#
(p.436)
Appendix D *R* _{γ}, *Rl* AND *Rτ* FOR ARBITRARY COLOUR FACTORS

#
(p.436)
Appendix D *R* _{γ}, *Rl* AND *Rτ* FOR ARBITRARY COLOUR FACTORS

This chapter contains a compilation of the ingredients that go into the theoretical prediction for *R* _{l} and *Rτ*. All expressions are given for arbitrary colour factors, which allows to evaluate not only the QCD-SU(3) predictions, but also the predictions for alternative theories with an unbroken gauge symmetry based on a simple Lie group. This is needed, for example, by any analysis which aims at a measurement of the colour factors from *R* _{l} and *Rτ*. Keeping the colour factors, it is convenient to redefine the coupling constant such that the amplitude for gluon emission from a quark is independent of the gauge group of the theory. Absorbing a factor 2π as well yields the redefined coupling

*n*

_{f}quark degrees of freedom then can be expressed as function of the free parameter

*a*

_{s}and the variables

# D.1 The running coupling constant and masses

The variation of the strong coupling constant *a* _{s} and renormalized masses$\overline{m}$with the renormalization scale of the theory is described by a coupled system of differential equations,

*b*

_{i}and

*g*

_{i}depend on the specific theory. The leading coefficients (Jones, 1974; Caswell, 1974; Tarasov

*et al*., 1980; Tarrach, 1981; Nacht-mann and Wetzel, 1981) are, c.f. Section 3.4.5,

Equation (D.3) determines how the strong coupling constant evolves for a fixed number of active flavours, whereas in practical applications one often has to relate a value of *α* _{s} from a scale *µ* _{4} with *n* _{f} = 4 active quark flavours to the measurement at a scale *µ* _{5} with *n* _{f} = 5 flavours. The treatment of flavour thresholds is described in Bernreuther and Wetzel (1982), Bernreuther (1983), Marciano (1984), and Rodrigo and Santamaria (1993), c.f. Section 3.4.5. With *a* _{s} = *a* _{s}(*n, f*, *µ*), the coupling constant *a* _{s}(±) = *a* _{s}(*n* _{f}± 1, µ) for a different number of flavours, but at the same energy scale µ, can be expressed as a power series in the original coupling. To $\text{O(}{\alpha}_{\text{s}}^{\text{3}})$ the expansion is given by

*μ*close to$\overline{m}(\overline{m})$is the coupling continuous, as one would naively expect. The numerical value of the point of continuity depends on the order of the perturbative expansion. Up to NLO, it coincides with$\overline{m}(\overline{m})$

In the context of arbitrary colour factors it would be preferable to express eqn (D.10) as a function of the pole masses *M* of the quarks rather than the$\overline{\text{MS}}$running masses$\overline{m}$since the latter already absorb part of the radiative corrections of the specific theory. To leading order the pole mass *M* is related to the running mass according to

*L*= In

*M*/

*µ*.

With these ingredients, the evolution of the strong coupling constant from the 0(1 GeV) scale upwards can be realized by using *n* _{f} = 3 up to µ = 2*M* _{c}, then *nf* = 4 up to µ = 2*M* _{b} and *n* _{f} = 5 until the top mass threshold µ = 2*M* _{t}. At each flavour threshold the matching condition eqn (D.15) has to be applied. The theoretical error of the procedure may be estimated by varying the matching scale between *M* and 2*M*.

The QCD corrections both for *Rτ* and *Rl* are related to the QCD correction *δ* _{0} of *R* _{γ},

*al*., 1991; Surguladze and Samuel, 1991),

_{3}≈ 1.2020569 and ζ

_{5}≈ 1.0369278. The coefficients

*d*

^{abc}are the symmetric structure constants of the gauge group. For SU(N) type theories one has${d}^{abc}{d}^{abc}/{C}_{F}^{3}=16{f}_{A}-6{f}_{A}^{2}$

# D.3 The theoretical prediction for *Rl*

The theoretical prediction for *Rl* is obtained from that for *R* _{γ} by taking into account quark mass effects and the fact that, in the coupling of the primary quarks to the Z, vector and axial-vector currents contribute differently (Hebbeker, 1991). The prediction can be written as

*δ*

_{0}is the QCD correction for the case of massless quarks which is common to the vector and the axial current, while

*δ*

_{v}is an additional term which only contributes to the vector current. The two remaining terms are mass corrections,

*δ*

_{m}, a contribution to the leading order coefficients which mainly comes from the axial couplings of the quarks to the Z, and δ

_{t}, a second order correction in the axial current due to the large mass splitting between top and bottom quark masses.

Using the effective parameterization of both the top and the Higgs mass dependence from the TOPAZ0 program (Montagna *et al*. 1993*a*, 1993*b*) given by Hebbeker *et al*., (1994), one obtains

*M*

_{t}= 150 GeV and

*M*

_{H}= 300 GeV this expression reproduces the value${R}_{l}^{\text{EW}}$given by Passarino (1993) based on TOPAZ0 for the same mass parameters. Note that the measured value of the top mass is M

_{t}≈ 175 GeV and that the mass of the Higgs particle is expected to be

*M*

_{H}< 200 GeV.

The dominant part of the QCD correction is the same as for *R* _{γ}, with the exception of the contribution proportional to *T* _{3} where only the vector current contributes,

*v*

_{q}and

*a*

_{q}, the vector and axial couplings of quarks q to the Z,

*T*

_{3}can be written as

^{2})β/2 for the vector current and β

^{2}for the axial coupling (Djouadi

*et al*., 1990), there is also a slight dependence on the quark masses. On the Z resonance these effects are small, and within the precision of these calculations, they can be ignored.

The leading order mass correction *δ* _{m}, expressed as a function of the pole mass of the quarks, is given by

*et al*., 1992). For a determination of the strong coupling from

*R*, however, the difference to the leading order term is negligible.

Details about the top mass correction *δ* _{t} can be found in the literature (Kniehl and Kühn 1989, 1990). The leading order term comes from an incomplete cancellation between two triangle diagrams Z → gg, where via the axial current the Z couples to a b-quark or a t-quark loop. The colour structure of this contribution is of the type$({T}_{ji}^{a}{T}_{ij}^{b})({T}_{lk}^{a}{T}_{kl}^{b})={T}_{F}^{2}{N}_{A}$or, because of the identity *N* _{A} = *N* _{F} *C* _{F}/*T* _{F}, equal to the product *T* _{F} *C* _{F} *N* _{F}. Setting the number of quark degrees of freedom to *N* _{F} = 3, the colour factor dependent correction becomes

*δ*

_{t}. The next-to-leading order correction to

*δ*

_{t}, proportional to${a}_{\text{s}}^{\text{3}}$, is known and amounts to 15% of this leading order correction (Chetyrkin and Tarasov, 1994).

# D.4 The theoretical prediction for *Rτ*

The theoretical prediction for *Rτ* is also related to *R* _{γ}. Detailed discussions can be found in the literature (Braaten *et al*. 1992; Le Diberder and Pich 1992*a*,
(p.441)
1992b; Pich 1992). Here, we will present only a short summary. Similar to the cases of *R*γ and *R* _{l} the starting point is

*δ*

_{EW}= 0.001. The dominant correction is

*δ*

_{0}, which can be calculated in perturbative QCD. The additional term

*δ*

_{np}covers the non-perturbative corrections.

The main difference from the case of *R* _{l} is the fact that the hadronic system produced in τ decays is not at a fixed mass, but rather exhibits a mass spectrum ranging from *M* _{π} to *M* _{τ}. As a consequence, the QCD correction to the hadronic width is obtained by integrating the correction to *R* _{γ} over the mass spectrum. Expressing the running coupling constant through its value at the scale *M _{τ}* and turning the integral over the mass spectrum into a contour integral, one obtains (Le Diberder and Pich 1992a, 1992b; Pich 1992)

*a*

_{s}(-

*s*) and${a}_{\text{s}}({M}_{\tau}^{2})$are related via eqn (D.3). From the experimental data, there are indications (Braaten

*et al*., 1992; ALEPH Collab., 1998d; OPAL Collab., 1999c) that the non-perturbative corrections are slightly negative and below 1%, δ

_{np}= -0.005 ± 0.005.