World Average Values for Semi-leptonic D decays
(updated 21 Sep 2022)

People working on this:   Hailong Ma

Semileptonic decays of DD mesons involve the interaction of a leptonic current with a hadronic current. The latter is nonperturbative and cannot be calculated from first principles; thus it is usually parameterized in terms of form factors. The transition matrix element is written

=iGF2VcqLμHμ,{\mathcal{M}} = -i\,\frac{G_F}{\sqrt{2}}\,V^{}_{cq}\,L^\mu H_\mu\,,

where GFG_F is the Fermi constant and VcqV^{}_{cq} is a CKM matrix element. The leptonic current LμL^\mu is evaluated directly from the lepton spinors and has a simple structure; this allows one to extract information about the form factors (in HμH^{}_\mu) from data on semileptonic decays [Phys. Lett. B 633 (2006) 61]. Conversely, because there are no strong final-state interactions between the leptonic and hadronic systems, semileptonic decays for which the form factors can be calculated allow one to determine |Vcq||V^{}_{cq}| [Prog. Theor. Phys. 49 (1973) 652].

1. DP𝓁ν𝓁D\rightarrow P \mathcal{l} \nu_{\mathcal{l}} decays

When the final state hadron is a pseudoscalar, the hadronic current is given by [eConf C 060409 (2006) 027] Hμ=P(p)|qγμc|D(p)=f+(q2)[(p+p)μmD2mP2q2qμ]+f0(q2)mD2mP2q2qμ,H_\mu = \left< P(p) | \bar{q}\gamma_\mu c | D(p') \right> \ =\ f_+(q^2)\left[ (p' + p)_\mu -\frac{m_D^2-m_P^2}{q^2}q_\mu\right] + f_0(q^2)\frac{m_D^2-m_P^2}{q^2}q_\mu\,, where mDm_D and pp' are the mass and four momentum of the parent DD meson, mPm_P and pp are those of the daughter meson, f+(q2)f_+(q^2) and f0(q2)f_0(q^2) are form factors, and q=ppq = p' - p. Kinematics require that f+(0)=f0(0)f_+(0) = f_0(0). The contraction qμLμq_\mu L^\mu results in terms proportional to m𝓁m^{}_\mathcal{l}[Phys. Rev. D 41 (1990) 142], and thus for 𝓁=e\mathcal{l}=e the terms proportional to qμq_\mu in the previous equation are negligible and only the f+(q2)f_+(q^2) vector form factor is relevant. The corresponding differential partial width is dΓ(DPeνe)dq2dcosθe=GF2|Vcq|232π3p*3|f+(q2)|2sinθe2,\frac{d\Gamma(D \to P e \nu_e)}{dq^2\, d\cos\theta_e} = \frac{G_F^2|V_{cq}|^2}{32\pi^3} p^{*\,3}|f_{+}(q^2)|^2\sin\theta^2_e\,, where p*=[(mD2(mP+q))2(mD2(mPq))]1/22mD{p^*}=\frac{\left [\left (m_D^2-(m_P+q)\right )^2\left (m_D^2-(m_P-q)\right )\right ]^{1/2}}{2 m_D} is the magnitude of the momentum of the final state hadron in the DD rest frame, and θe\theta_e is the angle of the electron in the eνe\nu rest frame with respect to the direction of the pseudoscalar meson in the DD rest frame.

1.1 Form factor parameterizations

The form factor is traditionally parameterized with an explicit pole and a sum of effective poles:

f+(q2)=f+(0)(1α)[(11q2/mpole2)+k=1Nρk1q2/(γkmpole2)],f_+(q^2) = \frac{f_+(0)}{(1-\alpha)} [(\frac{1}{1- q^2/m^2_{\mathrm{pole}}})+\sum_{k=1}^{N}\frac{\rho_k}{1- q^2/(\gamma_k\,m^2_{\mathrm{pole}})} ]\,,

where ρk\rho_k and γk\gamma_k are expansion parameters and α\alpha is a parameter that normalizes the form factor at q2=0q^2=0, f+(0)f_+(0). The parameter mpolem_{{\mathrm{pole}}} is the mass of the lowest-lying cqc\bar{q} resonance with the vector quantum numbers; this is expected to provide the largest contribution to the form factor for the cqc\rightarrow q transition. The sum over NN gives the contribution of higher mass states. For example, for DπD\to\pi transitions the dominant resonance is expected to be the D*(2010)D^*(2010), and thus mpole=mD*(2010)m^{}_{\mathrm{pole}}=m^{}_{D^*(2010)}. For DKD\to K transitions, the dominant resonance is expected to be the Ds*(2112)D^{*}_s(2112), and thus mpole=mDs*(2112)m^{}_{\mathrm{pole}}=m^{}_{D^{*}_{s}(2112)}.

(1) Simple pole

The equation can be simplified by neglecting the sum over effective poles, leaving only the explicit vector meson pole. This approximation is referred to as “nearest pole dominance” or “vector-meson dominance.” The resulting parameterization is f+(q2)=f+(0)(1q2/mpole2).f_+(q^2) = \frac{f_+(0)}{(1-q^2/m^2_{\mathrm{pole}})}\,. \label{SimplePole} However, values of mpolem_{{\mathrm{pole}}} that give a good fit to the data do not agree with the expected vector meson masses [eConf C 060409 (2006) 027]. To address this problem, the “modified pole” or Becirevic-Kaidalov (BK) parameterization Phys. Lett. B478 (2000) 417 was introduced. In this parameterization mpole/αBKm_{\mathrm{pole}} /\sqrt{\alpha_{\mathrm{BK}}} is interpreted as the mass of an effective pole higher than mpolem_{\mathrm{pole}}, i.e., it is expected that αBK<1\alpha_{\mathrm{BK}}<1. The parameterization takes the form f+(q2)=f+(0)(1q2/mpole2)1(1αBKq2mpole2),f_+(q^2) = \frac{f_+(0)}{(1-q^2/m^2_{\mathrm{pole}})} \frac{1}{\left(1-\alpha^{}_{\mathrm{BK}}\frac{q^2}{m^2_{\mathrm{pole}}}\right)}\,, where αBK\alpha^{}_{\mathrm{BK}} is a free parameter that takes into account contributions from higher states in the form of an additional effective pole. This parameterization is used by several experiments to determine form factor parameters.

(2) zz expansion

Alternatively, a power series expansion around some value q2=t0q^2=t_0 can be used to parameterize f+(q2)f^{}_+(q^2)[Phys. Lett. B 633 (2006) 61,Phys. Rev. Lett. 74 (1995) 4603,Phys. Rev. D 56 (1997) 303,Phys. Rev. Lett. 95 (2005)] This parameterization is model-independent and satisfies general QCD constraints. The expansion is given in terms of a complex parameter zz, which is the analytic continuation of q2q^2 into the complex plane: z(q2,t0)=t+q2t+t0t+q2+t+t0,z(q^2,t_0) = \frac{\sqrt{t_+ - q^2} - \sqrt{t_+ - t_0}}{\sqrt{t_+ - q^2} + \sqrt{t_+ - t_0}}\,, where t0=t+(11t/t+)t_{0}= t_{+} (1-\sqrt{1-t_{-}/t_{+}}) and t±(mD±mP)2t_\pm \equiv (m_D \pm m_P)^2. In this parameterization, q2=t0q^2=t_0 corresponds to z=0z=0, and the physical region extends in either direction up to ±|z|max=±0.051\pm|z|_{\mathrm{max}} = \pm 0.051 for DK𝓁ν𝓁D\to K \mathcal{l} \nu_\mathcal{l} decays, and up to ±0.17\pm 0.17 for Dπ𝓁ν𝓁D\to \pi \mathcal{l} \nu_\mathcal{l} decays.

The form factor is expressed as f+(q2)=1P(q2)ϕ(q2,t0)k=0ak(t0)[z(q2,t0)]k,f_+(q^2) = \frac{1}{P(q^2)\,\phi(q^2,t_0)}\sum_{k=0}^\infty a_k(t_0)[z(q^2,t_0)]^k\,, \label{z_expansion} where the Blaschke factor P(q2)P(q^2) is used to remove sub-threshold poles, for instance, P(q2)=1P(q^2)=1 for DπD\to \pi and P(q2)=z(q2,MDs*2)P(q^2)=z(q^2,M^2_{D^*_s}). The “outer” function ϕ(t,t0)\phi(t,t_0) can be any analytic function, but a preferred choice (see  Phys. Rev. Lett. 74 (1995) 4603,Phys. Rev. D 56 (1997) 303 and Nucl. Phys. B 189 (1981) 157.), obtained from the Operator Product Expansion (OPE), is ϕ(q2,t0)=α(t+q2+t+t0)×t+q2(t+t0)1/4 (t+q2+t+t)3/2(t+q2+t+)5, with α=πmc2/3\alpha = \sqrt{\pi m_c^2/3}. The OPE analysis provides a constraint upon the expansion coefficients, k=0Nak21\sum_{k=0}^{N}a_k^2 \leq 1. These coefficients receive 1/MD1/M_D corrections, and thus the constraint is only approximate. However, the expansion is expected to converge rapidly since |z|<0.051(0.17)|z|<0.051\ (0.17) for DKD\rightarrow K (DπD\rightarrow\pi) over the entire physical q2q^2 range, and the expanded equation remains a useful parameterization. The main disadvantage as compared to phenomenological approaches is that there is no physical interpretation of the fitted coefficients aKa_K.

(3) Three-pole formalism

An update of the vector pole dominance model has been developed for the Dπ𝓁ν𝓁D \to \pi \mathcal{l} \nu_\mathcal{l} channel [arXiv:1407.1019]. It uses information of the residues of the semileptonic form factor at its first two poles, the D*(2010)D^\ast(2010) and D*(2600)D^{\ast '}(2600) resonances. The form factor is expressed as an infinite sum of residues from JP=1J^P =1^- states with masses mDn*m^{}_{D^\ast_n}: f+(q2)=n=0Resq2=mDn*2f+(q2)mDn*2q2,f_+(q^2) = \sum_{n=0}^\infty \frac{\displaystyle{\underset{q^2= m_{D^\ast_n}^2}{\mathrm{Res}}} f_+(q^2)}{m_{D^\ast_n}^2-q^2} \,, \label{ThreePole} with the residues given by Resq2=mDn*2f+(q2)=12mDn*fDn*gDn*Dπ.\displaystyle{\underset{q^2=m_{D_n^\ast}^2}{\mathrm{Res}}} f_+(q^2)= \frac{1}{2} m^{}_{D_n^\ast}\,f^{}_{D_n^\ast}\,g^{}_{D_n^\ast D\pi}\,. \label{Residua} Values of the fD*f_{D^\ast} and fD*f_{D^{\ast '}} decay constants have been calculated relative to fDf_{D} via lattice QCD, with 2%\% and 28%\% precision, respectively [arXiv:1407.1019]. The couplings to the DπD\pi state, gD*Dπg^{}_{D^\ast D\pi} and gD*Dπg^{}_{D^{\ast '} D\pi}, are extracted from measurements of the D*(2010)D^\ast(2010) and D*(2600)D^{\ast '}(2600) widths by the BaBar and LHCb experiments [Phys. Rev. D 88 (2013) 052003,Phys. Rev. D 82 (2010)111101,JHEP 09 (2013) 145]. This results in the contribution from the first pole being determined with 3%3\% accuracy. The contribution from the D*(2600)D^{\ast '}(2600) pole is determined with poorer accuracy, 30%\sim 30\%, mainly due to lattice uncertainties. A superconvergence condition [Phys. Rev. D 55 (1997) 2817] n=0Resq2=mDn*2f+(q2)=0\sum_{n=0}^\infty {\displaystyle{\underset{q^2=m_{D^\ast_n}^2}{\mathrm{Res}}} f_+(q^2) }= 0 \label{superconvergence} is applied, protecting the form factor behavior at large q2q^2. Within this model, the first two poles are not sufficient to describe the data, and a third effective pole needs to be included.

One of the advantages of this phenomenological model is that it can be extrapolated outside the charm physical region, providing a method to extract the magnitude of the CKM matrix element VubV_{ub} using the ratio of the form factors of the Dπ𝓁νD\to \pi\mathcal{l} \nu and Bπ𝓁νB\to \pi\mathcal{l} \nu decay channels. It will be used once lattice calculations provide the form factor ratio fBπ+(q2)/fDπ+(q2)f^{+}_{B\pi}(q^2)/f^{+}_{D\pi}(q^2) at the same pion energy.

This form factor description can be extended to the DK𝓁νD\to K \mathcal{l} \nu decay channel, considering the contribution of several csc\bar s resonances with JP=1J^P = 1^-. The first two pole masses contributing to the form factor correspond to the Ds*(2112)D^{*}_s(2112) and Ds1*(2700)D^{*}_{s1}(2700) resonant states [PTEP 2020]. A constraint on the first residue can be obtained using information of the fKf_K decay constant [PTEP 2020] and the gg coupling extracted from the D*+D^{\ast +} width [Phys. Rev. D 88 (2013) 052003]. The contribution from the second pole can be evaluated using the decay constants from [Phys. Rev. D 87 (2013) 054007], the measured total width, and the ratio of D*KD^{\ast} K and DKD K decay branching fractions [PTEP 2020].

1.2 Combined results for the DP𝓁ν𝓁D\to P\mathcal{l}\nu_\mathcal{l} channels

Results and world averages for the products f+K(0)|Vcs|f_+^K(0)|V_{cs}| and f+π(0)|Vcd|f_+^\pi(0)|V_{cd}| as measured by CLEO-c, Belle, BaBar, and BESIII are summarized in Tables 1 and 2, and plotted in Fig. 1 (left) and Fig. 1 (middle), respectively. When calculating these world averages, the systematic uncertainties of the BESIII analyses are conservatively taken to be fully correlated.

The results and world averages of the products f+Dη(0)|Vcd|f_+^{D\to\eta}(0)|V_{cd}|, which have been measured by CLEO-c and BESIII, are summarized in Tables 3 and plotted in Fig. 1 (right). In averaging, the systematic uncertainties of the two BESIII analyses are conservatively taken to be fully correlated.

1.3 Determinations of|Vcs||V_{cs}|and|Vcd||V_{cd}|

Assuming unitarity of the CKM matrix, the values of the CKM matrix elements entering in charm semileptonic decays are evaluated as [PTEP 2020] |Vcs|=0.97320±0.00011,|Vcd|=0.22636±0.00048.\label{eq:charm:ckm} \begin{aligned} |V_{cs}| & = 0.97320\pm 0.00011 \, ,\\ |V_{cd}| & = 0.22636\pm 0.00048 \, . \end{aligned} Using the world average values of f+K(0)|Vcs|f_+^K(0)|V_{cs}| and f+π(0)|Vcd|f_+^{\pi}(0)|V_{cd}| from Tables 1 and 2 leads to the form factor values f+K(0)=0.7361±0.0034,f+π(0)=0.6351±0.0081,% \begin{aligned} f_+^K(0) & = 0.7361 \pm 0.0034 \, , \\ f_+^{\pi}(0) &= 0.6351 \pm 0.0081 \,, \end{aligned} where the former one deviates with the present average of lattice QCD calculations by 2.1σ2.1\sigma while good consistency is found for the latter one. Table 4 summarizes f+Dπ(0)f^{D\to\pi}_+(0) and f+DK(0)f^{D\to K}_+(0) results based on Nf=2+1+1N_f=2+1+1 flavour lattice QCD of the ETM collaboration [Phys. Rev. D 96 (2017) 054514], and earlier results based on Nf=2+1N_f=2+1 flavour lattice QCD of the HPQCD collaboration [Phys. Rev. D 82 (2010) 114506,Phys. Rev. D 84 (2011) 114505]. Recently, the Fermilab Lattice and MILC collaborations released their preliminary results of f+DK(0)f^{D\to K}_+(0) and f+Dπ(0)f^{D\to \pi}_+(0) based on Nf=2+1+1N_f=2+1+1 flavour lattice QCD calculations [Lattice 2018 (2019)]. The weighted averages are f+Dπ(0)=0.634±0.015f^{D\to\pi}_+(0)=0.634\pm0.015 and f+DK(0)=0.760±0.011f^{D\to K}_+(0)=0.760\pm0.011, respectively. The experimental accuracy is at present better than that from lattice calculations.

Alternatively, if one assumes the lattice QCD form factor values, the averages in Tables 1 and 2 give |Vcs|=0.9447±0.0043(exp.)±0.0137(LQCD),|Vcd|=0.2249±0.0028(exp.)±0.0055(LQCD).\begin{aligned} |V_{cs}| &= 0.9447 \pm 0.0043({\mathrm{exp}.})\pm 0.0137({\mathrm{LQCD}}) \, , \\ |V_{cd}| &= 0.2249 \pm 0.0028({\mathrm{exp}.})\pm 0.0055({\mathrm{LQCD}})\,. \end{aligned} Here, the uncertainties are dominated by the lattice QCD calculations. These values are consistent within 1.9σ1.9\sigma and 0.1σ0.1\sigma, respectively, with those obtained from the PDG global fit assuming CKM unitarity [PTEP 2020].

Tab.1 Results for f+K(0)|Vcs|f_+^K(0)|V_{cs}| from various experiments. BaBar 2007 [Phys. Rev. D76 (2007) 052005] and Belle 2006 [Phys. Rev. Lett. 97 (2006) 061804] only reported f+K(0)f_{+}^{K}(0) values. The listed |Vcs|f+K(0)|V_{cs}| f_{+}^{K}(0) values of these two experiments are obtained by multiplying f+K(0)f_{+}^{K}(0) with their quoted |Vcs||V_{cs}|.
DK𝓁ν𝓁D\to K \mathcal{l}\nu_{\mathcal{l}} Measurement Mode |Vcs|f+K(0)|V_{cs}| f_{+}^{K}(0) Comment
BESIII 2019 [Phys. Rev. Lett. 122 (2019) 011804] (D0;𝓁=μD^0;\mathcal{l}=\mu) 0.7133(38)(30) zz expansion, 2 terms
BESIII 2017 [Phys. Rev. D 96 (2017) 012002] (D+;𝓁=eD^+;\mathcal{l}=e) 0.6983(56)(112) zz expansion, 3 terms
BESIII 2015B [Phys. Rev. D 92 (2015) 112008] (D+;𝓁=eD^+;\mathcal{l}=e) 0.7370(60)(90) zz expansion, 3 terms
BESIII 2015A [Phys. Rev. D92 (2015) 072012] (D0;𝓁=eD^0;\mathcal{l}=e) 0.7195(35)(41) zz expansion, 3 terms
CLEO-c 2009 [Phys. Rev. D 80 (2009) 032005] (D0D^0, D+;𝓁=eD^+;\mathcal{l}=e) 0.7189(64)(48) zz expansion, 3 terms
BaBar 2007 [Phys. Rev. D76 (2007) 052005] (D0;𝓁=eD^0;\mathcal{l}=e) 0.7211(69)(85) Fitted pole mass +
modified pole ansatze;
|Vcs|=0.9729±0.0003|V^{}_{cs}| = 0.9729\pm 0.0003;
corrected for (D0Kπ+)\mathcal{B}(D^0\to K^-\pi^+)
Belle 2006 [Phys. Rev. Lett. 97 (2006) 061804] (D0;𝓁=e,μD^0;\mathcal{l}=e,\,\mu) 0.6762(68)(214) |Vcs|=0.97296±0.00024|V^{}_{cs}| = 0.97296\pm 0.00024
(PDG 2006 w/unitarity)
World average 0.7180(33) BESIII syst. fully correlated
Tab.2 Results for f+π(0)|Vcd|f_+^\pi(0)|V_{cd}| from various experiments.
Dπ𝓁ν𝓁D\to \pi \mathcal{l}\nu_{\mathcal{l}} Measurement Mode |Vcd|f+π(0)|V_{cd}| f_{+}^{\pi}(0) Comment
BESIII 2017 [Phys. Rev. D 96 (2017) 012002] (D+;𝓁=eD^+;\mathcal{l}=e) 0.1413(35)(12) zz expansion, 3 terms
BESIII 2015A [Phys. Rev. D92 (2015) 072012] (D0;𝓁=eD^0;\mathcal{l}=e) 0.1420(24)(10) zz expansion, 3 terms
CLEO-c 2009 [Phys. Rev. D 80 (2009) 032005] (D0D^0, D+;𝓁=eD^+;\mathcal{l}=e) 0.1500(40)(10) zz expansion, 3 terms
BaBar 2015 [Phys. Rev. D 91 (2015) 052022] (D0;𝓁=eD^0;\mathcal{l}=e) 0.1374(38)(24) zz expansion, 3 terms
Belle 2006 [Phys. Rev. Lett. 97 (2006) 061804] (D0;𝓁=e,μD^0;\mathcal{l}=e,\,\mu) 0.1417(45)(68) |Vcd|=0.2271±0.0010|V^{}_{cd}| = 0.2271\pm 0.0010
(PDG 2006 w/unitarity)<
World average 0.1426(18) BESIII syst. fully correlated
Tab.3 Results for f+Dη(0)|Vcd|f_+^{D\to\eta}(0)|V_{cd}| from various experiments.
D+η𝓁ν𝓁D^+\to \eta \mathcal{l}\nu_{\mathcal{l}} Measurement Mode |Vcd|f+Dη(0)|V_{cd}| f_{+}^{D\to\eta}(0) Comment
BESIII 2017 [Phys. Rev. Lett. 124 (2020) 231801] (D+;𝓁=eD^+;\mathcal{l}=e) 0.087(8)(2) zz expansion, 2 terms
BESIII 2015A [Phys. Rev. D 97 (2018) 092009] (D0;𝓁=eD^0;\mathcal{l}=e) 0.079(6)(2) zz expansion, 2 terms
CLEO-c 2009 [Phys. Rev. D 84 (2011) 032001] (D0D^0, D+;𝓁=eD^+;\mathcal{l}=e) 0.085(6)(1) zz expansion, 2 terms
World average 0.083(4) BESIII syst. fully correlated
Tab.4 Results of calculated from factors from various lattice QCD groups.
Collaboration f+Dπ(0)f^{D\to\pi}_+(0) f+DK(0)f^{D\to K}_+(0)
Fermilab Lattice and MILC [Lattice 2018 (2019) ] 0.625±0.017±0.0130.625\pm0.017\pm0.013 0.768±0.012±0.0110.768\pm0.012\pm0.011
ETM(2+1+1) [Phys. Rev. D 96 (2017) 054514] 0.612±0.0350.612\pm0.035 0.765±0.0310.765\pm0.031
HPQCD(2+1) [Phys. Rev. D 82 (2010) 114506,Phys. Rev. D 84 (2011) 114505] 0.666±0.0290.666\pm0.029 0.747±0.0190.747\pm0.019
Average 0.634±0.0150.634\pm0.015 0.760±0.0110.760\pm0.011

Fig.1 Comparison of the results of f+K(0)|Vcs| measured by the Belle [Phys. Rev. Lett. 97 (2006) 061804,], BaBar [Phys. Rev. D 76 (2007) 052005], CLEO-c [Phys. Rev. D 80 (2009) 032005], and BESIII [Phys. Rev. D 92 (2015) 112008, Phys. Rev. D92 (2015) 072012,Phys. Rev. D 96 (2017) 012002, Phys. Rev. Lett. 122 (2019) 011804] experiments.

1.4 Test of ee-μ\mu lepton flavour universality

In the Standard Model (SM), the couplings between the three families of leptons and gauge bosons are expected to be equal; this is known as lepton flavour universality (LFU). The semileptonic decays of pseudoscalar mesons are well understood in the SM and thus offer a robust way to test LFU and search for new physics. Various tests of LFU with BB semileptonic decays have been reported by BaBar, Belle, and LHCb. The average of the ratio of the branching fractions BD(*)τ+ντ/BD(*)𝓁+ν𝓁{\mathcal B}_{B\to \bar D^{(*)}\tau^+\nu_\tau}/ {\mathcal B}_{B\to \bar D^{(*)}\mathcal{l}^+\nu_\mathcal{l}} (𝓁=μ,e\mathcal{l}=\mu,\,e) deviates from the SM prediction by 3.4σ3.4\sigma. Precision measurements of the semileptonic DD decays also test LFU, and in a manner complimentary to that of BB decays [Phys. Rev. D 91 (2015) 094009]. Within the SM, the ratios DKμ+νμ/DKe+νe{\mathcal B}_{D\to \bar K\mu^+\nu_\mu}/{\mathcal B}_{D\to \bar K e^+\nu_e} and Dπμ+νμ/Dπe+νe{\mathcal B}_{D\to \pi\mu^+\nu_\mu}/{\mathcal B}_{D\to \pi e^+\nu_e} are predicted to be 0.975±0.0010.975\pm0.001 and 0.985±0.0020.985\pm0.002, respectively [Eur. Phys. J. C 78 (2018) 501]. The ratios are expected to be close to unity with negligible uncertainty mainly due to high correlation of the corresponding hadronic form factors [Eur. Phys. J. C 78 (2018) 501].

In the SM, the semimuonic DD decays are expected to have lower branching fraction than their semielectronic counterparts. Before BESIII, however, the information related to the semimuonic DD decays is relatively poor, mainly due to higher backgrounds caused due to difficulty of distinguishing muon and charged pions. In the charmed meson sector, only D0Kμ+νμD^0\to K^-\mu^+\nu_\mu, D0K*μ+νμD^0\to K^{*-}\mu^+\nu_\mu, D0πμ+νμD^0\to \pi^-\mu^+\nu_\mu, D+K0μ+νμD^+\to \bar K^0\mu^+\nu_\mu, D+ρ0μ+νμD^+\to \rho^0\mu^+\nu_\mu, and D+K*0μ+νμD^+\to \bar K^{*0}\mu^+\nu_\mu have been investigated in experiments previously. Except for D+K*0μ+νμD^+\to \bar K^{*0}\mu^+\nu_\mu, all measurements of the other decays are dominated by FOCUS and Belle experiments and the existing measurements suffer large uncertainties.

Since 2016, BESIII performed a series of studies of semimuonic DD decays, including improved measurements of D+K0μ+νμD^+\to\bar K^0\mu^+\nu_\mu [Eur. Phys. J. C 76 (2016) 369], D0πμ+νμD^0\to \pi^-\mu^+\nu_\mu [Phys. Rev. Lett. 121 (2018) 171803], and D0Kμ+νμD^0\to K^-\mu^+\nu_\mu [Phys. Rev. Lett. 122 (2019) 011804], and the first observations of D+π0μ+νμD^+\to \pi^0\mu^+\nu_\mu [Phys. Rev. Lett. 121 (2018) 171803], D+ωμ+νμD^+\to \omega\mu^+\nu_\mu [Phys. Rev. D 101 (2020) 072005], D+ημ+νμD^+\to \eta\mu^+\nu_\mu [Phys. Rev. Lett. 124 (2020) 231801]. All these analyses used the tagged method and 2.93 fb1^{-1} of data taken at 3.773 GeV. The reported branching fractions areB(D+K¯0μ+νμ)=(8.72±0.07±0.18)%,B(D0πμ+νμ)=(0.272±0.008±0.006)%,B(D+π0μ+νμ)=(0.350±0.011±0.010)%,B(D0Kμ+νμ)=(3.413±0.019±0.035)%,B(D+ωμ+νμ)=(0.177±0.018±0.011)%,B(D+ημ+νμ)=(0.104±0.010±0.005)%.

Combining these results with previous BESIII measurements of their counterparts of the semielectronic decays using the same data sample, the ratios of branching fractions are B(D0πμ+νμ)B(D0πe+νe)=0.922±0.030±0.022,B(D+π0μ+νμ)B(D+π0e+νe)=0.964±0.037±0.026,B(D0Kμ+νμ)B(D0Ke+νe)=0.974±0.007±0.012,B(D+ωμ+νμ)B(D+ωe+νe)=1.05±0.14,B(D+ημ+νμ)B(D+ηe+νe)=0.91±0.13.In addition, using the world average for (D+K0e+νe)\mathcal{B}(D^+\to \bar K^0 e^+\nu_e) [PTEP 2020] gives (D+K0μ+νμ)(D+K0e+νe)=1.00±0.03.\frac{\mathcal{B}(D^+\to \bar K^0\mu^+\nu_\mu)}{\mathcal{B}(D^+\to \bar K^0 e^+\nu_e)} = 1.00\pm0.03\,. These results indicate that any ee-μ\mu LFU violation in DD semileptonic decays has to be at the level of a few percent or less. BESIII also tested ee-μ\mu LFU in separate q2q^2 intervals using D0(+)π(0)𝓁+ν𝓁D^{0(+)}\to \pi^{-(0)}\mathcal{l}^+\nu_\mathcal{l} [Phys. Rev. Lett. 121 (2018) 171803] and D0K𝓁+ν𝓁D^0\to K^-\mathcal{l}^+\nu_\mathcal{l} [Phys. Rev. Lett. 122 (2019) 011804] decays. No indication of LFU above the 2σ2\sigma level was found.

In 2018, using 0.482 fb1^{-1} of data taken at a center-of-mass energy of 4.009 GeV, BESIII reported measurements of the branching fractions for semileptonic decays Ds+ϕμ+νμD^+_s\to \phi\,\mu^+\nu_\mu, Ds+ημ+νμD^+_s\to \eta \mu^+\nu_\mu, and Ds+ημ+νμD^+_s\to \eta^\prime \mu^+\nu_\mu[Phys. Rev. D 97 (2018) 012006]. Combining these results with previous measurements of Ds+ϕe+νeD^+_s\to \phi\,e^+\nu_e [Phys. Rev. D 97 (2018) 012006], Ds+ηe+νeD^+_s\to \eta e^+\nu_e, and Ds+ηe+νeD^+_s\to \eta^\prime e^+\nu_e [Phys. Rev. D 94 (2016) 112003] gives the ratios B(Ds+ϕμ+νμ)B(Ds+ϕe+νe)=0.86±0.29,B(Ds+ημ+νμ)B(Ds+ηe+νe)=1.05±0.24,B(Ds+ημ+νμ)B(Ds+ηe+νe)=1.14±0.68. These values are all consistent with unity. The uncertainties include both statistical and systematic uncertainties, the former of which dominates.

2. DV𝓁ν𝓁D\rightarrow V \mathcal{l} \nu_\mathcal{l} decays

When the final state hadron is a vector meson, the decay can proceed through both vector and axial vector currents, and four form factors are needed. The hadronic current is Hμ=Vμ+AμH^{}_\mu = V^{}_\mu + A^{}_\mu, where [Phys. Rev. D 41 (1990) 142] Vμ=V(p,ε)|q¯γμc|D(p)=2V(q2)mD+mVεμνρσενpρpσAμ=V(p,ε)|q¯γμγ5c|D(p)=i(mD+mV)A1(q2)εμ+iA2(q2)mD+mV(εq)(p+p)μ+ i2mVq2(A3(q2)A0(q2))[ε(p+p)]qμ. In this expression, mVm_V is the invariant mass of the daughter particles of the VV meson and A3(q2)=mD+mV2mVA1(q2)mDmV2mVA2(q2).A_3(q^2) = \frac{m_D + m_V}{2m_V}A_1(q^2)- \frac{m_D - m_V}{2m_V}A_2(q^2)\,. To avoid divergence of the i2mVq2(A3(q2)A0(q2))i\,\frac{2m_V}{q^2} \left (A_3(q^2)-A_0(q^2) \right ) item, kinematics require that A3(0)=A0(0)A_3(0) = A_0(0). Terms proportional to qμq_\mu are negligible for 𝓁=e\mathcal{l}=e. Thus, only the three form factors A1(q2)A_1(q^2), A2(q2)A_2(q^2) and V(q2)V(q^2) are relevant for charm decays.

The differential decay rate is dΓ(DVl¯νl)dq2dcosθl=GF2|Vcq|2128π3mD2pq2×[(1cosθl)22|H|2+(1+cosθl)22|H+|2+sin2θl|H0|2],where H±H^{}_\pm and H0H^{}_0 are helicity amplitudes, corresponding to helicities of the vector (VV) meson. The helicity amplitudes can be expressed in terms of the form factors as H±=1mD+mV[(mD+mV)2A1(q2)  2mDpV(q2)]H0=1|q|mD22mV(mD+mV) × [(1mV2q2mD2)(mD+mV)2A1(q2)4p2A2(q2)]. Here p*=[(mD2(mV+q))2(mD2(mVq))]1/22mDp^*=\frac{\left [\left (m_D^2-(m_V+q)\right )^2\left (m_D^2-(m_V-q)\right )\right ]^{1/2}}{2 m_D} is the magnitude of the three-momentum of the VV system as measured in the DD rest frame, and θ𝓁\theta_\mathcal{l} is the angle of the lepton momentum with respect to the direction opposite that of the DD in the WW rest frame (see Fig. 2 for the electron case, θe\theta_e). The left-handed nature of the quark current manifests itself as |H|>|H+||H_-|>|H_+|. The differential decay rate for DV𝓁νD\rightarrow V\mathcal{l}\nu followed by the vector meson decaying into two pseudoscalars is dΓ(DVl¯ν,VP1P2)dq2dcosθVdcosθldχ=3GF22048π4|Vcq|2p(q2)q2mD2B(VP1P2) ×{(1+cosθl)2sin2θV|H+(q2)|2+ (1cosθl)2sin2θV|H(q2)|2+ 4sin2θlcos2θV|H0(q2)|2 4sinθl(1+cosθl)sinθVcosθVcosχH+(q2)H0(q2)+ 4sinθl(1cosθl)sinθVcosθVcosχH(q2)H0(q2) 2sin2θlsin2θVcos2χH+(q2)H(q2)},where the helicity angles θ𝓁\theta^{}_\mathcal{l}, θV\theta^{}_V, and acoplanarity angle χ\chi are defined as shown in Fig. 2. Usually, the ratios of the form factors at q2=0q^2=0 are defined as rVV(0)A1(0),r2A2(0)A1(0). From the experimental point of view, these ratios can be obtained without any assumption about the total decay rates or the CKM matrix elements.

Table 5 lists measurements of rVr_V and r2r_2 from several experiments. Most of the measurements assume that the q2q^2 dependence of the form factors is given by the simple pole ansatz. Some of these measurements do not consider a separate SS-wave contribution; in this case such a contribution is implicitly included in the measured values.

Tab.5 Results for rVr_V and r2r_2 from various experiments. Experiments marked with *^* did not consider a separate SS-wave contribution.
Experiment Ref. rVr_V r2r_2
D+K¯*0𝓁+ν𝓁D^+\to \overline{K}^{*0}\mathcal{l}^+\nu_\mathcal{l}
E691*^* Phys. Rev. Lett. 65 (1990) 2630 2.0±\pm 0.6±\pm 0.3 0.0±\pm 0.5±\pm 0.2
E653*^* Phys. Lett. B274 (1992) 246 2.00±\pm 0.33±\pm 0.16 0.82±\pm 0.22±\pm 0.11
E687*^* Phys. Lett. B307 (1993) 262 1.74±\pm 0.27±\pm 0.28 0.78±\pm 0.18±\pm 0.11
E791 (e)*^* Phys. Rev. Lett. 80 (1998) 1393 1.90±\pm 0.11±\pm 0.09 0.71±\pm 0.08±\pm 0.09
E791 (μ\mu)*^* Phys. Lett. B440 (1998) 435 1.84±\pm0.11±\pm0.09 0.75±\pm0.08±\pm0.09
Beatrice*^* Eur. Phys. J. C6 (1999) 35 1.45±\pm 0.23±\pm 0.07 1.00±\pm 0.15±\pm 0.03
FOCUS Phys. Lett. B544 (2002) 89 1.504±\pm0.057±\pm0.039 0.875±\pm0.049±\pm0.064
BESIII (ee) Phys. Rev. D 94 (2016)032001 1.406±0.058±0.0221.406\pm0.058\pm0.022 0.784±0.041±0.0240.784\pm0.041\pm0.024
D0K¯0π𝓁+ν𝓁D^0\to \overline{K}^0\pi^-\mathcal{l}^+\nu_\mathcal{l}
FOCUS (μ\mu) Phys. Lett. B607 (2005) 67 1.706±\pm0.677±\pm0.342 0.912±\pm0.370±\pm0.104
BaBar (μ\mu) Phys. Rev. D83 (2011) 072001 1.493±0.014±0.0211.493 \pm 0.014 \pm 0.021 0.775±0.011±0.0110.775 \pm 0.011 \pm 0.011
BESIII (ee) Phys. Rev. D 99 (2019) 011103 1.46±0.07±0.021.46\pm0.07\pm0.02 0.67±0.06±0.010.67\pm0.06\pm0.01
D+ωe+νeD^+\to \omega e^+\nu_e
BESIII Phys. Rev. D 92 (2016) 071101 1.24±0.09±0.061.24\pm0.09\pm0.06 1.06±0.15±0.051.06\pm0.15\pm0.05
D0,+ρeνeD^{0,+}\to \rho\,e \nu_e
CLEO-c Phys. Rev. Lett. 110 (2013) 131802 1.48±0.15±0.051.48\pm0.15\pm0.05 0.83±0.11±0.040.83\pm0.11\pm0.04
BESIII Phys. Rev. Lett. 122 (2019) 062001 1.695±0.083±0.0511.695\pm0.083\pm0.051 0.845±0.056±0.0390.845\pm0.056\pm0.039
Ds+ϕe+νeD_s^+ \to \phi\,e^+ \nu_e
BaBar Phys. Rev. D 78 (2008) 051101 1.849±\pm0.060±\pm0.095 0.763±\pm0.071±\pm0.065
Ds+K*0e+νeD_s^+ \to K^{*0}\,e^+ \nu_e
BESIII*^* Phys. Rev. Lett. 122 (2019) 061801 1.67±0.34±0.161.67\pm0.34\pm0.16 0.77±0.28±0.070.77\pm0.28\pm0.07

3. DS𝓁ν𝓁D\to S \mathcal{l} \nu_{\mathcal{l}}decays

In 2018, BESIII reported measurements of semileptonic DD decays into a scalar meson, DS𝓁νD\to S \mathcal{l}\nu. The experiment measured D0(+)a0(980)e+νeD^{0(+)}\to a_0(980) e^+\nu_e, with a0(980)ηπa_0(980)\to\eta\pi. Signal yields of 25.75.7+6.425.7^{+6.4}_{-5.7} events for D0a0(980)e+νeD^0\to a_0(980)^-e^+\nu_e, and 10.24.1+5.010.2^{+5.0}_{-4.1} events for D+a0(980)0e+νeD^+\to a_0(980)^0e^+\nu_e, were obtained, resulting in statistical significances of greater than 6.5σ\sigma and 3.0σ\sigma, respectively [Phys. Rev. Lett. 121 (2018) 081802]. As the branching fraction for a0(980)ηπa_0(980)\to\eta\pi is not well-measured, BESIII reports the product branching fractions B(D0a0(980)e+νe)×B(a0(980)ηπ)=(1.330.29+0.33±0.09)×104,B(D+a0(980)0e+νe)×B(a0(980)0ηπ0)=(1.660.66+0.81±0.11)×104. The ratio of these values can be compared to a prediction based on QCD light-cone sum rules [Phys. Rev. D 96 (2017) 033002], after relating the a0(980)ηπa_0(980)\to\eta\pi branching fractions via isospin. The result is a difference of more than 2σ2\sigma. Taking the lifetimes of the D0D^0 and D+D^+ into account, and assuming (a0(980)ηπ)=(a0(980)0ηπ0)\mathcal{B}(a_0(980)^-\to\eta\pi^-) = \mathcal{B}(a_0(980)^0\to\eta\pi^0), the ratio of the partial widths is Γ(D0a0(980)e+νe)Γ(D+a0(980)0e+νe)=2.03±0.95±0.06.\frac{\Gamma(D^0\to a_0(980)^-e^+\nu_e)}{\Gamma(D^+\to a_0(980)^0e^+\nu_e)} = 2.03\pm 0.95\pm 0.06 \,. This value is consistent with the prediction based on isospin symmetry.

Recently, BESIII searched for the semileptonic decay of Ds+a0(980)e+νeD^{+}_s\to a_0(980) e^+\nu_e, with a0(980)0ηπ0a_0(980)^0\to\eta\pi^0. No significant signal is observed. The product branching fraction upper limit at the 90% confidence level is (Ds+a0(980)e+νe)×(a0(980)0ηπ0)<1.2×104\mathcal{B}(D^{+}_s\to a_0(980) e^+\nu_e)\times\mathcal{B}(a_0(980)^0\to\eta\pi^0)<1.2\times 10^{-4} [Phys. Rev. D 103 (2021) 092004].

4. DA𝓁ν𝓁D\to A \mathcal{l} \nu_{\mathcal{l}} decays

Experimental studies of semileptonic DD decays into a Axial-vector meson DA𝓁νD\to A \mathcal{l}\nu are challenging due to low statistics and high backgrounds. In 2007, CLEO-c reported first evidence for the Cabibbo-favored decay D0K1(1270)e+νeD^0\to K_1(1270)^-e^+\nu_e with a statistical significance of 4σ4\sigma [Phys. Rev. Lett. 99 (2007) 191801]. The branching fraction was measured to be (D0K1(1270)e+νe)=(7.63.0+4.1±0.6±0.7)×104\mathcal{B}(D^0\to K_1(1270)^-e^+\nu_e) = (7.6^{+4.1}_{-3.0}\pm0.6\pm0.7)\times 10^{-4}. In 2019, BESIII reported the first observation of D+K1(1270)0e+νeD^+\to \bar K_1(1270)^0e^+\nu_e, with statistical significance greater than 10σ10\sigma [Phys. Rev. Lett. 123 (2019) 231801]. The branching fraction was measured to be (D+K1(1270)0e+νe)=(23.0±2.62.1+1.8±2.5)×104\mathcal{B}(D^+\to \bar K_1(1270)^0e^+\nu_e) = (23.0\pm2.6^{+1.8}_{-2.1}\pm2.5)\times 10^{-4}. In 2021, the D0K1(1270)e+νeD^0\to K_1(1270)^-e^+\nu_e decay was observed for the first time by BESIII with a statistical significance greater than 10σ10\sigma [Phys. Rev. Lett. 127 (2021) 131801]. The reported branching fraction is (D+K1(1270)0e+νe)=(10.9±1.31.3+0.9±1.2)×104\mathcal{B}(D^+\to \bar K_1(1270)^0e^+\nu_e) = (10.9\pm1.3^{+0.9}_{-1.3}\pm1.2)\times 10^{-4}. Here, the third errors listed arise from the branching fraction for K1(1270)KππK_1(1270)\to K\pi\pi. The obtained branching fractions are consistent with the theoretical calculations with the K1K_1 mixing angle of 3333^\circ or 5757^\circ. Taking the lifetimes of D0D^0 and D+D^+ into account, the ratio of the partial widths is Γ(D+K1(1270)0e+νe)Γ(D0K1(1270)e+νe)=1.20±0.20±0.15.\frac{\Gamma(D^+\to \bar K_1(1270)^0e^+\nu_e)}{\Gamma(D^0\to K_1(1270)^-e^+\nu_e)} = 1.20\pm0.20\pm0.15. This value agrees with unity as predicted by isospin symmetry.

In addition, BESIII has searched for the Cabibbo-suppressed semileptonic decays D+b1(1235)0e+νeD^+\to b_1(1235)^0e^+\nu_e and D0b1(1235)0e+νeD^0\to b_1(1235)^0e^+\nu_e. No significant signal is observed. The product branching fraction upper limits at the 90% confidence level are (D+b1(1235)0e+νe)×(b1(1235)0ωπ0)<1.12×104\mathcal{B}(D^+\to b_1(1235)^0e^+\nu_e)\times\mathcal{B}( b_1(1235)^0\to\omega\pi^0)<1.12\times 10^{-4} and (D0b1(1235)e+νe)×(b1(1235)0ωπ0)<1.75×104\mathcal{B}(D^0\to b_1(1235)^-e^+\nu_e)\times \mathcal{B}( b_1(1235)^0\to\omega\pi^0)<1.75\times 10^{-4}, respectively [Phys. Rev. D 102 (2020) 112005].