Bi-local baryon interpolating fields with two flavours

0.19 MB·English

by  V. Dmitrasinovic

EPJ manuscript No.
(will be inserted by the editor)
Bi-local baryon interpolating fields with two flavours
V. Dmitraˇsinovi´c1 and Hua-Xing Chen2,3
1 Instituteof Physics, Belgrade University,Pregrevica 118, Zemun,P.O.Box 57, 11080 Beograd, Serbia,
2 DepartamentodeF´ısicaTe´oricaandIFIC,CentroMixtoUniversidaddeValencia-CSIC,InstitutosdeInvestigacio´ndePaterna,
Apartado 22085, 46071 Valencia, Spain,
1 3 Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871,
1 China
0
2
Received: date/ Revised version: date

n
a
Abstract. We construct bi-local interpolating field operators for baryons consisting of three quarks with
J
two flavors, assuming good isospin symmetry. We use the restrictions following from the Pauli principle
1
toderiverelations/identities amongthebaryonoperators withidenticalquantumnumbers.Suchrelations
3
that follow from the combined spatial, Dirac, color, and isospin Fierz transformations may be called the

(total/complete)Fierzidentities.Theserelationsreducethenumberofindependentbaryonoperatorswith
]
h any given spin and isospin. We also study the Abelian and non-Abelian chiral transformation properties
p of these fields and place them into baryon chiral multiplets. Thus we derive the independent baryon
- interpolating fields with given values of spin (Lorentz group representation), chiral symmetry (UL(2)×
p
e UR(2) group representation) and isospin appropriate for thefirst angular excited states of the nucleon.
h
[ PACS. 11.30.Rd Chiral symmetries– 12.38.-t Quantumchromodynamics –14.20.Gk Baryon resonances

1
v 1 Introduction
6
0
9 QCD is at present our best theoreticalframework for the description of hadrons and the chiralsymmetry is one of its
5 globalsymmetries that plays a key role in hadronphysics.Interpolating fields ofhadrons andof baryonsin particular
. havebeenpartandparceloflatticeQCDandQCDsumrulecalculationsforalmostthreedecades.Manysuchstudies
1
suggest that the minimal local structure, i.e. three quark fields without derivatives, baryon operators successfully
0
1 describe properties of the lowest-lying baryon ground state(s). Moving beyond the ground states to describe even the
1 lowest-lying excited states turns out to be a challenge in the local operator approximation, however. Interpolators B
: for baryons with spin largerthan 3/2 consisting of three quarks cannot be local operatorsin the continuum limit, see
v
Ref. [1].
i
X Indeed, for baryons with total angular momenta larger than 3/2, the “orbital” angular momentum contribution
r mustbenon-zero,thatcanonlybeintroducedbymeansofanadditionalfour-vector,seeRef.[1].Withlocaloperators,
a
there is only one such four-vector: the four-derivative, or equivalently the four-momentum of the baryon. Manifestly,
anapplicationofthefour-momentumoperatortothebaryonfieldcannotchangeitsangularmomentum.Onetherefore
needs another four-vector to “excite” the orbital angular momentum, and the two independent separations between
the quarks (the two Jacobi relative coordinate four-vectors)are precisely what one needs. In plain English, one needs
to have at least two quark fields at two different locations, that leads to a non-local baryon field.
In this paper we address the question of bi-local baryon fields in the continuum limit, as the first and simplest
extension beyond the local approximation. The properties of interpolating fields, such as their Fierz identities [2,3,4]
and chiral properties [5,6,7] have been explored in detail only in the lower spin and local operator limit, whereas the
study of higher spin fields has only recently begun, and that exclusively on the discrete space-time lattice(s), see e.g.
Ref. [8]; the higher spin fields in the continuum space-time have not been dealt with, as yet, to our knowledge.
Weshallconstructbi-localbaryoninterpolatingfieldsinsuchawaythattheybelongtoreduciblerepresentationsof
theLorentzgroupdescribedbytwo(half-)integers(p,q)andtoirreduciblerepresentationsofthe isospinSU(2)group,
describedbytheisospin,a(half-)integerI,wherethequarkfieldisexpressedastheiso-doubletfield.Itwasnotapriori
obvious,however,thattheyalsobelongtothesame(irreducible)representationsofthechiralgroupSU(2) SU(2) ,
R L
×
where I label the representations of the right-, and left- isospin groups SU(2) . The “ordinary” (vector) isospin
R,L R,L
I is the quantummechanicalsumofthe right-andleft- isospins:I = I I ,...,I +I .We haveshowninRef. [9,
L R L R
| − |
2 V.Dmitraˇsinovi´c, Hua-XingChen: Bi-local baryon interpolating fields with two flavours
10,11] that the Fierz identities among local baryonoperators also determine their chiralmultiplet structure and shall
show here that the same holds for bi-local fields.
This should not be surprising as the Fierz identities form an implementation of the Pauli principle, and different
permutationsymmetryclassesformdistinctmultipletsofcompositeparticles.Henceitisnecessarytocarefullytakeinto
accounttheFierzidentitiesalsoamongbi-localbaryonoperators.ThetwoJacobirelativecoordinatesformthebasisof
thetwo-dimensionalirreduciblerepresentationofthepermutationgroupS ,whichfactleadstogeneralized/composite
3
Fierz identities and further simplifies the classification of the resulting bi-local fields under the Pauli principle.
The standard isospin formalism greatly facilitates derivation of the Fierz identities and chiral transformations of
baryonoperators,due tothe factthatboth the quarksandthe nucleonsbelongto the iso-doubletrepresentation.The
compositeFierzidentities (i.e.inthe spatial,Dirac,isospinandcolorspace)andthechiraltransformationsofbaryons
are then straightforwardlyderived using the iso-doublet representation.
Wegiveanexplicitderivationoftheseidentitiesfortworeasons:a)thisisthefirstsuchderivation,toourknowledge;
and b) because of its relative simplicity, we hope that it will show the way to the chiral SU(3) SU(3) extension,
R L
×
that is (substantially) more complicated, and encourage others to attack this and the tri-local field problem. This
framework can be applied to other extensions, such as the inclusion of multi-quark configurations, and/or of gluon
fields into the baryon interpolators.
This paper is organizedas follows. In section 2, we firstly define all possible bi-local baryonoperators.We classify
the baryonoperatorsaccordingto the representationsof the Lorentzand the isospingroups.Then we apply the Fierz
transformationtoobtainFierzidentitiesamongthebaryonoperatorsforeachrepresentationoftheLorentzandchiral
isospin group. In section 3, we derive the Abelian and non-Abelian chiral transformations of the baryon operators
as functions of the quarks’ chiral transformation parameters, using the iso-doublet representation. All possible chiral
multiplets for the bi-localbaryonoperatorsaredisplayedby taking into accountthe Fierz identities. The finalsection
is a summary and an outlook to possible future extensions and applications. In Appendix A we define all possible
quarkbi-linearfieldswithatmostonederivativeandsummarizetheirchiraltransformations.InAppendixBwedefine
the Fierz transformations in the color, flavor and spatial spaces.
2 Baryon Field Operators
We start with some general comments about three-quark baryon interpolating operators. An interpolating operator
B for baryons consisting of three quarks cannot be local in general: Indeed, for baryons with total angular momenta
larger than 3/2, the “orbital” angular momentum contribution must be non-zero, and that can only be described by
meansofadditionalfour-vectoroperators.Withlocaloperators,thereisonlyonesuchfour-vector:thefour-derivative,
or equivalently the four-momentum of the baryon. Manifestly, application of this operator to the baryon field cannot
change its angular momentum. One, therefore needs another four-vector to “excite” the orbital angular momentum,
and the separations between the quarks (two Jacobi relative coordinates) are precisely what one needs.
So, in general one must write1
B(x,y,z) ǫ qT(x)Γ q (y) Γ q (z), (1)
∼ abc a 1 b 2 c
(cid:0) (cid:1)
where q(x)=(u(x), d(x))T is an iso-doubletquark field at locationx, the superscript T represents the transpose and
the indicesa, bandcrepresentthecolor.Herethe antisymmetrictensorincolorspaceǫ ensuresthebaryons’being
abc
color singlets. From now on, we shall omit the color indices always assuming that the system is a color singlet, which
further implies that any pair of quarks (a “diquark”) is in a colour anti-triplet state.
The matrices Γ are tensor products of Dirac and isospin matrices. With a suitable choice of Γ , the baryon
1,2 1,2
operators are defined so that they form an irreducible representation of the Lorentz and isospin groups, as we shall
show in this section.
Note that we use the iso-doublet form for the quark field q, although the explicit expressions in terms of up and
down quarks are usually employed in lattice QCD and QCD sum rule studies. We have shown in Ref. [11] that the
iso-doublet formulation leads to a simple classification of baryons into isospin multiplets and to a straightforward
derivation of Fierz identities and chiral transformations of baryon operators.
As the tri-local fields are substantially more complicated than the bi-local ones, and neither have been considered
in the literature, as yet, we shall proceed with an analysis of the latter.
1 Of course one must include the color-dependent and path-dependent “gauge factors”. We shall drop them henceforth, to
keep thenotation simple.
V.Dmitraˇsinovi´c, Hua-XingChen: Bi-local baryon interpolating fields with two flavours 3
2.1 Bi-local Baryon Fields
A bi-local interpolating operator B(x,y) for baryons consisting of three quarks can be generally written as
B(x,y) qT(x)Γ q(y) Γ q(x)+ qT(x)Γ q(x) Γ q(y)
1 2 3 4
∼
=D(cid:0) i(x,y)Γiq(x(cid:1))+Dj(x,x(cid:0))Γjq(y), (cid:1) (2)
where Di(x,y) are bi-local diquark fields at location x,y, see Appendix A. The Pauli principle relates the two terms
in Eq. (2). Here we shall consider the Pauli principle in two steps.
The first step is to apply the Pauli principle to the first and second quarks, i.e. to the diquarks, as discussed in
Appendix A. Second, an additional constraint comes from the permutation of the second and the third quark, which
corresponds to the usual Fierz transformation.
NotethattheFierztransformationconnectsonlybaryonoperatorsbelongingtothesameLorentzandisospingroup
multiplets. We may, therefore, classify the baryon operators according to their Lorentz and isospin representations
following Chung et al [3]. It has been known that such baryon operators may couple either to the even- or to the
odd-parity states. In the following discussion all the baryon operators will be defined as having even parity. We note,
however,thattwodifferentisospinbaryonoperatorsbelongingtothesamechiralmultipletmayhaveoppositeparities.
Toconstructthebi-localbaryonfields,wefollowthesameapproachweusedbefore,andweclassifythemaccording
to their spin and isospin. It is convenient to introduce a “tilde-transposed” quark field q˜as follows
q˜=qTCγ (iτ ), (3)
5 2
where C =iγ γ is the Dirac field charge conjugation operator, τ is the second isospin Pauli matrix, whose elements
2 0 2
form the antisymmetric tensor in isodoublet space.
2.2 J = 1 and I = 1 fields
2 2
Firstly, we consider the simplest case D(12,0)I=21, where D(21,0) denotes the representation of the Lorentz group and
I = 1 denotes the isospin. There are twenty bi-local nucleon operators of J = 1 and I = 1
2 2 2
N (x,y)=(q˜(x)q(y))q(x),
1
N (x,y)=(q˜(x)γ q(y))γ q(x),
2 5 5
N (x,y)=(q˜(x)γ q(y))γµq(x),
3 µ
N4(x,y)=(q˜(x)γµγ5τiq(y))γµγ5τiq(x),
N (x,y)=(q˜(x)σ τiq(y))σµντiq(x),
5 µν
NN6((xx,,yy))==((qq˜˜((xx))γτiqτ(iyq)()yτ)i)qγ(xτ)iq,(x),
7 5 5
N (x,y)=(q˜(x)γ τiq(y))γµτiq(x),
N98(x,y)=(q˜(x)γµµγ5q(y))γµγ5q(x),
N (x,y)=(q˜(x)σ q(y))σµνq(x),
10 µν
N11(x,y)=(q˜(x)q(x))q(y),
N (x,y)=(q˜(x)γ q(x))γ q(y),
12 5 5
N (x,y)=(q˜(x)γ q(x))γµq(y),
13 µ
N14(x,y)=(q˜(x)γµγ5τiq(x))γµγ5τiq(y),
N (x,y)=(q˜(x)σ τiq(x))σµντiq(y),
15 µν (4)
NN16((xx,,yy))==((qq˜˜((xx))γτiqτ(ixq)()xτ)i)qτ(iyγ)q,(y),
17 5 5
N (x,y)=(q˜(x)γ τiq(x))γµτiq(y),
N1198(x,y)=(q˜(x)γµµγ5q(x))γµγ5q(y),
N (x,y)=(q˜(x)σ q(x))σµνq(y).
20 µν

4 V.Dmitraˇsinovi´c, Hua-XingChen: Bi-local baryon interpolating fields with two flavours
Among them N -N vanish due to the Pauli principle. Using the Fierz identities for the Dirac spin and isospin
16 20
indices we obtain the following identities:
N 1 1 1 1 1 1 1 1 1 1 N
N1 1 1 1−1 12 1 1 1−1 21 N11
 N2   4 4 −2 2 02 4 4 −2 2 02 N12
3 13
− − − − − −
NNNNNNN14567890= 18−−31133162242−31133416222−−−−3006362−−1020212−−−200022211 −−−−−141211442−−−−144124112−−2012012−−−−0306362−−000223362NNNNNNN11111124567890 ,
and
N 1 1 1 1 1 1 1 1 1 1 N
N11 1 1 1−1 12 1 1 1−1 21 N1
N12  4 4 −2 2 02 4 4 −2 2 02  N2 
13 3
− − − − − −
NNNNNNN11111124567890= 81−−31133162242−31133416222−−−−3006362−−1020212−−−200022211 −−−−−141211442−−−−144124112−−2012012−−−−0306362−−000223362NNNNNNN14567890 .
Solving these equations, we obtain the following solutions
N 6 6 6 2 1
6
− − − −
N 6 6 6 2 1
7
− −
 N   24 24 12 4 0 
N8 −8 8 4 −4 0 N1
NNNNN111190123= 81−−8222644−−−−2224864−−2011222−−20424 10410 NNNN2345 . (5)
 14  
N  −72 72 −0 −0 4
 15  − 
   
Therefore, only these five (N ,N ,N ,N ,N ) of the original twenty operators survive the Pauli principle.
1 2 3 4 5
2.3 J = 1 and I = 3 fields
2 2
Next we consider D(12,0)I=23 fields. Baryon operators with I = 23 must contain either the axial-vector or the tensor
diquark, so there are ten bi-local baryon fields with J = 1 and I = 3 left
2 2
∆i(x,y)=(q˜(x)γ γ τjq(y))γµγ Pij q(x),
4 µ 5 5 3/2
(∆i5(x,y)=(q˜(x)σµντjq(y))σµνP3i/j2q(x),
∆i(x,y)=(q˜(x)τjq(y))Pij q(x),
6 3/2
∆i(x,y)=(q˜(x)γ τjq(y))γ Pij q(x),
 7 5 5 3/2
∆i8(x,y)=(q˜(x)γµτjq(y))γµP3i/j2q(x),
∆i14(x,y)=(q˜(x)γµγ5τjq(x))γµγ5P3i/j2q(y),
∆i (x,y)=(q˜(x)σ τjq(x))σµνPij q(y),
( 15 µν 3/2
∆i (x,y)=(q˜(x)τjq(x))Pij q(y), (6)
16 3/2
∆i (x,y)=(q˜(x)γ τjq(x))γ Pij q(y),
 17 5 5 3/2
∆i18(x,y)=(q˜(x)γµτjq(x))γµP3i/j2q(y).

V.Dmitraˇsinovi´c, Hua-XingChen: Bi-local baryon interpolating fields with two flavours 5
Among these operators ∆i -∆i vanishes due to the Pauli principle. Here Pij is the isospin-projection operator for
16 18 3/2
I = 3, which is defined, together with an isospin-projectionoperator Pij for I = 1, as
2 1/2 2
1 1
Pij =δij τiτj, Pij = τiτj. (7)
3/2 − 3 1/2 3
The I = 3 projection operatorsatisfies τiPij =0, which ensures τi∆i =0. Again the Fierz transformationprovides
2 3 4,5
2
the following relations
∆i 1 1 1 1 1 ∆i
∆6i 1 1 1−1 21 ∆1i6
∆7i = 1 4 4−2 2 02 ∆1i7 , (8)
∆8i 4 4−4 −2−2 0 ∆1i8
∆4i  −12 12 −0 −0 2∆1i4
 5  −  15
    
and
∆i 1 1 1 1 1 ∆i
∆1i6 1 1 1−1 21 ∆6i
∆1i7= 1 4 4−2 2 02 ∆7i  . (9)
∆1i8 4 4−4 −2−2 0 ∆8i
∆1i4 −12 12 −0 −0 2∆4i 
 15  −  5
    
Solving these equations, we obtain the following solutions
∆i 2 1
6
∆i 2−1
 ∆7i = 1−4 −0  ∆i4 . (10)
8 4 ∆i
∆i14 −8 0 (cid:18) 5(cid:19)
∆i   0 8
 15  − 
   
Therefore, only two (∆i,∆i) of the ten ∆ operators are independent under the Pauli principle.
4 5
2.4 J = 3 and I = 1 fields
2 2
There are two possible fields/Lorentz group representations with J = 3: 1) the D(1,1) and 2) the D(3,0).
2 2 2
2.4.1 D(1,1) and I = 1
2 2
For J = 3 fields one of the allowed Lorentz representations is D(1,1). In this case, baryonoperators may containthe
2 2
vector and the axial-vector,or the tensor diquark. So we altogether have twelve bi-local baryon fields
N (x,y)=(q˜(x)γ q(y))Γµνγ q(x),
3µ ν 3/2 5
N (x,y)=(q˜(x)γ γ τiq(y))Γµντiq(x),
 4µ ν 5 3/2
N5µ(x,y)=(q˜(x)σαβτiq(y))Γ3µ/α2γβγ5τiq(x),
N (x,y)=(q˜(x)γ τiq(y))Γµνγ τiq(x),
N8µ(x,y)=(q˜(x)γνγ q(y))Γ3µ/ν2q5(x),
 9µ ν 5 3/2
N10µ(x,y)=(q˜(x)σαβq(y))Γ3µ/α2γβγ5q(x),
N13µ(x,y)=(q˜(x)γνq(x))Γ3µ/ν2γ5q(y),
N (x,y)=(q˜(x)γ γ τiq(x))Γµντiq(y),
 14µ ν 5 3/2
N15µ(x,y)=(q˜(x)σαβτiq(x))Γ3µ/α2γβγ5τiq(y),
N (x,y)=(q˜(x)γ τiq(x))Γµνγ τiq(y),
N18µ(x,y)=(q˜(x)γνγ q(x))Γ3µ/ν2q5(y),
 19µ ν 5 3/2
N20µ(x,y)=(q˜(x)σαβq(x))Γ3µ/α2γβγ5q(y).

6 V.Dmitraˇsinovi´c, Hua-XingChen: Bi-local baryon interpolating fields with two flavours
Among them N -N vanishes due to the Pauli principle. Similarly to the isospin projection operators, Γµν is the
18µ 20µ 3/2
spin-projection operator for J = 3 states, which is defined, together with the J = 1 projection operator Γµν, by
2 2 1/2
1 1
Γµν =gµν γµγν, Γµν = γµγν. (11)
3/2 − 4 1/2 4
Owing to this projection operator, the J = 3 baryon operators satisfy the Rarita-Schwinger condition γ Nµ = 0.
2 µ 3,4,5
The Fierz transformation provides the following relations
N 1 1 1 1 1 1 N
3µ 13µ
N 3 1 1 1 3 3 N
4µ 14µ
 N  16−2 0 −2 6−0 N 
5µ = − − 15µ , (12)
 N8µ  43−1−1−1 3 3 N18µ
 N  1 1 1 1 1 1N 
 9µ   − −  19µ
N  2 2 0 2 2 0 N 
 10µ  − −  20µ
    
and
N 1 1 1 1 1 1 N
13µ 3µ
N 3 1 1 1 3 3 N
14µ 4µ
N  16−2 0 −2 6−0  N 
15µ = − − 5µ . (13)
N18µ 43−1−1−1 3 3  N8µ 
N  1 1 1 1 1 1 N 
 19µ  − −  9µ 
N  2 2 0 2 2 0 N 
 20µ  − −  10µ
    
Solving these equations, we obtain the following linear relations
N 3 1 1
8µ
−
N 1 1 1
N9µ  1−2−2 0  N3µ
NN1130µµ= 2−−31 13 11 NN45µµ . (14)
N14µ  6 −2 2 
 15µ  − 
   
Thus, we take (N ,N ,N ) as the independent fields.
3µ 4µ 5µ
2.4.2 D(3,0) and I = 1
2 2
There are four other fields with J = 3 and I = 1 in the D(3,0) Lorentz group representation, i.e. that have two
2 2 2
Lorentz indices
N (x,y) =(q˜(x)σ τiq(y))Γµναβτiq(x), (15)
5µν αβ 3/2
N (x,y) =(q˜(x)σ q(y))Γµναβq(x), (16)
10µν αβ 3/2
N (x,y) =(q˜(x)σ τiq(x))Γµναβτiq(y), (17)
15µν αβ 3/2
N (x,y) =(q˜(x)σ q(x))Γµναβq(y). (18)
20µν αβ 3/2
Among them N vanishes due to the Pauli principle. Here Γµναβ is another J = 3 projection operator defined as
20µν 2
1 1 1
Γµναβ = gµαgνβ gνβγµγα+ gµβγνγα+ σµνσαβ , (19)
− 2 2 6
(cid:18) (cid:19)
The Fierz transformation provides the following relations
N 1 1 1 N
10µν = 20µν , (20)
N5µν 2 3 1 N15µν
(cid:18) (cid:19) (cid:18) − (cid:19)(cid:18) (cid:19)
and
N 1 1 1 N
20µν = 10µν . (21)
N15µν 2 3 1 N5µν
(cid:18) (cid:19) (cid:18) − (cid:19)(cid:18) (cid:19)
V.Dmitraˇsinovi´c, Hua-XingChen: Bi-local baryon interpolating fields with two flavours 7
Solving these equations, we obtain
N = N ,N = 2N . (22)
10µν 5µν 15µν 5µν
− −
Thus, we take N as the independent field.
5µν
We have, therefore, the grand total of four independent bi-local baryon fields (N ,N ,N ,N ) with J = 3
3µ 4µ 5µ 5µν 2
and I = 1.
2
2.5 J = 3 and I = 3 fields
2 2
There are two possible fields/Lorentz group representations with J = 3: 1) the D(1,1) and 2) the D(3,0).
2 2 2
2.5.1 D(1,1) and I = 3
2 2
For D(1,21)I=32, there are six operators
∆i (x,y)=(q˜(x)γ γ τjq(y))ΓµνPij q(x),
4µ ν 5 3/2 3/2 (23)
(∆i5µ(x,y)=(q˜(x)σαβτjq(y))Γ3µ/α2γβγ5P3i/j2q(x),
∆i (x,y)=(q˜(x)γ τjq(y))Γµνγ Pij q(x), (24)
8µ ν 3/2 5 3/2
∆i (x,y)=(q˜(x)γ γ τjq(x))ΓµνPij q(y),
14µ ν 5 3/2 3/2 (25)
∆i (x,y)=(q˜(x)σ τjq(x))Γµαγβγ Pij q(y),
( 15µ αβ 3/2 5 3/2
∆i (x,y)=(q˜(x)γ τiq(x))Γµνγ Pij q(y). (26)
18µ ν 3/2 5 3/2
Among them ∆i vanishes due to the Pauli principle. The Fierz transformation provides the following relations
18µ
∆i 1 1 1 ∆i
8µ 1 18µ
∆i = 1 1 1 ∆i , (27)
 4µ 2 −  14µ
∆i 2 2 0 ∆i
5µ − 15µ
    
and
∆i 1 1 1 ∆i
18µ 1 8µ
∆i = 1 1 1 ∆i . (28)
 14µ 2 −  4µ
∆i 2 2 0 ∆i
15µ − 5µ
    
Solving these equations, we obtain
∆i 1 1
∆i8µ = −0 −1 ∆i4µ , (29)
 14µ  −  ∆i
∆i 2 1 (cid:18) 5µ(cid:19)
15µ − −
   
i.e. there are two independent ∆ fields: ∆i ,∆i .
4µ 5µ
2.5.2 D(3,0) and I = 3
2 2
Finally in the D(32,0)I=23 Lorentz representation, there are only two ∆ operators
∆i (x,y)= (q˜(x)σ τjq(y))ΓµναβPij q(x), (30)
5µν αβ 3/2 3/2
∆i (x,y)= (q˜(x)σ τjq(x))ΓµναβPij q(y).
15µν αβ 3/2 3/2
The Fierz transformation provides the following relation
∆i =∆i . (31)
5µν 15µν
i.e. there is one independent ∆ field: ∆i .
5µν
We have, therefore, the grand total of three independent bi-local ∆ baryon fields, (∆i ,∆i ,∆i ) with J = 3
4µ 5µ 5µν 2
and I = 3.
2
8 V.Dmitraˇsinovi´c, Hua-XingChen: Bi-local baryon interpolating fields with two flavours
3 Chiral Transformations
In this section, we investigate the chiral transformations of bi-local baryon operators. The chiral mixing of baryon
operatorsiscausedbytheirdiquarkcomponents,soitisconvenienttoclassifythebaryonoperatorsaccordingtotheir
diquark chiral multiplets: D ,D (0, 0), Dµ,Dµi (1, 1) and Dµνi (1, 0)+(0, 1). The analysis in the section is
1 2 ∈ 3 4 ∈ 2 2 5 ∈
similar to our previous papers [9,11] about local fields, so we simply list the results.
3.1 J = 1
2
Under the Abelian chiral transformation the rule, we have
δ N =iaγ (N +2N ), (32)
5 1 5 1 2
δ N =iaγ (2N +N ), (33)
5 2 5 1 2
δ N = iaγ N , (34)
5 3 5 3
−
δ N = iaγ N , (35)
5 4 5 4
−
δ N =3aγ N , (36)
5 5 5 5
and
δ ∆i = iaγ ∆i , (37)
5 4 − 5 4
δ ∆i = 3iaγ ∆i . (38)
5 5 5 5
We can diagonalize N and N and obtain
1 2
δ (N +N )=3iaγ (N +N ), (39)
5 1 2 5 1 2
δ (N N )= iaγ (N N ). (40)
5 1 2 5 1 2
− − −
Under the SU(2) chiral transformation the rule, we have
A
a
δ N =ia τγ N , (41)
5 1 · 5 1
δ N =ia τγ N , (42)
5 2 5 2
·
2
a
δ N = ia τγ N ia τγ N 2iγ a ∆ , (43)
5 3 − · 5 3− 3 · 5 4− 5 · 4
1
a
δ N = 2ia τγ N + ia τγ N 2iγ a ∆ , (44)
5 4 − · 5 3 3 · 5 4− 5 · 4
a
δ N =ia τγ N , (45)
5 5 · 5 5
and
2 2
δa∆i = 2iγ ajPij N iγ ajPij N + iτiγ a ∆ ia τγ ∆i , (46)
5 4 − 5 3/2 3− 3 5 3/2 4 3 5 · 4− · 5 4
δa∆i = 2iτiγ a ∆ +3ia τγ ∆i .
5 5 − 5 · 5 · 5 5
WefindthatN ,N and∆i canbereducedtoirreduciblecomponentsbytakingtheantisymmetriclinearcombination
3 4 4
of the two nucleon fields:
a
δ (N N ) =ia τγ (N N ), (47)
5 3− 4 · 5 3− 4
5
a
δ (3N +N ) = iγ a τ(3N +N )+8a ∆ , (48)
5 3 4 − 5 3 · 3 4 · 4
(cid:20) (cid:21)
2 2
δa∆i = iγ ajPij (3N +N ) τia ∆ +a τ∆i . (49)
5 4 − 5 3 3/2 3 4 − 3 · 4 · 4
h i
Thuswe findthat(N N ),(N N ), andN formfour independent(1,0)chiralmultiplets, N + 1N ,∆i form
1± 2 3− 4 5 2 3 3 4 4
one (1,1) chiral multiplet and ∆i forms one (3,0) chiral multiplet.
2 5 2 (cid:0) (cid:1)
V.Dmitraˇsinovi´c, Hua-XingChen: Bi-local baryon interpolating fields with two flavours 9
3.2 J = 3
2
Under the Abelian chiral transformation rule, we have
δ N = iaγ N , (50)
5 3µ 5 3µ
δ N = iaγ N , (51)
5 4µ 5 4µ
δ N = iaγ N , (52)
5 5µ 5 5µ
and
δ ∆i = iaγ ∆i , (53)
5 4µ 5 4µ
δ ∆i = iaγ ∆i . (54)
5 5µ 5 5µ
δ N =3iaγ N , (55)
5 5µν 5 5µν
δ ∆i =3iaγ ∆i . (56)
5 5µν 5 5µν
Under the SU(2) chiral transformation rule, we have
A
2
δaN = ia τγ N + ia τγ N +2iγ a ∆µ, (57)
5 3µ · 5 3µ 3 · 5 4µ 5 · 4
1
δaN = 2ia τγ N ia τγ N +2iγ a ∆µ, (58)
5 4µ · 5 3µ− 3 · 5 4µ 5 · 4
5
a
δ N = iτ aγ N 4iγ a ∆ , (59)
5 5µ 3 · 5 5µ− 5 · 5µ
2 2
δa∆µi = 2iγ ajPijN + iγ ajPijN iτiγ a ∆µ+ia τγ ∆µi, (60)
5 4 5 32 3µ 3 5 23 4µ− 3 5 · 4 · 5 4
4 2
δa∆µi = iγ ajPij Nµ iτiγ a ∆µ+ia τγ ∆µi, (61)
5 5 −3 5 3/2 5 − 3 5 · 5 · 5 5
and
a
δ N =iτ aγ N , (62)
5 5µν · 5 5µν
δa∆i = 2iτiγ a ∆ +3ia τγ ∆i .
5 5µν − 5 · 5µν · 5 5µν
Thus we find that (N N ) form one (1,0) chiral multiplet; N + 1N ,∆i and N ,∆i form two inde-
3µ− 4µ 2 3µ 3 4µ 4µ 5µ 5µ
pendent (1,1) chiral multiplets; N (1,0), ∆i (3,0) are also independent chiral multiplets.
2 5µν ∈ 2 5µν ∈ 2 (cid:0) (cid:1) (cid:0) (cid:1)
4 Summary and Conclusions
We have investigated the chiral multiplets consisting of bi-local three-quark baryon operators, where we took into
account the Pauli principle by way of the Fierz transformation. All spin 1 and 3 baryon operators were classified ac-
2 2
cordingtotheirLorentzandisospingrouprepresentations,wherespinandisospinprojectionoperatorswereemployed
in Tables 1, 2, 3.
We derived the non-trivial relations among various baryon operators due to the Fierz transformations, and thus
found the independent baryonfields.Then we found that(N N ), (N N ), and N formfour independent (1,0)
1± 2 3− 4 5 2
chiral multiplets, whereas N + 1N ,∆i form one (1,1) chiral multiplet and the independent field ∆i also forms
3 3 4 4 2 5
a separate (3,0) chiral multiplet. Thus five nucleons fields, and two ∆s, with J = 1 are independent in the bi-local
2 (cid:0) (cid:1) 2
limit, in stark contrast with the local limit where there are two nucleons and no ∆, see Ref. [11].
IntheJ = 3 sector,the(N N )formanindependent(1,0)chiralmultiplet; N + 1N ,∆i and N ,∆i
2 3µ− 4µ 2 3µ 3 4µ 4µ 5µ 5µ
form two independent (1,1) chiral multiplets; N (1,0) and ∆i (3,0) are also independent chiral multiplets,
2 5µν ∈ 2 5µν ∈ 2 (cid:0) (cid:1) (cid:0) (cid:1)
again in contrast with the local limit where there is only independent nucleon field and two independent ∆’s [11].
This increase of the number of independent fields is in line with our expectations from the non-relativistic quark
model,wherethenumberofPauli-allowedthree-quarkstatesintheLP =1 shellsharplyrisesfromthecorresponding
−
number in the ground state.
10 V.Dmitraˇsinovi´c, Hua-XingChen: Bi-local baryon interpolating fields with two flavours
Table 1. TheAbelian and thenon-Abelianaxial charges (+sign indicates “naive”, -sign “mirror” transformation properties)
and the non-Abelian chiral multiplets of JP = 1, Lorentz representation (1,0) nucleon and ∆ fields. All of the fields are
2 2
independentand Fierz invariant.
gA(0) gA(1) SUL(2)×SUR(2)
N −N −1 +1 (1,0)⊕(0,1)
1 2 2 2
N +N +3 +1 (1,0)⊕(0,1)
1 2 2 2
N −N −1 +1 (1,0)⊕(0,1)
3 4 2 2
N + 1N −1 −5 (1,1)⊕(1,1)
3 3 4 3 2 2
∆ −1 −1 (1,1)⊕(1,1)
4 3 2 2
N +3 +1 (1,0)⊕(0,1)
5 2 2
∆ +3 +1 (3,0)⊕(0,3)
5 2 2
Table2.TheAbelianandthenon-Abelianaxialchargesandthenon-AbelianchiralmultipletsofJP = 3,Lorentzrepresentation
2
(1,1) nucleon and ∆ fields. Allof thefields are independentand Fierz invariant.
2
gA(0) gA(1) SUL(2)×SUR(2)
Nµ−Nµ +1 −1 (0,1)⊕(1,0)
3 4 2 2
Nµ+ 1Nµ +1 +5 (1,1)⊕(1,1)
3 3 4 3 2 2
∆µ +1 +1 (1,1)⊕(1,1)
4 3 2 2
Nµ +1 +5 (1,1)⊕(1,1)
5 3 2 2
∆µ +1 +1 (1,1)⊕(1,1)
5 3 2 2
Table3.TheAbelianandthenon-Abelianaxialchargesandthenon-AbelianchiralmultipletsofJ = 3,Lorentzrepresentation
2
(3,0) nucleon and ∆ fields. Allof thefields are independentand Fierz invariant.
2
UA(1) SUA(2) SUV(2)×SUA(2)
Nµν +3 +1 (1,0)⊕(0,1)
5 2 2
∆µν +3 +1 (3,0)⊕(0,3)
5 2 2
As in the case of local operators, we showed that the Fierz transformation connects only the baryon operators
with identical group-theoreticalproperties, i.e., belonging to the same chiral multiplet. Then we studied chiral trans-
formations of the bi-local baryon operators. We found that baryons with different isospins may mix under the chiral
transformations, i.e., they may belong to the same chiral multiplet. The parity does not play an apparent role in the
chiral properties of the baryon operators at this (non-dynamical) level.
One of potential applications of our results should be in attempts to determine the baryons’ chiral mixing coeffi-
cients/angles, such as Refs. [12,13,7,14]. This is by no means straightforward business, as there is no guarantee that
these angles are observables.In the case of the ground state one is fortunate to have the flavor-singletand octet axial
couplingsasanexternalinputintothemixingformalismthatleadstosatisfactoryfitstobaryon/hyperonmasseswith
reasonable subsequent physical conclusions [7,14].
TheframeworkpresentedhereholdsinstandardapproachestoQCD,suchaslatticeQCDandtheQCDsumrules,
under the proviso that chiral symmetry is observed by the approximation used. There is another (sub-)field where
it ought to make an impact: on the class of fully relativistic approaches, such as those based on the Bethe-Salpeter
equation, to chiral quark models [15,16,17,18,19,20,21].
We have employed the standard isospin formalism instead of the explicit expressions in terms of different flavored
quarks in the flavor components of the baryon fields that are commonplace in this line of work. By using the isospin
formalism, we have been able to derive all Fierz identities and chiral transformations of the baryons systematically.
The extension to SU(3) is not as straightforward as one might have imagined, however, so we leave it for another
occasion.
Acknowledgments
We wish to thank Dr. K. Nagata and and Prof A. Hosaka, for valuable conversations and correspondence about the
Fierz transformationoftriquarkfields and(in)dependence ofthe nucleoninterpolatingfields.One ofus (V.D.) wishes
to thank Profs.H. Tokiand A. Hosaka,for hospitality at RCNP onseveraloccasion,where this workwas startedand
continued. This work was financed by the Serbian Ministry of Science and Technological Development under Grant
No. 141025.