1. Weak Interacting Holographic QCD
Doron Gazit
Institute for Nuclear Theory
University of Washington
&
Ho-Ung Yee
ICTP, Trieste
Phys. Lett. B (2008), doi: /j.physletb , arXiv:

2. Large N QCD has a dual classical theory in 5-D?!
Large N factorization of gauge invariant operators:
Implies a classical theory for gauge invariant operators (AKA master fields).
RG running survives the large N limit, thus the master field is a function of the energy scale:
The RG equations constrain flow in this scale
Holographic QCD is a gravitational theory of gauge invariant fields in 5 dimensions.
5th dimension corresponds roughly to the energy scale.
PANIC08 - Weak interacting holographic QCD - Gazit & Yee

3. PANIC08 - Weak interacting holographic QCD - Gazit & Yee
Things that we know
AdS/CFT Duality proposal N=4 Super Yang-Mills theory in (3+1)D for Nc∞, gYM0 and fixed but large is equivalent to Type IIB Supergravity in AdS5×S5 with size l¼
Mandalcena 1998
PANIC08 - Weak interacting holographic QCD - Gazit & Yee

4. We thus expect the dual theory of QCD…
In the UV regime: highly nonlocal, corresponding to asymptotic freedom.
In the IR regime: local, corresponding to the strongly correlated QCD.
Thus, current models of Holographic QCD model the gravitational dual as a local theory.
Properties of existing models of Holographic QCD:
Chiral symmetry.
Confinement.
Explain experimental observables to 20%.
PANIC08 - Weak interacting holographic QCD - Gazit & Yee

5. Motivation for Weak Interacting Holographic QCD
Many important processes include weak interaction and low-energy QCD.
Numerical calculation of observables from first principles QCD possible only using lattice calculations.
Weak interacting Holographic QCD will provide an additional tool for estimating Hadronic processes.
PANIC08 - Weak interacting holographic QCD - Gazit & Yee

6. Low-energy Weak interaction
SU(2)L EM

7. How to perturb the QCD Lagrangian?
Gauge
Gravity
Perturbation to the Lagrangian.
Single trace operator O.
A Lagrangian pertutbation:
Deforming boundary conditions of field near UV boundary.
A 5D field, such that:
Boundary conditions:
However, weak perturbation is NOT a single trace operator
PANIC08 - Weak interacting holographic QCD - Gazit & Yee

8. For a general functional perturbation of a single trace operator
Witten, Yee, Marolf-Ross
PANIC08 - Weak interacting holographic QCD - Gazit & Yee

9. Implementation
The idea is general enough to implement in any Holographic Model. We demonstrated on two models:
Top – Down Model: Sakai-Sugimoto Model [Sakai and Sugimoto, Witten]
Bottom – Up Model: Hard/Soft Wall Model [Erlich-Katz-Son-Stephanov, Da Rold -Pomarol].
Both models lead to essentially the same results, as the 5D gauge field includes pions and spin-1 massive bosons when we expand it in terms of 4D modes.
PANIC08 - Weak interacting holographic QCD - Gazit & Yee

10. PANIC08 - Weak interacting holographic QCD - Gazit & Yee
Sakai-Sugimoto Model
Set-up of D8 and branes in UV, joined at IR boundary of NC D4 branes.
Geometrical realization of U(Nf)R×U(Nf)LU(Nf)I
The dynamics resides in the U(Nf) gauge field A(x,z).
Values near UV couple to the QCD Nöther currents.
Near UV boundary
PANIC08 - Weak interacting holographic QCD - Gazit & Yee

11. PANIC08 - Weak interacting holographic QCD - Gazit & Yee
Hard/Soft Wall Model
A phenomenological model, with an artificial IR cutoff, embedded in the dilaton field.
The chiral symmetry is due to an expectation value of the bifundamental field – broken hidden symmetries.
Mesons come from the 5D gauge fields, as well as the field X, upon KK mode expansion to 4D.
Including non-vanishing quark-mass – natural.
Near UV boundary
PANIC08 - Weak interacting holographic QCD - Gazit & Yee

12. Implementing the prescription:
For quarks with U(1) charge:
Treating the lepton current as an external current.
The current is decomposed:
Sakai- Sugimoto
Hard/Soft wall model
PANIC08 - Weak interacting holographic QCD - Gazit & Yee

13. PANIC08 - Weak interacting holographic QCD - Gazit & Yee
Charged Pion Decay
Treating the lepton current as an external source.
Thus:
We define the pion field so that there will be no 5th dimension mixing term in the kinetic energy:
SS:
As a result –
Same result as old current algebra, and without free parameters (due to the chiral symmetry)!
Same result is achieved in the Hard/Soft wall model.
PANIC08 - Weak interacting holographic QCD - Gazit & Yee

14. How to calculate different reactions?
Write equation of motion for the global gauge field (i.e. the U(NF) current).
Solve it with the prescribed boundary conditions.
If you’d like pions to be involved, do it in the previous gauge fixing for Az.
For reactions that include nucleons, choose a model for baryons, and calculate baryon-pion coupling from the kinetic term, and from magnetic type of couplings:
PANIC08 - Weak interacting holographic QCD - Gazit & Yee

15. PANIC08 - Weak interacting holographic QCD - Gazit & Yee
Nucleons
Sakai-Sugimoto:
Nucleons are instantons in 5D. We take only lowest excitation, such that spin connections can be cancelled –
Hard/Soft Wall model:
Nucleons are spin ½ doublets.
Nucleon mass
Hong, Rho, Yee, Yi, Phys. Rev. D 77, (2008).
PANIC08 - Weak interacting holographic QCD - Gazit & Yee

16. PANIC08 - Weak interacting holographic QCD - Gazit & Yee
Neutron b-decay
Sakai-Sugimoto
Hard/Soft wall model
With:
With:
The axial constant is measured using the neutron beta decay 
PANIC08 - Weak interacting holographic QCD - Gazit & Yee

17. Parity non-conserving pion-nucleon coupling
First example without an external source.
We are interested in parity violating couplings of mesons to the nucleons.
To this end, we consider only charged pion-nucleon coupling.
In both models, the result in the zero q limit is identical to the current algebra result:
Still, a lot to be done!
PANIC08 - Weak interacting holographic QCD - Gazit & Yee

18. PANIC08 - Weak interacting holographic QCD - Gazit & Yee
Conclusions
We have outlined a prescription to include weak interactions in the framework of holographic QCD.
Applicable up to energies of a few GeV, when strong coupling is still valid.
We have shown its strength by using Sakai-Sugimoto and Hard/Soft wall models to calculate few exemplar reactions.
The current approach, contrary to other approaches (such as cPT), gives not only the operator structure, but the numerical coefficients, to about 20%, and valid for energies above the chiral limit.
PANIC08 - Weak interacting holographic QCD - Gazit & Yee