The $a$-theorem and the Asymptotics of 4D Quantum Field Theory (1204.5221v2)

Abstract: We study the possible IR and UV asymptotics of 4D Lorentz invariant unitary quantum field theory. Our main tool is a generalization of the Komargodski-Schwimmer proof for the $a$-theorem. We use this to rule out a large class of renormalization group flows that do not asymptote to conformal field theories in the UV and IR. We show that if the IR (UV) asymptotics is described by perturbation theory, all beta functions must vanish faster than $(1/|\ln\mu|)^{1/2}$ as $\mu \to 0$ ($\mu \to \infty$). This implies that the only possible asymptotics within perturbation theory is conformal field theory. In particular, it rules out perturbative theories with scale but not conformal invariance, which are equivalent to theories with renormalization group pseudocycles. Our arguments hold even for theories with gravitational anomalies. We also give a non-perturbative argument that excludes theories with scale but not conformal invariance. This argument holds for theories in which the stress-energy tensor is sufficiently nontrivial in a technical sense that we make precise.

• The paper extends the KS a-theorem to constrain 4D QFT RG flows, showing that non-conformal asymptotics require a vanishing stress-energy tensor trace.
• It employs dilaton-dilaton scattering amplitude analysis to demonstrate that bounded, perturbative couplings enforce trivial or conformal behavior in both UV and IR.
• It provides non-perturbative arguments ruling out scale invariance without conformal invariance, narrowing the viable endpoints of RG flows.

The $a$-Theorem and Asymptotics of 4D Quantum Field Theory

The paper by Markus A. Luty, Joseph Polchinski, and Riccardo Rattazzi addresses a fundamental question in four-dimensional Lorentz invariant quantum field theory (QFT) concerning renormalization group (RG) flows and the asymptotic behavior of theories in the ultraviolet (UV) and infrared (IR) limits. The paper investigates whether these asymptotics always lead to conformal field theories (CFTs) and employs a generalized proof of the $a$-theorem to explore this.

The $a$-theorem, originally proved by Komargodski and Schwimmer (KS), is a central tool in this analysis. The theorem asserts that for any RG flow connecting two CFTs in four dimensions, the anomaly coefficient $a$, which encodes the trace anomaly related to the Euler density, decreases from the UV to the IR. The authors extend the KS argument to demonstrate that perturbative QFTs which asymptote to non-conformal theories are constrained by the vanishing trace of the stress-energy tensor, leaving only conformal, or trivial, behavior.

This work imposes restrictions on RG flows by considering the convergence properties of integrals over the dilaton-dilaton scattering amplitude, $A(s)$, where $s$ is the square of the center-of-mass energy. The dilaton is introduced as a Goldstone boson of spontaneously broken scale invariance, and the scattering amplitude serves as a measure of conformality. The authors show that the imaginary part of $A(s)$, being a sum of positive terms (corresponding to probabilities of scattering into intermediate states), must be vanishing in both the UV and IR if the renormalized couplings remain bounded and perturbative.

Moreover, they present a non-perturbative argument suggesting that any scale-invariant theory with a non-zero stress-energy tensor trace is inherently contradictory unless the trace itself trivializes. This effectively rules out theories with only scale invariance (SFTs) and consolidates the understanding that scale invariance without conformal invariance is improbable.

The implications of this paper are significant for both theoretical research and practical applications. It strengthens the hypothesis that conformity is the natural IR fate for most four-dimensional QFTs, thereby streamlining efforts towards classifying possible endpoint behaviors of RG flows. The research also underpins the theoretical expectation that a broader class of phenomena across physics could be captured within the framework of CFTs or trivial theories, simplifying the landscape of possible physical theories.

The paper sets a foundation for future exploration, suggesting that robust constraints such as these could potentially apply to more complex frameworks or higher-dimensional theories. Additionally, it invites further investigation into the geometric analogs via holographic principles, which may offer additional insights into the structure of the RG flows and the nature of conformality in various dimensions.

In summary, this work provides an in-depth analysis of the asymptotic behavior of 4D QFTs using the $a$-theorem, contributing significantly to the understanding of scale and conformal invariance and effectively narrowing the scope of viable endpoints for RG flows.