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In this paper we establish some new results similar to Lagrange's four-square theorem. For example, we prove that any integer $n>1$ can be written $w(5w+1)/2+x(5x+1)/2+y(5y+1)/2+z(5z+1)/2$ with $w,x,y,z\in\mathbb Z$. Here we state two general results: (1) If $a$ and $b$ are odd integers with $a>0$, $b>-a$ and $\gcd(a,b)=1$, then all sufficient large integers can be written as $$\frac{w(aw+b)}2+\frac{x(ax+b)}2+\frac{y(ay+b)}2+\frac{z(az+b)}2$$ with $w,x,y,z$ nonnegative integers. (2) If $a$ and $b$ are integers with $a>0$, $b>-a$, $2\mid a$ and $\gcd(a,b)=1$, then all sufficient large integers can be written as $$w(aw+b)+x(ax+b)+y(ay+b)+z(az+b)$$ with $w,x,y,z$ nonnegative integers., Comment: 11 pages

In this talk, we present a new interpretation for the recently observed $\Xi(1820)$ resonance. We recall that the chiral unitary approach for the interaction of pseudoscalar mesons with the baryons of the decuplet predicts two states for the $\Xi(1820)$ resonance, one with a narrow width and the other one with a large width. We contrast this fact with the recent BESIII measurement of the $K^- \Lambda$ mass distribution in the $\psi(3686)$ decay to $K^- \Lambda \bar\Xi^+ $, which demands a width much larger than the average of the PDG, and show how the consideration of the two $\Xi(1820)$ states provides a natural explanation to this apparent contradiction., Comment: Proceeding of the QNP 2024 Conference. 6 pages, 3 figures, 1 table. arXiv admin note: substantial text overlap with arXiv:2309.03618

Among the key enabling 6G techniques, multiple-input multiple-output (MIMO) and non-orthogonal multiple-access (NOMA) play an important role in enhancing the spectral efficiency of the wireless communication systems. To further extend the coverage and the capacity, the simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) has recently emerged out as a cost-effective technology. To exploit the benefit of STAR-RIS in the MIMO-NOMA systems, in this paper, we investigate the analysis and optimization of the downlink dual-user MIMO-NOMA systems assisted by multiple STAR-RISs under the generalized singular value decomposition (GSVD) precoding scheme, in which the channel is assumed to be Rician faded with the Weichselberger's correlation structure. To analyze the asymptotic information rate of the users, we apply the operator-valued free probability theory to obtain the Cauchy transform of the generalized singular values (GSVs) of the MIMO-NOMA channel matrices, which can be used to obtain the information rate by Riemann integral. Then, considering the special case when the channels between the BS and the STAR-RISs are deterministic, we obtain the closed-form expression for the asymptotic information rates of the users. Furthermore, a projected gradient ascent method (PGAM) is proposed with the derived closed-form expression to design the STAR-RISs thereby maximizing the sum rate based on the statistical channel state information. The numerical results show the accuracy of the asymptotic expression compared to the Monte Carlo simulations and the superiority of the proposed PGAM algorithm.

Different from the previous treatment in a two-body framework, we introduce the dimeson distribution amplitudes (DAs) to describe the strong dynamics between the S-wave resonances $a_0(980, 1450)$ and the $K\bar K (\pi\eta)$ pair, where the Gegenbauer coefficient required is determined from the experimental data on the time-like form factors involved. The branching ratios and direct CP asymmetries of the decays $B \to a^{(\prime)}_0 h \to K\bar K(\pi\eta) h$, with $a_0=a_0(980)$, $a^{\prime}_0=a_0(1450)$ and $h$ referring to a pion or a kaon, are then calculated in the perturbative QCD (PQCD) approach. We find that the branching ratios of the corresponding quasi-two-body decays $B\to a^{(\prime)}_0 K$ obtained with the narrow width approximation are closer to those predicted in the QCD factorization (QCDF) approach compared to the previous PQCD calculations, no matter a three-body or a two-body framework is assumed. Furthermore, all our predictions for these $B\to a^{(\prime)}_0 K$ decays are below the current experimental upper limits except for those of decays $B^0\to a^{(\prime)-}_0K^+$, which are (slightly) larger than the upper limits. Under the narrow width approximation, the branching ratios of the decays $B^+\to a^{(\prime)+}_0\pi^0$, $B^0\to a^{(\prime)+}_0\pi^-$ and $B^0\to a^{(\prime)0}_0\pi^0$ are comparable to or agree well with the previous PQCD and the QCDF calculations. While for the decays $B^+\to a^{(\prime)0}_0\pi^+$ and $B^0\to a^{(\prime)-}_0\pi^+$, their branching ratios are predicted to be unexpectedly large, for example, the obtained branching ratio of decay $B^+\to a^0_0\pi^+$ is even higher than the current experimental upper limit., Comment: 22 pages, 2 figures

An integer-valued polynomial $P(x,y,z)$ is said to be universal (over $\mathbb Z$) if each nonnegative integer can be written as $P(x,y,z)$ with $x,y,z\in\mathbb Z$. In this paper, we mainly introduce a new technique to determine the universality of some sums in the form $x(a_1x+a_2)/2+y(b_1y+b_2)/2+z(c_1z+c_2)/2$ (with $a_1-a_2,b_1-b_2,c_1-c_2$ all even) conjectured by Sun, using various identities of Ramanujan's theta functions. For example, we prove that $x(3x+1)+y(3y+2)+2z(3z+2)$ and $x(4x+r)+y(3y+1)/2+z(7z+1)/2\ (r=1,3)$ are universal., Comment: 22 pages, refined version with some remarks added

The semileptonic and nonleptonic decays of the b-flavor vector mesons $B^{*}_{u,d,s}$ and $B_{c}^{*}$ are investigated within the covariant light-front quark model (CLFQM). By calculating the form factors of the transitions $B_{u, d, s, c}^{*}\to P$ under the CLFQM, with $P$ denoting a pseudoscalar meson, i.e., $\pi, K, \eta_c(1S,2S), D_{(s)}, B_{(s)}$, we predict and discuss several physical observables, including the branching ratios, polarization fractions $f_{L}, f_{\|}$, and forward-backward asymmetries $A_{FB}$. The total widths of the single-photon radiative decay channels for these b-flavor vector mesons are estimated using their partial widths. In these considered decays, one can find that the semileptonic decays $B_{s}^{*0}\to D_{s}^{-}\ell^{\prime+}{\nu}_{\ell^\prime}$ and $B_{c}^{*+}\to B_{s}^{0}\ell^{\prime+}{\nu}_{\ell^\prime}, \eta_{c}\ell^{\prime+}{\nu}_{\ell^\prime}$, with $\ell^\prime$ being $e$ or $\tau$, and the nonleptonic channels $B_{c}^{*+}\to B^0_{s} \pi^{+}, B^0_{s} \rho^{+}$ have the largest branching ratios, which can reach up to the $10^{-7}$ order, and are most likely to be accessible at the future high-luminosity LHCb and Belle-II experiments., Comment: 30 pages, 4 figures,accepted for publication in Chin. Phys. C

In the present work, we construct the Lagrangians including three-meson, meson-diquark-antidiquark vertices, such that the diquark-antidiquark component as well as the molecular component are introduced within the effective field theory. With the obtained effective potentials projecting to spin 0, 1 and 2, we solve the Bethe-Salpeter equation with the on-shell approximation, and find that $X(4500)$ can be explained as the mixture of components $D_{s}^{*+}D_{s}^{*-}$, ${A}_{cq}\bar{A}_{cq}$ and ${A}_{cs}\bar{A}_{cs}$ with $I^G(J^{PC})=0^+(0^{++})$. In addition, another two resonances with quantum numbers $I^G(J^{PC})=0^+(1^{++})$ and $I^G(J^{PC})=0^+(2^{++})$ are predicted., Comment: 6 pages, 2 figures

Let $p=2n+1$ be an odd prime. In this paper, we mainly evaluate determinants involving $(\frac {j+k}p)\pm(\frac{j-k}p)$, where $(\frac{\cdot}p)$ denotes the Legendre symbol. When $p\equiv1\pmod4$, we determine the characteristic polynomials of the matrices $$\left[\left(\frac{j+k}p\right)+\left(\frac{j-k}p\right)\right]_{1\le j,k\le n}\ \ \text{and}\ \ \left[\left(\frac{j+k}p\right)-\left(\frac{j-k}p\right)\right]_{1\le j,k\le n},$$ and also prove that \begin{align*} &\ \left|x+\left(\frac{j+k}p\right)+\left(\frac{j-k}p\right)+\left(\frac jp\right)y+\left(\frac kp\right)z+\left(\frac{jk}p\right)w\right|_{1\le j,k\le n} \\=&\ (-p)^{(p-5)/4}\left(\left(\frac{p-1}2\right)^2wx-\left(\frac{p-1}2y-1\right)\left(\frac{p-1}2z-1\right)\right), \end{align*} which was previously conjectured by the second author., Comment: 16 pages. Add Theorem 1.3 and its proof

Let $\{A'_n\}$ be the Ap\'ery numbers given by $A'_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k.$ For any prime $p\equiv 3\pmod 4$ we show that $A'_{\frac{p-1}2}\equiv \frac{p^2}3\binom{\frac{p-3}2}{\frac{p-3}4}^{-2}\pmod {p^3}$. Let $\{t_n\}$ be given by $$t_0=1,\ t_1=5\quad\hbox{and}\quad t_{n+1}=(8n^2+12n+5)t_n-4n^2(2n+1)^2t_{n-1}\ (n\ge 1).$$ We also obtain the congruences for $t_p\pmod {p^3},\ t_{p-1}\pmod {p^2}$ and $t_{\frac{p-1}2}\pmod {p^2}$, where $p$ is an odd prime., Comment: 16 pages

Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. In this paper, we study the determinant $$\det\left[\left(\frac{j^2-k^2}p\right)+\left(\frac{jk}p\right)w\right]_{\delta\le j,k\le (p-1)/2}$$ with $\delta\in\{0,1\}$. For example, we prove that the determinant does not depend on $w$ if $p\equiv3\pmod4$ and $\delta=0$., Comment: 7 pages

Recently, using modular forms F. Beukers posed a unified method that can deal with a large number of supercongruences involving binomial coefficients and Ap\'ery-like numbers. In this paper, we use Beukers' method to prove some conjectures of the first author concerning the congruences for $$\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k^3}{m^k}, \ \sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{4k}{2k}}{m^k}, \ \sum_{k=0}^{p-1}\frac{\binom{2k}k\binom{3k}k\binom{6k}{3k}}{m^k}, \ \sum_{n=0}^{p-1}\frac{V_n}{m^n},\ \sum_{n=0}^{p-1}\frac{T_n}{m^n},\ \sum_{n=0}^{p-1}\frac{D_n}{m^n} $$ and $\sum_{n=0}^{p-1}(-1)^nA_n$ modulo $p^3$, where $p$ is an odd prime representable by some suitable binary quadratic form, $m$ is an integer not divisible by $p$, $V_n=\sum_{k=0}^n\binom{2k}k^2\binom{2n-2k}{n-k}^2$, $T_n=\sum_{k=0}^n\binom nk^2\binom{2k}n^2$, $D_n=\sum_{k=0}^n\binom nk^2\binom{2k}k\binom{2n-2k}{n-k}$ and $A_n$ is the Ap\'ery number given by $A_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2$., Comment: 59 pages

In this paper, we prove a conjecture of the second author by evaluating the determinant $$\det\left[x+\left(\frac{i-j}p\right)+\left(\frac ip\right)y+\left(\frac jp\right)z+\left(\frac{ij}p\right)w\right]_{0\le i,j\le(p-3)/2}$$ for any odd prime $p$, where $(\frac{\cdot}p)$ denotes the Legendre symbol. In particular, the determinant is equal to $x$ when $p\equiv 3\pmod4$., Comment: 16 pages. Make the main result more general

This work aims to tackle the labor-intensive and resource-consuming task of indoor positioning by proposing an efficient approach. The proposed approach involves the introduction of a semi-supervised learning (SSL) with a biased teacher (SSLB) algorithm, which effectively utilizes both labeled and unlabeled channel data. To reduce measurement expenses, unlabeled data is generated using an updated channel simulator (UCHS), and then weighted by adaptive confidence values to simplify the tuning of hyperparameters. Simulation results demonstrate that the proposed strategy achieves superior performance while minimizing measurement overhead and training expense compared to existing benchmarks, offering a valuable and practical solution for indoor positioning., Comment: ACCEPTED for presentation at 2024 IEEE Global Communications Conference

For any prime $p>3$ and rational $p$-integers $b,c$ with $c(b^3-27c)\not\equiv 0\pmod p$ let $V_p(x^2+bx+\frac cx)$ be the residue-counts of $x^2+bx+\frac cx$ modulo $p$ as $x$ runs over $1,2,\ldots,p-1$. In this paper, we reveal the connection between $V_p(x^2+bx+\frac cx)$ and the number of points on certain elliptic curve over the field $\Bbb F_p$., Comment: 15 pages

We report on the first measurement of the elemental fragmentation cross sections (EFCSs) of $^{29-33}\mathrm{Si}$ on a carbon target at $\sim$230~MeV/nucleon. The experimental data covering charge changes of $\Delta Z$ = 1-4 are reproduced well by the isospin-dependent quantum molecular dynamics (IQMD) coupled with the evaporation GEMINI (IQMD+GEMINI) model. We further explore the mechanisms underlying the single-proton removal reaction in this model framework. We conclude that the cross sections from direct proton knockout exhibit a overall weak dependence on the mass number of $\mathrm{Si}$ projectiles. The proton evaporation induced after the projectile excitation significantly affects the cross sections for neutron-deficient $\mathrm{Si}$ isotopes, while neutron evaporation plays a crucial role in the reactions of neutron-rich $\mathrm{Si}$ isotopes. It is presented that the relative magnitude of one-proton and one-neutron separation energies is an essential factor that influences evaporation processes., Comment: 7 pages, 4 figures

In this study, we explore the interactions between mesons and baryons in the open heavy sectors to identify potential triple-heavy molecular pentaquarks. We derive the meson-baryon interaction potentials using the vector meson exchange mechanism within the extended local hidden gauge formalism. The scattering amplitudes are computed by solving the coupled-channel Bethe-Salpeter equation, revealing several bound systems. By analyzing the poles of these amplitudes in the complex plane, we determine the masses and widths of these bound states. Additionally, we evaluate the couplings and compositeness of different channels within each bound system to assess their molecular characteristics. Our predictions include four $\Omega_{ccc}$-like states, four $\Omega_{bbb}$-like states, fourteen $\Omega_{bcc}$-like states, and ten $\Omega_{bbc}$-like states, which could be targets for future experimental investigations., Comment: 14 pages, 3 figures, 8 tables, accepted by Phys. Rev. D

In this paper, we study arithmetic properties of certain determinants involving powers of $i^2+cij+dj^2$, where $c$ and $d$ are integers. For example, for any odd integer $n>1$ with $(\frac dn)=-1$ we prove that $\det [ (\frac{i^2+cij+dj^2}{n})]_{0\le i,j\le n-1}$ is divisible by $\varphi(n)^2$, where $(\frac{\cdot}{n})$ is the Jacobi symbol and $\varphi$ is Euler's totient function. This confirms a previous conjecture of the second author., Comment: 13 pages

In this paper we investigate determinants whose entries are linear combinations of Legendre symbols. We deduce some new results in this direction; for example, we prove that for any prime $p\equiv3\pmod4$ we have $$\det\left[x+\left(\frac{j-k}p\right)+\left(\frac jp\right)-\left(\frac kp\right)\right]_{0\le j,k\le(p-1)/2}=4,$$ where $(\frac{\cdot}p)$ is the Legendre symbol. We also pose many conjectures for further research. For example, for any prime $p>3$ we conjecture that \begin{align*}&\ \det\left[\left(\frac{j+k}p\right)+\left(\frac{j-k}p\right)+\left(\frac{jk}p\right)\right]_{1\le j,k\le(p-1)/2} \\=&\ \begin{cases}(\frac 2p)p^{(p-5)/4}&\text{if}\ p\equiv1\pmod4, \\(-1)^{(h(-p)-1)/2}(1-(2-(\frac 2p))h(-p))p^{(p-3)/4}&\text{if}\ p\equiv3\pmod4, \end{cases}\end{align*} where $h(-p)$ is the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p})$., Comment: 24 pages. Refined version with typos corrected

In this work, we make a study of $T_{cc}^+$ state observed by the LHCb collaboration in 2021. In obtaining the effective potentials using the One-Boson-Exchange Potential Model we use an exponential form factor, and find that in the short and medium range, the contributions of the $\pi$, $\rho$ and $\omega$ exchanges are comparable while in the long range the pion-exchange contribution is dominant. Based on the assumption that $T_{cc}^+$ is a loosely bound state of $D^*D$, we focus on its three-body decay using the meson-exchange method. Considering that the difference between the thresholds of $D^{*+}D^0$ and $D^{*0}D^+$ is even larger than the binding energy of $T_{cc}^+$, the isospin-breaking effect is amplified by the small binding energy of $T_{cc}^+$. Explicitly including such an isospin-breaking effect we obtain, by solving the Schr\"{o}dinger equation, that the probability of the isoscalar component is about $91\%$ while that of the isovector component is around $9\%$ for $T_{cc}^+$. Using the experimental value of the mass of $T_{cc}^+$ as an input, we obtain the wave function of $T_{cc}^+$ and further obtain its width via the three-body hadronic as well as the radiative decays. The total width we obtain is in agreement with the experimental value of the LHCb measurement with a unitarised Breit-Wigner profile. Conversely, the current results support the conclusion that $T_{cc}^+$ is a hadronic molecule of $D^*D$., Comment: 10 pages, 2 figures

In this work, we investigate the form factors of the transitions $B_{(s)} \to D_{0}^{\ast}(2300),D_{s0}^{\ast}(2317),$ $ D_{s1}(2460) $ and $ D_{s1}(2536)$ in the covariant light-front quark model (CLFQM), where these final states are considered as P-wave excited charmed mesons. In order to obtain the form factors for the physical transition processes, we need to extend these form factors from the space-like region to the time-like region. The $q^{2}$-dependence for each transition form factor is also plotted. Then, combined with those form factors, the branching ratios of the two-body nonleptonic decays $B_{(s)}\to D^*_{(s)0}(2300,2317)M, D_{s1}(2460,2536)M$ with $M$ being a light pseudoscalar (vector) meson or a charmed meson are calculated by considering the QCD radiative corrections to the hadronic matrix elements with the QCD factorization approach. Most of our predictions are comparable to the results given by other theoretical approaches and the present available data., Comment: 30 pages, 4 figures

Let $p>3$ be a prime, and let $d\in\mathbb Z$ with $p\nmid d$. For the determinants $$S_m(d,p)=\det\left[(i^2+dj^2)^{m}\right]_{1\leqslant i,j \leqslant (p-1)/2}\ \ \left(\frac{p-1}2\leqslant m\leqslant p-1\right),$$ Sun recently determined $S_m(d,p)$ modulo $p$ when $m\in\{p-2,p-3\}$ and $(\frac {-d}p)=-1$. In this paper, we obtain $S_{p-2}(d,p)$ modulo $p$ in the remaining case $(\frac{-d}p)=1$, and determine the Legendre symbols $(\frac{S_{p-3}\,(d,p)}p)$ and $(\frac{S_{p-4}\,(d,p)}p)$ in some special cases., Comment: 14 pages

In this study, we reconsider the states $Z_b(10610)$ and $Z_b(10650)$ by investigating the presence of diquark-anti-diquark components as well as the hadronic molecule components in the framework of effective field theory. The different masses of pseudoscalar mesons such as $\pi^{0}$, $\eta_{8}$, and $\eta_{0}$, as well as vector mesons like $\rho^{0}$ and $\omega$ violate the OZI rule that is well depicted under the $[U(3)_L\otimes U(3)_R]_{global}\otimes [U(3)_V]_{local}$ symmetry. To account for the contribution of intermediate bosons of heavy masses within the OBE model, we introduce an exponential form factor instead of the commonly used monopole form factor in the past. By solving the coupled-channel Schr\"{o}dinger equation with the Gaussian expansion method, our numerical results indicate that the $Z_b(10610)$ and $Z_b(10650)$ states can be explained as hadronic molecules slightly mixing with diquark-anti-diquark states., Comment: 9 pages, 2 figures

In this paper we evaluate several determinants involving quadratic residues modulo primes. For example, for any prime $p>3$ with $p\equiv3\pmod4$ and $a,b\in\mathbb Z$ with $p\nmid ab$, we prove that $$\det\left[ 1+\tan\pi\frac{aj^2+bk^2}p \right]_{1\le j,k \le \frac{p-1}2} = \begin{cases}-2^{(p-1)/2}p^{(p-3)/4}&\text{if}\ (\frac{ab}p)=1, \\p^{(p-3)/4}&\text{if}\ (\frac{ab}p)=-1, \end{cases}$$ where $(\frac{\cdot}p)$ denotes the Legendre symbol. We also pose some conjectures for further research., Comment: 25 pages. Mainly add Theorem 1.1(i) and its proof

Using cyclotomic multiple zeta values of level $8$, we confirm and generalize several conjectural identities on infinite series with summands involving $\binom{2k}k8^k/(\binom{3k}k\binom{6k}{3k})$. For example, we prove that \[\sum_{k=0}^\infty\frac{(350k-17)\binom{2k}k8^k} {\binom{3k}k\binom{6k}{3k}}=15\sqrt2\,\pi+27\] and \[\sum_{k=1}^\infty\frac{\left\{(5k-1)\left[16\mathsf H_{2k-1}^{(2)}-3\mathsf H_{k-1}^{(2)}\right]-\frac{12(6k-1)}{(2k-1)^2}\right\}\binom{2k}k8^k} {k(2k-1)\binom{3k}k\binom{6k}{3k}}=\frac{\pi^3}{12\sqrt2},\] where $\mathsf H^{(2)}_m$ denotes the second-order harmonic number $\sum_{0

We perform polylogarithmic reductions for several classes of infinite sums motivated by Z.-W. Sun's related works in 2022--2023. For certain choices of parameters, these series can be expressed by cyclotomic multiple zeta values of levels $4$, $5$, $6$, $7$, $8$, $9$, $10$, and $12$. In particular, we obtain closed forms of the series $$\sum_{k=0}^\infty\frac{x_0^k}{(k+1)\binom{3k}k} \ \ \text{and}\ \ \sum_{k=1}^\infty\frac{x_0^k}{k^2\binom{3k}k}$$ for any $x_0\in(-27/4,27/4)$., Comment: 23 pages

A novel H3N3-2$_\sigma$ interpolation approximation for the Caputo fractional derivative of order $\alpha\in(1,2)$ is derived in this paper, which improves the popular L2C formula with (3-$\alpha$)-order accuracy. By an interpolation technique, the second-order accuracy of the truncation error is skillfully estimated. Based on this formula, a finite difference scheme with second-order accuracy both in time and in space is constructed for the initial-boundary value problem of the time fractional hyperbolic equation. It is well known that the coefficients' properties of discrete fractional derivatives are fundamental to the numerical stability of time fractional differential models. We prove the related properties of the coefficients of the H3N3-2$_\sigma$ approximate formula. With these properties, the numerical stability and convergence of the difference scheme are derived immediately by the energy method in the sense of $H^1$-norm. Considering the weak regularity of the solution to the problem at the starting time, a finite difference scheme on the graded meshes based on H3N3-2$_\sigma$ formula is also presented. The numerical simulations are performed to show the effectiveness of the derived finite difference schemes, in which the fast algorithms are employed to speed up the numerical computation.

In this work, we investigate the double-heavy molecular pentaquark states with the quark contents $ccss\bar{s}$, $bbss\bar{s}$, and $bcss\bar{s}$ by using the coupled channel approach. The extended local hidden gauge Lagrangians are used to obtain the meson-baryon interactions by exchanging the vector mesons. We predict some candidates for the molecular states with the quantum numbers $I(J^{P}) = 0(1/2^{-}, 3/2^{-}, 5/2^{-})$, whose binding energies are of the order of $20-30$ MeV and whose widths are all less than $8$ MeV. These predicted exotic double-heavy molecular pentaquark states may be accessible in future experiments such as LHCb., Comment: Minor corrections are made

In addition to the strong and electromagnetic decay modes, the $\psi(1S,2S)$ and $\eta_{c}(1S,2S)$ can also decay via the weak interaction. Such weak decays can be detected by the high-luminosity heavy-flavor experiments. At present, some of the semileptonic and nonleptonic $J/\Psi$ weak decays have been measured at BESIII. Researching for these charmonium weak decays to $D_{(s)}$ meson can provide a platform to check of the standard model (SM) and probe new physics (NP). So we investigate the semileptonic and nonleptonic weak decays of $\psi(1S,2S)$ and $\eta_{c}(1S,2S)$ to $D_{(s)}$ within the covariant light-front quark model (CLFQM). With form factors of the transitions $\psi(1S,2S)\to D_{(s)}$ and $\eta_{c}(1S,2S)\to D_{(s)}$ calculated under the CLFQM, we predict and discuss some physical observables, such as the branching ratios, the longitudinal polarizations $f_{L}$ and the forward-backward asymmetries $A_{FB}$. One can find that the Cabibbo-favored semi-leptonic decay channels $\psi(1S,2S)\to D_{s}^{-}\ell^{+}\nu_{\ell}$ with $\ell=e,\mu$ and the nonleptonic decay modes $\psi(1S,2S)\to D_{s}^{-}\rho^{+}$ have relatively large branching ratios of the order $\mathcal{O}(10^{-9})$, which are most likely to be accessible at the future high-luminosity experiments., Comment: 26 pages, 4 figures

We propose an exactly solvable lattice model, motivated by the significance of the extended Hubbard model ($t-U-V$ model) and inspired by the work of Hatsugai and Kohmoto. The ground state exhibits a diverse array of phases, including the charge-$4e$ condensed phase, the charge-$2e$ superconducting phase, the half-filled insulating phase, the quarter-filled insulating phase, the metallic phase, and an unconventional metallic phase. Among them, the unconventional metallic phase could be of particular significance, for the coexistence of electrons and pairs at zero energy. These findings are poised to advance our understanding and exploration of strongly correlated physics., Comment: 34pages, 15 figures

During 2022--2023 Z.-W. Sun posed many conjectures on infinite series with summands involving generalized harmonic numbers. Motivated by this, we deduce $31$ series identities involving harmonic numbers, three of which were previously conjectured by the second author. For example, we obtain that \[ \sum_{k=1}^{\infty} \frac{(-1)^k}{k^2{2k \choose k}{3k \choose k}} \big( \frac{7 k-2}{2 k-1} H_{k-1}^{(2)}-\frac{3}{4 k^2} \big)=\frac{\pi^4}{720}. \] and \[ \sum_{k=1}^\infty \frac{1}{k^2 {2k \choose k}^2} \left( \frac{30k-11}{k(2k-1)} (H_{2k-1}^{(3)} + 2 H_{k-1}^{(3)}) + \frac{27}{8k^4} \right) = 4 \zeta(3)^2, \] where $H_n^{(m)}$ denotes $\sum_{0

For a prime $p>3$ and $a\in \Bbb Z$ with $p\nmid a$ let $V_p(x^2+\frac ax)$ be the residue-counts of $x^2+\frac ax$ modulo $p$ as $x$ runs over $1,2,\ldots,p-1$. In this paper, we obtain an explicit formula for $V_p(x^2+\frac ax)$, which is concerned with cubic residues and binary quadratic forms., Comment: 8 pages

We recall that the chiral unitary approach for the interaction of pseudoscalar mesons with the baryons of the decuplet predicts two states for the $\Xi(1820)$ resonance, one with a narrow width and the other one with a large width. We contrast this fact with the recent BESIII measurement of the $K^- \Lambda$ mass distribution in the $\psi(3686)$ decay to $K^- \Lambda \bar\Xi^+ $, which demands a width much larger than the average of the PDG, and show how the consideration of the two $\Xi(1820)$ states provides a natural explanation to this apparent contradiction., Comment: 9 pages, 3 figures

In this work, we investigate the semileptonic decays of $B_{c}$ meson to $\eta_{c}(1S,2S,3S)$, $\psi(1S,2S,3S)$ and $X(3872)$ within the framework of covariant light-front quark model (CLFQM). We combine the helicity amplitudes via the corresponding form factors to obtain the branching ratios of the semileptonic decays $B_{c}\to \eta_{c}(1S, 2S, 3S)\ell\nu_{\ell}$, $B_{c}\to \psi(1S, 2S, 3S))\ell\nu_{\ell}$ and $B_{c}\to X(3872)\ell\nu_{\ell}$ with $\ell=e,\mu,\tau$. In view of the $R_{J/\Psi}$ anomaly released by the LHCb collaboration, it is necessary to calculate the ratios $R_X$ with $X=\psi(1S,2S,3S),\eta_c(1S,2S,3S),X(3872)$ systematically, which are helpful to check the lepton flavor universality (LFU). Furthermore, we also take into account another two physical observables, one is the longitudinal polarization fraction $f_{L}$ and the other is the forward-backward asymmetry $A_{FB}$, which can provide new clues to understand the $R_{J /\Psi}$ anomaly. Such theoretical predictions are necessary and interesting, which can be tested in the future LHCb experiments., Comment: 23 pages, 4 figures

In this work we study the quasi-two-body decays $B_c \to \ K^{*} h \to K \pi h (h = D, D_s, K, \pi, \eta, \eta')$ in the perturbative QCD (PQCD) approach. The two-meson distribution amplitudes (DAs) $\Phi^{\text{P-wave}}_{K\pi}$ are introduced to describe the final state interactions of the K \pi pair, which involve the time-like form factors F_{K\pi}(s) parameterized by the relativistic Breit-Wigner function and the Gegenbauer polynomials. We calculate the branching ratios for these quasi-two-body decays, from which one can obtain the branching raios for the corresponding two-body decays under the narrow width approximation relation. We find that $B^+_c\to K^{*+}D^0$ and $B^+_c\to K^{*0}D^+$ have the largest branching ratios, which can reach up to $10^{-6}$, while the branching ratios for other two-body decays are very small and only about $10^{-8}\sim10^{-7}$. As we expected that the branching ratios of the pure annihilation decays are usually small, while in our considered such type of decays, the channel $B_c^+ \to \bar K^{*0}K^{+}$ has the largest branching ratio, which is near $10^{-6}$. These results are consistent with the previously PQCD calculations obtained in the two-body framework, which can be tested by the future LHCb experiments. For the decays $B_c^+ \to K^{*+} D^{0}\to K^{0}\pi^+D^{0} , B_c^+ \to K^{*0}D^{+}\to K^{+}\pi^-D^{+}$ and $B_c^+ \to \bar K^{*0}D_s^{+}\to K^{-}\pi^+D_s^{+}$, we calculate their direct CP violations and find that $A_{CP}(B_c^+ \to K^{*+}D^{0}\to K^{0}\pi^+D^{0})=(-14.6_{-1.12}^{+9.19})\%$ is the largest one, which is possible measured by the present LHCb experiments. For the pure annihilation type decays, there is no CP violations because only the tree operators are involved. Furthermore, we also give the differential distributions of the branching ratios and the direct CP violations for the decays $B_c\to K^* D_{(s)}\to K \pi D_{(s)}$., Comment: 15 pages, 3 figures, 2 tables

In this paper, we deduce several new identities on infinite series with denominators of summands containing both binomial coefficients and linear parts. For example, we evaluate the sums $$\sum_{k=1}^\infty\frac{x_0^k}{(2k-1)\binom{3k}k},\ \sum_{k=0}^\infty\frac{x_0^k}{(3k+1)\binom{3k}k} \ \text{and}\ \sum_{k=0}^\infty\frac{x_0^k}{(3k+2)\binom{3k}k}$$ for any $x_0\in(-27/4,27/4)$. For any $1

With the chiral unitary approach, we evaluate the hidden strange $B_{c}$-like molecular states of $b\bar{c}s\bar{s}$ systems $\bar{B}_{s}\bar{D}_{s}$, $\bar{B}_{s}^{*}\bar{D}_{s}$, $\bar{B}_{s}\bar{D}_{s}^{*}$, and $\bar{B}_{s}^{*}\bar{D}_{s}^{*}$ coupled to the non-strange channels. The $S$-wave scattering amplitudes are calculated based on the vector meson exchange, four pseudoscalar mesons contact interactions, and four vector mesons contact interactions obtained from the extended local hidden gauge approach. We find six states below the threshold of the most relevant channel. The binding energies of these states are around $1-10$ MeV and the widths are around $0.2-0.7$ MeV. Our research is a supplement to the mass spectra of $B_{c}$-like states, which may be useful for the experimental search in the future., Comment: 15 pages, 8 figures

Charge-changing cross section ($\sigma_{\text{cc}}$) measurements via the transmission method have made important progress recently aiming to determine the charge radii of exotic nuclei. In this work, we report a new $\sigma_{\text{cc}}$ measurement of 304(9) MeV/nucleon $^{28}$Si on carbon at the second Radioactive Ion Beam Line in Lanzhou (RIBLL2) and describe the data analysis procedure in detail. This procedure is essential to evaluate the systematic uncertainty in the transmission method. The determined $\sigma_{\mathrm{cc}}$ of 1125(11) mb is found to be consistent with the existing data at similar energies. The present work will serve as a reference in the $\sigma_{\text{cc}}$ determinations at RIBLL2., Comment: 9 pages, 13 figures, to be published in Chinese Physics C