Non quadratic potentials and quantization in QFT (home exercise)

In summary, the conversation discusses finding the stationary points and solutions for the potential ##V(\phi)##, and using these solutions to approximate the potential and find the mass spectrum for a fermion field ##\Psi## and a neutral scalar particle ##\chi##. It is noted that the process $$\pi\pi\rightarrow\pi\pi$$ cannot be described at tree level using only Yukawa vertices, leading to confusion about whether expanding the potential loses interactions. The expert suggests that the Taylor expansion is unnecessary and that the mass terms are enough to determine the masses of the fields. Dropping terms in the expansion will result in losing interactions.
  • #1
Homework Statement
I'm given a field theory described by the lagrangian density
$$L=i\bar{\Psi}\gamma^{\mu}\partial_{\mu}\Psi-{{1}\over{2}}\partial_{\mu}\phi\partial^{\mu}\phi-g\phi\bar{\Psi}\Psi-{{\lambda}\over{4}}\phi^4+{{3m^2}\over{2}}\phi^2+{{2m^3}\over{\sqrt{\lambda}}}\phi$$


I'm asked to (1) find all the particles described by this theory, (2) find the Feynman rules and then (3) compute the differential cross section at tree level for the process $$\pi\pi\rightarrow\pi\pi$$, where ##\pi## is the particle described by excitations of ##\phi##
Relevant Equations
I wrote the potential for the ##\phi## field as
$$V(\phi)={{\lambda}\over{4}}\phi^4-{{3m^2}\over{2}}\phi^2-{{2m^3}\over{\sqrt{\lambda}}}\phi$$
I noticed that ##V(\phi)## has nonzero minima, therefore I found the stationary points as ##{{\partial{V}}\over{\partial\phi}}=0##, and found the solutions:
$$\phi^0_{1,2}=-{{m}\over{\sqrt{\lambda}}}\quad \phi^0_3={{2m}\over{\sqrt{\lambda}}}$$
of these, only ##\phi^0_3## is a stable minimum, while the other two solutions are inflection points of ##V##. (I attached a plot of such potential)

Now, I expect that expanding ##V(\phi)## around ##\phi^0_3## should give me the so called "mass spectrum" (because I was told so), so what I did was to approximate ##V(\phi)## around the minimum configuration and substitute such approximated potential in the lagrangian density ##L## as follows:

$$V(\phi)=V(\phi^0_3)+{{1}\over{2}}V^{\prime\prime}(\phi-\phi^0_3)^2+O(\Delta\phi^2)$$

dropping constant terms and higher orders, I found the approximated potential as:

$$V(\phi)={{9m^2}\over{2}}(\phi-{{2m}\over{\sqrt{\lambda}}})^2$$
Then, I defined a new field ##\chi## as the oscillation from the equilibrium position: ##\chi=\phi-{{2m}\over{\lambda}}## and by substituting ##\chi## in ##L## I found a new lagrangian in terms of the fields ##\Psi## and ##\chi##:
$$\tilde{L}=\bar{\Psi}(i\gamma^{\mu}\partial_{\mu}-{{2mg}\over{\sqrt{\lambda}}})\Psi-{{1}\over{2}}(\partial_{\mu}\chi)^2-g\chi\bar{\Psi}\Psi-{{9m^2}\over{2}}\chi^2$$
So I conclude that this lagrangian describes a fermion field ##\Psi## of mass ##m_{\Psi}=2mg/\sqrt{\lambda}## and a neutral scalar particle ##\chi## of mass ##m_{\chi}=3m##. The Feynman rules consist of those from Yukawa theory and come from the ##g\chi\bar{\Psi}\Psi## term.

This is where I get confused. Assuming that the ##\pi## particle is the one associated with the field ##\chi##, the process $$\pi\pi\rightarrow\pi\pi$$ cannot be described at tree level using only Yukawa vertices. This makes me believe that I got the interaction terms wrong, and that I should not have expanded ##V##, however I remember hearing something about "fields are just excitations around stable configurations" and "we can only quantize around the minimum" which was the initial motivation to do all of the above.

I guess my question is if it is always sensible to expand the potential at quadratic order, or if in doing so I loose interactions?
Am I missing something?
 

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  • #2
You don't need the Taylor expansion part. You get the masses by looking at the mass terms, ie the ones quadratic in fields. Where did you see that you have to do the Taylor expansion? Perhaps they didn't mean Taylor expansion but rather what you did with the χ field. After all doing a Taylor expansion doesn't make that much sense here, it is already a polynomial. Dropping terms will cost you interactions.
 
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1. What is a non-quadratic potential in quantum field theory (QFT)?

A non-quadratic potential in QFT refers to a potential energy function that is not proportional to the square of the field. In other words, it is a potential that cannot be written in the form of V(φ) = m²φ², where m is the mass of the field and φ is the field itself. Non-quadratic potentials are important in QFT because they allow for the study of more complex and realistic physical systems.

2. How do non-quadratic potentials affect the quantization process in QFT?

Non-quadratic potentials can significantly complicate the quantization process in QFT. In the standard approach, known as canonical quantization, the Hamiltonian of a system is written in terms of the fields and their conjugate momenta. However, with a non-quadratic potential, the conjugate momenta cannot be expressed solely in terms of the fields, making the quantization process more challenging.

3. What are some examples of non-quadratic potentials in QFT?

One example of a non-quadratic potential is the quartic potential, V(φ) = λφ⁴, which is commonly used in scalar field theories. Another example is the Higgs potential, V(φ) = μ²φ² - λφ⁴, which is used in the Standard Model of particle physics to give particles mass. Non-quadratic potentials can also arise in theories involving fermions, such as the Yukawa potential, V(φ) = gφψψ, which describes the interaction between a scalar field and a fermion field.

4. How do non-quadratic potentials affect the behavior of particles in QFT?

The behavior of particles in QFT is determined by the potential energy function of the system. Non-quadratic potentials can lead to more complex and interesting behavior, such as the creation of new particles or the breaking of symmetries. They can also affect the scattering amplitudes and cross sections of particles, leading to different predictions for experimental results.

5. Are there any techniques for dealing with non-quadratic potentials in QFT?

Yes, there are several techniques for dealing with non-quadratic potentials in QFT. One approach is to use perturbation theory, which involves expanding the potential in a series and calculating the effects of each term. Another technique is to use numerical methods, such as lattice QFT, which allows for the simulation of non-quadratic potentials on a discrete grid. Additionally, there are alternative quantization methods, such as path integral quantization, which can be used to handle non-quadratic potentials more easily.

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