## Wonderful QCD!

28/11/2008

On Science this week appeared a milestone paper showing two great achievements by lattice QCD:

• QCD gives a correct description of low energy phenomenology of strong interactions.
• Most of the ordinary mass (99%) is due to the motion of quarks inside hadrons.

The precision reached has no precedent. The authors are able to get a control of involved  errors in such a way to reach an agreement of about 1% into the computation of nucleon masses. Frank Wilczek gives here a very readable account of these accomplishments and is worthwhile reading. These results open a new era into this kind of method to extract results to be compared with experiments for QCD and give an important confirmation to our understanding of strong interactions. But I would like to point out Wilczek’s concern: Until we will not have a theoretical way to obtain results from QCD in the low energy limit, we will miss a great piece of understanding of physics. This is a point that I discussed largely with my posts in this blog but it is worthwhile repeating here coming from such an authoritative voice.

An interesting point about these lattice computations can be made by observing that again no $\sigma$ resonance is seen. I would like to remember that in these computations entered just u, d and s quarks as the authors’ aims were computations of bound states of such quarks. Some authoritative theoretical physicists are claiming that this resonance should be a tetraquark, that is a combination of u and d quarks and their antiparticles. What we can say about from our point of view? As I have written here some time ago, lattice computations of the gluon propagator in a pure Yang-Mills theory prove that this can be fitted with a Yukawa form

$G(p)=\frac{A}{p^2+m^2}$

being $m\approx 500 MeV$. This is given in Euclidean form. This kind of propagators says to us that the potential should be Yukawa-like, that is

$V(r)=-A\frac{e^{-mr}}{r}$

if this is true no tetraquark state can exist for lighter quarks. The reason is that a Yukawa-like potential heavily damps any van der Waals kind of residual potential. But, due to asymptotic freedom, this is no more true for heavier quarks c and b  as in this case the potential is Coulomb-like and, indeed, such kind of states could have been seen at Tevatron.

We expect that the glueball spectrum should display itself in the observed hadronic spectrum. This means that a major effort in lattice QCD computations should be aimed in this direction now that such a deep understanding of known hadronic states has been reached.

## Gradient expansions, strong perturbations and classicality

24/11/2008

It is a common view that when in an equation appears a very large term we cannot use any perturbation approach at all. This is a quite common prejudice and forced physicists, for a lot of years, to invent exotic approaches with very few luck to unveil physics behind equations. The reason for this relies on a simple trick generally overlooked by mathematicians and physicists and here is my luck. This idea can be easily exposed for the Schroedinger equation. So, let us consider the case

$(H_0+\lambda V)|\psi\rangle=i\hbar\frac{\partial|\psi\rangle}{\partial t}$

with $\lambda\rightarrow\infty$. This is a very unlucky case both for a physicist and a mathematician as the only sure approach that come to our rescue is a computer program with all the difficulties this implies. Of course, it would be very nice if we could find a solution in the form of an asymptotic series like

$|\psi\rangle=|\psi_0\rangle+\frac{1}{\lambda}|\psi_1\rangle+\frac{1}{\lambda^2}|\psi_2\rangle+\ldots$

but we know quite well that if we insert such a solution into the Schroedinger equation we get meaningless results. But there is a very smart trick that can get us out of this dark and can produce the required result. I have exposed this since 1992 on Physical Review A (see here) and this paper was not taken too seriously by the community so that I had time enough to be able to apply this idea to all fields of physics. The paper producing the turning point has been published on Physical Review A (thank you very much, Bernd Crasemann!). You can find it here and here. The point is that when you have a strong perturbation, an expansion is not enough. You also need a rescaling in time like $\tau=\lambda t$. If you do this and insert the above expansion into the original Schroedinger equation, this time you will get meaningful results: A dual Dyson series that, being now the perturbation independent of time, becomes a well-known gradient expansion: Wigner-Kirkwood series. But this series is a semiclassical one and you get the striking result that a strongly perturbed quantum system is a semiclassical system! So, if you want to change a quantum system into a classical one just perturb it strongly. This is something that happens when one does a measurement in quantum mechanics using just electromagnetic fields that are the only means we know to accomplish such a task.

This result about strong perturbations and semiclassicality has been published on a long time honored journal: Proceedings of the Royal Society A (see here and here). I am pleased of this also because of my estimation for Michael Berry, the Editor. I have met him at a Garda lake’s Conference some years ago and I have listened a beautiful talk by him about the appearance of a classical world out of the quantum conundrum. I remember he asked me how to connect to internet from the Conference site but there there was just a not so cheap machine from Telecom Italia and then my help was quite limited.

So, I just removed a prejudice and was lucky enough to give sound examples in all branches of physics. Sometime, looking in some dusty corners of physics and mathematics can be quite rewarding!

## 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.

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.

## Wikipedia and physics

18/11/2008

Wikipedia is doing a relevant service to our community. I have contributed to some voices both for the english and the italian version and I have used it a lot of time to make myself acquainted with some unknown parts of physics. I think that this is worthwhile a support:

As you will know since now I am a strong supporter of any kind of revolution and this is a big one.

So, long life to Wikipedia!

## First photo of an exoplanet

14/11/2008

I think that is worthwhile to put here this first photo of an exoplanet. This is the result of a joint effort of three research groups, one of them using Hubble of course. Here is the photo

but this is more explicative (via Corriere della Sera)

A fine article about, as always on Physics World, can be found here.

## Quantum mechanics and gravity

11/11/2008

Reading the daily by arxiv today I cannot overlook a quite interesting paper that will appear soon on Physical Review Letters. This paper (see here), written by Saurya Das and Elias Vagenas, presents some relevant conclusions about the effects of gravity in quite common quantum mechanical systems. The authors rely their conclusions on an acquired result, due mostly to string theory, that a fundamental length must exist and this fundamental length modifies in a well defined way the indeterminacy principle. So, one can quantify this effect on whatever quantum mechanical system through a correcting Hamiltonian term and evaluating the effect of gravity on this system. In this way one can obtain an estimation on how relevant is the effect and how far can be an experimental measurement of this. The conclusions the authors reached are quite interesting. Of course, all of the cases imply a too small effect to be in the reach of a laboratory observation but, the most not trivial conclusion is that could exist an intermediate fundamental length that could be observed e.g. at LHC. This intermediate length should be placed between the electroweak and the Planck scale.

It is the first time that I see such estimations on quite simple quantum mechanical models and I would expect more extended analysis on a similar line. Surely, it would be striking to see in laboratory such a tiny effect correcting the Lamb shift. But, working in quantum optics, I learned that progress experimentalists are able to put out can be very impressive in a very short time. So, I would not be surprised if in some years Physical Review Letters should publish some experimental letter about this matter being the first evidence of a quantum gravity effect in a laboratory.

## Screening masses in SU(3) Yang-Mills theory

10/11/2008

Thanks to a useful comment by Rafael Frigori (see here) I become aware of a series of beautiful papers by an Italian group at Universita’ della Calabria. I was mostly struck by a recent paper written by R. Fiore, R. Falcone, M. Gravina and A. Papa (see here) that appeared in Nuclear Physics B (see here). This paper belongs to a long series of works about the behavior of Yang-Mills theory at non-null temperature and its critical behavior. Indeed, using high-temperature expansion and Polyakov loops one arrives at the main conclusion that the ratio between the lowest and the higher state of the theory must be 3/2. This ratio depends on the universality class the theory belongs to and so, on the kind of effective theory one has in the proper temperature limit (below or above $T_c$). It should be said that, in order to get a proper verification of the above prediction, people use lattice computations. Fiore et al. use a lattice of $16^3 \times 4$ points and, as all this kind of computations are done on lattices having such a dimension, one can cast some doubt about the fact that the true ground state of the theory is really hit. Indeed, this happens in all this kind of computations done to get a glueball spectrum that seem at odd with those giving the gluon propagator producing a lower screening mass at about 500 MeV (see my post here). A state at about 500 MeV is seen at accelerator facilities as $\sigma$ resonance or f0(600) but is not predicted by any lattice computation. One of the reasons to reduce lattice volume is that one can reach higher values of $\beta$ granting the reaching of a non-perturbative regime, the one interesting for us.

What can we say about this ratio with our theory? We have put on arxiv a paper that answer this question (see here). These results were also presented at QCD 08 in Montpellier (see here). We assume that the $\sigma$ cannot be seen at such small volumes but its excited state $\sigma^*$ can be obtained. This implies that one can exchange the $\sigma^*$ with the lowest state and $0^+$ as the higher one. Then this ratio gives exactly 3/2 as expected. We can conclude on the basis of this analysis that this ratio is the same independently on the temperature but, the one to be properly measured is given in the paper of Craig McNeile (see here) that gives close agreement between lattice and theoretical predictions.

So, we would like to see lattice computations of Yang-Mills spectra at lower lattice spacing and increased volumes granting in this way the proper value of the ground state. This is overwhelming important in view of the fact that no real understanding exists of the existence of the $\sigma$ resonance with lattice computations. This will implies, as discussed above, a deeper understanding of the spectrum of the theory also at higher temperatures.