Distribution amplitudes of Σ and Λ and their electromagnetic form factors

Transcription

1 Distribution amplitudes of Σ and Λ and their electromagnetic form factors Yong-Lu Liu 1 and Ming-Qiu Huang arxiv: v1 [hep-ph] 12 Nov 28 Department of Physics, National University of Defense Technology, Hunan 4173, China Abstract Based on QCD conformal partial wave expansion to the leading order conformal spin accuracy, we present the light-cone distribution amplitudes (DAs) of Σ and Λ baryons to twist 6. We conclude that fourteen independent DAs are needed to describe the valence three-quark states of the baryons at small transverse separations. The nonperturbative parameters relevant to the DAs are determined in the framework of QCD sum rule method. With the obtained DAs, a simple investigation on the electromagnetic form factors of these baryons are given. The magnetic moments of the baryons are estimated by fitting the magnetic form factor by the dipole formula. PACS: Hf, Hx, 13.4.Gp, 14.2.Jn. Keywords: Conformal expansion; Distribution amplitude; Light cone sum rules; Electromagnetic form factor. 1 E-mil: yongluliu@nudt.edu.cn

2 1 Introduction The theory of the hard exclusive process in QCD has been studied extensively for several decades, for the investigation of the exclusive reactions provides unmatched opportunities to understand the hadron structure. The theoretical calculation of the processes was developed early in 197 s [1, 2]. In the model of the hard exclusive process, the concept of the distribution amplitudes (DAs), which are the fundamental nonperturbative functions describing the hadronic structure, was introduced. The DAs, physically speaking, describe the decomposition of the hadron momentum in parton configurations, which is important to make the QCD description of hard exclusive reactions quantitative. The DAs of mesons have been investigated extensively in the past, some of which have been done to high twist accuracy [3, 4, 5, 6, 7]. However the corresponding studies on baryons received less attention because of the complex structure of the baryon, and the existing investigations of DAs were mainly focused on the nucleon (See [8] for a review). The DAs of the nucleon and other octet baryons were firstly calculated in the QCD sum rule framework on the moments in Ref. [9]. A systematic study on the nucleon DAs were provided in Ref. [1], in which the DAs of the nucleon are investigated up to twist 6. In the paper [11], the axial vector higher twist DAs of the Λ baryon were given to the leading order conformal spin accuracy. Recently Ref. [12] gave the DAs of the Λ b in the heavy quark limit, and Ref. [13] offered a complete analysis of the one-loop renormalization of twist-4 operators of the nucleon. For the DAs of baryons with other quantum number, the literature [14] presented a description of helicity λ = 3/2 baryons with a new approach. The investigation of Σ and Λ baryon DAs, which was firstly done by V.L. Chernyak et al in Ref. [9], only gave the leading twist DAs of them. Whereas more detailed description of the internal structures of these baryons needs information on higher order twist DAs. This work is devoted to give an investigation on Σ and Λ DAs up to twist 6. The higher twist contributions come from several physical origins. The first contribution is from bad components in the wave function and in particular of components 1

3 with wrong spin projection. The second contribution comes from transverse motion of quarks in the leading twist components. Finally, higher Fock states with additional gluons or quark/antiquark pairs also contribute to the DAs. It has been known that for mesons, contributions due to bad components in the quark-antiquark DAs can be described in terms of higher Fock states by equations of motion [3, 5]. Since quark-antiquark-gluon matrix elements between the vacuum and the meson state are numerically small, contributions of bad components to mesonic DAs are small enough. However things are different for baryons because equations of motion are not sufficient to eliminate the higher-twist three-quark system with additional gluons. At the same time, matrix elements of higher twist three-quark operators are large compared with the leading one. Thus the first contribution is assumed to dominate the DAs rather than the other two origins. For this reason we only consider contributions coming from bad components in the decomposition of the Lorentz structure in this paper. The usual description of the DAs is based on the conformal symmetry of the QCD Lagrangian for the dynamics dominated on the light-cone. The DAs with definite twist can be expanded by the partial wave functions with the specific conformal spin j. The conformal spin of a quark is defined as j = (l +s)/2, where l is the canonical dimension of the quark and s is its spin. For a hadron, contributions of the higher order conformal spin j+n (n =, 1, 2,...) are given by the leading contribution multiplied by polynomials which are orthogonal over the leading weight function. In this paper the DAs of Σ and Λ baryons are investigated by utilizing the method proposed in Ref. [1]. We find the description of the Λ and Σ baryons needs fourteen independent DAs. This is different from that of the nucleon, whose isospin is 1/2. The isospin of the nucleon gives symmetry relationships to reduce the number of the independent DAs to 8. The fourteen independent DAs can be expanded in operators with increasing conformal spin. Then with the equation of motion, the parameters of the conformal expansion are expressed by the local nonperturbative parameters that can be determined on the QCD sum rule method. In this work, we expand the DAs to leading order conformal spin accuracy, and use QCD sum rule to 2

4 determine the nonperturbative parameters. We get the DAs of Σ and Λ up to twist 6. Results on the DAs provide large opportunities to investigate processes connected with the baryons in the framework of light-cone sum rule (LCSR) [11, 15, 16, 17, 18, 19], for DAs are the fundamental input parameters in this framework. LCSR is a developed nonperturbative QCD method that include the traditional à la SVZ sum rule [2] technique and the theory of hard exclusive processes. The main idea of LCSR is to expand the products of currents near the light-cone, and the nonperturbative effects are described by the DAs rather than condensates in traditional QCD sum rule [21, 22, 23]. As a simple application, the electromagnetic (EM) form factors of the baryons are examined in the framework of LCSR with the obtained DAs. EM form factors are fundamental objects for understanding the inner structure of the hadron. As there are little experimental data available on Σ and Λ baryon EM form factors, it is instructive and necessary to give an investigation on them theoretically. In the experimental point of view, the EM form factors can be described by the electric and magnetic Sachs form factors, and the magnetic Sachs form factor at zero momentum transfer defines the magnetic moment of the baryon. We assume that the dependence of the magnetic form factor on the momentum transfer can be expressed by the dipole formula, therefore after fitting the magnetic form factor by the dipole formula, we give estimations on the magnetic moments of the Σ baryons. The paper is organized as follows. Section 2 presents the notations and the definitions of the baryon DAs to twist 6 by the matrix elements of the three-quark operator between the baryon state and the vacuum. This section also gives the properties of the DAs from the symmetry relationships. Section 3 gives the leading-order conformal expansion of the DAs based on the conformal invariance of the Lagrangian. The DAs are simplified to the nonperturbative parameters which can be calculated in the framework of the QCD sum rule. Section 4 is devoted to give the QCD sum rules for the nonperturbative parameters related to the DAs and then present numerical analysis. Section 5 is a simple application of the obtained DAs to investigate the EM form factors of the baryons. We give in this section the dependence of the EM form factors on the momentum transfer. After fitting 3

5 the results by the dipole formula, we also give an estimation on the magnetic moments of the baryons. Summary and conclusion are given in section 6. 2 Definitions of the light-cone distribution amplitudes 2.1 General classification Light-cone distribution amplitudes of the baryon can be expressed in terms of matrix element of the gauge-invariant operators sandwiched between the vacuum and the baryon state: ǫ ijk q 1 i α (a 1 z)q 2 j β (a 2z)s k γ (a 3z) X(P), (1) where the letters α, β, γ refer to Lorentz indices and i, j, k refer to color indices, z is a light-like vector which satisfies z 2 = and a i are real numbers denoting coordinates of valence quarks. In view of the Lorentz covariance, spin and parity of the baryons, the general decomposition of the matrix element is written as [1, 11]: 4 ǫ ijk q 1 i α (a 1 z)q 2 j β (a 2z)s k γ (a 3z) X(P) = S 1 MC αβ (γ 5 X) γ + S 2 M 2 C αβ ( zγ 5 X) γ + P 1 M (γ 5 C) αβ X γ + P 2 M 2 (γ 5 C) αβ ( zx) γ +V 1 ( PC) αβ (γ 5 X) γ + V 2 M ( PC) αβ ( zγ 5 X) γ + V 3 M (γ µ C) αβ (γ µ γ 5 X) γ +V 4 M 2 ( zc) αβ (γ 5 X) γ + V 5 M 2 (γ µ C) αβ (iσ µν z ν γ 5 X) γ + V 6 M 3 ( zc) αβ ( zγ 5 X) γ +A 1 ( Pγ 5 C) αβ X γ + A 2 M ( Pγ 5 C) αβ ( zx) γ + A 3 M (γ µ γ 5 C) αβ (γ µ X) γ +A 4 M 2 ( zγ 5 C) αβ X γ + A 5 M 2 (γ µ γ 5 C) αβ (iσ µν z ν X) γ + A 6 M 3 ( zγ 5 C) αβ ( zx) γ +T 1 (P ν iσ µν C) αβ (γ µ γ 5 X) γ + T 2 M (z µ P ν iσ µν C) αβ (γ 5 X) γ +T 3 M (σ µν C) αβ (σ µν γ 5 X) γ + T 4 M (P ν σ µν C) αβ (σ µ z γ 5 X) γ +T 5 M 2 (z ν iσ µν C) αβ (γ µ γ 5 X) γ + T 6 M 2 (z µ P ν iσ µν C) αβ ( zγ 5 X) γ +T 7 M 2 (σ µν C) αβ (σ µν zγ 5 X) γ + T 8 M 3 (z ν σ µν C) αβ (σ µ z γ 5 X) γ, (2) 4

6 where X γ is the spinor of the baryon with the quantum number I(J P ) = 1( 1+ ) for Σ ± 2 and I(J P ) = ( 1 2+ ) for Λ (I is the isospin, J is the total angular momentum and P is the parity), and C is the charge conjugation matrix and σ µν = i 2 [γ µ, γ ν ]. All the functions S i, P i, A 1, V i and T i depend on the scalar product P z. The calligraphic invariant functions in Eq. (2) do not have a definite twist, thus the twist classification need to be carried out in the infinite momentum frame. Here we introduce the second auxiliary light-like vector: M 2 p µ = P µ 1 2 z µ p z, p2 =, (3) so that P p if the baryon mass can be neglected. In the infinite momentum frame, the baryon is assume to move in the positive e z direction, then p + and z are the only nonvanishing components of p and z. In this frame the terms on twist can be classified in powers of p + and the baryon spinor X γ is decomposed into large and small components X γ + and X γ : Xγ(P, λ) = 1 2p z ( p z+ z p) = X+ γ (P, λ) + X γ (P, λ), (4) where the projection operators Λ + = p z 2p z, Λ = z p 2p z (5) project the spinor onto the plus and minus components. From the Dirac equation PX(P) = MX(P), we get the following useful relations: px(p) = MX + (P), zx(p) = 2p z M X (P). (6) Using the explicit expressions for X(P), it is easy to see that Λ + X = X + p + while Λ X = X 1/ p +. By the twist counts in terms of 1/p + the definition of light-cone DAs with a definite twist is given as 4 ǫ ijk q 1 i α (a 1 z)q 2 j β (a 2z)s k γ(a 3 z) X(P) = S 1 MC αβ ( γ5 X +) γ + S 2MC αβ ( γ5 X ) γ + P 1M (γ 5 C) αβ X + γ + P 2M (γ 5 C) αβ X γ 5

7 +V 1 ( pc) αβ ( γ5 X +) γ + V 2 ( pc) αβ ( γ5 X ) γ + V 3 2 M (γ C) αβ ( γ γ 5 X +) γ M 2 + V 4 2 M (γ ( C) αβ γ γ 5 X ) + V γ 5 2pz ( zc) αβ ( γ5 X +) γ + M2 2pz V 6 ( zc) αβ ( γ5 X ) γ +A 1 ( pγ 5 C) αβ X + γ + A 2 ( pγ 5 C) αβ X γ + A 3 2 M (γ γ 5 C) αβ ( γ X +) γ M 2 + A 4 2 M (γ ( γ 5 C) αβ γ X ) + A γ 5 2pz ( zγ 5C) αβ X γ + + M2 2pz A 6 ( zγ 5 C) αβ Xγ ( +T 1 (iσ p C) αβ γ γ 5 X +) + T ( γ 2 (iσ p C) αβ γ γ 5 X ) + T M γ 3 pz (iσ ( p zc) αβ γ5 X +) γ M 2 M +T 4 pz (iσ ( z pc) αβ γ5 X ) + T γ 5 2pz (iσ ( zc) αβ γ γ 5 X +) γ + M2 2pz T ( 6 (iσ z C) αβ γ γ 5 X ) + M T ) 7 γ 2 (σ C) αβ (σ γ 5 X + γ +M T ) 8 2 (σ C) αβ (σ γ 5 X, (7) γ where an obvious notation σ pz = σ µν p µ z ν etc. is used as a shorthand and stands for the projection transverse to z, p, e.g. γ γ = γ µ g µνγ ν with g µν = g µν (p µ z ν +z µ p ν )/pz. The DAs F = S i, P i, V i, A i, T i with a definite twist are classified in Tab. 1. Each distribution amplitude F i can be represented as F(a i p z) = Dxe ipz P i x ia i F(x i ), (8) where the dimensionless variables x i, which satisfy the relation < x i < 1, i x i = 1, correspond to the longitudinal momentum fractions carried by the quarks inside the baryon. The integration measure is defined as Dx = 1 dx 1 dx 2 dx 3 δ(x 1 + x 2 + x 3 1). (9) Then the invariant functions S i, P i, V i, A i, T i can be expressed in terms of the DAs S i, P i, V i, A i, T i with a definite twist. For scalar and pseudo-scalar distributions the following relations hold: S 1 = S 1, 2p z S 2 = S 1 S 2, P 1 = P 1, 2p z P 2 = P 2 P 1, (1) 6

8 for vector distributions: V 1 = V 1, 2p zv 2 = V 1 V 2 V 3, 2V 3 = V 3, 4p zv 4 = 2V 1 + V 3 + V 4 + 2V 5, (11) 4p zv 5 = V 4 V 3, (2p z) 2 V 6 = V 1 + V 2 + V 3 + V 4 + V 5 V 6, for axial vector distributions: A 1 = A 1, 2p za 2 = A 1 + A 2 A 3, 2A 3 = A 3, 4p za 4 = 2A 1 A 3 A 4 + 2A 5, (12) 4p za 5 = A 3 A 4, (2p z) 2 A 6 = A 1 A 2 + A 3 + A 4 A 5 + A 6, and, finally, for tensor distributions: T 1 = T 1, 2p zt 2 = T 1 + T 2 2T 3, 2T 3 = T 7, 2p zt 4 = T 1 T 2 2T 7, 2p zt 5 = T 1 + T 5 + 2T 8, (2p z) 2 T 6 = 2T 2 2T 3 2T 4 + 2T 5 + 2T 7 + 2T 8, (13) 4p zt 7 = T 7 T 8, (2p z) 2 T 8 = T 1 + T 2 + T 5 T 6 + 2T 7 + 2T 8. For Σ +( ), the identity of the two u(d) quarks in the baryon gives symmetry properties of the DAs. The Lorentz decomposition on the γ-matrix structure implies that the vector and tensor DAs are symmetric, whereas the scalar, pseudo-scalar and axial-vector DAs are antisymmetric under the interchange of the two u(d) quarks: V i (1, 2, 3) = V i (2, 1, 3), T i (1, 2, 3) = T i (2, 1, 3), S i (1, 2, 3) = S i (2, 1, 3), P i (1, 2, 3) = P(2, 1, 3), A i (1, 2, 3) = A(2, 1, 3). (14) The calligraphic structures in Eq. (2) have the similar relationships. For Λ, the similar relationships hold from the isospin symmetry that the vector and tensor DAs are antisymmetric, while the scalar, pseudoscalar and axial-vector DAs are symmetric: V i (1, 2, 3) = V i (2, 1, 3), T i (1, 2, 3) = T i (2, 1, 3), 7

9 S i (1, 2, 3) = S i (2, 1, 3), P i (1, 2, 3) = P(2, 1, 3), A i (1, 2, 3) = A(2, 1, 3). (15) 2.2 Representation in terms of chiral fields This subsection gives the DAs representation in terms of chiral fields. The discussion is mainly about Σ + baryon, and the counterparts of the others are similar. In terms of quark fields with definite chirality: q ( ) = 1 2 (1 ± γ 5)q, (16) the DAs can be interpreted transparently. Projection on the state where the spins of the two u quarks are antiparallel, that is u u, singles out vector and axial vector amplitudes, while the two u quarks are parallel, that are u u and u u, single out scalar, pseudo-scalar and tensor structures. Similar as expressions in Ref. [1], the DAs can be defined in terms of chiral fields. The leading twist-3 distribution amplitude can be defined as: ) ǫ (u ijk i (a 1z)C zu j (a 2z) zs k (a 3z) P = 1 2 pz zσ+ Dxe ipz P x i a i Φ 3 (x i ), (17) and the distributions for twist-4 are: ) ǫ (u ijk i (a 1z)C zu j (a 2z) ps k (a 3z) P = 1 2 pz pσ+ Dxe ipz P x i a i Φ 4 (x i ), ) ǫ (u ijk i (a 1z)C zγ pu j (a 2z) γ zs k (a 3z) P = pzm zσ + Dxe ipz P x i a i Ψ 4 (x i ), ) ǫ (u ijk i (a 1z)C p zu j (a 2z) zs k (a 3z) P = 1 2 pzm zσ+ Dxe ipz P x i a i Ξ 4 (x i ), (18) and the distributions for twist-5 are similarly written as: ) ǫ (u ijk i (a 1z)C pu j (a 2z) zs k (a 3z) P = 1 4 M2 zσ + Dxe ipz P x i a i Φ 5 (x i ), ) ǫ (u ijk i (a 1z)C pγ zu j (a 2z) γ p s k (a 3z) P = pzm p Σ + Dxe ipz P x i a i Ψ 5 (x i ), 8

10 ) ǫ (u ijk i (a 1z)C z p u j (a 2z) ps k (a 3z) P = 1 2 pzm p Σ+ Dxe ipz P x i a i Ξ 5 (x i ), (19) and finally the twist-6 one can be expressed as: ) ǫ (u ijk i (a 1z)C pu j (a 2z) ps k (a 3z) P = 1 4 M2 pσ + Dxe ipz P x i a i Φ 6 (x i ). (2) Different from the nucleon, here the isospin of the Σ baryon is 1. Thus there are not similar relationships as that in Ref. [1] from the isospin symmetry to reduce the number of the independent DAs. So the following additional chiral fields representations are needed to get all the DAs: ) ǫ (u ijk i (a 1z)C p zu j (a 2z) zs k (a 3z) P = 1 2 pzm zσ+ Dxe ipz P x i a i (S 1 P 1 + T 3 + T 7 ), ) ǫ (u ijk i (a 1z)C z p u j (a 2z) ps k (a 3z) P = 1 2 pzm p Σ+ Dxe ipz P x i a i (S 2 P 2 T 4 + T 8 ), (21) and ) ǫ (u ijk i (a 1z)Ciσ z u j (a 2z) γ zs k (a 3z) P = 2pz zσ + Dxe ipz P x i a i T 1 (x i ), ) ǫ (u ijk i (a 1z)Ciσ z u j (a 2z) γ ps k (a 3z) P = 2pz pσ + Dxe ipz P x i a i T 2 (x i ), ) ǫ (u ijk i (a 1z)Ciσ p u j (a 2z) γ zs k (a 3z) P = M 2 zσ + Dxe ipz P x i a i T 5 (x i ), ) ǫ (u ijk i (a 1z)Ciσ p u j (a 2z) γ ps k (a 3z) P = M 2 pσ + Dxe ipz P x i a i T 6 (x i ). (22) The twist classification of these additional DAs are shown in Tab. 1. The following denotations (S 1 P 1 + T 3 + T 7 )(x i ) Ξ 4(x i ), and (S 2 P 2 T 4 + T 8 )(x i ) Ξ 5(x i ) are adopted for convenience. The similar relationships hold for the Λ baryon under the exchange u, u, s to u, d, s, and for the Σ baryon under the exchange u, u, s to d, d, s. 9

11 3 Conformal expansion The spirit of the conformal expansion of distribution amplitudes is similar to the partial wave expansion of a wave function in quantum mechanics. The idea is to use the conformal symmetry of the QCD Lagrangian to study the DAs, which allows to separate longitudinal degrees of freedom from transverse ones [1]. The transverse coordinates are replaced by the renormalization scale, which is determined by the renormalization group. While the dependence on longitudinal momentum fractions, which is living on the light cone, is taken into account by a set of orthogonal polynomials that form an irreducible representation of the collinear subgroup SL(2, R) of the conformal group. As to leading logarithmic accuracy, the renormalization group equations are driven by tree-level counter terms, they have the conformal symmetry. This leads to the fact that the components of the DAs with different conformal spin do not mix under renormalization to this accuracy. The SL(2, R) group is governed by four generators P +, M +, D and K, where the definition is used for a vector A: A + = A µ z µ and A = A µ p µ /p z. The four generators P µ, K µ, D and M µν are the translation, special conformal transformation, dilation and Lorentz generators, respectively. The generators of the collinear subgroup SL(2, R) can be described by the following four operators: L + = ip +, L = i 2 K, L = i 2 (D M +), E = i(d + M + ). (23) For a field living on the light cone Φ(z), the acting of the above generators on it yields the following relations: [L 2, Φ(z)] = j(j 1)Φ(z), [E, Φ(z)] = (l s)φ(z), [E,L 2 ] =, [E,L ] =, (24) and L 2 = L 2 L + L + L, (25) where j = (l + s)/2 is called the conformal spin and t = l s is the twist. In the above notations, l is the canonical dimension of the quark field and s is the quark spin projection 1

12 on the light-cone. The role of the generator E is analogous to the Hamiltonian in quantum mechanics, and the twist corresponds to the eigenvalue of the Hamiltonian. For a given twist distribution amplitude, it can be expanded by the conformal partial wave functions that are the eigenstates of L 2 and L. For multi-quark states we need to deal with the problem of summation of conformal spins, and here the group is non-compact. The distribution amplitude with the lowest conformal spin j min = j 1 + j 2 + j 3 of a three-quark state is [3, 4] Φ as (x 1, x 2, x 3 ) = Γ[2j 1 + 2j 2 + 2j 3 ] Γ[2j 1 ]Γ[2j 2 ]Γ[2j 3 ] x 1 2j 1 1 x 2 2j 2 1 x 3 2j 3 1. (26) Contributions with higher conformal spin j = j min + n (n = 1, 2,...) are given by Φ as multiplied by polynomials that are orthogonal over the weight function (26). In this paper the calculation just considers DAs to the leading order conformal spin accuracy. For DAs in Tab. 1, we give their conformal expansion: for twist 3 and Φ 3 (x i ) = 12x 1 x 2 x 3 φ 3 (µ), T 1(x i ) = 12x 1 x 2 x 3 φ 3 (µ), (27) Φ 4 (x i ) = 24x 1 x 2 φ 4 (µ), Ψ 4(x i ) = 24x 1 x 3 ψ 4 (µ), Ξ 4 (x i ) = 24x 2 x 3 ξ 4(µ), Ξ 4(x i ) = 24x 2 x 3 ξ 4 (µ), T 2 (x i ) = 24x 1 x 2 φ 4 (µ), (28) for twist 4 and Φ 5 (x i ) = 6x 3 φ 5 (µ), Ψ 5(x i ) = 6x 2 ψ 5 (µ), Ξ 5 (x i ) = 6x 1 ξ 5 (µ), Ξ 5 (x i) = 6x 1 ξ 5 (µ), T 5 (x i ) = 6x 3 φ 5(µ), (29) for twist 5 and Φ 6 (x i ) = 2φ 6 (µ), T 6(x i ) = 2φ 6 (µ). (3) 11

13 for twist 6. There are altogether 14 parameters which can be determined by the equations of motion. 3.1 DAs of the Σ baryon The normalization of the DAs of Σ + are determined by matrix element of the local threequark operator. The Lorentz decomposition of the matrix element can be expressed explicitly as follows: 4 ǫ ijk u i α ()uj β ()sk γ () Σ+ (P) = V 1 ( PC) αβ(γ 5 Σ + ) γ + V 3 (γ µc) αβ (γ µ γ 5 Σ + ) γ +T 1 (P ν iσ µν C) αβ (γ µ γ 5 Σ + ) γ + T 3 M(σ µν C) αβ (σ µν γ 5 Σ + ) γ. (31) Similar to definitions in Ref. [1], the above four parameters can be expressed by the following matrix elements: ǫ [ ijk u i ()C zu j () ] γ 5 zs k () P = f Σ +pz zσ + (P), ǫ [ ijk u i ()Cγ µ u j () ] γ 5 γ µ s k () P = λ 1 MΣ + (P), ǫ [ ijk u i ()Cσ µν u j () ] γ 5 σ µν s k () P = λ 2 MΣ + (P), ǫ [ ijk u i ()Ciq ν σ µν u j () ] γ 5 γ µ s k () P = λ 3 M qσ + (P). (32) As is mentioned above, there is no relations derived from isospin symmetry, so four but not three matrix elements are needed to determine the four parameters V 1, V 3, T 1, and T 3. After a simple calculation, we arrive at the following expressions of V 1, V 3, T 1, and T 3 with the four parameters defined in Eqs. (32): V 1 = f Σ +, V 3 = 1 4 (f Σ + λ 1), T 1 = 1 18 (λ 2 + 4λ 3 ), T 3 = 1 36 (2λ 3 λ 2 ). (33) At the same time, the coefficients of the operators in Eqs. (27)-(3) can be expressed to the leading order conformal spin accuracy as φ 3 = φ 6 = f Σ +, ψ 4 = ψ 5 = 1 2 (f Σ + λ 1), 12

14 φ 4 = φ 5 = 1 2 (f Σ + + λ 1), φ 3 = φ 6 = ξ 5 = 1 18 (λ 2 + 4λ 3 ), φ 4 = ξ 4 = 1 18 (8λ 3 λ 2 ), φ 5 = ξ 5 = 1 6 λ 2, ξ 4 = 1 6 (4λ 3 λ 2 ). (34) 3.2 DAs of the Λ baryon The Lorentz decomposition of the local matrix element of Λ can be expressed explicitly as follows: 4 ǫ ijk u i α()d j β ()sk γ() Λ(P) = S 1C αβ (γ 5 Λ) γ + P 1(γ 5 C) αβ Λ γ +A 1 ( Pγ 5C) αβ Λ γ + A 3 M(γ µγ 5 C) αβ (γ µ Λ) γ. (35) In order to get the above parameters, the following matrix elements are used: ǫ [ ijk u i ()Cγ 5 zd j () ] zs k () P = f Λ (pz) zλ(p), ǫ [ ijk u i ()Cγ 5 γ µ d j () ] γ µ s k () P = λ 1 MΛ(P), ǫ [ ijk u i ()Cγ 5 d j () ] s k () P = λ 2 MΛ(P), ǫ [ ijk u i ()Cd j () ] γ 5 s k () P = λ 3 M 2 Λ(P). (36) A simple calculation leads to the following relationships: A 1 = f Λ, A 3 = 1 4 (f Λ λ 1 ), P 1 = λ 2, S 1 = λ 3. (37) To the leading order accuracy of the conformal spin expansion, the coefficients of the operators in Eqs. (27)-(3) for Λ can be expressed as φ 3 = φ 6 = f Λ, φ 4 = φ 5 = 1 2 (f Λ + λ 1 ), ψ 4 = ψ 5 = 1 2 (f Λ λ 1 ), ξ 4 = ξ 5 = λ 2 + λ 3, ξ 4 = ξ 5 = λ 3 λ 2. (38) 13

15 4 Determination of the parameters in QCD sum rule 4.1 QCD sum rules for Σ baryon The determination of the nonperturbative parameters f Σ, λ 1, λ 2 and λ 3 of the Σ baryon can be done in the framework of QCD sum rule. The method is carried out from the following correlation functions for Σ + : Π i (q 2 ) = d 4 xe iq x Tj i (x) j i (), (39) with the definitions of the currents: j 1 (x) = ǫ ijk [u i (x)c zu j (x)]γ 5 zs k (x), (4) j 2 (x) = ǫ ijk [u i (x)cγ µ u j (x)]γ 5 γ µ s k (x), (41) j 3 (x) = ǫ ijk [u i (x)cσ µν u j (x)]γ 5 σ µν s k (x), (42) and the forth current: j 4 (x) = ǫ ijk [u i (x)ciq ν σ µν u j (x)]γ 5 γ µ s k (x). (43) Inserting the complete set of states with the same quantum numbers as those of Σ +, the hadronic representations of the correlation functions are given as Π 1 (q 2 ) = 2fΣ 2 +(q 1 z)3 z M 2 q + 2 Π 2 (q 2 ) = M 2 λ 2 q + M 1 M 2 q + 2 Π 3 (q 2 ) = M 2 λ 2 q + M 2 M 2 q + 2 Π 4 (q 2 ) = q 2 M 2 λ 2 q + M 3 M 2 q + 2 s s s s ρ h 2 (s) s q 2ds, ρ h 3 (s) s q 2ds, ρ h 1 (s) s q 2ds, ρ h 4 (s) s q2ds. (44) On the operator product expansion (OPE) side the calculation takes into account condensates up to dimension 8. To give the sum rules, we utilize the dispersion relationship and assume the quark-hadron duality. After taking Borel transformation on both sides of 14

16 the hadronic representation and QCD expansion and equating the two sides, the final sum rules are given as follows: and and and 4(2π) 4 fσ 2 M 2 +e M B 2 = 1 5 s 4(2π) 4 λ 2 1M 2 e M 2 s M B 2 = + b 6 s(1 x) 5 e s M B 2 ds b s 6 s x(1 x)(1 2x) s e s 2 [(1 x)(1 + x)(1 8x + x 2 ) 12x 2 ln x]e (1 x) 2 e s M 2 B ds a2 e m2 M 2 B + 2a s m s s s 2(2π) 4 λ 2 2 M2 e M2 /MB 2 = s 2 ( 1 + 8x 8x 3 + x x 2 ln x)e + b 3 s (1 x)(4 7x)e s M 2 B ds 12m s a s s s (4π) 4 λ 2 3 M2 e M2 /MB 2 = s 2 {[(1 x)(1 + x)(1 8x + x 2 ) 12x 2 ln x] (1 x)5 }e s M B2 ds + b a 2 e m 2 s s M B2 + 8m s a s s e s e e s M 2 B ds, (45) s M B 2 ds s M 2 B ds, (46) s M B 2 ds s M 2 B ds, (47) (1 x)(11 5x 4x 2 )e s M B2 ds M 2 B ds. (48) where the notation x = /s is used, and the other parameters employed are the standard values: a = (2π) 2 ūu =.55 GeV 3, b = (2π) 2 α s G 2 /π =.47 GeV 4, a s = (2π) 2 ss = m 2 a, ūg c σ Gu = m 2 ūu, and m 2 =.8 GeV 2. As the usual way of the sum rule, the auxiliary Borel parameter M 2 B should have a proper range in which the results of the sum rules vary mildly with it. On the one hand the Borel parameter is expected to be large so that the higher order dimensional contributions are suppressed, and on the other hand the Borel parameter needs to be small enough to suppress the higher resonance contributions. Fig. 1 shows the dependence of the above parameters on the Borel parameter MB 2. The window of Borel parameter is choose as 1 GeV 2 M 2 B 2 GeV2, in which our results are acceptable. 15

17 To determine the relative sign of f Σ + and λ 1, we give the sum rule of f Σ +λ 1: 4(2π) 4 f Σ +λ 1Me M 2 M B 2 = m s 3 b 6 m s s s s[(1 x)(3 + 13x 5x 2 + x 3 ) + 12x lnx]e 1 (1 x)(2 5x) (1 x)[1 + ]e s M B 2 ds + 4 s 3x 3 a s s e s s M B 2 ds M 2 B ds. (49) It is similar that the relative sign of λ 2 and λ 3 to λ 1 can be given by the following two sum rules: and (2π) 4 (λ 1 λ 2 + λ 1 λ 2)M 3 e M 2 M B2 = 12m s a s (2π) 4 (λ 1 λ 3 + λ 1λ 3 )M 3 e M 2 M B2 = (1 + m 2 /s)(1 x)2 e s M B2 ds, (5) s {as(1 x) 2 (2 + x) + m2 a 2 [1 3 2 (1 x)(1 + x) + (1 x)(13 25x + 2x2 )]}e s M B2 ds. (51) Fig. 2 gives the dependence of the above sum rules on the Borel parameter M 2 B. The final numerical values of the coupling constants of Σ + are: f Σ + = (9.3 ±.3) 1 3 GeV 2, λ 1 = (2.9 ±.1) 1 2 GeV 2, λ 2 = (2.8 ±.3) 1 2 GeV 2, λ 3 = (2.6 ±.3) 1 2 GeV 2, (52) and parameters of Σ are: f Σ = (9.4 ±.3) 1 3 GeV 2, λ 1 = (2.9 ±.1) 1 2 GeV 2, λ 2 = (2.8 ±.3) 1 2 GeV 2, λ 3 = (2.6 ±.3) 1 2 GeV 2. (53) 4.2 QCD sum rules for Λ baryon The sum rules of the Λ baryon parameters begins with the following correlation functions: Π i (q 2 ) = d 4 xe iq x Tj i (x) j i (), (54) 16

18 with the definitions of the currents: j 1 (x) = ǫ [ ijk u i (x)cγ 5 zd j (x) ] zs k (x), j 2 (x) = ǫ [ ijk u i (x)cγ 5 γ µ d j (x) ] γ µ s k (x), j 3 (x) = ǫ [ ijk u i (x)cγ 5 d j (x) ] s k (x), j 4 (x) = ǫ [ ijk u i (x)cd j (x) ] γ 5 s k (x). (55) The similar processes as in the above subsection lead to the following results: and and and (4π) 4 fλ 2 M 2 e M B 2 = 2 s s(1 x) 5 e s M B 2 ds b s 5 m 2 3 4(2π) 4 λ 2 1 M2 e M 2 M B 2 = 1 s 2 + b 12 s 4(4π) 4 λ 2 2 M2 e M 2 s M B 2 = + b 3 s 4(4π) 4 λ 2 3M 2 e M 2 s M B 2 = + b 3 s and the sum rule of f Λ λ 1 is (4π) 4 f Λ λ M 2 1 Me M B 2 = 2 s 3 m s + b 3 m s 1 s M x(1 x)(1 2x)e B 2 ds, (56) s 2 [(1 x)(1 + x)(1 8x + x 2 ) 12x 2 lnx]e (1 x) 2 e s M 2 B ds 4 3 a2 e m2 s M 2 B + m s a s s s 2 [(1 x)(1 + x)(1 8x + x 2 ) 12x 2 ln x]e (1 x)(1 + 5x)e s M 2 B ds a2 e M 2 B + 4m s a s s s s M B 2 ds e sm2 B ds, (57) e s 2 [(1 x)(1 + x)(1 8x + x 2 ) 12x 2 ln x]e (1 x)(1 + 5x)e s M 2 B ds + 32 s 3 a2 e M 2 B + 4m s a s s e s[(1 x)(3 + 13x 5x 2 + x 3 ) + 12x ln x]e 1 (1 x)(2 5x) (1 x)[1 + ]e s M B 2 ds a s s 3x 6 17 s M B 2 ds s M 2 B ds, (58) s M B 2 ds s s e s M 2 B ds, (59) s M B 2 ds M 2 B ds. (6)

19 Note that the sum rules of λ 2 and λ 3 are the same. In Fig. 3, the sum rules of the parameters on the Borel parameter M 2 B are shown. The final numerical results for the parameters of Λ are: f Λ = (6. ±.3) 1 3 GeV 2, λ 1 = (1. ±.4) 1 2 GeV 2, λ 2 = (1. ±.1) 1 2 GeV 2, λ 3 = (1. ±.1) 1 2 GeV 2. (61) In the above results, f Λ and λ 1 have the same sign, which is different from that shown in Ref. [11]. The relative sign of λ 1 and λ 2 can not be determined by the method presented above. Here we only list the absolute values of the two parameters. 5 Application:electromagnetic form factors of the baryons with light-cone QCD sum rule 5.1 LCSR for the electromagnetic form factors The EM form factors of hadrons are the fundamental objects for understanding their internal structures. There were a lot of investigations on various hadrons both experimentally and theoretically, including meson [24, 25, 26, 27, 28, 29] and baryon [3, 31, 32, 33, 34, 35, 36, 37, 38, 39]. While as there were few experimental data and theoretical investigations, the EM form factors of the baryons such as Σ and Λ and so on have not received much attention in the past years. In the previous work [4] we gave an investigation on the EM form factors of the Λ baryon. In this section, the EM form factors of the Σ baryon is investigated in the frame work of light-cone QCD sum rule method with the help of the DAs obtained in this paper, and the magnetic moments of the same baryons are estimated by comparing our results with the existing dipole formula. The matrix element of the magnetic current of a baryon between the baryon states can be expressed as the Dirac and Pauli form factors F 1 (Q 2 ) and F 2 (Q 2 ), respectively: Σ(P, s) j em µ () Σ(P, s ) = Σ(P, s)[γ µ F 1 (Q 2 ) i σ µνq ν 2M F 2(Q 2 )]Σ(P, s ), (62) where j em µ = e u ūγ µ u + e s sγ µ s is the electromagnetic current relevant to the hadron. And P, s and P, s are the four-momenta and the spins of the initial and the final Σ baryon 18

20 states, respectively. From the experimental viewpoint, the EM form factors can be expressed by the electric G E (Q 2 ) and magnetic G M (Q 2 ) Sachs form factors: G M (Q 2 ) = F 1 (Q 2 ) + F 2 (Q 2 ), G E (Q 2 ) = F 1 (Q 2 ) Q2 4M 2F 2(Q 2 ), (63) and at the point Q 2 = the magnetic G M (Q 2 ) form factor gives the magnetic moment of the baryon: G M () = µ Σ. (64) To evaluate the magnetic moment of the baryon from its EM form factors, the magnetic form factor G M (Q 2 ) is assumed to be described by the dipole formula: 1 µ Σ G M (Q 2 ) = 1 (1 + Q 2 /m 2 ) 2 = G D(Q 2 ). (65) As there is no information about the parameter m 2 from experimental data, the two parameters m 2 and µ Σ are estimated simultaneously by fitting the magnetic form factor by the dipole formula (65). The calculation mainly focus on Σ + baryon and it is similar for the calculation of Σ. The process of the derivation begins with the correlation function: T µ (P, q) = i d 4 xe iq x T {j Σ +()j em µ (x)} Σ+ (P, s). (66) The hadronic representation of the correlation function is acquired by inserting a complete set of states with the same quantum numbers as those of Σ + : z µ T µ (P, q) = 1 M 2 Σ P 2f Σ +(P z)[2(p zf 1 (Q 2 ) q z 2 F 2(Q 2 )) z + +(P zf 2 (Q 2 ) + q z 2 F 2(Q 2 z q )) M Σ + ]Σ + (P, s) +..., (67) where P = P q, and the dots stand for the higher resonances and continuum contributions. Here the correlation function is contracted with z µ to get rid of contributions proportional to z µ which is subdominant on the light cone. While on the theoretical side, 19

21 the correlation function (66) can be calculated to the leading order of α s as { 1 z µ T µ = (P z) 2 ( zσ + 1 ) γ 4e u dα 2 s p {B (α 2 2 ) + M2 (s p 2 ) B 1(α 2 ) M (s P 2 ) 2B 2(α 2 )} + 2e s dα 3 s 2 P {C (α 2 3 ) + M2 (s 2 P 2 ) C M 4 } 1(α 3 ) 2 (s 2 P 2 ) 2C 2(α 3 )} { 1 +(P z) 2 M( z qσ + ) γ 4e u M 2 1 dα 2 1 α 2 (s P 2 ) 2 { D 1(α 3 ) +2 (s P 2 ) B 2(α 2 )} + e s M 2 } +2 (s 2 P 2 ) C 2(α 3 )}, (68) dα 3 1 α 3 (s 2 P 2 ) 2 { E 1(α 3 ) where s = (1 α 2 )M 2 + (1 α 2) α 2 Q 2 and s 2 = (1 α 3 )M 2 + (1 α 3) Q 2 + m2 s α 3. Here the following notations are used for convenience: B (α 2 ) = 1 α2 α 3 dα 1 V 1 (α 1, α 2, 1 α 1 α 2 ), B 1 (α 2 ) = (2Ṽ1 Ṽ2 Ṽ3 Ṽ4 Ṽ5)(α 2 ), B 2 (α 2 ) = ( Ṽ 1 + Ṽ 2 + Ṽ 3 + Ṽ 4 + Ṽ 5 Ṽ 6 )(α 2 ), C (α 3 ) = 1 α3 dα 1 V 1 (α 1, 1 α 1 α 3, α 3 ), C 1 (α 3 ) = (2Ṽ1 Ṽ2 Ṽ3 Ṽ4 Ṽ5)(α 3 ), C 2 (α 3 ) = ( Ṽ 1 + Ṽ 2 + Ṽ 3 + Ṽ 4 + Ṽ 5 Ṽ 6 )(α 3 ), D 1 (α 2 ) = (Ṽ1 Ṽ2 Ṽ3)(α 2 ), E 1 (α 3 ) = (Ṽ1 Ṽ2 Ṽ3)(α 3 ), (69) where Ṽ i (α 2 ) = Ṽ i (α 2 ) = α2 1 α dα 2 2 α2 α dα 2 2 dα 1 V i (α 1, α 2, 1 α 1 α 2 ), 1 α dα 2 2 dα 1 V i (α 1, α 2, 1 α 1 α 2 ), 2

22 α3 1 α Ṽ i (α 3 ) = dα 3 3 dα 1 V i (α 1, 1 α 1 α 3, α 3), α3 α Ṽ i (α 3 ) = dα α dα 3 3 dα 1 V i (α 1, 1 α 1 α 3, α 3). (7) Then equating both sides of the Borel transformed version of hadronic and theoretical representations with the assumption of quark-hadron duality, the final sum rules are given as follows: 2f Σ +F 1 (Q 2 )e M 2 1 M B2 = 4e u α 2 dα 2 e s } M 2 B {B (α 2 ) + M2 2 M B 1(α 2 ) M4 4 B M B 2(α 2 ) B +4e u e s α 2 { M 2 B 2 M2 B α2m Q 2 1 (α 2 ) M2 M 2 B } B 2 (α 2 ) α e u e s α 2 M 2 B 2M 4 d B α2 2 M2 + Q 2 2 (α 2 ) dα 2 α2 2 M2 + Q e s dα 3 e s 2 } M 2 B {C (α 3 ) + M2 2 α 3 M C 1(α 3 ) M4 4 B M C 2(α 3 ) B +2e s e s M B 2 α 2 3M 2 α 2 3 M2 + Q e s e s M B 2 α 2 3 M4 } {C 1 (α 3 ) M2 C 2 (α 3 ) M 2 B α 2 3 d C α3m Q 2 + m 2 2 (α 3 ) s dα 3 α3m Q 2 + (71) for F 1 (Q 2 ) and f Σ +F 2 (Q 2 )e M 2 { 1 M B2 = M 2 4e u α 2 dα 2 e s M B 2 1 α 2 M B 2 { D 1(α 2 ) + M2 M B 2 D 2(α 2 )} 4e u e s α M 2 2 B α2m Q {D 1(α 2 2 ) M2 4e u e s M B 2 α 2 2M 2 α 2 2 M2 + Q 2 +2e s 1 2e s e s M 2 B 2e s e s M 2 B α 3 dα 3 e s 2 1 M 2 B α 3 M 2 B B 2 (α 2 )} α 2 d B 2 (α 2 ) dα 2 α2 2 M2 + Q 2 2 α 3 M { E 1(α 3 ) + M2 2 B M B 2(α 3 )} B {E 1 (α 3 ) M2 C MB 2 2 (α 3 )} α3 2 M2 + Q 2 + α3m 2 2 d α } 3 C α3m Q 2 + m 2 2 (α 3 ) s dα 3 α3m Q

23 (72) for F 2 (Q 2 ). 5.2 Numerical analysis In the numerical analysis for the form factors, the continuum threshold is chosen as s = ( ) GeV 2. The masses of the Σ baryons from [41] are M Σ + = GeV and M Σ = GeV. The parameters f Σ and λ 1 are used as the central values in Eqs. (52) and Eqs. (53). For the auxiliary Borel parameter MB 2, there should be a region where the sum rules are almost independent of it. To choose a platform for the Borel parameter, we should suppress both resonance contributions and the higher twist contributions simultaneously. Fig. 4 shows the dependence of the magnetic form factors on the Borel parameter at different points of Q 2. Our results are acceptable in the range 2. GeV 2 M 2 B 4. GeV 2. The estimation on the magnetic moment of the baryon comes from the fitting of the magnetic form factor by the dipole formula (65). In the following analysis the Borel parameter is chosen to be MB 2 = 3 GeV2. Fig. 5 gives the dependence of the magnetic form factor G M (Q 2 ) on the momentum transfer at different points of the threshold s. The figure shows that G M (Q 2 ) decreases with Q 2, which is in accordance with the assumption in Eq. (65). To estimate the magnetic moment numerically, the magnetic form factor G M (Q 2 ) is fitted by the formula µ Σ /(1 + Q 2 /m 2 )2, which is described by the dashed lines in Fig. 5. From the figures the magnetic moment of Σ + is given as µ Σ + = (3.35 ±.11)µ N, and the estimation of the other parameter is m 2 = (.89 ±.4) GeV2. The similar process is carried out for the numerical analysis of Σ. The estimation of the Σ magnetic moment is shown in Fig. 6, which are the magnetic form factors of the baryon at different threshold and the fittings by the dipole formula. The numerical values from the fittings are µ Σ = (1.59 ±.2)µ N and m 2 = (.79 ±.3) GeV2. Tab. 2 lists magnetic moments of the two baryons from various approaches: data from Particle Data Group (PDG) [41]; QCD sum rules [42] (SR(1) for χ = 3.3 and SR(2) 22

24 for χ = 4.5); QCD string approach (QCDSA) [43]; chiral perturbation theory (χpt) [44]; Skyrme model (SKRM) [45]; light cone sum rules [46] (LCSR(1) for χ = 3.3 and LCSR(2) for χ = 4.5). The table shows that our results are larger in absolute values than the others. This may partly lie in the factor that more detailed information on DAs calls for higher order conformal expansion. At the same time, the choice of the interpolating currents may affect the results, for they couple both to the spin s = 1 and spin s = baryons [15, 16]. The estimation is expected to be better if more information about the DAs are known and higher order QCD coupling O(α s ) effect are included. Finally, the dependence of G M (Q 2 )/µ Σ G D (Q 2 ) on Q 2 is given in Fig. 7. In the numerical analysis, the input parameters m 2 used in the dipole formula (65) are the central values obtained above, which are m 2 =.89 GeV2 for Σ + and m 2 =.79 GeV2 for Σ, while the magnetic moments used come from [41], which are µ Σ + = 2.458µ N and µ Σ = 1.16µ N. 6 Summary In this paper we present the DAs of baryons with quantum number I(J P ) = 1( 2+ 1 ) (for Σ ± ) and I(J P ) = ( 1 2+ ) (for Λ) up to twist 6. We find that fourteen independent DAs are needed to describe the structure of the baryons. The method employed is based on the conformal partial wave expansion, and the parameters of the nonlocal nonperturbative parameters are determined in the QCD sum rule framework. Our calculation on the conformal expansion of the DAs is to the leading order conformal spin accuracy. Compared with the previous work [11], the calculation on the Λ baryon gives DAs of all other Lorentz structures besides axial-like vector structures. Another new result is that the relative sign of the two parameters f Λ and λ 1 are positive. With the DAs obtained in this work, the EM form factors of Σ are investigated in the range 1 GeV 2 Q 2 7 GeV 2. We assume that the magnetic form factor can be described by the dipole formula. By fitting the result by the dipole formula, the magnetic moments of the baryons are estimated, which are µ Σ + = (3.35±.11)µ N, and µ Σ = (1.59±.2)µ N. Compared with values given by Particle Data Group [41], our results are larger in absolute 23

25 values. This shows that our calculation needs more detailed information on the DAs, which may come from higher order conformal spin contributions, and at the same time our choice of the interpolating currents, which couple both to the spin s = 1 and spin s = 3 baryon, 2 2 may also affect our estimation. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Contract No Appendices In the appendices we give our results on the DAs of Σ and Λ explicitly. As the definition in (7), our results are list in the following subsections. A DAs of the Σ baryon Twist-3 distribution amplitudes of Σ are: Twist-4 distribution amplitudes are: V 1 (x i ) = 12x 1 x 2 x 3 φ 3, A 1(x i ) =, T 1 (x i ) = 12x 1 x 2 x 3 φ 3. (73) S 1 (x i ) = 6(x 2 x 1 )x 3 (ξ 4 + ξ 4 ), P 1 (x i ) = 6(x 2 x 1 )x 3 (ξ 4 ξ 4 ), V 2 (x i ) = 24x 1 x 2 φ 4, A 2(x i ) =, V 3 (x i ) = 12x 3 (1 x 3 )ψ 4, A 3(x i ) = 12x 3 (x 1 x 2 )ψ 4, T 2 (x i ) = 24x 1 x 2 φ 4, T 3 (x i ) = 6x 3 (1 x 3 )(ξ 4 + ξ 4 ), T 7 (x i ) = 6x 3 (1 x 3 )(ξ 4 ξ4 ). (74) Twist-5 distribution amplitudes are: S 2 (x i ) = 3 2 (x 1 x 2 )(ξ 5 + ξ 5 ), P 2 (x i ) = 3 2 (x 1 x 2 )(ξ 5 ξ 5 ), 24

26 V 4 (x i ) = 3(1 x 3 )ψ 5, A 4 (x i ) = 3(x 1 x 2 )ψ 5, V 5 (x i ) = 6x 3 φ 5, A 5(x i ) =, T 4 (x i ) = 3 2 (x 1 + x 2 )(ξ 5 + ξ 5 ), T 5(x i ) = 6x 3 φ 5, T 8 (x i ) = 3 2 (x 1 + x 2 )(ξ 5 ξ 5). (75) And finally twist-6 distribution amplitudes are: B DAs of the Λ baryon V 6 (x i ) = 2φ 6, A 6 (x i ) =, T 6 (x i ) = 2φ 6. (76) Twist-3 distribution amplitudes of Λ are: V 1 (x i ) =, A 1 (x i ) = 12x 1 x 2 x 3 φ 3, T 1 (x i ) =. (77) Twist-4 distribution amplitudes are: S 1 (x i ) = 6x 3 (1 x 3 )(ξ 4 + ξ 4 ), P 1 (x i ) = 6(1 x 3 )(ξ 4 ξ 4 ), V 2 (x i ) =, A 2 (x i ) = 24x 1 x 2 φ 4, V 3 (x i ) = 12(x 1 x 2 )x 3 ψ 4, A 3 (x i ) = 12x 3 (1 x 3 )ψ 4, T 2 (x i ) =, T 3 (x i ) = 6(x 2 x 1 )x 3 ( ξ 4 + ξ 4 ), T 7 (x i ) = 6(x 1 x 2 )x 3 (ξ 4 + ξ 4 ). (78) Twist-5 distribution amplitudes are: S 2 (x i ) = 3 2 (x 1 + x 2 )(ξ 5 + ξ 5 ), P 2 (x i ) = 3 2 (x 1 + x 2 )(ξ 5 ξ 5 ), V 4 (x i ) = 3(x 2 x 1 )ψ 5, A 4(x i ) = 3(1 x 3 )ψ 5, 25

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31 Fig. 1. Dependence of the four parameters f Σ +, λ 1, λ 2 and λ 3 on the Borel parameter MB 2, where the lines correspond to the threshold s = GeV 2 from bottom up. Fig. 2. Sum rules of the relative signs of the parameters on the Borel parameter, where the threshold is used as s = GeV 2. Fig. 3. Dependence of the four parameters f Λ, λ 1, λ 2 and f Λ λ 1 of Λ on the Borel parameter MB 2, where the lines correspond to the threshold s = GeV 2 from bottom up. Fig. 4. The dependence of the form factor G M (Q 2 ) of Σ on the Borel parameter at different momentum transfer, where the lines correspond to the points Q 2 = 1, 2, 3, 5, 6 GeV 2 from the up down (left Σ + ) and from the bottom up(right Σ ), respectively. Fig. 5. Fittings of the form factor G M (Q 2 ) by the dipole formula µ Σ /(1 + Q 2 /m 2 ) 2, where the dashed lines are the fittings. And figures (a), (b) correspond to the threshold s = 2.7, 2.9 GeV 2, respectively. Fig. 6. Fittings of the form factor G M (Q 2 ) by the dipole formula µ Σ /(1 + Q 2 /m 2 )2, where the dashed lines are the fittings. And figures (a), (b) correspond to the threshold s = 2.7, 2.9 GeV 2, respectively. Fig. 7. The dependence of the form factor G M (Q 2 )/(µ Σ G D (Q 2 )) on Q 2, where the lines correspond to the threshold s = 2.7, 2.8, 2.9 GeV 2 from bottom up. The left corresponds to Σ + and the right corresponds to Σ. 3

32 Table 1: Independent baryon distribution amplitudes that enter the expansion in ((17) to (22)). Lorentz-structure Light-cone projection nomenclature twist-3 (C z) z u + u+ s+ Φ 3 (x i ) = [V 1 A 1 ] (x i ) (Ciσ z ) γ z u + u+ s+ T 1 (x i ) twist-4 (C z) p u + u+ s Φ 4 (x i ) = [V 2 A 2 ] (x i ) (C zγ p ) γ z u + u s+ Ψ 4 (x i ) = [V 3 A 3 ](x i ) (C p z) z u u+ s+ Ξ 4 (x i ) = [T 3 T 7 + S 1 + P 1 ] (x i ) (C p z) z u u+ s+ Ξ 4 (x i) = [T 3 + T 7 + S 1 P 1 ] (x i ) (Ciσ z ) γ p u + u+ s T 2 (x i ) twist-5 (C p) z u u s+ Φ 5 (x i ) = [V 5 A 5 ] (x i ) (C pγ z ) γ p u u+ s Ψ 5 (x i ) = [V 4 A 4 ](x i ) (C z p) p u + u s Ξ 5 (x i) = [ T 4 T 8 + S 2 + P 2 ](x i ) (C z p) p u + u s Ξ 5 (x i ) = [S 2 P 2 T 4 + T 8 ] (x i ) (Ciσ p ) γ z u u s+ T 5 (x i ) twist-6 (C p) p u u s Φ 6 (x i ) = [V 6 A 6 ] (x i ) (Ciσ p ) γ p u u s T 6 (x i ) 31

33 Table 2: Magnetic moments of the Σ baryons from various models µ(µ N ) PDG SR(1) SR(2) QCDSA χpt SKRM LCSR(1) LCSR(2) Ours µ Σ µ Σ

34 f Σ (GeV 2 ) λ 2 (GeV 2 ) x M B 2(GeV 2 ) M B 2(GeV 2 ) λ 1 (GeV 2 ) λ 3 (GeV 2 ) M B 2(GeV 2 ) M B 2(GeV 2 ) Figure 1: 33

35 f Σ λ 1 *(GeV 2 ) x M B 2(GeV 2 ) (λ 1 λ 2 *+λ 1 *λ 2 )(GeV 4 ) M B 2(GeV 2 ) (λ 1 λ 3 *+λ 1 *λ 3 )(GeV 4 ) M B 2(GeV 2 ) Figure 2: 34

36 f Λ (GeV 2 ) λ 2 (GeV 2 ) x M B 2(GeV 2 ) M B 2(GeV 2 ) λ 1 (GeV 2 ) f Λ λ 1 *(GeV 2 ) x M B 2(GeV 2 ) M B 2(GeV 2 ) Figure 3: 35

37 1 G M (Q 2 )(GeV 2 ) M B 2(GeV 2 ) G M (Q 2 )(GeV 2 ) M B 2(GeV 2 ) Figure 4: 36

38 G M (Q 2 ).6.4 G M (Q 2 ) Q 2 (GeV 2 ) Q 2 (GeV 2 ) (a) (b) Figure 5: 37

39 G M (Q 2 ) G M (Q 2 ) Q 2 (GeV 2 ) (a) Q 2 (GeV 2 ) (b) Figure 6: 38

40 µ Σ G M (Q 2 )/G D (Q 2 ) Q 2 (GeV 2 ) µ Σ G M (Q 2 )/G D (Q 2 ) Q 2 (GeV 2 ) Figure 7: 39