How Accurate is the Equation of Motion Derived from This Lagrangian?

[Hypo86]

Hi!

I have the following problem with some old lecture notes I recently had a look on.

I have two different fermions (1 and 2) with masses m1 and m2
and the following Lagrangian (where the mass term for fermion 2 is dropped, because we are only interested in the dynamics of fermion 1) of the form:

$$\mathcal{L} = \bar{\Psi}_{1} \left( i {\not}{\partial} - {\not}{A} \left( 1 - \gamma_{5} \right) - m_{1} \right) \Psi_{1},$$

where

$$A^{\mu} = \dfrac{G_{F}}{\sqrt{2}} \left[ \bar{\Psi}_{2} \gamma^{\mu} \left( 1 - \gamma_{5} \right) \Psi_{2} \right] =: (\varphi, \vec{A})$$

If we compute the equation of motion from that (Euler-Lagrange), one finds:

$$\left( i \dfrac{\partial}{\partial t} - \varphi \right) \Psi_{1} = \left[ \vec{\alpha} \cdot \left( \dfrac{1}{i} \nabla - \vec{A} \right) + \beta m_{1} \right] \Psi_{1},$$

where $\beta = \gamma^{0}$ and $\vec{\alpha} = \beta \vec{\gamma}$

I think this is not correct, because there are two $(1 - \gamma_{5})$ factors missing.
I find:

$$\left( i \dfrac{\partial}{\partial t} - \varphi ( 1 - \gamma_{5} ) \right) \Psi_{1} = \left[ \vec{\alpha} \cdot \left( \dfrac{1}{i} \nabla - \vec{A} ( 1 - \gamma_{5} \right) + \beta m_{1} \right] \Psi_{1}$$

Did I miss anything? I would be glad if someone could have a short look on it.
Thanks a lot!

 

[nrqed]

Hi!

I have the following problem with some old lecture notes I recently had a look on.

I have two different fermions (1 and 2) with masses m1 and m2
and the following Lagrangian (where the mass term for fermion 2 is dropped, because we are only interested in the dynamics of fermion 1) of the form:

$$\mathcal{L} = \bar{\Psi}_{1} \left( i {\not}{\partial} - {\not}{A} \left( 1 - \gamma_{5} \right) - m_{1} \right) \Psi_{1},$$

where

$$A^{\mu} = \dfrac{G_{F}}{\sqrt{2}} \left[ \bar{\Psi}_{2} \gamma^{\mu} \left( 1 - \gamma_{5} \right) \Psi_{2} \right] =: (\varphi, \vec{A})$$

Hi, welcome to PF!Can you double check what you wrote above? It does not make sense as stated. A is a gauge field, it should not be expressed in terms of the lagrangian of the second fermion. It looks like what you wrote for A^mu should be a term in the lagrangian, not A^mu