Abstract
We address the problem of extrapolating the vector form factor f + Bπ , which is relevant to B → πℓν ℓ decays, from the region of small to the region of large momentum transfer. As input, we use the QCD light-cone sum rule at small momentum transfer. We carry out a comprehensive Bayesian uncertainty analysis and obtain correlated uncertainties for the normalization and shape parameters of the form factor. The z-series parametrization for f + Bπ is employed to extrapolate our results to large momentum transfer, and to compare with the lattice QCD results. To test the validity of our extrapolation we use the upper and lower bounds from the unitarity and positivity of the two-point correlator of heavy-light quark currents. This correlator is updated by including the NNLO perturbative term and the NLO correction to the quark condensate contribution. We demonstrate that an additional input including the form factor, its first and second derivative calculated at one value of momentum transfer from the light-cone sum rules, considerably improves the bounds. This only holds when the correlations between the form factor parameters are taken into account. We further combine our results with the latest experimental measurements of B → πℓν ℓ by the BaBar and Belle collaborations, and obtain |V ub | = (3. 32 + 0.26− 0.22 ) ⋅ 10− 3 from a Bayesian analysis.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
BaBar collaboration, P. del Amo Sanchez et al., Study of B → πℓν and B → ρℓν decays and determination of |V ub |, Phys. Rev. D 83 (2011) 032007 [arXiv:1005.3288] [INSPIRE].
BaBar collaboration, P. del Amo Sanchez et al., Measurement of the B 0 → π ℓℓ+ ν and B + → η (′)ℓ+ ν branching fractions, the B 0 → π −ℓ+ ν and B + → ηℓ+ ν form-factor shapes and determination of |V ub |, Phys. Rev. D 83 (2011) 052011 [arXiv:1010.0987] [INSPIRE].
BELLE collaboration, H. Ha et al., Measurement of the decay B 0 → π −ℓ+ ν and determination of |V ub |, Phys. Rev. D 83 (2011) 071101 [arXiv:1012.0090] [INSPIRE].
BaBar collaboration, J.P. Lees et al., Branching fraction and form-factor shape measurements of exclusive charmless semileptonic B decays and determination of |V ub |, Phys. Rev. D 86 (2012) 092004 [arXiv:1208.1253] [INSPIRE].
Belle collaboration, A. Sibidanov et al., Study of exclusive B → X u ℓν Decays and Extraction of |V ub | using full reconstruction tagging at the Belle experiment, Phys. Rev. D 88 (2013) 032005 [arXiv:1306.2781] [INSPIRE].
Particle Data Group collaboration, J. Beringer et al., Review of particle physics, Phys. Rev. D 86 (2012) 010001 [INSPIRE].
E. Dalgic et al., B meson semileptonic form-factors from unquenched lattice QCD, Phys. Rev. D 73 (2006) 074502 [Erratum ibid. D 75 (2007) 119906] [hep-lat/0601021] [INSPIRE].
J.A. Bailey et al., The B → πℓν semileptonic form factor from three-flavor lattice QCD: a model-independent determination of |V ub |, Phys. Rev. D 79 (2009) 054507 [arXiv:0811.3640] [INSPIRE].
QCDSF collaboration, A. Al-Haydari et al., Semileptonic form factors D → pi, K and B→pi, K from a fine lattice, Eur. Phys. J. A 43 (2010) 107 [arXiv:0903.1664] [INSPIRE].
A. Khodjamirian, T. Mannel, N. Offen and Y.-M. Wang, B → πℓν l width and |V ub | from QCD light-cone sum rules, Phys. Rev. D 83 (2011) 094031 [arXiv:1103.2655] [INSPIRE].
A. Bharucha, Two-loop corrections to the B → π form factor from QCD sum rules on the light-cone and |V ub |, JHEP 05 (2012) 092 [arXiv:1203.1359] [INSPIRE].
C. Bourrely, I. Caprini and L. Lellouch, Model-independent description of B → πlν decays and a determination of |V ub |, Phys. Rev. D 79 (2009) 013008 [Erratum ibid. D 82 (2010) 099902] [arXiv:0807.2722] [INSPIRE].
G. Duplancic, A. Khodjamirian, T. Mannel, B. Melic and N. Offen, Light-cone sum rules for B → π form factors revisited, JHEP 04 (2008) 014 [arXiv:0801.1796] [INSPIRE].
P. Ball and R. Zwicky, New results on B → π, K, η decay formfactors from light-cone sum rules, Phys. Rev. D 71 (2005) 014015 [hep-ph/0406232] [INSPIRE].
I.I. Balitsky, V.M. Braun and A.V. Kolesnichenko, Σ+ → P γ decay in QCD (in Russian), Sov. J. Nucl. Phys. 44 (1986) 1028 [INSPIRE].
V.M. Braun and I.E. Filyanov, QCD sum rules in exclusive kinematics and pion wave function, Z. Phys. C 44 (1989) 157 [INSPIRE].
V.L. Chernyak and I.R. Zhitnitsky, B meson exclusive decays into baryons, Nucl. Phys. B 345 (1990) 137 [INSPIRE].
V.M. Belyaev, A. Khodjamirian and R. Ruckl, QCD calculation of the B → pi, K form-factors, Z. Phys. C 60 (1993) 349 [hep-ph/9305348] [INSPIRE].
V.M. Belyaev, V.M. Braun, A. Khodjamirian and R. Ruckl, D ∗ Dπ and B ∗ Bπ couplings in QCD, Phys. Rev. D 51 (1995) 6177 [hep-ph/9410280] [INSPIRE].
A. Khodjamirian, R. Ruckl, S. Weinzierl and O.I. Yakovlev, Perturbative QCD correction to the B → π transition form-factor, Phys. Lett. B 410 (1997) 275 [hep-ph/9706303] [INSPIRE].
E. Bagan, P. Ball and V.M. Braun, Radiative corrections to the decay B → πeν and the heavy quark limit, Phys. Lett. B 417 (1998) 154 [hep-ph/9709243] [INSPIRE].
P. Ball, V.M. Braun and A. Lenz, Higher-twist distribution amplitudes of the K meson in QCD, JHEP 05 (2006) 004 [hep-ph/0603063] [INSPIRE].
A. Khodjamirian, C. Klein, T. Mannel and N. Offen, Semileptonic charm decays D → πlν(ℓ) and D → Klν(ℓ) from QCD light-cone sum rules, Phys. Rev. D 80 (2009) 114005 [arXiv:0907.2842] [INSPIRE].
P. Gelhausen, A. Khodjamirian, A.A. Pivovarov and D. Rosenthal, Decay constants of heavy-light vector mesons from QCD sum rules, Phys. Rev. D 88 (2013) 014015 [Erratum ibid. D 89 (2014) 099901] [arXiv:1305.5432] [INSPIRE].
H. Leutwyler, The Ratios of the light quark masses, Phys. Lett. B 378 (1996) 313 [hep-ph/9602366] [INSPIRE].
B.L. Ioffe, Condensates in quantum chromodynamics, Phys. Atom. Nucl. 66 (2003) 30 [hep-ph/0207191] [INSPIRE].
S.S. Agaev, V.M. Braun, N. Offen and F.A. Porkert, Light cone sum rules for the π 0 -γ∗ -γ form factor revisited, Phys. Rev. D 83 (2011) 054020 [arXiv:1012.4671] [INSPIRE].
S.S. Agaev, V.M. Braun, N. Offen and F.A. Porkert, BELLE data on the π 0γ ∗ γ form factor: a game changer?, Phys. Rev. D 86 (2012) 077504 [arXiv:1206.3968] [INSPIRE].
A. Gelman et al., Bayesian data analysis, 3rd edition, Chapman & Hall/CRC Texts in Statistical Science 106, Chapman and Hall/CRC, U.S.A. (2013).
D. van Dyk et al., EOS: a HEP program for flavor observables, the version used for this publication is available at http://project.het.physik.tu-dortmund.de/source/eos/tag/btopi-ff.
C. Bouchard, B → πℓν and B s → Kℓν from FNAL/HPQCD, talk given at the 8th International Workshop on the CKM Unitarity Triangle , September 8-12, Vienna University of Technology, Vienna, Austria (2014).
A. Khodjamirian, R. Rückl, S. Weinzierl and O.I. Yakovlev, Perturbative QCD correction to the light cone sum rule for the B ∗ Bπ and D ∗ Dπ couplings, Phys. Lett. B 457 (1999) 245 [hep-ph/9903421] [INSPIRE].
W. Detmold, C.J.D. Lin and S. Meinel, Calculation of the heavy-hadron axial couplings g 1 , g 2 and g 3 using lattice QCD, Phys. Rev. D 85 (2012) 114508 [arXiv:1203.3378] [INSPIRE].
L. Lellouch, Lattice constrained unitarity bounds for \( {\overline{B}}^0\to {\pi}^{+}{\ell}^{-}{\overline{\nu}}_{\ell } \) decays, Nucl. Phys. B 479 (1996) 353 [hep-ph/9509358] [INSPIRE].
C. Bourrely, B. Machet and E. de Rafael, Semileptonic decays of pseudoscalar particles (M → M ′ℓν ℓ ) and short distance behavior of quantum chromodynamics, Nucl. Phys. B 189 (1981) 157 [INSPIRE].
T. Mannel and B. Postler, Improved unitarity bounds for \( {\overline{B}}^0\to {\pi}^{+}\ell {\overline{\nu}}_{\ell}\left(\overline{\ell}\right) \) decays, Nucl. Phys. B 535 (1998) 372 [hep-ph/9805425] [INSPIRE].
G. Abbas, B. Ananthanarayan, I. Caprini, I. Sentitemsu Imsong and S. Ramanan, Theory of unitarity bounds and low energy form factors, Eur. Phys. J. A 45 (2010) 389 [arXiv:1004.4257] [INSPIRE].
F. Beaujean, A Bayesian analysis of rare B decays with advanced Monte Carlo methods, Ph.D. thesis, Fakultät für Physik, Technische Universität München, München, Germany (2012).
Heavy Flavor Averaging Group collaboration, Y. Amhis et al., Averages of B-hadron, C-hadron and τ -lepton properties as of early 2012, arXiv:1207.1158 [INSPIRE], see the PDG 2013 update at http://www.slac.stanford.edu/xorg/hfag.
P. Gambino, P. Giordano, G. Ossola and N. Uraltsev, Inclusive semileptonic B decays and the determination of |V ub |, JHEP 10 (2007) 058 [arXiv:0707.2493] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1409.7816
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Imsong, I.S., Khodjamirian, A., Mannel, T. et al. Extrapolation and unitarity bounds for the B → π form factor. J. High Energ. Phys. 2015, 126 (2015). https://doi.org/10.1007/JHEP02(2015)126
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2015)126