Conformal Symmetry Breaking Research Articles

KdV conformal symmetry breaking in nearly AdS2

We study the gauge theory formulation of Jackiw-Teitelboim gravity and propose Korteweg-de Vries asymptotic conditions that generalize the asymptotic dynamics of the theory. They permit to construct an enlarged set of boundary actions formed by higher order generalizations of the Schwarzian derivative that contain the Schwarzian as lower term in a tower of SL(2, ℝ) invariants. They are extracted from the KdV Hamiltonians and can be obtained recursively. As a result, the conformal symmetry breaking observed in nearly AdS2 is characterized by a much larger set of dynamical modes associated to the stationary KdV hierarchy. We study quantum perturbation theory for the generalized Schwarzian action including the symplectic measure and compute the one-loop correction to the partition function. We find that despite the non-linear nature of the higher-Schwarzian contribution, it acquires a manageable expression that renders a curious leading temperature dependence on the entropy S = #Ta for a an odd integer.

Mass ladder operators and quasinormal modes of the static BTZ-like black hole in the Einstein-bumblebee gravity

Abstract We investigate the mass ladder operators for the static BTZ-like black hole in the Einstein-bumblebee gravity, and probe the quasinormal frequencies of the mapped modes by the mass ladder operators for a scalar perturbation under the Dirichlet and Neumann boundary conditions. It is found that the mass ladder operators depend on the Lorentz symmetry breaking parameter, and the imaginary parts of the frequencies shifted by the mass ladder operators increase with the increase of the Lorentz symmetry breaking parameter under two boundary conditions. It should be noted that, under the Neumann boundary condition, the mapped modes caused by the mass ladder operator $D_{0,k_+}$ are unstable. Moreover, the mass ladder operators do not change the Breitenlohner-Freedman bound for the scalar modes, just as in the usual BTZ black hole. These results could help us to further understand the conformal symmetry and Lorentz symmetry breaking in the Einstein-bumblebee gravity.

Moduli spaces in CFT: large charge operators

Using the large-charge expansion, we prove a necessary condition for a CFT to exhibit conformal symmetry breaking, under the assumption that a continuous global symmetry is also broken on the moduli space: there must be a tower of charged local operators whose scaling dimensions are asymptotically linear in the charge. In supersymmetric theories with a continuous R-symmetry and a holomorphic moduli space, the existence of such a tower of operators follows trivially from a BPS condition: their scaling dimensions are then exactly linear in the R-charge. We illustrate the more general statement in several examples of three-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 CFTs, where the leading linear behavior receives nontrivial corrections. By considering a suitable scaling limit, we also relate the spectrum of states with large charge on the cylinder (isomorphic to local operators) to the spectrum of massive particles on the moduli space.

Pion gravitational form factors in the QCD instanton vacuum. I

The pion form factors of the QCD energy-momentum tensor (EMT) are studied in the instanton liquid model (ILM) of the QCD vacuum. In this approach, the breaking of conformal symmetry is encoded in the form of stronger-than-Poisson fluctuations in the number of instantons. For the trace of the EMT, it is shown that the gluonic trace anomaly term contributes half the pion mass, with the other half coming from the quark-mass-dependent σ term. The Q2 dependence of the form factors is governed by glueball and scalar meson exchanges. For the EMT, the spin-0 (trace) and spin-2 (traceless rank-2 tensor) form factors are computed at next-to-leading order in the instanton density using effective quark operators. Relations between the gluon and quark contributions to the EMT form factors are derived. The form factors are also expressed in terms of the pion light-front wave functions in the ILM. The results at the low resolution scale of the inverse instanton size are evolved to higher scales using the renormalization group equation. The ILM results compare well with those of recent lattice QCD calculations. Published by the American Physical Society 2024

Trace anomaly form factors from lattice QCD

The hadron mass can be obtained through the calculation of the trace of the energy-momentum tensor in the hadron which includes the trace anomaly and sigma terms. The anomaly due to conformal symmetry breaking is believed to be an important ingredient for hadron mass generation and confinement. In this work, we will present the calculation of the glue part of the trace anomaly form factors of the pion up to Q2∼4.3 GeV2 and the nucleon up to Q2∼1 GeV2. The calculations are performed on a domain wall fermion ensemble with overlap valence quarks at seven valence pion masses varying from ∼250 to ∼540 MeV, including the unitary point ∼340 MeV. We calculate the radius of the glue trace anomaly for the pion and the nucleon from the z expansion. By performing a two-dimensional Fourier transform on the glue trace anomaly form factors in the infinite momentum frame with no energy transfer, we also obtain their spatial distributions for several valence quark masses. The results are qualitatively extrapolated to the physical valence pion mass with systematic errors from the unphysical sea quark mass, discretization effects in the renormalization sum rule, and finite-volume effects to be addressed in the future. We find the pion’s form factor changes sign, as does its spatial distribution, for light quark masses. This explains how the trace anomaly contribution to the pion mass approaches zero toward the chiral limit. Published by the American Physical Society 2024

A new braneworld with conformal symmetry breaking

We explore the conformal 5D braneworld, where warping emerges through conformal symmetry breaking. Our scenario seamlessly aligns with conventional brane approaches if conformal symmetry remains unbroken. It is shown that a model with a single conformal breaking parameter effectively localizes gravity on the brane, but it falls short in trapping gauge bosons. However, in scenarios with two parameters, gravity is localized, and the model also achieves the localization of zero modes for both gauge and Dirac fields.

Hadrons, superconductor vortices, and cosmological constant

We explore the roles of the trace anomaly in several hadron properties. We derive the scale invariant expression for the pressure from the gravitational form factors (GFF) of QCD which results in consistent results for the mass and rest energy from the GFF and those from the trace and the Hamiltonian of the energy-momentum tensor (EMT) operators. It is shown that the energy-equilibrium correspondence of hadrons infers an equation of state where the trace anomaly matrix element, emerging from the glue condensate in the vacuum, gives a negative constant pressure that leads to confinement, much like the confinement mechanism for the vortices in type II superconductors where the negative constant pressure is due to the cost of depleting the superconducting condensate. We also note that both the trace anomaly in the QCD energy-momentum tensor and the cosmological constant in Einstein's equation are associated with the metric term which contributes to both energy and pressure. Their difference in terms of the role the pressure plays is discussed. Finally, we note that a lattice calculation of the trace anomaly distribution in the pion has addressed a question about the trace anomaly contribution to the pion mass and suggests that there might be a connection between the conformal symmetry breaking and chiral symmetry breaking in this case.

Magnetogenesis in Non-Local Models during Inflation

The generation of magnetic fields during the inflation in an electromagnetic model with a non-local form factor in Maxwell’s action is studied. The equations of motion for the electromagnetic field are derived and solved. It is found that the conformal symmetry breaking due to the non-local form factor does not lead to the generation of magnetic fields during the inflation in the absence of an interaction with the inflaton field. If such a coupling takes place, then the presence of the form factor inhibits the generation of primordial magnetic fields compared to the case where the non-local form factor is absent.

Gravitational reheating at strong coupling

We use the gauge/gravity correspondence to study the gravitational reheating of strongly coupled gauge theories in the rapid exit from the long inflationary (de Sitter) phase. We estimate the maximal reheating temperature of holographic models in d-spatial dimensions, whose scale invariance is explicitly broken by a source term for an operator \U0001d4aa∆ of dimension frac{d}{2}+frac{1}{2}<Delta <d+1 , as well as the top-down \U0001d4a9 = 2* and the cascading gauge theories. The reheating is most efficient when the inflationary phase Hubble constant is much larger than the mass scale of the conformal symmetry breaking of the theory, and for theories with the conformal symmetry breaking operators that are close to marginal.

Relevant dilaton stabilization

We propose a simple modification of the Goldberger-Wise mechanism for stabilizing the scale of spontaneously broken conformal theories. The source of explicit conformal symmetry breaking is a relevant operator with a small coefficient, as opposed to the usual mechanism of an almost marginal operator with an order-one coefficient. In the warped 5D picture this relevant stabilization corresponds to a small tadpole for the bulk scalar on the UV brane, which can be technically natural if it is the only source for the breaking of a symmetry (for example, a discrete Z2). This modification of the stabilization mechanism has significant consequences for the nature of the conformal phase transition, since the radion/dilaton potential is no longer shallow. The bounce action is significantly reduced, leading to a weaker first-order phase transition instead of the supercooled and strongly first-order transition seen in Goldberger-Wise stabilization. This also leads to reduction of gravitational wave signals which, however, may still be observable at future detectors. We present numerical and analytical studies of the phase transition and the resulting gravitational wave signal strength, assuming that the effective dilaton potential provides a good leading approximation. While the dilaton is not expected to be generically light in this setup, in order to keep perturbative control over the effective theory one needs to mildly tune the dilaton quartic to be somewhat small.

Supercool exit: Gravitational waves from QCD-triggered conformal symmetry breaking

Classically conformal Standard Model extensions predict an intriguing thermal history of the early universe. In contrast to the common paradigm, the onset of the electroweak phase transition can be significantly delayed while the universe undergoes a period of thermal inflation. Then, a first-order chiral phase transition could not only trigger electroweak symmetry breaking but also initiate the exit from supercooling. To study the dynamics of this scenario, we focus on low-energy quark-based QCD effective models that exhibit a first-order transition. While a large amount of latent heat is naturally involved if thermal inflation ends, we find that a supercooling period prior to the QCD scale considerably enhances the timescale of the transition. This enhancement implies great observational prospects at future gravitational wave observatories. Our results are readily applicable to a wide class of scale-invariant SM extensions, as well as strongly coupled dark sectors.

Novel features of attractors and transseries in nonconformal Bjorken flows

In this work we investigate the impact of conformal symmetry breaking on hydrodynamization of a far-from-equilibrium fluid. We find a new kind of transseries solutions for the nonconformal hydrodynamic equations of a longitudinal boost invariant expanding plasma. The new transseries solutions unveil a rich physical structure which arises due to the interplay of different physical scales. In the perfect fluid case the nonconformal speed of sound slows down the cooling of the temperature due to the emergence of logarithmic corrections that depends on the mass of the particle. These terms propagate into the perturbative and nonperturbative sectors of the transseries once viscous corrections are included. The logarithmic mass contributions increase the asymptotic value of the Knudsen number while decreasing the damping rate of the transient nonhydrodynamic modes and thus, yielding to an extremely slow hydrodynamization process where flow lines merge to their forward attractor at extremely late times. The early time free streaming expansion is modified and receives logarithmic mass corrections induced by the shear-bulk couplings. The global flow structure and numerical analyses carried out in our work demonstrate the existence of local attraction at early and late times for the shear viscous tensor and bulk viscous pressure.

EFT of conformal symmetry breaking

Conformal symmetry can be spontaneously broken due to the presence of a defect or other background, which gives a symmetry-breaking vacuum expectation value (VEV) to some scalar operators. We study the effective field theory of fluctuations around these backgrounds, showing that it organizes as an expansion in powers of the inverse of the VEV, and computing some of the leading corrections. We focus on the case of spacelike defects in a four-dimensional Lorentzian theory relevant to the pseudoconformal universe scenario, although the conclusions extend to other kinds of defects and to the breaking of conformal symmetry to Poincar\'e symmetry.

Hidden conformal symmetry for dyonic Kerr-Sen black hole and its gauged family

Motivated by advanced progress in the holographic theory between rotating black holes and CFT, we explore the conformal invariance on dyonic Kerr-Sen black hole and its gauged family. We consider a neutral massless scalar probe on the black holes’ background within the low-frequency limit and exhibit that the solution space possesses SL(2,R)times SL(2,R) isometry. The periodic identification of the azimuthal angle corresponds to the spontaneous conformal symmetry breaking by temperatures T_L,T_R. Using the computation of the central charges on Sakti and Burikham (Phys Rev D 106:106006, 2022) that we recalculate by considering the contributions of all associated fields, we successfully derive the Bekenstein-Hawking entropy from Cardy entropy formula. Furthermore, we also calculate the absorption cross-section from gravity side for generic non-extremal dyonic Kerr-Sen black hole and near-extremal gauged dyonic Kerr-Sen black hole. Our calculations show that those quantities are in a perfect match with the calculation from CFT. Therefore, our findings further support the duality between rotating black holes and CFTs.

Residue families, singular Yamabe problems and extrinsic conformal Laplacians

Let (X,g) be a compact Riemannian manifold with boundary Mn and σ a defining function of M. To these data, we associate natural conformally covariant polynomial one-parameter families of differential operators C∞(X)→C∞(M). They arise through a residue construction which generalizes an earlier construction in the framework of Poincaré-Einstein metrics and are referred to as residue families. Residue families may be viewed as curved analogs of conformal symmetry breaking differential operators. The main ingredient of the definition of residue families are eigenfunctions of the Laplacian of the singular metric σ−2g. We prove that if σ is an approximate solution of a singular Yamabe problem, i.e., if σ−2g has constant scalar curvature −n(n+1), up to a sufficiently small remainder, these families can be written as compositions of certain degenerate Laplacians (Laplace-Robin operators). This result implies that the notions of extrinsic conformal Laplacians and extrinsic Q-curvature introduced in recent works by Gover and Waldron can naturally be rephrased in terms of residue families. This spectral theoretical perspective allows easy new proofs of several results of Gover and Waldron. Moreover, it allows us to relate the extrinsic conformal Laplacians and the critical extrinsic Q-curvature to the scattering operator of the asymptotically hyperbolic metric σ−2g, extending the work of Graham and Zworski. The relation to the scattering operators implies that the extrinsic conformal Laplacians are self-adjoint. We describe the asymptotic expansion of the volume of a singular Yamabe metric in terms of Laplace-Robin operators (reproving results of Gover and Waldron). We also derive new local holographic formulas for all extrinsic Q-curvatures (critical and sub-critical ones) in terms of renormalized volume coefficients, the scalar curvature of the background metric, and the asymptotic expansions of eigenfunctions of the Laplacian of the singular metric σ−2g. These results naturally extend earlier results in the Poincaré-Einstein case. Furthermore, we prove a new formula for the singular Yamabe obstruction Bn. The simple structure of these formulas shows the benefit of the systematic use of so-called adapted coordinates. We use the latter formula for Bn to derive explicit expressions for the obstructions in low-order cases (confirming earlier results). Finally, we relate the obstruction Bn to the supercritical Q-curvature Qn+1.

Spin Sum Rule of the Nucleon in the QCD Instanton Vacuum

We briefly review some essential aspects of the QCD instanton vacuum in relation to the quantum breaking of conformal symmetry, the spontaneous breaking of chiral symmetry, and the axial U(1) anomaly. The anomaly causes the intrinsic nucleon spin to transmute to the vacuum topological charge by quantum tunneling. We use Ji′s invariant spin decomposition to discuss the spin budget of the nucleon as a quark–diquark state in the QCD instanton vacuum. A measure of the intrinsic quark spin of the nucleon is a measure of the quenched topological susceptibility of the QCD instanton vacuum.

The Extended Diffusive Sachdev–Ye–Kitaev Model as a Sort of “Strange Metal”

The linear dependence on temperature of the resistivity in metals undergoing high‐temperature superconducting transition is a puzzle, which is generally attributed to non‐Fermi liquid behavior on a large range of temperature values. No particle and momentum conservation characterizes the 0 + 1 − d Sachdev–Ye–Kitaev model as a tractable non‐Fermi liquid starting paradigm, when extended to higher space dimensions. Due to the conformal symmetry breaking in the incoherent phase, Goldstone bosons arise, which provide energy diffusion in the lattice. The linear response to energy diffusion in the model is discussed, which is in principle charge neutral and non‐momentum conserving, and the temperature dependence of the thermal and electric conductivity in the incoherent prechaotic, intermediate temperature phase is obtained.

Aspects of $$(d+D)$$-dimensional anisotropic conformal gravity

We discuss various aspects of anisotropic gravity in \((d+D)\)-dimensional spacetime, where D dimensions are treated as extra dimensions. It is based on the foliation preserving diffeomorphism invariance and anisotropic conformal invariance. The anisotropy is embodied by introducing a factor z which discriminates the scaling degree of the extra D dimensions against the d-dimensional base spacetime and Weyl scalar field which mediates the anisotropic scaling symmetry. There is no intrinsic scale but a physical scale \(M_*\) emerges as a consequence of spontaneous conformal symmetry breaking. We discuss interesting lower dimensional gravity theories obtained from our model. In a \((d,D)=(2,2)\) case, we suggest a UV-complete unitary quantum gravity which might become Einstein gravity in IR. In a certain (2,1) case, we obtain CGHS model. Some vacuum solutions are also obtained and we discuss an issue of ‘size separation’ between the base spacetime and the extra dimensions. The size separation means large distinction between the scales (e.g. cosmological constant) appearing in the base spacetime and the extra dimensions respectively.

Mass sum rule of hadrons in the QCD instanton vacuum

We briefly review the key aspect of the QCD instanton vacuum in relation to the quantum breaking of conformal symmetry and the trace anomaly. We use Ji$^\prime s$ invariant mass decomposition of the energy momentum tensor together with the trace anomaly, to discuss the mass budget of the nucleon and pion in the QCD instanton vacuum. A measure of the gluon condensate in the nucleon, is a measure of the compressibility of the QCD instanton vacuum as a dilute topological liquid.

Curvature perturbations and anomaly explain dark energy

Abstract We investigate the history of dark energy to explain the present magnitude. We assume the dark energy is the residual cosmological constant. The most important channel in the reheating process is gluon pair production by the quantumchromodynamic trace anomaly. We argue that dark energy decays rapidly by gluon pair emissions during the reheating and after the big bang. The reheating temperature is determined by the decay width of dark energy, $\Gamma$, and the Planck mass, $M_\mathrm{p}$, as $\sqrt{M_\mathrm{P}\Gamma} \sim 10^6 \, GeV$. This is a consequence of Friedmann’s equation and the equilibrium condition $\Gamma\sim H$. As the Universe cools below the hadronic scale, the dark energy density is almost frozen. Nevertheless, the dark energy further decreases by emitting two photons. We have estimated the current decay rate of dark energy from the quantum electrodynamic trace anomaly. The consistent solution of the Friedmann equation is in excellent agreement with the observations. The suppression factor of the dark energy scale is the product of the fine structure constant, $\alpha$, and the curvature perturbation, $P$: $10^{-30}=(\alpha^2P/4\pi)^2$. We argue that the conformal symmetry breaking in both ultraviolet and infrared are necessary unless dark energy is subtracted. We also investigate leptogenesis by adding massive right-handed neutrinos: realistic leptogenesis takes place during the reheating process.