Dynamical implications of gluonic excitations in meson-meson systems

We study meson-meson interactions using an extended $q^2\bar{q}^2(g)$ basis that allows calculating coupling of an ordinary meson-meson system to a hybrid-hybrid one. We use a potential model matrix in this extended basis which at quark level is known to provide a good fit to numerical simulations of a $q^2\bar{q}^2$ system in pure gluonic theory for static quarks in a selection of geometries. We use a combination of resonating group method formalism and Born approximation to include the quark motion using wave functions of a $q\bar{q}$ potential within a cluster. This potential is taken to be quadratic for ground states and has an additional smeared $\frac{1}{r}$ (Gaussian) for the matrix elements between hybrid mesons. For the parameters of this potential, we use values chosen to 1) minimize the error resulting from our use of a quadratic potential and 2) best fit the lattice data for differences of $\Sigma_{g}$ and $\Pi_{u}$ configurations of the gluonic field between a quark and an antiquark. At the quark (static) level, including the gluonic excitations was noted to partially replace the need for introducing many-body terms in a multi-quark potential. We study how successful such a replacement is at the (dynamical) hadronic level of relevance to actual hard experiments. Thus we study effects of both gluonic excitations and many-body terms on mesonic transition amplitudes and the energy shifts resulting from the second order perturbation theory (i.e. from the respective hadron loops). The study suggests introducing both energy and orbital excitations in wave functions of scalar mesons that are modelled as meson-meson molecules or are supposed to have a meson-meson component in their wave functions.Comment: 26 pages, 10 figure

Connections between the stability of a Poincare map and boundedness of certain associate sequences

Let $m\ge 1$ and $N\ge 2$ be two natural numbers and let ${\mathcal{U}}=\{U(p, q)\}_{p\ge q\ge 0}$ be the $N$-periodic discrete evolution family of $m\times m$ matrices, having complex scalars as entries, generated by ${\mathcal{L}}(\mathbb{C}^m)$-valued, $N$-periodic sequence of $m\times m$ matrices $(A_n).$ We prove that the solution of the following discrete problem $$y_{n+1}=A_ny_n+e^{i\mu n}b,\quad n\in\mathbb{Z}_+,\quad y_0=0$$ is bounded for each $\mu\in\mathbb{R}$ and each $m$-vector $b$ if the Poincare map $U(N, 0)$ is stable. The converse statement is also true if we add a new assumption to the boundedness condition. This new assumption refers to the invertibility for each $\mu\in\mathbb{R}$ of the matrix $V_{\mu}:=\sum\nolimits_{\nu=1}^NU(N, \nu)e^{i\mu \nu}.$ By an example it is shown that the assumption on invertibility cannot be removed. Finally, a strong variant of Barbashin's type theorem is proved

Wave function dependent Form Factors and Radii of Mesons

In this work, meson wave functions are selected to investigate the properties of charmonium, bottomonium, and charmed-bottom mesons. To find the masses of the ground, radial and orbital excited states of mesons, variational method is applied on the selected meson wave functions. In addition, mesons are treated non-relativistically by considering the chromodynamic potential model in the linear plus coulombic form along with the incorporation of spin. Our mass predictions show good agreement with the available experimental data and the theoretical predictions found by different methods. Moreover, RMS radii, form factors and charged radii are calculated by using the selected trial wave functions. Momentum dependance of the form factors is shown graphically. Predicted RMS radii and charged radii are compared with the theoretical results, wherever available. Results show that RMS radii and charged radii have inverse relation with the masses of mesons, i.e., heavier mesons have smaller radii and vice versa.Comment: 9 pages, 3 figures, 6 table