Pseudoscalars and the η ′ in the holographic soft-wall model

. Pseudoscalar mesons and glueballs are studied in a model based on the AdS / QCD correspondence. Glueball masses are obtained in the pure-gauge case, while the mixing between pseudoscalar glueballs and singlet pseudoscalar mesons produces the expected mass of the η ′ meson. The topological susceptibility is computed as a function of the quark mass. The model reproduces partial conservation of axial current, the anomaly equation, and the Witten-Veneziano relation.


Introduction
The η ′ meson has a mass of 958 MeV, too high for it to be a pseudo-Goldstone boson. Indeed, while S U(2) A symmetry is spontaneously and explicitly broken generating an octet of pseudo-Goldstone bosons, U(1) A is anomalous. We describe the U(1) A anomaly and reproduce the η ′ mass in a bottom-up holographic model by considering coupled singlet pseudoscalar mesons and glueballs.

Model and results
In this work, the η ′ mass and the topological susceptibility are computed in the soft-wall model [1], one of the bottom-up approaches that can be followed when applying the AdS/CFT correspondence to QCD. This is motivated by the fact that in AdS/CFT the nonperturbative regime of the gauge theory corresponds to the supergravity limit of the dual string theory, characterized by small coupling and a large radius of the spacetime. In a bottom-up approach, a 5d effective field theory is properly constructed in a anti-de Sitter (AdS) space with metric with R the AdS curvature radius and z ⩾ 0 the additional bulk coordinate; z = 0 corresponds to the AdS boundary. The AdS/CFT correspondence links a gravity theory to a conformal gauge theory, but QCD is only asymptotically conformal, so in the soft-wall model conformal invariance is broken by inserting a background dilaton field e −ϕ in the action, behaving as e −c 2 z 2 , where c sets the mass scale of the model. The 5d theory contains the fields dual to the QCD operators under study, whose masses are fixed by the conformal dimension of their dual operator according to the relation m 2 5 R 2 = (∆ − p)(∆ + p − 4); their boundary values (i.e. their values at z = 0) are the sources of the corresponding QCD operators. A crucial feature of AdS/CFT is that the partition function of the 5d model is equal to the generating functional of the correlators in the gauge theory. The former has a simple form in the supergravity limit, given by the exponential of the on-shell action. This means that the correspondence provides a straightforward way to compute correlation functions of a strongly-coupled gauge theory, simply deriving the on-shell action of the bulk theory with respect to the boundary values of the fields. To this aim, it is useful to write a field in Fourier space as ϕ(q, z) = K(q, z)ϕ 0 (q), where ϕ 0 (q) is the source of the operator and K(q, z) is a the so-called bulk-to-boundary propagator.
The soft-wall model was introduced in [1] to study chiral symmetry breaking, so the involved fields and the Lagrangian are well known. The fields dual to the left and right currents are the vector fields 2n f I n f and Tr(T a T b ) = δ ab /2 for a, b = 1, ..., n 2 f − 1. Their masses vanish, so they are 5d gauge fields. Indeed, the global chiral symmetry of QCD is promoted to a local symmentry in the dual 5d theory. The axial field is then defined as A = (A L − A R )/2. We choose the gauge A 5 = 0, while the other components of the axial field can be written as a sum A A µ = A A ⊥ µ +∂ µ φ A of a transverse part, describing axial-vector mesons, and a longitudinal part, contributing to the descritpion of pseudoscalar mesons. The 5d field dual to the QCDq R q L operator is a scalar field that has a vacuum expectation value (vev) X 0 (z), and a phase η A describing pseudoscalar mesons: We consider a model with n f = 3 identical quarks, so X 0 is proportional to the identity matrix: We also assume [2]: where m q is the quark mass, and σ is an independent coefficient related to the chiral condensate: ⟨qq⟩ = − N c 2π 2 σ. Finally, the field Y(x, z) = Y 0 (z)e 2ia(x,z) is a scalar dual to the square of the gluon field strength. The phase a(x, z) is dual to the α s 8π G µνG µν operator describing the pseudoscalar glueballs, and it mixes with the singlet pseudoscalar meson fields. The vev is given by: a form obtained solving the equation of motion assuming no potential term. It contains two additional parameters, y 0 and y 1 . We assume a( 2 is a potential term which vanishes in the pure-gauge limit [3]. The Lagrangian describing these fields is: The first term describes the octet of pseudoscalar mesons [1], while the second deals with the Y field and is invariant under U(1) A transformations: The term in Eq. (5) has also been obtained from the Wess-Zumino action in another bottomup model [4]. After expanding the Lagrangian, we obtain the following expression: containing the known terms for η 8 and ϕ 8 , which describe the octet of pseudoscalar mesons, plus new terms involving a, η 0 and ϕ 0 , describing the singlet pseudoscalar mesons. Notice that the a field is only coupled to singlet pseudoscalar mesons, since we are considering identical quarks. Some of the parameters appearing in the Lagrangian are already known from previous studies in the soft-wall model: R/k = N c /16π 2 and g 2 5 = 3/4 are fixed from scalar and vector meson two-point functions [5]; c = 388 MeV is obtained by fitting the ρ meson mass [1]. There are four unknown parameters, m q ,qq, y 0 , y 1 , that we shall fix from phenomenology.

Pseudoscalar glueballs in pure gauge
Let us first consider the pure-gauge case, where the action only contains the a field describing pseudoscalar glueballs: The bulk-to-boundary propagator and the two-point function of the GG operator can be computed analytically [2]. The two-point function has an infinite number of poles, the positions of which correspond to the masses of the pseudoscalar glueballs m 2 n = 4c 2 (n + 2), while the residues R n are related to the decay constants: R n = m 4 n f 2 n . The lightest state has mass m GG,0 = 1.1 GeV while the first radial excitation has mass m GG,1 = 1.34 GeV, suggesting it could correspond to the η(1405). The first term in the high Q 2 = −q 2 expansion of the two-point function can be matched to QCD [6], thus fixing y 0 = 1 . This shows that the Y field appears in the Lagrangian at a lower order in the large N c expansion with respect to the other fields. Indeed, in the limit N c → ∞ the anomaly turns off, and the η ′ is degenerate with pions, while the degeneracy is lifted by qq annihilation diagrams, which are suppressed by 1/N c . Assuming y 0 = 1 √ N c π , the decay constant of the ground state is 9.8 MeV. The q 2 → 0 limit of the two-point function gives the topological susceptibility [7]: In pure gauge we find χ PG = α s y 1 /π 3 . Assuming χ PG ∼ (191 MeV) 4 , we can fix y 1 = 0.041 GeV 4 .

Nonsinglet pseudoscalar mesons
Away from pure gauge, a tower of singlet and nonsinglet pseudoscalar mesons appears. The equations of motion for the fields describing each nonsinglet meson in the octet are: A combination of the two equations gives a constraint equation: To study correlation functions it is convenient to use the matrix formalism [8], since the two fields are given by a linear combination of the sources in Fourier space: where Φ 0 (q 2 ) = (φ 8 0 (q 2 ), −η 8 0 (q 2 )) T is the vector containing the sources of the two operators. Then, the two one-point functions are contained in a vector: where ⟨J 8 φ ⟩ and ⟨J 8 η ⟩ are the one-point functions of the longitudinal axial current (∂ µψ γ 5 γ µ T 8 ψ) and the pseudoscalar current (2m qψ γ 5 T 8 ψ), respectively, in the nonsinglet sector. Using Eq. (14), we find which establishes the partial conservation of axial current. The two-point functions are collected in the matrix We have checked that in the large Q 2 = −q 2 limit they behave as in QCD [2]. The pion decay constant in the chiral limit is: a value obtained after setting |⟨qq⟩| = (0.281 GeV) 3 . The strange quark mass can be set requiring that the first pole of the two-point function is located at the mass of the η meson, obtaining m η 8 = 548 MeV if m q = m s = 59.5 MeV.

Singlet pseudoscalar mesons
Next, we have solved the three coupled equations of motion for the singlet states η 0 , ϕ 0 and a: These equations can be properly combined and integrated to get a constraint equation: In the matrix formalism, the three fields are aggregated in a vector where Ψ 0 (q 2 ) = (φ 0 0 (q 2 ), −η 0 0 (q 2 ), a 0 (q 2 )) T is the vector containing the sources of the three operators. The masses of the singlet states can be obtained from the poles of the two-point functions: The lightest state, which we indentify with the η ′ meson, has mass 958 MeV, while in the chiral limit its mass is 903 MeV. In Fig. 1 we show the values of the squared mass of the lightest state and of the first radial excitation as a function of the quark mass, together with the result for the lighest state in the octet. The one-point functions are: where ⟨J 0 φ ⟩ and ⟨J 0 η ⟩ are the one-point functions of the longitudinal axial current (∂ µψ γ 5 γ µ T 0 ψ) and of the pseudoscalar current (2m qψ γ 5 T 0 ψ) in the singlet sector, respectively, while ⟨J 0 a ⟩ is the one-point function of the topological charge density (⟨ α s 8π G µνG µν ⟩). The constraint equation (23) establishes a relation between the three one-point functions: a Ward identity representing the anomaly equation in this formalism. From Π aa , the two-point function of the GG operator, we can compute the topological susceptibility as in Eq. (11). Expanding the equations of motion in the chiral limit for large N c [2], at lowest order the constraint equation reproduces the Witten-Veneziano relation χ PG = m 2 η ′ f 2 π 2n f . At q 2 = 0 the equations of motion can be integrated and the following relation holds [2]: so the full (inverse) topological susceptibility gets a contribution from the pure-gauge one and another one accounting for finite quark masses. The numerical result is represented by the black curve in Fig. 2. As m q → ∞ it approaches the pure-gauge value, while for small m q it depends linearly on the quark mass, in agreement with chiral perturbation theory, which predicts χ f ∼ ⟨qq⟩ n f m q [9], shown by the blue dashed line in Fig. 2. χ t vanishes at m q = 0, as expected. The relation represented by the orange curve in Fig. 2, was proposed in [10] [11]. Finally, in Fig. 3 we  have compared our results for n f = 2 with some lattice data [12][13] [14]. It is not possible to get a quantitative comparison since the various points have been obtained with different values of the chiral condensate, producing different initial slopes. Ref. [12] Ref. [13] Ref. [14] Eq. (30)

Conclusions
The masses of singlet and nonsinglet pseudoscalar mesons have been computed in the softwall holographic model of QCD. The mixing among the fields dual to the axial current, pseudoscalar current and the GG operator can explain the large mass of the η ′ . Partial conservation of axial current, anomaly equation and Witten-Veneziano relation have been derived from the constraint equations (14) and (23), obtained by a combination of the equations of motion of the involved fields. The topological susceptibility χ t has been computed for any value of the quark mass. The correction χ f to the pure-gauge value depends linearly on the quark mass for small values of m q , as in chiral perturbation theory, while it diverges as m 4 q at high m q .