# Quantum chromodynamics with advanced computing

Andreas S Kronfeld  Fermi National Accelerator Laboratory, Batavia, IL 60510,
USA
(for the USQCD Collaboration<EMAIL_ADDRESS>
###### Abstract

We survey results in lattice quantum chromodynamics from groups in the USQCD
Collaboration. The main focus is on physics, but many aspects of the
discussion are aimed at an audience of computational physicists.

##  1 Introduction and background

Quantum chromodynamics (QCD) is the modern theory of the strong nuclear force.
The physical degrees of freedom are quarks and gluons. The latter are the
quanta of gauge fields, in some ways analogous to photons. The crucial
difference is that gluons couple directly to each other, whereas photons do
not. A consequence of the self-coupling is that the force between quarks does
not vanish at large distances (as the Coulomb force $e^{2}/(4\pi r^{2})$ does)
but becomes a constant $F_{\rm QCD}\approx
800~{}\textrm{MeV}\,\textrm{fm}^{-1}\approx 15,000~{}\textrm{N}$. It would,
thus, require a vast amount energy to separate quarks out to a macroscopic
distance. Indeed, long before the separation becomes large enough to measure,
the energy stored in the gluon field “sparks” into quark-antiquark pairs. This
phenomenon of QCD explains why freely propagating quarks are not observed in
nature, and it is called confinement.

QCD is a quantum field theory, which means that from the outset one must deal
with mathematical objects that are unfamiliar even to many physicists. The
most widely used theoretical tool for quantum field theories is relativistic
perturbation theory, as developed for quantum electrodynamics (QED) by
Feynman, Schwinger, Tomonaga and others sixty years ago Schwinger:QED .
Perturbation theory provides the key connection between the mathematical
theory with experiments in QED Kinoshita:1996vz  and the Glashow-Weinberg-
Salam theory merging QED with the weak nuclear force Alcaraz:2007ri .

A textbook example of perturbative quantum field theory is to calculate how
virtual pairs induce a distance dependence on the coupling in quantum gauge
theories. In QED, one considers the fine structure constant,

$\alpha=\frac{e^{2}}{4\pi\hbar c},$ (1)

where $-e$ is the charge of the electron. In QCD, one has the strong coupling
$\alpha_{s}$, related to the gauge coupling $g$ by analogy with equation (1).
When virtual pairs are taken into account, $\alpha$ and $\alpha_{s}$ are not
constant but depend on distance; they are said to “run.” The running, depicted
in figure 1, is completely different in the two theories.

(a) (b)

Figure 1: Running of (a) the QED and (b) the QCD coupling with distance. For
QCD the red curve shows the perturbative result, and the (higher) blue show
the effect of confining forces.

In QED $\alpha$ becomes constant at distances $r>\hbar/m_{e}c$ (the electron’s
Compton wavelength), but at short distances it grows. In QCD $\alpha_{s}$
grows at long distance; perturbative running no longer makes sense once $r$ is
in the confining regime. At short distances—or, equivalently, high
energies—the QCD coupling is small enough that perturbation theory can again
be used. For example, the cross section for an electron-positron pair to
annihilate and produce a quark-antiquark pair can be reliably computed as a
power series in $\alpha_{s}$. The quark and antiquark each fragment into a jet
of hadrons—mostly pions, but some protons and neutrons, too—and general
properties such as the energy flow and angular distributions of these jets can
be traced back to the quark and antiquark. In this way, the experiments can
measure quark properties, and these are found to agree with theoretical
calculations Dixon:2007hh .

QCD and the electroweak theory form the foundation of the Standard Model of
elementary particles. They are so well established that many particle
physicists consider the chromodynamic and electroweak gauge symmetries, and
the associated quantum-number assignments of the quarks and leptons, to be
laws of nature. We cannot conceive of a more comprehensive theory that does
not encompass them. But there is more to the Standard Model. We know that
something must spontaneously break the electroweak symmetry and that something
(perhaps the same thing) must generate masses for the quarks and leptons. The
Standard Model contains interactions that _model_ these phenomena. The model
interactions are consistent with observations yet also incomplete. It is hoped
that insights obtained with the Large Hadron Collider (LHC), to commence
operations at CERN later this year, will enable particle physicists to solve
many of the puzzles raised by the Standard Model.

The purpose of this paper is to cover QCD, particularly in the strong-coupling
regime where perturbation theory is insufficient. It is worth bearing in mind,
however, that there is a close connection between many of the results
discussed below and some of the big questions in particle physics. A simple
example is the masses of the quarks. To determine the quark masses (which
because of confinement cannot be measured), we have to calculate the relation
between quark masses and measurable quantities, such as hadron masses. Quark
masses are interesting for many reasons, the most unsettling of which is as
follows.

Quark flavors in the Standard Model:
$\begin{array}[]{ccc}\left(\begin{array}[]{c}u\\\
d\end{array}\right)&\left(\begin{array}[]{c}c\\\
s\end{array}\right)&\left(\begin{array}[]{c}t\\\
b\end{array}\right)\\\\[10.00002pt] m_{u}<m_{d}&m_{c}>m_{s}&m_{t}>m_{b}\\\
!!!&&\end{array}$ What explains the pattern of masses? Figure 2: The six
_flavors_ of quarks come the three _generations_ with a doublet in each
generation. The doublet structure plays a role in the electroweak
interactions. For example, the upper (lower) entries have electric charge
$+2/3$ ($-1/3)$.

The mass of the top quark is larger than that of the bottom quark; see figure
2. Similarly, the mass of the charmed quark is larger than that of the strange
quark. If this simple pattern held with the up and down quarks—the ones that
make up proton and neutrons—then the up quark would be more massive than the
down. But then protons would decay to neutrons, positrons, and neutrinos
($p\to ne^{+}\nu$). The positrons would find electrons and annihilate to
photons.

This universe would consist of neutron stars surrounded by a swarm photons and
neutrinos, and nothing else. Our universe is not at all like this, because
neutrons are more massive than protons. The allowed decay reaction is $n\to
pe^{-}\bar{\nu}$, leading to an abundance of protons and electrons, and making
possible chemistry and biochemistry. With QCD we trace the neutron-proton mass
difference back to the down-up quark masses and then to the deeper origin of
quark masses. The pattern of quark masses is necessarily not simple, and the
search for simple explanation of a messy pattern occupies many particle
theorists. Below we give further examples where QCD is essential for working
out the mysteries of particle physics.

QCD is also the cornerstone of modern nuclear physics. The simplest nucleus is
nothing but the proton, a bound state of two up quarks and one down quark. The
neutron is a bound state of two down quarks and one up quark. The confining
force is very, very strong. By comparison, the force traditionally called the
strong nuclear force is a residue of the fundamental chromodynamic force. This
is similar to van der Waals forces among molecules, which are electromagnetic
forces between neutral objects with structure leading to a distribution of
electric charge. Some basic problems are, therefore, to understand nucleon
structure directly from QCD, to study the excitation spectrum of nucleons (and
their cousins with strangeness and other flavors), and to see how few-nucleon
(or, more generally, few-hadron) systems interact.

Another active area of research in nuclear QCD is the properties of quark
matter at high temperature and density. This regime is important to cosmology,
because an epoch of the early universe consisted of hot quark matter, and to
astrophysics, because, for example, neutron stars are composed of dense quark
matter. In the laboratory, nuclear physicists create hot, dense quark matter
by colliding heavy ions, at present with Brookhaven’s RHIC, soon with (the
heavy-ion mode of) CERN’s LHC, and someday with GSI’s FAIR.

At strong coupling it is necessary to go beyond perturbation theory. In a
relativistic quantum field theory, such as QCD, this is not easy. An important
obstacle is that a quantum field associates one (or perhaps a few) degrees of
freedom with every point in space-time. Hence, the number of degrees of
freedom is uncountably infinite. Furthermore, the fluctuations at short
distances, if added up naively, contribute an ultraviolet divergence to
anything of interest. These fluctuations are treated by a set of ideas known
as _renormalization_ Lepage:1989hf . In short, one must find a tool that
(correctly) handles both strongly coupled fields and the renormalization of
short-distance fluctuations.

This tool is lattice field theory or, for gauge theories like QCD, lattice
gauge theory. The idea, which goes back at least as far as Heisenberg, is to
replace continuous space (or, in practice, space-time) with a discrete grid,
or lattice. Then the number of degrees of freedom is countable. If the lattice
is of finite extent, it occupies a finite physical volume, and, moreover, the
number of degrees of freedom is finite. In 1974, Kenneth Wilson showed how to
maintain gauge symmetry with a discrete lattice Wilson:1974sk . In this and
subsequent work, he and others elaborated how the lattice provides a
mathematically rigorous definition of quantum field theory via functional
integrals Glimm:1987ng .

The functional integral for quantum fields on a finite lattice is a familiar
object, a multi-dimensional integral, that can be evaluated on a computer.
Before downloading some publicly available code usqcd  and trying to run it on
a laptop, let us pause to contemplate how many variables of integration are
reasonable (for physics). One wants to have the spacing between sites on the
lattice small enough to resolve hadron structure, and one wants the volume
large enough that the boundary conditions do not modify the structure. If one
supposes that the ratio of these two lengths is 32, then the dimensions of the
integrals of interest are

${\tt gluon~{}d.o.f.}=8\times 4\times 32^{3}\times 128>10^{8},$ (2)

because gluons come in 8 colors with 4 polarization states each. The factor
128 presumes a space-time lattice with a time extent long (in relativistic
units) compared to the spatial extent; this is common practice.

To cope with integrals of such large dimension, the only practical technique
is Monte Carlo integration with importance sampling. The weight guiding the
importance sampling is $e^{-S}$, where $S$ is the classical action for the
random sample of gluon variables. In functional integral formalisms for
quantum mechanics, the expression for the action is the defining equation of
the physical system. The development of algorithms to generate these samples
is a vibrant subject, covered in part by Bálint Joó’s poster here at the
SciDAC conference. This paper will not dwell on algorithms, but it is
important to mention one more complication.

Quarks are fermions and, as such, must satisfy the Pauli exclusion principle.
In the functional integral formalism, this is handled by introducing
anticommuting Grassmann variables for fermions. The integration is no longer
Riemann (or Lebesgue) integration but a formal procedure called Berezin
integration. To make a long story short, in lattice QCD we always can and do
carry out the Berezin integration by hand. The outcome is that the weight for
the importance sampling becomes $\det M\exp(-S_{\rm gluons})$, where $M$ is a
sparse matrix with space-time indices. The matrix $M$ is $N\times N$, where

$N={\tt quark~{}d.o.f.}=n_{f}\times 3\times 2\times 2\times 32^{3}\times
128>10^{8}.$ (3)

The quark field represents $n_{f}$ flavors with 3 colors and 2 spin states of
quark and antiquark for every flavor and color. The matrix $M$ is sparse
because it is a lattice version of a differential operator, the Dirac operator
(generalized to QCD). Incorporating $\det M$ into the weight is
computationally demanding, increasing by two or three orders of magnitude the
amount of floating-point operations needed to generate a new configuration of
gluons.

The lattice is a helpful device mathematically and computationally, but it is
not physical. To obtain a real result of the chromodynamics of continuum
spacetime, one must work out the integrals for a sequence of lattices, and
take the limit $a\to 0$ in a way that respects renormalization. Taking $a$
smaller and smaller increases the computing burden as onerously as
$a^{-(4+z)}$. The exponent $z$ depends on the algorithm and is typically
around 1 or 2. The 4 in the exponent is the key reason why numerical lattice
QCD requires the most advanced computing facilities available. No algorithm
can reduce it, because it is the dimension of space-time.

Most lattice-QCD calculations are carried out in two steps. The first is to
generate an ensemble of gluon fields, with a certain lattice spacing and quark
masses. This requires the computationally super-demanding $\det M$. To
generate an ensemble of useful size, one needs several hundred, perhaps even a
few thousand, samples of the gluon field. Computers of the highest available
capability are needed for this step. In recent years, lattice gauge theorists
have carried out this step with the special-purpose computer QCDOC, large
clusters of PCs, and, more recently, leadership-class machines constructed
under the auspices of the DOE’s Office of Advanced Scientific Computing
Research (ASCR). These ensembles of gluon fields are valuable: within the U.S.
they are generated by and for particle and nuclear physicists and shared under
various agreements. File-sharing is facilitated by formats and software
developed by the International Lattice Data Grid DeTar:2007au .

The second step is to mine these ensembles for physics. A simple example is to
compute the quantum mechanical amplitude for a proton to propagate from one
point to another, and study the behavior of the amplitude as the separation
varies: this is what one does to compute the proton mass, in fact. The key
ingredients are a few rows of the inverse of the matrix $M$, introduced above.
With the same ensemble one can study protons, pions, and more uncommon hadrons
such as the charmed strange pseudoscalar meson (or $D_{s}$ for short).
Different rows of $M^{-1}$ (i.e., different flavors, colors, spins, and space-
time separations) are combined in various ways to obtain the quantum numbers
of the hadrons in the problem at hand. Because each problem is different, and
there are so many of them, we must attack this problem with the highest
capacity computers. The job mix here is heterogeneous, with many users whose
job streams range from many large jobs to extremely many medium-sized jobs.
The challenge is to develop systems that scale well (for the large jobs),
while maintaining the flexibility needed to handle the heterogeneity. The
USQCD Collaboration has designed clusters of PCs that deliver this capacity in
a cost-effective way Holmgren:2004nk .

The remainder of this paper focuses on physics. It is organized according to
four principal themes of lattice gauge theory:

  * •

Determination of fundamental parameters of the Standard Model of particle
physics

  * •

Study of nucleon structure, the hadron spectrum, and hadron interactions

  * •

Simulation of the thermodynamic properties of QCD at nonzero temperature and
density

  * •

Lattice gauge theories beyond QCD, such as those appearing in models of
electroweak symmetry breaking

Roughly speaking, the first and last are particle physics, and the second and
third are nuclear physics. But the lines are blurry: for example, some of the
simplest calculations of proton structure (the second item) may play a role
more accurate calculations of $pp$ scattering cross sections at the LHC. The
way that scientific goals drive the computing needs of lattice gauge theory
are discussed further, along the lines of these themes, in public whitepapers
of the USQCD Collaboration usqcd .

##  2 Standard Model particle physics

In quantum electrodynamics, the fundamental parameters are the masses of
electrically charged particles and the fine structure constant $\alpha$.
Similarly, in quantum _chromo_ dynamics, the fundamental parameters are the
masses of colored particles (the quarks) and the strong coupling $\alpha_{s}$.
Just as $\alpha$ can be determined in many ways Kinoshita:1996vz ,
$\alpha_{s}$ can be determined at high energies from jet cross sections with
perturbative QCD, and at low energies from the hadron spectrum with lattice
QCD. Two recent results are (evaluated at distance $r=\hbar/m_{Z^{0}}c$)

$\alpha_{s}=\left\\{\begin{array}[]{llll}0.1172\pm 0.0022&\textrm{perturbative
QCD}&\textrm{jet shapes}&\textrm{Ref.~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Becher:2008cf}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
0.1170\pm 0.0012&\textrm{lattice
QCD}&\textrm{hadrons}&\textrm{Ref.~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Mason:2005zx}{\@@citephrase{(}}{\@@citephrase{)}}}}\end{array}\right.,$
(4)

where both error bars reflect experimental and theoretical uncertainties. The
first result is determined at high energies, around 100 GeV, in the regime
where quarks can be treated as nearly free. The second is determined at low
energies, of a few GeV and lower, where quarks are confined. Both are linked
soundly to the defining equations of QCD. The agreement demonstrates the
richness and broad validity of QCD and should make any enthusiastic scientist
say “Wow!”

A straightforward application of lattice QCD is to work out how hadron masses
depend on quark masses. Quark masses are interesting for several reasons. As
fundamental constants of nature, they are interesting in their own right. As
discussed in the introduction, the pattern of quark masses is puzzling and,
hence, a motivation to search for extensions of the Standard Model that would
contain a simple explanation.

Three flavors of quarks—top, bottom, and charm—have masses large enough to
influence some high-energy scattering and decay processes. Their masses can be
determined with perturbative QCD as well as lattice QCD. (The results agree
Yao:2006px .) The other three quarks—up, down, and strange—have masses so
small that the full bound-state problem must be considered. Lattice QCD,
therefore, provides the best information. Some recent results are the
following Mason:2005bj ; Bernard:2007ps :

$\displaystyle m_{u}=1.9$ $\displaystyle\pm$ $\displaystyle
0.2~{}\textrm{MeV}/c^{2},$ (5) $\displaystyle m_{d}=4.6$ $\displaystyle\pm$
$\displaystyle 0.3~{}\textrm{MeV}/c^{2},$ (6) $\displaystyle m_{s}=88$
$\displaystyle\pm$ $\displaystyle 5~{}\textrm{MeV}/c^{2}.$ (7)

One remarkable aspect of these results is that the strange quark’s mass is
somewhat smaller than had been thought on the basis of less reliable
techniques for attacking strongly-coupled QCD. This point is, perhaps, of
interest mostly to particle physicists, but a further one is of broader
interest. The up and down masses—the constituents of protons and neutrons and,
thus, everyday matter—are very small, within an order of magnitude of the
electron mass ($0.511~{}\textrm{MeV}/c^{2}$). Only about 1–2% of the proton
mass ($938.3~{}\textrm{MeV}/c^{2}$) or neutron mass
($939.6~{}\textrm{MeV}/c^{2}$) is accounted for by the quark masses.
Consequently, most of their masses—and the mass of everyday matter—arises from
confinement, namely, the binding energy of the gluons and the (relativistic)
kinetic energy of bound quarks. This is a stunning insight, to be revisited in
Section 5.

The caption of figure 2 states that the pairings of quarks in each doublet
stem from the electroweak interactions. That is not the full story. There is
no reason for the eigenstates of mass to be the eigenstates of the interaction
term with $W$ gauge bosons. The two are related by a unitary transformation,
$V_{\rm CKM}$, namely,

$\left(\begin{array}[]{c}d\\\ s\\\ b\end{array}\right)_{W}=V_{\rm
CKM}\left(\begin{array}[]{c}d\\\ s\\\ b\end{array}\right)_{\rm mass},\quad
V_{\rm CKM}=\left(\begin{array}[]{ccc}V_{ud}&V_{us}&V_{ub}\\\
V_{cd}&V_{cs}&V_{cb}\\\ V_{td}&V_{ts}&V_{tb}\end{array}\right).$ (8)

Here the initials CKM stand for Cabibbo-Kobayashi-Maskawa; Cabibbo introduced
equations (8) for two generations, and Kobayashi and Maskawa were the first to
study the ramifications for three generations.

The left-hand side of the first equation denotes the electroweak basis. The
labels on the elements of $V_{\rm CKM}$ are the weak isopartner and the mass
eigenstate, because the transition amplitude for $W\to\bar{b}u$, for example,
is proportional to $V_{ub}$. The study of physical processes related to
$V_{\rm CKM}$ is called flavor physics, because the central issues are to
understand why there are several flavors of quarks and what lends them their
separate identities.

As a unitary matrix, $V_{\rm CKM}$ contains complex entries. Some of the
phases can be absorbed into unobservable phases of the quark wave functions.
For only two generations, one physical parameter remains, and it is real. For
three generations, as in nature, three physical parameters are real and one
phase remains. The complex couplings imparted by the phase lead to physical
processes that proceed at different rates for particles and the corresponding
antiparticles. Such reactions are needed to explain the abundance of matter
and the dearth of antimatter in the universe. Among particle physicists, this
phenomenon is called $CP$ violation. It is clearly intriguing to know whether
the $CP$ violation of the CKM matrix is enough to account for the matter-
antimatter asymmetry of the universe. According to present measurements and
theoretical understanding, it is insufficient, so one would like to know what
other interactions violate $CP$.

In the Standard Model, there is a scalar boson called the Higgs boson. Section
5 will explain more about it. For now let us note that both the quark masses
and the CKM matrix have their origin in matrices of couplings between quark
fields and the Higgs field. Therefore, the puzzles of $CP$ violation and the
pattern of the quark masses are tied together by the Standard Model.

To make progress in flavor physics, lattice-QCD calculations are needed to
interpret the experimental measurements. In a schematic form the rate $\Gamma$
is given by

$\Gamma=\left(\begin{array}[]{c}\textrm{known}\\\
\textrm{factor}\end{array}\right)\left(\begin{array}[]{c}\textrm{CKM}\\\
\textrm{factor}\end{array}\right)\left(\begin{array}[]{c}\textrm{QCD}\\\
\textrm{factor}\end{array}\right),$ (9)

where the “known factor” consists of well-measured physical constants and
numerical factors like $4\pi$. The paradigm is to measure as many flavor-
changing processes as possible, and use the Standard-Model formulae to over-
determine the CKM matrix. If all the determinations are consistent, then one
can set limits on non-Standard sources of flavor and $CP$ violation. To do so,
the QCD factor must be computed, and often the only way to do so is with
lattice QCD.

There are dozens of relevant measurements, but because the CKM matrix is
unitary, the elements of the CKM matrix must satisfy constraints. A particular
vivid one is the unitarity triangle that stems from the orthogonality of
columns of a unitary matrix:

$V_{ud}^{*}V_{ub}+V_{cd}^{*}V_{cb}+V_{td}^{*}V_{tb}=0,$ (10)

which traces out a triangle in the complex plane. The triangle and the present
uncertainties from various constraints is shown in figure 3, dividing each
side by $V_{cd}^{*}V_{cb}$.

Figure 3: Constraints on the CKM unitarity triangle from experiments and QCD
calculations, highlighting those where lattice QCD plays a role. The upper
panel shows the current status, and the lower a forecast of how the
uncertainties can be reduced with lattice QCD VandeWater:2007zz . In the lower
panel, one can envision tension between the lime-green hyperbola and the blue
wedge, depending on how the central values evolve and measurements and
lattice-QCD computations improve. The current status of the angles $\alpha$
and $\gamma$ is omitted, because better measurements are needed; they are
expected from LHC$b$.

Except for the wedge labelled $\sin\beta$ the uncertainties can be reduced
with lattice calculations of the QCD factor.

Lattice calculations for a wide range of observables relevant to the CKM
matrix have appeared over the past few years, and further improvements are
under way. These include calculations of properties of strange kaons
Bernard:2007ps ; Aubin:2004fs ; Beane:2006kx ; Gamiz:2006sq ; Antonio:2007pb ;
Boyle:2007qe , charmed $D$ and $D_{s}$ mesons Aubin:2004ej ; Aubin:2005ar ;
Follana:2007uv , and beautiful $B$ and $B_{s}$ mesons Okamoto:2004xg ;
Dalgic:2006dt ; Gray:2005ad ; Bernard:2006zz ; Dalgic:2006gp ; Laiho:2007pn .
For recent reviews, including references to work in Asia, Australia, and
Europe, see DellaMorte:2007ny ; Juttner:2007sn .

The past several months witnessed an unexpected development in flavor physics,
in the field of leptonic decays of pseudoscalar mesons. (Pseudoscalar bosons
have spin 0 and negative parity; the most common examples are the pion and
kaon.) One can consider the measured decay rates to determine a QCD matrix
element called the decay constant, and denoted $f_{\pi}$ (for the pion),
$f_{K}$ (for the kaon), and so on. Table 1 shows the results of some SciDAC-
supported calculations of decay constants and the corresponding measurements.

Table 1: Decay constants $f_{P}$ of pseudoscalar mesons (here $P\in\\{\pi,K,D,D_{s}\\}$). Values in MeV. Experimental determinations assume a suitable element of the CKM matrix element consistent with all flavor data. Meson | MILC Bernard:2007ps  | FNAL Aubin:2005ar  | HPQCD Follana:2007uv  | Experiment | (Ref.) | Deviation
---|---|---|---|---|---|---
$\pi$ | $128\pm 3$ | – | $132\pm 2$ | $130.7\pm 0.4$ | Yao:2006px  | $0.4\sigma$
$K$ | $154\pm 3$ | – | $157\pm 2$ | $159.8\pm 1.5$ | Yao:2006px  | $1.7\sigma$
$D^{+}$ | – | $201\pm 19$ | $207\pm 4$ | $206\pm 9$ | Eisenstein:2008sq  | $0.1\sigma$
$D_{s}$ | – | $249\pm 17$ | $241\pm 3$ | $277\pm 9$ | Dobrescu:2008er  | $3.8\sigma$

The calculations agree very well with the measurements, except in the case of
the charmed strange pseudoscalar meson $D_{s}$.

This set of circumstances is odd, because the computation is easiest for the
$D_{s}$. Algorithms for the propagators of light quarks slow down as the quark
mass $m_{q}$ is decreased. As a consequence, properties of hadrons containing
up or down quarks are reached via a (controlled) extrapolation in $m_{q}$.
This introduces a source of uncertainty that is absent for the $D_{s}$, which,
as the lightest bound state of a charmed quark and a strange antiquark, can be
computed by working directly at the physical quark masses.

There is no obvious explanation for this discrepancy within the Standard
Model. Unless there is a real blunder somewhere, the discrepancy points to a
non-Standard particle mediating the decays $D_{s}\to\mu\nu$ and
$D_{s}\to\tau\nu$, thereby changing the interpretation of the “measured”
$f_{D_{s}}$. It turns out to be rather easy to devise models of such
interactions Dobrescu:2008er . Given the agreement of CLEO’s new measurement
of $f_{D^{+}}$ Eisenstein:2008sq , new $W^{\prime}$ bosons and charged Higgs
bosons seem unlikely, making leptoquarks—particles with baryon and lepton
number—are the most plausible agents of decay.

##  3 Nucleon structure and nuclear physics

Let us turn now to nuclear physics, where the importance of lattice QCD is
difficult to overstate. Many nuclear laboratories, such as Jefferson
Laboratory in the U.S., have turned their attention to the structure of the
proton (the simplest nucleus!) Hagler:2007hu , the excitation spectrum of
baryons McNeile:2007fu , and processes that may produce glueballs. On the
theoretical side, the previous decade witnessed dramatic growth in the
application of effective field theories to understand nuclear interactions
Kaplan:2006sv . These techniques connect a wealth of nuclear and hadronic data
to the QCD Lagrangian. Lattice QCD is an irreplaceable tool to interpret the
new and upcoming experiments at JLab and to compute the so-called low-energy
constants of the effective theory Beane:2008dv .

A topic at the interface between nuclear and particle physics concerns the
distributions of partons (a generic term for quarks and gluons) inside the
proton. In high-energy collisions of $p\bar{p}$ (at the Tevatron) or $pp$ (at
the LHC), the transverse motion of partons can be neglected. To compute cross
sections (to produce, say, a Higgs boson), the key property needed is the
distribution of the longitudinal momentum. From a nuclear perspective, this is
also one of the most basic aspects of proton structure. Usually one studies
parton distributions $q$ as a function of the fraction $x$ of longitudinal
momentum carried by a parton, $0<x<1$. Fits to scattering data can determine
$q(x)$ only over a range, $x_{\rm min}<x<x_{\rm max}$, limited by kinematics
and statistics. Lattice QCD can compute moments of $q$: $\langle
x^{n}\rangle_{q}=\int_{0}^{1}dx\,x^{n}q(x)$. Clearly the two kinds of
information are complementary. The experimental range can be extended with _ad
hoc_ functions possessing the right asymptotic forms at the endpoints, and in
this way an Ansatz-guided experimental determination of $\langle
x^{n}\rangle_{q}$ is possible. One finds good agreement with lattice
calculations Hagler:2007xi . To explore nucleon structure further, the next
step is to probe the transverse structure and compute the so-called
generalized parton distributions Hagler:2007xi .

The axial charge of the nucleon is another topic of keen interest, because it
appears in the description of $\beta$ decay. It is also related to the
relative spin polarization of up and down quarks in the proton. Some recent
results with lattice QCD are

$g_{A}=\left\\{\begin{array}[]{rl}1.21\pm
0.08&\textrm{Ref.~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Edwards:2005ym}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
1.20\pm 0.07&\textrm{Ref.~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Yamazaki:2008py}{\@@citephrase{(}}{\@@citephrase{)}}}}\end{array}\right.,$
(11)

showing good agreement with experiment ($g_{A}=1.269\pm 0.003$). Work is under
way to reduce the theoretical uncertainty.

In all fields of physics, spectroscopy is a time-honored approach to learn
about the dynamics of the underlying phenomena of interest. In nuclear and
hadronic physics, therefore, an important area of investigation is the
excitation spectrum of hadrons, particularly baryons. Excited states present
additional challenges for lattice QCD, but the tools needed are available and
have been demonstrated to work Sasaki:2001nf ; Basak:2007kj . This subject
extends to include topics such as the intriguing Roper resonance Mathur:2003zf
and electromagnetic transitions of the form $\gamma N\to N^{*}$, where $N^{*}$
is an excited nucleon Lin:2008qv .

Because the gluons couple to each other, it is conceivable that there are
hadrons consisting essentially of gluons, with no valence quarks. (Virtual
quark-antiquark pairs appear in all hadrons.) The most persuasive evidence
that glueballs do indeed exist come from lattice-QCD calculations. In an
approximation omitting virtual quark pairs, the evidence is clear that stable
gluon-only bound states arise Vaccarino:1999ku ; Morningstar:1999rf . There
are two challenges with glueballs. One, obviously, is to extend the lattice
calculations to full QCD with virtual quark-antiquark pairs. The other is to
identify them unambiguously in the laboratory Sexton:1995kd . Experiments with
$\gamma p$ collisions are expected to produce such states and are planned at
the 12 GeV upgrade of CEBAF, the accelerator at Jefferson Lab. To plan this
program it is essential to know the photocouplings of the mesons (exotic or
not). It is possible to compute these photocouplings in lattice QCD, as has
been demonstrated for charmonium ($\bar{c}c$ mesons) Dudek:2006ej . An
extension of this work to light mesons, including exotics and glueballs, is
under way.

A long-term goal of nuclear physics is to compute many-body interactions from
QCD. In addition to providing a better understanding of the nucleus, these
calculations are relevant to mesonic atoms, namely, those containing a
positively and negatively charged meson, and to strangeness in neutron stars.
Although in some cases lattice QCD results can be compared to measurements,
the more interesting situation is where this cannot be done. The aim here is
to gain confidence in the reliability of these difficult calculations, so that
one can use them (with robust uncertainty estimates) in applications where it
is otherwise impossible to acquire the needed information any other way.

An interesting example is the lattice calculation of the $\pi K$ scattering
lengths, of which there are two, $a_{1/2}$ and $a_{3/2}$, depending on whether
the $\pi K$ system has isospin $I=\frac{1}{2}$ or $\frac{3}{2}$. A recent
lattice-QCD calculation finds Beane:2006gj

$\displaystyle a_{1/2}m_{\pi}c/\hbar$ $\displaystyle=$
$\displaystyle+0.1725\pm 0.0017^{+0.0023}_{-0.0156},$ (12) $\displaystyle
a_{3/2}m_{\pi}c/\hbar$ $\displaystyle=$ $\displaystyle-0.0574\pm
0.0016^{+0.0024}_{-0.0058},$ (13)

in convenient dimensionless units. These are in rough, but not spectacular,
agreement with other determinations using chiral perturbation theory or the
Roy-Steiner equations. For example, an analysis based on the Roy-Steiner
method Buettiker:2003pp  finds

$\displaystyle a_{1/2}m_{\pi}c/\hbar$ $\displaystyle=$ $\displaystyle+0.224\pm
0.022,$ (14) $\displaystyle a_{3/2}m_{\pi}c/\hbar$ $\displaystyle=$
$\displaystyle-0.0448\pm 0.0077,$ (15)

employing an extrapolation from high energies down to the threshold (where the
scattering length is defined); see Beane:2008dv  for more discussion. A more
direct experimental determination is proposed by the DiRAC Collaboration,
which would form $\pi^{-}K^{+}$ atoms DiRAC . In this sense, Eqs. (12) and
(13) represent something exciting: a theoretical prediction awaiting
definitive experimental confirmation.

Multimeson systems are proving to be a fruitful area of research, with further
work on $\pi\pi$ Beane:2007xs , $KK$ Beane:2007uh , and even states with as
many as twelve pions Detmold:2008fn . Once methods have been vetted in mesonic
systems, the next step is to consider nucleon-nucleon Beane:2006mx  and
nucleon-hyperon Beane:2006gf  systems. This is more difficult, because the
signal-to-noise ratio in the Monte Carlo estimate of baryonic correlation
functions is worse. Here the methods of effective field theories can be used
to extend feasible lattice-QCD calculations to a wider range of nuclear
phenomena. The hyperon-nucleon interaction is of interest, because there is
speculation that the large Fermi energy may make it energetically favorable
for some nucleons to transmute to hyperons Kaplan:1986yq .

##  4 QCD thermodynamics

The discussion in the introduction of the computational scope of numerical
lattice QCD mentions that, but does not explain why, the spacetime lattice
usually has a time extent much longer than the spatial dimension, for example,
$32^{3}\times 128$. The reason is that the functional integral actually
describes a grand canonical ensemble (in the sense of thermodynamics) at
temperature $T$,

$k_{B}T=\hbar c/N_{t}a,$ (16)

where $N_{t}$ is the number of sites in the time direction, $a$ is the lattice
spacing, and $k_{B}$ is Boltzmann’s constant. In the applications to particle
and nuclear physics discussed in Sections 2 and 3, one wants $T=0$ and, hence,
takes $N_{t}$ large enough for the Boltzmann suppression
$e^{-mc^{2}/k_{B}T}=e^{-N_{t}amc/\hbar}$ to eliminate thermal effects, where
$m$ is a hadron mass.

There are, however, many areas of research where a nonzero temperature and a
nonzero chemical potential arise. For nonzero temperature, one simply makes
$N_{t}$ smaller, typically $4\leq N_{t}\leq 12$ in current work. Such lattices
take somewhat less memory, but not less computing because thermodynamics
obviously requires calculations at several temperatures. It is conceptually
easy to introduce a nonzero chemical potential into lattice gauge theory, but
the conceptually clean approach is computationally beyond present resources.
For small chemical potential, several approaches are available and efficient
Schmidt:2006us .

The physical phenomena of interest are summarized by the phase diagram in
figure 4.

Figure 4: Phase diagram of QCD, showing conjectured phases and phenomena of
physical interest. Accelerator experiments at RHIC and LHC probe small
chemical potential, around the crossover regime. At the planned Facility for
Antiproton and Ion Research (FAIR) experiments can reach larger chemical
potential. Image from the CBM Collaboration cbm .

The key feature is a phase transition from hadronic matter to a substance
called the quark-gluon plasma. The transition is expected to be first order
and end at a critical point at chemical potential $\mu\neq 0$. Between the
critical point and the $\mu=0$ axis the transition is a rapid crossover.
Lattice-QCD calculations, as well as measurements of heavy-ion collisions at
RHIC and LHC, probe the crossover, which is relevant to the cooling of the
early universe. Neutron stars appear at high baryonic density, namely, high
$\mu$. At even higher densities, several phases with color superconductivity
are thought to exist. Apart from heavy-ion collisions (so far at small $\mu$)
much of this phase diagram is unexplored, and lattice gauge theory offers the
only _ab initio_ , quantitative, theoretical tool to see if it really reflects
QCD.

The central concept in QCD thermodynamics is the equation of state (EoS),
namely, the relationship between the pressure $p$ and energy density
$\epsilon$ as the temperature $T$ and chemical potential $\mu$ are varied. The
EoS reveals the relevant degrees of freedom. In QCD, are they hadrons? Or
quarks and gluons? At low temperatures, hadrons are clearly present, and
phenomenological models of hadron gases can describe QCD thermodynamics. At
very high temperatures, perturbation theory in $\sqrt{\alpha_{s}(\hbar
c/k_{B}T)}$ can be used. But near the transition, nonperturbative methods are
needed.

The first step is to find the transition temperature, which has now been done
with 2+1 flavors of realistic sea quarks Bernard:2004je ; Cheng:2006qk . The
different physics below, near, and above the critical temperature $T_{c}$ also
has ramifications for controling systematic errors. Discretization effects
from the lattice, for example, depend on the basic degrees of freedom. (For a
crossover it is, strictly speaking, incorrect to speak of a critical
temperature. But the critical point in the phase diagram is close enough to
the $\mu=0$ axis that thermodynamic variables change rapidly at $T_{c}$.)

The next step is to map out the thermal contribution to the trace of the
energy-momentum tensor, $\tr\Theta=\epsilon-3p$. From the temperature
dependence of $\epsilon-3p$ other thermodynamic variables, such as the entropy
density, can be worked out. This also has now been done with 2+1 flavors of
realistic sea quarks Bernard:2006nj ; Cheng:2007jq , including at small $\mu$
Bernard:2007nm . As the system is heated, even to $2T_{c}$, the behavior of
$\epsilon-3p$ shows that the system does not turn rapidly into weakly
interacting quarks and gluons.

To make contact with heavy-ion collisions, the last step would be to take the
output of lattice QCD as input to hydrodynamic models. With these models one
would then compute experimental observables, such as elliptic flow and
quarkonium suppression. With current computing resources, the EoS has been
pinned down with 20–25% accuracy. To make a big impact on hydrodynamic
modeling, however, it will be necessary to reduce the error to 5%. This is
conceptually straightforward to achieve, but it will be computationally
challenging. Part of the strategy is to reduce discretization effects, which
scale as $a^{2}$ ($a$ is the lattice spacing), while the net cost scales as
$a^{-11}$.

##  5 Non-Standard particle physics

As mentioned above, particle physics awaits a new era when the LHC starts
producing physics results. Commencing later this year, the LHC will collide
two beams of 7 TeV (TeV = teraelectronvolt) protons. (Amusingly, some particle
physicists now call this energy scale the terascale, deliberately alluding to
terascale computing. It would be unkind to tell them that in computing we are
moving on to the petascale.) In particular, the LHC will address the central
problem of particle physics, which is the origin of the $W$ and $Z$ boson
masses. The fact that these masses do not vanish demonstrates that some
mechanism breaks the electroweak gauge symmetry spontaneously. “Spontaneous
symmetry breaking” means simply that, although the equations of motion are
symmetric, the solution chosen is not symmetric. For example, in the Standard
Model a scalar field breaks electroweak symmetry, via a technique invented by
Higgs. Interactions between the Higgs field and the fermions also generate (in
the Standard Model) the quark and charged lepton masses and the CKM matrix.
Therefore, flavor physics will remain interesting in the LHC era, and the
physics program discussed in Section 2 will need to be continued and extended.

In this section, however, I would like to consider applications of lattice
gauge theory to understanding the mechanism of electroweak symmetry breaking.
Since we suspect that the Standard Higgs sector is far from the full story, it
is possible (some would say likely) that strongly coupled fields are involved.
For example, instead of being an elementary field, the Higgs boson (and its
siblings, the longitudinal polarization states of the $W$ and $Z$ bosons)
could be composite. The constituents would then form a whole spectrum of
states, just as quarks bind into many different hadrons. Even if the Higgs is
elementary at the terascale, other agents of electroweak symmetry breaking
could be strongly coupled.

Numerical work on theoretical bounds on the Higgs mass are a long-standing
topic in lattice field theory Dashen:1983ts ; Heller:1993yv . The issue is
that the Higgs mass is proportional to its self-coupling, which runs at short
distances in a way similar to $\alpha$ in QED; see figure 1. The perturbative
running becomes invalid once the coupling is too large, and nonperturbative
methods are required. A subtle relation arises between the Higgs mass and a
scale where new phenomena imply that the Standard Model breaks down. To make a
long story short, one can in this way bound the Higgs mass from above. A
similar analysis applied to the Yukawa coupling between the Higgs field and
the top quark leads to a lower bound on the Higgs mass, as discussed in a
recent comprehensive analysis Fodor:2007fn .

A popular class of non-Standard models is to replace the Higgs sector with a
strongly interacting sector that is, in some respects at least, analogous to
QCD. In fact, absent a stronger source of symmetry breaking, the pions of QCD
would break the electroweak symmetry, leading to $W$ and $Z$ masses around 100
MeV. What one needs, then, is a new quantum number, called technicolor, and a
gauge force to bind the corresponding particles, called techniquarks,
together. These dynamics are posited to take place at terascale energies and,
by analogy with QCD, generate vector boson masses around 100 GeV, as observed.
The earliest models of technicolor followed the example of QCD too closely and
have been ruled out by experimental measurements. One way around the
experimental constraints is to assume that the coupling evolves more slowly
than in QCD. This is called “walking” (i.e., slower than running) technicolor.
For a wide range of scales below the confining scale, the coupling is too
strong for perturbation theory to be relied on, so nonperturbative work with
lattice gauge theory is called for. In the past year, the walking hypothesis
has indeed been tested, by examining the scaling behavior of the technipion
mass, the vector meson mass, and the technipion decay constant
Catterall:2007yx . Even more recently, the evolution of $\alpha_{s}$ in (a
TeV-scale version of) QCD with several flavors has been computed
Appelquist:2007hu , to see how many flavors are needed to having a walking
coupling, and how a walking coupling changes the physics.

The most popular way to relieve the theoretical problems of the Standard Model
is to introduce supersymmetry, a symmetry that transforms fermions into
bosons, and vice versa. In these models, every known particle (quarks,
leptons, gauge bosons) comes with another state known as a superpartner. In
the minimal supersymmetric model, the Higgs sector contains three neutral
bosons, a charged pair, and their superpartners. No superpartner has been
observed, so at the terascale and below supersymmetry must be broken somehow.
Usually supersymmetric models are conceived of as weakly coupled, but, since
they are gauge theories, strong coupling is not to be dismissed out of hand.
It is therefore of great interest to formulate and simulate supersymmetric
gauge theories on a spacetime lattice. This has been an extremely fruitful
theoretical field during the past several years Kaplan:2003uh ; Giedt:2007hz ,
and now first numerical simulations are being carried out Catterall:2007fp ;
Catterall:2008yz . This rapid development yields hope that nonperturbative
studies of supersymmetric models will be possible, including studies of
supersymmetry breaking.

##  6 Perspective

Every now and then one hears the question, “How much computing would it really
take to solve QCD on a computer?” The questioner is (usually) well meaning,
but most experts would agree that the question is not well conceived. The
calculations for some of the simplest hadronic quantities are finally becoming
mature, with analyses that forthrightly address all sources of uncertainty.
This class of observable, sometimes called “gold-plated” Davies:2003ik ,
consists of masses of particles that are stable (with respect to the strong
interactions), not too close to thresholds, and of hadronic transition matrix
elements with one or zero of these stable hadrons in the final and initial
states. Many, but not all, aspects flavor physics (Section 2), and some
aspects of hadronic structure (Section 3), lie in this class, and the first
solid round of serious computation is nearing its end.

But this is the end of the beginning. In these nearly mature fields, it turns
out that precision is important, at least for the foreseeable future. For
example, to gain persuasive evidence of new phenomena in flavor physics, 5–25%
accuracy is not enough. A better target (given the stakes and the quality of
the corresponding experiments) is 1–3%. If the history of kaon physics is a
guide, it will eventually be important to go further. Happily, the need for
and clear path to precision has persuaded the particle-physics community that
the resources needed for lattice flavor physics are worth the cost.

Even a quick glance at Section 3 reveals that the large majority of
interesting problems in nuclear physics require much more computing resources
than gold-plated observables. By definition the excited hadrons are not stable
under the strong interaction. They will be harder to pin down, and it is
possible that we will not know how hard until the work currently under way
reaches some of its milestones. The calculations of scattering lengths, while
impressive, must be repeated with several values of the physical volume to
ensure that the mathematical formalism used to extract them works as it
should. Given the importance of nuclear lattice QCD in guiding nuclear
experimentation, one would hope for full support from experimenters for the
needed computing resources.

The “how much?” question should be rephrased. It is really a class of
questions, “How much computing would it take to compute my favorite
observable?” Of course, the answer depends on your observable and its wider
context. It should be clear, however, that physics with either precision and
complexity leads to questions whose answers will need one or two orders of
magnitude more computing than what is available now.

Another reason the “how much?” question is ill-conceived is that it often
overlooks the heterogeneous job mix needed to attack many problems in lattice
QCD. It is not enough to generate huge ensembles of lattice gauge fields, with
ever smaller lattice spacing and ever smaller quark masses, on the world’s
highest-capability computer. This step is necessary, creating a gold mine of
information. It is not enough, however, merely to open a mine; the gold must
be extracted and, here, we need a large capacity of computing that is flexible
in every way (while having significant capability of its own).

We do not know how broad the future of lattice gauge theory will be. If the
LHC discovers a strongly interacting sector beyond the Standard Model, it is
likely that chiral fermions play a role. This is simply because, at the
terascale, we already know that left-handed fermions and right-handed fermions
are different fields. But chiral fermions possess some conceptual and
computational challenges not found in QCD. A strongly coupled terascale will
bring many more particle theorists to lattice gauge theory, and this larger
community will clamor for computing that is up to the task.

This work has been supported in part by the United States National Science
Foundation and the Office of Science of the U.S. Department of Energy (DOE).
Software development and hardware prototyping within the USQCD Collaboration
are supported by SciDAC. Fermilab is operated by Fermi Research Alliance, LLC,
under Contract DE-AC02-07CH11359 with the US DOE.

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