Bhabha scattering - AbsoluteAstronomy.com

Feynman diagrams
Annihilation
Scattering
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**.

In quantum electrodynamics

Quantum electrodynamics

Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

, Bhabha scattering is the electron

Electron

The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

-positron

Positron

The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron...

scattering

Scattering

Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...

process:

There are two leading-order Feynman diagram

Feynman diagram

Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...

s contributing to this interaction: an annihilation process and a scattering process. The Bhabha scattering rate is used as a luminosity

Luminosity

Luminosity is a measurement of brightness.-In photometry and color imaging:In photometry, luminosity is sometimes incorrectly used to refer to luminance, which is the density of luminous intensity in a given direction. The SI unit for luminance is candela per square metre.The luminosity function...

monitor in electron-positron colliders.

Bhabha scattering is named after Indian physicist Homi J. Bhabha

Homi J. Bhabha

Homi Jehangir Bhabha, FRS was an Indian nuclear physicist and the chief architect of the Indian atomic energy program...

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

Mandelstam variables

In theoretical physics, the Mandelstam variables are numerical quantities that encode the energy, momentum, and angles of particles in a scattering process in a Lorentz-invariant fashion...

is the fine-structure constant

Fine-structure constant

In physics, the fine-structure constant is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction. Being a dimensionless quantity, it has constant numerical value in all systems of units...

, and

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; for energies not too small compared to this mass, the contribution from Z boson exchange also becomes important.

Mandelstam variables

In this article, the Mandelstam variables

Mandelstam variables

In theoretical physics, the Mandelstam variables are numerical quantities that encode the energy, momentum, and angles of particles in a scattering process in a Lorentz-invariant fashion...

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

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: are the Gamma matrices, are the four-component spinors for fermions, while are the four-component spinors for anti-fermions (see Four spinors).
(scattering)	(annihilation)

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

Square of matrix element

To calculate the unpolarized cross section

Cross section (physics)

A cross section is the effective area which governs the probability of some scattering or absorption event. Together with particle density and path length, it can be used to predict the total scattering probability via the Beer-Lambert law....

, 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. That is,
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First, calculate

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Magnitude squared of M

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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.
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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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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

Crossing symmetry

In quantum field theory, a branch of theoretical physics, crossing is the property of scattering amplitudes that allows antiparticles to be interpreted as particles going backwards in time....

, and the initial and final state particles are the same, it is sufficient to permute the momenta, yielding
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Solution

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

Completeness relations

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

where

(see Feynman slash notation)

Trace identities

Main article: Trace identities

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

The Trace of any product of an odd number of 's is zero

Using these two one finds that, for example,
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Uses

Bhabha scattering has been used as a luminosity

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%.
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
Belle experiment
The Belle experiment is a particle physics experiment conducted by the Belle Collaboration, an international collaboration of more than 400 physicists and engineers investigating CP-violation effects at the High Energy Accelerator Research Organisation in Tsukuba, Ibaraki Prefecture, Japan.The...

and BaBar
BaBar experiment
The BaBar experiment, or simply BaBar, is an international collaboration of more than 500 physicists and engineers studying the subatomic world at energies of approximately ten times the rest mass of a proton . Its design was motivated by the investigation of CP violation...

"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
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....

.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
Anomalous magnetic dipole moment
In quantum electrodynamics, the anomalous magnetic moment of a particle is a contribution of effects of quantum mechanics, expressed by Feynman diagrams with loops, to the magnetic moment of that particle...

of the muon
Muon
The muon |mu]] used to represent it) is an elementary particle similar to the electron, with a unitary negative electric charge and a spin of ½. Together with the electron, the tau, and the three neutrinos, it is classified as a lepton...

, which is used to constrain supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

and other models of physics beyond the Standard Model
Beyond the Standard Model
Physics beyond the Standard Model refers to the theoretical developments needed to explain the deficiencies of the Standard Model, such as the origin of mass, the strong CP problem, neutrino oscillations, matter–antimatter asymmetry, and the nature of dark matter and dark energy...

.

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