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