Transcript Διαφάνεια 1

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
• Calculation of the N-D transition matrix element
• Results:
Quenched QCD
Dynamical Quarks included
• Outlook
Nucleon Deformation
1
S
2
Spectroscopic quadrupole moment vanishes

1
2
|Q|
1
2
 0
Intrinsic quadrupole moment w.r.t. body-fixed frame exists
Q0   dr  (r )(3z  r )
2
2
Q0  0
prolate
Q0  0
oblate
modelling required !
N  D(1232)
γ* Μ1 , Ε2 , C2
Δ(qqq)
p(qqq)
I=
1
2
J=
1
2
u
d
u
u
938 MeV
d
u
Μ1+ , Ε1+ , S1+
Spherical  M1
Deformed  M1 , E2 , C2
Deformation signal
πo
I=
3
2
J=
1232 MeV
 E13/ 2 
EMR=Re  3/ 2 
 M 1 
 S13/ 2 
CMR=Re  3/ 2 
 M 1 
3
2
EMR & CMR
Experimental Status
uncertainties in modelling final state interactions
2
BAT ES
LEGS
MAM I
HALL C
BATES
MAMI
CLAS
-4
1
HALL C
CLAS
BONN
0
CMR
EMR
-8
-1
-12
-2
-3
-16
-4
-20
-5
0
1
2
2
Q
3
4
5
0
1
2
3
Q
4
5
2
Thanks to N. Sparveris (Athens, IASA)
Lattice QCD
• Rotate to Euclidean time: t  -i t
|  (t )  e
 Ht
|  (0) 
• Discretize space-time
Fermions
on sites
 ( x)
igT a Ama ( x )
U x ,m  e
Gauge fields
on links
X
(m1,2,3)
t (m4)
Wilson formulation (1974)
1
m
TrFm F
2
3
1
1  {Tr[U1U 2U 3U 4 ]  h.c.}
6
a
2
4
1
Plaquette gauge action
1
2
m
 (i m D  m)
1
(m  4) 1 1  { 1[(1   m )U m  (1   m )U m ]} 2
2
Wilson-Dirac operator DW
Generate an ensemble of gauge fields {U}
distributed with the Boltzmann weight
Z   D D DUe
  DUe
  S gauge [U ]
  S gauge [U ]
 x DW y
x,y
det[ DW (U )]

1
6g 2
Calculate any n-point function of QCD
  | Oˆ1 (U , , )....Oˆ n (U , , ) |  
1
  S gauge [U ]
=  DUe
det[ DW (U )]Oˆ1 (U , , )....Oˆ n (U , , )
Z
Limitations
• finite lattice spacing a ~ 0.1 fm
(momentum cutoff ~ p/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 approximation
ignore quark loops
• DW breaks chiral symmetry
heavy quarks ; mp > 400 MeV
Overlap or Domain-Wall maintain chiral symmetry
but very CPU expensive
The Transition Matrix Element
1/ 2
2  mD mN 
tm


 D( p, s) | J | N ( p, s)  i
u
(
p
,
s
)
O
u ( p, s )
t


3  ED EN 
m
H.F.Jones and M.C.Scadron, Ann. Phys. (N.Y.) 81,1 (1973)
Otm  GM 1 (q 2 ) K Mtm1  GE 2 (q 2 ) K Etm2  GC 2 (q 2 ) KCtm2
magnetic dipole
electric quadrupole
scalar quadrupole
static D frame
2
G E2 (q )
EMR=2
G M1 (q )
q
2
G C2 (q )
CMR=2M D G M1 (q 2 )
Hadrons and transitions in Lattice QCD
B p ( x)   abc [ua ( x)C 5d b ( x)]uc ( x)
u
B(x)
u
d
B(0)
• 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
N ( p,t ) N ( p, 0)   dxe  ip x  B( x ,t ) B(0) 
Z N e  EN t

D
x
N
• 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
C. Alexandrou, Ph. de Forcrand, H. Neff, J. Negele,
W. Schroers and A. Tsapalis PRL, 94, 021601 (2005)
323 x 64 lattice
β = 6.0
200 gauge fields
Wilson quarks
La = 3.2 fm
EMR (%)
CMR (%)
V. Pascalutsa & M. Vanderhaeghen, hep-ph/0508060
eN  eNp
In Chiral Effective
Field Theory
dexpansion scheme
MD  MN
is small
d

 ~ 1 GeV
fit low energy constants
NLO results at Q2 = 0.1 GeV2
Non-analyticities in mp
reconcile the heavy
quark lattice results
with experiment
Full QCD
C. Alexandrou, R. Edwards, G. Koutsou, Th. Leontiou,
H. Neff, J. Negele, W. Schroers and A. Tsapalis
Hybrid scheme
valence quarks
sea quarks
‘domain wall’ quarks
• 2 light + 1 heavy flavour
• action with small
good chiral properties;
lighter pions
discretization error
very CPU expensive
V
mp (GeV)
203 x 32
0.60
203 x 32
0.50
283 x 32
0.36
}
a=0.125 fm
GM1 : dynamical vs quenched @ mπ = 0.50 GeV
GM1
conclusions
• The N to Delta transition form factors can be studied
efficiently using Lattice QCD
• accurate determination of GM1 in quenched theory ;
deviation from fitted experimental data (MAID)
• 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)