Directly Calculating the Glue Component of the Nucleon in Lattice - - PowerPoint PPT Presentation

Directly Calculating the Glue Component of the Nucleon in Lattice QCD

CHEP 2019 Tomas L. Howson,

– QCDSF-UKQCD–CSSM Collaboration – Adelaide – Edinburgh – Liverpool – RIKEN (Kobe) – Leipzig – FZ (Jülich) – DESY (Hamburg)

November 5, 2019

Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 1 / 19

Hadron Composition

What’s in a Proton? It’s well understood that hadrons are composed of smaller constituents; quarks and gluons However, how a hadron’s properties are distributed amongst the components is still foggy E.g. how the spins of the various quarks and gluons sum to the 1/2 of a baryon is still an open area of investigation Today we are interested in the momentum distribution of nucleons

https://www.quantamagazine.org/physicists-finally-nail-the-protons-size-and-hope-dies-20190911/ Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 2 / 19

Momentum Fractions

Momentum Fractions The quantities we are interested in are the momentum fractions of a hadron. Denoted xp, give the fraction of the hadrons momentum contained in each constituent type p xg for gluons xq =

f xf for quarks, f denoting each flavour of quark

These quantities give good signposts of the composition of hadrons ("How much

• f the proton is quark?")

Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 3 / 19

Momentum Fractions

Sum Rule As these momentum fractions are fractions, we expect xg + xq = 1 Call this statement the sum rule Provides a good test to check validity of a method

Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 4 / 19

The Lattice

A Brief Overview of Lattice QCD Provides a method of calculating quantities otherwise impossible Same purpose as perturbation theory Perturbation theory breaks down for strongly coupled forces Lattice can work where perturbation theory fails (e.g. QCD at the scale of hadrons)

Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 5 / 19

The Lattice

How it Works Turns the infinities finite Full Theory O = N

• DφOe−S[φ]

Infinitely many field configurations Infinitely large space Infinitely dense spacial points

Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 6 / 19

The Lattice

How it Works Turns the infinities finite Full Theory O = N

• DφOe−S[φ]

Infinitely many field configurations Infinitely large space Infinitely dense spacial points → Lattice O ≈ 1 N

• i

O[φi], P(φi) = e−S[φi]. Finite sample of field configurations Finitely bound region of space Finitely discretised points within

Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 6 / 19

The Lattice

Further Approximations To make calculations further practical, we make two further approximations: – Quenched gluons, i.e. disregard sea quarks, gluons don’t create quark lines – Heavier than physical quarks (κ = 0.1320), makes calculating 2-point functions much quicker

Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 7 / 19

Gluonic Quantities

For some, the approximation is not enough Gluonic quantities tend to still be extremely noisy, entirely disconnected contributions Large number of measurements needed to obtain sensible results An alternative method to direct calculations may lead to stronger signals for cheaper calculations We utilise the Feynman-Hellmann Method to obtain these calculations Wish to test the method for a relevant physical quantity

Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 8 / 19

Gluonic Quantities

Previous Attempts On O(5000) configurations Very weak signal, leads to an uncertain result Quoted xg = 0.53(23) arXiv:hep-lat/9608017

Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 9 / 19

Feynman-Hellmann Method

Outlining the Method The Feynman-Hellmann method produces 3-point functions by only calculating 2-point functions The method is centred around considering a modification to the relevant action S → Sλ = S + λO, so X(x)X(y)λ = N

• DUX(x)X(y) e−S[U]−λO

Then ∂ ∂λ X(x)X(y)

• λ=0 = N
• DUX(x)X(y)(−O) e−S[U] = − X(x)OX(y)

I.e. obtain 3-point functions by only calculating 2-point functions

Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 10 / 19

Feynman-Hellmann Method

Our formula We have for a chosen operator Ob,

• X(p)|Ob|X(p)
• = 2(m2

X + 4

3 p) xg . Plugging the operator we want into the previous formula and knowing that X(t)X(0) = Ae−mX t + higher order terms, We get xg = − 2 3mX ∂mX ∂λ

• λ=0

Looking at the shift in the mass will give us our momentum fractions

Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 11 / 19

Results

xg

0.03 0.02 0.01 0.00 0.01 0.02 0.03 0.96 0.97 0.98 0.99 1.00 amN

Using previous formula, we get xLat

g

= 0.480(46)

Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 12 / 19

Renormalisation

What’s the purpose Quantities calculated on a lattice won’t have correct properties, i.e. won’t obey sum rule Can convert to continuous space quantity by rescaling Define Z the renormalisation factor, so OR = ZO When operators mix, Z becomes a matrix We’ll need Zgg to renormalise xg Can find this by looking at how the EMT of a lone gluon rescales between schemes

Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 13 / 19

Renormalisation

Formula Due to quenched approx, xR

g = Zgg xLat g

Use Feynman-Hellmann method to obtain

• A(p)ObA(−p)
• from A(p)A(−p)

Considering amputated vertex functions Γ = D−1(p)

• A(p)ObA(−p)
• D−1(p),

from operator insertion on gluon lines, Zgg = Z3 Tr(ΓBornΓBorn) Tr(ΓBornΓLat)

Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 14 / 19

Results

Zgg

2 4 6 8 10 12 14 16 (ap)2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Zg 0.658±0.108 Data

From this, we find Zgg = 0.65(11)

Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 15 / 19

Analysis

Checking the Sum Rule xR

g = Zgg xg = 0.32(5)

Also have xMS

q

= 0.630(6), arXiv:hep-lat/0410187 So xR

g + xMS q

= 0.94(6) A bit small! But not the full story...

Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 16 / 19

Analysis

Sources of error Quark-gluon mixing neglected, must be included, will increase sum Excited state contamination Quenched / heavy quark approximations Inconsistent factors in definitions Conflicting definitions of factors/operators Conflicting sign conventions

Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 17 / 19

Next Steps

Where to go? The obvious next step is to improve on approximations made here Rigorously lay out conventions, clear up any conflicts A dynamical run will likely provide more physically significant results

Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 18 / 19

Closing

Summary Due to noisy quantities in the gluonic sector of hadrons, this is not often examined directly Wish to find methods of calculation that lead to cleaner results The Feynman-Hellmann Method provides a promising avenue for investigation Further work required to properly normalise results Wish to apply the method to other gluonic quantities (e.g. spin composition)

Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 19 / 19