Exploring Nonperturbative Behaviour of Moments and Cumulants in Quantum Theories

TL;DR

This work analyzes nonperturbative, large-$n$ behavior of correlation functions in massive scalar theories with $φ^{2p}$ self-interactions by employing a double scaling limit $g^2\to0$, $n\to\infty$ with $\epsilon=g^2 n$ fixed and resumming the effective saddle point in zero and one dimensions. It shows that the moments $\gamma_n$ grow exponentially in the nonperturbative regime, with the growth tempered by higher $p$, while cumulants $G_n$ exhibit factorial growth largely independent of $p$, indicating a universal large-$n structure for fully-connected correlations. A semiclassical effective action $S_{\mathrm{eff}}=S_E-\varepsilon\log\varphi(0)$ with $\varepsilon=g^2 n$ enables a controlled resummation, producing an exponent $F(\epsilon)$ that captures infinite-loop effects and reveals turning points and asymptotic suppression at large $\epsilon$ for larger $p$ in the QM analogue. The results imply that fully-connected information encoded in cumulants carries universal nonperturbative dynamics, while lower-point moments remain sensitive to UV operator structure; these insights pave the way for extending to higher dimensions and exploring connections to EFT operator universality and large-charge regimes.

Abstract

The dynamics of quantum fields become nonperturbative when their interactions are probed by a large number of particles. To explore this regime we study correlation functions which involve a large number of fields, focussing on massive scalar theories that feature arbitrary self-interactions, $φ^{2p}$. Treating quantum fields as operator-valued distributions, we investigate $n$-point correlation functions at ultra-short distances and compute moments and cumulants of fields, using a semiclassical saddle point approximation in the double scaling limit of weak coupling, $λ\to 0$, large quantum number, $n \to \infty$, while keeping $λn$ constant. Addressing the nonperturbative regime, where $λn \gtrsim 1$, requires a resummation of the effective saddle point to all orders in $λn$. We perform this resummation in zero and one dimensions, and show that the moments, corresponding to correlation functions including disconnected contributions, grow exponentially with $n$. This growth is significantly reduced for higher-order self-interactions, i.e. for larger $p$. On the other hand, we argue that the cumulants, which represent connected correlation functions, grow even more rapidly and are mostly independent of $p$.

Figures (4)

• Figure 1: Moments $\gamma_n$ normalised to $Z_0$, as a function of the number of field insertions $n$ in zero dimensions. The different colours represent different self-interaction terms of the theory, $\phi^{2p}$. For simplicity, the coupling is set to one, $g^2=1$.
• Figure 2: Cumulants $G_n$ (left) and their rescaled ratios (right), as a function of the number of fields $n$ in zero dimensions. The different colours represent different self-interaction terms of the theory, $\phi^{2p}$. For simplicity, the coupling is set to one, $g^2=1$.
• Figure 3: Resummed exponent $F$ of the moments $\gamma_n$, as a function of $\epsilon$ in the double scaling limit $g^2 \to 0$ and $n \to \infty$, where $\epsilon = g^2 n$ is kept fixed. The different colours represent different self-interaction terms of the theory, $\phi^{2p}$.
• Figure 4: Normalised moments $\gamma_n$ (left) and cumulants $G_n$ (right), as a function of $n$ in one dimension. The different colours represent the self-interaction terms of the theory, $\phi^{2p}$. For simplicity, both $\gamma_n$ and $G_n$ are considered at $g^2 = 1$.