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Gravitational form factors of the $Z$-boson

P. Beißner, J. Yu. Panteleeva, B. -D. Sun, E. Epelbaum, J. Gegelia

TL;DR

The paper investigates the gravitational structure of the Z-boson by computing one-loop electroweak corrections to its energy-momentum tensor matrix elements, parameterized by gravitational form factors. Using an effective field theory framework with general-coordinate invariance, the authors extract the form factors from the EMT three-point function and show that ultraviolet divergences cancel via a counterterm from a non-minimal coupling $c R \Phi^\dagger \Phi$. They find normalizations $A_0(0)=1$ and $J(0)=1$, and obtain a finite mean-square energy radius $r_0^2 \approx (9.5\times 10^{-9} + 8.2\times 10^{-12} i)\, \mathrm{fm}^2$, indicating the Z-boson possesses a nonzero gravitational size with an imaginary part arising from the particle’s instability. The work demonstrates how gravitational structure can be studied in the electroweak sector within EFT and provides numerical estimates for the Z-boson’s gravitational dressing, with implications for neutral particle probes of gravity at the quantum level.

Abstract

Matrix elements of the energy-momentum tensor for one-particle states of the $Z$-boson are parameterized in terms of gravitational form factors. One-loop order electroweak corrections to these quantities are calculated. Renormalization and physical interpretation of the obtained results are discussed.

Gravitational form factors of the $Z$-boson

TL;DR

The paper investigates the gravitational structure of the Z-boson by computing one-loop electroweak corrections to its energy-momentum tensor matrix elements, parameterized by gravitational form factors. Using an effective field theory framework with general-coordinate invariance, the authors extract the form factors from the EMT three-point function and show that ultraviolet divergences cancel via a counterterm from a non-minimal coupling . They find normalizations and , and obtain a finite mean-square energy radius , indicating the Z-boson possesses a nonzero gravitational size with an imaginary part arising from the particle’s instability. The work demonstrates how gravitational structure can be studied in the electroweak sector within EFT and provides numerical estimates for the Z-boson’s gravitational dressing, with implications for neutral particle probes of gravity at the quantum level.

Abstract

Matrix elements of the energy-momentum tensor for one-particle states of the -boson are parameterized in terms of gravitational form factors. One-loop order electroweak corrections to these quantities are calculated. Renormalization and physical interpretation of the obtained results are discussed.
Paper Structure (6 sections, 18 equations, 3 figures)

This paper contains 6 sections, 18 equations, 3 figures.

Table of Contents

  1. Introduction
  2. One-loop corrections to the gravitational form factors of the $Z$-boson
  3. Summary
  4. Definition of loop integrals
  5. Contributions to the pole position and the residue of the Higgs propagator
  6. Gravitational form factors at $q^2=0$

Figures (3)

  • Figure 1: Topologies of one-loop diagrams contributing to the self-energy of the $Z$-boson. Wiggly lines correspond to $Z$-bosons while dashed lines represent vector bosons, fermions, Higgs bosons, Goldstone bosons and Faddeev-Popov ghosts.
  • Figure 2: Topologies of one-loop diagrams contributing to the three-point vertex function of the EMT operator and two $Z$-boson fields. The crosses stand for the EMT insertions and the wiggly lines refer to $Z$-bosons. The dashed lines represent vector bosons, fermions, Higgs bosons, Goldstone bosons and Faddeev-Popov ghosts.
  • Figure 3: Counter term diagram cancelling the divergence of the one-loop contribution to the matrix element of the EMT operator. The vertex with cross stands for the counter term, and the wiggly and straight solid lines refer to $Z$- and Higgs bosons, respectively.