Nuclear Modification to Parton Distribution Functions and Parton Saturation

We introduce a generalized definition of parton distribution functions (PDFs) for a more consistent all-order treatment of power corrections. We present a new set of modified DGLAP evolution equations for nuclear PDFs, and show that the resummed $\alpha_s A^{1/3}/Q^2$-type of leading nuclear size enhanced power corrections significantly slow down the growth of gluon density at small-$x$. We discuss the relation between the calculated power corrections and the saturation phenomena.Comment: 4 pages, to appear in the proceedings of QM200

Role of the nonperturbative input in QCD resummed Drell-Yan QTQ_T-distributions

We analyze the role of the nonperturbative input in the Collins, Soper, and Sterman (CSS)'s $b$-space QCD resummation formalism for Drell-Yan transverse momentum ($Q_T$) distributions, and investigate the predictive power of the CSS formalism. We find that the predictive power of the CSS formalism has a strong dependence on the collision energy $\sqrt{S}$ in addition to its well-known $Q^2$ dependence, and the $\sqrt{S}$ dependence improves the predictive power at collider energies. We show that a reliable extrapolation from perturbatively resummed $b$-space distributions to the nonperturbative large $b$ region is necessary to ensure the correct $Q_T$ distributions. By adding power corrections to the renormalization group equations in the CSS formalism, we derive a new extrapolation formalism. We demonstrate that at collider energies, the CSS resummation formalism plus our extrapolation has an excellent predictive power for $W$ and $Z$ production at all transverse momenta $Q_T\le Q$. We also show that the $b$-space resummed $Q_T$ distributions provide a good description of Drell-Yan data at fixed target energies.Comment: Latex, 43 pages including 15 figures; typos were correcte

The structures of Hausdorff metric in non-Archimedean spaces

For non-Archimedean spaces $X$ and $Y,$ let $\mathcal{M}_{\flat } (X), \mathfrak{M}(V \rightarrow W)$ and $\mathfrak{D}_{\flat }(X, Y)$ be the ballean of $X$ (the family of the balls in $X$), the space of mappings from $X$ to $Y,$ and the space of mappings from the ballen of $X$ to $Y,$ respectively. By studying explicitly the Hausdorff metric structures related to these spaces, we construct several families of new metric structures (e.g., $\widehat{\rho } _{u}, \widehat{\beta }_{X, Y}^{\lambda }, \widehat{\beta }_{X, Y}^{\ast \lambda }$) on the corresponding spaces, and study their convergence, structural relation, law of variation in the variable $\lambda,$ including some normed algebra structure. To some extent, the class $\widehat{\beta }_{X, Y}^{\lambda }$ is a counterpart of the usual Levy-Prohorov metric in the probability measure spaces, but it behaves very differently, and is interesting in itself. Moreover, when $X$ is compact and $Y = K$ is a complete non-Archimedean field, we construct and study a Dudly type metric of the space of $K-$valued measures on $X.$Comment: 43 pages; this is the final version. Thanks to the anonymous referee's helpful comments, the original Theorem 2.10 is removed, Proposition 2.10 is stated now in a stronger form, the abstact is rewritten, the Monna-Springer is used in Section 5, and Theorem 5.2 is written in a more general for

Minimum-error discrimination between mixed quantum states

We derive a general lower bound on the minimum-error probability for {\it ambiguous discrimination} between arbitrary $m$ mixed quantum states with given prior probabilities. When $m=2$, this bound is precisely the well-known Helstrom limit. Also, we give a general lower bound on the minimum-error probability for discriminating quantum operations. Then we further analyze how this lower bound is attainable for ambiguous discrimination of mixed quantum states by presenting necessary and sufficient conditions related to it. Furthermore, with a restricted condition, we work out a upper bound on the minimum-error probability for ambiguous discrimination of mixed quantum states. Therefore, some sufficient conditions are obtained for the minimum-error probability attaining this bound. Finally, under the condition of the minimum-error probability attaining this bound, we compare the minimum-error probability for {\it ambiguously} discriminating arbitrary $m$ mixed quantum states with the optimal failure probability for {\it unambiguously} discriminating the same states.Comment: A further revised version, and some results have been adde