The Belle II Collaboration

# Measurement of the $\tau$-lepton mass with the Belle II experiment

I. Adachi  0000-0003-2287-0173 K. Adamczyk  0000-0001-6208-0876 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. 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 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 M. Barrett  0000-0002-2095-603X J. Baudot
0000-0001-5585-0991 M. Bauer  0000-0002-0953-7387 A. Baur  0000-0003-1360-3292
A. Beaubien  0000-0001-9438-089X F. Becherer  0000-0003-0562-4616 J. Becker
0000-0002-5082-5487 P. K. Behera  0000-0002-1527-2266 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 T. Bilka  0000-0003-1449-6986 D. Biswas
0000-0002-7543-3471 A. Bobrov  0000-0001-5735-8386 D. Bodrov
0000-0001-5279-4787 A. Bolz  0000-0002-4033-9223 A. Bondar
0000-0002-5089-5338 J. Borah  0000-0003-2990-1913 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 V. Chekelian  0000-0001-8860-8288 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 L. M. Cremaldi  0000-0001-5550-7827 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 A. De Yta-Hernandez  0000-0002-2162-7334 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 S. Dreyer  0000-0002-6295-100X S. Dubey
0000-0002-1345-0970 G. Dujany  0000-0002-1345-8163 P. Ecker
0000-0002-6817-6868 M. Eliachevitch  0000-0003-2033-537X 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 A. Garmash
0000-0003-2599-1405 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 P. Grace
0000-0001-9005-7403 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 S. Halder  0000-0002-6280-494X 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 I. Heredia de la Cruz  0000-0002-8133-6467 M. Hernández
Villanueva  0000-0002-6322-5587 A. Hershenhorn  0000-0001-8753-5451 T. Higuchi
0000-0002-7761-3505 E. C. Hill  0000-0002-1725-7414 M. Hoek
0000-0002-1893-8764 M. Hohmann  0000-0001-5147-4781 C.-L. Hsu
0000-0002-1641-430X 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 S. Ito
0000-0003-2737-8145 R. Itoh  0000-0003-1590-0266 M. Iwasaki
0000-0002-9402-7559 P. Jackson  0000-0002-0847-402X 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 A. Johnson  0000-0002-8366-1749 K. K. Joo
0000-0002-5515-0087 H. Junkerkalefeld  0000-0003-3987-9895 D. Kalita
0000-0003-3054-1222 A. B. Kaliyar  0000-0002-2211-619X 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. Ketter  0000-0002-5161-9722 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 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 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 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  0000-0002-9073-5689 P. Leitl
0000-0002-1336-9558 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
Y. Li  0000-0002-4413-6247 J. Libby  0000-0002-1219-3247 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 L. Martel  0000-0001-8562-0038 C. Martellini
0000-0002-7189-8343 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 K. Matsuoka
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 M. Milesi
0000-0002-8805-1886 C. Miller  0000-0003-2631-1790 M. Mirra
0000-0002-1190-2961 K. Miyabayashi  0000-0003-4352-734X R. Mizuk
0000-0002-2209-6969 G. B. Mohanty  0000-0001-6850-7666 N. Molina-Gonzalez
0000-0002-0903-1722 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 Y. Nakazawa  0000-0002-6271-5808 A. Narimani Charan
0000-0002-5975-550X M. Naruki  0000-0003-1773-2999 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 C. Niebuhr  0000-0002-4375-9741 N. K. Nisar
0000-0001-9562-1253 S. Nishida  0000-0001-6373-2346 H. Ono
0000-0003-4486-0064 Y. Onuki  0000-0002-1646-6847 P. Oskin
0000-0002-7524-0936 F. Otani  0000-0001-6016-219X P. Pakhlov
0000-0001-7426-4824 G. Pakhlova  0000-0001-7518-3022 A. Paladino
0000-0002-3370-259X A. Panta  0000-0001-6385-7712 E. Paoloni
0000-0001-5969-8712 S. Pardi  0000-0001-7994-0537 K. Parham
0000-0001-9556-2433 H. Park  0000-0001-6087-2052 S.-H. Park
0000-0001-6019-6218 A. Passeri  0000-0003-4864-3411 S. Patra
0000-0002-4114-1091 S. Paul  0000-0002-8813-0437 T. K. Pedlar
0000-0001-9839-7373 I. Peruzzi  0000-0001-6729-8436 R. Peschke
0000-0002-2529-8515 R. Pestotnik  0000-0003-1804-9470 F. Pham
0000-0003-0608-2302 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 N. Rad  0000-0002-5204-0851 P. Rados  0000-0003-0690-8100
G. Raeuber  0000-0003-2948-5155 S. Raiz  0000-0001-7010-8066 M. Reif
0000-0002-0706-0247 S. Reiter  0000-0002-6542-9954 M. Remnev
0000-0001-6975-1724 I. Ripp-Baudot  0000-0002-1897-8272 G. Rizzo
0000-0003-1788-2866 S. H. Robertson  0000-0003-4096-8393 M. Roehrken
0000-0003-0654-2866 J. M. Roney  0000-0001-7802-4617 A. Rostomyan
0000-0003-1839-8152 N. Rout  0000-0002-4310-3638 G. Russo  0000-0001-5823-4393
D. Sahoo  0000-0002-5600-9413 S. Sandilya  0000-0002-4199-4369 A. Sangal
0000-0001-5853-349X L. Santelj  0000-0003-3904-2956 Y. Sato
0000-0003-3751-2803 V. Savinov  0000-0002-9184-2830 B. Scavino
0000-0003-1771-9161 C. Schmitt  0000-0002-3787-687X 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 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 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 Z. S. Stottler  0000-0002-1898-5333 R. Stroili
0000-0002-3453-142X M. Sumihama  0000-0002-8954-0585 K. Sumisawa
0000-0001-7003-7210 W. Sutcliffe  0000-0002-9795-3582 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 A. Thaller
0000-0003-4171-6219 O. Tittel  0000-0001-9128-6240 R. Tiwary
0000-0002-5887-1883 D. Tonelli  0000-0002-1494-7882 E. Torassa
0000-0003-2321-0599 N. Toutounji  0000-0002-1937-6732 K. Trabelsi
0000-0001-6567-3036 I. Tsaklidis  0000-0003-3584-4484 I. Ueda
0000-0002-6833-4344 T. Uglov  0000-0002-4944-1830 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 Y. Ushiroda
0000-0003-3174-403X S. E. Vahsen  0000-0003-1685-9824 R. van Tonder
0000-0002-7448-4816 G. S. Varner  0000-0002-0302-8151 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 B. Wach  0000-0003-3533-7669
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 M. Watanabe
0000-0001-6917-6694 S. Watanuki  0000-0002-5241-6628 M. Welsch
0000-0002-3026-1872 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. H. Yin  0000-0002-1479-9349 K. Yoshihara
0000-0002-3656-2326 C. Z. Yuan  0000-0002-1652-6686 Y. Yusa
0000-0002-4001-9748 L. Zani  0000-0003-4957-805X V. Zhilich
0000-0002-0907-5565 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 measurement of the $\tau$-lepton mass using a sample of about 175
million $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events collected with the
Belle II detector at the SuperKEKB $e^{+}e^{-}$ collider at a center-of-mass
energy of $10.579\,\mathrm{Ge\kern-1.00006ptV}$. This sample corresponds to an
integrated luminosity of $190\,\mathrm{fb^{-1}}$. We use the kinematic edge of
the $\tau$ pseudomass distribution in the decay
${\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}}$ and measure the $\tau$
mass to be $1777.09\pm 0.08\pm 0.11\,\mathrm{Me\kern-1.00006ptV\\!/c^{2}}$,
where the first uncertainty is statistical and the second systematic. This
result is the most precise to date.

The $\tau$-lepton mass, $m_{\tau}$, is one of the fundamental parameters of
the standard model. An experimental determination of $m_{\tau}$ with the
lowest possible uncertainty has important consequences for tests of lepton-
flavor-universality between $\tau$ and lighter leptons where the $\tau$ mass
enters to the fifth power Amhis et al. (2021) . Precise knowledge of
$m_{\tau}$ is also important for the predictions of leptonic and hadronic
branching fractions of the $\tau$ Tsai (1971) and the determination of the
strong-interaction coupling $\alpha_{s}$ at the $\tau$-mass scale Davier et
al. (2006, 2014).

The most precise measurements of the $\tau$ mass currently available are
reported in Table 1. The BES Bai et al. (1996), KEDR Anashin et al. (2007),
and BES III Ablikim et al. (2014) collaborations measured the $\tau$ mass by
analyzing the $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ cross-section near the
$\tau^{+}\tau^{-}$ production threshold. The Belle Abe et al. (2007) and BaBar
Aubert et al. (2009a) measurements use the pseudomass endpoint method Albrecht
et al. (1992) at a center-of-mass energy near the $\Upsilon(4S)$ mass. The
highest precision to date has been achieved by the BES III collaboration.
While the statistical and systematic uncertainties for the BES III measurement
are of similar size, the precision of the Belle and BaBar measurements is
limited by systematic uncertainties. For both experiments the largest
systematic uncertainties arose from the knowledge of the beam energy and the
momentum reconstruction of the $\tau$-decay products.

Experiment |  $m_{\tau}$ [${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ]
---|---
BES Bai et al. (1996) | $1776.96^{\;\;+\,\;0.18}_{\;\;-\,\;0.21}\ {}^{\;\;+\;\;0.25}_{\;\;-\;\;0.17}$
KEDR Anashin et al. (2007) | $1776.80^{\;\;+\,\;0.25}_{\;\;-\,\;0.23\;\;}\pm 0.15$
BES III Ablikim et al. (2014) | $1776.91\pm 0.12^{\;\;+\;\;0.10}_{\;\;-\;\;0.13}$
Belle Abe et al. (2007) | $1776.61\pm 0.13\pm 0.35$
BaBar Aubert et al. (2009a) | $1776.68\pm 0.12\pm 0.41$
Table 1: Summary of most precise measurements of the $\tau$ mass to date.

In this paper, we report a measurement of $m_{\tau}$ using a sample of about
175 million $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events recorded with the
Belle II detector Abe (2010) at the asymmetric-energy $e^{+}e^{-}$ SuperKEKB
collider Akai et al. (2018). The data, collected between March 2019 and July
2021 near the nominal center-of-mass energy of
$\sqrt{s}=10.579$$\mathrm{\,Ge\kern-1.00006ptV}$, correspond to an integrated
luminosity of 190$\mbox{\,fb}^{-1}$. We determine $m_{\tau}$ from the hadronic
decays $\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ using the
pseudomass endpoint method. Charge-conjugate modes are implied throughout.
Assuming zero mass for the neutrino, the $\tau$ mass is given by

$m_{\tau}=\sqrt{M_{3\pi}^{2}+2(E_{\tau}^{*}-E^{*}_{3\pi})(E^{*}_{3\pi}-p^{*}_{3\pi}\cos\alpha^{*})}.$
(1)

Here, and throughout the paper, quantities in the $e^{+}e^{-}$ center-of-mass
frame are indicated by the asterisk. The mass, energy, and momentum of the
three-pion system are denoted by $M_{3\pi}$, $E^{*}_{3\pi}$, and
$p^{*}_{3\pi}$, respectively. The energy of the $\tau$ is given by
$E_{\tau}^{*}$; $\alpha^{*}$ is the angle between the momenta of the three-
pion system and the neutrino. The energy $E_{\tau}^{*}$ is half of the
$e^{+}e^{-}$ center-of-mass energy $\sqrt{s}/2$ up to corrections from initial
state radiation (ISR) from the $e^{\pm}$ beams and final state radiation (FSR)
from the $\tau$ and its decay products. The pseudomass $M_{\rm min}$ is
defined by setting $\alpha^{*}$ equal to 0 and therefore minimizing Eq. 1 as

$M_{\rm
min}=\sqrt{M^{2}_{3\pi}+2(\sqrt{s}/2-E^{*}_{3\pi})(E^{*}_{3\pi}-p^{*}_{3\pi})}\leq
m_{\tau}.$ (2)

In the absence of ISR and FSR, and assuming a perfect measurement of the four-
momentum of the three-pion system, the $M_{\rm min}$ distribution extends up
to $m_{\tau}$, where it has a sharp edge. The momentum resolution of the
detector and the energy loss through radiation smear the endpoint position and
introduce a tail towards larger $M_{\rm min}$ values. However, as seen in Fig.
1, an edge remains in the observed $M_{\rm min}$ distribution and is used to
measure the $\tau$ mass. One challenge is to precisely measure the inputs to
Eq. 2, namely the $e^{+}e^{-}$ center-of-mass energy $\sqrt{s}$ and the
momenta of the $\tau$-decay products. Another challenge is to develop an
empirical model to describe the $M_{\rm min}$ distribution. Any inaccuracy in
either directly impacts the $\tau$-mass determination.

Figure 1:  Spectrum of $M_{\rm min}$ in experimental data (dots), along with
simulated background contributions from
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events with decays other than
$\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ (orange area with solid
line), $e^{+}e^{-}\rightarrow q\bar{q}$ events (blue area with dashed line),
and other background sources (gray area with dotted line).

##  I The Belle II detector and simulation

The Belle II detector consists of several subdetectors arranged in a
cylindrical structure around the $e^{+}e^{-}$ interaction point Abe (2010).
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% 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, respectively. The electromagnetic
calorimeter (ECL), divided into the forward endcap, barrel, and backward
endcap, fills the remaining volume inside a 1.5 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.

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 $u$, $d$, $c$, or $s$ quark; $e^{+}e^{-}\rightarrow
e^{+}e^{-}(\gamma)$ and $\mu^{+}\mu^{-}(\gamma)$ events;
$e^{+}e^{-}\rightarrow\ell^{+}\ell^{-}\ell^{+}\ell^{-}$ events, where $\ell$
is a charged lepton; $e^{+}e^{-}\rightarrow e^{+}e^{-}h^{+}h^{-}$ events,
where $h$ indicates a pion, kaon, or proton; and $e^{+}e^{-}\rightarrow
e^{+}e^{-}nh$ events with $n>2$. We use simulated events to identify
discriminating features effective to suppress these backgrounds. The
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ process is generated using the KKMC
generator Jadach et al. (2000, 2001). The $\tau$ decays are simulated by
TAUOLA Jadach et al. (1990) and their FSR by PHOTOS Barberio et al. (1991). We
use KKMC to simulate $\mu^{+}\mu^{-}(\gamma)$ and $q\bar{q}$ production;
PYTHIA Sjöstrand et al. (2015) for the fragmentation of the $q\bar{q}$ pair;
BabaYaga@NLO Balossini et al. (2006, 2008); Carloni Calame et al. (2004);
Carloni Calame (2001); Carloni Calame et al. (2000) for $e^{+}e^{-}\rightarrow
e^{+}e^{-}(\gamma)$ events; and AAFH Berends et al. (1985a, b); Berends et al.
(1986) and TREPS Uehara (1996) for the production of non-radiative final
states $\ell^{+}\ell^{-}\ell^{+}\ell^{-}$ and $e^{+}e^{-}h^{+}h^{-}$. There is
no generator to simulate the $e^{+}e^{-}\rightarrow e^{+}e^{-}nh$ process. The
Belle II analysis software Kuhr et al. (2019); bas  uses the GEANT4
Agostinelli et al. (2003) package to simulate the response of the detector to
the passage of the particles.

##  II Event selection

In the $e^{+}e^{-}$ center-of-mass frame, the $\tau$ leptons are produced in
opposite directions. Thus, the decay products of one $\tau$ are isolated from
those of the other $\tau$, and they are 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 $i$th final-state particle in the
$e^{+}e^{-}$center-of-mass frame Brandt et al. (1964); Farhi (1977).

We define the _signal_ hemisphere as that containing three charged particles,
which are assumed to originate from the
$\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ decay, and require that
the other hemisphere, named _tag_ , contains only one charged particle and up
to one neutral pion. Thus, the tag side contains leptonic
($\tau^{+}\rightarrow e^{+}\nu_{e}\bar{\nu}_{\tau}$ and
$\tau^{+}\rightarrow\mu^{+}\nu_{\mu}\bar{\nu}_{\tau}$) and hadronic
(predominantly $\tau^{+}\rightarrow h^{+}\bar{\nu}_{\tau}$ and
$\tau^{+}\rightarrow h^{+}\pi^{0}\bar{\nu}_{\tau}$ ) $\tau$ decays.

We select $\tau$-pair candidates by requiring the event to contain exactly
four charged particles with zero total charge, each having a trajectory
displaced from the average interaction point by less than 3 cm along the $z$
axis and less than 1 cm in the transverse plane to reduce the contamination of
tracks originated from beam-background interactions. No particle-
identification requirements are imposed on any of the charged particles. The
momenta of charged particles are scaled with factors that range from $0.99660$
to $1.00077$ depending on the charge and $\cos\theta$ to correct for
imperfections in the magnetic-field description used in the event
reconstruction, misalignment of the detector, and material mismodeling. The
correction factors are evaluated by measuring the mass-peak position of high-
yield samples of $D^{0}\rightarrow K^{-}\pi^{+}$ decays reconstructed in data
and comparing them to the known value Workman et al. (2022).

Neutral pions are identified as photon pairs with masses between
115${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
152${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Those photons are identified
from ECL energy deposits (clusters), reconstructed within the CDC acceptance,
$17^{\circ}<\theta<150^{\circ}$, to ensure they are not matched to any charged
particle. Depending on whether the photons are reconstructed in the forward,
barrel, or backward region of the detector, requirements are different.
Photon-energy thresholds ranging from 60$\mathrm{\,Me\kern-1.00006ptV}$ to
600$\mathrm{\,Me\kern-1.00006ptV}$ suppress the beam-induced backgrounds,
which are larger in the endcaps compared to the barrel region. Requirements on
the cosine of the angle between the momenta of the two photons and on the
momentum of the reconstructed neutral pion reduce the combinatorial background
from low-energy photons.

The online event selection, _trigger_ , is based on the energy deposits and
their topologies in the ECL. The trigger efficiency is driven by the
requirements of at least three clusters with a topology inconsistent with a
Bhabha event and one of the clusters having an energy larger than 300 MeV. The
trigger efficiency in the experimental data is approximately $92\%$.

Background from $q\bar{q}$ production is suppressed by rejecting events
containing neutral pions in the signal hemisphere or more than one neutral
pion in the tag hemisphere. Events with photons of energy greater than
0.2$\mathrm{\,Ge\kern-1.00006ptV}$ that are not used in neutral-pion
reconstruction are also rejected. In order to suppress $e^{+}e^{-}(\gamma)$,
$\mu^{+}\mu^{-}(\gamma)$, $\ell^{+}\ell^{-}\ell^{+}\ell^{-}$, and
$e^{+}e^{-}h^{+}h^{-}$ events, which are characterized by low-momentum charged
particles, we rank the three charged particles in the signal hemisphere in
decreasing order of transverse momenta and ensure that their values exceed
0.6, 0.2, and 0.1${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. We further reject
events from $q\bar{q}$, $e^{+}e^{-}(\gamma)$, and $e^{+}e^{-}(nh)$ processes
by restricting the thrust value and the visible center-of-mass energy of the
event, $E^{*}_{\rm vis}$, that is the sum of all reconstructed particles’
energies, to be within ranges of [0.87,0.97] and [2.5,
9.0]$\mathrm{\,Ge\kern-1.00006ptV}$, respectively. We remove most of the
remaining background with requirements on the magnitude of the missing
momentum, $0.05<p^{*}_{\rm miss}<3.5$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$,
on its polar angle, $0.5<\theta^{*}_{\rm miss}<2.7$$\rm\,rad$, and on the
square of the missing mass, $0<M^{2}_{\rm
miss}<54{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. 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 missing mass is
defined as $M^{2}_{\rm miss}=(\sqrt{s}-E^{*}_{\rm vis})^{2}-(p^{*}_{\rm
miss})^{2}$.

After all requirements, we observe 583192 events in the experimental data in
the $M_{\rm min}$ range
$[1.70,1.85]$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The signal-
reconstruction efficiency in this region is $2.3\%$, and the purity of the
sample is $90\%$. Among the signal events, around $56\%$ are lepton tagged and
the remaining events are hadron tagged. The dominant backgrounds are from the
$e^{+}e^{-}\rightarrow q\bar{q}$ processes ($6.4\%$), followed by
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ with decays other than
$\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ in the signal hemisphere
($2.0\%$).

##  III Beam energy

The computation of $M_{\rm min}$ relies on the knowledge of the $e^{+}e^{-}$
center-of-mass energy. We exploit the fact that the collision energy is just
slightly above the kinematic-production threshold for $B\bar{B}$ pairs and
measure the $B$-meson energy, $E_{B}^{*}$, using fully reconstructed neutral
and charged $B$-meson decays,

$E_{B}^{*}=\sqrt{m_{B}^{2}+(p^{*}_{B})^{2}}\approx
m_{B}+\frac{1}{2m_{B}}(p^{*}_{B})^{2}.$ (3)

Here $p_{B}^{*}$ and $m_{B}$ are the momentum and mass of the $B$ meson. A
maximum-likelihood fit is performed to the $E_{B}^{*}$ distribution to
determine its peak position. The collision energy $\sqrt{s}$ is obtained from
$E_{B}^{*}$ after correcting for the effect of ISR and by accounting for the
energy dependence of the $e^{+}e^{-}\rightarrow B\bar{B}$ cross-section Aubert
et al. (2009b); Mizuk et al. (2021) using the following procedure.

We use a model where the center-of-mass energy of the colliding particles is
described by a Gaussian of width $\sigma_{\sqrt{s}}$ about a mean value
$\sqrt{s}$, where $\sqrt{s}$ and $\sigma_{\sqrt{s}}$ vary slowly during data-
taking. The $e^{+}e^{-}\rightarrow B\bar{B}$ cross-section is then given by

$\frac{d^{2}\sigma}{dx\,d\sqrt{s^{\prime}}}=G(\sqrt{s^{\prime}}-\sqrt{s},\sigma_{\sqrt{s}})\,W(s^{\prime},x)\,\sigma_{0}(s^{\prime}(1-x)),$
(4)

where $\sqrt{s^{\prime}}$ is the event-by-event center-of-mass energy, $x$ is
the fraction of energy carried by the ISR photon Benayoun et al. (1999), and
$G$ is the Gaussian distribution. The photon-emission probability is described
by the function $W$, and $\sigma_{0}$ is the Born cross-section for
$e^{+}e^{-}\rightarrow B\bar{B}$ Mizuk et al. (2021). In terms of these
quantities, the $B$-meson energy is given by
$E^{*}_{B}=\frac{1}{2}\sqrt{s^{\prime}(1-x)}$. We use events simulated
according to Eq. 4 to establish a mapping between the corrected quantities
($\sqrt{s},\sigma_{\sqrt{s}}$) and observed quantities
($E^{*}_{B},\sigma_{E^{*}_{B}}$). The inverse of this mapping is used to
obtain corrected values for the center-of-mass energy $\sqrt{s}$ as a function
of data-taking time. The results are shown in Fig. 2. During the 2019 and
early 2020 data-taking periods, the $\sqrt{s}$ value is around
2$\mathrm{\,Me\kern-1.00006ptV}$ above the nominal value $\sqrt{s}_{\rm
nominal}=10.579$$\mathrm{\,Ge\kern-1.00006ptV}$. It then drifts to lower
values where it stabilizes at around 6$\mathrm{\,Me\kern-1.00006ptV}$ below
the nominal value by the middle of the 2021 data-taking period.

Figure 2: Corrected center-of-mass energy $\sqrt{s}$ (solid) and center-of-
mass energy of $B\bar{B}$ pair $2E_{B}^{*}$ (dashed blue line) as functions of
data-taking time, expressed in terms of chronologically ordered event numbers.
The horizontal dashed line represents the nominal center-of-mass energy,
$\sqrt{s}_{\rm nominal}=10.579\mathrm{\,Ge\kern-1.00006ptV}$, and the vertical
dashed lines indicate the start of different data-taking periods. The 68.3%
confidence level band of $\sqrt{s}$ is displayed as shaded area.

##  IV Method

To reduce experimenter’s bias, we validate the method of the $\tau$-mass
measurement and estimate the statistical and systematic uncertainties before
looking at the central value of the result. The $M_{\rm min}$ distribution
around the edge can be empirically described as a Heaviside step function
multiplied by second-order polynomials, and convoluted with a double-Gaussian
resolution function to account for the ISR, FSR, and detector resolution. We
approximate this function with the following expression:

$\displaystyle F(M_{\rm min})$ $\displaystyle=$ $\displaystyle
1-P_{3}\cdot\arctan\bigg{(}\frac{M_{\rm min}-P_{1}}{P_{2}}\bigg{)}$ (5)
$\displaystyle+$ $\displaystyle P_{4}(M_{\rm min}-P_{1})+P_{5}(M_{\rm
min}-P_{1})^{2}\;.$

To determine the $\tau$ mass, we perform an unbinned maximum-likelihood fit
Verkerke and Kirkby (2003) to the $M_{\rm min}$ distribution in the range
$[1.70,1.85]$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ using Eq. 5. The
parameter $P_{1}$ determines the edge position and therefore is an estimator
of the $\tau$ mass. The $P_{2}$ parameter modifies the slope of the threshold,
while the rest of the parameters describe the shape away from the edge. Fits
to simulated events in which the generated value of the $\tau$ mass is varied
in the range [1772, 1782]${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, show that
on average $P_{1}$ overestimates the $\tau$ mass with a constant offset of
$0.40\pm 0.03$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. This bias results
from the empirical parametrization of the $M_{\rm min}$ distribution.

Figure 3 shows the $M_{\rm min}$ distribution in the range of
$[1.70,1.85]$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ in data with the
background predicted from simulation and the fit projection overlaid. While
the $\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ events show a clear
threshold, the background processes are featureless around the endpoint. Their
contribution is described by the parameters $P_{3}$, $P_{4}$, and $P_{5}$. The
observed value $P_{1}=1777.49\pm
0.08$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is then corrected for the
estimator bias to obtain the measured $\tau$ mass,

$m_{\tau}=1777.09\pm 0.08\;{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},$ (6)

where the uncertainty is the statistical uncertainty of the $P_{1}$ parameter.
The statistical precision of the result is validated by generating simplified
simulated experiments based on Poisson statistics, as well as by re-sampling
the data based on bootstrapping techniques Efron (1979), and repeating the
measurement on them.

Figure 3: Top: Spectrum of $M_{\rm min}$ in experimental data (dots) with fit
result (solid blue line) and background contribution (gray-filled area)
overlaid. Bottom: Differences between data and fit result divided by the
statistical uncertainties.

##  V Systematic uncertainties and consistency checks

The systematic uncertainties are grouped into categories associated with the
knowledge of the colliding beams, the reconstruction of the charged particles,
the fit model, and imperfections in the simulated data. Table 2 summarizes the
sources that contribute to the total uncertainty. The largest uncertainty
arises from the beam-energy correction, followed by the uncertainty on the
charged-particle momentum. The various systematic uncertainties are added in
quadrature, resulting in a total systematic uncertainty of
$0.11$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

Source | Uncertainty
---|---
| [${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ]
Knowledge of the colliding beams: |
Beam-energy correction | $0.07$
Boost vector | $<0.01$
Reconstruction of charged particles: |
Charged-particle momentum correction | $0.06$
Detector misalignment | 0.03
Fit model: |
Estimator bias | 0.03
Choice of the fit function | 0.02
Mass dependence of the bias | $<0.01$
Imperfections of the simulation: |
Detector material density | 0.03
Modeling of ISR, FSR and $\tau$ decay | 0.02
Neutral particle reconstruction efficiency |  $\leq$ 0.01
Momentum resolution | $<0.01$
Tracking efficiency correction | $<0.01$
Trigger efficiency | $<0.01$
Background processes | $<0.01$
Total | $0.11$
Table 2: Summary of systematic uncertainties in the $\tau$-mass measurement.

###  V.1 Knowledge of the colliding beams

The uncertainty on the $\sqrt{s}$ measurement, as indicated by the red band in
Fig. 2, is on average around 0.75$\mathrm{\,Me\kern-1.00006ptV}$ and is
dominated by systematic uncertainties. The estimation of $\sqrt{s}$ from the
$B$-meson energy relies on the knowledge of the energy dependence of the
$e^{+}e^{-}\rightarrow B\bar{B}$ cross section, whose uncertainty is driven by
the beam-energy uncertainty of the BaBar measurement Mizuk et al. (2021);
Aubert et al. (2009b). An additional uncertainty originates from the
uncertainties in the average values of the charged
($0.26$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) and neutral
($0.20$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) $B$-meson masses Workman et
al. (2022). Since the $e^{+}e^{-}\rightarrow B\bar{B}$ sample is dominated by
charged $B$ mesons, the weighted average of the two uncertainties results in
$0.24$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, assuming the uncertainties
of the $B^{+}$ and $B^{0}$ masses are fully correlated. In the experimental
data the $B$-meson energy spread has an average value of
5.4$\mathrm{\,Me\kern-1.00006ptV}$. A systematic uncertainty of
0.25$\mathrm{\,Me\kern-1.00006ptV}$ is assigned to the measurement of
$\sigma_{E_{B}^{*}}$ based on simulation. The impact of the various systematic
uncertainties on the $\tau$ mass is added in quadrature and results in
$0.07$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

The computation of $M_{\rm min}$ relies on the knowledge of the boost vector
of the center-of-mass frame. The boost vector is measured in experimental data
using a dimuon sample. Its uncertainty is found to contribute negligibly to
the systematic uncertainty on the $\tau$ mass.

###  V.2 Reconstruction of charged particles

Several sources of systematic uncertainty impact the determination of the
charged-particle momentum corrections derived from the $D^{0}\rightarrow
K^{-}\pi^{+}$ sample. Different sources contribute to different $\cos\theta$
regions. In particular, some residual dependence of the scale factors on the
transverse momentum $p_{\rm T}$ is observed. This effect is included as a
source of systematic uncertainty by measuring the variation in the scale
factors after splitting the $D^{0}\rightarrow K^{-}\pi^{+}$ sample in $p_{\rm
T}$ at its median value of 1.3 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The
small deviation from the known value of the $D^{0}$ mass-peak observed in
simulation is also included as systematic uncertainty. Other important sources
of uncertainties include the modeling of the $D^{0}$ mass peak, the
uncertainty of the known $D^{0}$ mass, and a bias due to differences in the
$\cos\theta$ distributions of the charged particles in the
$\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ and $D^{0}\rightarrow
K^{-}\pi^{+}$ samples. As a consistency check, the scale factors are tested in
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$, $D^{0}\rightarrow
K^{-}\pi^{+}\pi^{-}\pi^{+}$, and $J/\psi\rightarrow\mu^{-}\mu^{+}$ decays. The
$D^{+}$, $D^{0}$, and $J/\psi$ peak positions match the known values within
the uncertainties, as shown in Fig. 4 for the $D^{\pm}$. Assuming that the
individual systematic effects are independent, the impact of the variations of
the momentum correction on the $\tau$ mass are added in quadrature, resulting
in 0.06${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

Figure 4: Deviation of the $D^{\pm}$ invariant-mass peak position from the
known value before (blue) and after (red) momentum corrections as a function
of the cosine of the kaon polar angle $\theta_{K}$. The vertical error bars on
the blue points indicate the statistical uncertainties in determining the peak
position while the vertical error bars on the red points indicate the
statistical and total uncertainties of the applied momentum corrections.

The correction of tracking misalignment uses cosmic-ray tracks, and di-muon
and hadronic collision-events Bilka et al. (2020). To estimate the impact of a
residual misalignment in the determination of $m_{\tau}$, various misalignment
configurations are used in the simulated data. The maximum deviation with
respect to the nominal configuration,
0.03${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, is assigned as the systematic
uncertainty due to the residual misalignment of the sub-detectors.

###  V.3 Fit model

The uncertainty of the estimator bias directly propagates to the precision of
the $\tau$ mass, resulting in an uncertainty of
0.03${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. To test the independence of
the estimator bias on the $\tau$ mass, an alternative assumption of a linear
dependence is used, which results in the same bias. Thus no additional
systematic uncertainty is assigned.

The dependence of the result on the choice of the edge parametrization is
investigated by repeating the measurement with alternative functions used
previously by the Belle and BaBar collaborations Abe et al. (2007); Aubert et
al. (2009a). The largest deviation with respect to the main result is
0.02${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and is assigned as a systematic
uncertainty.

###  V.4 Imperfections of the simulation

We study possible simulation mismodelings that might lead to an incorrect
estimation of the fit-bias correction. Differences between the properties of
material used in the simulation and those in the detector may have an impact
on the correction of the fit bias. Studies of the interaction of photons with
the detector material indicate a deficit of around $10\%$ in the density of
the beam-pipe in simulation. The impact of this deficit is tested by
increasing by $10\%$ the beam-pipe density in the simulation of a signal
sample corresponding to $4\mbox{\,ab}^{-1}$. The statistical uncertainty of
the difference between the results using the nominal simulation and the
simulation with the modified material density is
0.03${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, which is seen to be
significantly larger than the actual difference between the two models. Hence,
we assign the statistical precision of the difference as the uncertainty for
this effect.

The modeling of ISR and FSR as well as the kinematic properties of the
$\tau$-decay products may be different in simulated and experimental data. The
simulated $M_{\rm min}$ distribution is weighted according to the observed
differences between the experimental and simulated distributions in
$p^{*}_{3\pi}$. The impact on the result is found to be
0.02${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

Systematic uncertainties due to the simulation mismodeling of photon and
neutral-pion reconstruction, transverse-momentum resolution, track-finding,
trigger efficiencies, and background processes are found to be below or equal
to 0.01${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ each.

###  V.5 Consistency checks

We check the stability of the result throughout various data-taking periods
and observe no evidence for a time dependence. To exclude a potential
dependence of the measured $\tau$ mass on the kinematic properties of the
three-pion system or the $\tau$-decay products, we divide the data into sub-
regions of various kinematic variables. Specifically, we use the cosine of the
polar angle of the three-pion system and the individual pions, $M_{3\pi}$ and
$p_{3\pi}$, and the momentum of the highest-momentum decay product. We obtain
consistent results, indicating no significant unaccounted-for systematic
effects. Finally, we explicitly test for a dependence of the measurement on
the modeling of the $\tau$ decay. In the version of the TAUOLA program used
for the simulation of $\tau$ decays Jadach et al. (1993) the modeling of the
three-pion mass distribution in the
$\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ channel is based on form
factors from Ref. Kuhn and Mirkes (1992). As an alternative we use a sample
simulated with form factors based on resonance chiral-Lagrangian currents for
the hadronic $\tau$ decays Banerjee et al. (2021); Shekhovtsova et al. (2012);
Chrzaszcz et al. (2018); Nugent et al. (2013). Using 6.6$\mbox{\,ab}^{-1}$ of
simulated samples, the fit to the generator-level $M_{\rm min}$ distributions
of $\tau$ decays simulated with the two models show negligible variation in
the resulting $P_{1}$ values. The $P_{1}$ values from fits to the
reconstructed distributions are in agreement within $1.7\sigma$. Therefore no
additional source of systematic uncertainty is considered.

##  VI Summary

We measure the mass of the $\tau$ lepton to be

$m_{\tau}=1777.09\pm 0.08\pm 0.11{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
(7)

using $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ data collected with the Belle II
detector at a center-of-mass energy of
$\sqrt{s}=10.579$$\mathrm{\,Ge\kern-1.00006ptV}$ and corresponding to an
integrated luminosity of 190$\mbox{\,fb}^{-1}$. The statistical uncertainty
per unit sample size is smaller compared to the previous results Abe et al.
(2007); Aubert et al. (2009a) owing to the improved event selection and
momentum resolution of the Belle II detector, which result in a steeper slope
of the $M_{\rm min}$ distribution in the threshold region. The main sources of
systematic uncertainty arise from the knowledge of the beam energy and from
the uncertainty of the charged-particle momentum correction. As shown in Fig.
5, our result is consistent with previous measurements Ablikim et al. (2014);
Bai et al. (1996); Anashin et al. (2007); Abe et al. (2007); Aubert et al.
(2009a) and is the most precise to date.

Figure 5: Summary of the most precise $\tau$-mass measurements Bai et al.
(1996); Abe et al. (2007); Anashin et al. (2007); Aubert et al. (2009a);
Ablikim et al. (2014) compared with the result of this work. The vertical gray
band indicates the average value of previous measurements Workman et al.
(2022). The inner bars represent the statistical uncertainties, while the
outer bars indicate the total uncertainties.

This work, based on data collected using the Belle II detector, which was
built and commissioned prior to March 2019, was supported by Science Committee
of the Republic of Armenia Grant No. 20TTCG-1C010; 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 Ministry of
Education, Youth, and Sports of the Czech Republic under Contract No. LTT17020
and Charles University Grant No. SVV 260448 and 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 (France); BMBF, DFG, HGF, MPG, and AvH
Foundation (Germany); Department of Atomic Energy under Project Identification
No. RTI 4002 and Department of Science and Technology (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.
22H00144, 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. 2018R1D1A1B07047294, No.
2019R1I1A3A01058933, 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 No. 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, Agreement No. 14.W03.31.0026, 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 Ministry of Science and Technology and
research Grants No. MOST106-2112-M-002-005-MY3 and No.
MOST107-2119-M-002-035-MY3, and the 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-SC0022350, No. DE-SC0023470; and the Vietnam Academy of
Science and Technology (VAST) under Grant No. DL0000.05/21-23.

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 offsite computing support.

## References

  * Amhis et al. (2021) Y. S. Amhis et al. (Heavy Flavor Averaging Group), Eur. Phys. J. C 81, 226 (2021), updated results and plots available at https://hflav.web.cern.ch/.
  * Tsai (1971) Y.-S. Tsai, Phys. Rev. D 4, 2821 (1971).
  * Davier et al. (2006) M. Davier, A. Hocker, and Z. Zhang, Rev. Mod. Phys. 78, 1043 (2006).
  * Davier et al. (2014) M. Davier, A. Höcker, B. Malaescu, C. Z. Yuan, and Z. Zhang, Eur. Phys. J. C 74, 2803 (2014).
  * Bai et al. (1996) J. Z. Bai et al. (BES Collaboration), Phys. Rev. D 53, 20 (1996).
  * Anashin et al. (2007) V. V. Anashin et al. (KEDR Collaboration), Nucl. Phys. B Proc. Suppl. 169, 125 (2007).
  * Ablikim et al. (2014) M. Ablikim et al. (BES III Collaboration), Phys. Rev. D 90, 012001 (2014).
  * Abe et al. (2007) K. Abe et al. (Belle Collaboration), Phys. Rev. Lett. 99, 011801 (2007).
  * Aubert et al. (2009a) B. Aubert et al. (BaBar Collaboration), Phys. Rev. D 80, 092005 (2009a).
  * Albrecht et al. (1992) H. Albrecht et al. (ARGUS Collaboration), Phys. Lett. B 292, 221 (1992).
  * Abe (2010) T. Abe (Belle II Collaboration), Tech. Rep. (2010), eprint 1011.0352.
  * Akai et al. (2018) K. Akai, K. Furukawa, and H. Koiso, Nucl. Instrum. Meth. A907, 188 (2018).
  * Jadach et al. (2000) S. Jadach, B. F. L. Ward, and Z. Wa̧s, Comput. Phys. Commun. 130, 260 (2000).
  * Jadach et al. (2001) S. Jadach, B. F. L. Ward, and Z. Wa̧s, Phys. Rev. D 63, 113009 (2001).
  * Jadach et al. (1990) S. Jadach, J. H. Kuhn, and Z. Wa̧s, Comput. Phys. Commun. 64, 275 (1990).
  * Barberio et al. (1991) E. Barberio, B. van Eijk, and Z. Wa̧s, Comput. Phys. Commun. 66, 115 (1991).
  * Sjöstrand et al. (2015) T. Sjöstrand, S. Ask, J. R. Christiansen, R. Corke, N. Desai, P. Ilten, S. Mrenna, S. Prestel, C. O. Rasmussen, and P. Z. Skands, Comput. Phys. Commun. 191, 159 (2015).
  * Balossini et al. (2006) G. Balossini, C. M. Carloni Calame, G. Montagna, O. Nicrosini, and F. Piccinini, Nucl. Phys. B 758, 227 (2006).
  * Balossini et al. (2008) G. Balossini, C. Bignamini, C. M. C. Calame, G. Montagna, O. Nicrosini, and F. Piccinini, Phys. Lett. B 663, 209 (2008).
  * Carloni Calame et al. (2004) C. M. Carloni Calame, G. Montagna, O. Nicrosini, and F. Piccinini, Nucl. Phys. B Proc. Suppl. 131, 48 (2004).
  * Carloni Calame (2001) C. M. Carloni Calame, Phys. Lett. B 520, 16 (2001).
  * Carloni Calame et al. (2000) C. M. Carloni Calame, C. Lunardini, G. Montagna, O. Nicrosini, and F. Piccinini, Nucl. Phys. B 584, 459 (2000).
  * Berends et al. (1985a) F. Berends, P. Daverveldt, and R. Kleiss, Nucl. Phys. B 253, 421 (1985a).
  * Berends et al. (1985b) F. Berends, P. Daverveldt, and R. Kleiss, Nucl. Phys. B 253, 441 (1985b).
  * Berends et al. (1986) F. Berends, P. Daverveldt, and R. Kleiss, Comp. Phys. Commun. 40, 285 (1986).
  * Uehara (1996) S. Uehara (1996), eprint 1310.0157.
  * Kuhr et al. (2019) T. Kuhr, C. Pulvermacher, M. Ritter, T. Hauth, and N. Braun (Belle II Framework Software Group), Comput. Softw. Big Sci. 3, 1 (2019).
  * (28) _Belle II Analysis Software Framework (basf2)_ , https://doi.org/10.5281/zenodo.5574115.
  * Agostinelli et al. (2003) S. Agostinelli et al. (GEANT4 Collaboration), Nucl. Instrum. Meth. A 506, 250 (2003).
  * Brandt et al. (1964) S. Brandt, C. Peyrou, R. Sosnowski, and A. Wroblewski, Phys. Lett. 12, 57 (1964).
  * Farhi (1977) E. Farhi, Phys. Rev. Lett. 39, 1587 (1977).
  * Workman et al. (2022) R. L. Workman et al. (Particle Data Group), PTEP 2022, 083C01 (2022).
  * Aubert et al. (2009b) B. Aubert et al. (BaBar Collaboration), Phys. Rev. Lett. 102, 012001 (2009b).
  * Mizuk et al. (2021) R. Mizuk et al. (Belle Collaboration), JHEP 06, 137 (2021).
  * Benayoun et al. (1999) M. Benayoun, S. I. Eidelman, V. N. Ivanchenko, and Z. K. Silagadze, Mod. Phys. Lett. A 14, 2605 (1999).
  * Verkerke and Kirkby (2003) W. Verkerke and D. P. Kirkby, eConf C0303241, MOLT007 (2003).
  * Efron (1979) B. Efron, The Annals of Statistics 7, 1 (1979).
  * Bilka et al. (2020) T. Bilka et al. (Belle II PXD, SVD, Software Collaboration), EPJ Web Conf. 245, 02023 (2020).
  * Jadach et al. (1993) S. Jadach, Z. Wa̧s, R. Decker, and J. H. Kuhn, Comput. Phys. Commun. 76, 361 (1993).
  * Kuhn and Mirkes (1992) J. H. Kuhn and E. Mirkes, Z. Phys. C 56, 661 (1992).
  * Banerjee et al. (2021) S. Banerjee, D. Biswas, T. Przedzinski, and Z. Wa̧s (2021), eprint 2111.05914.
  * Shekhovtsova et al. (2012) O. Shekhovtsova, T. Przedzinski, P. Roig, and Z. Wa̧s, Phys. Rev. D 86, 113008 (2012).
  * Chrzaszcz et al. (2018) M. Chrzaszcz, T. Przedzinski, Z. Wa̧s, and J. Zaremba, Comput. Phys. Commun. 232, 220 (2018).
  * Nugent et al. (2013) I. M. Nugent, T. Przedzinski, P. Roig, O. Shekhovtsova, and Z. Wa̧s, Phys. Rev. D 88, 093012 (2013).


hypothesis that $\operatorname{c-Ulam}[y^{\prime}](x^{\prime},z^{\prime})\neq
1.$ It is then evident that
$\operatorname{c-Ulam}[y^{\prime}](x^{\prime},z^{\prime})\widehat{\cdot}\operatorname{E_{2}}[d](x^{\prime\prime},z^{\prime\prime})\neq
1$ as well. Additionally, we have $2^{k-d}\leq y^{\prime}-1<2^{k-d+1}$ as
well, since $y,$ and thus $y^{\prime},$ are not powers of $2.$ Hence, we also
have $\operatorname{c-Ulam}[y^{\prime}-1](x^{\prime},z^{\prime})\neq 1$ and
thus
$\operatorname{c-Ulam}[y^{\prime}-1](x^{\prime},z^{\prime})\widehat{\cdot}\operatorname{E_{1}}[d](x^{\prime\prime},z^{\prime\prime})\neq
1.$ From this, we immediately obtain that $\operatorname{c-Ulam}[y](x,z)\neq
1.$

Hence, in any case, if $x+z<2^{k+1}-1,$ then
$\operatorname{c-Ulam}[y](x,z)\neq 1,$ completing the inductive step.

  2. 2.

Translating to $\operatorname{c-Ulam}[y],$ we recall that we can have
exceptional aperiodic points $(x,z)\in\operatorname{Ulam}[y]$ only if
$x+z\in\\{y,y+1,y+2\\},$ hence the second part of the statement. Otherwise, we
once again know and assume that $(x,z)\in\operatorname{Ulam}[y]$ if and only
if $\operatorname{c-Ulam}[y](x,z)=1$ and $x+z\equiv 1\pmod{2}.$ It then
suffices to prove specifically that
$\operatorname{c-Ulam}[y](x,z)=\operatorname{c-Ulam}[y](\tilde{x},\tilde{z})=1$
implies $\tilde{x}+\tilde{z}\leq 2^{k+2}-y-2,$ by induction. Note that we have
already proved the other inequality $2^{k+1}-1\leq\tilde{x}+\tilde{z}$ in the
first part.

The base cases $y=2^{k}$ for some $k$ are trivial; the upper bound for $2^{k}$
is $2^{k+2}-2^{k}-2,$ which can only be violated if $\tilde{x},\tilde{z}\geq
2^{k},$ in which case we can set
$x^{\prime}=\tilde{x}-2^{k},z^{\prime}=\tilde{z}-2^{k}$ so that
$x^{\prime}+z^{\prime}>2^{k}-2.$ Then we recall by Lemma 6.5 that

$\operatorname{c-Ulam}[2^{k}](x,z)=\operatorname{c-Ulam}[2^{k}](\tilde{x},\tilde{z})=\operatorname{c-Ulam}_{3}[2^{k}](x^{\prime},z^{\prime})=\operatorname{E_{1}}[k](x^{\prime},z^{\prime})=0.$

Suppose now that $\operatorname{c-Ulam}[Y](x,z)=1$ implies
$\tilde{x}+\tilde{z}\leq 2^{K+2}-Y-2$ for all $Y<y$ (where
$K=\left\lfloor\log_{2}(Y)\right\rfloor$). Now, fix $x$ and $z$ and say that
$\tilde{x}+\tilde{z}>2^{k+2}-y-2.$ If $y$ is odd and
$\operatorname{c-Ulam}[y](x,z)=1,$ then $x+z$ must be odd and also
$\operatorname{c-Ulam}[y-1](x,z)=1,$ so for any such $x,z,$ we have

$\tilde{x}+\tilde{z}\leq 2^{k+2}-(y-1)-2=2^{k+2}-y-1.$

But since $\tilde{x}+\tilde{z}$ is odd and $2^{k+2}-y-1$ is even, we must in
fact have $\tilde{x}+\tilde{z}\leq 2^{k+2}-y-2,$ contradiction. Thus, we may
assume that $y=2^{d}y^{\prime}$ is even. Then we have

$\operatorname{c-Ulam}[y](x,z)=\operatorname{c-Ulam}[y^{\prime}-1](x^{\prime},z^{\prime})\widehat{\cdot}\operatorname{E_{1}}[d](x^{\prime\prime},z^{\prime\prime})\widehat{+}\operatorname{c-Ulam}[y^{\prime}](x^{\prime},z^{\prime})\widehat{\cdot}\operatorname{E_{2}}[d](x^{\prime\prime},z^{\prime\prime}).$

Now, observe that $2^{k-d}\leq y^{\prime}<2^{k-d+1}.$ Moreover, we also have
$2^{k-d}\leq y^{\prime}-1<2^{k-d+1},$ since $y^{\prime}$ is not a power of
$2.$ Since

$\tilde{x}+\tilde{z}=2^{d}(\tilde{x^{\prime}}+\tilde{z^{\prime}})+x^{\prime\prime}+z^{\prime\prime}>2^{k+2}-y-2,$

we must have either
$\tilde{x^{\prime}}+\tilde{z^{\prime}}>2^{k-d+2}-y^{\prime}-1$ or
$x^{\prime\prime}+z^{\prime\prime}>2^{d}-2.$ If, indeed, the first inequality
holds, then by the inductive hypothesis we in fact have that neither
$\operatorname{c-Ulam}[y^{\prime}-1](x^{\prime},z^{\prime})$ nor
$\operatorname{c-Ulam}[y^{\prime}](x^{\prime},z^{\prime})$ can equal $1.$
Thus, neither can the respective summands, nor their sum ($1$ is only a
product of two $1$’s and only a sum of a $1$ and a $0$). Hence, in this case
$\operatorname{c-Ulam}[y](x,z)\neq 1.$

If $\tilde{x^{\prime}}+\tilde{z^{\prime}}\leq 2^{k-d+2}-y^{\prime}-2,$ then we
see that $x^{\prime\prime}+z^{\prime\prime}>2^{d+1}-2,$ which is impossible
since $x^{\prime\prime},z^{\prime\prime}<2^{d}.$ Thus, the only remaining case
is that $\tilde{x^{\prime}}+\tilde{z^{\prime}}=2^{k-d+2}-y^{\prime}-1$ and
then correspondingly $x^{\prime\prime}+z^{\prime\prime}=2^{d}-2.$ It is easy
to see from here, since
$\tilde{x^{\prime}}+\tilde{z^{\prime}}>2^{k-d+2}-y^{\prime}-2,$ that
$\operatorname{c-Ulam}[y^{\prime}](x^{\prime},z^{\prime})\neq 1.$ Also,
$\tilde{x^{\prime}}+\tilde{z^{\prime}}$ is even, and thus
$\operatorname{c-Ulam}[y^{\prime}-1](x^{\prime},z^{\prime})\neq 1$ as well.
Thus, the same conclusion as before holds, implying that
$\operatorname{c-Ulam}[y](x,z)\neq 1.$

In particular, we have $\operatorname{c-Ulam}[y](x,z)\neq 1$ whenever
$x+z>2^{k+2}-y-2$ and thus the inductive step is proved.

  3. 3.

The result follows in the exact stated form for all Zumkeller $y$ by Theorem
5.1. Thus, we may assume that $y$ is not Zumkeller, so
$(x,z)\in\operatorname{Ulam}[y]$ if and only if
$\operatorname{c-Ulam}[y](x,z)=1$ and $x+z$ is odd. Since we are assuming
$x+z\equiv-1\pmod{2^{k+1}},$ the odd condition is tautological. Thus, it
suffices to prove, for any $x+z\equiv-1\pmod{2^{k+1}},$ that
$\operatorname{c-Ulam}[y](x,z)=1.$ Once again, we prove this by induction.

First, note that $x+z\equiv-1\pmod{2^{k+1}}$ is in fact equivalent to
$\tilde{x}+\tilde{z}=2^{k+1}-1.$ We will use this instead.

The base cases are $y=2^{k}$ for any $k.$ Note that we must have either
$\tilde{x}\geq 2^{k}$ or $\tilde{z}\geq 2^{k},$ but not both. Without loss of
generality, the second is true, so we have by Lemma 6.5 that

$\displaystyle\operatorname{c-Ulam}[2^{k}](x,z)$
$\displaystyle=\operatorname{c-Ulam}[2^{k}](\tilde{x},\tilde{z})=\operatorname{c-Ulam}_{2}[2^{k}](\tilde{x},\tilde{z}-2^{k})$
$\displaystyle=\operatorname{E_{1}}[k](\tilde{x},\tilde{z}-2^{k})\widehat{+}\operatorname{E_{2}}[k](\tilde{x},\tilde{z}-2^{k}).$

But we know that $\operatorname{E_{2}}[k](\tilde{x},\tilde{z}-2^{k})=1$ and
$\operatorname{E_{1}}[k](\tilde{x},\tilde{z}-2^{k})=0$ because
$\tilde{x}+\tilde{z}-2^{k}=2^{k}-1.$ Thus,
$\operatorname{c-Ulam}[2^{k}](x,z)=1,$ as desired.

Now, suppose that $\operatorname{c-Ulam}[Y](x,z)=1$ whenever
$\tilde{x}+\tilde{z}=2^{K+1}-1$ for all $Y<y$ (where
$K=\left\lfloor\log_{2}(Y)\right\rfloor$). Fix $x$ and $z$ such that
$\tilde{x}+\tilde{z}=2^{k+1}-1.$ If $y$ is odd, then

$\operatorname{c-Ulam}[y](x,z)=\operatorname{c-Ulam}[y-1](x,z)=1,$

so we may assume without loss of generality that $y=2^{d}y^{\prime}$ is even.
Then

$\operatorname{c-Ulam}[y](x,z)=\operatorname{c-Ulam}[y^{\prime}-1](x^{\prime},z^{\prime})\widehat{\cdot}\operatorname{E_{1}}[d](x^{\prime\prime},z^{\prime\prime})\widehat{+}\operatorname{c-Ulam}[y^{\prime}](x^{\prime},z^{\prime})\widehat{\cdot}\operatorname{E_{2}}[d](x^{\prime\prime},z^{\prime\prime}).$

Now, we have

$\tilde{x}+\tilde{z}=2^{d}(\tilde{x^{\prime}}+\tilde{z^{\prime}})+x^{\prime\prime}+z^{\prime\prime}=2^{k+1}-1,$

from which it is easy to deduce that
$\tilde{x^{\prime}}+\tilde{z^{\prime}}=2^{k-d+1}-1$ and
$x^{\prime\prime}+z^{\prime\prime}=2^{d}-1.$ Therefore, since $2^{k-d}\leq
y^{\prime}-1,y<2^{k-d+1}$ (as before), we see that
$\operatorname{c-Ulam}[y^{\prime}-1](x^{\prime},z^{\prime})=\operatorname{c-Ulam}[y^{\prime}](x^{\prime},z^{\prime})=1.$
Additionally, $\operatorname{E_{1}}[d](x^{\prime\prime},z^{\prime\prime})=0$
and $\operatorname{E_{2}}[d](x^{\prime\prime},z^{\prime\prime})=1,$ from which
we immediately obtain $\operatorname{c-Ulam}[y](x,z)=1,$ as desired.

  4. 4.

Once again, the result holds for all Zumkeller $y$ by Theorem 5.1. Thus, we
may assume that $(x,z)\in\operatorname{Ulam}[y]$ if and only if
$\operatorname{c-Ulam}[y](x,z)=1$ and $x+z$ is odd. Since we are assuming
$\tilde{x}+\tilde{z}=2\left\lceil 2^{k+1}-\frac{y}{2}-1\right\rceil-1,$ the
odd condition is tautological. Thus, it suffices to prove, for any
$\tilde{x}+\tilde{z}=2\left\lceil 2^{k+1}-\frac{y}{2}-1\right\rceil-1,$ that
$\operatorname{c-Ulam}[y](x,z)=1.$ We prove this by induction. We need to in
fact strengthen our inductive statement—that also
$\operatorname{c-Ulam}[y](x,z)=1$ if $y$ is even and $x+z=2^{k+2}-y-2$ (if $y$
is odd, then $2^{k+2}-y-2=2\left\lceil 2^{k+1}-\frac{y}{2}-1\right\rceil-1$).

The base cases are once again $y=2^{k}$ for any $k.$ We may assume $k>0,$ as
the case $k=0$ devolves into a triviality. Now, we have
$\tilde{x}+\tilde{z}\in\\{2^{k+2}-2^{k}-3,2^{k+2}-2^{k}-2\\}.$ Note that we
must have at least one of $\tilde{x}\geq 2^{k}$ or $\tilde{z}\geq 2^{k}.$ If
only one is true, without loss of generality the second, then we have

$\displaystyle\operatorname{c-Ulam}[2^{k}](x,z)$
$\displaystyle=\operatorname{c-Ulam}[2^{k}](\tilde{x},\tilde{z})=\operatorname{c-Ulam}_{2}[2^{k}](\tilde{x},\tilde{z}-2^{k})$
$\displaystyle=\operatorname{E_{1}}[k](\tilde{x},\tilde{z}-2^{k})\widehat{+}\operatorname{E_{2}}[k](\tilde{x},\tilde{z}-2^{k}).$

But $\tilde{x}+\tilde{z}-2^{k}\geq 2^{k+1}-3\geq 2^{k}-1$ (since $k>0$), and
thus $\operatorname{E_{1}}[k](\tilde{x},\tilde{z}-2^{k})=0.$ But
$\operatorname{E_{2}}[k]=1$ and therefore indeed
$\operatorname{c-Ulam}[2^{k}](x,z)=1.$ If, on the other hand,
$\tilde{x},\tilde{z}\geq 2^{k},$ then

$\operatorname{c-Ulam}[2^{k}](x,z)=\operatorname{c-Ulam}[2^{k}](\tilde{x},\tilde{z})=\operatorname{c-Ulam}_{3}[2^{k}](\tilde{x}-2^{k},\tilde{z}-2^{k})=\operatorname{E_{1}}[k](\tilde{x}-2^{k},\tilde{z}-2^{k}).$

Now, we have $(\tilde{x}-2^{k})+(\tilde{z}-2^{k})\in\\{2^{k}-3,2^{k}-2\\}$
(which is notably impossible for $k=1$). It thus remains to show that
$\operatorname{E_{1}}[k](\tilde{x}-2^{k},\tilde{z}-2^{k})=1,$ which is
relatively tedious, but we provide the proof here for completeness. Let
$x^{\prime}=\tilde{x}-2^{k}=x_{k-1}x_{k-2}\cdots x_{1}x_{0}$ in binary and
$z^{\prime}=\tilde{z}-2^{k}=z_{k-1}z_{k-2}\cdots z_{1}z_{0},$ and say that
$e<k$ is the largest satisfying $x^{\prime},z^{\prime}\pmod{2^{e+1}}<2^{e}.$
Then $x_{i}+z_{i}=1$ for all $i>e,$ and $x_{e}+z_{e}=0.$ Say
$x^{\prime\prime}=x\pmod{2^{e}}$ and $z^{\prime\prime}=z\pmod{2^{e}}.$ Then we
have

$2^{k}-3\leq
x^{\prime}+z^{\prime}=2^{k-1}+\cdots+2^{e+1}+x^{\prime\prime}+z^{\prime\prime},$

so $x^{\prime\prime}+z^{\prime\prime}\geq 2^{e+1}-3\geq 2^{e}-1$ unless $e=0,$
in which case $2^{e}-1=0$ and still $x^{\prime\prime}+z^{\prime\prime}\geq
2^{e}-1.$ In particular, $\operatorname{E_{1}}[k](x^{\prime},z^{\prime})=1,$
and so $\operatorname{c-Ulam}[2^{k}](x,z)=1,$ as desired.

Now, suppose that $\operatorname{c-Ulam}[Y](x,z)=1$ whenever
$\tilde{x}+\tilde{z}=2\left\lceil 2^{K+1}-\frac{Y}{2}-1\right\rceil-1$ or
$2^{K+2}-Y-2$ for all $Y<y$ (where $K=\left\lfloor\log_{2}(Y)\right\rfloor$).
Fix $x$ and $z$ such that $\tilde{x}+\tilde{z}=2\left\lceil
2^{k+1}-\frac{y}{2}-1\right\rceil-1$ or $2^{k+2}-y-2.$ If $y$ is odd, then
$\tilde{x}+\tilde{z}=2\left\lceil 2^{k+1}-\frac{y-1}{2}-1\right\rceil-1$ as
well and so by induction

$\operatorname{c-Ulam}[y](x,z)=\operatorname{c-Ulam}[y-1](x,z)=1,$

so we may assume without loss of generality that $y=2^{d}y^{\prime}$ is even.
Then

$\operatorname{c-Ulam}[y](x,z)=\operatorname{c-Ulam}[y^{\prime}-1](x^{\prime},z^{\prime})\widehat{\cdot}\operatorname{E_{1}}[d](x^{\prime\prime},z^{\prime\prime})\widehat{+}\operatorname{c-Ulam}[y^{\prime}](x^{\prime},z^{\prime})\widehat{\cdot}\operatorname{E_{2}}[d](x^{\prime\prime},z^{\prime\prime}).$

Now, we have

$\tilde{x}+\tilde{z}=2^{d}(\tilde{x^{\prime}}+\tilde{z^{\prime}})+x^{\prime\prime}+z^{\prime\prime}\in\\{2^{k+2}-y-3,2^{k+2}-y-2\\},$

from which we either have
$\tilde{x^{\prime}}+\tilde{z^{\prime}}=2^{k-d+2}-y^{\prime}-2$ and
$x^{\prime\prime}+z^{\prime\prime}\in\\{2^{d+1}-3,2^{d+1}-2\\}$ or
$\tilde{x^{\prime}}+\tilde{z^{\prime}}=2^{k-d+2}-y^{\prime}-1$ and
$x^{\prime\prime}+z^{\prime\prime}\in\\{2^{d}-3,2^{d}-2\\}.$ In both cases,
note that $2^{k-d}\leq y^{\prime}-1,y<2^{k-d+1}$ (as before). In the former
case, we see that, since $y^{\prime}$ is odd,
$\operatorname{c-Ulam}[y^{\prime}](x^{\prime},z^{\prime})=\operatorname{c-Ulam}[y^{\prime}-1](x^{\prime},z^{\prime})=1.$
Moreover, $2^{d+1}-2,2^{d+1}-3\geq 2^{d}-1$ (since $d\geq 1$) and so
$\operatorname{E_{1}}[d](x^{\prime\prime},z^{\prime\prime})=0.$ Since
$\operatorname{E_{2}}[d]=1,$ we immediately obtain
$\operatorname{c-Ulam}[y](x,z)=1\widehat{\cdot}0\widehat{+}1\widehat{\cdot}1=1.$
In the latter case, we use Item 2 to deduce that
$\operatorname{c-Ulam}[y^{\prime}](x^{\prime},z^{\prime})=0$ and the inductive
step to see that
$\operatorname{c-Ulam}[y^{\prime}-1](x^{\prime},z^{\prime})=1.$ Moreover, we
showed above that
$\operatorname{E_{1}}[d](x^{\prime\prime},z^{\prime\prime})=1$ in either case
($x^{\prime\prime}+z^{\prime\prime}$ is either $2^{d}-3$ or $2^{d}-2$) and so
$\operatorname{c-Ulam}[y](x,z)=1\widehat{\cdot}1\widehat{+}0=1,$ completing
the inductive step.∎

## Acknowledgements

The research was conducted at the 2022 University of Minnesota Duluth REU and
fully supported by the generosity of Jane Street Capital, the National
Security Agency (grant number H98230-22-1-0015), and the Stanford Mathematics
Research Center. The author is deeply grateful to Joe Gallian for organizing
this REU and for his comments on this paper. A great thank you to Noah Kravitz
for suggesting to study Ulam structures and providing a plethora of guidance
and support throughout. Thank you also to Amanda Burcroff and Colin Defant, as
well as the students and visitors of the Duluth REU, for a great program and
research environment.

##  ?refname?

  * [1] Tej Bade, Kelly Cui, Antoine Labelle, and Deyuan Li. Ulam sets in new settings. _Integers_ , 20, 2020.
  * [2] Philip Gibbs and Judson McCranie. The Ulam numbers up to one trillion. 2017. Available at https://vixra.org/abs/1711.0134.
  * [3] Noah Kravitz and Stefan Steinerberger. Ulam sequences and Ulam sets. _Integers_ , 18, 2018.
  * [4] OEIS Foundation Inc. The On-Line Encyclopedia of Integer Sequences, 2022. Published electronically at http://oeis.org.
  * [5] Vladimir Shevelev. Binomial coefficient predictors. _Journal of Integer Sequences_ , 14, 2011.
  * [6] Stefan Steinerberger. A hidden signal in the Ulam sequence. _Experimental Mathematics_ , (23):460–467, 2017.
  * [7] S. Ulam. Combinatorial analysis in infinite sets and some physical theories. _SIAM Review_ , 6(4):343–355, 1964.