nLab loop order

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 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 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 perturbative solutions of the classical equations of motion (see 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 Cacciari 05, slide 5, e.g. in Higgs field computation, see ADDHM 15). In rare cases higher loop orders are used (for instance in the computation of the anomalous magnetic moments AHKN 12, but this is not a scattering experiment).

This usefulness of low loop order is fortunate because

1. the S-matrix formal power series for all theories of interest has vanishing radius of convergence (Dyson 52), hence is at best an asymptotic series for which the sum of more than some low order terms is meaningless;

2. the computational effort increases immensely with loop order.

Relation to powers in Planck’s constant

The following is taken from geometry of physics – A first idea of quantum field theory. See there for more background.

Proposition

(loop order and tree level of Feynman perturbation series)

The effective action (this def.) contains no negative powers of Planck's constant $\hbar$, hence is indeed a formal power series also in $\hbar$:

$$S_{eff}(g,j) \;\in\; PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ] \,.$$

and in particular

$$\left\langle S_{eff}(g,j) \right\rangle \;\in\; \mathbb{C}[ [ \hbar, g, j] ] \,.$$

Moreover, the contribution to the effective action in the classical limit $\hbar \to 0$ is precisely that of Feynman amplitudes of those finite multigraphs (this prop.) which are trees (this def.); thus called the tree level-contribution:

$$S_{eff}(g,j)\vert_{\hbar = 0} \;=\; i \hbar \underset{\Gamma \in \mathcal{G}_{conn} \cap \mathcal{G}_{tree}}{\sum} \Gamma\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma)} \right) \,.$$

Finally, a finite multigraph $\Gamma$ (this def.) which is planar and connected contributes to the effective action precisely at order

$$\hbar^{L(\Gamma)} \,,$$

where $L(\Gamma) \in \mathbb{N}$ is the number of faces of $\Gamma$, here called the number of loops of the diagram; here usually called the loop order of $\Gamma$.

(Beware the terminology clash with graph theory, see the discussion of tadpoles here)

Proof

By this def. the explicit $\hbar$-dependence of the S-matrix is

$$\mathcal{S} \left( S_{int} \right) \;=\; \underset{k \in \mathbb{N}}{\sum} \frac{1}{k!} \frac{1}{(i \hbar)^k} T( \underset{k \, \text{factors}}{\underbrace{S_{int}, \cdots, S_{int}}} )$$

and by this prop. the further $\hbar$-dependence of the time-ordered product $T(\cdots)$ is

$$T(S_{int}, S_{int}) \;=\; prod \circ \exp\left( \hbar \left\langle \Delta_F, \frac{\delta}{\delta \mathbf{\Phi}} \otimes \frac{\delta}{\delta \mathbf{\Phi}} \right\rangle \right) ( S_{int} \otimes S_{int} ) \,,$$

By the Feynman rules (this prop.) this means that

1. each vertex of a Feynman diagram contributes a power $\hbar^{-1}$ to its Feynman amplitude;

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

If we write

$$E(\Gamma), V(\Gamma) \;\in\; \mathbb{N}$$

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

(1)$$\hbar^{E(\Gamma) - V(\Gamma)} \,.$$

So far this holds for arbitrary $\Gamma$. If however $\Gamma$ is connected and planar, then Euler's formula asserts that

(2)$$E(\Gamma) - V(\Gamma) \;=\; L(\Gamma) - 1 \,.$$

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 (this equation), has the same contributions of Feynman amplitudes, but multiplied by another power of $\hbar^1$, hence it contributes at order

$$\hbar^{E(\Gamma) - V(\Gamma) + 1} = \hbar^{L(\Gamma)} \,.$$

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 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 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 planar graph with $L(\Gamma') = 0$. Therefore Euler's formula (2) implies that that $E(\Gamma') = V(\Gamma') -1$.

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

$$\hbar^{E(\Gamma) - V(\Gamma)} = \hbar^{ \overset{\geq 0}{\overbrace{E(\Gamma) - E(\Gamma')}} + \overset{= -1}{\overbrace{E(\Gamma') - V(\Gamma)}} }$$

and precisely if it is a tree its Feynman amplitude is of order $\hbar^{-1}$.

Related concepts

• radiative correction

References

General discussion includes

• Stanley J. Brodsky, Paul Hoyer, The $\hbar$-Expansion in Quantum Field Theory, Phys.Rev.D83:045026, 2011 (arXiv:1009.2313)

Discussion of tree level Feynman diagrams as perturbative solutions in classical field theory is in

• Robert Helling, Solving classical field equations (pdf)

Discussion of loop orders of relevance in comparison to experiment cited above includes for instance the following

• Matteo Cacciari, (Theoretical) review of heavy quark production, BNL 14/12/2005 (pdf)

• Tatsumi Aoyama, Masashi Hayakawa, Toichiro Kinoshita, Makiko Nio, Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant, 10.1103/PhysRevLett.109.111807 (arXiv:1205.5368)

• Charalampos Anastasiou, Claude Duhr, Falko Dulat, Franz Herzog, Bernhard Mistlberger, Higgs boson gluon-fusion production in N3LO QCD, Phys. Rev. Lett. 114, 212001 (2015) (arXiv:1503.06056)