Φ4 Theory Research Articles

Simple non-perturbative resummation schemes beyond mean-field II: Thermodynamics of scalar ϕ4 theory in 1 + 1 dimensions at arbitrary coupling

Recently, non-perturbative approximate solutions were presented that go beyond the well-known mean-field resummation. In this work, these non-perturbative approximations are used to calculate finite-temperature equilibrium properties for scalar [Formula: see text] theory in two dimensions such as the pressure, entropy density and speed of sound. Unlike traditional approaches, it is found that results are well-behaved for arbitrary temperature/coupling strength, are independent of the choice of the renormalization scale [Formula: see text], and are apparently converging as the resummation level is increased. Results also suggest the presence of a possible analytic cross-over from the high-temperature to the low-temperature regime based on the change in the thermal entropy density.

Transcendental structure of multiloop massless correlators and anomalous dimensions

We give a short account of recent advances in our understanding of the π- dependent terms in massless (Euclidean) 2-point functions as well as in generic anomalous dimensions (ADs) and β-functions. We extend the considerations of [1] by two more loops, that is for the case of 6- and 7-loop correlators and 7- and 8-loop renormalization group (RG) functions. Our predictions for the (π-dependent terms) of the 7-loop RG functions for the case of the O(n) \U0001d7194 theory are in full agreement with the recent results from [2]. All available 7- and 8-loop results for QCD and the scalar O(n) ϕ4 theory obtained within the large Nf approach to the quantum field theory (see, e.g. [3]) are also in full agreement with our results.

Field theories for loop-erased random walks

Self-avoiding walks (SAWs) and loop-erased random walks (LERWs) are two ensembles of random paths with numerous applications in mathematics, statistical physics and quantum field theory. While SAWs are described by the n→0 limit of ϕ4-theory with O(n)-symmetry, LERWs have no obvious field-theoretic description. We analyse two candidates for a field theory of LERWs, and discover a connection between the corresponding and a priori unrelated theories. The first such candidate is the O(n)-symmetric ϕ4 theory at n=−2 whose link to LERWs was known in two dimensions due to conformal field theory. Here it is established in arbitrary dimension via a perturbation expansion in the coupling constant. The second candidate is a field theory for charge-density waves pinned by quenched disorder, whose relation to LERWs had been conjectured earlier using analogies with Abelian sandpiles. We explicitly show that both theories yield identical results to 4-loop order and give both a perturbative and a non-perturbative proof of their equivalence. This allows us to compute the fractal dimension of LERWs to order ε5 where ε=4−d. In particular, in d=3 our theory gives zLERW(d=3)=1.6243±0.001, in excellent agreement with the estimate z=1.62400±0.00005 of numerical simulations.

Tensor network analysis of critical coupling in two dimensional \u03d54 theory

We make a detailed analysis of the spontaneous Z2-symmetry breaking in the two dimensional real ϕ4 theory with the tensor renormalization group approach, which allows us to take the thermodynamic limit easily and determine the physical observables without statistical uncertainties. We determine the critical coupling in the continuum limit employing the tensor network formulation for scalar field theories proposed in our previous paper. We obtain [λ/μc2]cont. = 10.913(56) with the quartic coupling λ and the renormalized critical mass μc. The result is compared with previous results obtained by different approaches.

Non-Equilibrium ϕ4 theory for networks: towards memory formations with quantum brain dynamics

We investigate the time evolution of quantum fields in neutral scalar ϕ4 theory for open systems with the central region and the multiple reservoirs (networks) as a toy model of quantum field theory of the brain. First we investigate the Klein–Gordon (KG) equations and the Kadanoff–Baym (KB) equations in open systems in d + 1 dimensions. Next, we introduce the kinetic entropy current and provide the proof of the H-theorem for networks. Finally, we solve the KG and the KB equations numerically in spatially homogeneous systems in 1 + 1 dimensions. We find that decoherence, entropy saturation and chemical equilibration all occur during the time evolution in the networks. We also show how coherent field transfer takes place in the networks.

NLO radiative correction to the Casimir energy in Lorentz-violating scalar field theory

Violation of the Lorentz symmetry has important effects on physical quantities including field propagators. Therefore, in addition to the leading order, the sub-leading order of quantities may be modified. In this paper, we calculate the next to leading (NLO) radiative corrections to the Casimir energy in the presence of two perfectly conducting parallel plates for ϕ4 theory with a Lorentz-breaking extension. We do the renormalization and investigate these NLO corrections for three distinct directions of the Lorentz violation; temporal direction, parallel and perpendicular to the plates.

\u03ba-Poincar\xe9 invariant orientable field theories at one-loop

We consider a family of κ-Poincaré invariant scalar field theories on 4-d κ-Minkowski space with quartic orientable interaction, that is for which ϕ and its conjugate ϕ† alternate in the quartic interaction, and whose kinetic operator is the square of a Uκ(iso(4))-equivariant Dirac operator. The formal commutative limit yields the standard complex ϕ4 theory. We find that the 2-point function receives UV linearly diverging 1-loop corrections while it stays free of IR singularities that would signal occurrence of UV/IR mixing. We find that all the 1-loop planar and non-planar contributions to the 4-point function are UV finite, stemming from the existence of the particular estimate for the propagator partly combined with its decay properties at large momenta, implying formally vanishing of the beta-functions at 1-loop so that the coupling constants stay scale-invariant at 1-loop.

Non-equilibrium [formula omitted] theory in open systems as a toy model of quantum field theory of the brain

We investigate non-equilibrium relativistic ϕ4 theory in open systems (the relevant system and the two reservoirs) as a toy model of quantum field theory of the brain, Quantum Brain Dynamics (QBD). We give time evolution equations of quantum fields, that is the Klein–Gordon (KG) equations for background coherent scalar fields and the Kadanoff–Baym (KB) equations for quantum fluctuations. Next we introduce a relativistic kinetic entropy current with the gradient expansion technique and show the H-theorem in the presence of Next-to-Leading-Order self-energy of the coupling expansion and tunneling between systems in d+1 dimensions. Finally we demonstrate numerical simulations with KG and KB equations in 1+1 dimensions. We find that the decoherence (field–particle conversion), the entropy production and the chemical equilibration occur in time evolution of the relevant system and the two reservoirs. We also demonstrate how the background coherent fields transfer between systems. The presented formalism is proposed as a mathematical representation of QBD.

The Dirichlet Casimir energy for ϕ4 theory in a rectangular waveguide

In this paper, we presented the zero- and first-order radiative corrections to the Casimir energy for a massive scalar field confined with Dirichlet boundary condition in an open-ended rectangular waveguide. In the calculation procedure, we applied a systematic renormalization program that allows all influences imposed by dominant boundary conditions in a problem be automatically reflected in the counterterms, leading the counterterms to be obtained in a position-dependent manner. To remove the appeared divergences in the computation task, the box subtraction scheme as a regularization technique was used. In this regularization technique, usually, two similar configurations were introduced. Then, to find the Casimir energy, the zero-point energies of these two configurations were subtracted from each other via defining appropriate limits. In the present work, first, the leading order Casimir energy for the massive scalar field in a waveguide is briefly presented. Next, by applying the renormalization and regularization procedures, the first-order radiative correction to the Casimir energy in the waveguide is calculated. Finally, all the necessary limits of the obtained answers for massive and massless cases are computed and the consistency of the obtained results are discussed.

Symmetry, Chaos and Temperature in the One-dimensional Lattice φ4 Theory

The symmetries of the minimal $\phi^4$ theory on the lattice are systematically analyzed. We find that symmetry can restrict trajectories to subspaces, while their motions are still chaotic. The chaotic dynamics of autonomous Hamiltonian systems are discussed, in relation to the thermodynamic laws. Possibilities of configurations with non-equal ideal gas temperatures in the steady state, in Hamiltonian systems, are investigated, and examples of small systems in which the ideal gas temperatures are different within the system are found. The pairing of local (finite-time) Lyapunov exponents are analyzed, and their dependence on various factors, such as the energy of the system, the characteristics of the initial conditions are studied, and discussed. We find that for the $\phi^4$ theory, higher energies lead to faster pairing times. We also find that symmetries can impede the pairing of local Lyapunov exponents, and the convergence of Lyapunov exponents.

Phase transition for the system of finite volume in the ϕ4 theory in the Tsallis nonextensive statistics

We studied the effects of nonextensivity on the phase transition for the system of finite volume [Formula: see text] in the [Formula: see text] theory in the Tsallis nonextensive statistics of entropic parameter [Formula: see text] and temperature [Formula: see text], when the deviation from the Boltzmann–Gibbs (BG) statistics, [Formula: see text], is small. We calculated the condensate and the effective mass to the order [Formula: see text] with the normalized [Formula: see text]-expectation value under the free particle approximation with zero bare mass. The following facts were found. The condensate [Formula: see text] divided by [Formula: see text], [Formula: see text], at [Formula: see text] ([Formula: see text] is the value of the condensate at [Formula: see text]) is smaller than that at [Formula: see text] for [Formula: see text] as a function of [Formula: see text] which is the physical temperature [Formula: see text] divided by [Formula: see text]. The physical temperature [Formula: see text] is related to the variation of the Tsallis entropy and the variation of the internal energies, and [Formula: see text] at [Formula: see text] coincides with [Formula: see text]. The effective mass decreases, reaches minimum, and increases after that, as [Formula: see text] increases. The effective mass at [Formula: see text] is lighter than the effective mass at [Formula: see text] at low physical temperature and heavier than the effective mass at [Formula: see text] at high physical temperature. The effects of the nonextensivity on the physical quantity as a function of [Formula: see text] become strong as [Formula: see text] increases. The results indicate the significance of the definition of the expectation value, the definition of the physical temperature, and the constraints for the density operator, when the terms including the volume of the system are not negligible.

Three ways to solve critical ϕ4 theory on 4 − 𝜖 dimensional real projective space: Perturbation, bootstrap, and Schwinger–Dyson equation

In this paper, we solve the two-point function of the lowest dimensional scalar operator in the critical [Formula: see text] theory on [Formula: see text] dimensional real projective space in three different methods. The first is to use the conventional perturbation theory, and the second is to impose the cross-cap bootstrap equation, and the third is to solve the Schwinger–Dyson equation under the assumption of conformal invariance. We find that the three methods lead to mutually consistent results but each has its own advantage.

Time-dependent observables in heavy ion collisions. Part I. Setting up the formalism

We adapt the Schwinger-Keldysh formalism to study heavy-ion collisions in perturbative QCD. Employing the formalism, we calculate the two-point gluon correlation function G22aμ, bν due to the lowest-order classical gluon fields in the McLerran-Venugopalan model of heavy ion collisions and observe an interesting transition from the classical fields to the quasi-particle picture at later times. Motivated by this observation, we push the formalism to higher orders in the coupling and calculate the contribution to G22aμ, bν coming from the diagrams representing a single rescattering between two of the produced gluons. We assume that the two gluons go on mass shell both before and after the rescattering. The result of our calculation depends on which region of integration over the proper time of the rescattering τZ gives the correct correlation function at late proper time τ when the gluon distribution is measured. For (i) τZ ≫ 1/Qs and τ − τZ ≫ 1/Qs (with Qs the saturation scale) we obtain the same results as from the Boltzmann equation. For (ii) τ − τZ ≫ τZ ≫ 1/Qs we end up with a result very different from kinetic theory and consistent with a picture of “free-streaming” particles. Due to the approximations made, our calculation is too coarse to indicate whether the region (i) or (ii) is the correct one: to resolve this controversy, we shall present a detailed diagrammatic calculation of the rescattering correction in the φ4 theory in the second paper of this duplex.

Time-dependent observables in heavy ion collisions. Part II. In search of pressure isotropization in the \u03c64 theory

To understand the dynamics of thermalization in heavy ion collisions in the perturbative framework it is essential to first find corrections to the free-streaming classical gluon fields of the McLerran-Venugopalan model. The corrections that lead to deviations from free streaming (and that dominate at late proper time) would provide evidence for the onset of isotropization (and, possibly, thermalization) of the produced medium. To find such corrections we calculate the late-time two-point Green function and the energy-momentum tensor due to a single 2 → 2 scattering process involving two classical fields. To make the calculation tractable we employ the scalar φ4 theory instead of QCD. We compare our exact diagrammatic results for these quantities to those in kinetic theory and find disagreement between the two. The disagreement is in the dependence on the proper time τ and, for the case of the two-point function, is also in the dependence on the space-time rapidity η: the exact diagrammatic calculation is, in fact, consistent with the free streaming scenario. Kinetic theory predicts a build-up of longitudinal pressure, which, however, is not observed in the exact calculation. We conclude that we find no evidence for the beginning of the transition from the free-streaming classical fields to the kinetic theory description of the produced matter after a single 2 → 2 rescattering.

Functional renormalization group and Kohn–Sham scheme in density functional theory

Deriving accurate energy density functional is one of the central problems in condensed matter physics, nuclear physics, and quantum chemistry. We propose a novel method to deduce the energy density functional by combining the idea of the functional renormalization group and the Kohn–Sham scheme in density functional theory. The key idea is to solve the renormalization group flow for the effective action decomposed into the mean-field part and the correlation part. Also, we propose a simple practical method to quantify the uncertainty associated with the truncation of the correlation part. By taking the φ4 theory in zero dimension as a benchmark, we demonstrate that our method shows extremely fast convergence to the exact result even for the highly strong coupling regime.

Physics beyond the standard model may be described by a massless QFT

I argue that the work of Zimmermann with Lowenstein on the massless ϕ4 theory [1] can play an important role in going beyond the standard model of elementary particles.

Statistical nature of infrared dynamics on de Sitter background

In this study, we formulate a systematic way of deriving an effective equation of motion(EoM) for long wavelength modes of a massless scalar field with a general potential V(ϕ) on de Sitter background, and investigate whether or not the effective EoM can be described as a classical stochastic process. Our formulation gives an extension of the usual stochastic formalism to including sub-leading secular growth coming from the nonlinearity of short wavelength modes. Applying our formalism to λ ϕ4 theory, we explicitly derive an effective EoM which correctly recovers the next-to-leading secularly growing part at a late time, and show that this effective EoM can be seen as a classical stochastic process. Our extended stochastic formalism can describe all secularly growing terms which appear in all correlation functions with a specific operator ordering. The restriction of the operator ordering will not be a big drawback because the commutator of a light scalar field becomes negligible at large scales owing to the squeezing.

Quantum spaces, central extensions of Lie groups and related quantum field theories11Based on a talk presented by Poulain T at the XXVth International Conference on Integrable Systems and Quantum symmetries (ISQS-25), Prague, June 6-10 2017.

Quantum spaces with su(2) noncommutativity can be modelled by using a family of SO(3)-equivariant differential *-representations. The quantization maps are determined from the combination of the Wigner theorem for SU(2) with the polar decomposition of the quantized plane waves. A tracial star-product, equivalent to the Kontsevich product for the Poisson manifold dual to su(2) is obtained from a subfamily of differential *-representations. Noncommutative (scalar) field theories free from UV/IR mixing and whose commutative limit coincides with the usual ϕ4 theory on ℛ3 are presented. A generalization of the construction to semi-simple possibly non simply connected Lie groups based on their central extensions by suitable abelian Lie groups is discussed.

Unimodular Gravity and General Relativity UV divergent contributions to the scattering of massive scalar particles

We work out the one-loop and order κ2 mϕ2 UV divergent contributions, coming from Unimodular Gravity and General Relativity, to the S matrix element of the scattering process ϕ + ϕ→ ϕ + ϕ in a λ ϕ4 theory with mass mϕ. We show that both Unimodular Gravity and General Relativity give rise to the same UV divergent contributions in Dimensional Regularization. This seems to be at odds with the known result that in a multiplicative MS dimensional regularization scheme the General Relativity corrections, in the de Donder gauge, to the beta function, βλ, of the λ coupling do not vanish, whereas the Unimodular Gravity corrections, in a certain gauge, do vanish. Actually, by comparing the UV divergent contributions calculated in this paper with those which give rise to the non-vanishing gravitational corrections to βλ, one readily concludes that the UV divergent contributions that yield the just mentioned non-vanishing gravitational corrections to βλ do not contribute to the UV divergent behaviour of the S matrix element of ϕ + ϕ→ ϕ + ϕ. This shows that any physical consequence—such as the existence of asymptotic freedom due to gravitational interactions—drawn from the value of βλ is not physically meaningful.

Regularizing Feynman path integrals using the generalized Kontsevich-Vishik trace

A fully regulated definition of Feynman’s path integral is presented here. The proposed re-formulation of the path integral coincides with the familiar formulation whenever the path integral is well defined. In particular, it is consistent with respect to lattice formulations and Wick rotations, i.e., it can be used in Euclidean and Minkowski space-time. The path integral regularization is introduced through the generalized Kontsevich-Vishik trace, that is, the extension of the classical trace to Fourier integral operators. Physically, we are replacing the time-evolution semi-group by a holomorphic family of operators such that the corresponding path integrals are well defined in some half space of C. The regularized path integral is, thus, defined through analytic continuation. This regularization can be performed by means of stationary phase approximation or computed analytically depending only on the Hamiltonian and the observable (i.e., known a priori). In either case, the computational effort to evaluate path integrals or expectations of observables reduces to the evaluation of integrals over spheres. Furthermore, computations can be performed directly in the continuum and applications (analytic computations and their implementations) to a number of models including the non-trivial cases of the massive Schwinger model and a φ4 theory.