Hadrons in AdS/QCD models

in Hadron Physics: a Challenge to Holography, iiP-UFRN-Natal –RN, July 29 – August 09 2013

Henrique Boschi Filho Instituto de Física – Universidade Federal do Rio de Janeiro* Work done in collaboration with: Nelson Braga (IF-UFRJ), Alfonso Ballón-Bayona (IFT-UNESP), Marcus Torres (SPhT, Saclay) and Matthias Ihl (DIAS, Dublin) * Supported also by CNPq and CAPES

1

Summary of the talk: • Brief Review: hard wall AdS/QCD model, hadronic spectrum,… • Deep inelastic scattering in AdS/QCD. • Hadronic form factors from D4-D8 brane model. • Other Results

2

AdS/CFT correspondence, J. Maldacena, 1997 (simplified version of a particular useful case)

Remarks.: String theory space = AdS5 X S5

AdS = anti-de Sitter; S = sphere

Gauge theory: SU(N) with very large N

(supersymmetric and conformal). 3

Exact equivalence between String Theory in a 10-dimensional space and a gauge theory on the 4-dimensional boundary.

At low energies string theory is represented by an effective supergravity theory → gauge / gravity duality

4

Anti-de Sitter space (Poincaré patch)

bulk ↔ boundary mapping

E. Witten and L. Suskind (1998) ; A. Peet and J. Polchinski (1998)

The 4-dim boundary is at z = 0 → z

States of the boundary

gauge theory with energy E

↔ Region of AdS space

x

Holographic relation suggests:

5

Hadronic masses from AdS/CFT H.B.-F and N. Braga, JHEP2003, EPJC2004 • Masses of scalar glueballs JPC = 0++ , 0++* , 0++** , … with good agreement with lattice results.

Cut off in AdS space: 0 < Z < Zmax ↔ infrared cut off in gauge theory.

This idea was introduced by Polchinski and Strassler (PRL 2002) to find the scaling of high energy Glueball scattering amplitudes at fixed angles.

This kind of model was then called the Hard wall AdS/QCD model and extended to other hadrons

Glueballs ↔ Normalizable modes of a scalar field in an AdS slice with size Zmax = 1/ ΛQCD

Deep inelastic scattering in AdS/QCD.

Inclusive cross section characterized by the hadronic tensor

qP

qq

2 x ,

22

Dynamical variables: Bjorken variable

The Structure functions F1,2 (x, q2 ) contain informations about the distribution of constituents inside the hadron.

Polschinski and Strassler (2003) found prescriptions for calculating the structure functions, using the hard wall model (depending on the kinematical regime; here we just review the case when supergravity approximation holds).

7

Hadronic tensor (spin independent case)

8

The matrix element of the hadronic current is given by a 10-dimensional supergravity interaction action. For scalars:

Gauge theory Supergravity ( ~ low energy String theory)

↕ ↕

Result for fermions (τ is the twist = d - s):

(For spin ½ in the hard wall, Polchinski & Strassler)

↔

(Just one hadron in the final state)

Hadronic structure functions at small x. C.A.Ballon Bayona, H.B.-F. and N.Braga, 2008

Center of mass energy is larger at small x:

So, we expect more hadrons in the final state.

∆ = scaling (conformal) dimension of hadronic operator.

↔ minimum number of constituents of the state.

Bulk/Boundary relation: Δ is “regulated” by the 5-

dimensional mass of the dual bulk field

9

s

10

So, we summed over final hadronic states with all allowed values of ∆ (in Polchinski Strassler article ∆initial = ∆final ), finding a behaviour similar to GEOMETRIC SCALING :

Geometric Scaling: Stásto, Golec-Biernat, Kwiecinski; PRL 2001

Our AdS/QCD result imply a similar scaling with (without summing all values of ∆ the result is a negative )

For Bjorken parameter x < 0.01 the observed total cross section: depends on x and q2 only through the combination:

with:

11

D4-D8 brane model for hadrons Sakai and Sugimoto (2005)

•D8 (probe) branes embedded in D4 brane space. •Holographic model for (large NC , strongly coupled) QCD. D4 brane background: With: , τ is a compact dimension. Period of τ is related to minimum value of U → mass scale.

12

In this model mesons correspond to fluctuations of the D8 brane solutions in the D4 background.

13

Vector and axial vector mesons are described by U(NF ) gauge field fluctuations. 4-dim effective action (after field redefinitions, ...)

→ The effective actions show up with a set of prescritions for calculating masses and couplings. (everything is solved numerically)

• Important: D4-D8 model realizes vector meson dominance.

Vector and axial-vector mesons:

14

Form factors for vector and axial-vector mesons in the D4-D8 model: C.A.Ballon Bayona, H.B.-F., N.Braga, M.A.C.Torres, JHEP 2010

VMD (Vector meson dominance)

Interaction with a photon mediated by the exchange of vector mesons

15

Generalized form factors for vector mesons:

is the coupling between the photon and the vector meson, Is the 3 vertex on vector mesons, .... .

Where, in the model:

... and similar expressions for the axial-vector mesons.

16

Results: appropriate decrease with q- 4 for large q.

17

Interesting quantities in the elastic case: • Form factors for vector mesons with transversal and longitudinal polarizations

In the D4-D8 model we found:

That imply the large q2 behaviour expected from QCD:

18

Other Results:

A. Ballón-Bayona, H.B.-F., N. Braga, M. Ihl, M. Torres

19

Other Results:

A. Ballón-Bayona, H.B.-F., M. Ihl, M. Torres, JHEP (2010)