Generalized Ehrhart polynomials

Let $P$ be a polytope with rational vertices. A classical theorem of Ehrhart states that the number of lattice points in the dilations $P(n) = nP$ is a quasi-polynomial in $n$. We generalize this theorem by allowing the vertices of P(n) to be arbitrary rational functions in $n$. In this case we prove that the number of lattice points in P(n) is a quasi-polynomial for $n$ sufficiently large. Our work was motivated by a conjecture of Ehrhart on the number of solutions to parametrized linear Diophantine equations whose coefficients are polynomials in $n$, and we explain how these two problems are related.Comment: 18 pages, no figures; v2: Sections 4 and 5 added, proofs and exposition have been expanded and clarifie

CPCP violation in charmed hadron decays into neutral kaons

We find a new $CP$ violating effect in charmed hadron decays into neutral kaons, which is induced by the interference between the Cabibbo-favored and doubly Cabibbo-suppressed amplitudes with the $K^{0}-\overline K^{0}$ mixing. It is estimated to be of order of $\mathcal{O}(10^{-3})$, much larger than the direct $CP$ asymmetry, but missed in the literature. To reveal this new $CP$ violation effect, we propose a new observable, the difference of the $CP$ asymmetries in the $D^{+}\to \pi^{+}K_S^0$ and $D_{s}^{+}\to K^{+} K_S^0$ modes. Once the new effect is determined by experiments, the direct $CP$ asymmetry then can be extracted and used to search for new physics.Comment: 6 pages, 3 figures. Contribution to the proceeding of The 15th International Conference on Flavor Physics & CP Violation, 5-9 June 2017, Prague, Czech Republi

Implications on the first observation of charm CPV at LHCb

Very recently, the LHCb Collaboration observed the $CP$ violation (CPV) in the charm sector for the first time, with $\Delta A_{CP}^{\rm dir}\equiv A_{CP}(D^0\to K^+K^-)-A_{CP}(D^0\to \pi^+\pi^-)=(-1.54\pm0.29)\times10^{-3}$. This result is consistent with our prediction of $\Delta A_{CP}^{\rm SM}=(-0.57\sim -1.87)\times 10^{-3}$ obtained in the factorization-assisted topological-amplitude (FAT) approach in [PRD86,036012(2012)]. It implies that the current understanding of the penguin dynamics in charm decays in the Standard Model is reasonable. Motivated by the success of the FAT approach, we further suggest to measure the $D^+\to K^+K^-\pi^+$ decay, which is the next potential mode to reveal the CPV of the same order as $10^{-3}$.Comment: 10 page

Branching ratios and direct CP asymmetries in D→PPD\to PP decays

We propose a theoretical framework for analyzing two-body nonleptonic $D$ meson decays, based on the factorization of short-distance (long-distance) dynamics into Wilson coefficients (hadronic matrix elements of four-fermion operators). The parametrization of hadronic matrix elements in terms of several nonperturbative quantities is demonstrated for the $D\to PP$ decays, $P$ denoting a pseudoscalar meson. We consider the evolution of Wilson coefficients with energy release in individual decay modes, and the Glauber strong phase associated with the pion in nonfactorizable annihilation amplitudes, that is attributed to the unique role of the pion as a Nambu-Goldstone boson and a quark-anti-quark bound state simultaneously. The above inputs improve the global fit to the branching ratios involving the $\eta'$ meson, and resolves the long-standing puzzle from the $D^0\to\pi^+\pi^-$ and $D^0\to K^+K^-$ branching ratios, respectively. Combining short-distance dynamics associated with penguin operators and the hadronic parameters determined from the global fit to branching ratios, we predict direct CP asymmetries, to which the quark loops and the scalar penguin annihilation give dominant contributions. In particular, we predict $\Delta A_{\rm CP}\equiv A_{\rm CP}(K^+K^-)-A_{\rm CP}(\pi^+\pi^-)=-1.00\times 10^{-3}$, lower than the LHCb and CDF data.Comment: 17 pages, 3 figures, matches published versio

Branching ratios and direct CP asymmetries in D→PVD\to PV decays

We study the two-body hadronic $D\to PV$ decays, where $P$ ($V$) denotes a pseudoscalar (vector) meson, in the factorization-assisted topological-amplitude approach proposed in our previous work. This approach is based on the factorization of short-distance and long-distance dynamics into Wilson coefficients and hadronic matrix elements of four-fermion operators, respectively, with the latter being parametrized in terms of several nonperturbative quantities. We further take into account the $\rho$-$\omega$ mixing effect, which improves the global fit to the branching ratios involving the $\rho^0$ and $\omega$ mesons. Combining short-distance dynamics associated with penguin operators and the hadronic parameters determined from the global fit to branching ratios, we predict direct $CP$ asymmetries. In particular, the direct $CP$ asymmetries in the $D^0\to K^0\overline{K}^{*0},~\overline{K}^0K^{*0}$, $D^+\to\pi^+\rho^0$, and $D_s^+\to K^+\omega,~K^+\phi$ decays are found to be of ${\cal O}(10^{-3})$, which can be observed at the LHCb or future Belle II experiment. We also predict the $CP$ asymmetry observables of some neutral $D$ meson decays.Comment: 16 pages, 2 figure