Møller scatter

Feynman diagrams
t channel
u channel

As Møller scattering refers to the scattering of two electrons to one another. With the exception of very high-energy collisions, such as those occurring e.g. B. are generated artificially in modern particle accelerators, the interaction between the electrons can be assumed to be purely electromagnetic . With this assumption - which is also the basis of Christian Møller's original publication - the scattering can be described with quantum electrodynamics (QED). At higher energies, measurable corrections occur through other interactions, the exchange of Z bosons in the standard model of elementary particle physics (SM) or other exchange particles in exotic physics models (“physics beyond the standard model”).

Quantum electrodynamics

The only interaction that exists in quantum electrodynamics is the electromagnetic interaction . In the parlance of quantum field perturbation theory , the momentum exchange between the participating electrons takes place via the exchange of virtual photons .

The incident electrons carry the impulses and before the collision . After the collision there are two electrons with the pulses and . As a first approximation of perturbation theory, there are two Feynman diagrams that describe the process (see figure on the right). The two diagrams, named t-channel and u-channel after the Mandelstam variables appearing in the denominator of the respective mathematical expression , differ only in the relationship between the pulses. One can visualize this in such a way that the outgoing electron with momentum may have had the momentum or the momentum before the scattering , and therefore both possibilities contribute to the scattering probability (the cross section ). ${\ displaystyle p_ {1}}$ ${\ displaystyle p_ {2}}$ ${\ displaystyle p_ {3}}$ ${\ displaystyle p_ {4}}$ ${\ displaystyle p_ {3}}$ ${\ displaystyle p_ {1}}$ ${\ displaystyle p_ {2}}$

In the center of gravity system , in the relativistic limit case, i.e. when the energies of the electrons are significantly greater than the rest energy of the electron (511 keV ), the differential cross-section results

{\ displaystyle {\ frac {d \ sigma} {d \ Omega}} = {\ frac {\ alpha ^ {2}} {8E ^ {2}}} (\ hbar c) ^ {2} \ left [{ \ frac {1+ \ cos ^ {4} (\ theta / 2)} {\ sin ^ {4} (\ theta / 2)}} + {\ frac {2} {\ sin ^ {2} (\ theta / 2) \ cos ^ {2} (\ theta / 2)}} + {\ frac {1+ \ sin ^ {4} (\ theta / 2)} {\ cos ^ {4} (\ theta / 2) }} \ right]}

,

where E is the energy of an electron, the coupling constant of the QED , and the scattering angle. ${\ displaystyle \ alpha}$ ${\ displaystyle \ theta}$

Other interactions

Even in a first approximation, the momentum exchange of the electrons does not necessarily have to take place via an intermediate photon. For example, the standard model of particle physics also allows a momentum exchange via an intermediate Z boson . The corresponding Feynman diagrams are similar to the diagrams from electrodynamics, with the inner line now being a Z boson. However, the associated terms differ in two important ways:

The contribution of the intermediate particle to the scattering probability depends on its mass. In particular, the contribution of exchanged Z bosons for center of gravity energies well below 91 GeV / c², the mass of a Z boson, is negligible compared to the contribution from photon exchange. At correspondingly high energies, as can be generated in particle accelerators , for example, the contribution of the Z boson can be measured.
In contrast to the photon exchange, the exchange of a Z boson is sensitive to the polarization of the electrons. This leads to a measurable dependence of the total effective cross-section on the electron polarization.

In addition to the contribution from the exchange of Z bosons, other direct contributions from previously unknown elementary particles are also conceivable. Since such contributions have not yet been measured, these particles must either have a slight interaction with the electrons, such as B. a possible graviton, or have a high mass, so that their contributions for center of gravity energies below this mass are strongly suppressed.

Individual evidence

↑ James Bjorken , Sidney Drell : Relativistic quantum mechanics ("Relativistic quantum mechanics"). Akademischer Verlag Spektrum, Heidelberg 1998, ISBN 3-86025-595-9 .
↑ Hans Frauenfelder , Ernest M. Henley Particles and Cores . R. Oldenbourg Verlag, Munich 1979, ISBN 3-486-20591-9
↑ Imran Younus: First results from E158 Measurimng Parity Violation in Moller scattering . In: Adam Para (Ed.): Neutrino factories and superbeams. 5th International Workshop on Neutrino Factories and Superbeams, New York 5–11. June 2003 (AIP Conference proceedings; 721). American Institute of Physics, New York 2004, ISBN 0-7354-0201-9 , pp. 367-370, doi : 10.1063 / 1.1818436 , bibcode : 2004AIPC..721..367Y .

Web links

Christian Møller: On the theory of fast electrons passing through matter . In: Annalen der Physik , Vol. 406, Issue 5, pp. 531-585. bibcode : 1932AnP ... 406..531M .

[1] James Bjorken , Sidney Drell : Relativistic quantum mechanics ("Relativistic quantum mechanics"). Akademischer Verlag Spektrum, Heidelberg 1998, ISBN 3-86025-595-9 .

[2] Hans Frauenfelder , Ernest M. Henley Particles and Cores . R. Oldenbourg Verlag, Munich 1979, ISBN 3-486-20591-9

[3] Imran Younus: First results from E158 Measurimng Parity Violation in Moller scattering . In: Adam Para (Ed.): Neutrino factories and superbeams. 5th International Workshop on Neutrino Factories and Superbeams, New York 5–11. June 2003 (AIP Conference proceedings; 721). American Institute of Physics, New York 2004, ISBN 0-7354-0201-9 , pp. 367-370, doi : 10.1063 / 1.1818436 , bibcode : 2004AIPC..721..367Y .