Isospin Symmetry Breaking and Octet Baryon Masses due to Their

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

Isospin Symmetry Breaking and Octet Baryon Masses due to Their Mixing with Decuplet Baryons

arXiv:1312.1451v2 [hep-ph] 14 Mar 2014

Hua-Xing Chen School of Physics and Nuclear Energy Engineering and International Research Center for Nuclei and Particles in the Cosmos, Beihang University, Beijing 100191, China Received: date / Revised version: date Abstract. We study the isospin symmetry breaking and mass splittings of the eight lowest-lying baryons. We consider three kinds of baryon mass terms, including the bare mass term, the electromagnetic terms and the spontaneous chiral symmetry breaking terms. We include the mixing term between flavor-octet and flavor-decuplet baryons. This assumes that the lowest-lying Σ and Ξ baryons contain a few decuplet components and so are not purely flavor-octet. We achieve a good fitting that the difference between every fitted mass and its experimental value is less than 0.2 MeV. PACS. 14.20.-c Baryons – 11.30.Rd Chiral symmetries – 11.30.Hv Flavor symmetries

1 Introduction Isospin symmetry breaking in mass splittings of the lowest-lying flavor-octet baryons is of fundamental importance [1, 2]. Lots of efforts and many methods have been devoted to study it, such as the chiral perturbation theory (χPT) [3, 4], the chiral soliton model [5,6], the Cottingham’s sum rule [7], the QCD sum rule [8,9,10,11,12], the Dashen’s theorem [13,14,15] and the Skyrme Model [16,17], etc.. Isospin symmetry breaking originates from two different sources: electromagnetic (EM) self-energies and the current quark mass differences. Moreover, due to their influences, ¯ 6= h¯ the quark condensates are also different, i.e., h¯ uui = 6 hddi ssi, and so the SU (3)L ⊗ SU (3)R chiral symmetry and the isospin symmetry are both explicitly and spontaneously broken in a dissymmetric way. In general we can separate baryon masses into hadronic and electromagnetic parts, which are contributed by QCD and electromagnetic effects, respectively. In recent years, lattice QCD is developing very fast and several lattice groups have determined these two contributions [18,19,20,3,21,22,23,24,25,26,27,28,29] (reviewed in Refs. [31,30,32]). In this paper we shall study the isospin symmetry breaking and mass splittings of the eight lowest-lying baryons. We assume that their masses originate from three different sources: the bare mass term, the electromagnetic terms and the spontaneous chiral symmetry breaking terms. Particularly, we shall use group theoretical methods to study the spontaneous chiral symmetry breaking terms [33,34,35,36,37,38,39,40,41,42,43,44,45]. Using this method we have studied local and non-local three-quark baryon fields, and found there are five chiral multiplets [46,47,48,49]: [(3, 1) ⊕ (1, 3)]3 = [(1, 1)] ⊕ [(8, 1) ⊕ (1, 8)] ⊕ [(10, 1) ⊕ (1, 10)] ⊕[(¯ 3, 3) ⊕ (3, ¯ 3)] ⊕ [(6, 3) ⊕ (3, 6)] ,

(1)

as well as their mirror partners. Because the chiral symmetry is broken, these five multiplets do not actually exist. They mix and compose flavor-singlet, flavor-octet and flavor-decuplet baryons, and then compose our real world: [(1, 1)]&[(¯ 3, 3) ⊕ (3, ¯ 3)] → 1F , ¯ ¯ [(8, 1) ⊕ (1, 8)]&[(3, 3) ⊕ (3, 3)]&[(6, 3) ⊕ (3, 6)] → 8F ,

[(10, 1) ⊕ (1, 10)]&[(6, 3) ⊕ (3, 6)] → 10F .

Usually the eight lowest-lying baryons are considered to be “purely” flavor-octet. However, it might be possible that flavor-octet and flavor-decuplet baryons can mix through their common [(6, 3) ⊕ (3, 6)] chiral components, and so they are not pure any more. For example, the lowest-lying Σ(1193) baryons can mix with the higher Σ baryons having the same quantum numbers J P = 1/2+. Several candidates are Σ(1660), Σ(1770) and Σ(1880), which may contain

2

Hua-Xing Chen: Isospin Symmetry Breaking and Octet Baryon Masses due to Their Mixing with Decuplet Baryons

flavor-decuplet components. We note that the [(6, 3) ⊕ (3, 6)] chiral multiplet is important to explain the experimental value of the axial charge gA = 1.267 [50,51]. In this paper we assume the mixing of flavor-octet and flavor-decuplet baryons can contribute to flavor-octet baryon ¯ B obtained in Refs. [52,53], where B ¯ masses. To study this effect, we shall use the chiral invariant Lagrangians BM and B denote baryon fields, and M denotes meson fields. However, these Lagrangians contain too many parameters to be solvable, and so we shall first use them to obtain flavor-singlet Lagrangians and then obtain the spontaneous chiral symmetry breaking mass terms. By assuming this mixing the lowest-lying Σ and Ξ baryons contain a few decuplet components and so are not purely flavor-octet. Although these contributions may be quite small, they can still be important, because masses of the lowest-lying flavor-octet baryons have been measured so accurately nowadays. We note that flavor-singlet and flavor-octet baryons can also mix through their common [(¯ 3, 3)⊕ (3, ¯ 3)] chiral components, which we shall not study in this paper [54]. This paper is organized as follows. In Sec. 2 we transform from chiral-singlet Lagrangians to flavor-singlet ones. In Sec. 3 we introduce the formulae used to calculate the eight lowest-lying baryon masses. We consider three kinds of terms: the bare mass term, the electromagnetic terms and the spontaneous chiral symmetry breaking terms. The spontaneous chiral symmetry breaking terms are discussed in Sec. 3.1 and Sec. 3.2, other terms are discussed in Sec. 3.3, and their summation is shown in Sec. 3.4. In Sec. 4 we use these formulae to perform numerical analyses and do the fitting. Sec. 5 is a summary.

2 From Chiral-Singlet Lagrangians to Flavor-Singlet Lagrangians ¯ B have been obtained in Refs. [52,53], where B ¯ and B denote baryon fields, The chiral invariant Lagrangians BM and M denotes meson fields. However, they contain too many parameters to be solvable. Therefore, in this section we transform from these chiral-singlet Lagrangians to flavor-singlet ones, which will be used to calculate the spontaneous chiral symmetry breaking terms in the next section. In Refs. [46,47,48,49] we use group theoretical methods to study local and non-local three-quark baryon fields, and found there are five chiral multiplets: [(1, 1)], [(8, 1) ⊕ (1, 8)], [(10, 1) ⊕ (1, 10)], [(¯ 3, 3) ⊕ (3, ¯ 3)] and [(6, 3) ⊕ (3, 6)] as well as their mirror partners. We note that only non-local three-quark baryon fields can belong to the [(1, 1)] chiral multiplet [47,48,49]. Because the chiral symmetry is broken, these five multiplets do not actually exist. These chiral multiplets mix and compose the physical flavor-singlet (Λ), flavor-octet (N ) and flavor-decuplet (∆) baryons, and then compose our real world: |Λi = α1 |Λ[(1,1)] i + α2 |Λ[(¯3,3)] i ,

|N i = β1 |N[(8,1)] i + β2 |N[(¯3,3)] i + β3 |N[(6,3)]i ,

(2)

|∆i = γ1 |∆[(10,1)] i + γ2 |∆[(6,3)]i .

We use the [(6, 3) ⊕ (3, 6)] chiral multiplet as an example to show how to obtain flavor-singlet Lagrangians from chiral-singlet ones. The chiral-singlet Lagrangian has been obtained in Ref. [52]: ¯ a (σ c + iγ5 π c )(Dc )ab N b , g(18) N (18) (18) (18)

(3)

where a, b = 1 · · · 18, c = 0 · · · 8, g(18) is the coupling constant and σ c + iγ5 π c are meson fields belonging to the a [(¯ 3, 3) ⊕ (3, ¯ 3)] chiral representation. The baryon field N(18) belongs to the chiral representation [(6, 3) ⊕ (3, 6)]. It contains both flavor-octet and flavor-decuplet baryons: T P T N a = (NµN , ∆P N(18) µ ) = (N[(6,3)] , ∆[(6,3)] ) ,

(4)

where N = 1 · · · 8 and P = 1 · · · 10; ǫabc and SPABC are the totally anti-symmetric tensor and totally symmetric tensor, respectively; NµN and ∆P µ are the octet and decuplet baryons belonging to the [(6, 3) ⊕ (3, 6)] chiral multiplet [46]. The matrices D(18) are: 1 18×8 0 0 , (5) D(18) = √ 0 −2 × 110×10 6 ! Da(8) + 23 Fa(8) − √13 Ta(8/10) , Da(18) = − 23 Fa(10) − √13 T†a (8/10) where Da(8) , Fa(8) , Ta(8/10) and Fa(10) are listed in Ref. [51]. From this chiral-singlet Lagrangian we obtain five flavorsinglet Lagrangians: 1 0 0 N M ¯ g(18) N[(6,3)] (σ + iγ5 π )( √ δN M ) N[(6,3)] , 6


3

2 c M c c c ¯N N[(6,3)] , F ) (σ + iγ π )(D + g(18) N N M 5 (8) [(6,3)] 3 (8) 2 0 0 √ g(18) ∆¯P δ ) ∆Q (6) (σ + iγ π )(− P Q 5 [(6,3)] [(6,3)] , 6 2 c c c ∆Q F ) (σ + iγ π )(− g(18) ∆¯P P Q 5 [(6,3)] [(6,3)] , 3 (10) 1 c c c P ¯N g(18) N [(6,3)] (σ + iγ5 π )(− √ T(8/10) )N P ∆[(6,3)] + h.c. . 3 where N, M = 1 · · · 8, P, Q = 1 · · · 10 and c = 1 · · · 8. It seems that it is much more complicated to use flavor-singlet Lagrangians than chiral-singlet ones, because there are much more flavor-singlet Lagrangians. However, it turns out to be that using these flavor-singlet Lagrangians is ¯ B are identity matrices (11×1 , 18×8 and 110×10 ), much simpler, because the only possible matrices to connect BM a a a a a D(8) , F(8) , F(10) , T(1/8) and T(8/10) [51]. Summarizing all these flavor-singlet Lagrangians, we obtain six diagonal terms ¯ 0Λ , (7) g1 Λσ ¯ N σ0 N N , g2 N ¯ N σ c (Dc )N M N M , g3 N

(8)

¯ N σ c (Fc )N M N M , g4 N (8) g5 ∆¯P σ 0 ∆P ,

(10)

(9)

(8)

¯P

g6 ∆ σ

c

(Fc(10) )P Q ∆Q

(11)

,

(12)

and two off-diagonal terms ¯ c (Tc )1N N N + h.c. , g7 Λσ 1/8 ¯ N σ c (Tc )N P ∆P + h.c. , g8 N

(13) (14)

(8/10)

where N, M = 1 · · · 8, P, Q = 1 · · · 10 and c = 1 · · · 8. Some readers may be quite familiar with these Lagrangians because they are similar to the lowest-order meson-baryon χPT Lagrangian in the flavor space. In these equations ¯ N σ c (Dc )N M N M every flavor-singlet Lagrangian is the summation of several chiral components [51,52], for example, N (8) is the summation of ¯N ¯ N σ c (Dc )N M N M ∼ a1 N (15) σ c (Dc )N M N M N (8)

[(6,3)]

(8)

[(6,3)]

¯ N¯ σ c (Dc )N M N M¯ + a2 N (8) [(3,3)] [(3,3)]

M c c ¯N + a3 N [(8,1)] σ (D(8) )N M N[(6,3)] ¯N σ c (Dc )N M N M¯ + a4 N [(8,1)]

(8)

[(3,3)]

+ h.c. + · · · .

Here we only include flavor-singlet terms containing scalar meson fields σ a , which have nonzeron condensates hσ0,3,8 i. We note that the Lagrangians containing π a can also contribute, and their contributions can be calculated using many methods, such as the chiral perturbation theory [3,4,55,56,57,58,59,60,61,62,63]. However, they are beyond the scope of this paper, and we shall omit their contributions. In this paper we only consider the mixing of flavor-octet and flavor-decuplet baryons, and so Eqs. (7) and (13) are irrelevant. Moreover, we assume all the decuplet baryons have the same mass m∆ = 2 GeV, and so we only need + Eqs. (8), (9), (10), (11) and (14). This is because that the lowest ∆ baryon having quantum numbers J P = 21 is + ∆(1910) [50]; besides it, there can be other decuplet baryons having the same quantum numbers J P = 12 but having larger masses, which can also mix with the lowest-lying flavor-octet baryons. The dependence of our results on m∆ will be studied in Sec. 4.6.

3 Mass Formulae 3.1 Flavor-Octet Baryons Using Lagrangians (8), (9) and (10) we can investigate the SU (3) flavor symmetry breaking effects through the terms containing nonzero condensate hσ8 i, and the SU (2) isospin symmetry breaking effects through the terms containing

4


nonzero condensate hσ3 i. Their explicit forms after the spontaneous chiral symmetry breaking are ¯ N + ΣΣ ¯ + ΞΞ ¯ + Λ¯0 Λ0 Lafter = g2 hσ0 i × N N 1 ¯ 1 ¯ 1 1 ¯ N + √ ΣΣ − √ ΞΞ − √ Λ¯0 Λ0 + g3 hσ8 i × − √ N 2 3 3 2 3 3 √ √3 ¯ N − 3 ΞΞ ¯ + g4 hσ8 i × N 2 2 1 1 ¯ 0 0 1 1 0 0 1 ¯0 0 1 ¯− − Λ + g3 hσ3 i × p¯+ p+ − n ¯ n − Ξ Ξ + Ξ Ξ + √ Λ¯0 Σ 0 + √ Σ 2 2 2 2 3 3 1 1 0 0 ¯ 0 Ξ 0 − 1 Ξ¯ − Ξ − , ¯ +Σ + − Σ ¯ −Σ− + 1 Ξ + g4 hσ3 i × p¯+ p+ − n ¯ n +Σ 2 2 2 2

(16)

where N = (p+ , n0 )T , Σ = (Σ + , Σ 0 , Σ − )T and Ξ = (Ξ 0 , Ξ − )T . We note that Σ 0 and Λ0 can mix with each other when g3 hσ3 i 6= 0. 3.2 Mixing of Flavor-Octet and Flavor-Decuplet Baryons Using the Lagrangian (14) we can investigate the isospin symmetry breaking effects through mixing terms containing nonzero condensates hσ3 i and hσ8 i. Its explicit form after the spontaneous chiral symmetry breaking is 1 ¯ ∗ − √1 ΞΞ ¯ ∗ + √1 Σ ¯ ∗ Σ − √1 Ξ ¯ ∗Ξ √ ΣΣ (17) Lafter = g hσ i × 8 8 N∆ 2 2 2 2 r r 2 2 0 0 1 ¯ − ∗− 1 ¯ 0 ∗0 1 ¯ − ∗− 1 1 ¯ + ∗+ + + + g8 hσ3 i × Σ −√ Σ Σ −√ Ξ Ξ +√ Ξ Ξ − √ Λ¯0 Σ ∗0 p¯ ∆ + n ¯ ∆ +√ Σ 3 3 6 6 6 6 2 r r 1 ¯ ∗− − 1 ¯ ∗0 0 1 ¯ ∗− − 1 ¯ ∗0 0 1 ¯ ∗+ + 2 ¯+ + 2 ¯0 0 ∆ p + ∆ n +√ Σ Σ −√ Σ Σ −√ Ξ Ξ +√ Ξ Ξ −√ Σ Λ , + 3 3 6 6 6 6 2 where Σ ∗ = (Σ ∗+ , Σ ∗0 , Σ ∗− )T and Ξ ∗ = (Ξ ∗0 , Ξ ∗− )T . We find that the effects of terms containing hσ3 i and hσ8 i have some overlaps: the former is used to decrease Σ and Ξ masses, while the latter is used to decrease the masses of p+ , n0 , Σ + , Σ − , Ξ 0 and Ξ − . 3.3 Other Contributions Besides the spontaneous chiral symmetry breaking terms listed in previous subsections, octet baryon masses are contributed by two other sources: ¯ N N N . Numerically it is equivalent to the spontaneous chiral symmetry breaking 1. The “bare” mass term mbare N N N ¯ N N N . It conserves the SU (3) flavor symmetry. ¯ term g2 hσ0 iN N . We write them together as m0 N 2. The electromagnetic terms, which break the isospin symmetry. We can not calculate these terms directly, but just use two simple schemes to evaluate them. They both contain only one free parameter, which will be used to do fitting in Sec. 4. The first scheme, “Scheme A”, simply assumes masses of charged and unchanged baryons are different, for A A A examples, EMpA+ = EMΣ + = EMA and EMn0 = EMΣ 0 = 0. The second scheme, “Scheme B”, simply calculates electromagnetic interactions among the three valence quarks, for example, EMpB+ = EMB × 32 23 − 13 32 − 13 32 = 0 and EMnB0 = EMB × − 32 13 − 32 13 + 31 31 = − 31 EMB . These two assumptions are listed in Tab. 1. We note that the Scheme A considers baryons as a whole, while the Scheme B considers baryons as three-quark objects. They both satisfy the well-known Coleman-Glashow mass formula Mp+ − Mn0 EM + (MΞ 0 − MΞ − )EM = (MΣ + − MΣ − )EM [64]. 3. We note that the decuplet baryon masses are all assumed to be m∆ = 2 GeV. 3.4 Flavor-Octet Baryon Mass Formulae Summarizing all the possible terms we arrive at the formulae to calculate octet baryon masses. There are diagonal and off-diagonal terms.


5

Table 1. The electromagnetic terms under the two assumptions, the Scheme A and the Scheme B. They both contain only one free parameter, which will be used to do fitting in Sec. 4. Scheme A Scheme B

p+ EMA 0

n0 0 − 31 EMB

Σ+ EMA 0

Σ0 0 − 31 EMB

Σ− EMA 1 EMB 3

Ξ0 0 − 31 EMB

Ξ− EMA 1 EMB 3

Λ0 0 − 31 EMB

The diagonal terms are: mp+ = mn0 mΣ + mΣ 0 mΣ − mΞ 0 mΞ − mΛ0

√ 3 1 m0 + F − √ D+ 2 2 3 √ 1 3 F − √ D− m0 + 2 2 3

1 1 f + d + EMp+ 2 2

!

p¯+ p+ ,

! 1 1 ¯ 0 n0 , f − d + EMn0 n = 2 2 1 ¯ +Σ + , = m0 + √ D + f + EMΣ + Σ 3 1 ¯ 0Σ 0 , = m0 + √ D + EMΣ 0 Σ 3 1 ¯ −Σ − , = m0 + √ D − f + EMΣ − Σ 3 ! √ 1 1 1 3 ¯ 0Ξ 0 , F − √ D + f − d + EMΞ 0 Ξ = m0 − 2 2 2 2 3 ! √ 3 1 1 1 ¯ −Ξ − , = m0 − F − √ D − f + d + EMΞ − Ξ 2 2 2 2 3 1 = m0 − √ D + EMΛ0 Λ¯0 Λ0 , 3

(18)

where m0 = mbare + g2 hσ0 i , D ≡ g3 hσ8 i , F ≡ g4 hσ8 i , d ≡ g3 hσ3 i , f ≡ g4 hσ3 i .

(19)

We can easily verify that these diagonal hadronic parts satisfy the well-known Gell-Mann-Okubo mass formula 2 (MN + MΞ ) = 3MΛ + MΣ [65,66]. The off-diagonal terms are: 1 ¯0 0 mΣ 0 Λ0 = √ d Σ Λ + Λ¯0 Σ 0 , 3 1 ¯ ∗0 Λ0 + Λ¯0 Σ ∗0 , mΣ ∗0 Λ0 = − √ mN ∆ Σ 2 r 2 ¯ + p+ , mN ∆ p¯+ ∆+ + ∆ mp+ ∆+ = 3 r 2 mn0 ∆0 = ¯ 0 ∆0 + ∆¯0 n0 , mN ∆ n 3 1 1 ¯ + Σ ∗+ + Σ ¯ ∗+ Σ + , Σ mΣ + Σ ∗+ = √ MN ∆ + √ mN ∆ 2 6 1 ¯ 0 Σ ∗0 + Σ ¯ ∗0 Σ 0 , mΣ 0 Σ ∗0 = √ MN ∆ Σ 2 1 1 ¯ − Σ ∗− + Σ ¯ ∗− Σ − , mΣ − Σ ∗− = √ MN ∆ − √ mN ∆ Σ 2 6 1 1 ¯ 0 Ξ ∗0 + Ξ ¯ ∗0 Ξ 0 , Ξ mΞ 0 Ξ ∗0 = − √ MN ∆ − √ mN ∆ 2 6

(20)

6


mΞ − Ξ ∗− = where

1 1 ¯ − Ξ ∗− + Ξ ¯ ∗− Ξ − , Ξ − √ M N ∆ + √ mN ∆ 2 6 MN ∆ ≡ g8 hσ8 i , mN ∆ ≡ g8 hσ3 i .

(21)

We use m∆ to denote the decuplet baryon masses, which are all assumed to be 2 GeV: m∆ = 2 GeV .

(22)

Summarizing all these diagonal and off-diagonal terms, we arrive at the following mass formulae:   q √ 2 1 1 1 + m0 + 23 F − 2√ D + f + d + EM m p 2 2 3 N∆  3 q Mp+ ∈ eigenvalues of  , 2 m m N ∆ ∆ 3  q  √ 2 1 1 1 0 D − f − d + EM m m0 + 23 F − 2√ N ∆ n 2 2 3 3 , q Mn0 ∈ eigenvalues of  2 m m N ∆ ∆ 3 ! 1 m0 + √3 D + f + EMΣ + √12 MN ∆ + √16 mN ∆ MΣ + ∈ eigenvalues of , √1 MN ∆ + √1 mN ∆ m∆ 2 6   √1 d √1 MN ∆ m0 + √13 D + EMΣ 0 3 2  √1 d m0 − √13 D + EMΛ0 − √12 mN ∆  MΣ 0 &MΛ0 ∈ eigenvalues of  , 3 1 √ MN ∆ √1 mN ∆ − m ∆ 2 2 ! 1 1 m0 + √3 D − f + EMΣ − √2 MN ∆ − √16 mN ∆ , MΣ − ∈ eigenvalues of √1 MN ∆ − √1 mN ∆ m∆ 2 6 MΞ 0 ∈ eigenvalues of MΞ − ∈ eigenvalues of

m0 − m0 −

√ 3 2 F

(23)

(24)

(25)

(26)

(27) !

,

(28)

!

.

(29)

+ 12 f − 21 d + EMΞ 0 − √12 MN ∆ − 1 m∆ − √2 MN ∆ − √16 mN ∆

√1 mN ∆ 6

1 D − 12 f + 21 d + EMΞ − − √12 MN ∆ + − 2√ 3 − √12 MN ∆ + √16 mN ∆ m∆

√1 mN ∆ 6

−

1 √ D 2 3

√ 3 2 F

In these equations there are altogether nine parameters: m0 , D, F , d, f , m∆ , MN ∆ , mN ∆ and EMA (EMB ) with two constraints d hσ3 i f mN ∆ = = = , D F MN ∆ hσ8 i

(30)

and so there are altogether seven free parameters. Moreover, in Sec. 4.6 we shall find that we can always achieve a good fitting no matter which value of m∆ is chosen, and so there are only six free parameters involved in the mass fitting. In the next section we shall use these equations to fit the eight lowest-lying flavor-octet baryon masses whose measurements are very accurate nowadays [50]: Mpphy = 938.27MeV , Mnphy = 939.57MeV , 0 + phy phy phy MΣ + = 1189.37MeV , MΣ 0 = 1192.64MeV , MΣ − = 1197.45MeV ,

MΞphy 0 phy MΛ0

=

1314.86MeV , MΞphy −

(31)

= 1321.71MeV ,

= 1115.68MeV .

4 Numerical Analyses In this section we perform numerical analyses and fit the eight lowest-lying flavor-octet baryon masses using the equations obtained in the previous section. To do the fitting, we use the following variance equation:  v u phy 2 u 1 X (MN − MN )  , (32) var = M inimize t 2 8 1 MeV N


constrained by

7

mN ∆ = Ff = M N∆ , m∆ = 2 GeV d D

phy where MN are the experimental values of the eight lowest-lying flavor-octet baryon masses, and MN are our fitted masses calculated using Eqs. (23)-(29). Our fitting is separated into four steps. The first step will be done in Sec. 4.1 where we only consider the SU (3) flavor symmetry breaking, and m0 , D and F are assumed to be nonzero; the second step will be done in Sec. 4.2 where we do not consider the contributions of electromagnetic effects and decuplet baryons, and m0 , D, F , d and f are assumed to be nonzero; the third step will be done in Sec. 4.3 where the contributions of electromagnetic effects are included while the contributions of decuplet baryons are not included; the fourth step will be done in Sec. 4.4 where the contributions of decuplet baryons are included while the contributions of electromagnetic effects are not included; the fifth step will be done in Sec. 4.5 where all the nine parameters are assumed to be nonzero. In Sec. 4.6 we shall discuss the dependence of our fittings on the decuplet baryon mass m∆ . In Appendix. A we shall further discuss the contributions of the current quark masses mu,d,s .

4.1 Step 1: m0 , D and F In this subsection we only consider the SU (3) flavor symmetry breaking, and m0 , D and F are assumed to be nonzero. In this case there are three free parameters involved. To do the fitting we use the following variance equation:  v u phy 2 u 1 X (MN − MN ) , (33) var1 = M inimize t 2 8 1 MeV N d = f = MN ∆ = mN ∆ = EMA = EMB = 0 constrained by . m∆ = 2 GeV Very quickly we can obtain that var1 = 3.66, when m0 = 1151.19 MeV, D = 71.56 MeV and F = −219.03 MeV. The results are shown in Table. 2. In this step the difference between every fitted mass and its experimental value is less phy than 5.8 MeV, i.e., |MN − MN | < 5.8 MeV. The maximum is due to the Λ baryon that |MΛ − MΛphy | = 5.8 MeV. Table 2. Step 1: m0 , D and F are assumed to be nonzero. Step 1, var1 = 3.66 Parameters (MeV) Masses (MeV) Mp+ = 940.85 m0 = 1151.19 Mn0 = 940.85 D = 71.56 MΣ + = 1192.51 F = −219.03 MΣ 0 = 1192.51 d=0 MΣ − = 1192.51 f =0 MΞ 0 = 1320.22 MN∆ = 0 MΞ − = 1320.22 mN∆ = 0 MΛ0 = 1109.88 EMA = EMB = 0 m∆ = 2000

4.2 Step 2: m0 , D, F , d and f In this subsection we do not consider the contributions of electromagnetic effects and decuplet baryons, and m0 , D, F , d and f are assumed to be nonzero. There is one constraint among them and so there are four free parameters. To do the fitting we use the following variance equation:  v u phy 2 u 1 X (MN − MN )  , (34) var2 = M inimize t 2 8 1 MeV N  d D = Ff constrained by MN ∆ = mN ∆ = EMA = EMB = 0 .  m∆ = 2 GeV

8


The results are shown in Table. 3, where var2 = 2.56. Although var2 < var1 , in this step the maximum difference phy between every fitted mass and its experimental value is still 5.8 MeV, i.e., |MN − MN | < 5.8 MeV. This is because phy that the mass of the Λ baryon is still not well fitted that |MΛ − MΛ | = 5.8 MeV. Table 3. Step 2: m0 , D, F , d and f are assumed to be nonzero. Step 2, var2 = 2.56 Parameters (MeV) Masses (MeV) Mp+ = 939.44 m0 = 1151.19 Mn0 = 942.27 D = 71.57 MΣ + = 1188.32 F = −219.02 MΣ 0 = 1192.52 d = 1.37 MΣ − = 1196.71 f = −4.20 MΞ 0 = 1317.43 MN∆ = 0 MΞ − = 1323.00 mN∆ = 0 MΛ0 = 1109.87 EMA = EMB = 0 m∆ = 2000

4.3 Step 3: m0 , D, F , d, f and EMA (EMB ) In this subsection we include the contributions of electromagnetic effects, and m0 , D, F , d, f and EMA (EMB ) are assumed to be nonzero. There is one constraint among them and so there are five free parameters. To do the fitting we use the following variance equation:  v u phy 2 u 1 X (MN − MN )  , (35) var3 = M inimize t 2 8 1 MeV N  d D = Ff constrained by MN ∆ = mN ∆ = 0 .  m∆ = 2 GeV The results are shown in Table. 4. We find that both the Scheme A and the Scheme B lead to similar results having var3 = 2.55, and in both schemes the difference between every fitted mass and its experimental value is less than 5.7 phy MeV, i.e., |MN − MN | < 5.7 MeV. Still the maximum is due to the Λ baryon that |MΛ − MΛphy | = 5.7 MeV. Table 4. Step 3: m0 , D, F , d, f and EMA (EMB ) are assumed to be nonzero. Step 3, Scheme A, var3 = 2.55 Parameters (MeV) Masses (MeV) Mp+ = 939.20 m0 = 1151.38 Mn0 = 942.43 D = 71.71 MΣ + = 1188.17 F = −219.02 MΣ 0 = 1192.79 d = 1.39 MΣ − = 1196.64 f = −4.24 MΞ 0 = 1317.55 MN∆ = 0 MΞ − = 1322.80 mN∆ = 0 MΛ0 = 1109.98 EMA = −0.38 m∆ = 2000

Step 3, Scheme B, var3 = 2.55 Parameters (MeV) Masses (MeV) Mp+ = 939.14 m0 = 1151.11 Mn0 = 942.46 D = 71.76 MΣ + = 1188.12 F = −219.12 MΣ 0 = 1192.89 d = 1.45 MΣ − = 1196.61 f = −4.42 MΞ 0 = 1317.57 MN∆ = 0 MΞ − = 1322.74 mN∆ = 0 MΛ0 = 1110.02 EMB = −1.04 m∆ = 2000

In this step the two electromagnetic parameters EMA and EMB are both negative, producing an electromagnetic γ mass difference between the proton and the neutron, Mp−n ≡ EMp+ − EMn0 < 0. This is not consistent with almost all theoretical calculations [2,7,3,23,27,24,26,25], suggesting that this fitting is not well done.


9

4.4 Step 4: m0 , D, F , d, f , m∆ , MN ∆ and mN ∆ In this subsection we include the contributions of decuplet baryons, and m0 , D, F , d, f , m∆ , MN ∆ and mN ∆ are assumed to be nonzero. There are two constraints among them and so there are six free parameters (including m∆ ). To do the fitting we fix m∆ = 2 GeV and use the following variance equation:  v u phy 2 u 1 X (MN − MN ) , (36) var4 = M inimize t 2 8 1 MeV N  mN ∆ d D = Ff = M N∆ constrained by EMA = EMB = 0 .  m∆ = 2 GeV The results are shown in Table. 5, where var4 = 0.44. This is significantly smaller than var1 and var2 . In this step the phy difference between every fitted mass and its experimental value is less than 0.61 MeV, i.e., |MN − MN | < 0.61 MeV. phy + This happens to the proton that |Mp −Mp+ | = 0.61 MeV, and now the Λ baryon is well fitted that |MΛ −MΛphy | = 0.03 MeV. Table 5. Step 4: m0 , D, F , d, f , m∆ , MN∆ and mN∆ are assumed to be nonzero. Step 4, var4 = 0.44 Parameters (MeV) Masses (MeV) Mp+ = 937.66 m0 = 1163.95 Mn0 = 940.16 D = 83.52 MΣ + = 1188.99 F = −232.01 MΣ 0 = 1193.17 d = 1.40 MΣ − = 1197.33 f = −3.90 MΞ 0 = 1315.28 MN∆ = −175.16 MΞ − = 1321.26 mN∆ = −2.94 MΛ0 = 1115.71 EMA = EMB = 0 m∆ = 2000

4.5 Step 5: m0 , D, F , d, f , m∆ , MN ∆ , mN ∆ and EMA (EMB ) In this subsection we include all the nine parameters, m0 , D, F , d, f , m∆ , MN ∆ , mN ∆ and EMA (EMB ), and assume that they are nonzero. There are two constraints among them and so there are seven free parameters (including m∆ ). To do the fitting we fix m∆ = 2 GeV and use the following variance equation:  v u phy 2 u 1 X (MN − MN ) , (37) var5 = M inimize t 2 8 1 MeV N d mN ∆ = Ff = M N∆ constrained by D . m∆ = 2 GeV The results are shown in Table. 6. When using the Scheme A, we obtain var5 = 0.06, and in this case the difference phy between every fitted mass and its experimental value is less than 0.11 MeV, i.e., |MN −MN | < 0.11 MeV. When using the Scheme B, we obtain var5 = 0.10, and in this case the difference between every fitted mass and its experimental phy value is less than 0.16 MeV, i.e., |MN − MN | < 0.16 MeV. Hence, the Scheme A seems slightly better than the Scheme B. We can solve the eigenvectors of Eqs. (23), (25) and (28) and estimate the decuplet components contained in the N , Σ and Ξ baryons, which are around 5 × 10−6 , 2.4% and 3.4%, respectively, using both schemes. We can also estimate the Σ 0 − Λ0 mixing [67,68,69,70,71], which is at the level of 10−4 : |Λ0 iphy = 0.9999|Λ0i − 0.0137|Σ 0i − 0.0043|Σ ∗0i .

(38)

10

Hua-Xing Chen: Isospin Symmetry Breaking and Octet Baryon Masses due to Their Mixing with Decuplet Baryons Table 6. Step 5: all the nine parameters are assumed to be nonzero. Step 5, Scheme A, var5 = 0.06 Parameters (MeV) Masses (MeV) Mp+ = 938.20 m0 = 1164.03 Mn0 = 939.64 D = 83.70 MΣ + = 1189.42 F = −232.59 MΣ 0 = 1192.53 d = 1.36 MΣ − = 1197.52 f = −3.78 MΞ 0 = 1314.90 MN∆ = −179.02 MΞ − = 1321.67 mN∆ = −2.91 MΛ0 = 1115.68 EMA = 0.98 m∆ = 2000

Step 5, Scheme B, var5 = 0.10 Parameters (MeV) Masses (MeV) Mp+ = 938.16 m0 = 1164.74 Mn0 = 939.68 D = 83.73 MΣ + = 1189.43 F = −232.44 MΣ 0 = 1192.48 d = 1.25 MΣ − = 1197.55 f = −3.47 MΞ 0 = 1314.93 MN∆ = −179.34 MΞ − = 1321.63 mN∆ = −2.68 MΛ0 = 1115.69 EMB = 2.10 m∆ = 2000

γ The two assumptions to calculate electromagnetic terms, the Scheme A and the Scheme B, give different Mp−n ≡ EMp+ − EMn0 . This suggests that to well separate hadronic and electromagnetic effects, one needs more reasonable γ γ γ (reliable) calculations. The Scheme A produces Mp−n = 0.98 MeV, MΣ + −Σ − = 0, MΞ 0 −Ξ − = −0.98 MeV, and γ γ MΣ + +Σ − −2Σ 0 = 1.96 MeV. These results are just consistent with the data estimated in Ref. [2] that Mp−n = EMA = γ γ γ 0.76 ± 0.30 MeV, MΣ + −Σ − = −0.17 ± 0.30, MΞ 0 −Ξ − = −0.86 ± 0.30 MeV, and MΣ + +Σ − −2Σ 0 = 1.78 ± 0.14. The γ γ γ γ Scheme B produces Mp−n = 0.70 MeV, MΣ + −Σ − = −0.70 MeV, MΞ 0 −Ξ − = −1.40 MeV, and MΣ + +Σ − −2Σ 0 = 1.40 MeV. These results are not consistent with Ref. [2]. Again the Scheme A seems to be better than the Scheme B.

4.6 Dependence on the decuplet baryon mass m∆ In this subsection we study the dependence of our results on the decuplet baryon mass m∆ . We change m∆ from 1350 MeV to 2500 MeV. To do the fitting we use the following variance equation:  v u phy 2 u 1 X (MN − MN ) , (39) var6 = M inimize t 8 1 MeV2 N d f mN ∆ , = = D F MN ∆ changing from 1800 MeV to 2500 MeV .

constrained by and m∆

phy We show var6 and Max|MN − MN | (the maximum difference between every fitted mass and its experimental value) as functions of m∆ in Fig. 1. We clearly see that we can always achieve a good fitting no matter which value of m∆ is chosen. Hence, we have proved that there are only six free parameters involved in our fittings. However, the mixing parameters MN ∆ and mN ∆ do depend on m∆ , and so the decuplet components contained in the Σ and Ξ baryons also depend on m∆ . We show them in Fig. 2. Here we would like to note again that the Lagrangians containing π a can also contribute, as discussed in Sec. 2. In this subsection we have found that we can use m∆ as small as 1500 MeV to achieve a good fitting, which value is smaller than the total masses of the J P = 0− pseudoscalar mesons π and the J P = (3/2)+ decuplet baryons Σ(1385) and Ξ(1530). This may suggest that the contributions of the J P = (1/2)+ decuplet baryons, which are investigated in this paper, may be equivalently estimated using the Lagrangians containing these J P = 0− pseudoscalar mesons and J P = (3/2)+ decuplet baryons. However, these are beyond the scope of this paper, and so we did not investigate them.

5 Summary We studied the isospin symmetry breaking and mass splittings of the lowest-lying flavor-octet baryons. We included three kinds of baryon mass terms: the bare mass term, the electromagnetic terms and the spontaneous chiral symmetry breaking terms. Particulary, we included the mixing term between flavor-octet and flavor-decuplet baryons. This assumes that the lowest-lying Σ and Ξ baryons contain a few decuplet components and so are not purely flavor-octet.

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 1350 1500

1750

2000

2250

1 Max|M N -M phy N | [MeV]

var6 [MeV]

Hua-Xing Chen: Isospin Symmetry Breaking and Octet Baryon Masses due to Their Mixing with Decuplet Baryons 1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 2500

0 1350 1500

11

1750

m [MeV]

2000

2250

0 2500

m [MeV]

phy Fig. 1. The variance var6 and the maximum difference between every fitted mass and its experimental value, Max|MN − MN |, as functions of the decuplet baryon mass m∆ . We use the solid and dashed curves to denote the results obtained using the Scheme A and the Scheme B, respectively.

5

5

4

4

3

3

2

2

1

1

0 1350 1500

1750

2000 m [MeV]

2250

0 2500

20

20

15

15

10

10

5

5

0 1350 1500

1750

2000

2250

0 2500

m [MeV]

Fig. 2. The decuplet components contained in the Σ and Ξ baryons, as functions of the decuplet baryon mass m∆ . We use the solid and dashed curves to denote the results obtained using the Scheme A and the Scheme B, respectively, while they are almost the same.

Summarizing all these terms, we obtained the formulae to calculate octet baryon masses, and we used them to fit the eight lowest-lying flavor-octet baryon masses whose measurements are very accurate nowadays. Our fittings was separated into five steps. We summarize all the results and show them in Fig. 3. In the last step we included all the nine parameters, m0 , D, F , d, f , m∆ , MN ∆ , mN ∆ and EMA (EMB ). There are two constraints among these parameters, and so there are seven free parameters (including m∆ ). After fixing m∆ = 2 GeV, we obtain the results: when using the Scheme A, the difference between every fitted mass and its experimental value is less phy than 0.11 MeV, i.e., |MN − MN | < 0.11 MeV; when using the Scheme B, this value is less than 0.16 MeV, i.e., phy |MN − MN | < 0.16 MeV. Moreover, we have shown that a good fitting can be always achieved no matter which value of m∆ is chosen, and so there are six free parameters involved in the mass fitting. We estimated the decuplet components contained in the Σ and Ξ baryons, which are around 2.4% and 3.4%, respectively, when using m∆ = 2 GeV. We also estimated the Σ 0 − Λ0 mixing, which is at the level of 10−4 . To end this paper we would like to note again that the flavor-singlet and flavor-octet baryons can also mix through their common [(¯ 3, 3) ⊕ (3, ¯ 3)] chiral components. We have studied this mixing and the results are still in preparations [54].

Acknowledgments

This work is partly supported by the National Natural Science Foundation of China under Grant No. 11205011.

12


Step 1 (m 0 , D and F) phy

var 1 = 3.66, |MN - MN | < 5.8 MeV

Step 2 (m 0 , D, F, d and f) phy

var 2 = 2.56, |MN - MN | < 5.8 MeV

Step 3 (m 0 , D, F, d, f and EM A /EM B ) phy Scheme A: var 3 = 2.55, |M N - M N | < 5.7 MeV phy Scheme B: var 3 = 2.55, |M N - M N | < 5.7 MeV

Step 4 (m 0 , D, F, d, f, m , M

and m )

phy

var 4 = 0.44, |MN - MN | < 0.61 MeV

Step 5 (m 0 , D, F, d, f, m , M , m and EMA /EM B ) phy Scheme A: var5 = 0.06, |M N - M N | < 0.11 MeV phy Scheme B: var 5 = 0.10, |M N - M N | < 0.16 MeV

Fig. 3. Our fittings are separated into five steps.

A The Quark Mass Terms In this appendix we discuss the quark mass terms, mu , md and ms . We shall prove that they are numerically equivalent ¯ N N M , i.e., mu,d,s and m0 , ¯ N N N and g4 hσ3,8 i(F3,8 )N M N to the spontaneous chiral symmetry breaking terms g2 hσ0 iN (8) F , f are numerically equivalent. In the following we show how to relate these six parameters. Assuming the flavor-octet baryon masses come from the following quark mass terms: mp+ = 2mu + md , mn0 = mu + 2md , mΣ + = 2mu + ms , mΣ 0 = mu + md + ms , mΣ − = 2md + ms , mΞ 0 = mu + 2ms , mΞ − = md + 2ms , mΛ0 = mu + md + ms .

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Inserting 1 m0 + 3 1 md = m0 + 3 1 ms = m0 − 3

mu =

1 √ F+ 2 3 1 √ F− 2 3 1 √ F, 3

1 f, 2 1 f, 2

(41)

these equations are changed to √ 3 F+ 2 = m0 + f , mΣ 0

mp+ = m0 + mΣ +

√ 3 1 1 f , mn0 = m0 + F − f, 2 2 2 = m0 , mΣ − = m0 − f ,

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13

√ √ 1 1 3 3 F + f , mΞ − = m0 − F − f, mΞ 0 = m0 − 2 2 2 2 mΛ0 = m0 , which are just Eqs. (18) keeping only m0 , F and f . Therefore, we have proved that mu,d,s and m0 , F , f are numerically equivalent.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.

R. P. Feynman and G. Speisman, Phys. Rev. 94, 500 (1954). J. Gasser and H. Leutwyler, Phys. Rept. 87, 77 (1982). P. E. Shanahan, A. W. Thomas and R. D. Young, Phys. Lett. B 718, 1148 (2013). P. E. Shanahan, A. W. Thomas and R. D. Young, Phys. Rev. D 87, 114515 (2013). M. Praszalowicz, A. Blotz and K. Goeke, Phys. Rev. D 47, 1127 (1993). G. -S. Yang, H. -C. Kim and M. V. Polyakov, Phys. Lett. B 695, 214 (2011). A. Walker-Loud, C. E. Carlson and G. A. Miller, Phys. Rev. Lett. 108, 232301 (2012). C. Adami, E. G. Drukarev and B. L. Ioffe, Phys. Rev. D 48, 2304 (1993) [Erratum-ibid. D 52, 4254 (1995)]. H. Forkel and M. Nielsen, Phys. Rev. D 55, 1471 (1997). E. M. Henley and T. Meissner, Phys. Rev. C 55, 3093 (1997). S. Narison, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 17, 1 (2002). B. L. Ioffe, V. S. Fadin, V. A. Khoze and L. N. Lipatov, Quantum Chromodynamics: Perturbative and Nonperturbative Aspects, Cambridge University Press, 2010. R. F. Dashen, Phys. Rev. 183, 1245 (1969). J. F. Donoghue, B. R. Holstein and D. Wyler, Phys. Rev. D 47, 2089 (1993). J. Bijnens, Phys. Lett. B 306, 343 (1993). B. -A. Li, M. -L. Yan and K. -F. Liu, Phys. Lett. B 177, 409 (1986). P. Jain, R. Johnson, N. W. Park, J. Schechter and H. Weigel, Phys. Rev. D 40, 855 (1989). S. Durr, Z. Fodor, J. Frison, C. Hoelbling, R. Hoffmann, S. D. Katz, S. Krieg and T. Kurth et al., Science 322, 1224 (2008). A. Duncan, E. Eichten and R. Sedgewick, Phys. Rev. D 71, 094509 (2005). S. Basak et al. [MILC Collaboration], PoS CD 12, 030 (2013). T. Bhattacharya, S. D. Cohen, R. Gupta, A. Joseph and H. -W. Lin, arXiv:1306.5435 [hep-lat]. J. R. Green, J. W. Negele, A. V. Pochinsky, S. N. Syritsyn, M. Engelhardt and S. Krieg, Phys. Rev. D 86, 114509 (2012). S. R. Beane, K. Orginos and M. J. Savage, Nucl. Phys. B 768, 38 (2007). R. Horsley et al. [QCDSF and UKQCD Collaborations], Phys. Rev. D 86, 114511 (2012). S. .Borsanyi, S. Drr, Z. Fodor, J. Frison, C. Hoelbling, S. D. Katz, S. Krieg and T. .Kurth et al., Phys. Rev. Lett. 111, 252001 (2013). G. M. de Divitiis, R. Frezzotti, V. Lubicz, G. Martinelli, R. Petronzio, G. C. Rossi, F. Sanfilippo and S. Simula et al., Phys. Rev. D 87, 114505 (2013). T. Blum, R. Zhou, T. Doi, M. Hayakawa, T. Izubuchi, S. Uno and N. Yamada, Phys. Rev. D 82, 094508 (2010). W. Bietenholz, V. Bornyakov, M. Gockeler, R. Horsley, W. G. Lockhart, Y. Nakamura, H. Perlt and D. Pleiter et al., Phys. Rev. D 84, 054509 (2011). S. Aoki et al. [PACS-CS Collaboration], Phys. Rev. D 79, 034503 (2009). Z. Fodor and C. Hoelbling, Rev. Mod. Phys. 84, 449 (2012). G. Colangelo, S. Durr, A. Juttner, L. Lellouch, H. Leutwyler, V. Lubicz, S. Necco and C. T. Sachrajda et al., Eur. Phys. J. C 71, 1695 (2011). A. Portelli, PoS KAON 13, 023 (2013). S. Weinberg, Phys. Rev. 177, 2604 (1969). S. Weinberg, Phys. Rev. Lett. 65, 1177 (1990). D. B. Leinweber, Phys. Rev. D 51, 6383 (1995). B. L. Ioffe, Nucl. Phys. B 188, 317 (1981) [Erratum-ibid. B 191, 591 (1981)]. Y. Chung, H. G. Dosch, M. Kremer and D. Schall, Nucl. Phys. B 197 (1982) 55. D. Espriu, P. Pascual and R. Tarrach, Nucl. Phys. B 214 (1983) 285. T. D. Cohen and X. D. Ji, Phys. Rev. D 55, 6870 (1997). K. Nagata, A. Hosaka and V. Dmitrasinovic, Mod. Phys. Lett. A 23, 2381 (2008). T. D. Cohen and L. Y. Glozman, Int. J. Mod. Phys. A 17, 1327 (2002). D. Jido, M. Oka and A. Hosaka, Prog. Theor. Phys. 106, 873 (2001). H. -X. Chen, Eur. Phys. J. C 72, 2204 (2012). H. -X. Chen, Adv. High Energy Phys. 2013, 750591 (2013). H. -X. Chen, Eur. Phys. J. C 73, 2628 (2013). H. -X. Chen, V. Dmitrasinovic, A. Hosaka, K. Nagata and S. -L. Zhu, Phys. Rev. D 78, 054021 (2008). V. Dmitrasinovic and H. -X. Chen, Eur. Phys. J. C 71, 1543 (2011). H. -X. Chen, Eur. Phys. J. C 72, 2129 (2012).

14 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71.

Hua-Xing Chen: Isospin Symmetry Breaking and Octet Baryon Masses due to Their Mixing with Decuplet Baryons H. -X. Chen and V. Dmitrasinovic, Phys. Rev. D 88, 036013 (2013). J. Beringer et al. [Particle Data Group Collaboration], Phys. Rev. D 86, 010001 (2012). H. -X. Chen, V. Dmitrasinovic and A. Hosaka, Phys. Rev. D 81, 054002 (2010). H. X. Chen, V. Dmitrasinovic and A. Hosaka, Phys. Rev. D 83, 014015 (2011). H. -X. Chen, V. Dmitrasinovic and A. Hosaka, Phys. Rev. C 85, 055205 (2012). H. -X. Chen, Isospin Symmetry Breaking and Octet Baryon Masses due to Their Mixing with the Singlet Baryon, in preparations. S. Weinberg, Physica A 96, 327 (1979). J. Gasser and H. Leutwyler, Annals Phys. 158, 142 (1984). V. Bernard, N. Kaiser and U. -G. Meissner, Int. J. Mod. Phys. E 4, 193 (1995). A. Pich, Rept. Prog. Phys. 58, 563 (1995). G. Ecker, Prog. Part. Nucl. Phys. 35, 1 (1995). S. Scherer and M. R. Schindler, Lect. Notes Phys. 830, pp.1 (2012). V. Pascalutsa, Phys. Rev. D 58, 096002 (1998). V. Pascalutsa and R. Timmermans, Phys. Rev. C 60, 042201 (1999). X. -L. Ren, L. -S. Geng and J. Meng, arXiv:1307.1896. S. R. Coleman and S. L. Glashow, Phys. Rev. Lett. 6, 423 (1961). M. Gell-Mann, Phys. Rev. 125, 1067 (1962). S. Okubo, Prog. Theor. Phys. 27, 949 (1962). S. -L. Zhu, W. Y. P. Hwang and Z. -s. Yang, Phys. Rev. D 57, 1524 (1998). H. Muller and J. R. Shepard, J. Phys. G 26, 1049 (2000). N. Yagisawa, T. Hatsuda and A. Hayashigaki, Nucl. Phys. A 699, 665 (2002). S. Shinmura, K. S. Myint, T. Harada and Y. Akaishi, J. Phys. G 28, L1 (2002). K. Sasaki, T. Inoue and M. Oka, Few Body Syst. Suppl. 14, 277 (2003).