Who fears a non-perturbative Higgs field?

28/07/2009

One of my preferred readings in the blogosphere is Tommaso Dorigo’s blog. I think this is a widely known blog for people interested about physics and got some citation also at New York Times. Quite recently he published a very interesting post (see here) about the fate of our loved Standard Model taking the move from a very nice paper by J.Ellis, J.R.Espinosa, G.F.Giudice, A.Hoecker, and A.Riotto (see here). These authors are well known and really smart at their work and, indeed, I have noticed this paper as it appeared in arxiv. My readers know that I work on a small part (QCD) of the whole picture arisen in sixties and seventies and I have never taken a look from outside. So, while I appreciated this paper I thought it was not the case to comment on it  in my blog. But reading Tommaso’s post some thoughts come to my mind and these are really pertinent.

People put out two kind of constraints on the Higgs part of the standard model to have an idea of what to expect. I give you here the Higgs potential for your needs

$V_H=\frac{1}{4}\lambda(\phi^\dagger\phi-v^2)^2$

and one immediately realizes that it introduces two free parameters. The critical one is $\lambda$ and let me explain why. When one does quantum field theory, the only real tool that she has to do any meaningful computation is small perturbation theory. The word “small” is never said but it should be said in any circumstance as this technique only works if you have a small parameter in your theory (a coupling) to use as a development parameter. Otherwise we are lost and all starts to become foggy and not so well-defined. Today, nobody knows how to manage a theory with a strong coupling. Parameter $\lambda$ is exactly such a coupling and we are able to manage a Higgs field when this parameter is small. But when you do small perturbation theory in quantum field theory you realize immediately that infinities come out and you are not able to obtain meaningful results going beyond the first order. For the most interesting theories around we are lucky:  Schwinger, Tomonaga, Feynman and Dyson invented renormalization and this works to remove infinities at each order of perturbation theory in the Standard Model and also for the Higgs, if the coupling is small. We are so accustomed to such a situation that we think that this is all one needs to know to understand quantum field theory: Perturbation theory and renormalization. We think that small perturbation theory is the perturbation theory and nothing else. So, we hope also the Higgs field should fulfill such requirements. Indeed, we are already in trouble in QCD for these same reasons but I have discussed at lengthy such a situation before here and I do not want to repeat myself.

There is no reason whatsoever to believe that we know all one has to know to manage a quantum field theory. Higgs could as well be not that light and strong coupled and there is no reason to think that Nature chose the small coupling case to favor us. Of course, if things will not stay this way I will be happy as a light Higgs is favored by supersymmetry and I like supersymmetry. But I would like also to emphasize that we already have all we need to manage analytically a strong coupled Higgs field. This matter I have discussed widely here and in my published papers.

So, while we all agree that a light Higgs is favored my view is that we should not have any fear of a non-perturbative Higgs field.

Exact solutions on arxiv

24/07/2009

As promised in my preceding post (see here), I have posted yesterday a preprint on this matter on arxiv (see here). I have presented all the exact solutions I was able to obtain at a classical level and I have given a formulation of the quantum field theory for a massless quartic theory. The key point in this case is the solution of the equation for the propagator

$-\Box\Delta+3\lambda\phi_c^2(x)\Delta=\delta^D(x)$

being $\phi_c$ the given exact classical solution. As usual, I have used a gradient approximation and the solution of the equation

$\ddot\phi(t)+3\lambda\phi_c^2(t,0)\phi(t)=\delta(t)$

that I know when the phase in $\phi_c(t,0)$ is quantized as $(4n+1)K(i)$, being $n$ an integer and $K(i)$ an elliptic integral. This gives back a consistent result in the strong coupling limit, $\lambda\rightarrow\infty$, with my preceding paper on Physical Review D (see here).

The conclusion is rather interesting as quantum field theory, given from such subset of classical solutions, is trivial when the coupling becomes increasingly large as one has a Gaussian generating functional and the spectrum of a harmonic oscillator. This is in perfect agreement with common wisdom about this scalar theory. So, in some way, Jacobi elliptical functions that describe nonlinear waves behave as plane waves for a quantum field theory in a regime of a strong coupling.

NASA shows leftovers of missions on the moon

18/07/2009

There has always been a lot to laugh about skepticism on lunar missions. Today, I apprehend from an article in the New York Times (see here) that NASA released a set of photos from Lunar Reconnaissance Orbiter (LRO) clearly showing leftovers from Apollo missions. They released photos for missions 11 and 14 to 17 (Apollo 13 never landed for an explosion on board and photos on Apollo 12 will be released in the near future). I show here the photo on Apollo 14 taken from New York Times clearly showing the lunar module and its shadow and the most used path by astronauts. I can still remember that night with my father waiting for the landing of the lunar module and Neil Armstrong making his first step on the moon. Most of the choices I have done after in my professional life are due to that exciting night.

Thank you very much, folks!

Exact solutions of nonlinear equations

15/07/2009

Recently, I have posted on the site of Terry Tao and Jim Colliander, Dispersive Wiki. I am a regular contributor to this beautiful effort to collect all available knowledge about differential equations and dispersive phenomena. Of course, I can give contributions as a solver of differential equations in the vein of a pure physicist. But mathematicians are able to give rigorous theorems on the behavior of the solutions without really solving them. I invite you to take some time to look at this site and, if you are an expert, to register and contribute to it.

My recent contribution is about exact solutions of nonlinear equations. This is a really interesting field and most of the relevant results come from soliton theory.  Terry posted on his blog about Liouville equation (see here). This equation is exactly solvable and is widely known to people working in string theory. But also one of the most known equations in physics literature can be solved exactly. My preprint shows this. Indeed I have to update it as, working on KAM theorem, I have obtained the exact solution to the following equation (check here on Dispersive Wiki):

$\Box\phi +\mu_0^2\phi+\lambda\phi^3 = 0$

that can be written as

$\phi(x) = \pm\sqrt{\frac{2\mu^4}{\mu_0^2 + \sqrt{\mu_0^4 + 2\lambda\mu^4}}}{\rm sn}\left(p\cdot x+\theta,\sqrt{\frac{-\mu_0^2 + \sqrt{\mu_0^4 + 2\lambda\mu^4}}{-\mu_0^2 - \sqrt{\mu_0^4 + 2\lambda\mu^4}}}\right)$

being now the dispersion relation

$p^2=\mu_0^2+\frac{\lambda\mu^4}{\mu_0^2+\sqrt{\mu_0^4+2\lambda\mu^4}}.$

As always $\mu$ is an arbitrary parameter with the dimension of a mass. You can see here an example of mass renormalization due to interaction. Indeed, from the dispersion relation we can recognize the following renormalized mass

$m^2(\mu_0,\mu,\lambda)=\mu_0^2+\frac{\lambda\mu^4}{\mu_0^2+\sqrt{\mu_0^4+2\lambda\mu^4}}$

that depends on the coupling. This class of solutions clearly show how the nonlinearities produce contributions to mass. Either by modifying it or by generating it. So, it is not difficult to imagine that Nature may have adopted them to display mass wherever there is not.

As a by-product, I am now able to give a consistent quantum field theory in the infrared for the scalar field (always thank to my work on KAM theorem), obtaining the needed corrections to the propagator and the spectrum. I hope to find some time in the next days to add all this new material to my preprint. Meanwhile, enjoy Dispersive Wiki!

Problems at arxiv

14/07/2009

Today I received the following message from arxiv at Cornell:

Sadly, you do not currently appear to have permission to access http://arxiv.org/

If you believe this determination to be in error, see http://arxiv.org/denied.html for additional information.

All the mirrors seem to work well.

Back to Soverato

03/07/2009

Next three days I will not be able to manage my blog. I take a holiday. Please, look below to understand why I am unplugging myself.

Have a nice week-end!