CKM global fit and constraints on New Physics in the meson mixing
Abstract
An update profile of the CKM matrix is given, providing numerical and graphical constraints on the CKM parameters in the Standard Model. Constraints on additional parameters accounting for possible new physics contribution in a modelindependent analysis are also reported with emphasis on the leptonic decay of the and on the mixing phase.
I Introduction
Within the Standard Model (SM), the quark flavor mixing is described by the CabibboKobayashiMaskawa (CKM) matrix CKM (). The 3x3 unitary CKM matrix can be fully parametrized by four independent parameters among which a single nonvanishing complex phase accounts for the violation of the CP symmetry.
Inspired from the one proposed by Wolfenstein Wolfenstein () the following parametrization, phaseconvention independent and unitaryexact to all orders in thePapII (),
is used throughout this document:
While the parameters and are accurately determined through the measurement of from the superallowed nuclear transitions, from the semileptonic kaon decay and from the semileptonic decay with charm, the complex parameter, being the apex coordinates of the unitarity triangle (UT) related to the first and third quark families, is less constrained. The metrology of the UT apex via the determination of its sides and angles allows to measure the size of the CP violation and to validate the overall consistency of the KM scenario within the Standard Model. Any inconsistency would suggest contributions from physics beyond the Standard Model.
Ii The Ckm Fit Inputs
s The CKM global fit is performed within a frequentist statistical approach including a specific treatment to handle the theoretical uncertainties on some of the inputs (RFit thePapII ()). The inputs used for the fit are the most recent available value of the observables from the and meson physics for which the theoretical conversion into CKM parameters is under control. The most relevant input observables of the global CKM fit, with their value and reference, are listed in the table 1. The observable values that have been updated for this analysis with respect to the previous similar analysis CKMMoriond08 () are indicated.
input  source  value  (ref)  update  UT constraint  associated theory [parameters] 
Nuclear decays  (Towner ())  [ form factor]  
SL kaon decays  (Flavia ())  [ form factor]  
SL charmed decays  (HFAG ())  [ form factor and/or OPE]  
SL charmless decays  (HFAG ())  [ form factor and/or OPE]  
BR  Leptonic B decays  (TauNu ())  *  leptonic amplitude [, ]  
mixing  (Dms ())  =2 amplitude [, , , ]  
mixing  (HFAG ())  =2 amplitude [ , ]  
mixing  (PDG ())  *  )  =2 amplitude [, , , ]  
sin  Charmonium decays  ()  (HFAG ())  *    
BR &  decays  Bfactories average  (angle ())  *  SU(2)  
BR &  decays  Bfactories average  (angle ())  *  GLW/ADS/GGSZ 
The right side of the table addresses the associated theory, possibly including phenomenological parameters, needed to derive a constraint on the UT parameter(s) from the input observable.
The value coming from the lattice QCD calculation (LQCD) for some of the phenomenological quantities used in this analysis are listed in the second column of the table 2. For comparison purpose, some alternative and more recent values proposed in the reference Lubicz () are quoted in the last column (note that the statistical and theoretical uncertainties have been assumed quadratically summable by the authors).
A more detailed review of the CKM global fit inputs can be found in CKMfitter ().
Iii The Sm Ckm Fit Results
The left part of the figure 1 displays the result of the global fit together with the 95% CL contours of the individual constraints.
A satisfactory agreement is observed from the various individual contributions at the 2 level, establishing the KM mechanism as the dominant source of the CP violation in the meson system.
As shown on the right side of the figure 1, a slight tension is however revealed when comparing the global constraint coming from the observables dominated by the experimental measurements (UT angles) with the constraint derived from the observables inducing larger theoretical uncertainties (, BR(), , /). This classification of the observables is for illustration purpose and should not be used to draw any conclusion about the possible origin of the tension: a similar discrepancy is also visible when comparing the constraint from the treelevel processes with the mixing loopinduced or the constraint from the CPconserving observables versus the CPviolating ones.
As illustrated on the left plot of the figure 2, the bulk of the tension is located in the correlations between the CPconserving, UTside related, theorydependent observable, BR (central value of which slightly increases with the summer 2008 update TauNu ()) and the CPviolating, UTangle related, theoryfree observable, sin (central value of which slightly decreases HFAG ()).
Quantitatively, the minimal of the global fit decreases by 2.9 (2.6 ) when removing BR (sin) from the list of the fit inputs. Obviously, the disagreement is not large enough to exclude the possibility of a statistical origin of the tension.
However, it is worth mentioning that a similar discrepant pattern is also observed for the leptonic decays (see for instance Kronfeld (); Gamiz ()), suggesting a common origin.
As an illustration, the expected and measured branching fraction of the leptonic and decays to final state are displayed on the right side of the figure 2.
The possibility of a generic problem with the lattice prediction for the theoretical parameters related to the meson has been investigated.
Although the treelevel expression for the amplitude of the the annihilation decay is directly proportional to the decay constant , it can be shown that an underestimation of the lattice prediction for the parameters product as used in this analysis, can not by itself explain the observed tension.
Indeed, the BR together with the observables provide a independent constraint in the () plane in which a clear tension remains as illustrated on the figure 3 for the two set of the LQCD parameters values quoted in the table 2.
More precisely, the independent ratio:
allows for a theory parameterfree prediction of the bag parameter, , from the experimental measurement of , (or ), , BR and . The resulting prediction is 2.4 away from the lattice calculation Tantalo (); Nierste () as shown in the right plot of the figure 3.
This deviation is essentially dominated by the experimental uncertainties on the branching ratio and on the UT angles and .
A lattice origin of the tension would thus involve the bag parameter that
controls the correlations between BR and the angle , as explicitely shown in the above formula. Further investigations are needed to check whether the possible correlations between the theoretical errors affecting the lattice parameters, generally not provided by the lattician community and therefore neglected in this analysis, could account for the observed tension.
Eventually, the hypothesis of new contributions beyond the SM is discussed in the next section.
Iv Constraints on New Physics in the Mesons Mixing
iv.1 Fit procedure
New Physics (NP) is expected to affect the amplitude of the neutral mesons mixing in many scenarii (see for instance Fleisher ()). Assuming NP to contribute mostly to the shortdistance part of the processes, a modelindependent parametrization has been proposed Lenz (); Grossman ():
where the label stands for the or flavor of the neutral meson and the complex parameter accounts for the NP contribution. Assuming in addition that the treelevel mediated decays proceeding through a Four Flavor Change get only SM contributions (SM4FC hypothesis SM4FC (); CKMfitter ()), the observables , and ^{1}^{1}1where means the UT angle extracted from the analysis of the charmonium meson decays. are not affected by the NP contribution and can be used in a (SM+NP) global fit to fix the SM parameters. The oscillations parameters, the weak phases, the semileptonic asymmetries and the meson lifetimedifferences are affected by the phase and/or the amplitude of the NP contribution as quoted in the table 3 and allow to constrain the NP deviation to SM parametrized with .
parameter  prediction in the presence of NP 

The resulting constraints in the (Re,Im) plane are summarized in the left and right side of the figure 4 for the and case, respectively. For a more complete review of the constraints from the global fit of NP in the B meson mixing see Capri ().
iv.2 The case
For the meson case, the dominant constraints come from the mixing phase, , and the oscillation frequency . Both reasonably agree with their SM prediction. A 2.1 deviation is obtained for the 2dimensional SM hypothesis . The 1dimension hypothesis based on the phase only, =0, corresponding the phase expectation in the Standard Model as well as in the Minimal Flavour Violation scenarii, results in a 1.5 deviation. This slight disagreement with the SM hypothesis is essentially the translation of the tension in the SM global fit discussed in the previous section (obviously the discrepancy in the leptonic decays mentionned in the previous section is unaffected by the NP scenario in the mixing assumed here).
When removing BR observable from the fit, the deviation reduces down to 0.9 for both the 1D null NPphase hypothesis and the 2D SM hypothesis.
iv.3 The case
The fit for New Physics in the mixing also exhibits a 2.1 deviation for the SM 2D hypothesis, and a 2.5 for the 1D null NPphase hypothesis, =0 ^{2}^{2}2A deviation in excess of 3 in the fit for NP in the mixing had previously been reported by the UTFit collaboration UTfit () and has recently been updated to a lower value Maurizio (). Contrarily to the case, the source of the discrepancy is obvious: the deviation is fully dominated by the direct measurement by the CDF and the D0 experiments of the correlated weak mixing phase and lifetime difference, () through the timedependent angular analysis of the decay . Both collaborations report a large phase with respect to the SM expectation Phis (). The Heavy Flavor Averaging Group HFAG () obtains a 2.2 deviation for the SM hypothesis in the D0/CDF () combination. The other contributions to the NP fit only provide weak constraint:

the oscillation frequency, , only sensitive to the module of is consistent with its SM prediction.

the semileptonic asymmetries, , and the FlavorSpecific propertime, do not display any significant sensitivity to the NP fit parameters in the current state of their determination.

the theoretical lifetime difference proportional to the cosine of the weak phase: slightly tends to push the NP phase toward the zero value as the measured is larger than the SM expectation for Lenz ().
V Summary and Conclusions
The current data successfully fit the KM mechanism establishing it as the dominant source of CP violation in the sector. The current limitation of the CKM consistency tests are of three origins: LQCD quantities, the UT angle and the measurements. With the recent update of the inputs, a slight tension appears in the global CKM fit within the SM hypothesis. This tension is mainly driven by the larger BR updated value and the smaller sin value that Bfactories recently reported and results from a nontrivial correlation that we have explicited in the ratio BR/.
Both updates are consistent with the previous measurements but slightly go away from their SM prediction.
Besides the possible statistical fluctuation of the experimental measurements, further investigations are necessary to understand the role LQCD quantities may play in the origin of the tension.
Assuming the presence of New Physics in the mixing, a modelindependent fit procedure has been applied.
As a translation of the observed tension in the SM CKM fit, the SM hypothesis is 2.1 away from the preferred NP fit solution in the case.
A 2.1 deviation to SM is also observed in the sector, fully dominated by the direct experimental measurement of the mixing phase by the Tevatron experiments.
Within a joint fit for NP in the and mixing, the 4dimension SM hypothesis () results in a 2.9 deviation with respect to the prefered fit solution.
A more complete update with all the data eventually available at the end of the summer 2008 is to appear in CKMfitter (). In the near future, significant improvements in the lattice QCD predictions as well as more precise experimental measurements can be expected in both the and the sectors. These data will be scrutinized accurately to confirm or not the appearing tension in the sector and the large mixing phase observed in the mixing. Eventually a major step in the consistency tests of the KM mechanism will arise with the imminent LHC era and in particular with the dedicated LHCb experiment.
Acknowledgements.
My warmest thanks go to my collaborators of the CKMfitter group for their support in the preparation of this talk. I also would like to thank the organizers of the ICHEP08 conference for their kind hospitality in Philadelphia, PA.References
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