Predictions by string theory? f 0 (1500) → 4π 0 : Suppressed Charge radius of Roper = 0.73[fm] [w/ C.I.Tan and S.Terashima, ] [w/ T.Sakai and S.Sugimoto, to appear]

ID of QCD glueball? → Scalar glueball （ ex ： → meson, → baryon ） Lattice prediction of lightest glueball mass: 1600MeV 0++ state Perturbative QCD Lattice QCD We need theoretical description of glueball decay! Chiral perturbation Mixings? Difficult

Gauge/Gravity correspondence A Solution : Holographic QCD String theory 4D gauge large Nc, large λ 10D classical gravity on curved background large Nc, strong coupling 10D classical gravity + “flavor D-brane” on curved background Holographic QCD [Step 1] [Step 2]

= Source of closed strings = Source of gravity = Extended blackhole “blackbrane” in 10D D-branes giving the duality Quantizing Strings defined in 10D spacetime Open string → massless gauge field Closed string → massless graviton D-branes = Object on which open strings can end Deform Open string theory on the Dp-brane is : SU(Nc) gauge theory in p+1 dimensions Nc open strings Nc parallel Dp-branes 2 [Step 1]

Gauge/Gravity correspondence Black brane Nc D-branes Propagation of SU(Nc) gauge theory composite states Propagation of graviton in near-horizon geometry of black p-brane (glueball)

Nf “flavor D-brane” Propagation of meson SU(Nf) gauge fields in higher dim. D-brane on curved background Introducing quarks [Step 2]

We get interacting hadrons from strings [Einstein action + higher dim. YM action] on curved background in 10D Decomposition by fields in 4 dimensions [Glueball + Meson] interacting action No mixing, Coefficients computed explicitly

Power of holography Decay branchHolographic QCD Decay width No decay Experiments f 0 (1500) [PDG] not seen f 0 (1500) can be identified as a pure glueball

Decay width and branching ratios Decay width of each branch If we tune the glueball mass and eta mass by hand so that it can fit the experimental data, then Comparison ： In experiments, f 0 (1500) decays as ： not produced Our results are consistent

What is the ID of Glueballs? Glueballs ： Bound state consisting only of Gluons, among many hadrons appearing at low energy of QCD. → Scalar glueball （ ex ： → meson, → baryon ） Mystery of Glueballs Which is the glueball, among states observed with ?

Diff. （３） There are many hadrons having the same quantum number, so the glueball is mixed among mesons. Generally mass eigen states are mixed. Diff. （１） Perturbative analysis is impossible due to strong coupling at low energy of QCD. This is related to the mass-gap problem in YM, which is a millenium problem! Diff. （２） Lattive QCD predicts that the lightest glueball has and its mass should be around 1600 MeV. But it cannot calculate dynamical decay process, so no more comparison is available. Difficulties in identifying the glueball Diff. （４） Chiral perturbation cannot be applied to glueball dynamics, since glueballs live not really in low energy.

Glueball Decay in Holographic QCD Koji Hashimoto (RIKEN) 27 th Nov., KEK arXiv/ (hep-th) w/ Seiji Terashima （ YITP ） and Chung-I Tan （ Brown U ）

Plan of today’s talk 1. Mystery of Glueballs 2. Holographic QCD ： From String Theory to Hadrons 3. Glueball Decay

1. Mystery of Glueballs What is the ID of Glueballs? Present status of Glueball searches Power of Holographic QCD

What is the ID of Glueballs? Glueballs ： Bound state consisting only of Gluons, among many hadrons appearing at low energy of QCD. → Scalar glueball （ ex ： → meson, → baryon ） Mystery of Glueballs Which is the glueball, among states observed with ?

Ref ： PDG data of decay of glueball candidates

Diff. （３） There are many hadrons having the same quantum number, so the glueball is mixed among mesons. Generally mass eigen states are mixed. Diff. （１） Perturbative analysis is impossible due to strong coupling at low energy of QCD. This is related to the mass-gap problem in YM, which is a millenium problem! Diff. （２） Lattive QCD predicts that the lightest glueball has and its mass should be around 1600 MeV. But it cannot calculate dynamical decay process, so no more comparison is available. Difficulties in identifying the glueball Diff. （４） Chiral perturbation cannot be applied to glueball dynamics, since glueballs live not really in low energy.

PDG prediction is f 0 (1500) ＝ glueball f 0 (1370) can be produced by 2γ and thus composed by charged quarks. f 0 (1710) decays mainly to ２ K, thus strangeness should be the main component. Reason ： Present status of glueball ID proposal But, no one could compute these…. Not decisive! Holographic QCD enables us to compute spectra and couplings of/among composite states of stronly coupled QCD! Decay products, width, branching ratios : Computable! （ but at large Nc ） Our motivation, and Result Conclusion: f 0 (1500) is the glueball Holographic QCD ＝ Application of AdS/CFT to QCD

2. Holographic QCD AdS/CFT From AdS/CFT to QCD Sakai-Sugimoto model The model and experimental data

Very brief history of string theory and AdS/CFT 1960’s ～ 70’s ： String theory was born in hadron physics Regge trajectory, s-t channel duality, ’tHooft large N, …. 1970’s ～ 80’s ： String as quantum gravity and unification Standard model and superstrings, supergravity Late in 1990’s ～： Revolution by D-branes and duality Toward non-perturbative definition AdS/CFT (gauge/string duality ） ？

Low energy effective action of open strings on N D3-branes AdS/CFT ： Equivalence of two ways to describe D-branes Open string （ gauge theory ） Closed string （ gravity ） Closed string in blackbrane background of N D3-branes 4dim. gauge theory 10 dim. supergravity in curved backgrounds Corresp. [Maldacena(97)]

with the low energy limit super Yang-Mills Theories in correspondence Open string side: Near horizon geometry of BPS black 3-brane solution in 10 dimensions : Closed string side: Supergravity on This classical geometry is valid when is required

Closed string Closed string Black brane Physical quantities in correspondence ＝ Closed string Closed string Open string D-branes Correlation functions in gauge theory can be computed by gravity theory. [Gubser-Klebanov-Polyakov] [Witten] Gauge invariant composite operators in YM theory Bulk fields in supergravity

AdS/CFT deals with strongly coupled gauge theories; Then why not QCD? Philosophy Present status ： It reproduces various characteristics of low energy hadron physics and provides a new viewpoint (paradigm), though with some difficulties Difficulties ： ・ Large N ・ Decoupling of higher dimensional DoF Holographic QCD (“AdS/QCD”)

・ Nc D4-branes wrap, and gauginos satisfy anti-periodic boundary condition → 4d pure Yang-Mills at low energy ・ Nf D8-branes intersect with the D4s → Nf left-handed massless quarks ・ Nf anti-D8-branes intersect with the D4s → Nf right-handed massless quarks Massless QCD is brane-engineered at low energy [Witten] Open string side (D-branes) Sakai-Sugimoto model (hep-th/ )

Closed string side (gravity) [Witten] D8s are put there as probes (probe approximation, valid at ) Near horizon geometry of black 4-brane solution on which fermions satisfy the anti-periodic b.c. is Once the correspondence is applied ・・・ ・ D8 → bound states of quarks （ Mesons, Baryons ） KK modes of gauge fields on the D8 → Meson D8-brane action → Chiral lagrangian ・ gravity → bound states of gluons （ Glueballs ）

The role of the D8-branes ＝ D4 D8

Massless QCD (weak coupling description) Spontaneous chiral SB at strong coupling Spontaneous chiral symmetry breaking Chiral symmetry Replacing the D4 by its gravity solution D8s are connected, and gauge symmetry is ＝ Gauge symmetries on the D8 and anti D8

KK decomposition of this YM theory on the curved background is the meson lagrangian Metric induced on the D8 by the D4-brane graviy solution is KK modes of → Vector mesons A KK mode of → Pion Meson sector in the SS model D8-brane action on the curved background AdS/CFT : Redefinition of coordinates ：

Action is evaluated as ・ KK modes of gauge field : Eigen equation for the modes : ・ Pion is given by the zeroth mode of the decomposition : ・ Higher modes are absorbed by field redefinition :

Final lagrangian quadratic in KK modes is Eigenvalues correpond to masses of vector mesons. Comparison with observed data Meson interactions can be computed from YM interactions.

3. Glueball Decay Summary We describe decay of lightest glueball in QCD by using AdS/CFT. The computed decay products and width are consistent with a hadron f 0 (1500) which is a glueball candidate. arXiv/ (hep-th) Seiji Terashima （ YITP ） and Chung-I Tan （ Brown ）

Our strategy Using holographic QCD, we compute analytically interactions between Glueballs and Mesons / photons, calculate the decay products and widths, and compare them with experimental data. Gravity fluctuations around near horizon geometry of non -BPS black 4-brane （ Witten ） Gauge （ QCD ） side Gravity side Gluon sector （ Glueballs ） Quark sector （ Mesons ） Gauge fluctuation on the probe D8 （ Sakai-Sugimoto ） Csaki,Ooguri,Oz,Terning(98) Brower,Mathur,Tan(00) Sakai,Sugimoto(04,05) Our work We compute couplings between the two sectors in the gravity side, and describe the glueball decay

Review: Computing Glueball spectrum via AdS/CFT Wrap a D4-brane around a circle, and impose anti-periodic boundary condition for the fermions to break the SUSY Supergravity fluctuation ocrresponding to the lightest glueball （ Constable ・ Myers, Brower ・ Mathur ・ Tan ） Gravity background dual to 4d pure YM （ Witten ） ： Glueball field ： eigenfunction in higher dim

（ Brower ・ Mathur ・ Tan,2003 ） Glueball spectrum obtained in AdS/CFT Lattice calculation (SU(3) pure YM) （ Morningstar ・ Peardon,1999 ）

Computing the coupling between glueballs and mesons Glueball → Gravity Meson → Gauge (on D8) ② In string theory, all the interactions between gravity and D8 gauge fields are encoded in D8-brane action We substitute gravity and D8 gauge fluctuations representing the mesons and glueballs, and perform the integration of higher dimensional space We obtain interacting lagrangian of glueball / meson fields ① correspondence:

Glueball, Pion,ρmeson （ This expression is for a single flavor, for simplicity ） Result Kinetic terms No mixing between mesons and lightest glueball Interaction terms

Possible decay process of the lightest glueball Interaction terms obtained via AdS/CFT : YM ← CS Among these, （ ii ）（ iii ） includes more than 5 pions after the decay and so are negligible. Possible decay processes are These reproduces decay products of f 0 (1500)

Decay width and branching ratios Decay width of each branch If we tune the glueball mass and eta mass by hand so that it can fit the experimental data, then Comparison ： In experiments, f 0 (1500) decays as ： not produced Our results are consistent

4. Summary and Prediction

Summary Holographic QCD can really help computing interactions among hadrons. It enables us to compute glueball interactions analytically, and f 0 (1500) can be identified as a scalar glueball. Other results ・ Other decay process ： No decay to ２ γ → f 0 (1370) is not a glueball Small decay width to ２ K→ f 0 (1710) is not a glueball ・ Prediction ： No decay of f 0 (1500) to 4 π ０ Many more can be computed similarly Interactions of heavier glueballs (with different spins etc) Glueball-glueball interactions Glueballs in finite temperature, finite baryon density, ・・・・