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 � x � p , give the fraction of the hadrons momentum contained in each constituent type p � x � g for gluons � x � q = � f � x � f for quarks, f denoting each flavour of quark These quantities give good signposts of the composition of hadrons ("How much of 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 � x � g + � x � q = 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 � D φ O e − S [ φ ] �O� = N 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 Lattice Full Theory �O� ≈ 1 � � P ( φ i ) = e − S [ φ i ] . O [ φ i ] , D φ O e − S [ φ ] �O� = N N → i Infinitely many field configurations Finite sample of field configurations Infinitely large space Finitely bound region of space Infinitely dense spacial points 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 � x � g = 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 � D UX ( x ) X ( y ) e − S [ U ] − λ O S → S λ = S + λ O , so � X ( x ) X ( y ) � λ = N Then ∂ � D UX ( x ) X ( y )( −O ) e − S [ U ] = − � X ( x ) O X ( y ) � � ∂λ � X ( x ) X ( y ) � λ = 0 = N � 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 O b , X + 4 � � = 2 ( m 2 X ( p ) |O b | X ( p ) p ) � x � g . 3 � Plugging the operator we want into the previous formula and knowing that � X ( t ) X ( 0 ) � = Ae − m X t + higher order terms , We get � 2 ∂ m X � � x � g = − � 3 m X ∂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 � x � g 1.00 0.99 Using previous formula, we get am N 0.98 � x � Lat = 0 . 480 ( 46 ) g 0.97 0.96 0.03 0.02 0.01 0.00 0.01 0.02 0.03 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 O R = Z O When operators mix, Z becomes a matrix We’ll need Z gg to renormalise � x � g 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, � x � R g = Z gg � x � Lat g � A ( p ) O b A ( − p ) � from � A ( p ) A ( − p ) � Use Feynman-Hellmann method to obtain Considering amputated vertex functions Γ = D − 1 ( p ) D − 1 ( p ) , � A ( p ) O b A ( − p ) � from operator insertion on gluon lines, Tr(Γ Born Γ Born ) Z gg = Z 3 Tr(Γ Born Γ Lat ) Tomas L. Howson, (U of Adelaide) Directly Calculating the Glue Component of the Nucleon in Lattice QCD, CHEP 2019 14 / 19

Results Z gg 0.658±0.108 Data 1.2 1.0 0.8 Z g 0.6 From this, we find Z gg = 0 . 65 ( 11 ) 0.4 0.2 0.0 0 2 4 6 8 10 12 14 16 ( ap ) 2 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 � x � R g = Z gg � x � g = 0 . 32 ( 5 ) Also have � x � MS = 0 . 630 ( 6 ) , arXiv:hep-lat/0410187 q So � x � R g + � x � MS = 0 . 94 ( 6 ) q 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