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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. Large N QCD has a dual classical theory in 5-D?!.

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weak interacting holographic qcd

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

large n qcd has a dual classical theory in 5 d
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

things that we know
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

we thus expect the dual theory of qcd
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

motivation for weak interacting holographic qcd
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

how to perturb the qcd lagrangian
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

for a general functional perturbation of a single trace operator
For a general functional perturbation of a single trace operator

Witten, Yee, Marolf-Ross

PANIC08 - Weak interacting holographic QCD - Gazit & Yee

implementation
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

sakai sugimoto model
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

hard soft wall model
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

implementing the prescription
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

charged pion decay
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

how to calculate different reactions
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

nucleons
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

neutron b decay
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

parity non conserving pion nucleon coupling
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

conclusions
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