Section 1.1: The Landau-Ginzburg Model

Physical systems are typically described using a Lagrangian, which combines a system's kinetic and potential energy. Often, this Lagrangian maintains symmetry under certain transformations. For example, if the potential of a Lagrangian resembles a cylindrical shape, it exhibits rotational symmetry, as rotations would not alter its appearance. Conversely, in high-energy scenarios, the symmetry of the Lagrangian tends to be preserved.

To delve deeper into this concept, let's consider a mechanism governing a collection of atoms within a magnetic material, initially proposed by physicists Ginzburg and Landau. They formulated a Lagrangian for the magnetic material to encapsulate its essential characteristics. This Lagrangian serves as a compact representation of a physical system's information, often based on its symmetry properties. Below, I've provided a qualitative version of the Lagrangian to enhance comprehension.

Since the aim was to describe the collective behavior of numerous particles, the Lagrangian represents a field rather than a single particle. This field is denoted by the Greek letter phi in the expression above. Essentially, a field refers to a physical entity that possesses attributes at every location in space, akin to temperature in a room. The first term of the Lagrangian accounts for the kinetic energy of the field alone. Additionally, a term reflecting a 'mass' property is included. The last term captures interactions among particles, with the interaction strength represented by a specific number. Typically, this interaction term must remain small to accurately depict the physics involved.

When the field exhibits a positive mass, the Lagrangian resembles a potential well, with the 'stable state' of the field located at the bottom of this well. Physicists refer to this as the ground state, a crucial aspect of analysis. In instances of positive mass, the ground state occurs when the field value is zero. Most inquiries in this domain revolve around the ground state.

What happens when the temperature decreases?

Landau and Ginzburg posed intriguing questions to derive exciting physics from this Lagrangian. One such question was how the system behaves as its temperature drops, resulting in reduced energy. Given that the Lagrangian does not explicitly include a temperature term, how can we account for it? Their initial assumption was to make the mass variable depending on temperature, leading them to develop a model where mass becomes negative if the temperature dips low enough.

What does it imply when mass is negative? According to Einstein's special relativity, there is a relationship between an object's mass and its permissible speed. This relationship, known as the energy-momentum relation, can be articulated both verbally and mathematically.

This relationship reveals several key insights. First, it indicates that particles with positive mass cannot exceed the speed of light within any inertial reference frame—these are the particles we encounter in our everyday lives. Additionally, it states that massless particles must travel at the speed of light, which explains why photons, as packets of light, possess no mass. Lastly, it suggests that particles with negative mass must move faster than light, leading to the concept of tachyons, which challenge causality in physics and are thus problematic for any coherent physical theory.

Consequently, when the temperature drops sufficiently, tachyons may emerge, presenting an issue in this theoretical framework. To address this, Landau and Ginzburg modified the Lagrangian by reversing the mass's sign. The temperature at which mass transitions to a negative value is termed the critical temperature. When the temperature is low enough, the potential effectively inverts, altering the position of minimum points. Previously, the system had a single minimum point; now, there are two.

The diagram illustrates two possible scenarios. On the left, the temperature is high, and the mass is positive. On the right, the temperature is low. What occurs at the zero point? This area atop the hill is unstable—if a particle resting there is nudged, it will naturally descend to one of the ground states on either side, as depicted by the black dot on the right diagram. Thus, the former ground state associated with higher temperatures becomes unstable, replaced by two distinct lower ground states.

The transition from a single ground state to two potential ground states exemplifies what is meant by ‘symmetry breaking.’ At sufficiently low temperatures, the physical system must now select a solution: either the positive ground state on the right or the negative ground state on the left. Previously, the Lagrangian was indifferent to 'positive' or 'negative' signs. Now, with the symmetry broken at lower temperatures, the system must choose between these two ground states, which lies at the core of the symmetry breaking mechanism.

Section 1.2: Goldstone Bosons

Now that we understand the implications of broken symmetry in our system, how do we produce a particle from it? Since the Lagrangian was originally symmetric, Noether's theorem asserts that a charge is linked to this symmetry. For instance, symmetries in quantum electrodynamics give rise to what we now recognize as electric charge. When one of these ground states is excited through this charge, a massless particle can emerge, akin to a photon! The massless particles resulting from symmetry breaking are referred to as Goldstone Bosons. These bosons do not exist at the unstable state but only at the ground states where symmetry has been disrupted.

Wrap-Up

In conclusion, I hope this exploration of the principles of symmetry breaking and its connection to particle physics has been enlightening. Keep an eye out for future discussions on this intriguing topic!

References

[1] Miransky, Vladimir A. (1993). Dynamical Symmetry Breaking in Quantum Field Theories. p. 15. ISBN 9810215584.