Embed Size (px)
Citation preview
Quarkonium spin structure in lattice NRQCD
Howard D. Trottier*
Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6~Received 5 December 1996!
Numerical simulations of the quarkonium spin splittings are done in the framework of lattice nonrelativisticquantum chromodynamics. At leading order in the velocity expansion the spin splittings are ofO(MQv
4),where MQ is the renormalized quark mass andv2 is the mean squared quark velocity (vc
2'0.3 andvY2'0.1). A systematic analysis is done of all next-to-leading order corrections. This includes the addition of
O(MQv6) relativistic interactions, and the removal ofO(a2MQv
4) discretization errors in the leading-orderinteractions. Simulations are done for bothS- andP-wave mesons, with a variety of heavy quark actions andover a wide range of lattice spacings. Two prescriptions for the tadpole improvement of the action are alsostudied in detail: one using the measured value of the average plaquette, the other using the mean linkmeasured in the Landau gauge. Next-to-leading order interactions result in a very large reduction in thecharmonium splittings, down by about 60% from their values at leading order. There are further indicationsthat the velocity expansion may be poorly convergent for charmonium. Preliminary results show a smallcorrection to the hyperfine splitting in theY system.@S0556-2821~97!01609-3#
PACS number~s!: 12.38.Gc, 12.39.Jh, 14.40.Gx
I. INTRODUCTION
Quarkonium physics has been the subject of renewed the-oretical interest in recent years. The rich phenomenology ofthe charmonium andY families has spurred the developmentof nonrelativistic quantum chromodynamics~NRQCD!, aneffective field theory that relies on an expansion of the actionin the mean-squared velocityv2 of the heavy quarks(vc
2'0.3 andvY2'0.1).
NRQCD has been formulated both on the lattice@1,2# andin the continuum@3#. Lattice simulations of theY and char-monium systems have recently been done by the NRQCDCollaboration @4–8#. Results have also been reported forheavy-light @9# and bc spectra@7#, and some unquenchedsimulations have also been done@8#.
There are two key theoretical ingredients underlying lat-tice NRQCD calculations. One is an expansion of the effec-tive action to a sufficient order in the heavy quark velocity,so as to obtain results of the desired accuracy. The other keyingredient is the use of tadpole renormalization@10# of theoperators in the lattice action, which may then allow forreliable tree-level matching of the lattice theory with con-tinuum QCD. A related problem is the development of suf-ficiently accurate discretizations of the relevant operators.
The quarkonium spin structure is particularly sensitive tothe details of the NRQCD Hamiltonian. TheY @5,7# andcharmonium@6# spin splittings were recently analyzed toleading order inv2 by the NRQCD Collaboration. At leadingorder in the velocity expansion the spin splittings are ofO(MQv
4), whereMQ is the renormalized quark mass. In thecharmonium system, however, appreciable next-to-leading-order effects are expected, given the large mean-squared ve-locity. Indeed a recent analysis of the charmonium hyperfinesplitting using the relativistic Fermilab action gives a result
('70 MeV! @11# that is significantly smaller than is obtainedfrom leading-order NRQCD ('96 MeV! @6#.
In this paper a systematic analysis is done of all next-to-leading-order corrections to the spin splittings. This includesthe addition ofO(MQv
6) relativistic interactions, as well asthe removal ofO(a2MQv
4) discretization errors that arepresent in the leading-order spin-dependent operators consid-ered in Refs.@5–7#.
Results from simulations with a variety of heavy quarkactions and over a wide range of lattice spacings are pre-sented for the charmoniumS-wave hyperfine splitting. Somepreliminary results are also reported for the charmoniumP-wave fine structure and for theY hyperfine splitting.
Furthermore, two prescriptions for defining the tadpoleimprovement of the action are studied in detail: one using themeasured value of the average plaquette, as considered inRefs. @5,6#, the other using the mean link measured in theLandau gauge. Landau gauge tadpole improvement has re-cently been shown to yield smaller discretization errors inthe gluonic action~as measured by violations of rotationalinvariance in the heavy quark potential!, compared to calcu-lations using the average plaquette as input@12#. It is alsointeresting to note that the mean link is maximized in theLandau gauge, and so this prescription provides a lowerbound on the tadpole renormalization compared to mean linkdeterminations using other gauge fixings.
An important aspect of these simulations is the removal ofleading discretization errors in the gluonic action as well asin the NRQCD action. Specifically, anO(a4)-accurate glu-onic action is used, together withO(a4)-accurate cloverfields and covariant derivatives in the heavy quark action.Tadpole improvement of both the gluonic and the heavyquark actions has recently been shown to give a good de-scription of the spin-averaged charmonium spectrum even oncoarse lattices with spacingsa as large as 0.4 fm@13#. In thispaper the charmonium spin splittings are computed on lat-tices with spacings in the range of about 0.17–0.39 fm.
The next-to-leading-order interactions are shown to result*Electronic address: [email protected]
PHYSICAL REVIEW D 1 JUNE 1997VOLUME 55, NUMBER 11
550556-2821/97/55~11!/6844~8!/$10.00 6844 © 1997 The American Physical Society
in a very large reduction in the charmonium hyperfine split-ting, down by about 60% from the leading-order result re-ported in Ref.@6# on a lattice of comparable spacing~whenthe same tadpole improvement scheme is used!; results forthe tripletP-wave meson masses show that the fine structuresplittings are also reduced by about 60%. The next-to-leading-order charmonium hyperfine splitting is about(5565) MeV, compared to the relativistic Fermilab actionresult of about (7063) MeV @11#. While the two calcula-tions have different systematic errors, this comparison sug-gests that further relativistic corrections, beyond the next-to-leading order considered here, are again very large.
These results indicate that the NRQCD velocity expan-sion may be poorly convergent for charmonium, with thefirst three terms in the expansion for the hyperfine splittingapparently oscillating in sign. Another possibility is thatthere are large radiative corrections to the coefficients of thespin-dependent operators in the effective action. Preliminaryresults for theY hyperfine splitting at next-to-leading orderin v2 ~and with leading discretization errors removed! showlittle change from the leading-order result reported in Ref.@7#.
The dependence of the hyperfine splitting on lattice spac-ing is also analyzed. The splitting shows very little depen-dence ona when Landau gauge tadpole renormalization isused. The average plaquette tadpole scheme, on the otherhand, has large discretization errors, but the results are notinconsistent with an extrapolation to the same splitting atzero lattice spacing in the two schemes. These results pro-vide further support for the use of Landau gauge tadpolerenormalization@12#.
II. QUARK AND GAUGE-FIELD ACTIONS
The NRQCD effective action is based on power-countingrules for the magnitude of heavy quark and gauge-field op-erators in quarkonium states. The expansion parameters arethe mean-squared velocity of the heavy quarks in the boundstate and the strong coupling constant. The coefficients of theoperators in the effective action, to a given order inv2, aredetermined by matching the predictions of NRQCD withthose of full QCD@1,2#.
The heavy quark lattice Hamiltonian is conveniently de-composed into the leading covariant kinetic energy operatorH0, plus relativistic and discretization correctionsdH. Fol-lowing Refs.@5,6#, the quark Green function is given by
Gt115S 12aH0
2n D nU4†S 12
aH0
2n D n~12adH !Gt ~ t.1!,
~1!
where the initial evolution is set by
G15S 12aH0
2n D nU4†S 12
aH0
2n D ndxW ,0 . ~2!
The kinetic energy operatorH0 is of O(v2) and is given by
H052D~2!
2Mc0 , ~3!
whereMc0 is the bare quark mass andD (2) is the lattice
Laplacian. Relativistic corrections are organized in powers ofthe heavy quark velocity, with terms up toO(v6) consideredhere:
dH5dH ~4!1dH ~6!. ~4!
Simulations with the complete set ofO(v4) correctionswere reported in Refs.@5,6#:
dH ~4!52c1~D~2!!2
8~Mc0!3
1c2ig
8~Mc0!2
~D•E2E•D!
2c3g
8~Mc0!2
s•~D3E2E3D!2c4g
2Mc0s•B
1c5a2D~4!
24Mc0 2c6
a~D~2!!2
16n~Mc0!2
. ~5!
The first two terms indH (4) are spin-independent relativisticcorrections, and the last two terms come from finite latticespacing corrections to the lattice Laplacian and the latticetime derivative, respectively. The parametern is introducedto remove instabilities in the heavy quark propagator causedby the highest momentum modes of the theory@2#.
While dH (4) yields spin-averaged spectra to next-to-leading-order in the velocity expansion, it contains only lead-ing order spin-dependent interactions. In potential model lan-guage the third term above (c3) generates the spin-orbit andtensor potentials which drive theP-wave fine structure,while the fourth term (c4) generates the color-magneticS-wave hyperfine splitting.
Spin-dependent interactions atO(v6) were derived inRef. @2#:
dH ~6!52c7g
8~Mc0!3
$D~2!,s•B%
2c83g
64~Mc0!4
$D~2!,s•~D3E2E3D!%
2c9ig2
8~Mc0!3
s•E3E. ~6!
Notice the field strength bilinear (c9), which is peculiar tothe non-Abelian theory. There are additionalO(v6) termswhich contribute to spin-averaged spectra; these are not con-sidered here.
The derivative operators and the fields are evaluated withtheir leading discretization errors removed, in order to mini-mize the effects of lattice artifacts on the spin splittings. Thisis indicated by the tilde on these operators in Eqs.~5! and~6!. At leading order ina the action of the symmetric latticederivativeD i is defined by
aD iG~x
Di5D i2a2
6D i
~1 !D iD i~2 ! , ~8!
whereD i(1) andD i
(2) are leading-order forward and back-ward covariant finite differences:
aD i~1 !G~x![Ui~x!G~x1a ı!2G~x!,
aD i~2 !G~x![G~x!2Ui
†~x2a ı!G~x2a ı!. ~9!
At leading order ina the lattice LaplacianD (2) is expressedin terms of covariant second order differences
D~2!5(i
D i~2! , ~10!
where
a2D i~2!G~x!5Ui~x!G~x1a ı!22G~x!
1Ui†~x2a ı!G~x2a ı!. ~11!
TheO(a4)-accurate LaplacianD(2) is used in Eq.~6!, and atthe tree level is given by@2#
D~2!5D~2!2a2
12D~4!, ~12!
where
D~4!5(i
~D i~2!!2 ~13!
is a lattice representation of the continuum operator( iDi4 .
Note that discretization errors inH0 have been removed inthis way by the addition of thec5 term in Eq.~5!. What isnew in the present work is the removal of discretization er-rors in the leading-orderP-wave interaction (c3) and theDarwin term (c2), as well as the addition of theO(v6) spinterms in Eq.~6!.
To complete the removal of discretization errors from thespin-dependent interactionsO(a4)-accurate chromoelectricand chromomagnetic fields have been used in these simula-tions. In Refs.@5,6# the leading-order field strengthFmn wasevaluated using the standard traceless clover operator:
Fmn~x!51
2i@Vmn~x!2Vmn
† ~x!#21
3Im@TrVmn~x!#, ~14!
where Vmn is an average over the four counterclockwiseplaquettes in the (m,n) plane containing the sitex;
Vmn521
4 (a56m
(b56n
Ua~x!Ub~x1a !
3U2a~x1a1b !U2b~x1b !. ~15!
The O(a4)-accurate field strengthFmn used here is com-puted following the analysis of Ref.@2#:
Fmn~x!5 53Fmn~x!2 1
6 @Um~x!Fmn~x1m !Um† ~x!
1Um† ~x2m !Fmn~x2m !Um~x2m !2~m↔n!#.
~16!
The ‘‘improved’’ chromoelectric and chromomagnetic fieldsare defined in terms of the field strength,Ei5F4i and Bi512e i jk F jk .It should be noted that some of the discretization errors
that are removed by usingD, D(2), and Fmn in Eqs.~5! and~6!, in place of their leading-order counterparts, may in factbe comparable to or smaller than higher-order relativisticcorrections that are not included here. This includes, for ex-ample, the use ofD(2) andFmn in theO(v
6) terms in Eq.~6!and the use of the improved operators in the Darwin term(c2). On the other hand, the use of improved operators in theleading spin-dependent interactions (c3 andc4) corrects forerrors ofO(a2Mcv
4) in the spin splittings which, for therange of lattice spacings studied here, may be comparable totheO(Mcv
6) contributions fromdH (6).At the tree level all of the coefficientsci in Eqs.~5! and
~6! are 1. However, very large radiative corrections in thelattice theory can arise from tadpoles that are induced by thenonlinear connection between the link variablesUm and thecontinuum gauge fields. Most of the effects of tadpoles canbe removed by a mean-field renormalization of the link@10#:
Um~x!→Um~x!
u0. ~17!
The links are rescaled in the simulation before they are inputto the quark propagator subroutine, to be sure that Eq.~17! iscorrectly implemented in all terms in the heavy quark action.
In most previous work the fourth root of the averageplaquette has been used to set the value ofu0:
u0,P[^ 13ReTrUpl&
1/4. ~18!
Simulations were done here with this renormalization pre-scription. In addition, simulations were also done using themean link in the Landau gauge to setu0, as recently sug-gested by Lepage@12#:
u0,L[^ 13ReTrUm&, ]mAm50, ~19!
where a standard lattice implementation of the continuumLandau gauge fixing is used@14# ~it was found that the re-moval of leading discretization errors in the lattice version of]mAm50 results in a negligible change to the value ofu0,L).
Finally, the gauge-field configurations were generated us-ing anO(a4)-accurate tadpole-improved action@13#:
S@U#5b(pl
13ReTr~12Upl!2
b
20u02(
rt
13ReTr~12U rt!,
~20!
where the sums are over all oriented 131 plaquettes~pl! and132 rectangles~rt!.
6846 55HOWARD D. TROTTIER
III. MESON PROPAGATORS
In order to increase the overlap of the meson propagatorswith the ground states of interest here, a gauge-covariantsmearing procedure has been used@15#. A meson creationoperator is constructed from quark and antiquark creationoperatorsc† andx† @1,5,6#:
(xW
c†~xW !G~xW !x†~xW !, ~21!
with
G~xW 
lattices are given in Tables I and II. In order to distinguishbetween the two sets of simulations,bL is used to denote thelattice coupling when Landau gauge tadpole renormalizationis used andbP when the average plaquette is used.
A standard Cabbibo-Marinari pseudo heat bath was usedto generate the gauge-field configurations. Integrated auto-correlation timest int were checked for all correlation func-tions and were found to be remarkably short; ten updatesbetween measurements yieldt int&0.5 on the three latticeswith a&0.2 fm, and on the coarser lattices five updates werefound to be sufficient.
Smeared-smeared correlators were used for theP waves,while local sources and smeared sinks were used for theSwaves. Ten smearing iterations@ns510 in Eq. ~23!# wereused for the three lattices witha&0.2 fm, five iterations forthe lattices witha near 0.28 fm, and two iterations for thelattices witha'0.39 fm. A smearing weighte51/12 wasused in all cases.
The lattice spacings are determined from the spin-averaged 1P21S mass difference, following Refs.@5,6#.This mass difference is known to be independent of thequark mass in the charm to bottom region. For this purposethe singlet1P1 and the spin-averaged
3S1,1S0 masses were
used. The simulation results for the splitting were fixed to theexperimental value for charmonium of 458 MeV.
After the lattice spacing was extracted, the kinetic massM kin ~in physical units! of the 1S0 state was determined byfitting the energyEP of the boosted state to the form
EP2E05P2
2M kin. ~26!
Fits were made to the state with momentum components(1,0,0) in units of 2p/(Na); in some cases simultaneous fitsincluding states with momentum components (1,1,0) and(1,1,1) were also done, with little change to the fit values ofM kin . A dispersion relation including relativistic corrections@5,6# was also tried, and the resulting changes to the fit val-ues ofM kin were within a few percent, as expected on theselattices.
The correct values of the bare charm massMc0 were de-
termined by tuning so thatM kin agrees with the experimentalvalue of the mass of thehc ~2.98 GeV!. The bare masses arelisted in Tables I and II, and all yieldM kin52.9(1) GeV. Forbare massesaMc
0,1.2 a stability parametern54 was usedin the quark propagators, Eqs.~1! and ~2!; for1.2,aMc
0,1.5, n53 was used, and for the larger baremassesn52 was used.
Effective mass plotsmeff(T)52 ln@G(T)/G(T21)# forseveral lattices are shown in Figs. 1–4, using jackknife er-rors. Single-exponential fits to the correlation functions areused to get the best estimates of the masses of the individualstates. The fitting procedure included the full covariance ma-trix for the data, using the singular value decomposition~SVD! algorithm@16#. The correlation functions for states of a
FIG. 3. Effective mass plot forbP57.2 (a50.17 fm!: 1P1 state(h) and 1S0 state (s).
FIG. 4. Effective mass plot forbP56.25 (a50.39 fm!: 1P1
state (h) and 1S0 state (s).
TABLE III. Examples of fits to Landau gauge tadpole simula-tion atbL57.4 (a50.18 fm!. Single-exponential fits were used foreach individual state, and a correlateddE fit for the 3S1 and
1S0was used to get the hyperfine splitting.
tmin /tmax1P1
3S11S0
3S1-1S0
2/16 0.735~5! 0.298~2! 0.242~1! 0.0537~6!
3/16 0.715~6! 0.293~2! 0.239~1! 0.0533~5!
4/16 0.709~9! 0.289~2! 0.237~1! 0.0518~6!
5/16 0.694~12! 0.286~2! 0.235~1! 0.0506~6!
6/16 0.688~16! 0.285~2! 0.235~1! 0.0497~7!
7/16 0.672~23! 0.284~2! 0.234~1! 0.0500~8!
8/16 0.661~30! 0.284~2! 0.234~1! 0.0498~9!
9/16 0.700~51! 0.284~2! 0.234~1! 0.0494~10!
TABLE IV. Examples of fits to Landau gauge tadpole simula-tion atbL56.6 (a50.39 fm!. Fits were done as in Table III.
tmin /tmax1P1
3S11S0
3S121S0
2/10 1.280~8! 0.404~1! 0.304~1! 0.0980~6!
3/10 1.268~15! 0.395~1! 0.299~1! 0.0969~6!
4/10 1.279~35! 0.393~1! 0.298~1! 0.0950~7!
5/10 1.198~71! 0.392~2! 0.298~1! 0.0945~7!
6/10 1.18~14! 0.392~2! 0.298~1! 0.0946~10!7/10 1.44~43! 0.392~2! 0.298~1! 0.0944~12!
6848 55HOWARD D. TROTTIER
given partial wave are highly correlated; following Refs.@5,6# a spin splittingdE was obtained from a correlated fit ofthe form
Gmeson,A~ t !5cAe2EAt, Gmeson,B~ t !5cBe
2~EA1dE!t.~27!
Detailed fit results for several lattices are reported inTables III–VI. The statistical errors were estimated usingbootstrap ensembles of 1000 samples. Final estimates of thedimensionless energies are obtained from these fits by find-ing two or three successivetmin /tmax intervals for which thefit results overlap within statistical errors; acceptableQ val-ues were obtained in all cases at thesetmin /tmax values. Es-timates of the systematic errors in the final fit results aretaken from the largest statistical errors in the overlappingintervals.
The final fit results are shown in Tables VII and VIII,where the resulting lattice spacings and hyperfine splittingsin physical units are also given. The dominant error in thesplitting comes from the systematic error in the determina-tion of the bare quark mass. The error in the mass comes inpart from the uncertainty ina21, which has been included inthe error estimates for the splittings in physical units; how-ever, there is a further systematic error of order 10% in thequark mass determination, coming from higher-order~spin-independent! relativistic corrections@6#.
V. DISCUSSION
The hyperfine splittings are plotted as a function of latticespacing in Fig. 5, where the results from the relativistic Fer-milab action@11# and the leading-order NRQCD calculation@6# are included. Some coarse lattice results from thetadpole-improved relativistic D234@17# action are alsoshown.
The next-to-leading-order corrections result in a verylarge reduction in the hyperfine splitting, down by about60% from the leading-order result on a lattice of comparablespacing, when the same plaquette tadpole renormalizationscheme (u0,P) is used in both cases.
The hyperfine splitting shows very littlea dependencewhen the Landau gauge tadpole schemeu0,L is used. Theresults withu0,P , on the other hand, have large discretizationerrors, which prevents a reliable extrapolation to zero latticespacing in this case; however, the results are not inconsistentwith an extrapolation to the same splitting as is obtained withLandau gaugeu0,L .
From these results the hyperfine splitting at next-to-leading order in the velocity expansion, and at zero latticespacing, can be estimated at roughly (5565) MeV. This canbe compared with the Fermilab action result of approxi-mately (7063) MeV @11#. While the two calculations havedifferent systematic errors, this suggests that further relativ-istic corrections, beyond the next-to-leading order consideredhere, are again very large.@The experimental value is(11862) MeV, which indicates that there are significant ef-fects due to quenching@11,6#.#
These considerations are supported by results obtainedhere for the tripletP-wave spectra (3P0,
3P1,3P2). The next-
to-leading-order fine structure splittings are apparently verysmall, and much better statistics are required for an accuratedetermination. Simulations withu0,L give a 3P2-
3P0 split-ting of about (30615) MeV, down by about 60% from theleading-order result of (110610) MeV reported in Ref.@6#@the experimental value is (141610) MeV#. The next-to-leading-order fine structure splittings withu0,P are evenharder to measure, but the results suggest that the splittingsmay actually be in the wrong order with that tadpole scheme~as least in the range of lattice spacings analyzed here!.
These results indicate that the NRQCD velocity expan-sion for charmonium may be poorly convergent, with thefirst three terms in the expansion for the hyperfine splitting
TABLE V. Examples of fits to average plaquette tadpole simu-lation atbP57.2 (a50.17 fm!. Fits were done as in Table III.
tmin /tmax1P1
3S11S0
3S1-1S0
2/16 1.120~6! 0.715~2! 0.675~2! 0.0399~5!
3/16 1.101~7! 0.708~2! 0.670~2! 0.0395~5!
4/16 1.089~9! 0.706~2! 0.669~2! 0.0369~5!
5/16 1.083~12! 0.704~2! 0.667~2! 0.0363~6!
6/16 1.087~15! 0.702~2! 0.667~2! 0.0357~7!
7/16 1.083~21! 0.700~2! 0.666~2! 0.0352~7!
8/16 1.093~29! 0.701~2! 0.666~2! 0.0352~8!
TABLE VI. Examples of fits to average plaquette tadpole simu-lation atbP56.25 (a50.39 fm!. Fits were done as in Table III.
tmin /tmax1P1
3S11S0
3S1-1S0
2/10 1.773~7! 0.858~1! 0.801~1! 0.0575~3!
3/10 1.754~14! 0.853~1! 0.797~1! 0.0570~3!
4/10 1.754~26! 0.852~1! 0.795~1! 0.0565~3!
5/10 1.771~60! 0.851~1! 0.795~1! 0.0561~4!
6/10 0.851~1! 0.795~1! 0.0560~4!
7/10 0.850~1! 0.795~1! 0.0556~6!
TABLE VII. Final fit results for the dimensionless energiesfrom Landau gauge tadpole simulations; the resulting lattice spac-ings and hyperfine splittings in physical units are also shown. Thequoted errors in the hyperfine splittings in physical units include thesystematic errors ina.
a HyperfinebL
1P11S0
3S-1S0 ~fm! ~MeV!
7.4 0.68~2! 0.234~1! 0.0497~7! 0.176~9! 55.7~27!7.0 1.01~2! 0.305~1! 0.0772~8! 0.278~9! 54.6~18!6.6 1.27~3! 0.298~1! 0.0945~7! 0.388~13! 48.0~17!
TABLE VIII. Final fit results from average plaquette simula-tions.
a HyperfinebL
1P11S0
3S-1S0 ~fm! ~MeV!
7.2 1.09~1! 0.666~2! 0.0352~7! 0.171~4! 40.5~13!7.0 1.23~2! 0.728~2! 0.0354~7! 0.205~9! 34.1~16!6.8 1.42~2! 0.790~2! 0.0427~6! 0.257~9! 32.7~12!6.25 1.75~3! 0.795~1! 0.0560~6! 0.393~13! 28.1~10!
55 6849QUARKONIUM SPIN STRUCTURE IN LATTICE NRQCD
apparently oscillating in sign. However, there are othersources of systematic error in the NRQCD action which mustalso be considered. Radiative corrections to the operator co-efficientsc3 andc4 in Eq. ~5! are of particular importance.
It is worthwhile to assess the relative importance of thevarious next-to-leading-order corrections that have been con-sidered here. Within a given tadpole renormalization schemethe most important correction for the charmonium systemcomes from the relativistic spin-dependent interactionsdH (6) @Eq. ~6!#, and these drive the large reduction in thesplittings. There is some indication that the field strengthbilinear ~term c9) plays a relatively small role in these ef-fects. The use ofO(a4)-accurate clover fields increases thespin splittings, a correction amounting to about 20% of theO(a2)-accurate splittings on the coarsest lattices consideredhere ~this correction falls below about 10% at the smallestspacings!.
The effect of a change in the tadpole renormalizationscheme is very significant. The spin splittings vary as 1/u0
4
due to the renormalization of the clover field interactions~except for the field strength bilinear, which varies as1/u0
8). This renormalization causes most of the change due totadpole scheme, as can be seen from the values ofu0,L andu0,P in Tables I and II~additional changes in the splittingsare presumably caused by the renormalization of the gluonicaction!. For example, ata'0.18 fm the ratio (u0,P /u0,L)
4 isabout 1.2, and ata'0.39 fm the ratio is about 1.5.
Relativistic corrections are expected to be much smaller
for theY system. Preliminary results from a next-to-leadingorder calculation atbP57.2 give anY-hb hyperfine splittingof (22.461.3) MeV, using a bare massaMb
053.15. This iswithin errors of the leading-order result reported in Ref.@7#on a lattice with comparable spacing~usingu0,P), which isconsistent with the velocity expansion, given the fact thatvY2'0.1. The small size of the net correction, relative to the
charmonium system, also appears to be driven by a nearcancellation of theO(a2MQv
4) discretization corrections~which tend to raise the splittings! and theO(MQv
6) relativ-istic corrections~which tend to lower them!, which are muchcloser in magnitude in theY system. This was demontratedby doing a calculation with a Wilson gauge field action atb55.7, without removing discretization errors in the cloverfield, but including theO(MQv
6) relativistic interactions;this results in a reduction of theY hyperfine splitting byabout 15% compared to the leading-order calculation@7# atthe sameb.
It is also interesting to note that the lattice spacing asdetermined from the 1P-1S splitting is different forY andcharmonium. At bP57.2 preliminary results giveaY
50.146(9) fm, compared toac50.171(4) fm. This is com-parable to the difference between the two determinations ofthe spacing in Ref.@7#.
VI. SUMMARY AND OUTLOOK
It has been shown that spin-dependent interactions atnext-to-leading order in the NRQCD velocity expansionyield very large corrections to the charmonium spin split-tings, down by about 60% from their values at leading order~when the same tadpole improvement scheme is used on lat-tices with comparable spacings!. There are indications thatfurther relativistic corrections for charmonium are also verylarge. The corrections to the hyperfine splitting in theY sys-tem are small. More work needs to be done in order to assessthe validity of the NRQCD effective action in simulations ofcharmonium, including better measurements of the tripletP-wave spectra. Estimates of the radiative corrections to theoperator coefficients in the heavy quark action are alsoneeded. More complete calculations of theY splittingswould also provide useful information. The results obtainedhere provide further support for the use of Landau gaugetadpole renormalization.
ACKNOWLEDGMENTS
I am indebted to G. P. Lepage and R. M. Woloshyn formany helpful discussions and suggestions. I also thank C.Davies, J. Shigemitsu, and J. Sloan for useful conversations.This work was supported in part by the Natural Sciences andEngineering Research Council of Canada.
@1# G. P. Lepage and B. A. Thacker, inField Theory on the Lat-tice, Proceedings of the International Symposium, Seillac,France, 1987, edited by A. Billoireet al. @Nucl. Phys. B~Proc.Suppl.! 4, 199~1988!#; B. A. Thacker and G. P. Lepage, Phys.Rev. D43, 196 ~1991!.
@2# G. P. Lepage, L. Magnea, C. Nakhleh, U. Magnea, and K.Hornbostel, Phys. Rev. D46, 4052~1992!.
@3# G. T. Bodwin, E. Braaten, and G. P. Lepage, Phys. Rev. D51,1125 ~1995!.
@4# C. T. H. Davies, K. Hornbostel, A. Langnau, G. P. Lepage, A.
FIG. 5. Hyperfine splittings versus lattice spacing squared. Thenext-to-leading-order NRQCD results with Landau gauge tadpoles(j) and with average plaquette tadpoles (s) are shown, as well asthe leading-order result (3) from Ref. @6#. Also shown are resultsobtained with the relativistic Fermilab action (h) @11# and coarselattice results from the relativistic tadpole-improved D234 action(n) @17#. The experimental value is (11862) MeV.
6850 55HOWARD D. TROTTIER
Lidsey, C. J. Morningstar, J. Shigemitsu, and J. Sloan, Phys.Rev. Lett.73, 2654~1994!.
@5# C. T. H. Davies, K. Hornbostel, A. Langnau, G. P. Lepage, A.Lidsey, J. Shigemitsu, and J. Sloan, Phys. Rev. D50, 6963~1994!.
@6# C. T. H. Davies, K. Hornbostel, G. P. Lepage, A. Lidsey, J.Shigemitsu, and J. Sloan, Phys. Rev. D52, 6519~1995!.
@7# C. T. H. Davies, K. Hornbostel, A. Langnau, G. P. Lepage, A.Lidsey, J. Shigemitsu, and J. Sloan, inLattice ’95, Proceedingsof the International Symposium, Melbourne, Australia, editedby T. D. Kieu et al. @Nucl. Phys. B~Proc. Suppl.! 47, 421~1996!#.
@8# For a recent review, see J. Shigemitsu, inLattice ’96, Proceed-ings of the International Symposium, St. Louis, Missouri, ed-ited by T. Golterman@Nucl. Phys. B~Proc. Suppl.! ~in press!#,Report No. hep-lat/9608058~unpublished!.
@9# A. Ali Khan, C. T. H. Davies, S. Collins, J. Sloan, and J.Shigemitsu, Phys. Rev. D53, 6433~1996!.
@10# G. P. Lepage and P. B. Mackenzie, Phys. Rev. D48, 2250~1993!.
@11# A. El-Khadra and B. Mertens, inLattice ’94, Proceedings ofthe International Symposium, Bielefeld, Germany, edited by F.Karschet al. @Nucl. Phys. B~Proc. Suppl.! 42, 406~1995!#; A.El-Khadra, in Lattice ’91, Proceedings of the InternationalSymposium, Tsukuba, Japan, edited by M. Fukugitaet al.@ibid. 26, 372~1992!#. More recent results are reviewed in Ref.@8#.
@12# G. P. Lepage~private communication!.@13# M. Alford, W. Dimm, G. P. Lepage, G. Hockney, and P. B.
Mackenzie, Phys. Lett. B361, 87 ~1995!.@14# See, for example, C. T. H. Davieset al., Phys. Rev. D37,
1581 ~1988!.@15# G. P. Lepage~private communication!.@16# I thank R. M. Woloshyn for providing me with a copy of his
SVD fitting routines.@17# M. G. Alford, T. R. Klassen, and G. P. Lepage, Report No.
hep-lat/9608113~unpublished!.
55 6851QUARKONIUM SPIN STRUCTURE IN LATTICE NRQCD
![Heavy quark potentials in effective string theory fileFrom NRQCD to pNRQCD Potential NRQCD: e ective theory forheavy quarkonium [Brambilla, Pineda, Soto, Vairo, Nucl. Phys. B (2000)]](https://img.pdfslide.net/doc/110x75/5d4aeb7c88c99394798b5137/heavy-quark-potentials-in-effective-string-theory-nrqcd-to-pnrqcd-potential-nrqcd.jpg)
