Rahul Sinha
This paper has been temporarily withdrawn by the author. A modified version will be resubmitted. One of the trivial solutions was missed with crucial bearing on conclusions.
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The magnitudes of the CKM matrix elements $V_{td}$, $V_{ts}$, and $V_{tb}$ are determined for the first time without any assumptions of unitarity. The implications for the unitarity of the CKM matrix as a whole are discussed.
The unitarity triangles of the $3\times 3$ Cabibbo-Kobayashi-Maskawa (CKM) matrix are studied in a systematic way. We show that the phases of the nine CKM rephasing invariants are indeed the outer angles of the six unitarity triangles and measurable in the $CP$-violating decay modes of $B_{d}$ and $B_{s}$ mesons. An economical notation system is introduced for describing properties of the unitarity triangles. To test unitarity of the CKM matrix we present some approximate but useful relations among the sides and angles of the unitarity triangles, which can be confronted with the accessible exp…
Concise variable transformations between the four angles of the CKM matrix in the standard representation advocated by the Particle Data Group and the angles of the unitarity triangles are derived. The behavior of these transformations in various limits is explored. The straightforward extension of this calculation to other representations and more generations is indicated.
The unitarity of the CKM matrix is examined in the light of the latest available accurate data. The analysis shows that for the elements in the first row, there are two types of experimental measurements leading to different results for the violation of unitarity. A conclusive result cannot be derived at present. Only more precise data can determine whether the CKM matrix opens new vistas beyond the standard model or not. The experimental data for the elements in the second row are not as precise as for those in the first row, and within the measured uncertainty, the unitarity is not violated.…
Hierarchy of CKM matrix elements and implications of unitarity
本文在幺正性约束下重新审视CKM矩阵元素的层级结构，并比较9种等价参数化方案对CP破坏参数ε_k的计算，发现并非所有方案都能重现该参数值，揭示了层级结构对参数化选择的影响。
We explore the unitarity violation of the CKM matrix in the model in which the third generation fermions are subjected to the separate SU(2)$_L$ gauge interaction. With the recent LEP and SLC data at $Z$-pole and low-energy neutral current interaction data, the analysis on the parameter space of the model is updated, and the unitary violation is predicted under the constraint.
Within the Standard Model, a review of the current determination of the CKM unitarity triangle parameters is presented, using experimental constraints from the measurements of |epsilon_K|, V_ub/V_cb, Delta m_d and from the limit on Delta m_s, available by end 1999.
In this presentation, I review the status of selected Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and their role in the Unitarity Triangle (UT). Since this conference concluded, many new results have been finalized and are included in the new world averages (WAs) in the 2002 edition of the PDG. I will focus here on some outstanding issues in the measurements of the CKM elements in the third row and column.
We point out that $R \equiv Br(b\rightarrow d γ)/Br(b\rightarrow s γ)$ is a sensitive probe of possible violation of CKM unitarity. We compute $R$ in a minimal extension of the Standard Model containing an additional isosinglet charge ($-1/3$) quark, which leads to a deviation from CKM unitarity.
Motivated by the possibility of the low value of sin2βin the measurements of BABAR and BELLE collaborations, a reference unitarity triangle is constructed using the unitarity of the CKM matrix and the experimental values of the well known CKM elements, without involving any inputs from the processes which might include the new physics effects. The angles of the triangle are evaluated by finding the CP violating phase δthrough the Jarlskog's rephasing invariant parameter J. The present data and the unitarity of the CKM matrix gives for δthe range 28^o to 152^o, which for sin2βtranslates to the …
The aim of the paper is to propose one paradigm change of CKM global fits on experimental data from electroweak sector. The change refers to using in fits the exact unitarity constraints expressed in terms of four invariant parameters, such as moduli of the CKM matrix, and to take into account an important set of the available experimental data. In the paper we use data from nuclear beta decays, and from leptonic and semileptonic decays, in order to find the most probable numerical form of the CKM matrix, as well as the determination of decay constants, $f_{P}$, and of various form factors $f_…
We perform a new numerical analysis of hyperon semileptonic decays emphasizing the systematic uncertainties. The poor understanding of SU(3) symmetry breaking effects at second order in the vector form factor translates into a large error of |V(us)|. Using our determination |V(us)| = 0.226 +/- 0.005 together with those coming from other sources we test the unitarity of the CKM matrix.
The Jarlskog rephasing invariant parameter $|J|$ is evaluated using one of the six Unitarity Triangles involving well known CKM matrix elements \vud, \vus,~\rub, ~\vcd, ~\vcs~ and ~\vcb. With PDG2000 values of \vud~ etc. as input, we obtain $|J|=(2.71 \pm 1.12) \times 10^{-5}$, which in the PDG representation of CKM matrix leads to the range $21^o~to~159^o$ for the CP violating phase $δ$. The CKM matrix elements evaluated using this range of $δ$ are in agreement with the PDG CKM matrix. The implications of refinements in the input on $|J|$, $δ$ and CKM matrix elements have also been studied.
In the Standard Model of elementary particles, quark-mixing is expressed in terms of a 3 x 3 unitary matrix V, the so called Cabibbo-Kobayashi-Maskawa (CKM) matrix. Significant unitarity checks are so far possible for the first row of this matrix. This article reviews the experimental and theoretical information on these matrix elements. On the experimental side, we find a 2.2 sigma to 2.7 sigma deviation from unitarity, which conflicts with the Standard Model.
We discuss radiative corrections to K_{e3} and pi_{e3} decays within chiral perturbation theory with virtual photons and leptons. We then present the extraction of the CKM elements V_{us} and V_{ud} using the presently available experimental input. Finally, we discus the test of CKM unitarity based on the present independent knowledge of V_{ud} and V_{us}, and describe the prospects of improvement with the upcoming B.R. measurements.
Decay rates of $B^0(t)\toπ^+π^-,~B^0\toπ^- K^+,~B^+\toπ^+ K^0~ (K_S\toπ^+π^-$) and of charge-conjugate processes are studied within flavor SU(3) symmetry and first-order SU(3) breaking. We show that these measurements can determine with a reasonable accuracy the two angles, $α$ and $γ$, of the Cabibbo-Kobayashi-Maskawa unitarity triangle.
Unitarity along with precision measurements of sin2β, V_{us} and V_{cb} allows one to find a lower bound V_{ub}\geq 0.0035 which, on using the recently measured angle αof the unitarity triangle, translates to V_{ub}= 0.0035\pm 0.0002. This precise value, stable for a good deal of changes in α, along with CP violating phase δfound from unitarity allows the construction of a `precise' CKM matrix. The above unitarity based value of V_{ub} is in agreement with the latest exclusive value used as input by UTfit, CKMfitter, HFAG, however underlines the so called `tension' faced by the latest inclusiv…
We discuss possible signatures of the tree level FCNC, which results from the non-unitarity of CKM matrix. We first define the unitaity step-by-step, and possible test of the non-unitaity through the 4-value-KM parametrization. We, then, show how the phase angle of the unitary triangle would change in case of the vector-like down quark model. As another example of tree FCNC, we investigate the leptophobic $Z'$ model and its application to the recent $B_s$ mixing measurements.
We present a detailed numerical study of the K_{e3} decays to O(p^6,(m_d-m_u)p^2,e^2p^2) in chiral perturbation theory with virtual photons and leptons. We describe the extraction of the CKM matrix element |V_{us}| from the experimental K_{e3} decay parameters. We propose a consistency check of the K^+_{e3} and K^0_{e3} data that is largely insensitive to the dominating theoretical uncertainties, in particular the contributions of order p^6. Our analysis is highly relevant in view of the recent high statistics measurement of the K^+_{e3} branching ratio by E865 at Brookhaven which does not ind…
Top flavour-changing neutral decays are extremely suppressed within the Standard Model (SM) by the GIM mechanism, but can reach observable rates in some of its extensions. We compute the branching ratios for t -> c gamma and t -> c g in minimal SM extensions where the addition of a vector-like up or down quark singlet breaks the unitarity of the 3 x 3 Cabibbo-Kobayashi-Maskawa (CKM) matrix. The maximum rates obtained indicate to what extent present experimental data allow 3 x 3 CKM unitarity to be broken in these models, and are too small to be observed in the near future. As a by-product, we …
Determinations of $\vud$ and $\vus$ along with their implications for the unitarity test $\vud^2+\vus^2+\vub^2=1$ are discussed. The leading two loop radiative corrections to neutron $β$-decay are evaluated and used to derive a refined relationship $\vud^2 τ_n (1+3g_A^2)=4908(4)$sec. Employing $\vud =0.9740(5)$ from superallowed nuclear decays and the measured neutron lifetime $τ_n = 885.7(7)$sec, leads to the precise prediction $g_A=1.2703(8)$ which is compared with current direct experimental values. Various extractions of $\vus$ are described and updated. The long accepted Particle Data Gro…
This report contains the results of the Workshop on the CKM Unitarity Triangle, held at CERN on 13-16 February 2002 to study the determination of the CKM matrix from the available data of K, D, and B physics. This is a coherent document with chapters covering the determination of CKM elements from tree level decays and K and B meson mixing and the global fits of the unitarity triangle parameters. The impact of future measurements is also discussed.
We point out that hints of deviations from unitarity in the first row of the CKM matrix may be explained by the presence of a single vector-like top. We study how the stringent experimental constraints arising from CP Violation in the kaon sector and from meson mixing such as $D^0$-$\overline{D}^0$, $K^0$-$\overline{K}^0$ and $B^0_{d,s}$-$\overline{B}^0_{d,s}$ can be satisfied in the proposed framework. In order for the deviations from unitarity to be of the required size while keeping the theory perturbative, the new top quark should have a mass $m_T \lesssim 7$ TeV which could be probed in u…
In the standard model (SM) the lowest order contribution to the quark electric dipole moment (EDM) occurs at the three loop level. We show that the non-unitarity of the CKM matrix in models with an extended quark sector typically gives rise to a quark EDM at the two loop level which has no GIM-like suppression factors except the external quark mass. The induced neutron EDM is of order 10^{-29} e cm and can be well within the reach of the next generation of experiments if it is further enhanced by long distance physics as happens in the SM.
Using an effective field theory approach, we study the new physics (NP) corrections to muon and beta decays and their effects on the extractions of V_ud and V_us. Assuming nearly flavor blind NP interactions we find that the CKM-unitarity test is the only way to expose NP. The four short-distance operators that can generate a deviation are strongly constrained by the phenomenological bound |Vud|^2 + |Vus|^2 + |Vub|^2 - 1 = (-0.0001 \pm 0.0006), corresponding to an effective scale > 11 TeV (90% CL). Depending on the operator, this constraint is at the same level or better than that generated by…
Unitarity Relations in the Presence of Vector-Like Quarks
系统研究含类矢量夸克扩展模型中CKM幺正性关系，分析其对模量差异和虚部影响，并探讨上型VLQ解决第一行幺正性问题的特殊情形。
The form factors for the $K_{\ell 3}$ semileptonic decay are computed to order $O(p^4)$ in generalized chiral perturbation theory. The main difference with the standard $O(p^4)$ expressions consists in contributions quadratic in quark masses, which are described by a single divergence-free low-energy constant, $A_3$. A new simultaneous analysis is presented for the CKM matrix element $V_{us}$, the ratio $F_K/F_π$, $K_{\ell 3}$ decay rates and the scalar form factor slope $λ_0$. This framework easily accommodates the precise value for $V_{ud}$ deduced from superallowed nuclear $β$-decays.
The unitarity based analysis carried out in hep-ph/0106161, involving the evaluation of Jarlskog's rephasing invariant parameter J and the angles of the unitarity triangle is extended to include the effects of K^0-\bar K^0 and B^0-\bar B^0 mixings. The present analysis attempts to evaluate CP violating phase δ, V_{td} and the angles of the unitarity triangle by invoking the constraints imposed by unitarity, K^0 -\bar K^0 and B^0 -\bar B^0 mixings separately as well as collectively, using the "full scanning" of input parameters. Interestingly, our results indicate that the lower limit of δis mo…
The unitarity triangle can be determined by means of two measurements of its sides or angles. Assuming the same relative errors on the angles $(α,β,γ)$ and the sides $(R_b,R_t)$, we find that the pairs $(γ,β)$ and $(γ,R_b)$ are most efficient in determining $(\bar\varrho,\barη)$ that describe the apex of the unitarity triangle. They are followed by $(α,β)$, $(α,R_b)$, $(R_t,β)$, $(R_t,R_b)$ and $(R_b,β)$. As the set $\vus$, $\vcb$, $R_t$ and $β$ appears to be the best candidate for the fundamental set of flavour violating parameters in the coming years, we show various constraints on the CKM m…
We present the status of the Unitarity Triangle analysis focused on the analyses connected to the CKM angles extraction. The angle values are found to be $α= (90.6\pm 6.6)^{\circ}$, $\sin(2β) = 0.68\pm 0.023$, and $γ=(72.2\pm 9.2)^{\circ}$.
With these proceedings, we present both a review of the experimental and theoretical information on quark-mixing with focus on the first quark generation. The papers present new findings on these topics in the context of what is known so far. Besides this, about half a dozen new neutron-decay instruments being planned or under construction are presented. Better neutron sources, in particular for high fluxes of cold and high densities of ultra-cold neutrons will boost fundamental studies in these fields. The workshop included invited talks and a panel discussion. The results of the panel discus…
We investigate the impact of extra leptons on observed tensions in the muon $g-2$ and the first-row CKM unitarity. By introducing a new SU(2)$_L$ doublet lepton and a SU(2)$_L$ triplet lepton, we find that both of the tensions can be explained simultaneously under constraints from electroweak precision observables and Higgs-boson decays. Our model could be tested by measurements of $h\toμμ$ at future collider experiments.
Using a technique that is not subject to first-order SU(3) symmetry breaking effects, we determine the $V_{us}$ element of the CKM matrix from data on semileptonic hyperon decays. We obtain $V_{us}$ =0.2250(27). This value is of similar precision to the one derived from $K_{l3}$, but higher and in better agreement with the unitarity requirement, $|V_{ud}|^2+|V_{us}|^2+|V_{ub}|^2=1$.
Direct Evaluation of CP Phase of CKM matrix, General Perturbative Expansion and Relations with Unitarity Triangles
本文通过重相位不变公式评估CKM矩阵的CP相，并探讨其与单位arity三角形的关系。
Motivated by the possibility of the low value of sin2βin the measurements of BABAR and BELLE collaborations, we have explored the possibilty of construction of reference unitarity triangle using the unitarity of the CKM matrix, the existence of nonzero CP violating phase δand the experimental values of the well known CKM elements, without involving any inputs from the processes which might include the new physics effects. The angles of the reference triangle are evaluated by finding δthrough the Jarlskog's rephasing invariant parameter J. The present data and the unitarity of the CKM matrix gi…
Due to a recent more precise evaluation of $V_{ud}$ and $V_{us}$, the unitarity condition of the first row in the Cabibbo-Kobayashi-Maskawa (CKM) matrix: $|V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 = 0.99798 \pm 0.00038$ now stands at a deviation more than $4σ$ from unity. Furthermore, a mild excess in the overall Higgs signal strength appears at about $2σ$ above the standard model (SM) prediction, as well as the long-lasting discrepancy in the forward-backward asymmetry ${\cal A}_{\rm FB}^b$ in $Z\to b\bar b$ at LEP. Motivated from the above three anomalies we investigate an extension of the SM wit…
We propose a minimal extension of the Standard Model where an up-type vector-like quark, denoted $T$, is introduced and provides a simple solution to the CKM unitarity problem. We adopt the Botella-Chau parametrization in order to extract the $4\times 3$ quark mixing matrix which contains the three angles of the $3\times 3$ CKM matrix plus three new angles denoted $θ_{14}$, $θ_{24}$, $θ_{34}$. It is assumed that the mixing of $T$ with standard quarks is dominated by $θ_{14}$. Imposing a recently derived, and much more restrictive, upper-bound on the New Physics contributions to $ε_{K}$, we fin…
We review the present status of the CKM matrix and we offer some visions of its future. After a brief presentation of the theoretical framework for weak decays we discuss the following topics: i) CKM matrix from tree level decays, ii) Standard analysis of the unitarity triangle, iii) CKM matrix from rare and CP violating K- and B-decays, iv) CKM matrix from CP violation in two-body B-decays. In particular we compare the CKM potentials of the standard analysis of the unitarity triangle, of the very clean K --> πν\barνdecays and of CP asymmetries in B-decays.
In presenting an up-to-date account of the experimental knowledge of the CKM matrix, special emphasis is placed on the exceptional degree of consistency shown by the current Standard Model determination of the unitarity triangle; some implications in the question of how the theoretical nature of the dominant uncertainties affects the Standard Model predictions are discussed. Recent experimental results include improved determinations of |V_{cb}| by OPAL and CLEO, the confirmation of rare (charmless hadronic and electromagnetic penguin) B decays and preliminary sin(2 beta) measurements by BaBar…
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 model-independent analysis are also reported with emphasis on the leptonic decay of the Bd and on the Bs mixing phase.
In this talk I briefly review the precision test of the first row CKM unitarity, focusing mainly on $V_{ud}$ and $V_{us}$ extracted from pion, kaon, neutron and superallowed nuclear beta decays. I will discuss the current status of several important Standard Model theory inputs to these processes, and the need for future improvements.
The 3 X 3 CKM matrix defines six unitarity triangles in the complex plane, which will be carefully explored in the LHCb experiment and at the Super-B factory. We calculate the running effects of nine different inner angles and eighteen different sides of the six triangles from the electroweak scale to a superhigh-energy scale by using the one-loop renormalization-group equations, and demonstrate that all the nine angles are stable against radiative corrections. In particular, we find that the CP-violating angle αis most insensitive to the changes of energy scales.
We report on a new method to a global fit of the CKM matrix by using the necessary and sufficient condition the data have to satisfy in order to find a unitary matrix compatible with them. This condition writes as $-1\le \cosϕ\le 1$ where $ϕ$ is the phase that accounts for CP violation. By using it we get that the experimental data are to a high degree compatible to unitarity and that $ϕ$ takes values around $90^0$, in contrast to the previous determinations. Numerical results are provided for the CKM matrix entries, the mixing angles between generations and all the angles of the standard unit…
Testing the CKM unitarity at high energy via the $W^+W^-$ production at the LHC and future colliders
本文提出利用W⁺W⁻产额直接探测CKM矩阵的单位性，通过高能行为异常验证单位性破坏，结合ATLAS数据约束当前最大单位性破坏，并展望LHC高亮度阶段和未来100 TeV对撞机结果。
The aim of the paper is to make a comparison between the unitarity condition method and the standard version of the unitarity triangle approach by using as parameters four independent moduli $|U_{ij}|$. This choice is motivated by the measurability property and leads to a simple criterion for the separation of unistochastic matrices from the double stochastic ones, whose fulfillment is the key point of any global fit for the CKM entries. In our formulation both the methods are exact, do not depend on any assumption as the smallness of some parameter and both can be used to global fits in the q…
The possibility to test the unitarity triangle of CKM matrix is examined by using the semileptonic rare $B$ decays, $B \rightarrow X_q \, ł^+ \, ł^-$ with $q = d, s$ and $ł$'s are leptons. The discussion is emphasized on the CP asymmetry in the framework of standard and beyond the standard models. For the beyond standard model, a minimal extension of the standard model containing an additional isosinglet charge (-1/3) charge quark, which leads to a deviation from CKM unitarity, is considered. As the results, in the standard model the CP asymmetry is visible only for $B \rightarrow X_d \, ł^+ \…
This article surveys the important questions attached to unitarity tests of the Cabibbo-Kobayashi-Maskawa matrix, with a focus on the direct determinations of the magnitude of CKM matrix elements. The current CKM anomalies are discussed and the clear-cut prospects at LHCb and Belle II envisaged. The anticipated precision on $C\!P$-conserving and $C\!P$-violating observables at the projected upgrades of the LHCb and Belle~II experiments is examined from the point of view of the search for New Physics in $B$-meson mixing, addressing the bottlenecks in precision for the interpretation of the meas…
The parameter $ε_K$ describes CP violation in the neutral kaon system and is one of the most sensitive probes of new physics. The large uncertainties related to the charm-quark contribution to $ε_K$ have so far prevented a reliable standard-model prediction. We show that CKM unitarity enforces a unique form of the $|ΔS = 2|$ weak effective Lagrangian in which the short-distance theory uncertainty of the imaginary part is dramatically reduced. The uncertainty related to the charm-quark contribution is now at the percent level. We present the updated standard-model prediction $ε_K = 2.16(6)(8)(1…
Several methods exist to measure $CP$ violation observables related to the CKM unitarity triangle angle $γ$ using $B$ meson decays. These observables are different for every $B$ meson decay considered, although the information they contain on $γ$ is encoded in a similar way for all of them. This paper describes a strategy for a simultaneous measurement of $γ$ using several $B$ meson decays that takes into account possible correlations between them. Sensitivity studies demonstrate that the simultaneous analysis of several $B$ meson decay modes results in smaller uncertainties and improved stati…
The influence of nucleon-nucleon short-ranged correlations (SRC) on nuclear super-allowed $β$ decay is examined. The need for this is driven by the observed depletion of spectroscopic strength obtained in studies of $(e,e')$ and $(d,^3{\rm He})$ reactions on a wide variety of nuclei. We show that the influence of SRC is model-dependent, but may be very substantial. The $^{46}$V nucleus is used as an example. The resulting impact on studies of the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element is discussed.
This talk gives an introduction to the Standard Model Cabibbo-Kobayashi-Maskawa matrix and methods to extract information on the parameters of the unitarity triangle from B decays.
We investigate a flexible method in which we can test the unitarity of the quark flavor mixing matrix step-by-step. Singular-Value-Decomposition (SVD) techniques are used in analyzing the mixing matrix over a broader parameter region than the unitarity region. Unitary constraints make us extract CP violating properties without any specific parametrization when the magnitudes of at least three mixing matrix elements in three generation quark mixing are given. This method can also be applied to the analysis of lepton flavor mixing, in which only a few moduli are presently measured.
Planned as a review of CPV in B-mesons which covered recent B-factories results these lectures appeared to be a bit wider. It is not natural to be limited by direct CPV and that in mixing in B-mesons and not to speak about the analogous phenomena in K-mesons since it is very useful and interesting to study what is common and what is different in these systems and why. CKM matrix elements are extracted from K and B mixings and decays and the deviation from unitarity of CKM matrix may become a place in which New Physics will show up. So we discuss this simple and elegant piece of Standard Model …
We have investigated the present renormalization prescriptions of Cabibbo-Kobayashi-Maskawa (CKM) matrix at one-loop level. We emphasize at one prescription which is formulated with reference to the case of no mixing of quark's generations and point out that it doesn't make the physical amplitude involving quark mixing ultraviolet finite. We then propose a revised prescription to solve this problem and simultaneously keep the unitarity of the CKM matrix. Through explicit calculations we also prove that in order to keep the CKM matrix gauge independent the unitarity of the CKM matrix must be pr…
We explore the possibility of resolving simultaneously the $W$-boson mass shift reported by the CDF collaboration and the apparent deviation from unitarity in the first row of the CKM mixing matrix in a class of left-right symmetric models. The fermion masses are generated in these models through a universal seesaw mechanism, utilizing vector-like partners of the usual fermions. We find a unique solution to the two anomalies where the mixing of the down quark with a vector-like quark (VLQ) resolves the CKM unitarity puzzle, and the mixing of the top quark with a VLQ partner explains the $W$ bo…
Assuming that the Kobayashi-Maskawa mechanism gives the dominant contribution to CP violation at low energies, we propose a novel way of testing the flavour sector of the Standard Model which has the potencial for discovering New Physics. Using 3 x 3 unitarity of the V_{CKM} matrix and choosing a complete set of rephasing invariant phases, we derive a set of exact relations in terms of measurable quantities, namely moduli of V_{CKM} and arguments of rephasing invariant quartets. These tests complement the usual analysis in the ρ, ηplane and, if there is New Physics, may reveal its source.
We report on an exact method for global fits of the CKM matrix by using the necessary and sufficient conditions the data have to satisfy in order to find a unitary matrix compatible with them, and this method can be applied to both quark and lepton CKM matrices. The key condition writes $-1\le\cosδ\le 1$ where $δ$ is the phase that encodes the CP violation, and it is obtained when one describes the CKM matrix in terms of four moduli of its entries. We use it to reconstruct many approximate unitary matrices from the PDG data, and by using the double stochasticity property of the unitary matrice…
Some fine differences between the twin $b$-flavored unitarity triangles are calculated by means of a generalized Wolfenstein parametrization of the CKM matrix, and a possibility of experimentally establishing the second triangle is briefly discussed. We find that the apexes of these two triangles, characterized respectively by $(\overlineρ, \overlineη)$ and $(\widetildeρ, \widetildeη)$, are located on the same circular arc in the complex plane. This observation provides us with a new way to test consistency of the CKM picture of CP violation in the quark sector and probe possible new physics. …
Noting the hierarchy between three mixing angles, $θ_{2,3}={\cal O}(θ_1^2)$, we present an exact form of the quark mixing matrix, replacing Wolfenstein's approximate form. In addition, we suggest to rotate the unitarity triangle, using the weak CP phase convention where the phase is located at the (31) element $\sinθ_1\sinθ_2 e^{iδ}$ while the (13) element $\sinθ_1\sinθ_3$ is real. For the $(ab)$ unitarity triangle, the base line (x-axis) is defined from the product of the first row elements, $V_{1a}V^*_{1b}$, and the angle between two sides at the origin is defined to be the phase $δ$. This i…
We present the update of the Unitarity Triangle (UT) analysis within the Standard Model (SM) and beyond. Within the SM, combining the direct measurements on sides and angles, the UT turns out to be overconstraint in a consistent way, showing that the CKM matrix is the dominant source of flavour mixing and CP-violation and that New Physics (NP) effects can appear at most as small corrections to the CKM picture. Generalizing the UT analysis to investigate NP effects, constraints on b -> s transitions are also included and both CKM and NP parameters are fitted simultaneously. While no evidence of…
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