Møller scattering

Short description: Electron-electron scattering

Feynman diagrams (time from left to right)

t-channel
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u-channel
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Møller scattering is the name given to electron-electron scattering in quantum field theory, named after the Danish physicist Christian Møller who derived it in 1932.[1] The electron interaction that is idealized in Møller scattering forms the theoretical basis of many familiar phenomena such as the repulsion of electrons in the helium atom. While formerly many particle colliders were designed specifically for electron-electron collisions, more recently electron-positron colliders have become more common. Nevertheless, Møller scattering remains a paradigmatic process within the theory of particle interactions.

We can express this process in the usual notation, often used in particle physics:e−e−⟶e−e−,

In quantum electrodynamics, there are two tree-level Feynman diagrams describing the process: a t-channel diagram in which the electrons exchange a photon and a similar u-channel diagram. Crossing symmetry, one of the tricks often used to evaluate Feynman diagrams, in this case implies that Møller scattering should have the same cross section as Bhabha scattering (electron-positron scattering).

In the electroweak theory the process is instead described by four tree-level diagrams: the two from QED and an identical pair in which a Z boson is exchanged instead of a photon. The weak force is purely left-handed, but the weak and electromagnetic forces mix into the particles we observe. The photon is symmetric by construction, but the Z boson prefers left-handed particles to right-handed particles. Thus the cross sections for left-handed electrons and right-handed differ. The difference was first noticed by the Russian physicist Yakov Zel'dovich in 1959, but at the time he believed the parity violating asymmetry (a few hundred parts per billion) was too small to be observed. This parity violating asymmetry can be measured by firing a polarized beam of electrons through an unpolarized electron target (liquid hydrogen, for instance), as was done by an experiment at the Stanford Linear Accelerator Center, SLAC-E158.[2] The asymmetry in Møller scattering is

APV=−meEGF2πα16sin2Θcm(3+cos2Θcm)2(14−sin2θW),

where me is the electron mass, E the energy of the incoming electron (in the reference frame of the other electron), GF is Fermi's constant, α is the fine structure constant, Θcm is the scattering angle in the center of mass frame, and θW is the weak mixing angle, also known as the Weinberg angle.

## QED computation

The Møller scattering can be calculated from the QED point-of-view, at the tree-level, with the help of the two diagrams shown on this page. These two diagrams are contributing at leading order from the QED point-of-view. If we are taking in account the weak force, which is unified with the electromagnetic force at high energy, then we have to add two tree-level diagram for the exchange of a Z0 boson. Here we will focus our attention on a strict tree-level QED computation of the cross section, which is rather instructive but maybe not the most accurate description from a physical point-of-view.

Before the derivation, we write the 4-momenta as (p1and p2for incoming electrons, p3and p4for outgoing electrons, and m=me):

p1=(E,0,0,p),p2=(E,0,0,−p),

p3=(E,psin⁡θ,0,pcos⁡θ),p4=(E,−psin⁡θ,0,−pcos⁡θ).

The Mandelstam variables are:

s=(p1+p2)2=(p3+p4)2

t=(p1−p3)2=(p4−p2)2

u=(p1−p4)2=(p3−p2)2

These Mandelstam variables satisfy the identity: s+t+u≡∑mj2=4m2.

According to the two diagrams on this page, the matrix element of t-channel is

iℳt=(−ie)2u¯(p3)γμu(p1)−itu¯(p4)γμu(p2),

the matrix element of u-channel is

iℳu=(−ie)2u¯(p3)γμu(p2)−iuu¯(p4)γμu(p1).

So the sum is

iℳ=i(ℳt−ℳu)=−i(−ie)2[1tu¯(p3)γμu(p1)u¯(p4)γμu(p2)−1uu¯(p3)γμu(p2)u¯(p4)γμu(p1)].

Therefore,

|ℳ|2=e4{1t2[u¯(p3)γμu(p1)][u¯(p1)γνu(p3)][u¯(p4)γμu(p2)][u¯(p2)γνu(p4)]+1u2[u¯(p3)γμu(p2)][u¯(p2)γνu(p3)][u¯(p4)γμu(p1)][u¯(p1)γνu(p4)]−1tu[u¯(p3)γμu(p1)][u¯(p2)γνu(p3)][u¯(p4)γμu(p2)][u¯(p1)γνu(p4)]−1tu[u¯(p3)γμu(p2)][u¯(p1)γνu(p3)][u¯(p4)γμu(p1)][u¯(p2)γνu(p4)]}.

To calculate the unpolarized cross section, we average over initial spins and sum over final spins, with the factor 1/4 (1/2 for each incoming electron):

14∑spins|ℳ|2=e44{1t2Tr[γμ(p1+m)γν(p3+m)]Tr[γμ(p2+m)γν(p4+m)]+1u2Tr[γμ(p2+m)γν(p3+m)]Tr[γμ(p1+m)γν(p4+m)]−2tuTr[(p3+m)γμ(p1+m)γν(p4+m)γμ(p2+m)γν]}

where we have used the relation ∑sus(p)u¯s(p)=p+m=γμpμ+m. We would next calculate the traces.

The first term in the braces is

1t2Tr[γμ(p1+m)γν(p3+m)]Tr[γμ(p2+m)γν(p4+m)]=16t2(p1μp3ν+p3μp1ν+(−p13+m2)gμν)(p2μp4ν+p4μp2ν+(−p24+m2)gμν)=32t2(p12p34+p23p14−m2p13−m2p24+2m4)=32t2(p122+p142+2m2(p14−p12))=8t2(s2+u2−8m2(s+u)+24m4)

Here pij≡pi⋅pj, and we have used the γ-matrix identity

Tr[γμγνγργσ]=4(ημνηρσ−ημρηνσ+ημσηνρ)

and that trace of any product of an odd number of γμ is zero.

Similarly, the second term is

1u2Tr[γμ(p2+m)γν(p3+m)]Tr[γμ(p1+m)γν(p4+m)]=32u2(p12p34+p13p24−m2p23−m2p14+2m4)=8u2(s2+t2−8m2(s+t)+24m4)

Using the γ-matrix identities

Tr[γμγνγμγν]=−32,

Tr[γργμγσγνγμγν]=Tr[γργμγνγσγμγν]=16gρσ,

Tr[γργμγσγνγλγμγτγν]=−32gρλgστ,

and the identity of Mandelstam variables: s+t+u≡∑mj2, we get the third term

−2tuTr[(p3+m)γμ(p1+m)γν(p4+m)γμ(p2+m)γν]=−32tu(−2p12p34+2m2(p12+p13+p14)−2m4)=16tu(s2−8m2s+12m4)

Therefore,

|ℳ|2‾≡14∑spins|ℳ|2=2e4{1t2(s2+u2−8m2(s+u)+24m4)+1u2(s2+t2−8m2(s+t)+24m4)+2tu(s2−8m2s+12m4)}.

Substitute in the momenta we have set here, which are

s=4E2=ECM2,

t=2p2(cos⁡θ−1),

u=2p2(−cos⁡θ−1).

Finally we get the unpolarized cross section dσdΩ=164π2ECM2|p→f||p→i||ℳ|2‾=α22ECM2{1t2(s2+u2−8m2(s+u)+24m4)+1u2(s2+t2−8m2(s+t)+24m4)+2tu(s2−8m2s+12m4)}=α2ECM2p4sin4θ[4(m2+2p2)2+(4p4−3(m2+2p2)2)sin2θ+p4sin4θ].

with E2=m2+p2 and ECM=2E.

In the nonrelativistic limit, m≫p,

dσdΩ=m4α2ECM2p4sin4θ(4−3sin2θ)=m4α2ECM2p4sin4θ(1+3cos2θ).

In the ultrarelativistic limit, m≪p,

dσdΩ=α2ECM2p4sin4θ(16p4−8p4sin2θ+p4sin4θ)=α2ECM2sin4θ(3+cos2θ)2.

## References

1. ↑ Møller, C. (1932). "Zur Theorie des Durchgangs schneller Elektronen durch Materie". Annalen der Physik 406 (5): 531–585. doi:10.1002/andp.19324060506. Bibcode: 1932AnP...406..531M. https://doi.org/10.1002/andp.19324060506.
2. ↑ Anthony, P. L. (Aug 2005). "Precision Measurement of the Weak Mixing Angle in Møller Scattering". Physical Review Letters (American Physical Society) 95 (8). doi:10.1103/PhysRevLett.95.081601. PMID 16196849. Bibcode: 2005PhRvL..95h1601A. http://link.aps.org/doi/10.1103/PhysRevLett.95.081601.

## External links

* SLAC E158: Measuring the Electron's WEAK Charge

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Quantum electrodynamics

* Anomalous magnetic dipole moment
* Bhabha scattering
* Bremsstrahlung
* Compton scattering
* Dual photon
* Electron
* Feynman diagram
* Feynman slash notation
* Gupta–Bleuler formalism
* Lamb shift
* Møller scattering
* Path integral formulation
* Photon
* Positron
* Positronium
* Probability amplitude
* Propagator
* QED vacuum
* Self-energy
* Vacuum polarization
* Vertex function
* Virtual particles
* Ward–Takahashi identity
* ξ gauge

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