Non-Leptonic Decays

Fig. 4 shows schematically various processes which contribute to weak nonleptonic decays of mesons (21). The relative importance of these mechanisms varies widely between different decays, in principle allowing their effects to be disentangled. In addition, for sufficiently large , it should be possible to distinguish directly between the processes of Fig. 4 by the structure of the final states that they yield [31] (for example, Fig. 4(a) would lead to predominantly three-jet final states. while Fig. 4(c) would usually give rise to two-jets). It will turn out that all the mechanisms in Fig. 4 should contribute to decays. Non-leptonic decays are probably dominated by Fig. 4(b); in decays, Figs. 4(a) and 4(b) are probably competitive, and in decays Fig. 4(a) probably dominates. For heavier mesons, Figs. 4(b, c) should progressively become less important.

[ Figure 4 ] Schematic diagrams for nonleptonic decay of a meson .

In the free quark approximation, Fig. 4(a) implies an decay width , where gives the effective number of generations to which the intermediate may decay and corrections from non-zero final quark masses have been ignored. Any mixing angle factors appearing at the vertex have been absorbed into an effective . Note that existing examples (c, b) suggest that these mixing angles arrange all but of all decays to be to the closest in mass to (although it appears that . Combining with the semileptonic decay rate implied by the analogous diagram Fig. 3(a) (which should always dominate over Fig. 3(b), except perhaps for some decays, as discussed above) yields in the free quark approximation a semileptonic branching ratio given just by the semileptonic branching ratio . Thus Fig. 4(a) implies that the semileptonic branching ratios for all mesons containing a given heavy quark should be equal. The large experimental violation of this equality in and decays demonstrates that these receive important contributions from processes other than Fig. 4(a) (see below); for heavier quarks, however, Fig. 4(a) should dominate. If mixing angles always arrange to be closest in mass to , then decay of a very heavy through Fig. 4(a) would involve many emissions: the probability that a lepton would be produced in one of these decays is quite high for an th generation ). Gluon emissions in QCD modify the total width for Fig. 4(a) obtained in the free quark approximation. Just as for Fig. 3(a), the corrections are infrared finite by virtue of the non-zero initial mass. To color averaging decouples the contributions from gluons emitted by the decay products and by the (interference diagrams are proportional to the color matrix , since carries no color). Hence, to , remains proportional to the semileptonic branching ratio . This corresponds to an correction (Ref. [31], p. 447; Ref. [32]), to the non-leptonic decay rate. Note that to this order, the correction is finite as : in higher orders, terms may appear (although all ultraviolet divergences must be renormalizable within QCD alone, since is colorless). The decoupling of contributions from gluon emissions from and does not persist beyond : the correction must be obtained by an explicit (rather complicated) calculation (22). The result will determine the relevant scale for the effective coupling appearing in to : this scale is presumably rather than , but may involve large numerical constants. In going beyond one must presumably account for gluon exchanges between the (or its decay products) and the ``spectator'' in the original . To , color averaging again cancels diagrams involving a gluon exchanged between the decay products and the : hence to , spectator effects do not affect the semileptonic branching ratio. The role of the ``spectator'' is similar to that of atomic electrons in nuclear beta decay: so long as the momenta of the Q decay products are much greater than the inverse size of the initial state, the spectator's effects will probably be negligible (23). If one of the decay products is correlated (in flavor, color and momentum) with the spectator, then interference (e.g., Pauli exclusion) effects may occur (24). Since the average momentum of the decay products , they should lie in a different region of phase space from the spectator, with negligible interference, unless . When occurs in a meson, the two final antiquarks have the same flavor; in a meson, they do not. The nature of the interference between the two in decay depends on their colors: if the properties of the decay are unrestricted, then the Pauli exclusion between with identical colors enhances the total decay rate (cf., Ref. [33]). The same result obtains if one treats the final state as two color singlet systems, and allows interference between production of the color singlet systems, even if the colors of their constituent differ (i.e., and systems are treated as indistinguishable, although their internal quarks have different colors). If now, the properties of the final state are restricted, other patterns of interference are obtained: for example, if the effective Hamiltonian transforms as 6 under (so that it is antisymmetric under the interchange , then destructive interference occurs [34] if the quark momenta are similar. Similarly, in decays, the components may be suppressed by such interference effects. Up to corrections the three final quarks in Fig. 4(a) should give rise at sufficiently high to three separated hadron jets: the spectator carries only a small fraction of the total energy for large , and so cannot initiate a jet. In the free quark approximation, production of an pair (30) at rest (threshold) with decays according to Fig. 4(a) gives the shape parameters [31] . For comparison, a two-particle final state gives , while an isotropic final state has . (A three-body final state with uniform matrix element over available phase space gives . Two-jet decays, as may result from Fig. 4(c), would give . Subsequent decays of heavy quarks produced in these decays should usually not modify the significantly (since , is typically , the secondary decay products do not subtend large angles),although identification of the decays would reveal the flavor of the quark jet, and thus aid in discriminating between the processes of Fig. 4. The in the free quark approximation quoted above are modified by gluon emissions from the final partons, and by hadronic effects. Simulations [31] of these corrections suggest that for , processes near threshold should be clearly distinguished from the usual two-jet processes by the comparative isotropy of their final states, as revealed for example in the distributions . For , it appears that the three-jet final states of Fig. 4(a) should also be distinguished from the two-jet ones of Fig. 4(c), allowing direct discrimination between these decay mechanisms for mesons containing quarks.

Consider now the process illustrated in Fig. 4(b), which is analogous to (though much more effective than) the semileptonic decay mechanism of Fig. 2(b). In contrast to Fig. 4(a), the contributions of Fig. 4(b1) and Fig. 4(b2) depend critically on the quantum numbers of the initial state. At momenta , the Lorentz structures of the diagrams (b1) and (b2) are identical (they are related in this limit by Fierz reordering of the effective four-fermion vertices). The diagrams differ in flavor, and also in that Fig. 4(b2) requires the annihilating to be in a color singlet state, while Fig. 4(b1) does not. As discussed above for Fig. 2(b), the inevitable presence of soft gluons from the initial state means that the initial populate all color states according to their statistical weights. (Summing over final colors in Fig. 4(b2) gives a factor 3; the necessary averaging over the initial colors introduces a cancelling factor 1/3, so that Figs. 4(b1) and Fig. 4(b2) do not differ in their color combinatoric weights (24) ). For -quark decays, Fig. 4(b1) contributes in sd mesons and Fig. 4(b2) in mesons (25). For -quark decays, Fig. 4(b1) can contribute in mesons, but Fig. 4(b2) is Cabibbo suppressed by a factor in mesons; it can nevertheless contribute to meson decays. For -quarks, a similar pattern obtains: and mesons may decay through Fig. 4(b2) without additional mixing angle suppression relative to in Fig. 4(a); only and not mesons may decay without suppression through Fig. 4(b1). This pattern would continue for quarks. In the free quark approximation, the rate for decay of a pseudoscalar meson by the processes Fig. 4(b1) and Fig. 4(b2) may be estimated in analogy to Fig. 1 as and , where mixing angle factors have again been absorbed into The factor arises through helicity suppression, as in Fig. 1. These decay rates are to be compared with the rate for the three-body decay Fig. 4(a). The ratio of to this is : the factor 200 results from suppression of Fig. 4(a) relative to Fig. 4(b) by virtue of its three-body final state. As discussed above, for mesons containing a light spectator probably . Thus, even if always ; would fall below for : however, for comparatively small , Fig. 4(b) may well dominate over Fig. 4(a). In decays, Fig. 4(b) should be at least comparable in importance to Fig. 4(a) (the naive free quark estimate suggests that it should dominate). As mentioned above, the contribution of Fig. 4(b2) is Cabibbo suppressed in decay (so that it is unimportant in the total width, but should be significant in the partial width for final states, as discussed below). In decays, , and . Even with pessimistic choices for and this ratio is , and it is very possibly . In this case, the decay width would be dominated by Fig. 4(b1), and be larger than the width, which is dominated by Fig. 4(a). The experimental observation (discussed below) supports this picture. For decays, , so that there is slightly smaller than in decays. In decays, : if in this case, then . In decays, there presumably is again . When , final states of decays will consist predominantly of two jets . QCD effects will modify the free quark estimate . To however, no terms appear, so long as the colors of the initial and final quarks are summed over without restriction. The reason is as for Fig. 3(a): color averaging cancels diagrams in which the exchange is accompanied by a gluon; for the remaining diagrams the exchange may be approximated by a local four-fermion vertex, since only the invariant mass (and not itself) is relevant. If the colors of initial or final quarks are restricted, then terms do appear: these will be crucial in consideration of the rule below. In higher orders, corrections may appear even for color-averaged widths. Exchanges between the incoming lead to infrared divergences: at least in as far as terms are absent, these are, however, subsumed in the definition of or (28). Gluon emissions and exchanges in the final state modify the total width through their effect on the decay width. Another potentially important effect of QCD corrections is in modifying or circumventing the helicity suppression factor in . The possibility for the final quarks to be produced with large invariant masses before radiating gluons might be expected to affect the helicity suppression. In fact, these effects lead only to corrections: the divergence of the weak current, to which a spin-0 must couple, remains zero to all orders in if the quark masses vanish, irrespective of the invariant mass. (Nevertheless, confinement effects can contribute terms from the effective quark masses they produce). If the helicity suppression is genuinely to be avoided, then, as discussed above for and Fig. 3(b), gluon emissions must produce a spin-l, rather than spin-0, state. As for Fig. 2(b), however, such effects are probably not large (29). In analogy with Fig. 3(b), one may perhaps estimate by perturbation theory the rate for a gluon emission to produce a spin-1 state as (30). Here, the initial light quark mass enters through its magnetic moment, and thus appears in the denominator: in , however, the final mass appeared in the numerator, providing helicity suppression. Thus : taking (as appropriate for quarks), then if , (or perhaps ) is necessary before . (If , then is required). Estimating gives , so that . Thus, while gluon emission can avoid helicity suppression, it cannot render Fig. 4(b) competitive with Fig. 4(a) at large .

I now discuss the diagrams of Fig. 4(c) (sometimes referred to by the inappropriate name of ``penguins'' [35]). In these diagrams, the final has the same electric charge as , and therefore must lie in a different weak isomultiplet: the dominant contributions come when the intermediate is in either the or the isomultiplet. For -quark decays, the intermediate is predominantly or : in either case Fig. 4(c) is proportional to the mixing angle factor . However, for -quark decays, Fig.4(a) is also proportional to so that in this ease, Fig. 4(c) may be competitive with Fig. 4(a). For -quark decays, the intermediate in Fig. 4(c) is or , and the final is , so that again a mixing angle factor enters. On the other hand, in -quark decays, Fig. 4(a) gives an final state, with mixing angle factor : in this ease, therefore, Fig. 4(c) may only be significant in suppressed final state decays. For -quark decays, intermediate and in Fig. 4(c) yield transitions; these involve essentially the same mixing angle factors as in Fig. 4(a), but lead to rather than final states. -quark decays are similar to -quark ones: Fig. 4(c) suffers mixing angle suppression with respect to Fig. 4(a), and may only be important in suppressed decay channels. This pattern would probably be repeated for any further generations of quarks: the heavier member of each weak isodoublet would decay predominantly to the lighter member, with little contribution from Fig.4(c), while in decays of the lighter member, Figs. 4(a) and 4(c) may be competitive.

In -quark decays, the contributions of intermediate and contain mixing angle factors with opposing signs (, and , respectively) by virtue of the GIM arrangement. If , these contributions would cancel exactly, and Fig. 4(c) would vanish. Typically, if a momentum flows through the virtual intermediate line in Fig. 4(c), such a cancellation is effective: the magnitude of Fig. 4(c) is thus dominated by the region where the intermediate has a small momentum , which is particularly sensitive to hadronic effects. For decays, the and contributions typically cancel for , while in decays, the and contributions only cancel for . These GIM cancellations also relegate the real gluon emission process to subleading order, and allow only virtual gluon emission. Consider the decay , which is in many respects analogous to . If is on its mass shell, this must be a magnetic moment transition (31) ( is the momentum), even if , are off-shell: it is therefore proportional to the ``s-d weak transition color magnetic moment''. This is analogous to the weak contribution to the anomalous magnetic moment (32). However, in , an intermediate gives , while an intermediate gives , which cancels the contribution up to terms . The one-loop vertex diagram for (with momentum for ) involves the numerator factor , where is the left-hand projection operator from the couplings, and is the intermediate , mass: this factor contains no terms linear in . Thus the mass parameter appearing in the decay amplitude cannot depend on the intermediate masses (except through corrections), but only on the external masses, so that the contributions differ (to leading order) only by mixing angle factors, and cancel as described above. This cancellation may be avoided, however, if one of the two possible intermediate has a mass so that its effects are : in that case, the other possible is effectively freed from the GIM arrangement, and may mediate a decay at a substantial rate , (omitting mixing angle factors). If this interesting possibility would be realized in -quark decays (33). The cancellation would also be avoided if -coupled not only to left-handed, but also to right-handed currents. In that case, one of the in the numerator factor is replaced by , and a term linear in the intermediate quark mass appears [36,37,38] (if such right-handed currents contributed in they would give a large amplitude . Any quarks coupled to such right-handed currents should decay at a rate (dropping mixing angle factors), where is roughly mass of the heaviest contributing intermediate . Charged Higgs scalars in place of the usual may also lead to significant decays [39]. In known cases, however, such processes are absent and decays are relegated to subheading order. The cancellation between and intermediate states in, for example, , is not exact, but is violated by , yielding a negligible decay amplitude [38] , assuming . The cancellation between and intermediate states in at relies on the absence of terms from divergences in the loop integration. At , gluon corrections to introduce ultraviolet divergences and terms (similarly, the weak contribution to presumably receives electromagnetic corrections : these destroy the cancellation and lead to a decay amplitude [40] . Again this is in most cases probably too small to be of practical relevance. Just as independent higher-order corrections to can avoid the cancellation, so also can exchanges between the intermediate and spectator , by providing an additional spin-flip amplitude in the effective propagator. Soft vector exchanges are helicity-conserving and thus inadequate. However, the presumably propagates in an effective external color field generated by the and other constituents of the original . This may contain a spin-flip component, which may act over the propagation time of the virtual . For , this may give a decay amplitude perhaps , which could be significant (34). Insofar as is small, so also the weak radiate decay process should be small, as discussed below.

The GIM cancellation in Fig. 4(c) does not occur for if the radiated gluon is off its mass shell: the `` weak transition charge radius'' is non-zero at leading order. Since the process via a one-loop vertex correction exhibits no additional ultraviolet divergences in the limit , one may approximate the exchange by a local four-fermi on interaction. Then, performing a Fierz transformation, may be approximated by ``'', followed by ``'' via a virtual loop. Thus the rate for is proportional to the one-loop vacuum polarization diagram [41]: taking as the momentum, the amplitude for, say, becomes , with the form factor given by the difference between one-loop vacuum polarization with a and a loop. For (35); for ; and for , (36). The virtual gluon produced in may either be spacelike, and be absorbed by a spectator (or ) from the initial meson, or be timelike, and ``decay'' into a pair. The presence of an explicit factor in the amplitude for small (which is necessary to maintain gauge invariance, since there can be no static ``transition charge'') cancels the pole in the virtual propagator, leading to an effective local or interaction. An important difference from Figs. 4(b) and 4(a) is that the coupling is independent of the helicities, so that the effective local four fermion interaction contains terms with as well as Lorentz structure (37). Using the vertex given above, one may estimate the rate for the decay (integrating over the intermediate invariant mass) as contains (assuming , which is much smaller than the analogous rate from Fig. 4(a). For quark decays, is again not sufficiently large to render this decay mechanism important. On the other hand, the process may well be important for small . The intermediate coupling is independent of the helicity. The and must be left-handed in order to participate in the weak vertex: however, the may be right-handed, so that the initial and final systems have helicity zero. The spin of the decaying pseudoscalar usually constrains the to have spin 0. In Figs. 2 and 4(b) the coupling required both and to be left-handed giving a helicity state with a helicity suppressed amplitude to have spin 0. This helicity suppression is absent in Fig. 4(c) both for the initial and final systems. Not only is the explicit factor in the rates for Figs. 2 and 4(b) absent for Fig. 4(c); in addition, the effective for Fig. 4(c) should be larger than for Figs. 2 and 4(b) since it involves no helicity suppression factor: naively (38) , [42]. For heavy , there should be no significant enhancement of over . For light , the magnitude of the relevant is difficult to estimate [43], but may be enhanced by perhaps even a factor overly (a naive guess would take , suggesting , which would render Fig. 4(c) dominant over Figs. 4(a, b) in decays. To estimate the rate for , one must determine the invariant mass of the -channel exchanged . Assuming the initial to be at rest, the momenta of the final are completely determined, and (39) (cf. [41]). The magnitude of the vertex form factor depends on the relative size of and the intermediate virtual mass. In quark decays, while . Hence , yielding a naive free quark estimate : the small numerical factors appearing in render this estimate, like that for above, slightly below the estimated from Fig. 4(a). In quark decays, contributes only to Cabibbo suppressed final state decays. Here and (for decays) (for decays) [41] so that , yielding a naive estimate : this compares favorably to the free quark estimate of . In quark decays, with intermediate is not Cabibbo suppressed With respect to . In meson decays, and , suggesting : presumably , so that (for ) this rate is small compared to from Fig. 4(b), and Fig. 4(c) is probably unimportant.

The estimates of Fig. 4(c) given above were all to lowest order in : I now discuss the higher order corrections they receive. Consider, to be definite, the example of with intermediate or . If (and or ) then QCD corrections to the and exchange contributions must be identical: the mixing angles yield opposing signs for them, so that they cancel to all orders in (but lowest order in ) If, for example, , then in higher orders, there exist diagrams involving annihilation of the intermediate with the spectator (corresponding to the same amplitudes as ``box diagram'' corrections to Fig. 4(b)), which destroy cancellation between the and contributions. The local four-fermion form for the interaction in Fig. 4(c) remains unmodified by higher order corrections: gauge invariance prevents the appearance of nonlocal terms involving exchanges with uncanceled propagators (39). As for Fig. 4(b), however, infrared divergences remain even after summing over all possible real and virtual gluon corrections with (40). AS mentioned above, at terms from Fig. 4(c) are always canceled by the GIM mechanism to ones. Any corrections appearing in higher orders must usually have coefficients which vanish in the exact GIM limit . It seems probable that the GIM cancellation removes all ultraviolet divergences which potentially occur as , so that only logarithmic factors appear in higher orders (as indicated by explicit calculations [47]). As mentioned for above, the intermediate may interact with the ``external field'' present in the original meson: this field is presumably unable to absorb sufficient momentum for it to replace the perturbative exchange.

Nonleptonic weak baryon decays should proceed by processes analogous to those for mesons in Fig. 4 (41). The independent Q decay mechanism of Fig. 4(a) (and of Fig. 4(c) if dominates) should have the same characteristics as in mesons. The exchange diagram in Fig. 4(b1) induces the reactions , etc., but does not occur with , etc., initial states. When these processes are embedded in baryons, the initial may have total angular momentum or , usually with roughly equal probabilities. The helicity suppression encountered for processes from Fig. 4(b) in spin 0 mesons is therefore absent for processes in baryons. The single decay process analogous to Fig. 4(a) suggests that the weak decay rates of all baryons containing should be equal. The analogue of Fig. 4(b1) contributes only in baryons containing particular spectator quarks: its presence would, for example, imply different lifetimes for different baryons in an isomultiplet. The analogue of Fig. 4(c) with exchange to a spectator should behave in baryons much as in mesons. since in baryons Fig. 4(b1) suffers no helicity suppression, naive estimates suggest that Fig. 4(c) is always smaller than Fig. 4(b1) whenever mixing angles allow the latter to contribute. If Fig. 4(a) and its analogues dominate all nonleptonic decays, then the lifetimes of baryons and of mesons containing a given heavy quark should be approximately equal. The absolute importance of Fig. 4(b) compared to Fig. 4(a) depends on Presumably is similar to wavefunction at the origin which determines the magnitude of hyperfine splittings between, e.g., and baryons. In the nonrelativistic approximation, . For the decay of the charmed baryon , Fig. 4(b) may contribute; then (42).

I now summarize the discussion of inclusive nonleptonic weak decays based on Fig. 4 given above, and relate it to some relevant experimental data. For hadrons containing quarks ) the independent decay process of Fig. 4(a) should be dominant, and the final states of the decays should consist of three resolvable hadron jets. (Even for , helicity suppression renders Fig. 4(b) insignificant). Production of mesons containing quarks close to threshold has recently been observed [48]: detailed data on their decay properties is not yet available, but will presumably soon be forthcoming. (Analysis of distributions nevertheless indicates the expected [31] spherical event structure [48] which was hinted at by higher energy data on inclusive production [49]). In these decays, Fig. 4(a) should again dominate. Mixing angles do not suppress Fig. 4(b) in and mesons: however, the smallness of prevents a substantial contribution except in the case, where the process may be roughly comparable to . Figure 4(c) can give without mixing angle suppression (relative to ): again, however, the decay mechanism is rendered insignificant by the fact that and by small numerical factors entering in the loop diagram. Assuming that Fig. 4(a) dominates, one expects a semileptonic branching ratio where . If, as indirect phenomenological evidence suggests, [50], , so that about half of all hadronic decay final states potentially contain two charmed hadrons. With Fig. 4(a) dominant, the lifetime of mesons containing quarks would be sec (corresponding to a track length at a laboratory energy ).

Several thousand decays of pairs produced near threshold (at ) in annihilation have been analyzed [25]. (In a few years, the Mark III detector at SPEAR should collect some million such decays, allowing a more precise phenomenological investigation). A few decay events have been observed directly in (triggered/hybrid) emulsion experiments [51] (and more are expected in new high-resolution detectors (e.g., [52])). Several experiments observe production in hadronic collisions through specific exclusive or inclusive decay channels: in most cases, these give no additional information on decays. production in annihilation still awaits confirmation (43); a few candidate decay events were found in emulsion [51] and there is some indication of hadronic production [54]. Charmed baryon pair production has been observed in annihilation [53], but only with the specific decay channel . A few candidates have been found in emulsion experiments [52], and there have been several measurements of production in hadronic reactions (some indicating a rather large forward production cross-section). Nonleptonic decays should be dominated by Fig. 4(a); in nonleptonic decays Fig. 4(b1) may also contribute, and naive estimates given above indicate that it could well dominate over Fig. 4(a). Semileptonic and decays should proceed through Fig. 3(a). Thus the semileptonic branching ratio should be , and the (proper) lifetime would be sec. Figure 3(a) implies ; Fig. 4(b1) can yield) , and hence and . Experimental results from production in annihilation [25] give , , supporting the hypothesis that Fig. 4(a) dominates nonleptonic decay, and indicating a significant contribution from Fig. 4(b1) to nonleptonic decay. Assuming these experimental results suggest that Direct measurements of emulsion events give [51] sec (5 events), sec (7 events), sec (2 events), sec (6 events), indicating that . The near agreement of this result with that deduced from the semileptonic branching ratios indicates that the assumption is approximately correct. As discussed above, nonleptonic decays may receive contributions from Figs. 4(a) and 4(b2): the final state in this case should make the helicity suppression of Fig. 4(b) more effective (by a factor which perhaps than in decay, yielding intermediate between and , as suggested by the lifetime measurement quoted above. Semileptonic decays should again be dominated by Fig. 3(a). Since the decay may also occur: if indeed Fig. 4(b) dominates nonleptonic decay, the naive estimate of it given above implies . (A direct estimate using the guess and the ``measured'' F lifetime suggests a still larger branching ratio, perhaps For the charmed baryon one expects significant contributions from the analogue of Fig. 4(b1), suggesting , in conflict with the experimental lifetime determination quoted above [51]. (This discrepancy would be removed if a large fraction of the experimental candidates were, in fact, , for which Fig. 4(b) can give no contribution, leaving Fig. 4(a) dominant, and yielding a lifetime comparable to that of ). An important consequence of Figs. 4(a) and 4(b) in decays is that only a fraction of the final states should involve no s quark, and thus have . Important contributions from Fig. 4(c) (which are not expected according to the naive estimates given above) would enhance this ratio. Experimentally, only the two body decay modes and have been measured [25]; their rate relative to is roughly consistent with (see below) (44). In decays, Fig. 4(a) should give predominantly , presumably yielding hadron final states containing or , while Fig. 4(b) would give , yielding hadron final states containing , but much fewer or . The scanty experimental data on production do not yet allow discrimination between these two cases (45). The final states of decays experimentally appear to be predominantly , as expected from Fig. 4.

For strange particle decays, the inclusive treatment of decay rates given above is largely inappropriate: the energy released in the decays is so low that the detailed types and masses of the final hadrons are crucial. Approximating the final quarks in weak decays as free, one expects (46): in practice . The origin of this failure is clarified by consideration of the partial decay widths to particular pion final states. Experimental results show that ; . Taking a matrix element uniform in the available phase space (which is found experimentally to be a reasonable approximation) suggests : the actual rate for is enhanced by a factor over this estimate. A uniform matrix element (scaled by ) suggests (47) (e.g., [56]): in practice, while . The difference between semileptonic and nonleptonic decay rates even after accounting for phase space effects would result (as in the case of mesons) from contributions of Figs. 4(b) and 4(c) as well as 4(a). The suppression of compared to by a factor indicates that the effective nonleptonic weak Hamiltonian for decays transforms under strong isospin predominantly as . In , the final must have , while in , they may have : the ratio of to is thus explained if the effective weak Hamiltonian responsible is predominantly , with only (or component. The basic weak vertex has , while involves . In the free quark approximation, the and , processes of Fig. 4(a, b) contain and terms (but no component) in equal concentrations (up to Clebsch-Gordan coefficients). It will turn out (see below) that the different isospin components of these reactions correspond to different color components: perturbative QCD corrections thus affect the relative concentrations of the two isospin components (probably enhancing . The process of Fig. 4(c) is purely . Note that in the present considerations of exclusive hadron final states, the diagrams of Fig. 4 must be added coherently, and the processes illustrated there may interfere. Only quark subprocesses with the correct isospin transformation properties can contribute in hadronic decays to hadron final states of definite isospin. Note, however, that isospin invariance is respected in the development of the hadron final state only inasmuch as the and are indistinguishable: electromagnetic interactions and effects may modify the isospin of the final state. In hadronic terms, a virtual may be produced initially, and may then mix through isospin violation (Such as is responsible for decay) to with an amplitude at the level [57]. Such effects could almost account for even if were pure : they should also lead to contributions. The hypothesis that transforms approximately as yields relations between rates for other decays. A final state may have or : the rule requires . Writing the amplitude for transitions as , one has , implying . In , both and transitions may occur, and may interfere. One finds in this case (ignoring 0(1%) phase space corrections) where is the strong interaction final state phase shift for a system with isospin , and ; (48) , [58]. Experimentally , again implying . Thus there appears to be a universal contribution to decays, with an amplitude that of the dominant term. Assuming that any contribution is small, the experimental imply that its amplitude that of the term. final states may have are probably much suppressed by centrifugal barrier effects, since they must involve : the part of the weak Hamiltonian gives only , while can be reached only by . If terms are absent, , , These relations receive corrections from final state isospin violation effects. Two decays with equal amplitudes uniform in the available phase space have widths , which differ by virtue of . Taking such a phase space correction in gives a correction to the relation . If instead, the measured phase distribution is used, the correction . (Final state Coulomb interactions between provide a further correction). Experimentally, dividing out all kinematic isospin violation [60], , : uncertainties in removal of final state isospin violation effects preclude definite conclusion of terms in . Ignoring final state isospin violation, (both final states are dominantly , so that no phase shift difference enters). Experimentally, , probably indicating , in rough agreement with the contribution deduced for . Further measures of terms involving the distribution of final energies in also indicate similar concentrations, but are severely hampered by kinematic isospin violations. Of four particle decay modes, only has been measured. Assuming a matrix element uniform in available phase space (and scaled by ) suggests ; experimentally, . The relative strengths of semileptonic and nonleptonic and contributions to decays thus appear to be roughly independent of the specific decay considered, and to have amplitudes in the ratio .

Semileptonic decay presumably occurs basically through the mechanism of Fig. 3(a). However, since the energy released is small, the final hadron system usually consists only of a single pion, with a definite isospin . To form this pion, the final from and the spectator must be in an rather than state. In the case of decay, the must have . In decay, the initially have equal amplitudes to be in an or an state. Because of the small energy release, hadronic effects force the to have , and thus suppress with respect to by a factor 2. (If were larger, so that many were produced in decays, the could have either isospin with eventually equal amplitudes, and the rates for semileptonic and decay would become equal.) In nonleptonic decays, the small number of final pions again introduces constraints on the total isospin of the quark systems from which they form. The total amplitude for a decay may be considered roughly as a product of the amplitude for a quark system with particular isospin to be produced and the amplitude for that system to form the final pions, summed over all possible isospin states. Because of the isospin invariance of strong interactions, there exist direct relations between both amplitudes for systems of specific and different , as exploited above. In most cases, comparisons of either amplitude for different are difficult. (An exception is the example of (see above) where symmetries do not allow an on state, so that an quark system has zero amplitude to transform into the final state and induce the decay). Even after the final have been ``produced'', they still undergo strong final state interactions, which may modify the amplitudes for different . Interactions between outgoing on-shell final pions have an amplitude (by unitarity) of unit modulus. Because the pions propagate on-shell between successive rescatterings (see (49) ), the amplitude attains an imaginary part proportional to the rescattering amplitude, and gives a phase to the amplitude for a decay with final state isospin . This phase difference between the amplitudes for and was accounted for in the comparison of these processes above. However, in addition to these pure phase factors arising from interactions between on-shell outgoing pions, the modulus of the total decay amplitude is modified by final state interactions acting between off-shell pions: these distort the usual outgoing plane waves and alter the ``wavefunction at the origin'' for the system (50). A quantitative estimate of such effects is very difficult: results are inevitably sensitive to the composite structure of the pion, and it is impossible to disentangle effects of ``final state, interactions'' from features of the ``primary interaction''. Nevertheless, there are some qualitative indications that ``final state'' interactions should enhance the rate for production of over systems. At low energies (below threshold) the relevant -wave elastic scattering phase shifts are well-fit by a scattering length approximation ; in the (nonexotic) channel the interaction is strongly attractive, with , while in the (exotic) channel, it is slightly repulsive, with [58]. Final state interactions should thus tend to enhance final state) processes relative to ones. (Attempts to obtain a quantitative estimate of this effect from, e.g., a comparison of the factors are thwarted by sensitivity to high behavior, where unknown inelastic contributions are presumably important. A very rash guess is provided by : not a large factor compared to the observed ratio 400 of to rates).

In addition to such ``large distance'' effects, ``short distance'' phenomena, best considered in the framework of the quark diagrams Fig. 4, may also contribute to the suppression of relative to processes. Recall that the simple comparisons between measured decay rates discussed above indicated that the ratio of semileptonic to non-leptonic decay amplitudes . The ratio here may be obtained directly, e.g., from . The deduction of the relative size of semileptonic and nonleptonic amplitudes requires some assumptions regarding the phase space structure of the decay rates: it remains possible (although unlikely) that the relevant semileptonic amplitude . Figures 4(a,b) contain both and components; the process of Fig. 4(c) is, however, pure . The simple free-quark Estimate for from Fig. 4(c) given above suggested that numerical factors associated with loop integration render it slightly smaller than Fig. 4(a). A serious quantitative estimate would, however, require greater information on the structure of hadrons than is yet available: it is still certainly conceivable that , with and dominated by Fig. 4(a), and dominated by a larger term from Fig. 4(c). In the free quark approximation, the processes of Fig. 4(a) and or of Fig. 4(b) give essentially equal and amplitudes. The different isospin channels correspond to amplitudes with different symmetries under interchange: the overall symmetry of the amplitudes then requires quark pairs in different SU(3) color representations. Thus gluon exchange corrections depend on the isospin properties of the amplitude, and may enhance relative to parts (51). Consider at first, for simplicity, the weak reaction : this is directly relevant in nonleptonic weak hyperon decays; results for the cases of Fig. 4(a, b) will be obtained by crossing. The process proceeds at lowest order by -channel exchange: it receives corrections from real gluon emission and virtual gluon exchanges. The real gluon emission terms introduce much infrared divergence, but (at ) exhibit no ultraviolet divergences as , and thus can generate no terms. Virtual gluon exchanges do involve ultraviolet divergences, and thus may produce terms. Nevertheless, when all possible color quantum numbers of the initial and final states are averaged over, these terms cancel, as discussed above. However, if the color quantum numbers are restricted by requiring a specific isospin state, the terms no longer cancel, and serve to enhance the rates for production of some isospin states at the cost of others. At the energies of concern here, the reaction must occur with essentially zero impact parameter, and thus involve no orbital angular momentum. The nature of the coupling requires the interacting to have oppositely-directed helicities, so that the reaction occurs in a total angular momentum channel, so that the spatial and spin parts of the final state wavefunction are antisymmetric under the interchange . The initial state clearly has (strong) isospin . The final state may have : if , then the complete reaction is purely ; if , then it may involve a component. When , the isospin part of the final wavefunction is antisymmetric under the interchange ; when , it is symmetric . Assuming that the final obey Fermi-Dirac statistics, their total wavefunction must be antisymmetric under (52). Thus if , the must be antisymmetric in their color quantum numbers, while if , they must be symmetric. The initial and final may transform under according to the representations : the representation is antisymmetric in the quark indices, while the 6 is symmetric. (For , the possible representations are , which are respectively antisymmetric and symmetric). reaction thus requires the initial and final to transform according to the symmetric, 6 representation of ; when , the may also transform under the antisymmetric representation (53). The amplitude for virtual gluon exchange corrections to depends on the color representation: it will turn out that one gluon exchange is attractive (leading to an enhanced scattering amplitude) for the representation, and repulsive for the 6. The (averaged) amplitude for one gluon exchange between in a color symmetric state is proportional to (the accounts for the absence of colorless gluons); in a color antisymmetric state, the amplitude is proportional instead to (54). For the case, while : one gluon exchange yields terms which enhance the amplitude for , and suppress the amplitude. (Note that summing over all possible initial and final colors yields the required vanishing coefficient for . . terms are not the only part of the one-gluon exchange amplitude which may depend on the color representation. Soft gluon emission and exchange occur coherently from the two quarks, and are thus potentially very sensitive to their total color. However, as mentioned above, QCD processes occurring at distances presumably neutralize the color, but do not affect their isospin (although they may modify the amplitudes for different , e.g., through the ``final state interactions'' discussed in the previous paragraph). The connection between the isospin and color derived above holds only at short distances: at larger distances, one must account for gluon radiation; it seems probable that no significant dependence of the scattering amplitude on the original color (and isospin) will survive. Thus . terms may plausibly be the only component of the amplitude which depend significantly on . The discussion above indicates that the relevant infrared cutoff on the one-gluon exchange amplitude (or the inverse size of the initial meson). In the simplest approximation, one may consider a sequence of independent gluon exchanges between the incoming and outgoing . The invariant masses of the exchanged gluons are as usual kinematically constrained to be ordered. The maximum invariant mass of the gluon closest to the exchange is : for larger invariant masses the exchange would cease to act as a point interaction, and the amplitude would be damped. The amplitude for gluon exchange then , where is proportional to the color factors derived above, and accounts for integration over longitudinal kinematic parameters for the exchanges (cf., e.g., [66]). Summing the contributions from all possible numbers of exchanged gluons (55) then gives a correction factor : this suggests that the amplitude is suppressed with respect to the amplitude by a factor . Lack of knowledge regarding the infrared cutoff prevents a satisfactory quantitative conclusion from this result. Taking , it suggests 0.3--0.5. This estimate was based solely on a leading log approximation in which successive gluon exchanges are assumed (statistically) independent. To improve the approximation one must account for interference between successive emissions. The color factors for the corresponding diagrams (which involve, e.g., crossed gluon ``rungs'') exhibit no simple behavior in the relevant color symmetric and antisymmetric channels (even in the limit : an explicit calculation of all contributing diagrams is thus required (56). Having considered the process , it is a matter of crossing to apply the results to the processes and , of Figs. 4(a, b): perturbative QCD corrections should again provide some enhancement of over terms.

It is at present not possible to make a convincing quantitative conclusion on the origin of the isospin dependence of decay rates. Three qualitative phenomena nevertheless suggest effects in the observed direction, but each alone is probably not of a sufficient magnitude. First, any contributions from Fig. 4(c) must be pure , and thus tend to enhance this component over and semileptonic decays. Second, final state strong interactions in the systems produced by the decays may depend on the final isospin in such a way as to enhance decays relative to ones. Third, gluon exchange effects at distances as estimated by a leading log approximation probably provide some enhancement of over amplitudes.

Weak hyperon decays in many respects parallel kaon decays. The only nonleptonic baryon decays (except , allowed by phase space constraints are , while of semileptonic decays, only have been observed. No meaningful quantitative conclusions on the relative sizes of the nonleptonic and semileptonic decay amplitudes may be drawn from comparisons between the measured rates for the two-body decays and the three-body decays (very naive estimates based on matrix elements uniform in available phase space and scaled by indicate that the amplitudes are equal to within an order of magnitude, with that for nonleptonic decays often the smaller). Semileptonic hyperon decays are presumably dominated by the process of Fig. 3(a): their rates are roughly independent of the types of the ``spectator'' quarks in the initial and final baryons (except through the initial and final wavefunction factors). The amplitudes for nonleptonic decays are of the form : the term yields -wave final states, while the term gives -wave ones, in which the final polarization is opposite to the initial B polarization. The assumption of a pure effective weak Hamiltonian yields several relations between the various nonleptonic hyperon decay rates (which should hold separately for the - and -wave amplitudes). For the isoscalar hyperons and , the assumption implies the relations . Experimentally the first of these relations is valid to within , while from the experimental measurement it appears that the second is considerably violated, implying a amplitude in this case 0.1--0.2. Making the (perhaps questionable) assumption that the matrix elements of the effective weak Hamiltonian for baryons in the same isomultiplet are related by isospin rotations yields as a further consequence of the transformation of the effective Hamiltonian the relations and . Experimentally, these relations are also violated by at most in amplitude. decays are forbidden by energetic constraints). The Lee-Sugawara relation requires the further assumption that the matrix elements of the effective weak Hamiltonian for states is related to that for by an unbroken rotation. It is again found to be valid experimentally to better than 10% (or a factor of 2 for -wave amplitudes). All nonleptonic hyperon decays may receive contributions from the processes (cf., Fig. 4(a)) and (cf., Fig. 4(c)). The exchange process (cf., Fig. 4(b1)) can give a significant contribution only if the initial baryon contains a (``valence'') quark. This mechanism may therefore contribute to and decays, but not to or decays. The observed validity of the `` rule'' relations between and decay rates mentioned above indicates, however, that this mechanism is probably not important, despite encouraging estimates made above (and, e.g., [68]). Just as in decays, the isospin transformation properties of the effective weak Hamiltonian for hyperon decays should be affected by both final state hadronic interactions and by short distance perturbative QCD effects. The phase of the amplitudes may again be determined by the elastic scattering phase shifts in the relevant - or -wave orbital angular momentum state. The modification to the modulus of the amplitude due to final state hadronic effects is incalculable as in the decay case. Qualitatively, however, it seems likely that will be enhanced by such effects when the interaction is attractive at energies (as revealed by the presence of resonances in the system). (For example, dominance in decays may in part be accounted for by the quantum numbers of resonance in the system: the or even pole is much closer to than the lowest-lying resonance . However, decays should be dominated by the pole and thus predominantly , since there are no resonances, whereas experimental measurements mentioned above indicate a significant component). The process analogous to Fig. 4(c) is pure , and may be dominant in hyperon decays. The process (analogous to Fig. 4(a)) contains both , and , components, as does (cf., Fig. 4(b)). It was shown above that if the system in the final states of these latter processes is constrained to transform as a representation under , then the processes are pure . In the absence of color interactions (or gluons), this would be achieved if the final in or the initial in were contained in a single baryon [69,57]. In practice, as discussed above, the colors of the initial (and final) quark states at large distances are probably irrelevant: they are arbitrarily modified by radiation of very soft gluons. Thus the enhancement of over amplitudes in hyperon decays should occur through the same effects, and to a roughly equal extent as in decays.

The treatment of charmed meson decays given above was concerned primarily with purely inclusive final states. The rates for charmed meson decays to specific exclusive hadron final states presumably exhibit enhancements and suppressions analogous to those found in strange particle decays. The first important source of such modifications would be strong interactions between the final state hadrons. As in the case of decays, one may estimate the relative phases of various decay amplitudes using Watson's theorem [70]: the probably important effect of final state interactions on the moduli of the decay amplitudes remains entirely incalculable. For two (and perhaps three) body final states, experimental phase shifts may be used to estimate the phases of the decay amplitudes [70]: the effects of these phases alone give numerically-important corrections. For weak decays of mesons with masses , inelastic final state scatterings render Watson's theorem inapplicable to the phase of a particular decay amplitude: as , the random phases of the increasing number of contributing scattering amplitudes will presumably yield a decreasing phase difference between any two decay amplitudes. Just as in decays, perturbative QCD effects at short distances may modify the rates for decays to hadron final states with different transformation properties under interchanges of quark flavors. In the process , gluon exchanges should enhance final states antisymmetric under the interchange , corresponding to ; similarly, in such effects should enhance final states antisymmetric under compared to those symmetric in this interchange. The magnitude of this effect in decays should be of the same order as in decays: the infrared cutoff introduced above is determined by the inverse size of the initial meson, which is similar for the and cases (the effect may be slightly smaller in than in decays). If such short distance phenomena are the primary cause of enhancement in decays, then the enhancement of antisymmetric over symmetric final states in decays should be by the same large factor. The antisymmetric part of the effective Hamiltonian for decay transforms as a under rotations (contained in 20 of ); the symmetric part transforms as a 15 under (contained in 84 of ) (e.g., [71]). 6 dominance has many consequences for exclusive decay rates (e.g., [72])). For example, all Cabibbo-favored two body decays (e.g., are forbidden. Experimentally (e.g., [25]), such decays are observed with branching ratios relative to three-body modes roughly as expected from available phase space. Further consequences of dominance for differential widths in three-body decay modes await experimental investigation. It seems unlikely, however, that the suppression of 15 with respect to 6 amplitudes for decays will be as marked as in decays. If this is the case, then it indicates that perturbative QCD effects connected with symmetries between final state quarks are not particularly significant: the dominance of the , component in decays is thus presumably a consequence of important contributions from final state hadronic effects and Fig. 4(c).

Precise predictions even for inclusive decays of (e.g., ) mesons are rendered impossible by the presence of the light in the initial state, and the inevitable infrared divergences which accompany it. One case in which such difficulties are absent is for initial quarkonium states (e.g., ; denoted generically ). Weak contributions in decays may arise either through a -channel exchange process , or via an -channel . The presence of such weak amplitudes may be detected through the small parity-violating correlations which they induce [73] by interference with the dominant -conserving electromagnetic or strong amplitude. In a free quark approximation, the weak amplitude for decay is given roughly by : for , this ratio is 0(2%). When gluon exchange and emission effects are included, both weak and electromagnetic amplitudes are modified: however, as for Fig. 4(b) above, in the leading log approximation, no terms appear in the weak amplitude, and the ratio suffers no large corrections. violation may induce correlations, where is a spin vector, and is a three momentum. If longitudinally-polarized are produced by collisions of longitudinally-polarized beams, then such correlations between the outgoing (rather than ) momentum and the spin may occur. (Correlations between and the incoming spin are strictly not -violating; however, the -conserving exchange background is negligible on resonance). The momentum direction may be determined statistically by measurement of high-energy . For , such -violating weak effects should occur at the level, and eventually be observable. For heavier (e.g., states, the effects should be much larger.

I now comment briefly on radiative weak decays. These might occur either through single quark decay process such as , or as radiative corrections to the mechanisms of Fig. 4. As discussed above in connection with Fig. 4(c), however, the vertex vanishes for on-shell photons (up to : thus the single quark decay mechanism should not be important (it is found to be inadequate phenomenologically [74]). In radiative two-body hyperon decays such as , it seems likely that the process occurring as a radiative correction to Fig. 4(b) or 4(c) dominates: the must have energy , so that the are directed opposite to it into the same final baryon. The rate for this kinematic configuration is and small compared to the pure process (the relevant numerical coefficient has not yet been calculated, but is probably small). The fact that is as large as is thus somewhat surprising. Nevertheless, final state hadronic effects are undoubtedly very important in such decays, and may largely determine, for example, the ratio of -violating and -conserving amplitudes (e.g., [75])). If radiative corrections to diagrams analogous to Fig. 4(c) dominate radiative hyperon decays, then the rates for all possible such decays should be similar. However, if radiative corrections to the analogue of Fig. 4(b) are important, then the radiative decay rates of and should be smaller than those of and (since Fig. 4(b) cannot contribute in the former baryons). The experimental result that perhaps suggests that the analogue of Fig. 4(b) is indeed important. Radiative decays involving more than two final particles (e.g., are usually dominated by Bremsstrahlung from the participating hadrons, and do not directly probe the weak interaction.

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