Connecting B_d and B_s decays through QCD factorisation and flavour symmetries
aa r X i v : . [ h e p - ph ] O c t Connecting B d and B s decays through QCDfactorisation and flavour symmetries S´ebastien Descotes-Genon
Laboratoire de Physique Th´eorique,CNRS/Univ. Paris-Sud 11 (UMR 8627), 91405 Orsay Cedex, FranceE-mail: [email protected]
Abstract.
We analyse B d,s → K ( ∗ )0 K ( ∗ )0 modes within the SM, relating them in a controlledway through SU (3)-flavour symmetry and QCD-improved factorisation. We propose a set ofsum rules for such penguin-mediated decays to constrain some CKM angles. We determine B s → KK branching ratios and CP-asymmetries as functions of A dir ( B d → K ¯ K ). Applyingthe same techniques to B d,s → K ∗ K ∗ , we outline strategies to determine the B s mixing angle. Non-leptonic two-body B d - and B s -decays provide many interesting ways of testing theCKM mechanism of CP-violation, but the effects of strong interaction often hinder quantitativepredictions. The relevant hadronic quantities can be estimated through flavour symmetries,such as U -spin, but with a sizeable uncertainty. QCD factorisation (QCDF) provides acomplementary tool, specially for short distances, but this expansion in α s and 1 /m b cannotpredict some 1 /m b -suppressed long-distance effects. Recently, it was proposed to improvetheoretical predictions by combining QCDF and U -spin in particular classes decays [3, 4].
1. Sum rules
In the Standard Model (SM), we can always split a B -decay amplitude into its tree andpenguin contributions ¯ A ≡ A ( ¯ B q → M ¯ M ) = λ ( q ) u T qM + λ ( q ) c P qM according to the CKM factors λ ( q ) p = V pb V ∗ pq . One can compute these contributions for ¯ B s → K ¯ K within QCDF [1, 2]:ˆ T s = ¯ α u −
12 ¯ α u EW + ¯ β u + 2 ¯ β u −
12 ¯ β u EW − ¯ β u EW , (1)ˆ P s = ¯ α c −
12 ¯ α c EW + ¯ β c + 2 ¯ β c −
12 ¯ β c EW − ¯ β c EW , (2)where ˆ P sC = P sC /A sKK , ˆ T sC = T sC /A sKK and A qKK = M B q F ¯ B q → K (0) f K G F / √ β ’s denoteweak-annihilation contributions whereas α ’s collect remaining terms (vertex and hard-spectatorinteractions). A similar structure occurs fro the tree and penguin contributions T d and P d for ¯ B d → K ¯ K , and for longitudinally polarised K ∗ ¯ K ∗ [2, 4]. As exemplified in eqs. (1)-(2), for penguin-mediated decays, T and P are actually generated only by penguin topologies,and thus share the same long-distance dynamics: the difference comes from the ( u or c ) quarkrunning in the loops [3]. Thus, ∆ = T − P is midly affected by annihilation and hard-spectatorcontributions, and it can be computed with smaller uncertainties than T or P individually withinCDF: ∆ d ≡ T d − P d = A dKK [ α u − α c + β u − β c + 2 β u − β c ]. The 1 /m b suppressed long-distance dynamics, modelled in QCDF, cancels in the differences between u and c contributions.These theoretically well-behaved differences are related to the CP-averaged branching ratio BR and the direct and mixed CP-asymmetries A dir and A mix (see [3, 4] for the exact definitions).For a B d meson decaying through a b → D process ( D = d, s ) [such as B d → K ∗ ¯ K ∗ or B d → φ ¯ K ∗ (with a subsequent decay into a CP eigenstate)], one extracts α [5] and β from:sin α = g BR/ (2 | λ ( D ) u | | ∆ | ) (cid:18) − q − ( A dir ) − ( A mix ) (cid:19) , (3)sin β = g BR/ (2 | λ ( D ) c | | ∆ | ) (cid:18) − q − ( A dir ) − ( A mix ) (cid:19) , (4)where g BR is the CP-averaged branching ratio, up to a trivial kinematic factor [4]. Similaridentities can be used for a B s meson decaying through b → D ( D = d, s ) [such as B s → K ∗ ¯ K ∗ or B s → φ ¯ K ∗ ] to extract the angles β s and γ , assuming no New Physics in the decay. B d → K ¯ K and B s → K ¯ K These penguin-mediated decays are related by U -spin, with a small breaking: few processes(weak annhilation and spectator interaction) probe the spectator quark, as confirmed by QCDF: P s = f P d h A dKK /P d ) n δα c − δα c EW / δβ c + 2 δβ c − δβ c EW / − δβ c EW oi , (5) T s = f T d h A dKK /T d ) n δα u − δα u EW / δβ u + 2 δβ u − δβ u EW / − δβ u EW oi , (6)These ratios involve the U -spin breaking differences δα pi ≡ ¯ α pi − α pi (id. for β ). Apart from theratio f = M B s F ¯ B s → K (0) / [ M B d F ¯ B d → K (0)], U -spin arises only through 1 /m b -suppressed termsin which most long-distance effects have cancelled out. In agreement with this observation,QCDF [2] yields tiny uncertainties: | P s / ( f P d ) − | ≤
3% and | T s / ( f T d ) − | ≤ /m b -suppressed contributions than thepredictions for indivdiual tree or penguin contributions, and thus they provide an interestingalternative to a pure QCDF computation. One can also relate the penguin contributions to¯ B d → K ¯ K and ¯ B s → K + K − (see [3] for the treatment of tree contributions).For B d → K ¯ K , the branching ratio BR d = (0 . ± . · − [6] has been measured. If A d dir becomes available, we may exploit the theoretically well-controlled value of ∆ d ≡ T d − P d to get the two moduli and the relative phase of T d and P d from BR d , A d dir and ∆ d . Thenwe can use the previous bounds to compute the tree and penguin contributions for B s → KK decays, leading to the SM predictions for the corresponding observables (see Table I in ref. [3]): Br ( B s → K ¯ K ) = (18 ± ± ± · − and Br ( B s → K + ¯ K − ) = (20 ± ± ± · − , thelatter being in very good agreement with the latest CDF measurement [7]. B d → K ∗ ¯ K ∗ and B s → K ∗ ¯ K ∗ We focus on observables for mesons with a longitudinal polarisation which can be measuredexperimentally and predicted theoretically with a good accuracy. B s → K ∗ ¯ K ∗ is in principlea clean mode to extract the mixing angle φ s . An expansion in powers of λ ( s ) u /λ ( s ) c yields A longmix ( B s → K ∗ ¯ K ∗ ) ≃ sin φ s +2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ( s ) u λ ( s ) c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Re T s ∗ P s ∗ ! sin γ cos φ s + · · · = sin φ s +∆ S ( B s → K ∗ ¯ K ∗ )(7)A significant value of T s ∗ /P s ∗ could spoil the extraction of sin φ s . One can use our knowledgeon T s ∗ − P s ∗ to bound ∆ S ( B s → K ∗ ¯ K ∗ ), as illustrated in fig. 1. H B s ® K * K * L ´ D S H B s ® K * K * L -15 -10 -5 0 5 10 15phis-0.2-0.100.10.20.3 Amix
Figure 1.
The absolute bounds on ∆ S ( B s → K ∗ ¯ K ∗ ) as functions of BR long ( B s → K ∗ ¯ K ∗ )(on the left) and the relation between A longmix ( B s → K ∗ ¯ K ∗ ) and the B s − ¯ B s mixing angle φ s (on the right), assuming BR long ( B d → K ∗ ¯ K ∗ ) ≥ × − and γ = 62 ◦ .One can also relate the observables in B d,s → K ∗ ¯ K ∗ through the same combinationof U -spin symmetry and QCDF. Once again, U -spin is mainly broken through the ratioof relevant form factors f ∗ , whereas most of the long-distance annihilation and spectatorscattering contributions cancel in P s ∗ / ( f ∗ P d ∗ ) and T s ∗ / ( f ∗ T d ∗ ). Indeed, QCDF yields | P s ∗ / ( f ∗ P d ∗ ) − | ≤
12% and | T s ∗ / ( f ∗ T d ∗ ) − | ≤ B s → K ∗ ¯ K ∗ observables. The ratio of branching ratios BR long ( B s → K ∗ ¯ K ∗ ) /BR long ( B d → K ∗ ¯ K ∗ ) and the asymmetries as predicted in the SM turn out to be quite insensitive to theexact value of BR long ( B d → K ∗ ¯ K ∗ ) as long as BR long ( B d → K ∗ ¯ K ∗ ) ≥ × − : BR long ( B s → K ∗ ¯ K ∗ ) /BR long ( B d → K ∗ ¯ K ∗ ) = 17 ± A longdir ( B s → K ∗ ¯ K ∗ ) = 0 . ± . A longmix ( B s → K ∗ ¯ K ∗ ) = 0 . ± .
018 (9)If one assumes no New Physics in the decay B s → K ∗ ¯ K ∗ , this method relates directly A longmix ( B s → K ∗ ¯ K ∗ ) and φ s as indicated in fig.1.
4. Conclusions
We have combined experimental data, flavour symmetries and QCDF to gain control on penguin-mediated B d,s decays. The difference between tree and penguin contributions can be assessedwith a good accuracy. The U -spin breaking between B d and B s modes arises in few factorisablecorrections (ratio of form factors) and non-factorisable corrections (weak annihilation andspectator scattering). QCDF confirms these expectations, and provides predictions with alimited model dependence on 1 /m b -suppressed long-distance contributions. We outlined theimplications for B s → K ¯ K in pseudoscalar and vector channels. Sizeable NP effects would breakthese SM correlations between B d and B s decays, leading to departure from our predictions. Acknowledgments
Work supported in part by EU Contract No. MRTN-CT-2006-035482, “FLAVIAnet”. [1] M.Beneke et al. , Nucl.Phys.B (2000) 313; Nucl.Phys.B (2001) 245[2] M. Beneke and M. Neubert, Nucl.Phys.B (2003) 333[3] S. Descotes-Genon, J. Matias and J. Virto, Phys. Rev. Lett. (2006) 061801[4] S. Descotes-Genon, J. Matias and J. Virto, accepted for publication in Phys. Rev. D, arXiv:0705.0477.[5] A. Datta, M. Imbeault, D. London and J. Matias, Phys. Rev. D , 093004 (2007)[6] B. Aubert et al. [BaBar], Phys. Rev. Lett. (2005) 221801; H. C. Huang [Belle], hep-ex/0205062.[7] M. Morello, CP asymmetries at CDF , this conference, and Nucl. Phys. Proc. Suppl.170