Holographic Estimate of Electromagnetic Mass

Using the gauge/gravity duality, we calculate the electromagnetic contributions to hadron masses, where mass generates dynamically by strong QCD interactions. Based on the Sakai-Sugimoto model of holographic QCD we find that the electromagnetic mass of proton is $0.48~{\rm MeV}$ larger than that of neutron, which is in agreement with recent lattice results. Similarly for pions we obtain $m_{\pi^{\pm}}-m_{\pi^0}=1.8~{\rm MeV}$, roughly half of the experimental value. The electromagnetic mass of pions is found to be independent of $N_c$ and 't Hooft coupling and its scale is set only by the Kaluza-Klein scale of the model, $M_{\rm KK}=949~{\rm MeV}$.


INTRODUCTION
Quantum chromodynamics (QCD) is now firmly accepted as the theory of strong interactions. It is well tested not only at high energy, where QCD couples weakly, but also at low energy, where QCD becomes strongly interacting. In practice, however, QCD is notoriously hard to solve directly to describe the properties of hadrons such as mass spectrum, form factors, or other matrix elements, which are intrinsically nonperturbative. Understanding the low-energy physics of hadrons in terms of QCD still remains to be a great challenge. In recent years, however, the lattice QCD has progressed enormously to simulate with parameters such as the light quark masses very close to physical values in describing the properties of hadrons, whose analytic understanding is therefore pressing.
Another development to understand low-energy QCD has been made recently in string theory. Recent study of string theory has shown that a strongly coupled conformal field theory (CFT) is dual to a weakly interacting gravity in anti-de Sitter (AdS) space in the limit of large degrees of freedom, which is known as AdS/CFT correspondence. This finding leads to a new possibility of solving QCD in a certain limit, if one finds the holographic dual of QCD.
Several models for holographic QCD were proposed and were shown to be quite successful in describing the properties of hadrons [1][2][3][4][5]. Holographic QCD deals directly with hadrons as basic degrees of freedom and the dynamics of hadrons is determined by the gauge/gravity duality prescription, found in string theory when both the number of colors, N c , and the 't Hooft coupling, λ = g 2 N c , are very large.
In this paper we calculate the electromagnetic contributions to the hadron masses in the Sakai-Sugimoto model of holographic QCD [1], which is constructed from a type IIA string theory with D4-D8 branes to provide a successful effective theory of hadrons, consistent with QCD [2]. Calculating accurately the electromagnetic mass is quite important, since it is vital to determine the precise value of the light quark masses from the hadron spectrum.
In this paper we provide a holographic estimate of the electromagnetic mass of hadrons, which will be complementary to the lattice results.

MESON MASSES
One of the successes of QCD is that it naturally explains why low-lying mesons like pions and kaons are much lighter than baryons. QCD with three (light) flavors has a chiral symmetry, SU(3) L × SU(3) R in the massless limit, which is predicted to be spontaneously broken to SU(3) V due to the strong dynamics of QCD. Pions and kaons are then identified as the Nambu-Goldstone bosons associated with the spontaneously broken chiral symmetry. By Goldstone theorem, the Nambu-Goldstone bosons are massless when the chiral symmetry is exact. However, the chiral symmetry is only approximate in QCD due to the current quark mass and also electroweak interactions, which then makes pions and kaons massive, but much lighter than baryons.
When a continuous global symmetry, G f , is spontaneously broken to its subgroup H and also intrinsically broken to S w ⊂ G f , the orientation of the true vacuum should be selected or aligned to minimize the vacuum energy, induced by the intrinsic breaking. Provided that the intrinsic breaking is perturbative, the potential for the vacuum orientation g ∈ G f is given to the lowest order as where |0, g = U(g) |0 is a G f -rotated vacuum of QCD and H ′ is the Hamiltonian that breaks G f intrinsically. Instead of rotating the vacuum, one might rotate the external perturbation, then the minimization of the vacuum energy becomes tantamount to finding the direction of the symmetry-breaking Hamiltonian in the basis of the unperturbed vacuum. Finding the vacuum alignment due to the intrinsic breaking was studied by Dashen [12] for the chiral symmetry breaking in strong interactions and subsequently by Weinberg [13] in a generalized context of dynamical electroweak symmetry breaking. Later it has been applied to calculate masses of Nambu-Goldstone bosons in technicolor [14,15].
There are two sources for the intrinsic breaking of chiral symmetry in QCD. One is current quark mass and the other is the electroweak interaction. At the leading order in perturbation the vacuum alignment by both current quark mass and electroweak interactions can be treated independently. In this paper we study the vacuum alignment due to the electromagnetic interaction and calculate the electromagnetic corrections to the pion mass in holographic QCD. The vacuum alignment in holographic QCD due to the current quark mass has been studied previously [16][17][18].
The vacuum energy density due to the electromagnetic interaction is given in the leading order as (See Fig. 1) where U = e 2iπ/fπ describes the pion fields, J Qem µ the electromagnetic currents of quarks, and D µν (x) denotes the photon propagator. The electric charge operator, Q em , in the basis of the strong-interaction vacuum, is defined as a sum of isospin and hypercharge, Q em = I 3 + 1 2 Y up to a G f -rotation. For simplicity, we consider two-flavor QCD, where G f = SU(2) L × SU(2) R × U(1) V and Y becomes the baryon number. The electromagnetic mass of pions is now given as [14,15] where with the vector and axial vector flavor currents, V a µ and A a µ (a = 1, 2, 3) respectively. The vacuum correlator of (vector) flavor currents is given as which is nonperturbative and has precluded any analytic calculations, though several attempts were made to estimate the vacuum correlator by using QCD sum rules or dispersion relations among others. (The vacuum correlator of axial vector currents is similarly defined.) However, in the large N c and large λ limit, where QCD is described by its gravity dual, called holographic QCD, the correlators of flavor currents are easily calculated by gauge/gravity duality. The holographic QCD is described in general by a five-dimensional Chern-Simons-Yang-Mills theory for the flavor symmetry of QCD, the boundary gauge theory, where S CS = N c /24π 2 ω 5 (A) is the 5D Chern-Simons action, that reproduces the QCD flavor anomaly, and the fifth coordinate z corresponds to the holographic direction with a warp factor, parameterized by g(z) and f (z). In the case of Sakai-Sugimoto model, that we will focus on in the paper, g 3 = f = √ 1 + z 2 , taking the Kaluza-Klein scale M KK = 1, and . By the AdS/CFT correspondence the 5D action becomes the generating functional for one-particle irreducible Green's function of 4D operators in the limit of large number of color (N c ≫ 1) and large 't Hooft coupling (λ ≫ 1), if evaluated on-shell for the bulk fields dual to the 4D operators, whose ultra-violet values are identified as the sources of the operators.
By the AdS/CFT prescription the (axial) vector current-current correlators in the momentum space are given by the bulk action evaluated on-shell as where z m is the UV boundary of the holographic coordinate that goes to infinity in the Sakai-Sugimoto model and A(z, q 2 ) is the non-normalizable bulk gauge field, up to the polarization vector, that satisfies the equation of motion, with (anti-) symmetric UV boundary conditions for (axial) vector currents. If we expand the non-normalizable modes, corresponding to the source, in the basis of normalizable modes of bulk solutions as a sum of infinite tower of (axial) vector mesons (see Fig. 2), using the decomposition formula [19], we get for the vector and axial vector correlators, respectively, following the notations in [2], where g v n (g a n ) and m v n (m a n ) are the decay constants and mass of n-th (axial) vector mesons, respectively. After Wick-rotation, the electromagnetic mass becomes, generalizing the formula by Das et al. [20], where δm 2 π is a counter term to the pion mass, coming from the counter term to the vacuum energy (see Fig. 1), and Λ is the UV cutoff. The Weinberg first sum rule on spectral functions is used in the second line, since (axial) vector currents are conserved in hQCD 1 . Being a radiative correction to the pion mass term, the EM corrections are quadratically divergent, which should be removed by the counter term, δm 2 π . We note the UV divergent piece in the EM corrections is independent of the pion decay constant, f π and present even when pions are massless. Since, however, the pions become massless and decouple in the limit f π → ∞, we demand that the counter term δm 2 π cancels the UV divergence exactly 2 . Since the sum rule for the Kawarabayashi-Suzuki-Riazuddin-Fayyazuddin (KSRF) relations saturates rather quickly after 4-th vector mesons [2], we sum up to the first four 1 We checked numerically that lim Q 2 →0 Q 2 Π A (−Q 2 ) = f 2 π and lim Q 2 →0 Q 2 Π V (−Q 2 ) = 0 in the Sakai-Sugimoto model. Therefore the Weinberg first sum rule should hold in the Sakai-Sugimoto model. 2 Each term in the sum of Eq. (12) is logarithmically divergent, but if the Weinberg second sum rule on the spectral functions, n (g 2 v n − g 2 a n ) = 0, holds in the Sakai-Sugimoto model, the logarithmic divergence is absent in the sum and the EM mass is insensitive to the UV cutoff, when the quadratic divergence is removed.
vectors with Λ 2 = 16.45 M 2 KK , after subtracting out the quadratic divergence, to find m π ± − m π 0 ≃ 4.8 MeV, which is close to the experimental value 4.6 MeV. One can also calculate the mass difference numerically without the mode expansion, using the prescription, Eq. (8), from the numerical solutions for the bulk gauge field A(z, q 2 ). For the latter we find m π ± − m π 0 ≃ 3.5 ∼ 7.2 MeV by taking a hard UV cutoff in the integration, Eq (11),

BARYONS IN HOLOGRAPHIC QCD
Baryons in the large N QCD should be realized as solitons [21]. In holographic QCD, being a 5D gauge theory, there is a topologically conservered current where the 5d coordinates M, N, L, P, Q = 0, 1, 2, 3, z . As the 5d current couples to the U(1) bulk gauge field, corresponding to the quark number, through the 5d Chern-Simons term, one can define the 4d baryon current as which becomes the Skyrme current upon the Kaluza-Klein reduction of the bulk gauge fields [22]. Baryons are hence realized as topological solitons in holographic QCD [23][24][25], whose topological charge is nothing but the instanton number, identified as baryon numbers; where F a IJ are the field strength tensors of the bulk flavor gauge fields andF a IJ are their dual with I, J = 1, 2, 3, z .
In holographic QCD the size of instanton is not a zero mode but has a potential, due to the warped geometry and to the U(1) Coulomb repulsion, which has a minimum away from the origin. The energy of the soliton from the DBI part of the 5d action (7) is given as where the SU(2) index a = 1, 2, 3 and the spatial index i, j = 1, 2, 3 . Since the instanton soliton satisfies the duality condition, where ǫ is the fully antisymmetric tensor in the flat space and g 4 is the determinant of the induced 4d metric of the z = constant hyper surface, we find that The DBI energy can be then written as where E a i = F a iz and B a i = 1 2 ǫ ijk F a jk . For the small size soliton, located at the origin z = 0, we may use the flat-space instanton soluton with an instanton density for N B = 1 where ρ is the size of the instanton. For ρ ≪ 1 (≡ M −1 KK ) the DBI energy may be expanded in powers of ρ to get where m Without the Coulomb repulsion the potential for the size ρ will have minimum at the origin so that the instanton soliton is unstable against shrinking to a zero size. However, since the instanton sources through the 5D Chern-Simons term the U(1) gauge field in the bulk, the Coulomb repulsion balances the gravitational attraction to stabilize the instanton soliton. The action for the bulk U(1) can be written as where the electrostatic potential energy is coming from the 5d Chern-Simons term of holographic QCD action (7). For a given, static U(1) charge distribution J 0 (x i , z), its Coulomb energy is given as where we used the Gauss's law in the second line. For the small size soliton located at the the electric field can be obtained by the Gauss's law from which we get Therefore, the Coulomb energy becomes in the expansion of powers of ρ The total energy of the instanton soliton becomes for ρ ≪ 1 We see that the size of the instanton soliton, that minimizes the total energy (29), becomes which is quite small in the large λ limit, as the Coulomb potential energy is subdominant in the 1/λ expansion, compared to the DBI energy [23][24][25] . Being a small-sized soliton, the supergravity approximation of Sakai-Sugimoto model might be invalid to describe holographic baryons. However, it has been shown that the stringy effects to the instanton solitons are numerically suppressed [26].
Since the size of instanton solitons turns out to be much smaller than the characteristic length of holographic QCD, the holographic baryons can be described by point-like bulk spinor fields, assuming N c is odd. Similarly to the bottom-up model of holographic baryons [5], a 5D effective action for the bulk baryon field has been derived in the 5D momentum expansion to be consistent with the bulk gauge symmetry [23]. Especially the coefficient of the Pauli term has been calculated for the baryons sitting at the origin, z = 0, at the leading order, which is essential to correctly reproduce the axial couplings of nucleons among others.
The (low-energy) effective action for the holographic baryons is given in powers of momentum as [5,23], where B is the 5d bulk spinor with a position-dependent mass m b (z), e A M is the vielbein, satisfying g M N = e A M e B N η AB , and the Dirac matrices satisfy Γ A , Γ B = 2η AB . The covariant derivative where the Lorentz generator Γ AB = 1 2i Γ A , Γ B . The coupling of Pauli term is determined by the fact that it should produce the correct long distance tail of instanton solitons that the spinor sources. For the spinor located at the origin, µ 5 (0) = 4π 2 ρ 2 b /(3κ) [23]. The effective action gives couplings of baryons to mesons, upon the Kaluza-Klein reduction of the bulk fields, which turn out to be in good agreement with data.

ELECTROMAGNETIC MASSES OF BARYONS
At the leading order in the α em expansion the electromagnetic (EM) contributions to baryon masses consist of two parts. One is through the vertex correction (shown in Fig. 3) and the other is through the corrections to the baryon self energy, Fig. 4  correction to the baryon self energy is in general present because baryons have the nonminimal EM couplings, Fig. 4 (b), as well as the minimal coupling with form factors.
The QCD corrections to the EM vertex of nucleons are in general non-perturbative and difficult to calculate. But, in holographic QCD they are rather easily calculated from the wave-function overlap integration by the AdS/CFT prescription. As was shown in [23], the Pauli term in the non-minimal coupling of bulk spinors in (31) does not contain a U(1) part, as the spinor sources the instanton solitons, which have only a SU(2) long-distance tail. An immediate consequence of this is that the two-spinor two-photon vertex, Fig 4 (b), vanishes.
The EM corrections to the baryon self energy therefore vanish as well, not just at the leading order but at all orders in the α em expansions. In holographic QCD the EM corrections to the baryon mass come only from the vertex corrections. To By the Lorentz invariance and the current conservation we get for where M N ≃ 940 MeV is the nucleon mass. From the effective action (7) we can easily obtain the EM form factors, given as the overlap of the wave functions in the holographic direction [27]: the Dirac and Pauli form factors are given respectively as V,min Q em + g where m 2k+1 is the mass of k-th (proper) vector meson and g of nucleons become in the Landau gauge, which is consistent with the Ward-Takahashi identity for the γ matrix structure of the vertex in our approximation [28], We find, keeping the first four vector mesons (k = 1, · · · , 4) in the form factors [27], that δM p = 0.494 MeV, δM n = 0.018 MeV.
To conclude we have calculated the electromagnetic contributions to the hadron mass in holographic QCD. For pions, the vacuum alignment due to the electromagnetic interactions has been calculated holographically to obtain the mass difference m π ± −m π 0 = 3.6 ∼ 7. the latest lattice result [11] prefers a bit larger value than ours.