The Schwinger model revisited

We evaluate Polyakov loops and string tension in two-dimensional QED with both massless and massive N -flavor fermions at zero and finite temperature. External charges, or external electric fields, induce phases in fermion masses and shift the value of the vacuum angle parameter θ, which in turn alters the chiral condensate. In particular, in the presence of two sources of opposite charges, q and −q, the shift in θ is 2π(q/e) independent of N . The string tension has a cusp singularity at θ = ±π for N ≥ 2 and is proportional to m 2N/(N +1) at T = 0.Two-dimensional QED, the Schwinger model, with massive N -flavor fermions resembles fourdimensional QCD in various aspects, including confinement, chiral condensates, and θ vacua [1]- [11]. Much progress has been made recently in evaluating chiral condensates and string tension in the massive theory [12]-[16]. In this paper we shall show that the three phenomena, confinement, chiral condensates, and θ vacua, are intimately related to each other. In particular, the string tension in the confining potential is determined by the θ dependence of chiral condensates ψ − ψ .The behavior of the model is distinctively different, depending on whether N = 1 (one-flavor) or N ≥ 2 (multi-flavor), and on whether fermions are massless or massive. The massless (m = 0) theory is exactly solvable. ψ − ψ θ = 0 for N = 1, but ψ − ψ θ = 0 for N ≥ 2. [17,18] In either cases the string tension between two external sources of opposite charge vanishes [4,8,12]. In the massive (m = 0) theory ψ − ψ θ is proportional to m (N −1)/(N +1) cos 2N/(N +1) (θ/N ) at , 13]. For N ≥ 2 the dependence on m is non-analytic. It also has a cusp singularity at θ = ±π. A perturbation theory in fermion masses is not valid at low temperature.

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“…In general a mass-eigenstate field χ α with a mass µ α is related to φ a by an orthogonal transformation χ α = U αa φ a . In (7) we have…”

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confidence: 99%

Confinement and chiral condensates in 2d QED with massive N-flavor fermions

Rodriguez¹,

Hosotani²

1996

Physics Letters B

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“…The mapping here provided relates two local models each with its physical Hilbert space reconstructed from the Wightmann functions of its own polynomial algebra [10,11]. Since the mapping involves non-local functions it should be clear that within the pure BF model there are two Hilbert spaces to be obtained.…”

Section: Discussionmentioning

confidence: 99%

A new picture on the (3+1)D topological mass mechanism

Ventura¹,

Amaral²,

Costa³

et al. 2004

J. Phys. A: Math. Gen.

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We present a class of mappings between the fields of the Cremmer-Sherk and pure BF models in 4D. These mappings are established by two distinct procedures. First a mapping of their actions is produced iteratively resulting in an expansion of the fields of one model in terms of progressively higher derivatives of the other model fields. Secondly an exact mapping is introduced by mapping their quantum correlation functions. The equivalence of both procedures is shown by resorting to the invariance under field scale transformations of the topological action. Related equivalences in 5D and 3D are discussed. A cohomological argument is presented to provide consistency of the iterative mapping.

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“…We have already pointed out that introducing a functional integration over θ renders the extended partition function with the Lagrangian (19) symmetric under the gauge transformation (20). At first sight, this may appear surprising: After all, the fermionic functional measure is precisely not invariant under chiral rotations such as the one of eq.…”

Section: A Toy Model: An Effective Lagrangian For the Schwinger Modelmentioning

confidence: 99%

“…Usually, they are defined as an ensemble of systems carrying integer instanton number, thus exhibiting a non-trivial vacuum structure. At least for the Schwinger model, this is a necessary ingredient in order to have well-defined asymptotic states [19]; in particular, this guarantees the cluster decomposition property for (vector) gauge-invariant objects. The most interesting aspect with respect to the topics we are discussing in the following is the chiral anomaly in the U(1) sector.…”

Section: A Toy Model: An Effective Lagrangian For the Schwinger Modelmentioning

confidence: 99%

Gauge-symmetric approach to effective lagrangians: The η′ meson from QCD

Damgaard

Nielsen²,

Sollacher³

1994

Nuclear Physics B

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We present a general scheme for extracting effective degrees of freedom from an underlying fundamental Lagrangian, through a series of well-defined transformations in the functional integral of the cut-off theory. This is done by introducing collective fields in a gauge-symmetric manner. Through appropriate gauge fixings of this symmetry one can remove long-distance degrees of freedom from the underlying theory, replacing them by the collective fields. Applying this technique to QCD, we set out to extract the long-distance dynamics in the pseudoscalar flavour singlet sector through a gauging (and subsequent gauge fixing) of the U (1) A flavour symmetry which is broken by the anomaly. By this series of exact transformations of a cut-off generating functional for QCD, we arrive at a theory describing the long-distance physics of a pseudoscalar flavour singlet meson coupled to the residual quark-gluon degrees of freedom. As examples of how known low-energy physics can be reproduced in this formulation, we rederive the Witten-Veneziano relation between the η ′ mass and the topological susceptibility, now for any value of the number of colours N c . The resulting effective Lagrangian contains an axial vector field, which shares the relevant features with the Veneziano ghost. This field is responsible for removing the η ′ degree of freedom from the fundamental QCD Lagrangian.

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The Schwinger model revisited

Cited by 43 publications

References 26 publications

Confinement and chiral condensates in 2d QED with massive N-flavor fermions

Confinement and chiral condensates in 2d QED with massive N-flavor fermions

A new picture on the (3+1)D topological mass mechanism

Gauge-symmetric approach to effective lagrangians: The η′ meson from QCD

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