Bhabha Scattering
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In
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
, Bhabha scattering is the
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
-
positron The positron or antielectron is the particle with an electric charge of +1''elementary charge, e'', a Spin (physics), spin of 1/2 (the same as the electron), and the same Electron rest mass, mass as an electron. It is the antiparticle (antimatt ...
scattering In physics, scattering is a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiat ...
process: ::e^+ e^- \rightarrow e^+ e^- There are two leading-order
Feynman diagram In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced ...
s contributing to this interaction: an annihilation process and a scattering process. Bhabha scattering is named after the Indian physicist
Homi J. Bhabha Homi Jehangir Bhabha, FNI, FASc, FRS (30 October 1909 – 24 January 1966) was an Indian nuclear physicist who is widely credited as the "father of the Indian nuclear programme". He was the founding director and professor of physics at the ...
. The Bhabha scattering rate is used as a luminosity monitor in electron-positron colliders. Due to crossing symmetry, Bhabha scattering has the same amplitude as Møller scattering.


Differential cross section

To leading order, the spin-averaged differential cross section for this process is ::\frac = \frac \left( u^2 \left( \frac + \frac \right)^2 + \left( \frac \right)^2 + \left( \frac \right)^2 \right) \, where ''s'',''t'', and ''u'' are the
Mandelstam variables In theoretical physics, the Mandelstam variables are numerical quantities that encode the energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical ...
, \alpha is the
fine-structure constant In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Alpha, Greek letter ''alpha''), is a Dimensionless physical constant, fundamental physical constant that quantifies the strength of the el ...
, 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 In theoretical physics, the Mandelstam variables are numerical quantities that encode the energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical ...
are defined by :: 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 In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced ...
one can show the following diagrams give these matrix elements: : 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. That is, :: First, calculate , \mathcal, ^2 \,: ::


Scattering term (t-channel)


Magnitude squared of M

::


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. :: 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: ::


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 :: (This is proportional to (1 + \cos^2\theta) 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 ::\frac = \frac + \frac + \frac \,


Simplifying steps


Completeness relations

The completeness relations for the four-spinors ''u'' and ''v'' are ::\sum_ = p\!\!\!/ + m \, ::\sum_ = p\!\!\!/ - m \, :where ::p\!\!\!/ = \gamma^\mu p_\mu \,      (see
Feynman slash notation In the study of Dirac fields in quantum field theory, Richard Feynman introduced the convenient Feynman slash notation (less commonly known as the Dirac slash notation). If ''A'' is a covariant vector (i.e., a 1-form), : \ \stackrel\ \gamma^ ...
) ::\bar = u^ \gamma^0 \,


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 \gamma_\mu \,'s is zero #\operatorname (\gamma^\mu\gamma^\nu) = 4\eta^ #\operatorname\left( \gamma_\rho \gamma_\mu \gamma_\sigma \gamma_\nu \right) = 4 \left( \eta_\eta_-\eta_\eta_+\eta_\eta_ \right) \, Using these two one finds that, for example, ::


Uses

Bhabha scattering has been used as a
luminosity Luminosity is an absolute measure of radiated electromagnetic radiation, electromagnetic energy per unit time, and is synonymous with the radiant power emitted by a light-emitting object. In astronomy, luminosity is the total amount of electroma ...
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–Positron Collider II The Beijing Electron–Positron Collider II (BEPC II) is a Chinese electron–positron collider, a type of particle accelerator, located in Shijingshan District, Beijing, People's Republic of China. It has been in operation since 2008 and has a ...
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. 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 A muon ( ; from the Greek letter mu (μ) used to represent it) is an elementary particle similar to the electron, with an electric charge of −1 '' e'' and a spin of  ''ħ'', but with a much greater mass. It is classified as a ...
, which is used to constrain
supersymmetry Supersymmetry is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It propo ...
and other models of physics beyond the Standard Model.


References

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Bhabha scattering on arxiv.org
{{QED Quantum electrodynamics Eponyms in physics