Relativistic Coulomb Scattering

“Relativistic Coulomb scattering” can mean a scalar or spinor wave in a fixed electrostatic potential, an exact external-field scattering solution, or a process with a recoiling dynamical target. These problems share a Coulomb tail but have different amplitudes and approximation parameters. This page derives the scalar leading Born result, compares it with the Mott result, and identifies the long-range structure that neither result removes.

Required background. Mott Scattering supplies the spinor result; Klein–Gordon Theory in External Potentials supplies the scalar equation; Coulomb Scattering owns the nonrelativistic long-range scattering construction. Helpful background. The Klein–Gordon Coulomb Problem explains the point-origin boundary condition.

Scalar electrostatic Born scattering

Use $\hbar=c=1$, $m>0$, and a positive incoming energy $E>m$. Let $p=\sqrt{E^2-m^2}$, $\beta=p/E$, and $V(r)=g/r$, with signed $g=qQ/(4\pi)$. The source is fixed and infinitely heavy. Writing the scalar field as $e^{-iEt}\phi(\mathbf x)$ gives

$$(\nabla^2+p^2)\phi=(2EV-V^2)\phi.$$

The outgoing Helmholtz kernel

$$G_p^{(+)}(\mathbf r)=\frac{e^{ipr}}{4\pi r}$$

obeys $(\nabla^2+p^2)G_p^{(+)}=-\delta^3$. The minus source therefore gives the integral equation

$$\phi(\mathbf x)=e^{i\mathbf p\cdot\mathbf x} -\int d^3y\,G_p^{(+)}(\mathbf x-\mathbf y) [2EV(\mathbf y)-V(\mathbf y)^2]\phi(\mathbf y).$$

For this short-range integral construction, first screen or regularize the Coulomb potential. Its unscreened scattering limit requires the long-range phases discussed below. At leading order in $g$, replace the wave under the integral by the incident plane wave and omit $V^2$, which is second order. The large-$r$ outgoing coefficient is

$$f_{\rm KG}^{(1)}(\theta) =-\frac{E}{2\pi}\widetilde V(\mathbf p'-\mathbf p).$$

Taking the screening length to infinity at fixed $\theta\ne0$ gives

$$f_{\rm KG}^{(1)} =-\frac{2Eg}{|\mathbf p'-\mathbf p|^2} =-\frac{g}{2p\beta\sin^2(\theta/2)}.$$

The asymptotic incoming and outgoing kinetic energies are the same, so their KG flux prefactors cancel. With this amplitude normalization, $d\sigma_{\rm KG}^{(1)}/d\Omega=|f_{\rm KG}^{(1)}|^2$. The negative sign in the amplitude follows from the source equation, even though it disappears from this leading cross section.

At the next amplitude order one must include both the explicit $-V^2$ term in the differential equation and the iteration of $2EV$. Keeping only one of those contributions does not define the second-order KG Coulomb amplitude.

What spin changes at leading order

For equal $p,\beta,g$, define the scalar baseline

$$B(\theta)= \left[\frac{g}{2p\beta\sin^2(\theta/2)}\right]^2.$$

Then the leading unpolarized results are

$$\begin{aligned} \frac{d\sigma_{\rm KG}^{(1)}}{d\Omega}&=B(\theta),\\ \frac{d\sigma_{\rm Dirac}^{(1)}}{d\Omega} &=B(\theta)\left[1-\beta^2\sin^2(\theta/2)\right]. \end{aligned}$$

The spinor trace producing the second line belongs to Mott Scattering. Both results approach Rutherford at low speed. At high speed the Dirac factor suppresses large-angle scattering; the scalar result has no corresponding helicity-overlap factor.

For a central potential the spin-half scattering operator can be parameterized, after flux normalization, as $f(\theta)I+i h(\theta)\boldsymbol\sigma\cdot\widehat{\mathbf n}$, where $\widehat{\mathbf n}$ is normal to the scattering plane. Its unpolarized differential cross section is $|f|^2+|h|^2$. Relative complex phases also control polarization observables. Thus matching a single unpolarized angular curve does not establish equality of the scalar and spinor scattering operators.

In SI units replace $g$ by $\zeta=qQ/(4\pi\epsilon_0\hbar c)$ and restore $\hbar^2$ in the squared prefactor: $B=[\zeta\hbar/(2p\beta\sin^2(\theta/2))]^2$. Here $p$ is physical momentum.

The common logarithmic phase at large distance

Separate the scalar solution as $\phi=u_\ell(r)Y_{\ell m}/r$. Its exact exterior radial equation is

$$u_\ell''+ \left[p^2-\frac{2Eg}{r} +\frac{g^2-\ell(\ell+1)}{r^2}\right]u_\ell=0.$$

At large $r$, the local radial wave number has expansion $p-Eg/(pr)+O(r^{-2})$. Integrating fixes the leading outgoing phase,

$$u_\ell(r)\ \propto\ \exp\!\left\{i\left[pr-\eta\ln(2pr)+\delta_\ell\right]\right\}, \qquad \eta=\frac{Eg}{p}=\frac g\beta,$$

up to subleading asymptotic corrections and an incoming component. The additive constant inside the logarithm can be absorbed into the phase convention. A repulsive potential has positive $\eta$ in the present signed convention.

The leading long-range Dirac radial phase has the same $\eta$: its local energy is also $E-V$, while spin-dependent exterior terms enter below the $1/r$ energy term. Its channel phases and full amplitudes are nevertheless different.

The logarithm does not approach a constant as $r\to\infty$. Consequently an exact unscreened solution cannot simply be compared with a free plane wave plus an unmodified spherical wave at infinity. The accepted Coulomb Scattering page develops the modified asymptotic convention, Coulomb phases, and screened-limit caveats.

Why the Born baseline is not an exact relativistic formula

The scalar radial equation contains $g^2/r^2$. Even before considering spin, the relativistic problem is not obtained by replacing the nonrelativistic mass by $E$ in every exact Coulomb formula. For example its local exponents satisfy

$$\nu(\nu-1)=\ell(\ell+1)-g^2, \qquad \nu_\pm=\frac12\pm\sqrt{(\ell+\tfrac12)^2-g^2}.$$

They differ from $\ell+1$ and $-\ell$. An exact partial-wave construction must specify the point-origin condition and the long-range normalization as well as solve the radial equation. Strong-coupling or finite-size problems cannot silently inherit a weak-coupling point-source boundary choice.

The nonrelativistic Coulomb amplitude happens to have the same Rutherford magnitude as its leading Born expression, while retaining a nontrivial Coulomb phase. This special magnitude equality neither proves convergence of a Born amplitude expansion nor extends automatically to the relativistic scalar and spinor problems.

For weak $|g|/\beta$ and fixed nonforward angle, the displayed Born comparison is useful. At large coupling, near the forward limit, or when polarization phases matter, a controlled calculation must specify additional treatment. The unscreened total cross section diverges in both leading cases.

External field or dynamical target

An exact solution of a one-particle external-field equation resums interactions with the prescribed potential. It still does not include target recoil, target excitation, a dynamical photon field, or radiative corrections. Those are distinct physical additions.

For a target initially at rest, energy conservation in a dynamical elastic process reads

$$E+M_{\rm target} =E'+\sqrt{M_{\rm target}^2+|\mathbf k|^2}.$$

Thus $E'=E$ is only the infinite-mass limit. At large but finite mass, the leading recoil energy is $|\mathbf k|^2/(2M_{\rm target})$. Its importance depends on momentum transfer and the desired accuracy. Replacing $m$ by a reduced mass inside the Mott formula does not derive the relativistic two-body flux, spin structure, or recoil kinematics.

Exercises

1. Derive the scalar Born amplitude for $V(r)=g e^{-\mu r}/r$ and take its fixed-angle unscreened limit.

Solution

$\widetilde V=4\pi g/(\mathbf k^2+\mu^2)$ gives $f^{(1)}=-2Eg/(\mathbf k^2+\mu^2)$. At $\theta\ne0$, $\mu\downarrow0$ gives the stated Coulomb amplitude. Taking the forward limit first instead produces a dependence on $\mu$ that becomes singular as screening is removed.

2. Why is a small pointwise ratio $|V|/E$ at large radius insufficient to neglect the Coulomb phase?

Solution

The local wave-number correction is proportional to $1/r$. Its integral grows as $\ln r$. A term that is small locally can therefore accumulate an unbounded phase over an infinite propagation distance. Cross sections and phases have different accuracy tests.

3. In the scalar radial equation, find the leading small-$g$ shift of the exponent $\nu_+$.

Solution

Expanding the square root gives $\nu_+=\ell+1-g^2/(2\ell+1)+O(g^4)$ at fixed $\ell$. The change begins at second order and is invisible in the leading linear Born amplitude.

References

• Bjorken, James D., and Sidney D. Drell. Relativistic Quantum Mechanics. McGraw–Hill, 1964. External-field Coulomb scattering.
• Greiner, Walter. Relativistic Quantum Mechanics: Wave Equations. 3rd edition, Springer, 2000. doi:10.1007/978-3-662-04275-5. Scalar and spinor Coulomb equations.
• Taylor, John R. Scattering Theory: The Quantum Theory of Nonrelativistic Collisions. Wiley, 1972; Dover reprint, 2006. Long-range Coulomb scattering and asymptotic conventions.