slides PDF - Theory Group

Πανεπιστήµιο Κύπρου 

Κ. Αλεξάνδρου 

Hadron Physics on the Lattice 

C. Alexandrou, University of Cyprus PSI, December 18th 2007

Outline 

The Highlight of Lattice 2007- Reaching the chiral regime 

- Results using light dynamical fermions 

Twisted mass - ETMC (Cyprus, France, Germany, Italy, Netherlands, UK, Spain, 

Switzerland) 

Clover improved dynamical fermions - QCDSF/UKQCD 

Overlap fermions - JLQCD 

Hybrid approach - LHPC and Cyprus 

Staggered fermions:Charm Physics - HPQCD 

- Chiral extrapolation of lattice results 


Lattice 2007: Highlights 

• Impressive progress in unquenched simulations using various 

fermions discretization schemes 


ψ(n) 

Basics: 

a 

µ 

U µ (n)=e igaAµ(n) 

S = S G + S F Difficult to simulate --> 

S F = ! (k)D(k,n)!(n) 

!O" = 

Lattice fermions 

C. Alexandrou, University of Cyprus PSI, December 18th 2007 

" 

k,n 

Integrate out 

# dUnd$ nd$ nO[U,$ ,$ ]e %S 

& 

n 

Z 

It is well known that naïve discretization of Dirac action 

S = d F 4 x !(x) # " ! + m% 

$ µ µ & !(x) 

' ( 

!(x + µ)- !(x -µ) 

! !(x) = µ 

2a 

= 

90’s Det[D]=1 : quenched 

# dUndet[D]O[U,D %1 ]e %SG & 

n 

Z 

leads to 16 species 

A number of different discretization schemes have been developed to avoid doubling: 

Wilson : add a second derivative-type term --> breaks chiral symmetry for finite a 

Staggered: “distribute” 4-component spinor on 4 lattice sites --> still 4 times more species, 

take 4th root, non-locality 

Nielsen-Ninomiya no go theorem: impossible to have doubler-free, chirally 

symmetric, local, translational invariant fermion lattice action BUT…

Chiral fermions 

Instead of a D such that {γ 5 , D}=0 of the no-go theorem, find a D such that 

{γ 5 ,D}=2a D γ 5 D Ginsparg - Wilson relation 

Kaplan: Construction of D in 5-dimensions Domain Wall fermions 

Luescher 1998: realization of chiral symmetry but NO no-go theorem! 


L 5 

Left-handed fermion 

right-handed fermion 

Neuberger: Construction of D in 4-dimensions but with use of sign function 

Overlap fermions 

D = 1 + !(D w ) 

, 

2 

!(D ) = w Dw + 

D Dw 

w 

Equivalent formulations of chiral fermions on the lattice 

But still expensive to simulate despite progress in algorithms 

Compromise: - Use staggered sea (MILC) and domain wall valence (hybrid) - LHPC 

Rely on new improved algorithms 

- Twisted mass Wilson - ETMC 

- Clover improved Wilson - QCDSF, UKQCD, PACS-CS

Simulation of dynamical fermions 

Results from different discretization schemes must agree in the continuum 

limit 


Clark


Impressive progress in unquenched simulations using various 


With staggered, Clover improved and Twisted mass dynamical 

fermions results are reported with pion mass of ~300 MeV 

PACS-CS reported simulations with 210 MeV pions! 

But no results shown yet 

RBC-UKQCD: Dynamical N F =2+1 domain wall simulations reaching ~300 MeV on ~3 fm 

volume are underway 

JLQCD: Dynamical overlap N F =2 and N F =2+1, m l ~m s /6, small volume (~1.8 fm) 

Progress in algorithms for calculating D -1 as m π physical value 

Deflation methods: project out lowest eigenvalues 

Lüscher, Wilcox, Orginos/Stathopoulos 


Deflation methods in fermion inverters -Wilcox 

Solve linear system 

Twisted mass 20 3 x32 at κ cr 



Impressive progress in unquenched simulations using various 


With staggered, Clover improved and Twisted mass dynamical 

fermions results are reported with pion mass of ~300 MeV 

PACS-CS reported simulations with 210 MeV pions! 

But no results shown yet 

RBC-UKQCD: Dynamical N F =2+1 domain wall simulations reaching ~300 MeV on ~3 fm 

volume are underway 

JLQCD: Dynamical overlap N F =2 and N F =2+1, m l ~m s /6, small volume (~1.8 fm) 

Progress in algorithms for calculating D -1 as m π physical value 

Deflation methods: project out lowest eigenvalues 

Lüscher, Wilcox, Orginos/Stathopoulos 

Chiral extrapolations 

Using lattice results on f π /m π yield LECs to unprecedented accuracy 

Begin to address more complex observables e.g. excited states and resonances, 

decays, finite density 


Chiral extrapolations 

Controlled extrapolations in volume, lattice spacing and quark masses: still needed 

for reaching the physical world 

Quark mass dependence of observables teaches about QCD - can not be obtained 

from experiment 

Lattice systematics are checked using predictions from χPT 

Chiral perturbation theory in finite volume: 

Correct for volume effects depending on pion mass and lattice volume 

Like in infinite volume 


Pion sector 

Applications to pion and nucleon mass and decay constants 

e.g. 

m ! " m ! (L) 

m ! 

f ! - f ! (L) 

f ! 

= - 

= - 

( 

x 

$ 

% 

2# Im ! 

| ! n|!0 

' 

x 

# 

$ 

! If " 

| ! n|"0 

(2) (#) + xIm! 

(2) (!) + xIf" 

x = m 2 

0 

0 2 

(4!f ) " 

, ! = m " | ! n | L 

Applied by QCDSF to correct lattice data 

r 0 =0.45(1)fm 

(4) (#) 

(4) (!) 


% 

& 

& 

' 

G. Colangelo, St. Dürr and Haefeli, 2005 

Pion decay constant 

NLO continuum χPT

Twisted mass dynamical fermions 

Consider two degenerate flavors of quarks: Action is 

$ 

( ) 

S F = a 4 !(x) D w [U]+ m 0 + iµ" 5 # 3 

x 

!(x) 

Wilson Dirac operator untwisted mass 

Advantages: - Automatic O(a) improvement (at maximal twist) 

- Only one parameter to tune (zero PCAC mass at smallest µ) 

- No additional operator improvement 

Disadvantages: - explicit chiral symmetry breaking (like in all Wilson) 


Twisted mass µ 

τ 3 acts in flavor space 

- explicit breaking of flavour symmetry appears only at O(a 2 ) 

in practice only affects π 0 

Physics results reported for ~300 MeV pions on spatial size of about 2.7 fm with a < 0.1fm

Pion decay constant with twisted mass fermions 

Simultaneous chiral fits to m π and f π 

r 0 =0.441(14)fm 

N F =2 


Extract low energy constants 

l ! log 3,4 " $ 2 ' 

3,4 

& 2 ) 

% m ± 

# ( 

l 3 = 3.44(28), l 4 = 4.61(9) 

S. Necco, 

Lat07

ππ s-wave length a 0 0 and a0 2 

Leutwyler (hep-ph/0612112) 

ππ-scattering lengths 

ETMC 


π-π scattering 

Lüscher’s finite volume approach: 

Calculate the energy levels of two particle states in a finite box 

!En = En - 2m = 2 p 2 2 

+ m - 2m 

n 

Expand ΔE 0 about p~0: 

with p n given by 

!E = - 0 4!a 

mL 3 1 + c ( 

* 1 

) 

* 

a 

L + c 2 

" a % 

# 

$ L& 

' 

where we use particle definition for the scattering length: 


2 

p cot !(p) = 1 

!L 

+ 

- 

, 

- + O 1 

" 

L 6 

# 

$ 

a = lim 

p!0 

% 

& 

' 

) ! 

n 

tan"(p) 

p 

f( ! n) 

! 

n 2 " pL # & 

$ 

% 2! ' 

( 

Applied to I=2 two-pion system within the hybrid approach (staggered sea and domain 

wall fermions) 

m π a 0 2 =-0.0426(27) 

S. Beans et al. PRD73 (2006) 

2

More statistics 

mixed action χPT formula 

no new LECs 

Other applications: I=1 KK scattering 

π-π scattering - improved 

m π a 0 2 =-0.04330(42) 

A. Walker-Loud, arXiv:0706.3026 

form independent of sea quarks if valence are chiral 

N-N scattering - too noisy 


Dynamical overlap, L~1.8 fm 

Pion form factor 

F ! (q 2 ) = 


1 

1 " q 2 2 

/ m # 

+ d1q2 + ... 

F ! (q 2 ) = 1 " 1 

6 < r2 > q 2 + ... 

< r 2 1 

>= c " ln 

0 2 

(4!f ) ! m # 

% 

$ 

Sh. Hashimoto 

2 

! 

(4!f ) ! 2 

& 

( 

' 

+ c1m 2 

!

4-point function: first application of the one-end trick (C. Michael): 

q 

G. Koutsou, Lat07 

-q 


S. Wilson Simula, ETM Collaboration 

one-trick for 3pt function 

VMD : 


1 

1! q 2 2 

/ m " 

Set initial and final pion momentum to zero 

--> no disadvantage in applying the oneend 

trick


Comparison with experiment 


S. Simula , 

Lat07

Rho form factors 

decomposed into three form factors Gc (Q2 ), GM (Q2 ), GQ (Q2 ) 

Decreasing quark mass 


Related to the ρ quadrupole 

moment G Q (0)=m ρ 2 Qρ 

Adelaide group: Q ρ non-zero --> rho-meson deformed in agreement with wave 

function results using 4pt functions 

One-end trick improves signal, application to 

baryons is underway 

G. Koutsou, Lat07 

Non-relativistic approx. 

|Ψ(y)| 2

Baryon sector 

Twisted mass and Staggered results 

Two fit parameters: m 0 =0.865(10), c 1 =-1.224(17) GeV -1 to be compared with ~-0.9 

GeV -1 (π-N sigma term) 


Delta mass splitting 

Symmetry: u d --> Δ ++ is degenerate with Δ - and Δ + with Δ 0 

Check for flavour breaking by computing mass splitting between the two degenerate 

pairs 

Splitting consistent with zero in agreement with theoretical expectations that 

isospin breaking only large for neutral pions (~16% on finest lattice) 


Hybrid actions 

Wilson fermions 

Coupling constants: g A , g πNN , g πΝΔ , … 

Baryon structure 

Form factors: G A (Q 2 ), G p (Q 2 ), F 1 (Q 2 ),…. 

Parton distribution functions: q(x), Δq(x),… 

Generalized parton distribution functions: H(x,ξ,Q 2 ), E(x,ξ,Q 2 ),… 

Masses: two-point functions 

Need to evaluate three-point functions: 

In addition: 

O(t, ! q) = d 3 x e !i! x. ! q 

" 


#(x) $ D µ 1 µ 2 ( D ... )#(x) 

Four-point functions: yield detailed info on quark distribution 

only simple operators used up tp now - need all-to-all propagators

Nucleon axial charge 

Forward nucleon axial vector matrix element: < N | u! µ ! 5 u - d! µ ! 5 d | N >= g A v(p)! µ ! 5 v(p) 

Pleiter, Lat07 

Improved Wilson (QCDSF) 

HBχPT with Δ 

Hemmert et al. , 

Savage & Beane 


accurately measured 

no-disconnected diagrams 

chiral PT 

Finite volume dependence

Isovector form factors 

obtained from 

connected diagram 

Nucleon form factors 

Disconnected diagram not evaluated 

Isovector Sachs form factors: 

p n 

G = G ! G E E E 

p n 

G = G ! G M M M 


In last couple of years 

better methods have 

become available, 

dilution, one-end trick,… 

with G E and G M given in terms of F 1 and F 2 

G E (q 2 ) = F 1 (q 2 ) + 

G M (q 2 ) = F 1 (q 2 ) + F 2 (q 2 ) 

q 2 

(2m N ) 2 F 2 (q2 ) 

Evaluation of 3pt function using the fixed 

sink approach --> any current to be inserted 

with no additional computational cost

Cyprus/MIT Collaboration, C. A., 

G. Koutsou, J. W. Negele, A. Tsapalis 

q 2 =- Q 2 

Results using dynamical Wilson fermions 

and domain wall valence on staggered sea 

(from LHPC) are consistent 

Nucleon EM form factors 

F p = 2 

3 Fu ! 1 

3 Fd 

F n = ! 1 

3 Fu + 2 

3 Fd 


QCDSF/UKQCD 

Dirac FF

Nucleon axial form factors 

Within the fixed sink method for 3pt function we can evaluate the nucleon matrix 

element of any operator with no additional computational cost 

A a 

= !" " µ µ 5 

2 ! 

use axial vector current to obtain the axial form factors 

pseudoscalar density to obtain GπN (q2 ) 

Pa # 

= !i" 5 

a 

2 ! 

< N( ! 2 

3 ! " m % 

N 

p ! , s ! ) | A | N( p,s) >= i 

µ 

$ 

# E ( p ! )E (p) 

' 

N N & 

2m < N( q ! p ! , s ! ) | P 3 | N( ! 2 " m % 

N 

p, s) >= $ 

# E ( p ! )E (p) 

' 

N N & 

PCAC relates G A and G p to G πNN : 

Pion pole dominance: 


# a 

1/2 

G (Q A 2 ) - Q2 

2 

4mN Gp (Q2 ) = 1 

2mN 1 

2m N 

u( ! 

1/2 

) 

p ) G (Q A 2 )( ( + µ 5 q µ ( 5 

+ 

2mN * + 

2 

f m G!NN (Q ! ! 

2 ) 

2 2 

m + Q ! 

2 

f m G!NN (Q ! ! 

2 ) 

2 2 

m + Q ! 

G (Q p 2 ) ~ 2G !NN (Q2 )f ! 

2 2 

m + Q ! 

u( p ! ) i( 5 

G (Q p 2 , 

) . 

-. 

) 3 

2 u(p) 

/ 3 

Th. Korzec 

2 u(p) 

Generalized Goldberger- 

Treiman relation 

! G !NN (Q 2 )f ! = m N G A (Q 2 ) Goldberger-Treiman relation

monopole fit 

G A and G p 

monopole fit to 

experimental results 

pion pole dominance 


Results in the hybrid approach 

by the LPHC in agreement with 

dynamical Wilson results 

Unquenching effects large 

at small Q 2 in line with 

theoretical expectations that 

pion cloud effects are 

dominant at low Q 2 

Cyprus/MIT collaboration

N to Δ transition form factors 


Electromagnetic N to Δ: Write in term of Sachs form 

factors: 

< !( ! p " , s " ) | j | N( µ ! p,s) >= i 2 

1/2 

# m m & 

! N 

3 

% 

$ E ( p " )E (p) 

( u 

! N ' 

) ( ! p " , s " )O u( )µ ! p,s) 

O = G (q )µ M1 2 M1 

)K + G (q )µ E2 2 E2 

)K + G (q )µ C2 2 C2 

)K )µ 

Magnetic dipole: 

dominant 

Electric 

quadrupole 

Coloumb 

quadrupole 

Axial vector N to Δ: four additional form factors, C 3 A (Q 2 ), C4 A (Q 2 ), C5 A (Q 2 ), C6 A (Q 2 ) 

PCAC relates C5 A and C6 A to GπΝΔ : 

Pion pole dominance: 

A 2 Q 

C (Q ) ! 5 

2 

2 

mN C A 2 1 

(Q ) = 6 

2m N 

1 

m N 

A 2 1 

C (Q ) ! 6 

2 

2 

f m G (Q " " "N# 2 ) 

2 2 

m + Q " 

G !N" (Q 2 )f ! 

2 2 

m + Q ! 

# G (Q !N" 2 A 2 

)f = 2m C (Q ) 

! N 5 

Dominant: C 5 A GA 

C 6 A G p 

Generalized non-diagonal 

Goldberger-Treiman relation 

Non-diagonal Goldberger-Treiman 

relation

Q 2 =0.127 GeV 2 

Quadrupole form factors and deformation 

C. N. Pananicolas, Eur. Phys. J. A18 (2003) 


Quantities measured in the lab frame of 

the Δ are the ratios: 

R EM (EMR) = ! G E2 (Q2 ) 

G M1 (Q 2 ) 

R SM (CMR) = ! | ! q | 

2m " 

G C2 (Q 2 ) 

G M1 (Q 2 ) 

Precise experimental data strongly suggest 

deformation of Nucleon/Δ 

First conformation of non-zero EMR and 

CMR in full QCD 

Large pion effects

Axial N to Δ 

Bending seen in HBχPT 

with explicit Δ - Procura et al. 


A 

C6 SSE to O(ε 3 ): 

C 5 A (Q 2 ,mπ )=a 1 +a 2 m π 2 +a3 Q 2 +loop 

2 

mN = a 4 ! 

1 

2 2 

m + Q " 

$ 

& 

% & 

2 

a + a m + a7Q 5 6 " 

2 

+loop(m ,c ,g ,g ,f ,#) 

" A A 1 " 

Again large unquenching 

effects at small Q 2 

' 

) 

( )

Curves refer to the quenched data 

G πΝ =a(1-bQ 2 /m π 2 ) 

G πΝΔ =1.6G πΝ 

GTR: G πΝ =G A m N /f π 

G πΝN and G πΝΔ 


Q 2 -dependence for G πΝ (G πΝΔ ) 

extracted from G A (C 5 A ) using the 

Goldberger-Treiman relation deviates at 

low Q 2 

Small deviations from linear Q 2 - 

dependence seen at very low Q 2 in the 

case of G πΝΔ 

G πΝΔ /G πΝ = 2C 5 A /GA as predicted by 

taking ratios of GTRs 

G πΝΔ /G πΝ =1.60(1)

Goldberger-Treiman Relations 

Deviations from GTR 

Improvement when using 

chiral fermions 


1 

2m N 

1 

m N 

G (Q p 2 ) ! 2G !N (Q2 )f ! 

2 2 

m + Q ! 

" G !N (Q 2 )f ! = m N G A (Q 2 ) 

A 2 1 

C (Q ) ! 6 

2 

G !N" (Q 2 )f ! 

2 2 

m + Q ! 

# G (Q !N" 2 A 2 

)f = 2m C (Q ) 

! N 5

Four form factors: G E0 , G E2, G M1, G M3 

given in terms of a 1 , a 2 ,c 1 ,c 2 

Like for the nucleon form factors 

only the connected diagram is 

evaluated --> isovector part 

G E0 

Δ electromagnetic form factors 

Q 2 (GeV 2 ) 

< !( ! p " , s " ) | j µ | !( ! p, s) >= i 

O #µ$ = % #$ a 1 i& µ + a 2 

m π =0.56 GeV 

m π =0.49GeV 

m π =0.41 GeV 

2m ! 


' 

) 

( 

( " 

2 

m ! 

E ( " ! ! 

p )E ( ! ! p) u ( p " , " # 

* 

p µ + p µ ) , - q# q $ 

2 

+ 

G M1 

2m ! 

' 

) 

( 

Q 2 (GeV 2 ) 

s )O#µ$ u $ (p,s) 

c 1 i& µ + c 2 

2m ! 

( ! " 

p µ + ! p µ ) 

m π =0.56 GeV 

m π =0.49 GeV 

m π =0.41 GeV 

Accurate results for the dominant form factors calculated in the quenched approximation. 

Unquenched evaluation is in progress - Cyprus/MIT Collaboration, C. A., Th. Korzec, Th. Leontiou J. W. 

Negele, A. Tsapalis 

* 

, 

+

Electric quadrupole Δ form factor 

G E2 connected to deformation of the Δ: 

G E2

Moments of parton distributions 

Parton distributions measured in deep inelastic scattering 

Forward matrix elements: 

< N(p, s) | O { µ 1 ...µ n} 

| N(p, s) >~ dx 

q 


! x n-1 q(x) 

< x n > = dx x q n q(x) + (-1) n+1 1 

% ! 

" 

q(x) # 

$ 

q = q + q & ' 

-1 

< x n > = dx x (q n 

1 

% (q(x) + (-1) 

-1 

n ! 

" 

(q(x) # 

$ (q = q& - q' < x n > = dx x )q n 

! ! µ unpolarized moment 

1 ...µ n µ 1 µ µ 

O = q " 

2 ! i D ...i D n} $ 

q # 

% 

polarized moment 

transversity 

1 

% 

-1 

q 

µµ 1 ...µ n O = q ! ! 5q 

5 µ i ! D µ ! µ " 

1 ...i D n} $ 

# 

% q 

µ&µ 1 ...µ n O = q ! ' Tq 

5 µ& i ! D µ ! µ " 

2 ...i D n} $ 

# 

% q 

! 

D = 1 

2 ( " D ( # Moments of parton distributions 

3 types of operators: 

D) 

)q(x) + (-1) n+1 ! 

" 

)q(x) # 

$ )q = qT + q *

Moments of Generalized parton distributions 

Off diagonal matrix element Genelarized parton distributions 

$ n-1 

µ 1 ...µ n 

< N(p',s') | O 

{ } q 

| N(p,s) >= u( p ! ) & 

A q B 

Ani (t)K + Bni (t)K q 

&# 

ni 

ni 

i=0 

% &even 

GPD measured in Deep Virtual Compton Scattering 

p = 1 

p + ! 

2 p ( ), 

t = #Q 

q = (x + ")p, q ! = (x # ")p 

2 = $ 2 

q C 

{ } + " C (t)K n,even ni n 


Generalized FF 

H n (!, t) " dx 

E n (!, t) " dx 

' 

) u(p) 

) 

( ) 

1 

# x 

-1 

n-1 H(x, !, t) 

1 

# x 

-1 

n-1 E(x, !, t) 

H n (0, t) = A n0 (t) A 10 (t) = F 1 (t) 

E n (0, t) = B n0 (t) B 10 (t) = F 2 (t) 

t=0 (forward) yield moments of 

parton distributions 

GPDs

Transverse quark densities 

Fourier transform of form factors/GPDs gives probability 

Transverse quark distributions: 

u-d 

NLO: g A,0 --> g A,lat , same with f π 

Self consistent HBχPT 

LHPC 


1 

dx x n-1 

! q(x, ! b ) = " d2 # " 

(2$ ) 2 ! e -i! b " . ! # ! 

" q 2 

An0 (- #" ) 

-1 

q(x % 1, ! b " ) & ' 2 (b " ) 

M. Burkardt, PRD62 (2000) 

q(x, ! 2 d 

b ) = ! 

2 " ! 

(2# ) 2 $ e -i! b ! . ! " ! 2 

H(x,% = 0, -"! ) 

as n increases slope of A n0 decreases

Total spin of quark: 

Spin of the nucleon: 

Spin structure of the nucleon 

J q 

q ( (0) ) L = J - q q 1 

1 

= 

2 < x > q +B20 QCDSF/UKQCD Dynamical Clover 

1 

2 = Lu+d + 1 

2 !" u+d + J g 

A20 (0) 

For m π ~340 MeV: J u =0.279(5) J d =-0.006(5) 

2 !" q 

L u+d =-0.007(13) ΔΣ u+d =0.558(22) 

In agreement with LPHC, Ph. Hägler et al., hep-lat/07054295 


!" = ! q 

A (0) 10

Unpolarized 

d 

u 

Transverse spin density 


q unpolarized N unpolarized 

N 

N 

u 

d


HPQCD: C. Davies et al. 

Charm Physics 

Determination of CKM matrix, Test unitarity triangle 

Weak decays - calculate in lattice QCD 

0.9 1.0 1.1 0.9 1.0 1.1 

Quenched 

Bench mark 

First message: Use unquenched QCD 

N f =2+1 dynamical 

staggered fermions 

Full QCD results show impressive agreement with 

experiment across the board - from light to heavy quarks 


Lattice QCD

Weak Decays and CKM matrix 

For charm use highly improved staggered action remove further cut off effects at 

α(amc ) 

Leptonic D-meson decays: 

2 and (amc ) 4 

Semi-leptonic: 

in progress 

Lattice results versus experiment - CLEO-c 


f Ds 

c 

d 

l 

ν 

f Ds /f D =1.164(11)

B-decays 

Use non-relativistic QCD for b-quark with m b fixed from m Υ 

f Bs /f Bq 

Log- dependence on 

light quark mass 

m q/ m s 

f B = 216(22) MeV, 


f Bs 

f B 

= 1.20(3) 

to be compared with experiment 

ICHEP 06: f B =229(36)(34)MeV 

Semileptonic B-decays 

B-->πlν

Unitarity triangle 


Using lattice input 

Expectation: errors on lattice results 

should halve in the next two years

Conclusions 

• Lattice QCD is entering an era where it can make significant contributions in 

the interpretation of current experimental results. 

• A valuable method for understanding hadronic phenomena 

Accurate results on coupling constants, Form factors, moments of 

generalized parton distributions are becoming available close to the chiral 

regime (m π ~300 MeV). 

More complex observables are evaluated e.g excited states, 

polarizabilities, scattering lengths, resonances, finite density 

First attempts into Nuclear Physics e.g. nuclear force 

• Computer technology and new algorithms will deliver 100´s of Teraflop/s in 

the next five years 

Provide dynamical gauge configurations in the chiral regime 

Enable the accurate evaluation of more involved matrix elements

slides PDF - Theory Group

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?