Semileptonic D (s) -meson decays in the light of recent data
Nakul R. Soni, Mikhail A. Ivanov, Jürgen G. Körner, Jignesh N. Pandya, Pietro Santorelli, Chien-Thang Tran
aa r X i v : . [ h e p - ph ] J a n Semileptonic D ( s ) -meson decays in the light of recent data N. R. Soni, ∗ M. A. Ivanov, † J. G. K¨orner, ‡ J. N. Pandya, § P. Santorelli,
4, 5, ¶ and C. T. Tran
6, 4, ∗∗ Applied Physics Department, Faculty of Technology and Engineering,The Maharaja Sayajirao University of Baroda, Vadodara 390001, Gujarat, India Bogoliubov Laboratory of Theoretical Physics,Joint Institute for Nuclear Research, 141980 Dubna, Russia PRISMA Cluster of Excellence, Institut f¨ur Physik,Johannes Gutenberg-Universit¨at, D-55099 Mainz, Germany Dipartimento di Fisica “E. Pancini”, Universit`a di Napoli Federico II,Complesso Universitario di Monte S. Angelo, Via Cintia, Edificio 6, 80126 Napoli, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, 80126 Napoli, Italy Institute of Research and Development, Duy Tan University, 550000 Da Nang, Vietnam (Dated: January 28, 2019)
Abstract
Inspired by recent improved measurements of charm semileptonic decays at BESIII, we studya large set of D ( D s )-meson semileptonic decays where the hadron in the final state is one of D , ρ , ω , η ( ′ ) in the case of D + decays, and D , φ , K , K ∗ (892) , η ( ′ ) in the case of D + s decays. Therequired hadronic form factors are computed in the full kinematical range of momentum transfer byemploying the covariant confined quark model developed by us. A detailed comparison of the formfactors with those from other approaches is provided. We calculate the decay branching fractionsand their ratios, which show good agreement with available experimental data. We also givepredictions for the forward-backward asymmetry and the longitudinal and transverse polarizationsof the charged lepton in the final state. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] ∗∗ Electronic address: [email protected]; corresponding author . INTRODUCTION Semileptonic D ( D s )-meson decays provide a good platform to study both the weak andstrong interactions in the charm sector (for a review, see e.g., Ref. [1]). Measurements of theirdecay rates allow a direct determination of the Cabibbo-Kobayashi-Maskawa (CKM) matrixelements | V cs | and | V cd | . In particular, the average of the measurements of BABAR [2, 3],Belle [4], BESIII [5], and CLEO [6] of the decays D → π ( K ) ℓν was used to extract theelements | V cd ( s ) | , as recently reported by the Particle Data Group (PDG) [7]. Such extractionof the CKM matrix elements from experiments requires theoretical knowledge of the hadronicform factors which take into account the nonperturbative quantum chromodynamics (QCD)effects.The elements | V cs | and | V cd | can also be determined indirectly by using the unitarityconstraint on the CKM matrix. This method was very useful in the past when the directmeasurements still suffered from large uncertainties, both experimental and theoretical.Once these matrix elements are determined, whether directly or indirectly, one can in reversestudy the strong interaction effects in various charm semileptonic channels to reveal thedecay dynamics. One can also test the predictions of different theoretical approaches, suchas the form factors and the branching fractions. In this manner, the study of semileptoniccharm decays can indirectly contribute to a more precise determination of other CKM matrixelements such as | V ub | , in the sense that constraints provided by charm decays can improvethe theoretical inputs needed for extracting | V ub | from exclusive charmless B semileptonicdecays.Recent progresses in experimental facilities and theoretical studies have made more andmore stringent tests of the standard model (SM) available in the charm sector and haveopened a new window through which to look for possible new physics effects beyond theSM. These tests include the CKM matrix unitarity, CP violations, isospin symmetry, andlepton flavor universality (LFU). Notably, the BESIII collaboration has reported recentlymeasurements of many semimuonic charm decays [8–10], some for the first time and somewith much improved precision. This paves the way to the search for signals of LFU violationsin these channels. In addition, the study of the decays D s → η ( ′ ) ℓ + ν ℓ provides informationabout the η − η ′ mixing angle and helps probe the interesting η − η ′ -glueball mixing [11, 12].From the theoretical point of view, the calculation of hadronic form factors plays a crucial2ole in the study of charm semileptonic decays. This calculation is carried out by nonper-turbative methods including lattice QCD (LQCD) [13–15], QCD sum rules [16–18], light-cone sum rules (LCSR) [19–25], and phenomenological quark models. Regarding the quarkmodels used in studies of semileptonic D decays, one can mention the Isgur-Scora-Grinstein-Wise (ISGW) model [26] and its updated version ISGW2 [27], the constituent quark model(CQM) [28], the relativistic quark model based on the quasipotential approach [29], the chiralquark model [30], the light-front quark model (LFQM) [31–33], and the model based on thecombination of heavy meson and chiral symmetries (HM χ T) [34, 35]. Several semileptonicdecay channels of the D ( s ) mesons were also studied in the large energy effective theory [36],chiral perturbation theory [37], the so-called chiral unitary approach ( χ UA) [38], and anew approach assuming pure heavy quark symmetry [39]. Recently, a simple expression for D → K semileptonic form factors was studied in Ref. [40]. We also mention here earlyattempts to account for flavor symmetry breaking in pseudoscalar meson decay constantsby the authors of Ref. [41]. It is worth noting that each method has only a limited range ofapplicability, and their combination will give a better picture of the underlined physics [28].In this paper, we compute the form factors of the semileptonic D ( D s ) decays in theframework of the covariant confined quark model (CCQM) [42–45]. To be more specific,we study the decays D + → ( D , ρ , ω, η, η ′ ) ℓ + ν ℓ , D + s → ( D , φ, K , K ∗ (892) , η, η ′ ) ℓ + ν ℓ ,and D → ρ − ℓ + ν ℓ . This paper follows our previous study [46] in which some of us haveconsidered the decays D → K ( ∗ ) ℓ + ν ℓ and D → πℓ + ν ℓ in great detail. Our aim is to providea systematic and independent study of D ( s ) semileptonic channels in the same theoreticalframework. This will shed more light on the theoretical study of the charm decays, especiallyon the shape of the corresponding form factors, since the CCQM predicts the form factors inthe whole physical range of momentum transfer without using any extrapolations. Besides,many of the studies mentioned in the previous paragraph were done about a decade ago,with the main focus on the branching fraction. In light of recent data, more up-to-datepredictions are necessary, not only for the branching fraction but also for other physicalobservables such as the forward-backward asymmetry and the lepton polarization. Finally,such a systematic study is necessary to test our model’s predictions and to better estimateits theoretical error.The rest of the paper is organized as follows. In Sec. II, we briefly provide the definitionsof the semileptonic matrix element and hadronic form factors. Then we give the decay3istribution in terms of the helicity amplitudes. In Sec III, we introduce the essentialingredients of the covariant confined quark model and describe in some detail the calculationof the form factors in our approach. Numerical results for the form factors, the decaybranching fractions, and other physical observables are presented in Sec. IV. We compareour findings with other theoretical approaches as well as experimental data including recentLQCD calculations and BESIII data. Finally, the conclusion is given in Sec. V. II. MATRIX ELEMENT AND DECAY DISTRIBUTION
Within the SM, the matrix element for semileptonic decays of the D ( s ) meson to a pseu-doscalar ( P ) or a vector ( V ) meson in the final state is written as M ( D ( s ) → ( P, V ) ℓ + ν ℓ ) = G F √ V cq h ( P, V ) | ¯ qO µ c | D ( s ) i [ ℓ + O µ ν ℓ ] , (1)where O µ = γ µ (1 − γ ), and q = d, s . The hadronic part in the matrix element is parametrizedby the invariant form factors which depend on the momentum transfer squared q betweenthe two mesons as follows: h P ( p ) | ¯ qO µ c | D ( s ) ( p ) i = F + ( q ) P µ + F − ( q ) q µ , h V ( p , ǫ ) | ¯ qO µ c | D ( s ) ( p ) i = ǫ † α M + M h − g µα P qA ( q ) + P µ P α A + ( q ) (2)+ q µ P α A − ( q ) + iε µαP q V ( q ) i , where P = p + p , q = p − p , and ǫ is the polarization vector of the vector meson V , sothat ǫ † · p = 0. The mesons are on shell: p = m D ( s ) = M , p = m P,V = M .For later comparison of the form factors with other studies, we relate our form factorsdefined in Eq. (2) to the well-known Bauer-Stech-Wirbel (BSW) form factors [47], namely, F + , for D ( s ) → P and A , , and V for D ( s ) → V . Note that in Ref. [47] the notation F was used instead of F + . The relations read e A = A + , e V = V, e F + = F + , e A = M − M M + M A , e F = F + + q M − M F − , (3) e A = M − M M (cid:16) A − A + − q M − M A − (cid:17) . d Γ( D ( s ) → ( P, V ) ℓ + ν ℓ ) dq = G F | V cq | | p | q π M (cid:16) − m ℓ q (cid:17) × h(cid:16) m ℓ q (cid:17) ( | H + | + | H − | + | H | ) + 3 m ℓ q | H t | i , (4)where | p | = λ / ( M , M , q ) / M is the momentum of the daughter meson in the restframe of the parent meson. Here, the helicity amplitudes for the decays D ( s ) → V ℓ + ν ℓ aredefined as H ± = 1 M + M ( − P qA ± M | p | V ) ,H = 1 M + M M p q (cid:2) − P q ( M − M − q ) A + 4 M | p | A + (cid:3) , (5) H t = 1 M + M M | p | M p q (cid:2) P q ( − A + A + ) + q A − (cid:3) . In the case of the decays D ( s ) → P ℓ + ν ℓ one has H ± = 0 , H = 2 M | p | p q F + , H t = 1 p q ( P qF + + q F − ) . (6)In order to study the lepton-mass effects, one can define several physical observables suchas the forward-backward asymmetry A ℓF B ( q ) and the longitudinal P ℓL ( q ) and transverse P ℓT ( q ) polarization of the charged lepton in the final state. This requires the angulardecay distribution, which was described elsewhere [50]. In short, one can write down theseobservables in terms of the helicity amplitudes as follows: A ℓF B ( q ) = − | H + | −| H − | +4 δ ℓ H H t (1 + δ ℓ ) P | H n | +3 δ ℓ | H t | , (7) P ℓL ( q ) = − (1 − δ ℓ ) P | H n | − δ ℓ | H t | (1 + δ ℓ ) P | H n | +3 δ ℓ | H t | , (8) P ℓT ( q ) = − π √ √ δ ℓ ( | H + | −| H − | − H H t )(1 + δ ℓ ) P | H n | +3 δ ℓ | H t | , (9)5here δ ℓ = m ℓ / q is the helicity-flip factor, and the index n runs through (+ , − , q range is better suited for experimental measurementswith low statistics. To calculate the average one has to multiply the numerator and denom-inator of e.g. Eq. (7) by the phase-space factor C ( q ) = | p | ( q − m ℓ ) /q and integratethem separately. These observables are sensitive to contributions of physics beyond the SMand can be used to test LFU violations [51–57]. III. FORM FACTORS IN THE COVARIANT CONFINED QUARK MODEL
In this study, the semileptonic form factors are calculated in the framework of theCCQM [42, 43]. The CCQM is an effective quantum field approach to the calculationof hadronic transitions. The model is built on the assumption that hadrons interact viaconstituent quark exchange only. This is realized by adopting a relativistic invariant La-grangian that describes the coupling of a hadron to its constituent quarks. This approachcan be used to treat not only mesons [58–62], but also baryons [63–65], tetraquarks [66–68],and other multiquark states [69] in a consistent way. For a detailed description of the modeland the calculation techniques we refer the reader to the references mentioned above. Welist below only several key features of the CCQM for completeness.For the simplest hadronic system, i.e. a meson M , the interaction Lagrangian is given by(10) L int = g M M ( x ) Z dx dx F M ( x ; x , x )¯ q ( x )Γ M q ( x ) + H . c ., where g M is the quark-meson coupling and Γ M the Dirac matrix. For a pseudoscalar (vector)meson Γ M = γ (Γ M = γ µ ). The vertex function F M ( x, x , x ) effectively describes the quarkdistribution in the meson and is given by F M ( x, x , x ) = δ (cid:16) x − X i =1 w i x i (cid:17) · Φ M (( x − x ) ) , (11)where w q i = m q i / ( m q + m q ) such that w + w = 1. The function Φ M depends on theeffective size of the meson. In order to avoid ultraviolet divergences in the quark loopintegrals, it is required that the Fourier transform of Φ M has an appropriate falloff behaviorin the Euclidean region. Since the final results are not sensitive to the specific form of Φ M ,for simplicity, we choose a Gaussian form as follows: e Φ M ( − p ) = Z dxe ipx Φ M ( x ) = e p / Λ M , (12)6 IG. 1: Quark model diagram for the D ( s ) -meson semileptonic decay. where the parameter Λ M characterizes the finite size of the meson.The coupling strength g M is determined by the compositeness condition Z M = 0 [70],where Z M is the wave function renormalization constant of the meson. This conditionensures the absence of any bare quark state in the physical mesonic state and, therefore,helps avoid double counting and provides an effective description of a bound state.In order to calculate the form factors, one first writes down the matrix element of thehadronic transition. In the CCQM, the hadronic matrix element is described by the one-loop Feynman diagram depicted in Fig. 1 and is constructed from the convolution of quarkpropagators and vertex functions as follows: h P ( p ) | ¯ qO µ c | D ( s ) ( p ) i = N c g D ( s ) g P Z d k (2 π ) i e Φ D ( s ) (cid:0) − ( k + w p ) (cid:1) e Φ P (cid:0) − ( k + w p ) (cid:1) × tr[ O µ S ( k + p ) γ S ( k ) γ S ( k + p )] , (13) h V ( p , ǫ ) | ¯ qO µ c | D ( s ) ( p ) i = N c g D ( s ) g V Z d k (2 π ) i e Φ D ( s ) (cid:0) − ( k + w p ) (cid:1) e Φ V (cid:0) − ( k + w p ) (cid:1) × tr[ O µ S ( k + p ) γ S ( k ) ǫ † S ( k + p )] , (14)where N c = 3 is the number of colors, w ij = m q j / ( m q i + m q j ), and S , are quark propagators,for which we use the Fock-Schwinger representation S i ( k ) = ( m q i + k ) ∞ Z dα i exp[ − α i ( m q i − k )] . (15)It should be noted that all loop integrations are carried out in Euclidean space.7 ABLE I: Meson size parameters in GeV.Λ D Λ D s Λ K Λ K ∗ Λ φ Λ ρ Λ ω Λ q ¯ qη Λ s ¯ sη Λ q ¯ qη ′ Λ s ¯ sη ′ m u/d m s m c m b λ Using various techniques described in our previous papers, a form factor F can be finallywritten in the form of a threefold integral F = N c g D ( s ) g ( P,V ) 1 /λ Z dt t Z dα Z dα δ (cid:16) − α − α (cid:17) f ( tα , tα ) , (16)where f ( tα , tα ) is the resulting integrand corresponding to the form factor F , and λ isthe so-called infrared cutoff parameter, which is introduced to avoid the appearance of thebranching point corresponding to the creation of free quarks and taken to be universal forall physical processes.The model parameters, namely, the meson size parameters, the constituent quark masses,and the infrared cutoff parameter are determined by fitting the radiative and leptonic decayconstants to experimental data or LQCD calculations. The model parameters required forthe calculation in this paper are listed in Tables I and II. Other parameters such as the massand lifetime of mesons and leptons, the CKM matrix elements, and physical constants aretaken from the recent report of the PDG [7]. In particular, we adopt the following valuesfor the CKM matrix elements: | V cd | = 0 .
218 and | V cs | = 0 . mathematica as well as fortran code.In the CCQM, the form factors are calculable in the entire range of momentum transfer.The calculated form factors are very well represented by the double-pole parametrization F ( q ) = F (0)1 − a ˆ s + b ˆ s , ˆ s = q m D ( s ) . (17)Our results for the parameters F (0), a , and b appearing in the parametrization Eq. (17) aregiven in Table III. 8 ABLE III: Parameters of the double-pole parametrization Eq. (17) for the form factors.
F F (0) a b F F (0) a bA D → ρ + A D → ρ − − .
74 1.11 0.22 A D → ρ − . V D → ρ A D → ω + A D → ω − − .
69 1.17 0.26 A D → ω − . V D → ω A D s → φ + A D s → φ − − .
95 1.20 0.26 A D s → φ − . V D s → φ A D s → K ∗ + A D s → K ∗ − − .
82 1.32 0.34 A D s → K ∗ − . V D s → K ∗ F D → η + F D → η − − .
37 1.02 0.18 F D → η ′ + F D → η ′ − − .
064 2.29 1.71 F D → D + F D → D − − .
026 6.32 8.37 F D s → η + F D s → η − − .
42 0.74 0.008 F D s → η ′ + F D s → η ′ − − .
28 0.92 0.009 F D s → K + F D s → K − − .
38 1.14 0.24 F D s → D + F D s → D − − .
34 6.79 8.91
It is worth noting here that in the calculation of the D ( s ) → η ( ′ ) form factors one hasto take into account the mixing of the light and the s -quark components. By assuming m u = m d ≡ m q , the quark content can be written as ηη ′ = − sin δ cos δ − cos δ sin δ q ¯ qs ¯ s , q ¯ q ≡ u ¯ u + d ¯ d √ . (18)The angle δ is defined by δ = θ P − θ I , where θ I = arctan(1 / √
2) is the ideal mixing angle.We adopt the value θ P = − . ◦ from Ref. [71]. IV. RESULTS AND DISCUSSIONA. Form factors
In this subsection, we compare our form factors with those from other theoretical ap-proaches and from experimental measurements. For convenience, we relate all form factorsfrom different studies to the BSW form factors, as mentioned in Sec. II. In the SM, thehadronic matrix element between two mesons is parametrized by two form factors ( F + and F ) for the P → P ′ transition and four form factors ( A , , and V ) for the P → V one.9owever, in semileptonic decays of D and D s mesons, the form factors F and A are lessinteresting because their contributions to the decay rate vanish in the zero lepton-mass limit(the tau mode is kinematically forbidden). Therefore, we focus more on the form factors F + , A , A , and V . We note that the uncertainties of our form factors mainly come fromthe errors of the model parameters. These parameters are determined from a least-squaresfit to available experimental data and some lattice calculations. We have observed that theerrors of the fitted parameters are within 10%. We then calculated the propagation of theseerrors on the form factors and found the uncertainties on the form factors to be of order 20%at small q and 30% at high q . At maximum recoil q = 0, the form factor uncertaintiesare of order 15%.We start with the D ( s ) → P transition form factor F + ( q ). In Table IV, we compare themaximum-recoil values F + ( q = 0) with other theoretical approaches. It is observed thatour results are in good agreement with other quark models, especially with the CQM [28]and the LFQM [32]. Besides, quark model predictions for F + (0) of the D ( s ) → η ( ′ ) channelsare in general higher than those obtained by LCSR [22, 24] and LQCD [14]. This suggeststhat more studies of these form factors are needed. For example, a better LQCD calculationof F + (0) is expected. Note that the authors of Ref. [14] considered their LQCD calculationas a pilot study rather than a conclusive one. TABLE IV: Comparison of F + (0) for D ( s ) → P transitions. D → η D → η ′ D s → η D s → η ′ D s → K Present 0 . ± .
10 0 . ± .
11 0 . ± .
12 0 . ± .
11 0 . ± . . . . . . . . . . . . . M π =470 MeV [14] . . . . . . . . . . . LQCD M π =370 MeV [14] . . . . . . . . . . . LCSR [22] 0 . ± .
051 0 . ± .
105 0 . ± .
033 0 . ± . . . . LCSR [24] 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . . . Regarding the D ( s ) → V transition form factors A , A , and V , it is more interesting tocompare their ratios at maximum recoil. The ratios are defined as follows: r = A ( q = 0) A ( q = 0) , r V = V ( q = 0) A ( q = 0) . (19)In Table V, we compare these ratios with the world average given by the PDG [7] and withother theoretical results obtained in CQM [28], LFQM [32], HM χ T [35], and LQCD [13]. Our10
ABLE V: Ratios of the D ( s ) → V transition form factors at maximum recoil.Channel Ratio Present PDG [7] LQCD [13] CQM [28] LFQM [32] HM χ T [35] D → ρ r . ± .
19 0 . ± . . . . r V . ± .
25 1 . ± . . . . D + → ω r . ± .
19 1 . ± . . . . . . . r V . ± .
25 1 . ± . . . . . . . D + s → φ r . ± .
20 0 . ± .
11 0 . r V . ± .
27 1 . ± .
08 1 . D + s → K ∗ r . ± . . . . . . . r V . ± . . . . . . . results for the form factor ratios r and r V agree well with the PDG data within uncertaintyexcept for the ratio r V ( D + s → φ ), for which our prediction is much lower than that fromPDG. Note that our prediction r V ( D + s → φ ) = 1 .
34 is close to the value 1 .
42 from the
FIG. 2: Form factor F + ( q ) for D + s → K in our model, LFQM [32], LCSR [20], and CQM [28]. LFQM [32]. It is also seen that for most cases, the HM χ T predictions [35] for the ratios at q = 0 are largely different from the PDG values, demonstrating the fact that this model ismore suitable for the high q region.In order to have a better picture of the form factors in the whole q range 0 ≤ q ≤ q max = ( m D ( s ) − m P/V ) we plot in Figs. 2–5 their q dependence from various studies. It isvery interesting to note that, in all cases, our form factors are close to those obtained in thecovariant LFQM [32], and this is not for the first time such a good agreement is observed.In a previous study of the semileptonic decays B c → J/ψ ( η c ) ℓν [72] it was seen that thecorresponding form factors agree very well between our model and the covariant LFQM [73].This suggests that a comparison of the two models in more detail may be fruitful. It is also11 IG. 3: Form factor F + ( q ) for D +( s ) → η ( ′ ) in our model, LCSR [20, 22, 24], and CQM [28]. worth noting that the HM χ T [35] prediction for the form factor A ( q ) is systematicallymuch higher than that from other theoretical calculations.Very recently, the ETM collaboration has provided the lattice determination [74] forthe full set of the form factors characterizing the semileptonic D → π ( K ) ℓν and rare D → π ( K ) ℓℓ decays within and beyond the SM, when an additional tensor coupling is considered.As mentioned before, the decays D → π ( K ) ℓν have been studied in our model already [46].However, we compute the D → π ( K ) ℓν form factors including the tensor one in this paper,in order to compare with the recent ETM results. This demonstrates the fidelity of theCCQM predictions for the hadronic form factors and helps us better estimate the theoreticaluncertainties of our model. Moreover, the tensor and scalar form factors are essential for thestudy of possible new physics in these decays [for more detail we refer to a similar calculationof the full set of B → D ( ∗ ) and B → π ( ρ ) form factors in our model [75, 76]].The new tensor form factor is defined by h P ( p ) | ¯ qσ µν (1 − γ ) c | D ( p ) i = iF T ( q ) M + M (cid:16) P µ q ν − P ν q µ + iε µνP q (cid:17) . (20)12 IG. 4: Form factors for D + s → φ (left) and D + s → K ∗ (892) (right) in our model, LFQM [32],HM χ T [35], and CQM [28]. IG. 5: Form factors for D → ρ (left) and D + → ω (right) in our model, LFQM [32], HM χ T [35],CQM [28], and CLEO data [77]. F ( q ) by using the form factors F + ( q ) and F − ( q ) defined in Eq. (2),with the help of the relation F ( q ) = F + ( q ) + q M − M F − ( q ) . (21)Meanwhile, the ETM collaboration directly calculated the scalar matrix element h P ( p ) | ¯ qc | D ( p ) i and then determined F ( q ) using the equation of motion. In this way,the final result becomes sensitive to the quark mass difference.In Fig. 6 we compare the form factors F ( q ), F + ( q ), and F T ( q ) of the D → π ( K ) ℓν transitions with those obtained by the ETM collaboration. It is seen that our F ( q ) agreeswell with the ETM only in the low q region. However, our results for F + ( q ) are very closeto those of the ETM. Note that the determination of F + ( q ) by the ETM is dependent on F ( q ). It is interesting that the tensor form factors between the two studies are in perfectagreement. Even though this form factor does not appear within the SM, this agreement hasan important meaning because, in both approaches, the tensor form factor is determineddirectly from the corresponding matrix element without any additional assumptions. InTable VI, we present the values of the form factors and their ratios at maximum recoil. Onesees that our results agree with the ETM calculation within uncertainty. TABLE VI: D → π ( K ) ℓν form factors and their ratios at q = 0. f Dπ + (0) f DK + (0) f DπT (0) f DKT (0) f DπT (0) /f Dπ + (0) f DKT (0) /f DK + (0)Present 0.63 0.78 0.53 0.70 0.84 0.90ETM [74] 0 . . . . . . B. Branching fractions and other observables
In Tables VII and VIII, we summarize our predictions for the semileptonic branchingfractions of the D and D s mesons, respectively. For comparison, we also list results ofother theoretical calculations and the most recent experimental data given by the CLEOand BESIII collaborations. Note that the uncertainties of our predictions for the branchingfractions and other polarization observables are of order 50%, taking into account only themain source of uncertainties related to the form factors.15 IG. 6: D → π ( K ) ℓν form factors obtained in our model (solid lines) and in lattice calculation(dots with error bars) by the ETM collaboration [74]. In general, our results for the branching fractions are consistent with experimental dataas well as with other theoretical calculations. It is worth mentioning that, for such a large setof decays considered in this study, our branching fractions agree very well with all availableexperimental data except for one channel, the D + s → K ℓ + ν ℓ . In this case, our prediction isnearly twice as small as the CLEO central value [78] and about 30% smaller than the LFQMprediction [33]. 16 ABLE VII: Branching fractions of D + ( D )-meson semileptonic decays.Channel Unit Present Other Reference Data Reference D → ρ − e + ν e − χ UA [38] 1 . ± . ± .
039 BESIII [79]1 . +0 . − . ± .
006 LCSR [25] 1 . ± . ± .
10 CLEO [77]2.0 HM χ T [35] D → ρ − µ + ν µ − χ UA [38] D + → ρ e + ν e − χ UA [38] 1 . ± . ± .
061 BESIII [79]2 . +0 . − . ± .
015 LCSR [25] 2 . ± . +0 . − . CLEO [77]2.5 HM χ T [35] D + → ρ µ + ν µ − χ UA [38] 2 . ± . D + → ωe + ν e − χ UA [38] 1 . ± . ± .
08 BESIII [80]2.5 HM χ T [35] 1 . ± . ± .
07 CLEO [77]2.1 ± D + → ωµ + ν µ − χ UA [38]2.0 ± D + → ηe + ν e − ± . ± . ± .
51 BESIII [81]24 . ± .
26 LCSR [22] 11 . ± . ± . . ± .
98 LCSR [24] D + → ηµ + ν µ − ± D + → η ′ e + ν e − ± . ± . ± .
13 BESIII [81]3 . ± .
77 LCSR [22] 2 . ± . ± .
07 CLEO [82]1 . ± .
17 LCSR [24] D + → η ′ µ + ν µ − ± We also give prediction for the ratio Γ( D → ρ − e + ν e ) / D + → ρ e + ν e ) which shouldbe equal to unity in the SM, assuming isospin invariance. Our calculation yields 0 . . ± . +0 . − . [77]. Besides, our ratio of branchingfractions B ( D + s → η ′ e + ν e ) / B ( D + s → ηe + ν e ) = 0 .
37 coincides with the result 0 . ± . . ± .
14 by BESIII [84]. Finally,we predict B ( D + → η ′ e + ν e ) / B ( D + → ηe + ν e ) = 0 .
21, which agrees very well with thevalues 0 . ± .
05 and 0 . ± .
05 we got from experimental data by CLEO [82] and BE-SIII [81], respectively. It is worth mentioning here that very recently, the BESIII collabora-tion has reported their measurement of B ( D → K − µ + ν µ ) [85] with significantly improvedprecision. In their paper, they also approved the prediction of our model for the ratio B ( D → K − µ + ν µ ) / B ( D → K − e + ν e ) provided in Ref. [46].In Table IX, we present our results for the semileptonic decays D +( s ) → D e + ν e , whichare rare in the SM due to phase-space suppression. These decays are of particular interest17 ABLE VIII: Branching fractions of D s -meson semileptonic decays (in %).Channel Present Other Reference Data Reference D + s → φe + ν e χ UA [38] 2 . ± . ± .
09 BESIII [9]3.1 ± . ± . ± . ± . BABAR [86]2.4 HM χ T [35] 2 . ± . ± .
08 CLEO [78] D + s → φµ + ν µ χ UA [38]2.9 ± . ± . ± .
09 BESIII [9] D + s → K e + ν e ± . ± . ± .
03 CLEO [78] D + s → K µ + ν µ ± D + s → K ∗ e + ν e χ UA [38] 0 . ± . ± .
01 CLEO [78]0.19 ± χ T [35] D + s → K ∗ µ + ν µ χ UA [38]0.19 ± D + s → ηe + ν e ± . ± . ± .
08 BESIII [84]2 . ± .
32 LCSR [22] 2 . ± . ± .
19 CLEO [78]2 . ± .
28 LCSR [24] D + s → ηµ + ν µ ± . ± . ± .
11 BESIII [9] D + s → η ′ e + ν e ± . ± . ± .
05 BESIII [84]0 . ± .
23 LCSR [22] 0 . ± . ± .
06 CLEO [78]0 . ± .
14 LCSR [24] D + s → η ′ µ + ν µ ± . ± . ± .
07 BESIII [9]TABLE IX: Semileptonic branching fractions for D +( s ) → D ℓ + ν ℓ .Channel Present Other Reference Data Reference D + → D e + ν e . × − . × − [87] < . × − BESIII [88]2 . × − [89] D + s → D e + ν e . × − (2 . ± . × − [87] . . . . . . . × − [89] since they are induced by the light quark decay, while the heavy quark acts as the spectator.Besides, the small phase space helps reduce the theoretical errors. The first experimentalconstraint on the branching fraction B ( D + → D e + ν e ) was recently obtained by the BESIIIcollaboration [88]. However, the experimental upper limit is still far above the SM predic-tions. The branching fractions obtained in our model are comparable with other theoreticalcalculations using the flavor SU(3) symmetry in the light quark sector [87, 89].Finally, in Table X we list our predictions for the forward-backward asymmetry hA ℓF B i ,18 ABLE X: Forward-backward asymmetry and lepton polarization components. hA eF B i (cid:10) A µF B (cid:11) h P eL i (cid:10) P µL (cid:11) h P eT i (cid:10) P µT (cid:11) D → ρ − ℓ + ν ℓ .
21 0 . − . − .
92 1 . × − D + → ρ ℓ + ν ℓ .
22 0 . − . − .
92 1 . × − D + → ωℓ + ν ℓ .
21 0 . − . − .
92 1 . × − D + → ηℓ + ν ℓ − . × − − . − . − .
83 2 . × − D + → η ′ ℓ + ν ℓ − . × − − . − . − .
70 4 . × − D + → D ℓ + ν ℓ − . . . . − . . . . . . .D + s → φℓ + ν ℓ .
18 0 . − . − .
91 1 . × − D + s → K ∗ ℓ + ν ℓ .
22 0 . − . − .
92 1 . × − D + s → K ℓ + ν ℓ − . × − − . − . − .
86 2 . × − D + s → ηℓ + ν ℓ − . × − − . − . − .
84 2 . × − D + s → η ′ ℓ + ν ℓ − . × − − . − . − .
75 3 . × − D + s → D ℓ + ν ℓ − . × − . . . − . . . . . . . the longitudinal polarization h P ℓL i , and the transverse polarization h P ℓT i of the charged leptonin the final state. It is seen that, for the P → V transitions, the lepton-mass effect in hA ℓF B i is small, resulting in a difference of only 10%–15% between the corresponding electron andmuon modes. For the P → P ′ transitions, hA µF B i are about 10 times larger than hA eF B i .This is readily seen from Eq. (7): for P → P ′ transitions the two helicity amplitudes H ± vanish and the forward-backward asymmetry is proportional to the lepton mass squared.Regarding the longitudinal polarization, the difference between h P µL i and h P eL i is 10%–30%.One sees that the lepton-mass effect in the transverse polarization is much more significantthan that in the longitudinal one. This is true for both P → P ′ and P → V transitions. Notethat the values of hA eF B i and h P eL ( T ) i for the rare decays D +( s ) → D e + ν e are quite different incomparison with other P → P ′ transitions due to their extremely small kinematical regions. V. SUMMARY AND CONCLUSION
We have presented a systematic study of the D and D s semileptonic decays within theframework of the CCQM. All the relevant form factors are calculated in the entire rangeof momentum transfer squared. We have also provided a detailed comparison of the formfactors with other theoretical predictions and, in some cases, with available experimentaldata. In particular, we have observed a good agreement with the form factors obtained inthe covariant LFQM, for all decays. It is worth noting that our tensor form factors for the19 → π ( K ) ℓν decays are in perfect agreement with the recent LQCD calculation by theETM collaboration [74].We have given our predictions for the semileptonic branching fractions and their ratios. Ingeneral, our results are in good agreement with other theoretical approaches and with recentexperimental data obtained by BABAR , CLEO, and BESIII. In all cases, our predictionsfor the branching fractions agree with experimental data within 10%, except for the D + s → K ℓ + ν ℓ channel. Our predictions for the ratios of branching fractions are in full agreementwith experimental data. To conclude, we have provided the first ever theoretical predictionsfor the forward-backward asymmetries, and lepton longitudinal and transverse polarizations,which are important for future experiments. Acknowledgments
J. N. P. acknowledges financial support from University Grants Commission of Indiaunder Major Research Project F.No.42-775/2013(SR). P. S. acknowledges support from Is-tituto Nazionale di Fisica Nucleare, I.S. QFT HEP. M. A. I., J. G. K., and C. T. T. thankHeisenberg-Landau Grant for providing support for their collaboration. M. A. I. acknowl-edges financial support of PRISMA Cluster of Excellence at University of Mainz. N. R. S.thanks Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Researchfor warm hospitality during Helmholtz-DIAS International Summer School “Quantum FieldTheory at the Limits: from Strong Field to Heavy Quarks” where this work was initiated.C. T. T. acknowledges support from Duy Tan University during the beginning stage of thiswork. M. A. I. and C. T. T. appreciate warm hospitality of Mainz Institute for TheoreticalPhysics at University of Mainz, where part of this work was done.
Note added. —Recently, we became aware of the paper [90] where the BESIII collaborationreported their new measurements of the branching fractions for the decays D + s → K e + ν e and D + s → K ∗ e + ν e with improved precision. They also obtained for the first time the valuesof the form factors at maximum recoil. Our predictions for the branching fraction B ( D + s → K ∗ e + ν e ) as well as the form factor parameters f D s K + (0), r D s K ∗ V (0), and r D s K ∗ (0) agree withthe new BESIII results. Regarding their result B ( D + s → K e + ν e ) = (3 . ± . × − , thecentral value is closer to our prediction, in comparison with the CLEO result [78]. However,20he BESIII result is still at 1 σ larger than ours. [1] J. D. Richman and P. R. Burchat, Rev. Mod. Phys. , 893 (1995) [hep-ph/9508250].[2] J. P. Lees et al. ( BABAR
Collaboration), Phys. Rev. D , 052022 (2015) [arXiv:1412.5502].[3] B. Aubert et al. ( BABAR
Collaboration), Phys. Rev. D , 052005 (2007) [arXiv:0704.0020].[4] L. Widhalm et al. (Belle Collaboration), Phys. Rev. Lett. , 061804 (2006) [hep-ex/0604049].[5] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D , 072012 (2015) [arXiv:1508.07560].[6] D. Besson et al. (CLEO Collaboration), Phys. Rev. D , 032005 (2009) [arXiv:0906.2983].[7] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D , 030001 (2018).[8] M. Ablikim et al. (BESIII Collaboration), Eur. Phys. J. C , 369 (2016) [arXiv:1605.00068].[9] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D , 012006 (2018) [arXiv:1709.03680].[10] M. Ablikim et al. (BESIII Collaboration), arXiv:1802.05492 [hep-ex].[11] V. V. Anisovich, D. V. Bugg, D. I. Melikhov, and V. A. Nikonov, Phys. Lett. B , 166(1997) [hep-ph/9702383].[12] C. Di Donato, G. Ricciardi, and I. Bigi, Phys. Rev. D , 013016 (2012) [arXiv:1105.3557].[13] G. C. Donald, C. T. H. Davies, J. Koponen, and G. P. Lepage (HPQCD Collaboration), Phys.Rev. D , 074506 (2014) [arXiv:1311.6669].[14] G. S. Bali, S. Collins, S. D¨urr, and I. Kanamori, Phys. Rev. D , 014503 (2015)[arXiv:1406.5449].[15] S. Aoki et al. , Eur. Phys. J. C , 112 (2017) [arXiv:1607.00299].[16] P. Ball, Phys. Rev. D , 3190 (1993) [hep-ph/9305267].[17] P. Colangelo and F. De Fazio, Phys. Lett. B , 78 (2001) [hep-ph/0107137].[18] D. S. Du, J. W. Li, and M. Z. Yang, Eur. Phys. J. C , 173 (2004) [hep-ph/0308259].[19] A. Khodjamirian, R. Ruckl, S. Weinzierl, C. W. Winhart, and O. I. Yakovlev, Phys. Rev. D , 114002 (2000) [hep-ph/0001297].[20] Y. L. Wu, M. Zhong, and Y. B. Zuo, Int. J. Mod. Phys. A , 6125 (2006) [hep-ph/0604007].[21] K. Azizi, R. Khosravi, and F. Falahati, J. Phys. G , 095001 (2011) [arXiv:1011.6046].[22] N. Offen, F. A. Porkert, and A. Sch¨afer, Phys. Rev. D , 034023 (2013) [arXiv:1307.2797].[23] U. G. Meißner and W. Wang, Phys. Lett. B , 336 (2014) [arXiv:1312.3087].[24] G. Duplancic and B. Melic, J. High Energy Phys. 11 (2015) 138 [arXiv:1508.05287].
25] H. B. Fu, X. Yang, R. L¨u, L. Zeng, W. Cheng, and X. G. Wu, arXiv:1808.06412.[26] N. Isgur, D. Scora, B. Grinstein, and M. B. Wise, Phys. Rev. D , 799 (1989).[27] D. Scora and N. Isgur, Phys. Rev. D , 2783 (1995) [hep-ph/9503486].[28] D. Melikhov and B. Stech, Phys. Rev. D , 014006 (2000) [hep-ph/0001113].[29] R. N. Faustov, V. O. Galkin, and A. Y. Mishurov, Phys. Rev. D , 1391 (1996)[hep-ph/9506220].[30] T. Palmer and J. O. Eeg, Phys. Rev. D , 034013 (2014) [arXiv:1306.0365].[31] Z. T. Wei, H. W. Ke, and X. F. Yang, Phys. Rev. D , 015022 (2009) [arXiv:0905.3069].[32] R. C. Verma, J. Phys. G , 025005 (2012) [arXiv:1103.2973].[33] H. Y. Cheng and X. W. Kang, Eur. Phys. J. C , 587 (2017); , 863(E) (2017)[arXiv:1707.02851].[34] S. Fajfer and J. F. Kamenik, Phys. Rev. D , 014020 (2005) [hep-ph/0412140].[35] S. Fajfer and J. F. Kamenik, Phys. Rev. D , 034029 (2005) [hep-ph/0506051].[36] J. Charles, A. Le Yaouanc, L. Oliver, O. Pene, and J. C. Raynal, Phys. Rev. D , 014001(1999) [hep-ph/9812358].[37] J. Bijnens and I. Jemos, Nucl. Phys. B846 , 145 (2011) [arXiv:1011.6531].[38] T. Sekihara and E. Oset, Phys. Rev. D , 054038 (2015) [arXiv:1507.02026].[39] L. R. Dai, X. Zhang, and E. Oset, Phys. Rev. D , 036004 (2018) [arXiv:1806.09583].[40] T. N. Pham, Int. J. Mod. Phys. A , 1850160 (2018) [arXiv:1801.09534].[41] S. S. Gershtein and M. Y. Khlopov, Pisma Zh. Eksp. Teor. Fiz. , 374 (1976); M. Y. Khlopov,Sov. J. Nucl. Phys. , 583 (1978) [Yad. Fiz. , 1134 (1978)].[42] G. V. Efimov and M. A. Ivanov, Int. J. Mod. Phys. A , 2031 (1989); The Quark ConfinementModel of Hadrons (CRC Press, Boca Raton, 1993).[43] T. Branz, A. Faessler, T. Gutsche, M. A. Ivanov, J. G. K¨orner, and V. E. Lyubovitskij, Phys.Rev. D , 034010 (2010) [arXiv:0912.3710].[44] M. A. Ivanov, J. G. K¨orner, S. G. Kovalenko, P. Santorelli, and G. G. Saidullaeva, Phys. Rev.D , 034004 (2012) [arXiv:1112.3536].[45] T. Gutsche, M. A. Ivanov, J. G. K¨orner, V. E. Lyubovitskij, and P. Santorelli, Phys. Rev. D , 074013 (2012) [arXiv:1207.7052].[46] N. R. Soni and J. N. Pandya, Phys. Rev. D , 016017 (2017) [arXiv:1706.01190].[47] M. Wirbel, B. Stech, and M. Bauer, Z. Phys. C , 637 (1985).
48] J. G. K¨orner and G. A. Schuler, Z. Phys. C , 511 (1988); , 690(E) (1989); Phys. Lett. B , 306 (1989); Z. Phys. C , 93 (1990).[49] T. Gutsche, M. A. Ivanov, J. G. K¨orner, V. E. Lyubovitskij, P. Santorelli, and N. Habyl, Phys.Rev. D , 074001 (2015); , 119907(E) (2015) [arXiv:1502.04864].[50] M. A. Ivanov, J. G. K¨orner, and C. T. Tran, Phys. Rev. D , 114022 (2015)[arXiv:1508.02678].[51] S. Bifani, S. Descotes-Genon, A. Romero Vidal and M. H. Schune, arXiv:1809.06229 [hep-ex].[52] M. A. Ivanov, J. G. K¨orner, and C. T. Tran, Phys. Rev. D , 036021 (2017)[arXiv:1701.02937].[53] Q. Y. Hu, X. Q. Li and Y. D. Yang, arXiv:1810.04939 [hep-ph].[54] P. Asadi, M. R. Buckley and D. Shih, arXiv:1810.06597 [hep-ph].[55] N. Rajeev and R. Dutta, Phys. Rev. D , 055024 (2018) [arXiv:1808.03790].[56] F. Feruglio, P. Paradisi, and O. Sumensari, arXiv:1806.10155 [hep-ph].[57] R. Alonso, J. Martin Camalich, and S. Westhoff, arXiv:1811.05664 [hep-ph].[58] M. A. Ivanov and P. Santorelli, Phys. Lett. B , 248 (1999) [hep-ph/9903446].[59] A. Faessler, T. Gutsche, M. A. Ivanov, J. G. K¨orner, and V. E. Lyubovitskij, Eur. Phys. J.direct , 18 (2002) [hep-ph/0205287].[60] M. A. Ivanov, J. G. K¨orner, and P. Santorelli, Phys. Rev. D , 054024 (2006)[hep-ph/0602050].[61] M. A. Ivanov and C. T. Tran, Phys. Rev. D , 074030 (2015) [arXiv:1701.07377].[62] S. Dubniˇcka, A. Z. Dubniˇckov´a, M. A. Ivanov, A. Liptaj, P. Santorelli, and C. T. Tran,arXiv:1808.06261.[63] T. Gutsche, M. A. Ivanov, J. G. K¨orner, V. E. Lyubovitskij, and P. Santorelli, Phys. Rev. D , 074031 (2013) [arXiv:1301.3737].[64] T. Gutsche, M. A. Ivanov, J. G. K¨orner, and V. E. Lyubovitskij, Phys. Rev. D , 054013(2017) [arXiv:1708.00703].[65] T. Gutsche, M. A. Ivanov, J. G. K¨orner, V. E. Lyubovitskij, P. Santorelli, and C. T. Tran,Phys. Rev. D , 053003 (2018) [arXiv:1807.11300].[66] S. Dubnicka, A. Z. Dubnickova, M. A. Ivanov, J. G. K¨orner, P. Santorelli, and G. G. Saidul-laeva, Phys. Rev. D , 014006 (2011) [arXiv:1104.3974].[67] F. Goerke, T. Gutsche, M. A. Ivanov, J. G. K¨orner, V. E. Lyubovitskij, and P. Santorelli, hys. Rev. D , 094017 (2016) [arXiv:1608.04656].[68] F. Goerke, T. Gutsche, M. A. Ivanov, J. G. K¨orner, and V. E. Lyubovitskij, Phys. Rev. D ,054028 (2017) [arXiv:1707.00539].[69] T. Gutsche, M. A. Ivanov, J. G. K¨orner, V. E. Lyubovitskij, and K. Xu, Phys. Rev. D ,114004 (2017) [arXiv:1710.02357].[70] A. Salam, Nuovo Cimento , 224 (1962); S. Weinberg, Phys. Rev. , 776 (1963).[71] T. Feldmann, P. Kroll, and B. Stech, Phys. Rev. D , 114006 (1998) [hep-ph/9802409].[72] C. T. Tran, M. A. Ivanov, J. G. K¨orner, and P. Santorelli, Phys. Rev. D , 054014 (2018)[arXiv:1801.06927].[73] W. Wang, Y. L. Shen, and C. D. Lu, Phys. Rev. D , 054012 (2009), [arXiv:0811.3748].[74] V. Lubicz, L. Riggio, G. Salerno, S. Simula, and C. Tarantino (ETM Collaboration),Phys. Rev. D , 054514 (2017) [arXiv:1706.03017]; Phys. Rev. D , 014516 (2018)[arXiv:1803.04807].[75] M. A. Ivanov, J. G. K¨orner, and C. T. Tran, Phys. Rev. D , 094028 (2016)[arXiv:1607.02932].[76] M. A. Ivanov, J. G. K¨orner, and C. T. Tran, Phys. Part. Nucl. Lett. , 669 (2017).[77] S. Dobbs et al. (CLEO Collaboration), Phys. Rev. Lett. , 131802 (2013) [arXiv:1112.2884].[78] J. Hietala, D. Cronin-Hennessy, T. Pedlar, and I. Shipsey, Phys. Rev. D , 012009 (2015)[arXiv:1505.04205].[79] M. Ablikim et al. (BESIII Collaboration), arXiv:1809.06496 [hep-ex].[80] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D , 071101 (2015) [arXiv:1508.00151].[81] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D , 092009 (2018) [arXiv:1803.05570].[82] J. Yelton et al. (CLEO Collaboration), Phys. Rev. D , 032001 (2011) [arXiv:1011.1195].[83] J. Yelton et al. (CLEO Collaboration), Phys. Rev. D , 052007 (2009) [arXiv:0903.0601].[84] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D , 112003 (2016) [arXiv:1608.06484].[85] M. Ablikim et al. (BESIII Collaboration), arXiv:1810.03127 [hep-ex].[86] B. Aubert et al. ( BABAR
Collaboration), Phys. Rev. D , 051101 (2008) [arXiv:0807.1599].[87] H. B. Li and M. Z. Yang, Eur. Phys. J. C , 841 (2009) [arXiv:0709.0979].[88] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D , 092002 (2017) [arXiv:1708.06856].[89] S. Faller and T. Mannel, Phys. Lett. B , 653 (2015) [arXiv:1503.06088].[90] M. Ablikim et al. (BESIII Collaboration), arXiv:1811.02911 [hep-ex].(BESIII Collaboration), arXiv:1811.02911 [hep-ex].