## QCD and lattice computations

30/08/2008

QCD in the infrared limit is generally not manageable for computations. We are not able to derive from it masses and other properties of hadrons. So, people thought to use computers to solve it in order to get exact results from it. Since the start, many difficulties were met by people working with this approach pioneered by K. G. Wilson (the Nobel prize winner for the introduction of renormalization group in statistical mechanics). The most serious ones are implied into the limitations of the resources of the computer one uses. On a lattice you have a spacing and you are interested in the continuum limit when the spacing goes to zero. But having the spacing going to zero implies more and more computational resources that are difficult to be found still today. The other question originates on how large is the volume you are using. One should be sure that small volume effects do not enter into our computations so that one is still not into the asymptotic limit is interested on. This latter problem is not so severe even if it has been advocated in computations of gluon and ghost propagators being theoretical expecations seriously at odd with those coming out from lattice.

In a comment about my analysis of quarkonia (see here) it was questioned by James Amundson at Fermilab that my potential does not seem to agree with the one emerging from lattice. People at Fermilab is doing a very good job for lattice QCD and so this comment should be taken rather seriously. Indeed, there is a point I did not emphaise in my answer. I have got an interquark potential

$V(r)=-\frac{\alpha_s}{r}+0.8762499705\alpha_s\sqrt{\sigma}$

but I do not take $\alpha_s$ to be a constant. Rather, it depends on the energy scale where I am doing computations and this is the key trick that does the job and I get the right answers.

But let me comment about the present situation of lattice QCD. I think that currently the most striking results are given in the following figure

This figure is saying to us that introducing quark sea the precision of computations improves dramatically. This computation was carried on by MILC, HPQCD, UKQCD and Fermilab Lattice Collaborations. In their paper they declare a spacing 1/8 fm and 1/11 fm. Quark masses are generally taken somewhat different from those of PDG as the proper ones require more computational resources. Indeed, the reached volumes are never that large for the reasons seen above. One can look at Gauge Connection to have an idea of the configurations generally involved. So, if volume effects enter in some physical quantity we cannot be aware of them. This is the situation seen on the computations of gluon and ghost propagators (see here). The situation in this case if far more simpler as there are no quarks. This is pure Yang-Mills theory and so people was able to reach volumes till $(128)^4$ that is a really huge volume.

But for the spectrum of pure Yang-Mills we are not that lucky as computations do not seem to hit the true ground state of the theory. Besides, we have $\sigma$ resonance seen at accelerator facilities but not with lattice computations at any level. Indeed, we know that there is an incongruence between lattice computations of pure Yang-Mills spectra and computations of the gluon propagator (see here and my paper for QCD 08). So, where is the $\sigma$ resonance on the lattice? Full QCD or pure Yang-Mills? In the latter case is a glueball. I think this is one of the main problem to be addressed in the very near future together with the computation of golden-plated quantities. There is too much involved in this to ignore it.

## Farewell to Bell Labs

30/08/2008

I would like to share my dislike with others in the blogosphere about the fate of Bell Labs (see Lubos and Doug “Sad. Just sad.”). People working in these labs changed the World and represents much better than other what doing science means for all.

Lucent is also in Italy being a wide-world corporation. Some years ago I tried to be hired by them as a small part of Bell Labs was also in Italy but I get no answer back. They were already in the crisis that changed so deeply this company till the status we see today. One should understand that, for industry, research can be a luxury, what really matters today is to find as rapidly as possible solutions to make money. This does not mean that it is the right way to proceed but is the way I see applied everywhere,

In some way, money overcome the idea-leaded path toward the future we were all accustomed to and we have to forget it. We do not need solutions, we need more and more convinced people that buys our products. These are not time for Einstein, it is time to sell refrigerators to Eskimos.

If this is our future, money rather than ideas, bad times are waiting us. In Italy the question of industry doing research is dead many years ago. In US there are a lot of company doing this. The hope is that they do not give up.

## Infrared QCD

29/08/2008

Today the Editor of International Journal of Modern Physics E, the World Scientific journal on nuclear physics, communicated to me that my paper has been accepted for publication. Such kind of communications represent a moment of great accomplishment for us doing research. The reason is that a publication is a recognition of our work as worth to appear in an archival journal. This paper is very important as I show, given the gluon propagator, how a Nambu-Jona-Lasinio model can be straightforwardly derived with all the constants fixed by QCD. This is a substantial starting point to compute all properties of particles in hadronic physics. Indeed, in our recent posts about quarkonia  we have seen how masses for ground state resonances can be obtained with a very small error. These results represent a sound proof that our propagator is the right one if lattice and numerical computations would not be enough.

Finally, Nora Brambilla communicated to me that I am enlisted intto the mail list of Quarkonia Working Group. This group is composed by some of the most important physicists working in the area of hadronic physics and Nora and her husband Antonio Vairo are two of them. I hope to give some significant contributions in the near future to this group.

## Quarkonia and Dirac spectra

28/08/2008

In these days we are discussing at length the question of heavy quarkonia, that is bound states of heavy quark-antiquark and we have got a perfect agreement for their ground states assuming a potential in the form

$V(r)=-\frac{\alpha_s}{r}+0.8762499705\alpha_s\sqrt{\sigma}$

being $\sigma=(0.44GeV)^2$ the string tension for Yang-Mills theory. This potential was derived here and here. We derived it in the limit of small distances and this means that excited states and states with higher angular momentum can fail to be recovered and the full potential without any approximation should be used instead. Anyhow, our derivation of ground states was in the non-relativistic approximation. We want to check here the solution of Dirac equation to get a complete confirmation of our results and, as an added bonus, we will derive also the mass of  $B_c$ that is a bottom-charm meson. As said we cannot do better as to go higher excited states we need to solve Dirac equation with the full potential, an impossible task unless we recur to numerical computations.

So, let us write down the Dirac spectrum for a heavy quark-antiquark state:

$M(n,j)=\frac{3}{2}m_q+\frac{m_q}{2}\frac{1}{\sqrt{1+\frac{\alpha_s^2}{(n-\delta_j)^2}}}+0.8762499705\alpha_s\sqrt{\sigma}$

being

$\delta_j=j+\frac{1}{2}-\sqrt{(j+\frac{1}{2})^2-\alpha_s^2}$.

We apply this formula to charmonium, bottomonium and toponium obtaining

$m_{\eta_c}=2977$ MeV

against the measured one $m_{\eta_c}=2979.8\pm 1.2$ MeV and

$m_{\eta_b}=9387.5$ MeV

against the measured one $9388.9 ^{+3.1}_ {-2.3} (stat) +/- 2.7(syst)$ MeV and, finally

$m_{\eta_t}=344.4$ GeV

that confirms our preceding computation. The agreement is absolutely striking. But we can do better. We consider a bottom-charm meson $B_c$ and the Dirac formula

$M(n,j)=m_c+m_b-\frac{m_cm_b}{m_c+m_b}+\frac{m_cm_b}{m_c+m_b}\frac{1}{\sqrt{1+\frac{\alpha_s^2}{(n-\delta_j)^2}}}$

$+0.8762499705\alpha_s\sqrt{\sigma}$

obtaining

$m_{B_c}=6.18$ GeV

against the PDG average value $6.286\pm 0.005$ GeV the error being about 2%!

Our conclusion is that, at least for the lowest states, our approximation is extremely good and confirms the constant originating from our form of gluon propagator that was the main aim of all these computations. The implications are that quarkonia could be managed with our full potential and Dirac equation on a computer, a task surely easier than solving full QCD on a lattice.

## The interquark potential

27/08/2008

In our initial post about quarkonia we have derived the interquark potential from the gluon propagator. In this post we want to deepen this matter being this central to all hadronic bound states. The gluon propagator is given by

$G(p^2)=\sum_{n=0}^\infty\frac{B_n}{p^2-m_n^2+i\epsilon}$

being

$B_n=(2n+1)\frac{\pi^2}{K^2(i)}\frac{(-1)^{n+1}e^{-(n+\frac{1}{2})\pi}}{1+e^{-(2n+1)\pi}}$

and

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

with $\sqrt{\sigma}=0.44$ GeV being $\sigma$ the string tension that is just an integration constant of Yang-Mills theory, arising from conformal invariance, to be fixed experimentally. We have obtained this propagator in a series of papers starting from a massless scalar theory. The most relevant of them is here. It is immediate to recognize that this propagator is just an infinite superposition of Yukawa propagators. But the expectations from effective theories are quite different (see Brambilla’s CERN yellow report here). Indeed, a largely used interquark potential is given by

$V(r)=-\frac{a}{r}+\sigma r +b$

but this potential is just phenomenological and not derived from QCD. Rather, as pointed out by Gocharov (see here) this potential is absolutely not a solution of QCD. We note that it would be if the linear term is just neglected as happens at very small distances where

$V_C(r)\approx -\frac{a}{r}+b$.

We can derive this potential from the gluon propagator imposing $p_0=0$ and Fourier transforming in space obtaining

$V(r)=-\frac{\alpha_s}{r}\sum_{n=0}^\infty B_n e^{-m_n r}$

and we can Taylor expand the exponential in r obtaining

$V(r)\approx -\frac{\alpha_s}{r}+Ar+b$

but we see immediately that $A=0$ and so no linear term exists in the potential for heavy quarkonia! This means that we can formulate a relativistic theory of heavy quarkonia by solving the Dirac equation for the corresponding Coulombic-like potential whose solution is well-known and adding the b constant. We will discuss such a spectrum in future posts.

For lighter quarks the situation is more involved as we have to take into account the full potential and in this case no solution is known and one has to use numerical computation. But solving Dirac equation on a computer should be surely easier than treating full QCD.

## Ground state of toponium

27/08/2008

Following the series of posts I started after the beautiful result of BABAR collaboration, now I try to get a prevision for a new resonance, i.e. the ground state of $\bar t t$ quarkonium that is known in literature as toponium. This resonance has a large mass with respect to the others due to t quark being about 37 times more massive than b quark. In this case we have a theoretical reference by Yuri Goncharov (see here) published in Nuclear Physics A (see here). Goncharov assumes a mass for the top quark of 173.25 GeV and gets $\alpha_s(m_t)\approx 0.12$. He has a toponium ground state mass of 347.4 GeV. How does our formula compare with these values?

Let us give again this formula as

$\eta_t(1S)=2m_t-\frac{1}{4}\alpha_s^2m_t+0.876\alpha_s\sqrt{\sigma}$

being $\sqrt{\sigma}=0.44$ GeV. We obtain

$m_{\eta_t}=345.9$ GeV

that is absolutely good being the error of about 0.4% compared to Goncharov’s paper! Now, using the value of PDG $m_t=172.5$ GeV we get our final result

$m_{\eta_t}=344.4$ GeV.

This is the next quarkonium to be seen. Alhtough we can theoretically do this computation, we want just to point out that no toponium could be ever observed due to the large mass and width of the t quark (see here for a computation).

## Ground state of charmonium

26/08/2008

After the satisfactory derivation of the bottomonium ground state mass (see here) we would like to apply similar concepts to charmonium. Before we go on I would like to mention here the beautiful paper by Nora Brambilla and a lot of other contributors that any serious researcher in the field of heavy quark physics should read (see here). This paper has been published as a yellow report by CERN. What we want to prove here is that the knowledge of the gluon propagator can give a nice understanding of the ground state of quarkonia. Anyhow, for charmonium we could not be that lucky as relativistic effects are more important here than for bottomonium. Besides, if we would like to expand to higher order in r the quark potential we would be no more able to treat the Schroedinger equation unless we treat these terms as a perturbation but this approach is not successful giving at best slowing convergence of the series for bottomonium and an useless result for charmonium.

PDG gives us the data for the ground state of charmonium $\eta_c(1S)$:

$m_{\eta_c}=2979.8\pm 1.2$ MeV

$m_c=1.25\pm 0.09$ GeV ($\bar{MS}$ scheme)

$\alpha_s(m_c)=0.39$

and then, our computation gives

$m_{\eta_c}=2m_c-\frac{1}{4}\alpha_s^2m_c+0.876\alpha_s\sqrt{\sigma}\approx 2602.8$ MeV

that has an error of about 13%. With a quark mass of 1.44 GeV we would get a perfect agreement with $\eta_c(1S)$ mass that makes this computation quite striking together with the analogous computations for the ground state of the bottomonium.

As said at the start, heavy quarkonia are a well studied matter and whoever interested to deepen the argument should read the yellow report by Brambilla and others.

Update:I would like to point out the paper by Stephan Narison (see here and here) that obtains the pole masses of c and b quarks being these the ones I use to obtain the right ground state of charmonium and bottomonium. Striking indeed!

## Another success at SLAC

25/08/2008

Yesterday I was reading a copy of August of Physics World, that I receive being member of Institute of Physics, and come to a small piece about a recent measure at SLAC. BABAR collaboration was able to identify and measure the mass of the ground state of $\bar b b$ meson also called bottomonium and identified as $\eta_b(1S)$ (their paper is here). In order to reach their aims, they used the process $\Upsilon(3S)\rightarrow \gamma\eta_b(1S)$ and they were successful in obtaining a very precise value of the mass, $9388.9 ^{+3.1}_ {-2.3} (stat) +/- 2.7(syst)$ MeV, collecting about 20000 photons produced in the process.

We want to build on this beautiful result at SLAC by deriving the mass of the ground state of bottomonium. We already know, since the studies of charmonium, that a Coulomb-like potential does most of the job but not all. This is a great intuition by Politzer, one of the discoverers of asymptotic freedom (the others being Gross and Wilczek). The reason why this approximation works so well is that these quarks are really massive and so the interaction happens at very short distances and also a non-relativistic approximation does hold.

In order to verify how good is this approximation we consider our gluon propagator. This is given by

$G(p^2)=\sum_{n=0}^\infty\frac{B_n}{p^2-m_n^2+i\epsilon}$

being

$B_n=(2n+1)\frac{\pi^2}{K^2(i)}\frac{(-1)^{n+1}e^{-(n+\frac{1}{2})\pi}}{1+e^{-(2n+1)\pi}}$

and

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

with $\sqrt{\sigma}=0.44$ GeV. At this stage we can derive the potential between quarks setting $p_0=0$ and Fourier transforming in space coordinates giving

$V(r)=-\alpha_s\sum_{n=0}^\infty B_n \frac{e^{-m_n r}}{r}$.

So, using the small distance approximation we get finally

$V(r)\approx -\frac{\alpha_s}{r}+\alpha_s\sum_{n=0}^\infty B_nm_n$

and we can estimate the constant to be $\epsilon=0.876\alpha_s\sqrt{\sigma}$. Finally, from PDG we have $\alpha_s(m_b)=0.22$ being $m_b=4.68$ GeV the mass of the bottom quark. $\eta_b(1S)$ is a singlet state and so no spin-orbit effect is present. Using standard formula for hydrogen atom we have finally

$m_{\eta_b}=2m_b-\frac{1}{4}\alpha_s^2 m_b+\epsilon=9.388$ GeV

that is the mass measured at SLAC. We just note the relevance of the constant term $\epsilon=0.0848$ GeV to reach the agreement, a very nice confirmation of our gluon propagator and the approximations used for heavy quark bound states.

As a final consideration we note as a good theory permits to do calculations to be compared with experiments. Bad theories do not have such a property proving themselves ugly already at the start.

## Dyson-Schwinger equations and Mathematica

22/08/2008

As always I read the daily arxiv sends to me and I have found a beatiful work due to Alkofer and collaborators. An important reason to mention it too here is that it gives an important tool to work with that can be downloaded. This tool permits to obtain Dyson-Schwinger equations for any field theory. Dyson-Schwinger equations are a tower of equations giving all the correlators of a quantum field theory so, if you know how to truncate this tower you will be able to get a solution to a quantum field theory in some limit.

The paper is here. The link to download the tool for Mathematica version 6.0 and higher is here.

I hope to have some time to study it and try a conversion for Maple. Currently I was not able to test it as on my laptop I have an older version of Mathematica but is just few hours away from testing on my desktop.

## Sorry but your paper is wrong!

19/08/2008

In our preceding posts we have largely discussed what are the results emerging from lattice about the gluon and ghost propagators and the running coupling and how functional methods, in the way they are currently adopted, fail to reach agreement with lattice computations at very large volumes. But we want to resume here what are the main conclusions that are obtained from such applications of functional methods. People working in this way fix the gluon propagator as

$D(p^2)=(p^2)^{k_D-1}F(p^2)$

and for the ghost

$G(p^2)=(p^2)^{k_G-1}H(p^2)$

then the claim is made that the relation $k_D+2k_G=0$ does hold while the functions $F,H$ are taken to be regular as momenta go to zero. From the relation between the exponents $k_D,k_G$ we can conclude that, excluding the trivial solution, if the gluon propagator goes to zero at lower momenta, that is $k_D>0$, than we must have $k_G<0$ that means that the ghost propagator must go to infinity at lower momenta. What they get is that the ghost propagator should go to infinity faster than a free particle. If this would be true all the confining scenarios (Zwanzinger-Gribov and Kugo-Ojima) hold true. The ghost holds a prominent role and, last but not least, a proper defined running coupling goes to a fixed point to lower momenta.

Lattice computations say that all this is blatantly wrong. Indeed, we have learned from them that

• Gluon propagator reaches a finite non-null value at lower momenta.
• Ghost propagator is that of a free particle and so ghosts play no role at lower momenta.
• Running coupling is seen to approach zero at lower momenta.

From this we can easily derive our exponents as defined by people working with functional methods: $k_D=1$ and $k_G=0$ so that $k_D+2k_G=1\ne 0$ and no relation between exponents is seen to exist. So we have got a clear cut criterion to say when a published paper about infrared behavior of Yang-Mills theory is blatantly wrong independently on the prestige of the journal that publishes it. This happens all the times the relation $k_D+2k_G=0$ is assumed to hold. I can grant that there are a lot around of these wrong papers published on the highest ranked journals. If you have time and you need fun try to search for them.

I would like to say that this belongs to dynamics of science. We are presently in a transition situation about our matter, a situation similar at that happened after the discovery of the $J/\psi$ resonance that took some time before people agreed on its nature. So, there is nothing to say to editors or referee and also to authors as mistakes are the most common facts in physics and very few people hit the right track after a wide cemetery of mistakes and wrong theories.