Semileptonic Decays

Having discussed purely leptonic decays, I now turn to semileptonic decays of the form where denotes any hadronic system, and is a pseudoscalar meson containing a heavy (unstable) quark and a lighter antiquark (e.g., . Such decays are of particular significance because the leptons they produce are in many reactions the only signals for heavy flavor production. The basic diagrams for semileptonic decays are shown schematically in Fig. 3. In Fig. 3(a), the heavy quark Q undergoes an independent semileptonic decay, while in Fig. 3(b), it disappears through annihilation with a ``spectator'' antiquark in the initial into a pair: in most cases, the mechanism (a) probably dominates. The rough size of the incoming state is if (or otherwise): hence its constituents are typically off-shell by an amount , giving the , a Fermi momentum . So long as (as for , it should probably be adequate to estimate the process of Fig. 3(a) by approximating the initial as free. For sufficiently large , inclusive decays in which all possible final hadron systems are summed over, may be estimated to , by treating the final as free. In this approximation (and for simplicity taking at first , the energy spectrum is analogous to the spectrum in decay, and is given by (for negatively charged ) or (for positively-charged ), where (10). Thus or 0.35 (note that is conveniently a Lorentz invariant for massless and unpolarized ): for finite is softened by a factor . In QCD perturbation theory, the produced by the emission need not be on-shell, but may have a spectrum of invariant masses extending up to the kinematic boundary , thus softening the spectrum produced in the decay. Assuming , the production of a lepton with energy fraction x requires that the total transverse momentum emitted in gluons by the outgoing be . This restriction forces any gluons emitted to be both soft and nearly collinear with the , yielding double logarithmic terms. Keeping only leading terms at , the correction becomes [19,20]

(which implies a correction ) to (11). Emissions of multiple soft and collinear gluons are independent, and their effects thus exponentiate when summed to all orders in (in the leading double log approximation), yielding

(Since this approximation is formally valid only for , modifications e.g., to by polynomials in cannot be distinguished). Note that whereas the form (2) exhibited an unphysical divergence at , the exponentiated form (3) goes to zero as (after a peak at , corresponding to a produced invariant mass ). This radiation damping results from the impossibility of producing with no gluon emission. Eq. (3) assumes : for , divergences from collinear gluon radiation are regulated, and the correction factor becomes

(An analogous result obtains in decay with (e.g., Ref. [22]). The double log exponentiation (3) is known only to leading order: exponentiation of the single log terms appearing when is proven to all orders. For large , one may also estimate the total semileptonic decay rate from Fig. 3(a) using perturbation theory. To , the correction due to gluon emissions is analogous to that in decay and given by : integration of the approximate differential decay spectrum (2) yields (12). The excellence of this approximation suggests an estimate of the correction summed to all orders in by integration of (3), yielding [20] , with and co erf (13). Note that no infrared divergences appear at any order in : they are cancelled by summation of all possible gluon configurations (real and virtual corrections) in the final state. (If the incoming were massless, then it could be degenerate with states containing in addition collinear gluons; uncancelled infrared divergences would remain unless the contributions of such states were included. The cancellation of infrared divergences associated with scattering or decay of a single incoming massive quark on summation only over possible final states has been proved to all orders in [23]. The absence of infrared divergent terms sensitive to the structure of the initial state supports the approximation of an isolated initial . It is also noteworthy that the found here contains no uncancelled ultraviolet divergences or terms: the exchange is thus safely approximated by a four-fermion interaction. The reason for this is that gluon corrections to act only at the vertex: they are ultraviolet finite by virtue of the QCD Ward identity (since is color singlet) (14) , and cannot depend on (but only on the invariant mass . The ultraviolet behavior of electromagnetic corrections to (e.g., is somewhat more complicated [24]: because is charged, virtual photons may be exchanged between and/or and , so that a box diagram occurs. Such a diagram is ultraviolet divergent when so (local four-fermion interaction). However, in the particular case of decay with a coupling, the divergences and terms cancel. This may be proved by showing that no divergences appear as . In this limit the local interaction may freely be Fierz transformed and for couplings, may be written as with the same couplings. This interaction may be pictured as occurring by exchange of an (infinite mass) fictitious neutral vector between and . Then photon corrections occur only at the vertex, and are ultraviolet finite because the current is (approximately (14) ) conserved (and clearly cannot depend on the fictitious . If, however, the original coupling is not of the form, then Fierz rearrangement will introduce non-vector currents, which are not conserved, and usually lead to ultraviolet divergences (hence electromagnetic corrections to neutron decay with phenomenological fields and A coupling are not ultraviolet finite). Further, only if in the electric charges is Fierz rearrangement to neutral currents possible. Electromagnetic corrections to with e.g., are not ultraviolet finite as : to obtain renormalizable corrections, one must use the complete Weinberg-Salam model in this case, and include, for example, photon emissions from the intermediate . After renormalization, electromagnetic corrections to may well contain terms.

[ Figure 3 ] Schematic diagrams for semileptonic decay of a meson .

The considerations of the previous paragraph were based on perturbation theory. Hadronic effects presumably become important when gluon emissions have degraded the final invariant mass to within of its mass shell. The behavior of for is particularly sensitive to this region, so that perturbative estimates become unreliable (and, for example, the peak at from (3) with an effective coupling may be swamped by hadronic effects). Nevertheless, experimental electron spectra from decays may be adequately fit by the perturbative estimates but including a Fermi smearing for the initial (and taking , (15) ): setting leads to a slightly worse fit. For example, the experimental [25] while with one obtains and with to and by estimating higher orders as in (3). For -quark decays, hadronic effects should be comparatively unimportant: assumed a dominant coupling, one may therefore make a firm prediction of for this case (if , it would become ).

The discussion of semileptonic decays above has assumed the mechanism of Fig. 3(a). The process of Fig. 3(b) would yield different results. In direct analogy to the discussion of purely leptonic pseudoscalar meson decays above, the rate for Fig. 3(b) (with vanishes by helicity suppression when the energy of the final hadrons goes to zero. To determine the recoil hadron energy spectrum, one must estimate what fraction of the initial energy is effectively carried by the and (when the is ``probed'' by a of invariant mass ). A rough guess would be that the gluons in effectively carry an energy (19). In that case, Fig. 3(b) would suffer a helicity suppression . In addition to such ``primordial'' gluons, the presence of the interaction may induce gluon radiation whose rate may perhaps be estimated by perturbation theory. Starting from a color singlet pure system two gluons must be emitted to conserve color (so that occurs), although one of these gluons may be arbitrarily soft. In practice, the initial state must always contain some gluons albeit arbitrarily soft, which allow the annihilating to be in a color 8 state, and thus require no second gluon emission. (Formally, the process exhibits an uncancelled infrared divergence (16); this is presumably cancelled when the appropriate composite initial state is included). I shall assume that the process may be estimated by considering only single hard gluon emission, and that the amplitude for the soft gluon processes which account for color conservation is one. Then the single gluon energy spectrum [26] , and the total rate is given in analogy with by , where the effective color magnetic moment is taken as (19) for . This rate is to be compared with the (free quark approximation) result for Fig. 3(a), after accounting for the different mixing angles (hence effective ) sampled in the two cases. For decays, Fig. 3(b) is Cabibbo suppressed, while Fig. 3(a) is not. In decays, on the other hand, neither Fig. 3(b) nor Fig. 3(a) suffers Cabibbo suppression. Taking the estimates given above suggests , which is not competitive with Fig. 3(a). An evident consequence of the mechanism of Fig. 3(a) is that semileptonic and decays should be similar. (Experimental results support the similarity of lepton spectra [25] and probably total rates in semileptonic and decays). If Fig. 3(b) were important in decay, then its semileptonic decay width would be larger than those of and ; (although if were important, the energy spectrum would fortuitously probably be quite similar, with . For mesons containing quarks, only states may undergo semileptonic decays through Fig. 3(b) without additional mixing angle suppression: but there the ratio of Fig. 3(b) to Fig. 3(a) is probably still smaller than for decays.

One may be tempted to apply the basically perturbative analysis of semileptonic decays given above to decays. The mass is, however, far too small for such a procedure to be profitable: one should instead consider explicit exclusive hadronic modes, such as . In describing exclusive semileptonic decays, one must sum coherently (20) over possible quark subprocesses: by construction, all subprocesses yield the same final state. Decays of the form (which for zero invariant mass occur only through the vector weak current) may often be related to each other and to electromagnetic processes using approximate SU(3) invariance. For example, isospin invariance constrains (up to 0(1%) phase space and electromagnetic radiative corrections). However, for decays such as the number of possible contributing amplitudes precludes such relations. The mechanism of Fig. 3(a) suggests that : on the other hand, the process of Fig. 3(b) can occur (with ``valence'' quarks) only for (yielding an state), suggesting that the rates for the and decay may differ. Fig. 3(b) probably suffers little helicity suppression with respect to Fig. 3(a) in this case. (Experimentally, has a measured branching ratio , while the ev branching ratio is ; it would be expected at a level .

In Fig. 3(a), is not necessarily the heaviest quark in . For example, a meson may decay either through (to ) or through (to , where in the latter case, follows. Unless mixing angles strongly inhibit decays of the heavier quark, these will, however, probably dominate (for the () case, .

The discussion of semileptonic decays above has implicitly assumed that the decaying meson mass . When approaches , the semileptonic decay rate (from Fig. 3(a)) will increase, as the intermediate becomes closer to its mass shell: in the limit , is enhanced by a factor six through this effect. When , decays to real should dominate (since they involve fewer powers of the semiweak coupling constant): the resulting three hadron jets should provide a spectacular signature for production of such heavy , e.g., in collisions [28]. A simple calculation of the rate for the decay indicates that it grows , due to the production of longitudinally-polarized so that for , a produced would undergo weak decay before being confined into a pseudoscalar meson: it could therefore, for example, transmit polarization information from its production to its decay products. However, the presence of fermions with is not consistent [29] with the standard weak interaction model used to calculate the decay rate. The semileptonic decays considered above were all of pseudoscalar mesons. However, for , vector (i.e., ) states should decay through with an branching ratio [30]. As mentioned above, with masses close to should exhibit large branching ratios.

previous  l  next