Higgs mechanism is essential

30/10/2009

As the readers of my blog know, I have developed, in a series of papers, the way to manage massive solutions out of massless theories, both in classical and quantum cases. You can check my latest preprints here and here. To have an idea, if we consider an equation

\Box\phi+\lambda\phi^3=0

then a solution is

\phi(x)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p\cdot x+\theta,i)

provided

p^2=\mu^2\left(\frac{\lambda}{2}\right)^\frac{1}{2}

being \mu and \theta two arbitrary constants and \rm sn a Jacobi elliptical function. We see that a massless theory has massive solutions arising just from a strong nonlinearity into the equation of motion. The question one may ask is: Does this mechanism work to give mass to particles in the Standard Model? The answer is no and this can already be seen at a classical level. To show this, let us consider the following Yukawa model

L=\bar{\psi}(i\gamma\cdot\partial-g\phi)\psi+\frac{1}{2}(\partial\phi)^2-\frac{\lambda}{4}\phi^4

being g a Yukawa coupling. Assuming \lambda very large, one is reduced to the solution of the following Dirac equation that holds at the leading order

\left(i\gamma\cdot\partial-g\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p\cdot x+\theta,i)\right)\psi=0

and this equation is exactly solved in a closed form, provided the fermion has exactly the same mass of the boson, that is g=\sqrt{\lambda/2}. So, we see that the massless fermion acquires mass too but it must be degenerate with respect to the bosonic field. This would imply that one needs a different scalar field for each fermion and such bosons would have the same masses of the fermions. This is exactly what happens in a supersymmetric theory but the theory we are considering is not. So, it would be interesting to reconsider all this with supersymmetry, surely something to do in the near future.

This means that Higgs mechanism is essential yet in the Standard Model to understand how to achieve a finite mass for all particles in the theory. We will see in the future what Nature reserved us about.


A quite effective QCD theory

21/11/2008

As far as my path toward understanding of QCD is concerned, I have found a quite interesting effective theory to work with that is somewhat similar to Yukawa theory. Hideki Yukawa turns out to be more in depth in his hindsight than expected.

yukawa Indeed, I have already showed as the potential in infrared Yang-Mills theory is an infinite sum of weighted Yukawa potentials with the range, at each order, decided through a mass formula for glueballs that can be written down as

m_n=(2n+1)\frac{\pi}{2K(i)}\sqrt{\sigma}

being \sigma the string tension, an experimental parameter generally taken to be (440 MeV)^2, and K(i) is an elliptic integral, just a number.

The most intriguing aspect of all this treatment is that an effective infrared QCD can be obtained through a scalar field. I am about to finish a paper with a calculation of the width of the \sigma resonance, a critical parameter for our understanding of low energy QCD. Here I put the generating functional if someone is interested in doing a similar calculation (time is rescaled as t\rightarrow\sqrt{N}gt)

Z[\eta,\bar\eta,j] \approx\exp\left\{i\int d^4x\sum_q \frac{\delta}{i\delta\bar\eta_q(x)}\frac{\lambda^a}{2\sqrt{N}}\gamma_i\eta_i^a\frac{\delta}{i\delta j_\phi(x)}\frac{i\delta}{\delta\eta_q(x)}\right\} \times
\exp\left\{-\frac{i}{Ng^2}\int d^4xd^4y\sum_q\bar\eta_q(x)S_q(x-y)\eta_q(y)\right\}\times
\exp\left\{\frac{i}{2}(N^2-1)\int d^4xd^4y j_\phi(x)\Delta(x-y)j_\phi(y)\right\}.

As always, S_q(x-y) is the free Dirac propagator for the given quark q=u,d,s,\ldots and \Delta(x-y) is the gluon propagator that I have discussed in depth in my preceding posts. People seriously interested about this matter should read my works (here and here).

For a physical understanding of this you have to wait my next posting on arxiv. Anyhow, anybody can spend some time to manage this theory to exploit its working and its fallacies. My hope is that, anytime I post such information on my blog, I help the community to have an anticipation of possible new ways to see an old problem with a lot of prejudices well grounded.


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