Model independent analysis of a class of $\bar{B}_s^0 $ decay modes

The widths of a class of two-body $\bar{B}_s^0 $ decays induced by $b \to c \bar u d$ and $b \to c \bar u s$ transitions are determined in a model-independent way, using $SU(3)_F$ symmetry and existing information on $\bar B \to D_{(s)} P$ and $\bar B \to D_{(s)} V$ decays, with $P$ and $V$ a light pseudoscalar or vector meson. The results are relevant for the $B_s$ physics programmes at the hadron colliders and at the $e^+ e^-$ factories running at the peak of $\Upsilon(5S)$.

In the next few years an intense B s physics programme will be pursued at the hadron colliders, the Fermilab Tevatron and the CERN LHC, and at the e + e − factories running at Υ(5S). The programme includes precise determination of the B s − B s mixing parameters and search for CP violating asymmetries in B s decays, with the aim of providing new tests of the Standard Model (SM) and searching for physics beyond SM. The analysis of rare B s transitions is another aspect of the research programme, with the same aim of looking for deviations from SM expectations.
The knowledge of non leptonic B s decay rates is of prime importance for working out the research programme. For example, B s − B s mixing can be studied using B s twobody hadronic decay modes in addition to semileptonic modes. It is noticeable that the widths of a set of two-body transitions can be predicted in a model independent way, using the symmetries of QCD and available information on B decays. We are referring in particular to a class of decay modes induced by the quark transitions b → cūd and b → cūs, for example those collected in Table 1. The key observation is that the various Table 1: SU(3) decay amplitudes for B 0 s → D (s) P decays, with P a light pseudoscalar meson. In the last column the corresponding branching fractions predicted using the method described in the text are reported.
decay modes are governed, in the SU(3) F limit, by few independent amplitudes that can be constrained, both in moduli and in phase differences, from corresponding B decay processes.
Considering transitions with a light pseudoscalar meson belonging to the octet in the final state, the scheme where the correspondence can be established involves the three different topologies in B 0 s decays induced by b → cūd(s), namely the color allowed topology T , the color suppressed topology C and the W -exchange topology E. The transition in the SU(3) singlet η 0 involves another amplitude D in principle not related to the previous ones. Notice that the identification of the different amplitudes is not graphical, it is based on SU(3) [1]. Since B → DP decays induced by the quark processes b → cuq (q = d or s) involve a weak Hamiltonian transforming as a flavor octet, using de Swart's notation T (µ) ν for the ν = (Y, I, I 3 ) component of an irreducible tensor operator of rank (µ) [2], one can write:  Table 1, i.e. the color suppressed, color enhanced and W-exchange diagrams, respectively. The SU(3) representation for B decays is reported in Table 2.
Considering Table 2 one realizes that the threeB → DK experimental rates could allow to obtain |T |, |C| and the phase difference δ C − δ T . This was already observed in [4], and can be recast in the determination of the two independent isospin amplitudes A 1 and A 0 for I = 1 and I = 0 isospin DK final states: the difference of the B − andB 0 lifetimes: τ B − = 1.671 ± 0.018 ps and τ B 0 = 1.537 ± 0.014 ps, but neglecting the tiny phase space correction due to the difference between p D 0 K − = p D 0K 0 = 2280 MeV and p D +K − = 2279 MeV, with p the modulus of the three-momentum of one of the two final mesons in the B rest frame, one would obtain allowed region for A 0 /A 1 at various confidence levels by minimizing the χ 2 function for the three branching ratios and plotting the χ 2 contours that correspond to a given confidence level, as done in fig.1. Due to the quality of the experimental data and to the correlation between |A 0 /A 1 | and δ 0 −δ 1 , the allowed region is not tightly constrained, in particular the phase difference could be zero.
We pause here, since we can elaborate once more about factorization approximations sometimes adopted for computing non leptonic decays, in this case for B mesons [5]. In fig.1 we have shown the predictions by, e.g., naive factorization, where the decay amplitudes are written in terms of K and D meson leptonic constants f K and f D , and the B → D and B → K form factors F 0 : . The result of this approach corresponds to vanishing phase difference; using a 1 = c 1 + c 2 /3 and a 2 = c 2 +c 1 /3, with c 1 and c 2 the Wilson coefficients appearing in the effective hamiltonian inducing the decays (for their numerical values we quote a 1 = (1.036, 1.017, 1.025) and a 2 = (0.073, 0.175, 0.140) at LO and at NLO (in NDR and HV renormalization schemes) accuracy, respectively [6]) we obtain results corresponding to the dots along the horizontal axis in fig. 1, which do not belong to the region permitted by experimental data at 95% CL. In generalized factorization, where a 1 and a 2 are considered as parameters, the phase difference is constrained to be zero, too. This is allowed by the experimental data on these three channels, but excluded if one considers all channels, as we shall see below.
Coming to bounding the decay amplitudes, the fourB → Dπ andB → D s K decay rates cannot determine C, T , E and their phase differences [7].B → D s K only fixes the modulus of E, which is not small at odds with the expectations by factorization, where Wexchange processes are suppressed by ratios of decay constants and form factors and are usually considered to be negligible. Moreover, the presence of E does not allow to directly relate color favoured T or color suppressed C decay amplitudes in Dπ and DK final states. What can be done, however, is to use all the information onB → Dπ, D s K and DK (7 experimental data) to determine T , C and E (5 parameters). A similar attitude has been recently adopted in [8]. Noticeably, the combined experimental information is enough accurate to tightly determine the ranges of variation for all these quantities.
In fig. 2 we have depicted the allowed regions in the C/T and E/T planes, obtained fixing the other variables to their fitted values, with the corresponding confidence levels.
It is worth noticing that the phase differences between the various amplitudes are close to be maximal; this signals again deviation from naive (or generalized) factorization, provides contraints to QCD-based approaches proposed to evaluate non leptonic B decay amplitudes [9,10,11] and points towards sizeable long-distance effects in C and E [12,13].
To obtain the amplitudes we have fixed the ratio |V us /V ud | to the experimental result: |V us /V ud | = 0.226 ± 0.003, and we have taken into account the phase space correction due to p DK , p Dπ = 2306 MeV and p DsK − = 2242 MeV. We obtain | C T | = 0.53 ± 0.10, | E T | = 0.115 ± 0.020, δ C − δ T = (76 ± 12) • and δ E − δ T = (112 ± 46) • . We have to mention that the accuracy of the fit is not particularly high since χ 2 /dof = 2.3, i.e. a fit probability of 10%. This is entirely due to a single entry in Table 2 Table 1. The uncertainties in the predicted rates are small; in particular, the W -exchange induced processes B 0 s → D + π − , D 0 π 0 are precisely estimated [14].
Considering the decays with η or η ′ in the final state, they involve the amplitude D corresponding to the transition in a SU(3) singlet η 0 , and the η − η ′ mixing angle θ (in a one angle mixing scheme): If we use the value θ = −15.4 0 for the mixing angle [15], we obtain | D T | = 0.41 ± 0.11 without sensibly constraining the D − T phase difference, δ D − δ T = −(25 ± 51) • . Corresponding B 0 s decay rates are predicted consequently. The key of the success of the programme of predicting B s decay rates is the small number of amplitudes in comparison to the available data, a feature which is not common to all processes. Considering b → cūd(s) induced transitions, one could look at the case of one light vector meson in the final state, with the same SU(3) decomposition reported in Tables 1, 2 (we denote by a prime the amplitudes involved in this case). B decay data are collected in Table 3. The difference with respect to the previous case is that the W-exchange modeB 0 → D + s K * − has not been observed, yet, therefore the E ′ amplitude is poorly determined considering only the other modes. Taking into account phase space corrections due to p Dρ = 2235 MeV and p DK * = 2211 MeV, we obtain | C ′ T ′ | = 0.36 ± 0.10, predictions for B 0 s decay rates are collected in Table 3: as anticipated, the accuracy is not high for W −exchange induced decays. On the other hand, the prediction for the rate of Table 3.

is compatible with the upper bound in
Considering other decay modes induced by the same quark transitions, namelyB → D * (s) P andB → D * (s) V decays, the present experimental data are not precise enough to sensibly constrain the independent amplitudes and to provide stringent predictions for B s . As soon as the experimental accuracy will improve, a similar analysis will be possible to describe B 0 s → D * (s) P modes, while the three helicityB → D * (s) V amplitudes will be needed to determine the corresponding B s decays.  Let us finally comment on the possible role of SU(3) F breaking terms that can modify our predictions. Those effects are not universal, and in general cannot be reduced to well defined and predictable patterns without new assumptions. Their parametrization would introduce additional quantities [16] that at present cannot be sensibly bounded since their effects seem to be smaller than the experimental uncertainties. Therefore they can be neglected until the experimental errors remain at the present level. It will be interesting to investigate their role when the B s decay rates will be measured and more precise B branching fractions will be available.

Acnowledgments
We thank F. De Fazio for discussions. We acknowledge partial support from the EC Contract No. HPRN-CT-2002-00311 (EURIDICE).