Christos Kallidonis gave a seminar on exploring nucleon parton distribution functions (PDFs) from lattice QCD. He outlined the theoretical framework for calculating PDFs on the lattice using the pseudo-PDF approach, which involves evaluating matrix elements in Euclidean space-time. Preliminary results for unpolarized PDFs were presented and future directions discussed. The talk provided motivation for studying PDFs from first principles using lattice QCD and an overview of the pseudo-PDF methodology.

1. Christos Kallidonis
JLab Cake Seminar | Feb. 8, 2021
The Nucleon Parton Distribution Functions
from Lattice QCD
An exploratory study of distillation in the pseudo-PDF framework

2. Outline
lattice
evaluation
PDF results
summary
& outlook
theoretical
framework
intro &
motivation
HadStruct Collaboration
Robert Edwards, C.K., Jianwei Qiu, David Richards, Frank Winter [a]
Balint Joó [b]
Carl Carlson, Colin Egerer, Tanjib Khan, Christopher Monahan,
Kostas Orginos, Eloy Romero [c]
Wayne Morris, Anatoly Radyushkin [d]
Joe Karpie [e]
Savvas Zafeiropoulos [f]
Yan-Qing Ma [g]
[a] Jefferson Lab, [b] Oak Ridge, [c] William & Mary,
[d] Old Dominion University, [e] Columbia University,
[f] Aix Marseille University, [g] Peking University

3. How can we study hadron structure?
• Hadrons are complicated systems made of quarks, antiquarks, gluons (
• Dynamics of partons encoded in e.g.:
Form factors Parton distribution
functions (PDFs) d
A. Scapellato (Adam Mickiewicz University) Nucleon PDFs and GPDs from lattice QCD
Introduction
Motivation - why the PDFs?
}
Generalized Parton Distributions (GPDs)
Transverse-momentum dependent (TMD) PDFs
complement 3D picture of hadrons
Parton Distribution Functions (PDFs):
‣ part of ongoing efforts for deep understanding of internal quark and gluon structure of hadrons
‣ number density of “partons” with a longitudinal momentum fraction of the nucleon’s momentum
1D object, necessary in 3D imaging of hadrons
‣ accessible in deep inelastic and semi-inclusive deep inelastic scattering
x
→
Important LHC applications
‣ predictions of various cross sections
‣ W-boson mass
‣ Strong coupling constant
‣ Higgs boson couplings and cross-sections (probe BSM physics)
Gao et.al., arXiv: 1709.04922

4. Introduction
Motivation - why on the lattice?
Hadrons have an immensely rich composition: Experimental extraction of information becomes
more complex the deeper we look
Accessing PDFs from experiments is an intricate procedure
Still, relatively good precision exists for unpolarized PDFs
Kovarik et.al., arXiv: 1905.06957
7
x*f(x,Q)
x
CT18 at 2 GeV
s
g/5
u
d
–
d
–
u
c
0.0
0.5
1.0
1.5
2.0
10
-6
10
-4
10
-3
10
-2
10
-1 0.2 0.5 0.9
x*f(x,Q)
x
CT18 at 100 GeV
s
g/5
u
d
–
d
–
u
c
b
0.0
0.5
1.0
1.5
2.0
10
-6
10
-4
10
-3
10
-2
10
-1 0.2 0.5 0.9
x,Q)
CT18Z at 2 GeV
s
g/5
u
d
–
d
–
1.0
1.5
2.0
x,Q)
CT18Z at 100 GeV
s
g/5
u
d
–
d
–
1.0
1.5
2.0
Hou et.al. arXiv: 1912.10053
Lattice Evaluation
‣ PDFs: “hot” topic, deserve theoretical attention
‣ unpolarized PDFs benchmark quantity
‣ transversity PDFs not well constrained
‣ GPDs and TMDs even more difficult to access
‣ first principles calculation
→

5. PDFs on the lattice
pseudo-PDF framework
Unpolarized PDF: non-local matrix element on the light-cone in Minkowski space
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q(x) =
Z 1
1
d⇠
4⇡
e ixP +
⇠
hN(P)| ¯(⇠ ) +
W(⇠ , 0) (0)|N(P)i
Inaccessible on the lattice!
<latexit sha1_base64="QxkCXaXXpq/dgqIIFLlU2fXxLRI=">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</latexit>
v±
=
v0
± v3
p
2
: light-cone coordinates
ξ+
, ξ−
x0
x3
ξ+
ξ−
light-cone distances shrink to the
origin in Euclidean space-time
Methods for accessing -dependence on the lattice:
‣ hadronic tensor
‣ auxiliary quarks
‣ “good lattice cross sections”
‣ “OPE without OPE”
‣ quasi-distributions
‣ pseudo-distributions
x
K.F. Liu et.al. arXiv: hep-ph/9306299
U. Aglietti et.al. arXiv: hep-ph/9806277, W. Detmold et al. arXiv: hep-lat/0507007,
V. Braun et. al. arXiv: 0709.1348
Y-Q. Ma and J. Qiu arXiv: 1404.6860,
1412.2688, 1709.03018
A.J. Chambers et.al. arXiv: 1703.01153
X. Ji arXiv: 1305.1539
A. Radyushkin arXiv: 1705.01488, 1612.05170, 1710.08813,
1711.06031, 1801.02427, 1912.04244
Review: K. Cichy, M. Constantinou arXiv: 1811.07248

6. PDFs on the lattice
pseudo-PDF framework
Matrix element in Euclidean space:
Lorentz decomposition
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M↵
(P, z) = 2P↵
M(⌫, z2
) + 2z↵
N(⌫, z2
)
leading twist
+ 𝒪(z2
Λ2
QCD)
higher twist
kinematics
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P = (E, 0, 0, Pz)
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z = (0, 0, 0, z3)
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↵ = 0
vector current: unpolarized PDFs
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M↵
(P, z) ⌘ hN(P)| ¯(z) ↵
W(z, 0) (0)|N(P)i
Wilson line boosted nucleon states
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M(⌫, z2
3) =
Z 1
1
dxei⌫x
P(x, z2
3) ) P(x, z2
3) =
1
2⇡
Z 1
1
d⌫e i⌫x
M(⌫, z2
3)
pseudo-PDF
FT: Along constant line
z3
<latexit sha1_base64="4hHFANeHviVkkMWgywy+i0Pv3qI=">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</latexit>
q(x) = lim
z3!0
P(x, z2
3)
PDF
(ignoring divergences)
: “Ioffe time”
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⌫ ⌘ Pz · z3
B. Ioffe, Phys.Lett. 30B, 123 (1969)
Ioffe-time distributions
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M(⌫, z2
3) ⇠ hN(Pz)| ¯(z3) 0
W(z3, 0) (0)|N(Pz)i
Lorentz invariant!
, : Lorentz scalars
ν z3
features:
‣ standard log. divergence
‣ power divergence (Wilson line)
‣ mult. renormizable to all orders in PT
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M(⌫, z2
3)
T. Ishikawa et.al. arXiv: 1609.02018
J. W. Chen et. al. arXiv: 1707.03107

7. PDFs on the lattice
pseudo-PDF framework
Steps towards -dependence
x
i) cancel divergences - renormalization
reduced-ITD
iii) reconstruction of -dependence
x
F. T. in -space:
ν
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Q(⌫, µ2
) =
Z 1
1
dxei⌫x
q(x, µ2
)
<latexit sha1_base64="oT+0KHaIr/RM7hjHKLPJQjIX8Uw=">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</latexit>
Re Q(⌫, µ2
) =
Z 1
0
dx cos(⌫x)qv(x, µ2
)
<latexit sha1_base64="7nvsVyySzfeYOANVvx1Y3OQSdK0=">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</latexit>
Im Q(⌫, µ2
) =
Z 1
0
dx sin(⌫x)qv2s(x, µ2
)
Inverse problem…!
K. Orginos et.al. arXiv: 1706.05373
J. Karpie et. al. arXiv: 1710.08288
J. Karpie et. al. arXiv: 1807.10933
A. Radyushkin arXiv: 1705.01488
<latexit sha1_base64="ufDO2bwQ0PTVfOHwZZ1L74vdHXM=">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</latexit>
M(⌫, z2
3) =
✓
M(Pzz3, z2
3)
M(Pzz3, z2
3)|z3=0
◆
⇥
✓
M(Pzz3, z2
3)|Pz=0,z3=0
M(Pzz3, z2
3)|Pz=0
◆
<latexit sha1_base64="EarU8i0j3TGPacGrbQ6T+iUZP9o=">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</latexit>
M(⌫, z2
3) =
M(Pzz3, z2
3)
M(Pzz3, z2
3)|Pz=0
renormalization
V0
<latexit sha1_base64="V3t2LdZcGXusZQxUs9iZAc9pXKo=">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</latexit>
✓
ZV (Pz) =
1
M(Pzz3, z2
3)|z3=0
◆
valence quark
q + q̄
ii) matching to light-cone ITDs
1-loop matching equation:
evolution to common scale 1/z0
conversion to scheme
MS
light-cone ITD
<latexit sha1_base64="2o+yCBDc1TlCsbx2scqzyN2Wt3c=">AAAC6XicdVFbaxNBFJ5drdb10lQffRkMQkpL2F3a2DwIwaL4oNCCaQOZzTI7mU2GzF6YixCH/RG+ia/+Hf+A/8bZi9CIHhj45jvnfHPmO0nJmVS+/8tx79zdu3d//4H38NHjJwe9w6fXstCC0CkpeCFmCZaUs5xOFVOczkpBcZZwepNsLur8zWcqJCvyT2pb0ijDq5yljGBlqbj3E2VYrVOBN+ZjNUC5PvmyCI/ga9jwBHNz1dIo03XiGCJbTAzCvFzjWMKL+F1lQlSyCrFcxf4igEuNTnb69S0BxGmq5ojnLRrY55pMK0sXVmuFswzHb4+DqjKnFRJstVZHjV6SmjfVXEf1GN39Q31va6K41/eHwXg0DkPoD/0mLBiNz8/ORjDomD7o4jI+dPbQsiA6o7kiHEs5D/xSRQYLxQinlYe0pCUmG7yicwtznFEZmcb2Cr60zBKmhbAnV7Bhb3cYnEm5zRJbWQ8r/87V5L9yc63S88iwvNSK5qR9KNUcqgLWO4RLJihRfGsBJoLZWSFZY+ufspv2dqQacSIiQ7Vdxs6HLFXvvfI8zxr3xx34f3AdDoPR0L867U8mnYX74Dl4AQYgAK/ABLwHl2AKiBM6Mwc7ibtxv7rf3O9tqet0Pc/ATrg/fgOsfukf</latexit>
M(⌫, z2
) = Q(⌫, µ2
) +
↵sCF
2⇡
Z 1
0
du Q(u⌫, µ2
)

ln
✓
z2
µ2 e2 E +1
4
◆
B[u] + L[u]
A. Radyushkin arXiv: 1801.02427
J.-H.Zhang et. al. arXiv: 1801.03023
T. Izubuchi et. al. arXiv: 1801.3917

8. Lattice evaluation
Correlation functions: “Distillation” as a smearing method
Smearing: Increase overlap of hadron interpolating fields with ground state
Distillation: Smearing operator low-rank eigenvector representation
→
<latexit sha1_base64="LNf1EkghLGCuYHBLsx67coqgZvY=">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</latexit>
⇤~
x,~
y(t) ⌘
N
X
k=1
vk(~
x, t)v†
k(~
y, t) = V (t)V †
(t) , r2
vk = kvk
<latexit sha1_base64="lLuBnZavVOmEY3iLkQOaidYwexs=">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</latexit>
qsm(~
x, t) = ⇤~
x,~
y(t)q(~
y, t)
<latexit sha1_base64="TFql/Z4y3xzWSgwX8jrRyFS5mAs=">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</latexit>
XB(~
x, t) = ✏abc
S 1 2 3
(D1q1)a
1
(~
x, t)(D2q2)b
2
(~
x, t)(D1q3)c
3
(~
x, t)
<latexit sha1_base64="9y2zSXOQS1TTlL6Rj9X+owRfsRM=">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</latexit>
qsm(~
x, t) = F[U(~
x, t); ~
y]q(~
y, t)
<latexit sha1_base64="HF5ZxH7U+Ine0XsBMepGzMSLNHI=">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</latexit>
r2
~
x,~
y(t) = 6 ~
x~
y
3
X
j=1
h
Uj(~
x, t) ~
x+ĵ,~
y + U†
j (~
x ĵ, t) ~
x ĵ,~
y
i
”Standard” techniques: Gauge-covariant, based on 3D lattice Laplacian
perambulators:
<latexit sha1_base64="1Gh0B8X7KF3PjWMVVLalM+rUG+8=">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</latexit>
⌧(tf , t0) = V †
(tf )M 1
(tf , t0)V (t0)
elementals:
<latexit sha1_base64="7Nuc1HuoPJnp5qRYba7d+cW7fpw=">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</latexit>
i,j,k
1 2 3
(t) = ✏abc
S 1 2 3
(D1vi(t))a
(D2vj(t))b
(D1vk(t))c
gen. perambulators:
<latexit sha1_base64="4C5I9UwcIo+CCLLVGYm4Q4zed3E=">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</latexit>
˜
⌧(tf , tins, t0) = V †
(tf )M 1
(tf , tins) (tins)M 1
(tins, t0)V (t0)
Pros:
‣ factorization of correlation functions
‣ various spin structures without additional inversions
‣ mom. projection at the sink and the source
‣ reusability of building blocks
Cons:
‣ extra step: compute eigenvectors of
‣ big files/storage
∇2
inversions:
<latexit sha1_base64="p5Omw89Ua4LwxSaq1yWeHjMqQ9g=">AAACuHicdVFdb9MwFHUyBiN8dfDIi0WFaCWonK2jGwipEi+8IA1Bt0lNFm5cp7XqfMi+qVSi/BJ+Gf8GJ+2kFY0r2bo+59xz7eu4UNIgY38cd+/e/v0HBw+9R4+fPH3WOXx+YfJSczHhucr1VQxGKJmJCUpU4qrQAtJYict4+bnhL1dCG5lnP3BdiDCFeSYTyQEtFHV+f48CUMUCrque7Ne99UeMWJ9+ol+vq3d+cw5Wgle/6rfYj6qNNIgFQr2KZG/L9YOZUAiWb5h2i1h9A6J1rO90tJ12PK3utmvU6bIBOz4dsRFlg+HZERu+pw1ycjwcUX/A2uiSbZxHh85+MMt5mYoMuQJjpj4rMKxAo+RK1F5QGlEAX8JcTG2aQSpMWLVTrOlri8xokmu7MqQteruigtSYdRpbZQq4MP9yDXgXNy0xOQ0rmRUlioxvGiWlopjT5kvoTGrBUa1tAlxLe1fKF6CBo/04b8eqNec6rETJQe08yEKJhmXteZ4d3M106P+Ti6OBfzJg34bd8Xg7wgPykrwiPeKTERmTL+ScTAh3HOeNwxzf/eD+dOeu3EhdZ1vzguyEq/8CEabWMw==</latexit>
S(i)
↵ (y; t0) = M 1
(y; ~
z, t)↵ vi(~
z) 0 tt0
= M 1
(y; ~
z, t0)↵ 0
vi(~
z)
<latexit sha1_base64="Ec16og68vhuG9SbHwaiiQEtK+LU=">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</latexit>
C2pt(tf , t0) =
X
~
x,~
y
hX(~
x, tf )X̄(~
y, t0)i
t0
tf
t0
tf
tins
W(z3)
<latexit sha1_base64="Umwm2f9JJtZZfaCxTOYf7ULsBnQ=">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</latexit>
C3pt(tf , tins, t0; ) =
X
~
x,~
z,~
y
hX(~
x, tf ) [q̄ 0
Wq](~
z, t) X̄(~
y, t0)i
M. Peardon et. al. arXiv: 0905.2160
<latexit sha1_base64="RbJdi9FM93Q0Ih66kTr/LKalENI=">AAACN3icdVBNSyNBFOxRd9W4H1GPXhqD4LJLmAlrNAchIIgnUTAqZOLwptOTNPb0DN1vhDDMv/Li3/CmFw+KePUf2BPjsrtoQUN11Xt0V4WpFAZd98aZmp759Hl2br6y8OXrt+/VxaVjk2Sa8Q5LZKJPQzBcCsU7KFDy01RziEPJT8LzndI/ueDaiEQd4SjlvRgGSkSCAVopqO77MeCQgcx3iyD3QaZDCMyv/cAUdJv6kke47v30Iw3sj+srCCWcNYq8HPO1GAzxx9n4ElRrbt1rNVuNBnXr7hiWNFtbGxtN6k2UGpngIKhe+/2EZTFXyCQY0/XcFHs5aBRM8qLiZ4anwM5hwLuWKoi56eXj3AVds0qfRom2RyEdq39v5BAbM4pDO1mmNP97pfie180w2urlQqUZcsVeH4oySTGhZYm0LzRnKEeWANPC/pWyIdiO0FZdsSW8JaUfk+NG3WvW3cPftXZ7UsccWSGrZJ14ZJO0yR45IB3CyCW5Jffkwbly7pxH5+l1dMqZ7CyTf+A8vwCBJ62Q</latexit>
F↵s,Ns
=
✓
1 +
↵sr2
Ns
◆Ns

9. Lattice evaluation
Simulation details
Lattice action / configurations:
‣ Wilson-Clover fermion action,
‣ lattice volume:
‣
‣ ,
‣ ,
Nf = 2 + 1
323
× 64
β = 6.3 , a = 0.091 fm (a−1
= 2.16 GeV)
mπ = 0.350 GeV mN = 1.160 GeV
L = 2.9 fm mπL = 5.15
W&M / JLab
Parameters and statistics:
‣ 350 gauge configurations
‣ distillation vectors:
‣ 6
‣ 4 time sources
‣ boost momentum
‣ displacements
‣ 5600 statistics
Nev = 64
tsep ≡ (tf − t0) = [4a, 6a, …, 14a] → 0.4…1.3 fm
t0/a = [ti
0, + T/4, + T/2, + 3T/4]
Pz = 2π/L × [0, ± 1,…, ± 6] → 0…2.55 GeV
z3 = 0, ± 1a, …, ± 8a → 0…0.73 fm
Runs performed at JLab computer clusters and at
Frontera (TACC) using the Chroma software suite and
in-house analysis packages
t0
tf
tins
W(z3)
<latexit sha1_base64="Umwm2f9JJtZZfaCxTOYf7ULsBnQ=">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</latexit>
C3pt(tf , tins, t0; ) =
X
~
x,~
z,~
y
hX(~
x, tf ) [q̄ 0
Wq](~
z, t) X̄(~
y, t0)i
Pz

10. Matrix elements: ground state dominance
i) Summation method
ii) Plateau method (ask for more!)
Ratio of 3pt/2pt functions:
<latexit sha1_base64="2ky8d8js9KtiZKQAZr1JfrTdLRE=">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</latexit>
R(Pz, z3; tsep, tins) =
C3pt(Pz, z3; tsep, tins)
C2pt(Pz, tsep)
tsep tins a
!
tins t0 a
M(⌫, z2
3)
1
2
3
4
5
6
7
Re[R
sum
(P
z
=
2;
z
3
=
2)]
Linear fit tlow
sep = 4
Linear fit tlow
sep = 6
Linear fit tlow
sep = 8
0.46
0.48
0.50
0.52
0.54
0.56
Re[R
r°sum
(P
z
=
2;
z
3
=
2)]
2 4 6 8 10 12 14 16
tsep/a
0.2
0.4
0.6
0.8
1.0
Im[R
sum
(P
z
=
2;
z
3
=
2)]
2 4
0.055
0.060
0.065
0.070
Im[R
r°sum
(P
z
=
2;
z
3
=
2)]
1
2
3
4
5
6
7
Re[R
sum
(P
z
=
2;
z
3
=
2)]
Linear fit tlow
sep = 4
Linear fit tlow
sep = 6
Linear fit tlow
sep = 8
0.46
0.48
0.50
0.52
0.54
0.56
Re[R
r°sum
(P
z
=
2;
z
3
=
2)]
0.6
0.8
1.0
(P
z
=
2;
z
3
=
2)]
0.065
0.070
(P
z
=
2;
z
3
=
2)]
1
2
3
4
5
6
7
Re[R
sum
(P
z
=
2;
z
3
=
2)]
Linear fit tlow
sep = 4
Linear fit tlow
sep = 6
Linear fit tlow
sep = 8
0.46
0.48
0.50
0.52
0.54
0.56
Re[R
r°sum
(P
z
=
2;
z
3
=
2)]
2 4 6 8 10 12 14 16
tsep/a
0.2
0.4
0.6
0.8
1.0
Im[R
sum
(P
z
=
2;
z
3
=
2)] 2 4 6 8 10 12 14
tsep/a
0.055
0.060
0.065
0.070
Im[R
r°sum
(P
z
=
2;
z
3
=
2)]
<latexit sha1_base64="xrvQzIEPHvfGUDzeXrOv1TxZg2A=">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</latexit>
= c1 + M(Pz, z3) ⇥ tsep + O e Etsep
<latexit sha1_base64="MDeB3dEcH/2OyxZi1NTkz+5hqpo=">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</latexit>
Rsum(Pz, z3; tsep) =
tsep a
X
tins=a
R(Pz, z3; tsep, tins)
Perform linear fits, increase lower value of
until convergence
tsep
L. Maiani et. al. Nucl.Phys. B293 (1987)
Lattice evaluation
0.040
0.042
0.044
0.046
0.048
°8 °6 °4 °2 0 2 4 6 8
[tins ° tsep/2]/a
0.060
0.062
0.064
0.066
0.068
0.070
0.072
0.074
Pz = 2 , z3 = 2
4
Im[R(P
z
,
z
3
)]
0.040
0.042
0.044
0.046
0.048
0.050
Pz = 2 , z3 = 1
tsep = 4a
tsep = 6a
tsep = 8a
tsep = 10a
tsep = 12a
tsep = 14a Plateau fits
°8 °6 °4 °2 0 2 4 6 8
[tins ° tsep/2]/a
0.060
0.062
0.064
0.066
0.068
0.070
0.072
0.074
Pz = 2 , z3 = 2
4 6 8
t
Im[R(P
z
,
z
3
)]
0.040
0.042
0.044
0.046
0.048
0.050
Pz = 2 , z3 = 1
tsep = 4a
tsep = 6a
tsep = 8a
tsep = 10a
tsep = 12a
tsep = 14a Platea
°8 °6 °4 °2 0 2 4 6 8
[tins ° tsep/2]/a
0.060
0.062
0.064
0.066
0.068
0.070
0.072
0.074
Pz = 2 , z3 = 2
4
Im[R(P
z
,
z
3
)]

11. 0.82
0.83
0.84
0.85
0.86
0.87
Pz = 2 , z3 = 0
tsep = 4a
tsep = 6a
tsep = 8a
tsep = 10a
tsep = 12a
tsep = 14a
Summ. linear
0.485
0.490
0.495
0.500
0.505
0.510
0.515
Pz = 2 , z3 = 2
0.112
0.114
0.116
0.118
0.120
0.122
0.124
0.126 Pz = 2 , z3 = 5
°8 °6 °4 °2 0 2 4 6 8
[tins ° tsep/2]/a
0.016
0.018
0.020
0.022
0.024 Pz = 2 , z3 = 8
4 6 8
tlow
sep/a
Re[R(P
z
,
z
3
)]
0.040
0.042
0.044
0.046
0.048
0.050 Pz = 2 , z3 = 1
tsep = 4a
tsep = 6a
tsep = 8a
tsep = 10a
tsep = 12a
tsep = 14a Summ. linear
0.0600
0.0625
0.0650
0.0675
0.0700
0.0725
0.0750
Pz = 2 , z3 = 2
0.042
0.044
0.046
0.048
0.050
0.052
0.054
Pz = 2 , z3 = 5
°8 °6 °4 °2 0 2 4 6 8
[tins ° tsep/2]/a
0.012
0.014
0.016
0.018
Pz = 2 , z3 = 8
4 6 8
tlow
sep/a
Im[R(P
z
,
z
3
)]
Lattice evaluation
Matrix elements: select data

12. PDF results
Pseudo-ITDs
0 1 2 3 4 5 6 7 8 9 10
∫
°0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Re[M
Lsumm
(∫,
z
2
3
)]
Pz = 0
Pz = 1
Pz = 2
Pz = 3
Pz = 4
Pz = 5
Pz = 6
<latexit
sha1_base64="x4YNioFrma8b5rW3tPEcqbq+dPo=">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</latexit>
Re
⇥
M(⌫,
z
2
3
)
⇤
0 1 2 3 4 5 6 7 8 9 10
∫
°0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Im[M
Lsumm
(∫,
z
2
3
)]
Pz = 0
Pz = 1
Pz = 2
Pz = 3
Pz = 4
Pz = 5
Pz = 6
<latexit
sha1_base64="Ytv6KAbYVi9nG4xqfS7e/mR/+z0=">AAACSXicbVBNSyNBEO2J667Ofhj16KUxLChImImyehS86EFQMCpkxtDTqUmadPcM3TVCHOZP+Gu8utf9Bfsz9iae7Ik5GN0HDY/3qqqrXpJLYTEI/nqNhU+Ln78sLftfv33/sdJcXbu0WWE4dHkmM3OdMAtSaOiiQAnXuQGmEglXyfio9q9uwViR6Quc5BArNtQiFZyhk/rNnTIyip6oKpKQYi9SDEepYePytNqKdLFz19+96WxHRgxHGPebraAdTEE/knBGWmSGs/6qtxgNMl4o0Mgls7YXBjnGJTMouITKjwoLOeNjNoSeo5opsHE5PauiP50yoGlm3NNIp+rbjpIpaycqcZX12va9V4v/83oFpgdxKXReIGj++lFaSIoZrTOiA2GAo5w4wrgRblfKR8wwji5Jf27UdDg3cQkFZ3LuICfVSVa+77vgwvcxfSSXnXb4qx2c77UOO7MIl8gG2SRbJCT75JAckzPSJZzckwfySH57f7x/3pP3/Fra8GY962QOjYUX956xWQ==</latexit>
Im
⇥
M(⌫,
z
2
3
)
⇤
‣ get rid of logarithmic divergences
‣ renormalization of vector current
‣ higher-twist contaminations partially suppressed
𝒪(z2
3Λ2
QCD)
<latexit sha1_base64="ufDO2bwQ0PTVfOHwZZ1L74vdHXM=">AAAC8HicjVFNj9MwEHUCC0v46sKRi0WF1JVWlVPYsj0UrcSFC1KR6O5KTYgc12mtOk5kO0jd4P/BDXHl73Di3zDJBokiVjCSpfF7b8aeN2kphbGE/PD8Gzf3bt3evxPcvXf/wcPewaMzU1Sa8TkrZKEvUmq4FIrPrbCSX5Sa0zyV/DzdvG74849cG1Go93Zb8jinKyUywagFKOl9j3Jq15mmm/qtG0SqOrpMnn8YHeIpjiTP7CACjtWtilHZiGbJJWg6nbue+pTUkEyJc5EWq7U9jKzIufm/vlAM9yk56lr8W+kc7t5Jen0yDCfjyWiEyZC0Acl4cnJ8PMZhh/RRF7PkwNuLlgWrcq4sk9SYRUhKG9dUW8Ekd0FUGV5StqErvoBUUZgjrlv3HX4GyBJnhYajLG7R3ytqmhuzzVNQNjOYP7kG/Bu3qGx2EtdClZXlil09lFUS2wI3q8RLoTmzcgsJZVrAXzFbU7DVwsKDnVZtc6bjmlfg4c5AADXrd0EQgHG/3MHXJ2ejYTgekncv+qevOgv30RP0FA1QiF6iU/QGzdAcMW/iJd7aE772P/tf/K9XUt/rah6jnfC//QTi8OuL</latexit>
M(⌫, z2
3) =
✓
M(Pzz3, z2
3)
M(Pzz3, z2
3)|z3=0
◆
⇥
✓
M(Pzz3, z2
3)|Pz=0,z3=0
M(Pzz3, z2
3)|Pz=0
◆

13. Evolution and matching to : Light-cone ITDs
MS
evolution to common scale 1/z0
conversion to scheme
MS
light-cone ITD
<latexit sha1_base64="6I2IXBKK/mWdKBImIj3ojFH7TM8=">AAACRXicdZDdahNBFMdnq9W69SOtl94MhkKkuMyuiakXhaA3vaxg2kJ2DbOTs8mQ2Q9mzhTCsq/g03hbb30GH8K70ls7SSMY0QMDf/7na84vrZQ0yNgPb+ve/e0HD3ce+buPnzx91trbPzOl1QKGolSlvki5ASULGKJEBReVBp6nCs7T+Ydl/vwStJFl8QkXFSQ5nxYyk4Kjs8atzvuOfUWPaawgw1GcaS7q8NB+jpo6fG0bGms5nWEyPhy32iyI2Jt3UURZwPrdXsic6L1l/aMuDQO2ijZZx+l4z9uOJ6WwORQoFDdmFLIKk5prlEJB48fWQMXFnE9h5GTBczBJvTqpoQfOmdCs1O4VSFfunx01z41Z5KmrzDnOzN+5pfmv3MhidpTUsqgsQiHuFmVWUSzpkg+dSA0C1cIJLrR0f6Vixh0WdBT9jVGr4UInNVjB1cZBznIo543v+w7cbzr0/+IsCsJewD5224PBGuEOeUFekg4JSZ8MyAk5JUMiyBfylVyRb95376d37d3clW55657nZCO8X7cxNa95</latexit>
B(u) =

1 + u2
1 u +
<latexit sha1_base64="+3l2zM+HYCOdXsduRi0NJzyHHwI=">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</latexit>
L(u) =

4
ln(1 u)
1 u
2(1 u)
+
‣ treat each independently
‣ need polynomial parametrization w.r.t.
‣ evolution/matching processes separately or combined
can give rise to different systematics
‣ 1-loop matching: ~10% effect to pseudo-ITD data
‣ small higher-twist effects
z3
ν
<latexit sha1_base64="bUpnZ+DHs1y67TuCAYb3jWu66Ow=">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</latexit>
M(u⌫, µ2
) !
<latexit sha1_base64="n3hk3Z2isDnt98tfGQCNOjveN/w=">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</latexit>
Q(⌫, µ2
) = M(⌫, z2
)
↵sCF
2⇡
Z 1
0
du M(u⌫, z2
)

ln
✓
z2
µ2 e2 E +1
4
◆
B[u] + L[u]
<latexit sha1_base64="30bquwHjvCKgXjdaoVM2+yOL7uY=">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</latexit>
Z 1
0
M(u⌫)[f(u)]+ =
Z 1
0
(M(u⌫) M(⌫)) f(u)
Common scale:
<latexit sha1_base64="2wZvWsh4IXg2kLWFd/Nh6LwksVQ=">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</latexit>
z0(µ = 2GeV) =
2e E 1/2
µ
= 0.067 fm ,
1
z0
= 2.94GeV
PDF results
0 1 2 3 4 5 6 7 8 9 10
∫
°0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Re[Q(∫,
µ
2
)]
M(∫, z2
3)
Pz = 0
Pz = 1
Pz = 2
Pz = 3
Pz = 4
Pz = 5
Pz = 6
<latexit sha1_base64="kck6YtsZYdhnTQsjPFVQMizaNeQ=">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</latexit>
M(⌫, z2
3)

14. Evolution and matching to : Light-cone ITDs
MS
evolution to common scale 1/z0
conversion to scheme
MS
light-cone ITD
<latexit sha1_base64="6I2IXBKK/mWdKBImIj3ojFH7TM8=">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</latexit>
B(u) =

1 + u2
1 u +
<latexit sha1_base64="+3l2zM+HYCOdXsduRi0NJzyHHwI=">AAACUXicdZBNa9swGMdlb107dy9pd9xFLAxSRo3sOU17GAR22WGHDpa2EJsgK3IiIstGelQIxl9kn6bX7brTPspuU9wMlrE9IPHX/3mR9MtrKQwQ8sPzHzzce7R/8Dg4fPL02fPe0fGVqaxmfMIqWembnBouheITECD5Ta05LXPJr/PV+03++pZrIyr1GdY1z0q6UKIQjIKzZr3k48Ce4Hc4lbyAaZIWmrImlWoQndqTtnF7i0/j7oRTLRZLyGZvZr0+CWPy9iKOMQnJKBlGxInhGRmdJzgKSRd9tI3L2ZG3l84rZkuugElqzDQiNWQN1SCY5G2QWsNrylZ0wadOKlpykzXd91r82jlzXFTaLQW4c//saGhpzLrMXWVJYWn+zm3Mf+WmForzrBGqtsAVu7+osBJDhTes8FxozkCunaBMC/dWzJbUEQJHNNgZ1Q1nOmu4ZVTufMhZjuqqDYLAgftNB/9fXMVhNAzJp6Q/Hm8RHqCX6BUaoAiN0Bh9QJdoghj6gu7QV/TN++799JHv35f63rbnBdoJ//AXOmCvzg==</latexit>
L(u) =

4
ln(1 u)
1 u
2(1 u)
+
<latexit sha1_base64="n3hk3Z2isDnt98tfGQCNOjveN/w=">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</latexit>
Q(⌫, µ2
) = M(⌫, z2
)
↵sCF
2⇡
Z 1
0
du M(u⌫, z2
)

ln
✓
z2
µ2 e2 E +1
4
◆
B[u] + L[u]
<latexit sha1_base64="30bquwHjvCKgXjdaoVM2+yOL7uY=">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</latexit>
Z 1
0
M(u⌫)[f(u)]+ =
Z 1
0
(M(u⌫) M(⌫)) f(u)
Common scale:
<latexit sha1_base64="2wZvWsh4IXg2kLWFd/Nh6LwksVQ=">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</latexit>
z0(µ = 2GeV) =
2e E 1/2
µ
= 0.067 fm ,
1
z0
= 2.94GeV
PDF results
0 1 2 3 4 5 6 7 8 9 10
∫
°0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Im[Q(∫,
µ
2
)]
M(∫, z2
3)
Pz = 0
Pz = 1
Pz = 2
Pz = 3
Pz = 4
Pz = 5
Pz = 6
<latexit sha1_base64="kck6YtsZYdhnTQsjPFVQMizaNeQ=">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</latexit>
M(⌫, z2
3)
‣ treat each independently
‣ need polynomial parametrization w.r.t.
‣ evolution/matching processes separately or combined
can give rise to different systematics
‣ 1-loop matching: ~10% effect to pseudo-ITD data
‣ small higher-twist effects
z3
ν
<latexit sha1_base64="bUpnZ+DHs1y67TuCAYb3jWu66Ow=">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</latexit>
M(u⌫, µ2
) !

15. Full picture: -dependence
x
<latexit sha1_base64="ec+jDU0k1TeZXl/5C2j/fnt2hyI=">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</latexit>
Q(⌫, µ2
) =
Z 1
1
dxei⌫x
q(x, µ2
)
discrete data in -space
ν
continuous function
in -space
x
<latexit sha1_base64="s43+tV+ttccgFKZTQDnTszK+27k=">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</latexit>
Re Q(⌫, µ2
) =
Z 1
0
dx cos(⌫x)qv(x, µ2
)
valence quark distribution:
Inverse problem! Way around:
‣ choose a physically meaningful functional form for
‣ introduce model bias in determination, but…
‣ general form:
i) 2-parameter fit:
ii) 3-parameter fit:
q(x, μ2
)
α, β ≠ 0, γ, δ = 0
α, β, δ ≠ 0, γ = 0
<latexit sha1_base64="NIw3nN6MJgQRlsFY51X2P+E1QmU=">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</latexit>
qv(x) =
x↵
(1 x) (1 +
p
x + x)
B(↵ + 1, + 1) + B(↵ + 3/2, + 1) + B(↵ + 2, + 1)
: beta-function
B(i, j)
<latexit sha1_base64="21L9iKzfhLn/lpdw4rNnqmdi5e0=">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</latexit>
2
=
⌫max
X
⌫=0
Q(⌫, µ2
) QFT(⌫, µ2
)
2
Q
, QFT(⌫, µ2
) =
Z 1
0
dx cos(⌫x)qv(x)
PDF results
0 1 2 3 4 5 6 7 8 9 10
∫
°0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Re[Q(∫,
µ
2
)]
2-param
3-param
Pz = 0
Pz = 1
Pz = 2
Pz = 3
Pz = 4
Pz = 5
Pz = 6
χ2
∼ 2.3

16. Full picture: -dependence
x
<latexit sha1_base64="ec+jDU0k1TeZXl/5C2j/fnt2hyI=">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</latexit>
Q(⌫, µ2
) =
Z 1
1
dxei⌫x
q(x, µ2
)
discrete data in -space
ν
continuous function
in -space
x
<latexit sha1_base64="s43+tV+ttccgFKZTQDnTszK+27k=">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</latexit>
Re Q(⌫, µ2
) =
Z 1
0
dx cos(⌫x)qv(x, µ2
)
valence quark distribution:
0.0 0.2 0.4 0.6 0.8 1.0
x
0
1
2
3
4
5
6
q
v
(x)
CJ15
MSTW
NNPDF
q
(3)
v (x)
0.0 0.2 0.4 0.6 0.8 1.0
x
0
1
2
3
4
5
6
q
v
(x)
q
(2)
v (x)
q
(3)
v (x)
CJ15: A. Accardi et. al. arXiv: 1602.03154
MSTW: A.D.Martin etl al. arXiv: 0901.0002
NNPDF: R.D.Ball et. al. arXiv: 1706.00428
Inverse problem! Way around:
‣ choose a physically meaningful functional form for
‣ introduce model bias in determination, but…
‣ general form:
i) 2-parameter fit:
ii) 3-parameter fit:
q(x, μ2
)
α, β ≠ 0, γ, δ = 0
α, β, δ ≠ 0, γ = 0
<latexit sha1_base64="NIw3nN6MJgQRlsFY51X2P+E1QmU=">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</latexit>
qv(x) =
x↵
(1 x) (1 +
p
x + x)
B(↵ + 1, + 1) + B(↵ + 3/2, + 1) + B(↵ + 2, + 1)
: beta-function
B(i, j)
<latexit sha1_base64="21L9iKzfhLn/lpdw4rNnqmdi5e0=">AAACt3icdVFda9swFJW9deu8r3R73MtlYSOBLLXNmjWMjsJg7LGBpi1EtlEUORGxZFeSS4LxH9k/27+ZnGTQjO6C0NG5515J506LjGvj+78d99HjgydPD595z1+8fPW6dfTmSuelomxM8yxXN1OiWcYlGxtuMnZTKEbENGPX0+X3Jn99x5Tmubw064JFgswlTzklxlJJ6xemCx6HcAZYlyKpsCzP/DpudntQAgRZ1TXgVBFajTqW7mFRxmEXPsFoq/hxWd/j6wprPhckGcVhjXs9/PVhXXMjlybx4wBmK8A019CkYdWF2+Sus+omrbbfD4aDYRiC3/c3YcFgeHpyMoBgx7TRLi6SI+cAz3JaCiYNzYjWk8AvTFQRZTjNWO3hUrOC0CWZs4mFkgimo2pjYg0fLDODNFd2SQMb9n5FRYTWazG1SkHMQv+ba8iHcpPSpKdRxWVRGibp9qK0zMDk0EwEZlwxarK1BYQqbt8KdEGs38bOzdtrtWlOVVSxkpJs70OWsjNa1p7nWeP+ugP/B1dhPxj0/dHn9vm3nYWH6B16jzooQF/QOfqJLtAYUQc5H51jx3eHbuKm7mIrdZ1dzVu0F+7tHyo+0X4=</latexit>
2
=
⌫max
X
⌫=0
Q(⌫, µ2
) QFT(⌫, µ2
)
2
Q
, QFT(⌫, µ2
) =
Z 1
0
dx cos(⌫x)qv(x)
PRELIMINARY
PDF results

17. Conclusions
Summary & Outlook
‣ Lattice QCD via pseudo-PDF framework: in good position to give reliable insight on nucleon PDFs
‣ Distillation: valuable alternative to standard smearing techniques
‣ controllable statistical uncertainties and excited state effects
‣ Some stretch between lattice and phenomenological -dependence exists:
‣ lattice artifacts finite volume and lattice spacing, non-physical simulations
‣ parametrization of PDFs be systematic: try different approaches, compare with other lattice evaluations
On the horizon:
‣ results from two other ensembles, one at almost physical pion mass study volume effects, remove systematics
‣ plans for analyzing ensembles with smaller lattice spacing continuum extrapolation
‣ helicity, transversity distributions & GPDs on the way!
‣ coming soon: disconnected diagrams
x
→
→
→
→

18. Conclusions
Summary & Outlook
‣ Lattice QCD via pseudo-PDF framework: in good position to give reliable insight on nucleon PDFs
‣ Distillation: valuable alternative to standard smearing techniques
‣ controllable statistical uncertainties and excited state effects
‣ Some stretch between lattice and phenomenological -dependence exists:
‣ lattice artifacts finite volume and lattice spacing, non-physical simulations
‣ parametrization of PDFs be systematic: try different approaches, compare with other lattice evaluations
On the horizon:
‣ results from two other ensembles, one at almost physical pion mass study volume effects, remove systematics
‣ plans for analyzing ensembles with smaller lattice spacing continuum extrapolation
‣ helicity, transversity distributions & GPDs on the way!
‣ coming soon: disconnected diagrams
x
→
→
→
→
Thank you

19. Bonus!

20. (Bonus!) Matrix elements
Ensuring ground state dominance
i) Plateau method
0.695
0.700
0.705
0.710
0.715 Pz = 1 , z3 = 1
tsep = 4a
tsep = 6a
tsep = 8a
tsep = 10a
tsep = 12a
tsep = 14a Plateau fits
°8 °6 °4 °2 0 2 4 6 8
[tins ° tsep/2]/a
0.210
0.212
0.214
0.216
0.218
Pz = 1 , z3 = 4
4 6 8 10 12 14
tsep/a
Re[R(P
z
,
z
3
)]
Ratio of 3pt/2pt functions:
<latexit sha1_base64="B0WpVQZDUMfB2zKuvzkTsak6BfQ=">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</latexit>
R(Pz, z3; tsep, tins) =
C3pt(Pz, z3; tsep, tins)
C2pt(Pz, tsep)
‣ for each , perform constant fits for multiple ranges
‣ decrease each fit range by one point on each side
‣ best fit: longest for which
tsep
χ2
≤ 0.9

21. (Bonus!) Matrix elements
Ensuring ground state dominance
0.75
0.80
0.85
0.90
0.95
Pz = 3 , z3 = 0
ts = 4a
ts = 6a
ts = 8a
ts = 10a
ts = 12a
ts = 14a
Plateau fits Summ. linear
0.425
0.450
0.475
0.500
0.525
0.550
0.575
Pz = 3 , z3 = 2
0.08
0.09
0.10
0.11
0.12
0.13 Pz = 3 , z3 = 5
°8 °6 °4 °2 0 2 4 6 8
[tins ° ts/2]/a
0.005
0.010
0.015
0.020
0.025
Pz = 3 , z3 = 8
4 6 8 10 12 14
ts/a
4 6 8
tlow
s /a
Re[R(P
z
,
z
3
)]
0.022
0.023
0.024
0.025
0.026
Pz = 1 , z3 = 1
ts = 4a
ts = 6a
ts = 8a
ts = 10a
ts = 12a
ts = 14a
Plateau fits Summ. linear
0.033
0.034
0.035
0.036
0.037
0.038
0.039
Pz = 1 , z3 = 2
0.024
0.025
0.026
0.027
0.028
0.029 Pz = 1 , z3 = 5
°8 °6 °4 °2 0 2 4 6 8
[tins ° ts/2]/a
0.0085
0.0090
0.0095
0.0100
0.0105
0.0110
Pz = 1 , z3 = 8
4 6 8 10 12 14
ts/a
4 6 8
tlow
s /a
Im[R(P
z
,
z
3
)]