the n to delta transition form factors from lattice qcd
DESCRIPTION
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- D transition form factors LATTICE QCD: Hadronic states and transitions between them Limitations - PowerPoint PPT PresentationTRANSCRIPT
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)