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Holographic Baryons : Static Properties and Form Factors from Gauge/String Duality

In this paper, we study properties of baryons by using a holographic dual of QCD on the basis of the D4/D8-brane configuration, where baryons are described by a soliton. We first determine the asymptotic behavior of the soliton solution, which allows us to evaluate well-defined currents associated with the U(N_f)_L \times U(N_f)_R chiral symmetry. Using the currents, we compute static quantities of baryons such as charge radii and magnetic moments, and make a quantitative test with experiments. It is emphasized that not only the nucleon but also excited baryons, such as \Delta, N(1440), N(1535) etc., can be analyzed systematically in this model. We also investigate the form factors and find that our form factors agree well with the results that are well-established empirically. With the form factors, the effective baryon-baryon-meson cubic coupling constants among their infinite towers in the model can be determined. Some physical implications following from these results are discussed.


§1. Introduction
The gauge/string duality has opened up new technology to analyze strongly coupled gauge theories. Although it has not been proven yet, there is a lot of highly nontrivial evidence suggesting the duality between four-dimensional gauge theories and string theory in higher-dimensional curved backgrounds, at least for the cases with supersymmetry and conformal symmetry 1)-3) (for a review, see Ref. 4)). It is obviously important to see if this idea can be applied to realistic QCD in many respects. One of the advantages of QCD is that we can use experimental data to check if the duality really works. In general, the analysis of non-supersymmetric strongly coupled gauge theories is very difficult, and hence it is almost impossible to establish such duality. However, for QCD, we can omit the complicated calculations on the gauge theory side and simply compare the results on the string theory side with the experimental data to obtain nontrivial evidence of the duality.
If the gauge/string duality is really applicable to QCD, it will provide us with new deep insight into the theory of strong interaction in both conceptual and practical terms. It will tell us that both QCD and string theory in a higher-dimensional curved background (holographic QCD) can be a fundamental theory of strong interaction at the same time. Of course, in the high-energy weakly coupled regime, QCD is a better description. However, at least in the large-N c strongly coupled regime, the string theory can be used as a powerful tool to calculate various physical quantities.
In Refs. 5) and 6), it was proposed that QCD with N f massless quarks is dual to type IIA string theory with N f -probe D8-branes in the D4-brane background used in Ref. 7) at low energy. In this model, mesons appear as open strings on the D8-branes and baryons are expressed as D4-branes wrapped on the nontrivial four-cycle of the background, as was anticipated in Refs. 8) and 9) in the context of the AdS/CFT correspondence. The effective action for the open strings turned out to be a five-dimensional U(N f ) Yang-Mills-Chern-Simons (YM-CS) theory in a curved background. Via the equivalence between the wrapped D4-branes and the instantons in the D8-brane world-volume gauge theory, 10) the baryons were obtained as instanton configurations in the Euclidean four-dimensional space in this five-dimensional gauge theory 5) (see also Ref. 11)).
The idea of realizing baryons as solitons was pioneered by Skyrme in the early 1960s. 12) He started with an effective action for a pion with a fourth-derivative term (Skyrme term) and constructed a soliton solution (Skyrmion) in which the U(N f )-valued pion field carries a nontrivial winding number interpreted as the baryon number. In the model of holographic QCD, 5), 6) this construction is naturally realized. It can be shown that the five-dimensional gauge theory is equivalent to a theory of mesons including pions as well as infinitely many vector and axial-vector mesons. The pion sector is given by the Skyrme model and the instanton configuration reduces to a Skyrmion, realizing the old idea of Atiyah and Manton 13) who proposed expressing a Skyrmion using a gauge configuration with a non-zero instanton number.
The Skyrme model was further developed by Adkins et al., 14) who analyzed various static properties of the nucleon and ∆ by quantizing the Skyrmion. It is natural to expect that the analysis can be improved by applying their strategy to the holographic model of QCD, since the model automatically contains the contribution from the vector and axial-vector mesons such as ρ, ω, a 1 , etc. * ) In fact, the quantization of the instanton representing a baryon was studied in Refs. 16)-18). It was found in Ref. 16) that the spectrum of the baryons obtained in the model, including nucleons, ∆, N(1440), N(1535), etc, is qualitatively similar to that observed in nature, although the prediction of the mass differences tended to be larger than the observed differences. In Refs. 17) and 18), a five-dimensional fermionic field was introduced into the five-dimensional gauge theory to incorporate the baryon degrees of freedom, where the interaction terms between the two fields were analyzed by quantizing the instanton. Using this five-dimensional system, the interaction of the baryons with the electromagnetic field and mesons was computed, which was found to be in good agreement with the experimental data.
In this paper, we analyze the static properties of baryons, such as the magnetic moments, charge radii, axial couplings, etc. of nucleons, ∆, N(1440), and N(1535), generalizing Refs. 14) and 16). These quantities are obtained by calculating the currents corresponding to the chiral symmetry that contain the information of the couplings to the external electromagnetic gauge field. As we will see, most of our results are closer to the experimental values than the results given in Ref. 14) for the Skyrme model. For some quantities of the excited baryons, our results are predictions. We also consider the form factors of the spin 1/2 baryons and study how they behave as functions of the momentum transfer.
Note that there are some closely related works that address the same subject. As mentioned above, Refs. 17) and 18) analyze the properties of baryons using the five-dimensional gauge theory with a five-dimensional fermion. On the other hand, our analysis is based on the quantum mechanics obtained through the usual procedure of quantizing solitons. These two approaches should in principle give the same results, and in fact we will find that some of our results are consistent with those in the studies, although the relation is not completely transparent. The work of Hata et al. 19) is more directly related to ours. The main difference is the definition of the currents. As discussed in their paper, their currents are problematic, since they are not gauge-invariant and have some ambiguities in the definition.
The paper is organized as follows. In §2, after briefly reviewing the main points of the model and the construction of the baryon, we calculate the currents corresponding to the chiral symmetry. The applications are given in §3. We calculate various physical quantities including charge radii, magnetic moments, axial coupling, etc., and compare them with the experimental values for nucleons. We also present these values for excited baryons as our predictions. In §4, we compute the form factors of the spin 1/2 baryons and study their properties including cubic baryon-baryon-meson couplings. Section 5 is devoted to a summary and discussion with tables summarizing our results including our predictions for future experiments. Two appendices summarize some technical details. §2. Currents

The model
In this subsection, we give a brief review of Refs. 5), 6), and 16) with an emphasis on the construction of baryons. * ) Based on the idea of gauge/string duality, it was proposed that the meson effective theory is given by a five-dimensional U(N f ) YM-CS theory in a curved background. The action of the model is Here, µ, ν = 0, 1, 2, 3 are four-dimensional Lorentz indices, and z is the coordinate of the fifth dimension. The quantity A = A α dx α = A µ dx µ + A z dz (α = 0, 1, 2, 3, z) is the fivedimensional U(N f ) gauge field and F = 1 2 F αβ dx α ∧ dx β = dA + iA ∧ A is its field strength. The constant κ is related to the 't Hooft coupling λ and the number of colors N c as Although it is not explicitly written in (2 . 1), the mass scale of the model is given by a parameter M KK , which is the only dimensionful parameter of the model. In Refs. 5) and 6), these two parameters are chosen as to fit the experimental values of the ρ meson mass and the pion decay constant f π ≃ 92.4 MeV. In this paper, we mainly work with M KK = 1 unit. The functions h(z) and k(z) are given by and ω 5 (A) is the CS 5-form defined as * ) The action (2 . 1) is obtained in Ref. 5) as the effective action of N f -probe D8-branes placed in the D4-brane background studied in Ref. 7) and is thought to be an effective theory of mesons in four-dimensional (large N c ) QCD with N f massless quarks.
In this paper, we consider the N f = 2 case, and the U(2) gauge field A is decomposed as where τ a (a = 1, 2, 3) are Pauli matrices and τ 0 = 1 2 is a unit matrix of size 2. Then, the equations of motion are where D α = ∂ α + iA α is the covariant derivative. The baryon in this model corresponds to a soliton with a nontrivial instanton number on the four-dimensional space parameterized by x M (M = 1, 2, 3, z). The instanton number is interpreted as the baryon number N B , where Unfortunately, because the equations of motion are complicated nonlinear differential equations in a curved space-time, it is difficult to find an analytic solution corresponding to the baryons. However, as observed in Refs. 16) and 17), the center of the instanton solution is located at z = 0 and its size is of order λ −1/2 , and hence we can focus on a tiny region * ) Here we omit the symbol '∧ ′ for the wedge products of A and F (e.g. AF 2 = A ∧ F ∧ F ) in the CS 5-form. around z = 0 for large λ, in which the warp factors h(z) and k(z) can be approximated by 1. Then it follows that the static baryon configuration is given by the BPST instanton solution with the U(1) electric field of the form (2 . 12) Here with X M = (X 1 , X 2 , X 3 , Z) = ( X, Z) being the position of the soliton in the spatial R 4 direction and the instanton size ρ. Substituting this configuration into the action and taking the nontrivial z dependence of the background into account as a 1/λ correction, we find that ρ and Z have a potential of the form which is minimized at In order to quantize the soliton, we use the moduli space approximation method 20), 21) and then the system is reduced to the quantum mechanics on the instanton moduli space. The SU(2) one-instanton moduli space is simply given by M ≃ R 4 × R 4 /Z 2 , which is parameterized by ( X, Z) and y I (I = 1, 2, 3, 4) with the Z 2 action y I → −y I . The size of the instanton ρ is related to y I by ρ = y 2 1 + · · · + y 2 4 , and a I ≡ y I /ρ represent the SU(2) orientations of the instanton. The Lagrangian of the collective motion of the soliton was obtained in Ref. 16) as where M 0 = 8π 2 κ. The Hamiltonian is given by with the canonical momenta This system is manifestly invariant under SO(4) rotation acting on y I . As argued in Refs. 14) and 16) the generators of SO(4) ≃ (SU(2) I × SU(2) J )/Z 2 symmetry correspond to the isospin and spin operators given by respectively.
The explicit eigenfunctions of the Hamiltonian (2 . 17) are obtained in Ref. 16). They are characterized by the quantum numbers (l, I 3 , s, n ρ , n z ) as well as the momentum p. Here l = 1, 3, 5, · · · are positive odd integers related to isospin I and spin J by I = J = l/2. Note that (2 . 19) implies I 2 = J 2 , and hence only the states with I = J appear in the spectrum as in Ref. 14) for the Skyrme model. I 3 and s denote the eigenvalues of the third component of the isospin and spin, respectively. n ρ and n z are nonnegative integers corresponding to the excitations with respect to ρ and Z, respectively. The species of the baryon are specified by B ≡ (l, I 3 , n ρ , n z ) and the wavefunctions are of the form with | p = 1 (2π) 3/2 e i p· X . For later use, we normalize the baryon state as For example, the proton and neutron are interpreted as the particles with B = (1, 1/2, 0, 0) and B = (1, −1/2, 0, 0), respectively, and the corresponding wavefunctions with s = 1/2 are

Currents
As in the usual effective field theory approach to the hadrons, it is useful to introduce the external gauge fields A Lµ and A Rµ associated with the chiral symmetry U(N f ) L × U(N f ) R . The currents J µ L and J µ R associated with the chiral symmetry are then read off from the terms linear with respect to the external gauge fields in the effective action as In terms of the vector and axial-vector current are external vector and axial-vector fields.
We can apply this idea to compute the currents in the present model, which leads us to a prescription analogous to that used in the AdS/CFT correspondence, 2), 3) as first explored in Ref. 11) and discussed recently in Ref. 19). As studied in Refs. 5) and 6), the chiral symmetry is identified with the constant gauge transformation at infinity z = ±∞, and the external gauge fields A Lµ and A Rµ can be introduced by considering the five-dimensional gauge field with the boundary conditions respectively. To compute the currents, we substitute a solution of the equations of motion with this boundary condition into the action and keep only the linear terms in the external gauge fields A L and A R assuming that the external fields are infinitesimal. Consider the gauge configuration of the form where A cl α is a classical solution of the equations of motion with A cl α (z = ±∞) = 0 and δA α (x, z) is an infinitesimal deviation from it satisfying (2 . 30) Substituting this configuration into the action (2 . 1), we obtain where ψ 0 (z) = 2 π arctan z. Here only the surface terms at z = ±∞ remain in (2 . 31) because of the equations of motion.
Note that the currents in (2 . 32) are consistent with the four-dimensional effective action obtained in Ref. 6). It was shown in Ref. 6) * ) that the four-dimensional effective action derived from our model has the following terms: g a n a n µ + f π ∂ µ Π , (2 . 34) where Π(x), v n µ (x), and a n µ (x) are the pion, vector meson, and axial-vector meson fields, respectively, and f π , g v n , and g a n are the decay constants of these mesons, respectively. These mesons are related to the five-dimensional gauge field as We can show that ψ 2n−1 (z) and ψ 2n (z) are even and odd functions of z, respectively, if we arrange ψ n (z) such that the eigenvalues satisfy λ 1 < λ 2 < λ 3 < · · · . The decay constants are given by Note that the functions ψ n (z) behave as O(z −1 ) in the z → ±∞ limit and the decay constants g v n and g a n are determined as the coefficients in front of 1/z as , ψ 2n (z → ±∞) ≃ g a n 2κz .
(2 . 39) The following expressions are also useful: g a n a n µ .
( As in (2 . 6), we decompose the currents as Then the baryon number current is given by As a check, the baryon number density is computed as where we have used the equation of motion (2 . 7). This is consistent with (2 . 11) as expected.

Asymptotic solution
In order to calculate the currents (2 . 32), we have to know how the field strength F µz behaves at z = ±∞. Unfortunately, we cannot directly use the solution (2 . 12) since it is only valid in the region ξ ≪ 1. * ) In this subsection, we study how to extend the solution to the 1 ≪ ξ region. * ) Here we are assuming Z ∼ Z cl = 0, since the expectation value of f (Z) is approximated by its classical value f (Z cl ) for the large N c and large λ limit.
Let us first summarize the gauge configuration in the ξ ≪ 1 region. Here we include the time-dependent moduli parameters, which are treated as operators in quantum mechanics (2 . 17). As explained in Ref. 16), the SU(2) gauge field takes the form where A cl M is the solution (2 . 12) and V satisfies which follows from the Gauss law constraint. Here where y = y 4 + iy a τ a , a = a 4 + ia a τ a = y/ρ, and J a is the spin operator defined in (2 . 19).
It is convenient to perform the gauge transformation This choice of gauge is useful in considering the asymptotic behavior, as we will see in the following, while a singularity develops at ξ = 0.
The U(1) part is treated as a perturbation in the background given by the SU(2) gauge configuration obtained above. The leading contribution to the U(1) part is obtained by solving the following linearized equations of motion (in the Lorenz gauge): Here we substituted (2 . 49) and (2 . 50) into the equations of motion (2 . 7) and (2 . 9) with the warp factors h(z) and k(z) approximated by 1. We neglect the terms including ∂ 2 0 , because we are interested in slowly moving solitons. * ) * ) The time derivative squared such asZ andŻ 2 can be traded by Z or Z 2 via its Schrödinger equation or the relationŻ = P Z /M 0 ; thus for large N c and λ, it is approximated by its classical value, which vanishes.

The regular solution is found to be
(2 . 54) Note that (2 . 55) and (2 . 56) are neglected in Ref. 16), since the energy contributions from them are subleading in the 1/λ expansion given in Ref. 16). However, here we keep them because they give the leading contribution to the isoscalar current density. It can also be shown that the F 2 terms in the equations of motion (2 . 7) and (2 . 9) are subleading in the 1/λ expansion, justifying the above perturbative treatment for the U(1) part.
So far, we have established a solution that is valid in the region ξ ≪ 1. We now consider how to find the solution in the 1 ≪ ξ region. The key observation is that all the components of the gauge field in (2 . 49), (2 . 50), (2 . 54), (2 . 55), and (2 . 56) are suppressed in the ρ ≪ ξ ≪ 1 region in the large λ limit. This implies that the nonlinear terms in the equations of motion can be neglected in this region for large λ. Our strategy is to find a solution of the linearized equations of motion in the ρ ≪ ξ region that smoothly connects the previous solution in the overlapping region ρ ≪ ξ ≪ 1.
For this purpose, we note that for ρ ≪ ξ ≪ 1, the gauge field (2 . 49), (2 . 50), (2 . 54), (2 . 55), and (2 . 56) are approximated as is the Green's function in the flat R 4 that satisfies We can easily check that the gauge configuration (2 . 57)-(2 . 62) satisfies the Maxwell and the linearized YM equations without sources: as well as the gauge condition In order to connect this solution to the large ξ region, we have to take into account the effect of the curved background. The linearized equations of motion and the gauge condition that generalize (2 . 65) and (2 . 66) to the case with nontrivial warp factors h(z) and k(z) are respectively.
To solve these equations, we define Green's functions in the curved space as where {ψ n (z)} n=1,2,··· is the complete set defined in (2 . 36) and (2 . 37), {φ n (z)} n=0,1,··· is another complete set given by and Y n (r) is the Yukawa potential with meson mass m n = √ λ n , Note that the normalization of the functions {φ n } n=0,1,2,··· is fixed by Using (2 . 36), (2 . 72), (2 . 74), and the completeness conditions it is easy to verify Then, the solution of equations (2 . 67)-(2 . 69) is obtained by replacing the Green's function G flat in (2 . 57)-(2 . 62) with G or H as follows: Here, we neglected the terms including ∂ 2 0 as before. Since the Green's functions G and H approach G flat for ρ ≪ ξ ≪ 1, this solution is smoothly connected with the previous solution (2 . 57)-(2 . 62) in this region, as expected. Note also that the Green's functions G and H vanish at ξ → ∞, and hence the linear approximation of the equations of motion does not break down all the way to infinity. Therefore, we can read off the behavior of the gauge field at z → ±∞, which is needed to calculate the currents, from (2 . 80)-(2 . 85).

Computation of the current
Since the wavefunctions of the low-lying baryon states, such as (2 . 22), are dominant for Using the asymptotic solution (2 . 80)-(2 . 85) and the relations (2 . 87), we obtain The following formulas are also useful: Using the relation (2 . 87), it is easy to check that these currents are conserved (up to terms including ∂ 2 0 ). * ) §3. Static properties of baryons In this section, we study the static properties of baryons as applications of the currents obtained in the previous section.

Baryon number density, isoscalar mean square radius
The baryon number density is obtained from (2 . 43) and (2 . 96) as As a check, the baryon number charge is calculated using this expression as where O = B, s|O|B, s is the expectation value with respect to a baryon state |B, s . Here we have used (2 . 40) and (2 . 76).
The baryon number density per unit r is given by the integrand of (3 . 2), Then the isoscalar mean square radius is In the large N c and large λ limit, the baryon wavefunction is localized at Z = 0, as we can see from the wavefunction (2 . 22) for the nucleon states, and hence the expectation * ) Here we neglect the effect of the U (1) A anomaly, since it is subleading in the 1/N c expansion.
Note that the function and the boundary condition F (z) → 0 (z → ±∞). The first relation is obtained from (2 . 36), (2 . 40), and (2 . 76). The solution of (3 . 7) is given by The constant F 0 is fixed by the boundary condition and we obtain Therefore (3 . 5) can be evaluated as and if we use the value (2 . 3) for M KK , we obtain The experimental value is r 2 1/2 I=0 | exp ≃ 0.806 fm 23) and the prediction of the Skyrme model in Ref. 14) is 0.59 fm.
It is interesting to note that this value is independent of λ and N c . The N c independence is consistent with the analysis of baryons in large N c QCD. 24) The λ independence suggests that the size of the baryon number distribution is governed by the scale of the vector meson mass rather than the size of the soliton ρ cl ∼ O(λ −1/2 ) for large λ.
Given the wavefunction of the baryon state, it is also possible to evaluate the expectation value F (Z) numerically. * ) For the nucleon wavefunction given by (2 . 22), we obtain (3 . 12) * ) Note that in the previous section the chiral currents were obtained using the approximation Z ∼ 0, and thus the classical value for Z was used. Here, we simply assume the same chiral current and evaluate it using the quantum states of the baryons. This value is the same for the other states with n z = 0, such as ∆(1232) and N(1440), since Z dependence of the wavefunction is the same for these states. For the states with n z = 1, such as N(1535), we obtain See Appendix A.2 for more details.

Isoscalar magnetic moment
The isoscalar magnetic moment is defined as 14) It is easy to see that only the last term in (2 . 97) contributes to the integral. Then by evaluating the angular integral in (3 . 14) we obtain Using (2 . 47), this can be evaluated as where J i is the spin operator (2 . 19). For example, for nucleon states with up spin, this gives The g factor is defined as where σ i is the Pauli matrix that acts on a spin doublet, and M N is the nucleon mass. If we use (2 . 3) and the experimental value for the nucleon mass M N ≃ 940 MeV, we have * ) The experimental value is g I=0 | exp ≃ 1.76 23) and the prediction obtained from the Skyrme model in Ref. 14) is g I=0 | ANW ≃ 1.11. * * ) * ) Note that M N is not calculated in Ref. 16) because the total contribution from the zero point energy of the fluctuations around the soliton solution is difficult to evaluate. Here the g factor is computed simply to express the magnetic moments in the unit of 1/(4M N )| exp . * * ) The same expressions as (3 . 17) and (3 . 19) were obtained using the current in Ref. 19). However, the numerical values were different, because the values of M KK and κ used in Ref. 19) were different from ours.

Isovector charge density and charge radii
The isovector charge density is given by (2 . 98) and the isovector charge is evaluated using A relation similar to (2 . 47), where I a is the isospin operator defined in (2 . 19), implies as expected.
The isovector charge density per unit r is proportional to the angular integral of the isovector charge density with the normalization condition ∞ 0 dr ρ I=1 (r) = 1. This turns out to be identical to the baryon number density: (3 . 23) Therefore, the isovector mean square charge radius r 2 I=1 = ∞ 0 r 2 ρ I=1 (r)dr is the same as the isoscalar mean square radius r 2 I=0 evaluated in §3.1. This result is somewhat puzzling since it is known that the isovector mean square radius is divergent in the chiral limit. 25) There is however no contradiction. This divergence is due to the IR divergence of pion loops. 26) Our analysis only involves a string world-sheet with disk topology, and hence the pion loops are not included. Therefore it will be important to include the quark mass in the model * ) and estimate the contribution from the annulus diagram to make a comparison with the experimental value, which is beyond the scope of the present paper.
The electric charge is defined by which gives Q em = 1 for a proton (I 3 = 1/2) and Q em = 0 for a neutron (I 3 = −1/2). Then, because of identity (3 . 23), the electric charge density is given by ρ E = ρ I=0 for a proton and ρ E = 0 for a neutron, and we obtain r 2 E,p = r 2 I=0 (for a proton) , r 2 E,n = 0 (for a neutron) .  Although our calculation does not reproduce the experimental value of the electric charge radius of the neutron r 2 E,n , the vanishing of the neutron electric charge density seems to be a good approximation for reproducing the observed behavior of the electric form factor, as we will study in §4.
The charge radius for the excited baryons is similarly found to be the same as their isoscalar mean square radius r 2 I=0 obtained in §3.1. In particular, our analysis predicts that the Roper excitation N(1440) has a charge radius equal to that of a proton (3 . 12), while that of N(1535), (3 . 13), is smaller.

Magnetic moment
The isovector magnetic moment is defined as Here the additional factor of 2 in the integrand is due to our normalization of the current.

Substituting (2 . 99) into this expression gives
For the baryon states of I = J = 1/2, we can use the identity 14) with σ i and τ a being the Pauli matrices corresponding to spin and isospin, respectively. Here we have used the notation where χ (s) and ψ I 3 are defined as Then, we obtain where ρ 2 is the expectation value of ρ 2 for the nucleon states. Using the wavefunction (2 . 22), ρ 2 is obtained as In the large N c limit, it can be approximated by its classical value ρ 2 cl in (2 . 15). However, for N c = 3, (3 . 33) implies ρ 2 nρ=0 ≃ 1.62 ρ 2 cl , suggesting that the 1/N c corrections are relatively large for this quantity.
Consequently, we obtain Here we have recovered the M KK dependence by dimensional analysis. The isovector g factor, defined as in (3 . 18), is then for N c = 3, M N ≃ 940 MeV, and (2 . 3). The experimental value is g I=1 | exp ≃ 9.41 23) and the prediction obtained from the Skyrme model in Ref. 14) is g I=1 | ANW ≃ 6.38. If we approximate ρ 2 by its classical value ρ 2 cl , we have g I=1 ≃ 4.34, which is considerably smaller than the experimental value. It is interesting to note that going to ρ 2 from ρ 2 cl by multiplying by the ratio (N c + √ 5/2)/N c for large N c , as can be seen from (3 . 33), has an effect similar to the "N c → N c + 2" rule discussed in Ref. 17). The value of the anomalous magnetic moment obtained using this rule from the five-dimensional effective spinor field theory approach in Ref. 17) is close to ours. * ) The magnetic moments for the proton and neutron (measured in units of the Bohr magneton µ N = 1/(2M N )) are given as the contribution of the isoscalar component will be buried in the 1/N c corrections of the isovector component in linear combinations such as (3 . 36). Therefore, it is more meaningful to consider g I=0 and g I=1 rather than g p and g n in our analysis.
from which we see that the agreement with our result (3 . 43) is not very good, particularly for µ ∆ 0 . This is because the contributions of µ I=1 ∼ O(N c ) and µ I=0 ∼ O(1/N c ) roughly cancel each other out for ∆ 0 and hence the 1/N c corrections cannot be neglected. To make a more reasonable comparison, we should compare µ I=0 and µ I=1 with experimental or lattice results as explained above. Then, the average of the magnetic moments in (3 . 45), should be compared with our result for the isoscalar component 1 2 µ I=0 (∆) = 3 2 µ I=0 (p) ≃ 1.26. The isovector components are extracted by considering the differences: These values are compared with our result 3c 5 µ I=1 (p) ≃ 2.82.

Axial coupling
As explained in Ref. 14), the axial coupling g A of the baryon states of I = J = 1/2 is given by (3 . 48) From (2 . 99), the integral of the axial-vector current becomes Although this is an integral of a total derivative, it does not vanish because of the term proportional to 1/r in (2 . 95). In fact, the integral can be performed using the Gauss' divergence theorem as where Y n (r) is defined in (2 . 73) with λ 0 = 0 for n = 0. This implies that only the n = 0 component of the mesons, that is the pion, contributes to the integral. Again using (3 . 29), we obtain where ρ 2 /k(Z) is the expectation value with respect to the spin 1/2 baryon states. Comparing this with (3 . 48), we obtain * ), * * ) .
(3 . 52) If we approximate ρ and Z by their classical values, we obtain We can see that the axial coupling for N(1440) is large while that for the negative parity baryon N(1535) is small compared with the proton/neutron. There is another more direct way to perform the integral in (3 . 49), which will be useful in §3.7. Using (2 . 95) and (2 . 74), the integrand of (3 . 49) is The sign of this equation is taken to be positive such that the axial coupling is defined to be positive. This sign can be flipped if one exchanges the definitions of "left" and "right" chiral sectors in the two asymptotes z → ±∞, i.e. the positive sign is a convention. * * ) This expression, once the classical value k(Z) = 1 is imposed, is equal to that obtained in Ref. 19). Furthermore, if we use ρ 2 = ρ 2 cl , it agrees with g A,mag in Ref. 17).

Goldberger-Treiman relation
The axial coupling obtained in the previous subsection is related to the πNN coupling g πN N by the Goldberger-Treiman relation We can derive this relation in our context following the argument given in Ref. 14) for the Skyrme model.
It is argued in Ref. 14) that the pion field behaves asymptotically as in the presence of a nucleon. Here, g πN N is the πNN coupling and the expectation value is taken for the nucleon. In the present case, (2 . 85) provides us with information on the asymptotic pion field. The pion field can be read from the n = 0 component of (2 . 71) substituted in (2 . 85). We find (3 . 67) Comparing (3 . 64) and (3 . 67), we obtain where we have used (2 . 38) and (3 . 52). This is nothing but the Goldberger-Treiman relation (3 . 63).

Axial radius
We define ρ A (r) as a function proportional to the expectation value of the integrand of (3 . 62) with the normalization ∞ 0 dr ρ A (r) = 1, The axial radius is obtained as From (2 . 36) and (3 . 59), this function satisfies the differential equation Integrating this equation, we obtain where f 0 is a constant. In order to fix f 0 , we use the identity (3 . 75) Therefore, if we approximate the expectation value by its classical value, we obtain If we numerically evaluate the expectation value of F A (Z) using the wavefunction (2 . 22), we obtain The experimental value is r 2 1/2 A | exp ≃ 0.674 fm. 36), * ) We can compute the axial radius for excited baryons in the same manner as for the other quantities evaluated before. The computation of the axial radius is independent of the profile of R(ρ) in the wavefunction; thus, for the N(1440) (Roper) with (n ρ , n z ) = (1, 0), it gives the same result as that for the proton/neutron (3 . 77). For the N(1535) with (n ρ , n z ) = (0, 1), we obtain (see Appendix A.2 for details) This is smaller than the axial radius of the proton/neutron. §4. Form factors In this section, we compute the form factors of spin 1/2 baryons associated with the currents J µ V,A obtained in the previous section. To this end, we first give a brief review of how to compute the form factors from the matrix elements of the currents. The present model enables us to compute the matrix elements easily with the tools formulated in §2.

Formalism
Let us first consider the matrix elements of a vector current for a baryon of spin 1/2, This value is obtained by applying the formula r 2 A = −6d/(dk 2 ) log g A (k 2 )| k 2 =0 to the axial form factor g A (k 2 ) in Ref. 36), which is fitted by a dipole.
where k = p − p ′ and m B is the baryon mass. On the right-hand side of (4 . 2), no summation is taken for the index C. F 1 (k 2 ) and F 2 (k 2 ) are the scalar functions of k 2 called the Dirac and Pauli form factors, respectively, whose dependence on n ρ and n z is indicated implicitly. They can be computed by evaluating the matrix elements of the current. u( p , s) and u( p ′ , s ′ ) are the Dirac spinors associated with the initial and final states of the baryon of mass m B , respectively. The normalization condition * ) is given by It is useful to write the matrix elements in the Breit frame with p = − p ′ = k/2 and Here G E,M ( k 2 ) are the Sachs form factors, related to the Dirac and Pauli form factors by The formulae of the Dirac spinor needed for this manipulation are summarized in Appendix B.2. The Sachs form factors can be obtained by evaluating the left-hand side of (4 . 6) using the baryon wavefunctions given in §2.1 and Appendix A.1. * ) With this normalization, we assign 2p 0 u( p , s) and 2p 0 u( p , s) to an incoming and outgoing external line, respectively, in the computation of the Lorentz invariant matrix elements.
The electromagnetic form factors are defined by considering the matrix elements of the electromagnetic current, Then, for the states with I 3 = +1/2 (p) and I 3 = −1/2 (n), the Sachs form factors associated with the electromagnetic current are given by respectively. G p,n E and G p,n M are called the electric and magnetic Sachs form factors, and their Fourier transformation provides the distribution of the electric charge density and the magnetic current density, respectively. (See Refs. 37)-39) for reviews.) Next we study the axial form factor associated with the axial current J µ A (x). As in the vector current case, we consider the matrix elements of the axial current for a spin 1/2 baryon, The matrix elements consistent with the symmetries can be written in terms of the axial form factor g A (k 2 ) and the induced pseudoscalar form factor g P (k 2 ) as On the right-hand side of (4 . 11), no summation is taken for the index C. The current conservation law yields (4 . 14) In the nonrelativistic limit, the spatial component becomes Again, the axial form factor g A (k 2 ) can be computed by evaluating the matrix elements in (4 . 15) from the baryon wavefunctions given in §2.1 and Appendix A.1.
In order to derive the form factors from the above formalism, it is useful to perform the Fourier transformation of the currents defined by show that g v n ψ 2n−1 (Z) g v n ψ 2n−1 (Z) g a n ∂ Z ψ 2n (Z) J cj A ( k) = e −i k· X 2π 2 κρ 2 tr τ c aτ a a −1 δ aj − k a k j k 2 n≥1 g a n ∂ Z ψ 2n (Z) where we have used (2 . 18), (2 . 47), and (3 . 21). Here '· · · ' denotes the terms that are odd with respect to Z; * ) they do not contribute to the result for the matrix elements. Useful formulas here are g a n ∂ Z ψ 2n (Z) The latter can be shown by using (3 . 60). In addition, it is important to note that the operator ordering in the Fourier transformation of the currents is fixed uniquely by the requirement 25) * ) More precisely, J V j includes a term that is even in Z and proportional toŻ. We discard this term, since it is negligible for large N c and λ.
which is equivalent to the Hermiticity of the current in the x-representation.
Using these expressions for the Fourier transformed currents together with the relation , the matrix elements can be calculated as

Form factors
Comparing (4 . 17)-(4 . 20) with (4 . 6), we find g v n ψ 2n−1 (Z) g v n ψ 2n−1 (Z) being the isoscalar and isovector g factors of the baryon B, as derived in § §3.2 and 3.4, respectively. Here we have used the formula (3 . 29). We note also Using (4 . 9), the electric and magnetic Sachs form factors for nucleons and their excited states with I = J = 1/2 are obtained as g v n ψ 2n−1 (Z) where are the g factors of the nucleons (and their excitations). It follows that they satisfy the Experimentally, the Sachs form factors for the proton and neutron are known to be well described as with Λ 2 = 0.71 GeV 2 . That is, the three form factors G p E , G p M , and G n M are proportional to each other and characterized by the dipole behavior. Furthermore, the electric charge density of the neutron can be well approximated to be flat. It turns out that our result (4 . 33) is in accord with these experimental results. In particular, the infinite sum in (4 . 33) can be approximated by a single dipole factor, showing the agreement as a function of k 2 . To see this, we expand our result (4 . 33) as a Taylor series in k 2 , n≥1 g v n ψ 2n−1 (Z) where the coefficients are obtained as the expectation values of As seen in (3 . 3), we have f 0 (Z) = 1. Let us evaluate these coefficients using the classical approximation ψ 2n−1 (Z) ≃ ψ 2n−1 (0). The first nontrivial coefficient, f 1 (0), is in fact equal to F 0 /6 given in (3 . 10). The higher coefficients can be obtained in the same way as F 0 , as they satisfy the recursive relation We thus obtain On the other hand, the dipole expression (4 . 34) can be expanded using Λ 2 = 0.758 GeV 2 as where we have chosen the parameter Λ 2 in such a way that the first coefficient reproduces the f 1 (0) of (4 . 38) with the unit M KK = 949 MeV = 1. We find a good agreement for the latter coefficients, suggesting that our form factors (4 . 33) exhibit the dipole behavior indicated by experiments. It is also useful to note that the relation (4 . 37) implies that the function The solution of this equation evaluated at z = 0 is plotted in Fig. 1. The electric and magnetic charge radii can be computed from the first coefficient of the form factor expanded in powers of k 2 . Namely, they are given by except for the neutron charge radius, which is defined by .

(4 . 43)
Since all the form factors are proportional to each other (except for G n E , which vanishes) as in (4 . 33), we conclude that It is easy to check that the definition of the charge radii in (4 . 42) is consistent with our previous calculation in §3, and using our result for the electric charge radius of the proton The values observed in experiments are r 2 1/2 E,p | exp ≃ 0.875 fm, 23) r 2 1/2 M,p | exp ≃ 0.855 fm, 40) and r 2 1/2 M,n | exp ≃ 0.873 fm, 41) which are reasonably close to our result.
The axial form factors are given by Note that the values at k 2 = 0 are obtained by using (3 . 60) as .
It is empirically known that the axial form factor g A (k 2 ) can also be well fitted by a dipole profile. Using the same technique as above, the Taylor expanded axial form factor with the classical approximation ∂ Z ψ 2n (Z) ≃ ∂ z ψ 2n (0) is found to take the form These coefficients are close to those obtained from the dipole profile with M A ≃ 1.26 GeV.

Cubic coupling
The form factors computed above are composed of an infinite tower of poles that correspond to the vector and axial-vector meson exchange. From the residues, we can extract information on cubic couplings among baryons and (axial-)vector mesons. We assume that there exist cubic couplings of the form in the four-dimensional baryon-meson effective action. Here v n µ (x) and v na µ (x) are the U(1) and SU(2) parts of the nth vector meson v n µ (x), respectively, and B(x) is the baryon field. Note that we do not include the direct interaction among baryons and the background gauge potential V We recall that the vector and axial-vector mesons couple with the background gauge potentials as in (2 . 34). Using these interactions, the Dirac and Pauli form factors are computed to be and a similar relation holds for the SU(2) sector. On the other hand, in the present model, (4 . 27) and (4 . 28) together with (4 . 7) yield the Dirac and Pauli form factors of the form g v n ψ 2n−1 (Z) (4 . 52) Here we have kept only the leading terms in the large N c and large λ limit. * ) By comparing these results, we obtain It is interesting to note the relation In particular, for the case with n = 1 and B being a nucleon, this means the g ωN N = N c g ρN N , which coincides with the constituent quark model prediction. This relation is found also in Ref. 17).
The axial form factors g A and g A can be obtained from the effective cubic couplings with pion and axial-vector mesons of the form L a int = n≥1 g a n BB a n µ Biγ 5 γ µ τ 0 2 B + g a n BB a na µ Biγ 5 γ µ τ a 2 B where a n µ (x) and a na µ (x) are the U(1) and SU(2) parts of the nth axial vector meson a n µ (x), respectively, and π(x) and π a (x) are the U(1) and SU(2) parts of the pion field Π(x), respectively. As in (4 . 50), no direct coupling between the baryons and V (−) µ is assumed. It follows from this and (2 . 34) that the invariant amplitude with the incoming and outgoing baryons in the presence of the external gauge field V (−) µ is given by * ) g a n g a n BB k 2 + λ 2n g a n g a n BB λ 2n 1 k 2 + λ 2n u( p , s) By comparing this with the matrix elements (4 . 11)-(4 . 13), we obtain g A , h A , g A , and h A as functions of the cubic coupling constants in (4 . 55): g a n g a n BB k 2 + λ 2n , g A (k 2 ) = n≥1 g a n g a n BB k 2 + λ 2n , (4 . 57) g a n g a n BB λ 2n g a n g a n BB λ 2n Equating these results with (4 . 46) then using (4 . 14) and (3 . 60) leads to the relations g a n BB = N c 32π 2 κ ∂ Z ψ 2n (Z) , g a n BB = 8π 2 κ 3 ρ 2 ∂ Z ψ 2n (Z) , (4 . 59) .

details) can be estimated as * )
n λ n ψ n (Z) nz=0 ψ n (Z) nz=1 ∂ Z ψ n (Z) nz=0 ∂ Z ψ n (Z) nz=1 As an example, consider the nucleon with n ρ = n z = 0. Using this table, we obtain n g v n N N g v n N N g a n N N g a n N N In particular, for the ρ meson we thus obtain This is consistent with the experimental data g ρN N | exp = 4.2 − 6.5 42) (see also Ref. 17)).
Note that the functional form of (4 . 53) shows the universality of the ρ meson couplings among the spin 1/2 baryons with any n ρ and n z = 0. Moreover, for large N c and large λ, To see if the ρ meson universality holds extensively in the meson sector, we list the meson cubic couplings g ρππ , g ρv n v n , and g ρa n a n computed in Ref. 6): g ρππ = 4.81 n g ρv n v n g ρa n a n It seems that the relation g ρππ ∼ g ρρρ ∼ g ρN N roughly holds within 20% error, although g ρv n v n with n > 1 and g ρa n a n are not close enough to ensure the universality.  In this paper, baryons have been described as a soliton, and the baryon physics has been analyzed by a standard semiclassical quantization of the soliton. This approach appears to be completely different from the treatment of Ref. 17), in which a key step is to introduce a five-dimensional spinor field into the five-dimensional YM-CS system to represent the baryons. In order to relate the two approaches, consider the vBB coupling, for simplicity, in the four-dimensional meson-baryon effective Lagrangian (4 . 50): If we substitute g v n BB from (4 . 53), this can be written as Here A µ is the U(1) part of the five-dimensional gauge potential in (2 . 35), and B(x, z) ≡ B(x)ψ Z (z) is a five-dimensional spinor field constructed by the four-dimensional baryon field B(x) and the suitably normalized wavefunction ψ Z (Z) corresponding to the baryon B. This term can be summarized as a five-dimensional gauge interaction by regarding ψ Z (Z) as an eigenmode of a wave equation that reproduces the baryon spectrum found in Ref. 16). Likewise, the interaction terms given in (4 . 50) and (4 . 55) are sufficient to reconstruct the five-dimensional spinor field action with the Pauli-type interaction in Ref. 17).
We have concentrated on a one-point function of mesons and the electromagnetic field in the presence of a single quantized soliton, mainly to extract the static properties of baryons.
There are other interesting aspects of the force associated with the baryons, for example, interactions between baryons, in particular. When baryons are far from each other, we can use our results and compute the nuclear force based on a one-meson-exchange picture of the nuclear force. Resolving the issue of the nuclear force at short distance is interesting and will be reported in our forthcoming paper. 45) moduli degrees of freedom of the soliton are X i (t) , Z(t) , ρ(t) , a I (t) .
(A . 1) X i and Z describe the center-of-mass motion of the soliton, while ρ and a I (I = 1, 2, 3, 4) describe the size of the soliton, which is instanton-like in the SU(2) sector, and the orientation of the instanton in the SU(2) group space, respectively, with (a I ) 2 = 1. The Hamiltonian for the moduli (2 . 17) provides the quantized energy eigenstates of the baryon, specified by the quantum number B = (l, I 3 , n ρ , n z ) and its spin s. The wavefunctions for the quantized states can be written explicitly for low-lying states. The nucleon wavefunctions, B = (1, ±1/2, 0, 0) and s = 1/2, are written as (2 . 22), The rest of the ∆ wavefunctions with s = 3/2 are obtained by letting the isospin lowering operator act on (A . 6): (a 1 + ia 2 ) 2 (a 4 + ia 3 ) , (for ∆ + ) The wavefunction R(ρ) for ∆ (states with l = 3 and n ρ = 0) is given as We use these wavefunctions to compute the static quantities of excited baryons.

A.2. Expectation values relevant to N(1440) and N(1535)
First, we summarize what quantities are necessary for computing various static quantities presented in §3: The first quantity F (Z) is necessary for computing r 2 I=0 , and thus r 2 E,M . The second, third, and fourth quantities, are for the magnetic moment µ, the axial coupling g A , and the axial radius r 2 1/2 A , respectively. First, we compute ρ 2 . The integral to be evaluated is The former equation is for the proton/neutron, (3 . 33). For N c = 3, these are numerically given by ρ 2 nρ=0 ≃ 1.62 × ρ 2 cl , (A . 13) ρ 2 nρ=1 ≃ 2.37 × ρ 2 cl . (A . 14) Next, we compute 1 k(Z) , F (Z) , and f A (Z) , that is, the quantities relevant to the Z directions. This is necessary for the N(1535) excitation. The classical values are given with In the large λ expansion, the wavefunction for Z is localized at the origin Z = 0. Inclusion of the subleading terms causes differences in baryon states.
Numerical evaluation using the wavefunction (A . 5) is performed as follows: where we have used κ = 0.00745.
For F (Z) and F A (Z), we first solve the differential equations satisfied by them, (3 . 7) and (3 . 72), and then use the solutions to numerically evaluate the normalized integrals. We First of all, the third component of the spin of these states is chosen to be s = +3/2. Therefore, using (2 . 47), we can see that the value of ρ 2 χ i=3 is three times that of the proton/neutron for which s = 1/2.
(A . 29) The denominators originate from the normalization of the states (A . 6) and (A . 7). It is also easy to see that