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Register a new userApplications of the Feynman-Hellmann theorem in hadron structure
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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.
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The forward Compton amplitude describes the process of virtual photon scattering from a hadron and provides an essential ingredient for the understanding of hadron structure. As a physical amplitude, the Compton tensor naturally includes all target mass corrections and higher twist effects at a fixed virtuality, $Q^2$. By making use of the second-order Feynman-Hellmann theorem, the nucleon Compton tensor is calculated in lattice QCD at an unphysical quark mass across a range of photon momenta $3 lesssim Q^2 lesssim 7$ GeV$^2$. This allows for the $Q^2$ dependence of the low moments of the nucleon structure functions to be studied in a lattice calculation for the first time. The results demonstrate that a systematic investigation of power corrections and the approach to parton asymptotics is now within reach.
We perform a Nf = 2 + 1 lattice QCD simulation to determine the quark spin fractions of hadrons using the Feynman-Hellmann theorem. By introducing an external spin operator to the fermion action, the matrix elements relevant for quark spin fractions are extracted from the linear response of the hadron energies. Simulations indicate that the Feynman-Hellmann method offers statistical precision that is comparable to the standard three-point function approach, with the added benefit that it is less susceptible to excited state contamination. This suggests that the Feynman-Hellmann technique offers a promising alternative for calculations of quark line disconnected contributions to hadronic matrix elements. At the SU(3)-flavour symmetry point, we find that the connected quark spin fractions are universally in the range 55-70% for vector mesons and octet and decuplet baryons. There is an indication that the amount of spin suppression is quite sensitive to the strength of SU(3) breaking.
The Feynman-Hellmann theorem can be derived from the long Euclidean-time limit of correlation functions determined with functional derivatives of the partition function. Using this insight, we fully develop an improved method for computing matrix elements of external currents utilizing only two-point correlation functions. Our method applies to matrix elements of any external bilinear current, including nonzero momentum transfer, flavor-changing, and two or more current insertion matrix elements. The ability to identify and control all the systematic uncertainties in the analysis of the correlation functions stems from the unique time dependence of the ground-state matrix elements and the fact that all excited states and contact terms are Euclidean-time dependent. We demonstrate the utility of our method with a calculation of the nucleon axial charge using gradient-flowed domain-wall valence quarks on the $N_f=2+1+1$ MILC highly improved staggered quark ensemble with lattice spacing and pion mass of approximately 0.15 fm and 310 MeV respectively. We show full control over excited-state systematics with the new method and obtain a value of $g_A = 1.213(26)$ with a quark-mass-dependent renormalization coefficient.
We present a simple derivation of the Hellmann-Feynman theorem at finite temperature. We illustrate its validity by considering three relevant examples which can be used in quantum mechanics lectures: the one-dimensional harmonic oscillator, the one-dimensional Ising model and the Lipkin model. We show that the Hellmann-Feynman theorem allows one to calculate expectation values of operators that appear in the Hamiltonian. This is particularly useful when the total free-energy is available, but there is not direct access to the thermal average of the operators themselves.
Study of the hadronic matrix elements can provide not only tests of the QCD sector of the Standard Model (in comparing with existing experiments) but also reliable low-energy hadronic quantities applicable to a wide range of beyond-the-Standard Model scenarios where experiments or theoretical calculations are limited or difficult. On the QCD side, progress has been made in the notoriously difficult problem of addressing gluonic structure inside the nucleon, reaching higher-$Q^2$ region of the form factors, and providing a complete picture of the proton spin. However, even further study and improvement of systematic uncertainties are needed. There are also proposed calculations of higher-order operators in the neutron electric dipole moment Lagrangian, which would be useful when combined with effective theory to probe BSM. Lattice isovector tensor and scalar charges can be combined with upcoming neutron beta-decay measurements of the Fierz interference term and neutrino asymmetry parameter to probe new interactions in the effective theory, revealing the scale of potential new TeV particles. Finally, I revisit the systematic uncertainties in recent calculations of $g_A$ and review prospects for future calculations.
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