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Wetterich Equation

physicsparticle physicsquantum physicsquantum field theoryrenormalizationcetztikz

The Wetterich eqn. is a non-linear functional integro-differential equation of one-loop structure that determines the scale-dependence of the flowing action Γk\Gamma_k in terms of fluctuations of the fully-dressed regularized propagator [Γk(2)+Rk]−1[\Gamma_k^{(2)} + R_k]^{-1}. It admits a simple diagrammatic representation as a one-loop equation as shown in this diagram.


Wetterich Equation

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  Code

  LaTeX

wetterich-equation.tex (24 lines)

\documentclass[tikz,border={0 3}]{standalone}

\usetikzlibrary{patterns,decorations.markings}

\def\lrad{1}
\def\mrad{0.175*\lrad}
\def\srad{0.15*\lrad}

\begin{document}
\begin{tikzpicture}[
    pin edge={shorten <=5*\lrad},
    cross/.style={fill=white,path picture={\draw[black] (path picture bounding box.south east) -- (path picture bounding box.north west) (path picture bounding box.south west) -- (path picture bounding box.north east);}},
    dressed/.style={fill=white,postaction={pattern=north east lines}},
    momentum/.style 2 args={->,semithick,yshift=5pt,shorten >=5pt,shorten <=5pt},
    loop/.style 2 args={thick,decoration={markings,mark=at position {#1} with {\arrow{>},\node[anchor=\pgfdecoratedangle-90,font=\footnotesize,] {$p_{#2}$};}},postaction={decorate}}
  ]

  \draw[loop/.list={{0.25}{1},{0.75}{2}}] (0,0) circle (\lrad);
  \draw[cross] (-\lrad,0) circle (\srad) node[left=2pt] {$\partial_k R_{k,ij}(p_1,p_2)$};
  \draw[dressed] (\lrad,0) circle (\mrad) node[right=2pt] {$\bigl[\Gamma_k^{(2)} + R_k\bigr]_{ji}^{-1}(p_2,p_1)$};

\end{tikzpicture}
\end{document}

  Typst

wetterich-equation.typ (81 lines)

#import "@preview/cetz:0.3.4": canvas, draw
#import "@preview/modpattern:0.1.0": modpattern
#import draw: line, content, circle, mark

#set page(width: auto, height: auto, margin: 8pt)

// Define styles and constants
#let radius = 1.25 // \lrad in original
#let med-rad = 0.175 * radius // \mrad
#let small-rad = 0.15 * radius // \srad

// Create hatched pattern for vertices
#let hatched = modpattern(
  (.1cm, .1cm),
  std.line(start: (0%, 100%), end: (100%, 0%), stroke: 0.5pt),
  background: white,
)

// Helper functions
#let cross(pos, label: none, rel-label: (0, 0), name: none, ..rest) = {
  let txt = text(size: 16pt, baseline: -0.2pt)[$times.circle$]
  content(pos, txt, stroke: none, fill: white, frame: "circle", padding: -2.75pt, name: name)
  if label != none {
    let label-pos = if rel-label != none { (rel: rel-label, to: pos) } else { pos }
    content(label-pos, $#label$, ..rest)
  }
}

#let dressed-vertex(pos, label: none, rel-label: none, name: none, radius: med-rad, ..rest) = {
  circle(pos, radius: radius, fill: hatched, name: name, stroke: 0.5pt)
  if label != none {
    let label-pos = if rel-label != none { (rel: rel-label, to: pos) } else { pos }
    content(label-pos, $#label$, ..rest)
  }
}

#canvas({
  // Main loop
  circle((0, 0), radius: radius, stroke: 1pt, name: "loop")

  // Add momentum arrows and labels around loop
  for (ii, pos) in ((1, "0.25"), (2, "0.75")) {
    let angle = float(pos) * 360
    let label-angle = (angle - 3) * 1deg

    // Add momentum labels
    let rel-pos = (0.75 * radius * calc.cos(label-angle), 0.75 * radius * calc.sin(label-angle))
    content(
      (rel: rel-pos, to: "loop"),
      $p_#ii$,
      size: 8pt,
    )

    // Add arrow marks
    mark(
      symbol: "stealth",
      (name: "loop", anchor: angle * 1deg),
      (name: "loop", anchor: (angle + 1) * 1deg),
      ..(width: .25, length: .15, stroke: .7pt, angle: 60deg, scale: .7, fill: black),
    )
  }

  // Add regulator cross with label
  cross(
    (-radius, 0),
    label: $partial_k R_(k,i j)(p_1,p_2)$,
    rel-label: (-0.25, 0),
    name: "regulator",
    anchor: "east",
  )

  // Add dressed vertex with inverse propagator
  dressed-vertex(
    (radius, 0),
    label: $[Gamma_k^((2)) + R_k]_(j i)^(-1)(p_2,p_1)$,
    rel-label: (0.25, 0),
    name: "vertex",
    anchor: "west",
  )
})