The weak couplings of the tau may be studied by investigating the energy spectra of various tau decay products. These spectra are determined by the spin polarization of the taus and the nature of the decay .
At the , taus are produced with spin polarization due to the inherent parity violation of the couplings. At the SLC, this is enhanced due to the electron beam polarization. We can write the production cross sections for left and right handed taus as follows:
where is the polar angle of the () with respect to the incident electron (positron) direction, is the electron beam polarization, and and are the electron and tau parity-violation asymmetry parameters defined as:
where is the ratio between the effective vector and axial-vector couplings to the leptons.
This results in the following expression for the tau polarization as a function of angle and beam polarization:
Due to the large beam polarization at SLC, tau polarizations are much higher than at LEP, especially at high (see Fig. ). Also, the polarization of the tau is largely determined by the beam polarization and the production angle, and is relatively unaffected by the parity violation.
In this paper, we study the energy spectra of taus decaying to pions, electrons and muons. Taus produced at the are highly boosted, and as a result, angular information about the decays is lost due to the collimation of the decay. In the case of a two body decay, such as , the decay spectrum at rest is mono-energetic, so the boosted energy spectrum merely reflects the angle of the pion in the rest frame. In the case of a three body decay, such as or , the boosted energy spectrum is a combination of both the rest frame angular distribution and energy spectra. In all of these cases, we can parameterize the decay spectrum in two parts, a constant part that is unaffected by the handedness of the tau, and a polarization dependent part that changes sign depending on the handedness of the tau:
In the case of the pion (kaon), we can describe this with one parameter, , which characterizes the polarization dependent term. We get the following decay spectrum for :
where and are the masses of the and the decayed hadron respectively, and is the hadron's energy scaled by the energy ().
In the case of the leptons, neglecting mass differences, we can describe the energy spectrum with the three Michel parameters , and . Here the parameter describes the non-polarization-dependent term, and and describe the polarization dependent terms. We get the following spectrum for :
These decay spectra are combined with the production cross sections to get theoretical decay distributions:
These can be rewritten as:
which clearly illustrates the importance of high tau polarization for measuring the polarization dependent terms. For decays, the sign of the polarization dependent term is reversed due the opposite helicity of the anti-neutrino. However, since at the taus are produced in pairs with opposite helicities, if the direction is used as the lab frame variable the and decay spectra are identical.