Four-fermion deformations of the massless Schwinger model and confinement

Aleksey Cherman, Theodore Jacobson, Mikhail Shifman, Mithat Ünsal, Arkady Vainshtein

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3 Scopus citations


We consider the massless charge-N Schwinger model and its deformation with two four-fermion operators. Without the deformations, this model exhibits chiral symmetry breaking without confinement. It is usually asserted that the massless Schwinger model is always deconfined and a string tension emerges only when a mass for the fermion field is turned on. We show that in the presence of these four-fermion operators, the massless theory can in fact confine. One of the four-fermion deformations is chirally neutral, and is a marginal deformation. The other operator can be relevant or irrelevant, and respects a ℤ2 subgroup of chiral symmetry for even N, hence forbidding a mass term. When it is relevant, even the exactly massless theory exhibits both confinement and spontaneous chiral symmetry breaking. The construction is analogous to QCD(adj) in 2d. While the theory without four-fermion deformations is deconfined, the theory with these deformations is generically in a confining phase. We study the model on ℝ2 using bosonization, and also analyze the mechanism of confinement on ℝ × S1, where we find that confinement is driven by fractional instantons.

Original languageEnglish (US)
Article number87
JournalJournal of High Energy Physics
Issue number1
StatePublished - Jan 2023

Bibliographical note

Publisher Copyright:
© 2023, The Author(s).


  • Anomalies in Field and String Theories
  • Confinement
  • Wilson, ’t Hooft and Polyakov loops


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