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The Minimal Set of Electroweak Precision Parameters

G. Cacciapaglia, C. Csáki, G. Marandella, A. Strumia

TL;DR

This work identifies a minimal, physically transparent set of parameters to capture heavy new-physics effects in electroweak precision data, using an EFT approach that separates oblique corrections from vertex corrections. By applying equations of motion to eliminate charged-lepton currents, the authors isolate seven oblique parameters ($\hat{S},\hat{T},\hat{U},W,Y,X,V$) plus two key quark-coupling combinations ($\delta\varepsilon_q,\delta C_q$), with an additional $\delta\varepsilon_b$ for the third generation when needed. Global fits show that only about 9 of 18 operators are truly constrained, and a simple oblique-only approximation already yields reasonable bounds; adding the two quark-parameters improves accuracy to about 10% in most cases. The framework is demonstrated on generic heavy $Z'$ scenarios, deriving analytic expressions for the parameters in terms of $Z'$ charges and showing leptonic data largely drive the bounds. A positivity result for $W$ and $Y$ is established, reinforcing the physical consistency of the oblique parameter set. Overall, the method offers a practical, scalable way to translate EW precision data into robust constraints on a wide class of heavy new physics.

Abstract

We present a simple method for analyzing the impact of precision electroweak data above and below the Z-peak on flavour-conserving heavy new physics. We find that experiments have probed about ten combinations of new physics effects, which to a good approximation can be condensed into the effective oblique parameters Shat, That, Uhat, V, X, W, Y (we prove positivity constraints W, Y >= 0) and three combinations of quark couplings (including a distinct parameter for the bottom). We apply our method to generic extra Z' vectors.

The Minimal Set of Electroweak Precision Parameters

TL;DR

This work identifies a minimal, physically transparent set of parameters to capture heavy new-physics effects in electroweak precision data, using an EFT approach that separates oblique corrections from vertex corrections. By applying equations of motion to eliminate charged-lepton currents, the authors isolate seven oblique parameters () plus two key quark-coupling combinations (), with an additional for the third generation when needed. Global fits show that only about 9 of 18 operators are truly constrained, and a simple oblique-only approximation already yields reasonable bounds; adding the two quark-parameters improves accuracy to about 10% in most cases. The framework is demonstrated on generic heavy scenarios, deriving analytic expressions for the parameters in terms of charges and showing leptonic data largely drive the bounds. A positivity result for and is established, reinforcing the physical consistency of the oblique parameter set. Overall, the method offers a practical, scalable way to translate EW precision data into robust constraints on a wide class of heavy new physics.

Abstract

We present a simple method for analyzing the impact of precision electroweak data above and below the Z-peak on flavour-conserving heavy new physics. We find that experiments have probed about ten combinations of new physics effects, which to a good approximation can be condensed into the effective oblique parameters Shat, That, Uhat, V, X, W, Y (we prove positivity constraints W, Y >= 0) and three combinations of quark couplings (including a distinct parameter for the bottom). We apply our method to generic extra Z' vectors.

Paper Structure

This paper contains 13 sections, 40 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The minimal set of constrained parameters
  3. The oblique parameters
  4. Vertex corrections
  5. A simple approximation
  6. Global fit
  7. Example: a generic $Z'$
  8. Proof for $W,Y\ge 0$
  9. Conclusions
  10. The minimal parameters for a general Lagrangian
  11. Oblique corrections and neutrinos
  12. Vertex corrections
  13. The universal limit

Figures (3)

  • Figure 1: The red dots are the ordered eigenvalues of the full error matrix, that describe the sensitivity of present data (upper dots correspond to more precise combinations). Precision data significantly constrain only about 10 new-physics effects. The blue circles show the same eigenvalues recomputed making our simplifying approximation.
  • Figure 2: Distibutions of the ratios between the approximate over true bound in various approximations. In the first "oblique" panel we include in the fit only $\hat{S}$, $\hat{T}$, $\hat{U}$, $W$, $Y$, $V$ and $X$. In the second panel we add the two parameters $\delta C_q$ and $\delta \varepsilon_q$ for the quarks. Finally we include all the parameters except $\delta C_q$ and $\delta \varepsilon_q$.
  • Figure 3: Bounds on $M_{Z'}/g_{Z'}$ in $\,{\rm TeV}$ at $99\%$ CL for different $Z'$ models. Their effect dominantly depends on the charge of the Higgs and the leptons: we here assume the normalization $Z_L^2 + Z_E^2 + Z_H^2 = 2$ such that $Z_H=0$ at the boundary of the circles. The three plots, done assuming different sets of quark charges (zero, universal-like and SU(5)-unified) are almost identical, confirming the validity of an approximate analysis. The dashed line corresponds to a universal $Z'$, and the dashed ellipse to a $Z'$ compatible with SM Yukawa couplings. The dots show some well-known $Z'$s.