Invited seminar talk that I gave at The Cyprus Institute in Oct. 2018.

1. The nucleon electromagnetic form factors at high
momentum transfer from Lattice QCD
Christos Kallidonis
Department of Physics and Astronomy
Stony Brook University
The Cyprus Institute Seminar Series
Oct. 16, 2018
With:
S. Syritsyn (Stony Brook University and RIKEN-BNL)
M. Engelhardt (New Mexico State University)
J.Green (DESY)
S. Meinel (University of Arizona and RIKEN-BNL)
J. Negele, A. Pochinsky (MIT)
based on C. Kallidonis et al. [arXiv: 1810.04294]

2. • Introduction
• Lattice QCD
• Motivation
• Methodology
• Results
• Conclusions
!2
Outline

3. One of the four fundamental interactions, the others being
Gravitation, Electromagnetism, Weak interactions
Strongest of all, dominant in subatomic level
Confines quarks into hadrons (proton, neutron, etc)
• quarks: fundamental particles of matter (6 in total)
• gluons: force mediators (8 in total)
• quarks and gluons carry color charge
Binds protons and neutrons to create the atomic nucleus
Most of the proton mass (99%) comes from strong interactions
between its constituent quarks and gluons
!3
Introduction
The Strong interaction

4. !4
Fascinating theory that describes the strong interactions
• parameters: quark masses and coupling constant
quark part gluon part
LQCD =
X
f
¯f (i µ
Dµ mf ) f
1
4
Ga
µ⌫Gµ⌫
a
Dµ = @µ + igGµ
Gµ⌫ = @µG⌫ @⌫Gµ gfabc
Gb
µGc
⌫
Confinement: low energies: coupling constant becomes large
• we need infinite amount of energy to break the bonds that hold
quarks together
• there are no free quarks in nature
Asymptotic freedom: high energies: coupling constant becomes small
• quarks interact weakly (theories based on approximations)
Coupling constant varies with the energy
Introduction
Quantum Chromodynamics (QCD)

5. !5
Lattice QCD
• only method available which solves QCD with no approximations
• calculation from first principles, QCD Lagrangian is the only input
• insight to the properties of hadrons via numerical simulations than run
on powerful supercomputers
• discretize the 4-d space-time, introduce a lattice spacing
• various discretization schemes exist, each with pros and cons
• quark fields “live” on the lattice sites
• gluon fields “live” between lattice sites
• when the size of the lattice is taken infinitely large and the lattice
sites infinitesimally close to each other, the continuum theory is
recovered

6. • use Monte Carlo methods to create ensembles of gluon configurations
• statistical interpretation of the theory
Physical processes on the lattice
• associated with operators coupling with quarks
• hadron observables obtained from matrix elements
• correlation functions expressed in terms of quark propagators
• typically 103 of configurations are required
• average over configurations, error scales as ⇠ 1/
p
N
!6
Lattice QCD
Lattice QCD is an extremely computationally intensive application
most computational resources are spent on calculation of
gluon configurations and quark propagators

7. !7
Lattice QCD
Image courtesy of LPC Clermont
Factors affecting the computational resources required:
• discretization scheme
• lattice spacing: smaller is more expensive
• lattice volume: larger is more expensive
• quark mass: smaller is more expensive

8. !8
Lattice QCD
With the availability of improved algorithms and more capable supercomputers, simulations
at realistic quark masses, large volumes and small lattice spacings are now feasible
SUMMIT Supercomputer, Oak Ridge
IBM Power9 CPU
Nvidia Volta GV100
GPU
14
Advances in Lattice QCD
Huge computational power needed
&
Algorithmic improvements
323
×64 L=2.1fm
5000 configs 1000 configs
Cost of 1000 configurations at physical mq is currently O(10) TFlops × year
5000 configs, 323 x 64
1e-02
1e-01
1e+00
1e+01
1e+02
1e+03
1e+04
0 0.05 0.1 0.15 0.2
Cost(TFlop-yrs)
m2
⇡ (GeV2
)
Nucleon mass, 1% error
Axial charge, 1% error
Conventional laptop CPU: ~ 0.1 Tflop
Nvidia GV100 GPU: 7.9 Tflop/double, 14.8 Tflop/single

9. !9
Lattice QCD
Systematic uncertainties & challenges
2.2
2.3
2.4
2.5
2.6
0 0.05 0.1 0.15 0.2 0.25
m⇤+
c
(GeV)
m2
⇡ (GeV2
)
= 1.90
= 1.95
= 2.10
Continuum limit
• finite lattice spacing - need to take continuum limit
• simulate with fine lattices
• Finite volume - need to extrapolate to infinite volume
• simulate large volumes
• Contamination from non-desirable hadron states
• proper analysis methods to extract/isolate correct information
• Unrealistic simulations due to input quark mass
• perform extrapolations to physical quark mass
• Renormalization and mixing

10. !10
Elastic electron-proton scattering
Tool to extract the nucleon’s electric and magnetic form factors
• an electron (incident particle) collides with a proton (target)
• electromagnetic interaction via exchange of a virtual photon
• elastic: total conservation of kinetic energy (no energy loss)
e-
proton
~p , E
e-
~p 0
, E0
e-
e-
γ
Q2
= (~p ~p 0
)2
(E E0
)2
Euclidean momentum transfer:
• in an experiment, the scattering cross section is measured
• cross section is expressed in terms of two form factors, F1 and F2
• functions of Q2
GE(Q2
) = F1(Q2
)
Q2
(2mN )2
F2(Q2
) GM (Q2
) = F1(Q2
) + F2(Q2
)
electric
form factor
magnetic
form factor

11. !11
Why the nucleon?
proton neutron
electric charge: +1
mass: 938.3 MeV
stable particle
electric charge: 0
mass: 939.6 MeV
stable within atomic nucleus,
decays via weak β-decay when free
Acted upon equally by the strong interactions, considered as two states of the same nucleon
• mass difference about 0.15%
• partners of an Isospin (I=1/2) doublet
• components of the atomic nuclei, binding blocks of all visible matter
• lightest baryons
Can be studied easier in experiments and theory compared to heavier, non-stable hadrons

12. !12
Motivation
The challenge of understanding the properties and
structure of hadrons via QCD has always occupied a
central place in nuclear physics
Goals of Lattice QCD calculations:
• make contact with well-known quantities
• provide input for quantities not easily
accessible in experiments
• explore new physics from first principles

13. !13
Motivation
Why the electric and magnetic form factors?
• among the most important observables characterizing the nucleon
• description of the spatial distributions of electric charge and magnetization
• information on how different the nucleon is from a point-like particle
• value at GE(Q2 = 0), GM(Q2 = 0): electric charge, magnetic moment
• slope of GE , GM at Q2 = 0: electric and magnetic radius
High-momentum transfer calculation on the lattice:
• test validity of theoretical predictions, e.g. perturbative QCD, quark models, phenomenology
• input to DVCS measurements, probing GPDs
• nucleon FFs: good framework to test high-momentum region on the lattice
Rich experimental activity
• Super-BigBite Spectrometer at JLab Hall A
• elastic ep scattering experiments up to
• dependence
• scaling of at
• individual contributions from up- and down-quarks
• finalized/published results in ~ 5yr
Q2
! 1
GE/GM
F1/F2
Q2
⇠ 18 GeV2
Jefferson Lab

14. !14
Simulation details
• two Nf=2+1 Wilson-clover ensembles, produced by JLab lattice group
• different lattice volumes, similar lattice spacing
• Computational resources: BNL Institutional Cluster, USQCD 2017 allocation
• Calculation: Qlua interface: QUDA-MG for propagators, contractions on GPU
A.V. Pochinksy S. Syritsyn, C.K.
D5-ensemble: = 6.3, a = 0.094 fm, a 1
= 2.10 GeV
323
⇥ 64, L = 3.01 fm
aµl -0.2390
aµs -0.2050
 0.132943
Csw 1.205366
m⇡ (MeV) 280
m⇡L 4.26
Statistics 86144
D6-ensemble: = 6.3, a = 0.091 fm, a 1
= 2.17 GeV
483
⇥ 96, L = 4.37 fm
aµl -0.2416
aµs -0.2050
 0.133035
Csw 1.205366
m⇡ (MeV) 170
m⇡L 3.76
Statistics 50176

15. !15
Correlation functions
Matrix element of the vector current: Vµ(x) = ¯(x) µ (x)
hN(p0
, s)|Vµ|N(p, s)i = ¯uN (p0
, s)

µF1(q2
) +
i µ⌫q⌫
2mN
F2(q2
) uN (p, s)
Dirac
form factor
Pauli
form factor
e-
e-
γ
proton
Three-point correlation function
¯N(~x0, t0)N(~xs, ts)
Vµ(~xins, tins)
Gµ( , ~p0
, ~q, ts, tins) =
X
~xs,~xins
e i~p0
·(~xs ~x0)
ei~q·(~xins ~x0)
↵hN↵(~xs, ts)Vµ(~xins, tins) ¯N (~x0, t0)i
• (ts-t0) ~ 0.55 fm - 0.95 fm
• consider only connected contributions
On the lattice:
¯N(~x0, t0)
N(~xs, ts)
Two-point correlation function
C(~p0
, ts) =
X
~xs
e i~p0
·(~xs ~x0)
( 4) ↵hN↵(~xs, ts) ¯N (~x0, t0)i

16. !16
Kinematics: Accessing the Breit Frame
we incorporate boosted nucleon states
to access the high- region, keeping the energy of
the states as low as possible
Q2 N↵(~p 0
, t) =
X
~x
✏abc
⇥
ua
µ(x)(C 5)µ⌫db
⌫(x)
⇤
uc
↵(x)e i~p 0
·~x
Breit frame: ~p = ~p 0
, E = E0
Q2
= 4~p2
Q2
= (~p ~p 0
)2
(E E0
)2
S~kb
(x) ⌘
1
1 + 6↵
"
(x) + ↵
3X
µ=±1...
Uµ(x)ei~kb·ˆµ
(x + ˆµ)
#
Gaussian “momentum” smearing:
~kb = 0.5 ~p 0
G. Bali et al. [arXiv: 1602.05525]
N(~p 0
, ts)
¯N( ~p 0
, t0)
Vµ(~q, tins)
• boosting in single direction
D5 ~P 0
= ( 4, 0, 0) ! Q2
⇠ 10.9 GeV2
D6 ~P 0
= ( 5, 0, 0) ! Q2
⇠ 8.1 GeV2
• diagonal boosting in x-y plane
D5 ~P 0
= ( 3, 3, 0) ! Q2
⇠ 12.2 GeV2
N(~p 0
, ts)
¯N( ~p 0
, t0)
Vµ(~q, tins)

17. !17
Extracting the matrix element
Ratio of 2pt and 3pt functions
Rµ
( , ~q, ~p0
; ts, tins) =
Gµ( , ~p0
, ~q, ts, tins)
C(~p0, ts t0)
⇥
s
C(~p, ts tins)C(~p0, tins t0)C(~p0, ts t0)
C(~p0, ts tins)C(~p, tins t0)C(~p, ts t0)
C =
s
2m2
N
EN (EN + mN )
k = i 5 k 4polarized
4 =
1 + 4
4
unpolarized
Projectors:
Q2
⌘ q2
⇧0
( 4, ~q) = C
EN + mN
2mN
GE(Q2
) ⇧i
( 4, ~q) = C
qi
2mN
GE(Q2
)
⇧i
( k, ~q) = C
✏ijkqj
2mN
GM (Q2
)
S =
NX
n
⇣P
m=E,M AnmGm ⇧n
⌘2
2
n
Rµ tins t0 1
!
ts tins 1
⇧µ
( , ~q)1. Plateau method:
2. Two-state fit method:
Anm(~p, ~p0
) = hN|n, ~p0
ihm, ~p|Nihn, ~p0
|Vµ|m, ~pi/[2
p
En(~p)En(~p0)]
cn(~p0
) = |hN|n, ~p0
i|2
/2En(~p0
)
h0, ~p0
|Vµ|0, ~pi =
A00(~p, ~p0
)
p
c0(~p)c0(~p0)
C(~p0
, ts) ' e E(~p0
)ts
h
c0(~p0
) + c1(~p0
)e E1(~p0
)ts
i
Gµ( , ~p0
, ~p, ts, tins) ' e E0(~p0
)(ts tins)
e E0(~p)(tins t0)
⇥
⇥ [A00(~p, ~p0
) + A01(~p, ~p0
)e E1(~p)(tins t0)
+
+ A10(~p, ~p0
)e E1(~p0
)(ts tins)
+
+ A11(~p, ~p0
)e E1(~p0
)(ts tins)
e E1(~p)(tins t0)
i
0.4 0.2 0.0 0.2 0.4
(tins ts/2) [fm]
0.015
0.020
0.025
0.030
0.035
0.040
R(ts,tins)!Fp
1(Q2
)
Q2
= 10.9 GeV2
2-state fit
ts = 0.56 fm
ts = 0.66 fm
ts = 0.75 fm
ts = 0.85 fm
ts = 0.94 fm
Need to make sure that we get the nucleon ground state
ts
t0
tins

18. !18
Effective Energy
0 2 4 6 8 10 12 14 16
ts/a
0.4
0.6
0.8
1.0
1.2
1.4
aEe↵
[-1, 0, 0]
[0, 0, 0]
[1, 0, 0]
[2, 0, 0]
[3, 0, 0] [4, 0, 0]
0 2 4 6 8 10 12 14 16
ts/a
0.4
0.6
0.8
1.0
aEe↵
[-1, 0, 0]
[0, 0, 0]
[1, 0, 0]
[2, 0, 0]
[3, 0, 0]
[4, 0, 0]
[5, 0, 0]
D5
D6
EN
e↵ (ts) = log
✓
C(~p, ts)
C(~p, ts + 1)
◆
ts 1
! EN
• two-state fits to our lattice data are of
good quality
• horizontal line from
using lattice value of mN
• ground state energy slightly
overestimates cont. dispersion relation
• excited states faint after ~ ts/a = 9
E = m2
N + p2
¯N(~x0, t0)
N(~xs, ts)

19. the scaling rules of Brodsky and Farrar should scale like 1/Q2
, were not available until JLab came on
line. With the publication by Jones et al. of Ref. [30], however, it became clear that both G
p
E
/G
p
M
and
similarly Fp
2 /Fp
1 did not behave as expected, and showed no evidence of scaling over the range of Q2
for which accurate measurements were made. The failure of this scaling, and equivalently the failure
of G
p
E
/G
p
M
to remain constant, came as a great surprise, and is viewed by many as the most striking
result to date to come out of JLab.
Figure 6.2: Shown in (a) are early SLAC data in which Q2Fp
1 is plotted versus Q2. Scaling is clearly visible
above something like 10 GeV2. In (b), data on Q2Fp
2 /Fp
1 are plotted, and scaling is clearly not visible. In both
plots, the solid lines represent calculations involving model-independent sum rules of GPDs. In (b), the lines
show pQCD calculations by Belitsky, Ji, and Yuan [38] that predict Q2 F2/F1 / ln2
(Q2/⇤2), where ⇤ is a
non-perturbative mass scale.
Clearly, a more sophisticated description is needed to describe some of the new FF data. One
approach involves calculations of the FFs using model-independent sum rules of generalized parton
distributions (GPDs). The results of such calculations are shown in both Figs. 6.2a and b with the solid
lines. We will discuss this approach further in a subsequent section. Another approach is a pQCD
calculation performed by Belitsky, Ji, and Yuan [38]. In their calculation, they relaxed the assump-
tion of Brodsky and Farrar that the quarks were moving collinearly with the proton. They included
52
!19
Form Factor Results I: RatioF2/F1
• - dependence compares well with exp. data and phenom. parametrization
• scaling reproduced
• consistency between on-axis / x-y diagonal boost momentum for D5
W. M. Alberico et al. [arXiv: 0812.3539]
A.V. Belitsky et al. [arXiv: hep-ph/0212351]
Q2
S.B.S Program report
0 2 4 6 8 10 12
Q2
[GeV2
]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Q2
Fp
2/Fp
1(Q2
)[GeV2
]
Phenom.
Plat., D5, ~p 0
= ( 4, 0, 0)
2-state, D5, ~p 0
= ( 4, 0, 0)
2-state, D5, ~p 0
= ( 3, 3, 0)
2-state, D6, ~p 0
= ( 5, 0, 0)
Q2
Fp
2 /Fp
1 (Q2
) ⇠ log2
[Q2
/⇤]

20. Form Factor Results II: Ratio
!20
• consistency between our lattice data
• good agreement with experiment / phenomenology for proton up to Q2 ~ 6 GeV2
• lattice data support smoother approach towards zero
2
in GeV2
Q
0 5 10 15
p
M/G
p
EGp
µ
-0.5
0.0
0.5
1.0
GEp(1)
GEp(2)
GEp(3) (prelim, stat only)
GEp(5) E12-07-109, SBS
VMD - E. Lomon (2002)
VMD - Bijker and Iachello (2004)
RCQM - G. Miller (2002)
DSE - C. Roberts (2009)
= 300 MeVΛ,2
)/Q2
Λ/2
(Q2
ln∝1
/F2F
0
n
M/Gn
EGn
0.0
0.5
1.0
Figure 6.1: Shown are existing data and projected errors for measurem
netic form factors of the proton (left panel) and neutron (right panel). T
made within the Super Bigbite project are shown by the open red sq
published results of GEp(1) [30], GEp(2) [31], preliminary results fro
of GEp(5) in a 60-day run [6]. The various theoretical curves are discu
for G
n
E
/G
n
M
, we show published data including those due to Madey and
of GEn(1) (E02-013). We also show the projected errors of GEn(2), w
and E12-09-006 with SHMS (open blue points).
the virtual photon. For the case of ⇤
= 0 or 180 and ✓⇤
= 90 , t
proportional to GE/GM . In fact, even if ⇤
is close to 0 and ✓⇤
is
data to Eq. 6.7.
Whether working with a polarized target or a recoil polarimete
has only a negligible contribution from two-photon effects, and
standard for any form-factor measurement.
6.2 Physics content of form factors
S.B.S Program report
0 2 4 6 8 10 12
Q2
[GeV2
]
0.0
0.2
0.4
0.6
0.8
1.0
1.2
µpGp
E/Gp
M(Q2
)
Phenom.
Plat., D5, ~p 0
= ( 4, 0, 0)
2-state, D5, ~p 0
= ( 4, 0, 0)
2-state, D5, ~p 0
= ( 3, 3, 0)
2-state, D6, ~p 0
= ( 5, 0, 0)
GE/GM

21. Form Factor Results II: Ratio
!21
GE/GM
• neutron: out lattice data underestimate experiment / phenomenology
• same qualitative behavior
2
in GeV2
Q
0 5 10 15
p
M
/G
p
E
Gp
µ
-0.5
0.0
0.5
1.0
GEp(1)
GEp(2)
GEp(3) (prelim, stat only)
GEp(5) E12-07-109, SBS
VMD - E. Lomon (2002)
VMD - Bijker and Iachello (2004)
RCQM - G. Miller (2002)
DSE - C. Roberts (2009)
= 300 MeVΛ,2
)/Q2
Λ/2
(Q2
ln∝1
/F2F
2
in GeV2
Q
0 5 10
n
M/Gn
EGn
0.0
0.5
1.0
VMD - E. Lomon (2002)
RCQM - G. Miller (2002)
DSE - C. Roberts (2009)
- Schiavilla & Sick
20
d(e,e’d) T
= 300 MeV,2
)/Q2
/2
(Q2
ln1
/F2
F
Galster fit (1971)
Madey, Hall C
E02-013
E12-09-006, Hall C
E12-09-016, Hall A
Figure 6.1: Shown are existing data and projected errors for measurements of the ratios of the electric and mag-
netic form factors of the proton (left panel) and neutron (right panel). The projected errors for the measurements
made within the Super Bigbite project are shown by the open red squares. On the left panel are shown the
published results of GEp(1) [30], GEp(2) [31], preliminary results from GEp(3) [32] and the projected results
of GEp(5) in a 60-day run [6]. The various theoretical curves are discussed in the text. On the right-hand panel,
for G
n
E
/G
n
M
, we show published data including those due to Madey and co-workers [33], and preliminary results
of GEn(1) (E02-013). We also show the projected errors of GEn(2), which is part of the Super Bigbite project,
and E12-09-006 with SHMS (open blue points).
the virtual photon. For the case of ⇤
= 0 or 180 and ✓⇤
= 90 , the asymmetry A = A?, and is nearly
proportional to GE/GM . In fact, even if ⇤
is close to 0 and ✓⇤
is close to 90 , it is quite easy to fit the
data to Eq. 6.7.
Whether working with a polarized target or a recoil polarimeter, the double-polarization asymmetry
has only a negligible contribution from two-photon effects, and for high Q2
, it has become the gold
standard for any form-factor measurement.
6.2 Physics content of form factors
0 2 4 6 8 10 12
Q2
[GeV2
]
0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
µnGn
E/Gn
M(Q2
)
Phenom.
Plat., D5, ~p 0
= ( 4, 0, 0)
2-state, D5, ~p 0
= ( 4, 0, 0)
2-state, D5, ~p 0
= ( 3, 3, 0)
2-state, D6, ~p 0
= ( 5, 0, 0)

22. !22
Form Factor Results III: ,F1 F2
• shallow trend towards phenom. with increasing source-sink separation
• similar qualitative behavior, overestimation of phenom. prediction
W. M. Alberico et al. [arXiv: 0812.3539]
0.00
0.25
0.50
0.75
1.00
Fp
1(Q2
)
Phenom.
tsep = 0.56 fm
tsep = 0.66 fm
tsep = 0.75 fm
tsep = 0.85 fm
tsep = 0.94 fm
2-state
0 2 4 6 8 10 12
Q2
[GeV2
]
0.00
0.25
0.50
0.75
1.00
Fp
2(Q2
)
0.10
0.05
0.00
Fn
1(Q2
)
0 2 4 6 8 10 12
Q2
[GeV2
]
1.5
1.0
0.5
0.0
Fn
2(Q2
)
Phenom.
tsep = 0.56 fm
tsep = 0.66 fm
tsep = 0.75 fm
tsep = 0.85 fm
tsep = 0.94 fm
2-state

23. !23
Form Factor Results III: ,F1 F2
0.15
0.10
0.05
0.00
Q2
Fn
1(Q2
)[GeV2
]
0 2 4 6 8 10 12
Q2
[GeV2
]
0.6
0.4
0.2
0.0
Q2
Fn
2(Q2
)[GeV2
]
Phenom.
2-state, D5, ~p 0
= ( 4, 0, 0)
2-state, D5, ~p 0
= ( 3, 3, 0)
2-state, D6, ~p 0
= ( 5, 0, 0)
0.0
0.1
0.2
0.3
0.4
Q2
Fp
1(Q2
)[GeV2
]
0 2 4 6 8 10 12
Q2
[GeV2
]
0.0
0.2
0.4
0.6
Q2
Fp
2(Q2
)[GeV2
]
Phenom.
2-state, D5, ~p 0
= ( 4, 0, 0)
2-state, D5, ~p 0
= ( 3, 3, 0)
2-state, D6, ~p 0
= ( 5, 0, 0)
• discrepancies for individual form factors
• a thorough investigation is needed

24. !24
Form Factor Results III: ,F1 F2
0 2 4 6 8 10 12
Q2
[GeV2
]
0
1
2
3
4
5
6
7
Q4
FU
1,2.5⇥Q4
FD
1
FU
1
FD
1
FU
1 [Alberico]
FD
1 [Alberico]
ts = 0.56 fm
ts = 0.66 fm
ts = 0.75 fm
ts = 0.85 fm
ts = 0.94 fm
2-state
0 2 4 6 8 10 12
Q2
[GeV2
]
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
Q4
FU
2,(0.75)⇥Q4
FD
2
FU
1
FD
1
FU
2 [Alberico]
FD
2 [Alberico]
ts = 0.56 fm
ts = 0.66 fm
ts = 0.75 fm
ts = 0.85 fm
ts = 0.94 fm
2-state
• discrepancies observed for form factors of up- and down- quarks

25. !25
Systematics I: Parity mixing for boosted states
0
B
B
B
B
B
B
B
B
B
B
B
@
0¯0 0¯1 0¯2 0¯3 0¯00
0¯10
0¯20
0¯30
1¯0 1¯1 1¯2 1¯3 1¯00
1¯10
1¯20
1¯30
2¯0 2¯1 2¯2 2¯3 2¯00
2¯10
2¯20
2¯30
3¯0 3¯1 3¯2 3¯3 3¯00
3¯10
3¯20
3¯30
00¯0 00¯1 00¯2 00¯3 00¯00
00¯10
00¯20
00¯30
10¯0 10¯1 10¯2 10¯3 10¯00
10¯10
10¯20
10¯30
20¯0 20¯1 20¯2 20¯3 20¯00
20¯10
20¯20
20¯30
30¯0 30¯1 30¯2 30¯3 30¯00
30¯10
30¯20
30¯30
1
C
C
C
C
C
C
C
C
C
C
C
A
• At non-zero momentum, correlators projected with include
parity contaminations
• need to make sure that correlators from states at non-zero momentum correspond to the same
zero-momentum states
±
⌘
1
2
(1 + 4)
O((E m)/2E)
0
B
B
B
B
B
B
B
B
B
B
@
0¯0 0¯1 0¯2 0¯3 0¯00
0¯10
0¯20
0¯30
1¯0 1¯1 1¯2 1¯3 1¯00
1¯10
1¯20
1¯30
2¯0 2¯1 2¯2 2¯3 2¯00
2¯10
2¯20
2¯30
3¯0 3¯1 3¯2 3¯3 3¯00
3¯10
3¯20
3¯30
00¯0 00¯1 00¯2 00¯3 00¯00
00¯10
00¯20
00¯30
10¯0 10¯1 10¯2 10¯3 10¯00
10¯10
10¯20
10¯30
20¯0 20¯1 20¯2 20¯3 20¯00
20¯10
20¯20
20¯30
30¯0 30¯1 30¯2 30¯3 30¯00
30¯10
30¯20
30¯30
1
C
C
C
C
C
C
C
C
C
C
A
0
B
B
B
B
B
B
B
B
B
B
@
0¯00¯10¯20¯30¯00
0¯10
0¯20
0¯30
1¯01¯11¯21¯31¯00
1¯10
1¯20
1¯30
2¯02¯12¯22¯32¯00
2¯10
2¯20
2¯30
3¯03¯13¯23¯33¯00
3¯10
3¯20
3¯30
00¯000¯100¯200¯300¯00
00¯10
00¯20
00¯30
10¯010¯110¯210¯310¯00
10¯10
10¯20
10¯30
20¯020¯120¯220¯320¯00
20¯10
20¯20
20¯30
30¯030¯130¯230¯330¯00
30¯10
30¯20
30¯30
1
C
C
C
C
C
C
C
C
C
C
A
0
B
B
B
B
B
B
B
B
B
B
B
@
0¯0 0¯1 0¯2 0¯3 0¯00
0¯10
0¯20
0¯30
1¯0 1¯1 1¯2 1¯3 1¯00
1¯10
1¯20
1¯30
2¯0 2¯1 2¯2 2¯3 2¯00
2¯10
2¯20
2¯30
3¯0 3¯1 3¯2 3¯3 3¯00
3¯10
3¯20
3¯30
00¯0 00¯1 00¯2 00¯3 00¯00
00¯10
00¯20
00¯30
10¯0 10¯1 10¯2 10¯3 10¯00
10¯10
10¯20
10¯30
20¯0 20¯1 20¯2 20¯3 20¯00
20¯10
20¯20
20¯30
30¯0 30¯1 30¯2 30¯3 30¯00
30¯10
30¯20
30¯30
1
C
C
C
C
C
C
C
C
C
C
C
A
F. M. Stokes et al. [arXiv: 1302.4152]
Parity-Expanded Variational Analysis (PEVA): Isolates parity of boosted hadron states
Cij( ; ~p, t) = Tr
"
X
~x
h i
(x)¯j
(0)ie i~p·~x
#
, i, j = 1, . . . , Nexpand operator basis of correlation matrix
p ⌘
1
4
(1 + 4)(1 i 5 k ˆpk)
i
p ⌘ p
i
i0
p ⌘ p 5
i
Gij(~p, t) = Cij( p; ~p, t)
Gij0 (~p, t) = Cij( 5 p; ~p, t)
Gi0j(~p, t) = Cij( p 5; ~p, t)
Gi0j0 (~p, t) = Cij( 5 p 5; ~p, t)
GEVP: G(~p, t + t)uuu↵
(~p) = e E↵(~p) t
G(~p, t)uuu↵
(~p)

26. !26
Systematics I: Parity mixing for boosted states
0
B
B
B
B
B
B
B
B
B
B
B
@
0¯0 0¯1 0¯2 0¯3 0¯00
0¯10
0¯20
0¯30
1¯0 1¯1 1¯2 1¯3 1¯00
1¯10
1¯20
1¯30
2¯0 2¯1 2¯2 2¯3 2¯00
2¯10
2¯20
2¯30
3¯0 3¯1 3¯2 3¯3 3¯00
3¯10
3¯20
3¯30
00¯0 00¯1 00¯2 00¯3 00¯00
00¯10
00¯20
00¯30
10¯0 10¯1 10¯2 10¯3 10¯00
10¯10
10¯20
10¯30
20¯0 20¯1 20¯2 20¯3 20¯00
20¯10
20¯20
20¯30
30¯0 30¯1 30¯2 30¯3 30¯00
30¯10
30¯20
30¯30
1
C
C
C
C
C
C
C
C
C
C
C
A
Investigation:
• two-point functions from nucleon interpolating operators at four different values of
Gaussian smearing different overlap with nucleon ground state
• perform PEVA analysis for various sets of operators
• D5 ensemble, ~8000 statistics
effect due to parity mixing is
negligible within our statistics
0.50
0.54
0.58
0.62
0.66
Ee↵
~P0
= (1, 0, 0)
Non-PEVA
PEVA 2x2, 2-3
PEVA 3x3, 1-2-3
PEVA 4x4, 0-1-2-3
0 2 4 6 8 10 12 14
t/a
0.70
0.74
0.78
0.82
0.86
0.90
Ee↵
~P0
= (3, 0, 0)
0.50
0.54
0.58
0.62
0.66
Ee↵
~P0
= (1, 0, 0)
Non-PEVA
PEVA 3x3, 0-1-2
PEVA 3x3, 0-1-3
PEVA 3x3, 0-2-3
PEVA 3x3, 1-2-3
0 2 4 6 8 10 12 14
t/a
0.70
0.74
0.78
0.82
0.86
0.90
Ee↵
~P0
= (3, 0, 0)

27. !27
Systematics II: Momentum discretization
Common convention:
0 1 2 3 4 5 6 7
nx
0
1
2
3
4
5
6
7
8
Rxy
~n = (nx, 1, 0)
t = 5
t = 7
t = 10
t = 12
• take appropriate traces and ratios of two-point
function to isolate momentum components
C(~p, t)
t 1
= |Z(~p)|2
S(~p)e E(~p)t
S(~p) =
i/p + m
2E(~p)
Im{Tr[ kS(~p)]} = 4pk ! Rxy(~p, t) ⌘
Im{Tr[ xC(~p, t)]}
Im{Tr[ yC(~p, t)]}
cont.
!
px
py
lattice momentum form:
•
•
•
~p
?
= ~
~p
?
= ~
1
6
~(a~)2
~p
?
=
1
a
sin(a~)
~p = ~ , ~ =
2⇡
L
~n , nx, ny, nz =
1
a

L
2
,
L
2
◆
⇡/a ⇡/a
nx = 6 ! x = 3⇡/8a
0 1 2 3 4 5 6
px
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Rxynx
t = 5 t = 7
effect due to anisotropic quark (boosted) smearing??

28. !28
Conclusions and Outlook
To-do:
• understand/resolve disagreement for individual form factors F1, F2
• complete investigation of excited state effects (perhaps larger ts?)
• consider other systematic effects
• improvement
• physical pion mass
• continuum extrapolation
• disconnected diagrams
O(a)
• high-Q2 on the lattice: feasible, but need to control systematics, noise-to-signal ratio
• our lattice results overestimate phenom. Q2-dependence for F1, F2
• however: good agreement with experiment for F2/F1 and GE/GM ratios up to Q2 ~ 6 GeV2
• consistent results between mπ = 170 MeV (D5), mπ = 280 MeV (D6): small pion mass and volume
effects
Thank you