Notes

(1) The configuration space propagator for a zero rest mass particle whose invariant mass is required to exceed is given by

where is the complementary sine integral, which goes through its first zero at .

(2) For comparison, with scalar quarks, but vector gluons, , while for spinor quarks, but scalar gluons, the soft divergence disappears, and . (For these cases, should also be modified.)

(3) In the limit , this becomes the light cone momentum fraction , and is Lorentz invariant.

(4) With this choice, becomes fraction: some radiated partons will then travel backwards with respect to their parent partons, thereby populating the region of phase space usually associated with emissions from the impotent parton (with ). A table of the explicit contributions from individual diagrams in different gauges is given in the second paper of [1]. Note that in the ``planar'' gauge with no or term, the lowest-order interference diagram vanishes exactly, rather than being simply suppressed: perhaps this behavior continues in higher orders.

(5) As mentioned in the final section of these notes, the analogy with electromagnetic shower development is even closer for hadron production by mow- scatterings of high-energy incident partons.

(6) In the operator product expansion approach, integrals over provide corrections to the coefficient function. Integrals over the provide the leading log anomalous dimensions, and much of the subleading log anomalous dimensions. completes the subleading log anomalous dimensions.

(7) The differential cross-sections for are given in [30]; the loop corrections to necessary for a complete treatment are under investigation [31].

(8) Log() terms from vertex corrections cancel those from quark self-energy insertions by the axial gauge Ward identities.

(9) In leading log approximation estimates, it is often convenient to account for virtual corrections to parton propagation as possible off-shell parton decay modes, thereby introducing a negative (usually divergent) term in the . If , then the resulting satisfies , so that the sum of probabilities for all possible fates of the off-shell parton explicitly sums to one.

(10) The form is very similar to that obtained for massive electrons in QED, or for massless electrons in a magnetic field, or confined within a finite volume.

(11) These subtleties are not usually visible in the treatment of QED with massive massive electrons. The reason is that a particular renormalization prescription (momentum space subtraction at ) is overwhelmingly convenient, because it causes all higher order terms in to vanish exactly in the low-energy limit for various processes (e.g., Compton scattering), so that measurements of these processes may determine the precise value of to be used in this renormalization scheme, without the need to calculate higher order terms in the perturbation series. To deduce in this scheme from other processes (e.g., ) requires explicit calculation of higher order terms before comparison with experiment. (The Thomson limit of is exactly , but contains higher order terms .) Note that in massless QED, the removal of infrared divergences from incoming composites again spoils the simple scheme. In QCD, all such low energy limits of perturbation theory are entirely irrelevant, since the theory has a strong coupling in that domain.

(12) As discussed in the final section of these notes, in processes involving initial hadrons (e.g., or ), the incoming partons initially have small invariant masses . As they approach the collision, they may radiate timelike invariant mass partons, and themselves acquire progressively more spacelike invariant masses, up to (where is the momentum transferred in the hard scattering; typically momentum). In most cases, the appearing in will be positive, as in radiation from time-like mass final partons, and so no terms should be introduced between the two cases. Consider, however, the emissions from incoming partons just before the hard scattering. Integrating these over available phase space typically gives . On the other hand, the virtual exchanges which cancel the infrared divergence at give roughly , wielding a total . Clearly, comparisons between rates for hard scatterings of incoming partons involving positive (e.g., ) will differ from those with (e.g., by terms. (The exponentiation of the corresponding double log series demonstrates that such terms sum to a correction where is the color charge of the incoming partons.) For outgoing partons, the real emission term becomes , again allowing some differences with incoming parton processes. However, away from the hard scattering, the sign of has no effect on corrections to decay probabilities. Nevertheless, the decay probabilities for incoming and outgoing partons may differ by terms. For outgoing partons, imposition of the cutoff prevents any intermediate partons from reaching their mass shells. Incoming partons presumably have a spread of invariant masses with variance , extending both to timelike and spacelike values. If an incoming parton begins with timelike invariant mass, it must pass before reaching spacelike mass: between radiations it may propagate on shell, thereby introducing terms, not present for outgoing partons. It seems likely that this effect is a consequence of the infinite life of incoming hadrons, and is not an artifact of the initial parton mass spectrum considered.

(13) One possible method for estimating the contributions of multiloop diagrams would be to consider the diagrams in the limit that the spacetime dimensionality , so that no loop integrals remain, and the diagrams at a given order must merely be counted: unfortunately, the numerical results of this ``approximation'' are not even close to those obtained when . Another, more hopeful, but more complicated, method of approximation consists of performing a hyperspherical (Gegenbauer) expansion on each propagator (see [10]), but retaining only, say, the zeroth (spherically symmetric) term (e.g., in ). Then loop integrals reduce to scalar integrals over , etc., but with quite complicated integrands. For the diagram the method gives 3/2 compared to the exact result . Note that the contributions from large Gegenbauer index terms fall off like , where (number of loops). Another possible approach would be direct numerical evaluation of Euclidean momentum integrals: the necessary Monte Carlo integration would, however, be very time consuming because of the canceling divergent integrands.

(14) This approach would, of course, fail in the (seemingly unlikely) event that the constant factors systematically changed from one emission to the next.

(15) In particular, the multiplicity depends on the assumption (discussed below) that ``final partons'' which can radiate no further because of the cut have zero invariant mass (rather than masses ).

(16) Thus the fractal dimensions of the sets of points representing the directions of parton momenta are much smaller than 2. (The angular structure of the events is not, in fact, not exactly self-similar---the effective fractal dimension changes logarithmically with the angular scale considered.) These results should be contrasted with those for electromagnetic showers in matter initiated by very high energy electrons or photons. In that case, transverse momenta are imparted predominantly by Coulomb scattering from nuclei: the maximum from each collision is (where is a mixed mass determined by the inverse nuclear size); since the energies of shower particles decrease only very slowly, there is no clumping in the final state, and the momenta of the are spread roughly uniformly. (This behavior is evident in extended air showers initiated by high energy cosmic rays.)

(17) Clearly observables with this property must be linear in the energies of collinear sets of partons (so that sphericity does not qualify). Such observables are formally classified as ``infrared'' finite'', since divergences appearing in their mean value from decays with cancel just as in the total cross-section . (The final phase space is weighted uniformly in the calculation of ; it is slightly, but continuously, corrugated in the calculation of the average values of infrared finite observables.)

(18) is related to the eigenvalues , , of the matrix[16] by . (The existence of this relation was suggested to me by R. K. Ellis.) Thrust is given roughly by . The are zero for events with inversion symmetry. For a final state consisting of three massless particles . In practical experiments where only incomplete hadronic final states are measured, missing energy may be corrected for by dividing each measured by the observed : missing momentum significantly affects , but may perhaps be corrected for by boosting to the measured rest frame. Note that with the definition (6), for a complete final state containing massive particles.

(19) The complete form, retaining a small mass for the gluon is [1] .

(20) For a planar event , while for an isotropic event, . In general, . In terms of the eigen-values . for planar events.

(21) As would lead, for example, to terms proportional to the sum of quark changes rather than in the total decay width (the former terms appear only at in perturbation theory).

(22) The decay widths may be obtained as the imaginary part of the ``vacuum'' energy density . For massless quarks and gluons, first-order calculations suggest that ; therefore has an absolute minimum at , indicating that the usual vacuum with should be stable. I am grateful to J. Sapirstein for discussions regarding these points.

(23) The separating gluons presumably interact by virtual gluon exchange. Self-energy corrections to the virtual gluon propagator yield an effective coupling which results in antiscreening of the charges. Spontaneous nonperturbative gluon production can roughly be considered as real production of the pairs responsible, through fluctuations in the exchanged gluon field. It is therefore possible that produced real pairs may also anitscreen the separating gluons, thereby increasing, rather than decreasing the reinteraction cross-section.

(24) The effective QED coupling strengthens the usual Coulomb potential at , and if screens the potential for . A deeply bound state with may therefore exist (but be totally unobservable in practical systems).

(25) The probabilities for formation of clusters with different masses may perhaps be estimated from the total annihilation cross-section at that mass, thereby providing a weighting for each event, which may possibly be used to infer the behavior of the high energy total cross-section.

(26) Note that below threshold, the photon is dominantly , ; final states consisting of an odd number of pions are therefore suppressed. This effect is the result of interference between and amplitudes, and cannot be obtained by classical considerations. It results in a considerable enhancement in the fraction of charged hadrons produced at small , and must be corrected for in comparisons of models with experimental charged multiplicities.

(27) In any confining potential, the spontaneous nonperturbative parton production discussed above should occur (as in atoms (e.g., [23]), typically leading to low-mass decay products. On the other hand, the statistical bootstrap model [24] favors unsymmetrical decays, with constant energy release, and one light, one heavy, product. It thus implies , in gross disagreement with data.

(28) For this and other experimental annihilation results, see the contributions from PETRA groups to this conference.

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