Φ4 Theory Research Articles

Three-body resonances in the φ4 theory

We study the properties of three-body resonances using a lattice complex scalar φ4 theory with two scalars, with parameters chosen such that one heavy particle can decay into three light ones. We determine the two- and three-body spectra for several lattice volumes using variational techniques, and then analyze them with two versions of the three-particle finite-volume formalism: the Relativistic Field Theory approach and the Finite-Volume Unitarity approach. We find that both methods provide an equivalent description of the energy levels, and we are able to fit the spectra using simple parametrizations of the scattering quantities. By solving the integral equations of the corresponding three-particle formalisms, we determine the pole position of the resonance in the complex energy plane and thereby its mass and width. We find very good agreement between the two methods at different values of the coupling of the theory.

ANEC on stress-tensor states in perturbative λ ϕ4 theory

We evaluate the Average Null Energy Condition (ANEC) on momentum eigenstates generated by the stress tensor in perturbative λ ϕ4 and general spacetime dimension. We first compute the norm of the stress-tensor state at second order in λ; as a by-product of the derivation we obtain the full expression for the stress tensor 2-point function at this order. We then compute the ANEC expectation value to first order in λ, which also depends on the coupling of the stress-tensor improvement term ξ. We study the bounds on these couplings that follow from the ANEC and unitarity at first order in perturbation theory. These bounds are stronger than unitarity in some regions of coupling space.

Non-Equilibrium ϕ4 Theory in a Hierarchy: Towards Manipulating Holograms in Quantum Brain Dynamics

We describe non-equilibrium ϕ4 theory in a hierarchical manner to develop a method for manipulating coherent fields as a toy model of introducing control into Quantum Field Theory (QFT) of the brain, which is called Quantum Brain Dynamics (QBD). We begin with the Lagrangian density of ϕ4 model, where we adopt 2-Particle-Irreducible (2PI) effective action, and derive the Klein–Gordon equation of coherent fields with a damping term as an input–output equation proposed in areas of morphological computation or reservoir computing. Our analysis is extended to QFT in a hierarchy representing multiple layers covering cortex in a brain. We find that the desired target function is achieved via time-evolution in the Klein–Gordon equations in a hierarchy of numerical simulations when a signal in both the input and output prevails over noise in the intermediate layers. Our approach will be applied to control coherent fields in the systems (in a hierarchy) described in the QFT framework, with potential applications allowing the manipulation of quantum fields, especially holograms in QBD. We could then provide realistic physical degrees of freedom of a light–matter system in the contexts of quantum cognition and the associated free-energy principle.

Nonperturbative bounds on scattering of massive scalar particles in d ≥ 2

We study two-to-two scattering amplitudes of a scalar particle of mass m. For simplicity, we assume the presence of ℤ2 symmetry and that the particle is ℤ2 odd. We consider two classes of amplitudes: the fully nonperturbative ones and effective field theory (EFT) ones with a cut-off scale M. Using the primal numerical method which allows us to impose full non-linear unitarity, we construct novel bounds on various observables in 2 ≤ d ≤ 4 space-time dimensions for both classes of amplitudes. We show that our bounds are much stronger than the ones obtained by using linearized unitarity or positivity only. We discuss applications of our bounds to constraining EFTs. Finally, we compare our bounds to the amplitude in ϕ4 theory computed perturbatively at weak coupling, and find that they saturate the bounds.

Integrating out beyond tree level and relativistic superfluids

We revisit certain subtleties of renormalization that arise when one derives a low-energy effective action by integrating out the heavy fields of a more complete theory. Usually these subtleties are circumvented by matching some physical observables, such as scattering amplitudes, but a more involved procedure is required if one is interested in deriving the effective theory to all orders in the light fields (but still to fixed order in the derivative expansion). As a concrete example, we study the U(1) Goldstone low-energy effective theory that describes the spontaneously broken phase of a ϕ4 theory for a complex scalar. Working to lowest order in the derivative expansion, but to all orders in the Goldstones, we integrate out the radial mode at one loop and express the low-energy effective action in terms of the renormalized couplings of the UV completion. This yields the one-loop equation of state for the superfluid phase of (complex) ϕ4. We perform the same analysis for a renormalizable scalar SO(N) theory at finite chemical potential, integrating out the gapped Goldstones as well, and confirm that the effective theory for the gapless Goldstone exhibits no obvious sign of the original SO(N) symmetry.

Recursive computation of Feynman periods

Feynman periods are Feynman integrals that do not depend on external kinematics. Their computation, which is necessary for many applications of quantum field theory, is greatly facilitated by graphical functions or the equivalent conformal four-point integrals. We describe a set of transformation rules that act on such functions and allow their recursive computation in arbitrary even dimensions. As a concrete example we compute all subdivergence-free Feynman periods in ϕ3 theory up to six loops and 561 of 607 Feynman periods at seven loops analytically. Our results support the conjectured existence of a coaction structure in quantum field theory and suggest that ϕ3 and ϕ4 theory share the same number content.

Conformal triangles and zig-zag diagrams

A convenient operator representation for zig-zag four-point and two-point planar Feynman diagrams relevant to the bi-scalar D-dimensional “fishnet” field theory is obtained. This representation gives a possibility to evaluate exactly diagrams of the zig-zag series in special cases. In particular, we give a fairly simple proof of the Broadhurst-Kreimer conjecture about the values of zig-zag multi-loop two-point diagrams which make a significant contribution to the renormalization group β-function in the 4-dimensional ϕ4 theory.

Form factors and spectral densities from Lightcone Conformal Truncation

We use the method of Lightcone Conformal Truncation (LCT) to obtain form factors and spectral densities of local operators \U0001d4aa in ϕ4 theory in two dimensions. We show how to use the Hamiltonian eigenstates from LCT to obtain form factors that are matrix elements of a local operator \U0001d4aa between single-particle bra and ket states, and we develop methods that significantly reduce errors resulting from the finite truncation of the Hilbert space. We extrapolate these form factors as a function of momentum to the regime where, by crossing symmetry, they are form factors of \U0001d4aa between the vacuum and a two-particle asymptotic scattering state. We also compute the momentum-space time-ordered two-point functions of local operators in LCT. These converge quickly at momenta away from branch cuts, allowing us to indirectly obtain the time-ordered correlator and the spectral density at the branch cuts. We focus on the case where the local operator \U0001d4aa is the trace Θ of the stress tensor.

Bootstrapping 2d ϕ4 theory with Hamiltonian truncation data

We combine the methods of Hamiltonian Truncation and the recently proposed generalisation of the S-matrix bootstrap that includes local operators to determine the two-particle scattering amplitude and the two-particle form factor of the stress tensor at s > 0 in the 2d ϕ4 theory. We use the form factor of the stress tensor at s ≤ 0 and its spectral density computed using Lightcone Conformal Truncation (LCT), and inject them into the generalized S-matrix bootstrap set-up. The obtained results for the scattering amplitude and the form factor are fully reliable only in the elastic regime. We independently construct the “pure” S-matrix bootstrap bounds (bootstrap without including matrix elements of local operators), and find that the sinh-Gordon model and its analytic continuation the “staircase model” saturate these bounds. Surprisingly, the ϕ4 two-particle scattering amplitude also very nearly saturates these bounds, and moreover is extremely close to that of the sinh-Gordon/staircase model.

Convexity, large charge and the large-N phase diagram of the φ4 theory

In this note we discuss the phase space of the O(2N) vector model in the presence of a quadratic and a quartic interaction by writing the large-N effective potential using large charge methods in dimensions 2 < D < 4 and 4 < D < 6. Based on a simple discussion of the convexity properties of the grand potential, we find very different behavior in the two regimes: while in 2 < D < 4, the theory is well-behaved, the model in 4 < D < 6 leads to a complex CFT in the UV, consistently with earlier results. We also find a new metastable massive phase in the high-energy regime for the theory on the cylinder.

Self-similar sequence transformation for critical exponents

Self-similar sequence transformation is an original type of nonlinear sequence transformations allowing for defining effective limits of asymptotic sequences. The method of self-similar factor transformations is shown to be regular. This method is applied for calculating the critical exponents of the O(N)-symmetric φ4 theory in three dimensions by summing asymptotic ε expansions. It is shown that this method is straightforward and essentially simpler than other summation techniques involving complicated numerical calculations, while enjoying comparable accuracy.

Numerical Study of Multiparticle Production in ϕ4 Theory: Comparison with Analytical Results

We develop a numerical method to compute the probabilities of multiparticle production in weakly coupled scalar theories. Our technique is based on D.T. Son’s semiclassical method of singular solutions. Applying it to the process $$1 \to n$$ in the unbroken four-dimensional $$\lambda {{\phi }^{4}}$$ theory, we reproduce the known results at $$1 \ll n \ll {{\lambda }^{{ - 1}}}$$ .

Hamiltonian truncation in Anti-de Sitter spacetime

Quantum Field Theories (QFTs) in Anti-de Sitter (AdS) spacetime are often strongly coupled when the radius of AdS is large, and few methods are available to study them. In this work, we develop a Hamiltonian truncation method to compute the energy spectrum of QFTs in two-dimensional AdS. The infinite volume of constant timeslices of AdS leads to divergences in the energy levels. We propose a simple prescription to obtain finite physical energies and test it with numerical diagonalization in several models: the free massive scalar field, ϕ4 theory, Lee-Yang and Ising field theory. Along the way, we discuss spontaneous symmetry breaking in AdS and derive a compact formula for perturbation theory in quantum mechanics at arbitrary order. Our results suggest that all conformal boundary conditions for a given theory are connected via bulk renormalization group flows in AdS.

Critical behavior of the 2d scalar theory: resumming the N8LO perturbative mass gap

We apply the optimized perturbation theory (OPT) to resum the perturbative series describing the mass gap of the bidimensional ϕ4 theory in the ℤ2 symmetric phase. Already at NLO (one loop) the method is capable of generating a quite reasonable non-perturbative result for the critical coupling. At order-g7 we obtain gc = 2.779(25) which compares very well with the state of the art N8LO result, gc = 2.807(34). As a novelty we investigate the supercritical region showing that it contains some useful complimentary information that can be used in extrapolations to arbitrarily high orders.

Real scalar phase transitions: a nonperturbative analysis

We study the thermal phase transitions of a generic real scalar field, without a Z2-symmetry, referred to variously as an inert, sterile or singlet scalar, or ϕ3 + ϕ4 theory. Such a scalar field arises in a wide range of models, including as the inflaton, or as a portal to the dark sector. At high temperatures, we perform dimensional reduction, matching to an effective theory in three dimensions, which we then study both perturbatively to three-loop order and on the lattice. For strong first-order transitions, with large tree-level cubic couplings, our lattice Monte-Carlo simulations agree with perturbation theory within error. However, as the size of the cubic coupling decreases, relative to the quartic coupling, perturbation theory becomes less and less reliable, breaking down completely in the approach to the Z2-symmetric limit, in which the transition is of second order. Notwithstanding, the renormalisation group is shown to significantly extend the validity of perturbation theory. Throughout, our calculations are made as explicit as possible so that this article may serve as a guide for similar calculations in other theories.

Kink-antikink collisions and multi-bounce resonance windows in higher-order field theories

We study collisions of coherent structures in higher-order field-theoretic models, such as the ϕ8,ϕ10 and ϕ12 ones. The main distinguishing feature, of the example models considered herein, is that the collision arises due to the long-range interacting algebraic tails of these solitary waves. We extend the approach to suitably initialize the relevant kinks, in the additional presence of finite initial velocity, in order to minimize the dispersive wave radiation potentially created by their slow spatial decay. We find that, when suitably initialized, these models still feature the multi-bounce resonance windows earlier found in models in which the kinks bear exponential tails, such as the ϕ4 and ϕ6 field theories among others. Also present is the self-similar structure of the associated windows with three- and more-bounce windows at the edges of two- and lower-bounce ones. Moreover, phenomenological but highly accurate (and predictive), scaling relations are derived for the dependence of the time between consecutive collisions and, e.g., the difference in kinetic energy between the incoming one and the critical one for one-bounces. Such scalings are traced extensively over two-bounce collision windows throughout the three models, hinting at the possibility of an analytical theory in this direction.

Critical exponents from the weak-coupling, strong-coupling and large-order parametrization of the hypergeometric ([formula omitted]) approximants

In this work, we suggest a new parametrization for the hypergeometric (k+1Fk) approximants introduced by Mera et al. (2015). The new parametrization enables the approximants to accommodate all perturbative and non-perturbative information of the divergent series as input. Also, the parametrization has been shown to account for the n! growth factor of the given perturbation series provided that one of the denominator parameters of the hypergeometric approximant takes large values. The algorithm with the new parametrization has been tested using two quantum mechanical problems where one can incorporate the weak-coupling, strong-coupling and large-order information. Accurate results have been obtained in using a relatively low order from the perturbation series. Since strong-coupling behavior is not yet known for the renormalization group functions of the O(N)-symmetric ϕ4 theory, we used weak-coupling and large-order parametrization to resum the seven-loop critical exponents ν,η and ω for N=0,1,2,3,4. In view of the recent results from six-loop resummation as well as Monte Carlo simulations and conformal bootstrap calculations, our results show a clear improvement to the six-loop results.

Exciting the domain wall soliton

Many solitonic configurations in field theory have localized bound states in their spectrum of linear perturbations. This opens up the possibility of having long lived excitations of these solitons that could affect their dynamics. We start the study of these effects in the simplest configuration of a domain wall kink solution in the λ ϕ4 theory in 1+1 dimensions. We show that this solution has a single bound state and numerically study its slow decay rate in flat space. We then investigate the amplitude of this excitation by simulating a cosmological phase transition that leads to the formation of these kinks in an expanding universe. We find that kinks get formed with a 20% excess of energy with respect to their lowest energy configuration. We also explore the kink solution interacting with a thermal bath and extract the amplitude of the localized excitation as a function of temperature. We note that this amplitude increases with temperature but again the extra energy in the kink never goes over the 20% level. Finally, we argue that this extra energy may have important consequences in the subsequent evolution of defects in numerical simulations.

Perturbative renormalization of the lattice regularized ϕ44 with flow equations

The flow equations of the renormalization group allow one to analyze the perturbative n-point functions of renormalizable quantum field theories. Rigorous bounds implying renormalizability permit one to control large momentum behavior, infrared singularities, and large order behavior in a number of loops and a number of arguments n. In this paper, we analyze the Euclidean four-dimensional massive ϕ4 theory using lattice regularization. We present a rigorous proof that this quantum field theory is renormalizable to all orders of the loop expansion based on the flow equations. The lattice regularization is known to break Euclidean symmetry. Our main result is the proof of the restoration of rotation and translation invariance in the renormalized theory using flow equations.

Investigation of Complex \u03d54 Theory at Finite Density in Two Dimensions Using TRG

We study the two-dimensional complex ϕ4 theory at finite chemical potential using the tensor renormalization group. This model exhibits the Silver Blaze phenomenon in which bulk observables are independent of the chemical potential below the critical point. Since it is expected to be a direct outcome of an imaginary part of the action, an approach free from the sign problem is needed. We study this model systematically changing the chemical potential in order to check the applicability of the tensor renormalization group to the model in which scalar fields are discretized by the Gaussian quadrature. The Silver Blaze phenomenon is successfully confirmed on the extremely large volume V = 10242 and the results are also ensured by another tensor network representation with a character expansion.