Holographic QCD predictions for production and decay of pseudoscalar glueballs

The top-down holographic Witten-Sakai-Sugimoto model for low-energy QCD, augmented by finite quark masses, has recently been found to be able to reproduce the decay pattern of the scalar glueball candidate f0(1710) on a quantitative level. In this Letter we show that this model predicts a narrow pseudoscalar glueball heavier than the scalar glueball and with a very restricted decay pattern involving eta or eta' mesons. Production should be either in pairs or in association with eta(') mesons. We discuss the prospect of discovery in high-energy hadron collider experiments through central exclusive production by comparing with eta' pair production.


INTRODUCTION
Quantum chromodynamics, the established theory of the strong interactions, predicts [1] the existence of flavor singlet mesons beyond those required by the quark model, because in the absence of quarks gluons by themselves can form bound states. However, the status of such "gluonium" or "glueball" states in the observed meson spectrum is still unclear and controversal [2][3][4][5].
In 1980, an isoscalar pseudoscalar with mass around 1.44 GeV which is copiously produced in the gluon-rich radiative decays of J/ψ was proposed as the first glueball candidate [6]. Originally named ι(1440) [7], this is now listed by the Particle Data Group [8] as the two states η(1405) and η(1475). Together with η(1295), this indeed would give rise to a supernumerary state beyond the first radial excitations of the η and η mesons, with η(1405) singled out as glueball candidate [9].
The situation thus appears to be analogous to the case of the scalar glueball, which is generally considered to give rise to a supernumerary state in the set of isoscalar scalar resonances f 0 (1370), f 0 (1500), and f 0 (1710), where only two are expected from the quark model (namely linear combinations ofūu +dd andss).
Here the discussion is divided on the question which of the two heavier resonances has the larger glueball contribution [10][11][12][13][14].
However, only the case of the scalar glueball candidates is supported by existing lattice QCD calculations [15,16] which consistently find that the lowest-lying glueball state has mass around 1.7 GeV and quantum numbers J P C = 0 ++ . The lowest-lying pseudoscalar glueball state is instead found to have a mass around 2.6 GeV, somewhat higher than the 2 ++ tensor glueball with mass around 2.4 GeV. Most lattice results have been obtained in the quenched approximation [17], i.e. without dynamical quarks, but recent unquenched lattice calculations [18,19] have found no evidence for significant unquenching effects, which however should be expected if the pseudoscalar glueball were to mix strongly with radi-ally excited η( ) mesons. Moreover, it is still a controversial issue whether all three states η(1295), η(1405), and η(1475) really exist [20].
We thus assume that the pseudoscalar glueball still has to be discovered and that it should be searched for in the mass range 2-3 GeV. Unfortunately, lattice QCD does not (yet) give information on the production and decay patterns of a pseudoscalar glueball, whereas phenomenological models are weakly constrained with regard to the particular form of pseudoscalar glueball interactions. [21] In this work we show that rather specific predictions can be obtained from the Witten-Sakai-Sugimoto (WSS) model for low-energy QCD, which is a top-down stringtheoretic construction in the large color number (N c ) limit with only one free dimensionless parameter. Extrapolated to N c = 3, it reproduces several experimental results in hadron physics to within 10-30% [22,23]. In Ref. [23] we have applied this model to calculate decay rates of scalar and tensor glueballs in the chiral limit, and in [24,25] with quark masses included. In the latter case we found a strong "nonchiral enhancement" of the decay of a predominantly dilatonic glueball into kaons and η mesons which quantitatively agrees remarkably well with the data for the glueball candidate f 0 (1710) as far as presently known (provided the not-yet-measured decay rate into η-η pairs is sufficiently small [25]). This suggests that f 0 (1710) could be a nearly pure glueball, in agreement with recent phenomenological models that favor f 0 (1710) as the scalar glueball [11,12] with comparatively small admixture of light quarkonia.
In this Letter we explore the implications of the WSS model for production and decay of the long-sought-after pseudoscalar glueball under the assumption that this very predictive approach based on the large-N c limit with largely unmixed glueballs can indeed give the right hints for real QCD.
While in Ref. [23] our WSS model prediction for the width of the tensor glueball of mass 2 GeV was very large, perhaps too large to be clearly observable, here we arrive at the prediction of a narrow pseudoscalar glue-ball state with a very restricted decay pattern which will be a conspicuous feature as long as mixing with quarkonia is small. The specific interactions also suggest that the pseudoscalar glueball may be difficult to produce in radiative charmonium decay, but could be a very interesting object for glueball searches in central exclusive production (CEP) experiments at sufficiently high energies.

EFFECTIVE LAGRANGIAN FOR PSEUDOSCALAR GLUEBALL INTERACTIONS
The WSS model is an extension of the Witten model [26] for nonsupersymmetric and nonconformal low-energy QCD based on D4 branes in type-IIA supergravity compactified on a circle and subjected to a consistent truncation of Kaluza-Klein states. It possesses an interesting spectrum of glueball states with J P C = 0 ++ , 2 ++ , 0 −+ , 1 +− , 1 −− [27] whose mass scale is set by the Kaluza-Klein mass M KK . Sakai and Sugimoto [28,29] showed that N f N c chiral quarks can be added through probe D8 and anti-D8 branes separated on the compactification circle, which introduces a purely geometric realization of nonabelian chiral symmetry break- The resulting effective chiral theory involves Goldstone pseudoscalars and a tower of vector and axial vector mesons.
Fixing M KK through the experimental value of the ρ meson mass and varying the 't Hooft coupling λ = 16.63 . . . 12.55 such that either the pion decay constant (as originally done in [28,29]) or the string tension in large-N c lattice simulations [30] is matched leads to quantitative predictions which are in the right ballpark when extrapolated to N c = 3 QCD [22]. In particular, it reproduces remarkably well the observed hadronic decay rates of the ρ and the ω mesons, which motivates the use of the WSS model also as a model for glueball decay [23,31]. In Ref. [23] we argued, however, that the lightest scalar glueball mode considered in Ref. [31] which comes from an "exotic polarization" of the dual graviton along the compactified direction (denoted by G E in the following) should be discarded and that instead the predominantly dilatonic mode (G D ) be identified with the glueball ground state. The chiral WSS model correctly incorporates the chiral anomaly and the Witten-Veneziano mechanism for giving mass to the flavor singlet pseudoscalar η 0 with [28,32,33] Introducing explicit quark mass terms in the effective Lagrangian such that physical pion and kaon masses are matched leads to η and η masses that agree with real QCD to within 10% [24,25]. As mentioned above, the flavor-asymmetric decay pattern observed for the scalar glueball candidate f 0 (1710) can be reproduced quantitatively with G D , if the (as yet undetermined) parameter for scalar glueball couplings to explicit quark mass terms is chosen such that the rate of decay into mixed ηη pairs remains small.
The interaction Lagrangian of the pseudoscalar glueballs is the same for both, the chiral and the massive version of the WSS model. The pseudoscalar glueball modes are provided by a Ramond 1-form field C 1 which plays the central role in producing the Witten-Veneziano mass m 0 . Following the notation of Ref. [28], the action for C 1 is given by S C1 ∝ √ −g|F 2 | 2 . Cancellation of the U(1) A anomaly requires thatF 2 is a gauge invariant combination of F 2 = dC 1 and the field η 0 = (f π / 2N f ) dz TrA z (z, x) with z parametrizing the radial extent of the joined D8 and anti-D8 branes on which the flavor gauge field A lives.
Inserting a mode expansion of the Ramond 1-form field C 1 with 4-dimensional pseudoscalar glueball fields G (n) (x), n = 1, . . ., together with scalar and tensor glueball fields entering through the metric in S C1 leads to the effective 4-dimensional Lagrangian (suppressing the summation over the mode number index (n)). Here O(G 2 D,E,T ) denotes higher-order interactions involving terms quadratic inG, η 0 and quadratic or higher in the glueball fields arising from metric fluctuations (the tensor glueball field T µν appears at most linearly, but also has interactions involving arbitrarily high powers of the scalar glueball field).
The mass of the lowest pseudoscalar glueball mode (n = 1) is [27]  Note that Eq. (1) contains a mass term for the flavor singlet η 0 [28], but no mixing of the pseudoscalar glueball modesG (n) with η 0 . Terms proportional to η 0G (n) vanish in the unperturbed background geometry, but appear in the presence of metric fluctuations dual to scalar glueballs. In the WSS model, such terms are the only ones which can mediate a decay of pseudoscalar glueballs. Explicitly they read (keeping the exotic glueball mode G E for completeness) with the numerical results for the coupling constants for the lowest pseudoscalar glueball mode listed in Table I (their integral representations will be given elsewhere).
coeff. valuē d0 17 The part of the action which leads to the Witten-Veneziano mass term also gives rise to interactions with scalar glueballs which were obtained (on-shell) in [24]. To linear order in glueball fields the corresponding interaction Lagrangian reads (also including an extra off-shell contribution for the exotic mode G E ) There are also interaction terms of the form (∂η 0 ) 2 G D,E,T coming from the DBI action of the D8 branes, which can be found in Ref. [23], as well as natural-parity violating terms η 0 G 2 T from Chern-Simons action of the D8 branes, which have been obtained in Ref. [34].
Interaction terms involving pairs of pseudoscalar glueballs and a scalar or tensor glueball are given by (The more unwieldy expression LG 2 G E will be given elsewhere.)

DECAY PATTERN OF THE PSEUDOSCALAR GLUEBALL
The only interaction terms arising within the WSS model that are relevant for the decay of pseudoscalar glueballs are contained in (2). They differ strongly from the leading interaction terms that have been assumed previously in phenomenological models.
Rosenzweig et al. [35,36] have assumed that the chiral anomaly is not saturated by η 0 alone, but involves a further physical pseudoscalar field (G 2 ) [37], which couples to the imaginary part of log det Σ, where Σ is the matrix of qq states (which is unitary in the nonlinear sigma model, involving only the pseudoscalars, but unrestricted in linear sigma models [38] so that it also accommodates scalar mesons). While a natural possibility [35] would be to identifyG 2 with the radial excitation of η 0 , it was proposed to identifyG 2 with the pseudoscalar glueball instead. Originally used in the context of the glueball candidate ι(1440), this approach was also adopted in the extended linear sigma model of Ref. [39] for pseudoscalar glueballs with a mass suggested by lattice QCD. The dominant decay mode of a pseudoscalar glueball in this approach turns out to be KKπ (branching ratio B ≈ 1/2) followed by ηππ (B ≈ 1/6) and η ππ (B ≈ 1/10).
Using large-N c chiral Lagrangians, Gounaris et al. [40] argued that there should be no coupling of the pseudoscalar glueball to Im log det Σ. Instead, a coupling to Im tr M q Σ was considered so that the pseudoscalar glueball is stable in the limit of massless quarks (M q being the quark mass matrix). This again gives a dominant decay mode KKπ, but with ηππ being more strongly suppressed (parametrically by a factor m 2 π /m 2 K ).
In agreement with the considerations of Ref. [40], the WSS model, which also corresponds to a large-N c chiral Lagrangian, does not lead to a coupling of the pseudoscalar glueball to Im log det Σ. However, its extension to finite quark masses (either through world-sheet instantons [41] or open-string tachyon condensation [42]) does not naturally lead to a coupling to Im tr M q Σ, because Ramond fields do not couple directly to fundamental strings. In the WSS model, the only coupling linear inG is to η 0 G. This suggests that the pseudoscalar glueball should decay dominantly in η( ) and the f 0 meson which corresponds to the scalar glueball, or η( ) and decay products of the latter. According to the WSS model, the decay mode KKπ that is obtained as the dominant one in the approaches mentioned above should instead be strongly suppressed.
When the mass of the pseudoscalar glueball is larger than the mass of the scalar glueball plus the η( ) mass, the scalar glueball can be produced on-shell. The resulting decay width is displayed in Fig. 1 as a function of the pseudoscalar glueball mass for the glueball mode G D with mass 1.5 GeV and also when raised in mass to match f 0 (1710), which in Ref. [24] we found to be favored by the WSS model [43]. For the latter case, Fig. 2 shows the (not necessarily resonant) dimensionless partial decay widths Γ i /M P forG → Gη( ) → P P η( ) where P = K, π, η, η with the decay pattern for the scalar glueball G = f 0 (1710) obtained in Ref. [24]. With m P ∼ 2.6 GeV as predicted by lattice QCD, the pseudoscalar glueball is predicted to be a rather narrow state; for m P 2.3 GeV it would be extremely narrow.  Partial widths of resonant and non-resonant decays G → Gη( ) → P P η( ) where P = K, π, η, η assuming the decay pattern for the scalar glueball G = f0(1710) obtained in Ref. [24]. (The two cases P P η and P P η are plotted in the same color but can be distinguished easily by the later onset of P P η which dominates at sufficiently high values of MP .)

PRODUCTION OF PSEUDOSCALAR GLUEBALLS
While scalar and tensor glueballs couple directly to qq mesons, pseudoscalar glueballs do so only through the former in the WSS model. This suggests that pseudoscalar glueballs are not as easily formed in radiative decays of J/ψ as the other glueballs, but they would have to arise from excited scalar or tensor glueballs decaying into η( )G orGG pairs. The thresholds for these processes are thus above the mass of the J/ψ so that excited ψ mesons or Υ would be required.
Another possibility is central exclusive production (CEP) in high-energy hadron collisions through double Pomeron or Reggeon exchange (corresponding to G T and (ρ, ω) trajectories; pion and scalar glueball exchanges are subdominant at high energies). The parametric orders of the corresponding amplitudes are shown in Fig. 3. Here production ofGη 0 occurs only via virtual scalar glueballs, whereas production ofGG can additionally proceed through virtual tensor glueballs. Also shown is the possibility of GG production through the natural-parity violating coupling of η 0 to two tensor glueballs (Pomerons), which is provided by the Chern-Simons part of the action of the D8 branes and which was recently studied within the WSS model in Ref. [34]. [44] Associated production of pseudoscalar glueballs with either η( ) or other glueballs is presumably beyond the reach of the older fixed-target experiments searching for glueballs, but seem to be an exciting possibility for the new generation of CEP experiments at the LHC.
Calculation of the corresponding production cross sections within the WSS model could be attempted by employing the techniques used in Ref. [34] for η and η production, but will be left for future work. In this Letter we only present results for the ratio of production rates ofGη andGG pairs over η η pairs [45], when both are produced through a virtual G D glueball. This ratio is fixed by the vertices obtained above together with the results obtained in Ref. [24], and the result is shown in Fig. 4 for the range of 't Hooft coupling discussed above. The amplitude M(G * →G 2 ) ∼ λ −1/2 N −1 c is parametrically of the same order as M(G * → η 2 ) so that the ratio N (GG)/N (η η ) is particularly well determined (at least for fixed meson masses in the scenario of Ref. [24]) [46]. The results in Fig. 4 indicate that CEP of η G is only one order of magnitude below CEP of η η , while above the threshold forGG pairs, production of the latter is even up to one order of magnitude larger than CEP of η η .
Central exclusive production of η pairs has been studied in the Durham model in Ref. [47], where its production cross section was estimated. For example, at √ s = 1.96 TeV this work obtained σ(η η )/σ(π 0 π 0 ) ∼ 10 3 . . . 10 5 assuming sufficiently high transverse momentum such that a perturbative approach becomes justified.
Since small transverse momentum is expected to pro- vide a glueball filter [48,49] and the production ofG together with anotherG or η( ) according to the present model proceeds through virtual scalar glueballs, the kinematical regime of small transverse momentum (small azimuthal angle φ pp ) would be particularly interesting for the search of pseudoscalar glueballs. [50] To summarize, the WSS model suggests very specific production and decay mechanisms of pseudoscalar glueballs that make them very interesting for CEP experiments at high-energy hadron colliders. All this of course assumes that pseudoscalar glueballs do not mix strongly with qq states. At large N c this mixing is suppressed, but it is uncertain whether this feature extends to real QCD. However the smallness of unquenching effects in glueball studies in lattice QCD found in Ref. [18,19] could indicate that nearly pure glueballs are possible after all.
We thank Paolo Gandini, Nelia Mann, Denis Parganlija, and Ulrich Wiedner for discussions and correspondence. This work was supported by the Austrian Science Fund FWF, project no. P26366, and the FWF doctoral program Particles & Interactions, project no. W1252.