Likelihood analysis of the sub-GUT MSSM in light of LHC 13-TeV data

We describe a likelihood analysis using MasterCode of variants of the MSSM in which the soft supersymmetry-breaking parameters are assumed to have universal values at some scale \(M_\mathrm{in}\) below the supersymmetric grand unification scale \(M_\mathrm{GUT}\), as can occur in mirage mediation and other models. In addition to \(M_\mathrm{in}\), such ‘sub-GUT’ models have the 4 parameters of the CMSSM, namely a common gaugino mass \(m_{1/2}\), a common soft supersymmetry-breaking scalar mass \(m_0\), a common trilinear mixing parameter A and the ratio of MSSM Higgs vevs \(\tan \beta \), assuming that the Higgs mixing parameter \(\mu > 0\). We take into account constraints on strongly- and electroweakly-interacting sparticles from \(\sim 36\)/fb of LHC data at 13 TeV and the LUX and 2017 PICO, XENON1T and PandaX-II searches for dark matter scattering, in addition to the previous LHC and dark matter constraints as well as full sets of flavour and electroweak constraints. We find a preference for \(M_\mathrm{in}\sim 10^5\) to \(10^9 \,\, \mathrm {GeV}\), with \(M_\mathrm{in}\sim M_\mathrm{GUT}\) disfavoured by \(\Delta \chi ^2 \sim 3\) due to the \(\mathrm{BR}(B_{s, d} \rightarrow \mu ^+\mu ^-)\) constraint. The lower limits on strongly-interacting sparticles are largely determined by LHC searches, and similar to those in the CMSSM. We find a preference for the LSP to be a Bino or Higgsino with \(m_{\tilde{\chi }^0_{1}} \sim 1 \,\, \mathrm {TeV}\), with annihilation via heavy Higgs bosons H / A and stop coannihilation, or chargino coannihilation, bringing the cold dark matter density into the cosmological range. We find that spin-independent dark matter scattering is likely to be within reach of the planned LUX-Zeplin and XENONnT experiments. We probe the impact of the \((g-2)_\mu \) constraint, finding similar results whether or not it is included.     

Likelihood analysis of the pMSSM11 in light of LHC 13-TeV data

We use MasterCode to perform a frequentist analysis of the constraints on a phenomenological MSSM model with 11 parameters, the pMSSM11, including constraints from \(\sim 36\)/fb of LHC data at 13 TeV and PICO, XENON1T and PandaX-II searches for dark matter scattering, as well as previous accelerator and astrophysical measurements, presenting fits both with and without the \((g-2)_\mu \) constraint. The pMSSM11 is specified by the following parameters: 3 gaugino masses \(M_{1,2,3}\), a common mass for the first-and second-generation squarks \(m_{\tilde{q}}\) and a distinct third-generation squark mass \(m_{\tilde{q}_3}\), a common mass for the first-and second-generation sleptons \(m_{\tilde{\ell }}\) and a distinct third-generation slepton mass \(m_{\tilde{\tau }}\), a common trilinear mixing parameter A, the Higgs mixing parameter \(\mu \), the pseudoscalar Higgs mass \(M_A\) and \(\tan \beta \). In the fit including \((g-2)_\mu \), a Bino-like \(\tilde{\chi }^0_{1}\) is preferred, whereas a Higgsino-like \(\tilde{\chi }^0_{1}\) is mildly favoured when the \((g-2)_\mu \) constraint is dropped. We identify the mechanisms that operate in different regions of the pMSSM11 parameter space to bring the relic density of the lightest neutralino, \(\tilde{\chi }^0_{1}\), into the range indicated by cosmological data. In the fit including \((g-2)_\mu \), coannihilations with \(\tilde{\chi }^0_{2}\) and the Wino-like \(\tilde{\chi }^\pm _{1}\) or with nearly-degenerate first- and second-generation sleptons are active, whereas coannihilations with the \(\tilde{\chi }^0_{2}\) and the Higgsino-like \(\tilde{\chi }^\pm _{1}\) or with first- and second-generation squarks may be important when the \((g-2)_\mu \) constraint is dropped. In the two cases, we present \(\chi ^2\) functions in two-dimensional mass planes as well as their one-dimensional profile projections and best-fit spectra. Prospects remain for discovering strongly-interacting sparticles at the LHC, in both the scenarios with and without the \((g-2)_\mu \) constraint, as well as for discovering electroweakly-interacting sparticles at a future linear \(e^+ e^-\) collider such as the ILC or CLIC.     

CP violations in predictive neutrino mass structures

We study the CP-violation effects from two types of neutrino mass matrices with (i) \((M_\nu )_{ee}=0\), and (ii) \((M_\nu )_{ee}=(M_\nu )_{e\mu }=0\), which can be realized by the high-dimensional lepton number violating operators \(\bar{\ell }_R^c\gamma ^\mu L_L (D_\mu \Phi )\Phi ^2\) and \(\bar{\ell }_R^c l_R (D_\mu {\Phi })^2\Phi ^2\), respectively. In (i), the neutrino mass spectrum is in the normal ordering with the lightest neutrino mass within the range \(0.002\,\mathrm{eV}\lesssim m_0\lesssim 0.007\,\mathrm{eV}\). Furthermore, for a given value of \(m_0\), there are two solutions for the two Majorana phases \(\alpha _{21}\) and \(\alpha _{31}\), whereas the Dirac phase \(\delta \) is arbitrary. For (ii), the parameters of \(m_0\), \(\delta \), \(\alpha _{21}\), and \(\alpha _{31}\) can be completely determined. We calculate the CP-violating asymmetries in neutrino–antineutrino oscillations for both mass textures of (i) and (ii), which are closely related to the CP-violating Majorana phases.     

Gluino reach and mass extraction at the LHC in radiatively-driven natural SUSY

Radiatively-driven natural SUSY (RNS) models enjoy electroweak naturalness at the 10% level while respecting LHC sparticle and Higgs mass constraints. Gluino and top-squark masses can range up to several TeV (with other squarks even heavier) but a set of light Higgsinos are required with mass not too far above \(m_h\sim 125\) GeV. Within the RNS framework, gluinos dominantly decay via \(\tilde{g}\rightarrow t\tilde{t}_1^{*},\ \bar{t}\tilde{t}_1 \rightarrow t\bar{t}\widetilde{Z}_{1,2}\) or \(t\bar{b}\widetilde{W}_1^-+c.c.\), where the decay products of the higgsino-like \(\widetilde{W}_1\) and \(\widetilde{Z}_2\) are very soft. Gluino pair production is, therefore, signaled by events with up to four hard b-jets and large \(\not \!\!{E_T}\). We devise a set of cuts to isolate a relatively pure gluino sample at the (high-luminosity) LHC and show that in the RNS model with very heavy squarks, the gluino signal will be accessible for \(m_{\tilde{g}} < 2400 \ (2800)\) GeV for an integrated luminosity of 300 (3000) fb\(^{-1}\). We also show that the measurement of the rate of gluino events in the clean sample mentioned above allows for a determination of \(m_{\tilde{g}}\) with a statistical precision of 2–5% (depending on the integrated luminosity and the gluino mass) over the range of gluino masses where a 5\(\sigma \) discovery is possible at the LHC.     

$$\varvec{B^0\rightarrow K^{*0}\mu ^+\mu ^-}$$ 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.     

Fully constrained Majorana neutrino mass matrices using $$\varvec{\varSigma (72\times 3)}$$

In 2002, two neutrino mixing ansatze having trimaximally mixed middle (\(\nu _2\)) columns, namely tri-chi-maximal mixing (\(\text {T}\chi \text {M}\)) and tri-phi-maximal mixing (\(\text {T}\phi \text {M}\)), were proposed. In 2012, it was shown that \(\text {T}\chi \text {M}\) with \(\chi =\pm \,\frac{\pi }{16}\) as well as \(\text {T}\phi \text {M}\) with \(\phi = \pm \,\frac{\pi }{16}\) leads to the solution, \(\sin ^2 \theta _{13} = \frac{2}{3} \sin ^2 \frac{\pi }{16}\), consistent with the latest measurements of the reactor mixing angle, \(\theta _{13}\). To obtain \(\text {T}\chi \text {M}_{(\chi =\pm \,\frac{\pi }{16})}\) and \(\text {T}\phi \text {M}_{(\phi =\pm \,\frac{\pi }{16})}\), the type I see-saw framework with fully constrained Majorana neutrino mass matrices was utilised. These mass matrices also resulted in the neutrino mass ratios, \(m_1:m_2:m_3=\frac{\left( 2+\sqrt{2}\right) }{1+\sqrt{2(2+\sqrt{2})}}:1:\frac{\left( 2+\sqrt{2}\right) }{-1+\sqrt{2(2+\sqrt{2})}}\). In this paper we construct a flavour model based on the discrete group \(\varSigma (72\times 3)\) and obtain the aforementioned results. A Majorana neutrino mass matrix (a symmetric \(3\times 3\) matrix with six complex degrees of freedom) is conveniently mapped into a flavon field transforming as the complex six-dimensional representation of \(\varSigma (72\times 3)\). Specific vacuum alignments of the flavons are used to arrive at the desired mass matrices.     

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.     

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 \(\Omega =2^{-4},...,2^{7}\).     

Scattering of kinks of the sinh-deformed $$\varphi ^4$$ model

We consider the scattering of kinks of the sinh-deformed \(\varphi ^4\) model, which is obtained from the well-known \(\varphi ^4\) model by means of the deformation procedure. Depending on the initial velocity \(v_\mathrm {in}\) of the colliding kinks, different collision scenarios are realized. There is a critical value \(v_\mathrm {cr}\) of the initial velocity, which separates the regime of reflection (at \(v_\mathrm {in}>v_\mathrm {cr}\)) and that of a complicated interaction (at \(v_\mathrm {in}<v_\mathrm {cr}\)) with kinks’ capture and escape windows. Besides that, at \(v_\mathrm {in}\) below \(v_\mathrm {cr}\) we observe the formation of a bound state of two oscillons, as well as their escape at some values of \(v_\mathrm {in}\).     

Signals of leptophilic dark matter at the ILC

Adopting a model independent approach, we constrain the various effective interactions of leptophilic DM particles with the visible world from the WMAP and Planck data. The thermally averaged indirect DM annihilation cross section and the DM–electron direct-detection cross section for such a DM candidate are observed to be consistent with the respective experimental data. We study the production of cosmologically allowed leptophilic DM in association with \(Z\, (Z\rightarrow f\bar{f})\), \(f\equiv q,\,e^-,\, \mu ^-\) at the ILC. We perform the \(\chi ^2\) analysis and compute the 99% C.L. acceptance contours in the \(m_\chi \) and \(\varLambda \) plane from the two-dimensional differential distributions of various kinematic observables obtained after employing parton showering and hadronisation to the simulated data. We observe that the dominant hadronic channel provides the best kinematic reach of 2.62 TeV (\(m_\chi \) = 25 GeV), which further improves to \(\sim \)3 TeV for polarised beams at \(\sqrt{s} = 1\) TeV and an integrated luminosity of 1 ab\(^{-1}\).     

Noncommutative products of Euclidean spaces

We present natural families of coordinate algebras on noncommutative products of Euclidean spaces \({\mathbb {R}}^{N_1} \times _{\mathcal {R}} {\mathbb {R}}^{N_2}\). These coordinate algebras are quadratic ones associated with an \(\mathcal {R}\)-matrix which is involutive and satisfies the Yang–Baxter equations. As a consequence, they enjoy a list of nice properties, being regular of finite global dimension. Notably, we have eight-dimensional noncommutative euclidean spaces \({\mathbb {R}}^{4} \times _{\mathcal {R}} {\mathbb {R}}^{4}\). Among these, particularly well behaved ones have deformation parameter \(\mathbf{u} \in {\mathbb {S}}^2\). Quotients include seven spheres \({\mathbb {S}}^{7}_\mathbf{u}\) as well as noncommutative quaternionic tori \({\mathbb {T}}^{{\mathbb {H}}}_\mathbf{u} = {\mathbb {S}}^3 \times _\mathbf{u} {\mathbb {S}}^3\). There is invariance for an action of \({{\mathrm{SU}}}(2) \times {{\mathrm{SU}}}(2)\) on the torus \({\mathbb {T}}^{{\mathbb {H}}}_\mathbf{u}\) in parallel with the action of \(\mathrm{U}(1) \times \mathrm{U}(1)\) on a ‘complex’ noncommutative torus \({\mathbb {T}}^2_\theta \) which allows one to construct quaternionic toric noncommutative manifolds. Additional classes of solutions are disjoint from the classical case.     

Photoreflectance spectroscopy of the chalcopyrite semiconductor <Emphasis Type="Bold">AgInS</Emphasis><Subscript>2</Subscript> for ordinary and extraordinary rays

Photoreflectance spectra have been measured on the chalcopyrite semiconductor silver indium disulfide (\(\hbox {AgInS}_{2}\)) for light polarization \({\varvec{E}}\) perpendicular (\({\varvec{E}} \bot {c}\)) and parallel to the c-axis (\({\varvec{E}} \vert \vert {c}\)) at temperature between 10 and 300 K. The measured photoreflectance spectra revealed distinct structures at 1.8–2.1 eV. The lowest bandgap energies \(E_{0A}\), \(E_{0B}\), and \(E_{0C}\) of \(\hbox {AgInS}_{2}\) show unusual temperature dependence at low temperatures (\(\le\)140 K). The \(E_{0\alpha }\) (\(\alpha =A, B, C\)) is found to increase with increasing temperature from 10 to \(\sim\)140 K and decreases with a further increase in temperature. This result has been successfully explained by taking into account the effects of thermal expansion and electron–phonon interaction. The spin–orbit and crystal-field splitting parameters of \(\hbox {AgInS}_{2}\) are determined to be \(\Delta _{{\mathrm{so}}}=38\) meV and \(\Delta _{{\mathrm{cr}}}=-168\) meV at T = 10 K, respectively, and are discussed from an aspect of the electronic energy band structure consequences. The temperature dependence of spin–orbit and crystal-field splitting parameters of \(\hbox {AgInS}_{2}\) was also presented.     

Stochastic Averaging Principle for Spatial Birth-and-Death Evolutions in the Continuum

We study a spatial birth-and-death process on the phase space of locally finite configurations \({\varGamma }^+ \times {\varGamma }^-\) over \({\mathbb {R}}^d\). Dynamics is described by an non-equilibrium evolution of states obtained from the Fokker-Planck equation and associated with the Markov operator \(L^+(\gamma ^-) + \frac{1}{\varepsilon }L^-\), \(\varepsilon > 0\). Here \(L^-\) describes the environment process on \({\varGamma }^-\) and \(L^+(\gamma ^-)\) describes the system process on \({\varGamma }^+\), where \(\gamma ^-\) indicates that the corresponding birth-and-death rates depend on another locally finite configuration \(\gamma ^- \in {\varGamma }^-\). We prove that, for a certain class of birth-and-death rates, the corresponding Fokker-Planck equation is well-posed, i.e. there exists a unique evolution of states \(\mu _t^{\varepsilon }\) on \({\varGamma }^+ \times {\varGamma }^-\). Moreover, we give a sufficient condition such that the environment is ergodic with exponential rate. Let \(\mu _{\mathrm {inv}}\) be the invariant measure for the environment process on \({\varGamma }^-\). In the main part of this work we establish the stochastic averaging principle, i.e. we prove that the marginal of \(\mu _t^{\varepsilon }\) onto \({\varGamma }^+\) converges weakly to an evolution of states on \({\varGamma }^+\) associated with the averaged Markov birth-and-death operator \({\overline{L}} = \int _{{\varGamma }^-}L^+(\gamma ^-)d \mu _{\mathrm {inv}}(\gamma ^-)\).     

Induced-gravity GUT-scale Higgs inflation in supergravity

Models of induced-gravity inflation are formulated within Supergravity employing as inflaton the Higgs field which leads to a spontaneous breaking of a \(U(1)_{B-L}\) symmetry at \(M_\mathrm{GUT}=2\cdot 10^{16}~{\mathrm{GeV}}\). We use a renormalizable superpotential, fixed by a U(1) R symmetry, and Kähler potentials which exhibit a quadratic non-minimal coupling to gravity with or without an independent kinetic mixing in the inflaton sector. In both cases we find inflationary solutions of Starobinsky type whereas in the latter case, others (more marginal) which resemble those of linear inflation arise too. In all cases the inflaton mass is predicted to be of the order of \(10^{13}~{\mathrm{GeV}}\). Extending the superpotential of the model with suitable terms, we show how the MSSM \(\mu \) parameter can be generated. Also, non-thermal leptogenesis can be successfully realized, provided that the gravitino is heavier than about \(10~{\mathrm{TeV}}\).     

NLO+NLL collider bounds,Dirac fermion and scalar dark matter in the B–L model

Baryon and lepton numbers being accidental global symmetries of the Standard Model (SM), it is natural to promote them to local symmetries. However, to preserve anomaly-freedom, only combinations of B–L are viable. In this spirit, we investigate possible dark matter realizations in the context of the \(U(1)_\mathrm{B{-}L}\) model: (i) Dirac fermion with unbroken B–L; (ii) Dirac fermion with broken B–L; (iii) scalar dark matter; (iv) two-component dark matter. We compute the relic abundance, direct and indirect detection observables and confront them with recent results from Planck, LUX-2016, and Fermi-LAT and prospects from XENON1T. In addition to the well-known LEP bound \(M_{Z^{\prime }}/g_\mathrm{BL} \gtrsim 7\) TeV, we include often ignored LHC bounds using 13 TeV dilepton (dimuon + dielectron) data at next-to-leading order plus next-to-leading logarithmic accuracy. We show that, for gauge couplings smaller than 0.4, the LHC gives rise to the strongest collider limit. In particular, we find \(M_{Z^{\prime }}/g_\mathrm{BL} > 8.7\) TeV for \(g_\mathrm{BL}=0.3\). We conclude that the NLO+NLL corrections improve the dilepton bounds on the \(Z^{\prime }\) mass and that both dark matter candidates are only viable in the \(Z^{\prime }\) resonance region, with the parameter space for scalar dark matter being fully probed by XENON1T. Lastly, we show that one can successfully have a minimal two-component dark matter model.     

Searching for cosmological preferred axis using cosmographic approach

Recent released Planck data and other astronomical observations show that the universe may be anisotropic on large scales. This hints a cosmological privileged axis in our anisotropic expanding universe. This paper proceeds a modified redshift in anisotropic cosmological model as \( 1+\tilde{z}(t,\hat{\mathbf{p }})=\frac{a(t_{0)}}{a(t)}(1-A(\hat{\mathbf{n }}.\hat{\mathbf{p }}))\) (where A is the magnitude of anisotropy, \(\hat{\mathbf{n }}\) is the direction of privileged axis, and \(\hat{\mathbf{p }}\) is the direction of each SNe Ia sample to galactic coordinates) along with anisotropic parameter \(\delta =\frac{A(\hat{\mathbf{n }}.\hat{\mathbf{p }})}{1+A(\hat{\mathbf{n }}.\hat{\mathbf{p }})}\). The luminosity distance is expanded with model-independent cosmographic parameters as a function of modified redshift \(\tilde{z}\). As the transformation matrix \(M(n\times n)\) is obtained to convert the Taylor series coefficients of isotropic luminosity distance to corresponding anisotropic parameters. These results culminate the magnitude of anisotropy about \(\mid A\mid \simeq 10^{-3}\) and the direction of preferred axis as \((l,b)=\left( 297^{-34}_{+34},3^{-28}_{+28}\right) \), which are consistent with other studies in \(1-\sigma \) confidence level.     

Isospin analysis of charmless <Emphasis Type="Italic">B</Emphasis>-meson decays

We discuss the determination of the CKM angle \(\alpha \) using the non-leptonic two-body decays \(B\rightarrow \pi \pi \), \(B\rightarrow \rho \rho \) and \(B\rightarrow \rho \pi \) using the latest data available. We illustrate the methods used in each case and extract the corresponding value of \(\alpha \). Combining all these elements, we obtain the determination \(\alpha _\mathrm{dir}={({86.2}_{-4.0}^{+4.4} \cup {178.4}_{-5.1}^{+3.9})}^{\circ }\). We assess the uncertainties associated to the breakdown of the isospin hypothesis and the choice of the statistical framework in detail. We also determine the hadronic amplitudes (tree and penguin) describing the QCD dynamics involved in these decays, briefly comparing our results with theoretical expectations. For each observable of interest in the \(B\rightarrow \pi \pi \), \(B\rightarrow \rho \rho \) and \(B\rightarrow \rho \pi \) systems, we perform an indirect determination based on the constraints from all the other observables available and we discuss the compatibility between indirect and direct determinations. Finally, we review the impact of future improved measurements on the determination of \(\alpha \).     

Light deflection and Gauss–Bonnet theorem: definition of total deflection angle and its applications

In this paper, we re-examine the light deflection in the Schwarzschild and the Schwarzschild–de Sitter spacetime. First, supposing a static and spherically symmetric spacetime, we propose the definition of the total deflection angle \(\alpha \) of the light ray by constructing a quadrilateral \(\varSigma ^4\) on the optical reference geometry \({\mathscr {M}}^\mathrm{opt}\) determined by the optical metric \(\bar{g}_{ij}\). On the basis of the definition of the total deflection angle \(\alpha \) and the Gauss–Bonnet theorem, we derive two formulas to calculate the total deflection angle \(\alpha \); (1) the angular formula that uses four angles determined on the optical reference geometry \({\mathscr {M}}^\mathrm{opt}\) or the curved \((r, \phi )\) subspace \({\mathscr {M}}^\mathrm{sub}\) being a slice of constant time t and (2) the integral formula on the optical reference geometry \({\mathscr {M}}^\mathrm{opt}\) which is the areal integral of the Gaussian curvature K in the area of a quadrilateral \(\varSigma ^4\) and the line integral of the geodesic curvature \(\kappa _g\) along the curve \(C_{\varGamma }\). As the curve \(C_{\varGamma }\), we introduce the unperturbed reference line that is the null geodesic \(\varGamma \) on the background spacetime such as the Minkowski or the de Sitter spacetime, and is obtained by projecting \(\varGamma \) vertically onto the curved \((r, \phi )\) subspace \({\mathscr {M}}^\mathrm{sub}\). We demonstrate that the two formulas give the same total deflection angle \(\alpha \) for the Schwarzschild and the Schwarzschild–de Sitter spacetime. In particular, in the Schwarzschild case, the result coincides with Epstein–Shapiro’s formula when the source S and the receiver R of the light ray are located at infinity. In addition, in the Schwarzschild–de Sitter case, there appear order \({\mathscr {O}}(\varLambda m)\) terms in addition to the Schwarzschild-like part, while order \({\mathscr {O}}(\varLambda )\) terms disappear.     

Asymptotic behavior of the Schrödinger–Debye system with refractive index of quadratic wave amplitude

We obtain local well-posedness for the one-dimensional Schrödinger–Debye interactions in nonlinear optics in the spaces \(L^2\times L^p,\; 1\le p < \infty \). When \(p=1\) we show that the local solutions extend globally. In the focusing regime, we consider a family of solutions \(\{(u_{\tau }, v_{\tau })\}_{\tau >0}\) in \( H^1\times H^1\) associated to an initial data family \(\{(u_{\tau _0},v_{\tau _0})\}_{\tau >0}\) uniformly bounded in \(H^1\times L^2\), where \(\tau \) is a small response time parameter. We prove that \(\left( u_{\tau }, v_{\tau }\right) \) converges to \(\left( u, -|u|^2\right) \) in the space \(L^{\infty }_{[0, T]}L^2_x\times L^1_{[0, T]}L^2_x\) whenever \(u_{\tau _0}\) converges to \(u_0\) in \(H^1\) as long as \(\tau \) tends to 0, where u is the solution of the one-dimensional cubic nonlinear Schrödinger equation with the initial data \(u_0\). The convergence of \(v_{\tau }\) for \(-|u|^2\) in the space \(L^{\infty }_{[0, T]}L^2_x\) is shown under compatibility conditions of the initial data. For non-compatible data, we prove convergence except for a corrector term which looks like an initial layer phenomenon.     

Two-body decays of gluino at full one-loop level in the quark-flavour violating MSSM

We study the two-body decays of the gluino at full one-loop level in the Minimal Supersymmetric Standard Model with quark-flavour violation (QFV) in the squark sector. The renormalisation is done in the \(\overline{\mathrm{DR}}\) scheme. The gluon and photon radiations are included by adding the corresponding three-body decay widths. We discuss the dependence of the gluino decay widths on the QFV parameters. The main dependence stems from the \(\tilde{c}_R \)–\( \tilde{t}_R\) mixing in the decays to up-type squarks, and from the \(\tilde{s}_R \)–\( \tilde{b}_R\) mixing in the decays to down-type squarks due to the strong constraints from B-physics on the other quark-flavour-mixing parameters. The full one-loop corrections to the gluino decay widths are mostly negative and of the order of about ?10%. The QFV part stays small in the total width but can vary up to ?8% for the decay width into the lightest \(\tilde{u}\) squark. For the corresponding branching ratio the effect is somehow washed out by at least a factor of two. The electroweak corrections can be as large as 35% of the SUSY QCD corrections.