The Belle II Collaboration

# Search for an invisible $Z^{\prime}$ in a final state with two muons and
missing energy at Belle II

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 T. Aushev  0000-0002-6347-7055 V. Aushev
0000-0002-8588-5308 H. Bae  0000-0003-1393-8631 S. Bahinipati
0000-0002-3744-5332 P. Bambade  0000-0001-7378-4852 Sw. Banerjee
0000-0001-8852-2409 J. Baudot  0000-0001-5585-0991 M. Bauer
0000-0002-0953-7387 A. Baur  0000-0003-1360-3292 A. Beaubien
0000-0001-9438-089X J. Becker  0000-0002-5082-5487 P. K. Behera
0000-0002-1527-2266 J. V. Bennett  0000-0002-5440-2668 E. Bernieri
0000-0002-4787-2047 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 D. Bodrov  0000-0001-5279-4787 A. Bolz
0000-0002-4033-9223 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 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 C. Chen  0000-0003-1589-9955 Y. Q. Chen
0000-0002-7285-3251 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 D. Cinabro
0000-0001-7347-6585 L. Corona  0000-0002-2577-9909 S. Cunliffe
0000-0003-0167-8641 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 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 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 C. Finck
0000-0002-5068-5453 G. Finocchiaro  0000-0002-3936-2151 A. Fodor
0000-0002-2821-759X F. Forti  0000-0001-6535-7965 B. G. Fulsom
0000-0002-5862-9739 E. Ganiev  0000-0001-8346-8597 M. Garcia-Hernandez
0000-0003-2393-3367 V. Gaur  0000-0002-8880-6134 A. Gaz  0000-0001-6754-3315
A. Gellrich  0000-0003-0974-6231 G. Ghevondyan  0000-0003-0096-3555 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 P. Goldenzweig  0000-0001-8785-847X 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 J. Guilliams
0000-0001-8229-3975 H. Haigh  0000-0003-1567-0907 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 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. 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 K. K. Joo  0000-0002-5515-0087 H. Junkerkalefeld
0000-0003-3987-9895 H. Kakuno  0000-0002-9957-6055 A. B. Kaliyar
0000-0002-2211-619X K. H. Kang  0000-0002-6816-0751 S. Kang
0000-0002-5320-7043 R. Karl  0000-0002-3619-0876 G. Karyan
0000-0001-5365-3716 C. Kiesling  0000-0002-2209-535X C.-H. Kim
0000-0002-5743-7698 D. Y. Kim  0000-0001-8125-9070 K.-H. Kim
0000-0002-4659-1112 Y.-K. Kim  0000-0002-9695-8103 H. Kindo
0000-0002-6756-3591 K. Kinoshita  0000-0001-7175-4182 P. Kodyš
0000-0002-8644-2349 T. Koga  0000-0002-1644-2001 S. Kohani
0000-0003-3869-6552 K. Kojima  0000-0002-3638-0266 T. Konno
0000-0003-2487-8080 A. Korobov  0000-0001-5959-8172 S. Korpar
0000-0003-0971-0968 E. Kovalenko  0000-0001-8084-1931 R. Kowalewski
0000-0002-7314-0990 T. M. G. Kraetzschmar  0000-0001-8395-2928 P. Križan
0000-0002-4967-7675 P. Krokovny  0000-0002-1236-4667 T. Kuhr
0000-0001-6251-8049 J. Kumar  0000-0002-8465-433X 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 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 C. Li  0000-0002-3240-4523 L. K. Li  0000-0002-7366-1307
J. Libby  0000-0002-1219-3247 K. Lieret  0000-0003-2792-7511 Z. Liptak
0000-0002-6491-8131 Q. Y. Liu  0000-0002-7684-0415 D. Liventsev
0000-0003-3416-0056 S. Longo  0000-0002-8124-8969 A. Lozar
0000-0002-0569-6882 T. Lueck  0000-0003-3915-2506 T. Luo  0000-0001-5139-5784
C. Lyu  0000-0002-2275-0473 M. Maggiora  0000-0003-4143-9127 R. Maiti
0000-0001-5534-7149 S. Maity  0000-0003-3076-9243 R. Manfredi
0000-0002-8552-6276 E. Manoni  0000-0002-9826-7947 S. Marcello
0000-0003-4144-863X C. Marinas  0000-0003-1903-3251 L. Martel
0000-0001-8562-0038 A. Martini  0000-0003-1161-4983 T. Martinov
0000-0001-7846-1913 L. Massaccesi  0000-0003-1762-4699 M. Masuda
0000-0002-7109-5583 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 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 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. Moneta
0000-0003-2184-7510 H.-G. Moser  0000-0003-3579-9951 M. Mrvar
0000-0001-6388-3005 R. Mussa  0000-0002-0294-9071 I. Nakamura
0000-0002-7640-5456 K. R. Nakamura  0000-0001-7012-7355 M. Nakao
0000-0001-8424-7075 H. Nakayama  0000-0002-2030-9967 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 N. K. Nisar
0000-0001-9562-1253 S. Nishida  0000-0001-6373-2346 S. Ogawa
0000-0002-7310-5079 H. Ono  0000-0003-4486-0064 Y. Onuki  0000-0002-1646-6847
P. Oskin  0000-0002-7524-0936 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 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 I. Peruzzi  0000-0001-6729-8436 R. Peschke
0000-0002-2529-8515 R. Pestotnik  0000-0003-1804-9470 M. Piccolo
0000-0001-9750-0551 L. E. Piilonen  0000-0001-6836-0748 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 L. Polat
0000-0002-2260-8012 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 A. Ramirez
Morales  0000-0001-8821-5708 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 D. Rodríguez Pérez  0000-0001-8505-649X 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 J. Schueler  0000-0002-2722-6953 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 C. P. Shen  0000-0002-9012-4618 T. Shillington
0000-0003-3862-4380 J.-G. Shiu  0000-0002-8478-5639 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 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 S. Stefkova  0000-0003-2628-530X Z. S. Stottler
0000-0002-1898-5333 R. Stroili  0000-0002-3453-142X J. Strube
0000-0001-7470-9301 Y. Sue  0000-0003-2430-8707 M. Sumihama
0000-0002-8954-0585 K. Sumisawa  0000-0001-7003-7210 W. Sutcliffe
0000-0002-9795-3582 S. Y. Suzuki  0000-0002-7135-4901 H. Svidras
0000-0003-4198-2517 M. Takizawa  0000-0001-8225-3973 U. Tamponi
0000-0001-6651-0706 K. Tanida  0000-0002-8255-3746 H. Tanigawa
0000-0003-3681-9985 N. Taniguchi  0000-0002-1462-0564 F. Tenchini
0000-0003-3469-9377 A. Thaller  0000-0003-4171-6219 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 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 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 A. Vinokurova
0000-0003-4220-8056 L. Vitale  0000-0003-3354-2300 V. Vobbilisetti
0000-0002-4399-5082 H. M. Wakeling  0000-0003-4606-7895 E. Wang
0000-0001-6391-5118 M.-Z. Wang  0000-0002-0979-8341 X. L. Wang
0000-0001-5805-1255 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 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 H. Ye  0000-0003-0552-5490 J. H. Yin  0000-0002-1479-9349
Y. M. Yook  0000-0002-4912-048X 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
Y. Zhang  0000-0003-2961-2820 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

###### Abstract

The $L_{\mu}-L_{\tau}$ extension of the standard model predicts the existence
of a lepton-flavor-universality-violating $Z^{\prime}$ boson that couples only
to the heavier lepton families. We search for such a $Z^{\prime}$ through its
invisible decay in the process
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}Z^{\prime}$. We use a sample of electron-
positron collisions at a center-of-mass energy of 10.58 GeV collected by the
Belle II experiment in 2019–2020, corresponding to an integrated luminosity of
79.7 fb-1. We find no excess over the expected standard-model background. We
set 90$\%$-confidence-level upper limits on the cross section for this process
as well as on the coupling of the model, which ranges from $3\times 10^{-3}$
at low $Z^{\prime}$ masses to 1 at $Z^{\prime}$ masses of 8
${\mathrm{\,Ge\kern-1.00006ptV\\!/}\textit{c}^{2}}$.

Recent experimental observations are in tension with the standard model (SM)
of particle physics. A notable example is the difference between the measured
and expected values of the muon anomalous magnetic moment Bennett et al.
(2006); Abi et al. (2021). In addition, the SM is known to provide an
incomplete description of nature since, among other prominent issues, it does
not address the phenomenology related to the existence of dark matter Bertone
et al. (2005), specifically the prediction of the observed relic density. A
simple way to explain both phenomena is the $L_{\mu}-L_{\tau}$ extension of
the SM He et al. (1991); Shuve and Yavin (2014); Altmannshofer et al. (2016).
This model gauges the difference of the muon and tau lepton numbers, giving
rise to a massive, electrically neutral, vector boson, the $Z^{\prime}$. This
particle would couple to the SM only through $\mu$, $\tau$, $\nu_{\mu}$, and
$\nu_{\tau}$ with coupling $g^{\prime}$. The $Z^{\prime}$, with such a lepton-
flavor-universality-violating coupling, would contribute to the muon magnetic
moment and, for certain values of $g^{\prime}$ and $M_{Z^{\prime}}$, would
explain the observed anomaly Altmannshofer et al. (2014). This model also
reproduces the observed dark-matter relic density, assuming dark matter is
charged under $L_{\mu}-L_{\tau}$ Shuve and Yavin (2014); Curtin et al. (2015),
and resolves the tensions in $b\rightarrow s\ell^{+}\ell^{-}\,(\ell=e,\mu)$
decays reported by the LHCb collaboration Crivellin et al. (2022); Aaij et al.
(2014, 2022); Sala and Straub (2017).

In this Letter, we report a search for the $Z^{\prime}$ in the process
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}Z^{\prime}$ with
$Z^{\prime}\rightarrow\text{invisible}$, where the $Z^{\prime}$ is radiated
off a muon. If the $Z^{\prime}$ couples only to SM particles, a model
henceforth referred to as the “vanilla” $L_{\mu}-L_{\tau}$ model, the
branching fraction to neutrinos
$\cal{B}\rm(Z^{\prime}\rightarrow\nu\bar{\nu})$ varies between 33% and 100%
depending on the $Z^{\prime}$ mass Araki et al. (2017); Bauer et al. (2018).

If the $Z^{\prime}$ can decay directly into a pair of dark matter particles
$\chi\bar{\chi}$ with a coupling constant $\alpha_{\rm D}={{g^{\prime}_{\rm
D}}^{2}}/{4\pi}$ one expects $g^{\prime}_{\rm D}\gg g^{\prime}$, which implies
$\cal{B}\rm(Z^{\prime}\rightarrow\chi\bar{\chi})\approx 1$: we henceforth
refer to this possibility as the “fully invisible” $L_{\mu}-L_{\tau}$ model.
We provide results for each scenario. In the vanilla model, the intrinsic
width $\Gamma_{Z^{\prime}}$ of the $Z^{\prime}$ is negligible compared to the
experimental resolution. In the fully invisible model, $\Gamma_{Z^{\prime}}$
depends on $\alpha_{\rm D}$: it is negligible for values of $\alpha_{\rm D}$
smaller than 1 for $M_{Z^{\prime}}$ $\approx$ 1.5
${\mathrm{\,Ge\kern-1.00006ptV\\!/}\textit{c}^{2}}$, and smaller than 0.1 for
$M_{Z^{\prime}}$ $\approx$ 4.5
${\mathrm{\,Ge\kern-1.00006ptV\\!/}\textit{c}^{2}}$. We focus our analysis on
the case in which $\cal{B}\rm(Z^{\prime}\rightarrow\chi\bar{\chi})\approx 1$
and $\Gamma_{Z^{\prime}}$ is negligible. We study separately one example in
which $\Gamma_{Z^{\prime}}$ is not negligible and assume one benchmark value
such that $\Gamma_{Z^{\prime}}=0.1M_{Z^{\prime}}$, corresponding to
$\alpha_{\rm D}=2.9$.

Searches for a $Z^{\prime}$ decaying to muons have been performed by the BABAR
Lees et al. (2016), Belle Czank et al. (2022), and CMS Sirunyan et al. (2019)
experiments, and for low-mass invisible $Z^{\prime}$ decays by the NA64-$e$
experiment Andreev et al. (2022). Belle II has previously conducted a search
for a $Z^{\prime}$ decaying invisibly using data collected during the
commissioning run in 2018, with a luminosity of 0.276 fb-1 Adachi et al.
(2020).

We use a sample of $e^{+}e^{-}$ collisions collected by Belle II at the
center-of-mass (c.m.) energy of the $\mathchar 28935\relax$(4S) resonance,
$10.58\mathrm{\,Ge\kern-1.00006ptV}$, in 2019–2020, corresponding to a total
integrated luminosity of 79.7 fb-1 Abudinén et al. (2020). This search
supersedes that in Ref. Adachi et al. (2020), with an integrated luminosity
nearly 300 times larger, improved muon identification, and the use of refined
analysis algorithms.

The invisible-$Z^{\prime}$ signature is a narrow enhancement in the
distribution of the mass $M_{\rm{recoil}}$ of the system recoiling against a
muon pair. In the following, recoil quantities are computed against the two
muons. These quantities coincide with $Z^{\prime}$ properties for signal
events and typically correspond to undetected SM particles for background. In
the analysis, we use the recoil mass squared $M^{2}_{\rm{recoil}}$ since this
quantity is smoother than $M_{\rm{recoil}}$ for low masses, where negative
$M^{2}_{\rm{recoil}}$ values may be observed due to resolution effects. We
select events with exactly two charged particles identified as muons, and
negligible additional activity in the detector. The dominant backgrounds are
final states with two muons and missing energy. These are primarily
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}(\gamma)$ events with one or more
undetected photons, $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}(\gamma)$ events
with both $\tau$ leptons decaying to muons and neutrinos, and
$e^{+}e^{-}\rightarrow e^{+}e^{-}\mu^{+}\mu^{-}$ events with electrons outside
the detector acceptance.

We extract the signal yield from a fit to the two-dimensional distribution of
the polar angle of the recoil momentum $\theta_{\rm{recoil}}$ and
$M^{2}_{\rm{recoil}}$. Control samples are used to check simulation
predictions and to infer correction factors. Selections are optimized using
simulated events prior to examining data. However, once data were examined, we
found an unexpected background source in the signal region that was taken into
account with a rescaling procedure.

The Belle II detector Abe et al. (2010) operates at the interaction region of
the SuperKEKB electron-positron collider Akai et al. (2018), located at KEK in
Tsukuba, Japan. The energies of the electron and positron beams are
$7\mathrm{\,Ge\kern-1.00006ptV}$ and $4\mathrm{\,Ge\kern-1.00006ptV}$ with a
boost of the c.m. frame $\beta\gamma=0.28$ relative to the laboratory frame.
The detector consists of several subdetectors arranged around the beam pipe in
a cylindrical structure. Subdetectors relevant for this analysis are briefly
described here in order from innermost out; more details are given in Ref. Abe
et al. (2010); Kou et al. (2019). The innermost component is the vertex
detector, consisting of two inner layers of silicon pixels and four outer
layers of silicon strips. The second pixel layer is partially installed,
covering one sixth of the azimuthal angle. The main tracking subdetector is a
large helium-based small-cell drift chamber. An electromagnetic calorimeter
(ECL) consists of a barrel and two endcaps made of CsI(Tl) crystals. A
superconducting solenoid provides a 1.5 T magnetic field. A $K^{0}_{L}$ and
muon subdetector is made of iron plates providing the magnetic flux-return
yoke, alternated with resistive-plate chambers and plastic scintillators in
the barrel, and with plastic scintillators only in the endcaps. The
longitudinal and transverse directions, and polar angle $\theta$ are defined
with respect to the detector’s cylindrical axis in the direction of the
electron beam. In the following, quantities are defined in the laboratory
frame unless otherwise specified.

Identification of muons relies mostly on charged-particle penetration depth in
the muon detector for momenta larger than 0.7 GeV/$c$ and on information from
the drift chamber and ECL otherwise. The ratio between the muon likelihood and
the sum of the likelihoods of all particle hypotheses is required to be
greater than 0.5. This retains 93–99% of muons, and rejects 80–97% of pions,
depending on their momenta. Electrons, used in control-sample studies, are
identified primarily by comparing momenta with energies of associated ECL
depositions, with a similar likelihood-ratio method. Photons are reconstructed
from ECL depositions with energy greater than 100 MeV that are not associated
with any track.

Signal events are simulated using MadGraph 5 Alwall et al. (2014) with initial
state radiation. The signal $M^{2}_{\rm{recoil}}$ resolution ranges from a
minimum of 0.06 $\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}\textit{c}^{4}$ at 80
$\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}\textit{c}^{4}$ to a maximum of 0.23
$\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}\textit{c}^{4}$ at 9
$\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}\textit{c}^{4}$. We generate 582
$Z^{\prime}$ samples, with negligible $\Gamma_{Z^{\prime}}$, corresponding to
mass hypotheses ranging from 0.01 to
$8.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}\textit{c}^{2}}$ in steps of 3 to 202
${\mathrm{\,Me\kern-1.00006ptV\\!/}\textit{c}^{2}}$, following the resolution.
Background events are simulated using the following generators:
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}(\gamma)$ with KKMC Jadach et al. (2000),
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}(\gamma)$ with KKMC Jadach et al. (2000)
in combination with TAUOLA Davidson et al. (2012), $e^{+}e^{-}\rightarrow
e^{+}e^{-}\mu^{+}\mu^{-}$ and $e^{+}e^{-}\rightarrow e^{+}e^{-}e^{+}e^{-}$
with AAFH Berends et al. (1985), $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}(\gamma)$
with PHOKHARA Czyż et al. (2013), and $e^{+}e^{-}\rightarrow
e^{+}e^{-}(\gamma)$ with BabaYaga@NLO Balossini et al. (2008). Backgrounds
coming from the final states $q\bar{q}$ ($q=u,d,s,c,b$), $J/\psi\gamma$ and
$\psi(2S)\gamma$ with $J/\psi,\psi(2S)\rightarrow\mu^{+}\mu^{-}$, and
$\mu^{+}\mu^{-}\nu\bar{\nu}$ are found to be negligible. Detector geometry and
interactions of final-state particles with the detector material are simulated
using Geant4 Agostinelli et al. (2003) and the Belle II Analysis Software
Framework Kuhr et al. (2019); The Belle II Collaboration .

The search uses an online event selection (trigger) that requires events with
at least one pair of tracks in a restricted polar-angle acceptance,
$\theta\in[37,120]^{\circ}$, and an azimuthal opening angle larger than
$90^{\circ}$ or $30^{\circ}$ for data collected in 2019 and 2020, respectively
(two-track trigger). Events consistent with Bhabha scattering are rejected by
a dedicated trigger veto.

In the offline analysis, we require that tracks originate from the interaction
point, with transverse and longitudinal projections of their distance of
closest approach smaller than 0.5 and 2.0 cm, respectively, to reject spurious
and beam-induced background tracks. We require events to have exactly two
oppositely charged particles identified as muons, that pass the trigger
requirements and have transverse momenta larger than 0.4 GeV/$c$. We reject
events with opening angles between the muons in the c.m. frame larger than
$179^{\circ}$, to suppress $\mu^{+}\mu^{-}(\gamma)$ backgrounds. For
$M^{2}_{\rm{recoil}}$ $<4$
$\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}\textit{c}^{4}$, most of the background
comes from $\mu^{+}\mu^{-}(\gamma)$ events with single-photon emission: we
require $\theta_{\rm{recoil}}$ to be within the ECL barrel acceptance
$[34,123]^{\circ}$ so as to exclude regions where photons can escape
undetected. Additionally, we reject events with
$\theta_{\rm{recoil}}\in[89,91]^{\circ}$, where there is a 1.5${\rm\,mm}$-wide
gap in the ECL instrumentation. For $M^{2}_{\rm{recoil}}$ $>4$
$\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}\textit{c}^{4}$, most of the background
comes from $\mu^{+}\mu^{-}(\gamma$) events with multiple photon emission and
the correlation between the photon direction and $\theta_{\rm{recoil}}$ is
lost: we require $\theta_{\rm{recoil}}<123^{\circ}$ because the signal is
dominantly produced in the forward direction due to the c.m. boost. To
suppress $\mu^{+}\mu^{-}(\gamma)$ backgrounds, we impose a photon veto: we
require the total energy of all photons to be less than 0.5 GeV and no photon
be within $15^{\circ}$ of the recoil momentum. To suppress
$\mu^{+}\mu^{-}(\gamma)$ and $e^{+}e^{-}\mu^{+}\mu^{-}$ backgrounds, we
require that the transverse recoil momentum in the c.m. frame exceed 0.5
GeV/$c$.

The remaining background comes dominantly from $\tau^{+}\tau^{-}(\gamma)$
events with $\tau$ leptons decaying to muons or pions misidentified as muons
in the region $M^{2}_{\rm{recoil}}$ $<$ 50
$\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}\textit{c}^{4}$ and from
$e^{+}e^{-}\mu^{+}\mu^{-}$ events elsewhere. The background from
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}(\gamma)$ is subleading across the entire
mass range.

The final selection uses an artificial neural network, denoted as Punzi-net
Abudinén et al. (2022), trained on simulated signal and background events, and
specifically designed to optimize a figure of merit Punzi (2003) for all
$Z^{\prime}$ mass hypotheses simultaneously. We use as inputs the four
kinematic variables, all defined in the c.m. frame, with the highest
discriminating power: the transverse momentum of a muon with respect to the
dimuon thrust axis Brandt et al. (1964); Farhi (1977); the transverse momentum
of the higher-energy muon with respect to the momentum direction of the lower-
energy muon; the longitudinal momentum of the higher-energy muon with respect
to the momentum direction of the lower-energy muon; and the transverse
momentum of the dimuon system. The first three variables exploit mostly the
kinematic properties of $Z^{\prime}$ production through radiation from a final
state muon, compared to $\tau^{+}\tau^{-}(\gamma)$ events, in which the recoil
momentum arises from neutrinos from $\tau$ decays. The fourth variable
exploits the kinematic features of $\mu^{+}\mu^{-}(\gamma)$ and
$e^{+}e^{-}\mu^{+}\mu^{-}$ backgrounds, typically produced with low transverse
momenta. Additional details are given in Ref. Abudinén et al. (2022).

The resulting signal efficiency is typically 5%, nearly uniform as a function
of $M^{2}_{\rm{recoil}}$. The Punzi-net selection removes nearly all
$\tau^{+}\tau^{-}(\gamma)$ background, except for a small contribution in the
region $M^{2}_{\rm{recoil}}$ $>$ 50
$\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}\textit{c}^{4}$. The residual background
comes dominantly from $\mu^{+}\mu^{-}(\gamma)$ in the region
$M^{2}_{\rm{recoil}}$ $<$ 50
$\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}\textit{c}^{4}$ and from a large
contribution of $e^{+}e^{-}\mu^{+}\mu^{-}$ elsewhere. In particular, the
region $M^{2}_{\rm{recoil}}$ $<$ 1
$\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}\textit{c}^{4}$ is dominated by the
$\mu^{+}\mu^{-}(\gamma)$ process with a single photon emission. Above 1
$\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}\textit{c}^{4}$ the
$\mu^{+}\mu^{-}(\gamma)$ process contributes mostly with events containing two
radiated photons. Typically, one photon is collinear with the beams and
outside the acceptance, while the other is emitted in the direction of one of
the gaps between the barrel and the forward or backward ECL endcaps. For
$M^{2}_{\rm{recoil}}$ in the 1–50
$\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}\textit{c}^{4}$ range, this produces two
distinctive bands in the
$\theta^{\rm{\,c.m.}}_{\rm{recoil}}$–$M^{2}_{\rm{recoil}}$ plane, where
$\theta^{\rm{\,c.m.}}_{\rm{recoil}}$ is the polar angle of the recoil momentum
in the c.m. frame. This feature is exploited in a two-dimensional fitting
procedure, which incorporates the expected background shapes due to
$\mu^{+}\mu^{-}(\gamma)$ events, doubling the sensitivity as compared to one-
dimensional fitting.

We fit to data by maximizing a binned likelihood based on signal and
background two-dimensional templates obtained from simulation. The parameter
of interest is the signal cross section while the background normalization is
determined by the fit. The $M^{2}_{\rm{recoil}}$ bin widths vary across the
spectrum and are set to the signal $M^{2}_{\rm{recoil}}$ resolution. The
binning in $\theta^{\rm{\,c.m.}}_{\rm{recoil}}$ reflects the distribution of
$\mu^{+}\mu^{-}(\gamma)$ events, which depends on $M^{2}_{\rm{recoil}}$. The
number of bins varies from one (for high $M^{2}_{\rm{recoil}}$) to five (for
low $M^{2}_{\rm{recoil}}$). We perform a squared-mass scan in steps
corresponding to one unit of signal $M^{2}_{\rm{recoil}}$ resolution, testing
all simulated mass hypotheses. Each fit is performed in search windows of 20
$M^{2}_{\rm{recoil}}$ bins centered around each hypothesis. Uncorrelated
systematic uncertainties on the signal and background shapes (see below) are
included in the respective templates by introducing in each bin of the
template Gaussian nuisance parameters constrained by the corresponding
uncertainties. A frequentist procedure based on the profile-likelihood ratio
$\tilde{t}_{\mu}$ Cowan et al. (2011) is used to obtain 90% confidence-level
(CL) intervals on the cross section. We use the pyhf software package Heinrich
et al. (2021) for inference and check the consistency of our results with
simplified simulated samples.

We also consider the scenario in which $\Gamma_{Z^{\prime}}$ is not
negligible, as expected for large $\alpha_{\rm D}$ values Crivellin et al.
(2022); Crivellin and Hoferichter (2022), and study one benchmark case that
assumes $\Gamma_{Z^{\prime}}=0.1M_{Z^{\prime}}$. We account for the non-zero
width in the fitting procedure by changing the shape of the signal templates
to Breit-Wigner distributions with the widths $\Gamma_{Z^{\prime}}$ convolved
with Gaussian resolution functions. We use only one-dimensional
$M^{2}_{\rm{recoil}}$ templates and enlarge the search windows to cover the
sizable signal width.

Three control samples are used to validate the analysis and estimate
systematic uncertainties. The $\mu\mu\gamma$ control sample is obtained by
reversing the photon-veto criteria thus requiring a photon of energy greater
than $1\mathrm{\,Ge\kern-1.00006ptV}$ within 15∘ of the recoil momentum
direction. The $e\mu$ and $ee$ control samples are obtained by requiring one
or both tracks to be identified as electrons and then applying all the other
selection criteria.

The efficiency of the two-track trigger is studied with the $e\mu$ control
sample with events collected by an ECL-based trigger, which is used as a
reference. This requires that the total energy deposition in the barrel and
forward endcap exceed 1 GeV. The two-track trigger efficiency increases from
$89$ to $93\%$ as a function of $M^{2}_{\rm{recoil}}$.

We use the $ee$ control sample to study the photon-veto performance. We select
events with $M^{2}_{\rm{recoil}}<1$ GeV2/$c^{4}$, since they come dominantly
from Bhabha scattering with a single radiated photon. The results indicate
that the photon-veto inefficiency in the backward barrel ECL is larger than
that estimated in simulation. This study was performed after observing a large
data-simulation disagreement in the signal region compatible with photon-veto
inefficiency. The photon-veto inefficiencies measured with the $ee$ control
sample are used to correct the expected $\mu\mu$ background.

We estimate systematic uncertainties on the signal efficiency and on the
signal and background template shapes. The uncertainties on the template
shapes independently affect each of the bins contained within the templates.

Uncertainties in selection efficiencies due to data-simulation mismodeling are
studied by comparing data and simulation in the $\mu\mu\gamma$ and $e\mu$
control samples in three $M^{2}_{\rm{recoil}}$ ranges: [$-0.5,9$], [$9,36$],
[$36,81$] GeV2/$c^{4}$. The two control samples provide complementary coverage
of the $M^{2}_{\rm{recoil}}$ range, with $\mu\mu\gamma$ addressing the lower
region and $e\mu$ covering the higher. Systematic uncertainties due to data-
simulation mismodeling in the trigger, luminosity, tracking efficiency, muon
identification, background cross sections, and effect of the selections are
collectively evaluated through data-simulation comparison before the
application of the Punzi-net. Systematic uncertainties due to the Punzi-net
are evaluated by studying its efficiencies, as they are indicators of the
performances for the signal-like background component. The differences from
unity of the data-to-simulation ratios of event yields before the Punzi-net
application, and between the Punzi-net efficiencies in the three
$M^{2}_{\rm{recoil}}$ ranges are summed in quadrature and found to be 2.7%,
6.5%, and 8.3%, respectively. These differences are assigned as systematic
uncertainties on the signal efficiency.

The recoil mass resolution is studied using the $\mu\mu\gamma$ sample. The
width of the $M^{2}_{\rm{recoil}}$ distribution is 8% larger in data than in
simulation. This translates to a systematic uncertainty of 10% on the signal
template shape.

Systematic uncertainties due to background shapes are evaluated on the
$\mu\mu\gamma$ and $e\mu$ samples. We compute the average data-to-simulation
ratios for the number of events in each search window. We assign twice the
largest standard deviation on these averages in each of the three
$M^{2}_{\rm{recoil}}$ ranges as an uncertainty for the shape in the respective
$M^{2}_{\rm{recoil}}$ ranges. We use the $\mu\mu\gamma$ control sample for
$M^{2}_{\rm{recoil}}$ up to
56$\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}\textit{c}^{4}$ and the $e\mu$ control
sample above. The resulting uncertainties are 3.2%, 8.6%, and 25% in the three
$M^{2}_{\rm{recoil}}$ ranges.

Uncertainties on the background template shape from the photon-veto
inefficiency are studied on the $ee$ control sample and are on average 34% for
$M^{2}_{\rm{recoil}}$$<$ 1
$\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}\textit{c}^{4}$ decreasing to 5% above
1$\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}\textit{c}^{4}$. We assign a systematic
uncertainty of 1% to the measured integrated luminosity Abudinén et al.
(2020).

The observed and expected $M^{2}_{\rm{recoil}}$ distributions are shown in
Fig. 1. We find no significant excess of data above the expected background.
The $\chi^{2}$ value describing the goodness of the two-dimensional fit is
acceptable for each test $Z^{\prime}$ mass with the largest incompatibility
corresponding to a p-value of 0.05. The largest local significance is
2.8$\sigma$ for $M_{Z^{\prime}}=$ 2.352 GeV/$c^{2}$. The global significance
of this excess after correcting for the look-elsewhere effect Gross and
Vitells (2010) is 0.7$\sigma$.

[width=0.9]./figures/recoilM2_eecorr_v3_preliminary.pdf

Figure 1: Squared recoil mass spectrum of the $\mu^{+}\mu^{-}$ sample,
compared with the stacked contributions from the various simulated background
samples normalized to the integrated luminosity.

The 90% CL upper limits on the cross section for the process
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}Z^{\prime}$ with $Z^{\prime}$ invisible,
$\sigma({e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}Z^{\prime},\,\,Z^{\prime}\rightarrow\rm{invisible}})$
$=$
$\sigma({e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}Z^{\prime})\times\mathcal{B}(Z^{\prime}\rightarrow\rm{invisible}})$,
are shown in Fig. 2 as functions of $M_{Z^{\prime}}$, along with the expected
1$\sigma$ and 2$\sigma$ bands of expected limits (median limits from
background-only simulated samples). We set upper limits as small as 0.2 fb. In
addition, we show upper limits for the benchmark scenario in which we assume
non-negligible $\Gamma_{Z^{\prime}}$.

Cross section results are translated into 90% CL upper limits on the coupling
$g^{\prime}$ and shown in Fig. 3 in the fully invisible $L_{\mu}-L_{\tau}$
model for the cases of negligible and non-negligible $\Gamma_{Z^{\prime}}$.
For the case of negligible $\Gamma_{Z^{\prime}}$, these constraints hold for
$M_{Z^{\prime}}$ $\lesssim$ 6.5
${\mathrm{\,Ge\kern-1.00006ptV\\!/}\textit{c}^{2}}$. For $M_{Z^{\prime}}$ $>$
6.5 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}\textit{c}^{2}}$, the assumption
$\cal{B}\rm(Z^{\prime}\rightarrow\chi\bar{\chi})\approx 1$ is not valid:
numerical values in Fig. 3 can still be used, but need to be rescaled by
$1/\sqrt{\cal{B}\rm(Z^{\prime}\rightarrow\chi\bar{\chi})}$, which depends on
$\alpha_{\rm D}$. We also show limits from NA64-$e$ Andreev et al. (2022) and
the previous Belle II search Adachi et al. (2020). Figure 4 presents limits in
the vanilla $L_{\mu}-L_{\tau}$ model, together with limits from Belle Czank et
al. (2022), BABAR Lees et al. (2016), and CMS Sirunyan et al. (2019) searches
for $Z^{\prime}\rightarrow\mu^{+}\mu^{-}$ and the constraints from NA64-$e$
and Belle II Adachi et al. (2020). In both fully invisible and vanilla models,
we focus on the direct-search results and do not show constraints obtained
from reanalyses of data from neutrino experiments Altmannshofer et al. (2014);
Bellini et al. (2011); Kamada et al. (2018). Additional plots, including
indirect constraints from neutrino experiments and detailed numerical results,
are provided in the supplemental material sup .

[width=0.99]./figures/cross_section_gamma_preliminary.pdf

Figure 2: Observed 90$\%$ CL upper limits on the cross section
$\sigma({e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}Z^{\prime},\,\,Z^{\prime}\rightarrow\rm{invisible}})$
as functions of the $Z^{\prime}$ mass for the cases of negligible
$\Gamma_{Z^{\prime}}$ and for $\Gamma_{Z^{\prime}}=0.1M_{Z^{\prime}}$. Also
shown are previous limits from Belle II Adachi et al. (2020).

[width=0.99]./figures/gprime_only_inv_gamma_no_neutrino_preliminary.pdf

Figure 3: Observed 90$\%$ CL upper limits on the coupling $g^{\prime}$ for the
fully invisible $L_{\mu}-L_{\tau}$ model as functions of the $Z^{\prime}$ mass
for the cases of negligible $\Gamma_{Z^{\prime}}$ and for
$\Gamma_{Z^{\prime}}=0.1M_{Z^{\prime}}$. Also shown are previous limits from
NA64-$e$ Andreev et al. (2022) and Belle II Adachi et al. (2020) searches. The
red band shows the region that explains the muon anomalous magnetic moment
$(g-2)_{\mu}\pm 2\sigma$ Abi et al. (2021). The vertical dashed line indicates
the limit beyond which the hypothesis
$\cal{B}\rm(Z^{\prime}\rightarrow\chi\bar{\chi})\approx 1$ is not respected in
the negligible $\Gamma_{Z^{\prime}}$ case.

[width=0.99]./figures/gprime_lmultau_no_neutrino_preliminary.pdf

Figure 4: Observed 90$\%$ CL upper limits on the coupling $g^{\prime}$ for the
vanilla $L_{\mu}-L_{\tau}$ model as functions of the $Z^{\prime}$ mass. Also
shown are previous limits from Belle II Adachi et al. (2020) and NA64-$e$
Andreev et al. (2022) searches for invisible $Z^{\prime}$ decays, and from
Belle Czank et al. (2022), BaBar Lees et al. (2016), and CMS Sirunyan et al.
(2019) searches for $Z^{\prime}$ decays to muons (at 95% CL). The red band
shows the region that explains the muon anomalous magnetic moment
$(g-2)_{\mu}\pm 2\sigma$ Abi et al. (2021).

In summary, we search for an invisibly decaying $Z^{\prime}$ boson in the
process $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}Z^{\prime}$ using data
corresponding to 79.7$\mbox{\,fb}^{-1}$ collected by Belle II at SuperKEKB in
2019–2020. We find no significant excess above the expected background and set
90% CL upper limits on the coupling $g^{\prime}$ ranging from $3\times
10^{-3}$ at low $Z^{\prime}$ masses to 1 for masses at 8
${\mathrm{\,Ge\kern-1.00006ptV\\!/}\textit{c}^{2}}$. These are world-leading
direct-search results for $Z^{\prime}$ masses above 11.5
${\mathrm{\,Me\kern-1.00006ptV\\!/}\textit{c}^{2}}$ in the fully invisible
$L_{\mu}-L_{\tau}$ model and for masses in the range 11.5 to 211
${\mathrm{\,Me\kern-1.00006ptV\\!/}\textit{c}^{2}}$ in the vanilla
$L_{\mu}-L_{\tau}$ model. These limits are the first direct-search results
that exclude at 90% CL a fully invisible $Z^{\prime}$ boson as an explanation
of the $(g-2)_{\mu}$ anomaly for $0.8<M_{Z^{\prime}}<5.0$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}\textit{c}^{2}}$.

We thank Andreas Crivellin for useful discussions during the preparation of
this manuscript. 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. DE220100462, No.
DP180102629, No. DP170102389, No. DP170102204, No. DP150103061, No.
FT130100303, No. FT130100018, and No. FT120100745; 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; Chinese Academy of Sciences and research Grant No. QYZDJ-SSW-
SLH011, National Natural Science Foundation of China and research Grants No.
11521505, No. 11575017, No. 11675166, No. 11761141009, No. 11705209, and No.
11975076, LiaoNing Revitalization Talents Program under Contract No.
XLYC1807135, Shanghai Pujiang Program under Grant No. 18PJ1401000, and the CAS
Center for Excellence in Particle Physics (CCEPP); 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; 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; the raw-data centers at BNL, DESY, GridKa, IN2P3, INFN, and
the University of Victoria for offsite computing support.

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  * (43) See Supplemental Material at http://XYZ for additional plots.