I. Adachi ,

# Test of light-lepton universality in $\tau$ decays with the Belle II
experiment

0000-0003-2287-0173 K. Adamczyk  0000-0001-6208-0876 L. Aggarwal
0000-0002-0909-7537 H. Aihara ,  0000-0002-1907-5964 N. Akopov
0000-0002-4425-2096 A. Aloisio  0000-0002-3883-6693 N. Anh Ky ,
0000-0003-0471-197X D. M. Asner  0000-0002-1586-5790 H. Atmacan
0000-0003-2435-501X V. Aushev ,  0000-0002-8588-5308 M. Aversano
0000-0001-9980-0953 R. Ayad  0000-0003-3466-9290 V. Babu ,
0000-0003-0419-6912 H. Bae ,  0000-0003-1393-8631 S. Bahinipati
0000-0002-3744-5332 P. Bambade ,  0000-0001-7378-4852 Sw. Banerjee
0000-0001-8852-2409 S. Bansal  0000-0003-1992-0336 M. Barrett
0000-0002-2095-603X J. Baudot ,  0000-0001-5585-0991 A. Baur ,
0000-0003-1360-3292 A. Beaubien  0000-0001-9438-089X F. Becherer
0000-0003-0562-4616 J. Becker  0000-0002-5082-5487 J. V. Bennett ,
0000-0002-5440-2668 F. U. Bernlochner ,  0000-0001-8153-2719 V. Bertacchi
0000-0001-9971-1176 M. Bertemes  0000-0001-5038-360X E. Bertholet
0000-0002-3792-2450 M. Bessner  0000-0003-1776-0439 S. Bettarini
0000-0001-7742-2998 F. Bianchi ,  0000-0002-1524-6236 L. Bierwirth
0009-0003-0192-9073 T. Bilka  0000-0003-1449-6986 S. Bilokin
0000-0003-0017-6260 D. Biswas  0000-0002-7543-3471 A. Bobrov ,
0000-0001-5735-8386 D. Bodrov  0000-0001-5279-4787 A. Bolz
0000-0002-4033-9223 J. Borah ,  0000-0003-2990-1913 A. Boschetti
0000-0001-6030-3087 A. Bozek  0000-0002-5915-1319 M. Bračko
0000-0002-2495-0524 P. Branchini  0000-0002-2270-9673 T. E. Browder ,
0000-0001-7357-9007 A. Budano ,  0000-0002-0856-1131 S. Bussino
0000-0002-3829-9592 M. Campajola ,  0000-0003-2518-7134 L. Cao
0000-0001-8332-5668 G. Casarosa ,  0000-0003-4137-938X C. Cecchi
0000-0002-2192-8233 J. Cerasoli  0000-0001-9777-881X M.-C. Chang
0000-0002-8650-6058 P. Chang  0000-0003-4064-388X R. Cheaib
0000-0001-5729-8926 P. Cheema  0000-0001-8472-5727 B. G. Cheon ,
0000-0002-8803-4429 K. Chilikin  0000-0001-7620-2053 K. Chirapatpimol
0000-0003-2099-7760 H.-E. Cho  0000-0002-7008-3759 K. Cho  0000-0003-1705-7399
S.-J. Cho  0000-0002-1673-5664 S.-K. Choi  0000-0003-2747-8277 S. Choudhury
0000-0001-9841-0216 L. Corona ,  0000-0002-2577-9909 J. X. Cui ,
0000-0002-2398-3754 S. Das ,  0000-0001-6857-966X F. Dattola
0000-0003-3316-8574 E. De La Cruz-Burelo  0000-0002-7469-6974 S. A. De La
Motte  0000-0003-3905-6805 G. de Marino  0000-0002-6509-7793 G. De Nardo
0000-0002-2047-9675 M. De Nuccio  0000-0002-0972-9047 G. De Pietro
0000-0001-8442-107X R. de Sangro  0000-0002-3808-5455 M. Destefanis ,
0000-0003-1997-6751 S. Dey  0000-0003-2997-3829 R. Dhamija ,
0000-0001-7052-3163 A. Di Canto  0000-0003-1233-3876 F. Di Capua
0000-0001-9076-5936 J. Dingfelder  0000-0001-5767-2121 Z. Doležal
0000-0002-5662-3675 I. Domínguez Jiménez  0000-0001-6831-3159 T. V. Dong
0000-0003-3043-1939 M. Dorigo  0000-0002-0681-6946 D. Dorner
0000-0003-3628-9267 K. Dort  0000-0003-0849-8774 D. Dossett
0000-0002-5670-5582 S. Dreyer ,  0000-0002-6295-100X S. Dubey
0000-0002-1345-0970 K. Dugic ,  0009-0006-6056-546X G. Dujany
0000-0002-1345-8163 P. Ecker  0000-0002-6817-6868 M. Eliachevitch
0000-0003-2033-537X D. Epifanov  0000-0001-8656-2693 P. Feichtinger
0000-0003-3966-7497 T. Ferber  0000-0002-6849-0427 D. Ferlewicz
0000-0002-4374-1234 T. Fillinger  0000-0001-9795-7412 C. Finck
0000-0002-5068-5453 G. Finocchiaro  0000-0002-3936-2151 A. Fodor ,
0000-0002-2821-759X F. Forti  0000-0001-6535-7965 A. Frey  0000-0001-7470-3874
B. G. Fulsom ,  0000-0002-5862-9739 A. Gabrielli  0000-0001-7695-0537 E.
Ganiev ,  0000-0001-8346-8597 M. Garcia-Hernandez  0000-0003-2393-3367 G.
Gaudino ,  0000-0001-5983-1552 V. Gaur  0000-0002-8880-6134 A. Gaz
0000-0001-6754-3315 A. Gellrich ,  0000-0003-0974-6231 G. Ghevondyan
0000-0003-0096-3555 D. Ghosh  0000-0002-3458-9824 H. Ghumaryan
0000-0001-6775-8893 G. Giakoustidis  0000-0001-5982-1784 R. Giordano
0000-0002-5496-7247 A. Giri  0000-0002-8895-0128 A. Glazov
0000-0002-8553-7338 B. Gobbo  0000-0002-3147-4562 R. Godang
0000-0002-8317-0579 O. Gogota  0000-0003-4108-7256 P. Goldenzweig
0000-0001-8785-847X W. Gradl ,  0000-0002-9974-8320 T. Grammatico ,
0000-0002-2818-9744 S. Granderath  0000-0002-9945-463X E. Graziani ,
0000-0001-8602-5652 D. Greenwald  0000-0001-6964-8399 Z. Gruberová
0000-0002-5691-1044 T. Gu  0000-0002-1470-6536 Y. Guan  0000-0002-5541-2278 K.
Gudkova  0000-0002-5858-3187 Y. Han ,  0000-0001-6775-5932 T. Hara ,
0000-0002-4321-0417 K. Hayasaka ,  0000-0002-6347-433X H. Hayashii
0000-0002-5138-5903 S. Hazra  0000-0001-6954-9593 C. Hearty
0000-0001-6568-0252 M. T. Hedges  0000-0001-6504-1872 A. Heidelbach
0000-0002-6663-5469 I. Heredia de la Cruz  0000-0002-8133-6467 M. Hernández
Villanueva  0000-0002-6322-5587 T. Higuchi ,  0000-0002-7761-3505 M. Hoek ,
0000-0002-1893-8764 M. Hohmann  0000-0001-5147-4781 P. Horak ,
0000-0001-9979-6501 C.-L. Hsu ,  0000-0002-1641-430X T. Humair ,
0000-0002-2922-9779 T. Iijima  0000-0002-4271-711X K. Inami
0000-0003-2765-7072 G. Inguglia  0000-0003-0331-8279 N. Ipsita
0000-0002-2927-3366 A. Ishikawa ,  0000-0002-3561-5633 R. Itoh ,
0000-0003-1590-0266 M. Iwasaki  0000-0002-9402-7559 W. W. Jacobs ,
0000-0002-9996-6336 D. E. Jaffe  0000-0003-3122-4384 E.-J. Jang
0000-0002-1935-9887 Q. P. Ji  0000-0003-2963-2565 S. Jia  0000-0001-8176-8545
Y. Jin  0000-0002-7323-0830 H. Junkerkalefeld ,  0000-0003-3987-9895 M. Kaleta
,  0000-0002-2863-5476 D. Kalita  0000-0003-3054-1222 A. B. Kaliyar
0000-0002-2211-619X J. Kandra  0000-0001-5635-1000 S. Kang ,
0000-0002-5320-7043 G. Karyan ,  0000-0001-5365-3716 T. Kawasaki ,
0000-0002-4089-5238 F. Keil  0000-0002-7278-2860 C. Kiesling ,
0000-0002-2209-535X C.-H. Kim  0000-0002-5743-7698 D. Y. Kim
0000-0001-8125-9070 K.-H. Kim ,  0000-0002-4659-1112 Y.-K. Kim ,
0000-0002-9695-8103 H. Kindo ,  0000-0002-6756-3591 K. Kinoshita
0000-0001-7175-4182 P. Kodyš ,  0000-0002-8644-2349 T. Koga
0000-0002-1644-2001 S. Kohani  0000-0003-3869-6552 K. Kojima
0000-0002-3638-0266 T. Konno  0000-0003-2487-8080 A. Korobov ,
0000-0001-5959-8172 S. Korpar  0000-0003-0971-0968 E. Kovalenko ,
0000-0001-8084-1931 R. Kowalewski  0000-0002-7314-0990 T. M. G. Kraetzschmar
0000-0001-8395-2928 P. Križan  0000-0002-4967-7675 P. Krokovny ,
0000-0002-1236-4667 T. Kuhr ,  0000-0001-6251-8049 Y. Kulii
0000-0001-6217-5162 J. Kumar  0000-0002-8465-433X M. Kumar
0000-0002-6627-9708 K. Kumara ,  0000-0003-1572-5365 T. Kunigo ,
0000-0001-9613-2849 A. Kuzmin ,  0000-0002-7011-5044 Y.-J. Kwon ,
0000-0001-9448-5691 S. Lacaprara  0000-0002-0551-7696 Y.-T. Lai
0000-0001-9553-3421 K. Lalwani ,  0000-0002-7294-396X T. Lam
0000-0001-9128-6806 L. Lanceri  0000-0001-8220-3095 J. S. Lange
0000-0003-0234-0474 M. Laurenza  0000-0002-7400-6013 K. Lautenbach
0000-0003-3762-694X R. Leboucher ,  0000-0003-3097-6613 F. R. Le Diberder
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0000-0003-3833-2900 C. Lemettais  0009-0008-5394-5100 D. Levit ,
0000-0001-5789-6205 P. M. Lewis  0000-0002-5991-622X C. Li
0000-0002-3240-4523 L. K. Li  0000-0002-7366-1307 S. X. Li
0000-0003-4669-1495 Y. Li  0000-0002-4413-6247 Y. B. Li  0000-0002-9909-2851
J. Libby  0000-0002-1219-3247 Z. Liptak ,  0000-0002-6491-8131 M. H. Liu
0000-0002-9376-1487 Q. Y. Liu  0000-0002-7684-0415 Y. Liu  0000-0002-8374-3947
Z. Q. Liu ,  0000-0002-0290-3022 D. Liventsev  0000-0003-3416-0056 S. Longo
0000-0002-8124-8969 T. Lueck ,  0000-0003-3915-2506 C. Lyu ,
0000-0002-2275-0473 Y. Ma  0000-0001-8412-8308 M. Maggiora
0000-0003-4143-9127 S. P. Maharana  0000-0002-1746-4683 R. Maiti ,
0000-0001-5534-7149 S. Maity  0000-0003-3076-9243 G. Mancinelli
0000-0003-1144-3678 R. Manfredi  0000-0002-8552-6276 E. Manoni
0000-0002-9826-7947 M. Mantovano ,  0000-0002-5979-5050 D. Marcantonio
0000-0002-1315-8646 S. Marcello  0000-0003-4144-863X C. Marinas
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0000-0003-1706-9365 D. Matvienko  0000-0002-2698-5448 S. K. Maurya
0000-0002-7764-5777 J. A. McKenna  0000-0001-9871-9002 R. Mehta ,
0000-0001-8670-3409 F. Meier  0000-0002-6088-0412 M. Merola
0000-0002-7082-8108 F. Metzner ,  0000-0002-0128-264X C. Miller ,
0000-0003-2631-1790 M. Mirra  0000-0002-1190-2961 S. Mitra
0000-0002-1118-6344 K. Miyabayashi  0000-0003-4352-734X H. Miyake
0000-0002-7079-8236 R. Mizuk  0000-0002-2209-6969 G. B. Mohanty
0000-0001-6850-7666 S. Mondal ,  0000-0002-3054-8400 S. Moneta
0000-0003-2184-7510 H.-G. Moser ,  0000-0003-3579-9951 M. Mrvar
0000-0001-6388-3005 R. Mussa ,  0000-0002-0294-9071 I. Nakamura
0000-0002-7640-5456 K. R. Nakamura  0000-0001-7012-7355 M. Nakao ,
0000-0001-8424-7075 H. Nakazawa ,  0000-0003-1684-6628 Y. Nakazawa
0000-0002-6271-5808 A. Narimani Charan  0000-0002-5975-550X M. Naruki ,
0000-0003-1773-2999 D. Narwal  0000-0001-6585-7767 Z. Natkaniec
0000-0003-0486-9291 A. Natochii  0000-0002-1076-814X L. Nayak
0000-0002-7739-914X M. Nayak  0000-0002-2572-4692 G. Nazaryan
0000-0002-9434-6197 M. Neu  0000-0002-4564-8009 C. Niebuhr
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0000-0002-7310-5079 Y. Onishchuk ,  0000-0002-8261-7543 H. Ono
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0000-0003-2948-5155 S. Raiz  0000-0001-7010-8066 N. Rauls  0000-0002-6583-4888
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0000-0001-6975-1724 I. Ripp-Baudot  0000-0002-1897-8272 G. Rizzo ,
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0000-0003-0654-2866 J. M. Roney  0000-0001-7802-4617 A. Rostomyan ,
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0000-0003-1771-9161 C. Schmitt  0000-0002-3787-687X C. Schwanda ,
0000-0003-4844-5028 M. Schwickardi ,  0000-0003-2033-6700 Y. Seino
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0000-0002-4824-101X C. Sfienti  0000-0002-5921-8819 W. Shan
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0000-0002-8253-641X R. Žlebčík  0000-0003-1644-8523

###### Abstract

We present a measurement of the ratio $R_{\mu}={\cal
B}(\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau})/{\cal
B}(\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau})$ of branching fractions
$\cal B$ of the $\tau$ lepton decaying to muons or electrons using data
collected with the Belle II detector at the SuperKEKB $e^{+}e^{-}$ collider.
The sample has an integrated luminosity of at a centre-of-mass energy of
$10.58\text{\,}\text{$\mathrm{Ge\kern-1.00006ptV}$}$. Using an optimised event
selection, a binned maximum likelihood fit is performed using the momentum
spectra of the electron and muon candidates. The result, $R_{\mu}=0.9675\pm
0.0007\pm 0.0036$, where the first uncertainty is statistical and the second
is systematic, is the most precise to date. It provides a stringent test of
the light-lepton universality, translating to a ratio of the couplings of the
muon and electron to the $W$ boson in $\tau$ decays of $0.9974\pm 0.0019$, in
agreement with the standard model expectation of unity.

##  1 Introduction

In the standard model (SM) of particle physics, lepton flavour universality
(LFU) refers to an intrinsic property under which the electroweak gauge bosons
have the same couplings to the three generations of leptons $e$, $\mu$, and
$\tau$ Tsai:1971vv . This symmetry originates from the fact that the only
difference between lepton generations is derived from their distinct masses. A
broad class of SM extensions postulates the existence of new particles, such
as mediators of new interactions, that couple to the three leptons
differently, making searches of LFU violation compelling Jung:2010ik ;
Crivellin:2020klg . In the last decades, experimental tests of LFU have been
carried out using various processes, but no significant deviation from the SM
expectation has been observed. For example, tests in charged currents have
been carried out in the decays of pions PiENu:2015seu , kaons NA62:2012lny ;
KLOE:2009urs , $B$ mesons Belle:2019rba ; LHCb:2023zxo , on-shell $W$ bosons
ATLAS:2020xea ; CMS:2022mhs , and $\tau$ leptons CLEO:1996oro ; BaBar:2009lyd
. Among these, the most precise LFU test is achieved in pion decays, followed
by the $\tau$-lepton decays, which are sensitive not only to charged currents
but also to non-SM contributions of weak neutral currents
Altmannshofer:2016brv ; Bryman:2021teu . The LFU tests in $\tau$ decays rely
on measurements of the $\tau$ mass, lifetime, and branching fractions of
$\tau$ decays to lighter leptons or hadrons. The $e$-$\mu$ universality is
tested by comparing the measured rates of leptonic $\tau$ decays, whose
branching fractions are denoted as ${\cal
B}(\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau})$ and ${\cal
B}(\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau})$. Charge-conjugate modes
are implied throughout the paper. The ratio $R_{\mu}$ of the branching
fractions,

$R_{\mu}=\frac{{\cal
B}(\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau})}{{\cal
B}(\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau})},$ (1)

in turn, constrains the ratio of the effective coupling strengths $g_{e}$ and
$g_{\mu}$ of the electron and muon to the $W^{\pm}$,

$\displaystyle\Bigg{|}\frac{g_{\mu}}{g_{e}}\Bigg{|}_{\tau}=\sqrt{R_{\mu}\frac{f(m_{e}^{2}/m_{\tau}^{2})}{f(m_{\mu}^{2}/m_{\tau}^{2})}}.$
(2)

Here $f(x)=1-8x+8x^{3}-x^{4}-12x^{2}\ln{x}$ under the assumption of negligible
neutrino masses Tsai:1971vv , and $m_{e}$, $m_{\mu}$, and $m_{\tau}$ are the
masses of the corresponding leptons. Thus, the challenge to test the $e$-$\mu$
universality is to accurately determine the ratio of branching fractions
$R_{\mu}$. This deviates from unity in the SM due to the difference in mass of
the final state leptons and is predicted to be $0.9726$. Previous measurements
of $R_{\mu}$ were reported by the CLEO CLEO:1996oro  and BaBar BaBar:2009lyd
collaborations. The most precise single determination of $R_{\mu}$ to date
comes from the direct measurement of $R_{\mu}$ by the BaBar collaboration,
$R_{\mu}=0.9796\pm 0.0016\pm 0.0036$ BaBar:2009lyd , where the first
uncertainty is statistical and the second is systematic. The measurement has
$0.4\text{\,}\mathrm{\char 37\relax}$ precision limited by the systematic
uncertainties associated mainly with the lepton identification. This
uncertainty propagates to $0.2\text{\,}\mathrm{\char 37\relax}$ precision on
$|g_{\mu}/g_{e}|_{\tau}=1.0036\pm 0.0020$. The current world average value of
$|g_{\mu}/g_{e}|_{\tau}=1.0019\pm 0.0014$ HeavyFlavorAveragingGroup:2022wzx
is consistent with unity, as predicted by the SM. In this paper, we report a
measurement of $R_{\mu}$ using data collected with the Belle II detector
Abe:2010gxa  at the energy-asymmetric $e^{+}e^{-}$ SuperKEKB collider
Akai:2018mbz . The data were recorded between 2019 and 2022 at a centre-of-
mass energy of $10.58\text{\,}\text{$\mathrm{Ge\kern-1.00006ptV}$}$ and
correspond to an integrated luminosity of which translates to about
$333\text{\times}{10}^{6}$ $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events. We
determine $R_{\mu}$ from events in which one $\tau$ decays either to
$\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau}$ or
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$, and the other $\tau$
decays hadronically. First, we optimise a selection that is common to both
modes, after which we perform a binned maximum likelihood fit using the
momentum spectra of the lepton candidates. All systematic effects are modelled
directly in the likelihood function. Finally, we measure the value of
$R_{\mu}$ from the fit and translate it into $|g_{\mu}/g_{e}|_{\tau}$, testing
$e$-$\mu$ universality in charged current interactions.

##  2 The Belle II detector and simulation

The Belle II detector comprises several subdetectors arranged in a cylindrical
structure around the $e^{+}e^{-}$ interaction point Abe:2010gxa . Charged-
particle trajectories (tracks) are reconstructed by a two-layer silicon-pixel
detector, surrounded by a four-layer double-sided silicon-strip detector and a
central drift chamber (CDC). Only $15\text{\,}\mathrm{\char 37\relax}$ of the
second pixel layer was installed when the data were collected. Outside the
CDC, a time-of-propagation detector and an aerogel ring-imaging Cherenkov
detector cover the barrel and forward endcap regions. The electromagnetic
calorimeter (ECL), divided into the forward endcap, barrel, and backward
endcap, fills the remaining volume inside a $1.5\text{\,}\mathrm{T}$
superconducting solenoid and is used to reconstruct photons and electrons. A
$K_{L}^{0}$ and muon detection system is installed in the iron flux return of
the solenoid. The $z$ axis of the laboratory frame is defined as the detector
solenoid axis, with the positive direction along the electron beam. The polar
angle $\theta$ and the transverse plane are defined relative to this axis.
Simulated samples are used for studying sample composition and optimising the
analysis selections. We further rely on simulated samples to study and
determine efficiencies and to define fit templates for the extraction of
$R_{\mu}$. Several processes contribute to the
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ sample as backgrounds, including
$e^{+}e^{-}\rightarrow q\bar{q}$ events, where $q$ indicates a $u$, $d$, $s$,
or $c$ quark; $e^{+}e^{-}\rightarrow e^{+}e^{-}(\gamma)$ and
$\mu^{+}\mu^{-}(\gamma)$ events; $e^{+}e^{-}\rightarrow l^{+}l^{-}l^{+}l^{-}$
events, where $l$ is a charged lepton; $e^{+}e^{-}\rightarrow
e^{+}e^{-}h^{+}h^{-}$ events, where $h$ indicates a pion or kaon; and
$e^{+}e^{-}\rightarrow e^{+}e^{-}Nh$ events with multiplicity $N>2$. We use
several software packages to generate the simulated particles. The
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ process is generated using KKMC
Jadach:1999vf ; Jadach:2000ir , $\tau$ decays are simulated by TAUOLA
Banerjee:2021rtn ; Shekhovtsova:2012ra ; Chrzaszcz:2016fte ; Nugent:2013hxa
and their radiative corrections by PHOTOS Barberio:1990ms . We use KKMC to
simulate $\mu^{+}\mu^{-}(\gamma)$ and $q\bar{q}$ production; PYTHIA
Sjostrand:2014zea  for the fragmentation of the $q\bar{q}$ pair; BabaYaga@NLO
Balossini:2006wc ; Balossini:2008xr ; CarloniCalame:2003yt ;
CarloniCalame:2001ny ; CarloniCalame:2000pz  for $e^{+}e^{-}\rightarrow
e^{+}e^{-}(\gamma)$ events; and AAFH BERENDS1985421 ; BERENDS1985441 ;
BERENDS1986285  and TREPS Uehara:1996bgt  for the production of non-radiative
final states $l^{+}l^{-}l^{+}l^{-}$ and $e^{+}e^{-}h^{+}h^{-}$. Currently,
there is no generator to simulate $e^{+}e^{-}\rightarrow e^{+}e^{-}Nh$
processes. The Belle II analysis software Kuhr:2018lps ; basf2-zenodo  uses
the GEANT4 Agostinelli:2002hh  package to simulate the response of the
detector to the passage of the particles.

##  3 Event selection

The trigger is based on ECL energy deposits (clusters) and their topologies in
the ECL. The efficiency of the trigger system for this measurement is driven
by the condition that the combined energy deposit of all ECL clusters exceeds
$1\text{\,}\text{$\mathrm{Ge\kern-1.00006ptV}$}$. Events with two back-to-back
clusters in the centre-of-mass system, one of which exceeds
$4.5\text{\,}\text{$\mathrm{Ge\kern-1.00006ptV}$}$ and the other
$3\text{\,}\text{$\mathrm{Ge\kern-1.00006ptV}$}$, are vetoed by the trigger
system to reject Bhabha events. In the $e^{+}e^{-}$ centre-of-mass frame, the
$\tau$ leptons from $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ are produced in
opposite directions and with a significant boost. Thus, the decay products of
one $\tau$ are isolated from those of the accompanying $\tau$ and contained in
opposite hemispheres. The boundary between those hemispheres is the plane
perpendicular to the $\tau$ flight direction, which is experimentally
approximated by the thrust axis. The thrust axis is the unit vector $\hat{t}$
that maximizes the thrust value
$\sum{|\hat{t}\cdot\vec{p}_{i}^{\,*}|}/\sum{|\vec{p}_{i}^{\,*}|}$, where
$\vec{p}_{i}^{\,*}$ is the momentum of the $i$th final state particle in the
$e^{+}e^{-}$ centre-of-mass frame Brandt:1964sa ; Farhi:1977sg . Throughout
this paper, quantities in the $e^{+}e^{-}$ centre-of-mass frame are indicated
by an asterisk. We define the _signal_ hemisphere as the one containing a
charged particle originating either from $\tau^{-}\rightarrow
e^{-}\bar{\nu}_{e}\nu_{\tau}$ or
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ decays. We also require
that the opposite hemisphere, labelled with _tag_ , contains only one charged
particle and at least one neutral pion. Thus, the tag side contains
predominantly $\tau^{+}\rightarrow h^{+}n\pi^{0}\bar{\nu}_{\tau}$ decays with
multiplicity $n=1,2$. We select $\tau$-pair candidates by requiring the event
to contain exactly two charged particles with zero total charge, each having a
trajectory displaced from the average interaction point by less than
$3\text{\,}\cm$ along the $z$ axis and less than $1\text{\,}\cm$ in the
transverse plane to reduce the contribution of misreconstructed or poorly
constrained tracks. The charged particle on the tag side must satisfy the
condition $E_{\rm ECL}/p\leq 0.8\,c$ to suppress electron contamination. Here,
$E_{\rm ECL}$ denotes the energy deposit in the ECL, and $p$ is the magnitude
of the momentum vector of the associated particle. Muons are identified using
the discriminator $P_{\mu}={\cal L}_{\mu}/({\cal L}_{e}+{\cal L}_{\mu}+{\cal
L}_{\pi}+{\cal L}_{K}+{\cal L}_{p}+{\cal L}_{d})$ where the likelihood ${\cal
L}$ for each charged-particle hypothesis combines particle identification
information from all detectors except the silicon trackers. Electrons are
identified using a boosted decision tree classifier that is trained to
separate electrons from all other charged particles eIDbdt . This approach
gives improved results compared to the purely likelihood-based approach, in
particular for separating pions from electrons. Inputs to the classifier are
the likelihoods from each sub-detector, as well as additional ECL observables,
such as variables that characterise the cluster’s spatial structure. The most
discriminating variable in the momentum range relevant to this analysis is
$E_{\rm ECL}/p$. We use the output of the classifier, $P_{e}$, as a
discriminator for electron identification. We retain lepton candidates with
requirements $P_{e}>0.5$ and $P_{\mu}>0.9$. We require that each lepton
candidate, $\ell=e,\mu$ in the signal side satisfies the conditions
$1.5<p_{\ell}<$5\text{\,}\text{{\mathrm{Ge\kern-1.00006ptV\\!/}c}}$$, and
$0.82<\theta_{\ell}<$2.13\text{\,}\rm\mathrm{r}\mathrm{a}\mathrm{d}$$ to
ensure accurate particle identification information. Here $\theta_{\ell}$
refers to the polar angle of the momentum vector in the laboratory frame.
While inefficiencies of the particle identification system are taken into
account in our simulation, we correct for imperfections in the simulation
using data-driven factors from calibration channels, as functions of momentum,
polar angle, and charge. These channels are
$J/\psi\rightarrow\ell^{+}\ell^{-}$,
$e^{+}e^{-}\rightarrow\ell^{+}\ell^{-}\gamma$, and $e^{+}e^{-}\rightarrow
e^{+}e^{-}\ell^{+}\ell^{-}$ events for efficiency, and
$K^{0}_{S}\rightarrow\pi^{+}\pi^{-}$ and
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events for misidentification rates.
Due to limitations in calibration-sample sizes, events with candidate leptons
$p_{\ell}>$4\text{\,}\text{{\mathrm{Ge\kern-1.00006ptV\\!/}c}}$$ and
$\theta_{\ell}>$1.78\text{\,}\rm\mathrm{r}\mathrm{a}\mathrm{d}$$, or
$p_{\ell}>$4.5\text{\,}\text{{\mathrm{Ge\kern-1.00006ptV\\!/}c}}$$ and
$\theta_{\ell}>$1.16\text{\,}\rm\mathrm{r}\mathrm{a}\mathrm{d}$$ are vetoed.
Electron and muon identification efficiencies are $99.7\text{\,}\mathrm{\char
37\relax}$ and $93.9\text{\,}\mathrm{\char 37\relax}$, respectively. The rates
for misidentifying pions as electrons or muons are $0.9\text{\,}\mathrm{\char
37\relax}$ and $3.1\text{\,}\mathrm{\char 37\relax}$, respectively. The
momenta of charged particles are corrected for imperfections in the magnetic
field description used for event reconstruction, misalignment of the detector,
and material mismodelling. The corresponding correction factor is evaluated by
measuring the mass-peak position of a high-yield sample of $D^{0}\rightarrow
K^{-}\pi^{+}$ decays reconstructed in data and comparing this to the known
value ParticleDataGroup:2022pth . Neutral pions are identified as photon pairs
with an invariant mass between
$120\text{\,}\text{${\mathrm{Me\kern-1.00006ptV\\!/}c^{2}}$}$ and
$145\text{\,}\text{${\mathrm{Me\kern-1.00006ptV\\!/}c^{2}}$}$, which is within
two units of mass resolution from the known value. Those photons are
identified from clusters, reconstructed within the CDC acceptance,
$17^{\circ}<\theta<150^{\circ}$, to ensure they are not matched to any charged
particle. The energy threshold for selecting photon candidates varies based on
the polar detector region, aiming to suppress beam-induced backgrounds. This
adjustment is important for the endcaps where such backgrounds are more
prominent. Specifically, we use
$80\text{\,}\text{$\mathrm{Me\kern-1.00006ptV}$}$ for the forward region with
$$17\text{\,}\mathrm{\SIUnitSymbolDegree}$<\theta<$31.4\text{\,}\mathrm{\SIUnitSymbolDegree}$$,
$30\text{\,}\text{$\mathrm{Me\kern-1.00006ptV}$}$ for the barrel with
$$32.2\text{\,}\mathrm{\SIUnitSymbolDegree}$<\theta<$128.7\text{\,}\mathrm{\SIUnitSymbolDegree}$$,
and $60\text{\,}\text{$\mathrm{Me\kern-1.00006ptV}$}$ for the backward region
with
$$130.7\text{\,}\mathrm{\SIUnitSymbolDegree}$<\theta<$150\text{\,}\mathrm{\SIUnitSymbolDegree}$$.
The requirements
$\alpha_{\gamma\gamma}<$1.4\text{\,}\rm\mathrm{r}\mathrm{a}\mathrm{d}$$ and
$|\delta\phi|<$1.5\text{\,}\rm\mathrm{r}\mathrm{a}\mathrm{d}$$ on the angle
$\alpha$ between the momenta of the two photons, and on the difference of
azimuthal angles $\phi$ of the two photons reduce the combinatorial background
from low-energy photons. The efficiency for identifying neutral pions with
these criteria is $30\text{\,}\mathrm{\char 37\relax}$. The main background
contamination comes from $e^{+}e^{-}\rightarrow e^{+}e^{-}(\gamma)$,
$e^{+}e^{-}\rightarrow e^{+}e^{-}l^{+}l^{-}$ and $e^{+}e^{-}\rightarrow
e^{+}e^{-}Nh$ processes. The $e^{+}e^{-}\rightarrow e^{+}e^{-}Nh$ events are
suppressed using a data-driven selection. For these processes, the events are
concentrated at low thrust values, at large values of the squared missing mass
$M^{2}_{\rm miss}$, and at low missing transverse momentum $p^{*}_{T,\;\rm
miss}$. The missing momentum is the difference between the momenta of the
initial $e^{+}e^{-}$ and that of all reconstructed particles in the event,
while the square of the missing mass is defined as $M^{2}_{\rm
miss}=(\sqrt{s}/c^{2}-E^{*}_{\rm vis}/c^{2})^{2}-(p^{*}_{\rm miss}/c)^{2}$,
where $E^{*}_{\rm vis}$ is the energy of all reconstructed particles in the
event and $p^{*}_{\rm miss}$ is the magnitude of the missing momentum vector.
The $e^{+}e^{-}\rightarrow e^{+}e^{-}Nh$ background can be discriminated with
these variables since the $e^{+}e^{-}$ pair is not reconstructed and has a
momentum vector parallel to the beam axis, while the remaining particles are
not collinear. The requirements that the thrust value exceeds $0.85$ and
$M^{2}_{\rm
miss}<($20\text{\,}\text{{\mathrm{Ge\kern-1.00006ptV^2\\!/}c^4}}$+40\,\text{$\mathrm{Ge\kern-1.00006ptV}$}/c^{3}\cdot
p^{*}_{T,\,\rm{miss}})$ are chosen to remove the contribution from the
$e^{+}e^{-}\rightarrow e^{+}e^{-}Nh$ events. For all other background
processes, simulated signal and background events are used to train a neural
network event classifier using cross-entropy as a loss function Good:logloss ;
DBLP . The signal sample is defined as the combination of the
$\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau}$ and
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ samples to have a common
selection for both decays and, as a result, a cancellation of most of the
systematic uncertainties on $R_{\mu}$ that are associated with the selection.
Seven variables are used in the training: the thrust value, the polar angle of
the thrust vector, $E^{*}_{\rm{vis}}$, the transverse component of the missing
momentum direction in the centre-of-mass frame, the momentum of the tag side
charged particle and the invariant mass and polar angle of the $h^{+}n\pi^{0}$
system on the tag side in the centre-of-mass frame. We map them to four output
nodes — one representing the combined electron and muon signal sample, the
other three representing the various background events:
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events with a misidentified signal or
tag side, $e^{+}e^{-}\rightarrow e^{+}e^{-}$ events, and a combined sample of
all remaining background events. The most discriminating variable is
$E^{*}_{\rm{vis}}$, which is twice as important as the least discriminating
variable. Consistency checks with validation samples show no indication of
overtraining. Figure 1 shows the output distribution of the neural network
used to discriminate between $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events
with $\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau}$ or
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ decays on the signal
side and all other events. The figure uses simulated training samples.

Figure 1: Distributions of the output of the neural network, trained to
classify the combined sample of $\tau^{-}\rightarrow
e^{-}\bar{\nu}_{e}\nu_{\tau}$ and
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ decays. We show
simulated background contributions from
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events with decays other than
$\tau^{+}\rightarrow h^{+}n\pi^{0}\bar{\nu}_{\tau}$ on the tag side, with
decays other than $\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau}$ and
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ on the signal side, and
Bhabha events separately. The remaining background processes contributing to
the spectrum are combined and collectively called ‘Others’. The dashed
vertical line indicates the threshold used in this analysis.

To select signal events, we require the output value of the neural network to
be greater than 0.9, which yields the smallest total uncertainty on $R_{\mu}$
in simulation. The selected events are separated into electron and muon
samples. These samples will be simultaneously fit in bins of the lepton
candidate momentum in order to extract $R_{\mu}$ from the data. The momentum
distributions of the electron and muon candidates in the simulated
$\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau}$ and
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ samples, along with
simulated background contributions, are shown in Figure 2. The distributions
are corrected for imperfections in the simulation, in particular particle
identification, the reconstruction of neutral particles, and trigger. The
discontinuities at $4\text{\,}\text{${\mathrm{Ge\kern-1.00006ptV\\!/}c}$}$ and
$4.5\text{\,}\text{${\mathrm{Ge\kern-1.00006ptV\\!/}c}$}$ in the distributions
reflect the veto of events in certain lepton identification correction bins as
well as the impact of the correction factors themselves.

Figure 2: Momentum spectra of muon (left) and electron (right) candidates from
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ and $\tau^{-}\rightarrow
e^{-}\bar{\nu}_{e}\nu_{\tau}$, along with simulated background contributions.
The legend uses a similar nomenclature as Figure 1.

The trigger efficiency is measured with a reference sample selected by
independent triggers based on the number of particles reconstructed in the
CDC. The trigger efficiency in data is $99.8\text{\,}\mathrm{\char 37\relax}$
for $\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau}$ and
$96.6\text{\,}\mathrm{\char 37\relax}$ for
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ decays. In simulation,
the corresponding efficiencies are $98.6\text{\,}\mathrm{\char 37\relax}$ and
$95.4\text{\,}\mathrm{\char 37\relax}$, respectively. To account for
imperfection in the simulation of the trigger, we apply correction factors as
ratios of efficiencies in data and simulation to our simulated samples. The
correction factors are stable as a function of the lepton candidate momentum
and independent of the flavour of the signal lepton. The signal-reconstruction
efficiency in simulation after all selection requirements and applied
corrections is $9.602\text{\,}\mathrm{\char 37\relax}$ for
$\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau}$ and
$9.551\text{\,}\mathrm{\char 37\relax}$ for
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$. The purity is
$96\text{\,}\mathrm{\char 37\relax}$ for the $e$-sample and
$92\text{\,}\mathrm{\char 37\relax}$ for the $\mu$-sample. The dominant
backgrounds are from $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events with
$\tau^{-}\rightarrow h^{-}\nu_{\tau}$ or $\tau^{-}\rightarrow
h^{-}n\pi^{0}\nu_{\tau}$ decays in the signal hemisphere, which makes up
$1.3\text{\,}\mathrm{\char 37\relax}$ of the $e$-sample and
$5.2\text{\,}\mathrm{\char 37\relax}$ of the $\mu$-sample. This is followed by
$2.3\text{\,}\mathrm{\char 37\relax}$ of
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events with a misidentified tag side
decay and $0.2\text{\,}\mathrm{\char 37\relax}$ of $e^{+}e^{-}\rightarrow
e^{+}e^{-}\tau^{+}\tau^{-}$ events, for both samples. In data, we observe
$4\,371\,737$ events for which the lepton candidate is identified as a muon
and $4\,358\,376$ events for which it is identified as an electron.

##  4 Method

To determine signal yields we use the _pyhf_ package Heinrich:2021gyp , which
constructs a binned likelihood following the HistFactory Cranmer:2012sba
formalism. The templates for the signal and background momentum distributions
are derived from simulation and use $21$ bins spanning the lepton candidate
momentum. This binning choice is derived from the lepton identification
correction bins. The momentum range is covered by seven bins with a bin width
of $0.5\text{\,}\text{${\mathrm{Ge\kern-1.00006ptV\\!/}c}$}$, which are
further split into three equal-sized bins to define the templates. The events
are separated into channels according to the lepton identification criteria
$P_{e}>0.5$ and $P_{\mu}>0.9$. For each channel, one template represents the
signal component, and two templates describe the backgrounds, separated into
events with correctly identified leptons and all other background events.
Systematic uncertainties are incorporated with multiplicative or additive
event-count modifiers in the likelihood. The likelihood function $f$ is a
product of Poisson probability density functions $\cal P$ that combines the
information from all bins of signal and background samples,

$f(R_{\mu},\vec{\chi})=\prod_{b\in{\rm bins}}{\cal
P}(n_{b}^{e}\mid\nu_{b}^{e}(\vec{\chi}))\prod_{b\in{\rm bins}}{\cal
P}(n_{b}^{\mu}\mid\nu_{b}^{\mu}(R_{\mu},\vec{\chi}))\prod_{\chi\in\vec{\chi}}c_{\chi}(a_{\chi}\mid\chi).$
(3)

Here, $n_{b}^{e}$ and $n_{b}^{\mu}$ are the number of observed
$\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau}$ and
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ candidate events in data
for each bin $b$ of the template. The corresponding expected number of events
from simulations are denoted as $\nu_{b}^{e}$ and $\nu_{b}^{\mu}$. The
systematic uncertainties discussed below are included in the likelihood as a
set of nuisance parameters $\vec{\chi}$ that are event-count modifiers,
constrained by normal probability density functions $c_{\chi}$ associated with
auxiliary data $a_{\chi}$. We model each momentum spectrum as a sum of
contributions from the signal $\tau^{-}\rightarrow
e^{-}\bar{\nu}_{e}\nu_{\tau}$ or
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ decays and all other
sources of background events. Since the majority of the background originates
from $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events, we do not split the
background templates according to the physics processes. Instead, we
distinguish them by the signal side particle type in order to separate the
effects of the uncertainties associated with lepton identification and
misidentification on the templates. The resulting expressions for the expected
events for each bin are

$\displaystyle\nu_{b}^{e}(\vec{\chi})$ $\displaystyle=$
$\displaystyle\kappa_{e}\,\nu_{b}^{e\text{-sig}}(\vec{\chi})+\nu_{b}^{e\text{-bkg(true)}}(\vec{\chi})+\nu_{b}^{e\text{-bkg(fake)}}(\vec{\chi})\,\text{and}$
(4) $\displaystyle\nu_{b}^{\mu}(R_{\mu},\vec{\chi})$ $\displaystyle=$
$\displaystyle R_{\mu}\,\kappa_{e/\mu}^{\rm
gen}\,\kappa_{e}\,\nu_{b}^{\mu\text{-sig}}(\vec{\chi})+\nu_{b}^{\mu\text{-bkg(true)}}(\vec{\chi})+\nu_{b}^{\mu\text{-bkg(fake)}}(\vec{\chi})\;.$
(5)

Here $\nu_{b}^{e\text{-sig}}$ and $\nu_{b}^{\mu\text{-sig}}$ are the signal
yields, $\nu_{b}^{e\text{-bkg(true)}}$ and $\nu_{b}^{\mu\text{-bkg(true)}}$
are the yields of background events with the particle in the signal side
properly identified as either an electron or a muon, and
$\nu_{b}^{e\text{-bkg(fake)}}$ and $\nu_{b}^{\mu\text{-bkg(fake)}}$ stand for
the remaining background events, mostly with pions misidentified as leptons.
The factor $\kappa_{e}$, which is free in the fit, sets the overall
normalisation of the signal templates; the constant scaling factor
$\kappa_{e/\mu}^{\rm gen}\equiv{{\cal B}^{\rm gen}_{e}}/{{\cal B}^{\rm
gen}_{\mu}}$ takes the assumed branching fractions from the simulation into
account. This allows the ratio $R_{\mu}$ to be estimated directly from the
fit: it is determined simultaneously along with the nuisance parameters by
maximising the likelihood function. Figure 3 shows the yields from
experimental data, superimposed with the expected yields from simulation. The
hatched area in the lower panel indicates the possible variation of the yields
due to systematic effects, with the dominant contribution being the $\pi^{0}$
efficiency uncertainty.

Figure 3: Observed momentum distribution for muon (left) and electron (right)
candidates with simulation expectations overlaid. The lower panel shows the
ratio between data and expectations with systematic uncertainties (hatched).

To minimise the observer-expectancy effect, we validate the $R_{\mu}$
measurement method using simulation, and estimate the statistical and
systematic uncertainties using data without examining the central value of the
result. This is achieved by randomly adjusting the values of branching
fractions ${\cal B}_{e}^{\rm gen}$ and ${\cal B}_{\mu}^{\rm gen}$ used in the
generation within a $2\text{\,}\mathrm{\char 37\relax}$ range. This allows us
to validate the fit and the shapes of all distributions without observing the
actual value.

##  5 Systematic uncertainties and consistency checks

The systematic uncertainties are grouped into categories associated with
charged-particle identification, imperfections of the simulation, trigger,
size of the simulated samples, and luminosity. We include one or more nuisance
parameters for each systematic source depending on the correlations between
the individual template bins. The nuisance parameters modify the event-counts
of the templates to capture the impact of each systematic uncertainty on the
observed momentum distribution of the leptons. The total number of nuisance
parameters in the fit amounts to $194$. Table 1 summarises the sources
contributing to the total uncertainty on $R_{\mu}$. To estimate the
uncertainty of each contribution, we generate $2000$ simulated replicas of
data by sampling the likelihood, where only auxiliary data associated with the
respective nuisance parameters are varied according to the associated
constraint terms. The standard deviation of the resultant distribution of
fitted $R_{\mu}$ values is quoted as the size of the corresponding systematic
effect. The largest uncertainty on $R_{\mu}$ arises from the particle
identification, followed by the uncertainty on the trigger efficiency. The
total absolute systematic uncertainty on $R_{\mu}$ is $0.0036$.

Source | Uncertainty [%]
---|---
Charged-particle identification: | 0.32
Electron identification | 0.22
Muon misidentification | 0.19
Electron misidentification | 0.12
Muon identification | 0.05
Imperfections of the simulation: | 0.14
Modelling of FSR | 0.08
Normalisation of individual processes | 0.07
Modelling of the momentum distribution | 0.06
Tag side modelling | 0.05
$\pi^{0}$ efficiency | 0.02
Particle decay-in-flight | 0.02
Tracking efficiency | 0.01
Modelling of ISR | 0.01
Photon efficiency | $<0.01$
Photon energy | $<0.01$
Detector misalignment | $<0.01$
Momentum correction | $<0.01$
Trigger | 0.10
Size of the simulated samples | 0.06
Luminosity | 0.01
Total | 0.37
Table 1: Fractional systematic uncertainty on $R_{\mu}$, split into the
contributing sources. The total fractional systematic uncertainty is
$0.37\text{\,}\mathrm{\char 37\relax}$, translating into an absolute
uncertainty of $0.0036$.

###  5.1 Charged-particle identification

Systematic uncertainties associated with charged-particle identification are
obtained from data-driven corrections to the lepton identification
efficiencies obtained in simulation and particle misidentification rates. The
corrections and their uncertainties are summarised in Table 2. The
uncertainties of the correction factors have statistical and systematic
components. The statistical uncertainties are dominant for the muon
identification corrections and are included in the fit model as fully
independent across all correction bins. Systematic components dominate the
uncertainties of the electron identification corrections. The systematic
uncertainties are assumed to be fully correlated across the correction bins,
which is demonstrated in simplified simulated experiments to be the most
conservative approach. If the uncertainty is asymmetric, the larger of the
signed variations is considered for both. The resulting correlations between
the individual template bins are shown in Figure 4. The efficiency correction
uncertainties are applied to the templates with true leptons of the same type.
Meanwhile, misidentification rate uncertainties are applied to the templates
with the fake leptons. Since the corrections are binned in steps of
$0.5\text{\,}\text{$\mathrm{Ge\kern-1.00006ptV}$}$ in momentum, only 7
parameters are required to model the uncertainties associated with
efficiencies and misidentification rates, respectively. This results in a
total of 28 nuisance parameters. The relative uncertainty on $R_{\mu}$
associated with charged-particle identification is $0.32\text{\,}\mathrm{\char
37\relax}$, shared between the muon and electron identification and
misidentification uncertainties.

| Correction | Average | Uncertainty | Average
---|---|---|---|---
| range | correction | range [%] | uncertainty [%]
Electron identification | $0.99\text{\,}1.01\text{\,}$ | $1.00$ | $0.05\text{\,}1.35\text{\,}$ | $0.22$
Muon identification | $0.87\text{\,}0.99\text{\,}$ | $0.97$ | $0.10\text{\,}6.00\text{\,}$ | $0.38$
Electron misidentification | $1.00\text{\,}13.0\text{\,}$ | $2.61$ | $3.54\text{\,}162\text{\,}$ | $41.4$
Muon misidentification | $0.25\text{\,}1.10\text{\,}$ | $0.82$ | $3.11\text{\,}76.9\text{\,}$ | $11.7$
Table 2: Ranges and average values for the correction factors and associated
total uncertainties of the lepton identification efficiencies, and the rates
of pions misidentified as leptons. The uncertainties are given relative to the
correction factors. Figure 4: Correlations between the correction
uncertainties of the lepton efficiencies ($\mu^{\pm}\rightarrow\mu^{\pm}$,
$e^{\pm}\rightarrow e^{\pm}$) and misidentification rates
($h^{\pm}\rightarrow\mu^{\pm}$, $h^{\pm}\rightarrow e^{\pm}$). Each block for
every category shows the correlation between the individual momentum bins.

###  5.2 Imperfections of the simulation

Any mismodelling of the simulation that affects both the electron and the muon
momentum distributions equally would cancel in measuring $R_{\mu}$. We examine
potential simulation mismodellings that could impact the momentum
distributions of electrons and muons in $\tau^{-}\rightarrow
e^{-}\bar{\nu}_{e}\nu_{\tau}$ and
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ decays or one of the
backgrounds differently. Specifically, modelling of initial and final state
radiation (ISR and FSR), and the reconstruction efficiency of radiated photons
may differ in simulated samples and data. We modify the number of events with
an ISR photon by varying them from $5\text{\,}\mathrm{\char
37\relax}20\text{\,}\mathrm{\char 37\relax}$. The resulting effect on
$R_{\mu}$ is approximately $0.01\text{\,}\mathrm{\char 37\relax}$ for all
variations, since ISR largely cancels because it has a similar effect on all
templates. This is not true for FSR, which also includes bremsstrahlung
photons, since the effect on the momentum of the electron in
$\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau}$ is expected to be larger
than that of the muon in
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$. To account for the
possibility of misestimating the number of FSR photons in the simulation
compared to data, we randomly select $5\text{\,}\mathrm{\char 37\relax}$ of
the generated FSR photons in simulation and add their momenta to the momentum
of the corresponding lepton. This fraction of added FSR momenta is based on a
study with $J/\psi\rightarrow e^{+}e^{-}$ and
$J/\psi\rightarrow\mu^{+}\mu^{-}$ samples, with the second as a control mode
not sensitive to radiative effects in the final state. Agreement within
$5\text{\,}\mathrm{\char 37\relax}$ is observed between simulations and data
for the control samples in the distribution of the energy recovered as FSR
from electrons. The resulting template variations of the lepton momentum
distributions are symmetrised to also account for the possible underestimation
of the number of FSR photons in the simulation. The systematic uncertainty
arising from the potential mismodelling of FSR propagated to $R_{\mu}$ is
estimated to be $0.08\text{\,}\mathrm{\char 37\relax}$. Systematic effects may
emerge due to the differences in the normalisations of the underlying physics
processes. Here, we independently consider variations in the yields of
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events with a misidentified signal
side, with a misidentified tag side, as well as the yields of non-taupair
background processes. In addition, we vary the total
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ yield given the uncertainty of the
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ cross section and the uncertainties of
the tag side branching fractions. We assess the effect of these normalisation
differences by recomputing the templates with the varied yields. The resulting
variations are included in the model, where each independent variation is
parametrised by one nuisance parameter. The corresponding systematic
uncertainty on $R_{\mu}$ is $0.07\text{\,}\mathrm{\char 37\relax}$. To account
for possible simulation mismodelling that could alter the shape of the
templates, we add uncertainties to each template bin that are fully correlated
across all six templates but only affect one momentum bin at a time, resulting
in $21$ nuisance parameters. The size of this uncertainty is estimated in a
data-driven manner by analysing the spread of the ratio between data and
simulations of a sample without any particle identification requirement
imposed on the signal side. We use the standard deviation of this ratio,
computed in the $21$ momentum bins of the templates, which amounts to
$0.84\text{\,}\mathrm{\char 37\relax}$. This results in an overall effect on
$R_{\mu}$ of $0.06\text{\,}\mathrm{\char 37\relax}$. Using an _embedding_
technique on the combined sample of $\tau^{-}\rightarrow
e^{-}\bar{\nu}_{e}\nu_{\tau}$ and
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ events, both in
simulations and in data, we study possible simulation mismodellings of the tag
side that might affect the event selection. We select the electron or muon
candidate in each event and replace it with a randomly selected lepton from
the simulated $\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau}$ or
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ events. To ensure a
reasonable match, we perform these replacements in quintiles of the lepton
momentum, selecting charged particles with momentum magnitudes similar to the
original lepton. We then adjust the momentum direction of the replaced
particle to align with the original momentum direction. Additionally, all
variables affected by the replaced lepton momentum are recalculated
accordingly. The observed differences of the resulting data-over-simulation
ratios between the embedded $\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau}$
and $\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ events are around
$0.2\text{\,}\mathrm{\char 37\relax}$, depending on the momentum of the lepton
candidate. Since the same $\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau}$
or $\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ events are embedded
in data and simulated samples, any differences are expected to arise solely
from the tag side. These differences are included in the fit model as fully
uncorrelated variations of the template bins with $42$ nuisance parameters and
contribute a $0.05\text{\,}\mathrm{\char 37\relax}$ uncertainty on $R_{\mu}$.
The correction of neutral pion reconstruction efficiency, to account for
differences between simulation and data, is determined as a function of the
pion momentum from auxiliary measurements in calibration channels of $D$ and
$\tau$ decays. The impact of the uncertainty of this correction on $R_{\mu}$
is estimated to be $0.02\text{\,}\mathrm{\char 37\relax}$. To estimate the
impact of mismodelled charged-particle decay-in-flight rates in simulations,
we modify the number of tracks reconstructed from the decay products of other
particles, primarily pions, by $5\text{\,}\mathrm{\char 37\relax}$. The
mismodelling of tag side pion decays affects the muon and electron templates
similarly, leading to a cancellation of the systematic effects. However, the
signal side muon templates would be affected significantly by the modification
in pion decay rates, and the effect does not cancel out. The associated
systematic uncertainty on $R_{\mu}$ is $0.02\text{\,}\mathrm{\char 37\relax}$.
Differences between the track finding efficiencies in simulation and data have
been measured in $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events with one of
the $\tau$ leptons decaying to three charged hadrons. A per-track systematic
uncertainty of $0.24\text{\,}\mathrm{\char 37\relax}$ is included as a
normalisation uncertainty of the templates to account for these differences.
The associated systematic uncertainty on $R_{\mu}$ is
$0.01\text{\,}\mathrm{\char 37\relax}$. Systematic uncertainties due to
simulation mismodelling of the detection efficiency of photons and their
energies, detector misalignment, and charged-particle momentum correction are
each found to be below $0.01\text{\,}\mathrm{\char 37\relax}$.

###  5.3 Trigger

The template yields in each bin are corrected to account for differences in
trigger efficiency between data and simulation. The average of the correction
factors that are applied to the simulated events is $1.0122$ for the muon
channel and $1.012\,52$ for the electron channel. Several uncertainties are
associated with this correction, summarised in Table 3. They include
statistical uncertainties from the measured efficiency in data and simulation,
and systematic uncertainties associated with the efficiency estimation method.
For the systematic effects, we consider the potential bias introduced if the
CDC triggers used to define the reference sample in which the efficiency is
measured are not completely independent from the ECL triggers. This potential
bias is obtained from simulation by comparing the efficiency calculated in the
reference sample to the absolute efficiency for the entire sample. In
addition, we consider the observed differences in the correction factors when
computed with an alternative reference sample with additional CDC triggers
used to obtain it. The latter is the dominant source of uncertainty. The
uncertainties from the various sources are combined in quadrature for each bin
and included in the fit model with 44 nuisance parameters under the assumption
of independence across momentum bins and between the electron and muon
channels. This assumption is conservative, since most systematic effects
originate from a common source for both channels, which causes a cancellation
of the effect when measuring $R_{\mu}$. In particular, the changes in the
correction factors when computed with alternative reference triggers are
similar for both channels, which indicates a high correlation. However, while
the trigger efficiency is primarily driven by the tag hemisphere, some
independent effects can still arise due to differences in lepton flavour on
the signal hemisphere. This motivates the more conservative choice of
independent uncertainties. The systematic uncertainty propagating to $R_{\mu}$
is determined to be $0.10\text{\,}\mathrm{\char 37\relax}$.

|  $\mu$ channel |  $e$ channel
---|---|---
| Range [%] | Average [%] | Range [%] | Average [%]
Sample size |  $0.040.23\text{\,}$ | $0.05$ |  $0.010.08\text{\,}$ | $0.02$
Absolute efficiency |  $0.000.10\text{\,}$ | $0.07$ |  $-0.110.01\text{\,}$ | $-0.01$
Alternative reference triggers |  $0.090.20\text{\,}$ | $0.15$ |  $0.090.15\text{\,}$ | $0.13$
Table 3: Range and average of the uncertainties associated with the trigger
correction. The first row shows the statistical uncertainties, while the other
two rows show the uncertainties attributed to systematic effects, as mentioned
in the text. The uncertainties are given relative to the correction factors.
The systematic uncertainties are obtained from differences with respect to the
nominal correction, the direction of which is given by the sign.

###  5.4 Size of the simulated samples and luminosity

The templates are allowed to vary due to the limited size of the simulated
samples, which is parametrised by one nuisance parameter per bin. The extent
of this variation is mainly affected by the size of the simulated
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ sample, which is . The resulting
systematic uncertainty on $R_{\mu}$ is $0.06\text{\,}\mathrm{\char 37\relax}$.
Since the uncertainty on the luminosity measurement affects all distributions,
its impact on $R_{\mu}$ largely cancels. It does not entirely vanish since the
overall change in normalisation of the templates can also be related to a
combination of other systematic effects, which in turn can affect $R_{\mu}$.
This is a very small effect and of the order of $0.01\text{\,}\mathrm{\char
37\relax}$.

###  5.5 Consistency checks

We check the stability of the result throughout various data-taking periods
and observe no evidence of time dependence. We also determine that the value
of $R_{\mu}$ is unchanged by variations in the neural network threshold around
the nominal requirement. To exclude a potential dependence of the measured
$R_{\mu}$ on the properties of the electrons and muons, we divide the data
into sub-regions of various variables. Specifically, we use the lepton charge
($q_{\ell}$), momentum, and polar angle. Another split of the sample is
performed using the polar angle of the missing momentum vector
($\theta_{\text{miss}}$). To rule out potential dependence on the particle
identification requirements, we check the stability of the results using
different requirements on the particle identification discriminators $P_{e}$
and $P_{\mu}$. Namely, we use $P_{e}=0.90,0.95,0.99$ and
$P_{\mu}=0.50,0.95,0.99$ and compare the results to those obtained with the
nominal requirements. We also check the stability of the result when changing
the assumed correlation of lepton identification systematic uncertainties
($\rho_{\>\\!\text{LID}}$) and the number of bins used to define the
templates. The outcome of some of these checks is shown in Figure 5, where the
shaded area indicates the statistical uncertainty, while the error cap marks
the systematic uncertainty originating only from lepton identification, and
the error bars show the total uncertainties. We obtain consistent results from
all performed checks, indicating no significant unaccounted-for systematic
effects.

Figure 5: Stability of the fitted value of $R_{\mu}$. Shown are deviations
$\Delta R_{\mu}$ from the nominal value for different sub-regions of selected
variables, for different requirements on the particle identification
discriminators, and for different fit conditions. The measurements in each
sub-region are statistically independent in the first four groups, separated
by dashed vertical lines.

##  6 Results

The distribution of the muon and electron candidate momentum with fit results
overlaid is shown in Figure 6. We measure the ratio of branching fractions

$R_{\mu}=\frac{{\cal
B}(\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau})}{{\cal
B}(\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau})}=0.9675\pm 0.0007\pm
0.0036.$ (6)

The first uncertainty is statistical, while the second uncertainty is
systematic. The statistical uncertainty is determined by performing the fit
with all nuisance parameters fixed to their best-fit values, and the
systematic component is calculated by subtraction in quadrature from the total
uncertainty obtained with the nominal fit. This is consistent with the
systematic uncertainty obtained in Table 1. The dominant source of systematic
uncertainty is lepton identification.

Figure 6:  Observed momentum distribution for muon (left) and electron (right)
candidates with fit results overlaid. The lower panel shows the ratio between
data and fit results. The hatched area indicates the possible variation of the
fitted yields due to systematic effects, with the constraints of the nuisance
parameters reduced to their fit uncertainties and correlations taken into
account.

The measured normalisation yields are $4\,156\,500(99\,650)$ for
$\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau}$ and $4\,000\,190(99\,260)$
for $\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$, with a correlation
of $0.988$. The dominant source of uncertainty in the yield estimates is
related to the $\pi^{0}$ reconstruction efficiency, which only affects the tag
side. The value of $R_{\mu}$ obtained is shown in the left panel of Figure 7
and compared to previous measurements performed by CLEO CLEO:1996oro  and
BaBar BaBar:2009lyd  as well as the global determination from a fit to all
$\tau$ branching fractions HeavyFlavorAveragingGroup:2022wzx . Our result is
consistent with previous measurements and is the most precise measurement from
a single experiment to date. Using Equation 2, we translate the measured
$R_{\mu}$ value into the most stringent test of LFU in $\tau$-lepton decays
from a single experiment, obtaining $|g_{\mu}/g_{e}|_{\tau}=0.9974\pm 0.0019$
(see the right panel of Figure 7).

Figure 7: Determinations of $R_{\mu}$ (left) and $|g_{\mu}/g_{e}|_{\tau}$
(right) from previous individual measurements CLEO:1996oro ; BaBar:2009lyd
and the fit from the Heavy Flavor Averaging Group
HeavyFlavorAveragingGroup:2022wzx , compared with the result of this work. The
shaded areas represent the statistical uncertainties, while the error bars
indicate the total uncertainties. The vertical dashed line indicates the SM
prediction, including mass effects.

##  7 Summary

We report a test of light-lepton universality in leptonic $\tau$ decays using
a sample of data collected by the Belle II detector at the SuperKEKB
$e^{+}e^{-}$ collider at a centre-of-mass energy of
$10.58\text{\,}\text{$\mathrm{Ge\kern-1.00006ptV}$}$. Our result is currently
the world’s most precise test of light-lepton universality in $\tau$ decays
performed by a single experiment and is consistent with the SM.

###### Acknowledgements.

This work, based on data collected using the Belle II detector, which was
built and commissioned prior to March 2019, was supported by Higher Education
and Science Committee of the Republic of Armenia Grant No. 23LCG-1C011;
Australian Research Council and Research Grants No. DP200101792, No.
DP210101900, No. DP210102831, No. DE220100462, No. LE210100098, and No.
LE230100085; Austrian Federal Ministry of Education, Science and Research,
Austrian Science Fund No. P 34529, No. J 4731, No. J 4625, and No. M 3153, and
Horizon 2020 ERC Starting Grant No. 947006 “InterLeptons”; Natural Sciences
and Engineering Research Council of Canada, Compute Canada and CANARIE;
National Key R&D Program of China under Contract No. 2022YFA1601903, National
Natural Science Foundation of China and Research Grants No. 11575017, No.
11761141009, No. 11705209, No. 11975076, No. 12135005, No. 12150004, No.
12161141008, and No. 12175041, and Shandong Provincial Natural Science
Foundation Project ZR2022JQ02; the Czech Science Foundation Grant No.
22-18469S and Charles University Grant Agency project No. 246122; European
Research Council, Seventh Framework PIEF-GA-2013-622527, Horizon 2020 ERC-
Advanced Grants No. 267104 and No. 884719, Horizon 2020 ERC-Consolidator Grant
No. 819127, Horizon 2020 Marie Sklodowska-Curie Grant Agreement No. 700525
“NIOBE” and No. 101026516, and Horizon 2020 Marie Sklodowska-Curie RISE
project JENNIFER2 Grant Agreement No. 822070 (European grants); L’Institut
National de Physique Nucléaire et de Physique des Particules (IN2P3) du CNRS
and L’Agence Nationale de la Recherche (ANR) under grant ANR-21-CE31-0009
(France); BMBF, DFG, HGF, MPG, and AvH Foundation (Germany); Department of
Atomic Energy under Project Identification No. RTI 4002, Department of Science
and Technology, and UPES SEED funding programs No. UPES/R&D-SEED-
INFRA/17052023/01 and No. UPES/R&D-SOE/20062022/06 (India); Israel Science
Foundation Grant No. 2476/17, U.S.-Israel Binational Science Foundation Grant
No. 2016113, and Israel Ministry of Science Grant No. 3-16543; Istituto
Nazionale di Fisica Nucleare and the Research Grants BELLE2; Japan Society for
the Promotion of Science, Grant-in-Aid for Scientific Research Grants No.
16H03968, No. 16H03993, No. 16H06492, No. 16K05323, No. 17H01133, No.
17H05405, No. 18K03621, No. 18H03710, No. 18H05226, No. 19H00682, No.
20H05850, No. 20H05858, No. 22H00144, No. 22K14056, No. 22K21347, No.
23H05433, No. 26220706, and No. 26400255, and the Ministry of Education,
Culture, Sports, Science, and Technology (MEXT) of Japan; National Research
Foundation (NRF) of Korea Grants No. 2016R1D1A1B02012900, No.
2018R1A2B3003643, No. 2018R1A6A1A06024970, No. 2019R1I1A3A01058933, No.
2021R1A6A1A03043957, No. 2021R1F1A1060423, No. 2021R1F1A1064008, No.
2022R1A2C1003993, and No. RS-2022-00197659, Radiation Science Research
Institute, Foreign Large-Size Research Facility Application Supporting
project, the Global Science Experimental Data Hub Center of the Korea
Institute of Science and Technology Information and KREONET/GLORIAD;
Universiti Malaya RU grant, Akademi Sains Malaysia, and Ministry of Education
Malaysia; Frontiers of Science Program Contracts No. FOINS-296, No. CB-221329,
No. CB-236394, No. CB-254409, and No. CB-180023, and SEP-CINVESTAV Research
Grant No. 237 (Mexico); the Polish Ministry of Science and Higher Education
and the National Science Center; the Ministry of Science and Higher Education
of the Russian Federation and the HSE University Basic Research Program,
Moscow; University of Tabuk Research Grants No. S-0256-1438 and No.
S-0280-1439 (Saudi Arabia); Slovenian Research Agency and Research Grants No.
J1-9124 and No. P1-0135; Agencia Estatal de Investigacion, Spain Grant No.
RYC2020-029875-I and Generalitat Valenciana, Spain Grant No.
CIDEGENT/2018/020; National Science and Technology Council, and Ministry of
Education (Taiwan); Thailand Center of Excellence in Physics; TUBITAK ULAKBIM
(Turkey); National Research Foundation of Ukraine, Project No. 2020.02/0257,
and Ministry of Education and Science of Ukraine; the U.S. National Science
Foundation and Research Grants No. PHY-1913789 and No. PHY-2111604, and the
U.S. Department of Energy and Research Awards No. DE-AC06-76RLO1830, No. DE-
SC0007983, No. DE-SC0009824, No. DE-SC0009973, No. DE-SC0010007, No. DE-
SC0010073, No. DE-SC0010118, No. DE-SC0010504, No. DE-SC0011784, No. DE-
SC0012704, No. DE-SC0019230, No. DE-SC0021274, No. DE-SC0021616, No. DE-
SC0022350, No. DE-SC0023470; and the Vietnam Academy of Science and Technology
(VAST) under Grants No. NVCC.05.12/22-23 and No. DL0000.02/24-25. These
acknowledgements are not to be interpreted as an endorsement of any statement
made by any of our institutes, funding agencies, governments, or their
representatives. We thank the SuperKEKB team for delivering high-luminosity
collisions; the KEK cryogenics group for the efficient operation of the
detector solenoid magnet; the KEK Computer Research Center for on-site
computing support; the NII for SINET6 network support; and the raw-data
centers hosted by BNL, DESY, GridKa, IN2P3, INFN, and the University of
Victoria.

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