# Presentation: 016080

## Moebius DW Fermions&Ward-Takahashi identities*

** *****Brookhaven Nat’l Lab.  Ides of March,  2007***

****RCB, Hartmut Neff and Kostas Orginos*
*hep-lat/0409118  & hep-lat/0703XXX*

***Richard C. Brower***

**Notes:**
Lectures:
Ancient Lore
 Before QCD
	                String theory was discovered before QCD  Why?
            	Field theory appeared to fail 3 out 4 time:
           	Maybe Hadronic and Gravity needs a new approach
            	Nuclear Democracy and search for small parameter
            	Regge and Dolan-Horn-Schmid Duality
            	The Pion amplitude and Chiral L
After QCD
                     1/N is string theory:  weak and strong coupling 
                      Super strings & Failure of the QCD string.
String/Gauge Duality
           AdS/CFT Maldacena conjecture
                   Dbranes and the strong coupling (gravity) limit
           Breaking Conformal and SUSY 
                  Deconfinement and Confinement
                   Hard vs Soft Scattering
                  Regge/Large Angles/DIS
            Non-perturbative Physics in the 5th dimension
                  Finite T
                  Instantons/
                  Quarks/Baryons
                  Friossart Bound.

QCD String Spectrum vs. Lattice Data
         Glueball Spectra: 
               M theory Spectra at strong coupling
               Comparison with Bag and Lattice Spectra
         Stretched String
               Luscher term and corrections
               Seeing the 5th dimension 
               Lattice sees the QCD string 

## Need to implement Overlap Operator(aka Ginsparg Wilson Relation)

- Need to implement Overlap Operator*(aka Ginsparg Wilson Relation)*

- Algorithm: Choose Rational approx:
-    L[x] ' x/|x|

*Physics: Choose 4-d “kernel”:*
- H  ́ 5 D(M5)

***Two steps***

## L[x]| at  Ls = 16 (polar) vs 8 (Zolotarev)

- L[x]| at  Ls = 16 (polar) vs 8 (Zolotarev)

## Mobius generalization of Shamir/Borici

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- Shamir: b5 = a5, c5 = 0

- Borici: b5 = c5 = a5

- s = 1

- s = 2

- s = 3

- s = L_s

- x

- x  - 

- x  + 

- D-

- D+

- D-

## Modified Even/Odd 4-d Checkerboard 

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| --- | --- | --- | --- | --- |
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- b5

- a5

- 1+5

- 1-5

- b5

- x

- x  - 

- x  + 

- s = 1

- s = 2

- s = 3

- s = L_s

##  Even/Odd Partition of  Matrix

- On 163 x 32,  = 6.0 lattice
- For Shamir: Both 4-d & 5-d  Even/Odd give »  2.7 speed up.
- For Borici/Moebius: Even/Odd gives » 2.7 speed up

- Iee and Ioo SHOULD be simple to invert

##  DW Construction: Shamir vs Borici

## Moebius Generalization 

- Parameters: M5 ,  a5 = b5 – c5 and  scale: a = b5 + c5

***Since  ***
***Moebius is an new (scaled) polar algorithm***

## 163 x 32 Gauge Lattice @  = 6 & m = 0.44

***16******3****** x 32 Gauge Lattice @ ************ = 6 & m************ = 0.44***

## Domain Wall Implementation

*L**s**  **£** L**s**  DW Matrix:  *

- ( need s dependence for Chiu’s Zolotarev:
- b5(s) + c5(s) = (s), b5(s) – c5(s) = a5)

## To get all the nice identities for Borici, Chiu and Mobius

- Generalized 5 Hermiticity and All That

## G-W error operator: 

- Chiral violation for Overlap Action        (Kikukawa & Noguchi hep-lat/9902022)

##  Edwards & Heller use “Standard” UDL decomposition 

-  Edwards & Heller use “Standard” UDL decomposition

- Step #1: Prepare the Pivots by Permute Columns

## Step #3 Back substitution to get L matrix

- Step #2: Do Gaussian Elimination to get U matrix

- where

## LUD =>   

## DW/Overlap Equivalence:

***note: ******Standard approach****** ***

***where***

## Application of DW/overlap equ  to currents 

- Application of DW/overlap equ  to currents

***z***

***z,1***

***y,1***

***y***

## Split Screen Correlators: 5-d Vector => 4-d Axial

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- s = 1

- s = 2

- s = M

- s = L_s

**q****L**

**q****R**

**Q****L**

**Q****R**

**q****L**

**q****R**

**Q****R**

**Q****L**

- LEFT

- RIGHT

## Bulk to Boundary  Propagators

- where s = M plane

- (See Kikukawa and Noguchi, hep-lat/99902022)

**y**

**y**

## Ward Takashi: DW =>  Overlapy  

***implies***

***where***

*( **y** For single current   disconnected diagram gives anomaly)*

## Measuring the Operator Ls

- (use Plateau region away from sources)

- Sum over t  Measure Matrix element of  L operator

*|**** > in the Eigen basis of H = ****5** D(-M) *

## where

- Derivation:

## Model for mres dependence on  & Ls 

***(****) has negligible dependence on **** and L**s*

*(Parameterize and fit m**res** data)*