The N to Delta transition form factors from Lattice QCD

Antonios Tsapalis

University of Athens, IASA

EINN05, Milos, 21-9-2005

outline

• Nucleon Deformation &

N- transition form factors

• LATTICE QCD:

Hadronic states and transitions between them

Limitations

• Calculation of the N-transition matrix element

• Results:

Quenched QCD

Dynamical Quarks included

• Outlook

1

2S

1 1

2 2| | 0Q

2 20 ( )(3 )Q dr r z r

Spectroscopic quadrupole moment vanishes

Intrinsic quadrupole moment w.r.t. body-fixed frame exists

0 0Q

0 0Q

prolate

oblate

modelling required !

Nucleon Deformation

u d

u

u d

u

γ* Μ1 , Ε2 , C2

Μ1+ , Ε1+ , S1+ πo

p(qqq)

I = J =

938 MeV

2

1

2

1

Δ(qqq)

I = J =

1232 MeV

2

3

2

3

Spherical M1Deformed M1 , E2 , C2

Deformation signal

(1232)N

3/ 213/ 21

EMR=ReE

M

3/ 21

3/ 21

CMR=ReS

M

EMR & CMREMR & CMRExperimental StatusExperimental Status

-20

-16

-12

-8

-4

0 1 2 3 4 5

BATESMAMI

HALL CCLAS

BONN

CM

R

Q2

-5

-4

-3

-2

-1

0

1

2

0 1 2 3 4 5

BATES

MAMI

LEGS

HALL C

CLAS

EM

R

Q2

Thanks to N. Sparveris (Athens, IASA)

uncertainties in modelling final state interactions

Lattice QCD

• Rotate to Euclidean time: t -i

| ( ) | (0)He

( )x

• Discretize space-time

X (

( ),

a aigT A xxU e

Fermions on sites

Gauge fields on links

1

2

3

4

1 2 3 4

11 { [ ] . .}

6Tr U U U U h c

1

2TrF F

( )i D m

1 2

1 11 2

1( 4) [(1 ) (1 ) ]

2{ }m U U

Wilson-Dirac operator DW

Plaquette gauge action

Wilson formulation (1974)

a

,

[

[ ]

]det[ ( )]

W yxx y

gauge

gauge D

W

S U

S U

Z D D DUe

DUe D U

1

1

[ ]

ˆ ˆ | ( , , ).... ( , , ) |

1d ˆ ˆ= ( , , ).... (et[ ( )] , , )gauge

W

n

n

S U

O U O U

DU O U O UZ

e D U

Generate an ensemble of gauge fields {U} distributed with the Boltzmann weight

Calculate any n-point function of QCD

2

1

6g

Limitations

• finite lattice spacing a ~ 0.1 fm (momentum cutoff ~ /a)

• finite lattice volume La ~ 2-3 fm

• finite ensemble of gauge fields U

• det(DW) very expensive to include

set det(DW) = 1 quenched approximationignore quark loops

• DW breaks chiral symmetry

heavy quarks ; m > 400 MeVOverlap or Domain-Wall maintain chiral symmetry

but very CPU expensive

The Transition Matrix Element1/ 2

2( , ) | | ( , ) ( , ) ( , )

3N

N

m mp s J N p s i u p s O u p s

E E

2E2

2M1

G (q )EMR=-

G (q )

2 21 2 2 2

21 2( ) ( ) ( )M C CE EMG q G qO K K q KG

magnetic dipole electric quadrupole scalar quadrupole

2C2

2M1

q G (q )CMR=-

2 G (q )M

static frame

H.F.Jones and M.C.Scadron, Ann. Phys. (N.Y.) 81,1 (1973)

Hadrons and transitions in Lattice QCD

• generate a baryon at t=0

• annihilate the baryon at time t

• measure the 2-pt function • extract the energy from the exponential

decay of the state in Euclidean time

5( ) [ ( ) ( )] ( )p abc a b cB x u x C d x u x

( , ) ( ,0) ( , ) (0) Nip x ENN p N p dxe B x B Z e

B(x) B(0)

u

ud

Nx

• generate a nucleon at t=0

• inject a photon with momentum q at t=t1

• annihilate a Delta at time t=t2

• measure the 3-pt function • extract the form factors from suitable ratios

of 3-pt and 2-pt functions

Quenched Results

323 x 64 lattice

β = 6.0

200 gauge fields

Wilson quarks

La = 3.2 fm

C. Alexandrou, Ph. de Forcrand, H. Neff, J. Negele, W. Schroers and A. Tsapalis PRL, 94, 021601 (2005)

EMR (%)

CMR (%)

V. Pascalutsa & M. Vanderhaeghen, hep-ph/0508060

NLO results at Q2 = 0.1 GeV2

In Chiral Effective Field Theory

eN eN

expansion scheme

NM M

is small

~ 1 GeV

Non-analyticities in m reconcile the heavy quark lattice results

with experiment

fit low energy constants

Full QCD

Hybrid schemevalence quarks

‘domain wall’ quarkssea quarks

• 2 light + 1 heavy flavour

• action with small

discretization error

203 x 32 0.60

203 x 32 0.50

283 x 32 0.36

V m(GeV)

a=0.125 fm}

C. Alexandrou, R. Edwards, G. Koutsou, Th. Leontiou, H. Neff, J. Negele, W. Schroers and A. Tsapalis

good chiral properties; lighter pions

very CPU expensive

GM1 : dynamical vs quenched @ mπ = 0.50 GeV

GM1

conclusions

• accurate determination of GM1 in quenched theory ;

deviation from fitted experimental data (MAID)

• The N to Delta transition form factors can be studied

efficiently using Lattice QCD

• EMR & CMR negative ; nucleon deformation

• calculation with dynamical quarks in progress ;

smaller volumes increased noise

• higher statistics is required in order to reach the

level of precision necessary for the detection of

unquenching effects (pion cloud)