1 Introduction

CERN-PH-TH/2008-167

SI-HEP-2008-15

Precision Physics with at the LHC:

[0.2cm] The Quest for New Physics

Sven Faller, Robert Fleischer and Thomas Mannel

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

Theoretische Physik 1, Fachbereich Physik, Universität Siegen,

D-57068 Siegen, Germany

Dipartimento di Fisica, Università di Roma “La Sapienza”,

I-00185 Roma, Italy

CP-violating effects in the time-dependent angular distribution of the decay products play a key rôle for the search of new physics. The hadronic Standard-Model uncertainties are related to doubly Cabibbo-suppressed penguin contributions and are usually assumed to be negligibly small. In view of recent results from the Tevatron and the quickly approaching start of the data taking at the LHC, we have a critical look at the impact of these terms, which could be enhanced through long-distance QCD phenomena, and explore the associated uncertainty for the measurement of the CP-violating mixing phase. We point out that these effects can actually be controlled by means of an analysis of the time-dependent angular distribution of the decay products, and illustrate this through numerical studies. Moreover, we discuss -breaking effects, which limit the theoretical accuracy of our method, and suggest internal consistency checks of .

October 2008

1 Introduction

The exploration of CP-violating effects in -meson decays offers a particularly promising probe for the search of New Physics (NP). In this respect, a key channel is , which is the counterpart of the “golden” decay to measure the angle in the unitarity triangle (UT) of the Cabibbo–Kobayashi–Maskawa (CKM) matrix. Since the decay involves two vector mesons in the final state, the time-dependent angular distribution of the decay products of the vector mesons, and , has to be measured in order to disentangle the admixture of different CP eigenstates [1, 2].

Within the Standard Model (SM), the CP-violating effects in the time-dependent angular distribution are expected to be small. On the other hand, a preferred mechanism to accommodate a measurement of non-vanishing CP asymmetries would be given by CP-violating NP contributions to mixing (see, for instance, [3]). Recent results from the first tagged, time-dependent analyses performed by the CDF [4] and DØ [5] collaborations at the Tevatron (FNAL) may actually point towards this direction, and have led to quite some attention [6]. The decay is a main target of the LHCb experiment (CERN), which will soon start taking data and will allow us to explore the CP-violating phenomena in this transition with impressive accuracy [7]: already with of data, corresponding to one nominal year of operation, the experimental uncertainty for the mixing phase is expected to be , and an upgrade of LHCb with an integrated luminosity of would eventually allow us to even reach a sensitivity of [8].

In view of these exciting prospects, we have a closer look at the CP-violating effects in the time-dependent angular distribution that arise within the SM and limit the theoretical accuracy of the benchmark for the search for NP. Here the key rôle is played by penguin topologies, which are doubly Cabibbo suppressed and hence usually assumed to be negligible. However, these contributions cannot be calculated reliably from QCD, and could mimic CP-violating effects which might be misinterpreted as signals of NP in mixing with a small but sizeable CP-violating NP phase.

In the present paper, we point out that the penguin effects can actually be controlled by means of an analysis of the angular distribution of and its CP conjugate. Applying flavour-symmetry arguments and neglecting penguin annihilation and exchange topologies (which can be probed through ), the relevant hadronic parameters entering the observables can be determined, thereby allowing us to take them into account in the extraction of . We suggest to perform a simultaneous analysis of the and channels at LHCb, and encourage the CDF and DØ collaborations to search for signals of this transition, as these would allow us to give first constraints on the penguin effects in and their impact on the extraction of the CP-violating mixing phase. Further information can be obtained from the decay, in particular for the resolution of a discrete ambiguity through experimental data.

As pointed out in Ref. [9], the data for CP violation in and the branching ratio of this channel signal that such effects are sizeable and soften the tension in the fit of the UT between its angle and side as determined through CP violation in decays and semileptonic transitions, respectively. In particular, the measurement of has already reached a level of precision where subleading effects, i.e. doubly Cabibbo-suppressed penguin contributions, have to be included in order to match the experimental accuracy (see also Ref. [10]). This feature strengthens the need to deal with such effects in analyses of CP violation in as well. In particular, we expect that the penguin effects interfere constructively with mixing-induced CP violation and could lead to CP asymmetries as large as , which would be significantly larger than the naive SM estimate of and could be well detected at LHCb.

The outline of this paper is as follows: in Section 2, we give an overview of the analysis and explore the impact of the penguin effects on the measurement of , while we discuss the strategy to include the hadronic penguin contributions with the help of in Section 3. This strategy is illustrated in Section 4. A detailed discussion of -breaking effects and internal consistency checks that are offered by the observables of our decays into two vector mesons are given in Section 5. Finally, we summarize our conclusions in Section 6.

2 Review of

2.1 Structure of the Angular Distribution

In contrast to the decay , we have to deal with two vector mesons in the final state of , which is an admixture of CP-odd and CP-even eigenstates. Using the angular distribution of the decay products of the vector mesons, the CP eigenstates can be disentangled. To this end, we introduce linear polarization states of the vector mesons, which are longitudinal () or transverse to their directions of motion. In the latter case, the polarization states may be parallel () or perpendicular () to one another [11]. The time-dependent angular distribution of takes the following general form [1]:

(1)

where the decay kinematics is described by the , and the time-dependent coefficients are given as

(2)

with linear polarization amplitudes , where is the relevant low-energy effective Hamiltonian. Here describes a CP-odd final-state configuration, whereas and correspond to CP-even final-state configurations.

In the case of the CP-conjugate decay , we may write the angular distribution as

(3)

Since the meson content of the state is the same whether it results from the or decays, we may use the same angles , and as in (1) to describe the kinematics of the decay products. Following these lines, the effects of CP transformations relating to are then taken into through the CP eigenvalues of the final-state configuration . Therefore the same functions are present in (1) and (3). For the explicit form the of these quantities, see Ref. [1].

Figure 1: Decay topologies contributing to in the SM.

2.2 Structure of the Decay Amplitudes

As can be seen in Fig. 1, colour-suppressed tree-diagram-like and penguin topologies contribute to the decay within the SM. For a given final-state configuration , the decay amplitude can therefore be written as

(4)

where the are CKM factors, while and are CP-conserving strong amplitudes related to tree-diagram-like and penguin topologies (with internal quarks), respectively. Using the appropriate low-energy effective Hamiltonian, the latter quantities can be expressed in terms of linear combinations of perturbatively calculable Wilson coefficient functions and non-perturbative hadronic matrix elements of the corresponding four-quark operators, which are associated with large uncertainties. Using the CKM unitarity relation to eliminate the factor, we obtain

(5)

where

(6)

and

(7)

are CP-conserving hadronic parameters, while

(8)
(9)
(10)
(11)

are CKM parameters [9, 12], and the UT angle flips its sign when considering CP-conjugate processes:

(12)

Here is the CP eigenvalue of the final-state configuration .

2.3 Time-dependent Observables

If we neglect CP violation in oscillations, which can be probed through wrong-charge lepton asymmetries and is a tiny effect in the SM, the formalism of mixing yields the following expressions [13]:

(13)
(14)

where and are the decay widths of the “light” and “heavy” mass eigenstates, respectively, is their average, and the difference of the mass eigenvalues. The labels “D” and “M” remind us that non-vanishing values of and are generated through direct and mixing-induced CP-violating effects, respectively.

Since the hadronic parameters , which are essentially unknown, enter (5) and (12) in combination with the doubly Cabibbo-suppressed parameter , they are usually neglected. In this limit, we obtain

(15)
(16)

where we have introduced the abbreviation , and is the CP-violating mixing phase. In the ratio of the CP-violating rate difference (16), which requires the “tagging” of whether we had an initially, i.e. at time , present or meson, and the “untagged” rate (15), the overall normalization cancels, so that can be extracted. For the corresponding time-dependences of the other observables provided by the angular distribution, see Ref. [2].

In the SM, we have , where the numerical value follows from the current CKM fits [14]. However, since mixing is a strongly suppressed flavour-changing neutral-current (FCNC) process in the SM, it is a sensitive probe for NP effects in the TeV regime. Should new particles actually contribute to this phenomenon, the off-diagonal mass element of the mixing matrix is modified as follows [3]:

(17)

where measures the strength of the NP contribution with respect to the SM, and is a CP-violating NP phase. Consequently, we have

(18)
(19)

As discussed in Ref. [3], the values of and can be converted into contours in the plane, which sets the parameter space for NP contributions to mixing.

For many years, only lower bounds on were available from the LEP (CERN) experiments and SLD (SLAC) [15]. In 2006, the value of could eventually be pinned down at the Tevatron [16]. The current status can be summarized as follows:

(20)

In order to determine the parameter from these measurements, the SM value of is required, involving a hadronic parameter , which can be determined by means of lattice QCD techniques and introduces the corresponding uncertainties into the analysis. The HPQCD collaboration finds [19], which yields .

Figure 2: The situation in the plane of the NP parameters for mixing arising from current data and theory input, as discussed in the text.

Recently, following Refs. [1, 2], the CDF [4] and DØ [5] collaborations have reported the first results from tagged, time-dependent analyses of the full three-angle distribution of the decay products. In an analysis by the UTfit collaboration [6], taking also other constraints into account, it is argued that these results may indicate CP-violating NP contributions to mixing, which would immediately rule out models with minimal flavour violation (MFV). Recently, a first average of the CDF and DØ data was presented by the Heavy Flavour Averaging Group (HFAG) [20], corresponding to the following twofold solution:

(21)

In Fig. 2, we show – as an update of the analysis performed in Ref. [3] – the corresponding situation in the plane: the central hill-like region corresponds to , i.e. the mass difference , while the two branches represent the twofold solution for ; the overlap of the and constraints results in the two shaded allowed regions. It will be very interesting to monitor these measurements in the future. Fortunately, the analyses are very accessible at the LHCb experiment [7], which will soon start taking data.

2.4 Impact of Penguin Contributions

The experimental results discussed in the previous section were obtained by assuming that the doubly Cabibbo-suppressed parameters , which describe – sloppily speaking – the ratio of penguin to tree contributions, play a negligible rôle. In view of the search for NP signals, which requires a solid control of the SM effects, and the tremendous accuracy that can be achieved at LHCb, we generalize here the formulae to take also these contributions into account.

Let us first have a look at the untagged observables. Following Ref. [13], we have

so that

(24)

On the other hand, the CP-violating observables are given as follows:

(25)
(26)

Note that (2.4) and (2.4) are not independent from (25) and (26), as

(27)
Figure 3: Impact of the penguin parameter on the mixing-induced CP asymmetry for as function of the mixing phase .

The ratio of the “tagged” rate difference (14) and the “untagged” rate (13) can be written as

(28)

where , and

(29)

If we introduce

(30)

the corresponding observables take the following forms:

(31)
(32)
(33)

The measurement of relies on a sizeable value of the width difference . Moreover, we have

(34)

For the extraction of , the key observables are the ; in Fig. 3, we illustrate the impact of the penguin parameter . Since is defined in (7) in such a way that is given by if we assume factorization, we have used this value in order to calculate the curves shown in Fig. 3. For this strong phase the penguin effects are actually maximal in since only enters. On the other hand, the direct CP asymmetries would then vanish, as they are proportional to .

We observe that in order to accommodate a value of , as given in (21), we would need , which appears completely unrealistic. However, since suffers from large uncertainties, values as large as can a priori not be excluded. Should take a value on the small side, these hadronic SM contributions would lead to a significant uncertainty in the extraction of the mixing phase.

Figure 4: Left panel: the dependence of on for various values of ; right panel: the dependence of on for various values of .

In order to explore this effect in more detail, we use (31) and (32) to derive the following expression:

(35)

where

(36)

and

(37)

so that

(38)

It should be stressed that the shift of the mixing phase does not depend on the value of itself. In Fig. 4, we show the dependence of on the penguin parameter for various values of , and give – in order to monitor the corresponding direct CP asymmetries – a similar plot for . We observe that is of the same size as for , and that a value of would induce a shift of . As can be seen in the left panel of Fig. 4, we have for and values of as large as . Interestingly, as we expect , the shift of is expected to be negative as well, i.e. it would interfere constructively with . These features are fully supported by our recent analysis of the channel [9]. Consequently, it is important to get a handle on the penguin effects in the decay.

3 The Control Channel

3.1 Structure of the Decay Amplitudes

In Fig. 5, we show the decay topologies contributing to the channel. The key difference with respect to the decay shown in Fig. 1 is that is caused by quark-level processes, whereas originates from transitions. Consequently, the CKM factors are different in these channels. In analogy to (5), we may write

(39)

where and are the counterparts of the parameters introduced in (6) and (7), respectively. In contrast to (5), the latter parameter does not enter (39) in a doubly Cabibbo-suppressed way. Consequently, the channel offers a sensitive probe for this quantity. If we apply the flavour symmetry of strong interactions, we obtain

(40)

as well as

(41)

In addition to flavour-symmetry arguments we have here also assumed that penguin annihilation () and exchange () topologies, which contribute to but have no counterpart in , play a negligible rôle. Fortunately, the importance of these topologies can be probed with the help of the channel, which has amplitudes proportional to . The Belle collaboration has recently reported the new upper bound of ( C.L.) [21], which does not show any anomalous enhancement. The theoretical uncertainties associated with the application of the flavour symmetry will be discussed separately in Section 5.

Figure 5: Decay topologies contributing to in the SM.

3.2 Observables

In contrast to the decay, the final states of and its CP conjugate are flavour-specific, i.e. the charges of the pions and kaons depend on whether we had a or meson in the initial state. Consequently, the time-dependent angular distributions do not show CP violation due to interference between mixing and decay, i.e. the observables introduced in (14) have no counterparts, and do not depend on the mixing phase. However, untagged observables, as well as direct CP-violating asymmetries provide actually sufficient information to determine and . In Appendix A, we give the expressions for the time-dependent angular distributions, which allow the determination of the relevant observables.

Let us first discuss the untagged case, and introduce

(42)

where is the counterpart of (24). Using (40), we may extract from the untagged observables. Moreover, using also (41), we can determine as a function of with the help of the following formulae:

(43)

where

(44)

and

(45)

Here the main uncertainty is associated with the determination of , which relies on (40). In Subsection 5.2, we have a closer look at the corresponding -breaking corrections, and give numerical results for the extraction of the from the untagged observables. On the other hand, thanks to the terms in (42), the impact of corrections to (41) is tiny.

Another useful quantity is offered by the direct CP asymmetry

(46)

which can be extracted from a rate difference; it takes the same form as (28) for . In analogy to , also allows us to determine as a function of . To this end, we may again use (43), with the following replacements:

(47)

It should be emphasized that the corresponding curve in the plane is theoretically clean, whereas that described by (43) is affected in particular by the -breaking effects entering the determination of .

The intersection of the and contours allows us then to extract and from the data. Finally, applying (41) and the results derived in Section 2.4, we can include the penguin effects in the determination of the mixing phase. Let us first illustrate this method in the next section by discussing a numerical example before giving a detailed discussion of the relevant -breaking effects in Section 5.

4 A Numerical Example

For the illustration of the strategy discussed above, we assume , and hadronic parameters given by and , yielding the observables and . These input values are consistent with the ranges of the parameters and found in Ref. [9]; we expect a picture for and that is similar to the one for their counterparts.

Figure 6: Illustration of the contours in the plane for an example, as discussed in the text.

In Fig. 6, we show the contours in the plane that arise in this example. We observe that a twofold solution emerges for , which can be resolved through the sign of . Theoretically, we expect a negative value of this quantity, which is also supported by the data. In order to resolve this ambiguity experimentally, we need an additional observable, which would be provided by mixing-induced CP violation. Since the processes have flavour-specific final states, they do not show this phenomenon. On the other hand, mixing-induced CP violation would arise in modes, in analogy to processes [1]. Unfortunately, it is essentially impossible to study the corresponding experimental signatures in a hadronic environment, i.e. at the Tevatron or LHC.

However, we may alternatively use the channel [13], which can be obtained from by replacing the strange spectator quark through a down quark, as can be seen in Fig. 5. In this case, the final state is an admixture of different CP eigenstates, in analogy to , and we can extract the following mixing-induced CP asymmetry from the time-dependent angular distribution:

(48)

where is the CP eigenvalue of the final-state configuration , i.e., , and , whereas denotes the mixing phase [9]; for simplicity, we have also denoted the hadronic parameters by and , as we expect them to be approximately equal to those of thanks to the flavour symmetry. Using (43) with the replacements

(49)
(50)

the measurement of the mixing-induced CP asymmetry allows us to fix another contour in the plane. If we consider the example given above with , we obtain , which results in the contours shown in Fig. 7. We see that the twofold ambiguity in the determination of the hadronic parameters can now be resolved, thereby leaving us with our input values.

Figure 7: Illustration of the resolution of the twofold ambiguity in Fig. 6 through the mixing-induced CP violation in .

Since the width of the is three-times larger than that of the , the control channel should be experimentally better accessible than . Moreover, if we neglect -breaking effects due to the different spectator quarks, we expect the simple relation [20]. However, already a rather crude measurement of the mixing-induced CP-violating observables of would be sufficient to resolve the ambiguity in the extraction of and . In particular, the expected negative value of would be indicated by values of