Review of Unquenched Results

10 Pages·0.15 MB·English

by  Robert D. Mawhinney

CU–TP–965
Review of Unquenched Results
Robert D. Mawhinneya∗
aDepartment of Physics, Columbia University, New York, NY 10027,USA
One of the major frontiers of lattice field theory is the inclusion of light fermions in simulations, particularly
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in pursuit of accurate, first principles predictions from lattice QCD. With dedicated Teraflops-scale computers
0
currently simulating QCD, another step towards precision full QCD simulations is underway. In addition to
0
ongoing staggered and Wilson fermion simulations, first results from full QCD with domain wall fermions are
2
available. Aftersomediscussionofworktowardbetteralgorithms,simulationscompletedtodatewillbediscussed.
n
a
J
1. INTRODUCTION and is a useful testing ground for exploring fi-
0
3 nite volume effects and simulation lengths. Also,
Amajorchallengeinrealizingthefullpotential
hadronic states differing only by their parity are
1 of the non-perturbative regularization provided degenerate unless chiral symmetry is broken, so
v by lattice field theory is the presence of fermion
studyingtheseotherparitystatesinthespectrum
2 fieldsinsimulations. Inmanycases,suchasQCD
mayshowsensitivitytotheinclusionofthequark
3
at finite density and the Hubbard model, the
0 determinant.
fermions require an improvement in algorithms
1 Most of the techniques developed in the
0 before large systems can be simulated. For QCD quenched approximationfor measuring other ob-
0 at zero chemical potential, simulations with dy-
servables can be easily adapted to full QCD sim-
0 namical fermions will continue to become bet-
ulations. Many groups are reporting on effects
/
t ter controlled by faster computers, even without ofdynamicalfermionsoutsideofthelighthadron
a
(hoped for) theoretical improvements.
-l There is over a decade of experience with full spectrum (fB, etc.). Please see the reviews of
p these areas for information.
QCD simulations, although only recently has
e
h enough data become available for some extrap-
2. ALGORITHMS
: olations to the continuum [1]. As with quenched
v
i simulationsquestionsofwhataresatisfactoryvol- Three groups reported work on the algorithms
X
umes and lattice spacings, what are reliable sim- used to produce the Markov chain in numerical
r ulation lengths and what are the best extrapola- simulations.
a
tions need to be answered. Removing the uncon-
2.1. Meron Cluster Algorithm
trolledtruncationofthequenchedapproximation
Wiese and collaborators (MIT) [3], have tack-
isvitaltoachievingprecisionresults,butanswers
led the long standing problem of algorithms for
to the above are very important.
fermion systems where the weights in the path
Afterabriefoverviewoftheworkonalgorithms
integralarecomplex. Computationaltimesexpo-
presented at this conference, I will focus on re-
nential in the volume are needed if the weight is
sultsforQCD,particularlythelowlyinghadrons.
made real by taking its absolute value and mov-
Thereisevidencethequenchedhadronspectrum,
ing the phase to an observable. By working with
in the continuum and chiral limits, differs from
a continuum Euclidean time formulation, where
nature [2]. For full QCD, the light hadron spec-
fermions are represented by their world lines,
trum is interesting in its own right, provides the
their algorithms form clusters whose flip gives a
basis for a determination of light quark masses
definite sign. Asecondimprovementinsures that
∗SupportedinpartbytheUSDepartmentofEnergyand clusterswhichcancelthroughasignflipandthose
theRIKEN-BNLResearchCenter. that don’t are generated with similar probabili-
2
ties. Thisalgorithm,withitsattendantimproved 3. SIMULATION KEY
estimator, has allowed them to simulate various
To distinguish the different actions, the abbre-
fermionic systems that are intractable with con-
viations below will be used (following the CP-
ventional approaches. Application of these ideas
PACS collaboration).
to coupled gauge-fermion systems is being inves-
tigated. gauge action
P plaquette
2.2. Truncated Determinant Approach R RG improved
Duncan, et. al. [4] have been studying an al-
gorithm where the fermion determinant is split, fermion action
in a gauge-invariantway,into aninfraredandul- S staggered
traviolet part. The ultraviolet part is modeled W Wilson
by an effective action made up of small Wilson C Clover
loops, while the infrared part is determined pre- D domain wall
cisely using a Lanczos procedure. To date, they
The RG improved gauge action is the one pro-
have generated a sequence of gauge fields using
posed by Iwasaki, et. al. [7].
justtheinfraredpartofthedeterminantandhave
In the tables of simulations parameters, three
found that the exact contribution of the ultravi-
results are generally listed: m a, m /m and
ρ π ρ
olet part is well fit by an expansion in a small
m L, where L is the spatial lattice extent and
ρ
number of Wilson loops. A somewhat surprising
a is the lattice spacing. The value of m a (where
ρ
resultalsocomesfromthegenerationofthegauge
available) is for the quark mass which gives the
fields: they first update links and then put in a
quoted m /m . Also note that a 770 MeV ρ in
π ρ
global accept/reject step based on the infrared
a 3 fm box has m L = 11.6. Since these tables
ρ
part of the determinant. They find reasonable
should only serve as a guide to the general fea-
acceptance for this, even for large volumes.
tures of the data sets, errors are not listed and
only2significantfiguresaregiven. Pleaseseethe
2.3. Multiboson Algorithm
original work for more details.
de Forcrand and collaborators [5] have been
The hybridMonteCarlo(HMC)algorithmhas
testing an evolved version of Luscher’s multibo-
been used for the 2 flavor Wilson and domain
sonalgorithm[6]inarealQCDcontextandcom-
wall and 4 flavor staggered simulations. For 2
paring its performance to a standard, but highly
flavor staggered simulations the ‘R’ algorithm of
tuned, hybrid Monte-Carlo code of the SESAM
Gottlieb, et. al. [8] has been employed. The tra-
collaboration. This new work uses only 24 bo-
jectory counts in the tables are from the groups
son fields for a simulation with m /m = 0.833,
π ρ themselves and the definition of a trajectory dif-
a dramatic improvement. This is achieved by an
fers between the groups.
“ultraviolet filtering” of the determinant, where
thedeterminantisfilteredbyanexponentialterm
4. STAGGERED FERMIONS
made up of weighted small Wilson loops, and a
cancelling term enters the gauge action. (This Recent staggered simulations have been done
filtering is similar to that used by [4], but here bytheMILCandColumbiagroupsandarelisted
is part of an exact algorithm.) An optimized in Table 1. MILC reported hadron spectrum re-
quasi-heatbath is then applied to the combined sults last year [9] and new results for decay con-
boson-gauge system. They conclude that for a stantsthisyear[10]. Theyfoundacontinuumex-
163×24 lattice, with β =5.6 and the equivalent trapolation of m /m for staggered fermions in
N ρ
to κ=0.156for Wilsonfermions,the multiboson fullQCDgivesalargervaluethaninthequenched
algorithm decorrelates the plaquette better than continuum. The Columbia group has new data
HMC, measuring both algorithms in units of ap- for 2 and 4 flavor QCD with staggered fermions,
plications of the Dirac operator. with simulations still underway [11].
3

Table 1
0.8
Summary of staggered simulations.
MILC - PS action - 243×64 - 2 flavors
β m mρa mρL mπ/mρ traj. s
s 0.6
5.6 0.08 0.98 23 0.76 2000 a
m
5.6 0.06 0.87 21 0.74 2000

5.6 0.04 0.75 18 0.71 2000 n
o
5.6 0.02 0.59 14 0.63 2000 r
d
0.4
5.6 0.01 0.50 12 0.53 2000 a
h
CU QCDSP - PS action - 163×32 - 2 flavors
β m m a m L m /m traj.
ρ ρ π ρ 0.2
5.7 0.015 0.48 7.7 0.63 5000 0.0 2 4 6 8 10k 3k 1k
CU QCDSP - PS action - 163×32 - 4 flavors
β m m a m L m /m traj.
ρ ρ π ρ
Figure 1. A comparison of hadron masses for
5.4 0.02 0.50 8.0 0.71 5180
a 10,000 trajectory run (10k), a 3,000 trajectory
5.4 0.015 0.48 7.7 0.67 4750
run (3k) and a 1,000 trajectory run (1k).
CU QCDSP - PS action - 243×32 - 4 flavors
β m m a m L m /m traj.
ρ ρ π ρ
5.4 0.02 0.49 12 0.72 5000
andcouplings,achievingreliableerrorsatthefew
5.4 0.01 0.37 8.9 0.66 5000
percentlevellikelyrequiresmorethan10,000tra-
jectory run.
Another important question for full QCD sim-
ulations is the volume required. We now investi-
There are existing results from 2 flavor stag- gate this question through the splitting between
geredsimulationsatβ =5.7 andm=0.01which parity partners in the hadron spectrum. We will
give some information about run lengths. Figure see that, at least for staggeredfermions, this is a
1shows,startingatthetop,m ,m andm from sensitive indicator of finite volume effects.
N ρ π
theColumbiagroup[12]andFukugita,et.al.[13]. MILCstudiesoffinitevolumeeffectsin2flavor
The masseslabeled “1k”arefrom1,000trajecto- dynamicalstaggeredsimulationsforcouplingsup
ries on a 204 lattice, “3k” from 3,000 trajectories toβ =5.6[15]showlatticevolumesof2.5-3fermi
on 163 ×32 and “10k” from 10,000 trajectories remove finite volume effects for m /m ∼ 0.5.
π ρ
on 163×40. The points to the left of “10k” are Their most recent β = 5.6 results for a 244 ×
the 10,000 trajectory run broken up in to nine 48 lattice [16] are shown in Figure 2. There is
1,000 trajectory runs (plus thermalization) [14]. no sign of parity doubling, consistent with their
The solid horizontal lines are the “10k” masses statement about finite volume effects.
and the dashed horizontal lines give ±5% of the Fukugita, et. al. also studied finite volume ef-
central value. fects but at weaker coupling (β = 5.7). Their
The similar size for the errors on the 3k and data is plotted in Figure 3, revealing parity dou-
10k runs (the same analysis was used for both) bling in the m → 0 limit for the N and N′.
′
indicates that long correlation times were likely (The N is the staggered fermion parity partner
missed in the 3,000 trajectory run. The larger of N.) Finite volume effects are likely distort-
error bars for the 1,000 trajectory runs may be ing the baryons,but not the mesons. Which way
due to short term noise effects. For these masses mN and mN′ move for larger volume is an open
4
2.0 1.2
N’ N’
N 1.0 N
1.5 a a
s 1 s 1
s ρ s 0.8 ρ
a a
m m
n 1.0 n 0.6
o o
r r
d d
0.4
a a
h 0.5 h
0.2
0.0 0.0
0.0 0.02 0.04 0.06 0.08 0.0 0.01 0.02
m m
q q
Figure 2. Some of the MILC 243×48, 2 flavor Figure 3. Staggered 2 flavor simulations on 204
staggered spectrum plotted versus quark mass. lattices showing parity doubling in mN and mN′
The lines merely connect the points. as m→0.
question, currently being addressed by 243 simu- 7.0 for 163.
lations underway at Columbia. Fourflavorslikelymakethisfinitevolumeeffect
The Columbia group has previously reported more pronounced, but it is a warning for 3 flavor
parity doubling for 4 flavor QCD on a 163 ×32 simulations. We are currently simulating with 2
volume with β = 5.4. (These parameters give a flavors on a 243 × 32 volume to see how large
rhomasswithinafewpercentoftherhomassfor the effects are there. Unfortunately, these large
2flavorsona163×32lattice.) Figure4showsour finite volumeeffects aremaskinganyinformation
results,wherethem=0.01pointisfromthe256- about the role of the determinant in the parity
node Columbia machine, the m = 0.015 point is splittings.
from QCDSP and the m=0.02 result was calcu- A final message about run lengths from the 4
latedonboththe256-nodemachineandQCDSP, flavor staggered simulations is shown in Figure
whichagreewithinerrors. Paritydoublingisclear 6. The upper line is the pion propagator at a
for both mesons and baryons as m→0. distance 10 lattice sites from the source for the
The Columbia group has now completed a m = 0.01 simulation, plotted against trajectory
5,000 trajectory simulation on a 243×32 volume number. The lower line is the same propagator
using QCDSP. Figure 5 shows that the degener- forthem=0.02simulation. Them=0.01simu-
acy between the hadrons in the m → 0 has gone lation shows fluctuations on a few thousand tra-
away. There is very little change in the m=0.02 jectorytimescale. Thisisclearevidencethatvery
masses, but for m = 0.01 m and m have long runs are needed as the quark mass is made
ρ N
dropped by ∼ 20% (m : 0.438(8) → 0.373(6), smaller. (Longautocorrelationtimes fortopolog-
ρ
m : 0.690(21) → 0.574(9)). This clearly shows ical charge for 4 flavor staggered simulations on
N
finite volume effects distorting the parity split- a 163×32 volume with β = 5.35 and m = 0.01
tingsandhereitprimarilyeffectsthenucleonand have also been seen [17].)
rho masses. Also, giventhe largedecreasein m ,
ρ
the larger243 lattice hasm L=8.9verycloseto
ρ
5
1.2
N’ 1.2
1.0 N
N’
s a1 1.0 N
s 0.8 ρ
a a
m ss 0.8 ρ1
n 0.6 ma
o
dr n 0.6
a 0.4 o
h dr
0.4
a
0.2 h
0.2
0.0
0.0 0.01 0.02
0.0
m 0.0 0.01 0.02
q
m
q
Figure4. Staggered4flavorsimulationson163×
32 lattices showing parity doubling in mN, mN′ Figure5. Staggered4flavorsimulationson243×
and m , m as m→0.
ρ a1 32latticesshowingparitydoublingeliminatedfor
larger volumes.
5. WILSON FERMIONS
The SESAM [18], UKQCD [1] and CP-PACS
160
[2] collaborations reported on full QCD simula-
0
tions last year and UKQCD [19] [20] and CP- 1 140
PACS [21] [22] have new results this year. The =

run parameters are given in Table 2 and 3. Both t 120
t
UKQCDandCP-PACSareusingcloverimproved a

r
Wilson fermions; UKQCD uses CSW determined to 100
by the Alpha collaboration and CP-PACS uses a a
g
tadpole improved value. a 80
p
UKQCD chooses β and κ to keep the lattice o
spacing, as determined using the Sommer scale pr 60

r andthestaticquark–anti-quarkpotential,con- π
0
stant. The physical volume then corresponds to 40
0 2000 4000 6000
1.7 fm for all lattice spacings they consider. As
discussed above for staggered fermions, this vol- trajectory number
ume can be expected to be rather small. They
do report evidence that the potential at small r
Figure 6. The pion propagator at separation
forthedynamicalsimulationsliesbelowthevalue
10 for 4 flavor simulations on 243 × 32 lattices
for quenched simulations. In addition, plotting
plotted versus trajectory number. Much longer
vector meson masses against pseudoscalar me-
autocorrelation times seem visible for the upper
son masses, they see a trend toward the physical
∗ plot (m = 0.01) when compared to the lower
(K,K ) value.
(m=0.02).
6
Table 2 Table 3
Summary of UKQCD and SESAM 2 flavor Wil- Summary of CP-PACS 2 flavor Wilson fermion
son fermion simulations. simulations.
SESAM - PW action - β =5.6 - 163×32 CP-PACS - RC action - β =1.80 - 123×24
κ m a m L m /m traj. κ c m /m traj.
ρ ρ π ρ sw π ρ
0.156 0.53 8.5 0.83 5000 0.1409 1.60 0.81 6250
0.1565 0.50 8.0 0.81 5000 0.1430 1.60 0.75 5000
0.157 0.46 7.4 0.76 5000 0.1445 1.60 0.70 7000
0.1575 0.41 6.6 0.68 5000 0.1464 1.60 0.55 5250
UKQCD - PC action - 163×32 CP-PACS - RC action - β =1.95 - 163×32
β =5.29, 5.26, 5.2, 5.2, respectively κ c m /m traj.
sw π ρ
κ csw mρa mρL mπ/mρ conf. 0.1375 1.53 0.80 7000
0.1340 1.92 0.70 11 0.83 101 0.1390 1.53 0.75 7000
0.1345 1.95 0.65 10 0.78 101 0.1400 1.53 0.69 7000
0.1350 2.02 0.59 9.4 0.69 150 0.1410 1.53 0.59 7000
0.1355 2.02 0.58 102
CP-PACS - RC action - β =2.10 - 243×48
κ c m /m traj.
sw π ρ
0.1357 1.47 0.81 2000
0.1367 1.47 0.76 2000
The CP-PACScollaborationhas been involved 0.1374 1.47 0.69 2000
in extensive simulations of full QCD with a va- 0.1382 1.47 0.58 2000
riety of lattice spacings, quark masses and vol-
umes. Their parameter choices keep the spatial CP-PACS - RC action - β =2.20 - 243×48
size fixed at ∼ 2.4 fm, using the ρ mass at the
κ c m /m traj.
sw π ρ
physicalm /m valuetosetthescale. Withtheir
π ρ 0.1351 1.44 0.80 2000
full QCD data set, they can extrapolate to the
0.1358 1.44 0.75 2000
continuum, with fixed finite volume effects. To
0.1363 1.44 0.70 2000
date, they only have 2,000 trajectories for their
0.1368 1.44 0.63 2000
β = 2.10 point, which experience with staggered
fermions suggestsmay notbe long enough. They
are addressing this issue.
CP-PACS found that a major difficulty with
the quenched hadron spectrum is the failure of φ masses, they have also determined the a → 0
a unique strange quark mass to give the physi- strangequarkmassatµ=2GeV.Oneoftheirre-
cal values for the K and φ masses. They have sultsisshowninFigure8. Theyfindvariousways
addressed this question with their new data and of determining the strange quark mass all agree
one of their results is shown in Figure 7. They in the a → 0 limit and there is a large difference
find evidence that the a → 0 extrapolation of betweenthe quenchedandunquenchedvaluesfor
the latticeresultismuchclosertothe physicalK this mass. Their final result is m =87(11) MeV
s
mass(⋄)thanthequencheda→0value. Extrap- (φinput)andm =84(7)MeV(K input). These
s
olations of the octet and decuplet baryonmasses values are considerably lower than those from
[21] have larger errors and definitive conclusions phenomenology. They also find m = 3.3(4)
u,d
cannot be drawn. MeV.
Given their evidence that a single value of the The CP-PACS collaboration also reported a
strange quark mass determines both the K and value for the flavor singlet pseudo-scalar meson
7
φ input
RC (combined fits) 180
quenched PW (qχPT fits) RC, mq with sea quark Kc
1.00 K* quenched PW (polynomial fits) RC, mq with PQ Kc
RC, AWI m
q
160 qPW, mq=(1/K−1/Kc)/2
qPW, AWI m
q
0.80 eV]
M 140
m [GeV] 0.60 K µ=2GeV) [ 120
m(s
100
0.40
80
0.0 0.2 0.4 0.6 0.8 1.0 1.2
a [GeV−1] 0.0 0.2 0.4 0.6 0.8 1.0 1.2
a [GeV−1]
Figure 7. CP-PACS meson masses using the φ
mass to set the strange quark mass. Figure 8. CP-PACS strange quark mass at µ=
2GeV as determined from the φ mass.
for 2 flavor QCD, referred to as the η [22]. Us-
ing a volume source without gauge fixing (the produced a great deal of excitement for both an-
Kuramashimethod)tomeasurethedisconnected alytical and numerical studies. These formula-
quark diagrams, the mass difference between the tions offer a way of separating the chiral limit
π and η can be calculated, at least when the from the continuum limit and may lead to a lat-
disconnected diagrams are not far separated in ticeformulationofnon-abelianchiralgaugetheo-
time. They find m non-zero in the chiral limit ries [26]. For QCD-like theories, quenched simu-
η
and a subsequent continuum extrapolation gives lationswithdomainwallfermionswerefirstdone
m =863(86)MeV. by [27]and were reviewedlastyear [28]. Concur-
η
UKQCD has also reported a preliminary value rently, simulations of the Schwinger model [29],
for the η mass [20] for their set of dynamical lat- including dynamical fermions, showed that do-
tices. They employ a variance reduction tech- mainwallfermionsproducedtheexpectedphysics
nique in finding the propagator for the discon- and were compatible with standard HMC algo-
nected diagrams and get an η mass around 800 rithms.
MeVinthechirallimit,withanuncontrolledsys- Two areas where using fermions with better
tematic error. chiral properties are of particular interest are in
Both groups see little problem with autocor- simulations studying QCD thermodynamics [30]
relation times for these topologically sensitive and matrix elements [31][32] [33]. The character
measurements. UKQCDfinds an autocorrelation of the QCD phase transition is governed by the
time smaller than that found by the SESAM col- chiral symmetries of the theory. For matrix ele-
laboration. ment calculations, chiral symmetry can be vital
for controlling operator mixing and using chiral
perturbation theory as a guide. Both of these
6. DOMAIN WALL FERMIONS
areas are under active study with the QCDSP
The development of domain wall and overlap computers, using domain wall fermions.
[23] [24] [25] formulations of lattice fermions has The Columbia group has been studying
8
full QCD thermodynamics with domain wall ral condensate undergo a rapid crossover [30] is
fermions. The studies to locate the transition re- seenandscalesettingcalculationshavebeendone
gionandsettheparameterstousearedetailedin there. An important question is how the resid-
[30]. Aspartofthis work,zerotemperaturescale ual quark mass, m (due to mixing of the light
res
setting calculations are also required, which we modes between the two walls), depends on the
will focus on here [34]. The full QCD, zero tem- extent of the fifth dimension.
peraturedomainwallsimulationsdonetodateare For the PD action, the critical coupling on
detailed in Table 4. For these dynamical simula- an N = 4 lattice is β = 5.325, for L = 24,
t s
tions, a heavy bosonic field, frequently called the m = 0.02 and M = 1.9. Figure 9 shows hadron
f
Pauli-Villars field, is needed to remove the bulk masses for this β determined on 83 × 32 lat-
infinity that occurs when the extent of the fifth tices for dynamical fermion masses of 0.02 and
dimension is sent to infinity. 0.06. By extrapolating m2 to zero, one finds
π
m = 0.059(2). (A similar value, within errors,
res
can also be extracted from the Ward-Takahashi
Table 4 identity [35].) This is clearly a large mass, com-
Asummaryof2flavordomainwallfermionsimu- pared to the input mass of 0.02.
lations. All simulations use a domain wall height Extrapolatingthe rhoandnucleonmassto the
of 1.9. point where m2π vanishes gives mρ = 1.02(7),
m = 1.14(9) and m /m = 1.10(12). (The
N ρ N
errors are calculated by simple propagation of
CU QCDSP - PD action - β =5.325 - 83×32 errors.) While the length of the simulations
m L m a m L m /m traj. (∼ 1,000 trajectories) may be sufficient at these
s ρ ρ π ρ
0.06 24 1.3 10.4 0.64 1170 strongcouplings,onlyasinglevolumeandlattice
0.02 24 1.2 9.6 0.55 955 spacing has been studied. However, it is encour-
agingthatthisratioismuchclosertoitsphysical
CU QCDSP - PD action - β =5.325 - 163×16 value than for other fermion methods at this lat-
tice spacing.
m L m a m L m /m traj.
s ρ ρ π ρ
Itisimportanttostudytheeffectsofincreasing
0.02 24 1.2 9.6 0.57 560
the fifth dimension on the residual mass. Figure
CU QCDSP - RD action - β =1.9 - 83×32 10 shows the residual mass versus Ls (the length
of the fifth dimension) for simulations on 83×4
m L m a m L m /m traj.
s ρ ρ π ρ lattices with the PD action at β = 5.2 with a
0.02 24 1.2 9.6 0.52 875
quark mass of 0.02 and a domain wall height of
1.9. Theselatticesareintheconfinedphase. The
CU QCDSP - RD action - β =2.0 - 83×32
residual mass is determined from the axial ward
m L m a m L m /m traj.
s ρ ρ π ρ identity, asdiscussedin[35]. Thefunctionshown
0.06 24 1.1 8.8 0.66 960 ism(GMOR) =0.17(2)×exp(−0.026(6)×L ). The
res s
0.02 24 0.95 7.6 0.50 1010
data showsavanishing residualmassinthe large
0.02 48 1.0 8.0 0.43 760
L limit, but the falloff is very slow. Increasing
s
the fifth dimensionby24dropsthe residualmass
by about a factor of 2.
In an effort to decrease m for a fixed L ,
res s
Scale setting calculations to support thermo- the Columbia grouphasstudiedthe renormaliza-
dynamics studies are necessarily at fairly strong tion groupimprovedactionproposed by Iwasaki.
coupling. A first positive result is that even for For quenched simulations, m is made smaller
res
thesecourselattices,theHMCalgorithmexhibits for a given L , but the behavior with L may
s s
no problems thermalizing and evolving lattices. not be improved [34]. Results for hadron masses
A clearregionwhere the Wilson line andthe chi-
9
2.0 N
0.16
ρ
π
N 1.5
m
0.12

,
ρ
m 1.0
2, π mres0.08
m
0.5
0.04
0.0
−0.10 −0.05 0.00 0.05 0.10 0.15
m
0.00
f 0 8 16 24 32 40 48
L
s
Figure 9. Columbia QCDSP hadron masses for
simulationswithdynamicaldomainwallfermions
with the PD action. The scale of the phase tran-
Figure 10. m from the Ward-Takahashi iden-
sitiononanNt =4latticedeterminestheparam- tityasmeasurreeds on83×4latticesintheconfined
eters.
phase with the PD action and β =5.2.
fordynamicalsimulationswiththeRDactionare
shown in Figure 11. These were carried out on 7. CONCLUSIONS
83×32 lattices for β = 2.0 with L = 24. From
s
the behavior of m2 one finds m = 0.013(2). With the current-Teraflops scale computers,
π res
At the point where m2 vanishes m =0.855(49), full QCD simulations have reached the point
π ρ
where the finite volume, long simulation time,
m = 1.07(14) and m /m = 1.25(14). (Once
N ρ N
small quark mass and continuum limits can be
again, these are naive errors.)
probed, but not concurrently. The HMC and
Whilem issmallerfortheRDsimulationsat
res
HMD algorithms continue to be the techniques
β = 2.0 than the PD simulations at β = 5.325,
of choice, but the multiboson algorithmhas been
the physical lattice scales are different. For the
evolvedtoa competitivelevel. Staggered4flavor
RD action, the N =4 phase transition is at β =
t
QCD shows large finite volume effects, even for
1.9. Wedonothavetwodynamicalmassesforthe
m /m > 0.5 and the parity splittings for light
RD action at β = 1.9, but we can compare the π ρ
hadronsarequitevolumesensitive. Forcurrently
pion masses at m = 0.02. For the PD action at
accessible weaker coupling simulations, runs of
β =5.325, m =0.654(3) and for the RD action
π
length greater than 10,000 trajectories are prob-
atβ =1.9,m =0.604(3),a slightimprovement.
π
ably needed.
(However, the uncertainty in the determination
CP-PACS has improved their results on the
of the critical temperature could easily account
continuumlimitof2flavorQCDandfindcontin-
for this difference.)
uum meson masses closer to the physical values
A direct comparison of the pion mass for the
RD action (β = 2.0, 83 × 32) for L = 24 than for the quenched case. This leads them to
s
a determination of the strange quark mass which
gives m = 0.475(7), while for L = 48, m =
π s π
is lower than that expected phenomenologically.
0.420(10). m is decreasing as L is increased,
res s
For staggered fermions, the new 2 flavor points
although the rate is slow.
being produced by Columbia will augment the
existing continuum extrapolation of the MILC
10
6. M. Lu¨scher, Nucl. Phys. B418 (1994) 637.
2.0 N
ρ 7. Y. Iwasaki, Nucl. Phys. B258 (1985) 141.
π 8. S. Gottlieb, et. al., Phys. Rev. D35 (1987)
mN 1.5 2531.
9. C. Bernard, et. al., Nucl. Phys. B (Proc.
,
ρ
m 1.0 Suppl.) 73 (1999) 198.
, π 10. C. Bernard, . al., these proceedings.
2
m 11. C.Sui,Nucl.Phys.B(Proc.Suppl.)73(1999)
0.5
228andPhDThesis,ColumbiaUniversity,in
preparation.
0.0
12. D. Chen and R. Mawhinney, Nucl. Phys. B
−0.10 −0.05 0.00 0.05 0.10 0.15
(Proc. Suppl.) 53 (1997) 216.
m
f 13. M. Fukugita, et. al., Phys. Rev. D47 (1993)
4739.
14. Dong Chen, PhD Thesis, Columbia Univer-
Figure 11. Columbia QCDSP hadron masses for
sity, 1995.
dynamical domain wall simulations with the RD
15. C. Bernard, et. al., Nucl. Phys. B (Proc.
action for β =2.0.
Suppl.) 60A (1998) 297.
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for other fermion formulations. Further studies 21. A. Ali Khan, et. al., these proceedings and
will be needed to check the scaling properties for hep-lat/9909050.
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lattices can offset the extra calculational cost of hep-lat/9909045.
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