Fu Func nctions tions on on th the Lat attic tice Huey-Wen - - PowerPoint PPT Presentation

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Fu Func nctions tions on on th the Lat attic tice Huey-Wen - - PowerPoint PPT Presentation

Nov 22, 2022 244 likes •565 views

Pa Part rton on Di Dist stribution ribution Fu Func nctions tions on on th the Lat attic tice Huey-Wen Lin University of Washington Huey-Wen Lin Lattice Workshop @ NTU 1 Part rton on Dis istri ributio tion n Functio

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SLIDE 1

Pa Part rton

  • n Di

Dist stribution ribution Fu Func nctions tions

  • n
  • n th

the Lat attic tice

Huey-Wen Lin University of Washington

Huey-Wen Lin — Lattice Workshop @ NTU 1

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SLIDE 2

Part rton

  • n Dis

istri ributio tion n Functio ions ns

§ Important fundamental QCD property

 Exploration of the valence and sea-quark content of the nucleon

§ Important for BSM searches

 Provides SM cross-section prediction for LHC new-physics search  IceCube PeV neutrinos can be explained by PDF uncertainties  Proton weak charge (medium-modification effects)

§ Structure function/distribution functions

 Deep inelastic scattering (1960s)

Huey-Wen Lin — Lattice Workshop @ NTU 2

𝑋

𝜈𝜉 = 𝑗 𝑒4𝑦 𝑓𝑗𝑟𝑦 𝑂 𝑈 𝐾𝜈 𝑦 , 𝐾𝜉 0

𝑂 𝜏 ∼ 𝑀𝜈𝜉𝑋

𝜈𝜉

𝑦 = 𝑅2/2𝑟 ⋅ 𝑄

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SLIDE 3

Part rton

  • n Dis

istri ributio tion n Functio ions ns

§ Still limited knowledge

 Many on-going/planned experiments (EIC, LHeC, … facilities)

§ Structure function/distribution functions

 Deep inelastic scattering (1960s)

Huey-Wen Lin — Lattice Workshop @ NTU 3

𝑋

𝜈𝜉 = 𝑗 𝑒4𝑦 𝑓𝑗𝑟𝑦 𝑂 𝑈 𝐾𝜈 𝑦 , 𝐾𝜉 0

𝑂 𝜏 ∼ 𝑀𝜈𝜉𝑋

𝜈𝜉

𝑦 = 𝑅2/2𝑟 ⋅ 𝑄

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SLIDE 4

§ Lattice QCD is an ideal theoretical tool for investigating strong-coupling regime of quantum field theories

 Ideal tool for studying nonperturbative hadron structure, but…

Part rton

  • n Dis

istri ributio tion n Functio ions ns

§ Structure function/distribution functions

 Deep inelastic scattering (1960s)

Huey-Wen Lin — Lattice Workshop @ NTU 4

§ Lattice calculations rely on operator product expansion

For example, the unpolarized structure  c1, c2 are Wilson coefficients

2 𝑒𝑦 𝑦𝑜−1𝐺

1 𝑦, 𝑅2 = 𝑟=𝑣,𝑒

𝑑1,𝑜

𝑟

𝜈2 𝑅2 , 𝑕 𝜈 𝑦𝑜 𝑟 𝑒𝑦 𝑦𝑜−2𝐺

2 𝑦, 𝑅2 = 𝑟=𝑣,𝑒

𝑑2,𝑜

𝑟

𝜈2 𝑅2 , 𝑕 𝜈 𝑦𝑜 𝑟

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SLIDE 5

§ Leading moment x, hypercubic decomposition

 4141 = 11  31  61  63: O44−(O11+O22+O33)/3 O14+O41, (requires p≠0)  Both operators go to same continuum limit

§ No mixing with operators of same or lower dimension § To improve to O(a)

 Consider all irrelevant operators of same symmetry:

§ Higher moments x2

 41: O111 mixes with q

―γ1q with coefficient ~ 1/a2

 42: O{123} requires all momentum components to be nonzero  81: O{441}−(O{221}+O{331})/2 mixes under renormalization

§ For higher spin, all ops mix with lower-dimension ops

xn Moments

Huey-Wen Lin — Lattice Workshop @ NTU 5

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SLIDE 6

§ Leading moment x, hypercubic decomposition

 4141 = 11  31  61  63: O44−(O11+O22+O33)/3 O14+O41, (requires p≠0)  Both operators go to same continuum limit

§ No mixing with operators of same or lower dimension § To improve to O(a)

 Consider all irrelevant operators of same symmetry:

§ Higher moments x2

 41: O111 mixes with q

―γ1q with coefficient ~ 1/a2

 42: O{123} requires all momentum components to be nonzero  81: O{441}−(O{221}+O{331})/2 mixes under renormalization

§ For higher spin, all ops mix with lower-dimension ops

xn Moments

Huey-Wen Lin — Lattice Workshop @ NTU 6

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SLIDE 7

§ For higher spin, all ops mix with lower-dimension ops

 Tricks: subtraction to remove divergent terms, heavy fields, four-point functions… None is practical enough

§ Relative error grows in higher moments

 Calculation would be costly

xn Moments

x2q x3q

Dolgov et al., PRD66, 034506 (2002) Göckeler et al. PRD71, 114511 (2005) LHPC (SCRI, SESAM): 2f, Wilson and clover QCDSF: 0f

Huey-Wen Lin — Lattice Workshop @ NTU 7

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SLIDE 8

Limited Access

§ What can we learn about the x-distribution?

 Make an ansätz of some smooth form for the distribution and fix the parameters by matching to the lattice moments

  • W. Detmold et al, Eur.Phys.J.direct C3 (2001) 1–15

Cannot separate valence- quark contribution from sea

New idea needed to access the sea!

Huey-Wen Lin — Lattice Workshop @ NTU 8

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SLIDE 9

The Idea

§ Approaching lightcone with large P

 Just another limit to take, like taking a→0

Xiangdong Ji, Phys. Rev. Lett. 111, 039103 (2013)

Huey-Wen Lin — Lattice Workshop @ NTU 9

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SLIDE 10

§ Lightcone quark distribution

The Idea

Nucleon momentum P

µ

Lightcone coordinate ξ±=(t±z)/√

―

2

Renormalization scale µ Gluon potential A+

§ Approaching lightcone with large P

 Just another limit to take, like taking a→0

Xiangdong Ji, Phys. Rev. Lett. 111, 039103 (2013)

Huey-Wen Lin — Lattice Workshop @ NTU 10

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SLIDE 11

§ Finite-momentum quark distribution

The Idea

Lattice z coordinate x=kz/P

z

Product of lattice gauge links  In P

z limit, parton distribution is

recovered  For finite P

z, corrections are needed

Nucleon momentum P

µ={P 0,0,0,P z}

Xiangdong Ji, Phys. Rev. Lett. 111, 039103 (2013)

Huey-Wen Lin — Lattice Workshop @ NTU 11

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SLIDE 12

§ Exploratory study

 Nf = 2+1+1 clover/HISQ lattices (MILC) Mπ ≈ 310 MeV, a ≈ 0.12 fm (L ≈ 2.88 fm)  Isovector only (“disconnected” suppressed)

gives us flavor asymmetry between up and down quark

 2 source-sink separation (tsep≈0.96 and 1.2 fm) used

Some Lattice Details

§ Properties known on these lattices

 Lattice ZΓ for bilinear operator ~ 1 (with HYP-smearing)  Mπ L ≈ 4.6 large enough to avoid finite-volume effects

tsep

§ Feasible with today’s computational resources!

 O(hour) rewriting three-point insertion code (Chroma)  8/16 nodes on UW Hyak cluster Hyak @ UW

Huey-Wen Lin — Lattice Workshop @ NTU 12

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SLIDE 13

§ Exploratory study

Quark Distribution

P

z {1, 2, 3} 2π⁄L

 Lattice momenta discretized by finite size of volume  How many links are needed?

Huey-Wen Lin — Lattice Workshop @ NTU 13

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SLIDE 14

§ Exploratory study

Quark Distribution

Uncorrected bare lattice results

Preliminary

x=kz/P

z

P

z {1, 2, 3} 2π⁄L

Huey-Wen Lin — Lattice Workshop @ NTU 14

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SLIDE 15

§ Exploratory study

Quark Distribution

Preliminary

Distribution gets sharper as P

zincreases

Artifacts due to finite P

z

  • n the lattice

Improvement?

Work out leading-P

z

corrections

P

z {1, 2, 3} 2π⁄L

Huey-Wen Lin — Lattice Workshop @ NTU 15

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SLIDE 16

§ Back to the continuum

Quark Distribution

What we want What we calculate

  • n the lattice

P

z  {1, 2, 3} 2π⁄L

Dominant correction (for nucleon); known scaling form

J.-W. Chen

Not included yet; O(20%) systematic

J.-H. Zhang, Y . Zhao, J.-W. Chen et

  • al. (in preparation)

Xiangdong Ji, Phys. Rev. Lett. 111, 039103 (2013)

Huey-Wen Lin — Lattice Workshop @ NTU 16

Smaller P

z correction but

complicated twist-4

  • perator

(extrapolate it away)

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SLIDE 17

§ Back to the continuum

Quark Distribution

What we want What we calculate

  • n the lattice

P

z  {1, 2, 3} 2π⁄L

Smaller P

z correction but

complicated twist-4

  • perator

(extrapolate it away) Dominant correction (for nucleon); known scaling form

J.-W. Chen

Not included yet; O(20%) systematic

J.-H. Zhang, Y . Zhao, J.-W. Chen et

  • al. (in preparation)

Xiangdong Ji, Phys. Rev. Lett. 111, 039103 (2013)

Huey-Wen Lin — Lattice Workshop @ NTU 17

§ Changes in x, and q(x)

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SLIDE 18

Quark Distribution

§ Exploratory study

 Take ratios (partially cancel statistical and systematic errors)

Removing O(MN

2/4P z 2) errors

No significant finite-momentum effect seen for P

z>1

§ Renormalization needed

Preliminary

Huey-Wen Lin — Lattice Workshop @ NTU 18

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SLIDE 19

Quark Distribution

§ Compare with experiments

  • K. Ackersta et al. (HERMES Collaboration),

Phys.Rev.Lett. 81, 5519 (1998)

Huey-Wen Lin — Lattice Workshop @ NTU 19

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SLIDE 20

Quark Distribution

§ Compare with experiments Compared with E866

Too good to be true?

Lost resolution in small-x region

Future improvement to have larger lattice volume

  • R. Towell et al. (E866/NuSea), Phys.Rev. D64, 052002 (2001)

Huey-Wen Lin — Lattice Workshop @ NTU 20

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SLIDE 21

Helicity Distribution

§ Exploratory study

Preliminary

Huey-Wen Lin — Lattice Workshop @ NTU 21

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SLIDE 22

Helicity Distribution

§ Exploratory study Corrected to O(Pz

−2)

Gray band shows extrapolation of Pz

−4 terms

Large O(Pz

−4) seen but

well fit by extrapolation

Preliminary

Huey-Wen Lin — Lattice Workshop @ NTU 22

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SLIDE 23

Helicity Distribution

§ Experimental comparison

  • A. Airapetian et al. (HERMES),

Phys.Rev. D71, 012003 (2005)

§ Model: e.g. chiral quark-soliton model

  • B. Dressler et al,hep-ph/9809487
  • D. De Florian et al.,

PRL 101 (2008) 072001

Huey-Wen Lin — Lattice Workshop @ NTU 23

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SLIDE 24

Helicity Distribution

§ Experimental comparison

  • A. Airapetian et al. (HERMES),

Phys.Rev. D71, 012003 (2005)

§ Model: e.g. chiral quark-soliton model

  • B. Dressler et al,hep-ph/9809487
  • D. De Florian et al.,

PRL 101 (2008) 072001

Huey-Wen Lin — Lattice Workshop @ NTU 24

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SLIDE 25

Transversity Distribution

§ Exploratory study § Renormalization needed

Preliminary

Uncorrected bare lattice results

Huey-Wen Lin — Lattice Workshop @ NTU 25

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SLIDE 26

Transversity Distribution

§ Exploratory study Removing O(MN

2/4P z 2) errors

§ Renormalization needed

Preliminary

Huey-Wen Lin — Lattice Workshop @ NTU 26

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SLIDE 27

Transversity Distribution

§ Exploratory study

 We found δu

― < δd ―with

large sea asymmetry  Chiral quark-soliton model

  • B. Dressler et al.,

hep-ph/9809487 P . Schweitzer et al. PRD 64, 034013 (2001)

CQS model

Preliminary

Huey-Wen Lin — Lattice Workshop @ NTU 27

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SLIDE 28

Pion Distribution Amplitude

§ Exploratory study

Preliminary

P

z {1, 2, 3} 2π⁄L

Leading mass correction applied Dominated by O(Λ2

QCD/P z 2) errors

Huey-Wen Lin — Lattice Workshop @ NTU 28

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SLIDE 29

§ Overcoming longstanding obstacle to x-distribution

 New idea by Ji for studying full x dependence of PDFs  Promising results on unpolarized and polarized sea asymmetry compared with experiments, even at non-physical pion mass

§ Caveats

 Not a precision calculation yet  Need to complete the other pz corrections (on-going; possibly done in a couple weeks)  Systematics due to large momenta (some ideas to improve it)

§ Need improvement for large-momentum sources

 Better overlapping boosted hadron smearing (asymmetric source)  Applications: large-q form factors, hadronic and flavor physics, …

Exciting time for hadron structure on the lattice

Huey-Wen Lin — Lattice Workshop @ NTU 29

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SLIDE 30

§ Overcoming longstanding obstacle to x-distribution

 New idea by Ji for studying full x dependence of PDFs  Promising results on unpolarized and polarized sea asymmetry compared with experiments, even at non-physical pion mass

§ Caveats

 Not a precision calculation yet  Need to complete the other Pz corrections (ongoing; possibly done in a couple weeks)  Systematics due to large momenta (some ideas to improve it)

§ Need improvement for large-momentum sources

 Better overlap with boosted hadron (asymmetric smearing)  Applications: large-q form factors, hadronic and flavor physics, …

Summary and Outlook

Exciting time for hadron structure on the lattice

Huey-Wen Lin — Lattice Workshop @ NTU 30

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SLIDE 31

§ We hope this exploratory study motivates others to give Ji’s method a try

 More details in the upcoming paper(s), such as P

z corrections from various sources

Xiangdong Ji, J.-H. Zhang, Y . Zhao, J.-W. Chen, S. D. Cohen, ....

§ Hope to see many calculations and more ideas

 Like gS,T calculations with PNDME  Many more quantities to study: strange/charm/beauty sea distributions, gluons, TMD…

Summary and Outlook

Exciting time for hadron structure on the lattice

§ “Working” workshop in Shanghai Nov. 6–8, 2013 Frontiers of Parton Distribution Functions on the Lattice

http://www.phys.washington.edu/users/hwlin/LPDF2013

Huey-Wen Lin — Lattice Workshop @ NTU 31