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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2011-172 LHCb-PAPER-2011-018

Measurement of $b$ hadron production fractions in 7 TeV $pp$ collisions

The LHCb Collaboration 111Authors are listed on the following pages.

Measurements of $b$ hadron production ratios in proton-proton collisions at a
centre-of-mass energy of 7 TeV with an integrated luminosity of 3 pb-1 are
presented. We study the ratios of strange $B$ meson to light $B$ meson
production $f_{s}/(f_{u}+f_{d})$ and $\mathchar 28931\relax_{b}^{0}$ baryon to
light $B$ meson production $f_{\Lambda_{b}}/(f_{u}+f_{d})$ as a function of
the charmed hadron-muon pair transverse momentum $p_{\rm T}$ and the $b$
hadron pseudorapidity $\eta$, for $p_{\rm T}$ between 0 and 14 GeV and $\eta$
between 2 and 5. We find that $f_{s}/(f_{u}+f_{d})$ is consistent with being
independent of $p_{\rm T}$ and $\eta$, and we determine $f_{s}/(f_{u}+f_{d})$
= 0.134$\pm$0.004${}^{+0.011}_{-0.010}$, where the first error is statistical
and the second systematic. The corresponding ratio
$f_{\Lambda_{b}}/(f_{u}+f_{d})$ is found to be dependent upon the transverse
momentum of the charmed hadron-muon pair,
$f_{\Lambda_{b}}/(f_{u}+f_{d})=(0.404\pm 0.017{\rm(stat)}\pm
0.027{\rm(syst)}\pm 0.105{\rm(Br)})\times[1-(0.031\pm 0.004{\rm(stat)}\pm
0.003{\rm(syst)})\times p_{\rm T}{\rm(GeV)}]$, where Br reflects an absolute
scale uncertainty due to the poorly known branching fraction ${\cal
B}(\mathchar 28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+})$. We extract the
ratio of strange $B$ meson to light neutral $B$ meson production $f_{s}/f_{d}$
by averaging the result reported here with two previous measurements derived
from the relative abundances of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\pi^{-}$ to
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}K^{-}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$. We
obtain $f_{s}/f_{d}=0.267^{+0.021}_{-0.020}$.

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Robbe7, E. Rodrigues47, F. Rodrigues2, P. Rodriguez Perez36, G.J. Rogers43, S.
Roiser37, V. Romanovsky34, M. Rosello35,n, J. Rouvinet38, T. Ruf37, H. Ruiz35,
G. Sabatino21,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail47, B.
Saitta15,d, C. Salzmann39, M. Sannino19,i, R. Santacesaria22, C. Santamarina
Rios36, R. Santinelli37, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C.
Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina30, P. Schaack49, M.
Schiller24, S. Schleich9, M. Schmelling10, B. Schmidt37, O. Schneider38, A.
Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A. Sciubba18,l, M.
Seco36, A. Semennikov30, K. Senderowska26, I. Sepp49, N. Serra39, J. Serrano6,
P. Seyfert11, B. Shao3, M. Shapkin34, I. Shapoval40,37, P. Shatalov30, Y.
Shcheglov29, T. Shears48, L. Shekhtman33, O. Shevchenko40, V. Shevchenko30, A.
Shires49, R. Silva Coutinho54, H.P. Skottowe43, T. Skwarnicki52, A.C. Smith37,
N.A. Smith48, E. Smith51,45, K. Sobczak5, F.J.P. Soler47, A. Solomin42, F.
Soomro49, B. Souza De Paula2, B. Spaan9, A. Sparkes46, P. Spradlin47, F.
Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone52,37, B.
Storaci23, M. Straticiuc28, U. Straumann39, N. Styles46, V.K. Subbiah37, S.
Swientek9, M. Szczekowski27, P. Szczypka38, T. Szumlak26, S. T’Jampens4, E.
Teodorescu28, F. Teubert37, E. Thomas37, J. van Tilburg11, V. Tisserand4, M.
Tobin39, S. Topp-Joergensen51, N. Torr51, E. Tournefier4,49, M.T. Tran38, A.
Tsaregorodtsev6, N. Tuning23, M. Ubeda Garcia37, A. Ukleja27, P. Urquijo52, U.
Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36,
S. Vecchi16, J.J. Velthuis42, M. Veltri17,g, K. Vervink37, B. Viaud7, I.
Videau7, X. Vilasis-Cardona35,n, J. Visniakov36, A. Vollhardt39, D. Voong42,
A. Vorobyev29, H. Voss10, S. Wandernoth11, J. Wang52, D.R. Ward43, A.D.
Webber50, D. Websdale49, M. Whitehead44, D. Wiedner11, L. Wiggers23, G.
Wilkinson51, M.P. Williams44,45, M. Williams49, F.F. Wilson45, J. Wishahi9, M.
Witek25, W. Witzeling37, S.A. Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z.
Xing52, Z. Yang3, R. Young46, O. Yushchenko34, M. Zavertyaev10,a, F. Zhang3,
L. Zhang52, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A.
Zvyagin 37.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Cracow, Poland
26Faculty of Physics & Applied Computer Science, Cracow, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam

##  1 Introduction

The fragmentation process, in which a primary $b$ quark forms either a
$b\bar{q}$ meson or a $bq_{1}q_{2}$ baryon, cannot be reliably predicted
because it is driven by strong dynamics in the non-perturbative regime. Thus
fragmentation functions for the various hadron species must be determined
experimentally. The LHCb experiment at the LHC explores a unique kinematic
region: it detects $b$ hadrons produced in a cone centered around the beam
axis covering a region of pseudorapidity $\eta$, defined in terms of the polar
angle $\theta$ with respect to the beam direction as $-\ln(\tan{\theta/2})$,
ranging approximately between 2 and 5. Knowledge of the fragmentation
functions allows us to relate theoretical predictions of the $b\bar{b}$ quark
production cross-section, derived from perturbative QCD, to the observed
hadrons. In addition, since many absolute branching fractions of $B^{-}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays have been well
measured at $e^{+}e^{-}$ colliders [1], it suffices to measure the ratio of
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ production to either
$B^{-}$ or $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ production to
perform precise absolute $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ branching fraction
measurements. In this paper we describe measurements of two ratios of
fragmentation functions: $f_{s}/(f_{u}+f_{d})$ and
$f_{\Lambda_{b}}/(f_{u}+f_{d})$, where $f_{q}\equiv{\cal B}(b\rightarrow
B_{q})$ and $f_{\Lambda_{b}}\equiv{\cal B}(b\rightarrow\Lambda_{b})$. The
inclusion of charged conjugate modes is implied throughout the paper, and we
measure the average production ratios.

Previous measurements of these fractions have been made at LEP [2] and at CDF
[3]. More recently, LHCb measured the ratio $f_{s}/f_{d}$ using the decay
modes $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}$, $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}K^{-}$, and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\pi^{-}$
[4] and theoretical input from QCD factorization [5, 6]. Here we measure this
ratio using semileptonic decays without any significant model dependence. A
commonly adopted assumption is that the fractions of these different species
should be the same in high energy $b$ jets originating from $Z^{0}$ decays and
high $p_{\rm T}$ $b$ jets originating from $p\bar{p}$ collisions at the
Tevatron or $pp$ collisions at LHC, based on the notion that hadronization is
a non-perturbative process occurring at the scale of $\Lambda_{\rm QCD}$.
Nonetheless, the results from different experiments are discrepant in the case
of the $b$ baryon fraction [2].

The measurements reported in this paper are performed using the LHCb detector
[7], a forward spectrometer designed to study production and decays of hadrons
containing $b$ or $c$ quarks. LHCb includes a vertex detector (VELO),
providing precise locations of primary $pp$ interaction vertices, and of
detached vertices of long lived hadrons. The momenta of charged particles are
determined using information from the VELO together with the rest of the
tracking system, composed of a large area silicon tracker located before a 4
Tm dipole magnet, and a combination of silicon strip and straw drift chamber
detectors located after the magnet. Two Ring Imaging Cherenkov (RICH)
detectors are used for charged hadron identification. Photon detection and
electron identification are implemented through an electromagnetic calorimeter
followed by a hadron calorimeter. A system of alternating layers of iron and
chambers provides muon identification. The two calorimeters and the muon
system provide the energy and momentum information to implement a first level
(L0) hardware trigger. An additional trigger level is software based, and its
algorithms are tuned to the experiment’s operating condition.

In this analysis we use a data sample of 3 pb-1 collected from 7 TeV centre-
of-mass energy $pp$ collisions at the LHC during 2010. The trigger selects
events where a single muon is detected without biasing the impact parameter
distribution of the decay products of the $b$ hadron, nor any kinematic
variable relevant to semileptonic decays. These features reduce the systematic
uncertainty in the efficiency. Our goal is to measure two specific production
ratios: that of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
relative to the sum of $B^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$, and that of $\mathchar
28931\relax_{b}^{0}$, relative to the sum of $B^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$. The sum of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$, $B^{-}$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\mathchar
28931\relax_{b}^{0}$ fractions does not equal one, as there is other $b$
production, namely a very small rate for $B_{c}^{-}$ mesons, bottomonia, and
other $b$ baryons that do not decay strongly into $\mathchar
28931\relax_{b}^{0}$, such as the $\Xi_{b}$. We measure relative fractions by
studying the final states $D^{0}\mu^{-}\overline{\nu}X$,
$D^{+}\mu^{-}\overline{\nu}X$, $D^{+}_{s}\mu^{-}\overline{\nu}X$, $\mathchar
28931\relax_{c}^{+}\mu^{-}\overline{\nu}X$,
$D^{0}K^{+}\mu^{-}\overline{\nu}X$, and $D^{0}p\mu^{-}\overline{\nu}X$. We do
not attempt to separate $f_{u}$ and $f_{d}$, but we measure the sum of $D^{0}$
and $D^{+}$ channels and correct for cross-feeds from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\mathchar
28931\relax_{b}^{0}$ decays. We assume near equality of the semileptonic decay
width of all $b$ hadrons, as discussed below. Charmed hadrons are
reconstructed through the modes listed in Table 1, together with their
branching fractions. We use all $D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$
decays rather than a combination of the resonant $\phi\pi^{+}$ and
$\overline{K}^{*0}K^{+}$ contributions, because these $D^{+}_{s}$ decays
cannot be cleanly isolated due to interference effects of different
amplitudes.

Table 1: Charmed hadron decay modes and branching fractions. Particle | Final state | Branching fraction (%)
---|---|---
$D^{0}$ | $K^{-}\pi^{+}$ | 3.89$\pm$0.05 [1]
$D^{+}$ | $K^{-}\pi^{+}\pi^{+}$ | 9.14$\pm$0.20 [19]
$D_{s}^{+}$ | $K^{-}K^{+}\pi^{+}$ | 5.50$\pm$0.27 [20]
$\Lambda_{c}^{+}$ | $pK^{-}\pi^{+}$ | 5.0$\pm$1.3 [1]

Each of these different charmed hadron plus muon final states can be populated
by a combination of initial $b$ hadron states. $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mesons decay semileptonically into
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 $D^{+}_{s}K$ meson pairs. $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mesons decay predominantly
into $D_{s}^{+}$ mesons, but can also decay into $D^{0}K^{+}$ and
$D^{+}K_{S}^{0}$ mesons; this is expected if the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays into a $D_{s}^{**}$
state that is heavy enough to decay into a $DK$ pair. In this paper we measure
this contribution using $D^{0}K^{+}X\mu^{-}\overline{\nu}$ events. Finally,
$\mathchar 28931\relax_{b}^{0}$ baryons decay mostly into $\mathchar
28931\relax_{c}^{+}$ final states. We determine other contributions using
$D^{0}pX\mu^{-}\overline{\nu}$ events. We ignore the contributions of
$b\rightarrow u$ decays that comprise approximately 1% of semileptonic $b$
hadron decays [8], and constitute a roughly equal portion of each $b$ species
in any case.

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

$\displaystyle n_{\rm corr}(B\rightarrow D^{0}\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{{\cal{B}}(D^{0}\rightarrow
K^{-}\pi^{+})\epsilon(B\rightarrow D^{0})}\times$
$\displaystyle\left[n(D^{0}\mu)-n(D^{0}K^{+}\mu)\frac{\epsilon(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{0})}{\epsilon(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{0}K^{+})}-n(D^{0}p\mu)\frac{\epsilon(\mathchar
28931\relax_{b}^{0}\rightarrow D^{0})}{\epsilon(\mathchar
28931\relax_{b}^{0}\rightarrow D^{0}p)}\right],$

where we use the shorthand $n(D\mu)\equiv n(DX\mu^{-}\overline{\nu})$. An
analogous abbreviation $\epsilon$ is used for the total trigger and detection
efficiencies. For example, the ratio $\epsilon(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{0})/\epsilon(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+})$ gives
the relative efficiency to reconstruct a charged $K$ in semi-muonic $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays producing a $D^{0}$
meson. Similarly

$\displaystyle n_{\rm corr}(B\rightarrow D^{+}\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{\epsilon(B\rightarrow
D^{+})}\left[\frac{n(D^{+}\mu^{-})}{{\cal{B}}(D^{+}\rightarrow
K^{-}\pi^{+}\pi^{+})}-\right.$ (2)
$\displaystyle\left.\frac{n(D^{0}K^{+}\mu^{-})}{{\cal{B}}(D^{0}\rightarrow
K^{-}\pi^{+})}\frac{\epsilon(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+})}{\epsilon(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+})}\right.$
$\displaystyle\left.-\frac{n(D^{0}p\mu^{-})}{{\cal{B}}(D^{0}\rightarrow
K^{-}\pi^{+})}\frac{\epsilon(\mathchar 28931\relax_{b}\rightarrow
D^{+})}{\epsilon(\mathchar 28931\relax_{b}\rightarrow D^{0}p)}\right].$

Both the $D^{0}X\mu^{-}\overline{\nu}$ and the $D^{+}X\mu^{-}\overline{\nu}$
final states contain small components of cross-feed from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays to
$D^{0}K^{+}X\mu^{-}\overline{\nu}$ and to $D^{+}K^{0}X\mu^{-}\overline{\nu}$.
These components are accounted for by the two decays $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}^{+}X\mu^{-}\overline{\nu}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s2}^{*+}X\mu^{-}\overline{\nu}$ as reported in a recent LHCb publication
[9]. The third terms in Eqs. 1 and 2 are due to a similar small cross-feed
from $\mathchar 28931\relax_{b}^{0}$ decays.

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

$\displaystyle n_{\rm corr}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\mu)$
$\displaystyle=$ $\displaystyle\frac{1}{\epsilon(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s})}\left[\frac{n(D^{+}_{s}\mu)}{{\cal B}(D^{+}_{s}\rightarrow
K^{+}K^{-}\pi^{+})}-\right.$ $\displaystyle\left.N(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}+B^{-}){\cal B}(B\rightarrow
D^{+}_{s}K\mu)\epsilon(\bar{B}\rightarrow D^{+}_{s}K\mu)\right],$

where the last term subtracts yields of $D^{+}_{s}KX\mu^{-}\overline{\nu}$
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 branching fraction ${\cal
B}(B\rightarrow D_{s}^{(*)+}K\mu\nu)=(6.1\pm 1.2)\times 10^{-4}$ [10] measured
by the BaBar experiment. In addition, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays semileptonically into
$DKX\mu^{-}\overline{\nu}$, and thus we need to add to Eq. 1

$n_{\rm corr}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
DK\mu)=2\frac{n(D^{0}K^{+}\mu)}{{\cal B}(D^{0}\rightarrow
K^{-}\pi^{+})\epsilon(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+}\mu)},$
(4)

where, using isospin symmetry, the factor of 2 accounts for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{0}X\mu^{-}\overline{\nu}$ semileptonic
decays.

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

$\frac{f_{s}}{f_{u}+f_{d}}=\frac{n_{\rm corr}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D\mu)}{n_{\rm
corr}(B\rightarrow D^{0}\mu)+n_{\rm corr}(B\rightarrow
D^{+}\mu)}\frac{\tau_{B^{-}}+\tau_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}}{2\tau_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}}}.$ (5)

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 final charmed hadron, given by the sum of the contributions shown
in Eqs. 1 and 4, and the symbols $\tau_{B_{i}}$ indicate the $B_{i}$ hadron
lifetimes, that are all well measured [1]. We use the average $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime, 1.472$\pm$0.025 ps
[1]. This equation assumes equality of the semileptonic widths of all the $b$
meson 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,
of the order of 1% [11, 12, 13].

The $\mathchar 28931\relax_{b}^{0}$ corrected yield is derived in an analogous
manner. We determine

$n_{\rm corr}(\mathchar 28931\relax_{b}^{0}\rightarrow
D\mu)=\frac{n(\Lambda_{c}^{+}\mu^{-})}{{\cal{B}}(\Lambda_{c}^{+}\rightarrow
pK^{-}\pi^{+})\epsilon(\mathchar 28931\relax_{b}^{0}\rightarrow\mathchar
28931\relax_{c}^{+})}+2\frac{n(D^{0}p\mu^{-})}{{\cal{B}}(D^{0}\rightarrow
K^{-}\pi^{+})\epsilon(\mathchar 28931\relax_{b}^{0}\rightarrow D^{0}p)},$ (6)

where $D$ represents a generic charmed hadron, and extract the
$\Lambda_{b}^{0}$ fraction using

$\frac{f_{\Lambda_{b}}}{f_{u}+f_{d}}=\frac{n_{\rm
corr}(\Lambda_{b}^{0}\rightarrow D\mu)}{n_{\rm corr}(B\rightarrow
D^{0}\mu)+n_{\rm corr}(B\rightarrow D^{+}\mu)}\frac{\tau_{B^{-}}+\tau_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}}{2\tau_{\Lambda_{b}^{0}}}(1-\xi).$
(7)

Again, we assume near equality of the semileptonic widths of different $b$
hadrons, but we apply a small adjustment $\xi=4\pm$2%, to account for the
chromomagnetic correction, affecting $b$-flavoured mesons but not $b$ baryons
[11, 12, 13]. The uncertainty is evaluated with very conservative assumptions
for all the parameters of the heavy quark expansion.

##  2 Analysis method

To isolate a sample of $b$ flavoured hadrons with low backgrounds, we match
charmed hadron candidates with tracks identified as muons. Right-sign (RS)
combinations have the sign of the charge of the muon being the same as the
charge of the kaon in $D^{0}$, $D^{+}$, or $\mathchar 28931\relax_{c}^{+}$
decays, or the opposite charge of the pion in $D^{+}_{s}$ decays, while wrong-
sign (WS) combinations comprise combinations with opposite charge
correlations. WS events are useful to estimate certain backgrounds. This
analysis follows our previous investigation of $b\rightarrow
D^{0}X\mu^{-}\overline{\nu}$ [14]. We consider events where a well-identified
muon with momentum greater than 3 GeV and transverse momentum greater than 1.2
GeV is found. Charmed hadron candidates are formed from hadrons with momenta
greater than 2 GeV and transverse momenta greater than 0.3 GeV, and we require
that the average transverse momentum of the hadrons forming the candidate be
greater than 0.7 GeV. Kaons, pions, and protons are identified using the RICH
system. The impact parameter (IP), defined as the minimum distance of approach
of the track with respect to the primary vertex, is used to select tracks
coming from charm decays. We require that the $\chi^{2}$, formed by using the
hypothesis that each track’s IP is equal to 0, is greater than 9. Moreover,
the selected tracks must be consistent with coming from a common vertex: the
$\chi^{2}$ per number of degrees of freedom of the vertex fit must be smaller
than 6. In order to ensure that the charm vertex is distinct from the primary
$pp$ interaction vertex, we require that the $\chi^{2}$, based on the
hypothesis that the decay flight distance from the primary vertex is zero, is
greater than 100.

Charmed hadrons and muons are combined to form a partially reconstructed $b$
hadron by requiring that they come from a common vertex, and that the cosine
of the angle between the momentum of the charmed hadron and muon pair and the
line from the $D\mu$ vertex to the primary vertex be greater than 0.999. As
the charmed hadron is a decay product of the $b$ hadron, we require that the
difference in $z$ component of the decay vertex of the charmed hadron
candidate and that of the beauty candidate be greater than 0. We explicitly
require that the $\eta$ of the $b$ hadron candidate be between 2 and 5. We
measure $\eta$ using the line defined by connecting the primary event vertex
and the vertex formed by the $D$ and the $\mu$. Finally, the invariant mass of
the charmed hadron and muon system must be between 3 and 5 GeV for
$D^{0}\mu^{-}$ and $D^{+}\mu^{-}$ candidates, between 3.1 and 5.1 GeV for
$D^{+}_{s}\mu^{-}$ candidates, and between 3.3 and 5.3 GeV for $\mathchar
28931\relax_{c}^{+}\mu^{-}$ candidates.

We perform our analysis in a grid of 3 $\eta$ and 5 $p_{\rm T}$ bins, covering
the range $2<\eta<5$ and $p_{\rm T}\leq 14$ GeV. The $b$ hadron signal is
separated from various sources of background by studying the two dimensional
distribution of charmed hadron candidate invariant mass and ln(IP/mm). This
approach allows us to determine the background coming from false combinations
under the charmed hadron signal mass peak directly. The study of the ln(IP/mm)
distribution allows the separation of prompt charm decay candidates from
charmed hadron daughters of $b$ hadrons [14]. We refer to these samples as
Prompt and Dfb respectively.

###  2.1 Signal extraction

We describe the method used to extract the charmed hadron-$\mu$ signal by
using the $D^{0}X\mu^{-}\overline{\nu}$ final state as an example; the same
procedure is applied to the final states $D^{+}X\mu^{-}\overline{\nu}$,
$D^{+}_{s}X\mu^{-}\overline{\nu}$, and $\mathchar
28931\relax_{c}^{+}X\mu^{-}\overline{\nu}$. We perform unbinned extended
maximum likelihood fits to the two-dimensional distributions in $K^{-}\pi^{+}$
invariant mass over a region extending $\pm$80 MeV from the $D^{0}$ mass peak,
and ln(IP/mm). The parameters of the IP distribution of the Prompt sample are
found by examining directly produced charm [14] whereas a shape derived from
simulation is used for the Dfb component.

Figure 1: The logarithm of the IP distributions for (a) RS and (c) WS $D^{0}$
candidate combinations with a muon. The dotted curves show the false $D^{0}$
background, the small red-solid curves the Prompt yields, the dashed curves
the Dfb signal, and the larger green-solid curves the total yields. The
invariant $K^{-}\pi^{+}$ mass spectra for (b) RS combinations and (d) WS
combinations are also shown.

An example fit for $D^{0}\mu^{-}\overline{\nu}X$, using the whole $p_{\rm T}$
and $\eta$ range, is shown in Fig. 1. The fitted yields for RS are
27666$\pm$187 Dfb, 695$\pm$43 Prompt, and 1492$\pm$30 false $D^{0}$
combinations, inferred from the fitted yields in the sideband mass regions,
spanning the intervals between 35 and 75 MeV from the signal peak on both
sides. For WS we find 362$\pm$39 Dfb, 187$\pm$18 Prompt, and 1134$\pm$19 false
$D^{0}$ combinations. The RS yield includes a background of around 0.5% from
incorrectly identified $\mu$ candidates. As this paper focuses on ratios of
yields, we do not subtract this component. Figure 2 shows the corresponding
fits for the $D^{+}X\mu^{-}\overline{\nu}$ final state. The fitted yields
consist of 9257$\pm$110 Dfb events, 362$\pm$34 Prompt, and 1150$\pm$22 false
$D^{+}$ combinations. For WS we find 77$\pm$22 Dfb, 139$\pm$14 Prompt and
307$\pm$10 false $D^{+}$ combinations.

Figure 2: The logarithm of the IP distributions for (a) RS and (c) WS $D^{+}$
candidate combinations with a muon. The grey-dotted curves show the false
$D^{+}$ background, the small red-solid curves the Prompt yields, the blue-
dashed curves the Dfb signal, and the larger green-solid curves the total
yields. The invariant $K^{-}\pi^{+}\pi^{+}$ mass spectra for (b) RS
combinations and (d) WS combinations are also shown.

The analysis for the $D_{s}^{+}X\mu^{-}\overline{\nu}$ mode follows in the
same manner. Here, however, we are concerned about the reflection from
$\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+}$ where the proton is taken to be a
kaon, since we do not impose an explicit proton veto. Using such a veto would
lose 30% of the signal and also introduce a systematic error. We choose to
model separately this particular background. We add a probability density
function (PDF) determined from simulation to model this, and the level is
allowed to float within the estimated error on the size of the background. The
small peak near 2010 MeV in Fig. 3(b) is due to
$D^{*+}\rightarrow\pi^{+}D^{0},D^{0}\rightarrow K^{+}K^{-}$. We explicitly
include this term in the fit, assuming the shape to be the same as for the
$D^{+}_{s}$ signal, and we obtain 4$\pm$1 events in the RS signal region and
no events in the WS signal region. The measured yields in the RS sample are
2192$\pm$64 Dfb, 63$\pm$16 Prompt, 985$\pm$145 false $D^{+}_{s}$ background,
and 387$\pm$132 $\Lambda_{c}^{+}$ reflection background. The corresponding
yields in the WS sample are 13$\pm$19, 20$\pm$7, 499$\pm$16, and 3$\pm$3
respectively. Figure 3 shows the fit results.

Figure 3: The logarithm of the IP distributions for (a) RS and (c) WS
$D^{+}_{s}$ candidate combinations with a muon. The grey-dotted curves show
the false $D^{+}_{s}$ background, the small red-solid curves the Prompt
yields, the blue-dashed curves the Dfb signal, the purple dash-dotted curves
represent the background originating from $\mathchar 28931\relax_{c}^{+}$
reflection, and the larger green-solid curves the total yields. The invariant
$K^{-}K^{+}\pi^{+}$ mass spectra for RS combinations (b) and WS combinations
(d) are also shown.  Figure 4: The logarithm of the IP distributions for (a)
RS and (c) WS $\Lambda_{c}^{+}$ candidate combinations with a muon. The grey-
dotted curves show the false $\Lambda_{c}^{+}$ background, the small red-solid
curves the Prompt yields, the blue-dashed curves the Dfb signal, and the
larger green-solid curves the total yields. The invariant $pK^{-}\pi^{+}$ mass
spectra for RS combinations (b) and WS combinations (d) are also shown.

The last final state considered is $\mathchar
28931\relax_{c}^{+}X\mu^{-}\overline{\nu}$. Figure 4 shows the data and fit
components to the ln(IP/mm) and $pK^{-}\pi^{+}$ invariant mass combinations
for events with $2<\eta<5$. This fit gives 3028$\pm$112 RS Dfb events,
43$\pm$17 RS Prompt events, 589$\pm$27 RS false $\mathchar
28931\relax_{c}^{+}$ combinations, 9$\pm$16 WS Dfb events, 0.5$\pm$4 WS Prompt
events, and 177$\pm$10 WS false $\mathchar 28931\relax_{c}^{+}$ combinations.

Figure 5: (a) Invariant mass of $D^{0}p$ candidates that vertex with each
other and together with a RS muon (black closed points) and for a
$\overline{p}$ (red open points) instead of a $p$; (b) fit to $D^{0}$
invariant mass for RS events with the invariant mass of $D^{0}p$ candidate in
the signal mass difference window; (c) fit to $D^{0}$ invariant mass for WS
events with the invariant mass of $D^{0}p$ candidate in the signal mass
difference window.

The $\mathchar 28931\relax_{b}^{0}$ may also decay into
$D^{0}pX\mu^{-}\overline{\nu}$. We search for these decays by requiring the
presence of a track well identified as a proton and detached from any primary
vertex. The resulting $D^{0}p$ invariant mass distribution is shown in Fig. 5.
We also show the combinations that cannot arise from $\Lambda_{b}^{0}$ decay,
namely those with $D^{0}\overline{p}$ combinations. There is a clear excess of
RS over WS combinations especially near threshold. Fits to the $K^{-}\pi^{+}$
invariant mass in the $[m(K^{-}\pi^{+}p)-m(K^{-}\pi^{+})+m(D^{0})_{\rm PDG}]$
region shown in Fig. 5(a) give 154$\pm$13 RS events and 55$\pm$8 WS events. In
this case, we use the WS yield for background subtraction, scaled by the RS/WS
background ratio determined with a MC simulation including $(B^{-}+\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{0}X\mu^{-}\overline{\nu})$ and generic $b\overline{b}$ events. This ratio
is found to be 1.4$\pm$0.2. Thus, the net signal is 76$\pm$17$\pm$11, where
the last error reflects the uncertainty in the ratio between RS and WS
background.

###  2.2 Background studies

Apart from false $D$ combinations, separated from the signal by the two-
dimensional fit described above, there are also physical background sources
that affect the RS Dfb samples, and originate from $b\overline{b}$ events,
which are studied with a MC simulation. In the meson case, the background
mainly comes from $b\rightarrow DDX$ with one of the $D$ mesons decaying semi-
muonically, and from combinations of tracks from the $pp\rightarrow b\bar{b}X$
events, where one $b$ hadron decays into a $D$ meson and the other $b$ hadron
decays semi-muonically. The background fractions are (1.9$\pm$0.3)% for
$D^{0}X\mu^{-}\overline{\nu}$, (2.5$\pm$0.6)% for
$D^{+}X\mu^{-}\overline{\nu}$, and (5.1$\pm$1.7)% for
$D_{s}^{+}X\mu^{-}\overline{\nu}$. The main background component for
$\mathchar 28931\relax_{b}^{0}$ semileptonic decays is $\mathchar
28931\relax_{b}^{0}$ decaying into $D^{-}_{s}\mathchar 28931\relax_{c}^{+}$,
and the $D^{-}_{s}$ decaying semi-muonically. Overall, we find a very small
background rate of (1.0$\pm$0.2)%, where the error reflects only the
statistical uncertainty in the simulation. We correct the candidate $b$ hadron
yields in the signal region with the predicted background fractions. A
conservative 3% systematic uncertainty in the background subtraction is
assigned to reflect modelling uncertainties.

Figure 6: Projections of the two-dimensional fit to the $q^{2}$ and
$m(D^{+}_{s}\mu)$ distributions of semileptonic decays including a $D^{+}_{s}$
meson. The $D_{s}^{*}/D_{s}$ ratio has been fixed to the measured $D^{*}/D$
ratio in light $B$ decays (2.42$\pm$0.10), and the background contribution is
obtained using the sidebands in the $K^{+}K^{-}\pi^{+}$ mass spectrum. The
different components are stacked: the background is represented by a black
dot-dashed line, $D^{+}_{s}$ by a red dashed line, $D_{s}^{*+}$ by a blue
dash-double dotted line and $D_{s}^{**+}$ by a green dash-dotted line.

###  2.3 Monte Carlo simulation and efficiency determination

In order to estimate the detection efficiency, we need some knowledge of the
different final states which contribute to the Cabibbo favoured semileptonic
width, as some of the selection criteria affect final states with distinct
masses and quantum numbers differently. Although much is known about the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $B^{-}$ semileptonic
decays, information on the corresponding $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\mathchar
28931\relax_{b}^{0}$ semileptonic decays is rather sparse. In particular, the
hadronic composition of the final states in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays is poorly known [9],
and only a study from CDF provides some constraints on the branching ratios of
final states dominant in the corresponding $\mathchar 28931\relax_{b}^{0}$
decays [15].

In the case of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}$
semileptonic decays, we assume that the final states are $D^{+}_{s}$,
$D^{*+}_{s}$, $D_{s0}^{*}(2317)^{+}$, $D_{s1}(2460)^{+}$, and
$D_{s1}(2536)^{+}$. States above $DK$ threshold decay predominantly into
$D^{(*)}K$ final states. We model the decays to the final states
$D^{+}_{s}\mu^{-}\overline{\nu}$ and $D^{*+}_{s}\mu^{-}\overline{\nu}$ with
HQET form factors using normalization coefficients derived from studies of the
corresponding $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $B^{-}$
semileptonic decays [1], while we use the ISGW2 form factor model [16] to
describe final states including higher mass resonances.

In order to determine the ratio between the different hadron species in the
final state, we use the measured kinematic distributions of the quasi-
exclusive process $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\mu^{-}\overline{\nu}X$. To reconstruct the squared invariant mass of
the $\mu^{-}\overline{\nu}$ pair ($q^{2}$), we exploit the measured direction
of the $b$ hadron momentum, which, together with energy and momentum
conservation, assuming no missing particles other than the neutrino, allow the
reconstruction of the $\nu$ 4-vector, up to a two-fold ambiguity, due to its
unknown orientation with respect to the $B$ flight path in its rest frame. We
choose the solution corresponding to the lowest $b$ hadron momentum. This
method works well when there are no missing particles, or when the missing
particles are soft, as in the case when the charmed system is a $D^{*}$ meson.
We then perform a two-dimensional fit to the $q^{2}$ versus $m(\mu D^{+}_{s})$
distribution. Figure 6 shows stacked histograms of the $D^{+}_{s}$,
$D^{*+}_{s}$, and $D_{s}^{**+}$ components. In the fit we constrain the ratio
${\cal B}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}\mu^{-}\overline{\nu})/{\cal B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\mu^{-}\overline{\nu})$ to be equal to the average
$D^{*}\mu^{-}\overline{\nu}/D\mu^{-}\overline{\nu}$ ratio in semileptonic
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $B^{-}$ decays
(2.42$\pm$0.10) [1]. This constraint reduces the uncertainty of one $D^{**}$
fraction. We have also performed fits removing this assumption, and the
variation between the different components is used to assess the modelling
systematic uncertainty.

A similar procedure is applied to the $\mathchar 28931\relax_{c}^{+}\mu^{-}$
sample and the results are shown in Fig. 7. In this case we consider three
final states, $\mathchar 28931\relax_{c}^{+}\mu^{-}\overline{\nu}$,
${\it\Lambda_{c}}(2595)^{+}\mu^{-}\overline{\nu}$, and
${\it\Lambda_{c}}(2625)^{+}\mu^{-}\overline{\nu}$, with form factors from the
model of Ref. [17]. We constrain the two highest mass hadrons to be produced
in the ratio predicted by this theory.

Figure 7: Projections of the two-dimensional fit to the $q^{2}$ and
$m(\mathchar 28931\relax_{c}^{+}\mu^{-})$ distributions of semileptonic decays
including a $\mathchar 28931\relax_{c}^{+}$ baryon. The different components
are stacked: the dotted line represents the combinatoric background, the
bigger dashed line (red) represents the $\mathchar
28931\relax_{c}^{+}\mu^{-}\overline{\nu}$ component, the smaller dashed line
(blue) the ${\it\Lambda_{c}}(2595)^{+}$, and the solid line represents the
${\it\Lambda_{c}}(2625)^{+}$ component. The
${\it\Lambda_{c}}(2595)^{+}/{\it\Lambda_{c}}(2625)^{+}$ ratio is fixed to its
predicted value, as described in the text. Figure 8: Measured proton
identification efficiency as a function of the $\mathchar
28931\relax_{c}^{+}\mu^{-}$ $p_{\rm T}$ for $2<\eta<3$, $3<\eta<4$, $4<\eta<5$
respectively, and for the selection criteria used in the $\mathchar
28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+}$ reconstruction.

The measured pion, kaon and proton identification efficiencies are determined
using $K_{\rm S}^{0}$, $D^{*+}$, and $\Lambda^{0}$ calibration samples where
$p$, $K$, and $\pi$ are selected without utilizing the particle identification
criteria. The efficiency is obtained by fitting simultaneously the invariant
mass distributions of events either passing or failing the identification
requirements. Values are obtained in bins of the particle $\eta$ and $p_{\rm
T}$, and these efficiency matrices are applied to the MC simulation.
Alternatively, the particle identification efficiency can be determined by
using the measured efficiencies and combining them with weights proportional
to the fraction of particle types with a given $\eta$ and $p_{\rm T}$ for each
$\mu$ charmed hadron pair $\eta$ and $p_{\rm T}$ bin. The overall efficiencies
obtained with these two methods are consistent. An example of the resulting
particle identification efficiency as a function of the $\eta$ and $p_{\rm T}$
of the $\mathchar 28931\relax_{c}^{+}\mu^{-}$ pair is shown in Fig. 8.

As the functional forms of the fragmentation ratios in terms of $p_{\rm T}$
and $\eta$ are not known, we determine the efficiencies for the final states
studied as a function of $p_{\rm T}$ and $\eta$ within the LHCb acceptance.
Figure 9 shows the results.

##  3 Evaluation of the ratios ${f_{s}/(f_{u}+f_{d})}$ and
${f_{\Lambda_{b}}/(f_{u}+f_{d})}$

Perturbative QCD calculations lead us to expect the ratios
$f_{s}/(f_{u}+f_{d})$ and $f_{\Lambda_{b}}/(f_{u}+f_{d})$ to be independent of
$\eta$, while a possible dependence upon the $b$ hadron transverse momentum
$p_{\rm T}$ is not ruled out, especially for ratios involving baryon species
[18]. Thus we determine these fractions in different $p_{\rm T}$ and $\eta$
bins. For simplicity, we use the transverse momentum of the charmed
hadron-$\mu$ pair as the $p_{\rm T}$ variable, and do not try to unfold the
$b$ hadron transverse momentum.

In order to determine the corrected yields entering the ratio
$f_{s}/(f_{u}+f_{d})$, we determine yields in a matrix of three $\eta$ and
five $p_{\rm T}$ bins and divide them by the corresponding efficiencies. We
then use Eq. 5, with the measured lifetime ratio $(\tau_{B^{-}}+\tau_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}})/2\tau_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}}=1.07\pm 0.02$ [1] to derive
the ratio $f_{s}/(f_{u}+f_{d})$ in two $\eta$ bins. The measured ratio is
constant over the whole $\eta$-$p_{\rm T}$ domain. Figure 10 shows the
$f_{s}/(f_{u}+f_{d})$ fractions in bins of $p_{\rm T}$ in two $\eta$
intervals.

Figure 9: Efficiencies for $D^{0}\mu^{-}\overline{\nu}X$,
$D^{+}\mu^{-}\overline{\nu}X$, $D^{+}_{s}\mu^{-}\overline{\nu}X$, $\mathchar
28931\relax_{c}^{+}\mu^{-}\overline{\nu}X$ as a function of $\eta$ and $p_{\rm
T}$.

$\begin{array}[]{cc}\includegraphics[width=433.62pt]{fig10}\end{array}$

Figure 10: Ratio between $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and light $B$ meson production fractions as a function of the transverse momentum of the $D^{+}_{s}\mu^{-}$ pair in two bins of $\eta$. The errors shown are statistical only. Table 2: Systematic uncertainties on the relative $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ production fraction. Source | Error (%)
---|---
Bin-dependent errors | 1.0
${\cal B}(D^{0}\rightarrow K^{-}\pi^{+})$ | 1.2
${\cal B}(D^{+}\rightarrow K^{-}\pi^{+}\pi^{+})$ | 1.5
${\cal B}(D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+})$ | 4.9
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ semileptonic decay modelling | 3.0
Backgrounds | 2.0
Tracking efficiency | 2.0
Lifetime ratio | 1.8
PID efficiency | 1.5
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+}X\mu^{-}\overline{\nu}$ | ${}^{+4.1}_{-1.1}$
${\cal B}((B^{-},\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0})\rightarrow D_{s}^{+}KX\mu^{-}\overline{\nu})$ | 2.0
Total | ${}^{+8.6}_{-7.7}$
Figure 11: $f_{+}/f_{0}$ as a function of $p_{\rm T}$ for $\eta$=(2,3) (a) and
$\eta$=(3,5) (b). The horizontal line shows the average value. The error shown
combines statistical and systematic uncertainties accounting for the detection
efficiency and the particle identification efficiency.

By fitting a single constant to all the data, we obtain
$f_{s}/(f_{u}+f_{d})=0.134\pm 0.004^{+0.011}_{-0.010}$ in the interval
$2<\eta<5$, where the first error is statistical and the second is systematic.
The latter includes several different sources listed in Table 2. The dominant
systematic uncertainty is caused by the experimental uncertainty on
${\cal{B}}(D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+})$ of 4.9%. Adding in the
contributions of the $D^{0}$ and $D^{+}$ branching fractions we have a
systematic error of 5.5% due to the charmed hadron branching fractions. The
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ semileptonic modelling
error is derived by changing the ratio between different hadron species in the
final state obtained by removing the SU(3) symmetry constrain, and changing
the shapes of the less well known $D^{**}$ states. The tracking efficiency
errors mostly cancel in the ratio since we are dealing only with combinations
of three or four tracks. The lifetime ratio error reflects the present
experimental accuracy [1]. We correct both for the bin-dependent PID
efficiency obtained with the procedure detailed before, accounting for the
statistical error of the calibration sample, and the overall PID efficiency
uncertainty, due to the sensitivity to the event multiplicity. The latter is
derived by taking the kaon identification efficiency obtained with the method
described before, without correcting for the different track multiplicities in
the calibration and signal samples. This is compared with the results of the
same procedure performed correcting for the ratio of multiplicities in the two
samples. The error due to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{0}K^{+}X\mu^{-}\overline{\nu}$ is obtained by changing the RS/WS background
ratio predicted by the simulation within errors, and evaluating the
corresponding change in $f_{s}/(f_{u}+f_{d})$. Finally, the error due to
$(B^{-},\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0})\rightarrow
D_{s}^{+}KX\mu^{-}\overline{\nu}$ reflects the uncertainty in the measured
branching fraction.

Isospin symmetry implies the equality of $f_{d}$ and $f_{u}$, which allows us
to compare $f_{+}/f_{0}\equiv n_{\rm corr}(D^{+}\mu)/n_{\rm corr}(D^{0}\mu)$
with its expected value. It is not possible to decouple the two ratios for an
independent determination of $f_{u}/f_{d}$. Using all the known semileptonic
branching fractions [1], we estimate the expected relative fraction of the
$D^{+}$ and $D^{0}$ modes from $B^{+/0}$ decays to be $f_{+}/f_{0}=0.375\pm
0.023,$ where the error includes a 6% theoretical uncertainty associated to
the extrapolation of present experimental data needed to account for the
inclusive $b\rightarrow c\mu^{-}\overline{\nu}$ semileptonic rate. Our
corrected yields correspond to $f_{+}/f_{0}=0.373\pm 0.006$ (stat) $\pm$ 0.007
(eff) $\pm$ 0.014, for a total uncertainty of 4.5%. The last error accounts
for uncertainties in $B$ background modelling, in the
$D^{0}K^{+}\mu^{-}\overline{\nu}$ yield, the $D^{0}p\mu^{-}\overline{\nu}$
yield, the $D^{0}$ and $D^{+}$ branching fractions, and tracking efficiency.
The other systematic errors mostly cancel in the ratio. Our measurement of
$f_{+}/f_{0}$ is not seen to be dependent upon $p_{\rm T}$ or $\eta$, as shown
in Fig. 11, and is in agreement with expectation.

Figure 12: Fragmentation ratio $f_{\mathchar 28931\relax_{b}}/(f_{u}+f_{d})$
dependence upon $p_{\rm T}(\mathchar 28931\relax_{c}^{+}\mu^{-})$. The errors
shown are statistical only.

We follow the same procedure to derive the fraction $f_{\mathchar
28931\relax_{b}}/(f_{u}+f_{d})$, using Eq. 7 and the ratio
$(\tau_{B^{-}}+\tau_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}})/(2\tau_{\mathchar
28931\relax_{b}^{0}})=1.14\pm 0.03$ [1]. In this case, we observe a $p_{\rm
T}$ dependence in the two $\eta$ intervals. Figure 12 shows the data fitted to
a straight line

$\frac{f_{\mathchar 28931\relax_{b}}}{f_{u}+f_{d}}=a[1+b\times p_{\rm T}({\rm
GeV})].$ (8)

Table 3 summarizes the fit results. A corresponding fit to a constant shows
that a $p_{\rm T}$ independent $f_{\mathchar 28931\relax_{b}}/(f_{u}+f_{d})$
is excluded at the level of four standard deviations. The systematic errors
reported in Table 3 include only the bin-dependent terms discussed above.

Table 4 summarizes all the sources of absolute scale systematic uncertainties,
that include several components. Their definitions mirror closely the
corresponding uncertainties for the $f_{s}/(f_{u}+f_{d})$ determination, and
are assessed with the same procedures. The term $\mathchar
28931\relax_{b}\rightarrow D^{0}pX\mu^{-}\overline{\nu}$ accounts for the
uncertainty in the raw $D^{0}pX\mu^{-}\overline{\nu}$ yield, and is evaluated
by changing the RS/WS background ratio (1.4$\pm$0.2) within the quoted
uncertainty. In addition, an uncertainty of 2% is associated with the
derivation of the semileptonic branching fraction ratios from the
corresponding lifetimes, labelled $\Gamma_{\rm sl}$ in Table 4. The
uncertainty is derived assigning conservative errors to the parameters
affecting the chromomagnetic operator that influences the $B$ meson total
decay widths, but not the $\mathchar 28931\relax_{b}^{0}$. By far the largest
term is the poorly known ${\cal B}(\mathchar 28931\relax_{c}^{+}\rightarrow
pK^{-}\pi^{+}$); thus it is quoted separately.

Table 3: Coefficients of the linear fit describing the $p_{\rm T}(\mathchar 28931\relax_{c}^{+}\mu^{-})$ dependence of $f_{\mathchar 28931\relax_{b}}/(f_{u}+f_{d})$. The systematic uncertainties included are only those associated with the bin-dependent MC and particle identification errors. $\eta$ range | $a$ | $b$
---|---|---
2-3 | 0.434$\pm$0.040$\pm$0.025 | -0.036$\pm$0.008$\pm$0.004
3-5 | 0.397$\pm$0.020$\pm$0.009 | -0.028$\pm$0.006$\pm$0.003
2-5 | 0.404$\pm$0.017$\pm$0.009 | -0.031$\pm$0.004$\pm$0.003
Table 4: Systematic uncertainties on the absolute scale of $f_{\Lambda_{b}}/(f_{u}+f_{d})$. Source | Error (%)
---|---
Bin dependent errors | 2.2
${\cal B}(\mathchar 28931\relax_{b}^{0}\rightarrow D^{0}pX\mu^{-}\overline{\nu})$ | 2.0
Monte Carlo modelling | 1.0
Backgrounds | 3.0
Tracking efficiency | 2.0
$\Gamma_{\rm sl}$ | 2.0
Lifetime ratio | 2.6
PID efficiency | 2.5
Subtotal | 6.3
${\cal B}(\mathchar 28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+})$ | 26.0
Total | $26.8$

In view of the observed dependence upon $p_{\rm T}$, we present our results as

$\left[\frac{f_{\mathchar 28931\relax_{b}}}{f_{u}+f_{d}}\right](p_{\rm
T})=(0.404\pm 0.017\pm 0.027\pm 0.105)\times[1-(0.031\pm 0.004\pm 0.003)\times
p_{\rm T}{\rm(GeV)}],$ (9)

where the scale factor uncertainties are statistical, systematic, and the
error on ${\cal B}(\mathchar 28931\relax_{c}\rightarrow pK^{-}\pi^{+})$
respectively. The correlation coefficient between the scale factor and the
slope parameter in the fit with the full error matrix is $-0.63$. Previous
measurements of this fraction have been made at LEP and the Tevatron [3]. LEP
obtains 0.110$\pm$0.019 [2]. This fraction has been calculated by combining
direct rate measurements with time-integrated mixing probability averaged over
an unbiased sample of semi-leptonic $b$ hadron decays. CDF measures
$f_{\mathchar 28931\relax_{b}}/(f_{u}+f_{d})=0.281\pm
0.012^{+0.011+0.128}_{-0.056-0.086}$, where the last error reflects the
uncertainty in ${\cal B}(\mathchar 28931\relax_{c}^{+}\rightarrow
pK^{-}\pi^{+})$. It has been suggested [3] that the difference between the
Tevatron and LEP results is explained by the different kinematics of the two
experiments. The average $p_{\rm T}$ of the $\mathchar
28931\relax_{c}^{+}\mu^{-}$ system is 10 GeV for CDF, while the $b$-jets, at
LEP, have $p\approx 40$ GeV. LHCb probes an even lower $b$ $p_{\rm T}$ range,
while retaining some sensitivity in the CDF kinematic region. These data are
consistent with CDF in the kinematic region covered by both experiments, and
indicate that the baryon fraction is higher in the lower $p_{\rm T}$ region.

##  4 Combined result for the production fraction $f_{s}/f_{d}$ from LHCb

From the study of $b$ hadron semileptonic decays reported above, and assuming
isospin symmetry, namely $f_{u}=f_{d}$, we obtain

$\left(\frac{f_{s}}{f_{d}}\right)_{\rm sl}=0.268\pm 0.008({\rm
stat})^{+0.022}_{-0.020}({\rm syst}),$

where the first error is statistical and the second is systematic.

Measurements of this quantity have also been made by LHCb by using hadronic
$B$ meson decays [4]. The ratio determined using the relative abundances of
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}$ to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}K^{-}$ is

$\left(\frac{f_{s}}{f_{d}}\right)_{\rm h1}=0.250\pm 0.024(\rm stat)\pm
0.017({\rm syst})\pm 0.017({\rm theor}),$

while that from the relative abundances of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\pi^{-}$ to
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$ [4]
is

$\left(\frac{f_{s}}{f_{d}}\right)_{\rm h2}=0.256\pm 0.014({\rm stat})\pm
0.019({\rm syst})\pm 0.026({\rm theor}).$

The first uncertainty is statistical, the second systematic and the third
theoretical. The theoretical uncertainties in both cases include non-
factorizable SU(3)-breaking effects and form factor ratio uncertainties. The
second ratio is affected by an additional source, accounting for the
$W$-exchange diagram in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$ decay.

In order to average these results, we consider the correlations between
different sources of systematic uncertainties, as shown in Table 5. We then
utilise a generator of pseudo-experiments, where each independent source of
uncertainty is generated as a random variable with Gaussian distribution,
except for the component $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{0}K^{+}\mu^{-}\overline{\nu}_{\mu}X$, which is modeled with a bifurcated
Gaussian with standard deviations equal to the positive and negative errors
shown in Table 5. This approach to the averaging procedure is motivated by the
goal of proper treatment of asymmetric errors [21]. We assume that the
theoretical errors have a Gaussian distribution.

Table 5: Summary of the systematic and theoretical uncertainties in the three LHCb measurements of $f_{s}/f_{d}$. Source | Error (%) |
---|---|---
| $(f_{s}/f_{d})_{\rm sl}$ | $(f_{s}/f_{d})_{\rm h1}$ | $(f_{s}/f_{d})_{\rm h2}$ |
Bin dependent error | 1.0 | - | - | Uncorrelated
Semileptonic decay modelling | 3.0 | - | - | Uncorrelated
Backgrounds | 2.0 | - | - | Uncorrelated
Fit model | - | 2.8 | 2.8 | Uncorrelated
Trigger simulation | - | 2.0 | 2.0 | Uncorrelated
Tracking efficiency | 2.0 | - | - | Uncorrelated
${\cal B}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+}X\mu^{-}\overline{\nu})$ | ${}^{+4.1}_{-1.1}$ | - | - | Uncorrelated
${\cal B}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}/B^{-}\rightarrow D^{+}_{s}KX\mu^{-}\overline{\nu})$ | 2.0 | - | - | Uncorrelated
Particle identification calibration | 1.5 | 1.0 | 2.5 | Correlated
$B$ lifetimes | 1.5 | 1.5 | 1.5 | Correlated
${\cal B}(D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+})$ | 4.9 | 4.9 | 4.9 | Correlated
${\cal B}(D^{+}\rightarrow K^{-}\pi^{+}\pi^{-})$ | 1.5 | 1.5 | 1.5 | Correlated
SU(3) and form factors | - | 6.1 | 6.1 | Correlated
$W$-exchange | - | - | 7.8 | Uncorrelated

We define the average fraction as

$f_{s}/f_{d}=\alpha_{1}(f_{s}/f_{d})_{\rm sl}+\alpha_{2}(f_{s}/f_{d})_{\rm
h1}+\alpha_{3}(f_{s}/f_{d})_{\rm h2},$ (10)

where

$\alpha_{1}+\alpha_{2}+\alpha_{3}=1.$ (11)

The RMS value of $f_{s}/f_{d}$ is then evaluated as a function of $\alpha_{1}$
and $\alpha_{2}$.

We derive the most probable value $f_{s}/f_{d}$ by determining the
coefficients $\alpha_{i}$ at which the RMS is minimum, and the total errors by
computing the boundaries defining the 68% CL, scanning from top to bottom
along the axes $\alpha_{1}$ and $\alpha_{2}$ in the range comprised between 0
and 1. The optimal weights determined with this procedure are
$\alpha_{1}=0.73$, and $\alpha_{2}=0.14$, corresponding to the most probable
value

$f_{s}/f_{d}=0.267^{+0.021}_{-0.020}.$

The most probable value differs slightly from a simple weighted average of the
three measurements because of the asymmetry of the error distribution in the
semileptonic determination. By switching off different components we can
assess the contribution of each source of uncertainty. Table 6 summarizes the
results.

Table 6: Uncertainties in the combined value of $f_{s}/f_{d}$. Source | Error (%)
---|---
Statistical | 2.8
Experimental systematic (symmetric) | 3.3
${\cal B}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+}X\mu^{-}\overline{\nu})$ | ${}^{+3.0}_{-0.8}$
${\cal B}(D^{+}\rightarrow K^{-}\pi^{+}\pi^{-})$ | 2.2
${\cal B}(D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+})$ | 4.9
$B$ lifetimes | 1.5
${\cal B}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}/B^{-}\rightarrow D^{+}_{s}KX\mu^{-}\overline{\nu})$ | 1.5
Theory | 1.9

##  5 Conclusions

We measure the ratio of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ production fraction to the sum
of those for $B^{-}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$
mesons $f_{s}/(f_{u}+f_{d})$ = 0.134$\pm$0.004${}^{+0.011}_{-0.010}$, and find
it consistent with being independent of $\eta$ and $p_{\rm T}$. Our results
are more precise than, and in agreement with, previous measurements in
different kinematic regions. We combine the LHCb measurements of the ratio of
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ production fractions obtained
using $b$ hadron semileptonic decays, and two different ratios of branching
fraction of exclusive hadronic decays to derive
$f_{s}/f_{d}=0.267^{+0.021}_{-0.020}$. The ratio of the $\Lambda_{b}^{0}$
baryon production fraction to the sum of those for $B^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mesons varies with the $p_{\rm T}$
of the charmed hadron muon pair. Assuming a linear dependence up to $p_{\rm
T}=14$ GeV, we obtain

$\frac{f_{\mathchar 28931\relax_{b}}}{f_{u}+f_{d}}=(0.404\pm 0.017\pm 0.027\pm
0.105)\times[1-(0.031\pm 0.004\pm 0.003)\times p_{\rm T}{\rm(GeV)}],$ (12)

where the errors on the absolute scale are statistical, systematic and error
on ${\cal B}(\mathchar 28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+})$
respectively. No $\eta$ dependence is found.

## 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 CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (the Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.

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