Nonchiral enhancement of scalar glueball decay in the Witten-Sakai-Sugimoto model
aa r X i v : . [ h e p - ph ] J u l Nonchiral enhancement of scalar glueball decay in the Witten-Sakai-Sugimoto model
Frederic Br¨unner and Anton Rebhan
Institut f¨ur Theoretische Physik, Technische Universit¨at Wien,Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria (Dated: July 31, 2015)We estimate the consequences of finite masses of pseudoscalar mesons on the decay rates ofscalar glueballs in the Witten-Sakai-Sugimoto model, a top-down holographic model of low-energyQCD, by extrapolating from the calculable vertex of glueball fields and the η ′ meson which followsfrom the Witten-Veneziano mechanism for giving mass to the latter. Evaluating the effect on therecently calculated decay rates of glueballs in the Witten-Sakai-Sugimoto model, we find a strongenhancement of the decay of scalar glueballs into kaons and η mesons, in fairly close agreement withexperimental data on the glueball candidate f (1710). PACS numbers: 11.25.Tq,13.25.Jx,14.40.Be,14.40.Rt
The fundamental theory of the strong interactions,quantum chromodynamics (QCD), which has quarks con-fined in color-neutral bound states, admits also boundstates whose valence constituents are all gluons, the non-abelian gauge bosons of QCD. This prediction of addi-tional mesons called gluonia or glueballs dates back tothe early 1970’s [1] and has been substantiated by latticeQCD [2], which estimates the mass of the lowest glueballstate to be around 1600–1800 MeV. Experimentally, how-ever, their status remains unclear and controversial [3].The lowest scalar glueball state has quantum numbersof the vacuum and can be expected to mix with scalarmesons made from quarks and antiquarks. To disentan-gle the contributions, information on decay processes isneeded. Theoretical expectations vary greatly—the low-est glueball state may be even so broad that it forms amere background for the isoscalar meson spectrum [4].QCD in the limit of a large number of colors N c [5, 6],which in many cases turns out to be a remarkably suc-cessful approximation to real QCD with N c = 3, predictsa parametric suppression of decay rates of glueballs com-pared to light quarkonia by a factor 1 /N c as well as asuppression of mixing [7]. If glueballs are indeed narrowand not strongly mixed, one should be able to identify oneof the isoscalar-scalar mesons below 2 GeV as a predom-inantly glueball state. In phenomenological studies theexperimentally well-established [8] mesons f (1500) and f (1710) have been identified alternatingly as possibleglueball candidates [9–11]. Both are comparatively nar-row states, but their decay patterns are rather different: f (1500) decays primarily into four pions and secondlyinto two pions, with decays into kaons and η mesonssuppressed, whereas f (1710) instead decays predomi-nantly into two kaons, with a ratio [8] Γ(2 π ) / Γ( K ¯ K ) =0 . +0 . − . , much lower than 3:4 expected from a flavor-blind glueball. In the case of f (1500) the strong devi-ation from flavor-blindness is usually attributed to mix-ing, while for f (1710) it has been suggested that glue-balls couple more strongly to the more massive pseu-doscalar mesons, a mechanism termed “chiral suppres- sion” [12, 13], which could make it possible that f (1710)is a nearly unmixed glueball as most recently argued forin [10, 11] (see also Ref. [14]).Since lattice QCD results on glueballs in interactionwith quarks are still sparse, in particular concerning de-cay patterns, it is of interest to employ (top-down) gauge-gravity duality, a string-theoretic approach to studystrongly coupled large- N c gauge theories, to obtain newinsights from first principles [15]. In fact, the spectrumof glueballs has been one of the first applications of anonsupersymmetric holographic model derived by Wit-ten [16] from type-IIA superstring theory [17–19]. TheWitten model has subsequently been extended by Sakaiund Sugimoto to include chiral quarks through D8-D8probe branes [20, 21]. With only one free coupling con-stant at a (Kaluza-Klein) mass scale M KK ∼ f (1500), although the mass ofthe lowest holographic glueball mode comes out at 855MeV. In Ref. [24] we have recently revisited this calcu-lation with the result that the decay width of the lowestmode is much higher than the one obtained in [23]. Sincethe lowest mode corresponds to an “exotic polarization”[18] of the gravitational fields of the Witten model, wehave proposed to discard the latter and to instead con-sider the next-lowest, predominantly dilatonic mode withmass 1487 MeV as corresponding to the glueball in QCD.Despite the closeness of its mass to that of the f (1500)meson, we have found that the decay pattern into twoand four pseudoscalar mesons is not reproduced: the de-cay rate of f (1500) into two pions is underestimated byabout a factor of 2, while the prediction for the domi-nant decay mode of f (1500), which is decay into fourpions, is an order of magnitude too small. Extrapolatingthe mass of the holographic glueball to that of the glue-ball candidate f (1710) (which is within 16% of the massof the dilatonic mode) [25], we have instead found closeagreement with the decay rate into two pions. Since theWitten-Sakai-Sugimoto model is chiral, pions, kaons, and η mesons are predicted simply in flavor-symmetric ratios3:4:1, thus failing to explain the much stronger decaysinto two kaons and two η mesons.In this Letter we study the possible effect of finitequark masses on the decay rate of glueballs in the Witten-Sakai-Sugimoto model in order to see whether a suffi-ciently strong enhancement of the decay into kaons andalso η mesons could result. In Refs. [26, 27], it hasbeen shown that nonlocal mass terms implementing Gell-Mann-Oakes-Renner relations can be induced by eitherworldsheet instantons or a deformation by a bifundamen-tal field related to the open string tachyon that arisesbetween (parallel) D and anti-D branes. Since no com-plete calculation along these lines exists, the additionalcoupling of glueballs and pseudoscalar mesons induced bythe nonlocal mass term is not known. However, the addi-tional coupling of glueballs to η ′ mesons due to the part ofits mass term which arises from the anomalous breakingof U(1) A flavor symmetry can be calculated exactly. Wepropose to use that as a simple and also plausible modelfor how the totality of nonlocal mass terms for the pseu-doscalar meson depend on the scalar glueball fields, andto thereby extrapolate the results for the glueball decaypattern obtained in [24] to finite pseudoscalar masses.In the chiral Witten-Sakai-Sugimoto model [20, 21],the coupling of pseudoscalar Goldstone bosons as wellas vector and axial vector mesons to scalar and tensorglueball fields are determined by the dependence of theDirac-Born-Infeld part of the D8 brane action on metricand dilaton fields, S DBID8 = − T D8 Tr Z d xe − Φ p − det (˜ g MN + 2 πα ′ F MN ) , (1)where Φ and ˜ g MN are the dilaton and the nine-dimensional induced metric in the ten-dimensional back-ground given by ds = (cid:16) uR (cid:17) / (cid:2) η µν dx µ dx ν + f ( u ) dx (cid:3) + (cid:18) Ru (cid:19) / (cid:20) du f ( u ) + u d Ω (cid:21) , f ( u ) = 1 − u u , (2) e Φ = g s (cid:16) uR (cid:17) / , (3)with circle-compactified x ≃ x + 2 π/M KK and M KK = u / R − / . The stacks of N f D8 and anti-D8 branesare assumed to be localized at antipodal points, givingrise to trivial embeddings x = const. , which extend fromthe holographic boundary at u = ∞ to the minimal point u KK where branes and antibranes connect. This breaksthe chiral group U( N f ) L × U( N f ) R down to its diagonal group, leading to a nonet (for N f = 3) of pseudoscalarGoldstone bosons described by U ( x ) = P exp i Z ∞−∞ dzA z ( z, x ) = e i Π a λ a /f π , (4)where z/u KK = p ( u/u KK ) − f π to 92.4 MeV and the mass ofthe lowest vector meson mode A µ ( z, x ) = ρ µ ( x ) ψ (1) ( z )to the ρ meson mass m ρ ≈
776 MeV fixes M KK = 949MeV and λ = g N c = 16 .
63 [20, 21]; matching in-stead m ρ / √ σ with σ the string tension of the model tolarge- N c lattice results [22, 24] gives a somewhat lowervalue of the ’t Hooft coupling, λ = 12 .
55, which we usewith the higher value to give a band of variation for theholographic predictions.In Ref. [20, 28] (see also [29]) it was shown thatthe U(1) A anomaly requires to combine the Ramond-Ramond 2-form field strength F with the isoscalar η that is localized on the D8 branes in a gauge-invariantcombination ˜ F with bulk action S C = − π (2 πl s ) Z d x √− g | ˜ F | (5)with ˜ F = 6 πu M − u θ + p N f f π η ! du ∧ dx , (6)where θ is the QCD theta angle and η ( x ) = f π p N f Z dz Tr A z ( z, x ) . (7)This gives rise to a Witten-Veneziano [30, 31] mass termfor η that is local with respect to the effective 3+1-dimensional boundary theory but nonlocal in the bulk,with mass squared m = N f π N c λ M . (8)For N f = N c = 3, M KK = 949 MeV, and λ varied from16.63 to 12.55 one finds m = 967 . . .
730 MeV.The other pseudoscalar mesons described by (4) aremassless in the Witten-Sakai-Sugimoto model. Currentquark masses can in principle be introduced through adeformation by a bulk field T in the bifundamental rep-resentation of the chiral symmetry group [26] that is re-lated to tachyon condensation or alternatively throughworlsheet instantons [27]. Both introduce nonlocal massterms for the pseudoscalar mesons, which one may qual-itatively write as Z d x Z ∞ u KK du h ( u ) Tr (cid:16) T ( u ) P e − i R dzA z ( z,x ) + h.c. (cid:17) , (9)where h ( u ) includes metric fields. Choosing appropriateboundary conditions for T , the quark mass matrix arisesthrough Z ∞ u KK du h ( u ) T ( u ) ∝ M = diag( m u , m d , m s ) , (10)thereby realizing a Gell-Mann-Oakes-Renner relation.Integration over u leads to mass terms for all Goldstonebosons including one for the flavor singlet η in additionto the Witten-Veneziano mass term. The flavor octet η and η can be diagonalized to mass eigenstates η and η ′ . With M = diag( m, m, m s ), m = ( m u + m d ) /
2, fixing m π = 140 MeV and m K = 497 MeV, this diagonalizationyields for λ = 16 . . . . . m η = 518 . . .
476 MeV , m η ′ = 1077 . . .
894 MeV , (11) θ P = − . ◦ . . . − . ◦ , (12)with θ P the octet-singlet mixing angle, which shows thatthe above holographic result for m is in the right ball-park [32].In order to determine how mass terms affect the cou-pling of glueballs and mesons worked out in Ref. [24],we would need to know the dependence on dilaton andmetric fields of h ( u ) as well as the profile of the bifun-damental field T ( u ). Absent those, we turn to the fullyknown nonlocal mass term produced by (5) and (6) for η . Inserting the mode expansion of glueball fields G D and G E defined in Ref. [24], we find S eff .η = − Z d x m η (1 − d G D + 5˘ c G E ) + . . . (13)with d = 3 u Z ∞ u KK H D ( u ) u − du ≈ . λ / N c M KK , (14)˘ c = 34 u Z ∞ u KK H E ( u ) u − du ≈ . λ / N c M KK , (15)where the latter is given for completeness only, since weare going to discard the “exotic” mode G E given theresults in [24]. Here H D,E ( u ) are the radial profile func-tions of the glueball modes, normalized such as to give acanonical kinetic term for G D,E ( x ).Given the similarity to how a nonlocal mass termis generated through worldsheet instantons, this resultseems to be a reasonable first guess of how nonlocalmass terms couple in general. As far as a bifundamen-tal field T associated with tachyon condensation is con-cerned, a plausible guess would be that the metric depen-dence derives from the integration measure of D8 branes, d x e − Φ √− ˜ g. For the predominantly dilatonic glueballfield [33], this turns out to have exactly the same depen-dence on terms linear in G D ( x ) H D ( u ) as follows from (5)and (6), namely a factor (1 − G D ( x ) H D ( u )). In order tocalculate the coupling constant analogous to d in (14), one would need to know the holographic profile of T , ofwhich we only know that it will be concentrated around u = u KK . As a simplistic guess one could try a func-tion that mimics the profile of the term A z ∂ µ A z in theD-brane action when A z equals the zero mode describingthe Goldstone bosons. This would simply determine theanalog of d to be equal to the coupling d appearing inthe chiral G D ππ term [24], L chiral G D ππ = 12 d Tr ( ∂ µ π∂ ν π ) (cid:18) η µν − ∂ µ ∂ ν M (cid:19) G D , (16)where d ≈ . λ − / N − c M − . This differs from d by a mere 4%.We shall therefore continue with the working hypoth-esis that the overall coupling of the glueball field to themass term for the pseudoscalar mesons is universal. Thisessentially assumes that the mixing of singlet and octetmesons is invariant under a holographic renormalizationgroup evolution, which in particular implies the absenceof a direct coupling Gηη ′ .With this assumption, i.e. adding − X i m i P i (1 − d G D ) (17)with P i the mass eigenstates of the pseudoscalar mesonsto (16), we obtain the following modification factor forthe decay rate to two pseudoscalar mesons of mass m P : (cid:18) − m P M (cid:19) / (cid:18) α m P M (cid:19) (18)with α = 4(3 d /d −
1) = 8 . . . . for G D . (19)An analogous calculation for the “exotic” scalar glueballusing the results of [24] for the chiral contributions gives α = 4(5˘ c − ˘ c ) / ( c + 2˘ c ) = 2 . . . . for G E . (20)[Note that to leading order the dependence on λ and M KK drops out in (19) and (20).] In (18) the first fac-tor represents a simple kinematical suppression which isovercome by the coupling of the glueball field to the massterm of the pseudoscalar fields. A similar result, but with α = 1 was obtained in Ref. [34] for a simple effective fieldtheory where the scalar glueball field is identified with thedilaton of QCD (a scalar field with a potential matched tothe QCD trace anomaly). With α = 1 the nonchiral en-hancement is cancelled by the kinematical suppression toorder m P /M , thus restoring approximate flavor symme-try, while for larger values m P the net effect is a (slight)reduction of the decay rate.In Table I we compare the deviations from flavor sym-metry as they are reported by the Particle Data Group[8] with the modification resulting from (18) and (19).Remarkably, the experimental ratio Γ( ππ ) / Γ( K ¯ K ) is re-produced within the experimental error bar, whereas theprediction for Γ( ηη ) / Γ( K ¯ K ) remains within 1.33 stan-dard deviations.In Table II we compare our complete set of predic-tions for the decay rates for a scalar glueball with masscorresponding to either 1505 MeV ( f (1500)) or 1722Mev ( f (1710)), with and without inclusion of massesfor the pseudoscalar mesons, to experiment. In the caseof f (1500), the (experimentally well-known) decay pat-tern is neither matched qualitatively nor quantitatively[24]. Inclusion of the pseudoscalar masses helps for thetotal width but modifies the decay pattern adversely.For f (1710), branching ratios are less accurately known.The prediction for the total width is increased slightlyabove the experimental value when masses are included,but the decay pattern into two pseudoscalar mesons isimproved markedly, as we have already shown in TableI. In fact, the experimental results quoted for the partialwidths should be considered as upper values as they ig-nore the possibility of decay into four or more pions, sinceat least decay of f (1710) into two ω mesons and furtherto six pions has been seen [8] in J/ψ → γf (1710) (in factat the level of 75% of the rate into two pions [8], so thatthe holographic prediction may not be very far off) [35]The only remaining major mismatch between existing ex-perimental data for f (1710) [36] and the prediction ofthe Witten-Sakai-Sugimoto model thus appears to be therather high rate for decay into four pions, which is pre-dicted by the latter to proceed through 2 ρ and ρππ atthe level of about twice the rate into two ω mesons [24],and this prediction is not modified by the introductionof quark masses at the level of our approximation.To summarize, by extrapolating the exactly calculablecoupling of scalar glueballs to the mass term of the isos-inglet pseudoscalar meson in the originally chiral Witten-Sakai-Sugimoto model, we found a significantly enhanceddecay of scalar glueballs into kaons and η mesons com-pared to flavor-symmetric ratios. This is in line withthe previously proposed mechanism of “chiral suppres-sion” of scalar glueball decay which has been posited asexplanation how the isoscalar meson f (1710) with itspreferred decay into two kaons could be predominantlygluonic rather than an s ¯ s state [12, 13]. From this weconclude that the top-down holographic Witten-Sakai-Sugimoto model may well be consistent with a glueballinterpretation of f (1710) while disfavoring the otherpopular glueball candidate f (1500). In this case, thesuccessful reproduction of the branching ratios given inTable I is correlated to a sufficiently small rate for thedecay G → ηη ′ , for which only upper limits exist so far[37]. Moreover, the Witten-Sakai-Sugimoto model pre-dicts significant partial widths for the decay of f (1710)into four and six pions, of which so far only the latterhave been confirmed experimentally according to [8].We thank Denis Parganlija and Timm Wrase for many f (1710) exp.(PDG) WSS massive · Γ( ππ ) / Γ( K ¯ K ) 0.55 +0 . − . · Γ( ηη ) / Γ( K ¯ K ) 1.92 ± .
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