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spaghetti3451

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Will it make any difference to your Feynman rules if you expand the Lagrangian around different minima of the potential?

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In summary, spontaneous symmetry breaking involves expanding the Lagrangian around a potential minimum and using this to write Feynman rules. Whether different minima are chosen will affect the basis in which the rules are written, and in turn, the physical predictions for the theory. This can be seen in a simpler theory with a ##\phi \rightarrow -\phi## symmetry, where the cubic self-interaction term and the masses of the electron and muon are dependent on the chosen minimum. Therefore, expanding the Lagrangian around different minima will result in different physical predictions.

- #1

spaghetti3451

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Will it make any difference to your Feynman rules if you expand the Lagrangian around different minima of the potential?

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If you chose another basis you will just get SU(2) rotated Feynman rules.

- #3

spaghetti3451

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How about a simpler theory:

Let's say we have the following theory with ##\phi \rightarrow -\phi## symmetry under ##\psi_{i}\rightarrow \gamma_{5}\psi_{i}##:

$$\mathcal{L} = \bar{\psi}_{e}(i\gamma^{\nu}{\partial_{\nu}}-y_{\mu}\phi)\psi_{\mu}+\frac{1}{2}(\partial_{\mu}\phi)^{2}-V(\phi)$$

with $$V(\phi) = -\frac{1}{2}|\kappa^{2}|\phi^{2} + \frac{\lambda}{24}\phi^{4}$$

It can be shown that this theory has two true vacua at ##\phi = \pm \nu##, and after expanding about ##\phi(x) = \nu + h(x)##, we get

$$\underbrace{\bar{\psi}_{e}(i\gamma^{\nu}{\partial_{\nu}}-y_{e}\nu)\psi_{e}}_{\text{Dirac Lagrangian for field $\psi_{e}$ with mass $y_{e}\nu$}} \qquad \underbrace{-y_{e}\bar{\psi}_{e}h\psi_{e}}_{\text{interaction term coupling field $\psi_{e}$ with field $h$ with Yukawa coupling $y_{e}$}} +\underbrace{\frac{1}{2}(\partial_{\mu}h)(\partial^{\mu}h)-\frac{1}{2}\left(2|\kappa^{2}|\right)h^{2}}_{\text{Klein-Gordon Lagrangian for field $h$ with mass $\displaystyle{\sqrt{2|\kappa^{2}|}}$}}\\ \\ \underbrace{-\frac{\lambda}{6}\nu h^{3}}_{\text{cubic self-interaction term for field $h$ with coupling constant $\displaystyle{\frac{\lambda}{6}\nu}$}}\qquad \underbrace{-\frac{\lambda}{24}h^{4}}_{\text{quartic self-interaction term for field $h$ with coupling constant $\displaystyle{\frac{\lambda}{24}}$}}$$

I can see clearly that the cubic self-interaction term as well as the masses of the electron and the muon depend on the value of ##\nu##. Does this not mean that different minima give different physical predictions?

Let's say we have the following theory with ##\phi \rightarrow -\phi## symmetry under ##\psi_{i}\rightarrow \gamma_{5}\psi_{i}##:

$$\mathcal{L} = \bar{\psi}_{e}(i\gamma^{\nu}{\partial_{\nu}}-y_{\mu}\phi)\psi_{\mu}+\frac{1}{2}(\partial_{\mu}\phi)^{2}-V(\phi)$$

with $$V(\phi) = -\frac{1}{2}|\kappa^{2}|\phi^{2} + \frac{\lambda}{24}\phi^{4}$$

It can be shown that this theory has two true vacua at ##\phi = \pm \nu##, and after expanding about ##\phi(x) = \nu + h(x)##, we get

$$\underbrace{\bar{\psi}_{e}(i\gamma^{\nu}{\partial_{\nu}}-y_{e}\nu)\psi_{e}}_{\text{Dirac Lagrangian for field $\psi_{e}$ with mass $y_{e}\nu$}} \qquad \underbrace{-y_{e}\bar{\psi}_{e}h\psi_{e}}_{\text{interaction term coupling field $\psi_{e}$ with field $h$ with Yukawa coupling $y_{e}$}} +\underbrace{\frac{1}{2}(\partial_{\mu}h)(\partial^{\mu}h)-\frac{1}{2}\left(2|\kappa^{2}|\right)h^{2}}_{\text{Klein-Gordon Lagrangian for field $h$ with mass $\displaystyle{\sqrt{2|\kappa^{2}|}}$}}\\ \\ \underbrace{-\frac{\lambda}{6}\nu h^{3}}_{\text{cubic self-interaction term for field $h$ with coupling constant $\displaystyle{\frac{\lambda}{6}\nu}$}}\qquad \underbrace{-\frac{\lambda}{24}h^{4}}_{\text{quartic self-interaction term for field $h$ with coupling constant $\displaystyle{\frac{\lambda}{24}}$}}$$

I can see clearly that the cubic self-interaction term as well as the masses of the electron and the muon depend on the value of ##\nu##. Does this not mean that different minima give different physical predictions?

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Spontaneous symmetry breaking is a phenomenon in which a system that is initially symmetrical or uniform chooses a particular state or direction, breaking its symmetry. This can occur in physical systems such as magnets or subatomic particles, as well as in abstract systems such as mathematical equations.

A potential minimum is a state of lowest energy in a system. In the context of spontaneous symmetry breaking, it refers to the state that a system settles into after breaking its symmetry. This state is often referred to as the "ground state" or "vacuum state" of the system.

Spontaneous symmetry breaking occurs when the potential energy of a system has multiple minima, and the system is in a state of unstable equilibrium. Small fluctuations or disturbances can cause the system to shift to a new, lower-energy minimum, breaking its initial symmetry.

Potential minima play a crucial role in spontaneous symmetry breaking. They represent the different possible states that a system can exist in, and the system will tend to settle into the state with the lowest energy. As the system undergoes symmetry breaking, it transitions from one potential minimum to another, ultimately settling into its lowest-energy state.

Spontaneous symmetry breaking occurs in many physical systems, such as ferromagnets, superconductors, and liquid crystals. It also plays a role in the formation of crystals, where atoms arrange themselves into a regular lattice structure. In the field of particle physics, the Higgs mechanism is an example of spontaneous symmetry breaking, which gives particles mass by breaking the symmetry between the weak and electromagnetic forces.

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