A fresh look into m¯c,b(m¯c,b) and precise fD(s),B(s) from heavy–light QCD spectral sum rules

Using recent values of the QCD (non-)perturbative parameters given in Table 1 and an estimate of the N3LO QCD perturbative contributions based on the geometric growth of the PT series, we re-use QCD spectral sum rules (QSSR) known to N2LO PT series and including all dimension-six NP condensate contributions in the full QCD theory, for improving the existing estimates of ${\overline{m}}_{c,b}$ and $f_{D_{(s)},B_{(s)}}$ from the open charm and beauty systems. We especially study the effects of the subtraction point on “different QSSR data” and use (for the first time) the Renormalization Group Invariant (RGI) scale-independent quark masses in the analysis. The estimates [rigourous model-independent upper bounds within the SVZ framework] reported in Table 8: $f_D/f_{\pi }=1.56(5)$ $[?1.68(1)]$, $f_B/f_{\pi }=1.58(5)$ $[?1.80(3)]$ and $f_{D_s}/f_K=1.58(4)$ $[?1.63(1)]$, $f_{B_s}/f_K=1.50(3)$ $[?1.61(3.5)]$, which improve previous QSSR estimates, are in perfect agreement (in values and precisions) with some of the experimental data on $f_{D,D_s}$ and on recent lattice simulations within dynamical quarks. These remarkable agreements confirm both the success of the QSSR semi-approximate approach based on the OPE in terms of the quark and gluon condensates and of the Minimal Duality Ansatz (MDA) for parametrizing the hadronic spectral function which we have tested from the complete data of the $J/\psi$ and ? systems. The values of the running quark masses ${\overline{m}}_c(m_c)=1286(66)\text{MeV}$ and ${\overline{m}}_b(m_b)=4236(69)\text{MeV}$ from $M_{D,B}$ are in good agreement though less accurate than the ones from recent $J/\psi$ and ? sum rules.     

On the non-integrability of a generalized Darboux Halphen system

In this paper we study a generalized Darboux Halphen system given by ${\dot{x}}_1=x_2{x}_3-x_1(x_2+x_3)+{\tau }^2({\alpha }_1,{\alpha }_2,{\alpha }_3,x_1,x_2,x_3)\text{,}$${\dot{x}}_2=x_3{x}_1-x_2(x_3+x_1)+{\tau }^2({\alpha }_1,{\alpha }_2,{\alpha }_3,x_1,x_2,x_3)\text{,}$${\dot{x}}_3=x_1{x}_2-x_3(x_1+x_2)+{\tau }^2({\alpha }_1,{\alpha }_2,{\alpha }_3,x_1,x_2,x_3)\text{,}$ where $x_1$, $x_2$, $x_3$ are real variables, ${\alpha }_1,{\alpha }_2,{\alpha }_3$ are real constants and ${\tau }^2({\alpha }_1,{\alpha }_2,{\alpha }_3,x_1,x_2,x_3)={\alpha }_1^2(x_1-x_2)(x_3-x_1)+{\alpha }_2^2(x_2-x_3)(x_1-x_2)+{\alpha }_3^2(x_3-x_1)(x_2-x_3)\text{.}$ We prove that, for any $({\alpha }_1,{\alpha }_2,{\alpha }_3)\in {\mathbb{R}}^3\setminus \left\{(0,0,0)\right\}$, this system does not admit any non-constant global first integral that can be described by a formal power series. Furthermore, restricting the values of $({\alpha }_1,{\alpha }_2,{\alpha }_3)$ to a full Lebesgue measure set, we prove that this system does not admit any non-constant rational or Darbouxian global first integral. This is a first step toward proving that this system is chaotic.