Asymptotic limits and sum rules for the quark propagator

Abstract

For the structure functions of the quark propagator, the asymptotic behavior is obtained for general, linear, covariant gauges, and in all directions of the complex k 2 -plane. Asymptotic freedom is assumed. Corresponding previous results for the gauge field propagator are important in the derivation. Except for coefficients, the leading asymptotic terms are determined by one-loop or by two-loop information, and are gauge independent. Various sum rules are derived.

Analytic properties of the Landau gauge gluon and quark propagators

“…−→ 0 in all directions of the complex p 2 -plane [77]. Additionally, the theory of complex functions tells us that if σ v (p 2 ) and σ s (p 2 ) are not constant, they cannot be analytic over the whole complex plane: non-constant, entire functions which are analytic at all finite points in the complex plane are already excluded by the asymptotic properties of the propagator functions.…”

“…The most rigorous constraint on the non-perturbative quark propagator is that it must reduce to a free fermion propagator at large momenta because of asymptotic freedom. This entails that the propagator functions, σ s,v (p 2 ) |p 2 |→∞ −→ 0 in all directions of the complex p 2plane [77]. Additionally, the theory of complex functions tells us that if σ v (p 2 ) and σ s (p 2 ) are not constant, they cannot be analytic over the whole complex plane: non-constant, entire functions which are analytic at all finite points in the complex plane are already excluded by the asymptotic properties of the propagator functions.…”

Pion observables calculated in Minkowski and Euclidean spaces from Ansatz quark propagators

“…The first of the above constraints follows from the consideration of the large-momentum limit of σ V (x); the second one arises from the requirement that M (x) must vanish for large spacelike real momenta. 1 The Ansatz (6) guarantees that the quark dressing functions σ S,V (z) → 0 for all |z| → ∞ in the complex z plane [35]. For the given set of parameters the functions x → A(−x) and x → B(−x) have poles at x = 0.488784 GeV 2 and x = 2.65383 GeV 2 .…”