Diquark Representations for Singly Heavy Baryons with Light Staggered Quarks

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by  Steven Gottlieb

Diquark Representations for Singly Heavy Baryons
with Light Staggered Quarks
Steven Gottlieb,∗ Heechang Na,† and Kazuhiro Nagata‡
Department of Physics, Indiana University
Bloomington IN 47405 USA
(Dated: February 1, 2008)
Inthestaggered fermion formulation oflattice QCD,weconstructandclassify thediquarkoper-
atorstobeembeddedinthesinglyheavybaryons,qqQ. Alongthesamemannerasinthestaggered
mesonclassificationsgivenin[10],weestablishthegrouptheoreticalconnectionsbetweencontinuum
and lattice staggered diquarkrepresentations.
PACSnumbers: 11.15.Ha
8
0
0 I. INTRODUCTION Baryon JP Cont.(SU(2)S,SU(2)I)Z(SU(2)S,SU(8)xy,SU(4)z)Z
2 ΛQ 12+ (ll)Q (1A,1A)0 (1A,28A,1)0
n There have been a number of works addressing the ΣΞQ(Q∗) 12+12(+32+)((llsl))QQ ((13AS,,32S)−)01 ((31SA,,386,S4,)1−)10
Ja horeeatvicyablaproyionntsofrfovmietwh.e eIxnpelaritmtiecnetaQlCaDs,wesellvaersatlhceatlhcue-- ΞΩ′QQ((∗∗)) 1212++((3322++))((slss))QQ ((33SS,,12))−−21 (3(3SS,1,,81,04S)−)−12
1 lations have been performed in the quenched regime
3 [1, 2, 3, 4, 5, 6, 7]. As for the inclusion of dynami- TABLE I: Quantum numbers for singly heavy baryons for
cal quarks, two of the authors have been studying the 2+1 flavor symmetry : Z denotes strangeness. The states
] with asterisks represent the spin 3 states. The fourth and
t charmed and bottomed baryon mass spectrum using the 2
a fifthcolumnrepresentthelightdiquarkirrepsforsingletaste
dataof2+1flavorsdynamicalimprovedstaggeredquarks
l and four tastes, respectively.
- [8]. Since the staggeredfermion provides very fast simu-
p
e lations and much less statistical errors compared to the
h other available frameworks, (see, for example, Ref [9]),
[ these studies should more extensively be performed in
lattice QCD are performed.
terms of finer lattices. Accordingly, from the theoretical
2 Letusbeginwithreviewingthediquarkirrepsformass
v point of view, it is desired to establish the group the- degenerate light quarks of single taste, namely physical
7 oretical classification of pairs of light staggered quarks valence quarks. According to the non-relativistic SU(6)
3 (staggered diquarks) in order to extract the desired spin
quark model, the diquarks should belong to the irreps
35 and parity state in the singly heavy baryon analyses. In 21S, the symmetric part of 6⊗6. It has the following
this short report we construct all the possible time-local
. decompositionintoSU(2)S×SU(3)F,thedirectproduct
7 staggered diquarks embedded in singly heavy baryons.
of the spin and flavor,
0 Along the similar manner as in the meson classification
7 [10], we establish the group theoretical connection be-
SU(6)⊃SU(2) ×SU(3)
0 S F
tween lattice and continuum representations w.r.t. spin,
: 21S →(3S,6S)⊕(1A,3A). (1)
v taste and 2+1 flavor symmetries.
i
X Thelabeling indicatesthe dimensionofeachirrepswhile
r the subscripts S and A indicate the symmetric and anti-
a II. STAGGERED DIQUARK symmetricpart, respectively. Inthe caseof2+1flavors,
REPRESENTATIONS IN THE CONTINUUM
SU(3) is decomposed into SU(2) isospin group. We
F I
then have
Asinglyheavybaryonconsistsoftwolightquarks(up,
downorstrange)andoneheavyquark(charmorbottom SU(2) ×SU(3) ⊃SU(2) ×SU(2)
S F S I
(ortop)). Thequantumnumbersofsinglyheavybaryons (3S,6S)→(3S,3S)0⊕(3S,2)−1⊕(3S,1)−2, (2)
are listed in Table I. In this section, we classify the ir-
reps of the staggered diquarks w.r.t. spin, flavor and (1A,3A)→(1A,1A)0⊕(1A,2)−1, (3)
taste symmetry group in the continuum spacetime. We
wheresubscripts0,−1,−2denotethestrangenessassoci-
especially take 2+1 as the flavor symmetry group un-
atedwitheachirrep. Eachofthese irrepshas one-to-one
der which the recent dynamical simulations of staggered
correspondence to the physical diquark state in singly
heavy baryons,as listed in the fourth column of Table I.
As for the light staggered quarks having four tastes
with degenerate mass, the above SU(3) flavor symme-
∗Electronicaddress: [email protected] F
†Electronicaddress: [email protected] try is extended to SU(12)f flavor-taste symmetry [11].
‡Electronicaddress: [email protected] Correspondingly, the staggered diquarks belong to the
2
symmetric irreps of SU(24) which has the following de- successively applied to classifying staggered baryons as
composition, well as mesons [10, 14, 15]. The important symmetries
of staggered fermions we need to look at in our study
SU(24)⊃SU(2)S ×SU(12)f arethe90◦ rotationsR(ρσ),theshifttransformationsSµ,
300S →(3S,78S)⊕(1A,66A). (4) and the space inversion Is. Since the shift operations
S contain taste matrices, pure translations T may be
µ µ
For 2 +1 flavor staggered quarks, the SU(12)f flavor- representedby the squareof Sµ, Tµ =Sµ2. Discrete taste
taste symmetry group is broken to SU(8)xy ×SU(4)z, transformations in Hilbert space are readily defined by
where we follow the notationgivenin [11]. The SU(8)xy −1
Ξ ≡S T 2.TheΞ generate32elementCliffordgroup
denotesthesymmetrygroupfortwolightvalencequarks µ µ µ µ
while SU(4)z the one for a strange valence quark. The which is isomorphic to the discrete subgroup of SU(4)T
decomposition of SU(12)f into SU(8)xy×SU(4)z gives, in the continuum spacetime. The representations of Ξµ,
D (Ξ ), for a given quark number q, obey
q µ
SU(2)S×SU(12)f ⊃SU(2)S×SU(8)xy×SU(4)z
(3S,78S)→(3S,36S,1)0⊕(3S,8,4)−1⊕(3S,1,10S)−2, (5) Dq(Ξµ)Dq(Ξν)=eiπqDq(Ξν)Dq(Ξµ). (14)
(1A,66A)→(1A,28A,1)0⊕(1A,8,4)−1⊕(1A,1,6A)−2. (6)
Sincethespaceinversioncontainsatastetransformation
Weassumethetastesymmetryrestoresinthecontinuum Ξ , the parity should be defined by
4
limit andall the four tastes become equivalent. We then
see that all the irreps except (1A,1,6A)−2 are to be de- P =Ξ4Is. (15)
generate with physical diquarks under this assumption.
We list the staggered irreps for the physical diquarks in Note that the parity is non-locally defined in time di-
the last column of Table I. The strangeness −2 spin sin- rection since Ξ4 is non-local in time. For the purpose
glet diquark state (1A,1,6A)−2 in (6) is obviously pe- of spectroscopy, we are particularly interested in a sym-
culiar to the staggered representation and it does not metrygroupgeneratedbythe transformationswhichare
correspondto any physical state in the continuum limit. local in time and commuting with T4. Such a group is
In order to make contact with the lattice symmetry calledgeometricaltime slicegroup(GTS)whichisgiven
group, we further need to categorize the physical states by
w.r.t. SU(2) × SU(4) , the spin and taste symme-
S T GTS = G(R(kl),Ξ ,I ) (16)
tries, since the lattice rotation group for the staggered m s
fermion is embedded in the diagonal subgroup of space-
wherek,l,m=1∼3[10,14,15]. Thedefining represen-
time and taste rotation [12, 13]. We first decompose the
tationofGTS isgivenbythe staggeredquarkfieldspro-
entire flavor-taste symmetry group SU(12) into the di-
f jected on zero spatial momentum. It is an eight dimen-
rect product of SU(3) and SU(4) ,
F T sional representation denoted as 8. The anti-staggered
quark fields also belong to the representation 8. The
SU(12) ⊃ SU(3) ×SU(4)
f F T
GTS representation of staggered diquark is accordingly
78S → (6S,10S)⊕(3A,6A), (7) expressedby 8×8. The decomposition of 8×8 into the
66A → (6S,6A)⊕(3A,10S). (8) bosonic irreps is given in [10],
and successively into the 2+1 flavors. We then have 8×8=X{1σsσ+3σsσ+3′′σsσ+3′′′′σsσ+6σsσ} (17)
SU(2)S×SU(8)xy×SU(4)z⊃SU(2)S×SU(2)I×SU(4)T σs,σ
Σ(Q∗) :(3S,36S,1)0 →(3S,3S,10S)0⊕(3S,1A,6A)0, (9) where 1, 3, 3′′, 3′′′′ and 6 are representing the bosonic
representations of GTS. The σ and σ are denoting the
Ξ′Q(∗) :(3S,8,4)−1 →(3S,2,10S)−1⊕(3S,2,6A)−1, (10) eigenvalue of I and D(Ξ Ξ Ξs), respectively. In Eq.
s 1 2 3
Ω(Q∗) :(3S,1,10S)−2 →(3S,1,10S)−2, (11) (17),theσs andσ aresummedoverσs =±1andσ =±1
ΛQ :(1A,28A,1)0 →(1A,1A,10S)0⊕(1A,3S,6A)0, (12) respectively.
The irreducibly transforming diquark operators are
ΞQ :(1A,8,4)−1 →(1A,2,10S)−1⊕(1A,2,6A)−1. (13)
listed in Table II and III. As in the meson case, all
The main goal of this report is to construct the lattice the irreps are categorized into four classes from 0 to 3,
staggereddiquarkoperatorscategorizedintothephysical depending on how far the two staggered quarks are dis-
irreps given in the right hand sides of (9)-(13). placed each other. The third column of the tables gives
the operator form of the diquarks. The fourth column
gives the corresponding GTS irreps. The η and ζ de-
µ µ
III. STAGGERED DIQUARK note the sign factors defined by ηµ(x)=(−1)x1+···+xµ−1
REPRESENTATIONS ON THE LATTICE and ζµ(x) = (−1)xµ+1+···+x4, respectively, while ǫ is de-
fined as ǫ(x)=(−1)x1+x2+x3+x4. The Dk represents the
The symmetry group of staggered fermion action on symmetric shift operators defined by D φ(x) = 1[φ(x+
k 2
the Euclidean lattice was first elaborated in [12, 13] and a )+φ(x−a )]. Fornotationalsimplicity,thesumoverx,
k k
3
class No. operator GTS (rσsσ) σ4 ΓS⊗ΓT JP order (SU(2)S,SU(4)T)
0 1 χχ 1++ −+ γγ54⊗⊗γγ54 00−+ p/1E (1A,6A)
2 η4ζ4χχ 1+− + γ4γ5⊗γ4γ5 0+ 1 (1A,6A)
− 1⊗1 0− p/E
3 ηkǫζkχχ 3′′′′+− + γk⊗γk 1+ 1 (3S,10S)
− γlγm⊗γlγm 1− p/E
4 η4ζ4ηkǫζkχχ 3′′′′++ −+ γγkkγγ54⊗⊗γγkkγγ54 11−+ p/1E (3S,10S)
1 5 χηkDkχ 3−+ + γkγ5⊗γ5 1− p/E
6 η4ζ4χηkDkχ 3−− −+ γlγγkmγ4⊗⊗γγ44γ5 11−+ p/1E (3S,10S)
− γk⊗1 1+ 1 (3S,6A)
7 χǫζkDkχ 3′′−− + 1⊗γk 0− p/E
− γ4γ5⊗γlγm 0+ 1 (1A,10S)
8 η4ζ4χǫζkDkχ 3′′−+ + γ4⊗γkγ4 0− p/E
9 ηkǫζkχηlDlχ 6−− −+ γγ5kγ⊗l⊗γkγγk5 10−+ p/1E (1A,6A)
− γm⊗γlγm 1+ 1 (3S,10S)
10 η4ζ4ηkǫζkχηlDlχ 6−+ + γmγ5⊗γkγ4 1− p/E
− γmγ4⊗γkγ5 1+ 1 (3S,6A)
TABLEII:GTS irrep., σ4, ΓS⊗ΓT andcontinuumstates for staggered diquarkoperators uptoclass 1. (k,l,m=1∼3, k6=
l6=m6=k). The summation over x,flavor and color indices are omitted.
class No. operator GTS (rσsσ) σ4 ΓS⊗ΓT JP order (SU(2)S,SU(4)T)
2 11 χηkDk{ηlDlχ} 3++ + γmγ4⊗γ5 1+ 1 (3S,6A)
12 η4ζ4χηkDk{ηlDlχ} 3+− +− γγmmγ⊗5γ⊗4γγ54 11−+ p/1E (3S,6A)
− γkγl⊗1 1− p/E
13 χζkDk{ζlDlχ} 3′′++ −+ γγ45⊗⊗γγmmγγ54 00−+ p/1E (1A,10S)
14 η4ζ4χζkDk{ζlDlχ} 3′′+− + γ4γ5⊗γm 0+ 1 (1A,10S)
− 1⊗γkγl 0− p/E
15 ηmζmχηkDk{ζlDlχ} 6++ + γlγ4⊗γkγ4 1+ 1 (3S,10S)
− γlγ5⊗γkγ5 1− p/E
16 η4ζ4ηmζmχηkDk{ζlDlχ} 6+− −+ γkγγml⊗⊗γγklγm 11−+ p/1E (3S,10S)
3 17 χη1D1{η2D2{η3D3χ}} 1−+ + γ4⊗γ5 0− p/E
18 η4ζ4χη1D1{η2D2{η3D3χ}} 1−− +− 1γ⊗5⊗γ4γγ45 00−+ p/1E (1A,10S)
− γ4γ5⊗1 0+ 1 (1A,6A)
19 ηkǫζkχη1D1{η2D2{η3D3χ}} 3′′′′−− + γlγm⊗γk 1− p/E
− γk⊗γlγm 1+ 1 (3S,10S)
20 η4ζ4ηkǫζkχη1D1{η2D2{η3D3χ}} 3′′′′−+ + γkγ5⊗γkγ4 1− p/E
− γkγ4⊗γkγ5 1+ 1 (3S,6A)
TABLEIII:GTS irrep.,σ4,ΓS⊗ΓT andcontinuumstatesforstaggereddiquarkoperatorsclass2and3. (k,l,m=1∼3, k6=
l6=m6=k). The summation over x,flavor and color indices are omitted.
the colorandflavorindices aresuppressed. Forexample, case, where the operators are given by ψ(Γ ⊗(Γ )T)ψ
S T
χη D χ stands for χa (x,t)η (x)D χb (x,t). As far in the spin-taste basis.
k k Px f1 k k f2
as the lattice symmetry group GTS is concerned, each
diquark operator is formally corresponding to the me-
sonoperatorgivenin [10] throughreplacingthe leftmost IV. CONNECTION BETWEEN LATTICE AND
χ by χ. This is because the staggered quark and anti- CONTINUUM IRREPS
quark belong to the same GTS irrep for each color and
flavor. The σ inthe fifth columndenotes the eigenvalue ConsultingtherelationsbetweenlatticeRF irrepsand
4
of X with which the parity is given by P = σ σ . The continuum spin irreps given in [10] and assuming that
4 s 4
sixth column gives the spin and taste matrices Γ ⊗Γ the groundstates of lattice irreps correspondto the low-
S T
which come into the diquark operators in the spin-taste est possible spin in the continuum limit, one could make
basis, ψT(CΓ ⊗(Γ C−1)T)ψ, where the superscript T anassignmentofspinandparityJP foreachGTS irrep.
S T
denotes transpose andC denotes the chargeconjugation SeetheseventhcolumnofTablesIIandIII. Onealsosee
matrix. ThepresenceofC andC−1ensuresthecovariant that the combinations, CΓ = Cγ ,Cγ γ ,Cγ γ ,Cγ ,
S k k 4 4 5 5
properties under the spin and taste rotations in the con- generate upper × upper products of the Dirac spinors
tinuum limit. Notice that the assignmentof Γ ⊗Γ for in Dirac representation for each taste and then give rise
S T
eachGTSirrepissystematicallydifferentfromthemeson to O(1) contributions, while the combinations, CΓ =
S
4
No. ΣQ(∗):(3S,3S,10S)0 Ξ′Q(∗):(3S,2,10S)−1 ΩQ(∗):(3S,1,10S)−2
3 ηkǫζkll ηkǫζkls+ηkǫζksl ηkǫζkss
4 η4ζ4ηkǫζkll η4ζ4ηkǫζkls+η4ζ4ηkǫζksl η4ζ4ηkǫζkss
5 lηkDkl lηkDks+sηkDkl sηkDks
9 ηkǫζklηlDll ηkǫζklηlDls+ηkǫζksηlDll ηkǫζksηlDls
15 ηmζmlηkDk{ζlDll} ηmζmlηkDk{ζlDls}+ηmζmsηkDk{ζlDll} ηmζmsηkDk{ζlDls}
16 η4ζ4ηmζmlηkDk{ζlDll} η4ζ4ηmζmlηkDk{ζlDls}+η4ζ4ηmζmsηkDk{ζlDll} η4ζ4ηmζmsηkDk{ζlDls}
19 ηkǫζklη1D1{η2D2{η3D3l}}ηkǫζklη1D1{η2D2{η3D3s}}+ηkǫζksη1D1{η2D2{η3D3l}}ηkǫζksη1D1{η2D2{η3D3s}}
No. ΣQ(∗):(3S,1A,6A)0 Ξ′Q(∗):(3S,2,6A)−1
6 η4ζ4l1ηkDkl2−η4ζ4l2ηkDkl1 η4ζ4lηkDks−η4ζ4sηkDkl
10 η4ζ4ηkǫζkl1ηlDll2−η4ζ4ηkǫζkl2ηlDll1 η4ζ4ηkǫζklηlDls−η4ζ4ηkǫζksηlDll
11 l1ηkDk{ηlDll2}−l2ηkDk{ηlDll1} lηkDk{ηlDls}−sηkDk{ηlDll}
12 η4ζ4l1ηkDk{ηlDll2}−η4ζ4l2ηkDk{ηlDll1} η4ζ4lηkDk{ηlDls}−η4ζ4sηkDk{ηlDll}
20 η4ζ4ηkǫζkl1η1D1{η2D2{η3D3l2}}−η4ζ4ηkǫζkl2η1D1{η2D2{η3D3l1}}η4ζ4ηkǫζklη1D1{η2D2{η3D3s}}−η4ζ4ηkǫζksη1D1{η2D2{η3D3l}}
No. ΛQ:(1A,1A,10S)0 ΞQ:(1A,2,10S)−1
7 l1ǫζkDkl2−l2ǫζkDkl1 lǫζkDks−sǫζkDkl
13 l1ζkDk{ζlDll2}−l2ζkDk{ζlDll1} lζkDk{ζlDls}−sζkDk{ζlDll}
14 η4ζ4l1ζkDk{ζlDll2}−η4ζ4l2ζkDk{ζlDll1} η4ζ4lζkDk{ζlDls}−η4ζ4sζkDk{ζlDll}
17 l1η1D1{η2D2{η3D3l2}}−l2η1D1{η2D2{η3D3l1}}lη1D1{η2D2{η3D3s}}−sη1D1{η2D2{η3D3l}}
No. ΛQ:(1A,3S,6A)0 ΞQ:(1A,2,6A)−1
1 ll ls+sl
2 η4ζ4ll η4ζ4ls+η4ζ4sl
8 η4ζ4lǫζkDkl η4ζ4lǫζkDks+η4ζ4sǫζkDkl
18 η4ζ4lη1D1{η2D2{η3D3l}}η4ζ4lη1D1{η2D2{η3D3s}}+η4ζ4sη1D1{η2D2{η3D3l}}
TABLEIV:Latticestaggereddiquarkoperatorscategorizedintothecontinuumirreps(SU(2)S,SU(2)I,SU(4)T)Z : Theland
s denote light and strange quark, respectively. No. indicates the operator numberin Table II and III. The summation over x
and color indices are omitted.
C,Cγ ,Cγ γ ,Cγ γ , induce upper × lower products, manner. Thedecompositionofcontinuumspin,2+1fla-
4 k l k 5
so that they are suppressed by O(p/E) in the non- vorand taste symmetry groupinto the lattice symmetry
relativistic limit. See the second last column of Ta- group is given by,
ble II and III. It is important to notice that only
the positive parity states survive in the non-relativistic SU(2)S ×SU(2)I ×SU(4)T ⊃ SU(2)I ×GTS. (18)
limit, although every GTS irrep contains both parities.
This is in accordance with the nature of physical di- Table IV summarizes all the local-time lattice di-
quarks. As fortheSU(4) irreps,oneseesthatthe com- quark operators categorized into each continuum irrep
T
binations, ΓTC−1 = γkC−1,γ4C−1,γkγlC−1,γkγ4C−1, (SU(2)S,SU(2)I,SU(4)T)Z given in (9)-(13).
are symmetric so that they altogether correspond to
10S of SU(4)T, while the anti-symmetric six combina-
tions Γ C−1 = C−1,γ γ C−1,γ γ C−1,γ C−1, belong Acknowledgments
T k 5 4 5 5
to 6A of SU(4)T. The assignments of non-relativistic
SU(2) ×SU(4) for the lattice irreps are readily given We would like to thank S. Basak, C. Bernard and C.
S T
fortheO(1)operators. Theyarelistedinthelastcolumn DeTar for useful discussions and comments. We thank
of Table II and III. J.Baileyforimportantcomments onthe physicalstates.
The final step is to take into account the 2+ 1 fla- This work has been supported by U.S. Department of
vor symmetry, which could be done in a straightforward Energy, Grant No. FG02-91ER40661.
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