Optimized perturbation theory

Theoretically, it turns out that the second-order corrections to the 3-jet rate are large. It is therefore not unreasonable to expect large third-order corrections to the 4-jet rate. Indeed, the experimental 4-jet rate is much larger than second order predicts (when fragmentation effects have been included), if $\alpha_{\mathrm{s}}$ is determined based on the 3-jet rate [Sjö84a,JAD88].

The only consistent way to resolve this issue is to go ahead and calculate the full next order. This is a tough task, however, so people have looked at possible shortcuts. For example, one can try to minimize the higher-order contributions by a suitable choice of the renormalization scale [Ste81] -- `optimized perturbation theory'. This is equivalent to a different choice for the $Q^2$ scale in $\alpha_{\mathrm{s}}$, a scale which is not unambiguous anyway. Indeed the standard value $Q^2 = s = E_{\mathrm{cm}}^2$ is larger than the natural physical scale of gluon emission in events, given that most gluons are fairly soft. One could therefore pick another scale, $Q^2 = f s$, with $f < 1$. The ${\cal O}(\alpha_{\mathrm{s}})$ 3-jet rate would be increased by such a scale change, and so would the number of 4-jet events, including those which collapse into 3-jet ones. The loop corrections depend on the $Q^2$ scale, however, and compensate the changes above by giving a larger negative contribution to the 3-jet rate.

The possibility of picking an optimized scale $f$ is implemented as follows [Sjö89]. Assume that the differential 3-jet rate at scale $Q^2 = s$ is given by the expression

$$R_3 = r_1 \alpha_{\mathrm{s}}+ r_2 \alpha_{\mathrm{s}}^2 ~,$$ (34)

where $R_3$, $r_1$ and $r_2$ are functions of the kinematical variables $x_1$ and $x_2$ and the $y$ cut, as implied by the second-order formulae above, see e.g. eq. (). When the coupling is chosen at a different scale, $Q'^2 = f s$, the 3-jet rate has to be changed to

$$R_3' = r_1' \alpha_{\mathrm{s}}' + r_2 \alpha_{\mathrm{s}}'^2 ~,$$ (35)

where $r_1' = r_1$,

$$r_2' = r_2 + r_1 \frac{33-2n_f}{12\pi} \ln f ~,$$ (36)

and $\alpha_{\mathrm{s}}' = \alpha_{\mathrm{s}}(fs)$. Since we only have the Born term for 4-jets, here the effects of a scale change come only from the change in the coupling constant. Finally, the 2-jet cross section can still be calculated from the difference between the total cross section and the 3- and 4-jet cross sections.

If an optimized scale is used in the program, the default value is $f=0.002$, which is favoured by the studies in ref. [Bet89]. (In fact, it is also possible to use a correspondingly optimized $R_{\mathrm{QCD}}$ factor, eq. (), but then the corresponding $f$ is chosen independently and much closer to unity.) The success of describing the jet rates should not hide the fact that one is dabbling in (educated, hopefully) guesswork, and that any conclusions based on this method have to be taken with a pinch of salt.

One special problem associated with the use of optimized perturbation theory is that the differential 3-jet rate may become negative over large regions of the $(x_1, x_2)$ phase space. This problem already exists, at least in principle, even for a scale $f = 1$, since $r_2$ is not guaranteed to be positive definite. Indeed, depending on the choice of $y$ cut, $\alpha_{\mathrm{s}}$ value and recombination scheme, one may observe a small region of negative differential 3-jet rate for the full second-order expression. This region is centred around $\mathrm{q}\overline{\mathrm{q}}\mathrm{g}$ configurations, where the $\mathrm{q}$ and $\overline{\mathrm{q}}$ are close together in one hemisphere and the $\mathrm{g}$ is alone in the other, i.e. $x_1 \approx x_2 \approx 1/2$. It is well understood why second-order corrections should be negative in this region [Dok89]: the $\mathrm{q}$ and $\overline{\mathrm{q}}$ of a $\mathrm{q}\overline{\mathrm{q}}\mathrm{g}$ state are in a relative colour octet state, and thus the colour force between them is repulsive, which translates into a negative second-order term.

However, as $f$ is decreased below unity, $r_2'$ receives a negative contribution from the $\ln f$ term, and the region of negative differential cross section has a tendency to become larger, also after taking into account related changes in $\alpha_{\mathrm{s}}$. In an event-generator framework, where all events are supposed to come with unit weight, it is clearly not possible to simulate negative cross sections. What happens in the program is therefore that no 3-jet events at all are generated in the regions of negative differential cross section, and that the 3-jet rate in regions of positive cross sections is reduced by a constant factor, chosen so that the total number of 3-jet events comes out as it should. This is a consequence of the way the program works, where it is first decided what kind of event to generate, based on integrated 3-jet rates in which positive and negative contributions are added up with sign, and only thereafter the kinematics is chosen.

Based on our physics understanding of the origin of this negative cross section, the approach adopted is as sensible as any, at least to that order in perturbation theory (what one might strive for is a properly exponentiated description of the relevant region). It can give rise to funny results for low $f$ values, however, as observed by OPAL [OPA92] for the energy-energy correlation asymmetry.