Staggered Diquarks for the Singly Heavy Baryons
aa r X i v : . [ h e p - l a t ] O c t Staggered Diquarks for Singly Heavy Baryons
Steven Gottlieb
Department of Physics, Indiana University, Bloomington, 47405, IN, U.S.A.E-mail: [email protected]
Heechang Na
Department of Physics, Indiana University, Bloomington, 47405, IN, U.S.A.E-mail: [email protected] x Kazuhiro Nagata ∗ Department of Physics, Indiana University, Bloomington, 47405, IN, U.S.A.E-mail: [email protected]
In the staggered fermion formulation of lattice QCD, we construct diquark operators which areto be embedded in singly heavy baryons. The group theoretical connections between continuumand lattice staggered diquark representations are established.
The XXV International Symposium on Lattice Field TheoryJuly 30 - August 4 2007Regensburg, Germany ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ taggered Diquarks for Singly Heavy Baryons
Kazuhiro Nagata
Baryon J P Z Content ( SU ( ) S , SU ( ) I ) Z ( SU ( ) S , SU ( ) x , y , SU ( ) z ) Z L Q + ( ll ) Q ( A , A ) ( A , A , ) X Q + − ( ls ) Q ( A , ) − ( A , , ) − S ( ∗ ) Q + ( + ) ( ll ) Q ( S , S ) ( S , S , ) X ′ ( ∗ ) Q + ( + ) − ( ls ) Q ( S , ) − ( S , , ) − W ( ∗ ) Q + ( + ) − ( ss ) Q ( S , ) − ( S , , S ) − Table 1:
Quantum numbers for singly heavy baryons for 2 + Z denotes strangeness.The states with asterisks represent the spin states. The fifth and sixth column represent the light diquarkirreps for single taste and four tastes, respectively.
1. Introduction
There have been a number of attempts to investigate heavy baryons in terms of experimentalas well as theoretical methods. In lattice QCD, several calculations have been performed in thequenched regime [1, 2, 3, 4, 5, 6, 7] and given a fair agreement with the experimental results. In[8], two of the authors have reported the results of preliminary study for the singly charmed baryonmass spectrum using the data of 2 + +
2. Staggered Diquarks in the Continuum Spacetime
A singly heavy baryon operator consists of two light quarks (up, down or strange) and oneheavy quark (charm or bottom (or top)). The quantum numbers of singly heavy baryons are listedin Table 1. In this section, we classify the irreps of the staggered diquarks w.r.t. spin, flavor andtaste symmetry group in the continuum spacetime. We especially take 2 + SU ( ) quark model suggests thatthe diquarks should belong to the irreps S , the symmetric part of ⊗ , which has the followingdecomposition into SU ( ) S × SU ( ) F , the direct product of non-relativistic spin and SU ( ) flavor, SU ( ) ⊃ SU ( ) S × SU ( ) F S → ( S , S ) ⊕ ( A , A ) , (2.1)2 taggered Diquarks for Singly Heavy Baryons Kazuhiro Nagata where the labeling represents the dimension of each irreps while the subscripts S and A indicatethe symmetric and anti-symmetric part, respectively. For 2 + SU ( ) F is decomposed into SU ( ) I isospin group. Accordingly, we have SU ( ) S × SU ( ) F ⊃ SU ( ) S × SU ( ) I ( S , S ) → ( S , S ) ⊕ ( S , ) − ⊕ ( S , ) − (2.2) ( A , A ) → ( A , A ) ⊕ ( A , ) − , (2.3)where subscripts 0 , − , − SU ( ) F flavor symmetry is extended to SU ( ) f flavor-taste symmetry [11]. Correspondingly, the stag-gered diquarks belong to the symmetric irreps of SU ( ) which has the following decomposition, SU ( ) ⊃ SU ( ) S × SU ( ) f S → ( S , S ) ⊕ ( A , A ) . (2.4)For 2 + SU ( ) f flavor-taste symmetry group is broken to SU ( ) x , y × SU ( ) z , where SU ( ) x , y denotes the symmetry group for two light valence quarks while SU ( ) z theone for a strange valence quark. The decomposition of SU ( ) f into SU ( ) xy × SU ( ) z gives, SU ( ) S × SU ( ) f ⊃ SU ( ) S × SU ( ) x , y × SU ( ) z ( S , S ) → ( S , S , ) ⊕ ( S , , ) − ⊕ ( S , , S ) − , (2.5) ( A , A ) → ( A , A , ) ⊕ ( A , , ) − ⊕ ( A , , A ) − , (2.6)We assume in the continuum limit that the taste symmetry restores and all the four tastes becomeequivalent. Then we see all the irreps except ( A , , A ) − are to be degenerate with physicaldiquarks under this assumption. We list the staggered irreps for the physical diquarks in the lastcolumn of Table 1. The strangeness − ( A , , A ) − in (2.6) does notcorrespond to any physical state in continuum limit.In order to make contact with the lattice symmetry group, we further decompose the physicalstates into SU ( ) S × SU ( ) I × SU ( ) T as follows, SU ( ) S × SU ( ) x , y × SU ( ) z ⊃ SU ( ) S × SU ( ) I × SU ( ) T S ( ∗ ) Q : ( S , S , ) → ( S , S , S ) ⊕ ( S , A , A ) (2.7) X ′ ( ∗ ) Q : ( S , , ) − → ( S , , S ) − ⊕ ( S , , A ) − (2.8) W ( ∗ ) Q : ( S , , S ) − → ( S , , S ) − (2.9) L Q : ( A , A , ) → ( A , A , S ) ⊕ ( A , S , A ) (2.10) X Q : ( A , , ) − → ( A , , S ) − ⊕ ( A , , A ) − . (2.11)The main goal of this article is to construct the lattice staggered diquark operators categorized intothe physical irreps given in the right hand sides of (2.7)-(2.11).3 taggered Diquarks for Singly Heavy Baryons Kazuhiro Nagata
3. Staggered Diquarks on the Lattice
The symmetry group of staggered fermion action on Euclidean lattice was first elaborated in[15, 16] and successively applied to classifying staggered baryons as well as mesons [12, 13, 14].The important symmetries of staggered fermions in our study are 90 ◦ rotations R ( rs ) , shift trans-formations S m , space inversion I s . Since the shift operations S m contain taste matrices, pure transla-tions T m may be represented by the square of S m , T m = S m . Discrete taste transformations in Hilbertspace are readily defined by X m ≡ S m T − m . The X m generate 32 element Clifford group which isisomorphic to the discrete subgroup of SU ( ) T in the continuum spacetime. Since the space in-version contains a taste transformation X , the parity should be defined by P = X I s . Note that theparity is non-locally defined in time direction since X is non-local in time. For the purpose ofspectroscopy, we are particularly interested in a symmetry group generated by the transformationswhich are local in time and commuting with T . Such a group is called geometrical time slice group( GT S ) which is given by
GT S = G ( R ( kl ) , X m , I s ) , (3.1)where k , l , m = ∼ GT S is given by the staggeredquark fields projected on zero spatial momentum. It is an eight dimensional representation denotedas . The anti-staggered quark fields also belong to the representation . The GT S representation ofstaggered diquark is accordingly expressed by × . The decomposition of × into the bosonicirreps is given in [13], × = (cid:229) s s = ± , s = ± { s s s + s s s + ′′ s s s + ′′′′ s s s + s s s } , (3.2)where , , ′′ , ′′′′ and are representing the bosonic representations of GT S with s s , the eigen-value of I s and s , the eigenvalue of D ( X X X ) .The irreducibly transforming diquark operators are listed in Table 2 and 3. As in the me-son case, all the irreps are categorized into four classes from 0 to 3, depending on how far thetwo staggered quarks are displaced each other. The third column of the tables gives the operatorform of the diquarks. The fourth column gives the corresponding GT S irreps. The h m and z m de-note the sign factors defined by h m ( x ) = ( − ) x + ··· + x m − and z m ( x ) = ( − ) x m + + ··· + x , respectively,while e is defined as e ( x ) = ( − ) x + x + x + x . The D k represents the symmetric shift operatorsdefined by D k f ( x ) = [ f ( x + a k ) + f ( x − a k )] . For notational simplicity, the sum over x , thecolor and flavor indices are suppressed without any confusion. For example, ch k D k c stands for (cid:229) x c af ( x , t ) h k ( x ) D k c bf ( x , t ) . As far as the lattice symmetry group GT S is concerned, each diquarkoperator is formally corresponding to the meson operator given in [13] through replacing the left-most c by c . This is because the staggered quark and anti-quark belong to the same GT S irrep foreach color and flavor. The s in the fifth column denotes the eigenvalue of X with which the parityis given by P = s s s . The sixth column gives the spin and taste matrices G S ⊗ G T which comeinto the diquark operators in the spin-taste basis, y T ( C G S ⊗ ( G T C − ) T ) y , where the superscript T denotes transpose and C denotes the charge conjugation matrix. The presence of C and C − ensuresthe covariant properties under the spin and taste rotations in the continuum limit. Notice that theassignment of G S ⊗ G T for each GT S irrep is systematically different from the meson case, wherethe operators are given by y ( G S ⊗ ( G T ) T ) y in the spin-taste basis.4 taggered Diquarks for Singly Heavy Baryons Kazuhiro Nagata class No. operator
GT S ( r s s s ) s G S ⊗ G T J P order ( SU ( ) S , SU ( ) T ) cc ++ + g ⊗ g + ( A , A ) − g ⊗ g − p / E h z cc + − + g g ⊗ g g + ( A , A ) − ⊗ − p / E h k ez k cc ′′′′ + − + g k ⊗ g k + ( S , S ) − g l g m ⊗ g l g m − p / E h z h k ez k cc ′′′′ ++ + g k g ⊗ g k g + ( S , S ) − g k g ⊗ g k g − p / E ch k D k c − + + g k g ⊗ g − p / E − g k g ⊗ g + ( S , S ) h z ch k D k c −− + g l g m ⊗ g g − p / E − g k ⊗ + ( S , A ) cez k D k c ′′−− + ⊗ g k − p / E − g g ⊗ g l g m + ( A , S ) h z cez k D k c ′′− + + g ⊗ g k g − p / E − g ⊗ g k g + ( A , A ) h k ez k ch l D l c −− + g k g l ⊗ g k − p / E − g m ⊗ g l g m + ( S , S ) h z h k ez k ch l D l c − + + g m g ⊗ g k g − p / E − g m g ⊗ g k g + ( S , A ) Table 2:
GT S irrep., s , G S ⊗ G T and continuum states for staggered diquark operators up to class 1.( k , l , m = ∼ , k = l = m = k ). The summation over x , flavor and color indices are omitted. class No. operator GTS ( r s s s ) s G S ⊗ G T J P order ( SU ( ) S , SU ( ) T ) ch k D k { h l D l c } ++ + g m g ⊗ g + ( S , A ) − g m g ⊗ g − p / E h z ch k D k { h l D l c } + − + g m ⊗ g g + ( S , A ) − g k g l ⊗ − p / E cz k D k { z l D l c } ′′ ++ + g ⊗ g m g + ( A , S ) − g ⊗ g m g − p / E h z cz k D k { z l D l c } ′′ + − + g g ⊗ g m + ( A , S ) − ⊗ g k g l − p / E h m z m ch k D k { z l D l c } ++ + g l g ⊗ g k g + ( S , S ) − g l g ⊗ g k g − p / E h z h m z m ch k D k { z l D l c } + − + g l ⊗ g k + ( S , S ) − g k g m ⊗ g l g m − p / E ch D { h D { h D c }} − + + g ⊗ g − p / E − g ⊗ g + ( A , S ) h z ch D { h D { h D c }} −− + ⊗ g g − p / E − g g ⊗ + ( A , A ) h k ez k ch D { h D { h D c }} ′′′′−− + g l g m ⊗ g k − p / E − g k ⊗ g l g m + ( S , S ) h z h k ez k ch D { h D { h D c }} ′′′′− + + g k g ⊗ g k g − p / E − g k g ⊗ g k g + ( S , A ) Table 3:
GT S irrep., s , G S ⊗ G T and continuum states for staggered diquark operators class 2 and 3.( k , l , m = ∼ , k = l = m = k ). The summation over x , flavor and color indices are omitted. taggered Diquarks for Singly Heavy Baryons Kazuhiro Nagata
4. Connection between lattice and continuum irreps
Consulting the relations between lattice RF irreps and continuum spin irreps given in [13] andassuming that the ground states of lattice irreps correspond to the lowest possible spin in the con-tinuum limit, one could make an assignment of spin and parity J P for each GT S irrep. See the sev-enth column of Tables 2 and 3. One also see that the combinations, C G S = C g k , C g k g , C g g , C g , generate upper × upper products of the Dirac spinors in Dirac representation for each taste andthen give rise to O ( ) contributions, while the combinations, C G S = C , C g , C g k g l , C g k g , generate upper × lower products, so that they are suppressed by O ( p / E ) in the non-relativistic limit. See thesecond last column of Table 2 and 3. An important notice here is that only the positive parity statessurvive in the non-relativistic limit, which is in accordance with the property of physical diquarks.As for the SU ( ) T irreps, one sees that the combinations, G T C − = g k C − , g C − , g k g l C − , g k g C − , are symmetric so that they altogether belong to S irrep of SU ( ) T , while the anti-symmetric sixcombinations G T C − = C − , g k g C − , g g C − , g C − , belong to A irrep of SU ( ) T . The assign-ments of non-relativistic SU ( ) S × SU ( ) T for the lattice irreps are readily given for the O ( ) operators. They are listed in the last column of the tables. The final step is to take into account the2 + + SU ( ) S × SU ( ) I × SU ( ) T ⊃ SU ( ) I × GT S . (4.1)In Table 4, we list all the lattice diquark operators which are local in time and categorize them intoeach continuum irrep ( SU ( ) S , SU ( ) I , SU ( ) T ) Z previously given in (2.7)-(2.11).
5. Summary
Continuum and lattice irreps of staggered diquarks with SU ( ) taste symmetry in 2 + SU ( ) symmetry group which is the SU ( ) taste extensionof ordinary SU ( ) non-relativistic quark model. This procedure has been also taken in the studyof staggered baryon classifications [11]. As for the lattice representations, we have consulted thelattice symmetry group of staggered fermion action elaborated in [15, 16]. Although the irreps oflattice symmetry group GT S cannot have any definite parity, we have explicitly shown that onlythe positive parity state contributes in the non-relativistic limit, which is in accordance with theproperty of physical diquarks.
Acknowledgments
We would like to thank S. Basak, C. Bernard and C. DeTar for useful discussions and com-ments. We thank J. Bailey for important comments on the physical states. This work has beensupported by U.S. Department of Energy, Grant No. FG02-91ER 40661.
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No. S ( ∗ ) Q : ( S , S , S ) X ′ ( ∗ ) Q : ( S , , S ) − W ( ∗ ) Q : ( S , , S ) − h k ez k ll h k ez k ls + h k ez k sl h k ez k ss h z h k ez k ll h z h k ez k ls + h z h k ez k sl h z h k ez k ss l h k D k l l h k D k s + s h k D k l s h k D k s h k ez k l h l D l l h k ez k l h l D l s + h k ez k s h l D l l h k ez k s h l D l s h m z m l h k D k { z l D l l } h m z m l h k D k { z l D l s } + h m z m s h k D k { z l D l l } h m z m s h k D k { z l D l s } h z h m z m l h k D k { z l D l l } h z h m z m l h k D k { z l D l s } + h z h m z m s h k D k { z l D l l } h z h m z m s h k D k { z l D l s } h k ez k l h D { h D { h D l }} h k ez k l h D { h D { h D s }} + h k ez k s h D { h D { h D l }} h k ez k s h D { h D { h D s }} No. S ( ∗ ) Q : ( S , A , A ) X ′ ( ∗ ) Q : ( S , , A ) − h z l h k D k l − h z l h k D k l h z l h k D k s − h z s h k D k l h z h k ez k l h l D l l − h z h k ez k l h l D l l h z h k ez k l h l D l s − h z h k ez k s h l D l l l h k D k { h l D l l } − l h k D k { h l D l l } l h k D k { h l D l s } − s h k D k { h l D l l } h z l h k D k { h l D l l } − h z l h k D k { h l D l l } h z l h k D k { h l D l s } − h z s h k D k { h l D l l } h z h k ez k l h D { h D { h D l }} − h z h k ez k l h D { h D { h D l }} h z h k ez k l h D { h D { h D s }} − h z h k ez k s h D { h D { h D l }} No. L Q : ( A , A , S ) X Q : ( A , , S ) − l ez k D k l − l ez k D k l l ez k D k s − s ez k D k l l z k D k { z l D l l } − l z k D k { z l D l l } l z k D k { z l D l s } − s z k D k { z l D l l } h z l z k D k { z l D l l } − h z l z k D k { z l D l l } h z l z k D k { z l D l s } − h z s z k D k { z l D l l } l h D { h D { h D l }} − l h D { h D { h D l }} l h D { h D { h D s }} − s h D { h D { h D l }} No. L Q : ( A , S , A ) X Q : ( A , , A ) − ll ls + sl h z ll h z ls + h z sl h z l ez k D k l h z l ez k D k s + h z s ez k D k l h z l h D { h D { h D l }} h z l h D { h D { h D s }} + h z s h D { h D { h D l }} Table 4:
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