Complete QCD Corrections to Hadronic Z–Decays in Order α_{s}^{4}

P. A. Baikov

Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University 1(2), Leninskie gory, Moscow 119234, Russian Federation

K. G. Chetyrkin,^{∗} J. H. Kühn, and J.Rittinger

Institut für Theoretische Teilchenphysik, Universität Karlsruhe, D-76128 Karlsruhe, Germany

Corrections of order α_{s}^{4} for the axial singlet contributions for the decay rate of the Z-boson into hadrons are evaluated in the limit of the heavy top quark mass. Combined with recently finished O(α_{s}^{4}) calculations of the non-singlet corrections, the new results directly lead us to the first complete O(α_{s}^{4}) prediction for the total hadronic decay rate of the Z-boson.

The new O(α_{s}^{4}) term in Z-decay rate lead to a significant stabilization of the perturbative series, to a reduction of the theory uncertainty in the strong coupling constant α_{s}, as extracted from these measurements, and to a small shift of the central value.

PACS numbers: 12.38.-t 12.38.Bx 12.20.-m

The precise determination of the Z-boson decay rate into hadrons at LEP [1] has led to one of the most precise determinations of the strong coupling constant α_{s}. From the experimental side, in view of the fully inclusive nature of this measurement, the result is fairly robust, in particular since it is insensitive to simulations of the hadronic final state. Hence the error is essentially dominated by the statistical uncertainty. From the theory side the advantage of the measurement is its high energy, and as a result, the irrelevance of nonperturbative and power-law suppressed terms. The smallness of α_{s} at high energies then leads to a rapid decrease of higher order corrections in the perturbative series and, correspondingly, to a significant reduction of the theory error.

A variety of methods has been suggested to estimate the remaining uncertainty in the theory prediction. Using the last calculated term is probably the most conservative approach, varying the renormalization scale μ within an energy range characteristic for the problem (e.g. M_{Z}/3 < μ < 3 M_{Z}) is frequently used, albeit with considerable ambiguity in the actual choice of the region of the μ-variation. In order to reduce the theoretical uncertainty in the extraction of α_{s} to a level significantly smaller than the experimental one (which amounts to ±0.0026 at present [1]), the knowledge of the corrections of O(α_{s}^{4}) is necessary. At the same time this calculation opens the window for a considerable improvement in the α_{s}-determination at Giga Z, the project of a highluminosity linear collider operating at the Z-resonance (see e.g. [2], where a precision of 0.0005 to 0.0007 has been advertised). The dominant part of the α_{s}^{4}- corrections, the ”non-singlet”-piece, has been evaluated in [3]. This has lead to a slight shift of the central value of α_{s} upwards from 0.1185±0.0026 to 0.1190±0.0026 [3] and a reduction of the theory error far below the error of 0.0026 from experiment. However, as noted already in [3], for a complete evaluation of the decay rate in O(α_{s}^{4}) an additional set of corrections, namely those for the “singlet” contributions, is required. For the axial current correlator these start at O(α_{s}^{2}) [4, 5], for the vector correlator at O(α_{s}^{3}). Both of them are presently known to third order in α_{s} only [6–10]. Hence, for a completely consistent O(α_{s}^{4}) extraction of the strong coupling the extension of these results by one order in α_{s} is required. Before describing this calculation in detail, let us briefly recall the basic structure of QCD corrections to the correlator of the electromagnetic and the neutral current, respectively, their similarities and their main differences. After splitting off inessential kinematic factors, the absorptive part of the current-current correlator of the electromagnetic current is expressed by the familiar R-ratio

R^{em} = 3[ ∑_{f} q_{f}^{2}r_{NS}^{V} + (∑_{f} q_{f})^{2}r_{S}^{V} ] ,  (1)

where r_{NS}^{V} and r_{S}^{V} stand for the (numerically dominant) non-singlet and the singlet part respectively. The corresponding decomposition for the correlator of the neutral current involves the following four terms

R^{nc} = 3[ ∑_{f}v_{f}^{2}r_{NS}^{V} + ( ∑_{f}v_{f})^{2}r_{S}^{V} + ∑_{f}a_{f}^{2}r_{NS}^{A} + r_{S;t,b}^{A}] ,

(2)

with v_{f} ≡ 2I_{f} − 4q_{f}s_{W}^{2} , a_{f} ≡ 2I_{f} and s_{W} defined as effective weak mixing angle. Here all but the top quark are assumed to be massless.

(Mass corrections to both vector and axial vector correlator due to other massive quarks are dominated by the bottom quark and can be classified by orders in m_{b}^{2}/M_{Z}^{2} and α_{s}. Up to O(α_{s}^{2}m_{b}^{2}/M_{Z}^{2}) and O(α_{s}^{2}m_{b}^{4}/M_{Z}^{4}) they can be found in [11], as well terms of order α_{s}^{2}m_{b}^{2}/M_{Z}^{2} (const + log m_{b}^{2}/M_{Z}^{2}) and α_{s}^{2}m_{b}^{2}/M_{t}^{2} (const + log m_{b}^{2}/M_{Z}^{2}) that arise from axial vector singlet contributions. Terms of order α_{s}^{3}m_{b}^{4}/M_{Z}^{4} and α_{s}^{4}m_{b}^{2}/M_{Z}^{2} can be found in [12] and [13] respectively. Corrections of order α_{s}^{2}m_{Z}^{2}/m_{t}^{2} and α_{s}^{3}m_{Z}^{2}/m_{t}^{2} from singlet and non-singlet terms are known from [4, 5, 14] and [15] respectively. These are important for the actual α_{s}-determination, but will not be discussed further in the present paper.)

From the prefactors of the non-singlet contributions in electromagnetic, vector and axial correlator it is evident that different quark flavours contribute incoherently, hence additive to the rate. Thus their contribution is significantly enhanced in comparison with the singlet terms where amplitudes from different flavours interfere destructively, with prefactors (∑_{f} q_{f})^{2} and (∑_{f} v_{f})^{2} for the electromagnetic and neutral current respectively.

Non-singlet contributions are present at the parton level and the QCD corrections are known in second [16], third [6, 7] and fourth [3] order in α_{s}. In terms of Feynman diagrams, non-singlet contributions are characterized by the fact that one quark loop connects the two external currents (Fig. 1a). In the absorptive part of this fermion loop no top quark is present due to kinematic reasons, whence the non-singlet functions are identical r_{NS}^{V} = r_{NS}^{A} ≡ r_{NS}.

In the case of singlet contributions of the vector current the two currents couple to two different quark loops (Fig. 1b) requiring a three-gluon intermediate state. Correspondingly the leading term is of O(α_{s}^{3}) and has been obtained already long time ago [6, 7]. The NLO corrections to this result are of O(α_{s}^{4}). They serve to soften the strong scale dependence of the O(α_{s}^{3}) result, stabilize the theory prediction and will be the subject of this paper.

The situation is different in the case of the singlet axial vector current correlator. The axial couplings of the two members of an isospin doublet are opposite equal. Hence their singlet contributions vanishes, if the corresponding quark masses are equal. This approximation is valid for the two lightest quark doublets. The only remaining contribution originates from the combination of bottom and top quarks with their specific mass hierarchy m_{b}^{2} << M_{Z}^{2} << m_{t}^{2} (Fig. 1c). In this case the contribution starts at O(α_{s}^{2}) and is further enhanced by the ”large” logarithm log(m_{t}^{2}/M_{Z}^{2}) [4, 5]. Corrections of O(α_{s}^{3}) have been calculated in [8–10], those of O(α_{s}^{4}) will be the subject of this paper.

The evaluation of the NLO terms of r_{V}^{S} requires the calculation of the absorptive parts of five-loop diagrams with massless propagators which, with the help of some complicated combinatorics based on the R^{∗}-operation [17], can be boiled down to the calculation of four-loop

(a)  (b)  (c)

FIG. 1: Different contributions to r-ratios: (a) non-singlet,

(b) vector singlet and (c) axial vector singlet.

propagator diagrams. The latter have been computed via reduction to 28 master integrals, based on evaluating sufficiently many terms of the 1/D expansion [18] of the corresponding coefficient functions [19]. This direct procedure required huge computing resources and was performed using a parallel version [20] of FORM [21]. The master integrals are reliably known from [22–24]. The details of the calculation, the results in analytic form and their relation to the Gross-Llewellyn Smith sum rule will be given in [27].

The evaluation of the NNLO terms of R_{S;t,b}^{A} involves again absorptive parts of five-loop diagrams with massless propagators, however, in addition also absorptive parts of four-loop diagrams combined with one-loop massive tadpoles, etc. down to one-loop massless diagrams together with four-loop massive tadpoles. The latter have been computed with the help of the Laporta algorithm [25] implemented in Crusher [26]. The methods employed in our calculations, together with the results will be described in more detail in [27].

The result is valid in the limit M_{Z}^{2} ≪ 4 M_{t}^{2}, an excellent approximation as evident from the lower orders. The relative importance of the various terms is best seen from the results for the various r-ratios introduced above, expressed in numerical form

r_{NS} =1 + a_{s} + 1.4092 a_{s}^{2} − 12.7671 a_{s}^{3} − 79.9806 a_{s}^{4} ,
r_{S}^{V} = − 0.4132 a_{s}^{3} − 4.9841 a_{s}^{4} ,
r_{S:t,b}^{A} = (−3.0833 + l_{t}) a_{s}^{2}  (3)
+ (−15.9877 + 3.7222 l_{t} + 1.9167 l_{t}^{2}) a_{s}^{3}

+ (49.0309 − 17.6637 l_{t} + 14.6597 l_{t}^{2} + 3.6736 l_{t}^{3}) a_{s}^{4} ,

with a_{s} = α_{s}(M_{Z})/π and l_{t} = ln(M_{Z}^{2}/M_{t}^{2}). Since all three r-ratios are separately scale invariant, the corresponding results for a generic value of α_{s}(μ) can easily be reconstructed. Using for the pole mass M_{t} the value 172 GeV the axial singlet contribution is given in numerical form by

r_{S;t,b}^{A} = −4.3524 a_{s}^{2} − 17.6245 a_{s}^{3} + 87.5520 a_{s}^{4} . (4)

Collecting now all QCD terms, the decay rate of the Zboson into hadrons can be cast into the following form

Γ_{Z} = Γ_{0} R^{nc} = \frac{G_{F} M_{Z}^{3}}{24π\sqrt{2}} R^{nc} .  (5)

Here all electroweak corrections are assumed to be collected in the prefactor Γ_{0}, and the forementioned mass corrections are ignored as well as electroweak and mixed QCD-electroweak corrections [28–30]. Thus the R-ratio is now known up to O(a_{s}^{4})

R^{nc} =20.1945 + 20.1945 a_{s}
+ (28.4587 − 13.0575 + 0) a_{s}^{2}
+ (−257.825 − 52.8736 − 2.12068) a_{s}^{3}
+ (−1615.17 + 262.656 − 25.5814) a_{s}^{4} , (6)

with s_{W}^{2} = 0.231. The three terms in the brackets display separately non-singlet, axial singlet and vector singlet contributions.

Let us now evaluate the impact of the newly calculated terms on the α_{s}-determination from Z-decays. Following our approach for the non-singlet terms (where a shift δα_{s} = 0.0005 had been obtained [3], consistent with an analysis [31] based on results of the electroweak working group [1] and a modified interface to ZFITTER v. 6.42 [32, 33] and confirmed by the G-fitter collaboration [32,30,31]), we consider the quantity R^{nc} as “pseudo-observable”. With a starting value R^{nc} = 20.9612, if evaluated for α_{s} = 0.1190 and without the α_{s}^{4} singlet terms, a shift δα_{s} = −0.00008 is obtained after including the newly calculated contributions.

As discussed in [3], the non-singlet α_{s}^{4} term leads to a considerable stabilization of the theory prediction, and, correspondingly, to a reduction of the theory error. A similar statement holds true for the singlet contribution. To illustrate this aspect, the dependence on the renormalization scale μ is shown in Fig. 2 for r_{NS}, r_{S}^{V} and r_{S;t,b}^{A}. The relative variation is significantly reduced in all three cases. In particular for the vector singlet case we observe a shift of the result by about a factor 1.45 (for μ = M_{Z}) and a considerable flattening of the result. Using for example the Principle of Minimal Sensitivity (PMS) [35] as a guidance for the proper choice of scale, μ = 0.3 M_{Z} seems to be favoured, leading to an amplification of the LO result by a factor 1.68 (if the latter is evaluated for μ = M_{Z}, as done traditionally).

Let us assume that the remaining theory uncertainties from r_{NS}, r_{S}^{V} and r_{S;t,b}^{A} can be estimated by varying μ between M_{Z}/3 and 3 M_{Z} and using the maximal variation as twice the uncertainty δr. This leads to δΓ_{NS} = 0.101 MeV, δΓ_{S}^{V} = 0.0027 MeV and δΓ_{S}^{A} = 0.042 MeV. Even adding these terms linearly, they are far below the experimental error of δΓ_{exp} = 2.0 MeV [36]. In combination with the quadratic and quartic mass terms, which are known to O(α_{s}^{4}) and O(α_{s}^{3}) respectively, this analysis completes the QCD corrections to the Z decay rate.

Let us also comment on the impact of the α_{s}^{4} singlet result on the measurement of R^{em} at low energies, i.e. in the region accessible at BESS or at B-factories, say between 3 GeV and 10 GeV. Considering the large luminosities collected at these machines, a precise α_{s} determination from R^{em} seems possible [38]. In the low energy region only r_{S}^{V} and r_{NS}^{V} contribute. Since ∑_{f=u,d,s} q_{f} = 0, the singlet contribution vanishes in the three flavour case. If we consider the region above charm and below bottom threshold, say at 10 GeV, only u, d, s and c quarks contribute, the relative weight of the r_{S}^{V} in eq. (1) is given by (∑ q_{f})^{2}/(∑ q_{f}^{2}) = 2/5, and thus is fairly suppressed. At energy of 10 GeV, in the absence of open bottom quark contribution, it seems appropriate to analyze the results in an effective four flavour theory with

r_{S}^{V} = −0.41318 a_{s}^{3}(μ) − (5.1757 + 2.5824 ln μ^{2}/s) a_{s}^{4}(μ).

As shown in Fig. 3, it is evident that the scale dependence is softened in NLO. Again a scale μ around 0.3 \sqrt{s}

(a)

(b)

(c)

FIG. 2: Scale dependence of (a) non-singlet r_{NS}, (b) vector singlet r_{S}^{V} and (c) axial vector singlet r_{S;t,b}^{A}. Dotted, dashdotted, dashed and solid curves refer to O(α_{s}) up to O(α_{s}^{4}) predictions. α_{s}(M_{Z}) = 0.1190 and n_{l} = 5 is adopted in all these curves.

minimizes the NLO corrections.

FIG. 3: Scale dependence of the vector singlet r_{S}^{V} around 10 GeV. Dashed and solid curves refer to O(α_{s}^{3}) and O(α_{s}^{4}) predictions. n_{l} = 4 and α_{s}(10GeV) = 0.1806 as obtained with the use of package RunDec [37] have been assumed.

In conclusion we want to mention that all our calculations have been performed on a SGI ALTIX 24-node IB-interconnected cluster of 8-cores Xeon computers using parallel MPI-based [20] as well as thread-based [39] versions of FORM [21]. For evaluation of color factors we have used the FORM program COLOR [40]. The diagrams have been generated with QGRAF [41]. This work was supported by the Deutsche Forschungsgemeinschaft in the Sonderforschungsbereich/Transregio SFB/TR-9 “Computational Particle Physics”, by Graduiertenkolleg 1694 “Elementarteilchenphysik bei höchster Energie und höchster Präzision” and by RFBR grants 11-02-01196 and 10-02-00525.

We thank P. Marquard for his friendly help with the package Crusher.

∗ Permanent address: Institute for Nuclear Research, Russian Academy of Sciences, Moscow 117312, Russia.

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