Determining the glue component of the nucleon
R. Horsley, T. Howson, W. Kamleh, Y. Nakamura, H. Perlt, P.E.L. Rakow, G. Schierholz, H. Stüben, R.D. Young, J. M. Zanotti
PPoS(LATTICE2019)220ADP-20-4/T1114DESY 20-009Liverpool LTH 1224
Determining the glue component of the nucleon
R. Horsley ∗ a , T. Howson b , W. Kamleh b , Y. Nakamura c , H. Perlt d , P. E. L. Rakow e ,G. Schierholz f , H. Stüben g , R. D. Young b and J. M. Zanotti b a School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, UK b CSSM, Department of Physics, University of Adelaide, Adelaide SA 5005, Australia c RIKEN Advanced Institute for Computational Science, Kobe, Hyogo 650-0047, Japan d Institut für Theoretische Physik, Universität Leipzig, 04109 Leipzig, Germany e Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool,Liverpool L69 3BX, UK f Deutsches Elektronen-Synchrotron DESY, 22603 Hamburg, Germany g Universität Hamburg, Regionales Rechenzentrum, 20146 Hamburg, GermanyE-mail: [email protected]
QCDSF-UKQCD-CSSM Collaborations
Computing the gluon component of momentum in the nucleon is a difficult and computation-ally expensive problem, as the matrix element involves a quark-line-disconnected gluon operatorwhich suffers from ultra-violet fluctuations. But also necessary for a successful determination isthe non-perturbative renormalisation of this operator. As a first step we investigate here this renor-malisation in the RI − MOM scheme. Using quenched QCD as an example, a statistical signal isobtained in a direct calculation using an adaption of the Feynman-Hellmann technique. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] J a n etermining the glue component . . . R. Horsley
1. Introduction
How the nucleon’s momentum is distributed among its constituents is a question that has beendiscussed for many years. Indeed the fact that the measurement of the fraction of the nucleonmomentum carried by quarks did not sum up to one gave early indications for the existence of thegluon and QCD. If (cid:104) x (cid:105) f is the fraction of nucleon momentum carried by parton f (quark, q , orgluon, g ) then we have ∑ q (cid:104) x (cid:105) q + (cid:104) x (cid:105) g = , (1.1)where experimentally (cid:104) x (cid:105) g ∼ . This talk will describe our progress in a determination of therenormalisation of (cid:104) x (cid:105) g using lattice gauge theory techniques. Previous work includes [1, 2, 3, 4, 5,6]. The aim here will be to compare with the previous QCDSF-UKQCD result [3] but now usingthe Feynman–Hellmann (FH) theorem to also determine the Z g renormalisation constant using the RI − MOM renormalisation procedure, [7], rather than imposing the sum rule, eq. (1.1).The relevant operators that we consider here ( (cid:104) x (cid:105) ≡ v n with n =
2, for both quark/gluon) are (cid:104) N ( (cid:126) p ) | (cid:98) O ( b ) f | N ( (cid:126) p ) (cid:105)(cid:104) N ( (cid:126) p ) | N ( (cid:126) p ) (cid:105) = − E (cid:18) − m E (cid:19) (cid:104) x (cid:105) f , where O ( b ) f = O ( f ) − O ( f ) ii , (1.2)and with O ( g ) µν = F a µα F a να , O ( b ) g = ( E a i − B a i ) , O ( q ) µν = ¯ q γ µ ↔ D ν q , O ( b ) q = ¯ q γ ↔ D q − ¯ q γ i ↔ D i q , (1.3)(using the Euclidean metric) with the notation O ( τ ) = ∑ (cid:126) x O ( τ ,(cid:126) x ) with ↔ D = ( ← D − → D ) /
2. This rep-resentation for the gluon, O ( b ) g , allows for (cid:126) p = (cid:126) O ( a ) g ∼ (cid:126) E × (cid:126) B , but now (cid:126) p = (cid:126) (cid:104) x (cid:105) g is related (and equivalent) to the decomposi-tion of the nucleon mass via the energy–momentum tensor, [8]. As O ( b ) f = ¯ T ( f ) where ¯ T µν is thetraceless energy-momentum tensor, then for example the gluon contribution to the nucleon mass, m , is ∼ m (cid:104) x (cid:105) g . As is well known this can be generalised and used (for higher n ) in the OPE, forexample for DIS.
2. Lattice (cid:104) x (cid:105) g Rather than forming ratios of 3-point to 2-point correlation functions which are very noisy,[1], we choose instead to add the operator of interest to the action, [3] S → S ( λ ) = S + λ ∑ τ O ( τ ) , (2.1)and perform subsidiary runs at different λ s. E ( λ ) is then determined and the Feynman–Hellmann(FH) theorem is then used to find the matrix element of interest ∂ E ( λ ) ∂ λ (cid:12)(cid:12)(cid:12)(cid:12) λ = = (cid:68) N (cid:12)(cid:12)(cid:12) : (cid:98) O : (cid:12)(cid:12)(cid:12) N (cid:69) (cid:104) N | N (cid:105) , (2.2) We have slightly changed our convention for O ( g ) µν compared to [3]. etermining the glue component . . . R. Horsley (where : . . . : means that the vacuum term has been subtracted). For quark operators this methodincludes both quark-line -connected and -disconnected terms. We shall illustrate here the purelyquark-line-disconnected (cid:104) x (cid:105) dis g ≡ (cid:104) x (cid:105) g for quenched QCD.Using the Wilson gluonic action as Re tr C [ − U plaq µν ( x )] = a g F a µν ( x ) + . . . motivates thesimplest definition of electric and magnetic fields on each time slice as E a ( τ ) = β ∑ (cid:126) xi Re tr c (cid:2) − U plaq i ( (cid:126) x , τ ) (cid:3) , B a ( τ ) = β ∑ (cid:126) xi < j Re tr c (cid:104) − U plaq i j ( (cid:126) x , τ ) (cid:105) , (2.3)( β = / g ). The modified action in this case is S ( λ ) = ∑ τ (cid:0) [ E a ( τ ) + B a ( τ )] + λ O ( b ) ( τ ) (cid:1) , (2.4)with O ( b ) ( τ ) = [ E a ( τ ) − B a ( τ )] . This can be implemented by generating anisotropic lattices.In [3] we have described the determination of (cid:104) x (cid:105) lat g using this method.
3. Renormalisation
We now discuss some aspects of our renormalisation procedure, the main goal of this talk.We shall only consider here renormalisation for the quenched case, [1, 2] – in the conclusion andoutlook section we shall comment on the case when dynamical quarks are included.
We expect the renormalisation pattern to be for the gluon and (two) valence quarks (cid:104) x (cid:105) g (cid:104) x (cid:105) con u (cid:104) x (cid:105) con d R = Z gg Z gq Z gq Z qq
00 0 Z qq (cid:104) x (cid:105) g (cid:104) x (cid:105) con u (cid:104) x (cid:105) con d lat . (3.1)where con , (connected) or valence here means only for quark-line connected terms in the correlationfunction. In the quenched limit, we have no disconnected quark-line terms, so we shall drop thisindex here. For the bottom two rows of the renormalisation matrix, the zeroes are justified becauseif you don’t put in a valence (cid:104) x (cid:105) q ‘by hand’ then it remains zero.Due to the momentum sum rule, we must have ( (cid:104) x (cid:105) g + (cid:104) x (cid:105) u + (cid:104) x (cid:105) d ) R = Z g (cid:104) x (cid:105) lat g + Z q ( (cid:104) x (cid:105) u + (cid:104) x (cid:105) d ) lat = , (3.2)where Z g , Z q just depend on the coupling (and so in the quenched limit does Z gg ). Hence we have Z g = Z gg , Z q = Z MS gq + Z MS qq . (3.3)We now discuss our procedure for estimating Z g from RI − MOM and FH. The standard procedureis used here for RI − MOM . We first define the 2- and 1-particle-irreducible (or 1 PI ) correlationfunctions, D λ , Γ ( b ) ( p ) respectively, as (cid:104) A ( p ) A ( − p ) (cid:105) λ = D λ ( p ) and (cid:104) A ( p ) O ( b ) A ( − p ) (cid:105) = − ∂∂ λ D λ ( p ) (cid:12)(cid:12)(cid:12)(cid:12) λ = = D ( p ) Γ ( b ) ( p ) D ( p ) . (3.4)2 etermining the glue component . . . R. Horsley
We expect their structures to be of the form D ( p ) = D Born ( p ) ∆ ( p ) , Γ ( b ) ( p ) = Γ ( b ) Born ( p ) Λ ( b ) ( p ) , (3.5)where D Born ( p ) , Γ ( b ) Born ( p ) are the tree level or Born terms. The renormalisation constants arespecified by A R = Z / A and O ( b ) R = Z g O ( b ) ⇒ D R = Z D , Γ ( b ) R = Z g Z − Γ ( b ) . (3.6)To define Z , Z g we take the renormalisation conditions as D R ( p ) | p = µ = D Born ( p ) (cid:12)(cid:12) p = µ Γ ( b ) R ( p ) (cid:12)(cid:12) p = µ = Γ ( b ) Born ( p ) (cid:12)(cid:12) p = µ ⇒ Z = ∆ , Z g = Λ ( b ) ∆ . (3.7)So effectively we have to determine ∆ , Λ ( b ) . This thus first necessitates a determination of theBorn correlation functions. After some algebra, we find that the Born propagator for arbitrary λ and general gauge fixingparameter, ξ , is given by D Born λ ( p ) ab µν = (cid:18) a µν p + λ ( p − (cid:126) p ) + b µν ( + λ ) p + ξ c µν p (cid:19) δ ab , (3.8)where a µν = δ µν − p µ p ν p − b µ b ν b , b µν = b µ b ν b , c µν = p µ p ν p , (3.9)and b = ( (cid:126) pp , − (cid:126) p ) . Note that b µ thus satisfies b · p = b = p (cid:126) p . Furthermore a , b and c are orthogonal projectors, which simplifies calculations considerably. Using D Born − λ , which iswell defined and can be immediately found from eq. (3.8) gives upon generalising the definition ineq. (3.4) to arbitrary λ , Γ ( b ) Born ( p ) ab µν = (cid:2) a µν ( p − (cid:126) p ) + p b µν (cid:3) δ ab , (3.10)which is independent of λ and also independent of ξ . We are now in a position to compute ∆ , Λ ( b ) from eq. (3.5) and hence Z g from eq. (3.7). Usingthe results of section 3.2 and eq. (3.5) we have the equations D ( p ) = D Born ( p ) ∆ ( p ) , − ∂∂ λ D λ ( p ) (cid:12)(cid:12)(cid:12)(cid:12) λ = = ( p ) Λ ( b ) ( p ) ∆ ( p ) Γ ( b ) Born ( p ) . (3.11)There are now many possibilities. We can simply take the trace of these equations. This gives Z g = (cid:18) − p p (cid:19) tr D ( p ) ∂∂λ tr D λ ( p ) (cid:12)(cid:12)(cid:12) λ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = µ , (3.12)3 etermining the glue component . . . R. Horsley (where tr X ≡ X aa µµ ). Another possibility might be to first multiply by Γ ( b ) Born before taking the trace.This gives Z g =
43 13 (cid:32) + (cid:18) − p p (cid:19) (cid:33) p tr D ( p ) ∂∂λ tr D λ ( p ) Γ ( b ) Born (cid:12)(cid:12)(cid:12) λ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = µ . (3.13) Practically to reduce lattice artifacts, if the gluon propagator is defined in the natural way, withdistances measured from the mid-point of the link, i.e. A µ ( x + ˆ µ / ) = ( / ig )(( U µ ( x ) − U µ ( x ) † ) − tr C ( U µ ( x ) − U µ ( x ) † )) and A a µ ( p ) = ∑ x e ip · ( x + ˆ µ / ) A a µ ( x + ˆ µ / ) , which is important if µ (cid:54) = ν , then weget the tree-level results by the substitution p µ → ( p µ / ) . For simplicity of notation we shallcontinue to write p µ .In Fig. 1 we plot p tr D λ ( p ) (for β = . ×
48 lattice in the Landau gauge, ξ = p p D ( p ) = 0.0333=0 =0.0333 D ( p ) p =1.8 p =2.2 p =4.4 p =7.0 p =10.2 Figure 1:
Left panel: p tr D λ ( p ) versus p for λ = ± . D λ ( p ) versus λ for selected values of p , as given in the figure together with a linear fit (in λ ). O ( ) configurations per λ value were generated. against p where p = ( π / )( n , n , n , ) i.e. with a ‘cylinder’ cut (left panel) and against λ (rightpanel). From the gradients of the fits for each p (some selected values are given in the right panelof Fig. 1) we can determine Z g as given in eq. (3.12). In Fig. 2 we we compare Z g determined from p Z gg Fitted Value Z / Lat
Figure 2: Z g as given in eq. (3.12) versus p , together with a linear fit. etermining the glue component . . . R. Horsley eq. (3.12) together with a linear fit Z g = A + Bp , where the gradient term is taken to representresidual lattice effects. We find A = . ( ) for Z g . Work is in progress to try to reduce theerrors.As a benchmark comparison from [9] we have Z g | g = = ( − . g + . g ) / ( − . g ) | g = = . ( ) . (This follows from setting Z g = − g / ( c σ − c τ ) , together witha non-perturbative determination of the anisotropic coefficients c σ and c τ , see also [10].) Thiscomparison to [9] is the main result given here.
4. Conclusions and outlook
In conclusion (cid:104) x (cid:105) R g is a notoriously difficult quantity to compute as it is a short distance quantitywith numerically large fluctuations – it is a ‘disconnected quantity’. A straightforward determina-tion requires hundreds of thousands of configurations. We have developed a FH technique, nowincluding the renormalisation, which although several runs are required each run is only moderatelyexpensive.We note that it is also possible to determine Z gq in the same way using the FH theorem aftersuitably modifying the quark propagator, when Z gq ∝ tr Γ ( b ) Born ( S − λ − S − ) . (4.1)Finally a few comments about a more realistic computation with 2 + (cid:104) x (cid:105) g (cid:104) x (cid:105) con u (cid:104) x (cid:105) con d (cid:104) x (cid:105) con s (cid:104) x (cid:105) dis u (cid:104) x (cid:105) dis d (cid:104) x (cid:105) dis s (cid:104) x (cid:105) con q v R = Z gg Z gq Z gq Z gq Z gq Z gq Z gq Z gq Z a − Z b Z a − Z b Z a − Z b Z a − Z b Z a − Z b Z a − Z b Z qg Z b Z b Z b Z a Z b Z b Z b Z qg Z b Z b Z b Z b Z a Z b Z b Z qg Z b Z b Z b Z b Z b Z a Z b Z a − Z b (cid:104) x (cid:105) g (cid:104) x (cid:105) con u (cid:104) x (cid:105) con d (cid:104) x (cid:105) con s (cid:104) x (cid:105) dis u (cid:104) x (cid:105) dis d (cid:104) x (cid:105) dis s (cid:104) x (cid:105) con q v lat , (4.2)where we consider here the case of n f = n f v = (cid:104) x (cid:105) q = (cid:104) x (cid:105) con q + (cid:104) x (cid:105) dis q with for example (cid:104) x (cid:105) conR q = Z NS qq (cid:104) x (cid:105) conlat q . All Z s depend on scheme and renormali-sation scale µ . The non-singlet (e.g. (cid:104) x (cid:105) u − (cid:104) x (cid:105) d ) and singlet (i.e. (cid:104) x (cid:105) u + (cid:104) x (cid:105) d + (cid:104) x (cid:105) s ) renormalisationconstants are thus Z NS qq = Z a − Z b , Z S qq = Z NS qq + n f Z b , (4.3)respectively. As before we have (cid:32) (cid:104) x (cid:105) g + ∑ q (cid:104) x (cid:105) q + ∑ q v (cid:104) x (cid:105) q v (cid:33) R = Z g (cid:104) x (cid:105) lat g + Z q (cid:32) ∑ q (cid:104) x (cid:105) q + ∑ q v (cid:104) x (cid:105) q v (cid:33) lat = , etermining the glue component . . . R. Horsley giving Z g = Z MS gg + n f Z MS qg , Z q = Z MS gq + Z NSMS qq , with as before Z g , Z q just depending on the coupling, g , but individual terms depend on the chosenscheme, e.g. MS . A similar FH scheme for the renormalisation is being developed here. Acknowledgements
The numerical configuration generation (using the BQCD lattice QCD program [12])) and dataanalysis (using the Chroma software library [13]) was carried out on the IBM BlueGene/Q and HPTesseract using DIRAC 2 resources (EPCC, Edinburgh, UK), the IBM BlueGene/Q (NIC, Jülich,Germany) and the Cray XC40 at HLRN (The North-German Supercomputer Alliance), the NCINational Facility in Canberra, Australia (supported by the Australian Commonwealth Government)and Phoenix (University of Adelaide). RH was supported by STFC through grant ST/P000630/1.HP was supported by DFG Grant No. PE 2792/2-1. PELR was supported in part by the STFCunder contract ST/G00062X/1. GS was supported by DFG Grant No. SCHI 179/8-1. RDY andJMZ were supported by the Australian Research Council Grant No. DP190100297. We thank allfunding agencies.
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