Determinations of form factors for semileptonic decays and leptoquark constraints
By analyzing all existing measurements for ( ) decays, we find the determinations of both the vector form factor and scalar form factor for semileptonic decays from these measurements are feasible. With the analysis results on these measurements together with magnitude of CKM matrix element , , we obtain these form factors. Both and determined from experiments are consistent within error with those in recent lattice calculations. With our analysis results in conjunction with the average of form factor in lattice calculations, we extract , which is in good agreement within error with the value obtained from SM global fit. With obtianed in this analysis together with the value obtained from leptonic decays, we obtain the average of ,which is in good agreement with the one from PDG’2016. Moreover, using the analysis results in the context of the SM as input parameters, we re-analyze these measurements in the context of new physics. Constraints on scalar leptoquarks are obtained for different final states of semileptonic .
Semileptonic decays have long been of great interest in the field of flavor physics, because they are important in validating the lattice QCD (LQCD) calculations, extracting the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, and searching for New Physics (NP) beyond the Standard Model (SM) of particle physics.
For ( ) decays, strong and weak interaction portions can be well separated and the effects of strong interactions can be parameterized by form factors. In the SM, the differential decay rate for is given by
where is the Fermi constant, p represents the three momentum of the meson in the rest frame. is the four momentum transferred to and ranges from where the daughter meson has the maximum possible momentum to where the meson at rest. The vector form factor and the scalar form factor are defined via
At the maximal recoil point, kinematic constraints lead .
In the last 30 years, various measurements for decays were performed at more than ten experiments. For and/or decays, branching fractions and/or decay rates in different bins were measured at experiments such as the E691 ,Mark-III ,CLEO , CLEO-II , BaBar , BES-II [7, 8],CLEO-c  and BES-III [10, 11, 12]. The FOCUS Collaboration measured non-parametric relative form factor for decays at the FOCUS experiment in 2005 . And in 2006, vector form factor for decays was measured by the Belle Collaboration at the Belle experiment. By comprehensive analysis of these measurements, one can obtain, the product of the hadronic form factor at and the magnitude of CKM matrix element , . Then with together with from SM global fit one can determine the form factor , or with together with calculated in lattice QCD one can extract . In 2014, Ref.  reported their determination of and extraction of by comprehensively analyzing all the existed measurements for decays before 2014. In these experimental and theoretical studies, the contributions of term are neglected, since they are suppressed by the mass squared of lepton.
With the increase of measurements for decays and the improvement of measurement accuracy, we believe that both the vector and scalar form factors can be determined by comprehensive analysis of these measurements. Here, for the first time, we try to determine both the vector form factor and scalar form factor by comprehensively analyzing all the existing measurements for decays. As the result of this analysis, we report the product and shape parameters, and , of vector and scalar form factors , which are the best fitting results to the experimental data in the context of the SM. Then we determine by considering the product in conjunction with from SM global fit. With the determined and shape parameters and , we chick the lattice calculations for the vector form factor and the scalar form factor . With the product together with calculated in lattice QCD, we extract .
In addition, a comprehensive analysis of these measurements is also important for searching or constraining the contributions of non-Standard interactions beyond the Standard weak to decays. One candidate of the non-Standard interactions is the exchange a scalar leptoquark [16, 17, 18]. Leptoquarks are hypothetical color-triplet bosons that carry both baryon number and lepton number, and can thus couple directly to a quark and a lepton [19, 20]. Leptoquark can be of either vector (spin-1) or scalar (spin-0) nature according to their properties under the Lorentz transformations. Some scalar leptoquarks can lead to the effective vertex. Searching the signals or obtaining more constraints of scalar leptoquarks from decays is thus one of the goals of this article. Using the analysis results in the context of the SM and from SM global fit as input parameters, we re-analyze existing measurements for decays in the context of new physics and obtain constraints on scalar leptoquark from these measurements.
The article is organized as follows: We first review the parametrization of form factors in Section 2. The details of experimental data for decays are presented in Section 3. In Section 4, we first discuss the feasibility of the scalar form factor obtained from analyzing these experimental data. Then we describe our analysis procedure of fitting to the experimental data in the context of the SM. At the last of this section, we present the analysis result of these measurements. In Section 5, we re-analyze these measurements in the context of new physics, and give the bounds for leptoquarks. Finally, conclusions for this work are given in Section 6.
2 Parameterization of the form factors
The form factors and can be parameterized according to the constraints of their general properties of analyticity, cross symmetry, and unitarity. Various parameterizations for these form factors exist in literatures such as the single pole model, the modified pole model, the model and the . However, experimental data seems do not support previous three models . Thus we use the in this article. In this parameterization, form factors transformed from -space to -space, where
with and . The form factors is then expressed as
where are real coefficients. The function is for the vector form factor or for the scalar form factor . And is
where is the mass of the charm quark.
To some extent, a two-parameter series expansion of the form factors is fine. Using the relation deduced from Eq.(5) and setting , one can obtain a vector (scalar) form factor with two free parameters and
|E691||()||= 0.91 0.07 0.11|
|CLEO||()||= 0.90 0.06 0.06|
|CLEO-II||()||= 0.978 0.027 0.044|
|CLEO-II||()||= 2.60 0.35 0.26|
|BaBar||()||= 0.927 0.007 0.012|
The existing measurements for and can be divided into three categories:
(i) Ratio of decay rates or of branching fractions , where or , and . The ratios measured at different experiments are shown in Table 1.
(ii) Decay branching fraction , where is the absolute branching fraction for decay and is the absolute branching fraction for . The absolute branching fractions measured at different experiments are shown in Tab. 2.
(iii) Decay rate , where represents the partial decay rate for or decay in a certain bin.
Measurements of the first two categories could not be used directly to determine and the shapes of form factors. To use these measurements, we should first transfer them into absolute decay rates in certain ranges .
One can obtain the absolute decay rates from the categories (i) and (ii) measurements respectively by
where is the branching fraction for or decays, and is the lifetime of or meson. To avoid the possible correlations, we use , (the sum of and ), s, s from PDG.
After these transformations, the absolute decay rates obtained from categories (i) and (ii) measurements together with category (iii) measurements, partial decay rates in different bins for , are shown in Tabs. 3 and 4.
|CLEO-c||(, 0.2)||17.82 0.43|
|(0.2, 0.4)||15.83 0.39|
|(0.4, 0.6)||13.91 0.36|
|(0.6, 0.8)||11.69 0.32|
|(0.8, 1.0)||9.36 0.28|
|(1.0, 1.2)||7.08 0.24|
|(1.2, 1.4)||5.34 0.21|
|(1.4, 1.6)||3.09 0.16|
|(1.6, )||1.28 0.11|
|BES-III||(, 0.1)||8.812 0.187|
|(0.1, 0.2)||8.743 0.162|
|(0.2, 0.3)||8.295 0.159|
|(0.3, 0.4)||7.567 0.153|
|(0.4, 0.5)||7.486 0.152|
|(0.5, 0.6)||6.446 0.138|
|(0.6, 0.7)||6.200 0.134|
|(0.7, 0.8)||5.519 0.126|
|(0.8, 0.9)||5.028 0.119|
|(0.9, 1.0)||4.525 0.111|
|(1.0, 1.1)||3.972 0.103|
|(1.1, 1.2)||3.326 0.093|
|(1.2, 1.3)||2.828 0.085|
|(1.3, 1.4)||2.288 0.077|
|(1.4, 1.5)||1.737 0.068|
|(1.5, 1.6)||1.314 0.058|
|(1.6, 1.7)||0.858 0.050|
|(1.7, )||0.379 0.039|
|CLEO-c||(, 0.2)||17.79 0.65|
|(0.2, 0.4)||15.62 0.59|
|(0.4, 0.6)||14.02 0.54|
|(0.6, 0.8)||12.28 0.49|
|(0.8, 1.0)||8.92 0.41|
|(1.0, 1.2)||8.17 0.37|
|(1.2, 1.4)||4.96 0.27|
|(1.4, 1.6)||2.67 0.19|
|(1.6, )||1.19 0.13|
|BES-III||(, 0.2)||16.97 0.60|
|(0.2, 0.4)||15.29 0.53|
|(0.4, 0.6)||13.57 0.47|
|(0.6, 0.8)||11.65 0.40|
|(0.8, 1.0)||9.33 0.34|
|(1.0, 1.2)||7.06 0.28|
|(1.2, 1.4)||4.96 0.20|
|(1.4, 1.6)||2.97 0.14|
|(1.6, )||1.01 0.07|
We also consider the non-parametric relative form factors for decays measured by the FOCUS Collaboration at the FOCUS experiment in 2005 . The relative form factors at the central values of nine bins were obtained in assuming has been normalized to 1 and the ratio , where . We list these measurements in Tab. 5.
In 2006, the Belle Collaboration reported their results on the measurements of the vector form factors for . Both considering the signal events for decays and decays, they obtained the form factors in 27 bins with bin size of 0.067 GeV. It is worthy to note that these measurements were obtained in the case of the masses of leptons neglected in the definition of differential decay rate for decays. Thus the vector form factor is different from the one defined in this article. In the following, to make a distinction between these vector form factors, we use to represent the vector form factor in the case of neglecting the lepton mass. These measured at the Belle experiment can’t be used directly in this analysis. To use these measurements, we should translate them into products by using the magnitude of CKM matrix element , , which was used by the Belle Collaboration to obtain these in their article. The measurements , collected in Ref. , are listed in Tab. 6.
The non-parametric relative form factors measured at the FOCUS experiment and the vector form factor measured at the Belle experiment are very important for the determination of scalar form factor for semileptonic decays. We will discuss this issue in the next section.
4 Fits to experimental data in the context of the SM
Our goal is to obtain the product and the shapes of the vector and scalar form factors for semileptonic decays from the existing measurements. The first task we should deal with is to confirm the feasibility of obtaining the scalar form factor by analyzing these experimental data, which is depending on the relative errors of these measurements and the contribution ratios of term to these measurements. If the contributions of the scalar form factor term to these measurements are much smaller than the errors of these measurements, then the fitting result of the scalar form factor will not be credible. Thus the confirmation of the feasibility is very important. The second task is to construct a Chi-squared function , by minimizing this function we can arrive to the results of this fit.
4.1 The contribution ratios of scalar form factor
The contribution ratio of term to the decay rate for decay at a certain can be described by the ratio
where represents and the other portion of the first item of the denominator is coefficient of the , the numerator is the coefficient of . These coefficients can be easily obtained from Eq. 1.
The contributions of term to the relative form factor defined in the FOCUS Collaboration’s article  and the vector form factor defined in the Belle Collaboration’s article  can be described by
where , , , and is a constant, where is for the relative form factor and is for the vector form factor .
and varying with were shown in Figs. 1 (a) and (b), respectively. The in Eqs. 10 and 11 is set to 1.0, 1.5 or 2.0, which is according to previous lattice calculations. 1.0 is a good approximation at low range, 2.0 can be used for high range, and 1.5 is suitable for the middle range. Fig. 1 (a) shows that the contributions of term to the partial decay rates for decay are only in the most range except and . The are much smaller than the relative errors of the partial decay rates for measured at experiments, these relative errors (0.02) can be calculated from Tabs. 3 and 4. Fig. 1 (b) shows the contribution rates of term to the relative form factor of () and the contribution rates of term to the vector form factor of (). As comparison, relative errors of measured by the FOCUS Collaboration and relative errors of measured by the Belle Collaboration, and , are also presented in Fig.1(b). We can find that all these relative errors and are smaller than the contribution rates of term to the relative form factor for () and the contribution rates of term to the vector form factor for (), except at the range of GeV measured at the Belle experiment.
Through the above analysis we can come to the conclusion that relative form factors measured at the FOCUS experiment and vector form factors measured at the Belle experiment can be used for the determination of the scalar form factor for semileptonic decays, while the measurements of partial decay rates for can not. But note that all these measurements are important for the determination of the vector form factor for semileptonic decays.
4.2 Construct Chi-squared function
To obtain and shapes of the vector and scalar form factors, we perform our fit to these experimental measurements by minimizing the Chi-squared function
where is constructed for measurements of partial decay rates in different ranges for as shown in Tabs. 3 and 4, is for the non-parametric form factors measured at the FOCUS experiment, and corresponds to the Belle Collaboration measured products .
Since there are correlations between the measurements of partial decay rates for decays and/or decays, the is given by
where is the partial decay rate measured in experiment, denotes its theoretical expectation, and is the inverse of the covariance matrix , which is a matrix. To compute the covariances of these 54 partial decay rates measured in different ranges and at different experiments, we adopt the concept proposed in Ref. : (a) at the same experiment, the statistical and systematic errors of these partial decay rates, and corresponding correlations between these partial decay rates are used to compute their covariances; (b) the systematic uncertainties caused by the lifetime of meson are fully correlated among all of the partial decay rates for decays measured at different experiments. (c) the systematic uncertainties related to are full correlated among all of the measurements of category (i) in Section 3 for decays.
Due to the correlations between measurements of the non-parametric form factors at the FOCUS experiment, the in Eq. 12 is defined as
where and are the measured value from the FOCUS experiment and the theoretical expectation of at the center of -th bin, respectively. It is worth noting that vector form factor in Eq. 7 can not be used as the theoretical form factor directly, even has been normalized to 1, because some assumptions about the expression of differential decay rate for process are different between this article and Ref. . By comparing Eq. (1) in this article and Eq. (2) in Ref. , we can obtain
where , , , and . To obtain Eq. 15 we make an approximation
The in Eq. (14) is the inverse of the covariance matrix , which is a matrix. We can construct the covariance matrix by the relation , where () is the standard error of at the central value of the -th (-th) bin measured at the FOCUS experiment, and is the correlation coefficient of measurements of at -th bin and -th bin.
The in Eq. (12) is built for the products measured at the Belle experiment. The is defined as
where and are experimental and theoretical values of in the -th bin respectively, and represents the standard deviation of . In Eq. 17, we neglect some possible correlations among the measurements of . Similar to the analysis of the measurements at the FOCUS experiment above, by comparing Eq. (1) in this article and Eq. (1) in Ref. , the expression of is