## A set of exact classical solutions of Yang-Mills equations

28/02/2009

Following the discussion with Rafael Frigori (see here) and returning to sane questions, I discuss here a class of exact classical solutions that must be considered in the class of Maximal Abelian Gauge.  As usual, we consider the following Yang-Mills action

$S=\int d^4x\left[\frac{1}{2}\partial_\mu A^a_\nu\partial^\mu A^{a\nu}+\partial^\mu\bar c^a\partial_\mu c^a\right.$

$-gf^{abc}\partial_\mu A_\nu^aA^{b\mu}A^{c\nu}+\frac{g^2}{4}f^{abc}f^{ars}A^b_\mu A^c_\nu A^{r\mu}A^{s\nu}$

$\left.+gf^{abc}\partial_\mu\bar c^a A^{b\mu}c^c\right]$

being $c,\ \bar c$ the ghost field, $g$ the coupling constant and, for the moment we omit the gauge fixing term. Let us fix the gauge group being SU(2). We choose the following (Smilga’s choice, see the book):

$A_1^1=A_2^2=A_3^3=\phi$

being $\phi$ a scalar field. The other components are taken to be zero. It easy to see that the action becomes

$S=-6\int d^4x\left[\frac{1}{2}\partial_\mu\phi\partial^\mu\phi+\partial\bar c\partial c\right]+6\int d^4x\frac{2g^2}{4}\phi^4.$

This is a very nice result as if we have a solution of the scalar field theory we get immediately a classical solution of Yang-Mills equations while the ghost field decouples and behaves as that of a free particle. But such solutions do exist. We can solve exactly the  equation

$\partial_t^2\phi-\partial_x^2\phi+2g^2\phi^3=0$

by

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

being sn Jacobi snoidal function, $\mu,\ \theta$ two arbitrary constants, if holds

$p^2=\mu^2\left(\frac{2g^2}{2}\right)^\frac{1}{2}.$

We see that the field acquired a mass notwithstanding it was massless and the same happens to the Yang-Mills field. These are known as non-linear waves. These solutions do not represent a new theoretical view. A new theoretical view is given when they are used to build a quantum field theory. This is the core of the question.

What happens when we keep the gauge fixing term as

$\frac{1}{\xi}(\partial\cdot A)^2?$

If you substitute Smilga’s choice in this term you will find a correction to the kinematic term implying a rescaling of space variables. This is harmless for the obtained solutions resulting in the end into a multiplicative factor for the action of the scalar field.

The set of Smilga’s choice is very large and increases with the choice of the gauge group. But such solutions always exist.

Yang-Mills propagators and QCD to appear in Nucl. Phys. B: Proc. Suppl.

Update: Together with Terry Tao, we agreed that these solutions hold in a perturbative sense, i.e.

$A_\mu^a(x)=\eta_\mu^a\phi(x)+O(1/g)$

being $\eta_\mu^a$ a constant and $g$ the coupling taken to be very large. These become exact solutions when just time dependence is retained. So, the theorem contained in the above papers is correct for this latter case and approximate in the general case. As the case of interest is that of a large coupling, these results permit to say that all the conclusions drawn in the above papers are correct. This completed proof will appear shortly in Modern Physics Letters A (see http://arxiv.org/abs/0903.2357 ).

Thanks a lot to Terry for the very helpful criticism.

## Quote of the day

27/02/2009

Today let me quote Einstein:

Great spirits have often encountered violent opposition from weak minds

My dedication is for people working on string theory.

## Osaka and Berlin merge their data!

26/02/2009

Today in arxiv appeared a relevant paper by Osaka and Berlin groups (see here). This is a really important paper as these two groups merged their data for the lattice computation of the gluon and ghost propagators for SU(3) in the Coulomb gauge. As usual I give here a picture summing up their results about gluon propagator

As you can see one has again the propagator reaching a finite, non-null value in the infrared limit. About the ghost propagator they obtain again a result very near to the case of a free particle. In this case the agreement was perfect for SU(2). For SU(3) there is a tiny disagreement.

I would like to emphasize a couple of points that should be discussed with these results at hand. There is a paper, published on Physical Review Letters, that was claiming that the gluon propagator in the Coulomb gauge should take the Gribov form going to zero at lower momenta. You can find this paper here and here. I think that authors should reconsider their computations as the disagreement with lattice is really serious. All the research lines aimed at a proof of confinement scenarios heavily relying on Gribov ideas seem to have reached a failure point. There could be a lot of reasons for this but it seems to me that, as lattice computations improve, we are left with the only option that the starting points of all these studies are to be reconsidered.

A second point to be made is the completely missing link between people working on the computation of propagators and those working on the spectrum of QCD. I think this is the moment to try to connet these two relevant areas as times are mature to try a consistency check between them. After the failure in view of some functional methods do we have to believe yet that Kaellen-Lehman formula does not apply in the infrared limit?

## Scholarpedia

24/02/2009

I would like to point out to my readers Scholarpedia. This represents a significant effort of the scientific community to grant a wiki-like resource with the benefit of peer-review. This means that articles are written on invitation and reviewed by referees chosen by the Editorial Board. This resource is important as correctness of information is granted by the review process and by the choice of the authors that are generally main contributors to the considered fields. It is interesting to point out that, currently, there are articles written by 15 Nobelists and 4 Fields medalists. The most relevant aspect to be emphasized is that the information is freely accessible to everybody exactly in the spirit of Wikipedia.

## Quantum field theory and gradient expansion

21/02/2009

In a preceding post (see here) I showed as a covariant gradient expansion can be accomplished maintaining Lorentz invariance during computation. Now I discuss here how to manage the corresponding generating functional

$Z[j]=\int[d\phi]e^{i\int d^4x\frac{1}{2}[(\partial\phi)^2-m^2\phi^2]+i\int d^4xj\phi}.$

This integral can be computed exactly, the theory being free and the integral is a Gaussian one, to give

$Z[j]=e^{\frac{i}{2}\int d^4xd^4yj(x)\Delta(x-y)j(y)}$

where we have introduced the Feynman propagator $\Delta(x-y)$. This is well-knwon matter. But now we rewrite down the above integral introducing another spatial coordinate and write down

$Z[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-(\partial\phi)^2-m^2\phi^2]+i\int d\tau d^4xj\phi}.$

Feynman propagator solving this integral is given by

$\Delta(p)=\frac{1}{p_\tau^2-p^2-m^2+i\epsilon}$

and a gradient expansion just means a series into $p^2$ of this propagator. From this we learn immeadiately two things:

• When one takes $p=0$ we get the right spectrum of the theory: a pole at $p_\tau^2=m^2.$
• When one takes $p_\tau=0$ and Wick-rotates one of the four spatial coordinates we recover the right Feynman propagator.

All works fine and we have kept Lorentz invariance everywhere hidden into the Euclidean part of a five-dimensional theory. Neglecting the Euclidean part gives us back the spectrum of the theory. This is the leading order of a gradient expansion.

So, the next step is to see what happens with an interaction term. I have already solved this problem here and was published by Physical Review D (see here). In this paper I did not care about Lorentz invariance as I expected it would be recovered in the end of computations as indeed happens. But here we can recover the main result of the paper keeping Lorentz invariance. One has

$Z[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-(\partial\phi)^2-m^2\phi^2-\frac{\lambda}{2}\phi^4]+i\int d\tau d^4xj\phi}$

and if we want something not trivial we have to keep the interaction term into the leading order of our gradient expansion. So we will break the exponent as

$Z[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-\frac{\lambda}{2}\phi^4]-i\int d\tau d^4x\frac{1}{2}[(\partial\phi)^2+m^2\phi^2]+i\int d\tau d^4xj\phi}$

and our leading order functional is now

$Z_0[j]=\int[d\phi]e^{i\int d\tau d^4x\frac{1}{2}[(\partial_\tau\phi)^2-\frac{\lambda}{2}\phi^4]+i\int d\tau d^4xj\phi}.$

This can be cast into a Gaussian form as, in the infrared limit, the one of our interest, one can use the following small time approximation

$\phi(x,\tau)\approx\int d\tau' d^4y \delta^4(x-y)\Delta(\tau-\tau')j(y,\tau')$

being now

$\partial_\tau^2\Delta(\tau)+\lambda\Delta(\tau)^3=\delta(\tau)$

that can be exactly solved giving back all the results of my paper. When the Gaussian form of the theory is obtained one can easily show that, in the infrared limit, the quartic scalar field theory is trivial as we obtain again a generating functional in the form

$Z[j]=e^{\frac{i}{2}\int d^4xd^4yj(x)\Delta(x-y)j(y)}$

being now

$\Delta(p)=\sum_n\frac{A_n}{p^2-m^2_n+i\epsilon}$

after Wick-rotated a spatial variable and having set $p_\tau=0$. The spectrum is proper to a trivial theory being that of an harmonic oscillator.

I think that all this machinery does work very well and is quite robust opening up a lot of possibilities to have a look at the other side of the world.

## Most extreme gamma-ray blast yet

20/02/2009

As my blog’s readers know, I follow as far as I can space missions that can have a deep impact on our knowledge of universe. Most of them are from NASA. One of these missions is Fermi-GLAST that has produced a beautiful result quite recently.  It has seen the greatest gamma-ray burst ever (see here). The paper with the results is appeared on Science (see here). The burst was seen in Carina constellation.  These explosions are the most energetic processes in the universe and were uncovered by chance with military satellites named Vela used to find nuclear explosions in the atmosphere in the sixties of the last century. Understanding gamma-ray bursts implies a deeper understanding of stellar explosions.