Fermions and massless scalar field

As I always do to take trace of some computations I am carrying on here and there, I put them on the blog. This time I devised to solve Dirac equation using the massive solution of the massless scalar field given here and here:

$\phi(x)=\mu\left(2 \over \lambda\right)^{1 \over 4}{\rm sn}(p\cdot x,i).$

$\mu$ is an arbitrary parameter, $\lambda$ the coupling and ${\rm sn}$ the snoidal Jacobi elliptical function. An arbitrary phase $\varphi$ can be added but we take it to be zero in order to keep formulas simpler. Now, we couple this field to a massless fermion field and one has to solve Dirac equation

$(i\gamma\partial+\beta \phi)\psi=0$

where we have used the following Yukawa model for the coupling

$L_{int}=\Gamma\bar\psi\psi\phi$

so that $\beta=\Gamma\mu\left(2 \over \lambda\right)^{1 \over 4}$ . Dirac equation with such a field can be solved exactly to give

$\psi(x)=e^{-iq\cdot x}e^{-i\frac{p\cdot q}{m_0^2}p\cdot x-\beta\frac{\gamma\cdot p}{m_0^2}[\ln({\rm dn}(p\cdot x,i)-i{\rm cn}(p\cdot x,i))-\ln(1-i)]}u_q$

being $m_0=\mu (\lambda /2)^{1\over 4}$ the mass acquired by the scalar field and ${\rm dn}$ and ${\rm cn}$ two other Jacobi elliptical functions. This formula says us an interesting thing, that is there is a fermion excitation with zero mass unless a mass is initially given to the fermion. Such a conclusion is reminiscent of the pion status in QCD. So, the computation may seem involved but the conclusion is quite rewarding!

This entry was posted on Wednesday, July 30th, 2008 at 4:38 pm and is filed under Physics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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