Operator Product Expansion

All quantum field theories with well-behaved ultra violet behavior are believed to have an operator product expansion (OPE), as first proposed by Wilson, and Zimmermann. This means that the product of any two local fields located at nearby points $x$ and $y$ can be expanded in the form

\mathcal{O}_A(x)\mathcal{O}_B(y) \sim \sum_C \mathcal{C}_{AB}^C(x-y) \, \mathcal{O}_C(y),

where $A,B,C$ are labels for the various local fields in the given theory (incorporating also their tensor character, spin etc.), and where $\mathcal{C}_{AB}^C$ are certain numerical coefficient functions—or rather distributions—that depend on the theory under consideration, the coupling constants, etc. The sign “$\sim$” indicates that this can be understood as an asymptotic expansion: If the sum on the right side is carried out to a sufficiently large but finite order, then the remainder goes to zero fast as $x \to y$ in the sense of operator insertions into a quantum state, i.e. inside a correlation function. This assumption, together with practical rules for calculating the coefficients in the expansion are an important tool in high energy physics calculations related to quantum chromodynamics. It also underlies much of the formalism of conformal field theory in 2 spacetime dimensions. In the latter context, it has been formalized by the concept of a “vertex algebra”, which itself has deep connections to several areas of mathematics. The theme of the latter approach is to view the coefficients as some kind of “structure constants” of some algebraic structure.

In the context of 4-dimensional quantum field theories without conformal invariance, not so much had been known about the precise properties of this expansion, and the underlying mathematical structures underlying it. We have been looking into these properties, and in particular the question to what extent the coefficients really define some sort of algebraic structure. The first issue to be considered were the convergence properties of the expansion. Rather surprisingly, a concrete analysis of the error term showed that the expansion is not only asymptotic, but even converges at finite (!) distances, to arbitrary loop orders, in a perturbative Euclidean quantum field theory. The precise mathematical result has been obtained in the Euclidean setting (see below) but its physical significance is maybe best explained in the Minkowskian context. There, the analogue of our result would be that correlation functions such as the two-point function $\langle \mathcal{O}_A(x)\mathcal{O}_B(y) \rangle_\Psi$ in a state (Note: The state should have a well-behaved high energy behavior. In the Minkowskian context, it should e.g. have bounded energy $E$, see below for an appropriate replacement in the Euclidean context) $\Psi$ are entirely determined by the collection of OPE coefficients which are  state independent, together with the 1-point functions $\langle\mathcal{O}_C(y) \rangle_\Psi$:

\langle\mathcal{O}_A(x) \mathcal{O}_B(y) \rangle_\Psi = \sum_C \mathcal{C}_{AB}^C(x-y) \
\langle \mathcal{O}_C(y) \rangle_\Psi \, ,

where the infinite sum over “$C$” would be convergent, and $(x-y)^2$ would not necessarily have to be small (Note however that one expects convergence to hold in the relativistic context only for spacelike distances, $(x-y)^2>0$, because of locality). An analogous statement would apply to the higher $n$-point functions. Thus, the OPE coefficients capture the state-independent algebraic structure of QFT, while all the information about the quantum state, i.e. $n$-point functions, is contained in the 1-point functions (“form factors”) only.

Concretely, we proved convergence of the OPE in the context of perturbative Euclidean QFT, to arbitrary  loop orders. The model that we considered is a hermitian scalar field with self-interaction $g \varphi^4$ and mass $m > 0$ on flat 4-dimensional Euclidean space. The composite fields $\mathcal{O}_A$ in this model are simply the linear combinations of monomials in the basic field $\varphi$ and its derivatives and are denoted by

\mathcal{O}_A = \partial^{w_1} \varphi \cdots \partial^{w_n} \varphi \, ,
\qquad A = \{n,w\} \, ,

where each $w_i$ is a 4-dimensional multi-index representing the various possible combinations of partial derivatives. One defines the “engineering dimension” of such a field as usual by

[A] = n + \sum_i |w_i|  \, .

Each OPE coefficient $\mathcal{C}^C_{AB}(x-y)$ is itself a formal power series in $\hbar$ (“loop expansion”). In perturbation theory, one is not concerned with the convergence of these expansions in $\hbar$. Instead, one is concerned here with the convergence of the OPE (i.e. the expansion in “$C$”) at arbitrary but fixed order $l$ in $\hbar$. To analyze this issue, one must insert the left- and right sides of \eqref{ope1} into a correlation function containing suitable “spectator fields”, which play the role of a quantum state in the Euclidean context. A simple and natural choice for the spectator fields is e.g.

\varphi(f_{p_i}) := \int d^4x\ \varphi(x) \ f_{p_i} (x) \ ,

where $p_i$ is a 4-momentum, and where $f_{p_i}$ is a smooth function whose Fourier transform $\hat f_{p_i}(q)$ has compact support for $q$ in a ball of radius $\epsilon$ around $p_i$. One result concerning the OPE is the following:

Theorem: Let the sum $\sum_C$ in the operator product expansion (1) be over all $C$ such that

[C] – [A] – [B] \le \Delta

where $\Delta$ is some positive integer. Then for each such $\Delta$, we have the following bound for the “remainder” in the OPE in loop order $l$:

\Bigg| \bigg\langle
\mathcal{O}_A(x) \mathcal{O}_B(0) \, \varphi(f_{p_1}) \cdots \varphi(f_{p_n}) \bigg\rangle
– \sum_{C} \mathcal{C}_{AB}^C(x) \,
\bigg\langle \mathcal{O}_C(0) \, \varphi(f_{p_1}) \cdots \varphi(f_{p_n})
\bigg\rangle \Bigg|\\
\hspace{1.5cm} \ \ \le \ \ m^{[A]+[B]+n}\
\sqrt{[A]![B]!} \ K^{[A]+[B]}
\ \prod_i \sup |\hat f_{p_i}|
\hspace{1cm} \times  \ \
\sup(1,\frac{|\vec p|_n}{m})^{2([A]+[B])(n+2l+1)+3n}
\frac{\log^\lambda \sup(1,\frac{|\vec p|_n}{m})}
{2^\lambda \lambda!} \nonumber\\
\times \ \ \frac{1}{\sqrt{\Delta!}} \ \Bigg( K \ m \ |x| \
\sup (1, \frac{|\vec p|_n}{m})^{n+2l+1} \Bigg)^{\Delta}\nonumber \ .

Here, $\langle \, . \, \rangle$ denote correlation functions, and $K$ is a constant depending on $n,l$. Furthermore, $|\vec p|_n$ is the larges momentum among the $p_i$, and $f_{p_i}$ are smooth test functions in position space, whose support in momentum space is contained in a ball of radius $\epsilon$ around $p_i$.

This result establishes the convergence of the OPE, i.e. the sum over $C$, at each fixed order in perturbation theory, because the remainder evidently goes to zero as $\Delta \to \infty$. There are no conditions on $x$, so the OPE converges even at arbitrarily large distances! But we note that such conditions could arise if we were to allow, a wider class of spectator fields, for example, if we were to replace $f_{p_i}$ by test-functions whose Fourier transforms are only decaying in momentum space, but are not of compact support. This type of behavior can be understood in a way by the fact that $|\vec p|_n$ gives a measure for the “typical energy” of the “state” in which we try to carry out the OPE. As the high energy behavior of the “state” becomes worse, so do the convergence properties of the OPE.

To prove the theorem, one first has to give a prescription for defining the Schwinger functions and OPE coefficients in renormalized perturbation theory. There are several options; we found it convenient to use the Wilson-Wegner-Polchinski flow equation method, and the substantial refinements/improvements due to Kopper et al. In this method, one first introduces an infrared cutoff called $\Lambda$, and an ultraviolet cutoff called $\Lambda_0$. One then defines the quantities of interest for finite values of the cutoffs, and derives for them a flow equation as a function of $\Lambda$. For suitable  boundary conditions its solutions may be bounded inductively and uniformly  in the ultraviolet cutoff $\Lambda_0$.  In our case, we need bounds for the remainder in the OPE.  Again, such bounds are verified inductively.

Using these methods, one can also establish other properties of the OPE. An obvious general expectation is that, since these coefficients resemble the structure constants of an algebra, there should hold certain associativity conditions. For example, consider a product of three fields $\mathcal{O}_{A_1}(x_1)\mathcal{O}_{A_2}(x_2)\mathcal{O}_{A_3}(x_3)$, which we may expand as in eq. \eqref{ope1}. Now, suppose that $|x_1-x_2|$ is smaller than $|x_2-x_3|$. Then it seems natural to first expand the product $\mathcal{O}_{A_1}(x_1) \mathcal{O}_{A_2}(x_2)$, regarding $\mathcal{O}_{A_3}(x_3)$ as merely a spectator, and then to expand the result of this OPE times the spectator in a subsequent OPE. One would expect these expansions to agree, and our theorem shows that this expectation is correct:

Theorem: Up to any arbitrary but fixed loop order $l$ in $g\varphi^4$-theory, the identity

\mathcal{C}_{A_1 A_2 A_3}^B(x_1, x_2, x_3) = \sum_C \mathcal{C}_{A_1 A_2}^C(x_1, x_2) \ \mathcal{C}_{CA_3}^B(x_2, x_3)

holds for all configurations satisfying

0<\frac{|x_{1}-x_2|}{|x_2-x_{3}|} < \frac{1}{K}

for some (sufficiently large) constant $K>0$ (depending on $l, B$).

In particular, implicit in this statement is the claim that the infinite sum over $C$ on the right side of formula (8) converges.  In the free field theory, our result would be fairly trivial and we may in fact take $ K=1$. However, in the presence of interaction the result seems rather non-trivial and we can only show that convergence occurs if $ K$ is a rather large numerical constant, i.e. it appears that $|x_1-x_2|$ must be much smaller than $|x_2-x_3|$. The result can also be generalized to a larger number of factors in the product, and it also leads to a recursion procedure for calculating the coefficients. Based on such properties of the OPE, one can attempt to base the definition of a quantum field theory on the algebras that can be defined by these coefficients. Although the theorems mentioned above were (for simplicity) obtained in flat (Euclidean) spacetime, one may also prove them for convex normal neighborhoods of Riemannian spacetimes.

Some references are: