Catastrophe theoretic approach to the Higgs Mechanism

Abstract

A geometric perspective of the Higgs Mechanism is presented. Using Thom's Catastrophe Theory, we study the emergence of the Higgs Mechanism as a discontinuous feature in a general family of Lagrangians obtained by varying its parameters. We show that the Lagrangian that exhibits the Higgs Mechanism arises as a first-order phase transition in this general family. We find that the Higgs Mechanism (as well as Spontaneous Symmetry Breaking) need not occur for a different choice of parameters of the Lagrangian, and further analysis of these unconventional parameter choices may yield interesting implications for beyond standard model physics.

Figures (5)

• Figure 1: Variation in the number of critical points of the cusp potential in the $b-a$ plane.
• Figure 2: The catastrophe manifold for the Cusp Catastrophe. We can see how the critical points (number and values) vary across the $b-a$ plane.
• Figure 3: The first-order phase transition in the cusp potential is shown. We fix $a = -1$, vary $b$ from $\frac{-1}{4}$ to $\frac{1}{4}$ and plot the cusp potential. The ground states for the potentials are indicated by red dots. a) For $b = -\frac{1}{4}$, we have two negative critical points, and the ground state is at the positive critical point (a minimum). b) At $b = 0$, the minima become degenerate, and the ground state can be at either minimum. Till now, the ground state energy has been decreasing with increasing $b$. c) As soon as $b > 0$, the ground state shifts abruptly to the negative minimum. The ground state energy now suddenly starts decreasing with increasing $b$. Since the ground state was never at $x = 0$, the rate of change of energy given by Eq.(\ref{['deriv']}) changes from a strictly positive value to a strictly negative value. Thus the rate of change of energy has shifted discontinuously, completing the first-order phase transition.
• Figure 4: The Higgs potential defined by Eq.(\ref{['potential']}) is plotted in the $\phi_1-\phi_2$ plane for $\lambda = \mu = 4$. Note the resemblance to Fig.\ref{['phase transition_b=0']}.
• Figure 5: The new Higgs potential (Eq.(\ref{['new_pot']}) is plotted for different values of $b$, with fixed $\lambda = \mu = 4$. The axes are hidden due to a different scaling of each figure to highlight relevant features. Note the cone-like structure at the origin for the non-zero values of $b$, and how the origin acts as a minimum for $b>0$ despite the first derivative with respect to $r$ being non-zero.