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The Nielsen Identities of the SM and the definition of mass

Paolo Gambino, Pietro Antonio Grassi

TL;DR

The paper develops and applies Nielsen identities, derived from an extended BRST framework, to the Standard Model to control gauge-parameter dependence of Green functions and to define gauge-invariant mass parameters via complex pole positions. It proves, to all orders, that the complex poles corresponding to physical fields (including W, Z, CP-violating scalars, and fermions with mixing) are gauge independent when physical renormalization conditions are used (i.e., $\beta_i^\xi=0$). The authors provide explicit NI for two-point functions across bosons, fermions, and scalars, and demonstrate how gauge cancellations proceed in physical amplitudes, clarifying renormalization and infrared issues (notably the IR finiteness of the W pole mass). The framework offers practical, cross-check-ready tools for higher-order calculations and for relating theoretical parameters to physical observables, even in the presence of mixing and CP violation.

Abstract

In a generic gauge theory the gauge parameter dependence of individual Green functions is controlled by the Nielsen identities, which originate from an enlarged BRST symmetry. We give a practical introduction to the Nielsen identities of the Standard Model (SM) and to their renormalization and illustrate the power of this elegant formalism in the case of the problem of the definition of mass.We prove to all orders in perturbation theory the gauge-independence of the complex pole of the propagator for all physical fields of the SM, in the most general case with mixing and CP violation. At the amplitude level, the formalism provides an intuitive and general understanding of the gauge recombinations which makes it particularly useful at higher orders. We also include in an appendix the explicit expressions for the fermionic two-point functions in a generic R_ξgauge.

The Nielsen Identities of the SM and the definition of mass

TL;DR

The paper develops and applies Nielsen identities, derived from an extended BRST framework, to the Standard Model to control gauge-parameter dependence of Green functions and to define gauge-invariant mass parameters via complex pole positions. It proves, to all orders, that the complex poles corresponding to physical fields (including W, Z, CP-violating scalars, and fermions with mixing) are gauge independent when physical renormalization conditions are used (i.e., ). The authors provide explicit NI for two-point functions across bosons, fermions, and scalars, and demonstrate how gauge cancellations proceed in physical amplitudes, clarifying renormalization and infrared issues (notably the IR finiteness of the W pole mass). The framework offers practical, cross-check-ready tools for higher-order calculations and for relating theoretical parameters to physical observables, even in the presence of mixing and CP violation.

Abstract

In a generic gauge theory the gauge parameter dependence of individual Green functions is controlled by the Nielsen identities, which originate from an enlarged BRST symmetry. We give a practical introduction to the Nielsen identities of the Standard Model (SM) and to their renormalization and illustrate the power of this elegant formalism in the case of the problem of the definition of mass.We prove to all orders in perturbation theory the gauge-independence of the complex pole of the propagator for all physical fields of the SM, in the most general case with mixing and CP violation. At the amplitude level, the formalism provides an intuitive and general understanding of the gauge recombinations which makes it particularly useful at higher orders. We also include in an appendix the explicit expressions for the fermionic two-point functions in a generic R_ξgauge.

Paper Structure

This paper contains 9 sections, 64 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Nielsen identities in the SM
  3. Tadpoles
  4. W boson
  5. The $Z-\gamma$ system
  6. The scalar sector
  7. Fermions
  8. Application to physical amplitudes
  9. Summary

Figures (4)

  • Figure 1: One-loop diagram contributing to ${\bf \Gamma}_{\chi\gamma_{ H}}$.
  • Figure 2: One-loop diagrams contributing to ${\bf \Gamma}_{{\chi_{ W} \gamma_{ W}} { W}}$.
  • Figure 3: Schwinger-Dyson equation for the $W$ two-point function. The blobs on the internal lines represent connected propagators (chains of bubbles), while the blob on the vertex represents a one-particle irreducible Green function.
  • Figure 4: Nielsen identity for the two-point function ${\bf \Gamma}_{\phi_i\phi_j}$.