3. Infrared Stability

Graphs in which gluons or quarks may be soft, or may become collinear, receive divergent contributions from these kinematic configurations. The Kinoshita-Lee-Nauenberg (KLN) theorem guarantees, however, that in the total cross section such divergences cancel, leaving a finite result [8]. The divergences encountered in the calculation of the moments of the ,

are very similar to those involved in the calculation of the total cross section. It seems very probable, therefore, that they should cancel in the same way.

In eq. (3.1), we are implicitly calculating the moments of the for final states consisting of free quarks and gluons. We discuss the procedure in detail in subsect. 3.2. In reality, one must calculate the for final states consisting of hadrons. We discuss this in subsect. 3.3. Our discussion there will require precise definition of a jet, and since this problem provides a useful introduction, we shall consider it first.

3.1. Jets

A typical quark or gluon jet is shown schematically in fig. 14. Such a jet would be indistinguishable from a single particle by a detector with energy resolution greater than and angular resolution greater than . ( and are normalized so that a detector which only counts the total cross section and cannot distinguish any details of the final state has .) To order ( is the QCD coupling constant) the only diagram for production of two jets by a virtual photon is that of fig. 15. To order , there are two types of diagram which lead to two-jet final states. Examples are given in figs. 16a, b. For the diagram of fig. 16a to contribute to the 2-jet cross section, either the gluon or quark must have an energy , or they must be traveling in the same direction to within . If these conditions are not satisfied, then the detector will be able to tell that three, rather than two, particles were produced, and the event will be classified as a three-jet one. In order to regularize the calculation of diagrams such as those in fig. 16, one must assign the gluon a fictitious mass, . Then the probability corresponding to diagrams like fig. 16b will contain terms of the form , . Diagrams like fig. 16a will contain terms of the form , .

[ Figure 4 ] Schematic form of a typical quark or gluon jet. Particles within the jet must either have an energy less than a fraction of the energy of the primary particle, or must travel at an angle less than with respect to it.

[ Figure 5 ] The lowest order diagram for the process .

[ Figure 6 ] Examples of terms at in the amplitude squared for the process (G). The first diagram will contribute either to the two- or three-jet cross section, depending on the momentum of the gluon, while the second will contribute only to two-jet production.

If and are both taken to be one, that is, if the detector is never able to distinguish a `three-jet' event, so that it measures only the total cross section, then the KLN theorem guarantees that it will measure a finite cross section to any order in . Specifically, to order , the terms in in the loop and tree graphs will cancel, so long as . It turns out that the cancellation of -dependent terms occurs, at every order in , even if [25]. (This result was verified explicitly to in ref. [2] and essentially to in ref. [9].) This means that the cross section for production of two or three jets at is free of divergences, as long as and are both finite. If either of and is set to zero, then a divergent cross section will be obtained. The cross section would be for production of, say, a quark, with no associated gluons. Since any quark produced will always radiate some gluons, such a cross section is not physically meaningful.

The cross section at for inclusive production of a quark from a virtual photon is shown in fig. 17, as a function of the fractional energy , of the quark. For the quark to have its maximum energy , the kinematics of the process (see the appendix) require no gluons, of any energy, to have been produced. However, processes like that in fig. 16b can occur, and give rise to a divergent cross section just at . That is, they contribute terms to the differential cross section like , where is a positive constant which diverges when . The integral of shown in fig. 17 over from 0 to 1 is rendered finite by the presence of the function at (to see this, one must first regularize by taking ; the relevant formulas are given in sect. 4). This corresponds to the total cross section at , which is known to be finite from the KLN theorem.

[ Figure 7 ] The differential cross section for the process anything calculated to order , as a function of the fractional energy, , of the quark. The final state is considered to contain two jets if , and to contain three jets if the quark has . Only the process G (fig. 16a) contributes to 3-jet production, but both G (fig. 16a) and (figs. 15 and 16b) can give rise to two-jet final states. The infrared divergences in the cross sections for these processes cancel when the total two-jet production cross section is computed.

Let us now introduce the energy resolution . Quarks with will be assumed to belong to 2-jet events, while those with will be distinguished as belonging to three-jet events. If is taken to be 0, then all of the sharply rising 3-particle production cross section will be included in the 3-jet cross section, while the negative delta function associated with the two-particle production process will not be included. Clearly, with this choice for , the `3-jet' production cross section will be infinite. As discussed above, however, such a choice for is not physically sensible. If a finite value for is chosen, then the two- and three-jet configurations will be defined as indicated in fig. 17, and their cross sections will be separately finite.

The above discussion is incomplete because it ignores divergences associated with collinear quarks and gluons. In fact, one must only include in the 3-jet sample events in which all the quarks and gluons are separated by angles greater than . (Only if the quarks are taken to be massless does the collinear quark and gluon configuration lead to a divergence but collinear gluon pairs inevitably give divergent contributions.) Instead of making cuts in both angle and energy, one may cut only in or in order to isolate three-jet events.

The conclusion of this discussion is, therefore, that, while it is sensible to ask for the total cross section to any order in , it is not sensible to ask for the total probability that, say, two jets are produced to that order, unless one has specified reasonable values for the resolution parameters and which delineate two- and three-jet production.

3.2. Moments of the for Final States of Quarks and Gluons

Now let us return to the evaluation of the . We shall begin by ignoring the fact that the quarks and gluons produced by the virtual photon will eventually be combined into hadrons. At first we simply calculate for final states consisting of free quarks and gluons. To do this, we must evaluate the expression (3.1), where the integral is over all possible kinematic configurations for the final state, and in a sum is done over all the particles in each final-state configuration. The numerator of (3.1) may be written as

First consider evaluating this for . Clearly, in every kinematic configuration, will simply be 1. Then (3.2) reduces to an expression for the total cross section, which is known to be finite. Now consider evaluating . For the result to be sensible, it must be independent of the gluon mass (infrared cut-off) , when the integral over all kinematic configurations is done. To order , one may construct explicitly, and check that the integral (3.2) is independent of . This is done in sect. 4. The integral is found to divide into two parts. The first is identical to the total cross section, while the second contains a weighting function which vanishes in all kinematic configurations for which divergences occur in the differential cross section (4). Note that the particularly simple form for in the case of 3-particle final states (see eq. (4.9)) will not persist in higher orders: only at O is the divergent term encountered in the calculation of the moments of exactly the total cross section.

Perturbation theory can only be valid if successive terms in the perturbation expansion are, on average, progressively smaller. In QED, the breakdown of the perturbation expansion in the infrared region is well-known. The same phenomenon occurs in QCD. If the main contribution to the expectation value of an observable comes from a kinematic region close to an infrared divergence, then its value deduced from perturbation theory must be suspect. The parameter which governs the applicability of perturbation theory to the process G is presumably , where are the fractional energies of the and is the QCD (running) coupling constant evaluated at the c.m. energy under consideration. As , this parameter becomes large and the G final state becomes indistinguishable from with . When the parameter is large, higher orders in the perturbation expansion will be no less important than the low orders under consideration. The results for the total cross section and (as we shall argue below) for the moments of the are finite at each order in the perturbation expansion. However, the actual finite numerical values may be modified significantly by the inclusion of higher orders. It is believed that this phenomenon does not occur for the total cross section. This is exemplified by the O contribution to the total cross section, which is smaller than the lowest-order term by a factor of . Some moments of the cross section will, while remaining finite, probe kinematic regions close to infrared divergences to a greater extent than the total cross section. These moments should remain finite, but may well receive numerically important contributions from higher-order effects. This phenomenon occurs for some of the high-order moments of, for example, , in the process G (see subsect 4.4).

Cancellations of -dependent terms in the total cross section come when a tree graph at one order in gives rise to (almost) the same final-state kinematic configuration as loop graphs of the same order. So long as an observable does not distinguish between the canceling graphs (that is, it gives them all the same weight in an integral of the type (3.2)), the integrals for its moments should experience the same cancellations of -dependent terms as occur in the integral for the total cross section. The have this property. The decomposition of the integral for mentioned above is one consequence. In a divergent kinematic configuration, different graphs are treated the same, just as in the calculation of the total cross section.

Any variable which is proportional to the energies of the final particles will have the same value for kinematic configurations which differ simply by the addition of a gluon of indefinitely low energy. However, one must also demand that the variables do not distinguish between configurations containing a single particle and two (or more) nearly collinear particles with the same total energy [5,6]. The fulfill both requirements. It, therefore, seems very reasonable to expect that the infrared divergences associated with the calculation of their moments to every order in will cancel in the manner described above. The same cancellation should occur for the moments of `spherocity' but not for those of `sphericity', since the latter is not linear in the momenta of nearly collinear final particles (5).

The linearity in momenta of the is advantageous at the hadron as well as the quark and gluon level. For example, the values of the are essentially unaffected by decays with small internal transverse momenta; they have the same value whether the parent particles or their decay products are measured.

3.3. Moments of the for Hadronic Final States

Further problems appear when one attempts to calculate the moments of the (or `spherocity', `thrust' and so on) for genuine processes involving the production of hadrons rather than of quarks and gluons. For example, at O, one must decide whether a quark and soft gluon will fragment into hadrons as a quark or like a quark and a gluon. This means that in order to estimate the production of hadrons, one must divide up the O cross section into 2- and 3-jets parts by using sensible values for the parameters and which distinguish the two. Then, in the two-jet region, we assume that it is a two-quark state which fragments to hadrons, while in the three-jet region, the gluon as well as the two quarks are taken to fragment separately into hadrons.

It is as if the formation of the final hadron state provides a measuring apparatus with finite resolution for the subprocess involving quarks and gluons. The configuration of the final hadron state is unaffected by small changes in the energies and momenta of the quarks and gluon which generate them. Only a large deviation in the energies of the quarks from those of the process is reflected by the appearance of three hadron jets in the final state; smaller deviations are beyond the `resolution' of the hadrons in the final state. The `resolution' of the hadrons will be determined by the model for their production by the `fragmentation' of quarks and gluons. To be exact, this fragmentation should be taken to depend on the total c.m. energy, . Such dependence, which involves mixing between quark and gluon fragmentation functions, is believed to be given by simple renormalization group equations. However, the necessary formalism has so far been developed only for single-hadron inclusive distributions [10] (6). In order to make realistic estimates for the moments of , we require a model for the generation of a complete hadronic final state. For this purpose, we use the model developed by Field and Feynman [18], despite the fact that it exhibits exact scaling at asymptotic energies. Fortunately, the model turned out to be adequate, and our final results are essentially independent of the arbitrary cut-offs between two- and three-jet events.

3.4. The Infrared Instability of

From the formula (2.20), one finds that the coefficient of the term in is , which is not infrared stable against collinear divergences. Since , this divergent contribution must be absorbed by a compensating divergence at other values of . The presence of divergences in is to be expected, since events containing two particles which are arbitrarily close in angle will be weighted differently in the computation of from those containing a single particle which carries the sum of their energies, because the two particles cannot pass through the same point detector. It is clear that to the lowest order in for any process, there will be no infrared divergences in , and that in the next order, divergences will appear only at those values of for which the lowest-order is non-zero (e.g., for (G) to O, divergences appear only at ). In higher orders, divergences will occur at all values of .

The energy correlation defined by eq. (2.22) is formally infrared stable so long as the are non-zero (7). However, as the angular size of the detectors tends to zero, will become infrared unstable. For final states consisting of free quarks and gluons, might serve as a satisfactory measure. However, for genuine hadronic final states, the formation of the final state from quarks and gluons introduces a finite angular resolution, whose size cannot at present be deduced directly from QCD. Unless is very large, the unknown resolution associated with the formation of hadrons will be the most important quantity in determining , thereby rendering it useless. If one chooses a large value of , very little information will be obtained unless the integration over the areas is performed using a non-trivial weight function, in which case essentially the will be obtained.

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