nucleon axial and nucleon-to-delta axial transition form factors from lattice qcd
DESCRIPTION
Nucleon Axial and Nucleon-to-Delta Axial Transition Form Factors from Lattice QCD. A. Tsapalis Institute of Accelerating Systems and Applications University of Athens. in collaboration with C. Alexandrou (Univ. of Cyprus) G. Koutsou (Univ. of Cyprus) Th. Leontiou (Univ. of Cyprus) - PowerPoint PPT PresentationTRANSCRIPT
Nucleon Axial and Nucleon-to-Delta Axial Transition Form Factors from
Lattice QCD
A. Tsapalis
Institute of Accelerating Systems and Applications
University of Athens
in collaboration with
C. Alexandrou (Univ. of Cyprus)
G. Koutsou (Univ. of Cyprus)
Th. Leontiou (Univ. of Cyprus)
J. W. Negele (MIT)
outline
• Nucleon Axial Form Factors GA and GP
• PCAC and pion pole dominance • Nucleon-to-Delta Axial Transition FFs• Lattice Evaluation of the FFs • Results – Checking the Pion Pole dominance &
Goldberger-Treiman (GT) relations in N-N & N-Δ• Conclusions
arXiv:0706.3011, to appear in PRD
Nucleon Axial Form Factors
axial isovector current ψ(x)τ
γγ(x)ψ(x)Aa
μaμ 25
),(2
)(2
)(),(),(||),( 35
25
23 spuqGm
qqGspuspNAspN p
NA
axial vector form factor induced pseudoscalar
GA (q2) – from neutrino scattering & pion electroproduction
GP (q2) – from muon capture experiments
• theoretically studied in chiral effective theories
axial charge GA (0) = 1.2695 (29) from nuclear β decay
• pioneering lattice study in PRL 74 2172 (1995) (K.F. Liu, S.J. Dong, T. Draper, W. Wilcox)
• recent study by LHPC+MILC , arXiv:0705.4295
Pseudoscalar Form Factor & PCAC
aa mfA 2PCAC in hadron world
aq
a PmA 2 Axial WT identity in QCD
ψ(x)τ
γ(x)ψ(x)Pa
a
25 pseudoscalar current
πΝΝ form factor defined via
),(),()(
),(||),(2 522
223 spuispu
qm
qGmfspNPspNm NN
q
connected to πΝΝ strong coupling constant gπΝΝ = GπΝΝ(mπ2)
)(2
2
1)(
4)( 2
22
22
2
22 qG
qm
mf
mqG
m
qqG NN
NP
NA
PCAC
Pion Pole dominance & GT relations
)(2
2
1)(
4)( 2
22
22
2
22 qG
qm
mf
mqG
m
qqG NN
NP
NA
Pion pole on RHS constraints the induced pseudoscalar
)(2
~)(2
1 222
2 qGqm
fqG
m NNPN
and leads to Goldberger-Treiman relation
)(~)( 22 qGm
fqG NN
NA
at q2 = 0 satisfy NN
NA g
m
fg
to 5% accuracy
)1(2.13NNg
from low energy πΝΝ dynamics
also fixes the ratio
22
24/
qm
mGG NAP
Nucleon to Δ(1232) Axial Transition Form Factors
transverse part
),()(
)()()(
),(3
2
),(||),(
2
262
52
24
23
3
spuqqm
qCgqCqggggp
m
qC
m
qCspu
spNAsp
N
AA
N
A
N
A
Adler parameterization
small ≈ 0 dominant FFs
C5A
analogous to GA (q2) C6A
analogous to GP (q2)
• not much known experimentally
• electroproduction experiments at JLab will measure N to Δ parity violating asymmetry
connected to C5A
• theoretical arguments indicate that C3A, C4
A are small
• Lattice study in PRL 98,052003 (2006) established smallness of C3
A and C4A , predicted q2 dependence
of dominant form factors C5A and C6
A
Pseudoscalar πΝΔ Form Factor & PCAC
πΝΔ form factor defined via
connected to πΝΔ strong coupling constant gπΝΔ = GπΝΔ(mπ2)
PCAC
Non-diagonal Goldberger-Treiman relation
),(2),(
)(
3
2),(||),(2
22
223 spu
m
qspu
qm
qGmfspNPspm
N
Nq
)(2
1)()( 2
22
22
62
22
5 qGqm
mf
mqC
m
qqC N
N
A
N
A
)()(2
~)(1 2
222
6 qGqm
fqC
m NA
N
Pion pole dominance relates:
)(2
)( 225 qG
m
fqC N
N
A
..and fixes the ratio
22
2
56 / qm
mCC NAA
Evaluating Form Factors from Lattice QCD
• measure 3-point-functions of axial & pseudoscalar currents
• form ratios where t- and Z- dependence cancels
|)0,0(),(),(|
))(exp()exp();,;,(
113
22
,1212
12
3
txAtxT
xppixpippttGxx
NAN
);;,(););(2();;2(
);,;,();;,;,( 1,1
41241
12
12121
3
pp
pttGptG
ppttGppttR Attt
NNNN
NAN
A
• determine the optimal linear combination of 3pts
00
0
2
1,
00
0
2
14
Iii
kinematics: qpp
,0
)( pN )( pN
),( 22 tx
),( 11 tx
)0,0(
X
5
]
[
)(2)(
)())((4
);;,0();(
2321
2,3,2,1
3
1
QGm
qqqq
QGmEm
CijqjqS
PN
j
AjjjNNNk
kAA
maximal number of momentum vectors contribute in rotationally symmetric fashion
optimizing the measurement
• sequential inversions through the sink
)( pN )( pN
),( 22 tx )0,0(
),( 11 txX
)( pN )( pN
),( 22 tx )0,0(
15
xqie
• only one sequential inversion for GA(Q2), GP(Q2), GπNN(Q2)
• all operators and momenta q
measured at small cost
• look for plateau in t1 / Smear source & sink quarks to damp
fast the excited states
)()(22
);;,0();( 222
2321
5
3
15 QG
Qmm
mf
m
qqqCqqS NN
qNkk
PP
• simultaneous overconstrained analysis of all data
maximal accuracy for the form factors – Q2 dependence
Lattice parameters
Wilson NF = 0 β=6.0
323x64 a=0.09 fm
mπ = 0.56 GeV
mπ = 0.49 GeV
mπ = 0.41 GeV
Wilson NF = 2 β=5.6 a=0.08 fm
243x40 mπ = 0.69 GeV (TXL)
243x40 mπ = 0.51 GeV (TXL)
243x32 mπ = 0.38 GeV (DESY)
Nucleon Axial
N-to-Δ Axial +
Hybrid scheme
MILC NF = 2 + 1
Domain Wall valence (L5=16)
a=0.125 fm
ams amu mπ 0.05 0.03 0.59 GeV 203x64 0.05 0.02 0.50 GeV 203x64 0.05 0.01 0.36 GeV 283x64
plateaus for GπΝΝ
Wilson NF=0, 323x64, mπ=0.49 GeV
NF=2, 243x40, mπ=0.69 GeV
Wilson NF=0, 323x64, mπ=0.41 GeV
GA(Q2), GP(Q2), GπNN(Q2) C5A(Q2), C6
A(Q2), GπNΔ(Q2)
MILC(DWF) 0.01/0.05, mπ=0.36 GeV 203x64 vs 283x64,
source-sink distance 11a vs 13a
Volume (2.5fm)3 vs (3.5fm)3
Ground state dominance
Checking the parameters
Results (I) – Nucleon Axial Form Factors
Hybrid results from 0705.4295 LHPC & MILC (Hägler etal)
222
02
1/)(
A
AmQ
gQG
• dipole fit describes well GA
mA >=1.5 GeV (solid / fit) mA=1.1 GeV (dotted / exp)
• pion pole dominates Gp
22
24/
mQ
mGG NAP
(dash)
monopole fit (solid)
Results (II) – N to Δ Axial Transition FFs
• dipole fit describes well CA5
mA >=1.5 GeV (solid / fit) mA=1.28 GeV (dotted /
exp) • pion pole dominates CA
6
22
2
56 /mQ
mCC NAA
(dash: wilson)
(dot: MILC)
monopole fit (solid)
Results (III) – Checking Ratios of GT relations
)(
)(8
)(
)(2
)(
)(2
26
2
25
2
2
QG
QC
QG
QC
QG
QG
p
A
A
A
NN
N
pion pole dominance
renormalization constants, fπ , mq cancel
1.63(1)
1.60(2)
1.73(3)
weak Q2 and mq dependence
Conclusions• momentum dependence of the NN & NΔ axial form factors is evaluated optimally in Lattice QCD
• dipole dependence of GA and C5A is verified –
requires larger axial mass at the 410 MeV pion lattices
• monopole behavior of Gp and C6A is verified
• unquenching effects are visible at low Q2 and mπ = 360 MeV in the Hybrid scheme (MILC+DWF) – GA approaches expected behavior
• ratios of GT relations in NN & NΔ systems are satisfied – show very weak quark mass and Q2 dependence