The Belle II Collaboration

# Determination of $|V_{ub}|$ from simultaneous measurements of untagged
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ decays

I. Adachi  0000-0003-2287-0173 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. Althubiti  0000-0003-1513-0409 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 M. Bauer
0000-0002-0953-7387 A. Baur  0000-0003-1360-3292 A. Beaubien
0000-0001-9438-089X F. Becherer  0000-0003-0562-4616 J. Becker
0000-0002-5082-5487 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 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. Boschetti  0000-0001-6030-3087 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 Q. Campagna
0000-0002-3109-2046 M. Campajola  0000-0003-2518-7134 L. Cao
0000-0001-8332-5668 G. Casarosa  0000-0003-4137-938X C. Cecchi
0000-0002-2192-8233 J. Cerasoli  0000-0001-9777-881X M.-C. Chang
0000-0002-8650-6058 P. Chang  0000-0003-4064-388X R. Cheaib
0000-0001-5729-8926 P. Cheema  0000-0001-8472-5727 B. G. Cheon
0000-0002-8803-4429 K. Chilikin  0000-0001-7620-2053 K. Chirapatpimol
0000-0003-2099-7760 H.-E. Cho  0000-0002-7008-3759 K. Cho  0000-0003-1705-7399
S.-J. Cho  0000-0002-1673-5664 S.-K. Choi  0000-0003-2747-8277 S. Choudhury
0000-0001-9841-0216 L. Corona  0000-0002-2577-9909 J. X. Cui
0000-0002-2398-3754 F. Dattola  0000-0003-3316-8574 E. De La Cruz-Burelo
0000-0002-7469-6974 S. A. De La Motte  0000-0003-3905-6805 G. De Nardo
0000-0002-2047-9675 M. De Nuccio  0000-0002-0972-9047 G. De Pietro
0000-0001-8442-107X R. de Sangro  0000-0002-3808-5455 M. Destefanis
0000-0003-1997-6751 S. Dey  0000-0003-2997-3829 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 D. Dorner
0000-0003-3628-9267 K. Dort  0000-0003-0849-8774 D. Dossett
0000-0002-5670-5582 S. Dreyer  0000-0002-6295-100X S. Dubey
0000-0002-1345-0970 K. Dugic  0009-0006-6056-546X G. Dujany
0000-0002-1345-8163 P. Ecker  0000-0002-6817-6868 M. Eliachevitch
0000-0003-2033-537X P. Feichtinger  0000-0003-3966-7497 T. Ferber
0000-0002-6849-0427 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 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 S. Granderath
0000-0002-9945-463X D. Greenwald  0000-0001-6964-8399 Z. Gruberová
0000-0002-5691-1044 T. Gu  0000-0002-1470-6536 K. Gudkova  0000-0002-5858-3187
I. Haide  0000-0003-0962-6344 S. Halder  0000-0002-6280-494X Y. Han
0000-0001-6775-5932 T. Hara  0000-0002-4321-0417 C. Harris
0000-0003-0448-4244 K. Hayasaka  0000-0002-6347-433X H. Hayashii
0000-0002-5138-5903 S. Hazra  0000-0001-6954-9593 C. Hearty
0000-0001-6568-0252 M. T. Hedges  0000-0001-6504-1872 A. Heidelbach
0000-0002-6663-5469 I. Heredia de la Cruz  0000-0002-8133-6467 M. Hernández
Villanueva  0000-0002-6322-5587 T. Higuchi  0000-0002-7761-3505 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 S. Jia
0000-0001-8176-8545 Y. Jin  0000-0002-7323-0830 A. Johnson
0000-0002-8366-1749 K. K. Joo  0000-0002-5515-0087 H. Junkerkalefeld
0000-0003-3987-9895 D. Kalita  0000-0003-3054-1222 A. B. Kaliyar
0000-0002-2211-619X J. Kandra  0000-0001-5635-1000 K. H. Kang
0000-0002-6816-0751 S. Kang  0000-0002-5320-7043 G. Karyan
0000-0001-5365-3716 T. Kawasaki  0000-0002-4089-5238 F. Keil
0000-0002-7278-2860 C. Kiesling  0000-0002-2209-535X C.-H. Kim
0000-0002-5743-7698 D. Y. Kim  0000-0001-8125-9070 K.-H. Kim
0000-0002-4659-1112 Y.-K. Kim  0000-0002-9695-8103 H. Kindo
0000-0002-6756-3591 K. Kinoshita  0000-0001-7175-4182 P. Kodyš
0000-0002-8644-2349 T. Koga  0000-0002-1644-2001 S. Kohani
0000-0003-3869-6552 K. Kojima  0000-0002-3638-0266 T. Konno
0000-0003-2487-8080 A. Korobov  0000-0001-5959-8172 S. Korpar
0000-0003-0971-0968 E. Kovalenko  0000-0001-8084-1931 R. Kowalewski
0000-0002-7314-0990 P. Križan  0000-0002-4967-7675 P. Krokovny
0000-0002-1236-4667 T. Kuhr  0000-0001-6251-8049 Y. Kulii  0000-0001-6217-5162
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 K. Lalwani
0000-0002-7294-396X 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 P. Leo  0000-0003-3833-2900 C. Lemettais
0009-0008-5394-5100 D. Levit  0000-0001-5789-6205 P. M. Lewis
0000-0002-5991-622X L. K. Li  0000-0002-7366-1307 S. X. Li
0000-0003-4669-1495 Y. Li  0000-0002-4413-6247 Y. B. Li  0000-0002-9909-2851
J. Libby  0000-0002-1219-3247 Z. Liptak  0000-0002-6491-8131 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 C. Martellini
0000-0002-7189-8343 A. Martens  0000-0003-1544-4053 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 S. Mitra  0000-0002-1118-6344 K. Miyabayashi
0000-0003-4352-734X R. Mizuk  0000-0002-2209-6969 G. B. Mohanty
0000-0001-6850-7666 S. Mondal  0000-0002-3054-8400 S. Moneta
0000-0003-2184-7510 H.-G. Moser  0000-0003-3579-9951 M. Mrvar
0000-0001-6388-3005 R. Mussa  0000-0002-0294-9071 I. Nakamura
0000-0002-7640-5456 M. Nakao  0000-0001-8424-7075 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 M. Niiyama
0000-0003-1746-586X S. Nishida  0000-0001-6373-2346 S. Ogawa
0000-0002-7310-5079 Y. Onishchuk  0000-0002-8261-7543 H. Ono
0000-0003-4486-0064 G. Pakhlova  0000-0001-7518-3022 S. Pardi
0000-0001-7994-0537 K. Parham  0000-0001-9556-2433 H. Park
0000-0001-6087-2052 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 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 I. Prudiiev
0000-0002-0819-284X 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 L. Reuter
0000-0002-5930-6237 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 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 S. Schneider  0009-0002-5899-0353 M. Schnepf
0000-0003-0623-0184 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 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 N. Suwonjandee
0009-0000-2819-5020 H. Svidras  0000-0003-4198-2517 M. Takahashi
0000-0003-1171-5960 M. Takizawa  0000-0001-8225-3973 U. Tamponi
0000-0001-6651-0706 S. Tanaka  0000-0002-6029-6216 K. Tanida
0000-0002-8255-3746 F. Tenchini  0000-0003-3469-9377 A. Thaller
0000-0003-4171-6219 O. Tittel  0000-0001-9128-6240 R. Tiwary
0000-0002-5887-1883 D. Tonelli  0000-0002-1494-7882 E. Torassa
0000-0003-2321-0599 K. Trabelsi  0000-0001-6567-3036 M. Uchida
0000-0003-4904-6168 I. Ueda  0000-0002-6833-4344 T. Uglov  0000-0002-4944-1830
K. Unger  0000-0001-7378-6671 Y. Unno  0000-0003-3355-765X K. Uno
0000-0002-2209-8198 S. Uno  0000-0002-3401-0480 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 A. Vossen
0000-0003-0983-4936 B. Wach  0000-0003-3533-7669 M. Wakai  0000-0003-2818-3155
S. Wallner  0000-0002-9105-1625 E. Wang  0000-0001-6391-5118 M.-Z. Wang
0000-0002-0979-8341 Z. Wang  0000-0002-3536-4950 A. Warburton
0000-0002-2298-7315 M. Watanabe  0000-0001-6917-6694 S. Watanuki
0000-0002-5241-6628 C. Wessel  0000-0003-0959-4784 E. Won  0000-0002-4245-7442
X. P. Xu  0000-0001-5096-1182 B. D. Yabsley  0000-0002-2680-0474 S. Yamada
0000-0002-8858-9336 S. B. Yang  0000-0002-9543-7971 J. Yelton
0000-0001-8840-3346 J. H. Yin  0000-0002-1479-9349 Y. M. Yook
0000-0002-4912-048X K. Yoshihara  0000-0002-3656-2326 C. Z. Yuan
0000-0002-1652-6686 L. Zani  0000-0003-4957-805X F. Zeng  0009-0003-6474-3508
B. Zhang  0000-0002-5065-8762 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 present a measurement of $|V_{ub}|$ from a simultaneous study of the
charmless semileptonic decays $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$, where $\ell=e,\mu$. This
measurement uses a data sample of 387 million $B\overline{B}$ meson pairs
recorded by the Belle II detector at the SuperKEKB electron-positron collider
between 2019 and 2022. The two decays are reconstructed without identifying
the partner $B$ mesons. We simultaneously measure the differential branching
fractions of $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ decays as functions of $q^{2}$
(momentum transfer squared). From these, we obtain total branching fractions
$\mathcal{B}(B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell})=(1.516\pm
0.042(\mathrm{stat})\pm 0.059(\mathrm{syst}))\times 10^{-4}$ and
$\mathcal{B}(B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell})=(1.625\pm
0.079(\mathrm{stat})\pm 0.180(\mathrm{syst}))\times 10^{-4}$. By fitting the
measured $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ partial branching
fractions as functions of $q^{2}$, together with constraints on the non-
perturbative hadronic contribution from lattice QCD calculations, we obtain
$|V_{ub}|$ = $(3.93\pm 0.09\pm 0.13\pm 0.19)\times 10^{-3}$. Here, the first
uncertainty is statistical, the second is systematic, and the third is
theoretical.

##  I Introduction

The Cabibbo-Kobayashi-Maskawa (CKM) CKM2 ; CKM  matrix elements are related to
four fundamental parameters of the Standard Model (SM) of particle physics.
The magnitude of the matrix element $V_{ub}$ can be determined by measuring
the rate of $B\rightarrow X_{u}\ell\nu_{\ell}$ decays, which is proportional
to $|V_{ub}|^{2}$, where $X_{u}$ is a charmless hadronic final state and
$\ell$ is a light charged lepton. Two methods can be used to measure
$|V_{ub}|$. In the inclusive method, no specific $X_{u}$ final state is
reconstructed, and the sum of all possible final states is analyzed. The
theoretical description involves calculation of the total semileptonic rate.
In the exclusive method, a specific final state is reconstructed and the
theoretical description takes the form of parametrizations of the low-energy
strong interactions (form factors). These two methods have complementary
uncertainties introduced by their theoretical descriptions. Determinations of
$|V_{ub}|$ from experimental data obtained by the inclusive and exclusive
methods currently differ by approximately 2.5 standard deviations HFLAV .

The most experimentally and theoretically reliable exclusive measurements of
$|V_{ub}|$ come from $B\rightarrow\pi\ell\nu_{\ell}$ decays. The current world
average of $|V_{ub}|$ from this mode, taking all of the experimental and
theoretical information into consideration, is $(3.67\pm 0.09\pm 0.12)\times
10^{-3}$ HFLAV , where the first uncertainty is experimental and the second is
theoretical. A simultaneous measurement of
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ with
$B\rightarrow\rho\ell\nu_{\ell}$ decays allows for an additional measurement
of $|V_{ub}|$ while accounting for decays of one type that are reconstructed
as the other. Therefore, we simultaneously reconstruct
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ (with charge conjugation implied
throughout), focusing on decays with entirely charged final states in order to
avoid backgrounds associated with neutral particles. We measure the
differential branching fractions of these decays as a function of $q^{2}$, the
squared momentum transferred from the $B$ meson to the hadron. We use data
from the Belle II detector located at the SuperKEKB electron-positron collider
at KEK, in Japan. The $B$-meson decays are reconstructed in $e^{+}e^{-}$
$\rightarrow$ $\mathchar 28935\relax{(4S)}$ $\rightarrow$ $B\overline{B}$
events. We use an “untagged” reconstruction method in which the signal lepton
and hadron candidates are selected without prior reconstruction (tagging) of
the partner $B$ meson. This leads to a high signal efficiency, but also a low
purity due to the large combinatorial background from the partner $B$ meson
and increased backgrounds from $e^{+}e^{-}$ collisions that produce light
quark pair (continuum) events.

We reconstruct $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ events and make signal selections
that reduce backgrounds. For both modes we build three-dimensional histograms
of two kinematic variables and the reconstructed $q^{2}$. From simulation, we
define $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ signal categories based on 13 and
10 disjoint intervals (bins) of simulated $q^{2}$, respectively. Signal
templates defined in this way, which we refer to as true $q^{2}$ binning, can
overlap in reconstructed $q^{2}$ due to resolution effects. We extract signal
yields in all 23 signal categories with a simultaneous extended maximum-
likelihood fit to both histograms. From these, we obtain differential
branching fractions as functions of true $q^{2}$. We then combine constraints
from theoretical form-factor predictions from unquenched lattice QCD (LQCD)
FLAG21  and light-cone sum rule (LCSR) LCSR_pi ; BSZ  calculations with the
measured differential branching fractions to determine $|V_{ub}|$.

##  II Theoretical Framework

In contrast to hadronic decays, in semileptonic $B\rightarrow
X_{u}\ell\nu_{\ell}$ decays, the leptonic and hadronic parts of the decay
matrix element factorize at leading electroweak order, which allows for a
straightforward determination of $|V_{ub}|$. However, the description of the
hadronic contribution to the matrix element has many uncertainties, since
higher-order perturbative corrections and hadronization processes need to be
included. Form factors parametrize these QCD processes in the full phase space
and are constrained in specific phase-space regions by theoretical predictions
based on, for example, LQCD and LCSR calculations JochenReview . These
predictions rely on the $V{-}A$ structure of the process and are usually
formulated as functions of $q^{2}$. For the semileptonic decay $B\rightarrow
X\ell\nu_{\ell}$,

$q^{2}=(p_{B}-p_{X})^{2},$ (1)

where $p_{B}$ and $p_{X}$ are the four-momenta of the $B$ meson and the
hadron, respectively. Here we use natural units with $c=1$ throughout. LQCD
predictions rely on first principles to predict the form factors for
$B\rightarrow\pi\ell\nu_{\ell}$, and are only available in the high-$q^{2}$
region. LCSR predictions, on the other hand, are only applicable in the
low-$q^{2}$ region, but are available for $B\rightarrow\pi\ell\nu_{\ell}$ and
$B\rightarrow\rho\ell\nu_{\ell}$ decays.

In the theoretical description of the kinematic processes as a function of
$q^{2}$, it is necessary to distinguish between decays into pseudoscalar and
vector mesons, such as $B\rightarrow\pi\ell\nu_{\ell}$ and
$B\rightarrow\rho\ell\nu_{\ell}$, respectively. The hadronic matrix element of
the decay $B\rightarrow\pi\ell\nu_{\ell}$ can be written in terms of two form
factors $f_{+}(q^{2})$ and $f_{0}(q^{2})$, where $f_{0}(q^{2})$ is negligible
for $\ell=e,\mu$. The differential decay rate can then be expressed as a
function of $f_{+}(q^{2})$ and $|V_{ub}|$ JochenReview :

$\frac{\mathrm{d}\Gamma(B\rightarrow\pi\ell\nu_{\ell})}{\mathrm{d}q^{2}\mathrm{d}\cos{\theta_{W\ell}}}=|V_{ub}|^{2}\frac{G^{2}_{F}|\vec{p}_{\pi}|^{3}}{32\pi^{3}}\sin^{2}{\theta_{W\ell}}|f_{+}(q^{2})|^{2},$
(2)

where $G_{F}$ is the Fermi constant, $\vec{p}_{\pi}$ is the momentum of the
pion in the $B$-meson rest frame, and $\theta_{W\ell}$ is the angle between
the $W$ boson momentum in the $B$ rest frame and the lepton momentum in the
$W$ rest frame.

The hadronic matrix element of $B\rightarrow\rho\ell\nu_{\ell}$ is described
by four form factors, which can be reduced to three, $A_{1}(q^{2})$,
$A_{2}(q^{2})$, and $V(q^{2})$, for decays to light leptons ($\ell=e,\mu$).
The form factors can be rewritten in terms of the helicity amplitudes,
$H_{+}(q^{2})$, $H_{-}(q^{2})$, and $H_{0}(q^{2})$, of the $\rho$ meson. The
differential decay rate as a function of $H_{\pm}$, $H_{0}(q^{2})$, and
$|V_{ub}|$ then becomes JochenReview :

$\frac{\mathrm{d}\Gamma(B\rightarrow\rho\ell\nu_{\ell})}{\mathrm{d}q^{2}\mathrm{d}\cos{\theta_{W\ell}}}=|V_{ub}|^{2}\frac{G^{2}_{F}|\vec{p}_{\rho}|q^{2}}{128\pi^{3}m_{B}^{2}}\\\
\times\biggr{[}\sin^{2}{\theta_{W\ell}}|H_{0}(q^{2})|^{2}+(1-\cos{\theta_{W\ell}})^{2}\frac{|H_{+}(q^{2})|^{2}}{2}\\\
+(1+\cos{\theta_{W\ell}})^{2}\frac{|H_{-}(q^{2})|^{2}}{2}\biggr{]},$ (3)

where $m_{B}$ is the mass of the $B$ meson, and $\vec{p}_{\rho}$ is the
momentum of the $\rho$ meson in the $B$ rest frame.

Due to the relationship between the differential decay rate of
$B\rightarrow\pi\ell\nu_{\ell}$ (or $B\rightarrow\rho\ell\nu_{\ell}$),
$|V_{ub}|$, and the respective form factors, $|V_{ub}|$ can be extracted from
measurements of the differential decay rates of
$B\rightarrow\pi\ell\nu_{\ell}$ (or $B\rightarrow\rho\ell\nu_{\ell}$) if the
$q^{2}$ shape and normalizations of the form factors are known. The
normalizations are provided by theoretical QCD calculations of the form
factors. Since these calculations are not available across the entire $q^{2}$
range, the $q^{2}$ dependence of the form factors is interpolated between the
high and low-$q^{2}$ regions using analyticity and unitarity arguments. One
technique employs dispersion relations to expand in powers of the variable
$z(q^{2},q_{0}^{2})$ defined as

$z(q^{2},q_{0}^{2})=\frac{\sqrt{m^{2}_{+}-q^{2}}-\sqrt{m^{2}_{+}-q_{0}^{2}}}{\sqrt{m^{2}_{+}-q^{2}}+\sqrt{m^{2}_{+}-q_{0}^{2}}},$
(4)

where $m_{+}=m_{B}+m_{X}$ is the sum of the masses of the $B$ meson and the
hadron. According to Ref. BCL  the optimal choice of the parameter $q_{0}^{2}$
is given by $q_{0}^{2}=m_{+}(\sqrt{m_{B}}-\sqrt{m_{X}})^{2}$.

Two expansion parametrizations are considered in this work. The first
parametrization is the Bourrely-Caprini-Lellouch (BCL) parametrization BCL ,
which expands the form factor $f_{+}(q^{2})$ using form-factor coefficients
$b^{+}_{k}$ up to expansion order $K$ as:

$f_{+}(q^{2})=P(q^{2})\sum^{K-1}_{k=0}b^{+}_{k}\Bigr{[}z^{k}-(-1)^{k-K}\frac{k}{K}z^{K}\Bigr{]}.$
(5)

Here $P(q^{2})=(1-q^{2}/m^{2}_{R})^{-1}$ is called the inverse Blaschke
factor, where the mass of the resonance $m_{R}$ depends on the allowed angular
momentum and parity. The expansion of the form factor $f_{0}(q^{2})$ using
form-factor coefficients $b^{0}_{k}$ takes the form:

$f_{0}(q^{2})=f_{+}(0)\Bigr{[}1+\sum^{K-1}_{k=0}b^{0}_{k}z^{k}\Bigr{]}.$ (6)

The Bharucha-Straub-Zwicky (BSZ) parametrization BSZ  instead is a series
expansion around $q^{2}=0$, using form-factor coefficients $b^{i}_{k}$, so
that the form factors
$f_{i}(q^{2})\in\\{A_{1}(q^{2}),A_{2}(q^{2}),V(q^{2})\\}$ take the form

$f_{i}(q^{2})=P(q^{2})\sum^{K-1}_{k=0}b^{i}_{k}\Bigr{(}z(q^{2})-z(0)\Bigr{)}^{k}.$
(7)

By fitting the form-factor parametrizations given in Equations 5, 6 and 7 to
measured partial branching-fraction spectra, the form-factor coefficients
$b^{i}_{k}$ can be extracted. In addition, by adding theoretical input from
LCSR or LQCD calculations, we can determine $|V_{ub}|$.

##  III Detector, data set, and simulation

###  III.1 Detector

The Belle II experiment b2tdr  is located at SuperKEKB, which collides
electrons and positrons at and near the $\Upsilon(4S)$ resonance superkekb .
The Belle II detector b2tdr  has a cylindrical geometry and includes a two-
layer silicon-pixel detector (PXD) surrounded by a four-layer double-sided
silicon-strip detector (SVD) SVD  and a 56-layer central drift chamber (CDC).
These detectors reconstruct trajectories (tracks) of charged particles. Only
one sixth of the second layer of the PXD was installed for the data analyzed
here. The symmetry axis of these detectors, defined as the $z$ axis, is almost
coincident with the direction of the electron beam. Surrounding the CDC, which
also provides $\mathrm{d}E/\mathrm{d}x$ energy-loss measurements and has a
polar angle acceptance of 17–150∘, is a time-of-propagation counter (TOP) TOP
in the central region and an aerogel-based ring-imaging Cherenkov counter
(ARICH) in the forward region. These detectors provide charged-particle
identification. Surrounding the TOP and ARICH is an electromagnetic
calorimeter (ECL) based on CsI(Tl) crystals that primarily provides energy and
timing measurements for photons and electrons. Outside of the ECL is a
superconducting solenoid magnet. Its flux return is instrumented with
resistive-plate chambers and plastic scintillator modules to detect muons,
$K^{0}_{L}$ mesons, and neutrons. The solenoid magnet provides a 1.5 T
magnetic field that is parallel to the $z$ axis.

Using $\mathrm{d}E/\mathrm{d}x$ energy-loss data from the CDC and information
from the two particle-identification detectors, the ARICH and TOP, as well as
data from the ECL, the SVD, and the $K^{0}_{L}$ and muon detector (KLM),
charged particles of different masses are identified via particle-
identification (PID) likelihood ratios. Each of these is a ratio of the
likelihood $\mathcal{L}$ for one charged-particle hypothesis $\alpha$ to the
sum of the likelihoods for all hypotheses:
$\mathcal{R}_{\alpha}=\mathcal{L}_{\alpha}/(\mathcal{L}_{e}+\mathcal{L}_{\mu}+\mathcal{L}_{\pi}+\mathcal{L}_{K}+\mathcal{L}_{p}+\mathcal{L}_{d})$,
where $\alpha\in\\{e,\mu,\pi,K$, proton ($p$), deuteron ($d$)$\\}$. The
identification efficiencies and misidentification probabilities of the pion,
kaon, electron, and muon likelihood ratios are determined separately for both
charges in bins of momentum and polar angle using data samples of control
modes, such as $J/\psi\rightarrow\ell^{+}\ell^{-}$ and $D^{*+}\rightarrow
D^{0}[\rightarrow K^{-}\pi^{+}]\pi^{+}$.

###  III.2 Experimental data

The primary data set used in this analysis consists of ($387\pm 6$) million
$\mathchar 28935\relax{(4S)}$ $\rightarrow$ $B\overline{B}$ events from
$(364\pm 2)$ fb-1 of electron-positron collisions collected at a center-of-
mass (c.m.) energy of $\sqrt{s}=$ 10.58 GeV, corresponding to the
$\Upsilon(4S)$ resonance. We use an additional sample corresponding to
$(42.6\pm 0.3)$ fb-1 of off-resonance collision data, collected at a c.m.
energy 60 MeV below the $\Upsilon$(4S) resonance, to describe background from
continuum processes. These include $q\overline{q}$ production $e^{+}e^{-}$
$\rightarrow$ $u\bar{u},\,d\bar{d},\,s\bar{s}$, and $c\bar{c}$, QED processes
such as $e^{+}e^{-}$ $\rightarrow$ $\tau^{+}\tau^{-}$, and two-photon
processes such as $e^{+}e^{-}$ $\rightarrow$ $e^{+}e^{-}\ell^{+}\ell^{-}$ and
$e^{+}e^{-}$ $\rightarrow$ $e^{+}e^{-}\pi^{+}\pi^{-}$.

###  III.3 Simulation

We use simulated data sets referred to as Monte Carlo (MC) samples to identify
efficient background-discriminating variables and to form fit templates for
signal extraction. The MC samples for $B\overline{B}$ background events
correspond to an integrated luminosity of 3 ab-1 and contain semileptonic and
hadronic $B$ decays, generated using the EvtGen EvtGen  software package.

We use generated signal MC samples containing 10 million events of
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ decays to obtain reconstruction
efficiencies and study the key kinematic distributions. In order to describe
the remaining $B\rightarrow X_{u}\ell\nu_{\ell}$ decays, we simulate four
samples, each containing 50 million events, of resonant and nonresonant
$B^{0}\rightarrow X_{u}^{-}\ell^{+}\nu_{\ell}$, and $B^{+}\rightarrow
X_{u}^{0}\ell^{+}\nu_{\ell}$. The resonant sample contains $B\rightarrow
X_{u}\ell\nu_{\ell}$ decays, where
$X_{u}\in\\{\pi,\rho,\omega,\eta,\eta^{\prime}\\}$. We simulate the
hadronization of the $X_{u}$ state to multiple hadrons in nonresonant
$B\rightarrow X_{u}\ell\nu_{\ell}$ events with PYTHIA Pythia . The second $B$
meson decays in the same way as the generic $B\overline{B}$ background
described above.

We also generate 1 ab-1 of $q\overline{q}$ and $\tau\tau$ continuum-event
samples with KKMC KKMC , and simulate $q\overline{q}$ fragmentation with
PYTHIA Pythia  and $\tau$ decays with TAUOLA TAUOLA . Additionally, we
simulate 2 ab-1 of two-photon production with AAFH AAFH . Final-state
radiation of photons from stable charged particles is simulated with PHOTOS
PHOTOS  and we overlay simulated beam-induced backgrounds to all generated
events BeamBKG . The propagation of particles through the detector and the
resulting interactions are simulated using Geant4 GEANT4 . We reconstruct and
analyze simulated and experimental data with the Belle II analysis software
framework, basf2 basf2 ; basf2Repo .

In the simulation of $B\overline{B}$ backgrounds, we use world-average
$B\rightarrow X\ell\nu_{\ell}$ branching fractions PDG  in combination with an
isospin-symmetry assumption, following the procedure in Ref. BFcalc  for
$B\rightarrow X_{c}\ell\nu_{\ell}$ decays. We assume that the remaining
difference between the sum of the exclusive $B\rightarrow X_{c}\ell\nu_{\ell}$
decay branching fractions and the measured total branching fraction,
accounting for approximately 4% of the $B\rightarrow X_{c}\ell\nu_{\ell}$
decays, is saturated by $B\rightarrow D^{(*)}\eta\ell\nu_{\ell}$ decays, which
corresponds to the procedure in Ref. GapMethod . For the form factors of
$B\rightarrow D\ell\nu_{\ell}$ and $B\rightarrow D^{*}\ell\nu_{\ell}$ decays,
we use the Boyd-Grinstein-Lebed parametrization (BGL) BGL  with central values
from Ref. D_FF  and Dst_FF , respectively.

For $B\rightarrow X_{u}\ell\nu_{\ell}$ decays, the sum of the measured
exclusive branching fractions is approximately 20% of the inclusively measured
branching fraction. We model the remaining difference with nonresonant
$B\rightarrow X_{u}\ell\nu_{\ell}$ decays. An important component of these
nonresonant $B\rightarrow X_{u}\ell\nu_{\ell}$ decays are events in which the
$X_{u}$ system consists of a $\pi^{+}\pi^{-}$ pair. The partial-branching-
fraction spectrum of $B^{+}\rightarrow\pi^{+}\pi^{-}\ell^{+}\nu_{\ell}$ as a
function of the di-pion invariant mass $m_{\pi\pi}$ has been measured in Ref.
Nonres . We compare this experimental spectrum to the simulated spectrum of
nonresonant $B^{+}\rightarrow\pi^{+}\pi^{-}\ell^{+}\nu_{\ell}$ events.
Assuming that nonresonant $B\rightarrow X_{u}\ell\nu_{\ell}$ events have no
structure below the $\rho^{0}$ mass peak, we interpolate the measured spectrum
to the $\rho^{0}$ mass peak window by fitting a straight line with floating
slope and intercept to the spectrum surrounding the peak. We then assign event
weights to the simulated $B^{+}\rightarrow\pi^{+}\pi^{-}\ell^{+}\nu_{\ell}$
events in order to recover the measured partial branching fractions as a
function of $m_{\pi\pi}$ in the simulation.

We describe the remaining nonresonant $B\rightarrow X_{u}\ell\nu_{\ell}$
decays with the De Fazio and Neubert (DFN) model DFN , which combines a
prediction of the triple-differential rate in $X_{u}$ particle mass $m_{X}$,
the $B$ rest-frame lepton energy $E_{\ell}$, and $q^{2}$, with a non-
perturbative shape function using an exponential model. In the simulation we
use central values for the two relevant parameters provided in Ref. DFNparam .
The total $B\rightarrow X_{u}\ell\nu_{\ell}$ composition is described by
implementing a hybrid model Hyb , following closely the method in Ref. munu .
This approach combines the exclusive and nonresonant decay rates in bins of
$m_{X}$, $E_{\ell}$, and $q^{2}$, in order to reproduce the inclusive rates.

We describe $B\rightarrow\pi\ell\nu_{\ell}$ decays using the BCL
parametrization BCL  with central values for the parameters $b^{+}_{k}$ and
$b^{0}_{k}$ in Equations 5 and 6, respectively, from Ref FLAG21 , and
$B\rightarrow\rho\ell\nu_{\ell}$ and $B\rightarrow\omega\ell\nu_{\ell}$ decays
using the BSZ parametrization BSZ  with central values for $b^{i}_{k}$ in
Equation 7 from Ref. rho_om_FF . The lineshape of the $\rho$ meson is modeled
following the description in Ref. RhoFlorian  neglecting interference between
the $\rho$ and the $\omega$ meson, which is included as a systematic
uncertainty and described in Section VII. For the form-factor description of
$B\rightarrow\eta^{(\prime)}\ell\nu_{\ell}$ decays we use a LCSR calculation
eta_FF .

##  IV Event reconstruction and selection

###  IV.1 Signal reconstruction and selection

We begin signal reconstruction by identifying track candidates that pass
certain quality criteria. The extrapolated trajectories must originate from a
cylindrical region of length $3$ cm along the $z$ axis and radius $0.5$ cm in
the transverse plane, centered on the $e^{+}e^{-}$ interaction point.
Furthermore, charged particles must have transverse momenta greater than 0.05
GeV and polar angles within the CDC acceptance. We discard events with fewer
than five tracks satisfying the above criteria.

In the remaining events, we select signal-lepton candidates from among the
selected tracks by requiring that their c.m. momenta $p^{*}_{\ell}$ are in the
range $[1.0,2.85]$ GeV and $[1.4,2.85]$ GeV in the
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ modes, respectively. These
selections significantly reduce the number of events in which the lepton
originates from $B\rightarrow X_{c}\ell\nu_{\ell}$ decays, where $X_{c}$ is a
hadronic final state containing a charm quark. We choose a higher lepton
momentum threshold for $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$
candidates, since, due to the different spin structure, the lepton momentum
spectrum of $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ peaks at a higher
momentum than that of $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$. In order to
reduce sensitivity to modeling of the detector response in the extreme forward
and backward directions of the lepton polar angle $\theta_{\ell}$, we exclude
events with $\cos\theta_{\ell}<-0.55$ and $\cos\theta_{\ell}>0.85$.

We require that electron and muon candidates have PID likelihood ratios
greater than 0.9. The electron likelihood ratio combines information from the
CDC, ECL, ARICH and the KLM, while the muon likelihood ratio also includes
information from the TOP. The average electron (muon) efficiency is 92 (93)%.
The hadron misidentification rates are 0.2% for the electron and 3.2% for the
muon selection, respectively. The four-momenta of the electron candidates are
corrected for bremsstrahlung by adding the four-momenta of photons with an ECL
cluster energy below $1.0$ GeV for $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$
and $0.5$ GeV for $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ found within a
cone of 0.05 rad around the electron-momentum vector.

In the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ mode we select pion
candidates from the remaining tracks and require that they have a charge
opposite to that of the lepton candidate. In the
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ mode we require that the two
selected pion candidates that compose the $\rho$ candidate have opposite
charges and have a mass $m_{\pi\pi}$ in the range $[0.554,0.996]$ GeV. This
selection reduces combinatorial background, but is loose enough to reduce
sensitivity to modeling of the $\rho$ mass distribution. We reduce sensitivity
to the modeling of the detector response in the extreme backward region of
pion polar angle $\theta_{\pi}$ by selecting pions with
$\cos\theta_{\pi}>-0.65$. All pion candidates are required to have a PID
likelihood ratio greater than $0.1$. The pion likelihood, in addition to data
from the CDC, ECL, ARICH and the KLM, includes information from the SVD and
the TOP. To improve the particle-identification performance, we require that
the pion candidates have at least 20 measurement points in the CDC. The
resulting average pion efficiency is 86% with kaon and lepton
misidentification rates of 7% and 0.4%, respectively.

The selections described below are designed to reduce backgrounds and enhance
signal purity. We remove candidates with kinematic properties inconsistent
with the signal $B$ meson decay. Assuming that only a single massless particle
is not included in the event reconstruction, the angle between the $B$ meson
and the combination of the signal lepton and hadron candidates, denoted $Y$,
is given by

$\cos\theta_{\mathrm{BY}}=\frac{2E^{*}_{B}E^{*}_{Y}-m_{B}^{2}-m_{Y}^{2}}{2|\vec{p}^{\,*}_{B}||\vec{p}^{\,*}_{Y}|},$
(8)

where $E^{*}_{Y}$, $|\vec{p}^{\,*}_{Y}|$, and $m_{Y}$ are the energy,
magnitude of the three-momentum, and invariant mass of the $Y$ in the c.m.
frame, respectively. The energy $E^{*}_{B}$ and the magnitude of the three-
momentum $|\vec{p}^{\,*}_{B}|$ of the $B$ meson are calculated from the beam
properties, and $m_{B}$ is the mass of the $B$ meson PDG . For correctly
reconstructed signal decays and assuming perfect resolution, we expect
$\cos\theta_{\mathrm{BY}}$ to lie between $-1$ and 1. However, to retain
enough background events to train the classifiers discussed in Section V.2, we
choose a less restrictive requirement, $|\cos\theta_{\mathrm{BY}}|<1.6$.
Additionally, we perform vertex fits TreeFit  to the hadron and lepton
candidates and require that they converge.

###  IV.2 Missing momentum reconstruction

We estimate the momentum of the signal neutrino by attributing the sum of the
remaining tracks and electromagnetic energy depositions (clusters) in the
event, called the rest of event (ROE), to the partner $B$. From energy and
momentum conservation, we construct the missing four-momentum in the c.m.
frame,

$(E^{*}_{\mathrm{miss}},\vec{p}^{\,*}_{\mathrm{miss}})=(E^{*}_{\Upsilon(4S)},\vec{p}^{\,*}_{\Upsilon(4S)})-\left(\sum_{i}E^{*}_{i},\sum_{i}\vec{p}^{\,*}_{i}\right),$
(9)

where $E^{*}_{i}$ and $\vec{p}^{\,*}_{i}$ correspond to the c.m. energy and
momentum of the $i$th track or cluster in the event, respectively. We
determine $E^{*}_{i}$ using the momentum derived from the reconstructed track
and select the mass hypothesis $\alpha$ with the highest value of the
likelihood ratio $\mathcal{R}_{\alpha}$. We attribute the missing four-
momentum to the signal neutrino, with momentum
$\vec{p}^{\,*}_{\nu}=\vec{p}^{\,*}_{\mathrm{miss}}$, and energy,
$E^{*}_{\nu}=|\vec{p}^{\,*}_{\nu}|=|\vec{p}^{\,*}_{\mathrm{miss}}|$. Taking
the magnitude of $\vec{p}^{\,*}_{\mathrm{miss}}$, instead of
$E^{*}_{\mathrm{miss}}$, to approximate the neutrino energy, leads to an
improvement in resolution of 15%. While reconstruction losses add up linearly
in the calculation of $E^{*}_{\mathrm{miss}}$, this is not the case for the
vector sum calculation of $\vec{p}^{\,*}_{\mathrm{miss}}$.

Since all reconstructed tracks and clusters contribute to the resolution of
the neutrino momentum estimation, obtaining an ROE as pure and complete as
possible is critical. To reduce the impact of clusters from beam-induced
backgrounds, acceptance losses, or other effects, we impose quality
requirements for objects to be included in the ROE. We only consider clusters
that are within the CDC acceptance with energies in the forward, barrel, and
backward directions greater than 0.060, 0.050, and 0.075 GeV, respectively. We
require that the clusters contain more than one calorimeter crystal and are
detected within 200 ns of the collision time, which is approximately five
times the mean timing resolution of the calorimeter. In addition to removing
background particles from the ROE, we must account for particles that escape
undetected. To reduce the impact of events with undetected particles, we
require that the polar angle of the missing momentum in the laboratory frame
$\theta_{\mathrm{miss}}$ is within the CDC acceptance.

###  IV.3 Signal extraction variables

We reconstruct $q^{2}$ from Equation 1, and thus need to estimate the $B$
momentum vector. One existing method, called the Diamond Frame DiamondFrame ,
takes the weighted average of four possible $\vec{p}^{\,*}_{B}$ vectors
uniformly distributed in azimuthal angle on the cone defined by
$\cos\theta_{\mathrm{BY}}$, weighting by the $\sin^{2}\theta_{B}$
distribution, which expresses the prior probability of the $B$ flight
direction in $\mathchar 28935\relax{(4S)}$ decays with respect to the
electron-positron beam axis. A second method, called the ROE method ROEmethod
, assumes the signal $B$ momentum vector to be the vector on the
$\cos\theta_{\mathrm{BY}}$ cone that is closest to antiparallel to the ROE
momentum vector $\vec{p}^{\,*}_{\mathrm{ROE}}$. There is a third method
CombMethod  that combines these two by multiplying the Diamond Frame weights
by $\frac{1}{2}(1-\hat{p}^{\,*}_{B}\cdot\hat{p}^{\,*}_{\mathrm{ROE}})$ and
averaging over ten vectors uniformly distributed on the cone, where
$\hat{p}^{\,*}_{B}$ and $\hat{p}^{\,*}_{\mathrm{ROE}}$ denote the unit vectors
of $\vec{p}^{\,*}_{B}$ and $\vec{p}^{\,*}_{\mathrm{ROE}}$, respectively. We
adopt this combined method because, in simulation, it assigns reconstructed
signal candidates to the correct $q^{2}$ bin more often than other methods do,
leading to a reduction in the bin migrations of up to 2%. The resolutions in
$q^{2}$ decrease with increasing $q^{2}$ and vary from 0.09–0.60 GeV2 in the
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ mode, and from 0.16–0.84 GeV2 in
the $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ mode.

We divide $B$ candidates into 13 reconstructed $q^{2}$ bins in the
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ mode and into 10 bins in the
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ mode. The lowest bin boundary is
at zero, and the first 12 (9) bins have uniform bin widths of 2 GeV2 in the
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$
($B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$) mode. The last bins extend to
the kinematic limits of 26.4 GeV2 in the
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ mode and 20.3 GeV2 in the
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ mode. The following are the
labels and bin edges for the $q^{2}$ bins: $q1:q^{2}\in[0,2]$, $q2:[2,4]$,
$q3:[4,6]$, $q4:[6,8]$, $q5:[8,10]$, $q6:[10,12]$, $q7:[12,14]$, $q8:[14,16]$,
$q9:[16,18]$, $q10:[18,20(20.3)]$, $q11:[20,22]$, $q12:[22,24]$,
$q13:[24,26.4]$ GeV2.

Two additional variables that test the kinematic consistency of a candidate
with a signal $B$ decay using ROE information are the beam-constrained mass,
defined as

$M_{\mathrm{bc}}=\sqrt{E^{*2}_{\mathrm{beam}}-|\vec{p}^{\,*}_{B}|^{2}}=\sqrt{\left(\frac{\sqrt{s}}{2}\right)^{2}-|\vec{p}^{\,*}_{B}|^{2}}$
(10)

and the energy difference, defined as

$\Delta E=E^{*}_{B}-E^{*}_{\mathrm{beam}}=E^{*}_{B}-\frac{\sqrt{s}}{2},$ (11)

where $E^{*}_{\mathrm{beam}}$, $E^{*}_{B}$ and $\vec{p}^{\,*}_{B}$ are the
single-beam energy, the reconstructed $B$ energy, and the reconstructed $B$
momentum, all determined in the $\mathchar 28935\relax{(4S)}$ rest frame,
respectively. The reconstructed $B$ energy (momentum) is given by the sum of
the reconstructed energies (momenta) of the signal lepton and hadron
candidates and the inferred neutrino energy (momentum) described above. We
define a fit region in $\Delta E$ and $M_{\mathrm{bc}}$, corresponding to
$-0.95<\Delta E<1.25$ GeV and $5.095<M_{\mathrm{bc}}<5.295$ GeV. This region
is enriched in signal, but at the same time includes background-enhanced
regions to allow sufficient discrimination between signal and background.

##  V Background suppression

###  V.1 Signal and background categories

As explained in Section III.3, the simulated samples can be separated into two
main categories: $B\overline{B}$ events and continuum events. For the
$B\overline{B}$ events we define a subcategory that combines true signal and
combinatorial signal events. In combinatorial signal either the signal hadron
or lepton candidate is incorrectly chosen. In addition, we define an isospin-
conjugate signal category, where the lepton originates from the isospin
conjugate of the signal decay. The branching fraction of the isospin-conjugate
signal events scales with the branching fraction of true signal events under
the assumption of isospin symmetry:

$\mathcal{B}(B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell})=2\tau_{0}/\tau_{+}\times\mathcal{B}(B^{+}\rightarrow\pi^{0}\ell^{+}\nu_{\ell})$,

$\mathcal{B}(B^{0}\rightarrow\rho^{-}\ell^{+}\nu_{\ell})=2\tau_{0}/\tau_{+}\times\mathcal{B}(B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell})$,

where $\tau_{+}/\tau_{0}$ is the $B$ lifetime ratio.

A fourth signal category, called cross-feed signal, includes events in which
the lepton originates from the reconstructed signal decay mode, but was
reconstructed in another signal mode, i.e. $B\rightarrow\rho\ell\nu_{\ell}$
events reconstructed in the $B\rightarrow\pi\ell\nu_{\ell}$ sample, and vice
versa. Since the number of true, combinatorial, and isospin-conjugate signal
events scales with the same branching fraction as the cross-feed signal events
from the other reconstructed mode, we combine the four signal categories into
a total signal category.

We further split the remaining $B\overline{B}$ events into the two largest
semileptonic backgrounds, $B\rightarrow X_{c}\ell\nu_{\ell}$ and $B\rightarrow
X_{u}\ell\nu_{\ell}$. The remaining $B\overline{B}$ events are combined into
the _other_ $B\overline{B}$ category, which is mainly composed of candidates
with misidentified leptons, or with leptons from secondary decays.

###  V.2 Background suppression using boosted decision trees

In order to further reduce the $B\overline{B}$ and continuum backgrounds, we
train boosted decision trees (BDTs) to separate signal from these two
background categories using the FastBDT package FastBDT . Since the background
composition is different in the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$
and $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ modes and depends on the
$q^{2}$ bin, we train the classifier and optimize the selection separately for
each $q^{2}$ bin and mode. In addition, to increase the sensitivity in the
highest $q^{2}$ bin in the $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ mode,
we split the last bin into two bins during training and optimization of the
selection. Since separate classifiers are trained for the suppression of
continuum and $B\overline{B}$ backgrounds, we train a total of
$2\times(13+11)=48$ BDTs. In each training we use equal amounts of simulated
true signal and background events, corresponding to 400 fb-1 of simulated
continuum data, and 1 ab-1 of simulated $B\overline{B}$ data.

We use event-shape, kinematic and topological variables as BDT input
variables. The event-shape input variables includes four normalized Fox-
Wolfram moments FWM  and variables based on the thrust axis, which is the axis
that maximizes the sum of the projected momenta of a collection of particles
in the event Thrust . Distinct thrust axes can be defined for the signal $B$
and the ROE. Their magnitudes, the angle between the two axes
$\cos\theta_{T}$, and the angle between the signal $B$ thrust axis and the
beam direction serve as input variables. Three cones with opening angles of
10∘, 20∘, and 30∘ centered around the signal $B$ thrust axis are defined, and
the momentum flows into each of the three cones are added as input variables
CLEOCones .

The kinematic and topological variables include the p-value from the
$\chi^{2}$ of the vertex fit to the pion and lepton candidates, and the cosine
of the angle between the signal $B$ momentum vector and the vector connecting
its fitted vertex to the interaction point, both in the plane parallel and
perpendicular to the beam axis. Other input variables are
$\cos\theta_{\mathrm{BY}}$, $\theta_{\mathrm{miss}}$, the number of tracks,
the momentum of the ROE, and the polar angle of the lepton candidate. The
remaining variables are the angle between the direction of the lepton in the
$W$ frame and the $W$ in the $B$ frame, and for the
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ mode, the angle between the
$\rho$ in the $B$ frame and the pion in the $\rho$ rest frame.

We train each of the 48 BDTs using the twelve input variables that provide the
highest discriminating power in the corresponding $q^{2}$ bin, and ensure that
none of the selected input variables are strongly correlated with $\Delta E$
or $M_{\mathrm{bc}}$. The input variable with the most discriminating power
for the suppression of continuum events is $\cos\theta_{T}$, while
$\cos\theta_{\mathrm{BY}}$ and the $\chi^{2}$ vertex fit probability are the
input variables with the most discriminating power for the suppression of
$B\overline{B}$ events. We identify the optimal selection criterion on the
combination of the continuum and $B\overline{B}$ output classifiers within
each $q^{2}$ bin by maximizing the ratio between the number of signal events
and the square root of the sum of the number of signal and background events,
as predicted by simulation. In the
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$($B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$)
mode these selections retain 55%(48%) of the signal, 4% (1%) of the
$B\overline{B}$ background, and 1% (0.6%) of the continuum background.

###  V.3 Continuum reweighting

Because of the small size of the off-resonance data, especially after
suppressing continuum events, using it directly to construct fit templates
results in large statistical fluctuations in the individual bins. Instead, we
weight the simulated continuum candidates using off-resonance data in order to
use the resulting template during signal extraction. To obtain the weights we
initially compare the $q^{2}$ shapes of simulated continuum data and off-
resonance data, where we account for the difference in cross section between
the on- and the off-resonance data sets. We observe similar normalization
differences in the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ modes, corresponding to a ratio
between the number of events in off-resonance to simulated data of 1.2. After
correcting the simulated continuum data for the total normalization,
differences in the $q^{2}$ and $\Delta E$ spectra remain. These residual
differences are removed by correcting the simulated two-dimensional $q^{2}$
and $\Delta E$ spectra using bin-by-bin event weights, which are the ratios
between the number of off-resonance events and the number of simulated
continuum events in each bin. This approach relies on the assumption that the
difference between off-resonance data and the simulated continuum sample is
independent of $M_{\mathrm{bc}}$. We validate this assumption and verify the
reweighting procedure by obtaining reasonable p-values from $\chi^{2}$ tests
on the distributions of $\Delta E$ and $M_{\mathrm{bc}}$ in the reweighted
simulated and off-resonance data. The systematic uncertainty due to the
limited size of the off-resonance data sample is described in Section VII.

###  V.4 Selection summary

At this stage of the analysis, an average of $1.08$ ($1.18$) candidates remain
for each selected event in the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$
($B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$) mode. Some of the events are
also reconstructed in both modes. In events with multiple candidates in both
or either of the modes, we randomly select one and discard the rest. In this
way we ensure that a single event cannot contribute multiple times to either
or both of the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ modes.

We divide the simulated signal events into categories based on their true
$q^{2}$ values, using the same intervals as the reconstructed $q^{2}$ bins
described in Section IV.3. We define the signal efficiency as the fraction of
signal events generated in a given true $q^{2}$ interval that survive all
selections, regardless of whether the generated events are reconstructed in
the same $q^{2}$ interval. In simulation, these efficiencies vary from 9% to
19% in the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ mode and from 3% to 9%
in the $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ mode. We call the ratio
between the number of true and combinatorial signal events and the total
number of signal events the signal strength. Depending on the true $q^{2}$
bin, the signal strength varies between 69% and 99% in the
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ mode and between 23% and 57% in
the $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ mode. The distributions of
$\Delta E$ and $M_{\mathrm{bc}}$ integrated over the 13 (10) reconstructed
$q^{2}$ bins for $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$
($B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$) decays after all selections are
shown in Fig. 1.

Figure 1: Distributions of $\Delta E$ (left) and $M_{\mathrm{bc}}$ (right)
reconstructed in Belle II data integrated over the $q^{2}$ bins for
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ decays (top) and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ decays (bottom) with expected
distributions from simulation overlaid. The simulated samples are weighted
according to luminosity. The hatched areas include statistical and systematic
uncertainties on the simulated distributions, discussed in Section VII. The
expected signal distributions (scaled by a factor two) are also shown. The
panels below the histograms show the difference between collision and
simulated data divided by the combined uncertainty.

##  VI Signal extraction

###  VI.1 Fit method

We extract the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ signal yields by performing a
simultaneous extended maximum likelihood fit to the binned three-dimensional
distributions of $\Delta E$, $M_{\mathrm{bc}}$ and reconstructed $q^{2}$. We
divide the $\Delta E$ and $M_{\mathrm{bc}}$ distributions into 5 and 4 bins,
respectively. The corresponding bin widths vary, with smaller bins being used
in the signal-rich regions. The binning is chosen so that a clear separation
of signal and background is achieved, while ensuring a sufficient number of
events populate each bin. This results in 20 bins per reconstructed $q^{2}$
bin, and therefore a total of 20 $\times[$13
($B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$) + 10
($B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$)$]=$ 460 bins.

The background in each signal mode is treated separately and split into four
components according to the description in Section V.1. The fit parameters
include unconstrained scale factors $b_{p}^{(\pi/\rho)}$ for each mode
$(\pi/\rho)\ell\nu$ and each background component $p$. The background
templates are constructed from simulated $B\overline{B}$ data and the
reweighted continuum data. We introduce Gaussian penalty factors to constrain
the continuum yields to the scaled off-resonance yields.

In addition to the background templates, we also have one independent total
signal template for each true $q^{2}$ bin $i$ and each mode, resulting in 23
signal templates with unconstrained scale factors $s_{i}^{\pi}$ and
$s_{i}^{\rho}$. Because these templates are based on true $q^{2}$ categories,
they naturally account for resolution effects that could cause events that are
generated in one $q^{2}$ bin to be reconstructed in another. The composition
of the total signal component is described in Section V.1. In this way, the
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ background in the
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ mode is linked with the
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ signal in the
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ mode, and vice versa. The signal
templates are constructed from simulated signal events. The contribution of
true and combinatorial signal to the total signal templates, the signal
strength, is inferred from simulation. A summary of the 31 templates and scale
factors is given in Table 1.

Table 1: Summary of the templates and corresponding scale factors determined from the fit for the different background sources and signal samples. There is one signal scale factor for each true $q^{2}$ bin $i$ and each signal decay, where $i\in[1,13]$ for $s_{i}^{\pi}$ and $i\in[1,10]$ for $s_{i}^{\rho}$. All fit parameters are free, with $b^{\pi}_{\mathrm{cont}}$ and $b^{\rho}_{\mathrm{cont}}$ constrained by off-resonance data. Component | Reconstructed mode
---|---
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ | $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$
Signal: |  |
True signal | $s_{i}^{\pi}$ | $s_{i}^{\rho}$
Combinatorial signal | $s_{i}^{\pi}$ | $s_{i}^{\rho}$
Isospin-conjugate signal | $s_{i}^{\pi}$ | $s_{i}^{\rho}$
Cross-feed | $s_{i}^{\rho}$ | $s_{i}^{\pi}$
Background: |  |
$B\rightarrow X_{u}\ell\nu_{\ell}$ | $b_{X_{u}\ell\nu}^{\pi}$ | $b_{X_{u}\ell\nu}^{\rho}$
$B\rightarrow X_{c}\ell\nu_{\ell}$ | $b_{X_{c}\ell\nu}^{\pi}$ | $b_{X_{c}\ell\nu}^{\rho}$
Other $B\overline{B}$ | $b_{B\overline{B}}^{\pi}$ | $b_{B\overline{B}}^{\rho}$
Continuum | $b_{\mathrm{cont}}^{\pi}$ | $b_{\mathrm{cont}}^{\rho}$

The likelihood to be maximized is

$\mathcal{L}(\vec{S},\vec{B}\,)=\prod_{l}\mathrm{Poisson}(N_{l}|\sum_{j}S_{lj}+\sum_{k}B_{lk}),$
(12)

where $N_{l}$ is the observed number of events in bin $l$, $\vec{S}$ and
$\vec{B}$ are the vectors of signal and background templates, respectively,
$S_{lj}$ is the number of events in bin $l$ of signal fit template $j$, and
$B_{lk}$ is the number of events in bin $l$ of background fit template $k$.

###  VI.2 Fit results

The fit projections of $\Delta E$ and $M_{\mathrm{bc}}$ in each $q^{2}$ bin
are shown in Fig. 2 for the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ modes. The $\chi^{2}$ per degree
of freedom of the fit is $468.5/429=1.09$. The correlations between the
component yields are all smaller than $0.75$. The highest observed
correlations occur between the $B\rightarrow X_{c}\ell\nu_{\ell}$ and
$B\overline{B}$ background yields in the
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ mode. In the higher $q^{2}$ bins,
the signal scale factor becomes increasingly correlated to the $B\rightarrow
X_{u}\ell\nu_{\ell}$ scale factor.

Figure 2: Distributions of $\Delta E$ (top) and $M_{\mathrm{bc}}$ (bottom) in
the $q^{2}$ bins for $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ candidates reconstructed in Belle
II data with fit projections overlaid. The difference between collision and
simulated data divided by the collision data uncertainty is shown in the
panels below the histograms. The boundaries of the $q^{2}$ bins are provided
in the text above.

Using the expected number of signal events from simulation, the fitted signal
scale factors, and the signal strengths, we obtain the signal yields in each
true $q^{2}$ bin, corresponding to the number of true and combinatorial signal
events. The signal yields with statistical and systematic uncertainties are
given in Table 2. The sources of systematic uncertainty and their estimation
is described in Section VII.

Table 2: Signal yields for the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ modes in each true $q^{2}$ bin with statistical and systematic uncertainties. The boundaries of the $q^{2}$ bins are given in the text above. | Yield
---|---
$q^{2}$ bin | $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ | $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$
$q1\phantom{0}$ |  0869 $\pm$ 095 $\pm$ 139 |  0332 $\pm$ 100 $\pm$ 118
$q2\phantom{0}$ | 1406 $\pm$ 123 $\pm$ 172 |  0651 $\pm$ 114 $\pm$ 178
$q3\phantom{0}$ | 1426 $\pm$ 112 $\pm$ 124 |  0630 $\pm$ 131 $\pm$ 131
$q4\phantom{0}$ | 1714 $\pm$ 120 $\pm$ 139 | 1028 $\pm$ 147 $\pm$ 240
$q5\phantom{0}$ | 1617 $\pm$ 120 $\pm$ 113 | 1273 $\pm$ 158 $\pm$ 236
$q6\phantom{0}$ | 2167 $\pm$ 138 $\pm$ 151 | 1207 $\pm$ 164 $\pm$ 244
$q7\phantom{0}$ | 1817 $\pm$ 143 $\pm$ 172 |  0962 $\pm$ 136 $\pm$ 206
$q8\phantom{0}$ | 1921 $\pm$ 147 $\pm$ 181 | 1141 $\pm$ 118 $\pm$ 218
$q9\phantom{0}$ | 1640 $\pm$ 149 $\pm$ 174 |  0936 $\pm$ 114 $\pm$ 186
$q10$ | 1328 $\pm$ 142 $\pm$ 156 |  0821 $\pm$ 096 $\pm$ 220
$q11$ | 1472 $\pm$ 140 $\pm$ 239 |
$q12$ |  0819 $\pm$ 120 $\pm$ 211 |
$q13$ |  0295 $\pm$ 066 $\pm$ 122 |

The partial branching fraction in true $q^{2}$ bin $i$ is calculated using the
signal yield, $N_{i}$, and the corresponding signal efficiency,
$\epsilon_{i}$, from

$\displaystyle\Delta\mathcal{B}_{i}(B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell})=\frac{N_{i}(1+f_{+0})}{4\epsilon_{i}\times
N_{B\overline{B}}},$ (13a)
$\displaystyle\Delta\mathcal{B}_{i}(B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell})=\frac{N_{i}(1+f_{+0})}{4\epsilon_{i}\times
N_{B\overline{B}}}\times\frac{1}{f_{+0}},$ (13b)

where $f_{+0}=\mathcal{B}(\Upsilon(4S)\rightarrow
B^{+}{B^{-}})/\mathcal{B}(\Upsilon(4S)\rightarrow
B^{0}\overline{B^{0}})=1.065\pm 0.052$ fp0 , and $N_{B\overline{B}}$ is the
number of $B\overline{B}$ pairs. The partial branching-fraction
$\Delta\mathcal{B}$ results for $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$
and $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ decays are given in Table 3
and shown as functions of $q^{2}$ in Fig. 3. The entries in the total
correlation matrices of the partial branching fractions of
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ are presented in Tables 9 and 10
in Appendix A. The central values of the partial branching fractions and the
statistical and systematic covariance matrices combining both modes will be
made available on HEPData.

Table 3: Partial branching fractions $\Delta\mathcal{B}$ ($\times 10^{4}$) in each $q^{2}$ bin for the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ modes. The first uncertainty is statistical and the second is systematic. The boundaries of the $q^{2}$ bins are provided in the text above. |  $\Delta\mathcal{B}$ ($\times 10^{4}$)
---|---
$q^{2}$ bin | $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ | $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$
$q1\phantom{0}$ | 0.117 $\pm$ 0.013 $\pm$ 0.019 | 0.109 $\pm$ 0.033 $\pm$ 0.039
$q2\phantom{0}$ | 0.142 $\pm$ 0.013 $\pm$ 0.018 | 0.140 $\pm$ 0.025 $\pm$ 0.038
$q3\phantom{0}$ | 0.119 $\pm$ 0.009 $\pm$ 0.011 | 0.113 $\pm$ 0.024 $\pm$ 0.024
$q4\phantom{0}$ | 0.137 $\pm$ 0.010 $\pm$ 0.012 | 0.162 $\pm$ 0.023 $\pm$ 0.038
$q5\phantom{0}$ | 0.129 $\pm$ 0.010 $\pm$ 0.010 | 0.193 $\pm$ 0.024 $\pm$ 0.036
$q6\phantom{0}$ | 0.170 $\pm$ 0.011 $\pm$ 0.013 | 0.183 $\pm$ 0.025 $\pm$ 0.037
$q7\phantom{0}$ | 0.139 $\pm$ 0.011 $\pm$ 0.014 | 0.161 $\pm$ 0.023 $\pm$ 0.035
$q8\phantom{0}$ | 0.146 $\pm$ 0.011 $\pm$ 0.015 | 0.225 $\pm$ 0.023 $\pm$ 0.044
$q9\phantom{0}$ | 0.119 $\pm$ 0.011 $\pm$ 0.013 | 0.182 $\pm$ 0.022 $\pm$ 0.037
$q10$ | 0.096 $\pm$ 0.010 $\pm$ 0.012 | 0.158 $\pm$ 0.019 $\pm$ 0.043
$q11$ | 0.109 $\pm$ 0.010 $\pm$ 0.018 |
$q12$ | 0.065 $\pm$ 0.010 $\pm$ 0.017 |
$q13$ | 0.028 $\pm$ 0.006 $\pm$ 0.011 |

The total branching fractions determined from the sums of the partial
branching fractions are

$\mathcal{B}(B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell})=(1.516\pm 0.042\pm
0.059)\times 10^{-4}$

$\mathcal{B}(B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell})=(1.625\pm 0.079\pm
0.180)\times 10^{-4}$

where the first uncertainties are statistical and the second are systematic.
The results are consistent with world averages PDG , and the full experimental
correlation between the two values is $-0.16$.

We perform additional fits to test the stability of the results. We divide the
data set by lepton flavor, by lepton charge, and by $\theta_{\mathrm{miss}}$
region. We then fit separately for each subsample and check that the results
agree within statistical uncertainties.

##  VII Systematic uncertainties

The fractional uncertainties on the partial branching fractions in each
$q^{2}$ bin from various sources of systematic uncertainty are given in Tables
4 and 5. All systematic uncertainties are evaluated using the same approach.
For each source of uncertainty, we vary the templates 1000 times by sampling
from Gaussian distributions of the central values fully taking correlations
into account. For example, to evaluate the uncertainties due to the
$B\rightarrow\omega\ell\nu_{\ell}$ form factors, we sample 1000 alternative
$B\rightarrow X_{u}\ell\nu_{\ell}$ distributions by assuming the form-factor
parameter uncertainties follow Gaussian distributions. We create 1000
simplified simulated data (toy) distributions by adding the resulting
variations to the remaining nominal templates. Finally, we fit the nominal
templates to the toy distributions and obtain a covariance matrix of the
fitted yields for each source of uncertainty using Pearson correlations
Pearson . Covariance matrices for the signal strengths and efficiencies are
evaluated using a similar approach. The systematic uncertainties on the
partial branching fractions are evaluated by propagating the covariance
matrices of the fitted yields, the signal strengths and efficiencies.

Table 4: Summary of fractional uncertainties in % on the
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ partial branching fractions
$\Delta\mathcal{B}$ in each $q^{2}$ bin. The boundaries of the $q^{2}$ bins
are provided in the text above. $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$
---
Source | $q1$ | $q2$ | $q3$ | $q4$ | $q5$ | $q6$ | $q7$ | $q8$ | $q9$ | $q10$ | $q11$ | $q12$ | $q13$
Detector effects |  02.0 |  00.9 |  01.1 |  01.0 |  01.0 |  01.1 |  01.1 |  01.0 |  00.9 |  01.2 |  02.3 |  04.1 |  05.8
Beam energy |  00.6 |  00.8 |  00.7 |  00.8 |  00.7 |  00.6 |  00.6 |  00.6 |  00.5 |  00.5 |  00.5 |  00.6 |  00.7
Simulated sample size |  04.7 |  03.8 |  03.3 |  03.2 |  03.2 |  02.9 |  03.8 |  03.7 |  04.0 |  04.5 |  05.9 |  08.0 | 13.6
BDT efficiency |  01.3 |  01.3 |  01.3 |  01.3 |  01.3 |  01.3 |  01.3 |  01.3 |  01.3 |  01.3 |  01.3 |  01.3 |  01.3
Physics constraints |  02.9 |  02.9 |  02.9 |  02.9 |  02.9 |  02.9 |  02.9 |  02.9 |  02.9 |  02.9 |  02.9 |  02.9 |  02.9
Signal model |  00.1 |  00.1 |  00.2 |  00.1 |  00.0 |  00.2 |  00.2 |  00.4 |  00.3 |  00.8 |  00.9 |  00.2 |  04.9
$\rho$ lineshape |  00.1 |  00.1 |  00.3 |  00.3 |  00.2 |  00.1 |  00.3 |  00.1 |  00.3 |  00.1 |  00.2 |  00.2 |  00.6
Nonresonant $B\rightarrow\pi\pi\ell\nu_{\ell}$ |  00.5 |  00.6 |  00.4 |  00.4 |  00.5 |  01.0 |  01.2 |  01.0 |  00.8 |  01.8 |  01.2 |  02.3 | 14.3
DFN parameters |  00.8 |  00.4 |  01.5 |  01.6 |  01.4 |  01.7 |  01.2 |  00.1 |  00.7 |  01.2 |  02.9 |  03.5 |  03.7
$B\rightarrow X_{u}\ell\nu_{\ell}$ model |  00.2 |  00.4 |  00.3 |  00.4 |  00.2 |  00.9 |  01.1 |  01.2 |  01.0 |  01.3 |  01.6 |  00.7 |  08.7
$B\rightarrow X_{c}\ell\nu_{\ell}$ model |  01.4 |  02.0 |  01.7 |  01.3 |  01.3 |  01.4 |  01.8 |  01.6 |  01.3 |  01.4 |  01.1 |  00.5 |  01.7
Continuum | 15.1 | 11.3 |  07.6 |  07.1 |  05.8 |  05.7 |  08.1 |  08.3 |  09.6 | 10.4 | 14.5 | 23.8 | 34.4
Total systematic | 16.4 | 12.6 |  09.3 |  08.7 |  07.7 |  07.7 | 10.0 |  09.9 | 11.1 | 12.2 | 16.6 | 26.0 | 41.6
Statistical | 11.0 |  08.8 |  07.9 |  07.0 |  07.5 |  06.4 |  07.9 |  07.7 |  09.1 | 10.7 |  09.6 | 14.6 | 22.6
Total | 19.7 | 15.4 | 12.2 | 11.2 | 10.7 | 10.0 | 12.7 | 12.6 | 14.4 | 16.3 | 19.1 | 29.8 | 47.3
Table 5: Summary of fractional uncertainties in % on the
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ partial branching fractions
$\Delta\mathcal{B}$ in each $q^{2}$ bin. The boundaries of the $q^{2}$ bins
are provided in the text above. $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$
---
Source | $q1$ | $q2$ | $q3$ | $q4$ | $q5$ | $q6$ | $q7$ | $q8$ | $q9$ | $q10$
Detector effects |  02.8 |  02.0 |  01.6 |  01.1 |  01.7 |  01.9 |  02.4 |  01.4 |  01.4 |  01.6
Beam energy |  02.1 |  01.9 |  01.9 |  01.5 |  01.3 |  01.1 |  01.0 |  00.9 |  00.8 |  00.5
Simulated sample size | 14.1 |  07.8 |  07.4 |  06.3 |  06.3 |  05.2 |  06.4 |  05.6 |  06.2 |  07.3
BDT efficiency |  01.6 |  01.6 |  01.6 |  01.6 |  01.6 |  01.6 |  01.6 |  01.6 |  01.6 |  01.6
Physics constraints |  02.8 |  02.8 |  02.8 |  02.8 |  02.8 |  02.8 |  02.8 |  02.8 |  02.8 |  02.8
Signal model |  00.7 |  00.2 |  00.2 |  00.2 |  00.3 |  00.4 |  00.5 |  00.3 |  01.8 |  02.4
$\rho$ lineshape |  01.7 |  01.6 |  02.0 |  01.0 |  01.9 |  01.8 |  01.4 |  00.9 |  01.6 |  01.7
Nonresonant $B\rightarrow\pi\pi\ell\nu_{\ell}$ |  05.6 |  06.3 |  06.7 |  08.6 |  09.3 | 10.7 | 10.1 |  07.0 |  07.8 | 11.8
DFN parameters |  03.6 |  05.5 |  04.1 |  03.5 |  01.1 |  01.2 |  02.7 |  01.7 |  01.9 |  02.3
$B\rightarrow X_{u}\ell\nu_{\ell}$ model |  01.7 |  03.0 |  03.8 |  05.0 |  05.8 |  06.1 |  06.3 |  01.9 |  07.2 | 12.4
$B\rightarrow X_{c}\ell\nu_{\ell}$ model |  01.8 |  01.9 |  01.7 |  01.1 |  01.4 |  01.7 |  00.9 |  00.9 |  01.9 |  02.6
Continuum | 31.5 | 24.3 | 17.0 | 19.6 | 13.2 | 14.8 | 16.0 | 16.6 | 15.2 | 18.7
Total systematic | 35.6 | 27.5 | 21.0 | 23.5 | 18.8 | 20.5 | 21.6 | 19.4 | 20.2 | 27.0
Statistical | 30.0 | 17.5 | 20.8 | 14.4 | 12.4 | 13.6 | 14.1 | 10.4 | 12.2 | 11.8
Total | 46.6 | 32.6 | 29.6 | 27.6 | 22.6 | 24.6 | 25.8 | 22.0 | 23.6 | 29.5

###  VII.1 Detector and beam-energy effects

The detector uncertainties include uncertainties arising from the tracking
efficiency and the corrections to the lepton- and pion-identification
efficiencies. All of these were estimated from studies of independent data
control samples.

In addition, we observe a dependence of the reconstructed $q^{2}$ resolution
on the c.m. energy. Since the c.m. energy in the simulated sample differs from
the mean c.m. energy in data, we account for the effect on the shape of the
signal template. We investigate the effect using a control mode, in which we
fully reconstruct $B^{+}\rightarrow J/\psi[\rightarrow\mu^{+}\mu^{-}]K^{+}$
events. By removing one of the muons, we obtain events that are similar to
signal decays with a single missing neutrino.

We find a 4% difference in $q^{2}$ resolution between measured and simulated
data in this control mode. We scale the resolutions in each $q^{2}$ bin
obtained from the simulated sample, and using the true $q^{2}$ values, in
combination with Gaussian smearing according to the new resolutions, produce
1000 pseudo-reconstructed $q^{2}$ distributions. By combining these with the
remaining unaffected templates we obtain 1000 varied toy distributions, which
are then fit using the nominal templates.

###  VII.2 Simulated sample size

The effect of having limited samples of simulated data is considered. The
largest uncertainty contribution comes from the limited size of the simulated
continuum sample. In addition to shape variations due to the number of events
within each bin, we also account for migration effects in the true $q^{2}$
distribution, which results in signal-template migrations. To estimate this
uncertainty, we sample, with replacement, true $q^{2}$ values from the total
signal component 1000 times, split these into the true-$q^{2}$ templates, then
fit these templates to the sum of the nominal templates.

###  VII.3 BDT efficiency

We estimate an uncertainty to account for possible disagreements between the
signal efficiencies in experimental and simulated data of the selection on the
48 BDT output classifiers. For each BDT output classifier selection, we use
the $B^{+}\rightarrow J/\psi[\rightarrow\mu^{+}\mu^{-}]K^{+}$ control mode
discussed above to determine the ratio between the efficiency in experimental
and simulated data. The ratios are in agreement with unity within
uncertainties.

To account for these uncertainties we separately evaluate the standard
deviations of the ratios for each type of BDT and mode and assign these as
uncertainties on the signal efficiencies. We obtain uncertainties of 1.1% and
0.6% on the efficiencies of the selection on the continuum suppression BDTs in
the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ modes, respectively. For the
$B\overline{B}$ suppression BDTs the uncertainties on the efficiencies are
0.7% and 1.5% for the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ modes, respectively.

###  VII.4 Physics constraints

We consider additional systematic uncertainties from the number of
$B\overline{B}$ pairs $N_{B\overline{B}}$ and the branching fraction ratio of
$\mathchar 28935\relax{(4S)}$ $\rightarrow$ $B\overline{B}$, $f_{+0}$. These
affect the calculation of the partial branching fractions from the yields. The
uncertainty on $N_{B\overline{B}}$ results in a relative uncertainty of 1.4%,
while the uncertainty on $f_{+0}$ contributes uncertainties of 2.5% and 2.4%
to the uncertainties on the partial branching fractions for
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$, respectively.

In addition, we assign an uncertainty to the assumption of isospin symmetry.
The assumption discussed in Section V.1 relies on the $B$ lifetime ratio
$\tau_{+}/\tau_{0}=1.076\pm 0.004$ PDG . To estimate the effect on the signal
yields, we vary the relative fraction of neutral and charged $B$ modes in the
signal templates by sampling Gaussian variations of the fractions within the
relative uncertainty of 0.4%.

###  VII.5 Signal model and $\rho$ lineshape

There are three ways in which the signal form-factor and branching-fraction
uncertainties may affect our results. The first is the residual signal form-
factor model dependence of the signal templates. Since the signal is extracted
in multiple bins of true $q^{2}$, and therefore the fit is allowed to
determine the $q^{2}$ spectrum, this contribution is small but not negligible.
The second effect comes from signal form-factor and branching-fraction
uncertainties on the composition of the background $B\rightarrow
X_{u}\ell\nu_{\ell}$ template, which propagate through the fit to
uncertainties on the signal yields.

The third effect accounts for bin migrations between the true $q^{2}$ bins due
to signal form-factor uncertainties. These are reflected in uncertainties on
the signal strengths. The assigned uncertainties are smaller than the ones
originating from the dependence of the background $B\rightarrow
X_{u}\ell\nu_{\ell}$ template on the signal model, but they are larger than
the ones due to the residual signal-template dependence. The combination of
these uncertainties is included in the signal model category in Tables 4 and
5.

There is an additional uncertainty source concerning the modeling of the
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ signal related to the lineshape
of the $\rho$ meson. We account for possible $\rho$-$\omega$ interference
resulting in a change in the $\rho$ lineshape. In Ref. RhoFlorian  an
amplitude fit incorporating the interference term is performed to the di-pion
invariant mass spectrum measured in Ref. Nonres  and 1$\sigma$ variations of
the diagonalized fit-parameter uncertainties are obtained. We vary the
lineshape of all true $\rho$ mesons for each variation, repeat the fit, and
take the largest change in fit results from the model without $\rho$-$\omega$
interference as the systematic uncertainty.

###  VII.6 $B\rightarrow X_{u}\ell\nu_{\ell}$ background

####  VII.6.1 Nonresonant $B\rightarrow\pi\pi\ell\nu_{\ell}$ component

One contribution to the $B\rightarrow X_{u}\ell\nu_{\ell}$ background
uncertainties is related to the treatment of the nonresonant
$B\rightarrow\pi\pi\ell\nu_{\ell}$ component, which was described in Section
III.3. In addition to the measured partial branching-fraction spectrum, Ref.
Nonres  also provides the corresponding covariance matrix. We therefore vary
the partial branching fractions according to this covariance matrix to
determine the uncertainty on our signal yields. Overall, this results in the
largest contribution to the uncertainties originating from the $B\rightarrow
X_{u}\ell\nu_{\ell}$ background.

####  VII.6.2 DFN parameters

A second uncertainty component comes from the uncertainty in the DFN shape
function parameters that define the shape of the nonresonant $B\rightarrow
X_{u}\ell\nu_{\ell}$ background. We follow the procedure of Ref. HFLAV  to
evaluate the uncertainties on the relevant parameters provided in Ref.
DFNparam . We then generate simulated samples of nonresonant $B\rightarrow
X_{u}\ell\nu_{\ell}$ events based on these varied parameters and reweight our
nominal samples to match the varied samples. We then estimate the
uncertainties by sampling 1000 toy distributions in agreement with Gaussian
variations of these parameters. We also compare to a different nonresonant
$B\rightarrow X_{u}\ell\nu_{\ell}$ model BLNP  but refrain from adding any
additional uncertainties due to this, since the current uncertainties already
cover any difference introduced by the change in model.

####  VII.6.3 $B\rightarrow X_{u}\ell\nu_{\ell}$ model

Finally, we evaluate the uncertainties due to the
$B\rightarrow\omega\ell\nu_{\ell}$, $B\rightarrow\eta\ell\nu_{\ell}$, and
$B\rightarrow\eta^{\prime}\ell\nu_{\ell}$ form factors, and obtain
uncertainties listed in the $B\rightarrow X_{u}\ell\nu_{\ell}$ category. This
category also includes the effects of uncertainties of the exclusive and
inclusive $B\rightarrow X_{u}\ell\nu_{\ell}$ branching fractions, except for
the $B\rightarrow\pi\ell\nu_{\ell}$ and $B\rightarrow\rho\ell\nu_{\ell}$
branching fractions.

###  VII.7 $B\rightarrow X_{c}\ell\nu_{\ell}$ model

The $B\rightarrow X_{c}\ell\nu_{\ell}$ model category in Tables 4 and 5
includes the effects of the uncertainties of the $B\rightarrow
D\ell\nu_{\ell}$ and $B\rightarrow D^{*}\ell\nu_{\ell}$ form-factor
parameters, and the exclusive and inclusive $B\rightarrow X_{c}\ell\nu_{\ell}$
branching fractions. For the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ mode
this contribution to the total systematic uncertainty is larger than that of
the $B\rightarrow X_{u}\ell\nu_{\ell}$ model at low $q^{2}$, but smaller at
high $q^{2}$. It is subdominant over the entire $q^{2}$ range for the
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ mode.

###  VII.8 Continuum reweighting

The limited off-resonance sample size affects the continuum weights obtained
during the reweighting procedure, since the weights are calculated using the
number of off-resonance events within each bin. To account for the
uncertainties of these numbers, we produce a set of 1000 continuum weights by
recalculating the weights for each bin using off-resonance event numbers drawn
from the corresponding Poisson distributions. We then proceed with the usual
procedure for determining systematic uncertainties described above. The
resulting uncertainties dominate both in the
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ mode, and are especially large in
those $q^{2}$ bins where the continuum background component is largest.

(a)

(b)

(c)

Figure 3: Measured partial branching fractions as a function of $q^{2}$ for
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ (a,b) and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ (c). The fitted differential
rates are shown together with the one, two, and three standard-deviation
uncertainty bands for fits using constraints on the form factors from (a)
LQCD, (b) LQCD and LCSR, and (c) LCSR predictions.

##  VIII $|V_{ub}|$ determination

We extract $|V_{ub}|$ separately from
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ using $\chi^{2}$ fits to the
measured $q^{2}$ spectra. The $\chi^{2}$ is defined as

$\displaystyle\chi^{2}=\sum_{i,j=1}^{N}(\Delta\mathcal{B}_{i}-\Delta\Gamma_{i}\tau)$
$\displaystyle C^{-1}_{ij}(\Delta\mathcal{B}_{j}-\Delta\Gamma_{j}\tau)$
$\displaystyle+\sum_{m}\chi^{2}_{\mathrm{Theory},m},$ (14)

where $C^{-1}_{ij}$ is the inverse total covariance matrix of the measured
partial branching fractions $\Delta\mathcal{B}_{i}$ in bin $i$, and $N$ is the
number of $q^{2}$ bins. The quantities $\Delta\Gamma_{i}$ contain the
predictions for the differential decay rates in bin $i$, $\tau$ is the $B$
lifetime, and $\chi^{2}_{\mathrm{Theory},m}$ incorporates the constraints from
theory calculation $m$. The predictions for the differential decay rates
provided in Equations 2 and 3 include the form factors and $|V_{ub}|$.

For $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ we include LQCD constraints on
three $f_{+}(q^{2})$ BCL form-factor coefficients $b^{+}_{k}$ and two
$f_{0}(q^{2})$ BCL form-factor coefficients $b^{0}_{k}$ given in Equations 5
and 6, respectively, as nuisance parameters. These nuisance parameters
constrain the shape and normalization of the relevant form factors entering
the differential decay rate and allow for a determination of $|V_{ub}|$. In
the evaluation of the inverse Blaschke factors for the expansion of
$f_{+}(q^{2})$ in Equation 5, $m_{R}$ takes the value of 5.325 GeV BCL . The
$\chi^{2}_{\mathrm{LQCD}}$ term takes the form:

$\displaystyle\chi^{2}_{\mathrm{LQCD}}=\sum_{k,l=1}^{5}(b_{k}-b^{\mathrm{LQCD}}_{k})$
$\displaystyle C^{-1}_{\mathrm{LQCD},kl}(b_{l}-b^{\mathrm{LQCD}}_{l}),$ (15)

where the constraints on the form-factor coefficients $b^{\mathrm{LQCD}}_{k}$
and the corresponding inverse covariance matrix $C^{-1}_{\mathrm{LQCD},kl}$
are taken from the latest version of the FLAG 21 review (February 2023) FLAG21
, and combine results from the FNAL/MILC FermilabMILC , RBC/UKQCD RBC_UKQCD ,
and JLQCD JLQCD  collaborations.

In addition to the LQCD constraints, we may also add LCSR constraints from
Ref. LCSR_pi , which determine $f_{+}(q^{2})$ and $f_{0}(q^{2})$ at five
points in $q^{2}$. In this case the additional $\chi^{2}_{\mathrm{LCSR}}$ term
takes the form:

$\displaystyle\chi^{2}_{\mathrm{LCSR}}=\sum_{k,l=1}^{10}(f_{k}-f^{\mathrm{LCSR}}_{k})$
$\displaystyle C^{-1}_{\mathrm{LCSR},kl}(f_{l}-f^{\mathrm{LCSR}}_{l}).$ (16)

Here we implement direct constraints on the form factors $f_{k}$
($f_{+}(q^{2})$ and $f_{0}(q^{2})$) from LCSR predictions
$f^{\mathrm{LCSR}}_{k}$, taking the corresponding inverse covariance matrix
$C^{-1}_{\mathrm{LCSR},kl}$ into account.

The result for $|V_{ub}|$ from the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$
mode using only the LQCD constraints is:

$|V_{ub}|_{B\rightarrow\pi\ell\nu_{\ell}}=(3.93\pm 0.09\pm 0.13\pm 0.19)\times
10^{-3}$,

where for all quoted $|V_{ub}|$ results the first uncertainty is statistical,
the second is systematic and the third is theoretical. Upon adding the LCSR
constraints, the result for $|V_{ub}|$ from
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ becomes:

$|V_{ub}|_{B\rightarrow\pi\ell\nu_{\ell}}=(3.73\pm 0.07\pm 0.07\pm 0.16)\times
10^{-3}$.

The measured central values of $|V_{ub}|$ and the BCL form-factor coefficients
from the fits to the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ spectrum are
provided in Table 6. The full correlation matrices corresponding to these
values are provided in Tables 11 and 12 in Appendix A.

Table 6: Measured central values of $|V_{ub}|$ and the BCL form-factor
coefficients with total uncertainties from the fits to the
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ spectrum.
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$
---
|  | LQCD | LQCD + LCSR
$|V_{ub}|$ ($10^{-3}$) |  $-$3.93 $\pm$ 0.25 |  $-$3.73 $\pm$ 0.19
$f_{+}(q^{2})$ | $b_{0}^{+}$ |  $-$0.42 $\pm$ 0.02 |  $-$0.45 $\pm$ 0.02
$b_{1}^{+}$ |  $-$0.52 $\pm$ 0.05 |  $-$0.52 $\pm$ 0.05
$b_{2}^{+}$ |  $-$0.81 $\pm$ 0.21 |  $-$1.02 $\pm$ 0.18
$f_{0}(q^{2})$ | $b_{0}^{0}$ |  $-$0.02 $\pm$ 0.25 |  $-$0.59 $\pm$ 0.02
$b_{1}^{0}$ |  $-$1.43 $\pm$ 0.08 |  $-$1.39 $\pm$ 0.07
$\chi^{2}$/ndf | 8.39/7 | 8.36/7

For $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ we include LCSR constraints
on six BSZ coefficients $b^{i}_{k}$ given in Equation 7 from Ref. BSZ . These
correspond to constraints on two coefficients each for $A_{1}(q^{2})$,
$A_{2}(q^{2})$, and $V(q^{2})$. The $\chi^{2}_{\mathrm{LCSR}}$ term for the
fit to the measured $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ $q^{2}$
spectrum takes the form:

$\displaystyle\chi^{2}_{\mathrm{LCSR}}=\sum_{k,l=1}^{6}(b_{k}-b^{\mathrm{LCSR}}_{k})$
$\displaystyle C^{-1}_{\mathrm{LCSR},kl}(b_{l}-b^{\mathrm{LCSR}}_{l}),$ (17)

where $b^{\mathrm{LCSR}}_{k}$ are the constraints on the coefficients and
$C^{-1}_{\mathrm{LCSR},kl}$ is the corresponding inverse covariance matrix
predicted by LCSR calculations. In the evaluation of the inverse Blaschke
factors for the expansion of $A_{1}(q^{2})$ and $A_{2}(q^{2})$ in Equation 7,
$m_{R}$ takes the value of 5.724 GeV, while it is 5.325 GeV for the expansion
of $V(q^{2})$ BSZ . The $|V_{ub}|$ result obtained from
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ using LCSR constraints is:

$|V_{ub}|_{B\rightarrow\rho\ell\nu_{\ell}}=(3.19\pm 0.12\pm 0.17\pm
0.26)\times 10^{-3}$.

The measured central values of $|V_{ub}|$ and the BSZ form-factor coefficients
from the fit to the $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ spectrum are
provided in Table 7. The full correlation matrix corresponding to these values
is provided in Table 13 in Appendix A. Fig. 3 shows the measured and fitted
differential rates of $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$, as well as the one, two, and
three standard-deviation uncertainty bands from the fits.

Table 7: Measured central values of $|V_{ub}|$ and the BSZ form-factor
coefficients with total uncertainties from the fit to the
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ spectrum.
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$
---
|  | LCSR
$|V_{ub}|$ ($10^{-3}$) |  $-$3.19 $\pm$ 0.33
$A_{1}(q^{2})$ | $b_{0}^{A_{1}}$ |  $-$0.27 $\pm$ 0.03
$b_{1}^{A_{1}}$ |  $-$0.34 $\pm$ 0.13
$A_{2}(q^{2})$ | $b_{0}^{A_{2}}$ |  $-$0.29 $\pm$ 0.03
$b_{1}^{A_{2}}$ |  $-$0.66 $\pm$ 0.17
$V(q^{2})$ | $b_{0}^{V}$ |  $-$0.33 $\pm$ 0.03
$b_{1}^{V}$ |  $-$0.93 $\pm$ 0.17
$\chi^{2}$/ndf | 3.85/3

The $|V_{ub}|$ results obtained from
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ are consistent with previous
exclusive measurements HFLAV . The result obtained from
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ is lower, but consistent with
previous $|V_{ub}|$ determinations from $B\rightarrow\rho\ell\nu_{\ell}$
decays rho_om_FF . The $\chi^{2}$ per degree of freedom for the fits vary from
1.19 to 1.28, and are provided in Tables 6 and 7 for
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$, respectively. The extracted
central values of $|V_{ub}|$ and the coefficients, with the corresponding full
covariance matrices, for the fits to the
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ spectra will be provided on
HEPData. We confirm the stability of the $|V_{ub}|$ results by repeating the
fits using different $q^{2}$ cut-off values. The results are presented in Fig.
4 in Appendix B.

The fractional uncertainties on the $|V_{ub}|$ results from various sources of
systematic uncertainty are shown in Table 8. For both
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ the largest contribution to the
systematic uncertainty comes from the limited off-resonance data sample. In
addition, for $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ the systematic
uncertainty from nonresonant $B\rightarrow\pi\pi\ell\nu_{\ell}$ is
significant.

Table 8: Summary of fractional uncertainties in % on the extracted $|V_{ub}|$ values. | $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ | $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$
---|---|---
| LQCD | LQCD | LCSR
| \+ LCSR
Detector effects | 0.64 | 0.24 |  00.44
Beam energy | 0.05 | 0.03 |  00.09
Simulated sample size | 1.51 | 0.78 |  01.41
BDT efficiency | 0.31 | 0.21 |  00.28
Physics constraints | 0.61 | 0.43 |  00.88
Signal model | 0.38 | 0.13 |  00.41
$\rho$ lineshape | 0.26 | 0.21 |  00.13
Nonres. $B\rightarrow\pi\pi\ell\nu_{\ell}$ | 0.43 | 0.11 |  01.97
DFN parameters | 0.64 | 0.32 |  00.88
$B\rightarrow X_{u}\ell\nu_{\ell}$ model | 0.61 | 0.40 |  01.56
$B\rightarrow X_{c}\ell\nu_{\ell}$ model | 0.51 | 0.43 |  00.50
Continuum | 2.39 | 1.37 |  04.91
Total systematic | 3.26 | 1.91 |  05.33
Statistical | 2.31 | 1.82 |  03.76
Theory | 4.83 | 4.29 |  08.15
Total | 6.40 | 5.13 | 10.34

##  IX Summary

We extract partial branching fractions of
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ decays reconstructed in a 364
fb-1 electron-positron collision data sample collected by the Belle II
experiment. The branching-fraction of
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ is found to be $(1.516\pm
0.042(\mathrm{stat})\pm 0.059(\mathrm{syst}))\times 10^{-4}$, and the
branching fraction of $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ is found to
be $(1.625\pm 0.079(\mathrm{stat})\pm 0.180(\mathrm{syst}))\times 10^{-4}$.
These results are consistent with the current world averages PDG .

We extract values of the CKM matrix-element magnitude $|V_{ub}|$ from
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ decays using LQCD constraints
provided by the FLAG 21 review (updated February 2023) FLAG21 . We obtain
$(3.93\pm 0.09(\mathrm{stat})\pm 0.13(\mathrm{syst})\pm
0.19(\mathrm{theo}))\times 10^{-3}$. Using additional constraints from LCSR
LCSR_pi  this result becomes $(3.73\pm 0.07(\mathrm{stat})\pm
0.07(\mathrm{syst})\pm 0.16(\mathrm{theo}))\times 10^{-3}$. The $|V_{ub}|$
result obtained from the measurement of
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ with constraints from LCSR BSZ
is $(3.19\pm 0.12(\mathrm{stat})\pm 0.18(\mathrm{syst})\pm
0.26(\mathrm{theo}))\times 10^{-3}$.

Currently our results are limited by the size of the off-resonance data set
and the description of the nonresonant $B\rightarrow X_{u}\ell\nu_{\ell}$
background. The first uncertainty could be reduced by improvements in the
simulation of continuum backgrounds.

This work, based on data collected using the Belle II detector, which was
built and commissioned prior to March 2019, 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 34529, No. J 4731, No. J 4625, and No. M 3153, 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 and Charles University Grant Agency project No. 246122; 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; Agencia Estatal de Investigacion, Spain Grant No.
RYC2020-029875-I and Generalitat Valenciana, Spain Grant No.
CIDEGENT/2018/020; The Knut and Alice Wallenberg Foundation (Sweden),
Contracts No. 2021.0174 and No. 2021.0299; 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
and IBBelle on site; 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, and the University of Victoria.

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##  Appendix A Correlation matrices

Tables 9 and 10 show the full experimental covariance matrices for the
measurements of the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ partial branching fractions
$\Delta\mathcal{B}$, respectively. In addition, Tables 11 and 12 contain the
full correlation matrices of the measurements of $|V_{ub}|$ and the form-
factor coefficients from $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ using
constraints from LQCD or from LQCD in combination with LCSR. Table 13 gives
the full correlation matrix of the $|V_{ub}|$ and form-factor coefficient
measurement from $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ using LCSR
constraints.

Table 9: Total correlation matrix of the partial branching fractions
$\Delta\mathcal{B}_{i}$

for $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$.  $q^{2}$ bin $q1$ $q2$ $q3$
$q4$ $q5$ $q6$ $q7$ $q8$ $q9$ $q10$ $q11$ $q12$ $q13$ $q1$ $-$1.000 $q2$
$-$0.021 $-$1.000 $q3$ $-$0.105 $-$0.193 $-$1.000 $q4$ $-$0.018 $-$0.019
$-$0.139 $-$1.000 $q5$ $-$0.031 $-$0.052 $-$0.202 $-$0.053 $-$1.000 $q6$
$-$0.065 $-$0.058 $-$0.034 $-$0.097 $-$0.004 $-$1.000 $q7$ $-$0.097 $-$0.160
$-$0.069 $-$0.226 $-$0.223 $-$0.090 $-$1.000 $q8$ $-$0.067 $-$0.097 $-$0.026
$-$0.026 $-$0.194 $-$0.255 $-$0.213 $-$1.000 $q9$ $-$0.088 $-$0.035 $-$0.019
$-$0.027 $-$0.053 $-$0.170 $-$0.108 $-$0.110 $-$1.000 $q10$ $-$0.007 $-$0.007
$-$0.001 $-$0.053 $-$0.067 $-$0.100 $-$0.050 $-$0.058 $-$0.196 $-$1.000 $q11$
$-$0.075 $-$0.001 $-$0.059 $-$0.005 $-$0.021 $-$0.056 $-$0.028 $-$0.035
$-$0.148 $-$0.236 $-$1.000 $q12$ $-$0.050 $-$0.080 $-$0.014 $-$0.004 $-$0.035
$-$0.044 $-$0.038 $-$0.101 $-$0.074 $-$0.187 $-$0.297 $-$1.000 $q13$ $-$0.030
$-$0.053 $-$0.115 $-$0.024 $-$0.041 $-$0.048 $-$0.011 $-$0.078 $-$0.092
$-$0.129 $-$0.212 $-$0.355 $-$1.000

Table 10: Total correlation matrix of the partial branching fractions $\Delta\mathcal{B}_{i}$ for $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$. $q^{2}$ bin | $q1$ | $q2$ | $q3$ | $q4$ | $q5$ | $q6$ | $q7$ | $q8$ | $q9$ | $q10$
---|---|---|---|---|---|---|---|---|---|---
$q1$ |  $-$1.000 |  |  |  |  |  |  |  |  |
$q2$ |  $-$0.340 |  $-$1.000 |  |  |  |  |  |  |  |
$q3$ |  $-$0.146 |  $-$0.322 |  $-$1.000 |  |  |  |  |  |  |
$q4$ |  $-$0.023 |  $-$0.241 |  $-$0.241 |  $-$1.000 |  |  |  |  |  |
$q5$ |  $-$0.052 |  $-$0.131 |  $-$0.275 |  $-$0.060 |  $-$1.000 |  |  |  |  |
$q6$ |  $-$0.017 |  $-$0.139 |  $-$0.183 |  $-$0.464 |  $-$0.148 |  $-$1.000 |  |  |  |
$q7$ |  $-$0.021 |  $-$0.197 |  $-$0.068 |  $-$0.184 |  $-$0.428 |  $-$0.030 |  $-$1.000 |  |  |
$q8$ |  $-$0.149 |  $-$0.018 |  $-$0.054 |  $-$0.216 |  $-$0.205 |  $-$0.311 |  $-$0.063 |  $-$1.000 |  |
$q9$ |  $-$0.095 |  $-$0.101 |  $-$0.050 |  $-$0.115 |  $-$0.136 |  $-$0.156 |  $-$0.235 |  $-$0.005 |  $-$1.000 |
$q10$ |  $-$0.004 |  $-$0.187 |  $-$0.083 |  $-$0.153 |  $-$0.151 |  $-$0.133 |  $-$0.188 |  $-$0.341 |  $-$0.241 |  $-$1.000
Table 11: Full correlation matrix of $|V_{ub}|$ and the BCL form-factor coefficients from the fit to the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ spectrum with LQCD constraints. | $|V_{ub}|$ | $b_{0}^{+}$ | $b_{1}^{+}$ | $b_{2}^{+}$ | $b_{0}^{0}$ | $b_{1}^{0}$
---|---|---|---|---|---|---
$|V_{ub}|$ |  $-$1.000 |  |  |  |  |
$b_{0}^{+}$ |  $-$0.806 |  $-$1.000 |  |  |  |
$b_{1}^{+}$ |  $-$0.053 |  $-$0.273 |  $-$1.000 |  |  |
$b_{2}^{+}$ |  $-$0.062 |  $-$0.319 |  $-$0.338 |  $-$1.000 |  |
$b_{0}^{0}$ |  $-$0.315 |  $-$0.409 |  $-$0.073 |  $-$0.204 |  $-$1.000 |
$b_{1}^{0}$ |  $-$0.142 |  $-$0.048 |  $-$0.150 |  $-$0.258 |  $-$0.775 |  $-$1.000
Table 12: Full correlation matrix of $|V_{ub}|$ and the BCL form-factor coefficients from the fit to the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ spectrum with LQCD and LCSR constraints. | $|V_{ub}|$ | $b_{0}^{+}$ | $b_{1}^{+}$ | $b_{2}^{+}$ | $b_{0}^{0}$ | $b_{1}^{0}$
---|---|---|---|---|---|---
$|V_{ub}|$ |  $-$1.000 |  |  |  |  |
$b_{0}^{+}$ |  $-$0.791 |  $-$1.000 |  |  |  |
$b_{1}^{+}$ |  $-$0.007 |  $-$0.339 |  $-$1.000 |  |  |
$b_{2}^{+}$ |  $-$0.243 |  $-$0.375 |  $-$0.448 |  $-$1.000 |  |
$b_{0}^{0}$ |  $-$0.376 |  $-$0.430 |  $-$0.065 |  $-$0.190 |  $-$1.000 |
$b_{1}^{0}$ |  $-$0.003 |  $-$0.164 |  $-$0.127 |  $-$0.244 |  $-$0.830 |  $-$1.000
Table 13: Full correlation matrix of $|V_{ub}|$ and the BSZ form-factor coefficients from the fit to the $B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ spectrum with LCSR constraints. | $|V_{ub}|$ | $b_{0}^{A_{1}}$ | $b_{1}^{A_{1}}$ | $b_{0}^{A_{2}}$ | $b_{1}^{A_{2}}$ | $b_{0}^{V}$ | $b_{1}^{V}$
---|---|---|---|---|---|---|---
$|V_{ub}|$ |  $-$1.000 |  |  |  |  |  |
$b_{0}^{A_{1}}$ |  $-$0.464 |  $-$1.000 |  |  |  |  |
$b_{1}^{A_{1}}$ |  $-$0.035 |  $-$0.542 |  $-$1.000 |  |  |  |
$b_{0}^{A_{2}}$ |  $-$0.735 |  $-$0.241 |  $-$0.117 |  $-$1.000 |  |  |
$b_{1}^{A_{2}}$ |  $-$0.126 |  $-$0.007 |  $-$0.023 |  $-$0.472 |  $-$1.000 |  |
$b_{0}^{V}$ |  $-$0.473 |  $-$0.894 |  $-$0.493 |  $-$0.255 |  $-$0.056 |  $-$1.000 |
$b_{1}^{V}$ |  $-$0.064 |  $-$0.538 |  $-$0.946 |  $-$0.144 |  $-$0.127 |  $-$0.558 |  $-$1.000

##  Appendix B $|V_{ub}|$ stability test

Fig. 4 presents the values of $|V_{ub}|$ extracted when different $q^{2}$ cut-
off values are used during the $\chi^{2}$ fits.

(a)

(b)

(c)

Figure 4: Measured $|V_{ub}|$ values from
$B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell}$ (a,b) and
$B^{+}\rightarrow\rho^{0}\ell^{+}\nu_{\ell}$ (c) for different cut-off values
of $q^{2}$ in the fit. In (a), where only LQCD constraints are used, minimum
cut-off values are tested, while in (b,c), where LCSR constraints are used,
maximum cut-off values are tested. We also show a comparison to the world-
average $|V_{ub}|$ values from $B\rightarrow\pi\ell\nu_{\ell}$ and inclusive
$B\rightarrow X_{u}\ell\nu_{\ell}$ from Ref. HFLAV . In (c) we further compare
to the $|V_{ub}|$ result obtained in Ref. rho_om_FF  from
$B\rightarrow\rho\ell\nu_{\ell}$ measurements.