arXiv subjects

P. Kielanowski

Publications and source records attributed to P. Kielanowski.

16 recordsLinked to original sources

arXiv2021

Test of the 4-th quark generation from the Cabibbo-Kobayashi-Maskawa matrix

The structure of the mixing matrix, in the electroweak quark sector with four generations of quarks is investigated. We conclude that the area of the unitarity quadrangle is not a good choice as a possible measure of the CP~violation. In search of new physics we analyze how the existence of the 4-th quark family may influence on the values of the Cabibbo-Kobayashi-Maskawa matrix and we show that one can test for the existence of the 4-th generation using the Jarlskog invariants of the known quarks only. The analysis based on the measured unitary triangle exhibits some tension with the assumption of three quark generations. The measurement of the unitarity triangle obtained from the scalar product of the second row/column of the CKM matrix by the complex conjugate of third row/column can provide information about the existence of the fourth generation of quarks.

hep-ph↗

arXiv2015

Cabibbo-Kobayashi-Maskawa matrix: parameterizations and rephasing invariants

The paper is devoted to a discussion of general properties of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. First we propose a general method of a recursive construction of the CKM matrix for any number of generations. This allows to set up a parameterization with desired properties. As an application we generalize the Wolfenstein parameterization to the case of 4 generations and obtain restrictions on the CKM suppression of the fourth generation. Motivated by the rephasing invariance of the CKM observables we next consider the general phase invariant monomials built out of the CKM matrix elements and their conjugates. We show, that there exist 30 fundamental phase invariant monomials and 18 of them are a product of 4 CKM matrix elements and 12 are a product of 6 CKM matrix elements. In the Main Theorem we show that all rephasing invariant monomials can be expressed as a product of at most 5 factors: 4 of them are fundamental phase invariant monomials and the fifth factor consists of powers of squares of absolute values of the CKM matrix elements.

hep-ph↗

arXiv2012

Outlook on the Higgs particles, masses and physical bounds in the Two Higgs-Doublet Model

The Higgs sector of models beyond the standard model requires special attention and study, since through them, a natural explanation can be offered to current questions such as the big differences in the values of the masses of the quarks (hierarchy of masses), the possible generation of flavor changing neutral currents (inspired by the evidence about the oscillations of neutrinos), besides the possibility that some models, with more complicated symmetries than those of the standard model, have a non standard low energy limit. The simplest extension of the standard model known as the two-Higgs-doublet-model (2HDM) involves a second Higgs doublet. The 2HDM predicts the existence of five scalar particles: three neutral ($A^{0}$), ($h^{0}$, $H^{0}$) and two charged ($H^{\pm}$). The purpose of this work is to determine in a natural and easy way the mass eigenstates and masses of these five particles, in terms of the parameters $λ_{i}$ introduced in the minimal extended Higgs sector potential that preserves the CP symmetry. We discuss several cases of Higgs mixings and the one in which two neutral states are degenerate. As the values of the quartic interactions between the scalar doublets are not theoretically determined, it is of great interest to explore and constrain their values, therefore we analize the stability and triviality bounds using the Lagrange multipliers method and numerically solving the renormalization group equations. Through the former results one can establish the region of validity of the model under several circumstances considered in the literature.

hep-ph↗

arXiv2008

Renormalization Group Equations for the CKM matrix

We derive the one loop renormalization group equations for the Cabibbo-Kobayashi-Maskawa matrix for the Standard Model, its two Higgs extension and the minimal supersymmetric extension in a novel way. The derived equations depend only on a subset of the model parameters of the renormalization group equations for the quark Yukawa couplings so the CKM matrix evolution cannot fully test the renormalization group evolution of the quark Yukawa couplings. From the derived equations we obtain the invariant of the renormalization group evolution for three models which is the angle $α$ of the unitarity triangle. For the special case of the Standard Model and its extensions with $v_{1}\approx v_{2}$ we demonstrate that also the shape of the unitarity triangle and the Buras-Wolfenstein parameters $\barρ=(1-{1/2}λ^{2})ρ$ and $\barη=(1-{1/2}λ^{2})η$ are conserved. The invariance of the angles of the unitarity triangle means that it is not possible to find a model in which the CKM matrix might have a simple, special form at asymptotic energies.

hep-th↗

arXiv2005

Complex Energies and Beginnings of Time Suggest a Theory of Scattering and Decay

Many useful concepts for a quantum theory of scattering and decay (like Lippmann-Schwinger kets, purely outgoing boundary conditions, exponentially decaying Gamow vectors, causality) are not well defined in the mathematical frame set by the conventional (Hilbert space) axioms of quantum mechanics. Using the Lippmann-Schwinger equations as the takeoff point and aiming for a theory that unites resonances and decay, we conjecture a new axiom for quantum mechanics that distinguishes mathematically between prepared states and detected observables. Suggested by the two signs $\pm iε$ of the Lippmann-Schwinger equations, this axiom replaces the one Hilbert space of conventional quantum mechanics by two Hardy spaces. The new Hardy space theory automatically provides Gamow kets with exponential time evolution derived from the complex poles of the $S$-matrix. It solves the causality problem since it results in a semigroup evolution. But this semigroup brings into quantum physics a new concept of the semigroup time $t=0$, a beginning of time. Its interpretation and observations are discussed in the last section.

quant-ph↗

arXiv2003

Precise bounds on the Higgs boson mass

We study the renormalization group evolution of the Higgs quartic coupling $λ_{H}$ and the Higgs mass $m_{H}$ in the Standard Model. The one loop equation for $λ_{H}$ is non linear and it is of the Riccati type which we numerically and analytically solve in the energy range $[m_{t},E_{GU}]$ where $m_{t}$ is the mass of the top quark and $E_{GU}=10^{14}$ GeV. We find that depending on the value of $λ_{H}(m_{t})$ the solution for $λ_{H}(E)$ may have singularities or zeros and become negative in the former energy range so the ultra violet cut off of the standard model should be below the energy where the zero or singularity of $λ_{H}$ occurs. We find that for $0.369\leqλ_{H}(m_{t})\leq0.613$ the Standard Model is valid in the whole range $[m_{t},E_{GU}]$. We consider two cases of the Higgs mass relation to the parameters of the standard model: (a) the effective potential method and (b) the tree level mass relations. The limits for $λ_{H}(m_{t})$ correspond to the following Higgs mass relation $150\leq m_{H}\lessapprox 193$ GeV. We also plot the dependence of the ultra violet cut off on the value of the Higgs mass. We analyze the evolution of the vacuum expectation value of the Higgs field and show that it depends on the value of the Higgs mass. The pattern of the energy behavior of the VEV is different for the cases (a) and (b). The behavior of $λ_{H}(E)$, $m_{H}(E)$ and $v(E)$ indicates the existence of a phase transition in the standard model. For the effective potential this phase transition occurs at the mass range $m_{H}\approx 180$ GeV and for the tree level mass relations at $m_{H}\approx 168$ GeV.

hep-ph↗

arXiv2003

Renormalization group evolution of the CKM matrix

We present here the most important ideas, equations and solutions for the running of all the quark Yukawa couplings and all the elements of the Cabibbo-Kobayashi-Maskawa matrix, in the approximation of one loop, and up to order $λ^{4}$, where $λ\sim 0.22$ is the sine of the Cabibbo angle. Our purpose is to determine what the evolution of these parameters may indicate for the physics of the standard model (SM), the minimal supersymmetric standard model (MSSM) and for the Double Higgs Model (DHM).

hep-ph↗

arXiv2003

Scale dependence of the quark masses and mixings: leading order

We consider the Renormalization Group Equations (RGE) for the couplings of the Standard Model and its extensions. Using the hierarchy of the quark masses and of the Cabibbo-Kobayashi-Maskawa (CKM) matrix our argument is that a consistent approximation for the RGE should be based on the parameter $λ= |\hat{V}_{ud}| \approx0.22$. We consider the RGE in the approximation where we neglect all the relative terms of the order $\simλ^{4}$ and higher. Within this approximation we find the exact solution of the evolution equations of the quark Yukawa couplings and of the vacuum expectation value of the Higgs field. Then we derive the evolution of the observables: quark masses, CKM matrix, Jarlskog invariant, Wolfenstein parameters of the CKM matrix and the unitarity triangle. We show that the angles of the unitarity triangle remain constant. This property may restrict the possibility of new symmetries or textures at the grand unification scale.

hep-ph↗

arXiv2001

Time Asymmetric Boundary Conditions and the Definition of Mass and Width for Relativistic Resonances

The definition of mass and width of relativistic resonances and in particular of the $Z$-boson is discussed. For this we use the theory based on time asymmetric boundary conditions given by Hardy class spaces ${\mathbf Φ}_-$ and ${\mathbf Φ}_+$ for prepared in-states and detected out-states respectively, rather than time symmetric Hilbert space theory. This Hardy class boundary condition is a mathematically rigorous form of the singular Lippmann-Schwinger equation. In addition to the rigorous definition of the Lippmann-Schwinger kets $|[j,{\mathsf s}]^{\pm}>$ as functionals on the spaces ${\mathbf Φ}_{\mp}$, one obtains Gamow kets $|[j,{\mathsf s}_R]^- >$ with complex centre-of-mass energy value ${\mathsf s}_R=(M_R-iΓ_R/2)^2$. The Gamow kets have an exponential time evolution given by $\exp{(-iM_Rt-Γ_Rt/2)}$ which suggests that $(M_R,Γ_R)$ is the right definition of the mass and width of a resonance. This is different from the two definitions of the $Z$-boson mass and width used in the Particle Data Table and leads to a numerical value of $M_R=(91.1626\pm 0.0031) {\rm GeV}$ from the $Z$-boson lineshape data.

hep-th↗

arXiv2000

Energy dependence of the quark masses and mixings

The one loop Renormalization Group Equations for the Yukawa couplings of quarks are solved. From the solution we find the explicit energy dependence on $t=\ln E/μ$ of the evolution of the {\em down} quark masses $q=d,s,b$ from the grand unification scale down to the top quark mass $m_{t}$. These results together with the earlier published evolution of the {\em up} quark masses completes the pattern of the evolution of the quark masses. We also find the energy dependence of the absolute values of the Cabibbo-Kobayashi-Maskawa (CKM) matrix $|V_{ij}|$. The interesting property of the evolution of the CKM matrix and the ratios of the quark masses: $m_{u,c}/m_{t}$ and $m_{d,s}/m_{b}$ is that they all depend on $t$ through only one function of energy $h(t)$.

hep-ph↗

arXiv2000

Theorems on the Renormalization Group Evolution of Quark Yukawa Couplings and CKM Matrix

We analyze the two loop renormalization group equations in the standard model and its extensions for the coupling constants and the quark Yukawa couplings. The key point of our analysis is the observed hierarchy of the quark masses and the CKM matrix. For the one loop evolution we find the explicit solution for the evolution of the Yukawa couplings and show the following: 1. the CKM matrix depends on the energy on only one function of energy; 2. the ratios of the down quark masses depend on the energy through the same function as the CKM matrix; 3. the diagonalizing matrices of the biunitary transformation of the up quarks are energy independent. Next we give the explicit form of the two loop corrections to the evolution of the quark Yukawa couplings and show that the relative corrections are of the order (λ^{4}) for the quark Yukawa couplings and of the order (λ^{5}) for the CKM matrix. Finally we give the equations of the one loop evolution of the squares of the matrix elements of the CKM matrix and their explicit solution.

hep-ph↗

arXiv2000

Relativistic Gamow Vectors II

Motivated by the debate of possible definitions of mass and width of resonances for $Z$-boson and hadrons, we suggest a definition of unstable particles by ``minimally complex'' semigroup representations of the Poincaré group characterized by $(j,{\mathsf s}=(m-iΓ/2)^{2})$ in which the Lorentz subgroup is unitary. This definition, though decidedly distinct from those based on various renormalization schemes of perturbation theory, is intimately connected with the first order pole definition of the $S$-matrix theory in that the complex square mass $(m-iΓ/2)^{2}$ characterizing the representation of the Poincaré semigroup is exactly the position ${\mathsf s}_R$ at which the $S$-matrix has a simple pole. Wigner's representations $(j,m)$ are the limit case of the complex representations for $Γ=0$. These representations have generalized vectors (Gamow kets) which have, in addition to the $S$-matrix pole at ${\mathsf s}=(m-iΓ/2)^{2}$, all the other properties that heuristically the unstable states need to possess: a Breit-Wigner distribution in invariant square mass and a lifetime $τ=\frac{1}Γ$ defined by the exactly exponential law for the decay probability ${\cal P}(t)$ and rate $\dot{\cal P}(t)$ given by an exact Golden Rule which becomes Dirac's Golden Rule in the Born-approximation. In addition and unintended, they have an asymmetric time evolution.

hep-th↗

arXiv1999

Semigroup Representations of the Poincare Group and Relativistic Gamow Vectors

Gamow vectors are generalized eigenvectors (kets) of self-adjoint Hamiltonians with complex eigenvalues $(E_{R}\mp iΓ/2)$ describing quasistable states. In the relativistic domain this leads to Poincaré semigroup representations which are characterized by spin $j$ and by complex invariant mass square ${\mathsf{s}}={\mathsf{s}}_{R}=(M_{R}-\frac{i}{2}Γ_{R})^{2}$. Relativistic Gamow kets have all the properties required to describe relativistic resonances and quasistable particles with resonance mass $M_{R}$ and lifetime $\hbar/Γ_{R}$.

hep-th↗

arXiv1999

Relativistic Gamow Vectors

Gamow vectors in non-relativistic quantum mechanics are generalized eigenvectors (kets) of self-adjoint Hamiltonians with complex eigenvalues. Like the Dirac kets, they are mathematically well defined in the Rigged Hilbert Space. Gamow kets are derived from the resonance poles of the S-matrix. They have a Breit-Wigner energy distribution, an exponential decay law, and are members of a basis vector expansion whose truncation gives the finite dimensional effective theories with a complex Hamiltonian matrix. They also have an asymmetric time evolution described by a semigroup generated by the Hamiltonian, which expresses a fundamental quantum mechanical arrow of time. These Gamow kets are generalized to relativistic Gamow vectors by extrapolating from the Galilei group to the Poincare group. This leads to semigroup representations of the Poincare group which are characterized by spin j and complex invariant mass square. In these non-unitary representations the Lorentz subgroup is unitarily represented and the four-momenta are "minimally complex" in the sense that the four-velocity is real. The relativistic Gamow vectors have all the properties listed above for the non-relativistic Gamow vectors and are therefore ideally suited to describe relativistic resonances and quasistable particles.

hep-th↗

arXiv1997

Quark mixings as a test of a new symmetry of quark Yukawa couplings

Based on the hierarchy exhibited by quarks masses at low energies, we assume that Yukawa couplings of up and down quarks are related by $Y_u\propto Y_d^2$ at grand unification scales. This ansatz gives rise to a symmetrical CKM matrix at the grand unification (GU) scale. Using three specific models as illustrative examples for the evolution down to low energies, we obtain the entries and asymmetries of the CKM matrix which are in very good agreement with their measured values. This indicates that the small asymmetry of the CKM matrix at low energies may be the effect of the renormalization group evolution only.

hep-ph↗

arXiv1995

Time-Reversal and Irreversibility

The time reversal and irreversibility in conventional quantum mechanics are compared with those of the rigged Hilbert space quantum mechanics. We discuss the time evolution of Gamow and Gamow-Jordan vectors and show that the rigged Hilbert space case admits a new kind of irreversibility which does not appear in the conventional case. The origin of this irreversibility can be traced back to different initial-boundary conditions for the states and observables. It is shown that this irreversibility does not contradict the experimentally tested consequences of the time-reversal invariance of the conventional case but instead we have to introduce a new time reversal operator.

quant-ph↗