Useful trick to compute correlation functions of composite operators

TL;DR

The paper tackles how to efficiently compute correlation functions of gauge-invariant composite operators by embedding the problem into an extended theory with an auxiliary non-dynamical field $B(x)$ coupled to the target operator $O(x)$ via a parameter $\alpha$. It derives a master equation, $\langle O(x_1)\dots O(x_r)\rangle_c^S = \frac{1}{r!} \left. \frac{\partial^r}{\partial \alpha^r} \langle B(x_1)\dots B(x_r) \rangle_c^{S_B} \right|_{\alpha=0}$, linking $O$-correlators to derivatives of $B$-correlators in the augmented theory $S_B$ and enabling the reuse of standard elementary-field computational tools. The method is demonstrated for $O(x)=\frac{\phi^2(x)}{2}$ in $\lambda\phi^4$ theory, showing that the $\alpha^2$-order reproduces the expected $O$-correlator and that higher-order terms map consistently to familiar diagrammatic structures through contraction identities. The authors further argue for all-orders consistency and discuss renormalization considerations, indicating that the approach preserves the UV structure of the original theory while offering a practical computational route for composite operators.

Abstract

In general, in gauge field theories, physical observables are represented by gauge-invariant composite operators, such as the electromagnetic current. As we recently demonstrated in the context of the $U\left(1\right)$ and $SU\left(2\right)$ Higgs models \cite{Dudal:2019pyg,Dudal:2020uwb,Maas:2020kda}, correlation functions of gauge-invariant operators exhibit very nice properties. Besides the well-known gauge independence, they do not present unphysical cuts, and their Källén-Lehmann representations are positive, at least perturbatively. Despite all these interesting features, they are not employed as much as elementary fields, mainly due to the additional complexities involved in their computation and renormalization. In this article, we present a useful trick to compute loop corrections to correlation functions of composite operators. This trick consists of introducing an additional field with no dynamics, coupled to the composite operator of interest. By using this approach, we can employ the traditional algorithms used to compute correlation functions of elementary fields.

Figures (7)

• Figure 1: Diagram with one contraction between fields of the same $O\left(x\right)$.
• Figure 2: From the diagrammatic perspective, a local composite operator can be consider as a vertex with an additional external momenta $p$, which can be associated to an imaginary external line.
• Figure 3: Diagrammatic representation of (a) the propagator of $\phi$ and (b) the quadratic vertex.
• Figure 4: Feynman diagrams that contribute to $\left\langle O\left(x\right)O\left(y\right)\right\rangle _{c}^{S}$.
• Figure 5: Diagrammatic representation of (a) the propagator of $B$ and (b) the $B\phi$-vertex.