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- 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?

$$\phi^0_{1,2}=-{{m}\over{\sqrt{\lambda}}}\quad \phi^0_3={{2m}\over{\sqrt{\lambda}}}$$

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?