QCD Spectroscopy

18 Pages·0.22 MB·

by  D. Weingarten

1
QCD spectroscopy
a
Don Weingarten
a
IBM Research, P.O.B 218,
Yorktown Heights, NY 10598,USA
4 I review recent work on the calculation of hadron masses, decay constants and wave functions.
9

n
a 1. INTRODUCTION real improvement in algorithms, are needed to
J
get reliable estimates for the zero lattice spac-

8 Exactly what set of topics is supposed to go
ing, in(cid:12)nite volume limits of most properties of
1 intothe spectroscopy review talkseems to varya
light hadrons. For full QCD calculations and

bit from one lattice conference to the next. My valence approximation calculations tuned to the
1 de(cid:12)nition of the territory this year is masses, de-
same values of m(cid:25)=m(cid:26), lattice spacing in physi-
2 cay constants and wave functions for both the
calunits andlattice period inphysicalunits, and
0
valence(quenched) approximationandfullQCD.
1 with reasonably large lattice period, correspond-
I will include glueballs but not hadrons contain-
0 ing hadron masses, at the parameter values for
4 ing heavy quarks and will restrict myself mainly which calculations have been done so far, agree
9 to work done during the last year.
within statistics. Full QCD and valence approx-
/ The spectroscopy oflowlyingquarkstates and
t imation calculations, with parameters tuned to
a
glue states I think remains a critical part of the
l agree at su(cid:14)ciently large volume, yield masses
- QCD.Anunavoidablecomponentofshowingthat
p which disagree if the the lattice volume is then
QCD actually does explain the strong interac-
e mademuchsmaller. Thedi(cid:11)erence involumede-
h tions is to show that it accounts for the masses
pendence between full QCD and the valence ap-
oflowlyinghadrons. QCDhas been assignedthe
proximationhas been shownto be aconsequence
homeworkproblemofhadronspectroscopyjustas
of the nonzero vacuum expectation value which
earlyquantummechanicswasassignedtoworkon
occurs infullQCDfor gaugeloopsclosed around
the spectrum of hydrogen and other lightatoms.
a space-direction lattice period. Except for very
Inaddition,whateveralgorithmscanprovethem-
small lattices, these loops have zero expectation
selvesinreproducingtheknownfeaturesoflowly-
in the valence approximation.
ingspectroscopy canthenbeappliedwithgreater
Forspectroscopy inthevalenceapproximation,
con(cid:12)dence to extract predictions from QCD for
on the other hand, there are now calculations for
things we do not know yet. The masses of the
Wilsonquarks ofmasses and decay constants ex-
lightestglueballs,ofcourse, arenotknown. Are-
trapolated to physical quark mass, zero lattice
liable lattice calculation of these numbers seems
spacing and in(cid:12)nite volume. Ratios of several
likely to be a required (cid:12)rst step toward the ex-
light hadron masses found in these limits dif-
perimentalidenti(cid:12)cation of glueballs.
fer from experiment by less than 6%, with sta-
The overall picture of the present state of
tistical errors of 8% and less, and are statisti-
hadron spectroscopy is roughly as follows. For
cally consistent with experiment. Although chi-
spectroscopy including quark-antiquark vacuum
ralperturbation theory suggests that the valence
polarization, life is still pretty di(cid:14)cult. Calcu-
approximation will behave anomalously at suf-
lations are restricted to fairly small statistical
(cid:12)ciently small quark mass, it now appears that
ensembles and comparatively large lattice spac-
thisproblemwouldonlydisrupt thehadronmass
ing and quark mass. Signi(cid:12)cantly more machine
ratios which have been calculated if they were
power, perhaps a factor of 100 or more, or a
2
evaluated at quark masses below the light quark (cid:15) Masses
masses. Valenceapproximationmesondecaycon-
(cid:15) Decay constants
stants extrapolated to physical quark mass, zero
(cid:15) Glueball masses
lattice spacing and in(cid:12)nite volume fall typically
about 15% below experiment. This underesti-
mate ranges in signi(cid:12)cance from about 1 to 3 2. TECHNICAL ISSUES
standard deviations. The valence approximation
2.1. Extracting Masses from Propagators
isexpected,however,tobemostreliableforquan-
At Lattice 92 the QCDPAX collaboration [1]
titiesdeterminedprimarilybythelowmomentum
presentedcalculationscheckingforhighmasscon-
behavior of the chromoelectric (cid:12)eld. Decay con-
tamination in the values of hadron masses re-
stants are proportional to the absolute square of
portedbyseveralothergroups. Thelightestmass
meson wave functions at the origin and are thus
contributing toa propagatorshould be extracted
sensitive to the high momentumbehavior of the
from the propagator's fall o(cid:11) at asymptotically
chromoelectric (cid:12)eld. A simple renormalization
large time separations, and picks up high mass
group estimate suggests that the error in the va-
contamination from excited intermediate states
lence approximation'streatment of the high mo-
if it is obtained at time separations which are
mentumbehavior of the chromoelectric (cid:12)eld will
too small. The value which QCDPAX found
lead todecay constants fallingbelow those ofthe
for the rho mass with Wilson quarks in the va-
full theory.
lence approximation using 200 gauge con(cid:12)gura-
Two independent, in(cid:12)nite volume, continuum 3
tions on a lattice 24 (cid:2)54 at (cid:12) of 6.0 and k of
limit valence approximation calculations have
0.1550was3standard deviationsbelowthe value
now also been reported for the scalar glueball
which the APE collaboration found at the same
mass. The two data sets agree within statistical
(cid:12) and k using 78 gauge con(cid:12)gurations on a lat-
errors before extrapolation, but the two groups 3
tice 24 (cid:2)32 [2]. Both groups have now run new
extrapolate di(cid:11)erentlytozero latticespacingand
calculationswithlargerensembles[3,4]. The new
get answers which are not quite consistent. The
calculations of the rho mass are both in agree-
calculation with higher statistical weight, using
ment with the QCDPAX value from Lattice 92.
about eight times as many con(cid:12)gurations, pre-
The originaldisagreement appears to meto have
dicts a massof1740(cid:6)70MeV, favoringf0(1710)
beencausedbyastatistical(cid:13)uctuationinthe(cid:12)rst
as the scalar glueball.
APE ensemble. The (cid:13)uctuation caused the ap-
The subjects I am going to cover will be orga-
pearance of an e(cid:11)ective mass plateau at a some-
nized as follows:
what smaller time separation and therefore with
a somewhatlarger mass value than occurs in the
1. Technical issues
new data.
(cid:15) Extracting hadron masses from prop- At Lattice 92 Ukawa [5] questioned whether
agators high mass contamination might not be present
(cid:15) Volumedependence in some of the masses which I discussed from
calculations on GF11 [6]. A comparison of
(cid:15) Valence approximation versus full
masses found from propagators for several dif-
QCD
ferent choices of hadron sink operators, however,
(cid:15) Wave functions, state vectors
provides evidence that our numbers probably do
2. Calculations with (cid:12)xed lattice spacing not have signi(cid:12)cant high mass contamination.
This subject willbe discussed below in Sect. 4.1.
(cid:15) Valence approximation
2.2. Volume Dependence
(cid:15) Full QCD
New data, extending earlier results [7], on the
3. Valence approximationzero lattice spacing volumedependence ofhadronmassesinfullQCD
limits was presented by the MILC collaboration [8].
3
(cid:12) mqa S m(cid:25)=m(cid:26) m(cid:26)a
5.415 0.050 4 (cid:0) 16 0.597(1) 1.023(2)
5.445 0.025 8 (cid:0) 16 0.489(2) 0.918(4)
5.470 0.0125 12 (cid:0) 16 0.367(3) 0.883(6)
Table 1
Parameters of MILC collaboration runs, on lat-
3
tices S (cid:2)24 with m(cid:25)=m(cid:26) in each case given for
the largest lattice.
(cid:12) mqa S m(cid:25)=m(cid:26) m(cid:26)a
valence approximation
6.0 0.02 6 (cid:0) 24 0.643(4) 0.520(3)
6.0 0.01 6 (cid:0) 24 0.513(7) 0.465(6)
full QCD
5.7 0.02 8 (cid:0) 20 0.692(5) 0.492(3)
5.7 0.01 8 (cid:0) 20 0.586(11) 0.418(7)
Figure 1. Volume dependence of hadron masses
found by the MILC collaboration at (cid:12) of 5.415. Table 2
3.5 ParametersofrunsbytheKyoto-Tsukubacollab-
3
oration,onlatticesS (cid:2)24withm(cid:25)=m(cid:26) andm(cid:26)a
3.0 ma=0.01 full QCD
q quenched in each case given for the largest lattice.
2.5
V)2.0 (cid:3)
Ge equal to the observed value of mK=mK.
m(1.5 Data on the volumedependence of valence ap-
N proximationmasses[9]andacomparisonwiththe
1.0
ρ
volume dependence of full QCD masses [10] was
0.5 π reported by the Kyoto-Tsukuba collaboration .
0.0 The parameters of these runs are shown in Ta-
0.5 1.0 1.5 2.0 2.5 3.0
La(fm) ble 2. For the largest values of space direction
period, the values ofm(cid:25)=m(cid:26) and m(cid:26)a for the va-
lence approximation run at mqa of 0.01 appear
Figure 2. Volume dependence of hadron masses
roughlycomparableto those ofthe fullQCD run
found by the Kyoto-Tsukuba collaboration at (cid:12)
at the same quark mass, and a similar approxi-
of 5.7.
mate equality holds for the runs at mqa of 0.02.
DataforfullQCDandforthevalenceapproxima-
These calculations were done with full QCD for tionwithmqaof0.01isshowninFigure2. Inthe
two (cid:13)avors of Kogut-Susskind fermions. A sum- valenceapproximation,however,m(cid:25),m(cid:26) andmN
mary of the run parameters is shown in Table 1. are found to approach their in(cid:12)nite volume val-
For the range of parameters examined, values ues more rapidly than they do in full QCD. The
of mN and m(cid:26) within 2% of their in(cid:12)nite vol- origin of this di(cid:11)erence is shown to be related to
umelimitsrequiredspacedirectionlatticeperiods the di(cid:11)erence between the behavior in full QCD
greater than about 2.5 fm. The volume depen- andinthe valenceapproximationofthevariables
dence of hadron masses for mqa of 0.05 is shown Pi, the trace of the product of gauge links closed
in Figure 1. The horizontal lines indicate inter- aroundthelatticeperiodinspacedirectioni. The
valswithin2%ofin(cid:12)nitevolumemasses. Thelat- valuesofPi a(cid:11)ectthecontributiontoquarkprop-
tice spacing was chosen by setting the pion mass agators of quark paths wrapping around the lat-
to 500 MeV, since in this case m(cid:25)=m(cid:26) is nearly tice period. In the valence approximation, for
4
lattice periods larger than the inverse of the de- 2.3. Valence Approximation versus Full
con(cid:12)ning temperature, the familiarZ3 symmetry QCD
of the QCD action without fermions prevents Pi The valence approximation may be viewed
from acquiring a vacuum expectation value. In as replacing the momentum-dependent color di-
fullQCDthissymmetryisbrokenbythe fermion electric constant arising from quark-antiquark
termsintheaction,andPihasanonzerovacuum vacuum polarization with its zero-momentum
expectationvaluewhichfallscontinuouslytozero limit[13]. Thisapproximationmightbeexpected
asthelatticeperiodbecomeslarger. Bychanging to be fairlyreliable for low-lyingbaryon and me-
the boundary conditions of the fermionsentering sonmassesdeterminedbythelowmomentumbe-
the action fromperiodic to antiperiodic, the sign havior of the chromoelectric (cid:12)eld.
of the vacuum expectation of Pi can be changed. Missing from the valence approximation,how-
TheKyoto-Tsukubacollaborationshowsthatthis ever, are couplings of vector mesons to their de-
change causes hadron masses to gofromdecreas- cay channels. For the spin-3/2 baryon multiplet,
ing with volumeto increasing with volume. This these couplingsarepresent butaltered fromtheir
result, along with similar information obtained valuesinfullQCD[14]. AcalculationbyLeinwe-
varying the boundary conditions on the quarks berandCohen[15]attemptstoestimatethee(cid:11)ect
entering hadronpropagators,establishes thatthe of this omission on m(cid:26). These authors consider
di(cid:11)erence between the vacuumexpectation of Pi also the e(cid:11)ect of obtaining m(cid:26) by the linear ex-
in the valence approximationand in the full the- trapolation down to light quark masses of values
ory accounts for the di(cid:11)erences in volumedepen- of vector masses calculated with heavier quarks.
dence. Leinweber and Cohen use a simplelocalcoupling
The volume dependence in hadron masses in between the rho and two pions, combined with a
fullQCDreportedbytheMILCcollaborationand momentumcuto(cid:11)intheotherwise divergentpion
by the Kyoto-Tsukuba collaboration are consis- loop integral entering the rho propagator. The
tentwitheachother. Forallbutthelargestvalues cuto(cid:11) is intended to model the momentum de-
of space direction lattice period, both data sets pendence ofthecouplingbetween the rhoandpi-
shown signi(cid:12)cantly stronger volume dependence onswhich,intherealworld,wouldcausetheloop
than predicted by Lu(cid:127)scher's rigorous asymptotic integraltoconverge. The error inthe valenceap-
formula [11]. Both sets of results are consistent proximationvalueform(cid:26) arisingfromthemissing
with the behavior loopcombinedwithextrapolationfromthe range
ofquarkmassesusedinRef.[6],isestimatedtobe
c between 0 and (cid:0)25 MeV, thus less than 3%. For
m(L) =m(1)+ (1)
L3 a linear extrapolation of the vector meson mass
as a function of valence quark mass in full QCD,
predicted for intermediate values of L by the workingfromaquarkmassintervalextendingup-
model of Ref. [12]. Lu(cid:127)scher's formula embodies ward fromthe point corresponding toa 500MeV
the e(cid:11)ect of meson exchange between a hadron pion,theerror arisingfromextrapolationaloneis
and its nearest periodic image. The model of estimated to be (cid:0)10 to (cid:0)20 MeV.
Ref. [12] is intended to take into account in ad- A discussion in Ref. [6] of the extrapolation of
dition the (cid:12)nite extension of a hadron's wave hadron masses produces as a byproduct an addi-
function, or equivalently the momentum depen- tional piece of evidence suggesting that the cou-
dent form factor appearing in a hadron's cou- plingoflighthadronstodecaychannelshasarel-
pling to exchanged mesons. For lattice peri- atively small e(cid:11)ect on their masses in full QCD
ods signi(cid:12)cantly larger than the extension of a and therefore givesrise to arelativelysmallerror
hadron's wave function, the model of Ref. [12] is when omitted or altered in the valence approx-
designed to reproduce Lu(cid:127)scher's formula. Fits of imation. Take the up and down quark masses
the Kyoto-Tsukuba data for full QCD to Eq. (1) to be equal, for simplicity, to a single normal
are shown in Figure 2. quark mass, mn. Then expand the masses of
5
3 lations for the vector mesons, spin-1/2 baryons
and spin-3/2 baryons. Leinweber and Cohen's
∆ calculationforfullQCD maybe viewed as an es-
)n N timate of the error of the extrapolation for the
m
m(ρ 2 vector multiplet. In place of their error value of
m] / ∆ ρ p(cid:0)l1y0intgot(cid:0)hi2s0eMxterVap,othlaetieornrotroacthtueamllyasosbtdaaitnaedfraopm-
,N experiment is (cid:0)4 MeV, or about 0.5%. For the
m
1
m,ρ spin-1/2 baryon multipletthe error is +13 MeV,
[ about 1.4%, and for the spin-3/2 baryon multi-
plet the error is +10 MeV, about 0.8%. In two
of these multiplets, however, decay widths vary
0
0.0 0.5 1.0 1.5 drastically among the strange hadrons to which
m / m the linear (cid:12)ts are made and are generally much
q s
smaller than the decay width of the nonstrange
hadronwhosemassisfoundbytheextrapolation.
Figure 3. Linear extrapolations to determine
In the vector meson multiplet, the rho width is
nonstrange vector, spin-1/2 and spin-3/2 baryon (cid:3)
149 MeV, while the K and phi used to (cid:12)nd the
massesfromthe massesoftheir strange partners.
rho mass have widths of 49.8MeV and 4.4MeV,
respectively. Themembersofthespin-1/2baryon
the hadrons in the vector, spin-1/2 and spin-3/2
multipletarestable withrespect tothestrongin-
baryon multiplets as power series in the mass
teraction. In the spin-3/2 baryon multiplet, the
of the strange quark, ms, around the point at (cid:1) has a width of about 115 MeV, while (cid:6)(cid:3), (cid:4)(cid:3)
which the strange quark mass and normal quark
and (cid:10) used to (cid:12)nd the (cid:1) mass have widths of
mass are equal, keeping only the leading linear
36 MeV, 10 MeV and 0 MeV, respectively. The
term. Applying (cid:13)avor SU(3) symmetry at the
(cid:10) does not decay by the strong interaction. It is
point with ms equal to mn then gives relations hard to imaginehow the masses ofthe broadrho
betweenthemassesofhadronscomposedofasin-
and (cid:1) could be extrapolated so accurately from
gle (cid:13)avor of quark and hadrons composed of two
their strange partners, whose widths are much
di(cid:11)erent (cid:13)avors of quarks. For the vector multi-
narrower and vary signi(cid:12)cantly,if hadron masses
plet we obtain, for example,
werealteredbymorethanapercentorsobytheir
m(cid:26)[(mn+ms)=2;(mn+ms)=2]= decay widths.
Calculations of the di(cid:11)erence between the va-
mK(cid:3)(mn;ms): (2)
lence approximation and full QCD in the limit
In addition, in the valence approximation,m(cid:30) is of small quark masses have been reported by
given by m(cid:26)(ms;ms). Corresponding sets of re- Bernard and Golterman[16] and by Sharpe [17].
lations hold in the baryon multiplets. For hadrons composed of quarks which are su(cid:14)-
Using these relations, the mass of the member ciently light, these authors show that chiral per-
of each multiplet containing no strange quarks turbationtheorysuggeststhevalenceapproxima-
can be obtained by linear extrapolation fromthe tionwillbehavepathologically. Asaconsequence
0
masses of the multiplet members containing one oftheabsenceofvacuumpolarization,the(cid:17) mass
or more strange quark. Extrapolating from ex- goes to zero along with the pion mass as quark
0
perimentallyobservedmassesinthiswayrequires masses are made small. Virtual (cid:17) loops become
only the quark content of each hadron involved divergent atlowmomentumas the quark massis
and does not depend on the actual values of madesmalland contribute additionallogarithms
quark masses. These extrapolations are, essen- ofthe quark massnot present in chiralperturba-
tially, just a reinterpretation of the Gell-Mann- tion theory for full QCD and not multiplied by
Okubo mass formulas. Figure 3 shows extrapo- as manypowers of m(cid:25) as would occur in the full
6
theory. Forthebehaviorofthepionmassatsmall lattice (cid:12) m(cid:25)=m(cid:26) (cid:14)
quark mass,which in fullQCD is Wilson
3
16 (cid:2)24 5.70 0.49 (cid:0) 0.69 0.04(6)
2
m(cid:25) =cmq; (3) 243(cid:2)36 5.93 0.47 (cid:0) 0.74 (cid:0)0:03(3)
2
30(cid:2)32 (cid:2)40 6.17 0.48 (cid:0) 0.74 (cid:0)0:07(3)
Sharpe has summed a collection of such singu-
Kogut-Susskind
lar diagrams for the valence approximation and
3
32 (cid:2)64 6.00 0.31 (cid:0) 0.50 (cid:25)0:03
obtained
3
32 (cid:2)64 6.50 0.40 (cid:0) 0.66 0:02(1)
1
m2(cid:25) = c0mq1+(cid:14); (4) Table 3
2 Values of the parameter (cid:14) obtained from (cid:12)ts to
(cid:22)
(cid:14) = : (5) data for the pion mass as a function of quark
Nf(4(cid:25)f(cid:25))2
mass.
Here (cid:22) is a measure of the amplitude for the
0
quark-antiquark pair in the (cid:17) to annihilate into hadronmassratioswouldbesmallerthanthesta-
gluons. In full QCD (cid:22) becomes the mass of the tistical errors.
0
(cid:17) in the limit of zero quark mass and would be Labrenz and Sharpe [19] have also attempted
about 900 MeV. Using this estimate, along with to use chiral perturbation theory to estimate the
93MeV forf(cid:25), the valueof(cid:14) becomes about0.2. e(cid:11)ectofthevalenceapproximationonthenucleon
There is, however, no reason to assume (cid:22) has mass, mN. While I believe that the basic idea of
the same value in full QCD and in the valence trying to use chiral perturbation theory to make
approximation. A recent Monte Carloevaluation such comparisons is very promising, the particu-
of (cid:22) in the valence approximation [18] yielded lar results which they presented in Dallasappear
0:700(cid:6)0:050GeV,giving(cid:14) ofabout0.12. Fits of to me to be unconvincing. For the behavior of
Eq. (4) to data to be discussed in Sect. 4.1 for mN at small pion mass, chiral perturbation the-
Wilson quarks [6] and for Kogut-Susskind [25] ory for full QCD gives an expression of the form
quarks, give values of (cid:14) shown in Table 3. For
Wilsonquarks,mq inthese(cid:12)tsis(2k)(cid:0)1(cid:0)(2kc)(cid:0)1 mN =a+b(cid:25)m2(cid:25) +bKm2K +c(cid:25)m3(cid:25) +cKm3K: (6)
where kc isthecriticalhoppingconstantatwhich
Chiral perturbation theory for the valence ap-
m(cid:25) becomes zero. The value of kc was taken as
proximation gives an expression with an addi-
one ofthe adjustableparameters ineach (cid:12)t. The
tionalpion term and no K terms, since these en-
negativevaluesof(cid:14) fortwooutofthreeofthe(cid:12)ts
tail vacuumstrange quark loops,
withWilsonquarksimplythatthecorresponding
mq are above the range in which the argument mN =a0+a00m(cid:25) +b0(cid:25)m2(cid:25) +c0(cid:25)m3(cid:25): (7)
leading to Eq. (4) is applicable. The values of
00
(cid:14) found with Kogut-Susskind quarks are much Theextra a m(cid:25) termarisesfromthe samemech-
00
smaller than 0.12 but consistently positive. The anism leading to Eq. (4), and the coe(cid:14)cient a
Kogut-Susskind (cid:12)ts are done at smaller m(cid:25)=m(cid:26) is proportional to (cid:14). Our earlier discussion of (cid:14)
00
than those with Wilson fermions, and therefore suggests that the a coe(cid:14)cient will be small.
would correspond to smallervalues ofmq ifmea- Although it might seem that a comparison of
sured in common units. Thus there is some ev- Eqs.(6)and(7)couldbe used toestimatetheer-
idence for a slow approach to the behavior of ror invalence approximationcalculations of mN,
Eq. (4), perhaps with (cid:14) of 0.12, for m(cid:25)=m(cid:26) suf- itturnsoutthiscannotbedone. Onedi(cid:14)cultyis
(cid:12)ciently below 0.3. In any case, if the extrap- thatEq.(6)describes thevariationofmN asboth
olations to small quark mass to be discussed in seaandvalencequarkmassesarevariedtogether,
Sect. 4.1 were done using Eq. (3) for m(cid:25)=m(cid:26) while Eq. (7) describes only the result of vary-
above 0.3 and using Eq. (4) with (cid:14) of 0.12 for ing the valence quark mass with the sea quark
m(cid:25)=m(cid:26) below 0.3, in place of the actual (cid:12)ts us- mass kept (cid:12)xed. As the sea quark mass varies
ing Eq. (3) everywhere, the changes in predicted in full QCD, the e(cid:11)ective low-momentum gauge
7
couplingvaries, andtherefore the gaugecoupling
to be put into the corresponding valence approx-
imation must vary. Thus for each choice of m(cid:25)
and mK in the full QCD relation Eq. (6), a dif-
0
ferent and a priori unknown value of a must be
used in the corresponding valence approximation
relation Eq. (7). For lattice spacing su(cid:14)ciently
small, a change in the gauge coupling entering
thevalenceapproximationcauses achangeinlow
lyinghadron masses by a single overallscale fac-
tor. Thus the ratio of Eq. (7) to a corresponding
equation for some other massparameter, such as
f(cid:25), will be free of the scale factor and perhaps
mightbe compared to the ratio of Eqs. (6) to an
expression for f(cid:25) in full QCD.
A useful comparison for the ratio mN=f(cid:25) still
can not be made. The di(cid:14)culty at this point
is that the valence approximation chiral pertur-
bation theory expression for mN=f(cid:25) depends on
Figure 4. Monte Carlo data for the shift in beta
0
an unknown additive constant, comparable to a
of the decon(cid:12)ning transition from the valence
in Eq. (7), giving the valence approximation to
approximation to full QCD, for Kogut-Susskind
mN=f(cid:25) in the limit of zero quark mass. I know
fermions, in comparison to the one loop weak-
of no way to (cid:12)x this number from chiral pertur-
coupling prediction.
bationtheory. In the limitof very heavy m(cid:25) and
mK, mN=f(cid:25) will become equal in full QCD and
in the valence approximation. This limit, how- the valence approximation depends on unknown
ever,isbeyondtherangeofapplicabilityofchiral higher terms in the chiral lagrangians in both
perturbation theory and provides no help in de- cases. With plausible guesses for these terms,
terminingthe missingparameter. One mighttry Sharpe [20] obtains the result that this quantity
todeterminethemissingcoe(cid:14)cientbya(cid:12)ttothe inthevalenceapproximationwillbe smallerthan
data of Ref. [6]. Unfortunately, as mentioned al- in full QCD by about 0.12. The calculation of
ready in the discussion of Eq. (4), this data does Ref.[24](cid:12)ndsfK=f(cid:25) belowitsexperimentallyob-
not appear to extend to su(cid:14)ciently small quark served value by 0.06(9).
massfor the fullapparatus ofchiralperturbation Still another method of calculating di(cid:11)erences
theory to be applicable. between thevalenceapproximationandfullQCD
If, nonetheless, the expression of Eq. (7) is (cid:12)t- has been considered by Hasenfratz and De-
ted [19] to the data of Ref. [6] at (cid:12) of 5.93, the Grand [21]. These authors evaluate the e(cid:11)ective
extrapolatedvalueofmN atthelightquarkmass shift in chromoelectric charge arising from vac-
di(cid:11)ers fromthe result of the linear extrapolation uum polarization due to heavy Kogut-Susskind
used in Ref. [6] by only about 3%. quark-antiquark pairs in full QCD. For su(cid:14)-
There are comparisons between full QCD and ciently heavy quarks, the determination of this
the valence approximation which have fewer dif- change is reduced to evaluating the one loop
(cid:12)culties. An example of one such comparison is quark-antiquark vacuum polarization term cou-
for fK=f(cid:25). In both full QCD and in the valence pling two gluons in the weak-coupling expansion
approximationthisquantitymustbecome1ifm(cid:25) for lattice QCD. The resulting shift in (cid:12) is linear
ismadeequalto mK. The required additivecon- inthenumberofquark(cid:13)avors. Thepredicted(cid:1)(cid:12)
stantsarethusdetermined. Acompletelyreliable is compared with Monte Carlo data for the dif-
comparison between fK=f(cid:25) in full QCD and in ference between the critical (cid:12)c of the decon(cid:12)ning
8
transitionforQCDwithoutquarksand(cid:12)c witha
number of quarks ranging from 2 to 24. For lat-
tices withtimedirectionperiodrangingfrom4to
8, Figure 4 shows the predicted shift in compari-
son to Monte Carlo data. The overall agreement
is quite good for mqa(cid:21)0:05.
2.4. Wave Functions, State Vectors
Iwouldliketomentionbrie(cid:13)ytworecentpieces
of work on hadron wave functions and state vec-
tors. Kieu and Negele [22] have examined a
Bethe-Salpeter wave function for the pion with
quark and antiquark joined, in e(cid:11)ect, by the
ground state chromoelectric (cid:12)eld con(cid:12)guration
which would occur between a static color charge
at the quark position and static anticolor charge
at the antiquark position. They found that this
wave function falls o(cid:11) with the quark-antiquark
Figure5. UKQCDdataforvectormesonsmasses.
separation signi(cid:12)cantly more slowly than does
For quark and antiquark hopping constants k1
the Coulomb gauge wave function. Thus for (cid:0)1 (cid:0)1 (cid:0)1
and k2, keff is (2k1) +(2k2) .
increasing quark-antiquark separation, a quark-
antiquark pair joined by the ground state (cid:12)eld
con(cid:12)guration becomes progressively closer to the
truepionstatevectorthandoesaCoulombgauge
quark-antiquark pair.
Liu [23] has compared hadron masses calcu- UKQCD collaboration used the clover action.
lated in the usual valence approximation with The hadron propagators were found with point
massesobtainedinthevalenceapproximationus- sources and point sinks. The clover action [29]
ing a quark coupling matrix that allows quarks consists of the Wilson fermion action with the
to propagate in only one direction of time. The simplest additional term added to cancel, to ze-
state vector of a hadron in the usual valence ap- roth order in the chromoelectric coupling con-
proximation, Liu shows, at any instant may in- stant, the O(a) irrelevant coupling of the Wilson
cludequark-antiquarkpairsinadditiontotheva- action.
lencequarks. Thesepairsdonotoccurinhadrons In addition to calculating pseudoscalar and
composed of quarks which can propagate in only vector masses with quark and antiquark masses
one time direction. As consequence of eliminat- equal, the UKQCD collaboration also found
ing these pairs, Liu (cid:12)nds, the rho and pion be- masses for mesons composed of a quark and an-
comenearlydegenerateandthenucleonanddelta tiquark with di(cid:11)erent masses. With this data a
baryon become nearly degenerate. direct test can be madeof Eq. (2) and the corre-
spondingequationforthesquaresofpseudoscalar
meson masses. Figure 5 shows a comparison of
3. Spectrum Calculations with Fixed Lat-
the masses of vector mesons composed of quark
tice Spacing
andantiquarkwiththesamemassandthemasses
3.1. Valence Approximation of vector mesons composed of quark and anti-
Three recent spectrum calculations using the quark with di(cid:11)erent masses. The data strongly
valence approximation are summarized in Ta- supports Eq. (2). Corresponding data for pseu-
ble 4. Kim and Sinclair used Kogut-Susskind doscalar masses supports the version of Eq. (2)
fermions with wall sources and point sinks. The for squared pseudoscalar masses.
9
action lattice (cid:12) mqa m(cid:25)=m(cid:26) con(cid:12)gs. ref.
3
Argonne valence, KS 32 (cid:2)64 6.0 0.0100 0.500(7) 66 [25]
3
32 (cid:2)64 6.0 0.0050 0.393(10) 66 [25]
3
32 (cid:2)64 6.0 0.0025 0.311(9) 66 [25]
3
32 (cid:2)64 6.5 0.0100 0.659(8) 100 [25]
3
32 (cid:2)64 6.5 0.0050 0.520(12) 100 [25]
3
32 (cid:2)64 6.5 0.0025 0.401(15) 100 [25]
action lattice (cid:12) k m(cid:25)=m(cid:26) con(cid:12)gs. ref.
3
UKQCD valence, clover 24 (cid:2)48 6.2 0.14144 0.77(1) 60 [26]
3
24 (cid:2)48 6.2 0.14226 0.62(2) 60 [26]
3
24 (cid:2)48 6.2 0.14262 0.52(2) 60 [26]
3
GF11 valence, Wilson 16 (cid:2)32 5.70 0.1600 0.856(3) 219 [6]
0.1650 0.690(6) 219 [6]
0.16625 0.612(6) 219 [6]
0.1675 0.491(8) 219 [6]
3
24 (cid:2)32 5.70 0.1600 0.854(2) 92 [6]
0.1650 0.693(4) 92 [6]
0.1663 0.610(5) 58 [6]
0.1675 0.502(6) 92 [6]
3
24 (cid:2)36 5.93 0.1543 0.830(5) 210 [6]
0.1560 0.737(6) 210 [6]
0.1573 0.603(6) 210 [6]
0.1581 0.466(8) 210 [6]
2
32 (cid:2)30(cid:2)40 6.17 0.1500 0.867(2) 219 [6]
0.1519 0.735(4) 219 [6]
0.1526 0.633(6) 219 [6]
0.1532 0.478(1) 219 [6]
action lattice (cid:12) k m(cid:25)=m(cid:26) time ref.
3
HEMCGC full QCD, Wilson 16 (cid:2)32 5.3 0.1670 0.722(4) 2400 [27]
3
16 (cid:2)32 5.3 0.1675 0.599(8) 1300 [27]
Table 4
Parameters of recent hadron spectrum calculations.
10
3.2. Full QCD
Data for full QCD with two (cid:13)avors of Wil-
son fermions collected by the HEMCGC collab-
oration [27] is also listed in Table 4. These
calculations use the hybrid Monte Carlo algo-
rithm. Hadron propagators were calculated for
wallsources andeitherwallorpointsinks,andin
addition for Gaussian sources with either Gauss-
ianorpointsinks. Theintegratedautocorrelation
time for the plaquette was found to be about 80
time units for the data at the heavier pion mass
and120timeunitsforthedataatthelighterpion
mass. An examination of the dispersion of pion
e(cid:11)ective masses calculated from propagators av-
eragedoverbinsgaveroughlyequivalentautocor-
relationestimates. Thusthedatacollectedatthe
largerpionmassconsisted ofperhaps 30indepen-
dentcon(cid:12)gurationsandthedatawiththesmaller
pion mass perhaps 10. Figures 6 and 7 show full
QCD results obtained with various combinations Figure 6. HEMCGC mass ratios with various
ofseaquarkandvalencequarkhoppingconstants combinationsofvalencequarkandseaquarkhop-
in comparison to valence approximation masses ping constants. The squares have sea quark k of
3
for Wilson fermions on a lattice 16 (cid:2)32 at (cid:12) of 0.1675, the crosses have sea quark k of 0.1670.
5.85 and 5.95 [28]. For the range of parameters The diamondsare valence approximationresults.
considered, the di(cid:11)erences between the valence The circle and question markshow expected val-
approximationand full QCD consistent with the ues for in(cid:12)nite quark massand for the real world
statistical uncertainties. value of the light quark masses.
4. Valence Approximation Zero Lattice
Spacing Limits
4.1. Masses
The (cid:12)rst systematic evaluation of the in(cid:12)nite
volume continuum limit of hadron masses ex-
trapolatedtophysicalquarkmasseswasreported
this year by the GF11 collaboration [6]. Hadron
masses were calculated with Wilson quarks on a
setoffourlattices. Theparametersenteringthese
calculations are listed in Table 4.
Each con(cid:12)guration was (cid:12)xed into Coulomb
gauge. Inallcases,hadronpropagatorswerethen
calculated using a Gaussian quark source with
2
mean radius squared <r > of 6 in lattice units,
and using point sinks and Gaussian sinks with
2
< r > of 1.5, 6, 13.5, and 24. Hadron masses
were(cid:12)ttedtoeachchannelbyexamininge(cid:11)ective
massplotsandselectingthelargestintervalwhich
Figure 7. HEMCGC hyper(cid:12)ne splittings. Sym-
might appear to be a plateau. An automatic (cid:12)t-
bols have same meaningas in preceding (cid:12)gure.