# Verifying the consistency of a quantum field theoretic cross-section formula

Consider the scattering $$e^-e^-\rightarrow e^-e^-$$ in QED. The cross-section is a Lorentz invariant quantity and therefore, given by $$\sigma=\sigma(m_e,e,s,\theta)$$ where electron mass $m_e$, electron charge $e$ etc are running mass and running charge. They change with the centre-of-mass energy. $\theta$ is the scattering angle and $s$ stands for the Mandelstam variable.

QED gives a finite formula for the RHS. LHS is measured directly from experiments by measuring the scattered flux of particles in a given direction $(\theta,\phi)$. In order to check the consistency i.e., whether LHS=RHS, we have to put values of $m_e$, $e$ on the RHS by independently measuring $m_e$ and $e$ (at the colliding energy).

$m_e$ can be measured by measuring four-momenta of the scattered electrons and using dispersion relation, and by similarly measuring $e$. Is that correct?

However, if a scattering was mediated by Z/W boson as the virtual particle, the expression for $\sigma$ would contain the mass of W boson at that energy. But I believe, Z/W boson mass cannot be recovered here by measuring the four-momenta of the scattered particles. This is because, being a virtual particle, it doesn't obey energy-momentum relation.

How does one verify then the consistency of the corresponding scattering cross-section formula?

• What does "if a scattering was mediated by Z/W boson as the virtual particle" mean? You cannot determine "which" of the diagrams describing a scattering "actually happened" because the diagrams do not really depict actual processes to begin with. It would be like asking "which path did the electron take?" after detecting it at the screen in a double-slit experiment. Dec 2 '16 at 19:37
• @ACuriousMind I meant, consider a process mediated by W boson as the internal line. May be many higher order diagrams also contribute. In any case, after you renormalize, you're left with a finite scattering cross-section formula that carries physical W boson at the colliding energy as an input on the RHS. My question is, if W is virtual, how do you measure its mass? that is also running. Right?
– SRS
Dec 2 '16 at 19:47
• @ACuriousMind- In other words consider a QFT in which a scattering occurs via exchange of a massive internal boson i.e., virtual. I renormalize, express the result in terms of physical masses. Now I need physical masses as the experimental input to match with the QFT formula. If it match, I verify the consistency of my prediction.
– SRS
Dec 2 '16 at 19:52
• @ACuriousMind $G^0$, Q-weak and possibly other experiments have measured the weak electron-proton scattering cross-section by isolating the parity violating component in polarized beam experiments. I'm not aware, off the top of my head, if the week electron-electron or electron-positron scattering cross-sections have been measured. To be sure this only insures that a parity violating weak process was part of the event, not that the leading $Z$-exchange diagram "actually happened", but it probes that physics none-the-less. Dec 2 '16 at 21:05

However, it is possible to see the effect of the $Z_0$-exchange. It should show up as a resonance. If the exchange energy is close to the mass of the $Z_0$ then the cross-section increases. So, after you have accumulated some data and then plot the cross-section as a function of the energy then the curve would show a bump at the energy that conrresponds to the mass of the $Z_0$.
• You can also isolate the weak interaction at low $Q^2$ (or $t$ if you like the Mandelstam variables) because the whole parity violating part of the cross-section involved a weak contribution (at tree order or in a correction diagram, but it has to be there). Dec 11 '16 at 22:03