# Form factors \( f_{ + }^{{B \to \pi }}(0) \) and \( f_{ + }^{{D \to \pi }}(0) \) in *QCD* and determination of |*V* _{ ub }| and |*V* _{ cd }|

- 72 Downloads
- 11 Citations

## Abstract

We present a QCD study on *B, D* → *π* semileptonic transitions at zero momentum transfer and an estimate of magnitudes of the associated CKM matrix elements. Light cone sum rules (LCSRs) with chiral correlator are applied to calculate the form factors \( f_{ + }^{{B \to \pi }}(0) \) and \( f_{ + }^{{D \to \pi }}(0) \). We show that there is no twist-3 and-5 component involved in the light-cone expansions such that the resulting sum rules have a good convergence and offer an understanding of these form factors at twist-5 level. A detailed \( \mathcal{O}\left( {{\alpha_s}} \right) \) computation is carried out in leading twist-2 approximation and the \( \overline {MS} \) masses are employed for the underlying heavy quarks. With the updated inputs and experimental data, we have \( f_{ + }^{{B \to \pi }}(0) = 0.28_{{ - 0.02}}^{{ + 0.05}} \) and \( \left| {{V_{{ub}}}} \right| = \left( {3.4_{{ - 0.6}}^{{ + 0.2}}\pm 0.1\pm 0.1} \right) \times {10^{{ - 3}}};f_{ + }^{{D \to \pi }}(0) = 0.62\pm 0.03 \) and \( \left| {{V_{{cd}}}} \right| = 0.244\pm 0.005\pm 0.003\pm 0.008 \). As a by-product, a numerical estimate for the decay constant *f* _{ D } is yielded as \( {f_D} = 190_{{ - 11}}^{{ + 15}}{\text{MeV}} \).

## Keywords

QCD Phenomenology NLO Computations## References

- [1]BABAR collaboration, P. del Amo Sanchez et al., Measurement of the
*B*^{0}→*π*^{ℓ}*ℓ*^{+}*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].ADSGoogle Scholar - [2]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].ADSGoogle Scholar - [3]CLEO collaboration, D. Besson et al.,
*Improved measurements of D meson semileptonic decays to π and K mesons*,*Phys. Rev.***D 80**(2009) 032005 [arXiv:0906.2983] [INSPIRE].ADSGoogle Scholar - [4]CLEO collaboration, B. Eisenstein et al.,
*Precision measurement of B*(*D*^{+}→*μ*^{+}*ν*)*and the pseudoscalar decay constant*\( f_D^{ + } \),*Phys. Rev.***D 78**(2008) 052003 [arXiv:0806.2112] [INSPIRE].ADSGoogle Scholar - [5]CKMfitter Group collaboration, J. Charles et al.,
*CP violation and the CKM matrix: assessing the impact of the asymmetric B factories*,*Eur. Phys. J.***C 41**(2005) 1 [hep-ph/0406184] [INSPIRE].ADSCrossRefGoogle Scholar - [6]UTfit collaboration, A. Bevan et al.,
*Update of the unitarity triangle analysis*, PoS(ICHEP 2010)270 [arXiv:1010.5089] [INSPIRE]. - [7]
- [8]
- [9]I. Balitsky, V.M. Braun and A. Kolesnichenko,
*Radiative decay*Σ^{+}→*pγ in quantum chromodynamics*,*Nucl. Phys.***B 312**(1989) 509 [INSPIRE].ADSCrossRefGoogle Scholar - [10]V.L. Chernyak and I.R. Zhitnitsky,
*B-meson exclusive decays into baryons*,*Nucl. Phys.***B 345**(1990) 137 [INSPIRE].ADSCrossRefGoogle Scholar - [11]V. Belyaev, A. Khodjamirian and R. Rückl,
*QCD calculation of the B*→*π*,*K form-factors*,*Z. Phys.***C 60**(1993) 349 [hep-ph/9305348] [INSPIRE].ADSGoogle Scholar - [12]V. Belyaev, V.M. Braun, A. Khodjamirian and R. Rückl,
*D*^{∗}*Dπ and B*^{∗}*Bπ couplings in QCD*,*Phys. Rev.***D 51**(1995) 6177 [hep-ph/9410280] [INSPIRE].ADSGoogle Scholar - [13]A. Khodjamirian, R. Rückl, 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].ADSGoogle Scholar - [14]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].ADSGoogle Scholar - [15]P. Ball,
*B*→*π*and*B*→*K transitions from QCD sum rules on the light cone*,*JHEP***09**(1998) 005 [hep-ph/9802394] [INSPIRE]. - [16]P. Ball and R. Zwicky,
*Improved analysis of B*→*πeν from QCD sum rules on the light cone*,*JHEP***10**(2001) 019 [hep-ph/0110115] [INSPIRE].ADSCrossRefGoogle Scholar - [17]A. Khodjamirian, R. Rückl, S. Weinzierl, C. Winhart and O.I. Yakovlev,
*Predictions on B*→*π*\( \overline \ell \)*ν*_{ℓ},*D*→*π*\( \overline \ell \)*ν*_{ℓ}*and D → K*\( \overline \ell \)*ν*_{ℓ}*from QCD light cone sum rules*,*Phys. Rev.***D 62**(2000) 114002 [hep-ph/0001297] [INSPIRE].ADSGoogle Scholar - [18]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].ADSGoogle Scholar - [19]P. Ball and R. Zwicky,
*|V*_{ub}*| and constraints on the leading-twist pion distribution amplitude from B*→*πℓν*,*Phys. Lett.***B 625**(2005) 225 [hep-ph/0507076] [INSPIRE].ADSGoogle Scholar - [20]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].ADSCrossRefGoogle Scholar - [21]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].ADSGoogle Scholar - [22]T. Huang, Z.-H. Li and X.-Y. Wu,
*Improved approach to the heavy to light form-factors in the light cone QCD sum rules*,*Phys. Rev.***D 63**(2001) 094001 [INSPIRE].ADSGoogle Scholar - [23]X.-G. Wu and T. Huang,
*Radiative corrections on the B*→*P form factors with chiral current in the light-cone sum rules*,*Phys. Rev.***D 79**(2009) 034013 [arXiv:0901.2636] [INSPIRE].ADSGoogle Scholar - [24]P. Ball,
*Testing QCD sum rules on the light-cone in D*→ (*π, K*)*ℓν decays*,*Phys. Lett.***B 641**(2006) 50 [hep-ph/0608116] [INSPIRE].ADSGoogle Scholar - [25]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].ADSGoogle Scholar - [26]G.P. Lepage and S.J. Brodsky,
*Exclusive processes in quantum chromodynamics: evolution equations for hadronic wave functions and the form-factors of mesons*,*Phys. Lett.***B 87**(1979) 359 [INSPIRE].ADSGoogle Scholar - [27]G.P. Lepage and S.J. Brodsky,
*Exclusive processes in perturbative quantum chromodynamics*,*Phys. Rev.***D 22**(1980) 2157 [INSPIRE].ADSGoogle Scholar - [28]P. Ball and A.N. Talbot,
*Models for light-cone meson distribution amplitudes*,*JHEP***06**(2005) 063 [hep-ph/0502115] [INSPIRE].ADSCrossRefGoogle Scholar - [29]S. Agaev, V. Braun, N. Offen and F. Porkert,
*Light cone sum rules for the π*^{0}*γ*^{∗}*γ form factor revisited*,*Phys. Rev.***D 83**(2011) 054020 [arXiv:1012.4671] [INSPIRE].ADSGoogle Scholar - [30]P. Ball, V. Braun and A. Lenz,
*Twist-*4*distribution amplitudes of the K*^{∗}*and ϕ mesons in QCD*,*JHEP***08**(2007) 090 [arXiv:0707.1201] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar - [31]J.H. Kühn, M. Steinhauser and C. Sturm,
*Heavy quark masses from sum rules in four-loop approximation*,*Nucl. Phys.***B 778**(2007) 192 [hep-ph/0702103] [INSPIRE].ADSCrossRefGoogle Scholar - [32]R. Boughezal, M. Czakon and T. Schutzmeier,
*Charm and bottom quark masses from perturbative QCD*,*Phys. Rev.***D 74**(2006) 074006 [hep-ph/0605023] [INSPIRE].ADSGoogle Scholar - [33]Particle Data Group collaboration, K. Nakamura et al.,
*Review of particle physics*,*J. Phys.***G 37**(2010) 075021 [INSPIRE].ADSGoogle Scholar - [34]BABAR collaboration, R. Barlow,
*Experimental status of B → τν and B → ℓν*(*γ*), arXiv:1102.1267 [INSPIRE]. - [35]T. Huang and Z.-H. Li,
*The binding energy of the excited heavy light mesons in HQET*,*Phys. Lett.***B 438**(1998) 159 [INSPIRE].ADSGoogle Scholar - [36]QCDSF collaboration, A. Al-Haydari et al.,
*Semileptonic form factors D → π, K and B → π, K from a fine lattice*,*Eur. Phys. J.***A 43**(2010) 107 [arXiv:0903.1664] [INSPIRE].ADSGoogle Scholar - [37]Fermilab Lattice, MILC and HPQCD collaborations, C. Aubin et al.,
*Semileptonic decays of D mesons in three-flavor lattice QCD*,*Phys. Rev. Lett.***94**(2005) 011601 [hep-ph/0408306] [INSPIRE].ADSCrossRefGoogle Scholar - [38]C. Bernard, C. DeTar, M. Di Pierro, A. El-Khadra, R. Evans, et al.,
*Visualization of semileptonic form factors from lattice QCD*,*Phys. Rev.***D 80**(2009) 034026 [arXiv:0906.2498] [INSPIRE].ADSGoogle Scholar - [39]A. Abada et al.,
*Heavy to light semileptonic decays of pseudoscalar mesons from lattice QCD*,*Nucl. Phys.***B 619**(2001) 565 [hep-lat/0011065] [INSPIRE].ADSCrossRefGoogle Scholar - [40]H. Na et al.,
*D → π, ℓν semileptonic decays*,*|V*_{cd}*| and 2*^{nd}*row unitarity from lattice QCD*,*Phys. Rev.***D 84**(2011) 114505 [arXiv:1109.1501] [INSPIRE].ADSGoogle Scholar - [41]S. Di Vita et al.,
*Form factors of the D → π and D → K semileptonic decays with N*_{f}= 2*twisted mass lattice QCD*, PoS(LATTICE 2010)301 [arXiv:1104.0869] [INSPIRE]. - [42]ETM collaboration, B. Blossier et al.,
*Pseudoscalar decay constants of kaon and D-mesons from N*_{f}= 2*twisted mass lattice QCD*,*JHEP***07**(2009) 043 [arXiv:0904.0954] [INSPIRE].Google Scholar - [43]Fermilab Lattice and MILC collaborations, A. Bazavov et al.,
*B- and D-meson decay constants from three-flavor lattice QCD*, arXiv:1112.3051 [INSPIRE]. - [44]C. Davies et al.,
*Update: precision D*_{s}*decay constant from full lattice QCD using very fine lattices*,*Phys. Rev.***D 82**(2010) 114504 [arXiv:1008.4018] [INSPIRE].ADSGoogle Scholar - [45]S. Narison,
*Light and heavy quark masses, flavor breaking of chiral condensates, meson weak leptonic decay constants in QCD*, hep-ph/0202200 [INSPIRE]. - [46]A. Penin and M. Steinhauser,
*Heavy light meson decay constant from QCD sum rules in three loop approximation*,*Phys. Rev.***D 65**(2002) 054006 [hep-ph/0108110] [INSPIRE].ADSGoogle Scholar - [47]J. Bordes, J. Penarrocha and K. Schilcher,
*D and D*_{S}*decay constants from QCD duality at three loops*,*JHEP***11**(2005) 014 [hep-ph/0507241] [INSPIRE].ADSCrossRefGoogle Scholar