The Belle and Belle II Collaborations

# Measurement of branching fractions, CP asymmetry, and isospin asymmetry for
$\bm{B\rightarrow\rho\gamma}$ decays using Belle and Belle II data

I. Adachi  0000-0003-2287-0173 K. Adamczyk  0000-0001-6208-0876 L. Aggarwal
0000-0002-0909-7537 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 R. Ayad
0000-0003-3466-9290 V. Babu  0000-0003-0419-6912 H. Bae  0000-0003-1393-8631
S. Bahinipati  0000-0002-3744-5332 P. Bambade  0000-0001-7378-4852 Sw.
Banerjee  0000-0001-8852-2409 S. Bansal  0000-0003-1992-0336 M. Barrett
0000-0002-2095-603X J. Baudot  0000-0001-5585-0991 A. Baur
0000-0003-1360-3292 A. Beaubien  0000-0001-9438-089X F. Becherer
0000-0003-0562-4616 J. Becker  0000-0002-5082-5487 J. V. Bennett
0000-0002-5440-2668 F. U. Bernlochner  0000-0001-8153-2719 V. Bertacchi
0000-0001-9971-1176 M. Bertemes  0000-0001-5038-360X E. Bertholet
0000-0002-3792-2450 M. Bessner  0000-0003-1776-0439 S. Bettarini
0000-0001-7742-2998 B. Bhuyan  0000-0001-6254-3594 F. Bianchi
0000-0002-1524-6236 L. Bierwirth  0009-0003-0192-9073 T. Bilka
0000-0003-1449-6986 S. Bilokin  0000-0003-0017-6260 D. Biswas
0000-0002-7543-3471 A. Bobrov  0000-0001-5735-8386 D. Bodrov
0000-0001-5279-4787 A. Bolz  0000-0002-4033-9223 A. Bondar
0000-0002-5089-5338 A. Bozek  0000-0002-5915-1319 M. Bračko
0000-0002-2495-0524 P. Branchini  0000-0002-2270-9673 R. A. Briere
0000-0001-5229-1039 T. E. Browder  0000-0001-7357-9007 A. Budano
0000-0002-0856-1131 S. Bussino  0000-0002-3829-9592 M. Campajola
0000-0003-2518-7134 L. Cao  0000-0001-8332-5668 G. Casarosa
0000-0003-4137-938X C. Cecchi  0000-0002-2192-8233 J. Cerasoli
0000-0001-9777-881X M.-C. Chang  0000-0002-8650-6058 P. Chang
0000-0003-4064-388X R. Cheaib  0000-0001-5729-8926 P. Cheema
0000-0001-8472-5727 B. G. Cheon  0000-0002-8803-4429 K. Chilikin
0000-0001-7620-2053 K. Chirapatpimol  0000-0003-2099-7760 H.-E. Cho
0000-0002-7008-3759 K. Cho  0000-0003-1705-7399 S.-K. Choi
0000-0003-2747-8277 S. Choudhury  0000-0001-9841-0216 L. Corona
0000-0002-2577-9909 S. Das  0000-0001-6857-966X F. Dattola
0000-0003-3316-8574 E. De La Cruz-Burelo  0000-0002-7469-6974 S. A. De La
Motte  0000-0003-3905-6805 G. De Nardo  0000-0002-2047-9675 M. De Nuccio
0000-0002-0972-9047 G. De Pietro  0000-0001-8442-107X R. de Sangro
0000-0002-3808-5455 M. Destefanis  0000-0003-1997-6751 R. Dhamija
0000-0001-7052-3163 A. Di Canto  0000-0003-1233-3876 F. Di Capua
0000-0001-9076-5936 J. Dingfelder  0000-0001-5767-2121 Z. Doležal
0000-0002-5662-3675 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 R. Garg
0000-0002-7406-4707 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 E. Graziani
0000-0001-8602-5652 D. Greenwald  0000-0001-6964-8399 Z. Gruberová
0000-0002-5691-1044 T. Gu  0000-0002-1470-6536 Y. Guan  0000-0002-5541-2278 K.
Gudkova  0000-0002-5858-3187 S. Halder  0000-0002-6280-494X Y. Han
0000-0001-6775-5932 T. Hara  0000-0002-4321-0417 H. Hayashii
0000-0002-5138-5903 S. Hazra  0000-0001-6954-9593 M. T. Hedges
0000-0001-6504-1872 A. Heidelbach  0000-0002-6663-5469 I. Heredia de la Cruz
0000-0002-8133-6467 M. Hernández Villanueva  0000-0002-6322-5587 T. Higuchi
0000-0002-7761-3505 M. Hoek  0000-0002-1893-8764 M. Hohmann
0000-0001-5147-4781 P. Horak  0000-0001-9979-6501 C.-L. Hsu
0000-0002-1641-430X T. Humair  0000-0002-2922-9779 T. Iijima
0000-0002-4271-711X K. Inami  0000-0003-2765-7072 N. Ipsita
0000-0002-2927-3366 A. Ishikawa  0000-0002-3561-5633 R. Itoh
0000-0003-1590-0266 M. Iwasaki  0000-0002-9402-7559 P. Jackson
0000-0002-0847-402X W. W. Jacobs  0000-0002-9996-6336 E.-J. Jang
0000-0002-1935-9887 Q. P. Ji  0000-0003-2963-2565 S. Jia  0000-0001-8176-8545
Y. Jin  0000-0002-7323-0830 K. K. Joo  0000-0002-5515-0087 H. Junkerkalefeld
0000-0003-3987-9895 D. Kalita  0000-0003-3054-1222 A. B. Kaliyar
0000-0002-2211-619X J. Kandra  0000-0001-5635-1000 K. H. Kang
0000-0002-6816-0751 G. Karyan  0000-0001-5365-3716 T. Kawasaki
0000-0002-4089-5238 F. Keil  0000-0002-7278-2860 C. Kiesling
0000-0002-2209-535X C.-H. Kim  0000-0002-5743-7698 D. Y. Kim
0000-0001-8125-9070 K.-H. Kim  0000-0002-4659-1112 Y.-K. Kim
0000-0002-9695-8103 H. Kindo  0000-0002-6756-3591 K. Kinoshita
0000-0001-7175-4182 P. Kodyš  0000-0002-8644-2349 T. Koga  0000-0002-1644-2001
S. Kohani  0000-0003-3869-6552 K. Kojima  0000-0002-3638-0266 A. Korobov
0000-0001-5959-8172 S. Korpar  0000-0003-0971-0968 E. Kovalenko
0000-0001-8084-1931 R. Kowalewski  0000-0002-7314-0990 T. M. G. Kraetzschmar
0000-0001-8395-2928 P. Križan  0000-0002-4967-7675 P. Krokovny
0000-0002-1236-4667 T. Kuhr  0000-0001-6251-8049 J. Kumar  0000-0002-8465-433X
M. Kumar  0000-0002-6627-9708 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 M. J. Lee
0000-0003-4528-4601 D. Levit  0000-0001-5789-6205 P. M. Lewis
0000-0002-5991-622X C. Li  0000-0002-3240-4523 L. K. Li  0000-0002-7366-1307
Y. Li  0000-0002-4413-6247 Y. B. Li  0000-0002-9909-2851 J. Libby
0000-0002-1219-3247 M. H. Liu  0000-0002-9376-1487 Q. Y. Liu
0000-0002-7684-0415 Z. Q. Liu  0000-0002-0290-3022 D. Liventsev
0000-0003-3416-0056 S. Longo  0000-0002-8124-8969 T. Lueck
0000-0003-3915-2506 C. Lyu  0000-0002-2275-0473 Y. Ma  0000-0001-8412-8308 M.
Maggiora  0000-0003-4143-9127 S. P. Maharana  0000-0002-1746-4683 R. Maiti
0000-0001-5534-7149 S. Maity  0000-0003-3076-9243 G. Mancinelli
0000-0003-1144-3678 R. Manfredi  0000-0002-8552-6276 E. Manoni
0000-0002-9826-7947 M. Mantovano  0000-0002-5979-5050 D. Marcantonio
0000-0002-1315-8646 S. Marcello  0000-0003-4144-863X C. Marinas
0000-0003-1903-3251 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 D. Matvienko  0000-0002-2698-5448 S. K. Maurya
0000-0002-7764-5777 J. A. McKenna  0000-0001-9871-9002 R. Mehta
0000-0001-8670-3409 F. Meier  0000-0002-6088-0412 M. Merola
0000-0002-7082-8108 F. Metzner  0000-0002-0128-264X 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 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 M. Neu
0000-0002-4564-8009 C. Niebuhr  0000-0002-4375-9741 S. Nishida
0000-0001-6373-2346 S. Ogawa  0000-0002-7310-5079 Y. Onishchuk
0000-0002-8261-7543 H. Ono  0000-0003-4486-0064 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. Panta  0000-0001-6385-7712 S. Pardi
0000-0001-7994-0537 K. Parham  0000-0001-9556-2433 H. Park
0000-0001-6087-2052 S.-H. Park  0000-0001-6019-6218 A. Passeri
0000-0003-4864-3411 S. Patra  0000-0002-4114-1091 S. Paul  0000-0002-8813-0437
T. K. Pedlar  0000-0001-9839-7373 R. Peschke  0000-0002-2529-8515 R. Pestotnik
0000-0003-1804-9470 M. Piccolo  0000-0001-9750-0551 L. E. Piilonen
0000-0001-6836-0748 G. Pinna Angioni  0000-0003-0808-8281 P. L. M. Podesta-
Lerma  0000-0002-8152-9605 T. Podobnik  0000-0002-6131-819X S. Pokharel
0000-0002-3367-738X C. Praz  0000-0002-6154-885X S. Prell  0000-0002-0195-8005
E. Prencipe  0000-0002-9465-2493 M. T. Prim  0000-0002-1407-7450 H. Purwar
0000-0002-3876-7069 P. Rados  0000-0003-0690-8100 G. Raeuber
0000-0003-2948-5155 S. Raiz  0000-0001-7010-8066 N. Rauls  0000-0002-6583-4888
M. Reif  0000-0002-0706-0247 S. Reiter  0000-0002-6542-9954 M. Remnev
0000-0001-6975-1724 I. Ripp-Baudot  0000-0002-1897-8272 G. Rizzo
0000-0003-1788-2866 S. H. Robertson  0000-0003-4096-8393 M. Roehrken
0000-0003-0654-2866 J. M. Roney  0000-0001-7802-4617 A. Rostomyan
0000-0003-1839-8152 N. Rout  0000-0002-4310-3638 G. Russo  0000-0001-5823-4393
D. A. Sanders  0000-0002-4902-966X S. Sandilya  0000-0002-4199-4369 L. Santelj
0000-0003-3904-2956 Y. Sato  0000-0003-3751-2803 V. Savinov
0000-0002-9184-2830 B. Scavino  0000-0003-1771-9161 C. Schmitt
0000-0002-3787-687X C. Schwanda  0000-0003-4844-5028 A. J. Schwartz
0000-0002-7310-1983 M. Schwickardi  0000-0003-2033-6700 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. P. Shen  0000-0002-9012-4618 X. D. Shi
0000-0002-7006-6107 T. Shillington  0000-0003-3862-4380 T. Shimasaki
0000-0003-3291-9532 J.-G. Shiu  0000-0002-8478-5639 D. Shtol
0000-0002-0622-6065 A. Sibidanov  0000-0001-8805-4895 F. Simon
0000-0002-5978-0289 J. B. Singh  0000-0001-9029-2462 J. Skorupa
0000-0002-8566-621X R. J. Sobie  0000-0001-7430-7599 M. Sobotzik
0000-0002-1773-5455 A. Soffer  0000-0002-0749-2146 A. Sokolov
0000-0002-9420-0091 E. Solovieva  0000-0002-5735-4059 S. Spataro
0000-0001-9601-405X B. Spruck  0000-0002-3060-2729 M. Starič
0000-0001-8751-5944 P. Stavroulakis  0000-0001-9914-7261 S. Stefkova
0000-0003-2628-530X R. Stroili  0000-0002-3453-142X M. Sumihama
0000-0002-8954-0585 K. Sumisawa  0000-0001-7003-7210 W. Sutcliffe
0000-0002-9795-3582 H. Svidras  0000-0003-4198-2517 M. Takizawa
0000-0001-8225-3973 U. Tamponi  0000-0001-6651-0706 S. Tanaka
0000-0002-6029-6216 K. Tanida  0000-0002-8255-3746 F. Tenchini
0000-0003-3469-9377 O. Tittel  0000-0001-9128-6240 R. Tiwary
0000-0002-5887-1883 D. Tonelli  0000-0002-1494-7882 E. Torassa
0000-0003-2321-0599 K. Trabelsi  0000-0001-6567-3036 I. Tsaklidis
0000-0003-3584-4484 M. Uchida  0000-0003-4904-6168 I. Ueda
0000-0002-6833-4344 Y. Uematsu  0000-0002-0296-4028 T. Uglov
0000-0002-4944-1830 K. Unger  0000-0001-7378-6671 Y. Unno  0000-0003-3355-765X
K. Uno  0000-0002-2209-8198 S. Uno  0000-0002-3401-0480 P. Urquijo
0000-0002-0887-7953 Y. Ushiroda  0000-0003-3174-403X S. E. Vahsen
0000-0003-1685-9824 R. van Tonder  0000-0002-7448-4816 K. E. Varvell
0000-0003-1017-1295 M. Veronesi  0000-0002-1916-3884 A. Vinokurova
0000-0003-4220-8056 V. S. Vismaya  0000-0002-1606-5349 L. Vitale
0000-0003-3354-2300 V. Vobbilisetti  0000-0002-4399-5082 R. Volpe
0000-0003-1782-2978 B. Wach  0000-0003-3533-7669 M. Wakai  0000-0003-2818-3155
S. Wallner  0000-0002-9105-1625 E. Wang  0000-0001-6391-5118 M.-Z. Wang
0000-0002-0979-8341 X. L. Wang  0000-0001-5805-1255 Z. Wang
0000-0002-3536-4950 A. Warburton  0000-0002-2298-7315 S. Watanuki
0000-0002-5241-6628 C. Wessel  0000-0003-0959-4784 J. Wiechczynski
0000-0002-3151-6072 E. Won  0000-0002-4245-7442 X. P. Xu  0000-0001-5096-1182
B. D. Yabsley  0000-0002-2680-0474 S. Yamada  0000-0002-8858-9336 W. Yan
0000-0003-0713-0871 S. B. Yang  0000-0002-9543-7971 J. Yelton
0000-0001-8840-3346 J. H. Yin  0000-0002-1479-9349 K. Yoshihara
0000-0002-3656-2326 C. Z. Yuan  0000-0002-1652-6686 L. Zani
0000-0003-4957-805X B. Zhang  0000-0002-5065-8762 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

###### Abstract

We present measurements of $B^{+}\rightarrow\rho^{+}\gamma$ and
$B^{0}\rightarrow\rho^{0}\gamma$ decays using a combined data sample of
$772\times 10^{6}$ $B\overline{B}$ pairs collected by the Belle experiment and
$387\times 10^{6}$ $B\overline{B}$ pairs collected by the Belle II experiment
in $e^{+}e^{-}$ collisions at the $\Upsilon(4S)$ resonance. After an optimized
selection, a simultaneous fit to the Belle and Belle II data sets yields
$114\pm 12$ $B^{+}\rightarrow\rho^{+}\gamma$ and $99\pm 12$
$B^{0}\rightarrow\rho^{0}\gamma$ decays. The measured branching fractions are
$(13.1^{+2.0+1.3}_{-1.9-1.2})\times 10^{-7}$ and $(7.5\pm
1.3^{+1.0}_{-0.8})\times 10^{-7}$ for $B^{+}\rightarrow\rho^{+}\gamma$ and
$B^{0}\rightarrow\rho^{0}\gamma$ decays, respectively, where the first
uncertainty is statistical and the second is systematic. We also measure the
isospin asymmetry $A_{\rm
I}(B\rightarrow\rho\gamma)=(10.9^{+11.2+7.8}_{-11.7-7.3})\%$ and the direct CP
asymmetry $A_{\it CP}(B^{+}\rightarrow\rho^{+}\gamma)=(-8.2\pm
15.2^{+1.6}_{-1.2})\%$.

††preprint: APS/123-QED

##  I Introduction

Flavor-changing neutral-current (FCNC) processes are sensitive probes of
physics beyond the standard model (SM). These decays are forbidden at tree
level, making them particularly sensitive to beyond-the-SM contributions,
which could be significant compared to loop-level SM amplitudes. Beyond-the-SM
physics searches with radiative decays of $B$ mesons, such as $B\rightarrow
K^{\ast}\gamma$ and $B\rightarrow\rho\gamma$, is attractive, as these decays
are FCNC processes dominated by the $b\rightarrow(s,d)\gamma$ transition,
which makes it easier to distinguish beyond-the-SM contributions Belle-
II:2018jsg .

The $B\rightarrow\rho\gamma$ decay involves a $b\rightarrow d$ transition at
the quark level and, within the SM, has a branching fraction one order of
magnitude smaller than radiative $B$ decays involving $b\rightarrow s$
transitions. However, the $B\rightarrow\rho\gamma$ decay mode can be affected
by beyond-the-SM processes not appearing in $b\rightarrow s$ processes. The
decay $B\rightarrow\rho\gamma$ was first observed by Belle experiment
Belle:2005grh  in 2006 and later confirmed by BaBar experiment BaBar:2006lms .
Both experiments subsequently made more precise measurements with larger data
sets: $657\times 10^{6}~{}B\overline{B}$ pairs for Belle Belle:2008imr  and
$465\times 10^{6}~{}B\overline{B}$ pairs for BaBar BaBar:2008txc . In the SM,
the $B\rightarrow\rho\gamma$ CP-averaged isospin asymmetry ($\bar{A}_{\rm
I}^{\rm SM}$) is predicted to be close to that of $B\rightarrow
K^{\ast}\gamma$ and to equal $(5.2\pm 2.8)\%$ Lyon:2013gba . This asymmetry is
defined as $\bar{A}_{\rm I}=\left(A_{\rm I}^{b}+A_{\rm I}^{\bar{b}}\right)/2$,
with

$\displaystyle A_{\rm
I}^{b}=\frac{c^{2}_{\rho}\Gamma\left(\overline{B}{}^{0}\rightarrow\rho^{0}\gamma\right)-\Gamma\left(B^{-}\rightarrow\rho^{-}\gamma\right)}{c^{2}_{\rho}\Gamma\left(\overline{B}{}^{0}\rightarrow\rho^{0}\gamma\right)+\Gamma\left(B^{-}\rightarrow\rho^{-}\gamma\right)},$
(1)

and $A_{\rm I}^{\bar{b}}$ defined similarly but with the CP-conjugate modes.
The factor $c_{\rho}=\sqrt{2}$ results from the quark content of the
$\rho^{0}$ meson (i.e., a Clebsch-Gordan coefficient). Experimentally, only
the isospin asymmetry with CP-averaged branching fractions, $A_{\rm I}$, has
been measured to date:

$\displaystyle A_{\rm
I}=\frac{c_{\rho}^{2}\Gamma(\overset{\textbf{(---)}}{B^{0}}\rightarrow\rho^{0}\gamma)-\Gamma(B^{\pm}\rightarrow\rho^{\pm}\gamma)}{c_{\rho}^{2}\Gamma(\overset{\textbf{(---)}}{B^{0}}\rightarrow\rho^{0}\gamma)+\Gamma(B^{\pm}\rightarrow\rho^{\pm}\gamma)},$
(2)

which equals $\bar{A}_{I}$ if the CP asymmetry

$\displaystyle A_{\it
CP}(B\rightarrow\rho\gamma)=\frac{\Gamma\left(\overline{B}\rightarrow\overline{\rho}\gamma\right)-\Gamma\left(B\rightarrow\rho\gamma\right)}{\Gamma\left(\overline{B}\rightarrow\overline{\rho}\gamma\right)+\Gamma\left(B\rightarrow\rho\gamma\right)}$
(3)

is the same for $B^{+}\rightarrow\rho^{+}\gamma$ and
$B^{0}\rightarrow\rho^{0}\gamma$ decays. The current world average of isospin
asymmetry measurements $A_{\rm
I}^{\mathrm{exp}}(\rho\gamma)=(30^{+16}_{-13})\%$ ParticleDataGroup:2022pth
is about two standard deviations from the SM expectation.

Here we report measurements of $B\rightarrow\rho\gamma$ decays performed using
both Belle and Belle II data sets. The signal decay modes are
$B^{+}\rightarrow\rho^{+}\left(\rightarrow\pi^{0}\pi^{+}\right)\gamma$ and
$B^{0}\rightarrow\rho^{0}\left(\rightarrow\pi^{+}\pi^{-}\right)\gamma$ charge-
conjugate . We use the full Belle data set corresponding to 711
${\mathrm{fb}^{-1}}$ taken at the $\Upsilon(4S)$ resonance energy (on-
resonance) containing $(772\pm 11)\times 10^{6}$ $B\overline{B}$ pairs. In
addition, we use a 362 ${\mathrm{fb}^{-1}}$ Belle II data set collected from
2019–2022, containing $(387\pm 6)\times 10^{6}$ $B\overline{B}$ pairs. We also
use off-resonance data sets, collected at an energy 60 MeV below the
$\Upsilon(4S)$ resonance, to study continuum background
($e^{+}e^{-}\rightarrow q\overline{q}$, where $q=u,d,s,c$). The off-resonance
data samples correspond to 89.5 ${\rm fb^{-1}}$ and 42.3 ${\rm fb^{-1}}$ for
Belle and Belle II, respectively. After applying selection criteria to
identify $B\to\rho\gamma$ candidates, we fit the data to determine $B^{0}$ and
$B^{+}$ branching fractions and asymmetries $A_{I}$ and $A_{CP}$.

This paper is organized as follows. Section II introduces the Belle and Belle
II detectors followed by the description of the data and simulated samples.
The event selection and reconstruction of the decays are described in section
III. The methods to suppress background are presented in section IV. Section V
explains the fitting procedure to extract signal yields. The systematic
uncertainties are discussed in section VI. The results are presented in
section VII.

##  II Detectors and data sets

The Belle detector Belle:2000cnh ; PTEP_belle  was a large-solid-angle
spectrometer that operated at the KEKB asymmetric-energy $e^{+}e^{-}$ (3.5 GeV
on 8.0 GeV) collider Kurokawa:2001nw ; PTEP_kekb . The detector consisted of a
silicon vertex detector and a central drift chamber (CDC) for reconstructing
trajectories of charged particles (tracks), an array of aerogel Cherenkov
counters (ACC) and time-of-flight scintillation counters (TOF) for identifying
charged particles, and an electromagnetic CsI(Tl) crystal calorimeter (ECL)
for identifying photons and electrons. These detectors were surrounded by a
superconducting solenoid coil providing a magnetic field of 1.5 T. An iron
flux return yoke located outside the coil was instrumented with resistive-
plate chambers to detect $K^{0}_{L}$ mesons and identify muons.

The Belle II detector Belle-II:2010dht  is an upgrade of the Belle detector
and operates at the SuperKEKB $e^{+}e^{-}$ collider. The energies of electron
and positron beams are 7.0 GeV and 4.0 GeV, respectively. The vertex detector
consists of pixel sensors and double-sided silicon strips. The Belle II CDC is
surrounded by two types of Cherenkov light detector systems used for particle
identification: a time-of-propagation detector (TOP) for the barrel region
(32.2∘ to 128.7∘), and an aerogel ring-imaging Cherenkov detector (ARICH) for
the forward endcap region (12.4∘ to 31.4∘). The Belle ECL is re-used in Belle
II along with the solenoid and the iron flux return yoke. For both Belle and
Belle II, the $z$ axis is defined as the central axis of the solenoid, with
the positive direction being very close to the direction of the electron beam.

Monte Carlo simulated samples are used to optimize selection criteria, study
sources of backgrounds, calculate reconstruction efficiencies, and determine
probability density functions (PDFs) used for fitting the data. We use EvtGen
Lange:2001uf  to generate $e^{+}e^{-}\rightarrow B\overline{B}$ pairs. In
addition, we use Pythia 8.2 Sjostrand:2014zea  for the Belle II sample. We
simulate continuum background events using Pythia 6.4 Sjostrand:2006za  for
the analysis of Belle data, and KKMC Ping:2008zz  for hard scattering followed
by Pythia 8.2 Sjostrand:2014zea  for hadronization in Belle II. We simulate
the effects of final state radiation by Photos photos  for both Belle and
Belle II. The detector response is based on geant3 geant3  and geant4 geant4
for Belle and Belle II data, respectively. We use simulated samples of ${\rm
2\times 10^{6}}$ signal events and more than ${\rm 1ab^{-1}}$ of background
events for our studies. The data and simulation are processed using the Belle
II analysis software framework basf2 ; basf2_repo ; Gelb:2018agf .

##  III Event selection

Most of the selection criteria used for Belle and Belle II are similar;
however, certain criteria are adjusted to account for the improved performance
of the upgraded detector.

The triggers of Belle and Belle II use either the number of tracks, or ECL
energy depositions (clusters), or the total energy of all clusters. The
trigger efficiency for signal events is almost 100%. In the offline analysis,
a high energy primary photon ($\gamma_{\rm prim}$) candidate is reconstructed
from an ECL cluster not associated with any track. Only ECL clusters whose
polar angle $\theta$ is in the barrel region are considered. The c.m. energy
of the primary photon candidate ($E_{\gamma}^{\ast}$) is required to satisfy
${\rm 1.8~{}GeV}<E_{\gamma}^{\ast}<{\rm 2.8~{}GeV}$. For Belle, we require
$E_{\rm 9}/E_{\rm 25}>0.95$ to select a cluster shape consistent with an
electromagnetic shower, where $E_{\rm 9}/E_{\rm 25}$ is defined as the ratio
of energy deposits within a $3\times 3$ array of CsI(Tl) crystals and within a
$5\times 5$ array, both centered on the crystal with the highest energy. For
Belle II, we require $E_{\rm 9}/E_{\rm 21}>0.95$, where $E_{21}$ is defined
similarly to $E_{25}$ but with energy deposits at the four corners of the
$5\times 5$ array removed. We also define for Belle II a cluster second moment
$S\equiv\Sigma_{i}w_{i}E_{i}r_{i}^{2}/\Sigma_{i}w_{i}E_{i}$, where $E_{i}$ is
the energy deposited in the $i$th crystal, and $r_{i}$ is the distance in the
plane perpendicular to the shower axis from the center of the cluster to the
center of the $i$th crystal. The energies are weighted by factors $w_{i}$
ranging from 0.0–1.0 that account for a crystal energy being shared among
overlapping clusters. The sum is taken over all crystals in a cluster. To
reject clusters resulting from hadron showers, we require that there be at
least eight crystals in a cluster. To better identify the primary photon
cluster, we require $S<{\rm 1.5}~{}{\rm cm^{2}}$.

Track candidates are required to satisfy $dr<0.5$ cm and $|dz|<2$ cm, where
$dr$ and $dz$ are transverse and longitudinal impact parameters, respectively.
We select $\pi^{+}$ candidates from tracks by requiring ${\cal
R}_{\pi/K}={\cal L}_{\pi}/({\cal L}_{\pi}+{\cal L}_{K})>0.6$, where ${\cal
L}_{\pi(K)}$ is the likelihood for a pion (kaon) hypothesis. These likelihoods
are based on information from the CDC, ACC, and TOF detectors in Belle, and
from the CDC, TOP, and ARICH detectors in Belle II.

Neutral $\pi^{0}$ mesons are reconstructed via
$\pi^{0}\rightarrow\gamma\gamma$ decays. In Belle data, $\pi^{0}$ mesons are
reconstructed using two photons, each with energy greater than 50 MeV. We
require the invariant mass of the two photons to satisfy ${\rm
119~{}MeV}/c^{2}<M(\gamma\gamma)<{\rm 151~{}MeV}/c^{2}$. In Belle II data, the
energy thresholds of the daughter photons depend on the region in the ECL. For
the barrel region, the daughter photons are required to have energies greater
than 30 MeV, while for the forward and backward endcap regions we use
thresholds of 80 MeV and 60 MeV, respectively. In addition, the daughter
photons must satisfy the condition $\Sigma w_{i}>{\rm 1.5}$. The invariant
mass of the two photons in Belle II is required to be in the range ${\rm
120~{}MeV}/c^{2}<M(\gamma\gamma)<{\rm 145~{}MeV}/c^{2}$.

We reconstruct $\rho^{0}\rightarrow\pi^{+}\pi^{-}$ and
$\rho^{+}\rightarrow\pi^{+}\pi^{0}$ candidates by requiring that the invariant
mass of the pion pair be in the range ${\rm[0.64,0.90]~{}GeV}/c^{2}$ for Belle
data, and the range ${\rm[0.65,0.90]~{}GeV}/c^{2}$ for Belle II data. A $B$
candidate is subsequently reconstructed by combining a $\rho$ candidate with a
$\gamma_{\rm prim}$ candidate.

To remove low multiplicity and $q\overline{q}$ events, we require at least two
tracks in the event and that the ratio of the second to zeroth Fox-Wolfram
moments fox  be less than 0.7. To fit for the signal yield, we define two
observables: the beam-energy-constrained mass $M_{\rm
bc}\equiv\sqrt{\left(E_{\rm beam}^{\ast}/c^{2}\right)^{2}-|{\bf
p}_{B}^{\ast}/c|^{2}}$, and the energy difference $\Delta E\equiv
E_{B}^{\ast}-E_{\rm beam}^{\ast}$, where $E_{\rm beam}$ is the beam energy and
$E_{B}$ and $p_{B}$ are the reconstructed energy and momentum, respectively,
of the signal $B$ candidate. All quantities are calculated in the $e^{+}e^{-}$
c.m. frame. For the $B^{0}\rightarrow\rho^{0}\gamma$ mode, we improve the
resolution in $M_{\rm bc}$ by substituting, for the magnitude of the photon
momentum, the difference between the beam energy and the energy of the
$\rho^{0}$ candidate:
$\mbox{\boldmath$\vec{p}$}_{B^{0}}\rightarrow\mbox{\boldmath$\vec{p}$}_{\rho^{0}}+\left(\mbox{\boldmath$\vec{p}$}_{\gamma}/|\mbox{\boldmath$\vec{p}$}_{\gamma}|\right)\left(E_{\rm
beam}-E_{\rho^{0}}\right)/c$, where $E_{\rho^{0}}$ and
$\mbox{\boldmath$\vec{p}$}_{\rho^{0}}$ are the energy and momentum,
respectively, of the $\rho^{0}$ candidate, and
$\mbox{\boldmath$\vec{p}$}_{\gamma}$ is the momentum of the $\gamma_{\rm
prim}$ candidate. According to MC simulation, this modification reduces the
width of the $M_{\rm bc}$ distribution by 18%. We retain signal candidates
that satisfy $M_{\rm bc}>5.2~{}{\rm GeV/}c^{2}$ and $|\Delta E|<0.3~{}{\rm
GeV}$. In addition, we define a variable $M(K\pi)$, which is the invariant
mass of the $\rho$ candidate calculated assuming that one of the charged pions
is a kaon. For $\rho^{0}\rightarrow\pi^{+}\pi^{-}$ decays, the substitution of
the kaon mass for the pion mass is applied to the pion with the lower value of
${\cal R_{\pi/{\it K}}}$. This allows us to distinguish
$B\rightarrow\rho\gamma$ from $B\rightarrow K^{\ast}\gamma$, which peaks in
the $K^{\ast}\left(892\right)^{0}$ mass region. We retain events satisfying
$M(K\pi)\in\left(0.80,1.50\right)~{}{\rm GeV}/c^{2}$. To reduce combinatorial
background due to $B\overline{B}$ events, the $B$ meson candidate must also
satisfy $|\cos{\theta_{h}}|<0.8$, where $\theta_{h}$ is the angle in the $B$
rest frame between the momentum of $\gamma_{\rm prim}$ and the negative of the
boost direction of the laboratory frame. For correctly reconstructed signal
decays, the $|\cos{\theta_{h}}|$ distribution is uniform, while for
combinatorial background, the distribution tends to peak at
$|\cos{\theta_{h}}|\approx 1$. After applying the above selection criteria,
12.3% (3.0%) of events have multiple $B^{+}$ ($B^{0}$) candidates. For such
multiple candidate events, the average multiplicity is 2.3 for both $B^{+}$
and $B^{0}$ decays. To minimize potential bias, we select one candidate
randomly.

To assess the quality of the simulation, we compare its predictions with data
using two control samples, $B\rightarrow\overline{D}\pi^{+}$ and $B\rightarrow
K^{\ast}\gamma$ decays. We reconstruct both $\overline{D}{}^{0}\rightarrow
K^{+}\pi^{-}$ and $D^{-}\rightarrow K^{+}\pi^{-}\pi^{-}$ decays, requiring
that the invariant masses be in the ranges ${\rm
1.85~{}GeV}/c^{2}<M(K^{+}\pi^{-})<{\rm 1.88~{}GeV}/c^{2}$ and ${\rm
1.86~{}GeV}/c^{2}<M(K^{+}\pi^{-}\pi^{-})<{\rm 1.88~{}GeV}/c^{2}$. The
$K^{\ast}$ mesons are reconstructed via $K^{\ast+}\rightarrow K^{+}\pi^{0}$
and $K^{\ast 0}\rightarrow K^{+}\pi^{-}$ decays, where the charged kaon is
required to have ${\cal R}_{K/\pi}>0.6$ and the $\pi^{0}$ and $\pi^{-}$ must
satisfy the same criteria as applied to signal $B\rightarrow\rho^{0}\gamma$
decays. The invariant mass of the $K\pi$ meson pair must satisfy ${\rm
0.817~{}GeV}/c^{2}<M(K^{+}\pi^{-(0)})<{\rm 0.967~{}GeV}/c^{2}$.

##  IV Background suppression

After applying the above selection criteria, the main remaining source of
background is $q\overline{q}$ events, where the candidate $\gamma_{\rm prim}$
results from an asymmetric $\pi^{0}\rightarrow\gamma\gamma$ or
$\eta\rightarrow\gamma\gamma$ decay. In this case, one of the photons has much
higher energy than the other. To reduce this background, we use two boosted
decision trees: a dedicated $\pi^{0}(\eta)$ veto and one for $q\overline{q}$
suppression.

###  IV.1 $\pi^{0}(\eta)$ veto

For the $\pi^{0}$ and $\eta$ veto, we pair $\gamma_{\rm prim}$ with other
photons in the event to reconstruct $\pi^{0}\rightarrow\gamma\gamma$ and
$\eta\rightarrow\gamma\gamma$ decays. If such a candidate is identified, the
$\gamma_{\rm prim}$ candidate is removed. The energy of the photon with which
$\gamma_{\rm prim}$ is paired ($\gamma_{\rm soft}$) must exceed a polar-angle-
dependent threshold. For Belle II data, we require that the sum of crystal
weights $w_{i}$ for the $\gamma_{\rm soft}$ cluster be at least two, and that
this cluster be reconstructed in time with the event trigger. The probability
of a correctly reconstructed $\pi^{0}(\eta)$ is calculated using a boosted
decision tree ($BDT_{v}$) trained using simulated events. The $BDT_{v}$ uses
up to eight input variables: (1) the invariant mass of the two photons
$M(\gamma_{\rm prim}\gamma_{\rm soft})$; (2) the $\gamma_{\rm soft}$ energy;
(3) the $\gamma_{\rm soft}$ polar angle; (4) $\Sigma w_{i}$ for the
$\gamma_{\rm soft}$ cluster; (5) the $\gamma_{\rm soft}$ $E_{9}/E_{25(21)}$;
(6) the cosine of the angle in the $\pi^{0}(\eta)$ rest frame between the
momentum of $\gamma_{\rm prim}$ and the negative of the boost direction of the
laboratory frame; (7) the distance between the ECL cluster of $\gamma_{\rm
soft}$ and the nearest track extrapolated to the ECL; and (8) the output of a
multivariate classifier based on Zernike moments of the $\gamma_{\rm soft}$
shower zernike . These moments are defined in the plane perpendicular to the
shower direction. Requirements (7) and (8) are not applied to Belle data,
while for Belle II data, variables (4), (5), and (8) are used only for the
$\eta$ veto. The $BDT_{v}$ value is calculated for all $\gamma_{\rm soft}$
candidates. Among all $\gamma_{\rm prim}\gamma_{\rm soft}$ pairs, the one with
the highest $BDT_{v}$ value is selected. The background in which the
$\gamma_{\rm prim}$ originates from $\pi^{0}(\eta)\rightarrow\gamma\gamma$
decays tends to peak at one. Selection criteria on the $BDT_{v}$ output are
optimized to maximize the signal significance, $n_{s}/\sqrt{n_{s}+n_{b}}$,
where $n_{s(b)}$ is the number of simulated signal (background) events after
the $BDT_{v}$ selection in a signal-enriched region defined as $M_{\rm
bc}>{\rm 5.27~{}GeV}/c^{2}$, ${\rm-0.20~{}GeV}<\Delta E<{\rm 0.10~{}GeV}$, and
$M(K\pi)>{\rm 0.92~{}GeV}/c^{2}$. The optimized selection for $BDT_{v}$
rejects about 75% (82%) of $q\overline{q}$ background for Belle (Belle II)
data, while retaining about 83% (89%) of signal decays.

The signal efficiency is checked using a $B\rightarrow\overline{D}\pi^{+}$
control sample. The $\pi^{+}$ momentum is scaled by a factor 1.087 to match
the momentum of $\gamma_{\rm prim}$, and the mass $M(\pi\gamma_{\rm soft})$ is
used in place of $M(\gamma_{\rm prim}\gamma_{\rm soft})$. Due to isolation
criteria applied to the $\gamma_{\rm prim}$ candidate, the distribution of
$M(\pi\gamma_{\rm soft})$ differs from that of $M(\gamma_{\rm prim}\gamma_{\rm
soft})$; to correct for this, the $B\rightarrow\overline{D}\pi^{+}$ events are
weighted such that these distributions match. After this weighting, the
$BDT_{v}$ distributions show improved agreement with MC-simulated
$B\rightarrow\rho\gamma$ events, as shown in Fig. 1. The difference in signal
efficiency before and after this weighting is taken as a systematic
uncertainty.

Figure 1: Distributions of $BDT_{v}$ for simulated data, for Belle (top) and
Belle II (bottom). The solid red histograms are $B\rightarrow\rho\gamma$, the
dotted blue histograms are $B\rightarrow\overline{D}\pi^{+}$ and the points
are the $B\rightarrow\overline{D}\pi^{+}$ with $M(\pi\gamma_{\rm soft})$
correction.

###  IV.2 $q\overline{q}$ suppression

To reduce the remaining contamination from continuum background, we introduce
another boosted decision tree classifier ($BDT_{q\overline{q}}$) that relies
on differences in event shapes between $B\overline{B}$ and continuum
$q\overline{q}$ events. The variables used in $BDT_{q\overline{q}}$ are as
follows: the cosine of the angle between the thrust axis of the signal $B$
candidate and the thrust axis of the rest of the event
$\cos{\theta_{T_{B}T_{O}}}$ Farhi:1977sg ; modified Fox-Wolfram moments fox ;
pi0pi0 ; the outputs of the $B$ flavor-tagging algorithms of Belle
Belle:2004uxp  and Belle II Belle-II:2021zvj ; the difference between the $z$
coordinates of the signal and the companion $B$ decay vertices; the distance
of closest approach between the signal $\pi^{+}$ track and the companion $B$
decay vertex; the sphericity of non-signal particles in the event sphericity ;
and the cosine of the polar angle of the signal $B$. The
$\cos{\theta_{T_{B}T_{O}}}$ variable is the most discriminating. The flavor-
tagging algorithm outputs two quantities: the flavor tag $q$ ($=\pm 1$), and a
quality factor $r$ that ranges from zero for no flavor discrimination to one
for unambiguous flavor assignment. The $BDT_{q\overline{q}}$ training uses
simulated $B\rightarrow\rho\gamma$ and continuum events. The training is
performed separately for $B^{+}\rightarrow\rho^{+}\gamma$ and
$B^{0}\rightarrow\rho^{0}\gamma$ decays, and for Belle and Belle II data. The
requirements on the $BDT_{q\overline{q}}$ output are chosen to maximize the
signal significance, separately for three ranges of the flavor-tagging quality
$r$: $(0,0.4)$, $(0.4,0.825)$, and $(0.825,1)$. All requirements are greater
than 0.95. The signal efficiency is estimated from simulation and corrected
for possible differences between data and simulation using the ratio of
efficiencies between data and simulation for the $B\rightarrow K^{\ast}\gamma$
control samples. These correction factors range from 0.96 to 1.10. The signal
efficiencies in Belle data for the $BDT_{q\overline{q}}$ requirement are 30%
and 40% for $B^{+}$ and $B^{0}$ decays, respectively. For Belle II data, the
corresponding efficiencies are 45% and 52%.

##  V Signal extraction

To measure the $B\rightarrow\rho\gamma$ branching fractions and asymmetries
$A_{\rm I}$ and $A_{\it CP}$, we perform an extended unbinned maximum
likelihood fit to the observables $M_{\rm bc}$, $\Delta E$, and $M(K\pi)$. We
simultaneously fit six independent data sets: $B^{+}$, $B^{-}$, and
$(B^{0}+\overline{B}^{0})$ candidates combined, for Belle and Belle II data.
In the fit, we model four components: signal, $B\rightarrow K^{\ast}\gamma$,
$B\overline{B}$ background, and $q\overline{q}$ background. The probability
density functions (PDFs) used for these components are as follows.

For signal decays, $M_{\rm bc}$, $\Delta E$, and $M(K\pi)$ are found to be
essentially uncorrelated and the PDF is taken to be the product of separate
functions for each. We model both the $M_{\rm bc}$ and $\Delta E$
distributions with Crystal Ball functions Oreglia:1981fx , and the $M(K\pi)$
distribution with a Novosibirsk function Belle:1999bhb  convolved with a
Gaussian. For $B\rightarrow K^{\ast}\gamma$ background, $M_{\rm bc}$, $\Delta
E$, and $M(K\pi)$ are correlated, and we use a three-dimensional histogram
PDF. For $B\overline{B}$ background, we again use the product of three one-
dimensional functions: the sum of a Crystal Ball function and ARGUS function
ARGUS:1990hfq  for $M_{\rm bc}$, an exponential function for $\Delta E$, and a
histogram PDF for $M(K\pi)$. Finally, for $q\overline{q}$ background, we use
the product of an ARGUS function for $M_{\rm bc}$, a quadratic Chebychev
polynomial for $\Delta E$, and a Novosibirsk function convolved with a
Gaussian for $M(K\pi)$.

The shape parameters of the $\Delta E$ distribution for $q\overline{q}$
background are floated in the fit, while all other PDF shape parameters are
fixed to values obtained from MC simulation. To account for possible
differences between data and simulation, the $M_{\rm bc}$ and $\Delta E$
distributions for signal decays are corrected according to small differences
observed between data and simulation for the $B\rightarrow K^{\ast}\gamma$
control samples. Similarly, the parameters for the $M_{\rm bc}$ and $M(K\pi)$
distributions of $q\overline{q}$ background are corrected according to small
differences observed between data and simulation in the sideband region
$BDT_{q\overline{q}}\in[0.6,0.9]$. In addition to these fixed shapes, we also
fix the yields of $B\rightarrow K^{\ast}\gamma$ and $B\overline{B}$
backgrounds to expectations based on MC simulation.

With the above PDFs, we perform two fits. We first fit directly for parameters
$D$, $A_{\rm I}$, and $A_{\it CP}$, defined as

$\displaystyle
D=c_{\rho}^{2}\frac{\tau_{B^{\pm}}}{\tau_{B^{0}}}\mathcal{B}(\overset{\textbf{(---)}}{B^{0}}\rightarrow\rho^{0}\gamma)+\mathcal{B}\left(B^{\pm}\rightarrow\rho^{\pm}\gamma\right),$
(4) $\displaystyle A_{\rm
I}=\frac{c_{\rho}^{2}\frac{\tau_{B^{\pm}}}{\tau_{B^{0}}}\mathcal{B}(\overset{\textbf{(---)}}{B^{0}}\rightarrow\rho^{0}\gamma)-\mathcal{B}\left(B^{\pm}\rightarrow\rho^{\pm}\gamma\right)}{D},$
(5) $\displaystyle{\rm and}$ $\displaystyle A_{\it
CP}=\frac{N_{B^{-}}-N_{B^{+}}}{N_{B^{-}}+N_{B^{+}}},$ (6)

where $\tau_{B^{\pm}}/\tau_{B^{0}}=1.076\pm 0.004$ ParticleDataGroup:2022pth
is the ratio of $B^{\pm}$ to $B^{0}$ lifetimes.

The relationship between the number of signal events ($N_{i}$) and the
branching fractions in Eq. 5 and 6 is given by

${\cal
B}(B^{\pm(0)}\rightarrow\rho^{\pm(0)}\gamma)=\frac{N_{B^{-}(\overline{B}{}^{0})}+N_{B^{+}(B^{0})}}{2N_{B\overline{B}}f_{+-(00)}\epsilon^{\pm(0)}}.$
(7)

Here, $N_{B\overline{B}}$ is the number of $B\overline{B}$ pairs and equals
$(771.6\pm 10.6)\times 10^{6}$ for Belle and $(378.5\pm 5.8)\times 10^{6}$ for
Belle II; $f_{+-}/f_{00}=1.065\pm 0.052$ Belle:2022hka  111For
$a=f_{+-}/f_{00}$, $f_{+-}=a/(1+a)$ and $f_{00}=1/(1+a)$. is the production
ratio of $B^{+}B^{-}$ to $B^{0}\overline{B}{}^{0}$ pairs at an $e^{+}e^{-}$
center-of-mass energy corresponding to the $\Upsilon(4S)$ resonance; and
$\epsilon^{\pm}$ and $\epsilon^{0}$ are reconstruction efficiencies for
$B^{\pm}\rightarrow\rho^{\pm}\gamma$ and $B^{0}\rightarrow\rho^{0}\gamma$,
respectively. The efficiencies of $B^{+}\rightarrow\rho^{+}\gamma$ and
$B^{-}\rightarrow\rho^{-}\gamma$ decays are consistent with each other and
taken to be identical. The signal efficiencies $\epsilon^{\pm}$ and
$\epsilon^{0}$ for the Belle (Belle II) sample are 5.5% and 10.3% (11.0% and
14.9%), respectively. The higher efficiency for Belle II is due mainly to the
improved performance of the $BDT_{v}$ and $BDT_{q\overline{q}}$ algorithms.
After this first fit, we perform a second fit directly for
$\mathcal{B}\left(B^{+}\rightarrow\rho^{+}\gamma\right)$ and
$\mathcal{B}\left(B^{-}\rightarrow\rho^{-}\gamma\right)$ to obtain their
uncertainties (rather than unfolding them from the first fit results,
accounting for correlations). The distributions of $M_{\rm bc}$, $\Delta E$,
and $M(K\pi)$ are shown in Figs. 2–4 along with projections of the fit result.

Figure 2: Distributions of $M_{\rm bc}$, $\Delta E$ and $M(K\pi)$ for
$B^{+}\rightarrow\rho^{+}\gamma$ candidates reconstructed in the signal-
enriched region of the other two variables. The signal-enriched region is
defined as $M_{\rm bc}>5.27~{}{\mathrm{GeV}}/c^{2}$, $|\Delta
E|<0.1~{}{\mathrm{GeV}}$ and $M(K\pi)>0.92~{}{\mathrm{GeV}}/c^{2}$. The points
with error bars are data, the solid red curves are the sum of signal and
background PDFs, the dashed red curves are signal, the dotted-dashed blue
curves are continuum background, the dashed magenta curves are
$K^{\ast}\gamma$ background, and the dotted green curves are $B\overline{B}$
background other than $K^{\ast}\gamma$. The discrete nature of the solid red
curves is due to the use of histogram PDFs.

Figure 3: Distributions of $M_{\rm bc}$, $\Delta E$ and $M(K\pi)$ for
$B^{+}\rightarrow\rho^{+}\gamma$ candidates reconstructed in the signal-
enriched region of the other two variables. The signal-enriched region is
defined as $M_{\rm bc}>5.27~{}{\mathrm{GeV}}/c^{2}$, $|\Delta
E|<0.1~{}{\mathrm{GeV}}$ and $M(K\pi)>0.92~{}{\mathrm{GeV}}/c^{2}$. The points
with error bars are data, the solid red curves are the sum of signal and
background PDFs, the dashed red curves are signal, the dotted-dashed blue
curves are continuum background, the dashed magenta curves are
$K^{\ast}\gamma$ background, and the dotted green curves are $B\overline{B}$
background other than $K^{\ast}\gamma$. The discrete nature of the solid red
curves is due to the use of histogram PDFs.

Figure 4: Distributions of $M_{\rm bc}$, $\Delta E$ and $M(K\pi)$ for
$B^{+}\rightarrow\rho^{+}\gamma$ candidates reconstructed in the signal-
enriched region of the other two variables. The signal-enriched region is
defined as $M_{\rm bc}>5.27~{}{\mathrm{GeV}}/c^{2}$, $|\Delta
E|<0.1~{}{\mathrm{GeV}}$ and $M(K\pi)>0.92~{}{\mathrm{GeV}}/c^{2}$. The points
with error bars are data, the solid red curves are the sum of signal and
background PDFs, the dashed red curves are signal, the dotted-dashed blue
curves are continuum background, the dashed magenta curves are
$K^{\ast}\gamma$ background, and the dotted green curves are $B\overline{B}$
background other than $K^{\ast}\gamma$. The discrete nature of the solid red
curves is due to the use of histogram PDFs.

Using Eqs. 4-7, we extract the signal yields; these are listed in Table 1
along with the corresponding reconstruction efficiencies and the fitted yields
of $q\overline{q}$ background.

Table 1: Efficiency ($\epsilon$), signal yield ($N_{\rho\gamma}$) and continuum background yield ($N_{q\overline{q}}$) from Belle (711 ${\rm fb^{-1}}$) and Belle II (362 ${\rm fb^{-1}}$) data. The uncertainties for $N_{\rho\gamma(q\overline{q})}$ are statistical only. Mode | $\epsilon~{}[\%]$ | $N_{\rho\gamma}$ | $N_{q\overline{q}}$
---|---|---|---
Belle $B^{+}\rightarrow\rho^{+}\gamma$ | $5.5\pm 0.5$ | $31\pm 6$ | $400\pm 21$
Belle $B^{-}\rightarrow\rho^{-}\gamma$ | $5.5\pm 0.5$ | $26\pm 6$ | $369\pm 21$
Belle $B^{0}\rightarrow\rho^{0}\gamma$ | $10.3\pm 0.4$ | $58\pm 10$ | $1307\pm 38$
Belle II $B^{+}\rightarrow\rho^{+}\gamma$ | $11.0\pm 1.1$ | $30\pm 6$ | $552\pm 25$
Belle II $B^{-}\rightarrow\rho^{-}\gamma$ | $11.0\pm 1.1$ | $26\pm 6$ | $549\pm 25$
Belle II $B^{0}\rightarrow\rho^{0}\gamma$ | $14.9\pm 0.5$ | $41\pm 7$ | $1114\pm 35$

The overall signal yields in the full fitted region are $114\pm 12$
$B^{\pm}\rightarrow\rho^{\pm}\gamma$ decays and $99\pm 12$
$B^{0}\rightarrow\rho^{0}\gamma$ decays. As a cross-check, we fit for the
branching fractions using only Belle data; our results are in agreement with
the previous Belle measurement Belle:2008imr .

##  VI Systematic uncertainties

The various sources of systematic uncertainty are listed in Table 2 and
estimated as follows.

The systematic uncertainty arising from particle reconstruction consists of
several parts. The uncertainty related to photon detection is measured in
Belle using $e^{+}e^{-}\rightarrow e^{+}e^{-}\gamma$ (radiative Bhabha) events
and in Belle II using $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}\gamma$ events; the
resulting uncertainties are 2% and 1%, respectively. The uncertainty due to
$\pi^{+}$ identification is measured in Belle (0.7%) and Belle II (0.3%) using
a sample of $D^{\ast+}\rightarrow D^{0}(\rightarrow K^{-}\pi^{+})\pi^{+}$
decays. The uncertainty due to track reconstruction is evaluated in Belle
(0.3%) using $D^{\ast+}\to D^{0}(\to K_{S}\,\pi^{+}\pi^{-})\pi^{+}$ decays,
and in Belle II (0.3%) using $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events.
The uncertainty due to $\pi^{0}$ reconstruction is estimated by comparing the
ratio of efficiencies for two decay channels with different numbers of
$\pi^{0}$’s, for both data and simulation. For Belle, the ratio is the
efficiency for $\eta\rightarrow\pi^{0}\pi^{0}\pi^{0}$ to that for
$\eta\rightarrow\pi^{+}\pi^{-}\pi^{0}$ or $\eta\rightarrow\gamma\gamma$; the
resulting uncertainty is 3%. For Belle II the ratio is the efficiency for
$D^{0}\rightarrow K^{-}\pi^{+}\pi^{0}$ to that for $D^{0}\rightarrow
K^{-}\pi^{+}$, and the resulting uncertainty is 5%.

Table 2: Systematic uncertainties on the branching fractions for $B^{+}\rightarrow\rho^{+}\gamma$ (${\cal B}_{\rho^{+}\gamma}$) and $B^{0}\rightarrow\rho^{0}\gamma$ (${\cal B}_{\rho^{0}\gamma}$), and on the isospin and CP asymmetries. Source | ${\cal B}_{\rho^{+}\gamma}\times 10^{8}$ | ${\cal B}_{\rho^{0}\gamma}\times 10^{8}$ | $A_{\rm I}$ | $A_{\it CP}$
---|---|---|---|---
Particle detection | 4.1 | 1.3 | 1.4% | 0.5%
Selection criteria | 9.0 | 3.4 | 4.0% | 0.5%
Fixed fit parameters | 1.1 | 2.7 | 1.8% | 0.2%
Signal shape | 4.7 | 3.0 | 3.1% | 0.5%
Histogram PDFs | 1.0 | 0.6 | 0.5% | 0.1%
Peaking $K^{\ast}\gamma$ bkg | 3.4 | 5.4 | 3.1% | 0.1%
Other peaking $B\overline{B}$ bkgs | 2.2 | 0.8 | 0.9% | 0.2%
Peaking $B\overline{B}$ $A_{\it CP}$ | 0.1 |  $<$0.1 | 0.1% | 1.0%
Number of $B\overline{B}$’s | 1.7 | 1.4 | 0.3% | 0.1%
$\tau_{B^{\pm}}/\tau_{B^{0}}$ | 0.1 |  $<$0.1 | 0.2% |  $<$0.1%
$f_{+-}/f_{00}$ | 4.0 | 3.6 | 3.8% |  $<$0.1%
Total | 12.5 | 8.6 | 7.5% | 1.4%

Systematic uncertainties due to selection criteria are evaluated as follows.
The uncertainty due to differences between data and simulation for the BDT
requirements is evaluated by varying the calibration factors obtained from the
$B\rightarrow\overline{D}\pi$ and $B\rightarrow K^{\ast}\gamma$ control
samples by one standard deviation. The resulting uncertainties arising from
the $\pi^{0}$ veto are 0.8% (0.7%) for the $B^{+}\to\rho^{+}\gamma$
($B^{0}\to\rho^{0}\gamma$) branching fraction, and 0.7% for the isospin
asymmetry. The uncertainties arising from $q\overline{q}$ suppression along
with the $\eta$ veto are 6.8% (4.2%) for the $B^{+}$ ($B^{0}$) branching
fraction, and 3.9% for $A_{I}$. The uncertainty due to possible mis-modeling
of the $B\rightarrow\overline{D}\pi^{+}$ control sample in Belle (see section
IV.1) is 0.9% (1.5%) for the $B^{+}$ ($B^{0}$) branching fraction and 0.3% for
$A_{I}$. The uncertainties on $A_{CP}$ are $O(0.1\%)$.

The uncertainty related to each fixed parameter in the fit is estimated by
varying each parameter by its uncertainty and repeating the fit. The
difference between the fit result and our nominal result is taken as the
systematic uncertainty.

The uncertainty arising from the shape of the signal PDF is evaluated by
varying the calibration factors obtained from the fit to the $B\rightarrow
K^{\ast}\gamma$ control sample by their uncertainties and repeating the fit.
The difference between the result and our nominal result is taken as the
systematic uncertainty.

The uncertainty due to modeling $B\rightarrow K^{\ast}\gamma$ is evaluated by
using a modified histogram PDF; this is obtained from simulated experiments
generated with the original histogram as a kernel function. The differences in
the fit results from the nominal values are taken as the systematic
uncertainties. The same procedure is adopted to estimate the uncertainty due
to the histogram PDF used to model the $M(K\pi)$ distribution of the
$B\overline{B}$ combinatorial background.

The systematic uncertainties due to $B\overline{B}$ background that peaks in
the signal-enriched region (“peaking background”) are estimated by varying
this background as follows. The $K^{\ast}\gamma$ background yield is estimated
as $N_{K^{\ast}\gamma}=2\cdot N_{B\overline{B}}\cdot{\cal B}(B\to
K^{\ast}\gamma)\cdot F_{K/\pi}\cdot\epsilon_{K^{\ast}\gamma}$, where
$F_{K/\pi}$ is the probability for a charged kaon to be misidentified as a
pion, and $\epsilon_{K^{\ast}\gamma}$ is the efficiency to select
$K^{\ast}\gamma$ background. The misidentification probability $F_{K/\pi}$ is
obtained from a study of $D^{\ast+}\to D^{0}(\to K^{-}\pi^{+})\pi^{+}$ decays;
the fractional uncertainty is 6.1% for Belle and 9.7% for Belle II. To
estimate the systematic uncertainty arising from $B\rightarrow K^{\ast}\gamma$
background, we vary both ${\cal B}(B\rightarrow K^{\ast}\gamma)$
ParticleDataGroup:2022pth  and $F_{K/\pi}$ by their uncertainties. Other
peaking background sources are $B\rightarrow X_{s(d)}\gamma$ decays, where
$X_{s(d)}$ is any final state of $s(d)$ quark hadronization with strangeness
equal to one (zero) except for $K^{\ast}\left(892\right)\gamma$
($\rho\gamma$), and hadronic $B$ decays with
$\pi^{0}(\eta)\rightarrow\gamma\gamma$ in the final states. The former is
varied according to its experimental uncertainty ParticleDataGroup:2022pth .
For the latter, we take the fractional uncertainty on the number of peaking
background events to be the weighted sum of the fractional uncertainties on
the branching fractions for the main decay modes that contribute. For the
$B^{+}\rightarrow\rho^{+}\gamma$ mode, the contamination from
$B^{+}\rightarrow\rho^{+}\pi^{0}(\eta)$ decays is dominant at 24% (16%),
according to the simulation. For the $B^{0}\rightarrow\rho^{0}\gamma$ mode,
the main contamination comes from $B^{0}\rightarrow\rho^{0}\eta$ (9%),
$B^{+}\rightarrow a_{1}^{+}\pi^{0}$ (8%), and
$B^{+}\rightarrow\rho^{+}\rho^{0}$ (8%). For modes that have not been
measured, a 50% uncertainty is assumed for the branching fractions.

For the $A_{\it CP}$ measurement, we calculate the effect of a possible CP
asymmetry of $B\overline{B}$ peaking background. The $A_{\it CP}$ values for
$B\rightarrow K^{\ast}\gamma$ and $B\rightarrow X_{s}\gamma$ are varied by
their measured uncertainties ParticleDataGroup:2022pth , while $A_{CP}$ for
$B\rightarrow X_{d}\gamma$ is varied by $\pm 60\%$. The latter corresponds to
a conservative uncertainty on the SM prediction Benzke:2010tq . The $A_{\it
CP}$ values for other modes such as $B\rightarrow\rho\pi^{0}$ are varied by
$\pm 10\%$. The uncertainties on the number of $B\overline{B}$ pairs, the
ratio $\tau_{B^{+}}/\tau_{B^{0}}$, and the ratio $f_{+-}/f_{00}$ Belle:2022hka
are all taken into account.

##  VII results

We determine the branching fractions, CP asymmetry, and isospin asymmetry
directly from the fit. The results are

$\displaystyle{\cal B}\left(B^{+}\rightarrow\rho^{+}\gamma\right)$
$\displaystyle=\left(13.1^{+2.0+1.3}_{-1.9-1.2}\right)\times 10^{-7}$ (8)
$\displaystyle{\cal B}\left(B^{0}\rightarrow\rho^{0}\gamma\right)$
$\displaystyle=\left(7.5\pm 1.3^{+1.0}_{-0.8}\right)\times 10^{-7}$ (9)
$\displaystyle A_{\it CP}\left(B^{+}\rightarrow\rho^{+}\gamma\right)$
$\displaystyle=\left(-8.2\pm 15.2^{+1.6}_{-1.2}\right)\%$ (10) $\displaystyle
A_{\rm I}\left(B\rightarrow\rho\gamma\right)$
$\displaystyle=\left(10.9^{+11.2+6.8+3.8}_{-11.7-6.2-3.9}\right)\%,$ (11)

where the first uncertainty is statistical, the second is systematic, and the
third for $A_{\rm I}$ is the uncertainty from $f_{+-}/f_{00}$ Belle:2022hka
combined with the uncertainty from the ratio of $B^{+}$ to $B^{0}$ lifetimes.

In summary, we have measured the branching fractions, direct CP asymmetry, and
isospin asymmetry of $B\rightarrow\rho\gamma$ decays using a total of 1073
${\mathrm{fb}^{-1}}$ of Belle and Belle II data. These results are the most
precise to date and supersede the previous Belle results Belle:2005grh . The
results for $A_{\it CP}$ and $A_{\rm I}$ are consistent with SM expectations.

This work, based on data collected using the Belle II detector, which was
built and commissioned prior to March 2019, and data collected using the Belle
detector, which was operated until June 2010, was supported by Higher
Education and Science Committee of the Republic of Armenia Grant No.
23LCG-1C011; Australian Research Council and Research Grants No. DP200101792,
No. DP210101900, No. DP210102831, No. DE220100462, No. LE210100098, and No.
LE230100085; Austrian Federal Ministry of Education, Science and Research,
Austrian Science Fund No. P 31361-N36 and No. J4625-N, and Horizon 2020 ERC
Starting Grant No. 947006 “InterLeptons”; Natural Sciences and Engineering
Research Council of Canada, Compute Canada and CANARIE; National Key R&D
Program of China under Contract No. 2022YFA1601903, National Natural Science
Foundation of China and Research Grants No. 11575017, No. 11761141009, No.
11705209, No. 11975076, No. 12135005, No. 12150004, No. 12161141008, and No.
12175041, and Shandong Provincial Natural Science Foundation Project
ZR2022JQ02; the Czech Science Foundation Grant No. 22-18469S; European
Research Council, Seventh Framework PIEF-GA-2013-622527, Horizon 2020 ERC-
Advanced Grants No. 267104 and No. 884719, Horizon 2020 ERC-Consolidator Grant
No. 819127, Horizon 2020 Marie Sklodowska-Curie Grant Agreement No. 700525
“NIOBE” and No. 101026516, and Horizon 2020 Marie Sklodowska-Curie RISE
project JENNIFER2 Grant Agreement No. 822070 (European grants); L’Institut
National de Physique Nucléaire et de Physique des Particules (IN2P3) du CNRS
and L’Agence Nationale de la Recherche (ANR) under grant ANR-21-CE31-0009
(France); BMBF, DFG, HGF, MPG, and AvH Foundation (Germany); Department of
Atomic Energy under Project Identification No. RTI 4002, Department of Science
and Technology, and UPES SEED funding programs No. UPES/R&D-SEED-
INFRA/17052023/01 and No. UPES/R&D-SOE/20062022/06 (India); Israel Science
Foundation Grant No. 2476/17, U.S.-Israel Binational Science Foundation Grant
No. 2016113, and Israel Ministry of Science Grant No. 3-16543; Istituto
Nazionale di Fisica Nucleare and the Research Grants BELLE2; Japan Society for
the Promotion of Science, Grant-in-Aid for Scientific Research Grants No.
16H03968, No. 16H03993, No. 16H06492, No. 16K05323, No. 17H01133, No.
17H05405, No. 18K03621, No. 18H03710, No. 18H05226, No. 19H00682, No.
20H05850, No. 20H05858, No. 22H00144, No. 22K14056, No. 22K21347, No.
23H05433, No. 26220706, and No. 26400255, and the Ministry of Education,
Culture, Sports, Science, and Technology (MEXT) of Japan; National Research
Foundation (NRF) of Korea Grants No. 2016R1D1A1B02012900, No.
2018R1A2B3003643, No. 2018R1A6A1A06024970, No. 2019R1I1A3A01058933, No.
2021R1A6A1A03043957, No. 2021R1F1A1060423, No. 2021R1F1A1064008, No.
2022R1A2C1003993, and No. RS-2022-00197659, Radiation Science Research
Institute, Foreign Large-Size Research Facility Application Supporting
project, the Global Science Experimental Data Hub Center of the Korea
Institute of Science and Technology Information and KREONET/GLORIAD;
Universiti Malaya RU grant, Akademi Sains Malaysia, and Ministry of Education
Malaysia; Frontiers of Science Program Contracts No. FOINS-296, No. CB-221329,
No. CB-236394, No. CB-254409, and No. CB-180023, and SEP-CINVESTAV Research
Grant No. 237 (Mexico); the Polish Ministry of Science and Higher Education
and the National Science Center; the Ministry of Science and Higher Education
of the Russian Federation and the HSE University Basic Research Program,
Moscow; University of Tabuk Research Grants No. S-0256-1438 and No.
S-0280-1439 (Saudi Arabia); Slovenian Research Agency and Research Grants No.
J1-9124 and No. P1-0135; Ikerbasque, Basque Foundation for Science, the State
Agency for Research of the Spanish Ministry of Science and Innovation through
Grant No. PID2022-136510NB-C33, Agencia Estatal de Investigacion, Spain Grant
No. RYC2020-029875-I and Generalitat Valenciana, Spain Grant No.
CIDEGENT/2018/020; the Swiss National Science Foundation; National Science and
Technology Council, and Ministry of Education (Taiwan); Thailand Center of
Excellence in Physics; TUBITAK ULAKBIM (Turkey); National Research Foundation
of Ukraine, Project No. 2020.02/0257, and Ministry of Education and Science of
Ukraine; the U.S. National Science Foundation and Research Grants No.
PHY-1913789 and No. PHY-2111604, and the U.S. Department of Energy and
Research Awards No. DE-AC06-76RLO1830, No. DE-SC0007983, No. DE-SC0009824, No.
DE-SC0009973, No. DE-SC0010007, No. DE-SC0010073, No. DE-SC0010118, No. DE-
SC0010504, No. DE-SC0011784, No. DE-SC0012704, No. DE-SC0019230, No. DE-
SC0021274, No. DE-SC0021616, No. DE-SC0022350, No. DE-SC0023470; and the
Vietnam Academy of Science and Technology (VAST) under Grants No.
NVCC.05.12/22-23 and No. DL0000.02/24-25.

These acknowledgements are not to be interpreted as an endorsement of any
statement made by any of our institutes, funding agencies, governments, or
their representatives.

We thank the SuperKEKB team for delivering high-luminosity collisions; the KEK
cryogenics group for the efficient operation of the detector solenoid magnet;
the KEK Computer Research Center for on-site computing support; the NII for
SINET6 network support; and the raw-data centers hosted by BNL, DESY, GridKa,
IN2P3, INFN, PNNL/EMSL, and the University of Victoria.

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