QCD condensates from tau-decay data: A functional approach

We study a functional method to extract the V-A condensate of dimension 6 from a comparison of tau-decay data with the asymptotic space-like QCD prediction. Our result is in agreement within errors with that from conventional analyses based on finite energy sum rules.


Introduction
Although QCD has been with us for three decades, the knowledge of the values of the various fundamental or effective parameters of the theory (with the possible exception of the coupling constant) such as quark masses and condensates is still astonishingly limited. The precise data on τ -decay obtained by the ALEPH [1] and OPAL [2] collaborations at CERN have offered an opportunity for new studies, which range from an extraction of the strange quark mass [3] to the determination of various condensate parameters. Of particular interest was the extraction of the dimension-6 condensate [4,5,6,7,8] which, in the chiral limit, determines, e.g., the K → ππ matrix elements of the relevant electroweak penguin operators.
It should be kept in mind, however, that the extraction of the condensate parameters of QCD constitute a so-called ill-posed inverse problem which is basically unstable with respect to errors in the input data. An example is the popular method used to obtain condensates from experimental spectral functions by QCD sum rules (finiteenergy sum rules, FESR). This procedure corresponds to an analytic continuation from a finite contour (the part of the positive real axis on which the data are given) which is known to be notoriously unstable with respect to data errors. The extraction of QCD condensates requires therefore a carefully chosen stabilization mechanism. In the case of QCD sum rules, this is achieved by an implicit and rather ad hoc assumption that the series of the operator product expansion essentially breaks off after a finite number of terms. Amazingly, a careful analysis of finite energy sum rules [7] of the chiral spectral function shows remarkable stability with respect to variations of the duality interval. To investigate further the reliability of the extraction of QCD parameters via sum rules, it would be prudent to develop alternative methods.
In this letter we study a functional method [9,10] which allows us to extract without prejudice the condensates from a comparison of the time-like data with the asymptotic space-like QCD results. We will see that the price to be paid for the increased credibility are possibly larger errors in the values of the extracted parameters.

QCD condensates
We consider the polarization operator of hadronic vector and axial-vector charged currents, J µ = V µ =ūγ µ d and J µ = A µ =ūγ µ γ 5 d, The conservation of the vector current implies Π and can be measured in hadronic τ -decays. We consider specifically the V − A component which is related to the branching ratios of τ decays through Here, V ud is the weak mixing CKM-matrix element, |V ud | 2 = 0.9752 ± 0.0007, the τ mass is denoted m τ = 1.777 GeV and S EW = 1.0194 ± 0.0040 accounts for electroweak radiative corrections [11]. The spin-0 axial vector contribution a 0 (s) is dominated by the one-pion state, a 0 (s) = 2π 2 f 2 π δ(s − m 2 π ), with the π-decay constant f π = 0.1307 GeV. Its contribution in the last term of (3) is tiny and will be neglected. The contribution of the pion pole to the first term in Eq. (3) is well identified in the data and concentrated at low s ≃ m 2 π ; thus it can be removed from the data and taken into account explicitly without introducing sizable additional uncertainties. The experimental data [1,2] * are given by binned and normalized event numbers related to the differential distribution dR τ,V −A /ds and can therefore be viewed as a measurement of the function The (V − A) correlator is special since it vanishes identically in the chiral limit (m q = 0) to all orders in QCD perturbation theory. Renormalon ambiguities are thus avoided. Non-perturbative terms can be calculated for large |s| by making use of the operator product expansion (OPE) of QCD are vacuum matrix elements of local operators of dimension D (so-called condensates). Their contribution is known up to dimension 8 and read, at leading order, The last two results hold in the vacuum dominance approximation. The numerical is very small and this condensate can be neglected in our analysis. * We use the ALEPH data [1] because of their smaller experimental errors.
The next-to-leading-order corrections to O V −A 6 have been calculated in [12,13]. They depend on the regularization scheme implying that the value of the condensate itself is a scheme-dependent quantity. Explicitly, where [12], 89 48 anticommuting γ 5 [13].
The renormalization scale µ 2 is conveniently chosen to be −s.
, Eq. (9), is taken from [6]. It involves a quark-gluon condensate for which various estimates exist. The typical scales determining the condensates are around 300 MeV, e.g. qq ≃ (250 MeV) 3 , (α s /π) G 2 ≃ (300 MeV) 3 . Assuming a similar scale for the condensate entering to be of order 10 −3 GeV 8 . This is small enough so that the OPE makes sense. If O V −A 8 would be larger, radiative corrections to higher-dimension condensates would mix significantly with the lower-dimension condensates through their imaginary parts. There exist a number of QCD sum rule extractions of the value of the D = 8 condensate. They range from This value corresponds to a scale of about 400 MeV which is comparable to Λ QCD . The variation of these results represents the ambiguities inherent in the QCD sum rule approach. In the next section we shall present our alternative rigorous functional method which allows us to extract the condensates from a comparison of the data with the asymptotic space-like QCD results in an unambiguous way.

An L 2 norm approach
We consider a set of functions F (s) (where F (s) relates to Π ii) The asymptotic behavior of F (s) is restricted by fixing the number of subtractions in the dispersion relation between F (s) and its imaginary part along the cut f (s) = ImF (s + i0)| s∈Γ R (for Π (0+1) V −A (s) no subtractions are needed): We have two sources of information which will be used to determine F (s) and f (s). First, there are experimental data in a time-like interval Γ exp = [s 0 , s max ] with s 0 > 0 for the imaginary part of the amplitude. Although these data are given on a sequence of adjacent bins, we describe them by a function f exp (s). We assert that f exp is a real, not necessarily continuous function. The experimental precision of the data is described by a covariance matrix V (s, s ′ ).
On the other hand, we have a theoretical model, in fact QCD. From perturbative QCD we can obtain a prediction for the amplitude in a space-like interval † Γ L = [s 2 , s 1 ]. This model amplitude F QCD (s) is a continuous function of real type, but does not necessarily conform to the analyticity property i). Since perturbative QCD is expected to be reliable for large energies, we expect that there is also useful information about the imaginary part of the amplitude provided that |s| is large, i.e. we can also use f QCD (s) = ImF QCD (s + i0)| s∈(smax,∞) . In order to compare the true amplitude with theory, we can therefore split the integral in the dispersion relation (12), and test the hypothesis whether the left-hand side can be described by QCD.
We also need an a-priori estimate of the accuracy of the QCD predictions. This will be described by a continuous, strictly positive function σ L (s) for s ∈ Γ L which should describe errors due to the truncation of the perturbative series and the operator product expansion and is expected to decrease as |s| → ∞ and diverge for s → 0. In the case of Π V −A which does not have perturbative contributions, we will take the contribution of the dimension D = 8 operator as an error and use σ L (s) = O 8 /s 4 with O 8 in the order of 10 −3 GeV 8 . If the perturbative part dominates, as is the case for the individual vector or axial vector correlators, the last known term of the perturbation series could be used as a sensible estimate of the error corridor.
The goal is to check whether there exists any function F (s) with the above analyticity properties, the true amplitude, which is in accord with both the data in Γ exp and the QCD model in Γ L . In order to quantify the agreement we will define functionals χ 2 L [f ] and χ 2 R [f ] using an L 2 norm. For the time-like interval we simply compare the true amplitude f (s) with the data and use the covariance matrix of the experimental data as a weight function: Experimental data correspond to cross sections measured in bins of s, so that we can calculate this integral in terms of a sum over data points. The ALEPH data which we use are given for 65 equal-sized bins of width ∆s = 0.05 GeV 2 between 0 and 3.25 GeV 2 . χ 2 R given in (14) is in fact the conventional definition of a χ 2 and has a probabilistic interpretation: for uncorrelated data obeying a Gaussian distribution we would expect † We do not exclude the case s 2 → −∞.
to obtain χ 2 R = N, where N is the number of data points. Since experimental data at different energies are correlated, we instead expect In order to define a measure for the agreement of the true function f (s) with theory, we use the left-hand side of (13) which is well-defined and expected to be a reliable prediction of QCD in the space-like interval for not too small |s|. This expression can be compared with the corresponding integral over the true function. Thus we define (16) where w L is the weight function for the space-like interval and identified with 1/σ 2 L (s). The integral is normalized to unity for the case where the difference within parentheses saturates the error σ L .
In order to find the true function f (s), we can combine the information contained in χ 2 R , (14), and χ 2 L , (16) in the following way [9,10]. We fix and minimize χ 2 L : These conditions are equivalent to finding the unrestricted minimum of the functional where µ is the Lagrange multiplier, which will be found later. The solution of the condition χ 2 R [f ] = χ 2 0 will be denoted by f (x; µ): To this end we require the Fréchet derivative of F to be zero for any function Y . This leads to the following integral equation for the imaginary part f (x; µ): where λ = 1/µ and Eq. (19) is a Fredholm equation of the second type which is stable against variations of its input. At this stage we should notice that if one had claimed that in the space-like region the function F (s) was given by some analytic expression (e.g., by some few QCD terms), this would be equivalent to saying that χ 2 L vanished identically. But then, from the definition of the functional F and the vanishing of its Fréchet derivative, it follows that µ = 1/λ is zero which will turn the integral equation (19) into a Fredholm equation of the first kind which is known to be unstable.
The integral equation will be solved numerically by expanding f (s) in terms of Legendre polynomials. The algorithm to determine an acceptable value for the condensate is then the following: and the corresponding α s qq 2 is the value for the condensate we are looking for.

Numerical results and discussion
A typical situation resulting from this algorithm is shown in Fig. 1. The left part of this figure shows χ 2 L which has the expected quadratic dependence of α s qq 2 . The values of α s qq 2 corresponding to the minimum of χ 2 L are listed in the Table below for various choices of O 8 as discussed above. The regularized function shown in the right part of Fig. 1 follows nicely the data points, except at large s. Here the experimental errors are large and hence, as it should happen, the regularizing effect by means of the functional (14) is not as effective.
In the numerical evaluation we have used the NLO expression for α s (s) with Λ The result for α s qq 2 is not sensitive to changing Λ within the present experimental error ±0.030 GeV. For the evaluation of χ 2 L we have restricted the range of integration within limits s 2 ≤ s ≤ s 1 < 0. We checked that our result is insensitive to changes of s 2 as soon as its absolute value is chosen larger than O(100) GeV 2 . Since the error channel defined by O 8 /s 4 diverges for s → 0, one could, in principle, choose the upper limit s 1 = 0. Numerical instabilities require a non-zero value. We observe a well-defined plateau for the result for α s qq 2 as a function of s 1 between −1.0 and −0.5 GeV 2 and quote the values for s 1 = −0.7 GeV 2 . which are not inconsistent with our number. The fact that we find agreement within errors is not trivial. Since our approach is based on less assumptions we may conclude that the sum rule results with their relatively small errors are indeed trustworthy.