Witten-Veneziano mechanism and pseudoscalar glueball-meson mixing in holographic QCD

We revisit the $U(1)_A$ anomaly in the holographic model of low-energy QCD by Witten, Sakai, and Sugimoto, presenting a new and direct derivation of the Witten-Veneziano mechanism for generating the mass of the $\eta'$ through an anomalous mixing of the Ramond-Ramond $C_1$ field with the singlet component of the pseudoscalar mesons. The latter turns out to have a kinetic mixing with the normalizable modes of the $C_1$ field representing pseudoscalar glueballs, yielding additional vertices for their production and their decay that dominate over those of the unmixed case considered previously in the Witten-Sakai-Sugimoto model. The leading channel is predicted to be decay into two vector mesons, followed in importance by decay into three pseudoscalar mesons. The issue of production of pseudoscalar glueballs in radiative $J/\psi$ decays and in double diffractive processes is also discussed briefly.


Introduction
In QCD, the U(1) A part of the tree-level flavor symmetry U(N f ) L × U(N f ) R is broken by the axial anomaly, leading to only N 2 f −1 pseudoscalar Goldstone bosons in the spontaneous breaking of U(1) V × SU(N f ) L × SU(N f ) R → U(N f ) V with masses determined by the Gell-Mann-Oakes-Renner relation, while one isoscalar pseudoscalar boson, the η , is found to be too heavy to be a Goldstone boson [1]. Its mass is determined by nonperturbative effects involving the axial anomaly, and it was suggested by 't Hooft [2,3] that these are due to instantons. However, in the limit of large number of colors N c the effects of a dilute gas of instantons is exponentially suppressed. A different mechanism was proposed by Veneziano [4] and Witten [5], who showed that the nontrivial θ-dependence of large-N c pure Yang-Mills theory implies a mass term for the singlet η 0 according to m 2 0 = 2N f χ g /f 2 π , where χ g is the topological susceptibility of pure Yang-Mills theory.
In [6,7] it was shown that this mechanism is indeed realized in the AdS/CFT framework. In the Witten model [8] of low-energy QCD, which is based on a supersymmetrybreaking circle compactification of N c D4 branes in type-IIA supergravity, this was initially discussed in a model with flavor D6 branes [7] and subsequently also in the chiral Witten-Sakai-Sugimoto (WSS) model [9,10], which is based on D8 branes and which realizes a fully nonabelian chiral symmetry breaking.
A more detailed discussion of the Witten-Veneziano mechanism in the WSS model was recently given by Bartolini et al. in [11]. In the present paper we present an alternative, more direct derivation, which moreover allows us to reanalyse the possible mixing of the singlet η 0 with the pseudoscalar glueballG. In the bottom-up V-QCD model of ref. [12], where the Veneziano limit of N c → ∞ with N f /N c fixed is employed, such mixing in the context of the Witten-Veneziano relation has been discussed in [13]. In the WSS model, one of us with F. Brünner has found [14] a vanishing mass mixing of η 0 andG which led to the conclusion of a very narrow pseudoscalar state, because to leading order in the WSS model only vertices involving jointly the scalar glueball and η 0 appeared to be present. Our new derivation reveals the presence of a kinetic (derivative) mixing which leads to additional decay modes of pseudoscalar glueballs that are dominating those considered in Ref. [14] or in the phenomenological model of Ref. [15].
The importance of gluonic contributions for the physics of η mesons is already evidenced by the Witten-Veneziano mechanism. It has also been emphasized by the analysis of ref. [16], who have however left open the question whether this involves a coupling to physical pseudoscalar glueball excitations (see also [17][18][19]). Possible pseudoscalar glueballmeson mixing scenarios have been discussed widely in the literature [20][21][22][23][24][25], but exclusively in the form of mass mixings. The mixing scenario that we obtain in the WSS model leads to a dominance of the couplings that are induced by the Chern-Simons term, in particular to two vector mesons. Inclusion of quark masses leads to additional couplings such as to three pseudoscalar mesons.
Assuming that the couplings obtained in the WSS model are more reliable than the results for the glueball masses, which for tensor and pseudoscalar glueballs are much lower than those predicted by lattice QCD, we consider a range of pseudoscalar glueball masses, with the result that a pseudoscalar glueball of around 2.6 GeV as indicated by (quenched) lattice QCD is found to be a rather broad resonance instead of a narrow state, which may hinder its experimental identification. On the other hand, its relatively strong coupling to vector mesons should enhance its production cross section in central exclusive production as well as in radiative J/ψ decays.
In sec. 2 we review the WSS model, in particular the role of the Ramond-Ramond C 1 which determines the θ parameter of the dual theory, and how the U(1) A anomaly and the Witten-Veneziano mechanism is realized. In sec. 3 we derive the mixing of η 0 and the pseudoscalar glueball modes contained in the normalizable C 1 modes, and we work out its consequences for glueball-meson vertices and decay patterns of the pseudoscalar glueball in sec. 4, followed by a discussion and an outlook in sec. 5.

The Witten model of low-energy QCD
The WSS model is based on the Witten model of low-energy QCD [8] provided by the near-horizon geometry of a large number (N c ) of coincident D4 branes in type-IIA superstring theory wrapped on a circle of circumference R 4 = 2πM −1 KK with anti-periodic boundary conditions for fermions. Since gauginos are massive at tree-level and adjoint scalars acquire masses through loops, the dual theory at energies much smaller than M KK is pure non-supersymmetric Yang-Mills theory at large 't Hooft coupling λ = N c g 2 YM . In the supergravity approximation, where the Kaluza-Klein mass scale M KK is kept finite, the background can be obtained from the dimensional reduction of an 11-dimensional doubly Wick-rotated black-hole geometry in AdS 7 × S 4 , where µ, ν = 0, . . . , 3, η µν = diag(−1, 1, 1, 1), together with a nonzero 4-form field strength where ω 4 is the volume form of a unit 4-sphere with volume V 4 = 8π 2 /3. A regular Euclidean black hole is produced by identifying τ ≡ x 4 . Dimensional reduction from 11-dimensional supergravity [26] with κ 2 11 = (2πl P ) 9 /(4π) and l P = g 1/3 s l s through x 11 x 11 + 2πR 11 , with R 11 = g s l s , and where e Φ = (r/L) 3/2 and hatted (unhatted) indices refer to 11 (10) dimensions, leads to type-IIA supergravity with string-frame action (2.4) In order to have a standard form of D-brane actions with prefactors µ p = (2π) −p l −(p+1) s for both the DBI and CS parts, we absorb the factor g 2 s originally contained in 2κ 2 10 = 2κ 2 11 /(2πR 11 ) by rescaling F 2,4 = g s F 10d 2,4 and e −Φ = g s e −Φ 10d , upon which we drop the 10d labels on the fields and redefine 2κ 2 10 = (2π) 7 l 8 s . In the coordinates used in [9,10], the 10-dimensional metric reads with f (U ) = 1 − (U KK /U ) 3 and R D4 ≡ L/2; the dilaton and 4-form field strength are given by e Φ = g s (U/R D4 ) 3/4 (2.7) and Here and in the following we stick to the usual normalization of the Ramond-Ramond fields in the string-theory literature [26]. In [9,10] these fields are rescaled according to C p = (2π) p−1 l p s C SS p .) The 4-form field strength is related to the number N c of D4 branes by which implies R 3 D4 ≡ (L/2) 3 = πg s N c l 3 s . The parameters 1 of the dual boundary theory which upon dimensional reduction through the circle S τ with radius M −1 KK becomes pure 3+1-dimensional Yang-Mills theory can be identified from the UV limit U → ∞ of the D4 brane action with µ 4 = (2π) −4 l −5 s and α = l 2 s . This gives with k an integer. The Witten background reviewed above has a vanishing one-form field C τ and thus corresponds to the theory with vanishing θ parameter, or sufficiently small θ/N c such that the backreaction on the background can be neglected, in the k = 0 branch. (The fully backreacted case has been worked out in Ref. [28].) The holographic dictionary thus relates non-normalizable fluctuations of C τ to a local, possibly x-dependent θ parameter. Normalizable modes are instead interpreted as pseudoscalar (J P C = 0 −+ ) glueball excitations [29].
The relevant quadratic action for these fields is The resulting field equations have the non-normalizable solution In the Witten model of pure Yang-Mills theory, ϑ = θ, but since this connection will be modified when quarks are included, we have providently introduced a different symbol.
Inserting (2.14) in the action gives with χ g being the topological susceptibility.
The normalizable solutions will be expanded in mass eigenfunctions, with radial eigenvalue equations

Inclusion of quarks and U (1) A anomaly
Sakai and Sugimoto have extended the Witten model by introducing left and right handed chiral quarks through N f pairs of D8 and D8 probe branes localized at separate points on the circle S τ at the holographic boundary. Chiral symmetry breaking emerges from the fact that the D8 and D8 branes have to join in the cigar-shaped background geometry. Choosing antipodal points on S τ leads to embedding functions with constant x 4 ≡ τ and a joining of the branes at the tip of the cigar at r KK (or U KK ). In this case one can extend the coordinate Z introduced in (2.5) to the range −∞ . . . ∞ in order to cover the radial extent of joined D8 and D8 branes.
The action for these flavor branes is given by where Tr denotes symmetrized trace, µ 8 = (2π) −8 l −(9) s , and F = dA + A ∧ A is the field strength tensor for the nonabelian flavor gauge fields living on the D8-branes. (B 2 is the bulk Kalb-Ramond 2-form field; its normalizable modes contain the pseudovector (1 +− ) glueballs of the dual gauge theory [29,30].) The even and odd radial mode functions of A µ are associated with the towers of vector and axial-vector meson fields, with the lowest mass eigenvalue (m 2 v 1 ≈ 0.669M 2 KK ) being identified as the ρ meson mass such that M KK = 949 MeV.
The massless pseudoscalar Goldstone bosons are described by where f 2 π = 1 54π 4 λN c M 2 KK and λ a are Gell-Mann matrices supplemented by λ 0 = (N f /2) −1/2 1. Setting f π = 92.4MeV fixes λ ≈ 16.63 for N c = 3. A smaller value of about 12.55 would be found by matching instead the large-N lattice result [31] for the string tension. As in [32] we shall consider the range λ = 16.63 . . . 12.55 in order to obtain a theoretical error band for our quantitative predictions.
The 9-dimensional Chern-Simons term of the flavor brane is given by where the so-called A-roof genus factor involvesÂ(R) = 1 + 1 The term involving C 3 contains the Wess-Zumino-Witten term of the dual gauge theory because F 4 = dC 3 is nonzero in the background [9,33]. The Chern-Simons term is also involved in the U (1) A anomaly of the dual gauge theory, because the C 7 term modifies the equations of motion of the C 1 field which is responsible for the θ parameter. Using Hodge duality, dC 7 = F 8 = F 2 , and integrating by parts, one finds that where ω τ = [δ (τ ) + δ (τ − π)] dτ has been introduced to extend the integration to the entire bulk spacetime. Thus the flavor probe branes induce a mixing term of the Abelian part of the flavor gauge fieldÂ := N −1 f Tr A living on the D8 brane with the field C τ . The linear equations of motion for C τ get an additional term localized on the D8 brane, which we solve by introducing the localized fluctuation C (δ) τ according to where we have assumed C (δ) τ = 0 with respect to Minkowski space coordinates and ∂ µÂ µ = 0. Switching to Z 2 = (r/r KK ) 6 − 1 and witĥ we thus obtain (2.24) The "anomalous" part C (δ) τ contributes to the θ parameter, In the presence of flavor branes, the θ parameter is therefore no longer given by ϑ alone, but also involves η 0 .
In the remainder of this work we will set the θ parameter to 0, which corresponds to non-vanishing ϑ(x) according to The meson field η 0 therefore also appears in the non-normalizable mode C (0) τ , which is a fundamental ingredient in the realization of the Witten-Veneziano mechanism in the WSS model.
Note that, in the chiral case, a constant θ can be absorbed simply in a field redefinition η 0 → η 0 +f π θ/ 2N f , since only derivatives of η 0 appear in the effective action produced by S D8 DBI . However, introducing mass terms for quarks either through world-sheet instantons or nonnormalizable modes of bifundamental fields corresponding to open-string tachyons [34][35][36][37][38][39] produces the additional term where such a redefinition changes the phase of the quark mass matrix M = diag(m u , m d , m s ) according to M → Me iθ/N f . Both, in the chiral limit and in the case with nonzero quark masses, the singlet η 0 receives an extra mass term that is determined by the topological susceptibility of pure Yang-Mills theory obtained in (2.15), in accordance with the Witten-Veneziano formula [4,5].

Pseudoscalar glueball-meson mixing
With the additional term C (δ) τ ∝ η 0 we have solved the anomalous equations of motion. 2 Now we will determine the field redefinitions that are necessary to obtain a diagonal action for the Minkowski space fields η 0 andG. In doing this we encounter divergent terms 2 As stated after (2.22), we had to assume C proportional to δ(0) in analogy to the Hořava-Witten calculation [40]. By adding to the Lagrangian terms beyond the probe approximation one would presumably be able to cancel these divergences. In the following we will however just drop terms proportional to δ(0), i.e., the (C (δ) τ ) 2 term in S R and the C (δ) τ term in S CS . Let us start with the effective kinetic terms coming from the first part of (2.13), where we have fixed the normalization of the radial mode functions in C (2) τ by requiring For the constant ζ 1 which corresponds to a wave function renormalization of η 0 we obtain whereζ 1 is divergent, since it is obtained by integrating non-normalizable modes, but ζ 1 is suppressed by a factor N c /N f compared to the kinetic term for η 0 contained in S D8 DBI . The constant ζ 2 which is associated with a kinetic mixing of η 0 and pseudoscalar glueball modes is finite and given by for the lowest pseudoscalar glueball. (The next-to-lightest, excited pseudoscalar glueball has The remaining terms from the background action (2.13) and the CS action (2.20) give the effective mass terms with the values already determined above, and (for the lightest pseudoscalar glueball mode) The terms involving ∂ r C τ ] cancel by the equation of motion (2.22) with the CS term. The mass mixing term η 0G vanishes because as already pointed out in [14]. However, the term involving C (δ) τ C (2) τ in (3.5) has produced a nontrivial kinetic mixing term of order (N f /N c ) 1/2 . To get rid of such mixing terms one has to perform a non-unitary field redefinition. In the present case, one has to make the substitutions where we have taken into account that ζ 2 ∝ N f /N c while ζ 1 , m 2 0 ∝ N f /N c . The field redefinitions (3.11) thus diagonalize the bilinear terms up to and including order N f /N c , yielding also a small positive contribution to the pseudoscalar glueball mass term, for λ = 16.63 . . . 12.55. Dropping all terms of order (N f /N c ) 3/2 and higher, we see that the divergent coefficient ζ 1 can be ignored and that (3.11) reduces to (3.14) Note that this is very different from the mixing scenario proposed by Rosenzweig et al. in [20,21] and also considered in [23]. In that scenario it is assumed that the chiral anomaly is not saturated by η 0 alone, but only together with the pseudoscalar glueball field 3 . This leads to a nondiagonal mass matrix for η 0 and the pseudoscalar glueball which can be diagonalized by an orthogonal matrix. The field definition (3.11), on the other hand, is nonunitary. It is needed to remove the kinetic mixing term (3.2), but originally there is no mass mixing term because of (3.10); a mass mixing term appears after the field redefinition in (3.12), but only at the order N 3/2 f /N 3/2 c which is beyond the probe approximation. 4 In Ref. [14], where (in view of the vanishing mass mixing term) the interactions of the unmixed pseudoscalar glueball were considered, it was found that those are given either by pairs of pseudoscalar glueballs interacting with scalar or tensor glueballs or by a vertex connecting a pseudoscalar glueball with η 0 and a scalar glueball. The latter, which is relevant for the decay of a pseudoscalar glueball, is due to the fact that the integral (3.10) for the η 0G mass mixing term no longer vanishes when metric fluctuations dual to a scalar glueball are inserted. Assuming that this is the dominant vertex, a very narrow pseudoscalar glueball was predicted whose decays had to involve η or η together with the decay products of a scalar glueball.
Through the shift (3.14) a pseudoscalar glueball acquires also all types of vertices that the singlet η 0 possesses. In the DBI part of the D8 brane action, such vertices are contained only in terms involving F to fourth and higher power, which are suppressed by higher powers of α . In the chiral limit, the dominant interactions come from the CS part of the D8 brane. With quark masses introduced according to ( The additional term in (3.15) also contains a bilinear term involvingG and η 0,8 , It implies a further mixing ofG with η 0,8 which turns out to be negligible as concerns the masses of η, η , andG when the mass matrix of theG-η 0 -η 8 sector is diagonalized. Due to the large mass of the pseudoscalar glueball, the corrections to the masses of η, η , and G are only about −0.015%, −0.008%, and +0.09%, respectively, and thus can be safely neglected. Also the vertices induced by the kinetic mixing ofG and η 0 receive only small corrections of the order ζ 2 m 2 K /m 2 G ζ 2 . However, the small mixing ofG with η 8 gives rise to new types of vertices from the DBI action.
In the next section we shall consider the consequences of all these additional interactions in turn.

Decay into two vector mesons
The dominant decay channel of the pseudoscalar glueball mode turns out to be a decay into two vector mesons. This decay arises from the Chern-Simons term and is mediated by the mixing of η 0 with the glueballG according to (3.14). Before the field redefinition we have the interaction term withÂ Z ∝ η 0 , which after the field redefinition yields the term with the coupling constant for the lowest vector meson mode ψ 1 . For the lowest axial vector mesons with radial mode function ψ 2 with mass square eigenvalue m 2 a 1 ≈ 1.57M 2 KK ≈ (1190MeV) 2 , which is quite close to the experimental value of 1230 MeV of the lightest a 1 axial vector meson, 5 we obtain instead k 2 = 4.8888 M −1 KK N −1 c λ − 1 2 . Performing the polarization sums we obtain the amplitude squared where M and m are the masses ofG and v (and similarly for the heavier vector and axial vector modes when M is large enough). The partial widths for decays into the various vector mesons are listed in table 1 for λ = 16.63 . . . 12.55 and glueball mass M G given alternatively by the WSS result (3.13) and by an (admittedly speculative) extrapolation to the lattice prediction of 2600 MeV, where we have assumed that the mixing parameter ζ 2 and thus k 1 (which has inverse mass dimension) scales like M −1 G when the glueball mass is raised. (Keeping the mixing parameter as is, the widths for a 2600 MeV glueball would all be a factor of about 2 larger.) Also with the assumed reduction of k 1 , a 2600 MeV pseudoscalar glueball is thus projected to be a rather broad resonance.

Decay into three pseudoscalar mesons
In Ref. [41], Gounaris and Neufeld have proposed that a pure pseudoscalar glueball decays predominantly through an interaction Lagrangian involving iG Tr (M (U − U † )) in order to explain that the pseudoscalar glueball candidate at the time, ι(1460), 6 seemed to decay mainly into KKπ. Decays into ηππ would thus be suppressed by a factor (m π /m K ) 4 . In sect. 3 we have found that such an interaction Lagrangian is in fact generated in the WSS model with nonzero quark masses by the kinetic mixing of η 0 and pseudoscalar glueball modes. Explicitly, it reads in the isospin symmetric case m u = m d . This contains vertices of the pseudoscalar glueball with KKπ, KKη( ), and η( ) 3 of the order of ζ 2 m 2 K /f 2 π , and vertices with η( )ππ proportional to ζ 2 m 2 π /f 2 π . In table 2 we list the resulting partial decay widths for a pseudoscalar glueball with mass M = 1813 ± 7 MeV according to (3.13) and, as above, with mass extrapolated to 2600 MeV as predicted by quenched lattice QCD.
In chiral perturbation theory, a well-known problem is that the leading-order Lagrangian severely underestimates the decay η → ηππ whose leading-order amplitude is proportional to m 2 π , to wit, |M(η → ηππ)| = |2 √ 2 cos(2θ P ) − sin(2θ P )|(m π /f π ) 2 /6. This accounts for only about 3% of the experimental result. Higher-order terms in chiral perturbation theory which are not present in the WSS model have been shown to give contributions which are much larger [45,46]. By the same token, the partial decay rates of a pseudoscalar glueball into three pseudoscalar mesons that result from its mixing with η 0 can be expected to be strongly underestimated by (4.5), in particular the amplitudes for decay into ηππ,  Table 2. Partial decay widths in MeV for the decay of a pseudoscalar glueball in three pseudoscalar mesons resulting from (4.5) with λ = 16.63 . . . 12.55. The results forG → KKπ do not yet include the decaysG → KK * → KKπ) discussed in sec. 4.3, and those forG → P P η( ) with P a pseudoscalar meson neglectG → f 0 (1710)η ( ) → P P η ( ) . For comparison, the decay modes for an unmixed pseudoscalar glueball viaG → f 0 (1710)η( ) as obtained in Ref. [14] are evaluated in the last entry of the table for the same set of parameters. These latter results are however incomplete in the mixed case (see text).
which are proportional to m 2 π . The other decay amplitudes are of the order of m 2 K , but since m K is small compared to m G , formally higher-order contributions could again be much more important. However, even if we assume that these partial decay widths are underestimated by an order of magnitude, they are still much smaller than the decays into two vector mesons.
In Ref. [14], one of us with F. Brünner has calculated the decays of an unmixed pseudoscalar glueball into three pseudoscalar mesons in the WSS model, which necessarily involves η 0 and a scalar glueball. If the latter is identified with f 0 (1710), which decays predominantly into kaons (as is predicted [47,48] by the WSS model when scalar glueballs have no or negligible decay width into ηη ), the dominant channel isG → f 0 (1710)η → KKη, whereas KKπ decays are excluded. With mixing, additional amplitudes forG → P P η( ) arise from G scalar η 2 0 and G scalar,tensor (∂ µ η 0 ) 2 terms in the DBI action and also from the additional terms involving C (δ) τ in S R . They would have to be included in a complete calculation of P P η( ) decays. To give an idea of the magnitude of the contributions from G → f 0 (1710)η( ) → P P η( ), in table 2 we have included the partial widths of these decay modes as obtained in the unmixed case in [14]. While such contributions are moderate for the pseudoscalar glueball mass (3.13), they become important for M G extrapolated to the mass predicted by lattice which is well above the threshold for f 0 (1710)+η decays. However, also these contributions are always much smaller than those for the decaysG → vv.

Decay into one pseudoscalar meson together with one or two vector mesons
The very small mixing of a pseudoscalar glueball with η 8 that is induced by the quark mass term (3.16) also gives rise to vertices with one pseudoscalar meson and one or two where KK * is short forKK * , KK * etc. The corresponding amplitudes are proportional to In table 3 the resulting partial decay widths are listed. While the partial width for a decay of the pseudoscalar glueball into K * , one further vector meson and one pseudoscalar meson are rather small, the partial width forG → KK * (and subsequently → KKπ) turns out to be comparable to that of the direct decayG → KKπ evaluated above. Since we assume that the latter may be underestimated significantly, we have not carried out a full calculation ofG → KKπ combining coherently both amplitudes.
In summary, we are finding a rich pattern of decays of the pseudoscalar glueball in three or more mesons, which involve vertices which are proportional to pseudoscalar meson masses. The by far dominant decay modes are however given by two vector mesons.

Discussion and outlook
If the mass of the pseudoscalar glueball is as given by the WSS model, (3.13), the pseudoscalar glueball is predicted to be a rather narrow resonance with relative width Γ/M ≈ 0.03 . . . 0.04. However, the WSS model appears to underestimate significantly the mass of heavier glueballs. The tensor glueball comes out as 1487 MeV, while it is predicated as around 2400 MeV by both quenched and unquenched lattice QCD [49][50][51][52][53]. Extrapolating our results to a pseudoscalar glueball mass as indicated by (quenched) lattice QCD, 2.6 GeV, the prediction is instead that of a very broad resonance. In fig. 1 we display two possible extrapolations, one with unchanged vertices and mixing, which leads to an extremely large total width, and the somewhat more moderate scenario (underlying the above tables) where the mixing parameter ζ 2 is assumed to be decreasing like M WSS /M .
The rather large decay width of a pseudoscalar glueball to two vector mesons that we obtained from the WSS model also implies a significant coupling to photons because of the vector meson dominance that is inherent in the WSS model [10]. This also applies to the scalar and tensor glueballs studied before in [32]. Radiative glueball decays will be studied in detail in forthcoming work, as well as the contribution of glueballs to hadronic lightby-light scattering, which is of relevance to the theoretical prediction for the muon g − 2, where holographic QCD has been argued to provide a promising framework for quantitative predictions [54]. An important implication of the mixing-induced vertices determined in this work is that contrary to the unmixed case [32] the WSS model predicts sizeable vertices for the production of single pseudoscalar glueballs in double diffractive scattering. While Regge regime hadronic scattering requires extensions and extrapolations of the WSS model [55][56][57], the structure of the vertices with Reggeons and Pomerons are completely determined by the Chern-Simons action and the mixing of the pseudoscalar glueball with η 0 . For Reggeons, the vertices have the form (4.2), while Pomeron vertices can be derived from the A-roof genus factor in (2.19) as in [55]. This leads to 7 The results for the unmixed case of ref. [32] instead implied that central exclusive production of pseudoscalar glueballs required the formation of pairsGG, η( )G, or G scalar,tensorG with vertices of order λ −1 N −2 c , N Similarly, the kinetic mixing of the pseudoscalar glueball with η 0 allows the production of single pseudoscalar glueballs in radiative J/ψ decays with rates of the parametric order N −1 c times that of the decay rate for J/ψ → γη c . However, the (semi-)quantitative results that we obtained within the WSS model suggest that the experimental identification of the pseudoscalar glueball may be made difficult by a very large decay width if its mass is around 2.6 GeV as indicated by (quenched) lattice results. In future work we intend to further explore these predictions in the context of radiative J/ψ decays and central exclusive production in the Regge regime.