Semileptonic Decays of Charmed Mesons to Light Scalar Mesons

PHYSICAL REVIEW D 102, 016013 (2020)

Semileptonic decays of charmed mesons to light scalar mesons

N. R. Soni * Department of Physics, Faculty of Science, The Maharaja Sayajirao University of Baroda, Vadodara 390002, Gujarat, India † A. N. Gadaria , J. J. Patel , and J. N. Pandya Applied Physics Department, Faculty of Technology and Engineering, The Maharaja Sayajirao University of Baroda, Vadodara 390001, Gujarat, India

(Received 4 February 2020; accepted 30 June 2020; published 20 July 2020)

Within the framework of covariant confined quark model, we compute the transition form factors of D and Ds mesons decaying to light scalar mesons f0ð980Þ and a0ð980Þ. The transition form factors are then utilized to compute the semileptonic branching fractions. We study the channels, namely, þ → ð980Þlþν → ð980Þlþν l ¼ μ DðsÞ f0 l and D a0 l for e and . For computation of semileptonic branching fractions, we consider the a0ð980Þ meson to be the conventional quark-antiquark structure and the f0ð980Þ meson as the admixture of ss¯ and light quark-antiquark pairs. Our findings are found to support the recent BESIII data.

DOI: 10.1103/PhysRevD.102.016013

I. INTRODUCTION the ss¯ counterpart. Recently, BESIII Collaboration reported the first ever experimental observation for the semileptonic Charmed semileptonic decays are important for the study þ branching fraction of D → a0ð980Þe ν [5]. The notable of open flavor hadron spectroscopy in general and heavy e results on these channels read quark decay properties in particular. More specifically, the scalar mesons below 1 GeV in final product can provide the þ þ þ − −5 BðD →f0ð980Þe ν ;f0ð980Þ→π π Þ<2.5×10 ½4 key information regarding their internal structure as well as e þ þ þ − the chiral symmetry in the low energy region of non- BðDs →f0ð980Þe νe;f0ð980Þ→π π Þ perturbative QCD [1]. The internal structure of these ¼ð0.130.040.01Þ% ½3 mesons is yet to be clearly understood from the theoretical 0 − þ − − studies attempted so far. BðD →a0ð980Þ e νe;a0ð980Þ →ηπ Þ ð980Þ Conventionally, the structure of f0 meson was ¼ð1.33þ0.33 0.09Þ×10−4 ½5 thought to be a bound state of quark-antiquark pair. CLEO −0.29 Bð þ → ð980Þ0 þν ð980Þ0 →ηπ0Þ was the first to have studied the semileptonic decays of D a0 e e;a0 þ D → f0ð980Þe ν [2,3] and recently, BESIII has reported þ0.81 −4 s e ¼ð1.66−0 66 0.11Þ×10 ½5: ð1Þ branching fractions for the semileptonic decays of Dþ → . ð980Þ þν f0 e e [4] suggesting that the internal structure of On experimental front, BESIII and other worldwide exper- ð980Þ bound state of f0 meson may not be the conventional imental facilities have reported the most precise results quark-antiquark pair. These experimental results also indi- on semileptonic decay of DðsÞ to pseudoscalar and vector f0ð980Þ cate that the internal structure of meson could be mesons. From theory point of view, these channels are ss¯ admixture of lighter quark state and state. For the straightforward to study because the internal structure/ D → f0ð980Þ, the dominant contribution is from the lighter → ð980Þ quark content of the daughter meson is a typical quark- quarks, whereas that for the Ds f0 channel is from antiquark system. But the quark structure of the scalar mesons below 1 GeV has varied explanations (see note *[email protected] on scalar mesons below 2 GeV in Particle Data Group † [email protected] Ref. [6]). The computation of branching fractions of DðsÞ mesons decaying to a0ð980Þ and f0ð980Þ is highly sensi- Published by the American Physical Society under the terms of tive to the internal structure of these mesons. The theo- the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to retical approaches so far include lattice quantum the author(s) and the published article’s title, journal citation, chromodynamics [7–12], QCD sum rules [13–17], chiral and DOI. Funded by SCOAP3. unitary approach [18], and different quark models [19–24].

2470-0010=2020=102(1)=016013(9) 016013-1 Published by the American Physical Society SONI, GADARIA, PATEL, and PANDYA PHYS. REV. D 102, 016013 (2020)

There are different ways in which these scalar mesons are fractions, and other physical observables such as forward studied globally, viz. conventional quark-antiquark states backward asymmetry and lepton polarization. The present [25–27] and compact multiquark states including diquark- study will help understand the essential dynamics of diantiquark [28], meson-meson composite molecule charmed semileptonic decays and the possible structure [29–37], as well as compact structure of tetra quarks of the scalar mesons below 1 GeV, namely, f0ð980Þ and [38–46]. In this mass range, one can also consider the a0ð980Þ. possibility of scalar glueball bound state [47,48]. This paper is organized in the following way: after In quark-antiquark picture, several theories have been introducing the requirement for the study of scalar mesons proposed including the constituent quark model [25]. The in the semileptonic decays with the literature in Sec. I,we semileptonic branching fractions for Ds → f0ð980Þ were briefly introduce the essential components of covariant considered in the light front quark model [26] and semi- confined quark model employed here for computation of leptonic as well as rare decays of the BðsÞ have also been the hadronic form factors in Sec. II. Using the transition studied in the formalism of covariant quark model [27].In form factors, we compute the semileptonic branching composite structure, there are several ways in which the fractions. In Sec. III, we present our numerical results of structure of scalar mesons is proposed. Fariborz et al. semileptonic branching fractions in comparison with other studied the light scalar mesons considering diquark- theoretical results and available experimental data. Finally, diantiquark state in the formalism of linear sigma model in Sec. IV, we summarize and conclude the present work. [28]. The scalar mesons have been considered to be the ¯ bound states of KK molecules in the potential model II. FORM FACTORS AND SEMILEPTONIC approach [29,30]. Achasov et al. considered the scalar BRANCHING FRACTIONS meson to be the KK¯ molecule in the radiative decays of ϕ meson [31]. The assignment of f0ð980Þ as the molecular Within standard model, the semileptonic decays are structure of KK¯ has also been used in the phenomeno- very well separated by strong and weak interactions. logical Lagrangian approach by studying the strong decays The charmed meson semileptonic decays to light scalar of f0ð980Þ to ππ and γγ channels [32]. The multiquark meson can be written as structure was also attempted in the unitarized meson model [33], effective field theory [34,35], as well as þ GF μ þ MðDð Þ → Sl νlÞ¼pffiffiffi V hSjqO¯ cjDð Þi½l Oμνl; chiral perturbation theory [36,37]. Maiani et al. studied s 2 cq s the diquark-diantiquark structure via strong decay of DðsÞ μ μ mesons [42,43]. Light scalar mesons are also studied in the with O ¼ γ ð1 − γ5Þ and q ∈ d; s. The matrix element in framework of tetraquark mixing [44–46]. this process is very well parametrized in terms of transition Lattice quantum chromodynamics investigations of these form factors given by scalar mesons are reported employing the four quark Mμ ¼ μ ð 2Þþ μ ð 2Þ ð Þ [49,50] and diquark-diantiquark pictures [51]. The tran- S P Fþ q q F− q : 2 sition form factors for the decays with scalar mesons as daughter products are also computed in the framework of Here P ¼ p1 þ p2 and q ¼ p1 − p2 with p1 and p2 to be – QCD sum rules [52 54] and light cone sum rules [55,56]. the momentum of DðsÞ meson of mass m1 and momentum Cheng et al. studied the transition form factors and semi- of scalar (S) meson of mass m2, respectively. The form → ð980Þ leptonic branching fractions for the channel D a0 factors Fþ and F− are computed in the entire accessible in the light cone sum rule approach where they considered physical range of momentum transfer in the formalism of a0ð980Þ to be the conventional quark-antiquark state [57]. CCQM. The Lagrangian describing the coupling of the The transition form factors are also determined in the light constituent quarks to the meson can be written as [58–61] front quark model considering them as a four quark state Z [22]. Jaffe considered the diquark-diantiquark structure of L ¼ ð Þ ð ; Þ¯ ð ÞΓ ð Þ these mesons in the formalism of MIT bag model [1]. int gM M x dx1dx2FM x x1;x2 q2 x2 Mq1 x1 The present work is focused on the semileptonic decay þ H:c: ð3Þ of D and Ds mesons to the light scalar mesons, namely, f0ð980Þ and a0ð980Þ in the framework of covariant Γ confined quark model (CCQM) [58–61]. The CCQM is Here, M is the Dirac matrix projecting onto spin of Γ ¼ the effective field theory approach with the built-in infrared corresponding mesonic state. It should read, i.e., M I, 5 confinement for the hadronic interactions to their constitu- γ , γμ for scalar, pseudoscalar, and vector mesons, respec- ents. Recently, we studied the semileptonic decays of tively. gM is the coupling strength of the meson with its D and Ds mesons to the pseudoscalar and vector mesons constituent quarks. FM, the translation invariant vertex in this formalism in great detail [62–66]. In these papers, function characterizing the effective physical size of the we investigated the transition form factors, branching hadron, is given by

016013-2 SEMILEPTONIC DECAYS OF CHARMED MESONS TO LIGHT … PHYS. REV. D 102, 016013 (2020) X2 ð Þ¼δ − Φ ðð − Þ2Þ ð Þ FM x; x1;x2 x wixi M x1 x2 ; 4 O i¼1

Φ with M as the correlation function of two constituent q k+p k+p q ¼ ð þ Þ 1 1 2 2 quarks with masses mq1 and mq2 and wqi mqi = mq1 mq2 D()s ()p1 S ()p such that w1 þ w2 ¼ 1. We choose Gaussian form for the 2 vertex function as q q 3 k 3 Φ˜ ð− 2Þ¼ ð 2 Λ2 Þ ð Þ M p exp p = M ; 5 where the model parameter ΛM characterizes the effective finite size of the mesons. Note that Eq. (5) is the Fourier 2 2 D ((kwp13 1 )) S ((kwp23 2 )) transform of the vertex function Eq. (4) for meson M. The ()s coupling strength gM can be determined using the renorm- FIG. 2. Quark model diagrams for the D-meson leptonic decay. alization of the one loop self-energy Feynman diagram. This is also known as the compositeness condition which ¯ μ ensures the absence of any bare quark state in the final hSðp2ÞjqO cjDðsÞðp1Þi Z mesonic state [67–69], 4 d k 2 ¼ N g g Φ ð−ðk þ w13p1Þ Þ c DðsÞ S ð2πÞ4i DðsÞ 3 2 gM ˜ 0 2 2 Z ¼ 1 − Π ðm Þ¼0: ð Þ × Φ ð−ðk þ w23p2Þ Þ M 4π2 M M 6 S μ 5 ×tr½S2ðk þ p2ÞO S1ðk þ p1Þγ S3ðkÞ 2 μ 2 μ The matrix element of self-energy diagram and semi- ¼ Fþðq ÞP þ F−ðq Þq ; ð8Þ leptonic decays are constructed from the S matrix using the interaction Lagrangian equation (2). The corresponding where Nc ¼ 3 is the number of flavors. The Fock- one loop Feynman diagram is drawn using the convolution Schwinger representation of the quark propagator (S1, S2 of quark propagator and vertex functions (Figs. 1 and 2). and S3) is used in computing the loop integral. This method The matrix element for self-energy diagram for any meson involves the conversion of the loop momenta into the can be written as exponential of function. The necessary loop integral can be evaluated analytically using the FORM code [70]. Finally, Z 4 the universal infrared cutoff parameter λ is used in Π˜ ð 2Þ¼ 2 d k Φ˜ 2 ð− 2Þ M p Ncg 4 k computation that guarantees the quark confinement within M ð2πÞ i M the hadrons. We take λ to be the same for all the physical ðΓ ð þ ÞΓ ð − ÞÞ ð Þ ×tr MS1 k w1p MS2 k w2p : 7 processes. This computation technique is quite general and can be used for Feynman diagrams with any numbers Π˜ 0 of loops. All the numerical calculations including the In Eq. (6), M is the derivative of meson mass operator Eq. (7). Similarly, the matrix element for the semileptonic multidimensional integrations are performed using a Mathematica code developed by us. For more detailed Dð Þ decays to scalar mesons can be written as s information regarding computation technique of the loop q integral, we suggest the reader to refer to Refs. [61,64]. The 2 necessary model parameters for computation of semileptonic branching fractions are given in Table I. These parameters have been determined for the basic electromag- p p netic properties like leptonic decay constants to match with the corresponding experimental data or lattice simu- lation results [71]. For present computations, we employ model parameters that are obtained using updated least square fit procedure performed in Refs. [27,64,72]. The parametrization was achieved to keep the deviation in the computed decay constants defined by the function q P ðyexperiment−ytheoryÞ2 1 χ2 ¼ i i – σ i σ2 to be minimum [73 75]. Here, i i FIG. 1. Diagram describing meson mass operator. are reported experimental standard deviations. After all, the

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TABLE I. Model parameters, namely, quark masses, size The form factors given in Eq. (8) are also very well parameters, and infrared cutoff parameter (all in GeV). represented in the double pole approximation as

qq¯ ¯ m m m Λ Λ Λ Λ Λss λ 2 u=d s c D Ds a0 f0 f0 Fð0Þ q Fðq2Þ¼ ;s¼ : ð9Þ 0.241 0.428 1.672 1.60 1.75 1.50 0.25 1.30 0.181 1 − as þ bs2 m2 DðsÞ parameters were fitted to get the best possible decay The parameters in the double pole approximation for the constant values; the uncertainties in the model parameters different decay channels are given in Table II. It is worth were determined by individually changing them to get the mentioning that the parametrization in the double pole exact experimental or lattice results. The difference approximation is quite precise and the deviation of the form between these two values of the parameters was considered factors from the actual data is less than 1% in the entire as uncertainty in the respective parameter. These uncer- range of momentum transfer. tainties are considered absolute for given parameters and Using the necessary model parameters (Table I) and are then transported to the form factors in the whole q2 computed form factors (Table II), we determine the semi- range. In Fig. 3, we present the spread of form factors Fþ in leptonic branching fractions in terms of helicity structure the whole q2 range due to propagation of uncertainty in the functions using the relation [76,77] parameters. It is observed that the uncertainties are of 2 þ the order of 4%–6% at the maximum recoil (q ¼ 0) and dΓðDðsÞ → Sl νlÞ – 2 ¼ 2 2 8% 10% at the minimum recoil (q qmax). Further, we dq compute their propagation in determination of uncertainty 2 2 2 2 G jV j jp2jq v in branching fractions using the generic method given in ¼ F cq ðð1 þ δ ÞjH j2 þ 3δ jH j2Þ; ð Þ 12ð2πÞ3 2 l 0 l t 10 the Appendix. m1

2 FIG. 3. q dependence of the DðsÞ → S form factors.

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TABLE II. Double pole parameters for the computation of form factors in Eq. (9).

FFð0Þ ab F Fð0Þ ab

þ þ D →f0ð980Þ 0 45 0 02 D →f0ð980Þ 0 40 0 02 Fþ . . 1.36 0.32 F− . . 0.71 0.24 þ þ Ds →f0ð980Þ 0 36 0 02 Ds →f0ð980Þ −0 39 0 02 Fþ . . 0.99 0.13 F− . . 1.13 0.18 0 − → ð980Þ D →a0ð980Þ 0.55 0.02 1.05 0.15 D a0 0.03 0.01 −0.04 32.81 Fþ F− þ 0 D →a0ð980Þ 0 55 0 02 D→a0ð980Þ 0 03 0 01 Fþ . . 1.06 0.16 F− . . 1.43 72.93

2 2 where δl ¼ ml=2q is the helicity flip factor, jp2j¼ Bennich et al where they considered the phenomenological 1 2 2 2 2 λ = ðm ;m ;q Þ=2m is the momentum of the daugh- analysis of two body decay of Dð Þ mesons and the form DðsÞ S DðsÞ s ter (Scalar) meson in the rest frame of the parent (DðsÞ) factors were computed using the dispersion relation (DR) 2 2 and covariant light front dynamics (CLFD) [83]. meson, and v ¼ 1 − ml=q is the velocity-type parameter. In the above Eq. (10), the bilinear combinations of the Furthermore, various other approaches have been reported helicity structure function are defined in terms of form considering the quark contribution. For example, Oset et al. ð980Þ factors as have studied the structure of f0 in various channels using chiral unitary approach and inferred that the f0ð980Þ 1 has major contribution from the strange quarks [18,84–87]. ¼ pffiffiffiffiffi ð þ 2 Þ Ht 2 PqFþ q F− ; Bediaga et al. studied the structure of f0ð980Þ in semi- q þ leptonic decay DðsÞ → f0ð980Þl νl using QCD sum rules 2m1jp2j H0 ¼ pffiffiffiffiffi Fþ: ð11Þ approach and inferred that the contribution of light quark q2 component is not negligible [13]. Motivated by these findings clearly suggesting considerable strange quark This helicity technique is formulated in Refs. [78–80] and contribution, we choose the mixing angle to be in the is also discussed recently in Refs. [76,77]. The computation range 25° < θ < 40° [81,82] for present computations and technique in CCQM is very general and can accommodate the computed branching fraction range turns out to be hadronic state with any number of constituent quarks. þ þ −5 BðD → f0ð980Þe ν Þ¼ð5.95 − 10.06Þ × 10 III. NUMERICAL RESULTS AND DISCUSSION e þ þ −5 BðD → f0ð980Þμ νμÞ¼ð5.92 − 10.01Þ × 10 After fixing the quark masses and meson size parame- Bð þ → ð980Þ þν Þ¼ð0 14 − 0 27Þ 10−2 ters, we compute the transition form factors for the semi- Ds f0 e e . . × leptonic decays of D and D mesons to the scalar mesons þ þ −2 s BðD → f0ð980Þμ νμÞ¼ð0.14 − 0.26Þ × 10 : below 1 GeV in the entire physical range of momentum s transfer. Looking at the literature, theoretically the internal structure of these scalar mesons is still not very much clear. In Table IV, we quoted the central values of branching In this study, we consider the internal structure of scalar fractions and the corresponding value of the mixing angle is 31°. Also, in Tables II and III, we quoted the values of form meson a0ð980Þ to be the conventional quark-antiquark state while f0ð980Þ meson as the admixture of qq¯ and ss¯ state factors and double pole parameters for mixing angle of 31°. characterized by a mixing angle θ. In terms of wave function, the internal structures of these scalar mesons are defined as TABLE III. Comparison of the form factor at the maximum recoil. j ð980Þi ¼ θj ¯iþ θj ¯i f0 cos ss sin qq Channel Present Other Reference − þ ja0ð980Þ i¼jdu¯i; D → f0ð980Þ 0.45 0.02 0.321 LCSR [56] 1 0.216 LFQM [26] 0 ¯ ja0ð980Þ i¼pffiffiffi juu¯ − ddi; 0.21 CLFD [83] 2 0.22 DR [83] pffiffiffi þ D → f0ð980Þ 0.39 0.02 0.30 0.03 LCSR [55] with qq¯ ¼ 1= 2ðuu¯ þ dd¯Þ and the mixing angle θ to be in s 25 θ 40 140 θ 165 0.434 LFQM [26] the range ° < < ° and ° < < ° [81,82]. 0.45 CLFD [83] Similar approach has been used earlier in the formalism 0.46 DR [83] of light front quark model and the authors obtained the 0 → ð980Þ− 0 55 0 02 1 75þ0.26 D a0 . . . −0.27 LCSR [57] mixing angle to be ð56 7Þ°orð124 7Þ° [26]. Another þ 0 D → a0ð980Þ 0.55 0.02 1.76 0.26 LCSR [57] approach was used in covariant quark model formalism by

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TABLE IV. Branching fractions of D → S semileptonic decay. IV. CONCLUSION

Channel Unit Present Theory Reference In this paper, we have considered the f0ð980Þ meson to ¯ þ þ −5 7 78 0 68 5 7 1 3 be the admixture of ss and light quark component with D → f0ð980Þe νe 10 . . . . LFQM [26] þ þ −5 the mixing angle of 25° < θ < 40° and a0ð980Þ meson to D → f0ð980Þμ νμ 10 7.87 0.67 þ → ð980Þ þν 10−2 0 21 0 02 0 2þ0.05 be the conventional quark-antiquark pair. We have Ds f0 e e . . . −0.04 LCSR [55] þ þ −2 employed the covariant confined quark model to study Ds → f0ð980Þμ νμ 10 0.21 0.02 0 → ð980Þ− þν 10−4 1 68 0 15 4 08þ1.37 the semileptonic branching fraction of charmed mesons D a0 e e . . . −1.22 LCSR [57] 0 − þ −4 decaying to the light scalar mesons. Our results are found D → a0ð980Þ μ νμ 10 1.63 0.14 þ → ð980Þ0 þν 10−4 2 18 0 38 5 40þ1.78 to be consistent with theoretical results as well as available D a0 e e . . . −1.59 LCSR [57] þ 0 þ −4 experimental data. The present study indicates that the D → a0ð980Þ μ νμ 10 2.12 0.37 0 − þ internal structure of f0ð980Þ has higher contribution of ss¯ ΓðD →a0ð980Þ e νeÞ 1 95 0 38 þ 0 þ . . ΓðD →a0ð980Þ e νeÞ state, suggesting the validity of chiral unitary approach, light cone sum rules analysis, and light front quark model approaches. We have also provided theoretical prediction In computing the transition form factors, we considered for the semileptonic branching fractions of charmed meson ¯ → ð980Þ contribution from qq for the channel D f0 meson to scalar meson in the muon channel for the first time. ¯ → ð980Þ and that from ss for Ds f0 . The form factors The present study can help in understanding the internal appearing in Eq. (8) are computed in the entire accessible structure of the scalar mesons below 1 GeV. As no absolute range of momentum transfer and associated double pole value of the branching fractions is available in the literature, parameters [Eq. (9)] are tabulated in Table II. We also we expect more accurate data coming from the worldwide compare our results of the form factors at the maximum upgraded experimental facilities to check the validity of 2 ¼ 0 recoil (q ) in Table III along with the other theoretical computed results in this study. models using light cone sum results data, covariant light front dynamics, and dispersion relations. Our results of fþð0Þ for D → f0ð980Þ are lower than those obtained ACKNOWLEDGMENTS using the light cone sum rules (LCSR) [56] from the We thank Professor Mikhail A. Ivanov for the continu- → ππlν¯ theoretical analysis of D decays. However, they ous support throughout this work and for providing match well with the quark-antiquark picture of scalar meson critical remarks for improvement of the paper. J. N. P. ¯ in light front quark model (LFQM) [26] and the mixing of ss acknowledges financial support from University Grants with light quark state of scalar mesons in CLFD/DR [83] Commission of India under Major Research Project → ð980Þ approach. For Ds f0 channel, our result of the F.No. 42-775/2013 (S. R.). N. R. S. and A. N. G. thank ð0Þ fþ matches well with the LCSR [55], but it is lower than Bogoliubov Laboratory of Theoretical Physics, Joint the LFQM and CLFD/DR approach. We also provide the Institute for Nuclear Research for warm hospitality during → ð980Þ form factors for the channel D a0 in comparison Helmholtz-DIAS International Summer School “Quantum with the LCSR results [57], where the structure of a0ð980Þ is ¯ Field Theory at the Limits: from Strong Field to Heavy considered to be the conventional qq state. Quarks” where this work was initiated. The computed form factors are then utilized for calculation of semileptonic branching fractions using Eq. (10). Our results of semileptonic branching fractions for both APPENDIX: PROPAGATION electron channel and muon channel are presented in OF UNCERTAINITY Table IV in comparison with other theoretical and available experimental data. No experimental results are available for The error propagation in the branching fraction can be þ → ð980Þ þν computed using the most common technique. In general, the absolute branching fractions of DðsÞ f0 e e. However, recently BESIII set the upper limit on the the differential branching fractions Eq. (10) can also be þ þ þ − −5 rewritten in terms of form factors as BðD → f0ð980Þe νe;f0ð980Þ → π π Þ < 2.8 × 10 with the confidence level of 90% [4]. Also, Hietala et al. B Bð þ → ð980Þ þν ð980Þ → πþπ−Þ¼ d ¼ ð 2 ð 2Þþ 2 ð 2Þþ ð 2Þ ð 2ÞÞ ð Þ predicted Ds f0 e e;f0 2 N aFþ q bF− q cFþ q F− q : A1 ð0.13 0.03 0.01Þ% using the CLEO-c data [88].For dq D → a0ð980Þ channel also, the absolute value of branching fraction is not available in the BESIII paper [5]. They Here N includes the terms involving the various constants 0 − þ such as Fermi coupling constant, Cabibbo-Kobayashi- ΓðD →a0ð980Þ e νeÞ predicted the ratio of the partial width þ 0 þ ¼ ΓðD →a0ð980Þ e νeÞ Maskawa matrix elements, meson masses, etc., and a, b, 2.03 0.95 0.06 and we obtained the ratios to be and c are the coefficients of form factors in Eq. (11). The 1.95 0.38, which is within the uncertainty limits pre- uncertainty in the measurement of branching fractions dicted by them [5]. because of the uncertainty in form factors can be written as

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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ∂ðdBÞ ∂ðdBÞ dðΔBÞ¼dq N ΔFþ þ ΔF− ; ðA2Þ ∂Fþ ∂F−

2 2 where ðΔFiÞ ¼ðFi · εiÞ with ε the relative error of all the form factors. The uncertainties in the form factors are also extracted using the same method. Finally, we determine the uncertainty in the branching fractions by integrating the above equation.

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