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{{short description|Electron-positron scattering}}{| border="1" cellpadding="5" cellspacing="0" align="right" style="width:220px; text-align:justify"! style="background:#ffdead;" | Feynman diagrams

Annihilation220px

Scattering220px

In quantum electrodynamics, Bhabha scattering is the electron-positron scattering process: There are two leading-order Feynman diagrams contributing to this interaction: an annihilation process and a scattering process. Bhabha scattering is named after the Indian physicist Homi J. Bhabha.The Bhabha scattering rate is used as a luminosity monitor in electron-positron colliders.

Differential cross section

To leading order, the spin-averaged differential cross section for this process is where s,t, and u are the Mandelstam variables, alpha is the fine-structure constant, and theta is the scattering angle.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; at higher energies the contribution from Z boson exchange also becomes important.

Mandelstam variables

In this article, the Mandelstam variables are defined by (k+p)^2= ,(k'+p')^2 approx ,2 k cdot p approx, 2 k' cdot p' ,        

t= ,	(k-k')^2= ,	(p-p')^2approx ,| -2 k cdot k' approx ,| -2 p cdot p' ,

u= ,	(k-p')^2= ,	(p-k')^2approx ,| -2 k cdot p' approx ,| -2 k' cdot p ,

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

Deriving unpolarized cross section

Matrix elements

Both the scattering and annihilation 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: 160pxGamma matrices,u, mathrm{and} bar{u}, are the four-component spinors for fermions, whilev, mathrm{and} bar{v}, are the four-component spinors for anti-fermions (see Dirac equation#Four spinor>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.

Square of matrix element

To calculate the unpolarized cross section, one must average over the spins of the incoming particles (se- and se+ possible values) and sum over the spins of the outgoing particles. That is, mathcal{M}|^2 ,|

mathcal{M}|^2 ,

First, calculate |mathcal{M}|^2 ,: frac{(bar{v}_{k} gamma^mu v_{k'} )( bar{u}_{p'} gamma_mu u_p)}{(k-k')^2} right|^2 , (scattering)||{}- 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^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)

|

frac{(bar{v}_{k} gamma^nu u_p )( bar{u}_{p'} gamma_nu v_{k'} )}{(k+p)^2} right|^2 ,	(annihilation)

Scattering term (t-channel)

Magnitude squared of M

||= 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^nu v_{k'})( bar{u}_{p'} gamma_nu 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^nu v_{k'} right) left( bar{u}_{p'} gamma_nu 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^nu v_{k'} right) left( bar{u}_{p} gamma_mu u_{p'} right) left( bar{u}_{p'} gamma_nu u_p right) ,|     (4) ,

Sum over spins

Next, we'd like to sum over spins of all four particles. Let s and s' be the spin of the electron and r and r' be the spin of the positron. ||= frac{e^4}{(k-k')^4}operatorname{Tr}left( Big(sum_{r'} v_{k'} bar{v}_{k'} Big) gamma^mu Big(sum_{r}v_{k} bar{v}_{k} Big) gamma^nu right) operatorname{Tr} left( Big(sum_{s} u_p bar{u}_{p} Big) gamma_mu Big( sum_{s'}{u_{p'} bar{u}_{p'}} Big) gamma_nu right) ,|     (6) ,|

(now use Completeness relations)|

||=frac{e^4}{(k-k')^4}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)|

||=frac{e^4}{(k-k')^4}left(4 left( {k'}^mu k^nu - (k' cdot k)eta^{munu} + k'^nu k^mu right) + 4 m^2 eta^{munu} right) left( 4 left( {p'}_mu p_nu - (p' cdot p)eta_{munu} + p'_nu p_mu right) + 4 m^2 eta_{munu} right) ,|     (8) ,||=frac{32{e^4}}{(k-k')^4}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, in the case of electrons one is usually interested in energy scales that far exceed the electron mass. Neglecting the electron mass yields the simplified form: |

Bhabha scattering#Mandelstam variables>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} ,

Annihilation term (s-channel)

The process for finding the annihilation term is similar to the above. Since the two diagrams are related by crossing symmetry, and the initial and final state particles are the same, it is sufficient to permute the momenta, yielding ||=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} ,(This is proportional to(1 + cos^2theta)where theta is the scattering angle in the center-of-mass frame.)

Solution

Evaluating the interference term along the same lines and adding the three terms yields the final result

Simplifying steps

Completeness relations

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

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^mugamma^nu) = 4eta^{munu}
3. operatorname{Tr}left( gamma_rho gamma_mu gamma_sigma gamma_nu right) = 4 left( eta_{rhomu}eta_{sigmanu}-eta_{rhosigma}eta_{munu}+eta_{rhonu}eta_{musigma} right) ,

Using these two one finds that, for example,