Conformal Anomaly Research Articles

Pair creation in electric fields, anomalies, and renormalization of the electric current

We investigate the Schwinger pair production phenomena in spatially homogeneous strong electric fields. We first consider scalar QED in four-dimensions and discuss the potential ambiguity in the adiabatic order assignment for the electromagnetic potential required to fix the renormalization subtractions. We argue that this ambiguity can be solved by invoking the conformal anomaly when both electric and gravitational backgrounds are present. We also extend the adiabatic regularization method for spinor QED in two-dimensions and find consistency with the chiral anomaly. We focus on the issue of the renormalization of the electric current $\langle j^\mu \rangle$ generated by the created pairs. We illustrate how to implement the renormalization of the electric current for the Sauter pulse.

A -theorem for Horndeski gravity at the critical point

We study holographic conformal anomalies and the corresponding $a$-theorem for Einstein gravity extended with Horndeski terms that involve up to and including linear curvature tensors. We focus on our discussion in $D=5$ bulk dimensions. For the generic Horndeski coupling, the $a$-charge is the same as that in Einstein gravity, but the inclusion of the Horndeski term violates the $a$-theorem. However, there exists a critical point of the Horndeski coupling, for which the theory admits nearly AdS spacetimes with non-vanishing Horndeski scalar. The full AdS isometry is broken down by the logarithmic scalar hair to the Poincar\'e group plus the scale invariance. We find that in this case the $a$-charge depends on the AdS radius $\ell$ and the integration constant $\chi_s$ of the Horndeski scalar. In addition, we find that two new central charges emerge, that are absent in gravities with minimally-coupled matter. We call them $b$-charges. These $b$-charges also depend on $\ell$ and $\chi_s$. We construct an $a$-function for fixed $\ell$ but with the running Horndeski scalar $\chi$ replacing the constant $\chi_s$, and establish the holographic $a$-theorem using the null energy condition in the bulk. Furthermore, we find that there exist analogous monotonous $b$-functions as well. We also obtain the $a$-charge and the $a$-theorem in general odd bulk dimensions.

Liouville conformal field theories in higher dimensions

We consider a generalization of the two-dimensional Liouville conformal field theory to any number of even dimensions. The theories consist of a log-correlated scalar field with a background mathcal{Q} -curvature charge and an exponential Liouville-type potential. The theories are non-unitary and conformally invariant. They localize semiclassically on solutions that describe manifolds with a constant negative mathcal{Q} -curvature. We show that CT is independent of the mathcal{Q} -curvature charge and is the same as that of a higher derivative scalar theory. We calculate the A-type Euler conformal anomaly of these theories. We study the correlation functions, derive an integral expression for them and calculate the three-point functions of light primary operators. The result is a higher-dimensional generalization of the two-dimensional DOZZ formula for the three-point function of such operators.

Generation of a Nernst Current from the Conformal Anomaly in Dirac and Weyl Semimetals.

We show that a conformal anomaly in Weyl and Dirac semimetals generates a bulk electric current perpendicular to a temperature gradient and the direction of a background magnetic field. The associated conductivity of this novel contribution to the Nernst effect is fixed by a beta function associated with the electric charge renormalization in the material. We discuss the experimental feasibility of the proposed phenomenon.

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.

Renormalization, conformal ward identities and the origin of a conformal anomaly pole

We investigate the emergence of a conformal anomaly pole in conformal field theories in the case of the TJJ correlator. We show how it comes to be generated in dimensional renormalization, using a basis of 13 form factors (the F-basis), where only one of them requires renormalization (F13), extending previous studies. We then combine recent results on the structure of the non-perturbative solutions of the conformal Ward identities (CWI's) for the TJJ in momentum space, expressed in terms of a minimal set of 4 form factors (A-basis), with the properties of the F-basis, and show how the singular behaviour of the corresponding form factors in both basis can be related. The result proves the centrality of such massless effective interactions induced by the anomaly, which have recently found realization in solid state, in the theory of topological insulators and of Weyl semimetals. This pattern is confirmed in massless abelian and nonabelian theories (QED and QCD) investigated at one-loop.

Conformal anomaly of generalized form factors and finite loop integrals

We reveal a new mechanism of conformal symmetry breaking at Born level. It occurs in generalized form factors with several local operators and an on-shell state of massless particles. The effect is due to hidden singularities on collinear configurations of the momenta. This conformal anomaly is different from the holomorphic anomaly of amplitudes. We present a number of examples in four and six dimensions. We find an application of the new conformal anomaly to finite loop momentum integrals with one or more massless legs. The collinear region around a massless leg creates a contact anomaly, made visible by the loop integration. The anomalous conformal Ward identity for an ℓ−loop integral is a 2nd-order differential equation whose right-hand side is an (ℓ − 1)−loop integral. It could serve as a new useful tool to find/test analytic expressions for conformal integrals. We illustrate this point with several examples of known integrals. We propose a new differential equation for the four-dimensional scalar double box.

2D BPS rings from sphere partition functions

We consider extremal correlation functions, involving arbitrary number of BPS (chiral or twisted chiral) operators and exactly one anti-BPS operator in 2D mathcal{N} = (2, 2) theories. These correlators define the structure constants in the rings generated by the BPS operators with their operator product expansions. We present a way of computing these correlators from the sphere partition function of a deformed theory using localization. Relating flat space and sphere correlators is nontrivial due to operator mixing on the sphere induced by conformal anomaly. We discuss the supergravitational source of this complication and a resolution thereof. Finally, we demonstrate the process for the Quintic GLSM and the Landau-Ginzburg minimal models.

On actions for (entangling) surfaces and DCFTs

The dynamics of surfaces and interfaces describe many physical systems, including fluid membranes, entanglement entropy and the coupling of defects to quantum field theories. Based on the formulation of submanifold calculus developed by Carter, we introduce a new variational principle for (entangling) surfaces. This principle captures all diffeomorphism constraints on surface/interface actions and their associated spacetime stress tensor. The different couplings to the geometric tensors appearing in the surface action are interpreted in terms of response coefficients within elasticity theory. An example of a surface action with edges at the two-derivative level is studied, including both the parity-even and parity-odd sectors. Its conformally invariant counterpart restricts the type of conformal anomalies that can appear in two-dimensional submanifolds with boundaries. Analogously to hydrodynamics, it is shown that classification methods can be used to constrain the stress tensor of (entangling) surfaces at a given order in derivatives. This analysis reveals a purely geometric parity-odd contribution to the Young modulus of a thin elastic membrane. Extending this novel variational principle to BCFTs and DCFTs in curved spacetimes allows to obtain the Ward identities for diffeomorphism and Weyl transformations. In this context, we provide a formal derivation of the contact terms in the stress tensor and of the displacement operator for a broad class of actions.

Virial expansion for the Tan contact and Beth-Uhlenbeck formula from two-dimensional SO(2,1) anomalies

The relationship between 2D $SO(2,1)$ conformal anomalies in nonrelativistic systems and the virial expansion is explored using recently developed path-integral methods. In the process, the Beth-Uhlenbeck formula for the shift of the second virial coefficient $\delta b_2$ is obtained, as well as a virial expansion for the Tan contact. A possible extension of these techniques for higher orders in the virial expansion is discussed.

One-loop renormalization of Lorentz and CPT -violating scalar field theory in curved spacetime

The one-loop divergences for the scalar field theory with Lorentz and/or CPT breaking terms are obtained in curved space-time. We analyze two separate cases: minimal coupled scalar field with gravity and nonminimal one. For the minimal case with real scalar field, the counterterms are evaluated in a nonperturbative form in the CPT-even parameter through a redefinition of space-time metric. In the most complicated case of complex scalar field nonminimally interacting with gravity, the solution for the divergences is obtained in the first order in the weak Lorentz violating parameter. The necessary form of the vacuum counterterms indicate the most important structures of Lorentz/CPT violations in the pure gravitational sector of theory. The conformal theory limit is also analyzed. It turns out that if we allow the violating fields to transform, the classical conformal invariance of massless scalar fields can be maintained in the $\,\xi = 1/6\,$ case. At quantum level the conformal symmetry is violated by trace anomaly. As a result, conformal anomaly and the anomaly-induced effective action are evaluated in the presence of extra Lorentz- and/or CPT-violating parameters. Such gravitational effective action is important for cosmological applications and can be used for searching of Lorentz violation in primordial universe in the cosmological perturbations, especially gravitational waves.

Non-local Effects of Conformal Anomaly

It is shown that the nonlocal anomalous effective actions corresponding to the quantum breaking of the conformal symmetry can lead to observable modifications of Einstein’s equations. The fact that Einstein’s general relativity is in perfect agreement with all observations including cosmological or recently observed gravitational waves imposes strong restrictions on the field content of possible extensions of Einstein’s theory: all viable theories should have vanishing conformal anomalies. It is shown that a complete cancellation of conformal anomalies in $$D=4$$ for both the $$C^2$$ invariant and the Euler (Gauss–Bonnet) invariant can only be achieved for N-extended supergravity multiplets with $$N \ge 5$$ .

Universality for shape dependence of Casimir effects from Weyl anomaly

We reveal elegant relations between the shape dependence of the Casimir effects and Weyl anomaly in boundary conformal field theories (BCFT). We show that for any BCFT which has a description in terms of an effective action, the near boundary divergent behavior of the renormalized stress tensor is completely determined by the central charges of the theory. These relations are verified by free BCFTs. We also test them with holographic models of BCFT and find exact agreement. We propose that these relations between Casimir coefficients and central charges hold for any BCFT. With the holographic models, we reproduce not only the precise form of the near boundary divergent behavior of the stress tensor, but also the surface counter term that is needed to make the total energy finite. As they are proportional to the central charges, the near boundary divergence of the stress tensor must be physical and cannot be dropped by further artificial renormalization. Our results thus provide affirmative support on the physical nature of the divergent energy density near the boundary, whose reality has been a long-standing controversy in the literature.

What can decadal variability tell us about climate feedbacks and sensitivity?

Radiative feedbacks are known to determine climate sensitivity. Global top-of-atmosphere radiation correlations with surface temperature performed here show that decadal variability in surface temperature is also reinforced by strong positive feedbacks in models, both in the long wave (LW) and short wave (SW), offsetting much of the Planck radiative damping. Net top-of-atmosphere feedback is correlated with the magnitude of decadal temperature variability, particularly in the tropics. This indicates decadal-timescale radiative reinforcement of surface temperature variability. Assuming a simple global ocean mixed layer response, the reinforcement is found to be of a magnitude comparable to that required for typical decadal global scale anomalies. The magnitude of decadal variability in the tropics is uncorrelated with LW feedbacks, but it is correlated with total SW feedbacks, which are, in turn, correlated with tropical SW cloud feedback. Globally, water vapour/lapse rate, surface albedo and cloud feedbacks on decadal timescales are, on average, as strong as those operating under climate change. Together these results suggest that some of the physical processes responsible for setting the magnitude of global temperature change in the twenty-first century and climate sensitivity also help set the magnitude of the natural decadal variability. Furthermore, a statistically significant correlation exists between climate sensitivity and decadal variability in the tropics across CMIP5 models, although this is not apparent in the earlier generation of CMIP3 models. Thus although the link to sensitivity is not conclusive, this opens up potential paths to improve our understanding of climate feedbacks, climate sensitivity and decadal climate variability, and has the potential to reduce the associated uncertainty.

Computational domain length and Reynolds number effects on large-scale coherent motions in turbulent pipe flow

ABSTRACTWe present results from direct numerical simulations (DNS) of turbulent pipe flow at shear Reynolds numbers up to Reτ = 1500 using different computational domains with lengths up to . The objectives are to analyse the effect of the finite size of the periodic pipe domain on large flow structures in dependency of Reτ and to assess a minimum required for relevant turbulent scales to be captured and a minimum Reτ for very large-scale motions (VLSM) to be analysed. Analysing one-point statistics revealed that the mean velocity profile is invariant for . The wall-normal location at which deviations occur in shorter domains changes strongly with increasing Reτ from the near-wall region to the outer layer, where VLSM are believed to live. The root mean square velocity profiles exhibit domain length dependencies for pipes shorter than 14R and 7R depending on Reτ. For all Reτ, the higher-order statistical moments show only weak dependencies and only for the shortest domain considered here. However, the analysis of one- and two-dimensional pre-multiplied energy spectra revealed that even for larger , not all physically relevant scales are fully captured, even though the aforementioned statistics are in good agreement with the literature. We found to be sufficiently large to capture VLSM-relevant turbulent scales in the considered range of Reτ based on our definition of an integral energy threshold of 10%. The requirement to capture at least 1/10 of the global maximum energy level is justified by a 14% increase of the streamwise turbulence intensity in the outer region between Reτ = 720 and 1500, which can be related to VLSM-relevant length scales. Based on this scaling anomaly, we found Reτ⪆1500 to be a necessary minimum requirement to investigate VLSM-related effects in pipe flow, even though the streamwise energy spectra does not yet indicate sufficient scale separation between the most energetic and the very long motions.

AdS7/CFT6 with orientifolds

AdS7 solutions of massive type IIA have been classified, and are dual to a large class of six-dimensional (1, 0) SCFT’s whose tensor branch deformations are described by linear quivers of SU groups. Quivers and AdS vacua depend solely on the group theory data of the NS5-D6-D8 brane configurations engineering the field theories. This has allowed for a direct holographic match of their a conformal anomaly. In this paper we extend the match to cases where O6 and O8-planes are present, thereby introducing SO and USp groups in the quivers. In all of them we show that the a anomaly computed in supergravity agrees with the holographic limit of the exact field theory result, which we extract from the anomaly polynomial. As a byproduct we construct special AdS7 vacua dual to nonperturbative F-theory configurations. Finally, we propose a holographic a-theorem for six-dimensional Higgs branch RG flows.

Conformal Ward Identities and the Coupling of QED and QCD to Gravity

We present a general study of 3-point functions of conformal field theory (CFT) in momentum space, following a reconstruction method for tensor correlators, based on the solution of the conformal Ward identities (CWIs), introduced in recent works. We investigate and detail the structure of the CWIs and their non-perturbative solutions, and compare them to perturbation theory, taking QED and QCD as examples. Exact solutions of CFT’s in the flat background limit in momentum space are matched by the perturbative realizations in free field theories, showing that the origin the conformal anomaly is related to effective scalar interactions, generated by the renormalization of the longitudinal components of the corresponding operators.

Anomaly cancellation in effective supergravity theories from the heterotic string: Two simple examples

We use Pauli–Villars regularization to evaluate the conformal and chiral anomalies in the effective field theories from Z3 and Z7 compactifications of the heterotic string without Wilson lines. We show that parameters for Pauli–Villars chiral multiplets can be chosen in such a way that the anomaly is universal in the sense that its coefficient depends only on a single holomorphic function of the three diagonal moduli. It is therefore possible to cancel the anomaly by a generalization of the four-dimensional Green–Schwarz mechanism. In particular we are able to reproduce the results of a string calculation of the four-dimensional chiral anomaly for these two models.

Heat kernel and Weyl anomaly of Schrödinger invariant theory

We propose a method inspired by discrete light cone quantization to determine the heat kernel for a Schr\"odinger field theory (Galilean boost invariant with $z=2$ anisotropic scaling symmetry) living in $d+1$ dimensions, coupled to a curved Newton-Cartan background, starting from a heat kernel of a relativistic conformal field theory ($z=1$) living in $d+2$ dimensions. We use this method to show that the Schr\"odinger field theory of a complex scalar field cannot have any Weyl anomalies. To be precise, we show that the Weyl anomaly ${\mathcal{A}}_{d+1}^{G}$ for Schr\"odinger theory is related to the Weyl anomaly of a free relativistic scalar CFT ${\mathcal{A}}_{d+2}^{R}$ via ${\mathcal{A}}_{d+1}^{G}=2\ensuremath{\pi}\ensuremath{\delta}(m){\mathcal{A}}_{d+2}^{R}$, where $m$ is the charge of the scalar field under particle number symmetry. We provide further evidence of the vanishing anomaly by evaluating Feynman diagrams in all orders of perturbation theory. We present an explicit calculation of the anomaly using a regulated Schr\"odinger operator, without using the null cone reduction technique. We generalize our method to show that a similar result holds for theories with a single time-derivative and with even $z>2$.

Boundary conformal anomalies on hyperbolic spaces and Euclidean balls

We compute conformal anomalies for conformal field theories with free conformal scalars and massless spin 1/2 fields in hyperbolic space ℍd and in the ball {mathbb{B}}^d , for 2≤d≤7. These spaces are related by a conformal transformation. In even dimensional spaces, the conformal anomalies on ℍ2n and {mathbb{B}}^{2n} are shown to be identical. In odd dimensional spaces, the conformal anomaly on {mathbb{B}}^{2n+1} comes from a boundary contribution, which exactly coincides with that of ℍ2n + 1 provided one identifies the UV short-distance cutoff on {mathbb{B}}^{2n+1} with the inverse large distance IR cutoff on ℍ2n + 1, just as prescribed by the conformal map. As an application, we determine, for the first time, the conformal anomaly coefficients multiplying the Euler characteristic of the boundary for scalars and half-spin fields with various boundary conditions in d = 5 and d = 7.