A Roadmap to Control Penguin Effects in and
Kristof De Bruyn and Robert Fleischer
Nikhef, Science Park 105, NL-1098 XG Amsterdam, Netherlands
Department of Physics and Astronomy, Vrije Universiteit Amsterdam,
NL-1081 HV Amsterdam, Netherlands
Measurements of CP violation in and decays play key roles in testing the quark-flavour sector of the Standard Model. The theoretical interpretation of the corresponding observables is limited by uncertainties from doubly Cabibbo-suppressed penguin topologies. With continuously increasing experimental precision, it is mandatory to get a handle on these contributions, which cannot be calculated reliably in QCD. In the case of the measurement of from , the -spin-related decay offers a tool to control the penguin effects. As the required measurements are not yet available, we use data for decays with similar dynamics and the flavour symmetry to constrain the size of the expected penguin corrections. We predict the CP asymmetries of and present a scenario to fully exploit the physics potential of this decay, emphasising also the determination of hadronic parameters and their comparison with theory. In the case of the benchmark mode used to determine the – mixing phase the penguin effects can be controlled through and decays. The LHCb collaboration has recently presented pioneering results on this topic. We analyse their implications and present a roadmap for controlling the penguin effects.
The data of the first run of the Large Hadron Collider (LHC) at CERN have led to the exciting discovery of the Higgs boson [1, 2] and are, within the current level of precision, globally consistent with the picture of the Standard Model (SM). The next run of the LHC at almost the double centre-of-mass energy of the colliding protons, which will start in spring 2015, will open various new opportunities in the search for New Physics (NP) . These will be both in the form of direct searches for new particles at the ATLAS and CMS experiments, and in the form of high-precision analyses of flavour physics observables at the LHCb experiment. Concerning the latter avenue, also the Belle II experiment at the KEK Super Factory will enter the stage in the near future . The current LHC data suggest that we have to prepare ourselves to deal with smallish NP effects, and it thus becomes mandatory to have a critical look at the theoretical assumptions underlying the experimental analyses.
Concerning measurements of CP violation, the and decays play outstanding roles as they allow determinations of the – mixing phases and , respectively. These quantities take the forms
The theoretical precision for the extraction of and from the CP asymmetries of the and decays is limited by doubly Cabibbo-suppressed penguin contributions. The corresponding non-perturbative hadronic parameters cannot be calculated in a reliable way within QCD. However, in the era of high-precision measurements, these effects have to be controlled with the final goal to match the experimental and theoretical precisions [9, 10, 11, 12, 13, 14, 15, 16].
As was pointed out in Ref. , is related to through the -spin symmetry of strong interactions, and allows a determination of the penguin corrections to the measurement of . Concerning the channel, an analysis of CP violation is more involved as the final state consists of two vector mesons and thus is a mixture of different CP eigenstates which have to be disentangled through an angular analysis of their decay products [17, 18]. In this case, the decays  and  are tools to take the penguin effects into account. The LHCb collaboration has very recently presented the first polarisation-dependent measurements of from in Ref. . We shall discuss the implications of these exciting new results in detail.
Since a measurement of CP violation in is not yet available, we use the flavour symmetry and plausible assumptions for various modes of similar decay dynamics to constrain the relevant penguin parameters. Following these lines, we assess their impact on the measurement of and predict the CP-violating observables of . In our benchmark scenario, we discuss also the determination of CP-conserving strong amplitudes, which will provide valuable insights into non-factorisable -spin-breaking effects through the comparison with theoretical form-factor calculations.
Concerning the channel, measurements of CP violation are also not yet available. However, in the case of , the LHCb collaboration has recently announced the first results of a pioneering study , presenting in particular a measurement of mixing-induced CP violation and constraints on the penguin effects. This new experimental development was made possible through the implementation of the method proposed by Zhang and Stone in Ref. . We shall have a detailed look at these exciting measurements and discuss important differences between the penguin probes and . We extract hadronic parameters from the data, allowing insights into -breaking and non-factorisable effects through a comparison with theory, and point out a new way to combine the information provided by the , system in a global analysis of the penguin parameters.
The outline of this paper is as follows: in Section 2, we introduce the general formalism to deal with the penguin effects. In Section 3, we explore the constraints of the currently available data for the penguin contributions to the system, while we turn to the discussion of the most recent LHCb results for and the penguin probes , in Section 4. In Section 5, we outline a roadmap for dealing with the hadronic penguin uncertainties in the determination of and . Finally, we summarise our conclusions in Section 6.
2 CP Violation and Hadronic Penguin Shifts
For the neutral decays () discussed in this paper, the transition amplitudes can be written in the following form :
Here is the CP eigenvalue of the final state , is a CP-conserving normalisation factor representing the dominant tree topology, parametrises the relative contribution from the penguin topologies, is the CP-conserving strong phase difference between the tree and penguin contributions, whereas their relative weak phase is given by the UT angle . The parameters and depend both on CKM factors and on hadronic matrix elements of four-quark operators entering the corresponding low-energy effective Hamiltonian.
In order to extract information on , CP-violating asymmetries are measured :
where the dependence on the decay time enters through – oscillations, and and denote the mass and decay width differences of the two mass eigenstates, respectively.
Using Eqs. (3) and (4), the direct and mixing-induced CP asymmetries and take the following forms :111Whenever information from both and decays is needed to determine an observable, as is the case for CP asymmetries or untagged branching ratios, we use the notation and .
while the observable is given by
For the discussion of the penguin effects, the following expression will be particularly useful (generalising the formulae given in Ref. ):
It should be emphasised that is a phase shift which depends on the non-perturbative parameters and and cannot be calculated reliably within QCD. In the case of , the following simple situation arises:
allowing us to determine directly from the mixing-induced CP asymmetry.
Since in the decays and the parameters corresponding to are doubly Cabibbo-suppressed, Eq. (13) is approximately valid. However, in the era of high-precision studies of CP violation, we nonetheless have to control these effects. As the corresponding penguin parameters are Cabibbo-allowed in the and , decays, these modes allow us to probe the penguin effects. Making use of the flavour symmetry, we may subsequently convert the penguin parameters into their and counterparts, where in the latter case also plausible dynamical assumptions beyond the are required.
3 The , System
3.1 Decay Amplitudes and CP Violation
In the SM, the decay into a CP eigenstate with eigenvalue originates from a colour-suppressed tree contribution and penguin topologies with -quark exchanges (), which are described by CP-conserving amplitudes and , respectively, and illustrated in Fig. 1. The primes are introduced to remind us that we are dealing with a quark-level process. Using the unitarity of the CKM matrix, the decay amplitude can be expressed in the following form :
are CP-conserving hadronic parameters. The Wolfenstein parameter takes the value , and
are combinations of CKM matrix elements. The parameter measures the size of the penguin topologies with respect to the tree contribution, and is associated with the CP-conserving strong phase . A key feature of the decay amplitude in Eq. (14) is the suppression of the term by the tiny factor . Consequently, can be extracted with the help of Eq. (13) up to corrections of .
As was pointed out in Ref. , the decay is related to through the -spin symmetry of strong interactions. It originates from transitions and therefore has a CKM structure which is different from . In analogy to Eq. (14), we write
where the hadronic parameters are defined as their counterparts. In contrast to Eq. (14), there is no factor present in front of the second term, thereby “magnifying” the penguin effects. On the other hand, the in front of the overall amplitude suppresses the branching ratio with respect to .
The -spin symmetry of strong interactions implies
In the factorisation approximation the hadronic form factors and decay constants cancel in the above amplitude ratios , i.e. -spin-breaking corrections enter through non-factorisable effects only. On the other hand, the relation
is already in factorisation affected by -breaking effects, entering through hadronic form factors as we will discuss in more detail below.
It is well known that the factorisation approximation does not reproduce the branching ratios of decays well, thereby requiring large non-factorisable effects. Furthermore, the QCD penguin matrix elements of the current–current tree operators, which are usually assumed to yield the potential enhancement for the penguin contributions, vanish in naive factorisation for decays. Consequently, large non-factorisable contributions may also affect the penguin parameters and , thereby enhancing them from the smallish values in factorisation, and Eq. (19) may receive sizeable corrections – despite the cancellation of form factors and decay constants in factorisation.
Making the replacements
we may apply the formalism introduced in Section 2, yielding the following phase shifts:
The expansions in terms of the penguin parameters show an interesting feature: the phase shifts are maximal for a strong phase difference around or . Conversely, the penguin shifts will be tiny for values around or , even for sizeable . The and enter
3.2 Branching Ratio Information
were performed, resulting in the world average 
Information on the penguin parameters is also encoded in this observable, thereby complementing the CP asymmetries. In view of the sizeable decay width difference of the -meson system, which is described by the parameter 
the “experimental” branching ratio (25) has to be distinguished from the “theoretical” branching ratio defined by the untagged decay rate at time . The conversion of one branching ratio concept into the other can be done with the help of the following expression :
The observable depends also on the penguin parameters, as can be seen in Eq. (8).
The effective lifetime
In order to utilise the branching ratio information, we construct the observable
where is the lifetime and denotes the phase-space function for these decays . In terms of the penguin parameters, we obtain
which corresponds to for .
The determination of from the experimentally measured branching ratios is affected by -spin-breaking corrections which enter through the ratio . Consequently, is not a particularly clean observable. On the other hand, the analysis of the direct and mixing-induced CP asymmetries does not require knowledge of .
3.3 Determination of and the Penguin Parameters
If we complement the ratio with the direct and mixing-induced CP asymmetries of the channel, we have sufficient information to determine and the penguin parameters and by means of the -spin relation in Eq. (19) . In this strategy, serves as an input, where we may either use its SM value in Eq. (2) or the value extracted from experimental data, as discussed in Section 4. We advocate the latter option since it takes possible CP-violating NP contributions to – mixing into account.
Although can be extracted with this method at the LHCb upgrade, the corresponding precision is not expected to be competitive with other strategies . It is therefore advantageous to employ as an input. Using data from pure tree decays of the kind , the following averages were obtained by the CKMfitter and UTfit collaborations:
For the numerical analysis in this paper, we shall use the CKMfitter result in view of the larger uncertainty. By the time of the LHCb upgrade and Belle II era, much more precise measurements of from pure tree decays will be available (see Section 3.5).
Once the direct and mixing-induced CP asymmetries of the channel have been measured, Eqs. (6) and (7) can be used with Eq. (21) to determine and in a theoretically clean way. Employing the -spin relation (19) allows us to convert these parameters into the phase shift , and thus to include the penguin effects in the determination of .
3.4 Constraining the Penguin Effects through Current Data
As a measurement of CP violation in is not yet available, the -spin strategy sketched above cannot yet be implemented in practice. However, in order to already obtain information on the size of the penguin parameters and and their impact on high-precision studies of CP violation, we may use experimental data for decays which have dynamics similar to .
If we replace the strange spectator quark with a down quark, as proposed in Ref. , we obtain the decay , which is the vector–pseudo-scalar counterpart of the vector–vector mode . The mode has contributions from penguin annihilation and exchange topologies, illustrated in Fig. 2, which have no counterpart in and are expected to be small. They can be probed through the decay (and for ) . First measurements of CP violation in were reported by the BaBar and Belle collaborations:
The results for the mixing-induced CP asymmetry are not in good agreement with each other, with the BaBar result lying outside the physical region. The Heavy Flavour Averaging Group (HFAG) has refrained from inflating the uncertainties in their average, giving . The Belle II experiment will hopefully clarify this unsatisfactory situation.
The charged counterpart of also has dynamics similar to but — as it is the decay of a charged meson — does not exhibit mixing-induced CP violation. It receives additional contributions from an annihilation topology, illustrated in Fig. 2, which arises with the same CKM factor as the penguin topologies with internal up-quark exchanges, contributing similarly to the penguin parameter (defined in analogy to Eq. (16)). If this parameter is determined from the charged , decays and compared with the other penguin parameters, footprints of the annihilation topology could be detected. In view of the present uncertainties, we neglect the annihilation topology, like the contributions from the exchange and penguin annihilation topologies in . In Appendix A, we give a more detailed discussion of the annihilation contribution and its importance based on constraints from current data, which do not indicate any enhancement.
We shall also add data for the (neglecting again the corresponding annihilation contribution) and modes to the global analysis, although the penguin contributions are doubly Cabibbo-suppressed in these decays.
Using the flavour symmetry and assuming both vanishing non-factorisable corrections and vanishing exchange and annihilation topologies, the decays listed above are characterised by a universal set of penguin parameters , which can be extracted from the input data through a global fit. The resulting picture extends and updates the previous analyses of Refs. [12, 13].
A first consistency check is provided by the ratios
involving decays which originate from the same quark-level processes but differ through their spectator quarks . Neglecting exchange and annihilation topologies and assuming perfect flavour symmetry of strong interactions, these ratios equal one. Within the uncertainties, this picture is supported by the data, as shown in Fig. 3. In this compilation, the -factory branching ratio measurements are corrected for the measured pair production asymmetry between and  at the resonance. Note that the branching ratios for decays into final states with or mesons have to be multiplied by a factor of two in Eq. (40) to take the and wave functions into account.
Let us now probe the penguin parameters through the various branching ratios. To this end, we use ratios defined in analogy to in Eq. (34). The extraction of these quantities from the data requires knowledge of the amplitude ratio , which is given in factorisation as follows :
The corresponding form factors have been calculated in the literature using a variety of techniques. For our analysis, we take the results from light cone QCD sum rules (LCSR), which are typically calculated at . The relevant form factors are ,  and , where the first two describe transitions for both the and the mesons. The dependence of these form factors is parametrised by means of the BGL method described in Ref. .
Using these form factors and neglecting non-factorisable -breaking effects,
we obtain the various observables compiled in Fig. 4. With exception
of the last entry, all observables share the same ratio .
Consequently, their central values and uncertainties are highly correlated.
However, even restricting the comparison to the statistical uncertainties shows an excellent
compatibility between the various results. The corresponding ratios are related to each
other through the isospin symmetry (neglecting additional topologies), and we obtain
a consistent experimental picture. The
agreement with the last entry, which involves the decay
instead of the modes, suggests that non-factorisable -breaking
effects and the impact of additional decay topologies
are small, thereby complementing the picture of Fig. 3.
The uncertainties are still too large to draw definite conclusions.
For the global fit to extract the penguin parameters and we use the input quantities summarised in Table 1, and add the CKMfitter result for in Eq. (37) as an asymmetric Gaussian constraint. As far as the observables are concerned, we employ the average of the combinations, which involve the same set of form factors (see Fig. 4), and the observable of the system. The branching ratios entering the observables are complemented by the corresponding direct CP asymmetries.
In order to add the mixing-induced CP asymmetry of the channel to the fit, the – mixing phase is needed as an input. However, the measured CP-violating asymmetries of the decay allow us to determine only the effective mixing phase222The numerical value in Eq. (42) actually corresponds to the mixing-induced CP asymmetry , which is an average of and data .
from Eq. (9). But — if we express the phase shift in terms of the penguin parameters — we may add this observable to our analysis.
The global fit yields for four degrees of freedom , indicating good agreement between the different input quantities. It results in the solutions
while is constrained to the input in Eq. (37). In Fig. 5, we show the correlation between and . The value of in Eq. (44) will serve as an input in Section 4. Following Ref. , we illustrate the various constraints entering the fit through contour bands of the individual observables in Fig. 6. For the range, we have used the value of in Eq. (44). In comparison with the analysis of Ref. , the penguin parameters are now constrained in a more stringent way. The penguin parameters in Eq. (43) result in the following penguin phase shift:
with confidence level contours shown in Fig. 7.
3.5 Benchmark Scenario for
Let us conclude the analysis of the penguin effects in by discussing a future benchmark scenario pointing to the LHCb upgrade era. Using the results in Eq. (43) and assuming the SM value for in Eq. (2), we obtain the following predictions:
in agreement with the experimental result in Eq. (32).
In order to illustrate the potential of the decay to extract the penguin parameters at the LHCb upgrade, let us assume that has been determined in a clean way from pure tree decays as
Let us assume that the CP-violating asymmetries of the channel have been measured as follows: