Quartic Interaction Research Articles

Matrix product states and variational methods applied to critical quantum field theory

We study the second-order quantum phase transition of massive real scalar field theory with a quartic interaction in ($1+1$) dimensions on an infinite spatial lattice using matrix product states. We introduce and apply a naive variational conjugate gradient method, based on the time-dependent variational principle for imaginary time, to obtain approximate ground states, using a related ansatz for excitations to calculate the particle and soliton masses and to obtain the spectral density. We also estimate the central charge using finite-entanglement scaling. Our value for the critical parameter agrees well with recent Monte Carlo results, improving on an earlier study which used the related density matrix normalization group method, verifying that these techniques are well-suited to studying critical field systems. We also obtain critical exponents that agree, as expected, with those of the transverse Ising model. Additionally, we treat the special case of uniform product states (mean field theory) separately, showing that they may be used to investigate noncritical quantum field theories under certain conditions.

Galileon forces in the Solar System

We consider the challenging problem of obtaining an analytic understanding of realistic astrophysical dynamics in the presence of a Vainshtein screened fifth force arising from infrared modifications of General Relativity. In particular, we attempt to solve -- within the most general flat spacetime galileon model -- the scalar force law between well separated bodies located well within the Vainshtein radius of the Sun. To this end, we derive the exact static Green's function of the galileon wave equation linearized about the background field generated by the Sun, for the minimal cubic and maximally quartic galileon theories, and then introduce a method to compute the general leading order force law perturbatively away from these limits. We also show that the same nonlinearities which produce the Vainshtein screening effect present obstacles to an analytic calculation of the galileon forces between closely bound systems within the solar system, such as that of the Earth and Moon. Within the test mass approximation, we deduce that a large enough quartic galileon interaction would suppress the effect on planetary perihelion precession below the level detectable by even the next-generation experiments.

Measuring SUSY quartics from stop decays

In supersymmetric theories, there is a scalar quartic related to the top and bottom yukawas that gives LHC observable decays of the stop or sbottom. After electroweak symmetry breaking, this quartic induces the decay of the stop (sbottom) to a charged Higgs and the sbottom (stop) without allowing for the corresponding channel involving the W boson. Unlike other scalar quartics, this scalar quartic is measurable at the LHC and for compressed spectra may even be the discovery channel for the stop or sbottom. Observing this decay and measuring the value of the quartic interaction would give direct evidence that the underlying theory is supersymmetric as only a fermionic symmetry can relate a scalar quartic to a yukawa interaction. This decay pathway is explored and situations where it can be the dominant decay mode for either the stop or the sbottom are highlighted. Measurement of this decay mode at the LHC would unambiguously show that the underlying theory of nature was supersymmetric.

Effective action and phase transitions of scalar field on the fuzzy sphere

Scalar field theory on the fuzzy two-sphere, represented as a hermitian matrix model that includes kinetic, mass and quartic interaction terms, is studied. The effective action in the symmetric large-N regime is analyzed using a self-consistent bootstrap method which fixes its form up to sixth order in the eigenvalues and gives a closed expression to all orders in the quadratic invariant of the matrix, valid close to semicircular distributions. Using this action the eigenvalue distribution is calculated for the interacting theory in the appropriate scaling limit and the phase transition from the disordered to the symmetric ordered phase is identified, including nonperturbative effects.

Double scaling in tensor models with a quartic interaction

Abstract In this paper we identify and analyze in detail the subleading contributions in the 1/N expansion of random tensors, in the simple case of a quartically interacting model. The leading order for this 1/N expansion is made of graphs, called melons, which are dual to particular triangulations of the D-dimensional sphere, closely related to the “stacked” triangulations. For D < 6 the subleading behavior is governed by a larger family of graphs, hereafter called cherry trees, which are also dual to the D-dimensional sphere. They can be resummed explicitly through a double scaling limit. In sharp contrast with random matrix models, this double scaling limit is stable. Apart from its unexpected upper critical dimension 6, it displays a singularity at fixed distance from the origin and is clearly the first step in a richer set of yet to be discovered multi-scaling limits.

Possible nematic order driven by magnetic fluctuations in iron pnictides

In this paper, instabilities of the isotropic metallic phase in iron pnictides are investigated. The relevant quartic fermionic interaction terms in the model are identified using phase space arguments. Using the functional integral formalism, a Hubbard-Stratonovich transformation is used to decouple these quartic terms. This procedure introduces several bosonic fields which describe the low-energy collective modes of the system. By studying the behavior of these collective modes, a possible instability is found in the forward scattering channel of the isotropic phase driven by magnetic fluctuations. Using mean field analysis, we obtain a static and homogeneous ground state. This ground state is metallic, but the electron Fermi pockets are distorted unequally at different pockets in momentum space. This results in a desirable nematic ordering which breaks the lattice ${C}_{4}$ symmetry but preserves translational symmetry and may explain several experimental observations.

Product approximations for a class of quantum anharmonic oscillators

In this article, we investigate analytically and numerically a class of non-autonomous Schrödinger equations in one space dimension describing the dynamics of quantum anharmonic oscillators driven by time-dependent quartic interactions. We do so within a suitably constructed Faedo–Galerkin scheme by analyzing several product approximations for their solutions, which involve various exponential operator splittings. Our main objective is to study the convergence rates and the accuracy of such approximations, among which there are extensions of the Trotter–Kato product formula and several other variants. Crucial to our analysis is the knowledge of the lowest energy level and of the corresponding eigensolution to the associated time-independent problem, which we also compute within the very same framework.

Quantum field theory in a multimetric background

By means of simple models in a flat spacetime manifold we examine some of the issues that arise when quantizing interacting quantum fields in multi-metric backgrounds. In particular we investigate the maintenance of a causal structure in the models. In this context we introduce and explain the relevance of an interpolating metric that is a superposition of the individual metrics in the models. We study the renormalisation of a model with quartic interactions and elucidate the structure of the renormalisation group and its implications for Lorentz symmetry breakdown.

Matrix-valued Boltzmann equation for the nonintegrable Hubbard chain

The standard Fermi-Hubbard chain becomes nonintegrable by adding to the nearest neighbor hopping additional longer range hopping amplitudes. We assume that the quartic interaction is weak and investigate numerically the dynamics of the chain on the level of the Boltzmann type kinetic equation. Only the spatially homogeneous case is considered. We observe that the huge degeneracy of stationary states in the case of nearest neighbor hopping is lost and the convergence to the thermal Fermi-Dirac distribution is restored. The convergence to equilibrium is exponentially fast. However for small next-nearest neighbor hopping amplitudes one has a rapid relaxation towards the manifold of quasistationary states and slow relaxation to the final equilibrium state.

Noncommutative field theories on $ \mathbb{R}_{\lambda}^3 $: towards UV/IR mixing freedom

Abstract We consider the noncommutative space $ \mathbb{R}_{\lambda}^3 $ , a deformation of the algebra of functions on $ {{\mathbb{R}}^3} $ which yields a “foliation” of $ {{\mathbb{R}}^3} $ into fuzzy spheres. We first construct a natural matrix base adapted to $ \mathbb{R}_{\lambda}^3 $ . We then apply this general framework to the one-loop study of a two-parameter family of real-valued scalar noncommutative field theories with quartic polynomial interaction, which becomes a non-local matrix model when expressed in the above matrix base. The kinetic operator involves a part related to dynamics on the fuzzy sphere supplemented by a term reproducing radial dynamics. We then compute the planar and non-planar 1-loop contributions to the 2-point correlation function. We find that these diagrams are both finite in the matrix base. We find no singularity of IR type, which signals very likely the absence of UV/IR mixing. We also consider the case of a kinetic operator with only the radial part. We find that the resulting theory is finite to all orders in perturbation expansion.

“Super”-dilatation symmetry of the top-Higgs system

The top-Higgs system, consisting of top quark (LH doublet, RH singlet) and Higgs boson kinetic terms, with gauge fields set to zero, has an exact (modulo total divergences) symmetry where both fermion and Higgs fields are shifted and mixed in a supersymmetric fashion. The full Higgs-Yukawa interaction and Higgs-potential, including additional \sim 1/\Lambda^2 NJL-like interactions, also has this symmetry to O(1/\Lambda^4), up to null-operators. Thus the interaction lagrangian can be viewed as a power series in 1/\Lambda^2. The symmetry involves interplay of the Higgs quartic interaction with the Higgs-Yukawa interaction and implies the relationship, \lambda = \half g^2 between the top--Yukawa coupling, g, and Higgs quartic coupling, \lambda, at a high energy scale \Lambda >= few TeV. We interpret this to be a new physics scale. The top quark is massless in the symmetric phase, satisfying the Nambu-Goldstone theorem. The fermionic shift part of the current is \propto (1-H^\dagger H/v^2), owing to the interplay of \lambda and g, and vanishes in the broken phase. Hence the Nambu-Goldstone theorem is trivially evaded in the broken phase and the top quark becomes heavy (it is not a Goldstino). We have m_t=m_h, subject to radiative corrections that can in principle pull the Higgs into concordance with experiment.

Comparison of mass renormalization schemes for simple model systems

We discuss an alternative method to mass renormalize a quantum field Hamiltonian based on the requirement that the vacuum and single-particle sectors are not self-scattering. We illustrate the feasibility of this method for the concrete example of bosonic quadratic and quartic interactions. The results are compared with those obtained by the standard renormalization technique based on a spectral analysis of the usual field-based propagators. We also discuss a potential method for renormalizing a quantum field theory in a non-perturbative way using computational methods.

A NEW, ANALYTIC, NON-PERTURBATIVE, GAUGE-INVARIANT, REALISTIC FORMULATION OF QCD

This Brief Review presents a new, analytic, gauge-invariant, non-perturbative and rigorous approach to the calculation of QCD phenomena. In this formulation a basic distinction between the conventional description of QCD and a more "realistic" description, taking into account that asymptotic quarks exist only in bound states, is brought into focus by the evaluation of the Schwinger solution for the QCD generating functional (GF) in terms of the exact Fradkin representations of the Green's functional G0(x, y|A) and the vacuum functional L[A]. A new and simplifying output called "Effective Locality" appears, in which the interactions between quarks by the exchange of a "gluon bundle" (GB) (which "bundle" contains an infinite number of gluons, including cubic and quartic gluon interactions) display an exact locality property that reduces the several functional integrals of the formulation down to a set of ordinary integrals, susceptible to desktop-computer computation. As an example of the power of these methods we offer as a first pencil-and-paper calculation the quark–antiquark binding potential of a pion, and the corresponding three-quark binding potentials of a nucleon, obtained in a simple way from relevant eikonal scattering approximations. This example is followed by the estimation of a nucleon–nucleon binding potential to form a model deuteron, which may have relevance to the origin of the nuclear shell model and nuclear binding in general.

SOFT GRAVITONS SCREEN COUPLING IN DE SITTER SPACE

On the de Sitter background, the gravitational propagator contains an infra-red logarithm term which breaks the de Sitter symmetry. We investigate how such soft gravitons affect the local dynamics of matter fields. We show that the infra-red logarithms do not spoil Lorentz invariance in scalar, Dirac and gauge field theories. In the interacting field theory with quartic, Yukawa and gauge interactions, we find that the effective couplings are time dependent and the evolutions are physical as their relative scaling exponents are gauge invariant.

SOFT WALL MODEL WITH INVERSE EXPONENTIAL PROFILE AS A MODEL FOR THE AXIAL AND PSEUDOSCALAR MESONS

According to the conjecture of the gauge/gravity duality, a strongly coupled 4D field theory may be described by a dual 5D theory which is a free field theory in the first approximation. The usual description of the axial-vector sector in the bottom–up AdS/QCD models includes, however, the quartic interaction with a bulk scalar field. A question arises, whether it is possible to describe some aspects of the chiral symmetry breaking (CSB) in a way more consistent with the original proposal, i.e. within a free 5D theory? We suggest that a natural candidate for this purpose is the soft wall (SW) model with inverse exponential profile.

Precision unification in λSUSY with a 125GeV Higgs

It is challenging to explain the tentative 125 GeV Higgs signal in the Minimal Supersymmetric Standard Model (MSSM) without introducing excessive fine-tuning, and this motivates the study of non-minimal implementations of low energy supersymmetry (SUSY). A term \lambda SH_uH_d involving a Standard Model (SM) singlet state S leads to an additional source for the quartic interaction raising the mass of the lightest SM-like Higgs. However, in order to achieve m_h \approx 125 GeV with light stops and small stop mixing, it is necessary for \lambda \gtrsim 0.7 and consequently \lambda may become non-perturbative before the unification scale. Moreover, as argued by Barbieri, Hall, et al. low fine-tuning prefers the region \lambda~1-2, leading to new or non-perturbative physics involving S below the GUT scale (`\lambda SUSY' models). This raises the concern that precision gauge coupling unification, the prime piece of indirect experimental evidence for low energy SUSY, may be upset. Using the NSVZ exact \beta-function along with well motivated assumptions on the strong coupling dynamics we show that this is not necessarily the case, but rather there exist classes of UV completions where the strong-coupling effects can naturally correct for the present ~3% discrepancy in the two-loop MSSM unification prediction for \alpha_s. Moreover, we argue that in certain scenarios a period of strong coupling can also be beneficial for t-b unification, while maintaining the small to moderate values of tan\beta preferred by the Higgs mass.

On the structure of quartic vertices for massless higher spin fields on Minkowski background

We consider in detail the structure of quartic vertices for massless higher spin fields on Minkowski background, and study the consistency conditions imposed on cubic and quartic interactions by symmetries of the S-matrix. We discuss the possibility of generalizing the construction of quartic vertices to D-dimensional anti-de Sitter space.

A new approach to analytic, non-perturbative and gauge-invariant QCD

Following a previous calculation of quark scattering in eikonal approximation, this paper presents a new, analytic and rigorous approach to the calculation of QCD phenomena. In this formulation a basic distinction between the conventional “idealistic” description of QCD and a more “realistic” description is brought into focus by a non-perturbative, gauge-invariant evaluation of the Schwinger solution for the QCD generating functional in terms of the exact Fradkin representations of Green’s functional Gc(x,y|A) and the vacuum functional L[A]. Because quarks exist asymptotically only in bound states, their transverse coordinates can never be measured with arbitrary precision; the non-perturbative neglect of this statement leads to obstructions that are easily corrected by invoking in the basic Lagrangian a probability amplitude which describes such transverse imprecision.The second result of this non-perturbative analysis is the appearance of a new and simplifying output called “Effective Locality”, in which the interactions between quarks by the exchange of a “gluon bundle”–which “bundle” contains an infinite number of gluons, including cubic and quartic gluon interactions–display an exact locality property that reduces the several functional integrals of the formulation down to a set of ordinary integrals. It should be emphasized that “non-perturbative” here refers to the effective summation of all gluons between a pair of quark lines–which may be the same quark line, as in a self-energy graph–but does not (yet) include a summation over all closed-quark loops which are tied by gluon-bundle exchange to the rest of the “Bundle Diagram”. As an example of the power of these methods we offer as a first analytic calculation the quark–antiquark binding potential of a pion, and the corresponding three-quark binding potential of a nucleon, obtained in a simple way from relevant eikonal scattering approximations. A second calculation, analytic, non-perturbative and gauge-invariant, of a nucleon–nucleon binding potential to form a model deuteron, will appear separately.

D-term enhancement in a spin-1 top partner model

Supersymmetric models with extended electroweak gauge groups have the potential to enhance the Higgs quartic interaction through nondecoupling D-terms. We consider the D-term enhancement effect in a vector top partner model, where the quadratic divergence to the Higgs mass from the virtual top quark is canceled by its corresponding spin-1 superpartners. We are going to show that the model can predict a Higgs mass beyond the LEP bound, and is consistent with the precision electroweak constraints.

Cross-talk dynamics of optical solitons in multichannel waveguide systems with a Ginzburg-Landau gain-loss profile

We investigate energy exchange in fast optical soliton collisions in Kerr nonlinear waveguides with weak linear loss, cubic gain, and quintic loss, i.e., with a Ginzburg-Landau (GL) gain-loss profile. We find that the amplitude shift in a single two-soliton collision in the presence of quintic loss is quartic in the initial soliton amplitudes, and that three-soliton interaction gives a significant contribution to the amplitude shift in a three-soliton collision. Moreover, we show that collision-induced dynamics of soliton amplitudes in a two-channel waveguide system with a GL gain-loss profile is described by a Lotka-Volterra (LV) model with quadratic and quartic interaction terms. Using the LV model, we show that linearly stable transmission with equal prescribed amplitudes can be achieved by a proper choice of the time-slot width and the ratio between the cubic gain and quintic loss coefficients. The linear stability of the transmission is confirmed by the numerical solution of a perturbed system of coupled nonlinear Schr\"odinger equations. Further numerical simulations show that transmission in waveguides with a GL gain-loss profile is less susceptible to radiation emission effects compared with waveguides with linear gain and cubic loss.