Canonical characters on quasi-symmetric functions and bivariate Catalan numbers

Every character on a graded connected Hopf algebra decomposes uniquely as a product of an even character and an odd character (Aguiar, Bergeron, and Sottile, math.CO/0310016). We obtain explicit formulas for the even and odd parts of the universal character on the Hopf algebra of quasi-symmetric functions. They can be described in terms of Legendre's beta function evaluated at half-integers, or in terms of bivariate Catalan numbers: $$C(m,n)=\frac{(2m)!(2n)!}{m!(m+n)!n!}.$$ Properties of characters and of quasi-symmetric functions are then used to derive several interesting identities among bivariate Catalan numbers and in particular among Catalan numbers and central binomial coefficients

Multigraded combinatorial Hopf algebras and refinements of odd and even subalgebras

We develop a theory of multigraded (i.e., $N^l$-graded) combinatorial Hopf algebras modeled on the theory of graded combinatorial Hopf algebras developed by Aguiar, Bergeron, and Sottile [Compos. Math. 142 (2006), 1--30]. In particular we introduce the notion of canonical $k$-odd and $k$-even subalgebras associated with any multigraded combinatorial Hopf algebra, extending simultaneously the work of Aguiar et al. and Ehrenborg. Among our results are specific categorical results for higher level quasisymmetric functions, several basis change formulas, and a generalization of the descents-to-peaks map.Comment: 49 pages. To appear in the Journal of Algebraic Combinatoric

T Violation in Baryonic B Decays

We study $T$ violation in the three-body charmless baryonic decay of \bar B^0 \ra \Lambda \bar p \pi^+ through the $T$-odd triple product correlation in the standard model. We find that the $T$ violating asymmetry is 10%, which is accessible to the current B factories at KEK and SLAC, while the CP violating one 1.1%. We emphasize that this triple product correlation would be the first measurable direct $T$ violating effect predicted in the standard model.Comment: 4 pages, 1 table, parallel talk given at International Conference on QCD and Hadronic Physics, Beijing, China, 16-20 Jun 200

Search for XYZXYZ states in Λb\Lambda_b decays at the LHCb

We consider $X(3872)$ and $Y(4140)$ as the vector tetraquark states of $X_c^0\equiv c\bar c u\bar u(d\bar d)$ and $c\bar c s\bar s$, respectively. By connecting $\Lambda_b\to X_c^0\Lambda$ to $B^-\to X_c^0 K^-$, we predict that the branching ratios of $\Lambda_b\to \Lambda(X(3872)^0\to) J/\psi \pi^+\pi^-$ and $\Lambda_b\to \Lambda(Y(4140)\to) J/\psi \phi$ are $(5.2\pm 1.8)\times 10^{-6}$ and $(4.7\pm 2.6)\times 10^{-6}$, which are accessible to the experiments at the LHCb, respectively. The measurements of these $\Lambda_b$ modes would be the first experimental evidences for the $XYZ$ states in baryonic decays.Comment: 7 pages, 1 figure

Radiative Baryonic BB Decays

We study the structure-dependent contributions to the radiative baryonic $B$ decays of $B \to {\bf B}{\bf \bar B'}\gamma$ in the standard model. We show that the decay branching ratios of $Br(B \to {\bf B}{\bf \bar B'}\gamma)$ are $O(10^{-7})$, which are larger than the estimated values of $O(10^{-9})$ induced from inner bremsstrahlung effects of the corresponding two-body modes. In particular, we find that $Br(B^- \to \Lambda \bar p \gamma)$ is around $1 \times 10^{-6}$, which is close to the pole model estimation but smaller than the experimental measurement from BELLE.Comment: 11 pages, 2 figures, Revtex4, new fitting is include

Pentaquarks from intrinsic charms in Λb\Lambda_b decays

We study the three-body $\Lambda_b$ decays of $\Lambda_b\to J/\psi pM$ with $M=K^-$ and $\pi^-$. The two new states ${\cal P}_{c1}\equiv {\cal P}_c(4380)^+$ and ${\cal P}_{c2}\equiv {\cal P}_c(4450)^+$ observed recently as the resonances in the $J/\psi p$ invariant mass spectrum of $\Lambda_b\to J/\psi p K^-$ can be identified to consist of five quarks, $uudc\bar c$, being consistent with the existence of the pentaquark states. We argue that, in the doubly charmful $\Lambda_b$ decays of $\Lambda_b\to J/\psi pK^-$ through $b\to c\bar c s$, apart from those through the non-resonant $\Lambda_b\to pK^-$ and resonant $\Lambda_b\to \Lambda^*\to pK^-$ transitions, the third contribution with the non-factorizable effects is not the dominant part for the resonant $\Lambda_b\to K^-{\cal P}_{c1,c2}, {\cal P}_{c1,c2}\to J/\psi p$ processes, such that we propose that the ${\cal P}_{c1,c2}$ productions are mainly from the charmless $\Lambda_b$ decays through $b\to \bar u u s$, in which the $c\bar c$ content in ${\cal P}_{c1,c2}$ arises from the intrinsic charms within the $\Lambda_b$ baryon. We hence predict the observables related to the branching ratios and the direct CP violating asymmetries to be ${\cal B}(\Lambda_b\to \pi^-({\cal P}_{c1,c2}\to) J/\psi p)/{\cal B}(\Lambda_b\to K^-({\cal P}_{c1,c2}\to) J/\psi p)=0.8\pm 0.1$, ${\cal A}_{CP}(\Lambda_b\to \pi^-({\cal P}_{c1,c2}\to)J/\psi p)=(-3.9\pm 0.2)\%$, and ${\cal A}_{CP}(\Lambda_b\to K^-({\cal P}_{c1,c2}\to)J/\psi p)=(5.8\pm 0.2)\%$, which can alleviate the inconsistency between the theoretical expectations from the three contributions in the doubly charmful $\Lambda_b$ decays and the observed data.Comment: 10 pages, 2 figures, revised version accepted by PL