CERN–TH/96290
Cavendish–HEP–96/19
hepph/9612353
NonPerturbative Corrections to Heavy Quark
[2mm] Fragmentation in Annihilation^{*}^{*}*Research supported in part by the U.K. Particle Physics and Astronomy Research Council and by the EC Programme “Training and Mobility of Researchers”, Network “Hadronic Physics with High Energy Electromagnetic Probes”, contract ERB FMRXCT960008.
P. Nason and B.R. Webber
Theory Division, CERN, CH1211 Geneva 23, Switzerland
Cavendish Laboratory, University of Cambridge,
Madingley Road, Cambridge CB3 0HE, U.K.
Abstract
We estimate the nonperturbative powersuppressed corrections to heavy flavour fragmentation and correlation functions in annihilation, using a model based on the analysis of oneloop Feynman graphs containing a massive gluon. This approach corresponds to the study of infrared renormalons in the large limit of QCD, or to the assumption of an infraredfinite effective coupling at low scales. We find that the leading corrections to the heavy quark fragmentation function are of order , where is a typical hadronic scale ( GeV) and is the heavy quark mass. The inclusion of higher corrections corresponds to convolution with a universal function of concentrated at values of its argument of order , in agreement with intuitive expectations. On the other hand, corrections to heavy quark correlations are very small, of the order of , where is the centreofmass energy and .
CERN–TH/96290
November 1996
1 Introduction
The production of hadrons containing heavy quarks in annihilation has proved to be a process of great importance for testing the Standard Model and searching for new physics. Heavy flavour processes are also valuable as a testingground for new techniques in QCD, because the heavy quark mass provides a second large momentum scale, in addition to the overall hard scale , set by the centreofmass energy in annihilation. This has led to a rather good understanding of heavy quark production and fragmentation within the context of perturbation theory [1–4]. On the other hand, it has become clear as a result of this understanding that the and quark masses are not large enough for a purely perturbative approach to provide a good description of the data for these flavours [2,3]. This is because nonperturbative effects can give rise to contributions of order , where is a typical soft physics scale. One may also worry about the possibility of nonperturbative effects of order , which could be significant at present energies. Thus it becomes important to study powersuppressed corrections in QCD from as many viewpoints as possible, and to apply the resulting insight to heavy flavour processes in particular.
One approach to the study of powersuppressed corrections, which has proved popular recently, arose from the study of infrared renormalons [5]. Here a divergent series of perturbative contributions gives rise to a powersuppressed renormalon ambiguity in the prediction of perturbation theory. One can then argue that a nonperturbative contribution with the same power behaviour should be present, with an ambiguity in its coefficient which cancels that associated with the renormalon. Conversely, when the renormalon ambiguity involves a high inverse power of the hard scale, one expects nonperturbative corrections to be especially small. This approach can be reformulated without reference to renormalons by postulating the existence of an infraredregular effective coupling at low scales (the ‘dispersive approach’ of ref. [6]). The expected powersuppressed corrections are consistent with those predicted by more rigorous approaches such as the operator product expansion where applicable, and have been found to agree fairly well with those suggested by experimental data [7–12].
The aim of the present paper is to study power suppressed effects in heavy flavour production in annihilation, using either the ‘renormalon’ or ‘dispersive’ approach, which are equivalent for the purpose of the present work. Our aim will be to test some standard assumption about the form of nonperturbative corrections that enter the fragmentation function. Furthermore, we will also examine the correlation between the quark and antiquark momenta. This topic is of practical interest, since this correlation affects the determination of , the partial width for the decay of the boson into flavoured hadrons.
1.1 Renormalons and powersuppressed corrections
In the renormalon approach, described in detail for shape variable calculations in ref. [13], one starts from the firstorder perturbative contributions to the process , involving emission of a single real or virtual gluon. One then identifies a factorially divergent series of perturbative contributions associated with light quark loop insertions on the gluon line. These contributions will be dominant at high orders for sufficiently large values of , the number of light flavours. One now argues [14–16] that the true highorder behaviour of perturbative QCD can be approximated by making the replacement in the large behaviour. There is support for this ‘naive nonAbelianization’ assumption in the highorder behaviour of the average plaquette in quenched () lattice QCD [17].
The resulting factorial divergences of the perturbation series can be of two types. Those associated with highmomentum regions of integration (ultraviolet renormalons) correspond to divergent series with alternating signs, which can be summed unambiguously using standard techniques such as Borel summation. We shall not be concerned with them in the present paper. Those due to low momenta flowing in loop integrals (infrared renormalons) produce samesign asymptotic series, which are intrinsically ambiguous. The ambiguity is of the order of the smallest term in the series, which turns out to be a powersuppressed quantity, of order where is a number that depends on the observable being computed.
In the full theory of QCD, the infrared renormalon ambiguities of perturbation theory must be cancelled by nonperturbative contributions. We shall assume that the presence of an infrared renormalon with a particular value of the power indicates that a comparable powersuppressed nonperturbative contribution is actually present in the full theory.
The dispersive approach of ref. [6] involves similar calculations, with a slightly different interpretation. A (formal) dispersion relation for the QCD running coupling of the form
(1.1) 
is assumed. In QCD, unlike QED, the running of the coupling cannot be associated with vacuum polarization effects alone. However, in the same spirit as the ‘naive nonAbelianization’ assumption, it is further assumed that the dominant effect on some QCD observable of the running of in oneloop graphs may be represented in terms of the spectral function and a characteristic function , as follows:
(1.2) 
where the relation (1.1) has been used to eliminate . The characteristic function is obtained by computing the relevant oneloop graphs with a nonzero gluon mass [14,8] and dividing by . Integrating Eq. (1.2) by parts, we can write
(1.3) 
where
(1.4) 
and we have introduced the effective coupling , defined in terms of by
(1.5) 
It follows from this definition and Eq. (1.1) that
(1.6) 
and therefore in the perturbative domain the standard and effective couplings are approximately the same. In the large renormalon approach^{†}^{†}†We stress that this model is not physically fully consistent, because of the presence of the Landau pole, which implies that the support of the spectral function of must be extended to negative values of the argument for Eq. (1.1) to be valid. we have the explicit expression, obtained by substituting the oneloop running coupling into Eq. (1.6):
(1.7) 
where .
The characteristic function is more precisely a function of the dimensionless ratio , where is the characteristic scale of the hard process. If the effective coupling in Eq. (1.3) has a nonperturbative component , with support limited to low values of , the corresponding correction to ,
(1.8) 
will therefore have a dependence determined by the low behaviour of .
A crucial point is that only those terms in that are nonanalytic at () can produce powersuppressed contributions to . This is because the integer moments of are required to vanish, for consistency with the operator product expansion. The same result may be seen in the renormalon analysis of ref. [13]: for any behaviour of of the form as , the renormalon contribution is proportional to and therefore vanishes for integer . On the other hand, a behaviour implies a nonvanishing correction proportional to , while gives , etc. Thus our objective is to identify the leading nonanalytic terms in the behaviour of the characteristic function at small values of the gluon masssquared, which will tell us the dependence of the leading powersuppressed corrections.
If one makes the additional assumption that the effective coupling modification in Eq. (1.8) is universal, one obtains a factorization property for powersuppressed corrections, which leads to relationships between the coefficients of the corrections to different observables. For variables like event shapes, this type of factorization is only approximate, due to the fact that a cut dressed gluon line is weighted differently, depending upon the value of the shape variable for the particular finalstate structure of the cut gluon [13]. In the present case, however, the dressed gluon is cut fully inclusively, without any weight, and therefore factorization may be more reliable.
In the case of heavy flavour processes, we also want to study corrections that are suppressed by powers of the heavy quark mass, . As long as we treat both and as large parameters, and keep track of the dependence on their ratio, this will be done automatically when we extract the nonanalytic terms in . Defining , a term will indicate a correction proportional to , implies , and so on.
In our terminology, mass corrections of the form will not be called powersuppressed, since we are always assuming that is not small.
2 Calculations
2.1 Massive gluon cross sections
Considering first the vector current contribution, the distribution of the heavy quark and antiquark energy fractions and with emission of a gluon of mass in the process is given by
(2.1) 
Here where , is the quark mass, ,
(2.2) 
is the heavy quark velocity, and
(2.3) 
is the Born cross section for heavy quark production by a vector current, being the massless quark Born cross section.
The phase space is determined by the triangle relation
(2.4) 
where and . This gives where
(2.5) 
and
(2.6) 
In the case of the axial current contribution, instead of Eq. (2.1) we have
(2.7) 
where
(2.8) 
being the Born cross section for heavy quark production by the axial current:
(2.9) 
2.2 Leading power corrections
Clearly the expressions (2.1) and (2.7) are analytic functions of at except possibly for and near 1. The phase space is also analytic in whenever the gluon momentum is large, since in this region one can always expand kinematic variables in powers of . As discussed above, this implies that there are no nonperturbative corrections of the type we are considering for . Nonanalytic behaviour may only arise in the region where (). In order to investigate the corrections associated with this region we take moments of the form
(2.10) 
and expand
(2.11) 
where and . The first term corresponds to the total heavy flavour cross section, whose dominant power correction is of order or smaller. However, the next two terms give contributions proportional to at small , which could give rise to and/or corrections. To evaluate them we note that their contribution to the difference for small can be written, for both the vector and axial current contributions, as where
(2.12) 
being the region between the phase space boundaries for and . Eq. (2.4) may be expanded in the region , so that the boundary of phase space in this region is given by
(2.13) 
which is the equation of a hyperbola. Changing variables to where and , the region may be written as
(2.14) 
where . Performing the integration one gets
(2.15) 
Therefore the leading power correction is of order rather than . The leading correction with an explicit dependence on is of order . This is consistent with the finding for light quark fragmentation functions [6,10,11,18]: the leading power correction in is of order .
The linear dependence on the moment index in the result (2.15) implies a behaviour in space of the form
(2.16) 
This means that, as far as the leading power correction is concerned, the twoparticle heavy quark distribution factorizes. In the dispersive approach of ref. [6], the nonperturbative correction is given in terms of the lowenergy modification to the effective coupling, , by Eq. (1.8). Defining the nonperturbative parameter
(2.17) 
Eqs. (1.8) and (2.16) then imply that
(2.18) 
where . Thus we see that the main nonperturbative effect is a shift in the heavyquark momentum fractions by an amount . Assuming approximate universality of , one may estimate from lightquark event shape data that GeV [6], which gives GeV. This agrees with the order of magnitude of the nonperturbative shift estimated from in heavy flavour fragmentation [2].
2.3 Higher power corrections
Powersuppressed effects in the heavy flavour fragmentation functions should be equivalent to a convolution with a nonperturbative initial condition of the form , and therefore should approach a delta function as . This can be inferred by intuitive reasoning, but can also be derived more rigorously in the context of the heavy quark mass expansion [19]. In this section we will show that this expectation is also fulfilled in our model.
First of all we note that Eq. (2.18) can be written as
(2.19) 
and therefore the expected form is indeed obtained when one includes only the leading power correction.
To go beyond the leading correction, we have to consider higher moments with respect to and in Eq. (2.12). We examine first the moments of the singleparticle distribution (). For moments weighted by with we have to define the integration region more carefully. Consider for simplicity the case that is small (i.e. ), so that . The upper limit of the integration becomes a constant of order unity when . Hence this region gives a term that is analytic in , which will not contribute to power corrections. The important region is . The leading nonanalytic term coming from this region is proportional to when is odd, and proportional to when is even. Hence for every value of there is a power correction of order .
In detail, for small and Eq. (2.12) becomes
(2.20) 
Now
(2.21) 
where the dots correspond to terms giving contributions that are analytic and/or higherorder in . Thus, keeping only the leading nonanalytic parts, we find
(2.22) 
which corresponds in space to
(2.23) 
We can therefore see the expected scaling of the fragmentation function. We also see that while the leading power correction, eq. (2.18), corresponds to a simple shift in the value of , this behaviour is not preserved by the higher power corrections.
3 Correlations
We consider now the higher power corrections to the twoparticle distribution, i.e. the inclusion of higher powers of in Eq. (2.16). We have to examine the double moments corresponding to Eq. (2.12) with general weights . Again we treat only the case of small . Then we find that for any such that , the leading nonanalytic term is suppressed by a factor of relative to that for weight . Therefore the leading terms in each order of remain of the form (2.16):
(3.1) 
where is as given in Eq. (2.23). Thus the twoparticle distribution including these terms still factorizes and can be expressed as a function of and , although beyond leading order in it differs from the simple product of delta functions given in Eq. (2.19).
Because of its possible impact on the determination of in decays, it is interesting to determine what is the leading power correction to the momentum correlation . As stated before, the correction to behaves as at small . The term gives zero at the order we are considering. In fact in the large limit it is subleading, and in the dispersive approach it is of second order in the effective coupling. Therefore the momentum correlation is of order .
In order to confirm this conclusion, we also calculated the difference using the exact phase space and matrix elements. We found that the leading term at small is proportional to . Thus, corrections to the momentum correlation in our model are suppressed by at least two powers of , and should therefore be completely negligible at LEP energies. Whether this result survives higherorder corrections is an open question, and in fact a very difficult one. We simply point out here that, while Monte Carlo models seem to indicate the presence of corrections to correlations (see ref. [4] and references therein), the simple model that we have adopted in this work does not provide support for the presence of such corrections. This is also consistent with the findings of ref. [10], where power corrections to fragmentation functions were computed in the strictly massless limit.
4 Discussion
We have examined the heavy flavour fragmentation function and correlations in a simple model, and found the following results.
At leading order, nonperturbative effects in the fragmentation function can be represented as a convolution with a function of the form
(4.1) 
which approaches a function as . The leading power correction has the form , where is a typical soft hadronic scale. An estimate of this correction based on the approach proposed in ref. [6] gives the correct sign and order of magnitude.
In the twoparticle distribution, corrections of the order of factorize and therefore no large correlations of this order arise. Since correlations are important for their possible impact on the determination of in decays, and since the perturbative value of the correlation is of the order of 1% [4], it is also important to understand whether corrections of the order of are present. In our analysis, consistently with ref. [10], terms of this order do not arise. In fact, we also verified numerically that the dependence of the correlation is of order , and therefore the leading power correction is less than order .
We end with a comment on the relationship between our results and those of ref. [3]. In that paper a resummed expression for the heavy quark spectrum was derived and numerical results were presented using various models for the behaviour of the QCD running coupling at low scales. The nonperturbative component of the coupling generates corrections which should correspond to those considered here, after convolution with the perturbative fragmentation function.
Acknowledgements
We would like to thank Yu. Dokshitzer for useful comments.
BRW is grateful to the CERN Theory Division for hospitality
during part of this work.
Appendix
We give here for reference the singleparticle inclusive distribution which results from integrating the vector twoparticle distribution (2.1) for :
(A.1) 
where and are given in Eq. (2.5) and
(A.2) 
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