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| | Hi there. Allow me begin by introducing the author, her title is Sophia. Office supervising is where her main earnings comes from but she's already utilized for an additional one. Playing badminton is a factor that he is completely addicted to. Ohio is where her home is.<br><br>My web-site: best psychics ([http://Cartoonkorea.com/ce002/1093612 Cartoonkorea.com]) |
| ! style="background:#ffdead;" | [[Feynman diagrams]]
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| |-
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| | align="center" | '''Annihilation'''<br>[[Image:Electron-positron-annihilation.svg|220px]]
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| | align="center" | '''Scattering'''<br>[[Image:Electron-positron-scattering.svg|220px]]
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| |Note: Time moves forward from the left side of the diagram to the right. '''The arrows are simply markers of particle motion, and are ''not'' the same as the arrows conventionally written into Feynman diagrams'''.
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| |}
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| In [[quantum electrodynamics]], '''Bhabha scattering''' is the [[electron]]-[[positron]] [[scattering]] process:
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| ::<math>e^+ e^- \rightarrow e^+ e^-</math>
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| There are two leading-order [[Feynman diagram]]s contributing to this interaction: an annihilation process and a scattering process. Bhabha scattering is named after the Indian physicist [[Homi J. Bhabha]].
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| The Bhabha scattering rate is used as a [[luminosity]] monitor in electron-positron colliders.
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| ==Differential cross section==
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| To [[leading-order|leading order]], the spin-averaged [[differential cross section]] for this process is
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| ::<math>\frac{\mathrm{d} \sigma}{\mathrm{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) \,</math>
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| where ''s'',''t'', and ''u'' are the [[Mandelstam variables]], <math>\alpha</math> is the [[fine-structure constant]], and <math>\theta</math> is the scattering angle. | |
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| 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.
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| ===Mandelstam variables===
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| In this article, the [[Mandelstam variables]] are defined by
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| ::{|
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| |align="right"|<math>s= \,</math>
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| |align="right"|<math>(k+p)^2= \,</math>
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| |align="right"|<math>(k'+p')^2 \approx \,</math>
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| |align="right"|<math>2 k \cdot p \approx\,</math>
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| |align="right"|<math> 2 k' \cdot p' \,</math>
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| |rowspan="3"| [[Image:Mandelstam01.png]]
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| |-
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| |align="right"|<math>t= \,</math>
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| |align="right"|<math>(k-k')^2= \,</math>
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| |align="right"|<math>(p-p')^2\approx \,</math>
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| |<math> -2 k \cdot k' \approx \,</math>
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| |<math> -2 p \cdot p' \,</math>
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| |-
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| |align="right"|<math>u= \,</math>
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| |align="right"|<math>(k-p')^2= \,</math>
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| |align="right"|<math>(p-k')^2\approx \,</math>
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| |<math> -2 k \cdot p' \approx \,</math>
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| |<math> -2 k' \cdot p \,</math>
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| |}
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| where the approximations are for the high-energy (relativistic) limit.
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| ==Deriving unpolarized cross section==
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| ===Matrix elements===
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| 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:
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| :{| border="0" cellpadding="5" cellspacing="0"
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| |align="center" | [[Image:Feynman-scattering.svg|160px]]
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| |align="center" | [[Image:Feynman-annihilation.svg|160px]]
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| |Where we use:<br><math>\gamma^\mu \,</math> are the [[Gamma matrices]],<br><math>u, \ \mathrm{and} \ \bar{u}\,</math> are the four-component spinors for fermions, while<br><math>v, \ \mathrm{and} \ \bar{v}\,</math> are the four-component spinors for anti-fermions (see [[Dirac equation#Four spinor|Four spinors]]).
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| |-
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| |align="center" | (scattering)
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| |align="center" | (annihilation)
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| |-
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| |<math>\mathcal{M} = \,</math>
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| |<math>-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) </math>
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| |<math>+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) </math>
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| |}
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| Notice that there is a relative sign difference between the two diagrams.
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| ===Square of matrix element===
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| To calculate the unpolarized [[Cross section (physics)|cross section]], one must ''average'' over the spins of the incoming particles (''s''<sub>e-</sub> and ''s''<sub>e+</sub> possible values) and ''sum'' over the spins of the outgoing particles. That is,
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| ::{|
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| |<math>\overline{|\mathcal{M}|^2} \,</math>
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| |<math> = \frac{1}{(2s_{e-} + 1)(2 s_{e+} + 1)} \sum_{\mathrm{spins}} |\mathcal{M}|^2 \,</math>
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| |-
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| |<math>= \frac{1}{4} \sum_{s=1}^2 \sum_{s'=1}^2 \sum_{r=1}^2 \sum_{r'=1}^2 |\mathcal{M}|^2 \,</math>
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| |}
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| First, calculate <math>|\mathcal{M}|^2 \,</math>:
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| ::{| cellpadding=4
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| |<math>|\mathcal{M}|^2 \,</math>=
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| |<math> e^4 \left| \frac{(\bar{v}_{k} \gamma^\mu v_{k'} )( \bar{u}_{p'} \gamma_\mu u_p)}{(k-k')^2} \right|^2 \,</math>
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| | align="center" | (scattering)
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| |-
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| |<math>{}- 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) \,</math>
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| | align="center" | (interference)
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| |-
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| |<math>{}- 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)^* \,</math>
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| | align="center" | (interference)
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| |-
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| |<math>{}+ e^4 \left| \frac{(\bar{v}_{k} \gamma^\nu u_p )( \bar{u}_{p'} \gamma_\nu v_{k'} )}{(k+p)^2} \right|^2 \,</math>
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| | align="center" | (annihilation)
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| |}
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| ===Scattering term (t-channel)===
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| ====Magnitude squared of M====
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| ::{|
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| |<math>|\mathcal{M}|^2 \,</math>
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| |<math>= \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) \,</math>
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| | <math>(1) \,</math>
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| |-
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| |<math>= \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) \,</math>
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| | <math>(2) \,</math>
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| |-
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| |align="center" | (complex conjugate will flip order)
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| |<math>= \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) \,</math>
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| | <math>(3) \,</math>
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| |-
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| |align="center" | (move terms that depend on same momentum to be next to each other)
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| |-
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| |<math>= \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) \,</math>
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| | <math>(4) \,</math>
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| |}
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| ====Sum over spins====
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| 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.
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| ::{|
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| |<math> \sum_{\mathrm{spins}} |\mathcal{M}|^2 \,</math>
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| |<math>= \frac{e^4}{(k-k')^4}\left(\sum_{r'} \bar{v}_{k'} \gamma^\mu (\sum_{r}v_{k} \bar{v}_{k}) \gamma^\nu v_{k'} \right) \left(\sum_{s} \bar{u}_{p} \gamma_\mu (\sum_{s'}{u_{p'} \bar{u}_{p'}}) \gamma_\nu u_p \right) \,</math>
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| | <math>(5) \,</math>
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| |-
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| |<math>= \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) \,</math>
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| | <math>(6) \,</math>
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| |-
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| |align="center" | (now use [[Bhabha scattering#Completeness relations|Completeness relations]])
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| |-
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| |<math>=\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) \,</math>
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| | <math>(7) \,</math>
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| |-
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| |align="center" | (now use [[Bhabha scattering#Trace identities|Trace identities]])
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| |-
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| |<math>=\frac{e^4}{(k-k')^4}\left(4 \left( {k'}^\mu k^\nu - (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 - (p' \cdot p)\eta_{\mu\nu} + p'_\nu p_\mu \right) + 4 m^2 \eta_{\mu\nu} \right) \,</math>
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| | <math>(8) \,</math>
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| |-
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| |<math>=\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) \,</math>
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| | <math>(9) \,</math>
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| |}
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| 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:
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| ::{|
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| |<math>\frac{1}{4} \sum_{\mathrm{spins}} |\mathcal{M}|^2 \,</math>
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| |<math> = \frac{32e^4}{4(k-k')^4} \left( (k' \cdot p') (k \cdot p) + (k' \cdot p) (k \cdot p') \right) \,</math>
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| |-
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| |(use the [[Bhabha scattering#Mandelstam variables|Mandelstam variables]] in this relativistic limit)
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| |-
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| |<math>=\frac{8e^4}{t^2} \left(\tfrac{1}{2} s \tfrac{1}{2}s + \tfrac{1}{2}u \tfrac{1}{2} u \right) \,</math>
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| |-
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| |<math>= 2 e^4 \frac{s^2 +u^2}{t^2} \,</math>
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| |}
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| ===Annihilation term (s-channel)===
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| 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
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| ::{|
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| |<math>\frac{1}{4} \sum_{\mathrm{spins}} |\mathcal{M}|^2 \,</math>
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| |<math> = \frac{32e^4}{4(k+p)^4} \left( (k \cdot k') (p \cdot p') + (k' \cdot p) (k \cdot p') \right) \,</math>
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| |-
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| |<math>=\frac{8e^4}{s^2} \left(\tfrac{1}{2} t \tfrac{1}{2}t + \tfrac{1}{2}u \tfrac{1}{2} u \right) \,</math>
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| |-
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| |<math>= 2 e^4 \frac{t^2 +u^2}{s^2} \,</math>
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| |}
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| (This is proportional to
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| <math>(1 + \cos^2\theta)</math>
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| where <math>\theta</math> is the scattering angle in the center-of-mass frame.) | |
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| ===Solution===
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| Evaluating the interference term along the same lines and adding the three terms yields the final result
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| ::<math>\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} \,</math>
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| ==Simplifying steps==
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| ===Completeness relations===
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| The completeness relations for the [[Dirac_spinor#Four-spinor_for_particles|four-spinors]] ''u'' and ''v'' are
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| ::<math>\sum_{s=1,2}{u^{(s)}_p \bar{u}^{(s)}_p} = p\!\!\!/ + m \,</math>
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| ::<math>\sum_{s=1,2}{v^{(s)}_p \bar{v}^{(s)}_p} = p\!\!\!/ - m \,</math>
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| :where
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| ::<math>p\!\!\!/ = \gamma^\mu p_\mu \,</math> (see [[Feynman slash notation#With four-momentum|Feynman slash notation]])
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| ::<math>\bar{u} = u^{\dagger} \gamma^0 \,</math>
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| ===Trace identities===
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| :''Main article:'' [[Gamma matrix#Identities|Trace identities]]
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| To simplify the trace of the [[Gamma matrices|Dirac gamma matrices]], one must use trace identities. Three used in this article are:
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| #The Trace of any product of an ''odd number'' of <math>\gamma_\mu \,</math>'s is zero
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| #<math>\operatorname{Tr} (\gamma^\mu\gamma^\nu) = 4\eta^{\mu\nu}</math>
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| #<math>\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) \,</math>
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| Using these two one finds that, for example,
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| ::{|
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| |<math>\operatorname{Tr}\left( (p\!\!\!/' + m) \gamma_\mu (p\!\!\!/ + m) \gamma_\nu \right) \,</math>
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| |<math> = \operatorname{Tr}\left( p\!\!\!/' \gamma_\mu p\!\!\!/ \gamma_\nu \right) + \operatorname{Tr}\left(m \gamma_\mu p\!\!\!/ \gamma_\nu \right) \,</math>
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| |-
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| | <math>+ \operatorname{Tr}\left( p\!\!\!/' \gamma_\mu m \gamma_\nu \right) + \operatorname{Tr}\left(m^2 \gamma_\mu \gamma_\nu \right) \,</math>
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| |-
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| |align="center"|(the two middle terms are zero because of (1))
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| |-
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| |<math> = \operatorname{Tr}\left( p\!\!\!/' \gamma_\mu p\!\!\!/ \gamma_\nu \right) + m^2 \operatorname{Tr}\left(\gamma_\mu \gamma_\nu \right) \,</math>
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| |-
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| |align="center"|(use identity (2) for the term on the right)
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| |-
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| |<math>= {p'}^{\rho} p^\sigma \operatorname{Tr}\left( \gamma_\rho \gamma_\mu \gamma_\sigma \gamma_\nu \right) + m^2 \cdot 4\eta_{\mu\nu} \,</math>
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| |-
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| |align="center"|(now use identity (3) for the term on the left)
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| |-
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| |<math>= {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} \,</math>
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| |-
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| |<math>=4 \left( {p'}_\mu p_\nu - \mathbf{p' \cdot p}\eta_{\mu\nu} + p'_\nu p_\mu \right) + 4 m^2 \eta_{\mu\nu} \,</math>
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| |}
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| ==Uses==
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| 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.
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| * 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%.<ref>[http://adsabs.harvard.edu/abs/1995PhDT.......160W A Study of Small Angle Radiative Bhabha Scattering and Measurement of the Luminosity at SLD]</ref>
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| * 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 experiment|Belle]] and [[BaBar experiment|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 [[renormalization|radiative corrections]].[http://arxiv.org/abs/hep-ph/0003268] 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]].
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| ==References==
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| <references/>
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| *{{cite book | author=Halzen, Francis; Martin, Alan | title=Quarks & Leptons: An Introductory Course in Modern Particle Physics | publisher=John Wiley & Sons | year=1984 | isbn=0-471-88741-2}}
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| *{{cite book | author=Peskin, Michael E.; Schroeder, Daniel V. | title=An Introduction to Quantum Field Theory | publisher=Perseus Publishing | year=1994 | isbn=0-201-50397-2}}
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| *[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=270563 Bhabha scattering on arxiv.org]
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| {{QED}}
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| [[Category:Quantum field theory]]
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| [[Category:Quantum electrodynamics]]
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