Electromagnetic form factors of octet baryons in QCD

The electromagnetic form factors of octet baryons are estimated within light cone QCD sum rules method, using the most general form of the interpolating current for baryons. A comparison of our predictions on the magnetic dipole and electric form factors with the results of other approaches is performed.

PACS numbers: 11.55.Hx, 14.20.–c, 14.20.Mr

## 1 Introduction

Electromagnetic form factors of nucleons provide information on their internal structures, i.e., about the spatial distribution of charge and magnetization of the nucleon. Nucleon electromagnetic form factors that are the functions of only four–momentum transfer squared are described by Dirac and Pauli form factors which are related to the electric and magnetic dipole form factors and as,

 GE = F1(Q2)−Q24m2BF2(Q2) , GM = F1(Q2)+F2(Q2) . (1)

Obviously, in the limit the form factors and correspond to the charge and magnetic moments of the nucleon, while and describe the charge and anomalous magnetic moments of the nucleon.

The study of electromagnetic form factors of hadrons receives a lot of attention during the past decade. Recent experiments on the nucleon form factors using the polarized electron beam and polarized protons, which are presented in detail in [1], allow more accurate measurements of the nucleon form factors at higher values of the momentum transfer. In the polarization measurements it is observed that the ratio can not be determined by the simple dipole form with [2, 3, 4]. The neutron form factors that are measured up to recently can provide detailed comparison of the proton and neutron form factors [5, 6].

Considerable progress has also been achieved on the electromagnetic excitation of nucleon resonances during last years. The cross sections and photon asymmetries for the photo production of the pion and mesons are measured at MAMI at Mainz, ELSA at Bonn, LEGS at Brookhaven, and GRAAL at Grenoble. Moreover, a large amount of data has already been collected for the excitation and single data points are obtained for the longitudinal and transversal form factors of the , , etc., whose results are all given in [7]. These progresses in experiments open a way to real possibility of measuring the electromagnetic form factors of the octet baryons in near future.

In the present work we calculate the electromagnetic form factors of the octet baryons within the light cone QCD sum rules (LCSR) method by employing the general form of the interpolating currents. It should be noted that this problem is studied in the same method for the Ioffe current alone in [8, 9, 10]. It should also be reminded to the interested reader that the nucleon electromagnetic form factors are calculated for the Ioffe and general currents in [11] and [12], respectively.

The plan of this work is as follows. In section 2 we introduce the correlation function which we shall use in calculating the electromagnetic form factors of the octet baryons, and discuss how the interpolating currents of the octet baryons are related to each other. In section 3, the light cone QCD sum rules for the electromagnetic form factors are obtained in the case when the correlation functions are calculated in terms of the main nonperturbative input parameters, namely in terms of distribution amplitudes (DAs) of the octet baryons. The last section contains the details of the numerical calculations of the electromagnetic form factors of the octet baryons.

## 2 LCSR for the electromagnetic form factors of octet baryons

In order to obtain the LCSR for the electromagnetic form factors of the octet baryons we start by considering the following vacuum–to–one–octet baryon correlation function,

 Πμ(p,q)=i∫d4xeiqx⟨0∣∣T{η(0)jelμ(x)}∣∣B(p)⟩ , (2)

where is the interpolating current of the octet baryon, is the electromagnetic current, is the vector Lorentz index, is the time ordering operation, and is the one particle baryon state with momentum .

The most general forms of the interpolating currents for the octet baryons can be written as,

 ηΣ0 = ηΣ+ = 2εabc(uaTCAℓ1sb)Aℓ2uc , ηΣ− = 2εabc(daTCAℓ1sb)Aℓ2dc , ηΞ0 = ηΣ+(u↔s) , ηΞ− = ηΣ−(d↔s) , (3)

where , , .

The interpolating current of the baryon can also be obtained from that of baryon in the following way [13]:

 2ηΣ0(d↔s)+ηΣ0=−√3ηΛ , or, 2ηΣ0(u↔s)−ηΣ0=−√3ηΛ . (4)

Our primary aim is the calculation of the phenomenological part of the correlation function (2). According to the standard procedure, in order to obtain the physical part of the correlation function of the octet baryons we insert a full set of baryons into Eq. (2). Separating the contribution of the ground state baryon we get,

 Πμ(p,q) = ⟨0|η|B(p−q)⟩⟨B(p−q)∣∣jelμ∣∣B(p)⟩m2B−(p−q)2+⋯ , (5)

where represents the contributions of the higher states and continuum.

The matrix element appearing in Eq. (5) are determined as,

 ⟨0|η|B(p−q)⟩ = λBu(p−q) , (6)

where is the residue of the members of the octet baryons. The hadronic matrix element with the electromagnetic current is determined in terms of three independent form factors , and in the following way,

 ⟨B(p−q)∣∣jelμ∣∣B(p)⟩ = ¯u(p−q)[F1(q2)γμ−iσμνqν2mBF2(q2)+qμ2mBF3(q2)]u(p) . (7)

From conservation of the electromagnetic current we get . Taking Dirac equation into account, one can show that the general decomposition of the correlation function (2) contains six independent amplitudes in the presence of the electromagnetic current,

 Πμ(p,q)=[Π1pμ+Π2pμto0.0pt/q+Π3γμ+Π4γμto0.0pt/q+Π5qμ+Π6qμto0.0pt/q]u(p) . (8)

Using the definitions given by Eqs. (6) and (7), we get the following expression for the hadronic part,

 Πμ(p,q) = λBm2B−(p−q)2{2F1(q2)pμ+F2(q2)mBpμto0.0pt/q+[F1(q2)+F2(q2)]γμto0.0pt/q (9) + [−2F1(q2)−F2(q2)]qμ−F2(q2)2mBqμto0.0pt/q} .

Equating the coefficients of each Lorentz structure in Eqs. (8) and (9) we get the sum rules for the form factors. In order to perform numerical analysis we need expressions of the invariant functions from QCD side.

The calculation of the invariant functions from QCD side is carried out when the external momenta and are taken in deep Eucledian space, i.e., and , which is necessary to perform operator product expansion (OPE) near the light cone . The OPE result can be obtained as the sum over octet baryon distribution amplitudes (DAs) of growing twist, which are the main non–perturbative inputs of the LCSR.

As has already been noted, the DAs of , and are investigated in [8, 9, 10]. The DAs of the octet baryons with are defined from the matrix element of the three–quark operator between the vacuum and the baryon state , whose form is given as,

 εabc⟨0∣∣qa1α(a1x)qb2β(a2x)qc3γ(a3x)∣∣B(p)⟩ , (10)

where are the Dirac indices, are the color indices, and are positive numbers satisfying . Using the Lorentz covariance, as well as spin and parity of the baryons under consideration, the matrix element (10) can be decomposed as,

 4εabc⟨0∣∣qa1α(a1x)qb2β(a2x)qc3γ(a3x)∣∣B(p)⟩=∑iFiΓαβ1i(Γ2iB(p))γ , (11)

where are certain Dirac matrices, and are the DAs which do not have definite twists. The DAs with definite twists are determined from,

 4εabc⟨0∣∣qa1α(a1x)qb2β(a2x)qc3γ(a3x)∣∣B(p)⟩=∑iFiΓ′αβ1i(Γ′2iB(p))γ , (12)

where and . The Relations among these two sets of DAs are given as,

 S1=S1 ,(2P⋅x)S2=S1−S2 ,P1=P1 ,(2P⋅x)P2=P2−P1 ,V1=V1 ,(2P⋅x)V2=V1−V2−V3 ,2V3=V3 ,(4P⋅x)V4=−2V1+V3+V4+2V5 ,(4P⋅x)V5=V4−V3 ,(2P⋅x)2V6=−V1+V2+V3+V4+V5−V6 ,A1=A1 ,(2P⋅x)A2=−A1+A2−A3 ,2A3=A3 ,(4P⋅x)A4=−2A1−A3−A4+2A5 ,(4P⋅x)A5=A3−A4 ,(2P⋅x)2A6=A1−A2+A3+A4−A5+A6 ,T1=T1 ,(2P⋅x)T2=T1+T2−2T3 ,2T3=T7 ,(2P⋅x)T4=T1−T2−2T7 ,(2P⋅x)T5=−T1+T5+2T8 ,(2P⋅x)2T6=2T2−2T3−2T4+2T5+2T7+2T8 ,(4P⋅x)T7=T7−T8 ,(2P⋅x)2T8=−T1+T2+T5−T6+2T7+2T8 .

The complete decomposition of the DAs in Eq. (11) in terms of and functions, as well as the explicit expressions of DAs, can all be found in [8, 9, 10, 11].

Omitting the details of calculations of the theoretical part and equating the coefficients of the Lorentz structures , from hadronic and theoretical parts, and performing the Borel transformation and continuum subtraction over the variable , we get the following sum rules for the form factors,

 F1(Q2) = L2λB{∫1x0dx(−ρ2(x)x+ρ4(x)x2M2−ρ6(x)2x3M4)e−¯xQ2xM2+xm2BM2 + [ρ4(x0)Q2+x20m2B−12x0ρ6(x0)(Q2+x20m2B)M2 + F2(Q2) = 2mBF1(Q2)(ρ2(x)→ρ′2(x), ρ4(x)→ρ′4(x), ρ6(x)→ρ′6(x)) , (14)

where is the Borel parameter, , is the solution of the equality , is the mass of the members of the octet baryons and . The factor in Eq. (2) is the normalization factor whose value for the members of the octet baryons is determined as,

 (15)

The explicit expressions of and for , and baryons are presented in the Appendix.

## 3 Numerical analysis

As has already been mentioned, the main nonperturbative inputs of LCSR are the baryon DAs. Here we would like to make the following remark about the expressions of the DAs for the , and baryons. In [14], the DAs for nucleons were extended up to next-to leading order in conformal spin and the expressions of the nucleon DAs of twist-3 up to next-to-next to leading conformal spin is found in [15]. As a result of these two works it is obtained that the nucleon form factors are sensitive to the higher conformal spin contributions. For other members of the octet baryons similar calculations are not yet done and deserves a detailed study. In present work, we consider the DAs for the , and baryons without these contributions, whose expressions can be found in [8, 9, 10, 11]. The parameters appearing in the expressions of the DAs are estimated from the analysis of the sum rules given in [10, 11, 12]:

 fΞ = (9.9±0.4)×10−3 GeV2 , λ1 = −(2.1±0.1)×10−2 GeV2 , λ2 = (5.2±0.2)×10−2 GeV2 , λ3 = (1.7±0.1)×10−2 GeV2 , fΣ = (9.4±0.4)×10−3 GeV2 , λ1 = −(2.5±0.1)×10−2 GeV2 , λ2 = (4.4±0.1)×10−2 GeV2 , λ3 = (2.0±0.1)×10−2 GeV2 , fΛ = (6.0±0.3)×10−3 GeV2 , λ1 = (1.0±0.3)×10−2 GeV2 , |λ2| = (0.83±0.05)×10−2 GeV2 , |λ3| = (0.83±0.05)×10−2 GeV2 .

The remaining input parameters of the LCSR are the continuum threshold , the Borel parameter and the auxiliary parameter that appears in the expressions of the interpolating currents of the octet baryons.

In our numerical calculations we use the values , and for the continuum threshold, obtained from mass sum rules analysis in [16], for the , and baryons, respectively.

The Borel mass parameter is another auxiliary parameter of the sum rules. Therefore the “working region” of should be found, where the form factors are practically independent of it. The lower limit of can be obtained by requiring that the higher states and continuum contributions to the sum rules constitute, maximally, about 40% of the total result. The upper limit of can be determined by demanding that the operator product expansion should be convergent. Our calculations show that the region in which the aforementioned two conditions are properly satisfied, are: for and baryons; and for baryons. In further numerical analysis, we use for , and baryons.

The residues of the octet baryons are calculated in [16] and we shall use these results in our numerical analysis. Furthermore, it should be noted that, from experimental point of view, it is more convenient to study the Sachs form factors and that are given in Eq. (1).

The dependence of the magnetic and electric form factors for , and , baryons are shown in Figs. (1)–(6). In order to get “good” convergence of the light cone expansion and reliable results from the LCSR, sufficiently large is needed. In our numerical calculations we consider the lower limit of as , where above this point the higher twist contributions are suppressed. On the other side, the higher resonance and continuum contributions become small enough when . For this reason, we perform numerical analysis in the region .

In odd–numbered Figs.: (1), (3), (5) (even–numbered Figs.: (2), (4), (6)) we present the dependence of the magnetic dipole form factors (electric form factor ) on , at fixed values of and chosen from their working regions, and at several fixed values of the arbitrary parameter , for , and baryons, respectively.

• In the case of , these form factors get positive (negative) values for negative (positive) values of the parameter .

• The situation for is contrary to the behaviors of the form factors of the , i.e., the values of and are positive (negative) when the parameter is positive (negative).

• In the case of baryon, the form factors exhibit similar behaviors as the form factors of baryon.

• It is observed that the form factor of the baryon changes its sign practically at all considered values of . The zero values of depend on the values of the arbitrary parameter . But the values of are quite small, whose maximum value is about .

• It is interesting to observe that at and , for the changes its sign, while for the other values of it is always negative.

• For positive (negative) values of the magnetic dipole form factor for baryon attains at positive (negative) values.

• The situation for electric form factor , however, is slightly different. Namely, in the case of Ioffe current for which , becomes negative whose magnitude is negligibly small.

• When the form factors of the baryon are considered we see that changes its sign only at , while for all other values of it gets at only negative values. On the other hand the form factor is positive (negative) for all positive and negative values of .

We now compare our results on dependence of the magnetic and charge form factors with the ones existing in the literature. These form factors are discussed within the LCSR method for the Ioffe current () in [9, 10], within the framework of the fully relativistic constituent quark model in [17], in the framework of the covariant spectator quark model [18] and lattice QCD [19]. When we compare our predictions on the form factors with the results of the above–mentioned works we obtain that, at , our predictions on are very close to the results of [9, 10], and [17], except for the baryon. Our predictions for the magnetic form factor agree within the errors with the existing results. The small differences among the predictions can be attributed to the different values of the input parameters used in the numerical analysis.

As has already been noted, the interpolating currents of the octet baryons contain also the auxiliary arbitrary parameter . Obviously, the physically measurable quantities should be independent of this parameter. In order to find the working region of the parameter we demand that the form factors are practically independent of it. As an example, in Figs. (7) and (8) we present the dependence of and on for baryon at fixed values of and , for two fixed choices of , namely, and . We see from these figures that, in the region , and show very weak dependence on . In other words, the working region of the parameter for the baryon is .

We perform similar analysis for all other members of the octet baryons and find out that the region is the common working region to them as well. It should be noted here that for and baryons exhibit stability in the range . Also note that point, which is the Ioffe current corresponding to , belongs to the region where the predictions for the form factors are not reliable. Choosing the values of and from the relevant working regions, and from a comparison of our predictions on the form factors with the results of the above–mentioned works we see that,

• Predictions of all works for are very close to each other within the error limits.

• Our predictions on agree with the results of other approaches, except for the and baryons. In these cases our results are very close to the predictions of the lattice QCD, while considerable disagreements are observed with those obtained in [17] and [18].

The results obtained in this work can be improved by taking into account the corrections to the distribution amplitudes, and more accurate values of the input parameters entering the sum rules.

In conclusion, in the present work we have studied the charge and magnetic dipole form factors within the LCSR method by using the most general for of the interpolating currents for the octet baryons. We have compared our predictions on these form factors with the results existing in literature that were obtained in framework of the relativistic quark model, lattice QCD and LCSR for the Ioffe current.

## Appendix

In thhis Appendix we present the explicit expressions of the functions , and entering to the sum rules for the form factors and ,

 ρΣ+6(x) = 4eum3Σ+(1+β)x(m2Σ+x2+Q2)ˇˇB6(x) + 4esm2Σ+{m2Σ+ms(1−β)x2ˆˆC6+(1+β)[mΣ+(m2Σ+x2+Q2)xˆˆB6 − ms(Q2ˆˆB6+2m2Σ+x2ˆˆB8)]}(x) , ρΣ+4(x) = eumΣ+{−2m2Σ+x[(1−β)(ˇˇC6+ˇˇD6)−(1+β)(2ˇˇB6−3ˇˇB8)](x) + (1−β)[m2Σ+x2(ˇD4−3ˇD5−ˇC4+3ˇC5)+2Q2(ˇD2+ˇC2)](x) + (1+β)[Q2(ˇB2+5ˇB4)−m2Σ+x2(2ˇH1−2ˇE1+ˇB2−ˇB4+6ˇB5+12ˇB7)](x) − 2m2Σ+x∫¯x0dx3[(1−β)(AM1−VM1)+3(1+β)TM1](x,1−x−x3,x3)} + edmΣ+{−2m2Σ+x[(1−β)(˜˜C6−˜˜D6)+2(1+β)˜˜B8](x) + (1−β)[−m2Σ+x2(˜D4−˜D5+˜C4−˜C5)](x) + (1+β)[2Q2(˜B2+˜B4)−4m2Σ+x2(˜B5+2˜B7)](x) − 2m2Σ+x∫¯x0dx1[(1−β)(AM1+VM1)+2(1+β)TM1](x1,x,1−x1−x)} + esmΣ+{2mΣ+(1+β)[mΣ+x(2ˆˆB6−ˆˆB8)−msˆˆB6](x) + (1−β)[2(m2Σ+x2ˆC5+Q2ˆC2)−mΣ+msx(2ˆC2−ˆC4−ˆC5)](x) − (1+β)[Q2(ˆB2−3ˆB4)+m2Σ+x2(ˆB2−ˆB4+2ˆB5+4ˆB7)−4mΣ+msx(ˆB4−ˆB5)](x) − 2m2Σ+(1+β)x∫¯x0dx1TM1(x1,1−x1−x,x)} , ρΣ+2(x) = 2eumΣ+x∫¯x0dx3[(1−β)(A1+2A3−V1+2V3) − (1+β)(P1+S1+3T1−6T3)](x,1−x−x3,x3) + 2edmΣ+{[(1−β)(˜D2−˜C2)+(1+β)(˜B2−˜B4)](x) − x∫¯x0dx1[(1−β)(A1+A3+V1−V3)+2(1+β)(T1−2T3)](x1,x,1−x1−x)} + 2es{mΣ+[(1−β)ˆC2+(1+β)(ˆB2−ˆB4)](x) + ∫¯x0dx1[(1−β)(mΣ+xV3+msV1) − (1+β)(mΣ+x(P1+S1+T1−2T3)+msT1)](x1,1−x1−x,x)} , ρ′Σ+6(x) = −4eum2Σ+(1+β)(m2Σ+x2+Q2)ˇˇB6(x) − 4esm2Σ+{mΣ+ms(1−β)xˆˆC6+(1+β)[(m2Σ+x2+Q2)ˆˆB6+mΣ+msx(ˆˆB6−2ˆˆB8)]}(x) , ρ′Σ+4(x) = eum2Σ+{−3(1+β)ˇˇB6(x) + x[(1−β)(2ˇD2−ˇD4+3ˇD5+2ˇC2+ˇC4−3ˇC5) + 2(1+β)(ˇH1−ˇE1+ˇB2+2ˇB4+3ˇB5+6ˇB7)](x)} + edm2Σ+{2(1+β)˜˜B6(x) + x[(1−β)(˜D4−˜D5+˜C4−˜C5)+2(1+β)(˜B2+˜B4+2˜B5+4˜B7)](x) + 2(1−β)∫¯x0(AM1+VM1)(x1,x,1−x1−x)} + esmΣ+{−5mΣ+(1+β)ˆˆB6(x) − 2[(1−β)(mΣ+xˆC5−(mΣ+x+ms)ˆC2)−(1+β)(mΣ+x(ˆB4+ˆB5+2ˆB7)−ms(ˆB2+ˆB4))](x) + 2mΣ+(1−β)∫¯x0dx1(AM1−VM1)(x1,1−x1−x,x)} ,