Quantum Rutherford scattering and the scattering of classical waves by black holes have similar formal structures and can be studied using the same mathematical techniques. In both contexts, the long-range nature of the interaction leads to a divergent total cross section, which has been interpreted and regularized in various ways in the past literature. We review in detail the origin of this divergence, in both real and multipole spaces, and show that it arises from the incorrect use of approximations outside their domain of validity. We also stress that although black hole and quantum Rutherford scattering share the same formalism, the natures of the associated physical observables differ. We comment on the role of interference: while interference may be safely neglected in the context of quantum Rutherford scattering (due to the fact that the observable quantity is a flux, and the incoming flux is collimated), it should not be neglected in the context of classical waves scattered by a black hole, where one expects to see a superposition of transmitted and scattered waves in a broad region downstream from the target and a cross section is not connected to any physically observable quantity.

## I. INTRODUCTION

*m*, frequency

*ω*, and energy $ E = \u210f \omega $ scattering off a potential

*V*is

*θ*relative to the incoming wave direction as the flux of the current of the scattered waves in a surface $ d S$ normalized to the initial flux,

*r*is the radial distance from the scattering potential.

In many physical situations the potential in Eq. (1) has poles (*r* = 0 in the case of the Coulomb/Newtonian potential). As a consequence, the first derivative of the solution cannot be continuous at the pole location but the solution may nonetheless be well-defined and continuous. Also, when the potential does not fall off rapidly enough, as is the case for the Coulomb potential on which we focus hereafter, the cross section diverges in the forward direction. Exact solutions for such Rutherford scattering^{1} problems do actually exist and have been known for a long time.^{2–9} Since these exact solutions are well-defined everywhere in space, they allow one to track back the origin of the divergence of approximate solutions.

We discuss some usual methods used in the literature in order to regularize or justify Rutherford-like $ \theta \u2192 0$ divergences. Those methods become, strictly speaking, unnecessary once the exact solution is considered, and they misleadingly suggest that $ r \u2212 1$ scattering problems are ill-defined. We provide a pedagogical example stressing the importance of correctly stating the regime of validity of an approximate solution in order to correctly draw a physical interpretation from it. We also show that great care has to be taken in the inversion of limits to avoid misinterpreting anomalous behaviors of results that are in fact simply a consequence of an incorrect extrapolation of the approximate solution itself.

This paper is structured as follows. First, in Sec. II, we briefly review standard concepts in scattering problems and apply the usual procedure to compute the scattering cross section of Rutherford scattering. We show that the amplitude of the resulting scattered wave is divergent in the forward direction (hence the cross section has the same type of divergence). In Sec. III A, we share the exact solution to the Rutherford scattering equation and discuss its behavior, noting that it is well-defined (not divergent) everywhere in space. For an incoming wave function with wave vector $k$, we then identify a variable $ k r ( 1 \u2212 cos \u2009 \theta )$ that allows us to split the total solution in terms of incoming and outgoing waves as in Eq. (2) thanks to the properties of the exact solution. The “large-*r*” limit appears as an approximation valid only for large values of this dimensionless variable. As a consequence, for any given radial distance *r*, one can define a small *θ* region around the forward direction inside which the expansion of Eq. (2) is not valid. Hence, the Rutherford formula for the scattering cross section only holds outside this region. The *θ* = 0 divergence of the usually reported Rutherford result arises from the extrapolation of the approximate solution in a regime where it never holds. The exact solution of Rutherford scattering can also be obtained using a multipolar decomposition, and this is introduced in Sec. IV. Finally, Sec. V turns to a different problem: scattering of classical waves off Coulomb-like potentials. Mathematically the equation governing this scattering problem is very similar to the Rutherford equation describing quantum scattering,^{1} but we stress some caveats of this formal analogy. In particular, we show that a definition of a classical cross section in this context does not correspond to any observable quantity and can be misleading when trying to interpret physical effects in this classical framework.

## II. REVIEW OF RUTHERFORD SCATTERING

*ψ*contains an incident plane wave incoming along the e

_{z}-axis,

*A*with value $ A = Q 1 Q 2 / 4 \pi \u03f5 0$ in the Rutherford case of a particle of electric charge

*Q*

_{1}and target charge

*Q*

_{2},

*ϵ*

_{0}being the vacuum permittivity. Defining

In the standard analysis, one considers the problem in a regime where the waveform can be split into an incoming plane wave and an outgoing spherical wave as in Eq. (2). This regime is commonly referred to as the $ r \u2192 + \u221e$ limit (however, as we will show in Sec. III, this definition induces misconceptions when considering the infinitesimal scattering angle limit).

*f*is obtained by solving Eq. (1), with proper boundary conditions. Since the potential in Eq. (5) is spherically symmetric, and having chosen that the incoming wave travels along the azimuthal direction

*z*, it is clear by symmetry that

*f*depends only on the angle

*θ*, i.e., $ f = f ( \theta )$.

*ψ*is given by

*f*by

*r*limit on the wave solution and extracted the scattered wave from the total wave before computing the current. Indeed $ J [ \psi ] \u2212 J [ \psi in ] \u2260 J [ \psi scat ]$, where the difference arises from the presence of interference terms. The relevance of these terms will be discussed at the end of Sec. III E.

*r*limit. Here, we provide the result based on the Born approximation. We define $k$ as the initial wavevector and $ k \u2032$ as the wavevector of the scattered wave. The scattered wave is spherical by assumption, so $ k \u2032 = | k \u2032 | e r$. We assume that the scattering potential does not absorb momentum (elastic collision, fixed target) so that $ | k | = | k \u2032 | = k$. Defining $ q \u2261 k \u2212 k \u2032$, the Born approximation yields

^{10–12}

*θ*is the angle between $k$ and $ k \u2032$. For the $ r \u2212 1$ Coulomb potential of our problem, it is obvious that the integral in Eq. (14) is, strictly speaking, ill-defined. This fact should not be interpreted as an ill definition of wave scattering problems in $ r \u2212 1$ potentials, but only as the inapplicability of the Born approximation to the latter case.

*μ*= 0 in the latter result, one obtains

*classical Rutherford scattering cross section*, which is usually obtained from the conservation of energy, angular momentum, and Runge–Lenz vectors.

This cross section is divergent for $ \theta \u2192 0$. In the literature, different explanations have been provided to justify the presence of such a divergence. This divergence is often claimed to be artificial due to the unphysicality of the Coulomb potential itself because of screening.^{11,13,14} For example, in situations with many charges at finite temperature, the potential of a point charge is suppressed due to Debye–Hückel screening^{15} and a Yukawa potential is one possible screening parametrization. Equation (15) is then regular as $ \theta \u2192 0$ for $ \mu > 0$. While being physically motivated, this argument relies on a particular screening model and misleadingly suggests that the pure Coulomb potential model leads to divergences, a statement which Sec. III A shows not to be true. It is important to emphasize this when dealing with formally similar $ r \u2212 1$ classical wave scattering (off black holes) for which there is no immediate analogous regularization.

We also stress that in the classical particle computation of Rutherford scattering, the divergence is present and has been addressed in the literature. In this case, there exists a well-defined map between deflection angle and impact parameter *b*, and considering $ \theta defl \u2192 0$ means considering $ b \u2192 \u221e$.^{16,17} In other words, only particles passing infinitely far away from the potential are undeflected, and precisely those give divergent contributions to the cross section. Thus, in this case, the divergence issue is alleviated by arguing that in a concrete experimental setup, $ b \u2192 \u221e$ never happens, hence neither does $ \theta defl \u2192 0$. Moreover, typically the classical situation is such that there are multiple scattering centers (e.g., in a classical gold foil model), and there is a maximum value for *b*, given by half of the inter atomic distance. This provides a lower bound on $ \theta defl$.^{17} However, one cannot rely on this classical analogy when considering scattering of waves, as a definition of impact parameter is ambiguous in this case.

In Sec. III, we show that the problem of Rutherford scattering for waves is actually inherently free of divergence: the divergence is just an artifact of the fact that we take the $ \theta \u2192 0$ limit after having taken the large distance limit (and the two limits do not commute). It follows that all the arguments mentioned above presented to justify or deal with the Rutherford divergence are rather unnecessary, if not formally artificial, when discussing scattering by a pure Coulomb potential.

## III. FROM EXACT SOLUTION TO THE RUTHERFORD SOLUTION

### A. Exact solution

^{2–9}

*imposing*that, in the absence of interactions, it reduces to an incoming plane wave $ e i k . z$, in such a way that it matches the desired asymptotic behavior from Eq. (4). The derivation of Eq. (17) may be found in Sec. A of the supplementary material.

### B. Approximate solution

To find the *Rutherford solution*, one usually expands the function $ 1 F 1$ for large values of its argument, i.e., $ \rho s \u226b 1$. In other words, for a fixed geometry and wave frequency (*r* and *k*), we identify an angular region $ s \u226a 1 / ( r k )$ and compute the approximate solution valid asymptotically *outside* this region.

*q*:

*q.*

^{18,19}In our case, $ q = i s$, implying that the positive sign has to be chosen.

^{19}In the latter expression, $ ( x ) k$ is the Pochhammer symbol, which can be defined as $ ( x ) k \u2261 \Gamma ( x + k ) / \Gamma ( x )$. It follows that at large values of $ \rho s = k ( r \u2212 z ) = \rho ( 1 \u2212 cos \u2009 \theta )$, using Eq. (20) with $ q = i s \rho $ in Eq. (19), one finds at leading order

*θ*= 0. In this equation, one usually identifies an incident component (first term) and a scattered spherical wave (second term). The tilde symbol has been used to stress that this is an asymptotic approximate solution, and to distinguish it from the exact one. We observe that in Eq. (21) the correction to the incoming wave proportional to $ \gamma 2$ can be relevant for large values of

*γ*, and can be seen as a backreaction of the scattered wave on the incoming wave (when going beyond the Born approximation lowest order). Hence, we define the distorted incoming plane wave as

Notice that the incoming wave is distorted from a pure plane wave by the presence of the interaction, which gives the appearance of logarithmic phase shifts. A similar statement can be made for the phase corrections to the scattered wave. Logarithmic corrections to the phase are a specificity of $ 1 / r$ potentials.^{20} For the latter, the radial decay of the potential is too slow to have standard plane waves as solutions of the wave equation. Even at asymptotic radial distance, where we expect the scattered waves to become vanishingly small in amplitude, these phase corrections remain present as part of the incoming wave. A logarithmic feature in the phase is also present in the context of scattering of classical waves off a $ 1 / r$ potential, as is briefly discussed in Sec. V.

We further notice that the approximate solution diverges in the scattered part for $ \theta \u2192 0$, unlike the exact solution (19), as Fig. 1 illustrates. It clearly follows that the divergence in the cross section is a direct consequence of using the approximate solution (21) outside its regime of validity.

*Rutherford solution*of Eq. (21) found by expanding the exact solution for $ \rho s \u226b 1$, one identifies the scattering amplitude $ f ( \theta )$ as the

*r*-independent constant multiplying the (distorted) spherical term

*ρs*is asymptotically large, which clearly excludes the region

*θ*= 0. Experimental conditions, however, typically satisfy $ \rho s \u226b 1$,

^{21}ensuring the validity of the usual cross section result of Eq. (16), even in an ideal case without screening.

*ρs*, the extra logarithmic phase in Eq. (21) does not affect the currents defining the cross section. To be more precise, if $ \psi in \u2261 e i k z + i \gamma \u2009 ln \u2009 k ( r \u2212 z )$, the associated current is given as

### C. Small angle behavior

*r*limit and then faces a singularity at

*θ*= 0 to the one in which we look at the exact solution for $ \rho s \u226a 1$, which includes $ \theta \u2192 0$ and arbitrary

*r*. From the expansion

*θ*= 0 direction, we have exactly

*s*= 0 and the solution reduces to

For small angles, the functional form of the total solution does not show a natural splitting between an incoming-like and a scattered-like term, but rather a full superposition of the two, corresponding to the forward direction interference. As a result, there is no natural notion of scattering differential cross section in the $ \rho s \u226a 1$ region, because $ d \sigma / d \Omega $ relies on the existence of well-defined incoming and scattered currents. However, the total current associated with the full solution (19) remains well defined and divergence-free everywhere in space, and smoothly transitions between the small and large angle regimes. Currents are further discussed in Sec. III E.

### D. Interpretation of the forward direction

The large *ρs* limit can always be fulfilled for every $ \theta \u2208 ( 0 , \pi ]$. Indeed, in this case we have *s* > 0, such that there always exists a distance *r* large enough to satisfy $ \rho s \u226b 1$.

Given that $ \rho s = k ( r \u2212 z )$ by definition, writing *r* in Cartesian coordinates explicitly allows to rewrite the condition $ \rho s = 1$ as $ k ( z + 1 / 2 k ) = 1 2 k 2 ( x 2 + y 2 )$, which defines a paraboloid of revolution. The transverse section of the paraboloid increases as we go toward larger positive *z*, but its angular size, seen from the origin, decreases. The limit $ \rho s \u226b 1$ corresponds to the region sufficiently far outside this paraboloid.

*z*, we follow a point on the wavefront identified by a set of coordinates (

*x*,

*y*). Such a wavefront point will

*enter*the paraboloid once

*x*,

*y*) that one is following, and is located at a further distance from the scattering center for larger $ | x | , | y |$. We illustrate this in Fig. 2.

We consider the exact solution of the Rutherford problem, Eq. (17), and we look at the evolution of a point in the (*x*, *z*) plane. As Eq. (21) shows, the exact solution is normalized such that *ψ* is a traveling wave with unit amplitude when $ z \u2192 \u2212 \u221e$. However, sufficiently far inside the paraboloid, Eq. (28) shows that *ψ* is oscillating with reduced amplitude $ \u223c e \u2212 \pi \gamma / 2 | \Gamma ( 1 + i \gamma ) |$ which tends to 0 for $ \gamma \u226b 1$. This corresponds to a damped-wave regime. This can be understood to mean that the initial plane wave has transferred a part of its initial magnitude to the scattered wave outside the paraboloid (or inside the paraboloid if $ \gamma < 0$). The paraboloid marks the transition between the two asymptotic regimes.

### E. Outgoing current and interference

*ρ*. In the middle panel of Fig. 3, we plot the various contributions to the radial component of the current given in Eq. (30), while in the bottom panel we compare the outgoing current obtained from the exact solution to the approximated one. In the bottom panel of Fig. 3, we compare the radial part of $ J out$ and $ J \gamma 2 \u200a out$.

We note that using the distorted plane wave of Eq. (22), which includes a $ \gamma 2$ correction, to define $ J \gamma 2 \u200a out$ makes the latter closer to $ J \u0303 \u200a scat$ in the sense that it removes a small oscillatory pattern. This pattern is due to the interference between the $ \gamma 2$ correction to the incoming wave and the outgoing part in $ J \u200a out$.

*δ*

_{0}is a phase shift defined by $ e 2 i \delta 0 = \Gamma ( 1 + i \gamma ) / \Gamma ( 1 \u2212 i \gamma )$. Though $ J \u0303 \xd7 , r$ decays as $ 1 / r$, and thus slower than $ J \u0303 scat , r$, its angular dependence is oscillatory due to the argument of the last factor. This oscillatory behavior is manifested in the middle panel of Fig. 3. The oscillation length (in the orthoradial direction) is typically

The interference contribution may be discarded for a further reason that is not specific to Rutherford scattering but rather lies in the scattering formalism itself. While it is standard to solve the Schrödinger equation with an incoming plane wave condition and find the associated scattered wave as in Eq. (2), the experimentally relevant quantities are wave packets. The steps passing from plane wave to wave packet scattering are carried out in Ref. 22, where it is shown that the scattering cross is identical in both contexts (the argument is explicitly outlined for short range potentials and subsequently generalized to the Coulomb case). An important point is that, while in the context of plane waves, interference may be expected at every point in space due to the overlap of the incoming and scattered parts, such an interference does not occur for wave packets of finite support. A region of interference may still exist around the forward direction, but only in a very restricted angular range of $ \theta \u2208 [ 0 , 10 \u2212 3 ]$ for a collimated beam with a 1 mm transverse size and detector placed at 1 m from the target.^{22}

This is to be contrasted with the case of classical wave scattering discussed in Sec. V, where a long wavelength signal may be of macroscopic size and, thus, produce a large region of interference, as do classical waves on a surface of water.

## IV. RUTHERFORD IN $\u2113$ SPACE

*ψ*as

*ρ*, where large means typically $ \rho \u226b \u2113 ( \u2113 + 1 ) + \gamma 2$ since the expansion of Eq. (20) is typically valid when $ | ( b \u2212 a ) ( 1 \u2212 a ) / q | \u226a 1$ and $ | a ( a \u2212 b + 1 ) / q | \u226a 1$. After some algebra, we get

*γ*= 0 (implying $ \delta \u2113 = 0$, as well as $ \rho = \rho c$). The result is $ C \u0303 ( \u2113 , \gamma = 0 ) = 2 \u2113 + 1$, which can be generalized for $ \gamma \u2260 0$ to

*ρ*instead of

_{c}*ρ*in the exponent. This reflects the fact that the scattering problem does not have solutions in the exact form of a superposition of a plane wave and a spherical one, but instead has logarithmic modifications in the phases with respect to the latter, as we have seen in Eq. (21).

In the supplementary material, we compute the scattering amplitude using Eqs. (47) and (39). The multipole summation is nontrivial, but we rigorously prove it to lead to the unique result of Eq. (23).

## V. CLASSICAL WAVE SCATTERING

Problems of scattering classical waves off matter or spacetime structures have the same mathematical form as quantum Rutherford scattering.^{5,7,8} It follows that several mathematical techniques developed in the framework of quantum scattering can be imported in this context, with a few caveats.

*M*is the black hole mass. The speed of light and the gravitational constant are conventionally set to $ c = 1 = G$ here. The problem is static and spherically symmetric, and we assume to have a monochromatic wave with frequency

*ω*. As in Sec. II, we can choose our coordinate system with the polar axis aligned with the propagation direction of the incoming wave. With this choice, the problem depends only on the polar angle

*θ*, hence we can introduce the decomposition

*u*for simplicity. It can be verified that

^{4}

In the literature, this analogy with a quantum scattering problem is often pushed forward, and a differential cross section is defined as the ratio of the outgoing and incoming flux. While this is a perfectly well-defined object from a mathematical point of view, it does not directly correspond to a physical observable for classical waves. Indeed, when considering scattering of classical waves, the quantity that we measure is often related to the wave itself rather than a cross section. Moreover, as we have seen, when introducing a cross section we implicitly neglect the effect of interference between the incoming and outgoing wave. While this is well-justified in the context of a quantum scattering experiment, in the context of the scattering of a classical wave, there is no reason to assume that interference is small because the detector is typically smaller than the wavelength. On the contrary, the wave observed coming out of the scattering will be the superposition of a transmitted wave and a diffused one, and the two will interfere in an extended region after the scattering center.

## VI. DISCUSSION AND CONCLUSIONS

In this paper, we have reviewed, with a pedagogical approach, the problem of quantum scattering off a Coulomb-like potential. We have shown that in real space it is possible to find an exact solution, and we have identified the regime of validity of the so-called *Rutherford solution*, used in the computation of the scattering cross section. We have stressed that the divergence of the scattering cross section in the forward direction is a consequence of extrapolating the Rutherford solution outside its regime of validity. As a consequence, there is no need to invoke physical mechanisms to explain such a divergence, which is a pure mathematical artifact. We have also presented Rutherford scattering in multipole space and made apparent the similarities that this problem shares with the scattering of a classical scalar wave by a black hole.

Finally, we have commented on the role of interference and explained that, while in a quantum context interference effects can be legitimately neglected, this is not the case in the context of classical scattering of waves off mass singularities. This leads to some important differences between the problems of quantum and classical scattering. Indeed, while from a formal point of view, the problem of classical scattering is very similar to a quantum scattering in $\u2113$-space, the physically observable quantities in the two contexts are not the same. For a Rutherford-like quantum scattering, it is a flux (incoming and outgoing) that is observed. Interference can be neglected since in any observational setting the incoming beam is typically sufficiently well collimated. As a consequence, the notion of a cross section can be introduced, and it corresponds to an observable quantity. However, in a classical context, what may be observed is a waveform, which consists of a superposition of transmitted and scattered waves in a broad region after the target. In this case, the notion of a cross section does not correspond to any physical observable; while, of course, one can formally introduce and compute it, some care has to be taken when assigning it a physical meaning and when using it to describe measurable effects.

## SUPPLEMENTARY MATERIAL

Please click on this link to access the supplementary material, which includes details on the derivation of the exact Rutherford scattering solution, and a rigorous treatment of the multipole expansion, at https://doi.org/10.60893/figshare.ajp.c.7241929.

## ACKNOWLEDGMENTS

The authors thank Ruth Durrer, Djibril Ben Achour, and Guillaume Faye for discussions. M.P. and G.C. acknowledge support from the Swiss National Science Foundation (Ambizione grant, *Gravitational wave propagation in the clustered universe*). The work of G.C., C.P., and J-P.U. is founded by CNRS.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

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