Recent Developments in Lattice QCD
Gunnar S. Bali
Department of Physics & Astronomy, The University of Glasgow, Glasgow G12 8QQ, UK
Abstract. Recent trends in lattice QCD calculations and latest results are reviewed, with particular
emphasis on multiquark-states and the nucleon structure.
Keywords: Lattice QCD, nucleon structure, confinement, exotic hadrons, pentaquarks.
PACS: 12.38.Gc, 13.40.Gp, 12.38.Aw, 12.39.Mk.
1. INTRODUCTION
While the simplicity and elegance of QCD is very appealing theoretically, the phenomenological
observations of spontaneous chiral symmetry breaking and of the con-
finement of colour charges turned it into a major calculational nightmare: it took almost
twenty years after the discovery of asymptotic freedom in 1973 to convincingly
demonstrate that the QCD Lagrangian indeed implies these highly non-trivial collective
phenomena. This was done by means of numerical simulation.
In such lattice simulations, QCD is regularised by introducing a lattice cut-off a.
Subsequently, the nf +1 parameters of the theory, i.e. QCD coupling and quark masses
are matched to reproduce nf +1 hadron masses. Everything else is a prediction and in
this sense lattice QCD is a first principles approach. The confinement of colour implies
that finite size effects are usually tiny, as long as the spatial box extent La ≫m−1
p . We
are also fortunate to find that lattice spacings a−1 = 1 – 4 GeV are sufficiently small to
allow for controlled continuum limit extrapolations, a→0. This means that L≪100 is
sufficient, which makes QCD tractable on computers.
This need not be so. If for instance we were to attempt a brute force calculation of the
deuteron binding energy of about 2 MeV then resolving the large scale difference with
respect to the nucleon mass of almost a factor of 50 would require both, a huge volume
and incredible statistical accuracy. Of course such scale hierarchies indicate that direct
calculations should be avoidable as they lend themselves to an effective field theory
treatment.
We will discuss progress on computational problems in Sec. 2, before summarising
recent lattice studies of multiquark hadron spectroscopy in Sec. 3 and of the nucleon
structure in Sec. 4.
2. CHALLENGES IN LATTICE CALCULATIONS
On a lattice with V = L3T points, the lattice Dirac operator is a huge matrix of dimension
12V. The inversion of this operator represents the major computational task of
lattice QCD and this makes simulations incorporating sea quarks notoriously expensive.
The algorithmic cost explodes with small p masses, µ 1/(mpa)≃3. A smaller mp also
requires a larger spatial lattice volume and the scaling behaviour, keeping mpL fixed, is
even worse: µ 1/(mpa)≃7.
In Fig. 1 we display recent results obtained by the LHP and SESAM Collaborations
[1] on the quark contribution DS to the proton spin, in the MS scheme at a scale
m= 2 GeV, for nf = 2. The normalisation is such that 1
2 = 1
2DS+Lq +Jg, where Lq
is the contribution from the quark angular momentum and Jg from the gluons. Similar
results have been obtained by the QCDSF Collaboration [2]. In these simulations,
mp > 550 MeV. We took the liberty to convert the mass scale into units of our choice.
Obviously, for infinite quark masses we expect DS = 1. It is therefore not surprising
that the experimental value is overestimated. However, it is clear from the figure that
smaller quark masses are absolutely essential to allow for a meaningful chiral extrapolation.
Fortunately, with the advent of new Fermion formulations [3] that respect an exact
lattice chiral symmetry, a reduction of the quark mass towards values mp ≈ 180 MeV
has become possible recently [4], albeit so far only in the quenched approximation.
In many lattice calculations it is sufficient to calculate quark propagators that originate
from a fixed source point. In these cases only one column of the inverse Dirac
matrix needs to be calculated, naïvely reducing the effort by a factor V. In some cases
diagrams with disconnected quark lines are needed. Examples are the physics of flavour
singlet mesons, strong decays as well as parton distributions (PDs). In the latter case the
complication can be avoided by assuming SU(2) isospin symmetry and only calculating
differences between u and d quark distributions. Disconnected quark lines require all-toall
propagators and hence a complete inversion of the Dirac matrix appears necessary.
This turns out to be prohibitively expensive in terms of memory and computer time.
Fortunately, sophisticated noise reduced stochastic estimator techniques have been developed
over the past few years and as a result tremendous progress was achieved. One
such benchmark is the QCD string breaking problem, Q(r)Q(0) ↔ B(r)B(0), where
q/p
B = Qq and Q is a static quark, which represents one of the cleanest examples of a
strong decay. Within both, the transition matrix element as well as the BB state all-to-all
propagators are required. In Fig. 2 we display the result of a recent SESAM Collaboration
study [5]. An extrapolation to physical light quark masses yields a string breaking
distance rc ≈ 1.16 fm. The gap between the two states in the string breaking region is
DE ≈50MeV and we are able to resolve this with a resolution of 10 standard deviations!
In conclusion, all the long standing killers m≪ms, nf > 0 and all-to-all propagators
have been successfully tackled. However, we are still a few years away from overcoming
two of them at the same time and possibly almost a decade separates us from simulations
of flavour singlet diagrams with realistically light sea quarks. In most cases not all these
ingredients are required simultaneously. While bigger computers are always needed,
most of the recent progress would have been impossible without novel methods. The
gain factor from faster computers was almost 5,000 over the past 15 years. The factor
from theoretical and algorithmic advances is harder to quantify. Two such examples have
been mentioned: chiral lattice Fermions and all-to-all propagator techniques.
3. PENTAQUARKS AND TETRAQUARKS
QCD goes beyond the quark model and hence hadronic states that do not fit into a
naïve quark model of q ¯ q mesons and qqq baryons are of particular interest. Many of
the observed hadrons will contain considerable non-quark model components, in particular
within the scalar sector. Obviously, quantum numbers that are incomprehensible
with a quark model meson or baryon interpretation provide us with the most clean-cut
distinction. Such examples do exist in the Review of Particle Properties, namely the
JPC = 1−+ mesons p1(1400) and p1(1600). The minimal configuration required to obtain
a vector state with positive charge either consists of two quarks and two antiquarks
tetraquark/molecule) or of quark, antiquark and a gluonic excitation (hybrid meson).
Unfortunately, these resonances are rather broad with a width G ≈ 300 MeV which
might be one of the reasons why they are often ignored. However, the ratio G/m is very
much the same as for the (quark model) r(770) meson.
Another clear indication of an nquark > 3 nature would for instance be a baryonic
state with strangeness S = +1. The minimal quark configuration in this case consists
of five quarks (pentaquark): uudd ¯ s. It is no secret that over the past two years several
experiments have presented evidence of a very narrow Q+(1530) resonance, with decay
Q+→pK0 and Q+→nK+. The parity has not yet been established. However themass is
about 100 MeV above the KN threshold and for JP = 1/2− an S-wave decay is possible,
which is difficult to reconcile with a width G≪10 MeV. For 1/2+ a P-wave is required,
still a bit puzzling but less so. As the main decay channel does not require sea quarks one
might hope to gain some insight from quenched lattice simulations and several attempts
have been made [6, 7, 8, 9, 10, 11].
Two groups [7, 8] also investigated charmed pentaquarks and two studies [6, 9]
incorporated the I = 1 sector, in addition to I = 0. Two groups [8, 9] employed chiral
overlap Fermions while the others used conventional Wilson-type lattice quarks. Only
the Kentucky group [9] went down to mp ≈ 180 MeV while all other p masses were
larger than 400 MeV. The Budapest-Wuppertal group [6] varied the lattice spacing and
attempted a continuum-limit extrapolation. In all studies the negative parity mass came
out to be lighter than the positive parity one, which is expected in the heavy quark limit.
There are two crucial questions to be asked: what happens when realistic light quark
masses are approached? Do we see resonant or scattering states? Resolving a resonance
sitting on top of a tower of KN scattering states with different relative momenta appears
rather hopeless at first. However, there are two discovery tools available: variation of
the lattice volume and of the creation operator. By varying the volume one will change
the spectrum of KN scattering states as well as the coupling of a given operator to KN
(the spectral weight). If the pentaquark really was such a narrow resonance as some
experiments suggest then maybe a lattice operator can be constructed that has a large
overlap with this state but only a very small coupling to KN. For the 1/2+ state which
can only decay into a P-wave, the mass of the scattering state will depend on the lattice
size since the smallest possible non-vanishing lattice momentum is p/(aL). For 1/2−
the volume dependence of the lowest scattering state mass will be weak, however, the
scaling of the spectral weight with the volume can still provide us with a hint.
It turns out that the situation on the lattice is at least as ambiguous as the one
encountered in experiment. To demonstrate this we display some Kentucky results [9]
in Fig. 3. It appears that the 1/2− state dominantly couples to an S-wave KN. The 1/2+
however displays the qualitative volume dependence of a P-wave but does not share its
mass. Interpretation as a P-wave plus some linear momentum is a likely possibility. At
very small mp the situation becomes further complicated by the fact that there is no axial
anomaly in the quenched approximation. Hence the flavour singlet h′ is degenerate with
the p. As this contribution comes in with a negative spectral weight, it is sometimes
labelled as a “ghost”. The 1/2+ state can contain such a KN h′ S-wave (dashed lines).
While the 1/2− state becomes indistinguishable form a KN, the P =1/2+ might contain
a resonant component. In order to arrive at more definite conclusions a variation of the
creation operator as well as of the volume appears necessary, which is a very ambitious
project. Nonetheless, the lattice results obtained so far are quite instructive.
1/2- state
1/2+ state
Lattice studies of diquark interactions in a simplified, more controlled environment
represent an alternative strategy to the brute force simulation of unstable states. A baryon
with one static and two light quarks constitutes one such arena. One can of course
also investigate multiquark interactions in the nonrelativistic limit of infinitely heavy
quark masses. Such tetra- and penta-quark potentials have been studied recently by two
groups [11, 12, 13] and the results might provide model builders with some insight.
However, it is not clear how to relate these findings to the light quark limit in which
chiral symmetry appears to play a bigger rôle than instantaneous confining forces.
There exist quite a few narrow resonances very close to strong decay thresholds like
the L(1405), the recently discovered X(3872) charmonium state and the a0/ f0(980)
system. It is very conceivable that such states contain a sizable multiquark component.
The question then arises if these constitute would-be quark model states or if these are
true molecules/multiquark-states, that appear in addition to the quark model spectrum. A
fantastic arena to address this was provided by the recently discovered (probably scalar)
D∗
s (2317) and (probably axialvector) D∗
s (2457) states. First lattice studies [14, 15] have
been performed, with somewhat contradictory interpretations of very compatible results.
One might hope that a similar effort will be dedicated on the comparatively cleaner and
easier question of tetraquarks as has been on the pentaquark studies.
4. THE NUCLEON: FORM AND STRUCTURE
The obvious strength of lattice studies lies in hadron spectroscopy. Calculations of
hadronic matrix elements are more involved. However, once light sea quarks are included,
many states become unstable and it turns out that calculating internal properties
of stable particles is easier than resolving the strong decay dynamics of resonances. This
should also be clear from the pentaquark discussion above. In view of this, the phenomenologically
most exciting lattice input to be expected in the next few years should
be calculations of the hadron structure.
Quite a few results on moments of generalised parton distributions (GPDs), most notably
from the QCDSF [16], SESAM and LHP Collaborations [17], exist and the lattice
method is already very competitive in this area which is experimentally very challeng-
ing, in particular when it comes to transversity distributions. These new simulations
build upon the methods that have been developed and successfully used in the context
of calculations of unpolarised and polarised PDs. Possibly the biggest systematic uncertainty
in these past studies was the chiral extrapolation, down to physical mp.
In particular the lowest moment of the unpolarised distribution,
hxiu−d = hxiu−hxid, hxniq = Z 1
0
dxxn [q(x)−(−)n ¯ q(x)] , (1)
turned out to be overestimated by almost 60 % if linearly extrapolated in m2
p, both without
and with sea quarks [18, 19]. Chiral perturbation theory predicts deviations from
such a linear behaviour and there is some room for inventive extrapolations [18]. However,
even at mp ≈ 300 MeV there is no sign of a bending towards the chiral limit [20]
as can be seen from Fig. 4. In the right part of the figure the study is repeated with
chiral overlap Fermions which protect against mixing with lower dimensional lattice
operators, down to mp ≈ 450 MeV. In this case no non-perturbative renormalisation to
an intermediate MOM scheme is available as yet. However, in the Wilson case the difference
between purely perturbative and (partly) non-perturbative renormalisation was
negligible. This is potentially exciting news which probably will receive clarification
during the coming year.
Finally, there has also been progress in resolving the momentum dependence of
electromagnetic g∗N → D transition form factors in a quenched study [21], in the
region 0.1GeV2 < Q2 < 1.4GeV2. The magnetic dipole form factor is significantly
overestimated at large Q2, due to the unrealistically small charge radii of D and N. One
might hope that such effects cancel in part from form factor ratios. REM = GE2/GM1
is fairly constant at -0.02(1), once extrapolated to the chiral limit. In contrast, RCM =
GC2/GM1 decreases monotonously from -0.01(1) at 0.1 GeV2 down to -0.09(3) at
Q2 >1 GeV2. This behaviour is in good agreement with Q2 >0.4 GeV2 CLAS data [22]
while it is hard to reconcile with the OOPS point [23], RSM = [−6.1±0.2±0.5]% at
Q2 ≈ 0.13GeV2.
5. CONCLUSION
Lattice calculations have made a lot of progress recently. Many studies now include sea
quarks. Within the quenched approximation, light quark masses close to the physical
limit have been realised and the lattice provides a powerful tool for exploring the validity
range of chiral expansions. There has also been tremendous progress in the calculation
of diagrams with disconnected quark lines which are for instance needed to understand
strong decay processes and flavour singlet contributions. A lot of work still needs to be
done. I was only able to present a tiny selection of latest lattice results and my apologies
go to all those whose work I failed to mention.