Bhabha scattering - Citizendia

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. Motivation and history When calculating Scattering cross sections in Particle physics, the interaction between particles can be described Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The positrons or antielectron is the Antiparticle or the Antimatter counterpart of the Electron. Scattering is a general physical process whereby some forms of Radiation, such as Light, Sound or moving particles for example are forced to deviate from Motivation and history When calculating Scattering cross sections in Particle physics, the interaction between particles can be described The Bhabha scattering rate is used as a luminosity monitor in electron-positron colliders. Luminosity has different meanings in several different fields of science

Bhabha scattering is named after Indian physicist Homi J. Bhabha. This page is about the physicist Homi J Bhabha For the postcolonialist theorist see Homi K

1 Differential cross section
- 1.1 Mandelstam variables
2 Deriving unpolarized cross section
3 Simplifying steps
- 3.1 Completeness relations
- 3.2 Trace identities
4 Uses
5 References

Differential cross section

To leading order, the spin-averaged differential cross section for this process is

$\frac{\mathrm{d} \sigma}{\mathrm{d} (\cos\theta)} = \frac{\pi \alpha^2}{s} \left( u^2 \left( \frac{1}{s} + \frac{1}{t} \right)^2 + \left( \frac{t}{s} \right)^2 + \left( \frac{s}{t} \right)^2 \right) \,$

where s,t, and u are the Mandelstam variables,

α

is the fine-structure constant, and

θ

is the scattering angle. In nuclear and Particle physics, the concept of a cross section is used to express the likelihood of interaction between particles In Theoretical physics, the Mandelstam variables are numerical quantities that encode the Energy, Momentum, and angles of particles in a scattering process The fine-structure constant or Sommerfeld fine-structure constant, usually denoted \alpha \ is the Fundamental physical constant characterizing

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. The W and Z bosons are the Elementary particles that mediate the Weak force.

Mandelstam variables

In this article, the Mandelstam variables are defined by

$s= \,$	$(k+p)^2= \,$	$(k'+p')^2 \approx \,$	$2 k \cdot p \approx\,$	$2 k' \cdot p' \,$
$t= \,$	$(k-k')^2= \,$	$(p-p')^2\approx \,$	$-2 k \cdot k' \approx \,$	$-2 p \cdot p' \,$
$u= \,$	$(k-p')^2= \,$	$(p-k')^2\approx \,$	$-2 k \cdot p' \approx \,$	$-2 k' \cdot p \,$

Where the approximations are for the high-energy (relativistic) limit. In Theoretical physics, the Mandelstam variables are numerical quantities that encode the Energy, Momentum, and angles of particles in a scattering process

Deriving unpolarized cross section

Matrix elements

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). Motivation and history When calculating Scattering cross sections in Particle physics, the interaction between particles can be described In Mathematical physics, the gamma matrices, {γ0 γ1 γ2 γ3} also known as the Dirac matrices, form a matrix-valued In Physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides
	(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 (s_e- and s_e+ possible values) and sum over the spins of the outgoing particles. In nuclear and Particle physics, the concept of a cross section is used to express the likelihood of interaction between 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)

Scattering term

Magnitude squared of M

$\|\mathcal{M}\|^2 \,$	$= \frac{e^4}{(k-k')^4} \Big( (\bar{v}_{k} \gamma^\mu v_{k'} )( \bar{u}_{p'} \gamma_\mu u_p) \Big)^* \Big( (\bar{v}_{k} \gamma^\mu v_{k'})( \bar{u}_{p'} \gamma_\mu u_p) \Big) \,$	$(1) \,$
	$= \frac{e^4}{(k-k')^4} \Big( (\bar{v}_{k} \gamma^\mu v_{k'} )^* ( \bar{u}_{p'} \gamma_\mu u_p)^* \Big) \Big( (\bar{v}_{k} \gamma^\mu v_{k'})( \bar{u}_{p'} \gamma_\mu u_p) \Big) \,$	$(2) \,$
	(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)
	$= \frac{e^4}{(k-k')^4} \left( \bar{v}_{k'} \gamma^\mu v_{k} \right) \left( \bar{v}_{k} \gamma^\mu v_{k'} \right) \left( \bar{u}_{p} \gamma_\mu u_{p'} \right) \left( \bar{u}_{p'} \gamma_\mu u_p \right) \,$	$(4) \,$

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.

$\frac{(k-k')^4}{e^4} \sum_{\mathrm{spins}} \|\mathcal{M}\|^2 \,$	$= \left(\sum_{r'} \bar{v}_{k'} \gamma^\mu (\sum_{r}v_{k} \bar{v}_{k}) \gamma^\mu v_{k'} \right) \left(\sum_{s} \bar{u}_{p} \gamma_\mu (\sum_{s'}{u_{p'} \bar{u}_{p'}}) \gamma_\mu u_p \right) \,$	$(5) \,$
	$= \left( \Big(\sum_{r'} v_{k'} \bar{v}_{k'} \Big) \gamma^\mu \Big(\sum_{r}v_{k} \bar{v}_{k} \Big) \gamma^\mu \right) \left( \Big(\sum_{s} u_p \bar{u}_{p} \Big) \gamma_\mu \Big( \sum_{s'}{u_{p'} \bar{u}_{p'}} \Big) \gamma_\mu \right) \,$	$(6) \,$
	(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, in the case of electrons one is usually interested in energy scales that far exceed the electron mass. Differential cross section To leading order the spin-averaged Differential cross section for this process is \frac{\mathrm{d} \sigma}{\mathrm{d} Differential cross section To leading order the spin-averaged Differential cross section for this process is \frac{\mathrm{d} \sigma}{\mathrm{d} Neglecting the electron mass yields the simplified form:

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

Annihilation term

The process for finding the annihilation term is similar to the above. Differential cross section To leading order the spin-averaged Differential cross section for this process is \frac{\mathrm{d} \sigma}{\mathrm{d} 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

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

Solution

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

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

Simplifying steps

Completeness relations

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)

$\bar{u} = u^{\dagger} \gamma^0 \,$

Trace identities

Main article: Trace identities

To simplify the trace of the Dirac gamma matrices, one must use trace identities. In Quantum field theory, a branch of theoretical physics crossing symmetry is a symmetry that relates S-matrix elements In Quantum field theory, Dirac spinor is the Bispinor in the plane-wave solution \psi = \omega_\vec{p}\e^{-ipx} \ of the In the study of Dirac fields in Quantum field theory, Richard Feynman invented the convenient Feynman slash notation (less commonly known as the '''Dirac''' In Mathematical physics, the gamma matrices, {γ0 γ1 γ2 γ3} also known as the Dirac matrices, form a matrix-valued In Mathematical physics, the gamma matrices, {γ0 γ1 γ2 γ3} also known as the Dirac matrices, form a matrix-valued Three used in this article are:

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

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)
	$= {p'}^{\rho} p^\sigma 4 \left( \eta_{\rho\mu}\eta_{\sigma\nu}-\eta_{\rho\sigma}\eta_{\mu\nu}+\eta_{\rho\nu}\eta_{\mu\sigma} \right) + 4 m^2 \eta_{\mu\nu} \,$
	$=4 \left( {p'}_\mu p_\nu - \mathbf{p' \cdot p}\eta_{\mu\nu} + p'_\nu p_\mu \right) + 4 m^2 \eta_{\mu\nu} \,$

Uses

Bhabha scattering has been used as a luminosity monitor in a number of e+e- collider physics experiments. Luminosity has different meanings in several different fields of science 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. The Belle experiment is a Particle physics experiment conducted by the Belle Collaboration an international collaboration of more than 400 physicists and engineers investigating The BaBar experiment is an international collaboration of more than 550 physicists and engineers studying the subatomic world at energy of approximately ten times the rest mass of a proton 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. In Quantum field theory, the Statistical mechanics of fields and the theory of self-similar geometric structures renormalization refers to a collection [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. In Quantum electrodynamics, the anomalous magnetic moment of a particle is a contribution of effects of Quantum mechanics, expressed by Feynman diagrams The muon (from the letter mu (μ--used to represent it is an Elementary particle with negative Electric charge and a spin of 1/2 In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that In Physics, the Standard Model of Particle physics is currently the best description of all experimental data

References

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.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents