The S-wave resonance contributions in the $$B^{0}_{s}$$ decays into $$\psi (2S,3S)$$ψ(2S,3S) plus pion pair

The three-body decays \(B^0_s \rightarrow \psi (2S,3S) \pi ^+ \pi ^-\) are studied based on the perturbative QCD approach. With the help of the nonperturbative two-pion distribution amplitudes, the analysis is simplified into the quasi-two-body processes. Besides the traditional factorizable and nonfactorizable diagrams at the leading order, the next-to-leading order vertex corrections are also included to cancel the scale dependence. The \(f_0(980)\), \(f_0(1500)\) resonance contributions as well as the nonresonant contributions are taken into account using the presently known \(\pi \pi \) time-like scalar form factor for the \(s\bar{s}\) component. It is found that the predicted \(B^0_s \rightarrow \psi (2S) \pi ^+ \pi ^-\) decay spectra in the pion pair invariant mass shows a similar behavior as the experiment. The calculated S-wave contributions to the branching ratio of \(B^0_s \rightarrow \psi (2S) \pi ^+ \pi ^-\) is \(6.0\times 10^{-5}\), which is in agreement with the LHCb data \(\mathcal {B}(B^0_s \rightarrow \psi (2S) \pi ^+ \pi ^-)=(7.2\pm 1.2)\times 10^{-5} \) within errors. The estimate of \(\mathcal {B}(B^0_s \rightarrow \psi (3S) \pi ^+ \pi ^-)\) can reach the order of \(10^{-5}\), pending the corresponding measurements.     

$$\varvec{B^0\rightarrow K^{*0}\mu ^+\mu ^-}$$ decay in the aligned two-Higgs-doublet model

In the aligned two-Higgs-doublet model, we perform a complete one-loop computation of the short-distance Wilson coefficients \(C_{7,9,10}^{(\prime )}\), which are the most relevant ones for \(b\rightarrow s\ell ^+\ell ^-\) transitions. It is found that, when the model parameter \(\left| \varsigma _{u}\right| \) is much smaller than \(\left| \varsigma _{d}\right| \), the charged scalar contributes mainly to chirality-flipped \(C_{9,10}^\prime \), with the corresponding effects being proportional to \(\left| \varsigma _{d}\right| ^2\). Numerically, the charged-scalar effects fit into two categories: (A) \(C_{7,9,10}^\mathrm {H^\pm }\) are sizable, but \(C_{9,10}^{\prime \mathrm {H^\pm }}\simeq 0\), corresponding to the (large \(\left| \varsigma _{u}\right| \), small \(\left| \varsigma _{d}\right| \)) region; (B) \(C_7^\mathrm {H^\pm }\) and \(C_{9,10}^{\prime \mathrm {H^\pm }}\) are sizable, but \(C_{9,10}^\mathrm {H^\pm }\simeq 0\), corresponding to the (small \(\left| \varsigma _{u}\right| \), large \(\left| \varsigma _{d}\right| \)) region. Taking into account phenomenological constraints from the inclusive radiative decay \(B\rightarrow X_{s}{\gamma }\), as well as the latest model-independent global analysis of \(b\rightarrow s\ell ^+\ell ^-\) data, we obtain the much restricted parameter space of the model. We then study the impact of the allowed model parameters on the angular observables \(P_2\) and \(P_5'\) of \(B^0\rightarrow K^{*0}\mu ^+\mu ^-\) decay, and we find that \(P_5'\) could be increased significantly to be consistent with the experimental data in case B.     

Star product on complex sphere $$\mathbb {S}^{2n}$$

We construct a \(U_q\bigl (\mathfrak {s}\mathfrak {o}(2n+1)\bigr )\)-equivariant local star product on the complex sphere \(\mathbb {S}^{2n}\) as a non-Levi conjugacy class \(SO(2n+1)/SO(2n)\).     

$$f_0(500)$$, $$f_0(980)$$f0(980), and $$a_0(980)$$a0(980) production in the $$\chi _{c1} \rightarrow \eta \pi ^+\pi ^-$$χc1→ηπ+π- reaction

We study the \(\chi _{c1} \rightarrow \eta \pi ^+ \pi ^-\) decay, paying attention to the production of \(f_0(500)\), \(f_0(980)\), and \(a_0(980)\) from the final state interaction of pairs of mesons that can lead to these three mesons in the final state, which is implemented using the chiral unitary approach. Very clean and strong signals are obtained for the \(a_0(980)\) excitation in the \(\eta \pi \) invariant mass distribution and for the \(f_0(500)\) in the \(\pi ^+ \pi ^-\) mass distribution. A smaller, but also clear signal for the \(f_0(980)\) excitation is obtained. The results are contrasted with experimental data and the agreement found is good, providing yet one more test in support of the picture where these resonances are dynamically generated from the meson–meson interaction.     

Study of the weak annihilation contributions in charmless $$\varvec{B_s\rightarrow VV}$$ decays

In this paper, in order to probe the spectator-scattering and weak annihilation contributions in charmless \(B_s\rightarrow VV\) (where V stands for a light vector meson) decays, we perform the \(\chi ^2\)-analyses for the endpoint parameters within the QCD factorization framework, under the constraints from the measured \(\bar{B}_{s}\rightarrow \) \(\rho ^0\phi \), \(\phi K^{*0}\), \(\phi \phi \) and \(K^{*0}\bar{K}^{*0}\) decays. The fitted results indicate that the endpoint parameters in the factorizable and nonfactorizable annihilation topologies are non-universal, which is also favored by the charmless \(B\rightarrow PP\) and PV (where P stands for a light pseudo-scalar meson) decays observed in previous work. Moreover, the abnormal polarization fractions \(f_{L,\bot }(\bar{B}_{s}\rightarrow K^{*0}\bar{K}^{*0})=(20.1\pm 7.0)\%,(58.4\pm 8.5)\%\) measured by the LHCb collaboration can be reconciled through the weak annihilation corrections. However, the branching ratio of \(\bar{B}_{s}\rightarrow \phi K^{*0}\) decay exhibits a tension between the data and theoretical result, which dominates the contributions to \(\chi _\mathrm{min}^2\) in the fits. Using the fitted endpoint parameters, we update the theoretical results for the charmless \(B_s\rightarrow VV\) decays, which will be further tested by the LHCb and Belle-II experiments in the near future.     

X-ray spectra and valence states of cations in nanostructured half-doped $$\hbox {La}_{0.5}\hbox {Ca}_{0.5}\hbox {MnO}_{3}$$ manganite

Soft X-ray absorption (XAS) and emission (XES) spectroscopies were applied to determine valence states of manganese ions in nanostructured powder of half-doped \(\hbox {La}_{0.5}\hbox {Ca}_{0.5}\hbox {MnO}_{3}\) manganite obtained by milling in a ball mill. XAS spectra were measured both in surface-sensitivity total electron-yield and in bulk-sensitivity total fluorescence-yield modes. O K\(_{\upalpha }\) XES and O 1s XAS spectra characterized the occupied and unoccupied partial O 2p densities of states are compared with band-structure calculations made using the TB-LMTO-ASA codes. Experimental Mn 2p, Ca 2p, and La 3\(d\) XAS spectra are compared with results of crystal field atomic multiplet calculations. For the nanostructured system of \(\hbox {La}_{0.5}\hbox {Ca}_{0.5}\hbox {MnO}_{3}\), concentrations of Mn\(^{4+}\) ions are found to be increased with increasing the time of milling.     

The decays of $$\bar{B}^0$$, $$\bar{B}^0_s$$B¯s0 and $$B^-$$B- into $$\eta _c$$ηc plus a scalar or vector meson

We investigate the decays of \(\bar{B}^0_s\), \(\bar{B}^0\) and \(B^-\) into \(\eta _c\) plus a scalar or vector meson in a theoretical framework by taking into account the dominant process for the weak decay of \(\bar{B}\) meson into \(\eta _c\) and a \(q\bar{q}\) pair. After hadronization of this \(q\bar{q}\) component into pairs of pseudoscalar mesons we obtain certain weights for the pseudoscalar meson-pseudoscalar meson components. In addition, the \(\bar{B}^0\) and \(\bar{B}^0_s\) decays into \(\eta _c\) and \(\rho ^0\), \(K^*\) are evaluated and compared to the \(\eta _c\) and \(\phi \) production. The calculation is based on the postulation that the scalar mesons \(f_0(500)\), \(f_0(980)\) and \(a_0(980)\) are dynamically generated states from the pseudoscalar meson-pseudoscalar meson interactions in S-wave. Up to a global normalization factor, the \(\pi \pi \), \(K \bar{K}\) and \(\pi \eta \) invariant mass distributions for the decays of \(\bar{B}^0_s \rightarrow \eta _c \pi ^+ \pi ^-\), \(\bar{B}^0_s \rightarrow \eta _c K^+ K^-\), \(\bar{B}^0 \rightarrow \eta _c \pi ^+ \pi ^-\), \(\bar{B}^0 \rightarrow \eta _c K^+ K^-\), \(\bar{B}^0 \rightarrow \eta _c \pi ^0 \eta \), \(B^- \rightarrow \eta _c K^0 K^-\) and \(B^- \rightarrow \eta _c \pi ^- \eta \) are predicted. Comparison is made with the limited experimental information available and other theoretical calcualtions. Further comparison of these results with coming LHCb measurements will be very valuable to make progress in our understanding of the nature of the low lying scalar mesons, \(f_0(500), f_0(980)\) and \(a_0(980)\).     

On Blow-up Profile of Ground States of Boson Stars with External Potential

We study minimizers of the pseudo-relativistic Hartree functional \({\mathcal {E}}_{a}(u):=\Vert (-\varDelta +m^{2})^{1/4}u\Vert _{L^{2}}^{2}+\int _{{\mathbb {R}}^{3}}V(x)|u(x)|^{2}\mathrm{d}x-\frac{a}{2}\int _{{\mathbb {R}}^{3}}(\left| \cdot \right| ^{-1}\star |u|^{2})(x)|u(x)|^{2}\mathrm{d}x\) under the mass constraint \(\int _{{\mathbb {R}}^3}|u(x)|^2\mathrm{d}x=1\). Here \(m>0\) is the mass of particles and \(V\ge 0\) is an external potential. We prove that minimizers exist if and only if a satisfies \(0\le a<a^{*}\), and there is no minimizer if \(a\ge a^*\), where \(a^*\) is called the Chandrasekhar limit. When a approaches \(a^*\) from below, the blow-up behavior of minimizers is derived under some general external potentials V. Here we consider three cases of V: trapping potential, i.e. \(V\in L_{\mathrm{loc}}^{\infty }({\mathbb {R}}^3)\) satisfies \(\lim _{|x|\rightarrow \infty }V(x)=\infty \); periodic potential, i.e. \(V\in C({\mathbb {R}}^3)\) satisfies \(V(x+z)=V(x)\) for all \(z\in \mathbb {Z}^3\); and ring-shaped potential, e.g. \( V(x)=||x|-1|^p\) for some \(p>0\).     

On quantum analogue of dynamical stabilisation of inverted harmonic oscillator by time periodical uniform field

Quantum analogue of stabilised forced oscillations around an unstable equilibrium position is explored by solving the non-stationary Schrödinger equation (NSE) of the inverted harmonic oscillator (IHO) driven periodically by spatial uniform field of frequency \(\Omega \), amplitude \(F_{0}\) and phase \(\phi \), i.e. the system with the Hamiltonian of \(\hat{{H}}=(\hat{{p}}^{2}/2m)-(m\omega ^{2}x^{2}/2)-F_0 x\sin \) \(\left( {\Omega t+\phi } \right) \). The NSE has been solved both analytically and numerically by Maple 15 in dimensionless variables \(\xi = x\sqrt{m\omega /\hbar }\hbox {, }f_0 =F_0 /\omega \sqrt{\hbar m\omega }\) and \(\tau =\omega t\). The initial condition (IC) has been specified by the wave function (w.f.) of a generalised Gaussian type which suits well the corresponding quantum IC operator. The solution obtained demonstrates the non-monotonous behaviour of the coordinate spreading \(\sigma \left( \tau \right) \hbox { =}\sqrt{\big ( {\overline{\Delta \xi ^{2}\big ( \tau \big )} } \big )}\) which decreases first from quite macroscopic values of \(\sigma _{0} =2^{12,\ldots ,25}\) to minimal one of \(\sim \!(1/\sqrt{2})\) at times \(\tau <\tau _0 =0.125\ln \!\left( {16\sigma _0^4 +1} \right) \) and then grows back unlimitedly. For certain phases \(\phi \) depending on the \(\Omega /\omega \) ratio and \(n=\log _2\!\sigma _0 \), the mass centre of the packet \(\xi _{\mathrm {av}}( \tau )= \overline{\hat{{x}}(\tau )} \cdot \sqrt{m\omega /\hbar }\) delays approximately two natural ‘periods’ \(\sim \!(4\pi /\omega )\) in the area of the stationary point and then escapes to ‘\(+\)’ or ‘?’ infinity in a bifurcating way.  For ‘resonant’ \(\Omega =\omega \), the bifurcation phases \(\phi \) fit well with the regression formula of Fermi–Dirac type of argument n with their asymptotic \(\phi ( {\Omega ,n\rightarrow \infty } )\) obeying the classical formula \(\phi _{\mathrm {cl}} ( \Omega )=-\hbox {arctg} \, \Omega \) for initial energy \(E = 0\) in the wide range of \(\Omega =2^{-4},...,2^{7}\).     

Rapid communication: $$K_{S}^{0}$$ Production from beryllium target using 120 $$\hbox {GeV}/\hbox {c}$$GeV/c protons beam interactions at the MIPP experiment

We have measured the cross-section for the \(K_{S}^{0}\) production from beryllium target using 120 \(\hbox {GeV}/\hbox {c}\) protons beam interactions at the main injector particle production (MIPP) experiment at Fermilab. The data were collected with target having a thickness of 0.94% of the nuclear interaction length. The \(K_{S}^{0}\) inclusive differential cross-section in bins of momenta is presented covering momentum range from \(0.4\,\hbox {GeV}/\hbox {c}\) to \(30\,\hbox {GeV}/\hbox {c}\). The measured inclusive \(K_{S}^{0}\) production cross-section amounts to \(39.54\pm 1.46\delta _{\mathrm {stat}}\pm 6.97\delta _{\mathrm {syst}}\) mb and the value is compared with the prediction of FLUKA hadron production model.     

Strong decay of $$\Lambda _c(2940)$$ as a 2 P state in the $$\Lambda _c$$Λc family

Considering the mass, parity and \(D^0 p\) decay mode, we tentatively assign the \(\Lambda _c(2940)\) as the \(P-\)wave states with one radial excitation. Then, via studying the strong decay behavior of the \(\Lambda _c(2940)\) within the \(^3P_0\) model, we obtain that the total decay widths of the \(\Lambda _{c1}(\frac{1}{2}^-,2P)\) and \(\Lambda _{c1}(\frac{3}{2}^-,2P)\) states are 16.27 and 25.39 MeV, respectively. Compared with the experimental total width \(27.7^{+8.2}_{-6.0}\pm 0.9^{+5.2}_{-10.4}~\mathrm {MeV}\) measured by LHCb Collaboration, both assignments are allowed, and the \(J^P=\frac{3}{2}^-\) assignment is more favorable. Other \(\lambda \)-mode \(\Sigma _c(2P)\) states are also investigated, which are most likely to be narrow states and have good potential to be observed in future experiments.     

The open-charm radiative and pionic decays of molecular charmonium <Emphasis Type="Italic">Y</Emphasis>(4274)

In this work, we investigate the decay widths and the line shapes of the open-charm radiative and pionic decays of Y(4274) with the \(D_{s}\bar{D}_{s0}(2317)\) molecular charmonium assignment. Our calculation indicates that the decay widths of \(Y(4274)\to D^{+}_{s}D^{*-}_{s}\gamma\) and \(Y(4274)\to D^{+}_{s}D^{-}_{s}\pi^{0}\) can reach up to 0.05 keV and 0.75 keV, respectively. In addition, the result of the line shape of the photon spectrum of \(Y(4274)\to D_{s}^{+} {D}_{s}^{*-} \gamma\) shows that there exists a very sharp peak near the large end point of photon energy. The line shape of the pion spectrum of \(Y(4274)\to D_{s}^{+} {D}_{s}^{*-} \pi^{0}\) is similar to that of the pion spectrum of \(Y(4274)\to D_{s}^{+} {D}_{s}^{*-} \gamma\), where we also find a very sharp peak near the large end point of pion energy. According to our calculation, we suggest further experiments to carry out the search for the open-charm radiative and pionic decays of Y(4274).     

Local-duality QCD sum rules for strong isospin breaking in the decay constants of heavy–light mesons

We discuss the leptonic decay constants of heavy–light mesons by means of Borel QCD sum rules in the local-duality (LD) limit of infinitely large Borel mass parameter. In this limit, for an appropriate choice of the invariant structures in the QCD correlation functions, all vacuum-condensate contributions vanish and all nonperturbative effects are contained in only one quantity, the effective threshold. We study properties of the LD effective thresholds in the limits of large heavy-quark mass \(m_Q\) and small light-quark mass \(m_q\). In the heavy-quark limit, we clarify the role played by the radiative corrections in the effective threshold for reproducing the pQCD expansion of the decay constants of pseudoscalar and vector mesons. We show that the dependence of the meson decay constants on \(m_q\) arises predominantly (at the level of 70–80%) from the calculable \(m_q\)-dependence of the perturbative spectral densities. Making use of the lattice QCD results for the decay constants of nonstrange and strange pseudoscalar and vector heavy mesons, we obtain solid predictions for the decay constants of heavy–light mesons as functions of \(m_q\) in the range from a few to 100 MeV and evaluate the corresponding strong isospin-breaking effects: \(f_{D^+} - f_{D^0}=(0.96 \pm 0.09) \ \mathrm{MeV}\), \(f_{D^{*+}} - f_{D^{*0}}= (1.18 \pm 0.35) \ \mathrm{MeV}\), \(f_{B^0} - f_{B^+}=(1.01 \pm 0.10) \ \mathrm{MeV}\), \(f_{B^{*0}} - f_{B^{*+}}=(0.89 \pm 0.30) \ \mathrm{MeV}\).     

Further studies on the exclusive productions of $$J/\psi +\chi _{cJ}$$ ($$J=0,1,2$$J=0,1,2) via $$e^+e^-$$e+e- annihilation at the B factories

By including the interference effect between the QCD and the QED diagrams, we carry out a complete analysis on the exclusive productions of \(e^+e^- \rightarrow J/\psi +\chi _{cJ}\) (\(J=0,1,2\)) at the B factories with \(\sqrt{s}=10.6\) GeV at the next-to-leading-order (NLO) level in \(\alpha _s\), within the nonrelativistic QCD framework. It is found that the \({\mathcal {O}} (\alpha ^3\alpha _s)\)-order terms that represent the tree-level interference are comparable with the usual NLO QCD corrections, especially for the \(\chi _{c1}\) and \(\chi _{c2}\) cases. To explore the effect of the higher-order terms, namely \({\mathcal {O}} (\alpha ^3\alpha _s^2)\), we perform the QCD corrections to these \({\mathcal {O}} (\alpha ^3\alpha _s)\)-order terms for the first time, which are found to be able to significantly influence the \({\mathcal {O}} (\alpha ^3\alpha _s)\)-order results. In particular, in the case of \(\chi _{c1}\) and \(\chi _{c2}\), the newly calculated \({\mathcal {O}} (\alpha ^3\alpha _s^2)\)-order terms can to a large extent counteract the \({\mathcal {O}} (\alpha ^3\alpha _s)\) contributions, evidently indicating the indispensability of the corrections. In addition, we find that, as the collision energy rises, the percentage of the interference effect in the total cross section will increase rapidly, especially for the \(\chi _{c1}\) case.     

Spin-label Order Parameter Calibrations for Slow Motion

Calibrations are given to extract orientation order parameters from pseudo-powder electron paramagnetic resonance line shapes of ¹⁴N-nitroxide spin labels undergoing slow rotational diffusion. The nitroxide z-axis is assumed parallel to the long molecular axis. Stochastic-Liouville simulations of slow-motion 9.4-GHz spectra for molecular ordering with a Maier–Saupe orientation potential reveal a linear dependence of the splittings, \(2A_{\hbox{max} }\) and \(2A_{\hbox{min} }\), of the outer and inner peaks on order parameter \(S_{zz}\) that depends on the diffusion coefficient \(D_{{{\text{R}} \bot }}\) which characterizes fluctuations of the long molecular axis. This results in empirical expressions for order parameter and isotropic hyperfine coupling: \(S_{zz} = s_{1} \times \left( {A_{\hbox{max} } - A_{\hbox{min} } } \right) - s_{o}\) and \(a_{o}^{{}} = \tfrac{1}{3}\left( {f_{\hbox{max} } A_{\hbox{max} } + f_{\hbox{min} } A_{\hbox{min} } } \right) + \delta a_{o}\), respectively. Values of the calibration constants \(s_{1}\), \(s_{\text{o}}\), \(f_{\hbox{max} }\), \(f_{\hbox{min} }\) and \(\delta a_{o}\) are given for different values of \(D_{{{\text{R}} \bot }}\) in fast and slow motional regimes. The calibrations are relatively insensitive to anisotropy of rotational diffusion \((D_{{{\text{R}}//}} \ge D_{{{\text{R}} \bot }} )\), and corrections are less significant for the isotropic hyperfine coupling than for the order parameter.     

Scaling Limit of Symmetric Random Walk in High-Contrast Periodic Environment

It is shown that the deterministic infinite trigonometric products

$$\begin{aligned} \prod _{n\in \mathbb {N}}\left[ 1- p +p\cos \left( \textstyle n^{-s}_{_{}}t\right) \right] =: {\text{ Cl }_{p;s}^{}}(t) \end{aligned}$$

with parameters \( p\in (0,1]\ \& \ s>\frac{1}{2}\), and variable \(t\in \mathbb {R}\), are inverse Fourier transforms of the probability distributions for certain random series \(\Omega _{p}^\zeta (s)\) taking values in the real \(\omega \) line; i.e. the \({\text{ Cl }_{p;s}^{}}(t)\) are characteristic functions of the \(\Omega _{p}^\zeta (s)\). The special case \(p=1=s\) yields the familiar random harmonic series, while in general \(\Omega _{p}^\zeta (s)\) is a “random Riemann-\(\zeta \) function,” a notion which will be explained and illustrated—and connected to the Riemann hypothesis. It will be shown that \(\Omega _{p}^\zeta (s)\) is a very regular random variable, having a probability density function (PDF) on the \(\omega \) line which is a Schwartz function. More precisely, an elementary proof is given that there exists some \(K_{p;s}^{}>0\), and a function \(F_{p;s}^{}(|t|)\) bounded by \(|F_{p;s}^{}(|t|)|\!\le \! \exp \big (K_{p;s}^{} |t|^{1/(s+1)})\), and \(C_{p;s}^{}\!:=\!-\frac{1}{s}\int _0^\infty \ln |{1-p+p\cos \xi }|\frac{1}{\xi ^{1+1/s}}\mathrm{{d}}\xi \), such that

$$\begin{aligned} \forall \,t\in \mathbb {R}:\quad {\text{ Cl }_{p;s}^{}}(t) = \exp \bigl ({- C_{p;s}^{} \,|t|^{1/s}\bigr )F_{p;s}^{}(|t|)}; \end{aligned}$$

the regularity of \(\Omega _{p}^\zeta (s)\) follows. Incidentally, this theorem confirms a surmise by Benoit Cloitre, that \(\ln {\text{ Cl }_{{{1}/{3}};2}^{}}(t) \sim -C\sqrt{t}\; \left( t\rightarrow \infty \right) \) for some \(C>0\). Graphical evidence suggests that \({\text{ Cl }_{{{1}/{3}};2}^{}}(t)\) is an empirically unpredictable (chaotic) function of t. This is reflected in the rich structure of the pertinent PDF (the Fourier transform of \({\text{ Cl }_{{{1}/{3}};2}^{}}\)), and illustrated by random sampling of the Riemann-\(\zeta \) walks, whose branching rules allow the build-up of fractal-like structures.

     

Uncertainties of the $$B\rightarrow D$$B→ D transition form factor from the D-meson leading-twist distribution amplitude

The \(B\rightarrow D\) transition form factor (TFF) \(f^{B\rightarrow D}_+(q^2)\) is determined mainly by the D-meson leading-twist distribution amplitude (DA) , \(\phi _{2;D}\), if the proper chiral current correlation function is adopted within the light-cone QCD sum rules. It is therefore significant to make a comprehensive study of DA \(\phi _{2;D}\) and its impact on \(f^{B\rightarrow D}_+(q^2)\). In this paper, we calculate the moments of \(\phi _{2;D}\) with the QCD sum rules under the framework of the background field theory. New sum rules for the leading-twist DA moments \(\left\langle \xi ^n\right\rangle _D\) up to fourth order and up to dimension-six condensates are presented. At the scale \(\mu = 2 \,\mathrm{GeV}\), the values of the first four moments are: \(\left\langle \xi ^1\right\rangle _D = -0.418^{+0.021}_{-0.022}\), \(\left\langle \xi ^2\right\rangle _D = 0.289^{+0.023}_{-0.022}\), \(\left\langle \xi ^3\right\rangle _D = -0.178 \pm 0.010\) and \(\left\langle \xi ^4\right\rangle _D = 0.142^{+0.013}_{-0.012}\). Basing on the values of \(\left\langle \xi ^n\right\rangle _D(n=1,2,3,4)\), a better model of \(\phi _{2;D}\) is constructed. Applying this model for the TFF \(f^{B\rightarrow D}_+(q^2)\) under the light cone sum rules, we obtain \(f^{B\rightarrow D}_+(0) = 0.673^{+0.038}_{-0.041}\) and \(f^{B\rightarrow D}_+(q^2_{\mathrm{max}}) = 1.117^{+0.051}_{-0.054}\). The uncertainty of \(f^{B\rightarrow D}_+(q^2)\) from \(\phi _{2;D}\) is estimated and we find its impact should be taken into account, especially in low and central energy region. The branching ratio \(\mathcal {B}(B\rightarrow Dl\bar{\nu }_l)\) is calculated, which is consistent with experimental data.     

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.     

Holography of massive M2-brane theory: non-linear extension

We investigate the gauge/gravity duality between the \(\mathcal{N} = 6\) mass-deformed ABJM theory with \(\hbox {U}_k(N)\times \hbox {U}_{-k}(N)\) gauge symmetry and the 11-dimensional supergravity on LLM geometries with SO(2,1)\(\times \)SO(4)/\({\mathbb {Z}}_k\) \(\times \)SO(4)/\({\mathbb {Z}}_k\) isometry, in terms of a KK holography, which involves quadratic order field redefinitions. We establish the quadratic order KK mappings for various gauge invariant fields in order to obtain the canonical 4-dimensional gravity equations of motion and to reduce the LLM solutions to an asymptotically AdS\(_4\) gravity solutions. The non-linearity of the KK maps indicates that we can observe the true purpose of the non-linear KK holography of the LLM solutions. We read the vacuum expectation value of conformal dimension two operator from the asymptotically AdS\(_4\) gravity solutions. For the LLM solutions which are represented by square-shaped Young diagrams, we compare the vacuum expectation value obtained from the holographic procedure with the result obtained from the field theory, which is given by \(\langle \mathcal{O}^{(\Delta =2)}\rangle =\sqrt{k}N^{\frac{3}{2}}f_{(\Delta =2)}+\mathcal{O}(N)\), where \(f_{\Delta }\) is independent of N. Based on this result, we examine the gauge/gravity duality in the large N limit and finite k. We also show that the vacuum expectation values of the massive KK graviton modes are vanishing as expected by the supersymmetry.