IPPP/05/37 DCPT/05/74 TPIMINN05/22
and Constraints on the LeadingTwist
[5pt]
Pion Distribution Amplitude from
Patricia Ball^{*}^{*}*Patricia.B and
Roman Zwicky^{†}^{†}†
IPPP, Department of Physics,
University of Durham, Durham DH1 3LE, UK
William I. Fine Theoretical Physics Institute,
University of
Minnesota, Minneapolis, MN 55455, USA
Abstract:
[10pt] Using new experimental data on the leptonic mass spectrum of , we simultaneously determine and constrain and , the first two Gegenbauer moments of the pion’s leadingtwist distribution amplitude. We find , where the first error is experimental, the second comes from the shape of the form factor in and the third is a 8% uncertainty from the normalisation of the form factor. We also find and .
submitted to Physics Letters B
is one of the least wellknown elements of the CKM quarkmixing matrix. A more precise determination of this parameter will not only greatly improve the constraints on the unitarity triangle, but also provide a stringent test of the CKM mechanism of flavour structure and CP violation. In this letter, we determine from the exclusive semileptonic decay , based on the invariant leptonmass spectrum recently reported by BaBar [1] and the lightcone sum rule calculations of the relevant form factor in Ref. [2].
The hadronic matrix element relevant for is given by
(1) 
where the form factors depend on , the invariant mass of the leptonpair, with GeVGeV. is the dominant form factor, i.e. the only one needed for calculating the spectrum in ,
(2) 
for massless leptons; is the usual phasespace factor. The determination of from requires theoretical input on , which has been the subject of many a calculation using various methods, in particular quark models [3], QCD sum rules on the lightcone (LCSRs) [2, 4, 5] and lattice simulations [6]. The challenge for theory is twofold: the region of applicability of theoretical calculations is, in most cases, restricted to part of the full physical phasespace; a calculation of for, say, small values of is, however, not sufficient, as experimental data on the decay spectrum are still very scarce, so that any meaningful extraction of necessitates the extrapolation of the form factor to all . LCSR calculations, for instance, are valid for large pion momentum, which translates into small to moderate GeV, whereas lattice calculations are restricted to small pion momentum, corresponding to large GeV. Extrapolations rely either on a model for the dependence of , like vector meson dominance or the parametrisation advocated by Becirevic and Kaidalov [7], or dispersive bounds on the form factor, which have been studied for instance in Ref. [8]. The BaBar collaboration has measured the spectral decay distribution in 5 bins in [1], which is a significant improvement over previous results reported for 3 bins [9], and allows one, for the first time, to assess the validity of various parametrisations of the dependence of like

vector meson dominance (VMD);

the parametrisation of Becirevic and Kaidalov (BK) [7];

the extended BK parametrisation used by Ball and Zwicky (BZ) [2].
All these parametrisations can be motivated from the exact representation of in terms of a dispersion relation,
(3) 
where GeV is the mass of the meson which induces a pole below the lowest multiparticle threshold at . Vector meson dominance assumes that the form factor is dominated, for all physical , by the first term in (3):
(4) 
is the only free parameter of the VMD parametrisation. Becirevic and Kaidalov suggested as an alternative parametrisation the replacement of the second term in (3) by an effective pole at higher mass,
(5) 
with the three parameters or , which are related by
(6) 
where parameterises the position of the effective pole. This parametrisation was used by BZ to describe the results from LCSRs [2], whereas Becirevic and Kaidalov, faced with the challenge to fit three independent parameters to lattice data with limited accuracy, implemented the additional constraint motivated from heavy quark expansion, and obtained the following expression in terms of two parameters, or :
(7) 
The BaBar collaboration has measured the integrated spectrum of in five bins in [1], which allows one to confront the above parametrisations with experiment. Fitting the data by the VMD formula (4), we find d.o.f. The BK parametrisation (7) fits the data with d.o.f. and . Fitting the BZ parametrisation (5) is slightly more subtle, as the data prefer or even larger, which is outside the allowed parameterspace.^{1}^{1}1For the rescaling (6) is no longer valid. Bounding , we find a minimum d.o.f. and , . While this implies that the VMD parametrisation is disfavoured,^{2}^{2}2Actually VMD is disfavoured not only from the experimental point of view, but also from the theoretical one, as the values of do not agree with indepedent determinations of the residue of the pole, see the discussion in Sec. 4 of Ref. [2]. both BK and BZ are viable parametrisations. Motivated by these results, we formulate the following strategy for extracting from the data: we

calculate from LCSRs for values of where the method is applicable;

extrapolate the results to all using the experimentally favoured BK and BZ parametrisations;

use the experimental information on the spectrum to constrain the input parameters of the LCSRs, in particular the leadingtwist distribution amplitude;

determine from the total branching ratio using the BK and BZ parametrisations of .
Let us start with the calculation of . In Ref. [2], we have presented a comprehensive analysis of decay form factors calculated from QCD sum rules on the lightcone, to accuracy for twist2 and the dominant twist3 contributions; earlier analyses can be found in Refs. [4, 5]. We refer to these papers for an explanation of the method. The main theoretical uncertainty of these analyses comes from the pion’s leadingtwist lightcone distribution amplitude (DA) ; other sources of uncertainty include the quark mass, the quark condensate and sum rule specific parameters (Borel parameter and continuum threshold). The resulting total uncertainty of is between 10% and 13% [2]. Whereas the other parameters mainly determine the normalisation of the form factor, affects also and in particular the dependence and hence can be constrained from the measured spectrum. The DA is usually expressed in terms of its conformal expansion,
(8) 
where is the momentum fraction of the quark in the and runs from 0 to 1. The are Gegenbauer polynomials and , the socalled Gegenbauer moments, are hadronic parameters which depend on the factorisation scale . The respective contributions of to are shown in Fig. 1, for a typical choice of input parameters. The plot reveals that the dependence of the form factor is mostly sensititive to and only to a lesser extent to higher Gegenbauermoments, which agrees with the findings of Ref. [10]. We hence decide against using the models for proposed in Ref. [10], but stick with the expansion (8), which we truncate after the contribution in .
The values of the lowest lying Gegenbauer moments have been constrained from various sources, cf. Ref. [11, 12, 13], but still come with rather large uncertainties. A very conservative range of allowed values consistent with all known constraints is
(9) 
In this letter, we aim to constrain from experimental data within the above range.
We obtain values for in dependence on , and using the following criteria for the evaluation of the LCSRs:

we calculate as a function of the Borel parameter and the continuum threshold for two different values of , GeV, five different values of , GeV, and 16 different values of : , ;^{3}^{3}3In the actual calculation, are scaled up to the factorisation scale GeV using NLO evaluation. we interpolate the results in and in order to obtain as a smooth function of these parameters;

for each value of the input parameters, we determine at the minimum in the Borel parameter , which implies that becomes mildly dependent on . As discussed in Ref. [2], this procedure ensures that a LCSR for , obtained from the derivative of the LCSR for in , yields the physical value GeV;

we choose the continuum threshold in such a way that the continuum contribution to the LCSR is constant for all ; this implies that also becomes (mildly) dependent on . For each value of the input parameters, we calculate for three different values of the continuum contribution, 15%, 20% and 25%;

the LCSR actually yields , being the leptonic decay constant of the meson; in order to extract , we divide the LCSR by as calculated from a QCD sum rule to the same accuracy in .
Some of these criteria differ from those applied in Ref. [2]. We chose to modify the criteria used in our previous work since we focus, in this letter, on the dependence of the form factor on , which can be meaningfully determined only if is calculated using exactly the same criteria for all values of , , and . It is for this reason that we require the continuum contribution to be the same for all input parameters. The drawback of this procedure is that the sum rule specific parameters and both become dependent on , so that, in order to keep the calculational effort at a manageable level, we have to restrict ourselves to a few points in . For each value of we calculate the theoretical uncertainty by varying

by 40% around the central value;

by GeV;

the central value 20% of the continuum contribution between 15% and 25%;

the central value of the quark condensate, , between .
The above ranges of sum rule parameters are rather conservative and account for the “systematic” uncertainty of QCD sum rule calculations. All errors are added linearly, which yields a typical theoretical uncertainty of 8%.
At this point we would like to comment on the treatment of . The motivation for calculating from a QCD sum rule instead of using, for instance, the current world average from lattice calculations [14], is that (a) receives large and corrections from gluonexchange diagrams that also enter the LCSR for and (b) is very sensitive to . By dividing the LCSR for by obtained from a QCD sum rule to the same accuracy in , and using the same value of , one expects those large contributions to cancel and to reduce the sensitivity to the value of . One can check the extent to which the cancellation takes place by comparing the results of our calculation with that at treelevel.
We calculate for GeV, , using the same criteria as for the full form factor including corrections. In Fig. 2 we plot the ratio for the respective optimum values as a function of . The minima in are around GeV. The ratio is nearly constant , whereas the ratio of the decay constants, is 1.26. This means that there is indeed a strong cancellation between the radiative corrections to the LCSR for and the QCD sum rule for . We have checked that similar cancellations also occur for other values of and nonzero . Nonetheless there is a residual uncertainty due to the treatment of which we estimate to be about half the difference between the value of calculated from a QCD sum rule to accuracy and the central lattice value, i.e. about 5%. The QCD sum rule results for are given, with errors, in Tab. 1. Adding these two errors linealy, we obtain an uncertainty of of 8% which is independent of and translates into a 8% uncertainty of the normalisation of , which we treat separately from the error of calculated as described above.^{4}^{4}4As the quark condensate is a common input parameter for both and , the effect of its variation is included only once, in the uncertainty of .
The next task is to compare the form factor predictions to data and to determine bestfit values of , using the experimentally favoured BK and BZ parametrisations in order to extrapolate the LCSR results to all physical . It turns out that the LCSR results are actually described extremely well by the BK and BZ parametrisations, to within better than 0.5%, as already noted in Ref. [2]. In Fig. 3 we show the difference between the BK and the BZ fit, both for the form factor and the integrated spectra in the th bin, , . The form factors start to noticeably deviate only for very large . For the fit of the experimental spectrum, we treat the experimental errors as uncorrelated, but allow for a correlation of theory errors and perform the least fits using the following function and error matrix :
where is the partial branching fractions th bin, the corresponding theory error (without the error of the overall normalisation of ) and is the correlation of theory errors which we vary within .
We first study the BK parametrisation which features one parameter that can be determined from the experimental spectrum: . As depends on actually three parameters, , and , only one of them can be constrained. In Tab. 3 we give the bestfit values of for and fixed. The table reveals that it is indeed possible to reproduce the experimental central value of for any given and . It is also obvious that the impact of the precise value of is much smaller than that of . Averaging over within and over , we find
(10) 
We would like to stress that this is a completely new determination of and agrees very well with other determinations of this parameter [11, 12, 13]. What is truly remarkable, however, is that, despite different bestfit values of , the resulting values of , and hence , agree within 3%. That is: the theory error due to and gets largely diminished by the constraints on the spectrum. Using the parameter sets in Tab. 3, we plot, in Fig. 4, the partially integrated spectra, normalised to the full branching ratio, together with the experimental data. All six parameter sets produce nearly the same curve which coincides with the best experimental fit using the BK parametrisation. Using the average value of the branching ratio as given by HFAG [15], , we obtain
(11) 
where the first error is experimental, the second comes from the uncertainty in the shape, due to the spread of values of , and the third is from the normalisation of the form factor.
Let us now turn to the BZ parametrisation, which featurs two parameters that can be determined from the shape of the spectrum, and , which allows one to constrain for instance and in dependence on . The resulting constraints are shown in Fig. 5. The minimum for within the range specified in (9) is reached for , , i.e. at the border of the parameter space; fixing , the bestfit value of is
(12) 
The contours shown in the figure include all for which the fit of the corresponding form factor to the data yields . We immediately read off the following constraints:
(13) 
In Tab. 3, we give the fit results for and . The bestfit parameters do not agree, for within the range specified in (9), with those favoured by experiment. We have already pointed out earlier that the experimentally favoured value is actually outside the theoretically allowed parameter space and is not supported by the results of the LCSR calculation. As before, we find that the different values for result in nearly the same values of . In Fig. 4 we plot the bestfit values for the partially integrated branching fractions, obtained from the parameter sets in Tab. 3, in comparison with the experimental results. For , we find
(14) 
which agrees with (11).
Combining the results from the BK and the BZ analyses, we get the following final result for :
(15) 
the first error in is experimental, the second comes from the shape of the form factor and the third from the overall normalisation of . As for the constraints on , our final results are
(16) 
The value of is the weighted average of (10) and (12). We would like to stress again that these values result from a new and independent determination of these parameters, which is both consistent with and complementary to the results found in Refs. [11, 12, 13].
To summarize, we have discussed the constraints posed by recent experimental data on the shape of the decay form factor . We have found that both the BK parametrisation Eq. (7) and the BZ parametrisation Eq. (5) can describe the data, whereas vector meson dominance is disfavoured. We have calculated from QCD sum rules on the lightcone for small to moderate and extrapolated the form factor to all physical using the BK and BZ parametrisations, respectively. We have then used the experimental data to constrain the pion distribution amplitude that enters the calculation of . We found that, although the bestfit values of the Gegenbauer moments depend on , the resulting predictions for the form factors and the partial branching fractions are largely independent of the input parameters. This is one of the main results of this letter: the experimental information on the shape of the spectrum reduces the theoretical uncertainty of the prediction for the total branching ratio and hence the extracted value of . In order to constrain even further, it will be necessary to perform a combined analysis including also other experimental constraints from e.g. the electromagnetic form factor [12] and the – transition form factor [13].
The determination of presented in this letter can be improved in the future from both the experimental and the theoretical side. As for the latter, we would like to stress that the normalisation of depends on the treatment of , the decay constant of the meson. We have determined from a QCD sum rule to accuracy, using the same values for quark mass and quark condensate as in the LCSR for , and we have shown, by comparison with the corresponding treelevel sum rules, that the individually large corrections to and cancel in the ratio to a large extent. We have estimated the uncertainty of this procedure to be , which enters the normalisation of the form factor. This overall uncertainty can be reduced by calculating e.g. corrections to the LCSRs, which is a formidable, but not impossible task. The full corrections to are known and actually also reduce the uncertainty coming from [16]. As for the experimental input, smaller bins in would help to further constrain the shape of the form factor and ultimately avoid the necessity for extrapolation in .
Acknowledgements
R.Z. is supported by the Swiss National Science Foundation. P.B. would like to thank K. Hamilton for enlightening discussions on the art of error correlation & propagation.
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