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The g-2 in the neutral Electroweak model with cutoff: convergent expansion, RG and the Jackiw-Weinberg formula

Vieri Mastropietro, Michele Bianchessi

TL;DR

The paper provides a rigorous treatment of the electron g-2 in the neutral Electroweak sector with a momentum cutoff by constructing a convergent Renormalization Group expansion. It shows that the regularized gyromagnetic factor matches the Jackiw–Weinberg weak contribution up to controllable regulator-induced corrections, with precise bounds that scale as $\mathcal{O}(m^2/M^2)$ and depend on the cutoff $\Lambda$ and mass scales $M$. The approach hinges on multiscale decomposition, localization, and Gallavotti–Nicolò tree expansions to obtain uniform bounds and to exploit symmetry-induced cancellations that suppress higher-order terms. The results establish a quantitative link between the formal perturbative truncation and a nonperturbative, cutoff-regulated theory, with potential implications for the interpretation of precision QED/W interactions in effective field theory contexts. Overall, the work provides a mathematically rigorous justification for the perturbative truncation of weak-sector contributions to g-2 in a controlled effective-theory framework, and it exposes a pathway to assess truncation errors in similar nonconvergent expansions as energy cutoffs are varied.

Abstract

The prediction of the anomalous gyromagnetic factor of the electron, started with the evaluation of the electromagnetic contribution by Schwinger (1948) and of the weak contribution by Jackiw and Weinberg (1972), is one of the major successes of Quantum Field Theory and the Standard Model. The results obtained truncating the series are in spectacular agreement with experiments. Yet, a mathematical justification and an estimate of the truncation error are problematic, being such series diverging and not asymptotic to any QFT. For a non perturbative result, one has to consider the Standard Model as an effective theory valid up to certain energy scales. In this paper we consider the neutral sector of the Electroweak model with a momentum cutoff; we rigorously prove that the anomalous gyromagnetic factor in the effective regularized theory coincides with the Jackiw-Weinberg result, obtained by the truncation of the formal expansion with no cutoffs (whose sum is not expected to exist), up to a regularization-dependent correction which is subdominant in the weak coupling regime if the cutoff is smaller than the inverse coupling and larger than the boson mass. The proof is based on a convergent expansions and Renormalization Group (RG) methods; cancellations based on exact and approximated symmetries are needed to get lowest order dominance.

The g-2 in the neutral Electroweak model with cutoff: convergent expansion, RG and the Jackiw-Weinberg formula

TL;DR

The paper provides a rigorous treatment of the electron g-2 in the neutral Electroweak sector with a momentum cutoff by constructing a convergent Renormalization Group expansion. It shows that the regularized gyromagnetic factor matches the Jackiw–Weinberg weak contribution up to controllable regulator-induced corrections, with precise bounds that scale as and depend on the cutoff and mass scales . The approach hinges on multiscale decomposition, localization, and Gallavotti–Nicolò tree expansions to obtain uniform bounds and to exploit symmetry-induced cancellations that suppress higher-order terms. The results establish a quantitative link between the formal perturbative truncation and a nonperturbative, cutoff-regulated theory, with potential implications for the interpretation of precision QED/W interactions in effective field theory contexts. Overall, the work provides a mathematically rigorous justification for the perturbative truncation of weak-sector contributions to g-2 in a controlled effective-theory framework, and it exposes a pathway to assess truncation errors in similar nonconvergent expansions as energy cutoffs are varied.

Abstract

The prediction of the anomalous gyromagnetic factor of the electron, started with the evaluation of the electromagnetic contribution by Schwinger (1948) and of the weak contribution by Jackiw and Weinberg (1972), is one of the major successes of Quantum Field Theory and the Standard Model. The results obtained truncating the series are in spectacular agreement with experiments. Yet, a mathematical justification and an estimate of the truncation error are problematic, being such series diverging and not asymptotic to any QFT. For a non perturbative result, one has to consider the Standard Model as an effective theory valid up to certain energy scales. In this paper we consider the neutral sector of the Electroweak model with a momentum cutoff; we rigorously prove that the anomalous gyromagnetic factor in the effective regularized theory coincides with the Jackiw-Weinberg result, obtained by the truncation of the formal expansion with no cutoffs (whose sum is not expected to exist), up to a regularization-dependent correction which is subdominant in the weak coupling regime if the cutoff is smaller than the inverse coupling and larger than the boson mass. The proof is based on a convergent expansions and Renormalization Group (RG) methods; cancellations based on exact and approximated symmetries are needed to get lowest order dominance.

Paper Structure

This paper contains 34 sections, 9 theorems, 215 equations, 7 figures.

Table of Contents

  1. Introduction and main results
  2. The anomalous gyromagnetic factor
  3. The regularized neutral Electroweak model
  4. The regularized gyromagnetic factor
  5. Sketch of the proof
  6. Open problems and remarks
  7. Renormalization Group analysis
  8. Multiscale decomposition
  9. Localization
  10. Single scale integration
  11. Feynman diagrams
  12. Bounds on the renormalized expansion
  13. Proof of \ref{['eq:th:r_bound']} on scale $h$.
  14. Proof of \ref{['eq:th:p_bound']} on scale $h$.
  15. The bounds \ref{['eq:th:r_bound']}, \ref{['eq:th:p_bound']} imply \ref{['eq:renormalization_bound']}, \ref{['eq:renormalization_bound_zero_mass']}.
  16. ...and 19 more sections

Key Result

Theorem 1

Suppose that $K=4$ and $M > 10m$. If the limit $L\to \infty$ of the correlations (ct) exists, it verifies (eq:conditions_correlations) for a suitable choice of $Z^s_N, Z^{J, s}_N, m^s_N$ and is such that the regularized anomalous gyromagnetic factor eq:g-2:def satisfies where $\bar{\mathfrak{a}}_{\textsc{z},1}$ is given by (eq:JW_result) and $C_0, C_1, C_2, C_3$ are positive constants that do no

Figures (7)

  • Figure 1: A Gallavotti-Nicolò tree. The root vertex is located at the extreme left of the graph and it is not explicitly marked. The endpoints with label $N+1$ or $j \le N$ respectively correspond to ${V}^{N}$ or $\mathcal{L} {V}^{j}$ factors.
  • Figure 2: Interaction vertices and their values in Fourier space. These vertices are respectively associated with $\lambda$ endpoints, $J$ endpoints and $\eta$ endpoints. Note that the value of a $J$ interaction vertex depends on the scale $h$ of the first nontrivial vertex that precedes the corresponding endpoint.
  • Figure 3: A multiscale Feynman diagram (on the right) together with the corresponding Gallavotti-Nicolò tree (on the left).
  • Figure 4: Nonvanishing multiscale Feynman diagrams arising from $\xi_1, \dots, \tau_4$. Every unlabeled fermion propagator is equal to ${g}^{h^\star}$ and a $\mathcal{R}$ operator acts on every evidenced cluster whose inner propagators live on strictly higher scales than its external legs.
  • Figure 5: Fourth order diagram with the same structure as Karplus_Kroll_1950, together with corresponding tree.
  • ...and 2 more figures

Theorems & Definitions (25)

  • Theorem 1
  • Remark 1
  • Lemma 1
  • proof
  • Remark 2
  • Definition 1: Gallavotti-Nicolò trees
  • Definition 2: Value of a tree
  • Definition 3: Weighted norm
  • Lemma 2: Renormalized norm bounds
  • proof
  • ...and 15 more