relaxed to the stochastic dominance condition (a) in Proposition 3.1. Consider
two true null test statistics ${T_{1}}$, ${T_{2}}\sim U[0,1]$, two internal
negative control test statistics ${T_{3}}$, ${T_{4}}\sim\text{Beta}(1,2)$, and
assume all variables are independent. Then ${T_{1}}$, ${T_{2}}$ uniformly
stochastically dominate ${T_{3}}$, ${T_{4}}$, but $({p_{i}})_{1\leq i\leq 2}$
is not PRD because
$\mathbb{P}\left({p_{2}}=1\mid{p_{1}}=\frac{1}{3}\right)=\frac{4}{9}>\frac{5}{12}=\mathbb{P}\left({p_{2}}=1\mid{p_{1}}=\frac{2}{3}\right)$.

###  B.2 Fisher’s method and permutation test

We show directly combining Fisher’s method with RANC p-values fails to control
the type I error. In Figure B.1 panel (a), the theoretical null distribution
$\chi_{2{n}}^{2}$ assuming independent p-values is inappropriate. In contrast,
the simulated null based on $1000$ permutations is reasonably accurate (Figure
B.1 panel (b)).

(a) theoretical null

(b) simulated null

Figure B.1: Combining Fisher’s method with RANC p-values. We generate $400$
test statistics for investigation and $400$ internal negative control test
statistics independently from $\mathcal{N}(0,1)$, compute the RANC p-values
and further the Fisher’s test statistic. Panel (a) displays the quantile-
quantile plot based on the theoretical null $\chi^{2}_{2{n}}$ and panel (b)
depicts the simulated null ($10^{3}$ permutations).

###  B.3 [3, thm. 2]

Theorem 2 in [3] states that if $({T_{i}})_{i\in{\mathcal{I}_{0}}}$ is jointly
independent and $({T_{i}})_{i\in{\mathcal{I}_{0}}}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}({T_{j}})_{j\in{\mathcal{I}_{\text{nc}}}}$,
then $({p_{i}})_{i\in{\mathcal{I}}}$ is PRDS on
$({p_{i}})_{i\in{\mathcal{I}_{0}}}$. Surprisingly, this theorem does not
impose any assumptions on $({T_{i}})_{i\in{\mathcal{I}_{1}}}$ (what [3] call
outliers). The next simple counter-example shows that some assumptions on
$({T_{i}})_{i\in{\mathcal{I}_{1}}}$ are indeed necessary. In this example,
${\mathcal{I}_{0}}=\\{1\\}$, ${\mathcal{I}_{1}}=\\{2\\}$,
${\mathcal{I}_{\text{nc}}}=\\{3\\}$, ${T_{1}}$, ${T_{3}}$ are i.i.d. $U(0,1)$,
and ${T_{2}}=\mathbbm{1}_{\\{{T_{1}}<0.5\\}}$. For the increasing set
$\mathcal{D}=\\{{\bm{p}}:{p_{2}}>0.5\\}$, we have

$\displaystyle\mathbb{P}({\bm{p}}\in\mathcal{D}\mid{p_{1}}=0.5)$
$\displaystyle=\mathbb{P}({T_{1}}<0.5\mid{T_{1}}<{T_{3}})=2\mathbb{P}({T_{1}}<0.5,{T_{1}}<{T_{3}})=\frac{3}{4},$
$\displaystyle\mathbb{P}({\bm{p}}\in\mathcal{D}\mid{p_{1}}=1)$
$\displaystyle=\mathbb{P}({T_{1}}<0.5\mid{T_{1}}\geq{T_{3}})=2\mathbb{P}({T_{1}}<0.5,{T_{1}}\geq{T_{3}})=\frac{1}{4}.$

Thus, the desired PRDS property is not true in this case.

A closer examination of Theorem 2 in [3] shows that a key step in their proof
(reproduced below using our notation) directly drops the conditioning event
$p_{i}=p$ without any justification:

$\displaystyle\mathbb{P}\left({\bm{p}}\in\mathcal{D}\mid{p_{i}}=p\right)=\mathbb{E}_{\bm{Z}|{p_{i}}=p}\left[\mathbb{P}\left({\bm{p}}\in\mathcal{D}\mid\bm{Z}\right)\right],$

where $\bm{Z}$ is the set of order statistics of
$({T_{j}})_{j\in{\mathcal{I}_{\text{nc}}}}$. This step can be rectified by
assuming that $({T_{i}})_{i\in{\mathcal{I}_{1}}}$ is mutually independent and
is independent of the remaining test statistics.

##  Appendix C Additional numerical results

We report in Figure C.1 the power of BH RANC for ${n}\in\\{100,200\\}$,
${n_{1}}\in\\{10,20\\}$, $q\in\\{0.1,0.2\\}$, and
$m\in\\{50,100,150,\dotsc,500\\}$. We adopt the setting of independent
p-values with exact baseline p-values in Section 6. We do not report the power
of BH, as it is numerically identical to BH oracle in this case.

(a) ${n}=100,{n_{1}}=20,q=0.2,{n}/({n_{1}}q)=25$

(b) ${n}=100,{n_{1}}=20,q=0.1,{n}/({n_{1}}q)=50$

(c) ${n}=100,{n_{1}}=10,q=0.2,{n}/({n_{1}}q)=50$

(d) ${n}=100,{n_{1}}=10,q=0.1,{n}/({n_{1}}q)=100$

(e) ${n}=200,{n_{1}}=20,q=0.2,{n}/({n_{1}}q)=50$

(f) ${n}=200,{n_{1}}=20,q=0.1,{n}/({n_{1}}q)=100$

(g) ${n}=200,{n_{1}}=10,q=0.2,{n}/({n_{1}}q)=100$

(h) ${n}=200,{n_{1}}=10,q=0.1,{n}/({n_{1}}q)=200$

Figure C.1: Power analysis of the BH RANC with different numbers of internal
negative control units. We vary ${n}\in\\{100,200\\}$,
${n_{1}}\in\\{10,20\\}$, and $q\in\\{0.1,0.2\\}$. The number of internal
negative controls varies from $50$ to $500$. The results are aggregated over
$10^{3}$ trials.

Next, we carry out this power analysis with weaker non-nulls. In particular,
we adopt the setting of independent p-values with exact baseline p-values in
Section 6 but set the marginal distribution of the non-nulls to $\Phi(Z-2)$.
We vary ${n}\in\\{100,200\\}$, ${n_{1}}\in\\{10,20\\}$, and
$q\in\\{0.1,0.2\\}$. We increase the internal negative control sample size
from $50$ to $1000$. The results are reported in Figure C.2. In this setting,
we find that the BH RANC requires $m\geq 5{n}/(q{n_{1}})$ internal negative
controls to achieve comparable power as BH oracle.

(a) ${n}=100,{n_{1}}=20,q=0.2,\frac{{n}}{{n_{1}}q}=25$

(b) ${n}=100,{n_{1}}=20,q=0.1,\frac{{n}}{{n_{1}}q}=50$

(c) ${n}=100,{n_{1}}=10,q=0.2,\frac{{n}}{{n_{1}}q}=50$

(d) ${n}=100,{n_{1}}=10,q=0.1,\frac{{n}}{{n_{1}}q}=100$

(e) ${n}=200,{n_{1}}=20,q=0.2,\frac{{n}}{{n_{1}}q}=50$

(f) ${n}=200,{n_{1}}=20,q=0.1,\frac{{n}}{{n_{1}}q}=100$

(g) ${n}=200,{n_{1}}=10,q=0.2,\frac{{n}}{{n_{1}}q}=100$

(h) ${n}=200,{n_{1}}=10,q=0.1,\frac{{n}}{{n_{1}}q}=200$

Figure C.2: Power analysis of the BH RANC with weaker non-nulls and different
numbers of internal negative control units. We vary ${n}\in\\{100,200\\}$,
${n_{1}}\in\\{10,20\\}$, and $q\in\\{0.1,0.2\\}$. The internal negative
control sample size varies from $50$ to $1000$. The results are aggregated
over $10^{3}$ trials. Figure C.3: A comparison of the distribution of p-values
for the proteomic application obtained from the empirical null distributions
described in Section 7.2. It is expected that, when using the correct null
distribution, p-values between 0.5 and 1 should be nearly uniformly
distributed. Figure C.4: A quantile-quantile plot of the negative control
statistics against the empirical null distribution $\text{N}(-0.02,0.18)$
obtained by Efron’s method. The solid line corresponds to the diagonal.


BELLE2-CONF-PH-2022-017

October 9, 2022

The Belle II Collaboration

The Belle II Collaboration

#
Determination of $|V_{ub}|$ from untagged $B^{0}\to\pi^{-}\ell^{+}\nu_{\ell}$
decays using 2019-2021 Belle II data

F. Abudinén  I. Adachi  K. Adamczyk  L. Aggarwal  P. Ahlburg  H. Ahmed  J. K.
Ahn  H. Aihara  N. Akopov  A. Aloisio  F. Ameli  L. Andricek  N. Anh Ky  D. M.
Asner  H. Atmacan  V. Aulchenko  T. Aushev  V. Aushev  T. Aziz  V. Babu  S.
Bacher  H. Bae  S. Baehr  S. Bahinipati  A. M. Bakich  P. Bambade  Sw.
Banerjee  S. Bansal  M. Barrett  G. Batignani  J. Baudot  M. Bauer  A. Baur
A. Beaubien  A. Beaulieu  J. Becker  P. K. Behera  J. V. Bennett  E. Bernieri
F. U. Bernlochner  V. Bertacchi  M. Bertemes  E. Bertholet  M. Bessner  S.
Bettarini  V. Bhardwaj  B. Bhuyan  F. Bianchi  T. Bilka  S. Bilokin  D. Biswas
A. Bobrov  D. Bodrov  A. Bolz  A. Bondar  G. Bonvicini  A. Bozek  M. Bračko
P. Branchini  N. Braun  R. A. Briere  T. E. Browder  D. N. Brown  A. Budano
L. Burmistrov  S. Bussino  M. Campajola  L. Cao  G. Casarosa  C. Cecchi  D.
Červenkov  M.-C. Chang  P. Chang  R. Cheaib  P. Cheema  V. Chekelian  C. Chen
Y. Q. Chen  Y. Q. Chen  Y.-T. Chen  B. G. Cheon  K. Chilikin  K. Chirapatpimol
H.-E. Cho  K. Cho  S.-J. Cho  S.-K. Choi  S. Choudhury  D. Cinabro  L. Corona
L. M. Cremaldi  S. Cunliffe  T. Czank  S. Das  N. Dash  F. Dattola  E. De La
Cruz-Burelo  S. A. De La Motte  G. de Marino  G. De Nardo  M. De Nuccio  G. De
Pietro  R. de Sangro  B. Deschamps  M. Destefanis  S. Dey  A. De Yta-Hernandez
R. Dhamija  A. Di Canto  F. Di Capua  S. Di Carlo  J. Dingfelder  Z. Doležal
I. Domínguez Jiménez  T. V. Dong  M. Dorigo  K. Dort  D. Dossett  S. Dreyer
S. Dubey  S. Duell  G. Dujany  P. Ecker  S. Eidelman  M. Eliachevitch  D.
Epifanov  P. Feichtinger  T. Ferber  D. Ferlewicz  T. Fillinger  C. Finck  G.
Finocchiaro  P. Fischer  K. Flood  A. Fodor  F. Forti  A. Frey  M. Friedl  B.
G. Fulsom  M. Gabriel  A. Gabrielli  N. Gabyshev  E. Ganiev  M. Garcia-
Hernandez  R. Garg  A. Garmash  V. Gaur  A. Gaz  U. Gebauer  A. Gellrich  J.
Gemmler  T. Geßler  G. Ghevondyan  G. Giakoustidis  R. Giordano  A. Giri  A.
Glazov  B. Gobbo  R. Godang  P. Goldenzweig  B. Golob  P. Gomis  G. Gong  P.
Grace  W. Gradl  S. Granderath  E. Graziani  D. Greenwald  T. Gu  Y. Guan  K.
Gudkova  J. Guilliams  C. Hadjivasiliou  S. Halder  K. Hara  T. Hara  O.
Hartbrich  K. Hayasaka  H. Hayashii  S. Hazra  C. Hearty  M. T. Hedges  I.
Heredia de la Cruz  M. Hernández Villanueva  A. Hershenhorn  T. Higuchi  E. C.
Hill  H. Hirata  M. Hoek  M. Hohmann  S. Hollitt  T. Hotta  C.-L. Hsu  K.
Huang  T. Humair  T. Iijima  K. Inami  G. Inguglia  N. Ipsita  J. Irakkathil
Jabbar  A. Ishikawa  S. Ito  R. Itoh  M. Iwasaki  Y. Iwasaki  S. Iwata  P.
Jackson  W. W. Jacobs  D. E. Jaffe  E.-J. Jang  M. Jeandron  H. B. Jeon  Q. P.
Ji  S. Jia  Y. Jin  C. Joo  K. K. Joo  H. Junkerkalefeld  I. Kadenko  J. Kahn
H. Kakuno  M. Kaleta  A. B. Kaliyar  J. Kandra  K. H. Kang  S. Kang  P.
Kapusta  R. Karl  G. Karyan  Y. Kato  H. Kawai  T. Kawasaki  C. Ketter  H.
Kichimi  C. Kiesling  C.-H. Kim  D. Y. Kim  H. J. Kim  K.-H. Kim  K. Kim
S.-H. Kim  Y.-K. Kim  Y. Kim  T. D. Kimmel  H. Kindo  K. Kinoshita  C.
Kleinwort  B. Knysh  P. Kodyš  T. Koga  S. Kohani  K. Kojima  I. Komarov  T.
Konno  A. Korobov  S. Korpar  N. Kovalchuk  E. Kovalenko  R. Kowalewski  T. M.
G. Kraetzschmar  F. Krinner  P. Križan  R. Kroeger  J. F. Krohn  P. Krokovny
H. Krüger  W. Kuehn  T. Kuhr  J. Kumar  M. Kumar  R. Kumar  K. Kumara  T.
Kumita  T. Kunigo  M. Künzel  S. Kurz  A. Kuzmin  P. Kvasnička  Y.-J. Kwon  S.
Lacaprara  Y.-T. Lai  C. La Licata  K. Lalwani  T. Lam  L. Lanceri  J. S.
Lange  M. Laurenza  K. Lautenbach  P. J. Laycock  R. Leboucher  F. R. Le
Diberder  I.-S. Lee  S. C. Lee  P. Leitl  D. Levit  P. M. Lewis  C. Li  L. K.
Li  S. X. Li  Y. B. Li  J. Libby  K. Lieret  J. Lin  Z. Liptak  Q. Y. Liu  Z.
A. Liu  D. Liventsev  S. Longo  A. Loos  A. Lozar  P. Lu  T. Lueck  F.
Luetticke  T. Luo  C. Lyu  C. MacQueen  M. Maggiora  R. Maiti  S. Maity  R.
Manfredi  E. Manoni  A. Manthei  S. Marcello  C. Marinas  L. Martel  A.
Martini  L. Massaccesi  M. Masuda  T. Matsuda  K. Matsuoka  D. Matvienko  J.
A. McKenna  J. McNeil  F. Meggendorfer  F. Meier  M. Merola  F. Metzner  M.
Milesi  C. Miller  K. Miyabayashi  H. Miyake  H. Miyata  R. Mizuk  K. Azmi  G.
B. Mohanty  N. Molina-Gonzalez  S. Moneta  H. Moon  T. Moon  J. A. Mora
Grimaldo  T. Morii  H.-G. Moser  M. Mrvar  F. J. Müller  Th. Muller  G.
Muroyama  C. Murphy  R. Mussa  I. Nakamura  K. R. Nakamura  E. Nakano  M.
Nakao  H. Nakayama  H. Nakazawa  A. Narimani Charan  M. Naruki  Z. Natkaniec
A. Natochii  L. Nayak  M. Nayak  G. Nazaryan  D. Neverov  C. Niebuhr  M.
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K. Ogawa  S. Ogawa  S. L. Olsen  Y. Onishchuk  H. Ono  Y. Onuki  P. Oskin  F.
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A. Panta  E. Paoloni  S. Pardi  K. Parham  H. Park  S.-H. Park  B. Paschen  A.
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R. Pestotnik  F. Pham  M. Piccolo  L. E. Piilonen  G. Pinna Angioni  P. L. M.
Podesta-Lerma  T. Podobnik  S. Pokharel  L. Polat  V. Popov  C. Praz  S. Prell
E. Prencipe  M. T. Prim  M. V. Purohit  H. Purwar  N. Rad  P. Rados  S. Raiz
A. Ramirez Morales  R. Rasheed  N. Rauls  M. Reif  S. Reiter  M. Remnev  I.
Ripp-Baudot  M. Ritter  M. Ritzert  G. Rizzo  L. B. Rizzuto  S. H. Robertson
D. Rodríguez Pérez  J. M. Roney  C. Rosenfeld  A. Rostomyan  N. Rout  M.
Rozanska  G. Russo  D. Sahoo  Y. Sakai  D. A. Sanders  S. Sandilya  A. Sangal
L. Santelj  P. Sartori  Y. Sato  V. Savinov  B. Scavino  M. Schnepf  M. Schram
H. Schreeck  J. Schueler  C. Schwanda  A. J. Schwartz  B. Schwenker  M.
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Sevior  C. Sfienti  V. Shebalin  C. P. Shen  H. Shibuya  T. Shillington  T.
Shimasaki  J.-G. Shiu  B. Shwartz  A. Sibidanov  F. Simon  J. B. Singh  S.
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K. Sumisawa  T. Sumiyoshi  W. Sutcliffe  S. Y. Suzuki  H. Svidras  M. Tabata
M. Takahashi  M. Takizawa  U. Tamponi  S. Tanaka  K. Tanida  H. Tanigawa  N.
Taniguchi  Y. Tao  P. Taras  F. Tenchini  R. Tiwary  D. Tonelli  E. Torassa
N. Toutounji  K. Trabelsi  I. Tsaklidis  T. Tsuboyama  N. Tsuzuki  M. Uchida
I. Ueda  S. Uehara  Y. Uematsu  T. Ueno  T. Uglov  K. Unger  Y. Unno  K. Uno
S. Uno  P. Urquijo  Y. Ushiroda  Y. V. Usov  S. E. Vahsen  R. van Tonder  G.
S. Varner  K. E. Varvell  A. Vinokurova  L. Vitale  V. Vobbilisetti  V.
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Abdullah  B. Wang  C. H. Wang  E. Wang  M.-Z. Wang  X. L. Wang  A. Warburton
M. Watanabe  S. Watanuki  J. Webb  S. Wehle  M. Welsch  C. Wessel  J.
Wiechczynski  P. Wieduwilt  H. Windel  E. Won  L. J. Wu  X. P. Xu  B. D.
Yabsley  S. Yamada  W. Yan  S. B. Yang  H. Ye  J. Yelton  J. H. Yin  M.
Yonenaga  Y. M. Yook  K. Yoshihara  T. Yoshinobu  C. Z. Yuan  Y. Yusa  L. Zani
Y. Zhai  J. Z. Zhang  Y. Zhang  Y. Zhang  Z. Zhang  V. Zhilich  J. Zhou  Q. D.
Zhou  X. Y. Zhou  V. I. Zhukova  V. Zhulanov  R. Žlebčík

###### Abstract

We present an analysis of the charmless semileptonic decay
$B^{0}\to\pi^{-}\ell^{+}\nu_{\ell}$, where $\ell=e,\mu$, from 198.0 million
pairs of $B\bar{B}$ mesons recorded by the Belle II detector at the SuperKEKB
electron-positron collider. The decay is reconstructed without identifying the
partner $B$ meson. The partial branching fractions are measured independently
for $B^{0}\to\pi^{-}e^{+}\nu_{e}$ and $B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}$ as
functions of $q^{2}$ (momentum transfer squared), using 3896
$B^{0}\to\pi^{-}e^{+}\nu_{e}$ and 5466 $B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}$
decays. The total branching fraction is found to be $(1.426\pm 0.056\pm
0.125)\times 10^{-4}$ for $B^{0}\to\pi^{-}\ell^{+}\nu_{\ell}$ decays, where
the uncertainties are statistical and systematic, respectively. By fitting the
measured partial branching fractions as functions of $q^{2}$, together with
constraints on the nonperturbative hadronic contribution from lattice QCD
calculations, the magnitude of the Cabibbo-Kobayashi-Maskawa matrix element
$V_{ub}$, $(3.55\pm 0.12\pm 0.13\pm 0.17)\times 10^{-3}$, is extracted. Here,
the first uncertainty is statistical, the second is systematic and the third
is theoretical.

###### pacs:

##  1 Introduction

The Cabibbo-Kobayashi-Maskawa (CKM) matrix elements are fundamental parameters
of the Standard Model (SM) of particle physics Kobayashi:1973fv . The
magnitude of the matrix element $V_{ub}$ can be determined by measuring the
differential decay rate of $B\rightarrow\ X_{u}\ell\nu$ events, which is
proportional to $|V_{ub}|^{2}$. Here, $X_{u}$ is a charmless hadronic final
state and $\ell$ is a light charged lepton. One method to measure $|V_{ub}|$
is inclusive, where no specific $X_{u}$ final state is reconstructed, but
rather the sum of all possible final states is analysed. This is in contrast
to the exclusive method in which a specific final state is studied. The two
methods have complementary uncertainties introduced by theoretical QCD
descriptions. In inclusive decays, these descriptions involve the calculation
of the total semileptonic rate, while in the exclusive case they take the form
of parameterizations of the low-energy strong interactions (form factors).
However, the results for $|V_{ub}|$ obtained by the two methods differ
significantly Amhis:2019ckw . In order to resolve this disagreement, further
measurements using novel approaches are beneficial.

The decay $B^{0}\to\pi^{-}\ell^{+}\nu_{\ell}$ (with charge conjugation implied
throughout) is experimentally and theoretically the most reliable mode to
measure $|V_{ub}|$ through an exclusive channel at the $B$ Factories. Here, we
present a study of this decay using data from the Belle II detector located at
the SuperKEKB electron-positron collider at KEK, in Japan. The signal decay is
reconstructed from decays of $B^{0}$ mesons in
$e^{+}e^{-}\rightarrow\Upsilon(4S)\rightarrow B^{0}\overline{B^{0}}$ events.
The reconstruction method employed here is called untagged since the signal
lepton and pion candidates are selected without prior reconstruction (tagging)
of the partner $B$ meson. This leads to a high signal efficiency, but also a
low purity due to higher combinatorial background from the partner $B$ meson
and increased backgrounds from $e^{+}e^{-}$ collisions that produce light
quark pair (continuum) events.

The events are separated into six disjoint intervals (bins) of squared
momentum transfer from the $B$ meson to the pion, $q^{2}$. After selecting
signal events and suppressing backgrounds, the signal yields are extracted
from an extended likelihood fit to the binned two-dimensional distribution of
the energy difference $\Delta E$, and the beam-constrained mass $M_{bc}$, both
defined below, in bins of $q^{2}$. Theoretical form-factor predictions from
lattice QCD are combined with the measured differential branching fractions to
determine $|V_{ub}|$ FermilabMILC .

##  2 Detector, data set and simulation

###  2.1 Belle II detector

The Belle II detector Belle-II:2010dht  is located at the SuperKEKB
Akai:2018mbz  asymmetric-energy electron-positron collider, running at or
slightly below the $\Upsilon(4S)$ resonance energy. The detector is composed
of subdetectors arranged around the collision point in a cylindrical geometry.
The longitudinal direction is defined by the $z$-axis, which points
approximately along the electron-beam direction, while the transverse plane is
defined by the $x$\- and $y$-axes, which point radially out of the detector
ring. A superconducting magnet producing a uniform 1.5 T magnetic field
oriented along the detector axis is installed outside all but the last
subdetector. Particles produced at the interaction point first traverse the
vertex detector composed of a two-layer pixel detector and a four-layer
silicon strip detector that measure positions of decays into charged
particles. Currently, however, the second layer of the pixel detector covers
only one sixth of the azimuthal angle.

Trajectories of charged particles (tracks) are further determined from a
central drift chamber (CDC) that has a polar angle acceptance of 17–150∘. We
use two Cherenkov particle-identification subdetectors: the time-of-
propagation counter is located around the barrel region of the detector and
the aerogel ring-imaging Cherenkov detector is located in the forward endcap.
Photon- and electron-energy measurements in the barrel region and the endcaps
are performed using the electromagnetic calorimeter, which is composed of
CsI(Tl) crystals. Surrounding the magnet is the resistive-plate-based
$K^{0}_{L}$ and muon detector, which also doubles as a flux return for the
magnet.

Using specific-ionization energy-loss information provided by the CDC and
information from the two particle-identification subdetectors and the
$K^{0}_{L}$ and muon detector, charged particles of different masses are
distinguished and particle-identification (ID) variables are constructed.
These are normalized ratios of likelihoods for one charged-particle hypothesis
to the sum of all possible charged-particle likelihoods.

The detector performance is monitored using control samples; we correct for
time-dependent changes such as shifts in the magnetic field map or the photon
energy scale. In addition, the efficiencies and misidentification
probabilities of pions, kaons, electrons and muons are monitored separately by
charge in bins of particle momentum and polar angle using well-known physics
control modes, such as $J/\psi\to\ell^{+}\ell^{-}$ and $D^{*}\rightarrow
D^{0}[K\pi]\pi$.

###  2.2 Data set

The primary data set used in this analysis is collected at a center-of-mass
(CM) energy of $\sqrt{s}=$ 10.58 GeV, corresponding to the mass of the
$\Upsilon$(4S) resonance. A data set corresponding to an integrated luminosity
of $189$ fb-1 is collected at this CM energy, equivalent to a sample of
($198.0\pm 3.0$) million $\Upsilon(4S)\rightarrow B\overline{B}$ events. In
addition, we use a sample corresponding to $18$ fb-1 of off-resonance
collision data, collected at a CM energy 60 MeV below the $\Upsilon$(4S)
resonance, to describe background from continuum processes. These include
$q\overline{q}$ continuum background, i.e., $e^{+}e^{-}\to
u\bar{u},\,d\bar{d},\,s\bar{s}$ and $c\bar{c}$. The off-resonance data also
contain other continuum backgrounds, such as $e^{+}e^{-}\to\tau^{+}\tau^{-}$,
and two-photon processes, where $e^{+}e^{-}\to e^{+}e^{-}\ell^{+}\ell^{-}$.

###  2.3 Monte Carlo simulation

Simulated Monte Carlo (MC) samples of continuum and $\Upsilon(4S)\rightarrow
B\overline{B}$ events, corresponding to an integrated luminosity of 1 ab-1 are
used. We use these to identify efficient background-discriminating variables
and to form fit templates for signal extraction. In addition, MC samples of
signal $B^{0}\to\pi^{-}\ell^{+}\nu_{\ell}$ decays are used to obtain the
reconstruction efficiencies and study the key kinematic distributions. In
order to describe the composition of the $B\to X_{u}\ell\nu$ decays, 50
million events each of resonant and nonresonant $B^{0}\to X_{u}\ell\nu$, and
$B^{\pm}\to X_{u}\ell\nu$ are simulated.

The $B\bar{B}$ events are generated using the event generator EvtGen
Lange:2001uf , while the $q\overline{q}$ continuum events are generated using
PYTHIA Sjostrand:2014zea . Tau-pair events are generated using KKMC
Jadach:1999vf , and their decays are handled using TAUOLA Jadach:1990mz . The
simulation of two-photon processes is performed using AAFH BERENDS200282 .
Backgrounds induced by the beams are mixed into the MC samples using
simulations of beam-induced backgrounds BeamBKG . The propagation of the
particles through the detector and the resulting interactions are simulated
using Geant4 Agostinelli:2002hh  and final-state radiation is modeled using
PHOTOS Barberio:1990ms . All recorded and simulated events are handled with
the analysis software framework basf2 Kuhr:2018lps .

The branching-fraction values of the $B\to X\ell\nu$ decays assumed in the
simulation are obtained from current world averages Zyla:2020zbs , in
combination with assumed isospin symmetry, following the procedure in Ref.
Bernlochner:2016bci  for $B\to X_{c}\ell\nu$ decays. The remaining difference
between the sum of the exclusive $B\to X_{c}\ell\nu$ decay branching fractions
and the measured total branching fraction accounts for approximately 4% of the
$B\to X_{c}\ell\nu$ decays and is assumed to be saturated by $B\rightarrow
D^{(*)}\eta\ell\nu$ decays. For $B\to X_{u}\ell\nu$ decays, the difference
between the sum of the exclusive branching fractions and the measured total
branching fraction is assumed to be saturated by nonresonant $B\to
X_{u}\ell\nu$ decays with multiple pions in the final state following Ref.
Lange:2005yw . The nonresonant $B\to X_{u}\ell\nu$ decays account for
approximately 80% of the $B\to X_{u}\ell\nu$ decays.

The nonresonant $B\to X_{u}\ell\nu$ composition is described by implementing a
hybrid model PhysRevD.41.1496 , following closely the method in Ref.
Belle:2019iji . This approach combines the exclusive and nonresonant decay
rates in bins of $X_{u}$ particle mass $m_{X}$, the rest-frame lepton energy
$E_{\ell}$, and $q^{2}$, in order to recover the inclusive rates.

For the form factors of $B\to D\ell\nu$ and $B\to D^{*}\ell\nu$ decays, we use
the Boyd-Grinstein-Lebed parameterization PhysRevLett.74.4603  with central
values from Ref. PhysRevD.93.032006  and Ferlewicz:2020lxm , respectively. For
the $B\rightarrow\rho\ell\nu$ and $B\rightarrow\omega\ell\nu$ form factors we
choose the Bourrely-Caprini-Lellouch (BCL) parameterization PhysRevD.79.013008
with central values from Ref. Bernlochner:2021rel . Finally, for the form-
factor description of the $B\rightarrow\eta^{(^{\prime})}\ell\nu$ decays a
light-cone sum rule calculation is chosen Duplancic:2015zna .

##  3 Event reconstruction and selection

###  3.1 Event reconstruction

We begin signal reconstruction by identifying track candidates that pass
certain quality criteria. The extrapolated trajectories must pass within 1 cm
(3 cm) of the interaction point transverse (parallel) to the beam.
Furthermore, charged particles must have a transverse momentum greater than
0.05 GeV (we use natural units with $c=1$ throughout) and polar angles within
the CDC acceptance, in order to suppress misreconstructed track candidates. We
discard events with fewer than five tracks satisfying the above criteria.

In the remaining events, we select lepton candidates from among the selected
tracks by requiring their CM momentum $p^{*}_{\ell}$ to satisfy 1.0
$<p^{*}_{\ell}<3.2$ GeV. Electron and muon candidates are required to have an
electron or muon likelihood greater than 0.9, respectively. The average
electron (muon) efficiency is 91 (93)%. The hadron misidentification rates are
0.2% for the electron and 3.3% for the muon selection, respectively. Electron-
candidate four-momenta are corrected for bremsstrahlung by adding to them the
four-momenta of photons identified within a cone around the electron
direction.

Signal-pion candidates are selected from all remaining tracks that pass within
2 cm (4 cm) of the interaction point transverse (parallel) to the beam. They
are required to have a charge opposite to the lepton candidate and to have a
pion likelihood greater than 0.1 in order to suppress misidentification of
kaons as pions. To improve the particle-identification performance, we require
the pion candidates to leave at least 20 random measurement points (hits) in
the CDC. The resulting average pion efficiency is 84% with a kaon
misidentification rate of 7%.

We reduce backgrounds by removing candidates with kinematic properties
inconsistent with the signal $B$ decay. Under the assumption that only a
single massless particle is not included in the event reconstruction, the
angle between the $B$ meson and the combination of the lepton and pion
candidates, denoted $Y$, is fully determined,

$\cos\theta_{BY}=\frac{2E_{B}E_{Y}-m_{B}^{2}-m_{Y}^{2}}{2p_{B}p_{Y}},$ (1)

where $E_{Y}$, $p_{Y}$, and $m_{Y}$ are the energy, magnitude of the three-
momentum, and invariant mass of the $Y$, respectively. The energy $E_{B}$ and
the magnitude of the three-momentum of the $B$ meson $p_{B}$ are calculated
from the beam properties, and $m_{B}$ is the mass of the $B$ meson
Zyla:2020zbs . For correctly reconstructed signal decays, $\cos\theta_{BY}$
must lie between $-1$ and 1. However, to allow for a background-dominated
region, we set a looser requirement of $|\cos\theta_{BY}|<2.2$. The event
shape discriminates between $B\overline{B}$ and continuum events. Thus we
require the second normalized Fox-Wolfram moment FWM  to be less than 0.4.

We obtain the kinematic properties of the neutrino by assuming that the sum of
the remaining tracks and electromagnetic energy-depositions (clusters) in the
event, called the rest of event (ROE), represents the partner $B$ meson. From
energy and momentum conservation, we construct a missing four-momentum in the
CM frame,

$(E^{*}_{\mathrm{miss}},\vec{p}^{*}_{\mathrm{miss}})=(E_{\Upsilon(4S)},\vec{p}_{\Upsilon(4S)})-\left(\sum_{i}E^{*}_{i},\sum_{i}\vec{p}^{*}_{i}\right),$
(2)

where $E^{*}_{i}$ and $\vec{p}^{*}_{i}$ correspond to the CM energy and
momentum of the $i$th track or cluster in the event, respectively. This yields
the neutrino momentum, $\vec{p}^{*}_{\nu}=\vec{p}^{*}_{\mathrm{miss}}$, and
energy, $E^{*}_{\nu}=|\vec{p}^{*}_{\nu}|=|\vec{p}^{*}_{\mathrm{miss}}|$.

Since all reconstructed tracks and clusters contribute to the resolution of
the neutrino momentum, obtaining a pure ROE is critical. To reduce the impact
of clusters from beam-induced backgrounds, acceptance losses, or other
effects, we impose further criteria. We only consider clusters that are within
the CDC acceptance and have transverse momenta in forward, barrel, and
backward directions greater than 0.03, 0.04, and 0.06 GeV, respectively. The
transverse momentum of a cluster is defined using the energy and the location
with respect to the interaction point of the energy deposition. The clusters
are required to be detected within 200 ns of the collision time, which is
approximately five times the mean timing resolution of the calorimeter. The
clusters also have to consist of more than a single calorimeter crystal. In
addition to removing background particles from the ROE, we must account for
particles that may escape undetected. To reduce the impact of these events, we
require that the polar angle of the missing momentum in the laboratory frame
$\theta_{\mathrm{miss}}$ is within the CDC acceptance.

###  3.2 Signal extraction variables

We calculate $q^{2}$ from $q^{2}=(p_{B}-p_{\pi})^{2}$, and thus need a way to
estimate the $B$ momentum vector. One existing method, called the Diamond
Frame DiamondFrame , takes the weighted average of four possible $\vec{p}_{B}$
vectors uniformly distributed in azimuthal angle on the cone defined by
$\cos\theta_{BY}$ using weights of $\sin^{2}\theta_{B}$, which expresses the
prior probability of the $B$ flight direction in $\Upsilon(4S)$ decays with
respect to the beam axis. A second method, called the ROE method
Waheed:2019lxm , assumes the signal $B$ vector to be the vector on the
$\cos\theta_{BY}$ cone that is closest to antiparallel to the ROE momentum
vector. We introduce a new method that combines these two by multiplying the
Diamond Frame weights by
$\frac{1}{2}(1-\vec{p}_{B}\cdot\vec{p}_{\mathrm{ROE}}/(|\vec{p}_{B}||\vec{p}_{\mathrm{ROE}}|))$
and averaging over ten vectors uniformly distributed on the cone. We adopt
this combined method because, in simulation, it assigns reconstructed signal
candidates to the correct $q^{2}$ bin more often than other methods do,
leading to a reduction in the bin migrations of up to 2% and resolutions in
$q^{2}$ ranging from 0.14–0.45 GeV2. We divide $B$ candidates into six $q^{2}$
bins with the following labels: $q1:q^{2}\in[0,4]$, $q2:[4,8]$, $q3:[8,12]$,
$q4:[12,16]$, $q5:[16,20]$, $q6:[20,\infty]$ GeV2.

Using ROE information, two further variables that test the consistency of a
candidate with a signal $B$ decay are the beam-constrained mass, defined as

$M_{bc}=\sqrt{E^{*2}_{\mathrm{beam}}-|\vec{p}^{\,*}_{B}|^{2}}=\sqrt{\left(\frac{\sqrt{s}}{2}\right)^{2}-|\vec{p}^{\,*}_{B}|^{2}}$
(3)

and the energy difference, defined as

$\Delta E=E^{*}_{B}-E^{*}_{\mathrm{beam}}=E^{*}_{B}-\frac{\sqrt{s}}{2},$ (4)

where $E^{*}_{\mathrm{beam}}$, $E^{*}_{B}$ and $\vec{p}^{\,*}_{B}$ are the
single-beam energy, reconstructed $B$ energy, and reconstructed $B$ momentum
all determined in the $\Upsilon(4S)$ rest frame, respectively. The
reconstructed $B$ energy (momentum) is given by the sum of the reconstructed
energies (momenta) of the signal lepton and pion and the inferred neutrino
energy (momentum) described above. We define a fit region in $\Delta E$ and
$M_{bc}$, corresponding to $|\Delta E|<0.95$ GeV and $5.095<M_{bc}<5.295$ GeV.

##  4 Background suppression

###  4.1 Background categories

The sample can be separated into two main categories: $B\overline{B}$ events
and non-$B\overline{B}$ events. For the $B\overline{B}$ events we define a
subcategory that combines signal and combinatorial signal events. In
combinatorial signal either the pion or lepton candidate is incorrectly
identified. We further split the $B\overline{B}$ background into the two
largest semileptonic backgrounds, $B\rightarrow X_{c}\ell\nu$ and
$B\rightarrow X_{u}\ell\nu$. The remaining $B\overline{B}$ events are combined
into an _other_ $B\overline{B}$ category, mainly composed of candidates with
misidentified leptons or with leptons from secondary decays. The
non-$B\overline{B}$ events are combined into a continuum background category,
which contains $q\overline{q}$ and other continuum events.

###  4.2 Boosted decision trees

In order to further reduce the $B\overline{B}$ and $q\overline{q}$ backgrounds
we train boosted decision trees (BDTs) using the FastBDT methodology FastBDT ,
which differentiate between signal and each background category separately.
Since the background composition is different in each $q^{2}$ bin, we train
the classifier and optimize the selection separately for each $q^{2}$ bin.
This results in a total of 12 BDTs. In each training we combine the electron
and muon modes, since they have similar background compositions. We use equal
amounts of signal and background simulation events, corresponding to 200 fb-1
of $q\overline{q}$ simulated data, and 600 fb-1 of $B\overline{B}$ simulated
data, due to the low $B\overline{B}$ background retention.

Two input variables, $\cos\theta_{BY}$ and $\theta_{\mathrm{miss}}$, are
common to the $q\overline{q}$ and $B\overline{B}$ suppression. Further input
variables in the $q\overline{q}$ suppression BDT are based on the thrust axis,
which is the axis that maximizes the sum of the projected momenta of all
charged particles in the event. Distinct thrust axes can be defined for the
signal $B$ and the ROE. Their magnitudes, and the angle between the two axes
$\cos\theta_{T}$ serve as input variables. In addition, three cones with
opening angles of 10, 20, and 30∘ centered around the thrust axis are defined,
and the momentum flow into each of the three cones is added as an input
variable CLEOCones . The input variables of the $B\overline{B}$ suppression
BDTs are the number of tracks, the angle between the lepton and pion momenta,
and the momentum of the ROE. In addition, a vertex fit to the pion and lepton
candidates is performed using TreeFit TreeFit  and the $\chi^{2}$ probability
is used as a discriminating feature. The remaining two variables are the
cosines of the angles between the signal $B$ momentum vector and the vector
connecting its fitted vertex to the interaction point in the plane parallel
and perpendicular to the beam axis, respectively. The shared input variable
$\cos\theta_{BY}$ is one of the most discriminating variables in both the
suppression of continuum and $B\overline{B}$ events. The $\cos\theta_{T}$ and
the $\chi^{2}$ vertex fit probability also provide high discriminating power
in the suppression of continuum and $B\overline{B}$ backgrounds, respectively.
We identify the optimal selection criterion on the output classifier within
each $q^{2}$ bin by maximizing the ratio between the number of signal events
and the square root of the sum of the number of signal and background events,
as predicted by simulation.

After selecting on the output classifiers, the continuum background contains a
significant amount of residual two-photon background, especially $ee\tau\tau$
events, in the electron mode. Therefore, for the electron mode only, we train
another BDT common to all $q^{2}$ bins, to separate signal from $ee\tau\tau$
events. We use a sample corresponding to 2 ab-1 of $ee\tau\tau$ simulated data
and an equivalent amount of signal events. The input variables include the
total charge in the event, $\theta_{\mathrm{miss}}$, the polar angle of the
lepton, the number of clusters, the mass of the ROE, and $\xi_{z}$. Here, the
variable $\xi_{z}$ is calculated as the sum of the momenta parallel to the
beam axis divided by the sum of the energies of all charged particles in the
laboratory frame. None of the selected input variables are strongly correlated
with $\Delta E$ or $M_{bc}$.

###  4.3 Best candidate selection and efficiencies

At this stage, an average of $1.02$ candidates remain per selected event. In
events with multiple candidates, we randomly select one and discard the rest
The overall signal efficiencies range from 8% to 17% in the electron mode and
from 13% to 23% in the muon mode, depending on the $q^{2}$ bin.

##  5 Signal extraction

###  5.1 Continuum treatment

Because of the small off-resonance sample size, using the data directly as a
fit template introduces large systematic uncertainties. Instead, we weight the
simulated continuum candidates in order to use the resulting template during
signal extraction. We therefore compare the $q^{2}$ shapes of simulated
continuum data and off-resonance data. We observe a normalization difference
in the electron mode, and to a lesser extent in the muon mode. After
correcting the simulated continuum data for the total normalization, any
residual difference in the $q^{2}$ spectrum is resolved by correcting the
$q^{2}$ spectrum bin-by-bin. This approach relies on the assumption that the
difference between off-resonance data and the simulated continuum sample is
independent of $\Delta E$ and $M_{bc}$. Any observed deviation from this
assumption is treated as a systematic uncertainty and explained in Section 6.

###  5.2 Fit setup

We extract the signal by performing an extended likelihood fit to the binned
two-dimensional distribution of $\Delta E$ and $M_{bc}$ in bins of $q^{2}$. In
total we have 5 ($\Delta E$) $\times$ 4 ($M_{bc}$) $\times$ 6 ($q^{2}$) $=$
120 bins. The distributions of $\Delta E$ and $M_{bc}$ integrated over the six
$q^{2}$ bins for $B^{0}\to\pi^{-}\ell^{+}\nu_{\ell}$ decays are shown in
Figure 1. The likelihood to be maximized is

$\mathcal{L}(s_{j},b_{k})=\prod_{i}\mathrm{Poisson}(N_{i}|\sum_{j}s_{ij}+\sum_{k}b_{ik}),$
(5)

where $N_{i}$ is the observed number of events in bin $i$, $s_{ij}$ is the
number of events in bin $i$ of signal fit template $j$, and $b_{ik}$ is the
number of events in bin $i$ of background fit template $k$. The combinatorial
signal is included in the signal yield. The backgrounds are split into four
templates according to the description in Section 4.1. In addition, there is
one independent signal template for each of the six $q^{2}$ bins, leading to a
total of ten templates. The templates are constructed from 1 ab-1 of simulated
data, where we use the weighted simulated continuum events to extract the
continuum template. We use a Gaussian penalty factor to constrain the
continuum yield to the scaled off-resonance yield.

Figure 1: Simulated distributions of (left) $\Delta E$ and (right) $M_{bc}$
integrated over the six $q^{2}$ bins for $B^{0}\to\pi^{-}\ell^{+}\nu_{\ell}$
decays.

###  5.3 Fit results

The fit projections of $\Delta E$ and $M_{bc}$ in each $q^{2}$ bin are shown
in Figure 2. The correlations between the component yields are all lower than
0.7. The highest observed correlations occur between the $B\to X_{c}\ell\nu$
and continuum yields. In the higher $q^{2}$ bins, the signal becomes
increasingly correlated to the $B\to X_{u}\ell\nu$ yield.

The $q^{2}$ bin migrations in signal range from 3.7–9.8%. For correctly
reconstructed signal they are largest in the lowest $q^{2}$ bin and decrease
at higher $q^{2}$. The inclusion of combinatorial signal, however, also
results in larger bin migrations in the highest $q^{2}$ bin. We correct the
signal yields obtained from the fit for the migrations by applying the inverse
detector response matrix cowan1998statistical . The corrected yields are given
in Table 1. The $B^{0}\to\pi^{-}e^{+}\nu_{e}$ signal yields are lower than the
$B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}$ signal yields. This can partly be attributed
to lower signal efficiencies due to the additional selection on the two-photon
background suppression BDT.

Table 1: Signal yields corrected for bin migrations in each $q^{2}$ bin with statistical and systematic uncertainties. The boundaries of the $q^{2}$ bins are given in the text above. $q^{2}$ bin | $B^{0}\to\pi^{-}e^{+}\nu_{e}$ | $B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}$
---|---|---
$q1$ | 426 $\pm$ 49 $\pm$ 73 | 749 $\pm$ 74 $\pm$ 400
$q2$ | 927 $\pm$ 56 $\pm$ 133 | 1076 $\pm$ 61 $\pm$ 164
$q3$ | 856 $\pm$ 64 $\pm$ 96 | 1238 $\pm$ 77 $\pm$ 128
$q4$ | 577 $\pm$ 64 $\pm$ 64 | 819 $\pm$ 77 $\pm$ 71
$q5$ | 497 $\pm$ 66 $\pm$ 60 | 775 $\pm$ 84 $\pm$ 75
$q6$ | 613 $\pm$ 73 $\pm$ 227 | 809 $\pm$ 86 $\pm$ 164

Figure 2: Distributions of (left) $\Delta E$ and (right) $M_{bc}$ in the six
$q^{2}$ bins for (top) $B^{0}\to\pi^{-}e^{+}\nu_{e}$ and (bottom)
$B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}$ candidates reconstructed in Belle II data
with fit projections overlaid.

##  6 Systematic uncertainties

The fractional uncertainties on the corrected signal yields in each $q^{2}$
bin from various sources of systematic uncertainty are shown in Table 2. All
systematic uncertainties are evaluated using the same approach. For each
source of uncertainty, we vary the templates 1000 times by sampling from
Gaussian distributions of the central values. For example, to evaluate the
uncertainties due to the $B\to\rho\ell\nu$ form factors, we sample 1000
alternative $B\to X_{u}\ell\nu$ distributions by assuming the form-factor
parameter uncertainties follow Gaussian distributions. We create 1000
simplified simulated data (toy) distributions by adding the resulting
variations to any unaffected templates. We then fit the nominal templates to
the toy distributions and obtain a covariance matrix for each source of
uncertainty using Pearson correlation benesty2009pearson .

The largest contribution to the uncertainty in the $B\to X_{u}\ell\nu$
background template comes from the uncertainty in the $B\to\rho\ell\nu$ form-
factor parameters. The effect of the $B\to\rho\ell\nu$ form-factor and
branching fraction uncertainties is included in the $B\to\rho\ell\nu$ category
in Table 2. We also evaluate the uncertainties due to the $B\to\pi\ell\nu$,
$B\rightarrow\omega\ell\nu$, $B\rightarrow\eta\ell\nu$, and
$B\rightarrow\eta^{\prime}\ell\nu$ form factors, and obtain uncertainties
shown under the $B\to X_{u}\ell\nu$ category. This category also includes the
effects of uncertainties of the exclusive and inclusive $B\to X_{u}\ell\nu$
branching fractions, except for the $B\rightarrow\rho\ell\nu$ branching
fraction. The $B\to X_{c}\ell\nu$ category in Table 2 includes the effects of
the uncertainties of the $B\to D\ell\nu$ and $B\to D^{*}\ell\nu$ form-factor
parameters, and the exclusive and inclusive $B\to X_{c}\ell\nu$ branching
fractions.

The detector uncertainties include uncertainties arising from the tracking
efficiency and the corrections to the lepton- and pion-identification
efficiencies. The effect of having a small sample of simulated data is also
considered.

Two sources of uncertainty are included in the continuum category in Table 2.
One is the uncertainty on the signal yields due to the limited off-resonance
sample size. The resulting uncertainties range from 3% to 11% in the
intermediate $q^{2}$ bins and dominate the uncertainties in the highest and
lowest $q^{2}$ bins, in which the continuum background is largest. The other
source of uncertainty originates from the assumption that the continuum
reweighting in $q^{2}$ bins is independent of $\Delta E$ and $M_{bc}$. To
estimate this, we parameterize the shape difference between simulated
continuum and off-resonance data as a linear function in $\Delta E$. We fit
this parameterization to the observed difference and obtain central values and
a covariance for the two parameters. We then follow the same procedure used in
the evaluation of the other systematic uncertainties. We sample 1000 varied
continuum templates, generate the corresponding toy distributions, and by
fitting the nominal templates to the toy distributions obtain refitted yields.
If the difference between the nominal fit central value and the mean value of
the refitted yields is significant, we take this difference as an additional
uncertainty on the yield. This introduces large uncertainties to the highest
and lowest $q^{2}$ bins in the electron and muon modes, respectively.

We consider additional systematic uncertainties from the number of
$B\overline{B}$ pairs $N_{B\overline{B}}$, and the branching fraction of
$\Upsilon(4S)\rightarrow B^{0}\overline{B^{0}}$, $f=0.486\pm 0.006$
Zyla:2020zbs . These uncertainties do not affect the yields, but they
contribute a 2% systematic uncertainty to the branching fraction results.

Table 2: Summary of fractional uncertainties on the yields. Source | $B^{0}\to\pi^{-}e^{+}\nu_{e}$ | $B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}$
---|---|---
| $q1$ | $q2$ | $q3$ | $q4$ | $q5$ | $q6$ | $q1$ | $q2$ | $q3$ | $q4$ | $q5$ | $q6$
Detector | 1.2 | 1.0 | 1.1 | 1.4 | 2.3 | 2.4 | 2.3 | 3.2 | 3.3 | 1.2 | 1.9 | 3.8
MC sample size | 4.0 | 2.0 | 2.4 | 2.8 | 3.9 | 5.6 | 3.9 | 2.0 | 2.3 | 2.7 | 3.4 | 4.8
Continuum | 13.1 | 5.5 | 4.4 | 7.8 | 10.5 | 33.9 | 53.3 | 8.8 | 3.2 | 4.5 | 8.0 | 11.4
$B\rightarrow\rho\ell\nu$ | 9.5 | 12.5 | 9.7 | 6.9 | 3.4 | 12.9 | 8.7 | 11.6 | 8.6 | 6.3 | 3.3 | 14.3
$B\rightarrow X_{u}\ell\nu$ | 3.3 | 1.9 | 2.1 | 2.1 | 1.8 | 3.7 | 3.4 | 2.3 | 2.0 | 2.3 | 2.1 | 6.0
$B\rightarrow X_{c}\ell\nu$ | 2.3 | 3.0 | 1.1 | 0.8 | 0.5 | 2.4 | 2.4 | 1.5 | 1.5 | 0.8 | 0.5 | 2.2
Total syst. | 17.2 | 14.3 | 11.2 | 11.1 | 12.0 | 37.0 | 53.4 | 15.2 | 10.3 | 8.7 | 9.7 | 20.3
Stat. | 10.2 | 6.01 | 6.86 | 8.08 | 10.3 | 13.2 | 10.4 | 6.0 | 6.4 | 7.8 | 9.7 | 13.4
Total | 20.2 | 15.5 | 13.2 | 13.7 | 15.9 | 39.2 | 54.5 | 16.4 | 12.2 | 11.6 | 13.7 | 24.3

##  7 Results

###  7.1 Branching fractions

The partial branching fraction in $q^{2}$ bin $i$ is calculated using the
corrected yield, $N_{i}$, and the corresponding signal efficiency,
$\epsilon_{i}$, from

$\Delta\mathcal{B}_{i}=\frac{N_{i}}{2f\epsilon_{i}\,N_{B\overline{B}}}.$ (6)

The results for $B^{0}\to\pi^{-}e^{+}\nu_{e}$ and
$B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}$ decays, $\Delta\mathcal{B}_{i,\mathrm{e}}$
and $\Delta\mathcal{B}_{i,\mathrm{\mu}}$, respectively, are given in Table 3.
We also provide an average over both decay channels, fully correlating common
systematic uncertainties in Table 311footnotemark: 1. The total branching
fraction determined from the sum of the averaged partial branching fractions
is

$\mathcal{B}(B^{0}\to\pi^{-}\ell^{+}\nu_{\ell})=(1.426\pm
0.056(\mathrm{stat})\pm 0.125(\mathrm{syst}))\times 10^{-4}$,11footnotemark: 1

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

Table 3: $\Delta\mathcal{B}$ ($\times 10^{4}$) in each $q^{2}$ bin calculated from the corrected signal yields for the $B^{0}\to\pi^{-}e^{+}\nu_{e}$ and $B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}$ modes. The averaged partial branching fractions for $B^{0}\to\pi^{-}\ell^{+}\nu_{\ell}$ are also shown. The first uncertainty is statistical and the second is systematic. $q^{2}$ bin | $B^{0}\to\pi^{-}e^{+}\nu_{e}$ | $B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}$ | $B^{0}\to\pi^{-}\ell^{+}\nu_{\ell}$
---|---|---|---
$q1$ | 0.261 $\pm$ 0.030 $\pm$ 0.045 | 0.281 $\pm$ 0.028 $\pm$ 0.150 | 0.272 $\pm$ 0.031 $\pm$ 0.044
$q2$ | 0.298 $\pm$ 0.018 $\pm$ 0.043 | 0.285 $\pm$ 0.016 $\pm$ 0.044 | 0.290 $\pm$ 0.013 $\pm$ 0.040
$q3$ | 0.268 $\pm$ 0.020 $\pm$ 0.030 | 0.290 $\pm$ 0.018 $\pm$ 0.030 | 0.279 $\pm$ 0.014 $\pm$ 0.028
$q4$ | 0.196 $\pm$ 0.022 $\pm$ 0.022 | 0.204 $\pm$ 0.019 $\pm$ 0.018 | 0.199 $\pm$ 0.015 $\pm$ 0.017
$q5$ | 0.180 $\pm$ 0.024 $\pm$ 0.022 | 0.194 $\pm$ 0.021 $\pm$ 0.019 | 0.188 $\pm$ 0.016 $\pm$ 0.015
$q6$ | 0.200 $\pm$ 0.024 $\pm$ 0.074 | 0.220 $\pm$ 0.023 $\pm$ 0.045 | 0.198 $\pm$ 0.031 $\pm$ 0.042
00footnotetext: Note: these results differ slightly from those presented at
the 2022 conferences.

###  7.2 $|V_{ub}|$ determination

We extract $|V_{ub}|$ using $\chi^{2}$ fits to the measured $q^{2}$ spectra.
We include lattice QCD constraints on the eight BCL parameters from Ref.
FermilabMILC  as nuisance parameters. These constrain the shape and
normalization of the relevant form factors entering the differential decay
rate and allow for a determination of $|V_{ub}|$.

The $\chi^{2}$ is defined as

$\chi^{2}=\sum_{i,j=1}^{6}(\Delta\mathcal{B}_{i}-\Delta\Gamma_{i}\tau)C^{-1}_{ij}(\Delta\mathcal{B}_{j}-\Delta\Gamma_{j}\tau)+\chi^{2}_{\mathrm{LQCD}},$
(7)

where $C^{-1}_{ij}$ is the inverse total covariance matrix of the measured
partial branching fractions in bin $i$, $\mathcal{B}_{i}$. The quantities
$\Delta\Gamma_{i}$ contain the predictions for the partial decay rates in bin
$i$, $\tau$ is the $B^{0}$ lifetime, and $\chi^{2}_{\mathrm{LQCD}}$
incorporates the constraints from the lattice calculation.

The obtained results are

$|V_{ub}|_{B^{0}\to\pi^{-}e^{+}\nu_{e}}=(3.60\pm 0.18(\mathrm{stat})\pm
0.14(\mathrm{syst})\pm 0.18(\mathrm{theo}))\times 10^{-3}$

$|V_{ub}|_{B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}}=(3.71\pm 0.16(\mathrm{stat})\pm
0.15(\mathrm{syst})\pm 0.17(\mathrm{theo}))\times 10^{-3}$,
$|V_{ub}|_{B^{0}\to\pi^{-}\ell^{+}\nu_{\ell}}=(3.55\pm 0.12(\mathrm{stat})\pm
0.13(\mathrm{syst})\pm 0.17(\mathrm{theo}))\times 10^{-3}$.11footnotemark: 1

The value of $|V_{ub}|$ determined by fitting the averaged
$B^{0}\to\pi^{-}\ell^{+}\nu_{\ell}$ partial branching fractions is lower than
the $|V_{ub}|$ results from the $B^{0}\to\pi^{-}e^{+}\nu_{e}$ and
$B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}$ samples. This feature of the fit arises
because the extraction of $|V_{ub}|$ is most sensitive to the high $q^{2}$
region, where the average partial branching fraction listed in Table 3 is also
lower than the $B^{0}\to\pi^{-}e^{+}\nu_{e}$ and
$B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}$ partial branching fractions due to
correlations between $q^{2}$ bins. The partial branching fractions of
$B^{0}\to\pi^{-}\ell^{+}\nu_{\ell}$ measured as a function of $q^{2}$ are
shown in Figure 3. The fitted differential rate is also shown, and the one,
two, and three standard-deviation uncertainty bands are given.

Figure 3: Measured averaged partial branching fractions as a function of
$q^{2}$. The fitted differential rate is shown together with the one, two, and
three standard-deviation uncertainty bands.

##  8 Summary

We extract partial branching fractions for $B^{0}\to\pi^{-}e^{+}\nu_{e}$ and
$B^{0}\to\pi^{-}\mu^{+}\nu_{\mu}$ decays reconstructed in an electron-positron
collision data sample corresponding to 189 fb-1 collected by the Belle II
experiment in 2019-2021. By averaging the results we obtain partial branching
fractions for $B^{0}\to\pi^{-}\ell^{+}\nu_{\ell}$. The total branching
fraction result of $B^{0}\to\pi^{-}\ell^{+}\nu_{\ell}$ is found to be
$(1.426\pm 0.056(\mathrm{stat})\pm 0.125(\mathrm{syst}))\times 10^{-4}$. This
is consistent with the current world average of $(1.50\pm 0.06)\times 10^{-4}$
Zyla:2020zbs . Currently our results are limited by the size of the off-
resonance data set. This uncertainty could be reduced by improvements in the
simulation of continuum backgrounds. We also extract values of the CKM matrix-
element magnitude $|V_{ub}|$ and from $B^{0}\to\pi^{-}\ell^{+}\nu_{\ell}$
decays obtain $(3.55\pm 0.12(\mathrm{stat})\pm 0.13(\mathrm{syst})\pm
0.17(\mathrm{theo}))\times 10^{-3}$. This agrees with the current value
obtained by HFLAV Amhis:2019ckw  of $(3.70\pm 0.10(\mathrm{exp})\pm
0.12(\mathrm{theo}))\times 10^{-3}$.

##  9 ACKNOWLEDGEMENTS

We thank the SuperKEKB group for the excellent operation of the accelerator;
the KEK cryogenics group for the efficient operation of the solenoid; the KEK
computer group for on-site computing support; and the raw-data centers at BNL,
DESY, GridKa, IN2P3, and INFN for off-site computing support.

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