The PQCD approach towards to next-to-leading order: A short review
Shan Cheng1(), Zhen-Jun Xiao2()
1. 1School of Physics and Electronics, Hunan University, Changsha 410082, China 2. 2Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing 210023, China
In this short review we elaborate the significance of resummation in kT factorization theorem, and summarize the recent progresses in the calculations of the next-to-leading order contributions to B meson decays from the perturbative QCD (PQCD) approach. We also comment on the current status of the PQCD approach and highlight some key issues to develop it in the near future for more phenomenological applications.
. [J]. Frontiers of Physics, 2021, 16(2): 24201.
Shan Cheng, Zhen-Jun Xiao. The PQCD approach towards to next-to-leading order: A short review. Front. Phys. , 2021, 16(2): 24201.
M. Bauer, B. Stech, and M. Wirbel, Exclusive nonleptonic decays of D, D(s), and Bmesons, Z. Phys. C 34, 103 (1987) https://doi.org/10.1007/BF01561122
4
A. Ali, G. Kramer, and C. D. Lü, Experimental tests of factorization in charmless nonleptonic two-body Bdecays, Phys. Rev. D 58, 094009 (1998) https://doi.org/10.1103/PhysRevD.58.094009
5
A. Ali, G. Kramer, and C. D. Lü, CP violating asymmetries in charmless nonleptonic decays B→ PP, PV, VVin the factorization approach, Phys. Rev. D 59, 014005 (1999) https://doi.org/10.1103/PhysRevD.59.014005
6
Y. H. Chen, H. Y. Cheng, B. Tseng, and K. C. Yang, Charmless hadronic two-body decays of Buand Bdmesons, Phys. Rev. D 60, 094014 (1999) https://doi.org/10.1103/PhysRevD.60.094014
7
M. J. Savage and M. B. Wise, SU(3) predictions for nonleptonic Bmeson decays, Phys. Rev. D 39, 3346 (1989), Erratum: [ Phys. Rev. D 40, 3127 (1989)] https://doi.org/10.1103/PhysRevD.39.3346
8
L. L. Chau, H. Y. Cheng, W. K. Sze, H. Yao, and B. Tseng, Charmless nonleptonic rare decays of Bmesons, Phys. Rev. D 43, 2176 (1991); Erratum: Phys. Rev. D 58, 019902 (1998) https://doi.org/10.1103/PhysRevD.58.019902
A. Khodjamirian, T. Mannel, and P. Urban, Gluonic penguins in B→ ππfrom QCD light cone sum rules, Phys. Rev. D 67, 054027 (2003) https://doi.org/10.1103/PhysRevD.67.054027
12
A. Khodjamirian, T. Mannel, and B. Melic, QCD light cone sum rule estimate of charming penguin contributions in B→ ππ, Phys. Lett. B 571, 75 (2003); Erratum: Phys. Lett. B 572, 171 (2003) https://doi.org/10.1016/j.physletb.2003.08.021
13
A. Khodjamirian, T. Mannel, M. Melcher, and B. Melic, Annihilation effects in B→ ππfrom QCD light-cone sum rules, Phys. Rev. D 72, 094012 (2005) https://doi.org/10.1103/PhysRevD.72.094012
14
M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, QCD factorization for B→ ππdecays: Strong phases and CP violation in the heavy quark limit, Phys. Rev. Lett. 83, 1914 (1999) https://doi.org/10.1103/PhysRevLett.83.1914
15
M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, QCD factorization for exclusive, nonleptonic Bmeson decays: General arguments and the case of heavy light final states, Nucl. Phys. B 591, 313 (2000) https://doi.org/10.1016/S0550-3213(00)00559-9
16
M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, QCD factorization in B→ πK, ππdecays and extraction of Wolfenstein parameters, Nucl. Phys. B 606, 245 (2001) https://doi.org/10.1016/S0550-3213(01)00251-6
17
M. Beneke and T. Feldmann, Symmetry breaking corrections to heavy to light Bmeson form-factors at large recoil, Nucl. Phys. B 592, 3 (2001) https://doi.org/10.1016/S0550-3213(00)00585-X
18
C. W. Bauer, S. Fleming, D. Pirjol, and I. W. Stewart, An effective field theory for collinear and soft gluons: Heavy to light decays, Phys. Rev. D 63, 114020 (2001) https://doi.org/10.1103/PhysRevD.63.114020
C. W. Bauer, D. Pirjol, and I. W. Stewart, Factorization and endpoint singularities in heavy to light decays, Phys. Rev. D 67, 071502 (2003) https://doi.org/10.1103/PhysRevD.67.071502
Y. Y. Keum, H. N. Li, and A. I. Sanda, Penguin enhancement and B→ Kπdecays in perturbative QCD, Phys. Rev. D 63, 054008 (2001) https://doi.org/10.1103/PhysRevD.63.054008
24
C. D. Lü, K. Ukai, and M. Z. Yang, Branching ratio and CP violation of B→ ππdecays in perturbative QCD approach, Phys. Rev. D 63, 074009 (2001) https://doi.org/10.1103/PhysRevD.63.074009
25
Y.Y. Keum and H. N. Li, Nonleptonic charmless B decays: Factorization versus perturbative QCD, Phys. Rev. D 63, 074006 (2001) https://doi.org/10.1103/PhysRevD.63.074006
26
T. Becher, A. Broggio, and A. Ferroglia, Introduction to soft-collinear effective theory, Lect. Notes Phys. 896, pp 1–206 (2015), arXiv: 1410.1892 [hep-ph] https://doi.org/10.1007/978-3-319-14848-9_1
A. Khodjamirian, T. Mannel, and N. Offen, Form-factors from light-cone sum rules with B-meson distribution amplitudes, Phys. Rev. D 75, 054013 (2007) https://doi.org/10.1103/PhysRevD.75.054013
29
N. Gubernari, A. Kokulu, and D. van Dyk, B→ Pand B→ Vform factors from B-meson light-cone sum rules beyond leading twist, J. High Energy Phys. 1901, 150 (2019) https://doi.org/10.1007/JHEP01(2019)150
C. D. Lü, Y. L. Shen, Y. M. Wang, and Y. B. Wei, QCD calculations of B→ π, K form factors with higher-twist corrections, J. High Energy Phys. 1901, 024 (2019) https://doi.org/10.1007/JHEP01(2019)024
G. Bell, M. Beneke, T. Huber, and X. Q. Li, Two-loop current–current operator contribution to the non-leptonic QCD penguin amplitude, Phys. Lett. B 750, 348–355 (2015) https://doi.org/10.1016/j.physletb.2015.09.037
35
G. Bell, M. Beneke, T. Huber, and X. Q. Li, Non-leptonic B-decays at two loops in QCD factorization, PoS RADCOR2019, 032 (2019), arXiv: 1912.08429 [hep-ph]
36
G. Bell, M. Beneke, T. Huber, and X. Q. Li, Two-loop non-leptonic penguin amplitude in QCD factorization, J. High Energy Phys. 04, 055 (2020) https://doi.org/10.1007/JHEP04(2020)055
37
T. Kurimoto, H. N. Li, and A. I. Sanda, Leading power contributions to B → π, ρtransition form-factors, Phys. Rev. D 65, 014007 (2002) https://doi.org/10.1103/PhysRevD.65.014007
38
C. D. Lü and M. Z. Yang, B to light meson transition form-factors calculated in perturbative QCD approach, Eur. Phys. J. C 28, 515 (2003) https://doi.org/10.1140/epjc/s2003-01199-y
39
H. N. Li, Y. L. Shen, and Y. M. Wang, Next-to-leadingorder corrections to B→ πform factors in kTfactorization, Phys. Rev. D 85, 074004 (2012) https://doi.org/10.1103/PhysRevD.85.074004
40
S. Cheng, Y. Y. Fan, X. Yu, C. D. Lü, and Z. J. Xiao, The NLO twist-3 contributions to B → π form factors in kTfactorization, Phys. Rev. D 89, 094004 (2014) https://doi.org/10.1103/PhysRevD.89.094004
41
E. Dalgic, A. Gray, M. Wingate, C. T. H. Davies, G. P. Lepage, and J. Shigemitsu, B meson semileptonic form-factors from unquenched lattice QCD, Phys. Rev. D 73, 074502 (2006); Erratum: Phys. Rev. D 75, 119906 (2007) https://doi.org/10.1103/PhysRevD.75.119906
42
R. R. Horgan, Z. Liu, S. Meinel, and M. Wingate, Lattice QCD calculation of form factors describing the rare decays B → K∗ℓ+ℓ− and Bs → ϕℓ+ℓ−, Phys. Rev. D 89, 9–094501 (2014) https://doi.org/10.1103/PhysRevD.89.094501
43
J. M. Flynn, T. Izubuchi, T. Kawanai, C. Lehner, A. Soni, R. S. Van de Water, and O. Witzel, B→ πℓν and Bs → Kℓν form factors and |Vub| from 2+1-flavor lattice QCD with domain-wall light quarks and relativistic heavy quarks, Phys. Rev. D 91(7) 074510 (2015) https://doi.org/10.1103/PhysRevD.91.074510
44
J. A. Bailey, et al. [Fermilab Lattice and MILC], |Vub| from B → πℓν decays and (2+1)-flavor lattice QCD, Phys. Rev. D 92(1) 014024 (2015) https://doi.org/10.1103/PhysRevD.92.014024
45
J. A. Bailey, et al. [MILC], B → Dlν form factors at nonzero recoil and |Vcb| from 2+1-flavor lattice QCD, Phys. Rev. D 92(3) 034506 (2015) https://doi.org/10.1103/PhysRevD.92.034506
46
T. Kaneko, Lattice QCD: Hadron spectroscopy and flavor physics, arXiv: 1807.04134 [hep-lat]
V. V. Sudakov, Vertex parts at very high-energies in quantum electrodynamics, Sov. Phys. JETP 3, 65 (1956), [Zh. Eksp. Teor. Fiz. 30, 87 (1956)]
49
J. C. Collins, In perturbative quantum chromodynamics, Ed. A. H. Mueller, World Scientific, Singapore, 1989
50
G. P. Lepage and S. J. Brodsky, Exclusive processes in perturbative quantum chromodynamics, Phys. Rev. D 22, 2157 (1980) https://doi.org/10.1103/PhysRevD.22.2157
51
P. V. Landshoff and D. J. Pritchard, Higher order QCD corrections to exclusive processes: The multiple scattering mechanism, Z. Phys. C 6, 69 (1980) https://doi.org/10.1007/BF01427923
A. Sen, Asymptotic behavior of the wide angle on-shell quark scattering amplitudes in nonabelian gauge theories, Phys. Rev. D 28, 860 (1983) https://doi.org/10.1103/PhysRevD.28.860
J. C. Collins, D. E. Soper, and G. F. Sterman, Factorization for one loop corrections in the Drell–Yan process, Nucl. Phys. B 223, 381 (1983) https://doi.org/10.1016/0550-3213(83)90062-7
T. Huang, Q. X. Shen, and P. Kroll, A study of the applicability of perturbative QCD to the pion form factor, Z. Phys. C 50, 139 (1991) https://doi.org/10.1007/BF01558568
A. V. Efremov and A. V. Radyushkin, Factorization and asymptotical behavior of pion form-factor in QCD, Phys. Lett. B 94, 245 (1980) https://doi.org/10.1016/0370-2693(80)90869-2
66
G. P. Lepage and S. J. Brodsky, Exclusive processes in quantum chromodynamics: the form-factors of baryons at large momentum transfer, Phys. Rev. Lett. 43, 545 (1979); Erratum: Phys. Rev. Lett. 43, 1625 (1979) https://doi.org/10.1103/PhysRevLett.43.1625.2
67
R. Akhoury, G. F. Sterman, and Y. P. Yao, Exclusive semileptonic decays of Bmesons into light mesons, Phys. Rev. D 50, 358 (1994) https://doi.org/10.1103/PhysRevD.50.358
Z. J. Xiao, Y. Y. Fan, W. F. Wang, and S. Cheng, The semileptonic decays of B/Bs meson in the perturbative QCD approach: A short review, Chin. Sci. Bull. 59, 3787 (2014) https://doi.org/10.1007/s11434-014-0320-8
71
M. S. Alam, et al. [CLEO Collaboration], Exclusive hadronic B decays to charm and charmonium final states, Phys. Rev. D 50, 43 (1994)
T. W. Yeh and H. N. Li, Factorization theorems, effective field theory, and nonleptonic heavy meson decays, Phys. Rev. D 56, 1615 (1997) https://doi.org/10.1103/PhysRevD.56.1615
H. Y. Cheng, H. N. Li, and K. C. Yang, Gauge invariant and infrared finite theory of nonleptonic heavy meson decays, Phys. Rev. D 60, 094005 (1999) https://doi.org/10.1103/PhysRevD.60.094005
78
Y. M. Wang and Y. L. Shen, Subleading power corrections to the pion-photon transition form factor in QCD, J. High Energy Phys. 12, 037 (2017) https://doi.org/10.1007/JHEP12(2017)037
79
J. Gao, C. D. Lü, Y. L. Shen, Y. M. Wang, and Y. B. Wei, Precision calculations of B → V form factors from softcollinear effective theory sum rules on the light-cone, Phys. Rev. D 101, 074035 (2020) https://doi.org/10.1103/PhysRevD.101.074035
Z. J. Xiao, The B Meson Physics, Beijing: Science Press, 2013 (in Chinese)
82
Ed. A. J. Bevan, B. Golob, Th. Mannel, S. Prell, and B. D. Yabsley, The physics of the B factories, Eur. Phys. J. C 74, 3026 (2014); SLAC-PUB-15968, KEK Preprint2014-3
83
Y. Amhis, et al. [Heavy Flavor Averaging group (HFLAV)], Averages of b-hadron, c-hadron and τ-lepton properties as summer of 2016, Eur. Phys. J. C 77, 895 (2017) https://doi.org/10.1140/epjc/s10052-017-5058-4
84
Y. Amhis, et al. [Heavy Flavor Averaging group (HFLAV)], Averages of b-hadron, c-hadron and τ-lepton properties as of 2018, arXiv: 1909.12524V2 [hep-ex]
85
P. A. Zyla, et al. [Particle Data Group], The review of particle physics, Prog. Theor. Exp. Phys. 2020, 083C01 (2020)
Y. L. Zhang, S. Cheng, J. Hua, and Z. J. Xiao, ργ∗ → π(ρ) transition form factors in the perturbative QCD factorization approach, Phys. Rev. D 92, 094031 (2015) https://doi.org/10.1103/PhysRevD.92.094031
95
S. Cheng, Y. L. Zhang, J. Hua, H. N. Li, and Z. J. Xiao, Revisiting the factorization theorem for ργ∗ → π(ρ) at twist 3, Phys. Rev. D 95, 076005 (2017) https://doi.org/10.1103/PhysRevD.95.076005
96
J. Hua, S. Cheng, Y. L. Zhang, and Z. J. Xiao, Rho-pion transition form factors in the kT factorization formulism revisited, Phys. Rev. D 97, 113002 (2018) https://doi.org/10.1103/PhysRevD.97.113002
97
S. Nandi and H. N. Li, Next-to-leading-order corrections to exclusive processes in kT factorization, Phys. Rev. D 76, 034008 (2007) https://doi.org/10.1103/PhysRevD.76.034008
98
H. N. Li, Y. L. Shen, Y. M. Wang, and H. Zou, Next-toleading-order correction to pion form factor in kT factorization, Phys. Rev. D 83, 054029 (2011) https://doi.org/10.1103/PhysRevD.83.054029
99
S. Cheng, Y. Y. Fan, and Z. J. Xiao, NLO twist-3 contribution to the pion electromagnetic form factors in kT factorization, Phys. Rev. D 89, 054015 (2014) https://doi.org/10.1103/PhysRevD.89.054015
100
B. O. Lange and M. Neubert, Renormalization group evolution of the B meson light cone distribution amplitude, Phys. Rev. Lett. 91, 102001 (2003) https://doi.org/10.1103/PhysRevLett.91.102001
H. N. Li, Y. L. Shen, and Y. M. Wang, Resummation of rapidity logarithms in B meson wave functions, J. High Energy Phys. 1302, 008 (2013) https://doi.org/10.1007/JHEP02(2013)008
105
H. N. Li, Y. L. Shen, and Y. M. Wang, Joint resummation for pion wave function and pion transition form factor, J. High Energy Phys. 1401, 004 (2014) https://doi.org/10.1007/JHEP01(2014)004
106
X. D. Ji, J. P. Ma, and F. Yuan, QCD factorization for semi-inclusive deep-inelastic scattering at low transverse momentum, Phys. Rev. D 71, 034005 (2005) https://doi.org/10.1103/PhysRevD.71.034005
107
H. N. Li and Y. M. Wang, Non-dipolar Wilson links for transverse-momentum-dependent wave functions, J. High Energy Phys. 1506, 013 (2015) https://doi.org/10.1007/JHEP06(2015)013
108
M. Bander, D. Silverman, and A. Soni, CP noninvariance in the decays of the heavy charged quark systems, Phys. Rev. Lett. 43, 242 (1979) https://doi.org/10.1103/PhysRevLett.43.242
109
H. C. Hu and H. N. Li, Next-to-leading-order time-like pion form factors in kT factorization, Phys. Lett. B 718, 1351 (2013)
110
S. Cheng and Z. J. Xiao, Time-like pion electromagnetic form factors in kT factorization with the next-to-leadingorder twist-3 contribution, Phys. Lett. B 749, 1 (2015) https://doi.org/10.1016/j.physletb.2015.07.038
111
S. Cheng, Z. J. Xiao, and Y. L. Zhang, The NLO contributions to the scalar pion form factors and the O(αs2) annihilation corrections to the B → ππ decays, Nucl. Phys. B 896, 255 (2015)
Y. Li and C. D. Lü, Branching ratio and polarization of B → ρ(ω)ρ(ω) decays in perturbative QCD approach, Phys. Rev. D 73, 014024 (2006) https://doi.org/10.1103/PhysRevD.73.074002
116
Z. J. Xiao, D. Q. Guo, and X. F. Chen, Branching ratio and CP asymmetry of B → η(′)η(′) decays in the perturbative QCD approach, Phys. Rev. D 75, 014018 (2007) https://doi.org/10.1103/PhysRevD.75.014018
117
Y. Y. Keum, T. Kurimoto, H. N. Li, C. D. Lü, and A. I. Sanda, Nonfactorizable contributions to B → D(∗)Mdecays, Phys. Rev. D 69, 094018 (2004) https://doi.org/10.1103/PhysRevD.69.094018
A. Ali, G. Kramer, Y. Li, C. D. Lü, Y. L. Shen, W. Wang, and Y. M. Wang, Charmless non-leptonic Bs decays to PP, PV and VV final states in the pQCD approach, Phys. Rev. D 76, 074018 (2007) https://doi.org/10.1103/PhysRevD.76.074018
120
Z. T. Zou, A. Ali, C. D. Lü, X. Liu, and Y. Li, Improved estimates of the B(s) → VV decays in perturbative QCD approach, Phys. Rev. D 91, 054033 (2015) https://doi.org/10.1103/PhysRevD.91.054033
121
Y. Li, C. D. Lü, Z. J. Xiao, and X. Q. Yu, Branching ratio and CP asymmetry of Bs → π+π− decays in the perturbative QCD approach, Phys. Rev. D 70, 034009 (2004) https://doi.org/10.1103/PhysRevD.70.034009
122
Z. J. Xiao, W. F. Wang, and Y. Y. Fan, Revisiting the pure annihilation decays Bs → π+π− and B0 → K+K−: The data and the perturbative QCD predictions, Phys. Rev. D 85, 094003 (2012) https://doi.org/10.1103/PhysRevD.85.094003
123
X. Liu, Z. J. Xiao, and C. D. Lü, Pure annihilation type Bc→ M2M3 decays in the perturbative QCD approach, Phys. Rev. D 81, 014022 (2010) https://doi.org/10.1103/PhysRevD.81.014022
Z. J. Xiao and X. Liu, The two-body hadronic decays of Bc meson in the perturbative QCD approach: A short review, Chin. Sci. Bull. 59, 3748 (2014), and references therein https://doi.org/10.1007/s11434-014-0418-z
126
X. Liu and Z. J. Xiao, Light scalar mesons and charmless hadronic Bc → SP, SVdecays in the perturbative QCD approach, Phys. Rev. D 82, 054029 (2010) https://doi.org/10.1103/PhysRevD.82.054029
127
Z. Rui, W. F. Wang, G. X. Wang, L. H. Song, and C. D. Lü, The Bc → ψ(2S)π, ηc(2S)πdecays in the perturbative QCD approach, Eur. Phys. J. C 75, 293 (2015)
128
Z. Rui, H. Li, G. X. Wang, and Y. Xiao, Semileptonic decays of Bc meson to S-wave charmonium states in the perturbative QCD approach, Eur. Phys. J. C 76, 564 (2016) https://doi.org/10.1140/epjc/s10052-016-4424-y
X. Liu, R. H. Li, Z. T. Zou, and Z. J. Xiao, Nonleptonic charmless decays of Bc → TP, TV in the perturbative QCD approach, Phys. Rev. D 96, 013005 (2017) https://doi.org/10.1103/PhysRevD.96.013005
131
Z. T. Zou and C. D. Lü, The nonleptonic two body B decays involving a light tensor meson in fnal states, Chin. Sci. Bull. 59, 3738 (2014) https://doi.org/10.1007/s11434-014-0423-2
W. F. Wang, H. C. Hu, H. N. Li, and C. D. Lü, Direct CP asymmetries of three-body B decays in perturbative QCD, Phys. Rev. D 89, 074031 (2014)
134
R. Aaij, et al. [LHCb Collaboration], Measurement of CP violation in the phase space of B±→ K±π+π−, and B±→ K±K+K− decays, Phys. Rev. Lett. 111, 101801 (2013)
135
R. Aaij, et al. [LHCb Collaboration], Measurement of CP violation in the phase space of B±→ K+K?π± and B±→ π+π−π± decays, Phys. Rev. Lett. 112, 011801 (2014)
136
Y. Li, A. J. Ma, W. F. Wang, and Z. J. Xiao, Quasi-twobody decays B(s) → Pρ→ Pππin the perturbative QCD approach, Phys. Rev. D 95, 056008 (2017) https://doi.org/10.1103/PhysRevD.95.056008
137
Y. Li, A. J. Ma, W. F. Wang, and Z. J. Xiao, Quasi-twobody decays B(s) → Pρ′ (1450), Pρ′′ (1700 → Pππ in the perturbative QCD approach, Phys. Rev. D 96, 036014 (2017) https://doi.org/10.1103/PhysRevD.96.036014
138
A. J. Ma, Y. Li, W. F. Wang, and Z. J. Xiao, S-wave resonance contributions to B(s)0→ ηc(2S)π+π− in the perturbative QCD factorization approach, Chin. Phys. C 41, 083105 (2017) https://doi.org/10.1088/1674-1137/41/8/083105
139
Y. Li, D. C. Yan, R. Zhou, and Z. J. Xiao, S, P and D-wave resonance contributions to B(s) → ηc(1S, 2S)Kπdecays in the perturbative QCD approach, Phys. Rev. D 101, 016015 (2020) https://doi.org/10.1103/PhysRevD.101.016015
140
W. F. Wang, Will the subprocesses ρ(770; 1450) → K+K? contribute large branching fractions for B → πK+K− decays ? Phys. Rev. D 101, 111901(R) (2020) https://doi.org/10.1103/PhysRevD.101.111901
S. Mishima and A. I. Sanda, Calculation of magnetic penguin amplitudes in B → ϕK decays using PQCD approach, Prog. Theor. Phys. 110, 549 (2003) https://doi.org/10.1143/PTP.110.549
143
Y. L. Zhang, X. Y. Liu, Y. Y. Fan, S. Cheng, and Z. J. Xiao, B → ππ decays and effects of the next-to-leading order contributions, Phys. Rev. D 90, 014029 (2014) https://doi.org/10.1103/PhysRevD.90.014029
144
J. J. Wang, D. T. Lin, W. Sun, Z. J. Ji, S. Cheng, and Z. J. Xiao, B¯s0 → Kπ, KKdecays and effects of the nextto- leading order contributions, Phys. Rev. D 89, 074046 (2014) https://doi.org/10.1103/PhysRevD.89.074046
145
Y. Y. Fan, W. F. Wang, S. Cheng, and Z. J. Xiao, Anatomy of B → Kη(′) decays in different mixing schemes and effects of NLO contributions in the perturbative QCD approach, Phys. Rev. D 87, 094003 (2013) https://doi.org/10.1103/PhysRevD.87.094003
X. Liu, H. N. Li, and Z. J. Xiao, Transverse-momentumdependent wave functions with Glauber gluons in B → ππ, ρρ decays, Phys. Rev. D 91, 114019 (2015) https://doi.org/10.1103/PhysRevD.91.114019
Z. J. Xiao, Z. Q. Zhang, X. Liu, and L. B. Guo, Branching ratios and CP asymmetries of B → Kη(′) decays in the perturbative PQCD approach, Phys. Rev. D 78, 114001 (2008) https://doi.org/10.1103/PhysRevD.78.114001
150
Z. Q. Zhang, X. Liu, and Z. J. Xiao, B → πη(′), η(′)η(′) decays and NLO contributions in the PQCD approach, Chin. Phys. C 33, 722 (2009) https://doi.org/10.1088/1674-1137/33/9/003
151
W. Bai, M Liu, Y. Y. Fan, W. F. Wang, S. Cheng, and Z. J. Xiao, Revisiting the Kπ puzzle in the PQCD factorization approach, Chin. Phys. C 38, 033101 (2014) https://doi.org/10.1088/1674-1137/38/3/033101
152
D. C. Yan, P. Yang, X. Liu, and Z. J. Xiao, Anatomy of Bs → PV decays and effects of next-to-leading order contributions in the perturbative QCD factorization approach, Nucl. Phys. B 931 , 79 (2018) https://doi.org/10.1016/j.nuclphysb.2018.04.007
153
D. C. Yan, X. Liu, and Z. J. Xiao, Anatomy of Bs → VV decays and effects of next-to-leading order contributions in the perturbative QCD factorization approach, Nucl. Phys. B 935 , 17 (2018) https://doi.org/10.1016/j.nuclphysb.2018.08.002
154
D. C. Yan, X. Liu, and Z. J. Xiao, Anatomy of Bs → PP decays and effects of the next-to-leading order contributions in the perturbative QCD approach, Nucl. Phys. B 946 , 114705 (2019) https://doi.org/10.1016/j.nuclphysb.2019.114705
Z. J. Xiao, D. C. Yan, and X. Liu, B(s) → ηc(P, V) decays and effects of the next-to-leading order contributions in the perturbative QCD approach, Nucl. Phys. B 953 , 114954 (2020) https://doi.org/10.1016/j.nuclphysb.2020.114954
157
J. P. Lees, et al. [BaBar Collaboration], Evidence for an excess of B¯→ D(∗)τ−ν¯τ decays, Phys. Rev. Lett. 109, 101802 (2012)
158
R. Aaij, et al. [LHCb collaboration], Differential branching fraction and angular analysis of the decay B0 → K∗0μ+μ−, J. High Energy Phys. 1308, 131 (2013)
159
R. Aaij, et al. [LHCb collaboration], Measurement of form-factor independent observables in the decay B0 → K∗0μ+μ−, Phys. Rev. Lett. 111, 191801 (2013)
160
G. Caria, et al. [Belle collaboration], Measurement of R(D) and R(D∗) with a semileptonic tagging method, Phys. Rev. Lett. 124, 161803 (2020)
161
R. Aaij, et al. [LHCb collaboration], Search for leptonuniversality violation in B+ → K+l+l−, Phys. Rev. Lett. 122, 191801 (2019)
J. A. Bailey, et al., Refining new-physics searches in B → Dτνdecay with lattice QCD, Phys. Rev. Lett. 109, 071802 (2012)
165
S. Fajfer, J. F. Kamenik, I. Nisandzic, et al., Implications of lepton flavor universality violations in Bdecays, Phys. Rev. Lett. 109, 161801 (2012) https://doi.org/10.1103/PhysRevLett.109.161801
166
A. Celis, M. Jung, X. Q. Li, et al., Sensitivity to charged scalars in B → Dτν and B → τν decays, J. High Energy Phys. 1301, 054 (2013) https://doi.org/10.1007/JHEP01(2013)054
167
F. U. Bemlochner, Z. Ligeti, M. Papucci, and D. J. Robinson, Combined analysis of semileptonic B decays to D and D∗: R(D(∗)), |Vcb|, and new physics, Phys. Rev. D 95, 115008 (2017) https://doi.org/10.1103/PhysRevD.95.115008
Y. Y. Fan, W. F. Wang, S. Cheng, and Z. J. Xiao, Semileptonic decays B → D(∗)lν in the perturbative QCD factorization approach, Chin. Sci. Bull. 59, 125 (2014) https://doi.org/10.1007/s11434-013-0049-9
170
Y. Y. Fan, Z. J. Xiao, R. M. Wang, and B. Z. Li, The B → D(∗)lνdecays in the pQCD approach with the Lattice QCD input, Chin. Sci. Bull. 60, 2009 (2015) https://doi.org/10.1007/s11434-015-0959-9
171
Y. M. Wang, Y. B. Wei, Y. L. Shen, and C. D. Lü, Perturbative corrections to B → D form factors in QCD, J. High Energy Phys. 06, 062 (2017) https://doi.org/10.1007/JHEP06(2017)062
172
Y. Y. Fan, W. F. Wang, and Z. J. Xiao, Study of B¯s0→ (Ds+, Ds*+)l−ν¯ldecays in the perturbative QCD factorization approach, Phys. Rev. D 89, 014030 (2013) https://doi.org/10.1103/PhysRevD.89.014030
173
X. Q. Hu, S. P. Jin, and Z. J. Xiao, Semileptonic decays B/Bs→ (D(∗), Ds(*))lνlin the PQCD approach with the lattice QCD input, Chin. Phys. C 44, 053102 (2020) https://doi.org/10.1088/1674-1137/44/5/053102
174
W. F. Wang and Z. J. Xiao, Semileptonic decays of B/Bs→ (π, K)(l+l− , lν, νν¯) in the perturbative QCD approach beyond the leading order, Phys. Rev. D 86, 114025 (2012) https://doi.org/10.1103/PhysRevD.86.114025
175
W. F. Wang, Y. Y. Fan, M. Liu, and Z. J. Xiao, Semileptonic decays of B/Bs→ (η, η′, G)(l+l − , l−ν¯, νν¯) in the perturbative QCD approach beyond the leading order, Phys. Rev. D 87, 097501 (2013)
176
C. Bourrely, I. Caprini, and L. Lellouch, Modelindependent description of B → πlν decays and a determination of |Vub|, Phys. Rev. D 79, 013008 (2009); Erratum: Phys. Rev. D 82, 099902 (2010) https://doi.org/10.1103/PhysRevD.82.099902
177
S. P. Jin, X. Q. Hu, and Z. J. Xiao, Study of Bs → K(∗)l+l− in the PQCD factorization approach with lattice QCD input , Phys. Rev. D 102, 013001 (2020) https://doi.org/10.1103/PhysRevD.102.013001
178
A. Khodjamirian, T. Mannel, A. A. Pivovarov, and Y. M. Wang, Charm-loop effect in B → K(∗)ℓ+ℓ− and B → K∗γ, J. High Energy Phys. 09, 089 (2010) https://doi.org/10.1007/JHEP09(2010)089
179
A. Khodjamirian, T. Mannel, and Y. M. Wang, B → Kℓ+ℓ− decay at large hadronic recoil, J. High Energy Phys. 02, 010 (2013) https://doi.org/10.1007/JHEP02(2013)010
180
W. F. Wang, X. Yu, C. D. Lü, and Z. J. Xiao, Semileptonic decays Bc+→ D(s)*+ (l+νl, l+l− , νν¯) in the perturbative QCD approach, Phys. Rev. D 90, 094018 (2014) https://doi.org/10.1103/PhysRevD.90.094018
181
X. Q. Hu, S. P. Jin, and Z. J. Xiao, Semileptonic decays Bc → (ηc, J/Ψ)lν¯l in the “Lattice+PQCD” approach, Chin. Phys. C 44, 023104 (2020) https://doi.org/10.1088/1674-1137/44/2/023104
182
R. Aaij, et al. [LHCb collaboration], Measuremnt of the ratio of branching fractions B(Bc+→ J/Ψτ+ντ)/B(Bc+ → J/Ψμ+νμ), Phys. Rev. Lett. 120, 121801 (2018)
183
M. Mulder, on behalf of the LHCb Collaboration, Experimental status of LFU in b→ slltransitions, talk given at FPCP 2020, June 8, 2020.
Y. M. Wang, Factorization and dispersion relations for radiative leptonic Bdecay, J. High Energy Phys. 09, 159 (2016) https://doi.org/10.1007/JHEP09(2016)159
186
Y. L. Shen, Z. T. Zou, and Y. B. Wei, Subleading power corrections to B→ γlνdecay in PQCD approach, Phys. Rev. D 99, 016004 (2019) https://doi.org/10.1103/PhysRevD.99.016004
187
Y. M. Wang and Y. L. Shen, Subleading-power corrections to the radiative leptonic B → γℓνdecay in QCD, J. High Energy Phys. 05, 184 (2018) https://doi.org/10.1007/JHEP05(2018)184
188
Y. L. Shen, Z. T. Zou, and Y. Li, Power corrections to pion transition form factor in perturbative QCD approach, Phys. Rev. D 100, 016022 (2019) https://doi.org/10.1103/PhysRevD.100.016022
189
S. Cheng, A. Khodjamirian, and J. Virto, B → ππ form factors from light-cone sum rules with B-meson distribution amplitudes, J. High Energy Phys. 1705, 157 (2017) https://doi.org/10.1007/JHEP05(2017)157
190
S. Descotes-Genon, A. Khodjamirian, and J. Virto, Lightcone sum rules for B → Kπ form factors and applications to rare decays,J. High Energy Phys. 1912, 083 (2019) https://doi.org/10.1007/JHEP12(2019)083
H. N. Li and B. Melic, Determination of heavy meson wave functions from Bdecays, Eur. Phys. J. C 11, 695 (1999) https://doi.org/10.1007/s100520050665
196
C. D. Lü, M. Matsumori, A. I. Sanda, and M. Z. Yang, CP asymmetry, branching ratios and isospin breaking effects in B → ργ and B → ωγ decays with the pQCD approach, Phys. Rev. D 72, 094005 (2005); Erratum: Phys. Rev. D 73, 039902 (2006)
197
M. Tanabashi, et al. [Particle Data Group], The review of particle physics, Phys. Rev. D 98(3), 030001 (2018)
198
Y. Li and C. D. Lü, Annihilation type radiative decays of B meson in perturbative QCD approach, Phys. Rev. D 74, 097502 (2006) https://doi.org/10.1103/PhysRevD.74.097502
C. H. Chang, C. D. Lü, G. L. Wang, and H. S. Zong, The pure leptonic decays of Bc meson and their radiative corrections, Phys. Rev. D 60, 114013 (1999) https://doi.org/10.1103/PhysRevD.60.114013
X. G. He, T. Li, X. Q. Li, and Y. M. Wang, PQCD calculation for Λb → Λγ in the standard model, Phys. Rev. D 74, 034026 (2006) https://doi.org/10.1103/PhysRevD.74.034026
204
C. D. Lü, Y. M. Wang, H. Zou, A. Ali, and G. Kramer, Anatomy of the pQCD approach to the baryonic decays Λb → pπ, pK, Phys. Rev. D 80, 034011 (2009) https://doi.org/10.1103/PhysRevD.80.034011
A. Khodjamirian, C. Klein, T. Mannel, and Y. M. Wang, Form factors and strong couplings of heavy baryons from QCD light-cone sum rules, J. High Energy Phys. 09, 106 (2011) https://doi.org/10.1007/JHEP09(2011)106
207
Y. M. Wang, Y. L. Shen, and C. D. Lu, Λb→ p, Λ transition form factors from QCD light-cone sum rules,Phys. Rev. D 80, 074012 (2009) https://doi.org/10.1103/PhysRevD.80.074012
208
Y. M. Wang and Y. L. Shen, Perturbative corrections to Λb→ Λ form factors from QCD light-cone sum rules, J. High Energy Phys. 02, 179 (2016) https://doi.org/10.1007/JHEP02(2016)179
