Baryon Tri-local Interpolating Fields

EPJ manuscript No. (will be inserted by the editor)

Baryon Tri-local Interpolating Fields Hua-Xing Chen School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191, China

arXiv:1205.5328v2 [hep-ph] 2 Oct 2012

Received: date / Revised version: date Abstract. We systematically investigate tri-local (non-local) three-quark baryon fields with UL (2) × UR (2) chiral symmetry, according to their Lorentz and isospin (flavor) group representations. We note that they can also be called as “nucleon wave functions” due to this full non-locality. We study their chiral transformation properties and find all the possible chiral multiplets consisting of J = 21 and J = 32 baryon fields. We find that the axial coupling constant |gA | = 35 is only for nucleon fields belonging to the chiral representation ( 12 , 1) ⊕ (1, 21 ), which contains both nucleon and ∆ fields. Moreover, all the nucleon fields belonging to this representation have |gA | = 35 . PACS. 11.30.Rd Chiral symmetries – 12.38.-t Quantum chromodynamics – 14.20.Gk Baryon resonances

1 Introduction

In this paper, We systematically investigate these baryon fields according to their Lorentz and isospin (flavor) group representations. We study their chiral transformation properties and find all possible chiral multiplets consisting of J = 12 and J = 32 baryon fields. Our procedures mostly follow Ref. [11], but the calculations are a bit more complicated. The obtained baryon currents can be used in the lattice QCD and the QCD sum rule studies [14]. We also study axial charges, which are important in understanding both electro-weak and strong interactions [15]. This paper is organized as follows. In Sect. 2 we systematically classify tri-local (non-local) three-quark baryon fields, according to their Lorentz and isospin (flavor) group representations. In Sect. 3 we study the behaviors of these fields under the Abelian U (1)A and non-Abelian SU (2)A chiral transformations. Finally, we summarize the obtained chiral multiplets in Sect. 4 with some discussions.

The group theory has been applied to particle physics since its beginning. Particularly, the SU (2)L ⊗ SU (2)R and SU (3)L ⊗ SU (3)R chiral groups play important roles in low-energy hadron physics, and the relevant physics is called chiral symmetry [1,2,3,4,5,6]. For five decades now, chiral symmetry and its spontaneous breaking have been studied using different methods because of their importance in elucidating the origin of hadron masses [7]. In Refs. [6], S. Weinberg pointed out that the algebraic aspects of chiral symmetry are worth studying. Previous studies found some QCD identities for baryon currents (fields) using an algebraic method [8]. Other scholars systematically investigated three-quark baryon fields with the SU (2)L ⊗ SU (2)R chiral symmetry, and we investigated those with the SU (3)L ⊗ SU (3)R chiral symmetry [9,10]. Until now, most algebraic studies investigate local hadron fields. However, non-local ones should also be studied. They are important in the study of excited hadron states 2 Baryon Fields since (in the continuum limit) the local three-quark baryon fields can not have a spin larger than 3/2 [2]. To extend A non-trivial interpolating field B(x, y, z) coupling to baryons the algebraic studies on chiral symmetry, we systemati- composed of three-quark fields can be generally written as
cally investigated bi-local three-quark baryon fields with aT b c (y) Γ2 qC (z) (1) (x)Γ1 qB B(x, y, z) ∼ ǫabc qA the SU (2)L ⊗ SU (2)R chiral symmetry, where two quark fields are at the same place but the third one is at a dif- where a, b, c denote the color and A, B, C the flavor inferent position [11]. In this paper, we investigate tri-local dices, C = iγ2 γ0 is the charge-conjugation operator. qA (x) = (non-local) three-quark baryon fields, where three quarks (u(x), d(x))T is an iso-doublet quark field at location are all at different positions. Due to this full non-locality x, and the superscript T represents the transpose of the they can also be called as “nucleon wave functions”, which Dirac indices only (the flavor and color indices are not plays an important role in QCD process predictions. For transposed). The antisymmetric tensor in color space ǫabc example, the wave functions of B and D mesons are stud- ensures the baryons are in a color singlet state. Since it ied in order to investigate the mechanism of heavy meson always exists, we shall omit it from now on. The matrices decays [12]. The baryon (nucleon) wave functions are also Γ1,2 are Dirac matrices which describe the Lorentz strucstudied in order to investigate the baryon structure as well ture. With a suitable choice of Γ1,2 and taking a comas their decay properties [13]. bination of indices of A, B and C, the baryon fields are

2

Hua-Xing Chen: Baryon Tri-local Interpolating Fields

 N11    N12  N13   N   14 N  15 N16     N17 N18     N19 N20  N21     N22 N23   N   24 N  25 N 26     N27 N28   N   29 N30

defined so that they form an irreducible representation of the Lorentz and flavor groups. Previous studies investigated on local and bi-local baryon fields [9,11]. The present study investigates a more general case, the tri-local three-quark baryon fields, where the positions of quarks are all different (x 6= y 6= z). We note that due to this full non-locality we can also call them “nucleon wave functions”. The procedures mostly follow our previous reference [11], but a bit more complicated. Basically, the Fierz transformation is used to change the positions of quark fields. Afterwhich, we can apply the Pauli principle to check the antisymmetric properties of every pair of quarks. We classify tri-local three-quark baryon fields according to their spin and isospin, both of which can be either 1/2 or 3/2. To perform the classification and simplify the notations, a “tilde-transposed” quark field q˜ should be introduced: q˜ = q T Cγ5 (iτ2 ) ,

(2)

Moreover, we absorb the flavor indices into quark fields and write the baryon field as   (3) B(x, y, z) ∼ q˜T (x)Γ˜1 q(y) Γ˜2 q(z) .

The matrices Γ˜1,2 also contain some flavor structure. Considering the presence of three different positions (x 6= y 6= z), the possible baryon fields are B(x, y, z), B(y, z, x), B(z, x, y), B(x, z, y), B(z, y, x), and B(y, x, z). However, we can simply change the positions of the first and second quark fields, for example, aT b qA (x)γ5 qB (y)

=

bT a −qB (y)γ5 qA (x) .

2.1 J =

1 2

and I =

1 2

In this subsection, we study the first case D( 12 , 0)I= 21 , where the representation of the Lorentz group is D( 12 , 0) and the isospin is I = 12 . We find 30 tri-local baryon fields of J = 21 and I = 12 :  N1 = (˜ q (x)q(y))q(z) ,    q (x)γ5 q(y))γ5 q(z) ,  N2 = (˜ N3 = (˜ q (x)γµ q(y))γ µ q(z) ,   q (x)γµ γ5 τ i q(y))γ µ γ5 τ i q(z) ,   N4 = (˜ N = (˜ q (x)σµν τ i q(y))σ µν τ i q(z) ,  5 N6 = (˜ q (x)τ i q(y))τ i q(z) ,     N7 = (˜ q (x)γ5 τ i q(y))γ5 τ i q(z) , N8 = (˜ q (x)γµ τ i q(y))γ µ τ i q(z) ,   N = (˜ q (x)γµ γ5 q(y))γ µ γ5 q(z) ,   9 N10 = (˜ q (x)σµν q(y))σ µν q(z) ,

= (˜ q (y)q(z))q(x) , = (˜ q (y)γ5 q(z))γ5 q(x) , = (˜ q (y)γµ q(z))γ µ q(x) , = (˜ q (y)γµ γ5 τ i q(z))γ µ γ5 τ i q(x) , = (˜ q (y)σµν τ i q(z))σ µν τ i q(x) , = (˜ q (y)τ i q(z))τ i q(x) , = (˜ q (y)γ5 τ i q(z))τ i γ5 q(x) , = (˜ q (y)γµ τ i q(z))γ µ τ i q(x) , = (˜ q (y)γµ γ5 q(z))γ µ γ5 q(x) , = (˜ q (y)σµν q(z))σ µν q(x) .

The latter 20 fields N11 · · · N30 are the Fierz transformed fields of the former ten fields N1 · · · N10 . Using the Fierz identities for the Dirac spin and isospin indices, we obtain the following identities: NA = A11 NB , NB = A11 NC , NC = A11 NA .

(5)

where NA , NB , and NC denote the baryon vectors NA = (N1 , N2 , · · · , N10 )T , NB = (N11 , N12 , · · · , N20 )T ,

(4)

Therefore, we only need to consider three of them. We find that B(x, y, z), B(y, z, x), and B(z, x, y) can be related using only one transformation matrix. Thus, we use these three fields in the following subsections.

= (˜ q (z)q(x))q(y) , = (˜ q (z)γ5 q(x))γ5 q(y) , = (˜ q (z)γµ q(x))γ µ q(y) , = (˜ q (z)γµ γ5 τ i q(x))γ µ γ5 τ i q(y) , = (˜ q (z)σµν τ i q(x))σ µν τ i q(y) , = (˜ q (z)τ i q(x))τ i q(y) , = (˜ q (z)γ5 τ i q(x))τ i γ5 q(y) , = (˜ q (z)γµ τ i q(x))γ µ τ i q(y) , = (˜ q (z)γµ γ5 q(x))γ µ γ5 q(y) , = (˜ q (z)σµν q(x))σ µν q(y) ,

(6)

T

NC = (N21 , N22 , · · · , N30 ) , and A11 is a 10 × 10 transformation matrix 

A11

1 1 1 −1  1 1 −1 1   4 −4 −2 −2   −12 12 −6 2 1  36 36 0 0 =  8  −3 −3 −3 −1   −3 −3 3 1  −12 12 6 −2   4 −4 2 2 −12 −12 0 0

1 2 1 2

0 0 2 1 2 1 2

0 0 2

 −1 −1 −1 1 − 21 −1 −1 1 −1 − 21   −4 4 2 2 0   −4 4 −2 6 0   12 12 0 0 6  3  . (7) −1 −1 −1 −3 2   −1 −1 1 3 32  −4 4 2 −6 0   −4 4 −2 −2 0  12 12 0 0 −2

This matrix is non-singular and it satisfies (A11 )3 = 1. The solution is NC = (A11 )−1 NB = (A11 )−2 NA .

(8)

Therefore, we obtain a “trivial” result that all the ten tri-local baryon fields N1 · · · N10 are non-zero as well as complete and independent. Other 20 fields N11 · · · N30 are related to these ten fields through the Fierz transformation.

Hua-Xing Chen: Baryon Tri-local Interpolating Fields

2.2 J =

1 2

and I =

3 2

2.3 J =

In this subsection, we study the case D( 12 , 0)I= 23 . We find 15 bi-local baryon fields of J = 21 and I = 32 : ( ij ∆i4 = (˜ q (x)γµ γ5 τ j q(y))γ µ γ5 P3/2 q(z) , ij i j µν ∆ = (˜ q (x)σµν τ q(y))σ P3/2 q(z) ,  5i ij ∆ = (˜ q (x)τ j q(y))P3/2 q(z) ,   6 ij i j ∆7 = (˜ q (x)γ5 τ q(y))γ5 P3/2 q(z) ,   ∆i = (˜ j µ ij q (x)γµ τ q(y))γ P3/2 q(z) , 8 ( ij i q (z)γµ γ5 τ j q(x))γ µ γ5 P3/2 q(y) , ∆14 = (˜ i j µν ij ∆ = (˜ q (z)σµν τ q(x))σ P3/2 q(y) ,  15 ij i q (z)τ j q(x))P3/2 q(y) ,   ∆16 = (˜ ij ∆i17 = (˜ q (z)γ5 τ j q(x))γ5 P3/2 q(y) ,   ∆i = (˜ j µ ij q (z)γµ τ q(x))γ P3/2 q(y) , 18 ( ij ∆i24 = (˜ q (y)γµ γ5 τ j q(z))γ µ γ5 P3/2 q(x) , ij ∆i25 = (˜ q (y)σµν τ j q(z))σ µν P3/2 q(x) ,  i ij q (y)τ j q(z))P3/2 q(x) ,   ∆26 = (˜ ij i j ∆27 = (˜ q (y)γ5 τ q(z))γ5 P3/2 q(x) ,   ∆i = (˜ j µ ij q (y)γ τ q(z))γ P µ 28 3/2 q(x) .

The latter ten fields ∆14 · · · ∆18 and ∆24 · · · ∆28 are the Fierz transformed fields of the former five fields ∆4 · · · ∆8 . ij is the isospin-projection operator for I = 32 . Here, P3/2 We define it together with the isospin-projection operator ij P1/2 for I = 12 : 1 1 ij ij P3/2 = δ ij − τ i τ j , P1/2 = τ iτ j , 3 3 which satisfy

(9)

τ i P 3ij = 0 .

(10)

2

Using the Fierz identities for the Dirac spin and isospin indices, we obtain the following identities: ∆iA = A13 ∆iB , ∆iB = A13 ∆iC , ∆iC = A13 ∆iA . (11)

=

(∆i26

, ∆i27

, ∆i28

, ∆i24

, ∆i25 )T

(12)

,

The latter 12 fields are the Fierz transformed fields of the former six fields N3µ · · · N5µ and N8µ · · · N10µ . Similar to µν µν the isospin projection operators, we define Γ3/2 and Γ1/2 , which are the spin-projection operators for the J = 32 and J = 21 states: 1 1 µν µν = γ µγ ν . Γ3/2 = g µν − γ µ γ ν , Γ1/2 4 4 They satisfy µν γµ Γ3/2 = 0.

(15)

(16)

NAµ = A31 NBµ , NBµ = A31 NCµ , NCµ = A31 NAµ (17) . where

and A13 is a 5 × 5 transformation matrix   1 1 1 1 − 21 1 1 1 −1 −1 − 2  1   A13 =  4 −4 −2 2 0  . 4  4 −4 2 −2 0  −12 −12 0 0 −2

(13)

∆iC = (A13 )−1 ∆iB = (A13 )−2 ∆iA .

(14)

NAµ = (N3µ , N4µ , N5µ , N8µ , N9µ , N10µ )T , NBµ = (N13µ , N14µ , N15µ , N18µ , N19µ , N20µ )T , (18)

Again, this matrix is non-singular and it satisfies (A13 )3 = 1. The solution is Therefore, five fields are complete and independent.

1 2

Using the Fierz identities for the Dirac spin and isospin indices, we obtain the following identities:

∆iA = (∆i6 , ∆i7 , ∆i8 , ∆i4 , ∆i5 )T , ∆iC

and I =

The three-quark baryon field of J = 23 contains either one free Lorentz index (Nµ ) or two antisymmetric Lorentz indices (Nµν = −Nνµ ). The former case has a Lorentz representation D(1, 12 )I= 12 , and we find 18 tri-local baryon fields  µν q (x)γν q(y))Γ3/2 γ5 q(z) ,   N3µ = (˜ µν i i N4µ = (˜ q (x)γν γ5 τ q(y))Γ3/2 τ q(z) ,  µα β i  N5µ = (˜ q (x)σαβ τ q(y))Γ3/2 γ γ5 τ i q(z) ,  µν  q (x)γν τ i q(y))Γ3/2 γ5 τ i q(z) ,  N8µ = (˜ µν N9µ = (˜ q (x)γν γ5 q(y))Γ3/2 q(z) ,  µα β  N10µ = (˜ q (x)σαβ q(y))Γ3/2 γ γ5 q(z) ,  µν q (z)γν q(x))Γ3/2 γ5 q(y) ,   N13µ = (˜ µν i i N14µ = (˜ q (z)γν γ5 τ q(x))Γ3/2 τ q(y) ,  µα i  N15µ = (˜ q (z)σαβ τ q(x))Γ3/2 γ β γ5 τ i q(y) ,  µν  q (z)γν τ i q(x))Γ3/2 γ5 τ i q(y) ,  N18µ = (˜ µν N19µ = (˜ q (z)γν γ5 q(x))Γ3/2 q(y) ,  µα β  N20µ = (˜ q (z)σαβ q(x))Γ3/2 γ γ5 q(y) ,  µν q (y)γν q(z))Γ3/2 γ5 q(x) ,   N23µ = (˜ µν i i N24µ = (˜ q (y)γν γ5 τ q(z))Γ3/2 τ q(x) ,  µα i  N25µ = (˜ q (y)σαβ τ q(z))Γ3/2 γ β γ5 τ i q(x) ,  µν  γ5 τ i q(x) , q (y)γν τ i q(z))Γ3/2  N28µ = (˜ µν N29µ = (˜ q (y)γν γ5 q(z))Γ3/2 q(x) ,  µα β  N30µ = (˜ q (y)σαβ q(z))Γ3/2 γ γ5 q(x) .

where ∆iB = (∆i16 , ∆i17 , ∆i18 , ∆i14 , ∆i15 )T ,

3 2

3

NCµ = (N23µ , N24µ , N25µ , N28µ , N29µ , N30µ )T , and A31 is a 6 × 6 transformation matrix   1 1 1 −1 −1 −1  3 −1 1 1 −3 3   1  6 2 0 2 6 0  31 A =  . 4  −3 1 1 −1 3 3   −1 −1 1 1 1 −1  −2 2 0 2 −2 0

(19)

4

Hua-Xing Chen: Baryon Tri-local Interpolating Fields

This matrix is non-singular and it satisfies (A31 )3 = 1. The solution is 31 −1

NCµ = (A )

31 −2

NBµ = (A )

NAµ .

representation D(1, 12 )I= 32 , we find nine baryon fields  i µν ij q (x)γν γ5 τ j q(y))Γ3/2 P3/2 q(z) ,   ∆4µ = (˜ µα β ij ∆i5µ = (˜ q (x)σαβ τ j q(y))Γ3/2 γ γ5 P3/2 q(z) ,  µν ij  ∆i = (˜ j q (x)γν τ q(y))Γ3/2 γ5 P3/2 q(z) , 8µ  i µν ij q (z)γν γ5 τ j q(x))Γ3/2 P3/2 q(y) ,   ∆14µ = (˜ µα β ij i j ∆15µ = (˜ q (z)σαβ τ q(x))Γ3/2 γ γ5 P3/2 q(y) , (27)  µν ij  ∆i = (˜ i q (z)γν τ q(x))Γ3/2 γ5 P3/2 q(y) , 18µ  i µν ij q (y)γν γ5 τ j q(z))Γ3/2 P3/2 q(x) ,   ∆24µ = (˜ µα β ij i j ∆25µ = (˜ q (y)σαβ τ q(z))Γ3/2 γ γ5 P3/2 q(x) ,  µν ij  ∆i = (˜ i q (y)γν τ q(z))Γ3/2 γ5 P3/2 q(x) . 28µ

(20)

Therefore, six fields are complete and independent. For two antisymmetric Lorentz indices with the Lorentz representation D( 23 , 0)I= 21 , we find six fields: (

µναβ i N5µν = (˜ q (x)σαβ τ i q(y))Γ3/2 τ q(z) , µναβ N10µν = (˜ q (x)σαβ q(y))Γ3/2 q(z) ,

(

µναβ i N15µν = (˜ q (z)σαβ τ i q(x))Γ3/2 τ q(y) , µναβ N20µν = (˜ q (z)σαβ q(x))Γ3/2 q(y) ,

(

µναβ i N25µν = (˜ q (y)σαβ τ i q(z))Γ3/2 τ q(x) , µναβ N30µν = (˜ q (y)σαβ q(z))Γ3/2 q(x) .

(21)

The latter four fields are the Fierz transformed fields of the former two fields N5µν and N10µν . Here, Γ µναβ is another J = 32 projection operator defined as Γ µναβ

The latter six fields are the Fierz transformed fields of the former three fields ∆i4µ , ∆i5µ , and ∆i8µ . Using the Fierz identities for the Dirac spin and isospin indices, we obtain the following identities: ∆iAµ = A33 ∆iBµ , ∆iBµ = A33 ∆iCµ , ∆iCµ = A33 ∆iAµ .

(28)

  1 νβ µ α 1 µβ ν α 1 µν αβ µα νβ . where = g g − g γ γ + g γ γ + σ σ 2 2 6 (22)

Using the Fierz identities for the Dirac spin and isospin indices, we obtain the following identities:

∆iAµ = (∆i8µ , ∆i4µ , ∆i5µ )T , ∆iBµ = (∆i18µ , ∆i14µ , ∆i15µ )T , ∆iCµ

=

(∆i28µ

, ∆i24µ

, ∆i25µ )T

,

NAµν = B 31 NBµν , NBµν = B 31 NCµν , NCµν = B 31 NAµν . and A33 is a 3 × 3 transformation matrix (23)   1 −1 −1 1 A33 = −  −1 1 −1  . where 2 −2 −2 0 NAµν = (N5µν , N10µν )T ,

NBµν = (N15µν , N20µν )T ,

(24)

NCµν = (N25µν , N30µν )T , and B

31

is a 2 × 2 transformation matrix   1 1 −1 B 31 = 2 −3 −1

This matrix is non-singular and it satisfies (B 31 )3 = 1. The solution is NCµν = (B 31 )−1 NBµν = (B 31 )−2 NAµν .

(26)

Therefore, two fields are complete and independent.

2.4 J =

3 2

and I =

3 2

In this subsection, we also need to consider the case of one Lorentz index (∆iµ ) and two antisymmetric Lorentz indices (∆iµν = −∆iνµ ). For the former one with the Lorentz

(30)

This matrix is non-singular and it satisfies (A33 )3 = 1. The solution is ∆iCµ = (A33 )−1 ∆iBµ = (A33 )−2 ∆iAµ .

(25)

(29)

(31)

Therefore, three fields are complete and independent. Finally, for two antisymmetric Lorentz indices with the Lorentz representation D( 32 , 0)I= 23 , we find three ∆ fields µναβ ij ∆i5µν = (˜ q (x)σαβ τ j q(y))Γ3/2 P3/2 q(z) , µναβ ij ∆i15µν = (˜ q (z)σαβ τ j q(x))Γ3/2 P3/2 q(y) ,

∆i25µν

= (˜ q (y)σαβ τ

j

(32)

µναβ ij q(z))Γ3/2 P3/2 q(x) .

The latter two fields are the Fierz transformed fields of the first one ∆i5µν . Using the Fierz identities for the Dirac spin and isospin indices, we obtain the following identities: ∆i5µν = ∆i15µν = ∆i25µν . Therefore, there is only one field, and itself is a complete basis.

Hua-Xing Chen: Baryon Tri-local Interpolating Fields

5

and the D( 21 , 0)I= 23 baryon fields transform as

3 Chiral Transformations In this section, we study the chiral transformations of trilocal baryon fields. The analysis and notations in this section are similar to our previous paper [11], and the results are listed here. We do this procedure according to their spin in the following subsections.

 a i ij ij  δ5 ∆4 = −2iγ5 aj P3/2 N3 − 32 iγ5 aj P3/2 N4 (38) + 32 iτ i γ5 a · ∆4 − ia · τ γ5 ∆i4 ,  a i δ5 ∆5 = −2iτ i γ5 a · ∆5 + 3ia · τ γ5 ∆i5 ,  ij N7 δ5a ∆i6 = − 34 iτ i γ5 a · ∆7 + 2ia · τ γ5 ∆i7 − 32 iγ5 aj P3/2     2 2 i i j ij  − 3 iτ γ5 a · ∆6 + ia · τ γ5 ∆6 + 3 iγ5 a P3/2 N6 ,     a i ij 3.1 J = 21 δ5 ∆7 = − 34 iτ i γ5 a · ∆6 + 2ia · τ γ5 ∆i6 − 32 iγ5 aj P3/2 N6 ij 2 2 i i j 1  − 3 iτ γ5 a · ∆7 + ia · τ γ5 ∆7 + 3 iγ5 a P3/2 N7 , Under the Abelian U (1)A chiral transformation, the D( 2 , 0)I= 21   ij ij  i a  three-quark baryon fields transform as δ5 ∆8 = −2iγ5 aj P3/2 N9 − 32 iγ5 aj P3/2 N8     2 i i iτ γ a · ∆ − ia · τ γ ∆ . + 5 8 5 δ5 N1 = iaγ5 (N1 + 2N2 ) , 8 3     δ5 N2 = iaγ5 (2N1 + N2 ) , δ5 N3 = −iaγ5 N3 , (33) We find that N3 , N4 , and ∆i4 can be reduced to irreducible   components by taking the following linear combinations:   δ5 N4 = −iaγ5 N4 , δ5 N5 = 3aγ5 N5 ,  δ5a (N3 − N4 ) = ia · τ γ5 (N3 − N4 ) , (39) δ5 N6 = iaγ5 (N6 + 2N7 ) ,   5 (  a  δ5 N7 = iaγ5 (2N6 + N7 ) , δ5 (3N3 + N4 )h= −iγ5 3 a · τ (3N3 + N4 ) + 8a · ∆4 ,  i 2 j ij 2 i a i i δ5 N8 = −iaγ5 N8 , δ ∆ = −iγ a P (3N + N ) − τ a · ∆ + a · τ ∆ . 5 3 4 4 5 4 4 3 3/2 3     δ5 N9 = −iaγ5 N9 , δ5 N10 = 3aγ5 N10 , The same can be done for N6 , N7 , ∆i6 , and ∆i7 : 1 and the D( 2 , 0)I= 32 baryon fields transform as  δ5a (N6 + N7 ) = ia · τ γ5 (N6 + N7 ) , (40) δ5 ∆i4 = −iaγ5 ∆i4 ,  a 5 (34)  δ5 (N6 − N7 ) = − 3 ia · τ γ5 (N6 − N7 ) + 4iγ5 a · (∆6 − ∆7 ) , δ5 ∆i5 = 3iaγ5 ∆i5 , i i  δ a (∆6 − ∆7 ) = 32 iτ i γ5 a · (∆6 − ∆7 ) − ia · τ γ5 (∆6 − ∆7 )  5 ij  δ5 ∆i6 = iaγ5 (∆i6 + 2∆i7 ) , 4 + 3 iγ5 aj P3/2 (N6 − N7 ) , δ5 ∆i7 = iaγ5 (2∆i6 + ∆i7 ) ,  i i δ5 ∆i8 = −iaγ5 ∆i8 . δ5a (∆6 + ∆7 ) = −2iτ i γ5 a · (∆6 − ∆7 ) + 3ia · τ γ5 (∆6 − ∆7 ) . We find that N1 and N2 can be reduced to irreducible components by taking the antisymmetric linear combination of the two nucleon fields: δ5 (N1 + N2 ) = 3iaγ5 (N1 + N2 ) , δ5 (N1 − N2 ) = −iaγ5 (N1 − N2 ) .

(35)

The same can be done for N6 , N7 , ∆i6 , and ∆i7 : δ5 (N6 + N7 ) = 3iaγ5 (N6 + N7 ) , δ5 (N6 − N7 ) = −iaγ5 (N6 − N7 ) , δ5 (∆i6 δ5 (∆i6

+ −

∆i7 ) ∆i7 )

= =

(36)

3iaγ5 (∆i6 + ∆i7 ) , −iaγ5 (∆i6 − ∆i7 ) .

Under the non-Abelian SU (2)A chiral transformation, the D( 12 , 0)I= 12 baryon fields transform as  a δ5 N1 = ia · τ γ5 N1 ,    δ5 N2 = ia · τ γ5 N2 ,  δ5a N3 = −ia · τ γ5 N3 − 23 ia · τ γ5 N4 − 2iγ5 a · ∆4 ,  1 a    δ5a N4 = −2ia · τ γ5 N3 + 3 ia · τ γ5 N4 − 2iγ5 a · ∆4 , δ5 N5 = ia · τ γ5 N5 ,  a δ5 N6 = − 13 ia · τ γ5 N6 + 34 ia · τ γ5 N7     + 2iγ5 a · ∆6 − 2iγ5 a · ∆7 ,     δ5a N7 = − 31 ia · τ γ5 N7 + 34 ia · τ γ5 N6 + 2iγ5 a · ∆7 − 2iγ5 a · ∆6 , (37)    δ5a N8 = 31 ia · τ γ5 N8 − 2ia · τ γ5 N9 − 2iγ5 a · ∆8 ,   2   δ5a N9 = − 3 ia · τ γ5 N8 − ia · τ γ5 N9 − 2iγ5 a · ∆8 ,  a δ5 N10 = ia · τ γ5 N10 ,

The same is also true for N8 , N9 , and ∆i8 :

δ5a (N8 − N9 ) = ia · τ γ5 (N8 − N9 ) , (41)  a 5  δ5 (N8 + 3N9 ) = − 3 ia · τ γ5 (N8 + 3N9 ) − 8iγ5 a · ∆8 , ij δ a ∆i = − 32 iγ5 aj P3/2 (N8 + 3N9 ) + 32 iτ i γ5 a · ∆8  5 8 − ia · τ γ5 ∆i8 . 3.2 J =

3 2

Under the Abelian U (1)A chiral transformation, the D(1, 12 )I= 12 three-quark baryon fields transform as   δ5 N3µ = iaγ5 N3µ , δ5 N4µ = iaγ5 N4µ ,  δ N = iaγ N , 5 5µ 5 5µ   δ5 N8µ = iaγ5 N8µ , δ5 N9µ = iaγ5 N9µ , (42) δ N 5 10µ = iaγ5 N10µ . The D(1, 12 )I= 32 baryon fields transform as   δ5 ∆i4µ = iaγ5 ∆i4µ , δ5 ∆i5µ = iaγ5 ∆i5µ ,  δ5 ∆i8µ = iaγ5 ∆i8µ .

(43)

6

Hua-Xing Chen: Baryon Tri-local Interpolating Fields

The D( 32 , 0)I= 12 and D( 32 , 0)I= 32 baryon fields transform as   δ5 N5µν = 3iaγ5 N3µν , δ5 N10µν = 3iaγ5 N10µν , (44)  δ ∆i = 3iaγ ∆i . 5 5µν 5 5µν

4 Conclusions and Summary

In this section, we summarize the chiral multiplets found in the previous section. For the J = 21 baryon fields, (N1 ± N2 ), (N3 − N4 ), N5 , (N6 + N7 ), (N8 − N9 ), and N10 form seven [( 12 , 0) ⊕ (0, 21 )] chiral multiplets; (N3 + 1 i i i i 3 N4 , ∆4 ), (N6 − N7 , ∆6 − ∆7 ), and (N8 + 3N9 , ∆8 ) form 1 1 1 i Under the SU (2)A chiral transformation, the D(1, 2 )I= 12 three [(1, 2 ) ⊕ ( 2 , 1)] chiral multiplets; ∆5 and (∆i6 + baryon fields transform as ∆i7 ) form two [( 32 , 0) ⊕ (0, 23 )] chiral multiplets. For the J = 32 baryon fields, (N3µ − N4µ ), (N8µ − N9µ ), N10µ ,  a µ 2  δ5 N3µ = ia · τ γ5 N3µ + 3 ia · τ γ5 N4µ + 2iγ5 a · ∆4 , N5µν , and N10µν form five [( 21 , 0)⊕(0, 12 )] chiral multiplets, δ5a N4µ = 2ia · τ γ5 N3µ − 31 ia · τ γ5 N4µ + 2iγ5 a · ∆µ4 , (N3µ + 31 N4µ , ∆i4µ ), (N5µ , ∆i5µ ), and (N8µ + 3N9µ , ∆i8µ )  a δ5 N5µ = 35 ia · τ γ5 N5µ − 4iγ5 a · ∆5µ , form three [(1, 12 ) ⊕ ( 12 , 1)] chiral multiplets; ∆i5µν forms  1 a  δ5 N8µ = − 3 ia · τ γ5 N8µ + 2ia · τ γ5 N9µ + 2iγ5 a · ∆8µ , one [( 3 , 0) ⊕ (0, 3 )] chiral multiplet. 2 2 δ a N = 32 ia · τ γ5 N8µ + ia · τ γ5 N9µ + 2iγ5 a · ∆8µ , We find a total of 31 independent chiral multiplets.  5a 9µ δ5 N10µ = −ia · τ γ5 N10µ . They are listed in Tables 1, 2, and 3 with their Abelian ax(0) (1) (45)ial charge gA and the non-Abelian axial charge gA . The i SU (2)A chiral transformation of ∆ fields always contain two diagonal terms: τ i a · ∆ and a · τ ∆i . To show “numThe D(1, 21 )I= 23 baryon fields transform as bers” in these tables, we simply add their coefficients and  (1) 2 j ij a µi j ij use brackets to denote this, such as “gA = (− 31 )” for the δ5 ∆4 = 2iγ5 a P 3 N3µ + 3 iγ5 a P 3 N4µ   i  2 2  ∆4 field. We note that some of the fields have already been   − 23 iτ i γ5 a · ∆µ4 + ia · τ γ5 ∆µi  4 , studied in the bi-local case [11].   δ a ∆µi = − 4 iγ aj P ij N µ − 2 iτ i γ a · ∆µ 5 5 5 5 3 5 3/2 5 3 We find that the chiral transformation properties of (46) µi  + ia · τ γ ∆ , every new field, which turns up in the tri-local case, is 5  5   a µi 2 j ij j ij  similar to some old one in the local case. This is also true = 2iγ a P N + iγ a P N δ ∆  5 3 9µ 5 3 8µ 3  2 2  5 8  for the bi-local baryon fields, i.e., the chiral transformation µ µi 2 i − 3 iτ γ5 a · ∆8 + ia · τ γ5 ∆8 . properties of every new field, which turns up in the bilocal case, is similar to some old one in the local case. The D( 23 , 0)I= 12 and D( 32 , 0)I= 32 baryon fields transform We find that the [( 1 , 0) ⊕ (0, 1 )] chiral multiplets (N1 − 2 2 as N2 ), (N3 − N4 ), and (N8 − N9 ) transform in the same 1 1  a way, whereas the [(0, 2 ) ⊕ ( 2 , 0)] chiral multiplets (N3µ −  δ5 N5µν = iτ · aγ5 N5µν , N 4µ ), (N8µ − N9µ ), and N10µν transform like their mirror δ5a N10µν = iτ · aγ5 N10µν , (47) fields; the [( 1 , 0) ⊕ (0, 1 )] chiral multiplets (N + N ), N , 1 2 5  δ a ∆i = −2iτ i γ a · ∆ i 2 2 5 5µν + 3ia · τ γ5 ∆5µν , 5 5µν (N6 + N7 ), N10 , N5µν , and N10µν transform in the same way; the [( 12 , 1) ⊕ (1, 12 )] chiral multiplets (N3 + 31 N4 , ∆i4 ), i We find that N3µ , N4µ , and ∆4µ can be reduced to irre- (N6 − N7 , ∆i − ∆i ), and (N8 + 3N9 , ∆i ) transform in the 8 7 6 ducible components by taking the following linear combi- same way, whereas the [(1, 1 ) ⊕ ( 1 , 1)] chiral multiplets 2 2 nations: (N3µ + 31 N4µ , ∆i4µ ), (N5µ , ∆i5µ ), and (N8µ + 3N9µ , ∆i8µ ) their mirror fields; the [( 32 , 0)⊕ (0, 23 )] chiral δ5a (N3µ − N4µ ) = −ia · τ γ5 (N3µ − N4µ ) , (48)transform like µν i i i  µ multiplets ∆5 , (∆6 + ∆7 ), and ∆5 transform in the same 5 a  ia · τ γ (3N + N ) + 8iγ a · ∆ , δ (3N + N ) = 5 3µ 4µ 5 3µ 4µ 4 way.  5 3 µ 2 i 2 j ij δ5a ∆µi In summary, we have systematically investigated tri4 = 3 iγ5 a P 3 (3N3µ + N4µ ) − 3 iτ γ5 a · ∆4 2   local three-quark baryon fields. Through the Fierz trans. + ia · τ γ5 ∆µi 4 formation, we find that B(x, y, z), B(y, z, x), and B(z, x, y) can be related using some transformation matrices (othThe same can be done for N8µ , N9µ , and ∆i8µ : ers like B(y, x, z) can be also related, see Eq. (4)). The Pauli principle is taken into account also by using the Fierz δ a (N8µ − N9µ ) = −ia · τ γ5 (N8µ − N9µ ) , (49) transformation. However, different from the local and bi5 5 a cases, in the case of tri-local baryon fields, the Pauli  δ (N + 3N ) = ia · τ γ (N + 3N ) + 8iγ a · ∆ , 9µ 5 8µ 9µ 5 8µlocal  5 8µ 3 µ 2 i 2 j ij a µi principle does not forbid any possible structure among the δ5 ∆8 = 3 iγ5 a P 3 (N8µ + 3N9µ ) − 3 iτ γ5 a · ∆8 2  three quark fields inside, whose structure can be carried  µi + ia · τ γ5 ∆8 . out by suitable Lorentz and flavor matrices. We classified all the J = 12 and J = 23 baryon fields and i We note that N5µ and ∆5µ are also related to each other studied their chiral transformation properties, according under the SU (2)A chiral transformation, and the linear to their Lorentz and isospin (flavor) group representations. combinations are not needed. Together with the local and bi-local three-quark baryon

Hua-Xing Chen: Baryon Tri-local Interpolating Fields

fields, which have been studied in Refs. [9,11], we arrive at our final conclusion that the axial coupling constant |gA | = 35 only exists for nucleon fields belonging to the chiral representation ( 21 , 1) ⊕ (1, 12 ) where the nucleon and ∆ fields are both inside. Moreover, all the nucleon fields belonging to this chiral representation have |gA | = 35 . The procedures in this paper can be straightforwardly applied to bi-local/tri-local three-quark baryon fields having SU (3)L ⊗ SU (3)R chiral symmetry, which is our next subject. Table 1. Chiral multiplets of Lorentz representation D( 12 , 0), (0) together with their Abelian and non-Abelian axial charges gA (1) and gA . Baryon Fields N1 − N2 N1 + N2 N3 − N4 N5 N6 + N7 N8 − N9 N10 N3 + 13 N4 ∆i4 N6 − N7 ∆i6 − ∆i7 N8 + 3N9 ∆i8 ∆i5 i ∆6 + ∆i7

(0)

gA −1 +3 −1 +3 +3 −1 +3 −1 −1 −1 −1 −1 −1 +3 +3

(1)

gA +1 +1 +1 +1 +1 +1 +1 − 35 (− 31 ) − 53 (− 31 ) − 35 (− 31 ) (+1) (+1)

SUL (2) × SUR (2) ( 12 , 0) ⊕ (0, 21 ) ( 12 , 0) ⊕ (0, 21 ) ( 12 , 0) ⊕ (0, 21 ) ( 12 , 0) ⊕ (0, 21 ) ( 12 , 0) ⊕ (0, 21 ) ( 12 , 0) ⊕ (0, 21 ) ( 12 , 0) ⊕ (0, 21 ) ( 12 , 1) ⊕ (1, 21 ) ( 12 , 1) ⊕ (1, 21 ) ( 12 , 1) ⊕ (1, 21 ) ( 12 , 1) ⊕ (1, 21 ) ( 12 , 1) ⊕ (1, 21 ) ( 12 , 1) ⊕ (1, 21 ) ( 32 , 0) ⊕ (0, 23 ) ( 32 , 0) ⊕ (0, 23 )

Table 2. Chiral multiplets of Lorentz representation D(1, 21 ), (0) together with their Abelian and non-Abelian axial charges gA (1) and gA . Baryon Fields N3µ − N4µ N8µ − N9µ µ N10 µ N3 + 13 N4µ ∆µi 4 N5µ ∆µ5 µ N8 + 3N9µ ∆µi 8

(0)

gA +1 +1 +1 +1 +1 +1 +1 +1 +1

(1)

gA −1 −1 −1 + 35 (+ 31 ) + 35 (+ 31 ) + 35 (+ 31 )

SUL (2) × SUR (2) (0, 21 ) ⊕ ( 21 , 0) (0, 21 ) ⊕ ( 21 , 0) (0, 21 ) ⊕ ( 21 , 0) (1, 21 ) ⊕ ( 21 , 1) (1, 21 ) ⊕ ( 21 , 1) (1, 21 ) ⊕ ( 21 , 1) (1, 21 ) ⊕ ( 21 , 1) (1, 21 ) ⊕ ( 21 , 1) (1, 21 ) ⊕ ( 21 , 1)

Acknowledgments This work is partly supported by the National Natural Science Foundation of China under Grant No. 11147140,

7

Table 3. Chiral multiplets of Lorentz representation D( 32 , 0), (0) together with their Abelian and non-Abelian axial charges gA (1) and gA . N5µν µν N10 µν ∆5

UA (1) +3 +3 +3

SUA (2) +1 +1 (+1)

SUV (2) × SUA (2) ( 21 , 0) ⊕ (0, 12 ) ( 21 , 0) ⊕ (0, 12 ) ( 23 , 0) ⊕ (0, 32 )

and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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