1 Introduction

CERN-PH-TH/2007-080

: Status and Prospects

Robert Fleischer

Theory Division, Department of Physics, CERN, CH-1211 Geneva 23, Switzerland

Several years ago, it was pointed out that the -spin-related decays , and , offer interesting strategies for the extraction of the angle of the unitarity triangle. Using the first results from the Tevatron on the decays and the -factory data on modes, we compare the determinations of from both strategies, study the sensitivity on -spin-breaking effects, discuss the resolution of discrete ambiguities, predict observables that were not yet measured but will be accessible at LHCb, explore the extraction of the width difference from untagged rates, and address the impact of new physics. The data for the , system favour the BaBar measurement of direct CP violation in , which will be used in the numerical analysis, and result in a fortunate situation, yielding , where the latter errors correspond to a generous estimate of -spin-breaking effects. On the other hand, the , analysis leaves us with , and points to a value of the branching ratio that is larger than the current Tevatron result. An important further step will be the measurement of mixing-induced CP violation in , which will also allow us to extract the mixing phase unambiguously with the help of at the LHC. Finally, the measurement of direct CP violation in will make the full exploitation of the physics potential of the modes possible.

May 2007

## 1 Introduction

Decays of mesons into two light pseudoscalar mesons offer interesting probes for the exploration of CP violation. The key problem in these studies is usually given by the hadronic matrix elements of local four-quark operators, which suffer from large theoretical uncertainties. In 1999 [1], it was pointed that the system of the and decays is particularly interesting in this respect. These transitions, which receive contributions from tree and penguin topologies, allow us to determine the angle of the unitarity triangle (UT) of the Cabibbo–Kobayashi–Maskawa (CKM) matrix [2] with the help of the -spin symmetry, which is a subgroup of the flavour symmetry of strong interactions, connecting the strange and down quarks in the same way through transformations as the isopsin symmetry connects the up and down quarks. As can be seen in Fig. 1, the and modes are related to each other through an interchange of all down and strange quarks. Consequently, the -spin flavour symmetry allows us to derive relations between their hadronic parameters so that the experimental observables offer sufficient information to extract them and the UT angle from the data. The advantage of this -spin strategy with respect to the conventional flavour-symmetry strategies [3] is twofold:

• no additional dynamical assumptions such as the neglect of annihilation topologies have to be made, which could be spoiled by large rescattering effects;

• electroweak (EW) penguin contributions, which are not invariant under the isospin symmetry because of the different up- and down-quark charges, can be included.

The theoretical accuracy is therefore only limited by non-factorizable -spin-breaking effects, as the factorizable corrections can be taken into account through appropriate ratios of form factors and decay constants. Moreover, we have key relations between certain hadronic parameters, where these quantities cancel. Interestingly, also experimental insights into -spin-breaking effects can be obtained, which do not indicate any anomalous enhancement.

The relevant observables are the CP-averaged branching ratios as well as the direct and mixing-induced CP asymmetries and , respectively, entering the following time-dependent rate asymmetries for decays into CP eigenstates [4]:

 ACP(t) ≡ Γ(B0q(t)→f)−Γ(¯B0q(t)→f)Γ(B0q(t)→f)+Γ(¯B0q(t)→f) (1.1) = [AdirCP(Bq→f)cos(ΔMqt)+AmixCP(Bq→f)sin(ΔMqt)cosh(ΔΓqt/2)−AΔΓ(Bq→f)sinh(ΔΓqt/2)],

where and are the mass and width differences of the mass eigenstates, respectively. Throughout this paper, we shall apply a sign convention for CP asymmetries that is similar to (1.1), also for the direct CP asymmetries of decays into flavour-specific final states.

As can be seen in Fig. 1, there is yet another pair of -spin-related decays that is mediated by the same quark transitions: and . In contrast to the , system, the final states are flavour-specific. Consequently, we have to rely on the direct CP-violating rate asymmetry as no mixing-induced CP violation arises. If additional information provided by the channel is used, together with plausible dynamical assumptions about final-state interaction effects and colour-suppressed EW penguin topologies, the -spin-related , decays also allow the extraction of the CKM angle [5].

Thanks to the factories with the BaBar (SLAC) and Belle (KEK) experiments, the and decays are now experimentally well established, with the following CP-averaged branching ratios, as compiled by the Heavy Flavour Averaging Group (HFAG) [6]:

 BR(Bd→π+π−) = (5.16±0.22)×10−6, (1.2) BR(Bd→π∓K±) = (19.4±0.6)×10−6, (1.3) BR(B±→π±K) = (23.1±1.0)×10−6. (1.4)

The channel led to the observation of direct CP violation in the -meson system [7], where the current HFAG average reads as

Concerning the measurements of CP violation in , the BaBar and Belle collaborations agree now perfectly on the mixing-induced CP asymmetry:

 AmixCP(Bd→π+π−)={0.60±0.11±0.03(BaBar \@@cite[cite]{[\@@bibref{}{BaBar-Bpi+pi-}{}{% }]})0.61±0.10±0.04(Belle \@@cite[cite]{[\@@bibref{}{Belle-Bpi+pi-}{}{% }]}), (1.6)

yielding the average of [6]. On the other hand, the picture of direct CP violation is still not experimentally settled, and the corresponding -factory measurements differ at the level:

 AdirCP(Bd→π+π−)={−0.21±0.09±0.02(BaBar \@@cite[cite]{[\@@bibref{}{BaBar-Bpi+pi-}{}% {}]})−0.55±0.08±0.05(Belle \@@cite[cite]{[\@@bibref{}{Belle-Bpi+pi-}{}% {}]}). (1.7)

In a recent paper [10], it was pointed out that the branching ratio and direct CP asymmetry of the mode favour actually the BaBar result. Following a different avenue, we will arrive at the same conclusion.

Since the factories are operated at the resonance, decays could not be studied at these colliders.111Recently, data were taken by Belle at , allowing also access to decays [11]. The exploration of the system is the territory of hadron colliders, i.e. of the Tevatron (FNAL), which is currently taking data, and of the LHC (CERN), which will start operation soon. In fact, signals for the and decays were recently observed at the Tevatron by the CDF collaboration at the and levels, respectively, which correspond to the following CP-averaged branching ratios: [12, 13]:

 BR(Bs→π±K∓) = (5.00±0.75±1.0)×10−6, (1.8) BR(Bs→K+K−) = (24.4±1.4±4.6)×10−6. (1.9)

Moreover, also a CP violation measurement is available:

whereas results for the CP-violating observables of were not yet reported.

In view of this progress, it is interesting to confront the , and , strategies with the measurements performed at the factories and the Tevatron. This is also an important analysis in view of the quickly approaching start of the LHC with its dedicated -decay experiment LHCb, where the physics potential of the -meson system can be fully exploited [14]. We will therefore give a detailed presentation, collecting also the relevant formulae, which should be helpful for the analysis of the future improved experimental data. The outline of this paper is as follows: in Section 2, we have a closer look at the , strategy, and move on to the , system in Section 3. Finally, we summarize our conclusions in Section 4. For analyses using QCD factorization, soft collinear effective theory or perturbative QCD, the reader is referred to Refs. [15, 16, 17].

## 2 The Bd→π+π−, Bs→K+K− Strategy

### 2.1 CP Violation in Bd→π+π−

In the Standard Model (SM), using the unitarity of the CKM matrix, the transition amplitude of the decay can be written as follows [1]:

 A(B0d→π+π−)=eiγ(1−λ22)C[1−deiθe−iγ], (2.1)

where is the corresponding angle of the UT, the parameter of the Wolfenstein expansion of the CKM matrix [18], denotes a CP-conserving strong amplitude that is governed by the tree contributions, while the CP-consering hadronic parameter measures – sloppily speaking – the ratio of penguin to tree amplitudes. The CP asymmetries introduced in (1.1) take then the following form:

 AdirCP(Bd→π+π−) = −[2dsinθsinγ1−2dcosθcosγ+d2] (2.2) AmixCP(Bd→π+π−) = +[sin(ϕd+2γ)−2dcosθsin(ϕd+γ)+d2sinϕd1−2dcosθcosγ+d2], (2.3)

where is the CP-violating mixing phase, which is given by in the SM, with denoting another UT angle. This phase has been measured at the factories with the help of the “golden” decay and similar modes, including and channels to resolve a twofold ambiguity, as follows [6]:

 ϕd=(42.6±2)∘. (2.4)

The general expressions in (2.2) and (2.3) allow us to eliminate the strong phase , and to calculate as a function of by using the formulae given in Ref. [1]. In Fig. 2, we show the corresponding contours for the central values of the BaBar and Belle results in (1.6) and (1.7). In order to guide the eye, we have also included the contour (dotted line) representing the central value of the HFAG average of the BaBar and Belle results for the direct CP violation in [6]. It should be emphasized that these contours are valid exaclty in the SM.

### 2.2 CP-Averaged Bs→K+K−, Bd→π+π− Branching Ratios

Let now enter the stage. In analogy to (2.1), the corresponding decay amplitude can be written as

 A(B0s→K+K−)=eiγλC′[1+1ϵd′eiθ′e−iγ], (2.5)

where

 ϵ≡λ21−λ2=0.05, (2.6)

and and are the counterparts of the parameters and , respectively. If we apply the -spin symmetry, we obtain the following relations [1]:

 d′=d,θ′=θ. (2.7)

As was also pointed out in Ref. [1], these relations are not affected by factorizable -spin-breaking corrections, i.e. the relevant form factors and decay constants cancel. This feature holds also for chirally enhanced contributions to the transition amplitudes.

Since the CP asymmetries of the decay have not yet been measured, we have to use the CP-averaged branching ratio of this mode, which also provides valuable information. For the determination of , it is useful to introduce the quantity

 (2.8)

where

 Φ(x,y)≡√[1−(x+y)2][1−(x−y)2] (2.9)

is the well-known phase-space function, and the are the lifetimes. Applying the relations in (2.7), we arrive at

 K=1ϵ2[ϵ2+2ϵdcosθcosγ+d21−2dcosθcosγ+d2]. (2.10)

If we combine with , which depend both on , we can fix another contour in the plane with the help of the formulae given in Ref. [1].

In order to determine from the CP-averaged branching ratios, the -spin-breaking corrections to the ratio , which equals 1 in the strict -spin limit, have to be determined. In contrast to the -spin relations in (2.7), involves hadronic form factors in the factorization approximation:

 ∣∣∣C′C∣∣∣fact=fKfπFBsK(M2K;0+)FBdπ(M2π;0+)⎛⎝M2Bs−M2KM2Bd−M2π⎞⎠, (2.11)

where and denote the kaon and pion decay constants, and and parametrize the hadronic quark-current matrix elements and , respectively [19]. These quantities were analyzed using QCD sum-rule techniques in detail in Ref. [20], yielding

 ∣∣∣C′C∣∣∣QCDSRfact=1.52+0.18−0.14. (2.12)

As we will see in Section 3, we can actually determine this quantity with the help of the data for the , system. Since the corresponding value agrees remarkably well with (2.12), large non-factorizable -spin-breaking effects are disfavoured, which gives us further confidence in applying (2.7).

### 2.3 Extraction of γ and Hadronic Parameters

If we use (1.2) and (1.9) with (2.12) and add the errors in quadrature, we obtain

 K=41.03±10.27. (2.13)

In Fig. 2, we have also included the contour following from the central values of and . We see that the intersections with the contour following from the BaBar data give a twofold solution for around and , whereas we obtain no intersection with the corresponding Belle curve. Consequently, the measured branching ratio disfavours the Belle result for the direct CP violation in . A similar observation was also made in Ref. [10], using, however, a different avenue. For the following analysis, we will therefore only use the BaBar measurement of , which covers also the prediction for this asymmetry made in Ref. [10] within the uncertainties.

In Fig. 3, we show impact of the uncertainties of and the CP asymmetries of . We obtain the following numerical results:

 γ=(40.6+1.6+1.1+2.3−1.3−0.6−2.4)∘=(40.6+3.0−2.8)∘,d=0.243+0.024+0.015+0.002−0.028−0.008−0.001=0.243+0.028−0.029,θ=(29.2+5.5+14.2+1.7−3.5−12.8−1.3)∘=(29.2+15.3−13.3)∘, (2.14)
 γ=(66.6+2.6+1.1+3.2−2.9−2.0−3.6)∘=(66.6+4.3−5.0)∘,d=0.410+0.053+0.001+0.010−0.060−0.003−0.009=0.410+0.054−0.061,θ=(155.9+2.5+10.8+0.8−3.8−2.1−1.2)∘=(155.9+11.1−4.5)∘, (2.15)

Here we show the errors arising from , and , and have finally added them in quadrature.

### 2.4 Impact of U-Spin-Breaking Effects

Let us now explore the impact of non-factorizable -spin-breaking corrections to (2.7) by introducing the following parameters [21, 22]:

 ξ≡d′/d,Δθ≡θ′−θ. (2.16)

The expression for in (2.10) is then modified as

 K=1ϵ2[ϵ2+2ϵξdcos(θ+Δθ)cosγ+ξ2d21−2dcosθcosγ+d2]. (2.17)

Since the numerator is governed by the term, the dominant -spin-breaking effects are described by , whereas plays a very minor rôle, as was also noted in Refs. [21, 22]. This behaviour can nicely be seen in Fig. 4, where we have considered and . In view of the comments given above, these parameters describe generous -spin-breaking effects. Their impact on the numerical solutions in (2.14) and (2.15) is given as follows:

 γ=(40.6+3.0+1.3+0.2−2.8−1.6−0.3)∘,d=0.243+0.028+0.030+0.006−0.029−0.023−0.003,θ=(29.2+15.3+4.5+0.5−13.3−4.3−0.8)∘, (2.18)
 γ=(66.6+4.3+4.0+0.1−5.0−3.0−0.2)∘,d=0.410+0.054+0.082+0.002−0.061−0.060−0.001,θ=(155.9+11.1+3.6+0.1−4.5−3.8−0.3)∘, (2.19)

where the second and third errors refer to and , respectively. Interestingly, is only moderately affected by these effects, which do not exceed the current experimental uncertainties for the parameter ranges considered above. Performing measurements of CP violation in , which will be possible with impressive accuracy at the LHCb experiment [23], the use of the -spin symmetry can be minimized in the extraction of , and internal consistency checks become available. Before turning to these asymmetries, let us first discuss the discrete ambiguities affecting the extraction of .

### 2.5 Discrete Ambiguities

So far, we have restricted the discussion to the range of , which follows from the SM interpretation of the measurement of , which describes the indirect CP violation in the neutral kaon system [24, 25]. However, if we allow for new physics (NP), we have to consider the whole range of . As can be seen by having a closer look at the expressions given in (2.2), (2.3) and (2.10), for each of the two solutions listed in (2.18) and (2.19), we obtain an additional one through the following transformation:

 γ→γ−180∘,d→d,θ→θ−180∘, (2.20)

i.e. we have to deal with a fourfold discrete ambiguity, which has to be resolved for the search of NP.

To this end, let us first have a look at for (2.18) and (2.19), given by

 cosθ=+0.873+0.092−0.168andcosθ=−0.913+0.047−0.064, (2.21)

respectively, where we have added all errors in quadrature. Although non-factorizable effects have a significant impact on , we do not expect that they will change the sign of the cosine of this strong phase, which is negative in the notation used above. Consequently, (2.18) can be excluded through this argument. As we will see in Subsection 2.6, the future measurement of mixing-induced CP violation in should allow us to rule out this solution in a direct way. Moreover, as will be discussed in Subsection 3.4, already the current data for the observables of the , system exclude (2.18) and its “mirror” solution around following from (2.20), where the sign of would be as in factorization. In the case of the remaining mirror solution of (2.19) around , the sign of would be positive, i.e. opposite to our expectation, so that it can be ruled out as well.

Consequently, we are finally left with the numbers in (2.19). It is interesting to note that the corresponding value of in is in excellent agreement with the SM fits of the UT obtained by the UTfit and CKMfitter collaborations [24, 25], yielding and , respectively.

### 2.6 CP Violation in Bs→K+K−

Using the expression for the decay amplitude in (2.5), the observables entering the CP-violating rate asymmetry in (1.1) take the following form:

 AdirCP(Bs→K+K−) = 2ϵd′sinθ′sinγd′2+2ϵd′cosθ′cosγ+ϵ2, (2.22) AmixCP(Bs→K+K−) = +[d′2sinϕs+2ϵd′cosθ′sin(ϕs+γ)+ϵ2sin(ϕs+2γ)d′2+2ϵd′cosθ′cosγ+ϵ2], (2.23) AΔΓ(Bs→K+K−) = −[d′2cosϕs+2ϵd′cosθ′cos(ϕs+γ)+ϵ2cos(ϕs+2γ)d′2+2ϵd′cosθ′cosγ+ϵ2d′2], (2.24)

where is the CP-violating mixing phase; in the SM, it is given in terms of the Wolfenstein parameters by , and takes a tiny value of . If we consider this SM case for the solution of (2.15) and use the -spin relations in (2.7), we arrive at the following predictions:

where the treatment and notation of the errors is as in (2.14) and (2.15), i.e. refers to the uncertainties of , and . The interesting feature that the error of the direct CP asymmetry is independent of that of is due to the following -spin relation [1]:

and provides a nice numerical test. Moreover, all observables satisfy the relation

The impact of the -spin-breaking corrections discussed in Subsection 2.4 is given as follows:

where the second and third errors refer to and , respectively, as in (2.18) and (2.19). Whereas and are pretty stable with respect to the -spin-breaking effects, the direct CP asymmetry is significantly affected by . As we will discuss in Subsection 3.4, the measurement of the direct CP violation in strongly disfavours such effects.

The next important step in the analysis of the , system is the measurement of the mixing-induced CP violation in . Applying the formulae given in Ref. [1], this observable can be combined with to fix another contour in the plane. In Fig. 5, we illustrate the corresponding situation for the central numerical values given above, and observe that the measurement of will in fact allow us to resolve the twofold ambiguity in the extraction of the UT angle , as we noted in Subsection 2.5.

Finally, if also the direct CP asymmetry is measured, we can combine it with to calculate as a function of for a given value of the mixing phase [1]. It should be emphasized that this contour is – in contrast to those involving theoretically clean, in analogy to the curve following from the CP-violating observables. Using the first of the -spin relations in (2.7), we can then extract and , where the information provided by allows us to resolve the discrete ambiguity. Since the strong phases and can be determined as well, we may actually perform a test of the second -spin relation in (2.7). Moreover, the impact of -spin-breaking corrections to corresponds to a relative shift of the and contours; the situation for the extraction of in Fig. 5 would actually be very stable in this respect. This would be the most refined implementation of the , strategy for the extraction of . For recent LHCb studies, which look very promising, see Ref. [23].

The last observable that is provided by is , which enters the following “untagged” rate [26]:

 ⟨Γ(Bs(t)→K+K−)⟩≡Γ(B0s(t)→K+K−)+Γ(¯B0s(t)→K+K−) (2.29) ∝e−Γst[e+ΔΓst/2RL(Bs→K+K−)+e−ΔΓst/2RH(Bs→K+K−)],

where

 Γs≡Γ(s)H+Γ(s)L2,ΔΓs≡Γ(s)H−Γ(s)L (2.30)

depend on the decay widths and of the “heavy” and “light” mass eigenstates of the system, respectively, and

 RL(Bs→K+K−) ≡ 1−AΔΓ(Bs→K+K−)=1.964+0.007−0.011, (2.31) RH(Bs→K+K−) ≡ 1+AΔΓ(Bs→K+K−)=0.036+0.011−0.007; (2.32)

the numerical values correspond to the SM prediction in (2.28). Concerning a practical measurement of (2.29), most data come from short times with :

 ⟨Γ(Bs(t)→K+K−)⟩∝e−Γst[1−AΔΓ(Bs→K+K−)(ΔΓst2)+O((ΔΓst)2)]. (2.33)

Moreover, if the two-exponential form of (2.29) is fitted to a single exponential, the corresponding decay width satisfies the following relation [27]:

 ΓK+K−=Γs+AΔΓ(Bs→K+K−)ΔΓs2+O((ΔΓs)2/Γs). (2.34)

First studies along these lines were recently performed by the CDF collaboration [28], yielding . Using flavour-specific decays, a similar analysis allows the extraction of up to corrections of [27]. With the help of the analysis discussed above, which allows the calculation of , the width difference can then be extracted.

### 2.7 Impact of New Physics

Because of the impressive agreement of the value of that we extracted from the , data with the fits of the UT and the overall consistency with the SM (see also Section 3), dramatic NP contributions to the corresponding decay amplitudes are already excluded, although the experimental picture has still to be improved considerably. In particular, accurate measurements of through pure tree-level decays are not yet available, but will be performed at LHCb [14]; imporant examples are and decays, where the -spin symmetry provides again a useful tool [29].

Similar conclusions about NP effects in modes were drawn in Refs. [10, 30]. The corresponding -factory data may indicate a modified EW penguin sector with a large CP-violating NP phase through the results for mixing-induced CP violation in , thereby complementing the pattern of such CP asymmetries observed in other penguin modes, where the channel is an outstanding example. Since EW penguin topologies contribute to the , (and the , ) system in colour-suppressed form, they play there a minor rôle. Consequently, NP effects entering through the EW penguin sector could not be seen in the analysis discussed in this paper.

On the other hand, mixing offers a nice avenue for NP to manifest itself in . The mass difference was recently measured at the Tevatron [31, 32], with a value that is consistent with the SM expectation. On the other hand, this result still allows for large CP-violating NP contributions to mixing (see, for instance, Refs. [33, 34]). In this case, the mixing phase , which can be extracted through the time-dependent angular distribution of the decay products [35, 27], would take a sizeable value. Interestingly, also the , system allows us to search for NP effects of this kind. Assuming a value of , which corresponds to a simple “translation” of the tension in the CKM fits between and the UT side [33], we arrive at the situation illustrated in Fig. 6. There we show the contours involving that would arise if we assume the SM value of . In this case, we would arrive at quite some discrepancy, in particular through the contour following from the CP-violating asymmetries. For larger values of , the discrepancy would be even more pronounced. In this case, the measured value of would also not lie on the SM surface in observable space that was calculated in Ref. [36].

It is instructive to expand (2.23) and (2.24) in powers of , yielding

 AmixCP(Bs→K+K−) = +sinϕs+2(ϵd′)cosθ′sinγcosϕs+O((ϵ/d′)2), (2.35) AΔΓ(Bs→K+K−) = −cosϕs+2(ϵd′)cosθ′sinγsinϕs+O((ϵ/d′)2), (2.36)

where . We observe two interesting features:

• is strongly affected if moves away from thanks to the term, and offers also information on through the hadronic piece.

• deviates slowly from its SM value around as moves away from , and the hadronic term is suppressed by