3. Correlations with Respect to the Beam Axis

3.1. Introduction

We have discussed above the rotationally invariant observable obtained by averaging , defined in (1.2), over all possible positions of the two detectors which preserve their relative orientation. This averaged characterizes the shapes of events and is probably the most direct probe of their dynamical mechanisms. However, QCD also makes unambiguous predictions for the dependence of the shapes of events on their orientation with respect to the beam axis, which dependence we have thus far brusquely averaged away. In this section we consider this angular dependence using both the energy correlation and its moments with respect to which represent rotationally-non-invariant analogues of the In Sect. 3.2 we analyze the general form of the angular dependence of and describe a particularly convenient choice of angular coordinates. In Sect. 3.3 we define the moments of which provide an infrared stable measure of the angular correlations, and in Sect. 3.4 we present results for the three basic processes and We consider these both in the free quark and gluon approximation and including the effects of hadron fragmentation. However, we shall not consider either heavy quark or lepton production events (9); further, our results are specialized to the case of unpolarized electron and positron beams. It is straightforward to generalize our results in these two areas.

3.2. General Form of the Angular Dependence of Energy Correlations

Consider two detectors fixed at particular positions in space separated by an angle The directions of these detectors from the point of interaction (and the normal to the plane defined by them) may be used to define an orthogonal set of axes with respect to which one can specify the direction of the beam axis. Temporarily we consider the detectors to be fixed and take the beam direction as variable. In the next paragraph, we shall use the alternative choice of a fixed beam direction and detectors at variable angles. With the detectors fixed, one assigns one of the axes to be the normal and the other two to be in the plane formed by the two detectors. The direction of the axes in the plane may be chosen arbitrarily. All choices, if consistently used, are, of course, equivalent; however, some may be more convenient than others. As we shall see, the best choice is to take the z axis to be the normal to the plane of the detectors, and the and axes to be in the plane, with the x axis defined to be on the line bisecting the angle between the two detectors. We denote this choice by the subscript ( axis Normal to plane). We shall also sometimes discuss the system ( axis in Plane) whose axis is normal to the plane and axis is in the same direction as the axis in the system. The careful reader will perceive that various signs are undefined in these definitions (e.g., one can reverse directions of and axes in the system). However, no parity-conserving observable is sensitive to the ambiguities.

Now consider the energy correlations between two detectors for an annihilation event. Clearly, the correlation depends on the direction of the beam referred to the axis sets we described above; this beam direction may be specified by spherical polar angles and so we are led to define a beam-orientated energy correlation that is a function of and To be precise, we shall actually not define exactly in this way but rather fix the direction and define a fixed set of axes with along the direction. Then we take the ``reference'' orientation of the two detectors so that coincides with the detector-defined axis set described in the paragraph above. Any orientation of the two detectors is then specified by a rotation R acting on the reference orientation. Let be in the standard Euler angle form

where is a rotation about the axis through angle and a rotation about the axis. and are the angles to be used in the specification of . (10) For unpolarized beams, is independent of The unit spin of the photon severely limits the possible dependence of on and ; we now turn to a discussion of these constraints.

Consider the general process

where we have specialized to the case of point detectors and consider 1 and 2 as particles heading in the directions of the two detectors. Working in the virtual photon rest frame, let be the spin component of the virtual photon referred to the direction as a quantization axis, and let be the spin component of the virtual photon with respect to the ``reference'' final state axes. Then if denotes the unmeasured momenta and helicities of the final particles, the amplitude for anything may be written as

where are vertex functions, which give the helicity amplitudes for and anything, respectively. Taking and using (for massless ), the square of the amplitude (3.3) becomes

which gives the inclusive cross-section anything. For point detectors, one must simply multiply it by to obtain One may expand the resulting expression and derive the general form for the and dependence of The parity invariance of the interaction places some constraints on this form. The constraints take a different form depending on whether the axis lies in the 12 plane (as in the system) or along the normal to it (N system). In a -type system, parity invariance implies

where is some phase and denotes the state obtained by application to of the symmetry operator parity operator, a rotation through about the direction. of course, leaves the directions of 1 and 2 invariant. Combining (3.5) and (3.4), one finds

where through (which, of course, depend on ) can be related to bilinear sums over and

If now the axis is taken along the normal to the 12 plane, the symmetry operator for the system becomes so that the constraints from parity invariance become

Combining (3.7) and (3.4), one finds

Comparing (3.6) and (3.8), we see that there are, in general, 4 independent beam-oriented energy correlation functions. The specific expansion coefficients in the complete energy correlation as a function of the directions of the detectors with respect to the beam (or, equivalently, at least for the point detector case considered here, the inclusive differential cross-section for anything) depend, however, on the choice of coordinate system. The results (3.6) and (3.8) hold for any choice of orthogonal axes in the 12 planes. However, they must be symmetric under the interchange This constraint may be expressed most simply if one chooses one of the axes in the plane along the bisector of the angle between the directions of 1 and 2. In this case, the terms and vanish and one may write the angular distribution in either the or system as

With the above choice, our previous rotational averaged is just the function while moments of with respect to the orthogonal functions and give and :

3.3. Moments of Angular Correlations

The observables discussed in [1] were of the form

where are unit vectors along the momenta of particles and and the were chosen to be the Legendre polynomials. The give a complete specification of the rotationally invariant two-point energy correlation function in an event. In [10] and in Sect. 3 below, we discuss the expansion of the three-point energy correlation defined as

In this section, we consider another extension of (3.11) in which now depends on the direction of the incoming beams, We take

We find that these observables provide information on the angular distributions of planar structures in events with respect to the beam direction. The general analysis of Sect. 3.2 allows us to write and in terms of the linearly independent functions

where is the angle between particles and and are the angles which specify the beam direction in the coordinate system. This set of axes is defined as described in Sect. 3.2, but with the directions of particles and replacing the detector directions of the previous discussion. The subscript indicates that the axis is normal to the plane defined by and The explicit factor in (3.14) is necessary to make the well defined in the limits or This can be seen from the expressions for the beam angular functions in terms of scalar products of and :

which illustrates the necessity of the factor in the definition of to avoid problems at or We define using (3.13) with as taken from (3.14) or (3.16). It is clear that the share the infrared stability of the when they are computed in QCD perturbation theory.

Another possible set of observables, which appears to be less sensitive to hadron fragmentation than the (see below), is defined in analogy with (3.13):

The are only independent of the for and 1. The definition (3.17) is singular at for or For the axis remains well defined (along the direction) but the and axes of the system are undefined. In this case, we take the to have the values obtained by averaging uniformly over all possible directions of the and axes in the plane perpendicular to so that

where is angle between beam direction and the particles or

In the case an analogous situation pertains and only the axis is well defined. We take to have the value obtained by averaging over all possible directions for the and directions, so that

Note that in both limits, is proportional to giving the only possible non-trivial dependence on

The cannot be written in a form analogous to the from which their infrared stability would be evident. Their values depend, of course, on the treatment of the singular case . We give evidence below that the prescription for this described above is correct and renders the infrared stable. It is clear that the energy weighting in (3.17) protects the from soft infrared divergences. Collinear quarks and gluons also give rise to divergences in the differential cross-section. The can only be infrared stable if they take on the same value for divergent configurations in which two separate particles are exactly collinear and in which a single particle carries their total momentum. This will be the case with our prescription for handling collinear particles only if, in configurations where the particles are nearly collinear, all potential divergences are independent of the azimuthal directions of the particles with respect to the axis defined by the vector sum of their momenta. Any dependence on the azimuthal direction will appear in the amplitude as terms proportional to where is the transverse momentum of one of the particles with respect to the total momentum axis. Divergences in the amplitude are (up to logarithms) of the form Hence, any contribution to the amplitude which is not independent of azimuthal angle will be finite as Thus the divergent parts of amplitudes for collinear production of particles are azimuthally-symmetrical, so that (with our prescription) the take on the same value for this configuration as when a single particle is produced in place of several collinear ones. Hence it appears that the moments of the should be infrared stable when computed in QCD perturbation theory.

Consider now an event in which three partons are produced. Then the angular distribution of the plane defined by their momenta with respect to the beam axis will be characterized by or . In actual events, where the final state consists of hadrons, the values of these ratios will approach the free quark and gluon results as increases. The therefore provide a method for determining the angular distributions of planes of particles with respect to the beam direction without requiring the plane to be found by minimizing an observable [8], which might well induce spurious effects. The give the moments of the angular distributions of planes just as the described in [2] describe the angular distributions of jets.

The angular distributions of planes in general depend both on their polar and azimuthal angles with respect to the beam direction. The polar and azimuthal dependences can, of course, be rearranged by making different choices of frame (e.g., or ). One might expect that in some frame two jet events should contribute on average only to the polar distribution. However, while this is clearly the case for pure final states, it is no longer possible after fragmentation to hadrons to choose a frame in which the azimuthal dependence vanishes. Hence both and should be considered; no particular feature of the angular distribution appears to be especially distinguished.

Finally we note that are the Legendre transforms (with respect to ) of the angular terms (3.10) in the energy correlation function Their relation to is, therefore, analogous to the relation of the to the rotationally-averaged

3.4. Some Analytical Results for

For the process is only nonzero at where the ``plane'' given by the two detectors is undefined. We use the same azimuthally symmetric prescription for defining the plane here as we described for at in the previous section. If is the angle between the and direction, then the angular distribution gives for the parameters in defined according to (3.10)

To estimate the modifications to our results when the and fragment into hadrons, we would usually simulate complete hadronic events. We do this in the next section but present here a simple estimate for the effects of fragmentation of and which agrees closely with the results from the more complicated model. Our simple estimate is obtained by assuming that away from is dominated by a complete jet entering one detector and a single hadron from the other jet being incident on the second detector. The first jet then has an angular distribution with respect to the beam direction, while we take the single hadron to be distributed uniformly in azimuth about the direction of the jet from which it came. This picture leads to the simple predictions

which are in remarkably good agreement with the full Monte Carlo simulation of fragmentation shown in Figs. 8 and 9. Of course, this simple picture predicts only the angular correlations and not the overall normalization

[ Figure 8 ] The beam angular correlation defined in (3.10) for the processes and in the free quark and gluon approximation (``idealized calculation'') and for simulated hadronic final states. We also show results for events obtained from the simple fragmentation model described in Sect. 3.4. There is a severe nonuniformity in the curves near as discussed at the end of Sect. 3.4 [cf., (3.22)]. The figures do not attempt to illustrate this phenomenon correctly

[ Figure 9 ] The second independent beam angular correlation for the same processes as given in Fig. 8. For both and the subscript on the angles denotes the use of the frame where the axis is perpendicular to plane defined by the two particles

For the function is given in (2.24). One can also calculate and K in the free quark and gluon approximation to find [at ]

where the coefficients of the delta functions are, as usual, infrared divergent constants. For one also finds

while for this case can be calculated using the results of [9]. The analytical results for and are shown in Figs. 8 and 9 for both and The result is a surprisingly common one [see (3.20)-(3.23)]. We know of no simple explanation for this (see Appendix A); it appears to be ``accidental''. An isotropic (phase space) model would, of course, give zero for and so the common nonzero value of for the low order QCD processes in annihilation should provide a useful method of identifying them.

Inspection of (3.22) reveals that (ignoring delta function terms) both and tend to the values given in (3.20) for the pure process (with our prescription for defining the final plane) as This supports our argument in the last section for the infrared stability of the equality of the and values for and at will allow the infrared divergences to cancel between the contributions of the two final states. Remembering that the coefficient of in is just we may form the infrared finite combinations of the coefficients of in and :

where is the angle between particle and the beam. The infrared finiteness of these combinations of terms on their own occurs only at In higher orders, only the complete moments integrated over all possible final state configurations will be infrared finite. It is amusing that the result for is only violated in the coefficient of to

3.5. Some Results on and its Moments for Hadronic Events

The observables and introduced in Sect.3.3 are shown in Figs. 8 and 9 as a function of for events of several types, both in the free quark and gluon approximation, and for simulated hadronic final states. and completely specify the dependence of on the orientation of the two detectors with respect to the beam direction. The most striking feature of the curves in Figs. 8 and 9 is the similarity between results from the different QCD processes. An isotropic model with would be easy to distinguish from the QCD reactions. These figures also confirm that our simple model for fragmentation reproduces rather well the hadron final state angular functions given by the full Monte Carlo calculation. The effects of hadron fragmentation are minor except for the final state and even there the hadron final states show beam correlations very different from the isotropic case.

[ Table 1 ] The beam moments for at

[ Table 2 ] The beam moments for at

[ Table 3 ] The beam moments for at

In Tables 1 through 3, we give the mean values of the moments defined in Sect.3.3. These are infrared stable and probably preferable to and discussed above. The latter have particular problems near which are properly averaged in the moments. The tables give and is the simplest moment that provides the cleanest tests of the theory with the smallest effects from hadron fragmentation. In fact, the effects of hadron fragmentation appear to be less severe in than in the analogous single energy correlation in discussed in [1]. We currently consider as the best way of measuring beam angular dependence of the energy distribution of annihilation final states. has a weighting function that is odd in and so is useful for investigating which is predicted to be approximately odd in Thus and should always be small while provides a measure of the absolute magnitude of We have not discussed the simplest odd function because of its sensitivity to missing particles in incomplete final states observed experimentally (recall that vanishes because of momentum conservation). Finally the tables show which is the simplest example of a moment observable with an explicit suppression of the regions.

Tables 1 and 2 also show results with an cut (applied to the final hadrons) which selects ``true'' 3 jet events. The fact that these results are similar to those without the cut is an indication that the beam correlations do not depend importantly on the ``shapes'' of the events.

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