Open Access

Global analysis of charm mixing parameters and determination of the CKM angle gamma $γ$

F. Betti ¹, M. Bona^2,*, M. Ciuchini ^3,*, D. Derkach ^4,*, R. Di Palma ^3,5, A. L. Kagan ⁶, V. Lubicz ^3,5,*, G. Martinelli ^7,*, M. Pierini^8,* et al.

L. Silvestrini ^7,*, C. Tarantino ^3,5,*, and V. Vagnoni ^9,*

¹School of Physics and Astronomy, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom
²Department of Physics and Astronomy, Queen Mary University of London, London, United Kingdom
³INFN, Sezione di Roma Tre, Via della Vasca Navale 84, I-00146 Roma, Italy
⁴HSE University, Myasnitskaya ulitsa, 20, 101000 Moscow, Russia
⁵Dipartimento di Matematica e Fisica, Università di Roma Tre, Via della Vasca Navale 84, I-00146 Roma, Italy
⁶Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221,USA
⁷INFN, Sezione di Roma, Piazzale A. Moro 2, I-00185 Roma, Italy
⁸CERN, CH-1211 Geneva 23, Switzerland
⁹INFN, Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, Italy

^*Member of the UTfit Collaboration.

PDF

Phys. Rev. D 112, 013004 – Published 8 July, 2025

DOI: https://doi.org/10.1103/pknz-r41n

Abstract

We present an updated global analysis of beauty decays sensitive to the angle $γ$ of the Cabibbo-Kobayashi-Maskawa matrix and of $D$ -meson mixing data in the framework of approximate universality, in which $C P$ violation in $D - \bar{D}$ mixing is described in terms of two universal weak phases corresponding to dispersive and absorptive contributions. We extract the fundamental theoretical parameters determining absorptive and dispersive contributions to $D$ meson mixing and $C P$ violation, together with the angle $γ$ . The results for the charm mixing parameters are $x_{12} ≃ x = (0.401 \pm 0.043) %$ and $y_{12} ≃ y = (0.610 \pm 0.017) %$ , while the two $C P$ -violating phases are given by $ϕ_{2}^{M} = (0.13 \pm 0.70) °$ and $ϕ_{2}^{Γ} = (2.1 \pm 1.6) °$ . The angle $γ$ is found to be $γ = (65.7 \pm 2.5) °$ , in excellent agreement with the indirect determination from the unitarity triangle analysis.

Physics Subject Headings (PhySH)

Article Text

References (147)

S. L. Glashow, J. Iliopoulos, and L. Maiani, Weak interactions with lepton-hadron symmetry, Phys. Rev. D 2, 1285 (1970).
N. Cabibbo, Unitary symmetry and leptonic decays, Phys. Rev. Lett. 10, 531 (1963).
M. Kobayashi and T. Maskawa, $C P$ violation in the renormalizable theory of weak interaction, Prog. Theor. Phys. 49, 652 (1973).
Y. Grossman, A. L. Kagan, and Y. Nir, New physics and $C P$ violation in singly Cabibbo suppressed $D$ decays, Phys. Rev. D 75, 036008 (2007).
M. Ciuchini, E. Franco, D. Guadagnoli, V. Lubicz, M. Pierini, V. Porretti, and L. Silvestrini, $D^{0} - {\bar{D}}^{0}$ mixing and new physics: General considerations and constraints on the MSSM, Phys. Lett. B 655, 162 (2007).
E. Golowich, J. Hewett, S. Pakvasa, and A. A. Petrov, Implications of $D^{0} - {\bar{D}}^{0}$ mixing for new physics, Phys. Rev. D 76, 095009 (2007).
S. Fajfer, N. Kosnik, and S. Prelovsek, Updated constraints on new physics in rare charm decays, Phys. Rev. D 76, 074010 (2007).
M. Bona et al. (UTfit Collaboration), Model-independent constraints on $Δ F = 2$ operators and the scale of new physics, J. High Energy Phys. 03 (2008) 049.
M. Artuso, B. Meadows, and A. A. Petrov, Charm meson decays, Annu. Rev. Nucl. Part. Sci. 58, 249 (2008).
J. Brod, A. L. Kagan, and J. Zupan, Size of direct $C P$ violation in singly Cabibbo-suppressed $D$ decays, Phys. Rev. D 86, 014023 (2012).
D. Pirtskhalava and P. Uttayarat, $C P$ violation and flavor $S U (3)$ breaking in $D$ -meson decays, Phys. Lett. B 712, 81 (2012).
B. Bhattacharya, M. Gronau, and J. L. Rosner, $C P$ asymmetries in singly-Cabibbo-suppressed $D$ decays to two pseudoscalar mesons, Phys. Rev. D 85, 054014 (2012).
H.-Y. Cheng and C.-W. Chiang, Direct $C P$ violation in two-body hadronic charmed meson decays, Phys. Rev. D 85, 034036 (2012); 85, 079903(E) (2012).
E. Franco, S. Mishima, and L. Silvestrini, The standard model confronts $C P$ violation in $D^{0} \to π^{+} π^{-}$ and $D^{0} \to K^{+} K^{-}$ , J. High Energy Phys. 05 (2012) 140.
J. Brod, Y. Grossman, A. L. Kagan, and J. Zupan, $D^{0} \to π^{+} π^{-}$ , $D^{0} \to K^{+} K^{-}$ , J. High Energy Phys. 10 (2012) 161.
G. F. Giudice, G. Isidori, and P. Paradisi, Direct $C P$ violation in charm and flavor mixing beyond the SM, J. High Energy Phys. 04 (2012) 060.
T. Feldmann, S. Nandi, and A. Soni, Repercussions of flavour symmetry breaking on $C P$ violation in D-meson decays, J. High Energy Phys. 06 (2012) 007.
B. Keren-Zur, P. Lodone, M. Nardecchia, D. Pappadopulo, R. Rattazzi, and L. Vecchi, On partial compositeness and the $C P$ asymmetry in charm decays, Nucl. Phys. B867, 394 (2013).
Y. Grossman and S. Schacht, The emergence of the $Δ U = 0$ rule in charm physics, J. High Energy Phys. 07 (2019) 020.
A. Dery and Y. Nir, Implications of the LHCb discovery of $C P$ violation in charm decays, J. High Energy Phys. 12 (2019) 104.
A. J. Buras, P. Colangelo, F. De Fazio, and F. Loparco, The charm of 331, J. High Energy Phys. 10 (2021) 021.
S. Schacht and A. Soni, Enhancement of charm $C P$ violation due to nearby resonances, Phys. Lett. B 825, 136855 (2022).
S. Schacht, A U-spin anomaly in charm $C P$ violation, J. High Energy Phys. 03 (2023) 205.
E. M. Aitala et al. (E791 Collaboration), Search for $D^{0} - {\bar{D}}^{0}$ mixing in semileptonic decay modes, Phys. Rev. Lett. 77, 2384 (1996).
C. Cawlfield et al. (CLEO Collaboration), Limits on neutral $D$ mixing in semileptonic decays, Phys. Rev. D 71, 077101 (2005).
B. Aubert et al. (BABAR Collaboration, Search for $D^{0} - {\bar{D}}^{0}$ mixing using semileptonic decay modes, Phys. Rev. D 70, 091102 (2004).
B. Aubert et al. (BABAR Collaboration, Search for $D^{0} - \bar{D^{0}}$ mixing using doubly flavor tagged semileptonic decay modes, Phys. Rev. D 76, 014018 (2007).
U. Bitenc et al. (Belle Collaboration), Improved search for $D^{0} - \bar{D^{0}}$ mixing using semileptonic decays at Belle, Phys. Rev. D 77, 112003 (2008).
B. Aubert et al. (BABAR Collaboration, Measurement of $D^{0} - {\bar{D}}^{0}$ mixing from a time-dependent amplitude analysis of $D^{0} \to K^{+} π^{-} π^{0}$ decays, Phys. Rev. Lett. 103, 211801 (2009).
P. del Amo Sanchez et al. (BABAR Collaboration, Measurement of $D^{0} - \bar{D^{0}}$ mixing parameters using $D^{0} \to K_{S} π^{+} π^{-}$ and $D^{0} \to K_{S} K^{+} K^{-}$ decays, Phys. Rev. Lett. 105, 081803 (2010).
R. Aaij et al. (LHCb Collaboration), Model-independent measurement of mixing parameters in $D^{0} \to K_{S}^{0} π^{+} π^{-}$ decays, J. High Energy Phys. 04 (2016) 033.
J. P. Lees et al. (BABAR Collaboration, Measurement of the neutral $D$ meson mixing parameters in a time-dependent amplitude analysis of the $D^{0} \to π^{+} π^{-} π^{0}$ decay, Phys. Rev. D 93, 112014 (2016).
D. M. Asner et al. (CLEO Collaboration), Updated measurement of the strong phase in $D^{0} \to K^{+} π^{-}$ decay using quantum correlations in $e^{+} e^{-} \to D^{0} {\bar{D}}^{0}$ at CLEO, Phys. Rev. D 86, 112001 (2012).
J. Insler et al. (CLEO Collaboration), Studies of the decays $D^{0} \to K_{S}^{0} K^{-} π^{+}$ and $D^{0} \to K_{S}^{0} K^{+} π^{-}$ , Phys. Rev. D 85, 092016 (2012); 94, 099905(E) (2016).
T. A. Aaltonen et al. (CDF Collaboration), Observation of $D^{0} - {\bar{D}}^{0}$ mixing using the CDF II detector, Phys. Rev. Lett. 111, 231802 (2013).
T. Peng et al. (Belle Collaboration), Measurement of $D^{0} - {\bar{D}}^{0}$ mixing and search for indirect $C P$ violation using $D^{0} \to K_{S}^{0} π^{+} π^{-}$ decays, Phys. Rev. D 89, 091103 (2014).
R. Aaij et al. (LHCb Collaboration), Measurement of the mass difference between neutral charm-meson eigenstates, Phys. Rev. Lett. 122, 231802 (2019).
R. Aaij et al. (LHCb Collaboration), Model-independent measurement of charm mixing parameters in $B^{-} \to D^{0} (\to K_{S} π^{+} π^{-}) μ^{-} {\bar{ν}}_{μ} X$ decays, Phys. Rev. D 108, 052005 (2023).
S. Malde, C. Thomas, G. Wilkinson, P. Naik, C. Prouve, J. Rademacker, J. Libby, M. Nayak, T. Gershon, and R. A. Briere, First determination of the $C P$ content of $D \to π^{+} π^{-} π^{+} π^{-}$ and updated determination of the $C P$ contents of $D \to π^{+} π^{-} π^{0}$ and $D \to K^{+} K^{-} π^{0}$ , Phys. Lett. B 747, 9 (2015).
M. Ablikim et al. (BESIII Collaboration), Measurements of the $C P$ -even fractions of $D^{0} \to π^{+} π^{-} π^{0}$ and $D^{0} \to K^{+} K^{-} π^{0}$ at BESIII, Phys. Rev. D 111, 012007 (2025).
M. Ablikim et al. (BESIII Collaboration), Model-independent determination of the strong-phase difference between $D^{0}$ and ${\bar{D}}^{0} \to π^{+} π^{-} π^{+} π^{-}$ decays, Phys. Rev. D 110, 112008 (2024).
R. Aaij et al. (LHCb Collaboration), Studies of the resonance structure in $D^{0} \to K_{S}^{0} K^{\pm} π^{\mp}$ decays, Phys. Rev. D 93, 052018 (2016).
M. Ablikim et al. (BESIII Collaboration), Measurement of the branching fractions of doubly Cabibbo-suppressed $D$ decays, arXiv:2503.19542.
M. Ablikim et al. (BESIII Collaboration), Measurement of the strong-phase difference between $D^{0}$ and $\bar{D^{0}} \to K^{+} K^{-} π^{+} π^{-}$ in bins of phase space, arXiv:2502.12873.
R. Aaij et al. (LHCb Collaboration), First observation of $D^{0} - {\bar{D}}^{0}$ oscillations in $D^{0} \to K^{+} π^{-} π^{+} π^{-}$ decays and measurement of the associated coherence parameters, Phys. Rev. Lett. 116, 241801 (2016).
F. Betti, A review of $C P$ violation measurements in charm at LHCb, Symmetry 13, 1482 (2021).
J. Libby et al., New determination of the $D^{0} \to K^{-} π^{+} π^{0}$ and $D^{0} \to K^{-} π^{+} π^{+} π^{-}$ coherence factors and average strong-phase differences, Phys. Lett. B 731, 197 (2014).
T. Evans, S. Harnew, J. Libby, S. Malde, J. Rademacker, and G. Wilkinson, Improved determination of the $D \to K^{-} π^{+} π^{+} π^{-}$ coherence factor and associated hadronic parameters from a combination of $e^{+} e^{-} \to ψ (3770) \to c \bar{c}$ and $p p \to c \bar{c} X$ data, Phys. Lett. B 757, 520 (2016); 765, 402(E) (2017).
M. Ablikim et al. (BESIII Collaboration), Measurement of the $D \to K^{-} π^{+} π^{+} π^{-}$ and $D \to K^{-} π^{+} π^{0}$ coherence factors and average strong-phase differences in quantum-correlated $D \bar{D}$ decays, J. High Energy Phys. 05 (2021) 164.
M. Ablikim et al. (BESIII Collaboration), Improved measurement of the strong-phase difference $δ_{D}^{K π}$ in quantum-correlated $D \bar{D}$ decays, Eur. Phys. J. C 82, 1009 (2022).
T. Pajero, Recent advances in charm mixing and $C P$ violation at LHCb, Mod. Phys. Lett. A 37, 2230012 (2022).
L. M. Zhang et al. (Belle Collaboration), Improved constraints on $D^{0} - \bar{D^{0}}$ mixing in $D^{0} \to K^{+} π^{-}$ decays at BELLE, Phys. Rev. Lett. 96, 151801 (2006).
B. Aubert et al. (BABAR Collaboration, Evidence for $D^{0} - \bar{D^{0}}$ mixing, Phys. Rev. Lett. 98, 211802 (2007).
R. Aaij et al. (LHCb Collaboration), Search for charge-parity violation in semileptonically tagged $D^{0} \to K^{+} π^{-}$ decays, J. High Energy Phys. 03 (2025) 149.
R. Aaij et al. (LHCb Collaboration), Measurement of $D^{0} - {\bar{D}}^{0}$ mixing and search for $C P$ violation with $D^{0} \to K^{+} π^{-}$ decays, Phys. Rev. D 111, 012001 (2025).
E. M. Aitala et al. (E791 Collaboration), Measurements of lifetimes and a limit on the lifetime difference in the neutral D meson system, Phys. Rev. Lett. 83, 32 (1999).
J. M. Link et al. (FOCUS Collaboration), A measurement of lifetime differences in the neutral D meson system, Phys. Lett. B 485, 62 (2000).
S. E. Csorna et al. (CLEO Collaboration), Lifetime differences, direct $C P$ violation and partial widths in $D^{0}$ meson decays to $K^{+} K^{-}$ and $π^{+} π^{-}$ , Phys. Rev. D 65, 092001 (2002).
A. Zupanc et al. (Belle Collaboration), Measurement of $y_{C P}$ in $D^{0}$ meson decays to the $K_{S}^{0} K^{+} K_{-}$ final state, Phys. Rev. D 80, 052006 (2009).
J. P. Lees et al. (BABAR Collaboration, Measurement of $D^{0} - {\bar{D}}^{0}$ mixing and $C P$ violation in two-body $D^{0}$ decays, Phys. Rev. D 87, 012004 (2013).
M. Ablikim et al. (BESIII Collaboration), Measurement of $y_{C P}$ in $D^{0} - {\bar{D}}^{0}$ oscillation using quantum correlations in $e^{+} e^{-} \to D^{0} {\bar{D}}^{0}$ at $\sqrt{s} = 3.773 GeV$ , Phys. Lett. B 744, 339 (2015).
M. Starič et al. (Belle Collaboration), Measurement of $D^{0} - {\bar{D}}^{0}$ mixing and search for $C P$ violation in $D^{0} \to K^{+} K^{-}, π^{+} π^{-}$ decays with the full Belle data set, Phys. Lett. B 753, 412 (2016).
R. Aaij et al. (LHCb Collaboration), Measurement of the charm-mixing parameter $y_{C P}$ , Phys. Rev. Lett. 122, 011802 (2019).
M. Nayak et al. (Belle Collaboration), Measurement of the charm-mixing parameter $y_{C P}$ in $D^{0} \to K_{S}^{0} ω$ decays at Belle, Phys. Rev. D 102, 071102 (2020).
R. Aaij et al. (LHCb Collaboration), Measurement of the charm mixing parameter $y_{C P} - y_{C P}^{K π}$ using two-body $D^{0}$ meson decays, Phys. Rev. D 105, 092013 (2022).
T. A. Aaltonen et al. (CDF Collaboration), Measurement of indirect $C P$ -violating asymmetries in $D^{0} \to K^{+} K^{-}$ and $D^{0} \to π^{+} π^{-}$ decays at CDF, Phys. Rev. D 90, 111103 (2014).
R. Aaij et al. (LHCb Collaboration), Measurement of indirect $C P$ asymmetries in $D^{0} \to K^{-} K^{+}$ and $D^{0} \to π^{-} π^{+}$ decays using semileptonic $B$ decays, J. High Energy Phys. 04 (2015) 043.
R. Aaij et al. (LHCb Collaboration), Measurement of the $C P$ violation parameter $A_{Γ}$ in $D^{0} \to K^{+} K^{-}$ and $D^{0} \to π^{+} π^{-}$ decays, Phys. Rev. Lett. 118, 261803 (2017).
R. Aaij et al. (LHCb Collaboration), Updated measurement of decay-time-dependent $C P$ asymmetries in $D^{0} \to K^{+} K^{-}$ and $D^{0} \to π^{+} π^{-}$ decays, Phys. Rev. D 101, 012005 (2020).
R. Aaij et al. (LHCb Collaboration), Search for time-dependent $C P$ violation in $D^{0} \to K^{+} K^{-}$ and $D^{0} \to π^{+} π^{-}$ decays, Phys. Rev. D 104, 072010 (2021).
R. Aaij et al. (LHCb Collaboration), Measurement of $C P$ asymmetry in $D^{0} \to K^{-} K^{+}$ decays, Phys. Lett. B 767, 177 (2017).
R. Aaij et al. (LHCb Collaboration), Measurement of the time-integrated $C P$ asymmetry in $D^{0} \to K^{+} K^{-}$ decays, Phys. Rev. Lett. 131, 091802 (2023).
R. Aaij et al. (LHCb Collaboration), Measurement of $C P$ asymmetry in $D^{0} \to K^{-} K^{+}$ and $D^{0} \to π^{-} π^{+}$ decays, J. High Energy Phys. 07 (2014) 041.
R. Aaij et al. (LHCb Collaboration), Measurement of the difference of time-integrated $C P$ asymmetries in $D^{0} \to K^{-} K^{+}$ and $D^{0} \to π^{-} π^{+}$ decays, Phys. Rev. Lett. 116, 191601 (2016).
R. Aaij et al. (LHCb Collaboration), Observation of $C P$ violation in charm decays, Phys. Rev. Lett. 122, 211803 (2019).
B. Aubert et al. (BABAR Collaboration, Search for $C P$ violation in the decays $D^{0} \to K^{-} K^{+}$ and $D^{0} \to π^{-} π^{+}$ , Phys. Rev. Lett. 100, 061803 (2008).
M. Staric et al. (Belle Collaboration), Measurement of $C P$ asymmetry in Cabibbo suppressed $D^{0}$ decays, Phys. Lett. B 670, 190 (2008).
A. Di Canto (CDF Collaboration), $C P$ violation in charm decays at CDF, Nuovo Cimento C 036, 26 (2013).
T. Aaltonen et al. (CDF Collaboration), Measurement of $C P$ -violating asymmetries in $D^{0} \to π^{+} π^{-}$ and $D^{0} \to K^{+} K^{-}$ decays at CDF, Phys. Rev. D 85, 012009 (2012).
A. J. Bevan et al. (UTfit Collaboration), The UTfit Collaboration average of D meson mixing data: Spring 2012, J. High Energy Phys. 10 (2012) 068.
A. J. Bevan et al. (UTfit Collaboration), The UTfit Collaboration average of D meson mixing data: Winter 2014, J. High Energy Phys. 03 (2014) 123.
A. L. Kagan and L. Silvestrini, Dispersive and absorptive $C P$ violation in $D^{0} - \bar{D^{0}}$ mixing, Phys. Rev. D 103, 053008 (2021).
K. Abe et al. (Belle Collaboration), Study of $B^{\pm} \to D_{C P} K^{\pm}$ and $D_{C P}^{*} K^{* \pm}$ decays, Phys. Rev. D 73, 051106 (2006).
Y. Horii et al. (Belle Collaboration), Evidence for the suppressed decay $B^{-} \to D K^{-}, D \to K^{+} π^{-}$ , Phys. Rev. Lett. 106, 231803 (2011).
M. Nayak et al. (Belle Collaboration), Evidence for the suppressed decay $B^{-} \to D K^{-}$ , $D \to K^{+} π^{-} π^{0}$ , Phys. Rev. D 88, 091104 (2013).
R. Aaij et al. (LHCb Collaboration), Study of $B^{-} \to D K^{-} π^{+} π^{-}$ and $B^{-} \to D π^{-} π^{+} π^{-}$ decays and determination of the CKM angle $γ$ , Phys. Rev. D 92, 112005 (2015).
R. Aaij et al. (LHCb Collaboration), Measurement of $C P$ observables in $B^{\pm} \to D K^{* \pm}$ decays using two- and four-body $D$ final states, J. High Energy Phys. 11 (2017) 156; 05 (2018) 067(E).
R. Aaij et al. (LHCb Collaboration), Constraints on the unitarity triangle angle $γ$ from Dalitz plot analysis of $B^{0} \to D K^{+} π^{-}$ decays, Phys. Rev. D 93, 112018 (2016); 94, 079902(E) (2016).
R. Aaij et al. (LHCb Collaboration), Measurement of $C P$ observables in $B^{\pm} \to D^{(*)} K^{\pm}$ and $B^{\pm} \to D^{(*)} π^{\pm}$ decays using two-body $D$ final states, J. High Energy Phys. 04 (2021) 081.
R. Aaij et al. (LHCb Collaboration), Measurement of $C P$ observables in $B^{\pm} \to {DK}^{\pm}$ and $B^{\pm} \to D π^{\pm}$ with $D \to K_{S}^{0} K^{\pm} π^{\mp}$ decays, J. High Energy Phys. 06 (2020) 058.
R. Aaij et al. (LHCb Collaboration), Constraints on the CKM angle $γ$ from $B^{\pm} \to D h^{\pm}$ decays using $D \to h^{\pm} h^{' \mp} π^{0}$ final states, J. High Energy Phys. 07 (2022) 099.
I. Adachi et al. (Belle and Belle-II Collaborations), Measurement of $C P$ asymmetries and branching-fraction ratios for $B^{\pm} \to {DK}^{\pm}$ and $D \to π^{\pm}$ with $D \to K_{S}^{0} K^{\pm} π^{\mp}$ using Belle and Belle II data, J. High Energy Phys. 09 (2023) 146.
I. Adachi et al. (Belle and Belle-II Collaborations), Measurement of branching-fraction ratios and $C P$ asymmetries in $B^{\pm} \to D_{C P \pm} K^{\pm}$ decays at Belle and Belle II, J. High Energy Phys. 05 (2024) 212.
R. Aaij et al. (LHCb Collaboration), Study of $C P$ violation in $B^{0} \to {DK}^{*} (892)^{0}$ decays with $D \to K π (π π)$ , $π π (π π$ ), and KK final states, J. High Energy Phys. 05 (2024) 025.
R. Aaij et al. (LHCb Collaboration), Measurement of the CKM angle $γ$ with $B^{\pm} \to D [K^{\mp} π^{\pm} π^{\pm} π^{\mp}] h^{\pm}$ decays using a binned phase-space approach, J. High Energy Phys. 07 (2023) 138.
R. Aaij et al. (LHCb Collaboration), Measurement of the CKM angle $γ$ in $B^{\pm} \to D K^{*} (892)^{\pm}$ decays, J. High Energy Phys. 02 (2025) 113.
A. Poluektov et al. (Belle Collaboration), Evidence for direct $C P$ violation in the decay $B \to D^{(*)} K$ , $D \to K_{S} π^{+} π^{-}$ and measurement of the CKM phase $ϕ_{3}$ , Phys. Rev. D 81, 112002 (2010).
P. K. Resmi et al. (Belle Collaboration), First measurement of the CKM angle $ϕ_{3}$ with $B^{\pm} \to D (K_{S}^{0} π^{+} π^{-} π^{0}) K^{\pm}$ decays, J. High Energy Phys. 10 (2019) 178.
R. Aaij et al. (LHCb Collaboration), Measurement of the CKM angle $γ$ in $B^{\pm} \to D K^{\pm}$ and $B^{\pm} \to D π^{\pm}$ decays with $D \to K_{S}^{0} h^{+} h^{-}$ , J. High Energy Phys. 02 (2021) 169.
F. Abudinén et al. (Belle and Belle-II Collaborations), Combined analysis of Belle and Belle II data to determine the CKM angle $ϕ_{3}$ using $B^{+} \to D (K_{S}^{0} h^{+} h^{-}) h^{+}$ decays, J. High Energy Phys. 02 (2022) 063; 12 (2022) 034(E).
R. Aaij et al. (LHCb Collaboration), A study of $C P$ violation in the decays $B^{\pm} \to [K^{+} K^{-} π^{+} π^{-}]_{D} h^{\pm}$ ( $h = K$ , $π$ ) and $B^{\pm} \to [π^{+} π^{-} π^{+} π^{-}]_{D} h^{\pm}$ , Eur. Phys. J. C 83, 547 (2023).
R. Aaij et al. (LHCb Collaboration), A model-independent measurement of the CKM angle $γ$ in partially reconstructed $B^{\pm} \to D^{*} h^{\pm}$ decays with $D \to K_{S}^{0} h^{+} h^{-}$ $(h = π, K)$ , J. High Energy Phys. 02 (2024) 118.
R. Aaij et al. (LHCb Collaboration), Measurement of the CKM angle $γ$ using the $B^{\pm} \to D^{*} h^{\pm}$ channels, J. High Energy Phys. 12 (2023) 013.
R. Aaij et al. (LHCb Collaboration), Measurement of the CKM angle $γ$ in the $B^{0} \to D K^{* 0}$ channel using self-conjugate $D \to K_{S}^{0} h^{+} h^{-}$ decays, Eur. Phys. J. C 84, 206 (2024).
R. Aaij et al. (LHCb Collaboration), Measurement of $C P$ violation in $B^{0} \to D^{\mp} π^{\pm}$ decays, J. High Energy Phys. 06 (2018) 084.
R. Aaij et al. (LHCb Collaboration), Measurement of the CKM angle $γ$ and $B_{s}^{0} - {\bar{B}}_{s}^{0}$ mixing frequency with $B_{s}^{0} \to D_{s}^{\mp} h^{\pm} π^{\pm} π^{\mp}$ decays, J. High Energy Phys. 03 (2021) 137.
R. Aaij et al. (LHCb Collaboration), Measurement of $C P$ asymmetry in $B_{s}^{0} \to D_{s}^{\mp} K^{\pm}$ decays, J. High Energy Phys. 03 (2025) 139.
B. Aubert et al. (BABAR Collaboration, Measurement of time-dependent $C P$ -violating asymmetries and constraints on $\sin (2 β + γ)$ with partial reconstruction of $B \to D^{* \mp} π^{\pm}$ decays, Phys. Rev. D 71, 112003 (2005).
B. Aubert et al. (BABAR Collaboration, Measurement of time-dependent $C P$ asymmetries in $B^{0} \to D^{(*) \pm} π^{\mp}$ and $B^{0} \to D^{\pm} ρ^{\mp}$ decays, Phys. Rev. D 73, 111101 (2006).
F. J. Ronga et al. (Belle Collaboration), Measurements of $C P$ violation in $B^{0} \to D^{* -} π^{+}$ and $B^{0} \to D^{-} π^{+}$ decays, Phys. Rev. D 73, 092003 (2006).
S. Bahinipati et al. (Belle Collaboration), Measurements of time-dependent $C P$ asymmetries in $B \to D^{* \mp} π^{\pm}$ decays using a partial reconstruction technique, Phys. Rev. D 84, 021101 (2011).
B. Aubert et al. (BABAR Collaboration, Measurement of $C P$ violation parameters with a Dalitz plot analysis of $B^{\pm} \to D (π^{+} π^{-} π^{0}) K^{\pm}$ , Phys. Rev. Lett. 99, 251801 (2007).
B. Aubert et al. (BABAR Collaboration, Measurement of ratios of branching fractions and $C P$ -violating asymmetries of $B^{\pm} \to D^{*} K^{\pm}$ decays, Phys. Rev. D 78, 092002 (2008).
B. Aubert et al. (BABAR Collaboration, Measurement of $C P$ violation observables and parameters for the decays $B^{\pm} \to D K^{* \pm}$ , Phys. Rev. D 80, 092001 (2009).
P. del Amo Sanchez et al. (BABAR Collaboration, Measurement of $C P$ observables in $B^{\pm} \to D_{C P} K^{\pm}$ decays and constraints on the CKM angle $γ$ , Phys. Rev. D 82, 072004 (2010).
P. del Amo Sanchez et al. (BABAR Collaboration, Search for $b \to u$ transitions in $B^{-} \to D K^{-}$ and $D^{*} K^{-}$ decays, Phys. Rev. D 82, 072006 (2010).
J. P. Lees et al. (BABAR Collaboration, Search for $b \to u$ transitions in $B^{\pm} \to [K^{\mp} π^{\pm} π^{0}]_{D} K^{\pm}$ decays, Phys. Rev. D 84, 012002 (2011).
T. Aaltonen et al. (CDF Collaboration), Measurements of branching fraction ratios and $C P$ asymmetries in $B^{\pm} \to D_{C P} K^{\pm}$ decays in hadron collisions, Phys. Rev. D 81, 031105 (2010).
T. Aaltonen et al. (CDF Collaboration), Measurements of branching fraction ratios and $C P$ -asymmetries in suppressed $B^{-} \to D (\to K^{+} π^{-}) K^{-}$ and $B^{-} \to D (\to K^{+} π^{-}) π^{-}$ decays, Phys. Rev. D 84, 091504 (2011).
A. Poluektov et al. (Belle Collaboration), Measurement of $ϕ_{3}$ with Dalitz plot analysis of $B^{+} \to D^{(*)} K^{(*) +}$ decay, Phys. Rev. D 73, 112009 (2006).
P. del Amo Sanchez et al. (BABAR Collaboration, Evidence for direct $C P$ violation in the measurement of the Cabibbo-Kobayashi-Maskawa angle $γ$ with $B^{\pm} \to D^{(*)} K^{(*) \pm}$ decays, Phys. Rev. Lett. 105, 121801 (2010).
K. Negishi et al. (Belle Collaboration), First model-independent Dalitz analysis of $B^{0} \to D K^{* 0}$ , $D \to K_{S}^{0} π^{+} π^{-}$ decay, Prog. Theor. Exp. Phys. 2016, 043C01 (2016).
R. Aaij et al. (LHCb Collaboration), Simultaneous determination of CKM angle $γ$ and charm mixing parameters, J. High Energy Phys. 12 (2021) 141.
LHCb Collaboration, Simultaneous determination of the CKM angle $γ$ and parameters related to mixing and $C P$ violation in the charm sector, Technical Report No. LHCb-CONF-2024-004, CERN, Geneva, 2024, https://cds.cern.ch/record/2905625.
I. Adachi et al. (Belle-II and Belle Collaborations), Determination of the CKM angle $ϕ_{3}$ from a combination of Belle and Belle II results, J. High Energy Phys. 10 (2024) 143.
Y. Amhis et al., Averages of $b$ -hadron, $c$ -hadron, and $τ$ -lepton properties as of 2021, Phys. Rev. D 107, 052008 (2023).
Y. Nir, $C P$ violation, Conf. Proc. C 9207131, 81 (1992).
G. C. Branco, L. Lavoura, and J. P. Silva, $C P$ violation, Int. Ser. Monogr. Phys. 103, 1 (1999).
Y. Grossman, Y. Nir, and G. Perez, Testing new indirect $C P$ violation, Phys. Rev. Lett. 103, 071602 (2009).
A. L. Kagan and M. D. Sokoloff, On Indirect $C P$ violation and implications for $D^{0} - \bar{D^{0}}$ and $B_{s} - \bar{B_{s}}$ mixing, Phys. Rev. D 80, 076008 (2009).
R. Aaij et al. (LHCb Collaboration), Search for time-dependent $C P$ violation in $D^{0} \to π^{+} π^{-} π^{0}$ decays, Phys. Rev. Lett. 133, 101803 (2024).
M. Rama, Effect of $D - \bar{D}$ mixing in the extraction of gamma with $B^{-} \to D^{0} K^{-}$ and $B^{-} \to D^{0} π^{-}$ decays, Phys. Rev. D 89, 014021 (2014).
M. Gronau and D. Wyler, On determining a weak phase from $C P$ asymmetries in charged $B$ decays, Phys. Lett. B 265, 172 (1991).
M. Gronau and D. London, How to determine all the angles of the unitarity triangle from $B_{d}^{0} \to D K_{s}$ and $B_{s}^{0} \to D^{0}$ , Phys. Lett. B 253, 483 (1991).
D. Atwood, I. Dunietz, and A. Soni, Enhanced $C P$ violation with $B \to K D^{0} ({\bar{D}}^{0})$ modes and extraction of the CKM angle $γ$ , Phys. Rev. Lett. 78, 3257 (1997).
D. Atwood, I. Dunietz, and A. Soni, Improved methods for observing $C P$ violation in $B^{\pm} \to K D$ and measuring the CKM phase $γ$ , Phys. Rev. D 63, 036005 (2001).
A. Giri, Y. Grossman, A. Soffer, and J. Zupan, Determining gamma using $B^{+ -} \to D K^{+ -}$ with multibody $D$ decays, Phys. Rev. D 68, 054018 (2003).
A. Poluektov et al. (Belle Collaboration), Measurement of $ϕ_{3}$ with Dalitz plot analysis of $B^{\pm} \to D^{(*)} K^{\pm}$ decay, Phys. Rev. D 70, 072003 (2004).
A. Bondar, in Proceedings of BINP Special Analysis Meeting on Dalitz Analysis (to be published).
I. Dunietz, R. Fleischer, and U. Nierste, In pursuit of new physics with $B_{s}$ decays, Phys. Rev. D 63, 114015 (2001).
M. Artuso, G. Borissov, and A. Lenz, $C P$ violation in the $B_{s}^{0}$ system, Rev. Mod. Phys. 88, 045002 (2016); 91, 049901(A) (2019).
M. Bona (UTfit Collaboration), Updates on unitarity triangle fits (to be published).
M. Ciuchini, M. Pierini, and L. Silvestrini, The effect of penguins in the $B_{d} \to J / Ψ K^{0} C P$ asymmetry, Phys. Rev. Lett. 95, 221804 (2005).
R. Di Palma and L. Silvestrini, silvest/GammaDDbar: Code used for Summer24 DDbar mixing and gamma global analysis, 10.5281/zenodo.13380849 (2024).
A. Caldwell, D. Kollar, and K. Kroninger, BAT: The Bayesian analysis toolkit, Comput. Phys. Commun. 180, 2197 (2009).
LHCb Collaboration, Simultaneous determination of the CKM angle $γ$ and parameters related to mixing and $C P$ violation in the charm sector, Technical Report No. LHCb-CONF-2022-003, CERN, Geneva, 2022, https://cds.cern.ch/record/2838029.
R. Di Palma (UTfit Collaboration), Simultaneous determination of the charm mixing and $C P$ -violating parameters together with the CKM angle $γ$ , Proc. Sci. EPS-HEP2023 (2024) 344.

Phys. Rev. D 112, 013004– Published 8 July, 2025

Vol. 112, Iss. 1 — 1 July 2025

Received 9 December 2024
Accepted 22 May 2025

DOI: https://doi.org/10.1103/pknz-r41n

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

Physical Review D

Reuse & Permissions

Export citation

Export citation

Global analysis of charm mixing parameters and determination of the CKM angle gamma $γ$

Abstract

Physics Subject Headings (PhySH)

Article Text

References (147)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation

Global analysis of charm mixing parameters and determination of the CKM angle gamma γ

Abstract

Physics Subject Headings (PhySH)

Article Text

References (147)

Global analysis of charm mixing parameters and determination of the CKM angle gamma $γ$