On renormalization group flows and the a-theorem in 6d

Abstract

We study the extension of the approach to the a-theorem of Komargodski and Schwimmer to quantum field theories in d=6 spacetime dimensions. The dilaton effective action is obtained up to 6th order in derivatives. The anomaly flow a_UV - a_IR is the coefficient of the 6-derivative Euler anomaly term in this action. It then appears at order p^6 in the low energy limit of n-point scattering amplitudes of the dilaton for n > 3. The detailed structure with the correct anomaly coefficient is confirmed by direct calculation in two examples: (i) the case of explicitly broken conformal symmetry is illustrated by the free massive scalar field, and (ii) the case of spontaneously broken conformal symmetry is demonstrated by the (2,0) theory on the Coulomb branch. In the latter example, the dilaton is a dynamical field so 4-derivative terms in the action also affect n-point amplitudes at order p^6. The calculation in the (2,0) theory is done by analyzing an M5-brane probe in AdS_7 x S^4. Given the confirmation in two distinct models, we attempt to use dispersion relations to prove that the anomaly flow is positive in general. Unfortunately the 4-point matrix element of the Euler anomaly is proportional to stu and vanishes for forward scattering. Thus the optical theorem cannot be applied to show positivity. Instead the anomaly flow is given by a dispersion sum rule in which the integrand does not have definite sign. It may be possible to base a proof of the a-theorem on the analyticity and unitarity properties of the 6-point function, but our preliminary study reveals some difficulties.

41 pages, 5 figures

References in corpus (5)

• Towards a derivation of holographic entanglement entropy
• Causality, Analyticity and an IR Obstruction to UV Completion
• Holographic c-theorems in arbitrary dimensions
• Spontaneous Breaking of Conformal Invariance and Trace Anomaly Matching
• The Conformal Anomaly of M5-Branes

Cited by in corpus (95)

• Beyond the Cosmological Standard Model
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• Beyond Amplitudes' Positivity and the Fate of Massive Gravity
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• Consistency of the Standard Model Effective Field Theory
• Anomalies, Renormalization Group Flows, and the a-Theorem in Six-Dimensional (1,0) Theories
• Quantum Gravity Constraints from Unitarity and Analyticity
• On Scale and Conformal Invariance in Four Dimensions
• Top Down Approach to 6D SCFTs
• Constraints on RG Flow for Four Dimensional Quantum Field Theories
• 2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories
• Renormalization Group Flows on Line Defects
• Universal Entanglement and Boundary Geometry in Conformal Field Theory
• 6d holographic anomaly match as a continuum limit
• Recursion relations from soft theorems
• New soft theorems for the gravity dilaton and the Nambu-Goldstone dilaton at subsubleading order
• Spheres, Charges, Instantons, and Bootstrap: A Five-Dimensional Odyssey
• Scattering Amplitudes from Soft Theorems and Infrared Behavior
• Generalized $F$-Theorem and the $\epsilon$ Expansion
• Natural Selection Rules: New Positivity Bounds for Massive Spinning Particles
• Exact and Numerical Results on Entanglement Entropy in (5+1)-Dimensional CFT
• Aspects of Galileon Non-Renormalization
• Surface Defect, Anomalies and $b$-Extremization
• Stress Tensors from Trace Anomalies in Conformal Field Theories
• RG flows in d dimensions, the dilaton effective action, and the a-theorem
• Exploring soft constraints on effective actions
• Structures on the Conformal Manifold in Six Dimensional Theories
• A challenge to the $a$-theorem in six dimensions
• The F-Theorem and F-Maximization
• Consequences of Weyl Consistency Conditions
• Defect $a$-Theorem and $a$-Maximization
• Positivity Bounds without Boosts
• Non-linear Representations of the Conformal Group and Mapping of Galileons
• Limit Cycles in Four Dimensions
• Double-soft behavior of the dilaton of spontaneously broken conformal invariance
• Weyl Anomalies of Four Dimensional Conformal Boundaries and Defects
• Higher-Point Positivity
• Quantum Information Scrambling: From Holography to Quantum Simulators
• Comments on a-maximization from gauged supergravity
• M5-Brane and D-Brane Scattering Amplitudes
• On the conformal properties of topological terms in even dimensions
• The a-function for gauge theories
• Weyl versus Conformal Invariance in Quantum Field Theory
• Renormalization group flow of entanglement entropy on spheres
• The spectrum of IIB supergravity on AdS_5 x S^5/Z_3 and a 1/N^2 test of AdS/CFT
• Soft theorems from anomalous symmetries
• Back to Heterotic Strings on ALE Spaces: Part I -- Instantons, 2-groups and T-duality
• Swampland Conditions for Higher Derivative Couplings from CFT
• Hierarchy of RG flows in 6d $(1,0)$ orbi-instantons
• Scalar Fields Near Compact Objects: Resummation versus UV Completion
• On c-theorems in arbitrary dimensions
• Bootstrapping the $a$-anomaly in $4d$ QFTs
• RG flows on two-dimensional spherical defects
• A new 6d fixed point from holography
• Perturbative c-theorem in d-dimensions
• The a-theorem and conformal symmetry breaking in holographic RG flows
• Minimal Distances Between SCFTs
• Two-loop renormalization of multiflavor ${\varphi }^3$ theory in six dimensions and the trace anomaly
• The shape of Scalar Gauss-Bonnet Gravity
• Conformal Anomalies in Hydrodynamics
• Constraints on Perturbative RG Flows in Six Dimensions
• Evidence for a 5d F-theorem
• Spinning Sum Rules for the Dimension-Six SMEFT
• The a-function in six dimensions
• Spontaneous Symmetry Breaking, Conformal Anomaly and Incompressible Fluid Turbulence
• The a-function in three dimensions: beyond leading order
• All the entropies on the light-cone
• Quantum Corrections to Generic Branes: DBI, NLSM, and More
• The Dilaton Wess-Zumino Action in 6 Dimensions from Weyl Gauging: Local Anomalies and Trace Relations
• Implications of Conformal Symmetry in Quantum Mechanics
• An Effective Theory for Holographic RG Flows
• Weyl Anomaly induced Fermi Condensation and Holography
• $\mathcal{N}=(1,0)$ Anomaly Multiplet Relations in Six Dimensions
• Scheme invariants in phi^4 theory in four dimensions
• Free Energy on the Sphere for Non-Abelian Gauge Theories
• The dilaton Wess-Zumino action in higher dimensions
• The a-function for N=2 supersymmetric gauge theories in three dimensions
• Dilaton Effective Action with $\mathcal{N}=1$ Supersymmetry
• Renormalization Group Flows, the $a$-Theorem and Conformal Bootstrap
• Conformal Trace Relations from the Dilaton Wess-Zumino Action
• Boundary Anomalies and Correlation Functions
• Atomic Higgsings of 6D SCFTs
• Dilaton invading from infinitesimal extra dimension
• $c$-functions in Higher-derivative Flows Across Dimensions
• Comments on the Algebraic Properties of Dilaton Actions
• Atomic Higgsings of 6D SCFTs II: Induced Flows
• On the EFT of Conformal Symmetry Breaking
• On spontaneous breaking of conformal symmetry by probe flavour D-branes
• Holographic renormalization group flows in two-dimensional gravity and $AdS$ black holes
• Equivalence of EFTs and Time-Like Extra Dimensions
• Holography and Conformal Anomaly Matching
• Explicit Entropic Proofs of Irreversibility Theorems for Holographic RG Flows
• Running of the Number of Degrees of Freedom in Quantum Conformal Gravity