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\begin{document}

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\section*{loop order}

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

\hypertarget{algebraic_qunantum_field_theory}{}\paragraph*{{Algebraic Qunantum Field Theory}}\label{algebraic_qunantum_field_theory}

[[!include AQFT and operator algebra contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{RelationToPowersInPlancksConstant}{Relation to powers in Planck's constant}\dotfill \pageref*{RelationToPowersInPlancksConstant} \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 [[scattering amplitudes]] in the [[S-matrix]] are expressed as [[formal power series]] in (the [[coupling constant]] and) in [[Planck's constant]] $\hbar$. This formal power series may be expressed as a formal sum of contributions labeled by [[Feynman diagrams]]. The \emph{loop order} refers to something like the ``number of loops'' of [[edges]] in the [[Feynman diagram]] that contibutes to a given [[scattering amplitude]]. It turns out that the loop order corresponds to the order in $\hbar$ that is contributed by this diagram (see \hyperlink{RelationToPowersInPlancksConstant}{below}).

Therefore contributions of graphs at zero without loops (these are [[trees]], and hence these contributions are referred to as being at ``tree level'') correspond to the limit of [[classical field theory]] with $\hbar \to 0$. Indeed tree level Feynman diagrams yield [[perturbation theory|perturbative]] solutions of the [[classical field theory|classical]] [[equations of motion]] (see \hyperlink{Helling}{Helling}).

Most predictions of the [[standard model of particle physics]] have very good agreement with [[experiment]] already to very low loop order, first or second; inclusion of third loop order is used (at least in [[QCD]]) to constrain theoretical uncertainties of the result (see \hyperlink{Cacciari05}{Cacciari 05, slide 5}, e.g. in [[Higgs field]] computation, see \hyperlink{ADDHM15}{ADDHM 15}). In rare cases higher loop orders are used (for instance in the computation of the [[anomalous magnetic moments]] \hyperlink{AHKN12}{AHKN 12}, but this is not a scattering experiment).

This usefulness of low loop order is fortunate because

\begin{enumerate}%
\item the [[S-matrix]] [[formal power series]] for all [[theory (physics)|theories]] of interest has \emph{vanishing} [[radius of convergence]] (\href{perturbation+theory#Dyson52}{Dyson 52}), hence is at best an [[asymptotic series]] for which the [[sum]] of more than some low order terms is meaningless;


\item the computational effort increases immensely with loop order.



\end{enumerate}
\hypertarget{RelationToPowersInPlancksConstant}{}\subsection*{{Relation to powers in Planck's constant}}\label{RelationToPowersInPlancksConstant}

The following is taken from \emph{[[geometry of physics -- A first idea of quantum field theory]]}. See there for more background.

\begin{prop}
\label{FeynmanDiagramLoopOrder}\hypertarget{FeynmanDiagramLoopOrder}{}
\textbf{([[loop order]] and [[tree level]] of [[Feynman perturbation series]])}

The [[effective action]] (\href{A+first+idea+of+quantum+field+theory#InPerturbationTheoryActionEffective}{this def.}) contains no negative powers of [[Planck's constant]] $\hbar$, hence is indeed a [[formal power series]] also in $\hbar$:

\begin{displaymath}
S_{eff}(g,j)
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]
  \,.
\end{displaymath}
and in particular

\begin{displaymath}
\left\langle
    S_{eff}(g,j)
  \right\rangle
  \;\in\;
  \mathbb{C}[ [ \hbar, g, j] ]
  \,.
\end{displaymath}
Moreover, the contribution to the effective action in the [[classical limit]] $\hbar \to 0$ is precisely that of [[Feynman amplitudes]] of those [[finite multigraphs]] (\href{A+first+idea+of+quantum+field+theory#FeynmanPerturbationSeriesAwayFromCoincidingPoints}{this prop.}) which are [[trees]] (\href{A+first+idea+of+quantum+field+theory#GraphPlanar}{this def.}); thus called the \emph{[[tree level]]}-contribution:

\begin{displaymath}
S_{eff}(g,j)\vert_{\hbar = 0}
  \;=\;
  i \hbar
  \underset{\Gamma \in \mathcal{G}_{conn} \cap \mathcal{G}_{tree}}{\sum}
  \Gamma\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma)} \right)
  \,.
\end{displaymath}
Finally, a [[finite multigraph]] $\Gamma$ (\href{A+first+idea+of+quantum+field+theory#Graphs}{this def.}) which is [[planar graph|planar]] and [[connected graph|connected]] contributes to the effective action precisely at order

\begin{displaymath}
\hbar^{L(\Gamma)}
  \,,
\end{displaymath}
where $L(\Gamma) \in \mathbb{N}$ is the number of \emph{[[faces]]} of $\Gamma$, here called the \emph{number of loops} of the diagram; here usually called the \emph{[[loop order]]} of $\Gamma$.

(Beware the terminology clash with [[graph theory]], see the discussion of [[tadpoles]] \href{A+first+idea+of+quantum+field+theory#Tadpoles}{here})

\end{prop}
\begin{proof}
By \href{A+first+idea+of+quantum+field+theory#LagrangianFieldTheoryPerturbativeScattering}{this def.} the explicit $\hbar$-dependence of the [[S-matrix]] is

\begin{displaymath}
\mathcal{S}
  \left(
    S_{int}
  \right)
  \;=\;
  \underset{k \in \mathbb{N}}{\sum}
    \frac{1}{k!}
    \frac{1}{(i \hbar)^k}
    T( \underset{k \, \text{factors}}{\underbrace{S_{int}, \cdots,  S_{int}}} )
\end{displaymath}
and by \href{A+first+idea+of+quantum+field+theory#TimeOrderedProductAwayFromDiagonal}{this prop.} the further $\hbar$-dependence of the [[time-ordered product]] $T(\cdots)$ is

\begin{displaymath}
T(S_{int}, S_{int})
  \;=\;
  prod \circ
  \exp\left(
    \hbar
    \left\langle
      \Delta_F,
      \frac{\delta}{\delta \mathbf{\Phi}}
        \otimes
      \frac{\delta}{\delta \mathbf{\Phi}}
    \right\rangle
  \right)
 ( S_{int} \otimes S_{int} )
  \,,
\end{displaymath}
By the [[Feynman rules]] (\href{A+first+idea+of+quantum+field+theory#FeynmanPerturbationSeriesAwayFromCoincidingPoints}{this prop.}) this means that

\begin{enumerate}%
\item each [[vertex]] of a Feynman diagram contributes a power $\hbar^{-1}$ to its Feynman amplitude;


\item each [[edge]] of a Feynman diagram contributes a power $\hbar^{+1}$ to its Feynman amplitude.



\end{enumerate}
If we write

\begin{displaymath}
E(\Gamma), V(\Gamma) \;\in\; \mathbb{N}
\end{displaymath}
for the total number of [[vertices]] and [[edges]], respectively, in $\Gamma$, this means that a Feynman amplitude corresponding to some $\Gamma \in \mathcal{G}$ contributes precisely at order

\begin{equation}
\hbar^{E(\Gamma) - V(\Gamma)}
  \,.
\label{GeneralFeynmanDiagramhbarContribution}\end{equation}
So far this holds for arbitrary $\Gamma$. If however $\Gamma$ is [[connected graph|connected]] and [[planar graph|planar]], then \emph{[[Euler's formula]]} asserts that

\begin{equation}
E(\Gamma) - V(\Gamma)
  \;=\;
  L(\Gamma) - 1
  \,.
\label{ConnectedPlanarGraphEulerCharacteristic}\end{equation}
Hence $\hbar^{L(\Gamma)- 1}$ is the order of $\hbar$ at which $\Gamma$ contributes to the [[scattering matrix]] expressed as the [[Feynman perturbation series]].

But the [[effective action]], by definition (\href{A+first+idea+of+quantum+field+theory#eq:ExpansionEffectiveAction}{this equation}), has the same contributions of Feynman amplitudes, but multiplied by another power of $\hbar^1$, hence it contributes at order

\begin{displaymath}
\hbar^{E(\Gamma) - V(\Gamma) + 1} = \hbar^{L(\Gamma)}
  \,.
\end{displaymath}
This proves the second claim on [[loop order]].

The first claim, due to the extra factor of $\hbar$ in the definition of the effective action, is equivalent to saying that the Feynman amplitude of every [[connected graph|connected]] [[finite multigraph]] contributes powers in $\hbar$ of order $\geq -1$ and contributes at order $\hbar^{-1}$ precisely if the graph is a tree.

Observe that a [[connected graph|connected]] [[finite multigraph]] $\Gamma$ with $\nu \in \mathbb{N}$ vertices (necessarily $\nu \geq 1$) has at least $\nu-1$ edges and precisely $\nu - 1$ edges if it is a tree.

To see this, consecutively remove edges from $\Gamma$ as long as possible while retaining connectivity. When this process stops, the result must be a connected tree $\Gamma'$, hence a [[connected graph|connected]] [[planar graph]] with $L(\Gamma') = 0$. Therefore [[Euler's formula]] \eqref{ConnectedPlanarGraphEulerCharacteristic} implies that that $E(\Gamma') = V(\Gamma') -1$.

This means that the connected multigraph $\Gamma$ in general has a Feynman amplitude of order

\begin{displaymath}
\hbar^{E(\Gamma) - V(\Gamma)}
  =
  \hbar^{ \overset{\geq 0}{\overbrace{E(\Gamma) - E(\Gamma')}} + \overset{= -1}{\overbrace{E(\Gamma') - V(\Gamma)}}  }
\end{displaymath}
and precisely if it is a tree its Feynman amplitude is of order $\hbar^{-1}$.

\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[radiative correction]]

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

General discussion includes

\begin{itemize}%
\item Stanley J. Brodsky, Paul Hoyer, \emph{The $\hbar$-Expansion in Quantum Field Theory}, Phys.Rev.D83:045026, 2011 (\href{https://arxiv.org/abs/1009.2313}{arXiv:1009.2313})

\end{itemize}
Discussion of tree level Feynman diagrams as perturbative solutions in [[classical field theory]] is in

\begin{itemize}%
\item Robert Helling, \emph{Solving classical field equations} (\href{http://homepages.physik.uni-muenchen.de/~helling/classical_fields.pdf}{pdf})

\end{itemize}
Discussion of loop orders of relevance in comparison to [[experiment]] cited above includes for instance the following

\begin{itemize}%
\item Matteo Cacciari, \emph{(Theoretical) review of heavy quark production}, BNL 14/12/2005 (\href{https://www.phenix.bnl.gov/WWW/publish/xiewei/RBRC_Workshop_Dec/heavyworkshop/cacciari.pdf}{pdf})


\item Tatsumi Aoyama, Masashi Hayakawa, Toichiro Kinoshita, Makiko Nio, \emph{Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant}, 10.1103/PhysRevLett.109.111807 (\hyperlink{https://arxiv.org/abs/1205.5368}{arXiv:1205.5368})


\item Charalampos Anastasiou, Claude Duhr, Falko Dulat, Franz Herzog, Bernhard Mistlberger, \emph{Higgs boson gluon-fusion production in N3LO QCD}, Phys. Rev. Lett. 114, 212001 (2015) (\href{https://arxiv.org/abs/1503.06056}{arXiv:1503.06056})



\end{itemize}
[[!redirects loop orders]]

[[!redirects tree level]] [[!redirects tree levels]]

[[!redirects tree-level]] [[!redirects tree-levels]]

[[!redirects 1-loop]] [[!redirects 2-loop]] [[!redirects 3-loop]] [[!redirects 4-loop]]



\end{document}