Leptonic widths

An interesting empirical feature concerning the leptonic widths of all the vector mesons (ground states of each family), old and new, was noticed by Sakurai (1978). For the decay of a vector meson made of qjqj 

For an 5-state, V (0) 0 and (r)p decreases with r so, for a normalized wave function, we have approximately 

Although very crude, this method gives the correct dependence on the particle mass for essentially any non-pathological potential at least for the ground state. To the extent that we can further assume that the mass of the bound state is roughly twice the mass of the constituent quark and that the form of the potential does not change in ranging from the q to the T, we can compare the result (12.4.3) with the empirical form (12.4.1). 

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Let us turn now to the theoretical calculation of the leptonic widths in the SM in lowest order. 

Aside from its large mass (3097 MeV/c ), the most remarkable property of the J/ is its extreme narrowness (long lifetime) as compared with ordinary strong interaction resonances. In fact, while the latter tjrpically have widths of the order of few hundred MeV, and the widths seem to grow (linearly ) with mass, the J/ I (3097) while having a large mass, has a total width of only 86 6 keV The leptonic width, on the other hand, into the e+e channel is about 5.36 0.29 keV which is typical of vector mesons. 

The leptonic width for a non-relativistic qq system in a vector 1 state of mass My to decay into via one (virtual) photon,... 

Note that the u can be chosen as real if V is real. For computing the leptonic width or the hyperfine splitting of S-states, one often needs the probability of finding the quark and the antiquark at the same place, i.e.,... 

It led to a prediction that the number of different sorts of neutrino (equivalent in standard particle physics to the number of families of quarks and leptons) is less than 4 and probably no more than 3. This prediction was subsequently confirmed (subject to slight reservations about differences between effective numbers of neutrino species in the laboratory and in the early Universe) by measurements of the width or lifetime of the Z° boson at CERN in 1990. 

The purely leptonic hydrogen atom, muonium, consists of a positive muon and an electron. It is the ideal atom, free of the nuclear structure effects of H, D and T and also of the difficult, reduced mass corrections of positronium. An American-Japanese group has observed the 1S-2S transition in muonium to a precision somewhat better than a part in 107. [10] Because there were very few atoms available, the statistical errors precluded an accurate measurement. The "ultimate" value of this system is very great, being limited by the natural width of the 1S-2S line of 72 kHz, set by the 2.2 nsec lifetime of the muon. 

To obtain an estimate for the branching fraction we require the partial width r(W ud). As will be discussed in Chapter 9 quarks are assumed to couple to the W like the leptons (aside from the complication of Cabibbo mixing and its generalization, which we here ignore). To be precise we shall go through the calculation of r(W — in full detail... 

A summary of all two-body (leptonic, photonic, gluonic and partonic) Higgs decay widths is shown in Fig. 6.1 (from Pranzini et al, 1989) where one has taken... 

Fig. 6.1. The approximate partial decay widths of H to all two-body partonic decay modes. The dashed curves show decays to charged leptons, the solid curves the decay into hadrons as computed from the contributions to each quark separately. The dot-dashed curves illustrate the two-gluon and two-photon decays. See text.

Light neutrinos, i.e. those with mi, < Mz, will contribute to the Z , width via Z° vu and thus one can estimate the number of such neutrinos Nv (i.e. the number of leptonic doublet generations) from consideration of the total and partial Z widths. We shall discuss two rather different approaches. 

Table 11.4 summarizes the properties and main decays of the various levels. Note that the total width is in MeV, whereas the leptonic decay width Fe is in keV. 

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