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The gradient-flow coupling of three-and four-dimensional QED

Lars Georg, Robert V. Harlander, Robert H. Mason

Abstract

We evaluate the QED coupling in the gradient-flow scheme in three and four space-time dimensions. Our general result applies to any theory with a U(1) gauge field coupled to arbitary other fields via arbitrary interactions. As an example, we consider QED with $n_\text{f}$ flavors in three and four space-time dimensions and evaluate the corresponding $β$ functions. In four dimensions, we find that the perturbative expansion of the $β$ function behaves much better than the corresponding expression in QCD. In three dimensions, we recover both the ultraviolet as well as the infrared fixed points of the QED coupling in the large-$n_\text{f}$ limit.

The gradient-flow coupling of three-and four-dimensional QED

Abstract

We evaluate the QED coupling in the gradient-flow scheme in three and four space-time dimensions. Our general result applies to any theory with a U(1) gauge field coupled to arbitary other fields via arbitrary interactions. As an example, we consider QED with flavors in three and four space-time dimensions and evaluate the corresponding functions. In four dimensions, we find that the perturbative expansion of the function behaves much better than the corresponding expression in QCD. In three dimensions, we recover both the ultraviolet as well as the infrared fixed points of the QED coupling in the large- limit.
Paper Structure (10 sections, 53 equations, 4 figures)

This paper contains 10 sections, 53 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Flowed QED
  3. Four-dimensional QED
  4. Three-dimensional QED
  5. Naive perturbation theory
  6. Large-$n_\text{f}$ limit
  7. Conclusions
  8. Acknowledgments.
  9. Long- and short-flow-time limits of $E(t)$ in QED$_3$
  10. QED$_4$ gradient-flow coupling including QCD corrections

Figures (4)

  • Figure 1: The conversion of the coupling from the $\overline{\text{\scalefont{.9}MS}}$ scheme to the .9GF scheme (a,c) and back (b,d) for .9QED$_4$. Here we used $c_{t\mu}=e^{-\gamma_E/2}/\sqrt{2}$ and $n_\text{f}=1$ (a,b) or $n_\text{f}=5$ (c,d).
  • Figure 2: Comparison of the $\beta$ function in the $\text{GF}$ scheme (a,c) to the $\overline{\text{\scalefont{.9}MS}}$ scheme (b,d) for $n_\text{f}=1$ (a,b) and $n_\text{f}=5$ (c,d).
  • Figure 3: The conversion between the GF and $\overline{\text{\scalefont{.9}MS}}$ scheme together with the limiting cases for small/large $\hat{a}_\text{e}$.
  • Figure 4: The comparison of the $\beta$ function to the limiting cases: (a) small $\hat{a}^\text{\scalefont{.9}GF}_\text{e}$; (b) large $\hat{a}^\text{\scalefont{.9}GF}_\text{e}$.