Bhabha scattering

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Feynman diagrams
Annihilation
Scattering

In quantum electrodynamics, Bhabha scattering is the electron-positron scattering process:

e^+ e^- \rightarrow e^+ e^-

There are two leading-order Feynman diagrams contributing to this interaction: an annihilation process and a scattering process. The Bhabha scattering rate is used as a luminosity monitor in electron-positron colliders.

Bhabha scattering is named after Indian physicist Homi J. Bhabha.

Contents

The differential cross section for this process is

\frac{d \sigma}{d (\cos\theta)} = \frac{\pi \alpha^2}{s} \left( u^2 \left( \frac{1}{s} + \frac{1}{t} \right)^2 + \left( \frac{t}{s} \right)^2 + \left( \frac{s}{t} \right)^2 \right) \,
where s,t, and u are the Mandelstam variables, and
\alpha \, is the fine-structure constant.

This cross section is calculated neglecting the electron mass relative to the collision energy and including only the contribution from photon exchange. This is a valid approximation at collision energies small compared to the mass scale of the Z boson, about 91 GeV; for energies not too small compared to this mass, the contribution from Z boson exchange also becomes important.

In this article, the Mandelstam variables are defined by

s= \, (k+p)^2= \, (k'+p')^2 \approx \, 2 k \cdot p \approx\, 2 k' \cdot p' \,         Image:Mandelstam01.png
t= \, (k-k')^2= \, (p-p')^2\approx \, -2 k \cdot k' \approx \, -2 p \cdot p' \,
u= \, (k-p')^2= \, (p-k')^2\approx \, -2 k \cdot p' \approx \, -2 k' \cdot p \,

Where the approximations are for the high-energy (relativistic) limit.

Both diagrams contribute to the transition matrix element. By letting k and k' represent the four-momentum of the positron, while letting p and p' represent the four-momentum of the electron, and by using Feynman rules one can show the following diagrams give these matrix elements:

Where we use:
\gamma^\mu \, are the Gamma matrices,
u, \ \mathrm{and} \ \bar{u}\, are the four-component spinors for fermions, while
v, \ \mathrm{and} \ \bar{v}\, are the four-component spinors for anti-fermions (see Four spinors).
(scattering) (annihilation)
\mathcal{M} = \, -e^2 \left( \bar{v}_{k} \gamma^\mu v_{k'} \right) \frac{1}{(k-k')^2} \left( \bar{u}_{p'} \gamma_\mu u_p \right) +e^2 \left( \bar{v}_{k} \gamma^\nu u_p \right) \frac{1}{(k+p)^2} \left( \bar{u}_{p'} \gamma_\nu v_{k'} \right)

Notice that there is a relative sign difference between the two diagrams.

To calculate the cross section, one must average over the spins of the incoming particles (se- and se+) and sum over the spins of the outgoing particles. That is,

\overline{|\mathcal{M}|^2} \, = \frac{1}{(2s_{e-} + 1)(2 s_{e+} + 1)} \sum_{\mathrm{spins}} |\mathcal{M}|^2 \,
= \frac{1}{4} \sum_{s=1}^2 \sum_{s'=1}^2 \sum_{r=1}^2 \sum_{r'=1}^2 |\mathcal{M}|^2 \,

First, calculate |\mathcal{M}|^2 \,:

|\mathcal{M}|^2 \, = e^4 \left| \frac{(\bar{v}_{k} \gamma^\mu v_{k'} )( \bar{u}_{p'} \gamma_\mu u_p)}{(k-k')^2} \right|^2 \, (scattering)
- 2 e^4 \left( \frac{ (\bar{v}_{k} \gamma^\mu v_{k'} )( \bar{u}_{p'} \gamma_\mu u_p)}{(k-k')^2} \right)^* \left( \frac{ (\bar{v}_{k} \gamma^\nu u_p )( \bar{u}_{p'} \gamma_\nu v_{k'}) }{(k+p)^2} \right) \, (interference)
+ e^4 \left| \frac{(\bar{v}_{k} \gamma^\nu u_p )( \bar{u}_{p'} \gamma_\nu v_{k'} )}{(k+p)^2} \right|^2 \, (annihilation)

|\mathcal{M}|^2 \, = \frac{e^4}{(k-k')^4} \Big( (\bar{v}_{k} \gamma^\mu v_{k'} )( \bar{u}_{p'} \gamma_\mu u_p) \Big)^* \Big( (\bar{v}_{k} \gamma^\mu v_{k'})( \bar{u}_{p'} \gamma_\mu u_p) \Big) \,      (1) \,
= \frac{e^4}{(k-k')^4} \Big( (\bar{v}_{k} \gamma^\mu v_{k'} )^* ( \bar{u}_{p'} \gamma_\mu u_p)^* \Big) \Big( (\bar{v}_{k} \gamma^\mu v_{k'})( \bar{u}_{p'} \gamma_\mu u_p) \Big) \,      (2) \,
(complex conjugate will flip order)
= \frac{e^4}{(k-k')^4} \Big( \left(\bar{v}_{k'} \gamma^\mu v_{k} \right) \left( \bar{u}_{p} \gamma_\mu u_{p'} \right) \Big) \Big( \left( \bar{v}_{k} \gamma^\mu v_{k'} \right) \left( \bar{u}_{p'} \gamma_\mu u_p \right) \Big) \,      (3) \,
(move terms that depend on same momentum to be next to each other)
= \frac{e^4}{(k-k')^4} \left( \bar{v}_{k'} \gamma^\mu v_{k} \right) \left( \bar{v}_{k} \gamma^\mu v_{k'} \right) \left( \bar{u}_{p} \gamma_\mu u_{p'} \right) \left( \bar{u}_{p'} \gamma_\mu u_p \right) \,      (4) \,

Next, we'd like to sum over spins of all four particles. Let s and s' to be the spin of the electron and r and r' to be the spin of the positron.

\frac{e^4}{(k-k')^4} \sum_{\mathrm{spins}} |\mathcal{M}|^2 \, = \left(\sum_{r'} \bar{v}_{k'} \gamma^\mu (\sum_{r}v_{k} \bar{v}_{k}) \gamma^\mu v_{k'} \right) \left(\sum_{s} \bar{u}_{p} \gamma_\mu (\sum_{s'}{u_{p'} \bar{u}_{p'}}) \gamma_\mu u_p \right) \,      (5) \,
= \left( \Big(\sum_{r'} v_{k'} \bar{v}_{k'} \Big) \gamma^\mu \Big(\sum_{r}v_{k} \bar{v}_{k} \Big) \gamma^\mu \right) \left( \Big(\sum_{s} u_p \bar{u}_{p} \Big) \gamma_\mu \Big( \sum_{s'}{u_{p'} \bar{u}_{p'}} \Big) \gamma_\mu \right) \,      (6) \,
(now use Completeness relations)
=\operatorname{Tr}\left( (k\!\!\!/' - m) \gamma^\mu (k\!\!\!/ - m) \gamma^\nu \right) \cdot \operatorname{Tr}\left( (p\!\!\!/' + m) \gamma_\mu (p\!\!\!/ + m) \gamma_\nu \right) \,      (7) \,
(now use Trace identities)
=\left(4 \left( {k'}^\mu k^\nu - \mathbf{k' \cdot k}\eta^{\mu\nu} + k'^\nu k^\mu \right) + 4 m^2 \eta^{\mu\nu} \right) \left( 4 \left( {p'}_\mu p_\nu - \mathbf{p' \cdot p}\eta_{\mu\nu} + p'_\nu p_\mu \right) + 4 m^2 \eta_{\mu\nu} \right) \,      (8) \,
=32\left( (k' \cdot p') (k \cdot p) + (k' \cdot p) (k \cdot p') -m^2 p' \cdot p - m^2 k' \cdot k + 2m^4 \right) \,      (9) \,

Now that is the exact form, but often one makes the approximation that these particles are moving at high momentum so that the three terms involving mass may be neglected so:

\frac{1}{4} \sum_{\mathrm{spins}} |\mathcal{M}|^2 \, = \frac{32e^4}{4(k-k')^4} \left( (k' \cdot p') (k \cdot p) + (k' \cdot p) (k \cdot p') \right) \,
(use the Mandelstam variables in this relativistic limit)
=\frac{8e^4}{t^2} \left(\tfrac{1}{2} s \tfrac{1}{2}s + \tfrac{1}{2}u \tfrac{1}{2} u \right) \,
= 2 e^4 \frac{s^2 +u^2}{t^2} \,

The process for finding the annihilation term is similar to the above, just have to swap momentums, so that

\frac{1}{4} \sum_{\mathrm{spins}} |\mathcal{M}|^2 \, = \frac{32e^4}{4(k+p)^4} \left( (k \cdot k') (p \cdot p') + (k' \cdot p) (k \cdot p') \right) \,
=\frac{8e^4}{s^2} \left(\tfrac{1}{2} t \tfrac{1}{2}t + \tfrac{1}{2}u \tfrac{1}{2} u \right) \,
= 2 e^4 \frac{t^2 +u^2}{s^2} \,

It is not an easy process, but in the end

\frac{\overline{|\mathcal{M}|^2}}{2e^4} = \frac{u^2 + s^2}{t^2} + \frac{2 u^2}{st} + \frac{u^2 + t^2}{s^2} \,

The completeness relations for the four-spinors u and v are

\sum_{s=1,2}{u^{(s)}_p \bar{u}^{(s)}_p} = p\!\!\!/ + m \,
\sum_{s=1,2}{v^{(s)}_p \bar{v}^{(s)}_p} = p\!\!\!/ - m \,
where
p\!\!\!/ = \gamma^\mu p_\mu \,      (see Feynman slash notation)
\bar{u} = u^{\dagger} \gamma^0 \,

Main article: Trace identities

To simplify the trace of the Dirac gamma matrices, one must use trace identities. Three used in this article are:

  1. The Trace of any product of an odd number of \gamma_\mu \,'s is zero
  2. \operatorname{tr} (\gamma^\mu\gamma^\nu) = 4\eta^{\mu\nu}
  3. \operatorname{Tr}\left( \gamma_\rho \gamma_\mu \gamma_\sigma \gamma_\nu \right) = 4 \left( \eta_{\rho\mu}\eta_{\sigma\nu}-\eta_{\rho\sigma}\eta_{\mu\nu}+\eta_{\rho\nu}\eta_{\mu\sigma} \right) \,

Using these two one finds that, for example,

\operatorname{Tr}\left( (p\!\!\!/' + m) \gamma_\mu (p\!\!\!/ + m) \gamma_\nu \right) \, = \operatorname{Tr}\left( p\!\!\!/' \gamma_\mu p\!\!\!/ \gamma_\nu \right) + \operatorname{Tr}\left(m \gamma_\mu p\!\!\!/ \gamma_\nu \right) \,
         + \operatorname{Tr}\left( p\!\!\!/' \gamma_\mu m \gamma_\nu \right) + \operatorname{Tr}\left(m^2 \gamma_\mu \gamma_\nu \right) \,
(the two middle terms are zero because of (1))
= \operatorname{Tr}\left( p\!\!\!/' \gamma_\mu p\!\!\!/ \gamma_\nu \right) + m^2 \operatorname{Tr}\left(\gamma_\mu \gamma_\nu \right) \,
(use identity (2) for the term on the right)
= {p'}^{\rho} p^\sigma \operatorname{Tr}\left( \gamma_\rho \gamma_\mu \gamma_\sigma \gamma_\nu \right) + m^2 \cdot 4\eta_{\mu\nu} \,
(now use identity (3) for the term on the left)
= {p'}^{\rho} p^\sigma 4 \left( \eta_{\rho\mu}\eta_{\sigma\nu}-\eta_{\rho\sigma}\eta_{\mu\nu}+\eta_{\rho\nu}\eta_{\mu\sigma} \right) + 4 m^2 \eta_{\mu\nu} \,
=4 \left( {p'}_\mu p_\nu - \mathbf{p' \cdot p}\eta_{\mu\nu} + p'_\nu p_\mu \right) + 4 m^2 \eta_{\mu\nu} \,

Bhabha scattering has been used as a luminosity monitor in a number of e+e- collider physics experiments. The accurate measurement of luminosity is necessary for accurate measurements of cross sections.

  • Small-angle Bhabha scattering was used to measure the luminosity of the 1993 run of the Stanford Large Detector (SLD), with a relative uncertainty of less than 0.5%.[1]
  • Electron-positron colliders operating in the region of the low-lying hadronic resonances (about 1 GeV to 10 GeV), such as the Beijing Electron Synchrotron (BES) and the Belle and BaBar "B-factory" experiments, use large-angle Bhabha scattering as a luminosity monitor. To achieve the desired precision at the 0.1% level, the experimental measurements must be compared to a theoretical calculation including next-to-leading-order radiative corrections.[2] The high-precision measurement of the total hadronic cross section at these low energies is a crucial input into the theoretical calculation of the anomalous magnetic dipole moment of the muon, which is used to constrain supersymmetry and other models of physics beyond the Standard Model.

  • Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2. 
  • Peskin, Michael E.; Schroeder, Daniel V. (1994). An Introduction to Quantum Field Theory. Perseus Publishing. ISBN 0-201-50397-2. 


electron | positron | photon
self-energy | vacuum polarization | vertex function
Gupta-Bleuler formalism | ξ gauge | Ward-Takahashi identity
Compton scattering | Bhabha scattering | Møller scattering
anomalous magnetic dipole moment
bremsstrahlung | positronium
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