3. Pseudoscalar Meson Decays to Charged Lepton Pairs

3.1. Introduction

Gauge invariance forbids one-photon intermediate states in ; the largest electromagnetic contributions are of order . Thus NWI may be detectable here. In the past has been studied extensively [10], and stringent bounds set on the strength of NWI. The analysis of differs from that for since it has no electromagnetic component (except through higher-order weak interactions).

Experimentally, the only decay to have been observed is , and this in only one experiment [11] with few events. Measurements of are currently being attempted [12] which should be sensitive to , close to the theoretical lower bound. We find that pseudoscalar NWI could probably be detected if they occur in .

3.2. The Matrix Element

In order to treat the NWI component of , we ignore the neutral weak intermediate boson (Z) propagator and assume a general local first-order interaction. Thus

reducing to

The dimensions of the coupling constants are such as to be directly comparable with charged current ones: , and so on. There is no loop momentum integration in so we need know the weak form factor only on its mass shell. invariance implies .

An interesting test of invariance could be achieved by observing the electron asymmetries from the decays of in . Lee's model [13], however, predicts (and ) violation in Higgs' particle interactions, and thus violation here. Invariance under demands a final state, and, assuming no final-state interactions, violation of this symmetry would be manifest [46] in or in . There can be no violation (a signal for NWI) without violation. Nevertheless, net lepton helicities could be an important effect if scalar interactions take part in the decay. invariance provides no useful constraints in these processes.

The intermediate state in has been discussed by a number of authors; a clear survey is given by Quigg and Jackson [14]. They take ( model)

where and are the photon momenta and polarizations and is the mass of the vector meson which is assumed to saturate them. This yields ( is the pseudoscalar meson mass, the lepton mass)

For we have

where is the intermediate state component [14]

and the photon loop momentum, the pseudoscalar meson momentum, and and the lepton and antilepton momenta. We work in the frame is the photon loop momentum. The weak component is

In helicity representation,

where is the absorptive part, the dispersive part of the pure electromagnetic contribution. Hence

The first term is pure electromagnetic, the third pure weak, and the second an interference term. We discuss numerical estimates in subsect. 3.4. We find that for many values of , the interference term tends to be the smallest, since it involves a lepton mass factor relative to the pure weak term.

Since we have assumed , the in must be off-shell, and thus contributes only to the dispersive part of the amplitude. The rate (3.9) is given by

The form of this expression requires an accurate knowledge of , since any NWI can always be confused with errors in this quantity. The value of is nearly model-independent, since is known, but the calculation of necessitates a knowledge of the off-shell behaviour . We discuss these in the appendix.

3.3. Modifications to the Matrix Element

We discuss in appendix A.1 the effect of the intermediate state on in the case . Other decay products need not be considered, since they contribute to only at higher orders in . The intermediate state is forbidden by invariance. We shall discuss in subsect. 6.1. The and intermediate states may safely be ignored.

The Higgs scalar boson could lead to an interesting term. It appears in any theory in which the Higgs mechanism is used to break the gauge symmetry. This may also be achieved with quantum corrections to the classical potential (requiring no physical Higgs particle). The coupling of a single to other particles is given generally by

where denote particle types.

For a non-ghost can, in principle, be charged. It should then, however, contribute significantly to weak semileptonic decays, allowing a stringent bound to be put on its mass. We shall not do this here, since we know of no convincing models which involve a charged .

For invariance constrains or . Choosing a allows a pseudoscalar to contribute to pseudoscalar meson decays at first order. It is possible to construct models in which this occurs. A scalar , however, cannot affect meson decays, simply because of invariance. This is the case in the Weinberg-Salam model. We shall discuss the role of further in subsect. 6.5.

3.4. Numerical Estimates

Since the weak component of is purely dispersive, it cannot reduce any of the unitarity limits for these decays (values of (3.10) for ):

Experimental violation of any of these bounds does not appear to be comprehensible in the framework of present theory. It would require both that is violated, and that the decaying meson is not an eigenstate of (as in the case), in contradiction with, for example, the quark model.

The model which we have assumed above leads to , a value rather insensitive to , which is satisfactory. We also obtain . Ignoring NWI, we then have

Experimental information on these decays (given in sect. 2) is so far rather scanty, and so no good bounds on the strength of the NWI can be obtained: yields (1) . We should not, however, ignore weak corrections especially in and : both need only for weak and electromagnetic components to be comparable. Such a value would be expected for pseudoscalar NWI. Fig. 1 contains a plot of against .

A measurement of could yield much information on both electromagnetic and weak components in these decays. The model predicts for increasing by when MeV. Setting , however, yields (2) is quite sensitive to NWI.

[ Figure 1 ] Graph of against the dimensionless weak coupling constant . The upper full curve is for , the lower one for .

is an isovector interaction, an isoscalar one. Since vector NWI cannot contribute to , the Weinberg-Salam model predicts and . If this is the correct coupling constant, then we cannot expect to detect its presence in any conceivable experiment.

It is difficult to reconcile the theory (see ref. [16] for an exception) with the experimental value of . It could be explained by setting , but this would imply (if . It could also be explained by setting MeV in . Such a value of would lead to . We note, however, that standard form factor measurements indicate MeV in the case.

3.5. The Inverse Process

may also provide information on NWI. We have simply

which leads to

This yields

i.e. not completely unobservable. Experimentally this process will be difficult to detect, but by polarizing colliding beams, it might be possible to enhance it over one-photon processes sufficiently to allow detection. Background from could prove troublesome, however.

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