Spontaneous Symmetry Breaking: The Mexican Hat Potential

The Complex Scalar Field and the "Mexican Hat"

Let's consider a theory of a complex scalar field $\varphi (x)$ in $3+1$ dimensions. The Lagrangian is given by:

This Lagrangian is invariant under the global U(1) phase rotation $\varphi (x)\to e^{i\alpha }\varphi (x)$, where $\alpha$ is a constant. This symmetry, as we'll see, is responsible for charge conservation in theories like QED.

As in the real scalar case, we'll choose a potential that enables SSB. Let's take:

where $\lambda >0$ ensures the potential is bounded from below. If $m^2>0$, the potential has a unique, U(1)-symmetric minimum at $\varphi =0$. The interesting physics occurs when we take $m^2<0$. As before, we'll set $m^2=-{\mu }^2$ with ${\mu }^2>0$. The potential becomes:

If we write the complex field in terms of its real components, $\varphi ={\varphi }_1+i{\varphi }_2$, the potential is a function of two variables: $V({\varphi }_1,{\varphi }_2)=-{\mu }^2({\varphi }_1^2+{\varphi }_2^2)+\frac{\lambda }{2}({\varphi }_1^2+{\varphi }_2^2{)}^2$. This is the "Mexican hat" potential, a surface of revolution formed by rotating our previous double-well potential around the vertical axis.

The minimum of this potential is no longer at $\varphi =0$. Instead, we find the minima by setting the derivative with respect to $|\varphi |$ to zero:

This gives a local maximum at $|\varphi |=0$ and a circle of minima defined by:

This continuous circle of degenerate ground states, with radius $v$, is the vacuum manifold, ${\mathcal{M}}_0=S^1$. The system must settle on a single point on this circle, for example, the state represented by the red dot in the image. Any such choice, say $⟨\varphi ⟩=v$, is not invariant under the U(1) transformation $\varphi \to e^{i\alpha }\varphi$, which would move the system to a different point on the circle. Thus, the U(1) symmetry is spontaneously broken.

Excitations Around the Vacuum: Massive and Massless Modes

To understand the physics of the broken symmetry phase, we must analyze the small fluctuations (particles) around a chosen vacuum. Let's pick the vacuum to be at a real value $⟨\varphi ⟩=v$. We can parameterize the fluctuations in two physically intuitive ways.

A short video from ZAP Physics provides an excellent conceptual picture of what to expect.

The video highlights that fluctuations in the radial direction (climbing the "walls" of the hat) should be massive, while fluctuations in the azimuthal direction (moving along the "brim") should be massless. Let's now demonstrate this mathematically. We'll follow the clear presentation in David Tong's lecture notes.

Let's summarize and elaborate on the key steps from the reading. We parameterize the complex field $\varphi (x)$ using polar coordinates in field space:

Here, $r(x)$ is the radial field and $\theta (x)$ is the angular (or phase) field. The potential only depends on the radius, $V(r)=-{\mu }^2{r}^2+\frac{\lambda }{2}{r}^4$, and its minimum is at $r=v$. The kinetic term becomes:

The full Lagrangian is then:

Now, we expand around a chosen vacuum, which in these coordinates corresponds to $r=v$ and some constant value of $\theta$, say ${\theta }_0=0$. We define the fluctuation fields $\rho (x)$ and $\eta (x)$ (let's use $\eta$ for the angular field to avoid confusion with the radial field $\rho$ in some texts) as:

Substituting these into the Lagrangian and keeping terms up to quadratic order in the fluctuation fields gives us the theory of small excitations.

1. The Radial Mode ($\rho$):
   The potential can be written as $V(r)=\frac{\lambda }{2}(r^2-v^2{)}^2+\text{const}$.
   For the fluctuation $\rho$, this becomes:

   $$V(v+\rho )=\frac{\lambda }{2}((v+\rho {)}^2-v^2{)}^2+\text{const}=\frac{\lambda }{2}(2v\rho +{\rho }^2{)}^2+\text{const}\approx 2\lambda v^2{\rho }^2$$

   The Lagrangian for $\rho$ contains the terms $({\partial }_{\mu }\rho {)}^2-2\lambda v^2{\rho }^2$. This is the Lagrangian for a scalar particle with a mass-squared of:

   $$M_{\rho }^2=4\lambda v^2=4{\mu }^2$$

   This is the massive mode, corresponding to fluctuations in the radial direction, "up the hill" of the potential.

2. The Angular Mode ($\eta$):
   The potential $V(r)$ does not depend on $\theta$ at all. This is a direct consequence of the U(1) symmetry. Therefore, there is no potential term for the $\eta (x)$ field, such as ${\eta }^2$. The only term involving $\eta$ at the lowest order comes from the kinetic part:

   $$r^2({\partial }_{\mu }\theta {)}^2=(v+\rho {)}^2({\partial }_{\mu }\eta {)}^2\approx v^2({\partial }_{\mu }\eta {)}^2$$

   After rescaling the field to have a canonical kinetic term, the Lagrangian for the angular mode is simply that of a free, massless scalar field. This particle is called a Goldstone boson (or sometimes a Nambu-Goldstone boson). It corresponds to fluctuations along the degenerate circle of minima, which costs no potential energy.

This result is central: the spontaneous breaking of a continuous global symmetry has given rise to a massless particle in the theory's spectrum. This is not an accident of our U(1) model but a completely general feature.

For a more general case, consider the O(N) linear sigma model, which is a theory of N real scalar fields ${\varphi }^a$ with a potential symmetric under O(N) rotations. The logic is identical.

In the O(N) model, the vacuum manifold is an $(N-1)$-dimensional sphere, $S^{N-1}$. When we pick a vacuum, say along the N-th direction, we break the O(N) symmetry. However, a subgroup of rotations, O(N-1), that leaves this vacuum direction invariant remains unbroken. The fluctuations consist of:

• One massive particle, corresponding to oscillations radial to the $S^{N-1}$ sphere.
• $N-1$ massless Goldstone bosons, corresponding to fluctuations along the $N-1$ directions of the sphere.

Conclusion

In this lesson, we have successfully demonstrated spontaneous symmetry breaking in the canonical "Mexican hat" model. By extending our analysis from a discrete to a continuous U(1) symmetry, we uncovered a profound consequence that goes beyond simple mass generation.

Key Takeaways:

• Mexican Hat Potential and Vacuum Manifold: A scalar field theory with a continuous symmetry and a negative mass-squared term features a "Mexican hat" potential. The minima of this potential form a continuous set known as the vacuum manifold (${\mathcal{M}}_0$). For a complex scalar with U(1) symmetry, ${\mathcal{M}}_0=S^1$.
• SSB and Particle Spectrum: The system spontaneously chooses a single ground state from the vacuum manifold, breaking the continuous symmetry.
• Massive and Massless Excitations: Fluctuations around the chosen vacuum split into two types:
  • Massive modes: Excitations perpendicular to the vacuum manifold (the "radial" direction), which cost potential energy.
  • Massless modes (Goldstone bosons): Excitations along the directions of the vacuum manifold (the "angular" or "phase" direction), which cost no potential energy due to the original symmetry.

Preview of the Next Lesson:

The appearance of a massless boson in our model was no accident. It is a direct and necessary consequence of breaking a continuous global symmetry. In our next lesson, we will formalize this observation by stating and proving Goldstone's Theorem. This theorem provides a powerful link between symmetry, its spontaneous breaking, and the particle content of a physical theory, with deep implications for both condensed matter and particle physics.