Comments on the Parametrization of the Kobayashi-Maskawa Matrix

Chau, Ling‐Lie; Keung, Wai-Yee

doi:10.1103/physrevlett.53.1802

Cited by 617 publications

(526 citation statements)

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“…These may be parametrized in a variety of ways. The standard parametrization, proposed by Chau-Keung [3] and advocated by the Particle Data Group [4], uses three mixing angles θ 12 , θ 23 , θ 13 and one CP-odd phase δ 13 which generates CP violation. The fact that there is only one independent CP-violating parameter in the electroweak sector means that this is a very predictive and testable model.…”

Section: The Unitary-exact Wolfenstein Parametrization and Rephasing-mentioning

confidence: 99%

CKM Fits: What the Data Say (Focused on B Physics)

T’Jampens

2007

Nuclear Physics B - Proceedings Supplements

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An up-to-date profile of the Cabibbo-Kobayashi-Maskawa matrix is given with emphasis on the interpretation of recent results on CP violation from the B factories. We provide a review of the relevant experimental and theoretical inputs from the contributing domains of the electroweak interaction. We give numerical and graphical constraints on the CKM parameters and predictions of related physical observables. We discuss the impact of what the data actually say on model-independent studies of new physics. The Unitary-exact Wolfenstein Parametrization and Rephasing-invariant QuantitiesThe Standard Model (SM) of electroweak interactions describes CP violation in weak interactions as a consequence of a single non-vanishing complex phase in the three-generation Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix [1,2]. The CKM matrix V has four quantities which are fundamental constants. These may be parametrized in a variety of ways. The standard parametrization, proposed by Chau-Keung [3] and advocated by the Particle Data Group [4], uses three mixing angles θ 12 , θ 23 , θ 13 and one CP-odd phase δ 13 which generates CP violation. The fact that there is only one independent CP-violating parameter in the electroweak sector means that this is a very predictive and testable model. Following the observation of a hierarchy between different matrix elements, Wolfenstein [5] proposed an expansion of the CKM matrix in terms of four parameters, A, λ, ρ and η (λ being the expansion parameter). Defining [6] λ ≡ sin θ 12 , Aλ 2 ≡ sin θ 13 and Aλ 3 (ρ + iη) ≡ sin θ 13 e −iδ13 to all orders in λ (i.e., these expressions are exact by definition; they are not corrected by terms of higher order in λ) ensures a unitary-exact Wolfenstein parametrization. Moreover, physically meaningful quantities must be rephasing-invariant quantities. Such invariants are the moduli, |V ij |, and the quadri-products, V ij V kl V * il V * jk . The Wolfenstein parameters, being physical quantities, must be phase-convention invariant:. ρ and η with ρ + iη = V * ub /(Aλ 3 ) are not phaseconvention invariant, in contrast to ρ + iη [7]. The results given in this paper use the unitary-exact Wolfenstein parametrization and rephasing-invariant quantities. λ is determined from |V ud | (superallowed nuclear transitions) and |V us | (semileptonic kaon decays) to a combined precision of 0.5%. A is determined from |V cb | (inclusive and exclusive semileptonic B decays) to a combined precision of about 2%. While λ and A are well-known, the parameters ρ and η are much more uncertain (about 20% for ρ and 7% for η). The main goal of CP-violation experiments is to over-constrain these parameters by measuring both the three angles and the sides of the unitarity triangle (UT) and, possibly, to find inconsistencies suggesting the existence of physics beyond the SM. What is important is thus the capability of the CKM mechanism to describe flavor dynamics of many constraints from vastly different scales and not the measurement of the CKM phase's value per se. CKM Metrology:...

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Section: The Unitary-exact Wolfenstein Parametrization and Rephasing-mentioning

confidence: 99%

CKM Fits: What the Data Say (Focused on B Physics)

T’Jampens

2007

Nuclear Physics B - Proceedings Supplements

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“…In K decays, setting zero-isospin-change amplitude real is the Wu-Yang phase convention [6]. What Keung and I found by trials in [17] is the 3 × 3 forerunner of this standard construction and [18] extended it to a 4×4 case. An example of Corollary 1 is the phase-moving ST, V ′ = diag(1, 1, e −iα13 )Vdiag(1, 1, e iα13 ) and α ′ 23 =−α 13 .…”

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confidence: 99%

“…In [17], besides the standard parametrization of V, Keung and I found (by trials) a construction procedure for it: V = R (23) U (13) R (12), one factor for each independent plane. R (jk) is the rotation matrix in the jk-plane and U (jk) is R (jk) with ±s jk → ±s jk e ∓iα jk ,…”

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“…However, Majorana neutrino fields do not have phase freedom [15], so only the phase freedoms of the Dirac leptons can be used to strip away N phases from the Dirac-Majorana mixing matrix U ν M by one PT: (24) to Eq.(26). What has been adapted in neutrino research [15] is the 3 × 3 case of [17] for C ν M . All variations discussed here can also apply.…”

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confidence: 99%

“…(This was four years before [17], whose parametrization has been called by PDG [12] the standard parametrization for the CKM and the PMNS matrices, thanks to the "advocation" and use by [14,15].) The unique and ubiquitous | Im (z * 1 z 2 ) | found in ∆ cp were denoted by the symbol…”

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Phase direct CP violations and general mixing matrices

Chau

2007

Physics Letters B

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I formulate expressions for amplitudes suitable for quantifying both modulus and phase direct CP violations. They result in Möbius transformation (MT) relations, which provide encouraging information for the search of direct CP violations in general. I apply the formulation to calculate the measurements of phase direct CP violations and strong amplitudes in B ∓ → K ∓ π ± π ∓ by the Belle collaboration. For the formulation, I show a versatile construction procedure for N × N CabibboKobayashi-Maskawa (CKM) matrices, Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrices, and general unitary matrices. It clarifies the 3 × 3 cases and is useful for the beyond.

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CP Violating Phenomena and Theoretical Results

Grimus

1988

Fortschr. Phys.

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Comments on the Parametrization of the Kobayashi-Maskawa Matrix

Cited by 617 publications

References 19 publications

CKM Fits: What the Data Say (Focused on B Physics)

CKM Fits: What the Data Say (Focused on B Physics)

Phase direct CP violations and general mixing matrices

CP Violating Phenomena and Theoretical Results

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