Glueball Decay in Holographic QCD Seiji Terashima (YITP, Kyoto) Based on the work (arXiv:0709.2208) in collaboration with Koji Hashimoto (Riken) and Chung-I Tan (Brown) 2007 Dec. 18 at NCTS

1. Introduction

Excitations in QCD Mesons and Baryons found and identified in the experiments. Lattice QCD result and other theoretical result are consistent with those. Glueballs (exist in any (confined) gauge theory) ex. 2-point function <Tr (F^2)(x) Tr(F^2)(y) > Lattice QCD calculation predicted their spectra.

The (glueball) spectra of SU(3) Yang-Mills Lattice gauge theory from Morningstar-Peardon

Excitations in QCD Mesons and Baryons found and identified in the experiments. Lattice QCD result and other theoretical result are consistent with those. Glueballs (exist in any (confined) gauge theory) ex. 2-point function <Tr (F^2)(x) Tr(F^2)(y) > Lattice QCD calculation predicted their spectra. However, they are not confirmed by experiment although candidates for the glueballs are found.

Candidates for glueballs with J=I=0, P=C=+ (i. e. no charge Candidates for glueballs with J=I=0, P=C=+ (i.e. no charge. lowest mass.) A speculation is that f0(σ) : artifact of final state interaction f0(980): K K molecule f0(1370), f0(1500), f0(1710) are glueball and 2 scalar mesons. f0(1370) has 2-photon decay, and f0(1710) has large KK branching ratio. → f0(1500) might be the glueball. Not confirmed. Other possibilities. — — from Seth (2000), PDG, Armstrong et. al., Amsler

Pseudo scalar nonet and Scalar nonet (Nf=3) — — Pseudo scalar nonnet (=octet+singlet) ψγψ Scalar nonnet (=octet+singlet) ψψ 5 I³ = -1, - ½ , 0, ½, 1 I³ = -1, - ½ , 0, ½, 1 — — — — I= ½ , S=-1 d s (K0) u s (K-) I= ½ , S=-1 d s (K0*) u s (K-*) — — — d u (π-) — d u (a0-) — — I=1, S=0 — u d (π+) I=1, S=0 — u d (a0+) uu-dd/√ (π0) uu-dd/√ (a0 ) 2 2 — — — — s u (K+) s d (K0) s u (K+*) s d (K0*) I= ½ , S=1 I= ½ , S=1 — — — — — — — — uu+dd-2ss/√ (η) uu+dd-2ss/√ (f0) I=0, S=0 6 I=0, S=0 6 — — — — — — I=0, S=0, Singlet of SU(3) uu+dd+ss/√ (η’) I=0, S=0, Singlet of SU(3) uu+dd+ss/√ (f0) 6 6

Why it is difficult to identify the Glueballs? There are several mesons which have same charges and roughly same mass as the glueballs. → The branching ratio are needed to distinguish them. Experimentally we do not know much about the branching ratio of the glueball candidates. → We expect that LHC will give us a huge amounts of hadoronic data and improve the experimental situation drastically. However, on the theoretical side, it is very difficult to compute reliably couplings of glueballs to ordinary mesons in QCD. Actually, no reliable computations ever have done. → We need a way to compute the glueball decay reliably!

Problems for existing methods to compute glueball decay Chiral Lagrangian approach: The glueballs have relatively heavy (heavier than 1500 MeV). Thus no control by the derivative expansion. Moreover, glueballs are siglets of the flavor symmetry. Lattice QCD: Spectrum is easy. To compute the decay rate is possible, but very difficult because Lattice QCD is defined on the Euclidean space-time. Usual large N expansion: Weak t’Hooft coupling is needed to compute explicitly the decay rate. But confinement can not be seen by the weak coupling expansion. Thus, the Holographic QCD will be useful ! (though not so reliable now)

→ We explicitly compute the couplings between glueballs and mesons by using holographic QCD. Holographic QCD　=　application of AdS/CFT to QCD studies. AdS/CFT → large N gauge theory at strong t’Hooft coupling(g^2 N) = classical higher dimensional gravitational theory. This has been applied to (i) Glueball spectrum in large N pure Yang-Mills theory by D4-branes compactified on a circle. (ii) Meson spectrum/dynamics in large N QCD by adding D6-branes or pair of D8-anti D8 branes to D4-branes. We combine (i) and (ii) to compute glueball decay in large N gauge theory.

Summary of the result Decay of any glueball to 4 is suppressed. Prediction of the holographic QCD! “Vector meson dominance” for the glueball decay. (No direct 4 pion decay.) Decay of glueball to a pair of photons is suppressed. Mixing of the lightest glueball with mesons is small. The decay widths and branching ratios is consistent with the experimental data of the glueball candidate f0(1500).

Plan of the talk: Introduction Review of the Holographic QCD Glueball interaction in Holographic QCD Decay of lightest scalar glueball Conclusion

2. Review of Holographic QCD

Original AdS/CFT correspondence Low energy limit of the N D3-branes in IIB superstring theory → Supersymmetric and conformal, not like QCD Consider type IIA superstring theory compactified on S^1 and N D4-branes wrapping the S^1 with anti-periodic boundary conditions for fermions. (Witten) Then, we have (bosonic) pure 4-dim. Yang-Mills theory in the low energy limit. Close to QCD.

Gravity dual of the D4-brane on S^1 type IIA string = M-theory on S^1 D4-brane = M5-brane on S^1 type IIA string on S^1= M-theory on torus D4-brane wrapping S^1=M5-brane on torus Gravity dual = Near horizon limit of the M5-branes solutions in the IIA supergravity = doubly Wick-rotated AdS7-blackhole. (Euclidean time → anti-periodic b.c. for fermions) μ,ν run from 0 to 4. L and R are the parameters. x^4 is the M-theory circle. where τ= τ+

The metric fluctuations corresponds to glueballs in Yang-Mills theory. S^4 part is not expected to be necessary in the following, so integrating out it. Then we have 7-dimensional action: The metric fluctuations corresponds to glueballs in Yang-Mills theory. For example, the lightest state is the following fluctuations: g=ḡ+h Constable-Myers Brower-Mathur-Tan where is the glueball filed in the 3+1 dimension ( ) and is the mass squared of the glueball. The fluctuations does not depends on τ and x^4.

H(r) was given by the equation of motion which is written by new dimensionless coordinate Z: where and The boundary condition should be

In order to compute the glueball decay, not just spectra, we need to know the normalization of H(r), such that We computed the normalization of the H(r) numerically where we used the relations between IIA 10d sugra and Yang-Mills theory:

τ is periodic and its Kaluza-Klein mass is The solution of 11d sugra is equivalent to the solution of IIA 10d sugra. In the Sakai-Sugimoto notation, the solution of IIA sugra is μ,ν run from 0 to 3. which is equivalent to the previous solution by the following identification: τ is periodic and its Kaluza-Klein mass is

Other glueballs from 11d-sugra fields from Brower-Mathur-Tan (2000) (Note that the dilaton does not correspond to the lightest glueball.)

from Brower-Mathur-Tan (2000)

Comparison between the holographic and lattice calculation of glueball spectra Lattice (SU(3) gauge group) Holographic from Morningstar-Peardon from Brower-Mathur-Tan (2000) (We have dropped state)

Adding quarks in AdS/CFT: holographic QCD Adding Nf flavors → adding another kind of D-branes as probe. Here, we add Nf pairs of D8-brane and anti-D8-brane. This model has spontaneously broken chiral symmetry, so there is massless pion. Let us consider gravity dual, i.e. the D8-branes in the Witten’s background. (D8-brane and anti-D8-brane are connected and become smooth curved D8-branes as a result of the curved background.) The D8-brane action is Karch-Katz Myers et.al. Sakai-Sugimoto where F is the field strength of the 9-dim. gauge fields on the Nf D8-branes.

Integrating out the S^4 part, we have (Chern-Simons term is not relevant), Then the massless pions and the ρ-mesons appears as the lowest modes of KK-decomposition along z-direction: where Above pions and ρ-mesons are Nf xNf matrices. We will consider Nf=2 case in the followings.

So far, we have ignored (Nf x Nf) scalars corresponding to the transverse direction of the D8-brane. → They are the scalar nonet (for Nf=3) including chageless scalars. They will mix the lowest glueball. If this mixing is large, we have trouble to identify the glueball.

3. Glueball interaction in Holographic QCD

Computation of the interaction We would like to compute the couplings between the glueballs and mesons. In the gravity dual, they correspond to the supergravity fluctuations and the Yang-Mills fluctuations on the D8-branes, respectively. These two sectors are coupled in the combined system of supergravity plus D8-branes. The couplings between them are only in the D8-brane action through the background metric and dilaton, which includes the fluctuations corresponding to the glueball. We substitute the fluctuations of the supergravity fields (corresponding to the glueball) and the D8-brane massless fields (mesons) into the D8-brane action and integrate over the extra dimensions, to obtain the desired couplings in 4d. Basically just evaluate the D8-brane action. Very simple.

Generic feature of holographic glueball decay Glueballs are obviously flavor-blind. Thus couplings to mesons are universal against flavors. From the D8-brane action, we see that (1) No glueball interaction involving more than two pions. because Decay of any glueball to 4 is suppressed. Prediction of the holographic QCD! (2) Direct couplings of a glueball with more than five meson are suppressed. (implies “vector meson dominance”): No These are from “Holographic gauge” choice

Interaction of the lightest scalar glueball First, we rewrite the metric fluctuations in the 10d IIA sugra fields: Substituting them into the D8-brane action, we have where we have kept only the relevant terms for the decay of glueballs. The constants c are calculated numerically as

Mixing of glueball with mesons In large N expansion, we know the mixing between G (glueball) and X (meson) is . But we want to know the dependence on the t’Hooft coupling also. Actually, our leading order caluculation in the holographic QCD show that it is just : It is suppressed in large N limit. However, for a generic glueball, direct decay process is comparable to the decay through mixing.

As we have seen, the direct glueball decay: vs Direct meson decay: x times the mixing

But, we can show that no mixing between lightest glueball and meson Scalar mesons = transverse scalar of D8-branes, denoted by y, which is essentially τ. Terms linear in y in the D8-action is: All of these vanish for the lightest glueball. No mixing with mesons at order This is very important to distinguish the glueball and meson.

4. Decay of lightest scalar glueball

Lightest glueball mass is We have . Thus no 2 ρ-meson decay in holographic QCD (In the experiment, M=1507MeV, mρ=775MeV) We will use

Possible decay process (from kinematics) Branching ratio for f0(1500): (a) 35% (b)+(c) 49% (d) 7 %

From the effective action we have, we can compute the decay width. For Experimentally, Good agreement.

and Thus Experimentally, Consistent This is too small, but if we set to the experimental value by hand, we have Thus Experimentally, Consistent (In particular, taking into accont the masslessness of the pions)

5. Conclusion

First attempt in computing decays of glueballs to mesons using a holographic QCD (Sakai-Sugimoto model). The holographic QCD is, in principle, equivalent to QCD. We therefore expect that the holographic approach should provide interesting information on strong coupling physics of QCD. Explicit couplings between the lightest glueball and the mesons are given, and the associated decay products/widths are calculated. Our results are consistent with the experimental data of the decay for the f0(1500) which is thought to be the best candidate of a glueball in the hadronic spectrum. We have shown that there is no mixing with the mesons at the leading order. Decay of any glueball to 4 is suppressed. This is a prediction of the holographic QCD!

Other interesting directions • Multi-glueball couplings. Self-couplings of the glueballs can be computed in the supergravity sector. Emission of mesons from a propagating glueball can be described by the D8-brane action similarly. • Universally narrow width of glueballs. If one can show in the holographic QCD that the total decay width of any glueball state is narrow, that would provide support for this widely-held belief. We have shown the narrowness only for the lightest glueball. • Other glueballs. For example, the (J=1,P=+,C=−) glueballs reside in the NS-NS 2-form field, and it should have a large mixing with the meson fields because F always appeared in the action as a combination, F+B. • Thermal/dense QCD. • Computation of glueball couplings in other models of holographic QCD. For example, the flavor D6-branes enable one to introduce easily the quark mass.

Fin.