Unbiased analysis of CLEO data at NLO and pion distribution amplitude
Abstract
We discuss different QCD approaches to calculate the form factor of the transition giving preference to the lightcone QCD sum rules (LCSR) approach as being the most adequate. In this context we revise the previous analysis of the CLEO experimental data on by Schmedding and Yakovlev. Special attention is paid to the sensitivity of the results to the (strong radiative) corrections and to the value of the twistfour coupling . We present a full analysis of the CLEO data at the NLO level of LCSRs, focusing particular attention to the extraction of the relevant parameters to determine the pion distribution amplitude, i.e., the Gegenbauer coefficients . Our analysis confirms our previous results and also the main findings of Schmedding and Yakovlev: both the asymptotic, as well as the Chernyak–Zhitnitsky pion distribution amplitudes are completely excluded by the CLEO data. A novelty of our approach is to use the CLEO data as a means of determining the value of the QCD vacuum nonlocality parameter , which specifies the average virtuality of the vacuum quarks.
pacs:
11.10.Hi, 12.38.Bx, 12.38.Lg, 13.40.GpI Introduction
Recently, the CLEO collaboration CLEO98 has measured the form factor with high precision. This data has been processed by Schmedding and Yakovlev (SY) SchmYa99 using lightcone QCD sum rules (LCSR), taking also into account the perturbative QCD contributions in the nexttoleading order (NLO) approximation. In this way SY obtained useful constraints on the shape of the pion distribution amplitude (DA) in terms of confidence regions for the Gegenbauer coefficients and , the latter being the projection coefficients of the pion DA on the corresponding eigenfunctions. Note that SY have extended to the NLO the LCSR approach suggested before by Khodjamirian Kho99 for the leading order (LO) lightcone sum rule method.
The present analysis gives further support to the claim, expressed by the above mentioned authors, that LCSRs provide the most appropriate basis in describing the form factor of the transition. This is intimately connected with peculiarities of realphoton processes in QCD RR96 ; Kho99 . But the method of the CLEO data processing, adopted in SchmYa99 , seems to be not quite complete from our point of view. We think that an optimal analysis should take into account the correct ERBL evolution of the pion DA to the scale of the process (the latter not to be fixed at some average point, GeV, as done in SchmYa99 ) and to reestimate the contribution from the next twist term. The influence of both these effects appears to be important and it is examined here in detail. Furthermore, we are not satisfied with the error estimation performed in the SY analysis, for reasons to be explained later, and prefer therefore to use a more traditional treatment to determine the sensitivity to the input parameter and the construction of the 1 and 2 error contours.
Our main goal in the present work will be to obtain new constraints on the DA parameters from the CLEO data, taking into account all the remarks mentioned above, and then to compare them with the constraints following from QCD SRs with nonlocal condensates (NLC). We will not repeat here the derivation of the main results of LCSRs, as well as those related to NLC SRs, but we will refer the interested reader to Kho99 ; SchmYa99 and correspondingly to BMS01 ; BM02 and references therein. But for the sake of convenience we included a technical exposition of our approach in comprehensive appendices.
The paper is organized as follows. In Sec. II we review different QCD approaches to calculate the transition form factor , having recourse to QCD “factorization theorems” ERBL79 , encompassing both perturbation theory and LCSRs. The analysis of the CLEO data is discussed in Sec. III in conjunction with the SY approach in comparison with other approaches/approximations. In Sec. IV we present a complete NLO analysis of the CLEO data with a short discussion of the BLM setting procedure. Sec. V includes a comparison of the QCD SR pion DA models with the results obtained in Sec. IV from the CLEO data processing. In Sec. VI we summarize our conclusions. The paper ends with five appendices: in Appendix A we reestimate the value of the twistfour scale . In Appendix B the old Chernyak–Zhitnitsky (CZ) result for is discussed, paying attention to evolution effects. In Appendices C and D the twoloop results DaCh81 ; KMR86 for the purely perturbative part of the formfactor calculations and the ERBL evolution of the pion DA are outlined. Finally, in Appendix E, all needed calculation details for the NLO LCSR are presented.
Ii Transition form factor with LCSR
ii.1 Factorization of the form factor. Standard results
The form factor of the process is defined by the matrix element
(1) 
where , with are the virtualities of the photons, is the pion state with momentum . If both virtualities, and , are sufficiently large, the product of the currents in (1) can be expanded near the lightcone . This expansion results in factorization theorems, which control the structure of the form factor ERBL79 at leading twist. As a result, the form factor can be cast in the form of a convolution over the momentum fraction variable (of the total momentum ) to read is the quark electromagnetic current, and
(2) 
The hard amplitude – playing the role of the Wilson coefficient in the OPE – is calculable within QCD perturbation theory (PT):
(3) 
while the pion DA contains the longdistance effects and demands the application of nonperturbative methods. Due to factorization theorems, DAs enter as the central input of various QCD calculations of hard exclusive processes. The LO term in (3) depends only on kinematical variables, but the NLO amplitude , calculated in DaCh81 ; KMR86 , depends also on the factorization scale :
(4)  
(5) 
Here, is the QCD normalization factor, , and and denote, respectively, the scale of renormalization of the theory and the factorization scale of the process. and are, respectively, the perturbative expansion elements of the coefficient function of the process and the ERBL kernel in the LO (subscript 0) and NLO (subscript 1) approximation ^{1}^{1}1wherein is needed to evolve the pion DA in the NLO approximation.. The structure of is discussed in more detail in KMR86 ; MuR97 and in Appendix C.1. Here we only recall those features relevant for our discussion: Eq. (5), which specifies the definition of the coefficient function is a consequence of the QCD factorization theorems ERBL79 . Due to these theorems, the first logarithmic term in Eq. (5), originating from collinear divergences, can be absorbed into the renormalization of the DA, , following the ERBL equation (see App. D, Eq. (D.1)). Namely, the formal solution of the ERBL equation in the 1loop approximation is
(6) 
(here is the 1loop function)^{2}^{2}2and one should mean .. We adopt here as a factorization scale . For that case, , obtained in Eq. (2) at the scale , can be transformed into
(7) 
To fix the renormalization scale , one needs to go beyond the NLO approximation. In the absence of such information and in order to further simplify the NLO analysis, we also set . It is useful to expand over the eigenfunctions of the oneloop ERBL equation, i.e., in terms of the Gegenbauer polynomials ,
(8) 
In this representation, all the dependence on is contained in the coefficients and is dictated by the ERBL equation. Different reasons, explained in SchmYa99 and BMS01 ; BM02 , point to the possibility of retaining only the first 3 terms in this expansion. Following then this approximation, the form factor can be parameterized by only two variables and that accumulate the mesonic largedistance effects (at some scale ).
A special case of the OPE appears when one of the photons () is near the mass shell. It is instructive to present here an evident NLO perturbative expression for Eq. (7) in the formal limit (cf. Eq. (C.8) in Appendix C),
(9)  
with and
On the r.h.s. of Eq. (9), the twistfour contribution is included in terms of the twistfour asymptotic DAs, presented in Kho99 . The coefficients and the twistfour coupling are evolved to the scale using, correspondingly, the renormalization group (RG) equation to NLO and LO accuracy^{3}^{3}3The reasons for this treatment will be discussed in more detail in Sec. IV..
ii.2 Why do we need LCSRs for the transition form factor?
A straightforward calculation of in QCD is not possible. In particular, at the formal limit , it is not sufficient to retain only a few terms of the lightcone OPE of (1). One has, in addition, to take into account the interaction of the smallvirtuality photon at large distances proportional to , (for a recent discussion, consider section 4 in RR96 and MuR97 ). The LCSR method allows one to avoid the problem of the photon longdistance interaction by providing the means of performing all QCD calculations at sufficiently large and then use a dispersion relation to return to the massshell photon. To isolate the dangerous neighborhood of , one should apply an appropriate realistic model for the spectral density at low , based, for instance, on quarkhadron duality. Using this sort of analysis and by employing analyticity and duality arguments, the following expression was obtained in Kho99
(10) 
on the r.h.s. of (10) is taken from Eq. (2) and the Borel parameter is GeV, whereas is the meson mass and GeV denotes the effective threshold in the meson channel.
This program has first been suggested in general form in Kho99 and was realized there for the LO approximation of the process. Taking at LO in , the form factor was obtained from Eq. (10) with
(11) 
and
(12) 
Here, the numerically important twistfour contribution was also included using a simple asymptotic expression for the twistfour DA contribution Kho99
(13) 
Note that the coupling is determined from the matrix element
(14) 
The estimates for , including also its RGevolution, are presented in Appendix A.
ii.3 The framework of NLO LCSR for the form factor
An application of the above scheme to the NLO was more recently performed in SchmYa99 . The corrections to this process are expected to be rather large, of the order of 20% (see, e.g., MuR97 ). The size of these corrections can be roughly estimated from the perturbative expressions for their asymptotic parts by comparing to each other the values and in Eq. (9). Therefore, for a quantitative description of the form factor, this contribution should be taken into account. This important step was done by Schmedding and Yakovlev, who have used the NLO perturbative expression for in Eq. (10),
(15) 
where , to construct a NLO version of the LCSR for the form factor . The spectral density , based on Eq. (15), depends on the scale . In the original paper SchmYa99 , this scale was fixed by relating it to the mean value of with respect to the CLEO experimental data, i.e., by setting . Use of Eq. (15) implies that one accounts for the scale dependence in (e.g., for the different CLEO points) via the leading order perturbative formula
rather, than using the RG expression, given by Eq. (6). This seems to be a rather crude approximation, given that other reference points are evolved with by utilizing the NLO ERBL evolution equation.
Iii CLEO data analysis revisited
Let us scrutinize the form factor approaches, discussed above, in close comparison with the CLEO data CLEO98 .
iii.1 Theoretical approaches to the CLEO data
In this subsection – in order to be in close analogy with the original SY paper – we shall adopt their scale definition of the CZ DA. It is worth noting, however, that the genuine CZ DA differs from that definition because it is determined at the much lower normalization scale GeV (a discussion of this important point is relegated to Appendix B). To distinguish the two models in the present analysis, we will use in what follows a special notation: (i) CZ DA parameters originating from the SY prescription (), and by using a 2loop evolution to the scale , will be denoted with the superscripts CZSY, whereas (ii) those conforming with our prescription (, see Appendix B) and being 2loop evolved to the scale will be marked by the superscripts CZ.
The status of these approximations (NLO PT, LO LCSR and NLO LCSR) is illustrated in Fig. 1 by the relative positions of the corresponding admissible regions for these parameters in the plane. Here, the regions enclosed by the needlelike and ellipselike solid contours correspond to a deviation criterion (CL=68%) PDG2000 , while the broken contours refer to a deviation criterion (CL=95%). Note that these contours have been derived by taking into account only the statistical error bars in CLEO98 (see their Table 1). This marks a crucial difference between our processing of the data and that in SchmYa99 , where the “theoreticalsystematic uncertainties” have been involved in the statistical analysis together with the statistical ones. In other words, we do not “smear” the quantities (or ) over their corresponding (theoretical) errorbar intervals. In our opinion, such a manner would require an additional substantiation and further suggestions about the distribution of these errors that we want to avoid in our analysis. Hence, instead of that, we process the data at a few fixed values of in order to clarify the sensitivity of the results to this parameter.
It should be clear that a really admissible region might be somewhat larger than the presented “purely statistical” contours in Fig. 1, a price one has to pay for our strict way of data processing.

The needlelike contour on the left top corner of both parts of Fig. 1 corresponds to Eq. (9) and is stretched along the “diagonal” const. The weak dependence of Eq. (9) on slightly turns the angle of the diagonal . On the other hand, taking into account the evolution with of and for every makes the contour finite – much like a “diagonal” needlelike strip. As we have mentioned above, the formal limit of the NLO expression, Eq. (9), cannot give a reliable result for the form factor. In this context it is interesting to mention that the corresponding contour is located outside the regions determined by the LCSRs. Furthermore, all known DA models (see, e.g., Table 1) and the phenomenological predictions (BKM00 ; BMS01 and references therein) are located far away from this contour – clearly demonstrating the poor reliability of the corresponding perturbative approach.

At least the heavyline contours (enclosing also the SY point SchmYa99 ) correspond to the SY approximation. These contours do not overlap with those corresponding to the LO LCSR ones – even at the deviation level. Therefore, corrections are crucially important in extracting the DA parameters. Our bestfit point with respect to the NLO LCSR is close to but not coinciding with the one presented by SY (compare entries 3 and 4 in Table 1). In Fig. 1(a) the full circle inside the contour is just the SY point. The bestfit points, corresponding to different approximations and models, considered in the present analysis, are collected in Table 1, where the notation has been used (in correspondence to the number 15 of the CLEO experimental data points).
Bestfit points/models  

NLO PT (9) best fit  
LO LCSR best fit  
NLO LCSR best fit  
SY LCSR SchmYa99  
BMS model BMS01  
CZSY DA SchmYa99  
CZ DA CZ84  
Asympt. DA 
It should be stressed that the admissible region (heavyline contours), obtained with our dataprocessing procedure, differs from that in SchmYa99 . Our contours look slightly larger than theirs despite the fact that possible theoretical/systematic uncertainties were not included in our consideration. Just because of this latter reason, our contours possess another orientation relative to those of SY, as one appreciates by comparing Fig. 1(a) and Fig. 2 with Fig. 6 in SchmYa99 . Moreover, the CZSY DA model appears to be seemingly closer () to the bestfit point than the asymptotic one (). But the genuine CZ DA with and (consult the discussion in Appendix B) generates a value of , which is larger than that of the asymptotic DA.
The bestfitted linear combination^{4}^{4}4Dubbed “diagonal” in what follows. of , that determines the large axis of the NLO LCSR contour (see Fig. 1(a)) and parameterizes its orientation, is found to be
(16) 
instead of , reported in SchmYa99 . Note that the SY point also belongs to the diagonal: . The coefficient in (16) can be predicted without any fitting; it is solely determined by the structure of the NLO LCSR (10). Indeed, Eq. (10) can be rewritten as SchmYa99
(to be compared with the fit given in Eq. (16)) and the discussed coefficient expresses the average value of the ratio . Notice that this ratio, averaged over the CLEO data range , amounts to 0.31, while the r.h.s. of Eq. (16) is determined by the experimental data on the form factor. The coefficient , obtained in the SY fit SchmYa99 , can be associated with the same ratio at the mean value . The ratio is a concave function in and, therefore, its mean value, , is smaller than its value, , at the “mean point” .
iii.2 Sensitivity to input parameters
As it turns out, the location of the admissible regions is rather sensitive to the value of the input parameter . To illustrate this point, we have repeated the data processing with an admissible (near its low boundary) value (see Appendix A). All contours in Fig. 1(b) shift closer to the asymptotic point (◆), but their relative positions do not drastically change and, hence, the main conclusions (iiii) of the previous subsection remain valid. The results of this data processing are presented in Fig. 1(b) and in Table 2. One appreciates that the hierarchy of the different models (lower parts of both Tables) with respect to the NLO LCSR best fit does not change, though the values of can change significantly. Indeed, the point marking the BMS model moves from the deviation level at (see Table 1) inside the deviation region near the SY LCSR point at (cf. row 5 in Table 2). Therefore, the value of and, in general, also the value of the twistfour term can substantially affect the locations of the admissible regions. But all other options, like the CZ DA and the asymptotic DA, remain excluded at the –deviation level.
Bestfit points/models  

NLO PT (9) best fit  
LO LCSR best fit  
NLO LCSR best fit  
SY LCSR SchmYa99  
BMS model BMS01  
CZSY DA SchmYa99  
CZ DA CZ84  
Asympt. DA 
Let us pause for a moment and turn our attention to a recent paper by Diehl et al. DKV01 . The authors of this work employ a purely perturbative QCD approach to analyze the CLEO data without taking into account the twistfour contribution, i.e., using Eq. (9) with . They consider this treatment justified given the possible large uncertainties in estimating the twistfour contribution (which in the SY procedure is taken to be %). Comparing their results with those of Schmedding and Yakovlev SchmYa99 , Diehl et al. correctly note that the relative weights of and in display a much stronger dependence than in the leadingtwist case with the consequence that the allowed SY parameter region becomes much smaller than in their approach. However, the size of the twistfour contribution is crucial for accurately extracting the parameters and – as we have just demonstrated. Therefore in our analysis we use a different approach: the value of is connected with the parameter of the vacuum nonlocality. We first fix the value of and then we allow for the parameter to vary in a 10% range. The whole uncertainty in for the selected range of GeV amounts then to about 30% in accordance with Kho01 . This strategy enables us to use the CLEO data as a direct measure (a vacuum detector) to select that model for the QCD vacuum, which provides the best agreement between theory and experiment.
Iv Complete twoloop analysis of the CLEO data
In the previous section we have demonstrated the high sensitivity of the DA parameters to strong radiative corrections for the form factor, as well as to the scale of the twistfour contribution (see Fig. 1 a(b) and BMS01 ). Therefore, to obtain these parameters from the CLEO data in a reliable way, one should take into account the radiative corrections in the most accurate possible way. To this end, we want to improve in this section the accuracy of the extraction procedure of at the NLO level. A new estimate for , the magnitude of the twistfour contribution, is also introduced in the present analysis (see below). We also briefly discuss an attempt to go beyond the level of the NLO, having recourse to a recent calculation MNP01a of the radiative correction based on the BLM scale setting.
iv.1 Complete NLO analysis
Here we use the complete 2loop expression for the form factor , given by Eq. (7). For this reason, we put in (15) so that for the quantities
the NLO evolution is implied. Then, we substitute the spectral density , derived in SchmYa99 (see the text below Eq. (15)), in LCSR (10) to obtain in a regular manner and to fit the CLEO data over . The evolution is performed for every point , with the aim to return to the normalization scale and to extract the DA parameters at this reference scale. Stated differently, for every measurement, , its own factorization/renormalization scheme has been used so that the NLO radiative corrections are taken into account in a complete way.
The accuracy of the procedure is, nevertheless, still limited owing to the mixing of the NLO and the LO approximations. Indeed, the value of the twistfour coupling , as well as its RGevolution with , are estimated in the LO approximation. This quantity enters the LCSR formula (15) together with the NLOpart and can lead to an additional uncertainty. In order to improve the theoretical accuracy of the values of and , extracted from the CLEO data, one has to reestimate the twistfour contribution, , with a better accuracy.
To summarize, our data processing procedure differs from the SY one in the following points:

The value of the parameter has been reestimated to read (see Appendix A), and this value has been used in the data processing.
This processing of the CLEO data produces the admissible regions, one of which, corresponding to , is shown in Fig. 2(a), where the original SY regions (Fig. 6 in SchmYa99 ) are also presented in Fig. 2(d) for the ease of comparison.
To produce the complete  and contours, corresponding to , we need to unite three regions obtained for different values of the twistfour parameter: . This procedure is illustrated in Fig. 2(b) using as an example the contour. Let us remind the reader in this context that the SY contours are stretched along the “LO perturbative” diagonal const (the dashed straight line on the l.h.s. resembles exactly this diagonal) while the solid (dotted) contours correspond to the 1 (2 ) regions. This stretching of the contours appears here because of the SY manner of the data processing, namely, because the theoretical uncertainties of the input parameters were also involved in the statistical analysis.
Bestfit point/models  

New NLO LCSR best fit  
SY NLO LCSR SchmYa99  
BMS model BMS01  
Asymptotic model  
CZ model CZ84 
The new bestfit point (✚, “New”), as well as the whole contours themselves appear to be displaced in Fig. 2(a) (approximately) along the new diagonal (cf. Eq. (16)),
(17) 
In Fig. 2(c) we present for comparison the contours of the previous NLO analysis in the sense of SY (low right corner of Fig. 1(a)) drawn, however, at the scale of Fig. 2(a). The positions of the bestfit points and models are provided in Table 3.
It should be clear from our discussion that these new contours are somewhat smaller than the previous ones (Fig. 2(c)), but slightly larger than the original SY ones (Fig. 2(d)), and show another orientation along the diagonal Eq. (17). The difference between the new regions, determined in the present analysis, and those of the SY one is remarkable. For instance, the SY point appears now near the boundary and inside the region in Fig. 2(a). Moreover, the preliminary (i.e., for ) SY bestfit point, SchmYa99 , and the phenomenological estimates for (), presented in BKM00 , (), lie on the boundary of the united region (see Fig. 2(b)).
iv.2 Beyond the NLO approximation: effects from BLM scalesetting
The renormalization scale in Eq. (7) can be fixed by a NNLO calculation of following the BLM prescription. Recently, the NNLO contribution to , proportional to and required for the BLM scale setting, was obtained in MNP01a for a kinematics with . As an exercise, let us perform the new fit to obtain the scale and the bestfit point for the NLO expression given by Eq. (9). We follow the same procedure as in Sec. III.1, replacing this time in Eq. (9), where for Eq. (7.7) from MNP01a is used. As it turns out, practically for all points in the considered domain in the lower halfplane , the BLM setting leads to the condition , in conformance with the results of MNP01a . Therefore, for this region, the BLM setting seems to rule out predictions from the NLO perturbation theory.
Only for points within a rather thin strip in the upper halfplane (cf. Eqs. (7.2a), (7.7) in MNP01a ), the BLM setting gives . Interestingly, the case discussed in Sec. III.1 (ii) and based on Eq. (9) belongs just to this thin strip. The corresponding values are displayed below for comparison together with the initial result (second row) without the BLM setting.
(18)  
To calculate the imaginary part , used in Eq. (10), one needs to know the NNLO contribution proportional to for , which is still not computed. For this reason, the results obtained with the BLM scale setting fall out of the region of the NLO LCSR analysis. The calculation of the complete NNLO contribution or, at least, its convolution with is a very demanding task that has not been accomplished yet.
V Pion DA from QCD SR vs CLEO data
Let us now turn to the important topic of whether the CLEO data is consistent with the nonlocal QCD SR results for . We present in Fig. 3 the results of the data analysis for three central values of the coupling , which in turn correspond to three admissible values of the vacuum nonlocality parameter . For each value of from this ensemble, we define the corresponding central value of and its uncertainty (for details, see Appendix A). Then, we process the CLEO data as described in the previous section and obtain the complete  and contours on the plane , following from the CLEO experiment. An example of these regions is represented in Fig. 2(b) and is also displayed in Fig. 3(a), where the contour is shown as a solid line and the contour as a dashed one.
The task now is to compare these new constraints
with those following
from the QCD SRs with nonlocal condensates.
We have established in BMS01
that a twoparameter model
really enables one
to fit all the moment constraints for
that result from NLC QCD SRs (see MR89 for more details).
The only parameter entering the NLC SRs is the correlation scale
in the QCD vacuum, known from nonperturbative
calculations and lattice simulations (for a discussion and references,
see Appendix A).
The three slanted and shaded rectangles in Fig. 3 are the constraints on the Gegenbauer coefficients () resulting from the NLC QCD SRs at different values of GeV , BMS01 ; BM02 . The overlap of the displayed regions in Fig. 3 can serve as a means of determining the appropriate value of . In fact, one may conclude that the value GeV is more preferable relative to the higher values of . It should be noted, however, that even for this lowest value of that scale, the agreement with the constraints in Fig. 3 is rather moderate and of similar quality as using the SY constraints BMS01 ; BM02 . It is tempting to test even smaller values of than GeV in the NLC SR as an attempt to improve the agreement with the CLEO constraints in Fig. 3. But such values appear to be at the lower limits for the estimates from nonperturbative approaches (see Appendix A). Furthermore, the NLC SR becomes unstable at such low values of . As a result, the accuracy of the DA moments is rather poor and the final constraint on () becomes unreliable. Taking into account all these arguments, we think that an improvement of the ingredients of the NLC ansatz may provide a better agreement with the CLEO data than just using GeV .
Vi Conclusions
In this paper we have studied the theoretical predictions for the pion transition form factor in comparison with the CLEO experimental data CLEO98 on this form factor. We have presented a full analysis of this data and contrasted the results with those found in the context of QCD LCSR at the NLO level. In this way, we have revised and improved the procedure of analyzing the CLEO data, first performed by Schmedding and Yakovlev in SchmYa99 . The main goal has been to obtain constraints on the shape of the pion DA of twist2, in the most accurate way. The values of the crucial parameters, viz. the twistfour coupling and , involved in this procedure, have been treated more accurately than in previous approaches. The main findings may be summarized as follows.

We have tested different kinds of approximations to calculate and revealed how the size of and the twistfour corrections can affect the admissible regions of the DA parameters .

New admissible regions for the parameters, see Figs. 2(b), 3(a), – different from those in SchmYa99 – have been obtained, though the constraints do not change drastically in the sense that the initial SY bestfit point still belongs to a deviation region (CL=68%) in this space, whereas the CZ DA and also the asymptotic one are definitely excluded at the level of a deviation criterion (CL=95%). Moreover, one may appreciate by comparing Figs. 2(a, b) with Fig. 2(d) that this exclusion with respect to the asymptotic DA becomes even more pronounced using our data processing.

The bunch of admissible pion DAs, , corresponding to the estimate GeV, that was constructed within the framework of QCD SRs with nonlocal condensates in BMS01 , compares well (at the –level) with the new more restrictive constraints obtained in the present investigation as Fig. 3 demonstrates. In addition, half of the calculated admissible region intersects with the domain as well, see Fig. 3(a).
Acknowledgements.
This work was supported in part by the Russian Foundation for Fundamental Research (contract 000216696), INTASCALL 2000 N 587, the Heisenberg–Landau Programme (grant 200215), and the COSY Forschungsprojekt Jülich/Bochum. We are grateful to A. Kotikov, A. Nagaitsev, K. Passek, A. Radyushkin, D. V. Shirkov, A. Sidorov, M. Strikman, and O. Teryaev for discussions and O. Yakovlev for correspondence. One of us (A. B.) is indebted to Prof. Klaus Goeke for the warm hospitality at Bochum University, where this work was partially carried out.Appendix A Revision of the QCD SR results for
The coupling was originally estimated in NSVZ84 and found to be . Here, we reanalyze the QCD SR for , derived in OPiv88 , which is based on a nondiagonal correlator of the quarkgluon and quark (pseudoscalar) currents. This SR relates to and determines the value of the ratio . Evaluating the SR leads to the estimate and consequently to . Moreover, is rather sensitive to the size of the radiative corrections. In this work, we use , obtained recently in a DIS fit of the CCFR data in KPS02 that leads to . The same sort of analysis in the NLO approximation leads to the estimate ^{5}^{5}5Using the values and given in KPS02 for , we recalculated the values for , i.e., and . that is indeed not far from the standard value (Appendix C.2).
To determine , we first fix the parameter by employing the “conservative estimate” GeV. In QCD the value of this parameter was estimated in the QCD SR approach BI82 and also using lattice data BM02 :
(A.1) 
A brief review of the different estimates of is given in BM02 .
The evaluation of the SR for the quantity ,
(A.2) 
for the standard value of the gluon condensate , SVZ , and with the fitting parameters, i.e., the coupling to the , , and the duality interval yields