Massless Scalar Field Theory Research Articles

Beyond Wilson? Carroll from current deformations

At extreme energies, both low and high, the spacetime symmetries of relativistic quantum field theories (QFTs) are expected to change with Galilean symmetries emerging in the very low energy domain and, as we will argue, Carrollian symmetries appearing at very high energies. The formulation of Wilsonian renormalisation group seems inadequate for handling these changes of the underlying Poincare symmetry of QFTs and it seems unlikely that these drastic changes can be seen within the realms of relativistic QFT. We show that contrary to this expectation, changes in the spacetime algebra occurs at the very edges of parameter space. In particular, we focus on the very high energy sector and show how bilinears of U(1) currents added to a two dimensional (massless) scalar field theory deform the relativistic spacetime conformal algebra to conformal Carroll as the effective coupling of the deformation is dialed to infinity. We demonstrate this using both a symmetric and an antisymmetric current-current deformation for theories with multiple scalar fields. These two operators generate distinct kinds of quantum flows in the coupling space, the symmetric driven by Bogoliubov transformations and the antisymmetric by spectral flows, both leading to Carrollian CFTs at the end of the flow.

Massless Lifshitz field theory for arbitrary z

By using the notion of fractional derivatives, we introduce a class of massless Lifshitz scalar field theory in (1+1)-dimension with an arbitrary anisotropy index z. The Lifshitz scale invariant ground state of the theory is constructed explicitly and takes the form of Rokhsar-Kivelson (RK). We show that there is a continuous family of ground states with degeneracy parameterized by the choice of solution to the equation of motion of an auxiliary classical system. The quantum mechanical path integral establishes a 2d/1d correspondence with the equal time correlation functions of the Lifshitz scalar field theory. We study the entanglement properties of the Lifshitz theory for arbitrary z using the path integral representation. The entanglement measures are expressed in terms of certain cross ratio functions we specify, and satisfy the c-function monotonicity theorems. We also consider the holographic description of the Lifshitz theory. In order to match with the field theory result for the entanglement entropy, we propose a z-dependent radius scale for the Lifshitz background. This relation is consistent with the z-dependent scaling symmetry respected by the Lifshitz vacuum. Furthermore, the time-like entanglement entropy is determined using holography. Our result suggests that there should exist a fundamental definition of time-like entanglement other than employing analytic continuation as performed in relativistic field theory.

State dependence of Krylov complexity in 2d CFTs

We compute the Krylov Complexity of a light operator O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}L in an eigenstate of a 2d CFT at large central charge c. The eigenstate corresponds to a primary operator O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}H under the state-operator correspondence. We observe that the behaviour of K-complexity is different (either bounded or exponential) depending on whether the scaling dimension of O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}H is below or above the critical dimension hH = c/24, marked by the 1st order Hawking-Page phase transition point in the dual AdS3 geometry. Based on this feature, we hypothesize that the notions of operator growth and K-complexity for primary operators in 2d CFTs are closely related to the underlying entanglement structure of the state in which they are computed, thereby demonstrating explicitly their state-dependent nature. To provide further evidence for our hypothesis, we perform an analogous computation of K-complexity in a model of free massless scalar field theory in 2d, and in the integrable 2d Ising CFT, where there is no such transition in the spectrum of states.

Gaussian beam propagation in a Lorentz-violating vacuum in the presence of a semi-transparent mirror

In this paper we study the propagation of structured optical scalar beams in a Lorentz-violating (LV) vacuum parametrized by a constant 4-vector u μ and in the presence of a semi-transparent mirror. The two bosonic degrees of freedom of the electromagnetic field can be described by a LV extension of the massless scalar field theory, whose LV part is characterized by the term (u · ∂ϕ)2. The mirror at a surface Σ is modelled by a delta-type potential in the Lagrange density for the LV scalar field, i.e. λ δ(Σ)ϕ 2, where the parameter λ controls the degree of transparency of the mirror. Using Green’s function techniques, we investigate the propagation of a Gaussian beam in the presence of a mirror which is perpendicular to the propagation direction and for two particular choices of the background 4-vector: parallel and perpendicular to the propagation direction. To quantify the Lorentz-violating effects we introduce the fidelity as a measurement of the closeness of the propagated field distribution with respect to that in the conventional vacuum. In the absence of the mirror (λ = 0) the fidelity is found to be close to one, and hence LV effects are quite small. However in the presence of the mirror, there are regions where the fidelity drops to zero, thus implying that LV effects could be clearly differentiated from the propagation in vacuum. Within the paraxial approximation we determine analytically the LV effects upon the Rayleigh range, the radius of the beam, the Gouy phase and the radius of curvature of the wavefronts. We discuss possible scenarios where our results could apply, by using optically transparent multiferroic materials, which offer unprecedented opportunities to tailor structured beam propagation, as well as to simulate an LV vacuum.

Symmetry group at future null infinity: Scalar theory

We reduce the massless scalar field theory in Minkowski spacetime to future null infinity. We compute the Poincar\'e flux operators, which can be generalized and identified as the supertranslation and superrotation generators. These generators are shown to form a closed symmetry algebra with a divergent central charge. In the classical limit, we argue that the algebra may be interpreted as the geometric symmetry of a Carrollian manifold, i.e., the hypersurface of future null infinity. Our method may be used to find more physically interesting Carrollian field theories.

Loop corrections in Minkowski spacetime away from equilibrium. Part I. Late-time resummations

Loop corrections to unequal-time correlation functions in Minkowski spacetime exhibit secular growth due to a breakdown of time-dependent perturbation theory. This is analogous to secular growth in equal-time correlators on time-dependent backgrounds, except that in Minkowski the divergences must not signal a real IR issue. In this paper, we calculate the late-time limit of the two-point correlator for different massless self-interacting scalar quantum field theories on a Minkowski background. We first use a late-time version of the in-in path integral starting in the vacuum of the free theory; in this limit, the calculation, including UV renormalization, reduces to that in in-out. We find linear or logarithmic growth in time, depending on whether the interaction strength is dimension-one or dimensionless, respectively. We next develop the Weisskopf-Wigner resummation method, that proceeds by demanding unitarity within a truncated Hilbert space, to calculate the resummed correlator and find that it gives an exact exponentiation of the late-time perturbative result. The resummed (unequal-time) correlator thus decays with an exponential or polynomial time-dependence, which is suggestive of ‘universal’ behavior that depends on the dimensions of the interaction strength.

Loop corrections in Minkowski spacetime away from equilibrium. Part II. Finite-time results

Loop corrections to finite-time correlation functions in quantum field theories away from equilibrium can be calculated using the in-in path integral approach. In this paper, we calculate the unequal-time two-point correlator for different massless self-interacting scalar quantum field theories on a Minkowski background, starting the field evolution at an arbitrary initial time. We find the counterterms that need to be added to UV-renormalize the result, including usual in-out counterterms in the dynamics and additional initial state counterterms that are required to cancel all UV divergences. We find that the late-time limit of the renormalized correlation function exhibits a linear or logarithmic growth in time, depending on whether the interaction strength is dimension-one or dimensionless, respectively. The late-time correlations match those obtained in our companion paper and, as shown there, the divergences do not indicate a real IR issue, consistent with what one would expect in Minkowski.

Determining all thermodynamic transport coefficients for an interacting large N quantum field theory

Thermodynamic transport coefficients can be calculated directly from quantum field theory without requiring analytic continuation to real time. We determine all second-order thermodynamic transport coefficients for the uncharged N-component massless (critical) scalar field theory with quartic interaction in the large N limit, for any value of the coupling. We find that in the large N limit, all thermodynamic transport coefficients for the interacting theory can be expressed analytically in terms of the in-medium mass and sums over modified Bessel functions. We expect our technique to allow a similar determination of all thermodynamic transport coefficients for all theories that are solvable in the large N limit, including certain gauge theories.

Nonlinear dynamical Casimir effect at weak nonstationarity

We show that even small nonlinearities significantly affect particle production in the dynamical Casimir effect at large evolution times. To that end, we derive the effective Hamiltonian and resum leading loop corrections to the particle flux in a massless scalar field theory with time-dependent Dirichlet boundary conditions and quartic self-interaction. To perform the resummation, we assume small deviations from the equilibrium and employ a kind of rotating wave approximation. Besides that, we consider a quantum circuit analog of the dynamical Casimir effect, which is also essentially nonlinear. In both cases, loop contributions to the number of created particles are comparable to the tree-level values.

Hidden conformal symmetry on the black hole photon sphere

We consider a class of static and spherically symmetric black hole geometries endowed with a photon sphere. On the one hand, we show that close to the photon sphere, a massless scalar field theory exhibits a simple dynamical SL(2, ℝ) algebraic structure which allows to recover the discrete spectrum of the weakly damped quasinormal frequencies in the eikonal approximation, and the associated quasinormal modes from the algebra representations. On the other hand, we consider the non-radial motion of a free-falling test particle, in the equatorial plane, from spatial infinity to the black hole. In the ultrarelativistic limit, we show that the photon sphere acts as an effective Rindler horizon for the geodesic motion of the test particle in the (t, r)-plane, with an associated Unruh temperature Tc = ħΛc/2πkB, where Λc is the Lyapunov exponent that characterizes the unstable circular motions of massless particles on the photon sphere. The photon sphere then appears as a location where the thermal bound on chaos for quantum systems with a large number of degrees of freedom, in the form conjectured a few years ago by Maldacena et al., is saturated. The study developed in this paper could hopefully shed a new light on the gravity/CFT correspondence, particularly in asymptotically flat spacetimes, in which the photon sphere may also be considered as a holographic screen.

Renormalization and non-renormalization of scalar EFTs at higher orders

We renormalize massless scalar effective field theories (EFTs) to higher loop orders and higher orders in the EFT expansion. To facilitate EFT calculations with the R* renormalization method, we construct suitable operator bases using Hilbert series and related ideas in commutative algebra and conformal representation theory, including their novel application to off-shell correlation functions. We obtain new results ranging from full one loop at mass dimension twelve to five loops at mass dimension six. We explore the structure of the anomalous dimension matrix with an emphasis on its zeros, and investigate the effects of conformal and orthonormal operators. For the real scalar, the zeros can be explained by a ‘non-renormalization’ rule recently derived by Bern et al. For the complex scalar we find two new selection rules for mixing n- and (n− 2)-field operators, with n the maximal number of fields at a fixed mass dimension. The first appears only when the (n− 2)-field operator is conformal primary, and is valid at one loop. The second appears in more generic bases, and is valid at three loops. Finally, we comment on how the Hilbert series we construct may be used to provide a systematic enumeration of a class of evanescent operators that appear at a particular mass dimension in the scalar EFT.

Emergent geometry and path integral optimization for a Lifshitz action

Extending the background metric optimization procedure for Euclidean path integrals of two-dimensional conformal field theories, introduced by Caputa et al. [Phys. Rev. Lett. 119, 071602 (2017), J. High Energy Phys. 11 (2017) 097], to a $z=2$ anisotropically scale-invariant $(2+1)$-dimensional Lifshitz field theory of a free massless scalar field, we find optimal geometries for static and dynamic correlation functions. For the static correlation functions, the optimal background metric is equivalent to an AdS metric on a Poincar\'e patch, while for dynamical correlation functions, we find Lifshitz like metric. This results suggest that a MERA-like tensor network, perhaps without unitarity, would still be considered an optimal background spacetime configuration for the numerical description of this system, even though the classical action we start with is not a conformal field theory.

Geometric quantum information structure in quantum fields and their lattice simulation

An upper limit to distillable entanglement between two disconnected regions of massless non-interacting scalar field theory has an exponential decay defined by a geometric decay constant. When regulated at short distances with a spatial lattice, this entanglement abruptly vanishes beyond a dimensionless separation, defining a negativity sphere. In two spatial dimensions, we determine this geometric decay constant between a pair of disks and the growth of the negativity sphere toward the continuum through a series of lattice calculations. Making the connection to quantum field theories in three-spatial dimensions, assuming such quantum information scales appear also in quantum chromodynamics (QCD), a new relative scale may be present in effective field theories describing the low-energy dynamics of nucleons and nuclei. We highlight potential impacts of the distillable entanglement structure on effective field theories, lattice QCD calculations and future quantum simulations.

Unruh-DeWitt detector responses for complex scalar fields in de Sitter spacetime

We derive the response function for a comoving, pointlike Unruh-DeWitt particle detector coupled to a complex scalar field ϕ, in the (3 + 1)-dimensional cosmological de Sitter spacetime. The field-detector coupling is taken to be proportional to ϕ†ϕ. We address both conformally invariant and massless minimally coupled scalar field theories, respectively in the conformal and the Bunch-Davies vacuum. The response function integral for the massless minimal complex scalar, not surprisingly, shows divergences and accordingly we use suitable regularisation scheme to find out well behaved results. The regularised result also contains a de Sitter symmetry breaking logarithm, growing with the cosmological time. Possibility of extension of these results with the so called de Sitter α-vacua is discussed. While we find no apparent problem in computing the response function for a real scalar in these vacua, a complex scalar field is shown to contain some possible ambiguities in the detector response. The case of the minimal and nearly massless scalar field theory is also briefly discussed.

Imprints of the Janis-Newman-Winicour spacetime on observations related to shadow and accretion

The final fate of gravitational collapse of massive stars has been a subject of interest for a long time since such a collapse may lead to black holes and naked singularities alike. Since, the formation of naked singularities is forbidden by the cosmic censorship conjecture, exploring their observational differences from black holes may be a possible avenue to search for these exotic objects. The simplest possible naked singularity spacetime emerges from the Einstein massless scalar field theory with the advantage that it smoothly translates to the Schwarzschild solution by the variation of the scalar charge. This background, known as the Janis-Newman-Winicour spacetime is the subject of interest in this work. %A further advantage of working with this spacetime is that it smoothly translates to the Schwarzschild solution by the variation of the scalar charge. We explore electromagnetic observations around this metric which involves investigating the characteristics of black hole accretion and shadow. We compute the shadow radius in this spacetime and compare it with the image of M87*, recently released by the Event Horizon Telescope Collaboration. Similarly, we derive the expression for the luminosity from the accretion disk and compare it with the observed optical luminosity of eleven Palomar Green quasars. Our analysis indicates that the shadow of M87* and the quasar optical data consistently favor the Schwarzschild background over the Janis-Newman-Winicour spacetime. The implications of this result are discussed.

Partner formula for an arbitrary moving mirror in 1+1 dimensions

In the information loss problem in black hole evaporation, the investigation of the purification partner of a Hawking particle is crucial. It is a well-known fact that the $3+1$ dimensional spherically symmetric gravitational collapse can be mimicked by $1+1$ dimensional moving mirror models. Since a detected particle in field theory is defined by what a particle detector observes, the diversity of detector designs yields a variety of particles and their partners. We provide a formula of generalized partners of detected particles emitted out of mirrors in an arbitrary motion for any Gaussian state in a free massless scalar field theory. Using our formula, we directly demonstrate information storage about initial phase information in a pure state of a detected particle and its partner. The form of the partner drastically changes depending on the detailed designs of particle detectors for Hawking radiation. In the case of a detected particle and its partner sensitive to information about initial phase, spatial configurations of the partner has long tails in a stage where only zero-point fluctuation is emitted out of the mirror.

Accordiohedra as positive geometries for generic scalar field theories

We build upon the prior works of [1-3] to study tree-level planar amplitudes for a massless scalar field theory with polynomial interactions. Focusing on a specific example, where the interaction is given by $\lambda_3\phi^{3}\ +\lambda_4 \phi^{4}$, we show that a specific convex realization of a simple polytope known as the accordiohedron in kinematic space is the positive geometry for this theory. As in the previous cases, there is a unique planar scattering form in kinematic space, associated to each positive geometry which yields planar scattering amplitudes.

Influence of conformal symmetry on the amplitude ratios for O(N) models

In this Letter we compute analytically the effect of conformal symmetry on the radiative corrections to the amplitude ratios for O massless scalar field theories in curved spacetime for probing the two-scale-factor universality hypothesis. For that we employ three distinct and independent field-theoretic renormalization group methods. The amplitude ratios values obtained are identical when computed through the three distinct methods, thus showing their universal character. Furthermore, they are the same as that obtained in flat spacetime, then satisfying the two-scale-factor universality hypothesis.

Galilean field theories and conformal structure

We perform a detailed analysis of Galilean field theories, starting with free theories and then interacting theories. We consider non-relativistic versions of massless scalar and Dirac field theories before we go on to review our previous construction of Galilean Electrodynamics and Galilean Yang-Mills theory. We show that in all these cases, the field theories exhibit non-relativistic conformal structure (in appropriate dimensions). The surprising aspect of the analysis is that the non-relativistic conformal structure exhibited by these theories, unlike relativistic conformal invariance, becomes infinite dimensional even in spacetime dimensions greater than two. We then couple matter with Galilean gauge theories and show that there is a myriad of different sectors that arise in the non-relativistic limit from the parent relativistic theories. In every case, if the parent relativistic theory exhibited conformal invariance, we find an infinitely enhanced Galilean conformal invariance in the non-relativistic case. This leads us to suggest that infinite enhancement of symmetries in the non-relativistic limit is a generic feature of conformal field theories in any dimension.

Charged Compact Boson Stars in a Theory of Massless Scalar Field

In this work we present some new results obtained in a study of the phase diagram of charged compact boson stars in a theory involving a complex scalar field with a conical potential coupled to a U(1) gauge field and gravity. We obtain new bifurcation points in this model. We present a detailed discussion of the various regions of the phase diagram with respect to the bifurcation points. The theory is seen to contain rich physics in a particular domain of the phase diagram.