On renormalization group flows and the a-theorem in 6d (1205.3994v1)

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.

• The paper extends the 4d a-theorem for RG flows to six dimensions by constructing a dilaton effective action to capture anomaly flow information.
• The findings are validated using a free massive scalar field example and by analyzing the (2,0) theory via a holographic probe M5-brane approach.
• This work provides a framework for computing 6d anomaly coefficients and attempts to use dispersion relations for an analytic proof, setting a precedent for higher-dimensional theories.

Renormalization Group Flows and the $a$-Theorem in Six Dimensions

This paper explores the extension of the $a$-theorem, originally formulated for four-dimensional quantum field theories (QFTs), to six-dimensional scenarios. The $a$-theorem provides a deep insight into the flow of renormalization group (RG) trajectories between ultraviolet (UV) and infrared (IR) fixed points, positing that the Euler anomaly constant $a$ associated with the trace anomaly of conformal field theories satisfies the inequality $a_\text{UV} \geq a_\text{IR}$. This theorem has significant implications for understanding the irreversibility of RG flows in even-dimensional QFTs.

The authors build on the work of Komargodski and Schwimmer, who demonstrated the $a$-theorem in four dimensions using dilaton effective actions, to extend it to six-dimensional theories. In their six-dimensional analysis, the authors utilize the dilaton as a probe to assess changes in central charges associated with the Euler term, a key aspect of the trace anomaly.

Main Contributions and Findings

1. Dilaton Effective Action in 6D: The paper carefully constructs the dilaton effective action, highlighting how the dilaton captures the information about the anomaly flow $a_\text{UV} - a_\text{IR}$. This involves calculating scattering amplitudes of the dilaton at low energies and identifying terms that contribute to $O(p^6)$ amplitudes, rooted in both 4-derivative and 6-derivative terms, in contrast to only the latter in four dimensions.
2. Example of Explicit Breaking: The authors confirm their theoretical construction using a simple example involving a free massive scalar field, where conformality is broken explicitly. This provides a concrete avenue where theoretical predictions about anomaly coefficients can be tested against calculated scattering amplitudes, affirming that the constructed dilaton effective action accurately captures the anomaly contributions.
3. Holographic Approach: Leveraging holography, the paper analyzes the (2,0) theory using the dynamics of M5-branes, demonstrating the dilaton effective action's relevance by matching the DBI action of a probe M5-brane in AdS$_7 \times S^4$ with expected dilaton dynamics. This additional complexity in the dilaton interactions is indicative of the additional structure presented by six-dimensional models.
4. Dispersion Relations: A novel aspect of the paper is an attempt to derive dispersion relations to prove the $a$-theorem analytically. In six dimensions, the complications arising from non-positive definite terms and the need for higher-point correlation functions are considered.

Implications and Future Work

The successful extension of the $a$-theorem to six dimensions has theoretical implications for understanding higher-dimensional conformal theories and their RG flows. This research provides a framework for evaluating and computing the anomaly coefficients in diverse field theories. The paper sets a precedent for further exploration in higher even dimensions, proposing ways the $a$-theorem might be validated through various analytic techniques, possibly including entanglement entropy and holographic methods.

Future research directions suggested by this paper involve:

• Refining the framework for analyzing higher-order correlations, particularly exploring the dynamics in scenarios where dilaton terms become significant for understanding anomaly contributions.
• Extending the techniques to prove anomaly theorems in higher than six dimensions (e.g., eight-dimensional theories) where even more complex dilaton interactions and symmetry structures could be investigated.
• Investigating odd-dimensional QFTs to assess connections with related concepts like the $F$-theorem, potentially uncovering deeper universality across dimensions concerning conformal anomalies.

The intricacy of analyzing six-dimensional models, with novel contributions needed for higher-point amplitudes, presents both challenges and opportunities for theoretical physics in uncovering deeper connections between symmetry, anomaly, and dimension.