Determination of $f_+^K(0)$ and Extraction of $|V_{cs}|$ from Semileptonic $D$ Decays

By globally analyzing all existing measured branching fractions and partial rates in different four momentum transfer-squared $q^2$ bins of $D\to Ke^+\nu_e$ decays, we obtain the product of the form factor and magnitude of CKM matrix element $V_{cs}$ to be $f_+^K(0)|V_{cs}|=0.717\pm0.004$. With this product, we determine the $D\to K$ semileptonic form factor $f_+^K(0)=0.737\pm0.004\pm0.000$ in conjunction with the value of $|V_{cs}|$ determined from the SM global fit. Alternately, with the product together with the input of the form factor $f_+^K(0)$ calculated in lattice QCD recently, we extract $|V_{cs}|^{D\to Ke^+\nu_e}=0.962\pm0.005\pm0.014$, where the error is still dominated by the uncertainty of the form factor calculated in lattice QCD. Combining the $|V_{cs}|^{D_s^+\to\ell^+\nu_\ell}=1.012\pm0.015\pm0.009$ extracted from all existing measurements of $D^+_s\to\ell^+\nu_\ell$ decays and $|V_{cs}|^{D\to Ke^+\nu_e}=0.962\pm0.005\pm0.014$ together, we find the most precisely determined $|V_{cs}|$ to be $|V_{cs}|=0.983\pm0.011$, which improves the accuracy of the PDG'2014 value $|V_{cs}|^{\rm PDG'2014}=0.986\pm0.016$ by $45\%$.


I. INTRODUCTION
In the Standard Model (SM) of particle physics, the mixing between the quark flavours in weak interaction is parameterized by the Cabibbo-Kobayashi-Maskawa (CKM) matrix, which is a 3×3 unitary matrix. Since the CKM matrix elements are fundamental parameters of the SM, precise determinations of these elements are necessary and very important in testing the SM and searching for New Physics (NP).
Since the effects of strong interactions and weak interaction can be well separated in semileptonic D decays, these decays are excellent processes from which we can determine the magnitude of CKM matrix element V cs(d) .
In the SM, neglecting the lepton mass, the differential decay rate for D → Ke + ν e process is given by where G F is the Fermi constant, p is the three momentum of the K meson in the rest frame of the D meson, q 2 is the four momentum transfer-squared, i.e. the invariant mass of the lepton and neutrino system, and f K + (q 2 ) is the form factor which parameterizes the effect of strong interaction.
In addtion to extraction of |V cs |, the precise measurements of the D → K semileptonic form factor is also very important to validate the lattice QCD (LQCD) calculation of the form factor. If the LQCD calculation of the form factor pass the test with the precisely measured form factor for D → Ke + ν e decay, the uncertainty of the semileptonic B decay form factor calculated in LQCD would be reduced. This would help in reducing the uncertainty of the measured |V ub | from semileptonic B decays [1]. The improved measurement of |V ub | from semileptonic B decay will improve the determination of the B d unitrarity triangle, with which one can more precisely test the SM and search for NP.
In the past decades, copious measurements of branching fractions and/or decay rates for D → Ke + ν e decays were performed at more than ten experiments. By comprehensive analysis of these existing measurements together with |V cs | from SM global fit or together with form factor f K + (0) calculated in LQCD, one can precisely determine the form factor f K + (0) or extract |V cs |. In this article, we report the determination of f K + (0) or extraction of |V cs | by analyzing all of these existing measurements of the semileptonic D → Ke + ν e decays in conjunction with |V cs | from SM global fit or with the form factor f K + (0) calculated in lattice QCD. In the following sections, we first review the experimental measurements of branching fractions and decay rates for D → Ke + ν e decays in Section II. We then describe our comprehensive analysis procedure for dealing with these measurements to obtain the product of f K + (0) and |V cs | in Section III. In Section IV, we present the final results of our comprehensive analysis of these measurements. We finally give a summary for the determination of f K + (0) and the extraction of |V cs | in Section V.
In 2007, the BaBar Collaboration studied the D 0 → K − e + ν e decays by analyzing 75 fb −1 data collected at 10.6 GeV [5]. They selected D 0 → K − e + ν e decays from e + e − → cc events and divide the candidate events into ten q 2 bins. In each q 2 bin, the partial decay rate is measured relative to the normalization mode, D 0 → K − π + .
All above mentioned measurements are relative measurements which could not be used directly to determine the form factor f K + (0) or |V cs |. To use these measurements to determine f K + (0) or |V cs |, we should first transfer these measurements into absolute decay rates in certain q 2 range. The absolute decay rate ∆Γ can be obtained from the measured relative decay branching ratio R by where B(D → Kπ) is the branching fraction for D 0 → K − π + or D + →K 0 π + decays, and τ D is the lifetime of D meson. Using the lifetime of D meson, τ D 0 = (410.1±1.5)×10 −15 s, and τ D + = (1040±7)×10 −15 s, the branching fractions of B(D 0 → K − π + ) = (3.91 ± 0.05)% and B(D + →K 0 π + ) = (2.93 ± 0.09)% quoted from PDG [6], we translate these measurements of relative branching fractions and relative partial decay rates into absolute partial decay rates as shown in Tabs. I and II. In addition to the measurements of relative branching fractions and relative partial rates, the FOCUS Collaboration measured the non-parametric relative form factors f K + (q 2 )/f K + (0) at the central values of nine q 2 bins by analyzing the D 0 → K − µ + ν µ decays in 2005 [7]. These measured variations of f K + (q 2 )/f K + (0) at FOCUS experiment also provide useful information about the semileptonic decay form factor and are helpful to determine the product f K + (0)|V cs | and the shape parameters of the form factor. These measurements are listed in Tab. III and are used in the further analysis.

B. Absolute Measurements
In 1989, the Mark III Collaboration performed a measurement of absolute branching fraction for semileptonic D 0 → K − e + ν e decay by analyzing data taken at the peak of ψ(3770) resonance with the Mark III detector. They tagged 3636 ± 54 ± 195D 0 mesons and found 55 D 0 → K − e + ν e signal events in the system recoiling against theD 0 tags. With these events, they measured the absolute decay branching fraction B(D 0 → K − e + ν e ) = (3.4 ± 0.5 ± 0.4)% [8].
Using the similar method as the one used in Mark III, the BES-II Collaboration measured the branching fractions of D → Ke + ν e decays by analyzing about I: The partial rates ∆Γ of the D 0 → K − e + νe decays in q 2 ranges obtained from different experiments. q 2 max is the maximum value of q 2 .
The partial decay rate is related to the decay branching fraction by Using the lifetimes of D 0 and D + mesons quoted from PDG [6], τ D 0 = (410.1±1.5)×10 −15 s and τ D + = (1040± 7) × 10 −15 s, we translate these absolute measurements of branching fractions for D → Ke + ν e decays into the   partial decay rates, which are shown in Tabs. I and II. In 2009, the CLEO Collaboration studied the semileptonic decays of D 0 → K − e + ν e and D + →K 0 e + ν e decays by analyzing 818 pb −1 data collected at 3.773 GeV with the CLEO-c detector. Using double tag method, they measured the decay rates for semileptonic D 0 → K − e + ν e and D + →K 0 e + ν e decays in nine q 2 bins [12]. These measurements of decay rates are summarized in Tabs. I and II.
In 2006, the Belle Collaboration published the results on the D 0 → K − ℓ + ν ℓ decays. They accumulated 56461 ± 309 ± 830 inclusive D 0 mesons and found 1318 ± 37 ± 7 signal events for D 0 → K − e + ν e decays and 1249 ± 37 ± 25 signal events for D 0 → K − µ + ν µ decays from a 282 fb −1 data set collected around 10.58 GeV with the Belle detector [13]. Using these selected events from semileptonic D 0 decays, they obtained the form factors f K + (q 2 ) in 27 q 2 bins with the bin size of 0.067 GeV 2 /c 4 .
To obtain the product f K + (q 2 i )|V cs | which will be used in our comprehensive analysis in Section III, we extrapo-late these measurements of form factors at the Belle experiment to the product f K + (q 2 i )|V cs | using the PDG2006 value of |V cs | = 0.97296 ± 0.00024 [14] which was originally used in the Belle's paper published. Table IV lists the form factors f K + (q 2 i ) measured at the Belle experiment and our translated products f K + (q 2 i )|V cs |. These products will be used in our further analysis described in Section III.

III. ANALYSIS
To obtain the product of the hadronic form factor at four momentum transfer q = 0, f K + (0), and the magnitude of CKM matrix element |V cs |, we perform a comprehensive χ 2 fit to these experimental measurements of the partial decay rates. The object function to be minimized in the fit is defined as where χ 2 R is for these measurements of decay branching fraction and/or partial decay rates in different q 2 ranges, χ 2 P corresponds to the products of f K + (q 2 i )|V cs | measured at the Belle experiment, and χ 2 F is built for the measure- Taking into account the correlations between these measurements, the quantity χ 2 R is given by where ∆Γ ex denotes the experimentally measured partial decay rate, ∆Γ th is the theoretical expectation of the decay rate, and C −1 R is the inverse of the covariance matrix C R , which is a 36 × 36 matrix containing the correlations between the measured partial decay rates. The construction of C R is discussed in subsection III B. With the parametrization of the form factor, the theoretically predicted partial decay rate in a given q 2 bin is obtained by integrating Eq. (1) from the low boundary q 2 low to the up boundary q 2 up of the q 2 bin, In this analysis, we used several forms of the form factor parameterizations which are discussed in subsection III A. Ignoring some possible correlations of the measurements of the product f K + (q 2 i )|V cs | measured at the Belle experiment, the function χ 2 P in Eq. (4) is defined as wheref ex i is the measured product f K + (q 2 )|V cs | at the center of ith q 2 bin q 2 i with the standard deviation σ i , andf th i is the theoretical expectation of the product f K + (q 2 )|V cs | at q 2 i . Considering the correlations of the non-parametric form factors measured at the FOCUS experiment, the χ 2 F is constructed as F is the inverse of the covariance matrix C F . The construction of C F is described later in the subsection III B.

A. Form Factor Parameterizations
In general, the single pole model is the simplest approach to describe the q 2 dependent behavior of form factor. The single pole model is expressed as where f K + (0) is the value of form factor at q 2 = 0, m pole is the pole mass which is predicted to be the mass of the D * + s meson for semileptonic D → Ke + ν e decays. The so-called BK parameterization [15] is also widely used in lattice QCD calculations and experimental studies of this decay. In the BK parameterization, the form factor of the semileptonic D → Ke + ν e decays is written as where m D * + s is the mass of the D * + s meson, and α is a free parameter to be fitted. The value of α is assumed to be around 1.75 for D → Kℓ + ν ℓ in the BK parameterization.
The ISGW2 model [16] assumes where q 2 max is the kinematical limit of q 2 , and r is the conventional radius of the meson. In this model, the predictions of f K + (q 2 max ) and r for D → Kℓ + ν ℓ decays are 1.23 and 1.12 GeV −1 c, respectively.
The most general parameterization of the form factor is the series expansion [17], which is based on analyticity and unitarity. In this parametrization, the variable q 2 is mapped to a new variable z through The form factor is then expressed in terms of the new variable z as where P (q 2 ) = z(q 2 , m 2 D * + s ) which accounts for the presence of the pole, φ(q 2 , t 0 ) is an arbitrary function, and a k (t 0 ) are real coefficients. In this analysis, the choice of φ(q 2 , t 0 ) is taken to be where m c is the mass of charm quark, which is taken to be 1.2 GeV/c 2 . In practical use, one usually make a truncation on the above series. Actually, it is found that the current experimental data can be adequately described by only the first three terms in Eq. (13). In this analysis we will fit the measured decay rates to the three-parameter series expansion. After optimizing the form factor parameters, we obtain the form for the three-parameter series expansion: where r k ≡ a k (t 0 )/a 0 (t 0 ) (k = 1, 2).

B. Covariance Matrix
It's a little complicated to compute the covariances of these 36 partial decay rates measured in different q 2 ranges and at different experiments. To be clear, we separate the correlations among these ∆Γ measurements into two case: the one associated with the experimental status of each independent experiment, and the one related to the external inputs of parameters such as the lifetime of the D meson.
The statistical uncertainties in the ∆Γ measurements from the same experiment are correlated to some extent, while these are independent for the measurements from different experiments. The systematic uncertainties from tracking, particle identification, etc. are usually independent between different experiments. In this analysis, we treat the systematic uncertainties except the ones from D lifetimes and branching fractions as fully uncorrelated between the measurements performed at different experiments. We consider these below: • The covariances of the ∆Γ measured at the same experiment are computed using the statistical errors, the systematic errors, and the correlation coefficients, which are presented in their original papers published.
• For the measurements of D 0 → K − e + ν e decay, the lifetime of D 0 meson is used to obtain the partial decay rates in particular q 2 ranges. The systematic uncertainties due to imperfect knowledge of D 0 lifetime are fully correlated among all these measurements of the partial rates of D 0 → K − e + ν e decay. Similarly, the systematic uncertainties related to D + lifetime are fully correlated among all of the ∆Γ measurements for D + →K 0 e + ν e decay.
• An additional systematic uncertainty from B(D 0 → K − π + ) is fully correlated between these relative measurements of D 0 → K − e + ν e decay at the E691, CLEO, CLEO-II and BaBar experiments.
Since we only use one relative measurement of D + →K 0 e + ν e decay which is from the CLEO-II experiment, there are no correlations due to the normalization branching fraction B(D + →K 0 π + ) between this measurement and other measurements.
With these considerations mentioned above, we then construct a 36 × 36 covariance matrix C R which is necessary in the form factor fit.
The entry in i-th row and j-th column of the covariance matrix C R in Eq. (8) is given by (C R ) ij = σ i σ j ρ ij , where σ i and σ j are the errors of the f K + (q 2 )/f K + (0) at q 2 i and q 2 j measured at the FOCUS experiment, respectively, and ρ ij is the correlation coefficient of these two measurements of f K + (q 2 )/f K + (0). The values of the errors and correlation coefficients are directly quoted from Ref. [7].

C. Fits to Experimental Data
Four fits are applied to the experimental data with the form factor hypothesis of single pole model, modified pole model, ISGW2 model and series expansion. The fit to experimental data returns the normalization f K + (0)|V cs | and the shape parameters of the form factor which govern the behavior of form factor in high q 2 range.
The numerical results of the fit corresponding to each form of the form factor parameterization are summarized in Tab. V. As an example, figure 1 presents the result of the fit in the case of using the form factor parameterization of series expansion. In Fig. 1 (a) and (b), we compared the measured branching fractions of D 0 → K − e + ν e and D + →K 0 e + ν e decays from different experiments. Figure 1 (c) and (d) show the measured differential decay rates for D 0 → K − e + ν e and D + →K 0 e + ν e , respectively. Figure 1 (e) depicts the measurements of f K + (q 2 )|V cs | at different q 2 from the Belle experiments. The FOCUS measurements of the normalized form factor f K + (q 2 )/f K + (0) are illustrated in Fig. 1 (f). In these figures, the lines show the best fit to these measurements.
To check the fit quality and also the isospin invariance, the experimentally measured decay branching fractions and/or partial rates are mapped into the product and where B denotes the measured branching fraction, the differential decay rate (dΓ/dq 2 ) i is obtained by dividing measured decay rate in q 2 bin i by the corresponding bin size. The normalization N is given by The effective p 3 i in q 2 bin i is given by To calculate the integral in Eqs. (18) and (19), we use the shape parameters of form factor, which is obtained from the series expansion fit to the data. Figure 2 shows the product f K + (q 2 )|V cs | as a function of q 2 , where the blue curve corresponds to the best series expansion fit to the experimental data. In this fit, eight measurements of f K + (0)|V cs | locate at q 2 = 0, which overlap each other. To be clear, these f K + (0)|V cs | translated from the decay branching fractions measured at different experiments are also displayed in the insert plot in Fig. 2.

IV. RESULTS
In this analysis, we choose the result from the fit using series expansion as our primary results and use this to extract the form factor f K + (0) and the magnitude of the CKM matrix element V cs .
A. Form Factor f K + (0) Dividing the value of f K + (0)|V cs | = 0.717±0.004 shown in Tab. V from the series expansion fit by the |V cs | = 0.97343 ± 0.00015 obtained using unitarity constraints [6] yields the form factor where the first uncertainty is from the combined statistical and systematic uncertainties in the partial decay rate measurements, and the second is due to the uncertainty in the |V cs |. The result for the form factor determined in this analysis is compared with the theoretical calculations of the form factor from the lattice QCD [18][19][20] and from QCD light-cone sum rules [21] in Fig. 3. Our result of the form factor extracted by analyzing all existing experimental measurements is consistent with these values predicted by theory, but is with higher precision than the most accurate one from LQCD calculation by a factor of 2.8.

B. CKM Matrix Element |Vcs|
Using the product f K + (0)|V cs | = 0.717 ± 0.004 obtained from the comprehensive series expansion fit in conjunction with the form factor f K + (0) = 0.745 ± 0.011 [18] calculated in LQCD for the D → K transition, we determine the magnitude of the CKM matrix element V cs to be where the last uncertainty corresponds to the accuracy of the form factor f K + (0) calculated in LQCD. Combining with the value |V cs | D + s →ℓ + ν ℓ = 1.012 ± 0.015 ± 0.009, which is extracted from the measurements of leptonic D + s decays (see Appendix A), we obtain the magnitude of the CKM matrix element V cs to be |V cs | = 0.983 ± 0.011.
(22) Figure 4 shows the comparisons of the value of |V cs | which is determined with the |V cs | D→Ke + νe in this analysis together with the |V cs | D + s →ℓ + ν ℓ determined from leptonic D + s decays, and the value from a SM global fit [6]. Figure 5 shows a comparison of our extracted |V cs | from all existing measurements of D → Ke + ν e and D + s → ℓ + ν ℓ decays along with the PDG2014 value of the |V cs | determined with CLEO-c, BaBar and Belle's measurements of D → Ke + ν e and D + s → ℓ + ν ℓ decays [6].

C. Parameters of Form Factor
When these shape parameters of the form factor parameterization are left free in the fit, the form factor parametrizations of the single pole model, BK model, the ISGW2 model, and the series expansion model are all capable of describing the experimental data with almost identical χ 2 probability. However, for the physical interpretation of the shape parameters in the single pole model, BK model, the ISGW2 model, the values of the parameters obtained from the fits are largely deviated from those expected values by these models. This indicates that the experimental data do not support the physical interpretation of the shape parameters in these parametriziations. Figure 6  comparisons between the measured values and the theoretically expected values for the pole mass M pole in single pole model, α in BK model, and r in ISGW2 model. These measured parameters do not agree with the values predicted by these form factor models.

V. SUMMARY
By globally analyzing all existing branching fractions of the D → Ke + ν e decays measured at earlier experiments and recent BESIII experiment as well as the precise measurements of partial decay rates in q 2 bins performed at the BaBar and CLEO-c experiments together, we obtain the most precise product of form factor and the magnitude of CKM matrix element V cs from a comprehensive χ 2 fit. This obtained product reflects all of measurements for D → Ke + ν e decays in the world in the last 25 years. With the obtained f K + (0)|V cs | in conjunction with |V cs | from SM global fit, we determined the form factor f K + (0) = 0.737 ± 0.004 ± 0.000, which is in good agreement within error with LQCD calculations, but more precise than the most accurate LQCD calculation of the form factor by 2.8 factors. Alternately, with the recent most precise semileptonic D → Ke + ν e decay form factor calculated in LQCD, we obtain the |V cs | D→Ke + νe = 0.962 ± 0.005 ± 0.014, where the error is dominated by the uncertainties in LQCD calculation of the hadronic form factor. This determined | cs |V 0 |V cs | is in good agreement within error with the one from SM global fit, which indicates that no evidence of new physic effects involved in the semileptonic D → Ke + ν e decays is observed at present experimental accuracy level.
If combining the |V cs | D→Ke + νe = 0.962 ± 0.005 ± 0.014 determined from semileptonic D decays and |V cs | D + s →ℓ + ν ℓ = 1.012 ± 0.015 ± 0.009 determined from leptonic D + s decays together, we find  In SM of particle physics, the decay width of D + s → ℓ + ν ℓ is given by (A1) where m ℓ is the mass of lepton and m D + s is the mass of D + s meson. The parameter f D + s is the decay constant, which is associated with the strong interaction effects between the two initial-state quarks.