Spontaneous Symmetry Breaking and the Higgs Mechanism

TL;DR

The work offers a concise, pedagical account of spontaneous symmetry breaking and the Higgs mechanism, connecting global symmetry breaking and Goldstone modes to the Anderson-Higgs mechanism that generates gauge boson masses in the electroweak sector. It builds the general framework for non-Abelian gauge theories, illustrates the absorption of Goldstone modes by gauge bosons, and applies these concepts in detail to the electroweak Standard Model, including fermion masses via Yukawa couplings and the resulting W, Z, and photon spectra. The analysis highlights how the Higgs field provides mass generation without violating gauge invariance and discusses predictive aspects such as coupling structures and mass relations, as well as the role of the Weinberg angle in gauge mixing. Finally, it surveys open questions about the origin and naturalness of the Higgs sector and points to beyond-Standard-Model ideas, such as composite Higgs scenarios, as potential avenues to address these issues and guide future experiments.

Abstract

The Higgs sector of the standard model of particle physics plays a central role in the generation of all the masses of elementary particles known so far. Here we give a pedagogical introduction to all the elements leading ot the Higgs mechanism and the Higgs boson, starting with the spontaneous symmetry breaking of global symmetries and the Goldstone theorem. We then consider the case of gauge symmetries, i.e. the Higgs mechanism, and its application to the electroweak sector of the standard model. We close with a reflection on the possible open questions that the very introduction of the Higgs sector in the standard model posses.

Figures (7)

• Figure 1: The $U(1)$ rotation $\phi\to e^{i\alpha}\phi$ for an initially real field.
• Figure 2: The mexican hat potential for the $\mu^2<0$ case in Eq. (\ref{['potential1']}). The condition (\ref{['vdef2']}) corresponds to the circle at the botton of the potential, the circle of radius $v$.
• Figure 3: The red circle represents the locus points of the minimum of the potential (\ref{['potential1']}) for $\mu^2<0$. The radius is $v$, a real number. The phase is not determined by the minimization. Here $phi_1=Re[\phi$ and $\phi_2=Im[\phi]$.
• Figure 4: Feynman rule for the non-diagonal contribution to the two-point function in (\ref{['kinscal1']}).
• Figure 5: New contributions to the gauge boson two-point function at tree level in the presence of spontaneous symmetry breaking. The first diagram is the gauge boson mass term insertion. The second one corresponds to the massless NGB contribution.