Notes

(1) Recall that for free Dirac spinors such that , , while .

(2) For massless quarks, .

(3) The upper component of a Dirac spinor gives the amplitude for a fermion to populate a particular spin state, the lower component gives the amplitude for the opposite spin state in the nonrelativistic limit): in a bound state wavefunction, the opposite spin of lower components is compensated by orbital angular momentum (thereby, e.g., introducing -wave components into a nominally -wave hydrogen atom level).

(4) The relation remains valid to all orders in QCD perturbation theory: it receives and other corrections from non-perturbative effects.

(5) In fact, for a potential of the form (e.g., [9]) with .

(6) That is, . For a non-relativistic pair with relative momentum . For ultrarelativistic systems, this probably becomes [11]. (I am grateful to C. Llewellyn Smith for a discussion on this matter).

(7) These results were obtained by numerical solution of Schrodinger's equation, taking , and .

(8) Box diagrams for , in which a hard virtual photon accompanies the apparently also suffer helicity suppression [14].

(9) These effective magnetic moments would not exhibit the large isospin violation naively expected from magnetic moments.

(10) The charge determines whether or are emitted in decays: the two possibilities involve conjugate quark currents, and exchange the and spectra. The spectra given here assume couplings: for a V+A coupling and ``'' emission (e.g., [18]). Note that three-body phase space alone gives .

(11) The exact form is given in Ref. [19], but does not differ significantly from this approximation over the whole range of . Note that the physical picture of gluon emissions only from the outgoing is realized for the contributions of explicit Feynman diagrams in an axial gauge with and .

(12) This estimate goes far beyond the formal region of applicability of the double log approximation (2).

(13) This estimate contains no explicit reference to the prescription used in renormalizing , but assumes that . The complete form depends on renormalization prescription, and can be obtained only by explicit calculation.

(14) As discussed above, current conservation (and hence the Ward identity) is violated by the unequal quark masses : this violation is nevertheless too soft to affect ultraviolet behavior.

(15) Such a fit is shown in Ref. [19] using the lepton spectrum; use of the higher order estimate (3) changes slightly the optimal parameters. Note, however, that the statistical quality of available data [25] is as yet quite insufficient to allow serious quantitative comparison.

(16) Even if the initial were far off mass shell, the restriction to color singlet initial and final states forbids virtual corrections to one gluon emission and prevents cancellation of infrared divergences from radiation of an arbitrarily soft second gluon (which would cancel if no color restriction were imposed).

(17) The formal infrared divergence of corrections to color magnetic moments is presumably regularized by the color magnetic fields present in the state [27].

(18) That is, the diagrams of Fig. 3(a) and 3(b) must be added before squaring, and the interference between the two mechanisms included.

(19) As for nucleons probed by spacelike invariant mass , the effective partitioning of energy between quark and gluon species in depends on the invariant mass of the probe used to measure it. Typically, a large invariant mass probe effects a measurement in a short time , during which the constituents may be off their mass shells by an amount : they thus radiate with probabilities . Here, however, allowing little radiation, so that the probes the ``primordial'' wavefunction. (In the analogous case of non-leptonic decays discussed below, , so that for significant radiation may occur).

(20) As elsewhere, I assume that no flavor-changing neutral currents are present. If is a state, then in Fig. 4(b2), the intermediate may be replaced by .

(21) The methods used to obtain the guess (2) for Fig. 3(a) are inapplicable here.

(22) If the product momenta are not large enough, then there will also be significant back reactions with the initial , etc.: for small outgoing velocities , these modify the total rate by a factor

(23) With charge quarks, it is not possible for and in to have the same flavor. Note that all produced by interactions are left-handed, and are therefore in the same helicity state.

(24) Statements to the contrary in the literature result from neglect of soft gluons from the initial . That such gluons must be present may formally be seen from the fact that they are required to cancel infrared divergences associated with the massless incoming .

(25) Assuming that only the ``valence'' is present in a given state. The large W mass samples the at very short distances, at which significant ``sea'' pairs may be present: their effects will be mentioned below.

(26) Only if the invariant mass of the extracted from were varied would calculable corrections to result (as essentially occurs for the Drell-Yan process).

(27) In the semileptonic case of Fig. 2(b), color conservation formally required two gluons to be emitted , although one of the gluons could be arbitrarily soft, and may be subsumed into the initial M wavefunction. Two gluon emission is also formally required in Fig. 4(b2). In Fig. 4(b1), however, color may be conserved with only one gluon emission. Nevertheless, as discussed for Fig. 2(b), the presence of initial soft gluons undoubtedly renders these differences irrelevant.

(28) In this nonleptonic process, there exist further diagrams, not present for Fig. 2(b), in which gluons are exchanged in parallel with the . As discussed above, these are absent at . Analogy with ``box diagram'' contributions to suggests that they are always negligible.

(29) This is a component of the manifestly gauge invariant form , where is the gluon field strength. This interaction also induces decays with an amplitude related to the magnetic moment decays discussed here.

(30) Since the are taken to have spin 0, the directions of the and decay products are uncorrelated: the therefore obey linear superposition, and for an final state is given simply by [31] ).

(31) That this is finite and contains no terms is evident from the absence of a possible counterterm for it in the original Lagrangian. Note that could involve not only a magnetic dipole term , but also an electric dipole term . This is not -violating (as would be) because the and are distinct and have different masses. (Recall that while a nonzero static electric dipole moment requires violation, transition electric dipole moments do not: hence, for example, a molecule can effectively have an electric dipole moment through electric dipole transitions between its closely-spaced rotational levels). The coefficient of the term is constrained by equations of motion to be times the term: the ratio of the to terms determines the magnitude of violation in a decay, and thus the angular correlation of the momentum with respect to the initial spin direction.

(32) Recall that Fig. 4(c) yields while Fig. 4(a) gives . Failure to observe at the 0(20%) level would provide an upper limit on the quark mass. The generation of from spontaneous symmetry breakdown in the standard Weinberg-Salam model requires .

(33) Similar effects may occur in the presence of instanton background fields. In an operator product expansion analysis they would presumably appear through operators (e.g., proportional to which vanish in perturbation theory.

(34) This form is only valid for ``'' : the contribution of intermediate quarks with is presumably .

(35) The exact may be written in the Feynman parametric form .

(36) The gauge invariant form of the amplitude is where is a covariant derivative, and is the gluon field strength tensor. The QCD equations of motion relate the amplitude for the virtual gluon to the amplitude for its source: . (These equations of motion are respected to all orders in ordinary perturbation theory; they are modified by semiclassical (e.g., instanton) effects). Thus may always be considered as or . Note that processes such as , where the virtual ``decays'' into , do not occur at this order.

(37) This conclusion may also be reached by applying the equation of motion (cf., [1]) [42]

(38) The complete formula for the exchanged in the process with at rest is .

(39) The absence of additional forms for the interaction is evident from the derivation of the minimal operator basis (with dimension so as not to incur additional factors) for such processes in Ref. [37]. (An explicit diagrammatic verification is given in Ref. [44]). Note that formal operator product expansion analyses indicate mixing of gauge invariant operators ``into'' additional gauge noninvariant operators (explicit calculations require gauge noninvariant counterterms): these operators give no contribution because their matrix elements between on-shell physical (gauge invariant) states vanish [45]. The removal of some possible operators contributing to requires application of equations of motion (e.g., . These equations are valid to all orders in perturbation theory: however, with a modified vacuum state (e.g., an instanton), they are altered.

(40) If , one might expect that (as in QED) all infrared divergences should cancel. The calculation of Ref. [46] on a process similar to the one considered here suggests that this cancellation may not occur in QCD.

(41) Such weak decays are only significant if at least the lowest-lying baryon containing some heavy quark is below threshold for a strong decay into a meson containing and a light baryon. This condition is well satisfied for and (2.1). Simple pictures suggest that the effective constituent masses of will enforce this condition for any flavor .

(42) The wavefunction at the origin (squared) for a three-body system bound by harmonic forces differs from that for a two body system bound by the same forces by only a factor 3/4.

(43) Early reports of from DASP were not substantiated by Mark II results, while recent Crystal Ball results also fail to exhibit the increase in inclusive production at expected from production and decay.

(44) Figures 4(a, b) yield decays , rather than . A recent report [35] of with a branching ratio of a few percent (comparable to that of is thus surprising. (Strong decays of low-lying meson resonances very rarely produce or except from valence in the initial state).

(45) The two candidate decays in the emulsion experiment [51] were and . Hadronic production is reported with the decay modes and [54], but or final states are probably suppressed for experimental reasons. There is a very slight indication of in annihilation [53]; if, in fact, such modes dominated the lack of an increase in production could be accounted for.

(46) All mechanisms in Fig. 4 contribute equally to these decays (in the relevant local limit : the process through Fig. 4(c) (which could contribute for but not ) is probably unimportant.

(47) For decay to n massless particles distributed uniformly in available phase space, (the mass parameters appearing in matrix elements for different are taken to be simply the total decaying particle mass). (PCAC estimates suggest that the relevant mass parameters should instead be ).

(48) Time reversal invariance (valid to ) requires all weak amplitudes for to be relatively real. Imaginary parts may, however, be introduced by strong reinteractions between the outgoing on-shell pions. The amplitude for elastic scattering attains its imaginary component via unitarity through on-shell propagation of between successive strong interactions. Such final state interactions determine the relative phase of the and amplitude for (e.g., [59]). Their effect on the absolute magnitudes of these amplitudes is not calculable.

(49) An example of this effect is the Gamow/Sommerfeld factor which accounts for Coulomb interactions between outgoing charged particles (with relative velocity ). Reinteractions between on-shell final particles can give only phase factors (by unitarity): in the Coulomb case the relevant phase shift is divergent by virtue of the infinite range of the Coulomb interactions.

(50) The original suggestion of such an effect was made in Ref. [61] in the context of the operator product expansion. The necessary anomalous dimensions were then computed in Ref. [62].

(51) In the physically-irrelevant case of Bose quarks with no color degree of freedom, the symmetry of the total wavefunction allows only a final state, and implies zero amplitude [63].

(52) If the initial are externally constrained to lie in a color rather than 6 state, then processes are forbidden. This is perhaps the case, as discussed below, when the initial come from a single baryon.

(53) These results may easily be derived using the direct methods of Ref. [64]. Alternatively, one may write the color part of the amplitude as , where the are representative matrices for . Then the Fierz identity gives [65] : taking the parts of this product symmetric and antisymmetric under yields the required result.

(54) In a complete treatment of, say, decay, one must also account for processes such as and , although these give no qualitative modification to the results. In an operator product expansion analysis, they correspond to ``mixing'' of operators under renormalization. Through such effects the ``sea quark'' content of the initial meson is accounted for.

(55) Such a calculation is apparently underway [67].

(56) To obtain an system, each pair must transform as a .

previous