Analysis of $D^+\to\bar K^0e^+\nu_e$ and $D^+\to\pi^0e^+\nu_e$ Semileptonic Decays

TL;DR

This paper measures the branching fractions and form factors of specific semileptonic D+ decays using BESIII data, providing precise tests of CKM matrix elements and form factor calculations.

Contribution

It provides new precise measurements of branching fractions and form factors for D+ semileptonic decays, and extracts CKM matrix elements using experimental data and lattice QCD inputs.

Findings

Branching fraction for D+ to K0e+nu_e: (8.60±0.06±0.15)%

Branching fraction for D+ to pi0e+nu_e: (3.63±0.08±0.05)%

Determined CKM elements |V_cs| and |V_cd| with uncertainties

Abstract

Using 2.93~fb$^{-1}$ of data taken at 3.773 GeV with the BESIII detector operated at the BEPCII collider, we study the semileptonic decays $D^+ \to \bar K^0e^+\nu_e$ and $D^+ \to \pi^0 e^+\nu_e$. We measure the absolute decay branching fractions $\mathcal B(D^+ \to \bar K^0e^+\nu_e)=(8.60\pm0.06\pm 0.15)\times10^{-2}$ and $\mathcal B(D^+ \to \pi^0e^+\nu_e)=(3.63\pm0.08\pm0.05)\times10^{-3}$, where the first uncertainties are statistical and the second systematic. We also measure the differential decay rates and study the form factors of these two decays. With the values of $|V_{cs}|$ and $|V_{cd}|$ from Particle Data Group fits assuming CKM unitarity, we obtain the values of the form factors at $q^2=0$, $f^K_+(0) = 0.725\pm0.004\pm 0.012$ and $f^{\pi}_+(0) = 0.622\pm0.012\pm 0.003$. Taking input from recent lattice QCD calculations of these form factors, we determine values of the CKM matrix elements $|V_{cs}|=0.944 \pm 0.005 \pm 0.015 \pm 0.024$ and $|V_{cd}|=0.210 \pm 0.004 \pm 0.001 \pm 0.009$, where the third uncertainties are theoretical.

Tables

Table 1. Table 1: The Δ ​ E Δ 𝐸 \Delta E requirements, the M BC subscript 𝑀 BC M_{\rm BC} signal regions, the yields of the D − superscript 𝐷 D^{-} tags ( N tag subscript 𝑁 tag N_{\rm tag} ) reconstructed in data, and the reconstruction efficiency ( ε tag subscript 𝜀 tag \varepsilon_{\rm tag} ) of D − superscript 𝐷 D^{-} tags. The uncertainties are statistical only.

Tag mode	$\Delta E$ (MeV)	$M_{\mathrm{BC}}$ (GeV$/c^2$)	$N_{\mathrm{tag}}$	${\epsilon }_{\mathrm{tag}}$ (%)
$D^{-}\to K^{+}{\pi }^{-}{\pi }^{-}$	$(-45,45)$	$(1.8640,1.8770)$	$806830\pm 1070$	$51.8\pm 0.1$
$D^{-}\to K_S^0{\pi }^{-}$	$(-45,45)$	$(1.8640,1.8770)$	$102755\pm 372$	$56.2\pm 0.2$
$D^{-}\to K_S^0{K}^{-}$	$(-45,45)$	$(1.8650,1.8770)$	$19566\pm 185$	$52.1\pm 0.5$
$D^{-}\to K^{+}{K}^{-}{\pi }^{-}$	$(-50,50)$	$(1.8650,1.8780)$	$68216\pm 966$	$41.2\pm 0.3$
$D^{-}\to K^{+}{\pi }^{-}{\pi }^{-}{\pi }^0$	$(-78,78)$	$(1.8620,1.8790)$	$271571\pm 2367$	$27.3\pm 0.1$
$D^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{-}$	$(-45,45)$	$(1.8640,1.8770)$	$32150\pm 371$	$56.9\pm 0.7$
$D^{-}\to K_S^0{\pi }^{-}{\pi }^0$	$(-75,75)$	$(1.8640,1.8790)$	$245303\pm 1273$	$31.3\pm 0.1$
$D^{-}\to K^{+}{\pi }^{-}{\pi }^{-}{\pi }^{-}{\pi }^{+}$	$(-52,52)$	$(1.8630,1.8775)$	$30923\pm 733$	$22.1\pm 0.2$
$D^{-}\to K_S^0{\pi }^{-}{\pi }^{-}{\pi }^{+}$	$(-50,50)$	$(1.8640,1.8770)$	$125740\pm 1203$	$33.0\pm 0.2$
Sum			$1703054\pm 3405$

Table 2. Table 2: The reconstruction efficiencies for D + → K ¯ 0 ​ e + ​ ν e → superscript 𝐷 superscript ¯ 𝐾 0 superscript 𝑒 subscript 𝜈 𝑒 D^{+}\to\bar{K}^{0}e^{+}\nu_{e} and D + → π 0 ​ e + ​ ν e → superscript 𝐷 superscript 𝜋 0 superscript 𝑒 subscript 𝜈 𝑒 D^{+}\to\pi^{0}e^{+}\nu_{e} determined from MC simulation. The efficiencies include the branching fractions for K ¯ 0 superscript ¯ 𝐾 0 \bar{K}^{0} and π 0 superscript 𝜋 0 \pi^{0} . The uncertainties are statistical only.

Tag mode	${\epsilon }_{\mathrm{tag},\mathrm{SL}}\left(D^{+}\to {\overline{K}}^0{e}^{+}{\nu }_e\right)$ (%)	$\epsilon \left(D^{+}\to {\overline{K}}^0{e}^{+}{\nu }_e\right)$ (%)	${\epsilon }_{\mathrm{tag},\mathrm{SL}}\left(D^{+}\to {\pi }^0{e}^{+}{\nu }_e\right)$ (%)	$\epsilon \left(D^{+}\to {\pi }^0{e}^{+}{\nu }_e\right)$ (%)
$D^{-}\to K^{+}{\pi }^{-}{\pi }^{-}$	$9.21\pm 0.02$	$17.77\pm 0.04$	$28.44\pm 0.06$	$54.88\pm 0.13$
$D^{-}\to K_S^0{\pi }^{-}$	$10.14\pm 0.05$	$18.05\pm 0.11$	$31.15\pm 0.15$	$55.43\pm 0.34$
$D^{-}\to K_S^0{K}^{-}$	$9.30\pm 0.08$	$17.84\pm 0.22$	$28.68\pm 0.23$	$55.02\pm 0.67$
$D^{-}\to K^{+}{K}^{-}{\pi }^{-}$	$7.39\pm 0.06$	$17.92\pm 0.18$	$22.53\pm 0.16$	$54.66\pm 0.53$
$D^{-}\to K^{+}{\pi }^{-}{\pi }^{-}{\pi }^0$	$4.98\pm 0.02$	$18.25\pm 0.09$	$15.49\pm 0.06$	$56.72\pm 0.29$
$D^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{-}$	$10.44\pm 0.11$	$18.34\pm 0.30$	$32.93\pm 0.33$	$57.82\pm 0.94$
$D^{-}\to K_S^0{\pi }^{-}{\pi }^0$	$5.67\pm 0.01$	$18.11\pm 0.08$	$17.83\pm 0.04$	$56.92\pm 0.25$
$D^{-}\to K^{+}{\pi }^{-}{\pi }^{-}{\pi }^{-}{\pi }^{+}$	$3.50\pm 0.04$	$15.88\pm 0.25$	$11.74\pm 0.14$	$53.20\pm 0.81$
$D^{-}\to K_S^0{\pi }^{-}{\pi }^{-}{\pi }^{+}$	$5.55\pm 0.02$	$16.84\pm 0.14$	$18.12\pm 0.06$	$54.97\pm 0.45$
Average		$17.83\pm 0.03$		$55.52\pm 0.10$

Table 3. Table 3: Summary of the systematic uncertainties considered in the measurements of the branching fractions of D + → K ¯ 0 ​ e + ​ ν e → superscript 𝐷 superscript ¯ 𝐾 0 superscript 𝑒 subscript 𝜈 𝑒 D^{+}\rightarrow\bar{K}^{0}e^{+}\nu_{e} and D + → π 0 ​ e + ​ ν e → superscript 𝐷 superscript 𝜋 0 superscript 𝑒 subscript 𝜈 𝑒 D^{+}\rightarrow\pi^{0}e^{+}\nu_{e} decays.

	Systematic uncertainty (%)
Source	$D^{+}\to {\overline{K}}^0{e}^{+}{\nu }_e$	$D^{+}\to {\pi }^0{e}^{+}{\nu }_e$
Number of $D^{-}$ tags	0.5	0.5
Tracking for $e^{+}$	0.19	0.15
PID for $e^{+}$	0.16	0.14
$K_S^0$ reconstruction	1.62	$\cdots$
${\pi }^0$ reconstruction	$\cdots$	1.00
Requirement on $E_{\gamma ,\max }$	0.1	0.1
Fit to $U_{\mathrm{miss}}$ distribution	0.2	0.6
Form factor	0.2	0.2
FSR recovery	0.1	0.5
MC statistics	0.2	0.2
$K_S^0/{\pi }^0$ branching fraction	0.07	0.03
Total	1.76	1.41

Table 4. Table 4: Summary of the range of each q 2 superscript 𝑞 2 q^{2} bin, the number of the observed signal events for D + → K ¯ 0 ​ e + ​ ν e → superscript 𝐷 superscript ¯ 𝐾 0 superscript 𝑒 subscript 𝜈 𝑒 D^{+}\to\bar{K}^{0}e^{+}\nu_{e} and D + → π 0 ​ e + ​ ν e → superscript 𝐷 superscript 𝜋 0 superscript 𝑒 subscript 𝜈 𝑒 D^{+}\to\pi^{0}e^{+}\nu_{e} in data.

$D^{+}\to {\overline{K}}^0{e}^{+}{\nu }_e$
Bin No.	1	2	3	4	5	6	7	8	9
$q^2$ (GeV${}_{}{}^2/c^4)$	$[0.0,0.2)$	$[0.2,0.4)$	$[0.4,0.6)$	$[0.6,0.8)$	$[0.8,1.0)$	$[1.0,1.2)$	$[1.2,1.4)$	$[1.4,1.6)$	$[1.6,q_{\max }^2)$
$N_{\mathrm{obs}}$	$5842\pm 81$	$4935\pm 73$	$4180\pm 67$	$3515\pm 62$	$2818\pm 55$	$2120\pm 48$	$1460\pm 40$	$860\pm 31$	$302\pm 19$
$D^{+}\to {\pi }^0{e}^{+}{\nu }_e$
Bin No.	1	2	3	4	5	6	7
$q^2$ (GeV${}_{}{}^2/c^4)$	$[0.0,0.3)$	$[0.3,0.6)$	$[0.6,0.9)$	$[0.9,1.2)$	$[1.2,1.5)$	$[1.5,2.0)$	$[2.0,q_{\max }^2)$
$N_{\mathrm{obs}}$	$658\pm 29$	$562\pm 27$	$467\pm 25$	$448\pm 24$	$401\pm 24$	$470\pm 26$	$404\pm 30$

Table 5. Table 5: Efficiency matrices ε i ​ j subscript 𝜀 𝑖 𝑗 \varepsilon_{ij} given in percent for D + → K 0 ¯ ​ e + ​ ν e → superscript 𝐷 ¯ superscript 𝐾 0 superscript 𝑒 subscript 𝜈 𝑒 D^{+}\to\bar{K^{0}}e^{+}\nu_{e} and D + → π 0 ​ e + ​ ν e → superscript 𝐷 superscript 𝜋 0 superscript 𝑒 subscript 𝜈 𝑒 D^{+}\to\pi^{0}e^{+}\nu_{e} decays. The column gives the true q 2 superscript 𝑞 2 q^{2} bin j 𝑗 j , while the row gives the reconstructed q 2 superscript 𝑞 2 q^{2} bin i 𝑖 i . The statistical uncertainties in the least significant digits are given in the parentheses.

$D^{+}\to {\overline{K}}^0{e}^{+}{\nu }_e$
Rec. $q^2$	True $q^2$ (GeV${}_{}{}^2/c^4$)
(GeV${}_{}{}^2/c^4$)	$[0.0,0.2)$	$[0.2,0.4)$	$[0.4,0.6)$	$[0.6,0.8)$	$[0.8,1.0)$	$[1.0,1.2)$	$[1.2,1.4)$	$[1.4,1.6)$	$[1.6,q_{\max }^2)$
$[0.0,0.2)$	$18.53\left(6\right)$	$0.95\left(1\right)$	$0.07\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$
$[0.2,0.4)$	$0.37\left(1\right)$	$16.86\left(6\right)$	$1.03\left(2\right)$	$0.05\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$
$[0.4,0.6)$	$0.00\left(0\right)$	$0.40\left(1\right)$	$16.03\left(6\right)$	$1.03\left(2\right)$	$0.03\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$
$[0.6,0.8)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.46\left(1\right)$	$15.72\left(6\right)$	$0.95\left(2\right)$	$0.02\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$
$[0.8,1.0)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.01\left(0\right)$	$0.44\left(1\right)$	$15.78\left(7\right)$	$0.93\left(2\right)$	$0.01\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$
$[1.0,1.2)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.01\left(0\right)$	$0.46\left(1\right)$	$15.76\left(8\right)$	$0.80\left(2\right)$	$0.01\left(0\right)$	$0.00\left(0\right)$
$[1.2,1.4)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.42\left(1\right)$	$15.58\left(9\right)$	$0.74\left(3\right)$	$0.00\left(0\right)$
$[1.4,1.6)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.38\left(2\right)$	$15.45\left(12\right)$	$0.78\left(5\right)$
$[1.6,q_{\max }^2)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.28\left(2\right)$	$15.98\left(19\right)$
$D^{+}\to {\pi }^0{e}^{+}{\nu }_e$
Rec. $q^2$	True $q^2$ (GeV${}_{}{}^2/c^4$)
(GeV${}_{}{}^2/c^4$)	$[0.0,0.3)$	$[0.3,0.6)$	$[0.6,0.9)$	$[0.9,1.2)$	$[1.2,1.5)$	$[1.5,2.0)$	$[2.0,q_{\max }^2)$
$[0.0,0.3)$	$53.84\left(15\right)$	$2.27\left(3\right)$	$0.17\left(1\right)$	$0.01\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$
$[0.3,0.6)$	$4.00\left(5\right)$	$48.24\left(15\right)$	$2.31\left(4\right)$	$0.14\left(1\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$	$0.01\left(0\right)$
$[0.6,0.9)$	$0.14\left(1\right)$	$5.66\left(6\right)$	$46.15\left(15\right)$	$2.34\left(4\right)$	$0.10\left(1\right)$	$0.00\left(0\right)$	$0.00\left(0\right)$
$[0.9,1.2)$	$0.04\left(0\right)$	$0.22\left(1\right)$	$6.24\left(6\right)$	$44.51\left(16\right)$	$2.16\left(4\right)$	$0.05\left(0\right)$	$0.00\left(0\right)$
$[1.2,1.5)$	$0.04\left(0\right)$	$0.08\left(1\right)$	$0.31\left(1\right)$	$6.33\left(7\right)$	$43.33\left(17\right)$	$1.36\left(3\right)$	$0.02\left(0\right)$
$[1.5,2.0)$	$0.03\left(0\right)$	$0.08\left(1\right)$	$0.22\left(1\right)$	$0.58\left(2\right)$	$6.52\left(8\right)$	$45.48\left(16\right)$	$1.12\left(3\right)$
$[2.0,q_{\max }^2)$	$0.13\left(1\right)$	$0.21\left(1\right)$	$0.34\left(1\right)$	$0.68\left(2\right)$	$1.30\left(3\right)$	$5.52\left(6\right)$	$50.46\left(19\right)$

Table 6. Table 6: Summary of the measured partial decay rates, relative statistical uncertainties, systematic uncertainties and corresponding correlation matrices for D + → K ¯ 0 ​ e + ​ ν e → superscript 𝐷 superscript ¯ 𝐾 0 superscript 𝑒 subscript 𝜈 𝑒 D^{+}\to\bar{K}^{0}e^{+}\nu_{e} and D + → π 0 ​ e + ​ ν e → superscript 𝐷 superscript 𝜋 0 superscript 𝑒 subscript 𝜈 𝑒 D^{+}\to\pi^{0}e^{+}\nu_{e} .

$D^{+}\to {\overline{K}}^0{e}^{+}{\nu }_e$
$q^2$ bin No.	1	2	3	4	5	6	7	8	9
$\Delta \Gamma$ (ns-1)	$16.97$	$15.29$	$13.57$	$11.65$	$9.33$	$7.06$	$4.96$	$2.97$	$1.01$
stat. uncert. (%)	$1.45$	$1.61$	$1.75$	$1.91$	$2.12$	$2.44$	$2.92$	$3.77$	$6.56$
stat. correl.	$1.000$
	$-0.073$	$1.000$
	$0.001$	$-0.084$	$1.000$
	$0.000$	$0.003$	$-0.091$	$1.000$
	$0.000$	$0.000$	$0.004$	$-0.085$	$1.000$
	$0.000$	$0.000$	$0.000$	$0.004$	$-0.085$	$1.000$
	$0.000$	$0.000$	$0.000$	$0.000$	$0.004$	$-0.075$	$1.000$
	$0.000$	$0.000$	$0.000$	$0.000$	$0.000$	$0.004$	$-0.069$	$1.000$
	$0.000$	$0.000$	$0.000$	$0.000$	$0.000$	$0.000$	$0.003$	$-0.059$	$1.000$
syst. uncert. (%)	$3.24$	$3.10$	$2.95$	$2.88$	$3.02$	$3.05$	$2.85$	$2.54$	$2.93$
syst. correl.	$1.000$
	$0.981$	$1.000$
	$0.979$	$0.976$	$1.000$
	$0.979$	$0.977$	$0.973$	$1.000$
	$0.978$	$0.976$	$0.973$	$0.970$	$1.000$
	$0.974$	$0.972$	$0.970$	$0.970$	$0.965$	$1.000$
	$0.966$	$0.964$	$0.963$	$0.962$	$0.960$	$0.954$	$1.000$
	$0.932$	$0.930$	$0.929$	$0.929$	$0.926$	$0.923$	$0.911$	$1.000$
	$0.891$	$0.889$	$0.886$	$0.888$	$0.886$	$0.883$	$0.875$	$0.840$	$1.000$
$D^{+}\to {\pi }^0{e}^{+}{\nu }_e$
$q^2$ bin No.	1	2	3	4	5	6	7
$\Delta \Gamma$ (ns-1)	$0.664$	$0.578$	$0.474$	$0.477$	$0.432$	$0.503$	$0.372$
stat. uncert. (%)	$4.55$	$5.53$	$6.60$	$6.48$	$7.28$	$6.52$	$8.97$
stat. correl.	$1.000$
	$-0.122$	$1.000$
	$0.011$	$-0.171$	$1.000$
	$-0.002$	$0.019$	$-0.190$	$1.000$
	$0.000$	$-0.003$	$0.021$	$-0.190$	$1.000$
	$0.000$	$-0.001$	$-0.005$	$0.016$	$-0.167$	$1.000$
	$-0.002$	$-0.003$	$-0.003$	$-0.008$	$-0.004$	$-0.128$	$1.000$
syst. uncert. (%)	$1.53$	$1.52$	$1.51$	$1.61$	$1.88$	$1.92$	$1.73$
syst. correl.	$1.000$
	$0.739$	$1.000$
	$0.742$	$0.664$	$1.000$
	$0.758$	$0.737$	$0.650$	$1.000$
	$0.772$	$0.740$	$0.712$	$0.698$	$1.000$
	$0.781$	$0.749$	$0.711$	$0.760$	$0.772$	$1.000$
	$0.760$	$0.730$	$0.697$	$0.727$	$0.756$	$0.740$	$1.000$

Table 7. Table 7: Summary of results of form factor fits for D + → K ¯ 0 ​ e + ​ ν e → superscript 𝐷 superscript ¯ 𝐾 0 superscript 𝑒 subscript 𝜈 𝑒 D^{+}\to\bar{K}^{0}e^{+}\nu_{e} and D + → π 0 ​ e + ​ ν e → superscript 𝐷 superscript 𝜋 0 superscript 𝑒 subscript 𝜈 𝑒 D^{+}\to\pi^{0}e^{+}\nu_{e} , where the first errors are statistical and the second systematic.

Single pole model
Decay mode	$f_{+}\left(0\right)\left|V_{cq}\right|$	$m_{\mathrm{pole}}$ (GeV$/c^2$)
$D^{+}\to {\overline{K}}^0{e}^{+}{\nu }_e$	$0.7094\pm 0.0035\pm 0.0111$	$1.935\pm 0.017\pm 0.006$
$D^{+}\to {\pi }^0{e}^{+}{\nu }_e$	$0.1429\pm 0.0020\pm 0.0009$	$1.898\pm 0.020\pm 0.003$
Modified pole model
Decay mode	$f_{+}\left(0\right)\left|V_{cq}\right|$	$\alpha$
$D^{+}\to {\overline{K}}^0{e}^{+}{\nu }_e$	$0.7052\pm 0.0038\pm 0.0112$	$0.294\pm 0.031\pm 0.010$
$D^{+}\to {\pi }^0{e}^{+}{\nu }_e$	$0.1400\pm 0.0024\pm 0.0010$	$0.285\pm 0.057\pm 0.010$
ISGW2 model
Decay mode	$f_{+}\left(0\right)\left|V_{cq}\right|$	$r$ (GeV${}_{}{}^{-1}c_{}^2$)
$D^{+}\to {\overline{K}}^0{e}^{+}{\nu }_e$	$0.7039\pm 0.0037\pm 0.0111$	$1.587\pm 0.023\pm 0.007$
$D^{+}\to {\pi }^0{e}^{+}{\nu }_e$	$0.1381\pm 0.0023\pm 0.0007$	$2.078\pm 0.067\pm 0.011$
Two-parameter series expansion
Decay mode	$f_{+}\left(0\right)\left|V_{cq}\right|$	$r_1$
$D^{+}\to {\overline{K}}^0{e}^{+}{\nu }_e$	$0.7053\pm 0.0040\pm 0.0112$	$-2.18\pm 0.14\pm 0.05$
$D^{+}\to {\pi }^0{e}^{+}{\nu }_e$	$0.1400\pm 0.0026\pm 0.0007$	$-2.01\pm 0.13\pm 0.02$
Three-parameter series expansion
Decay mode	$f_{+}\left(0\right)\left|V_{cq}\right|$	$r_1$	$r_2$
$D^{+}\to {\overline{K}}^0{e}^{+}{\nu }_e$	$0.6983\pm 0.0056\pm 0.0112$	$-1.76\pm 0.25\pm 0.06$	$-13.4\pm 6.3\pm 1.4$
$D^{+}\to {\pi }^0{e}^{+}{\nu }_e$	$0.1413\pm 0.0035\pm 0.0012$	$-2.23\pm 0.42\pm 0.06$	$1.4\pm 2.5\pm 0.4$

Equations

$$\frac{d\Gamma }{d{q}^2}=X\frac{G_F^2}{24{\pi }^3}|V_{cs(d)}{|}^2{p}^3|f_{+}(q^2){|}^2,$$

$$\mathcal{B}(D^{+}\to P{e}^{+}{\nu }_e)=\frac{N_{\mathrm{obs}}(D^{+}\to P{e}^{+}{\nu }_e)}{N_{\mathrm{tag}}\epsilon (D^{+}\to P{e}^{+}{\nu }_e)},$$

$$\mathcal{B}(D^{+}\to {\bar{K}}^0{e}^{+}{\nu }_e)=(8.60\pm 0.06\pm 0.15)\times 10^{-2}$$

$$\mathcal{B}(D^{+}\to {\pi }^0{e}^{+}{\nu }_e)=(3.63\pm 0.08\pm 0.05)\times 10^{-3},$$

$$I_K\equiv \frac{\Gamma (D^0\to K^{-}{e}^{+}{\nu }_e)}{\Gamma (D^{+}\to {\bar{K}}^0{e}^{+}{\nu }_e)}=1.03\pm 0.01\pm 0.02$$

$$I_{\pi }\equiv \frac{\Gamma (D^0\to {\pi }^{-}{e}^{+}{\nu }_e)}{2\Gamma (D^{+}\to {\pi }^0{e}^{+}{\nu }_e)}=1.03\pm 0.03\pm 0.02,$$

$$N_{\mathrm{obs}}^i=\sum _{j=1}^{N_{\mathrm{bins}}}{\epsilon }_{ij}{N}_{\mathrm{prd}}^j,$$

$${\epsilon }_{ij}=\frac{n_{ij}^{\mathrm{rec}}}{n_j^{\mathrm{gen}}}\frac{1}{{\epsilon }_{\mathrm{tag}}}{f}_{ij},$$

$$\Delta {\Gamma }_i=\frac{N_{\mathrm{prd}}^i}{{\tau }_{D^{+}}{N}_{\mathrm{tag}}}=\frac{1}{{\tau }_{D^{+}}{N}_{\mathrm{tag}}}\sum _j^{N_{\mathrm{bins}}}({\epsilon }^{-1}{)}_{ij}{N}_{\mathrm{obs}}^j,$$

$$C_{ij}={\left(\frac{1}{{\tau }_{D^{+}}{N}_{\mathrm{tag}}}\right)}^2\sum _{\alpha }{ϵ}_{i\alpha }^{-1}{ϵ}_{j\alpha }^{-1}[\delta (N_{\mathrm{obs}}^{\alpha }){]}^2,$$

$$C_{ij}={\left(\frac{1}{{\tau }_{D^{+}}{N}_{\mathrm{tag}}}\right)}^2\sum _{\alpha \beta }(N_{\mathrm{obs}}^{\alpha }{N}_{\mathrm{obs}}^{\beta }\mathrm{cov}[{ϵ}_{i\alpha }^{-1},{ϵ}_{j\beta }^{-1}]),$$

$$\mathrm{cov}[{ϵ}_{\alpha \beta }^{-1},{ϵ}_{ab}^{-1}]=\sum _{ij}({ϵ}_{\alpha i}^{-1}{ϵ}_{ai}^{-1})[{\sigma }^2({ϵ}_{ij}){]}^2({ϵ}_{j\beta }^{-1}{ϵ}_{jb}^{-1}).$$

$$f_{+}(q^2)=\frac{f_{+}(0)}{1-q^2/m_{\mathrm{pole}}^2},$$

$$f_{+}(q^2)=\frac{f_{+}(0)}{(1-q^2/m_{D_{(s)}^{\ast +}}^2)(1-\alpha q^2/m_{D_{(s)}^{\ast +}}^2)},$$

$$f_{+}(q^2)=f_{+}(q_{\max }^2){\left(1+\frac{r^2}{12}(q_{\max }^2-q^2)\right)}^{-2},$$

$$z(q^2,t_0)=\frac{\sqrt{t_{+}-q^2}-\sqrt{t_{+}-t_0}}{\sqrt{t_{+}-q^2}+\sqrt{t_{+}-t_0}},$$

$$f_{+}(q^2)=\frac{1}{P(q^2)\varphi (q^2,t_0)}\sum _{k=0}^{\infty }{a}_k(t_0)[z(q^2,t_0){]}^k,$$

$$\varphi (q^2,t_0)$$

$$f_{+}(q^2)=\frac{f_{+}(0)P(0)\varphi (0,t_0)(1+{\sum }_{k=1}^{k_{\max }}{r}_k[z(q^2,t_0){]}^k)}{P(q^2)\varphi (q^2,t_0)(1+{\sum }_{k=1}^{k_{\max }}{r}_k[z(0,t_0){]}^k)},$$

$${\chi }^2=\sum _{ij}(\Delta {\Gamma }_i-\Delta {\Gamma }_i^{\mathrm{th}}){\mathcal{C}}_{ij}^{-1}(\Delta {\Gamma }_j-\Delta {\Gamma }_j^{\mathrm{th}}),$$

$$\Delta {\Gamma }_i^{\mathrm{th}}={\int }_{q_{\min ,i}^2}^{q_{\max ,i}^2}X\frac{G_F^2}{24{\pi }^3}|V_{cs(d)}{|}^2{p}^3|f_{+}(q^2){|}^{2}d{q}^2,$$

$$f_{+}(q_i^2)=\sqrt{\frac{\Delta {\Gamma }_i}{\Delta q_i^2}\frac{24{\pi }^3}{X{G}_F^2{p_i^{\prime }}^3|V_{cq}{|}^2}}$$

$${p_i^{\prime }}^3=\frac{{\int }_{q_{\min ,i}^2}^{q_{\max ,i}^2}{p}^3|f_{+}(q^2){|}^{2}d{q}^2}{|f_{+}(q_i^2){|}^2(q_{\max ,i}^2-q_{\min ,i}^2)},$$

$$f_{+}^K(0)=0.725\pm 0.004\pm 0.012$$

$$f_{+}^{\pi }(0)=0.622\pm 0.012\pm 0.003,$$

$$|V_{cs}|=0.944\pm 0.005\pm 0.015\pm 0.024$$

$$|V_{cd}|=0.210\pm 0.004\pm 0.001\pm 0.009,$$

Full text

I Introduction

In the Standard Model (SM) of particle physics, the mixing between the quark flavours in the weak interaction is parameterized by the Cabibbo-Kobayashi-Maskawa (CKM) matrix, which is a $3\times 3$ unitary matrix. Since the CKM matrix elements are fundamental parameters of the SM, precise determinations of these elements are very important for tests of the SM and searches for New Physics (NP) beyond the SM.

Since the effects of strong and weak interactions can be well separated in semileptonic $D$ decays, these decays are excellent processes from which we can determine the magnitude of the CKM matrix element $V_{cs(d)}$. In the SM, neglecting the lepton mass, the differential decay rate for $D^{+}\to Pe^{+}\nu_{e}$ ($P=\bar{K}^{0}$ or $\pi^{0}$) is given by Z_Phys_C_46_93

[TABLE]

where $G_{F}$ is the Fermi constant, $V_{cs(d)}$ is the corresponding CKM matrix element, $p$ is the momentum of the meson $P$ in the rest frame of the $D$ meson, $q^{2}$ is the squared four momentum transfer, i.e., the invariant mass of the lepton and neutrino system, and $f_{+}(q^{2})$ is the form factor which parameterizes the effect of the strong interaction. In Eq. (1), $X$ is a multiplicative factor due to isospin, which equals to 1 for the decay $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ and $1/2$ for the decay $D^{+}\to\pi^{0}e^{+}\nu_{e}$.

In this article, we report the experimental study of $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ and $D^{+}\to\pi^{0}e^{+}\nu_{e}$ decays using a 2.93 fb*-1* lum data set collected at a center-of-mass energy of $\sqrt{s}=3.773$ GeV with the BESIII detector operated at the BEPCII collider. Throughout this paper, the inclusion of charge conjugate channels is implied.

The paper is structured as follows. We briefly describe the BESIII detector and the Monte Carlo (MC) simulation in Sec. II. The event selection is presented in Sec. III. The measurements of the absolute branching fractions and the differential decay rates are described in Sec. IV and V, respectively. In Sec. VI we discuss the determination of form factors from the measurements of decay rates, and finally, in Sec. VII, we present the determination of the magnitudes of the CKM matrix elements $V_{cs}$ and $V_{cd}$. A brief summary is given in Sec. VIII.

II BESIII detector

The BESIII detector is a cylindrical detector with a solid-angle coverage of $93\%$ of $4\pi$, designed for the study of hadron spectroscopy and $\tau$-charm physics. The BESIII detector is described in detail in Ref. bes3 . Detector components particularly relevant for this work are (1) the main drift chamber (MDC) with 43 layers surrounding the beam pipe, which performs precise determination of charged particle trajectories and provides a measurement of the specific ionization energy loss ($dE/dx$); (2) a time-of-flight system (TOF) made of plastic scintillator counters, which are located outside of the MDC and provide additional charged particle identification information; and (3) the electromagnetic calorimeter (EMC) consisting of 6240 CsI(Tl) crystals, used to measure the energy of photons and to identify electrons.

A geant4-based geant4 MC simulation software BOOST , which contains the detector geometry description and the detector response, is used to optimize the event selection criteria, study possible backgrounds, and determine the reconstruction efficiencies. The production of the $\psi(3770)$, initial state radiation production of $\psi(3686)$ and $J/\psi$, as well as the continuum processes of $e^{+}e^{-}\to\tau^{+}\tau^{-}$ and $e^{+}e^{-}\to q\bar{q}$ ($q=u,d,s$) are simulated by the MC event generator kkmc kkmc ; the known decay modes are generated by evtgen besevtgen with the branching fractions set to the world average values from the Particle Data Group (PDG) pdg2016 ; while the remaining unknown decay modes are modeled by lundcharm lundcharm . We also generate signal MC events consisting of $\psi(3770)\to D^{+}D^{-}$ events in which the $D^{-}$ meson decays to all possible final states and the $D^{+}$ meson decays to a hadronic or a semileptonic decay final state being investigated. In the generation of signal MC events, the semileptonic decays $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ and $D^{+}\to\pi^{0}e^{+}\nu_{e}$ are modeled by the the modified pole parametrization (see Sec. VI.1).

III Event reconstruction

The center-of-mass energy of 3.773 GeV corresponds to the peak of the $\psi(3770)$ resonance, which decays predominantly into $D\bar{D}$ ($D^{0}\bar{D}^{0}$ or $D^{+}D^{-}$) meson pairs. In events where a $D^{-}$ meson is fully reconstructed, the remaining particles must all be decay products of the accompanying $D^{+}$ meson. In the following, the reconstructed meson is called “tagged $D^{-}$” or “$D^{-}$ tag”. In a tagged $D^{-}$ data sample, the recoiling $D^{+}$ decays to $\bar{K}^{0}e^{+}\nu_{e}$ or $\pi^{0}e^{+}\nu_{e}$ can be cleanly isolated and used to measure the branching fraction and differential decay rates.

III.1 Selection of $D^{-}$ tags

We reconstruct $D^{-}$ tags in the following nine hadronic modes: $D^{-}\to K^{+}\pi^{-}\pi^{-}$, $D^{-}\to K^{0}_{S}\pi^{-}$, $D^{-}\to K^{0}_{S}K^{-}$, $D^{-}\to K^{+}K^{-}\pi^{-}$, $D^{-}\to K^{+}\pi^{-}\pi^{-}\pi^{0}$, $D^{-}\to\pi^{+}\pi^{-}\pi^{-}$ 111 We veto the $K^{0}_{S}\pi^{-}$ candidates when a $\pi^{+}\pi^{-}$ invariant mass falls within the $K_{S}^{0}$ mass window. , $D^{-}\to K^{0}_{S}\pi^{-}\pi^{0}$, $D^{-}\to K^{+}\pi^{-}\pi^{-}\pi^{-}\pi^{+}$, and $D^{-}\to K^{0}_{S}\pi^{-}\pi^{-}\pi^{+}$. The selection criteria of $D^{-}$ tags used here are the same as those described in Ref. BESIII_Dptomunu . Tagged $D^{-}$ mesons are identified by their beam-energy-constrained mass $M_{\rm BC}\equiv\sqrt{E_{\rm beam}^{2}/c^{4}-|\vec{p}_{\rm tag}|^{2}/c^{2}}$, where $E_{\rm beam}$ is the beam energy, and $\vec{p}_{\rm tag}$ is the measured 3-momentum of the tag candidate 222 In this analysis, all four-momentum vectors measured in the laboratory frame are boosted to the $e^{+}e^{-}$ center-of-mass frame. . We also use the variable $\Delta E\equiv E_{\rm tag}-E_{\rm beam}$, where $E_{\rm tag}$ is the measured energy of the tag candidate, to select the $D^{-}$ tags. Each tag candidate is subjected to a tag mode-dependent $\Delta E$ requirement as shown in Table 1. If there are multiple candidates per tag mode for an event, the one with the smallest value of $|\Delta E|$ is retained.

The $M_{\rm BC}$ distributions for the nine $D^{-}$ tag modes are shown in Fig. 1. A binned extended maximum likelihood fit is used to determine the number of tagged $D^{-}$ events for each of the nine modes. We use the MC simulated signal shape convolved with a double-Gaussian resolution function to represent the beam-energy-constrained mass signal for the $D^{-}$ daughter particles, and an ARGUS function Albrecht-1990am multiplied by a third-order polynomial BESII_D0toPenu ; BESIII_D0toPenu to describe the background shape for the $M_{\rm BC}$ distributions. In the fits all parameters of the double-Gaussian function, the ARGUS function, and the polynomial function are left free. The solid lines in Fig. 1 show the best fits, while the dashed lines show the fitted background shapes. The numbers of the $D^{-}$ tags ($N_{\rm tag}$) within the $M_{\rm BC}$ signal regions given by the two vertical lines in Fig. 1 are summarized in Table 1. In total, we find $1703054\pm 3405$ single $D^{-}$ tags reconstructed in data. The reconstruction efficiencies of the single $D^{-}$ tags, $\epsilon_{\rm tag}$, as determined with the MC simulation, are shown in Table 1.

III.2 Reconstruction of semileptonic decays

Candidates for semileptonic decays are selected from the remaining tracks in the system recoiling against the $D^{-}$ tags. The $dE/dx$, TOF and EMC measurements (deposited energy and shape of the electromagnetic shower) are combined to form confidence levels for the $e$ hypothesis ($CL_{e}$), the $\pi$ hypothesis ($CL_{\pi}$), and the $K$ hypothesis ($CL_{K}$). Positron candidates are required to have $CL_{e}$ greater than 0.1% and to satisfy $CL_{e}/(CL_{e}+CL_{\pi}+CL_{K})>0.8$. In addition, we include the 4-momenta of near-by photons within $5^{\circ}$ of the direction of the positron momentum to partially account for final-state-radiation energy losses (FSR recovery). The neutral kaon candidates are built from pairs of oppositely charged tracks that are assumed to be pions. For each pair of charged tracks, a vertex fit is performed and the resulting track parameters are used to calculate the invariant mass, $M(\pi^{+}\pi^{-})$. If $M(\pi^{+}\pi^{-})$ is in the range (0.484, 0.512) GeV$/c^{2}$, the $\pi^{+}\pi^{-}$ pair is treated as a $K_{S}^{0}$ candidate and is used for further analysis. The neutral pion candidates are reconstructed via the $\pi^{0}\to\gamma\gamma$ decays. For the photon selection, we require the energy of the shower deposited in the barrel (end-cap) EMC greater than 25 (50) MeV and the shower time be within 700 ns of the event start time. In addition, the angle between the photon and the nearest charged track is required to be greater than $10^{\circ}$. We accept the pair of photons as a $\pi^{0}$ candidate if the invariant mass of the two photons, $M(\gamma\gamma)$, is in the range (0.110, 0.150) GeV$/c^{2}$. A 1-Constraint (1-C) kinematic fit is then performed to constrain $M(\gamma\gamma)$ to the $\pi^{0}$ nominal mass, and the resulting 4-momentum of the candidate $\pi^{0}$ is used for further analysis.

We reconstruct the $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ decay by requiring exactly three additional charged tracks in the rest of the event. One track with charge opposite to that of the $D^{-}$ tag is identified as a positron using the criteria mentioned above, while the other two oppositely charged tracks form a $K_{S}^{0}$ candidate. For the selection of the $D^{+}\to\pi^{0}e^{+}\nu_{e}$ decay, we require that there is only one additional charged track consistent with the positron identification criteria and at least two photons that are used to form a $\pi^{0}$ candidate in the rest of the event. If there are multiple $\pi^{0}$ candidates, the one with the minimum $\chi^{2}$ from the 1-C kinematic fit is retained. In order to additionally suppress background due to wrongly reconstructed or background photons, the semileptonic candidate is further required to have the maximum energy of any of the unused photons, $E_{\gamma,\rm max}$, less than 300 MeV.

Since the neutrino is undetected, the kinematic variable $U_{\rm miss}\equiv E_{\rm miss}-c|\vec{p}_{\rm miss}|$ is used to obtain the information about the missing neutrino, where $E_{\rm miss}$ and $\vec{p}_{\rm miss}$ are, respectively, the total missing energy and momentum in the event. The missing energy is computed from $E_{\rm miss}=E_{\rm beam}-E_{P}-E_{e^{+}}$, where $E_{P}$ and $E_{e^{+}}$ are the measured energies of the pseudoscalar meson and the positron, respectively. The missing momentum $\vec{p}_{\rm miss}$ is given by $\vec{p}_{\rm miss}=\vec{p}_{D^{+}}-\vec{p}_{P}-\vec{p}_{e^{+}}$, where $\vec{p}_{D^{+}}$, $\vec{p}_{P}$ and $\vec{p}_{e^{+}}$ are the 3-momenta of the $D^{+}$ meson, the pseudoscalar meson and the positron, respectively. The 3-momentum of the $D^{+}$ meson is taken as $\vec{p}_{D^{+}}=-\hat{p}_{\rm tag}\sqrt{(E_{\rm beam}/c)^{2}-(m_{D^{+}}c)^{2}}$, where $\hat{p}_{\rm tag}$ is the direction of the momentum of the single $D^{-}$ tag, and $m_{D^{+}}$ is the $D^{+}$ mass. If the daughter particles from a semileptonic decay are correctly identified, $U_{\rm miss}$ is near zero, since only one neutrino is missing.

Figure 2 shows the $U_{\rm miss}$ distributions for the semileptonic candidates, where the potential backgrounds arise from the $D\bar{D}$ processes other than signal, $\psi(3770)\to$ non-$D\bar{D}$ decays, $e^{+}e^{-}\to\tau^{+}\tau^{-}$, continuum light hadron production, initial state radiation return to $J/\psi$ and $\psi(3686)$. The background for $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ is dominated by $D^{+}\to\bar{K}^{*}(892)^{0}e^{+}\nu_{e}$ and $D^{+}\to\bar{K}^{0}\mu^{+}\nu_{\mu}$. For $D^{+}\to\pi^{0}e^{+}\nu_{e}$, the background is mainly from $D^{+}\to K_{L}^{0}e^{+}\nu_{e}$ and $D^{+}\to K_{S}^{0}(\pi^{0}\pi^{0})e^{+}\nu_{e}$.

Following the same procedure described in Ref. BESIII_D0toPenu , we perform a binned extended maximum likelihood fit to the $U_{\rm miss}$ distribution for each channel to separate the signal from the background component. The signal shape is constructed from a convolution of a MC determined distribution and a Gaussian function that accounts for the difference of the $U_{\rm miss}$ resolutions between data and MC simulation. The background shape is formed from MC simulation. From the fits shown as the overlaid curves in Fig. 2, we obtain the yields of the observed signal events to be $N_{\rm obs}(D^{+}\to\bar{K}^{0}e^{+}\nu_{e})=26008\pm 168$ and $N_{\rm obs}(D^{+}\to\pi^{0}e^{+}\nu_{e})=3402\pm 70$, respectively.

To check the quality of the MC simulation, we examine the distributions of the reconstructed kinematic variables. Figure 3 shows the comparisons of the momentum distributions of data and MC simulation.

IV Branching fraction measurements

IV.1 Determinations of branching fractions

The branching fraction of the semileptonic decay $D^{+}\to Pe^{+}\nu_{e}$ is obtained from

[TABLE]

where $N_{\rm tag}$ is the number of $D^{-}$ tags (see Sec. III.1), $N_{\rm obs}(D^{+}\to Pe^{+}\nu_{e})$ is the number of observed $D^{+}\to Pe^{+}\nu_{e}$ decays within the $D^{-}$ tags (see Sec. III.2), and $\varepsilon(D^{+}\to Pe^{+}\nu_{e})$ is the reconstruction efficiency. Here the $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ efficiency includes the $K^{0}_{S}$ fraction of the $\bar{K}^{0}$ and $K^{0}_{S}\to\pi^{+}\pi^{-}$ branching fraction, the $D^{+}\to\pi^{0}e^{+}\nu_{e}$ efficiency includes the $\pi^{0}\to\gamma\gamma$ branching fraction pdg2016 .

Due to the difference in the multiplicity, the reconstruction efficiency varies slightly with the tag mode. For each tag mode $i$, the reconstruction efficiency is given by $\varepsilon^{i}=\varepsilon^{i}_{\rm tag,SL}/\varepsilon^{i}_{\rm tag}$, where the efficiency for simultaneously finding the $D^{+}\to Pe^{+}\nu_{e}$ semileptonic decay and the $D^{-}$ meson tagged with mode $i$, $\varepsilon^{i}_{\rm tag,SL}$, is determined using the signal MC sample, and $\varepsilon^{i}_{\rm tag}$ is the corresponding tag efficiency shown in Table 1. These efficiencies are listed in Table 2. The reconstruction efficiency for each tag mode is then weighted according to the corresponding tag yield in data to obtain the average reconstruction efficiency, $\bar{\varepsilon}=\sum_{i}(N_{\rm tag}^{i}\varepsilon^{i})/N_{\rm tag}$, as listed in the last row in Table 2.

Using the control samples selected from Bhabha scattering and $D\bar{D}$ events, we find that there are small discrepancies between data and MC simulation in the positron tracking efficiency, positron identification efficiency, $K_{S}^{0}$ and $\pi^{0}$ reconstruction efficiencies. We correct for these differences by multiplying the raw efficiencies $\varepsilon(D^{+}\to\bar{K}^{0}e^{+}\nu_{e})$ and $\varepsilon(D^{+}\to\pi^{0}e^{+}\nu_{e})$ determined in MC simulation by factors of 0.9957 and 0.9910, respectively. The corrected efficiencies are found to be $\epsilon^{\prime}(D^{+}\to\bar{K}^{0}e^{+}\nu_{e})=(17.75\pm 0.03)\%$ and $\epsilon^{\prime}(D^{+}\to\pi^{0}e^{+}\nu_{e})=(55.02\pm 0.10)\%$, where the uncertainties are only statistical.

Inserting the corresponding numbers into Eq. (2) yields the absolute decay branching fractions

[TABLE]

and

[TABLE]

where the first uncertainties are statistical and the second systematic.

IV.2 Systematic uncertainties

The systematic uncertainties in the measured branching fractions of $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ and $D^{+}\to\pi^{0}e^{+}\nu_{e}$ decays include the following contributions.

Number of $D^{-}$ tags. The systematic uncertainty of the number of $D^{-}$ tags is 0.5% BESIII_Dptomunu .

$e^{+}$* tracking efficiency.* Using the positron samples selected from radiative Bhabha scattering events, the $e^{+}$ tracking efficiencies are measured in data and MC simulation. Considering both the polar angle and momentum distributions of the positrons in the semileptonc decays, a correction factor of $1.0021\pm 0.0019$ ($1.0011\pm 0.0015$) is determined for the $e^{+}$ tracking efficiency in the branching fraction measurement of $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ ($D^{+}\to\pi^{0}e^{+}\nu_{e}$) decay. This correction is applied and an uncertainty of 0.19% (0.15%) is used as the corresponding systematic uncertainty.

$e^{+}$* identification efficiency.* Using the positron samples selected from radiative Bhabha scattering events, we measure the $e^{+}$ identification efficiencies in data and MC simulation. Taking both the polar angle and momentum distributions of the positrons in the semileptonic decays into account, a correction factor of $0.9993\pm 0.0016$ ($0.9984\pm 0.0014$) is determined for the $e^{+}$ identification efficiency in the measurement of $\mathcal{B}(D^{+}\to\bar{K}^{0}e^{+}\nu_{e})$ ($\mathcal{B}(D^{+}\to\pi^{0}e^{+}\nu_{e})$). This correction is applied, and an amount of 0.16% (0.14%) is assigned as the corresponding systematic uncertainty.

$K_{S}^{0}$* and $\pi^{0}$ reconstruction efficiency.* The momentum-dependent efficiencies for $K^{0}_{S}$ ($\pi^{0}$) reconstruction in data and in MC simulation are measured with $D\bar{D}$ events. Weighting these efficiencies according to the $K^{0}_{S}$ ($\pi^{0}$) momentum distribution in the semileptonic decay leads to a difference of $(-0.57\pm 1.62)\%$ ($(-0.85\pm 1.00)\%$) between the $K_{S}^{0}$ ($\pi^{0}$) reconstruction efficiencies in data and MC simulation. Since we correct for the systematic shift, the uncertainty of the correction factor, $1.62\%$ ($1.00\%$), is taken as the corresponding systematic uncertainty in the measured branching fraction of $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ ($D^{+}\to\pi^{0}e^{+}\nu_{e}$).

Requirement on $E_{\gamma,\rm max}$. By comparing doubly tagged $D\bar{D}$ hadronic decay events in the data and MC simulation, the systematic uncertainty due to this source is estimated to be 0.1%.

Fit to the $U_{\rm miss}$ distribution. To estimate the uncertainties due to the fits to the $U_{\rm miss}$ distributions, we refit the $U_{\rm miss}$ distributions by varying the bin size and the tail parameters (which are used to describe the signal shapes and are determined from MC simulation) to obtain the number of signal events from $D^{+}$ semileptonic decays. We then combine the changes in the yields in quadrature to obtain the systematic uncertainty (0.12% for $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$, 0.52% for $D^{+}\to\pi^{0}e^{+}\nu_{e}$). Since the background function is formed from many background modes with fixed relative normalizations, we also vary the relative contributions of several of the largest background modes based on the uncertainties in their branching fractions (0.12% for $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$, 0.01% for $D^{+}\to\pi^{0}e^{+}\nu_{e}$). In addition, we convolute the background shapes formed from MC simulation with the same Gaussian function in the fits (0.02% for $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$, 0.30% for $D^{+}\to\pi^{0}e^{+}\nu_{e}$). Finally we assign the relative uncertainties to be $0.2\%$ and $0.6\%$ for $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ and $D^{+}\to\pi^{0}e^{+}\nu_{e}$, respectively.

Form factor. In order to estimate the systematic uncertainty associated with the form factor used to generate signal events in the MC simulation, we re-weight the signal MC events so that the $q^{2}$ spectra agree with the measured spectra. We then remeasure the branching fraction (partial decay rates in different $q^{2}$ bins) with the newly weighted efficiency (efficiency matrix). The maximum relative change of the branching fraction (partial decay rates in different $q^{2}$ bins) is $0.2\%$ and is assigned as the systematic uncertainty.

FSR recovery. The differences between the results with FSR recovery and the ones without FSR recovery are assigned as the systematic uncertainties due to FSR recovery. We find the differences are $0.1\%$ and $0.5\%$ for $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ and $D^{+}\to\pi^{0}e^{+}\nu_{e}$, respectively.

MC statistics. The uncertainties in the measured branching fractions due to the MC statistics are the statistical fluctuation of the MC samples, which are $0.2\%$ for both of $D^{+}\rightarrow\bar{K}^{0}e^{+}\nu_{e}$ and $D^{+}\rightarrow\pi^{0}e^{+}\nu_{e}$ semileptonic decays.

$K_{S}^{0}$* and $\pi^{0}$ decay branching fractions.* We include an uncertainty of 0.07% (0.03%) on the branching fraction measurement of $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ ($D^{+}\to\pi^{0}e^{+}\nu_{e}$) to account for the uncertainty of the branching fraction of $K_{S}^{0}\to\pi^{+}\pi^{-}$ ($\pi^{0}\to\gamma\gamma$) decay pdg2016 .

Table 3 summarizes the systematic uncertainties in the measurement of the branching fractions. Adding all systematic uncertainties in quadrature yields the total systematic uncertainties of $1.76\%$ and $1.41\%$ for $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ and $D^{+}\to\pi^{0}e^{+}\nu_{e}$, respectively.

IV.3 Comparison

The comparisons of our measured branching fractions for $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ and $D^{+}\to\pi^{0}e^{+}\nu_{e}$ decays with those previously measured at the BES-II BESII_DptoK0enu , CLEO-c CLEOc_DtoPenu_818 and BESIII BESIII_DptoKLenu ; BESIII_DptoKSenu experiments as well as the PDG values pdg2016 are shown in Fig. 4. Our measured branching fractions are in agreement with the other experimental measurements, but are more precise. For $D^{+}\to\pi^{0}e^{+}\nu_{e}$, our result is lower than the only other existing measurement by CLEO-c CLEOc_DtoPenu_818 by $2.0\sigma$.

Using our previous measurements of $\mathcal{B}(D^{0}\to K^{-}e^{+}\nu_{e})$ and $\mathcal{B}(D^{0}\to\pi^{-}e^{+}\nu_{e})$ BESIII_D0toPenu , the results obtained in this analysis, and the lifetimes of $D^{0}$ and $D^{+}$ mesons pdg2016 , we obtain the ratios

[TABLE]

and

[TABLE]

which are consistent with isospin symmetry.

V Partial decay rate measurements

V.1 Determinations of partial decay rates

To study the differential decay rates, we divide the semileptonic candidates satisfying the selection criteria described in Sec. III into bins of $q^{2}$. Nine (seven) bins are used for $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ ($D^{+}\to\pi^{0}e^{+}\nu_{e}$). The range of each bin is given in Table 4. The squared four momentum transfer $q^{2}$ is determined for each semileptonic candidate by $q^{2}=(E_{e^{+}}+E_{\nu_{e}})^{2}/c^{4}-(\vec{p}_{e^{+}}+\vec{p}_{\nu_{e}})^{2}/c^{2}$, where the energy and momentum of the missing neutrino are taken to be $E_{\nu_{e}}=E_{\rm miss}$ and $\vec{p}_{\nu_{e}}=E_{\rm miss}\hat{p}_{\rm miss}/c$, respectively. For each $q^{2}$ bin, we perform a maximum likelihood fit to the corresponding $U_{\rm miss}$ distribution following the same procedure described in Sec. III.2 and obtain the signal yields as shown in Table 4.

To account for detection efficiency and detector resolution, the number of events $N^{i}_{\rm obs}$ observed in the $i$th $q^{2}$ bin is extracted from the relation

[TABLE]

where $N_{\rm bins}$ is the number of $q^{2}$ bins, $N_{\rm prd}^{j}$ is the number of semileptonic decay events produced in the tagged $D^{-}$ sample with the $q^{2}$ filled in the $j$th bin, and $\varepsilon_{ij}$ is the overall efficiency matrix that describes the efficiency and smearing across $q^{2}$ bins. The efficiency matrix element $\varepsilon_{ij}$ is obtained by

[TABLE]

where $n^{\rm rec}_{ij}$ is the number of the signal MC events generated in the $j$th $q^{2}$ bin and reconstructed in the $i$th $q^{2}$ bin, $n^{\rm gen}_{j}$ is the total number of the signal MC events which are generated in the $j$th $q^{2}$ bin, and $f_{ij}$ is the matrix to correct for data-MC differences in the efficiencies for $e^{+}$ tracking, $e^{+}$ identification, and $\bar{K}^{0}$ ($\pi^{0}$) reconstruction. Table 5 presents the average overall efficiency matrices for $D^{+}\to\bar{K^{0}}e^{+}\nu_{e}$ and $D^{+}\to\pi^{0}e^{+}\nu_{e}$ decays. To produce this average overall efficiency matrix, we combine the efficiency matrices for each tag mode weighted by its yield shown in Table 1. The diagonal elements of the matrix give the overall efficiencies for $D^{+}\to Pe^{+}\nu_{e}$ decays to be reconstructed in the correct $q^{2}$ bins in the recoil of the single $D^{-}$ tags, while the neighboring off-diagonal elements of the matrix give the overall efficiencies for cross feed between different $q^{2}$ bins.

The partial decay width in the $i$th bin is obtained by inverting the matrix Eq. (7),

[TABLE]

where $\tau_{D^{+}}$ is the lifetime of the $D^{+}$ meson pdg2016 . The $q^{2}$-dependent partial widths for $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ and $D^{+}\to\pi^{0}e^{+}\nu_{e}$ are summarized in Table 6. Also shown in Table 6 are the statistical uncertainties and the associated correlation matrices.

V.2 Systematic covariance matrices

For each source of systematic uncertainty in the measurements of partial decay rates, we construct an $N_{\rm bins}\times N_{\rm bins}$ systematic covariance matrix. A brief description of each contribution follows.

$D^{+}$* lifetime.* The systematic uncertainty associated with the lifetime of the $D^{+}$ meson (0.7%) pdg2016 is fully correlated across $q^{2}$ bins.

Number of $D^{-}$ tags. The systematic uncertainty from the number of the single $D^{-}$ tags (0.5%) is fully correlated between $q^{2}$ bins.

$e^{+}$, $K_{S}^{0}$, and $\pi^{0}$ reconstruction. The covariance matrices for the systematic uncertainties associated with the $e^{+}$ tracking, $e^{+}$ identification, $K_{S}^{0}$, and $\pi^{0}$ reconstruction efficiencies are obtained in the following way. We first vary the corresponding correction factors according to their uncertainties, then remeasure the partial decay rates using the efficiency matrices determined from the re-corrected signal MC events. The covariance matrix due to this source is assigned via $C_{ij}=\delta(\Delta\Gamma_{i})\delta(\Delta\Gamma_{j})$, where $\delta(\Delta\Gamma_{i})$ denotes the change in the partial decay rate measurement in the $i$th $q^{2}$ bin.

Requirement on $E_{\gamma,\rm max}$. We take the systematic uncertainty of $0.1\%$ due to the $E_{\gamma,\rm max}$ requirement on the selected events in each $q^{2}$ bin, and assume that this uncertainty is fully correlated between $q^{2}$ bins.

Fit to the $U_{\rm miss}$ distribution. The technique of fitting the $U_{\rm miss}$ distributions affects the number of signal events observed in the $q^{2}$ bins. The covariance matrix due to the $U_{\rm miss}$ fits is determined by

[TABLE]

where $\delta(N_{\rm obs}^{\alpha})$ is the systematic uncertainty of $N_{\rm obs}^{\alpha}$ associated with the fit to the corresponding $U_{\rm miss}$ distribution.

Form factor. To estimate the systematic uncertainty associated with the form factor model used to generate signal events in the MC simulation, we re-weight the signal MC events so that the $q^{2}$ spectra agree with the measured spectra. We then re-calculate the partial decay rates in different $q^{2}$ bins with the new efficiency matrices which are determined using the weighted MC events. The covariance matrix due to this source is assigned via $C_{ij}=\delta(\Delta\Gamma_{i})\delta(\Delta\Gamma_{j})$, where $\delta(\Delta\Gamma_{i})$ denotes the change of the partial width measurement in the $i$th $q^{2}$ bin.

FSR recovery. To estimate the systematic covariance matrix associated with the FSR recovery of the positron momentum, we remeasure the partial decay rates without the FSR recovery. The covariance matrix due to this source is assigned via $C_{ij}=\delta(\Delta\Gamma_{i})\delta(\Delta\Gamma_{j})$, where $\delta(\Delta\Gamma_{i})$ denotes the change of the partial decay rate measurement in the $i$th $q^{2}$ bin.

MC statistics. The systematic uncertainties due to the limited size of the MC samples used to determine the efficiency matrices are translated to the covariance via

[TABLE]

where the covariance of the inverse efficiency matrix elements are given by cov_inverse_matrix

[TABLE]

$K_{S}^{0}$* and $\pi^{0}$ decay branching fractions.* The systematic uncertainties due to the branching fractions of $K_{S}^{0}\to\pi^{+}\pi^{-}$ (0.07%) and $\pi^{0}\to\gamma\gamma$ (0.03%) are fully correlated between $q^{2}$ bins.

The total systematic covariance matrix is obtained by summing all these matrices. Table 6 summarizes the relative size of systematic uncertainties and the corresponding correlations in the measurements for the partial decay rates of the $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ and $D^{+}\to\pi^{0}e^{+}\nu_{e}$ semileptonic decays.

VI Form Factors

To determine the product $f_{+}(0)|V_{cs(d)}|$ and other form factor parameters, we fit the measured partial decay rates using Eq. (1) with the parameterization of the form factor $f_{+}(q^{2})$. In this analysis, we use several forms of the form factor parameterizations which are reviewed in Sec. VI.1.

VI.1 Form factor parameterizations

In general, the single pole model is the simplest approach to describe the $q^{2}$ dependence of the form factor. The single pole model is expressed as

[TABLE]

where $f_{+}(0)$ is the value of the form factor at $q^{2}=0$, and $m_{\rm pole}$ is the pole mass, which is often treated as a free parameter to improve fit quality.

The modified pole model BK is also widely used in Lattice QCD (LQCD) calculations and experimental studies of these decays. In this parameterization, the form factor of the semileptonic $D\to Pe^{+}\nu_{e}$ decays is written as

[TABLE]

where $m_{D^{*+}_{(s)}}$ is the mass of the $D^{*+}_{(s)}$ meson, and $\alpha$ is a free parameter to be fitted.

The ISGW2 model ISGW2 assumes

[TABLE]

where $q^{2}_{\rm max}$ is the kinematical limit of $q^{2}$, and $r$ is the conventional radius of the meson.

The most general parameterization of the form factor is the series expansion ff_zexpansion , which is based on analyticity and unitarity. In this parameterization, the variable $q^{2}$ is mapped to a new variable $z$ through

[TABLE]

with $t_{\pm}=(m_{D^{+}}\pm m_{P})^{2}$ and $t_{0}=t_{+}(1-\sqrt{1-t_{-}/t_{+}})$. The form factor is then expressed in terms of the new variable $z$ as

[TABLE]

where $a_{k}(t_{0})$ are real coefficients. The function $P(q^{2})$ is $P(q^{2})=z(t,m^{2}_{D^{*}_{s}})$ for $D\to K$ and $P(q^{2})=1$ for $D\to\pi$. The standard choice of $\phi(q^{2},t_{0})$ is

[TABLE]

where $m_{c}$ is the mass of the charm quark.

In practical use, one usually makes a truncation of the above series. After optimizing the form factor parameters, we obtain

[TABLE]

where $r_{k}\equiv a_{k}(t_{0})/a_{0}(t_{0})$. In this analysis we fit the measured decay rates to the two- or three-parameter series expansion, i.e., we take $k_{\rm max}=1$ or $2$. In fact, the $z$ expansion with only a linear term is sufficient to describe the data. Therefore we take the two-parameter series expansion as the nominal parameterization to determine $f_{+}^{K(\pi)}(0)$ and $|V_{cs(d)}|$.

VI.2 Fitting partial decay rates to extract form factors

In order to determine the form factor parameters, we fit the theoretical parameterizations to the measured partial decay rates. Taking into account the correlations of the measured partial decay rates among $q^{2}$ bins, the $\chi^{2}$ to be minimized in the fit is defined as

[TABLE]

where $\Delta\Gamma_{i}$ is the measured partial decay rate in the $i$th $q^{2}$ bin, $\mathcal{C}_{ij}^{-1}$ is the inverse matrix of the covariance matrix $\mathcal{C}_{ij}$. In the $i$th $q^{2}$ bin, the theoretical expectation of the partial decay rate is obtained by integrating Eq. (1),

[TABLE]

where $q^{2}_{{\rm min},i}$ and $q^{2}_{{\rm max},i}$ are the lower and upper boundaries of that $q^{2}$ bin, respectively.

In the fits, all parameters of the form factor parameterizations are left free. The central values of the form factor parameters are taken from the results obtained by fitting the data with the combined statistical and systematic covariance matrix together. The quadratic difference between the uncertainties of the fit parameters obtained from the fits with the combined covariance matrix and the uncertainties of the fit parameters obtained from the fits with the statistical covariance matrix only is taken as the systematic error of the measured form factor parameter. The results of these fits are summarized in Table 7, where the first errors are statistical and the second systematic.

Figure 5 shows the fits to the measured differential decay rates for $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ and $D^{+}\to\pi^{0}e^{+}\nu_{e}$. Figure 6 shows the projection of the fits onto $f_{+}(q^{2})$ for the $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ and $D^{+}\to\pi^{0}e^{+}\nu_{e}$ decays, respectively. In these two figures, the dots with error bars show the measured values of the form factors, $f_{+}(q^{2})$, in the center of each $q^{2}$ bin, which are obtained with

[TABLE]

in which

[TABLE]

where $|V_{cs}|=0.97351\pm 0.00013$ and $|V_{cd}|=0.22492\pm 0.00050$ are taken from the SM constraint fit pdg2016 . In the calculation of ${p^{\prime}_{i}}^{3}$, $f_{+}(q^{2})$ is computed using the two parameter series parameterization with the measured parameters.

VI.3 Determinations of $f_{+}^{K}(0)$ and $f_{+}^{\pi}(0)$

Using the $f_{+}^{K(\pi)}(0)|V_{cs(d)}|$ values from the two-parameter series expansion fits and taking the values of $|V_{cs(d)}|$ from the SM constraint fit pdg2016 as inputs, we obtain the form factors

[TABLE]

and

[TABLE]

where the first errors are statistical and the second systematic.

VII Determinations of $|V_{cs}|$ and $|V_{cd}|$

Using the values of $f_{+}^{K(\pi)}(0)|V_{cs(d)}|$ from the two-parameter $z$-series expansion fits and in conjunction with the form factor values $f_{+}^{K}(0)=0.747\pm 0.011\pm 0.015$ LQCD_fK and $f_{+}^{\pi}(0)=0.666\pm 0.020\pm 0.021$ LQCD_fpi calculated from LQCD, we obtain

[TABLE]

and

[TABLE]

where the first uncertainties are statistical, the second systematic, and the third are due to the theoretical uncertainties in the LQCD calculations of the form factors.

VIII Summary

In summary, by analyzing 2.93 fb*-1* of data collected at 3.773 GeV with the BESIII detector at the BEPCII, the semileptonic decays for $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ and $D^{+}\to\pi^{0}e^{+}\nu_{e}$ have been studied. From a total of $1703054\pm 3405$ $D^{-}$ tags, $26008\pm 168$ $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ and $3402\pm 70$ $D^{+}\to\pi^{0}e^{+}\nu_{e}$ signal events are observed in the system recoiling against the $D^{-}$ tags. These yield the absolute decay branching fractions to be $\mathcal{B}(D^{+}\to\bar{K}^{0}e^{+}\nu_{e})=(8.60\pm 0.06\pm 0.15)\times 10^{-2}$ and $\mathcal{B}(D^{+}\to\pi^{0}e^{+}\nu_{e})=(3.63\pm 0.08\pm 0.05)\times 10^{-3}$.

We also study the relations between the partial decay rates and squared 4-momentum transfer $q^{2}$ for these two decays and obtain the parameters of different form factor parameterizations. The products of the form factors and the related CKM matrix elements extracted from the two-parameter series expansion parameterization are selected as our primary results. We obtain $f_{+}(0)|V_{cs}|=0.7053\pm 0.0040\pm 0.0112$ and $f_{+}(0)|V_{cd}|=0.1400\pm 0.0026\pm 0.0007$. Using the global SM fit values for $|V_{cs}|$ and $|V_{cd}|$, we obtain the form factors $f^{K}_{+}(0)=0.725\pm 0.004\pm 0.012$ and $f^{\pi}_{+}(0)=0.622\pm 0.012\pm 0.003$. Furthermore, using the form factors predicted by the LQCD calculations, we obtain the CKM matrix elements $|V_{cs}|=0.944\pm 0.005\pm 0.015\pm 0.024$ and $|V_{cd}|=0.210\pm 0.004\pm 0.001\pm 0.009$, where the third errors are dominated by the theoretical uncertainties in the LQCD calculations of the form factors.

Acknowledgements.

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support. This work is supported in part by National Key Basic Research Program of China under Contract Nos. 2009CB825204, 2015CB856700; National Natural Science Foundation of China (NSFC) under Contracts Nos. 10935007, 11235011, 11305180, 11322544, 11335008, 11425524, 11635010; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the CAS Center for Excellence in Particle Physics (CCEPP); the Collaborative Innovation Center for Particles and Interactions (CICPI); Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contracts Nos. U1232201, U1332201, U1532257, U1532258; CAS under Contracts Nos. KJCX2-YW-N29, KJCX2-YW-N45; 100 Talents Program of CAS; National 1000 Talents Program of China; INPAC and Shanghai Key Laboratory for Particle Physics and Cosmology; German Research Foundation DFG under Contracts Nos. Collaborative Research Center CRC 1044, FOR 2359; Istituto Nazionale di Fisica Nucleare, Italy; Koninklijke Nederlandse Akademie van Wetenschappen (KNAW) under Contract No. 530-4CDP03; Ministry of Development of Turkey under Contract No. DPT2006K-120470; The Swedish Resarch Council; U. S. Department of Energy under Contracts Nos. DE-FG02-05ER41374, DE-SC-0010118, DE-SC-0010504, DE-SC-0012069; U.S. National Science Foundation; University of Groningen (RuG) and the Helmholtzzentrum fuer Schwerionenforschung GmbH (GSI), Darmstadt; WCU Program of National Research Foundation of Korea under Contract No. R32-2008-000-10155-0.