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The twisted fourth moment of the Riemann zeta-function was established by Hughes and Young [J. Reine Angew. Math. 641 (2010), 203--236] and later improved by Bettin, Bui, Li and Radziwill [J. Eur. Math. Soc. (JEMS) 22 (2020), 3953--3980]. In applications one would often like to take the Dirichlet polynomial to mimic either $1/ζ^r(s)$ (a mollifier) or $ζ(s)^r$ (an amplifier) for some $r>0$. Previous known results include the mean value of the fourth power of $ζ(s)$ times the square or the fourth power of a mollifier, or the square of an amplifier. In this paper we obtain the asymptotic formula for the fourth moment of the Riemann zeta-function times the fourth power of an amplifier. This has various applications to the theory of the Riemann zeta-function, e.g. gaps between zeros of $ζ(s)$ and lower bounds for moments.
Conventional array designs based on circular fourth-order cumulant typically adopt a single expression form of the fourth-order difference co-array (FODCA), which limits the achievable degrees of freedom (DOFs) and neglects the impact of mutual coupling among physical sensors. To address above issues, this paper proposes a novel scheme to design arrays with increased DOFs by combining different forms of FODCA while accounting for mutual coupling. A novel fourth-order hierarchical array (FOHA) based on different forms of FODCA is constructed using an arbitrary generator set. The analytical expression between the coupling leakage of the generator and the resulting FOHA is derived. Two specific FOHA configurations are presented with closed-form sensor placements. The arrays not only offer increased DOFs for resolving more sources in direction of-arrival (DOA) estimation but also effectively suppress mutual coupling. Moreover, the redundancy of FODCA is examined, and it is shown that arrays based on the proposed scheme achieve lower redundancy compared to existing arrays based on FODCA. Meanwhile, the necessary and sufficient conditions for signal reconstruction by FOHA are derived. Co
Let $(M,g)$ be a closed, smooth, Riemannian manifold of dimension $m \geq 1$. Let $η$ be a smooth $(0,1)$-tensor field on $M$. The divergence of $η$ is defined as $\text{div}_g(η):=g^{ij}( abla η)_{ij}$. Now let $Δ_g$ be a differential operator on $M$ that is given on functions by $Δ_g u = \text{div}_g abla u$. We will call $Δ_g$ the Laplace-Beltrami operator. With this definition in place, it is not difficult to produce an example of a formally self-adjoint elliptic differential operator on $M$ that has a sign-changing eigenfunction that is associated with the operator's lowest eigenvalue. Indeed, let $λ_2$ be the second lowest eigenvalue of $-Δ_g$, and let $L_g$ be a differential operator on $M$ that is given on functions by $L_g u = Δ^2_g u + λ_2 Δu$. Then $L_g$ will possess a sign-changing eigenfunction that is associated with $L_g$'s lowest eigenvalue.. The question that remains is given a smooth, closed manifold $M$ of dimension $m \geq 1$, how rare are formally self-adjoint elliptic differential operators on $M$ that have sign-changing eigenfunctions that are associated with the operators' lowest eigenvalues. In this paper, we will see that if $A = T -λg$, where $T$ is a sm
Given a graph sequence $\{G_n\}_{n\ge1}$ and a simple connected subgraph $H$, we denote by $T(H,G_n)$ the number of monochromatic copies of $H$ in a uniformly random vertex coloring of $G_n$ with $c \ge 2$ colors. In this article, we prove a central limit theorem for $T(H,G_n)$ with explicit error rates. The error rates arise from graph counts of collections formed by joining copies of $H$ that we call good joins. Counts of good joins are closely related to the fourth moment of a normalized version of $T(H,G_{n})$, and that connection allows us to show a fourth moment phenomenon for the central limit theorem. Precisely, for $c\ge 30$, we show that $T(H,G_n)$ (appropriately centered and rescaled) converges in distribution to $\mathcal{N}(0,1)$ whenever its fourth moment converges to 3 (the fourth moment of the standard normal distribution). We show the convergence of the fourth moment is necessary to obtain a normal limit when $c\ge 2$. The combination of these results implies that the fourth moment condition characterizes the limiting normal distribution of $T(H,G_n)$ for all subgraphs $H$, whenever $c\ge 30$.
Results are presented from a search for a fourth generation of quarks produced singly or in pairs in a data set corresponding to an integrated luminosity of 5 inverse femtobarns recorded by the CMS experiment at the LHC in 2011. A novel strategy has been developed for a combined search for quarks of the up and down type in decay channels with at least one isolated muon or electron. Limits on the mass of the fourth-generation quarks and the relevant Cabibbo-Kobayashi-Maskawa matrix elements are derived in the context of a simple extension of the standard model with a sequential fourth generation of fermions. The existence of mass-degenerate fourth-generation quarks with masses below 685 GeV is excluded at 95% confidence level for minimal off-diagonal mixing between the third- and the fourth-generation quarks. With a mass difference of 25 GeV between the quark masses, the obtained limit on the masses of the fourth-generation quarks shifts by about +/- 20 GeV. These results significantly reduce the allowed parameter space for a fourth generation of fermions.
We consider extensions of the standard model with fourth generation fermions (SM4) in which extra symmetries are introduced such that the transitions between the fourth generation fermions and the ones in the first three generations are forbidden. In these models, the stringent lower bounds on the masses of fourth generation quarks from direct searches are relaxed, and the lightest fourth neutrino is allowed to be stable and light enough to trigger the Higgs boson invisible decay. In addition, the fourth Majorana neutrino can be a subdominant but highly detectable dark matter component. We perform a global analysis of the current LHC data on the Higgs production and decay in this type of SM4. The results show that the mass of the lightest fourth Majorana neutrino is confined in the range $\sim 41-59$ GeV. Within the allowed parameter space, the predicted effective cross-section for spin-independent DM-nucleus scattering is $\sim 3\times 10^{-48}-6\times 10^{-46} \text{cm}^{2}$, which is close to the current Xenon100 upper limit and is within the reach of the Xenon1T experiment in the near future. The predicted spin-dependent cross sections can also reach $\sim 8\times 10^{-40}\text
We develop a fourth order simulation algorithm for solving the stochastic Langevin equation. The method consists of identifying solvable operators in the Fokker-Planck equation, factorizing the evolution operator for small time steps to fourth order and implementing the factorization process numerically. A key contribution of this work is to show how certain double commutators in the factorization process can be simulated in practice. The method is general, applicable to the multivariable case, and systematic, with known procedures for doing fourth order factorizations. The fourth order convergence of the resulting algorithm allowed very large time steps to be used. In simulating the Brownian dynamics of 121 Yukawa particles in two dimensions, the converged result of a first order algorithm can be obtained by using time steps 50 times as large. To further demostrate the versatility of our method, we derive two new classes of fourth order algorithms for solving the simpler Kramers equation without requiring the derivative of the force. The convergence of many fourth order algorithms for solving this equation are compared.
It is known that Flavor Democracy favors the existence of the fourth standard model (SM) family. In order to give nonzero masses for the first three family fermions Flavor Democracy has to be slightly broken. A parametrization for democracy breaking, which gives the correct values for fundamental fermion masses and, at the same time, predicts quark and lepton CKM matrices in a good agreement with the experimental data, is proposed. The pair productions of the fourth SM family Dirac $(ν_{4})$ and Majorana $(N_{1})$ neutrinos at future linear colliders with $\sqrt{s}=500$ GeV, 1 TeV and 3 TeV are considered. The cross section for the process $e^{+}e^{-}\toν_{4}\bar {ν_{4}}(N_{1}N_{1})$ and the branching ratios for possible decay modes of the both neutrinos are determined. The decays of the fourth family neutrinos into muon channels $(ν_{4}(N_{1})\toμ^{\pm}W^{\mp})$ provide cleanest signature at $e^{+}e^{-}$ colliders. Meanwhile, in our parametrization this channel is dominant. $W$ bosons produced in decays of the fourth family neutrinos will be seen in detector as either di-jets or isolated leptons. As an example we consider the production of 200 GeV mass fourth family neutrinos at $
We present a solution to the ghost problem in fourth order derivative theories. In particular we study the Pais-Uhlenbeck fourth order oscillator model, a model which serves as a prototype for theories which are based on second plus fourth order derivative actions. Via a Dirac constraint method quantization we construct the appropriate quantum-mechanical Hamiltonian and Hilbert space for the system. We find that while the second-quantized Fock space of the general Pais-Uhlenbeck model does indeed contain the negative norm energy eigenstates which are characteristic of higher derivative theories, in the limit in which we switch off the second order action, such ghost states are found to move off shell, with the spectrum of asymptotic in and out S-matrix states of the pure fourth order theory which results being found to be completely devoid of states with either negative energy or negative norm. We confirm these results by quantizing the Pais-Uhlenbeck theory via path integration and by constructing the associated first-quantized wave mechanics, and show that the disappearance of the would-be ghosts from the energy eigenspectrum in the pure fourth order limit is required by a hidden
Adding a fourth generation to the Standard Model and assuming it to be valid up to some cutoff Λ, we show that electroweak symmetry is broken by radiative corrections due to the fourth generation. The effects of the fourth generation are isolated using a Lagrangian with a genuine scalar without self-interactions at the classical level. For masses of the fourth generation consistent with electroweak precision data (including the B \rightarrow K π CP asymmetries) we obtain a Higgs mass of the order of a few hundreds GeV and a cutoff Λ around 1-2 TeV. We study the reliability of the perturbative treatment used to obtain these results taking into account the running of the Yukawa couplings of the fourth quark generation with the aid of the Renormalization Group (RG) equations, finding similar allowed values for the Higgs mass but a slightly lower cut-off due to the breaking of the perturbative regime. Such low cut-off means that the effects of new physics needed to describe electroweak interactions at energy above Λ should be measurable at the LHC. We use the minimal supersymmetric extension of the standard model with four generations as an explicit example of models realizing the dyna
If the fourth generation fermions exist, the new quarks could influence the branching ratio of the decay $B_s \to ν\barν γ$. We obtain two solutions of the fourth generation CKM factor $V^*_{t's} V_{t'b}$ from the decay of $B \to X_s γ$. With these two solutions we calculate the new contributions of the fourth generation quark to Wilson coefficients of the decay $B_s \to ν\barν γ$. The branching ratio of the decay $B_s \to ν\barν γ$ in the two cases are calculated. In one case, our results are quite different from that of SM, but almost same in another case. If a fourth generation should exist in nature and nature chooses the former case, this B meson decay could provide a possible test of the fourth generation existence.
I present a general exclusion bound for the Higgs in fourth generation scenarios with a general lepton sector. Recent Higgs searches in fourth generation scenarios rule out the entire Higgs mass region between 120 and 600 GeV. That such a large range of Higgs masses are excluded is due to the presence of extra heavy flavors of quarks, which substantially increase Higgs production from gluon fusion over the Standard Model rate. However, if heavy fourth generation neutrinos are less than half of the Higgs mass, they can dominate the Higgs decay branching fraction, overtaking the standard Higgs to WW* decay rate. The Higgs mass exclusion in a fourth generation scenario is shown most generally to be 155-600 GeV, and is highly dependent on the fourth generation neutrino mixing parameter.
The necessity of the fourth family follows from the SM basics. According to flavor democracy the Dirac masses of the fourth SM family fermions are almost equal with preferable value 450 GeV, which corresponds to common (for all fundamental fermions) Yukawa coupling equal to SU(2) gauge coupling gW. In principle, one expect u4 a little bit lighter than d4, while nu4 could be essentially lighter than l4 due to Majorana mass terms for right-handed components of neutrinos. Obviously, the fourth family quarks will be copiously produced at the LHC. However, the first indication of the fourth SM family may be provided by early Higgs boson observation due to almost an order enhancement of the gluon fusion to Higgs cross-section. For the same reason the Tevatron still has a chance to observe the Higgs boson before the LHC. Concerning the fourth family leptons, in general, best place will be NLC/CLIC. However, for some mass regions and MNS matrix elements double discovery of both the nu4 and H could be possible at the LHC.
We investigate the radial behavior of galactic rotation curves by a Fourth Order Gravity adding also the Dark Matter component. The Fourth Order Gravity is a Lagrangian containing the Ricci scalar, the Ricci and Riemann tensor, but the rotation curves are depending only on two free parameters. A systematic analysis of rotation curves, in the Newtonian Limit of theory, induced by all galactic sub-structures of ordinary matter is shown. This analysis is presented for Fourth Order Gravity with and without Dark Matter. The outcomes are compared with respect to classical outcomes of General Relativity. The gravitational potential of point-like mass is the usual potential corrected by two Yukawa terms. The rotation curve is higher or also lower than curve of General Relativity if in the Lagrangian the Ricci scalar square is dominant or not with respect to the contribution of the Ricci tensor square. The curves are compared with the experimental data for the Milky Way and the galaxy NGC 3198. Although the Fourth Order Gravity gives more rotational contributions, in the limit of large distances the Keplerian behavior is present, and it is missing if we add a Dark Matter component. By modif
In extensions of the standard model with a heavy fourth generation one important question is what makes the fourth-generation lepton sector, particularly the neutrinos, so different from the lighter three generations. We study this question in the context of models of electroweak symmetry breaking in warped extra dimensions, where the flavor hierarchy is generated by the localization of the zero-mode fermions in the extra dimension. In this setup the Higgs sector is localized near the infrared brane, whereas the Majorana mass term is localized at the ultraviolet brane. As a result, light neutrinos are almost entirely Majorana particles, whereas the fourth generation neutrino is mostly a Dirac fermion. We show that it is possible to obtain heavy fourth-generation leptons in regions of parameter space where the light neutrino masses and mixings are compatible with observation. We study the impact of these bounds, as well as the ones from lepton flavor violation, on the phenomenology of these models.
Existence of the fourth family follows from the basics of the Standard Model and the actual mass spectrum of the third family fermions. We discuss possible manifestations of the fourth SM family at existing and future colliders. The LHC and Tevatron potentials to discover the fourth SM family have been compared. The scenario with dominance of the anomalous decay modes of the fourth family quarks has been considered in details.
Imagine to discover a new fourth family of leptons at the Large Hadron Collider (LHC) but no signs of an associated fourth family of quarks. What would that imply? An intriguing possibility is that the new fermions needed to compensate for the new leptons gauge anomalies simultaneously address the big hierarchy problem of the Standard Model. A natural way to accomplish such a scenario is to have the Higgs itself be composite of these new fermions. This is the setup we are going to investigate in this paper using as a template Minimal Walking Technicolor. We analyze a general heavy neutrino mass structure with and without mixing with the Standard Model families. We also analyze the LHC potential to observe the fourth lepton family in tandem with the new composite Higgs dynamics. We finally introduce a model uniting the fourth lepton family and the technifermion sector at higher energies.
The resonant production of the fourth family slepton ~l_4 via R-parity violating interactions of supersymmetry at the Large Hadron Collider has been investigated. We study the decay mode of ~l_4 into the fourth family neutrino nu_4 and W boson. The signal will be a like-sign dimuon and dijet if the fourth family neutrino has Majorana nature. We discuss the constraints on the R-parity violating couplings lambda and lambda' of the fourth family charged slepton at the LHC with the center of mass energies of 7, 10 and 14 TeV.
We reanalyze constraints on the mass spectrum of the chiral fourth generation fermions and the Higgs bosons for the standard model (SM4) and the two Higgs doublet model (THDM). We find that the Higgs mass in the SM4 should be larger than roughly the fourth generation up-type quark mass, while the light CP even Higgs mass in the THDM can be smaller. Various mass spectra of the fourth generation fermions and the Higgs bosons are allowed. The phenomenology of the fourth generation models is still rich.
We consider a fourth order evolution equation involving a singular nonlinear term $\fracλ{(1-u)^{2}}$ in a bounded domain $Ω\subset\R^{n}$. This equation arises in the modeling of microelectromechanical systems. We first investigate the well-posedness of a fourth order parabolic equation which has been studied in \cite{Lau}, where the authors, by the semigroup argument, obtained the well-posedness of this equation for $n\leq2$. Instead of semigroup method, we use the Faedo-Galerkin technique to construct a unique solution of the fourth order parabolic equation for $n\leq7$, which improves and completes the result of \cite{Lau}. Besides, the well-posedness of the corresponding fourth order hyperbolic equation is obtained by the similar argument for $n\leq7$.