The Quantum Superalgebra {\mathfrak{osp}_{q}(1|2)} and a q-Generalization of the Bannai–Ito Polynomials

The Racah problem for the quantum superalgebra \({\mathfrak{osp}_{q}(1|2)}\) is considered. The intermediate Casimir operators are shown to realize a q-deformation of the Bannai–Ito algebra. The Racah coefficients of \({\mathfrak{osp}_q(1|2)}\) are calculated explicitly in terms of basic orthogonal polynomials that q-generalize the Bannai–Ito polynomials. The relation between these q-deformed Bannai–Ito polynomials and the q-Racah/Askey–Wilson polynomials is discussed.     

An embedding of the Bannai–Ito algebra in $$\mathscr {U}(\mathfrak {osp}(1,2))$$ and $$-1$$-1 polynomials

An embedding of the Bannai–Ito algebra in the universal enveloping algebra of \(\mathfrak {osp}(1,2)\) is provided. A connection with the characterization of the little \(-1\) Jacobi polynomials is found in the holomorphic realization of \(\mathfrak {osp}(1,2)\). An integral expression for the Bannai–Ito polynomials is derived as a corollary.     

Decay widths of the spin-2 partners of the <Emphasis Type="Italic">X</Emphasis>(3872)

We consider the X(3872) resonance as a \(J^\mathrm{{PC}}=1^{++}\) \(D\bar{D}^*\) hadronic molecule. According to heavy quark spin symmetry, there will exist a partner with quantum numbers \(2^{++}\), \(X_{2}\), which would be a \(D^*\bar{D}^*\) loosely bound state. The \(X_{2}\) is expected to decay dominantly into \(D\bar{D}\), \(D\bar{D}^*\) and \(\bar{D} D^*\) in d-wave. In this work, we calculate the decay widths of the \(X_{2}\) resonance into the above channels, as well as those of its bottom partner, \(X_{b2}\), the mass of which comes from assuming heavy flavor symmetry for the contact terms. We find partial widths of the \(X_{2}\) and \(X_{b2}\) of the order of a few MeV. Finally, we also study the radiative \(X_2\rightarrow D\bar{D}^{*}\gamma \) and \(X_{b2} \rightarrow \bar{B} B^{*}\gamma \) decays. These decay modes are more sensitive to the long-distance structure of the resonances and to the \(D\bar{D}^{*}\) or \(B\bar{B}^{*}\) final state interaction.     

Exotic tetraquark states with the $$qq\bar{Q}\bar{Q}$$ configuration

In this work, we study systematically the mass splittings of the \(qq\bar{Q}\bar{Q}\) (\(q=u\), d, s and \(Q=c\), b) tetraquark states with the color-magnetic interaction by considering color mixing effects and estimate roughly their masses. We find that the color mixing effect is relatively important for the \(J^P=0^+\) states and possible stable tetraquarks exist in the \(nn\bar{Q}\bar{Q}\) (\(n=u\), d) and \(ns\bar{Q}\bar{Q}\) systems either with \(J=0\) or with \(J=1\). Possible decay patterns of the tetraquarks are briefly discussed.     

Leptoquark mechanism of neutrino masses within the grand unification framework

We demonstrate the viability of the one-loop neutrino mass mechanism within the framework of grand unification when the loop particles comprise scalar leptoquarks (LQs) and quarks of the matching electric charge. This mechanism can be implemented in both supersymmetric and non-supersymmetric models and requires the presence of at least one LQ pair. The appropriate pairs for the neutrino mass generation via the up-type and down-type quark loops are \(S_3\)–\(R_2\) and \(S_{1,\,3}\)–\(\tilde{R}_2\), respectively. We consider two distinct regimes for the LQ masses in our analysis. The first regime calls for very heavy LQs in the loop. It can be naturally realized with the \(S_{1,\,3}\)–\(\tilde{R}_2\) scenarios when the LQ masses are roughly between \(10^{12}\) and \(5 \times 10^{13}\) GeV. These lower and upper bounds originate from experimental limits on partial proton decay lifetimes and perturbativity constraints, respectively. Second regime corresponds to the collider accessible LQs in the neutrino mass loop. That option is viable for the \(S_3\)–\(\tilde{R}_2\) scenario in the models of unification that we discuss. If one furthermore assumes the presence of the type II see-saw mechanism there is an additional contribution from the \(S_3\)–\(R_2\) scenario that needs to be taken into account beside the type II see-saw contribution itself. We provide a complete list of renormalizable operators that yield necessary mixing of all aforementioned LQ pairs using the language of SU(5). We furthermore discuss several possible embeddings of this mechanism in SU(5) and SO(10) gauge groups.     

Analysis of $$D_s D K^*$$ and $$D_s D^{*} K^*$$DsD∗K∗ vertices and branching ratio of $$B^+\rightarrow {K^*}^0 \pi ^+$$B+→K∗0π+

In this paper, the strong form factors and coupling constants of \(D_sDK^*\) and \(D_sD^*K^*\) vertices are investigated within the three-point QCD sum rules method with and without the \(SU_{f}(3)\) symmetry. In this calculation, the contributions of the quark–quark, quark–gluon, and gluon–gluon condensate corrections are considered. As an example of specific application of these coupling constants, the branching ratio of the hadronic decay \(B^+\rightarrow {K^*}^0 \pi ^+\) is analyzed based on the one-particle-exchange which is one of the phenomenological models. In this model, B decays into a \(D_s D^*\) intermediate state, and then these two particles exchange a \(D (D^*)\) producing the final \(K^*\) and \(\pi \) mesons. In order to compute the effect of these interactions, the \(D_s D K^*\) and \(D_s D^* K^*\) form factors are needed.     

Heuristic Relative Entropy Principles with Complex Measures: Large-Degree Asymptotics of a Family of Multi-variate Normal Random Polynomials

Let \(z\in \mathbb {C}\), let \(\sigma ^2>0\) be a variance, and for \(N\in \mathbb {N}\) define the integrals

$$\begin{aligned} E_N^{}(z;\sigma ) := \left\{ \begin{array}{ll} {\frac{1}{\sigma }} \!\!\!\displaystyle \int _{\mathbb {R}}\! (x^2+z^2) \frac{e^{-\frac{1}{2\sigma ^2} x^2}}{\sqrt{2\pi }}dx&{}\quad \text{ if }\, N=1,\\ {\frac{1}{\sigma }} \!\!\!\displaystyle \int _{\mathbb {R}^N}\! \prod \prod \limits _{1\le k<l\le N}\!\! e^{-\frac{1}{2N}(1-\sigma ^{-2}) (x_k-x_l)^2} \prod _{1\le n\le N}\!\!\!\!(x_n^2+z^2) \frac{e^{-\frac{1}{2\sigma ^2} x_n^2}}{\sqrt{2\pi }}dx_n &{}\quad \text{ if }\, N>1. \end{array}\right. \!\!\! \end{aligned}$$

These are expected values of the polynomials \(P_N^{}(z)=\prod _{1\le n\le N}(X_n^2+z^2)\) whose 2N zeros \(\{\pm i X_k\}^{}_{k=1,\ldots ,N}\) are generated by N identically distributed multi-variate mean-zero normal random variables \(\{X_k\}^{N}_{k=1}\) with co-variance \(\mathrm{{Cov}}_N^{}(X_k,X_l)=(1+\frac{\sigma ^2-1}{N})\delta _{k,l}+\frac{\sigma ^2-1}{N}(1-\delta _{k,l})\). The \(E_N^{}(z;\sigma )\) are polynomials in \(z^2\), explicitly computable for arbitrary N, yet a list of the first three \(E_N^{}(z;\sigma )\) shows that the expressions become unwieldy already for moderate N—unless \(\sigma = 1\), in which case \(E_N^{}(z;1) = (1+z^2)^N\) for all \(z\in \mathbb {C}\) and \(N\in \mathbb {N}\). (Incidentally, commonly available computer algebra evaluates the integrals \(E_N^{}(z;\sigma )\) only for N up to a dozen, due to memory constraints). Asymptotic evaluations are needed for the large-N regime. For general complex z these have traditionally been limited to analytic expansion techniques; several rigorous results are proved for complex z near 0. Yet if \(z\in \mathbb {R}\) one can also compute this “infinite-degree” limit with the help of the familiar relative entropy principle for probability measures; a rigorous proof of this fact is supplied. Computer algebra-generated evidence is presented in support of a conjecture that a generalization of the relative entropy principle to signed or complex measures governs the \(N\rightarrow \infty \) asymptotics of the regime \(iz\in \mathbb {R}\). Potential generalizations, in particular to point vortex ensembles and the prescribed Gauss curvature problem, and to random matrix ensembles, are emphasized.

     

Kaonic production of $ \Lambda$(1405) off deuteron target in chiral dynamics

The K^--induced production of \( \Lambda\)(1405) is investigated in K ^- d \( \rightarrow\) \( \pi\) \( \Sigma\) n reactions based on coupled-channels chiral dynamics, in order to discuss the resonance position of the \( \Lambda\)(1405) in the \( \bar{{K}}\) N channel. We find that the K ^- d \( \rightarrow\) \( \Lambda\)(1405)n process favors the production of \( \Lambda\)(1405) initiated by the \( \bar{{K}}\) N channel. The present approach indicates that the \( \Lambda\)(1405) -resonance position is 1420MeV rather than 1405MeV in the \( \pi\) \( \Sigma\) invariant-mass spectra of K ^- d \( \rightarrow\) \( \pi\) \( \Sigma\) n reactions. This is consistent with an observed spectrum of the K ^- d \( \rightarrow\) \( \pi^{{+}}_{}\) \( \Sigma^{{-}}_{}\) n with 686-844MeV/c incident K^- by bubble chamber experiments done in the 70s. Our model also reproduces the measured \( \Lambda\)(1405) production cross-section.     

The Planar Ising Model and Total Positivity

A matrix is called totally positive (resp. totally nonnegative) if all its minors are positive (resp. nonnegative). Consider the Ising model with free boundary conditions and no external field on a planar graph G. Let \(a_1,\dots ,a_k,b_k,\dots ,b_1\) be vertices placed in a counterclockwise order on the outer face of G. We show that the \(k\times k\) matrix of the two-point spin correlation functions

$$\begin{aligned} M_{i,j} = \langle \sigma _{a_i} \sigma _{b_j} \rangle \end{aligned}$$

is totally nonnegative. Moreover, \(\det M > 0\) if and only if there exist k pairwise vertex-disjoint paths that connect \(a_i\) with \(b_i\). We also compute the scaling limit at criticality of the probability that there are k parallel and disjoint connections between \(a_i\) and \(b_i\) in the double random current model. Our results are based on a new distributional relation between double random currents and random alternating flows of Talaska [37].

     

Superintegrability in Two Dimensions and the Racah–Wilson Algebra

The analysis of the most general second-order superintegrable system in two dimensions: the generic 3-parameter model on the 2-sphere is cast in the framework of the Racah problem for the \({\mathfrak{su}(1,1)}\) algebra. The Hamiltonian of the 3-parameter system and the generators of its quadratic symmetry algebra are seen to correspond to the total and intermediate Casimir operators of the combination of three \({\mathfrak{su}(1,1)}\) algebras, respectively. The construction makes explicit the isomorphism between the Racah–Wilson algebra, which is the fundamental algebraic structure behind the Racah problem for \({\mathfrak{su}(1, 1)}\) , and the invariance algebra of the generic 3-parameter system. It also provides an explanation for the occurrence of the Racah polynomials as overlap coefficients in this context. The irreducible representations of the Racah–Wilson algebra are reviewed as well as their connection with the Askey scheme of classical orthogonal polynomials.     

New feature of low $$p_{T}$$ charm quark hadronization in pp collisions at $$\sqrt{s}=7$$s=7 TeV

Treating the light-flavor constituent quarks and antiquarks whose momentum information is extracted from the data of soft light-flavor hadrons in pp collisions at \(\sqrt{s}=7\) TeV as the underlying source of chromatically neutralizing the charm quarks of low transverse momenta (\(p_{T}\)), we show that the experimental data of \(p_{T}\) spectra of single-charm hadrons \(D^{0,+}\), \(D^{*+}\) \(D_{s}^{+}\), \(\varLambda _{c}^{+}\) and \(\varXi _{c}^{0}\) at mid-rapidity in the low \(p_{T}\) range (\(2\lesssim p_{T}\lesssim 7\) GeV/c) in pp collisions at \(\sqrt{s}=7\) TeV can be well understood by the equal-velocity combination of perturbatively created charm quarks and those light-flavor constituent quarks and antiquarks. This suggests a possible new scenario of low \(p_{T}\) charm quark hadronization, in contrast to the traditional fragmentation mechanism, in pp collisions at LHC energies. This is also another support for the exhibition of the soft constituent quark degrees of freedom for the small parton system created in pp collisions at LHC energies.     

Conformal Galilei algebras,symmetric polynomials and singular vectors

We classify and explicitly describe homomorphisms of Verma modules for conformal Galilei algebras \({\mathfrak {cga}}_\ell (d,\mathbb {C})\) with \(d=1\) for any integer value \(\ell \in {\mathbb {N}}\). The homomorphisms are uniquely determined by singular vectors as solutions of certain differential operators of flag type and identified with specific polynomials arising as coefficients in the expansion of a parametric family of symmetric polynomials into power sum symmetric polynomials.     

Gibbs Measures Over Locally Tree-Like Graphs and Percolative Entropy Over Infinite Regular Trees

Consider a statistical physical model on the d-regular infinite tree \(T_{d}\) described by a set of interactions \(\Phi \). Let \(\{G_{n}\}\) be a sequence of finite graphs with vertex sets \(V_n\) that locally converge to \(T_{d}\). From \(\Phi \) one can construct a sequence of corresponding models on the graphs \(G_n\). Let \(\{\mu _n\}\) be the resulting Gibbs measures. Here we assume that \(\{\mu _{n}\}\) converges to some limiting Gibbs measure \(\mu \) on \(T_{d}\) in the local weak\(^*\) sense, and study the consequences of this convergence for the specific entropies \(|V_n|^{-1}H(\mu _n)\). We show that the limit supremum of \(|V_n|^{-1}H(\mu _n)\) is bounded above by the percolative entropy \(H_{\textit{perc}}(\mu )\), a function of \(\mu \) itself, and that \(|V_n|^{-1}H(\mu _n)\) actually converges to \(H_{\textit{perc}}(\mu )\) in case \(\Phi \) exhibits strong spatial mixing on \(T_d\). When it is known to exist, the limit of \(|V_n|^{-1}H(\mu _n)\) is most commonly shown to be given by the Bethe ansatz. Percolative entropy gives a different formula, and we do not know how to connect it to the Bethe ansatz directly. We discuss a few examples of well-known models for which the latter result holds in the high temperature regime.     

Noether symmetry approach in $$f(\mathcal {G},T)$$ gravity

We explore the recently introduced modified Gauss–Bonnet gravity (Sharif and Ikram in Eur Phys J C 76:640, 2016), \(f(\mathcal {G},T)\) pragmatic with \(\mathcal {G}\), the Gauss–Bonnet term, and T, the trace of the energy-momentum tensor. Noether symmetry approach has been used to develop some cosmologically viable \(f(\mathcal {G},T)\) gravity models. The Noether equations of modified gravity are reported for flat FRW universe. Two specific models have been studied to determine the conserved quantities and exact solutions. In particular, the well known deSitter solution is reconstructed for some specific choice of \(f(\mathcal {G},T)\) gravity model.     

Dispersion Estimates for the Discrete Laguerre Operator

We derive an explicit expression for the kernel of the evolution group \({\exp(-\mathrm{i} t H_0)}\) of the discrete Laguerre operator H₀ (i.e., the Jacobi operator associated with the Laguerre polynomials) in terms of Jacobi polynomials. Based on this expression, we show that the norm of the evolution group acting from \({\ell^1}\) to \({\ell^\infty}\) is given by \({(1+t^2)^{-1/2}}\).     

On <Emphasis Type="Italic">τ</Emphasis>-Compactness of Products of <Emphasis Type="Italic">τ</Emphasis>-Measurable Operators

Let \(\mathcal {M}\) be a von Neumann algebra of operators on a Hilbert space \(\mathcal {H}\), τ be a faithful normal semifinite trace on \(\mathcal {M}\). We obtain some new inequalities for rearrangements of τ-measurable operators products. We also establish some sufficient τ-compactness conditions for products of selfadjoint τ-measurable operators. Next we obtain a τ-compactness criterion for product of a nonnegative τ-measurable operator with an arbitrary τ-measurable operator. We construct an example that shows importance of nonnegativity for one of the factors. The similar results are obtained also for elementary operators from \(\mathcal {M}\). We apply our results to symmetric spaces on \((\mathcal {M}, \tau )\). The results are new even for the *-algebra \(\mathcal {B}(\mathcal {H})\) of all linear bounded operators on \(\mathcal {H}\) endowed with the canonical trace τ = tr.     

CGC/saturation approach: re-visiting the problem of odd harmonics in angular correlations

In this paper we demonstrate that the selection of events with different multiplicities of produced particles, leads to the violation of the azimuthal angular symmetry, \(\phi \rightarrow \pi - \phi \). We find for LHC and lower energies, that this violation can be so large for the events with multiplicities \(n \ge 2 \bar{n}\), where \(\bar{n}\) is the mean multiplicity, that it leads to almost no suppression of \(v_n\), with odd n. However, this can only occur if the typical size of the dipole in DIS with a nuclear target is small, or \(Q^2 \,>\,Q^2_s\left( A; Y_{\mathrm{min}},b\right) \), where \(Q_s\) is the saturation momentum of the nucleus at \(Y = Y_{\mathrm{min}}\). In the case of large sizes of dipoles, when \(Q^2 \,<\,Q^2_s\left( A; Y_{\mathrm{min}},b\right) \), we show that \(v_n =0\) for odd n. Hadron-nucleus scattering is discussed.     

Distribution of Singular Values of Random Band Matrices; Marchenko–Pastur Law and More

We consider the limiting spectral distribution of matrices of the form \(\frac{1}{2b_{n}+1} (R + X)(R + X)^{*}\), where X is an \(n\times n\) band matrix of bandwidth \(b_{n}\) and R is a non random band matrix of bandwidth \(b_{n}\). We show that the Stieltjes transform of ESD of such matrices converges to the Stieltjes transform of a non-random measure. And the limiting Stieltjes transform satisfies an integral equation. For \(R=0\), the integral equation yields the Stieltjes transform of the Marchenko–Pastur law.     

Exact holography of the mass-deformed M2-brane theory

We test the holographic relation between the vacuum expectation values of gauge invariant operators in \({\mathcal {N}} = 6\) U\(_k(N)\times \mathrm{U}_{-k}(N)\) mass-deformed ABJM theory and the LLM geometries with \({\mathbb {Z}}_k\) orbifold in 11-dimensional supergravity. To do so, we apply the Kaluza–Klein reduction to construct a 4-dimensional gravity theory and implement the holographic renormalization procedure. We obtain an exact holographic relation for the vacuum expectation values of the chiral primary operator with conformal dimension \(\Delta = 1\), which is given by \(\langle {\mathcal {O}}^{(\Delta =1)}\rangle = N^{\frac{3}{2}} \, f_{(\Delta =1)}\), for large N and \(k=1\). Here the factor \(f_{(\Delta )}\) is independent of N. Our results involve an infinite number of exact dual relations for all possible supersymmetric Higgs vacua and so provide a non-trivial test of gauge/gravity duality away from the conformal fixed point. We extend our results to the case of \(k\ne 1\) for LLM geometries represented by rectangular-shaped Young diagrams. We also discuss the exact mapping of the gauge/gravity at finite N for classical supersymmetric vacuum solutions in field theory side and corresponding classical solutions in gravity side.     

Excited state mass spectra of doubly heavy $$\Xi $$ baryons

In this paper, the mass spectra are obtained for doubly heavy \(\Xi \) baryons, namely, \(\Xi _{cc}^{+}\), \(\Xi _{cc}^{++}\), \(\Xi _{bb}^{-}\), \(\Xi _{bb}^{0}\), \(\Xi _{bc}^{0}\) and \(\Xi _{bc}^{+}\). These baryons consist of two heavy quarks (cc, bb, and bc) with a light (d or u) quark. The ground, radial, and orbital states are calculated in the framework of the hypercentral constituent quark model with Coulomb plus linear potential. Our results are also compared with other predictions, thus, the average possible range of excited states masses of these \(\Xi \) baryons can be determined. The study of the Regge trajectories is performed in (n, \(M^{2}\)) and (J, \(M^{2}\)) planes and their slopes and intercepts are also determined. Lastly, the ground state magnetic moments of these doubly heavy baryons are also calculated.