KEK preprint: 2022-32
Belle II preprint: 2022-006

The Belle II Collaboration

# Search for a $\tau^{+}\tau^{-}$ resonance in
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}\tau^{+}\tau^{-}$ events 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 S. Bansal  0000-0003-1992-0336 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 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 V. Bhardwaj  0000-0001-8857-8621 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 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 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 J. Cochran  0000-0002-1492-914X L. Corona
0000-0002-2577-9909 L. M. Cremaldi  0000-0001-5550-7827 S. Cunliffe
0000-0003-0167-8641 T. Czank  0000-0001-6621-3373 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 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 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 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 J. Guilliams  0000-0001-8229-3975 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 H. Hirata
0000-0001-9005-4616 M. Hoek  0000-0002-1893-8764 M. Hohmann
0000-0001-5147-4781 C.-L. Hsu  0000-0002-1641-430X T. Humair
0000-0002-2922-9779 T. Iijima  0000-0002-4271-711X K. Inami
0000-0003-2765-7072 G. Inguglia  0000-0003-0331-8279 N. Ipsita
0000-0002-2927-3366 A. Ishikawa  0000-0002-3561-5633 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 H. Kakuno
0000-0002-9957-6055 M. Kaleta  0000-0002-2863-5476 D. Kalita
0000-0003-3054-1222 A. B. Kaliyar  0000-0002-2211-619X J. Kandra
0000-0001-5635-1000 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 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 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 M. Kumar
0000-0002-6627-9708 R. Kumar  0000-0002-6277-2626 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. B. Li  0000-0002-9909-2851 J. Libby  0000-0002-1219-3247 K. Lieret
0000-0003-2792-7511 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 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 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 K. R. Nakamura  0000-0001-7012-7355 M. Nakao
0000-0001-8424-7075 H. Nakayama  0000-0002-2030-9967 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 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 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 J. Park
0000-0001-6520-0028 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 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 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 L. B. Rizzuto  0000-0001-6621-6646 S. H. Robertson
0000-0003-4096-8393 D. Rodríguez Pérez  0000-0001-8505-649X 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 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 M. Schnepf
0000-0003-0623-0184 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 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 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. 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 H. Tanigawa
0000-0003-3681-9985 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 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 G. S. Varner
0000-0002-0302-8151 K. E. Varvell  0000-0003-1017-1295 A. Vinokurova
0000-0003-4220-8056 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 A. Vossen  0000-0003-0983-4936 B. Wach
0000-0003-3533-7669 M. Wakai  0000-0003-2818-3155 H. M. Wakeling
0000-0003-4606-7895 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 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 H. Ye  0000-0003-0552-5490
J. Yelton  0000-0001-8840-3346 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. Zhai  0000-0001-7207-5122 Y. Zhang  0000-0003-2961-2820 V. Zhilich
0000-0002-0907-5565 J. S. Zhou  0000-0002-6413-4687 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

We report the first search for a non-standard-model resonance decaying into
$\tau$ pairs in $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}\tau^{+}\tau^{-}$ events
in the 3.6–10 GeV/$c^{2}$ mass range. We use a 62.8 fb-1 sample of
$e^{+}e^{-}$ collisions collected at a center-of-mass energy of 10.58 GeV by
the Belle II experiment at the SuperKEKB collider. The analysis probes three
different models predicting a spin-1 particle coupling only to the heavier
lepton families, a Higgs-like spin-0 particle that couples preferentially to
charged leptons (leptophilic scalar), and an axion-like particle,
respectively. We observe no evidence for a signal and set exclusion limits at
90% confidence level on the product of cross section and branching fraction
into $\tau$ pairs, ranging from 0.7 fb to 24 fb, and on the couplings of these
processes. We obtain world-leading constraints on the couplings for the
leptophilic scalar model for masses above 6.5 GeV/$c^{2}$ and for the axion-
like particle model over the entire mass range.

Despite its great successes, the standard model (SM) of particle physics is
known to provide an incomplete description of nature. For example, it does not
address the phenomenology related to the existence of dark matter Bertone  _et
al._ (2005), specifically in the prediction of the observed relic density. In
addition, recent experimental observations showed inconsistencies with the SM.
Prominent examples are the long-standing difference between the measured and
the expected value of the muon anomalous magnetic-moment Bennett  _et al._
(2006); Abi  _et al._ (2021); Aoyama  _et al._ (2020), and the tensions in
flavor observables reported by the BABAR, Belle, and LHCb experiments Lees
_et al._ (2013); Aaij  _et al._ (2018); Caria  _et al._ (2020); LHCb
Collaboration ; Crivellin  _et al._ (2022); Sala and Straub (2017); Chen and
Nomura (2018); Greljo  _et al._ (2021). Some of these observations can be
explained with the introduction of additional, possibly lepton-universality-
violating interactions mediated by non-SM neutral bosons. Examples are the
$L_{\mu}-L_{\tau}$ extension of the SM, a Higgs-like spin-0 particle
(leptophilic scalar), and axion-like particles (ALPs). The $L_{\mu}-L_{\tau}$
model gauges the difference between the muon and the $\tau$-lepton numbers
through the introduction of a neutral spin-1 boson $Z^{\prime}$ that couples
only to the second and third generations of leptons He  _et al._ (1991); Shuve
and Yavin (2014); Altmannshofer  _et al._ (2016). The $Z^{\prime}$ could also
mediate interactions between SM and dark matter. The leptophilic scalar $S$ is
an hypothetical particle that couples preferentially to charged leptons
through a parameter $\xi$ and Yukawa-like couplings to the individual families
proportional to the lepton masses Batell  _et al._ (2017). Axion-like
particles appear in many models with spontaneous breaking of global symmetries
as relics of high-energy extensions of the SM Bauer  _et al._ (2017, 2022). In
some models, they couple to charged leptons through parameters $C_{\ell\ell}$
with $\ell=e,\mu,\tau$, with a decay rate to leptons proportional to the
squared lepton-masses. The coupling to charged leptons is parametrized as
$|C_{\ell\ell}|/\Lambda$, where $\Lambda$ is the scale of the global symmetry
breaking Bauer  _et al._ (2022). For the ALP model we follow the approach of
Refs. Bauer  _et al._ (2017, 2022), in which the coupling of ALPs to charged
leptons is studied assuming no coupling to all the other particles, in
particular photons.

Searches for a $Z^{\prime}$ decaying to muons have been reported by the BABAR,
Belle, and CMS collaborations Lees  _et al._ (2016); Czank  _et al._ (2022);
Sirunyan  _et al._ (2019). An invisibly decaying $Z^{\prime}$ has been
searched for by the Belle II Adachi  _et al._ (2020, 2023a) and NA64-$e$
Andreev  _et al._ (2022) experiments. The leptophilic scalar decaying into
electrons and muons is constrained by BABAR for masses up to approximately 6.5
GeV/$c^{2}$ Lees  _et al._ (2020). Decays of ALPs into leptons are constrained
mostly through reinterpretations of other measurements Bauer  _et al._ (2017,
2022). For all these particles, decays into pairs of $\tau$ leptons are
unconstrained due to the experimental difficulties in fully reconstructing the
final state that has multiple neutrinos.

In this Letter, we search for a $X\to\tau^{+}\tau^{-}$ resonance, where
$X=Z^{\prime},S$, or ${\rm ALP}$, in
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}\tau^{+}\tau^{-}$ events by using the
knowledge of the initial-state total momentum such that signal forms a narrow
enhancement in the distribution of the recoil mass against two oppositely
charged muons $M_{\text{recoil}}(\mu\mu)$. We use a sample of $e^{+}e^{-}$
collisions produced at a center-of-mass (c.m.) energy $\sqrt{s}=10.58$ GeV in
2019-2020 by the SuperKEKB asymmetric-energy collider Akai  _et al._ (2018) at
KEK. The data, recorded by the Belle II detector Abe  _et al._ (2010); Kou
_et al._ (2019), correspond to an integrated luminosity of 62.8 fb-1 Abudinén
_et al._ (2020). The $L_{\mu}-L_{\tau}$ model is used as a benchmark to devise
the analysis selections through the process $e^{+}e^{-}\rightarrow
Z^{\prime}\,(\rightarrow\tau^{+}\tau^{-})\ \mu^{+}\mu^{-}$. We then check the
selection performance on the two additional models. In all the cases, the $X$
resonance is predominantly emitted as final-state radiation (FSR) off one of
the two muons. We restrict our analysis to $\tau$-lepton decays to one charged
particle and any number of neutral particles. We therefore select events with
exactly four charged particles, where at least two are identified as muons
(tagging muons). The main expected backgrounds are the processes
$e^{+}e^{-}\rightarrow q\bar{q}(\gamma)$ with $q=u,d,s,c,b$,
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}(\gamma)$ where one $\tau$ lepton decays
into a one-charged-particle state (one-prong) and the other to a three-
charged-particle state (three-prong), and four-lepton processes
$e^{+}e^{-}\rightarrow e^{+}e^{-}\mu^{+}\mu^{-}$,
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}\tau^{+}\tau^{-}$, and
$e^{+}e^{-}\rightarrow e^{+}e^{-}\tau^{+}\tau^{-}$. A multivariate analysis
exploits kinematic variables to enhance the signal-to-background ratio. A
control sample is used to compare data to simulation, from which the most
important systematic uncertainties are estimated. The signal yield is
extracted through fits to the $M_{\text{recoil}}(\mu\mu)$ distribution, which
allows an estimate of the background directly from data. To minimize bias, the
event selection and the analysis workflow are defined using simulated events
prior to examining data.

The Belle II detector Abe  _et al._ (2010); Kou  _et al._ (2019) consists of
several subdetectors arranged in a cylindrical structure around the
$e^{+}e^{-}$ interaction point. The longitudinal direction, the transverse
plane, and the polar angle $\theta$ are defined with respect to the detector’s
cylindrical axis in the direction of the electron beam. Charged-particle
trajectories (tracks) are reconstructed by a tracking system consisting of a
two-layer silicon-pixel detector, surrounded by a four-layer double-sided
silicon-strip detector and then a central drift chamber (CDC). The second
pixel layer was only partially installed for the data sample we analyze,
covering one sixth of the azimuthal angle. Outside the CDC, time-of-
propagation and aerogel ring-imaging Cherenkov detectors cover the
$\theta\in[31,128]^{\circ}$ and $\theta\in[14,30]^{\circ}$ polar ranges,
respectively, to provide charged-particle identification. The electromagnetic
calorimeter (ECL) reconstructs photons and identifies electrons. It fills the
remaining volume inside a superconducting solenoid that generates a 1.5-T
field. A $K^{0}_{L}$ and muon detection subsystem is installed in the iron
flux return of the solenoid.

The identification of muons relies mostly on charged-particle penetration in
the KLM for momenta larger than 0.7 GeV/$c$ and on information from the CDC
and ECL otherwise. Electrons are identified mostly by comparing measured
momenta with energies of the associated ECL depositions. We identify charged
hadrons as particles not compatible with both electrons and muons. Charged
pions are identified combining the information from all subdetectors except
the silicon detectors. Photons are reconstructed from ECL-energy depositions
greater than 100 MeV not associated with any track. Neutral pions are
identified as pairs of photons with an invariant mass within three standard
deviations from the known $\pi^{0}$ mass. Details of particle reconstruction
and identification are given in Refs. Kou  _et al._ (2019); Bertacchi  _et
al._ (2021).

Signal events are simulated using MadGraph5@NLO Alwall  _et al._ (2014),
including initial-state radiation (ISR). The $M_{\text{recoil}}(\mu\mu)$
resolution varies with the $Z^{\prime}$ mass: it is 30 MeV/$c^{2}$ at the
kinematic threshold 2$m_{\tau}$ and decreases smoothly to 10 MeV/$c^{2}$ at 6
GeV/$c^{2}$ and to 1 MeV/$c^{2}$ at 10 GeV/$c^{2}$. We generate events for
$Z^{\prime}$ masses ranging from 3.6 to 10 GeV/$c^{2}$ in steps of 25, 20, 10,
and 5 MeV/$c^{2}$, following the $M_{\text{recoil}}(\mu\mu)$ resolution. The
background processes are simulated using the following generators:
$e^{+}e^{-}\rightarrow u\bar{u},d\bar{d},s\bar{s},c\bar{c}$ with KKMC
interfaced with Pythia8 Sjöstrand  _et al._ (2015) and EvtGen Lange (2001);
$e^{+}e^{-}\rightarrow b\bar{b}$ with EvtGen;
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}(\gamma)$ with KKMC Jadach  _et al._
(2000) interfaced with Tauola Davidson  _et al._ (2012);
$e^{+}e^{-}\rightarrow e^{+}e^{-}\mu^{+}\mu^{-}$,
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}\tau^{+}\tau^{-}$, $e^{+}e^{-}\rightarrow
e^{+}e^{-}\tau^{+}\tau^{-}$,
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}\mu^{+}\mu^{-}$, and
$e^{+}e^{-}\rightarrow e^{+}e^{-}e^{+}e^{-}$ with AAFH Berends  _et al._
(1985); $e^{+}e^{-}\rightarrow e^{+}e^{-}\pi^{+}\pi^{-}$ with Treps Uehara
(2013); and $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}(\gamma)$ with KKMC.
Electromagnetic FSR is simulated with Photos Barberio  _et al._ (1991);
Barberio and Was (1994) for processes generated with EvtGen. All the four-
lepton processes generated with AAFH do not include ISR effects, which is a
source of unmodeled background. Additional sources of non-simulated
backgrounds include $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}\pi^{+}\pi^{-}$
processes and two-photon processes $e^{+}e^{-}\rightarrow e^{+}e^{-}h$, where
$h$ is typically a low-mass hadronic system. The detector geometry and
interactions of final-state particles with detector material are simulated
using Geant4 Agostinelli  _et al._ (2003) and the Belle II software Kuhr  _et
al._ (2019); The Belle II Collaboration .

The online event selection (trigger) is a logical OR of a three-track trigger
and a single-muon trigger. The former requires the presence of at least three
tracks in the $\theta\in[37,120]^{\circ}$ polar range. The latter is based on
the match between CDC tracks and signals in the $\theta\in[51,117]^{\circ}$
KLM polar range. The efficiency of both triggers is calculated using a
reference trigger, which requires that the total ECL-energy deposition in the
$\theta\in[22,128]^{\circ}$ polar range exceeds 1 GeV. The three-track-trigger
efficiency is measured in four-track events containing at least two pions and
one electron. The single-muon-trigger efficiency is measured in events with
one electron and one muon: the efficiency for events with multiple muons is
computed using the single-muon efficiency, assuming no correlation. The
overall trigger efficiency is 96% for $Z^{\prime}$ masses up to 8 GeV/$c^{2}$,
then it decreases smoothly to 90% at 9 GeV/$c^{2}$ and to 50% at 10
GeV/$c^{2}$.

To suppress misreconstructed and beam-induced background tracks, we require
that the transverse and longitudinal projections of their distance of closest
approach to the interaction point be smaller than 0.5 and 2.0 cm,
respectively. We require that events have exactly four charged particles with
zero net charge, with at least a pair of oppositely-charged particles
identified as muons and the remaining two particles separately identified as
electrons, muons, or charged hadrons, for which we assume the electron, muon
and pion mass hypotheses, respectively. Events with more than two identified
muons produce up to four candidates. We require that the four-track invariant
mass $M({\text{4\leavevmode\nobreak\ tracks}})$ be below 9.5 GeV/$c^{2}$ to
suppress the four-lepton backgrounds that peak at the c.m. energy, such as
$e^{+}e^{-}\rightarrow e^{+}e^{-}\mu^{+}\mu^{-}$,
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}\mu^{+}\mu^{-}$, and
$e^{+}e^{-}\rightarrow e^{+}e^{-}e^{+}e^{-}$. The remaining background is
largely dominated by $e^{+}e^{-}\rightarrow q\bar{q}(\gamma)$ and
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}(\gamma)$ and, to a lesser extent, by
$e^{+}e^{-}\rightarrow e^{+}e^{-}\mu^{+}\mu^{-}$ processes. The final signal-
to-background discrimination relies on signal-event properties: presence of a
resonance recoiling against the two tagging muons; FSR emission of the
resonance; and compatibility of the system recoiling against the tagging muons
with a $\tau^{+}\tau^{-}$ pair. We identify 14 variables, among which the most
discriminating are the following: the momenta of the two tagging muons in the
c.m. frame; the components of the recoil momentum (the $Z^{\prime}$ momentum,
for signal events) transverse to the momentum direction of each of the two
tagging muons in the c.m. frame; and topological variables defined in the rest
frame of the system recoiling against the two muons, such as the thrust value
Farhi (1977); Brandt  _et al._ (1964), the first Fox-Wolfram moment shape-
variable Fox and Wolfram (1978), and the angles between the thrust direction
and the directions of the muons. A description of the discriminating variables
is given in the Supplemental material sup . To reduce the dependence of some
variables on the $Z^{\prime}$ mass, they are subject to a preprocessing. For
example, the momentum variables are scaled by the maximum momentum of the
system with mass $M_{\text{recoil}}(\mu\mu)$ recoiling against the two tagging
muons.

We use multilayer perceptrons (MLPs) Hoecker  _et al._ (2009), trained on
simulated signal and background events, with 14 input nodes and one output
node for the signal-to-background discrimination. To improve performance, we
use eight separate MLPs in different $M_{\text{recoil}}(\mu\mu)$ intervals,
which we refer to as MLP ranges, approximately 1 GeV/$c^{2}$ wide. The
selection applied on the node output is optimized separately in each MLP range
with a figure of merit Punzi (2003) and then expressed as a function of
$M_{\text{recoil}}(\mu\mu)$ by interpolation. The resulting signal efficiency
varies with the $Z^{\prime}$ mass from 12% near the kinematic threshold
$2m_{\tau}$ to 2% at 10 GeV/$c^{2}$ sup . The background suppression reduces
the $e^{+}e^{-}\rightarrow q\bar{q}(\gamma)$ and
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}(\gamma)$ processes by two to three
orders of magnitude, so that the resulting expected background contains
significant contributions from the four-lepton
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}\tau^{+}\tau^{-}$ and
$e^{+}e^{-}\rightarrow e^{+}e^{-}\tau^{+}\tau^{-}$ processes, which were small
before the MLP selection. The fraction of surviving events with more than one
candidate is negligible.

We apply the full selection on signal events simulated according to the two
additional models, and compare the signal efficiencies with those estimated
for the $L_{\mu}-L_{\tau}$ model: relative differences are in the range
10%–20%.

The $M({\text{4\leavevmode\nobreak\ tracks}})$ distribution after all
selections applied is compared with the simulation in Fig. 1. The
discrepancies between data and simulation are due to large contributions from
non-simulated two-photon processes $e^{+}e^{-}\rightarrow e^{+}e^{-}h$ for
$M({\text{4\leavevmode\nobreak\ tracks}})$ $<4$ GeV/$c^{2}$, and to unmodeled
ISR in simulated four-lepton processes for $M({\text{4\leavevmode\nobreak\
tracks}})$ $>7$ GeV/$c^{2}$. Additional contributions to the observed
discrepancies come from the process
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}\pi^{+}\pi^{-}$. The origin of these
discrepancies is confirmed by specific studies on the
$M({\text{4\leavevmode\nobreak\ tracks}})$ distribution before the MLP
selection sup  and on a control sample after all the selections. A pion-tagged
control sample is selected by applying the analysis requirements with the two
tagging muons replaced by two charged pions. Both samples are dominated by
$e^{+}e^{-}\rightarrow q\bar{q}(\gamma)$ and
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}(\gamma)$ processes, which include ISR
in the respective generators. In both cases we observe good agreement for
dimuon or dipion masses greater than 2 GeV/$c^{2}$, where the two-photon
processes $e^{+}e^{-}\rightarrow e^{+}e^{-}h$ do not contribute sup .

The $M_{\text{recoil}}(\mu\mu)$ distribution after all the selections are
applied is shown in Fig. 2. Discrepancies induced by the lack of ISR effects
in four-lepton simulation appear mainly for $M_{\text{recoil}}(\mu\mu)$ below
approximately 6 GeV/$c^{2}$. Above 9 GeV/$c^{2}$ the discrepancies are due to
two-photon $e^{+}e^{-}\rightarrow e^{+}e^{-}h$ processes. Also visible is the
periodic structure originating from the eight MLP ranges. Neither of these
effects produce narrow peaking structures at the scale of the signal
resolution in the $M_{\text{recoil}}(\mu\mu)$ distribution, as shown by the
inset in Fig. 2.

Figure 1: Observed four-track invariant mass distribution compared to the
expectations of the simulation. Contributions from the various simulated
processes are stacked. Figure 2: Observed distribution of the recoil mass
against the two tagging muons, compared to the expectations of the simulation.
Contributions from the various simulated processes are stacked. The inset
shows an example fit at a signal mass hypothesis of 6.036 GeV/$c^{2}$, and the
difference between the number of observed and fitted events, divided by the
statistical uncertainty of the former.

The signal yields are obtained from a scan over the
$M_{\text{recoil}}(\mu\mu)$ spectrum through a series of unbinned maximum
likelihood fits. The signal $M_{\text{recoil}}(\mu\mu)$ distributions are
parametrized from the simulation as sums of two Crystal Ball functions
Skwarnicki (1986) sharing the same mean value. The scan step-size is half the
mass resolution. Each fit extends over an interval 40 times larger than the
$Z^{\prime}$ mass resolution. The background is described with a constant.
Higher-order polynomials for the background parametrization are investigated,
but their coefficients are compatible with zero over the full recoil-mass
spectrum. A total of 2384 fits are performed, covering the range 3.6–10
GeV/$c^{2}$. If a fitting interval extends over two different MLP ranges, we
use data selected by the MLP corresponding to the range where the central mass
value is located. The fit determines the signal and background yields using a
fixed signal shape. An example fit is shown in the inset of Fig. 2. We then
convert signal yields into cross sections, after correcting for signal
efficiency and luminosity.

Several sources of systematic uncertainties affecting the cross section
determination are taken into account: they are related to signal efficiency,
luminosity, and fit procedure. Uncertainties due to the trigger efficiency in
signal events are evaluated by propagating the uncertainties on the measured
trigger efficiencies. The relative uncertainty on the signal efficiency is
2.7% across the entire mass range. Uncertainties due to the tracking
efficiency are estimated in $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}(\gamma)$
events, in the one-prong against three-prong topology. The relative
uncertainty on the signal efficiency is 3.6%. Uncertainties due to the
particle identification requirement are studied using
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}\gamma$, $e^{+}e^{-}\rightarrow
e^{+}e^{-}\mu^{+}\mu^{-}$, $e^{+}e^{-}\rightarrow e^{+}e^{-}e^{+}e^{-}$,
$e^{+}e^{-}\rightarrow e^{+}e^{-}\pi^{+}\pi^{-}$events and final states with
either a $J/\psi$ or a $K^{0}_{S}$. The relative uncertainty on the signal
efficiency varies between 3.9% and 6.2%, depending on the $Z^{\prime}$ mass.
Uncertainties due to the MLP selection efficiency are evaluated on the pion-
tagged control sample. We compare MLP efficiencies in data and simulation in
signal-like regions of the control sample and assume that uncertainties
estimated in those conditions are representative of the signal conditions too.
We find good agreement between data and simulation and estimate a 2.8%
relative uncertainty on the signal efficiency from the uncertainty of the
data-simulation comparison. Uncertainties due to the interpolation of the
signal efficiency between simulated mass points are 2.5%, which is assigned as
a relative uncertainty on the signal efficiency. Uncertainties due to the fit
procedure are evaluated using a bootstrap technique. Signal events from
simulation are overlaid on simulated background with a yield corresponding to
the excluded 90% CL value and fitted for each $Z^{\prime}$ mass. The
distribution of the difference between the overlaid and the fitted yields,
divided by the fit uncertainty, has a negligible average bias with a width
that deviates from one by 4%, which is assigned as a relative uncertainty on
the signal-yield determination. Uncertainties due to differences in the
recoil-mass resolution between data and simulation are evaluated by
introducing an additional smearing on the simulated momenta of the two tagging
muons, which reflects the difference in momentum resolution measured with
cosmic rays and in $D^{\ast+}\to D^{0}\pi^{+}$ decays with respect to the
simulation predictions. The relative uncertainty on the signal-yield
determination is 3%. The relative uncertainty on the signal efficiency due to
the knowledge of the beam energy Adachi  _et al._ (2023b) is 1%. The
uncertainty due to the selection on the four-track invariant mass is
negligible. Finally, a relative uncertainty of 1% on the integrated luminosity
is considered Abudinén  _et al._ (2020).

All the systematic uncertainties are summed in quadrature: the final relative
systematic uncertainty on the cross section varies in the range 8.8%–10.0%
depending on the $Z^{\prime}$ mass. We account for systematic uncertainties by
approximating their effects as a Gaussian smearing of the signal efficiency.

The significance is evaluated as
$\sqrt{2\,\text{log}(\mathcal{L}/\mathcal{L}_{0})}$ where $\mathcal{L}$ and
$\mathcal{L}_{0}$ are the likelihoods of the fits with and without signal. The
largest local significance observed is $3.0\,\sigma$, corresponding to a
global significance of $1.8\,\sigma$, at a recoil mass of 9.695 GeV/$c^{2}$
sup . Since we do not observe any significant excess above the background, we
derive 90% CL upper limits on the process cross section $\sigma(e^{+}e^{-}\to\
X\ (\to\tau^{+}\tau^{-})\ \mu^{+}\mu^{-})$ $=$
$\sigma(e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}X)\times\mathcal{B}(X\rightarrow\tau^{+}\tau^{-})$
with $X=Z^{\prime},S,{\rm ALP}$, using the frequentist procedure $\rm CL_{S}$
Cowan  _et al._ (2011); Read (2002). The limits are shown in Fig. 3. Expected
limits are defined as median limits from background-only simulated samples
that use background yields observed from the fits to data. The combination of
the periodic structure originating from the MLP ranges and of the overlap
between the fit intervals induces an oscillatory behaviour. The resulting
upper limits are dominated by sample size, with systematic uncertainties
worsening them on average by 1% compared to the case in which they are
neglected.

The cross-section results are translated into upper limits on the coupling
constant $g^{\prime}$ of the $L_{\mu}-L_{\tau}$ model sup , on the coupling
strength $\xi$ of the leptophilic scalar $S$, and on the coupling
$|C_{\ell\ell}|/\Lambda$ for an ALP decaying to leptons: values as low as
2.5$\times 10^{-2}$, 51, and 200 TeV-1 are found, respectively. The last two
are shown in Fig. 4 as functions of the resonance mass. For the leptophilic
scalar model, we constrain the coupling $\xi$ to be smaller than approximately
200 for masses above 6.5 GeV/$c^{2}$, which are the first results in that
region. For the model with the ALP decaying to leptons, these are the first
results for the ALP-$\tau$ coupling.

Figure 3: Observed 90% CL upper limits and corresponding expected limits on
the cross section for the process $e^{+}e^{-}\rightarrow
X\,(\rightarrow\tau^{+}\tau^{-})\ \mu^{+}\mu^{-}$ with $X=Z^{\prime},S,{\rm
ALP}$ as functions of the $X$ resonance mass. The inset shows a magnification
of the region above 9 GeV/$c^{2}$.

In summary, we search for a resonance decaying to $\tau^{+}\tau^{-}$ in
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}\tau^{+}\tau^{-}$ events in a data sample
of $e^{+}e^{-}$ collisions at 10.58 GeV collected by Belle II in 2019–2020,
corresponding to an integrated luminosity of 62.8 fb-1. We find no significant
excess above the background and set upper limits on the cross section, ranging
from 0.7 fb to 24 fb, for masses between 3.6 and 10 GeV/$c^{2}$. We derive
exclusion limits on the couplings for three different models: the
$L_{\mu}-L_{\tau}$ model; a leptophilic scalar model, for which we probe for
the first time masses above 6.5 GeV/$c^{2}$; and a model with an ALP decaying
to leptons, for which we set world-leading limits over the entire mass range
considered.

Figure 4: Observed 90% CL upper limits and corresponding expected limits as
functions of mass on (top) the leptophilic scalar coupling $\xi$, and on
(bottom) the ALP coupling to leptons $|C_{\ell\ell}|/\Lambda$ in the
hypothesis of equal couplings to the three lepton families and zero couplings
to all other particles. Also shown are (top) constraints for $S$ decaying in
electrons or muons from a BABAR search Lees  _et al._ (2020) and (bottom)
constraints for an ALP decaying to leptons from a reinterpretation Bauer  _et
al._ (2017, 2022) of BABAR searches. The red band in the top plot shows the
region that explains the muon anomalous magnetic moment $(g-2)_{\mu}\pm
2\sigma$.

We thank Andrea Thamm for helpful conversations on the axion-like particle.

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.

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\begin{split} \label{E:PROOFSTEPJUSTBELOWTOPORDERENERGYINTEGRALIDENTITYWAVEWITHDIFFICULTTERM}
& \mathbb{E}_{(\textnormal{Wave})}[\Tanset^{\Ntop-1} \wavearray](t,\eik)
\mathbb{F}_{(\textnormal{Wave})}[\Tanset^{\Ntop-1} \wavearray](t,\eik)
\spacetimeintegralcontrolwave[\Tanset^{\Ntop-1} \wavearray](t,\eik)
\\
\leq
\mbox{\upshape data}
\mbox{\upshape data}
\cdot
\int_{t'=0}^t
\upmu_{\star}^{- \frac{A}{2} + \frac{1}{2}}(t',\eik)
\int_{t''=0}^{t'}
\upmu_{\star}^{-\frac{A}{2} - \frac{1}{2}}(t'',\eik)
\, \rmd t''
\, \rmd t'
\cdots.
\end{split}
\end{align}

Using (<ref>)
and twice using (<ref>),
we deduce that:

\begin{align}
\begin{split} \label{E:SECONDPROOFSTEPJUSTBELOWTOPORDERENERGYINTEGRALIDENTITYWAVEWITHDIFFICULTTERM}
& \mathbb{E}_{(\textnormal{Wave})}[\Tanset^{\Ntop-1} \wavearray](t,\eik)
\mathbb{F}_{(\textnormal{Wave})}[\Tanset^{\Ntop-1} \wavearray](t,\eik)
\spacetimeintegralcontrolwave[\Tanset^{\Ntop-1} \wavearray](t,\eik)
\\
\leq
\mbox{\upshape data}
\cdot
\upmu_{\star}^{-(A-2)}(t,\eik),
\end{split}
\end{align}

which is indeed consistent with the desired estimate (<ref>).
This concludes our proof sketch of (<ref>).

Using the same strategy, one can continue the descent in the energy estimate hierarchy, gaining two powers
of $\upmu_{\star}$ with each level of descent until one finally
one can use (<ref>) to reach the level (<ref>),
where the energies remain uniformly bounded.

§.§.§ Discussion of the proof of the top-order inequality (<ref>)

We now sketch some key ideas behind the proof of (<ref>).
We focus on the most difficult case
in which the operator $\Tanset^{\Ntop}$ in (<ref>)
is of the form[The case in which $\Tanset^{\Ntop}$ contains an $\Lunit$-differentiation is much easier because
in that case, one can directly use equation (<ref>) to estimate the $\Lunit$-derivatives
of $\mytr_{\gtorus} \upchi$. ]
$\Tanset^{\Ntop} = \Angularset^{\Ntop}$,
where $\Angularset^{\Ntop}$ is a string of elements of the $\ell_{t,\eik}$-tangent subset $\Angularset$ defined in (<ref>).
Using the wave equation energy-null flux identity (<ref>),
the commuted wave equation (<ref>),
the transport/elliptic estimates for the vorticity and entropy
described in Sects. <ref>–<ref>,
and the coerciveness of the spacetime integral from (<ref>),
one can show that for $\Psi \in \wavearray$ (which is defined in (<ref>)),
the following identity holds for $(t,\eik) \in [0,\Tboot) \times [-\rightu,\compactsupportu]$:

\begin{align}
\begin{split} \label{E:HIGHORDERENERGYINTEGRALIDENTITYWAVE}
& \mathbb{E}_{(\textnormal{Wave})}[\Angularset^{\Ntop} \Psi](t,\eik)
\mathbb{F}_{(\textnormal{Wave})}[\Angularset^{\Ntop} \Psi](t,\eik)
\spacetimeintegralcontrolwave[\Angularset^{\Ntop} \Psi](t,\eik)
\\
\mbox{\upshape data}
\int_{\mathcal{M}_{[0,t),[-\rightu,\eik]}}
(\multipliervectorfield \Angularset^{\Ntop} \Psi)
\cdot
(\muX \Psi)
\cdot
\Angularset^{\Ntop} \mytr_{\gtorus} \upchi
\, \voltorus \rmd \eik' \rmd t'
\\
& \ \
\int_{\mathcal{M}_{[0,t),[-\rightu,\eik]}}
\upmu
(\multipliervectorfield \Angularset^{\Ntop} \Psi)
\cdot
\Angularset^{\Ntop} \modVortVort
\, \voltorus \rmd \eik' \rmd t'
\\
& \ \
\int_{\mathcal{M}_{[0,t),[-\rightu,\eik]}}
\upmu
(\multipliervectorfield \Angularset^{\Ntop} \Psi)
\cdot
\Angularset^{\Ntop} \modDivGradEnt
\, \voltorus \rmd \eik' \rmd t'
\cdots,
\end{split}
\end{align}

where $\cdots$ denotes easier error terms.

We first consider the difficult case $\Psi = \RRiemann$.
We recall the definition (<ref>) of $\multipliervectorfield$.
We consider the part of the integral
\int_{\mathcal{M}_{[0,t),[-\rightu,\eik]}}
(\multipliervectorfield \Angularset^{\Ntop} \Psi)
\cdot
(\muX \Psi)
\cdot
\Angularset^{\Ntop} \mytr_{\gtorus} \upchi
\, \voltorus \rmd \eik' \rmd t'
on RHS (<ref>)
that is generated by the $2 \muX$ term in (<ref>).
Using (<ref>),
and using (<ref>) to replace $\muX \RRiemann$ with $\Lunit \upmu$
up to error terms,
we can express the $2 \muX$-involving
portion of the integral under consideration as follows, where we highlight the
crucial factors of $\frac{1}{\upmu}$ on RHS (<ref>),
and for simplicity, we have ignored the factors of $\smoothfunction(\wavearray)$ on RHS (<ref>):

\begin{align} \label{E:REEXPRESSINGDIFFICULTERRORINTEGRAL}
\begin{split}
\int_{\mathcal{M}_{[0,t),[-\rightu,\eik]}}
(\muX \Angularset^{\Ntop} \RRiemann)
\cdot
(\muX \RRiemann)
\cdot
\Angularset^{\Ntop} \mytr_{\gtorus} \upchi
\, \voltorus \rmd \eik' \rmd t'
\\
& =
\int_{\mathcal{M}_{[0,t),[-\rightu,\eik]}}
\frac{\Lunit \upmu}{\upmu}
\cdot
(\muX \Angularset^{\Ntop} \RRiemann)
\cdot
(\muX \Angularset^{\Ntop} \RRiemann)
\, \voltorus \rmd \eik' \rmd t'
\\
\ \
\int_{\mathcal{M}_{[0,t),[-\rightu,\eik]}}
\frac{\Lunit \upmu}{\upmu}
\cdot
(\muX \Angularset^{\Ntop} \RRiemann)
\cdot
\fullymodquant{\Angularset^{\Ntop}}
\, \voltorus \rmd \eik' \rmd t'
\cdots
\end{split}
\end{align}

From (<ref>), it follows that
the first integral on RHS (<ref>)
is bounded by the $A$-multiplied integral on RHS (<ref>)
(this integral contributes a “portion of” the “$A$”).
The $\fullymodquant{\Angularset^{\Ntop}}$-involving integral on RHS (<ref>)
can be bounded by a similar error term involving a double time-integral, where one of the time integrations
comes from integrating (<ref>);
we refer to the proof of <cit.>*Proposition 14.2 for the details.
The part of the first integral on RHS (<ref>)
that is generated by the $(1 + 2 \upmu) \Lunit$ term in (<ref>)
can be handled through an argument involving integration by parts in $\Lunit$, which leads to difficult
critical-strength boundary terms that make a similar contribution to the blowup of the top-order energies
as the $A$-multiplied integral on RHS (<ref>); the same arguments
given in the proof of <cit.>*Proposition 14.2 can be used to handle these terms.
The $\Angularset^{\Ntop} \modVortVort$- and $\Angularset^{\Ntop} \modDivGradEnt$-involving integrals
on RHS (<ref>) are easy to handle, thanks to the
helpful factor of $\upmu$ in the integrand and the
strategy for bounding $\Angularset^{\Ntop} \modVortVort$ and $\Angularset^{\Ntop} \modDivGradEnt$ via
transport-div-curl estimates that we described in Sect. <ref>.

We now consider the remaining cases, namely $\Psi \in \lbrace \LRiemann, \Ent, \fourvelocity^2, \fourvelocity^3 \rbrace$.
The identity (<ref>) holds in these cases too. However, the factor of
$\muX \Psi$ in the identity is now very small in $L^{\infty}$ in these cases, since we are considering perturbations of simple
isentropic plane symmetric solutions (for which these quantities all vanish). Hence, due to the smallness,
all error terms on RHS (<ref>) are much easier to treat and can in fact
be relegated to the error terms “$\cdots$” on RHS (<ref>);
see <cit.>*page 154 for further details.

§.§.§ Discussion of the proof of the just-below-top-order inequality (<ref>)

We now sketch some key ideas behind the proof of
the just-below-top-order inequality (<ref>).
First, one uses the identity (<ref>),
which, for any $\Psi \in \wavearray$, holds with $\Tanset^{\Ntop-1}$ in the role of $\Angularset^{\Ntop}$.
The key step, which is the one that is different compared to the proof of
(<ref>), is that
one bounds the error integral on LHS (<ref>), namely

\begin{align} \label{E:BELOWTOPORDERDIFFICULTERRORINTEGRAL}
\int_{\mathcal{M}_{[0,t),[-\rightu,\eik]}}
(\multipliervectorfield \Tanset^{\Ntop-1} \Psi)
\cdot
(\muX \Psi)
\cdot
\Tanset^{\Ntop-1} \mytr_{\gtorus} \upchi
\, \voltorus \rmd \eik' \rmd t',
\end{align}

in a different way. First, one uses the bootstrap assumptions to bound the factor
$|\muX \Psi|$ in (<ref>) by $\lesssim 1$.
Next, one uses (<ref>),
the Cauchy–Schwarz inequality, and (<ref>)
to bound the integral in (<ref>)

\begin{align} \label{E:FIRSTPROOFSTEPJUSTBELOWTOPORDERENERGYINTEGRALIDENTITYWAVEWITHDIFFICULTTERM}
\lesssim
\int_0^t
\frac{1}{\upmu_{\star}^{1/2}(t',\eik)}
\mathbb{E}_{(\textnormal{Wave})}^{1/2}[\Tanset^{\Ntop-1} \Psi](t',\eik)
\left\| \Tanset^{\Ntop-1} \mytr_{\gtorus} \upchi \right\|_{L^2(\Sigma_{t'}^{[-\rightu,\eik]})}
\, \rmd t'
\cdots,
\end{align}

where we have used that the $\nullhyparg{\eik}$-tangent derivative terms in the energy (<ref>) contain a $\upmu$-weight,
while there is an $\Lunit$ factor in the definition of $\multipliervectorfield$ that does not contain a $\upmu$ weight.
Next, to bound the factor
$\| \Angularset^{\Ntop-1} \mytr_{\gtorus} \upchi \|_{L^2(\Sigma_{t'}^{[-\rightu,\eik]})}$,
one commutes the Raychaudhuri equation (<ref>) with
$\Tanset^{\Ntop-1}$ and uses the schematic relation
$\Ricfour_{\Lunit \Lunit} = \Tanset^2 \wavearray + \cdots$
to deduce the following evolution equation, schematically depicted:
$\Lunit \Angularset^{\Ntop-1} \mytr_{\gtorus} \upchi = \Tanset^{\Ntop+1} \wavearray + \cdots$,
where $\cdots$ denotes lower order error terms.
This equation represents a loss of one derivative in the estimates and leads to the coupling between
the different order energies described below (<ref>).
However, such a loss is permissible below the top-order.
Since $\Lunit t = 1$ and $\Lunit \eik = 0$, we can integrate this evolution equation
and use (<ref>) to deduce that:

\begin{align}
\begin{split} \label{E:THIRDPROOFSTEPJUSTBELOWTOPORDERENERGYINTEGRALIDENTITYWAVEWITHDIFFICULTTERM}
\left\| \Tanset^{\Ntop-1} \mytr_{\gtorus} \upchi \right\|_{L^2(\Sigma_{t'}^{[-\rightu,\eik]})}
& \lesssim
\mbox{\upshape data}
\int_0^{t'}
\| \Tanset^{\Ntop+1} \wavearray \|_{L^2(\Sigma_{t''}^{[-\rightu,\eik]})}
\, \rmd t''
\cdots
\\
\lesssim
\mbox{\upshape data}
\int_0^{t'}
\frac{1}{\upmu_{\star}^{1/2}(t'',\eik)}
\mathbb{E}_{(\textnormal{Wave})}^{1/2}[\Tanset^{\Ntop} \Psi](t'',\eik)
\, \rmd t''
\cdots,
\end{split}
\end{align}

where to obtain the last line of (<ref>),
we have again used that the $\nullhyparg{\eik}$-tangent derivative terms in the energy (<ref>) contain a $\upmu$-weight.
Inserting (<ref>)
into (<ref>),
we conclude (<ref>).

§ OPEN PROBLEMS

In this section, we describe various open problems tied to shocks.

* Prove Conjecture <ref>.
In Sect. <ref>, we outlined how to achieve this.
We are confident that the conjecture can be proved, especially since
the formulation of relativistic Euler flow
provided by Theorem <ref>
is qualitatively similar to the formulation
of $3D$ non-relativistic compressible Euler flow derived in <cit.>,
a system for which shock formation results have been derived
(see Sect. <ref>).
Nonetheless, there are many non-trivial details that have to be checked,
and this would be a good project for someone who wants to learn the field.

* Prove Conjecture <ref>.
This problem is much more technically demanding than proving Conjecture <ref>.
However, for the same reasons
mentioned above, we expect that
this can be achieved by adapting the methods used in the non-relativistic work <cit.>
to the equations of Theorem <ref>.

* Prove Conjecture <ref>.
This problem is also much more technically demanding than proving Conjecture <ref>.
For the same reasons
mentioned above, we expect that
this can be achieved by adapting the methods that we are using in our forthcoming
non-relativistic work <cit.> to the equations of Theorem <ref>.

* The shock development problem.
Recall that in <cit.>, Christodoulou solved
the restricted shock development problem for the relativistic Euler equations
and the non-relativistic compressible Euler equations
in an arbitrary number of spatial dimensions; see Sect. <ref>.
The general problem, i.e., the shock development problem
with vorticity and entropy, remains an outstanding open problem for both systems in two or more spatial dimensions.

* The global behavior of solutions with shocks.
After the shock development problem (which is local-in-time) is solved,
a particularly compelling problem will be to understand the global-in-time-and-space
behavior of the corresponding weak solutions,
at least in a perturbative regime. Among the many outstanding
challenges in this problem is that of understanding the long-time behavior
of the vorticity and entropy; even in the context of smooth incompressible non-relativistic flows,
the long-time behavior of the vorticity is not well understood. We do, however, mention
that in the breakthrough work <cit.>, for axisymmetric incompressible non-relativistic flows in $3D$
with a boundary, smooth initial conditions were identified such that the
solution (including the vorticity) blows up in finite time in an approximately self-similar fashion;
such vorticity blowup, if present in multi-dimensional relativistic Euler flow,
would be a serious obstacle to finding a meaningful way to continue the solution weakly past the singularity.

* Extending the results to the coupled Einstein–Euler system.
It is of great interest to extend the above results to the Einstein–Euler system in multiple spatial dimensions.
The expectation is that while the fluid will exhibit shock wave phenomena (as in the uncoupled problem),
the gravitational metric will exhibit less singular behavior
As of present, the only known singularity formation result for the coupled system
is in the plane symmetric case <cit.>, in which the dynamics are described by $1+1$-dimensional hyperbolic PDEs.
In <cit.>, it was shown that for many equations of state and a large class of plane-symmetric initial data, the
Einstein-Euler solution breaks down in finite time. Although <cit.> provided heuristic arguments
suggesting that the singularity is of shock-type,
a precise description of the singularity was not given.
The multi-dimensional problem is a difficult PDE problem because surfaces that are
null or “barely spacelike” with respect to the acoustical metric are in fact timelike with respect to the gravitational
metric; the reason is that speed of sound is slower than the speed of propagation of gravitational waves
(at least when the speed of sound is less than unity). A corresponding
key difficulty that arises in the context of the PDE estimates is that
one must control the spacetime metric on various gravitationally timelike surfaces on which the fluid is singular; this is
difficult at the top derivative level because generally, energy estimates for the spacetime metric
are not available on surfaces that are timelike with respect to the spacetime metric.
In contrast, in multiple wave speed systems such that the shock forms in the fastest wave,
it is possible to close the top-order energy estimates and thus prove stable shock formation
for the coupled problem <cit.>.

* Extending the results to more complicated multiple speed systems. It is also of great physical and mathematical
interest to extend the above results to other multiple speed systems, such as the equations of
compressible magnetohydrodynamics, the GRMHD equations (i.e., the coupling of Einstein's equations to the equations
of relativistic magnetohydrodynamics), the equations of elasticity, the equations of nonlinear electromagnetism,
and the equations of crystal optics. A key difficulty of these systems is that their principal symbols
are more complicated than wave operators and transport operators. Relatedly, their corresponding geometry is more
complicated than Lorentzian geometry, and the characteristics comprise multiple sheets,
which can be singular even at the tangent space level.
These difficulties are a serious obstacle to implementing the kind of sharp version of
nonlinear geometric optics (i.e., eikonal functions) that has been so successfully employed
to study shocks for wave equations and fluid equations.

* Implosion singularities. In the breakthrough works fMpRiRjS2022a,fMpRiRjS2022b,
the authors proved implosion
singularity formation for some initially $C^{\infty}$ spherically symmetric solutions to the compressible Euler equations
and Navier–Stokes equations under adiabatic equations of state $p = \uprho^{\upgamma}$ for $\upgamma > 1$, outside of a
countable set of $\upgamma$-values.
In <cit.>, the results were extended to allow for all $\upgamma > 1$.
Implosion singularities are much more severe than shocks in the sense that the density and velocity themselves
blow up (in shock singularities, it is their gradient that blows up)
in finite time at the center of symmetry. The proof depends on the detailed structure of the equations
and in particular relies on a careful analysis of a phase portrait for self-similar solutions.
Hence, it is of interest to decide whether a similar result holds for the relativistic Euler equations.
We highlight the important related work <cit.>, in which the authors studied
spherically symmetric solutions
the Einstein–Euler equations under adiabatic equations of state $p = \varepsilon \uprho$ for $0 < \varepsilon \ll 1$ sufficiently small
and proved the existence of self-similar solutions that form a naked singularity.
See also <cit.>, in which the authors studied solutions
to the non-relativistic Euler–Poisson system under adiabatic equations of state $p = \uprho^{\upgamma}$ for $\upgamma \in (1,\frac{4}{3})$
and showed the existence of spherically symmetric
self-similar imploding solutions modeling gravitational collapse, i.e.,
initially smooth solutions such that the density blows up in finite time.

* Inviscid limits. In <cit.>, the authors studied the inviscid limit of solutions to the $1D$
viscous Burgers' equation all the way up to the time of first shock formation in the inviscid solution.
They decomposed the viscous solution into a singular piece and a smoother piece and proved that the viscous solution converges
to the singular piece in $L^{\infty}$ as the viscosity vanishes, where the $L^{\infty}$ norm is taken over the entire
slab of classical existence of the inviscid solution. This is the first result of its type that extends all the way to the time
of first singularity formation. Important open problems include extending this result to the $1D$ compressible Euler equations
(where the corresponding viscous equations are the Navier–Stokes equations) and, after that, to multi-dimensions.
It would also be of interest to extend the result to the relativistic Euler equations, but as of present,
it is not clear if there exist any well-posed relativistic viscous fluid models that suppress the singularity formation while
retaining physically desirable features such as causality.

* Rarefaction waves. In Sect. <ref>, we described the recent works tWLpY2023a,tWLpY2023b
on irrotational rarefaction wave solutions to the $2D$ compressible Euler equations. It is of interest to
eliminate the irrotationality assumption and to
extend these results to the relativistic Euler equations, and perhaps even the Einstein–Euler equations.

$'$ $'$ $'$

author=Abbrescia, Leo,
author=Speck, Jared,
title=The emergence of the singular boundary from the crease in $3D$
compressible Euler flow,
journal=arXiv e-prints,

author=Abbrescia, Leonardo,
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