, we can extract the contribution given by (II.2) which, as a sum over Regge poles, is only an approximation of , and which provides us with an approximation of the differential scattering cross section (1).

ii.3 CAM representation of the scattering amplitude for electromagnetic waves

In order to obtain the CAM representation of the scattering amplitudes (4) and (5), we use the Sommerfeld-Watson transformation in the form

(27)

Here with because we must take into account that the sum over defining the scattering amplitude (5) begins at . We then obtain

(28)

With the collect of Regge poles in mind, the contour is not as convenient as the contour used in Sec. II.2. However, it is possible to move to the left so that it coincides with but we then introduce a spurious double pole at (i.e., at ) corresponding to the term . It is necessary to remove the associated residue contribution and we obtain

(29)

By using the definition (13) and by noting that, for the electromagnetic field, we have formally [see Eqs. (7)-(9)], we can write

(30)

By now noting that

(31)

we can finally write

(32)

Here we have dropped terms which do not contribute to the scattering amplitude .

We now deform the contour in Eq. (II.3) in order to collect, by using Cauchy’s residue theorem, the Regge pole contributions. This is achieved by following, mutatis mutandis, the approach of Sec. II.2. We obtain

(33)

where

(34a)
with
(34b)
and
(34c)

is a background integral contribution and where

(35)

is a sum over the Regge poles lying in the first quadrant of the CAM plane.

Iii Reconstruction of differential scattering cross sections from Regge pole sums

In this section, we compare numerically the partial wave expansions of the differential scattering cross sections with their CAM representations or, more precisely, with their Regge pole approximations in order to highlight the benefits of working with Regge pole sums.

iii.1 Computational methods

In order to construct numerically the scattering amplitudes (2) and (4), the background integrals (24b), (24c), (34) and (34) as well as the Regge pole sums (II.2) and (II.3):

  1. We have to solve the problem (7)-(9) permitting us to obtain the function , the coefficients and and the -matrix elements . This must be achieved (i) for and as well as (ii) for and .

  2. We have to determine for the Regge poles , i.e., the solutions of (16) and to obtain the corresponding residues (26).

All these numerical results can be obtained by using, mutatis mutandis, the methods that have permitted us to provide in Ref. Folacci and Ould El Hadj (2018) a description of gravitational radiation from BHs based on CAM techniques (see, in particular, Secs. IIIB and IVA of this previous paper). It is moreover important to note that, due to the long-range nature of the fields propagating on a Schwarzschild BH, the scattering amplitudes (2) and (4) and the background integrals (24b) and (34) suffer of a lack of convergence [this is not the case for the background integrals (24c) and (34) because their integrands vanish exponentially as ]. In the Appendix, we explain how to overcome this problem, i.e., how to accelerate their convergence by employing an iterative method, the number of iterations being chosen to obtain stable numerical results. It should be noted that we have performed all the numerical calculations by using Mathematica Inc. ().

iii.2 Results and comments

In Figs. 1-3, we focus on the scalar field and we compare the differential scattering cross section (1) constructed from the partial wave expansion (2) with its Regge pole approximation obtained from the Regge pole sum (II.2). In Figs. 4-6 we focus on the electromagnetic field and we compare the differential scattering cross section (3) constructed from the partial wave expansion (4)-(6) with its Regge pole approximation obtained from the Regge pole sum (II.3). The comparisons are achieved for the reduced frequencies 1, 3 and 6 and, for these frequencies, we have displayed the lowest Regge poles and the associated residues in Table 1 (for the scalar field ) and in Table 2 (for the electromagnetic field). The higher Regge poles and their residues that have been necessary to obtain some of the results displayed in Figs. 1-6 are available upon request from the authors.

We can observe in Figs. 1-6 that the Regge pole approximation involving a small number of Regge poles permits us to describe very well the cross sections for intermediate and large values of the scattering angle and, in particular, the BH glory. Taking into account additional Regge poles improves the Regge pole approximation and we can see that, by summing over a large number of Regge poles, the whole scattering cross section is impressively described, this being valid even for small scattering angles.

It is important to note that it is not necessary to take into account the background integrals in order to reproduce the differential scattering cross sections. In fact, we can numerically obtain these contributions and observe that they are completely negligible for intermediate and large scattering angles. It seems they begin to play a role only for small angles. In Table 3, we have considered, for the electromagnetic field at , the various contributions to the CAM representation (33). We can see that, for the scattering angles and , the background integrals, although not completely negligible, play a minor role. It would be interesting to check if this remains valid even for scattering angles but, due to numerical instabilities when , we are not currently able to provide such a result.

Table 1: The lowest Regge poles for the scalar field and the associated residues . We assume .
Table 2: The lowest Regge poles for the electromagnetic field and the associated residues . We assume .
Figure 1: Scalar cross section of a Schwarzschild BH for and its Regge pole approximation.
Figure 2: Scalar cross section of a Schwarzschild BH for and its Regge pole approximation.
Figure 3: Scalar cross section of a Schwarzschild BH for and its Regge pole approximation.
Figure 4: Electromagnetic cross section of a Schwarzschild BH for and its Regge pole approximation.
Figure 5: Electromagnetic cross section of a Schwarzschild BH for and its Regge pole approximation.
Figure 6: Electromagnetic cross section of a Schwarzschild BH for and its Regge pole approximation.
Electromagnetic field at ()
(n=1,…, 60)
(n=1,…, 90)
(n=1,…, 90)
(n=1,…, 90)
Table 3: We compare, for the electromagnetic field at (), the exact value of the scattering amplitude (4) with the sum over Regge poles (II.3) and we highlight the minor role of the background integral contributions (34) and (34a).

Iv Conclusion

In this article, we have considered the scattering of scalar and electromagnetic waves by a Schwarzschild BH by focusing on the associated differential scattering cross sections and we have shown that these cross sections can be reconstructed, for arbitrary frequencies and scattering angles, in terms of Regge poles with great precision. This is really surprising and certainly due to the fact that BHs are very particular physical object. Indeed, in quantum mechanics de Alfaro and Regge (1965); Newton (1982), electromagnetism and optics Watson (1918); Sommerfeld (1949); Newton (1982); Nussenzveig (1992); Grandy (2000) and acoustics Überall (1992), a Regge pole sum alone is never sufficient to reconstruct a differential scattering cross section. From the Regge pole sum, we have been able to describe with a very good agreement not only the BH glory occurring in the backward direction but also the orbiting oscillations appearing on the differential scattering cross sections for small and intermediate scattering angles. Moreover, it is important to note that working with Regge pole sums has permitted us to overcome the difficulties linked to the lack of convergence of the partial wave expansions defining the cross sections which are due to the long-range nature of the fields propagating on a Schwarzschild background.

Here, it is important to recall that the CAM approach to scattering is usually combined with asymptotic methods in order to provide semiclassical interpretations of the phenomenons observed. This is the case in quantum mechanics de Alfaro and Regge (1965); Newton (1982), in electromagnetism and optics Watson (1918); Sommerfeld (1949); Newton (1982); Nussenzveig (1992); Grandy (2000) as well as in in acoustics and seismology Überall (1992); Aki and Richards (2002) where the association “surface wave-Regge pole” is of fundamental interest. This is also the case in BH physics where the Regge poles can be associated with surface waves trapped on the BH photon spheres (see, e.g., Refs. Andersson (1994); Decanini et al. (2003); Decanini and Folacci (2010); Dolan and Ottewill (2009); Decanini et al. (2010)). This point of view, which is valid in the high-frequency limit and has permitted us to understand the existence of the weakly damped QNMs as well as to describe analytically the high-energy absorption cross section for a wide class of BHs endowed with a photon sphere Decanini et al. (2011b, c, a), is not adopted in the present article. The CAM approach provides an exact representation of partial wave expansions valid for arbitrary frequencies and we have based our study on this point of view (see also our recent article Folacci and Ould El Hadj (2018) where we have previously adhered to such a point of view). In other words, we have here chosen to discard that part of the CAM machinery involving asymptotic methods. As a consequence, we have renounced to provide physical interpretations of the results obtained with, in return, impressive agreements between exact calculations and Regge pole approximations.

We hope in next works to extend our study to scattering of gravitational waves by a Schwarzschild BH as well as to scattering of waves by a Kerr BH (see Refs. Matzner and Ryan (1977, 1978); Handler and Matzner (1980); Glampedakis and Andersson (2001); Dolan (2008b, a) for articles concerning these two topics and which could serve as departure points for such works).

Acknowledgements.
MOEH wishes to thank Sam Dolan for discussions and for his kind invitation to the University of Sheffield where this work was completed.

*

Appendix A Iterative method to accelerate the convergence of the background integrals

Due to the long-range nature of the fields propagating on a Schwarzschild BH, the background integrals (24b) and (34) suffer of a lack of convergence. In this Appendix we explain how to overcome this problem, i.e., how to accelerate their convergence. In fact, the same problem occurs for the partial wave expansions (2), (5) and (4) (see, e.g., Refs. Anninos et al. (1992) where the case of the scalar field is discussed). For these partial wave expansions, the convergence can be obtained by employing an iterative method introduced a long time ago in the context of Coulomb scattering by Yennie, Ravenhall and Wilson Yennie et al. (1954) (see also Refs. Dolan (2008a); Crispino et al. (2009) where this method is used in the context of scattering by BHs and see, e.g., Ref. Anninos et al. (1992) for another approach based on the truncation of partial wave expansions and on the matching of the Schwarzschild -matrix elements with the Newtonian ones). In this Appendix, we briefly recall this iterative method because we use it in Sec. III in order to accelerate the convergence of the partial wave expansions (2) and (5) and we generalize it in order to deal with the background integrals (24b) and (34).

a.1 Acceleration of the convergence of partial wave expansions

In order to improve the convergence of the partial wave expansion

(36)

[see Eqs. (2) and (5)] we introduce the associated “reduced” series Yennie et al. (1954)

(37)

defined by

(38)

It should be noted that, by re-expressing the series (36) in the form (38), we isolate the pathological behavior of the partial wave expansions (2) and (5) occurring for . By using now the recursion relation Abramowitz and Stegun (1965)

(39)

in the form

(40)

we can show from (36)-(38) that the coefficients can be expressed in terms of the coefficients . We have, for ,

(41)

(here we take ). As noted in Ref. Yennie et al. (1954), we have for large values of ,

(42)

This explicitly shows that using the reduced series (37) greatly improves the convergence of the initial partial wave expansion (36).

a.2 Acceleration of the convergence of background integrals

In order to improve the convergence of the background integral

(43)

[see Eqs. (24b) and (34)], we first split it in the form

(44)

where

(45)

The choice of the truncation parameter will be discussed later. We then introduce the “reduced” integrals defined by

(46)

By using the relation

(47)

which is a consequence of the definition (II.2) and of the relation Abramowitz and Stegun (1965)

(48)

we can show that these reduced integrals can be written in the form

(49)

where is an integral over a finite integration domain which can be expressed in terms of the integral and where the function can be expressed in terms of the function . We have