Neutrino pair creation in electron positron scattering

[tom.stoer]

Hello,

does anybody know how to compare (at tree level) the two reactions

$$e^- + e^+ \to \gamma^\ast \to e^- + e^+$$

and

$$e^- + e^+ \to Z^\ast \to \nu + \bar{\nu} \;\;\; \text{and} \;\;\; e^- + e^+ \to W^\ast \to \nu_e + \bar{\nu}_e$$

The first process is the so-called Bhabha scattering whereas the second one is the neutrino-antineutrino pair creation from virtual W- and Z exchange.

In order to compare similar processes one should restrict to electron neutrinos in the final state of the Z boson; then both processes are rather similar in terms of their Feynman diagrams; the only difference is that in Bhabha scattering the exchanged particle (in both s- and t-channel) is a photon whereas in the second process the s-channel contribution comes from the Z, the t-channel contribution from the W, respectively.

My problems are the following:
1) the total cross section for Bhabha scattering diverges due to the long range Coulomb force; it has the well-known forward singularity

$$\frac{d\sigma_{e^-e^+ \to e^-e^+}}{d\Omega} \sim \frac{1}{\sin^4 \frac{\theta}{2}}$$

so one can't compare cross-sections directly

2) for the latter process I can't find any low-energy matrix element; in the literature only the high-energy regime is described.

Does anybody know
- how to compare the matrix elements, cross sections or the branching ratio for the two processes?
- whether there is a different possibility to compare photon- and W-/Z-boson-exchange at low energies?
- whether there are experimental results down to the MeV range?

 

[Vanadium 50]

Do you want a ballpark answer or does it have to be right?

To ballpark it replace every 1/q² with a 1/M_w² and every alpha with an alpha_w.

 

[tom.stoer]

Do you want a ballpark answer or does it have to be right?

The latter one :-)

To ballpark it replace every 1/q² with a 1/M_w² and every alpha with an alpha_w.

I agree; far away from M_Z and M_W this is the 4-fermion interaction.

 

[tom.stoer]

I checked arxiv for positronium decays (because for low energy this is the relevant system). One ref. is

http://arxiv.org/abs/hep-ph/9911410v1
Decays of Positronium
Andrzej Czarnecki, Savely G. Karshenboim
(Submitted on 19 Nov 1999)
Abstract: We briefly review the theoretical and experimental results concerning decays of positronium. Possible solutions of the "orthopositronium lifetime puzzle" are discussed. Positronium annihilation into neutrinos is examined and disagreement is found with previously published results.

The reference system is the para-positronium with dominant decay channel

$$(e^-+e^-)_{S=0} \to 2\gamma$$

and neutrino-antineutrino pair creation

$$(e^-+e^-)_{S=0} \to \nu_e + \bar{\nu}_e$$

The decay widths are

$$\Gamma_{e^-e^- \to 2\gamma} = \frac{\alpha^5m_e}{2}$$

and

$$\Gamma_{e^-e^- \to \nu_e\bar{\nu}_e} = \frac{G_F^2\alpha^3m_e^5}{24\pi^2}(1+4\sin^2\theta_W)^2$$

Replacing the Fermi constant $$G_F$$ via the W-boson mass $$M_W$$, Weinberg angle $$\sin^2\theta_W \simeq 0.23$$ and $$\alpha$$ one finds the branchung ratio

$$\frac{\Gamma_{e^-e^- \to \nu_e\bar{\nu}_e}}{\Gamma_{e^-e^- \to 2\gamma}} = \frac{f_W}{24}\left(\frac{m_e}{M_W}\right)^4$$

with

$$f_W = \frac{(1+4\sin^2\theta_W)^2}{\sin^4\theta_W}$$

The result

$$\frac{\Gamma_{e^-e^- \to \nu_e\bar{\nu}_e}}{\Gamma_{e^-e^- \to 2\gamma}} = 6.2 \cdot 10^{-18}$$

shows the suppression of the decay into a neutrino-antineutrino pair due to the W-boson mass.

My calculation differs by one order of magnitude which may be due to a missed numerical constant.