Disconnected 3-Point Functions Using Wilson Loops on the Lattice

Jacob Peyton

New Mexico State University

for 2024 Joint Photonuclear Reactions and Frontiers & Careers Workshop

Introduction: TMD PDFs

• Transverse Momentum Dependant Parton Distribution Functions
• Commonly Evaluated From SIDIS and Drell-Yan Processes
• Simulated by inserting Staple-Shaped Wilson Line between Hadron Propagators
• Width of Staple determines Transverse Displacement

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New Mexico State University

Jacob Peyton

Introduction: Boer et al

• Other Operators can be inserted that produce TMDs!
• These can potentially simulate other processes than SIDIS and Drell-Yan

Operator analysis of pT -widths of TMDs

Boer et al, 2015

arXiv 1503.03760v2

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New Mexico State University

Jacob Peyton

Introduction: Boer et al

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New Mexico State University

Jacob Peyton

Intro: The Disconnected 3pt Function

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• Proof of Concept of new Wilson Loop/Line shapes
• Simplest to Calculate
• Contribution to fully 3-Point Function (but least interesting)
• Dipole Approximation to Distribution Functions at small Longitudinal Momentum Fraction x

⟨C(t_s)U(t_i)⟩

⟨C(t_s)⟩⟨U(t_i)⟩

New Mexico State University

Jacob Peyton

Introduction: The Lattice

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c. 2004, Derek Leinweber

c. Forbes (r. 2024)

New Mexico State University

Jacob Peyton

Introduction: The Lattice

Creating a QCD Lattice in 3 Steps:

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New Mexico State University

Jacob Peyton

Introduction: The Lattice

Creating a QCD Lattice in 3 Steps:

1. Use a Wick Rotation to go to Euclidean Time.

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New Mexico State University

Jacob Peyton

Introduction: The Lattice

Creating a QCD Lattice in 3 Steps:

• Use a Wick Rotation to go to Euclidean Time.
• Implement Periodic Boundary Conditions to make Spacetime Finite but Continuous.

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c. Wikipedia (r. 2024)

New Mexico State University

Jacob Peyton

Introduction: The Lattice

Creating a QCD Lattice in 3 Steps:

• Use a Wick Rotation to go to Euclidean Time.
• Implement Periodic Boundary Conditions to make Spacetime Finite but Continuous.
• Discretize Spacetime.

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New Mexico State University

Jacob Peyton

Introduction: The Lattice

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Gluon Fields

Quark Fields

New Mexico State University

Jacob Peyton

Introduction: The Lattice

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Gluon Fields:

New Mexico State University

Jacob Peyton

Introduction: The Lattice

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Gluon Fields:

New Mexico State University

Jacob Peyton

Introduction: The Lattice

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Gluon Fields:

New Mexico State University

Jacob Peyton

Introduction: The Lattice

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Gluon Fields:

New Mexico State University

Jacob Peyton

Introduction: The Lattice

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• Size: 32³ × 96
• 0.114 Fermi Lattice Spacing
• 3.65 Fermi on a Side
• Pion Mass: 317 MeV
• Temperature: 55.5 KeV

New Mexico State University

Jacob Peyton

Methods: Wilson Loops

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Gluon Fields:

New Mexico State University

Jacob Peyton

Methods: Wilson Loops

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Gluon Fields:

New Mexico State University

Jacob Peyton

Methods: 2-Point Functions

​

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Quark Fields in 2pt Functions:

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Vary based on the simulation

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Quarks created at one point,

Annihilated at another.

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Feynman Path Integral

All Paths Used

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Proton’s Up Quarks are indistinguishable.

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⟨C(t_s)U(t_i)⟩

⟨C(t_s)⟩⟨U(t_i)⟩

New Mexico State University

Jacob Peyton

Methods: 2-Point Functions

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Start with quark propagators:

u d

• Propagate quarks from source to sink.
• Every site on the lattice is used as a sink.
• This includes sinks at the same time as the source.
• Sum over every possible and impossible path between source and sink.
• This includes time reversed paths and looping through whole lattice.

New Mexico State University

Jacob Peyton

Methods: 2-Point Functions

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For the pions:

u d

C₁(x,t) = Tr[ɣ⁵ud^†ɣ⁵]

C₂(x,t) = Tr[ɣ⁵uɣ³d^†ɣ⁵]

New Mexico State University

Jacob Peyton

Methods: 2-Point Functions

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For the pions:

u d

C₁(x,t) = Tr[ɣ⁵ud^†ɣ⁵]

C₂(x,t) = Tr[ɣ⁵uɣ³d^†ɣ⁵]

For the proton:

q₁ = (1/2)(1-iɣ⁰ɣ¹)(1+ɣ³)u

q₃ = (1/2)(ɣ⁰ɣ²)(1+ɣ³)u

q₂ = d(1/2)(ɣ⁰ɣ²)(1+ɣ³)

[q_D]_ɣɣ’^cc’ = 𝜖^a’b’c’𝜖^abc[q₂]_⍴ɣ^aa’[q₃]_⍴ɣ’^bb’

C(x,t) = Tr[Tr_spin(q₁)·Tr_spin(q_D)]+Tr[q₁·q_D]

New Mexico State University

Jacob Peyton

Methods: 2-Point Functions

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For the pions:

u d

C₁(x,t) = Tr[ɣ⁵ud^†ɣ⁵]

C₂(x,t) = Tr[ɣ⁵uɣ³d^†ɣ⁵]

For the anti-proton:

q₁ = (1/2)(1-iɣ⁰ɣ¹)(1-ɣ³)u

q₃ = (1/2)(ɣ⁰ɣ²)(1-ɣ³)u

q₂ = d(1/2)(ɣ⁰ɣ²)(1-ɣ³)

[q_D]_ɣɣ’^cc’ = 𝜖^a’b’c’𝜖^abc[q₂]_⍴ɣ^aa’[q₃]_⍴ɣ’^bb’

C(x,t) = Tr[Tr_spin(q₁)·Tr_spin(q_D)]+Tr[q₁·q_D]

New Mexico State University

Jacob Peyton

Methods: 2-Point Functions

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New Mexico State University

Jacob Peyton

Methods: Data Analysis

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New Mexico State University

Jacob Peyton

Methods: Data Analysis

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1. Average over 12 2pt Sources for all 3 Particles
2. Take 12 Portions of Wilson Loop Data based on 2-Point Function Source Times, and Average
3. Combine 2pt Source Data and Wilson Loop Portions for all 3 Particles and all 12 Sources per Configuration and then Average
4. Perform Jackknife Analysis on Wilson Loops over 968 Configurations
5. Perform Jackknife Analysis on 2pt Functions over 968 Configurations
6. Perform Jackknife Analysis on Combined Wilson Loops and 2pt Functions over 968 Configurations
7. Calculate Correlation Fraction
8. Perform Jackknife Analysis on Correlation Fraction over 968 Configurations

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⟨C(t_s)U(t_i)⟩

⟨C(t_s)⟩⟨U(t_i)⟩

New Mexico State University

Jacob Peyton

Results: Wilson Loops

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New Mexico State University

Jacob Peyton

Results: Wilson Loops

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New Mexico State University

Jacob Peyton

Results: 2-Point Functions

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Jacob Peyton

Results: Correlator > 1 (Proton)

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Jacob Peyton

Results: Correlator > 1 (Pion 1)

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Jacob Peyton

Results: Correlator > 1 (Pion 2)

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New Mexico State University

Jacob Peyton

Conclusion: We’ve Hit Something

• Boer et al proposed these new Wilson Line/Loop insertions
• Even when fully disconnected there is still a signal up to widths of 3 lattice spacings (0.342 fm)
• Signal is much stronger with pions than with protons
• Currently it’s not known how to extract p_T widths
• Nor how to relate new insertions to experiments

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New Mexico State University

Jacob Peyton

Conclusion: Future Work

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Quark Fields in 3pt Functions:

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One Quark’s path is ‘broken’ half way through.

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Quark is annihilated, operator inserted.

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Operator exists in a single time slice.

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The quark is recreated at the same time but displaced.

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New Mexico State University

Jacob Peyton

Conclusion: Future Work

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✔

One Down,

Three to Go!

New Mexico State University

Jacob Peyton

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Thank You!

New Mexico State University

Jacob Peyton

Bonus: Small-x Approximation

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Fourier

Conjugate

Relation

1/Px ∝

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Parton distribution function for quarks in an s-channel approach

Hautmann & Soper

arXiv 0702077v1

New Mexico State University

Jacob Peyton

Correlator > 1 (Antiproton)

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New Mexico State University

Jacob Peyton

Lattice QCD Crash Course

Creating a QCD Lattice in 3 Steps:

• Use a Wick Rotation to go to Euclidean Time.
• Implement Periodic Boundary Conditions to make Spacetime Finite but Continuous.
• Discretize Spacetime.

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New Mexico State University

Jacob Peyton

Lattice QCD Crash Course

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Gluon Fields:

New Mexico State University

Jacob Peyton

Lattice QCD Crash Course

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Gluon Fields:

New Mexico State University

Jacob Peyton