# Wilson-Fisher Fixed Point in 2+1 Dimensions

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In the paper "A Duality Web in 2+1 Dimensions and Condensed Matter Physics",

https://arxiv.org/abs/1606.01989

it is claimed on page 1 that the two theories

$$|D_{B}\phi|^{2}-g|\phi|^{4}\longleftrightarrow\frac{-1}{4e^{2}}\hat{f}_{\mu\nu}\hat{f}^{\mu\nu}+|D_{\hat{b}}\hat{\phi}|^{2}-\hat{g}|\hat{\phi}|^{4}+\frac{1}{2\pi}\epsilon^{\alpha\beta\gamma}\hat{b}_{\alpha}\partial_{\beta}B_{\gamma}$$

are dual and flow to Wilson-Fisher fixed point in the IR, where $\hat{f}_{\mu\nu}=\partial_{\mu}\hat{b}_{\nu}-\partial_{\nu}\hat{b}_{\mu}$.
The classical mass dimensions are $[ \phi ]=[ \hat{\phi} ]=1/2$, $[B]=[ \hat{b} ]=1$, $[g]=[ \hat{g} ]=1$, and $[e]=1/2$. In the IR limit, I expect that $e\rightarrow\infty$ and $g,\hat{g}\rightarrow\infty$, so I can drop the kinetic term of $\hat{b}$ field $d\hat{b}\wedge\ast d\hat{b}$, and the theories become strongly coupled.

However, from this bachelor thesis, the exact $\beta$-function of real $\phi^{4}$ scalar is computed via using the Wetterich's exact RG flow equation, which is widely used in the quantum gravity community.

https://www.ru.nl/publish/pages/760966/bachelorscriptie_arthur_vereijken.pdf

It shows that the theory

$$S=\int d^{D}x \left\{\frac{1}{2}\phi(x)\left(-\partial^{2}+m^{2}\right)\phi(x)+\frac{\lambda}{4!}\phi(x)^{4}\right\}$$

$$\equiv\int d^{D}x\left\{\frac{1}{2}\phi(x)\left(-Z_{\Lambda}\partial^{2}+\Lambda^{2}\tilde{m}_{\Lambda}^{2}\right)\phi(x)+\frac{\Lambda^{4-D}}{4!}\lambda_{\Lambda}\phi(x)^{4}\right\}$$

has a Wilson-Fisher fixed point at

$$Z_{\ast}=1$$

$$\tilde{m}_{\ast}=\frac{D-4}{16-D}$$

$$\tilde{\lambda}_{\ast}=\frac{9\cdot 2^{D+5}\pi^{D/2}\Gamma(D/2+1)(4-D)}{(16-D)^{3}}$$

In D=2+1, the Wilson-Fisher fixed point has a finite coupling, with negative mass-squared $-1/13$, and so has a spontaneous symmetry breaking.

However, in the paper by Nathan Seiberg, T. Senthil, Chong Wang, Edward Witten, it clearly says that the Wilson-Fisher fixed point in 2+1 dimensions is massless.

Am I misunderstanding anything here?

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