Predictions of selected flavour observables within the Standard Model
This letter gathers a selection of Standard Model predictions issued from the metrology of the CKM parameters performed by the CKMfitter group. The selection includes purely leptonic decays of neutral and charged , and mesons. In the light of the expected measurements from the LHCb experiment, a special attention is given to the radiative decay modes of mesons as well as to the -meson mixing observables, in particular the semileptonic charge asymmetries which have been recently investigated by the DØ experiment at Tevatron. Constraints arising from rare kaon decays are addressed, in light of both current results and expected performances of future rare kaon experiments. All results have been obtained with the CKMfitter analysis package, featuring the frequentist statistical approach and using Rfit to handle theoretical uncertainties.
|transitions||=||Hardy:2008gy ()||Nuclear matrix elements|
|semileptonic decays||=||Asner:2010qj ()||form factors, shape functions|
|semileptonic decays||=||Asner:2010qj ()||form factors, OPE matrix elements|
|, ,||branching ratios, CP asymmetries||Asner:2010qj ()||isospin symmetry|
|inputs for the 3 methods||Asner:2010qj ()||GGSZ, GLW, ADS methods|
In the Standard Model (SM), the weak charged-current transitions mix quarks of different generations, which is encoded in the unitary Cabibbo-Kobayashi-Maskawa (CKM) matrix Cabibbo:1963yz (); Kobayashi:1973fv (). In the case of three generations of quarks, the physical content of this matrix reduces to four real parameters, among which one phase, the only source of violation in the SM (we do not consider minute -violating effects from the strong-interaction -term or from the masses of neutrinos). One can define these four real parameters as:
and exploit the unitarity of the CKM matrix to determine all its elements (and when needed, to obtain their expansion in powers of ) Charles:2004jd ().
Extracting information on these parameters from data is a challenge for both experimentalists and theorists, since the SM depends on a large set of parameters which are not predicted within its framework, and must be determined experimentally. A further problem comes from the presence of the strong interaction that binds quarks into hadrons and is still difficult to tackle theoretically, leading to most of the theoretical uncertainties discussed when extracting the CKM matrix parameters. The CKMfitter group follows this goal using a standard -like frequentist approach, in addition to the Rfit scheme to treat theoretical uncertainties, aiming at combining a large set of constraints from flavour physics Charles:2004jd (); CKMfitterwebsite ().
Not all the observables in flavour physics can be used as inputs for these constraints, due to limitations on our experimental and/or theoretical knowledge on these quantities. The list of inputs of the global fit is indicated in table 1: they fulfill the double requirement of a satisfying control of the attached theoretical uncertainties and a good experimental accuracy of their measurements. In addition, we only take as inputs the quantities that provide reasonably tight constraints on the CKM parameters . This selects in particular leptonic and semileptonic decays of mesons yielding information on moduli of CKM matrix elements, non-leptonic two-body decays related to angles of the CKM matrix, and and -mixing parameters.
The current situation of the global fit in the is indicated in Fig. 1. Some comments are in order before discussing the metrology of the parameters. There exists a unique preferred region defined by the entire set of observables under consideration in the global fit. This region is represented by the yellow surface inscribed by the red contour line for which the values of and correspond to . The goodness of the fit can be addressed in the simplified case where all the inputs uncertainties are taken as Gaussian, with a -value found to be (i.e., 1.5 ; a rigorous derivation of the -value in the general case is beyond the scope of this letter wip ()). One obtains the following values (at 1 ) for the 4 parameters describing the CKM matrix:
At this stage, it is fair to say that the SM hypothesis has passed the statistical test of the global consistency of all observables embodied in the fit, although some discrepancies are detailed in the following sections. We are therefore allowed to perform the metrology of the CKM parameters and to give predictions for any CKM-related observable within the SM. Let us add that the existence of a region in the plane is not equivalent to the statement that each individual constraint lies in the global range of . One of the interest of SM predictions is that each comparison between the prediction issued from the fit and the corresponding measurement constitutes a null-test of the SM hypothesis. Indeed, we will see that discrepancies actually do exist among the present set of observables considered in this letter (the corresponding pulls are reported in Table 2).
We predict observables that were not used as input constraints, either because they are not measured with a sufficient accuracy yet, e.g., , or because the control on the theoretical uncertainties remains controversial, e.g., . The corresponding predictions can then be directly compared with their experimental measurements (when they are available). We also consider some particularly interesting observables used as an input of the fit, e.g., . In this last case, we must compare the measurement of the observable with the outcome of the fit without including the observable among the inputs, so that the experimental information is used only once.
Following this procedure, we do not take the following quantities as inputs, but we predict their values: the semileptonic asymmetries and , the weak phase in the mixing , the branching ratios of the dileptonic decays of neutral mesons ), the branching ratio of (exclusive and inclusive) radiative transitions, and rare decays. The first three observables have all in common to provide only loose constraints on the CKM parameters, while the two latter, though fulfilling the requirement of a good control of their related theoretical uncertainties, are so far out of reach of the current experiments. The LHCb experiment should bring a breakthrough in that respect very soon and these quantities will be included in the global fit once the required measurement accuracy is achieved LHCb:2009ny (). The experimental situation is pretty similar for the semileptonic asymmetries related to neutral-meson mixing, with the additional drawback that these observables suffer from large theoretical uncertainties. The exclusive radiative transitions suffer from significant uncertainties and are thus only consider for predictions. On the contrary, the inclusive , which have been measured and are well controlled theoretically, will be added as input of the global fit wip (), but are kept for the present letter among the predictions. Finally, rare kaon decays have not been measured yet or provide only loose constraints on the CKM matrix elements.
In the following sections, we first discuss the main sources of theoretical uncertainty, before spelling out some of the fundamental formulae used for our predictions within the SM. We then collect the results obtained and compare them with their measurements (when available).
Ii Strong interaction parameters
The first category of theoretical uncertainties in flavour analyses arise from matrix elements that encode the effects of strong interaction in the non-perturbative regime. These matrix elements boil down to decay constants, form factors and bag parameters for most of the observables under scrutiny in the present note, and all our predictions are subjected to and limited by the uncertainties in the determination of these observables. These uncertainties must be controlled with care since their misassessment or underestimation would affect the statements that we will make on flavour observables.
Among the different methods used to estimate non-perturbative QCD parameters, quark models, sum rules, and lattice QCD (LQCD) simulations are tools of choice. We opt for the latter whenever possible, as they provide well-established methods to compute these observables not only with a good accuracy at the present time, but also with a theoretical framework allowing for a systematic improvement on the theoretical control of the uncertainties. Over the last few years, many new estimates of the decay constants and the bag factors have been issued by different lattice collaborations, with different ways to address the uncertainties. A part of the uncertainties has a clear statistical interpretation: lattice simulations evaluate Green functions in an Euclidean metric expressed as path integrals using Monte Carlo methods, whose accuracy depends on the size of the sample of gauge configurations used for the computation in a straightforward way. But systematics are also present and depend on the strategies of computation chosen by competing lattice collaborations: discretisation methods used to describe gauge fields and fermions on a lattice, interpolating fields, parameters of the simulations, such as the size of the (finite) volumes and lattice spacing, the masses of the quarks that can be simulated, and the number of dynamical flavours included as sea quarks (2 and 2+1 being the most frequent, after a long period where only quenched simulations were available). These simulations must be extrapolated to obtain physical quantities (relying in particular on effective theories such as chiral perturbation theory and heavy-quark effective theory).
The combination of lattice values with different approaches to address the uncertainty budget is a critical point of most global analyses of the flavour physics data, even though the concept of the theoretical uncertainty for such quantities is ill-defined (and hence is the combination of them). Several approaches have been proposed to perform such a combination. We have collected the relevant LQCD calculations of the decay constants , , , , , , as well as the bag parameters , and , and the form factor . In addition we designed an averaging method aiming at providing a systematic, reproducible and to some extent conservative scheme Tisserand:2009ja (). These lattice averages are the input parameters used in the fits presented in this paper.
In the specific case of decay constants, the -flavour breaking ratios , , are better determined than the individual decay constants. We will therefore take these ratios as well as the strange-meson decay constants as reference quantities for our inputs. In the same spirit, it is more relevant to consider the predictions of the ratio of the kaon and pion leptonic partial widths, as well as instead of the individual branching ratios.
A comment is in order concerning the second category of theoretical uncertainties that are not directly related to LQCD. As far as global fit inputs are concerned, this is the case for the inclusive and exclusive determinations of and , which involve non-perturbative inputs of different nature. We use the latest HFAG results Asner:2010qj () for each of these determinations and combine inclusive and exclusive determinations following the same scheme as for the combination of lattice quantities. We refer the reader to refs. Lenz:2010gu (); Deschamps:2009rh () for a more detailed discussion of each constraint, whereas the related hadronic inputs can be found in ref. CKMfitterwebsite ().
Iii Neutral -meson leptonic decays
Dileptonic decays of and mesons are among the most appealing
laboratories to study scalar couplings in addition to the SM
couplings. The current experimental limits set by the Tevatron and LHCb
experiments on the dimuonic branching ratios TeVexp (); Aaij:2011rj () are
already giving significant constraints on scenarios beyond the Standard
Model. The main limitation in the current predictions arises from the
knowledge of the decay constants and .
The master formula for the branching ratios reads as:
where is the next-to-leading-order Inami-Lim function BBL (); bsmumuSM (). is the top quark mass defined in the scheme, related to the pole mass determined at the Tevatron as at next-to-leading order of QCD. is the Fermi constant, the electroweak mixing angle, and the electromagnetic coupling constant at the pole. We vary the renormalisation scale between and .
Iv Charged meson leptonic decays
The decay of a charged meson () into a leptonic pair is mediated in the SM by a charged weak boson, with the branching ratio:
where () stands for the up-like (down-like) valence quark of the meson respectively, is the relevant CKM matrix element, is the decay constant of the meson and its lifetime. A similar formula holds for decays into a single light meson (pion or kaon). The corrective factor stands for channel-dependent electromagnetic radiative corrections. They are taken into account in the case of the lighter mesons ( and ), where their impact is estimated to be at the level of Kl2pil2em (), and for the mesons, where the effect is Dl2em (). In the case of mesons, no dedicated corrections are supposed here, and we assume that all the corrections from soft photons will be taken into account through the Monte Carlo analyses of the experiments (see ref. Bl2em () for a discussion of the corrections due to soft-photon emission).
V Radiative -meson decays
The radiative transitions provide very interesting probes of New Physics, as they are mediated by penguin transitions which are directly related in the SM with and mixing (from the CKM point of view), but can be affected differently by additional particles/couplings. A convenient framework for their analysis is provided by the effective Hamiltonian where all degrees of freedom heavier than the -quark have been integrated out, leading to Wilson coefficients (encoding short-distance physics) multiplied by operators with light degrees of freedom (describing long-distance physics).
Hadronic uncertainties may be significant for the exclusive decays and due to the form factors and the long-distance gluon exchanges:
where is a Clebsch-Gordan coefficient, the electromagnetic coupling constant at vanishing momentum, the index corresponds to the photon polarisation, and is suppressed compared to where is the Wilson coefficient of the electromagnetic dipole operator (corrections can be estimated using a expansion). In eq. (6), denotes any up-type quark. We follow ref. Ball:2006eu () for the prediction of the branching ratio and the analysis of hadronic uncertainties (however, we use results from light-cone QCD sum rules and do not perform any rescaling of the tensor form factor ).
The inclusive transition can be treated relying on the quark-hadron duality and using a heavy-quark expansion, so that the prediction for this transition suffers from fewer theoretical uncertainties (mostly related to the precise values of the quark masses and the higher orders in both and expansions). However, this observable is not fully inclusive as a cut in the photon energy is required. The corresponding expression is bsgamma ():
where is a factor related to the choice of transition as a normalisation for the computation, collects the estimate from non-perturbative -suppressed contributions, and has been estimated up to next-to-next-to-leading order using an interpolation on the charm quark mass, leading to a formula of the form where the are perturbative kernels. For the present analysis, we use the parametrisation described in detail in ref. Deschamps:2009rh ().
The (exclusive and inclusive) radiative decays provide more observables, which are already experimentally accessible, but they are out of the scope of this short note wip ().
Vi -violating -mixing observables
The mixing of the and mesons can be described upon introducing the mass and decay matrices, and . These matrices are involved in the evolution operator for the quantum-mechanical description of the oscillations (with or ). Their diagonalisation defines the physical eigenstates and with masses and decay rates . One can reexpress these quantities in terms of three parameters: , and the relative phase .
Oscillation frequencies, which feature the CKM elements directly, can be predicted in a theoretically clean way, though the precision is severely limited by the knowledge of the decay constants and bag parameters Lenz:2010gu (). Here we would like to concentrate on the prediction of the four -violating observables (mixing phases) and (semileptonic asymmetries), with or . The two first observables are CKM angles of the and unitarity triangles, respectively, and read as functions of the CKM elements:
These angles (which should not be confused with the relative phases introduced above) measure violation arising in the interference between mixing and decay in and hence exhibit a strong hierarchy between the and quarks. On the contrary, the semileptonic asymmetry probes violation in the mixing, and can be written as:
The ingredients needed to predict these asymmetries are hence the matrix elements and . The dispersive term is mainly driven by box diagrams involving virtual top quarks, and it is related to the effective Hamiltonian as:
The SM expression for is BBL ():
with the four-quark operator and the Wilson coefficient :
and the Inami-Lim function is calculated from the box diagram with two internal top quarks. The absorptive term is dominated by on-shell charmed intermediate states, and it can be expressed as a two-point correlator of the Hamiltonian . By performing a -expansion of this two-point correlator, one can express in terms of and , where stands for “scalar” and are colour indices Lenz:2010gu (). The matrix elements are expressed in terms of the bag parameters:
One has also to consider further bag parameters which parameterise the matrix elements of the subleading operators in the heavy quark expansion of the -violating observables (only rough estimates are available for these bag parameters) Lenz:2010gu (). The SM predictions of the mixing phases and semileptonic asymmetries for the neutral mesons in the SM are collected in table 2.
|Charged Leptonic Decays|
|Neutral Leptonic decays|
|Asner:2010qj ()||( -6.5||0.8|
|0.678 0.020||Asner:2010qj ()||0.832||2.7|
|0.76 0.02||D0betas ()|
Vii The rare kaon decays and
Theoretically clean constraints on the CKM matrix can be obtained from rare kaon decays with neutrinos in the final state, as they can only arise via second-order weak transitions (-penguins and box) within the SM, and light-quark loops are strongly GIM-suppressed. Within the SM, the decay rate is given by Buchalla:1998ba (); Buras:2005gr (); Brod:2008ss (); Isidori:2005xm ():
where the isospin-breaking parameter can be extracted from semileptonic decays with a correction for the photon cut-off dependence, the functions comprise the top quark contributions, and the light quark contributions are given by the and parameters, which are the dominant theoretical uncertainties. Similarly for the mode, the SM decay rate is given by Buchalla:1998ba ():
with only small residual uncertainties from the isospin-breaking parameter and scale invariance. In terms of CKM parameters, a measurement of the provides a quasi-elliptical constraint in the plane, centered close to the vertex of the unitarity triangle located at . The measurement of the branching ratio for would provide a clean constraint on .
This letter collects a selection of SM predictions driven by the global fit of the CKM parameters, in view of related recent or foreseeable experimental measurements. The main outcome is summarised in Table 2, gathering the SM predictions using the inputs collected in Table 1. The third column of Table 2 shows the agreement between the measurement and the prediction as a pull. The latter is computed from the difference with and without the measurement of this observable, interpreted with the appropriate number of degrees of freedom, and converted in the number of equivalent standard deviations (the lack of an updated average for between the Tevatron experiments explain the presence of two distinct measurements as well as the absence of a pull).
The largest departures of the measurements from the SM predictions are found for two observables: and . It is remarkable that this discrepancy can be accommodated by a very simple extension of the SM allowing for the presence of New Physics in mixing, as discussed extensively in ref. Lenz:2010gu (). One can also notice that the decay exhibits only a mild discrepancy between prediction and measurement, due to the recent improvements in both lattice simulations and experimental measurements.
Concerning neutral-meson mixing, and following the outstanding success of the factories in their measurements of , one of the main goals and challenges for for the LHCb experiment will consist in characterising the -meson mixing properties through the measurement of all the relevant observables. Each of these measurements in LHCb but will provide a null test of the SM hypothesis. In the present experimental context, two of these observables are particularly interesting: the mixing weak phase and the difference of the semileptonic asymmetries for the and mesons. The former is predicted very accurately:
Significant constraints have already been set on this phase by the Tevatron experiments CDFbetas (); D0betas (). The LHCb experiment should in a near future settle its value, as suggested by the promising exploratory work with the first data described in ref. LHCbbetas ().
The semileptonic asymmetries are determined far less precisely by the global fit of the CKM parameters. Their prediction suffers from notable strong-interaction uncertainties (in particular bag parameters). Yet, following a recent DØ measurement of the dimuon asymmetry which departs from the SM by 3.2 Abazov:2010hv (), the measurement by the LHCb experiment of the difference of the semileptonic asymmetries is eagerly awaited. The prediction of the difference in the SM is:
Among the null tests of the SM hypothesis, the -penguin decay rate is specially appealing. Its next-to-leading order prediction from the global fit reads:
We would like to conclude this discussion with observables which can uniquely be measured at super- factories. The important role of onto the global fit has been already underlined in this letter, and its SM prediction is:
An improved precision of the measurement can only be achieved at high-luminosity factories. The branching ratio of the muonic mode, predicted to be:
is a further experimental target.
Let us finally add that this short letter has collected the SM predictions for some salient observables in flavour physics, in view of the running or foreseen experimental programmes here. This obviously does not exhaust the discussion of the inputs, predictions and methods dealt with the CKMfitter package, but we leave this subject for a more extensive forthcoming publication wip ().
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