Symmetry Restoration in QFT: Exploring V_{eff}(\phi)

[RedX]

In Zee's book on QFT he derives this equation for the effective potential in $$\phi^4$$ theory:

$$V_{eff}(\phi)=\frac{1}{4!}\lambda(M)\phi^4+\frac{\lambda(M)^2}{(16 \pi)^2}\phi^4[log\frac{\phi^2}{M^2}-\frac{25}{6}]+O(\lambda(M)^3)$$

The effective potential is basically the potential with quantum corrections built in and is basically what an experimentalist would measure.

I should note that there is no mass term proportional to $$\phi^2$$ in $$V_{eff}(\phi)$$ because Zee is enforcing that the mass term is zero by hand by choice of renormalization constants - the model he is considering really does have its physical mass equal to zero.

From the equation, symmetry is broken because of the $$\phi^4 log(\phi^2)$$ term, so that although the original potential is simply $$V(\phi)=\lambda \frac{\phi^4}{4!}$$ and hence has a minimum at $$\phi=0$$, the effective potential has a potential not at 0 because the log term becomes infinitely negative at small $$\phi$$ so the minimum of the potential occurs at small values of $$\phi$$ not quite zero, so symmetry is broken and all the consequences of symmetry-breaking such as particles getting mass hold.

My question is how is symmetry restored at high energies from this equation for the effective potential? If the effective potential is always of that form, then isn't there no way of escaping symmetry breaking? The only variable in the equation that can be adjusted is the mass scale M, so at high energies M should increase, but I don't see how an increase of M (and the accompanying increase in $$\lambda$$ by the beta function) changes the fact that the log term will cause symmetry breaking.

In Kaku's book on QFT there is a homework problem that asks you to prove that symmetry restoration occurs at a temperature $$T^2=-\frac{24m^2}{\lambda}$$. This takes a bit of thermal QFT and the imaginary-time formalism (Kaku's book is sometimes ridiculous in what it expects you to know) and the calculation is not very simple, but conceptually you can see how symmetry is restored at that temperature. But I can't see how varying the energy restores symmetry.

 

[eljose79]

Thank you for bringing up this interesting topic! It is true that at first glance, it may seem like the effective potential equation does not allow for symmetry restoration at high energies. However, there are a few key points to keep in mind when considering this issue:

1. The effective potential is a tool for calculating the vacuum expectation value (VEV) of the field, not the actual potential itself. The VEV is related to the physical mass of the particle, but it is not the same as the potential. So, while the effective potential may have a minimum at a non-zero value of \phi, this does not necessarily mean that the potential itself has a non-zero minimum.

2. The effective potential is calculated at a given energy scale, which is determined by the renormalization scale M. As you mentioned, changing this scale will change the value of the effective potential, but it does not change the underlying physics. In other words, the effective potential is not a physical quantity that can be directly measured, but rather a mathematical tool for understanding the behavior of the system at different energy scales.

3. The effective potential is only one part of the full picture. In order to fully understand the behavior of the system, we need to consider all possible interactions and their effects on the potential. This includes higher-order corrections, as well as the effects of temperature (as in the case of the homework problem you mentioned).

In summary, while the effective potential equation may seem to suggest that symmetry breaking is always present, it is important to keep in mind that this is just one aspect of the full picture. The behavior of the system at high energies is determined by a combination of factors, including the full potential, higher-order corrections, and the effects of temperature. Only by considering all of these factors can we fully understand the restoration of symmetry at high energies.

 

[Entropia]

Symmetry restoration in QFT is a complex and fascinating phenomenon, and the equation for the effective potential in Zee's book is just one aspect of it. To understand how symmetry is restored at high energies, we need to look at the concept of renormalization.

Renormalization is a mathematical technique used in QFT to remove infinities that arise in the theory due to quantum corrections. In the context of the effective potential, it involves adjusting the parameters of the theory, such as the mass scale M and the coupling constant \lambda, in order to cancel out the infinities that arise in the equation.

At low energies, where the log term in the effective potential dominates, the symmetry is broken and the minimum of the potential occurs at a non-zero value of \phi. However, as we increase the energy (or equivalently, decrease the distance scale), the log term becomes less important and the other terms in the equation start to dominate. At very high energies, the log term becomes negligible and the effective potential takes on a simpler form without the log term.

This means that at high energies, the symmetry is effectively restored because the effective potential becomes similar to the original potential without the quantum corrections. This is achieved by adjusting the parameters of the theory, such as M and \lambda, through the process of renormalization.

Furthermore, the concept of symmetry restoration at high energies is closely related to the concept of spontaneous symmetry breaking. In QFT, spontaneous symmetry breaking occurs when the ground state of the theory does not possess the same symmetries as the Lagrangian. This is similar to the concept you mentioned in your question, where the minimum of the potential occurs at a non-zero value of \phi instead of 0.

However, at high energies, the ground state of the theory is no longer the minimum of the effective potential, but rather the minimum of the original potential without the quantum corrections. This is why symmetry is restored at high energies.

In summary, symmetry restoration in QFT is a result of the interplay between renormalization and spontaneous symmetry breaking. At high energies, the parameters of the theory are adjusted through renormalization to effectively remove the quantum corrections and restore the original symmetry of the theory.

 

[Entropia]

Firstly, I would like to clarify that symmetry restoration in QFT is not a process that occurs at high energies, but rather at high temperatures. This is because at high temperatures, the thermal energy is enough to overcome the potential barrier and allow the field to fluctuate around the minimum of the potential, effectively restoring the symmetry.

Now, coming to the equation for the effective potential, it is important to note that the log term becomes infinitely negative only at very small values of \phi, and as the field increases, the log term becomes less and less negative. This means that at high energies, where the field values are larger, the log term becomes less significant and the potential essentially becomes dominated by the first term, which is symmetric.

Furthermore, as you have correctly pointed out, at high energies the mass scale M increases, which in turn leads to an increase in the coupling constant \lambda. This increase in \lambda counteracts the effects of the log term, making the potential more symmetric. In fact, at very high energies, the potential becomes completely symmetric and the symmetry is fully restored.

In summary, at high energies, the log term becomes less significant and the potential is dominated by the symmetric term. Additionally, the increase in the mass scale and coupling constant counteracts the effects of the log term, leading to complete symmetry restoration at very high energies. However, as mentioned earlier, symmetry restoration at high temperatures is a separate phenomenon that occurs due to thermal effects.