Use of semi-leptonic decays has become standard in constraining the Unitarity Triangle. Bearing in mind that precise calculations of these are very challenging, we propose an entirely new approach. In particular the |Vcb| + epsilon_K constraint, which depends extremely sensitively on Vcb in the traditional method, is replaced by the interplay between epsilon_K, BR(B ->tau nu) and Delta M_Bs. It is found that even in this method tensions with the Standard Model persist at the ~1.8 sigma level. Furthermore, improvements on the B -> tau nu branching ratio and on the lattice determination of f_Bs Bs^(1/2) can increase the effectiveness of this method significantly.
In this work we discuss in detail the non-perturbative determination of the momentum dependence of the form factors entering in semileptonic decays using unitarity and analyticity constraints. The method contains several new elements with respect to previous proposals and allows to extract, using suitable two-point functions computed non-perturbatively, the form factors at low momentum transfer $q^2$ from those computed explicitly on the lattice at large $q^2$, without any assumption about their $q^2$-dependence. The approach will be very useful for exclusive semileptonic $B$-meson decays, whe…
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, $|V_{ud}|^2+|V_{us}|^2+|V_{ub}|^2=1$.
We present the results of the application of the Dispersion Matrix approach to exclusive semileptonic $B$-meson decays. This method allows to determine the hadronic form factors in a non-perturbative and completely model-independent way. Starting from lattice results available at large values of the momentum transfer, the behaviour of the form factors in their whole kinematical range is obtained without introducing any parameterization of their momentum dependence. We will focus on the determination of the Cabibbo-Kobayashi-Maskawa matrix elements $\vert V_{cb} \vert$ and $\vert V_{ub} \vert$ …
We examine the semileptonic $B \to D^{(*)} \ell ν_\ell$ and $B \to π\ell ν_\ell$ decays adopting the unitarity-based Dispersive Matrix (DM) method, which allows to determine the shape of the relevant hadronic form factors (FFs) in their whole kinematical range, using only lattice QCD results available at large values of the 4-momentum transfer without making any assumption on their momentum dependence. Moreover, the experimental data are not used to constrain the shape of the FFs, but only to obtain our final exclusive determination of $\vert V_{cb} \vert$ and $\vert V_{ub} \vert$, namely: $\v…
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.
It has been clear for past many years that in low energy observables new physics can only appear as a perturbation. Therefore precise theoretical predictions and precise experimental measurements have become mandatory. Here we draw attention to the significant advances that have been made on the lattice in recent years in $K\to ππ$, $ΔM_K$, the long-distance part of $\varepsilon$ and rare K-decays. Thus, in conjunction with experiments, the construction of a unitarity triangle purely from Kaon physics should soon become feasible. Along with the B-unitarity triangle, this should allow for more …
We present a geometric description of lepton flavor mixing and CP violation in matter by using the language of leptonic unitarity triangles. The exact analytical relations for both sides and inner angles are established between every unitarity triangle in vacuum and its effective counterpart in matter. The typical shape evolution of six triangles with the terrestrial matter density is illustrated for a realistic long-baseline neutrino oscillation experiment.
We present a comprehensive analysis of leptonic unitarity triangles, using both current neutrino oscillation data and projections of next-generation oscillation measurements. Future experiments, sensitive to the degree of CP violation in the lepton sector, will enable the construction of precise triangles. We show how unitarity violation could manifest in the triangles and discuss how they serve as unitarity tests. We also propose the use of Jarlskog factors as a complementary means of probing unitarity. This analysis highlights the importance of testing the unitarity of the leptonic mixing ma…
The Cabibbo-Kobayashi-Maskawa (CKM) matrix element $\vert V_{ub}\vert$ is obtained from exclusive semileptonic $B \to π\ell ν_\ell$ and $B_s \to K \ell ν_\ell$ decays adopting the unitarity-based dispersion matrix approach for the determination of the hadronic form factors (FFs) in the whole kinematical range. We use lattice computations of the relevant susceptibilities and of the FFs in the large-$q^2$ regime in order to derive their behavior in the low-$q^2$ region without assuming any specific momentum dependence and without constraining their shape using experimental data. Then, we address…
We present a new calculation of the $D \rightarrow π, l ν$ semileptonic form factor $f^{D \rightarrow π}_+(q^2)$ at $q^2 = 0$ based on HISQ charm and light valence quarks on MILC $N_f = 2 +1$ lattices. Using methods developed recently for HPQCD's study of $D \rightarrow K, l ν$ decays, we find $f^{D \rightarrow π}_+(0) = 0.666(29)$. This signifies a better than factor of two improvement in errors for this quantity compared to previous calculations. Combining the new result with CLEO-c branching fraction data, we extract the CKM matrix element $|V_{cd}| = 0.225(6)_{exp.}(10)_{lat.}$, where the …
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 $γ=(72.2\pm 9.2)^{\circ}$.
Theoretical and computational advances in lattice calculations are reviewed, with focus on examples relevant to the unitarity triangle of the CKM matrix. Recent progress in semi-leptonic form factors for B -> pi l nu and B -> D* l nu, as well as the parameter ξin B-Bbar mixing, are highlighted.
In this time of transition to a new generation of quark flavor experiments we review both the theoretical and the experimental progress on the determination of unitarity triangle angles from penguin-dominated B decays. This summarizes the activities of the Working Group VI at the CKM2010 workshop.
I review several methods for determining the CKM phases $α\equiv ϕ_2$ and $γ\equiv ϕ_3$ through CP asymmetry measurements in hadronic B decays. The current status of these measurements and the near future feasibility of the methods are discussed.
We present a method of obtaining the entire unitarity triangle from measurements of B -> pi K decay rates alone. Electroweak penguin amplitudes are included, and are related to tree operators. Discrete ambiguities are removed by comparing solutions with independent experimental data. The theoretical uncertainty in this method is rather small, in the range 5--10%.
The decay B^0 -> DK^{*0} is well-known to provide excellent potential for a precise measurement of the Unitarity Triangle angle gamma in future experiments. It is noted that the sensitivity can be significantly enhanced by studying the amplitudes relative to those of the flavour-specific decay B^0 -> D_2^{*-}K^+, which can be achieved by analyzing the B^0 -> Dπ^-K^+ Dalitz plot.
I point out that $B \to X_q ł^+ ł^-$ decays $(q = s, d)$ are sensitive probes of possible violation of CKM unitarity. I compute the decay rates and asymmetries in a minimal extension of the Standard Model containing an additional isosinglet charge (-1/3) quark, which leads to a deviation from CKM unitarity. It is shown that even for small mixing ratios $|{z_{qb}}/{(V_{tq}^\ast V_{tb})}| \sim O({10}^{-2})$, the contribution of the tree-level $Z-$FCNC appearing in the model should change the rates and asymmetries significantly. Especially the CP asymmetry, $A_{CP} (B \to X_s ł^+ ł^-)$, can be en…
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.
The aim of the paper is to show that the nowadays experimental data from superallowed nuclear and neutron $β$ decays, and from leptonic and semileptonic decays allow the finding of the most probable numerical form of the KM matrix, as well as the determination of decay constants, $f_{P}$, and of various form factors $f_+(q^2)$, by using a genuine implementation of unitarity constraints. In particular this approach allows the determination of semileptonic form factors that is illustrated on the existing data of $D\rightarrow πl ν$ and $D\rightarrow K lν$ decays.
We derive a new set of sum rules between the neutrino mass and mixing parameters in vacuum and their effective counterparts in matter. The moduli of nine genuine lepton mixing matrix elements can then be calculated in terms of the matter-corrected ones, and the latter can directly be determined from a variety of long-baseline neutrino oscillations. We show that it is possible to reconstruct the leptonic unitarity triangles and CP violation in a similar parametrization-independent way.
In view of the observed strong hierarchy of quark masses, we propose a new description of flavor mixing which is particularly suited for models of quark mass matrices based on flavor symmetries. The necessary and sufficient conditions for CP violation are clarified. The emergence of CP violation is primarily linked to a large phase difference (near 90^{\circ}) in the light quark sector. The unitarity triangle is determined by the mass ratios of the light quarks. We conclude that the unitarity triangle should be close or identical to a rectangular triangle, and CP violation is maximal in this s…
The angles of all unitarity triangles of the Cabibbo-Kobayashi-Maskawa matrix are determined from the experimental data. Our analysis is independent of the parameterization of the CKM matrix and it is based on the predictions of the unitarity for the angles and the areas of the unitarity triangles. We note that the lengths of the sides of the four unitarity triangles determined from the experimental data do not form a triangle. We resolve this incompatibility by performing a constrained fit, assuming the equality of the area of the unitarity triangles. We demonstrate that the measured data are…
Given the best-fit results of six neutrino oscillation parameters, we plot the Dirac and Majorana unitarity triangles (UTs) of the 3\times 3 lepton flavor mixing matrix to show their real shapes for the first time. The connections of the Majorana UTs with neutrino-antineutrino oscillations and neutrino decays are explored, and the possibilities of right or isosceles UTs are discussed. In the neutrino mass limit of m_1 \to 0 or m_3 \to 0, which is allowed by current experimental data, we show how the six triangles formed by the effective Majorana neutrino masses \langle m\rangle_{αβ} (for α, β=…
Following analogy of the `db' triangle in the quark mixing case, we have explored the construction of the `ν_2.ν_3' leptonic unitarity triangle using the Maki-Nakagawa-Sakata matrix obtained by Bjorken {\it et al.} through generalization of the tribimaximal scenario. In particular, for the U_{e3} range 0.05-0.15, the existence of the leptonic unitarity triangle indicates a fairly good possibility of having non zero CP violation.
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…
We generalize the geometrical description of CP violation in the Standard Model in terms of a unitarity triangle. For three left-handed Majorana neutrinos CP violation in the lepton sector is determined by three unitarity triangles. With three additional right-handed neutrinos 15 quadrangles are required to characterize CP violation. We show the relation of the unitarity polygons with physical observables.
In this note I review the current determination of the unitarity triangle parameters by using the theoretical and experimental information available in summer 2000.
Flavour mixing and CP violation as measured in weak decays and mixing of neutral mesons are a fundamental tool to test the Standard Model (SM) and to search for new physics. New analyses performed at the LHC experiment open an unprecedented insight into the Cabibbo-Kobayashi-Maskawa (CKM) metrology and new evidence for rare decays. Important progress has also been achieved in theoretical calculations of several hadronic quantities with a remarkable reduction of the uncertainties. This improvement is essential since previous studies of the Unitarity Triangle did show that possible contributions…
The quark mixing matrix is parameterised such that its "Cabibbo substructure" is emphasised. One can choose one of the parameters to be an arbitrarily chosen angle of the unitarity triangle, for example the angle $β$ (also called $Φ_1$).
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. …
We describe the present status of the Unitarity Triangle and we give an outlook for its future determinations. We discuss new sets of fundamental flavour parameters and comment briefly on new physics beyond the Standard Model.
There are six leptonic unitarity triangles (LUTs) defined by six orthogonality conditions of the three-family lepton flavor mixing matrix in the complex plane. In the framework of the standard model or the minimal supersymmetric standard model, the evolutions of sides and inner angles of the six LUTs from a superhigh energy scale $Λ_{\rm H}^{}$ to the electroweak scale $Λ_{\rm EW}^{}$ due to the renormalization-group equation (RGE) running are derived in the integral form for both Dirac and Majorana neutrinos. Furthermore, the LUTs as an intuitively geometrical language are applied to the desc…
The unitarity triangle on the first quadrant in the $ρ-η$ plane is discussed in the framework of the quark mass matrices in the NNI basis. If the quark mass matrices of the up-type is the Fritzsch one and the down-type is the one proposed by Branco et al., respectively, one gets the unitarity triangle with the vertex on the first quadrant. This simple model may be a candidate of the quark mass matrix ansatze, if the allowed region of the vertex of the unitarity triangle is restricted in the first quadrant by future experiments.
The area of the unitarity triangle is a measure of CP-violation. We introduce the leptonic unitarity triangles and study their properties. We consider the possibility of reconstructing the unitarity triangle in future oscillation and non-oscillation experiments. A set of measurements is suggested which will, in principle, allow us to measure all sides of the triangle, and consequently to establish CP-violation. For different values of the CP-violating phase, $δ_D$, the required accuracy of measurements is estimated. The key elements of the method include determination of $|U_{e3}|$ and studies…
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.
By using the most recent determinations of the several theoretical and experimental input parameters, we update the Unitarity Triangle analysis in the Standard Model and discuss the sensitivity to New Physics effects. We investigate the interest of measuring with a better precision the various physical quantities entering the Unitarity Triangle analysis and study in a model independent way whether, despite the undoubted success of the CKM mechanism in the Standard Model, the Unitarity Triangle analysis still allows the presence of New Physics.
We review recent results of Lattice QCD calculations relevant for flavour physics. We discuss in particular the hadronic parameters entering the amplitudes of K0-K0bar, D0-D0bar and B0-B0bar mixing, the B- and D-meson decay constants and the form factors controlling B-meson semileptonic decays. On the basis of these lattice results, which are extensively collected in the paper, we also derive our averages of the relevant hadronic parameters.
After reviewing the present experimental constraints on the unitarity triangle, I discuss the various ways in which new physics can manifest itself in measurements of the parameters of the unitarity triangle. Apart from one exception, which I describe, new physics enters principally through new contributions to B0-B0(bar) mixing. Different models of new physics can be partially distinguished by looking at their effects on rare, flavour-changing $B$ penguin decays. (Invited talk given at the Symposium Twenty Beautiful Years of Bottom Physics, Chicago, IL, USA, June 29 -- July 2, 1997.)
There is no good reason to think that BSM-CP-odd phases(s) will necessarily cause large deviations in B-physics from predictions of the SM. Therefore, residual theory error in extraction of the unitarity triangle can undermine experimental efforts to search for BSM phase(s). We stress that final states containing D0 or D0-bar in decays of charged and neutral B's can yield all the angles of the unitarity triangle with negligble theory error (i.e. O(0.1%)).
The current status of lattice and dispersive bound calculations for exclusive semileptonic B decays is reviewed. Emphasis is placed on decays relevant for the measurement of the sides of the unitarity triangle determined by |V_{ub}| and |V_{cb}|.
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.
We examine the unitarity triangle of the KM matrix by using the general quark mass matrices on the NNI basis. The Fritzsch anzätz is modified by introducing four additional parameters. The KM matrix elements are expressed in terms of quark mass ratios, two phases and four additional parameters. It is found that the vertex of the unitarity triangle is predicted to be almost in the second quadrant on the $ρ-η$ plane as far as $V_{us}\simeq -{\displaystyle \sqrt{\frac{m_d}{m_s}}e^{ip}+ \sqrt{\frac{m_u}{m_c}}e^{iq}}$ is held.
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.
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…
We summarize the present knowledge on the non-perturbative hadronic inputs needed in the analysis of $B^0$-$\bar B^0$ mixing and the CP-violating parameter $\varepsilon$ of the $K^0$-$\bar K^0$ system. Using this information, together with the recently determined value of the top-quark mass, we update the phenomenological constraints on the unitarity triangle.
We consider the vector and scalar form factors of the charm-changing current responsible for the semileptonic decay D\rightarrow πl ν. Using as input dispersion relations and unitarity for the moments of suitable heavy-light correlators evaluated with Operator Product Expansions, including O(α_s^2) terms in perturbative QCD, we constrain the shape parameters of the form factors and find exclusion regions for zeros on the real axis and in the complex plane. For the scalar form factor, a low energy theorem and phase information on the unitarity cut are also implemented to further constrain the s…
We construct the CKM unitarity triangle from CP invariant quantities, using the coupling constant of weak decays with flavor change from b to u, and the particle - antiparticle mixing probability in the B_s and B_d systems. Also included are new measurements of the coupling V_us in Kaon decays. Of the two solutions, one agrees perfectly with the triangle constructed from CP violating processes in the K and B meson systems. The common solution yields a triangle with an area of J/2 = (1.51 +/- 0.09) x 10^{-5} and a CP violating phase gamma = 63.1^o +/- 4.0^o.
We perform a global fit to the CKM unitarity triangle using the latest experimental and theoretical constraints. Our emphasis is on the hadronic weak matrix elements that enter the analysis, which must be computed using lattice QCD or other nonperturbative methods. Realistic lattice QCD calculations which include the effects of the dynamical up, down, and strange quarks are now available for all of the standard inputs to the global fit. We therefore present lattice averages for all of the necessary hadronic weak matrix elements. We attempt to account for correlations between lattice QCD result…
We present an application of the unitarity-based dispersion matrix (DM) approach to the extraction of the CKM matrix element $|V_{cb}|$ from the experimental data on the exclusive semileptonic $B_{(s)} \to D_{(s)}^{(*)} \ell ν_\ell$ decays. The DM method allows to achieve a non-perturbative, model-independent determination of the momentum dependence of the semileptonic form factors. Starting from lattice results available at large values of the 4-momentum transfer and implementing non-perturbative unitarity bound, the behaviour of the form factors in their whole kinematical range is obtained w…
The potential for a precise measurement of the Unitarity Triangle angle $γ$ in future experiments from the decay $B^0 \to DK^{*0}$ is well-known. It has recently been suggested that the sensitivity can be significantly enhanced by analysing the $B^0 \to DK^+π^-$ Dalitz plot to extract amplitudes relative to those of the flavour-specific decay $B^0 \to D_2^{*-}K^+$. An extension to this method which includes the case where the neutral $D$ meson is reconstructed in suppressed final states is presented. The sensitivity to $γ$ is estimated using this method and compared to that obtained using the …
This is an opening talk for the workshop and is intended to be provocative. We take a stab at the following questions: How well do we really know the sides of the unitarity triangle? What else could we do to improve? (I propose several new measurements). What precision should we ultimately aim at in the determination of CKM elements? What have we learned so far about flavor physics? Where do we go from here?
With the help of the generalized Wolfenstein parametrization of quark flavor mixing and CP violation, we calculate fine differences between the twin $b$-flavored unitarity triangles defined by $V^{*}_{ub} V^{}_{ud} + V^{*}_{cb} V^{}_{cd} + V^{*}_{tb} V^{}_{td} = 0$ and $V^{*}_{ud} V^{}_{td} + V^{*}_{us} V^{}_{ts} + V^{*}_{ub} V^{}_{tb} = 0$ in the complex plane. We find that vertices of the rescaled versions of these two triangles, described respectively by $\barρ + {\rm i} \barη = -\left(V^{*}_{ub} V^{}_{ud}\right)/\left(V^{*}_{cb} V^{}_{cd}\right)$ and $\widetildeρ + {\rm i} \widetildeη = -\…
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.
