Decoupling the NLO coupled DGLAP evolution equations: an analytic solution to pQCD

Using repeated Laplace transforms, we turn coupled, integral-differential singlet DGLAP equations into NLO (next-to-leading) coupled algebraic equations, which we then decouple. After two Laplace inversions we find new tools for pQCD: decoupled NLO analytic solutions $F_{s}(x,Q^{2})={\mathcal{F}}_{s}(F_{s0}(x),G_{0}(x))$ , $G(x,Q^{2})={\mathcal{G}}(F_{s0}(x), G_{0}(x))$ . ${\mathcal{F}}_{s}$ , $\mathcal{G}$ are known NLO functions and $F_{s0}(x)\equiv F_{s}(x,Q_{0}^{2})$ , $G_{0}(x)\equiv G(x,Q_{0}^{2})$ are starting functions for evolution beginning at $Q^{2}=Q_{0}^{2}$ . We successfully compare our u and d non-singlet valence quark distributions with MSTW results (Martin et al., Eur. Phys. J. C 63:189, 2009).     

Parton Distribution Functions from QCD Analysis of HERA Combined Inclusive e ± p Scattering Cross Section Data

Utilizing very recent deep inelastic scattering measurements, a QCD analysis of proton structure function ${F_{2}^{p} (x,Q^2)}$ is presented. A wide range of the inclusive neutral-current deep-inelastic-scattering (NC DIS) data used in order to extract an updated set of parton distribution functions (PDFs). The HERA ‘combined’ data set on ${\sigma_{r,NC}^\pm (x,Q^2)}$ together with all available published data for heavy quarks ${F_2^{c,b}(x,Q^2)}$ , longitudinal F _L (x, Q ²) and also very recent reduced DIS cross section ${\sigma_{r,NC}^\pm (x,Q^2)}$ data from HERA experiments are the input in the present next-to-leading order (NLO) QCD analysis which determines a new set of parton distributions, called ${{\tt KKT11C}}$ . The extracted PDFs in the ‘fixed flavour number scheme’ (FFNS) are in very good agreement with the available theoretical models.     

35Cl NQR spectra of group 1 and silver dichloromethanesulfonates

The dichloromethanesulfonates of silver and other +1-charged cations, M ^?+?(Cl₂CHSO $_{3}^{-})$ (M = Ag, Tl, Li, Na, K, Rb, Cs) were synthesized and studied by ³⁵Cl NQR. Dichloromethanesulfonic acid was prepared by the methanolysis of dichloromethanesulfonyl chloride, and was then neutralized with the carbonates of the +1-charged cations to produce the corresponding dichloromethanesulfonate salt. This NQR study completed the investigation of the chloroacetates and chloromethanesulfonates of silver, Ag^?+?(Cl _x CH_3???x SO $_{3}^{-})$ and Ag^?+?(Cl _x CH_3???x CO $_{2}^{-})$ , and suggests (1) that the ability of organochlorine atoms to coordinate to silver decreases as the number of electron-withdrawing groups (Cl, SO $_{3}^{-}$ , CO $_{2}^{-})$ attached to the carbon atom increases; (2) that the unusually large NQR spectral width found among M ^?+?(Cl₂CHCO $_{2}^{-})$ salts is not present among M ^?+?(Cl₂CHSO $_{3}^{-})$ salts, and therefore is not generally characteristic of the dichloromethyl group in salts.     

Nuclear shadowing in Glauber–Gribov theory with Q 2-evolution

We consider deep inelastic scattering off nuclei in the Regge limit within the Glauber–Gribov model. Using unitarized parton distribution functions for the proton, we find sizeable shadowing effects on the nuclear total and longitudinal structure functions, $F_{2}^{A}$ and $F_{L}^{A}$ , in the low-x limit. Extending a fan-diagram analysis for the large-mass region of coherent diffraction off nuclei to high Q ², we also find significant shadowing effects in this kinematical regime. Finally, we discuss the shortcomings of our approach and possible extensions of the model to other kinematical regimes.     

The energy dependence of the hard exclusive diffractive processes in pQCD as the function of momentum transfer

We predict the dependence on energy of photo (electro) production processes: γ(γ ^*)+p→V+X with large rapidity gap at small x and large momentum ?t transferred to V in pQCD. Here V is a heavy quarkonium (J/ψ,?) or longitudinally polarized light vector meson (in the electroproduction processes), etc. In the kinematics of HERA we calculate the dependence on energy of cross sections of these processes as the function of momentum transfer t, photon virtuality Q ² and/or quarkonium mass. In the kinematical region $Q_{0}^{2}\le -t\ll Q^{2}+M^{2}_{V}$ the nontrivial energy dependence of the cross section for the vector meson production due to the photon scattering off a parton follows within QCD from the summing of the double logarithmic terms. In the second regime $-t\ge Q^{2}+M^{2}_{V}$ within DGLAP approximation in all orders of perturbation theory the $q\bar{q}$ -parton elastic cross section is energy independent. We show that the correct account of the double logarithmic terms and of the gluon radiation including kinematical constraints removes the disagreement between pQCD calculations and recent HERA experimental data. The explicit formula for the dependence of the differential cross section $\frac{d^{2}\sigma}{dt\,dx_{J}}$ of these processes on $s_{\gamma^{*}N}$ is obtained. We show that perturbative Pomeron type behavior may reveal itself only at energies significantly larger than those available at HERA. In addition we evaluate the energy dependence of DVCS processes.     

Quantum Sphere {\mathbb{S}^4} as a Non-Levi Conjugacy Class

We construct a ${U_\hbar(\mathfrak{sp}(4))}$ -equivariant quantization of the four-dimensional complex sphere ${\mathbb{S}^4}$ regarded as a conjugacy class, Sp(4)/Sp(2) ×?Sp(2), of a simple complex group with non-Levi isotropy subgroup, through an operator realization of the quantum polynomial algebra ${\mathbb{C}_\hbar[\mathbb{S}^4]}$ on a highest weight module of ${U_\hbar(\mathfrak{sp}(4))}$ .     

\mathfrak{Q}-, \mathfrak{P}-, and Weyl-ordering of Squeezing Operators: New Operator Identities and Applications

The basic operator ordering regarding to coordinate-momentum operator is discussed by virtue of the technique of integration within $\mathfrak{Q}$ -ordering (all Q are on the left of all P) and $\mathfrak{P}$ -ordering (all P are on the left of all Q). We derive new operator-ordering identities about $\mathfrak{Q}$ -ordering , $\mathfrak{P}$ -ordering and Weyl-ordering of both single-mode and two-mode squeezing operators. Its application in combinatorics is pointed out.     

Large N approach to Kaon decays and mixing 28 years later: \Delta I=1/2 rule, \hat{B}_K,and \Delta M_K

We review and update our results for $K\rightarrow \pi \pi $ decays and $K^0$ – $\bar{K}^0$ mixing obtained by us in the 1980s within an analytic approximate approach based on the dual representation of QCD as a theory of weakly interacting mesons for large $N$ , where $N$ is the number of colors. In our analytic approach the Standard Model dynamics behind the enhancement of $\hbox {Re}A_0$ and suppression of $\hbox {Re}A_2$ , the so-called $\Delta I=1/2$ rule for $K\rightarrow \pi \pi $ decays, has a simple structure: the usual octet enhancement through the long but slow quark–gluon renormalization group evolution down to the scales $\mathcal{O}(1\, {\hbox { GeV}})$ is continued as a short but fast meson evolution down to zero momentum scales at which the factorization of hadronic matrix elements is at work. The inclusion of lowest-lying vector meson contributions in addition to the pseudoscalar ones and of Wilson coefficients in a momentum scheme improves significantly the matching between quark–gluon and meson evolutions. In particular, the anomalous dimension matrix governing the meson evolution exhibits the structure of the known anomalous dimension matrix in the quark–gluon evolution. While this physical picture did not yet emerge from lattice simulations, the recent results on $\hbox {Re}A_2$ and $\hbox {Re}A_0$ from the RBC-UKQCD collaboration give support for its correctness. In particular, the signs of the two main contractions found numerically by these authors follow uniquely from our analytic approach. Though the current–current operators dominate the $\Delta I=1/2$ rule, working with matching scales $\mathcal{O}(1 \, {\hbox { GeV}})$ we find that the presence of QCD-penguin operator $Q_6$ is required to obtain satisfactory result for $\hbox {Re}A_0$ . At NLO in $1/N$ we obtain $R=\hbox {Re}A_0/\hbox {Re}A_2= 16.0\pm 1.5$ which amounts to an order of magnitude enhancement over the strict large $N$ limit value $\sqrt{2}$ . We also update our results for the parameter $\hat{B}_K$ , finding $\hat{B}_K=0.73\pm 0.02$ . The smallness of $1/N$ corrections to the large $N$ value $\hat{B}_K=3/4$ results within our approach from an approximate cancelation between pseudoscalar and vector meson one-loop contributions. We also summarize the status of $\Delta M_K$ in this approach.     

Pyrid-2-yl and 2-CyanoPhenyl fused heterocyclic compounds as human P2X_{3} inhibitors: a combined approach based on homology modelling,docking and QSAR analysis

P2X receptors are hetero-oligomeric proteins that function as membrane ion channels and are gated by extracellular ATP. The hP2X $_{3}$ subunit is a constituent of the channels on a subset of sensory neurons involved in pain signaling, where ATP released by damaged and inflamed tissue can initiate action potentials. Hence, the inhibition of ATP-activated P2X $_{3}$ receptor is an exciting approach for the treatment of inflammatory and neuropathic pain. Recently, the crystal structures of zebrafish P2X $_{4}$ (zP2X $_{4})$ were obtained in closed, apo state (PDB ID: 3I5D) and ATP-bound, open state (PDB ID: 4DW1). These structures were used to develop a homology model of human P2X $_{3}$ (hP2X $_{3})$ in order to identify through docking studies, the binding modes of known P2X $_{3}$ inhibitors and their key active site interactions, along with a pharmacophore-based 3D-QSAR model for a series of 136 Pyrid-2-yl and 2-CyanoPhenyl fused heterocyclic compounds. These 3D-QSAR models have been developed with different combinations of training and test set divisions obtained by random separation, Jarvis–Patrick clustering, K-means clustering and sphere exclusion methods. The best predictive 3D-QSAR model resulted in training set R $^{2 }$ of 0.75, internal test set Q $^{2}$ of 0.74, Pearson-R value of 0.87 and root mean square error of 0.37. The information generated by the pharmacophore model and docking analyses using the homology model provides valuable clues to design novel potent hP2X $_{3}$ inhibitors.     

Study of f 0(980) and f 0(1500) from B s →f 0(980)π,f 0(1500)π decays

In this paper, we analyze the scalar mesons f ₀(980) and f ₀(1500) from the decays $\bar{B}^{0}_{s}\to f_{0}(980)\pi^{0},\allowbreak f_{0}(1500)\pi^{0}$ within Perturbative QCD approach. From the leading-order calculations, we find that (a) in the allowed mixing angle ranges, the branching ratio of $\bar{B}^{0}_{s}\to f_{0}(980)\pi^{0}$ is about (1.0～1.6)×10^?7, which is smaller than that of $\bar{B}^{0}_{s}\to f_{0}(980)K^{0}$ (the difference is a few times even one order); (b) the decay $\bar{B}^{0}_{s}\to f_{0}(1500)\pi^{0}$ is better to distinguish between the lowest lying state or the first excited state for f ₀(1500), because the branching ratios for two scenarios have about one-order difference in most of the mixing angle ranges; and (c) the direct CP asymmetries of $\bar{B}^{0}_{s}\to f_{0}(1500)\pi^{0}$ for two scenarios also exists great difference. In scenario II, the variation range of the value ${\mathcal{A}}^{\mathrm{dir}}_{CP}(\bar{B}^{0}_{s}\to f_{0}(1500)\pi^{0})$ according to the mixing angle in scenario II is very small, except for the values for mixing angles near 90° or 270°, while the variation range of ${\mathcal{A}}^{\mathrm{dir}}_{CP}(\bar{B}^{0}_{s}\to f_{0}(1500)\pi^{0})$ in scenario I is very large. Compared with the future data for the decay $\bar{B}^{0}_{s}\to f_{0}(1500)\pi^{0}$ , it is easy to determine the nature of the scalar meson f ₀(1500).     

Shadowing correction to the gluon distribution behavior at small x

We determined the saturation exponent of the gluon distribution using the solution of the QCD nonlinear Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (NLDGLAP) evolution equation at small x . The very small-x behavior of the gluon distribution is obtained by solving the Gribov, Levin, Ryskin, Mueller and Qiu (GLR-MQ) evolution equation with the nonlinear shadowing term incorporated. The form of the initial condition for the equation is determined. We find, with decreasing x , the emergence of a singular behavior and the eventual taming (at R = 5 GeV^-1) and the essential taming (at R = 2 GeV^-1) of this singular behavior by the shadowing term. The nonlinear gluon density functions are calculated and compared with the results for the integrated gluon density from the Balitsky-Kovchegov (BK) equation for the different values of Q². It is shown that the results for the gluon density function are comparable with the results obtained from the BK equation solution. Also we show that for each x , the Q²-dependence of the data is well described by gluon shadowing corrections to the GLR-MQ equation. The resulting analytic expression allows us to predict the logarithmic derivative \( {\frac{{\partial F^{s}_{2} (x,Q^{2})}}{{\partial \ln Q^{2}}}}\) and to compare the results with H1 data and a QCD analysis fit.     

Non-linear QCD dynamics in two-photon interactions at high energies

Perturbative QCD predicts that the growth of the gluon density at high energies should saturate, forming a Color Glass Condensate (CGC), which is described in mean field approximation by the Balitsky–Kovchegov (BK) equation. In this paper we study the γγ interactions at high energies and estimate the main observables which will be probed at future linear colliders using the color dipole picture. We discuss in detail the dipole–dipole cross section and propose a new relation between this quantity and the dipole scattering amplitude. The total γγ, γ ^? γ ^? cross sections and the real photon structure function $F_{2}^{\gamma }(x,Q^{2})$ are calculated using the recent solution of the BK equation with running coupling constant and the predictions are compared with those obtained using phenomenological models for the dipole–dipole cross section and scattering amplitude. We demonstrate that these models are able to describe the LEP data at high energies, but predict a very different behavior for the observables at higher energies. Therefore we conclude that the study of γγ interactions can be useful to constrain the QCD dynamics.     

Preliminary studies of the Raman spectra of \hbox {Ag}_{2}\hbox {Te}\hbox { and }\hbox {Ag}_{5}\hbox {Te}_{3}

The theoretical calculations indicated that the monoclinic low-temperature phase of silver telluride $(\upbeta \hbox {-Ag}_{2}\hbox {Te})$ is a new binary topological insulator with highly anisotropic single Dirac cone surface. We obtained $\upbeta \hbox {-Ag}_{2}\hbox {Te}$ crystal ingots containing few grains by the Bridgman method. We also deposited thin films of tellurium, $\hbox {Ag}_{5}\hbox {Te}_{3}\hbox { and }(\hbox {Te+Ag}_{5}\hbox {Te}_{3})$ by thermal evaporation method. The Raman spectra of $\upbeta \hbox {-Ag}_{2}\hbox {Te}$ , tellurium and $\hbox {Ag}_{5}\hbox {Te}_{3}$ were measured at three excitation wave lengths: 633, 515 and 488 nm. The Raman active modes of $\upbeta \hbox {-Ag}_{2}\hbox {Te}$ , tellurium and $\hbox {Ag}_{5}\hbox {Te}_{3}$ are situated at frequencies below 300  $\hbox {cm}^{-1}$ while vibrations of other phases appear at higher frequencies.     

Nucleon and {\Delta} Elastic and Transition Form Factors

We present a unified study of nucleon and \({\Delta}\) elastic and transition form factors, and compare predictions made using a framework built upon a Faddeev equation kernel and interaction vertices that possess QCD-like momentum dependence with results obtained using a symmetry-preserving treatment of a vector \({\otimes}\) vector contact-interaction. The comparison emphasises that experiments are sensitive to the momentum dependence of the running couplings and masses in the strong interaction sector of the Standard Model and highlights that the key to describing hadron properties is a veracious expression of dynamical chiral symmetry breaking in the bound-state problem. Amongst the results we describe, the following are of particular interest: \({G_{E}^{p}(Q^{2})/G_{M}^{p}(Q^{2})}\) possesses a zero at Q ² = 9.5 GeV²; any change in the interaction which shifts a zero in the proton ratio to larger Q ² relocates a zero in \({G_{E}^{n}(Q^{2})/G_M^{n}(Q^{2})}\) to smaller Q ²; there is likely a value of momentum transfer above which \({G_{E}^{n} > G_{E}^{p}}\) ; and the presence of strong diquark correlations within the nucleon is sufficient to understand empirical extractions of the flavour-separated form factors. Regarding the \({\Delta(1232)}\) -baryon, we find that, inter alia: the electric monopole form factor exhibits a zero; the electric quadrupole form factor is negative, large in magnitude, and sensitive to the nature and strength of correlations in the \({\Delta(1232)}\) Faddeev amplitude; and the magnetic octupole form factor is negative so long as rest-frame P- and D-wave correlations are included. In connection with the \({N \to \Delta}\) transition, the momentum-dependence of the magnetic transition form factor, \({G_{M}^{*}}\) , matches that of \({G_{M}^{n}}\) once the momentum transfer is high enough to pierce the meson-cloud; and the electric quadrupole ratio is a keen measure of diquark and orbital angular momentum correlations, the zero in which is obscured by meson-cloud effects on the domain currently accessible to experiment. Importantly, within each framework, identical propagators and vertices are sufficient to describe all properties discussed herein. Our analysis and predictions should therefore serve as motivation for measurement of elastic and transition form factors involving the nucleon and its resonances at high photon virtualities using modern electron-beam facilities.     

Annihilation-type charmless radiative decays of B meson in non-universal Z′ model

We study charmless pure annihilation type radiative B decays within the QCD factorization approach. After adding the vertex corrections to the naive factorization approach, we find that the branching ratios of $\overline{B}^{0}_{d}\to\phi\gamma$ , $\overline{B}^{0}_{s}\to\rho^{0}\gamma$ and $\overline{B}^{0}_{s}\to\omega\gamma$ within the standard model are at the order of $\mathcal{O}(10^{-12})$ , $\mathcal{O}(10^{-10})$ and $\mathcal{O}(10^{-11})$ , respectively. The smallness of these decays in the standard model makes them sensitive probes of flavor physics beyond the standard model. To explore their physics potential, we have estimated the contribution of Z′ boson in the decays. Within the allowed parameter space, the branching ratios of these decay modes can be enhanced remarkably in the non-universal Z′ model: The branching ratios can reach to $\mathcal{O}(10^{-8})$ for $\overline{B}_{s}^{0}\to \rho^{0}(\omega)\gamma$ and $\mathcal{O}(10^{-10})$ for the $\overline{B}_{d}^{0}\to \phi \gamma$ , which are large enough for LHC-b and/or Super B-factories to detect those channels in near future. Moreover, we also predict large CP asymmetries in suitable parameter space. The observation of these modes could in turn help us to constrain the Z′ mass within the model.     

Supremum of the Airy2 Process Minus a Parabola on a Half Line

Let $\mathcal {A}_{2}(t)$ be the Airy₂ process. We show that the random variable $$\sup_{t\leq\alpha} \bigl\{\mathcal {A}_2(t)-t^2 \bigr\}+\min\{0,\alpha \}^2 $$ has the same distribution as the one-point marginal of the Airy_2→1 process at time α. These marginals form a family of distributions crossing over from the GUE Tracy-Widom distribution F _GUE(x) for the Gaussian Unitary Ensemble of random matrices, to a rescaled version of the GOE Tracy-Widom distribution F _GOE(4^1/3 x) for the Gaussian Orthogonal Ensemble. Furthermore, we show that for every α the distribution has the same right tail decay $e^{-\frac{4}{3} x^{3/2} }$ .     

Coset Constructions of Logarithmic (1, p) Models

One of the best understood families of logarithmic onformal field theories consists of the (1, p) models (p =  2, 3, . . .) of central charge c _{1, p} =1 ? 6(p ? 1)²/p. This family includes the theories corresponding to the singlet algebras ${\mathcal{M}(p)}$ and the triplet algebras ${\mathcal{W}(p)}$ , as well as the ubiquitous symplectic fermions theory. In this work, these algebras are realised through a coset construction. The ${W^{(2)}_n}$ algebra of level k was introduced by Feigin and Semikhatov as a (conjectured) quantum hamiltonian reduction of ${\widehat{\mathfrak{sl}}(n)_k}$ , generalising the Bershadsky–Polyakov algebra ${W^{(2)}_3}$ . Inspired by work of Adamovi? for p = 3, vertex algebras ${\mathcal{B}_p}$ are constructed as subalgebras of the kernel of certain screening charges acting on a rank 2 lattice vertex algebra of indefinite signature. It is shown that for p≤5, the algebra ${\mathcal{B}_p}$ is a quotient of ${W^{(2)}_{p-1}}$ at level ?(p ? 1)²/p and that the known part of the operator product algebra of the latter algebra is consistent with this holding for p> 5 as well. The triplet algebra ${\mathcal{W}(p)}$ is then realised as a coset inside the full kernel of the screening operator, while the singlet algebra ${\mathcal{M}(p)}$ is similarly realised inside ${\mathcal{B}_p}$ . As an application, and to illustrate these results, the coset character decompositions are explicitly worked out for p =  2 and 3.     

The Essence of Operators’ Weyl Ordering Scheme and New Operator Identities for Converting Q−P Ordering to Weyl Ordering

We find new operator formulas for converting Q?P and P?Q ordering to Weyl ordering, where Q and P are the coordinate and momentum operator. In this way we reveal the essence of operators’ Weyl ordering scheme, e.g., Weyl ordered operator polynomial ${_{:}^{:}}\;Q^{m}P^{n}\;{_{:}^{:}}$ , $$\begin{aligned} {_{:}^{:}}\;Q^{m}P^{n}\;{_{:}^{:}} =&\sum_{l=0}^{\min (m,n)} \biggl( \frac{-i\hbar }{2} \biggr) ^{l}l!\binom{m}{l}\binom{n}{l}Q^{m-l}P^{n-l} \\ =& \biggl( \frac{\hbar }{2} \biggr) ^{ ( m+n ) /2}i^{n}H_{m,n} \biggl( \frac{\sqrt{2}Q}{\sqrt{\hbar }},\frac{-i\sqrt{2}P}{\sqrt{\hbar }} \biggr) \bigg|_{Q_{\mathrm{before}}P} \end{aligned}$$ where ${}_{:}^{:}$ ${}_{:}^{:}$ denotes the Weyl ordering symbol, and H _m,n is the two-variable Hermite polynomial. This helps us to know the Weyl ordering more intuitively.     

On the Quasi-linear Elliptic PDE {-\nabla \cdot ( \nabla{u}/ \sqrt{1-| \nabla{u} |^2}) = 4 \pi \sum_k a_k \delta_{s_k}} in Physics and Geometry

It is shown that for each finite number N of Dirac measures ${\delta_{s_n}}$ supported at points ${s_n \in {\mathbb R}^3}$ with given amplitudes ${a_n \in {\mathbb R} \backslash\{0\}}$ there exists a unique real-valued function ${u \in C^{0, 1}({\mathbb R}^3)}$ , vanishing at infinity, which distributionally solves the quasi-linear elliptic partial differential equation of divergence form ${-\nabla \cdot ( \nabla{u}/ \sqrt{1-| \nabla{u} |^2}) = 4 \pi \sum_{n=1}^N a_n \delta_{s_n}}$ . Moreover, ${u \in C^{\omega}({\mathbb R}^3\backslash \{s_n\}_{n=1}^N)}$ . The result can be interpreted in at least two ways: (a) for any number N of point charges of arbitrary magnitude and sign at prescribed locations s _n in three-dimensional Euclidean space there exists a unique electrostatic field which satisfies the Maxwell-Born-Infeld field equations smoothly away from the point charges and vanishes as |s| ?? ??; (b) for any number N of integral mean curvatures assigned to locations ${s_n \in {\mathbb R}^3 \subset{\mathbb R}^{1, 3}}$ there exists a unique asymptotically flat, almost everywhere space-like maximal slice with point defects of Minkowski spacetime ${{\mathbb R}^{1, 3}}$ , having lightcone singularities over the s _n but being smooth otherwise, and whose height function vanishes as |s| ?? ??. No struts between the point singularities ever occur.