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In this work we discuss how femtoscopic analysis can shed light on the nature of the two lightest axial charmed mesons, denominated as $D_1(2430)$ and $D_1(2420)$, whose masses are similar but widths are different. Their properties are reasonably described taking into account meson-meson coupled channel dynamics and a bare quark-model pole constituting the lowest-order amplitudes. Two different bare quark-model states are used in order to accommodate the different scattering lengths coming from the lattice QCD calculations for the $D\pi$ and $D^*\pi$ systems and the data from the ALICE Collaboration on the $D\pi$ system. The amplitudes are employed as inputs to determine the correlation functions for the $D^{*}\pi$ and $D\rho$ channels and identify the signatures associated with the lowest-lying axial charmed mesons., Comment: 6 pages; 2 figures; Proceedings of The 10th International Conference on Quarks and Nuclear Physics, (QNP 2024) Jul 8-12, 2024, Barcelona, Spain

A generalized numerical semigroup is a submonoid $S$ of $\mathbb{N}^d$ with finite complement in it. We characterize isomorphisms between these monoids in terms of permutation of coordinates. Considering the equivalence relation that identifies the monoids obtained by the action of a permutation and establishing a criterion to select a representative from each equivalence class, we define some procedures for generating the set of all generalized numerical semigroups of given genus up to isomorphism. Finally, we present computational data and explore properties related to the number of generalized numerical semigroups of a given genus up to isomorphism., Comment: 18 pages

In this work we estimate the $N_{\psi(2S)} / N_{J/\psi}$ yield ratio in heavy-ion collisions, considering the interactions of the $\psi (2S) $ and $J/\psi$ states with light mesons in the hadron gas formed at the late stages of these collisions. Starting from the appropriate effective Lagrangians, we first compute the thermally-averaged cross sections for the production and absorption of the mentioned states, and then use them as input in the rate equations to determine the time evolution of $N_{\psi(2S)}$, $N_{J/\psi}$ and $N_{\psi(2S)} / N_{J/\psi}$. The main conclusion of our study is that the $\psi (2S) $ and $J/\psi$ multiplicities do not change much in the hadron gas phase and that the $\psi (2S)$ is more absorbed than the $J/\psi$. The obtained final ratio is in qualitative agreement with experimental data., Comment: 8 pages, 6 figures

In this article we investigate the shellability of the flag simplicial complexes attached to non-simple and thin polyominoes. As a consequence, we obtain the Cohen-Macaulayness and a combinatorial interepetation of the $h$-polynomial of the related coordinate rings., Comment: 18 pages, 13 figures. Every comment is very welcome!

We study low-energy nuclear physics at the future Electron-Ion Collider (EIC) at Brookhaven. By comparing the standard theory of electron-nucleus scattering with the equivalent photon method applied to Ultraperipheral Collisions (UPC) at the Large Hadron Collider (LHC) at CERN, we extract valuable insights into both processes. In the limit of extremely high beam energies and small energy transfers, very transparent equations emerge. We apply these equations to analyze nuclear fragmentation in UPCs at the LHC and eA scattering at the EIC, demonstrating that the EIC could facilitate unique photonuclear physics studies., Comment: typo corrected, figure 2 replaced with correct one

In this article, we study the squarefree powers of facet ideals associated with simplicial trees. Specifically, we examine the linearity of their minimal free resolution and their regularity. Additionally, we investigate when the first syzygy module of squarefree powers of a simplicial tree is generated by linear relations. Finally, we provide a combinatorial formula for the regularity of the squarefree powers of $t$-path ideals of path graphs., Comment: Any comments are welcome

In this work we study the multiplicity distributions (MDs) of charged particles within jets in proton-proton collisions, which were measured by the ATLAS collaboration in 2011, 2016 and 2019. The first data set refers to jets with lower transverse momenta ($4 < p_T < 40$ GeV ) whereas the other two refer to higher $p_T$ jets ($0.1 < p_T < 2.5$ TeV). We find that the first set shows no sign of KNO scaling whereas the higher $p_T$ sets gradually approach the scaling limit. In the first set the mean multiplicity as a function of $p_T$ can be well described by expressions derived from QCD with different approximation schemes. For higher ($> 500$ GeV) values of $p_T$ these expressions significantly overshoot the data. We show that the behavior of the MDs can be well represented by a Sub-Poisson distribution with energy dependent parameters. In the range $40 < p_T < 100$ GeV there is a transition from sub to super poissonian behavior and the MD evolves to a geometric distribution, which shows KNO scaling. In this way we fit the MDs in all transverse momentum intervals with one single expression. We discuss the implications of this phenomenological finding., Comment: 8 pages, 4 figures, discussion section expanded, version to appear in Phys. Rev. D

One of the hot topics in hadron physics is the study of the new exotic charmonium states and the determination of their internal structure. Another important topic is the search for effects of the magnetic field created in high energy nuclear collisions. In this note we show that we can use ultra-peripheral collisions to address both issues. We compute the cross section for the production of the $D^+ D^-$ molecular bound state in $\gamma-\gamma$ collisions. We also show how the magnetic field of the projectile can induce pion production in the target. Both processes have sizeable cross sections and their measurement would be very useful in the study of the topics mentioned above., Comment: 7 pages, 4 figures, Presented at UPC2023: International Workshop on the Physics of Ultra-Peripheral Collisions, Playa del Carmen, Mexico, 10-15.12.2023

In this work we present a calculation of exotic charmonium production in ultra-peripheral collisions, in which the exotic state is explicitly treated as a meson molecule. Our formalism is general but we focus on the lightest possible exotic charmonium state: a $D^+ D^-$ molecular bound state. It was proposed some time ago and it has been object of experimental searches. Here we study the production of the open charm pair in the process $\gamma \gamma \to D^+ D^-$. Then we use a prescription to project the free pair $ |D^+ D^- \rangle$ onto a bound state at the amplitude level and compute the cross section of the process $\gamma \gamma \to B$ (where $B$ is the bound state). Finally, we convolute this last cross section with the equivalent photon distributions coming from the projectile and target in an ultra-peripheral collision and find the $A A \to A A B$ cross section, which, for $Pb-Pb$ collisions at $\sqrt{s_{NN}} = 5.02$ TeV, is of the order of $3 \, \mu \mbox{b}$., Comment: 11 pages, 5 figures

Let $\mathcal{P}$ be a collection of cells and $I_\mathcal{P}$ be the associated ideal of inner $2$-minors as defined by A. A. Qureshi in 2012. In this paper, we provide a description of the package $\texttt{PolyominoIdeals}$ for the computer algebra software $\texttt{Macaulay2}$. More precisely, this package provides some functions that allow to define the ideal $I_{\mathcal{P}}$ in $\texttt{Macaulay2}$ and to compute its algebraic invariants or verifying its algebraic properties. We explain the usage of these functions and also give some examples. Refer to the tool provided in $\textit{https://rizwan.phd/polyo/}$ to get easier the desired input data for $\texttt{Macaulay2}$., Comment: In this draft we explain the usage of the package $\texttt{PolyominoIdeals}$ for $\texttt{Macaulay2}$ and we give some examples. Look at $\textit{https://rizwan.phd/polyo/}$ to get the input data for $\texttt{Macaulay2}$. In this initial phase, the aim is to provide some basic functions to study the polyomino ideals. Anyway the package is in progress and every contribution is very welcome as well!

In this work we evaluate the $X(3872)$ to $ \psi (2S) $ yield ratio ($N_X/N_{\psi(2S)}$) in Pb Pb collisions, taking into account the interactions of the $\psi (2S) $ and $ X(3872)$ states with light mesons in the hadron gas formed at the late stages of these collisions. We employ an effective Lagrangian approach to estimate the thermally-averaged cross sections for the production and absorption of the $\psi(2S)$ and use them in the rate equation to determine the time evolution of $N_{\psi(2S)}$. The multiplicity of these states at the end of mixed phase is obtained from the coalescence model. The multiplicity of $X(3872)$, treated as a bound state of $(D\bar D^{*} + c.c.)$ and also as a compact tetraquark, was already calculated in previous works. Knowing these yields, we derive predictions for the ratio ($N_X/N_{\psi(2S)}$) as a function of the centrality, of the center-of-mass energy and of the charged hadron multiplicity measured at midrapidity $[dN_{ch}/d\eta \,(\eta<0.5)]$. Finally, we make predictions for this ratio in Pb Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV to be measured by the ALICE Collaboration in the Run 3., Comment: 10 pages, 10 figures

We present a coupled channel treatment of meson-meson dynamics, for systems with spin-parity $1^+$, and determine the corresponding amplitudes by solving the Bethe-Salpeter equations, which lead to the generation of two axial resonances when mesons are considered as the degrees of freedom in the model. One of them is narrow and has properties in good agreement with those of $D_1(2420)$. The other pole is wider, but its real and imaginary parts do not match well with the mass and width of $D_1(2430)$. The situation improves when a bare quark-model state is included, indicating that the dynamics at the quark level as well as among hadrons can describe the two states simultaneously. Further, we discuss that there exists a divergence in the value of the $D^*\pi$ scattering length determined through data coming from lattice QCD calculations and from heavy ion collisions. Such different values can be accommodated in the model by making small changes in the parameters while producing two poles having properties compatible with the two lightest $D_1$ states. With these results, we proceed to calculate the correlation function for the $D^{*+(0)}\pi^{0(+)}$ system for different sizes of the source. We discuss which scenarios can be useful to shed some light on the issue., Comment: Version accepted for publication

We investigate the interactions of the charged exotic state $Z_c(3900)$ in a hadronic medium composed of light mesons. We study processes such as $Z_c \pi \to D\bar{D}$, $Z_c \pi \to D^*\bar{D}^*$, $Z_c \pi \to D\bar{D}^*$ and the inverse ones. Using effective Lagrangians and form factors calculated with QCD sum rules (treating the $Z_c(3900)$ as a tetraquark) we estimate the vacuum and thermally-averaged cross-sections of these reactions. We find that the $Z_c(3900)$ has relatively large interaction cross sections with the constituent particles of the hadronic medium. After that, we use the production and suppression cross sections in a rate equation to estimate the time evolution of the $Z_c$ multiplicity. We include the $Z_c$ decay and regeneration terms. The coalescence model is employed to compute the initial $Z_c$ multiplicity for the compact tetraquark configuration. Our results indicate that the combined effects of hadronic interactions, hydrodynamical expansion, decay and regeneration affect the final yield, which is bigger than the initial value. Besides, the dependence of the $Z_c$ final yield with the centrality, center-of-mass energy and the charged hadron multiplicity measured at midrapidity $[dN_{ch}/d\eta \,(\eta<0.5)]$ is also investigated., Comment: 9 pages; 8 figures

In a previous work we computed the thermally-averaged cross sections for the production and absorption of the $\chi_{c1}(4274)$ state in the hot hadron gas formed in heavy ion collisions. In the present work we estimate the final yield of this exotic state in these collisions. We use the coalescence model to fix the initial multiplicities. The state is is treated as a $P-$wave bound state of $D_s\bar D_{s0}$ and also as a compact tetraquark. The Bjorken picture is used to model the hydrodynamic expansion and cooling. Then, the kinetic equation is solved to evaluate the time evolution of the $\chi_{c1}(4274)$ yield during the hot hadron gas phase. Since the $\chi_{c1}(4274)$ decay width is large it might decay inside the hadron gas. Therefore we also include the $chi_{c1}(4274)$ decay and regeneration terms by means of an effective coupling, estimated from the available data. The combined effects of hadronic interactions and the $\chi_{c1}(4274)$ decay have a strong impact on the final yield. Also, predictions of the $\chi_{c1}(4274)$ multiplicity as a function of centrality and of the charged hadron multiplicity (measured at midrapidity) are presented. Finally, we calculate the yield of a proposed $P-$wave molecular state of $D_s \bar{D_{s0}}$, $Y^{\prime}(4274)$, characterized by a smaller width and smaller coupling constant obtained from the Weinberg compositeness condition., Comment: 7 pages, 5 figures

We investigate the algebraic properties of the coordinate ring of grid polyominoes. This class of non-simple and thin polyominoes was introduced by Mascia, Rinaldo, and Romeo, but not much is known about their polyomino ideals, $I_{\mathcal{P}}$, and their related coordinate ring, $K[\mathcal{P}]$. We give a formula for the Krull dimension of $K[\mathcal{P}]$ in terms of the combinatorics of the polyomino and we prove that $I_{\mathcal{P}}$ is of K\"onig type if and only if the polyomino has exactly one hole. In addition, we study a Conjecture of Rinaldo and Romeo which characterizes the thin polyominoes, and we confirm it for grid polyominoes. Namely, in the main result of this article, we prove that the $h$-polynomial of $K[\mathcal{P}]$ is equal to the rook polynomial of $\mathcal{P}$, and, as a consequence, the Castelnuovo-Mumford regularity of $K[\mathcal{P}]$ is equal to $r(\mathcal{P})$, the maximum number of rooks that can be placed in $\mathcal{P}$ in non-attacking positions. Our method to prove this result is based on the theory of simplicial complexes. We provide a suitable shelling order for $\Delta_{\mathcal{P}}$, the simplicial complex attached to $\mathcal{P}$ in relation to a generalized step of a facet, and we show that there is a one-to-one correspondence between the facets of $\Delta_{\mathcal{P}}$ with $k$ generalized steps and the $k$-rook configurations in $\mathcal{P}$., Comment: 28 pages, 42 figures. Comments are welcome!