The Belle II Collaboration

# Search for the $e^{+}e^{-}\to\eta_{b}(1S)\omega$ and
$e^{+}e^{-}\to\chi_{b0}(1P)\omega$ processes at
$\sqrt{s}=10.745\,\mathrm{GeV}$

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. 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 S. Bilokin
0000-0003-0017-6260 D. Biswas  0000-0002-7543-3471 A. Bobrov
0000-0001-5735-8386 D. Bodrov  0000-0001-5279-4787 A. Bolz
0000-0002-4033-9223 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 B. G. Cheon
0000-0002-8803-4429 K. Chilikin  0000-0001-7620-2053 K. Chirapatpimol
0000-0003-2099-7760 H.-E. Cho  0000-0002-7008-3759 K. Cho  0000-0003-1705-7399
S.-J. Cho  0000-0002-1673-5664 S.-K. Choi  0000-0003-2747-8277 S. Choudhury
0000-0001-9841-0216 L. Corona  0000-0002-2577-9909 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 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 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. 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 D. Epifanov  0000-0001-8656-2693 P. Feichtinger
0000-0003-3966-7497 T. Ferber  0000-0002-6849-0427 D. Ferlewicz
0000-0002-4374-1234 T. Fillinger  0000-0001-9795-7412 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 R. Garg
0000-0002-7406-4707 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 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 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 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 E. C. Hill  0000-0002-1725-7414 M. Hoek
0000-0002-1893-8764 M. Hohmann  0000-0001-5147-4781 P. Horak
0000-0001-9979-6501 C.-L. Hsu  0000-0002-1641-430X T. Humair
0000-0002-2922-9779 T. Iijima  0000-0002-4271-711X K. Inami
0000-0003-2765-7072 N. Ipsita  0000-0002-2927-3366 A. Ishikawa
0000-0002-3561-5633 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 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 H. Kakuno  0000-0002-9957-6055 D. Kalita
0000-0003-3054-1222 J. Kandra  0000-0001-5635-1000 K. H. Kang
0000-0002-6816-0751 S. Kang  0000-0002-5320-7043 G. Karyan
0000-0001-5365-3716 T. Kawasaki  0000-0002-4089-5238 F. Keil
0000-0002-7278-2860 C. Kiesling  0000-0002-2209-535X C.-H. Kim
0000-0002-5743-7698 D. Y. Kim  0000-0001-8125-9070 K.-H. Kim
0000-0002-4659-1112 Y.-K. Kim  0000-0002-9695-8103 H. Kindo
0000-0002-6756-3591 K. Kinoshita  0000-0001-7175-4182 P. Kodyš
0000-0002-8644-2349 T. Koga  0000-0002-1644-2001 S. Kohani
0000-0003-3869-6552 K. Kojima  0000-0002-3638-0266 T. Konno
0000-0003-2487-8080 A. Korobov  0000-0001-5959-8172 S. Korpar
0000-0003-0971-0968 E. Kovalenko  0000-0001-8084-1931 R. Kowalewski
0000-0002-7314-0990 T. M. G. Kraetzschmar  0000-0001-8395-2928 P. Križan
0000-0002-4967-7675 P. Krokovny  0000-0002-1236-4667 Y. Kulii
0000-0001-6217-5162 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 R. Leboucher  0000-0003-3097-6613 F. R. Le Diberder
0000-0002-9073-5689 M. J. Lee  0000-0003-4528-4601 P. Leitl
0000-0002-1336-9558 D. Levit  0000-0001-5789-6205 P. M. Lewis
0000-0002-5991-622X L. K. Li  0000-0002-7366-1307 Y. Li  0000-0002-4413-6247
Y. B. Li  0000-0002-9909-2851 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 A. Lozar
0000-0002-0569-6882 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 A. C. Manthei
0000-0002-6900-5729 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 H. Miyake
0000-0002-7079-8236 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 M. Nakao
0000-0001-8424-7075 H. Nakazawa  0000-0003-1684-6628 Y. Nakazawa
0000-0002-6271-5808 A. Narimani Charan  0000-0002-5975-550X M. Naruki
0000-0003-1773-2999 D. Narwal  0000-0001-6585-7767 Z. Natkaniec
0000-0003-0486-9291 A. Natochii  0000-0002-1076-814X L. Nayak
0000-0002-7739-914X M. Nayak  0000-0002-2572-4692 G. Nazaryan
0000-0002-9434-6197 C. Niebuhr  0000-0002-4375-9741 S. Nishida
0000-0001-6373-2346 S. Ogawa  0000-0002-7310-5079 Y. Onishchuk
0000-0002-8261-7543 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 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 B. Paschen
0000-0003-1546-4548 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 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 G. Pinna Angioni
0000-0003-0808-8281 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 P. Rados  0000-0003-0690-8100 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 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. A. Sanders  0000-0002-4902-966X 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. 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 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 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 H. Svidras  0000-0003-4198-2517 M. Takahashi
0000-0003-1171-5960 M. Takizawa  0000-0001-8225-3973 U. Tamponi
0000-0001-6651-0706 S. Tanaka  0000-0002-6029-6216 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. Tsaklidis  0000-0003-3584-4484 M. Uchida
0000-0003-4904-6168 I. Ueda  0000-0002-6833-4344 Y. Uematsu
0000-0002-0296-4028 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 K. E. Varvell  0000-0003-1017-1295 M. Veronesi
0000-0002-1916-3884 A. Vinokurova  0000-0003-4220-8056 V. S. Vismaya
0000-0002-1606-5349 L. Vitale  0000-0003-3354-2300 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 X. L. Wang  0000-0001-5805-1255 Z. Wang
0000-0002-3536-4950 A. Warburton  0000-0002-2298-7315 M. Watanabe
0000-0001-6917-6694 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 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
V. Zhilich  0000-0002-0907-5565 Q. D. Zhou  0000-0001-5968-6359 X. Y. Zhou
0000-0002-0299-4657 V. I. Zhukova  0000-0002-8253-641X R. Žlebčík
0000-0003-1644-8523

(December 20, 2023)

###### Abstract

We search for the $e^{+}e^{-}\to\eta_{b}(1S)\omega$ and
$e^{+}e^{-}\to\chi_{b0}(1P)\omega$ processes at a center-of-mass energy of
10.745 GeV, which is close to the peak of the $\Upsilon(10753)$ state. We use
data collected by the Belle II experiment during a special run, corresponding
to an integrated luminosity of $9.8\,\mathrm{fb}^{-1}$. We reconstruct
$\omega\to\pi^{+}\pi^{-}\pi^{0}$ decays and use the $\omega$ meson’s recoil
mass to search for the signals. We do not find evidence for either process,
and set upper limits on the corresponding Born-level cross sections of 2.5 pb
and 7.8 pb, respectively, at the 90% confidence level. The
$\chi_{b0}(1P)\omega$ limit is the result of a combination of this analysis
and a previous search using full reconstruction.

††preprint: APS/123-QED

##  I Introduction

Recently, the Belle experiment observed a new state, the $\Upsilon(10753)$, as
a narrow enhancement in the $e^{+}e^{-}\to\Upsilon(1S,2S,3S)\pi^{+}\pi^{-}$
cross sections [1]. Subsequently, the Belle II experiment observed a similar
structure in the $e^{+}e^{-}\to\chi_{b{1,2}}(1P)\omega$ cross sections, which
confirms the existence of the $\Upsilon(10753)$ in additional decay channels
[2]. This state has been interpreted in several ways: as an $\Upsilon(3D)$
bottomonium level mixed with nearby $S$-wave states via hadron loops [3, 4, 5,
6], a hadronic molecule with a small admixture of bottomonium [7], a hybrid
meson [8, 9, 10], or a compact tetraquark with a diquark-antidiquark structure
[11, 12]. Further studies of hadronic transitions from the $\Upsilon(10753)$
to lower bottomonia will help to understand its structure [6, 10, 11, 12, 13,
14, 15, 16, 17].

In this paper, we report on a search for the processes
$e^{+}e^{-}\to\eta_{b}(1S)\omega$ and $e^{+}e^{-}\to\chi_{b0}(1P)\omega$ at a
center-of-mass (c.m.) energy of $10.745\,\mathrm{GeV}$, which is close to the
peak of the $\Upsilon(10753)$ state. We use Belle II data corresponding to an
integrated luminosity of $9.8\,\mathrm{fb}^{-1}$ [18] collected during a
special run of the SuperKEKB collider at energies above the $\Upsilon(4S)$
resonance.

The $\eta_{b}(1S)$ and $\chi_{b0}(1P)$ mesons do not have exclusive decay
channels with a large product of efficiency and branching fraction. Thus, we
reconstruct only an $\omega$ meson in the $\pi^{+}\pi^{-}\pi^{0}$ decay and
use recoil mass,

$M_{\mathrm{recoil}}(\pi^{+}\pi^{-}\pi^{0})=\sqrt{\left(\frac{\sqrt{s}-E_{\omega}}{c^{2}}\right)^{2}-\left(\frac{p_{\omega}}{c}\right)^{2}},$
(1)

as the signal-extraction variable, where $E_{\omega}$ and $p_{\omega}$ are the
energy and momentum of the $\pi^{+}\pi^{-}\pi^{0}$ combination in the c.m.
frame.

In a previous study [2], we searched for the process
$e^{+}e^{-}\to\chi_{b0}(1P)\omega$ fully reconstructing the
$\chi_{b0}(1P)\to\Upsilon(1S)\gamma$ decay and found no significant signal.
The probability of the decay $\chi_{b0}(1P)\to\Upsilon(1S)\gamma$ is small,
thus, the sensitivity of partial reconstruction, applied in this analysis,
might be higher than that of full reconstruction.

We follow a blind approach, i.e., the analysis procedure is established before
examining the signal-variable distribution in data.

##  II Belle II detector and simulation

The analysis is based on a data sample collected with the Belle II detector
[19] at the asymmetric-energy $e^{+}e^{-}$ SuperKEKB collider [20].

The detector has a cylindrical geometry with $z$ axis approximately coincident
with the electron beam direction, which defines the forward direction. Belle
II includes a two-layer silicon-pixel detector (PXD) surrounded by a four-
layer double-sided silicon-strip detector and a 56-layer central drift chamber
(CDC). These detectors reconstruct tracks (trajectories of charged particles).
Only one sixth of the second layer of the PXD was installed for the data
analyzed here. Surrounding the CDC is a time-of-propagation counter (TOP) in
the central region, and an aerogel-based ring-imaging Cherenkov counter
(ARICH) in the forward region. These detectors provide charged-particle
identification. Surrounding the TOP and ARICH is an electromagnetic
calorimeter (ECL) made with CsI(Tl) crystals that provides energy and timing
measurements for photons and electrons. These sub-systems are surrounded by a
superconducting solenoid, providing an axial magnetic field of 1.5 T. An iron
flux return located outside the coil is instrumented with resistive plate
chambers and plastic scintillators to detect $K^{0}_{L}$ mesons and to
identify muons (KLM).

We generate signal events using EvtGen [21]. Events are generated with a
uniform distribution over angular variables and then are weighted according to
the theoretical expectations described in Appendix A. Initial-state radiation
(ISR) is simulated using the PHOKHARA generator [22] where processes at next-
to-leading order in the electromagnetic coupling are taken into account. We
assume that the signal processes proceed via the $\Upsilon(10753)$ state with
parameters from Ref. [1] and weight events accordingly. The GEANT4 package is
used to simulate the passage of the particles inside the detector and its
response [23]. All the data and simulated events are reconstructed and
analyzed using the Belle II analysis software [24, 25].

##  III Event selection

We reconstruct the $\omega$ meson using the $\omega\to\pi^{+}\pi^{-}\pi^{0}$
decay. Online event selection is based on the number of charged particles and
observed energy in an event, and is fully efficient for signal. For charged
pions, we require the distance from the beam-spot to be within 0.5 cm along
the $z$ axis and 0.3 cm in the transverse plane. Particle identification uses
the $dE/dx$ measurements in the CDC and information from TOP, ARICH, ECL, and
KLM. The corresponding likelihoods are calculated for each particle
hypothesis, and loose selections are applied to separate charged pions from
kaons, protons, and electrons.

Candidate $\pi^{0}$ mesons are reconstructed from pairs of photons, which are
energy deposits (clusters) in the ECL not matched to a track in the CDC. The
photon energy is required to exceed $50\,\mathrm{MeV}$ in the forward endcap
($12.4^{\circ}<\theta<31.4^{\circ}$) and barrel
($32.2^{\circ}<\theta<128.7^{\circ}$) regions of the ECL, and
$75\,\mathrm{MeV}$ in the backward endcap
($130.7^{\circ}<\theta<155.1^{\circ}$), because the latter region has higher
beam-induced background. To suppress beam-induced background, we require the
difference between the cluster time and the collision time to satisfy
$|\Delta{t}|<50\,\mathrm{ns}$. This requirement corresponds to approximately
two standard deviations in time resolution. To suppress background from
hadronic clusters, which are broader than electromagnetic ones, we require the
ratio of the energy deposit in a $3\times 3$ matrix of crystals to that in the
enclosing $5\times 5$ matrix in which the four corner crystals are excluded to
be greater than 0.8. The invariant mass of the selected photon pairs
$M(\gamma\gamma)$ is required to satisfy
$|M(\gamma\gamma)-m_{\pi^{0}}|<12\,\mathrm{MeV}/c^{2}$ for the
$\eta_{b}(1S)\omega$ channel and
$|M(\gamma\gamma)-m_{\pi^{0}}|<13\,\mathrm{MeV}/c^{2}$ for the
$\chi_{b0}(1P)\omega$ channel, where $m_{\pi^{0}}$ is the $\pi^{0}$ mass [26].
These requirements correspond to approximately two standard deviations in
$\pi^{0}$ mass resolution. To suppress combinatorial background from low-
energy photons, we require the momentum of the $\pi^{0}$ candidate in the c.m.
frame to exceed $260\,\mathrm{MeV}/c$ for the $\eta_{b}(1S)\omega$ channel and
$130\,\mathrm{MeV}/c$ for the $\chi_{b0}(1P)\omega$ channel. We perform a
mass-constrained kinematic fit for the $\pi^{0}$ candidates to improve the
$\pi^{0}$ momentum resolution.

The $\omega$ candidates are selected by combining $\pi^{+}$, $\pi^{-}$, and
$\pi^{0}$, and requiring the invariant mass to satisfy
$|M(\pi^{+}\pi^{-}\pi^{0})-m_{\omega}|<13\,\mathrm{MeV}/c^{2}$, where
$m_{\omega}$ is the $\omega$ mass [26]. In the Dalitz plot (DP) of the
$\omega\to\pi^{+}\pi^{-}\pi^{0}$ decay, the density of the signal decreases
towards the boundaries, while the combinatorial background is concentrated
near the boundaries. To suppress the combinatorial background, we use a
normalized distance $r$ to the center of the symmetrized DP [27] in our
candidate selection criteria. The variable $r$ takes the value $r=0$ at the DP
center and $r=1$ at its boundary. We require $r<0.84$ for the $\eta_{b}(1S)$
and $r<0.82$ for the $\chi_{b0}(1P)$.

To suppress background from continuum events $e^{+}e^{-}\to q\bar{q}$
($q=u,d,s,c$), which have a jet-like shape, we use the ratio $R_{2}$ of the
second to the zeroth order Fox Wolfram moments [28]. For $b\bar{b}$ events,
this variable peaks at approximately 0.1, while for continuum events it is
distributed almost uniformly between 0.0 and 1.0. The selection criteria are
$R_{2}<0.21$ for the $\eta_{b}(1S)$ and $R_{2}<0.28$ for the $\chi_{b0}(1P)$.

The above selection criteria are obtained by maximizing the figure of merit
(FoM), defined as $\epsilon_{\mathrm{sig}}/\sqrt{N_{\mathrm{bkg}}}$, where
$\epsilon_{\mathrm{sig}}$ is the signal efficiency determined using
simulation, and $N_{\mathrm{bkg}}$ is the background yield estimated using
data. This definition of the FoM takes into account the small signal-to-
background ratio for partial reconstruction. To find the global maximum of the
FoM, we scan each selection variable in an iterative way.

The $M_{\mathrm{recoil}}(\pi^{+}\pi^{-}\pi^{0})$ fit intervals are
$(9200,9600)\,\mathrm{MeV}/c^{2}$ and $(9780,9950)\,\mathrm{MeV}/c^{2}$ for
the $\eta_{b}(1S)$ and $\chi_{b0}(1P)$ candidates, respectively. For
reference, the $\eta_{b}(1S)$ mass is $(9398.7\pm 2.0)\,\mathrm{MeV}/c^{2}$,
and the $\chi_{bJ}(1P)$ $(J=0,1,2)$ masses are $(9859.4\pm
0.5)\,\mathrm{MeV}/c^{2}$, $(9892.8\pm 0.4)\,\mathrm{MeV}/c^{2}$, and
$(9912.2\pm 0.4)\,\mathrm{MeV}/c^{2}$, respectively [26]. The
$M_{\mathrm{recoil}}(\pi^{+}\pi^{-}\pi^{0})$ resolutions are
$15.0\,\mathrm{MeV}/c^{2}$ for the $\eta_{b}(1S)$ and
$8.4\,\mathrm{MeV}/c^{2}$ for the $\chi_{b0}(1P)$.

There are on average 1.6 and 4.0 candidates per event for the
$e^{+}e^{-}\to\eta_{b}(1S)\omega$ and $e^{+}e^{-}\to\chi_{b0}(1P)\omega$
processes, respectively. We verify that the additional candidates do not peak
in the $M_{\mathrm{recoil}}(\pi^{+}\pi^{-}\pi^{0})$ signal regions, and
therefore, we retain all candidates. The reconstruction efficiencies are 7.6%
and 7.8% for the $\eta_{b}(1S)\omega$ and $\chi_{b0}(1P)\omega$ channels,
respectively. These include corrections for the discrepancies between data and
simulation, which are discussed along with the systematic uncertainties in
Section V.

The $M(\pi^{+}\pi^{-}\pi^{0})$ distributions for the
$e^{+}e^{-}\to\eta_{b}(1S)\omega$ and $e^{+}e^{-}\to\chi_{b0}(1P)\omega$
candidates without the $\pi^{+}\pi^{-}\pi^{0}$ invariant mass requirement are
shown in Fig. 1. We perform a least-squares fit to these distributions, in
which the $\omega$-signal shape is modeled by a sum of a Gaussian function and
a double-sided Crystal Ball (CB) function [29], while the background is
described by second and third order Chebyshev polynomials for the
$\eta_{b}(1S)\omega$ and $\chi_{b0}(1P)\omega$ channels, respectively. The
parameters of the signal fit function are determined from simulation. To
account for possible data-simulation discrepancies, we introduce an overall
mass shift and a width scale factor that are determined from the fit. We find,
respectively, $(0.80\pm 0.16)\,\mathrm{MeV}/c^{2}$ and $1.07\pm 0.03$ for the
$\eta_{b}(1S)$, $(0.17\pm 0.18)\,\mathrm{MeV}/c^{2}$ and $0.99\pm 0.03$ for
the $\chi_{b0}(1P)$. The purity of the $\omega$-meson candidates is estimated
to be 12.9% and 5.3% for the $\eta_{b}(1S)\omega$ and $\chi_{b0}(1P)\omega$
channels, respectively.

Figure 1:  Distributions of $M(\pi^{+}\pi^{-}\pi^{0})$ for the
$e^{+}e^{-}\to\eta_{b}(1S)\omega$ (top) and $e^{+}e^{-}\to\chi_{b0}(1P)\omega$
(bottom) candidates. Points indicate the data; solid curves show the results
of the fit; dashed and dotted curves show the signal and background components
of the fit, respectively. Vertical lines indicate the $\omega$ signal region.

##  IV Yield measurement

The $M_{\mathrm{recoil}}(\pi^{+}\pi^{-}\pi^{0})$ distributions for the
$e^{+}e^{-}\to\eta_{b}(1S)\omega$ and $e^{+}e^{-}\to\chi_{b0}(1P)\omega$
candidates are shown in Figs. 2 and 3, respectively. We perform a $\chi^{2}$
fit to these distributions, in which the shapes of the $\eta_{b}(1S)$ and
$\chi_{bJ}(1P)$ signals are fixed to the simulation results. To model the
background, we use a 3rd-order Chebyshev polynomial for the $\eta_{b}(1S)$
channel, and the product of a 4th-order Chebyshev polynomial and a square-root
function for the $\chi_{b0}(1P)$ channel. Orders of the polynomial functions
are chosen to give the maximal $p$-value for the fit.

Based on the results of the full-reconstruction analysis [2], we find that the
expected ratio of the $\chi_{b1}(1P)$ and $\chi_{b2}(1P)$ yields with partial
reconstruction is $N_{1}/N_{2}=1.4\pm 0.7$. In an initial fit to the data we
fix $N_{1}/N_{2}=1.4$, and find $N_{1}+N_{2}=(5.5\pm 3.2)\times 10^{3}$, which
agrees with the expectation based on Ref. [2] of $(3.4\pm 1.0)\times 10^{3}$.
In the following, we fix $N_{1}+N_{2}$ to the expected value, which helps to
improve the sensitivity to the $\chi_{b0}(1P)$ signal. Thus, only the
$\eta_{b}(1S)$ and $\chi_{b0}(1P)$ yields, and background parameters, are free
in the fit. The fit results are shown in Figs. 2 and 3. We use
$1\,\mathrm{MeV}/c^{2}$ bins for fitting and $10\,\mathrm{MeV}/c^{2}$ or
$5\,\mathrm{MeV}/c^{2}$ bins for visualization. No significant signals are
observed; the obtained $\eta_{b}(1S)$ and $\chi_{b0}(1P)$ yields are given in
Table 1.

Figure 2:  Distribution of $M_{\mathrm{recoil}}(\pi^{+}\pi^{-}\pi^{0})$ for the $e^{+}e^{-}\to\eta_{b}(1S)\omega$ candidates. Top: data points with the fit function overlaid. Bottom: the same distributions with the background component of the fit function subtracted. The solid histogram shows the fit function for the best fit; the dashed histogram shows the same function with the yield fixed to the upper limit. Figure 3:  Distribution of $M_{\mathrm{recoil}}(\pi^{+}\pi^{-}\pi^{0})$ for the $e^{+}e^{-}\to\chi_{b0}(1P)\omega$ candidates. Symbols are the same as those in Fig. 2. The $\chi_{b1}(1P)$ and $\chi_{b2}(1P)$ contributions between 9.88 and $9.94\,\mathrm{GeV}/c^{2}$ are discussed in the text.  Table 1: Signal yields and Born-level cross sections (central values and upper limits at the 90% CL) for the processes $e^{+}e^{-}\to\eta_{b}(1S)\omega$ and $e^{+}e^{-}\to\chi_{b0}(1P)\omega$. | $\eta_{b}(1S)\omega$ | $\chi_{b0}(1P)\omega$
---|---|---
Yield ($10^{3}$) | $0.23\pm 0.49\pm 0.25$ | $1.2\pm 1.4\pm 0.9$
Born cross section (pb) | $0.5\pm 1.1\pm 0.6$ | $2.6\pm 3.1\pm 2.0$
Upper limit (pb) | $<2.5$ | $<8.7$

##  V Born cross sections and systematic uncertainties

The Born-level cross sections are calculated as

$\sigma_{\mathrm{B}}=\frac{N\,|1-\Pi|^{2}}{\epsilon\,\mathcal{L}\,(1+\delta_{\mathrm{ISR}})\,\mathcal{B}_{\mathrm{int}}},$
(2)

where $N$ is the signal yield, $\epsilon$ is the reconstruction efficiency,
$\mathcal{L}$ is the integrated luminosity; $|1-\Pi|^{2}=0.93$ is the vacuum
polarization [30], $\mathcal{B}_{\mathrm{int}}$ is a product of the
$\mathcal{B}(\omega\to\pi^{+}\pi^{-}\pi^{0})$ and
$\mathcal{B}(\pi^{0}\to\gamma\gamma)$ branching fractions,
$(1+\delta_{\mathrm{ISR}})=0.63$ is the radiative correction calculated using
the Kuraev-Fadin radiator [31] assuming production via $\Upsilon(10753)$.

The dominant contributions to the systematic uncertainty in the yields are
listed in Table 2. We vary the $\eta_{b}(1S)$ and $\chi_{b0}(1S)$ masses by
one standard deviation around their values [26]. The effect of variation of
the $\eta_{b}(1S)$ width is negligibly small. We vary the c.m. energy in Eq.
(1) by $\pm 1\,\mathrm{MeV}$ to account for the uncertainty in its calibration
[32]. We use the shifts and scale factors of the $M(\pi^{+}\pi^{-}\pi^{0})$
fits, determined in Section III, to calibrate the momentum resolution in the
signal simulation; the resulting change of the yields is negligibly small. The
peak position and the ISR tail of the signal function depend on the shape of
the cross section as a function of collision energy. We assume that the signal
cross section is constant in energy instead of considering the resonant
production via $\Upsilon(10753)$. Since $N$ and $(1+\delta_{\mathrm{ISR}})$
are correlated when the shape of the cross-section is varied, we calculate the
deviation of the ratio $N/(1+\delta_{\mathrm{ISR}})$. For the
$\chi_{b0}(1P)\omega$ channel, we vary the expected $\chi_{b1}(1P)$ and
$\chi_{b2}(1P)$ yields $N_{1}=(1.9\pm 0.4)\times 10^{3}$ and $N_{2}=(1.4\pm
0.6)\times 10^{3}$ according to their uncertainties considering their $-0.57$
correlation [2]. To estimate the contribution from the assumptions on the
background shape, we vary both boundaries of the $\eta_{b}(1S)$ fit interval
simultaneously by $\pm 50\,\mathrm{MeV}/c^{2}$, and the lower boundary of the
$\chi_{b0}(1P)$ fit interval by $\pm 50\,\mathrm{MeV}/c^{2}$. We also increase
the polynomial order by one. For the background-shape source, we consider the
root-mean-square spread of the deviations to be the corresponding systematic
uncertainty. For other sources, we use maximal deviations. The total
systematic uncertainty in the yields, shown in Table 2, is estimated as a sum
of the various contributions in quadrature.

Table 2: Systematic uncertainties in the yields for the processes $e^{+}e^{-}\to\eta_{b}(1S)\omega$ and $e^{+}e^{-}\to\chi_{b0}(1P)\omega$ (in units of $10^{3}$). |  $\eta_{b}(1S)\omega$ |  $\chi_{b0}(1P)\omega$
---|---|---
$\eta_{b}(1S)$ / $\chi_{b0}(1P)$ mass | $0.05$ | $0.08$
Collision-energy calibration | $0.02$ | $0.19$
Cross-section shape | $0.01$ | $0.13$
$\chi_{b1}(1P)$ and $\chi_{b2}(1P)$ yields | $-$ | $0.27$
Background shape | $0.24$ | $0.85$
Total | $0.25$ | $0.92$

A summary of the multiplicative uncertainties is presented in Table 3. The
possible discrepancies between data and simulation contribute to the
uncertainty in the reconstruction efficiency. The uncertainty in track
reconstruction efficiency is estimated using the
$\bar{B}^{0}\to{D}^{*+}(\to{D}^{0}\pi^{+})\pi^{-}$ decays for low-momentum
particles and using the $e^{+}e^{-}\to\tau^{+}\tau^{-}$ process for mid-to-
high momentum particles. The uncertainty in the efficiency of the PID
requirements is determined using ${K^{0}_{\rm S}}\to\pi^{+}\pi^{-}$,
$D^{*+}\to{}D^{0}(\to{}K^{-}\pi^{+})\pi^{+}$, and $\Lambda^{0}\to{}p\pi^{-}$
data samples. The uncertainty in the $\pi^{0}$ reconstruction efficiency is
estimated by comparing reconstructed $\eta\to\pi^{0}\pi^{0}\pi^{0}$,
$\eta\to\pi^{+}\pi^{-}\pi^{0}$, and $\eta\to\gamma\gamma$ decays. The
efficiencies of the $R_{2}$ requirements in the $\eta_{b}(1S)\omega$ and
$\chi_{b0}(1P)\omega$ channels are 82% and 91%, respectively. We assume that
the relative uncertainty in these efficiencies is 10%. The uncertainty in
luminosity is measured using Bhabha and $\gamma\gamma$ events [18]. The
uncertainties in the $\omega\to\pi^{+}\pi^{-}\pi^{0}$ and
$\pi^{0}\to\gamma\gamma$ branching fractions are taken from Ref. [26].

Table 3: Multiplicative systematic uncertainties for the measurement of the $e^{+}e^{-}\to\eta_{b}(1S)\omega$ and $e^{+}e^{-}\to\chi_{b0}(1P)\omega$ cross sections (in %). |  $\eta_{b}(1S)\omega$ |  $\chi_{b0}(1P)\omega$
---|---|---
Track reconstruction efficiency | 1.6 | 2.4
PID efficiency | 0.8 | 1.0
$\pi^{0}$ reconstruction efficiency | 3.2 | 7.3
$R_{2}$ efficiency | 10.0 | 10.0
Luminosity | 0.6 | 0.6
$\mathcal{B}(\omega\to\pi^{+}\pi^{-}\pi^{0})\,\mathcal{B}(\pi^{0}\to\gamma\gamma)$ | 0.7 | 0.7
Total multiplicative uncertainty | 10.7 | 12.7

The yield uncertainty $\delta_{N}$, obtained by adding the corresponding
statistical and systematic uncertainties in quadrature, is combined with the
multiplicative uncertainty $\delta$ using the following formula:

$(N\pm\delta_{N})\times(1\pm\delta)=N\pm(\delta_{N}\oplus{}N\delta\oplus\delta_{N}\delta),$
(3)

where the symbol $\oplus$ denotes addition in quadrature. The estimated Born-
level cross sections and upper limits at the 90% confidence level (CL) set
using the likelihood-ratio ordering method [33] are presented in Table 1.

##  VI Conclusions

We report a search for the $e^{+}e^{-}\to\eta_{b}(1S)\omega$ and
$e^{+}e^{-}\to\chi_{b}(1S)\omega$ processes at
$\sqrt{s}=10.745\,\mathrm{GeV}$. No significant signals are observed, and we
set the following 90% CL upper limits on Born-level cross sections:

$\begin{split}\sigma_{\mathrm{B}}(e^{+}e^{-}\to\eta_{b}(1S)\omega)&<2.5\,\mathrm{pb},\\\
\sigma_{\mathrm{B}}(e^{+}e^{-}\to\chi_{b0}(1P)\omega)&<8.7\,\mathrm{pb}.\\\
\end{split}$ (4)

The upper limit on the $e^{+}e^{-}\to\chi_{b0}(1P)\omega$ cross section is
comparable to the upper limit obtained using full reconstruction of 11.3 pb
[2]. We combine the two results, taking into account correlations, to obtain

$\sigma_{\mathrm{B}}(e^{+}e^{-}\to\chi_{b0}(1P)\omega)<7.8\,\mathrm{pb}.$ (5)

The tetraquark model of Ref. [11] predicts that the decay rate of
$\Upsilon(10753)\to\eta_{b}(1S)\omega$ is strongly enhanced compared to the
decay rates of $\Upsilon(10753)\to\Upsilon(nS)\pi^{+}\pi^{-}$. The obtained
upper limit on $\sigma_{\mathrm{B}}(\eta_{b}(1S)\omega)$ is close to the
measured values of $\sigma_{\mathrm{B}}(\Upsilon(nS)\pi^{+}\pi^{-})$, which
are in the range $(1-3)\,\mathrm{pb}$ [1]. Thus, our results do not support
the tetraquark-model prediction that the
$\Upsilon(10753)\to\eta_{b}(1S)\omega$ decay is enhanced [11]. In the $4S-3D$
mixing model, the decay rate of $\Upsilon(10753)\to\eta_{b}(1S)\omega$ is
smaller than the decay rate of $\Upsilon(10753)\to\Upsilon(nS)\pi^{+}\pi^{-}$
by a factor $0.2-0.4$ [17]; our upper limit is consistent with this
expectation.

The upper limit on the $\chi_{b0}(1P)\omega$ cross section is higher than the
measured $\chi_{b1}(1P)\omega$ and $\chi_{b2}(1P)\omega$ cross sections of
$(3.6\pm 0.9)\,\textrm{pb}$ and $(2.8\pm 1.3)\,\textrm{pb}$, respectively [2].
For a $4S-3D$ mixed state, the decay rate to $\chi_{b0}(1P)\omega$ is expected
to be comparable to the decay rates to $\chi_{b1}(1P)\omega$ and
$\chi_{b2}(1P)\omega$ [14]; our upper limit is consistent with this
expectation. In the charmonium sector, the decay of the $Y(4230)$ state to
$\chi_{c0}\omega$ is enhanced compared to the decays to $\chi_{c1}\omega$ and
$\chi_{c2}\omega$ [34]. We do not find an analogous enhancement in the decay
pattern of $\Upsilon(10753)$, which may indicate that $Y(4230)$ and
$\Upsilon(10753)$ have different structures.

##  VII Acknowledgements

We are grateful to A.I. Milstein for valuable discussions. 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 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. 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, 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; 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-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 the excellent operation of the accelerator and
their special efforts to accomplish the center-of-mass energy scan that made
these results possible; 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.

##  Appendix A Angular distributions

To estimate the angular distributions of the studied processes, we use a
nonrelativistic approximation. The amplitude of the
$e^{+}e^{-}\to\eta_{b}(1S)\omega(\to\pi^{+}\pi^{-}\pi^{0})$ process can be
written as

$T=[\textbf{e}_{1}\times\textbf{p}_{\omega}]\cdot\bm{\Lambda},$ (6)

where $\textbf{e}_{1}$ is the polarization vector of the virtual photon
produced in the $e^{+}e^{-}$ annihilation,
$\textbf{p}_{\omega}=\textbf{p}_{1}+\textbf{p}_{2}+\textbf{p}_{3}$ is the
total momentum of the three pions measured in the c.m. frame, and
$\bm{\Lambda}$ is defined as

$\bm{\Lambda}=E_{1}\,[\textbf{p}_{2}\times\textbf{p}_{3}]-E_{2}\,[\textbf{p}_{1}\times\textbf{p}_{3}]+E_{3}\,[\textbf{p}_{1}\times\textbf{p}_{2}],$
(7)

where $E_{i}$ are the energies of the pions measured in the c.m. frame. For
low-momentum $\omega$, the vector $\bm{\Lambda}$ is perpendicular to the
$\omega$ decay plane. Summation over $\textbf{e}_{1}$ gives the formula for
magnitude

$|T|^{2}=\bm{\Lambda}^{2}\textbf{p}_{\omega}^{2}-(\textbf{p}_{\omega}\cdot\bm{\Lambda})^{2}-([\textbf{n}_{1}\times\textbf{p}_{\omega}]\cdot\bm{\Lambda})^{2},$
(8)

where $\textbf{n}_{1}$ is the unit vector along the $e^{-}$ beam direction.
This formula corresponds to a $1+\cos^{2}\theta$ distribution in the $\omega$
production angle $\theta$ (the angle between the beam direction and the
$\omega$ momentum in the c.m. frame) and a $1+\cos^{2}\alpha$ distribution in
the angle $\alpha$ between the normal vector to the $\omega$ decay plane and
the momentum of the beams measured in the $\omega$ rest frame.

The amplitudes for the
$e^{+}e^{-}\to\chi_{bJ}\omega(\to\pi^{+}\pi^{-}\pi^{0})$ $(J=0,1,2)$ processes
can be written as

$\begin{split}T_{0}=&(\textbf{e}_{1}\cdot\bm{\Lambda})A_{0},\\\
T_{1}=&[\textbf{e}_{1}\times\bm{\Lambda}]\cdot\textbf{A}_{1},\\\
T_{2}=&e^{i}_{1}\Lambda^{j}A_{2}^{ij},\\\ \end{split}$ (9)

where $A_{0}$, $\textbf{A}_{1}$, and $A^{ij}_{2}$ are $\chi_{b0,1,2}$ wave
functions, respectively. After summing over the $\textbf{e}_{1}$ projections
the magnitudes are

$\begin{split}|T_{0}|^{2}=&\bm{\Lambda}^{2}-(\textbf{n}_{1}\cdot\bm{\Lambda})^{2},\\\
{|T_{1}|^{2}}=&\bm{\Lambda}^{2}+(\textbf{n}_{1}\cdot\bm{\Lambda})^{2},\\\
{|T_{2}|^{2}}=&7\bm{\Lambda}^{2}-(\textbf{n}_{1}\cdot\bm{\Lambda})^{2}.\\\
\end{split}$ (10)

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