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Register a new userOn the Feynman-Hellmann Theorem in Quantum Field Theory and the Calculation of Matrix Elements
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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.
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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.
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.
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.
Hadronic matrix elements of proton decay are essential ingredients to bridge the grand unification theory to low energy observables like proton lifetime. In this paper we non-perturbatively calculate the matrix elements, relevant for the process of a nucleon decaying into a pseudoscalar meson and an anti-lepton through generic baryon number violating four-fermi operators. Lattice QCD with 2+1 flavor dynamical domain-wall fermions with the {it direct} method, which is direct measurement of matrix element from three-point function without chiral perturbation theory, are used for this study to have good control over the lattice discretization error, operator renormalization, and chiral extrapolation. The relevant form factors for possible transition process from an initial proton or neutron to a final pion or kaon induced by all types of three quark operators are obtained through three-point functions of (nucleon)-(three-quark operator)-(meson) with physical kinematics. In this study all the relevant systematic uncertainties of the form factors are taken into account for the first time, and the total error is found to be the range 30%-40% for $pi$ and 20%-40% for $K$ final states.
The structure of the matrix elements of the energy-momentum tensor play an important role in determining the properties of the form factors $A(q^{2})$, $B(q^{2})$ and $C(q^{2})$ which appear in the Lorentz covariant decomposition of the matrix elements. In this paper we apply a rigorous frame-independent distributional-matching approach to the matrix elements of the Poincar{e} generators in order to derive constraints on these form factors as $q rightarrow 0$. In contrast to the literature, we explicitly demonstrate that the vanishing of the anomalous gravitomagnetic moment $B(0)$ and the condition $A(0)=1$ are independent of one another, and that these constraints are not related to the specific properties or conservation of the individual Poincar{e} generators themselves, but are in fact a consequence of the physical on-shell requirement of the states in the matrix elements and the manner in which these states transform under Poincar{e} transformations.
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