Talk given as part of the seminar series of the Nuclear Theory Group of the Dept. of Physics and Astronomy, Stony Brook University.

1. Computing the Nucleon Spin
from Lattice QCD
Christos Kallidonis
Postdoctoral Associate
Department of Physics and Astronomy, Stony Brook University
Nuclear Theory Group Seminar
January 11, 2018
Phys. Rev. Lett. 119 (2017) no. 14, 142002
[arXiv:1706.02973]
with
C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, G. Koutsou, A. Vaquero, C. Wiese

2. SBU Seminar
• Introduction - Motivation
• Lattice QCD
• Lattice methods
• Noise reduction techniques
• Matrix element extraction
• Results: Nucleon spin
• Summary and outlook
2
Outline

3. SBU Seminar 3
RHIC, BNL CEBAF, JLab
The “proton spin crisis”
famous experiment in 1987 showed that only a fraction of spin is carried by quarks
Motivation
• how much is the quark intrinsic spin?
• do gluons contribute?
• how about orbital angular momentum?
Lattice QCD is in good position nowadays to
provide reliable theoretical answers to these
questions
Rich experimental activity, BNL, JLab, CERN, DESY, SLAC
RHIC: World’s only polarized hadron collider
- explore partonic structure of proton
- evidence of gluons showing preferential alignment
of their spins with the proton’s spin
pDIS experiments, p-p collisions
ep collider: EIC@BNL
- study of structure & interactions
of gluon-dominated matter

4. SBU Seminar 4
Lattice QCD
• 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
4
freedom in choice of
• quark mass (heavier is cheaper)
• lattice spacing, (larger is cheaper)
• lattice volume L3 x T, (smaller is cheaper)
a ⇠ 0.1fm
L ⇠ 5 fm

5. SBU Seminar 5
Lattice QCD
Feynman path integral
hOiF =
R
D (x)D ¯(x)O[ , ¯, U]e
¯M
R
D (x)D ¯(x)e ¯M
hOi =
R
DU(x)hOiF eln[detM] SG
R
DU(x)eln[detM] SG
M: Wilson-Dirac operator
hOiF ⇠ M 1
[U]: quark propagator
Very large matrix! ⇠ 108
⇥ 108
• use Monte Carlo methods to create gluon configurations
• solve
• observables: correlation functions in terms of quark propagators
• average over configurations, error
• need 1000s of configurations
• very computationally intensive!!
• most computational resources used for configurations and propagators
⇠ 1/
p
Nc
M(x, y; U)s(y) = b(x) ) s(y) = M 1
(y, x; U)b(x)
SQCD =
X
f
X
x,y
¯f (x)M(x; y, U) f (y) + SG[U]

6. SBU Seminar 6
Lattice QCD
Systematic uncertainties
• Finite lattice spacing effects
need to take the continuum limit
-> simulate with small lattice spacing, typically
• Finite volume effects
need to extrapolate to infinite volume
-> simulate with large volumes, rule of thumb:
• Contamination from excited states
need to obtain the nucleon ground state
-> employ set of techniques to ensure ground state dominance
• Simulations with unphysical quark masses
as the physical point is approached simulations become more computationally intensive
-> simulations with physical parameters are feasible nowadays
a ⇠ 0.1fm
Lm⇡ ⇠ 3.5

7. SBU Seminar 7
Lattice QCD
Simulation “landscape”
Image courtesy of LPC Clermont

8. SBU Seminar 8
Lattice QCD
Simulation “landscape”
Advances in La
Huge computational
&
Algorithmic impr
323
×64
5000 configs
Lattice QCD at the physical point is a major breakthrough!
• new technologies - increased computational power
• improved algorithms
5000 configs, 323 x 64

9. SBU Seminar 9
Lattice QCD
this work:
• one gauge ensemble with physical quark masses
- Nf = 2 Twisted Mass fermion action with a clover term
- lattice spacing a = 0.093fm
- lattice volume L~4.5fm
SF,tm = 
X
x,y
X
µ
(x)
⇥
(1 µ)Uµ(x) x+ˆµ,y + (1 + µ)U†
µ(x aˆµ) x ˆµ,y
⇤
(y) +
+
X
x,y
(x) x,y + i2aµq 5⌧3
x,y (y)
Wilson TM at maximal twist:
• discretization effects are improved
• no further operator improvement required
O(a)

10. SBU Seminar 10
Lattice QCD
(~x0, t0)
(~xs, ts)
two-point function
C(ts) =
X
~xs
1
4
(1 + 0)hJB(~xs, ts) ¯JB(~x0, t0)i
(~x0, t0)
(xs, ts)
(xins, tins)
O
(~x0, t0)
(xs, ts)
(xins, tins)
O
three-point function
C3pt
(P, ts, tins) =
X
~xs,~xins
PhJ(~xs, ts)O (~xins, tins) ¯J(~x0, t0)i
Physical processes on the lattice
• associated with operators coupling with quarks
• observables obtained from matrix elements,
extracted from correlation functions
JN (x) = ✏abc
⇥
ua
(x)(C 5)db
(x)
⇤
uc
(x)

11. SBU Seminar 11
Noise reduction techniques
Computational requirements
• two-point function: one inversion
• connected three-point function: two inversions
requirements increase significantly as the
physical point is approached
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
(~x0, t0)
(~xs, ts)
(~x0, t0)
(xs, ts)
(xins, tins)
O

12. SBU Seminar 12
Noise reduction techniques
Computational requirements
(~x0, t0)
(xs, ts)
(xins, tins)
O
• disconnected three-point function: volume inversions
- propagator from all sites to all sites ⇠ 107
for physical point Nr ⇠ O(1000)
two-point function: one inversion
connected three-point function: two inversions( (
feasible, but still very expensive
L( ; t) =
X
~x
Tr
⇥
M 1
(x; x)
⇤ impractical!
Alternative: Stochastic method
noise sources, , solveNr |⇠ri = {1, 1, i, i} M| ri = |⇠ri
M 1
E =
1
Nr
X
r
| rih⇠r| = M 1
+ O
✓
1
p
Nr
◆

13. SBU Seminar 13
Noise reduction techniques
Lu±d
( ; t) = Tr
⇥
(M 1
u ± M 1
d )
⇤
One-end trick (TMF)
M 1
u M 1
d = 2iµ(M†
M) 1
5
M 1
u + M 1
d = 2 5DW (M†
M) 1
5
P. Boucaud et al. arXiv:0803.0224,
C. Michael et al. arXiv:0709.4564
Lu d
( ; t) = 2iµ Tr[(M†
M) 1
5 ]
Lu+d
( ; t) = 2 Tr[(M†
M) 1
5 5DW ]
standard one-end trick
generalized one-end trick
Exact Eigenvalue Reconstruction

14. SBU Seminar 14
Noise reduction techniques
Exact part
Eu d
( ; t) =
mX
j=1
Tr

1
j
hvj| 5 |vji Su d
( ; t) =
1
Nr
NrX
r=1
Tr [hsr| 5 |sri] + O
✓
1
p
Nr
◆
Stochastic part
M| ri = |⇠ri |sri = (1 P)| ri
Lu d
( ; t) = 2iµ
⇣
Tr
⇥
(M†
M) 1
P 5
⇤
| {z }
Eu d( ;t)
+ Tr
⇥
(M†
M) 1
(1 P) 5
⇤
| {z }
Su d( ;t)
⌘
P =
mX
i=1
|viihvi| ⌘ UU†
, M†
M|vii = i|vii , hvi|vji = ijIntroduce:
Lu d
( ; t) = 2iµ Tr[(M†
M) 1
5 ]
G. Bali et al. arXiv: hep-lat/0505012,
J. Foley et al. arXiv: hep-lat/0505023,
A. O’Cais et al. arXiv: hep-lat/0409069
Exact Eigenvalue Reconstruction

15. SBU Seminar 15
0 500 1000 1500 2000 2500 3000
Nst.
2.0e+05
4.0e+05
6.0e+05
8.0e+05
1.0e+06
1.2e+06
0 500 1000 1500 2000
2
stoch
Nev
• variance reduces as NeV increases
• have to compensate cost of calculating eigenvectors
• for Nev=100, variance reduced by x3
• 40% less computer time required
0
100
200
300
400
500
600
0 500 1000 1500 2000 2500 3000
GPUnode-hours
Nst.
EER, Nev = 100, Nst. = 750
Pure stochastic, Nst. = 2250
4.0e+05
6.0e+05
8.0e+05
1.0e+06
1.2e+06
2
stoch
Exact Eigenvalue Reconstruction
Noise reduction techniques

16. SBU Seminar 16
Noise reduction techniques
Exact Eigenvalue Deflation
0.02
0 5 10 15 20 25 30
Nrhs
0
1
10
100
1000
0 500 1000 1500 2000
483
⇥ 96, aµ = 0.0009
time(hrs)
Nrhs
CG
ARPACK-CG, NeV= 100
ARPACK-CG, NeV= 200
ARPACK-CG, NeV= 600
improvement of ~20x in
the inversion time for Nev = 600
• based on matrix eigen-decomposition
• use the eigenvectors of Dirac operator to speed-up the inversion
A =
NX
i=1
i|viihvi| ⌘ U⇤U†
! A 1
=
NX
i=1
1
i
|viihvi| ⌘ U⇤ 1
U†
, A: Hermitian
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

17. SBU Seminar 17
0
200
400
600
800
1000
0 250 500 750 1000 1250
483
⇥ 96, aµ = 0.0009
Piz-Daintnode-hrs
Nrhs
tmLQCD build-up + I/O
tmLQCD build-up
QUDA build-up
r.h.s. accum.
r.h.s. accum. (optimized)
• factor x4 gained in obtaining eigenvectors
GPU implementation: extension of QUDA
library
Noise reduction techniques
Transitioning from CPUs to GPUs
• graphics cards much more capable than processors
NVidia Tesla P100: “The most advanced data
center GPU ever built”

18. SBU Seminar
(~x0, t0)
(xs, ts)
(xins, tins)
O
18
Matrix elements
(~x0, t0)
(~xs, ts)
C(~q = ~0, ts) =
X
n
|hJ|ni|2
e Ents
C3pt
(P, ~q = ~0, ts, tins) =
X
n,n0
hJ|n0
ihn| ¯Jihn0
|O |nie En0 (ts tins)
e En tins
(~x0, t0)
(xs, ts)
(xins, tins)
O
ratio of 2pt and 3pt functions
R (tins, ts)
tins 1
!
ts 1
⇧ ! hN|O |Ni
have to make sure that we reach the ground state!

19. SBU Seminar 19
Ensuring ground state dominance
1. plateau method
• compute 3-point function for several source-sink separations, fit to the plateau region
• increase source-sink separation until plateau value does not change
-> statistical errors increase exponentially as separations becomes larger,
more statistics needed for larger separations
• excited states fall as e (E1 E0)(ts tins)
e (E1 E0)tins
Matrix elements

20. SBU Seminar 20
2. summation method
sum the ratio over the insertion time
Rsum
(ts) =
ts aX
tins=a
R (ts, tins)
Rsum
(ts) = c + ts M + O
⇣
e (E1 E0)ts
⌘
+ · · ·
excited states:
20
40
60
80
100
120
140
160
0.8 1 1.2 1.4 1.6 1.8
Rsum
(ts)
ts [fm]
tlow
s = 0.94 fm
tlow
s = 1.13 fm
Matrix elements
Ensuring ground state dominance
e (E1 E0)ts

21. SBU Seminar 21
3. two-state fit
C(~q = ~0, ts) =
X
n
|hJ|ni|2
e Ents
C3pt
(P, ~q = ~0, ts, tins) =
X
n,n0
hJ|n0
ihn| ¯Jihn0
|O |nie En0 (ts tins)
e En tins
consider first excited state in 3pt and
2pt functions
C3pt
= A00e E0 ts
+ A01e E0(ts tins)
e E1 tins
+ A10e E1(ts tins)
e E0 tins
+ A11e E1 ts
C = c0e E0 ts
+ c1e E1 ts
hN|O |Ni =
A00
c0
Matrix elements
Ensuring ground state dominance

22. SBU Seminar 22
Nucleon spin
Decoding the nucleon spin puzzle - Ji’s sum rule
1
2
⌘ Jtot = Jq + Jg
quark part gluon part
Jq,g =
1
2
(Aq,g
20 (0) + Bq,g
20 (0))
hxiq,g ⌘ Aq,g
20 (0)
hN(p, s0
)|Oµ
Aa |N(p, s)i , Oµ
Aa = ¯ 5 µ
⌧a
2
.
hN(p, s0
)|Oµ
Aa |N(p, s)i = i¯uN (p, s0
)

1
2
Ga
A(0) 5 µ uN (p, s)
hN(p, s0
)|Oµ⌫
V a |N(p, s)i , Oµ⌫
V a = ¯ {µ !
D ⌫} ⌧a
2
.
Renormalization functions: determined non-perturbatively
arXiv:1509.00213
arXiv:1705.03399
gluon contributions arXiv:1611.06901
hep-ph/9603249
hN(p0
, s0
)|Oµ⌫
V a |N(p, s)i =
= ¯uN (p0
, s0
)

Aa
20(q2
) {µ
P⌫}
+ Ba
20(q2
)
i {µ⇢
q⇢P⌫}
2m
+ Ca
20(q2
)
q{µ
q⌫}
m
1
2
uN (p, s)
2nd Mellin moment of unpolarized PDF
Jq =
1
2
⌃q + Lq
intrinsic spin
axial charge
⌃q = Ga
A(0)
G3
A(0) = gA
orbital angular
momentum
1st Mellin moment of polarized PDF

23. SBU Seminar 23
Nucleon spin
Connected contributions -Axial charge
• well-known experimentally from neutron
β-decay
• benchmark quantity in Lattice QCD
• our calculation: ~10000 measurements
Axial Charge
u-d
Volume Effects
Agreement within error bars

24. SBU Seminar 24
Nucleon spin
Connected contributions - Quark momentum fraction
• unpolarized PDFs: distribution of nucleon
momentum among its constituents
• measured in DIS experiments
• our calculation: ~70000 measurements
Quark Momentum Fraction
(p′
, s′
)|Oµν
DV|N(p, s)⟩=¯uN (p′
, s′
) A20(q2
) γ{µ
P ν}
+B20(q2
) iσ{µαqαP ν}
2m
+C20(q2
) 1
m q{µ
qν}
uN (p, s)
u-d
es must be assessed
MS(2GeV)

25. SBU Seminar 25
evaluation of disconnected gA, <x>
• considerable contributions
• ~500000 measurements
calculation not possible without
• graphics cards
• improved computational methods
all disconnected contributions are a first time calculation directly at the physical point
Nucleon spin
Disconnected contributions

26. SBU Seminar
4
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-
26
Nucleon spin
Intrinsic spin - lattice artifacts
Indirect assess of:
- cut-off effects
- volume effects
- quenching effects
need to examine these effects directly
open symbols: connected contribution
filled symbols: total contribution

27. SBU Seminar 27
Nucleon spin
Intrinsic spin - Orbital angular momentum
Collected Results
Orbital Angular momentum - Intrinsic spin
Largest contribution from up-quark
d-quark:
orbital angular momentum almost cancelled by its intrinsic spin
• up-quark: largest contribution
• down-quark: intrinsic spin cancels orbital angular momentum

28. SBU Seminar 28
Jq =
1
2
⌃q + LqJtot = Jq + Jg
C. Alexandrou et al. arXiv: 1611.06901
first direct evaluation
Nucleon spin
28
Summary of results
Jtot = 0.541(62)(49)
hxiR
g = 0.267(22)(27)
hxiR
q = 0.804(122)(95)
hxiR
q + hxiR
g = 1.07(12)(10)
1
2
⌃q = 0.201(17)(5)
Lq = 0.207(64)(45)
Jq = 0.408(61)(48)
Jg = 0.133(11)(14)

29. SBU Seminar 29
Summary
• Lattice QCD in good position to provide reliable benchmarks and predictions
• Nucleon structure from Lattice QCD at the physical point possible only with improved
computational methods and high-performance computing
• theoretical resolution to the spin puzzle from Lattice QCD
• computed value for total quark + gluon spin contributions consisted with experiment
• quark + gluon momentum fraction consistent with unity
Work to be done:
• large effort to directly assess lattice artifacts - ensemble with L=5.5fm, a = 0.086fm
• investigate novel noise reduction techniques
… standing by for experimental discoveries!
Thank you