# Presentation: 794427

## Slide 1

## Outline

** **Introduction
-  Holographic QCD
-  basic setup
-  masses and coupling constants
-  vector meson dominance/KSRF
-  tensor mesons
-  The U(1) problem
-  The problem
-  AdS/QCD construction
-  Predictions: masses, decay constants, mixing angles
-  Connection to instantons
-  Conclusions

## AdS/CFT

- N D-branes

- )       SU(N) yang-mills symmetry

- more D-branes, fluxes

**X**

- AdS5

- S5

-  some qualitative similarities to QCD

-  to get QCD, do we need the full string construction?

- warp factor

- ( for flavor, confinement, deviations from conformality, finite N, etc.)

## AdS/QCD

-   **Bottom up** approach

-   Fit to **QCD**, expanding around the conformal limit

-   Check internal consistency as **effective field theory**

- mass

- 

- 

- a1

- f2

******’**

- predicts spectrum

- other insights into **QCD**

- IR brane

## Why bottom up?

-  Fit some parameters, **predict** others
-  Has hope of computing **useful** non-perturbative quantites
-  light front wafefunctions, form factors, hadronic matrix elements
** ****Insights** into QCD sum rules, vector meson dominance, quark models, instantons, glueball spectra, etc
-  **Easier** to work with than the lattice

-  **Top** **down** string-model building approach may be too indirect

** ****Bottom up** appoach is directly related to **QCD** data from the start

-  The dual description of **QCD** may not be **simple** to descrbe

-  **Difficult** to find **correct** supergravity background, brane configurations, fluxes, etc...

-  **Understanding** dual to **QCD** may be relevant for **LHC**

** ****Randall-Sundrum** models are built the same way
-  May provide insight into more general **strong dynamics**

## Basic Idea

- external currents probe system

_**QCD**_

_**AdS**_

**operators **   $           bulk **fields**

**global** symmetries          $           **local** symmetries

- correlation functions        $           correlation functions

**hadrons**        $           **KK modes**

-  effective **low-energy **descriptions are the same

**coupling** constants        $           **overlap** integrals

-  quantitatively predicts features of dual (confined) theory

- IR brane (boudary conditions) models confinement

- J

- bulk **fields**

## Set-up

**QCD** Lagrangian

- has global SU(3)L x SU(3)R symmetry.

- 5D Gauge fields **A****L** and **A****R**

- Operators

- bulk fields  **X****i****j**

**A****L****, A****R**

**X****ij**

- mass term determined by scaling dimension

*i.e.*** X**** **** **=  z3 solves equations of motion

- ~  **X**** **** **is dimension 3

- AdS Lagrangian

## Gauge coupling from OPE

- In **QCD**, correlation function of vector **current**                           is

-  + power corrections

- In **AdS**, source fields on **UV brane**

**bulk** to **boundary** propogator
- (solution to eom with V(0)=1)

## Chiral symmetry breaking

- h **X****i****j** i = () = + 

- quark masses

- explicit breaking

- General solution to bulk equation of motion for **X ~**

- relevant in the UV

-  ~

- spontaneous breaking

- relevant in the IR

- quark condensate

- |DM**X**|2 !     ()2( **A****M**+dm****

**X **= v(z) exp( i ****

- mass term for axial gauge fields **A****M**** ****=**** ****(****A****L**** ****–**** A****R****)****M**

-  splits axial from vector
-  gives pion a mass

- Now just solve equations of motion!

## Connect to data

_**AdS Object**_

_**QCD Object**_

- IR cutoff zm        $              QCD

- first vector KK mass         $              m

- first axial KK mass         $              ma1

- A5 coupling     $              f

- first A5 mass          $              m

_**AdS Parameter**_

_**Observable**_

- m= 770 MeV         $             zm-1=323 MeV ~ QCD

- f= 93 MeV           $             = (333 MeV)3 ~

- m= 140 MeV       $            mq= 2.22 MeV  1⁄2 (mu+md)

- mK= 494 MeV         $            ms = 40.0 MeV

_**Predictions**_

- ma1  = 1363 MeV     (1230 MeV)

- f1⁄2  =  329  MeV     (345 MeV)

_**Data**_

- fa11⁄2 =  486  MeV     (433 MeV)

- mK*=  897  MeV     (892 MeV)

- m=  994  MeV     (1020 MeV)

- fK= 117   MeV     (113 MeV)

- RMS error ~9%

- mK1=  1290 MeV    (1270 MeV)

## Vector Meson Dominance

****

****

******'**

******‘**

- g

- g’

- g”

- =

- =

****

****

****

****

****

****

****

****

****

- +

- +

**+ . . .**

****

******'**

******‘**

****

****

****

****

_**KSRF II**_

**VMD** + assumptions

- )

_**AdS**_

-  **VMD** understood

** ****KSRF** II not reproduced

-  g is UV sensitive   (e.g. F3 contributes)

- ~ 1

- ~ 0

- ~ 0

- In AdS

## Spin 2 mesons

- What  does chiral perturbation theory say about higher spin mesons, such as the  f2?

- Start with free Fierz-Pauli Lagrangian for spin 2

- universal coupling to pions and photons

- = 1

- Naïve dimensional analysis does better:

- Suppose minimal coupling to energy momentum tensor (like graviton)

- 2 free parameters (mf and Gf) and still not predictive

- Experimental values are

## Spin 2 -- AdS

- quark contribution

** ****Graviton** excitation $  **spin 2** meson (f2)

-  Equation of motion is

-  Boundary conditions

- )          =  1236 MeV

-  5D coupling constant                fit from OPE

- AdS:

- QCD:

- gluon contribution

- ) solve

-  predict

-  keV

-  MeV

- completely UV safe

- UV sensitive

- (EXP: 2.60± 0.24 MeV)

- (EXP: 1275 MeV)

- (EXP: 156.9 ± 4 MeV)

- (3% off)

- (within error!)

- (off by a lot)

- satisfies

## Higher Spin Fields

-  **Higher spin** **mesons** dual to highr spin **fields** in **AdS**

- 3        $        massless **spin 3** field abc in AdS

- f4        $        massless **spin 4** field abcd in AdS

**... **

-  Linearized higher spin gauge invariance leads to

- )          satisfies s-1() =0

**data**

**AdS prediction**

****

******2**

******4**

******3**

- linear (regge) trajectory

- quadaratic KK trajectory

- only one input!

- predicts slope and intercept

********

******6**

## The U(1) problem

- The QCD Lagrangian

- has a global (classical) U(3) x U(3) symmetry

- ) expect 3 neutral pseudogoldstone bosons: , , and ’(957)

- Chiral Lagrangian

- Chiral anomaly breaks U(1)

- new term now **allowed**

-  = 0

-  = 1

- 686

-  = 850 MeV

- with = = ’, we can almost fit with 

- can fit exactly by tuning

*and* 

- chiral lagrangian can *accommodate* ’

- but does not predict anything about it

- 566

- 493

- m’=957

- m=139

- m=547

## How can the Anomaly lift the ’ mass?

- How can the Anomaly lift the ’ mass?

- The U(1) current                                     is anomalous

- =

******’**

- =

- +   . . .

- extract mass from p  0

- But in perturbative QCD, anomaly vanishes as p  0

- + . . .

- and

- factor of overall momentum

**vanishes** as p  0 anomaly cannot contribute in perturbation theory

- )

- The solution ... **instantons**!

- evaluated on a one instanton solution

- However, integral is divergent, so it cannot be used quantitatively

- »

## Lattice Calculations

- For currents with **flavor**, like J8, masses extracted from

- For U(1) currents, **other diagrams** are relevent

- OZI-rule violating

- disconnected diagrams

**Suppressed for large N****C**

-  In what sense is the ’ mass due to **instantons**?

- (like the anomaly, and the ’- mass splitting)

-  Lattice calculations are **difficult**

-  **quenched** approximation **fails**

-  need strange/up/down quark masses

-  is the p  0 limit **smooth**?

-  what do **OZI** suppressed diagrams do?

-  do we need to calculate in **Euclidean** space?

- m’ = 871 MeV

- m = 545 MeV

- _Lattice results_

## The ’ from AdS

- The ’ from AdS

- Recall we had a field  **X** ~

- Now introduce new field **Y**** **~

- dual to pions, i

- dual to “axion”, a

-  anomalous global U(1) is now gauged local U(1) in AdS
-  both a and 0 are charged

- h **X****i****j** i    =

- Recall

- Now,   h **Y** i    =

- power correction (neglect)

- (keep)

- scales like z9

- (strongly localized in IR)

## Match to QCD

- In perturbative QCD (for large Euclidean momentum)

- In AdS

- bulk to boundary solution

- We take

- QCD = -1

- That’s it – no new free parameters (except for )

## Solve differential equations

- 5 coupled differential equations

- a eom:

- q eom:

- s eom:

- 5(q) eom:

- 5(s) eom:

-  choose boundary conditions

- (0) = (0) =0

-  is stuckleberg field (longitudinal mode of A)

- ’(zm) = ’(zm) =0

-  we will scan over . But note

- correction to warp factor due to strange quark mass

-  forces ******0**() = ****()

-  as   1 forces******0**() = ****()

## Solution

**520** MeV

**867** MeV

**957**

**549**

_**EXP**_

_**AdS**_

_**error**_

- 9%

- 5%

- Also calculate decay constants

_**lattice**_

**871**

**545**

## Turn off anomaly

******’**

****

- Turn off 

- Turn off QCD

## Decay to photons

- ’

- 

- 

- Effective interaction described by Wess-Zumino-Witten term

- This is a total derivative, so

- (in units of TeV-1)

- A24.3

- A’48.1

_**AdS**_

_**EXP**_

- 24.9

- 31.3

- Mixing angle **varies **with z

******’**

****

- 

- 

- 

- 

- a

- a

##  dependence

-  dependence

******’**

****

**24.3**

**48.1**

_**AdS**_

_**EXP**_

**24.9**** **

**31.3**

****

**decay amplitudes**

**masses**

**520**

**867**

**957**

**549**

_**EXP**_

_**AdS**_

## Topological Susceptibility

- With **no quarks**

- _Lattice_
-  (191 Mev)4

- _large N_
-  (171 MeV)4

- _quarkless AdS _
-  (109 Mev)4

**with quarks**

- In **AdS**

- bulk to boundary propogator for a

-  numerical value strongly **depends** on C~s

-  For **C=0 **(no  term)  and **m****q ****≠ 0**  t(0) ≠ 0

-  For **C=0** (no  term)  and **m****q ****= 0**  t(0) = 0

- arguments about whether  and t  are physical produce the same qualtitative results

- ’2 ~ 1/Nc for large Nc

- Witten-Veneziano relation

## Instantons

- In the **conformal limit** (no IR brane, C~s const)

- With **IR brane** and z-dependent ****** = ********(******)

**Expanding** around the conformal solution, and integrating the action by parts on the bulk-to-boundary equation of motion gives

- This is the same as the **instanton** contribution

- with () ~ (z) and z~

## Conclusions

-   The **holographic** version of **QCD** works **really really well **
-  meson **spectrum**, coupling **constants** predicted
-  insights into **v**ector **m**eson **d**ominance, **KRSF**
-  **higher spin** fields can be understood
-  insights into **regge physics**
-  The **U(1) problem** is solved **quantitatively** and analytically (i.e. not with a lattice)
-  **** and ******’** masses predicted to **5%** and **9%** respectively
-  decay rate ****** **********predicted well ’ is IR-model-dependent
-  simple **mixing angle** interpretation for ’ decays fails
-  new handles to turn off the anomaly
-  **Witten-Veneziano** relation reproduced
-  Direct connection with **instantons**, with z ~  and an interpretation of the **instanton density**
-  Many **open questions**
-  **Why** does AdS/QCD work so well?
-  Can the expansion be made **systematic** to all orders?
-  Can we calculate **useful** non-perturbative observables: form factors, hadronic matrix elements, etc
-  Can lessons from AdS/QCD be applied to **other **strongly coupled** gauge theories**

- (e.g. RS/technicolor)