The Feynman-Hellmann (FH) relation offers an alternative way of accessing
hadronic matrix elements through artificial modifications to the QCD
Lagrangian. In particular, a FH-motivated method provides a new approach to
calculations of disconnected contributions to matrix elements and high-momentum
nucleon and pion form factors. Here we present results for the total nucleon
axial charge, including a statistically significant non-negative total
disconnected quark contribution of around $-5\%$ at an unphysically heavy pion
mass. Extending the FH relation to finite-momentum transfers, we also present
calculations of the pion and nucleon electromagnetic form factors up to
momentum transfers of around 7-8 GeV$^2$. Results for the nucleon are not able
to confirm the existence of a sign change for the ratio $\frac{G_E}{G_M}$, but
suggest that future calculations at lighter pion masses will provide
fascinating insight into this behaviour at large momentum transfers

Chambers, A. J.

Horsley, R.

Nakamura, Y.

Perlt, H.

Pleiter, D.

Rakow, P. E. L.

Schierholz, G.

Schiller, A.

Stüben, H.

Young, R. D.

Zanotti, J. M.

English

arXiv

The Feynman-Hellmann (FH) relation offers an alternative way of accessing hadronic matrix elements through artificial modifications to the QCD Lagrangian. In particular, a FH-motivated method provides a new approach to calculations of disconnected contributions to matrix elements and high-momentum nucleon and pion form factors. Here we present results for the total nucleon axial charge, including a statistically significant non-negative total disconnected quark contribution of around $-5\%$ at an unphysically heavy pion mass. Extending the FH relation to finite-momentum transfers, we also present calculations of the pion and nucleon electromagnetic form factors up to momentum transfers of around 7-8 GeV$^2$. Results for the nucleon are not able to confirm the existence of a sign change for the ratio $\frac{G_E}{G_M}$, but suggest that future calculations at lighter pion masses will provide fascinating insight into this behaviour at large momentum transfers

DESY Publication Database

Applications of the Feynman-Hellmann theorem in hadron structure

The Feynman-Hellmann (FH) relation offers an alternative way of accessing hadronic matrix elements through artificial modifications to the QCD Lagrangian. In particular, a FH-motivated method provides a new approach to calculations of disconnected contributions to matrix elements and high-momentum nucleon and pion form factors. Here we present results for the total nucleon axial charge, including a statistically significant non-negative total disconnected quark contribution of around 5% at an unphysically heavy pion mass. Extending the FH relation to finite-momentum transfers, we also present calculations of the pion and nucleon electromagnetic form factors up to momentum transfers of around 7–8 GeV2. Results for the nucleon are not able to confirm the existence of a sign change for the ratio GE/GM, but suggest that future calculations at lighter pion masses will provide fascinating insight into this behaviour at large momentum transfers

Chambers, A.

Zanotti, J.

Stuben, H.

Juelich Shared Electronic Resources

PoS(LATTICE 2015)125Applications of the Feynman-Hellmann theorem inhadron structureA. J. Chambers,a R. Horsley,b Y. Nakamura,c H. Perlt,d D. Pleiter,e f P. E. L. Rakow,gG. Schierholz,h A. Schiller,d H. Stüben,i R. D. Young∗a j and J. M. Zanottiaa CSSM, Department of Physics, University of Adelaide, Adelaide SA 5005, Australiab School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, UKc RIKEN Advanced Institute for Computational Science, Kobe, Hyogo 650-0047, Japand Institut für Theoretische Physik, Universität Leipzig, 04103 Leipzig, Germanye JSC, Jülich Research Centre, 52425 Jülich, Germanyf Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germanyg Theoretical Physics Division, Department of Mathematical Sciences University of Liverpool,Liverpool L69 3BX, UKh Deutsches Elektronen-Synchrotron DESY, 22603 Hamburg, Germanyi Regionales Rechenzentrum, Universität Hamburg, 20146 Hamburg, Germanyj CoEPP, Department of Physics, University of Adelaide, Adelaide SA 5005, AustraliaThe Feynman-Hellmann (FH) relation offers an alternative way of accessing hadronic matrix el-ements through artificial modifications to the QCD Lagrangian. In particular, a FH-motivatedmethod provides a new approach to calculations of disconnected contributions to matrix ele-ments and high-momentum nucleon and pion form factors. Here we present results for the totalnucleon axial charge, including a statistically significant non-negative total disconnected quarkcontribution of around −5% at an unphysically heavy pion mass. Extending the FH relation tofinite-momentum transfers, we also present calculations of the pion and nucleon electromagneticform factors up to momentum transfers of around 7–8 GeV2. Results for the nucleon are not ableto confirm the existence of a sign change for the ratio GEGM , but suggest that future calculationsat lighter pion masses will provide fascinating insight into this behaviour at large momentumtransfers.The 33rd International Symposium on Lattice Field Theory14 -18 July 2015Kobe International Conference Center, Kobe, Japan∗Speaker.c© 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). http://pos.sissa.it/PoS(LATTICE 2015)125Feynman-Hellmann & hadron structure R. D. Young1. IntroductionIn recent years, lattice calculations have seen much success in reproducing the hadron spec-trum, with increasing levels of precision. However with regards to investigations of hadron struc-ture, there are still many outstanding issues that have yet to be fully dealt with. In particular, openquestions include the details of the origin of hadronic spin and the short-distance distribution ofcharge within hadrons.In 1987, results obtained by the European Muon Collaboration (EMC) [1] (and more recentlyCOMPASS [2]) showed that the quark spin degrees of freedom carry only a small fraction of theoverall nucleon spin. Since then, a huge effort has been expended to reproduce this result theoret-ically, with lattice playing an important role. One challenge in lattice calculations of the nucleonaxial matrix elements is the difficulty of accessing fermion line disconnected contributions. Therehas been significant progress in this area, with the use of stochastic methods to attempt calculationsof these quantities [3, 4, 5, 6, 7] (also [8] for the vector matrix element). However in addition to this,there is still much ongoing discussion regarding the need to control potential excited-state contam-ination effects in three-point function calculations of the connected contributions [9, 10, 11, 12].Experimental determinations of high-momentum form factors for the nucleon and pion aredifficult due to the lack of free neutron or pion targets. The Rosenbluth separation method also hastrouble isolating the electric form factor of the proton at high momenta. In lattice, it is difficult toextract clean signals for the vector matrix element from the noise introduced by large momentumprojections. Additionally, lattice studies must address the same issues of excited-state contamina-tion control and disconnected contributions as in the axial case.Recently we have carried out work using a Feynman-Hellmann (FH) motivated method toaddress some of these issues. This approach uses the FH theorem to calculate hadronic matrixelements in lattice QCD through modifications to the QCD Lagrangian. The method has alreadyseen great success in calculations of connected contributions to axial matrix elements [13], gluonobservables [14], and singlet renormalisation factors [15]. Here we show recent determinations ofthe disconnected contributions to nucleon spin (discussed in detail in [16]), and present preliminarynew results for the nucleon and pion electromagnetic form factors.2. Feynman-Hellmann MethodsThe FH method relates shifts in the hadron spectrum, as an external field is applied, to hadronicmatrix elements. In this way, it allows quantities traditionally accessed using 3-point functions tobe calculated from 2-point functions alone. With an additional operator in the QCD Lagrangian,L →L +λO , (2.1)we have (up to choices of correlator projectors for states with non-trivial Dirac structure),∂E∂λ∣∣∣∣λ=0=12E〈H |O(0) |H 〉 , (2.2)for any hadron state H. Hence we see that on the lattice, a particular hadronic matrix element maybe calculated by performing hadron spectroscopy for multiple values of λ , and observing the linearbehaviour in the resulting energy shifts about λ = 0.2PoS(LATTICE 2015)125Feynman-Hellmann & hadron structure R. D. YoungThe modification in Eq. (2.1) may be made either during gauge field generation, or duringthe inversion of the Dirac operator matrix. The first case necessitates the generation of new fieldensembles for several values of λ , for each operator, and accesses fermion line disconnected con-tributions to the matrix element. The second case makes use of existing ensembles, and accessesfermion line connected contributions.3. Simulation DetailsFor this work we use gauge configurations with 2+ 1 flavours of non-perturbatively O(a)-improved Wilson fermions, where the lattice spacing a≈ 0.074 fm is set using a number of singletquantities [17, 18, 19]. The clover action comprises the tree-level Symanzik improved gluon actiontogether with a stout smeared fermion action, modified for the use of the FH technique [13].For the disconnected results in Sec. 4 we use ensembles with two sets of hopping parameters,(κl,κs) = (0.120900,0.120900) and (0.121095,0.120512), the first of these corresponding to theSU(3) flavour symmetric point. These both have a lattice volume of L3×T = 323× 64, and thecorresponding pion masses are approximately 470 and 310 MeV respectively (in the λ = 0 limit).The ensembles are generated with the modified quark action described in Eq. (4.1). For furtherdetails of these ensembles, and the simulated values of λ , see [16].Two volumes (L3× T = 243× 48 and 323× 64) are used for the form factor calculations inSec. 5, both with hopping parameters (κl,κs) = (0.120900,0.120900) corresponding to the SU(3)flavour symmetric point. The ensembles have been generated with an unmodified quark action,with the Dirac matrix modified during propagator calculation instead. We use two different valuesof λ , and several different sets of kinematics.4. Disconnected CalculationsThe FH approach offers an alternative technique for accessing disconnected contributions tomatrix elements. For an analysis of disconnected contributions to the axial operator, we modify thefermion part of the QCD Lagrangian during gauge field generation such thatL →L +λ ∑q=u,d,sq¯γ3γ5q . (4.1)Note that all three simulated flavours are modified simultaneously. It is expected that the largercombined signal will give a better signal-to-noise ratio than the individual flavour contributions.With the additional operator in Eq. (2.1), we expect the FH signal to appear in the imaginarypart of the nucleon correlation function. Introducing an additional imaginary factor in Eq. (4.1)would shift this signal into the real channel, however the resulting operator is not γ5-Hermitian,and hence introduces a sign problem. The nucleon correlator on the modified ensembles takes theform (to first order in λ ),G±(λ , t)large t−−−→ [A±λ (∆A+ i∆B)]e−[E±iλ∆Σdisc.]t , (4.2)where the disconnected contribution to the total quark spin fraction∆Σdisc. = ∆udisc.+∆ddisc.+∆s , (4.3)3PoS(LATTICE 2015)125Feynman-Hellmann & hadron structure R. D. Youngand ∆A and ∆B are arbitrary real numbers parameterising the shift in the correlation amplitude.The ± signs correspond to spin-up and down positive parity projections of the lattice nucleonstate respectively. To extract ∆Σdisc., we form a ratio of correlators on each ensemble, which isapproximately linear in t (to first order in λ ) for large lattice times,R(λ , t) =Im [G−(λ , t)−G+(λ , t)]Re [G−(λ , t)+G+(λ , t)]large t−−−→ λ∆Σdisc.t−λ ∆BA . (4.4)We introduce an effective phase shift, which at large times indicates ground state saturation,φeff. =1a[R(λ , t+a)−R(λ , t)] large t−−−→ λ∆Σdisc. . (4.5)Fig. 1 shows the change in the correlator phase with changing λ at the SU(3) flavour symmetricpoint, extracted using the described methods. By fitting the linear dependence of the phase shift,and repeating the analysis at the lighter mass, we calculate the quark disconnected contribution to∆Σ (renormalised according to determinations of the singlet and non-singlet factors in [15]) of∆ΣMS(2 GeV)disc. (mpi ≈ 470 MeV) =−0.055(18) , (4.6)∆ΣMS(2 GeV)disc. (mpi ≈ 310 MeV) = 0.026(14) . (4.7)The result at the heavier mass is consistent with existing results, however the unusual behaviour atthe lighter mass remains to be quantified.0.06 0.04 0.02 0.00 0.02 0.04 0.06aλ0.0100.0050.0000.0050.010aφFigure 1: Correlator phase on each ensemble asa function of λ , with the axial operator term in-cluded in the fermion action during gauge fieldgeneration with (κl ,κs)= (0.120900,0.120900).The slope is proportional to the quantity ∆Σdisc.0 2 4 6 8 10Q2  (GeV2 )0.00.20.40.60.81.01.2Q2Fpi(Q2) (GeV2)FHJLab ExperimentalVMD (mpi =470 MeV)VMD (physical)JLab Upgrade (Expected errors)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4(aQ)2Figure 2: Pion form factor as a function of mo-mentum transfer from both the FH method andJLab experimental data. The vector meson dom-inance is included to guide the eye.5. Form FactorsBy including a non-zero momentum injection in the operator insertion, we are able to apply theFH approach to calculate more general non-forward matrix elements. We make the modification tothe Lagrangian densityL (x)→L (x)+λei~q·(~x−~x0)q¯(x)γµq(x) , (5.1)4PoS(LATTICE 2015)125Feynman-Hellmann & hadron structure R. D. Youngwhere the modification is made to the u and d quarks separately. This change is only applied tothe Dirac fermion operator before inversion, and hence we only access connected contributions tothe vector matrix element. Note that the momentum insertion is made with respect to some spatiallocation~x0, and hence we restrict the space source location for quark propagators to this point. Ourapplication of the FH relation requires momentum projection at the propagator sink to enforce Breitframe kinematics. This choice acts to avoid an energy transfer through the current, which wouldlead to a non-trivial time dependence of the correlation function.We perform the operator insertion in Eq. (5.1) for two components of the vector current(µ = 2,4), and several different momentum transfers. For the majority of the chosen kinemat-ics, the total 3-vector momentum is zero, with ~p ′ = −~p. It is this simple case that is discussedhere, with more complicated momentum transfers to be discussed in a future publication. Thesekinematics, in addition to the Breit frame requirement, allow us to minimise momentum projectionat the propagator sink, which appears to reduce the noise introduced to the correlation function.5.1 PionIndividual quark contributions to the pion vector form factor Fpi are given by〈pi(~p ′) | q¯(0)γµq(0) |pi(~p)〉= (pµ + p′µ)Fpiq(Q2) . (5.2)Note that we work in the isospin symmetric limit, and consider only the modification to the u quarkin the pi+ state. The d quark result comes from charge conjugation. With the modification to theLagrangian in Eq. (5.1), considering only the temporal component of the vector current, we have∂E∂λ∣∣∣∣λ=0= Fpi(Q2) . (5.3)For the spatial component of the vector current, the first order shift in the energy vanishes when~p ′+~p=~0, and so we only use results from the temporal current insertion.Fig. 2 shows results obtained for Q2Fpi(Q2) at a variety of momentum transfers, compared withexperimental data from JLab. The form factor is anticipated to be quite sensitive to the quark mass,however the overall trend is in good agreement with naive vector meson dominance. A discussionof the asymptotic transition in the context of [20] will feature in an upcoming publication. Includedon the plot are the Q2 targets for the JLab upgrade, and expected error ranges. We note that latticeresults at such large momenta may be particularly complimentary to such experiments.5.2 NucleonIndividual quark contributions to the Dirac and Pauli form factors (F1 and F2 respectively) forthe nucleon are defined by the matrix elements〈N(~p ′,~s ′) | q¯(0)γµq(0) |N(~p,~s)〉= u¯(~p ′,~s ′)[γµF1q(Q2)+σµνqν2mF2q(Q2)]u(~p,~s) . (5.4)For the temporal component of the current insertion we choose projection matrices to project un-polarised positive parity states, and for the spatial component we project spin up and down positiveparity states individually. The resulting energy shifts from the FH relation are in general linearcombinations of F1 and F2, and the results from both current insertions can be combined to extract5PoS(LATTICE 2015)125Feynman-Hellmann & hadron structure R. D. Youngthe two form factors individually. However for the case ~p ′ = −~p, the FH analysis is greatly sim-plified. The shifts from the insertion of the temporal and spatial currents are directly proportionalto the Sachs electric and magnetic form factors respectively,∂E∂λ∣∣∣∣λ=0=mEGEq(Q2) and∂E∂λ∣∣∣∣λ=0=± q12EGM(Q2) . (5.5)Here q1 appears in the spatial case as a result of choosing the second spatial component of thevector current, and the z-axis as the spin quantisation axis.Fig. 3 shows results for the electric and magnetic form factors of the proton, determined bycombining results from the individual u and d contributions. We note very good agreement atlow Q2 between the standard 3-point function method and the FH approach, and particularly drawattention to the very promising results at high Q2. Although the current precision is not sufficientto confirm or deny the existence of a zero crossing in the ratio GEp/GMp, we are optimistic aboutthe possibilities of higher statistics calculations at lighter quark masses.0 2 4 6 8 10Q2  (GeV2 )0.00.20.40.60.81.0GEp(Q2)3-pt (243×48)FH (243×48)3-pt (323×64)FH (323×64)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4(aQ)20 2 4 6 8 10Q2  (GeV2 )0.00.51.01.5GMp(Q2)3-pt (243×48)FH (243×48)3-pt (323×64)FH (323×64)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4(aQ)2Figure 3: Electric and magnetic form factors for the proton, calculated using the FH method and a standard3-point function method, on two different lattice sizes.6. SummaryWe have shown how a FH approach can be used to perform a full high-precision calculationof the axial matrix element of the nucleon. We have also shown how the method may be extendedto the calculation of non-forward hadron matrix elements. The procedures described may easily beapplied to other hadron observables.Further extensions of the nucleon and pion form factor calculations will attempt to investigateeven higher momenta than those already accessed, in addition to explorations of the effect of latticesystematics on such simulations. These will be important for contributing understanding to ongoingexperimental efforts to access such scales.AcknowledgmentsThe numerical configuration generation was performed using the BQCD lattice QCD program,[21], on the IBM BlueGeneQ using DIRAC 2 resources (EPCC, Edinburgh, UK), the BlueGene6PoS(LATTICE 2015)125Feynman-Hellmann & hadron structure R. D. YoungP and Q at NIC (Jülich, Germany) and the Cray XC30 at HLRN (Berlin–Hannover, Germany).Some of the simulations were undertaken using resources awarded at the NCI National Facility inCanberra, Australia, and the iVEC facilities at the Pawsey Supercomputing Centre. These resourcesare provided through the National Computational Merit Allocation Scheme and the Universityof Adelaide Partner Share supported by the Australian Government. The BlueGene codes wereoptimised using Bagel [22]. The Chroma software library [23], was used in the data analysis. Thisinvestigation has been supported by the Australian Research Council under grants FT120100821,FT100100005 and DP140103067. HP was supported by DFG grant SCHI 422/10-1.References[1] J. Ashman et al., Phys. Lett. B206, 364 (1988).[2] V. Yu. Alexakhin et al., Phys. Lett. B647, 8 (2007), hep-ex/0609038.[3] R. Babich et al., Phys. Rev. D85, 054510 (2012), 1012.0562.[4] G. S. Bali et al., Phys. Rev. Lett. 108, 222001 (2012), 1112.3354.[5] M. Engelhardt, Phys. Rev. D86, 114510 (2012), 1210.0025.[6] A. Abdel-Rehim et al., Phys. Rev. D89, 034501 (2014), 1310.6339.[7] M. Deka et al., Phys. Rev. D91, 014505 (2015), 1312.4816.[8] J. Green et al., Phys. Rev. D92, 031501 (2015), 1505.01803.[9] B. J. Owen et al., Phys. Lett. B723, 217 (2013), 1212.4668.[10] S. Capitani et al., Phys. Rev. D86, 074502 (2012), 1205.0180.[11] S. Dinter et al., Phys. Lett. B704, 89 (2011), 1108.1076.[12] T. Bhattacharya et al., Phys. Rev. D89, 094502 (2014), 1306.5435.[13] A. J. Chambers et al., Phys. Rev. D90, 014510 (2014), 1405.3019.[14] R. Horsley et al., Phys. Lett. B714, 312 (2012), 1205.6410.[15] A. J. Chambers et al., Phys. Lett. B740, 30 (2015), 1410.3078.[16] A. J. Chambers et al., (2015), 1508.06856.[17] V. G. Bornyakov et al., (2015), 1508.05916.[18] W. Bietenholz et al., Phys. Lett. B690, 436 (2010), 1003.1114.[19] W. Bietenholz et al., Phys. Rev. D84, 054509 (2011), 1102.5300.[20] L. Chang et al., Phys. Rev. Lett. 111, 141802 (2013), 1307.0026.[21] Y. Nakamura and H. Stüben, PoS LATTICE2010, 040 (2010), 1011.0199.[22] P. A. Boyle, Comput. Phys. Commun. 180, 2739 (2009).[23] R. G. Edwards and B. Joo, Nucl. Phys. Proc. Suppl. 140, 832 (2005), hep-lat/0409003.7

Applications of the Feynman-Hellmann Theorem in Hadron Structure

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Applications of the Feynman-Hellmann theorem in hadron structure

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