Abstract
The Sommerfield model with a massive vector field coupled to a massless fermion in 1+1 dimensions is an exactly solvable analog of a Bank-Zaks model. The “physics” of the model comprises a massive boson and an unparticle sector that survives at low energy as a conformal field theory (Thirring model). We analyze generalizations of the Sommerfield model, and the corresponding generalizations of the Schwinger model, with more massless fermions and more vector fields.
Highlights
We will begin with a quick review of the Sommerfield model to set notation.2 The Sommerfield Lagrangian is LS
The Sommerfield model with a massive vector field coupled to a massless fermion in 1+1 dimensions is an exactly solvable analog of a Bank-Zaks model
The “physics” of the model comprises a massive boson and an unparticle sector that survives at low energy as a conformal field theory (Thirring model)
Summary
As in the Sommerfield model, to solve the model, it is convenient to decompose Aμ into scalar and pseudo-scalar fields. Where O is a real orthogonal nA × nA matrix We can use this freedom to take the physical vector boson mass matrix M to be diagonal and write [M ]jk = mj δjk [M02]jk m2j δjk [e]j [e]k π (3.15). The equality corresponds to the interesting case in which one of the eigenvalues of M0 goes to zero This is the “Schwinger point” [9] at which one linear combination of the vectors has zero mass so there is a gauge invariance. From (3.16), because the V are all massless, the mass matrix M0 only appears in the combination eT M0−2 e which from (3.25) is eT M0−2 e = eT M −1. Each of the massless fermions generates a contribution to the vector boson mass matrix like that in (3.7),. In correlators involving a single Aμj , a similar simplifcation obtains because the eT · M0−1 in (4.5) is multiplied by and M0−1 from (4.6) and correlator involves only the combination eT M0−2
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