## A light Higgs indeed!

02/08/2008

Tommaso Dorigo is shocking us in these days with a striking post after another. Today he posted this one where there is evidence that the Higgs is light indeed being between 115-135 GeV and there are reasons to regret. The most severe of these is the shutdown of LEP that Luciano Maiani was forced to order to start LHC construction. More time would have been given to this people and surely now we would not stay still waiting. But this was not Maiani’s fault. Luciano Maiani is a great physicist and has been my professor at “La Sapienza” where he tried to teach me quantum mechanics. Today I cannot say if he succeeded but I can hide myself behind Feynman’s view to be safe… Maiani was just forced to close LEP to respect scheduling and, I can guess, for the allocated budget at that time. This was the only logical choice. Now a great window is surely open for Fermilab to anticipate the discovery. We are eager to see. Meantime we can say that Lubos Motl is half right, we hope for the other half…

Update: For some guess about what to expect at LHC, Sean Carroll has posted this. We are all eager to see. Bets are on…

## Exact solution to a classical spontaneously broken scalar theory

02/08/2008

As promised in my preceding post I said that a classical spontaneously broken scalar theory can be exactly solved. This is true as I will show. Consider the equation

$\ddot\phi -\Delta\phi + \lambda\phi^3-m^2\phi=0.$

You can check by yourself that the exact solution is given by

$\phi(x)=v\cdot{\rm dn}(p\cdot x,i)$

being $v=\sqrt{2m^2/3\lambda}$ the v.e.v. of the field and ${\rm dn}$ an elliptical Jacobi function. As always the following dispersion relation must be true

$p^2=\frac{\lambda v^2}{2}$

giving a consistent classical solution. When one goes to see the spectrum of the theory, the Fourier series of the Jacobi dn function has a zero mass excitation, the Goldstone boson.

Update: A proper full solution is given by

$\phi(x)=v\cdot{\rm dn}(p\cdot x+\varphi,i)$

being $\varphi$ an integration constant.