On renormalization group flows and the \(a\)-theorem in 6d. (English) Zbl 1397.81179
Summary: 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 \(a_{\text{UV}}-a_{\text{IR}}\) 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 \(n\geq 4\). 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 \(\text{AdS}_7\times S^4\).
MSC:
| 81T17 | Renormalization group methods applied to problems in quantum field theory |
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