Non-conformal Theories Research Articles

Conformal anomalies and renormalized stress tensor correlators for nonconformal theories

We analyze the proposal of defining the Weyl anomaly for classically nonconformal theories as gmn⟨Tmn⟩−⟨gmnTmn⟩, originally put forward by Duff, in the case of a scalar field with quartic self-interaction in 4d. We work in the context of dimensional regularization in curved background to two loops (first order in the coupling). We review the original regularized but not renormalized prescription and its ambiguities; we argue that it cannot be extended to the interacting theory as it fails to provide a finite result. We then propose an alternative prescription via renormalized expectation values. At one loop, our candidate reproduces the local heat kernel result, while its extension to interacting theories contains nonlocal contributions. Published by the American Physical Society 2024

Timelike entanglement entropy and phase transitions in non-conformal theories

We propose a holographic formalism for a timelike entanglement entropy in non-conformal theories. This pseudoentropy is a complex-valued measure of information, which, in holographic non-conformal theories, receives contributions from a set of spacelike surfaces and a finite timelike bulk surface with mirror symmetry. We suggest a method of merging the surfaces so that the boundary length of the subregion is exclusively specified by holography. We show that in confining theories, the surfaces can be merged in the bulk at the infrared tip of the geometry and are homologous to the boundary region. The timelike entanglement entropy receives its imaginary and real contributions from the timelike and the spacelike surfaces, respectively. Additionally, we demonstrate that in confining theories, there exists a critical length within which a connected non-trivial surface can exist, and the imaginary part of the timelike entanglement entropy is non-zero. Therefore, the timelike entanglement entropy exhibits unique behavior in confining theories, making it a probe of confinement and phase transitions. Finally, we discuss the entanglement entropy in Euclidean spacetime in confining theories and the effect of a simple analytical continuation from a spacelike subsystem to a timelike one.

Averaged null energy and the renormalization group

We establish a connection between the averaged null energy condition (ANEC) and the monotonicity of the renormalization group, by studying the light-ray operator ∫ duTuu in quantum field theories that flow between two conformal fixed points. In four dimensions, we derive an exact sum rule relating this operator to the Euler coefficient in the trace anomaly, and show that the ANEC implies the a-theorem. The argument is based on matching anomalies in the stress tensor 3-point function, and relies on special properties of contact terms involving light-ray operators. We also illustrate the sum rule for the example of a free massive scalar field. Averaged null energy appears in a variety of other applications to quantum field theory, including causality constraints, Lorentzian inversion, and quantum information. The quantum information perspective provides a new derivation of the a-theorem from the monotonicity of relative entropy. The equation relating our sum rule to the dilaton scattering amplitude in the forward limit suggests an inversion formula for non-conformal theories.

Holographic RG flows from four-dimensional N = 2 gauged supergravities

In quantum field theories, the examination of particle interactions is facilitated through the application of Feynman diagrams. The computation of loop Feynman diagrams often yields divergent or infinite values. To tackle this issue, a method known as renormalization is employed, which entails the removal of these infinities. The phenomenon of a physical system undergoing changes under different scales gives rise to a concept referred to as the renormalization group. Certain renormalization group trajectories, denoted as RG flows, describe transformations of a conformal field theory (CFT) into other conformal or non-conformal theories. These transformations result in deformations of a UV conformal fixed point into another fixed point or into a non-conformal phase within the infrared (IR) regime. In this study, we investigate holographic RG flows originating from N = 2 gauged supergravity with an SO(2) × SO(6) gauge group. These solutions delineate RG flows from the N = 2 CFT to a three-dimensional non-conformal field theory driven by mass deformations. This transformation is elucidated through the AdS/CFT correspondence, also known as AdS/CFT holography.

Far-from-equilibrium attractors for massive kinetic theory in the relaxation time approximation

We investigate whether early and late time attractors for non-conformal kinetic theories exist by computing the time-evolution of a large set of moments of the one-particle distribution function. For this purpose we make use of a previously obtained exact solution of the 0+1D boost-invariant massive Boltzmann equation in relaxation time approximation. We extend prior attractor studies of non-conformal systems by using a realistic mass- and temperature-dependent relaxation time and explicitly computing the effect of varying both the initial momentum-space anisotropy and initialization time on the time evolution of a large set of integral moments. Our findings are consistent with prior studies, which found that there is an attractor for the scaled longitudinal pressure, but not for the shear and bulk viscous corrections separately. We further present evidence that both late- and early-time attractors exist for all moments of the one-particle distribution function that contain greater than one power of the longitudinal momentum squared.

Master equations for de Sitter DFPs

We develop master equations to study perturbative stability of de Sitter Dynamical Fixed Points (DFPs) of strongly coupled massive quantum field theories in d + 1 space-time dimensions with a holographic dual. The derived spectrum of linearized fluctuations characterizes the late-time dynamics of holographic strongly coupled non-conformal gauge theories in de Sitter background. Numerous checks and examples are presented.

Drag force in the vacuum of confining gauge theories

The complete absence of isolated quarks reaching particle detectors after high energy collisions suggests that some physical mechanism generates resistance to their propagation in the vacuum. In order to reveal such a mechanism, we analyze the fate of an infinitely heavy quark that is initially propagating in the vacuum with inertial motion. The non-perturbative structure of the vacuum is treated here using the gauge/gravity correspondence, the isolated quark on the boundary gauge theory is dual to a trailing string moving in the bulk of a higher dimensional curved space. We find that, for a large class of non-conformal gauge theories with a holographic dual, the geometrical structure of the bulk geometry induces a drag force on the quark that moves in the vacuum. In addition, we show that for these gauge theories there will be the presence of such a drag force due to its vacuum whenever the dual bulk geometry generates a linear potential for a qq¯ pair. The relation of the linear qq¯ potential with the drag force on the isolated quark is a holographic piece of evidence that both phenomena are different manifestations of the confinement of quarks.

Non-Kähler deformed conifold, ultraviolet completion and supersymmetric constraints in the baryonic branch

Gravity duals for a class of UV complete minimally supersymmetric nonconformal gauge theories require deformed conifolds with fluxes. However these manifolds do not allow for the standard Kähler or conformally Kähler metrics on them, instead the metrics are fully non-Kähler. We take a generic such configuration of a non-Kähler deformed conifold with fluxes and ask what constraints do supersymmetry impose in the baryonic branch. We study the supersymmetry conditions and show that for the correct choices of the vielbeins and the complex structure all the equations may be consistently solved. The constraints now lead not only to the known cases in the literature but also to some new backgrounds. We also show how geometric features of these backgrounds, including the overall warp-factor and the resolution parameters, can be seen on the field theory side from perturbative “probe-brane” type calculations by Higgsing the theory and studying one-loop 4-point functions of vector and chiral multiplets. Finally we discuss how UV completions of these gauge theories may be seen from our setup, both from type IIB as well as from the T-dual type IIA brane constructions.

On volume subregion complexity in non-conformal theories

We study the volume prescription of the holographic subregion complexity in a holographic 5-dimensional model consisting of Einstein gravity coupled to a scalar field with a non-trivial potential. The dual 4-dimensional gauge theory is not conformal and exhibits a RG flow between two different fixed points. In both zero and finite temperature we show that the holographic subregion complexity can be used as a measure of non-conformality of the model. This quantity exhibits also a monotonic behaviour in terms of the size of the entangling region, like the behaviour of the entanglement entropy in this setup. There is also a finite jump due to the disentangling transition between connected and disconnected minimal surfaces for holographic renormalized subregion complexity at zero temperature.

QGP universality in a magnetic field?

We use top-down holographic models to study the thermal equation of state of strongly coupled quark-gluon plasma in external magnetic field. We identify different conformal and non-conformal theories within consistent truncations of mathcal{N} = 8 gauged supergravity in five dimensions (including STU models, gauged mathcal{N} = 2* theory) and show that the ratio of the transverse to the longitudinal pressure PT /PL as a function of T / sqrt{B} can be collapsed to a ‘universal’ curve for a wide range of the adjoint hypermultiplet masses m. We stress that this does not imply any hidden universality in magnetoresponse, as other observables do not exhibit any universality. Instead, the observed collapse in PT /PL is simply due to a strong dependence of the equation of state on the (freely adjustable) renormalization scale: in other words, it is simply a fitting artifact. Remarkably, we do uncover a different universality in mathcal{N} = 2* gauge theory in the external magnetic field: we show that magnetized mathcal{N} = 2* plasma has a critical point at Tcrit/ sqrt{B} which value varies by 2% (or less) as m/ sqrt{B} ∈ [0, ∞). At criticality, and for large values of m/ sqrt{B} , the effective central charge of the theory scales as ∝ sqrt{B} /m.

Entanglement entropy of [formula omitted] de Sitter vacuum

de Sitter vacuum of nonconformal gauge theories is non-equilibrium, manifested by a nonvanishing rate of the comoving entropy production at asymptotically late times. This entropy production rate is related to the entanglement entropy of the de Sitter vacuum of the theory. We use holographic correspondence to compute vacuum entanglement entropy density sent of mass deformed N=4 supersymmetric Yang-Mills theory — the N=2⁎ gauge theory — for various values of the masses and the coupling constant to the background space-time curvature. For a particular choice of the curvature coupling, the Euclidean model can be solved exactly using the supersymmetric localization. We show that N=2⁎ de Sitter entanglement entropy is not the thermodynamic entropy of the localization free energy at de Sitter temperature. Neither it is related to the thermal entropy of de Sitter vacuum of pair-produced particles.

Chiral algebras from \u03a9-deformation

In the presence of an Ω-deformation, local operators generate a chiral algebra in the topological-holomorphic twist of a four-dimensional mathcal{N} = 2 supersymmetric field theory. We show that for a unitary mathcal{N} = 2 superconformal field theory, the chiral algebra thus defined is isomorphic to the one introduced by Beem et al. Our definition of the chiral algebra covers nonconformal theories with insertions of suitable surface defects.

Aspects of capacity of entanglement

Many quantum information theoretic quantities are similar to and/or inspired by thermodynamic quantities, with entanglement entropy being a well-known example. In this paper, we study a less well-known example, capacity of entanglement, which is the quantum information theoretic counterpart of heat capacity. It can be defined as the second cumulant of the entanglement spectrum and can be loosely thought of as the variance in the entanglement entropy. We review the definition of capacity of entanglement and its relation to various other quantities such as fidelity susceptibility and Fisher information. We then calculate the capacity of entanglement for various quantum systems, conformal and non-conformal quantum field theories in various dimensions, and examine their holographic gravity duals. Resembling the relation between response coefficients and order parameter fluctuations in Landau-Ginzburg theories, the capacity of entanglement in field theory is related to integrated gravity fluctuations in the bulk. We address the question of measurability, in the context of proposals to measure entanglement and R\'enyi entropies by relating them to $U(1)$ charges fluctuating in and out of a subregion, for systems equivalent to non-interacting fermions. From our analysis, we find universal features in conformal field theories, in particular the area dependence of the capacity of entanglement appears to track that of the entanglement entropy. This relation is seen to be modified under perturbations from conformal invariance. In quenched 1+1 dimensional CFTs, we compute the rate of growth of the capacity of entanglement. The result may be used to refine the interpretation of entanglement spreading being carried by ballistic propagation of entangled quasiparticle pairs created at the quench.

Nonlinearly realized conformal invariance in scale invariant field theories

Implications are explored of promoting non-conformal scale-invariant theories to conformal theories by nonlinearly realizing the missing symmetry. Properties of the associated Nambu-Goldstone mode imply that conformal invariance cannot be spontaneously broken to scale invariance in unitary theories and that, as well known, scale invariant unitary theories in two dimensions are also conformal. The promoted theories have only conformal primaries and no descendants. The (non-)decoupling of the Nambu-Goldstone mode is explicitly shown in examples of scale invariant theories that are actually (not) conformal.

Positive gravitational subsystem energies from CFT cone relative entropies

The positivity of relative entropy for spatial subsystems in a holographic CFT implies the positivity of certain quantities in the dual gravitational theory. In this note, we consider CFT subsystems whose boundaries lie on the lightcone of a point p. We show that the positive gravitational quantity which corresponds to the relative entropy for such a subsystem A is a novel notion of energy associated with a gravitational subsystem bounded by the minimal area extremal surface à associated with A and by the AdS boundary region  corresponding to the part of the lightcone from p bounded by ∂A. This generalizes the results of arXiv:1605.01075 for ball-shaped regions by making use of the recent results in arXiv:1703.10656 for the vacuum modular Hamiltonian of regions bounded on lightcones. As part of our analysis, we give an analytic expression for the extremal surface in pure AdS associated with any such region A. We note that its form immediately implies the Markov property of the CFT vacuum (saturation of strong subadditivity) for regions bounded on the same lightcone. This gives a holographic proof of the result proven for general CFTs in arXiv:1703.10656. A similar holographic proof shows the Markov property for regions bounded on a lightsheet for non-conformal holographic theories defined by relevant perturbations of a CFT.

A rederivation of the conformal anomaly for spin-

We rederive the conformal anomaly for spin- fermions by a genuine Feynman graph calculation, which has not been available so far. Although our calculation merely confirms a result that has been known for a long time, the derivation is new, and thus furnishes a method to investigate more complicated cases (in particular concerning the significance of the quantum trace of the stress tensor in non-conformal theories) where there remain several outstanding and unresolved issues.

Free energy of a heavy quark-antiquark pair in a thermal medium from AdS/CFT

We study the free energy of a heavy quark-antiquark pair in a thermal medium using the AdS/CFT correspondence. We point out that a commonly used prescription for calculating this quantity leads to a temperature dependence in conflict with general properties of the free energy. The problem originates from a particular way of subtracting divergences. We argue that the commonly used prescription gives rise to the binding energy rather than the free energy. We consider a different subtraction procedure and show that the resulting free energy is well-behaved and in qualitative agreement with results from lattice QCD. The free energy and the binding energy of the quark pair are computed for mathcal{N}=4 supersymmetric Yang-Mills theory and several non-conformal theories. We also calculate the entropy and the internal energy of the pair in these theories. Using the consistent subtraction, we further study the free energy, entropy, and internal energy of a single heavy quark in the thermal medium for various theories. Also here the results are found to be in qualitative agreement with lattice QCD results.

A one-dimensional theory for Higgs branch operators

We use supersymmetric localization to calculate correlation functions of half-BPS local operators in 3d mathcal{N}=4 superconformal field theories whose Lagrangian descriptions consist of vectormultiplets coupled to hypermultiplets. The operators we primarily study are certain twisted linear combinations of Higgs branch operators that can be inserted anywhere along a given line. These operators are constructed from the hypermultiplet scalars. They form a one-dimensional non-commutative operator algebra with topological correlation functions. The 2- and 3-point functions of Higgs branch operators in the full 3d mathcal{N}=4 theory can be simply inferred from the 1d topological algebra. After conformally mapping the 3d superconformal field theory from flat space to a round three-sphere, we preform supersymmetric localization using a supercharge that does not belong to any 3d mathcal{N}=2 subalgebra of the mathcal{N}=4 algebra. The result is a simple model that can be used to calculate correlation functions in the 1d topological algebra mentioned above. This model is a 1d Gaussian theory coupled to a matrix model, and it can be viewed as a gauge-fixed version of a topological gauged quantum mechanics. Our results generalize to non-conformal theories on S3 that contain real mass and Fayet-Iliopolous parameters. We also provide partial results in the 1d topological algebra associated with the Coulomb branch, where we calculate correlation functions of local operators built from the vectormultiplet scalars.

Conformal anomalies and gravitational waves

We argue that the presence of conformal anomalies in gravitational theories can lead to observable modifications to Einstein's equations via the induced anomalous effective actions, whose non-localities can overwhelm the smallness of the Planck scale. The fact that no such effects have been seen in recent cosmological or gravitational wave observations therefore imposes strong restrictions on the field content of possible extensions of Einstein's theory: all viable theories should have vanishing conformal anomalies. We then show that a complete cancellation of conformal anomalies in D=4 for both the C2 invariant and the Euler (Gauss–Bonnet) invariant E4 can only be achieved for N-extended supergravity multiplets with N⩾5, as well as for M theory compactified to four dimensions. Although there remain open questions, in particular concerning the true significance of conformal anomalies in non-conformal theories, as well as their possible gauge dependence for spin s⩾32, these cancellations suggest a hidden conformal structure of unknown type in these theories.

Paths to equilibrium in non-conformal collisions

We extend our previous analysis of holographic heavy ion collisions in non-conformal theories. We provide a detailed description of our numerical code. We study collisions at different energies in gauge theories with different degrees of non-conformality. We compare four relaxation times: the hydrodynamization time (when hydrodynamics becomes applicable), the EoSization time (when the average pressure approaches its equilibrium value), the isotropization time (when the longitudinal and transverse pressures approach each other) and the condensate relaxation time (when the expectation value of a scalar operator approaches its equilibrium value). We find that these processes can occur in several different orderings. In particular, the condensate can remain far from equilibrium even long after the plasma has hydrodynamized and EoSized. We also explore the rapidity distribution of the energy density at hydrodynamization. This is far from boost-invariant and its width decreases as the non-conformality increases. Nevertheless, the velocity field at hydrodynamization is almost exactly boost-invariant regardless of the non-conformality. This result may be used to constrain the initialization of hydrodynamic fields in heavy ion collisions.