# Mass terms in the SUSY gauged linear sigma model

Okay, I have a very basic question about the SUSY gauged linear sigma model which is driving me crazy. I am following Chapter $$15$$ of Mirror Symmetry by Hori et al. I am considering the SUSY gauged linear sigma model with gauge group $$U(1)$$ and $$N$$ chiral multiplets with the following Lagrangian: $$L=\sum_{j=1}^{N} \left[ -|D_{\mu}\phi_{j}|^{2}+i\bar{\psi}_{j-}(D_{0}+D_{1})\psi_{j-}+i\bar{\psi}_{j+}(D_{0}-D_{1})\psi_{j+} - |\sigma|^{2}|\phi_{j}|^{2} - \bar{\psi}_{j-}\sigma\psi_{j+}-\bar{\psi}_{j+}\bar{\sigma}\psi_{j-} + i (\bar{\phi}_{j}\lambda_{+}\psi_{j-}-\bar{\phi}_{j}\lambda_{-}\psi_{j+}-\bar{\psi}_{j-}\bar{\lambda}_{+}\phi_{j}+\bar{\psi}_{j+}\bar{\lambda}_{-}\phi_{j}) + \frac{1}{2e^{2}}\left( -|\partial_{\mu}\sigma|^{2}+i\bar{\lambda}_{-}(\partial_{0}+\partial_{1})\lambda_{-}+i\bar{\lambda}_{+}(\partial_{0}-\partial_{1})\lambda_{+}+v_{01}^{2} \right) +\theta v_{01} -\frac{e^{2}}{2}\left(\sum_{i=1}^{N}|\phi_{i}|^{2}-r\right)^{2} \right]$$ where the terms $$(\phi_{j},\psi_{j\pm})$$ form chiral supermultiplets and the terms $$(\sigma,v_{\mu},\lambda_{\pm})$$ form a vector supermultiplet.

For $$r>0$$, the scalar potential is clearly minimised by configurations: $$M_{vac}:=\{\phi_{i}:\sum_{i=1}^{N}|\phi_{i}|^{2}=r\}/U(1)\cong\mathbb{C}P^{N-1}$$ My problem centres around trying to figure out the (perturbative), masses of the various fields in this theory. The book claims that the massive modes constitute a massive supermultiplet of mass $$e\sqrt{2r}$$, in particular $$v_{\mu},\sigma$$, and $$\lambda_{\pm}$$ are claimed to have this mass, as are transverse modes of $$\phi_{i}$$ and certain modes of the $$\psi_{j}$$.

Clearly to find these masses, we want to pick a classical vacuum and expand around it. We expect the potential term: $$U= |\sigma|^{2}\sum_{i=1}^{N}|\phi_{i}|^{2} + \frac{e^{2}}{2}\left(\sum_{i=1}^{N}|\phi_{i}|^{2}-r\right)^{2}$$ to give mass to $$\sigma$$ and some modes of $$\phi_{j}$$. We expect the gauge kinetic term: $$\sum_{j=1}^{N} |D_{\mu}\phi_{j}|^{2}$$ to give mass to the gauge field (via the Higgs mechanism), and the Yukawa-like terms: $$L_{F}:= i (\bar{\phi}_{j}\lambda_{+}\psi_{j-}-\bar{\phi}_{j}\lambda_{-}\psi_{j+}-\bar{\psi}_{j-}\bar{\lambda}_{+}\phi_{j}+\bar{\psi}_{j+}\bar{\lambda}_{-}\phi_{j})$$ to give masses to the appropriate fermions.

We pick the vacuum $$\sigma=\phi_{1}=\ldots=\phi_{N-1}=0$$, $$\phi_{N}=\sqrt{r}$$. We let $$\phi_{N}=\phi_{N}'-\sqrt{r}$$, and begin to expand the Lagrangian around this vacuum.

Already, we see: $$U=|\sigma|^{2}(\phi_{N}'^{2}+\sqrt{r}\phi_{N}'+r)+\cdots=r|\sigma|^{2}+\cdots$$ Suggesting a mass of $$\sqrt{r}$$, not a mass of $$e\sqrt{2r}$$.

We now set $$\phi_{N}=\rho e^{i\theta}$$, i.e. switch to polar variables. Here our vaccuum is $$\rho=\sqrt{r}$$, $$\theta=0$$. I will discuss the gauge and fermion terms separately. Expanding $$-|D_{\mu}\phi_{N}|^{2}-U$$, and writing $$\rho=\sqrt{r}+\epsilon$$, we have: $$-(\partial_{\mu}\epsilon)^{2}+rv_{\mu}'^{2}-r|\sigma|^{2}-2e^{2}r\epsilon^{2}+\cdots$$ where $$\cdots$$ represents interaction terms, and where we have defined a new gauge field $$v_{\mu}'=v_{\mu}+\partial_{\mu}\theta$$. This shows that $$\epsilon$$ has the claimed mass, but $$v_{\mu}'$$ has an incorect mass of $$\sqrt{2r}$$.

What happens to $$L_{F}$$ is even worse. In the only term involving $$\phi_{N}$$, The Goldstone boson $$\theta$$ still appears (rather than having been eaten entirely), to yield: $$i\rho \left( e^{i\theta}(\bar{\psi}_{N+}\bar{\lambda}_{-}-\bar{\psi}_{N-}\bar{\lambda}_{+}) + e^{-i\theta}(\lambda_{+}\psi_{N-}-\lambda_{-}\psi_{N+}) \right)$$ I have no idea how to extract nice terms from this mess.

I know this should be basic stuff, but I have managed to thoroughly confuse myself, and any help would be much appreciated.

First, wouldn't hurt to specify spacetime dimension and number of supersymmetries. I guess $$D=2$$ $${\cal N}=(2,2)$$.
1. Notice that the kinetic terms of the vector multiplet have the factor $$1/(2e^2)$$. For canonical kinetic terms you need to multiply the vector multiplet components by $$\sqrt{2}e$$.
2. You can absorb the Goldstone by a phase rotation of the chiral fermion. Or just set $$\theta=0$$ as a gauge choice.