Accessing nucleon structure from Euclidean spacetime

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Accessing nucleon structure from Euclidean spacetime

Chris Monahan

Institute for Nuclear Theory

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HOW FAST DO PARTONS TRAVEL?

How is the momentum of a fast-moving nucleon distributed amongst its constituents?

WHERE DOES THE SPIN OF A PROTON COME FROM?

How do position and longitudinal momentum of

a parton correlate in a fast-moving nucleon?

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PDF UNCERTAINTIES

From J. Butterworth et al., J.P.G 43 (2016) 023001

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Nuclear landscape

Early universe

Neutron stars

LUX

LHCb 12 GeV

EIC @ JLab

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EXPERIMENTAL EXTRACTION

From PDG 2016

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EXPERIMENTAL EXTRACTION

PDF4LHC15 (NNLO), J. Phys. G 43 (2016) 023001

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PDFs FROM EUCLIDEAN SPACETIME

An unsolved almost-solved challenge

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Decompose cross-section

Hadronic contribution

in turn, expressed in terms of structure functions DIS

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Decompose cross-section

Hadronic contribution

in turn, expressed in terms of structure functions DIS

parton distribution functions (PDFs)

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Defined as

where

Renormalised PDFs

Satisfy DGLAP evolution

PDFs (GPDs)

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Mellin moments of PDFs

related to matrix elements

of local twist-two operators

Moving beyond three moments is very challenging

Cannot reconstruct PDFs from only three moments MOMENTS OF PDFs

Detmold et al., Eur. Phys. J. C 3 (2001) 1 Detmold et al., Phys. Rev. D 68 (2001) 034025 Detmold et al., Mod. Phys. Lett. A 18 (2003) 2681

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Nucleon axial charge

Controls:

nucleon-nucleon force

free neutron β-decay

early Universe composition Experimental value

cold neutron decay

MOMENTS OF PDFs:

AXIAL CHARGE

M.Constantinou, PoS(CD15) 009 2015

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Nucleon axial charge

MOMENTS OF PDFs:

AXIAL CHARGE

C.C.Chang et al (CalLat), 1710.06523 E.Berkowitz et al (CalLat), 1704.01114

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SPACELIKE DISTRIBUTIONS

Matrix elements of spacelike nonlocal operators

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● Quasi distributions

● Pseudo distributions

● Lattice “cross-sections”

SPACELIKE DISTRIBUTIONS

X. Ji, PRL 110 (2013) 262002 X. Ji, Sci.Ch. PMA 57 (2014) 1407

A.Radyushkin, PLB 767 (2017) 314 A.Radyushkin, PRD 96 (2017) 034025

Y.-Q. Ma & J.-W. Qiu, 1404.6860

Y.-Q. Ma & J.-W. Qiu, IJMP 37 (2015) 1560041 Y.-Q. Ma & J.-W. Qiu, 1709.03018

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SPACELIKE DISTRIBUTIONS

H.-W. Lin et al, PRD 91 (2015) 054510 C. Alexandrou et al., PRD 92 (2015) 014502 J.-H. Zhang et al., arXiv:1702.00008

J.-W. Chen et al., NPB 911 (2016) 246

See also:

H.-W. Lin et al (LP3), 1708.05301

C. Alexandrou et al (ETMC), NPB 923 (2017) 394

K. Orginos et al, 1706.05373

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Defined as

Recall

Related to light-front PDFs via

QUASI DISTRIBUTIONS

X. Ji, PRL 110 (2013) 262002 X. Ji, Sci.Ch. PMA 57 (2014) 1407

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GENERAL PROCEDURE

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GENERAL PROCEDURE

J.-W. Chen et al, NPB 12 (2016) 004 T. Ishikawa et al, arXiv:1609.02018 J.-W. Chen et al, 1706.01295 C. Alexandrou et al, NPB 923 (2017) 394

T. Ishikawa et al, 1707.03107 X. Ji et al, 1706.08962 J.-W. Chen et al, NPB 12 (2016) 004 T. Ishikawa et al, arXiv:1609.02018 X. Ji et al, PRD 92 (2015) 034006

C.E. Carlson, M. Freid, PRD 095 (2017) 094504 X. Xiong et al, 1705.00246 X. Ji et al, NPB 924 (2017) 366

X. Ji, PRL 110 (2013) 262002 X. Xiong et al, PRD 90 (2014) 014051 X. Ji et al, arXiv:1506.00248 H.-W. Lin et al, 1708.05301

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GENERAL PROCEDURE:

GENERAL CHALLENGES

Power-divergence must be controlled

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GENERAL PROCEDURE:

GENERAL CHALLENGES

Power-divergence must be controlled Large momentum required:

- discretised Fourier transform

- control power-suppressed corrections

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GENERAL PROCEDURE:

GENERAL CHALLENGES

Power-divergence must be controlled Large momentum required:

- discretised Fourier transform

- control power-suppressed corrections Renormalisation and continuum limit:

- perturbative truncation uncertainties - discretisation effects

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GENERAL PROCEDURE:

GENERAL CHALLENGES

Power-divergence must be controlled Large momentum required:

- discretised Fourier transform

- control power-suppressed corrections Renormalisation and continuum limit:

- perturbative truncation uncertainties - discretisation effects

Matrix elements extracted from Euclidean correlator - identical to that extracted from LSZ reduction

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GENERAL PROCEDURE:

GENERAL CHALLENGES

R. Briceno, M. Hansen & CJM, PRD 96 (2017) 014502

Matrix elements extracted from Euclidean correlator - identical to that extracted from LSZ reduction

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EUCLIDEAN CORRELATORS

Agnostic matrix elements

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Spacelike distributions assumed identical in Euclidean and Minkowski space First calculation to work strictly in Euclidean space found no IR divergence!

THE WORRY

C.E. Carlson, M. Freid, PRD 095 (2017) 094504

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Introduce a scalar, toy-model spacelike distribution

Momentum space correlation function:

perturbative QCD scalar toy model

Consider and compare:

1. LSZ reduction in Minkowski spacetime 2. Long time behaviour in Euclidean space

SCALAR TOY MODEL:

SPACELIKE DISTRIBUTION

R. Briceno, M. Hansen & CJM, PRD 96 (2017) 014502

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Spacelike distributions assumed identical in Euclidean and Minkowski space First calculation to work strictly in Euclidean space found no IR divergence!

No fundamental challenge to, or problem with, this whole approach THE WORRY

C.E. Carlson, M. Freid, PRD 095 (2017) 094504

R. Briceno, M. Hansen & CJM, PRD 96 (2017) 014502

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Power-divergence must be controlled GENERAL PROCEDURE:

GENERAL CHALLENGES

CJM & K. Orginos, JHEP 03 (2017) 116 CJM & K. Orginos, 1710.06466

CJM, 1710.04607

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SMEARING

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Deterministic evolution in new parameter - flow time - one-parameter mapping

- five-dimensional theory

Drives fields to minimise action - removes UV fluctuations Finite correlation functions remain finite

Correlation functions of “bulk” fields provide probe of underlying field theory GRADIENT FLOW

Narayanan & Neuberger, JHEP 0603 (2006) 064 Lüscher, Commun. Math. Phys. 293 (2010) 899

Lüscher & Weisz, JHEP 1102 (2011) 51 Luscher, JHEP 04 (2013) 123 Makino & Suzuki, arXiv:1410.7538

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Deterministic evolution in new parameter - flow time - one-parameter mapping

- five-dimensional theory

Drives fields to minimise action - removes UV fluctuations Finite correlation functions remain finite

Correlation functions of “bulk” fields provide probe of underlying field theory GRADIENT FLOW

CJM, PoS(Lattice2015) 052 Narayanan & Neuberger, JHEP 0603 (2006) 064

Lüscher, Commun. Math. Phys. 293 (2010) 899

Lüscher & Weisz, JHEP 1102 (2011) 51 Luscher, JHEP 04 (2013) 123 Makino & Suzuki, arXiv:1410.7538

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SMEARING

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GRADIENT FLOW

Gradient flow is a smearing (smoothing) tool that:

generates more continuum-like operators

provides a method to fix smearing length scale

Flow time serves as a nonperturbative, rotationally-invariant cutoff Matrix elements of operators at fixed flow time are finite

Fixing the flow time (physical units) allows a continuum limit In essence: exchange lattice regulator for gradient flow regulator

CJM & K. Orginos, PRD 91 (2015) 074513

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SMEARED QUASI DISTRIBUTIONS

Provides continuum limit

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Defined as

Related to light-front PDFs via

Provided

Matching kernel satisfies

SMEARED QUASI DISTRIBUTIONS

CJM & K. Orginos, JHEP 03 (2017) 116 CJM, 1710.04607

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Feynman diagrams at one loop in perturbation theory

Smeared quasi distribution Quasi distribution

MATRIX ELEMENTS IN PERTURBATION THEORY

CJM, 1710.04607

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At one loop

where

Two regimes:

1. Local vector-current limit

2. Small flow-time limit

MATRIX ELEMENTS IN PERTURBATION THEORY

CJM, 1710.04607

Hieda & Suzuki, MPLA 31 (2016) 1650214

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MATRIX ELEMENTS IN PERTURBATION THEORY

CJM, 1710.04607

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SUMMARY

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GENERAL PROCEDURE:

GENERAL CHALLENGES

Power-divergence must be controlled Large momentum required:

- discretised Fourier transform

- control power-suppressed corrections Renormalisation and continuum limit:

- perturbative truncation uncertainties - discretisation effects

Matrix elements extracted from Euclidean correlator - identical to that extracted from LSZ reduction

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PDFs FROM EUCLIDEAN SPACETIME

Quasi distributions

Most theoretical issues generally under control

SMEARED QUASI DISTRIBUTIONS Finite continuum distributions

Looking forward: study systematics THE GRADIENT FLOW

Nonperturbative, gauge-invariant regulator

Matrix elements finite at fixed flow time

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THANK YOU

cjm373@uw.edu

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Systematic uncertainties

finite lattice spacing

finite volume

unphysical pion masses

excited state contamination

Euclidean spacetime

nontrivial renormalisation

LATTICE QCD Nonperturbative gauge-invariant regulator Rigorous definition of the path integral

quarks

gluons

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SCALAR FIELD THEORY Scalar field theory

Exact solution possible with Dirichlet boundary conditions

Smearing radius

Interactions occur at zero flow time (i.e. in the original “boundary” theory):

guarantees that renormalised correlation functions remain finite.

CJM & K. Orginos, PRD 91 (2015) 074513

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QCD QCD

Exact solution no longer possible (even with Dirichlet boundary conditions)

Smearing radius

Interactions occur at non-zero flow time: generalised BRST symmetry guarantees renormalised correlation functions remain finite.

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EXPERIMENTAL EXTRACTION

From PDG 2016

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Implemented nonperturbatively via discretised diffusion equation

and

ON THE LATTICE

Lüscher & Weisz, JHEP 1102 (2011) 51 Luscher, JHEP 04 (2013) 123

lattice gauge action

covariant lattice Laplacian

Figure

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References

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