Higher Derivative Terms Research Articles

The zeroth law of black hole thermodynamics in arbitrary higher derivative theories of gravity

We consider diffeomorphism invariant theories of gravity with arbitrary higher derivative terms in the Lagrangian as corrections to the leading two derivative theory of Einstein’s general relativity. We construct a proof of the zeroth law of black hole thermo-dynamics in such theories. We assume that a stationary black hole solution in an arbitrary higher derivative theory can be obtained by starting with the corresponding stationary solution in general relativity and correcting it order by order in a perturbative expansion in the coupling constants of the higher derivative Lagrangian. We prove that surface gravity remains constant on its horizon when computed for such stationary black holes, which is the zeroth law. We argue that the constancy of surface gravity on the horizon is related to specific components of the equations of motion in such theories. We further use a specific boost symmetry of the near horizon space-time of the stationary black hole to constrain the off-shell structure of the equations of motion. Our proof for the zeroth law is valid up to arbitrary order in the expansion in the higher derivative couplings.

Spacetime instability and quantum gravity as low energy effective field theory

We discuss spacetime instability for effective field theories of quantum gravity. The effective action of gravity introduces infinite higher derivative curvature terms $R^{2}, { R }_{ \mu \nu }{ R }^{ \mu \nu }, R_{\mu\nu\kappa\lambda} R^{\mu\nu\kappa\lambda}\dots$. Although these higher derivative curvature terms are indispensable to construct the self-consistent renormalizable theory of quantum gravity, they lead to several pathologies. We clearly show that even if they are written as the Planck-suppressed operators they lead to serious consequences and and de Sitter or radiation-dominated Universe is highly unstable. We show that the couplings of these higher derivative curvatures must satisfy $\left|a_{1,2,3}\right|\gtrsim 10^{118}$ to be consistent with the cosmological observations. Thus, the standard effective field theories of quantum gravity fail to describe the observed Universe unless introducing a specific technique dealing with the higher derivative curvature terms.

Gravitational memory effects and higher derivative actions

We show that charges associated with the internal Lorentz symmetries of general relativity, with higher derivative boundary terms included in the action, capture observable gravitational wave effects. In particular, the Gauss-Bonnet charge measures the precession rate of a freely-falling gyroscope, while the Pontryagin charge encodes the relative radial acceleration of freely-falling test masses. This relation highlights the importance of the tetrad formalism and the physical significance of asymptotic internal Lorentz symmetries.

A -anomalous interactions of the holographic dilaton

We explore higher-derivative terms in the low-energy effective action for the dilaton, the Goldstone boson of spontaneously broken scale invariance. Focusing on the simplest holographic realization of spontaneously broken scale invariance, the Randall-Sundrum (RS) scenario, we identify the nonlinear action for the RS dilaton by integrating out Kaluza-Klein graviton modes. The coefficient of a particular fourth-derivative dilaton self-interaction can be identified with the Weyl $a$-anomaly of the dual conformal field theory, which we use to verify anomaly matching arguments. We also find novel, $a$-dependent couplings of the dilaton to light matter fields. These anomalous interactions can have a significant effect on the collider phenomenology and the cosmology, potentially allowing us to probe the structure of the underlying conformal sector via low-energy physics. The dilaton effective theory also serves as an interesting scalar analog of gravity, and we study solutions to the equation of motion that parallel black holes and cosmologies.

The second law of black hole mechanics in effective field theory

We investigate the second law of black hole mechanics in gravitational theories with higher derivative terms in the action. Wall has described a method for defining an entropy that satisfies the second law to linear order in perturbations around a stationary black hole. We show that this can be extended to define an entropy that satisfies the second law to quadratic order in perturbations, provided that one treats the higher derivative terms in the sense of effective field theory. We also address some outstanding issues with Wall’s method, in particular its gauge invariance and its relation to the Iyer-Wald entropy.

Semi-Lagrangian forward methods for some time-dependent nonlinear partial differential equations

In this article, we study one-step Semi-Lagrangian forward method for computing the numerical solutions of time-dependent nonlinear partial differential equations with initial and boundary conditions in one space dimension. Comparing with classic Semi-Lagrangian method, this method is more straight forward to analyze and implement. This method is based on Lagrangian trajectory from the departure points (regular nodes) to the arrival points by Runge-Kutta methods. The arrival points are traced forward from the departure points along the trajectory of the path. Most likely the arrival points are not on the regular grid nodes. However, it is convenient to approximate the high order derivative terms in spatial dimension on regular nodes. The convergence and stability are studied for the explicit methods. The numerical examples show that those methods work very efficient for the time-dependent nonlinear partial differential equations. For more information see https://ejde.math.txstate.edu/conf-proc/26/g1/abstr.html

String theory corrections to holographic black hole chemistry

The connection between the bulk and the boundary first law of thermodynamics in anti--de Sitter space has been discussed in generic higher derivative gravity. String theory corrections to supergravity render higher derivative terms in the bulk action, proportional to different powers of string theory parameter ${\ensuremath{\alpha}}^{\ensuremath{'}}$. A variation in the cosmological constant induces a variation in the 't Hooft coupling in the boundary theory. We show that in order to match the bulk first law and Smarr relation with the boundary side we need to include the variation of ${\ensuremath{\alpha}}^{\ensuremath{'}}$ in the bulk thermodynamics as a bookkeeping device. Accordingly, the boundary first law and Euler relation are modified with the inclusion of two central charges ($a$, $c$) and/or other chemical potentials as thermodynamic variables. We consider four- and six-derivative terms as well as the ${\text{Weyl}}^{4}$ terms (in type IIB) in bulk in support of our generic result.

Avoidance of Strong Coupling in General Relativity Solutions with a Timelike Scalar Profile in a Class of Ghost-Free Scalar-Tensor Theories

Modified gravity theories can accommodate exact solutions, for which the metric has the same form as the one in general relativity, i.e., stealth solutions. One problem with these stealth solutions is that perturbations around them exhibit strong coupling when the solutions are realized in degenerate higher-order scalar-tensor theories. We show that the strong coupling problem can be circumvented in the framework of the so-called U-DHOST theories, in which the degeneracy is partially broken in such a way that higher-derivative terms are degenerate only in the unitary gauge. In this sense, the scordatura effect is built-in in U-DHOST theories in general. There is an apparent Ostrogradsky mode in U-DHOST theories, but it does not propagate as it satisfies a three-dimensional elliptic differential equation on a spacelike hypersurface. We also clarify how this nonpropagating mode, i.e., the "shadowy" mode shows up at the nonlinear level.

Dual description of gauge theories from an iterative Noether approach

An iterative Noether scheme, advocated by Deser, is used to introduce gauge invariant couplings to nonrelativistic matter with global symmetries related to usual charge conservation and dipole conservation recently discussed in fractonic theories. No reference to any gauge principle, fractonic or otherwise, is required. A dual description is found where the theory is defined either in terms of the usual vector gauge field or, alternatively, in terms of higher derivatives of a symmetric tensor field given in fractonic theories. A connection between these two descriptions is obtained by providing an explicit map between the vector and tensor fields. This method yields a novel ‘minimal’ prescription involving tensor fields which is identified with the usual minimal prescription that involves vector fields, by using the mapping. It also spells out the structure of the pure gauge field action in both formulations. Extension of the abelian U(1) invariance to the nonabelian SU(N) invariance is done for the standard formulation.

Perturbative higher-derivative terms in mathcal{N} = 6 asymmetric orbifolds

We analyse the perturbative four-point amplitudes in the simplest string theory examples of T-fold backgrounds, which enjoy mathcal{N} = 6 supersymmetries in four dimensions. There are two theories defined as asymmetric orbifolds of order 2 and 3, respectively. The perturbative spectrum and the one-loop four-point amplitudes are shown to be invariant under an arithmetic symplectic group defined over the Hurwitz (respectively Eisenstein) quaternions. The supersymmetry constraints on the low energy effective couplings are determined and we conjecture the U-duality group of the theory and the non-perturbative completion of the six-derivative coupling as a rank one theta series. We discuss the non-perturbative spectrum of BPS states in the light of our analysis.

Stability of high order finite difference and local discontinuous Galerkin schemes with explicit-implicit-null time-marching for high order dissipative and dispersive equations

Time discretization is an important issue for time-dependent partial differential equations (PDEs). For the k-th (k≥2) order PDEs, the explicit time-marching method may suffer from a severe time step restriction τ=O(hk) for stability. The implicit and implicit-explicit (IMEX) time-marching methods can overcome this constraint. However, for the equations with nonlinear high derivative terms, the IMEX methods are not good choices either, since a nonlinear algebraic system must be solved (e.g. by Newton iteration) at each time step. The explicit-implicit-null (EIN) time-marching method is designed to cope with the above mentioned shortcomings. The basic idea of the EIN method discussed in this paper is to add and subtract a sufficiently large linear highest derivative term on one side of the considered equation, and then apply the IMEX time-marching method to the equivalent equation. The EIN method so designed does not need any nonlinear iterative solver, and the severe time step restriction for explicit methods can be removed. Coupled with the EIN time-marching method, we will discuss the high order finite difference and local discontinuous Galerkin schemes for solving high order dissipative and dispersive equations, respectively. By the aid of the Fourier method, we perform stability analysis for the schemes on the simplified equations with periodic boundary conditions, which demonstrates the stability criteria for the resulting schemes. Even though the analysis is only performed on the simplified equations, numerical experiments show that the proposed schemes are stable and can achieve optimal orders of accuracy for both one-dimensional and two-dimensional linear and nonlinear equations.

Non-trivial time crystal-like ground state for gravitational perturbation in quadratic gravity

We consider gravitational perturbation around maximally symmetric background of a theory of gravity involving quadratic curvature correction. This leads to a decoupled model of the standard transverse, traceless graviton mode and an additional scalar degree of freedom. Evolution of the latter is governed by a Lagrangian, which depends on higher derivative terms, inherited from the quadratic curvature correction in the action. It turns out that stability of the gravitational theory allows the scalar sector to choose between possible lowest energy states (or, ground states), that can sustain periodic form of inhomogeneity, either in space or in time. The latter, with a restricted form of invariance in time translation, is possibly a variant form of the time crystal. Existence of this novel ground state, depicting a time crystal, is solely related to the presence of higher curvature terms in the action. In particular, this non-trivial condensate-like ground state describe the spacetime fabric itself by introducing fundamental scales in the theory. Possible implications are also discussed.

Remarks on the Effects of Quantum Corrections on Moduli Stabilization and de Sitter Vacua in Type IIB String Theory

AbstractThe role of string loop corrections on the existence of de Sitter vacua and the moduli stabilization problem is examined in type IIB effective theories. The fundamental building blocks are a minimum of three intersecting D7 brane stacks, three Kähler moduli, and a novel Einstein‐Hilbert term associated with higher derivative terms of the 10‐dimensional effective action. It was shown in previous works that loop corrections appear which induce novel logarithmic volume‐dependent terms in the effective potential. When D‐term contributions are considered, all Kähler moduli are stabilized and de Sitter vacua are achieved. In the present work, a comprehensive study of multiple non‐perturbative terms in the superpotential is undertaken. The combined effects of the logarithmic loop corrections and two non‐perturbative superpotential Kaehler moduli dependent terms have been investigated. It is shown that a variety of fluxes exist for large as well as moderate volume compactifications which define a de Sitter space and stabilize the moduli fields. For large volumes, a generic simple form of the potential is achieved. The so obtained effective potential appears to be promising for cosmological applications.

Asymptotic nonlocality in gauge theories

Asymptotically nonlocal field theories represent a sequence of higher-derivative theories whose limit point is a ghost-free, infinite-derivative theory. Here we extend this framework, developed previously in a theory of real scalar fields, to gauge theories. We focus primarily on asymptotically nonlocal scalar electrodynamics, first identifying equivalent gauge-invariant formulations of the Lagrangian, one with higher-derivative terms and the other with auxiliary fields instead. We then study mass renormalization of the complex scalar field in each formulation, showing that an emergent nonlocal scale (i.e., one that does not appear as a fundamental parameter in the Lagrangian of the finite-derivative theories) regulates loop integrals as the limiting theory is approached, so that quadratic divergences can be hierarchically smaller than the lightest Lee-Wick partner. We conclude by making preliminary remarks on the generalization of our approach to non-Abelian theories, including an asymptotically nonlocal standard model.

Exact electromagnetic duality with nonminimal couplings

We study nonminimal extensions of Einstein-Maxwell theory with exact electromagnetic duality invariance. Any such theory involves an infinite tower of higher-derivative terms whose computation and summation usually represents a challenging problem. Despite that, we manage to obtain a closed form of the action for all the theories with a quadratic dependence on the vector field strength. In these theories we find that the Maxwell field couples to gravity through a curvature-dependent susceptibility tensor that takes a peculiar form, reminiscent of that of Born-Infeld Lagrangians. We study the static and spherically symmetric black hole solutions of the simplest of these models, showing that the corresponding equations of motion are invariant under rotations of the electric and magnetic charges. We compute the perturbative corrections to the Reissner-Nordstr\"om solution in this theory, and in the case of extremal black holes we determine exactly the near-horizon geometry as well as the entropy. Remarkably, the entropy only possesses a constant correction despite the action containing an infinite number of terms. In addition, we find there is a lower bound for the charge and the mass of extremal black holes. When the sign of the coupling is such that the weak gravity conjecture is satisfied, the area and the entropy of extremal black holes vanish at the minimal charge.

Fractons in effective field theories for spontaneously broken translations

We study the concomitant breaking of spatial translations and dilatations in Ginzburg-Landau-like models, where the dynamics responsible for the symmetry breaking is described by an effective Mexican hat potential for spatial gradients. We show that there are fractonic modes with either subdimensional propagation or no propagation altogether, namely, immobility. Such a class of effective field theories encompasses instances of helical superfluids and metafluids, where fractons can be connected to an emergent symmetry under higher-moment charges, leading in turns to the trivialization of some elastic coefficients. The introduction of a finite-charge density alters the mobility properties of fractons and leads to a competition between the chemical potential and the superfluid velocity in determining the gap of the dilaton. The mobility of fractons can also be altered at zero density upon considering additional higher-derivative terms.

The metric nature of matter

We construct a metric structure on a configuration space of gauge connections and show that it naturally produces a candidate for a non-perturbative, 3+1 dimensional Yang-Mills-Dirac quantum field theory on a curved background. The metric structure is an infinite-dimensional Bott-Dirac operator and the fermionic sector of the emerging quantum field theory is generated by the infinite-dimensional Clifford algebra required to construct this operator. The Bott-Dirac operator interacts with the HD(M) algebra, which is a non-commutative algebra generated by holonomy-diffeomorphisms on the underlying manifold, i.e. parallel-transforms along flows of vector fields. This algebra combined with the Bott-Dirac operator encode the canonical commutation and anti-commutation relations of the quantised bosonic and fermionic fields. The square of the Bott-Dirac operator produces both the Yang-Mills Hamilton operator and the Dirac Hamilton operator as well as a topological Yang-Mills term alongside higher-derivative terms and a metric invariant.

Natural higher-derivatives generalization for the Klein–Gordon equation

We propose a natural family of higher-order partial differential equations generalizing the second-order Klein–Gordon equation. We characterize the associated model by means of a generalized action for a scalar field, containing higher-derivative terms. The limit obtained by considering arbitrarily higher-order powers of the d’Alembertian operator leading to a formal infinite-order partial differential equation is discussed. The general model is constructed using the exponential of the d’Alembertian differential operator. The canonical energy–momentum tensor densities and field propagators are explicitly computed. We consider both homogeneous and non-homogeneous situations. The classical solutions are obtained for all cases.

Causality constraints in Quadratic Gravity

Classifying consistent effective field theories for the gravitational interaction has recently been the subject of intense research. Demanding the absence of causality violation in high energy graviton scattering processes has led to a hierarchy of constraints on higher derivative terms in the Lagrangian. Most of these constraints have relied on analysis that is performed in general relativistic backgrounds, as opposed to a generic solution to the equations of motion which are perturbed by higher curvature operators. Hence, these constraints are necessary but may not be sufficient to ensure that the theory is consistent. In this context, we explore the so-called CEMZ causality constraints on Quadratic Gravity in a space of shock wave solutions beyond GR. We show that the Shapiro time delay experienced by a graviton is polarization-independent and positive, regardless of the strength of the gravitational couplings. Our analysis shows that as far as the causality constraints are concerned, albeit inequivalent to General Relativity due to additional propagating modes, Quadratic Gravity is causal as per as the diagnostic proposed by CEMZ.

Holographic p-wave superconductivity from higher derivative theory

We construct a holographic SU(2) p-wave superconductor model with Weyl corrections. The high derivative (HD) terms do not seem to spoil the generation of the p-wave superconducting phase. We mainly study the properties of AC conductivity, which is absent in holographic SU(2) p-wave superconductor with Weyl corrections. The conductivities in superconducting phase exhibit obvious anisotropic behaviors. Along y direction, the conductivity sigma _{yy} is similar to that of holographic s-wave superconductor. The superconducting energy gap exhibits a wide extension. For the conductivity sigma _{xx} along x direction, the behaviors of the real part in the normal state are closely similar to that of sigma _{yy}. However, the anisotropy of the conductivity obviously shows up in the superconducting phase. A Drude-like peak at low frequency emerges in Resigma _{xx} once the system enters into the superconducting phase, regardless of the behaviors in normal state.