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PHYSICAL REVIEW D 101, 014006 (2020)

Witten-Veneziano mechanism and pseudoscalar - mixing in holographic QCD

Josef Leutgeb and Anton Rebhan Institut für Theoretische Physik, Technische Universität Wien, Wiedner Hauptstr. 8-10, A-1040 Vienna, Austria

(Received 4 November 2019; published 10 January 2020) ð1Þ We revisit the U 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 0 of the η through an anomalous mixing of the Ramond-Ramond C1 field with the singlet component of the pseudoscalar . The latter turns out to have a kinetic mixing with the normalizable modes of the C1 field representing pseudoscalar , 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 a decay into two vector mesons, followed in importance by decay into three pseudoscalar mesons. The issue of production of pseudoscalar glueballs in radiative J=ψ decays and in double diffractive processes is also discussed briefly.

DOI: 10.1103/PhysRevD.101.014006

I. INTRODUCTION Witten-Sakai-Sugimoto (WSS) model [9,10], which is based on D8-branes and which realizes a fully non-Abelian chiral In QCD, the Uð1Þ part of the tree-level flavor symmetry A symmetry breaking. UðN Þ ×UðN Þ is broken by the axial anomaly, leading f L f R A more detailed discussion of the Witten-Veneziano 2 − 1 to only Nf pseudoscalar Goldstone in the mechanism in the WSS model was recently given by ð1Þ ð Þ ð Þ → spontaneous breaking of U V ×SU Nf L ×SU Nf R Bartolini et al. in [11]. In the present paper we present ð Þ U Nf V with determined by the Gell-Mann-Oakes- an alternative, more direct derivation, which moreover Renner relation, while one isoscalar pseudoscalar , allows us to reanalyze the possible mixing of the singlet 0 ˜ the η , is found to be too heavy to be a [1]. η0 with the pseudoscalar glueball G. In the bottom-up Its mass is determined by nonperturbative effects involving V-QCD model of Ref. [12], where the Veneziano limit of ’ the axial anomaly, and it was suggested by t Hooft [2,3] Nc → ∞ with Nf=Nc fixed is employed, such mixing in that these are due to instantons. However, in the limit of the context of the Witten-Veneziano relation has been large number of colors Nc the effects of a dilute gas of discussed in [13]. In the WSS model, one of us with F. instantons is exponentially suppressed. A different mecha- Brünner has found [14] a vanishing mass mixing of η0 and nism was proposed by Veneziano [4] and Witten [5],who G˜ which led to the conclusion of a very narrow pseudo- showed that the nontrivial θ dependence of large-Nc pure scalar state, because to leading order in the WSS model Yang-Mills theory implies a mass term for the singlet η0 only vertices involving jointly the scalar glueball and η0 2 2 according to m0 ¼ 2Nfχg=fπ, where χg is the topological appeared to be present. Our new derivation reveals the susceptibility of pure Yang-Mills theory. presence of a kinetic (derivative) mixing which leads to In [6,7] it was shown that this mechanism is indeed additional decay modes of pseudoscalar glueballs that are realized in the AdS=CFT framework. In the Witten model dominating those considered in Ref. [14] or in the phe- [8] of low-energy QCD, which is based on a - nomenological model of Ref. [15]. breaking circle compactification of N D4-branes in type- The importance of gluonic contributions for the c 0 IIA supergravity, this was initially discussed in a model with of η mesons is already evidenced by the Witten-Veneziano flavor D6-branes [7] andsubsequentlyalsointhechiral 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 [17–19]). Possible pseudoscalar Published by the American Physical Society under the terms of glueball-meson mixing scenarios have been discussed the Creative Commons Attribution 4.0 International license. – Further distribution of this work must maintain attribution to widely in the literature [20 27], but exclusively in the the author(s) and the published article’s title, journal citation, form of mass mixings. The mixing scenario that we obtain and DOI. Funded by SCOAP3. in the WSS model leads to a dominance of the couplings

2470-0010=2020=101(1)=014006(12) 014006-1 Published by the American Physical Society JOSEF LEUTGEB and ANTON REBHAN PHYS. REV. D 101, 014006 (2020) that are induced by the Chern-Simons term, in particular to a large number (Nc) of coincident D4-branes in type-IIA two vector mesons. Inclusion of masses leads to superstring theory wrapped on a circle of circumference ¼ 2π −1 additional couplings such as to three pseudoscalar mesons. R4 MKK with antiperiodic boundary conditions for Assuming that the couplings obtained in the WSS model . Since are massive at tree level and are more reliable than the results for the glueball masses, adjoint scalars acquire masses through loops, the dual which for tensor and pseudoscalar glueballs are much lower theory at energies much smaller than MKK is pure non- than those predicted by lattice QCD, we consider a range of supersymmetric Yang-Mills theory at large ’t Hooft pseudoscalar glueball masses, with the result that a pseu- λ ¼ 2 coupling NcgYM. In the supergravity approximation, doscalar glueball of around 2.6 GeV as indicated by where the Kaluza-Klein mass scale MKK is kept finite, the (quenched) lattice QCD [28,29] appears to be a rather background can be obtained from the dimensional reduc- broad resonance instead of a narrow state, which may tion of an 11-dimensional doubly Wick-rotated black-hole hinder its experimental identification. On the other hand, its 4 geometry in AdS7 × S , relatively strong coupling to vector mesons should enhance 2 its production in central exclusive production 2 r 2 μ ν 2 ψ ds ¼ ½fðrÞdx þ ημνdx dx þ dx as well as in radiative J= decays. The decay patterns 11 L2 4 11 derived in this paper might help to identify a pseudoscalar L2 dr2 L2 glueball in experimental studies. þ þ Ω2 2 ð Þ 4 d 4; So far, the experimental search for glueballs in the r f r spectrum has led to two candidates for the lightest scalar r6 ð1500Þ ð1710Þ fðrÞ¼1 − KK ; ð1Þ glueball, f0 and f0 . Different phenomeno- r6 logical models are split on the question of which of the two would have the largest glueball component, and the WSS where μ; ν ¼ 0; …; 3, ημν ¼ diagð−1; 1; 1; 1Þ, together with 3 results for glueball decay patterns of Refs. [30,31] favor a nonzero 4-form field strength F4 ¼ 3ðL=2Þ ω4, where ω4 f0ð1710Þ. For the tensor glueball there are several candidates is the volume form of a unit 4-sphere with volume V4 ¼ above 1900 MeV, with the WSS model indicating [32] that it 8π2=3. A regular Euclidean black hole is produced by iden- τ ≡ ≃ þ 2π ¼ −1 ¼ 2 ð3 Þ should be a relatively broad resonance, if the mass is around tifying x4 x4 R4 with R4 MKK L = rKK . 2.4 GeV as indicated by both quenched and unquenched Dimensional reduction from 11-dimensional supergrav- – 2 9 1=3 lattice QCD [28,29,33 35]. Prior to the first lattice QCD ity [38] with κ11 ¼ð2πlPÞ =ð4πÞ and lP ¼ gs ls through studies, where the pseudoscalar glueball has been found to x11 ≃ x11 þ 2πR11, with R11 ¼ gsls, and be heavier than the tensor glueball, the ιð1440Þ ηð1405Þ ηð1475Þ 2 ¼ Mˆ Nˆ ¼ −2Φ=3 M N , now interpreted as two states and ds11 GMˆ Nˆ dx dx e gMNdx dx (see however [36]), was regarded as a strong glueball þ e4Φ=3ðdx11 þ A dxMÞ2; ð2Þ candidate; in fact, the ηð1405Þ is still often considered as M such [37]. In the Witten model, the pseudoscalar glueball is where eΦ ¼ðr=LÞ3=2 and hatted (unhatted) indices refer to heavier than the tensor glueball. The kinetic mixing obtained 11 (10) dimensions, leads to type-IIA supergravity with by us when are included increases the mass of the -frame action pseudoscalar glueball somewhat, while leaving unchanged 0 (to leading order) the ηð Þ mesons, which suggests that the ¼ þ þ ð Þ S SNS SR SCS; 3 pseudoscalar glueball is still to be discovered with a mass that is above that of the tensor glueball. where In Sec. II we review the WSS model, in particular the Z 1 pffiffiffiffiffiffi role of the Ramond-Ramond field C1 which determines the S ¼ d10x −ge−2Φ θ parameter of the dual theory, and how the Uð1Þ anomaly NS 2κ2 A 10  and the Witten-Veneziano mechanism is realized. In Sec. III 1 η þ 4∂ Φ∂MΦ − j j2 we derive the mixing of 0 and the pseudoscalar glueball × R M 2 dB2 ; modes contained in the normalizable C1 modes, and we Z 1 pffiffiffiffiffiffi work out its consequences for glueball-meson vertices and ¼ − 10 − ðj j2 þj j2Þ SR 2 d x g F2 F4 ; decay patterns of the pseudoscalar glueball in Sec. IV, 4κ10 followed by a discussion and an outlook in Sec. V. 1 j j2 ¼ M… Fp ! FM…F ; Z p II. WITTEN-SAKAI-SUGIMOTO MODEL AND 1 WITTEN-VENEZIANO MECHANISM ¼ − ∧ ∧ ð Þ SCS 2 B2 F4 F4: 4 4κ10 A. The Witten model of low-energy QCD The WSS model is based on the Witten model of low- In order to have a standard form of D-brane actions with −p −ðpþ1Þ energy QCD [8] provided by the near-horizon geometry of prefactors μp ¼ð2πÞ ls for both the Dirac-Born-Infeld

014006-2 WITTEN-VENEZIANO MECHANISM AND PSEUDOSCALAR … PHYS. REV. D 101, 014006 (2020) Z pffiffiffiffiffiffiffiffiffiffi (DBI) and Chern-Simons (CS) parts, we absorb D4 4 −Φ 2 2 2 S ¼ −μ4Tr d xdτe −gð5Þ the factor g originally contained in 2κ ¼ 2κ =ð2πR11Þ s 10 11   10D −Φ −Φ10 by rescaling F2 4 ¼ g F and e ¼ g e D ,upon 1 2 ; s 2;4 s 1 þ ð2πα0 j j2 þ …Þ which we drop the 10D labels on the fields and × 2 FYM 2 7 8 Z redefine 2κ10 ¼ð2πÞ ls . þ μ ð2πα0Þ2 ∧ ∧ þ … ð Þ In the coordinates used in [9,10], 4 C1 FYM FYM ; 11

r2 r6 U3 −4 −5 0 2 ¼ ð Þ ≡ 1 þ 2 ¼ ¼ ð Þ with μ4 ¼ð2πÞ ls and α ¼ ls. This gives U 2 ;KZ Z 6 3 ; 5 L r UKK Z KK 1 2 ¼ 2π θ þ 2π ¼ ð Þ gYM gslsMKK; k C1 12 the 10-dimensional metric reads ls Sτ   3=2 with k an integer. 2 U μ ν 2 ds10 ¼ ½ημνdx dx þ fðUÞdx4 The Witten background reviewed above has a vanishing RD4     one-form field Cτ and thus corresponds to the theory with 3 2 2 R 4 = dU D 2 2 vanishing θ parameter, or sufficiently small θ=Nc such that þ þ U dΩ4 ð6Þ U fðUÞ the backreaction on the background can be neglected, in the k ¼ 0 branch. (The fully backreacted case has been worked ð Þ¼1 − ð Þ3 ≡ 2 with f U UKK=U and RD4 L= ; the out in Ref. [40].) and 4-form field strength are given by The holographic dictionary thus relates nonnormalizable fluctuations of Cτ to a local, possibly x-dependent θ Φ ¼ ð Þ3=4 ð Þ parameter. Normalizable modes are instead interpreted as e gs U=RD4 7 pseudoscalar (JPC ¼ 0−þ) glueball excitations [41]. The relevant quadratic action for these fields is and Z 1 pffiffiffiffiffiffi ⊃− 10 − ðj j2Þ 3 3 SR 2 d x g F2 RD4 4κ F4 ¼ dC3 ¼ ω4: ð8Þ 10 Z Z g ∞ s 2πR4V4 ¼ − d4x dr 64κ2 [Here and in the following we stick to the usual normali-  10 rKK  3 7 zation of the Ramond-Ramond fields in the string-theory r L μν r 2 × η ∂μCτ∂νCτ þ ð∂ CτÞ : ð13Þ literature [38].In[9,10] these fields are rescaled according fðrÞ L3 r p−1 p SS to Cp ¼ð2πÞ ls Cp .] The 4-form field strength is related The resulting field equations have the nonnormalizable to the number Nc of D4-branes by solution Z 2 3 ð2πÞ ls F4 ¼ 2πNc; ð9Þ ð0Þ ls S4 Cτ ¼ fðrÞϑðxÞ for □ϑðxÞ¼0: ð14Þ 2πR4 3 ≡ ð 2Þ3 ¼ π 3 which implies RD4 L= gsNcls. In the Witten model of pure Yang-Mills theory, ϑ ¼ θ,but 1 The parameters of the dual boundary theory, which since this connection will be modified when quarks are upon dimensional reduction through the circle Sτ with included, we have providently introduced a different −1 3 þ 1 radius MKK becomes pure -dimensional Yang-Mills symbol. theory Inserting (14) in the action gives Z χ 3 4 1 θ g 4 2 λ M L ¼ − jF j2 þ F ∧ F ; ð Þ S ¼ − d xθ ; χ ¼ KK ; ð15Þ 2 2 Tr YM 8π2 Tr YM YM 10 R 2 g 4ð3πÞ6 gYM χ can be identified from the UV limit U → ∞ of the D4- with g being the topological susceptibility. brane action The normalizable solutions will be expanded in mass eigenfunctions, with radial eigenvalue equations   1Following the notation of [9,10,39], g2 differs by a 7 3 YM ∂ r ∂ ð2Þ þ r L 2 ð2Þ ¼ 0 ð Þ factor two from the usual physics convention so that r 3 rCτ M Cτ ; 16 α ¼ 2 ð2πÞ L fðrÞ G s gYM= .

014006-3 JOSEF LEUTGEB and ANTON REBHAN PHYS. REV. D 101, 014006 (2020) Z qffiffiffiffiffiffiffiffiffiffiffi ð2Þð ¼ ∞Þ¼ ð2Þð ¼ X subject to boundary conditions Cτ r Cτ r D8 ˆ 0 S ¼ μ8 AðRÞTr expð2πα F þ BÞ ∧ C ; ð19Þ Þ¼0 ∂ ð2Þð ¼ Þ ≠ 0 CS q rKK , rCτ r rKK . This determines the q D8 mass of the lightest pseudoscalar glueball as MG ¼ 1 885… ˆ . × MKK. where the so-called A-roof genus factor involves AðRÞ¼ 1þ 1 R ∧ Rþ… RMN ¼ 1 MN K L 192π2 Tr with 2RKL dx dx . The B. Inclusion of quarks and Uð1ÞA anomaly term involving C3 contains the Wess-Zumino-Witten term ¼ Sakai and Sugimoto have extended the Witten of the dual because F4 dC3 is nonzero in model by introducing left- and right-handed chiral quarks the background [9,44]. through N pairs of D8 and D8 probe branes localized at The Chern-Simons term is also involved in the f Uð1Þ anomaly of the dual gauge theory, because the separate points on the circle Sτ at the holographic boundary. A C7 term modifies the equations of motion of the C1 field UðN Þ ×UðN Þ →UðN Þ þ f L f R f L R which is responsible for the θ parameter. Using Hodge emerges from the fact that the D8- and D8-branes have to duality, dC7 ¼ F8 ¼ ⋆F2, and integrating by parts, one join in the cigar-shaped background geometry. Choosing finds that antipodal points on Sτ leads to embedding functions with constant x4 ≡ τ and a joining of the branes at Z the tip of the cigar at r (or U ). In this case one can ⊃ μ 2πα0 ðF ∧ Þ KK KK SCS 8 Tr 2 C7 extend the coordinate Z introduced in (5) to the range ZD8 −∞…∞ in order to cover the radial extent of joined D8- 0 ¼ μ82πα TrðA1Þ ∧⋆F2 ∧ ωτ and D8-branes. Z The action for these flavor branes is given by SD8 ¼ pffiffiffiffiffiffiffiffiffi 0 10 D8 þ D8 ¼ μ82πα dx jg10j½δðτÞþδðτ − πÞ SDBI SCS with Z rr ττ μν ττ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi × ðTrðArÞg g ∂rCτ þ g g TrðAμÞ∂νCτÞ; ð20Þ D8 ¼ −μ −Φ e − ð þ 2πα0F þ Þ ð Þ SDBI 8 e Tr det gð9Þ B2 ; 17 D8 where ωτ ¼½δðτÞþδðτ − πÞdτ has been introduced to extend the integration to the entire bulk spacetime. Thus the e μ ¼ð2πÞ−8 −ð9Þ where Tr denotes symmetrized trace, 8 ls , and flavor probe branes induce a mixing term of the Abelian F ¼ dA þ A ∧ A is the field strength tensor for the ˆ −1 part of the flavor gauge field A ≔ N TrA living on the non-Abelian flavor gauge fields living on the D8-branes. f D8-brane with the field Cτ. The linear equations of motion [B2 is the bulk Kalb-Ramond 2-form field; its normalizable modes contain the pseudovector (1þ−) glueballs of the dual for Cτ get an additional term localized on the D8-brane, gauge theory [41,42].]   A 1 r7 pffiffiffiffiffiffi The even and odd radial mode functions of μ are ∂ ∂ ¼ 2πα0μ ∂ ð − rr ττ ðA ÞÞ 2 r 3 rCτ 8 r gg g Tr r associated with the towers of vector and axial- 32κ10 L fields, with the lowest mass eigenvalue (m2 ≈ 0.669M2 ) v1 KK × ðδðτÞþδðτ − πÞÞ being identified as the ρ meson mass such that M ¼   KK 7 949 MeV. 0 r ˆ ¼ 2πα μ8∂ N A The massless pseudoscalar Goldstone bosons are r 24L3 f r described by × ðδðτÞþδðτ − πÞÞ; ð21Þ Z  ∞ a a UðxÞ¼P exp i dZ AZðZ; xÞ ¼ expðiΠ λ =fπÞ; which we solve by introducing the localized fluctuation −∞ ðδÞ Cτ according to ð18Þ ∂ ðδÞ ¼ 4πα0κ2 μ Aˆ ðδðτÞþδðτ − πÞÞ ð Þ 2 ¼ 1 λ 2 λa rCτ 10 8Nf r ; 22 where fπ 54π4 NcMKK and are Gell-Mann matrices 0 −1=2 supplemented by λ ¼ðNf=2Þ 1. Setting fπ ¼ 92.4 MeV □ ðδÞ ¼ 0 fixes λ ≈ 16.63 for Nc ¼ 3. A smaller value of about 12.55 where we have assumed Cτ with respect to ˆ μ would be found by matching instead the large-Nc lattice Minkowski space coordinates and ∂μA ¼ 0. Switching result [43] for the string tension. As in [32] we shall 2 ¼ð Þ6 − 1 to Z r=rKK and with consider the range λ ¼ 16.63…12.55 in order to obtain a pffiffiffi theoretical error band for our quantitative predictions. ˆ 2 1 The 9-dimensional Chern-Simons term of the flavor A ðZ; xÞ¼pffiffiffiffiffiffi η0ðxÞ; ð23Þ Z π 1 þ 2 brane is given by Nf fπ Z

014006-4 WITTEN-VENEZIANO MECHANISM AND PSEUDOSCALAR … PHYS. REV. D 101, 014006 (2020) we thus obtain III. PSEUDOSCALAR GLUEBALL-MESON Z MIXING Z ðδÞ ¼ ∂ ðδÞ ðδÞ Cτ dZ ZCτ With the additional term Cτ ∝ η0 we have solved the 0 pffiffiffi anomalous equations of motion.2 Now we will determine pffiffiffiffiffiffi 2 ¼ 4πα0κ2 μ ð Þη ð Þ the field redefinitions that are necessary to obtain a dia- 10 8 Nf π arctan Z 0 x ˜ fπ gonal action for the Minkowski space fields η0 and G.In × ½δðτÞþδðτ − πÞ: ð24Þ doing this we encounter divergent terms proportional to δð0Þ in analogy to the Horava-Wittenˇ calculation [51].By ðδÞ The “anomalous” part Cτ contributes to the θ adding to the Lagrangian terms beyond the probe approxi- parameter, mation one would presumably be able to cancel these Z Z Z divergences. In the following we will however just drop θ ¼ −1 ¼ −1 ¼ −1 τ∂ ð ð0Þ þ ðδÞÞ δð0Þ ð ðδÞÞ2 ls C1 ls F2 ls drd r Cτ Cτ terms proportional to , i.e., the Cτ term in SR and Sτ cigar ðδÞ pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi the Cτ term in S . 2πα04κ2 μ 2 CS 10 8 Nf Nf Let us start with the effective kinetic terms coming from ¼ ϑðxÞþ pffiffiffi η0ðxÞ¼ϑðxÞþ η0ðxÞ: 2fπls fπ the first part of (13), Z Z ð25Þ 3 ∞ 3 ðkinÞ π 4 r L μν S ¼ − dr d x η ∂μCτ∂νCτ: R 12κ2 M fðrÞ In the presence of flavor branes, the θ parameter is therefore 10 KK rKK no longer given by ϑ alone, but also involves η0. ð29Þ In the remainder of this work we will set the θ para- ð0Þ ðδÞ ð2Þ ð2Þ ˜ meter to 0, which corresponds to nonvanishing ϑðxÞ With Cτ ¼ Cτ þ Cτ þ Cτ and Cτ ðr; xÞ ∝ GðxÞ we according to write them as pffiffiffiffiffiffiffiffiffi Z   ð Þ 1 2Nf kin 4 μ μ ˜ ˜ μ ˜ S ¼ d x ζ1∂μη0∂ η0 þ ζ2∂μη0∂ G − ∂μG∂ G ; ϑðxÞ¼− η0ðxÞ: ð26Þ R 2 fπ ð30Þ The meson field η0 therefore also appears in the non- ð0Þ normalizable mode Cτ , which is a fundamental ingredient where we have fixed the normalization of the radial mode ð2Þ in the realization of the Witten-Veneziano mechanism in the functions in Cτ by requiring WSS model. Z θ π3 ∞ 3 1 Note that, in the chiral case, a constant canp beffiffiffiffiffiffiffiffiffi absorbed r L ð ð2Þ ˜ Þ2 ¼ ð Þ 2 dr Cτ =G : 31 simply in a field redefinition η0 → η0 þ fπθ= 2N , since 12κ ð Þ 2 f 10MKK rKK f r only derivatives of η0 appear in the effective action ζ produced by SD8 . However, introducing mass terms for For the constant 1 which corresponds to a wave function DBI η quarks either through world-sheet instantons or non- renormalization of 0 we obtain Z normalizable modes of bifundamental fields corresponding 3 ∞ 3 π r L ð0Þ ðδÞ ð0Þ 2 to open-string [45–50] produces the additional ζ1 ¼ − dr ðCτ þ 2Cτ ÞCτ =η0 12κ2 M fðrÞ term 10 KK rKK Z Nf ¯ ≕ ζ1; ð32Þ M 4 Nc Lm ∝ d xTrðMUðxÞþH:c:Þ; ð27Þ ¯ where ζ1 is divergent, since it is obtained by integrating where such a redefinition changes the phase of the non-normalizable modes, but ζ1 is suppressed by a factor η quark mass matrix M ¼ diagðmu;md;msÞ according to Nc=Nf compared to the kinetic term for 0 contained in iθ=N D8 ζ M → Me f . SDBI. The constant 2 which is associated with a kinetic Both, in the chiral limit and in the case with nonzero mixing of η0 and pseudoscalar glueball modes is finite and quark masses, the singlet η0 receives an extra mass term that given by is determined by the topological susceptibility of pure 2 ðδÞ Yang-Mills theory obtained in (15), As stated after (22), we had to assume □Cτ ¼ 0 with respect 2 to Minkowski coordinates. With η0 picking up the mass m , this 2 0 2 Nf assumption is violated, but only at higher order in N : m ¼ χ ; ð28Þ f 0 2 g □ ðδÞ ∼ 1=2□η ∝ 3=2 fπ Cτ Nf 0 Nf . (With nonzero quark masses, this seems safe as long as their contributions to the masses of the in accordance with the Witten-Veneziano formula [4,5]. pseudoscalar mesons are much smaller than m0.)

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3 Z 3 π ∞ r L ð0Þ ðδÞ ð2Þ ˜ which yield ζ2 ¼ − dr ðCτ þ Cτ ÞCτ =ðη0GÞ 12κ2 M fðrÞ 10 KK rKK 1 1 sffiffiffiffiffiffi bilin 2 ˜ 2 L ˜ ¼ − ð∂μη0Þ − ð∂μGÞ N η0;G 2 2 ¼ 0 011180… fλ ð Þ   . 33 1 1 Nc þ ∂ η ∂μ ˜ 2ζ ζ þ ζ3 − 2 ð1 þ ζ2Þ ˜ 2 μ 0 G 1 2 2 2 2 MG 2 G for the lowest pseudoscalar glueball. (The next-to-lightest, 1 2 2 2 ˜ 2 2 ζ ¼ −0 014314… − m η − m ζ2η0G þ OðN =N Þ pexcitedffiffiffiffiffiffiffiffiffiffiffiffiffiffi pseudoscalar glueball has 2 . 2 0 0 0 f c λ Nf=Nc .) 1 1 1 ¼ − ð∂ η Þ2 − ð∂ ˜ Þ2 − 2 ð1 þ ζ2Þ ˜ 2 The remaining terms from the background action (13) 2 μ 0 2 μG 2 MG 2 G and the CS action (20) 1 2 2 3=2 3=2 − m0η0 þ OðN =Nc Þ; ð40Þ Z 7 2 f ðmassÞ 1 10 pffiffiffiffiffiffi 1 r 2 ¼ − 4 ð∂ Þ SR 2 d x gS 4 3 rCτ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4κ10 2 L ζ ∝ Z where we have taken into account that 2 Nf=Nc 2 0 10 while ζ1, m0 ∝ N =N . The field redefinitions (39) thus þ μ82πα d xðδðτÞþδðτ − πÞÞ f c pffiffiffiffiffiffiffiffiffi diagonalize the bilinear terms up to and including order rr ττ Nf=Nc, yielding also a small positive contribution to the · jg10jTrðArÞg g ∂rCτ ð34Þ pseudoscalar glueball mass term, give the effective mass terms 2 ¼ð1789 0 Þ2 → ð1 þð0 01118λÞ2 Þ 2 MG . MeV . Nf=Nc MG 1 1 ðmassÞ 2 2 2 ˜ 2 ˜ ¼ð1819 7…1806 5 Þ2 ð Þ S ¼ − m η − M G þ ζ3η0G; ð Þ . . MeV 41 R 2 0 0 2 G 35 for λ ¼ 16.63…12.55. with the values already determined above, 3=2 Dropping all terms of order ðNf=NcÞ and higher, we Z ζ ∞ 7 λ2 see that the divergent coefficient 1 can be ignored and that 2 πV4 1 r ð0Þ 2 Nf 2 m ¼ dr ð∂ Cτ =η0Þ ¼ M (39) reduces to 0 κ2 M 24 L3 r 27π2N KK 10 KK rKK c sffiffiffiffiffiffi ð36Þ ˜ Nf ˜ η0 → η0 þ ζ2G ¼ η0 þ 0.01118 λG; N and (for the lightest pseudoscalar glueball mode) c ˜ ˜ Z G → G: ð42Þ π ∞ 1 7 2 V4 r ð2Þ ˜ 2 2 M ¼ dr ð∂ Cτ =GÞ ¼ 3.5539M : G κ2 24 3 r KK Note that this is very different from the mixing scenario 10MKK rKK L ð Þ proposed by Rosenzweig et al. in [20,21] and also con- 37 sidered in [25]. In that scenario it is assumed that the chiral ðδÞ ð0Þ ð2Þ anomaly is not saturated by η0 alone, but only together with The terms involving ∂ Cτ ∂ ½Cτ þ Cτ cancel by the 3 r r the pseudoscalar glueball field. This leads to a non- equation of motion (22) with the CS term. The mass mixing diagonal mass matrix for η0 and the pseudoscalar glueball η ˜ term 0G vanishes because which can be diagonalized by an orthogonal matrix. The Z Z ∞ ∞ field redefinition (39), on the other hand, is nonunitary. It is 7 ð0Þ ð2Þ ð2Þ ζ3 ∝ drr ∂rCτ ∂rCτ ∝ dr∂rCτ ¼ 0; ð38Þ needed to remove the kinetic mixing term (30),but rKK rKK originally there is no mass mixing term because of (38); a mass mixing term appears after the field redefinition in as already pointed out in [14]. 3 2 3 2 ðδÞ ð2Þ (40), but only at the order N = =N = which is beyond the However, the term involving Cτ Cτ in (33) has f c 4 produced a nontrivial kinetic mixing term of order probe approximation. 1=2 ðNf=NcÞ . To get rid of such mixing terms one has to perform a nonunitary field redefinition. In the present case, 3According to [20], this additional field could however also be 0 one has to make the substitutions interpreted as a radial excitation of η . 4In [22,25,26] a unitary mixing of η0 and pseudoscalar glue- ˜ balls has been considered from a phenomenological point of η0 → ð1 þ ζ1Þη0 þ ζ2G;   view, which has an important impact on the determination of the 1 pseudoscalar mixing angle θ when there is significant η0-G˜ ˜ 2 ˜ P G → 1 þ ζ2 G; ð39Þ mixing. By contrast, the nonunitary transformation (42) that we 2 obtained does not have this effect (to leading order).

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In Ref. [14], where (in view of the vanishing mass TABLE I. Partial decay widths in MeV for the decay of a mixing term) the interactions of the unmixed pseudoscalar pseudoscalar glueball in two vector mesons with λ ¼ 16 63…12 55 glueball were considered, it was found that those are given . . and MG given by the WSS result (41) and also either by pairs of pseudoscalar glueballs interacting with when extrapolated to the lattice prediction of 2600 MeV with scalar or tensor glueballs or by a vertex connecting a reduced mixing (see text). pseudoscalar glueball with η0 and a scalar glueball. The ΓpðMG ¼ 1813 7 MeVÞ ΓpðMG ¼ 2600 MeVÞ latter, which is relevant for the decay of a pseudoscalar ˜ ρρ 36.8…45.0 190…248 glueball, is due to the fact that the integral (38) for the η0G ωω 11.4…13.862…81 mass mixing term no longer vanishes when metric fluctua- KK¯ 2.7…1.8 189…246 tions dual to a scalar glueball are inserted. Assuming that ϕϕ 29…38 this is the dominant vertex, a very narrow pseudoscalar a1a1 3.1…4.0 η η0 P glueball was predicted whose decays had to involve or 51…61 473…618 together with the decay products of a scalar glueball. vv Through the shift (42) a pseudoscalar glueball acquires also all types of vertices that the singlet η0 possesses. In the decay mode arises from the Chern-Simons term and is DBI part of the D8-brane action, such vertices are con- ˜ mediated by the mixing of η0 with the glueball G according tained only in terms involving F to fourth and higher to (42). Before the field redefinition we have the interaction power, which are suppressed by higher powers of α0. In the term chiral limit, the dominant interactions come from the CS Z part of the D8-brane. With quark masses introduced ð2πα0Þ3 D8 according to (27), further interactions arise from (assuming S ⊃ μ8 TrðF ∧ F ∧ F ∧ C3Þ † CS 3! M ¼ M ) Z ð2πα0Þ3 3 2 μνρσ ˆ † † ⊃ μ8 L π ϵ TrðA ∂μAν∂ρAσÞ; ð45Þ TrðMðU þ U ÞÞ → TrðMðU þ U ÞÞ 2g Z qffiffiffiffiffiffiffiffiffiffiffi s −1 ˜ † þ iζ2fπ 2=NfGTrðMðU − U ÞÞ: ˆ with AZ ∝ η0, which after the field redefinition yields the ð43Þ term

D8 ˜ μνρσ The additional term in (43) also contains a bilinear term S ⊃ k1Gϵ Tr∂μvν∂ρvσ; ð46Þ ˜ CS involving G and η0;8,    rffiffiffi  with the coupling constant 2ζ 1 1 2 ð2Þ 2 ˜ 2 2 2 2 Z ΔLm ¼ −pffiffiffiG pffiffiffi m þ mπ η0 − ðm − mπÞη8 : 1 ð2πα0Þ3 3 3 K 2 3 K 3 2 2 k1 ¼ ζ2 pffiffiffiffiffiffi U μ8 L π dZϕ0ψ 1 N KK 2g ð44Þ f s −3=2 −1=2 −1=2 −1 ¼ ζ2 × 877.39λ N Nc M ˜ η f KK It implies a further mixing of G with 0;8 which turns out to −1 −1 −1 0 ¼ 9 8092M N λ 2; ð Þ be negligible as concerns the masses of η, η , and G˜ when . KK c 47 ˜ the mass matrix of the G-η0-η8 sector is diagonalized. Due for the lowest vector meson mode ψ 1. For the lowest axial- to the large mass of the pseudoscalar glueball, the correc- vector mesons with radial mode function ψ 2 and mass tions to the masses of η, η0, and G˜ are only about −0.015%, square eigenvalue m2 ≈ 1.57M2 ≈ ð1190 MeVÞ2, which −0.008%, and þ0.09%, respectively, and thus can be safely a1 KK is quite close to the experimental value of 1230 MeV neglected. Also the vertices induced by the kinetic mixing 5 ˜ of the lightest a1 axial-vector meson, we obtain instead of G and η0 receive only small corrections of the order 1 −1 −1 −2 2 2 ˜ k2 ¼ 4.8888M Nc λ . ζ2m =m ≪ ζ2. However, the small mixing of G with η8 KK K G Performing the polarization sums we obtain the ampli- gives rise to new types of vertices from the DBI action. tude squared In the next section we shall consider the consequences of   all these additional interactions in turn. X 2 jMð ˜ → Þj2 ¼ 8 2 2 M − 2 ð Þ G vv k1M 4 m ; 48 IV. PSEUDOSCALAR GLUEBALL DECAY MODES ϵ1;ϵ2

A. Decay into two vector mesons 5 In Table I we ignore the slightly heavier f1 and K1 axial- The dominant decay channel of the pseudoscalar glue- vector mesons, which would start contributing significantly only ball turns out to be a decay into two vector mesons. This for MG well above 2600 MeV.

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TABLE II. Partial decay widths in MeV for the decay of a pseudoscalar glueball in three pseudoscalar mesons resulting from (49) with λ ¼ 16.63…12.55. The results for G˜ → KK¯ π do not yet include the decays 0 G˜ → KK → KK¯ π) discussed in Sec. IV C, and those for G˜ → PPηð Þ with P a pseudoscalar meson neglect ˜ ð0Þ ð0Þ ˜ G → f0ð1710Þη → PPη . For comparison, the decay modes for an unmixed pseudoscalar glueball via G → 0 f0ð1710Þηð Þ 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).

ΓpðMG ¼ 1813 7 MeVÞ ΓpðMG ¼ 2600 MeVÞ KK¯ π (w/o KK) 0.398…0.387 0.558… 0.550 ¯ KKη (w/o f0ð1710Þη etc.) 0.0263…0.0064 0.1092…0.0285 KK¯ η0 0.3303…0.3570 ππη 0.0048…0.0061 0.0049…0.0061 ππη0 0.0011…0.0007 0.0021…0.0013 ηηη 0.0039…0.0007 0.0337…0.0064 ηηη0 0.0564…0.0277 ηη0η0 0.0047… 0.0048

ð0Þ ð0Þ PPη (w/o f0ð1710Þη ) 0.036…0.014 0.540…0.431

ð0Þ ð0Þ f0ð1710Þη → PPη [14] 0.0068…0.015 2.5…4.3

˜ where M and m are the masses of G and v (and similarly for pfffiffiffiffiffiffiffiffiffiπ ˜ ¯ † ΔLm ¼ iζ2 GTrMðU − U Þ; ð49Þ the heavier vector and axial-vector modes when M is large 2 2Nf enough). The partial widths for decays into the various vector with mesons are listed in Table I for λ ¼ 16.63…12.55 and M¯ ¼ ð 2 2 2 2 − 2ÞðÞ diag mπ;mπ; mK mπ 50 glueball mass MG given alternatively by the WSS result (41) and by an (admittedly speculative) extrapolation to the in the isospin symmetric case mu ¼ md. This contains lattice prediction of 2600 MeV, where we have assumed vertices of the pseudoscalar glueball with KK¯ π, KK¯ ηð0Þ, that the mixing parameter ζ2 and thus k1 (which has inverse ηð0Þ3 ζ 2 2 ηð0Þππ −1 and of the order of 2mK=fπ, and vertices with mass dimension) scales like M when the glueball mass is 2 2 G proportional to ζ2mπ=fπ. raised. (Keeping the mixing parameter as is, the widths for a In Table II we list the resulting partial decay widths for a 2600 MeV glueball would all be a factor of about 2 larger.) pseudoscalar glueball with mass M ¼ 1813 7 MeV Also with the assumed reduction of k1, a 2600 MeV according to (41) and, as above, with mass extrapolated pseudoscalar glueball is thus projected to be a rather broad to 2600 MeV as predicted by quenched lattice QCD. resonance. In chiral perturbation theory, a well-known problem is that the leading-order Lagrangian severely underesti- B. Decay into three pseudoscalar mesons mates the decay η0 → ηππ whose leading-order ampli- 2 0 In Ref. [52], Gounaris and Neufeld have proposed that a tudepffiffiffi is proportional to mπ, to wit, jMðη → ηππÞj ¼ 2 pure pseudoscalar glueball decays predominantly through j2 2 cosð2θ Þ − sinð2θ Þjðmπ=fπÞ =6. This accounts for ˜ ð ð − †ÞÞ P P an interaction Lagrangian involving iGTr M U U in only about 3% of the experimental result. Higher-order order to explain that the pseudoscalar glueball candidate terms in chiral perturbation theory which are not present in ιð1460Þ 6 ¯ π at the time, , seemed to decay mainly into KK . the WSS model have been shown to give contributions ηππ Decays into would thus be suppressed by a factor which are much larger [54,55]. By the same token, the ð Þ4 mπ=mK . partial decay rates of a pseudoscalar glueball into three In Sec. III we have found that such an interaction pseudoscalar mesons that result from its mixing with η0 can Lagrangian is in fact generated in the WSS model with be expected to be strongly underestimated by (49),in η nonzero quark masses by the kinetic mixing of 0 and particular the amplitudes for decay into ηππ, which are pseudoscalar glueball modes. Explicitly, it reads 2 proportional to mπ. The other decay amplitudes are of the 2 order of mK, but since mK is small compared to mG, 6This is now listed as two states ηð1405Þ and ηð1475Þ by the formally higher-order contributions could again be much Data Particle Group [53], of which the lighter one is still more important. However, even if we assume that these considered occasionally as a glueball candidate [37] in view of its supernumerary nature with respect to the , while partial decay widths are underestimated by an order of others question the existence of two separate states and argue for magnitude, they are still much smaller than the decays into a single ηð1440Þ [36]. two vector mesons.

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In Ref. [14], one of us with F. Brünner has calculated the In Table III the resulting partial decay widths are listed. decays of an unmixed pseudoscalar glueball into three While the partial widths for decays of the pseudoscalar pseudoscalar mesons in the WSS model, which necessarily glueball into K, one further vector meson and one involves η0 and a scalar glueball. If the latter is identified pseudoscalar meson are rather small, the partial width ˜ with f0ð1710Þ, which decays predominantly into for G → KK (and subsequently → KKπ) turns out to be (as is predicted [30,31] by the WSS model when scalar comparable to that of the direct decay G˜ → KKπ evaluated 0 glueballs have no or negligible decay width into ηη ), the above. Since we assume that the latter may be under- ˜ ¯ dominant channel is G → f0ð1710Þη → KKη, whereas estimated significantly, we have not carried out a full KKπ decays are excluded. With mixing, additional calculation of G˜ → KKπ combining coherently both ˜ → ηð0Þ η2 amplitudes for G PP arise from Gscalar 0 and amplitudes. ð∂ η Þ2 Gscalar;tensor μ 0 terms in the DBI action and also from In summary, we are finding a rich pattern of decays of the ðδÞ pseudoscalar glueball in three or more mesons, which the additional terms involving Cτ in SR. They would 0 involve vertices which are proportional to pseudoscalar have to be included in a complete calculation of PPηð Þ meson masses. The by far dominant decay modes are decays. To give an idea of the magnitude of the contribu- ˜ 0 0 however given by two vector mesons. tions from G → f0ð1710Þηð Þ → PPηð Þ, in Table II we have included the partial widths of these decay modes V. DISCUSSION AND OUTLOOK as obtained in the unmixed case in [14]. While such contributions are moderate for the pseudoscalar glueball If the mass of the pseudoscalar glueball is as given by mass (41), they become important for MG extrapolated to the WSS model, (41), the pseudoscalar glueball is the mass predicted by lattice which is well above the predicted to be a rather narrow resonance with relative threshold for f0ð1710Þþη decays. However, also these width Γ=M ≈ 0.03…0.04. However, the WSS model contributions are always much smaller than those for the appears to underestimate significantly the mass of heavier decays G˜ → vv. glueballs. The tensor glueball comes out as 1487 MeV, while it is predicated as around 2400 MeV by both quenched and unquenched lattice QCD [28,29,33–35]. C. Decay into one pseudoscalar meson together Extrapolating our results to a significantly higher pseu- with one or two vector mesons doscalar glueball mass as indicated by (quenched) lattice The very small mixing of a pseudoscalar glueball with η8 QCD, 2.6 GeV, the prediction is instead that of a very that is induced by the quark mass term (44) also gives rise broad resonance. In Fig. 1 we display two possible to vertices with one pseudoscalar meson and one or two extrapolations, one with unchanged vertices and mixing μ vector mesons due to the terms involving Trð∂μΠ; ½V ; ΠÞ when the glueball mass is increased in the final formulas, μ and Trð½Vμ; Π½V ; ΠÞ in the Yang-Mills part of the effec- which leads to an extremely large total width, and the tive action. When the glueball mass is above the respective somewhat more moderate scenario (underlying the above threshold, this gives rise to decay modes tables) where the mixing parameter ζ2 is assumed to be

˜ 0 G → KK ; πK K ;KK ρ;KK ω;KK ϕ; ηð ÞK K ; ð51Þ 0.5

¯ ¯ where KK is short for KK ;KK etc. The corresponding 0.4 ζ ð 2 − 2Þ 2 ∼ 10−2 amplitudes are proportional to 2 mK mπ =mG −1 2 −1=2 = −1 −1 0.3 times a factor gρππ ∝ λ Nc or gρρππ ∝ λ Nc . /M

0.2 TABLE III. Partial decay widths in MeV for the decay of a pseudoscalar glueball into one pseudoscalar meson and one or two vector mesons. 0.1 Γ ð ¼ 1813 7 Þ Γ ð ¼ 2600 Þ p MG MeV p MG MeV 0.0 1.8 2.0 2.2 2.4 2.6 KK 0 381…0 288 0 302…0 225 . . . . M[GeV] πKK 0.0113…0.0112 KKρ ð2.50…2.47Þ × 10−3 FIG. 1. Relative pseudoscalar glueball decay width Γ=M for −3 λ ¼ 16 63 KK ω ð0.799…0.787Þ × 10 . (blue) and 12.55 (orange) with two different extrap- KKϕ ð0.253…0.249Þ × 10−3 olations to larger masses: same vertices and mixing (full lines) or ζ ηKK ð0.097…0.026Þ × 10−3 when the mixing parameter 2 is assumed to be decreasing like MWSS=M (dashed lines).

014006-9 JOSEF LEUTGEB and ANTON REBHAN PHYS. REV. D 101, 014006 (2020) decreasing like MWSS=M. (The latter extrapolation could vertices can be derived from the A-roof genus also be interpreted as a corresponding increase of the factor in (19) as in [61].Thisleadsto8 mass parameter M which sets the scale of the glueball KK L ⊃ ζ ˜ ϵμνρσ½κ ∂ P ∂ Pα masses.7) CS 2G a ν αμ σ ρ The rather large decay width of a pseudoscalar glueball μνρσ α β β þ κbϵ ∂ν∂ Pμβð∂σ∂ Pαρ − ∂σ∂αPρÞ ð52Þ to two vector mesons that we obtained from the WSS ζ κ ζ κ 3 ∝ λ−1=2 −2 model also implies a significant coupling to with 2 aMKK, 2 bMKK Nc . The results for the because of the vector meson dominance that is inherent unmixed case of Ref. [32] instead implied that central in the WSS model [10]. This also applies to the scalar and exclusive production of pseudoscalar glueballs required the ˜ ˜ ηð0Þ ˜ ˜ tensor glueballs studied before in [32]. Radiative glueball formation of pairs G G, G,orGscalar;tensorG with vertices decays will be studied in detail in forthcoming work, as −1 −2 1=2 −5=2 −1 −3 of order λ N , N Nc ,orλ N N , respectively. well as the contribution of glueballs to hadronic light-by- c f f c The possibility of production of single pseudoscalar glue- light scattering, which is of relevance to the theoretical balls of course lowers the threshold significantly. prediction for the g − 2, where holographic QCD has Similarly, the kinetic mixing of the pseudoscalar glueball been argued to provide a promising framework for quanti- with η0 allows the production of single pseudoscalar glue- tative predictions [60]. balls in radiative J=ψ decays with rates of the parametric An important implication of the mixing-induced order N−1 times that of the decay rate for J=ψ → γη . vertices determined in this work is that contrary to c c However, the (semi-)quantitative results that we obtained the unmixed case [32] the WSS model predicts sizeable within the WSS model suggest that the experimental vertices for the production of single pseudoscalar glue- identification of the pseudoscalar glueball may be made balls in double diffractive scattering. While Regge difficult by a very large decay width if its mass is around regime hadronic scattering requires extensions and 2.6 GeVas indicated by (quenched) lattice results. In future extrapolations of the WSS model [61–63],thestructure work we intend to further explore these predictions in the of the vertices with Reggeons and are context of radiative J=ψ decays and central exclusive completely determined by the Chern-Simons action production in the Regge regime. and the mixing of the pseudoscalar glueball with η0. For Reggeons, the vertices have the form (46),while ACKNOWLEDGMENTS

7 We thank Francesco Bigazzi for useful discussions. J. L. The standard choice [9,10] MKK ¼ 949 MeV corresponds to using the ρ meson mass as input, which leads to a good match of has been supported by the doctoral programme DKPI of the hadronic decay widths of ρ and ω mesons for the range of λ Austrian Science Fund FWF, Project No. W1252-N27. considered here [32]. Lattice results for the glueball spectrum clearly favor larger values of MKK. The early studies of 8 in the WSS model [56,57] instead seemed to require M much Up to partial integrations, the tensorial structure of the two KK terms in (52) is in fact in one-to-one correspondence to the two smaller than 949 MeV [57], but more refined treatments [58,59] 0 00 have found a good fit of masses with the standard choice couplings gPPM˜ and gPPM˜ for two tensor pomerons with one of 949 MeV. pseudoscalar meson in [64].

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