16 March 2000

Ž .Physics Letters B 476 2000 309–315

Hyperfine effects in charmed baryons

R.M. WoloshynTRIUMF, 4004 Wesbrook Mall, VancouÕer, BC V6T 2A3, Canada

Received 12 January 2000; received in revised form 1 February 2000; accepted 4 February 2000Editor: H. Georgi

Abstract

Hadron masses are calculated in quenched lattice QCD with a fermion action of the D234 type on an anisotropic lattice.Hyperfine splittings for singly charmed baryons are found to be in agreement with expectations from the quark model andwith a magnitude slightly larger than experimental values. Masses of doubly charmed baryons are also calculated andcompared to a variety of model calculations. Hyperfine splittings in doubly charmed baryons are found to be slightly smallerthan in singly charmed states. q 2000 Published by Elsevier Science B.V. All rights reserved.

1. Introduction

Relatively little work has been done on heavybaryons using lattice QCD. The most complete studyin a relativistic framework is the one done in the

w xUKQCD collaboration 1 for both charm and bottomŽ .flavoured baryons using an O a improved fermion

action in quenched approximation. In that work hy-perfine splittings in both charm and bottom baryonswere found to be considerably smaller than those

w xpredicted by phenomenological models 2 and ob-served experimentally. A recent NRQCD calculationw x3 was able to resolve the hyperfine splittings forbaryons with b quarks and found values in accordwith phenomenological expectations. Since the hy-perfine splitting is an important feature of the baryonspectrum further investigation of charmed baryonsseems justified.

In this work we present the results of anotherquenched lattice simulation for charmed baryons.Due to limitations in computing resources it was not

possible to use the same lattice spacing and volumeas used in the UKQCD calculation. Rather we work

Ž .on a more coarse lattice ;0.2 fm with a highlyimproved action. Past experience has shown thatresults of reasonable accuracy may be obtained with

w xsuch lattices 4 . In order to check the calculation,Ž .the spectrum of baryons in the light quark u,d,s

sector was calculated at the same time. As well,meson masses for both heavy and light quarks werecalculated. The results of all these calculations are inreasonable agreement with experimental values andwith the results obtained at small lattice spacingw x5,6 .

w xThe simulation reported here differs from 1 inw xtwo other respects. In 1 the interpolating operators

used for the baryons were taken to have a formwhich emphasizes the heavy quark symmetry. Sec-ondly, the spin 1r2 and spin 3r2 S-like baryons areinterpolated by the same operator, a Rarita-Schwinger

w xspin-3r2 field. As is well known 7 , the correlatorof such a field propagates both Js1r2 and Js3r2

0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 00 00165-9

( )R.M. WoloshynrPhysics Letters B 476 2000 309–315310

states and it is the Js1r2 projection of this correla-w xtor that is identified with the S-like baryons in 1 .

The procedure used here is different. As has beendone in the context of a QCD sum rule calculationw x8 , the interpolating fields used for heavy baryonsare taken to have the same form as those used in thelight quark sector. The Js1r2 S baryon has aninterpolating operator which is distinct from thatused for the Js3r2 S ) , just as the nucleon isusually interpolated by a field that is distinct fromthe D. No assumption is made about heavy quarksymmetry. An advantage of this is that we can use

Ž .the same procedure and computer code at all masseswhich provides some check on the results.

The correlation function used to extract the massesof S ’s is not obtained from the projection of theJs3r2 field’s correlator. In fact, it seems that thespin 1r2 projection of this correlator has very pooroverlap with the Js1r2 ground state. Our findingis that the spin 1r2 projected correlator is very smalland noisy compared to the correlator calculated di-rectly using a spin 1r2 field.

The final conclusion of this study is that inquenched lattice QCD the hyperfine splittings forboth singly and doubly charmed baryons are inreasonable agreement with phenomenological expec-tations. Indications are that the splittings may beoverestimated compared to experiment which seemsto be a common tendency for quenched QCD simula-tions of baryons at all quark masses.

2. Method

The calculation is done on an anisotropic latticew x9 using the gauge field action

1S U sb c 1y Re Tr UŽ . Ž .ÝG p s p s3ps

1qc 1y Re Tr UŽ .Ýr s r s3rs

1qc 1y Re Tr UŽ .Ýp t p t3pt

1qc 1y Re Tr UŽ .Ýr st r st3rst

1qc 1y Re Tr U , 1Ž .Ž .Ýr t s r t s3rts

where ps and rs denote spatial plaquettes and spatialplanar 2=1 rectangles respectively. The plaquetteslying in the temporal-spatial planes are denoted by ptwhile rectangles with the long side in a spatialŽ . Ž .temporal direction are labeled by rst rts . The ccoefficients incorporate the aspect ratio jsa ras t

and gauge link renormalization factors u and u .s t

These renormalization factors are estimated using thelink expectation value in Landau gauge.

The fermion action is of the anisotropic D234w xtype 10 and has the form

S s c D x qc D xŽ . Ž .Ž .ÝF 1 i 1 i 2 i 2 ix , i

q c D x qc D xŽ . Ž .Ž .Ý 1 t 1 t 2 t 2 tx

q c c x s F x c xŽ . Ž . Ž .Ý 0 s i j i jx , i-j

q c c x s F x c xŽ . Ž . Ž .Ý 0 t 0 i 0 ix , i

y c x c x , 2Ž . Ž . Ž .Ýx

where

ˆD x sc x 1yjg U x c xq iŽ . Ž . Ž . Ž . Ž .1 i i i

†ˆqc xq i 1qjg U x c x ,Ž . Ž . Ž .Ž . i i

3Ž .

ˆD x sc x 1yg U x c xq tŽ . Ž . Ž . Ž . Ž .1 t 4 4

†ˆqc xq t 1qg U x c x , 4Ž . Ž . Ž . Ž . Ž .4 4

D xŽ .2 i

ˆ ˆsc x 1yjg U x U xq i c xq2 iŽ . Ž . Ž . Ž . Ž .i i i

ˆqc xq2 i 1qjgŽ .Ž . i

= † ˆ †U xq i U x c x , 5Ž . Ž . Ž .Ž .i i

D xŽ .2 t

ˆ ˆsc x 1yg U x U xq t c xq2 tŽ . Ž . Ž . Ž . Ž .4 4 4

ˆqc xq2 t 1qgŽ . Ž .4

= † ˆ †U xq t U x c x . 6Ž . Ž . Ž . Ž .4 4

The c coefficients in the fermion action include theaspect ratio, link renormalization and the hoppingparameter factors and are shown in Table 1.

( )R.M. WoloshynrPhysics Letters B 476 2000 309–315 311

Table 1Coefficients appearing in the gauge and fermion actions

s t st ts4 2 2c 5r3u j 5j r3u up . . . s s t

6 4 2 2 4c y1r12u j yj r12u u yj r12u ur . . . s s t s t4 2 2 2c 2kr3u j 2kr3u u j0 . . . s s t

2c 4kr3u j 4kr3u1 . . . s t2 2 2c ykr6u j ykr6u2 . . . s t

Hadron masses are calculated from zero-momen-tum correlation functions in the usual way. Formesons the interpolating fields were just the standardones. For baryons some discussion is needed since

w xthe procedure used here differs from that used in 1 .Ž .Start from the light quark u,d,s sector. A common

w xchoice 11 for the proton operator in terms of u andd quark fields is

abc Te u Cg d u , 7Ž .a 5 b c

where a,b,c are colour indices and Dirac indiceshave been suppressed. For D the operator

1abc T Te 2 u Cg d u q u Cg u d 8Ž .½ 5a m b c a m b c'3

w xis used. This choice of operators is not unique 12but it allows an easy generalization to other baryonsw x11 . The interpolating operators for the strange hy-perons S and S ) are obtained by the replacement

Ž . Ž .d™s in 7 and 8 respectively. Similarly the inter-polators for J and J ) are constructed by thereplacement u™s. The L hyperon is more of a

Ž .problem. In the SU 3 flavour limit it would benatural to use the octet lambda

1abc T TL s e 2 u Cg d s q u Cg s d�8 a 5 b c a 5 b c'6

Ty d Cg s u 9Ž .4a 5 b c

since it is degenerate with the nucleon and D. How-Ž .ever, since SU 3 flavour is broken, this choice is not

w xcompelling. For example, in 11 a combination L

Ž .with both SU 3 octet and singlet components wasdefined. In this work a ‘‘heavy’’ L with a form

Tu Cg d s , 10Ž .a 5 b c

which is natural in the heavy quark limit, is alsoused.

The operators used to calculate the masses of theeight ground state singly-charmed baryons are takento have the same structure as the operators givenabove. The S and S ) baryons are obtained by thec c

Ž . Ž . )substitution d™c in 7 and 8 while V and Vc c

are obtained by u™s, d™c. For the L and Jc cŽ . Ž .the operators 9 and 10 with the replacements

s™c and d™s, s™c are used. Finally, for theremaining two states we take

1X abc T TJ s e u Cg c s q s Cg c u 11Ž .� 4c a 5 b c a 5 b c'2

and

2) abc T TJ s e u Cg s c q s Cg c u½c a m b c a m b c'3

Tq c Cg u s . 12Ž .5a m b c

Rather than using the above operators which havean explicit relativistic form one could consider theoperators which survive in the limit of a static charmquark. The way to do this has been discussed insome detail in the context of QCD sum rule calcula-

w xtions 13,14 . However there is no particular advan-tage to taking this limit here. The operators we usecontain the leading heavy-quark components so thesimulation will decide by itself whether they aredominant. The advantage of using the explicit rela-tivistic forms is that it allows a unified analysis ofhyperfine effects over the whole mass range fromnucleon and D to charmed baryons.

Ž . Ž .The operators such as 8 and 12 propagate bothw xspin 1r2 and spin 3r2 states 7 . At zero momentum

the correlation function with spatial Lorentz indiceshas the general form

1 1C t s d y g g C t q g g C t ,Ž . Ž . Ž .Ž .i j i j i j 3r2 i j 1r23 3

13Ž .

where the subscripts 3r2 and 1r2 denoted the spinŽ .projections. The quantity C t was used to extract3r2

the mass of the spin 3r2 states. However, it wasŽ .found that the spin 1r2 projection C t was too1r2

noisy at large time separations to allow for thedetermination of a mass.

( )R.M. WoloshynrPhysics Letters B 476 2000 309–315312

Hadron correlators were calculated using interpo-lating operators in local form at both source and sinkand also applying a gauge invariant smearing to thequark propagators at the sink. The Gaussian smear-

Ž . w xing function, Eq. 13 of 15 , was used. Hadronmasses were obtained by a simultaneous fit to localand sink-smeared correlators.

3. Results

The calculations were carried out at bs2.1 onlattices with a bare aspect ratio j of 2. The Landaulink tadpole factors were first determined iterativelyto be u s0.7858 and u s0.9472 and these valuess t

were used in all subsequent calculations. The staticpotential was determined from both spatial and spa-tial-temporal Wilson loops. From this the latticespacing and renormalized anisotropy were obtained.

y1 ŽThe results for the lattice spacings are a s 0.977s. y1 Ž ."0.003 GeV and a s 1.914"0.017 GeV witht

a systematic uncertainty of 0.01 GeV coming fromuncertainty in the choice of parametrization of the

w xshort distance part of the potential 16 . The renor-malized anisotropy was found to be 1.95"0.02which is compatible with other studies done withimproved gluon actions at similar lattice spacingsw x17 .

Fermion propagators were calculated on a 103 =

30 lattice with Dirichlet boundary conditions on thefermion fields. A total of 420 configurations wereanalyzed. With some preliminary tuning it was foundthat ks0.182 and ks0.237 gave good values forthe Jrc and f meson masses so these were the k

values adopted for the charm and strange quarks forall calculations. Where necessary, masses were ex-trapolated to the physical up and down quark regionusing results from the set of hopping parametervalues 0.229,0.233,0.237 and 0.241. The value of the

Ž .critical k is 0.2429 2 .Ž .First consider the light quark u,d,s sector. The

pion and r-meson masses were fixed at 0.140 GeVand 0.770 GeV which determines the hopping pa-

Žrameter for up and down quarks taken to be degen-.erate and the lattice scale a . It was found thatr

y1 Ž .a s 1.99"0.12 GeV which is slightly largerr

than ay1 found from the static potential. This is ant

inevitable result of the quenched approximation. The

r-meson mass scale was used in all subsequentcalculations. The results in the light quark sector aregiven in Table 2 with statistical errors obtained by abootstrap procedure. The dominant systematic error,a 6% uncertainty in the scale determination, is notshown explicitly in this and subsequent tables butshould be kept in mind. For comparison, results from

Ž w xrecent calculations Table II in 5 and Table XVII inw x.6 done at small lattice spacing and extrapolated tothe continuum are also shown. The results of ourcoarse lattice simulation are seen to be quite reason-able.

In Table 2 the L mass calculated with the opera-Ž . Ž .tor 9 is given. The mass obtained using 10 was

essentially identical. This was found to be true for allquark masses.

As check on how well charmed quarks are beingsimulated we first show the results for charmoniumand D-mesons in Table 3. The hyperfine splitting

Ž .Ž .between h and Jrc is 78 2 5 MeV which is incw xvery good agreement with the results obtained in 19

using a completely different fermion action. It isconsiderably smaller than the experimental valuewhich is a well known feature of quenched QCD

w x )simulations for quarkonium 20 . The D yD split-tings are compatible with results from NRQCD on

w xsimilar lattices 21 and are also smaller than experi-mental values although the suppression of hyperfineeffects is less pronounced than in charmonium.

Table 2Hadron masses for light quarks. Masses are given in GeV, massdifferences are in MeV. The experimental values in this and other

w xtables are from 18

w x w xThis work CP-PACS 5 UKQCD 6 Experiment

Ž . Ž .K 0.485 6 0.553 10 0.498) q81Ž . Ž .K 0.902 26 0.889 3 0.748 0.896y4 6

Ž . Ž .N 0.942 67 0.878 25 0.940q1 9Ž . Ž .D 1.358 71 1.257 35 1.25 1.232y9

q20Ž . Ž .L 1.105 46 1.060 13 1.088 1.116y1 9q22Ž . Ž .S 1.184 35 1.176 11 1.091 1.193y1 1

) q15Ž . Ž .S 1.488 57 1.388 24 1.38 1.384y7q43Ž . Ž .J 1.283 28 1.288 8 1.242 1.318y2 4

) q11Ž . Ž .J 1.590 39 1.517 16 1.51 1.534y5) Ž .K y K 417 27 398

Ž .Dy N 416 93 292) Ž .S y S 304 64 191) Ž .J y J 307 43 216

( )R.M. WoloshynrPhysics Letters B 476 2000 309–315 313

Table 3Masses for mesons with charm quarks. Masses are given in GeV,mass differences are in MeV

This work Experiment

Ž .h 3.012 4 2.980cŽ .Jrc 3.087 4 3.097Ž .D 1.875 6 1.867

) Ž .D 2.007 9 2.008Ž .D 1.965 4 1.969s

) Ž .D 2.086 6 2.112sŽ .Jrc yh 78 2 117c

) Ž .D y D 136 7 141) Ž .D y D 122 4 143s s

As mentioned in Section 2 the interpolating opera-tors used for charmed baryons are taken to have thesame form as those used for the light baryons. Forexample, the correlator for S is calculated directlyc

using operators which are the same as used for thestrange S hyperon except the mass is increased tocharm. An alternative is to extract the masses of

Ž .S-like baryons e.g., S ,V from the spin-1r2 pro-c cŽjection of a correlation function of spin 3r2 fields see

Ž ..13 . Fig. 1 illustrates why this alternative is not

Table 4Masses of singly charmed baryons. Masses are given in GeV,mass differences are in MeV. The experimental values are taken

w x X w xfrom 18 except for J which is from 22c

w xThis work UKQCD 1 Experimentq4 q3Ž . Ž .L 2.304 39 2.27 2.285 1c y3 y3q7q5Ž . Ž .S 2.465 22 2.46 2.453 1c y3 y5

) q6q4Ž . Ž .S 2.557 30 2.44 2.518 2c y4 y5q3q4Ž . Ž .J 2.454 21 2.41 2.468 2c y3 y4

X q6q6Ž . Ž .J 2.579 14 2.51 2.575 3c y3 y6) q5q6Ž . Ž .J 2.672 16 2.55 2.645 2c y4 y5

q5q5Ž . Ž .V 2.664 12 2.68 2.704 4c y4 y6) q5q6Ž .V 2.757 14 2.66c y3 y7) q12q3Ž . Ž .S y S 91 25 y17 65 2c c y31 y2

X) q12q2Ž . Ž .J y J 94 13 y20 70 4c c y24 y3) q6q3Ž .V y V 94 10 y23c c y14 y2

used here. The correlation functions for V and thec

spin-3r2 projected V ) are plotted as a function ofc

lattice time. Also shown is the spin-1r2 projectedcorrelator. This spin-1r2 projected piece has a veryfast pre-asymptotic falloff and is therefore small andnoisy in the time region in which one would want to

Ž . ) Ž Ž . Ž ..Fig. 1. Correlation function for the V field squares and the V field spin 3r2 projection triangles , spin 1r2 projection circles as ac c

function of lattice time.

( )R.M. WoloshynrPhysics Letters B 476 2000 309–315314

Table 5Masses of doubly charmed baryons in GeV

) )J J V Vcc cc cc cc

Ž . Ž . Ž . Ž .this work 3.598 13 3.682 20 3.697 10 3.775 12w xpotential model 23 3.478 3.61

w x Ž . Ž . Ž . Ž .mass formulae 24 3.610 7 3.735 17 3.804 8 3.850 25w xbag model 25 3.511 3.630 3.664 3.764

w xHQET 26 3.610 3.680 3.710 3.760w xskyrmion model 27 3.752 3.793 3.934 3.953

extract the mass. This shows that the overlap of thespin-1r2 projection is small. It is also worth notingfrom Fig. 1 that even without any analysis one seesthat the V correlator has a less rapid falloff thanc

that of V ) i.e., V is less massive than V ).c c c

The results for singly charmed baryons are givenin Table 4. Overall the results are in reasonable closeto the experimental values where they are known.

The masses of doubly charmed baryons were alsocalculated. There are no experimental data but acomparison with a selection of model calculations is

Žgiven in Table 5 A more complete tabulation fromw x .various models may be found in 24 . As might

expected, without experimental constraints, the modelcalculations vary over a considerable range. Themass splittings from our quenched QCD simulation,Ž .Ž . ) Ž .Ž .84 13 5 MeV for J yJ and 78 7 5 MeV forcc cc

V ) yV , are substantial and lie in the middle ofcc cc

the range covered by the models listed in Table 5.They are only slightly smaller than the hyperfinesplittings found for singly charmed baryons.

4. Summary

Hadron masses were calculated with an improvedaction on an anisotropic lattice with a spatial latticespacing of about 0.2 fm. Comparison with simula-tions done at small lattice spacings and extrapolatedto the continuum indicate that lattice spacing errorsare less than 10%.

The focus of this study is charmed baryons. Forboth singly and doubly charmed baryons spin split-tings were found to be in agreement with expecta-tions of quark models and other phenomenologicalapproaches. The splittings for singly charmed baryonsare somewhat larger than experimental values. This

is in contrast to the small hyperfine effects for singlycharmed baryons reported by the UKQCD collabora-

w xtion 1 .The calculations reported here were done on a

small lattice at a relatively large lattice spacing. Bydoing a unified study for light and heavy quarkmasses and comparing to continuum results wherepossible we have some confidence that the correctqualitative pattern of hyperfine effects in charmedbaryons has been established for quenched latticeQCD. For a precise calculation, finite volume andlattice spacing issues have to be addressed. It ishoped that this can be done in the near future.

Acknowledgements

It is a pleasure to thank H.R. Fiebig, D.B. Lein-weber, R. Lewis, N.H. Shakespeare and H.D. Trot-tier for help and discussion and B.K. Jennings for theuse of his computers. This work is supported in partby the Natural Sciences and Engineering ResearchCouncil of Canada.

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