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I am interested in understanding the 1+1d Abelian Higgs model. Namely, the theory with action

$$S=\int d^2x \left[-\frac{1}{4e^2} F_{\mu\nu}F^{\mu\nu}-D_\mu \phi^* D^\mu \phi-V(|\phi|)+\frac{\theta}{4\pi}\epsilon^{\mu\nu}F_{\mu\nu}\right],$$

where $V$ is such that $|\phi|$ has a non-zero vev. I have read Coleman's lecture in Uses of Instantons, and I understand the following points:

  • perturbatively, we would find a massive vector boson and a massive Higgs field, so the spectrum would consist of two massive particles, a vector meson and a scalar meson. This is a familiar statement from higher-dimensional theories.

  • in 2d there are instanton corrections. We can calculate their contribution to the potential between two particles of charges $q$ and $-q$ separated by a distance $L$, and find $V(L)=LKe^{-S_0}[\cos\theta-\cos(\theta+2\pi q/e)]$. This means that particles of fractional charge are confined: if we try to separate them, we get a constant force pulling them together. However, this potential vanishes if $q$ is an integer multiple of $e$.

Coleman then asserts that this means the spectrum of the theory is the same as in the unbroken phase, in which particles are neutral bound states of the charged mesons.

This I do not understand at all. Can someone explain how? More precisely:

  • how does the instanton computation imply anything regarding the naive perturbative spectrum, which anyway contained only neutral particles?

  • and even if we had some charged particles, as long as $\phi$ has integer charge, doesn't the instanton contribution to the potential vanish for those?

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