[Paper Review] The need for the Higgs boson in the Standard Model

Jorge C. Romão|arXiv (Cornell University)|Mar 14, 2016

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper demonstrates that the Higgs boson is essential for maintaining unitarity in the Standard Model by analyzing scattering amplitudes involving longitudinally polarized W bosons. Using algebraic methods in quantum field theory, it shows that without the Higgs, amplitudes grow uncontrollably with energy, and precise cancellations—achievable only via Higgs exchange—prevent violation of unitarity, especially in processes like $\nu_e\bar{\nu}_e \to W_L^-W_L^+$ and $e^+e^- \to W_L^-W_L^+$. The Higgs ensures the theory remains predictive at high energies.

ABSTRACT

We review the role of the Higgs boson in preserving unitarity of the scattering amplitudes in the Standard Model (SM). We will look at the processes $ν_e + \barν_e ightarrow W^-_L +W^+_L $, $ W^-_L + W^+_L ightarrow W^-_L +W^+_L $ and $e^- + e^+ ightarrow W^-_L +W^+_L $ for longitudinally polarized gauge bosons. Special emphasis will be put in using algebraic methods to evaluate the amplitudes and cross sections. This note is based on Lectures given at the IDPASC Schools in Udine (2012) and Braga (2014).

Motivation & Objective

• To demonstrate that the Higgs boson is required to preserve unitarity in the Standard Model at high energies.
• To analyze scattering amplitudes involving longitudinally polarized W bosons in processes such as $\nu_e\bar{\nu}_e \to W_L^-W_L^+$, $W_L^-W_L^+ \to W_L^-W_L^+$, and $e^+e^- \to W_L^-W_L^+$.
• To show that without the Higgs, scattering amplitudes grow with $\sqrt{s}$, violating unitarity.
• To illustrate how algebraic techniques and symbolic computation (Mathematica, FORM) enable precise cancellation of divergent terms in amplitudes.
• To provide a pedagogical framework using FeynCalc and computational tools for high-energy amplitude calculations in the SM.

Proposed method

• Algebraic evaluation of scattering amplitudes using gauge boson self-couplings and vertex factors derived from the Standard Model Lagrangian.
• Use of polarization vectors for longitudinal and transverse W bosons, with exact and approximate expressions (e.g., $\varepsilon_L^\mu \approx p^\mu / M_W$) in high-energy limits.
• Application of the (V-A) current structure to determine helicity and polarization correlations in $e^+e^- \to W^-W^+$ processes.
• Employment of symbolic computation tools—Mathematica and FORM—for trace evaluations and simplification of $|\mathcal{M}|^2$ expressions.
• Implementation of numerical checks using double-precision Fortran, with emphasis on precision challenges at high energies (e.g., cancellations to $10^{-20}$).
• Hybrid computational strategy: use FORM for fast trace evaluation and Mathematica for simplification, achieving both speed and accuracy.

Experimental results

Research questions

• RQ1 Why is the Higgs boson necessary to maintain unitarity in the Standard Model at high energies?
• RQ2 How do scattering amplitudes for longitudinally polarized W bosons behave without Higgs exchange, and why do they violate unitarity?
• RQ3 What role does the Higgs play in canceling the $\mathcal{O}(s)$ growth of amplitudes in $W_L^-W_L^+ \to W_L^-W_L^+$ processes?
• RQ4 How can algebraic and symbolic computation techniques be used to achieve numerical precision in high-energy amplitude calculations?
• RQ5 What is the optimal computational workflow combining FORM and Mathematica to balance speed and simplification in $|\mathcal{M}|^2$ evaluations?

Key findings

• The Higgs boson is essential to cancel the $\mathcal{O}(s)$ growth of scattering amplitudes in processes involving longitudinally polarized W bosons, preserving unitarity.
• Without the Higgs, amplitudes for $\nu_e\bar{\nu}_e \to W_L^-W_L^+$ and $e^+e^- \to W_L^-W_L^+$ grow with $\sqrt{s}$, violating unitarity at high energies.
• At $\sqrt{s} = 10^7$ GeV, cancellations in the amplitude require precision better than one part in $10^{20}$, highlighting the necessity of exact Higgs contributions.
• The use of the approximation $\varepsilon_L^\mu \approx p^\mu / M_W$ is valid only up to $\mathcal{O}(1/\gamma^4)$ corrections, and more accurate expressions are needed for precision.
• A hybrid computational approach using FORM for trace evaluation and Mathematica for simplification reduces computation time from 350 s (Mathematica only) to ~20 s while maintaining high precision.
• The differential cross section for $e^+e^- \to W^-W^+$ is peaked in the forward direction, and the $\theta = 0$ peak in the TL+LT channel arises from helicity conservation in the (V-A) current.