Weak Interacting Holographic QCD

1. Weak Interacting Holographic QCD Doron Gazit Institute for Nuclear Theory University of Washington & Ho-Ung Yee ICTP, Trieste Phys. Lett. B (2008), doi:10.1016/j.physletb.2008.10.045, arXiv: 0807.0607

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. 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 Deforming boundary conditions of field near UV boundary. A 5D field, such that: Boundary conditions: • Perturbation to the Lagrangian. • Single trace operator O. • A Lagrangian pertutbation: 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. 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. 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/Softwall model PANIC08 - Weak interacting holographic QCD - Gazit & Yee

13. 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. 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. D77, 014030 (2008). PANIC08 - Weak interacting holographic QCD - Gazit & Yee

16. 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. 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