Effective field theory of 6D SUSY RG Flows

Motivated by its potential use in constraining the structure of 6D renormalization group flows, we determine the low energy dilaton-axion effective field theory of conformal and global symmetry breaking in 6D conformal field theories (CFTs). While our analysis is largely independent of supersymmetry, we also investigate the case of 6D superconformal field theories (SCFTs), where we use the effective action to present a streamlined proof of the 6D $a$-theorem for tensor branch flows, as well as to constrain properties of Higgs branch and mixed branch flows. An analysis of Higgs branch flows in some examples leads us to conjecture that in 6D SCFTs, an interacting dilaton effective theory may be possible even when certain four-dilaton four-derivative interaction terms vanish, because of large momentum modifications to four-point dilaton scattering amplitudes. This possibility is due to the fact that in all known $D>4$ CFTs, the approach to a conformal fixed point involves effective strings which are becoming tensionless.

Figure 1

A schematic plot of how the parameter $\widehat{b}$ depends on $f_t/f_H$ in a 6D mixed branch flow. The full $\widehat{b}$ interpolates between ${\widehat{b}}_{\text{Higgs}}$ and ${\widehat{b}}_{\text{tensor}}$ as we increase $f_t/f_H$. Three lines in the plot represent three special cases: ${\widehat{b}}_{\text{mixed}}<0$ (blue), ${\widehat{b}}_{\text{mixed}}=0$ (green), and ${\widehat{b}}_{\text{mixed}}>0$ (red). Note that $\widehat{b}$ has a nontrivial dependence on $f_t/f_H$ even when a mixed branch flow factorizes completely (${\widehat{b}}_{\text{mixed}}=0$).

Figure 2

Example of a one-loop diagram which contributes to four-dilaton four-derivative interaction terms in the dilaton effective action obtained from a Higgs branch flow. Here, we have indicated external scalars as associated with the hypermultiplets, with internal massive gauginos running in the loop. In 4D, this diagram scales as ${\mathcal{M}}_{2\to 2}\sim g^4{p}^4/{({\mathrm{\Lambda }}_{4D})}^4=p^4/f_H^4$, while in 6D we instead find ${\mathcal{M}}_{2\to 2}\sim g^4{p}^4/{({\mathrm{\Lambda }}_{6D})}^2=p^4/f_t^2{f}_H^4$, with $p$ the characteristic momentum of the external states.