Non-factorizable corrections and effective field theories

TL;DR

The paper develops an effective field theory framework to analyze higher-order radiative corrections in processes with unstable particles, revealing a natural separation between factorizable and non-factorizable effects across all orders. By exploiting a hierarchy of scales E^2 ≫ MΓ, it first integrates out hard modes (recovering a DPA-equivalent description), then resonant modes to separate production–decay from decay–decay nf, and finally soft modes to obtain an S-matrix in terms of external states. The authors derive explicit nf currents and operator structures in both QED and QCD, provide scaling estimates for nf corrections at high energy, and illustrate these ideas in neutral and charged unstable-particle processes, including γγ and e+e− collisions. The EFT approach offers gauge-invariant, systematic means to go beyond the DPA and to quantify nf effects, with potential applications to precise W, Z, and top-quark measurements at future colliders.

Abstract

We analyze the structure of higher-order radiative corrections for processes with unstable particles. By subsequently integrating out the various scales that are induced by the presence of unstable particles we obtain a hierarchy of effective field theories. In the effective field theory framework the separation of physically different effects is achieved naturally. In particular, we automatically obtain a separation of factorizable and non-factorizable corrections to all orders in perturbation theory. At one loop this treatment is equivalent to the double-pole approximation (DPA) but generalizes to higher orders and, at least in principle, to beyond the DPA. It is known that one-loop non-factorizable corrections to invariant mass distributions are suppressed at high energy. We study the mechanism of this suppression and obtain estimates of higher-order non-factorizable corrections at high energy.

Figures (7)

• Figure 1: Four-loop diagrams contributing to non-factorizable corrections. The diagrams are proportional to (a) $\sim\alpha_{\rm dipole}^4$, (b) $\sim\alpha_{\rm dipole}^3\alpha_{\rm prop}$, (c) $\sim\alpha_{\rm dipole}^2\alpha_{\rm prop}^2$, (d) $\sim\alpha_{\rm dipole}^2 \alpha_{\rm prop}^2$ and (e) $\sim\alpha_{\rm dipole} \alpha_{\rm prop}^3$.
• Figure 2: Relevant energy-momentum and virtuality scales with a list of corresponding fields contributing at each scale.
• Figure 3: Resonant (a) and background (b) Born contributions to pair production in the underlying theory. Resonant kinematics means that $D_1\sim D_2\sim \Lambda_1\Lambda_2$.
• Figure 4: Diagrams involving $\phi(P)$ that result in (a) $\alpha$ corrections to ${\cal P} X \phi\phi$ and (b) $\alpha$ corrections to the operator $X (\bar{\psi} \psi)(\bar{\psi} \psi)$. In diagram (a) all loops are hard but the decaying $\phi$ fields are resonant. In diagram (b) all internal fields are hard.
• Figure 5: One-loop radiative corrections in effective theory. (a) would be counted as non-factorizable and (b) as factorizable correction in the underlying theory. If the unstable particles are charged there are diagrams in the underlying theory that contain both factorizable and non-factorizable corrections. An example is given in (c).