Spin-1 Glueballs in the Witten-Sakai-Sugimoto Model

We consider the vector and the pseudovector glueball in the top-down holographic model of large-$N_c$ QCD of Witten and their decays into ordinary mesons described by the D8 brane construction due to Sakai and Sugimoto. At leading order, the relevant interactions are determined exclusively by the Chern-Simons action of the D8 branes and are thus rigidly connected to the chiral anomaly and the Wess-Zumino-Witten terms. As found in a previous study of the pseudovector glueball, which we revisit and complete, the resulting decay widths are surprisingly large, implying that both the pseudovector and the vector glueball are very broad resonances, with a conspicuous dominance of decays into $a_1\rho$ and $K_1(1400) K^*$ in the case of the vector glueball. We also obtain a certain weak mixing of vector glueballs with ordinary vector mesons, but we conclude that it does not provide an explanation for the so-called $\rho\pi$ puzzle in charmonium decays.

In spite of extensive theoretical and experimental studies, the status of glueballs in the hadron spectrum of QCD remains largely unsettled [2][3][4][5].While the spectrum of glueballs as obtained in lattice QCD [6][7][8][9][10] appears to be relatively stable when dynamical quarks are included, their interactions and the amount of mixing with ordinary mesons are difficult to pin down, so that no clear glueball state could be identified yet.
The next lightest glueball is the 2 ++ tensor glueball associated with the Pomeron [19], where lattice QCD indicates a mass around 2400 MeV, while Pomeron physics favors a somewhat smaller mass, followed by the 0 −+ pseudoscalar around 2600 MeV, which is expected to play a role in the chiral anomaly and the large η ′ mass.
In this work we continue the studies of Ref. [20][21][22][23][24][25][26] using the Witten-Sakai-Sugimoto (WSS) model [27,28] to derive predictions for the interactions of glueballs with ordinary mesons as well as their radiative decays.The Witten model [29] for low-energy large-N c QCD is based on a supersymmetry breaking background geometry provided by an N c ≫ 1 stack of circle compactified D4 branes in type-IIA supergravity, and it has a spectrum of spin-0 ±+ , spin-1 ±− , and spin-2 ++ glueballs with a mass hierarchy that is qualitatively in agreement with lattice findings [30,31].By adding stacks of N f ≪ N c D8 and anti-D8 probe branes, Sakai and Sugimoto have succeeded in constructing a top-down holographic model that provides a geometric model of nonabelian chiral symmetry breaking and reproduces numerous features of actual low-energy QCD qualitatively as well as semi-quantitatively, typically with 10-30% deviations, with a minimal number of free parameters.Because no further free parameters are involved to determine the interactions with glueballs, the WSS model is also very predictive with respect to interactions between glueballs and ordinary mesons, which are treated as (approximately) unmixed in the 't Hooft limit g 2 N c ≫ 1, N f ≪ N c , corresponding to a quenched approximation when we set N f = N c = 3 in the end.
In Ref. [26] we have recently revisited the predictions of the WSS model for meson decays upon including the η ′ mass from the U(1) A anomaly and adding a mass term for pseudoscalars induced by quark masses.Besides extending the decay patterns of scalar and tensor glueballs by radiative decay modes, we have also considered the pseudoscalar glueball, which is represented by a Ramond-Ramond 1-form field and whose interactions are determined by its anomaly-driven mixing with the η 0 meson.The interactions of the latter are uniquely given by the Chern-Simons (CS) term of the flavor branes, hence completely determined by the anomaly structure.
In this paper, we extend the analysis to spin-1 glueballs, where the quenched lattice QCD simulation of Ref. [8] predicts masses around 3000 MeV for the pseudovector (1 +− ) and around 3800 MeV for the vector (1 −− ) glueball, which is reproduced well by the WSS model as far as their ratio is concerned, while the overall scale is underestimated by about 30%.In the WSS model, the two spin-1 glueballs are represented by the Kalb-Ramond tensor field in conjunction with a Ramond-Ramond 3-form field.Their interactions with ordinary mesons are dominated by the unique CS action of the D8 branes; they are thus tied to the structure of the anomalous interactions of ordinary mesons.Moreover, through the Kalb-Ramond field, the 1 −− vector glueball mixes with the singlet component of ordinary vector mesons, which is interesting with regard to the proposal [32][33][34] that mixing with vector glueballs could explain the so-called ρπ puzzle in charmonium decays [35], which consists of a surprisingly strong suppression of ρπ and K * K in the decay of ψ(2S) compared to ψ(1S) = J/ψ.However, in the WSS model the decay pattern of the vector glueball turns out to have a strong enhancement in the a 1 ρ, K 1 K * , and f 1 ω channels, which are not seen in any of the ψ(nS) decays.The results of the WSS model thus do not support an explanation of the charmonium ρπ puzzle through vector glueball admixtures.
The couplings and decay patterns of vector and pseudovector glueballs are also of interest with regard to the physics of the Odderon [19,36], which recently has been claimed to have been discovered in joint experiments by the TOTEM and D0 collaborations [37].Brower et al. [38] have argued that in holographic QCD Odderons appear naturally as the Reggeized Kalb-Ramond modes in the Neveu-Schwarz sector of closed string theory, which contains both vector and pseudovector glueball modes whose interactions with ordinary hadrons are fixed in the WSS model without any additional free parameters.
However, as found in the previous study of the decays of the pseudovector glueball in Ref. [24], which we revisit and complete, the decay widths obtained in the WSS model are very large, making both spin-1 glueballs difficult to discover, albeit the peculiar decay pattern of the vector glueball may be helpful in this respect.
This paper is organized as follows.In Sec.II we recapitulate the WSS model as used in [26], but expanded to include all form fields relevant for spin-1 glueballs.In Sec.III we derive the bulk mode function of the vector glueball and describe its effects on the hadronic modes on the flavor branes, followed by a systematic evaluation of the hadronic and radiative decay modes, closing with a discussion of the implications for the ρπ puzzle in J/ψ and ψ ′ decays.In Sec.IV, we consider the pseudovector glueball, revisiting and completing the previous work of Ref. [24].Sec.V contains our conclusions and comments on phenomenological consequences.

II. QUICK REVIEW OF THE WITTEN-SAKAI-SUGIMOTO MODEL
The 10-dimensional background geometry corresponding to an N c ≫ 1 stack of D4 branes compactified with supersymmetry breaking boundary conditions in the circular fourth spatial coordinate is given by the metric with dilaton ϕ and Ramond-Ramond three-form field1 C 3 .Here x µ , µ = 0, 1, 2, 3, are the coordinates in the flat four-dimensional directions, U is the radial holographic direction, where regularity at U = U KK fixes the radius R D4 is related to the string coupling g s and the string length l s through R 3 D4 = πg s N c l 3 s , and the 't Hooft coupling of the dual four-dimensional Yang-Mills theory that arises after Kaluza-Klein reduction is given by This is a solution in type IIA supergravity, whose bosonic part reads [39] S IIA = S N S + S R + S CS , where (2.6) The probe (N f ≪ N c ) D8 and D8-branes extend along x µ , U , S 4 and are located in an antipodal configuration on the τ -circle, joining smoothly at U KK , thereby realizing spontaneous The action for the flavor D8-branes is given by the sum of the DBI action and the Chern-Simons action with F the nonabelian flavor field strength and Â(R) being the A-roof genus [39,40].The sum in the Chern-Simons term is a formal sum over the p-form gauge fields in the Ramond-Ramond sector of the theory.Following [27,28], the spectrum on the joined D8 and D8-brane is truncated to include only SO(5) invariant states.To this end, and to quadratic order, the DBI action in Eq.(2.8) reduces to2 with where z runs from −∞ to +∞ along the joined D8 branes.Performing a Kalzua-Klein(KK) decomposition for the five-dimensional flavor gauge fields yields a tower of massive vector and axial vector mesons corresponding to odd and even mode numbers n with even and odd z-parity, respectively (see our previous paper [26] for further details): Identifying the lightest vector mode with the ρ meson fixes M KK = 949 MeV [27,28], corresponding to m ρ = 776.4MeV.The scalar fields φ (n) can be absorbed by the fields B (n) µ except for φ (0) which corresponds to the massless pseudoscalar Goldstone multiplet of the broken chiral symmetry, with the Gell-Mann matrices λ a = 2T a and including the singlet term λ 0 = 2/N f 1.
To fix the 't Hooft coupling λ we use the resulting pion decay constant to get λ ≈ 16.63 from f π ≈ 92.4 MeV.To obtain an error estimate and following [21] we shall also consider the smaller value λ ≈ 12.55 obtained by matching the large-N c lattice result for the string tension obtained in Ref. [41].
The non-normalizable modes of the flavor gauge field A µ can be used to introduce the photon field as an external source via [28] with the quark charge matrix Q for N f = 3 given by where e is the electromagnetic charge.As reviewed in our previous paper [26], vector meson dominance (VMD) arises because the photon field couples exclusively through mixing with the tower of vector mesons.For on-shell photons, the corresponding holographic wave function entering the overlap integrals with the mode functions of hadronic fields reduces to unity; off-shell photons involve nontrivial bulk-to-boundary propagators.
For N f = 3, which we shall consider in the following, we also take into account that in the WSS model the U(1) A flavor symmetry is broken by an anomalous contribution of order 1/N c due to the C 1 Ramond-Ramond field, which gives rise to a Witten-Veneziano [42,43] mass term for the singlet η 0 pseudoscalar with [25,27] (2.16) For N f = N c = 3, one has m 0 = 967 . . .730 MeV for λ = 16.63 . . .12.55, which is indeed a phenomenologically interesting ballpark when finite quark masses are added to the model by the addition of an effective Lagrangian which can be motivated by either worldsheet instantons [44,45] or nonnormalizable modes of additional bifundamental fields corresponding to open-string tachyons [46][47][48][49].
In the following we shall consider this range of mixing angles in conjunction with the variation of λ, but we shall fix m η and m η ′ to their experimental values when evaluating phase space integrals.
Vector mesons remain unchanged by this introduction of quark masses.In the following we shall keep the (chiral) results for their couplings, but we will raise the masses of ω and ϕ mesons to their experimental values in phase space integrals, assuming ideal mixing.
In the WSS model, the axial vector meson a 1 is predicted with mass 1186.5 MeV, very close to the experimental result of 1230 (40) MeV.For the remaining axial vector mesons we again keep the chiral results for their couplings, but introduce phenomenological masses and mixing angles in phase space integrals.Here we use a mixing angle of θ f = 20.4 The physical strange axial vector mesons K 1 (1270) and K 1 (1400) are mixtures of K 1A (1 ++ ) and the excited axial vector meson K 1B (1 +− ) [50].Because in the WSS model, there is no 1 +− nonet of ordinary mesons, only K 1A is present, which couples to the physical K 1 mesons according to their mixing defined by In [50,51] the favored mixing angle is quoted as |θ K | ≈ 33 • , which we adopt in the following.Encouragingly, the WSS model predicts rather well the ballpark of several hadronic decays such as ρ → ππ, ω → πππ, a 1 → ρπ, and also various radiative decays, see Ref. [21,[26][27][28].

III. THE VECTOR GLUEBALL IN THE WSS
The mass spectra for the spin-1 fluctuations in the M-theory lift of the Witten model were first obtained in [31] by considering the fluctuations of A M N O and A M N 11 .In the 10D string frame, these fluctuations translate to C 3 and B 2 , respectively.
Treating contributions stemming from the D8-branes as perturbations later on, the relevant field equations are obtained by varying Eq.(2.5) with respect to B 2 and C 3 (3.1)

A. Ansatz, normalization and equations of motion
In [31] the 1 −− vector glueball mode is obtained from the A µντ and A µr11 components of the 11D gauge field A 3 which translates to C µντ and B µu in the 10D string frame.Note that including the B µu fluctuation is necessary to obtain a consistent solution of the equation of motion since these two fluctuations are tied by a topological mass term.Starting from the ansatz and neglecting backreactions from the DBI action, we obtain the mode equation for the vector glueball The relation to the notation used in [31] is and when using coordinates z along the D8 branes we have a(z) = zM 4 (z).By imposing the boundary conditions M ′ 4 (U KK ) = 1 and M 4 (∞) = 0 we obtain the mass spectrum M 2 V = λ V M 2 KK with the first three eigenvalues given by λ V = {9.22721,15.9535, 24.1552}.The lowest eigenvalue corresponds to the mass of M V = 2883 MeV which is below the (quenched) lattice result of ≈ 3850 MeV [7,8].
To fix the normalization we induce the fluctuations (3.2) in (2.5) and utilize the equation of motion (3.3) to get Requiring a kinetic term with canonical normalization after integrating over the holographic coordinate, the S 4 , and the S 1 , we set a(u leading to with for the ground-state vector glueball. When considering interactions with modes on the flavor branes, the integration variable z covers the holographic radial coordinate twice.The glueball modes are all even under z-parity.However the rescaling employed above corresponds to a(z) = zM 4 (z) and thus M 4 (z) has odd parity on the joint flavor branes.

B. Bilinear corrections due to the DBI action
Because the Kalb-Ramond field couples directly to the flavor branes through the DBI action, the latter gives rise to bilinear terms involving the vector glueball field and the singlet component of the vector meson field.

Mass correction
Integrating over the holographic direction and the S 4 , the DBI action gives rise to an additional mass term for the vector glueball proportional to N f /N c , given by where we projected out the spin-1 part of Cρσ (x µ ) with Cρσ ( Treating this contribution perturbatively we obtain for N f = 3, N c = 3, λ = 16.63 . . .12.55 an increase of the mass of the vector glueball of 100 . . .57 MeV, i.e., only 3.4 . . .2%.
Since this correction is of the same order as backreaction effects [52,53] that we otherwise ignore in the following, 3 and since it is numerically quite negligible, we shall later use only the leading order result for the vector glueball mass.

Mixing with vector mesons
A parametrically more important term of order N f /N c is given by a bilinear term involving the vector glueball and the singlet flavor gauge field v = v a=0 .Explicitly it is given by for the first three vector meson modes.Note that, as explained above, the integral over z involves M 4 as an odd function.Restricting to the ground-state singlet vector meson, the combined kinetic terms for singlet vector mesons and the vector glueball are then given by which would make the ω meson 2-3 MeV lighter than the ρ, while in reality it is roughly 12 MeV heavier.
Larger effects could however arise for vector mesons that are comparable in mass with the vector glueball, such as charmonia, but for those the WSS model does not provide a reasonable description, because their masses are dominated by the quark masses whereas the vector mesons in the WSS model are independent of quark masses.Nevertheless, we can study the additional decay modes of vector charmonia that would be contributed by a certain mixing with vector glueballs.We shall return to this question after having determined the decay modes and partial widths of vector glueballs.

C. Decays of the vector glueball
Except for the mixing term (3.10), all leading-order couplings of the vector glueball with ordinary mesons originating from the DBI action vanish, since they involve a trace of commutator terms.Hence to this order all couplings arise through the Chern-Simons term and are thus anomalous.Further we note that C 3 is dual to C 5 since F 6 = ⋆F 4 , leading to contributions from B 2 as well as C 3 .
From the Chern-Simons term of the D8-brane we obtain couplings to mesons, and through VMD also to photons, namely from (3.17) Looking at each term separately we have In the first term we can use the Hodge dual to fill the indices pertaining to the S 4 .In the second term we can distribute the indices to obtain the F 4 field strength from the background and B µz .Note that for the field strengths with p > 4 we have the twisted field strengths [55] but they are not dynamical [56].From (3.18) we obtain and from (3.19) where ⋆F V µν = √ □ Cµν .Furthermore we utilized the full antisymmetry to rewrite Interactions between the vector glueball, pseudoscalar mesons, and vector mesons are thus given by where and we explicitly pulled out the mass dependence in the Lagrangian and used A z = Π(x µ )K −1 / κπM 2 KK .The couplings to vector-and axial vector mesons are governed by where (3.27) Note that since M V ∝ M KK , (3.26) does not depend explicitly on the compactification scale.
The leading quartic couplings are obtained from the commutator terms in the non-abelian field strengths of the Chern-Simons interactions.To leading order we have with Finally, there are interactions with one axial vector meson and two vector mesons.With the masses obtained by the WSS model, these are however at the mass threshold of the vector glueball, and even above the mass threshold of the pseudovector glueball, which is why they will not be considered in the following.

Hadronic decays
From Eq.(3.24) we obtain the squared amplitude for the decay into one pseudoscalar and one vector meson with decay rate The resulting decay rates are collected in Table I  vector meson as For the three-body decays (3.28) yields where s ij is the center of mass energy of the vector meson and pseudoscalar subsystem.
Because a 1 decays into ρπ with a large decay width, which as mentioned above is in fact rather well reproduced by the WSS model, we should consider the decay channels a 1 ρ and ρρπ together (see Fig. 1), since these decays can interfere either positively or negatively.In fact, we find that there is almost maximal negative interference.In isolation, G V → a 1 ρ would have a partial width of 822. . .1089 MeV, whereas the resonant decay G V → a 1 ρ → ρρπ together with the nonresonant G V → ρρπ is only about 60% of that.
When extending these results to the axial vector mesons involving strange quarks, we instead treat those as narrow resonances and final decay products, neglecting the corresponding interference effects.In fact, in real QCD the axial vector mesons K 1 and f 1 have much smaller decay widths.Using their experimental widths indeed leads to comparatively minor changes of the combined resonant plus non-resonant three-body decays.
1. Feynman diagrams contributing to the hadronic three body decay of the vector glueball into ρρπ 2. Comparison with Ref. [1] In Ref. [1], Giacosa et al. have calculated branching ratios for the vector glueball resulting from three candidate interaction terms in a chiral Lagrangian inspired by the extended linear sigma model (eLSM) developed in [15,[57][58][59].Since there is no experimental information on the coupling constants in either of those terms, ratios of partial decay widths within each of the three possibilities have been worked out.Two of these terms involve dimension-4 operators and do not have a counterpart in the WSS model studied here, so the latter suggests that they may be subleading.A third one breaks dilatation invariance and involves the Levi-Civita tensor that appears also in all the interactions following from the Chern-Simons term in the WSS model, but the resulting interactions differ qualitatively from those considered in Ref. [1].In particular, there are terms in (3.26) which cannot be written in terms of the (dual) field strength tensor for the vector glueball field, whereas Ref. [1] considered only one term proportional to ⋆F V .
In Table II, our results for the ratios of the various partial decay widths and Γ(G V → ρπ) are compared with Ref. [1].In both models the dominant decay mode is Relative branching ratios of the hadronic decays of the vector glueball with WSS model mass M V = 2882 MeV and with quenched lattice QCD result [8] 3830 MeV, the latter for the sake of comparison with Ref. [1].
model this is a factor of 24 larger than Γ(G V → ρπ), while in the model of Ref. [1] this factor is 1.8, more than an order of magnitude smaller. 4The second strongest decay mode is K 1 K * , for which Ref. [1] does not list a result, followed by f 1 ω.The WSS model thus predicts a rather strong enhancement of decays into a pair of axial vector and vector compared to a pair of pseudoscalar and vector.

Radiative decays
From Eq.(3.24) we obtain the coupling to photons by utilizing VMD where Employing VMD in Eq.(3.26) we readily obtain the coupling between the vector glueball, an axial vector meson, and one photon as where and the Lagrangian is again independent of the compactification scale.The quartic coupling including one photon is obtained in a similar fashion from Eq.(3.28) where From Eq.(3.24) the squared amplitude for the decay into one pseudoscalar and one photon is obtained as with decay rate From Eq.(3.36) we obtain the squared amplitude for the decay into one axial vector meson and one photon The squared amplitude for the three-body decays resulting from Eq.(3.38) is given by There are no three-body decays with two external photons due to the appearance of the commutator in Eq. (3.38), but there are also decays into one photon, one vector meson, and one axial vector meson determined by with the same coupling f mn 1 as in (3.27) that dominated the hadronic decays.The various partial decay widths are collected in Table III.Again we combine ρπ decay products with resonant a 1 → ρπ contributions (see Fig. 2), although here the interference is of lesser importance.
Feynman diagrams contributing to the radiative three body decay of the vector glueball into πργ

Implications for the ρπ puzzle
A long-standing puzzle in charmonium physics is the experimental fact that the radial excitation ψ ′ = ψ(2S) = ψ(3686) of the vector meson J/ψ has decays into ρπ, K * K, and other hadronic channels with partial widths far below the expectation from their nature of a nonrelativistic bound state of c and c [35,60].
Early attempts to explain this are based on a mixing of the ground state J/ψ with a vector glueball that enhances the decay modes involved in the ρπ puzzle [32][33][34][61][62][63], for instance by assuming a narrow vector glueball with mass close to that of J/ψ so that a resonant enhancement of the mixing appears (cf.(3.13)).
The WSS model is certainly not suitable to describe the nonrelativistic cc bound states, but it makes concrete predictions for the decays of the vector glueball.Since the vector glueball is predicted to be a rather wide resonance, it does not fit the picture assumed in [33].Moreover, lattice QCD predicts a mass of the vector glueball about 700 MeV higher than that of J/ψ.Nevertheless, it is not excluded that the mixing of the vector glueball could be strongly different for the different vector charmonia.Indeed, in Sec.III B 2, we have found that the mixing of excited vector mesons depends strongly on the mode number, albeit the first two modes happened to be comparable, but that need not be the case for vector mesons far from the chiral limit.
However, the decay pattern that we have obtained for the vector glueball makes it rather unsuitable for an explanation of the ρπ puzzle.While the vector glueball has ρπ and K * K as important decay modes, decays into a 1 ρ and K 1 (1400)K * are much stronger, but have not been observed in the hadronic decays of J/ψ [64].

IV. REVISITING THE PSEUDOVECTOR GLUEBALL
After Kaluza-Klein reduction of the 3 form field A 3 of the 11D supergravity theory to 10D, the 1 +− glueball is identified with the fluctuations of B µν = A µν11 and C µτ r = A µτ r .In 10D notation the equations of motion are solved by Upon rescaling c(u) = (r/r KK ) 3 N 4 (r), u 3 = r 6 /r 6 KK , the radial mode corresponds to the one already obtained in [31].In terms of the z coordinate this rescaling amounts to c(z) = √ 1 + z 2 N 4 (z), hence the N 4 mode has even z parity.
In [24] only the Chern-Simons couplings arising from B 2 were considered.However, there is an additional coupling arising from the dualization of F 6 = ⋆F 4 which has been overlooked.Inducing the pseudovector fluctuation on this term we obtain additionally besides the couplings already computed in [24] A From this we obtain where the first term is the one already obtained in [24], and the second term involving − 1 3! arises through the dualization of C 3 , with This results in a reduction of the decay rates of roughly 30%.The corresponding coupling to the photon is readily obtained as where There is also a coupling between the pseudovector glueball and vector-and axial vector mesons present, which has not been considered in [24].Their masses are, however, at the threshold of the WSS model mass.Explicitly it is given by (4.9)This entails a coupling to photons and axial vector mesons given by

.11)
Three-body decays result from the interactions governed by where 5 b mn 2 = 81 8 for m = n = 1 differ from the ones in [24] by factors of 2 and 2 with a corresponding photon coupling given by The results for the hadronic decay rates are collected in Table IV.In this paper we have completed our previous study [26] of radiative and purely hadronic decays of glueballs in the WSS model by investigating the decay modes of spin-1 ±− glueballs.We have found that the latter are dominated by anomalous vertices involving the Levi-Civita symbol which are uniquely determined by the Chern-Simons action of the flavor branes.
In the case of the vector glueball, such anomalous decays have previously been studied by Giacosa et al. [1], however in the form of just one candidate term among others which are nonanomalous.While Ref. [1] also obtained a 1 ρ decays as dominant anomalous decay, the branching ratio for vector-axial vector decay modes is very much higher in the WSS prediction.For pseudovector glueballs we instead found a dominance of ρπ.
The WSS model also has direct vertices for the spin-1 glueballs with two vector mesons together with one pseudoscalar.In the case of the a 1 ρ channel we found that ρρπ interferes strongly and negatively with a 1 ρ → ρρπ.Whereas for vector glueballs a 1 ρ has a much larger amplitude, for pseudovector glueballs it is below the direct ρρπ channel.In Fig. 3 and 4 we display the corresponding Dalitz plots for the two spin-1 glueballs with mass given by the WSS model, which shows that in the case of the vector glueball, the resonant decay via a 1 should be visible.A clearer signal can however be expected for the decay channels G V → K 1 (1400)K * and G V → K 1 (1270)K * which arise in proportion to their K 1A content, since the strange axial vector mesons are more narrow resonances.When the vector glueball mass is extrapolated from the WSS model mass to the prediction of lattice QCD, K 1 (1400)K * becomes the leading mode.
The decay pattern of the vector glueball is thus conspicuously dominated by a 1 ρ and K 1 K * , which could help in finding its signatures in reactions such as those studied in [65] but also implies that a mixing of J/ψ with the vector glueball as proposed in [32][33][34][61][62][63] cannot explain the ρπ puzzle in J/ψ and ψ ′ decays.
We have also revisited the decay pattern of the pseudovector glueball of Ref. [24], confirming the conclusion of a very broad resonance, but correcting the result from Γ/M ∼ 0.92 . . .1.37 to 0.64. . .0.94.The heavier vector glueball has turned out to be only slightly less broad, with Γ/M ∼ 0.45 . . .0.60.The large widths probably make both spin-1 glueballs difficult to detect.On the other hand, their interactions, which are strongly dominated6 by anomalous vertices and which are numerically large, point to an important role in applications like those studied in Ref. [66].

I.
Introduction and Summary II.Quick review of the Witten-Sakai-Sugimoto model III.The Vector Glueball in the WSS A. Ansatz, normalization and equations of motion B. Bilinear corrections due to the DBI action 1.Mass correction 2. Mixing with vector mesons C. Decays of the vector glueball 1. Hadronic decays 2. Comparison with Ref. [1] 3. Radiative decays 4. Implications for the ρπ puzzle IV.Revisiting the Pseudovector Glueball V. Conclusion and Discussion I. INTRODUCTION AND SUMMARY

TABLE IV .
Hadronic decays of the pseudovector glueball with WSS model mass of M P V = 2311 MeV.

TABLE V .
3/2, respectively, due to the different normalization of the SU (N f ) generators.Hadronic decays of the pseudovector glueball with WSS model mass M P V = 2311 MeV and the quenched lattice value of 2980 MeV.

TABLE VI .
Radiative decays of the pseudovector glueball with WSS model mass of M P V = 2311 MeV.
Table V shows the change of the decay pattern when the WSS model mass M P V = 2311 MeV is replaced by the quenched lattice value of 2980 MeV.The radiative decays are displayed in Table VI; note that there is no analog of (3.44) and thus there are no avγ decays.