Numerical analysis of a baryon and its dilatation modes in holographic QCD

We investigate a baryon and its dilatation modes in holographic QCD based on the Sakai-Sugimoto model, which is expressed as a 1+4 dimensional U($N_f$) gauge theory in the flavor space. For spatially rotational symmetric systems, we apply a generalized version of the Witten Ansatz, and reduce 1+4 dimensional holographic QCD into a 1+2 dimensional Abelian Higgs theory in a curved space. In the reduced theory, the holographic baryon is described as a two-dimensional topological object of an Abrikosov vortex. We numerically calculate the baryon solution of holographic QCD using a fine and large lattice with spacing of 0.04 fm and size of 10 fm. Using the relation between the baryon size and the zero-point location of the Higgs field in the description with the Witten Ansatz, we investigate a various-size baryon through this vortex description. As time-dependent size-oscillation modes (dilatation modes) of a baryon, we numerically obtain the lowest excitation energy of 577 MeV and deduce the dilatational excitation of a nucleon to be the Roper resonance N$^*$(1440).


I. INTRODUCTION
Quantum chromodynamics (QCD) is established as the fundamental theory of strong interaction and characterized by SU(N c ) gauge symmetry and global SU(N f ) L × SU(N f ) R chiral symmetry.Owing to asymptotic freedom of QCD, high-energy hadron reactions can be analyzed using perturbative QCD.In low energy regions, however, the QCD coupling becomes strong, and the perturbative method is no more applicable.Therefore, for theoretical analyses of hadrons based on QCD, some nonperturbative methods are necessary such as lattice QCD.Based on gauge/gravity duality [1] for D branes [2] in superstring theory, holographic QCD is an interesting new tool to analyze the nonperturbative properties of QCD [3][4][5].
Around 1970, the string theory was originally proposed by Nambu, Goto, and Polyakov for the description of hadrons [6][7][8].After establishment of QCD, the string theory was not used for the main research of hadrons.Instead, this framework was reformulated as the superstring theory in 1980s [9] and has been studied as a plausible candidate of a grand unified theory including quantum gravity, and many studies have been constantly conducted to date.
The superstring theory is formulated in 10 dimensional space-time and includes D branes, on which open string endpoints exist [2].As a remarkable discovery by Polchinski, on the surface of N overlapped D branes, U(N ) gauge symmetry emerges, and the N D-branes system leads to U(N ) gauge theory [2].In fact, on the N D-branes, gauge fields A ab appear from the open string linking two D-branes labeled with a and b (a, b = 1, 2, ..., N ).On the other hand, around the D brane, a higher-dimensional supergravity theory is formed since the brane has mass, and multiple branes can be gravi-tational sources.Thus, there are two different theories relating to the D-brane, and the gauge theory on the Dbrane and the higher-dimensional gravity theory around the D-brane are conjectured to be equivalent [1], which is called AdS/CFT correspondence or gauge/gravity duality.This remarkable equivalence between gauge and gravity theories was first proposed by Maldacena from a detailed analysis of four-dimensional N = 4 supersymmetric (SUSY) SU(N ) Yang-Mills theory and fivedimensional Anti-de Sitter (AdS) supergravity theory [1], using large N argument.This equivalence was first called as AdS/CFT correspondence and is generalized as gauge/gravity duality in more general concept.In this correspondence, a strong-coupling gauge theory can be described with a weak-coupling gravity theory [1,[3][4][5].
With this correspondence, many researches have been performed, and one of the main category is to analyze strong-coupling QCD with a higher-dimensional classical gravity theory, which is called holographic QCD [3-5, 10, 11].In 1998, Witten formulated a non-SUSY version of holographic QCD for a four-dimensional puregluon Yang-Mills theory using S 1 -compactified N c D4 branes, where periodic and antiperiodic boundary conditions are imposed on boson and fermion fields, respectively [3].In 2005, Sakai and Sugimoto proposed holographic QCD for four-dimensional full QCD, including massless chiral quarks, by adding the N f pair of D8 and D8 branes to S 1 -compactified D4 branes [5].In fact, the D4/D8/D8 multi-D-brane system has SU(N c ) gauge symmetry and SU(N f ) L ×SU(N f ) R chiral symmetry and leads to massless QCD in infrared scale.Here, quarks and gluons appear from the massless modes of 4-8 and 4-4 strings, respectively.In the large N c limit, N c D4 branes are dominant as a gravity source and are converted into a gravitational field via the AdS/CFT correspondence or the gauge/gravity duality.The 't Hooft coupling λ ≡ N c g 2 is a control parameter of the gauge side, and strong-coupling QCD with a large λ corresponds to a weakly interacting gravitational theory.Then, this system is described by the D8 brane in the presence of a background gravitational field originating from the D4 brane.Nonperturbative properties of QCD can be analyzed with a classical gravity theory In the leading order of 1/N c expansion, the effective theory of the D8 brane in the D4-brane-induced background gravity is expressed with the Dirac-Born-Infeld (DBI) action and the Chern-Simons (CS) term.The leading order of 1/λ expansion is the DBI action, and the next leading of 1/λ is the CS term.Expanding the DBI action with 1/λ, the leading order becomes a fivedimensional Yang-Mills theory in flavor space in a curved space, where a background gravity appears only in the fifth dimension.The CS term is a topological term responsible for anomalies in QCD.In holographic QCD, the five-dimensional holographic field is decomposed into four-dimensional meson fields, and this theory is successful in the mesons sector, that is, it describes low-lying meson masses, inter-meson couplings, and phenomenological laws in hadron physics [5].Regarding baryonic degrees of freedom, baryons do not appear explicitly in the holographic action.In fact, the baryon does not appear as a system component in holographic QCD.
The absence of baryons in the effective action is a general result due to the large N c argument, because QCD is reduced into a weak-coupling theory of mesons and glueballs in the large N c limit [12], where the baryon mass increases as O(N c ) and the baryon do not appear in the effective action.In the large N c argument, the baryon is considered to appear as a soliton of mesons such as the Skyrmion in the Skyrme model [12][13][14].
Here, we briefly mention a historical overview of the Skyrme model.The Skyrme model is a low-energy effective theory in hadron physics, first proposed by Skyrme in 1961 [13].After the quark model was proposed in 1964 [15], main researches of strong interaction and hadrons were shifted to the quark theory, which was eventually developed into QCD.In 1979, Witten revived the Skyrme soliton picture of baryons from a large N c viewpoint of QCD [12], although the direct relation between the Skyrme model and QCD has not been clear.In 2005, Sakai and Sugimoto showed a theoretical explicit connection between massless QCD and the Skyrme Lagrangian, which is derived as the pion sector in holographic QCD, using the gauge/gravity duality for a D-brane system [5].(Actually, the Sakai-Sugimoto model reduces into the Skyrme model, when massive mesons are dropped off, leaving only light pions.) Now, let us concentrate on baryons in holographic QCD.Also for holographic QCD, which is derived with large N c and is described with only meson fields, the baryon is described as a chiral soliton of mesons, that is, a topological object like a brane-induces Skyrmion [16,17] or an instanton-like object [18,19].In fact, the baryon appears as an spatially extended object in holographic QCD.
To summarizes the above, holographic QCD is an analytical nonperturbative method for QCD and has direct connection with QCD, as a clearly strong advanced point.In holographic QCD, while mesons appear in the action and can be directly treated, the baryon is described as an extended soliton of mesons.Therefore, compared with the meson sector, the baryon sector is more difficult and has not been well studied in holographic QCD, in spite of several pioneering studies on the holographic baryon [16,[19][20][21][22][23][24].In addition, this baryonic soliton allows spatial dilatation modes as its excitation peculiar to the spatially-extended object, and thus we focus on this dilatation mode of the holographic baryon in the latter of our study.
In this study, we investigate a single baryon and its dilatation modes in holographic QCD, adopting the Sakai-Sugimoto model formulated as a 1+4 dimensional U(N f ) gauge theory in the flavor space.For spatially rotational symmetric systems, we apply a generalized Witten Ansatz and reduce 1+4 dimensional holographic QCD into a 1+2 dimensional Abelian Higgs theory, where the holographic baryon is expressed as an Abrikosov vortex.We numerically calculate the baryon solution of holographic QCD using a fine and large lattice, keeping background gravity from the N c D4-brane.
In addition, we investigate a various-size baryon and the dilatation mode (time-dependent size-oscillation) of a single baryon, using the relation between the baryon size and the zero-point location of the Higgs field in the reduced Abelian Higgs theory.
This size oscillation is physically considered as a collective motion and is difficult to be described in the quark model.Instead, the size oscillation mode has been studied in the Skyrmion research, and its lowest excitation mode is identified as the Roper resonance N * (1440) [25][26][27][28][29][30][31].
The Roper resonance is the first excited state of the nucleon N(940) with the positive parity and its energy being 1440 MeV.In the quark model, based on the single-particle picture, the first excited-state baryon is to have negative parity, and it contradicts the experimental data.In lattice QCD, the numerical results with overlap fermion well reproduce the Roper resonance in terms of the excitation energy and the positive parity [32,33].Here, lattice QCD is a powerful tool for the quantitative analysis of hadrons, but it is difficult to get state information of hadrons like the wave function due to path integral formalism, where all the states are integrated out.As an alternative method, the Skyrme model seems to succeed to reproduce the mass and parity of N * (1440) as the first excited state of N(940).In this chiral soliton picture, this first excited state is described as a dilatation or breathing mode of the ground-state soliton [26][27][28][29][30][31].
Therefore, we here investigate the dilatation mode of the baryon in holographic QCD and finally compare it with the Roper resonance N * (1440) in terms of its mass and parity.The dilatation mode of baryon can be described also in holographic QCD.Note again that the Sakai-Sugimoto model reduces into the Skyrme model, when massive mesons except for pions are dropped off.In fact, the holographic baryon appears as a soliton [16,19], i.e., a spatially extended object, and therefore has dilatation mode.Note, however, that holographic dilatation is four-dimensional spatial oscillation including the extra spatial dimension rather than ordinary three-dimensional one.
The organization of this paper is as follows.In Sec.II, we briefly review the Sakai-Sugimoto model as typical holographic QCD.In Sec.III, we apply the Witten Ansatz in holographic QCD.Owing to the Witten Ansatz, the 1+4 dimensional Yang-Mills theory reduces into a 1+2 dimensional Abelian Higgs theory.In Sec.IV, using the Witten Ansatz, we present the vortex description of baryons in holographic QCD, and we numerically obtain the ground-state solution of the holographic baryon using a fine and large-volume lattice.In Sec.V, we numerically analyze size dependence of the holographic baryon.In Sec.VI, we investigate timedependent dilatational modes of a single baryon in holographic QCD.Section VII is devoted for summary and conclusion.

II. HOLOGRAPHIC QCD ACTION IN THE SAKAI-SUGIMOTO MODEL
In this section, as a starting point, we introduce the Sakai-Sugimoto model, one of the most successful holographic QCD [5].In the Sakai-Sugimoto model, four-dimensional massless QCD is constructed using the D4/D8/D8 multi-D-brane system, which comprises spatially S 1 -compactified N c D4 branes attached with N f D8-D8 pairs.Here, N c means the color number, and N f the light flavor number.This compactification breaks SUSY due to the (anti)periodic conditions for bosons(fermions), as was demonstrated for a D4 brane system by Witten [3].The compactification radius is M −1 KK , and this model parameter physically corresponds to a UV cutoff in holographic QCD.This system is infrared equivalent to massless QCD, where chiral symmetry exists [5].Using AdS/CFT correspondence (gauge/gravity duality), the N c D4 branes are transformed into a gravitational source, and the system becomes N f D8 branes in the D4 gravity background, which leads to the DBI and CS action at the leading order of 1/N c within the probe approximation.In terms of 1/λ expansion, the DBI action includes its leading order, and the CS action is sub-leading.
From the multi-D-brane system which is infrared equivalent to massless QCD, the DBI action becomes 1+4 dimensional Yang-Mills theory on the flavor space of U(N f ) ≃ SU(N f )× U(1) at the leading order of 1/λ expansion [5]: In this paper, we use w for the extra fifth-coordinate in holographic QCD.Â denotes U(1) gauge field and F U(1) field strength [34].
For M, N = t, x, y, z, w, the field strengths are given by with the five-dimensional SU(N f ) gauge field A M (x µ , w) and U(1) gauge field ÂM (x µ , w), respectively.Throughout this paper, we take the M KK = 1 unit together with the natural unit, and κ is written as κ = λNc 216π 3 in this unit.Note that, as a relic of N c D4-branes, there appear background gravity k(w) and h(w) depending on the extra fifth-coordinate w, in the M KK = 1 unit.In Eq. ( 1) at the leading order of 1/λ expansion, SU(N f ) variables A and U(1) variables Â are completely separated, and hence we have divided and the U(1) sector S U(1) 5YM .The 1/N c -leading holographic QCD also has the CS term [5,19] as the next leading order of 1/λ.The CS term is a topological term responsible for anomalies in QCD, and its explicit form is where Â denotes the U(N f ) gauge field [5].Note that SU(N f ) variables A and U(1) variables Â are dynamically mixed in the CS term S CS in Eq. (4).
In this paper, to analyze baryons in holographic QCD, we consider both Yang-Mills and CS parts for the case of N f = 2.

III. WITTEN ANSATZ IN HOLOGRAPHIC QCD
Holographic QCD is formulated to be a 1+4 dimensional U(N f ) non-Abelian gauge theory with a gravitational background h(w) and k(w), which would be fairly difficult to analyze.To avoid the difficulty and to proceed analytic calculations, most previous works [19][20][21] were forced to take a flat background h(w) = k(w) = 1 and to use the simple 't Hooft instanton solution [35,36] in the flat space, although h(w) and k(w) are the trace of D4-branes and are to be relevant ingredients.
To deal with holographic QCD for N f = 2 without reduction the gravitational backgrounds h(w) and k(w), we adopt the Witten Ansatz [37] in this paper.The Witten Ansatz is generally applicable for spatially-rotational symmetric system in the SU(2) Yang-Mills theory.Applying this to holographic QCD, the 1+4 dimensional non-Abelian theory transforms to a 1+2 dimensional Abelian Higgs theory.Accordingly, relevant topological objects are changed from instantons to vortices, as will be shown in Sec.IV.In this section, we generalize the Witten Anstaz to be applicable to holographic QCD.

A. Witten Ansatz for Euclidean Yang-Mills theory
In this subsection, we briefly review the original Witten Ansatz [37] applied for the Euclidean fourdimensional SU(2) Yang-Mills theory, which is formulated on three spatial coordinates (x, y, z) and Euclidean time t.For spatially-rotational symmetric systems, the Witten Ansatz can be applied as with r ≡ (x i x i ) 1/2 , xi ≡ x i /r and δij ≡ δ ij − xi xj .Using the Witten Ansatz, the four-dimensional SU(2) Yang-Mills theory is reduced into a two-dimensional Abelian Higgs theory as where the complex Higgs field ϕ(t, r), Abelian gauge field a µ (t, r), its covariant derivative D µ , and field strength f µν in the Abelian Higgs theory are Here, we have used (0, 1) = (t, r) for the index of two dimensional coordinates.
B. Generalized Witten Ansatz for SU (2) f sector in holographic QCD In this subsection, we generalize the Witten Ansatz to be applicable for holographic QCD.
The SU (2) f sector in holographic QCD is expressed as a 1+4 dimensional Yang-Mills theory with gravitational backgrounds h(w) and k(w).Holographic QCD already includes four-dimensional Euclidean spatial coordinates (x, y, z, w) including the extra fifth-coordinate w, and instantons can be naturally introduced in holographic QCD without necessity of the Euclidean process or the Wick rotation.
Describing the SU(2) f gauge field A with the Pauli matrix τ a as A = A a τ a 2 ∈ su(2) f in holographic QCD, we take a generalized version of the Witten Ansatz for (x, y, z)-spatially rotational symmetric systems [23,24,34], A a w (t, x, y, z, w) = a w (t, r, w)x a , with r ≡ (x i x i ) 1/2 , xi ≡ x i /r and δij ≡ δ ij − xi xj .For 1+4 dimensional holographic QCD, we have extended the Witten Ansatz for A a 0 component, considering the (x, y, z)-rotational symmetry.Note that this Ansatz is a general form when (x, y, z)-rotational symmetry is imposed on gauge fields.
In the Witten Ansatz, the holographic field strength, F ij , F 0i , F wi and F 0w are expressed as With the Witten Ansatz, 1+4 dimensional SU(2) f Yang-Mills sector of holographic QCD is reduced into a 1+2 dimensional Abelian Higgs theory on a curved space.In fact, S where the complex Higgs field ϕ(t, r, w) ∈ C, Abelian gauge field a µ (t, r, w), its covariant derivative D µ , and field strength f µν in the Abelian Higgs theory are Here, we have used (0, 1, 2) = (t, r, w) for the index of 1+2 dimensional coordinates.Note that the factor k(w) appears in D 2 and F 12 associated with the index 2 (w), and otherwise the factor h(w) appears.From this action, the static energy of the Yang-Mills part is obtained as with ⃗ a = (a 1 , a 2 ) = (a r , a w ).This is similar with the 1+2 dimensional Ginzburg-Landau theory.As the different point, however, there appears a nontrivial metric of r 2 [37] in addition to holographic background gravity of h(w) and k(w) in the (r, w) half plane.To include all the gravitational effects exactly, we construct the lattice formalism, as shown in Appendix A, and perform the numerical calculation for holographic QCD.Finally in the subsection, we investigate the topological density ρ B in the Witten Ansatz.For the SU(2) f gauge configuration in the Witten Ansatz, the topological density ρ B in (x, y, z, w)-space is found to be [34] where θ ≡ arg ϕ and Roman small letters i, j take (1, 2) = (r, w).The topological density ρ B is expressed as a total derivative.Since we consider (x, y, z)-rotationally symmetric system, ρ B is independent of the spatial direction (x, ŷ, ẑ).In fact, ρ B takes an SO(3) rotationally symmetric form of ρ B (r, w) in the second line of Eq. ( 19).

C. U(1) sector in holographic QCD
In this subsection, we consider the U(1) sector in holographic QCD.Hereafter, the capital-letter index denotes the Euclidean spatial index as M = x, y, z, w.Also for the U(1) gauge field Â, we respect the spatial SO(3) rotational symmetry [23,24] as in the Witten Ansatz, and impose Âi (t, x, y, z, w) = âr (t, r, w)x i , while Â0 and Âw are treated to be arbitrary.In this case, one finds Fij = 0 and can take the âr = 0 gauge, which simplifies Âi = 0.Then, the 1/λ-leading term S U 5YM for the U(1) gauge field is written as [34] using the SO(3)-symmetric non-negative hermite kernel In this calculation, we take Âi = 0 using the gauge and rotational symmetry.We consider the CS term S CS as the next leading order of the 1/λ expansion.For the static SO(3)-rotationally symmetric configuration in the A 0 = 0 gauge, the CS term S CS in Eq. ( 4) is transformed as [19,23,24,34] up to total derivative.This is Coulomb-type interaction between the U(1) gauge potential Â0 and the topological density ρ B ≡ 1 16π 2 tr(F M N FMN ).Then, the total U(1) action depending on the U(1) gauge field Â is written as which leads to the field equations, For the static configuration, the additional energy E U (1)  from U(1) sector S U( 1) is simply given by For the static case, Âw is dynamically isolated in the field equation (25) and the last term is non-negative in the energy (26), and therefore we set Âw = 0, which satisfies the local energy-minimum condition.Then, one finds For the SO(3) rotationally symmetric solution, we eventually obtain the static energy of the U(1) part: using ρB (r, w) ≡ r 2 ρ B (r, w) and the hermite kernel K in (r, w)-space, For the numerical calculation of the U(1) sector, we mainly use this energy functional.(For another expression of E U(1) after Â0 path-integration, see Appendix B.) Note again that, at the leading order of 1/λ, the U(1) sector ( Â) completely decouples with the SU(2) f sector (⃗ a, ϕ) because the leading term is only the Yang-Mills action (1).However, at the next leading order of 1/λ, the U(1) term affects the SU(2) f part through the CS term as Eq.(23).
To summarize, the total energy E comprises two parts, and the total energy density E(r, w) is written as and we have to deal with coupled field equations of the SU(2) f and U(1) sectors and will perform the numerical calculation in a consistent manner.

IV. VORTEX DESCRIPTION OF BARYONS
In this section, we introduce vortex description of baryons in holographic QCD with applying the generalized Witten Ansatz.For a single baryon which is (x, y, z)-spatially rotational symmetric, applying the Witten Ansatz, we reduce the theory into a 1+2 dimensional Abelian Higgs theory in a curved space.In the reduced theory, the holographic baryon is expressed as a two-dimensional topological object of an Abrikosov vortex.We perform the numerical calculation of a B = 1 solution of holographic QCD as the single ground-state baryon, using a fine and large lattice with spacing of 0.04 fm and size of 10 fm.

A. Holographic baryons in Witten Ansatz
Large N c analyses of QCD indicate that explicit degree of freedoms are only mesons and glueballs, and baryons appear as solitons (topological objects) constructed with meson fields [12].Also, holographic QCD based on large N c becomes an effective theory of mesons, and baryons appear as chiral solitons composed of meson fields in this framework [5,16].Holographic QCD has four dimensional space (x, y, z, w), and instantons naturally appear as relevant topological objects in the four-dimensional space.The topological objects are physically identified as baryons in holographic QCD [19] and called holographic baryons.
Remarkably, the Witten Ansatz generally converts the topological description from a four-dimensional instanton into a two-dimensional vortex [37].Accordingly, the vortex number is interpreted as the baryon number in holographic QCD with the Witten Ansatz [34].
In fact, the baryon number B or the Pontryagin index is written by a contour integral in the (r, w)-plane where r≥0 denotes the contour integral around the whole half-plane of (r, w) with r ≥ 0. To keep the energy (18) finite, we have imposed the following boundary conditions: at the edge of the (r, w) half-plane.Thus, the baryon number B is converted into the vortex number in this formalism [34].
B. Abrikosov vortex solution for a baryon in holographic QCD In this subsection, we numerically calculate a groundstate baryon of holographic QCD through the Abrikosov vortex description in the 1+2 dimensional U(1) Abelian Higgs theory [34].
With imposing the global condition of B = 1, we numerically minimize the total energy E[ϕ, ⃗ a, Â0 ] in Eq. (30), which is equivalent to solving the equation of motion (EOM) of holographic QCD for the single groundstate baryon.
For the numerical calculation on the (r, w) plane, we adopt a fine and large-size lattice with spacing of 0.2 M −1 KK ≃ 0.04 fm and the extension of 0 ≤ r ≤ 250 and −125 ≤ w ≤ 125, that is, the system size is 250×250 grid corresponding to (50 M −1 KK ) 2 ≃ (10 fm) 2 in the physical unit.(In this numerical calculation, there appears subtle cancellation, and use of a coarse lattice might lead to an inaccurate result.Also, the lattice size should be increased until the volume dependence of physical quantities disappears.) On this lattice, starting from the 't Hooft solution [35,36] as a B = 1 topological configuration, we numerically perform minimization of the total energy E[ϕ, ⃗ a, Â0 ] keeping the topological charge by an iterative improvement, that is, many-time iterative local replacements of the field variable of ϕ, ⃗ a and Â0 .(See Appendix A for the detail.)During the update, the Higgs field ϕ(x) always has a zero point to ensure B = 1, and the Higgs zeropoint generally moves so as to realize the ground state.In this way, we eventually obtain the holographic fields ϕ(r, w), ⃗ a(r, w) and Â0 (r, w) for the ground-state baryon as the true solution of EOM in holographic QCD.
For the confirmation of numerical calculations, we also consider another different method, as shown in Appendix B. In this alternative method, we integrate out the U(1) gauge field Â0 and update only ϕ(r, w) and ⃗ a(r, w) on the lattice based on Eq. (B2).We have confirmed that both methods give the same numerical results for holographic baryons.
To visualize gauge and Higgs fields composing the Abrikosov vortex, we take the Landau gauge for the U(1) gauge degrees of freedom, ∂ i a i (r, w) = 0. Of course, main results including the total energy are gauge invariant and never depend on any gauge choices.
For the single ground-state baryon, we show field configurations ϕ(r, w) and ⃗ a(r, w) in Fig. 1, and also the U(1) gauge field Â0 (r, w) in Fig. 2. The Higgs field ϕ(r, w) indicates a clear topological structure characterizing the Abrikosov vortex, which is mapped into an instanton [37] via the Witten Ansatz and physically means a baryon in holographic QCD [23,34].In accordance with the field equation ( 25), Â0 is localized around the non-vanishing topological density ρ B , which will be shown in Fig. 3.  w for the ground-state baryon solution in holographic QCD.For the ground-state baryon, both profiles of ϕ and ⃗ a are found to be a little shrink in the w direction, compared with four-dimensional spherical 't Hooft solutions, which was also found in the previous numerical studies [23,24,34].

C. Properties of the ground-state baryon in holographic QCD
Now, we show the properties of the ground-state baryon in holographic QCD, which is numerically calculated by minimizing the total energy E[ϕ, ⃗ a, Â0 ] in Eq. (30).In Fig. 3, we show the topological density ρ B (r, w) and total energy density E(r, w) in the (r, w)plane for the Abrikosov vortex solution in the 1+2 dimensional Abelian Higgs theory.Both densities have a peak around (r, w) = (0, 0) and are extended in both the r and w directions.We show in Fig. 4 the densities multiplied by the integral measure factor r 2 , i.e., 4πr 2 ρ B (r, w) and 4πr 2 E(r, w), for the ground-state baryon, since ρB (r, w) ≡ r 2 ρ B (r, w) is a primary variable in this numerical calculation, as shown in Eq. ( 29).The non-zero size of the baryon is due to the repulsive force from the CS term [19].
In general, the integrated topological density is the baryon number B, and the baryon mass M B is given by integration of the total energy density E(r, w):

= 4π
By integration over the extra coordinate w, we obtain the ordinary densities in a three-dimensional space, The mass M B and size for the ground-state baryon are estimated as Here, some cautions are commented.When the selfdual BPS-saturated 't Hooft solution [35,36] is simply used, as was done in Ref. [19], the holographic baryon has an overestimated mass M BPS B ≃ 1.35 M KK ≃ 1.28 GeV and a smaller radius ⟨r 2 ⟩ BPS ρ B ≃ 2.2 M −1 KK ≃ 0.46 fm, as shown in Appendix C. (Because of such an overestimation, a small value of M KK = 500 MeV was adopted to adjust baryon masses in Ref. [19], which significantly differs from M KK ≃ 1 GeV for the meson sector.)Another caution is the numerical accuracy, and fine and large lattices are to be used for the numerical calculation.Owing to a relatively coarse and small-size lattice, the numerical results in the previous paper [34] include about 20% error for the baryon mass and size.Now, we investigate spatial distribution of the baryonnumber and energy densities for the ground-state baryon in holographic QCD. Figure 5 shows the baryon-number density ρ B (r) (i.e.topological density) and total energy density E(r), and their r 2 -multiplied values, 4πr 2 ρ B (r) and 4πr 2 E(r).One finds significant difference between the shapes of ρ B (r) and E(r) for the small r region.
Next, we investigate the energy contribution from the SU(2) f and U(1) parts, respectively.For the mass (total static energy) 1) of the ground-state holographic baryon, we obtain and hence the SU(2) f contribution (leading order of 1/λ expansion) is found to be quantitatively dominant.
As spatial distribution, the SU(2) f and U(1) energy densities are expressed as Figure 6 shows SU(2) f energy density E SU(2) f (r) and U(1) energy density E U(1) (r) of the ground-state baryon, together with the total energy density E(r) and baryon density ρ B (r).The dominant contribution is the SU(2) f part, and the total value E(r) is approximated by the SU(2) f energy density E SU(2) f (r), which seems to be consistent with 1/λ expansion.In particular, E SU(2) f (r) and E(r) has the same slope at the origin r = 0 and show enhancement for small r region, although it is masked by the integral measure r 2 .The shape of E U(1) (r) seems to follow ρ B (r), reflecting the direct coupling between Â0 and ρ B in the CS term (23).
Finally, we investigate self-duality breaking of the holographic baryon.The different shape between the topological and energy densities originates from the background FIG. 5. Baryon density ρB(r) and total energy density E(r) for the ground-state baryon.The lower panel shows 4πr 2 ρB(r) and 4πr 2 E(r), including the integral measure factor r 2 .Despite a significant difference between ρB(r) and E(r) around the origin, this difference is reduced by the r 2 multiplication.Here, the MKK = 1 unit is taken.
gravity, h(w) and k(w), and presence of the CS term.In the case of h(w) = k(w) = 1 without the CS term, the instanton solution has exact self-duality of F M N = FMN , leading to ρ B (r, w) ∝ E(r, w).Hence, the functional forms of ρ B (r) and E(r) are forced to be the same.In fact, between the shapes of ρ B (r) and E(r), the similarity indicates the self-dual tendency, and the different point indicates self-duality breaking.
For the single baryon case B = 1, we introduce the self-duality breaking parameter defined by which is normalized by the topological quantity.This is non-negative and becomes zero only in the exact selfdual case.The self-duality breaking of the ground-state baryon is found to be ∆ DB ≃ 0.17, which is non-zero but seems small.The small value might indicate that the true holographic configuration is close to be self-dual.Then, the ground-state baryon might be approximated by the self-dual 't Hooft instanton in holographic QCD.

V. SIZE-DEPENDENCE OF A HOLOGRAPHIC BARYON
As above mentioned, by using the Witten Ansatz, 1+4 dimensional holographic QCD is reduced into a 1+2 di-mensional Abelian Higgs theory.Accordingly, the topological description of a baryon is changed from an instanton to a vortex.In the previous section, we have numerically obtained the ground-state solution in holographic QCD, where the baryon size is automatically determined by minimizing the total energy.Now, let us consider a holographic baryon with various size.Note that the size is originally one of the moduli of an instanton in a flat space, and different size baryons are to be degenerate in holographic QCD, if one sets h(w) = k(w) = 1 and neglects the CS term.In the real holographic QCD, the size parameter is no more modulus, according to the background gravity and CS term.Nevertheless, the size might behave as a quasi-modulus in the holographic baryon, resulting in physical appearance of a soft vibrational mode as a low-lying excitation.Then, we investigate a baryon with various size and size dependence of the energy in holographic QCD in this section.
First, we consider the ordinary Yang-Mills theory and examine the moduli relation between an instanton and a vortex in the Witten Ansatz, as was originally shown by Witten [37].
Next, we proceed to the holographic baryon, and consider how to obtain an arbitrary-size baryon as a solution of holographic QCD, and investigate the size dependence of the baryon mass.

A. Instanton-vortex correspondence in Yang-Mills theory
In the four-dimensional Euclidean Yang-Mills theory, there exist topological solutions, instantons, where Euclidean time is necessary.A single instanton solution (BPST-'t Hooft solution [35,36]) is written as where the instanton center locates at x µ = X µ , and R denotes the instanton size.Together with color rotation, the location X µ and the size R are known as moduli, and they represent degrees of freedom for this topological solution, that is, their values do not affect the Yang-Mills action.Here, η a µν denotes the 't Hooft symbol [36] defined by Taking its center X µ = ( ⃗ 0, T ), there is SO(3) rotational symmetry in (x, y, z)-space, and the Witten Ansatz is applicable.With the Witten Ansatz, the four-dimensional Yang-Mills theory is reduced into a two-dimensional Abelian Higgs theory, and this instanton can be described by a single vortex.
To understand the relation between an instanton and a vortex, let us consider the 't Hooft solution in the form of the Witten Ansatz.This represents a single instanton solution and is rewritten as By comparing the functional form with Eqs. ( 5) and ( 6), SU(2) gauge fields can be converted into the fields of the reduced Abelian Higgs theory, and their forms are obtained as The complex Higgs field ϕ = (Reϕ, Imϕ) = (ϕ 1 , ϕ 2 ) takes zero at (r, t) = (R, T ) ≡ ζ.The vortex number is counted as the zero-point number in the Higgs field ϕ in the (r, t)-plane.Now, there is one zero point at ζ, and this configuration represents a single vortex.
Thus, in the Witten Ansatz, the vortex corresponding to a single instanton has a zero point ζ of the Higgs field ϕ.Remarkably, this Higgs-field zero point ζ relates to instanton parameters [37], In fact, the instanton size R and Euclidean fourthcoordinate T of the instanton center correspond to this zero point ζ in the Higgs field ϕ.In this configuration, the topological density is written by The topological density localizes around (r, t) = (0, T ), and its extension is about the size parameter R, which reflects the instanton size.

B. Various-size baryon mass in holographic QCD
In this subsection, using the above correspondence (53) between the Higgs zero-point ζ and the instanton size R in the Witten Ansatz, we try to control the baryon size by changing the location of the zero point ζ in the Higgs field ϕ in holographic QCD.Using this new viewpoint, we obtain various sizes of holographic baryons and investigate size dependence of the single baryon energy.
To begin with, we reformulate the above argument for 1+4 dimensional holographic QCD.We recall different points from the ordinary Yang-Mills theory.First, holographic QCD already has four-dimensional Euclidean space of (x, y, z, w), and the instanton can be naturally defined on this space including the fourth spatial coordinate w, without use of Euclidean process.Second, there appear the gravitational factor, h(w) and k(w), and the CS term, which results in a repulsive interaction among the baryon density ρ B in holographic QCD.Owing to these effects, the 't Hooft solution is no longer the exact solution but an approximate one in holographic QCD.
Therefore, we use the 't Hooft solution as a starting point for the B = 1 configuration, and search for the true solution of holographic QCD by a numerical iterative method with keeping the topological property unchanged, similarly in Sec.IV.
In 1+4 dimensional holographic QCD, taking the A 0 = 0 gauge, we use a holographic version of the 't Hooft solution (47), as a starting point of the topological configuration.Here, we locate the instanton center at the fourdimensional spatial origin (x, y, z, w) = (0, 0, 0, 0) for symmetry of (x, y, z)-spatial rotation and w-reflection, and then the initial configuration is set to be which can be rewritten as For the (x, y, z)-rotational symmetric system, through the Witten Ansatz, the non-Abelian gauge fields are converted into the Higgs field ϕ and Abelian gauge field ⃗ a, and they are expressed as In Eq. ( 59 In summary, to obtain various-size baryon solutions, we fix this zero-point location ζ = (ζ r , ζ w ) = (R, 0) of the Higgs field ϕ as a boundary condition and minimize the total energy E numerically.When the total energy E[ϕ, ⃗ a, Â0 ] is minimized against arbitrary local variation of ϕ, ⃗ a and Â0 , these fields satisfies the EOM of holographic QCD.
For the numerical calculation on the (r, w) plane, we use the same lattice used in Sec.IV, that is, a fine and large-size lattice with spacing of 0.2M −1 KK ≃ 0.04 fm and the extension of 0 ≤ r ≤ 250 and −125 ≤ w ≤ 125, of which physical size is (50  Figure 7 shows the static energy V (R) of the baryon with various size R.The potential minimum corresponds to the ground-state baryon.The numerical data are well fitted by a quadratic function, as shown in Fig. 7.Note that all the symbols represent the solution of holographic QCD under the constraint of size fixing, which can be regarded as a boundary condition.
We obtain the fitting quadratic function, in the M Kk = 1 unit.The value M 0 of potential minimum coincides with the previous calculation of the groundstate baryon mass M B ≃ 1.25 M KK in Eq. (39).
The size parameter R = R 0 which minimizes the static energy V (R) of the baryon is found to be R 0 ≃ 2.4 in the M KK = 1 unit.This result coincides with the groundstate result shown in Fig. 1, where the Higgs zero-point ζ r locates at about R = R 0 .If there were no non-trivial gravity (h(w), k(w)) and no CS term, this size dependence would disappear and the potential V (R) would become flat because the instanton size is originally a modulus, reflecting classical scale invariance of the Yang-Mills theory.In fact, this size dependence of holographic baryon mass originates from those gravitational effects and CS term.

C. Analysis of holographic baryon size
In this subsection, we investigate actual size of holographic baryons obtained in the previous subsection in terms of the size parameter R, which corresponds to the instanton size.
In our calculation, for each constraint of fixing the size R, we already obtain the topological density ρ B as Using this gauge-invariant local quantity, we investigate the size of the holographic baryon for each direction of r and w.Here, we define ρ B -weighted average of arbitrary (x, y, z)-rotational symmetric variable O(r, w) as In the ordinary four-dimensional Euclidean Yang-Mills theory, the Pontryagin density ρ of a single instanton with the size parameter R and its center at the origin is given by and the mean square radius weighted with ρ is evaluated as or equivalently for r ≡ (x 2 + y 2 + z 2 ) 1/2 and the Euclidean time t.
The holographic baryon is able to be dilatated simultaneously in both r and w direction, imposing the Higgs field zero-point constraints, and we obtain various size baryons.Based on the above relation, we consider size parameters of the holographic baryon in r and w directions, respectively.In a similar manner to Eq. ( 40) in Sec.IV, we define the radius weighted with the topological density ρ B as in the direction r and w, respectively.To compare with the instanton size parameter R, the factors, 2/3 and 2, have been introduced, considering Eq. ( 68).In fact, the ordinary self-dual Yang-Mills instanton satisfies Figure 8 shows the ρ B -weighted radius d r (R) and d w (R) as a function of R, weighted with the topological density.The solid line denotes d = R and is realized in the case of the ordinary four-dimensional YM theory (h(w) = k(w) = 1) without the CS term.In other words, if the holographic baryon were described with the 't Hooft solution, one would find d r (R) = d w (R) = R.
From Fig. 8, d r and d w are found to be monotonically increasing along with R.This monotonical increase reflects that the change of Higgs zero-point gives the change of holographic baryon (instanton) size.Later, we investigate the dilatation mode by regarding the size R as dynamical degree of freedom.
Around the ground state (R = R 0 ≃ 2.4), the value of d r (R) is larger than d w (R), i.e., d r (R) > d w (R), and this means an oblate-shaped instanton for the holographic baryon [24].
The slope of d r (R) is larger than d w (R), and d w (R) is almost flat against R. Therefore, the size change of holographic baryons is approximately regarded as threedimensional in r-direction rather than four-dimensional.(See Appendix C and D for four-dimensional size change using the BPS instanton.) These differences of the behavior for each direction would come from the nontrivial gravity fields, h(w) and k(w), because the CS term (23) equally acts in r and wdirection and does not break (x, y, z, w) O(4) symmetry.In fact, if h(w) = k(w) = 1, O(4) symmetry is exact and no (r, w)-asymmetry appears.Therefore, h(w) and k(w) are the very origin of (r, w)-asymmetry.
The flatness of d w (R) against R indicates that gravity fields h(w) and k(w) suppress the w-direction swelling.Approximately, one finds d r (R) ∼ d w (R), and then d r (R) seems to follow d w (R) and its soft R-dependence, which might imply that large deviation from the spherical shape is not favored energetically.As the result, the slope of both parameters become small.
In the original Yang-Mills theory, the (anti)instanton appears as the (anti)self-dual solution.In holographic QCD, owing to the presence of gravity (h(w) and k(w)) and the CS term, the self-duality of the solution is explicitly broken.
Similarly for the ground-state baryon in Sec.IV, we investigate self-duality breaking for various size holographic baryons.Figure 9 shows the self-duality breaking parameter ∆ DB in Eq. ( 46) as a function of the size parameter R.This quantity ∆ DB is non-negative and becomes zero only in the exact self-dual case.
One finds that the duality breaking parameter is minimized as ∆ DB ≃ 0.17 at R ≃ 2.2, which is close to the size R 0 ≃ 2.4 of the ground-state baryon.This value ∆ DB ≃ 0.17 seems to be small, and one might expect that the approximation of using the self-dual solution is not so bad.
However, the duality breaking parameter ∆ DB never becomes zero in holographic QCD, and this fact might have an important physical meaning for the baryonbaryon interaction as follows.
In the original four-dimensional Yang-Mills theory, the energy is classically bounded by BPS bound, and its minimum is achieved only if the configuration has selfduality.However, holographic QCD has a non-trivial gravity and the CS term, and thus they distort the selfduality.If there were multi instantons satisfying BPS saturation, its action would be determined only by the topological charge, indicating the "no interaction" between instantons.Then, as an interesting possibility, the self-duality breaking in holographic QCD might be related to the baryon-baryon interaction or the nuclear force [38].

VI. DILATATION MODE OF A SINGLE BARYON
In the previous section, we investigated size dependence of the static energy V (R) of a holographic baryon and showed that it seemed to be quadratic against the size R. Using this result, we numerically investigate timedependent size oscillation modes, i.e., dilatation modes, of a single baryon in holographic QCD in this section.
Since the instanton size R is a key parameter to determine the baryon size in holographic QCD, we describe the size oscillation of the holographic baryon by introducing time-dependence of size R, where R 0 denotes the size of the ground-state baryon and the size R(t) is expected to oscillate around this R 0 .Note again that the size of an instanton is originally a moduli, and, if there were only the Yang-Mills term in flat space h(w) = k(w) = 1, its value would not affect the energy of holographic QCD and the dilatation mode would appear as an exact zero mode.In reality, the gravity effect (h(w), k(w)) and CS term break this moduli property of the size; however, we expect that the dilatation mode is inherited to be a soft mode of the ground-state baryon and appears as a low-lying excitation in the single baryon spectrum in holographic QCD.As a characteristic property, the dilatation never changes the flavor and rotational properties and hence the dilatation excitation has the same quantum number as the ground state.
This dilatation differs from an ordinary (x, y, z)-spatial size oscillation, because holographic QCD has an extra dimension w and the holographic baryon extends also its direction.In fact, this extra-dimensional dilatation is peculiar to holographic QCD.
We consider time-dependent variation of R(t) around the ground-state size R 0 , which physically means a dilataion of the holographic baryon.The Lagrangian of the size variable R(t) is written as We have already calculated the potential term V (R) and have shown it to be almost quadratic in the previous section.We calculate the dilatation mode of the holographic baryon as a collective coordinate motion of the size R(t).Note here that, to estimate the frequency ω of the dilatational mode, one only has to calculate the mass parameter m R .(Of course, m R is not equal to the baryon mass M B .)Here, we use adiabatic approximation that time-dependence of the holographic fields is only through the size R(t).

A. Numerical calculation
In this subsection, we consider the baryon dilatation mode and investigate the excitation energy with keeping the gravity background, h(w) and k(w).We treat the size oscillation to be adiabatic and field motions are relatively faster than the size motion.Within this adiabatic treatment, the time dependence of holographic fields ϕ and a i is decided through only the baryon size R(t).Therefore, it is possible to write down field arguments symbolically as ϕ = ϕ(r, w; R(t)), a i = a i (r, w; R(t)). (72) Note that there is no contribution from Â0 in calculating the kinetic term of size variable R(t), because Â0 has no time-derivative and never accompanies Ṙ(t).
Based on this adiabatic treatment, time-derivative is converted to R-derivative as In our framework, O[R] is (numerically) calculable for arbitrary R, and the R-derivative is easily obtained numerically, We have already obtained holographic configurations ϕ(R(t)) with various size R, and the kinetic term of R(t) is expressed as and mass parameter m R on the size variation is given by Note that CS term also contains time derivative, but it is first order and no effect from the CS term for the kinetic term of dilatation mode.The numerical result is found to be The important point of this numerical calculation is that gravitational factors, h(w) and k(w), are exactly included.
With this value of the mass parameter m R and the quadratic fitting of Eq. (63) for the potential V (R), we obtain the dilatational excitation energy for the holographic baryon.
In terms of 1/N c expansion, the mass parameter m R ∝ κ in Eq.( 78) is O(N c ), and the potential V (R) of the holographic baryon is O(N c ), leading to A = O(N c ).Then, the dilatation excitation energy ω quantity.In Appendix D, we also consider a rough analytical estimation for the dilatation mode, when using 't Hooft instanton, i.e., the solution in the case of h(w) = k(w) = 1 without the CS term.This rough estimate gives a larger value of 771 MeV for the dilatation excitation energy, which seems consistent with 0.816 M KK ≃ 774 MeV of the excitation energy on instanton-size fluctuation in the previous research [19] using 't Hooft instanton.

B. Discussion
In the previous subsection, we numerically calculate the dilatation excitation energy, keeping the gravitational effect of h(w) and k(w).Note again that this background gravity is physically important because it is inherited from the N c D4 branes to express the original Yang-Mills theory.We have consistently performed a numerical calculation for both kinetic and potential terms to include the gravitational effect.We have eventually obtained the dilatation excitation energy of 577 MeV for the holographic baryon.
This dilatation mode appears as an excited baryon with the same quantum number as the ground state since this rotationally symmetric dilatation never changes the quantum numbers on the flavor and rotation.
Similar to the Skyrme soliton, for the description of definite spin/isospin states like N and ∆, semi-classical quantization by adiabatic rotation [19] is used for the holographic baryon.In this quantization process on the spin/isosipn, an O(1/N c ) mass correction is added to the O(N c ) baryon mass [19].
In terms of 1/N c expansion, the dilatation excitation energy is O(N 0 c ) as mentioned before, and thus the dilatation mode is more significant than the O(1/N c ) rotational energy, which leads to N-∆ mass splitting.Furthermore, the correction from this rotational effect to the dilatation mode is higher order of 1/N c expansion and then becomes negligible.) splitting, corresponding to dilatation and rotational effect, respectively.Therefore, the order of lowlying baryon mass is to be N < ∆ < N * < ∆ * in term of the 1/Nc expansion.
In holographic QCD, the dilatation excitation seems to appear as the first excitation of the ground-state baryon and has the same quantum number, e.g., positive parity.In addition to the qualitative properties, the dilatation excitation energy is estimated as 577 MeV in holographic QCD.From these results on the same quantum num-ber and the magnitude of the excitation energy, this dilatation mode of the nucleon N(940) would be identified as the Roper resonance N * (1440), which is positive parity and the first excitation of N(940).For the ∆(1232), ∆(1600) might be identified to be the dilatational excitation mode.
As an interesting possibility, the similar dilatation mode exists for every baryon as a universal phenomenon for a single baryon, because any extended stable soliton generally has such a dilatational mode.
The strange baryon mass is slightly larger, reflecting the non-zero strange quark mass; however, the SU(3) f flavor symmetry approximately holds.Here, we suppose that the mass excess of the strange quark is simply added to the holographic baryon, like the treatment of SU(3) f symmetry and its breaking in ordinary hadron physics [39].Then, also for strange baryons, the dilatation mode would appear, and its excitation energy is expected to take a similar value as the above result of 577 MeV.[40].Each groundstate baryon has the excitation with the same quantum number.Here, first-excited baryons are mainly listed, such as N * (1440) and ∆(1600).For each channel, the excitation energy seems to take a consistent value with the theoretical one, ω ≃ 577 MeV.Here, N * (1710) and ∆(1920) are identified to the second excitation mode.
For each channel with the same quantum number, we find that the first excitation energy seems to be a consistent similar value to the dilatational one, ω ≃ 577 MeV, theoretically obtained above.Also for multi-strange baryons in Ξ, Ξ * and Ω channel, we theoretically predict the dilatation modes with the excitation energy of about 577 MeV.However, the dilatational excited baryons for Ξ(1320), Ξ * (1530), and Ω(1673) are not yet observed experimentally because spin-parity information is not yet confirmed for their excited baryons.In this respect, further experimental analyses are much desired for excited baryons in terms of the dilatational mode in each baryon channel.

VII. SUMMARY AND CONCLUDING REMARKS
We have investigated a baryon and its dilatation modes in holographic QCD based on the Sakai-Sugimoto model, which is constructed with N c D4 branes and N f D8/ D8 branes in the superstring theory.This theory is expressed as a 1+4 dimensional U(N f ) gauge theory in the flavor space.
We have adopted a generalized version of the Witten Ansatz for spatially rotational symmetric systems and have reduced 1+4 dimensional holographic QCD into a 1+2 dimensional Abelian Higgs theory in a curved space.In this formulation, a four-dimensional instanton corresponding to a baryon is converted to a two-dimensional Abrikosov vortex.We have numerically calculated the baryon solution of holographic QCD using a fine and large-size lattice with spacing of 0.04 fm and size of 10 fm.
Using the relation between the baryon size and the zero-point location of the Higgs field in the Witten Ansatz, we have theoretically changed the size of holographic baryons and have investigated its properties, such as the energy and self-duality breaking, as function of the size parameter.Here, each configuration is a solution of EOM of holographic QCD under the constraint of fixing Higgs zero-point.
As time-dependent size-oscillation modes, we have investigated the dilatation modes of a baryon and have found that such a dilatational mode takes the excitation energy of 577 MeV.Since the dilatation does not change the quantum number including the parity, we have identified this dilatation mode for the nucleon N(940) as the Roper resonance N * (1440).We have conjectured that any baryons are expected to have such a dilatational excitation universally, and their excitation energy would be similar.
In this respect, further experimental analyses are much desired for excited baryons in terms of the dilatational mode in each baryon channel.In particular, the dilatational excited baryons for Ξ(1320), Ξ * (1530), and Ω(1673) have not yet been observed experimentally because spin-parity information is not yet confirmed for their excited baryons.
As a caution, the calculated values presented in this paper are to be regarded as semi-quantitative estimates in an idealized case of large N c in the chiral limit, and they have some deviation from experimental values in the real world.In the following, we mention some quantitative limitations and cautions on the approach used in this study.
This framework of holographic QCD has infrared equivalence with massless QCD and gives a useful analytical nonperturbative method to analyze QCD, which is a main reason to adopt holographic QCD in this study for baryons.On the quantitative accuracy, however, this framework includes some limitations.As an important caution, the present calculation is based on the 1/N c and 1/λ expansion, and the starting holographic action is up to the 1/N c -leading and 1/λ sub-leading order.Therefore, for more accurate estimation, it is desirable to check the contribution from 1/N c or 1/λ higher order terms.However, it is extremely difficult to extract the next order of 1/N c and 1/λ expansions in holographic QCD.
As a higher-order correction, there is an O(1/N c ) rotational effect of the hedgehog configuration, which is used for semi-classical analysis of the Skyrmion investigation [41].It is relatively 1/N 2 c smaller, compared with leading order O(N c ) of the baryon mass, although it is necessary for the N − ∆ splitting.Similarly, for the dilatational excitation, this correction appears as relatively 1/N 2 c -smaller order, and therefore we have ignored this higher order in this paper.However, this higher-order correction might be desired to reproduce real experimental data.
In addition, we have assumed spatially spherical shape of a baryon to apply the Witten Ansatz and adiabatic treatment for the dilatation dynamics.For more precise analysis of baryons, more sophisticated treatments might be desired.However, to go beyond the spherical symmetric solution, the Witten Ansatz is no more applicable, and then one has to deal with the four-dimensional analysis even for static baryons.To go beyond adiabatic approach is also a difficult problem widely appeared in theoretical physics, and one has to handle complicated local oscillation of all the holographic fields in the present case.
We have used holographic QCD based on N f = 2 in the chiral limit.Further extension including strangeness is an interesting subject in hadron physics, which can be done with N f = 3 holographic QCD [42].In the real world, u and d-quarks have small finite current mass of 2-5 MeV and s-quarks have current mass of about 93 MeV [40], and the non-zero quark mass explicitly breaks the chiral symmetry.In holographic QCD, however, it is difficult to introduce the finite quark mass or explicit chiralsymmetry breaking, and further theoretical development is required for quantitative argument of hyperons.
We have mainly presented the excitation energy for the dilatational mode of baryons, which appears in the same quantum numbers.It is desired for theoretical progress to show how to distinguish the dilatational mode from other excitation experimentally.To this end, we have to find out unique behavior for the dilatational mode, which is a future subject.The finding of such a peculiar quantity will lead to a better understanding of baryon spectra.
In Appendix C, we investigate the holographic baryon when the self-dual BPS instanton in Eq. ( 56) is used.The self-dual BPS instanton is the 't Hooft solution of the ordinary Yang-Mills theory, and hence, to be strict, this usage is justified in the case of flat space h(w) = k(w) = 1 and ignoring the CS term.Substituting the BPS instanton with the size R into the total energy E in Eq. ( 30) including the CS term and the background gravity, k(w) = 1 + w 2 and h(w) = k(w) −1/3 , one obtains the static baryon energy E(R) = V BPS (R) as the function of the instanton size R, and its minimum gives an approximate ground-state holographic baryon, which satisfies M BPS B ≃ 1.35 M KK ≃ 1.28 GeV and ⟨r 2 ⟩ BPS ρ B = 3 2 R BPS 0 ≃ 2.2 M −1 KK ≃ 0.46 fm, as shown in Fig. 14.Thus, when the self-dual BPS instanton is used, the holographic ground-state baryon has larger mass and smaller size [19] than the true solution numerically obtained in Sec.IV.
For the approximate ground-state holographic baryon, the corresponding Higgs field ϕ and gauge field ⃗ a are shown in Fig. 11.These fields give a topological density ρ B , and Â0 is obtained by solving EOM (25), as shown in Fig. 12. Figure 13 shows the r 2 -multiplied topological FIG.11.The Higgs field ϕ(r, w) (upper) and the Abelian gauge field a(r, w) = (a1, a2) (lower) for the BPS instanton in the Landau gauge in the MKK = 1 unit.For iterative improvement in the main sections, this configuration is used as the starting point, i.e., the initial configuration of the iteration.
and the energy densities.
The static baryon energy V BPS (R) for the BPS configuration is shown in Fig. 14, and it is approximately fit Thus, when the self-dual BPS instanton is used, the holographic baryon has larger mass and smaller size [19] than the true solution numerically obtained in Sec.IV.In Appendix D, we consider a rough analytical estimation of the dilation mode using the self-dual 't Hooft BPS instanton, which is justified in the flat space h(w) = k(w) = 1 and without the CS term.
In this case, when the flat space approximation h(w) = k(w) = 1 is used, the kinetic term T of the size variable R(t) is analytically expressed as which seems a larger value than the numerical result in Sec.VI.This estimation can be analytically done but

FIG. 1 .
FIG. 1. Vortex description for the single ground-state baryon: the Higgs field ϕ(r, w) = (Re[ϕ], Im[ϕ]) (upper) and the Abelian gauge field a(r, w) = (a1, a2) (lower) in the Landau gauge.The Kaluza-Klein unit of MKK = 1 is used.The Higgs field has a zero point at (r, w) ≃ (2.4,0), and its winding number around the zero point is equal to the baryon number, that is, B = 1.

FIG. 2 .
FIG. 2. Temporal component of the U(1) gauge field Â0(r, w) in the ground-state holographic baryon in the MKK = 1 unit.This U(1) gauge field Â0 directly interacts with the topological density ρB and leads to a repulsive force between topological densities, like the Coulomb interaction in QED.
), the zero point ζ of the Higgs field ϕ is found to locate at ζ = (ζ r , ζ w ) = (R, 0), and this ζ r determines the initial size of the holographic baryon.From this initial configuration, similarly in Sec.IV, we numerically search for the single baryon solution in holographic QCD by minimizing the total energy E[ϕ, ⃗ a, Â0 ], with fixing the Higgs-zero location ζ = (ζ r , ζ w ) = (R, 0), (61) where the former R corresponds to the instanton/baryon size.In fact, to consider various size baryons, we here use the correspondence between the instanton/baryon size and zero point ζ in the Higgs field ϕ composing the Abrikosov vortex [37].Note that the Higgs field must be zero at the center of the Abrikosov vortex to realize the finite energy, and therefore the Higgs field ϕ inevitably has a zero-point ζ in the presence of the vortex corresponding to B = 1 in holographic QCD.Here, to change the zero-point location ζ r = R corresponds to a baryonsize change, and its various changes lead to different-size baryons.

fm) 2 .
After an iterative improvement of holographic fields with the constraint of the Higgs zero-point location, we eventually obtain the Higgs and Abelian gauge fields, ϕ(r, w; R), ⃗ a(r, w; R), Â0 (r, w; R) (62) for the holographic baryon corresponding to the instanton size R.Note again that the presence of a zero point of the Higgs field indicates a B = 1 configuration, and the holographic fields obeying the local energy minimum condition also satisfy the EOM of holographic QCD.

FIG. 7 .
FIG. 7. Static energy V (R) of the baryon with various size R in the MKK = 1 unit.The symbol denotes numerical data of the baryon energy with different sizes.The line denotes a fit curve of the potential by a quadratic function.This potential has a minimum corresponding to the ground-state, and its value is consistent with the previous ground-state calculation.

FIG. 8 .
FIG. 8. ρB-weighted radius dr(R) and dw(R) of the holographic baryon in the direction r and w, respectively, as the function of the size parameter R in the MKK = 1 unit.The solid line denotes d = R, which is to be realized in the case of the ordinary four-dimensional YM theory (h(w) = k(w) = 1) without the CS term.If the holographic baryon were described with the 't Hooft solution, dr(R) = dw(R) = R would be satisfied.

FIG. 9 .
FIG. 9. Duality breaking parameter ∆DB for various baryon size R in the MKK=1 unit.The functional form of ∆DB(R) seems to be quadratic and it takes minimum at R ≃ 2.2.

Figure 10
FIG. 10.Schematic figure for mass splitting of the holographic baryon in terms of the 1/Nc expansion.(The symbol * denotes the excitation mode.)The holographic baryon has O(N 0 c ) and O(N −1 c) splitting, corresponding to dilatation and rotational effect, respectively.Therefore, the order of lowlying baryon mass is to be N < ∆ < N * < ∆ * in term of the 1/Nc expansion.

FIG. 14 .
FIG. 14. Static baryon energy V BPS (R) calculated from the BPS configurations in Eq. (56).The solid line is a fitting curve of quadratic function.

T ≃ κ d 3 x dwtr F 2 0M≃ κ d 3 x dw 24x 2 (
∂ t R. Here, h(w) = k(w) = 1 is used in the first line, and the instanton size R 0 appearing in the middle does not affect the result.In this way, the mass parameter m R on the size variation is estimated as m R = 16π 2 κ ≃ 1.18.(D2) Form this mass parameter m R and the potential V BPS (R) in Eq. (C1), one obtains a rough estimation of the excitation energy of the dilatation mode, ω = 2A BPS m R = 0.81 M KK ≃ 771 MeV, (D3)

Table I
presents the candidates of the dilational excitation in various channel of baryons, i.e., N, ∆ and Λ, Σ, Σ * channel.