Determination of $f_+^\pi(0)$ and Extraction of $|V_{cd}|$ from Semileptonic $D$ Decays

By globally analyzing all existing measured branching fractions for $D\to \pi e^+\nu_e$ decays, partial decay rates in different four momentum transfer-squared $q^2$ bins, as well as products of the decay form factor $f_+^\pi(q^2)$ and the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix element $|V_{cd}|$, we obtain $f_+^\pi(0)|V_{cd}|=0.1428\pm0.0019$. This product, in conjunction with $|V_{cd}|$ from a global Standard Model fit, implies a value for the $D\to\pi$ semileptonic form factor $f_+^\pi(0)=0.634\pm0.008\pm0.002$, which is consistent within error with those calculated in theory based on QCD, but with higher precision than the most accurate $f_+^\pi(0)_{\rm LQCD}=0.666\pm0.020\pm0.021$ calculated in LQCD by a factor of 3.3. Alternately, using this product together with the most accurate form factor calculated in LQCD, we find $|V_{cd}|^{D\to \pi e^+\nu_e}=0.2144\pm0.0029_{\rm exp}\pm 0.0093_{\rm LQCD}$. Combining this $|V_{cd}|^{D\to \pi e^+\nu_e}$ together with $|V_{cd}|^{D^+\to\mu^+\nu_\mu}=0.2160\pm0.0049\pm0.0014$ extracted from both the BESIII and CLEO-c measurements of $D^+\to\mu^+\nu_\mu$ decays, we find the most precisely extracted $|V_{cd}|$ to be $|V_{cd}|=0.2157\pm0.0045$ up to date, which improves the accuracy of the PDG'2014 value $|V_{cd}|_{\rm PDG'2014}=0.225\pm0.008$ by over $70%$. Using this $|V_{cd}|$ together with the PDG'2014 $|V_{ud}|$ and $|V_{td}|$, we check for first column unitarity and find $|V_{ud}|^2+|V_{cd}|^2+|V_{td}|^2-1=-0.004\pm0.002$, which deviates from unitarity by $2\sigma$. In addition, we find the ratio of $f_+^\pi(0)$ and $D^+$ decay constant $f_{D^+}$ to be $f_+^\pi(0)/f_{D^+}=(3.11\pm0.08)$ GeV$^{-1}$, which can be used to validate LQCD calculations for these two quantities.


I. INTRODUCTION
In the Standard Model (SM) of particle physics, the mixing between the quark flavors in weak interaction is parameterized by the Cabibbo-Kobayashi-Maskawa (CKM) matrix VCKM , which is a 3 × 3 unitary matrix.Since the CKM matrix elements are fundamental parameters of the SM, they should be measured as accurately as possible.Precise measurements of these elements are very important in testing the SM and searching for New Physics (NP) beyond the SM.Any improved measurement of these elements would be the important input for precision test of the SM.
Three generation unitarity can be checked to see whether VCKM * V † CKM = Î is satisfied, which leads to test first, second and third column/row unitarity.The unitarity also gives rise to unitarity triangle (UT) relation To check for this column/row unitarity and the UT relation, many experimental measurements and theoretical efforts have been made in flavor physics for many years.If any of these consistency checks significantly deviate from unitarity, it may indicate some evidence for NP effects.
Each matrix element can be extracted from measurements of different processes supplemented by theoretical calculations for corresponding hadronic matrix elements.Since the effects of strong interactions and weak interaction can be well separated in semileptonic D decays, these decays are excellent processes from which one can determine the magnitude of the CKM matrix element V cd(s) .In the SM, neglecting the lepton mass, the differential decay rate for D → πe + ν e process is given by where G F is the Fermi constant, p is the three momentum of the π 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 π + (q 2 ) is the form factor which parameterizes the effect of strong interaction in the decay.In Eq. (I.1), X is a multiplicative factor due to isospin, which equals to 1 for mode D 0 → π − e + ν e and 1/2 for mode D + → π 0 e + ν e .
In addition to extraction of |V cd |, precise measurements of the D → π 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 of D → πe + 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 UT, from which one can more precisely test the SM and search for NP beyond the SM.
In the past decades, copious measurements of decay branching fractions and/or decay rates for D → πe + ν e were performed at different experiments.To obtain the knowledge about f π + (0) and |V cd | as better as possible, we analyze all of these existing measurements.By a comprehensive analysis of these existing measurements together with |V cd | from a global SM fit or together with the form factor f π + (0) calculated in LQCD, we precisely determine f π + (0) and extract |V cd |.
In this article, we report the determination of f π + (0) and extraction of |V cd | by analyzing all of these existing measurements of the semileptonic D → πe + ν e decays in conjunction with |V cd | from a global SM fit or with the form factor f π + (0) calculated in LQCD.In the follow-ing sections, we first review the experimental measurements of decay branching fractions and decay rates for D → πe + ν e and pre-deal with these measurements to get decay rates to be used in the comprehensive analysis of all these existing measurements in Section II.We then describe our comprehensive analysis procedure for dealing with these measurements to obtain the product of f π + (0) and |V cd | 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 π + (0) and the extraction of |V cd | in Section V.

II. EXPERIMENTS
There are different kinds of measurements of D → πe + ν e decays performed at many experiments during last 25 years, some of which can not directly be used to determine f π + (0) and extract |V cd |.To determine these quantities from all of these existing measurements, some of these measurements are needed to be pre-processed.

A. Relative Measurements
In 1995, by analyzing 3.0 fb −1 data collected with the CLEO-II detector at the Cornell Electron Storage Ring (CESR), the CLEO Collaboration made a measurement of the branching ratio of the Cabibbo suppressed D 0 semileptonic decays.The CLEO Collaboration observed 87 ± 33 signal events for D 0 → π − e + ν e decays and obtained the ratio of branching fractions R 0 ≡ B(D 0 → π − e + ν e )/B(D 0 → K − e + ν e ) = 0.103 ± 0.039 ± 0.013 [2].
In 2005, the CLEO Collaboration measured the branching ratios of the semileptonic D 0 → π − ℓ + ν ℓ decays by analyzing about 7 fb −1 of data collected around the Υ(4S) resonance with the CLEO-III detector.Combining their measurements for electron mode and muon mode with considering the differences in phase spaces of these two decay modes, the CLEO Collaboration obtained the ratio of branching fractions to be R 0 ≡ B(D 0 → π − e + ν e ) /B(D 0 → K − e + ν e ) = 0.082 ± 0.006 ± 0.005 [5].
All above mentioned measurements are relative measurements which could not be used directly to determine f π + (0) or |V cd |.To use these measurements to determine f π + (0) or |V cd |, 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 In 2014, the BaBar Collaboration studied the D 0 → π − e + ν e decays by analyzing 347.2 fb −1 data collected at 10.6 GeV [7].They selected D 0 → π − 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 branching fraction is measured relative to the normalization mode, D 0 → K − π + .The partial decay rate in ith q 2 bin is given by where ∆B i is the branching fraction measured in ith q 2 bin.Inserting the lifetime of D 0 meson, τ D 0 = (410.1 ± 1.5) × 10 −15 s and the branching fraction values presented in Ref. [7] into Eq.(II.2), we translate these measurements of branching fractions in ten q 2 bins into absolute partial decay rates, which are shown in Tab.I.
In 2006, the Belle Collaboration published results on the D 0 → π − ℓ + ν ℓ decays.They accumulated 56461 ± 309 ± 830 inclusive D 0 mesons and found 126 ± 12 ± 3 signal events for D 0 → π − e + ν e decays and 106 ± 12 ± 6 signal events for D 0 → π − µ + ν µ decays from 282 fb −1 data collected around 10.58 GeV with the Belle detector [11].Using these selected events from semileptonic D 0 decays, the Belle Collaboration obtained the form factors f π + (q 2 ) in ten q 2 bins with the bin size of 0.3 GeV 2 /c 4 .To obtain the product f π + (q 2 i )|V cd | which will be used in our comprehensive analysis in Section III, we extrapolate these measurements of form factors at the Belle experiment to the product f π + (q 2 i )|V cd | using the PDG'2006 value of |V cd | = 0.2271 ± 0.0010 [12] which was originally used in the Belle's paper published.Table III lists the form factors f π + (q 2 i ) measured at the Belle experiment and our translated products f π + (q 2 i )|V cd |.These products will be used in our further analysis described in Section III.Recently, the BESIII Collaboration reported preliminary results of D 0 → π − e + ν e decays obtained by analyzing 2.92 fb −1 data taken at 3.773 GeV.The BE-SIII Collaboration accumulated (279.3 ± 0.4) × 10 4 D0 tags from five hadronic decay modes.In this sample of D0 tags, they observed 6297 ± 87 signal events for D 0 → π − e + ν e decays [13,14], and measured differential rates of D 0 → π − e + ν e decays at nine q 2 bins from 0.0 to 3.0 GeV 2 .By analyzing these differential decay rates the BESIII collaboration measured a value of the product [13,14] f π + (0)|V cd | = 0.1420 ± 0.0024 ± 0.0010, which is obtained from a fit to the data in the case of that the form of form factor is parameterized with threeparameters series expansion (see Section III).The last row of Tab.III lists this f π + (0)|V cd |, where the error is the combined statistical and systematic errors.

III. ANALYSIS
To obtain the product of the semileptonic form factor at four momentum transfer q = 0, f π + (0), and the magnitude of CKM matrix element |V cd |, we perform a comprehensive χ 2 fit to these experimental measurements of the partial decay rates and the products f π + (q 2 i )|V cd | listed in Tabs.I, II and III.The object function to be minimized in the fit is defined as where χ 2 R is for both the decay rates extrapolated from measurements of decay branching fraction and the partial decay rates measured in different q 2 ranges, and χ 2 P corresponds to the products of Taking into account the correlations between the measurements of the partial decay rates, the quantity χ 2 R is given by (III.2) 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 30 × 30 matrix containing the correlations between the measured partial decay rates listed in Tabs.I and II.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. (I.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 π + (q 2 i )|V cd | measured at the Belle and BESIII experiments, the function χ 2 P in Eq. (III.1) is defined as where f ex i is the measured product f π + (q 2 )|V cd | at q 2 i with the standard deviation σ i , and f th i is the theoretical expectation of the product f π + (q 2 )|V cd | at q 2 i .

A. Form Factor Parameterizations
Several model dependent calculations of form factor are often used in analysis of experimental measurements of semileptonic D decays.
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 π + (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 * + meson for semileptonic D → πℓ + ν ℓ 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 → πℓ + ν ℓ decays is written as where m D * + is the mass of the D * + meson, and α is a free parameter to be fitted.The value of α is assumed to be around 1.34 for D → πℓ + 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 prediction of r for D → πℓ + ν ℓ decays is 1.410 GeV −1 c.
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 ) = 1 for D → πℓ + ν ℓ , 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 , (III.10) 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. (III.9).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:

B. Covariance Matrix
It's a little complicated to compute the covariances of these 30 measurements of partial decay rates 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 for D → Ke + ν e 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 → π − 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 → π − e + ν e decay.Similarly, the systematic uncertainties related to D + lifetime are fully correlated among all of the ∆Γ measurements for D + → π 0 e + ν e decay.
• An additional systematic uncertainty from the branching fraction for D 0 → K − e + ν e decay is fully correlated between these relative measurements of D 0 → π − e + ν e decay at the CLEO-II, E687 and CLEO-III experiments.Since we only use one relative measurement of D + → π 0 e + ν e decay which is from the CLEO-II experiment, there are no correlations due to the branching fraction for D + → K0 e + ν e between this measurement and other measurements.
With these considerations mentioned above, we then construct a 30 × 30 covariance matrix C R which is necessary in the form factor fit.

C. Fits to Experimental Data
Four fits are applied to the experimental data with the form factor hypothesis of single pole model, BK model, ISGW2 model and series expansion.The fit to experimental data returns the normalization f π + (0)|V cd | 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.IV.As an example, Fig. 1 presents the result of the fit in the case of using the form factor parameterization of series expansion.In Fig. 1 (a), we compared the measured branching fractions of D → πe + ν e decays from different experiments.Figure 1 (b) depicts the measurements of f π + (q 2 )|V cd | at different q 2 from the Belle and BESIII experiments.Figure 1 (c) and (d) show the measured differential decay rates for D 0 → π − e + ν e and D + → π 0 e + ν e , respectively.In these figures, the lines show the best fit to these measurements of D → πe + ν e decays.
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 .

(III.15)
To calculate the integral in Eqs.(III.14) and (III.15),we use the shape parameters of the form factor, which is obtained from the series expansion fit to the data.Figure 2 shows the product f π + (q 2 )|V cd | as a function of q 2 , where the blue curve corresponds to the best series expansion fit to the experimental data.In this fit, seven measurements of f π + (0)|V cd | locate at q 2 = 0, which overlap each other.To be clear, these f π + (0)|V cd | 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 results from the fit using series expansion as our primary results and use this extracted f π + (0)|V cd | from the fit to determine the form factor f π + (0) and extract the magnitude of the CKM matrix element V cd .
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 cd |.The result of the form factor determined in this analysis is compared with the theoretical calculations of the form factor from the lattice QCD [18,19] and from QCD light-cone sum rules [20] in Fig. 3. Our result of the form factor determined by analyzing all existing experimental measurements of these decays is consistent within error with these values predicted by theory, but is with higher precision than the most accurate value of the form factor, f π + (0) LQCD = 0.666 ± 0.020 ± 0.021 calculated in LQCD [18], calculated in LQCD by a factor of 3.3.

B. 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 4 (a), (b) and (c) show the 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.
Our determined α = 0.252 ± 0.044 from this comprehensive analysis is 3.2σ smaller than α LQCD = 0.44±0.04calculated in LQCD [19].Figure 4  To check the consistency of the SM and validate the LQCD approach to the charm physics, we can compare the ratio of the D semileptonic decay form factor f π + (0) and D + decay constant f D + from both the measurements and the LQCD calculations of these quantities.With the most accurate f π + (0) LQCD = 0.666 ± 0.020 ± 0.021 calculated in LQCD [18] and most accurate f D + = (212.6± 0.4 +1.0 −1.2 ) MeV calculated in LQCD [21], we get From our determined f π + (0) given in Eq. (IV.1) and f D + given in Eq. (A.5) (see Appendix A), we find which is in very good agreement within error with the LQCD prediction of the ratio give in Eq. (IV.2).Using the product f π + (0)|V cd | = 0.1428 ± 0.0019 obtained from the comprehensive series expansion fit in conjunction with the form factor f π + (0) LQCD = 0.666 ± 0.020 ± 0.021 [18] calculated in LQCD for the D → π transition, we extract the magnitude of the CKM matrix element V cd from all existing measurements of semileptonic D decays to be where the first error is from the uncertainty in experimental measurements, the second uncertainty corresponds to the accuracy of the form factor f π + (0) calculated in LQCD.The experimental precision of this |V cd | D→πe + νe is 2 factor better than the PDG'2014 |V cd | D→πe + νe PDG ′ 2014 = 0.220±0.006±0.010[6] extracted from the average of the CLEO-c [10] and Belle [11] measurements of D → πℓ + ν ℓ decays in conjunction with f π + (0) LQCD = 0.666 ± 0.020 ± 0.021 [18].This big progress in improvement of the experimental accuracy of |V cd | D→πe + νe is mainly due to the recent BESIII measurement [13,14], due to BaBar measurement [7] as well as other earlier measurements [2-5, 8, 9]

V. SUMMARY
By globally analyzing all existing branching fractions of the D → πe + ν e decays measured at earlier experiments and products f π + (q 2 )|V cd | measured at the Belle and the BESIII experiments as well as the partial decay rates in q 2 bins measured at the BaBar and CLEO-c experiments together, we obtain the most precise product f π + (0)|V cd | = 0.1428 ± 0.0019.From this product we determined the form factor f π + (0) = 0.634 ± 0.008 ± 0.002, which is in good agreement within error with LQCD calculations of the form factor, but with more precision than the most accurate LQCD calculation of the form factor by 3.3 factors.We also determined the ratio of the semileptonic form factor f π + (0) and D + decay constant f D + to be

FIG. 1 :
FIG.1:(a) Comparisons of branching fraction measurements for D → πe + νe decays, (b) the product f π + (q 2 )|V cd | measured at the Belle and BESIII experiments, the differential decay rates as function of q 2 for (c) D 0 → π − e + νe measured at the BaBar and CLEO-c experiments, and (d) for D + → π 0 e + νe measured at the CLEO-c experiment.The blue lines show the fit to these measurements using the series expansion for the form factor.

2 FIG. 2 :
FIG.2:The product f π + (q 2 )|V cd | measured at different experiments as a function of q 2 .The blue curve represents the series expansion fit to these f π + (q 2 )|V cd |.The insert plot shows the comparison of the products f π + (0)|V cd | which are obtained using the branching fractions measured at different experiments.

FIG. 4 :
FIG. 4: Comparisons of the form factor parameters determined from experimental measurements and the theoretical expectations: (a) the pole mass M ploe in single pole model, (b) α in the BK model, and (c) r in the ISGW2 model.

Figure 5 FIG. 5 :FIG. 6 :
Figure 5 shows a comparison of the value of |V cd | which is determined with the |V cd | D→πe + νe obtained in this analysis together with the |V cd | D + →µ + νµ extracted from leptonic D + decays, and the value from the global SM fit [6]. Figure 6 shows a comparison of our extracted |V cd | from all existing measurements of D → πe + ν e and from both the BESIII and CLEO-c's measurements of D + → µ + ν µ decays along with the PDG'2014 value of the |V cd | determined with CLEO-c and Belle's measurements of D → πℓ + ν ℓ decays and neutrino interactions [6].Our extracted |V cd | = 0.2157 ± 0.0045 is in good agreement within error with the PDG'2014 value |V cd | PDG ′ 2014 = 0.225 ± 0.008, but improves the precision of the PDG'2014 value by over 70%.

1 FIG. 7 :
FIG.7: Unitarity checks for the first column and second row of the CKM matrix.
[f π + (0)/f D + ] exp = (3.11± 0.08) GeV −1 from experimental measurements and [f π + (0)/f D + ] LQCD = (3.13± 0.14) GeV −1from LQCD calculations.The experimental ratio is in excellent agreement within error with that determined from LQCD calculations of these two quantities.Alternately, with the recently most precise semileptonic D → πe + ν e decay form factor calculated in LQCD, we obtain |V cd | D→πe + νe = 0.2144 ± 0.0029 ± 0.0093, where the error is still dominated by the uncertainties in LQCD calculation of the semileptonic D → π form factor.This extracted |V cd | is consistent within 1.1σ with |V cd | = 0.22522 ± 0.00061 from the global SM fit.If combining this |V cd | D→πe + νe together with |V cd | D + →µ + νµ = 0.2160 ± 0.0049 ± 0.0014 extracted from leptonic D + decays together, we find |V cd | = 0.2157 ± 0.0045.This newly extracted |V cd | improves the accuracy of the PDG'2014 determination of |V cd | PDG ′ 2014 = 0.225±0.008by over 70%, and is the most precisely extracted |V cd | from all existing measurements of semileptonic D decays and from both the BESIII and CLEO-c's measurements of leptonic D + decays up to date.Combining the most precise |V cd | extracted in this work together with other updated |V ud | and |V td | given in PDG'2014, we find that the sum of the squared CKM matrix element in the first column deviates from unitarity by 2σ.
II.1) where B(D → Ke + ν e ) is the branching fraction for D 0 → K − e + ν e or D + → K0 e + ν e decays, and τ D is the lifetime of D meson.Using the lifetime of D meson,

TABLE III :
Measurements of form factors f π + (q 2 i ) at the Belle experiment and the products f π + (q 2 i )|V cd | obtained from the Belle and BESIII experiments.

TABLE IV :
Fitted parameters corresponding to different form factor parameterizations and χ 2 /d.o.f. of the fit.