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Charged Scalar Self-Mass during Inflation

E. O. Kahya, R. P. Woodard

TL;DR

This paper computes the one-loop self-mass $M^2(x;x')$ for a charged massless, minimally coupled scalar in a locally de Sitter background using two gauges: a noninvariant Feynman-type gauge and the de Sitter invariant Allen–Jacobson gauge. It presents a complete renormalization, detailing contributions from 4-point and two 3-point interactions, and shows that in the simple gauge the renormalized $M^2(x;x')$ contains local and nonlocal pieces that describe how inflationary particle production alters the scalar’s effective dynamics. The conformally coupled scalar serves as a consistency check, yielding a manifestly de Sitter invariant result in both gauges, confirming that the noninvariant gauge does not introduce physical de Sitter breaking at one loop. The Allen–Jacobson gauge, while de Sitter invariant, produces additional finite terms and gauge-dependent off-shell differences, prompting discussions about potential anti-sources at antipodal points to restore invariance and highlighting implications for stochastic formulations of SQED during inflation.

Abstract

We compute the one loop self-mass of a charged massless, minimally coupled scalar in a locally de Sitter background geometry. The computation is done in two different gauges: the noninvariant generalization of Feynman gauge which gives the simplest expression for the photon propagator and the de Sitter invariant gauge of Allen and Jacobson. In each case dimensional regularization is employed and fully renormalized results are obtained. By using our result in the linearized, effective field equations one can infer how the scalar responds to the dielectric medium produced by inflationary particle production. We also work out the result for a conformally coupled scalar. Although the conformally coupled case is of no great physical interest the fact that we obtain a manifestly de Sitter invariant form for its self-mass-squared establishes that our noninvariant gauge introduces no physical breaking of de Sitter invariance at one loop order.

Charged Scalar Self-Mass during Inflation

TL;DR

This paper computes the one-loop self-mass for a charged massless, minimally coupled scalar in a locally de Sitter background using two gauges: a noninvariant Feynman-type gauge and the de Sitter invariant Allen–Jacobson gauge. It presents a complete renormalization, detailing contributions from 4-point and two 3-point interactions, and shows that in the simple gauge the renormalized contains local and nonlocal pieces that describe how inflationary particle production alters the scalar’s effective dynamics. The conformally coupled scalar serves as a consistency check, yielding a manifestly de Sitter invariant result in both gauges, confirming that the noninvariant gauge does not introduce physical de Sitter breaking at one loop. The Allen–Jacobson gauge, while de Sitter invariant, produces additional finite terms and gauge-dependent off-shell differences, prompting discussions about potential anti-sources at antipodal points to restore invariance and highlighting implications for stochastic formulations of SQED during inflation.

Abstract

We compute the one loop self-mass of a charged massless, minimally coupled scalar in a locally de Sitter background geometry. The computation is done in two different gauges: the noninvariant generalization of Feynman gauge which gives the simplest expression for the photon propagator and the de Sitter invariant gauge of Allen and Jacobson. In each case dimensional regularization is employed and fully renormalized results are obtained. By using our result in the linearized, effective field equations one can infer how the scalar responds to the dielectric medium produced by inflationary particle production. We also work out the result for a conformally coupled scalar. Although the conformally coupled case is of no great physical interest the fact that we obtain a manifestly de Sitter invariant form for its self-mass-squared establishes that our noninvariant gauge introduces no physical breaking of de Sitter invariance at one loop order.

Paper Structure

This paper contains 12 sections, 147 equations.

Table of Contents

  1. Introduction
  2. Feynman Rules
  3. $M^2(x;x')$ in the Simplest Gauge
  4. Contributions from the 4-Point Interaction
  5. Contributions from Two 3-Point Interactions
  6. Contributions from $i\Delta_{\rm cf} \times i\Delta_{\rm cf}$
  7. Contributions from $i\Delta_{\rm cf} \times i\delta\!\Delta_A$
  8. Contributions from $i\delta\!\Delta_B \times i\Delta_{\rm cf}$
  9. Contributions from $i\delta\!\Delta_C \times i\Delta_{\rm cf}$
  10. Renormalization
  11. $M^2(x;x')$ in Allen-Jacobson Gauge
  12. Discussion