Nucleon Form Factors from Lattice QCD

Nucleon Form Factors from Lattice QCD
NUCLAT April 22, 2008
James Zanotti
University of Edinburgh
QCDSF & UKQCD Collaborations

Acknowledgements
QCDSF:
M. Goeckeler
Ph. Haegler
R. Horsley
Y. Nakamura
D. Pleiter
RBC/UKQCD:
Y.Aoki
T. Blum
H.W. Lin
M. Lin
P. Rakow
A. Schaefer
G. Schierholz
W. Schroers
H. Stueben
S. Ohta
S. Sasaki
R. Tweedie
T. Yamazaki

Outline
Electromagnetic Form Factors
Introduction & Motivation
Experimental Status
Lattice Techniques
q 2 Scaling
Static Quantities (Magnetic Moment, Charge Radii)
Accessing Small q 2
Twisted Boundary Conditions
Conclusions and Outlook

Introduction to Form Factors
F(q 2 )
q 2
If a nucleon was a point-like object with no internal structure,
a probe would simply measure its e.g. charge for all q 2

Introduction to Form Factors
F(q 2 )
q1
q 2
If a nucleon was a point-like object with no internal structure,
a probe would simply measure its e.g. charge for all q 2

Introduction to Form Factors
F(q 2 )
q2
q 2
If a nucleon was a point-like object with no internal structure,
a probe would simply measure its e.g. charge for all q 2

Introduction to Form Factors
F(q 2 )
q3
q 2
If a nucleon was a point-like object with no internal structure,
a probe would simply measure its e.g. charge for all q 2

Introduction to Form Factors
F(q 2 )
q3
q 2
But it’s not

Introduction to Form Factors
Quark (charge) distribution in transverse plane
∫
q(b 2 ⊥)= d 2 q ⊥ e −i⃗ b ⊥·q ⊥
F 1 (q 2 )
Distance of (active) quark to the centre of
momentum in a fast moving nucleon
R ⊥
b ⊥
P z

Introduction to Form Factors
Quark (charge) distribution in transverse plane
∫
q(b 2 ⊥)= d 2 q ⊥ e −i⃗ b ⊥·q ⊥
F 1 (q 2 )
Distance of (active) quark to the centre of
momentum in a fast moving nucleon
Provide information on the size and
internal charge densities
R ⊥
b ⊥
P z

Electromagnetic Form Factors
〈p ′ , s ′ |J µ (⃗q)|p, s〉 = ū(p ′ , s ′ )
( )
dσ dσ
dΩ = dΩ
point
Sachs Forms:
G E (q 2 ) = F 1 (q 2 ) − q2
(2m) 2 F 2(q 2 )
G M (q 2 ) = F 1 (q 2 ) + F 2 (q 2 )
Study
q 2
dependence
|F (q 2 )| 2
}
NP
Extract magnetic moments and charge radii
[
γ µ F 1 (q 2 ) + iσ µν q ]
ν
2m F 2(q 2 ) u(p, s)
k
q
k’
p
p’

Scaling of Form Factors
Naive expectation from dimensional counting
[Brodsky & Farrar, 1973]
F 1 ∝ 1 Q 4 (dipole?)
F (0)
(1 + Q 2 /M 2 ) p
F 2 ∝ 1 Q 6 (tripole?)
for
Q 2 > ζ pQCD
Q 2 F 2
F 1
∝ const
G E
G M
∝ const

Scaling of Form Factors
Experimental (Rosenbluth) data is found to satisfy:
G p E (Q2 ) =
G p,n
M (Q2 ) =
with
1
(1 + Q 2 /M 2 D )2
µ p,n
(1 + Q 2 /M 2 D )2
M D ≈ 0.71 GeV
µ p = 2.79 µ N
µ n = −1.91 µ N

Scaling of Form Factors
Reviews:
Gao [nucl-ex/0301002]
Arrington et al. [nucl-th/0611050]
JLab results suggest
QF 2
Q 2 F 2p
/F 1p
2
1.5
1
0.5
0
(a)
SU(6) + CQ ff
SU(6)
CQM
Soliton
PFSA
VMD
F 1
0.75
(b)
scaling
QF 2p
/F 1p
0.5
0.25
Jones et al.
Gayou et al.
0
0 1 2 3 4 5 6
Q 2 (GeV 2 )

Scaling of Form Factors
JLab results show deviation from µ p G E /G M ∼ 1
1
µ p
G Ep
/ G Mp
0.8
0.6
0.4
0.2
0
Jones et al.
Gayou et al.
Rosenbluth (Arrington)
0 2 4 6
Q 2 (GeV 2 )

G n E
Arrington et al. [nucl-th/0611050]
Galster: G
n
E (Q 2 )=
a G τ
(1 + b G τ) ·
1
(1 + Q 2 /M 2 ) 2 τ = Q 2 /4M 2 n

Baryon Three Point Functions
〈
Ω
∣ ∣T (χ(⃗x2 , t) O(⃗x 1 , τ) χ(0)) ∣ ∣ Ω
〉
Three-point function at the baryon level
G B OB(t, τ) = ∑ s,s ′ e −E p ′ (t−τ) e −E pτ 〈 Ω ∣ ∣ χ
∣ ∣p ′ , s ′〉〈 p ′ , s ′∣ ∣ O
∣ ∣p, s
〉〈
p, s
∣ ∣χ
∣ ∣Ω
〉
Eg. the matrix element of the electromagnetic current can be extracted
from a ratio of 3pt/2pt and has the general form
〈
p ′ , s ′∣ ∣ j
µ ∣ ∣ p, s
〉
= u(p ′ , s ′ )
(F 1 (q 2 )γ µ + iF 2 (q 2 µν qν
)σ
2M
)
u(p, s)

Lattice Techniques
¯N(0)
N(t, ⃗p ′ )

Lattice Techniques
τ
¯N(0)
N(t, ⃗p ′ )

Lattice Techniques
O(τ, ⃗q)
¯N(0)
N(t, ⃗p ′ )

Lattice Techniques
O(τ, ⃗q)
¯N(0,⃗p)
N(t, ⃗p ′ )

Lattice Techniques
Advantages: Free choice of
Momentum transfer
Operator (vector/axial/tensor)
Ideal for Form Factors, Structure Functions, GPDs
Disadvantages: Separate 3-pt inversion for each
Quark flavour
Hadron eg. p, Σ, ∆, π, N → γ∆
Polarisation
Sink momentum

Ignore Disconnected Terms
O(τ, ⃗q)
¯N(0,⃗p)
N(t, ⃗p ′ )

Form Factors:
F (v)
1
β = 5.25, κ sea = 0.13575, V = 24 3 × 48
Nf=2, Clover
(m π ≈ 550 MeV)
1.5
1
F 1
(v)
0.5
0
0 1 2 3 4
Q 2 [GeV 2 ]

Recall: Scaling of Form Factors
Naive expectation from dimensional counting
[Brodsky & Farrar, 1973] F (0)
(1 + Q 2 /M 2 ) p
F 1 ∝ 1 Q 4 (dipole?)
F 2 ∝ 1 Q 6 (tripole?)

Scaling of
F (v)
2
β = 5.25, κ sea = 0.13575, V = 24 3 × 48
(m π ≈ 550 MeV)
3
dipole
tripole
2
F 2
(v)
1
0
0 2 4
Q 2 [GeV 2 ]

Recall: Scaling of Form Factors
Naive expectation from dimensional counting
[Brodsky & Farrar, 1973]
F 1 ∝ 1 Q 4 (dipole?)
F 2 ∝ 1 Q 6 (tripole?)
for
Q 2 > ζ pQCD
Q 2 F 2
F 1
∝ const
G E
G M
∝ const

Scaling of F (v)
2 /F (v)
1
m π ≈ 550 MeV, a = 0.90, . . . , 0.75 fm
4
3
!=5.25, " sea
=0.13575
!=5.29, " sea
=0.13590
!=5.40, " sea
=0.13610
F 2
(v)
/ F 1
(v)
2
1
0 1 2 3 4
Q 2 [GeV 2 ]

Scaling of F (v)
2 /F (v)
1
Experimental data suggests
1/ √ Q 2
scaling
1.5
[GeV]
1
/ F 1
(p)
(Q 2 ) (1/2) F 2
(p)
0.5
!=5.25, " sea
=0.13575
!=5.29, " sea
=0.13590
!=5.40, " sea
=0.13610
0
0 1 2 3 4
Q 2 [GeV 2 ]

Scaling of
F (d)
1 /F (u)
1
Flavour dependence also observed in experimental data [Diehl,05]
0.6
!=5.25, " sea
=0.13575
!=5.29, " sea
=0.13590
!=5.40, " sea
=0.13610
0.4
F 1
(d)
/ F 1
(u)
0.2
0
0 1 2 3 4
Q 2 [GeV 2 ]

Form Factors:
F (p)
1
Comparison with experiment
1
0.8
0.6
experiment
400 MeV
550 MeV
700 MeV
F 1
p
0.4
0.2
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Q 2 [GeV] 2

Form Factor Radii
r 2 i = −6 dF i(q 2 )
dq 2 ∣ ∣∣q 2 =0
[
F i (q 2 ) = F i (0) 1 + 1 ]
6 r2 i q 2 + O(q 4 )

Form Factor Radii & Magnetic Moments
QCDSF, preliminary (2007)
1
6
0.8
4
(r 2
(v)
)
2
[fm
2
]
0.6
0.4
! (v)norm
2
0.2
0
0 0.5 1 1.5
m 2 PS [GeV]
0
0 0.5 1 1.5
m 2 PS [GeV]
[hep-lat/0303019]

Results RBC/UKQCD
Nf=2+1 Domain Wall Fermions
[Takeshi Yamazaki]
F V
(q 2 )/F V
(0)
1
0.9
0.8
0.7
0.6
0.5
m f
=0.005
m f
=0.01
m f
=0.02
m f
=0.03
experiment
1
0.4
0.3
0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.7
q 2 [GeV 2 ]
0.9
0.8
(〈r 1
2
〉)
1/2
[fm]
experiment
N f
=2+1 DWF (2.7fm)
N f
=2 DWF (1.9fm)
N f
=0 DWF (3.6fm)
N f
=2 Wilson (1.9fm)
N f
=0 Wilson (3.0fm)
0.6
0.5
0.4
0.3
0 0.1 0.2 0.3 0.4
2 2
m π[GeV ]

Results RBC/UKQCD
Nf=2+1 Domain Wall Fermions
[Takeshi Yamazaki]
1.1
1
0.9
0.8
(〈r 2
2
〉)
1/2
[fm]
experiment
N f
=2+1 DWF (2.7fm)
N f
=2 DWF (1.9fm)
N f
=0 DWF (3.6fm)
N f
=2 Wilson (1.9fm)
N f
=0 Wilson (3.0fm)
0.7
6
0.6
0.5
0.4
5
4
experiment
F 2
(0)
0.3
0 0.1 0.2 0.3 0.4
2 2
m π[GeV ]
3
2
1
N f
=2+1 DWF (2.7fm)
N f
=2 DWF (1.9fm)
N f
=0 DWF (3.6fm)
N f
=2 Wilson (1.9fm)
N f
=0 Wilson (3.0fm)
0
0 0.1 0.2 0.3 0.4
2 2
m π[GeV ]

Magnetic Moment
µ v (µ N )
5
4.5
4
3.5
3
2.5
2
1.5
1
N f =2 [210]
N f =0 [196]
N f =0 [207]
N f =2 [207]
N f =0 [206]
0 0.2 0.4 0.6 0.8 1 1.2
2
m ! [GeV
2 ]
[210,196]: Clover, [207]: Wilson, PRD74:034508, 2006, [206]: FLIC, PRD74:093005, 2006

G n E
β = 5.25, κ sea = 0.13575, V = 24 3 × 48
(m π ≈ 550 MeV)

Twisted
Boundary
Conditions
θ i
O(τ, ⃗q)
θ f
hep-lat/0411033, hep-lat/0703005
¯N(0,⃗p)
N(t, ⃗p ′ )
On a periodic lattice with spatial volume L 3 , momenta
are discretised in units of 2π/L
Modify boundary conditions on the valence quarks
ψ(x k + L) = e iθ k
ψ(x k ), (k = 1, 2, 3)
allows to tune the momenta continuously
q 2 = (p f − p i ) 2 =
{
[E f (⃗p f ) − E i (⃗p i )] 2 −
⃗p FT + ⃗ θ/L
[
(⃗p FT,f + θ ⃗ f /L) − (⃗p FT,i + θ ⃗ ] } 2
i /L)

Pion
Dispersion Relation

Twisted Boundary
Conditions
β =5.29, κ sea =0.13500, V= 16 3 × 32
(m π ≈ 850 MeV)

Twisted Boundary
Conditions
β =5.29, κ sea =0.13500, V= 16 3 × 32
(m π ≈ 850 MeV)

Twisted Boundary
Conditions
Recall: We need to extrapolate F2(q 2 ) to q 2 =0
Model dependence

Twisted Boundary
Conditions
Recall: We need to extrapolate F2(q 2 ) to q 2 =0
Model dependence

Pion Charge Radius
[Broemmel et al. (QCDSF), hep-lat/0608021]
F π (Q 2 )
1.2
1
0.8
0.6
0.4
0.2
0
F π (Q 2 )
1
0.8
0.6
0.4
0.2
0 0.2 0.4 0.6 0.8 1
Q 2 [GeV 2 ]
0 0.5 1 1.5 2 2.5 3 3.5 4
(r 0 M lat ) 2
m 2 π [GeV2 ]
0 0.2 0.4 0.6 0.8 1 1.2 1.4
10
9
8
7
6
5
4
3
0 1 2 3 4 5 6 7 8
(r 0 m π ) 2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
M 2 lat [GeV2 ]
Q 2 [GeV 2 ]
r 2 π = 6
M 2
[0.441(19)]
[0.452(11)]exp

Pion Charge Radius
[Boyle, Juettner, Flynn, Kelly, de Lima, Maynard, Sachradja, JZ.
DWF N f =2+1 (UKQCD), This week]
1
0.95
0.9
set A
set B
set C
pole fit
QCDSF/UKQCD
m π ≈ 330MeV
24 3 × 64
ylabel
0.85
0.8
0.75
0.7
0.65
0 0.05 0.1 0.15 0.2 0.25 0.3
xlabel
〈r 2 π〉 (2),NLO = − 12lr 6
f 2 − 1
8π 2 f 2 (
log m2 π
µ 2 +1 )
[0.418(28)]
[0.452(11)]exp

Conclusions
Lattice provides a useful tool for investigating FFs
Parameterisation of FFs
Lattice data not yet precise enough to fix q 2 dependence
Twisted boundary conditions may help
Qualitative agreement with experimental data
Search for chiral nonanalytic behaviour in charge radii/mag. mom.
Hint of strong effects at light quark masses
Lattice data much closer to the chiral limit is crucial
(and is starting to become available)
Reveal internal charge distribution of e.g. neutron