Towards new frontiers in CP violation in B decays
Abstract
CPviolating effects in decays of mesons offer a wide spectrum of probes for testing the phase structure of the quarkflavour sector of the Standard Model. After a brief discussion of the picture emerging from the current LHC data, the focus will be put on two specific topics: hadronic uncertainties from penguin topologies on measurements of the – mixing phases (), and the spinrelated decays and . Valuable new insights are expected from future studies of CP violation in decays. For the detection of possible new sources of CP violation, it will be crucial to match the experimental and theoretical precisions and to have a careful look at the underlying assumptions.

Nikhef, Science Park 105, NL1098 XG Amsterdam, Netherlands

Department of Physics and Astronomy, Vrije Universiteit Amsterdam, NL1081 HV Amsterdam, Netherlands

Email: Robert.Fleischer@nikhef.nl
1 Introduction
In the Standard Model (SM), the rich phenomenology of quarkflavour physics and CP violation is related to the Cabibbo–Kobayashi–Maskawa (CKM) quarkmixing matrix [1, 2]. Information on the phase structure and elements of this matrix are encoded in weak decays of , and mesons. Since the theory is formulated in terms of quarks while the mesons are bound states of strong interactions, we have to deal with processdependent, nonperturbative “hadronic” parameters in the calculation of the relevant transition amplitudes. This feature gives rise to the main challenge in studies of CP violation in decays: hadronic uncertainties.
In the presence of New Physics (NP), typically new sources of flavour and CP violation arise. Analyses of weak meson decays are facing an impressive hierarchy of scales:
(1) 
In order to deal with this situation, effective field theories offer the suitable theoretical tool. Within this framework, the heavy degrees of freedom (NP particles, top quark, and bosons) are integrated out from appearing explicitly and are described in shortdistance loop functions. Perturbative QCD corrections can be calculated in a systematic way, and renormalisation group techniques allow the summation of large logarithms. This machinery was applied to the SM and various popular NP scenarios, such as MSSM, models with universal and warped extra dimensions, little Higgs models, scenarios with extra bosons, etc., as reviewed in [3].
Following these lines, lowenergy effective Hamiltonians can be calculated for processes, taking the following general form [4]:
(2) 
where is Fermi’s constant and the denote combinations of CKM matrix elements. The shortdistance contribution to the decay amplitude is described by the Wilson coefficient functions , which can be calculated in perturbation theory for the SM and its extensions. On the other hand, the hadronic matrix elements are nonperturbative quantities, describing the longdistance contributions.
The key players for the exploration of CP violation in decays are nonleptonic channels. In the previous decade, there were interesting developments for calculations of such processes within QCD: QCD factorisation, the perturbative hardscattering (PQCD) approach, soft collinear effective theory (SCET) and QCD sum rules; the state of the art is discussed in [5]. There has recently been impressive progress in lattice QCD [6]. However, nonleptonic decays generally remain a theoretical challenge, which is also indicated by experimental data.
The outstanding feature of analyses of CP violation in decays is that the calculation of the hadronic matrix elements can be circumvented in fortunate cases (for a detailed discussion, see [7]). The corresponding strategies play a key role to “over constrain” the unitarity triangle (UT) of the CKM matrix. Detailed analyses and continuous updates are performed by the CKMfitter [8] and UTfit [9] collaborations. The current picture of the UT shows impressive consistency with the CKM sector of the SM, despite a few tensions.
The previous run of the Large Hadron Collider (LHC) has resulted in the exciting discovery of the Higgs boson. On the other hand, the ATLAS and CMS experiments have so far not seen signals of NP particles, and the SM flavour sector has been confirmed by the LHCb data, apart from a few discrepancies which are unfortunately not yet conclusive. The implications for the general structure of physics beyond the SM are a large characteristic NP scale, i.e. not just , or (and?) symmetries preventing large NP effects in the flavour sector, where models with “Minimal Flavour Violation” are the most prominent example.
Many more interesting results are expected from the next run of the LHC and its upgrade as well as highprecision flavour experiments. In view of the present situation, we have to prepare ourselves to deal with smallish NP effects. It will be crucial for resolving possible signals of NP in the data to have a careful look at the underlying theoretical assumptions and approximations. The challenge will be to match the experimental and theoretical precisions.
2 Penguin effects in benchmark probes of CP violation
Neutral mesons () show – mixing [10], which originates from box topologies in the SM but may well receive NP contributions. The CPviolating mixing phases are given by
(3) 
where is the usual angle of the UT, while and are parameters of the Wolfenstein parametrisation of the CKM matrix. The benchmark decays to measure the mixing phases through mixinginduced CP violation are given by , and . Decays of mesons play the key role at the LHC [11].
These determinations are affected by uncertainties from doubly Cabibbosuppressed penguin contributions [12–19], which cannot be calculated reliably and are usually neglected. In view of the current situation and the increasing experimental precision, the following questions arise: how important are the penguin contributions and how can they be controlled?
2.1 The system
In the SM, the decay originates from a coloursuppressed tree topology and penguin topologies with up, charm and top quarks running in the loops. Using the unitarity of the CKM matrix, the decay amplitude can be written as follows [12]:
(4) 
where and are CPconserving parameters, involving the relevant hadronic matrix elements. Whereas the former quantity is governed by the coloursuppressed tree contribution, the latter measures the ratio of penguin to tree topologies. The key feature of (2.1) is that the penguin parameter enters with the tiny ; is the usual UT angle.
CP violation is probed through the timedependent decay rate asymmetry
(5) 
where the direct CP asymmetry is proportional to . On the other hand, the mixinginduced CP asymmetry can be written in the following form [14]:
(6) 
where the hadronic phase shift is proportional to (and is usually neglected).
The decay is related to through the spin symmetry of strong interactions [12]. In the SM, its decay amplitude can be written as
(7) 
In contrast to (2.1), does not enter with , i.e. is not doubly Cabibbosuppressed. Consequently, the penguin effects are magnified in . It is useful to introduce a quantity , which complements the direct and mixinginduced CP asymmetries of the timedependent CPviolating rate asymmetry of .
The spin symmetry implies and , thereby allowing the determination of , and from and the two CPviolating observables of [12]. Since 1999, when this strategy was originally proposed, there has been a change of the main focus: a study [16] has shown that the extraction of will be feasible at LHCb but not competitive with other methods. As will be know by the time CP violation in can be detected, the corresponding CP asymmetries allow a clean determination of the penguin parameters and .
The channel was observed by CDF [20] and LHCb [21] but its CP asymmetries have not yet been measured. Using currently available data for decays with a CKM structure similar to , i.e. and , and complementing them with , data, the size of the penguin parameters and can be constrained. An analysis along these lines yields the following preliminary results [22]:
(8) 
2.2 CP violation in
The CKM structure of the channel is analogous to that of . However, as the final state is a mixture of CPeven and CPodd eigenstates , a timedependent angular analysis of the decay products has to be performed [23–25]. The impact of the SM penguin contributions is usually neglected. As in the case of , the expressions for the mixinginduced CP asymmetries are modified as follows [15]:
(9) 
where the hadronic phase shift depends on the finalstate configuration . The current average (neglecting the penguin effects) of the CDF, D0, ATLAS and LHCb data compiled by the Heavy Flavour Averaging Group is given by [26], which agrees with the SM value [8] of the – mixing phase.
A tool to control the penguin effects is offered by [15], which was observed by CDF [20] and LHCb [27]. Its branching ratio is found in agreement with the prediction following from , and its polarisation fractions agree well with those of . The channel, which shows also mixinginduced CP violation, is another interesting decay to shed light on the hadronic penguin effects [15].
2.3 CP violation in
Another interesting probe to study CP violation is provided by [29]. In contrast to , as the is a scalar state with quantum numbers , the final state is present in a wave and has the CP eigenvalue . Consequently, a timedependent angular analysis is not needed. On the other hand, the hadronic structure of the is still – after decades – not settled, with a variety of theoretical interpretations ranging from the quark–antiquark picture to tetraquarks. A detailed discussion of the implications of this feature for the extraction of was given in [18] (for decays, see [30]), while recent LHCb measurements related to this topic are reported in [31].
2.4 Effective decay lifetimes
The measurement of effective lifetimes of decays, which are defined as
(10) 
offers yet another way to obtain insights into CP violation [32]. Here it is particularly interesting to compare decays into CPodd final states, such as , with those into CPeven final states, such as and . The measured effective lifetimes can be converted into contours in the – plane, where is the decay width difference of the meson system (for an overview of the status of , see [33]). The lifetime contours are very robust with respect to hadronic uncertainties [32]. For an update with the most recent LHCb data, see [34]. The decay lifetimes result in a picture in agreement with the SM.
2.5 Comments for the LHCb upgrade era
In view of hadronic effects, it is important to give measurements of for the individual decay channels , i.e. and . The pattern of the may provide insights into the hadronic effects: differences in the values of would indicate hadronic effects. On the other hand, should no differences between the individual emerge, there would be evidence for negligible hadronic effects (within the errors) or a universal hadronic phase shift.
The timedependent analysis of CP violation in allows the clean determination of the corresponding penguin parameters (see Subsection 2.1). A sizeable penguin parameter would indicate a potential problem in the measurement of from the and channels. On the other hand, smallish penguin effects would give us confidence for the measurement of , although subtleties may arise due to the different final states.
3 CP Violation in and
The decays and receive contributions from tree and penguin topologies. In the SM, their decay amplitudes can be written as
(11) 
where , and their unprimed counterparts are CPconserving strong quantities [35]. The direct and mixinginduced CP asymmetries of the and decays allow the determination of theoretically clean contours in the – and – planes, respectively. Since these decays are related to each other through the interchange of all down and strange quarks, the spin symmetry of strong interactions implies and ; the former relation allows the extraction of and from the contours [35, 36]. Moreover, the strong phases and can be determined, allowing an internal consistency check of the spin symmetry. Further insights into the hadronisation dynamics are provided by and , which can be extracted from the ratio . This strategy is promising for the LHCb physics programme [28]. It will be particularly interesting to compare the resulting value of with those following from methods using only treediagramlike meson decays.
The picture resulting from the current data was explored in detail in [36, 37]; the numerical results given below refer to the update by Rob Knegjens in [38]. An interesting variant of the method was proposed in [39], as discussed in detail by Marco Ciuchini in [40].
Using information on , CP violation in and , and allowing for spinbreaking corrections , results in
(12) 
which agrees with the “treelevel” results [8] and [9] within the uncertainties. There are no indications for sizeable nonfactorisable breaking corrections in the corresponding data. In the SM, the mixinginduced CP asymmetry is predicted as
(13) 
while .
The first LHCb measurement [41] of the CPviolating observables yields
(14) 
and agrees with the SM predictions given above. In the future, once the experimental precision for the CP asymmetries improves, can be extracted exclusively from the – contours. The observable , which is affected by form factors and nonfactorisable effects, will then yield insights into hadronic physics. The current data point towards a fortunate situation for the determination of which is very robust with respect to spinbreaking corrections [36–38].
4 Outlook
The exploration of CP violation in decays is a very broad field, with many other interesting topics complementing those discussed above. I would like to briefly give two more examples:

A promising decay for Belle II at SuperKEKB is the decay . A correlation between the direct and mixinginduced CP asymmetries of this channel can be predicted in the SM, with current data showing an intriguing discrepancy [44]. It will be interesting to monitor the future measurements of the corresponding observables at Belle II.
We are moving towards new frontiers in particle physics. There are still no unambiguous signals for NP at the LHC, and it is impressive – also frustrating – to see how the SM stands more and more stringent tests, both at the highenergy and at the highprecision frontier. Much more is yet to come with the future running of the LHC and dedicated studies of flavour physics, including CP violation in decays. However, we have to prepare ourselves to deal with smallish NP effects in the data. In view of the increasing experimental precision, we have to be careful with respect to theoretical assumptions and approximations. The challenge will be the matching of the experimental and theoretical uncertainties in the future highprecision era. Interesting and fruitful years for the further testing of the SM and the search of NP are ahead of us!
Acknowledgements
I would like to thank Cristina Lazzeroni and her coorganizers for inviting me to this
most interesting and enjoyable conference and their kind hospitality in Birmingham.
References
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