Apollonian Equilateral Triangles

Given an equilateral triangle with $a$ the square of its side length and a point in its plane with $b$, $c$, $d$ the squares of the distances from the point to the vertices of the triangle, it can be computed that $a$, $b$, $c$, $d$ satisfy $3(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$. This paper derives properties of quadruples of nonnegative integers $(a,\, b,\, c,\, d)$, called triangle quadruples, satisfying this equation. It is easy to verify that the operation generating $(a,\, b,\, c,\, a+b+c-d)$ from $(a,\, b,\, c,\, d)$ preserves this feature and that it and analogous ones for the other elements can be represented by four matrices. We examine in detail the triangle group, the group with these operations as generators, and completely classify the orbits of quadruples with respect to the triangle group action. We also compute the number of triangle quadruples generated after a certain number of operations and approximate the number of quadruples bounded by characteristics such as the maximal element. Finally, we prove that the triangle group is a hyperbolic Coxeter group and derive information about the elements of triangle quadruples by invoking Lie groups. We also generalize the problem to higher dimensions

Fully open-flavor tetraquark states bcqˉsˉbc\bar{q}\bar{s} and scqˉbˉsc\bar{q}\bar{b} with JP=0+,1+J^{P}=0^{+},1^{+}

We have studied the masses for fully open-flavor tetraquark states $bc\bar{q}\bar{s}$ and $sc\bar{q}\bar{b}$ with quantum numbers $J^{P}=0^{+},1^{+}$. We systematically construct all diquark-antiquark interpolating currents and calculate the two-point correlation functions and spectral densities in the framework of QCD sum rule method. Our calculations show that the masses are about $7.1-7.2$ GeV for the $bc\bar{q}\bar{s}$ tetraquark states and $7.0-7.1$ GeV for the $sc\bar{q}\bar{b}$ tetraquarks. The masses of $bc\bar{q}\bar{s}$ tetraquarks are below the thresholds of $\bar{B}_{s}D$ and $\bar{B}_{s}^{*}D$ final states for the scalar and axial-vector channels respectively. The $sc\bar{q}\bar{b}$ tetraquark states with $J^{P}=1^{+}$ lie below the $B_{c}^{+}K^{*}$ and $B_{s}^{*}D$ thresholds. Such low masses for these possible tetraquark states indicate that they can only decay via weak interaction and thus are very narrow and stable.Comment: 17 pages, 4 figure