The Belle II Collaboration

#  Search for the baryon number and lepton number violating decays
$\tau^{-}\to\Lambda\pi^{-}$ and $\tau^{-}\to\mathrm{bar}{\Lambda}\pi^{-}$ at
Belle II

I. Adachi  0000-0003-2287-0173 L. Aggarwal  0000-0002-0909-7537 H. Ahmed
0000-0003-3976-7498 H. Aihara  0000-0002-1907-5964 N. Akopov
0000-0002-4425-2096 A. Aloisio  0000-0002-3883-6693 N. Althubiti
0000-0003-1513-0409 N. Anh Ky  0000-0003-0471-197X D. M. Asner
0000-0002-1586-5790 H. Atmacan  0000-0003-2435-501X T. Aushev
0000-0002-6347-7055 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 B. Bhuyan
0000-0001-6254-3594 F. Bianchi  0000-0002-1524-6236 L. Bierwirth
0009-0003-0192-9073 T. Bilka  0000-0003-1449-6986 D. Biswas
0000-0002-7543-3471 A. Bobrov  0000-0001-5735-8386 D. Bodrov
0000-0001-5279-4787 J. Borah  0000-0003-2990-1913 A. Boschetti
0000-0001-6030-3087 A. Bozek  0000-0002-5915-1319 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 Q. Campagna
0000-0002-3109-2046 M. Campajola  0000-0003-2518-7134 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 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 J. Cochran  0000-0002-1492-914X L. Corona
0000-0002-2577-9909 J. X. Cui  0000-0002-2398-3754 E. De La Cruz-Burelo
0000-0002-7469-6974 S. A. De La Motte  0000-0003-3905-6805 G. De Nardo
0000-0002-2047-9675 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 K. Dort  0000-0003-0849-8774 D. Dossett
0000-0002-5670-5582 S. Dubey  0000-0002-1345-0970 G. Dujany
0000-0002-1345-8163 P. Ecker  0000-0002-6817-6868 D. Epifanov
0000-0001-8656-2693 J. Eppelt  0000-0001-8368-3721 P. Feichtinger
0000-0003-3966-7497 T. Ferber  0000-0002-6849-0427 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 P. Gironella
0000-0001-5603-4750 A. Glazov  0000-0002-8553-7338 B. Gobbo
0000-0002-3147-4562 R. Godang  0000-0002-8317-0579 P. Goldenzweig
0000-0001-8785-847X W. Gradl  0000-0002-9974-8320 E. Graziani
0000-0001-8602-5652 D. Greenwald  0000-0001-6964-8399 Z. Gruberová
0000-0002-5691-1044 K. Gudkova  0000-0002-5858-3187 I. Haide
0000-0003-0962-6344 S. Halder  0000-0002-6280-494X K. Hara
0000-0002-5361-1871 C. Harris  0000-0003-0448-4244 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 R. Hoppe
0009-0005-8881-8935 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 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 J. Kandra
0000-0001-5635-1000 K. H. Kang  0000-0002-6816-0751 G. Karyan
0000-0001-5365-3716 T. Kawasaki  0000-0002-4089-5238 F. Keil
0000-0002-7278-2860 C. Kiesling  0000-0002-2209-535X D. Y. Kim
0000-0001-8125-9070 J.-Y. Kim  0000-0001-7593-843X K.-H. Kim
0000-0002-4659-1112 Y.-K. Kim  0000-0002-9695-8103 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 A. Korobov
0000-0001-5959-8172 S. Korpar  0000-0003-0971-0968 E. Kovalenko
0000-0001-8084-1931 R. Kowalewski  0000-0002-7314-0990 P. Križan
0000-0002-4967-7675 P. Krokovny  0000-0002-1236-4667 T. Kuhr
0000-0001-6251-8049 R. Kumar  0000-0002-6277-2626 K. Kumara
0000-0003-1572-5365 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 M. J. Lee
0000-0003-4528-4601 P. Leo  0000-0003-3833-2900 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 W. Z. Li  0009-0002-8040-2546 Y. Li  0000-0002-4413-6247
Y. B. Li  0000-0002-9909-2851 J. Libby  0000-0002-1219-3247 J. Lin
0000-0002-3653-2899 M. H. Liu  0000-0002-9376-1487 Q. Y. Liu
0000-0002-7684-0415 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
0000-0003-1903-3251 C. Martellini  0000-0002-7189-8343 A. Martens
0000-0003-1544-4053 A. Martini  0000-0003-1161-4983 T. Martinov
0000-0001-7846-1913 L. Massaccesi  0000-0003-1762-4699 M. Masuda
0000-0002-7109-5583 T. Matsuda  0000-0003-4673-570X 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 C. Miller
0000-0003-2631-1790 M. Mirra  0000-0002-1190-2961 S. Mitra
0000-0002-1118-6344 S. Mondal  0000-0002-3054-8400 S. Moneta
0000-0003-2184-7510 H.-G. Moser  0000-0003-3579-9951 R. Mussa
0000-0002-0294-9071 I. Nakamura  0000-0002-7640-5456 M. Nakao
0000-0001-8424-7075 Y. Nakazawa  0000-0002-6271-5808 M. Naruki
0000-0003-1773-2999 D. Narwal  0000-0001-6585-7767 Z. Natkaniec
0000-0003-0486-9291 A. Natochii  0000-0002-1076-814X M. Nayak
0000-0002-2572-4692 G. Nazaryan  0000-0002-9434-6197 M. Neu
0000-0002-4564-8009 C. Niebuhr  0000-0002-4375-9741 S. Nishida
0000-0001-6373-2346 S. Ogawa  0000-0002-7310-5079 H. Ono  0000-0003-4486-0064
P. Pakhlov  0000-0001-7426-4824 E. Paoloni  0000-0001-5969-8712 S. Pardi
0000-0001-7994-0537 J. Park  0000-0001-6520-0028 K. Park  0000-0003-0567-3493
S.-H. Park  0000-0001-6019-6218 B. Paschen  0000-0003-1546-4548 A. Passeri
0000-0003-4864-3411 S. Patra  0000-0002-4114-1091 T. K. Pedlar
0000-0001-9839-7373 R. Peschke  0000-0002-2529-8515 R. Pestotnik
0000-0003-1804-9470 M. Piccolo  0000-0001-9750-0551 L. E. Piilonen
0000-0001-6836-0748 P. L. M. Podesta-Lerma  0000-0002-8152-9605 T. Podobnik
0000-0002-6131-819X S. Pokharel  0000-0002-3367-738X C. Praz
0000-0002-6154-885X S. Prell  0000-0002-0195-8005 E. Prencipe
0000-0002-9465-2493 M. T. Prim  0000-0002-1407-7450 H. Purwar
0000-0002-3876-7069 G. Raeuber  0000-0003-2948-5155 S. Raiz
0000-0001-7010-8066 N. Rauls  0000-0002-6583-4888 M. Reif  0000-0002-0706-0247
S. Reiter  0000-0002-6542-9954 M. Remnev  0000-0001-6975-1724 L. Reuter
0000-0002-5930-6237 I. Ripp-Baudot  0000-0002-1897-8272 G. Rizzo
0000-0003-1788-2866 J. M. Roney  0000-0001-7802-4617 N. Rout
0000-0002-4310-3638 S. Sandilya  0000-0002-4199-4369 L. Santelj
0000-0003-3904-2956 V. Savinov  0000-0002-9184-2830 B. Scavino
0000-0003-1771-9161 M. Schnepf  0000-0003-0623-0184 C. Schwanda
0000-0003-4844-5028 Y. Seino  0000-0002-8378-4255 A. Selce
0000-0001-8228-9781 K. Senyo  0000-0002-1615-9118 J. Serrano
0000-0003-2489-7812 M. E. Sevior  0000-0002-4824-101X C. Sfienti
0000-0002-5921-8819 W. Shan  0000-0003-2811-2218 C. Sharma
0000-0002-1312-0429 C. P. Shen  0000-0002-9012-4618 X. D. Shi
0000-0002-7006-6107 T. Shillington  0000-0003-3862-4380 T. Shimasaki
0000-0003-3291-9532 J.-G. Shiu  0000-0002-8478-5639 D. Shtol
0000-0002-0622-6065 B. Shwartz  0000-0002-1456-1496 A. Sibidanov
0000-0001-8805-4895 F. Simon  0000-0002-5978-0289 J. B. Singh
0000-0001-9029-2462 J. Skorupa  0000-0002-8566-621X R. J. Sobie
0000-0001-7430-7599 M. Sobotzik  0000-0002-1773-5455 A. Soffer
0000-0002-0749-2146 A. Sokolov  0000-0002-9420-0091 E. Solovieva
0000-0002-5735-4059 W. Song  0000-0003-1376-2293 S. Spataro
0000-0001-9601-405X B. Spruck  0000-0002-3060-2729 M. Starič
0000-0001-8751-5944 P. Stavroulakis  0000-0001-9914-7261 S. Stefkova
0000-0003-2628-530X R. Stroili  0000-0002-3453-142X Y. Sue
0000-0003-2430-8707 M. Sumihama  0000-0002-8954-0585 K. Sumisawa
0000-0001-7003-7210 W. Sutcliffe  0000-0002-9795-3582 N. Suwonjandee
0009-0000-2819-5020 H. Svidras  0000-0003-4198-2517 M. Takahashi
0000-0003-1171-5960 M. Takizawa  0000-0001-8225-3973 U. Tamponi
0000-0001-6651-0706 K. Tanida  0000-0002-8255-3746 F. Tenchini
0000-0003-3469-9377 O. Tittel  0000-0001-9128-6240 R. Tiwary
0000-0002-5887-1883 D. Tonelli  0000-0002-1494-7882 E. Torassa
0000-0003-2321-0599 K. Trabelsi  0000-0001-6567-3036 I. Ueda
0000-0002-6833-4344 K. Unger  0000-0001-7378-6671 Y. Unno  0000-0003-3355-765X
K. Uno  0000-0002-2209-8198 S. Uno  0000-0002-3401-0480 P. Urquijo
0000-0002-0887-7953 S. E. Vahsen  0000-0003-1685-9824 R. van Tonder
0000-0002-7448-4816 K. E. Varvell  0000-0003-1017-1295 M. Veronesi
0000-0002-1916-3884 V. S. Vismaya  0000-0002-1606-5349 L. Vitale
0000-0003-3354-2300 V. Vobbilisetti  0000-0002-4399-5082 R. Volpe
0000-0003-1782-2978 M. Wakai  0000-0003-2818-3155 S. Wallner
0000-0002-9105-1625 E. Wang  0000-0001-6391-5118 M.-Z. Wang
0000-0002-0979-8341 Z. Wang  0000-0002-3536-4950 A. Warburton
0000-0002-2298-7315 S. Watanuki  0000-0002-5241-6628 C. Wessel
0000-0003-0959-4784 E. Won  0000-0002-4245-7442 X. P. Xu  0000-0001-5096-1182
B. D. Yabsley  0000-0002-2680-0474 S. Yamada  0000-0002-8858-9336 W. Yan
0000-0003-0713-0871 S. B. Yang  0000-0002-9543-7971 J. Yelton
0000-0001-8840-3346 J. H. Yin  0000-0002-1479-9349 K. Yoshihara
0000-0002-3656-2326 C. Z. Yuan  0000-0002-1652-6686 L. Zani
0000-0003-4957-805X B. Zhang  0000-0002-5065-8762 J. S. Zhou
0000-0002-6413-4687 Q. D. Zhou  0000-0001-5968-6359 V. I. Zhukova
0000-0002-8253-641X R. Žlebčík  0000-0003-1644-8523

###### Abstract

We present a search for the baryon number $B$ and lepton number $L$ violating
decays $\tau^{-}\rightarrow\Lambda\pi^{-}$ and
$\tau^{-}\rightarrow\mathrm{bar}{\Lambda}\pi^{-}$ produced from the
$e^{+}e^{-}\to\tau^{+}\tau^{-}$ process, using a 364 fb-1 data sample
collected by the Belle II experiment at the SuperKEKB collider. No evidence of
signal is found in either decay mode, which have $|\Delta(B-L)|$ equal to $2$
and $0$, respectively. Upper limits at 90% credibility level on the branching
fractions of $\tau^{-}\rightarrow\Lambda\pi^{-}$ and
$\tau^{-}\rightarrow\mathrm{bar}{\Lambda}\pi^{-}$ are determined to be
$4.7\times 10^{-8}$ and $4.3\times 10^{-8}$, respectively.

††preprint:

Baryon number violation (BNV) is necessary to explain a dynamical generation
of the asymmetry of matter and antimatter in the universe Sakharov:1967dj ;
Weinberg:1981wj ; Rubakov:1996vz ; Morrissey:2012db . For standard model (SM)
processes, at perturbative level, lepton number $L$ and baryon number $B$ are
conserved. However, nonperturbative effects, such as sphaleron processes,
could lead to violations of $B$ and $L$ separately at high temperatures, while
conserving their difference $B-L$ Phong:2020ybr ; Papaefstathiou:2019djz ;
Zhou:2019uzq ; Ho:2020ltr . Several beyond-the-SM theories predict BNV, such
as supersymmetry models with R-parity violation Sjostrand:2002ip , a black
hole model DeLuca:2021oer , superstring models Lazarides:1986th , and grand
unification models deGouvea:2014lva . Most of the models require $B-L$
conservation, while $|\Delta(B-L)|=2$ is allowed in some scenarios
deGouvea:2014lva ; Kamyshkov:1999yi ; Wilczek:1979et . Thus, BNV processes can
be accompanied by lepton number violation. Discoveries of such BNV processes
would reveal the existence of physics beyond the SM and shed light on matter-
antimatter asymmetry.

Over the last few decades, several experiments have searched for BNV, but no
evidence has been found PDG . Proton decays have been extensively studied by
Super-Kamiokande Super-Kamiokande:2009yit , with lower limits on the lifetime
of the proton on the order of $10^{33}$ years at the 90% confidence level,
while baryon-number-violating decays of charmed and bottom hadrons, $\tau$
leptons, and $Z$ bosons were studied by the CLEO CLEO:2009apb ; CLEO:1999emi ,
BaBar BaBar:2011yks ; BaBar:2011ouc , OPAL OPAL:1998gbn , Belle Belle:2005exq
; Belle:2023mao , and LHCb Collaborations LHCb:2013fsr , with upper limits on
the branching fractions in the range $10^{-8}-10^{-5}$ at the 90% confidence
level.

We use a sample of $e^{+}e^{-}$ collisions collected at a center-of-mass
(c.m.) energy $\sqrt{s}$ of 10.58 GeV corresponding to the mass of the
$\Upsilon(4S)$ resonance to search for BNV decays
$\tau^{-}\rightarrow\Lambda\pi^{-}$ and
$\tau^{-}\to\mathrm{bar}{\Lambda}\pi^{-}$. The data, recorded by the Belle II
detector Belle-II:2010dht  operating at the SuperKEKB AKAI2018188  asymmetric-
energy collider, have an integrated luminosity $\mathcal{L}$ of ($364\pm 2$)
fb-1 Lum . Searches for $\tau^{-}\rightarrow\Lambda\pi^{-}$ and
$\mathrm{bar}{\Lambda}\pi^{-}$ were performed by Belle Belle:2005exq  using a
154 fb-1 data sample yielding upper limits of $0.72\times 10^{-7}$ and
$1.4\times 10^{-7}$, respectively, at the 90% confidence level. With an
upgraded detector and a larger data sample, the Belle II experiment can
investigate these decays with improved sensitivity.

We select $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events in which one of the
two $\tau$ leptons decays into three charged particles (3-prong) and the other
into one charged particle (1-prong) and one or more neutrinos. We search for a
signal on the 3-prong (signal) side via the decay $\tau^{-}\to\Lambda(\to
p\pi^{-})\pi^{-}$ or
$\tau^{-}\to\mathrm{bar}{\Lambda}(\to\mathrm{bar}{p}\pi^{+})\pi^{-}$, while we
require that the other $\tau$ (tag) decays into
$e^{+}\nu_{e}\mathrm{bar}{\nu}_{\tau}$,
$\mu^{+}\nu_{\mu}\mathrm{bar}{\nu}_{\tau}$, $\pi^{+}\mathrm{bar}{\nu}_{\tau}$,
or $\pi^{+}\pi^{0}\mathrm{bar}{\nu}_{\tau}$ final states. Inclusion of charge-
conjugate states is always implied. A search region is defined in a plane
formed by the energy difference $\Delta E=E(\Lambda\pi)-\sqrt{s}/2$ and the
invariant mass $M(\Lambda\pi)$, where $\Lambda$ means $\Lambda$ or
$\mathrm{bar}{\Lambda}$, and signal is identified as an excess over background
expectations.

In the following, all variables are defined in the $e^{+}e^{-}$ c.m. system
unless otherwise specified. The analysis is optimized using simulated events,
and data events outside the signal region, before examining data in the signal
region.

The Belle II detector has a cylindrical geometry with the axis of symmetry
along the beamline ($z$ axis) Belle-II:2010dht . The innermost component is a
two-layer silicon-pixel detector surrounded by a four-layer double-sided
silicon-strip detector and a 56-layer central drift chamber (CDC). These
detectors reconstruct trajectories of charged particle (tracks). Only one
sixth of the second layer of the silicon-pixel detector was installed for the
data analyzed here. Surrounding the CDC, which also provides d$E$/d$x$ energy-
loss measurements, is a time-of-propagation detector in the central region and
an aerogel-based ring-imaging Cherenkov detector in the forward region. These
detectors provide particle identification (PID) for charged hadrons.
Surrounding them is an electromagnetic calorimeter (ECL) composed of CsI(Tl)
crystals that primarily provides energy and timing measurements for photons
and electrons. Outside of the ECL is a superconducting solenoid magnet. Its
flux return is instrumented with resistive-plate chambers and plastic
scintillator modules to detect muons, $K^{0}_{L}$ mesons, and neutrons. The
solenoid magnet provides a 1.5 T magnetic field parallel to the $z$ axis. The
longitudinal direction and the polar angle $\theta$ are defined with respect
to the $z$ axis, whose positive direction is that of the electron beam.

Simulated samples are used to estimate signal efficiencies and the number of
expected background events Zhou:2020ksj . We use the KKMC software package to
generate $2\times 10^{7}$ $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}(\gamma)$
signal events JADACH2000260 . The subsequent $\tau$ decays are simulated by
the TAUOLA software package JADACH1991275 , with one $\tau$ decaying to
$\Lambda\pi$ or $\mathrm{bar}{\Lambda}\pi$ according to a phase-space model
and the other decaying according to known branching fractions. PDG ;
JADACH2000260 . We use KKMC and TAUOLA also to simulate backgrounds. The KKMC
generator is also used for $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}(\gamma)$ and
$q\mathrm{bar}{q}$ processes, where $q$ indicates $u$, $d$, $s$, or $c$
quarks, with the $q\mathrm{bar}{q}$ pair fragmentation being simulated by the
PYTHIA8 software package SJOSTRAND2015159 . The PYTHIA8 and EvtGen
Lange:2001uf  software packages are used to simulate the $e^{+}e^{-}\to
b\mathrm{bar}{b}$ process. The BabaYaga@NLO software package is used to
simulate the $e^{+}e^{-}\rightarrow e^{+}e^{-}(\gamma)$ processes
Balossini:2006wc ; Balossini:2008xr ; CarloniCalame:2003yt ;
CarloniCalame:2001ny ; CarloniCalame:2000pz ; the AAFH Berends:1984ge ;
Berends:1984gf ; Berends:1986ig  and TREPS Uehara:1996bgt  software packages
are used for simulations of
$e^{+}e^{-}\rightarrow\ell^{+}\ell^{-}\ell^{+}\ell^{-}$ and
$e^{+}e^{-}h^{+}h^{-}$ processes, respectively, where $\ell$ indicates an
electron, muon, or $\tau$ lepton, and $h$ indicates a pion, kaon, or proton.
The size of each simulated sample is four times the size of the corresponding
component in data, except for the
$e^{+}e^{-}\to\ell^{+}\ell^{-}\ell^{+}\ell^{-}$ and $e^{+}e^{-}h^{+}h^{-}$
samples, which are the same size as expected in data, and the $e^{+}e^{-}\to
e^{+}e^{-}(\gamma)$ sample, which is 10% of that expected in data. EvtGen is
used to simulate the decay of hadrons Lange:2001uf . The PHOTOS software
package is used for the simulation of final state radiation Barberio:1990ms .
The Belle II software Kuhr:2018lps  uses the Geant4 GEANT4:2002zbu  software
package to simulate the response of the detector to the interactions of
particles.

The hardware trigger relies on energy deposits (clusters with energy larger
than 100 MeV) and their topologies in the ECL, and on the number of
reconstructed tracks in the CDC. Events are required to satisfy one of the
following criteria: three clusters with a topology inconsistent with a Bhabha
process, where at least one cluster has an energy greater than 500 MeV; the
sum of clusters exceeding 1 GeV; at least one charged particle with a momentum
greater than 0.7 GeV/$c$. The trigger efficiency for the
$\tau^{-}\to\Lambda\pi^{-}$ $(\tau^{-}\to\mathrm{bar}{\Lambda}\pi^{-})$
channel, estimated on simulated signal samples, is $99.5\%$ $(99.6\%)$.

Both charged and neutral particles are required to be within the acceptance of
the CDC, i.e., $-0.866<\cos\theta<0.956$. The transverse momentum of each
charged particle is required to exceed 0.1 GeV/$c$. Photons are identified as
clusters with energies greater than 0.1 GeV not associated to any track. The
thrust axis $\hat{t}$ is defined such that the value

$T=\frac{\sum_{i}\left|\vec{p}\,_{i}\cdot\hat{t}\right|}{\sum_{i}\left|\vec{p}\,_{i}\right|}$
(1)

is maximized. Here, $\vec{p}\,_{i}$ is the momentum of the $i$th particle and
the sum runs over all tracks and clusters. Events are geometrically split into
two opposite hemispheres with a plane perpendicular to the thrust axis. The
hemispheres corresponding to the signal and tag sides are required to contain
exactly three tracks and one track, respectively. In the following, we refer
to the pion from the $\Lambda$ or $\mathrm{bar}{\Lambda}$ decays as
$\pi_{\Lambda}$, the pion from the signal $\tau$ decays as $\pi_{\tau}$, and
the pion from the tag $\tau$ decays as $\pi_{\rm tag}$. We require that the
numbers of photons $N_{\gamma}^{\text{sig}}$ and $N_{\gamma}^{\text{tag}}$ on
the signal and tag sides each be less than four, and that the energy
$E_{\gamma}^{\text{sig}}$ of the most energetic photon on the signal side be
less than 1.0 GeV. These requirements take into account the possibility of
photons radiated from the initial state and photons from $\pi^{0}$ decays on
the tag side.

A signal region is defined in the plane of the $\Delta E$ versus
$M(\Lambda\pi)$, in which signal is expected to peak at $\Delta E\approx$ 0.0
and at $M(\Lambda\pi)\approx$ 1.78 GeV/$c^{2}$. The signal region, shown in
Fig. 1, has an elliptical shape whose size is defined so as to include 90% of
all reconstructed signal events. The distribution of signal in the $\Delta E$
versus $M(\Lambda\pi)$ plane is broadened by detector resolution and radiative
effects. Photons radiated from the intial state lead to a tail at low values
of $\Delta E$. A sideband region is also defined as the complements of the
signal regions enclosed in the ranges $1.70$ GeV/$c^{2}<M(\Lambda\pi)<1.85$
GeV/$c^{2}$ and $-0.35$ GeV $<\Delta E<0.25$ GeV, indicated by the rectangular
boxes in Fig. 1. Only candidates falling within the rectangular boxes in Fig.
1 are analyzed.

(a)(b)

Figure 1: Distributions of $\Delta E$ as a function of $M(\Lambda\pi)$ for
simulated (a) $\tau^{-}\rightarrow\Lambda\pi^{-}$ and (b)
$\tau^{-}\rightarrow\mathrm{bar}{\Lambda}\pi^{-}$ signal samples. The red
solid ellipses identify the signal regions, while the areas between the dashed
black boxes and the corresponding ellipses are the sideband regions.

We require that charged particles on the signal and tag sides be identified by
combining information from various subdetectors to form the likelihood ${\cal
L}_{i}$ for particle species $i$. The ratio ${\cal{R}}_{i}$ =
${\cal{L}}_{i}/{\Sigma_{j}{\cal{L}}_{j}}$ is used to identify protons and
muons, where $i$ indicates proton or muon, and $j$ indicates electron, moun,
pion, kaon, proton, and deuteron. We require ${\cal{R}}_{p}>$ 0.5 and
${\cal{R}}_{\mu}>$ 0.9, with efficiencies of 98.5% and 95%, respectively; 3.4%
(6.5%) probability for misidentifying a proton as a pion (kaon); and 4.1% for
misidentifying a muon as a pion. The electron identification relies on a
boosted decision tree classifier trained with information from all
subdetectors except the silicon vertex detectors. We require the boosted
decision tree output ${\cal{P}}^{e}>$ 0.9, with an efficiency of 99.4%, and
the rate for misidentifying an electron as a pion is 0.7%. Pion identification
relies on the ratio for particle types $m$ and $m^{\prime}$,
${\cal{R}}(m|m^{\prime})$ =
${\cal{L}}_{m}/({{\cal{L}}_{m}+{\cal{L}}_{m^{\prime}}})$, where $m$ and
$m^{\prime}$ represent proton, pion, and kaon. For $\pi_{\tau}$ and $\pi_{\rm
tag}$, we require ${\cal{R}}(p|\pi)<$ 0.6 and ${\cal{R}}(K|\pi)<$ 0.4, with
identification efficiencies of 98.2% and 99.9%, and the rates for
misidentifying a pion as a proton or a pion as a kaon are 9.4% and 8.6%,
respectively. The charged particle on the tag side is required to be
identified as $e$, $\mu$, or $\pi$.

We reconstruct $\Lambda$ and $\mathrm{bar}{\Lambda}$ candidates by combining
an identified $p$ or $\mathrm{bar}{p}$ with an oppositely charged particle for
which no particle identification is required. The resulting $p\pi^{-}$ and
$\mathrm{bar}{p}\pi^{+}$ invariant mass distributions are shown in Figs. 2(a)
and 2(b). Candidates within a $6$ MeV/$c^{2}$ range around the known $\Lambda$
mass are selected PDG . The flight significance of $\Lambda$ and
$\mathrm{bar}{\Lambda}$ candidates, defined as the ratio $L/\sigma$ of the
flight distance $L$ to its uncertainty $\sigma$, is required to be larger than
2.0, as shown in Figs. 2(c) and 2(d). After applying these selections, 83%
(88%) of the remaining candidates are correctly identified as $\Lambda$
($\mathrm{bar}{\Lambda}$), while the remaining 17% (12%) are due to random
combinations of tracks, according to simulation.

(a)(b)
(c)(d)

Figure 2: Distributions of (a) $M(p\pi^{-})$ for
$\tau^{-}\rightarrow\Lambda\pi^{-}$ candidates, (b)
$M(\mathrm{bar}{p}\pi^{+})$ for
$\tau^{-}\rightarrow\mathrm{bar}{\Lambda}\pi^{-}$ candidates, (c) $L/\sigma$
of $\Lambda$ candidates for $\tau^{-}\rightarrow\Lambda\pi^{-}$ and (d)
$L/\sigma$ of $\mathrm{bar}{\Lambda}$ candidates for
$\tau^{-}\rightarrow\mathrm{bar}{\Lambda}\pi^{-}$. The red open histograms
show the simulated signal distributions, the filled histograms are stacked to
show the simulated background distributions, with statistical uncertainties
displayed as hatched areas, and the points with error bars show the
distributions of the data in the sideband regions. The simulated signal
distribution is arbitrarily scaled. The blue arrows indicate the boundaries of
the selection criteria. Pull distributions show the difference between data
and simulation divided by the expected uncertainty on the model.

The missing momentum $p_{\rm miss}$ is defined as the difference between the
total initial momentum and the sum of the momenta of all charged particles and
photons. To suppress backgrounds from $e^{+}e^{-}\to e^{+}e^{-}(\gamma)$,
$e^{+}e^{-}\to\mu^{+}\mu^{-}(\gamma)$,
$e^{+}e^{-}\to\ell^{+}\ell^{-}\ell^{+}\ell^{-}$, and $e^{+}e^{-}\to
e^{+}e^{-}h^{+}h^{-}$ processes, the cosine of the angle between $p_{\rm
miss}$ and the track on the tag side is required to be greater than 0.15. The
angle $\theta_{\Lambda-\pi_{\tau}}$ between the momenta of the $\Lambda$
candidates and $\pi_{\tau}$ is required to be greater than 0.1 radians. We
also require $T>0.9$. After these selections are applied, contributions from
$e^{+}e^{-}\to\ell^{+}\ell^{-}\ell^{+}\ell^{-}$ and $e^{+}e^{-}\to
e^{+}e^{-}h^{+}h^{-}$ processes are negligible. At this stage of the analysis,
99.2% (99.3%) of the background is removed, with a signal efficiency of 74.3%
(74.7%) for the $\tau^{-}\to\Lambda(\mathrm{bar}{\Lambda})\pi^{-}$ decay
modes. The remaining background is dominated by
$e^{+}e^{-}\to\tau^{+}\tau^{-}(\gamma)$ and $e^{+}e^{-}\to q\mathrm{bar}{q}$
processes.

To further suppress the remaining background, a gradient boosted decision tree
(GBDT) classifier is used PSpeckmayer_2010 . We use 15 discriminating
observables defined at the event level, on the signal side, and on the tag
side as inputs to the classifier. The observables at event level are the sum
of the energies of all visible particles $E_{\rm vis}$; the magnitude of the
missing momentum; the square of the missing mass $M^{2}_{\rm miss}=E^{2}_{\rm
miss}-p^{2}_{\rm miss}$, where $E_{\rm miss}$ is the difference between the
total initial energy and $E_{\rm vis}$; the cosine of the polar angle of the
missing momentum; the cosine of the angle between the missing momentum and
$\hat{t}$; and the cosine of the angle between the missing momentum and the
$\Lambda\pi$ ($\mathrm{bar}{\Lambda}\pi$) system. The observables related to
the signal side are $N^{\rm sig}_{\gamma}$; $E_{\gamma}^{\rm sig}$;
$\theta_{\Lambda-\pi_{\tau}}$; the angle between the momenta of $\pi_{\tau}$
and $\pi_{\Lambda}$; and the momentum of the $\Lambda\pi$
($\mathrm{bar}{\Lambda}\pi$) system. The observables related to the tag side
are $N^{\rm tag}_{\gamma}$; the energy of the most energetic photon on the tag
side; the mass of the system recoiling against the track on the tag side; and
an identification code assigned to the track’s particle-identification
information to distinguish between $e$, $\mu$, and $\pi$. We train the GBDT
classifier with samples of simulated signal and background events satisfying
the selections described above. The GBDT for the two decay channels are
trained separately. Signal and background events are divided into training and
test samples in a 1:4 ratio.

The distributions of the GBDT outputs are shown in Fig. 3. To optimize the
requirements on the GBDT output values, we minimize the expected upper limits
at 90% credibility level CL1  on
${\mathcal{B}}(\tau^{-}\rightarrow\Lambda\pi^{-})$ and
${\mathcal{B}}(\tau^{-}\rightarrow\mathrm{bar}{\Lambda}\pi^{-})$ (as described
later in the paper) estimated with a simulated sample independent of those
used for training and testing. The optimized selections are
${\cal{P}}^{\text{GBDT}}>0.886$ $(0.803)$, resulting in a 78.8% (82.3%)
relative signal efficiency due to the GBDT only and 98.9% (98.8%) background
rejection for the $\tau^{-}\rightarrow\Lambda\pi^{-}$
$(\tau^{-}\rightarrow\mathrm{bar}{\Lambda}\pi^{-})$ decay mode. These values
are consistent for both the training and test samples. The final signal
efficiencies are 9.5% and 9.9% for $\tau^{-}\rightarrow\Lambda\pi^{-}$ and
$\tau^{-}\rightarrow\mathrm{bar}{\Lambda}\pi^{-}$ decays, respectively, which
include corrections for PID, trigger, and $\Lambda$ selection, as described
below.

(a)(b)

Figure 3: Distributions of GBDT outputs for (a)
$\tau^{-}\rightarrow\Lambda\pi^{-}$ and (b)
$\tau^{-}\rightarrow\mathrm{bar}{\Lambda}\pi^{-}$ samples. The red open
histograms show the simulated signal distributions, the filled histograms are
stacked to show the simulated background distributions, with statistical
uncertainties displayed as hatched areas, and the points with error bars show
the distributions of the data in the sideband regions. The simulated signal
distribution is arbitrarily scaled.

The branching fraction of the $\tau^{-}\rightarrow\Lambda\pi^{-}$ decay
channel is calculated using

${\mathcal{B}}(\tau^{-}\rightarrow\Lambda\pi^{-})=\frac{N_{\rm
sig}}{2\varepsilon_{\rm
sig}\mathcal{L}\sigma_{\tau\tau}{\mathcal{B}}(\Lambda\rightarrow p\pi^{-})},$
(2)

where $N_{\rm sig}$ is the signal yield, $\varepsilon_{\rm sig}$ is the signal
efficiency, $\mathcal{L}$ is the integrated luminosity,
$\sigma_{\tau\tau}=0.919\pm 0.003$ nb is the cross section of
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}(\gamma)$ Banerjee:2007is , and
${\mathcal{B}}(\Lambda\rightarrow p\pi^{-})=0.641\pm 0.005$ is the known
branching fraction of $\Lambda\rightarrow p\pi^{-}$ PDG . This formula also
applies to the $\tau^{-}\rightarrow\mathrm{bar}{\Lambda}\pi^{-}$ decay mode.

Several corrections and systematic uncertainties affect the terms in Eq. 2 and
therefore the determination of the branching fraction. The sources are from
signal efficiency, luminosity, $\sigma_{\tau\tau}$,
${\mathcal{B}}(\Lambda\rightarrow p\pi^{-})$, and signal yield.

Sources of systematic uncertainties for the signal efficiency include
uncertainties in the efficiencies of the $\Lambda$ selection, GBDT selection,
tracking, PID, and trigger. The uncertainty due to the requirement on the
flight significance for the $\Lambda$ selection is studied using a
$\Lambda_{c}^{+}\to\Lambda\pi^{+}$ control channel. The ratio of $\Lambda$
selection efficiencies in data and simulation is $0.977\pm 0.002$ $(0.976\pm
0.002)$ for $\tau^{-}\to\Lambda(\mathrm{bar}{\Lambda})\pi^{-}$ channel. These
values are used to correct the signal efficiencies, and the corresponding
uncertainties are taken as systematic uncertainties. The systematic
uncertainties on the signal efficiency from the GBDT selection are studied by
splitting the training samples into two equal size parts and training the
GBDTs separately. The differences between the resulting signal efficiencies
after applying the same requirements on the output values as in the nominal
analysis are both 0.50% for the two decay channels, which are taken as
systematic uncertainties due to the GBDT. The uncertainty due to tracking
efficiency is 0.24% per track, estimated with
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events, leading to a systematic
uncertainty of 0.96%. The systematic uncertainties from charged-particle
identification are studied in $\Lambda\rightarrow p\pi$, $D^{*+}\rightarrow
D^{0}(\rightarrow K^{-}\pi^{+})\pi^{+}$, and
$K_{S}^{0}\rightarrow\pi^{+}\pi^{-}$ decays for protons and pions, and in
$J/\psi\rightarrow e^{+}e^{-}(\gamma)$, $\mu^{+}\mu^{-}(\gamma)$,
$e^{+}e^{-}e^{+}e^{-}$, and $e^{+}e^{-}\mu^{+}\mu^{-}$ processes for muons and
electrons. Correction factor to the signal efficiency for the
$\tau^{-}\to\Lambda(\mathrm{bar}{\Lambda})\pi^{-}$ decay mode is $0.953\pm
0.021$ ($0.954\pm 0.022$) for the
$\tau^{-}\to\Lambda(\mathrm{bar}{\Lambda})\pi^{-}$ channel. The quadratic sum
of the individual charged-particle contributions is taken at the systematic
uncertainty from charged particle identification. The systematic uncertainty
on the trigger efficiency is studied using $e^{+}e^{-}\to\tau^{+}\tau^{-}$
events with $\tau^{-}\to\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ and $\tau^{+}$
decaying into the same final states as in the tag side of this analysis Tau3mu
. The trigger efficiency measured in data differs by 0.7% from that in
simulation; this difference is taken as the systematic uncertainty due to the
trigger.

The uncertainty in the branching fraction $\mathcal{B}(\Lambda\rightarrow
p\pi^{-})$ is 0.78% PDG . The uncertainty of the
$e^{+}e^{-}\to\tau^{+}\tau^{-}(\gamma)$ cross section, estimated with KKMC, is
0.33% Banerjee:2007is . The uncertainty in the integrated luminosity is 0.60%,
determined from the large-angle Bhabha scattering process Belle-II:2019usr .
The quadratic sum of all the above uncertainties, summarized in Table 1, is
$\sigma$ = 2.77% (2.82%) for the
$\tau^{-}\rightarrow\Lambda(\mathrm{bar}{\Lambda})\pi^{-}$ channel.

Table 1:  Summary of fractional systematic uncertainties. Source | Uncertainty (%)
---|---
$\tau^{-}\to\Lambda\pi^{-}$ | $\tau^{-}\to\mathrm{bar}{\Lambda}\pi^{-}$
$\Lambda$ selection | 0.20 | 0.20
GBDT selection | 0.50 | 0.50
Tracking efficiency | 0.96 | 0.96
Particle identification | 2.21 | 2.28
Trigger efficiency | 0.70 | 0.70
$\mathcal{B}(\Lambda\rightarrow p\pi^{-})$ | 0.78 | 0.78
$\tau$-pair cross section | 0.33 | 0.33
Luminosity | 0.60 | 0.60
Total | 2.77 | 2.82

The estimate of $N_{\rm sig}$ (see Eq. 2) depends on the expected background
yield. Systematic uncertainties affecting the expected background yield and
due to data-simulation discrepancies are evaluated from the sideband regions.
There are $N_{\rm SB}^{\rm data}=7$ $(6)$ data events in the sideband region
for the $\tau^{-}\rightarrow\Lambda(\mathrm{bar}{\Lambda})\pi^{-}$ channel,
while the corresponding number of events observed in simulated background
samples is $N_{\rm SB}^{\rm sim}=3.2^{+1.7}_{-1.2}$ $(5.5^{+2.1}_{-1.6})$,
where the uncertainties are due to the limited size of the simulated samples.
The data-simulation ratios $f_{\rm bkg}=N_{\rm SB}^{\rm data}/N_{\rm SB}^{\rm
sim}$ are used as correction factors for the background estimated in the
signal region: they are $2.2^{+1.7}_{-1.2}$ ($1.1^{+0.8}_{-0.5}$) for the
$\tau^{-}\rightarrow\Lambda(\mathrm{bar}{\Lambda})\pi^{-}$ channels, where the
uncertainties are statistical only. The relative uncertainties in $f_{\rm
bkg}$ are treated as systematic uncertainties on the expected background
yield, being ${}^{+77\%}_{-55\%}$ (${}^{+73\%}_{-45\%}$) for the
$\tau^{-}\rightarrow\Lambda(\mathrm{bar}{\Lambda})\pi^{-}$ channel. Some of
the sources of uncertainty listed in Table 1 also affect the expected
background yield. Their contributions are negligible compared to those
affecting the correction factors estimated in the sidebands and are not taken
into account.

The distributions of the events in the $\Delta E$ versus $M(\Lambda\pi)$ plane
are shown in Fig. 4. From simulation, the expected background yields for
$\tau^{-}\rightarrow\Lambda\pi^{-}$ and
$\tau^{-}\rightarrow\mathrm{bar}{\Lambda}\pi^{-}$ within the signal region,
after applying the correction factor $f_{\rm bkg}$, are $N_{\rm
exp}=1.0^{+1.3}_{-1.1}$ and $0.5\pm 0.6$, respectively, where the
uncertainties include both statistical and systematic uncertainties. No events
are observed in the signal region. The branching fractions are measured to be
$(-2.5^{+4.1+1.9}_{-3.7-1.4})\times 10^{-8}$ and $(-1.2\pm
2.8^{+0.9}_{-0.5})\times 10^{-8}$ for the $\tau^{-}\rightarrow\Lambda\pi^{-}$
and $\tau^{-}\rightarrow\mathrm{bar}{\Lambda}\pi^{-}$ channels, respectively,
where the first uncertainties are statistical and the second are systematic.

(a)
(b)

Figure 4: Distributions of the events for data and simulated background in the
$\Delta E$ versus $M(\Lambda\pi)$ plane for (a)
$\tau^{-}\rightarrow\Lambda\pi^{-}$ and (b)
$\tau^{-}\rightarrow\mathrm{bar}{\Lambda}\pi^{-}$ candidates after applying
all selections.

Since no signal is observed, we compute upper limits on the signal yields for
the $\tau^{-}\rightarrow\Lambda\pi^{-}$ and
$\tau^{-}\rightarrow\mathrm{bar}{\Lambda}\pi^{-}$ channels. We assume that the
signal yields $N_{\rm sig}$ and the expected background yields $N_{\rm exp}$
in the signal region follow a Poisson distribution. We use the prior
probability density function $1/\sqrt{N_{\rm sig}+N_{\rm exp}}$ to estimate
the expected upper limits of signal yields based on a Bayesian approach
ZHU2007322 . The uncertainty in $N_{\rm exp}$ and $\sigma$ are treated as two
independent parameters when estimating the upper limits on the signal yields
ZHU2007322 . The upper limits on the signal yields at 90% credibility level
are found to be 1.92 and 1.81, for the $\tau^{-}\rightarrow\Lambda\pi^{-}$ and
$\tau^{-}\rightarrow\mathrm{bar}{\Lambda}\pi^{-}$ channels, respectively. The
corresponding upper limits on the branching fractions are $4.7\times 10^{-8}$
and $4.3\times 10^{-8}$, respectively, while the expected upper limits based
on background simulated samples are $7.2\times 10^{-8}$ and $5.5\times
10^{-8}$, respectively. If we use the flat prior probability density function,
although it is not advised in Ref. ZHU2007322 , the corresponding upper limits
on the branching fractions will be $5.7\times 10^{-8}$ and $5.5\times
10^{-8}$. These upper limits do not appreciably change if we do not include
systematic uncertainties.

In summary, we present a search for the BNV and lepton number violation decays
$\tau^{-}\rightarrow\Lambda\pi^{-}$ and
$\tau^{-}\rightarrow\mathrm{bar}{\Lambda}\pi^{-}$ using a 364 fb-1 data sample
collected by the Belle II experiment at $\sqrt{s}$ = 10.58 GeV. No evidence of
signal is observed, and upper limits at 90% credibility level on the branching
fractions ${\mathcal{B}}(\tau^{-}\rightarrow\Lambda\pi^{-})$ and
${\mathcal{B}}(\tau^{-}\rightarrow\mathrm{bar}{\Lambda}\pi^{-})$ are estimated
to be $4.7\times 10^{-8}$ and $4.3\times 10^{-8}$, respectively. These are the
most stringent constraints to date on the branching fraction of
$\tau^{-}\rightarrow\Lambda\pi^{-}$ with $|\Delta(B-L)|=2$ and
$\tau^{-}\rightarrow\mathrm{bar}{\Lambda}\pi^{-}$ with $|\Delta(B-L)|=0$.

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 31361-N36 and No. J4625-N, 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; 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, the National Institute of
Informatics, and Science Information NETwork 5 (SINET5), 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 group and the NII for on-site computing support and SINET6
network support; and the raw-data centers at BNL, DESY, GridKa, IN2P3, INFN,
and the University of Victoria for off-site computing support.

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