Outline

2. Outline • Introduction • Holographic QCD • basic setup • masses and coupling constants • vector meson dominance/KSRF • tensor mesons • The U(1) problem • The problem • AdS/QCD construction • Predictions: masses, decay constants, mixing angles • Connection to instantons • Conclusions

3. AdS/CFT more D-branes, fluxes ( for flavor, confinement, deviations from conformality, finite N, etc.) • some qualitative similarities to QCD • to get QCD, do we need the full string construction? N D-branes ) SU(N) yang-mills symmetry AdS5 warp factor X S5

4. AdS/QCD • Bottom up approach • Fit to QCD, expanding around the conformallimit • Check internal consistency as effective field theory IR brane f2 a1 mass h’ r other insights into QCD predicts spectrum p

5. Why bottom up? • Topdown string-model building approach may be too indirect • Difficult to find correct supergravity background, brane configurations, fluxes, etc… • The dual description of QCD may not be simple to descrbe • Bottom up appoach is directly related to QCD data from the start • Fit some parameters, predict others • Has hope of computing useful non-perturbative quantites • light front wafefunctions, form factors, hadronic matrix elements • Insights into QCD sum rules, vector meson dominance, quark models, instantons, glueball spectra, etc • Easier to work with than the lattice • Understanding dual to QCD may be relevant for LHC • Randall-Sundrummodels are built the same way • May provide insight into more general strong dynamics

6. Basic Idea external currents probe system IR brane (boudary conditions) models confinement Jm bulk fields QCD AdS operators $ bulk fields global symmetries $ local symmetries correlation functions $ correlation functions • effective low-energy descriptions are the same • quantitativelypredicts features of dual (confined) theory hadrons $ KK modes coupling constants $ overlap integrals

7. Set-up QCD Lagrangian has global SU(3)L x SU(3)R symmetry. 5D Gauge fields AL and AR Operators AL, AR bulk fields Xij Xij AdS Lagrangian mass term determined by scaling dimension ~ Xis dimension 3 i.e. X= s z3 solves equations of motion

8. Gauge coupling from OPE In QCD, correlation function of vector current is + power corrections In AdS, source fields on UV brane bulk to boundary propogator (solution to eom with V(0)=1)

9. Chiral symmetry breaking General solution to bulk equation of motion for X ~ h Xij i = v(z) = mqz+ sz3 s ~ quark masses quark condensate explicit breaking relevant in the UV spontaneous breaking relevant in the IR X = v(z) exp( i p ) |DMX|2 ! v(z)2( AM+dmp + . . .)2 mass term for axial gauge fields AM=(AL– AR)M • splits axial from vector • gives pion a mass Now just solve equations of motion!

10. Connect to data AdS Object QCD Object Predictions Data ma1 = 1363 MeV (1230 MeV) IR cutoff zm $ LQCD mK*= 897 MeV (892 MeV) first vector KK mass $ mr mf= 994 MeV (1020 MeV) first axial KK mass $ ma1 mK1= 1290 MeV (1270 MeV) first A5 mass $ mp A5 coupling $ fp fr½ = 329 MeV (345 MeV) fa1½ = 486 MeV (433 MeV) fK= 117 MeV (113 MeV) Observable AdS Parameter RMS error ~9% mr= 770 MeV $ zm-1=323 MeV ~ LQCD fp= 93 MeV $ s = (333 MeV)3 ~ mp= 140 MeV $ mq= 2.22 MeV > ½ (mu+md) mK= 494 MeV $ ms = 40.0 MeV

11. Vector Meson Dominance p p p KSRF II p r = VMD + assumptions p p p p ) AdS p r • VMD understood • KSRF II not reproduced r' • grpp is UV sensitive (e.g. F3 contributes) r‘ In AdS gr’pp~ -0.044 gr”pp~ -0.039 grpp~ 0.48 p p = + + . . . + r r' r‘ p p ~ 1 ~ 0 ~ 0

12. Spin 2 mesons What does chiral perturbation theory say about higher spin mesons, such as the f2? Start with free Fierz-Pauli Lagrangian for spin 2 Suppose minimal coupling to energy momentum tensor (like graviton) universal coupling to pions and photons Experimental values are = 1 Naïve dimensional analysis does better: 2 free parameters (mf and Gf) and still not predictive

13. Spin 2 -- AdS • Graviton excitation $ spin 2 meson (f2) • Equation of motion is • Boundary conditions satisfies (EXP: 1275 MeV) ) mf2= 1236 MeV (3% off) • 5D coupling constant fit from OPE AdS: ) solve QCD: quark contribution gluon contribution • predict G(f2 T gg) = 2.70 keV G(f2 T pp) = 37.4 MeV (EXP: 2.60± 0.24 MeV) (EXP: 156.9 ± 4 MeV) (within error!) completely UV safe (off by a lot) UV sensitive

14. Higher Spin Fields • Higher spinmesons dual to highr spin fields in AdS w3 $ massless spin 3 field fabc in AdS f4 $ massless spin 4 field fabcd in AdS … • Linearized higher spin gauge invariance leads to ) ms satisfies Js-1(mszm) =0 quadaratic KK trajectory AdS prediction linear (regge) trajectory f6 f4 w3 r5 only one input! data f2 r predicts slope and intercept

15. The U(1) problem The QCD Lagrangian has a global (classical) U(3) x U(3) symmetry ) expect 3 neutral pseudogoldstone bosons: p (139), h(547), and h’(957) Chiral Lagrangian • Chiral anomaly breaks U(1) new term now allowed withfp=fh= fh’, we can almost fit withk can fit exactly by tuning mh’=957 chiral lagrangian can accommodateh’ 686 566 mh=547 493 but does not predict anything about it mp=139 k= 0 k= 850 MeV andk k= 1

16. How can the Anomaly lift the h’ mass? ~ F F The U(1) current is anomalous h’ = + . . . = + . . . extract mass from p T 0 But in perturbative QCD, anomaly vanishes as p T 0 » and vanishes as p T 0 anomaly cannot contribute in perturbation theory factor of overall momentum ) The solution … instantons! evaluated on a one instanton solution However, integral is divergent, so it cannot be used quantitatively

17. Lattice Calculations ( ( ) ) 0 8 8 J J ¹ ¹ q q ° ° ° ° ¿ q q = = i i 5 5 ¹ ¹ ¹ ¹ Lattice results For currents with flavor, like J8, masses extracted from mh’ = 871 MeV mh= 545 MeV For U(1) currents, other diagrams are relevent OZI-rule violating disconnecteddiagrams • Lattice calculations are difficult • quenched approximation fails Suppressed for large NC (like the anomaly, and the h’-p mass splitting) • need strange/up/down quark masses • In what sense is the h’ mass due to instantons? • is the p T 0 limit smooth? • what do OZI suppressed diagrams do? • do we need to calculate in Euclidean space?

18. The h’ from AdS dual to pions, pi Recall we had a field X ~ dual to “axion”, a Now introduce new field Y~ • anomalous global U(1) is now gauged local U(1) in AdS • both a and p0 are charged Recall h Xij i = Now, h Y i = power correction (neglect) (keep) scales like z9 (strongly localized in IR)

19. Match to QCD In perturbative QCD (for large Euclidean momentum) In AdS bulk to boundary solution We take LQCD = zm-1 That’s it – no new free parameters (except fork)

20. Solve differential equations 5 coupled differential equations a eom: hq eom: hs eom: A5(q) eom: A5(s) eom: • choose boundary conditions f is stuckleberg field (longitudinal mode of Am) p(0) = f(0) =0 p’(zm) = f’(zm) =0 correction to warp factor due to strange quark mass • we will scan over k. But note • forces p0(zm) = a(zm) • as kT1 forcesp0(z) = a(z)

21. Solution AdS EXP error lattice 867 MeV 957 9% 871 520 MeV 549 5% 545 Also calculate decay constants

22. Turn off anomaly Turn off LQCD h’ h Turn off k

23. Decay to photons gm h’ Effective interaction described by Wess-Zumino-Witten term gm (in units of TeV-1) This is a total derivative, so AdS EXP Ahgg= 24.3 24.9 Ah’gg= 48.1 31.3 Mixing anglevaries withz h h’ p8 a p0 p0 a p8

24. k dependence decay amplitudes AdS EXP h’ 48.1 h 31.3 24.9 24.3 k AdS EXP masses 957 867 549 520

25. Topological Susceptibility In AdS bulk to boundary propogator for a With no quarks quarkless AdS (109 Mev)4 Lattice (191 Mev)4 large N (171 MeV)4 with quarks Witten-Veneziano relation • numerical value strongly depends on C~as • For C=0 (no kterm) and mq ≠ 0ct(0) ≠ 0 arguments about whether q and ct are physical produce the same qualtitative results • For C=0 (no kterm) and mq = 0ct(0) = 0 mh’2~ 1/Nc for large Nc

26. Instantons In the conformal limit (no IR brane, C~as const) With IR brane and z-dependent k = k(z) Expanding around the conformal solution, andintegrating the action by parts on the bulk-to-boundary equation of motion gives This is the same as the instanton contribution with D(r) ~ k(z) and z~r

27. Conclusions • The holographic version of QCD works really really well • meson spectrum, coupling constants predicted • insights into vector meson dominance, KRSF • higher spin fields can be understood • insights into regge physics • The U(1) problem is solved quantitatively and analytically (i.e. not with a lattice) • h and h’ masses predicted to 5% and 9% respectively • decay rate G(hT gg)predicted well G(h’T gg) is IR-model-dependent • simple mixing angle interpretation for h h’ decays fails • new handles to turn off the anomaly • Witten-Veneziano relation reproduced • Direct connection with instantons, with z ~ r and an interpretation of the instanton density • Many open questions • Why does AdS/QCD work so well? • Can the expansion be made systematic to all orders? • Can we calculate useful non-perturbative observables: form factors, hadronic matrix elements, etc • Can lessons from AdS/QCD be applied to other strongly coupled gauge theories • (e.g. RS/technicolor)