A pot-pourri of results in QCD from large lattice simulations on the CM5

6 Pages·0.17 MB·English

by  Tanmoy Bhattacharya

1
(cid:3)
A pot-pourri of results in QCD from large lattice simulations on the CM5
a
Tanmoy Bhattacharyaand Rajan Gupta
a
T-8 Group, MS B285, Los Alamos National Laboratory,Los Alamos, New Mexico 87545 U. S. A.
3
We present a status report on simulationsbeing done on 32 (cid:2)64 latticesat (cid:12) = 6:0 using quenched Wilson
fermions. Results for the spectrum, decay constants, the kaon B-parameter BK, and semi-leptonicform factors
aregivenbased on the current statisticalsampleof28con(cid:12)gurations. These \Grand Challenge"calculationsare
being done on the CM5at Los Alamosin collaborationwith G. Kilcup,S. Sharpe and P. Tamayo. Weend with
a briefstatementofthe performanceof our SIMD code on the CM5.
1. LATTICE PARAMETERS [4] and Bernard and Golterman [3] have pointed
0
out that in the quenched approximation(cid:17) loops
Alltheresultsdescribedinthesetwotalkshave
give rise to unphysical termsin thechiral expan-
been obtained using the following lattice param-
sion for observables. These e(cid:11)ects have not been
4
eters. The 32 gauge lattices were generated at
includedintheanalysiswhenextrapolatingquan-
(cid:12) = 6:0 using the combination 10 over-relaxed
tities to the chiral limit.
(OR)sweepsfollowedby1Metropolissweep. We
We(cid:12)ndthatthedataforalmostallobservables
havestoredlatticesevery2500ORsweeps. Quark
show a signi(cid:12)cant curvature as a function of the
propagators,usingthesimpleWilsonaction,have
quark mass. So we make both a linear and a
been calculated after doubling the lattices in
quadratic(cid:12)tineachcaseandquote(cid:12)nalnumbers
3 3
the time direction, i:e: (32 (cid:2)32 ! 32 (cid:2) 64). 2
fromthe(cid:12)twithlower(cid:31) . Moredetailsaboutthe
This doubling was donein responseto the initial
(cid:12)ts are given in the individual sections.
hardware/operational constraints of the CM5 at
3
LANL.Currentlywearegenerating32 (cid:2)64. Pe-
riodicboundaryconditionsareusedinall4direc- 2. SPECTRUM
tions, bothduringlattice updateand propagator
We have analyzed three types of hadron cor-
calculation. Quark propagatorshave been calcu-
relators distinguished by the type of source/sink
latedusingtwokindsofextendedsources{Wup-
used to generate quark propagators. These are
pertal and Wall { at (cid:20) = 0:135 (C), 0:153 (S),
(i)wall sourceandpointsink(WL),(ii)Wupper-
0:155 (U1), 0:1558 (U2), and 0:1563 (U3). These
tal source and point sink (SL), and Wuppertal
quark massescorrespondto pseudoscalarmesons
source and sink (SS). With these three cases we
of mass 2800, 980, 700, 550 and 440 MeV re-
are able to make a consistency check since the
spectively where we have used 1=a=2:3GeV for
e(cid:11)ective mass meff(t) converges to the asymp-
the lattice scale. The three light quarks allow us
totic value from below for WL correlators and
to extrapolate the data to the physical u and d
fromaboveforSLandSScorrelatorsinallhadron
quarks,whiletheC andS (cid:20)valuesareselectedto
channels. Fig. 1 illustrates, using the pion U1U1
becloseto thephysical charmand strangequark
channel, noteworthy features of the convergence
masses. The C (S) quark mass is slightly lower
of meff(t). In the case of WL correlators, the
(higher) than the physical value. So far our sta-
approach of meff(t) to its asymptotic value is
tistical sampleconsistsof 28 con(cid:12)gurations,each
monotonic but quite slow and one needs to (cid:12)t
separatedby 5000OR sweeps. The goalis toan-
for t > 15. The statistical errors in meff(t) are
alyze 100 con(cid:12)gurations so the present analysis
smallest of the three cases. The SS correlators
should be considered preliminary. Also, Sharpe
reachtheasymptoticvalueatt(cid:25)8andthetime-
(cid:3)TalkspresentedbyT.BhattacharyaandR.Gupta. slice to time-slice (cid:13)uctuationsare of the order of
2
Figure 1. Comparison of the convergence of Figure 2. Quadratic extrapolation of (cid:1), nucleon
meff(t) for three types of pion (U1U1) correla- and (cid:26) mass data to the chiral limit.
tors. SL data has beenshifted by +0:05. 1
0:4
SL
0:8
0:35
SS
0:6
0:3
0:4
WL
0:25 0 0:2 0:4 0:6 0:8 1
0 10 20 30
the statistical errors. The SL correlators show
which gives a lattice scale of
correlated(cid:13)uctuationsin meff(t), wiggleswitha
period of about 12 time slices, which may be a (cid:0)1
a (m(cid:26))=2:26(27) GeV: (3)
feature of the Wuppertal source. Another pos-
sibility, which we will check on the new set of
We shall, for simplicity, use 1=a=2:3 GeV.
3
32 (cid:2) 64 lattices, is that this feature is due to
Wuppertalsourcequark propagatorsallow one
doublingthelatticein thetimedirection. Inspite
toconstructhadroncorrelatorswithnon-zeromo-
of these di(cid:11)erences in meff(t), the bottom line
menta. In Fig. 3 we show the pion propagatorat
is that all three kinds of correlators give consis-
4 di(cid:11)erent momenta. The signal in ~p = (1;0;0)
tent(within1(cid:27))resultsformassesforallmesons
and (1;1;0) correlators is quite good. Thus, as
and the error estimates are similar too. We give 3
we show later, using 32 lattices at (cid:12) =6:0 leads
the average values for masses in lattice units in
to a signi(cid:12)cantly improved signal in the calcula-
Table 1.
tion of phenomenologically interesting quantities
To extrapolate the data to the chiral limit in
2 2 like semi-leptonic formfactors and BK.
thelightquarkwe(cid:12)tasafunctionofm(cid:25)=m(cid:26) and
mq, where the latter is de(cid:12)ned either as log(1+ 2.1. Baryons
0:5(1=(cid:20)(cid:0)1=(cid:20)c))or0:5(1=(cid:20)(cid:0)1=(cid:20)c),ordetermined Inourpreviouswork[1] wehadfoundthatwall
non-perturbativelyasdiscussedinRef.[1]. Using
and Wuppertal sources give results that are sig-
2
a quadratic extrapolation of m(cid:25) data versus mq ni(cid:12)cantly di(cid:11)erent. In Fig. 4 we show a typical
(as it (cid:12)ts all the points)we get
example of the current status. We (cid:12)nd that on
(cid:20)c =0:15721(7): (1) the larger lattices WL and SL correlators give
consistent results. The plateau in meff(t) is not
This value is higher that given in Ref. [1], but
very cleanandany systematicdi(cid:11)erencebetween
consistentwithQCDPAXresult0.15717(3). Sim-
thetwocasesisanartifactofthe(cid:12)trangeand/or
ilarly,usingaquadratic(cid:12)t,theextrapolatedvalue
the statistical errors. The numbers given in Ta-
of m(cid:26) is (see Fig. 2)
ble 1 for the case of degenerate quarks are the
m(cid:26)(mq =0)=0:34(4) (2) mean of SL and WL results. Extrapolating the
3
Figure 3. E(cid:11)ective mass plot for pion (U1U1) at Table 1
di(cid:11)erent momenta. Masses in lattice units
m(cid:25) m(cid:26) Nucleon (cid:1)
CC 1:219(2) 1:232(02)
0:6
CS 0:857(2) 0:884(03)
CU1 0:818(3) 0:848(04)
CU2 0:803(4) 0:833(05)
0:5 CU3 0:794(6) 0:826(06)
p=(2,0,0)
SS 0:427(2) 0:509(03) 0:792(15) 0:854(19)
SU1 0:368(2) 0:467(04)
p=(1,1,0) SU2 0:344(2) 0:451(05)
SU3 0:328(2) 0:440(06)
0:4
U1U1 0:302(2) 0:423(05) 0:643(14) 0:721(22)
U2U2 0:240(2) 0:389(09) 0:572(12) 0:668(27)
U3U3 0:193(2) 0:364(12) 0:526(16) 0:636(30)
p=(1,0,0)
0:3
p=(0,0,0)
Table 2
0 10 20 30 f(cid:25) and 1=fV in lattice units
naive imp: naive imp:
f(cid:25) f(cid:25) 1=fV 1=fV
CC 0:126(3) 0:205(6) 0:097(2) 0:20(1)
data to the chiral limit (Fig. 2) gives
CS 0:103(3) 0:137(4) 0:121(3) 0:20(1)
mNa=0:44(20) CU1 0:094(3) 0:122(4) 0:116(3) 0:19(1)
CU2 0:090(3) 0:116(4) 0:115(4) 0:19(1)
m(cid:1)a=0:59(40); (4)
CU3 0:088(4) 0:113(5) 0:114(4) 0:18(1)
however, before comparing these numbers to ex- SS 0:090(3) 0:098(3) 0:228(6) 0:31(1)
perimental data the reader should note that we SU1 0:083(3) 0:089(3) 0:241(7) 0:32(1)
have not included the results of Labrenz and SU2 0:080(3) 0:084(3) 0:246(7) 0:33(1)
Sharpe[6] who predict additional non-analytic SU3 0:078(3) 0:081(3) 0:248(8) 0:33(1)
termsin the chiral behavior of quenched QCD. U1U1 0:077(3) 0:080(3) 0:261(7) 0:34(1)
U2U2 0:071(3) 0:072(3) 0:276(8) 0:35(1)
3. DECAY CONSTANTS U3U3 0:066(4) 0:067(4) 0:285(9) 0:36(1)
Mesondecayconstantscanbecalculatedusing
SL and SS correlators. We use the generic no- p
tation fPS for pseudoscalar mesons and fV for naive 2(cid:20) renormalization with Zv = 0:57 and
vector meson. There are many di(cid:11)erent ways of the 1-loop perturbative value for ZA. In both
combining these correlators to extract fPS and cases we use a boosted value for the coupling,
2
fV, some of which are described in Ref. [1]. We g =1:75. The results are displayed in Fig. 5.
(cid:0)1
(cid:12)ndthatall methodsgiveessentiallyidenticalre- As shown in Fig. 5, data for f(cid:26) is qualita-
sults, sowequotethemeanvaluein Table2. For tively di(cid:11)erent for cases with degenerate versus
the error estimate we give the largest individual non-degenerate quark masses. The large depen-
error. We also (cid:12)nd that the value of fPS is in- denceontherenormalizationconstant(naivever-
dependentofthepion'smomentum. Forcompar- susimproved)evenforsmallmq suggestspossibly
ison we give results using the local current with large O(a) corrections or e(cid:11)ects of quenching in
(cid:0)1
two di(cid:11)erent renormalizationfactors. Columns 2 f(cid:26) . To get f(cid:25) we can extrapolate the degener-
and4areusingtheLepage-Mackenziemean-(cid:12)eld ate combinations or all data with S and lighter
improvedscheme[5]whilecolumns1and3usethe quarks as they give qualitatively similar results.
4
Figure 4. E(cid:11)ective mass plot for nucleon with Figure 5. Meson decay constants fPS and fV(cid:0)1
WL and SL correlators. as a function of m2(cid:25)=m2(cid:26). Data for fV(cid:0)1 has been
shifted by +0:25 for clarity.
0:6
0:7 SL
0:4
0:65
0:2
0:6
WL
t
10 20
0
0 0:2 0:4 0:6 0:8 1
Usingthemean-(cid:12)eldimprovednormalizationswe
get (with all data)
one can remove most of these. The main limita-
f(cid:25)a=0:052(5) linear
tionofthatcalculationwasthatthespatiallattice
f(cid:25)a=0:057(24) quadratic: (5) size was small (163), consequently the momen-
2 tumgap (2(cid:25)=16)was large, and the signal in the
Both (cid:12)ts have comparable (cid:31) . From the linear
non-zero momentum correlators was marginal.
(cid:12)t we get 1=a=2:5(3)GeV. For fD we calculate
Both these limitations have been addressed on
theratiofD=fK usingS(U1)asthestrangequark
thepresentset of latticesand as a result thesta-
and get the result 1:38 (1:52), where the change
tisticalquality of the datais markedlyimproved.
duetothelightquarkmassis(cid:25)1inthelastdec-
In Fig. 6 we show the behavior of the lattice
imal place. Using fK = 160MeV and the mean
BK, evaluatedatmomentumtransferp=2(cid:25)=32,
of the above two numbers ((cid:25) physical s quark)
as a function of the time slice at which the op-
we get fD =232(20)MeV.
erator is inserted. The data shown are for two
4. B di(cid:11)erent kaon sources, i:e: (cid:22)s(cid:13)5 d and (cid:22)s(cid:13)4(cid:13)5 d
K
as they converge from opposite direction. The
The major problem in the calculation of the statistical errors are small and the two (cid:12)ts agree
kaon B-parameter with Wilson fermions is the withinerrors. Theresultsforp=2(cid:25)=32(cid:2)(1;1;0)
bad chiral behavior inducedby the mixing of the and p = 2(cid:25)=32(cid:2)(1;1;1) are also reliable. Full
(cid:1)S = 2 4-fermion operator with wrong chirality analysis will be presentedelsewhereRef. [7].
operators. This mixing arises as a result of the In Table 3 we present preliminary results for
r-term in the action and has been calculated to the B-parametersusing perturbatively improved
2
1-loop in perturbation theory. In Ref. [2] it was operators evaluated with a boosted g = 1:75
shown that the lattice artifacts completely over- and (cid:22)a = 1:0. In column 1 (2) we give the raw
whelmthesignalwhenusingthe1-loopimproved number for zero (one) unit of momentum trans-
operator, however, by calculating the matrix el- fer. The (cid:12)nal values of BK after removing the
ement at di(cid:11)erent values of momentum transfer leading bad chiral contributions by the method
5
Figure 6. BK at momentumtransferp~=(1;0;0) Table 3
asafunctionoftatwhichtheoperatorisinserted. BK, B7 and B8
Thetwo datasetscorrespondtothetwo kinds of BK B7 B8
operatorsused to producethe kaons. (p=0) (p=1) subtr.
0:6 CC 0:95(3) 0:95(4) 1:32(37) 0:97(3) 0:97(3)
CS 0:83(2) 0:84(3) 1:09(21) 0:94(3) 0:95(3)
CU1 0:79(2) 0:80(2) 0:94(19) 0:92(3) 0:94(3)
CU2 0:77(2) 0:77(2) 0:76(20) 0:91(3) 0:93(3)
0:5 CU3 0:76(2) 0:75(3) 0:50(24) 0:89(3) 0:92(3)
SS 0:50(1) 0:53(1) 0:82( 7) 0:87(2) 0:91(2)
SU1 0:36(1) 0:41(2) 0:72( 6) 0:84(1) 0:89(1)
SU2 0:29(1) 0:34(2) 0:66( 7) 0:83(1) 0:88(1)
0:4 SU3 0:23(1) 0:28(2) 0:60( 8) 0:82(1) 0:87(1)
U1U1 0:13(1) 0:21(2) 0:62( 6) 0:81(1) 0:86(1)
U2U2 (cid:0):26(2) (cid:0):08(3) 0:48( 8) 0:76(1) 0:82(1)
U3U3 (cid:0):78(3) (cid:0):42(4) 0:37(11) 0:72(1) 0:78(1)
0:3
Ext 1 0:48 0:80 0:85
Ext 2 0:43 0:74 0:80
t
0:2
0 10 20 30
a smaller p~. Very accurate and reliable results
for BK, obtained using staggered fermions, have
of momentum subtraction (Ref. [2]) are given in
beenpresentedatthisconferencebyS.Sharpe[8].
column 3. In columns 4 and 5 we give the re-
Wilson resultsareconsistentwith thesebut have
sults for the two left-right operators O7 and O8 largersystematicerrorsasaresultofmixingwith
(de(cid:12)nedin Ref. [2]).
operators having the wrong chiral behavior. By
The data in columns 1 and 2 show that the improving the quality of Wilson fermion results
statisticalerrorshavebeenreducedtothelevelof
one can, by comparison with staggered results,
a few percent. The reason that the errors in the
check our understandingof lattice artifacts.
(cid:12)nal B-parameters are large for heavy quarks is
mainly because E(p)(cid:25)m, while for light quarks
5. SEMILEPTONICMESONFORMFACTORS
the statistical (cid:13)uctuationsare large.
Thenumbersextrapolatedtothekaonmassare We have calculated the semileptonic pseudo-
givenin thelasttworowsinTable3. We usetwo scalar form factors of the D meson, both for
methodsforthis: inthe(cid:12)rstwekeepS mass(cid:12)xed D ! Ke(cid:23) and D ! (cid:25)e(cid:23). In calculating these
and extrapolate in the light quark mass using form factors we have held the D-meson at zero
S; U1; U2; U3. Inthesecond,thetwo quarksare spatialmomentum,and variedthe momentumof
taken to be degenerate. Both methodsgive sim- the other meson from 0 to (cid:25)=8. This provides a
ilar results for all three operators. This suggests large enough range (both positive and negative)
thattheSU(3)breakingdoesnotgivealargecon- in the invariant mass of the leptonic subsystem
tribution to these B-parameters. Also, there is ((cid:0)Q2)totestthepoledominancehypotheses. To
little evidence of unphysical contributions from controlsystematice(cid:11)ects,wehavedonethecalcu-
0
(cid:17) loopswhichareexpectedtobesigni(cid:12)cantnear lation using three di(cid:11)erent transcriptions (called
thechirallimitinthequenchedapproximation[8]. `local',`extended'and`conserved')ofthecurrent.
The values of BK, B7 and B8 are signi(cid:12)cantly ThedetailsofthemethodaregiveninRef.[9]and
lower than those found in Ref. [2]. We believe a completeanalysis for the presentset of lattices
that this is due to using a larger lattice and will be presentedin Ref. [10].
6
Table 4 Figure7. Comparisonof 1=f0 obtainedusing the
2
The mesonform factorsextrapolatedto Q =0 three vector currents. The data are for the case
K (cid:25) K (cid:25)
f+(0) f+(0) f0 (0) f0(0) where the spectator quark is U3. The (cid:12)ts show
U1 0.833(21) 0.787(24) 0.808(14) 0.735(14) comparison with pole dominance using the mea-
U2 0.837(24) 0.761(32) 0.808(15) 0.699(16) sured lattice value for the scalar mass, mDs.
U3 0.834(27) 0.737(41) 0.806(17) 0.670(18) 1:8
The renormalization of all three currents is 1:6
carried out using the Lepage-Mackenzie[5] mean
(cid:12)eld improved perturbationtheory. We (cid:12)nd that
1:4
the values obtained using the three currents are
very close. (See Fig. 7 for an example of con-
sistency between di(cid:11)erent currents.) We notice
that the statistical errors are very small when
thehadronscarryzerooroneunitofmomentum,
andsomesystematicdi(cid:11)erencebetweenthelocal 1
and the non-local currents become visible. For
~p=(1;1;1) and (2;0;0) thereis no clear plateau
0:8
in the data and the analysis is very sensitive to
the (cid:12)t range.
Except when ~p = (1;1;1), the data are con- 0:6
(cid:0)0:1 0 0:1 0:2
sistent (see Fig. 7 for an example) with the
hypothesis that the form factor is dominated by
thenearestpolewiththerightquantumnumbers
(cid:3) (cid:3) K (cid:25)
(Ds andD respectivelyforf+ andf+). Assum-
ing pole dominance we can extract f(Q2 = 0), 35mega(cid:13)ops/node. ThesetimingsincludeallI/O
and the results are summarized in Table 4. To andsetupoverhead. Forthepresentsetofphysics
withinstatisticalerrorsthenumbersareindepen- objectives we need 18000 node-hoursto generate
dent of the light quark mass for fK and there is and processone con(cid:12)guration.
aslight decreasein f(cid:25). Weexpect(cid:25)10%change We gratefully acknowledge the tremendous
in f(cid:25) on extrapolating the data to the chiral support provided by the ACL at Los Alamos.
limit and to the physical values of charm quark These calculations have been done as part of the
mass. Our (cid:12)nal numbers for f+K are close to the DOE HPCC Grand Challenge program.
K K K (cid:25)
phenomenologicalvalue, f+ (cid:25)f0 andf+ >f+.
REFERENCES
6. CODE PERFORMANCE ON THE CM5
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levelcodesisnotasigni(cid:12)cantoverheadduetothe 5. P. Lepageand P. Mackenzie, Phys. Rev. D48
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algorithmsandphysicsanalysiseasy. Ourpresent 8. S. Sharpe,these proceedings.
gauge update code runs at 25 mega(cid:13)ops/node 9. R. Gupta et al:, hep-lat/9310007.
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