Purely Leptonic Decays

I now consider direct couplings of mesons to virtual , as illustrated in Fig. 2. These couplings enter in the leptonic decays and in exclusive heavy lepton decays . Couplings of to vector mesons are related by (weak) isospin rotations to the corresponding couplings, which may be measured in . The coupling of Fig. 2 is proportional to the amplitude for the (valence) in to annihilate ``at a point'' into , and hence to the ``wavefunction at the origin'' . The complete coupling should presumably involve the product of this amplitude with the amplitude for the other constituents (or bag) of the state to disappear (into the ``vacuum''). A possible rationale for neglect of this latter term might be that without the ``valence'' , the contents of the meson are indistinguishable from fluctuations which would occur in the ``vacuum'' regardless of the presence of the meson [5]. For mesons in which the masses are sufficiently large compared to , the should be non-relativistic and nearly on-mass shell (so that admixtures of e.g., states into the wavefunction are negligible): in this case a non-relativistic may be defined and estimated using the Schrodinger equation from a potential (or bag) model. This treatment is probably suitable for heavy resonances (e.g., ; denoted generically throughout). (At high masses, interference between and the dominant leptonic decay mechanism may be revealed by longitudinal polarizations measured through decay product angular asymmetries [6]. Mesons such as will probably not have significant leptonic branching ratios, since their lowest-lying states will presumably be pseudoscalar). For mesons containing any light quarks, this approach probably fails. When constituent particles may be off their mass shells (as in any relativistic formulation), the meaning of becomes unclear: the relevant integral of the full Bethe-Salpeter wavefunctions depends on the invariant masses of the annihilating particles. In addition, the large size of a meson containing light quarks precludes any reliable estimate of a gluon exchange potential.

[ Figure 2 ] Schematic diagram for purely leptonic decay of a meson .

In most cases involving light quarks, Fig. 2 may be treated only by phenomenological means. The coupling may be deduced from by an isospin rotation; the result agrees with the measured decay rate. There is now also reasonable experimental evidence for a resonant decay [7]; its rate is consistent with a coupling the coupling, as expected in the chiral symmetry limit. On the other hand, in the non-relativistic limit, vanishes for the , since it is a -wave state. The experimental absence of such a suppression suggests that this is not a relevant limit: in a relativistic state, the "lower components'' of the Dirac spinors (which essentially correspond to an -wave state (3) ), become important, leaving no trace of the nominal -wave assignment in the chiral symmetric limit . Note that couplings to mesons, such as , would be ``second class'' (proportional to , and may occur only at the level of isospin violations [8] so that the rate for e.g., should be very small (perhaps times the rate).

Pseudoscalar mesons may couple to spin-0 through the divergence of the axial vector current. The rate for their resulting leptonic decays is given by , where parametrizes the coupling. The factor in this rate represents a ``helicity suppression'' which arises because the couples to the left-handed , but the spin of the original constrains the total angular momentum of the final state to be zero, thus requiring that the helicity of the be opposite to its spin, and introducing a factor into the decay amplitude. This factor renders leptonic decays miniscule for . Helicity suppression also affects the initial annihilation: the must couple to left-handed , but the total state has spin-0. This introduces a helicity suppression factor just as in the divergence of the (free quark (4) ) axial vector current : when , the axial current is conserved and the leptonic decay rate vanishes. The complete coupling is the product of the helicity suppression factor and (which gives roughly the inverse volume of the state). Typically, the value of depends on the reduced mass of the (5) , suggesting that all mesons containing light quarks should have similar (irrespective of the heavy quark mass). (The similarity of the wavefunctions for is supported by the similarity of their electromagnetic charge radii). In any non-relativistic treatment, the annihilation helicity suppression factor differs from one only by terms where is the binding energy, which cannot consistently be kept. If the mass of is dominated by the rest mass of one of its constituent quarks (as for mesons), then no significant helicity suppression should occur. Ignoring helicity suppression for and taking suggests (Ref. [10]), where . Experimentally, . Introducing a , suppression factor into gives if . Taking from the charge radius gives roughly the correct for the helicity unsuppressed coupling . Using the charge radii to estimate yields too large a value for : this suggests either that the helicity suppression is numerically important, or that the approximate relation between size and ``wavefunction at origin'' fails for the ultrarelativistic states. Admission of the ultrarelativistic nature of the could well allow is determined by the difference between the Bethe-Salpeter wavefunctions corresponding to the upper components of Dirac spinors and their lower components (6) (Ref. [11] which might, for example, be proportional to the larger quark mass. For heavy pseudoscalar mesons, the rates for purely leptonic decays to are probably rendered negligible by the helicity suppression factor. However, decays to leptons may well be significant. For charmed mesons, only the decay escapes Cabibbo suppression, while for -mesons, mixing angles probably suppress all but . Since these two states contain no (valence) quarks, non-relativistic estimates of their wavefunctions are plausibly adequate: taking a logarithmic interquark potential yields MeV, MeV, while a linear potential gives MeV, (7). To deduce the corresponding leptonic branching ratios, one must estimate the total nonleptonic decay rates as discussed below. decays may provide useful signatures for heavy meson production [12]. The estimation involves similar relativistic complications as for . One possible (but dubious) approach proceeds as follows: the behavior of the spin-0 axial vector spectral function (spin-0 decay rate through the axial vector current) may be estimated from QCD perturbation at high . This estimate may provide a finite energy sum rule for the integral of the actual spectral function, which certainly does not follow the perturbative form at low . Then the actual spectral function is approximated by a single , and the resulting integral compared with the perturbative estimate in the finite energy sum rules. A direct application of this procedure yields MeV [13]. It is very difficult to make a serious estimate of the errors in this result, but they are probably at least . This and the non-relativistic estimates given above suggest that in very heavy pseudoscalar mesons will not rise sufficiently to counteract the larger helicity suppressions in leptonic decays, so that their branching will become negligible. One possible circumstance in which this phenomenon is evaded would occur if the lowest-lying meson carrying a new flavor had rather than (this appears likely only if the new quark had : in that case decays would be significant. Note, however, that couplings could be measured in without helicity suppressions if a suitably placed heavy lepton exists. (Similar information could perhaps also be extracted from diffractive neutrino production . It is also possible that at high , the process (where is a pseudoscalar heavy meson) may occur through the divergence of the axial vector ; current: its presence would be revealed by isotropic angular distributions of decay products with respect to the original direction.

One effect which may in part overcome helicity suppression of leptonic pseudoscalar meson decays is emission of hard photons from the incoming . In the discussion of hadronic decays below, we will encounter similar considerations for gluon emission. The vector nature of the photon coupling prevents soft photon radiation from changing the spin of a state: however, hard photon emissions from an initial may carry away angular momentum, leaving effectively a spin-1 ``meson'' state, whose leptonic decays suffer no helicity suppression (8). If no photon emission occurs, then the leptonic decay rate for a pseudoscalar meson is given by , where quark mixing angles are absorbed into . Soft photon emissions yield (this correction exponentiates when summed to all orders in . Hard photon (``structure dependent'') radiation potentially provides a much larger correction, since it can overcome the helicity suppression. In practice, [Ref. [15]], while (Ref. [16]). For , the vector current contribution may be estimated using CVC from (although if is large there may be significant deviations from CVC [17]). In other cases, the rates are estimated by phenomenological Lagrangians involving intermediate fields. One may also attempt a static quark approximation [14]. The amplitude for a photon to reverse the spin of a quark in is proportional to its effective magnetic moment (9). Only photons emitted incoherently from the in may change the total spin. A naive calculation in the free static quark approximation then suggests : taking GeV, this overestimates and by about a factor ten. (If instead , a gross overestimate results). The decay is Cabibbo suppressed; is not, but probably has a branching ratio . For heavy ( mesons probably : thus , which becomes relatively smaller as .

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