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For a $(3+1)$-dimensional generalization of the Schwinger model, we computethe interaction energy between two test charges. The result shows that thestatic potential profile contains a linear term leading to the confinement ofprobe charges, exactly as in the original model in two dimensions. We furthershow that the same 4-dimensional model also appears as one version of the $ B\wedge F$ models in $(3+1)$ dimensions under dualization of Stueckelberg-likemassive gauge theories. Interestingly, this particular model is characterizedby the mixing between a $U(1)$ potential and an Abelian $3$-form field of thetype that appears in the topological sector of QCD.

The bosonized version of the Schwinger model in four dimensions: a blueprint for confinement?