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My question is regarding the following Gauged Linear Sigma Model (GLSM) in two dimensions.

$$\tag{1} S=\int d^2x\Big(-D_{\mu}\overline{\phi} D^{\mu}\phi +\frac{D^2}{2e'^2} +D(|\phi|^2-r)\Big).$$ Here $D_{\mu}\phi=\partial_{\mu}\phi+iQA_{\mu}\phi$, $D$ is an auxiliary field, and $e'$ is a coupling constant. (I have provided this action as a minimal working example in order to understand $\mathcal{N}=(2,2)$ supersymmetric GLSMs.)

Integrating out $D$ via its equation of motion, the action becomes
\begin{equation} \begin{aligned} S&=\int d^2x\Big(-D_{\mu}\overline{\phi} D^{\mu}\phi -\frac{e'^2(|\phi|^2-r)^2}{2}\Big)\\ &=\int d^2x\Big(-D_{\mu}\overline{\phi} D^{\mu}\phi -\frac{e'^2(|\phi|^4-2|\phi|^2r+r^2)}{2}\Big), \end{aligned}\tag{2} \end{equation} There is now a mass term in the action given by $e'^2r|\phi|^2$, and for $r>0$, a Higgs mechanism occurs, spontaneously breaking the gauge symmetry.

I am now interested in knowing how the Higgs mechanism can occur in the GLSM given in equation (1) at the QUANTUM level, where we CANNOT integrate out $D$ using its equation of motion. The following is my attempt: we can find the minimum of the potential energy in (1) to show that the vacuum expectation value $$ \tag{3} \langle D\rangle = -e'^2(\langle|\phi|^2\rangle-r)$$ According to Witten in http://arxiv.org/abs/hep-th/9301042 (page 21), this is only true at tree level, and there are further corrections at least at one-loop. Nevertheless, the leading term is that given in (3), and we should be able to plug this into the quantum EFFECTIVE action, whereby we can show that the Higgs mechanism occurs for $r>0$, as in (2). Is this correct?

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  • $\begingroup$ I should add that my main worry is that the quantum effects could be too large such that the derivation in (2) is modified substantially, and the Higgs mechanism does not occur. Witten also mentions on page 22, that if quantum corrections are large, then crucial properties cannot be read off from the classical Lagrangian. $\endgroup$ – Meer Ashwinkumar May 26 '16 at 7:15

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