Higher order corrections to the light-cone distribution amplitudes of the sigma baryon

# Higher order corrections to the light-cone distribution amplitudes 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 Department of Physics, School of Biomedical Engineering, Third Military Medical University, Chongqing 400038, China
July 13, 2019
###### Abstract

The light-cone distribution amplitudes (LCDAs) of the baryons up to twist- are 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 accuracy. The nonperturbative parameters relevant to the LCDAs are determined in the framework of the QCD sum rule method. The explicit expressions of the LCDAs are given as the main results.

###### pacs:
11.25.Hf,  11.55.Hx,  13.40.Gp,  14.20.Jn.

## I Introduction

Signals confirmed by ATLAS and CMSHiggsCMS (); HiggsATLAS () showed that the Higgs bosonHiggs () in the standard model (SM) have been found and the SM is most likely to be a precise theory at the present energy scale. New physics beyond the SM at higher energy scale is mostly concerned nowadays and in the near future. However, many difficulties are still alive in practical analysis of hadron physics involving nonperturbative QCD effect when we study hadronic phenomena at low energy, or scale. A typical method to solve the nonperturbative difficulties in QCD is factorization, in which the nonperturbative part is included into the wave function, such as the parton distribution functions for inclusive processes, fragmentation distribution functions for the hadronization, and the distribution amplitudes for exclusive processes. Specifically, in theoretical investigations of the hard exclusive processes exclusive (); exclusive2 () and hadronic physics with the QCD light-cone sum rule method lcsr1 (); lcsr2 (); lcsr3 (), the light-cone distribution amplitudes (LCDAs) are fundamental ingredients to be studied. Furthermore, when searching for new physics beyond the SM, it is an important way to study flavor physics, in which some processes that are sensitive to the new physics can be measured more precisely nowadays than any time before. All of these require detailed 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 mesonsmesondas () and the nucleonChernyak (); Braun1 (); Lenz (); dalattice (); Pasquini (); overviewofDAs (), whereas, theoretical studies of a large number of the hadron physics phenomena 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 determined experimentally at present. We have examined the LCDAs of the strange octet baryons in the previous workDAs () in the conformal spin expansion methodBraun2 (); Balitsky (); Lenz (). Our calculation concerns LCDAs to twist- to the accuracy of the leading order of conformal spin expansion. The obtained parameters are also used to analyze some hadronic physics processes as applicationsLYL (); LYL2 (); LYL3 (); Aliev (). However, some of the investigationsLYL (); LYL3 () have implied that corrections from the higher order conformal spin contributions may affect the results to some extent.

In the point view of applications, an important effect to the LCDAs is the correction of the higher twist distribution functions. The higher order twist contributions to LCDAs have several origins, among which the main one comes from “bad” components in the wave function and in particular of components with “wrong” spin projection for the case of baryons Braun1 (); DAs (). Compatible with the previous works, we focus on higher order twist contributions from bad components in the decomposition of the Lorentz structure in this paper. One of the general descriptions of LCDAs is based on the conformal symmetry of the massless QCD Lagrangian dominated on the light cone. The conformal partial wave expansion of the LCDAs can be carried out safely in the limit of the -flavor symmetry approach. However, when terms connected with the -quark mass are considered, the -flavor breaking effects need to be included. In the present work, effects from the -flavor symmetry breaking are considered as the corrections, which originate from two sources: isospin symmetry breaking and corrections to the nonperturbative parameters.

It is known that the leading order contribution with the conformal spin expansion approach comes from the properties of the matrix elements of the local three-quark operator between the vacuum and the baryon state. Thus, it is natural that higher order corrections should be related to the expansion of the matrix elements of the nonlocal three-quark operator at the zero point. However, we still need to estimate how much the contributions from four-particle effects will do on the result. Fortunately we have known that for processes whose dominant contribution is from the light cone the four-particle contributions can be safely omitted in the lower leading order. Thus, in the following analysis we only consider contributions from three-quark operator matrix element, whose higher moment is calculated with QCD sum rulesSVZ ().

As applications, the light-cone QCD sum rule method has been used to examine processes related to the strange octet baryons and give instructive estimates Aliev (); Wang (). In the previous works, we have analyzed some physical processes related to the final states about the baryon. The results are compatible with the experiments and(or) the other theoretical predicationsLYL2 (). Nevertheless, there are still some processes which are not well described LYL (); LYL3 (). We wish the higher order corrections from the higher conformal spin may give us more accurate estimates.

The rest of the paper is organized as follows. Section II is devoted to present the definitions of the higher order moment of the three-quark operators related to corrections of LCDAs from the higher conformal spin expansion. In Sec. II.2, the conformal partial wave expansion of the LCDAs is carried out by use of the conformal symmetry of the massless QCD Lagrangian. The nonperturbative parameters connected with the LCDAs are determined in Sec. III with the QCD sum rule method. Finally, we give the explicit expressions of the baryon LCDAs in Sec. IV. A summary is given is Sec. V. The equations of motion which are used to reduce the number of the free parameters are presented in Appendix A for the completeness of this paper. The sum rule of one coupling constant 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-cone distribution amplitudes of Σ

### ii.1 General definition

Matrix elements of the quark-quark or quark-gluon-quark field operator between vacuum or hadron states are the great important ingredients in analysis of processes in quantum field theory. Light-cone distribution amplitudes of the baryon are defined by the general Lorentz expansion of the matrix element of the nonlocal three-quark-operator between the vacuum and the baryon state

 ⟨0|ϵijkqiα(a1z)qjβ(a2z)skγ(a3z)|Σ(P)⟩, (1)

where represents or quark, which correspond to or baryon, respectively. The indices refer to Lorentz indices and represent color ones. It is noticed that to make the matrix element above gauge invariant, the gauge factor need to be inserted, whereas when fixed-point gauge 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 as

 4⟨0|ϵijkqiα(a1z)qjβ(a2z)skγ(a3z)|Σ(P)⟩=∑iFiΓαβ1i(Γ2iΣ)γ, (2)

where is the spinor of the baryon with the quantum number ( is the isospin, is the total angular momentum, and is the parity), are certain Dirac structures over which the sum is carried out, and are the independent distribution amplitudes which are functions of the scalar product DAs (). It is also noticed that and are the two light-cone vectors which satisfy and .

Functions defined above do not have definite twist. In order to classify the LCDAs according to the definite twist, we redefine the wave functions in the infinite momentum frame as:

 4⟨0|ϵijks1iα(a1z)s2jβ(a2z)qkγ(a3z)|Σ(P)⟩=∑iFiΓ′αβ1i(Γ′2iΣ)γ. (3)

A naive calculation shows that the invariant functions can be expressed in terms of the LCDAs . The two sets of definitions have the following relations:

 S1=S1, 2p⋅zS2=S1−S2, (4) P1=P1, 2p⋅zP2=P2−P1, V1=V1, 2p⋅zV2=V1−V2−V3, 2V3=V3, 4p⋅zV4=−2V1+V3+V4+2V5, 4p⋅zV5=V4−V3, (2p⋅z)2V6=−V1+V2+V3+V4+V5−V6

for scalar, pseudoscalar, and vector structure, and

 A1=A1, 2p⋅zA2=−A1+A2−A3, (5) 2A3=A3, 4p⋅zA4=−2A1−A3−A4+2A5, 4p⋅zA5=A3−A4, (2p⋅z)2A6=A1−A2+A3+A4−A5+A6

for axial-vector structure, and

 T1=T1, 2p⋅zT2=T1+T2−2T3, (6) 2T3=T7, 2p⋅zT4=T1−T2−2T7, 2p⋅zT5=−T1+T5+2T8, (2p⋅z)2T6=2T2−2T3−2T4+2T5+2T7+2T8, 4p⋅zT7=T7−T8, (2p⋅z)2T8=−T1+T2+T5−T6+2T7+2T8

for tensor structure.

The classifications of the LCDAs with a definite twist are listed in Table 1, where we take as an example. The explicit expressions of the definition can be found in Refs. Braun1 (); DAs (). Each distribution amplitude can be represented as

 F(aip⋅z)=∫Dxe−ipz∑ixiaiF(xi), (7)

where the dimensionless variables , which satisfy the relations and , correspond to the longitudinal momentum fractions along the light cone carried by the quarks inside the baryon. The integration measure is defined as

 ∫Dx=∫10dx1dx2dx3δ(x1+x2+x3−1). (8)

There exist some symmetry properties of the LCDAs from the identity of the two quarks in the baryon, which is useful to reduce the number of the independent functions. Taking into account the Lorentz decomposition of the -matrix structure, it is easy to see that the vector and tensor LCDAs are symmetric, whereas the scalar, pseudoscalar and axial-vector structures are antisymmetric under the interchange of the two quarks:

 Vi(1,2,3) = Vi(2,1,3),Ti(1,2,3)= Ti(2,1,3), Si(1,2,3) = −Si(2,1,3),Pi(1,2,3)=−P(2,1,3), Ai(1,2,3) = −A(2,1,3). (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 in terms of quark fields with definite chirality . Taking as an example, the classification of the LCDAs in this presentation can be interpreted transparently: projection on the state with the two -quarks antiparallel, i.e. , singles out vector and axial vector structures, while parallel ones, i.e. and , correspond to scalar, pseudoscalar and tensor structures. The explicit expressions of the LCDAs by chiral-field representations are presented in Table 1 as an example for . The counterparts of can be easily obtained under the exchange .

Note that in the case of the nucleon, the isospin symmetry can be used to reduce the number of the independent LCDAs to eightBraun1 (). However, there are no similar isospin symmetric relationships existing when the baryon is considered. Therefore, we need altogether chiral field representations to express all the LCDAs.

### ii.2 Conformal expansion

In this subsection we give the explicit form of the LCDAs with the aid of the conformal partial wave expansion approach. The main idea of this method is based on the conformal symmetry of the massless QCD Lagrangian. In this approach the longitudinal degrees of freedom can be separated from transverse ones. On the one hand, the properties of transverse coordinates are described by the renormalization scale that is determined by the renormalization group equation. On the other hand, the longitudinal momentum fractions that are living on the light cone are governed by a set of orthogonal polynomials, which form an irreducible representation of the collinear subgroup of the conformal group.

The algebra of the collinear subgroup is determined by the following four generators:

 L+=−iP+,L−=i2K−,L0=−i2(D−M−+),E=i(D+M−+), (10)

where , , , and correspond to the translation, special conformal transformation, dilation and Lorentz generators, respectively. The notations are used for a vector : and . Let , then a given distribution amplitude with a definite twist can be expanded by the conformal partial wave functions that are the eigenstates of and .

For the three-quark state, the distribution amplitude with the lowest conformal spin is Braun2 (); Balitsky ()

 Φas(x1,x2,x3)=Γ[2j1+2j2+2j3]Γ[2j1]Γ[2j2]Γ[2j3]x12j1−1x22j2−1x32j3−1, (11)

where represents the conformal spin of the quark field that is defined as half of the canonical dimension plus its spin . Contributions with higher conformal spin () are given by multiplied by polynomials that are orthogonal over the weight function (11). For LCDAs in Table 1, we give their conformal expansions:

 Φ3(xi) = 120x1x2x3[ϕ03+ϕ−3(x1−x2)+ϕ+3(1−3x3)+...], T1(xi) = 120x1x2x3[t01+t−1(x1−x2)+t+1(1−3x3)+...] (12)

for twist- and

 Φ4(xi) = 24x1x2[ϕ04+ϕ−4(x1−x2)+ϕ+4(1−5x3)+...], Ψ4(xi) = 24x1x3[ψ04+ψ−4(x1−x3)+ψ+4(1−5x2)+...], Ξ4(xi) = 24x2x3[ξ04+ξ−4(x2−x3)+ξ+4(1−5x1)+...], Ξ′4(xi) = 24x2x3[ξ′04+ξ′−4(x2−x3)+ξ′+4(1−5x1)+...], T2(xi) = 24x1x2[t02+t−2(x1−x2)+t+2(1−5x3)+...] (13)

for twist- and

 Φ5(xi) = 6x3[ϕ05+ϕ−5(x1−x2)+ϕ+5(1−2x3)+...], Ψ5(xi) = 6x2[ψ05+ψ−5(x1−x3)+ψ+5(1−2x2)+...], Ξ5(xi) = 6x1[ξ05+ξ−5(x2−x3)+ξ+5(1−2x1)+...], Ξ′5(xi) = 6x1[ξ′05+ξ′−5(x2−x3)+ξ′+5(1−2x1)+...], T5(xi) = 6x3[t05+t−5(x1−x2)+t+5(1−2x3)+...] (14)

for twist-, and

 Φ6(xi) = 2[ϕ06+ϕ−6(x1−x2)+ϕ+6(1−3x3)+...], T6(xi) = 2[t06+t−6(x1−x2)+t+6(1−3x3)+...] (15)

for twist-. Up to now there are altogether parameters which need to be determined.

To the next-to-leading order, the normalization of the baryon LCDAs is determined by the matrix element of the nonlocal three-quark operator expanded at the zero point. The decomposition of the matrix element is

 ⟨0|ϵijkuiα(a1z)ujβ(a2z)skγ(a3z)|Σ(P)⟩=⟨0|ϵijkuiα(a1z)ujβ(a2z)skγ(a3z)|Σ(P)⟩ +zλ⟨0|[ϵijkuiα(a1z)\lx@stackrel↔Dujβ(a2z)]skγ(a3z)|Σ(P)⟩ +zλ⟨0|ϵijkuiα(a1z)ujβ(a2z)[→Dskγ(a3z)]|Σ(P)⟩. (16)

The Lorentz decomposition of the matrix element can be expressed explicitly as

 4⟨0|ϵijksiα(0)sjβ(0)qkγ(0)|Σ(P)⟩=V01(⧸PC)αβ(γ5Σ)γ+V03(γμC)αβ(γμγ5Σ)γ +T01(PνiσμνC)αβ(γμγ5Σ)γ+T03M(σμνC)αβ(σμνγ5Σ)γ (17)

for the matrix element of the leading order, and

 4⟨0|ϵijkuiα(a1z)ujβ(a2z)[→Dskγ(a3z)]|Σ(P)⟩ =Vs1(⧸PC)αβ(γ5Σ)γ+V02M(⧸PC)αβ(γλγ5Σ)γ+Vs3PλM(γμC)αβ(γμγ5Σ)γ +V04M2(γλC)αβ(γ5Σ)γ+V05M2(γμC)αβ(iσμλγ5Σ)γ+Ts1Pλ(PνiσμνC)αβ(γμγ5Σ)γ +T02M(PνiσλνC)γ5Σγ+Ts3MPλ(σμνC)αβ(σμνγ5Σ)γ+T04M(PνσμνC)αβ(σμλγ5Σ)γ +T05M2(iσμλC)αβ(γμγ5Σ)γ+T07M2(σμνC)αβ(σμνγλγ5Σ)γ, (18)
 4⟨0|[ϵijkuiα(a1z)\lx@stackrel↔Dujβ(a2z)]skγ(a3z)|Σ(P)⟩ =Su1PλMCαβ(γ5Σ)γ+S02M2Cαβ(γλγ5Σ)γ+Pu1PλM(γ5C)αβΣγ+P02M2(γ5C)αβ(γλΣ)γ +Au1Pλ(⧸Pγ5C)αβΣγ+A02M(⧸Pγ5C)αβγλΣγ+A03PλM(γμγ5C)αβγμΣγ +A04M2(γλγ5C)αβΣγ+A05M2(γμγ5C)αβiσμλΣγ (19)

for the next leading order expansion. There are altogether nonperturbative parameters in the expressions. However, we need not so many free parameters because there are some constraints to reduce the freedom of the coefficients. It is noticed that all the parameters defined above are not independent and can be reduced with the help of the motion of equation, which can be seen in Appendix A.

Choosing as the independent parameters, the other ones can be expressed with them:

 V02=14(Vs1−2Vs3), V04=116(4V01−4V03−3Vs1+2Vs3), (20) V05=148(−4V01+4V03+3Vs1−50Vs3), T02=110(3Su1−3T01+6T03+2Ts1−2Ts3), T04=110(Su1−Ts1+2T03+4Ts1−14Ts3), T05=−Ts3, T07=130(5P02−Su1+T01−12T03−4Ts1+24Ts3), A02=14(4Au3−4V03−Vs1+6Vs3), A04=116(−4Au1−8Au3+4V03+Vs1−6Vs3), A05=148(4V01+20V03+3Vs1+14Vs3), S02=110(−10P02+3Su1+7T01+6T03+2Ts1−12Ts3), Pu1=15(−Su1+T01−12T03−4Ts1+24Ts3).

Recall the relations of the leading order, there are altogether parameters to be determined. To this end, we introduce the additional eight decay constants defined by the following matrix elements of a three-quark operator with a covariant derivative:

 ⟨0|ϵijk[ui(0)C⧸zγ5iz\lx@stackrel↔Duj(0)]⧸zsk(0)|Σ(P)⟩=−fΣAu1(P⋅z)2⧸zΣ(P)γ, ⟨0|ϵijk[ui(0)Cγuuj(0)]γ5⧸zγu(iz→Dsk)(0)|Σ⟩=λ1fs1(P⋅z)M⧸zΣ(P)γ, ⟨0|ϵijk[ui(0)Cσμνuj(0)]γ5⧸zσμν(iz→Dsk)(0)|Σ⟩=−λ2fs2(P⋅z)M⧸zΣ(P)γ, ⟨0|ϵijk[ui(0)Cγμγ5iz\lx@stackrel↔Duj(0)]⧸zγμsk(0)|Σ(P)⟩=−λ1fu1(P⋅z)M⧸zΣ(P)γ, ⟨0|ϵijk[ui(0)iPνCσμνuj(0)]γ5⧸z(iz→Dsk)(0)|Σ(P)⟩=−λ3fs3(P⋅z)M2⧸zΣ(P)γ, ⟨0|ϵijk[ui(0)Ciz\lx@stackrel↔Duj(0)]γ5sk(0)|Σ(P)⟩=Su1(P⋅z)MΣ(P)−S02M2(⧸zΣ(P))γ, ⟨0|ϵijk[ui(0)Ciz\lx@stackrel↔Dγ5uj(0)]sk(0)|Σ(P)⟩=Pu1(P⋅z)MΣ(P)+P02M2(⧸zΣ(P))γ. (21)

It is noticed that each of the last two matrix element have two different Lorentz structures which permit us to get two different sum rules; whereas the calculations also indicate that the sum rules from the last two ones are the same, so we can get the necessary equations from the two different sum rules.

We also need another four decay constant defined by the leading order local operator matrix element which has been calculated in the previous paper DAs ()

 ⟨0|ϵijk[u