The form factors from Analyticity and Unitarity
Analyticity and unitarity techniques are employed to obtain bounds on the shape parameters of the scalar and vector form factors of semileptonic decays. For this purpose we use vector and scalar correlators evaluated in pQCD, a low energy theorem for scalar form factor, lattice results for the ratio of kaon and pion decay constants, chiral perturbation theory calculations for the scalar form factor at the Callan-Treiman point and experimental information on the phase and modulus of form factors up to an energy . We further derive regions on the real axis and in the complex-energy plane where the form factors cannot have zeros.
decays are important for determining the matrix element , for recent reviews, see [1, 2] of the Cabibbo-Kobayashi-Maskawa (CKM) which in turn is crucial for testing the unitarity of the CKM matrix. Information on the experimental sector is rich, in particular for the decay rates, which were measured by a number of experiments BNL-E865 , KLOE , ISTRA+ , KTEV  and more recently a new analysis by NA48 . Recent lattice studies have also been carried out, see refs.[8, 9, 10, 11].
The decay of a kaon to a pion, a charged lepton and a neutrino is described by the matrix element
where is the vector form factor and the combination
is known as the scalar form factor. The expansion at
defines the slope and the curvature parameters where denotes the scalar and denotes the vector form factor. The precise determination of the element depends on how accurate the parametrization of the form factor is. To improve the precision and to provide bounds on the shape parameters of the form factors, we use inputs coming from certain low energy theorems, perturbative QCD, lattice computations and chiral perturbation theory. Our techniques allow us to incorporate the phase and modulus information of the form factors. We also apply the technique to find regions on the real axis and in the complex -plane where zeros are excluded. The knowledge of zeros is of interest, for instance, for the dispersive methods (Omnès-type representations) and for testing specific models of the form factors. For more details, see refs. [12, 13, 14, 15].
along the unitarity cut, whose upper bound is known from a dispersion relation, satisfied by a certain QCD correlator. For the scalar form factor this reads
with . Analogues expression, involving a suitable correlator denoted by , can be written down for the vector form factor. We can now use the conformal map
that maps the cut -plane onto the unit disc in the plane, with mapped onto , the point at infinity to and the origin to . Using this map, we cast eqn.(4) into a canonical form, incorporate phase and modulus information as well as the Callan-Treiman relations and finally employ a determinant for obtaining bounds on the shape parameters and for finding regions of excluded zeros in the complex -plane.
The essential inputs of our formalism, the vector and the scalar correlators, can be calculated in perturbative QCD up to order for [17, 18]. We get and , see also . An improvement can be achieved when we implement theoretical and experimental information into the formalism of unitarity bounds. The first improvement comes when we use the the value of vector form factor at zero momentum transfer. Recent determinations from the lattice give . We can also use two low energy theorems, namely soft pion and soft kaon theorems, for the improvement of the bounds on the slope and curvature parameters in the scalar case. The soft pion theorem relates the value of scalar form factor at first Callan-Treiman piont to the ratio of the decay constants [20, 21]:
At , a soft-kaon result  relates the value of the scalar form factor to
A calculation in ChPT to one-loop in the isospin limit  gives , but the higher order ChPT corrections are expected to be larger in this case. As discussed in , due to the poor knowledge of , the low-energy theorem eqn.(9) is not useful for further constraining the shape of the form factors at low energies. On the other hand, we obtain from the same machinery, the stringent bound on the quantity which is .
Further improvement of the bounds can be achieved if the phase of the form factor along the elastic part of the unitarity cut is known from an independent source. In our calculations we use below the phases from [27, 28] for the scalar form factor, and from [29, 30] for the vector form factor. Above we take Lipschitz continuous, i.e., a smooth function approaching at high energies. The results are independent of the choice of the phase for . We can further improve the bounds if the modulus of the form factor is known along the unitarity cut, : we can shift the branch point from to by subtracting the low energy integral from the integral Eq. (4). In order to estimate the low-energy integral, which is the value of the integral contribution from to , see for expressions ref., we use the Breit-Wigner parameterizations of and in terms of the resonances given by the Belle Collaboration  for fitting the rate of decay. The above leads to the value for the vector form factor and for the scalar form factor. By combining with the values , we obtain the new upper bound on the integral Eq. (4) from to , and .
In Figs. 1 and 2, the constraints for the scalar form factor are represented together with experimental information from various experiments. As shown in Fig. 1, the slope of the scalar form factor, predicted by NA48 (2007) is not consistent with our predictions (yellow band) which are obtained by taking into account the phase, modulus as well as the CT constraint. Nevertheless our predicted range for the slope is well-respected by the recent 2011 analysis by NA48  .
The value of for this new determination by the NA48 reads . On the theoretical side, the prediction of ChPT to two loops gives and which are consistent with our results within errors as shown in Fig. 2. For the central value of the slope given above, the range of is . The corresponding theoretical predictions are , obtained from dispersion relations.
Comparison of the experimental results with our constraints for the vector form factor is shown in Fig. 3. We find that except for the results from NA48 (2007) and KLOE, which have curvatures slightly larger than the allowed values, the experimental data satisfy our constraints. The new results from NA48  provide a curvature which overlaps with our constraints while the slope lies completely within our domain as seen in the figure. We also note that the theoretical predictions , obtained from ChPT to two loops, and , , and , obtained from dispersion relations are consistent with the constraint. For more results, see .
We can also extend our technique to derive regions in the complex plane where the form factors can not have zeros. For the form factors, the influence of possible zeros in the context of Omnès dispersive representations has been analyzed in . The absence of zeros is assumed in the recent analysis of KTeV data reported in . In Fig. 5 we show the region where zeros of the scalar form factors are excluded. The formalism rules out zeros in the physical region of the kaon semileptonic decay. In the case of complex zeros, we have obtained a rather large region where they cannot be present. In the case of the vector form factor, the analysis of  using data from decay concludes that complex zeros cannot be excluded, due to the lack of information on the phase of the form factor in the inelastic region. However our formalism is independent of phase information in the inelastic region and leads without any assumptions to a rather large domain where complex zeros are excluded. For the vector form factor, fig. 5 shows the region where complex zeros are excluded. For more results, see .
We have derived stringent constraints on the shape parameters of the form factors of decay which is the best source for the extraction of CKM matrix element . The results are promising and stringent especially in the case of the scalar form factor. The most recent results from NA48  is consistent with our prediction for the slope of scalar form factor and restricts the range of the slope to . We have also excluded zeros in a rather large domain at low energies both for the scalar and vector form factor. The Callan-Treiman input provides an additional constraint in the case of the scalar form factor and as a result excludes a larger domain of the energy plane where zeros can exist. Thus, this work represents a powerful application of the theory of unitarity bounds, which relies not so much on experimental information, but on theoretical inputs from perturbative QCD, low energy theorems and lattice calculations. It provides a powerful consistency check on determinations of shape parameters from phenomenology and experimental analyses.
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