- Email: [email protected]
The Unitarity Triangle Analysis in the Standard Model
Nuclear Physics B (Proc. Suppl.) 185 (2008) 5–10 www.elsevierphysics.com
The Unitarity Triangle Analysis in the Standard Model UTfit Collaboration: M. Bonaa , M. Ciuchinib∗ , E. Francoc , V. Lubiczdb , G. Martinelliec , F. Parodif , M. Pierinia , C. Schiavif , L. Silvestrinic , V. Sordinig , A. Stocchig , C. Tarantinodb , V. Vagnonih a
CERN, CH-1211 Geneva 23, Switzerland
b
INFN Sezione di Roma Tre, Via della Vasca Navale 84, I-00146 Roma, Italy
c
INFN Sezione di Roma, Piazzale A. Moro 2, I-00185 Roma, Italy
d e
Dipartimento di Fisica, Universit` a di Roma Tre, Via della Vasca Navale 84, I-00146 Roma, Italy
Dipartimento di Fisica, Universit` a di Roma “La Sapienza”, Piazzale A. Moro 2, I-00185 Roma, Italy
f
Dipartimento di Fisica, Universit` a di Genova and INFN Sezione di Genova, Via Dodecaneso 33, I-16146 Genova, Italy
g
Laboratoire de l’Acc´el´erateur Lin´eaire, IN2P3-CNRS and Universit´e de Paris-Sud, BP 34, F-91898 Orsay Cedex, France
h
INFN Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, Italy
1. Introduction The original mission of the UTfit Collaboration [1] was the determination of the coordinates ρ¯ and η¯ of the apex of the Unitarity Triangle (UT) and in general of the parameters describing the CKM matrix [2] in the Standard Model (SM). Nowadays the SM analysis encompasses many additional results, such as predictions for several flavour observables and measurements of hadronic parameters which can be compared with the lattice QCD predictions [3]. Furthermore, the UT analysis has been extended beyond the SM, allowing for a model-independent determination of ρ¯ and η¯ (assuming three-generation unitarity and negligible new physics (NP) contributions to tree-level processes) and a simultaneous evaluation of the size of NP contributions to ΔF = 2 amplitudes compatible with the flavour data [4,5]. Recently, the NP analysis has been expanded to include an effective field theory study of the allowed NP contributions to ΔF = 2 amplitudes. This allows to put model-independent bounds on ∗ talk
given by M.C.
0920-5632/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2008.10.001
the NP energy scale associated to flavour- and CP-violating phenomena [6]. In these proceedings we present a preliminary update of our UT analysis in the SM, including a set of fit predictions and a study of the compatibility between the fit results and some of the most interesting experimental constraints. The main difference with respect to previously published results comes from the use of updated set of lattice QCD results [7] and of some constraints (m ¯ t , α, γ, |Vub |) updated to the latest available measurements. A preliminary update of the NP analysis with the same novelties can be found in ref. [8]. Here we just comment the determination of ρ¯ and η¯ in the presence of NP and the perspectives for the UT analysis in the next years. The 2008 update of the UTfit analyses will be finalized after the Summer conferences. Final updated results will be available on the UTfit web site [1].
6
UTfit Collaboration / Nuclear Physics B (Proc. Suppl.) 185 (2008) 5–10
Table 1 Values of the input parameters used in the SM UT fit. A prior distributions for each parameter has been obtained by considering the first error as Gaussian and the second one, whenever present, as flat (the two distributions are convoluted to obtain the prior distribution). Entries marked with (†) are only indicative of the 68% probability ranges, as the full experimental likelihood has actually been used to obtain the prior distributions for these parameters. Entries without errors are considered as constants in the fit. αs (MZ ) 0.119 ± 0.003 1.16639 · 10−5 GeV−2 GF MW 80.425 GeV 91.1876 GeV MZ ¯ t) (162.8 ± 1.3) GeV m ¯ t (m ¯ b) (4.21 ± 0.08) GeV m ¯ b (m ¯ c) (1.3 ± 0.1) GeV m ¯ c (m (105 ± 15) MeV m ¯ s (2 GeV) MBd 5.279 GeV 5.375 GeV MBs (1.527 ± 0.008) ps τBd (1.643 ± 0.010) ps τB + (1.39 ± 0.12) ps τBs |Vcb | (excl.) (3.92 ± 0.11) · 10−2 |Vcb | (incl.) (4.168 ± 0.039 ± 0.058) · 10−2 |Vub | (excl.) (3.5 ± 0.4) · 10−3 |Vub | (incl.) (4.00 ± 0.15 ± 0.40) · 10−3 εK (2.232 ± 0.007) · 10−3 MK 497.648 MeV 160 MeV fK ˆK B 0.75 ± 0.07 Δmd (0.507 ± 0.005) ps−1 Δm (17.77 ± 0.12) ps−1 s ˆB fB B (270 ± 30) MeV s
s
ξ λ α(◦ ) sin 2β cos 2β γ(◦ ) (2β + γ)(◦ ) B(B → τ ντ ) fBd
1.21 ± 0.04 0.2258 ± 0.0014 92 ± 8 (†) 0.668 ± 0.028 (†) 0.88 ± 0.12 (†) (80 ± 13) ∪ (−100 ± 13) (†) (94 ± 53) ∪ (−90 ± 57) (†) (1.12 ± 0.45) · 10−4 (†) (200 ± 20) MeV
Figure 1. Result of the SM fit. The contours display the 68% and 95% probability regions selected by the fit in the ρ¯–¯ η plane. The 95% probability regions selected by the single constraints are also shown.
2. The Unitarity Triangle analysis in the Standard Model The basic idea of the UT analysis is to combine all the available theoretical and experimental information relevant to determine ρ¯ and η¯. To this end, we use the Bayesian approach described in ref. [9] 2 . The theoretical and experimental input values and errors are collected in Table 1 together with the most significant constants used in the analysis. The main results of the SM fit are given in Table 2. The ρ¯–¯ η plane is shown in Figure 1 where the 68% and 95% probability contours selected by the fit are plotted together with the 95% regions selected by the single constraints. The overall picture looks quite consistent. The CKM parameters ρ¯ and η¯ are determined in the SM with a relative errors of 14% and 4% respectively, an amazing 2 for
a different approach, see ref. [10].
UTfit Collaboration / Nuclear Physics B (Proc. Suppl.) 185 (2008) 5–10
result for the CP-violating parameter η¯ in particular. Within the precision of ∼ 5–10%, the CKM mechanism embedded in the SM seems able to describe the bulk of the CP-violating phenomena. In addition, flavour-changing CP-conserving and CP-violating processes select compatible regions in the ρ¯–¯ η plane, as predicted by the threegeneration unitarity. This is illustrated on the left side of fig. 2, while on the right side we show the constraining power of the CP-violating observables (namely the UT angles) in the Bd sector only. The Bayesian procedure produces posterior distributions for the observables included in the fit (and for derived ones, such as the SM phase of the Bs mixing amplitude βs ), from which we obtain the results displayed in Table 2. In order to check the compatibility of the various experimental measurements with the results of the fit, we compute the allowed range for a given observable from its posterior distribution obtained from a fit where that observable has not been used as a constraint. These “fit predictions” for a subset of observables are collected in Table 3. The two most significant tensions between measurements and fit predictions concern sin 2β and the inclusive determination of |Vub |. As can be seen in fig. 3, they are at the level of ∼ 1.5σ, showing the excellent overall compatibility of the measurements with the SM fit (with the remarkable exception of the Bs mixing phase, see refs. [5,8]). The tendency of |Vub |(inclusive) to be above the fit result (and the exclusive determination) is less pronounced than in the past, as updated theoretical analyses found smaller values for |Vub |(inclusive) [11,12]. The measured value of sin 2β is 1.5σ smaller than the fitted one. Comparing with the recent results of ref. [13], we find that the SM fit using constraints from |Vub |, εK and Δms /Δmd only is again 1.5σ larger than the measurement, using the input values of Table 1. This significance is not far from the value of 1.7σ quoted in ref. [13], although we compare with sin 2β as measured from B → J/ψK 0 modes only while ref. [13] uses the larger value sin 2β = 0.681 ± 0.025 obtained from the average of all charmonium modes. Fur-
7
Figure 2. On the left: constraints in the ρ¯–¯ η plane from the measurement of CP-conserving observables only. For comparison, the 95% regions selected by the CP-violating constraints are also shown. On the right: constraints in the ρ¯–¯ η plane from the measurement of the angles of the UT only. The 68% and 95% probability contours selected by the fit are plotted together with the 95% regions selected by the single constraints.
8
UTfit Collaboration / Nuclear Physics B (Proc. Suppl.) 185 (2008) 5–10
Table 2 Results of the SM fit obtained using the experimental constraints discussed in the text. We quote the 68% [95%] probability ranges obtained from the posterior distributions. λ 0.2259 ± 0.0015 [0.2228, 0.2288] A 0.809 ± 0.013 [0.783, 0.835] ρ¯ 0.155 ± 0.022 [0.112, 0.197] η¯ 0.342 ± 0.014 [0.316, 0.370] Rb 0.377 ± 0.013 [0.352, 0.403] 0.911 ± 0.022 [0.866, 0.953] Rt 92.1 ± 3.4 [85.7, 99.0] α(◦ ) 22.0 ± 0.8 [20.5, 23.7] β(◦ ) 65.6 ± 3.3 [58.9, 72.1] γ(◦ ) |Vcb | · 102 4.125 ± 0.045 [4.04, 4.21] 3.60 ± 0.12 [3.37, 3.85] |Vub | · 103 8.50 ± 0.21 [8.07, 8.92] |Vtd | · 103 0.209 ± 0.005 [0.199, 0.219] |Vtd /Vts | Reλt · 103 −0.32 ± 0.01 [−0.34, −0.30] 13.5 ± 0.5 [12.4, 14.6] Imλt · 105 JCP · 105 2.98 ± 0.12 [2.75, 3.22] Δms (ps−1 ) 17.75 ± 0.15 [17.4, 18.0] 0.0365 ± 0.0015 [0.0337, 0.0394] sin 2βs
thermore, when |Vub | is removed from the fit, we find sin 2β = 0.75 ± 0.09, quite smaller than the value 0.87 ± 0.09 of ref. [13], so that the significance of the difference is actually reduced to less than 1σ. As pointed out in ref. [14], there could be a tension also for εK once the full formula is used, including the term proportional to ImA0 /ReA0 which is usually neglected. This term produces a shift upward of about 4–8% of the εK constraint in the ρ¯–¯ η plane [14]. Together with the new small ˆK , this effect tends to make the sin 2β values of B tension more significant and could also have an impact on the agreement of |Vub | with the SM fit. At present, we are still using the approximate formula neglecting ImA0 /ReA0 , but we are considering the possibility to implement the full expression of εK in the near future, although the estimate of the additional term could be problematic, in particular beyond the SM.
Figure 3. Compatibility plots for |Vub | (left) and sin 2β (right). The average value of the measurement is plotted on the horizontal axis, its error is on the vertical one. The coloured bands delimit regions of values and errors which are less than a given number of σ from the fit result. The colour code is displayed on the right of each plot. For |Vub |, the exclusive (denoted by “+”) and inclusive (denoted by “∗”) measurements are shown separately.
UTfit Collaboration / Nuclear Physics B (Proc. Suppl.) 185 (2008) 5–10
9
Table 3 Fit predictions obtained without including the corresponding experimental constraints. We quote the 68% [95%] probability ranges obtained from the posterior distributions. α(◦ ) 92.5 ± 4.2 [84.3, 100.5] sin 2β 0.735 ± 0.034 [0.672, 0.800] 64.4 ± 3.4 [57.6, 71.3] γ(◦ ) |Vub | · 103 3.48 ± 0.16 [3.17, 3.80] Δms (ps−1 ) 17.0 ± 1.6 [14.0, 20.3] 0.0365 ± 0.0015 [0.0337, 0.0394] sin 2βs
3. Prospects for the Unitarity Triangle fit Once it is established that the CKM mechanism is the main source of CP-violation up to the Fermi scale, an accurate model-independent determination of ρ¯ and η¯ is crucial for identifying NP in the flavour sector. The generalized UT fit, using only ΔF = 2 processes and parametrizing generic NP contributions, allows for the model-independent determination of ρ¯ and η¯ under the assumptions of negligible tree-level NP contributions and threegenerations unitarity. Details of the method can be found in ref. [4]. Latest results are reported in ref. [8]. Here we just point out that the present relative error in the model-independent determination of ρ¯ (¯ η ) is about 26% (8%), roughly twice as large as the SM case. This implies that NP contributions to CP-violating observables below ∼ 10–20% can hardly be disentangled at present. We recently studied the impact of the future super flavour factory SuperB on the UT fits [15]. Two possible scenarios for the SM fit are shown in fig. 4, depending whether NP will make the constraints measured with the SuperB accuracy incompatible or not. In the latter case, the SM fit could determine ρ¯ and η¯ with a relative error of 2% and < 1% respectively. In any case, we found that a relative error of about 3% (1%) on the model-independent determination of ρ¯ (¯ η ) is attainable. This result would allow for searching NP contributions in the B sector at the percent level. In addition, it would greatly improve the NP sensitivity of
Figure 4. SM UT fit with future measurements as expected at SuperB. We assumed the SuperB errors and either central values tuned to be compatible as in the SM (on the left) or the present central values (on the right)
10
UTfit Collaboration / Nuclear Physics B (Proc. Suppl.) 185 (2008) 5–10
rare kaon decays, in particular of the golden mode KL → π 0 ν ν¯ [16]. Therefore, in spite of the impressive precision already achieved in the SM determination of η¯ (and to a lesser extent of ρ¯), there is still ample room for progresses. In particular, an order-ofmagnitude improvement of the UT fit results can be achieved at the next-generation flavour factories.
Acknowledgments We acknowledge partial support from RTN European contracts MRTN-CT-2004-503369 “The Quest for Unication”, MRTN-CT-2006035482 “FLAVIAnet” and MRTN-CT-2006035505 “Heptools”. M.C. is associated to the Dipartimento di Fisica, Universit` a di Roma Tre. E.F. and L.S. are associated to the Dipartimento di Fisica, Universit` a di Roma “La Sapienza”.
REFERENCES 1. UTfit Collaboration, http://www.utfit.org. 2. N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531; M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652. 3. M. Bona et al. [UTfit Collaboration], JHEP 0507 (2005) 028; JHEP 0610 (2006) 081. 4. M. Bona et al. [UTfit Collaboration], JHEP 0603 (2006) 080; Phys. Rev. Lett. 97 (2006) 151803. 5. M. Bona et al. [UTfit Collaboration], arXiv:0803.0659 [hep-ph]. 6. M. Bona et al. [UTfit Collaboration], JHEP 0803 (2008) 049. 7. V. Lubicz and C. Tarantino, arXiv:0807.4605 [hep-lat]. 8. L. Silvestrini, these proceedings. 9. M. Ciuchini et al., JHEP 0107 (2001) 013. 10. CKMfitter group, http://ckmfitter.in2p3.fr. 11. P. Gambino et al., JHEP 0710 (2007) 058. 12. M. Neubert, arXiv:0801.0675 [hep-ph]. 13. A. Soni and E. Lunghi, arXiv:0803.4340 [hepph].
14. A.J. Buras and D. Guadagnoli, arXiv:0805.3887 [hep-ph]. 15. M. Bona et al., arXiv:0709.0451 [hep-ph]. 16. see e.g. U. Haisch, PoS KAON (2008) 056.
The Unitarity Triangle Analysis in the Standard Model UTfit Collaboration: M. Bonaa , M. Ciuchinib∗ , E. Francoc , V. Lubiczdb , G. Martinelliec , F. Parodif , M. Pierinia , C. Schiavif , L. Silvestrinic , V. Sordinig , A. Stocchig , C. Tarantinodb , V. Vagnonih a
CERN, CH-1211 Geneva 23, Switzerland
b
INFN Sezione di Roma Tre, Via della Vasca Navale 84, I-00146 Roma, Italy
c
INFN Sezione di Roma, Piazzale A. Moro 2, I-00185 Roma, Italy
d e
Dipartimento di Fisica, Universit` a di Roma Tre, Via della Vasca Navale 84, I-00146 Roma, Italy
Dipartimento di Fisica, Universit` a di Roma “La Sapienza”, Piazzale A. Moro 2, I-00185 Roma, Italy
f
Dipartimento di Fisica, Universit` a di Genova and INFN Sezione di Genova, Via Dodecaneso 33, I-16146 Genova, Italy
g
Laboratoire de l’Acc´el´erateur Lin´eaire, IN2P3-CNRS and Universit´e de Paris-Sud, BP 34, F-91898 Orsay Cedex, France
h
INFN Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, Italy
1. Introduction The original mission of the UTfit Collaboration [1] was the determination of the coordinates ρ¯ and η¯ of the apex of the Unitarity Triangle (UT) and in general of the parameters describing the CKM matrix [2] in the Standard Model (SM). Nowadays the SM analysis encompasses many additional results, such as predictions for several flavour observables and measurements of hadronic parameters which can be compared with the lattice QCD predictions [3]. Furthermore, the UT analysis has been extended beyond the SM, allowing for a model-independent determination of ρ¯ and η¯ (assuming three-generation unitarity and negligible new physics (NP) contributions to tree-level processes) and a simultaneous evaluation of the size of NP contributions to ΔF = 2 amplitudes compatible with the flavour data [4,5]. Recently, the NP analysis has been expanded to include an effective field theory study of the allowed NP contributions to ΔF = 2 amplitudes. This allows to put model-independent bounds on ∗ talk
given by M.C.
0920-5632/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2008.10.001
the NP energy scale associated to flavour- and CP-violating phenomena [6]. In these proceedings we present a preliminary update of our UT analysis in the SM, including a set of fit predictions and a study of the compatibility between the fit results and some of the most interesting experimental constraints. The main difference with respect to previously published results comes from the use of updated set of lattice QCD results [7] and of some constraints (m ¯ t , α, γ, |Vub |) updated to the latest available measurements. A preliminary update of the NP analysis with the same novelties can be found in ref. [8]. Here we just comment the determination of ρ¯ and η¯ in the presence of NP and the perspectives for the UT analysis in the next years. The 2008 update of the UTfit analyses will be finalized after the Summer conferences. Final updated results will be available on the UTfit web site [1].
6
UTfit Collaboration / Nuclear Physics B (Proc. Suppl.) 185 (2008) 5–10
Table 1 Values of the input parameters used in the SM UT fit. A prior distributions for each parameter has been obtained by considering the first error as Gaussian and the second one, whenever present, as flat (the two distributions are convoluted to obtain the prior distribution). Entries marked with (†) are only indicative of the 68% probability ranges, as the full experimental likelihood has actually been used to obtain the prior distributions for these parameters. Entries without errors are considered as constants in the fit. αs (MZ ) 0.119 ± 0.003 1.16639 · 10−5 GeV−2 GF MW 80.425 GeV 91.1876 GeV MZ ¯ t) (162.8 ± 1.3) GeV m ¯ t (m ¯ b) (4.21 ± 0.08) GeV m ¯ b (m ¯ c) (1.3 ± 0.1) GeV m ¯ c (m (105 ± 15) MeV m ¯ s (2 GeV) MBd 5.279 GeV 5.375 GeV MBs (1.527 ± 0.008) ps τBd (1.643 ± 0.010) ps τB + (1.39 ± 0.12) ps τBs |Vcb | (excl.) (3.92 ± 0.11) · 10−2 |Vcb | (incl.) (4.168 ± 0.039 ± 0.058) · 10−2 |Vub | (excl.) (3.5 ± 0.4) · 10−3 |Vub | (incl.) (4.00 ± 0.15 ± 0.40) · 10−3 εK (2.232 ± 0.007) · 10−3 MK 497.648 MeV 160 MeV fK ˆK B 0.75 ± 0.07 Δmd (0.507 ± 0.005) ps−1 Δm (17.77 ± 0.12) ps−1 s ˆB fB B (270 ± 30) MeV s
s
ξ λ α(◦ ) sin 2β cos 2β γ(◦ ) (2β + γ)(◦ ) B(B → τ ντ ) fBd
1.21 ± 0.04 0.2258 ± 0.0014 92 ± 8 (†) 0.668 ± 0.028 (†) 0.88 ± 0.12 (†) (80 ± 13) ∪ (−100 ± 13) (†) (94 ± 53) ∪ (−90 ± 57) (†) (1.12 ± 0.45) · 10−4 (†) (200 ± 20) MeV
Figure 1. Result of the SM fit. The contours display the 68% and 95% probability regions selected by the fit in the ρ¯–¯ η plane. The 95% probability regions selected by the single constraints are also shown.
2. The Unitarity Triangle analysis in the Standard Model The basic idea of the UT analysis is to combine all the available theoretical and experimental information relevant to determine ρ¯ and η¯. To this end, we use the Bayesian approach described in ref. [9] 2 . The theoretical and experimental input values and errors are collected in Table 1 together with the most significant constants used in the analysis. The main results of the SM fit are given in Table 2. The ρ¯–¯ η plane is shown in Figure 1 where the 68% and 95% probability contours selected by the fit are plotted together with the 95% regions selected by the single constraints. The overall picture looks quite consistent. The CKM parameters ρ¯ and η¯ are determined in the SM with a relative errors of 14% and 4% respectively, an amazing 2 for
a different approach, see ref. [10].
UTfit Collaboration / Nuclear Physics B (Proc. Suppl.) 185 (2008) 5–10
result for the CP-violating parameter η¯ in particular. Within the precision of ∼ 5–10%, the CKM mechanism embedded in the SM seems able to describe the bulk of the CP-violating phenomena. In addition, flavour-changing CP-conserving and CP-violating processes select compatible regions in the ρ¯–¯ η plane, as predicted by the threegeneration unitarity. This is illustrated on the left side of fig. 2, while on the right side we show the constraining power of the CP-violating observables (namely the UT angles) in the Bd sector only. The Bayesian procedure produces posterior distributions for the observables included in the fit (and for derived ones, such as the SM phase of the Bs mixing amplitude βs ), from which we obtain the results displayed in Table 2. In order to check the compatibility of the various experimental measurements with the results of the fit, we compute the allowed range for a given observable from its posterior distribution obtained from a fit where that observable has not been used as a constraint. These “fit predictions” for a subset of observables are collected in Table 3. The two most significant tensions between measurements and fit predictions concern sin 2β and the inclusive determination of |Vub |. As can be seen in fig. 3, they are at the level of ∼ 1.5σ, showing the excellent overall compatibility of the measurements with the SM fit (with the remarkable exception of the Bs mixing phase, see refs. [5,8]). The tendency of |Vub |(inclusive) to be above the fit result (and the exclusive determination) is less pronounced than in the past, as updated theoretical analyses found smaller values for |Vub |(inclusive) [11,12]. The measured value of sin 2β is 1.5σ smaller than the fitted one. Comparing with the recent results of ref. [13], we find that the SM fit using constraints from |Vub |, εK and Δms /Δmd only is again 1.5σ larger than the measurement, using the input values of Table 1. This significance is not far from the value of 1.7σ quoted in ref. [13], although we compare with sin 2β as measured from B → J/ψK 0 modes only while ref. [13] uses the larger value sin 2β = 0.681 ± 0.025 obtained from the average of all charmonium modes. Fur-
7
Figure 2. On the left: constraints in the ρ¯–¯ η plane from the measurement of CP-conserving observables only. For comparison, the 95% regions selected by the CP-violating constraints are also shown. On the right: constraints in the ρ¯–¯ η plane from the measurement of the angles of the UT only. The 68% and 95% probability contours selected by the fit are plotted together with the 95% regions selected by the single constraints.
8
UTfit Collaboration / Nuclear Physics B (Proc. Suppl.) 185 (2008) 5–10
Table 2 Results of the SM fit obtained using the experimental constraints discussed in the text. We quote the 68% [95%] probability ranges obtained from the posterior distributions. λ 0.2259 ± 0.0015 [0.2228, 0.2288] A 0.809 ± 0.013 [0.783, 0.835] ρ¯ 0.155 ± 0.022 [0.112, 0.197] η¯ 0.342 ± 0.014 [0.316, 0.370] Rb 0.377 ± 0.013 [0.352, 0.403] 0.911 ± 0.022 [0.866, 0.953] Rt 92.1 ± 3.4 [85.7, 99.0] α(◦ ) 22.0 ± 0.8 [20.5, 23.7] β(◦ ) 65.6 ± 3.3 [58.9, 72.1] γ(◦ ) |Vcb | · 102 4.125 ± 0.045 [4.04, 4.21] 3.60 ± 0.12 [3.37, 3.85] |Vub | · 103 8.50 ± 0.21 [8.07, 8.92] |Vtd | · 103 0.209 ± 0.005 [0.199, 0.219] |Vtd /Vts | Reλt · 103 −0.32 ± 0.01 [−0.34, −0.30] 13.5 ± 0.5 [12.4, 14.6] Imλt · 105 JCP · 105 2.98 ± 0.12 [2.75, 3.22] Δms (ps−1 ) 17.75 ± 0.15 [17.4, 18.0] 0.0365 ± 0.0015 [0.0337, 0.0394] sin 2βs
thermore, when |Vub | is removed from the fit, we find sin 2β = 0.75 ± 0.09, quite smaller than the value 0.87 ± 0.09 of ref. [13], so that the significance of the difference is actually reduced to less than 1σ. As pointed out in ref. [14], there could be a tension also for εK once the full formula is used, including the term proportional to ImA0 /ReA0 which is usually neglected. This term produces a shift upward of about 4–8% of the εK constraint in the ρ¯–¯ η plane [14]. Together with the new small ˆK , this effect tends to make the sin 2β values of B tension more significant and could also have an impact on the agreement of |Vub | with the SM fit. At present, we are still using the approximate formula neglecting ImA0 /ReA0 , but we are considering the possibility to implement the full expression of εK in the near future, although the estimate of the additional term could be problematic, in particular beyond the SM.
Figure 3. Compatibility plots for |Vub | (left) and sin 2β (right). The average value of the measurement is plotted on the horizontal axis, its error is on the vertical one. The coloured bands delimit regions of values and errors which are less than a given number of σ from the fit result. The colour code is displayed on the right of each plot. For |Vub |, the exclusive (denoted by “+”) and inclusive (denoted by “∗”) measurements are shown separately.
UTfit Collaboration / Nuclear Physics B (Proc. Suppl.) 185 (2008) 5–10
9
Table 3 Fit predictions obtained without including the corresponding experimental constraints. We quote the 68% [95%] probability ranges obtained from the posterior distributions. α(◦ ) 92.5 ± 4.2 [84.3, 100.5] sin 2β 0.735 ± 0.034 [0.672, 0.800] 64.4 ± 3.4 [57.6, 71.3] γ(◦ ) |Vub | · 103 3.48 ± 0.16 [3.17, 3.80] Δms (ps−1 ) 17.0 ± 1.6 [14.0, 20.3] 0.0365 ± 0.0015 [0.0337, 0.0394] sin 2βs
3. Prospects for the Unitarity Triangle fit Once it is established that the CKM mechanism is the main source of CP-violation up to the Fermi scale, an accurate model-independent determination of ρ¯ and η¯ is crucial for identifying NP in the flavour sector. The generalized UT fit, using only ΔF = 2 processes and parametrizing generic NP contributions, allows for the model-independent determination of ρ¯ and η¯ under the assumptions of negligible tree-level NP contributions and threegenerations unitarity. Details of the method can be found in ref. [4]. Latest results are reported in ref. [8]. Here we just point out that the present relative error in the model-independent determination of ρ¯ (¯ η ) is about 26% (8%), roughly twice as large as the SM case. This implies that NP contributions to CP-violating observables below ∼ 10–20% can hardly be disentangled at present. We recently studied the impact of the future super flavour factory SuperB on the UT fits [15]. Two possible scenarios for the SM fit are shown in fig. 4, depending whether NP will make the constraints measured with the SuperB accuracy incompatible or not. In the latter case, the SM fit could determine ρ¯ and η¯ with a relative error of 2% and < 1% respectively. In any case, we found that a relative error of about 3% (1%) on the model-independent determination of ρ¯ (¯ η ) is attainable. This result would allow for searching NP contributions in the B sector at the percent level. In addition, it would greatly improve the NP sensitivity of
Figure 4. SM UT fit with future measurements as expected at SuperB. We assumed the SuperB errors and either central values tuned to be compatible as in the SM (on the left) or the present central values (on the right)
10
UTfit Collaboration / Nuclear Physics B (Proc. Suppl.) 185 (2008) 5–10
rare kaon decays, in particular of the golden mode KL → π 0 ν ν¯ [16]. Therefore, in spite of the impressive precision already achieved in the SM determination of η¯ (and to a lesser extent of ρ¯), there is still ample room for progresses. In particular, an order-ofmagnitude improvement of the UT fit results can be achieved at the next-generation flavour factories.
Acknowledgments We acknowledge partial support from RTN European contracts MRTN-CT-2004-503369 “The Quest for Unication”, MRTN-CT-2006035482 “FLAVIAnet” and MRTN-CT-2006035505 “Heptools”. M.C. is associated to the Dipartimento di Fisica, Universit` a di Roma Tre. E.F. and L.S. are associated to the Dipartimento di Fisica, Universit` a di Roma “La Sapienza”.
REFERENCES 1. UTfit Collaboration, http://www.utfit.org. 2. N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531; M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652. 3. M. Bona et al. [UTfit Collaboration], JHEP 0507 (2005) 028; JHEP 0610 (2006) 081. 4. M. Bona et al. [UTfit Collaboration], JHEP 0603 (2006) 080; Phys. Rev. Lett. 97 (2006) 151803. 5. M. Bona et al. [UTfit Collaboration], arXiv:0803.0659 [hep-ph]. 6. M. Bona et al. [UTfit Collaboration], JHEP 0803 (2008) 049. 7. V. Lubicz and C. Tarantino, arXiv:0807.4605 [hep-lat]. 8. L. Silvestrini, these proceedings. 9. M. Ciuchini et al., JHEP 0107 (2001) 013. 10. CKMfitter group, http://ckmfitter.in2p3.fr. 11. P. Gambino et al., JHEP 0710 (2007) 058. 12. M. Neubert, arXiv:0801.0675 [hep-ph]. 13. A. Soni and E. Lunghi, arXiv:0803.4340 [hepph].
14. A.J. Buras and D. Guadagnoli, arXiv:0805.3887 [hep-ph]. 15. M. Bona et al., arXiv:0709.0451 [hep-ph]. 16. see e.g. U. Haisch, PoS KAON (2008) 056.