Cp Violation Highlights: circa 2005

Recent highlights in CP violation phenomena are reviewed. B-factory results imply that CP-violation phase in the CKM matrix is the dominant contributor to the observed CP violation in K and B-physics. Deviations from the predictions of the CKM-paradigm due to beyond the Standard Model CP-odd phase are likely to be a small perturbation. Therefore, large data sample of clean B’s will be needed. Precise determination of the unitarity triangle, along with time dependent CP in penguin dominated hadronic and radiative modes are discussed. Null tests in B, K and top-physics and separate determination of the K-unitarity triangle are also emphasized. 1 B-factories help attain an important milestone: Good and bad news The two asymmetric B-factories at SLAC and KEK have provided a striking confirmation of the CKM paradigm 1). Existing experimental information from the indirect CP violation parameter, ǫ for the KL → ππ, semileptonic b → ueν and B − B̄ mixing along with lattice calculations predict that in the SM, (sin 2β) ≃ .70± .10 2, 3, 4). This is in very good agreement with the BELLE and BABAR result 5): ACP (B → ψK) = sin 2β = .726± .037 (1) This leads to the conclusion that the CKM phase of the Standard Model (SM) is the dominant contributor to ACP . That, of course, also means that CP-odd phase(s) due to beyond the Standard Model (BSM) sources may well cause only small deviations from the SM in B-physics. Actually, there are several reasons to think that BSM phase(s) may cause only small deviations in B-physics. In this regard, SM itself teaches a very important lesson. 2 Important lesson from the CKM-paradigm We know now that the CKM phase is 0(1) (actually, the CP violation parameter η is 0(.3) 2, 3, 4)). The CP effects that it causes on different observables though is quite different. In K-decays, the CP asymmetrics are ≤ 10. In charm physics, also there are good reasons to expect small observable effects. In top physics, the CKM phase causes completely negligible effects 6, 7). Thus only in B-decays, the large asymmetries (often 0(1)) are caused by the CKM phase. So even if the BSM phase(s) are 0(1) it is unlikely that again in B-physics they will cause large effects just as the SM does. 3 Remember the mν Situation with regard to BSM CP-odd phase(s) (χBSM ) is somewhat reminiscent of the neutrino mass (mν) 8). There was no good reason for mν to be zero; similarly, there are none for χBSM to be zero either. In the case of ν ′ s, there were the solar ν results that were suggestive for a very long time; similarly, in the case of χBSM , the fact that in the SM, baryogenesis is difficult to accomodate serves as the beacon. It took decades to show mν is not zero: ∆m 2 had to be lowered from ∼ O(1 − 10)eV 2 around 1983 down to O(10eV ) before mν 6= 0 was established via neutrino oscillations. We can hope for better luck with χBSM but there is no good reason to be too optimistic; therefore, we should not rely on luck but rather we should seriously prepare for this possibility. To recapitulate, just as the SM-CKM phase is 0(1), but it caused only 0(10) CP symmetries in Kdecays, an 0(1) BSM-CP-odd phase may well cause only very small asymmetries in B-physics. To search for such small effects: 1) We need lots and lots of clean B’s (i.e. 0(10) or more) 2) Intensive study of Bs mesons (in addition to B’s) becomes very important as comparison between the two types of B-mesons will teach us how to improve quantitative estimates of flavor symmetry breaking effects. 3) We also need clean predictions from theory (wherein item 2 should help). 4 Improved searches for BSM phase Improved searches for BSM-CP-odd phase(s) can be subdivided into the following main categories: a) Indirect searches with theory input b) Indirect searches without theory input c) Direct searches. 4.1 Indirect searches with theory input Among the four parameters of the CKM matrix, λ,A, ρ and η, λ = 0.2200 ± 0.0026, A ≈ 0.850 ± 0.035 9) are known quite precisely; ρ and η still need to be determined accurately. Efforts have been underway for many years to determine these parameters. The angles α, β, γ, of the unitarity triangle (UT) can be determined once one knows the 4-CKM parameters. A well studied strategy for determining these from experimental data requires knowledge of hadronic matrix elements. Efforts to calculate several of the relevant matrix elements on the lattice, with increasing accuracy, have been underway for past many years. A central role is played by the following four inputs 2, 3, 4): • BK from the lattice with ǫ from experiment • fB √ BB from the lattice with ∆md from experiment • ξ from the lattice with ∆ms ∆md from experiment • b→ulν b→clν from experiment, along with input from phenomenology especially heavy quark symmetry as well as the lattice. As mentioned above, for the past few years, these inputs have led to the important constraint: sin 2βSM ≈ 0.70±0.10 which is found to be in very good agreement with direct experimental determination, Eq. 1. Despite severe limitations (e.g. the so-called quenched approximation) these lattice inputs provided valuable help so that with B-Factory measurements one arrives at the very important conclusion that in B → J/ψ K the CKM-phase is the dominant contributor; any new physics (NP) contribution is unlikely to be greater than about 15%. What sort of progress can we expect from the lattice in the next several years in these (indirect) determination of the UT? To answer this it is useful to look back and compare where we were to where we are now. Perhaps this gives us an indication of the pace of progress of the past several years. Lattice calculations of matrix elements around 1995 10) yielded (amongst other things) sin 2β ≈ 0.59±0.20, whereas the corresponding error decreased to around±0.10 around 2001 2). In addition to β, such calculations also now constrain γ(≈ 60) with an error of around 10 2). There are three important developments that should help lattice calculations in the near future: 1. Exact chiral symmetry can be maintained on the lattice. This is especially important for light quark physics. 2. Relatively inexpensive methods for simulations with dynamical quarks (esp. using improved staggered fermions 11)) have become available. This should help overcome limitations of the quenched approximation. 3. About a factor of 20 increase in computing power is now being used compared to a few years ago. As a specific example one can see that the error on BK with the 1st use of dynamical domain wall fermions 12) now seems to be reduced by about a factor of two 13). In the next few years or so errors on lattice determination of CKM parameters should decrease appreciably, perhaps by a factor of 3. So the error in sin 2βSM ± 0.10 → ±0.03; γ ± 10 → 4 etc. While this increase in accuracy is very welcome, and will be very useful, there are good reasons to believe, experiment will move ahead of theory in direct determinations of unitarity angles in the next 5 years. (At present, experiment is already ahead of theory for sin 2β). 4.2 Indirect searches without theory input: Elements of a superclean UT One of the most exciting developments of recent years in B-physics is that methods have been developed so that all three angles of the UT can be determined cleanly with very small theory errors. This is very important as it can open up several ways to test the SM-CKM paradigm of CP violation; in particular, the possibility of searching for small deviations. Let us very briefly recapitulate the methods in question: • Time dependent CP asymmetry (TDCPA) measurements in B, B̄ → ψK type of final states should give the angle β very precisely with an estimated irreducible theory error (ITE) of ≤ O(0.1%) 14). • Direct CP (DIRCP) studies in B → “K”D, D̄ gives γ very cleanly 15, 16). • TDCPA measurements in B, B̄ → “K”D, D̄ gives (2β+ γ) and also β very cleanly 17, 18). • In addition, TDCPA measurements in Bs → KDs type modes also gives γ very cleanly 19). • Determination of the rate for the CP violating decay KL → πνν̄ is a very clean way to measure the Wolfenstein parameter η, which is indeed the CP-odd phase in the CKM matrix 20). It is important to note that the ITE for each of these methods is expected to be ≤ 1%, in fact perhaps even ≤ 0.1%. • Finally let us briefly mention that, TDCPA studies of B, B̄ → ππ or ρπ or ρρ gives α 21, 22, 23). However, in this case, isospin conservation needs be used and that requires, assuming that electro-weak penguins (EWP) make negligible contribution. This introduces some model dependence and may cause an error of order a few degrees, i.e. for α extraction the ITE may well end up being O(a few %). However, given that there are three types of final states each of which allows a determination of α, it is quite likely that further studies of these methods will lead to a reduction of the common source of error originating from isopsin violation due to the EWP. It is extremely important that we make use of these opportunities afforded to us by as many of these very clean redundant measurements as possible. In order to exploit these methods to their fullest potential and get the angles with errors of order ITE will, for sure, require a SUPER-B Factory(SBF) 8, 30, 31, 32). This in itself constitutes a strong enough reason for a SBF, as it represents a great opportunity to precisely nail down the important parameters of the CKM paradigm. 4.2.1 Prospects for precision determination of γ Below we briefly discuss why the precision extraction of γ seems so promising. For definiteness, let us recall the basic features of the ADS method 24). In this interference is sought between two amplitudes of roughly similar size i.e. B → KD and B → KD̄ where the D and D̄ decay to common final states such as the simple two body ones like Kπ, Kρ, Ka−1 , K π or they may also be multibody modes e.g. the Dalitz decayKππ, Kπππ etc. It is easy to see that the interference is between a colored allowed B decay followed by doubly Cabibbo suppressed D decay and a color-suppressed B decay followed by Cabibbo allowed D decay and consequently then interference tends to be maximal and should lead to large asymmetries. For a given (common) final state of D and D̄ the amplitude involves three unknowns: the color suppressed Br(B → KD̄), which is not directly accessible to experiment 24), the strong phase ξ fi and the weak phase γ. Corresponding to each such final state (FS) there are two observables: the rate for B decay and for the B decay. Thus, if you stick to just one common FS of D, D̄, you do not have enough information to solve for γ. If you next consider two common FS of D and D̄ then you have one additional unknown (a strong phase) making a total of 4-unknowns with also 4-observables. So with two final states the system becomes soluble, i.e. we can then use the experimental data to solve not only the value of γ but also the strong phases and the suppressed Br for B → KD̄. With N common FS ofD and D̄, you will have 2N observables and N + 2 unknowns. We need 2N ≥ (N + 2) i.e. N ≥ 2. The crucial point, though, is that there are a very large number of possible common modes of D and D̄ which can all be used to improve the determination of γ. Let us briefly mention some of the relevant common modes of D and D̄: • The CP-eigenstate modes, originally discussed by GLW 21): KS [π, η, η, ρ, ω]; ππ,.... • CP-non-eigenstates (CPNES), discussed by GLS 26): KK, ρπ... These are singly Cabibbo suppressed modes. • CPNES modes originally discussed by ADS 24, 25): K[π, ρ, a−1 ....] • There are also many multibody modes, such as the Dalitz D decays: KSπ π 27) or Kππ 25) etc; and also modes such as Kπππ, Kππππ, or indeed Kπ +nπ 17, 28, 29). Furthermore, multibody modes such as B → K i D → (Kπ)D or (Knπ)D 28, 33) can also be used. Fig. 1 and Fig. 2 show how combining different strategies helps a great deal. In the fig we show χ versus γ. As indicated above when you consider an individual final state of D and D̄ then of course there are 3 unknowns ( the strong phase, the weak phase (γ) and the “unmeasureable” Br) and only two observables (the rate for B and the rate for B). So in the figure, for a fixed value of γ, we search for the minimum of the χ by letting the strong phase and the “unmeasureable” Br take any value they want. Fig. 1 and Fig. 2 show situtation with regard to under determined and over determined cases respectively. The upper horizontal line corresponds roughly to the low luminosity i.e. comparable to the current B-factories 30, 32) whereas the lower horizontal curve is relevant for a super B-factory. In Fig. 1 in blue is shown the case when only the input from (GLW) CPES modes of D is used; note all the CPES modes are included here. You see that the resolution on γ then is very poor. In particular, this method is rather ineffective in giving a lower bound; its upper bound is better. In contrast, a single ADS mode (Kπ) is very effective in so far as lower bound is concerned, but it does not yield an effective upper bound (red). Note that in these two cases one has only two observables and 3 unknowns. In purple is shown the situation when these two methods are combined. Then at least at high luminosity there is significant improvement in attaining a tight upper bound; lower bound obtained by ADS alone seems largely unaffected. Shown in green is another under determined case consisting of the use of a single ADS mode, though it includes K as well D; this again dramatically improves the lower bound. From an examination of these curves it is easy to see that combining information from different methods and modes improves the determination significantly 28). Next we briefly discuss some over determined cases (Fig. 2). In purple all the CPES modes of D are combined with just one doubly Cabibbo-suppressed (CPNES) mode. Here there are 4 observables for the 4 unknowns and one gets a reasonable solution at least especially for the high luminosity case. The black curve is different from the purple one in only one respect; the black one also includes the D from B → KD where subsequently the D gives rise to a D. Comparison of the black one with the purple shows considerable improvement by including the D. In this case the number of observables (8) exceeds the number of unknowns (6). Figure 1: γ determination with incomplete input (i.e. cases when the number of observables is less than the number of unknown parameters). The upper horizontal line corresponds to low-luninosity i.e. around current B-factories whereas the lower horizontal curve is relevant for a SBF. Blue uses all CPES modes of D, red is with only Kπ and purple uses combination of the two. Green curve again uses on D, D̄ → Kπ but now includes K and D; see text for details. Actually, the D can decay to D via two modes: D → D + π or D + γ. Bondar and Gershon 34)have made a very nice observation that the strong phase for the γ emission is opposite to that of the π emission. Inclusion (blue curves) of both types of emission increases the number of observables to 12 with no increase in number of unknowns. So this improves the resolving power for γ even more. The orange curves show the outcome when a lot more input is included; not only K, K, D, D but also Dalitz and multibody decays of D are included. But the gains now are very modest; thus once the number of observables exceeds the number of unknowns by a few (say O(3)) further increase in input only has a minimal impact. Table 1: Projections for direct determination of UT. Now(0.2/ab) 2/ab 10/ab ITE sin 2φ1 0.037 0.015 0.015(?) 0.001 α(φ2) 13 ◦ 4(?) 2(?) 1(?) γ(φ3) ±20◦ ± 10 ± 10 5 to 2 < 1(?) 0.05 Let us briefly recall that another important way to get these angles is by studying time-dependent CP (TDCP) (or mixing-induced CP (MIXCP)) violation via B → D“K”. Once again, all the common decay modes of D and D̄ can be used just as in the case of direct CP studies involving B decays. Therefore, needless to say input from charm factory 29, 35, 36) also becomes desirable for MIXCP studies of B → D“K” as it is for direct CP using B. It is important to stress that this method gives not only the combinations of the angles (2β + γ ≡ α − β + π) but also in addition this is another way to get β cleanly 17, 18). In fact whether one uses B with DIRCP or B0−B̄0 with TDCP these methods are very clean with (as indicated above) the ITE of ≈ 0.1%. However, the TDCP studies for getting γ (with the use of β as determined from ψKs ) is less efficient than with the use of DIRCP involving B. Once we go to luminosities ≥ 1ab, though, the two methods for γ should become competitive. Note that this method for getting β is significantly less efficient than from the ψKs studies 17). Table 1 summarizes the current status and expectations for the near future for the UT angles. With the current O(0.4/ab) luminosity between the two B-factories, γ ≈ (69 ± 30) degrees. Most of the progress on γ determination so far is based on the use of the Dalitz mode, D0− > Ksππ 27). However, for now, this method has a disadvantage as it entails a a modelling of the resonances involved; though model independent methods of analysis, at least in principle, exist 17, 27, 28). The simpler modes (e.g. Kπ) require more statistics but they would not involve such modelling error as in the Dalitz method. Also the higher CP asymmetries in those modes should have greater resolving power for determination of γ. The table shows the statistical, systematic and the resonance-model dependent errors on γ separately. Note that for now i do not think the model dependent error (around 10 degrees) ought to be added in quadrature. That is why the combined error of ±30 degrees is somewhat inflated to reflect that. The important point to note is that as more B’s are accumulated, more and more decay modes can be included in determination of γ; thus for the next several years the accuracy on γ is expected to improve faster than 1/ √ (NB), NB being the number of B’s. 4.3 Direct searches: Two important illustrations B-decays offers a wide variety of methods for searching for NP or for BSM-CPodd phase(s). First we will elaborate a bit on the following two methods. • Penguin dominated hadronic final states in b → s transitions. • Radiative B-decays. Then we will provide a brief summary of the multitude of possibilities that a SBF offers, in particular, for numerous important approximate null tests (ANTs). 4.4 Penguin dominated hadronic final states in b → s transitions For the past couple of years, experiments at the two B-factories have been showing some indications of a tantalizing possibility i.e. a BSM-CP-odd phase in penguin dominated b → s transitions. Let us briefly recapitulate the basic idea. Fig. 3 show the experimental status 5). With about 250× 10 B-pairs in each of the B-factories, there are two related possible indications. In particular, BABAR finds about a 3σ deviation in B → η′Ks. Averaging over the two experiments, this is reduced to about 2.3σ. Secondly adding all such penguin dominated modes seems to indicate a 3.5σ effect. SinceB → η′Ks seems to be so prominently responsible for the indications of deviations in the current data sample, let us briefly discuss this particular FS. That the mixing induced CP in η ′ Ks can be used to test the SM was 1st proposed in 37). This was triggered in large part by the discovery of the unexpectedly large Br for B → η′Ks. Indeed ref. 37) emphasized that the large Br may be very useful in determining sin 2β with B → η′Ks and comparing it with the value obtained from B → ψKs. In fact it is precisely the large Br of B → η′Ks that is making the error of the TDCP measurement the smallest amongst all the penguin dominated modes presently studied. Note also that there is a corresponding proposal to use the large Br of the inclusive η ′ Xs for searching for NP with the use of direct CP 38, 39). Ref. 37) actually suggested use of TDCP studies not just in η ′ Ks but in fact also [η, π, ω, ρ, φ...]Ks to test the SM. These are, indeed most of the modes currently being used by BABAR & BELLE. Simple analysis in 37) suggested that in all such penguin dominated (b → s) modes Tree/Penguin is small, < 0.04. In view of the theoretical difficulties in reliably estimating these effects, Ref 37) emhasized that it would be very difficult in the SM to accomodate ∆S > 0.10, as a catious bound. 4.4.1 Final state interaction effects The original papers 40, 41, 37) predicting, ∆Sf = Sf − SψK ≈ 0 (2) used naive factorization; in particular, FSI were completly ignored. A remarkable discovery of the past year is that in several charmless 2-body B-decays direct CP asymmetry is rather large. This means that FSI (CP-conserving) phase(s) in exclusive B-decays need not be small 42). Since these are nonperturbative 43), model dependence becomes unavoidable. Indeed characteristically these FSI phase(s) arise formally from O(1/mB) corrections: • In pQCD 44) a phenomenological parameter kT , corresponding to the transverse momentum of partons, is introduced in order to regulate the end point divergences encountered in power corrections. This in turn gives rise to sizable strong phase difference from penguin induced annihilation. • In QCDF 45), in its nominal version, the direct CP asymmetry in many channels (e.g B → Kπ, ρπ, ππ.....) has the opposite sign compared to the experimental findings. Just like in the pQCD approach where the annihilation topology play an important role in giving rise to large strong phases, and for explaining the penguin-dominated VP modes, it has been suggested in 46) that in a specific scenario (S4), for QCDF to agree with the Br of penguin-dominated PV modes as well as with the measured sign of the direct asymmetry in the prominent channel B → Kπ, a large annihilation contribution be allowed by choosing ρA = 1, φA = −55◦ for PP, φA = −20◦ for PV and φA = −70◦ for VP modes. • In our approach 42), QCDF is used for short-distance (SD) physics; however, to avoid double-counting, we set the above two parameters [ρA, φA] as well as two additional parameters [ρH , φH ] that they have 46) to zero. Instead we try to include long-distance (1/mB) corrections by using on-shell rescattering of 2-body modes to give rise to the needed FS phases. So, for example, color-suppressed modes such as B → Kπ gets important contributions from color allowed processes: B → Kπ(ρ), D S D. The coupling strengths at the three vertices of such a triangular graph are chosen to give the known rates of corresponding physical processes such as B → D S D, D → D + π etc. Furthermore, since these vertices are not elementary and the exchanged particles are off-shell, form-factors have to be introduced so that loop integrals become convergent. Of course, there is no way to determine these reliably. We vary these as well as other parameters so that Br’s are in rough agreement with experiment, then we calculate the CP-asymmetries. Recall the standard form for the asymmetries: Γ(B(t) → f)− Γ(B(t) → f) Γ(B(t) → f) + Γ(B(t) → f) = Sf sin(∆mt) +Af cos(∆mt) (3) The TDCP asymmtery (Sf ) and direct CP asymmetry [Af ≡ −Cf (BaBar notation)] both depend on the strong phase. Thus measurements of direct CP asymmetry Af (in addition to Sf) allows tests of model calculations, though in practice its real use may be limited to those cases where the direct CP asymmetry is not small. This is the case, for example, for ρKS and ωKS 47). It is also important to realize that not only there is a correlation between Sf and Af for FS in B 0 decays, but also that the model entails specific predictions for direct CP in the charged counterparts. So, for example, in our model for FSI, large direct CP asymmetry is also expected in the charged counterparts of the above two modes. In addition to two body modes there are also very interesting 3-body modes such as B → KKKS(KL),KSKSKS(KL). These may also be useful to search for NP as they are also penguin dominated. We use resonancedominance of the relevant two body channels to extend our calculation of LD rescattering phases in these decays 48). Tables 2 and 3 summarize our results for ∆S and A for two body and 3-body modes. We find that 47, 48) B → η′KS , φKS and 3KS are cleanest 49), i.e. central values of ∆S as well as the errors are rather small, O(a few%). Indeed we find that even after including the effect of FSI, ∆S in most of these penguin-dominated modes, it is very difficult to get ∆S > 0.10 in the SM. Thus we can reiterate (as in 37)) that ∆S > 0.10 would be a strong evidence for NP. Having said that, it is still important to stress that genuine NP in these penguin dominated modes must show up in many other channels as well. Indeed, on completely model independent grounds 8), the underlying NP has to be either in the 4-fermi vertex (bsss̄) or (bsg, g = gluon). In either case, it has to materialize into a host of other reactions and phenomena and it is not possible that it only effects time dependent CP in say B → ηKs and/or φKs and/or 3Ks. For example, for the 4-fermi case, we should also expect non-standard effects in Bd → φ(η ′ )K, B+− > φ(η′ )K, Bs → φφ(η ′ )...In Table 2: Direct CP asymmetry parameter Af and the mixing-induced CP parameter ∆S f for various modes. The first and second theoretical errors correspond to the SD and LD ones, respectively (see 47) for details). The f0KS channel is not included as we cannot make reliable estimate of FSI effects on this decay; table adopted from 47). ∆Sf Af (%) Final State SD+LD Expt SD+LD Expt φKS 0.03 +0.01+0.01 −0.04−0.01 −0.38± 0.20 −2.6 −1.0−0.4 4± 17 ωKS 0.01 +0.02+0.02 −0.04−0.01 −0.17 −0.32 −13.2 −2.8−1.4 48± 25 ρKS 0.04 +0.09+0.08 −0.10−0.11 – 46.6 +12.9+3.9 −13.7−2.6 – ηKS 0.00 +0.00+0.00 −0.04−0.00 −0.30± 0.11 2.1 −0.2−0.1 4± 8 ηKS 0.07 +0.02+0.00 −0.05−0.00 − −3.7 −1.8−2.4 − πKS 0.04 +0.02+0.01 −0.03−0.01 −0.39 −0.29 3.7 −1.7−0.4 −8± 14 the second case not only there should be non-standard effects in these reactions but also in Bd(u) → Xsγ, Kγ, Bs → φγ..... and also in the corresponding ll modes. Unless corroborative evidence is seen in many such processes, the case for NP due to the non-vanishing of ∆S is unlikely to be compelling, especially if (say) ∆S <∼ 0.15. 4.4.2 Averaging issue As already emphasized in 37), to the extent that penguin contributions dominate in these many modes and tree/penguin is only a few percent testing the SM by adding Σ∆Sf , where f = KS + η ′ (φ, π, ω, ρ, η,KSKS ...), is sensible at least from a theoretical standpoint. At the same time it is important to emphasize that a convincing case for NP requires unambiguous demonstration of significant effects (i.e. ∆S > 0.10) in several individual channels. 4.4.3 Sign of ∆S For these penguin-dominated modes, ∆Sf is primarily proportional to the hadronic matrix element < f |ūΓbs̄Γ′u|B0 >. Therefore, in the SM for several of the final states (f), ∆Sf could have the same sign. So a systematic trend of ∆Sf being positive or negative (and small of O(a few %)) does not necessarily mean NP. The situation wrt to ηKS is especially interesting. As has been known for the past many years this mode has a very large Br, almost a factor of 7 larger than the similar two body K π mode. This large Br is of course also the reason why the statistical error is the smallest, about a factor of two less than any Table 3: Mixing-induced and direct CP asymmetries sin 2βeff (top) and Af (bottom), respectively, in B → KKKS and KSKSKS decays. Results for (KKKL)CP± are identical to those for (K KKS)CP∓; table taken from 48) . Final State sin 2βeff Expt. (KKKS)φKS excluded 0.749 +0.080+0.024+0.004 −0.013−0.011−0.015 0.57 +0.18 −0.17 (KKKS)CP+ 0.770 +0.113+0.040+0.002 −0.031−0.023−0.013 (KKKL)φKL excluded 0.749 +0.080+0.024+0.004 −0.013−0.011−0.015 0.09± 0.34 KSKSKS 0.748 +0.000+0.000+0.007 −0.000−0.000−0.018 0.65± 0.25 KSKSKL 0.748 +0.001+0.000+0.007 −0.001−0.000−0.018 Af (%) Expt. (KKKS)φKS excluded 0.16 +0.95+0.29+0.01 −0.11−0.32−0.02 −8± 10 (KKKS)CP+ −0.09 −0.00−0.27−0.01 (KKKL)φKL excluded 0.16 +0.95+0.29+0.01 −0.11−0.32−0.02 −54± 24 KSKSKS 0.74 +0.02+0.00+0.05 −0.06−0.01−0.06 31± 17 KSKSKL 0.77 +0.12+0.08+0.06 −0.28−0.11−0.07 other mode being used in the test. For this reason, it is gratifying that ηKS also happens to be theoretically very clean in several of the model calculations. This has the important repercussion that confirmation of a significant deviation from the SM, may well come 1st by using the ηKS mode, perhaps well ahead of the other modes 50). 4.4.4 Concluding remarks on penguin-dominated modes Concluding this section we want to add that while at present there is no clear or compelling deviation from the SM the fact still remains that this is a very important approximate null test (ANT). It is exceedingly important to follow this test with the highest luminosity possible to firmly establish that as expected in the SM, ∆Sf is really ∼ 0.05 and is not significantly different from this expectation. To establish this firmly, for several of the modes of interest, may well require a SBF. 5 Time dependent CP in exclusive radiative B-decays Br (B → γXs(d)) and direct CP asymmetry acp(B → γXs(d)) are well known tests of the SM 51, 52, 53, 54). Both of these use the inclusive reaction where the theoretical prediction for the SM are rather clean; the corresponding exclusive cases are theoretically problematic though experimentally more accessible. In 1997 another important test 55) of the SM was proposed which used mixing induced CP (MICP) or time-dependent CP (TDCP) in exclusive modes such as B → Kγ, ργ..... This is based on the simple observation that in the SM, photons produced in reactions such as B → Kγ,K 2γ, ργ... are predominantly right-handed whereas those in B̄ decays are predominantly left-handed. To the extent that FS of B and B̄ are different MICP would be suppressed in the SM. Recall, the LO Heff can be written as Heff = − √ 8GF emb 16π2 Fμν [ F q L qσ μν 1 + γ5 2 b+ F q R qσ μν 1− γ5 2 b ]