Abstract
We propose a stronger formulation of the dispersive (or unitarity) bounds à la Boyd-Grinstein-Lebed (BGL), which are commonly applied in analyses of the hadronic form factors for B decays. In our approach, the existing bounds are split into several new bounds, thereby disentangling form factors that are jointly bounded in the common approach. This leads to stronger constraints for these objects, to a significant simplification of our numerical analysis, and to the removal of spurious correlations among the form factors. We apply these novel bounds to \( \overline{B}\to {\overline{K}}^{\left(\ast \right)} \) and \( {\overline{B}}_s\to \phi \) form factors by fitting them to purely theoretical constraints. Using a suitable parametrization, we take into account the form factors’ below-threshold branch cuts arising from on-shell \( {\overline{B}}_s{\pi}^0 \) and \( {\overline{B}}_s{\pi}^0{\pi}^0 \) states, which so-far have been ignored in the literature. In this way, we eliminate a source of hard-to-quantify systematic uncertainties. We provide machine readable files to obtain the full set of the \( \overline{B}\to {\overline{K}}^{\left(\ast \right)} \) and \( {\overline{B}}_s\to \phi \) form factors in and beyond the entire semileptonic phase space.
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G. Isidori, Y. Nir and G. Perez, Flavor Physics Constraints for Physics Beyond the Standard Model, Ann. Rev. Nucl. Part. Sci. 60 (2010) 355 [arXiv:1002.0900] [INSPIRE].
C. Bobeth, G. Hiller and G. Piranishvili, Angular distributions of \( \overline{B}\to \overline{K}{\ell}^{+}{\ell}^{-} \) decays, JHEP 12 (2007) 040 [arXiv:0709.4174] [INSPIRE].
W. Altmannshofer et al., Symmetries and Asymmetries of B → K*μ+μ− Decays in the Standard Model and Beyond, JHEP 01 (2009) 019 [arXiv:0811.1214] [INSPIRE].
C. Bobeth, G. Hiller and D. van Dyk, The Benefits of \( \overline{B}\to {\overline{K}}^{\ast }{\ell}^{+}{\ell}^{-} \) Decays at Low Recoil, JHEP 07 (2010) 098 [arXiv:1006.5013] [INSPIRE].
J. Matias, F. Mescia, M. Ramon and J. Virto, Complete Anatomy of \( {\overline{B}}_d\to {\overline{K}}^{\ast 0}\left(\to K\pi \right){\ell}^{+}{\ell}^{-} \) and its angular distribution, JHEP 04 (2012) 104 [arXiv:1202.4266] [INSPIRE].
F. Beaujean, C. Bobeth, D. van Dyk and C. Wacker, Bayesian Fit of Exclusive \( b\to s\overline{\ell}\ell \) Decays: The Standard Model Operator Basis, JHEP 08 (2012) 030 [arXiv:1205.1838] [INSPIRE].
S. Descotes-Genon, J. Matias, M. Ramon and J. Virto, Implications from clean observables for the binned analysis of B → K*μ+μ− at large recoil, JHEP 01 (2013) 048 [arXiv:1207.2753] [INSPIRE].
LHCb collaboration, Measurement of Form-Factor-Independent Observables in the Decay B0 → K*0μ+μ−, Phys. Rev. Lett. 111 (2013) 191801 [arXiv:1308.1707] [INSPIRE].
S. Descotes-Genon, J. Matias and J. Virto, Understanding the B → K*μ+μ− Anomaly, Phys. Rev. D 88 (2013) 074002 [arXiv:1307.5683] [INSPIRE].
W. Altmannshofer and D.M. Straub, New Physics in B → K*μμ?, Eur. Phys. J. C 73 (2013) 2646 [arXiv:1308.1501] [INSPIRE].
F. Beaujean, C. Bobeth and D. van Dyk, Comprehensive Bayesian analysis of rare (semi)leptonic and radiative B decays, Eur. Phys. J. C 74 (2014) 2897 [Erratum ibid. 74 (2014) 3179] [arXiv:1310.2478] [INSPIRE].
R.R. Horgan, Z. Liu, S. Meinel and M. Wingate, Calculation of B0 → K*0μ+μ− and \( {B}_s^0\to \phi {\mu}^{+}{\mu}^{-} \) observables using form factors from lattice QCD, Phys. Rev. Lett. 112 (2014) 212003 [arXiv:1310.3887] [INSPIRE].
T. Hurth and F. Mahmoudi, On the LHCb anomaly in B → K*ℓ+ℓ−, JHEP 04 (2014) 097 [arXiv:1312.5267] [INSPIRE].
W. Altmannshofer and D.M. Straub, New physics in b → s transitions after LHC run 1, Eur. Phys. J. C 75 (2015) 382 [arXiv:1411.3161] [INSPIRE].
LHCb collaboration, Test of lepton universality using B+ → K+ℓ+ℓ− decays, Phys. Rev. Lett. 113 (2014) 151601 [arXiv:1406.6482] [INSPIRE].
S. Descotes-Genon, L. Hofer, J. Matias and J. Virto, Global analysis of b → sℓℓ anomalies, JHEP 06 (2016) 092 [arXiv:1510.04239] [INSPIRE].
M. Ciuchini et al., B → K*ℓ+ℓ− decays at large recoil in the Standard Model: a theoretical reappraisal, JHEP 06 (2016) 116 [arXiv:1512.07157] [INSPIRE].
LHCb collaboration, Angular analysis of the B0 → K*0μ+μ− decay using 3 fb−1 of integrated luminosity, JHEP 02 (2016) 104 [arXiv:1512.04442] [INSPIRE].
LHCb collaboration, Test of lepton universality with B0 → K*0ℓ+ℓ− decays, JHEP 08 (2017) 055 [arXiv:1705.05802] [INSPIRE].
N. Gubernari, M. Reboud, D. van Dyk and J. Virto, Improved theory predictions and global analysis of exclusive b → sμ+μ− processes, JHEP 09 (2022) 133 [arXiv:2206.03797] [INSPIRE].
P. Ball and V.M. Braun, Exclusive semileptonic and rare B meson decays in QCD, Phys. Rev. D 58 (1998) 094016 [hep-ph/9805422] [INSPIRE].
J. Charles et al., Heavy to light form-factors in the heavy mass to large energy limit of QCD, Phys. Rev. D 60 (1999) 014001 [hep-ph/9812358] [INSPIRE].
M. Beneke and T. Feldmann, Symmetry breaking corrections to heavy to light B meson form-factors at large recoil, Nucl. Phys. B 592 (2001) 3 [hep-ph/0008255] [INSPIRE].
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 (2001) 114020 [hep-ph/0011336] [INSPIRE].
M. Beneke and T. Feldmann, Factorization of heavy to light form-factors in soft collinear effective theory, Nucl. Phys. B 685 (2004) 249 [hep-ph/0311335] [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].
A. Khodjamirian, T. Mannel and N. Offen, Form-factors from light-cone sum rules with B-meson distribution amplitudes, Phys. Rev. D 75 (2007) 054013 [hep-ph/0611193] [INSPIRE].
G. Duplancic et al., Light-cone sum rules for B → π form factors revisited, JHEP 04 (2008) 014 [arXiv:0801.1796] [INSPIRE].
S. Jäger and J. Martin Camalich, On B → Vℓℓ at small dilepton invariant mass, power corrections, and new physics, JHEP 05 (2013) 043 [arXiv:1212.2263] [INSPIRE].
A. Bharucha, D.M. Straub and R. Zwicky, B → Vℓ+ℓ− in the Standard Model from light-cone sum rules, JHEP 08 (2016) 098 [arXiv:1503.05534] [INSPIRE].
N. Gubernari, A. Kokulu and D. van Dyk, B → P and B → V Form Factors from B-Meson Light-Cone Sum Rules beyond Leading Twist, JHEP 01 (2019) 150 [arXiv:1811.00983] [INSPIRE].
S. Descotes-Genon, A. Khodjamirian and J. Virto, Light-cone sum rules for B → Kπ form factors and applications to rare decays, JHEP 12 (2019) 083 [arXiv:1908.02267] [INSPIRE].
(HPQCD collaboration)§ and HPQCD collaborations, B → K and D → K form factors from fully relativistic lattice QCD, Phys. Rev. D 107 (2023) 014510 [arXiv:2207.12468] [INSPIRE].
S. Descotes-Genon, A. Khodjamirian, J. Virto and K.K. Vos, Light-Cone Sum Rules for S-wave B → Kπ Form Factors, JHEP 06 (2023) 034 [arXiv:2304.02973] [INSPIRE].
Flavour Lattice Averaging Group (FLAG) collaboration, FLAG Review 2021, Eur. Phys. J. C 82 (2022) 869 [arXiv:2111.09849] [INSPIRE].
M. Beneke, T. Feldmann and D. Seidel, Systematic approach to exclusive B → Vl+l−, Vγ decays, Nucl. Phys. B 612 (2001) 25 [hep-ph/0106067] [INSPIRE].
B. Grinstein and D. Pirjol, Exclusive rare B → K*ℓ+ℓ− decays at low recoil: Controlling the long-distance effects, Phys. Rev. D 70 (2004) 114005 [hep-ph/0404250] [INSPIRE].
M. Beylich, G. Buchalla and T. Feldmann, Theory of B → K(*)ℓ+ℓ− decays at high q2: OPE and quark-hadron duality, Eur. Phys. J. C 71 (2011) 1635 [arXiv:1101.5118] [INSPIRE].
A. Khodjamirian, T. Mannel, A.A. Pivovarov and Y.-M. Wang, Charm-loop effect in B → K(*)ℓ+ℓ− and B → K*γ, JHEP 09 (2010) 089 [arXiv:1006.4945] [INSPIRE].
A. Khodjamirian, T. Mannel and Y.M. Wang, B → Kℓ+ℓ− decay at large hadronic recoil, JHEP 02 (2013) 010 [arXiv:1211.0234] [INSPIRE].
C. Bobeth, M. Chrzaszcz, D. van Dyk and J. Virto, Long-distance effects in B → K*ℓℓ from analyticity, Eur. Phys. J. C 78 (2018) 451 [arXiv:1707.07305] [INSPIRE].
A. Arbey, T. Hurth, F. Mahmoudi and S. Neshatpour, Hadronic and New Physics Contributions to b → s Transitions, Phys. Rev. D 98 (2018) 095027 [arXiv:1806.02791] [INSPIRE].
H.M. Asatrian, C. Greub and J. Virto, Exact NLO matching and analyticity in b → sℓℓ, JHEP 04 (2020) 012 [arXiv:1912.09099] [INSPIRE].
N. Gubernari, D. van Dyk and J. Virto, Non-local matrix elements in B(s) → {K(*), ϕ}ℓ+ℓ−, JHEP 02 (2021) 088 [arXiv:2011.09813] [INSPIRE].
S. Okubo, Exact bounds for Kℓ3 decay parameters, Phys. Rev. D 3 (1971) 2807 [INSPIRE].
C.G. Boyd, B. Grinstein and R.F. Lebed, Constraints on form-factors for exclusive semileptonic heavy to light meson decays, Phys. Rev. Lett. 74 (1995) 4603 [hep-ph/9412324] [INSPIRE].
C.G. Boyd, B. Grinstein and R.F. Lebed, Precision corrections to dispersive bounds on form-factors, Phys. Rev. D 56 (1997) 6895 [hep-ph/9705252] [INSPIRE].
C. Bourrely, I. Caprini and L. Lellouch, Model-independent description of \( B\to \pi {\ell}^{-}\overline{\nu} \) decays and a determination of |Vub|, Phys. Rev. D 79 (2009) 013008 [Erratum ibid. 82 (2010) 099902] [arXiv:0807.2722] [INSPIRE].
D. Bigi, P. Gambino and S. Schacht, R(D*), |Vcb|, and the Heavy Quark Symmetry relations between form factors, JHEP 11 (2017) 061 [arXiv:1707.09509] [INSPIRE].
T. Blake, S. Meinel, M. Rahimi and D. van Dyk, Dispersive bounds for local form factors in Λb → Λ transitions, Phys. Rev. D 108 (2023) 094509 [arXiv:2205.06041] [INSPIRE].
Y. Amhis, M. Bordone and M. Reboud, Dispersive analysis of Λb → Λ(1520) local form factors, JHEP 02 (2023) 010 [arXiv:2208.08937] [INSPIRE].
G. Hiller and R. Zwicky, (A)symmetries of weak decays at and near the kinematic endpoint, JHEP 03 (2014) 042 [arXiv:1312.1923] [INSPIRE].
M.A. Shifman, Quark hadron duality, in the proceedings of the 8th International Symposium on Heavy Flavor Physics, Southampton, U.K., July 25–29 (1999), p. 1447–1494 [https://doi.org/10.1142/9789812810458_0032] [hep-ph/0009131] [INSPIRE].
I. Caprini, L. Lellouch and M. Neubert, Dispersive bounds on the shape of \( \overline{B}\to {D}^{\left(\ast \right)}{\ell}^{-}\overline{\nu} \) form-factors, Nucl. Phys. B 530 (1998) 153 [hep-ph/9712417] [INSPIRE].
M. Bordone, M. Jung and D. van Dyk, Theory determination of \( \overline{B}\to {D}^{\left(\ast \right)}{\ell}^{-}\overline{\nu} \) form factors at \( \mathcal{O}\left(1/{m}_c^2\right) \), Eur. Phys. J. C 80 (2020) 74 [arXiv:1908.09398] [INSPIRE].
M. Bordone, N. Gubernari, D. van Dyk and M. Jung, Heavy-Quark expansion for \( {\overline{B}}_s\to {D}_s^{\left(\ast \right)} \) form factors and unitarity bounds beyond the SU(3)F limit, Eur. Phys. J. C 80 (2020) 347 [arXiv:1912.09335] [INSPIRE].
A. Bharucha, T. Feldmann and M. Wick, Theoretical and Phenomenological Constraints on Form Factors for Radiative and Semi-Leptonic B-Meson Decays, JHEP 09 (2010) 090 [arXiv:1004.3249] [INSPIRE].
Particle Data Group collaboration, Review of Particle Physics, PTEP 2020 (2020) 083C01 [INSPIRE].
C.B. Lang, D. Mohler, S. Prelovsek and R.M. Woloshyn, Predicting positive parity Bs mesons from lattice QCD, Phys. Lett. B 750 (2015) 17 [arXiv:1501.01646] [INSPIRE].
ETM collaboration, Masses and decay constants of \( {D}_{(s)}^{\ast } \) and \( {B}_{(s)}^{\ast } \) mesons with Nf = 2 + 1 + 1 twisted mass fermions, Phys. Rev. D 96 (2017) 034524 [arXiv:1707.04529] [INSPIRE].
B. Pullin and R. Zwicky, Radiative decays of heavy-light mesons and the \( {f}_{H,{H}^{\ast },{H}_1}^{(T)} \) decay constants, JHEP 09 (2021) 023 [arXiv:2106.13617] [INSPIRE].
A. Bazavov et al., B- and D-meson leptonic decay constants from four-flavor lattice QCD, Phys. Rev. D 98 (2018) 074512 [arXiv:1712.09262] [INSPIRE].
I. Caprini, Functional Analysis and Optimization Methods in Hadron Physics, Springer (2019) [https://doi.org/10.1007/978-3-030-18948-8] [INSPIRE].
J.A. Bailey et al., B → Kl+l− Decay Form Factors from Three-Flavor Lattice QCD, Phys. Rev. D 93 (2016) 025026 [arXiv:1509.06235] [INSPIRE].
HPQCD collaboration, Rare decay B → Kℓ+ℓ− form factors from lattice QCD, Phys. Rev. D 88 (2013) 054509 [Erratum ibid. 88 (2013) 079901] [arXiv:1306.2384] [INSPIRE].
A. Khodjamirian and A.V. Rusov, Bs → Kℓνℓ and B(s) → π(K)ℓ+ℓ− decays at large recoil and CKM matrix elements, JHEP 08 (2017) 112 [arXiv:1703.04765] [INSPIRE].
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 (2014) 094501 [arXiv:1310.3722] [INSPIRE].
R.R. Horgan, Z. Liu, S. Meinel and M. Wingate, Rare B decays using lattice QCD form factors, PoS LATTICE2014 (2015) 372 [arXiv:1501.00367] [INSPIRE].
EOS Authors collaboration, EOS: a software for flavor physics phenomenology, Eur. Phys. J. C 82 (2022) 569 [arXiv:2111.15428] [INSPIRE].
D. Van Dyk et al., EOS version 1.0.7, https://doi.org/10.5281/ZENODO.7915652.
E. Higson, W. Handley, M. Hobson and A. Lasenby, Dynamic nested sampling: an improved algorithm for parameter estimation and evidence calculation, Stat. Comput. 29 (2018) 891.
J.S. Speagle, dynesty: a dynamic nested sampling package for estimating Bayesian posteriors and evidences, Mon. Not. Roy. Astron. Soc. 493 (2020) 3132 [arXiv:1904.02180] [INSPIRE].
S. Koposov et al., dynesty version 2.0.3, https://doi.org/10.5281/ZENODO.7388523.
N. Gubernari, M. Reboud, D. van Dyk and J. Virto, EOS/DATA-2023-01: Supplementary material for EOS/ANALYSIS-2023-02, https://doi.org/10.5281/ZENODO.7919635.
B. Simon, Orthogonal polynomials on the unit circle, American Mathematical Society (2005).
M. Bello Hernández and G. López Lagomasino, Ratio and relative asymptotics of polynomials orthogonal on an arc of the unit circle, J. Approx. Theor. 92 (1998) 216.
I.V. Krasovsky, Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle, Int. Math. Res. Not. 2004 (2004) 1249.
W.W. Buck and R.F. Lebed, New constraints on dispersive form-factor parameterizations from the timelike region, Phys. Rev. D 58 (1998) 056001 [hep-ph/9802369] [INSPIRE].
Acknowledgments
We thank Wolfgang Altmannshofer, Paolo Gambino and Peter Stangl for useful comments on our manuscript. N.G. and D.v.D thank Marzia Bordone and Martin Jung for helpful discussions about splitting the dispersive bounds by the polarisation in the context of exclusive b → c form factors. M.R. thanks Guillermo Lopez and Brian Simanek for useful inputs on the asymptotic behaviour of orthonormal polynomials as well as the IJCLab for its hospitality during the time the paper was written. D.v.D. acknowledges support by the U.K. Science and Technology Facilities Council (grant numbers ST/V003941/1 and ST/X003167/1). J.V. acknowledges funding from the Spanish MINECO through the “Ramón y Cajal” program RYC-2017-21870, the “Unit of Excellence María de Maeztu 2020-2023” award to the Institute of Cosmos Sciences (CEX2019-000918-M) and from the grants PID2019-105614GB-C21 and 2017-SGR-92, 2021-SGR-249 (Generalitat de Catalunya).
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Gubernari, N., Reboud, M., van Dyk, D. et al. Dispersive analysis of B → K(*) and Bs → ϕ form factors. J. High Energ. Phys. 2023, 153 (2023). https://doi.org/10.1007/JHEP12(2023)153
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DOI: https://doi.org/10.1007/JHEP12(2023)153