ABSTRACT
In conclusion of our excursus in QCD we summarize the quantitative verification of the theory. It is obvious, as precision tests of any interaction theory, we can use various measurements of its coupling constant. In QED, the best determinations of the coupling constant agree with the theoretical value to eight significant figures. However, in QCD the situation is rather different. Since the QCD perturbation theory works only for hard-scattering processes, with uncertainties specified by soft processes that are difficult to estimate, this theory has not been tested to such an extreme precision. Just the same, it is interesting to compare the best available determinations of αs to see how well they agree. Consider a typical QCD cross section that calculated perturbatively. If this process is
allowed even in the first order of the perturbation theory, the resulting cross section is given by the expression:
σ = ∑ i=0
A2i+1α 2i+1 s . (11.1)
When the process in question may begin with the second order of the perturbation theory, then, instead of (11.1), we have
σ = ∑ i=1
A2iα 2i s . (11.2)
The coefficients A1, A2, . . . come from calculating the appropriate Feynman diagrams. In performing such calculations, various divergences arise, and these must be regulated in a consistent way. This requires a particular renormalization scheme (RS). The most commonly used one is the modified minimal subtraction (MS) scheme (see Section 3.4). The cross sections and other physical quantities calculated to all orders in the perturbation theory do not depend on the RS. However, the finite coefficients An (with n ≥ 2) and the coupling constant αs depend implicitly on the renormalization convention employed and explicitly on the scale Q. Usually, the QCD cross sections are known to the leading order (LO), or to the next-to-
leading order (NLO), or in some cases, to the next-to-next-to-leading order (NNLO). Note, only the latter two cases, which have cancelled RS dependence, are useful for precision tests. At NLO the RS dependence is completely given by one condition that can be taken to be the value of the renormalization scale Q. At NNLO this is not sufficient, and Q is no longer equivalent to a choice of scheme; both must now be specified. Therefore, one has to address the question of what is the best choice for Q within a given scheme, usually MS. There is no definite answer to this question-higher-order corrections do not fix the scale, rather they give the theoretical predictions less sensitive to its variation. One should assume, choosing a scale Q characteristic of the typical energy scale (E) in
the process would be most appropriate. In general, a poor choice of scale generates terms of
Particle and
the order ln(E/µ) in the An’s. Various methods have been proposed including choosing the scale for which the NLO correction turns into zero (Fastest Apparent Convergence [225]); the scale for which the NLO prediction is stationary [226], (that is, the value of Q where dσ/dQ = 0); or the scale dictated by the effective charge scheme [228] or by the scheme proposed in [227]. By comparing the values of αs, which different reasonable schemes render, an estimate of theoretical errors can be obtained. One can also attempt to determine the scale from data by allowing it to vary and using a fit to define it. This method can allow a determination of the error due to the scale choice and can give more confidence in the end result [229]. It should be stressed that in many cases this scale uncertainty is the dominant error. An important practical consequence is that if the higher-order corrections are naturally
small, then the additional uncertainties caused by the Q dependence are likely to be small. However, there exist some processes for which the choice of scheme can influence the extracted value of αs. There is no resolution to this problem other than to try to calculate even more terms in the perturbation series. It is important to note that, since the perturbation series is an asymptotic expansion, there is a limit to the accuracy with which any theoretical quantity can be calculated. Experiments such as those from deep-inelastic scattering involve a range of energies.
After fitting to their measurements to QCD, the resulting fit can be expressed as a value of αs. Determinations of αs arise from fits to data employing NLO and NNLO: LO fits are not useful and their values will not be employed in the following. Care must be exerted when comparing results from NLO and NNLO. So as to compare the values of αs from various experiments, they must be evolved using the renormalization group to a common scale. For convenience, this is taken to be the mass of the Z-boson. The extrapolation is carried out using the same order in the perturbation theory as was used in the analysis. This evolution uses the third-order perturbation theory and can enter additional errors particularly if extrapolation from very small scales is employed. The variation in the charm and bottom quark masses can also introduce errors. For example, using
