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Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 1 / 46
Partonic structure of the nucleon from Lattice QCD
Krzysztof CichyAdam Mickiewicz University, Poznań, Poland
This project is supported by the National Science Center of
PolandSONATA BIS grant No 2016/22/E/ST2/00013 (2017-2022).
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Outline of the talk
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 2 / 46
1. Introduction
2. Quasi-PDFs and pseudo-PDFs
3. Results – pseudo-PDFs
4. Lattice impact on pheno?
5. New directions – twist-3, GPDs
6. Conclusions and prospects
Collaborators:• C. Alexandrou (Cyprus)• M. Bhat (Poznań)• S.
Bhattacharya (Temple)• M. Constantinou (Temple)• L. Del Debbio
(Edinburgh)• T. Giani (Edinburgh)• K. Hadjiyiannakou (Cyprus)• K.
Jansen (DESY)• A. Metz (Temple)• A. Scapellato (Poznań)• F.
Steffens (Bonn)
Based on:• M. Bhat, K. Cichy, M. Constantinou, A. Scapellato,
“Parton dis-
tribution functions from lattice QCD at physical quark masses
viathe pseudo-distribution approach”, arXiv:2005.02102
• S. Bhattacharya, K. Cichy, M. Constantinou, A. Metz, A.
Scapel-lato, F. Steffens, “New insights on proton structure from
lat-tice QCD: the twist-3 parton distribution function gT
(x)”,arXiv:2004.04130, “One-loop matching for the twist-3 parton
dis-tribution gT (x)”, arXiv:2005.10939 (accepted in PRD), “The
roleof zero-mode contributions in the matching for the twist-3
PDFse(x) and hL(x)”, arXiv:2006.12347
• C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou,K.
Jansen, A. Scapellato, F. Steffens, in preparation (GPDs)
• C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou,K.
Jansen, A. Scapellato, F. Steffens, “Systematic uncertaintiesin
parton distribution functions from lattice QCD simulations atthe
physical point”, Phys. Rev. D99 (2019) 114504
• K. Cichy, L. Del Debbio, T. Giani, “Parton distributions from
latticedata: the nonsinglet case”, JHEP 10 (2019) 137
• C. Alexandrou, K. Cichy, M. Constantinou, K. Jansen, A.
Scapel-lato, F. Steffens, “Light-Cone Parton Distribution Functions
fromLattice QCD”, Phys. Rev. Lett. 121 (2018)
112001,“Transversityparton distribution functions from lattice
QCD”, Phys. Rev. D98(2018) 091503 (Rapid Communications)
Review of the field:• K. Cichy, M. Constantinou, “A guide to
light-cone PDFs from Lat-
tice QCD: an overview of approaches, techniques and
results”,invited review article for a special issue of Advances in
High EnergyPhysics, Adv. High Energy Phys. 2019 (2019)
3036904,arXiv: 1811.07248 [hep-lat]
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Nucleon structure
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 3 / 46
The nucleon is a very complicated system. . .
-
Nucleon structure
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 3 / 46
The nucleon is a very complicated system. . .. . . and its
structure is more complexthe closer we look!
-
Nucleon structure
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 3 / 46
The nucleon is a very complicated system. . .. . . and its
structure is more complexthe closer we look!
Different aspects:
-
Nucleon structure
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 3 / 46
The nucleon is a very complicated system. . .. . . and its
structure is more complexthe closer we look!
Different aspects:• how the quarks and gluons move inside the
proton
-
Nucleon structure
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 3 / 46
The nucleon is a very complicated system. . .. . . and its
structure is more complexthe closer we look!
Different aspects:• how the quarks and gluons move inside the
proton• 3D imaging of the proton – “hadron tomography”
-
Nucleon structure
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 3 / 46
The nucleon is a very complicated system. . .. . . and its
structure is more complexthe closer we look!
Different aspects:• how the quarks and gluons move inside the
proton• 3D imaging of the proton – “hadron tomography”• role of
gluons and their emergent properties
-
Nucleon structure
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 3 / 46
The nucleon is a very complicated system. . .. . . and its
structure is more complexthe closer we look!
Different aspects:• how the quarks and gluons move inside the
proton• 3D imaging of the proton – “hadron tomography”• role of
gluons and their emergent properties• how is spin decomposed
-
Nucleon structure
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 3 / 46
The nucleon is a very complicated system. . .. . . and its
structure is more complexthe closer we look!
Different aspects:• how the quarks and gluons move inside the
proton• 3D imaging of the proton – “hadron tomography”• role of
gluons and their emergent properties• how is spin decomposed•
origin of proton mass
-
Nucleon structure
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 3 / 46
The nucleon is a very complicated system. . .. . . and its
structure is more complexthe closer we look!
Different aspects:• how the quarks and gluons move inside the
proton• 3D imaging of the proton – “hadron tomography”• role of
gluons and their emergent properties• how is spin decomposed•
origin of proton mass• . . .
-
Nucleon structure
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 3 / 46
The nucleon is a very complicated system. . .. . . and its
structure is more complexthe closer we look!
Different aspects:• how the quarks and gluons move inside the
proton• 3D imaging of the proton – “hadron tomography”• role of
gluons and their emergent properties• how is spin decomposed•
origin of proton mass• . . .
Different functions characterizing the behavior of partons:
-
Nucleon structure
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 3 / 46
The nucleon is a very complicated system. . .. . . and its
structure is more complexthe closer we look!
Different aspects:• how the quarks and gluons move inside the
proton• 3D imaging of the proton – “hadron tomography”• role of
gluons and their emergent properties• how is spin decomposed•
origin of proton mass• . . .
Different functions characterizing the behavior of partons:• 1D:
form factors• 1D: parton distribution functions (PDFs)
-
Nucleon structure
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 3 / 46
The nucleon is a very complicated system. . .. . . and its
structure is more complexthe closer we look!
Different aspects:• how the quarks and gluons move inside the
proton• 3D imaging of the proton – “hadron tomography”• role of
gluons and their emergent properties• how is spin decomposed•
origin of proton mass• . . .
Different functions characterizing the behavior of partons:• 1D:
form factors• 1D: parton distribution functions (PDFs)• 3D:
generalized parton distributions (GPDs)• 3D: transverse momentum
dependent PDFs (TMDs)
-
Nucleon structure
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 3 / 46
The nucleon is a very complicated system. . .. . . and its
structure is more complexthe closer we look!
Different aspects:• how the quarks and gluons move inside the
proton• 3D imaging of the proton – “hadron tomography”• role of
gluons and their emergent properties• how is spin decomposed•
origin of proton mass• . . .
Different functions characterizing the behavior of partons:• 1D:
form factors• 1D: parton distribution functions (PDFs)• 3D:
generalized parton distributions (GPDs)• 3D: transverse momentum
dependent PDFs (TMDs)• 5D: Wigner function
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PDFs and the lattice
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 4 / 46
Do we need to know partonic functions from the lattice?Maybe it
is not needed if we have huge expertise in fitting PDFs from
abundantexperimental data?
-
PDFs and the lattice
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 4 / 46
Do we need to know partonic functions from the lattice?Maybe it
is not needed if we have huge expertise in fitting PDFs from
abundantexperimental data?However:
• knowing something from first principles is always
desirable,
-
PDFs and the lattice
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 4 / 46
Do we need to know partonic functions from the lattice?Maybe it
is not needed if we have huge expertise in fitting PDFs from
abundantexperimental data?However:
• knowing something from first principles is always desirable,•
good knowledge only of unpolarized and helicity PDFs,• transversity
PDFs – not much constrained by experiment,
-
PDFs and the lattice
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 4 / 46
Do we need to know partonic functions from the lattice?Maybe it
is not needed if we have huge expertise in fitting PDFs from
abundantexperimental data?However:
• knowing something from first principles is always desirable,•
good knowledge only of unpolarized and helicity PDFs,• transversity
PDFs – not much constrained by experiment,• other kinds of
functions very difficult to extract solely from experiment:
GPDs, TMDs, twist-3, . . .
-
PDFs and the lattice
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 4 / 46
Do we need to know partonic functions from the lattice?Maybe it
is not needed if we have huge expertise in fitting PDFs from
abundantexperimental data?However:
• knowing something from first principles is always desirable,•
good knowledge only of unpolarized and helicity PDFs,• transversity
PDFs – not much constrained by experiment,• other kinds of
functions very difficult to extract solely from experiment:
GPDs, TMDs, twist-3, . . .
Hence, lattice extraction of partonic functions is
awell-justified aim!
-
PDFs and the lattice
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 4 / 46
Do we need to know partonic functions from the lattice?Maybe it
is not needed if we have huge expertise in fitting PDFs from
abundantexperimental data?However:
• knowing something from first principles is always desirable,•
good knowledge only of unpolarized and helicity PDFs,• transversity
PDFs – not much constrained by experiment,• other kinds of
functions very difficult to extract solely from experiment:
GPDs, TMDs, twist-3, . . .
Hence, lattice extraction of partonic functions is
awell-justified aim!• Need to start with reliable extraction of
PDFs.
-
PDFs and the lattice
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 4 / 46
Do we need to know partonic functions from the lattice?Maybe it
is not needed if we have huge expertise in fitting PDFs from
abundantexperimental data?However:
• knowing something from first principles is always desirable,•
good knowledge only of unpolarized and helicity PDFs,• transversity
PDFs – not much constrained by experiment,• other kinds of
functions very difficult to extract solely from experiment:
GPDs, TMDs, twist-3, . . .
Hence, lattice extraction of partonic functions is
awell-justified aim!• Need to start with reliable extraction of
PDFs.• PDFs non-perturbative, so very natural to try on the
lattice.
-
PDFs and the lattice
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 4 / 46
Do we need to know partonic functions from the lattice?Maybe it
is not needed if we have huge expertise in fitting PDFs from
abundantexperimental data?However:
• knowing something from first principles is always desirable,•
good knowledge only of unpolarized and helicity PDFs,• transversity
PDFs – not much constrained by experiment,• other kinds of
functions very difficult to extract solely from experiment:
GPDs, TMDs, twist-3, . . .
Hence, lattice extraction of partonic functions is
awell-justified aim!• Need to start with reliable extraction of
PDFs.• PDFs non-perturbative, so very natural to try on the
lattice.• But: PDFs given in terms of non-local light-cone
correlators – intrinsically
Minkowskian:q(x) =
1
2π
∫
dξ−e−ixp+ξ−〈N |ψ(ξ−)ΓA(ξ−, 0)ψ(0)|N〉,
where: ξ− = ξ0−ξ3√
2and A(ξ−, 0) is the Wilson line from 0 to ξ−.
-
PDFs and the lattice
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 4 / 46
Do we need to know partonic functions from the lattice?Maybe it
is not needed if we have huge expertise in fitting PDFs from
abundantexperimental data?However:
• knowing something from first principles is always desirable,•
good knowledge only of unpolarized and helicity PDFs,• transversity
PDFs – not much constrained by experiment,• other kinds of
functions very difficult to extract solely from experiment:
GPDs, TMDs, twist-3, . . .
Hence, lattice extraction of partonic functions is
awell-justified aim!• Need to start with reliable extraction of
PDFs.• PDFs non-perturbative, so very natural to try on the
lattice.• But: PDFs given in terms of non-local light-cone
correlators – intrinsically
Minkowskian:q(x) =
1
2π
∫
dξ−e−ixp+ξ−〈N |ψ(ξ−)ΓA(ξ−, 0)ψ(0)|N〉,
where: ξ− = ξ0−ξ3√
2and A(ξ−, 0) is the Wilson line from 0 to ξ−.
• inaccessible on the lattice...
-
PDFs and the lattice
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 4 / 46
Do we need to know partonic functions from the lattice?Maybe it
is not needed if we have huge expertise in fitting PDFs from
abundantexperimental data?However:
• knowing something from first principles is always desirable,•
good knowledge only of unpolarized and helicity PDFs,• transversity
PDFs – not much constrained by experiment,• other kinds of
functions very difficult to extract solely from experiment:
GPDs, TMDs, twist-3, . . .
Hence, lattice extraction of partonic functions is
awell-justified aim!• Need to start with reliable extraction of
PDFs.• PDFs non-perturbative, so very natural to try on the
lattice.• But: PDFs given in terms of non-local light-cone
correlators – intrinsically
Minkowskian:q(x) =
1
2π
∫
dξ−e−ixp+ξ−〈N |ψ(ξ−)ΓA(ξ−, 0)ψ(0)|N〉,
where: ξ− = ξ0−ξ3√
2and A(ξ−, 0) is the Wilson line from 0 to ξ−.
• inaccessible on the lattice...
Recently: new direct approaches to get x-dependence.
-
Approaches to light-cone PDFs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 5 / 46
• The common feature of all the approaches is that they rely to
some extent onthe factorization framework:
Q(x, µR) =
∫ 1
−1
dy
yC
(
x
y, µF , µR
)
q(y, µF ),some lattice observable
-
Approaches to light-cone PDFs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 5 / 46
• The common feature of all the approaches is that they rely to
some extent onthe factorization framework:
Q(x, µR) =
∫ 1
−1
dy
yC
(
x
y, µF , µR
)
q(y, µF ),some lattice observable
• Two classes of approaches:
⋆ generalizations of light-cone functions; direct x-dependence,⋆
hadronic tensor; decomposition into structure functions.
-
Approaches to light-cone PDFs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 5 / 46
• The common feature of all the approaches is that they rely to
some extent onthe factorization framework:
Q(x, µR) =
∫ 1
−1
dy
yC
(
x
y, µF , µR
)
q(y, µF ),some lattice observable
• Two classes of approaches:
⋆ generalizations of light-cone functions; direct x-dependence,⋆
hadronic tensor; decomposition into structure functions.
• Matrix elements: 〈N |ψ̄(z)ΓF (z)Γ′ψ(0)|N〉 with different
choices of Γ,Γ′ Diracstructures and objects F (z).
⋆ hadronic tensor – K.-F. Liu, S.-J. Dong, 1993⋆ auxiliary
scalar quark – U. Aglietti et al., 1998⋆ auxiliary heavy quark – W.
Detmold, C.-J. D. Lin, 2005⋆ auxiliary light quark – V. Braun, D.
Müller, 2007⋆ quasi-distributions – X. Ji, 2013⋆ “good lattice
cross sections” – Y.-Q. Ma, J.-W. Qiu, 2014,2017⋆
pseudo-distributions – A. Radyushkin, 2017⋆ “OPE without OPE” –
QCDSF, 2017
-
Overview of results from different approaches
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 6 / 46
Quasi-distributions
Nucleon twist-2 PDFs Nucleon twist-3 PDFs Nucleon GPDs ∆
PDFs
unpolarized helicity transversity
results @ physical pion mass
results extrapolated to physical pion mass
results @ non-physical pion mass
ETMC, PRL121(2018)112001ETMC, PRD99(2019)114504
LP3, arXiv:1803.04393LPC, PRD101(2020)034020BNL/MSU,
arXiv:2005.12015
ETMC, PRL121(2018)112001ETMC, PRD99(2019)114504LP3,
PRL121(2018)242003
BNL/MSU, arXiv:2005.12015
ETMC, PRD98(2018)091503(Rapid)ETMC, PRD99(2019)114504
LP3, arXiv:1810.05043
ETMC/Temple
arXiv:2004.04130ETMC
arXiv:1910.13229(paper soon!)
ETMC/Beijing
arXiv:2002.12044(accepted in PRD)
Pion/Kaon DA Pion PDF Pion GPD
LP3, PRD95(2017)094514LP3, NPB939(2019)429
LP3, PRD100(2019)034505BNL, PRD100(2019)034516
BNL, arXiv:2007.06590
Chen, NPB952(2020)114940
Pseudo-distributions
Nucleon PDFs Pion PDF
JLab, PRD96(2017)094503JLab, JHEP12(2019)081
JLab, arXiv:2004.01687ETMC, arXiv:2005.02102
JLab, PRD100(2019)114512
Good LCSs
Pion PDF
JLab, PRD99(2019)074507
JLab, arXiv:2001.04960
Auxiliary light quark
Pion DA
Regensburg, EPJC78(2018)217
Regensburg, PRD98(2018)094507
Auxiliary heavy quark
Pion DA
Taiwan/MIT, arXiv:1810.12194
OPE without OPE
Nucleon F1
QCDSF, PRL118(2017)242001
QCDSF, arXiv:2007.01523
Hadronic tensor
Nucleon W11
χQCD, PRD101(2020)114503
-
Overview of results from different approaches
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 6 / 46
Quasi-distributions
Nucleon twist-2 PDFs Nucleon twist-3 PDFs Nucleon GPDs ∆
PDFs
unpolarized helicity transversity
results @ physical pion mass
results extrapolated to physical pion mass
results @ non-physical pion mass
ETMC, PRL121(2018)112001ETMC, PRD99(2019)114504
LP3, arXiv:1803.04393LPC, PRD101(2020)034020BNL/MSU,
arXiv:2005.12015
ETMC, PRL121(2018)112001ETMC, PRD99(2019)114504LP3,
PRL121(2018)242003
BNL/MSU, arXiv:2005.12015
ETMC, PRD98(2018)091503(Rapid)ETMC, PRD99(2019)114504
LP3, arXiv:1810.05043
ETMC/Temple
arXiv:2004.04130ETMC
arXiv:1910.13229(paper soon!)
ETMC/Beijing
arXiv:2002.12044(accepted in PRD)
Pion/Kaon DA Pion PDF Pion GPD
LP3, PRD95(2017)094514LP3, NPB939(2019)429
LP3, PRD100(2019)034505BNL, PRD100(2019)034516
BNL, arXiv:2007.06590
Chen, NPB952(2020)114940
Pseudo-distributions
Nucleon PDFs Pion PDF
JLab, PRD96(2017)094503JLab, JHEP12(2019)081
JLab, arXiv:2004.01687ETMC, arXiv:2005.02102
JLab, PRD100(2019)114512
Good LCSs
Pion PDF
JLab, PRD99(2019)074507
JLab, arXiv:2001.04960
Auxiliary light quark
Pion DA
Regensburg, EPJC78(2018)217
Regensburg, PRD98(2018)094507
Auxiliary heavy quark
Pion DA
Taiwan/MIT, arXiv:1810.12194
OPE without OPE
Nucleon F1
QCDSF, PRL118(2017)242001
QCDSF, arXiv:2007.01523
Hadronic tensor
Nucleon W11
χQCD, PRD101(2020)114503
Adv. High Energy Phys. 2019 (2019)
3036904arXiv:1811.07248Special issue Transverse Momentum Dependent
Observables from Low toHigh Energy: Factorization, Evolution, and
Global Analyses
discusses in detail quasi-distributionsreviews also other
approaches
-
Quasi-PDFs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 7 / 46
Quasi-distribution approach:X. Ji, Parton Physics on a Euclidean
Lattice, Phys. Rev. Lett. 110 (2013) 262002
-
Quasi-PDFs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 7 / 46
Quasi-distribution approach:X. Ji, Parton Physics on a Euclidean
Lattice, Phys. Rev. Lett. 110 (2013) 262002
Main idea:
ξ3 ≡ z
ξ0 ≡ tξ+ξ−
-
Quasi-PDFs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 7 / 46
Quasi-distribution approach:X. Ji, Parton Physics on a Euclidean
Lattice, Phys. Rev. Lett. 110 (2013) 262002
Main idea:
ξ3 ≡ z
ξ0 ≡ tξ+ξ−
Correlation along the ξ−-direction:
q(x) = 12π
∫
dξ−e−ixp+ξ−〈N |ψ(ξ−)ΓA(ξ−, 0)ψ(0)|N〉
|N〉 – nucleon at rest in the light-cone frame
-
Quasi-PDFs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 7 / 46
Quasi-distribution approach:X. Ji, Parton Physics on a Euclidean
Lattice, Phys. Rev. Lett. 110 (2013) 262002
Main idea:
ξ3 ≡ z
ξ0 ≡ tξ+ξ−
Correlation along the ξ−-direction:
q(x) = 12π
∫
dξ−e−ixp+ξ−〈N |ψ(ξ−)ΓA(ξ−, 0)ψ(0)|N〉
|N〉 – nucleon at rest in the light-cone frame
Correlation along the ξ3 ≡ z-direction:
q̃(x) = 12π
∫
dz eixP3z〈N |ψ(z)ΓA(z, 0)ψ(0)|N〉
|N〉 – nucleon at rest in the standard frame
-
Quasi-PDFs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 7 / 46
Quasi-distribution approach:X. Ji, Parton Physics on a Euclidean
Lattice, Phys. Rev. Lett. 110 (2013) 262002
Main idea:
ξ3 ≡ z
ξ0 ≡ tξ+ξ−
Correlation along the ξ−-direction:
q(x) = 12π
∫
dξ−e−ixp+ξ−〈N |ψ(ξ−)ΓA(ξ−, 0)ψ(0)|N〉
|N〉 – nucleon at rest in the light-cone frame
Correlation along the ξ3 ≡ z-direction:
q̃(x) = 12π
∫
dz eixP3z〈N |ψ(z)ΓA(z, 0)ψ(0)|N〉
|N〉 – nucleon at rest in the standard frame
Correlation along the ξ3-direction:
q̃(x) = 12π
∫
dz eixP3z〈P |ψ(z)ΓA(z, 0)ψ(0)|P 〉
|P 〉 – boosted nucleon
-
Quasi-PDFs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 7 / 46
Quasi-distribution approach:X. Ji, Parton Physics on a Euclidean
Lattice, Phys. Rev. Lett. 110 (2013) 262002
Main idea:
ξ3 ≡ z
ξ0 ≡ tξ+ξ−
Correlation along the ξ−-direction:
q(x) = 12π
∫
dξ−e−ixp+ξ−〈N |ψ(ξ−)ΓA(ξ−, 0)ψ(0)|N〉
|N〉 – nucleon at rest in the light-cone frame
Correlation along the ξ3 ≡ z-direction:
q̃(x) = 12π
∫
dz eixP3z〈N |ψ(z)ΓA(z, 0)ψ(0)|N〉
|N〉 – nucleon at rest in the standard frame
Correlation along the ξ3-direction:
q̃(x) = 12π
∫
dz eixP3z〈P |ψ(z)ΓA(z, 0)ψ(0)|P 〉
|P 〉 – boosted nucleon
matching
Matching (Large Momentum Effective Theory (LaMET)X. Ji, Parton
Physics from Large-Momentum Effective Field Theory, Sci.China
Phys.Mech.Astron. 57 (2014) 1407
→ brings quasi-distribution to the light-cone distribution, up
to power-suppressed effects:
q̃(x, µ, P3) =∫ 1
−1dy|y| C
(
xy, µP3
)
q(y, µ) +O(
Λ2QCD/P23 ,M
2N/P
23
)
-
Quasi-PDFs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 7 / 46
Quasi-distribution approach:X. Ji, Parton Physics on a Euclidean
Lattice, Phys. Rev. Lett. 110 (2013) 262002
Main idea:
ξ3 ≡ z
ξ0 ≡ tξ+ξ−
Correlation along the ξ−-direction:
q(x) = 12π
∫
dξ−e−ixp+ξ−〈N |ψ(ξ−)ΓA(ξ−, 0)ψ(0)|N〉
|N〉 – nucleon at rest in the light-cone frame
Correlation along the ξ3 ≡ z-direction:
q̃(x) = 12π
∫
dz eixP3z〈N |ψ(z)ΓA(z, 0)ψ(0)|N〉
|N〉 – nucleon at rest in the standard frame
Correlation along the ξ3-direction:
q̃(x) = 12π
∫
dz eixP3z〈P |ψ(z)ΓA(z, 0)ψ(0)|P 〉
|P 〉 – boosted nucleon
matching
Matching (Large Momentum Effective Theory (LaMET)X. Ji, Parton
Physics from Large-Momentum Effective Field Theory, Sci.China
Phys.Mech.Astron. 57 (2014) 1407
→ brings quasi-distribution to the light-cone distribution, up
to power-suppressed effects:
q̃(x, µ, P3) =∫ 1
−1dy|y| C
(
xy, µP3
)
q(y, µ) +O(
Λ2QCD/P23 ,M
2N/P
23
)
quasi-PDF
-
Quasi-PDFs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 7 / 46
Quasi-distribution approach:X. Ji, Parton Physics on a Euclidean
Lattice, Phys. Rev. Lett. 110 (2013) 262002
Main idea:
ξ3 ≡ z
ξ0 ≡ tξ+ξ−
Correlation along the ξ−-direction:
q(x) = 12π
∫
dξ−e−ixp+ξ−〈N |ψ(ξ−)ΓA(ξ−, 0)ψ(0)|N〉
|N〉 – nucleon at rest in the light-cone frame
Correlation along the ξ3 ≡ z-direction:
q̃(x) = 12π
∫
dz eixP3z〈N |ψ(z)ΓA(z, 0)ψ(0)|N〉
|N〉 – nucleon at rest in the standard frame
Correlation along the ξ3-direction:
q̃(x) = 12π
∫
dz eixP3z〈P |ψ(z)ΓA(z, 0)ψ(0)|P 〉
|P 〉 – boosted nucleon
matching
Matching (Large Momentum Effective Theory (LaMET)X. Ji, Parton
Physics from Large-Momentum Effective Field Theory, Sci.China
Phys.Mech.Astron. 57 (2014) 1407
→ brings quasi-distribution to the light-cone distribution, up
to power-suppressed effects:
q̃(x, µ, P3) =∫ 1
−1dy|y| C
(
xy, µP3
)
q(y, µ) +O(
Λ2QCD/P23 ,M
2N/P
23
)
quasi-PDF PDF
-
Quasi-PDFs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 7 / 46
Quasi-distribution approach:X. Ji, Parton Physics on a Euclidean
Lattice, Phys. Rev. Lett. 110 (2013) 262002
Main idea:
ξ3 ≡ z
ξ0 ≡ tξ+ξ−
Correlation along the ξ−-direction:
q(x) = 12π
∫
dξ−e−ixp+ξ−〈N |ψ(ξ−)ΓA(ξ−, 0)ψ(0)|N〉
|N〉 – nucleon at rest in the light-cone frame
Correlation along the ξ3 ≡ z-direction:
q̃(x) = 12π
∫
dz eixP3z〈N |ψ(z)ΓA(z, 0)ψ(0)|N〉
|N〉 – nucleon at rest in the standard frame
Correlation along the ξ3-direction:
q̃(x) = 12π
∫
dz eixP3z〈P |ψ(z)ΓA(z, 0)ψ(0)|P 〉
|P 〉 – boosted nucleon
matching
Matching (Large Momentum Effective Theory (LaMET)X. Ji, Parton
Physics from Large-Momentum Effective Field Theory, Sci.China
Phys.Mech.Astron. 57 (2014) 1407
→ brings quasi-distribution to the light-cone distribution, up
to power-suppressed effects:
q̃(x, µ, P3) =∫ 1
−1dy|y| C
(
xy, µP3
)
q(y, µ) +O(
Λ2QCD/P23 ,M
2N/P
23
)
quasi-PDF PDFpert.kernel
-
Quasi-PDFs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 7 / 46
Quasi-distribution approach:X. Ji, Parton Physics on a Euclidean
Lattice, Phys. Rev. Lett. 110 (2013) 262002
Main idea:
ξ3 ≡ z
ξ0 ≡ tξ+ξ−
Correlation along the ξ−-direction:
q(x) = 12π
∫
dξ−e−ixp+ξ−〈N |ψ(ξ−)ΓA(ξ−, 0)ψ(0)|N〉
|N〉 – nucleon at rest in the light-cone frame
Correlation along the ξ3 ≡ z-direction:
q̃(x) = 12π
∫
dz eixP3z〈N |ψ(z)ΓA(z, 0)ψ(0)|N〉
|N〉 – nucleon at rest in the standard frame
Correlation along the ξ3-direction:
q̃(x) = 12π
∫
dz eixP3z〈P |ψ(z)ΓA(z, 0)ψ(0)|P 〉
|P 〉 – boosted nucleon
matching
Matching (Large Momentum Effective Theory (LaMET)X. Ji, Parton
Physics from Large-Momentum Effective Field Theory, Sci.China
Phys.Mech.Astron. 57 (2014) 1407
→ brings quasi-distribution to the light-cone distribution, up
to power-suppressed effects:
q̃(x, µ, P3) =∫ 1
−1dy|y| C
(
xy, µP3
)
q(y, µ) +O(
Λ2QCD/P23 ,M
2N/P
23
)
quasi-PDF PDFpert.kernel higher-twist effects
-
Pseudo-PDFs
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 8 / 46
The same matrix elements that are the basis for the
quasi-distributionapproach can also be used to define
pseudo-distributions.
-
Pseudo-PDFs
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 8 / 46
The same matrix elements that are the basis for the
quasi-distributionapproach can also be used to define
pseudo-distributions.
• Originated from Radyushkin’s pioneering studies of relations
betweenquasi-distributions, virtuality distribution functions
(VDFs) and “primordial”TMDs.A. Radyushkin, Phys. Lett. B767 (2017)
314
A. Radyushkin, Phys. Rev. D95 (2017) 056020
A. Radyushkin, Phys. Lett. B770 (2017) 514
-
Pseudo-PDFs
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 8 / 46
The same matrix elements that are the basis for the
quasi-distributionapproach can also be used to define
pseudo-distributions.
• Originated from Radyushkin’s pioneering studies of relations
betweenquasi-distributions, virtuality distribution functions
(VDFs) and “primordial”TMDs.A. Radyushkin, Phys. Lett. B767 (2017)
314
A. Radyushkin, Phys. Rev. D95 (2017) 056020
A. Radyushkin, Phys. Lett. B770 (2017) 514
• Radyushkin realized that quasi-PDFs may be treated as hybrids
of PDFs andthese “primordial” TMDs, which results in a rather
complicated convolutionnature of quasi-PDFs.
-
Pseudo-PDFs
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 8 / 46
The same matrix elements that are the basis for the
quasi-distributionapproach can also be used to define
pseudo-distributions.
• Originated from Radyushkin’s pioneering studies of relations
betweenquasi-distributions, virtuality distribution functions
(VDFs) and “primordial”TMDs.A. Radyushkin, Phys. Lett. B767 (2017)
314
A. Radyushkin, Phys. Rev. D95 (2017) 056020
A. Radyushkin, Phys. Lett. B770 (2017) 514
• Radyushkin realized that quasi-PDFs may be treated as hybrids
of PDFs andthese “primordial” TMDs, which results in a rather
complicated convolutionnature of quasi-PDFs.
• Thus, he proposed another approach, pseudo-distributions,
generalizinglight-cone PDFs onto spacelike intervals in a different
way.A. Radyushkin, Phys. Rev. D96 (2017) 034025
-
Pseudo-PDFs
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 8 / 46
The same matrix elements that are the basis for the
quasi-distributionapproach can also be used to define
pseudo-distributions.
• Originated from Radyushkin’s pioneering studies of relations
betweenquasi-distributions, virtuality distribution functions
(VDFs) and “primordial”TMDs.A. Radyushkin, Phys. Lett. B767 (2017)
314
A. Radyushkin, Phys. Rev. D95 (2017) 056020
A. Radyushkin, Phys. Lett. B770 (2017) 514
• Radyushkin realized that quasi-PDFs may be treated as hybrids
of PDFs andthese “primordial” TMDs, which results in a rather
complicated convolutionnature of quasi-PDFs.
• Thus, he proposed another approach, pseudo-distributions,
generalizinglight-cone PDFs onto spacelike intervals in a different
way.A. Radyushkin, Phys. Rev. D96 (2017) 034025
Central object: “Ioffe-time distribution” (ITD) – Q(ν,
µ2)Fourier-conjugate to PDF: Q(ν, µ2) =
∫ 1−1 dx e
iνxq(x, µ2)ν ≡ zP3 – “Ioffe time”
-
Pseudo-PDFs
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 9 / 46
• Later, the approach has been broadly investigated
theoretically...A. Radyushkin, Phys. Lett. B781 (2018) 433
A. Radyushkin, Phys. Rev. D98 (2018) 014019
J.-H. Zhang, J.-W. Chen, C. Monahan, Phys. Rev. D97 (2018)
074508
T. Izubuchi et al., Phys. Rev. D98 (2018) 056004
A. Radyushkin, Phys. Lett. B788 (2019) 380
A. Radyushkin, Phys. Rev. D100 (2019) 116011 (pseudo-GPDs)
I. Balitsky, W. Morris, A. Radyushkin, arXiv:1910.13963 (gluon
pseudo-PDFs)
S. Zhao, A. Radyushkin, arXiv:2006.05663 (B-meson pseudo-DA)
-
Pseudo-PDFs
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 9 / 46
• Later, the approach has been broadly investigated
theoretically...A. Radyushkin, Phys. Lett. B781 (2018) 433
A. Radyushkin, Phys. Rev. D98 (2018) 014019
J.-H. Zhang, J.-W. Chen, C. Monahan, Phys. Rev. D97 (2018)
074508
T. Izubuchi et al., Phys. Rev. D98 (2018) 056004
A. Radyushkin, Phys. Lett. B788 (2019) 380
A. Radyushkin, Phys. Rev. D100 (2019) 116011 (pseudo-GPDs)
I. Balitsky, W. Morris, A. Radyushkin, arXiv:1910.13963 (gluon
pseudo-PDFs)
S. Zhao, A. Radyushkin, arXiv:2006.05663 (B-meson pseudo-DA)
• ... and numerically on the lattice by the JLab group:K.
Orginos, A. Radyushkin, J. Karpie, S. Zafeiropoulos, Phys. Rev. D96
(2017)
094503 (quenched)
J. Karpie, K. Orginos, S. Zafeiropoulos, JHEP 11 (2018) 178
(moments)
B. Joó et al., JHEP 12 (2019) 081 (dynamical, N)
B. Joó et al., Phys. Rev. D100 (2019) 114512 (dynamical, π)
B. Joó et al., arXiv:2004.01687 (dynamical, N , approaching
physical point)
-
Pseudo-PDFs
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 9 / 46
• Later, the approach has been broadly investigated
theoretically...A. Radyushkin, Phys. Lett. B781 (2018) 433
A. Radyushkin, Phys. Rev. D98 (2018) 014019
J.-H. Zhang, J.-W. Chen, C. Monahan, Phys. Rev. D97 (2018)
074508
T. Izubuchi et al., Phys. Rev. D98 (2018) 056004
A. Radyushkin, Phys. Lett. B788 (2019) 380
A. Radyushkin, Phys. Rev. D100 (2019) 116011 (pseudo-GPDs)
I. Balitsky, W. Morris, A. Radyushkin, arXiv:1910.13963 (gluon
pseudo-PDFs)
S. Zhao, A. Radyushkin, arXiv:2006.05663 (B-meson pseudo-DA)
• ... and numerically on the lattice by the JLab group:K.
Orginos, A. Radyushkin, J. Karpie, S. Zafeiropoulos, Phys. Rev. D96
(2017)
094503 (quenched)
J. Karpie, K. Orginos, S. Zafeiropoulos, JHEP 11 (2018) 178
(moments)
B. Joó et al., JHEP 12 (2019) 081 (dynamical, N)
B. Joó et al., Phys. Rev. D100 (2019) 114512 (dynamical, π)
B. Joó et al., arXiv:2004.01687 (dynamical, N , approaching
physical point)
• Excellent review:A. Radyushkin, “Theory and applications of
parton pseudodistributions”,Int. J. Mod. Phys. A35 (2020)
2030002
-
Quasi-PDFs vs. pseudo-PDFs
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 10 / 46
spatial correlation in a boosted nucleon〈N(P3)|ψ(z)ΓA(z,
0)ψ(0)|N(P3)〉
renormalizationRI scheme (,other?)
renormalizationratios (,other?)
reconstruction ofx-dependenceF.T. in z
matching to light conein ν-space
matching to light conein x-space
reconstruction ofx-dependenceF.T. in ν
light-cone PDF
QUASI PSEUDO
-
Renormalization from a double ratio
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 11 / 46
The matrix element 〈N(P3)|ψ(z)γ0A(z, 0)ψ(0)|N(P3)〉 exhibits
twokinds of divergences:
• standard logarithmic divergence,• power divergence related to
the Wilson line.
Shown to be multiplicatively renormalizable to all orders in
PTT. Ishikawa et al., PRD96(2017)094019, X. Ji et al.,
PRL120(2017)112001
-
Renormalization from a double ratio
Outline of the talk
Lattice PDFs
PDFs
Approaches
Quasi-PDFs
Pseudo-PDFs
Results (pseudo)
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 11 / 46
The matrix element 〈N(P3)|ψ(z)γ0A(z, 0)ψ(0)|N(P3)〉 exhibits
twokinds of divergences:
• standard logarithmic divergence,• power divergence related to
the Wilson line.
Shown to be multiplicatively renormalizable to all orders in
PTT. Ishikawa et al., PRD96(2017)094019, X. Ji et al.,
PRL120(2017)112001
Both divergences can be canceled by forming a double ratio
withzero-momentum and local (z = 0) matrix elements:(also removes
part of HTE (generically O(z2Λ2QCD)))
M(ν, z2) =M(ν, z2) /M(ν, 0)
M(0, z2) /M(0, 0).
M(ν, z2) – “reduced” matrix elements (or pseudo-ITDs).
The double ratio defines a renormalization scheme
withrenormalization scale proportional to 1/z.
-
Matching to light-cone ITDs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 12 / 46
The reduced matrix elements, M(ν, z2), defined at different
scales 1/z, need to be:
-
Matching to light-cone ITDs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 12 / 46
The reduced matrix elements, M(ν, z2), defined at different
scales 1/z, need to be:
• evolved to a common scale,
-
Matching to light-cone ITDs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 12 / 46
The reduced matrix elements, M(ν, z2), defined at different
scales 1/z, need to be:
• evolved to a common scale,
• scheme-converted to the MS scheme −→ Q(ν, µ2).
-
Matching to light-cone ITDs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 12 / 46
The reduced matrix elements, M(ν, z2), defined at different
scales 1/z, need to be:
• evolved to a common scale,
• scheme-converted to the MS scheme −→ Q(ν, µ2).
The full 1-loop matching equation: A. Radyushkin,
PLB781(2018)433, PRD98(2018)014019;J.-H. Zhang et al.,
PRD97(2018)074508; T. Izubuchi et al., PRD98(2018)056004
M(ν, z2) = Q(ν, µ2)−αsCF2π
∫ 1
0du
[
ln
(
z2µ2e2γE+1
4
)
B(u) + L(u)
]
Q(uν, µ2)
with:
B(u) =
[
1 + u2
1− u
]
+
, L(u) =
[
4ln(1− u)
1− u− 2(1− u)
]
+
,
∫ 1
0[f(u)]+Q(uν) =
∫ 1
0f(u) (Q(uν)−Q(ν)) .
-
Matching to light-cone ITDs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 13 / 46
We invert the matching equation and look separately into the
effect of evolutionand scheme conversion:
-
Matching to light-cone ITDs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 13 / 46
We invert the matching equation and look separately into the
effect of evolutionand scheme conversion:• evolution:
M′(ν, z2, µ2) = M(ν, z2)− αsCF2π
∫ 10 du ln
(
z2µ2 e2γE+1
4
)
B(u)M(uν, z2),
The evolved ITD M′ has 3 arguments:the Ioffe time ν, the common
scale µ, the initial scale z.In principle, values should be
independent of the initial scale −→ test this.
-
Matching to light-cone ITDs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 13 / 46
We invert the matching equation and look separately into the
effect of evolutionand scheme conversion:• evolution:
M′(ν, z2, µ2) = M(ν, z2)− αsCF2π
∫ 10 du ln
(
z2µ2 e2γE+1
4
)
B(u)M(uν, z2),
The evolved ITD M′ has 3 arguments:the Ioffe time ν, the common
scale µ, the initial scale z.In principle, values should be
independent of the initial scale −→ test this.
• scheme conversion:Q(ν, z2, µ2) = M′(ν, z2, µ2)− αsCF2π
∫ 10 duL(u)M(uν, z
2).
Again 3 arguments and test of independence on the initial
scale.
-
Matching to light-cone ITDs
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 13 / 46
We invert the matching equation and look separately into the
effect of evolutionand scheme conversion:• evolution:
M′(ν, z2, µ2) = M(ν, z2)− αsCF2π
∫ 10 du ln
(
z2µ2 e2γE+1
4
)
B(u)M(uν, z2),
The evolved ITD M′ has 3 arguments:the Ioffe time ν, the common
scale µ, the initial scale z.In principle, values should be
independent of the initial scale −→ test this.
• scheme conversion:Q(ν, z2, µ2) = M′(ν, z2, µ2)− αsCF2π
∫ 10 duL(u)M(uν, z
2).
Again 3 arguments and test of independence on the initial
scale.
For the reconstruction of the final PDF−→ average the matched
ITDs Q(ν, z2, µ2) for cases where a given Ioffe time isachieved by
different combinations of (P3, z), denote such average by Q(ν,
µ
2).
-
Reconstruction of x-dependence
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 14 / 46
The ITDs, Q(ν, µ2), are related to PDFs, q(x, µ2), by a Fourier
transform:
Q(ν, µ2) =
∫ 1
−1dx eiνxq(x, µ2).
-
Reconstruction of x-dependence
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 14 / 46
The ITDs, Q(ν, µ2), are related to PDFs, q(x, µ2), by a Fourier
transform:
Q(ν, µ2) =
∫ 1
−1dx eiνxq(x, µ2).
Decomposing into real and imaginary parts:
ReQ(ν, µ2) =
∫ 1
0dx cos(νx)qv(x, µ
2),
ImQ(ν, µ2) =
∫ 1
0dx sin(νx)qv2s(x, µ
2),
where: qv = q − q̄, qv2s ≡ qv + 2q̄ = q + q̄.
-
Reconstruction of x-dependence
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 14 / 46
The ITDs, Q(ν, µ2), are related to PDFs, q(x, µ2), by a Fourier
transform:
Q(ν, µ2) =
∫ 1
−1dx eiνxq(x, µ2).
Decomposing into real and imaginary parts:
ReQ(ν, µ2) =
∫ 1
0dx cos(νx)qv(x, µ
2),
ImQ(ν, µ2) =
∫ 1
0dx sin(νx)qv2s(x, µ
2),
where: qv = q − q̄, qv2s ≡ qv + 2q̄ = q + q̄.
Inverse problem!
-
Reconstruction of x-dependence
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 14 / 46
The ITDs, Q(ν, µ2), are related to PDFs, q(x, µ2), by a Fourier
transform:
Q(ν, µ2) =
∫ 1
−1dx eiνxq(x, µ2).
Decomposing into real and imaginary parts:
ReQ(ν, µ2) =
∫ 1
0dx cos(νx)qv(x, µ
2),
ImQ(ν, µ2) =
∫ 1
0dx sin(νx)qv2s(x, µ
2),
where: qv = q − q̄, qv2s ≡ qv + 2q̄ = q + q̄.
Inverse problem!Discussed extensively in: J. Karpie, K. Orginos,
A. Rothkopf, S. Zafeiropoulos, JHEP 04 (2019) 057
-
Reconstruction of x-dependence
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 14 / 46
The ITDs, Q(ν, µ2), are related to PDFs, q(x, µ2), by a Fourier
transform:
Q(ν, µ2) =
∫ 1
−1dx eiνxq(x, µ2).
Decomposing into real and imaginary parts:
ReQ(ν, µ2) =
∫ 1
0dx cos(νx)qv(x, µ
2),
ImQ(ν, µ2) =
∫ 1
0dx sin(νx)qv2s(x, µ
2),
where: qv = q − q̄, qv2s ≡ qv + 2q̄ = q + q̄.
Inverse problem!Discussed extensively in: J. Karpie, K. Orginos,
A. Rothkopf, S. Zafeiropoulos, JHEP 04 (2019) 057
Ways out used in our work:
• Backus-Gilbert approach (with and without preconditioning),•
fitting ansatz reconstruction: q(x) = Nxa(1− x)b.
-
Lattice setup
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 15 / 46
• fermions: Nf = 2 twisted mass fermions + clover term
• gluons: Iwasaki gauge action, β = 2.1
• gauge field configurations generated by ETMC
β=2.10, cSW=1.57751, a=0.0938(3)(2) fm
483 × 96 aµ = 0.0009 mN = 0.932(4) GeV
L = 4.5 fm mπ = 0.1304(4) GeV mπL = 2.98(1)
P3 P3 [GeV] Nconfs Nmeas0 0 20 320
2π/L 0.28 19 18244π/L 0.55 18 17286π/L 0.83 50 48008π/L 1.11 425
3825010π/L 1.38 811 72990
-
Bare matrix elements
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 16 / 46
0
0.5
1
1.5
0 5 10 15
-0.2
-0.1
0
-
Reduced matrix elements
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 17 / 46
-1
0
1
0 2 4 6 8
0
0.5
1
1.5
-
Evolved and MS-converted matrix elements
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 18 / 46
0
0.5
1
1.5
0 2 4 6 8
0
0.5
1
1.5
-
Evolved and MS-converted matrix elements
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 18 / 46
0
0.5
1
1.5
0 2 4 6 8
0
0.5
1
1.5
-0.5
0
0.5
1
1.5
0 2 4 6 8
0
0.5
1
1.5
-
Averaged matrix elements
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 19 / 46
0
0.5
1
1.5
0 1 2 3 4 5
0
0.5
1
1.5
-
PDFs using ITDs with zmax = 4a
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 20 / 46
-
PDFs using ITDs with zmax = 8a
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 21 / 46
-
PDFs using ITDs with zmax = 12a
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 22 / 46
-
PDFs from naive FT – zmax-dependence
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 23 / 46
-
PDFs from BG – zmax-dependence
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 24 / 46
-
PDFs from fits – zmax-dependence
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 25 / 46
-
PDFs from fits – αs-dependence
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 26 / 46
-
BG with preconditioning vs. BG vs. fits
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 27 / 46
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:• Discretization effects
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:• Discretization effects: assume 20%
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:• Discretization effects: assume 20%
indirect support: no violation of continuum dispersion relation,
E2 = P 23 +m2N ,
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:• Discretization effects: assume 20%
indirect support: no violation of continuum dispersion relation,
E2 = P 23 +m2N ,
computations of moments of unpolarized PDFs by different groups:
deviations of O(5− 15%)from continuum at similar lattice
spacings.
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:• Discretization effects: assume 20%
indirect support: no violation of continuum dispersion relation,
E2 = P 23 +m2N ,
computations of moments of unpolarized PDFs by different groups:
deviations of O(5− 15%)from continuum at similar lattice
spacings.
• FVE
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:• Discretization effects: assume 20%
indirect support: no violation of continuum dispersion relation,
E2 = P 23 +m2N ,
computations of moments of unpolarized PDFs by different groups:
deviations of O(5− 15%)from continuum at similar lattice
spacings.
• FVE: assume 5%
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:• Discretization effects: assume 20%
indirect support: no violation of continuum dispersion relation,
E2 = P 23 +m2N ,
computations of moments of unpolarized PDFs by different groups:
deviations of O(5− 15%)from continuum at similar lattice
spacings.
• FVE: assume 5%indirect support: exp(−mπL) ≈ 0.05 for our
setup,
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:• Discretization effects: assume 20%
indirect support: no violation of continuum dispersion relation,
E2 = P 23 +m2N ,
computations of moments of unpolarized PDFs by different groups:
deviations of O(5− 15%)from continuum at similar lattice
spacings.
• FVE: assume 5%indirect support: exp(−mπL) ≈ 0.05 for our
setup,enhanced FVE? R. Briceño et al., Phys. Rev. D 98 (2018)
014511toy scalar model, relevant parameter for FVE: mN (L− z)
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:• Discretization effects: assume 20%
indirect support: no violation of continuum dispersion relation,
E2 = P 23 +m2N ,
computations of moments of unpolarized PDFs by different groups:
deviations of O(5− 15%)from continuum at similar lattice
spacings.
• FVE: assume 5%indirect support: exp(−mπL) ≈ 0.05 for our
setup,enhanced FVE? R. Briceño et al., Phys. Rev. D 98 (2018)
014511toy scalar model, relevant parameter for FVE: mN (L− z) −→
tiny,
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:• Discretization effects: assume 20%
indirect support: no violation of continuum dispersion relation,
E2 = P 23 +m2N ,
computations of moments of unpolarized PDFs by different groups:
deviations of O(5− 15%)from continuum at similar lattice
spacings.
• FVE: assume 5%indirect support: exp(−mπL) ≈ 0.05 for our
setup,enhanced FVE? R. Briceño et al., Phys. Rev. D 98 (2018)
014511toy scalar model, relevant parameter for FVE: mN (L− z) −→
tiny,worst case: relevant parameter for FVE in QCD: mπ(L− z)
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:• Discretization effects: assume 20%
indirect support: no violation of continuum dispersion relation,
E2 = P 23 +m2N ,
computations of moments of unpolarized PDFs by different groups:
deviations of O(5− 15%)from continuum at similar lattice
spacings.
• FVE: assume 5%indirect support: exp(−mπL) ≈ 0.05 for our
setup,enhanced FVE? R. Briceño et al., Phys. Rev. D 98 (2018)
014511toy scalar model, relevant parameter for FVE: mN (L− z) −→
tiny,worst case: relevant parameter for FVE in QCD: mπ(L− z)−→
still rather small for small z/a,
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:• Discretization effects: assume 20%
indirect support: no violation of continuum dispersion relation,
E2 = P 23 +m2N ,
computations of moments of unpolarized PDFs by different groups:
deviations of O(5− 15%)from continuum at similar lattice
spacings.
• FVE: assume 5%indirect support: exp(−mπL) ≈ 0.05 for our
setup,enhanced FVE? R. Briceño et al., Phys. Rev. D 98 (2018)
014511toy scalar model, relevant parameter for FVE: mN (L− z) −→
tiny,worst case: relevant parameter for FVE in QCD: mπ(L− z)−→
still rather small for small z/a,also indirectly no indication for
such effects in Z-factors for quasi-PDFs.
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:• Discretization effects: assume 20%
indirect support: no violation of continuum dispersion relation,
E2 = P 23 +m2N ,
computations of moments of unpolarized PDFs by different groups:
deviations of O(5− 15%)from continuum at similar lattice
spacings.
• FVE: assume 5%indirect support: exp(−mπL) ≈ 0.05 for our
setup,enhanced FVE? R. Briceño et al., Phys. Rev. D 98 (2018)
014511toy scalar model, relevant parameter for FVE: mN (L− z) −→
tiny,worst case: relevant parameter for FVE in QCD: mπ(L− z)−→
still rather small for small z/a,also indirectly no indication for
such effects in Z-factors for quasi-PDFs.
• Excited states
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:• Discretization effects: assume 20%
indirect support: no violation of continuum dispersion relation,
E2 = P 23 +m2N ,
computations of moments of unpolarized PDFs by different groups:
deviations of O(5− 15%)from continuum at similar lattice
spacings.
• FVE: assume 5%indirect support: exp(−mπL) ≈ 0.05 for our
setup,enhanced FVE? R. Briceño et al., Phys. Rev. D 98 (2018)
014511toy scalar model, relevant parameter for FVE: mN (L− z) −→
tiny,worst case: relevant parameter for FVE in QCD: mπ(L− z)−→
still rather small for small z/a,also indirectly no indication for
such effects in Z-factors for quasi-PDFs.
• Excited states: assume 10%
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:• Discretization effects: assume 20%
indirect support: no violation of continuum dispersion relation,
E2 = P 23 +m2N ,
computations of moments of unpolarized PDFs by different groups:
deviations of O(5− 15%)from continuum at similar lattice
spacings.
• FVE: assume 5%indirect support: exp(−mπL) ≈ 0.05 for our
setup,enhanced FVE? R. Briceño et al., Phys. Rev. D 98 (2018)
014511toy scalar model, relevant parameter for FVE: mN (L− z) −→
tiny,worst case: relevant parameter for FVE in QCD: mπ(L− z)−→
still rather small for small z/a,also indirectly no indication for
such effects in Z-factors for quasi-PDFs.
• Excited states: assume 10%evidence: ETMC, Phys. Rev. D 99
(2019) 114504 – suppressed below stat. precision
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:• Discretization effects: assume 20%
indirect support: no violation of continuum dispersion relation,
E2 = P 23 +m2N ,
computations of moments of unpolarized PDFs by different groups:
deviations of O(5− 15%)from continuum at similar lattice
spacings.
• FVE: assume 5%indirect support: exp(−mπL) ≈ 0.05 for our
setup,enhanced FVE? R. Briceño et al., Phys. Rev. D 98 (2018)
014511toy scalar model, relevant parameter for FVE: mN (L− z) −→
tiny,worst case: relevant parameter for FVE in QCD: mπ(L− z)−→
still rather small for small z/a,also indirectly no indication for
such effects in Z-factors for quasi-PDFs.
• Excited states: assume 10%evidence: ETMC, Phys. Rev. D 99
(2019) 114504 – suppressed below stat. precision
• Matching (truncation effects and HTE)
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:• Discretization effects: assume 20%
indirect support: no violation of continuum dispersion relation,
E2 = P 23 +m2N ,
computations of moments of unpolarized PDFs by different groups:
deviations of O(5− 15%)from continuum at similar lattice
spacings.
• FVE: assume 5%indirect support: exp(−mπL) ≈ 0.05 for our
setup,enhanced FVE? R. Briceño et al., Phys. Rev. D 98 (2018)
014511toy scalar model, relevant parameter for FVE: mN (L− z) −→
tiny,worst case: relevant parameter for FVE in QCD: mπ(L− z)−→
still rather small for small z/a,also indirectly no indication for
such effects in Z-factors for quasi-PDFs.
• Excited states: assume 10%evidence: ETMC, Phys. Rev. D 99
(2019) 114504 – suppressed below stat. precision
• Matching (truncation effects and HTE): assume 20%
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:• Discretization effects: assume 20%
indirect support: no violation of continuum dispersion relation,
E2 = P 23 +m2N ,
computations of moments of unpolarized PDFs by different groups:
deviations of O(5− 15%)from continuum at similar lattice
spacings.
• FVE: assume 5%indirect support: exp(−mπL) ≈ 0.05 for our
setup,enhanced FVE? R. Briceño et al., Phys. Rev. D 98 (2018)
014511toy scalar model, relevant parameter for FVE: mN (L− z) −→
tiny,worst case: relevant parameter for FVE in QCD: mπ(L− z)−→
still rather small for small z/a,also indirectly no indication for
such effects in Z-factors for quasi-PDFs.
• Excited states: assume 10%evidence: ETMC, Phys. Rev. D 99
(2019) 114504 – suppressed below stat. precision
• Matching (truncation effects and HTE): assume 20%indirect
support: little dependence on αs and on zmax
-
Systematics
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 28 / 46
Quantified systematics:
• zmax: ∆zmax(x) =|qzmax/a=12(x)−qzmax/a=4(x)|
2 ,• αs: ∆αs(x) = |qαs/π=0.129(x)− qαs/π=0.1(x)|.
Estimated systematics:• Discretization effects: assume 20%
indirect support: no violation of continuum dispersion relation,
E2 = P 23 +m2N ,
computations of moments of unpolarized PDFs by different groups:
deviations of O(5− 15%)from continuum at similar lattice
spacings.
• FVE: assume 5%indirect support: exp(−mπL) ≈ 0.05 for our
setup,enhanced FVE? R. Briceño et al., Phys. Rev. D 98 (2018)
014511toy scalar model, relevant parameter for FVE: mN (L− z) −→
tiny,worst case: relevant parameter for FVE in QCD: mπ(L− z)−→
still rather small for small z/a,also indirectly no indication for
such effects in Z-factors for quasi-PDFs.
• Excited states: assume 10%evidence: ETMC, Phys. Rev. D 99
(2019) 114504 – suppressed below stat. precision
• Matching (truncation effects and HTE): assume 20%indirect
support: little dependence on αs and on zmaxneeded: 2-loop
matching, explicit computation of HTE?
-
Final PDFs with systematics
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 29 / 46
-
Light-cone PDFs from pseudo and quasi
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 30 / 46
ETMC, Phys. Rev. Lett. 121 (2018) 112001ETMC, Phys. Rev. D 99
(2019) 114504
ETMC, arXiv:2005.02102
-
Comparison with JLab
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 31 / 46
B. Joó et al., arXiv:2004.01687 ETMC, arXiv:2005.02102
-
Pseudo-PDFs vs. quasi-PDFs
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 32 / 46
Is there an answer to the question whether
quasi-distributionsare “better” than pseudo-distributions or vice
versa?
-
Pseudo-PDFs vs. quasi-PDFs
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 32 / 46
Is there an answer to the question whether
quasi-distributionsare “better” than pseudo-distributions or vice
versa?
• Large nucleon boost: no doubt both need to give the same
answer.
-
Pseudo-PDFs vs. quasi-PDFs
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 32 / 46
Is there an answer to the question whether
quasi-distributionsare “better” than pseudo-distributions or vice
versa?
• Large nucleon boost: no doubt both need to give the same
answer.
• Practicioner’s view for realistically achievable
momenta:certainly different systematics, so worthwhile to use
both.
-
Pseudo-PDFs vs. quasi-PDFs
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 32 / 46
Is there an answer to the question whether
quasi-distributionsare “better” than pseudo-distributions or vice
versa?
• Large nucleon boost: no doubt both need to give the same
answer.
• Practicioner’s view for realistically achievable
momenta:certainly different systematics, so worthwhile to use
both.
• Both have certain practical advantages over the other:
-
Pseudo-PDFs vs. quasi-PDFs
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 32 / 46
Is there an answer to the question whether
quasi-distributionsare “better” than pseudo-distributions or vice
versa?
• Large nucleon boost: no doubt both need to give the same
answer.
• Practicioner’s view for realistically achievable
momenta:certainly different systematics, so worthwhile to use
both.
• Both have certain practical advantages over the other:
⋆ pseudo-distributions:fully utilize all nucleon boost datahave
canonical support in xmatching in ν-space might be more
controlled
reconstruction with a fitting ansatz natural
-
Pseudo-PDFs vs. quasi-PDFs
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 32 / 46
Is there an answer to the question whether
quasi-distributionsare “better” than pseudo-distributions or vice
versa?
• Large nucleon boost: no doubt both need to give the same
answer.
• Practicioner’s view for realistically achievable
momenta:certainly different systematics, so worthwhile to use
both.
• Both have certain practical advantages over the other:
⋆ pseudo-distributions:fully utilize all nucleon boost datahave
canonical support in xmatching in ν-space might be more
controlled
reconstruction with a fitting ansatz natural
⋆ quasi-distributions:longer on the market and much more
explored
-
Pseudo-PDFs vs. quasi-PDFs
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 32 / 46
Is there an answer to the question whether
quasi-distributionsare “better” than pseudo-distributions or vice
versa?
• Large nucleon boost: no doubt both need to give the same
answer.
• Practicioner’s view for realistically achievable
momenta:certainly different systematics, so worthwhile to use
both.
• Both have certain practical advantages over the other:
⋆ pseudo-distributions:fully utilize all nucleon boost datahave
canonical support in xmatching in ν-space might be more
controlled
reconstruction with a fitting ansatz natural
⋆ quasi-distributions:longer on the market and much more
explored
• Theoretical questions:
-
Pseudo-PDFs vs. quasi-PDFs
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 32 / 46
Is there an answer to the question whether
quasi-distributionsare “better” than pseudo-distributions or vice
versa?
• Large nucleon boost: no doubt both need to give the same
answer.
• Practicioner’s view for realistically achievable
momenta:certainly different systematics, so worthwhile to use
both.
• Both have certain practical advantages over the other:
⋆ pseudo-distributions:fully utilize all nucleon boost datahave
canonical support in xmatching in ν-space might be more
controlled
reconstruction with a fitting ansatz natural
⋆ quasi-distributions:longer on the market and much more
explored
• Theoretical questions:
⋆ How important is it that pseudo-distributions rely on a
small-z expansion?
-
Pseudo-PDFs vs. quasi-PDFs
Outline of the talk
Lattice PDFs
Results (pseudo)
Lattice setup
Bare ME
Reduced ME
Matched ME
PDFs
Systematics
Final PDFs
Results (other)
Summary
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 32 / 46
Is there an answer to the question whether
quasi-distributionsare “better” than pseudo-distributions or vice
versa?
• Large nucleon boost: no doubt both need to give the same
answer.
• Practicioner’s view for realistically achievable
momenta:certainly different systematics, so worthwhile to use
both.
• Both have certain practical advantages over the other:
⋆ pseudo-distributions:fully utilize all nucleon boost datahave
canonical support in xmatching in ν-space might be more
controlled
reconstruction with a fitting ansatz natural
⋆ quasi-distributions:longer on the market and much more
explored
• Theoretical questions:
⋆ How important is it that pseudo-distributions rely on a
small-z expansion?
⋆ Are there no obstacles to extract polarized distributions?
-
Impact of lattice data on phenomenology?
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 33 / 46
• Factorization relates experimental cross sections to PDFs.
-
Impact of lattice data on phenomenology?
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 33 / 46
• Factorization relates experimental cross sections to PDFs.•
Similarly: factorization relates lattice observables to PDFs,
e.g.:
q̃ (x, µ, P3) =
∫ 1
−1
dξ
|ξ|C
(
x
ξ, µ, P3
)
q(x, µ)
-
Impact of lattice data on phenomenology?
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 33 / 46
• Factorization relates experimental cross sections to PDFs.•
Similarly: factorization relates lattice observables to PDFs,
e.g.:
q̃ (x, µ, P3) =
∫ 1
−1
dξ
|ξ|C
(
x
ξ, µ, P3
)
q(x, µ)
• Question: can we treat lattice observables similarly to cross
sections?
-
Impact of lattice data on phenomenology?
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 33 / 46
• Factorization relates experimental cross sections to PDFs.•
Similarly: factorization relates lattice observables to PDFs,
e.g.:
q̃ (x, µ, P3) =
∫ 1
−1
dξ
|ξ|C
(
x
ξ, µ, P3
)
q(x, µ)
• Question: can we treat lattice observables similarly to cross
sections?• Recent attempt to learn something about this
question:
K.C., L. Del Debbio, T. Giani, JHEP 10 (2019) 137
-
Impact of lattice data on phenomenology?
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 33 / 46
• Factorization relates experimental cross sections to PDFs.•
Similarly: factorization relates lattice observables to PDFs,
e.g.:
q̃ (x, µ, P3) =
∫ 1
−1
dξ
|ξ|C
(
x
ξ, µ, P3
)
q(x, µ)
• Question: can we treat lattice observables similarly to cross
sections?• Recent attempt to learn something about this
question:
K.C., L. Del Debbio, T. Giani, JHEP 10 (2019) 137
• Using the robust NNPDF framework for fitting.
-
Impact of lattice data on phenomenology?
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 33 / 46
• Factorization relates experimental cross sections to PDFs.•
Similarly: factorization relates lattice observables to PDFs,
e.g.:
q̃ (x, µ, P3) =
∫ 1
−1
dξ
|ξ|C
(
x
ξ, µ, P3
)
q(x, µ)
• Question: can we treat lattice observables similarly to cross
sections?• Recent attempt to learn something about this
question:
K.C., L. Del Debbio, T. Giani, JHEP 10 (2019) 137
• Using the robust NNPDF framework for fitting.• Observables:
non-singlet distributions V3 and T3 (unpolarized):
V3 = u− ū− (d− d̄) = uV − dVT3 = u+ ū− (d+ d̄) = uV − dV +
2(uS − dS)
-
Impact of lattice data on phenomenology?
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 33 / 46
• Factorization relates experimental cross sections to PDFs.•
Similarly: factorization relates lattice observables to PDFs,
e.g.:
q̃ (x, µ, P3) =
∫ 1
−1
dξ
|ξ|C
(
x
ξ, µ, P3
)
q(x, µ)
• Question: can we treat lattice observables similarly to cross
sections?• Recent attempt to learn something about this
question:
K.C., L. Del Debbio, T. Giani, JHEP 10 (2019) 137
• Using the robust NNPDF framework for fitting.• Observables:
non-singlet distributions V3 and T3 (unpolarized):
V3 = u− ū− (d− d̄) = uV − dVT3 = u+ ū− (d+ d̄) = uV − dV +
2(uS − dS)
• We have:
ORe/Imγ0
(z, µ)=
∫ 1
0dx C
Re/Im3
(
x, z,µ
Pz
)
V3/T3 (x, µ) = CRe/Im3
(
z,µ
Pz
)
⊛V3/T3 (µ) ,
-
Impact of lattice data on phenomenology?
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 33 / 46
• Factorization relates experimental cross sections to PDFs.•
Similarly: factorization relates lattice observables to PDFs,
e.g.:
q̃ (x, µ, P3) =
∫ 1
−1
dξ
|ξ|C
(
x
ξ, µ, P3
)
q(x, µ)
• Question: can we treat lattice observables similarly to cross
sections?• Recent attempt to learn something about this
question:
K.C., L. Del Debbio, T. Giani, JHEP 10 (2019) 137
• Using the robust NNPDF framework for fitting.• Observables:
non-singlet distributions V3 and T3 (unpolarized):
V3 = u− ū− (d− d̄) = uV − dVT3 = u+ ū− (d+ d̄) = uV − dV +
2(uS − dS)
• We have:
ORe/Imγ0
(z, µ)=
∫ 1
0dx C
Re/Im3
(
x, z,µ
Pz
)
V3/T3 (x, µ) = CRe/Im3
(
z,µ
Pz
)
⊛V3/T3 (µ) ,
• The above equations implemented using FastKernel tables that
combine thematching and DGLAP evolution.
-
Impact of lattice data on phenomenology?
Krzysztof Cichy Partonic structure of the nucleon from Lattice
QCD – JLab Online Theory Seminar – 33 / 46
• Factorization relates experimental cross sections to PDFs.•
Similarly: factorization relates lattice observables to PDFs,
e.g.:
q̃ (x, µ, P3) =
∫ 1
−1
dξ
|ξ|C
(
x
ξ, µ, P3
)
q(x, µ)
• Question: can we treat lattice observables similarly to cross
sections?• Recent attempt to learn something about this
question:
K.C., L. Del Debbio, T. Giani, JHEP 10 (2019) 137
• Using the robust NNPDF framework for fitting.• Observables:
non-singlet distributions V3 and T3 (unpolarized):
V3 = u− ū− (d− d̄) = uV − dVT3 = u+ ū− (d+ d̄) = uV − dV +
2(uS − dS)
• We have:
ORe/Imγ0
(z, µ)=
∫ 1
0dx C
Re/Im3
(
x, z,µ
Pz
)
V3/T3 (x, µ) = CRe/Im3
(
z,µ
Pz
)
⊛V3/T3 (µ) ,
• The above equatio