Lepton polarization asymmetries in rare semi-tauonic $$ b \rightarrow s $$ exclusive decays at FCC- ee

We consider measurements of exclusive rare semi-tauonic b-hadron decays, mediated by the \(b \rightarrow s \tau ^+ \tau ^-\) transition, at a future high-energy circular electron–positron collider (FCC-ee). We argue that the high boosts of b-hadrons originating from on-shell Z boson decays allow for a full reconstruction of the decay kinematics in hadronic \(\tau \) decay modes (up to discrete ambiguities). This, together with the potentially large statistics of \(Z\rightarrow b\bar{b}\), opens the door for the experimental determination of \(\tau \) polarizations in these rare b-hadron decays. In the light of the current experimental situation on lepton flavor universality in rare semileptonic B decays, we discuss the complementary short-distance physics information carried by the \(\tau \) polarizations and suggest suitable theoretically clean observables in the form of single- and double-\(\tau \) polarization asymmetries.     

Probing the Higgs sector of $$$$ Higgs triplet model at LHC

In this paper, we investigate the Higgs Triplet Model with hypercharge \(Y_{\varDelta }=0\) (HTM0), an extension of the Standard model, caracterized by a more involved scalar spectrum consisting of two CP even Higgs \(h^0, H^0\) and two charged Higgs bosons \(H^\pm \). We first show that the parameter space of HTM0, usually delimited by combined constraints originating from unitarity and BFB as well as experimental limits from LEP and LHC, is severely reduced when the modified Veltman conditions at one loop are also imposed. Then, we perform an rigorous analysis of Higgs decays either when \(h^0\) is the SM-like or when the heaviest neutral Higgs \(H^0\) is identified to the observed 125 GeV Higgs boson at LHC. In these scenarios, we perform an extensive parameter scan, in the lower part of the scalar mass spectrum, with a particular focus on the Higgs to Higgs decay modes \(H^0 \rightarrow h^0h^0, H^\pm \,H^\mp \) leading predominantly to invisible Higgs decays. Finally, we also study the scenario where \(h^0, H^0\) are mass degenerate. We thus find that consistency with LHC signal strengths favours a light charged Higgs with a mass about 176–178 GeV. Our analysis shows that the diphoton Higgs decay mode and \(H \rightarrow Z \gamma \) are not always positively correlated as claimed in a previous study. Anti-correlation is rather seen in the scenario where h is SM like, while correlation is sensitive to the sign of the potential parameter \(\lambda \) when H is identified to 125 GeV observed Higgs.     

Chargino and neutralino production at $$e^+e^-$$ colliders in the complex MSSM: a full one-loop analysis

For the search for charginos and neutralinos in the Minimal Supersymmetric Standard Model (MSSM) as well as for future precision analyses of these particles an accurate knowledge of their production and decay properties is mandatory. We evaluate the cross sections for the chargino and neutralino production at \(e^+e^-\) colliders in the MSSM with complex parameters (cMSSM). The evaluation is based on a full one-loop calculation of the production mechanisms \(e^+e^- \rightarrow {\tilde{\chi }}_{c}^\pm {\tilde{\chi }}_{c^\prime }^\mp \) and \(e^+e^- \rightarrow {\tilde{\chi }}_{n}^0 {\tilde{\chi }}_{n^\prime }^0\)  including soft and hard photon radiation. We mostly restricted ourselves to a version of our renormalization scheme which is valid for \(|M_1| < |M_2|, |\mu |\) and \(M_2 \ne \mu \) to simplify the analysis, even though we are able to switch to other parameter regions and correspondingly different renormalization schemes. The dependence of the chargino/neutralino cross sections on the relevant cMSSM parameters is analyzed numerically. We find sizable contributions to many production cross sections. They amount to roughly \({\pm }15\%\) of the tree-level results but can go up to \({\pm }40\%\) or higher in extreme cases. Also the complex phase dependence of the one-loop corrections was found non-negligible. The full one-loop contributions are thus crucial for physics analyses at a future linear \(e^+e^-\) collider such as the ILC or CLIC.     

Electroweak vacuum stability in classically conformal $$$$ extension of the standard model

We consider the minimal U(1)\(_{B-L}\) extension of the standard model (SM) with the classically conformal invariance, where an anomaly-free U(1)\(_{B-L}\) gauge symmetry is introduced along with three generations of right-handed neutrinos and a U(1)\(_{B-L}\) Higgs field. Because of the classically conformal symmetry, all dimensional parameters are forbidden. The \(B-L\) gauge symmetry is radiatively broken through the Coleman–Weinberg mechanism, generating the mass for the \(U(1)_{B-L}\) gauge boson (\(Z^\prime \) boson) and the right-handed neutrinos. Through a small negative coupling between the SM Higgs doublet and the \(B-L\) Higgs field, the negative mass term for the SM Higgs doublet is generated and the electroweak symmetry is broken. In this model context, we investigate the electroweak vacuum instability problem in the SM. It is well known that in the classically conformal U(1)\(_{B-L}\) extension of the SM, the electroweak vacuum remains unstable in the renormalization group analysis at the one-loop level. In this paper, we extend the analysis to the two-loop level, and perform parameter scans. We identify a parameter region which not only solve the vacuum instability problem, but also satisfy the recent ATLAS and CMS bounds from search for \(Z^\prime \) boson resonance at the LHC Run-2. Considering self-energy corrections to the SM Higgs doublet through the right-handed neutrinos and the \(Z^\prime \) boson, we derive the naturalness bound on the model parameters to realize the electroweak scale without fine-tunings.     

Higgs–photon resonances

We study models that produce a Higgs boson plus photon (\(h^0\gamma \)) resonance at the LHC. When the resonance is a \(Z'\) boson, decays to \(h^0\gamma \) occur at one loop. If the \(Z'\) boson couples at tree level to quarks, then the \(h^0\gamma \) branching fraction is typically of order \(10^{-5}\) or smaller. Nevertheless, there are models that would allow the observation of \(Z'\rightarrow \,h^0\gamma \) at \(\sqrt{s}=13\) TeV with a cross section times branching fraction larger than 1 fb for a \(Z'\) mass in the 200–450 GeV range, and larger than 0.1 fb for a mass up to 800 GeV. The one-loop decay of the \(Z'\) into lepton pairs competes with \(h^0\gamma \), even if the \(Z'\) couplings to leptons vanish at tree level. We also present a model in which a \(Z'\) boson decays into a Higgs boson and a pair of collimated photons, mimicking an \(h^0\gamma \) resonance. In this model, the \(h^0\gamma \) resonance search would be the discovery mode for a \(Z'\) as heavy as 2 TeV. When the resonance is a scalar, although decay to \(h^0\gamma \) is forbidden by angular momentum conservation, the \(h^0\) plus collimated photons channel is allowed. We comment on prospects of observing an \(h^0\gamma \) resonance through different Higgs decays, on constraints from related searches, and on models where \(h^0\) is replaced by a nonstandard Higgs boson.     

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.     

Large-Time Behavior of Solutions to Vlasov-Poisson-Fokker-Planck Equations: From Evanescent Collisions to Diffusive Limit

The present contribution investigates the dynamics generated by the two-dimensional Vlasov-Poisson-Fokker-Planck equation for charged particles in a steady inhomogeneous background of opposite charges. We provide global in time estimates that are uniform with respect to initial data taken in a bounded set of a weighted \(L^2\) space, and where dependencies on the mean-free path \(\tau \) and the Debye length \(\delta \) are made explicit. In our analysis the mean free path covers the full range of possible values: from the regime of evanescent collisions \(\tau \rightarrow \infty \) to the strongly collisional regime \(\tau \rightarrow 0\). As a counterpart, the largeness of the Debye length, that enforces a weakly nonlinear regime, is used to close our nonlinear estimates. Accordingly we pay a special attention to relax as much as possible the \(\tau \)-dependent constraint on \(\delta \) ensuring exponential decay with explicit \(\tau \)-dependent rates towards the stationary solution. In the strongly collisional limit \(\tau \rightarrow 0\), we also examine all possible asymptotic regimes selected by a choice of observation time scale. Here also, our emphasis is on strong convergence, uniformity with respect to time and to initial data in bounded sets of a \(L^2\) space. Our proofs rely on a detailed study of the nonlinear elliptic equation defining stationary solutions and a careful tracking and optimization of parameter dependencies of hypocoercive/hypoelliptic estimates.     

Two-Higgs-doublet type-II and -III models and $$t\rightarrow c h$$ at the LHC

We study the constraints of the generic two-Higgs-doublet model (2HDM) type-III and the impacts of the new Yukawa couplings. For comparisons, we revisit the analysis in the 2HDM type-II. To understand the influence of all involving free parameters and to realize their correlations, we employ a \(\chi \)-square fitting approach by including theoretical and experimental constraints, such as the S, T, and U oblique parameters, the production of standard model Higgs and its decay to \(\gamma \gamma \), \(WW^*/ZZ^*\), \(\tau ^+\tau ^-\), etc. The errors of the analysis are taken at 68, 95.5, and \(99.7~\%\) confidence levels. Due to the new Yukawa couplings being associated with \(\cos (\beta -\alpha )\) and \(\sin (\beta -\alpha )\), we find that the allowed regions for \(\sin \alpha \) and \(\tan \beta \) in the type-III model can be broader when the dictated parameter \(\chi _F\) is positive; however, for negative \(\chi _F\), the limits are stricter than those in the type-II model. By using the constrained parameters, we find that the deviation from the SM in \(h\rightarrow Z\gamma \) can be of \(\mathcal{O}(10~\%)\). Additionally, we also study the top-quark flavor-changing processes induced at the tree level in the type-III model and find that when all current experimental data are considered, we get \(Br(t\rightarrow c(h, H) )< 10^{-3}\) for \(m_h=125.36\) and \(m_H=150\) GeV, and \(Br(t\rightarrow cA)\) slightly exceeds \(10^{-3}\) for \(m_A =130\) GeV.     

Two Higgs doublet models and $$b\rightarrow s$$ exclusive decays

We computed the leading order Wilson coefficients relevant to all the exclusive \(b\rightarrow s\ell ^+\ell ^-\) decays in the framework of the two Higgs doublet model (2HDM) with a softly broken \(\mathbb {Z}_2\) symmetry by including the \(\mathcal {O}(m_b)\) corrections. We elucidate the issue of appropriate matching between the full and the effective theory when dealing with the (pseudo-)scalar operators for which keeping the external momenta different from zero is necessary. We then make a phenomenological analysis by using the measured \({\mathcal {B}}(B_s\rightarrow \mu ^+\mu ^-)\) and \({\mathcal {B}}(B\rightarrow K \mu ^+\mu ^-)_{\mathrm {high}-q^2}\), for which the hadronic uncertainties are well controlled, and we discuss their impact on various types of 2HDM. A brief discussion of the decays with \(\tau \)-leptons in the final state is provided too.     

Compound Poisson Law for Hitting Times to Periodic Orbits in Two-Dimensional Hyperbolic Systems

We show that a compound Poisson distribution holds for scaled exceedances of observables \(\phi \) uniquely maximized at a periodic point \(\zeta \) in a variety of two-dimensional hyperbolic dynamical systems with singularities \((M,T,\mu )\), including the billiard maps of Sinai dispersing billiards in both the finite and infinite horizon case. The observable we consider is of form \(\phi (z)=-\ln d(z,\zeta )\) where d is a metric defined in terms of the stable and unstable foliation. The compound Poisson process we obtain is a Pólya-Aeppli distibution of index \(\theta \). We calculate \(\theta \) in terms of the derivative of the map T. Furthermore if we define \(M_n=\max \{\phi ,\ldots ,\phi \circ T^n\}\) and \(u_n (\tau )\) by \(\lim _{n\rightarrow \infty } n\mu (\phi >u_n (\tau ) )=\tau \) the maximal process satisfies an extreme value law of form \(\mu (M_n \le u_n)=e^{-\theta \tau }\). These results generalize to a broader class of functions maximized at \(\zeta \), though the formulas regarding the parameters in the distribution need to be modified.     

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.     

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 \(\Omega =2^{-4},...,2^{7}\).     

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.     

On Painlevé/gauge theory correspondence

We elucidate the relation between Painlevé equations and four-dimensional rank one \(\mathcal {N} = 2\) theories by identifying the connection associated with Painlevé isomonodromic problems with the oper limit of the flat connection of the Hitchin system associated with gauge theories and by studying the corresponding renormalization group flow. Based on this correspondence, we provide long-distance expansions at various canonical rays for all Painlevé \(\tau \)-functions in terms of magnetic and dyonic Nekrasov partition functions for \(\mathcal {N} = 2\) SQCD and Argyres–Douglas theories at self-dual Omega background \(\epsilon _1 + \epsilon _2 = 0\) or equivalently in terms of \(c=1\) irregular conformal blocks.     

<Emphasis Type="Italic">Z</Emphasis>′ discovery potential at the LHC in the minimal <Emphasis Type="Italic">B</Emphasis>–<Emphasis Type="Italic">L</Emphasis> extension of the standard model

We present the Large Hadron Collider (LHC) discovery potential in the Z′ sector of a \(U(1)_{B\mbox{--}L}\) enlarged Standard Model (that also includes three heavy Majorana neutrinos and an additional Higgs boson) for \(\sqrt{s}=7\), 10 and 14 TeV centre-of-mass (CM) energies, considering both the \(Z'_{B\mbox{--}L}\rightarrow e^{+}e^{-}\) and \(Z'_{B\mbox{--}L}\rightarrow\mu^{+}\mu^{-}\) decay channels. The comparison of the (irreducible) backgrounds with the expected backgrounds for the DØ experiment at the Tevatron validates our simulation. We propose an alternative analysis that has the potential to improve the DØ sensitivity. Electrons provide a higher sensitivity to smaller couplings at small \(Z'_{B\mbox{--}L}\) boson masses than do muons. The resolutions achievable may allow the \(Z'_{B\mbox{--}L}\) boson width to be measured at smaller masses in the case of electrons in the final state. The run of the LHC at \(\sqrt{s}=7\) TeV, assuming at most \(\int\mathcal{L} \sim1\) fb^?1, will be able to give similar results to those that will be available soon at the Tevatron in the lower mass region, and to extend them for a heavier M _Z′.     

The light CP-even MSSM Higgs mass resummed to fourth logarithmic order

We present the calculation of the light neutral CP-even Higgs mass in the MSSM for a heavy SUSY spectrum by resumming enhanced terms through fourth logarithmic order (N\(^{3}\)LL), keeping terms of leading order in the top Yukawa coupling \(\alpha _t\), and NNLO in the strong coupling \(\alpha _s\). To this goal, the three-loop matching coefficient for the quartic Higgs coupling of the SM to the MSSM is derived to order \(\alpha _t^2\alpha _s^2\) by comparing the perturbative EFT to the fixed-order expression for the Higgs mass. The new matching coefficient is made available through an updated version of the program Himalaya. Numerical effects of the higher-order resummation are studied using specific examples, and sources of theoretical uncertainty on this result are discussed.     

When is a Scale-Free Graph Ultra-Small?

In this paper we study typical distances in the configuration model, when the degrees have asymptotically infinite variance. We assume that the empirical degree distribution follows a power law with exponent \(\tau \in (2,3)\), up to value \(n^{{\beta _n}}\) for some \({\beta _n}\gg (\log n)^{-\gamma }\) and \(\gamma \in (0,1)\). This assumption is satisfied for power law i.i.d. degrees, and also includes truncated power-law empirical degree distributions where the (possibly exponential) truncation happens at \(n^{{\beta _n}}\). These examples are commonly observed in many real-life networks. We show that the graph distance between two uniformly chosen vertices centers around \(2 \log \log (n^{{\beta _n}}) / |\log (\tau -2)| + 1/({\beta _n}(3-\tau ))\), with tight fluctuations. Thus, the graph is an ultrasmall world whenever \(1/{\beta _n}=o(\log \log n)\). We determine the distribution of the fluctuations around this value, in particular we prove these form a sequence of tight random variables with distributions that show \(\log \log \)-periodicity, and as a result it is non-converging. We describe the topology and number of shortest paths: We show that the number of shortest paths is of order \(n^{f_n{\beta _n}}\), where \(f_n \in (0,1)\) is a random variable that oscillates with n. We decompose shortest paths into three segments, two ‘end-segments’ starting at each of the two uniformly chosen vertices, and a middle segment. The two end-segments of any shortest path have length \(\log \log (n^{{\beta _n}}) / |\log (\tau -2)|\)+tight, and the total degree is increasing towards the middle of the path on these segments. The connecting middle segment has length \(1/({\beta _n}(3-\tau ))\)+tight, and it contains only vertices with degree at least of order \(n^{(1-f_n){\beta _n}}\), thus all the degrees on this segment are comparable to the maximal degree. Our theorems also apply when instead of truncating the degrees, we start with a configuration model and we remove every vertex with degree at least \(n^{{\beta _n}}\), and the edges attached to these vertices. This sheds light on the attack vulnerability of the configuration model with infinite variance degrees.     

Scaling exponents of forced polymer translocation through a nanopore

We investigate several properties of a translocating homopolymer through a thin pore driven by an external field present inside the pore only using Langevin Dynamics (LD) simulations in three dimensions (3D). Motivated by several recent theoretical and numerical studies that are apparently at odds with each other, we estimate the exponents describing the scaling with chain length (Nof the average translocation time \(\ensuremath \langle\tau\rangle\) , the average velocity of the center of mass \(\ensuremath \langle v_{{\rm CM}}\rangle\) , and the effective radius of gyration \(\ensuremath \langle {R}_g\rangle\) during the translocation process defined as \(\ensuremath \langle\tau\rangle \sim N^{\alpha}\) , \(\ensuremath \langle v_{{\rm CM}} \rangle \sim N^{-\delta}\) , and \(\ensuremath {R}_g \sim N^{\bar{\nu}}\) respectively, and the exponent of the translocation coordinate (s -coordinate) as a function of the translocation time \(\ensuremath \langle s^2(t)\rangle\sim t^{\beta}\) . We find \(\ensuremath \alpha=1.36 \pm 0.01\) , \(\ensuremath \beta=1.60 \pm 0.01\) for \(\ensuremath \langle s^2(t)\rangle\sim \tau^{\beta}\) and \(\ensuremath \bar{\beta}=1.44 \pm 0.02\) for \(\ensuremath \langle\Delta s^2(t)\rangle\sim\tau^{\bar{\beta}}\) , \(\ensuremath \delta=0.81 \pm 0.04\) , and \(\ensuremath \bar{\nu}\simeq\nu=0.59 \pm 0.01\) , where \( \nu\) is the equilibrium Flory exponent in 3D. Therefore, we find that \(\ensuremath \langle\tau\rangle\sim N^{1.36}\) is consistent with the estimate of \(\ensuremath \langle\tau\rangle\sim\langle R_g \rangle/\langle v_{{\rm CM}} \rangle\) . However, as observed previously in Monte Carlo (MC) calculations by Kantor and Kardar (Y. Kantor, M. Kardar, Phys. Rev. E 69, 021806 (2004)) we also find the exponent α = 1.36 ± 0.01 < 1 + ν. Further, we find that the parallel and perpendicular components of the gyration radii, where one considers the “cis” and “trans” parts of the chain separately, exhibit distinct out-of-equilibrium effects. We also discuss the dependence of the effective exponents on the pore geometry for the range of N studied here.     

General one-loop formulas for decay $$h\rightarrow Z\gamma $$

Radiative corrections to the decay \(h\rightarrow Z\gamma \) are evaluated in the one-loop approximation. The unitary gauge is used. The analytic result is expressed in terms of the Passarino-Veltman functions. The calculations are applicable for the Standard Model as well for a wide class of its gauge extensions. In particular, the decay width of a charged Higgs boson \(H^{\pm }\rightarrow W^{\pm }\gamma \) can be derived. The consistence of our formulas and several specific earlier results is shown.