We employ a continuum approach to the three valence-quark bound-state problem in relativistic quantum field theory to predict a range of properties of the proton’s radial excitation and thereby unify them with those of numerous other hadrons. Our analysis indicates that the nucleon’s first radial excitation is the Roper resonance. It consists of a core of three dressed quarks, which expresses its valence-quark content and whose charge radius is 80% larger than the proton analogue. That core is complemented by a meson cloud, which reduces the observed Roper mass by roughly 20%. The meson cloud materially affects long-wavelength characteristics of the Roper electroproduction amplitudes but the quark core is revealed to probes with $Q^2\gtrsim 3m_N^2$.

We study $CP$ violation and the contribution of the strong kaon-pion interactions in the three-body $B\to K{\pi }^{+}{\pi }^{-}$ decays. We extend our recent work on the effect of the two-pion $S$- and $P$-wave interactions to that of the corresponding kaon-pion ones. The weak amplitudes have a first term derived in QCD factorization and a second one as a phenomenological contribution added to the QCD penguin amplitudes. The effective QCD coefficients include the leading order contributions plus next-to-leading order vertex and penguins corrections. The matrix elements of the transition to the vacuum of the kaon-pion pairs, appearing naturally in the factorization formulation, are described by the strange $K\pi$ scalar ($S$-wave) and vector ($P$-wave) form factors. These are determined from Muskhelishvili-Omnès coupled channel equations using experimental kaon-pion $T$-matrix elements, together with chiral symmetry and asymptotic QCD constraints. From the scalar form factor study, the modulus of the $K_0^{*}(1430)$ decay constant is found to be $(32\pm 5)\mathrm{MeV}$. The additional phenomenological amplitudes are fitted to reproduce the $K\pi$ effective mass and helicity angle distributions, the $B\to K^{*}(892)\pi$ branching ratios and the $CP$ asymmetries of the recent data from Belle and BABAR collaborations. We use also the new measurement by the BABAR group of the phase difference between the $B^0$ and ${\overline{B}}^0$ decay amplitudes to $K^{*}(892)\pi$. Our predicted $B^{\pm }\to K_0^{*}(1430){\pi }^{\pm }$, $K_0^{*}(1430)\to K^{\pm }{\pi }^{\mp }$ branching fraction, equal to $(11.6\pm 0.6)\times {10}^{-6}$, is smaller than the result of the analyzes of both collaborations. For the neutral $B^0$ decays, the predicted value is $(11.1\pm 0.5)\times {10}^{-6}$. In order to reduce the large systematic uncertainties in the experimental determination of the $B\to K_0^{*}(1430)\pi$ branching fractions, a new parametrization is proposed. It is based on the $K\pi$ scalar form factor, well constrained by theory and experiments other than those of $B$ decays.

We study the contribution of the strong interactions between the two pions in $S$ and $P$ waves to the weak $B\to \pi \pi K$ decay amplitudes. The interference between these two waves is analyzed in the $\pi \pi$ effective-mass range of the $\rho (770{)}^0$ and $f_0(980)$ resonances. We use a unitary $\pi \pi$ and $\overline{K}K$ coupled-channel model to describe the $S$-wave interactions and a Breit-Wigner function for the $P$-wave amplitude. The weak $B$-decay amplitudes, obtained from QCD factorization, are supplemented with charming penguin contributions in both waves. The four complex parameters of these long-distance terms are determined by fitting the model to the BABAR and Belle data on $B^{\pm ,0}\to {\pi }^{+}{\pi }^{-}{K}^{\pm ,0}$ branching fractions, $CP$ asymmetries, $\pi \pi$ effective-mass and helicity-angle distributions. This set of data, and, in particular, the large direct $CP$ asymmetry for $B^{\pm }\to \rho (770{)}^0{K}^{\pm }$ decays, is well reproduced. The interplay of charming penguin amplitudes and the interference of $S$ and $P$ waves describes rather successfully the experimental $\mathcal{S}$ and $\mathcal{A}$ values of the $CP$-violating asymmetry for both $B^0\to f_0(980)K_S^0$ and $B^0\to \rho (770{)}^0{K}_S^0$ decays.