Parity doubling structure of nucleon at non-zero density in the holographic mean field theory

We develope the holographic mean field theory approach in a bottom-up holographic QCD model including baryons and scalar mesons in addition to vector mesons and pions. We study the effect of parity doubling structure of baryons at non-zero density to the equation of state between the chemical potential and the baryon number density. We first show that we can adjust the amount of nucleon mass coming from the chiral symmetry breaking by changing the boundary value of the five-dimensional baryon fields. Then, introducing the mean field for the baryon fields, we calculate the equation of state between the baryon number density and its corresponding chemical potential. Then, comparing the predicted equation of state with the one obtained in a Walecka type model, we extract the density dependence of the effective nucleon mass. The result shows that the effective mass decreases with increasing density, and that the rate of decreasing is more rapid for the larger percentage of the mass coming from the chiral symmetry breaking.


I. INTRODUCTION
The spontaneous chiral symmetry breaking (χSB) is one of the most important features in low-energy QCD. This is considered to be the origin of several hadron masses, such as the lightest nucleon mass. However, there is a possibility that only a part of the lightest nucleon mass is generated by the spontaneous χSB and the remaining part is the chiral invariant mass. This structure is nicely expressed in so called parity doublet models (see, e.g. Refs. [1][2][3][4]).
It is an interesting question to ask how much amount of the nucleon mass is generated by the spontaneous χSB, or to investigate the origin of nucleon mass. Studying dense baryonic matter would give some clues to understand the origin of our mass, since a partial restoration of chiral symmetry will occur at high density region. We expect that the mass generated by the spontaneous χSB will become small near the chiral phase transition point.
It is not so an easy task to study the dense baryonic matter from the first principle, namely starting from QCD itself: It may not be good to use the perturbative analysis, and the lattice QCD is not applicable due to the sign problem at this moment. Then, instead of the analysis from the first principle, it may be useful to make an analysis based on effective models, especially for qualitative understanding.
Holographic QCD (hQCD) models (see, for reviews, e.g. Refs. [5,6] and references therein.) are constructed based on the AdS/CFT correspondence [7][8][9] and powerful tools to study the low-energy hadron physics. There exist several ways to apply hQCD models for dense baryonic matter (see e.g. Refs. [10][11][12][13][14][15][16][17]). Recently the * e-mail: he@hken.phys.nagoya-u.ac.jp † e-mail: harada@hken.phys.nagoya-u.ac.jp holographic mean field theory approach was proposed to study dense baryonic matter in Ref. [18]. This approach allows us to predict the equation of state between the chemical potential and the baryon number density. In Ref. [18], this approach was applied to a top-down model of hQCD [19] including the baryon fields in the framework of the Sakai-Sugimoto (SS) model [20]. It is known [21] that the SS model provides the repulsive force mediated by iso-singlet mesons such as ω meson among nucleons, while the attractive force mediated by the scalar mesons are not generated. As a result Ref. [18] shows that the chemical potential increases monotonically with the baryon number density. On the other hand, when the attraction mediated by the scalar meson is appropriately included, the chemical potential is expect to decrease up until the normal nuclear matter density, and then turn to increase (see e.g. Ref. [21]). Thus, it is interesting to study whether the chemical potential decreases with increasing density when the scalar degree of freedom is included. In this paper, for studying this, we adopt a bottom-up model given in Ref. [22] which includes five-dimensional baryon field included in the model proposed in Refs. [23,24]. There the five dimensional scalar field X is explicitly included to express the chiral symmetry breaking by its vacuum expectation value (VEV).
Yet another interest appears in a hQCD model of Ref. [22]. Since there is no chirality in five dimension, the hQCD model includes two baryon fields; one transforms linearly under U(2) R and another under U(2) L . The existence of two baryon fields naturally generates the parity doublet structure mentioned above. In Ref. [22], the boundary condition is adopted in such a way that all of the nucleon mass is generated by the chiral symmetry breaking.
In the present analysis, we will show that we can adjust the amount of nucleon mass coming from the chiral symmetry breaking by changing the boundary value of the five-dimensional baryon fields: The percentages of the chiral invariant mass in the nucleon mass is controlled by changing the boundary value. We study how the equation of state in the dense baryonic matter depends on the percentage of the nucleon mass originated from the spontaneous χSB in the holographic mean field theory approach. Our result shows that, larger the percentage of the mass coming from the spontaneous χSB is, more rapidly the effective nucleon mass, which is extracted from the equation of state by comparing it with the one obtained in a Walecka type model given in Ref. [26], with increasing baryon number density. This paper is organized as follows: In section II, we first review the model proposed in Ref. [22], and then show the parity doubling structure. We study the equation of state at non-zero baryon density in the model in section III. We also discuss the interpretation of our results in terms of a Walecka-type model. Finally, we give a summary and discussions in section IV. We summarize several intricate formulas needed in this paper in appendix A.

A. model
In this subsection we briefly review the holographic QCD model including baryons given in Ref. [22].
The fields relevant to the present analysis are the scalar meson field X and two baryon fields N 1 and N 2 , as well as the 5-dimensional gauge fields R A and L A , which transform under the 5-dimensional chiral symmetry as where g R,L ∈ U(2) R,L denote the transformation matrix of chiral symmetry, and A = µ , z with µ = 0, 1, 2, 3. By using these fields, the bulk action is given as where M labels the general space-time coordinate and A labels the local Lorentz space-time, with A, M ∈ (0, 1, 2, 3, z). By fixing the gauge for the Lorentz trans-formation, we take the vielbein as The Dirac matrices Γ A are defined as Γ µ = γ µ and Γ z = −iγ 5 which satisfy the anti-commutation relation The covariant derivatives for baryon and scalar meson are defined as . ω AB M is the spin connection given by In this subsection, we study the parity doubling structure of baryons in the model described in the previous subsection. Note that the analysis in this subsection is done for zero chemical potential, so that only the scalar field X has a mean field part, or 4-dimensional vacuum expectation value (VEV), expressed by X 0 . The equation of motion (EoM) for X 0 is read from the action S X in Eq. (2.10) as The solution for this EoM is obtained as [23,24] where M is the current quark mass and σ is the quark condensate qq . By using this vacuum solution, the EoMs for N 1 and N 2 are given by As done in Ref. [22], we decompose the bulk fields N 1 and N 2 as The mode expansions of N 1L,R and N 2L,R are performed as L,R (p) .

(2.24)
It is convenient to introduce f (n) which satisfy , and ± corresponding to mass eigenvalues. It should be noticed that Eq. (2.27) is rewritten as  +2 should be zero required by the normalizability. The value of f +1 at the IR boundary can be set 1 without loss of generality since the coupled differential equations in Eq. (2.26) are homogeneous equations. In Ref. [22], the value of f +2 at the IR boundary was taken as 0 in such a way that all of the mass of ground state baryon is generated by the chiral symmetry breaking expressed by the VEV of X 0 .
In the present analysis, we regard the IR value of f +2 , i.e. f +2 | z=zm = c 1 , as a parameter, which turns out to control the percentages of the chiral invariant mass included in the nucleon mass. We summarize the boundary condition in Table I for a convenience. In the remaining part of this subsection, we shall show the dependence of the percentage of the chiral invariant mass of the nucleon on the IR boundary value c 1 , for fixed value of z m = 1/0.3236 (GeV) −1 [23]. For a given value of c 1 , we first adjust the coupling G to ensure that the lowest eigenvalue becomes the nucleon mass of 0.94 GeV. We show how the value of G changes depending on the value of c 1 in Fig. 1. It should be We give an intricate discussion of the parity assignment in appendix A.
We next calculate the masses of higher excited nucleons using the value of G determined above for fixed c 1 . We show the c 1 -dependence of several masses in Fig. 2. Here, N (+) denotes the states with positive parity while N (−) stands for negative parity. This figure shows that, for c 1 > c * 1 ≈ 0.12, the first excited state carries the negative parity and the second the positive parity, and so on. For c 1 < c * 1 , on the other hand, the first excited state is the positive-parity excited nucleon, which seems consistent with the experimental data.
Finally in this subsection, we investigate the effect of dynamical chiral symmetry breaking on the nucleon mass. For quantifying this effect, we take σ = 0 and for several choices of c 1 . We consider the lowest eigenvalue m 0 , denoted as just m 0 , as the chiral invariant mass of nucleon. In Fig. 3, we plot the c 1 dependence of the value of 1 − m 0 /m N ≡ m(qq) mN which shows the percentage of the nucleon mass coming from the spontaneous chiral symmetry breaking. From Fig. 3 we conclude that, in the case of c 1 = 0, which is chosen in Ref. [22], all the nucleon mass comes from the spontaneous chiral symme-try breaking. On the other hand, when c 1 > 0.25, more than half of the nucleon mass is the chiral invariant mass.

III. EQUATION OF STATE IN THE HOLOGRAPHIC MEAN FIELD APPROACH TO THE MODEL
In this section, we study the finite density system using the holographic mean field theory proposed in Ref. [18]. In the holographic mean field theory, all the 5D fields are decomposed into the mean fields which depend only on the 5th coordinate z and the fluctuation fields. In the present analysis, we consider the symmetric nuclear matter, so that the proton and the neutron have the same mean fields. Furthermore, we assume that the mean fields for the vector and axial-vector gauge fields except the U(1) V gauge field and the traceless part of the scalar field are zero. Then, in the mean field analysis, we make the following replacements: where I is the 2 × 2 unit matrix in the flavor space, and the proton and the neutron have the same mean fields consistently with the symmetric nuclear matter. Note that both N 1 (z) and N 2 (z) are four-component spinors in the above expression.
The equations of motion for the mean fields, X(z), V M (z), N 1 (z), N 2 (z) are given by The equations of motions corresponding to Eq. (3.8) The derivative at IR boundary is taken to be zero: In the present analysis we do not include the effect of current quark mass, so that the value of X at UV boundary is taken to be zero: X(z = 0) = 0. There is an ambiguity for the IR value of X. In this analysis, following Ref. [13], we fix it to be the value determined at vacuum: X(z = z m ) = σ 0 z 3 m /2 with σ 0 = (318 MeV) 3 . This is based on the assumption that the IR values are not affected much by the chemical potential introduced at the UV boundary. For the baryon fields, we take the UV values of N + and N − to be zero following the holographic mean field theory [18]. The equations of motion for mean fields are not homogeneous equations, so that the normalization of the baryon fields become relevant. We change the IR values of N + and N − to control the baryon number density, which is written in terms of the baryon fields as 1 It should be noted that the ratio of two baryon fields at IR boundary is left free as in the previous section. So, we use N + (z = z m ) = c 2 and N − (z = z m ) = c 2 × c 1 , where c 2 determines the baryon number density while c 1 controls the percentage of the chiral invariant mass of nucleon. We summarize the boundary conditions in Table II. We solve the equations of motion in Eq. We first study the density dependence of the chiral condensate for checking the partial chiral restoration. Here we define the in-medium condensate through the holographic mean field X(z) as (3.10) We plot the density dependence of the σ normalized by the vacuum value σ 0 in Fig. 4. This shows that the quark condensate σ decreases with the increasing number density, which can be regarded as a sign of the partial chiral symmetry restoration. When the value of c 1 is decreased, the corresponding value of G becomes larger (see Fig. 1) to reproduce the nucleon mass. Since the larger G implies the larger correction to the scalar from the nucleon matter, the smaller c 1 we choose, the more rapidly the condensate σ decreases. The degreasing property of the chiral condensate is similar to the one obtained in Ref. [13].
We next show the resultant equation of state, a relation between the chemical potential and the baryon number density in Fig. 5. This figure shows that the chemical potential increases with the increasing baryon number density. This does not agree with the nature, in which the chemical potential decreases against the density in the low density region below the normal nuclear matter density. This decreasing property is achieved by the subtle cancellation between the repulsive and attractive forces. So this increasing property indicates that, in the present model, the repulsive force mediated by the U(1) gauge field is stronger than the attractive force mediated by the scalar degree included in X field. For studying the attractive force mediated by the scalar fields, we extract the density dependence of the effective nucleon mass using the Walecka type model (see e.g. Refs. [25,26]), in which the chemical potential µ is expressed as where ρ b is the baryon number density, g ω (n) N N is the coupling for nth eigenstate of the omega mesons, m ω (n) is its mass, k F is the Fermi momentum, and M * is the effective nucleon mass. Note that, in the free Fermi gas, k F is related to . In the present hQCD model, the ω (n) N N coupling is calculated in vacuum as g ω (n) N N = 15.5 ∼ 15.8, 8.9 ∼ 10.9 . . . depending on the value of c 1 . Using these couplings together with the masses of m ω (n) ∼ 780, 1794 . . . MeV, we convert the density dependence of µ obtained above into the one of the effective nucleon mass M * through Eq. (3.11). We plot the density dependence of the effective mass M * in Fig. 6. This shows that the effective mass decreases with increasing density. The decreasing rate is larger than the one obtained in Ref. [13], which is the reflection of the iterative corrections included through the holographic mean field theory. It should be noted that the decreasing of M * is more rapid for smaller value of c 1 . In other word, the larger the percentage of the mass coming from the chiral symmetry breaking is, more rapidly the effective mass M * decreases with density.
In Fig. 7, we plot the baryon charge distribution ρ(z) defined in Eq. is broader for larger value of ρ. This indicates that the distribution becomes more important for larger density, as shown in Ref. [18].

IV. A SUMMARY AND DISCUSSIONS
We develope the holographic mean field approach in a bottom-up holographic QCD model proposed in Ref. [22] which includes five-dimensional baryon field in the model proposed in Refs. [23,24]. We first study the mass spectrum of baryons with paying attention to the chiral invariant mass m 0 , which were formulated in parity doublet models (see, e.g. Refs. [1][2][3][4]). We found the parameter (c 1 ), which is one boundary value of two baryon fields, controls the percentage of the chiral invariant mass: for c 1 = 0 all of the mass of the ground-state nucleon is generated by the spontaneous chiral symmetry breaking, while for c 1 > 0.25, more than half of the nucleon mass is actually the chiral invariant mass.
We studied the density dependence of the chiral condensate. Our result shows that the quark condensate σ decreases with the increasing number density, which is consistent with the analysis done in Ref. [13]. Furthermore, we found that the σ decreases more rapidly for smaller value of c 1 . This is because the sigma coupling to the nucleon is larger for smaller c 1 .
We next calculated the equation of state between the baryon chemical potential and the baryon number density using the holographic mean field approach proposed in Ref. [18]. The resultant equation of state shows that the chemical potential increases with the increasing baryon number density. This indicates that, in the present model, the repulsive force mediated by the U(1) gauge field is stronger than the attractive force mediated by the scalar degree included in X field. For studying the attractive force mediated by the scalar fields, we extract the density dependence of the effective nucleon mass using a Walecka type model. Our result shows that the effective mass decreases with increasing density. Furthermore, the decreasing rate is more rapid for smaller value of c 1 . This is consistent with the fact that the percentage of the chiral invariant mass is larger for larger value of c 1 . In other word, the larger the percentage of the mass coming from the spontaneous χSB is, more rapidly the effective nucleon mass decreases with increasing baryon number density.
We also studied the baryon number distribution in the holographic direction. Our results show that the distribution is concentrated near the IR boundary for smaller ρ. This indicates that the distribution becomes more important for larger density.
In the present analysis, we made an analysis only at the mean field level. So a natural extension is to consider the fluctuations on the top of the mean field obtained here. It is also interesting to study the relation between the isospin chemical potential and the isospin density based on the approach developed in this paper, since the relation has a relevance to the symmetry energy. We leave these works to the future project.
In this appendix, we consider the parity transformation properties of the 5D fields. As in the 4-dimension, a parity transformation should flip the sign of normal three spatial coordinates. But the 5th coordinate z, as it is defined in the range z UV < z < z IR , does not participate in the parity transformation.
(A1) Using these conventions, we obtain the parity transformation of 5D fields as For the 5D spinors, their parity transformation properties are express as where η 1 and η 2 are arbitrary phases. For explicitly illustrating the parity invariance we rewrite the 5D Lagrangian in Eqs. (2.7)-(2.9) in terms of chiral basis as Rµ t a N 2L +N 2R izΓ µ ∂ µ + zΓ µ A a Rµ t a N 2R +N 2L (izΓ z ∂ z + zΓ z A a Rz t a − 2iΓ z + M 5 ) N 2R +N 2R (izΓ z ∂ z + zΓ z A a Rz t a − 2iΓ z + M 5 ) N 2L , (A5) Under parity transformation, they transform as It is easy to see that these actions are parity invariant if η 1 and η 2 satisfy