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).
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]
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
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