Bhabha scattering

In quantum electrodynamics, Bhabha scattering is theelectron-positron scattering process:

e+e ! e+e

There are two leading-order Feynman diagrams con-tributing to this interaction: an annihilation process and ascattering process. Bhabha scattering is named after theIndian physicist Homi J. Bhabha.The Bhabha scattering rate is used as a luminosity moni-tor in electron-positron colliders.

1 Dierential cross sectionTo leading order, the spin-averaged dierential cross sec-tion for this process is

dd(cos ) =

2

s

u21

s+

1

t

2+

t

s

2+st

2!

where s,t, and u are the Mandelstam variables, is thene-structure constant, and is the scattering angle.This cross section is calculated neglecting the electronmass relative to the collision energy and including onlythe contribution from photon exchange. This is a validapproximation at collision energies small compared to themass scale of the Z boson, about 91 GeV; at higher en-ergies the contribution from Z boson exchange also be-comes important.

1.1 Mandelstam variablesIn this article, the Mandelstam variables are dened by

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

2 Deriving unpolarized cross sec-tion

2.1 Matrix elementsBoth the scattering and annihilation diagrams contributeto the transition matrix element. By letting k and k' rep-resent the four-momentum of the positron, while lettingp and p' represent the four-momentum of the electron,and by using Feynman rules one can show the followingdiagrams give these matrix elements:

Notice that there is a relative sign dierence between thetwo diagrams.

2.2 Square of matrix elementTo calculate the unpolarized cross section, one must av-erage over the spins of the incoming particles (s- and spossible values) and sum over the spins of the outgoingparticles. That is,

First, calculate jMj2 :

2.3 Scattering term (t-channel)2.3.1 Magnitude squared of M

2.3.2 Sum over spins

Next, we'd like to sum over spins of all four particles. Lets and s be the spin of the electron and r and r' be thespin of the positron.

Now that is the exact form, in the case of electrons oneis usually interested in energy scales that far exceed theelectron mass. Neglecting the electron mass yields thesimplied form:

1

2 5 REFERENCES

2.4 Annihilation term (s-channel)The process for nding the annihilation term is similar tothe above. Since the two diagrams are related by crossingsymmetry, and the initial and nal state particles are thesame, it is sucient to permute the momenta, yielding

(This is proportional to (1 + cos2 ) where is the scat-tering angle in the center-of-mass frame.)

2.5 SolutionEvaluating the interference term along the same lines andadding the three terms yields the nal result

jMj22e4

=u2 + s2

t2+

2u2

st+

u2 + t2

s2

3 Simplifying steps

3.1 Completeness relationsThe completeness relations for the four-spinors u and vare

Xs=1;2

u(s)p u(s)p = p/ +m

Xs=1;2

v(s)p v(s)p = p/m

wherep/ = p (see Feynman slash no-tation)u = uy0

3.2 Trace identitiesMain article: Trace identities

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

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

2. Tr() = 4

3. Tr () =4 ( + )

Using these two one nds that, for example,

4 UsesBhabha scattering has been used as a luminosity monitorin a number of e+e collider physics experiments. Theaccurate measurement of luminosity is necessary for ac-curate measurements of cross sections.

Small-angle Bhabha scattering was used to measurethe luminosity of the 1993 run of the Stanford LargeDetector (SLD), with a relative uncertainty of lessthan 0.5%.[1]

Electron-positron colliders operating in the regionof the low-lying hadronic resonances (about 1 GeVto 10 GeV), such as the Beijing Electron Syn-chrotron (BES) and the Belle and BaBar B-factoryexperiments, use large-angle Bhabha scattering as aluminosity monitor. To achieve the desired preci-sion at the 0.1% level, the experimental measure-ments must be compared to a theoretical calcula-tion including next-to-leading-order radiative cor-rections. The high-precision measurement of the to-tal hadronic cross section at these low energies is acrucial input into the theoretical calculation of theanomalous magnetic dipole moment of the muon,which is used to constrain supersymmetry and othermodels of physics beyond the Standard Model.

5 References[1] A Study of Small Angle Radiative Bhabha Scattering and

Measurement of the Luminosity at SLD

Halzen, Francis; Martin, Alan (1984). Quarks &Leptons: An Introductory Course in Modern ParticlePhysics. John Wiley & Sons. ISBN 0-471-88741-2.

Peskin, Michael E.; Schroeder, Daniel V. (1994).An Introduction to Quantum Field Theory. PerseusPublishing. ISBN 0-201-50397-2.

Bhabha scattering on arxiv.org

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Differential cross sectionMandelstam variables

Deriving unpolarized cross sectionMatrix elementsSquare of matrix elementScattering term (t-channel)Magnitude squared of MSum over spins

Annihilation term (s-channel)Solution

Simplifying stepsCompleteness relationsTrace identities

UsesReferencesText and image sources, contributors, and licensesTextImagesContent license