In this short talk I present results on key quantities related to the structure of the nucleon, obtained from state-of-the-art Lattice QCD simulations. Results include the nucleon quark contents and the decomposition of the nucleon spin. Presented at the Early Career Research Symposium 2017 (ECRS 2017), Brookhaven National Laboratory

1. Probing Nucleon structure from
Lattice QCD simulations
Christos Kallidonis
Department of Physics and Astronomy
Stony Brook University
with
A. Abdel Rehim, C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, G. Koutsou, A. Vaquero, C. Wiese
(ETM Collaboration)
Early Career Research Symposium 2017, BNL
November 2, 2017

2. ECRS 2017, BNLC. Kallidonis 2
Outline
• Introduction - Strong force
• Lattice QCD
- Lattice methods
• Results
- Nucleon σ-terms
- Nucleon spin
• Summary and outlook
C. Alexandrou et al., Phys. Rev. Lett. 119, 2017, arXiv: 1706.02973
A. Abdel-Rehim, et al. Phys. Rev. Lett. 116, 2016, arXiv: 1601.01624

3. ECRS 2017, BNLC. Kallidonis 3
Introduction
Strong force: responsible for interactions between quarks and gluons
• confines quarks into hadrons (pion, proton, neutron, etc)
- binds protons and neutrons to form nuclei of the atoms
- formation of atoms: visible matter around us
• plethora of other phenomena
Quantum Chromodynamics (QCD):
fundamental theory that describes the strong interactions
parameters: quark masses and coupling constant
Dµ = @µ + igGµ
Gµ⌫ = @µG⌫ @⌫Gµ gfabc
Gb
µGc
⌫
SQCD =
Z
d4
x
X
f
¯f (x) (i µ
Dµ mf ) f (x)
1
4
Ga
µ⌫(x)Gµ⌫
a (x)

4. ECRS 2017, BNLC. Kallidonis 4
Lattice Quantum Chromodynamics
• Numerical simulations of QCD using Monte Carlo methods
• well-established framework for non-perturbative QCD
• Ab-initio calculations, QCD action only input
a
SQCD =
Z
d4
x
X
f
¯f (x) (i µ
Dµ mf ) f (x)
1
4
Ga
µ⌫(x)Gµ⌫
a (x)
• discretization of space-time, (K. Wilson, Phys. Rev. D10 2445, 1974)
• introduce lattice spacing, discretize on a 4-d lattice
• quarks: covariant derivative (nearest neighbor coupling)
• clover, twisted mass, domain wall, staggered,…
• gluons: link variables
• discretized forms must reduce to continuum form in the limits
SQCD
Uµ(x) = eiaGµ(x)
a ! 0 , L ! 1
SQCD =
X
f
X
x,y
¯f (x)M(x; y, U) f (y) + SG[U]

5. ECRS 2017, BNLC. Kallidonis 5
Lattice Quantum Chromodynamics
M: Wilson-Dirac operator
Very large matrix! ⇠ 108
⇥ 108
inverse of the Dirac operator (quark propagator): building block
of hadronic measurements on the lattice, most intensive part of
calculations
freedom in choice of
• quark mass (heavier is cheaper)
• lattice spacing, (larger is cheaper)
• lattice volume L3 x T, (smaller is cheaper)
each of these parameters is a new calculation
a ⇠ 0.1fm
L ⇠ 5 fm
a
SQCD =
X
f
X
x,y
¯f (x)M(x; y, U) f (y) + SG[U]
Monte Carlo methods: statistical treatment of the theory
• create gluon configurations
• observables: correlation functions in terms of quark propagators
• average over configurations, error
• need 1000s of configurations
⇠ 1/
p
Nc

6. ECRS 2017, BNLC. Kallidonis 6
Lattice Quantum Chromodynamics
Simulations “landscape”
simulations directly at the physical point: major breakthrough!
• improved algorithms
• Peta-scale supercomputers
Titan, OLCF, USA
this work:
- one gauge ensemble with twisted
mass clover-improved lattice action
- physical quark masses
- lattice spacing a = 0.093fm
- lattice volume L~4.5fm

7. ECRS 2017, BNLC. Kallidonis
(~x0, t0)
(xs, ts)
(xins, tins)
O
(~x0, t0)
(xs, ts)
(xins, tins)
O
three-point function
C3pt
(P, ts, tins) =
X
~xs,~xins
PhJN (~xs, ts)O (~xins, tins) ¯JN (~x0, t0)i
7
Probing nucleon structure from Lattice QCD
Physical processes on the lattice
• associated with operators coupling with quarks
• observables obtained from matrix elements,
extracted from correlation functions
(~x0, t0)
(~xs, ts)
two-point function
C(ts) =
X
~xs
1
4
(1 + 0)hJN (~xs, ts) ¯JN (~x0, t0)i
have to make sure that we reach
the ground state!
ratio of 2pt and 3pt functions
R (tins, ts)
tins 1
!
ts 1
⇧ ! hN|O |Ni

8. ECRS 2017, BNLC. Kallidonis
(~x0, t0)
(xs, ts)
(xins, tins)
O
• disconnected three-point function:
- volume inversions
- O(100000) statistics
⇠ 107
8
Probing nucleon structure from Lattice QCD
Computational requirements
• two-point function: one inversion
• connected three-point function: two inversions
requirements increase significantly as we go
towards the physical point
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
Calculation possible only with advanced computational methods and new technologies
- stochastic techniques
- improved methods based on matrix eigen-decomposition
- exploit power of graphics processing units (GPUs)
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

9. ECRS 2017, BNLC. Kallidonis 9
Probing nucleon structure from Lattice QCD
Searching for Dark matter
Dark matter candidates (WIMPs) interact
with nucleons via elastic scattering
cross-section of such interactions
directly related to nucleon σ-terms
σ-terms provide answers related to the
origin of the nucleon mass
Cryogenic Crystal Detectors
γ-ray emission,Andromeda galaxy

10. ECRS 2017, BNLC. Kallidonis 10
Probing nucleon structure from Lattice QCD
Searching for Dark matter - Nucleon σ-terms
0 20 40 60 80 100
ETMC Nf = 2 (2017)
QCD Nf = 2 + 1 (2015)
QCDSF Nf = 2 (2012)
BMW Nf = 2 + 1 (2015)
ETMC Nf = 2 + 1 + 1 (2014)
QCDSF-UKQCD Nf = 2 + 1 (2012)
Hoferichter (2015)
Alarc´on (2012)
Pavan (2002)
0 40 80 120 160
ETMC Nf = 2 (2017)
QCD Nf = 2 + 1 (2015)
QCD Nf = 2 + 1 (2013)
QCDSF Nf = 2 (2012)
BMW Nf = 2 + 1 (2015)
QCDSF-UKQCD Nf = 2 + 1 (2012)
⇡N (MeV)
s (MeV)
first direct evaluation of nucleon σ-terms at the physical point
A. Abdel-Rehim, C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G, Koutsou and A. Vaquero Aviles-Casco. Phys. Rev. Lett. 116, 2016
f ⌘ mf hN|¯qf qf |Ni
direct Feynman-Hellmann
f = mf
@mN
@mf
• no direct experimental measurements
• analyses of πN - scattering data
overestimate σ -terms
• direct evaluation more accurate than F-H
⇡N = 35.8(2.2)(1.5)(1.8
0.0) MeV
s = 35.8(7.4)(6.3)(1.8
0.0) MeV
c = 86.6(18.4)(6.9)(4.3
0.0) MeV

11. PhD DefenceC. Kallidonis 11
• how much is the quark intrinsic spin?
• do gluons contribute?
• how about orbital angular momentum?
Probing nucleon structure from Lattice QCD
Decoding the nucleon spin puzzle
1
2
⌘ Jtot = Jq + Jg
quark part gluon part
Jq,g =
1
2
(Aq,g
20 (0) + Bq,g
20 (0))
Aq,g
20 (0) ⌘ hxiq,g
Jq =
1
2
⌃q + Lq
intrinsic spin
axial charge
⌃q ⌘ gq
A
orbital angular
momentum
rich experimental activity, BNL, CERN, JLab, SLAC, ….
surprising result from EMC in 80’s: quark contributions to nucleon spin are small
arXiv: 1209.2803 for review
hN|Oµ
A|Ni
Oµ
A = ¯qf 5 µqf
hN |Oµ⌫
V |N i
Oµ⌫
V = ¯qf
{µ !
D ⌫}
qf
pDIS experiments
p-p collisions
ep collider: EIC@BNL

12. PhD DefenceC. Kallidonis
Probing nucleon structure from Lattice QCD
Decoding the nucleon spin puzzle - Summary
12
C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G, Koutsou, A. Vaquero Aviles-Casco, C. Wiese. Phys. Rev. Lett. 119, 2017
Jtot = Jq + Jg Jq =
1
2
⌃q + Lq
Jtot = 0.541(62)(49)
4
that the charm axial charge and momentum fraction, at
the physical point, is consistent with zero.
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
m [GeV]
0.03
0.02
0.01
Strange
Hybrid, Nf=2+1, a=0.124 fm
Clover: f=2+1, a=0.074 fm Nf=2, a=0.073 fmN
0.20
0.15
0.10
0.05
1
2q
Down
Nf=2: a=0.094 fm a=0.088 fm a=0.071 fm a=0.056 fm
0.3
0.4
0.5 Up
Nf=2+1+1: a=0.083 fm a=0.06 fm
Hybrid, Nf=2+1+1: a=0.090 fma=0.060 fm
TMF,
TMF,
FIG. 2: The up (upper), down (center) and strange (lower)
quark intrinsic spin contributions to the nucleon spin versus
the pion mass. Open symbols show results with only con-
nected contributions while filled symbols denote both con-
nected and disconnected contributions using the same ensem-
Jg = 0.133(11)(14) C. Alexandrou et al. arXiv: 1609.06901
first direct evaluation
1
2
⌃q = 0.201(17)(5)
Lq = 0.207(64)(45)
Jq = 0.408(61)(48)

13. ECRS 2017, BNLC. Kallidonis 13
Summary
• Lattice QCD provides a first principles calculation of key hadronic quantities
• Nucleon structure at the physical point
- possible with improved computational methods and high-performance computing
- valuable insight into important nucleon observables
- σ-terms -> implication with dark matter direct searches - valuable input
- spin -> theoretical resolution of the spin puzzle from Lattice QCD
Current/Future work
• lattice effects
• on-going developments for nucleon form factors at high-momentum transfer
• standing by for experimental discoveries!!
Thank you