Strange form factors of nucleon with nonlocal chiral effective Lagrangian

The strange form factors of nucleon are studied with the nonlocal chiral effective Lagrangian. One loop contributions from both octet and decuplet intermediate states are included. The relativistic regulator is obtained by the nonlocal Lagrangian where the gauge link is introduced to guarantee the local gauge symmetry. With the kaon loop, the calculated charge form factor is positive, while the magnetic form factor is negative. The strange magnetic moment is $-0.041^{+0.012}_{-0.014}$ with $\Lambda=0.9 \pm 0.1$ determined from the nucleon electromagnetic form factors. Our results are comparable with the recent lattice simulation.


II. CHIRAL EFFECTIVE LAGRANGIAN
The lowest order chrial lagrangian for baryons, pseudoscalar mesons and their interaction can be written as [40,41].
where D, F , C and H are the coupling constants. The chiral covariant derivative D µ is defined as The pseudoscalar meson octet couples to the baryon field through the vector and axial vector combinations as where Σ = ζ 2 = e 2iφ/f , f = 93 MeV.
The matrix of pseudoscalar fields φ is expressed as A µ is the photon field. The covariant derivative D µ in the decuplet part is defined as D ν T abc µ = ∂ ν T abc µ + (Γ ν , T µ ) abc , where Γ ν is the chrial connection [42] defined as (X, T µ ) = (X) a d T dbc µ + (X) b d T adc µ + (X) c d T abd µ . γ µνα ,γ µν are the antisymmetric matrices expressed as In the chiral SU (3) limit, the octet and decuplet baryons will have the same mass m B and m T . In our calculation, we use the physical masses for baryon octets and decuplets. The explicit form of the baryon octet is written as For the baryon decuplets, there are three indices, defined as The octet, decuplet and octet-decuplet transition magnetic moment operators are needed in the one loop calculation of nucleon electromagnetic form factors. The baryon octet anomalous magnetic Lagrangian is written as where At the lowest order, the Lagrangian will generate the following nucleon anomalous magnetic moments: The transition magnetic operator is where the matrix Q is defined as Q =diag{2/3, −1/3, −1/3}. The effective decuplet anomalous magnetic moment operator can be expressed as effective Lagrangian In quark model, the baryon magnetic moments can also be written in terms of quark magnetic moments. For example, and µ q can be written in terms of c 1 . For example, The strange quark contribution to the hyperons at tree level can be written as [43] Similarly, the strange quark contribution to the decuplet and transition magnetic moments at tree level can be written as [44] µ s Following the usual convention, the charge of the strange quark is taken to be 1. Now we construct the nonlocal Lagrangian which will generate the covariant regulator. The gauge invariant nonlocal Lagrangian can be obtained using the method in [38,39,45]. For instance, the local interaction including kaon can be written as where A s µ (x) is the external field interacting the strange quark. The nonlocal Lagrangian for this interaction is expressed as where F (x) is the correlation function. To guarantee the gauge invariant, the gauge link is introduced in the above Lagrangian. The regulator can be generated automatically with correlation function. With the same idea, the nonlocal interaction between baryons and A s µ (x) can also be obtained. For example, the local interaction between Λ and external field is written as The corresponding nonlocal Lagrangian is expressed as where F 1 (a) and F 2 (a) is the correlation function for the nonlocal strange electric and magnetic interactions. The form factors at tree level which are momentum dependent can be easily obtained with the Fourier transformation of the correlation function. The simplest choice is to assume that the correlation function of the strange electromagnetic vertex is the same as that of the lambda-kaon vertex, i.e. F 1 (a) = F 2 (a) = F (a). Therefore, the Dirac and Pauli form factors will have the same dependence on the momentum transfer at tree level. The better choice is to assume that the charge and magnetic form factors at tree level have the same the momentum dependence as lambda-kaon vertex, i.e. G tree is the Fourier transformation of the correlation function F (a) [39]. The corresponding function ofF 1 (q) andF 2 (q) is then expressed as From the above equations, one can see that in the heavy baryon limit, these two choices are equivalent. The nonlocal Lagrangian is invariant under the following gauge transformation where . From Eq. (18), two kinds of couplings between hadrons and external field can be obtained. One is the normal one expressed as This interaction is similar as the traditional local Lagrangian except the correlation function. The other one is the additional interaction obtained by the expansion of the gauge link, expressed as The additional interaction is important guarantee the local gauge symmetry resulting the net strangeness of nucleon zero.

III. STRANGE FORM FACTORS
The strange quark contribution to the Dirac and Pauli form factors are defined as where q = p ′ − p and Q 2 = −q 2 . The combination of the above form factors can generate electric and magnetic form factors contributed from strange quark According to the Lagrangian, the one loop Feynman diagrams which contribute to the strange form factors are plotted in Fig 1. In this section, we will only show the expressions for the intermediate octet baryon part. For the intermediate decuplet baryon part, the expressions are written in the Appendix. In Fig. 1a, the external field couples to the meson. The contribution of Fig. 1a to the matrix element in Eq. (25) is expressed as where the integral I Λ aK is expressed as and D K (k) is defined as m Λ and M k are the masses for the intermediate Λ hyperon and K meson, respectively. The integral I Σ aK is the same as I Λ aK except the intemediate hyperon mass m Λ is replaced by m Σ . Therefore, here we only show the expressions for Λ hyperon. In Fig.1b, the external field couples to the intermediate hyperons with electric interaction. The contribution of this diagram is expressed as where the integral I Λ bK is written as Fig.1c is similar as Fig.1b except for the magnetic interaction. The contribution of this diagram is written as where I Λ cK is expressed as . Fig. 1d and 1e are the Kroll-Ruderman diagrams. The contribution from these two diagrams is written as where These two diagrams only have contribution in the relativistic cases. In the heavy baryon limit, they have no contribution to either electric or magnetic form factors. Fig. 1f and 1g are the additional diagrams which generated from the expansion of the gauge link. The contribution of these two additional diagrams are expressed as where Using FeynCalc to simplify the γ matrix algebra, we can get the separate expressions for the Dirac and Pauli form factors.

IV. NUMERICAL RESULTS
In the numerical calculations, the parameters are chosen as D = 0.76 and F = 0.5 (g A = D + F = 1.26). The coupling constant C is chosen to be 1 which are the same as [47]. The off-shell parameter z is chosen to be z = −1 [48]. The covariant regulator is chosen to be of a dipole form where M j is the mass of the corresponding meson and it is zero for photon. Therefore, in this nonlocal Lagrangian, there are three parameters c 1 , c 2 and Λ to be determined. Λ is chosen to get the best description of the nucleon form factors up to relatively large Q 2 . By comparing with the experimental electromagnetic form factors of nucleon, the best Λ is found to be around 0.9 GeV. The other two parameters c 1 and c 2 are determined by the experimental magnetic moments of proton and neutron. With µ p = 2.79 and µ n = −1.91, we get c 1 = 2.081 and c 2 = 0.788.
Before present the results for strange form factors, we first show the electromagnetic form factors. In Fig. 2, the charge and normalized magnetic form factors of proton and neutron with Λ = 0.9 GeV are plotted. The solid line is for the empirical function 1/(1 + Q 2 /0.71GeV 2 ) 2 . The dashed, dotted and dash-dotted lines are for G p E , G p M /µ p and G n M /µ n , respectively. The dotted line is invisible because it coincides with the empirical line. The dashed line started from 0 is for G n E . The experimental data of neutron charge form factor are from Ref. [46]. From the figure, we can see that our calculated form factors are very close to the experimental data which is a great improvement compared with the results of the traditional chiral effective field theory [20,21]. Now we show the results for the strange form factors. The strange magnetic form factor G s M (Q 2 ) of nucleon versus Q 2 with different Λ is plotted in Fig. 3. The three solid lines from bottom to top, are for the results with Λ =1 GeV, 0.9 GeV and 0.8 GeV, respectively. The data with error bars from recent lattice simulation [49] are also shown in the figure. The strange magnetic form factors increases with the increasing momentum transfer Q 2 . At zero momentum transfer, when Λ = 0.9 ± 0.1 GeV, G s M (0) = −0.041 +0.012 −0.014 . The absolute value of strange magnetic moment in this relativistic chiral Lagrangian is smaller than that in heavy baryon approach, where G s M (0) = −0.058 +0.034 −0.053 [32]. The main reason for the difference is that rainbow diagrams (Fig. 1a and Fig. 1c) have much smaller contribution to G s M than that in the heavy baryon limit due to the covariant regulator. Though the Kroll-Ruderman and additional diagrams have sizeable contribution to G s M in this relativistic case, while in the heavy baryon limit such contribution is zero, the total absolute value of G s M (0) is a little smaller in relativistic case. The strange charge form factor G s E (Q 2 ) is plotted in Fig. 4. The three solid lines from top to bottom, are for the results with Λ = 1 GeV, 0.9 GeV and 0.8 GeV, respectively. When Q 2 = 0, G s E (0) = 0. This is true only when the additional diagrams generated from the expansion of the gauge link are included. The strange charge form factor first increases and then decreases with the increasing Q 2 . At finite Q 2 , G s E (Q 2 ) is always a small positive number. It is clear that for both strange charge and magnetic form factors, our result is comparable with the Lattice data. With the strange form factors, the strange radii can be obtained as With Λ = 0.9 ± 0.1 GeV, we have < (r s E ) 2 >= −0.004 ± 0.001 fm 2 and < (r s M ) 2 >= −0.028 ± 0.003 fm 2 . To see clearly the separate contribution from the octet and decuplet parts, and from the normal diagrams and additional diagrams, in Fig. 5, we plot each contribution to the strange magnetic form factor at Λ = 0.9 GeV separately. The solid, dashed and dotted lines are for total, octet and decuplet contribution to G s M (Q 2 ), respectively. The red lines are for the contribution from normal diagrams and the blue lines are for the contribution from additional diagrams. From the figure, one can see that, the octet contribution is dominant. Compared with the octet contribution, the decuplet part gives a smaller opposite number to G s M . The additional diagrams also provide important contributions to the total G s M . In Fig. 6, we plot the same curves, but for the strange charge form factor. At Q 2 = 0, the contributions from the normal and additional diagrams cancel each other. As a result, the net strangeness is zero. This is guaranteed by the gauge symmetry of strange quark. Similar as in the magnetic case, the octet contribution is dominant for the total G s E (Q 2 ). At small Q 2 , the contribution from the additional diagrams changes more quickly than that from the normal diagrams. Therefore, G s E first increases from 0 and then decreases smoothly with the increasing Q 2 . We studied the strange form factors of nucleon with the nonlocal chiral effective Lagrangian. Both the octet and decuplet intermediate states are included in the one loop calculation. The covariant form factors are derived from the nonlocal Lagrangian. This is the relativistic version of the finite-range-regularization, which make it possible to study the hadron structure at relatively large Q 2 . From the previous study of the nucleon electromagnetic form factors, it shows this nonlocal Lagrangian method is a great improvement compared with the traditional chiral effective theory. The gauge link is introduced to guarantee the local gauge symmetry. As a result, in addition to the normal diagrams which are generated from the minimal substitution, the additional diagrams appear which are generated from the expansion of the gauge link. These additional diagrams are crucial to get the net strangeness zero at Q 2 = 0 for nucleon. They also have important contribution to the magnetic form factors. For both G s E and G s M , the octet intermediated states provide more important contribution than decuplet intermediate states. In this nonlocal chiral effective Lagrangian, there are three free parameters. c 1 and c 2 are determined by the experimental magnetic moments of proton and neutron. Λ in the correlation function is determined by the best description of the nucleon electromagnetic form factors up to relatively large Q 2 . At finite momentum transfer, the strange charge form factor is positive, while the strange magnetic form factor is negative. At Q 2 = 0, the strange magnetic moment is −0.041 +0.012 −0.014 . Compared with the calculated G s M (0) in heavy baryon formalism with finite-range-regularization, the absolute value of G s M (0) calculated in this relativistic version is a little smaller. Our results are also comparable with the recent lattice simulation. As a summary, we list the contribution to the strange magnetic moment of each diagram in Table  I.