Holographic model for hadrons in deformed AdS 5 background K. Ghoroku (Fukuoka Tech.) N. Maru (U. Rome) M. Yahiro (Kyushu U.) M. Tachibana (Saga U.) Phys.Lett.B633(2006)606 1.Introduction and motivation 2.An example of holographic model for QCD 3.Analysis in deformed AdS 5 background 4.Summary and discussion Plan of talk

1. Introduction and motivation QCD, the theory of strong nuclear force, is nortoriouly intractable, due to the nature of strong coupling. Then one needs appropriate effective models for QCD. Chiral Lagrangian is one of them, based on chiral symmetry. Meanwhile, vector mesons play a significant role in hadron physics, though their interactions are not constrained by low-energy theorems. How to incorporate vector mesons into field theoretical frameworks? What is the EFT involving vector mesons? What symmetry behind? Holographic QCD via AdS/CFT

Original idea of AdS/CFT correspondence (Maldacena ‘98) N c D3-branes N=4 U(N c ) Super Yang-Mills in 4d Type IIB SUGRA on AdS 5 ×S 5 (gauge theory on the boundary) (gravity theory in the bulk) Gubser-Klebanov-Polyakov ‘98 Witten ‘98 The field operator correspondence Correlation functions in the gauge theory can be computed by differentiating with respect to a field living in the bulk (= source for the boundary operator) Field theory at strong coupling ⇔ Classical (super)gravity (string) Let’s apply this Idea to Hadron Physics!

First applications to QCD (pure Yang-Mills) 1. Wilson loop operator in N=4 SYM (Maldacena ‘98) 2. Glueball mass spectra (Csaki et al., Koch et al., Zyskin, Minahan, Ooguri et al ‘98 ) Two point function of glueball operators : (e.g., ) Solving the field eq. (e.g., for dilaton) in some curved background These are Yang-Mills staffs…..

Adding flavor (Karch-Katz ‘02) Applications into flavor physics N c D3-branes N f D7-branes (flavor brane) Probe approximation: Karch-Katz-Weiner(‘02), Myers et al.(‘03), Sakai-Sonnenschein(‘03), Nunez et al.(‘03), Babington et al(‘03), Ghoroku-Yahiro(‘04), Sakai-Sugimoto(‘04), etc. Chiral symmetry breaking Meson spectra Quark mass effect etc “flavor quark” Supersymmetric/Non-supersymmetric

“Bottom-Up approach” “One starts from four dimensional QCD, and attempts to guess its five dimensional holographic dual.” Erlich-Katz-Son-Stephanov (EKSS) (‘05) Da Rold-Pomarol (DRP) (‘05) Brodsky-Teramond (‘05) Hirn-Sanz (‘05) Katz-Lewandowski-Schwartz (‘05) Hambye et al. (‘05) (see also N. Evans, Nature 439, 921 (‘06)) AdS/QCD Various quantities being computed: Shifman, hep-ph/ Holographic models for hadrons meson and baryon spectra (e.g., vector, axial-vector mesons) decay constants coupling among mesons some QCD formula (sum rules) Non-supersymmetric models

2. An example of holographic model for QCD -- The EKSS/DRP model --

The model 5D geometry IR cutoff AdS 5 5D mass of the p-form field conformal dim. of operator O “A minimal set of operators associated with chiral dynamics”

The action Symmetry Gauged flavor symmetry Chiral symmetry breaking (Classical solution for X field)

Free parameters where Chiral condensate We take Quark mass (cf. compared to QCD: ) Below analysis is done in the case of N f =2 light flavors

taken from the talk by M.Stephanov vector/axial-vector mesons : pseudo scalar mesons * :input

3. Analysis in deformed AdS background

bulk scalar field warp factor Solutions of Einstein’s eq. Bulk action (AdS 5 ) EKSS/DRP Mass of the adjoint fermions of N=4 SYM

In this sense, λis the modification parameter, which corresponds to the supersymmetry breaking and gives some change to the system of our interest at the infrared. How this parameter affects the physical observables ( M v, F v ) including some higher excited states. Singlet scalar excitation (σ) which was not taken into account before (but see, Da Rold-Pomarol, hep-ph/ ). How much the results depend on the selected 5D background? In short, our interests here are

Scalar and vector meson part Chiral symmetry breaking quark mass chiral condensate thus Singlet scalar meson (σ) with

Vector meson ( gauge ) : the n-th excited vector meson mass The 1st excited vector meson mass,, is obtained as the function of and. By utilizing the experimental value of, can be expressed as the function of. As the result, the 2nd and all highly excited vector meson masses depend on only. We find that the 1st and the 2nd excited vector meson are fitted by the parameters and.

A comment (a bit long, but important) In the sector of scalar and vector mesons, the masses (also the decay constants) are independent with (or equivalently, the quark mass and the chiral condensate ). This is contrasted with the results from, e.g., the QCD sum rule, where the vector meson masses are the function of them. This is due to the fact that spontaneous chiral symmetry breaking is introduced by hand in this model, which was done by the choice of the profile of the scalar field X and this feature seems to be common to many models of holographic QCD. (For QSR, Krasnikov-Pivovarov-Tavkhelidze ‘83, Reinders-Rubinstein-Yazaki ‘85)

Axial vector sectorVector and axial vector sectors a: decay const. b: meson mass c: pion decay const. d=a, e=b, g=c h: ρ meson mass f: 2nd excited vector mass As for a) and b), the agreement of the theoretical results with experiment Becomes better as λ decreases. One then sees that λ=0 case, i.e., the AdS case, yields a best fit. As for d), e) and g), λ=0 is the best fit. But once we take into account as well as, the best value will be shifted to some small value of λ.

4. Summary and discussion Application of the idea of AdS/CFT duality to QCD dynamics “ Bottom-up” approach = holographic model for hadrons (gauged flavor symmetry) Some properties of scalar, vector and axial vector mesons (comment on v(z) independence of scalar and vector sector) Outlook Glueballs, baryons and U(1) vector mesons Strange mesons Finite temperature and/or density = phase transition (for finite T, Ghoroku-Yahiro, hep-ph/ )

Vector meson Linearlized eq. of motion ( gauge) The action on the solution source of the vector current then with UV cutoff

Current correlator since c.f. In QCD, Thus

“Hadrons” Normalizable modes of the 5D fields Eigenvalue of normalizable mode : Squared mass of meson Derivative of the mode : Decay constant of the meson and then decay constant Let be a solution of eq. of motion for vector meson with

Axial vector and pion Eqs. of motion GOR relation is reproduced.

and taken from the talk by M.Stephanov Input

What they have not done (i)Glueball spectrum (ii)OPE and higher order terms (iii) strange mesons (iv)Chiral anomaly (WZW) (v)Baryons (Brodsky-Teramond) (vi)running of the gauge coupling

3.2 Axial vector meson and pion part Unlike the previous case, the masses and decay constants of axial vector and pion depend on four parameters and. Gell-Mann-Oakes-Renner (GOR) quark mass dependence ・ meson mass pion mass GOR relation Solid line: direct calc. of Dotted circle: obtained from calculated through the GOR Two results agree with each other! In the right figure, when calculated pion mass and the decay constant well reproduces the experimental values.