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

People working on this:   Hailong Ma

Semileptonic decays of $D$ 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

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

where $G_F$ is the Fermi constant and $V^{}_{cq}$ is a CKM matrix element. The leptonic current $L^\mu$ is evaluated directly from the lepton spinors and has a simple structure; this allows one to extract information about the form factors (in $H^{}_\mu$) 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 $|V^{}_{cq}|$ [Prog. Theor. Phys. 49 (1973) 652].

1. $D\rightarrow P \mathcal{l} \nu_{\mathcal{l}}$ decays

When the final state hadron is a pseudoscalar, the hadronic current is given by [eConf C 060409 (2006) 027] $$H_\mu = \left< P(p) | \bar{q}\gamma_\mu c | D(p') \right> \ =\ f_+(q^2)\left[ (p' + p)_\mu -\frac{m_D^2-m_P^2}{q^2}q_\mu\right] + f_0(q^2)\frac{m_D^2-m_P^2}{q^2}q_\mu\,,$$ where $m_D$ and $p'$ are the mass and four momentum of the parent $D$ meson, $m_P$ and $p$ are those of the daughter meson, $f_+(q^2)$ and $f_0(q^2)$ are form factors, and $q = p' - p$. Kinematics require that $f_+(0) = f_0(0)$. The contraction $q_\mu L^\mu$ results in terms proportional to $m^{}_\mathcal{l}$[Phys. Rev. D 41 (1990) 142], and thus for $\mathcal{l}=e$ the terms proportional to $q_\mu$ in the previous equation are negligible and only the $f_+(q^2)$ vector form factor is relevant. The corresponding differential partial width is $$\frac{d\Gamma(D \to P e \nu_e)}{dq^2\, d\cos\theta_e} = \frac{G_F^2|V_{cq}|^2}{32\pi^3} p^{*\,3}|f_{+}(q^2)|^2\sin\theta^2_e\,,$$ where ${p^*}=\frac{\left [\left (m_D^2-(m_P+q)\right )^2\left (m_D^2-(m_P-q)\right )\right ]^{1/2}}{2 m_D}$ is the magnitude of the momentum of the final state hadron in the $D$ rest frame, and $\theta_e$ is the angle of the electron in the $e\nu$ rest frame with respect to the direction of the pseudoscalar meson in the $D$ rest frame.

1.1 Form factor parameterizations

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

$$f_+(q^2) = \frac{f_+(0)}{(1-\alpha)} [(\frac{1}{1- q^2/m^2_{\mathrm{pole}}})+\sum_{k=1}^{N}\frac{\rho_k}{1- q^2/(\gamma_k\,m^2_{\mathrm{pole}})} ]\,,$$

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

(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 $$f_+(q^2) = \frac{f_+(0)}{(1-q^2/m^2_{\mathrm{pole}})}\,. \label{SimplePole}$$ However, values of $m_{{\mathrm{pole}}}$ 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 $m_{\mathrm{pole}} /\sqrt{\alpha_{\mathrm{BK}}}$ is interpreted as the mass of an effective pole higher than $m_{\mathrm{pole}}$, i.e., it is expected that $\alpha_{\mathrm{BK}}<1$. The parameterization takes the form $$f_+(q^2) = \frac{f_+(0)}{(1-q^2/m^2_{\mathrm{pole}})} \frac{1}{\left(1-\alpha^{}_{\mathrm{BK}}\frac{q^2}{m^2_{\mathrm{pole}}}\right)}\,,$$ where $\alpha^{}_{\mathrm{BK}}$ 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) $z$ expansion

Alternatively, a power series expansion around some value $q^2=t_0$ can be used to parameterize $f^{}_+(q^2)$[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 $z$, which is the analytic continuation of $q^2$ into the complex plane: $$z(q^2,t_0) = \frac{\sqrt{t_+ - q^2} - \sqrt{t_+ - t_0}}{\sqrt{t_+ - q^2} + \sqrt{t_+ - t_0}}\,,$$ where $t_{0}= t_{+} (1-\sqrt{1-t_{-}/t_{+}})$ and $t_\pm \equiv (m_D \pm m_P)^2$. In this parameterization, $q^2=t_0$ corresponds to $z=0$, and the physical region extends in either direction up to $\pm|z|_{\mathrm{max}} = \pm 0.051$ for $D\to K \mathcal{l} \nu_\mathcal{l}$ decays, and up to $\pm 0.17$ for $D\to \pi \mathcal{l} \nu_\mathcal{l}$ decays.

The form factor is expressed as $$f_+(q^2) = \frac{1}{P(q^2)\,\phi(q^2,t_0)}\sum_{k=0}^\infty a_k(t_0)[z(q^2,t_0)]^k\,, \label{z_expansion}$$ where the Blaschke factor $P(q^2)$ is used to remove sub-threshold poles, for instance, $P(q^2)=1$ for $D\to \pi$ and $P(q^2)=z(q^2,M^2_{D^*_s})$. The “outer” function $\phi(t,t_0)$ 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 $$\varphi (q^2,t_0)=\alpha (\sqrt{t_{+}-q^2}+\sqrt{t_{+}-t_0})\times \frac{t_{+}-q^2}{(t_{+}-t_0{)}^{1/4}}\frac{(\sqrt{t_{+}-q^2}+\sqrt{t_{+}-t_{-}}{)}^{3/2}}{(\sqrt{t_{+}-q^2}+\sqrt{t_{+}}{)}^5},$$ with $\alpha = \sqrt{\pi m_c^2/3}$. The OPE analysis provides a constraint upon the expansion coefficients, $\sum_{k=0}^{N}a_k^2 \leq 1$. These coefficients receive $1/M_D$ corrections, and thus the constraint is only approximate. However, the expansion is expected to converge rapidly since $|z|<0.051\ (0.17)$ for $D\rightarrow K$ ($D\rightarrow\pi$) over the entire physical $q^2$ 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 $a_K$.

(3) Three-pole formalism

An update of the vector pole dominance model has been developed for the $D \to \pi \mathcal{l} \nu_\mathcal{l}$ channel [arXiv:1407.1019]. It uses information of the residues of the semileptonic form factor at its first two poles, the $D^\ast(2010)$ and $D^{\ast '}(2600)$ resonances. The form factor is expressed as an infinite sum of residues from $J^P =1^-$ states with masses $m^{}_{D^\ast_n}$: $$f_+(q^2) = \sum_{n=0}^\infty \frac{\displaystyle{\underset{q^2= m_{D^\ast_n}^2}{\mathrm{Res}}} f_+(q^2)}{m_{D^\ast_n}^2-q^2} \,, \label{ThreePole}$$ with the residues given by $$\displaystyle{\underset{q^2=m_{D_n^\ast}^2}{\mathrm{Res}}} f_+(q^2)= \frac{1}{2} m^{}_{D_n^\ast}\,f^{}_{D_n^\ast}\,g^{}_{D_n^\ast D\pi}\,. \label{Residua}$$ Values of the $f_{D^\ast}$ and $f_{D^{\ast '}}$ decay constants have been calculated relative to $f_{D}$ via lattice QCD, with 2$\%$ and 28$\%$ precision, respectively [arXiv:1407.1019]. The couplings to the $D\pi$ state, $g^{}_{D^\ast D\pi}$ and $g^{}_{D^{\ast '} D\pi}$, are extracted from measurements of the $D^\ast(2010)$ and $D^{\ast '}(2600)$ 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 $3\%$ accuracy. The contribution from the $D^{\ast '}(2600)$ pole is determined with poorer accuracy, $\sim 30\%$, mainly due to lattice uncertainties. A superconvergence condition [Phys. Rev. D 55 (1997) 2817] $$\sum_{n=0}^\infty {\displaystyle{\underset{q^2=m_{D^\ast_n}^2}{\mathrm{Res}}} f_+(q^2) }= 0 \label{superconvergence}$$ is applied, protecting the form factor behavior at large $q^2$. 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 $V_{ub}$ using the ratio of the form factors of the $D\to \pi\mathcal{l} \nu$ and $B\to \pi\mathcal{l} \nu$ decay channels. It will be used once lattice calculations provide the form factor ratio $f^{+}_{B\pi}(q^2)/f^{+}_{D\pi}(q^2)$ at the same pion energy.

This form factor description can be extended to the $D\to K \mathcal{l} \nu$ decay channel, considering the contribution of several $c\bar s$ resonances with $J^P = 1^-$. The first two pole masses contributing to the form factor correspond to the $D^{*}_s(2112)$ and $D^{*}_{s1}(2700)$ resonant states [PTEP 2020]. A constraint on the first residue can be obtained using information of the $f_K$ decay constant [PTEP 2020] and the $g$ coupling extracted from the $D^{\ast +}$ 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 $D^{\ast} K$ and $D K$ decay branching fractions [PTEP 2020].

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

Results and world averages for the products $f_+^K(0)|V_{cs}|$ and $f_+^\pi(0)|V_{cd}|$ 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 $f_+^{D\to\eta}(0)|V_{cd}|$, 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$|V_{cs}|$and$|V_{cd}|$

Assuming unitarity of the CKM matrix, the values of the CKM matrix elements entering in charm semileptonic decays are evaluated as [PTEP 2020] $$\label{eq:charm:ckm} \begin{aligned} |V_{cs}| & = 0.97320\pm 0.00011 \, ,\\ |V_{cd}| & = 0.22636\pm 0.00048 \, . \end{aligned}$$ Using the world average values of $f_+^K(0)|V_{cs}|$ and $f_+^{\pi}(0)|V_{cd}|$ from Tables 1 and 2 leads to the form factor values $$\begin{aligned} f_+^K(0) & = 0.7361 \pm 0.0034 \, , \\ f_+^{\pi}(0) &= 0.6351 \pm 0.0081 \,, \end{aligned}$$ where the former one deviates with the present average of lattice QCD calculations by $2.1\sigma$ while good consistency is found for the latter one. Table 4 summarizes $f^{D\to\pi}_+(0)$ and $f^{D\to K}_+(0)$ results based on $N_f=2+1+1$ flavour lattice QCD of the ETM collaboration [Phys. Rev. D 96 (2017) 054514], and earlier results based on $N_f=2+1$ 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 $f^{D\to K}_+(0)$ and $f^{D\to \pi}_+(0)$ based on $N_f=2+1+1$ flavour lattice QCD calculations [Lattice 2018 (2019)]. The weighted averages are $f^{D\to\pi}_+(0)=0.634\pm0.015$ and $f^{D\to K}_+(0)=0.760\pm0.011$, 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 $$\begin{aligned} |V_{cs}| &= 0.9447 \pm 0.0043({\mathrm{exp}.})\pm 0.0137({\mathrm{LQCD}}) \, , \\ |V_{cd}| &= 0.2249 \pm 0.0028({\mathrm{exp}.})\pm 0.0055({\mathrm{LQCD}})\,. \end{aligned}$$ Here, the uncertainties are dominated by the lattice QCD calculations. These values are consistent within $1.9\sigma$ and $0.1\sigma$, respectively, with those obtained from the PDG global fit assuming CKM unitarity [PTEP 2020].

1.4 Test of $e$-$\mu$ 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 $B$ semileptonic decays have been reported by BaBar, Belle, and LHCb. The average of the ratio of the branching fractions ${\mathcal B}_{B\to \bar D^{(*)}\tau^+\nu_\tau}/ {\mathcal B}_{B\to \bar D^{(*)}\mathcal{l}^+\nu_\mathcal{l}}$ ($\mathcal{l}=\mu,\,e$) deviates from the SM prediction by $3.4\sigma$. Precision measurements of the semileptonic $D$ decays also test LFU, and in a manner complimentary to that of $B$ decays [Phys. Rev. D 91 (2015) 094009]. Within the SM, the ratios ${\mathcal B}_{D\to \bar K\mu^+\nu_\mu}/{\mathcal B}_{D\to \bar K e^+\nu_e}$ and ${\mathcal B}_{D\to \pi\mu^+\nu_\mu}/{\mathcal B}_{D\to \pi e^+\nu_e}$ are predicted to be $0.975\pm0.001$ and $0.985\pm0.002$, 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 $D$ decays are expected to have lower branching fraction than their semielectronic counterparts. Before BESIII, however, the information related to the semimuonic $D$ 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 $D^0\to K^-\mu^+\nu_\mu$, $D^0\to K^{*-}\mu^+\nu_\mu$, $D^0\to \pi^-\mu^+\nu_\mu$, $D^+\to \bar K^0\mu^+\nu_\mu$, $D^+\to \rho^0\mu^+\nu_\mu$, and $D^+\to \bar K^{*0}\mu^+\nu_\mu$ have been investigated in experiments previously. Except for $D^+\to \bar K^{*0}\mu^+\nu_\mu$, 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 $D$ decays, including improved measurements of $D^+\to\bar K^0\mu^+\nu_\mu$ [Eur. Phys. J. C 76 (2016) 369], $D^0\to \pi^-\mu^+\nu_\mu$ [Phys. Rev. Lett. 121 (2018) 171803], and $D^0\to K^-\mu^+\nu_\mu$ [Phys. Rev. Lett. 122 (2019) 011804], and the first observations of $D^+\to \pi^0\mu^+\nu_\mu$ [Phys. Rev. Lett. 121 (2018) 171803], $D^+\to \omega\mu^+\nu_\mu$ [Phys. Rev. D 101 (2020) 072005], $D^+\to \eta\mu^+\nu_\mu$ [Phys. Rev. Lett. 124 (2020) 231801]. All these analyses used the tagged method and 2.93 fb$^{-1}$ of data taken at 3.773 GeV. The reported branching fractions are$$\begin{array}{rl}\mathcal{B}(D^{+}\to {\overline{K}}^0{\mu }^{+}{\nu }_{\mu })& =(8.72\pm 0.07\pm 0.18)\mathrm{\%},\\ \mathcal{B}(D^0\to {\pi }^{-}{\mu }^{+}{\nu }_{\mu })& =(0.272\pm 0.008\pm 0.006)\mathrm{\%},\\ \mathcal{B}(D^{+}\to {\pi }^0{\mu }^{+}{\nu }_{\mu })& =(0.350\pm 0.011\pm 0.010)\mathrm{\%},\\ \mathcal{B}(D^0\to K^{-}{\mu }^{+}{\nu }_{\mu })& =(3.413\pm 0.019\pm 0.035)\mathrm{\%},\\ \mathcal{B}(D^{+}\to \omega {\mu }^{+}{\nu }_{\mu })& =(0.177\pm 0.018\pm 0.011)\mathrm{\%},\\ \mathcal{B}(D^{+}\to \eta {\mu }^{+}{\nu }_{\mu })& =(0.104\pm 0.010\pm 0.005)\mathrm{\%}.\end{array}$$

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 $$\begin{array}{rl}\frac{\mathcal{B}(D^0\to {\pi }^{-}{\mu }^{+}{\nu }_{\mu })}{\mathcal{B}(D^0\to {\pi }^{-}{e}^{+}{\nu }_e)}& =0.922\pm 0.030\pm 0.022,\\ \\ \frac{\mathcal{B}(D^{+}\to {\pi }^0{\mu }^{+}{\nu }_{\mu })}{\mathcal{B}(D^{+}\to {\pi }^0{e}^{+}{\nu }_e)}& =0.964\pm 0.037\pm 0.026,\\ \\ \frac{\mathcal{B}(D^0\to K^{-}{\mu }^{+}{\nu }_{\mu })}{\mathcal{B}(D^0\to K^{-}{e}^{+}{\nu }_e)}& =0.974\pm 0.007\pm 0.012,\\ \\ \frac{\mathcal{B}(D^{+}\to \omega {\mu }^{+}{\nu }_{\mu })}{\mathcal{B}(D^{+}\to \omega e^{+}{\nu }_e)}& =1.05\pm 0.14,\\ \\ \frac{\mathcal{B}(D^{+}\to \eta {\mu }^{+}{\nu }_{\mu })}{\mathcal{B}(D^{+}\to \eta e^{+}{\nu }_e)}& =0.91\pm 0.13.\end{array}$$In addition, using the world average for $\mathcal{B}(D^+\to \bar K^0 e^+\nu_e)$ [PTEP 2020] gives $$\frac{\mathcal{B}(D^+\to \bar K^0\mu^+\nu_\mu)}{\mathcal{B}(D^+\to \bar K^0 e^+\nu_e)} = 1.00\pm0.03\,.$$ These results indicate that any $e$-$\mu$ LFU violation in $D$ semileptonic decays has to be at the level of a few percent or less. BESIII also tested $e$-$\mu$ LFU in separate $q^2$ intervals using $D^{0(+)}\to \pi^{-(0)}\mathcal{l}^+\nu_\mathcal{l}$ [Phys. Rev. Lett. 121 (2018) 171803] and $D^0\to K^-\mathcal{l}^+\nu_\mathcal{l}$ [Phys. Rev. Lett. 122 (2019) 011804] decays. No indication of LFU above the $2\sigma$ level was found.

In 2018, using 0.482 fb$^{-1}$ of data taken at a center-of-mass energy of 4.009 GeV, BESIII reported measurements of the branching fractions for semileptonic decays $D^+_s\to \phi\,\mu^+\nu_\mu$, $D^+_s\to \eta \mu^+\nu_\mu$, and $D^+_s\to \eta^\prime \mu^+\nu_\mu$[Phys. Rev. D 97 (2018) 012006]. Combining these results with previous measurements of $D^+_s\to \phi\,e^+\nu_e$ [Phys. Rev. D 97 (2018) 012006], $D^+_s\to \eta e^+\nu_e$, and $D^+_s\to \eta^\prime e^+\nu_e$ [Phys. Rev. D 94 (2016) 112003] gives the ratios $$\begin{array}{rl}\frac{\mathcal{B}(D_s^{+}\to \varphi {\mu }^{+}{\nu }_{\mu })}{\mathcal{B}(D_s^{+}\to \varphi e^{+}{\nu }_e)}& =0.86\pm 0.29,\\ \frac{\mathcal{B}(D_s^{+}\to \eta {\mu }^{+}{\nu }_{\mu })}{\mathcal{B}(D_s^{+}\to \eta e^{+}{\nu }_e)}& =1.05\pm 0.24,\\ \frac{\mathcal{B}(D_s^{+}\to {\eta }^{\prime }{\mu }^{+}{\nu }_{\mu })}{\mathcal{B}(D_s^{+}\to {\eta }^{\prime }{e}^{+}{\nu }_e)}& =1.14\pm 0.68.\end{array}$$ These values are all consistent with unity. The uncertainties include both statistical and systematic uncertainties, the former of which dominates.

2. $D\rightarrow V \mathcal{l} \nu_\mathcal{l}$ 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 $H^{}_\mu = V^{}_\mu + A^{}_\mu$, where [Phys. Rev. D 41 (1990) 142] $$\begin{array}{rl}{V}_{\mu }=⟨V(p,\epsilon )|\overline{q}{\gamma }_{\mu }c|D(p^{\prime })⟩=& \frac{2V(q^2)}{m_D+m_V}{\epsilon }_{\mu \nu \rho \sigma }{\epsilon }^{\ast \nu }{p}^{\mathrm{\prime }\rho }{p}^{\sigma }\\ A_{\mu }=& ⟨V(p,\epsilon )|-\overline{q}{\gamma }_{\mu }{\gamma }_{5}c|D(p^{\prime })⟩=-i(m_D+m_V)A_1(q^2){\epsilon }_{\mu }^{\ast }\\ & +i\frac{A_2(q^2)}{m_D+m_V}({\epsilon }^{\ast }\cdot q)(p^{\prime }+p{)}_{\mu }\\ & +i\frac{2m_V}{q^2}(A_3(q^2)-A_0(q^2))[{\epsilon }^{\ast }\cdot (p^{\prime }+p)]q_{\mu }.\end{array}$$ In this expression, $m_V$ is the invariant mass of the daughter particles of the $V$ meson and $$A_3(q^2) = \frac{m_D + m_V}{2m_V}A_1(q^2)- \frac{m_D - m_V}{2m_V}A_2(q^2)\,.$$ To avoid divergence of the $i\,\frac{2m_V}{q^2} \left (A_3(q^2)-A_0(q^2) \right )$ item, kinematics require that $A_3(0) = A_0(0)$. Terms proportional to $q_\mu$ are negligible for $\mathcal{l}=e$. Thus, only the three form factors $A_1(q^2)$, $A_2(q^2)$ and $V(q^2)$ are relevant for charm decays.

The differential decay rate is $$\begin{array}{rl}\frac{d\mathrm{\Gamma }(D\to V\overline{\mathcal{l}}{\nu }_{\mathcal{l}})}{d{q}^{2}d\cos {\theta }_{\mathcal{l}}}=& \frac{G_F^2|V_{cq}{|}^2}{128{\pi }^3{m}_D^2}{p}^{\ast }{q}^2\times \\ & [\frac{(1-\cos {\theta }_{\mathcal{l}}{)}^2}{2}|H_{-}{|}^2+\frac{(1+\cos {\theta }_{\mathcal{l}}{)}^2}{2}|H_{+}{|}^2+{\sin }^2{\theta }_{\mathcal{l}}|H_0{|}^2],\end{array}$$where $H^{}_\pm$ and $H^{}_0$ are helicity amplitudes, corresponding to helicities of the vector ($V$) meson. The helicity amplitudes can be expressed in terms of the form factors as $$\begin{array}{rl}{H}_{\pm }=& \frac{1}{m_D+m_V}[(m_D+m_V{)}^2{A}_1(q^2)\mp 2m_D^{}{p}^{\ast }V(q^2)]\\ H_0=& \frac{1}{|q|}\frac{m_D^2}{2m_V(m_D+m_V)}\times \\ & [(1-\frac{m_V^2-q^2}{m_D^2})(m_D+m_V{)}^2{A}_1(q^2)-4{p^{\ast }}^2{A}_2(q^2)].\end{array}$$ Here $p^*=\frac{\left [\left (m_D^2-(m_V+q)\right )^2\left (m_D^2-(m_V-q)\right )\right ]^{1/2}}{2 m_D}$ is the magnitude of the three-momentum of the $V$ system as measured in the $D$ rest frame, and $\theta_\mathcal{l}$ is the angle of the lepton momentum with respect to the direction opposite that of the $D$ in the $W$ rest frame (see Fig. 2 for the electron case, $\theta_e$). The left-handed nature of the quark current manifests itself as $|H_-|>|H_+|$. The differential decay rate for $D\rightarrow V\mathcal{l}\nu$ followed by the vector meson decaying into two pseudoscalars is $$\begin{array}{rl}\frac{d\mathrm{\Gamma }(D\to V\overline{\mathcal{l}}\nu ,V\to P_1{P}_2)}{d{q}^{2}d\cos {\theta }_{V}d\cos {\theta }_{\mathcal{l}}d\chi }=& \frac{3G_F^2}{2048{\pi }^4}|V_{cq}{|}^2\frac{p^{\ast }(q^2)q^2}{m_D^2}\mathcal{B}(V\to P_1{P}_2)\times \\ & \{(1+\cos {\theta }_{\mathcal{l}}{)}^2{\sin }^2{\theta }_V|H_{+}(q^2){|}^2\\ & +(1-\cos {\theta }_{\mathcal{l}}{)}^2{\sin }^2{\theta }_V|H_{-}(q^2){|}^2\\ & +4{\sin }^2{\theta }_{\mathcal{l}}{\cos }^2{\theta }_V|H_0(q^2){|}^2\\ & -4\sin {\theta }_{\mathcal{l}}(1+\cos {\theta }_{\mathcal{l}})\sin {\theta }_V\cos {\theta }_V\cos \chi H_{+}(q^2)H_0(q^2)\\ & +4\sin {\theta }_{\mathcal{l}}(1-\cos {\theta }_{\mathcal{l}})\sin {\theta }_V\cos {\theta }_V\cos \chi H_{-}(q^2)H_0(q^2)\\ & -2{\sin }^2{\theta }_{\mathcal{l}}{\sin }^2{\theta }_V\cos 2\chi H_{+}(q^2)H_{-}(q^2)\},\end{array}$$where the helicity angles $\theta^{}_\mathcal{l}$, $\theta^{}_V$, and acoplanarity angle $\chi$ are defined as shown in Fig. 2. Usually, the ratios of the form factors at $q^2=0$ are defined as $$\begin{array}{rlr}{r}_V^{}& \equiv & \frac{V(0)}{A_1(0)},\\ r_2^{}& \equiv & \frac{A_2(0)}{A_1(0)}.\end{array}$$ 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 $r_V$ and $r_2$ from several experiments. Most of the measurements assume that the $q^2$ dependence of the form factors is given by the simple pole ansatz. Some of these measurements do not consider a separate $S$-wave contribution; in this case such a contribution is implicitly included in the measured values.

3. $D\to S \mathcal{l} \nu_{\mathcal{l}}$decays

In 2018, BESIII reported measurements of semileptonic $D$ decays into a scalar meson, $D\to S \mathcal{l}\nu$. The experiment measured $D^{0(+)}\to a_0(980) e^+\nu_e$, with $a_0(980)\to\eta\pi$. Signal yields of $25.7^{+6.4}_{-5.7}$ events for $D^0\to a_0(980)^-e^+\nu_e$, and $10.2^{+5.0}_{-4.1}$ events for $D^+\to a_0(980)^0e^+\nu_e$, were obtained, resulting in statistical significances of greater than 6.5$\sigma$ and 3.0$\sigma$, respectively [Phys. Rev. Lett. 121 (2018) 081802]. As the branching fraction for $a_0(980)\to\eta\pi$ is not well-measured, BESIII reports the product branching fractions $$\begin{array}{rl}\mathcal{B}(D^0\to a_0(980{)}^{-}{e}^{+}{\nu }_e)\times \mathcal{B}(a_0(980{)}^{-}\to \eta {\pi }^{-})& =({1.33}_{-0.29}^{+0.33}\pm 0.09)\times {10}^{-4},\\ \mathcal{B}(D^{+}\to a_0(980{)}^0{e}^{+}{\nu }_e)\times \mathcal{B}(a_0(980{)}^0\to \eta {\pi }^0)& =({1.66}_{-0.66}^{+0.81}\pm 0.11)\times {10}^{-4}.\end{array}$$ 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 $a_0(980)\to\eta\pi$ branching fractions via isospin. The result is a difference of more than $2\sigma$. Taking the lifetimes of the $D^0$ and $D^+$ into account, and assuming $\mathcal{B}(a_0(980)^-\to\eta\pi^-) = \mathcal{B}(a_0(980)^0\to\eta\pi^0)$, the ratio of the partial widths is $$\frac{\Gamma(D^0\to a_0(980)^-e^+\nu_e)}{\Gamma(D^+\to a_0(980)^0e^+\nu_e)} = 2.03\pm 0.95\pm 0.06 \,.$$ This value is consistent with the prediction based on isospin symmetry.

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

4. $D\to A \mathcal{l} \nu_{\mathcal{l}}$ decays

Experimental studies of semileptonic $D$ decays into a Axial-vector meson $D\to A \mathcal{l}\nu$ are challenging due to low statistics and high backgrounds. In 2007, CLEO-c reported first evidence for the Cabibbo-favored decay $D^0\to K_1(1270)^-e^+\nu_e$ with a statistical significance of $4\sigma$ [Phys. Rev. Lett. 99 (2007) 191801]. The branching fraction was measured to be $\mathcal{B}(D^0\to K_1(1270)^-e^+\nu_e) = (7.6^{+4.1}_{-3.0}\pm0.6\pm0.7)\times 10^{-4}$. In 2019, BESIII reported the first observation of $D^+\to \bar K_1(1270)^0e^+\nu_e$, with statistical significance greater than $10\sigma$ [Phys. Rev. Lett. 123 (2019) 231801]. The branching fraction was measured to be $\mathcal{B}(D^+\to \bar K_1(1270)^0e^+\nu_e) = (23.0\pm2.6^{+1.8}_{-2.1}\pm2.5)\times 10^{-4}$. In 2021, the $D^0\to K_1(1270)^-e^+\nu_e$ decay was observed for the first time by BESIII with a statistical significance greater than $10\sigma$ [Phys. Rev. Lett. 127 (2021) 131801]. The reported branching fraction is $\mathcal{B}(D^+\to \bar K_1(1270)^0e^+\nu_e) = (10.9\pm1.3^{+0.9}_{-1.3}\pm1.2)\times 10^{-4}$. Here, the third errors listed arise from the branching fraction for $K_1(1270)\to K\pi\pi$. The obtained branching fractions are consistent with the theoretical calculations with the $K_1$ mixing angle of $33^\circ$ or $57^\circ$. Taking the lifetimes of $D^0$ and $D^+$ into account, the ratio of the partial widths is $$\frac{\Gamma(D^+\to \bar K_1(1270)^0e^+\nu_e)}{\Gamma(D^0\to K_1(1270)^-e^+\nu_e)} = 1.20\pm0.20\pm0.15.$$ This value agrees with unity as predicted by isospin symmetry.

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