Sea quark contributions to nucleon electromagnetic form factors with the nonlocal chiral effective Lagrangian

The sea quark contributions to the nucleon electromagnetic form factors from up, down and sea quarks are studied with the nonlocal chiral effective Lagrangian. Both octet and decuplet intermediate states are included in the one loop calculation. Compared with the strange form factors, though their signs are the same, the absolute value of the light quark form factors are much larger. For both electric and magnetic form factors, the contribution from $d$ quark is larger than that from $u$ quark. The current lattice data for the light-sea quark form factors are between our sea quark results for $u$ and $d$.


I. INTRODUCTION
The study of electromagnetic form factors of nucleon is of crucial importance to understand the non-perturbative properties of QCD. It is well known that a complete description of nucleon substructure must go beyond three valence quarks. One of the great challenges of modern hadron physics is to unravel the precise contribution of sea quark to the nucleon structure. Role of the sea remains a central issue in QCD, especially with respect to lattice QCD, where such terms involve so-called disconnected graphs, i.e. quark loops are connected only by gluons to the valence quarks.
Strange quark contribution to the nucleon form factors has attracted a lot of interest because it is purely from the sea quark. There are a lot of experimental and theoretical efforts to get the precise number of the strange form factors [1][2][3][4]. For the light quark, the sea and valence contributions to the nucleon properties are always combined together. With the help of quenched chiral perturbation theory, one can separate the sea and valence contributions of light quark. Though it is hard to get, the sea contribution of light quark to the nucleon form factors has been simulated in lattice as well as the strange quark. [5][6][7]. Therefore, it is very interesting to calculate the light sea quark contribution and compare results with the lattice data.
Though QCD is the fundamental theory to describe strong interactions, it is difficult to study hadron physics using QCD directly. There are many phenomenological models, such as the cloudy bag model [8], the constituent quark model [9], the 1/Nc expansion approach [10], the perturbative chiral quark model [11], the extended vector meson dominance model [12], the SU(3) chiral quark model [13], the quark-diquark model [14], etc. Besides the above phenomenological models, heavy baryon and relativistic chiral perturbation theory have been widely applied to study the hadron spectrum and structure. Historically, most formulations of ChPT are based on dimensional or infrared regularization. Though ChPT is a successful and systematic approach, for the nucleon electromagnetic form factors, it is only valid for Q 2 < 0.1 GeV 2 [15]. When vector mesons are included, the result is close to the experiments with Q 2 less than 0.4 GeV 2 [16].
An alternative regularization method, namely finite-range-regularization (FRR) has been proposed. Inspired by quark models that account for the finite-size of the nucleon as the source of the pion cloud, effective field theory with FRR has been widely applied to extrapolate the vector meson mass, magnetic moments, magnetic form factors, strange form factors, charge radii, first moments of GPDs, nucleon spin, etc [17][18][19][20][21][22][23][24][25][26]. In the finite-range-regularization, there is no cut for the energy integral. The regulator is not covariant and is in three-dimensional momentum space. This non-relativistic regulator can only be applied with the heavy baryon ChPT. Recently, we proposed a nonlocal chiral effective Lagrangian which makes it possible to study the hadron properties at relatively large Q 2 [27][28][29][30]. The nonlocal interaction generates both the regulator which makes the loop integral convergent and the Q 2 dependence of form factors at tree level. The obtained electromagnetic form factors and strange form factors of nucleon are very close to the experimental data [28,29].
In this paper, we will apply the nonlocal Lagrangians to investigate the light sea quark contribution to the nucleon factors. With the quark flow method same as in Ref. [31], we can separate the sea and valence quark contribution respectively. This method is equivalent to the quenched chiral perturbation theory. In section II, we will introduce the nonlocal chiral Lagrangian. The sea quark contributions to the nucleon form factors are derived in section III. Numerical results are shown in section IV and finally, section V is a short summary.

II. CHIRAL EFFECTIVE LAGRANGIAN
The lowest order chiral Lagrangian for baryons, pseudo-scalar mesons and their interaction can be written as [28,30], where D, F and C are the coupling constants. The chiral covariant derivative D µ is defined as The pseudo-scalar meson octet couples to the baryon field through the vector and axial vector combinations as where Q c can be the real charge matrix diag(2/3, −1/3, −1/3). The matrix of pseudo-scalar fields φ is expressed as A µ is the photon field. The covariant derivative D µ in the decuplet sector is defined as D ν T abc 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 decuplet, there are three indexes 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 quark contributions to the nucleon anomalous magnetic moments are expressed as Comparing with the results of constituent quark model where we can get The transition magnetic operator is The effective decuplet anomalous magnetic moment operator can be expressed as For each decuplet baryon, its moment F T 2 can also be written in terms of c 1 and c 2 . For example, for ∆ ++ , the magnetic moment µ ∆ ++ = 3µ u = 3c 2 . Therefore, F ∆ ++ 2 = 3c 2 − 2. Now we construct the nonlocal Lagrangian which will generate the covariant regulator. The gauge invariant nonlocal Lagrangian can be obtained using the method in [27][28][29]. For instance, the local interaction including π meson can be written as The corresponding nonlocal Lagrangian is expressed as where F (x) is the correlation function.
To guarantee the gauge invariance, the gauge link is introduced in the above Lagrangian. The regulator can be generated automatically with correlation function.With the same idea, the nonlocal electromagnetic interaction can also be obtained. For example, the local interaction between proton and photon is written as The corresponding nonlocal Lagrangian is expressed as where F 1 (a) and F 2 (a) are the correlation functions for the nonlocal electric and magnetic interactions. The form factors at tree level which are momentum dependent can be easily obtained with the Fourier transformation. The simplest choice is to assume that the correlation function of the nucleon electromagnetic vertex is the same as that of the nucleon-pion 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. As a result, the obtained charge form factor of proton decreases very quickly with increasing Q 2 and it will become negative after some Q 2 .
A better choice is to assume that the charge and magnetic form factors at tree level have the same the momentum dependence as nucleon-pion vertex, i.e. G tree is the Fourier transformation of the correlation function F (a). 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, π + (y) → e iα(y) π + (y), where α(x) = daα (x − a)F (a). From Eq. (15), two kinds of couplings between hadrons and one photon 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 to get the renormalized proton (neutron) charge 1 (0).

III. ELECTROMAGNETIC FORM-FACTORS
The contribution from the quark flavor f (f = u, d, s) to the Dirac and Pauli form factors of nucleon are defined as where q = p − p, Q 2 = −q 2 . The electromagnetic form-factors are defined as the combinations of the above ones for each flavor as In this manuscript, we will investigate the sea quark contribution to the nucleon electromagnetic form factors from u, d and s. According to the Lagrangian, the one loop Feynman diagrams which contribute to the nucleon electromagnetic form factors are plotted in Fig. 1. In full QCD, the coefficients are determined from the Lagrangian. In order to obtain the pure sea quark contribution, we need to get the coefficients for the quenched and loop diagrams separately as in Ref. [31] using the quark flows of Fig. 2. The obtained coefficients are the same as those extracted within the graded symmetry formalism in quenched chiral perturbation theory [32]. In Fig. 2, we plot the diagram for the π + rainbow diagram using quark flows as an example to show the method of separating the quenched and sea quark contribution. The coefficients for the π + loop diagram in full QCD is (D + F ) 2 . The coefficient of Fig. 2b for the sea quark contribution is the same as that of Fig. 2c for the K + loop. The coefficient for quenched sector can be obtained by subtracting the coefficient of sea diagram from the total one. The coefficients of u, d and s quark for both quenched and sea quark flow diagram are listed in Table. I. For π 0 case, the first and second rows are for uū and dd, respectively.
With the Lagrangian we can get the matrix element of Eq. (22). In this section, we will only show the expressions for the intermediate octet baryon part. For the intermediate decuplet baryon, the expressions are similar but more complicated.  Fig. 1a with π + and K + . The sea quark contributions of Fig. 1a for quark u d and s are written as, where the integral I BM a is expressed as where the integral I BM b is written as Fig.1c is similar as Fig.1b except for the magnetic interaction. The contributions of this diagram are written as where I ΛK c is expressed as Fig. 1d and 1e are the Kroll-Ruderman diagrams. The contributions from these two diagrams are written as where where m j is the meson mass for the baryon-meson interaction and it is zero for the hadron-photon interaction. It was found that when Λ was around 0.90 GeV, the results are very close to the experimental nucleon form factors. In Fig. 3, we plot the sea contribution of u quark with unity charge to the proton electric form factor. Three blue lines from up to bottom are for Λ = 1.0, 0.9 and 0.8 GeV, respectively. As a comparison, the central result for the strange quark is also plotted in the figure with red line. The solid dots with error bars are lattice data from Ref. [6]. Since we did not include the valence contribution of u quark in proton, the electric form factor of u quark is zero when Q 2 = 0. It then increases with the increasing Q 2 . When Q 2 is larger than about 0.3 GeV 2 , it deceases with Q 2 . From the figure, one can see the strange form factor can be described very well. The u quark result is a little smaller than the lattice data. We should mention that lattice data are for the light quark and it was assumed that u and d had the same sea contribution. Therefore, lattice data for the light quark can be approximately treated as an average of the u and d contribution. The sea contribution to the proton electric form factor from d is plotted in Fig. 4. Similar as in the u quark case, the electric form factor first increases from 0 and then decreases with the increasing Q 2 . It can be seen clearly that our calculated sea quark contribution is obviously larger than the lattice data. The larger sea contribution from d quark than that from u quark is due to the fact that there is no intermediate octet contribution for u quark sea. The only contribution for u quark sea is from the decuplet intermediate states. Similar results can be found for thed −ū asymmetry in proton, whered is excessū [36][37][38][39]. Though there is obviously difference of the sea quark contribution between u and d, both of them are much larger than that of strange quark contribution. The strange electric form factor is about 5-10 times smaller due to the suppression of the K meson loop.
The sea quark contribution to the proton magnetic form factor from u and d quark with unit charge are plotted in Fig. 5 and Fig. 6. Again, the calculated strange magnetic form factor is in good agreement with the lattice data. All From the above figures, one can see that the sea quark contribution of u and d quark is quite different. In both electric and magnetic form factor case, the absolute value of sea quark contribution of d is much larger than that of u. This is because for proton, there are two up quarks and one down quark. The u quark in the loop diagram can only form a decuplet state and there is no intermediate octet contribution to the sea quark form factors of u. We should mention this difference between light-sea quark form factors is not due to the mass difference of u and d quark. In fact, in our calculation the masses of π 0 , π + and π − are degenerate. It is straightforward that the sea quark form factor of u (d) in proton is the same as that of d (u) in neutron if the masses of proton and neutron are the same. The mass difference between proton and neutron will lead to a small charge symmetry violation, i.e. a small difference between G u p and G d n (G d p and G u n ). The large difference between G u p and G d p is because of the effect of non-perturbative valence quark environment instead of mass difference between u and d.
If the mass of the three u quark states is taken to be degenerate with nucleon mass, and η mass taken to be degenerate with π mass. the sea contribution from u and d quark in proton will be the same. This is the artifact of the current lattice simulation. Physically, three u quark can not form a octet baryon and the mass of η is much heavier than that of π. Therefore, it is very interesting and challenging to get the flavor asymmetry from the lattice with the full-QCD simulation.

V. SUMMARY
In this work, we applied the nonlocal chiral effective Lagrangian to study the sea quark contribution of light quark to the proton electromagnetic form factors. Since the sign of the sea quark form factors are the same for u, d and s quark, this calculation is helpful to understand the experimental values of strange form factors. It is also interesting to compare our result with that from the lattice simulation. In our calculation, the parameter Λ in the regulator is the same as the previous one which is determined by fitting the nucleon form factors. The low energy constants c 1 and c 2 are determined by the experimental magnetic moments of proton and neutron. Therefore, in calculating the sea quark form factors, there is no free parameters to be adjusted. Our results show the electric form factor of light sea quark with unity charge is positive, while the magnetic form factor is negative. Compared with the strange form factors, the absolute value of the light quark form factors are much larger. For both electric and magnetic form factors, the contribution from d quark is larger than that from u quark. The current lattice data for the light-sea quark form factors are between our u and d results. Therefore, it is interesting if this flavor asymmetry can be obtained from lattice with full-QCD simulation.