Higher-derivative Kinetic Terms Research Articles

Scale and conformal invariance in higher derivative shift symmetric theories

The critical behavior of infinite families of shift symmetric interacting theories with higher derivative kinetic terms (non unitary) is considered. Single scalar theories with shift symmetry are classified according to their upper critical dimensions and studied at the leading non trivial order in perturbation theory. For two infinite families, one with quartic and one with cubic interactions, beta functions, criticality conditions and universal anomalous dimensions are computed. At the order considered, the cubic theories enjoy a one loop non renormalization of the vertex, so that the beta function depends non trivially only on the anomalous dimension. The trace of the energy momentum tensor is also investigated and it is shown that these two families of QFTs are conformally invariant at the fixed point of the RG flow.

$S$ -matrix unitarity and renormalizability in higher-derivative theories

Abstract We investigate the relation between $S$-matrix unitarity ($SS^\dagger=1$) and renormalizability in theories with negative-norm states. The relation has been confirmed in many field theories, including gauge theories and Einstein gravity, by analyzing the unitarity bound, which follows from the $S$-matrix unitarity and the norm positivity. On the other hand, renormalizable theories with a higher-derivative kinetic term do not necessarily satisfy the unitarity bound because of the negative-norm states. In these theories it is not known whether the $S$-matrix unitarity provides a nontrivial constraint related to the renormalizability. In this paper, by relaxing the assumption of norm positivity we derive a bound on scattering amplitudes weaker than the unitarity bound, which may be used as a consistency requirement for $S$-matrix unitarity. We demonstrate in scalar field models with a higher-derivative kinetic term that the weaker bound and the renormalizability imply identical constraints.

Uncovering novel phase structures in Box ^k scalar theories with the renormalization group

We present a detailed version of our recent work on the RG approach to multicritical scalar theories with higher derivative kinetic term phi (-Box )^kphi and upper critical dimension d_c = 2nk/(n-1). Depending on whether the numbers k and n have a common divisor two classes of theories have been distinguished. For coprime k and n-1 the theory admits a Wilson-Fisher type fixed point. We derive in this case the RG equations of the potential and compute the scaling dimensions and some OPE coefficients, mostly at leading order in epsilon . While giving new results, the critical data we provide are compared, when possible, and accord with a recent alternative approach using the analytic structure of conformal blocks. Instead when k and n-1 have a common divisor we unveil a novel interacting structure at criticality. Box ^2 theories with odd n, which fall in this class, are analyzed in detail. Using the RG flows it is shown that a derivative interaction is unavoidable at the critical point. In particular there is an infrared fixed point with a pure derivative interaction at which we compute the scaling dimensions and, for the particular example of Box ^2 theory in d_c=6, also some OPE coefficients.

Multicritical scalar theories with higher-derivative kinetic terms: A perturbative RG approach with the ε -expansion

We employ perturbative RG and $\epsilon$-expansion to study multi-critical single-scalar field theories with higher derivative kinetic terms of the form $\phi(-\Box)^k\phi$. We focus on those with a $\mathbb{Z}_2$-symmetric critical point which are characterized by an upper critical dimension $d_c=2 n k/(n-1)$ accumulating at even integers. We distinguish two types of theories depending on whether or not the numbers $k$ and $n-1$ are relatively prime. When they are, the theory admits a local potential approximation. In this case we present the beta functional of the potential and use this to calculate some anomalous dimensions and OPE coefficients. These confirm some CFT data obtained using conformal block techniques, while giving new results. In the second case where $k$ and $n-1$ have a common divisor, the theories show a much richer structure induced by the presence of derivative operators. We study the case $k=2$ with odd values of $n$, which fall in the second class, and calculate the functional flows and spectrum. These theories have a phase diagram characterized at leading order in $\epsilon$ by four fixed points which apart from the Gaussian UV fixed point include an IR fixed point with purely derivative interactions.

Higher dimensional higher derivative ϕ4 theory

We construct several towers of scalar quantum field theories with an $O(N)$ symmetry which have higher derivative kinetic terms. The Lagrangians in each tower are connected by lying in the same universality class at the $d$-dimensional Wilson-Fisher fixed point. Moreover the universal theory is studied using the large $N$ expansion and we determine $d$-dimensional critical exponents to $O(1/N^2)$. We show that these new universality classes emerge naturally as solutions to the linear relation of the dimensions of the fields deduced from the underlying force-matter interaction of the universal critical theory. To substantiate the equivalence of the Lagrangians in each tower we renormalize each to several loop orders and show that the renormalization group functions are consistent with the large $N$ critical exponents. While we focus on the first two new towers of theories and renormalize the respective Lagrangians to $16$ and $18$ dimensions there are an infinite number of such towers. We also briefly discuss the conformal windows and the extension of the ideas to theories with spin-$\frac{1}{2}$ and spin-$1$ fields as well as the idea of lower dimension completeness.

NEC violation in mimetic cosmology revisited

In the context of Einstein gravity, if the null energy condition (NEC) is satisfied, the energy density in expanding space-times always decreases while in contracting space-times the energy density grows and the universe eventually collapses into a singularity. In particular, no non-singular bounce is possible. It is, though, an open question if this energy condition can be violated in a controlled way, i.e., without introducing pathologies, such as unstable negative-energy states or an imaginary speed of sound. In this paper, we will re-examine the claim that the recently proposed mimetic scenario can violate the NEC without pathologies. We show that mimetic cosmology is prone to gradient instabilities even in cases when the NEC is satisfied (except for trivial examples). Most interestingly, the source of the instability is always the Einstein-Hilbert term in the action. The matter stress-energy component does not contribute spatial gradient terms but instead makes the problematic curvature modes dynamical. We also show that mimetic cosmology can be understood as a singular limit of known, well-behaved theories involving higher-derivative kinetic terms and discuss ways of removing the instability.

On higher spin partition functions

We observe that the partition function of the set of all free massless higher spins s = 0, 1, 2, 3,... in flat space is equal to one: the ghost determinants cancel against the ‘physical’ ones or, equivalently, the (regularized) total number of degrees of freedom vanishes. This reflects large underlying gauge symmetry and suggests analogy with supersymmetric or topological theory. The Z = 1 property extends also to the AdS background, i.e. the 1-loop vacuum partition function of Vasiliev theory is equal to 1 (assuming a particular regularization of the sum over spins); this was noticed earlier as a consistency requirement for the vectorial AdS/CFT duality. We find that Z = 1 is true also in the conformal higher spin theory (with higher-derivative kinetic terms) expanded near flat or conformally flat S4 background. We also consider the partition function of free conformal theory of symmetric traceless rank s tensor field which has 2-derivative kinetic term but only scalar gauge invariance in flat 4d space. This non-unitary theory has Weyl-invariant action in curved background and it corresponds to ‘partially massless’ field in AdS5. We discuss in detail the special case of s = 2 (or ‘conformal graviton’), compute the corresponding conformal anomaly coefficients and compare them with previously found expressions for generic representations of conformal group in 4 dimensions.

Imaginary supergravity or Virial supergravity?

When a globally supersymmetric theory is scale invariant, it must possess a Virial supercurrent supermultiplet. The multiplet structure is analogous to the R-current supermultiplet in globally R-symmetric theories but we put extra “i”s in various formulae. We construct a novel type of supergravity from gauging the Virial supercurrent supermultiplet in d=1+3 dimensions. We give the full non-linear superspace action with the help of a covariantly linear unitary superconformal compensator. The resulting supergravity is peculiar: (1) no Einstein-like second order kinetic term is allowed without matter, (2) there exists a dynamical non-geometrical connection, (3) the metric is unimodular in the Wess–Zumino gauge and can be coupled only to scale invariant matter. Examples of matter couplings and higher derivative kinetic terms are studied from the superconformal approach. Our work completes the classification of the irreducible N=1 supergravities in d=1+3 dimensions.

Modified natural inflation: A small single field model with a large tensor to scalar ratio

In this paper we explored in detail a phenomenological model of modified single field natural inflation in light of recent cosmological experiments, BICEP2. Our main goal is to construct an inflationary model which not only predicts the important cosmological quantities such as $({n}_{s},r)$ compatible with experimental observation, but also is consistent with the low energy effective theory framework. Therefore, all fundamental scales apart from ${M}_{p}$ and quantities of our interest should be within the sub-Planckian region. In order to achieve our goal, we modify the usual single field natural inflationary model by a specific form of higher derivative kinetic terms called kinetic gravity braiding. One of our guiding principles to construct such a model is the constant shift symmetry of the axion. We have chosen the form of the kinetic gravity braiding term in such a way that it predicts the required value of ${n}_{s}\ensuremath{\simeq}0.96$ and a large tensor to scalar ratio, $r>0.1$. Importantly, for a wide range of parameter space our model has sub-Planckian axion decay constant $f$ and the scale of inflation $\mathrm{\ensuremath{\Lambda}}$. However, the reheating after the end of inflation limits the value of $f$ so that we are unable to get $f$ to be significantly lower than ${M}_{p}$. Furthermore, we find sub-Planckian field excursion for the axion field $\mathrm{\ensuremath{\Delta}}\ensuremath{\phi}\ensuremath{\simeq}f$ for the sufficient number of $e$-folding, $\mathcal{N}\ensuremath{\gtrsim}50$. We also discussed in detail about the natural preheating mechanism in our model based on the recently proposed Chern-Simon coupling. We found this gravity mediated preheating is very difficult to achieve in our model. With our general analytic argument, we also would like to emphasize that Chern-Simons mediated preheating is very unlikely to happen in any slow roll inflationary model.

Weighted power counting and chiral dimensional regularization

We define a modified dimensional-regularization technique that overcomes several difficulties of the ordinary technique, and is specially designed to work efficiently in chiral and parity violating quantum field theories, in arbitrary dimensions greater than 2. When the dimension of spacetime is continued to complex values, spinors, vectors and tensors keep the components they have in the physical dimension, therefore the $\gamma $ matrices are the standard ones. Propagators are regularized with the help of evanescent higher-derivative kinetic terms, which are of the Majorana type in the case of chiral fermions. If the new terms are organized in a clever way, weighted power counting provides an efficient control on the renormalization of the theory, and allows us to show that the resulting chiral dimensional regularization is consistent to all orders. The new technique considerably simplifies the proofs of properties that hold to all orders, and makes them suitable to be generalized to wider classes of models. Typical examples are the renormalizability of chiral gauge theories and the Adler-Bardeen theorem. The difficulty of explicit computations, on the other hand, may increase.

Non-canonical inflation in supergravity

We investigate the effect of non-canonical kinetic terms on inflation in supergravity. We find that the biggest impact of such higher-derivative kinetic terms is due to the corrections to the potential that they induce via their effect on the auxiliary fields, which now have a cubic equation of motion. This is in contrast to the usual (non-supersymmetric) effective field theory expansion which assumes that mass-suppressed higher-derivative terms do not affect the lower-derivative terms already present. We demonstrate with several examples that such changes in the potential can significantly modify the inflationary dynamics. Our results have immediate implications for effective descriptions of inflation derived from string theory, where higher-derivative kinetic terms are generally present. In addition we elucidate the structure of the theory in the parameter range where there are three real solutions to the auxiliary field's equation of motion, studying the resulting three branches of the theory, and finding that one of them suffers from a singularity in the speed of propagation of fluctuations.

Higher-derivative chiral superfield actions coupled toN=1supergravity

We construct $\mathcal{N}=1$ supergravity extensions of scalar field theories with higher-derivative kinetic terms. Special attention is paid to the auxiliary fields, whose elimination leads not only to corrections to the kinetic terms, but to new expressions for the potential energy as well. For example, a potential energy can be generated even in the absence of a superpotential. Our formalism allows one to write a supergravity extension of any higher-derivative scalar field theory and therefore has applications to both particle physics and cosmological model building. As an illustration, we couple the higher-derivative Dirac-Born-Infeld action describing a 3-brane in six dimensions to $\mathcal{N}=1$ supergravity. This displays a number of new features including the fact that in the regime where the higher-derivative kinetic terms become important, the potential tends to be everywhere negative.

Folded resonant non-Gaussianity in general single field inflation

We compute a novel type of large non-Gaussianity due to small periodic features in general single field inflationary models. We show that the non-Bunch-Davies vacuum component generated by features, although has a very small amplitude, can have significant impact on the non-Gaussianity. Three mechanisms are turned on simultaneously in such models, namely the resonant effect, non-Bunch-Davies vacuum and higher derivative kinetic terms, resulting in a bispectrum with distinctive shapes and running. The size can be equal to or larger than that previously found in each single mechanism. Our full results, including the resonant and folded resonant non-Gaussianities, give the leading order bispectra due to general periodic features in general single field inflation.

THE LEE-WICK STANDARD MODEL

This article reviews some recent work on a version of the standard model (the Lee-Wick standard model) that contains higher derivative kinetic terms that improve the convergence of loop diagrams removing the quadratic divergence in the Higgs boson mass. Naively higher derivative theories of this type are not acceptable since the higher derivative terms either cause instabilities (from negative energies) or a loss of unitarity (from negative norm states). Lee and Wick provided an interpretation for such theories arguing that theories with higher derivative kinetic terms can be unitary and stable if the states associated with the massive propagator poles, that arise from the higher derivatives, have widths and hence decay and are not in the spectrum of the theory.

Non-Gaussianities in single field inflation and their optimal limits from the WMAP 5-year data

Using the recently developed effective field theory of inflation, we argue that the size and the shape of the non-Gaussianities generated by single-field inflation are generically well described by two parameters: fNLequil, which characterizes the size of the signal that is peaked on equilateral configurations, and fNLorthog, which instead characterizes the size of the signal which is peaked both on equilateral configurations and flat-triangle configurations (with opposite signs). The shape of non-Gaussianities associated with fNLorthog is orthogonal to the one associated to fNLequil, and former analysis have been mostly blind to it. We perform the optimal analysis of the WMAP 5-year data for both of these parameters. We find no evidence of non-Gaussianity, and we have the following constraints: −125 ⩽ fNLequil ⩽ 435, −369 ⩽ fNLorthog ⩽ 71 at 95% CL. We show that both of these constraints can be translated into limits on parameters of the Lagrangian of single-field inflation. For one of them, the speed of sound of the inflaton fluctuations, we find that it is either bounded to be cs ⩾ 0.011 at 95% CL. or alternatively to be so small that the higher-derivative kinetic term dominate at horizon crossing. We are able to put similar constraints on the other operators of the inflaton Lagrangian.

Higher derivative operators in the noncommutative Schwinger model

A study of the noncommutative Schwinger model is presented. It is shown that the Schwinger mass is not modified by the noncommutativity of spacetime till the first nontrivial order in the noncommutative parameter. Instead, a higher derivative kinetic term is dynamically generated by the lowest-order vacuum polarization diagrams. We argue that in the framework of the Seiberg–Witten map the feature of non-unitarity for a field theory with spacetime noncommutativity is characterized by the presence of higher derivative kinetic terms. The θ-expanded version of a unitary theory will not generate the lowest-order higher derivative quadratic terms.

Tunneling with negative tension

We describe a new family of thin-wall instantons, with negative tension walls, that mediate tunneling between Minkowski or de Sitter vacua. Some of these instantons can have exponentially enhanced tunneling rates, and would dominate the Euclidean path integral, representing a severe non-perturbative instability in any theory which supports them. We give two constructions of these instantons in theories which are perturbatively stable, but violate the null energy condition. One construction uses a scalar field theory with higher-derivative kinetic term, and is similar to the Coleman-de Luccia positive tension instanton. The other construction employs a negative-tension boundary arising from Z_2 orbifolding: it resembles a bubble of nothing which nucleates and grows, consuming the surrounding de Sitter or Minkowski spacetime. We explain how the spectral flow for fluctutations about the thin-wall tunneling solutions automatically protects causality, for both positive and negative tension instantons. We comment briefly on the relation of these solutions to a Kalzua-Klein bubble of nothing, cosmological models relying on ghost condensates, and string theory orientifolds.

Renormalizable acausal theories of classical gravity coupled with interacting quantum fields

We prove the renormalizability of various theories of classical gravity coupled with interacting quantum fields. The models contain vertices with dimensionalities greater than 4, a finite number of matter operators and a finite or reduced number of independent couplings. An interesting class of models is obtained from ordinary power-counting renormalizable theories, letting the couplings depend on the scalar curvature R of spacetime. The divergences are removed without introducing higher derivative kinetic terms in the gravitational sector. The metric tensor has a non-trivial running, even if it is not quantized. The results are proved applying a certain map that converts classical instabilities, due to higher derivatives, into classical violations of causality, whose effects become observable at sufficiently high energies. We study acausal Einstein–Yang–Mills theory with an R-dependent gauge coupling in detail. We derive all-order formulae for the beta functions of the dimensionality-six gravitational vertices induced by renormalization, such as RRμνρσRμνρσ. Such beta functions are related to the trace-anomaly coefficients of the matter subsector.

Renormalization and causality violations in classical gravity coupled with quantum matter

I prove that classical gravity coupled with quantized matter can be renormalized with a finite number of independent couplings, plus field redefinitions, without introducing higher-derivative kinetic terms in the gravitational sector, but adding vertices that couple the matter stress-tensor with the Ricci tensor. The theory is called "acausal gravity", because it predicts the violation of causality at high energies. Renormalizability is proved by means of a map M that relates acausal gravity with higher-derivative gravity. The causality violations are governed by two parameters, a and b, that are mapped by M into higher-derivative couplings. At the tree level causal prescriptions exist, but they are spoiled by the one-loop corrections. Some ideas are inspired by the usual treatments of the Abraham-Lorentz force in classical electrodynamics.

Defining the Nambu-Jona-Lasinio Model by Higher Derivative Kinetic Term

Usual treatment of the Nambu--Jona-Lasinio (NJL) model using loop momentum cutoff suffers from ambiguities in choosing the loop momenta to be cut off and violation of (external) gauge invariance. We define the NJL model from the starting Lagrangian level by using a higher derivative fermion kinetic term with a cutoff parameter $\Lambda$. This definition is free from such ambiguities and manifestly keeps the chiral symmetry as well as the gauge invariance. Quantization of this higher derivative system, current operators and calculational method are discussed in some detail. Calculating the pion decay constant and $\pi^0\rightarrow 2\gamma$ decay amplitude, we explicitly demonstrate that the low energy theorem holds. It is observed that the NJL mass relation $m_\sigma = 2 m_0$ between the $\sigma$ meson and fermion masses no longer holds in this model. We also present a simplified calculational method which is valid when the cutoff parameter $\Lambda $ is much larger than the energy scale of physics.