Higher Order Corrections to the Light-cone Distribution Amplitudes of the Sigma baryon
aa r X i v : . [ h e p - ph ] J a n Higher order corrections to the light-cone distributionamplitudes of the sigma baryon
Yong-Lu Liu , Chun-Yu Cui , and Ming-Qiu Huang College of Science, National University of Defense Technology, Hunan 410073, China and Department of Physics, School of Biomedical Engineering,Third Military Medical University, Chongqing 400038, China (Dated: September 16, 2018)The light-cone distribution amplitudes (LCDAs) of the Σ ± baryons up to twist-6are investigated on the basis of the QCD conformal partial wave expansion approach.The calculations are carried out to the next-to-leading order of conformal spin accu-racy. The nonperturbative parameters relevant to the LCDAs are determined in theframework of the QCD sum rule method. The explicit expressions of the LCDAs aregiven as the main results. PACS numbers: 11.25.Hf, 11.55.Hx, 13.40.Gp, 14.20.Jn.
I. INTRODUCTION
Signals confirmed by ATLAS and CMS[1, 2] showed that the Higgs boson[3] in thestandard model (SM) have been found and the SM is most likely to be a precise theoryat the present energy scale. New physics beyond the SM at higher energy scale is mostlyconcerned nowadays and in the near future. However, many difficulties are still alivein practical analysis of hadron physics involving nonperturbative QCD effect when westudy hadronic phenomena at low energy, or Λ
QCD scale. A typical method to solve thenonperturbative difficulties in QCD is factorization, in which the nonperturbative part isincluded into the wave function, such as the parton distribution functions for inclusiveprocesses, fragmentation distribution functions for the hadronization, and the distributionamplitudes for exclusive processes. Specifically, in theoretical investigations of the hardexclusive processes [4, 5] and hadronic physics with the QCD light-cone sum rule method[6–8], the light-cone distribution amplitudes (LCDAs) are fundamental ingredients to bestudied. Furthermore, when searching for new physics beyond the SM, it is an importantway to study flavor physics, in which some processes that are sensitive to the new physicscan be measured more precisely nowadays than any time before. All of these requiredetailed information of the internal structure and the dynamical properties of the hadron,which are dominated by the nonperturbative QCD characters.In the past decades, many efforts have been made in the descriptions of mesons[9] andthe nucleon[10–15], whereas, theoretical studies of a large number of the hadron physicsphenomena require us to know LCDAs of many other hadrons such as the octet baryons,the decuplet baryons, and some excited hadron states that are difficult to be determinedexperimentally at present. We have examined the LCDAs of the strange octet baryons inthe previous work[16] in the conformal spin expansion method[12, 18, 19]. Our calculationconcerns LCDAs to twist-6 to the accuracy of the leading order of conformal spin expan-sion. The obtained parameters are also used to analyze some hadronic physics processesas applications[20–23]. However, some of the investigations[20, 22] have implied that cor-rections from the higher order conformal spin contributions may affect the results to someextent.In the point view of applications, an important effect to the LCDAs is the correctionof the higher twist distribution functions. The higher order twist contributions to LCDAshave several origins, among which the main one comes from “bad” components in thewave function and in particular of components with “wrong” spin projection for the caseof baryons [11, 16]. Compatible with the previous works, we focus on higher order twistcontributions from bad components in the decomposition of the Lorentz structure in thispaper. One of the general descriptions of LCDAs is based on the conformal symmetryof the massless QCD Lagrangian dominated on the light cone. The conformal partialwave expansion of the LCDAs can be carried out safely in the limit of the SU (3)-flavorsymmetry approach. However, when terms connected with the s -quark mass are considered,the SU (3)-flavor breaking effects need to be included. In the present work, effects from the SU (3)-flavor symmetry breaking are considered as the corrections, which originate fromtwo sources: isospin symmetry breaking and corrections to the nonperturbative parameters.It is known that the leading order contribution with the conformal spin expansionapproach comes from the properties of the matrix elements of the local three-quark operatorbetween the vacuum and the baryon state. Thus, it is natural that higher order correctionsshould be related to the expansion of the matrix elements of the nonlocal three-quarkoperator at the zero point. However, we still need to estimate how much the contributionsfrom four-particle effects will do on the result. Fortunately we have known that for processeswhose dominant contribution is from the light cone the four-particle contributions canbe safely omitted in the lower leading order. Thus, in the following analysis we onlyconsider contributions from three-quark operator matrix element, whose higher moment iscalculated with QCD sum rules[17].As applications, the light-cone QCD sum rule method has been used to examine pro-cesses related to the strange octet baryons and give instructive estimates [23, 24]. In theprevious works, we have analyzed some physical processes related to the final states aboutthe Σ baryon. The results are compatible with the experiments and(or) the other the-oretical predications[21]. Nevertheless, there are still some processes which are not welldescribed [20, 22]. We wish the higher order corrections from the higher conformal spinmay give us more accurate estimates.The rest of the paper is organized as follows. Section II is devoted to present thedefinitions of the higher order moment of the three-quark operators related to correctionsof LCDAs from the higher conformal spin expansion. In Sec. II B, the conformal partialwave expansion of the LCDAs is carried out by use of the conformal symmetry of themassless QCD Lagrangian. The nonperturbative parameters connected with the LCDAsare determined in Sec. III with the QCD sum rule method. Finally, we give the explicitexpressions of the Σ baryon LCDAs in Sec. IV. A summary is given is Sec. V. The equationsof motion which are used to reduce the number of the free parameters are presented inAppendix A for the completeness of this paper. The sum rule of one coupling constant V s is analyzed in Appendix B as an example to elucidate the principal process of this method,and the other sum rules can be carried out in the same way. II. HIGHER CONFORMAL EXPANSION OF THE LIGHT-CONEDISTRIBUTION AMPLITUDES OF Σ A. General definition
Matrix elements of the quark-quark or quark-gluon-quark field operator between vacuumor hadron states are the great important ingredients in analysis of processes in quantumfield theory. Light-cone distribution amplitudes of the Σ baryon are defined by the generalLorentz expansion of the matrix element of the nonlocal three-quark-operator between thevacuum and the baryon state h | ǫ ijk q iα ( a z ) q jβ ( a z ) s kγ ( a z ) | Σ( P ) i , (1)where q represents u or d quark, which correspond to Σ + or Σ − baryon, respectively. Theindices α, β, γ refer to Lorentz indices and i, j, k represent color ones. It is noticed that tomake the matrix element above gauge invariant, the gauge factor [ x, y ] = P exp[ ig s R dt ( x − y ) µ A µ ( tx +(1 − t ) y )] need to be inserted, whereas when fixed-point gauge ( x − y ) µ A µ ( x − y ) =0 is adopted, this factor is equal to unity. Thus in this paper we do not show them explicitly.Taking into account the Lorentz covariance, spin and parity properties of the baryons,the matrix element (1) is generally decomposed as4 h | ǫ ijk q iα ( a z ) q jβ ( a z ) s kγ ( a z ) | Σ( P ) i = X i F i Γ αβ i (cid:16) Γ i Σ (cid:17) γ , (2)where Σ γ is the spinor of the baryon with the quantum number I ( J P ) = 1(
12 + ) ( I is theisospin, J is the total angular momentum, and P is the parity), Γ i are certain Diracstructures over which the sum is carried out, and F i = S i , P i , A i , V i , T i are the independentdistribution amplitudes which are functions of the scalar product P · z [16]. It is also noticedthat z and p are the two light-cone vectors which satisfy z = 0 and p = 0.Functions defined above do not have definite twist. In order to classify the LCDAsaccording to the definite twist, we redefine the wave functions F i in the infinite momentumframe as: 4 h | ǫ ijk s iα ( a z ) s jβ ( a z ) q kγ ( a z ) | Σ( P ) i = X i F i Γ ′ αβ i (cid:16) Γ ′ i Σ (cid:17) γ . (3)A naive calculation shows that the invariant functions S i , P i , V i , A i , T i can be expressed interms of the LCDAs F i = S i , P i , V i , A i , T i . The two sets of definitions have the followingrelations: S = S , p · z S = S − S , P = P , p · z P = P − P , V = V , p · z V = V − V − V , V = V , p · z V = − V + V + V + 2 V , p · z V = V − V , (2 p · z ) V = − V + V + V + V + V − V (4)for scalar, pseudoscalar, and vector structure, and A = A , p · z A = − A + A − A , A = A , p · z A = − A − A − A + 2 A , p · z A = A − A , (2 p · z ) A = A − A + A + A − A + A (5)for axial-vector structure, and T = T , p · z T = T + T − T , T = T , p · z T = T − T − T , p · z T = − T + T + 2 T , (2 p · z ) T = 2 T − T − T + 2 T + 2 T + 2 T , p · z T = T − T , (2 p · z ) T = − T + T + T − T + 2 T + 2 T (6)for tensor structure.The classifications of the LCDAs F i with a definite twist are listed in Table I, where wetake Σ + as an example. The explicit expressions of the definition can be found in Refs.[11, 16]. Each distribution amplitude F i can be represented as F ( a i p · z ) = Z D xe − ipz P i x i a i F ( x i ) , (7)where the dimensionless variables x i , which satisfy the relations 0 < x i < P i x i = 1,correspond to the longitudinal momentum fractions along the light cone carried by thequarks inside the baryon. The integration measure is defined as Z D x = Z dx dx dx δ ( x + x + x − . (8)There exist some symmetry properties of the LCDAs from the identity of the two u/d quarks in the Σ baryon, which is useful to reduce the number of the independent functions. TABLE I: Independent baryon distribution amplitudes in the chiral expansion.Lorentz-structure Light-cone projection NomenclatureTwist-3 ( C z ) ⊗ 6 z u + ↑ u + ↓ s + ↑ Φ ( x i ) = [ V − A ] ( x i )( Ciσ ⊥ z ) ⊗ γ ⊥ z u + ↑ u + ↑ s + ↓ T ( x i )Twist-4 ( C z ) ⊗ 6 p u + ↑ u + ↓ s −↑ Φ ( x i ) = [ V − A ] ( x i )( C zγ ⊥ p ) ⊗ γ ⊥ z u + ↑ u −↓ s + ↓ Ψ ( x i ) = [ V − A ] ( x i )( C p z ) ⊗ 6 z u −↑ u + ↑ s + ↑ Ξ ( x i ) = [ T − T + S + P ] ( x i )( C p z ) ⊗ 6 z u −↓ u + ↓ s + ↑ Ξ ′ ( x i ) = [ T + T + S − P ] ( x i )( Ciσ ⊥ z ) ⊗ γ ⊥ p u + ↓ u + ↓ s −↓ T ( x i )Twist-5 ( C p ) ⊗ 6 z u −↑ u −↓ s + ↑ Φ ( x i ) = [ V − A ] ( x i )( C pγ ⊥ z ) ⊗ γ ⊥ p u −↑ u + ↓ s −↓ Ψ ( x i ) = [ V − A ] ( x i )( C z p ) ⊗ 6 p u + ↑ u −↑ s −↑ Ξ ( x i ) = [ − T − T + S + P ] ( x i )( C z p ) ⊗ 6 p u + ↓ u −↓ s −↑ Ξ ′ ( x i ) = [ S − P − T + T ] ( x i )( Ciσ ⊥ p ) ⊗ γ ⊥ z u −↓ u −↓ s + ↓ T ( x i )Twist-6 ( C p ) ⊗ 6 p u −↑ u −↓ s −↑ Φ ( x i ) = [ V − A ] ( x i )( Ciσ ⊥ p ) ⊗ γ ⊥ p u −↑ u −↑ s −↓ T ( x i ) Taking into account the Lorentz decomposition of the γ -matrix structure, it is easy to seethat the vector and tensor LCDAs are symmetric, whereas the scalar, pseudoscalar andaxial-vector structures are antisymmetric under the interchange of the two u/d quarks: V i (1 , ,
3) = V i (2 , , , T i (1 , ,
3) = T i (2 , , ,S i (1 , ,
3) = − S i (2 , , , P i (1 , ,
3) = − P (2 , , ,A i (1 , ,
3) = − A (2 , , . (9)The similar relationships hold for the “calligraphic” structures in Eq. (2).In order to expand the LCDAs by the conformal partial waves, we rewrite the LCDAs interms of quark fields with definite chirality q ↑ ( ↓ ) = (1 ± γ ) q . Taking Σ + as an example, theclassification of the LCDAs in this presentation can be interpreted transparently: projectionon the state with the two u -quarks antiparallel, i.e. u ↑ u ↓ , singles out vector and axial vectorstructures, while parallel ones, i.e. u ↑ u ↑ and u ↓ u ↓ , correspond to scalar, pseudoscalar andtensor structures. The explicit expressions of the LCDAs by chiral-field representations arepresented in Table I as an example for Σ + . The counterparts of Σ − can be easily obtainedunder the exchange u → d .Note that in the case of the nucleon, the isospin symmetry can be used to reduce thenumber of the independent LCDAs to eight[11]. However, there are no similar isospinsymmetric relationships existing when the Σ baryon is considered. Therefore, we needaltogether 14 chiral field representations to express all the LCDAs. B. Conformal expansion
In this subsection we give the explicit form of the LCDAs with the aid of the conformalpartial wave expansion approach. The main idea of this method is based on the conformalsymmetry of the massless QCD Lagrangian. In this approach the longitudinal degreesof freedom can be separated from transverse ones. On the one hand, the properties oftransverse coordinates are described by the renormalization scale that is determined by therenormalization group equation. On the other hand, the longitudinal momentum fractionsthat are living on the light cone are governed by a set of orthogonal polynomials, whichform an irreducible representation of the collinear subgroup SL (2 , R ) of the conformalgroup.The algebra of the collinear subgroup SL (2 , R ) is determined by the following fourgenerators: L + = − i P + , L − = i K − , L = − i D − M − + ) , E = i ( D + M − + ) , (10)where P µ , K µ , D , and M µν correspond to the translation, special conformal transforma-tion, dilation and Lorentz generators, respectively. The notations are used for a vector A : A + = A µ z µ and A − = A µ p µ /p · z . Let L = L − L + L + L − , then a given distributionamplitude with a definite twist can be expanded by the conformal partial wave functionsthat are the eigenstates of L and L .For the three-quark state, the distribution amplitude with the lowest conformal spin j min = j + j + j is [18, 19]Φ as ( x , x , x ) = Γ[2 j + 2 j + 2 j ]Γ[2 j ]Γ[2 j ]Γ[2 j ] x j − x j − x j − , (11)where j i represents the conformal spin of the quark field that is defined as half of thecanonical dimension plus its spin j = ( l + s ) /
2. Contributions with higher conformal spin j = j min + n ( n = 1 , , ... ) are given by Φ as multiplied by polynomials that are orthogonalover the weight function (11). For LCDAs in Table I, we give their conformal expansions:Φ ( x i ) = 120 x x x [ φ + φ − ( x − x ) + φ +3 (1 − x ) + ... ] ,T ( x i ) = 120 x x x [ t + t − ( x − x ) + t +1 (1 − x ) + ... ] (12)for twist-3 and Φ ( x i ) = 24 x x [ φ + φ − ( x − x ) + φ +4 (1 − x ) + ... ] , Ψ ( x i ) = 24 x x [ ψ + ψ − ( x − x ) + ψ +4 (1 − x ) + ... ] , Ξ ( x i ) = 24 x x [ ξ + ξ − ( x − x ) + ξ +4 (1 − x ) + ... ] , Ξ ′ ( x i ) = 24 x x [ ξ ′ + ξ ′− ( x − x ) + ξ ′ +4 (1 − x ) + ... ] ,T ( x i ) = 24 x x [ t + t − ( x − x ) + t +2 (1 − x ) + ... ] (13)for twist-4 and Φ ( x i ) = 6 x [ φ + φ − ( x − x ) + φ +5 (1 − x ) + ... ] , Ψ ( x i ) = 6 x [ ψ + ψ − ( x − x ) + ψ +5 (1 − x ) + ... ] , Ξ ( x i ) = 6 x [ ξ + ξ − ( x − x ) + ξ +5 (1 − x ) + ... ] , Ξ ′ ( x i ) = 6 x [ ξ ′ + ξ ′− ( x − x ) + ξ ′ +5 (1 − x ) + ... ] ,T ( x i ) = 6 x [ t + t − ( x − x ) + t +5 (1 − x ) + ... ] (14)for twist-5, and Φ ( x i ) = 2[ φ + φ − ( x − x ) + φ +6 (1 − x ) + ... ] ,T ( x i ) = 2[ t + t − ( x − x ) + t +6 (1 − x ) + ... ] (15)for twist-6. Up to now there are altogether 42 parameters which need to be determined.To the next-to-leading order, the normalization of the Σ baryon LCDAs is determinedby the matrix element of the nonlocal three-quark operator expanded at the zero point.The decomposition of the matrix element is h | ǫ ijk u iα ( a z ) u jβ ( a z ) s kγ ( a z ) | Σ( P ) i = h | ǫ ijk u iα ( a z ) u jβ ( a z ) s kγ ( a z ) | Σ( P ) i + z λ h | [ ǫ ijk u iα ( a z ) ↔ D u jβ ( a z )] s kγ ( a z ) | Σ( P ) i + z λ h | ǫ ijk u iα ( a z ) u jβ ( a z )[ ~Ds kγ ( a z )] | Σ( P ) i . (16)The Lorentz decomposition of the matrix element can be expressed explicitly as4 h | ǫ ijk s iα (0) s jβ (0) q kγ (0) | Σ( P ) i = V ( P C ) αβ ( γ Σ) γ + V ( γ µ C ) αβ ( γ µ γ Σ) γ + T ( P ν iσ µν C ) αβ ( γ µ γ Σ) γ + T M ( σ µν C ) αβ ( σ µν γ Σ) γ (17)for the matrix element of the leading order, and4 h | ǫ ijk u iα ( a z ) u jβ ( a z )[ ~Ds kγ ( a z )] | Σ( P ) i = V s ( P C ) αβ ( γ Σ) γ + V M ( P C ) αβ ( γ λ γ Σ) γ + V s P λ M ( γ µ C ) αβ ( γ µ γ Σ) γ + V M ( γ λ C ) αβ ( γ Σ) γ + V M ( γ µ C ) αβ ( iσ µλ γ Σ) γ + T s P λ ( P ν iσ µν C ) αβ ( γ µ γ Σ) γ + T M ( P ν iσ λν C ) γ Σ γ + T s M P λ ( σ µν C ) αβ ( σ µν γ Σ) γ + T M ( P ν σ µν C ) αβ ( σ µλ γ Σ) γ + T M ( iσ µλ C ) αβ ( γ µ γ Σ) γ + T M ( σ µν C ) αβ ( σ µν γ λ γ Σ) γ , (18)4 h | [ ǫ ijk u iα ( a z ) ↔ D u jβ ( a z )] s kγ ( a z ) | Σ( P ) i = S u P λ M C αβ ( γ Σ) γ + S M C αβ ( γ λ γ Σ) γ + P u P λ M ( γ C ) αβ Σ γ + P M ( γ C ) αβ ( γ λ Σ) γ + A u P λ ( P γ C ) αβ Σ γ + A M ( P γ C ) αβ γ λ Σ γ + A P λ M ( γ µ γ C ) αβ γ µ Σ γ + A M ( γ λ γ C ) αβ Σ γ + A M ( γ µ γ C ) αβ iσ µλ Σ γ (19)for the next leading order expansion. There are altogether 24 nonperturbative parametersin the expressions. However, we need not so many free parameters because there are someconstraints to reduce the freedom of the coefficients. It is noticed that all the parametersdefined above are not independent and can be reduced with the help of the motion ofequation, which can be seen in Appendix A.Choosing V , V , V s , V s , T , T , T s , T s , A u , A u , S u , P as the independent parameters,the other ones can be expressed with them: V = ( V s − V s ) , V = 116 (4 V − V − V s + 2 V s ) , V = ( − V + 4 V + 3 V s − V s ) , T = 110 (3 S u − T + 6 T + 2 T s − T s ) , T = ( S u − T s + 2 T + 4 T s − T s ) , T = −T s , T = (5 P − S u + T − T − T s + 24 T s ) , A = 14 (4 A u − V − V s + 6 V s ) , A = ( − A u − A u + 4 V + V s − V s ) , A = 148 (4 V + 20 V + 3 V s + 14 V s ) , S = ( − P + 3 S u + 7 T + 6 T + 2 T s − T s ) , P u = ( −S u + T − T − T s + 24 T s ) . (20)Recall the relations of the leading order, there are altogether 12 parameters to be de-termined. To this end, we introduce the additional eight decay constants defined by thefollowing matrix elements of a three-quark operator with a covariant derivative: h | ǫ ijk (cid:2) u i (0) C zu j (0) (cid:3) γ z ( iz ~Ds k )(0) | Σ( P ) i = f Σ V s ( P · z ) z Σ( P ) γ , h | ǫ ijk h u i (0) C zγ iz ↔ D u j (0) i zs k (0) | Σ( P ) i = − f Σ A u ( P · z ) z Σ( P ) γ , h | ǫ ijk (cid:2) u i (0) Cγ u u j (0) (cid:3) γ zγ u ( iz ~Ds k )(0) | Σ i = λ f s ( P · z ) M z Σ( P ) γ , h | ǫ ijk (cid:2) u i (0) Cσ µν u j (0) (cid:3) γ zσ µν ( iz ~Ds k )(0) | Σ i = − λ f s ( P · z ) M z Σ( P ) γ , h | ǫ ijk h u i (0) Cγ µ γ iz ↔ D u j (0) i zγ µ s k (0) | Σ( P ) i = − λ f u ( P · z ) M z Σ( P ) γ , h | ǫ ijk (cid:2) u i (0) iP ν Cσ µν u j (0) (cid:3) γ z ( iz ~Ds k )(0) | Σ( P ) i = − λ f s ( P · z ) M z Σ( P ) γ , h | ǫ ijk h u i (0) Ciz ↔ D u j (0) i γ s k (0) | Σ( P ) i = S u ( P · z ) M Σ( P ) − S M ( z Σ( P )) γ , h | ǫ ijk h u i (0) Ciz ↔ D γ u j (0) i s k (0) | Σ( P ) i = P u ( P · z ) M Σ( P ) + P M ( z Σ( P )) γ . (21)It is noticed that each of the last two matrix element have two different Lorentz structureswhich permit us to get two different sum rules; whereas the calculations also indicate thatthe sum rules from the last two ones are the same, so we can get the necessary equationsfrom the two different sum rules.We also need another four decay constant defined by the leading order local operatormatrix element which has been calculated in the previous paper [16] h | ǫ ijk (cid:2) u i (0) C zu j (0) (cid:3) γ zs k (0) | P i = f Σ P · z z Σ( P ) , h | ǫ ijk (cid:2) u i (0) Cγ µ u j (0) (cid:3) γ γ µ s k (0) | P i = λ M Σ( P ) , h | ǫ ijk (cid:2) u i (0) Cσ µν u j (0) (cid:3) γ σ µν s k (0) | P i = λ M Σ( P ) , h | ǫ ijk (cid:2) u i (0) Ciq ν σ µν u j (0) (cid:3) γ γ µ s k (0) | P i = λ M q Σ( P ) . (22)A simple calculation gives the relations between the local nonperturbative parameters V ,si , A ui , T ,si , P , S u and the decay constants defined in Eqs. (21) and (22): f Σ = V , λ = V − V ,λ = 6 T − T , λ = 3 T − T ,f Σ V s = V s , f Σ A u = A u ,λ f s = −V s + 4 V s + 2 V , λ f s = 6 T s − T − T s − T ,λ f u = A u + 2 A + 4 A u , λ f s = 3 T s + T − T s − T + 4 T + 12 T . (23)We also noticed that S u and P are defined directly by the matrix element and can bedetermined by the following method. Up to now we can express all the independentparameters by the nonperturbative decay constants defined in Eqs. (21) and (22): V = f Σ , V = 14 ( f Σ − λ ) , V s = f Σ V s , V s = 12 f s λ , T = 16 ( − λ + 4 λ ) , T = 112 ( − λ + 2 λ ) , T s = 6233 P − S u + 3199 λ − f s λ − λ − f s λ , T s = 411 P − S u + 233 λ − f s λ − λ − f s λ , A u = f Σ A u , A u = 112 ( f Σ − f A u + f Σ V s − λ − f s λ + 2 f u λ ) . (24)Further calculation shows that coefficients in Eqs. (12)-(15) can be expressed to thenext-to-leading order conformal spin accuracy as φ = φ = V , ψ = ψ = 2 V ,φ = φ = V − V , t = ξ ′ = − ξ = T ,t = t = ξ = − ξ ′ = T − T , − ξ = t = T (25)0for leading order and φ +3 = 216 V − V s , φ +6 = 2 V − V s + 12 V − V − V ,t +1 = 12 (7 T − T s ) , φ +4 = 32 ( V − V ) −
152 ( V s − V − V s ) ,t +6 = 2 T − T s + 12 T − T , t − = t − = t − = t − = 0 φ +5 = 5 V − V − V s − V + 20 V + 20 V s ,t +5 = 5 T + 20 T − T s − T − T ,t +2 = 32 ( T − T ) − T s − T − T s ) ,φ − = − A u , φ − = − A u + 2 A + 2 A + 2 A ) ,φ − = 152 ( A u + 2 A − A u ) , φ − = − A u + 2 A − A + 2 A u ) ,ψ +4 = 152 ( V s − A u ) − V , ψ − = 152 ( V − A u ) − V s ,ψ +5 = 40( V s + 2 V − A u + 2 A ) , ψ − = 10( V − V s − V − A u + 2 A ) ,ξ ′ +4 = 3(2 S u − P u − T s + 2 T ) , ξ ′− = − S u − P u ) − T + 9( T s − T ) ,ξ +4 = 6( S u + P u ) + 310 ( T − T + 10 T − T s + 8 T s ) ,ξ − = − S u + P u ) −
910 ( 1310 T − T + 10 T − T s + 8 T s ) ,ξ +5 = 20( T − T ) − T s + 2 T + T − T ) + 5( S u + P u − S + 2 P ) ,ξ − = − T − T − T ) + 45( T s + 2 T + T − T ) + 5( S u + P u − S + 2 P ) ,ξ ′ +5 = 40( T − T ) − T s + 2 T + T − T ) + 5( S u − P u − S − P ) ,ξ ′− = − T − T ) − T − T ) + 90( T s + 2 T + T − T )+5( S u − P u − S − P ) (26)for the next-to-leading order. III. NUMERICAL ANALYSIS OF THE SUM RULES FOR THENONPERTURBATIVE PARAMETERS
The nonperturbative parameters appearinf in the above section can be determined withQCD sum rules [17]. The QCD sum rule approach is a well-used tool to estimate unknownphysical parameters which are connected with the nonperturbative effects in low energyscale of strong interaction. Early in the 1980s the QCD sum rules were used to calculatethe moments of the meson and baryon LCDAs [25]. The detailed analysis of the sum rulesfor V s is presented in Appendix B as an example for the approach. Analysis of other sumrules are the same as the example. In this section we only present the explicit expressionsof the parameters from this method. It is noticed that the parameters related with the1leading order conformal spin expansion have been obtained in Ref. [16]. Herein we onlypresent the next-to-leading order ones. The sum rules are as follows: • The sum rule for V s is2 f V s e − M /M B = Z s m s e − s/M B ρ ( s ) ds + Π cond. , (27)where ρ ( s ) = 15 × × π s (1 − x ) (1 + 2 x ) + h g G i × π s x (1 − x )+ h g G i × π s x (1 − x ) (1 − x ) , (28)and Π cond. = m ( m − m s )3 × π h ¯ ss i M B − m s × π h ¯ sg · σGs i M B (1 + m s M B ) , (29)where x = m s /s , m s is the strange quark mass, M is the mass of Σ and M B is theBorel parameter. • The sum rule for A u is2 f A u e − M /M B = Z s m s e − s/M B ρ ( s ) ds + Π cond. , (30)where ρ ( s ) = h g G i × π s x (1 − x ) , (31)and Π cond. = h ¯ sσ · Gs i × π m s M B − h ¯ sσ · Gs i × π m s M B . (32) • The sum rule for f s is λ M f s e − M /M B = Z s m s e − s/M B ρ ( s ) ds + Π cond. , (33)where ρ ( s ) = − s × × π {
12 (1 − x )(9 − x + 119 x − x + 14 x )+30 x ln x } + h g G i × π (1 − x )(1 − x + 32 x ) , (34)Π cond. = m s M B π h ¯ ss i + m s ( m − m s )48 π h ¯ ss i − h ¯ qq i e − m s /M B (1 − m M B − m m s M B ) + m s h ¯ sgσ · Gs i × π (3 − m s M B ) . (35)2 • The independent sum rule for f s is − λ M f s e − M /M B = Z s m s e − s/M B ρ ( s ) ds + Π cond. , (36)where ρ ( s ) = − s × × π { [(1 − x )(16 − x + 31 x − x + 11 x ) − x ln x ] } − h g G i × π (1 − x )(19 + 223 x − x ) , (37)Π cond. = − m s l π h ¯ ss i ( M B −
16 ( m − m s )) + m s h ¯ sgσ · Gs i π (2 + m s M B ) . (38) • The independent sum rule for f u is − λ M f u e − M /M B = Z s m s e − s/M B ρ ( s ) ds + Π cond. , (39)where ρ ( s ) = s × × π { [(1 − x )(3 − x − x + 13 x − x ) − x ln x ] } + h g G i × π (1 − x ) (5 − x ) , (40)Π cond. = m s × π h ¯ ss i (2 M B − m + 2 m s ) − m s h ¯ sgσ · Gs i × π (3 + 2 m s M B ) . (41) • The independent sum rule for f s is − λ M f s e − M /M B = Z s m s e − s/M B ρ ( s ) ds + Π cond. , (42)where ρ ( s ) = − m s × π s { (1 − x )(9 − x − x − x + 4 x ) − x ln( x )] } + m s h g G i × π s (1 − x )(13 − x + 29 x − x ) − m s h g G i × π { (1 − x )(131 − x + 20 x ) + 72 ln x } , (43)Π cond. = ( m − m s ) M B π h ¯ ss i − m s h ¯ qq i e − m s /M B (1 + m M B − m m s M B )+ M B × π h ¯ sgσ · Gs i − m s × π h ¯ sgσ · Gs i . (44)3 • The sum rules of S u and S are f ∗ M S u e − M /M B = Z s m s e − s/M B ρ (1) ( s ) ds + Π (1) cond. , (45)where ρ (1) ( s ) = 15 × π s [ − (1 − x )(3 − x − x + 13 x − x ) + 60 x ln x ] − h g G i × π (1 − x ) (1 + 2 x ) + h g G i × π (1 − x ) , (46)Π (1) cond. = − m s π h ¯ ss i (4 M B − ( m − m s )) − m s h ¯ sgσ · Gs i × π (3 − m s M B ) , (47)and f ∗ M ( S u − S ) e − M /M B = Z s m s e − s/M B ρ (2) ( s ) ds + Π (2) cond. , (48)with ρ (2) ( s ) = 12 π s [(1 − x )(3 + 47 x + 11 x − x ) + 12 x (2 + 3 x ) ln x ]+ m s h g G i π [(1 − x ) − (1 − x )(2 + 5 x − x ) + 6 x ln xx ] − m s h g G i × π [2 (1 − x ) x + (1 − x )(3 − x ) + 2 ln x ] , (49)Π (2) cond. = M B π h ¯ ss i + 3( m − m s )32 π M B h ¯ ss i + h ¯ sgσ · Gs i × π ( m s + M B ) . (50)The calculation also shows that sum rules for P u and P are the same as that for S u and S . Therefore we do not show them explicitly in the text.In addition, we need to calculate the parameter f ∗ to get the numerical results of S u and S . The parameter f ∗ is defined by the following matrix element: h | ǫ ijk (cid:2) u i (0) Cu j (0) (cid:3) γ zs k (0) | Σ( P ) i = f ∗ z Σ . (51)In compliance with the standard procedure of QCD sum rules, we arrive at the final result:2 f ∗ e − M /M B = Z s m s e − s/M B ρ ( s ) ds + Π cond. , (52)with ρ ( s ) = − π [(1 − x )(3 − x ) + 2 ln x ] − h g G i × π (1 − x )( − x ) , (53)Π cond. = 23 h ¯ qq i e − m sM B + m s h ¯ ss i × π (3 M B − m + 2 m s ) . (54)4 TABLE II: Results from QCD sum rules of the nonperturbative parameters.Parameter V s A u f s f s M B ( GeV ) 0 . ∼ . ∼ . ∼ . . ∼ . . ± .
01 0 . ± . − . ± .
12 9 . ± . f u f s P ( GeV ) S u ( GeV ) M B ( GeV ) 0 . ∼ . . ∼ . . ∼ . . ∼ . − . ± .
01 1 . ± . . ± . − . ± . In fact, there are two different Lorentz structures which may give independent sum rulesfor most of the above parameters in the calculation. In practice we choose the proper oneswhich may contain more information of the hadrons and have good Borel working windows.Furthermore, in order to cancel uncertainties from auxiliary parameters such as Borel massand the threshold s as far as possible, we use the sum rules other than the central valuesin numerical analysis. For example, when analyzing Eq. (27), the parameter f is replacedby the sum rule obtained in Ref. [16].Before arriving at the final numerical values of the parameters from QCD sum rules, wefirst need to choose the working window of the Borel parameter, which is determined byrequiring that both the higher resonance contributions and the higher dimension contribu-tions are subdominant in comparison with the pole contributions. The choice of the Borelmass for different sum rules is presented in Table.II. Another important parameter in thesum rules is the threshold, by choosing which the higher resonance contribution can berepresented by the integration of the spectral density with the help of quark-hadron dual-ity. The threshold is usually connected with the first resonance having the same quantumnumber as the concerned composite particle. It is also required that the sum rule doesnot dependent on the threshold very much. With the above criterion, in the analysis weuse 2 .
65 GeV ≤ s ≤ .
85 GeV . Finally, the inputs of the vacuum condensates we usedare the standard values: a = − (2 π ) h ¯ uu i = 0 .
55 GeV , b = (2 π ) h α s G /π i = 0 .
47 GeV , a s = − (2 π ) h ¯ ss i = 0 . a , h ¯ ug c σ · Gu i = m h ¯ uu i , and m = 0 . . The mass of thestrange quark is used as m s = 0 .
15 GeV. In consideration of the isospin symmetry, theΣ mass is used the central value of Σ + presented by the Particle Data Group (PDG)[26]: M Σ + = 1 . IV. EXPLICIT EXPRESSIONS OF THE Σ LCDAS
In this section we present the explicit expressions of the Σ baryon LCDAs. By con-sidering expressions defined in (12) to (15), we first plot one of the twist-3 distribution5 - FIG. 1: Twist-3 distribution amplitudes Φ ( x i ) (left) and Twist-4 distribution amplitudesΦ ( x i ) (right). amplitude Φ ( x i ) and one of the twist-4 distribution amplitude Φ ( x ) in Fig. 1 as anexample.For the definition in (3), our results are listed as follows: Twist-3 distribution amplitudesof Σ are V ( x i ) = 120 x x x [ φ + φ +3 (1 − x )] , A ( x i ) = − x x x ( x − x ) φ − ,T ( x i ) = 120 x x x [ t + t − ( x − x ) + t +1 (1 − x )] . (55)Twist-4 distribution amplitudes are S ( x i ) = 6( x − x ) x ( ξ + ξ ′ + ξ +4 + ξ ′ +4 ) + 6( x − x ) x ( ξ − + ξ ′ − ) − x − x ) x ( ξ − + ξ ′ − ) ,P ( x i ) = 6( x − x ) x ( ξ − ξ ′ + ξ +4 − ξ ′ +4 ) + 6( x − x ) x ( ξ − − ξ ′ − ) − x − x ) x ( ξ − − ξ ′ − ) ,V ( x i ) = 24 x x [ φ + φ +4 (1 − x )] , A ( x i ) = − x x ( x − x ) φ − ,V ( x i ) = 12 x (1 − x )[ ψ + ψ +4 ] + 12[( x + x ) x − ( x + x ) x ] ψ − − x x x ψ +4 ,A ( x i ) = − x ( x − x )[ ψ + ψ − ] − x − x ) x ψ − + 12( x − x ) x ψ − ,T ( x i ) = 24 x x [ t + t − ( x − x ) + t +2 (1 − x )] ,T ( x i ) = 6 x (1 − x )( ξ + ξ ′ + ξ +4 + ξ ′ +4 ) + 6( x + x ) x ( ξ − + ξ ′ − ) − x + x ) x ( ξ − + ξ ′ − ) − x x x ( ξ +4 + ξ ′ +4 ) ,T ( x i ) = 6 x (1 − x )( − ξ + ξ ′ − ξ +4 + ξ ′ +4 ) + 6( x + x ) x ( − ξ − + ξ ′ − ) − x + x ) x ( − ξ − + ξ ′ − ) − x x x ( − ξ +4 + ξ ′ +4 ) . (56)6Twist-5 distribution amplitudes are S ( x i ) = 32 ( x − x )( ξ + ξ ′ + ξ +5 + ξ ′ +5 ) − x − x )( ξ +5 + ξ ′ +5 ) −
32 ( x − x ) x ( ξ − + ξ ′ − ) ,P ( x i ) = 32 ( x − x )( ξ − ξ ′ + ξ +5 − ξ ′ +5 ) − x − x )( ξ +5 − ξ ′ +5 ) −
32 ( x − x ) x ( ξ − − ξ ′ − ) ,V ( x i ) = 3(1 − x )[ ψ + ψ +5 ] + 6 x x ψ − − − x ) x ψ − − x + x ) ψ +5 ,A ( x i ) = 3( x − x )[ ψ + ψ +5 ] − x − x ) x ψ − + 6( x − x ) ψ +5 ,V ( x i ) = 6 x [ φ + φ +5 (1 − x )] , A ( x i ) = − x ( x − x ) φ − ,T ( x i ) = −
32 ( x + x )( ξ ′ + ξ + ξ ′ +5 + ξ +5 ) − x x ( ξ ′ − + ξ − )+ 32 (1 − x ) x ( ξ ′ − + ξ − ) + 3( x + x )( ξ ′ +5 + ξ +5 ) ,T ( x i ) = 6 x [ t + t − ( x − x ) + t +5 (1 − x )] ,T ( x i ) = 32 ( x + x )( ξ ′ − ξ + ξ ′ +5 − ξ +5 ) + 32 x x ( ξ ′ − + ξ − )+ 32 (1 − x ) x ( ξ ′ − − ξ − ) − x + x )( ξ ′ +5 − ξ +5 ) . (57)Finally twist-6 distribution amplitudes are V ( x i ) = 2[ φ + φ +6 (1 − x )] , A ( x i ) = − φ − ( x − x ) ,T ( x i ) = 2[ t + t − ( x − x ) + t +6 (1 − x )] . (58) V. SUMMARY
The main aim of this work is to present the explicit expressions of the Σ baryon light-conedistribution amplitudes. The LCDAs are examined up to twist-6 based on the conformalsymmetry of the massless QCD Lagrangian. The previous papers indicate that higherconformal spin expansion may contribute in some dynamical processes. Therefore we haveto deal with more nonperturbative parameters both at leading order and at next-to-leadingorder to give more detailed information of the LCDAs of the baryon.Although we can give a general definition of the LCDAs according to the Lorentz struc-ture of the nonlocal three-quark matrix element between vacuum and the baryon state, wefirst need to define the independent distribution amplitudes in a proper frame in order todescribe them with nonperturbative parameters of QCD. With the help of the conformalsymmetry, the LCDAs are redefined and expanded with the conformal spin to the next-to-leading (NL) order in terms of quark fields with definite chirality. In comparison withthe case of the nucleon, the number of the independent distribution functions of Σ is 14,7which come from the identity symmetry of the two u or d quarks. The NL corrections ofthe LCDAs come from the next-to-leading order expansion of the nonlocal three-quark op-erator matrix element. The matrix element is parametrized to the nonperturbative inputswhich are connected with the intrinsic properties of QCD. In the calculations, the requirednonperturbative inputs are determined in the QCD sum rule approach. We finally presentthe explicit expressions of the light-cone distribution amplitudes of the Σ baryon up totwist 6 as the main results of this paper. Acknowledgments
This work was supported in part by the National Natural Science Foundation of Chinaunder Contracts No.11105222 and No.11275268. One of the authors Y. L. Liu also thanksthe NSFC program (No.11391240184) and the International Centra of Theoretical Physics(ICTP) for financing the attendance of the summer school sm2463 and sm2466 held inItaly.
Appendix A: Equation of motion
There are altogether 24 coefficients when parametrizing the matrix element of the non-local three-quark operator. We wish to reduce the number of the independent parametersas far as possible. Fortunately there are relations from the equation of motion of the matrixelements of some different local composite operators. The same relations can be found inRef. [11]. In this paper we present them only for the completeness of the article and givethe direct results from the constraints of these equations. The constraints are: h | ǫ ijk u i (0) Cγ ρ u j (0) γ λ [ iD λ s γ ] k (0) | Σ , P i = 0 , h | ǫ ijk u i (0) Cγ λ u j (0)[ iD λ s γ ] k (0) | Σ , P i = P λ h | ǫ ijk u i (0) Cγ λ u j (0) s kγ (0) | Σ , P i , h | ǫ ijk u i (0) Cσ αβ u j (0) γ λ [ iD λ s γ ] k (0) | Σ , P i = 0 , h | ǫ ijk u i (0) Ciσ αβ u j (0)[ iD β s γ ] k (0) | Σ , P i = P β h ǫ ijk u i (0) Ciσ αβ u j (0) s k (0) | Σ , P i − h ǫ ijk [ u (0) Ci ↔ D α u (0)] ij s kγ (0) | Σ , P i , h | ǫ ijk u i (0) Ciγ σ αβ u j (0)[ iD β s γ ] k (0) | Σ , P i = P β h ǫ ijk u i (0) Cγ iσ αβ u j (0) s k (0) | Σ , P i − h | ǫ ijk [ u (0) Cγ i ↔ D α u (0)] ij s kγ (0) | Σ , P |i , h | ǫ ijk [ u (0) Cγ ρ γ ↔ D ρ u (0)] ij d kγ (0) | Σ , P i = 0 , h | ǫ ijk [ u (0) C { γ λ i ↔ D ρ − γ ρ i ↔ D λ } γ u (0)] ij s kγ (0) | Σ , P i = h | ǫ ijk [ u (0) C i { σ λρ γ α i ←− D α + γ α σ λρ i −→ D α } γ u (0)] ij s kγ (0) | Σ , P = − iǫ λραδ [ P α h | ǫ ijk u i (0) Cγ δ u j (0) d kγ (0) | Σ , P i − h | ǫ ijk u i (0) Cγ δ u j (0)[ iD α s γ ] k (0) | Σ , P ] . (A1)8A simple calculation leads to the following relationships: V s = 4 V + 2 V s , − V = 3 V s + V , V − V = V s − V s + 4 V − V , T s + T = 0 , T = −T s + 4 T s + 3 T , T − T − S = T s − T s − T + 3 T + 6 T T − T − S u = T s − T − T s − T , − T s + 2 T = − T − P u , T − P = 2 T s − T − T A u + A u + A + 4 A = 0 , A u − A = V + V − V s , A = V + V − V − V s . (A2) Appendix B: QCD sum rules of the nonperturbative parameters
In this Appendix we introduce the QCD sum rule method of the nonperturbative pa-rameters which are required in the paper. We take the process for the decay constant V s as an example. It starts from the following correlation function:Π( q ) = i Z d xe iq · z h| j s ( x )¯ j (0) |i , (B1)where j s ( x ) = ǫ ijk [ u i ( x ) C zu j ( x )] γ [ iz −→ D s ( x )] k , and j (0) = ǫ ijk [ u i (0) C zu j ( x )] γ s k (0).In compliance with the general process of the QCD sum rules, we need to calculate thecorrelation function both phenomenally and theoretically. On the phenomenon side, weinterpolate a complete set of states with the same quantum number as the Σ baryon to getthe hadronic representation Π( q ) = 2 f V s ( q · z ) zM − q + ..., (B2)where “...” represents contribution from higher resonances and continuum states. By mak-ing use of the dispersion relation, the equation above can be written as the integrationform, Π( q ) = 2 f V s ( q · z ) zM − q + Z ∞ s π Im Π( s ) s − q ds. (B3)On the theoretical side, we calculate the correlation function at the quark level by useof the operator product expansion (OPE). In the calculation we expand up to dimension 6accuracy. Then by hadron-quark duality approximation, the integration function in (B3)can be equalized by the spectral density calculated theoretically.As the two representations have the same content, they can be matched so as to getthe sum rule. Additionally, in order to make the numerical estimation more accurate, weuse the Borel transformation to suppress both higher resonances and higher dimensionalcontributions. The Borel transformation is defined asˆ B Q M B ≡ lim Q →∞ ,N →∞ ( − Q ) N ( ddQ ) N . (B4)9 M B2 (GeV ) V s ( G e V ) FIG. 2: Borel working window V s with threshold 2 .
65 GeV ≤ s ≤ .
85 GeV from up down. Before getting the numerical estimates of the hadronic parameter, we still have to de-termine the necessary input parameters such as the threshold s and the Borel mass M B .The threshold is the point from which the higher resonance contributions can be describedby the integration of the spectral density, so it is connected with the first resonance statethat has the same quantum number as the hadron we concern. In the numerical analysis,we use the values s = (2 . ∼ .
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