We present results on the nucleon valence quark distribution extracted from Lattice QCD simulations, using a gauge ensemble of $N_f=2+1$ Wilson-Clover fermions with a pion mass of $m_\pi = 350$ MeV and lattice spacing of about $a=0.093$ fm. We obtain reduced Ioffe Time Distributions (rITDs) by computing appropriate matrix elements on the lattice, and elaborate on the extraction of the desired quark distributions from the rITDs following the pseudo-PDF approach. A set of techniques are considered in order to ensure ground state dominance. Theoretical and experimental implications of our calculation are discussed.

1. Nucleon valence quark distribution
functions from Lattice QCD
An exploratory study of Distillation
in the pseudo-PDF framework
Christos Kallidonis (Jefferson Lab)
Apr. 14, 2021

2. Roadmap
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
lattice
evaluation
summary
& outlook
pseudo-PDF:
matrix elements
intro &
motivation
HadStruct Collaboration
full picture:
PDF -dependence
x

3. Introduction Why the PDFs? Why on the Lattice?
• 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
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)
0.0
0.5
1.0
1.5
2.0
10
-6
10
-4
10
-3
CT18Z at 2 GeV
s
2.0 2.0
experimental status from
Hou et.al. arXiv: 1912.10053
Parton Distribution Functions:
‣ -object: number density of “partons” with longitudinal momentum fraction of nucleon’s momentum
‣ accessible in deep inelastic and semi-inclusive deep inelastic scattering intricate procedure
‣ Still, good precision exists for unpolarized PDFs
1d x
→
Important LHC applications
‣ W-boson mass
‣ Strong coupling constant
‣ Higgs boson couplings and cross-sections (BSM physics)
Lattice Evaluation
‣ PDFs: “hot” topic
‣ first principles calculation
‣ unpolarized PDFs benchmark quantity
‣ transversity PDFs: not well constrained
‣ GPDs and TMDs: more difficult to access
→
Gao et.al., arXiv: 1709.04922

4. Unpolarized PDF: non-local matrix element on the light-cone in Minkowski space
<latexit sha1_base64="FJ8sLWusYBA9S14q/L9CMSzijNI=">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</latexit>
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
origin in Euclidean space-time
Methods for accessing -dependence on the lattice:
‣ hadronic tensor
‣ auxiliary quarks
‣ quasi-distributions
‣ “good lattice cross sections”
‣ “OPE without OPE”
‣ 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
Reviews: K. Cichy, M. Constantinou arXiv: 1811.07248,
M.Constantinou arXiv: 2010.02445
Steps towards -dependence within pseudo-PDF:
‣ compute appropriate matrix elements
‣ resolve divergences, renormalization
‣ perturbative evolution light-cone distribution
‣ Fourier Transform: reconstruct -dependence
x
→
x
pseudo-PDFs Matrix elements

5. Euclidean space
Lorentz decomposition
<latexit sha1_base64="E0sa7DU94HGbw6Z524UrPbuqsVc=">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</latexit>
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
<latexit sha1_base64="6W/Z/tWVyshbFjpuWYSsvssMUeM=">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</latexit>
M↵
(P, z) ⌘ hN(P)| ¯(z) ↵
W(z, 0) (0)|N(P)i
Wilson line boosted nucleon states
<latexit sha1_base64="zhH9sclxsIgEnWt4RQGNhATEj+I=">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</latexit>
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
F.T.: 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”
<latexit sha1_base64="D8xd+izXgjmnEW04K1HgA1thkBc=">AAACAHicbVDLSgMxFM34rPVVdeHCTbAIroaZVlvdSMGNywr2AZ0yZNJMG5pJxiRTaEs3/oobF4q49TPc+Tem7YBaPXDhcM693HtPEDOqtON8WkvLK6tr65mN7ObW9s5ubm+/rkQiMalhwYRsBkgRRjmpaaoZacaSoChgpBH0r6d+Y0CkooLf6WFM2hHqchpSjLSR/NyhxxPokfuEDmDVH3m4IzQc+UU/l3fs83LhslSEju3M8E3clORBiqqf+/A6AicR4RozpFTLdWLdHiOpKWZkkvUSRWKE+6hLWoZyFBHVHs8emMATo3RgKKQpruFM/TkxRpFSwygwnRHSPbXoTcX/vFaiw4v2mPI40YTj+aIwYVALOE0DdqgkWLOhIQhLam6FuIckwtpkljUhuIsv/yX1gu2WbOf2LF+5SuPIgCNwDE6BC8qgAm5AFdQABhPwCJ7Bi/VgPVmv1tu8dclKZw7AL1jvXxahlhI=</latexit>
⌫ ⌘ Pz · z3
B. Ioffe, Phys.Lett. 30B, 123 (1969)
Ioffe-time distributions
<latexit sha1_base64="7/9HqRNqJrcweS2cOYDmvTJVjbE=">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</latexit>
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
<latexit sha1_base64="BM6nsJQsx4bt4aKPHsCNCVauZq4=">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</latexit>
M(⌫, z2
3)
T. Ishikawa et.al. arXiv: 1609.02018
J.W. Chen et. al. arXiv: 1707.03107
pseudo-PDFs Matrix elements

6. Lattice evaluation Correlation functions: “Distillation” 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="9y2zSXOQS1TTlL6Rj9X+owRfsRM=">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</latexit>
qsm(~
x, t) = F[U(~
x, t); ~
y]q(~
y, t)
<latexit sha1_base64="xsxw936ozM1SDXZUZkcyi5I/paY=">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</latexit>
XB(~
x, t) = ✏abc
S 1 2 3
(D1q1)a
1
(~
x, t)(D2q2)b
2
(~
x, t)(D3q3)c
3
(~
x, t)
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="UFw9ad35MkYR2pmEArXZsenf5Dk=">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</latexit>
i,j,k
1 2 3
(t) = ✏abc
S 1 2 3
(D1vi(t))a
(D2vj(t))b
(D3vk(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 sink and source
‣ reusability of building blocks
Cons:
‣ extra step: compute eigenvectors of
‣ big files/storage
∇2
inversions:
<latexit sha1_base64="p5Omw89Ua4LwxSaq1yWeHjMqQ9g=">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</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="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
<latexit sha1_base64="RbJdi9FM93Q0Ih66kTr/LKalENI=">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</latexit>
F↵s,Ns
=
✓
1 +
↵sr2
Ns
◆Ns

7. 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
‣ 4 time sources
‣ boost momentum
‣ displacements
‣ 5600 statistics
‣ distillation vectors:
‣ 6
t0/a = [ti
0, + T/4, + T/2, + 3T/4]
Pz = 2π/L × [0, ± 1,…, ± 6] → 0…2.55 GeV
z3 = 0, ± 1a, …, ± 12a → 0…1.1 fm
Nev = 64
tsep ≡ (tf − t0) = [4a, 6a, …, 14a] → 0.4…1.3 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

8. Construct ratio:
<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)
Lattice evaluation Matrix elements: reaching the ground state
Two methods for ensuring ground state dominance
i) Plateau method
ii) Summation method
Perform fits from which the matrix element
is extracted (ask for more!)
Choose summation method results as final
0.82
0.83
0.84
0.85
0.86
0.87
Pz = 2 , z3 = 0
ts = 4a
ts = 6a
ts = 8a
ts = 10a
ts = 12a
ts = 14a
Plateau fits
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 ° ts/2]/a
0.016
0.018
0.020
0.022
0.024
Pz = 2 , z3 = 8
4 6 8 10 12 14
ts/a
4 6 8
tlow
s /a
Re[R(P
z
,
z
3
)]
<latexit sha1_base64="5+vVayv95XxZkryclb3iF5q9428=">AAACNnicdVDLSsNAFJ2o1Rpfra7EzWARKkhJqo3tRgpu3AgV7APaGCbTSTt08mBmIrSh+DVudeuvuHEnbv0Epw/Bih64cDj3fdyIUSEN41VbWl5Jra6l1/WNza3tnUx2tyHCmGNSxyELectFgjAakLqkkpFWxAnyXUaa7uBykm/eEy5oGNzKYURsH/UC6lGMpJKczH7HR7KPEUuux/maMxo5pycq7orHTiZnFMyKVSkWoVEwplDEqpRLJQuacyUH5qg5WS3V6YY49kkgMUNCtE0jknaCuKSYkbHeiQWJEB6gHmkrGiCfCDuZ/jCGR0rpQi/kKgIJp+rPjgT5Qgx9V1VOLha/cxPxr1w7ll7ZTmgQxZIEeLbIixmUIZwYAruUEyzZUBGEOVW3QtxHHGGpbNMXRk2HY24nJFaOLTykJI+jwVjXdWXctzvwf9IoFkyrYN6c5aoXcwvT4AAcgjwwwTmogitQA3WAwQN4BE/gWXvR3rR37WNWuqTNe/bAArTPL1k8qis=</latexit>
M(Pzz3, z2
3)
final value for
Valence quark distribution Real part
→

9. Cancel divergences - Renormalization
Divide with appropriate factor :
‣
‣ similar -dependence
‣ no -dependence ( )
‣ good choice:
D(z3)
D(0) = 1
z3
ν Pz = 0
<latexit sha1_base64="Dud1mLGib6nly/udWORn8I6yg9Y=">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</latexit>
D(z3) = M(0, z2
3)
<latexit sha1_base64="5+vVayv95XxZkryclb3iF5q9428=">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</latexit>
M(Pzz3, z2
3)
K. Orginos et.al. arXiv: 1706.05373
J. Karpie et. al. arXiv: 1710.08288
J. Karpie et. al. arXiv: 1807.10933
reduced Ioffe-time Distribution
<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=">AAACe3icdZFdS8MwFIaz+l2/pl56ExyCiox2urldCANvvBEmOBXWWtIs1bA0LUkqbLX/zj/hX/BWbwXTWsGJHgicvO85J8kTP2ZUKst6qRgzs3PzC4tL5vLK6tp6dWPzWkaJwKSPIxaJWx9JwignfUUVI7exICj0GbnxR2e5f/NIhKQRv1LjmLghuuc0oBgpLXnVOydE6iEQaJReZHsOTw4n3tFdYx+eQkerOC18jFhu97yJdsuK7H/ryUv1/tTKMq9as+p2p9VpNKBVt4rQSavTbjZb0C6VGiij521U5pxhhJOQcIUZknJgW7FyUyQUxYxkppNIEiM8QvdkoFOOQiLdtACRwV2tDGEQCb24goX6syNFoZTj0NeV+d3lby8X//IGiQrabkp5nCjC8ddBQcKgimBOFQ6pIFixsU4QFlTfFeIHpPkpzd6cGlUMx8JNSaLZTT1IS/lPZKZpanDfdOD/yXWjbrfq1uVxrdstES6CbbAD9oANTkAXnIMe6AMMnsEreAPvlQ+jZhwYh1+lRqXs2QJTYTQ/AYmmwcA=</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
◆
‣ cancel mult. factor from gauge link renormalization
‣ higher-twist contaminations partially suppressed
𝒪(z2
3Λ2
QCD)
Lattice evaluation Ioffe-time Distributions
<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 11 12 13 14 15
∫
°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

10. 32c64-280 48c96-170
Lattice volume
Pion mass
Physical volume
max Pz
323
× 64 483
× 96
mπ = 280 MeV mπ = 170 MeV
L = 2.9 fm L = 4.4 fm
2π/L × 3 2π/L × 3
<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
◆
Lattice evaluation Ioffe-time Distributions
Two more ensembles on the way
0 1 2 3 4 5
∫
°0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Re[M
Lsumm
(∫,
z
2
3
)]
32c64-350 32c64-280 48c96-170-all
PRELIMINARY
<latexit
sha1_base64="x4YNioFrma8b5rW3tPEcqbq+dPo=">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</latexit>
Re
⇥
M(⌫,
z
2
3
)
⇤
<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 11 12 13 14 15
∫
°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

11. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
∫
°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
evolve to common scale 1/z0
convert 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=">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</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)
‣ treat each independently
‣ need polynomial parametrization w.r.t.
‣ evolution/matching processes separately or combined can
give rise to different systematics
‣ residual discrepancies: evolution fails? higher twist effects?
z3
ν
<latexit sha1_base64="bUpnZ+DHs1y67TuCAYb3jWu66Ow=">AAACQHicdVBNa9tAEF25dZKqbeI0x16WmkIKRawUO05vhl5yKaRQJwZLNaP1yl68Won9SDDCP6C/ptf2mn+Rf5Bb6LWnrhwX6tA8WHi8NzM789JScG0IufEaT542t7Z3nvnPX7zc3WvtvzrXhVWUDWghCjVMQTPBJRsYbgQblopBngp2kc4/1v7FJVOaF/KLWZQsyWEqecYpGCeNW+04BzPLFMyrT8tDG0v7Ps7t1+hdrPh0ZkCp4spVkSAiRx+iCJOA9DrdkDjSPSa9kw4OA7JCG61xNt73mvGkoDZn0lABWo9CUpqkAmU4FWzpx1azEugcpmzkqISc6aRaXbPEb50ywVmh3JMGr9R/OyrItV7kqausd9cPvVr8nzeyJjtJKi5La5ik9x9lVmBT4DoaPOGKUSMWjgBV3O2K6QwUUOMC9DdGrYZTlVTMUhAbBzmpjnPp+74L7m86+HFyHgVhNyCfO+1+fx3hDnqN3qBDFKIe6qNTdIYGiKJv6Dv6gX56196td+f9ui9teOueA7QB7/cfeYSvVg==</latexit>
M(u⌫, µ2
) !
A. Radyushkin arXiv: 1801.02427
J.-H.Zhang et. al. arXiv: 1801.03023
T. Izubuchi et. al. arXiv: 1801.3917
Lattice evaluation Perturbative evolution: Light-cone ITDs
−
Common scale:
<latexit sha1_base64="gb5OHkZhbzISg/3r3qnC4X2338Y=">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</latexit>
z0(µ = 2GeV) = 0.067 fm ,
1
z0
= 2.94GeV
32c64 350
<latexit sha1_base64="kck6YtsZYdhnTQsjPFVQMizaNeQ=">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</latexit>
M(⌫, z2
3)

12. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
∫
°0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Re[Q(∫,
µ
2
)]
z3 ∑ 8
z3 ∑ 12
Pz = 0
Pz = 1
Pz = 2
Pz = 3
Pz = 4
Pz = 5
Pz = 6
<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:
A. Radyushkin arXiv: 1705.01488
Full picture PDF -dependence
x
32c64 350
−
Inverse problem! Way around:
‣ choose a physically meaningful functional form for
‣ introduce model bias in determination
‣ general form:
i) fit for
ii) fit for
q(x, μ2
)
z3 ≤ 8
z3 ≤ 12 : 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)
<latexit sha1_base64="4anW6mXL4wGAl+sgcH3m8NhQ4YY=">AAACfHicdZFdb9MwFIbdwGCEr25ccmNRkFqVVUnEsvViaIIbLodEt0lNFp24J6tV5wPbmVpF+Xn7EfsNu4VbhJt2E0VwJEuv3uf42H4dF4Ir7Tg3LevBw61Hj7ef2E+fPX/xsr2ze6ryUjIcsVzk8jwGhYJnONJcCzwvJEIaCzyLZ5+X/OwKpeJ59k0vCgxTuMx4whloY0Xt6Ht01Z336BENEgmsml8EIIopdN29ee8iiFEb2Q8mKDTQea+uPnVXDX33fUP7bu8O3yPvHtVRu+MM3KE/9DzqDJymjPCHh/v7PnXXToes6yTaaW0Fk5yVKWaaCVBq7DqFDiuQmjOBtR2UCgtgM7jEsZEZpKjCqkmipu+MM6FJLs3KNG3cP3dUkCq1SGPTmYKeqr/Z0vwXG5c6OQwrnhWlxoytDkpKQXVOl7HSCZfItFgYAUxyc1fKpmAS1SZ8e2NUM5zJsMKSgdh4kLHML8xq27ZNcHfp0P+LU2/g+gPv64fO8cd1hNvkNXlDusQlB+SYfCEnZEQYuSa35Af52fplvbX61t6q1Wqt97wiG2X5vwESPb9L</latexit>
qv(x) =
x↵
(1 x) (1 + x)
B(↵ + 1, + 1) + B(↵ + 2, + 1)

13. 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
qv(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:
PRELIMINARY
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
A. Radyushkin arXiv: 1705.01488
Full picture PDF -dependence
x
Inverse problem! Way around:
‣ choose a physically meaningful functional form for
‣ introduce model bias in determination
‣ general form:
i) fit for
ii) fit for
q(x, μ2
)
z3 ≤ 8
z3 ≤ 12 : 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)
<latexit sha1_base64="4anW6mXL4wGAl+sgcH3m8NhQ4YY=">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</latexit>
qv(x) =
x↵
(1 x) (1 + x)
B(↵ + 1, + 1) + B(↵ + 2, + 1)
0.0 0.2 0.4 0.6 0.8 1.0
x
0
1
2
3
4
5
6
q
v
(x)
z3 ∑ 8
z3 ∑ 12

14. 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, 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 model-independent approaches, compare with other lattice evaluations
On the horizon:
‣ finalize results from other ensembles study volume, pion mass effects
‣ analyze ensembles with smaller lattice spacing continuum extrapolation
‣ helicity, transversity distributions & GPDs on the way!
‣ coming soon: disconnected diagrams
x
→
→
→
→

15. Bonus!

16. (Bonus!) Lattice evaluation Matrix elements: reaching the ground state
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
)]
‣ 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
i) Plateau method
Construct ratio:
<latexit sha1_base64="2ky8d8js9KtiZKQAZr1JfrTdLRE=">AAACl3icdZFba9swFMdld5cuu2XrXsZexMIggdbIdtt0G2OBjNGXQTqWthCnRlZkR1S+IB1vS42/0j7M3vZtJqcO7HpA6PA/v8O5RYUUGgj5YdlbN27eur19p3P33v0HD7uPHp/qvFSMT1kuc3UeUc2lyPgUBEh+XihO00jys+hy3MTPPnOlRZ59glXB5ylNMhELRsFIYffbx/4kvNq9Cv3XEFaaF/Wu+UWm6wF+g4NYUVaNwypQKfYLqDcw/ouuN5i3wVpiUAdflUiWQJXKv8xafg9CEiQJpvOq5fbayFqtcZBSWDIqqw91P8jKpuiFNwi7PeJ4vusPfUwc4g33D1zjHLw8JMMj7DpkbT3U2iTsfg8WOStTngGTVOuZSwqYV1SBYJLXnaA0tSm7pAmfGTejKdfzar3XGr8wygLHuTIvA7xWf82oaKr1Ko0M2XSr/4w14r9isxLio7mZtiiBZ+y6UFxKDDlujoQXQnEGcmUcypQwvWK2pOYWYE7ZMUvYTIr/75x6jnvokJP93mjUrmMbPUPPUR+5aIhG6BhN0BQx64n1yhpb7+yn9lv7vX18jdpWm7ODfjP75CdN6sul</latexit>
R(Pz, z3; tsep, tins) =
C3pt(Pz, z3; tsep, tins)
C2pt(Pz, tsep)
tsep tins a
!
tins t0 a
M(⌫, z2
3)

17. ii) Summation method
<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
Im[R
sum
(P
z
=
1;
z
3
=
4)]
0.030
0.031
0.032
0.033
0.034
0.035
Im[R
r°sum
(P
z
=
1;
z
3
=
4)]
2 4 6 8 10 12 14 16
tsep/a
0.5
1.0
1.5
2.0
2.5
3.0
Re[R
sum
(P
z
=
1;
z
3
=
4)]
Linear fit tlow
sep = 4
Linear fit tlow
sep = 6
Linear fit tlow
sep = 8
2 4 6
0.208
0.210
0.212
0.214
0.216
0.218
Re[R
r°sum
(P
z
=
1;
z
3
=
4)]
Im[R
sum
(P
z
=
1;
z
3
=
4)]
0.030
0.031
0.032
0.033
0.034
0.035
Im[R
r°sum
(P
z
=
1;
z
3
=
4)]
2 4 6 8 10 12 14 16
tsep/a
0.5
1.0
1.5
2.0
2.5
3.0
Re[R
sum
(P
z
=
1;
z
3
=
4)]
Linear fit tlow
sep = 4
Linear fit tlow
sep = 6
Linear fit tlow
sep = 8
2 4 6 8 10 12 14
tsep/a
0.208
0.210
0.212
0.214
0.216
0.218
Re[R
r°sum
(P
z
=
1;
z
3
=
4)]
0.128
0.129
0.130
0.131
Re[R(P
z
°8 °6 °4 °2 0 2 4 6 8
[tins ° tsep/2]/a
0.210
0.212
0.214
0.216
0.218
Re[R(P
z
,
z
3
)]
Pz = 1 , z3 = 4
4 6 8
tsep/
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
Re[R(P
z
,
z
3
)]
(Bonus!) Lattice evaluation Matrix elements: reaching the ground state
Construct ratio:
<latexit sha1_base64="2ky8d8js9KtiZKQAZr1JfrTdLRE=">AAACl3icdZFba9swFMdld5cuu2XrXsZexMIggdbIdtt0G2OBjNGXQTqWthCnRlZkR1S+IB1vS42/0j7M3vZtJqcO7HpA6PA/v8O5RYUUGgj5YdlbN27eur19p3P33v0HD7uPHp/qvFSMT1kuc3UeUc2lyPgUBEh+XihO00jys+hy3MTPPnOlRZ59glXB5ylNMhELRsFIYffbx/4kvNq9Cv3XEFaaF/Wu+UWm6wF+g4NYUVaNwypQKfYLqDcw/ouuN5i3wVpiUAdflUiWQJXKv8xafg9CEiQJpvOq5fbayFqtcZBSWDIqqw91P8jKpuiFNwi7PeJ4vusPfUwc4g33D1zjHLw8JMMj7DpkbT3U2iTsfg8WOStTngGTVOuZSwqYV1SBYJLXnaA0tSm7pAmfGTejKdfzar3XGr8wygLHuTIvA7xWf82oaKr1Ko0M2XSr/4w14r9isxLio7mZtiiBZ+y6UFxKDDlujoQXQnEGcmUcypQwvWK2pOYWYE7ZMUvYTIr/75x6jnvokJP93mjUrmMbPUPPUR+5aIhG6BhN0BQx64n1yhpb7+yn9lv7vX18jdpWm7ODfjP75CdN6sul</latexit>
R(Pz, z3; tsep, tins) =
C3pt(Pz, z3; tsep, tins)
C2pt(Pz, tsep)
tsep tins a
!
tins t0 a
M(⌫, z2
3)

18. (Bonus!) Lattice evaluation Matrix elements: select data
0.836
0.838
0.840
0.842
0.844
0.846
0.848
0.850
Pz = 0 , z3 = 0
ts = 4a
ts = 6a
ts = 8a
ts = 10a
ts = 12a
ts = 14a Plateau fits
Summ. linear
0.502
0.504
0.506
0.508
0.510
Pz = 0 , z3 = 2
0.132
0.133
0.134
0.135
0.136
Pz = 0 , z3 = 5
°8 °6 °4 °2 0 2 4 6 8
[tins ° ts/2]/a
0.0275
0.0280
0.0285
0.0290
0.0295
Pz = 0 , z3 = 8
4 6 8 10 12 14
ts/a
4 6 8
tlow
s /a
Re[R(P
z
,
z
3
)]
0.835
0.840
0.845
0.850
0.855
Pz = 1 , z3 = 0
ts = 4a
ts = 6a
ts = 8a
ts = 10a
ts = 12a
ts = 14a Plateau fits
Summ. linear
0.4975
0.5000
0.5025
0.5050
0.5075
0.5100
0.5125
Pz = 1 , z3 = 2
0.128
0.129
0.130
0.131
0.132
0.133
0.134
Pz = 1 , z3 = 5
°8 °6 °4 °2 0 2 4 6 8
[tins ° ts/2]/a
0.0245
0.0250
0.0255
0.0260
0.0265
0.0270
0.0275
Pz = 1 , z3 = 8
4 6 8 10 12 14
ts/a
4 6 8
tlow
s /a
Re[R(P
z
,
z
3
)]

19. <latexit
sha1_base64="Ytv6KAbYVi9nG4xqfS7e/mR/+z0=">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</latexit>
Im
⇥
M(⌫,
z
2
3
)
⇤0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
∫
0.0
0.2
0.4
0.6
0.8
1.0
Im[M
Lsumm
(∫,
z
2
3
)]
48c96-170-all 32c64-350 32c64-280
PRELIMINARY
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=">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</latexit>
Im
⇥
M(⌫,
z
2
3
)
⇤
32c64-280 48c96-170
Lattice volume
Pion mass
Physical volume
max Pz
323
× 64 483
× 96
mπ = 280 MeV mπ = 170 MeV
L = 2.9 fm L = 4.4 fm
2π/L × 3 2π/L × 3
<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
◆
(Bonus!) Lattice evaluation Ioffe-time Distributions (Imag)
Two more ensembles on the way

20. light-cone ITD
<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]
A. Radyushkin arXiv: 1801.02427
J.-H.Zhang et. al. arXiv: 1801.03023
T. Izubuchi et. al. arXiv: 1801.3917
Lattice evaluation Perturbative evolution: Light-cone ITDs (Imag)
evolve to common scale 1/z0
convert to scheme
MS
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)
32c64 350
−
<latexit sha1_base64="6I2IXBKK/mWdKBImIj3ojFH7TM8=">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</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)
+
<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="gb5OHkZhbzISg/3r3qnC4X2338Y=">AAACZnicdZDRStxAFIZno1abtrpWpBfeDF0KCktIgkYXUYReaO8U3FXYhGUye7IOO5OEmUlhDXkmn8YLb+ytj+FsXKUr7YGBn++c/8zMH+ecKe26Dw1rYXHpw/LKR/vT5y+ra831rz2VFZJCl2Y8k9cxUcBZCl3NNIfrXAIRMYerePxz2r/6DVKxLL3UkxwiQUYpSxgl2qBB89ftwN0ORXHkl6EU+BR61Q4+wq7jBvthu2aJqMJ2OzzEYSIJLb2qNJ7KDPlOZxe/2QbNlut4naDj+1N7XUYEnYO9vQB7M9JCszofrDeWwmFGCwGpppwo1ffcXEclkZpRDpUdFgpyQsdkBH0jUyJARWX95wr/MGSIk0yak2pc078dJRFKTURsJgXRN+p9bwr/1esXOjmISpbmhYaUvlyUFBzrDE8DxEMmgWo+MYJQycxbMb0hJhltYrbnVtXLqYxKKCjhcx8yyKQ5rmzbNsG9poP/L3q+4wWOd7HbOjmeRbiCttB3tI08tI9O0Bk6R11E0R26R4/oT+PJWrU2rW8vo1Zj5tlAc2XhZ9VrthQ=</latexit>
z0(µ = 2GeV) = 0.067 fm ,
1
z0
= 2.94GeV
‣ treat each independently
‣ need polynomial parametrization w.r.t.
‣ evolution/matching processes separately or combined can
give rise to different systematics
‣ residual discrepancies: evolution fails? higher twist effects?
z3
ν
<latexit sha1_base64="bUpnZ+DHs1y67TuCAYb3jWu66Ow=">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</latexit>
M(u⌫, µ2
) !