3D Nucleon Tomography from LQCD INTRODUCTION Goal: Compute - - PowerPoint PPT Presentation

3D Nucleon Tomography from LQCD

Kostas Orginos (W&M/JLab) March 15-17, 2017 (JLab)

INTRODUCTION

Goal: Compute properties of hadrons from first principles Parton distribution functions (PDFs) and Generalized Parton distributions (GPDs) Transverse Momentum Dependent densities (TMDs) Form Factos … Lattice QCD is a first principles method For many years calculations focused on Mellin moments Can be obtained from local matrix elements of the proton in Euclidean space Breaking of rotational symmetry —> power divergences

• nly first few moments can be computed

Recently direct calculations of PDFs in Lattice QCD are proposed First lattice Calculations already available

• X. Ji, Phys.Rev.Lett. 110, (2013)

Y.-Q. Ma J.-W. Qiu (2014) 1404.6860 
 H.-W. Lin, J.-W. Chen, S. D. Cohen, and X. Ji, Phys.Rev. D91, 054510 (2015)

• C. Alexandrou, et al, Phys. Rev. D92, 014502 (2015)

PDFS: DEFINITION

· f(0)(ξ) = Z 1

1

dω 4π eiξP +ω− ⌧ P

• T ψ(0, ω, 0T)W(ω, 0)γ+ λa

2 ψ(0)

• P
• C

.

W(ω, 0) = P exp " ig0 Z ω− dyA+

α (0, y, 0T)Tα

#

Light-cone PDFs:

hP 0|Pi = (2π)32P +δ

• P + P 0 +

δ(2) PT P0

T

• a(n)

= Z 1 dξ ξn1 h f(0)(ξ) + (1)nf

(0)(ξ)

i = Z 1

1

dξ ξn1f(ξ),

Moments:

D P|O{µ1...µn} |P E = 2a(n) (P µ1 · · · P µn traces) .

Local matrix elements:

O{µ1···µn} = in−1ψ(0)γ{µ1Dµ2 · · · Dµn} λa 2 ψ(0) − traces

GPDS: DEFINITION

W(ω, 0) = P exp " ig0 Z ω− dyA+

α (0, y, 0T)Tα

#

GPDs:

hP 0|Pi = (2π)32P +δ

• P + P 0 +

δ(2) PT P0

T

• O{µ1···µn}

= in−1ψ(0)γ{µ1Dµ2 · · · Dµn} λa 2 ψ(0) − traces

¯ u(P 0) ✓ γ+H(x, ξ, t) + iσ+k∆k 2m E(x, ξ, t) ◆ = Z 1

1

dω 4π eiξP +ω− ⌧ P 0

• T ψ(0, ω, 0T)W(ω, 0)γ+ λa

2 ψ(0)

• P
• C

1

−1 dxxn−1

• H(x,ξ,t)

E(x,ξ,t)

• =

[(n−1)/2]

∑

k=0

(2ξ)2k

• An,2k(t)

Bn,2k(t)

• ±δn,even(2ξ)nCn(t).

generalized form factors A B and C are defined from the coefficients of this e

Moments:

Matrix elements of twist-2 operators

sfer ∆ = P′ −P as the virtuality t

lity t = ∆2. T e momentum

y x z

p xp

Q z 1 ~

⊥

δ

⊥

r

⊥

r

) , (

⊥

r x f

1

y x z

⊥

r

p

f x ( )

1

xp

Q z 1 ~

⊥

δ

x

y x z

⊥

r

) ( ⊥ r ρ

⊥

r

p

x

Form Factors Parton Distribution functions Generalized Parton Distribution functions

• X. Ji, D. Muller, A. Radyushkin (1994-1997)

Mellin moments are local matrix elements Can be evaluated in Euclidean space Lattice QCD calculations are possible Challenges: Renormalization and power divergent mixing Lattice breaks O(4) symmetry Only few moments can be computed

Z =

• DqD¯

qDAµ eS[¯

q,q,Aµ]

⇧O⌃ = 1 Z

• DqD¯

qDAµ O(¯ q, q, Aµ) eS[¯

q,q,Aµ]

In continuous Euclidian space:

U q

µ µ ν µν

P

Lattice regulator:

Uµ(x) = eiaAµ(x+ˆ

µ 2)

Gauge sector: Fermion sector:

LATTICE QCD

Ψ is now a vector whose components leave on the sites of the lattice

Sf = ¯ ΨDΨ

D is the Dirac matrix which is large and sparse

MONTE CARLO INTEGRATION

⟨O⟩ = 1 Z

µ,x

dUµ(x) O[U, D(U)−1] det

• D(U)†D(U)

nf /2 e−Sg(U)

O⇥ = 1 N

N

• i=1

O(Ui)

Monte Carlo Evaluation Statistical error

1 √ N

0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15 0.2 0.25 0.3

GE(Q2) Q2 [GeV2]

PACS Nf = 2 + 1 ETMC Nf = 2 SUMM ETMC Nf = 2, ts = 1.7 fm PNDME Nf = 2 + 1 + 1 Alberico et al.

1.5 2 2.5 3 3.5 4 4.5 5 0.05 0.1 0.15 0.2 0.25 0.3

GM(Q2) Q2 [GeV2]

PACS Nf = 2 + 1 ETMC Nf = 2 SUMM ETMC Nf = 2, ts = 1.7 fm PNDME Nf = 2 + 1 + 1 Alberico et al.

Form Factors PACS: Nf=2+1 mπ = 145 MeV 8.1 fm box ETMC: Nf=2+1 mπ = 131 MeV 4.5 fm box PNDME: mixed action mπ = 138 MeV 5.6 fm box

0.0 0.2 0.4 0.6 0.8 1.0 1.2 Q2 (GeV2) −0.004 −0.002 0.000 0.002 0.004 0.006 0.008 0.010 0.012 GE

strange light disconnected

0.0 0.2 0.4 0.6 0.8 1.0 1.2 Q2 (GeV2) −0.07 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0.00 GM

strange light disconnected

• J. Green et al. Phys.Rev. D92 (2015) no.3, 031501

Strange quark contribution to nucleon form factors

dynamical 2 + 1 flavors of Clover fermions 323 x 96 lattice of dimensions (3.6 fm)3 x (10.9 fm) a=0.115fm, pion mass 317 MeV

G(Q2) =

kmax

X

k

akzk, z = p tcut + Q2 − √tcut p tcut + Q2 + √tcut ,

z-expansion fit:

| |

Q 2 z

• R. J. Hill and G. Paz, Phys. Rev. D 84 (2011) 073006

0.0 0.2 0.4 0.6 0.8 1.0 1.2 Q2 (GeV2) −0.05 0.00 0.05 0.10 0.15 Gs

E +ηGs M

G0 HAPPEX A4 lattice

η = AQ2, A = 0.94 Comparison with experiments Gs

E + ηGs M

Experiment: forward-angle parity-violating elastic e-p scattering Prediction: very hard for such experiments to measure a non-zero result

Axial Charge

From M. Constantinou: arXiv:1701.02855

Moments of GPDs

LHPC: arXiv:0705.4295

Phys.Rev.D77:094502,2008

0.8 0.9 1 1.1 1.2

• ฀
• Moments of PDFs (Lattice vs Experiment)

LHPC: 2007

Can Lattice QCD go beyond moments? Lattice QCD can only compute time local matrix elements Euclidean space

QPDFS: MAIN IDEA

• X. Ji, Phys.Rev.Lett. 110, (2013)

• 2
• 1

1 2

• 2
• 1

1 2

lim

Pz→∞ q(0) (x, Pz) = f(x)

z t

Euclidean space time local matrix element is equal to the same matrix element in Minkowski space

Related ideas see: A more general point of view:

Y.-Q. Ma J.-W. Qiu (2014) 1404.6860

e σ(x, e µ2, Pz) = X

α={q,q,g}

Hα ✓ x, e µ Pz , e µ µ ◆ ⊗ fα(x, µ2) + O Λ2

QCD

e µ2 !

Minkowski space factorization: computable in perturbation theory

Detmold and Lin Phys.Rev.D73:014501,2006 K-F Liu Phys.Rev. D62 (2000) 074501

Practical calculations require a regulator (Lattice) Continuum limit has to be taken renormalization Momentum has to be large compared to hadronic scales to suppress higher twist effects Practical issue with LQCD calculations at large momentum … signal to noise ratio

q (x, Pz) = Z 1

−1

dξ ξ e Z ✓x ξ , µ Pz ◆ f(ξ, µ) + O(ΛQCD/Pz, MN/Pz)

• X. Xiong, X. Ji, J. H. Zhang, Y. Zhao, Phys. Rev. D 90, no. 1, 014051 (2014)
• T. Ishikawa et al. arXiv:1609.02018 (2016)

The matching kernel can be computed in perturbation theory

QUASI-PDFS

h(s) ✓ z √τ , √τPz, √τΛQCD, √τMN ◆ = 1 2Pz ⌧ Pz

• χ(z; τ)W(0, z; τ)γz

λa 2 χ(0; τ)

• Pz
• C

(2.

τ is the a regulator scale χ quark field W is the regulated gauge link

q (s) ξ, √τPz, √τΛQCD, √τMN

• =

Z ∞

−∞

dz 2πeiξzPzPz h(s)(√τz, √τPz, √τΛQCD, √τMN), (2.12)

At fixed flow time the quasi-PDF is finite in the continuum limit

Ji 2013

✓ i Pz ∂ ∂z ◆n−1 h(s) ✓ z √τ , √τPz, √τΛQCD, √τMN ◆ = Z ∞

−∞

dξ ξn−1e−iξzPzq (s) ξ, √τPz, √τΛQCD, √τMN

• b(s)

n

✓√τPz, ΛQCD Pz , MN Pz ◆ = Z ∞

−∞

dξ ξn−1q (s) ξ, √τPz, √τΛQCD, √τMN

• Using the previous definitions we have

By introducing the moments

b(s)

n

✓pτPz, ΛQCD Pz , MN Pz ◆ = c(s)

n (pτPz)

2P n

z

⌧ Pz

• 

χ(z; τ)γz(i

• Dz)(n−1) λa

2 χ(0; τ)

• z=0
• Pz
• C

. (3.7)

after removing MN/Pz effects

b(s)

n

pτPz, pτΛQCD

• = c(s)(pτPz)b(s,twist−2)

n

pτΛQCD

• + O

Λ2

QCD

P 2

z

!

Taking the limit of z going to 0 we obtain: i.e. the moments of the quasi-PDF are related to local matrix elements of the smeared fields These matrix elements are not twist-2. Higher twist effects enter as corrections that scale as powers of

[ H.-W. Lin, et. al Phys.Rev. D91, 054510 (2015)]

C(0)

n (pτµ, pτPz) =

Z ∞

−∞

dx xn−1 e Z(x, pτµ, pτPz)

q (s) x, pτΛQCD, pτPz

• =

Z 1

−1

dξ ξ e Z ✓x ξ , pτµ, pτPz ◆ f(ξ, µ) + O(pτΛQCD)

Introducing a kernel function such that: We can undo the Melin transform: Therefore regulated quasi-PDFs are related to PDFs if

ΛQCD, MN ⌧ Pz ⌧ τ −1/2,

Monahan and KO: arXiv:1612.01584

PROCEDURE OUTLINE

Compute equal time matrix elements in Euclidean space using Lattice QCD at sufficiently large momentum in

• rder to suppress higher twist effects

Take the continuum limit (renormalization) Equal time: Minkowski — Euclidean equivalence Perform the matching Kernel calculation in the continuum

• 1

1 2 3 0.2 0.4 0.6 0.8 1.0

x u - d Δu - Δd

• 1.0
• 0.5

0.5 1.0 1.5 0.5 1.0 1.5

x (u - d) / gV pz 2 3 Extrap.

Plots taken from: Chen et al. arXiv:1603.06664

Convergence with momentum extrapolation Including the 1-loop matching kernel

First Lattice results (Chen et. al)

Similar results have been achieved by Alexandrou et. al (ETMC)

Along these lines one can compute: TMDs (see Engelhardt et. al.) GPDs Distribution amplitudes Gluonic PDFs …..

CONCLUSIONS

Lattice QCD calculations have made a lot of progress and in some cases precision results are being obtained Physical quark masses, large volumes, large scale calculations Quasi-PDFs provide a novel way to study hadron structure in Lattice QCD Lattice calculations from several groups are on the way Several ideas for dealing with the continuum limit are now developing Promising new ideas: Stay tuned!

NUCLEON FORM FACTOR

Connected Disconnected Strange quark : disconnected only