| SCIENCE ACCOMPLISHMENTS |
| QCD: Results from Lattice
Quantum Chromodynamics |
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| Quantum chromodynamics (QCD) is the modern
theory of the strong force. In this theory, the main
objects are quarks and gluons, which are bound
by the strong force into protons, neutrons, and
other particles called hadrons. In the framework
of QCD, the strong nuclear force binding protons
and neutrons together into nuclei is actually only
a residue of the much stronger forces acting
between quarks and gluons. In fact, inside the
proton, even the concept of force is not very useful.
Within all hadrons we have a swirl of gluons
being exchanged back and forth as a manifestation
of the strong force. To make matters worse,
gluons can split into two, and then rejoin, or they
can split into a quark-antiquark pair. Even the
simplest hadron is a complex system hosting
constantly interacting components. |
| Despite this complexity, QCD is well established
experimentally. This is because at short distances
(or high energies), the coupling between the
particles is effectively small and particles move
around with relative freedom. This is called
asymptotic freedom and QCD is amenable to the
traditional methods of quantum field theory in
this regime. High-energy experiments have tested
and confirmed QCD in this realm, which led to
the 2004 Nobel Prize in Physics for Drs. David
Gross, David Politzer, and Frank Wilczek, the
theorists who provided the theory for short-range
QCD and asymptotic freedom. |
| On the other hand, the traditional methods
break down when gluons (and all the complexities
that they entail) travel over "long" distances (to particle
physicists, the nuclear size 10-15 m is a long
distance). There is a mathematically rigorous way
to handle the gluons in such cases, which is to
introduce a grid in space-time - the "lattice" of lattice
QCD - and use a computer to keep track of the
quarks and gluons on this grid. Of course, the computer
does not do everything; the person running
the computer has to make sure that the results do
not depend on how fine or coarse the grid is. Fortunately,
the grid spacing is considered a "short"
distance, so on this scale we can use the well-tested
methods of perturbative (short distance) QCD. |
| As is the case for all scientific and computational
problems, results need to be tested and validated
and we need to be sure that the techniques
employed are correct. Physicists from three lattice
QCD collaborations (Fermilab Lattice, MILC, and
HPQCD collaborations) decided to address these
concerns by making some specific predictions.
We used the same methods and input parameters
that were used previously with successful results,
and set out to calculate the properties of some
processes whose measurements were just around
the corner |
Our first prediction was for a process very similar to that shown in figure 1. Instead of a B meson
the initial state is a D meson, and the final hadron
is now a kaon (K). In this case the underlying
quark decay is charm to strange (c—>s), which is
better understood than b—>u. If we denote the
total momentum of the final state electron (e) and
neutrino ( ) in D—>Ke as q, then we predicted that the c—>s decay rate should depend on q2. The
q2 dependence had not been measured when our
paper was submitted. Two months later, an experiment
called FOCUS published the first measurement
of the q2 dependence of this decay and
confirmed our calculation. Since then, the Belle
collaboration has completed an even more precise
experimental measurement. A comparison
between measurement and prediction for the
QCD part of the decay (also called the form
factor) is shown in figure 2. Our calculation is
shown as a curve with an error band, and it is
clear that the data points from the Belle experiment
fall right on top of this curve. |
Another prediction that we made was for a
simpler decay of the D meson into just a positron
and a neutrino with no daughter hadron. Here a
single number, called the decay constant, summarizes
the influence of QCD on this process.
There are actually two mesons of interest, the D+
and the Ds; the latter has a strange (s) quark in it.
We finished our work on this decay in June 2005
and the CLEO experiment confirmed our calculation
at a conference that July. Since then,
another experiment named BaBar has released the
first precise measurement of the Ds decay constant.
This also agrees with our prediction. A nice way to
summarize these results is to take the ratio of the
two decay constants, because then several uncertainties
from the lattice QCD approach cancel.
We predicted that the ratio should be 0.786 ± 0.042,
using lattice QCD, and the experiments described
above have found a ratio of 0.779 ± 0.093
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Long Range Gluons of QCD |
Figure 1.
The decay of a neutral B meson. The weak interaction sector (particles connected to the
W boson) and the short-range gluons (short coiled lines) can be handled with traditional methods of quantum field theory. The long-range gluons (long coiled lines) need a numerical technique, known as lattice QCD. Recent advances in lattice QCD now include a realistic treatment of the subprocesses in which a gluon turns into a quark loop and back into gluons again. |
There are many problems in particle physics, nuclear physics, and astrophysics that require a quantitative understanding of the connection between quarks and gluons. One of the most basic problems in particle physics, for example, is the experimental quest to understand the interactions between each kind of quark. There are known to be six different types: down, up, strange, charm, bottom, and top. Most of the world is made of up and down quarks, which are confined inside protons and neutrons. But to particle physicists the others are more interesting, because their interactions are intimately related to the origin and evolution of the universe. |
| The situation described so far is depicted in figure 1. We would like to understand the decay of a bottom (or b, for short) quark, for example,
but the b quarks are usually found inside another
particle called the B meson. In this environment
the b quark will decay to an up (u) quark, and
emit (through a virtual W boson) an electron and
an antineutrino. No existing experimental
apparatus can detect a free quark; instead, we
detect the hadron (such as a pion, Õ) which
contains the quark resulting from the decay. |
| The principal challenge of QCD is to work out the strong, nonlinear interactions of long-range gluons. The best approach, which is already over 25 years old, is to use the lattice formulation of QCD and calculate the interactions on a computer. For many years, computing resources were not good enough to simulate one important aspect of these interactions: the tendency of gluons to turn into quark-antiquark pairs, then back into gluons. The situation finally changed at the beginning of this decade, when a combination of insightful ideas and gains in computing power permitted precise calculations of hadron properties. The predictions that resulted from this work agree with experimental measurements to an accuracy of about 1% or 2%. |
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The last prediction that we would like to discuss is the mass of an exotic particle called the charmed
beauty meson, or Bc. The calculation of its mass
tests two aspects of QCD: on one level, our methods
for treating the interaction between charmed
and bottom quarks; and on another level, our
methods for treating the conversion of gluons into
light quark pairs. Our calculation obtained a mass
of 6,304 ± 20 MeV/ c2, and was completed before
any experiment succeeded in the measurement.
In this case, the CDF experiment found a mass of
6,287 ± 5 MeV/ c2. Figure 3 shows an interesting
comparison with previous results. The point on
the left is an older calculation that did not include
the conversion of gluons into quark pairs and back
again. While its range includes the experimental
result, it is not constrained enough. It demonstrates
that the heavy quarks (charmed and bottom)
were being included correctly, but that some
other sections may be missing. The modern calculation
(middle point) corrects this discrepancy,
and is in good agreement with the measurement
from the CDF experiment (right-hand point). |
| Taken together, these results represent a stringent test of lattice QCD. As in any numerical simulation,
there are several ingredients going into
our computer code, and each of the three predictions
probes a different aspect of the contributing
physics. Furthermore, the D decay properties–
the leptonic decay constant and the semileptonic
form factors–can now be easily adapted to calculate
the analogous properties of B mesons, simply
by changing the mass of the heavy quark when
we run the simulation. |
Figure 2. Comparison of the q2 dependence of the form factor in the decay D–>Ken. The error band shows a calculation using lattice QCD and the points give an experimental measurement from the Belle Collaboration. |
Figure 3. Comparison of the Bc meson mass. From left to right: an old lattice calculation omitting the conversion of gluons into quarks and back; a modern calculation this feature of QCD; an experimental measurement from the CDF Collaboration. |
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| The B meson is one of hottest topics in all of elementary particle physics, and is being subjected to
intense scrutiny by experiments at several particle
accelerator laboratories. The reason for this intensive
study is that the underlying interactions of the
b quark are thought to provide important clues
about the nature of the universe. How do quarks
attain their mass? Or, an even more intriguing
question: which elementary processes are responsible
for the abundance of matter (and dearth of antimatter) in the universe? The properties of the different quark flavors are connected to yet another
mind-boggling question: why is the up quark's
mass a little bit smaller than the down quark's
mass? If it were the other way around (which
would fit the pattern seen in more exotic quarks),
then protons would decay to neutrons instead of
vice versa. A universe with this property would be
made only of neutrons, neutrinos, and photons;
there would be no atoms, no chemistry, and no life. |
| The answers to some of these questions may be close at hand. If any of the relevant effects of QCD on B meson decays are large enough, then terascale lattice QCD with an accuracy of 1% or 2% could begin to disentangle the underlying interactions. The harder questions in quark flavor physics–which are related to other overarching questions in particle physics, nuclear physics, and astrophysics–will require even more ambitious numerical calculations at the petascale. Our collaboration is well positioned to proceed to the challenges and promises of petascale computing. Quantum chromodynamics is one of the most powerful theories of the physical world and the combination of innovative minds and advanced computing resources can utilize its full potential to unravel more of the secrets of science at ultrasmall and ultra-large scales. |
| Contributor: Dr. Andreas S. Kronfeld, Theoretical Physics Department, Fermi National Accelerator Laboratory |
PI: Dr. Robert Suger, University of California–Santa Barbara; work done under the Lattice QCD SciDAC Project
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