Theoretical Motivation

One of the great triumphs of high-energy physics in the last decade is the impressive agreement between theoretical predictions of the Standard Model of electroweak interactions and experiments performed at LEP, SLC and elsewhere. Most spectacular was the accuracy of the prediction for the mass of the top quark before it had been directly measured at FNAL. This prediction took the measured value for the Fermi coupling constant, G_F, along with $\alpha$ and M_Z, as starting points and obtained a value for the top quark mass by fitting to electroweak data. The value obtained by this method is $m_t=177\pm20$GeV [5] to be compared with the current CDF and D0 average of $m_t=175\pm6$GeV. Indeed it is input from the charged-current sector of the theory, coming solely from G_F, that is the source of a quadratic sensitivity to the top quark mass and therefore drives the prediction [6]. The value of G_F, in turn, is best determined from the muon lifetime where it is often termed $G_\mu$. The relationship is

$$\Gamma_{\mu} = {1 \over \tau_{\mu}} = {G_{\mu}^2 m_{\mu}^5 \over 192 \pi^3} ( 1 + \delta)$$ (1)

where the term $\delta$ represents QED radiative corrections. Curiously, a precision measurement of the muon lifetime provided strong bounds on the top quark mass.

For many years, the errors on $\alpha$ and G_F were negligible compared to that of M_Z. Now the situation has changed and the error on M_Z is 1.8 MeV, or about 2 parts in 10⁵, which is comparable to that of $G_\mu$. As Stuart and von Ritbergen point out in their recent definitive paper [1] on the relation between the Fermi coupling constant and the muon lifetime, the expected error on the Z mass measurement at LEP was an order of magnitude larger than that which was ultimately achieved. Indeed, measurements at a muon collider may well improve our knowledge of the Z mass by another order of magnitude. It therefore makes sense to improve the measurement of $G_\mu$ to the extent that current facilities and technology allow, and to the extent that current theoretical understanding permits us to interpret the result.

Until recently, the relative error on the Fermi constant of 1.7  x  10^-5 was dominated by a theoretical error of 1.5  x  10^-5, which was an estimate of unknown 2-loop QED corrections in the $\delta$ term above. In their most recent paper [1], von Ritbergen and Stuart published results for the unknown corrections as well as confirmation of terms that had already been calculated by other authors. Completion of the 2-loop QED corrections required the calculation of matrix elements for the processes $\mu^-\rightarrow \nu{\mu} e^- {\bar \nu_{e}}$, $\mu^-\rightarrow \nu{\mu} e^- {\bar \nu_{e}} \gamma$, $\mu^-\rightarrow \nu{\mu} e^- {\bar \nu_{e} \gamma \gamma}$, $\mu^-\rightarrow \nu{\mu} e^- {\bar \nu_{e} e^+ e^-}$, with up to 2 virtual photons. The corrections are dominated by Feynman diagrams of two kinds: those which purely are photonic, containing no charged-fermion loops, and diagrams containing an electron loop or e⁺ e^- pairs in the final state. Diagrams involving muon loops and virtual hadrons are several orders of magnitude smaller. Overall, the residual theoretical uncertainty arising from missing higher order QED corrections are no larger than a few tenths of a ppm and thus the theoretical uncertainty on the extraction of the Fermi constant has been entirely eliminated (see also [7] and [8]).

Stuart and von Ritbergen also discuss the electroweak corrections to muon decay, which can be separated in a natural way from those involving QED alone. The electroweak corrections are contained in the term $\Delta r$, where the equation

$${G_{F} \over {\sqrt 2}} = {g^2 \over 8 M^2_W}(1+\Delta r)$$ (2)

defines the Fermi constant in terms of the Standard Model weak coupling constant g. The authors summarize their work by giving an unambiguous prescription which relates $\alpha$ (at low q²), the muon lifetime and the Z mass to renormalized Standard Model parameters.

The importance of G_F does not end with the top quark mass prediction. Now that the top quark mass has been directly measured and incorporated into the Standard Model, our knowledge of G_F will ultimately help in understanding in detail, the Higgs sector, through its radiative corrections. G_F is also very useful in analyzing extensions to the Standard Model as discussed recently by Marciano [9]. It is the input from G_F, through the $\rho$-parameter, that is responsible for the strongest constraints on sparticle mass splittings in SUSY models. But in the end, we simply do not know what arrangements have been made by nature, and it is our belief that the measurement of fundamental constants, that can be performed with great precision, will be of enduring value.