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!

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