\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{renormalization group flow}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory}

[[!include AQFT and operator algebra contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{ScalingTransformationsRGFlow}{Scaling transformations}\dotfill \pageref*{ScalingTransformationsRGFlow} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[perturbative quantum field theory]] the construction of the [[scattering matrix]] $\mathcal{S}$, hence of the [[interacting field algebra of observables]] for a given [[interaction]] $g S_{int}$ [[perturbation theory|perturbing]] around a given [[free field theory|free field]] [[vacuum]], involves choices of \emph{normalization} of [[time-ordered products]]/[[Feynman diagrams]] (traditionally called \emph{[[renormalization|``re''-normalizations]]}) encoding new [[interactions]] that appear where several of the original interaction vertices defined by $g S_{int}$ coincide.

Whenever a [[group]] $RG$ [[action|acts]] on the space of [[observables]] of the theory such that [[conjugation]] by this action takes [[renormalization scheme|(``re''-)normalization schemes]] into each other, then these choices of [[renormalization|(``re''-)normalization]] are parameterized by -- or ``flow with'' -- the elements of $RG$. This is called \emph{renormalization group flow} (prop. \ref{FlowRenormalizationGroup} below); often called \emph{RG flow}, for short.

The archetypical example here is the [[group]] $RG$ of [[scaling transformations]] on [[Minkowski spacetime]] (def. \ref{ScalingTransformations} below), which induces a [[renormalization group flow]] (prop. \ref{RGFlowScalingTransformations} below) due to the particular nature of the [[Wightman propagator]] resp. [[Feynman propagator]] on [[Minkowski spacetime]] (example \ref{ScalarFieldMassDimensionOnMinkowskiSpacetime} below). In this case the choice of [[renormalization|(``re''-)normalization]] hence ``flows with scale''.

Now the \emph{[[main theorem of perturbative renormalization]] states that (if only the basic [[renormalization condition]] called ``field independence'' is satisfied) any two choices of [[renormalization scheme|(``re''-)normalization schemes]] $\mathcal{S}$ and $\mathcal{S}'$ are related by a unique [[interaction vertex redefinition]] $\mathcal{Z}$, as}

\begin{displaymath}
\mathcal{S}' = \mathcal{S} \circ \mathcal{Z}
  \,.
\end{displaymath}
Applied to a parameterization/flow of renormalization choices by a group $RG$ this hence induces an [[interaction vertex redefinition]] as a function of $RG$. One may think of the shape of the interaction vertices as fixed and only their ([[adiabatic switching|adiabatically switched]]) [[coupling constants]] as changing under such an [[interaction vertex redefinition]], and hence then one has [[coupling constants]] $g_j$ that are parameterized by elements $\rho$ of $RG$:

\begin{displaymath}
\mathcal{Z}_{\rho_{vac}}^\rho
  \;\colon\;
  \{g_j\}
    \mapsto
  \{g_j(\rho)\}
\end{displaymath}
This dependendence is called \emph{running of the coupling constants} under the renormalization group flow (def. \ref{CouplingRunning} below).

One example of [[renormalization group flow]] is that induced by [[scaling transformations]] (prop. \ref{RGFlowScalingTransformations} below). This is the original and main example of the concept (\hyperlink{GellMannLow54}{Gell-Mann \& Low 54})

In this case the [[running of the coupling constants]] may be understood as expressing how ``more'' [[interactions]] (at higher energy/shorter [[wavelength]]) become visible (say to [[experiment]]) as the scale resolution is increased. In this case the dependence of the coupling $g_j(\rho)$ on the parameter $\rho$ happens to be [[differentiable function|differentiable]]; its [[logarithm|logarithmic]] [[derivative]] (denoted ``$\psi$'' in \hyperlink{GellMannLow54}{Gell-Mann \& Low 54}) is known as the \emph{[[beta function]]} (\hyperlink{Callan70}{Callan 70}, \hyperlink{Symanzik70}{Symanzik 70}):

\begin{displaymath}
\beta(g) \coloneqq \rho \frac{\partial g_j}{\partial \rho}
  \,.
\end{displaymath}
Notice that this is related to, but conceptually different from, \emph{[[Polchinski's flow equation]]} in the context of [[Wilsonian RG]].

The [[running of the coupling constants]] is not quite a [[representation]] of the [[renormalization group flow]], but it is a ``twisted'' representation, namely a [[group cocycle|group 1-cocycle]] (prop. \ref{CocycleRunningCoupling} below). For the case of [[scaling transformations]] this may be called the \emph{[[Gell-Mann-Low renormalization cocycle]]} (\hyperlink{BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09}).

For more see at

\begin{itemize}%
\item [[geometry of physics -- A first idea of quantum field theory]] the section \emph{\href{geometry+of+physics+--+A+first+idea+of+quantum+field+theory#Renormalization}{Renormalization}}

\end{itemize}
\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{prop}
\label{FlowRenormalizationGroup}\hypertarget{FlowRenormalizationGroup}{}
\textbf{([[renormalization group flow]])}

Let

\begin{displaymath}
vac
    \;\coloneqq\;
  (E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )
\end{displaymath}
be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to \href{S-matrix#VacuumFree}{this def.}) around which we consider [[interacting field theory|interacting]] [[perturbative QFT]].

Consider a [[group]] $RG$ equipped with an [[action]] on the [[Wick algebra]] of [[off-shell]] [[microcausal polynomial observables]] with formal parameters adjoined (as in \href{S-matrix#FormalParameters}{this def.})

\begin{displaymath}
rg_{(-)}
  \;\colon\;
  RG \times PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g, j ] ]
  \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ \hbar, g, j ] ]
  \,,
\end{displaymath}
hence for each $\rho \in RG$ a [[continuous linear map]] $rg_\rho$ which has an [[inverse]] $rg_\rho^{-1} \in RG$ and is a [[homomorphism]] of the [[Wick algebra]]-product (the [[star product]] $\star_H$ induced by the [[Wightman propagator]] of the given vauum $vac$)

\begin{displaymath}
rg_\rho( A_1 \star_H A_2 )
  \;=\;
  rg_\rho(A_1) \star_H rg_\rho(A_2)
\end{displaymath}
such that the following conditions hold:

\begin{enumerate}%
\item the action preserves the subspace of [[off-shell]] polynomial [[local observables]], hence it [[restriction|restricts]] as

\begin{displaymath}
rg_{(-)}
  \;\colon\;
  RG \times LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g,j\rangle
  \longrightarrow
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g,j\rangle
\end{displaymath}

\item the action respects the [[causal order]] of the spacetime support (\href{A+first+idea+of+quantum+field+theory#SpacetimeSupport}{this def.}) of local observables, in that for $O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]$ we have

\begin{displaymath}
\left(
    supp(O_1)
    \,{\vee\!\!\!\wedge}\,
    supp(O_2)
  \right)
  \phantom{A}
    \Rightarrow
  \phantom{A}
  \left(
    supp(rg_\rho(O_1))
    \,{\vee\!\!\!\wedge}\,
    supp(rg_\rho(O_2))
  \right)
\end{displaymath}
for all $\rho \in RG$.



\end{enumerate}
Then:

The operation of [[conjugation]] by this action on [[observables]] induces an [[action]] on the [[set]] of [[S-matrix]] [[renormalization schemes]] (\href{S-matrix#LagrangianFieldTheoryPerturbativeScattering}{this def.}, \href{S-matrix#calSFunctionIsRenormalizationScheme}{this remark}), in that for

\begin{displaymath}
\mathcal{S}
  \;\colon\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar , g, j] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})( (\hbar) )[ [ g, j] ]
\end{displaymath}
a perturbative [[S-matrix scheme]] around the given [[free field theory|free field]] [[vacuum]] $vac$, also the [[composition|composite]]

\begin{displaymath}
\mathcal{S}^\rho
  \;\coloneqq\;
  rg_\rho \circ \mathcal{S} \circ rg_{\rho}^{-1}
  \;\colon\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar , g, j] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})( (\hbar) )[ [ g, j] ]
\end{displaymath}
is an [[S-matrix]] scheme, for all $\rho \in RG$.

More generally, let

\begin{displaymath}
vac_\rho
    \;\coloneqq\;
  (E_{\text{BV-BRST}}, \mathbf{L}'_\rho, \Delta_{H,\rho} )
\end{displaymath}
be a collection of [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacua]] parameterized by elements $\rho \in RG$, all with the same underlying [[field bundle]]; and consider $rg_\rho$ as above, except that it is not an [[automorphism]] of any [[Wick algebra]], but an [[isomorphism]] between the [[Wick algebra]]-structures on various vacua, in that

\begin{equation}
rg_{\rho}( A_1 \star_{H, \rho^{-1} \rho_{vac}} A_2 )
  \;=\;
  rg_{\rho}(A_1) \star_{H, \rho_{vac}} rg_{\rho}(A_2)
\label{IntertwiningWickProductsActionRG}\end{equation}
for all $\rho, \rho_{vac} \in RG$

Then if

\begin{displaymath}
\{ \mathcal{S}_{\rho} \}_{\rho \in RG}
\end{displaymath}
is a collection of [[S-matrix schemes]], one around each of the [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacua]] $vac_\rho$, it follows that for all pairs of group elements $\rho_{vac}, \rho \in RG$ the [[composition|composite]]

\begin{equation}
\mathcal{S}_{\rho_{vac}}^\rho
  \;\coloneqq\;
  rg_\rho \circ \mathcal{S}_{\rho^{-1}\rho_{vac}} \circ rg_\rho^{-1}
\label{RGConjugateSmatrix}\end{equation}
is an [[S-matrix scheme]] around the vacuum labeled by $\rho_{vac}$.

Since therefore each element $\rho \in RG$ in the [[group]] $RG$ picks a different choice of [[renormalization|normalization]] of the [[S-matrix]] scheme around a given vacuum at $\rho_{vac}$, we call the assignment $\rho \mapsto \mathcal{S}_{\rho_{vac}}^{\rho}$ a \emph{[[renormalization group flow|re-normalization group flow]]}.

\end{prop}
(\hyperlink{BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09, sections 4.2, 5.1}, \hyperlink{Duetsch18}{Dütsch 18, section 3.5.3})

\begin{proof}
It is clear from the definition that each $\mathcal{S}^{\rho}_{\rho_{vac}}$ satisfies the axiom ``perturbation'' (in \href{S-matrix#LagrangianFieldTheoryPerturbativeScattering}{this def.}).

In order to verify the axiom ``[[causal additivity]]'', observe, for convenience, that by \href{S-matrix#CausalFactorizationAlreadyImpliesSMatrix}{this prop.} it is sufficient to check [[causal factorization]].

So consider $O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle$ two local observables whose spacetime support is in [[causal order]].

\begin{displaymath}
supp(O_1)
  \;{\vee\!\!\!\wedge}\;
  supp(O_2)
  \,.
\end{displaymath}
We need to show that the

\begin{displaymath}
\mathcal{S}_{\rho_{vac}}^{\rho}(O_1 + O_2)
  =
  \mathcal{S}_{\rho_{vac}}^\rho(O_1)
    \star_{H,\rho_{vac}}
  \mathcal{S}_{vac_e}^\rho(O_2)
\end{displaymath}
for all $\rho, \rho_{vac} \in RG$.

Using the defining properties of $rg_{(-)}$ and the [[causal factorization]] of $\mathcal{S}_{\rho^{-1}\rho_{vac}}$ we directly compute as follows:

\begin{displaymath}
\begin{aligned}
    \mathcal{S}_{\rho_{vac}}^\rho(O_1 + O_2)
    & =
    rg_\rho
      \circ
    \mathcal{S}_{\rho^{-1} \rho_{vac}}
      \circ
    rg_\rho^{-1}( O_1 + O_2 )
    \\
    & =
    rg_\rho
    \left(
      {\, \atop \,}
      \mathcal{S}_{\rho^{-1}\rho_{vac}}
      \left(
        rg_\rho^{-1}(O_1) + rg_\rho^{-1}(O_2)
      \right)
      {\, \atop \,}
    \right)
    \\
    & =
    rg_\rho
    \left(
      {\, \atop \,}
      \left(
        \mathcal{S}_{\rho^{-1}\rho_{vac}}\left(rg_\rho^{-1}(O_1)\right)
      \right)
        \star_{H, \rho^{-1} \rho_{vac}}
      \left(
        \mathcal{S}_{ \rho^{-1} \rho_{vac} }\left(rg_\rho^{-1}(O_2)\right)
      \right)
      {\, \atop \,}
    \right)
    \\
    & =
    rg_\rho
    \left(
      {\, \atop \,}
      \mathcal{S}_{\rho^{-1} \rho_{vac}}\left(rg_{\rho^{-1}}(O_1)\right)
      {\, \atop \,}
    \right)
    \star_{H, \rho_{vac}}
    rg_\rho
    \left(
      {\, \atop \,}
      \mathcal{S}_{\rho^{-1} \rho_{vac}}\left( rg_\rho^{-1}(O_2)\right)
      {\, \atop \,}
    \right)
    \\
    & =
    \mathcal{S}^\rho_{\rho_{vac}}( O_1 )
      \,
      \star_{H, \rho_{vac}}
      \,
     \mathcal{S}_{\rho_{vac}}^\rho(O_2)
    \,.
  \end{aligned}
\end{displaymath}
\end{proof}
\begin{defn}
\label{CouplingRunning}\hypertarget{CouplingRunning}{}
\textbf{([[running coupling constants]])}

Let

\begin{displaymath}
vac
  \coloneqq
  vac_e
    \;\coloneqq\;
  (E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )
\end{displaymath}
be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to \href{S-matrix#VacuumFree}{this def.}) around which we consider [[interacting field theory|interacting]] [[perturbative QFT]], let $\mathcal{S}$ be an [[S-matrix]] scheme around this vacuum and let $rg_{(-)}$ be a [[renormalization group flow]] according to prop. \ref{FlowRenormalizationGroup}, such that each re-normalized [[S-matrix scheme]] $\mathcal{S}_{vac}^\rho$ satisfies the [[renormalization condition]] ``field independence''.

Then by the [[main theorem of perturbative renormalization]] (\href{Stückelberg-Petermann+renormalization+group#AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition}{this prop.}) there is for every [[pair]] $\rho_1, \rho_2 \in RG$ a unique [[interaction vertex redefinition]]

\begin{displaymath}
\mathcal{Z}_{\rho_{vac}}^{\rho}
  \;\colon\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]
    \longrightarrow
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]
\end{displaymath}
which relates the corresponding two [[S-matrix]] schemes via

\begin{equation}
\mathcal{S}_{\rho_{vac}}^{\rho}
  \;=\;
  \mathcal{S}_{\rho_{vac}}
    \circ
  \mathcal{Z}_{\rho_{vac}}^\rho
  \,.
\label{SMatrixScemesRelatedByRunningFunction}\end{equation}
If one thinks of an [[interaction]] vertex, hence a [[local observable]] $g S_{int}+ j A$, as specified by the ([[adiabatic switching|adiabatically switched]]) [[coupling constants]] $g_j \in C^\infty_{cp}(\Sigma)\langle g \rangle$ multiplying the corresponding [[interaction]] [[Lagrangian densities]] $\mathbf{L}_{int,j} \in \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})$ as

\begin{displaymath}
g S_{int}
  \;=\;
  \underset{j}{\sum} \tau_\Sigma \left( g_j \mathbf{L}_{int,j} \right)
\end{displaymath}
(where $\tau_\Sigma$ denotes [[transgression of variational differential forms]]) then $\mathcal{Z}_{\rho_1}^{\rho_2}$ exhibits a dependency of the ([[adiabatic switching|adiabatically switched]]) [[coupling constants]] $g_j$ of the [[renormalization group flow]] parameterized by $\rho$. The corresponding functions

\begin{displaymath}
\mathcal{Z}_{\rho_{vac}}^{\rho}(g S_{int})
    \;\colon\;
   (g_j)
     \mapsto
   (g_j(\rho))
\end{displaymath}
are then called \emph{[[running coupling constants]]}.

\end{defn}
(\hyperlink{BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09, sections 4.2, 5.1}, \hyperlink{Duetsch18}{Dütsch 18, section 3.5.3})

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{prop}
\label{CocycleRunningCoupling}\hypertarget{CocycleRunningCoupling}{}
\textbf{([[running coupling constants]] are [[group cocycle]] over [[renormalization group flow]])}

Consider [[running coupling constants]]

\begin{displaymath}
\mathcal{Z}_{\rho_{vac}}^{\rho}
   \;\colon\;
   (g_j)
     \mapsto
   (g_j(\rho))
\end{displaymath}
as in def. \ref{CouplingRunning}. Then for all $\rho_{vac}, \rho_1, \rho_2 \in RG$ the following equality is satisfied by the ``running functions'' \eqref{SMatrixScemesRelatedByRunningFunction}:

\begin{displaymath}
\mathcal{Z}_{\rho_{vac}}^{\rho_1 \rho_2}
  \;=\;
  \mathcal{Z}_{\rho_{vac}}^{\rho_1}
    \circ
  \left(
    \sigma_{\rho_1}
      \circ
    \mathcal{Z}_{\rho^{-1} \rho_{vac}}^{\rho_2}
      \circ
    \sigma_{\rho_1}^{-1}
  \right)
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09 (69)}, \hyperlink{Duetsch18}{Dütsch 18, (3.325)})

\begin{proof}
Directly using the definitions, we compute as follows:

\begin{displaymath}
\begin{aligned}
    \mathcal{S}_{\rho_{vac}}
      \circ
    \mathcal{Z}_{\rho_{vac}}^{\rho_1 \rho_2}
    & =
    \mathcal{S}_{\rho_{vac}}^{\rho_1 \rho_2 }
    \\
    & =
    \sigma_{\rho_1}
      \circ
    \underset{
      =
      \mathcal{S}_{\rho_1^{-1} \rho_{vac}}^{\rho_2}
        =
      \mathcal{S}_{\rho_1^{-1} \rho_{vac}}
        \circ
      \mathcal{Z}_{\rho_1^{-1} \rho_vac}^{\rho_2}
    }{
    \underbrace{
      \sigma_{\rho_2}
        \circ
      \mathcal{S}_{\rho_2^{-1}\rho_1^{-1}\rho_{vac}}
        \circ
      \sigma_{\rho_2}^{-1}
    }}
      \circ
    \sigma_{\rho_1}^{-1}
    \\
    & =
    \underset{
      =
      \mathcal{S}_{\rho_{vac}}
        \circ
      \mathcal{Z}_{\rho_{vac}}^{\rho_1}
        \circ
      \sigma_{\rho_1}
    }{
    \underbrace{
      \sigma_{\rho_1}
        \circ
      \mathcal{S}_{\rho_1^{-1} \rho_{vac}}
        \circ
      \overset{ = id }{
        \overbrace{
          \sigma_{\rho_1}^{-1}
            \circ
          \sigma_{\rho_1}
        }
    }
    }}
      \circ
      \mathcal{Z}_{\rho_1^{-1} \rho_{vac}}^{\rho_2}
      \circ
      \sigma_{\rho_1}^{-1}
    \\
    & =
    \mathcal{S}_{\rho_{vac}}
      \circ
    \mathcal{Z}_{\rho_{vac}}^{\rho_1}
      \circ
    \underbrace{
      \sigma_{\rho_1}
        \circ
      \mathcal{Z}_{\rho_1^{-1} \rho_{vac}}^{\rho_2}
        \circ
      \sigma_{\rho_1}^{-1}
    }
  \end{aligned}
\end{displaymath}
This demonstrates the equation between vertex redefinitions to be shown after [[composition]] with an S-matrix scheme. But by the uniqueness-clause in the [[main theorem of perturbative renormalization]] the composition operation $\mathcal{S}_{\rho_{vac}} \circ (-)$ as a function from [[vertex redefinitions]] to S-matrix schemes is [[injective function|injective]]. This implies the equation itself.

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{ScalingTransformationsRGFlow}{}\subsubsection*{{Scaling transformations}}\label{ScalingTransformationsRGFlow}

We discuss (prop. \ref{RGFlowScalingTransformations} below) that, if the field species involved have well-defined [[mass dimension]] (example \ref{ScalarFieldMassDimensionOnMinkowskiSpacetime} below) then [[scaling transformations]] on [[Minkowski spacetime]] (example \ref{ScalingTransformations} below) induce a [[renormalization group flow]] (def. \ref{FlowRenormalizationGroup}). This is the original and main example of [[renormalization group flows]] (\hyperlink{GellMannLow54}{Gell-Mann\& Low 54}).

\begin{example}
\label{ScalingTransformations}\hypertarget{ScalingTransformations}{}
\textbf{([[scaling transformations]] and [[mass dimension]])}

Let

\begin{displaymath}
E \overset{fb}{\longrightarrow} \Sigma
\end{displaymath}
be a [[field bundle]] which is a [[trivial vector bundle]] over [[Minkowski spacetime]] $\Sigma = \mathbb{R}^{p,1} \simeq_{\mathbb{R}} \mathbb{R}^{p+1}$.

For $\rho \in (0,\infty) \subset \mathbb{R}$ a [[positive real number]], write

\begin{displaymath}
\itexarray{
    \Sigma
      &\overset{\rho}{\longrightarrow}&
    \Sigma
    \\
    x &\mapsto& \rho x
  }
\end{displaymath}
for the operation of multiplication by $\rho$ using the [[real vector space]]-[[structure]] of the [[Cartesian space]] $\mathbb{R}^{p+1}$ underlying [[Minkowski spacetime]].

By [[pullback of differential forms|pullback]] this acts on [[field histories]] ([[sections]] of the [[field bundle]]) via

\begin{displaymath}
\itexarray{
    \Gamma_\Sigma(E)
      &\overset{\rho^\ast}{\longrightarrow}&
    \Gamma_\Sigma(E)
    \\
    \Phi &\mapsto& \Phi(\rho(-))
  }
  \,.
\end{displaymath}
Let then

\begin{displaymath}
\rho
    \mapsto
  vac_\rho
    \;\coloneqq\;
  (E_{\text{BV-BRST}}, \mathbf{L}'_{\rho}, \Delta_{H,\rho} )
\end{displaymath}
be a 1-parameter collection of [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacua]] on that field bundle, according to \href{S-matrix#VacuumFree}{this def.}, and consider a decomposition into a set $Spec$ of field species (\href{S-matrix#VerticesAndFieldSpecies}{this def.}) such that for each $sp \in Spec$ the collection of [[Feynman propagators]] $\Delta_{F,\rho,sp}$ for that species \emph{scales homogeneously} in that there exists

\begin{displaymath}
dim(sp) \in \mathbb{R}
\end{displaymath}
such that for all $\rho$ we have (using [[generalized functions]]-notation)

\begin{equation}
\rho^{ 2 dim(sp) }  \Delta_{F, 1/\rho, sp}( \rho x )
  \;=\;
  \Delta_{F,sp, \rho = 1}(x)
  \,.
\label{FeynmanPropagatorScalingBehaviour}\end{equation}
Typically $\rho$ rescales a [[mass]] parameter, in which case $dim(sp)$ is also called the \emph{[[mass dimension]]} of the field species $sp$.

Let finally

\begin{displaymath}
\itexarray{
    PolyObs(E)
    &
      \overset{
        \sigma_\rho
      }{\longrightarrow}
    &
    PolyObs(E)
    \\
    \mathbf{\Phi}_{sp}^a(x)
      &\mapsto&
    \rho^{- dim(sp)} \mathbf{\Phi}^a( \rho^{-1} x )
  }
\end{displaymath}
be the [[function]] on [[off-shell]] [[polynomial observables]] given on [[field observables]] $\mathbf{Phi}^a(x)$ by [[pullback of differential forms|pullback]] along $\rho^{-1}$ followed by multiplication by $\rho$ taken to the negative power of the [[mass dimension]], and extended from there to all [[polynomial observables]] as an [[associative algebra|algebra]] [[homomorphism]].

This constitutes an [[action]] of the [[group]]

\begin{displaymath}
RG
    \coloneqq
  \left(
    \mathbb{R}_+, \cdot
  \right)
\end{displaymath}
of [[positive real numbers]] (under [[multiplication]]) on [[polynomial observables]], called the group of \emph{[[scaling transformations]]} for the given choice of field species and [[mass]] parameters.

\end{example}
(\hyperlink{Duetsch18}{Dütsch 18, def. 3.19})

\begin{example}
\label{ScalarFieldMassDimensionOnMinkowskiSpacetime}\hypertarget{ScalarFieldMassDimensionOnMinkowskiSpacetime}{}
\textbf{([[mass dimension]] of [[scalar field]])}

Consider the [[Feynman propagator]] $\Delta_{F,m}$ of the [[free field theory|free]] [[real scalar field]] on [[Minkowski spacetime]] $\Sigma = \mathbb{R}^{p,1}$ for [[mass]] parameter $m \in (0,\infty)$; a [[Green function]] for the [[Klein-Gordon equation]].

Let the group $RG \coloneqq (\mathbb{R}_+, \cdots)$ of [[scaling transformations]] $\rho \in \mathbb{R}_+$ on [[Minkowski spacetime]] (def. \ref{ScalingTransformations}) act on the mass parameter by inverse multiplication

\begin{displaymath}
(\rho , \Delta_{F,m})
    \mapsto
  \Delta_{F,\rho^{-1}m}(\rho (-))
  \,.
\end{displaymath}
Then we have

\begin{displaymath}
\Delta_{F,\rho^{-1}m}(\rho (-))
  \;=\;
  \rho^{-(p+1) + 2}
  \Delta_{F,1}(x)
\end{displaymath}
and hence the corresponding [[mass dimension]] (def. \ref{ScalingTransformations}) of the [[real scalar field]] on $\mathbb{R}^{p,1}$ is

\begin{displaymath}
dim(\text{scalar field})
  =
  (p+1)/2 - 1
  \,.
\end{displaymath}
\end{example}
\begin{proof}
By (\href{Feynman+propagator#FeynmanPropagatorAsACauchyPrincipalvalue}{this prop.}) the [[Feynman propagator]] in question is given by the [[Cauchy principal value]]-formula (in [[generalized function]]-notation)

\begin{displaymath}
\begin{aligned}
  \Delta_{F,m}(x)
  & =
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{+i}{(2\pi)^{p+1}}
  \int
  \int_{-\infty}^\infty
  \frac{
     e^{i k_\mu x^\mu}
  }{
    - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
  }
  \, d k_0 \, d^p \vec k
  \,.
  \end{aligned}
\end{displaymath}
By applying [[change of integration variables]] $k \mapsto \rho^{-1} k$ in the [[Fourier transform of distributions|Fourier transform]] this becomes

\begin{displaymath}
\begin{aligned}
  \Delta_{F,\rho^{-1}m}(\rho x)
  & =
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{+i}{(2\pi)^{p+1}}
  \int
  \int_{-\infty}^\infty
  \frac{
     e^{i k_\mu \rho x^\mu}
  }{
    - k_\mu k^\mu - \left( \rho^{-1} \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
  }
  \, d k_0 \, d^p \vec k
  \\
  & =
  \rho^{-(p+1)}
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{+i}{(2\pi)^{p+1}}
  \int
  \int_{-\infty}^\infty
  \frac{
     e^{i k_\mu x^\mu}
  }{
    - \rho^{-2} k_\mu k^\mu - \rho^{-2} \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
  }
  \, d k_0 \, d^p \vec k
  \\
  & =
  \rho^{-(p+1)+2}
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{+i}{(2\pi)^{p+1}}
  \int
  \int_{-\infty}^\infty
  \frac{
     e^{i k_\mu x^\mu}
  }{
    - k_\mu k^\mu -  \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
  }
  \, d k_0 \, d^p \vec k
  \\
  & =
  \rho^{-(p+1) + 2}
  \Delta_{F,m}(x)
  \end{aligned}
\end{displaymath}
\end{proof}
\begin{prop}
\label{RGFlowScalingTransformations}\hypertarget{RGFlowScalingTransformations}{}
\textbf{([[scaling transformations]] are [[renormalization group flow]])}

Let

\begin{displaymath}
vac \coloneqq vac_m
  \coloneqq (E_{\text{BV-BRST}}, \mathbf{L}', \Delta_{H,m})
\end{displaymath}
be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacua]] on that field bundle, according to \href{S-matrix#VacuumFree}{this def.} equipped with a decomposition into a set $Spec$ of field species (\href{S-matrix#VerticesAndFieldSpecies}{this def.}) such that for each $sp \in Spec$ the collection of [[Feynman propagators]] the corresponding field species has a well-defined [[mass dimension]] $dim(sp)$ (def. \ref{ScalingTransformations})

Then the [[action]] of the [[group]] $RG \coloneqq (\mathbb{R}_+, \cdot)$ of [[scaling transformations]] (def. \ref{ScalingTransformations}) is a [[renormalization group flow]] in the sense of \href{renormalization+group+flow#FlowRenormalizationGroup}{this prop.}.

\end{prop}
(\hyperlink{Duetsch18}{Dütsch 18, exercise 3.20})

\begin{proof}
It is clear that rescaling preserves [[causal order]] and the [[renormalization condition]] of ``field indepencen''.

The condition we need to check is that for $A_1, A_2 \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]$ two [[microcausal polynomial observables]] we have for any $\rho, \rho_{vac} \in \mathbb{R}_+$ that

\begin{displaymath}
\sigma_\rho
  \left(
    A_1 \star_{H, \rho^{-1} \rho_{vac} c} A_2
  \right)
  \;=\;
  \sigma_\rho(A_1)
    \star_{H,\rho_{vac}}
  \sigma_\rho(A_2)
  \,.
\end{displaymath}
By the assumption of decomposition into free field species $sp \in Spec$, it is sufficient to check this for each species $\Delta_{H,sp}$. Moreover, by the nature of the [[star product]] on [[polynomial observables]], which is given by iterated contractions with the [[Wightman propagator]], it is sufficient to check this for one such contraction.

Observe that the scaling behaviour of the [[Wightman propagator]] $\Delta_{H,m}$ is the same as the behaviour \eqref{FeynmanPropagatorScalingBehaviour} of the correspponding [[Feynman propagator]]. With this we directly compute as follows:

\begin{displaymath}
\begin{aligned}
    \sigma_\rho (\mathbf{\Phi}(x))
     \star_{F, \rho_{vac} m}
    \sigma_\rho (\mathbf{\Phi}(y)
    & =
    \rho^{-2 dim }
    \mathbf{\Phi}(\rho^{-1} x)
      \star_{F, \rho_{vac} m}
    \mathbf{\Phi}(\rho^{-1} y)
    \\
    & =
    \rho^{-2 dim }
    \Delta_{F, \rho_{vac} m}(\rho^{-1}(x-y))
    \\
    & =
    \Delta_{F, \rho^{-1}\rho_{vac}m }(x,y) \mathbf{1}
    \\
    & =
    rg_{\rho}\left(
     \Delta_{F, \rho^{-1}\rho_{vac}m }(x,y) \mathbf{1}
    \right)
    \\
    & =
    rg_{\rho}
    \left(
      \mathbf{\Phi}(x)
        \star_{F, \rho^{-1} \rho_{vac} m}
      \mathbf{\Phi}(y)
    \right)
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[gauge coupling unification]]


\item [[threshold correction]]


\item [[Wilsonian RG]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The original informal discussion for RG-flow along [[scaling transformations]] is due to

\begin{itemize}%
\item [[Murray Gell-Mann]] and F. E. Low, \emph{Quantum Electrodynamics at Small Distances}, Phys. Rev. 95 (5) (1954), 1300–1312 (\href{http://www.fafnir.phyast.pitt.edu/py3765/GellManLow.pdf}{pdf})


\item [[Curtis Callan]], \emph{Broken Scale Invariance in Scalar Field Theory}, Phys. Rev. D 2, 1541, 1970 (\href{https://doi.org/10.1103/PhysRevD.2.1541}{doi:10.1103/PhysRevD.2.1541})


\item [[Kurt Symanzik]], \emph{Small distance behaviour in field theory and power counting}, Communications in Mathematical Physics. 18 (3): 227–246 (\href{https://doi.org/10.1007/BF01649434}{doi:10.1007/BF01649434})



\end{itemize}
Formulation in the rigorous context of [[causal perturbation theory]]/[[pAQFT]], via the [[main theorem of perturbative renormalization]], is due to

\begin{itemize}%
\item [[Romeo Brunetti]], [[Michael Dütsch]], [[Klaus Fredenhagen]], \emph{Perturbative Algebraic Quantum Field Theory and the Renormalization Groups}, Adv. Theor. Math. Physics 13 (2009), 1541-1599 (\href{https://arxiv.org/abs/0901.2038}{arXiv:0901.2038})

\end{itemize}
reviewed in

\begin{itemize}%
\item [[Michael Dütsch]], section 3.5.3 of \emph{[[From classical field theory to perturbative quantum field theory]]}, 2018

\end{itemize}
In the context of [[factorization algebras]], this is given by the book [[Renormalization and Effective Field Theory]]

[[!redirects renormalization group flows]]

[[!redirects rg-flow]] [[!redirects rg-flows]]

[[!redirects rg flow]] [[!redirects rg flows]]

[[!redirects RG-flow]] [[!redirects RG-flows]]

[[!redirects RG flow]] [[!redirects RG flows]]

[[!redirects Gell-Mann-Low renormalization cocycle]] [[!redirects Gell-Mann-Low renormalization cocycles]]

[[!redirects Gell-Mann-Low cocycle]] [[!redirects Gell-Mann-Low cocycles]]

[[!redirects Gell-Mann-Low renormalization group]] [[!redirects Gell-Mann-Low renormalization groups]]

[[!redirects mass dimension]] [[!redirects mass dimensions]]

[[!redirects running coupling constant]] [[!redirects running coupling constants]]

[[!redirects running of the coupling constant]] [[!redirects running of the coupling constants]]



\end{document}