Determinations of form factors for semileptonic $D\rightarrow K$ decays and leptoquark constraints

By analyzing all existing measurements for $ D\rightarrow K \ell^+ \nu_{\ell} $ ( $\ell=e,\ \mu$ ) decays, we find that the determinations of both the vector form factor $f_+^K(q^2)$ and scalar form factor $f_0^K(q^2)$ for semileptonic $D\rightarrow K$ decays from these measurements are feasible. By taking the parameterization of the one order series expansion of the $f_+^K(q^2)$ and $f_0^K(q^2)$, $f_+^K(0)|V_{cs}|$ is determined to be $0.7182\pm0.0029$, and the shape parameters of $f_+^K(q^2)$ and $f_0^K(q^2)$ are $r_{+1}=-2.16\pm0.007$ and $r_{01}=0.89\pm3.27$, respectively. Combining with the average $f_+^K(0)$ of $N_f=2+1$ and $N_f=2+1+1$ lattice calculaltion, the $|V_{cs}|$ is extracted to be $0.964\pm0.004\pm0.019$ where the first error is experimental and the second theoretical. Alternatively, the $f_+^K(0)$ is extracted to be $0.7377\pm0.003\pm0.000$ by taking the $|V_{cs}|$ as the value from the global fit with the unitarity constraint of the CKM matrix. Moreover, using the obtained form factors by $N_f=2+1+1$ lattice QCD, we re-analyze these measurements in the context of new physics. Constraints on scalar leptoquarks are obtained for different final states of semileptonic $D \rightarrow K$ decays.


Introduction
Semileptonic D → P (P = K, π) 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 [1].
For D → Kl + ν l ( l = e, µ ) 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 D → Kl + ν l is given by where G F is the Fermi constant, p represents the three momentum of the K meson in the D rest frame. q ≡ p D − p K is the four momentum transferred to lν l and q 2 ranges from m 2 l where the daughter meson K has the maximum possible momentum to (m D − m K ) 2 where the K meson at rest. The vector form factor f K + (q 2 ) and the scalar form factor f K 0 (q 2 ) are defined via and At the maximal recoil point, kinematic constraints lead f K + (0) = f K 0 (0). In the last 30 years, various measurements for D → Kl + ν l decays were performed at more than ten experiments. For D 0 → K − e + ν e and/or D + →K 0 e + ν e decays, branching fractions and/or decay rates in different q 2 bins were measured at experiments such as the E691 [2],Mark-III [3],CLEO [4], CLEO-II [5], BaBar [6], BES-II [7,8],CLEO-c [9] and BES-III [10][11][12]. The FO-CUS Collaboration measured non-parametric relative form factor for D 0 → K − µ + ν µ decays at the FOCUS experiment in 2005 [13]. And in 2006, vector form factor for D 0 → K − l + ν l (l = e, µ) decays was measured by the Belle Collaboration at the Belle experiment [14]. By comprehensive analysis of these measurements, one can obtain, the product of the hadronic form factor at q 2 = 0 and the magnitude of CKM matrix element V cs , f K + (0)|V cs |. Then with f K + (0)|V cs | together with |V cs | from SM global fit one can determine the form factor f K + (0), or with f K + (0)|V cs | together with f K + (0) calculated in lattice QCD one can extract |V cs | [15]. In 2014, Ref. [15] reported their determination of f K + (0) and extraction of |V cs | by comprehensively analyzing all the existed measurements for D → Ke + ν e decays before 2014. In these experimental and theoretical studies, the contributions of f 0 term are neglected, since they are suppressed by the mass squared of lepton.
With the increase of measurements for D → Kl + ν l 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 f K + (q 2 ) and scalar form factor f K 0 (q 2 ) by comprehensively analyzing all the existing measurements for D → Kl + ν l decays. As the result of this analysis, we report the product f K + (0)|V cs | and shape parameters, r + and r 0 , 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 f K + (0) by considering the product f K + (0)|V cs | in conjunction with |V cs | from SM global fit. With the determined f K + (0) and shape parameters r + and r 0 , we chick the lattice calculations for the vector form factor f K + (q 2 ) and the scalar form factor f K 0 (q 2 ). With the product together with f K + (0) calculated in lattice QCD, we extract |V cs |. 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 D → Kl + ν l 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 effectivescνl vertex. Searching the signals or obtaining more constraints of scalar leptoquarks from D → Kl + ν l decays is thus one of the goals of this article. Using the analysis results in the context of the SM and V cs from SM global fit as input parameters, we re-analyze existing measurements for D → Klν l 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 D → Kl + ν l decays are presented in Section 3. In Section 4, we first discuss the feasibility of the scalar form factor f K 0 (q 2 ) 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.

Parameterization of the form factors
The form factors f K + (q 2 ) and f K 0 (q 2 ) can be parameterized according to the constraints of their general properties of analyticity, cross symmetry, and unitarity [21]. Various parameterizations for these form factors exist in literatures such as the single pole model [22], the modified pole model [22], the ISGW 2 model [23] and the series expansion [24]. However, experimental data seems do not support previous three models [15]. Thus we use the series expansion in this article. In this parameterization, form factors transformed from q 2 -space to z-space, where . The form factors is then expressed as where a k (t 0 ) are real coefficients. The function P(q 2 ) is P + (q 2 ) = z(q 2 , m 2 D * s ) for the vector form factor f K + (q 2 ) or P 0 (q 2 ) = z(q 2 , m 2 D * s0 ) for the scalar form factor f K 0 (q 2 ). And φ(q 2 , t 0 ) is where m c is the mass of the charm quark.
To some extent, a two-parameter series expansion of the form factors is fine. Using the deduced from Eq.(5) and setting k = 1, one can obtain a vector (scalar) form factor with two free parameters f K (0) and r +(0) where r +(0) = a

Experiments
The existing measurements for D 0 → K − e + ν e and D + →K 0 e + ν e can be divided into three categories:
(ii) Decay branching fraction B 0(+) , where B 0 is the absolute branching fraction for D 0 → K − e + ν e decay and B + is the absolute branching fraction for D + →K 0 e + ν e . The absolute branching fractions measured at different experiments are shown in Tab. 2.
(iii) Decay rate ∆Γ, where ∆Γ represents the partial decay rate for D 0 → K − e + ν e or D + →K 0 e + ν e decay in a certain q 2 bin.
Measurements of the first two categories could not be used directly to determine f K + (0)|V cs | and the shapes of form factors. To use these measurements, we should first transfer them into absolute decay rates in certain q 2 ranges [15].
After these transformations, the absolute decay rates obtained from categories (i) and (ii) measurements together with category (iii) measurements, partial decay rates in different q 2 bins for D → Ke + ν e , are shown in Tabs. 3

and 4.
We also consider the non-parametric relative form factors f K + (q 2 ) for D 0 → K − µ + ν µ decays measured by the FOCUS Collaboration at the FOCUS experiment in 2005 [13]. The relative Table 3: Partial decay rates ∆Γ of D 0 → K − e + ν e decays in q 2 ranges. Experiment q 2 (GeV) ∆Γ (ns −1 ) E691 [2] (m 2 e , q 2 max ) 86.32 ± 12.40 CLEO [4] (m 2 e , q 2 max ) 85.37 ± 8.10 CLEO-II [5] (m 2 e , q 2 max ) 92.77 ± 5.00 BaBar [6] (m 2 e , q 2 max ) 87  2.97 ± 0.14 (1.6, q 2 max ) 1.01 ± 0.07 form factors f K + (q 2 ) at the central values of nine q 2 bins were obtained in assuming f K + (0) has been normalized to 1 and the ratio f K We list these measurements in Tab. 5. In 2006, the Belle Collaboration reported their results on the measurements of the vector [14]. Both considering the signal events for D 0 → K − e + ν e decays and D 0 → K − µ + ν µ decays, they obtained the form factors f K + (q 2 ) in 27 q 2 bins with bin size of 0.067 GeV 2 . 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 D → Kl + ν l 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 f N L + (q 2 ) to represent the vector form factor in the case of neglecting the lepton mass. These f N L + (q 2 i ) measured at the Belle experiment can't be used directly in this analysis. To use these measurements, we should translate them into products f N L + (q 2 i )|V cs | by using the magnitude of CKM matrix element V cs , |V cs | = 0.97296 ± 0.00024, which was used by the Belle Collaboration to obtain these f N L + (q 2 i ) in their article. The measurements f N L + (q 2 i )|V cs |, collected in Ref. [15], are listed in Tab. 6. The non-parametric relative form factors f K + (q 2 ) measured at the FOCUS experiment and the vector form factor f N L + (q 2 ) measured at the Belle experiment are very important for the determination of scalar form factor f K 0 (q 2 ) for semileptonic D → K decays. We will discuss this issue in the next section.

Fits to experimental data in the context of the SM
Our goal is to obtain the product f K + (0)|V cs | and the shapes of the vector and scalar form factors for semileptonic D → K decays from the existing measurements. The first task we should deal with is to confirm the feasibility of obtaining the scalar form factor f K 0 (q 2 ) by analyzing these experimental data, which is depending on the relative errors of these measurements and the contribution ratios of f K 0 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 χ 2 , by minimizing this function we can arrive to the results of this fit. Table 6: f N L + (q 2 i )|V cs | measured at the Belle experiment, collected in Ref. [15].

The contribution ratios of scalar form factor
The contribution ratio of f 0 term to the decay rate for D → Ke + ν e decay at a certain q 2 can be described by the ratio where ρ represents |f K + (q 2 )/f K 0 (q 2 )| 2 and the other portion of the first item of the denominator is coefficient of the |f K + (q 2 )| 2 , the numerator is the coefficient of |f K 0 (q 2 )| 2 . These coefficients can be easily obtained from Eq. 1.
The contributions of f 0 term to the relative form factor f K + (q 2 ) defined in the FOCUS Collaboration's article [13] and the vector form factor f N L + (q 2 ) defined in the Belle Collaboration's article [14] can be described by where ρ =|f K , V e,µ (q 2 ) = 1 + m 2 e,µ /(2q 2 ) |p| 2 , and κ is a constant, where κ = 0 is for the relative form factor f K + (q 2 ) and κ = 1 is for the vector form factor f N L + (q 2 ).
(a) (b) Figure 1: The contribution ratio of f 0 term to the decay rate for D → Ke + ν e decay (a), the contributions of f 0 term to the relative form factor f K + (q 2 ) (κ = 0) and the vector form factor f N L + (q 2 ) (κ = 1) (b).
r Γ (q 2 ) and r f (q 2 ) varying with q 2 were shown in Figs. 1 (a) and (b), respectively. The ρ in Eqs. 10 and 11 is set to ρ = 1.0, 1.5 or 2.0, which is according to previous lattice calculations. ρ = 1.0 is a good approximation at low q 2 range, ρ = 2.0 can be used for high q 2 range, and ρ = 1.5 is suitable for the middle q 2 range. Fig. 1 (a) shows that the contributions of f K 0 term to the partial decay rates for D → Ke + ν e decay are only 10 −7 ∼ 10 −6 in the most q 2 range except q 2 → 0 and q 2 → (m D − m K ) 2 . The r Γ (q 2 ) are much smaller than the relative errors of the partial decay rates for D → Ke + ν e measured at experiments, these relative errors (>0.02) can be calculated from Tabs. 3 and 4. Fig. 1 (b) shows the contribution rates of f K 0 term to the relative form factor f K + (q 2 ) of D 0 → K − µ + ν µ (κ = 0) and the contribution rates of f K 0 term to the vector form factor f N L + (q 2 ) of D 0 → K − l + ν l (κ = 1). As comparison, relative errors of f K + (q 2 i ) measured by the FOCUS Collaboration and relative errors of f N L + (q 2 i ) measured by the Belle Collaboration, ∆ F OCU S and ∆ Belle , are also presented in Fig.1(b). We can find that all these relative errors ∆ F OCU S and ∆ Belle are smaller than the contribution rates of f K 0 term to the relative form factor for D 0 → K − µ + ν µ (κ = 0) and the contribution rates of f K 0 term to the vector form factor for D 0 → K − l + ν l (κ = 1), except ∆ Belle at the range of q 2 > 1.7 GeV 2 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 D → K decays, while the measurements of partial decay rates for D → Ke + ν e can not. But note that all these measurements are important for the determination of the vector form factor for semileptonic D → K decays.

Construct Chi-squared function
To obtain f K + (0)|V cs | and shapes of the vector and scalar form factors, we perform our fit to these experimental measurements by minimizing the Chi-squared function where χ 2 ∆Γ is constructed for measurements of partial decay rates in different q 2 ranges for D → Ke + ν e as shown in Tabs. 3 and 4, χ 2 F OC is for the non-parametric form factors f K + (q 2 ) measured at the FOCUS experiment, and χ 2 Bel corresponds to the Belle Collaboration measured products f N L + (q 2 i )|V cs |. Since there are correlations between the measurements of partial decay rates for D 0 → K − e + ν e decays and/or D + →K 0 e + ν e decays, the χ 2 ∆Γ is given by where ∆Γ is the partial decay rate measured in experiment, ∆Γ th denotes its theoretical expectation, and C −1 ∆Γ is the inverse of the covariance matrix C ∆Γ , which is a 54 × 54 matrix. To compute the covariances of these 54 partial decay rates measured in different q 2 ranges and at different experiments, we adopt the concept proposed in Ref. [15]: (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 D 0(+) meson are fully correlated among all of the partial decay rates for D 0(+) → K − (K 0 )e + ν e decays measured at different experiments. (c) the systematic uncertainties related to D 0 → K − π + are full correlated among all of the measurements of category (i) in Section 3 for D 0 → K − e + ν e decays.
Due to the correlations between measurements of the non-parametric form factors at the FOCUS experiment, the χ 2 F OC in Eq. 12 is defined as where f i and f th i are the measured value from the FOCUS experiment and the theoretical expectation of f K + (q 2 ) at the center of i-th q 2 bin, respectively. It is worth noting that vector form factor f K + (q 2 ) in Eq. 7 can not be used as the theoretical form factor f th i directly, even f K + (0) has been normalized to 1, because some assumptions about the expression of differential decay rate for D 0 → K − µ + ν µ process are different between this article and Ref. [13]. By comparing Eq. (1) in this article and Eq. (2) in Ref. [13], we can obtain To obtain Eq. 15 we make an approximation The C −1 F OC in Eq. (14) is the inverse of the covariance matrix C F OC , which is a 9 × 9 matrix. We can construct the covariance matrix C F OC by the relation ( is the standard error of f K + (q 2 ) at the central value of the i-th (j-th) q 2 bin measured at the FOCUS experiment, and ρ ij is the correlation coefficient of measurements of f K + (q 2 ) at i-th q 2 bin and j-th q 2 bin.
The χ 2 Bel in Eq. (12) is built for the products f N L + (q 2 i ) |V cs | measured at the Belle experiment. The χ 2 Bel is defined as where F i and F th i are experimental and theoretical values of f N L + (q 2 i )|V cs | in the i-th q 2 bin respectively, and σ i represents the standard deviation of F i . In Eq. 17, we neglect some possible correlations among the measurements of f N L + (q 2 i )|V cs |. Similar to the analysis of the measurements at the FOCUS experiment above, by comparing Eq. (1) in this article and Eq.

Fit to experimental data
We fit the experimental data with the vector and scalar form factors parameterized as Eq. (7). All of the SM input parameters such as the Fermi constant G F , the masses of mesons and charged leptons are taken from PDG [25]. The optimized f K + (0)|V cs |, the shape parameter of the vector form factor r + , the shape parameter of the scalar form factor r 0 and the minimized χ 2 value are shown in Tab. 7.
Figs. 2, 3 and 4 present the fitting results of the experimental data for semileptonic D → K decays. In Fig. 2 (a) and (b), we list the branching fractions for D 0 → K − e + ν e and D + → K 0 e + ν e measured at different experiments. The differential decay rates for D 0 → K − e + ν e and D + →K 0 e + ν e are shown in Fig. 2 (c) and (d), respectively. Fig. 3 shows the measurements of the relative form factors f K + (q 2 i ) in D 0 → K − µ + ν µ decay process from FOCUS experiment. The products f N L + (q 2 i )|V cs | measured at the Belle experiment for D 0 → K − l + ν l (l = e, µ) decays are present in Fig. 4 To check the quality of the fit to differential decay rates of D → Ke + ν e decays, these measurements are usually used to estimate the product f N L where the differential decay rate (dΓ/dq 2 ) i = ∆Γ i /∆q 2 i is obtained by the size of i-th q 2 bin dividing the partial decay rate measured in the q 2 bin, and p i is the momentum of K meson for i-th q 2 bin. From Eq. (1) one can obtain f N L + (q 2 )|V cs | as a function of q 2 together with f K + (q 2 i )|V cs | with errors measured at CLEO-c [9] and BES-III [10,12] experiments are shown in Fig. 5.

Determinations of f K + (0) and |V cs |
Note that the fit to experimental data returns just the product of the hadronic form factor f K + (0) and the magnitude of CKM matrix element V cs . To determine f K + (0) or extract |V cs |, we need more inputs. Since the CKM matrix element |V cs | can be obtained precisely from the SM global fit, by considering the product f K (0)|V cs | = 0.718 ± 0.003 shown in Tab. 7 together with |V cs | = 0.97351 ± 0.00013 obtained from global fit in the SM, one can obtain f K + (0) = 0.738 ± 0.003 ± 0.000, where the first error is from the uncertainties in the partial decay rate measurements, and the second is the contribution of the uncertainty of |V cs |. The form factor f K + (0) from the    recent lattice QCD calculations [26,27], the N f = 2 + 1 lattice average before 2017 [28] and experimental fit in 2014 [15] are compared in Fig. 6. Our fitting result is consistent with these theoretical calculations and presents a good consistency with the previous fitting result, but is with higher precision.
When f K + (0) has been determined, vector form factor f K + (q 2 ) and scalar form factor f K 0 (q 2 ) for semileptonic D → K decays are determined, since the shape parameters of vector and scalar form factors, r 1 and r 0 , have been obtained as returns in the fit. We show the comparisons of f K + (q 2 ) and f K 0 (q 2 ) obtained in this analysis with those obtained in the recent N f = 2 + 1 lattice calculations by JLQCD collaboration [26] and recent N f = 2 + 1 + 1 lattice calculations by V. Lubicz et al [27] in Fig. 7 (a) and (b), respectively. As shown in Fig. 7 (a), vector form factor f K + (q 2 ) obtain from the fit to experimental measurements presents more consistent within error with the prediction by JLQCD collaboration calculations, but with higher precision. Fig. 7 (b) shows the shapes and the error bands of scalar form factors f K 0 (q 2 ) obtained in this analysis and these values calculated by N f = 2 + 1 and N f = 2 + 1 + 1 lattice QCDs. The scalar form factors f K 0 (q 2 ) obtained in this analysis shows consistent within error with the predictions of both N f = 2 + 1 and N f = 2 + 1 + 1 lattice QCDs, but with less precision at high q 2 range. The comparisons of scalar form factors f K 0 (q 2 ) obtained from the fit to measurements for D → Klν l decays and these values calculated by lattice QCDs indicate that although the contributions of scalar form factor term to the decay of D → Kµν µ are small, the determination of scalar form factor from D → Kµν µ decays is feasible. And the precision of f K 0 (q 2 ) determined from the fit to experimental measurements will improved with the increase of measurements for D → Kµν µ decays and the improvement of measurement accuracy.
On the other hand, a comprehensive consideration of f K + (0)|V cs | = 0.718±0.003 and f K 0 (0) = 0.740 ± 0.018 (the average of N f = 2 + 1 lattice calculations f K 0 (0) = 0.698 ± 0.046 [26] and f K 0 (0) = 0.747 ± 0.019 [28]), the magnitude of the CKM matrix element V cs is determined to be where the first error is experimental and the second is theoretical. With comprehensive consideration of the |V cs | = 0.970 ± 0.004 ± 0.024 extracted in this analysis together with the |V cs | = 1.008 ± 0.021 determined from leptonic D s decay [25], the magnitude of the CKM  matrix element V cs is determined to be which is consistent with the average value |V cs | = 0.995 ± 0.016 from PDG'2016 [25]. Comparisons of |V cs | extracted from different analysis are shown in Fig. 8.

Leptoquark constraints from D → Kl + ν l decays
If there exist new interactions beyond the Standard W + mediating cs → vl, then they alter the decay rate and q 2 -distribution of D → Klν l decay processes. Since experimental data for D → Klν l decays, at some level, are in good agreement with the SM, the contributions of the new particles should be small and thus these new particles must be too heavy to directly detect. So an effective Lagrangian extended the SM is considered here. Omitting right-handed neutrinos, the effective Lagrangian is given by [16] where the terms parameterized by G F V * cs represent the effective SM Lagrangian, the other five arose by new type interactions.
A few possibilities can arise the non-Standard contributions to D s → lν decays which have been analyzed in Refs. [16,17]. Among the candidates they discussed, the most possible mechanism which can lead to an effectivescνl vertex is the u-channel exchange of a charge −1/3 scalar leptoquark S 0 , which transforms as color-triplet and weak-singlet with the U(1) [16,20]. The interactions between the leptoquark S 0 and the SM fermions can be described as [16] where the superscript C stands for charge conjugation, the subscript i denotes the generation of lepton, and λ LL 2i and λ RR 2i are complex Yukawa couplings. When the leptoquark mass satisfy m S 0 ≫ m D , one can obtain Then, there are only two unrelated coefficients left for two type fermion-leptoquark-fermion interactions in Eq. (25). We can name the one parameterized by |λ LL 2i | 2 /m 2 S 0 the LL type, which is caused by only the scalar leptoquark S L 0 , and the one parameterized by λ LL Because of these new interactions, the expression of differential decay rate for D → Klν l decays, Eq. (1), should be rewritten as where the new form factor, f 2 (q 2 ), is defined for describing the contribution of the tensorial operators via In both SCET and HQET limits, one can find f 2 (q 2 ) ≈ f + (q 2 ) [29].
To obtain constraints on LL-and LR-type fermion-leptoquark-fermion interactions from D → Klν l decays, using the central values of the product f K + (q 2 )|V cs | and the shape parameters r + and r 0 obtained from the fit to experimental measurements in the context of the SM (cf. Tab. 7) together with V * cs =|V cs | = 0.97351 (conventionally) obtained from the SM global fit as input parameters, we re-analyze the experimental data via Eq. (29) by one type interaction at a time. χ 2 varying with Λ/m 2 S 0 are shown in Figs. 9 (a) and (b) for the lepton-neutrino pairs are e-ν e and µ-ν µ , respectively. From Fig. 9 (a) one can find the bounds, at 95% C.L., for the case of the final states with e-ν e pair are And as Fig. 9 (b) shows, the bounds, at 95% C.L., for the case of the final states with µ-ν µ pair are  Recently, a search for pair production of secondgeneration leptoquarks is performed by the CMS Collaboration by using 35.9 fb −1 of data collected at √ s=13 TeV in 2016 with the CMS detector at the LHC [30]. By analyzing the final states with µµjj and µνjj, they exclude second-generation leptoquarks with masses less than 1530 GeV (1285 GeV) for β = 1(0.5) at 95% C.L., where β is the branching fraction of a leptoquark decaying to a charged lepton and a quark. Assuming lepton number conservation, with limits Eqs. (33) obtained from semileptonic D → K decays in conjunction with masses limits for second-generation scalar leptoquarks obtained by the CMS Collaboration, we show the combined limits on second-generation leptoquark S L 0 in Fig. 10.

Conclusions
By globally analyzing all existing measurements for D → Kl + ν l (l = e, µ) decays in the last 30 years, we find both the vector and scalar form factors of D → Kl + ν l decays from these experimental measurements. With two-parameter series expansion form factors, we obtain the product of form factor f K + (0) and the magnitude |V cs | and the shape parameters of both vector and scalar form factors f K + (0)|V cs | = 0.718 ± 0.003 ± 0.000, r + = −2.17 ± 0.007, 1.4 ± 3.5.
With the product f K + (0)|V cs | together with |V cs | obtained from SM global fit, we determine f K + (0) = 0.738 ± 0.003 ± 0.000, which is consistent within error with the lattice calculations, and presents a good consistency with the previous fitting result, but with higher precision.
Then we check the vector form factor f K + (q 2 ) and the scalar form factor f K 0 (q 2 ) calculated by lattice QCDs. Both f K + (q 2 ) and f K 0 (q 2 ) determined from the fit are consistent within error with recent N f = 2 + 1 lattice calculations. But the vector form factor f K + (q 2 ) determined from the fit is more precise than the value calculated by lattice QCD, and so is the scalar form factor f K 0 (q 2 ) at low q 2 range, but at high q 2 range f K 0 (q 2 ) determined from the fit is much less precise than that calculated by lattice QCD.
With the product f K + (0)|V cs | in conjunction with the recent average of form factor f K + (0) in N f = 2 + 1 lattice calculations, the magnitude of CKM matrix element V cs can be extracted |V cs | D→Klν l = 0.970 ± 0.004 ± 0.024, where the second error is from the lattice calculated form factor which is 6 times larger than the first error which is from experiments. The determined magnitude |V cs | presents a good consistency within error with the one from SM global fit. Then factoring in |V cs | Ds→lν l = 1.008 ± 0.021 determined from leptonic D s decay [25], the magnitude of the CKM matrix element V cs is determined to be |V cs | = 0.992 ± 0.016, which is in good agreement within error with the average value |V cs | = 0.995 ± 0.016 from PDG'2016 [25].
We re-analyze these experimental measurements in the context of new physics. Taking the returns of the fit in the context of the SM and |V cs | from SM global fit as input parameters, we constrain leptoquark S Considering recent mass constraints for second-generation leptoquarks obtained by the CMS Collaboration, we give a combined limits on second-generation leptoquark S L 0 .