Measurement of b-hadron fractions in 13 TeV pp collisions

TL;DR

This paper reports precise measurements of b-hadron production fractions in 13 TeV proton-proton collisions, revealing dependencies on transverse momentum and providing key data for understanding heavy-flavor production at the LHC.

Contribution

First measurement of b-hadron fractions in 13 TeV pp collisions using semileptonic decays, with detailed dependence on transverse momentum and pseudorapidity.

Findings

Measured B_s^0 and Lambda_b^0 fractions with uncertainties.

Lambda_b^0 ratio varies strongly with transverse momentum.

B_s^0 ratio shows mild dependence on transverse momentum.

Abstract

The production fractions of $\overline{B}_s^0$ and $\Lambda_b^0$ hadrons, normalized to the sum of $B^-$ and $\overline{B}^0$ fractions, are measured in 13 TeV pp collisions using data collected by the LHCb experiment, corresponding to an integrated luminosity of 1.67/fb. These ratios, averaged over the $b$-hadron transverse momenta from 4 to 25 GeV and pseudorapidity from 2 to 5, are $0.122 \pm 0.006$ for $\overline{B}_s^0$, and $0.259 \pm 0.018$ for $\Lambda_b^0$, where the uncertainties arise from both statistical and systematic sources. The $\Lambda_b^0$ ratio depends strongly on transverse momentum, while the $\overline{B}_s^0$ ratio shows a mild dependence. Neither ratio shows variations with pseudorapidity. The measurements are made using semileptonic decays to minimize theoretical uncertainties. In addition, the ratio of $D^+$ to $D^0$ mesons produced in the sum of $\overline{B}^0$ and $B^-$ semileptonic decays is determined as $0.359\pm0.006\pm 0.009$, where the uncertainties are statistical and systematic.

Tables

Table 1. Table 1: Branching fractions of semileptonic b 𝑏 b -hadron decays from direct measurements for B ¯ 0 {\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0} and B − superscript 𝐵 {{B}^{-}} mesons, ( ⟨ B ⟩ ≡ ⟨ B ¯ + 0 B − ⟩ \left<B\right>\equiv\left<{{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}+B^{-}\right> ), and derived for B ¯ s 0 {{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}} and Λ b 0 subscript superscript Λ 0 𝑏 {\mathchar 28931\relax}^{0}_{b} hadrons based on the equality of semileptonic widths and the lifetime ratios  [ 3 , 5 ] . Corrections to Γ SL subscript Γ SL \Gamma_{\rm SL} for B ¯ s 0 {\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s} ( − 1.0 ± 0.5 ) plus-or-minus 1.0 0.5 (-1.0\pm 0.5) % and Λ b 0 subscript superscript Λ 0 𝑏 {\mathchar 28931\relax}^{0}_{b} ( 3.0 ± 1.5 ) plus-or-minus 3.0 1.5 (3.0\pm 1.5) % are applied [ 5 ] . Correlations in the B ¯ 0 {\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0} and B − superscript 𝐵 {{B}^{-}} branching fraction measurements have been taken into account. See Ref.  [ 19 ] for more information.

Particle	$\tau$ (ps)	${ℬ}_{\mathrm{SL}}$ (%)	${ℬ}_{\mathrm{SL}}$ (%)
	measured	measured	used
$\overline{B}{}^0$	$1.520\pm 0.004$	$10.30\pm 0.19$	$10.30\pm 0.19$
$B^{-}$	$1.638\pm 0.004$	$11.08\pm 0.20$	$11.08\pm 0.20$
		$10.70\pm 0.19$	$10.70\pm 0.19$
$\overline{B}{}_s^0$	$1.526\pm 0.015$		$10.24\pm 0.21$
${\Lambda }_b^0$	$1.470\pm 0.010$		$10.26\pm 0.25$

Table 2. Table 2: Charm-hadron branching fractions for the decay modes used in this analysis. Note, the Λ c + subscript superscript Λ 𝑐 {\mathchar 28931\relax}^{+}_{c} branching fraction has been significantly improved since the previous analysis.

Decay	$ℬ$ (%)	Source
$D^0\to K^{-}{\pi }^{+}$	$3.93\pm 0.05$	PDG average [3]
$D^{+}\to K^{-}{\pi }^{+}{\pi }^{+}$	$9.22\pm 0.17$	CLEO-c [20]
$D_s^{+}\to K^{-}{K}^{+}{\pi }^{+}$	$5.44\pm 0.18$	PDG average [3]
${\Lambda }_c^{+}\to p{K}^{-}{\pi }^{+}$	$6.23\pm 0.33$	From Refs. [21, 22]

Table 3. Table 3: Global systematic uncertainties. The D 0 superscript 𝐷 0 D^{0} and D + superscript 𝐷 D^{+} branching fraction uncertainties are scaled by the fraction of each decay, f 0 subscript 𝑓 0 f_{0} and f + subscript 𝑓 f_{+} for f s / ( f u + f d ) subscript 𝑓 𝑠 subscript 𝑓 𝑢 subscript 𝑓 𝑑 f_{s}/(f_{u}+f_{d}) and f Λ b 0 / ( f u + f d ) subscript 𝑓 subscript superscript Λ 0 𝑏 subscript 𝑓 𝑢 subscript 𝑓 𝑑 f_{{{\mathchar 28931\relax}^{0}_{b}}}/(f_{u}+f_{d}) uncertainties.

Source	Value (%)
	$f_s/\left(f_u+f_d\right)$	$f_{{\Lambda }_b^0}/\left(f_u+f_d\right)$	$f_{+}/f_0$
Simulation	1.7	2.4	–
Backgrounds	0.9	0.3	–
Cross-feeds	1.2	0.4	0.2
$ℬ$($D^0\to K^{-}{\pi }^{+}$)	1.0	1.0	1.3
$ℬ$($D^{+}\to K^{+}{\pi }^{-}{\pi }^{-}$)	0.6	0.6	1.8
$ℬ$($D_s^{+}\to K^{+}{K}^{-}{\pi }^{+}$)	3.3	–	–
$ℬ$(${\Lambda }_c^{+}\to p{K}^{+}{\pi }^{-}$)	–	5.3	–
Measured lifetime ratio	1.2	0.7	–
${\Gamma }_{SL}$ correction	0.5	1.5	–
Total	4.3	6.1	2.2

Table 4. Table 4: Values of f s / ( f u + f d ) subscript 𝑓 𝑠 subscript 𝑓 𝑢 subscript 𝑓 𝑑 f_{s}/(f_{u}+f_{d}) and f Λ b 0 / ( f u + f d ) subscript 𝑓 subscript superscript Λ 0 𝑏 subscript 𝑓 𝑢 subscript 𝑓 𝑑 f_{{{\mathchar 28931\relax}^{0}_{b}}}/(f_{u}+f_{d}) in each p T ​ ( H b ) subscript 𝑝 T subscript 𝐻 𝑏 p_{\mathrm{T}}(H_{b}) bin. The first uncertainty is statistical and incorporates both the uncertainties due to the data sample size and the finite amount of simulated events, while the second is the overall systematic uncertainty, including global and bin-dependent systematic uncertainties.

$p_T\left(H_b\right)[\text{ GeV}]$	$f_s/\left(f_u+f_d\right)$	$f_{{\Lambda }_b^0}/\left(f_u+f_d\right)$
4–5	$0.125\pm 0.001\pm 0.007$	$0.324\pm 0.001\pm 0.025$
5–6	$0.125\pm 0.001\pm 0.007$	$0.281\pm 0.001\pm 0.018$
6–7	$0.122\pm 0.001\pm 0.006$	$0.257\pm 0.001\pm 0.017$
7–8	$0.125\pm 0.001\pm 0.006$	$0.245\pm 0.001\pm 0.017$
8–9	$0.116\pm 0.001\pm 0.006$	$0.227\pm 0.001\pm 0.015$
9–10	$0.120\pm 0.001\pm 0.006$	$0.210\pm 0.001\pm 0.015$
10–11	$0.121\pm 0.001\pm 0.006$	$0.194\pm 0.001\pm 0.013$
11–12	$0.116\pm 0.001\pm 0.006$	$0.191\pm 0.001\pm 0.014$
12–13	$0.116\pm 0.001\pm 0.006$	$0.172\pm 0.001\pm 0.013$
13–14	$0.122\pm 0.001\pm 0.007$	$0.159\pm 0.001\pm 0.012$
14–16	$0.112\pm 0.001\pm 0.006$	$0.165\pm 0.001\pm 0.012$
16–18	$0.107\pm 0.001\pm 0.006$	$0.136\pm 0.001\pm 0.010$
18–20	$0.115\pm 0.001\pm 0.008$	$0.126\pm 0.001\pm 0.010$
20–25	$0.111\pm 0.001\pm 0.007$	$0.109\pm 0.001\pm 0.009$

Table 5. Table 5: Covariance matrix for f s / ( f u + f d ) subscript 𝑓 𝑠 subscript 𝑓 𝑢 subscript 𝑓 𝑑 f_{s}/(f_{u}+f_{d}) in p T ​ ( H b ) subscript 𝑝 T subscript 𝐻 𝑏 p_{\mathrm{T}}(H_{b})  [GeV] bins; it accounts for statistical and bin-dependent systematic uncertainties, but not the global systematic uncertainties.

$p_T\left(H_b\right)$	4–5	5–6	6–7	7–8	8–9	9–10	10–11	11–12	12–13	13–14	14–16	16–18	18–20	20–25
4–5	2.30E-05	3.80E-06	1.84E-06	1.53E-06	9.28E-07	1.54E-06	1.77E-06	9.05E-07	7.47E-07	6.99E-07	6.46E-07	6.84E-07	5.47E-07	5.55E-07
5–6	3.80E-06	1.96E-05	2.37E-06	2.02E-06	1.23E-06	2.05E-06	2.36E-06	1.22E-06	1.04E-06	9.64E-07	8.94E-07	9.85E-07	8.04E-07	8.00E-07
6–7	1.84E-06	2.37E-06	1.10E-05	1.08E-06	6.71E-07	1.14E-06	1.32E-06	6.94E-07	5.83E-07	5.46E-07	5.18E-07	5.60E-07	4.67E-07	4.89E-07
7–8	1.53E-06	2.02E-06	1.08E-06	1.08E-05	6.46E-07	1.10E-06	1.27E-06	6.75E-07	5.73E-07	5.43E-07	5.06E-07	5.48E-07	4.80E-07	4.86E-07
8–9	9.28E-07	1.23E-06	6.71E-07	6.46E-07	8.73E-06	7.75E-07	9.09E-07	5.07E-07	4.26E-07	4.04E-07	3.79E-07	4.16E-07	3.77E-07	3.88E-07
9–10	1.54E-06	2.05E-06	1.14E-06	1.10E-06	7.75E-07	1.09E-05	1.66E-06	9.39E-07	8.03E-07	7.62E-07	7.24E-07	7.77E-07	7.01E-07	7.34E-07
10–11	1.77E-06	2.36E-06	1.32E-06	1.27E-06	9.09E-07	1.66E-06	1.33E-05	1.13E-06	9.90E-07	9.33E-07	9.37E-07	1.00E-06	9.03E-07	9.58E-07
11–12	9.05E-07	1.22E-06	6.94E-07	6.75E-07	5.07E-07	9.39E-07	1.13E-06	1.33E-05	5.94E-07	5.64E-07	5.74E-07	6.24E-07	5.73E-07	6.17E-07
12–13	7.47E-07	1.04E-06	5.83E-07	5.73E-07	4.26E-07	8.03E-07	9.90E-07	5.94E-07	1.60E-05	5.42E-07	5.62E-07	6.33E-07	5.73E-07	6.08E-07
13–14	6.99E-07	9.64E-07	5.46E-07	5.43E-07	4.04E-07	7.62E-07	9.33E-07	5.64E-07	5.42E-07	2.31E-05	5.31E-07	6.33E-07	5.98E-07	6.13E-07
14–16	6.46E-07	8.94E-07	5.18E-07	5.06E-07	3.79E-07	7.24E-07	9.37E-07	5.74E-07	5.62E-07	5.31E-07	1.32E-05	7.11E-07	6.66E-07	7.27E-07
16–18	6.84E-07	9.85E-07	5.60E-07	5.48E-07	4.16E-07	7.77E-07	1.00E-06	6.24E-07	6.33E-07	6.33E-07	7.11E-07	1.96E-05	8.08E-07	9.33E-07
18–20	5.47E-07	8.04E-07	4.67E-07	4.80E-07	3.77E-07	7.01E-07	9.03E-07	5.73E-07	5.73E-07	5.98E-07	6.66E-07	8.08E-07	3.71E-05	9.58E-07
20–25	5.55E-07	8.00E-07	4.89E-07	4.86E-07	3.88E-07	7.34E-07	9.58E-07	6.17E-07	6.08E-07	6.13E-07	7.27E-07	9.33E-07	9.58E-07	2.93E-05

Table 6. Table 6: Covariance matrix for f Λ b 0 / ( f u + f d ) subscript 𝑓 subscript superscript Λ 0 𝑏 subscript 𝑓 𝑢 subscript 𝑓 𝑑 f_{{{\mathchar 28931\relax}^{0}_{b}}}/(f_{u}+f_{d}) in p T ​ ( H b ) subscript 𝑝 T subscript 𝐻 𝑏 p_{\mathrm{T}}(H_{b})  [GeV] bins; it accounts for statistical and bin-dependent systematic uncertainties, but not the global systematic uncertainties.

$p_T\left(H_b\right)$	4–5	5–6	6–7	7–8	8–9	9–10	10–11	11–12	12–13	13–14	14–16	16–18	18–20	20–25
4–5	2.40E-4	3.21E-5	2.08E-5	5.03E-5	3.36E-5	4.21E-5	1.60E-5	3.50E-5	2.47E-5	1.30E-5	6.20E-6	3.10E-6	2.60E-6	3.46E-6
5–6	3.21E-5	4.34E-5	5.05E-6	1.30E-5	8.90E-6	1.12E-5	4.47E-6	9.65E-6	7.12E-6	3.73E-6	1.80E-6	9.11E-7	7.82E-7	1.12E-6
6–7	2.08E-5	5.05E-6	2.99E-5	9.39E-6	6.55E-6	8.41E-6	3.35E-6	7.43E-6	5.44E-6	2.86E-6	1.43E-6	7.04E-7	6.25E-7	8.96E-7
7–8	5.03E-5	1.30E-5	9.39E-6	5.32E-5	1.97E-5	2.56E-5	1.05E-5	2.26E-5	1.72E-5	9.14E-6	4.53E-6	2.33E-6	2.04E-6	3.01E-6
8–9	3.36E-5	8.90E-6	6.55E-6	1.97E-5	3.96E-5	2.10E-5	8.88E-6	1.95E-5	1.47E-5	7.91E-6	3.91E-6	2.03E-6	1.75E-6	2.75E-6
9–10	4.21E-5	1.12E-5	8.41E-6	2.56E-5	2.10E-5	5.44E-5	1.27E-5	2.81E-5	2.14E-5	1.21E-5	5.88E-6	3.11E-6	2.69E-6	4.44E-6
10–11	1.60E-5	4.47E-6	3.35E-6	1.05E-5	8.88E-6	1.27E-5	3.03E-5	1.33E-5	1.10E-5	6.15E-6	3.07E-6	1.67E-6	1.43E-6	2.48E-6
11–12	3.50E-5	9.65E-6	7.43E-6	2.26E-5	1.95E-5	2.81E-5	1.33E-5	6.41E-5	2.54E-5	1.42E-5	7.23E-6	3.98E-6	3.44E-6	5.86E-6
12–13	2.47E-5	7.12E-6	5.44E-6	1.72E-5	1.47E-5	2.14E-5	1.10E-5	2.54E-5	5.71E-5	1.36E-5	6.78E-6	3.85E-6	3.41E-6	6.10E-6
13–14	1.30E-5	3.73E-6	2.86E-6	9.14E-6	7.90E-6	1.21E-5	6.15E-6	1.42E-5	1.36E-5	4.37E-5	4.23E-6	2.47E-6	7.22E-6	3.88E-6
14–16	6.20E-6	1.80E-6	1.43E-6	4.53E-6	3.91E-6	5.88E-6	3.07E-6	7.23E-6	6.78E-6	4.23E-6	3.01E-5	1.35E-6	1.21E-6	2.24E-6
16–18	3.10E-6	9.11E-7	7.04E-7	2.33E-6	2.03E-6	3.11E-6	1.67E-6	3.98E-6	3.85E-6	2.47E-6	1.35E-6	3.17E-5	8.03E-7	1.54E-6
18–20	2.61E-6	7.82E-7	6.25E-7	2.04E-6	1.75E-6	2.67E-6	1.43E-6	3.44E-6	3.41E-6	2.22E-6	1.21E-6	8.03E-7	4.21E-5	1.62E-6
20–25	3.46E-6	1.12E-6	8.97E-7	3.01E-6	2.75E-6	4.44E-6	2.48E-6	5.83E-6	6.10E-6	3.88E-6	2.24E-6	1.54E-6	1.62E-6	3.09E-5

Equations

$$\frac{f_s}{f_u+f_d}(p_{\mathrm{T}})=A\left[p_1+p_2\times \left(p_{\mathrm{T}}-⟨p_{\mathrm{T}}⟩\right)\right],$$

$$\frac{f_s}{f_u+f_d}=0.122\pm 0.006,$$

$$\frac{f_{{\text{mathchar}28931}_b^0}}{f_u+f_d}(p_{\mathrm{T}})=A\left[p_1+\exp \left(p_2+p_3\times p_{\mathrm{T}}\right)\right],$$

$$\frac{f_{{\text{mathchar}28931}_b^0}}{f_u+f_d}=0.259\pm 0.018,$$

$$n_{\mathrm{corr}}(B\to D^0{\mu }^{-})$$

$$\left[n(D^0{\mu }^{-})-n(D^0{K}^{+}{\mu }^{-})\frac{ϵ(\overline{B}{}_s^0\to D^0)}{ϵ(\overline{B}{}_s^0\to D^0{K}^{+})}-n(D^{0}p{\mu }^{-})\frac{ϵ({\text{mathchar}28931}_b^0\to D^0)}{ϵ({\text{mathchar}28931}_b^0\to D^{0}p)}\right],$$

$$n_{\mathrm{corr}}(B\to D^{+}{\mu }^{-})$$

$$n_{\mathrm{corr}}(\overline{B}{}_s^0\to D_s^{+}{\mu }^{-})$$

$$n_{\mathrm{corr}}(\overline{B}{}_s^0\to DK{\mu }^{-})=\kappa \frac{n(D^0{K}^{+}{\mu }^{-})}{\mathcal{B}(D^0\to K^{-}{\pi }^{+})ϵ(\overline{B}{}_s^0\to D^0{K}^{+}{\mu }^{-})},$$

$$\frac{f_s}{f_u+f_d}=\frac{n_{\mathrm{corr}}(\overline{B}{}_s^0\to D{\mu }^{-})}{n_{\mathrm{corr}}(B\to D^0{\mu }^{-})+n_{\mathrm{corr}}(B\to D^{+}{\mu }^{-})}\frac{{\tau }_{B^{-}}+{\tau }_{\overline{B}{}^0}}{2{\tau }_{\overline{B}{}_s^0}}(1-{\xi }_s)$$

$$-\frac{\mathcal{B}(B\to D_s\overline{K}{\mu }^{-})}{⟨{\mathcal{B}}_{\mathrm{SL}}⟩}\frac{ϵ(\overline{B}\to D_s^{+})}{ϵ(\overline{B}{}_s^0\to D_s^{+})},$$

$$n_{\mathrm{corr}}({\text{mathchar}28931}_b^0\to H_c{\mu }^{-})=\frac{n({\text{mathchar}28931}_c^{+}{\mu }^{-})}{\mathcal{B}({\text{mathchar}28931}_c^{+}\to p{K}^{-}{\pi }^{+})ϵ({\text{mathchar}28931}_b^0\to {\text{mathchar}28931}_c^{+})}+2\frac{n(D^{0}p{\mu }^{-})}{\mathcal{B}(D^0\to K^{-}{\pi }^{+})ϵ({\text{mathchar}28931}_b^0\to D^{0}p)},$$

$$\frac{f_{{\text{mathchar}28931}_b^0}}{f_u+f_d}=\frac{n_{\mathrm{corr}}({\text{mathchar}28931}_b^0\to H_c{\mu }^{-})}{n_{\mathrm{corr}}(B\to D^0{\mu }^{-})+n_{\mathrm{corr}}(B\to D^{+}{\mu }^{-})}\frac{{\tau }_{B^{-}}+{\tau }_{\overline{B}{}^0}}{2{\tau }_{{\text{mathchar}28931}_b^0}}(1-{\xi }_{{\text{mathchar}28931}_b^0}).$$

Full text

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

​​​

CERN-EP-2019-016

LHCb-PAPER-2018-050

February 18, 2019

Measurement of $b$-hadron fractions in $13\text{\,Te\kern-2.07413ptV}$ $pp$ collisions

LHCb collaboration†††Authors are listed at the end of this paper.

The production fractions of ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ and ${\mathchar 28931\relax}^{0}_{b}$ hadrons, normalized to the sum of ${{B}^{-}}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$ fractions, are measured in $13\text{\,Te\kern-1.00006ptV}$ $pp$ collisions using data collected by the LHCb experiment, corresponding to an integrated luminosity of 1.67$\text{\,fb}^{-1}$. These ratios, averaged over the $b$-hadron transverse momenta from 4 to 25 GeV and pseudorapidity from 2 to 5, are $0.122\pm 0.006$ for ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$, and $0.259\pm 0.018$ for ${\mathchar 28931\relax}^{0}_{b}$, where the uncertainties arise from both statistical and systematic sources. The ${\mathchar 28931\relax}^{0}_{b}$ ratio depends strongly on transverse momentum, while the ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ ratio shows a mild dependence. Neither ratio shows variations with pseudorapidity. The measurements are made using semileptonic decays to minimize theoretical uncertainties. In addition, the ratio of ${D}^{+}$ to ${D}^{0}$ mesons produced in the sum of ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$ and ${{B}^{-}}$ semileptonic decays is determined as $0.359\pm 0.006\pm 0.009$, where the uncertainties are statistical and systematic.

To be published in Physical Review D Rapid Communications

© 2024 CERN for the benefit of the LHCb collaboration. CC-BY-4.0 licence.

Knowledge of the fragmentation fractions of ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ ($f_{s}$) and ${\mathchar 28931\relax}^{0}_{b}$ ($f_{{{\mathchar 28931\relax}^{0}_{b}}}$) hadrons is essential for determining absolute branching fractions (${\cal{B}}$) of decays of these hadrons at the LHC, allowing measurements, for example, of ${\cal{B}}({{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\rightarrow\mu^{+}\mu^{-})$ [1] and the future evaluation of $|V_{cb}|$ from ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}\mu^{-}\overline{\nu}_{\mu}$ decays [2].111Mention of a particular decay mode implies the use of the charge-conjugate one as well. Once these fractions are determined, measurements of absolute branching fractions of ${{B}^{-}}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$ mesons performed at $e^{+}e^{-}$ colliders operating at the $\mathchar 28935\relax(4S)$ resonance can be used to determine the ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ and ${\mathchar 28931\relax}^{0}_{b}$ branching fractions [3].

In this Letter we measure the ratios $f_{s}/(f_{u}+f_{d})$ and $f_{{{\mathchar 28931\relax}^{0}_{b}}}/(f_{u}+f_{d})$, where the denominator is the sum of ${{B}^{-}}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$ contributions, in the LHCb acceptance of pseudorapidity $2<\eta<5$ and transverse momentum $4<p_{\mathrm{T}}<25\text{\,Ge\kern-1.00006ptV}$,222We use natural units where $c=\hbar=1$. in 13 TeV $pp$ collisions. These ratios can depend on $p_{\mathrm{T}}$ and $\eta$; therefore, we perform the analysis using two-dimensional binning.

Much of the analysis method adopted in this study is an evolution of our previous $b$-hadron fraction measurements for 7 TeV $pp$ collisions[4]. We use the inclusive semileptonic decays $H_{b}\rightarrow H_{c}X\mu^{-}\overline{\nu}_{\mu}$, where $H_{b}$ indicates a $b$ hadron, $H_{c}$ a charm hadron, and $X$ possible additional particles. Each of the different $H_{c}$ plus muon final states can originate from the decay of different $b$ hadrons. Semileptonic decays of ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$ mesons usually result in a mixture of $D^{0}$ and $D^{+}$ mesons, while $B^{-}$ mesons decay predominantly into $D^{0}$ mesons with a smaller admixture of $D^{+}$ mesons. Both include a tiny component of ${{D}^{+}_{s}}{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}}$ meson pairs. Similarly, ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$ mesons decay predominantly into $D_{s}^{+}$ mesons, but can also decay into $D^{0}K^{+}$ and $D^{+}{{K}^{0}}$ meson pairs; this is expected if the ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$ meson decays into an excited $D_{s}^{+}$ state that is heavy enough to decay into a $DK$ pair. We measure this contribution using $D^{0}K^{+}X\mu^{-}\overline{\nu}_{\mu}$ events. Finally, ${{\mathchar 28931\relax}^{0}_{b}}$ baryons decay semileptonically mostly into ${{\mathchar 28931\relax}^{+}_{c}}$ final states, but can also decay into $D^{0}p$ and $D^{+}n$ pairs. We ignore the contributions of $b\rightarrow u$ decays that comprise approximately 1% of semileptonic $b$-hadron decays, and contribute almost equally to all $b$-hadron species. The detailed equations relating these yields to the final results are given in Ref. [4] and in the Supplemental material.

The theoretical basis for this measurement is the near equality of semileptonic widths, $\Gamma_{\rm SL}$, for all $b$-hadron species [5] whose differences are predicted to precisions of about 1%. The values we use for the individual $H_{b}$ semileptonic branching fractions (${\cal{B}}_{\rm SL}$) are listed in Table 1. The $H_{c}$ decay modes used and their branching fractions are given in Table 2.

The ratio of ${D}^{+}$ to ${D}^{0}$ meson production in the sum of semileptonic ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$ and ${{B}^{-}}$ decays, ${f_{+}}/{f_{0}}$, is used to check the analysis method. This result can be related to models of the hadronic final states in ${{{B}^{-}}}$ and ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$ semileptonic decays [6].

The data sample corresponds to 1.67$\text{\,fb}^{-1}$ of integrated luminosity obtained with the LHCb detector in 13 TeV $pp$ collisions during 2016. The LHCb detector [7, 8] is a single-arm forward spectrometer covering the pseudorapidity range $2<\eta<5$, designed for the study of particles containing $b$ or $c$ quarks. The detector elements that are particularly relevant to this analysis are: a silicon-strip vertex detector surrounding the $pp$ interaction region that allows $c$ and $b$ hadrons to be identified from their characteristically long flight distance from the primary vertex (PV); a tracking system that provides a measurement of the momentum, $p$, of charged particles, two ring-imaging Cherenkov detectors that are able to discriminate between different species of charged hadrons, and a muon detection system.

The online event selection is performed by a trigger [9] which consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction. At the hardware trigger stage, events are required to have a muon with large $p_{\mathrm{T}}$ or a hadron, photon or electron with high transverse energy in the calorimeters. For hadrons, the transverse energy threshold is 3.5 GeV. The software trigger requires a two-, three- or four-track secondary vertex with a significant displacement from any primary $pp$ interaction vertex. At least one charged particle must have $p_{\mathrm{T}}>1.6\text{\,Ge\kern-1.00006ptV}$ and be inconsistent with originating from a PV. A multivariate algorithm [10] is used for the identification of secondary vertices consistent with the decay of a $b$ hadron.

Simulation is required to model the effects of the detector acceptance and the imposed selection requirements. Here $pp$ collisions are generated using Pythia [11, *Sjostrand:2006za] with a specific LHCb configuration [13]. Decays of unstable particles are described by EvtGen [14], in which final-state radiation is generated using Photos [15]. The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit [16, *Agostinelli:2002hh] as described in Ref. [18].

Selection criteria are applied to muons and $H_{c}$ decay particles. The transverse momentum of each hadron must be greater than 0.3 GeV, and that of the muon larger than 1.3 GeV. Each track cannot point to any PV, implemented by requiring $\chi^{2}_{\text{IP}}>9$ with respect to any PV, where $\chi^{2}_{\text{IP}}$ is defined as the difference in the vertex-fit $\chi^{2}$ of a given PV reconstructed with and without the track under consideration being included. All final state particles are required to be positively identified using information from the RICH detectors (PID). Particles from $H_{c}$ decay candidates must have a good fit to a common vertex with $\chi^{2}$/ndof $<9$, where ndof is the number of degrees of freedom. They must also be well separated from the nearest PV, with the flight distance divided by its uncertainty greater than 5.

Candidate $b$ hadrons are formed by combining $H_{c}$ and muon candidates originating from a common vertex with $\chi^{2}$/ndof $<9$ and an $H_{c}\mu^{-}$ invariant mass, $m_{H_{c}\mu^{-}}$, in the range 3.0–5.0 GeV for $D^{0}$ and $D^{+}$, 3.1–5.1 GeV for ${D}^{+}_{s}$ and 3.3–5.3 GeV for ${\mathchar 28931\relax}^{+}_{c}$ candidates. In addition, we define $m_{\rm corr}\equiv\sqrt{m_{H_{c}\mu}^{2}+p_{\perp}^{2}}+p_{\perp}$, where $p_{\perp}$ is the magnitude of the combination’s momentum component transverse to the $b$-hadron flight direction; we require that $m_{\rm corr}>4.2$ or $4.5\text{\,Ge\kern-1.00006ptV}$ for ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ or ${\mathchar 28931\relax}^{0}_{b}$ candidates, respectively. For the ${{D}^{+}_{s}}\rightarrow K^{+}K^{-}\pi^{+}$ decay mode, vetoes are employed to remove backgrounds from real $D^{+}$ or ${\mathchar 28931\relax}^{+}_{c}$ decays where the particle assignments are incorrect.

Background from prompt $H_{c}$ production at the PV needs to be considered. We use the natural logarithm of the $H_{c}$ impact parameter, IP, with respect to the PV in units of mm. Requiring ln(IP/mm)$>-3$ is found to reduce the prompt component to be below 0.1%, while preserving 97% of all signals. This restriction allows us to perform fits only to the $H_{c}$ candidate mass spectra to find the $b$-hadron decay yields.

The $H_{c}$ candidates mass distributions integrated over $p_{\mathrm{T}}(H_{b})$ and $\eta$ are shown in Fig. 1. They consist of a prominent peak resulting from signal, and a small contribution due to combinatorial background from random combinations of particles that pass the selection. They are fit with a signal component comprised of two Gaussian functions, and a combinatorial background component modeled as a linear function. The total signal yields for $D^{0}X\mu^{-}\overline{\nu}_{\mu}$, $D^{+}X\mu^{-}\overline{\nu}_{\mu}$, $D_{s}^{+}X\mu^{-}\overline{\nu}_{\mu}$ and ${{\mathchar 28931\relax}^{+}_{c}}\mu^{-}X\overline{\nu}_{\mu}$ are 13 775 000, 4 282 700, 845 300, and 1 753 600, respectively.

Background contributions to the $b$-hadron candidates include hadrons faking muons, false combinations of charm hadrons and muons from the two $b$ hadrons in the event, as well as real muons and charm hadrons from $B\rightarrow{D}{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}}X$ decays, where one of the $D$ mesons decays into a muon. All the backgrounds are evaluated in two-dimensional $\eta$ and $p_{\mathrm{T}}$ intervals. The first two backgrounds are evaluated using events where the $H_{c}$ is combined with a muon of the wrong-sign (e.g. $D^{0}\mu^{+}$), forbidden in a semileptonic $b$-hadron decay. The wrong-sign backgrounds are $<1$% for each $H_{c}$ species. The background from $B\rightarrow D{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}}X$ decays is determined by simulating a mixture of these decays using their measured branching fractions [3]. The only decay mode significantly affected is ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\rightarrow{{D}^{+}_{s}}X\mu^{-}\overline{\nu}_{\mu}$ with contributions varying from 0.1% for $D^{0}D_{s}^{-}X$ to 1.8% for ${{D}^{+}_{s}}D_{s}^{-}X$ due to the large ${{D}^{+}_{s}}\rightarrow\mu^{+}\nu$ decay rate. The total $B\rightarrow D{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}}X$ background is $(5.8\pm 0.9)\%$.

The dominant component in ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ semileptonic decays is ${{D}^{+}_{s}}X\mu^{-}\overline{\nu}_{\mu}$, where $X$ contains possible additional hadrons. However, the ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ meson also can decay into $D^{0}K^{+}$ or $D^{+}K^{0}$ instead of ${D}^{+}_{s}$, so we must add this component to the ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ rate and subtract it from the $f_{u}+f_{d}$ fraction. Similarly, in ${\mathchar 28931\relax}^{0}_{b}$ semileptonic decays we find a $D^{0}pX$ component. The selection criteria for these final states are similar to those for the $D^{0}X\mu^{-}\overline{\nu}_{\mu}$ and ${{\mathchar 28931\relax}^{+}_{c}}X\mu^{-}\overline{\nu}_{\mu}$ final states described above with the addition of a kaon or proton with $p_{\mathrm{T}}>300\text{\,Me\kern-1.00006ptV}$ that has been positively identified. A veto is also applied to reject $D^{*+}\rightarrow\pi^{+}D^{0}$ decays where the pion mimics a kaon or a proton.

These samples contain background, resonant and nonresonant decays. Separation of these components is achieved by using both right-sign ($H_{c}$ with $\mu^{-}$) and wrong-sign ($H_{c}$ with $\mu^{+}$) candidates. In addition, the logarithm of the difference between the vertex $\chi^{2}$ formed by the added hadron track and the $D\mu$ system and the vertex $\chi^{2}$ of the $D\mu$ system, ${\rm ln}(\Delta\chi^{2}_{\rm V})$, provides separation between combinatorial background and nonresonant semileptonic decays. True resonant and nonresonant ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\rightarrow{{D}^{0}}{{K}^{+}}\mu^{-}\overline{\nu}_{\mu}$ or ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{D}^{0}}p\mu^{-}\overline{\nu}_{\mu}$ decays peak in the ${\rm ln}(\Delta\chi^{2}_{\rm V})$ distribution at a value of unity while the background is smooth and rises at higher values as the added track is generally not associated with the ${{D}^{0}}{\mu^{-}}$ vertex. To distinguish signal from background we define $m(D^{0}h)_{C}\equiv m(D^{0}h)-m(D^{0})+m(D^{0})_{\rm PDG}$, and perform two-dimensional fits to the $m(D^{0}h)_{C}$ and ${\rm ln}(\Delta\chi^{2}_{\rm V})$ distributions, where $h=K^{+}(p)$ for right-sign ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ (${\mathchar 28931\relax}^{0}_{b}$) decays.

The wrong-sign shapes are used to model the backgrounds. The resonant structures are modeled with relativistic Breit–Wigner functions convoluted with Gaussians to take into account the experimental resolution, except for the narrow $D_{s1}(2536)^{+}$ which is modeled with the sum of two Gaussians with a fixed mean. The nonresonant shape for the ${\rm ln}(\Delta\chi^{2}_{\rm V})$ distribution is taken as the same as the resonant one. Figure 2 shows the data and result of the fits for ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ and ${\mathchar 28931\relax}^{0}_{b}$ candidates.

For the ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ case, we find $22~{}\!610\pm 210$ $D_{s1}(2536)^{+}$, $14~{}\!290\pm 260$ $D_{s2}^{*}(2573)^{+}$, and $38~{}\!140\pm 460$ nonresonant decays, confirming the existence of both the $D_{s1}^{+}$ [23, 24] and $D_{s2}^{*+}$ [24] particles in semileptonic ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ decays with substantially more data, and showing the existence of the nonresonant component. To account for the unmeasured $D^{+}K^{0}$ channel we take different mixtures of $D^{*}$ and $D$ final states for the different resonant and nonresonant components. The $D_{s1}^{+}$ decays dominantly into $D^{*}$, while the $D_{s2}^{*+}$ decays dominantly into $D$ mesons [3]. For the nonresonant part we assume equal $D^{*}$ and $D$ yields.

In the ${\mathchar 28931\relax}^{0}_{b}$ case, we find $6120\pm 460$ ${{\mathchar 28931\relax}^{+}_{c}}(2860)$, $2200\pm 200$ ${{\mathchar 28931\relax}^{+}_{c}}(2880)$, $1200\pm 260$ ${{\mathchar 28931\relax}^{+}_{c}}(2940)$, and $29~{}\!770\pm 690$ nonresonant events. The decay rate into $D^{0}p$ is assumed to be equal to that into $D^{+}n$ using isospin conservation. All decays with an extra hadron have lower detection efficiencies than the sample without.

Efficiencies for all the samples are determined using data in two-dimensional $p_{\mathrm{T}}$ and $\eta$ bins. Trigger efficiencies are determined using a sample of $B^{-}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{-}$, with ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow{\mu^{+}}{\mu^{-}}$ decays where only one muon track is positively identified, in conjunction with viewing the effects of combinations of different triggers [25]. This sample is also used to determine muon identification efficiencies. Decays of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons to muons reconstructed using partial information from the tracking system, e.g. eliminating the vertex locator information, are also used to determine tracking efficiencies using data and to correct the simulation. Finally, the PID efficiencies are evaluated using kaons and pions from $D^{*+}\rightarrow\pi^{+}D^{0}$ decays, with $D^{0}\rightarrow K^{-}\pi^{+}$, and protons from $\mathchar 28931\relax\rightarrow p\pi^{-}$ and ${{\mathchar 28931\relax}^{+}_{c}}\rightarrow pK^{-}\pi^{+}$ decays [26]. In the measurement of $b$-hadron fraction ratios many of the efficiencies cancel and we are left with only residual effects to which we assign systematic uncertainties.

The $b$-hadron $\eta$ and $p_{\mathrm{T}}$, $p_{\mathrm{T}}(H_{b})$, must be known because the $b$ fractions can depend on production kinematics. While $\eta$ can be evaluated directly using the measured primary and secondary $b$ vertices, the value of $p_{\mathrm{T}}(H_{b})$ must be determined to account for the missing neutrino plus extra particles. The correction factor $k$ is given by the ratio of the average reconstructed to true $p_{\mathrm{T}}(H_{b})$ as a function of $m(H_{c}\mu^{-})$ and is determined using simulation. It varies from 0.75 for $m(H_{c}\mu^{-})$ equals 3 GeV to unity at $m(H_{c}\mu^{-})=m(H_{b})$.

The distribution of $f_{s}/(f_{u}+f_{d})$ as a function of $p_{\mathrm{T}}(H_{b})$ is shown in Fig. 3. We perform a linear $\chi^{2}$ fit incorporating a full covariance matrix which takes into account the bin-by-bin correlations introduced from the kaon kinematics, and PID and tracking systematic uncertainties. The factor $A$ in Eq. 1 incorporates the global systematic uncertainties described later, which are independent of $p_{\mathrm{T}}(H_{b})$. The resulting function is

[TABLE]

where $p_{\mathrm{T}}$ here refers to $p_{\mathrm{T}}(H_{b})$, $A=1\pm 0.043$, $p_{1}=0.119\pm 0.001$, $p_{2}=(-0.91\pm 0.25)\cdot 10^{-3}~{}\!{\rm GeV}^{-1}$, and $\langle p_{\mathrm{T}}\rangle=10.1\text{\,Ge\kern-1.00006ptV}$. The correlation coefficient between the fit parameters is 0.20. After integrating over $p_{\mathrm{T}}(H_{b})$, no $\eta$ dependence is observed (see the Supplemental material).

We determine an average value for $f_{s}/(f_{u}+f_{d})$ by dividing the yields of ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$ semileptonic decays by the sum of ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$ and ${{{B}^{-}}}$ semileptonic yields, which are all efficiency-corrected, between the limits of $p_{\mathrm{T}}(H_{b})$ of 4 and 25 GeV and $\eta$ of 2 and 5, resulting in

[TABLE]

where the uncertainty contains both statistical and systematic components, with the latter being dominant, and discussed subsequently. The total relative uncertainty is 4.8%.

Figure 3 also shows the ${\mathchar 28931\relax}^{0}_{b}$ fraction as a function of $p_{\mathrm{T}}(H_{b})$ demonstrating a large $p_{\mathrm{T}}$ dependence. The distribution in $\eta$ is flat. We perform a similar fit as in the ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ fraction case, using

[TABLE]

where $p_{\mathrm{T}}$ here refers to $p_{\mathrm{T}}(H_{b})$, $A=1\pm 0.061$, $p_{1}=(7.93\pm 1.41)\cdot 10^{-2}$, $p_{2}=-1.022\pm 0.047$, and $p_{3}=-0.107\pm 0.002~{}\!{\rm GeV}^{-1}$. The correlation coefficients among the fit parameters are 0.40 $(\rho_{12})$, –0.95 $(\rho_{13})$, and –0.63 $(\rho_{23})$.

The average value for $f_{{{\mathchar 28931\relax}^{0}_{b}}}/(f_{u}+f_{d})$ is determined using the same method as in the ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ case. The result is

[TABLE]

where the dominant uncertainty is systematic, and the statistical uncertainty is included. The overall uncertainty is 6.9%.

As a systematic check of the analysis method, and a useful measurement to test the knowledge of known semileptonic branching fractions and extrapolations used to saturate the unknown portion of the inclusive hadron spectrum, we measure the ratio of the $D^{0}X\mu^{-}\overline{\nu}_{\mu}$ to $D^{+}X\mu^{-}\overline{\nu}_{\mu}$ corrected yields $f_{+}/f_{0}$. We subtract the small contributions from ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ and ${\mathchar 28931\relax}^{0}_{b}$ decays, and a very small contribution from $B\rightarrow{{D}^{+}_{s}}{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}}\mu^{-}X$ decays has been taken into account [27], as in all the fractions measured above.

Assuming $f_{u}$ equals $f_{d}$, Ref. [6] estimates the fraction of $D^{+}\mu$ with respect to $D^{0}\mu$ modes in the sum of ${{{B}^{-}}}$ and ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$ decays as $0.387\pm 0.012\pm 0.026$. The first uncertainty comes from the uncertainties on known measurements. The second uncertainty comes from the different extrapolations from excited $D$ mesons used to saturate the remaining portion of the inclusive rate.

The $f_{+}/f_{0}$ ratio must be independent of $\eta$ and $p_{\mathrm{T}}$. To derive an overall value for $f_{+}/f_{0}$, the $p_{\mathrm{T}}(H_{b})$ distribution is fit to a constant. Only the PID and tracking systematic uncertainties on the second pion in the $D^{+}$ decay need be considered. Performing a $\chi^{2}$ fit using the full covariance matrix we find $f_{+}/f_{0}=0.359\pm 0.006\pm 0.009$, where the first uncertainty is from bin-by-bin statistical and systematic uncertainties, including correlations, and the second is systematic. The $\chi^{2}$/ndof is 0.63, in agreement with a flat spectrum. The measurement is consistent with the prediction and places some constraints on the $D^{**}$ content of semileptonic $B$ decays [6].

The dominant global systematic uncertainties are listed in Table 3. Simulation uncertainties are due to the modeling of excited charm states for the $f_{s}/(f_{u}+f_{d})$ determination and the weighting required for the $f_{{{\mathchar 28931\relax}^{0}_{b}}}/(f_{u}+f_{d})$ ratio, due to differences between the simulated and measured $p_{\mathrm{T}}$ spectra. Background uncertainties arise from $D{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}}X$ final states with uncertain branching fractions. Cross-feed uncertainties come from errors on efficiency estimates and the assumed $D^{*}$ to $D$ mixtures. Other smaller uncertainties depend on $p_{\mathrm{T}}(H_{b})$ and include tracking (0.2–1.8)%, particle identification (0.4–3.0)%, trigger (0.3–3.9)% and $k$-factor (0.2–1.8)%.

In conclusion, we measure the ratios of ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ and ${\mathchar 28931\relax}^{0}_{b}$ production to the sum of ${{B}^{-}}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$ to be $p_{\mathrm{T}}(H_{b})$ dependent (see Eqs. 1 and 2). The averages in the ranges $4<p_{\mathrm{T}}(H_{b})<25\text{\,Ge\kern-1.00006ptV}$, and $2<\eta<5$ are $f_{s}/(f_{u}+f_{d})=0.122\pm 0.006$, and $f_{{{\mathchar 28931\relax}^{0}_{b}}}/(f_{u}+f_{d})=0.259\pm 0.018$, respectively. Using 7 TeV data, LHCb determined $f_{s}/(f_{u}+f_{d})=0.1295\pm 0.0075$ with a $p_{\mathrm{T}}(H_{b})$ slope larger than, but consistent with these 13 TeV results [28]; no dependence on $\eta$ was observed. For the ${{\mathchar 28931\relax}^{0}_{b}}$ baryon, the fraction ratio is consistent with the 7 TeV measurements after taking into account the different $p_{\mathrm{T}}(H_{b})$ ranges used [29, 30, 4]. We observe no rapidity dependence over a similar $p_{\mathrm{T}}(H_{b})$ range as in Ref. [30].

These results are crucial for determining absolute branching fractions of ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ and ${\mathchar 28931\relax}^{0}_{b}$ hadron decays in LHC experiments. We also determine the ratio of $D^{0}$ to $D^{+}$ mesons produced in the sum of ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$ and ${{B}^{-}}$ semileptonic decays as $f_{+}/f_{0}=0.359\pm 0.006\pm 0.009$.

Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC (United Kingdom); NSF (USA). We acknowledge the computing resources that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID (Poland) and OSC (USA). We are indebted to the communities behind the multiple open-source software packages on which we depend. Individual groups or members have received support from AvH Foundation (Germany); EPLANET, Marie Skłodowska-Curie Actions and ERC (European Union); ANR, Labex P2IO and OCEVU, and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China); RFBR, RSF and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society and the Leverhulme Trust (United Kingdom); Laboratory Directed Research and Development program of LANL (USA).

1 Supplemental material

1.1 Relationships between raw $H_{c}\mu^{-}X$ measured yields and corrected yields

The corrected yields for ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$ or $B^{-}$ mesons decaying into $D^{0}\mu^{-}\overline{\nu}_{\mu}X$ or $D^{+}\mu^{-}\overline{\nu}_{\mu}X$, $n_{\rm corr}$, can be expressed in terms of the measured yields, $n$, as

[TABLE]

where we use the shorthand $n(D{\mu^{-}})\equiv n(DX\mu^{-}\overline{\nu}_{\mu})$. An analogous abbreviation $\epsilon$ is used for the total trigger and detection efficiencies. For example, the ratio $\epsilon({{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\rightarrow D^{0}K^{+})/\epsilon({{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\rightarrow D^{0})$ gives the relative efficiency to reconstruct a charged kaon in semimuonic ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$ decays producing a ${{D}^{0}}$ meson. The second term in this equation accounts for the ${{D}^{0}}{\mu^{-}}$ pairs originating from a ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$ decay, such as ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\rightarrow{{D}^{0}}{{K}^{+}}{\mu^{-}}$, while the third term accounts for the ${{D}^{0}}{\mu^{-}}$ pairs originating from ${{\mathchar 28931\relax}^{0}_{b}}$ semileptonic decays. These components are determined from the study of the final states ${{D}^{0}}{{K}^{+}}{\mu^{-}}$ and ${{D}^{0}}p{\mu^{-}}$respectively. The branching fraction ${\cal B}({{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}})$ appears because this decay mode is used in this study. Similarly

[TABLE]

Both the ${{D}^{0}}X\mu^{-}\overline{\nu}_{\mu}$ and the ${{D}^{+}}X\mu^{-}\overline{\nu}_{\mu}$ final states contain small components of cross-feed from ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$ decays to ${{D}^{0}}K^{+}X\mu^{-}\overline{\nu}_{\mu}$ and to $D^{+}{{K}^{0}}X\mu^{-}\overline{\nu}_{\mu}$, and from ${{\mathchar 28931\relax}^{0}_{b}}$ decays to ${{D}^{0}}pX\mu^{-}\overline{\nu}_{\mu}$ and to $D^{+}nX\mu^{-}\overline{\nu}_{\mu}$. Here we use isospin symmetry and infer the contributions by ${{D}^{+}}{\mu^{-}}$ pairs originating from a ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$ decay, such as ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\rightarrow{{D}^{+}}{{K}^{0}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ from the ${{D}^{0}}{{K}^{+}}{\mu^{-}}$ final states, and the contributions from ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{D}^{+}}n{\mu^{-}}{{\overline{\nu}}_{\mu}}$ from the ${{D}^{0}}p{\mu^{-}}$ yields.

The number of ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\rightarrow{{D}^{+}_{s}}X\mu^{-}\overline{\nu}_{\mu}$ decays in the final state is given by

[TABLE]

In addition, the ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$ meson decays semileptonically into $DKX\mu^{-}\overline{\nu}_{\mu}$, and thus we need to add to Eq. 5 the term

[TABLE]

where $\kappa$ accounts for the unmeasured ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\rightarrow D^{+}KX\mu^{-}\overline{\nu}_{\mu}$ semileptonic decays. The correction $\kappa$ is evaluated using the known decay modes of the $D_{s1}(2536)^{+}$ and $D^{*}_{s2}(2573)^{+}$ states and assuming that the nonresonant component of the hadronic mass spectrum decays in equal portions into $D$ or ${{D}^{*}}$ final states. The last term in Eq. 5 accounts for ${{D}^{+}_{s}}KX\mu^{-}\overline{\nu}_{\mu}$ final states originating from ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$ or $B^{-}$ semileptonic decays, and $N({{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}+B^{-})$ indicates the total number of ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$ and $B^{-}$ produced. We derive this correction using the PDG value for the branching fraction ${\cal B}(B^{-}\rightarrow D_{s}^{(*)+}K^{-}\mu^{-}\nu)=(6.1\pm 1.0)\times 10^{-4}$, and assuming the same rate for ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ decays using isospin invariance [3].

The equation for the ratio $f_{s}/(f_{u}+f_{d})$ is

[TABLE]

where ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\rightarrow D\mu$ represents ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$ semileptonic decays to a charmed hadron, given by the sum of the contributions shown in Eqs. 5 and 6, and the symbols $\tau_{B_{i}}$ indicate the $B_{i}$ hadron lifetimes, that are all well measured [3]. We use the average ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$ lifetime, $1.526\pm 0.015$ ps. This equation assumes equality of the semileptonic widths of all the $b$-hadron species. This is a reliable assumption, as corrections in HQET arise only to order 1/$m_{b}^{2}$ and the SU(3) breaking correction is quite small, $(-1.0\pm 0.5)\%$ [5]. The parameter $\xi_{s}$ accounts for this small adjustment. The second term is the subtraction of the $B^{-,0}\rightarrow{{D}^{+}_{s}}{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}}X\mu^{-}\overline{\nu}_{\mu}$ component that is reconstructed in the signal sample as described in Eq. 5. The $\mathcal{B}_{\rm SL}$ term in the denominator is the semileptonic branching fraction of the ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$ derived using the equality of the semileptonic widths and the measured lifetime of the ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$, listed in Table 1.

The ${{\mathchar 28931\relax}^{0}_{b}}$ corrected yield is derived in an analogous manner

[TABLE]

where $H_{c}$ represents a generic charmed hadron. The second term includes the cross-feed channel and the factor of two accounts for the isospin ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow D^{+}n{\mu^{-}}$ decay. The ${{\mathchar 28931\relax}^{0}_{b}}$ fraction is written as

[TABLE]

While we assume near equality of the semileptonic widths of different $b$ hadrons, we apply a small adjustment $\xi_{{{\mathchar 28931\relax}^{0}_{b}}}=(3.0\pm 1.5)$%, to account for the chromomagnetic correction, affecting $b$-flavored mesons but not $b$ baryons [5]. The uncertainty is evaluated with conservative assumptions for all the parameters of the heavy quark expansion.

1.2 Table of $b$-fractions versus $p_{\mathrm{T}}(H_{b})$

1.3 Fraction ratios as functions of $\eta$

Figure 4 shows measurements of the fraction ratios $f_{s}/(f_{u}+f_{d})$ and $f_{{{\mathchar 28931\relax}^{0}_{b}}}/(f_{u}+f_{d})$ as functions of $\eta$, integrated over $p_{\mathrm{T}}$. No $\eta$ dependence is visible with the current data sample.

1.4 Correlation matrices for the fits to $f_{s}/(f_{u}+f_{d})$ and $f_{{{\mathchar 28931\relax}^{0}_{b}}}/(f_{u}+f_{d})$

Table 5 shows the covariance matrix among the different $p_{\mathrm{T}}(H_{b})$ bins for $f_{s}/(f_{u}+f_{d})$, while Table 6 shows the covariance matrix among the different $p_{\mathrm{T}}(H_{b})$ bins for $f_{{{\mathchar 28931\relax}^{0}_{b}}}/(f_{u}+f_{d})$.

[FIGURE:]

[FIGURE:]