## Quantum field theory and triviality

15/07/2008

In the comments on the post where Peter Woit kindly pointed to my blog (see here) there is some point about when a quantum field theory can be considered “trivial”.  Indeed, this question is fairly well acquired for the expert of quantum field theory as there exists a theorem due to Michael Aizenman (see here) for the scalar field theory but this result only applies to theory with dimensions greater or equal 5. If the scalar theory is trivial for D=4 is an open and hotly debated question still far from being settled.

So, what should one mean by a “trivial” field theory? We can say that a quantum field theory becomes trivial when in some limit is a free theory. To become free a theory should have the coupling going to zero in the considered limit and only this that is meaningful. As it is largely known, for a gauge theory a propagator can depend on the choice of the gauge. But it is also true that for some theories the appearence of a free propagator and the corresponding behavior of higher n-point funtions is already a proof of triviality.

What does it happen to the running coupling of the Yang-Mills theory at lower momenta?  A long time expectation has been that the running coupling of Yang-Mills theory should reach a non-trivial infrared fixed point, that is, it should have had a fixed value as the energy goes to zero. Lattice results said the opposite. But there is a glitch here. The problem is that we have not a clear understanding of what should be a proper definition of the running coupling in the infrared limit. People working on the lattice have chosen the definition taken from functional methods arosen with Alkofer and von Smekal. The computation on the lattice of this quantity shows that it goes to zero and no fixed point emerges. Different definitions have been proposed (see here for a review) but there are also experimental evidences, due to Giovanni Prosperi and his group and presented here, that the coupling at low momenta goes to zero.

All this gives a strong clue that Yang-Mills theory is a trivial theory in the infrared and shares a similar fate with the scalar theory.

## Classical Yang-Mills theory and mass gap

15/07/2008

Yesterday I posted a paper on arxiv (see here). In this paper I have given a solution to the classical equations of motion of Yang-Mills theory. Indeed, as already seen for the quartic scalar field (see here), one can show, with undergraduate level arguments, that Yang-Mills theory has a massive exact solution at least at a classical level. This can be seen as a simple proof that a massless theory can indeed produce a mass gap and this mass gap is simply a dynamical effect arising from the self-interaction term of the equations of motion.

As obtained from standard textbooks, the equations of motion for the Yang-Mills potential $A^a_\nu$ are

$\small \ddot A^a_\nu-\Delta_2A^a_\nu-\partial_\nu(\partial^\mu A^a_\mu)+gf^{abc}A^{b\mu}(\partial_\mu A^c_\nu-\partial_\nu A^c_\mu)+$$gf^{abc}\partial^\mu(A^b_\mu A^c_\nu)+g^2f^{abc}f^{cde}A^{b\mu}A^d_\mu A^e_\nu = 0.$

We use latin letters for group indeces and greek letters for space-time indeces. As it is already seen for Maxwell equations, we fix properly the gauge to obtain a solution and we choose the Lorentz gauge $\partial^\mu A^a_\mu=0$ following a standard textbook procedure already known for the electromagnetic field. The coupling constant $g$ is adimensional and no other constants enter into the theory. $f^{abc}$ are structure constants of the gauge group.

In order to find a solution we make the choice to take all the components of the field equal. So we write $A^a_\mu=(A_\mu,A_\mu,\ldots,A_\mu)$ with $N^2-1$ elements for SU(N) and $A_\mu=(\phi,\phi,\phi,\phi)$ and so we have a total of $4(N^2-1)$ degrees of freedom. With this substitution the above equation assumes a more familiar form

$\ddot\phi-\Delta_2\phi+Ng^2\phi^3=0$

that is just the equation of the quartic scalar field and we know its solution! Checking back here we can write down the solution

$\phi=\Lambda\left(\frac{2}{Ng^2}\right)^{\frac{1}{4}}{\rm sn}(px+\varphi,i)$

where we have changed the name to the integration constant calling it $\Lambda$ and introduced another one, a phase, $\varphi$. The incredible interesting new is that the above solution holds only if the following dispersion relation holds

$E^2=p^2+\Lambda^2\left(\frac{Ng^2}{2}\right)^{\frac{1}{2}}$

and we see that the classical Yang-Mills field has acquired a mass and this mass goes to zero as the coupling $g$ goes to zero. So stronger is the coupling in the self-interacting term and larger is the mass gap. But this says us something more. Indeed, as already seen for the quartic scalar field, we have a mass spectrum that we can write down as

$m_n=(2n+1)\frac{\pi}{2K(i)}\left(\frac{Ng^2}{2}\right)^{\frac{1}{4}}\Lambda$

that as we have seen here agrees fairly well with lattice computations but we cannot pursue further this comparison as we are working at a classical level while to compare with lattice and experimental data a quantum field theory is needed. But already at this level the agreement is striking and worthing further analysis.

The lesson to be learnt is that an exact classical solution can give a lot of information about the physics one should expect for a quantum field theory. This is also true for the existence of a mass gap of Yang-Mills theory.