30 | July | 2008 | The Gauge Connection

Fermions and massless scalar field

30/07/2008

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!