NNLO moments of event shapes in e^{+}e^{-} annihilation

NNLO moments of event shapes in e+e- annihilation


We compute the next-to-next-to-leading order (NNLO) QCD corrections to the first five moments of six event shape variables related to three-particle final states in electron-positron annihilation; the thrust, the heavy jet mass, the -parameter, the wide and total jet broadenings and the three-to-two-jet transition parameter in the Durham algorithm . The NNLO corrections to the first moment are moderate for all event shapes, while the renormalisation scale dependence of the theoretical prediction is substantially reduced compared to the previously existing NLO results. From a comparison with data from JADE and OPAL, we observe that the energy dependence of the moments of the wide jet broadening and can be largely explained without any non-perturbative power corrections, while the other observables exhibit a clear need for power-like contributions at low centre-of-mass energy.

QCD, Jets, LEP and ILC Physics, NLO and NNLO Computations

1 Introduction

Event shape variables in annihilation provide an ideal testing ground to study Quantum Chromodynamics (QCD) and have been measured and studied extensively in the last two decades. In particular, event shape variables are interesting for studying the interplay between perturbative and non-perturbative dynamics. Apart from distributions of these observables, to which the NNLO corrections recently have become available [1, 2, 3, 4, 5, 6], one can also study mean values and higher moments. The th moment of an event shape observable is defined by


where is the kinematically allowed upper limit of the observable. Moments were measured for a variety of different event shape variables in the past. The most common observables of three-jet type are: thrust  [7] (where moments of are taken), the heavy jet mass  [8], the -parameter [9], the wide and total jet broadenings and  [10], and the three-to-two-jet transition parameter in the Durham algorithm  [11]. Definitions for all observables are given in, for example, Ref. [1]. Moments with have been measured by several experiments, most extensively by JADE [12] and OPAL [13], but also by DELPHI [14] and L3 [15]. A combined analysis of JADE and OPAL results has been performed in Ref. [16].

As the calculation of moments involves an integration over the full phase space, they offer a way of comparing to data which is complementary to the use of distributions, where in general cuts on certain kinematic regions are applied. Furthermore, the two extreme kinematic limits – two-jet-like events and multi-jet-like events – enter with different weights in each moment: the higher the order of the moment, the more it becomes sensitive to the multi-jet region. Therefore it is particularly interesting to study the NNLO corrections to higher moments of event shapes, as these corrections should offer a better description of the multi-jet region due to the inclusion of additional radiation at parton level.

Moments of event shape variables can be divided into a perturbative and a non-perturbative part,


where the non-perturbative part accounts for hadronisation effects. The non-perturbative part is suppressed by powers of , where is the centre of mass energy and is of the order of . The functional form of has been discussed quite extensively in the literature, but as this parameter is closely linked to non-perturbative effects, it cannot be fully derived from first principles.

The power corrections can be related to infrared renormalons in the perturbative QCD expansion for the event shape variable [17, 18, 19, 20, 21, 23, 22, 24]. The analysis of infrared renormalon ambiguities suggests power corrections of the form , but cannot make unique predictions for : it is only the sum of perturbative and non-perturbative contributions in (2) that becomes well-defined [25]. Different ways to regularize the IR renormalon singularities have been worked out in the literature. One approach is to introduce an IR cutoff and to replace the strong coupling constant below the scale by an effective coupling such that the integral of the coupling below has a finite value[23, 24]


This dispersive model for the strong coupling leads to a shift in the distributions


where the numerical factor depends on the event shape, while is believed to be universal and scales with the CMS energy like . We refer the interested reader to Refs. [24, 26, 27] for further details.

Another ansatz is that of Korchemsky et al. [28, 29] who suggest a shape function valid equally for thrust, -parameter and heavy jet mass , which is independent of the energy scale and takes into account the energy flow into two hemispheres of the final state. In the region , their predictions for the mean values of and coincide with those of renormalon based models, i.e. the leading power corrections are parametrised by a single non-perturbative scale . In the case of the -parameter however, they find an additional contribution modifying the size of the correction. This formalism also allows one to study the region , where power corrections of the form for arbitrary become important.

Gardi et al. [30, 31, 32, 33] make an analysis using the formalism of the so-called “dressed gluon exponentiation” [32], which resums the renormalon contribution to the Sudakov exponent, taking into account factorially enhanced subleading logarithms. Within this formalism, power law corrections beyond the leading term are predicted which probe the region . In particular, this formalism predicts that power corrections of the form are suppressed, which is also indicated by the data. Ref. [33] contains a detailed analysis of the power corrections for the thrust and heavy jet mass distributions, taking also hadron mass effects into account and deducing from renormalon analysis that the non-perturbative corrections to both distributions can be described by a single shape function. In Ref. [31], characteristic functions describing the leading and subleading power corrections to the first four moments of the thrust variable are derived.

In Ref. [34], a renormalisation group improved (RGI) treatment of the first moment of event shape observables is suggested and compared to data. A simultaneous fit of three parameters – , a parameter describing uncalculated perturbative higher order corrections and a parameter controlling the power corrections – showed for the cases of and that the data are consistent with within the RGI approach without leading to unreasonably large values of , but do not allow definite conclusions.

Overviews on the various approaches to estimate non-perturbative corrections can be found in Refs. [25, 35]. Recently, significant progress in the theoretical description of power corrections has also been made using Soft-Collinear Effetive Theory (SCET [36, 37, 38]), see e.g. Ref. [39].

An extensive comparison of data on events shapes with NLO+NLLA QCD predictions, supplemented by power corrections within the approach of [23, 24], has been performed in [40]. A determination of and from fits to distributions and mean values of five event shapes has been carried out, and indications were found that uncalculated higher orders may contribute significantly to the non-perturbative corrections. A determination of based on moments of event shapes calculated up to NLO has recently been reported in [41].

Based on the NNLO corrections published in [1], several new determinations of the strong coupling constant have become available recently. In Ref. [42], fits of six event shape distributions to ALEPH data have been made. In [43], the same perturbative results have been fitted to JADE data, including also matching [44] onto the resummation of large logarithms [45] in the next-to-leading log approximation (NLLA).

A very recent study of non-perturbative contributions to the thrust distribution, based on the NNLO corrections published in [1] and including NLLA resummation, as well as power corrections within the low-scale effective coupling formalism [23, 24], can be found in [46]. This work also provides a determination of and based on fits to a wealth of experimental data for the thrust distribution and confirms the conjecture of [40] that the explicit corrections become smaller as higher perturbative orders are included. This corroborates the hypothesis that the divergent renormalon contribution can be regularized by modifying the strong coupling at low scales, leading to the effective coupling used in eq. (3).

In this paper, we provide the NNLO corrections to the first five moments of the event shapes thrust , the normalised heavy jet mass , the -parameter, the wide and total jet broadenings and and the transition from three-jet to two-jet final states in the Durham jet algorithm . After reviewing the theoretical framework and fixing the notation in Section 2, we present results for the NNLO coefficients of the first five moments of the above-mentioned event shape distributions in Section 3. The numerical impact of these corrections and implications for the description of experimental event shape data are discussed in Section 4.

2 Theoretical framework

The perturbative expansion for the distribution of a generic event shape observable up to NNLO at centre-of-mass energy for renormalisation scale and is given by


Here the event shape distribution is normalised to the total hadronic cross section . With the assumption of massless quarks, then at NNLO we have,


where the Born cross section for is


The constant is given by [47, 48, 49]


where the QCD colour factors are


for colours and light quark flavours.

The perturbative QCD expansion of is then given by


In practice, we compute the perturbative coefficients , and , which are all normalised to :


, and are straightforwardly related to , and by


These coefficients are computed at a renormalisation scale fixed to the centre-of-mass energy, and are therefore just dimensionless numbers for each observable and each value of .

The computation of the coefficients is carried out using the parton-level event generator program EERAD3 [3, 50], which contains the relevant matrix elements with up to five external partons [51, 52, 53, 54], combined using an infrared antenna subtraction method [55]. A recently discovered inconsistency in the treatment of large-angle soft radiation terms [6] in the original EERAD3 implementation has been corrected. They account for an initially observed discrepancy between the EERAD3 results and the logarithmic contributions (computed within SCET) to the thrust distribution to NNLO [56], which are now in full agreement.

In terms of the running coupling , the NNLO expression for an event shape moment measured at centre-of-mass energy squared becomes:


3 NNLO contributions

The perturbative LO and NLO coefficients , have been known for many years [57, 58, 59, 60], and have been used extensively in the experimental studies of LEP data. Our major new results presented here are the perturbative NNLO coefficients . These coefficients receive contributions from six different colour factors:

We have computed these six different contributions separately.

The evaluation of the perturbative coefficients , and requires the numerical integration of the weighted perturbative event shape distributions down to the exact two-jet boundary . At this boundary, all event shape distributions diverge like , and are further enhanced by large logarithmic corrections at NLO and NNLO. Since the integrand for the -th moment is the distribution weighted by , the moments themselves are finite. The first moment does however contain an integrable logarithmic singularity, and thus receives sizable contributions from the region .

The numerical convergence of the perturbative coefficients computed by EERAD3 deteriorates for for the following reason. The four-particle and five-particle phase spaces in EERAD3 are generated with a lower cut-off on all dimensionless invariants . This cut-off is required to avoid sampling kinematical regions where the numerical evaluation of the four-parton one-loop matrix element becomes unstable due to the presence of large inverse Gram determinants, and also to avoid regions where the antenna subtraction procedure induces cancellations over too many orders of magnitude for the remainder to be evaluated reliably. The independence of the results on serves as a check for the proper implementation of the subtraction. In our calculation of the event shape distributions [1], we used as a default, observing that this value allowed a reliable evaluation of all contributions while ensuring numerical stability and convergence. In approaching from above, the unresolved invariants are no longer small compared to the resolved invariants, and the calculation is likely to miss parts of the finite remainder from the subtraction. For , no reliable evaluation is possible. Consequently, we have to impose a lower cut on when evaluating the perturbative coefficients. This cut is likely to affect especially the first moment. Due to the strong suppression of the region in the higher moments, this cut does not play a role for .

Therefore, in evaluating the perturbative coefficients, especially for the first moment, we carried out systematic studies to determine the values for the technical parameters and . It is observed that is sufficient for all observables. In approaching this value, one observes already a reasonable convergence of all colour factor contributions to the first moments, and from studies with lower values of , we estimate the contribution from this region to each colour factor not to exceed five per cent of the total. The choice of depends on the event shape variable under consideration. Already at NLO, one observes that is sufficiently small for a reliable evaluation of the first moment of , , and , while and require much smaller . In the distribution of both broadenings, one observes that a cut induces a modification of the distributions up to , thereby rendering the numerical evaluation of the first moment of the broadenings much more challenging. This effect is illustrated in Figure 1 for the NLO-contribution to the distribution, evaluated for different values of . For the NNLO contributions to the first moment of the broadenings, we chose , which is clearly a compromise between the accuracy of the evaluation and the numerical convergence. We have verified for both broadenings that for fixed , the numerical values of first moments do stabilize at , and we estimate that the individual colour factor contributions do not change by more than three per cent for lower . Full studies with much lower were however not possible because of rapid deterioration of the numerical convergence. For these reasons, a systematic error of about 50% on the first moments of the broadenings should be taken into account.

In Tables 1 and 2, we list the individual colour factor contributions to the first five moments of the different event shape variables. Table 3 contains the total LO, NLO and NNLO coefficients. The moments given in Tables 1 to 3 have been calculated with for and , where studies of other values lead us to a systematic error estimate of 10% (except for the first moment, where the systematic error is estimated at 50%). For the other event shape variables, the default for the first moments is with a systematic error of 6%, while the values for the higher moments have been calculated with and studies of other values result in a systematic error estimate of 3% for , , and .

The behaviour in the different event shape variables is best seen by considering the ratios




which we display for in Figures 24. Although substantially above the world average, it turned out that this value of is the best-fit result [42] of fitting the pure NNLO predictions to event shape data from ALEPH. Consequently, we also employ this value in the event shape studies here. We observe that the perturbative corrections affect the different shape variables in a substantially different manner, which was already evident at NLO (see for example [13]). While the size of the corrections is largely independent of for , and (and roughly at NNLO for all three variables), it increases with for , and (amounting to for for all three variables). This picture is consistent with the observations made on the perturbative stability of the distributions [1], where the former three displayed uniform and moderate NNLO corrections, resulting in a stabilization of the perturbative expansion, while the latter three suffered large corrections.

4 Numerical results and comparison to data

Moments of event shapes have been measured by various collider experiments at centre-of-mass energies ranging from 10 GeV to 206 GeV [61]. These data are particularly interesting in view of the determination of non-perturbative power corrections. Since the power correction contributions scale like inverse powers of the centre-of-mass energy , one needs a large range in energies to empirically disentangle their contribution from the perturbative contribution, which scales logarithmically with the energy.

However, even based on data covering about a factor twenty in energy range, missing higher order contributions may be misidentified as power corrections. The accuracy on the extraction of power correction contributions from data is therefore inherently linked to the precision of the perturbative prediction. We therefore expect the NNLO corrections computed here to lead to a considerable improvement of the power correction extraction from event shape moments.

In Figure 5, we computed the energy dependence of the first moments of all event shape variables for at LO, NLO and NNLO. The theory uncertainty at each order is estimated by varying the renormalization scale in the strong coupling constant in the range . The theory predictions are compared to data from JADE and OPAL [61]. It can be clearly seen that the inclusion of the NNLO corrections leads to a substantial reduction (by a factor two to three compared to NLO) of the theoretical uncertainty in all variables. The theory error bands at NLO and NNLO generally overlap (although the overlap is only marginal for , and , which have the largest corrections, see above). We observe that the energy dependence of the first moment is well described by the purely perturbative NNLO QCD contribution for , while all other observables require the presence of additional non-perturbative contributions at low centre-of-mass energies. Comparing NLO and NNLO predictions, we observe that the NNLO contribution comes closer to the experimental data at low by about 30% for and , while only little improvement over NLO is observed for the other observables. The low- discrepancy between data and NNLO theory is most pronounced in , and . It is especially interesting to note that the required size of the power correction contribution is uncorrelated with size of the perturbative higher order corrections: both and receive only moderate NLO and NNLO corrections; to describe the low-energy data on the first moment of , one must postulate substantial (larger than in any other observable) power corrections, while data on are well described without power corrections.

The higher moments of the six standard event shapes are displayed in Figures 6 (), 7 (), 8 (), 9 (), 11 () and 10 (). The predictions are compared to JADE and OPAL data [61]. The qualitative behaviour of the higher moments of the different shape variables is similar to what was observed for the first moments. For , and , we again observe that the NLO and NNLO theory error bands merely overlap. Despite the increase in the size of the NNLO corrections to the higher moments, this situation does not deteriorate for higher , mainly because of the equally large increase of the NLO corrections. This behaviour should however be taken as an indication of the poor perturbative convergence of these three observables. For all , we observe a substantial discrepancy between NNLO theory and experimental data at low- in , and and , indicating large power corrections to these observables. The relative size of the discrepancy does not decrease with , illustrating the relevance of power corrections throughout the full kinematical range of the event shape variable (and not only in the two-jet region). Data on the higher moments of and are well described by NNLO theory, leaving only little room for power correction contributions to these observables.

With the new NNLO corrections, the theory uncertainty is reduced to the few per cent level for the moments of all event shape observables. It should therefore now be possible to perform precise studies of power corrections on these data sets.

5 Conclusions and Outlook

In this paper, we have computed the NNLO corrections to moments of the six most commonly studied event shape variables: , , , , and . Especially the first moment receives substantial contributions from kinematical configurations deep inside the two-jet region, well beyond the region probed by event shape distributions. Consequently, the evaluation of the first moment is technically demanding, in particular for the jet broadenings.

The NNLO corrections to the first moment are moderate for all event shapes, leading to a -factor of order unity (in the range 0.95 to 1.4 for the different shapes). For higher moments, we observed that the NNLO corrections to the moments of , and increase with increasing moment number , resulting in large NNLO -factors of up to 3. For , and , we observe that the corrections remain moderate towards higher , indicating the better perturbative stability of these three variables.

The energy dependence of the event shape moments can be used for the empirical determination of power corrections due to non-perturbative effects. For these studies, a reliable description of the perturbative contributions is mandatory. We observe that the newly computed NNLO corrections lead to a substantial reduction of the theoretical uncertainty on these predictions. From a comparison with data from JADE and OPAL, we observe that the energy dependence of the moments of and can be largely described without any power corrections, while the other observables show clear indications for power-like contributions at low centre-of-mass energy.

In view of the newly computed NNLO perturbative corrections, it will be very interesting to revisit the existing non-perturbative power correction models. Given the considerably reduced uncertainty on the perturbative prediction, a more precise extraction of hadronization parameters can now be envisaged.


AG and TG would like to acknowledge the Aspen Center of Physics, where part of this work was carried out. GH would like to thank the ITP, University of Zürich for hospitality. EWNG gratefully acknowledges the support of the Wolfson Foundation and the Royal Society. This research was supported in part by the Swiss National Science Foundation (SNF) under contracts PP0022-118864 and 200020-117602, by the UK Science and Technology Facilities Council and by the European Commission’s Marie-Curie Research Training Network under contract MRTN-CT-2006-035505 “Tools and Precision Calculations for Physics Discoveries at Colliders”.

Table 1: Individual colour factor contributions to event shape moments at NNLO: , , . The given errors are the statistical errors from the numerical integration. Systematic errors originating from technical cut parameters (see text) are not shown in the table.
Table 2: Individual colour factor contributions to event shape moments at NNLO: , , . The given errors are the statistical errors from the numerical integration. Systematic errors originating from technical cut parameters (see text) are not shown in the table.                                   
2.1035 2.1035 8.6379 4.0674 4.0674 0.8942
0.1902 0.1902 2.4317 0.3369 0.3369 0.08141
0.02988 0.02988 1.0792 0.04755 0.04755 0.01285
0.005858 0.005858 0.5685 0.008311 0.008311 0.002523
0.001295 0.001295 0.3272 0.001630 0.001630 0.0005570
Table 3: Contributions to event shape moments at LO, NLO, NNLO.
Figure 1: The NLO contribution to the wide jet broadening distribution evaluated for different technical cut-offs on all phase space variables.
Figure 2: The ratios () of the NNLO (NLO) corrections relative to LO for the first five moments of the thrust and heavy mass distributions evaluated at with .
Figure 3: The ratios () of the NNLO (NLO) corrections relative to LO for the first five moments of the -parameter and total jet broadening distributions evaluated at with .
Figure 4: The ratios () of the NNLO (NLO) corrections relative to LO for the first five moments of the wide jet broadening and distributions evaluated at with .
Figure 5: First moments of six event shape variables. The bands show the scale variations obtained by using and . The data are from the JADE and OPAL experiments, taken from [61].
Figure 6: Perturbative corrections to the higher moments of . The bands show the scale variations obtained by using and . The data are taken from [61].
Figure 7: Perturbative corrections to the higher moments of the normalised heavy jet mass . The bands show the scale variations obtained by using and . The data are taken from [61], which only contains data for the moments of , such that no data are shown for .
Figure 8: Perturbative corrections to the higher moments of the -parameter. The bands show the scale variations obtained by using and . The data are taken from [61].
Figure 9: Perturbative corrections to the higher moments of the 2-jet to 3-jet transition parameter . The bands show the scale variations obtained by using and . The data are taken from [61].
Figure 10: Perturbative corrections to the higher moments of the wide jet broadening . The bands show the scale variations obtained by using and . The data are taken from [61].
Figure 11: Perturbative corrections to the higher moments of the total jet broadening . The bands show the scale variations obtained by using and . The data are taken from [61].


  1. preprint: ZU-TH 04/09, IPPP/09/15


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