Asymptotic Long-Distance Expansion of Euclidean Correlators in Lattice Parton Applications

Abstract

Bilinear Euclidean quark and gluon correlators with Wilson links have been used widely for applications of large-momentum effective field theories to computing non-perturbative collinear and soft parton physics. Due to color confinement, these correlators decay exponentially at large spatial distances, a behavior crucial for computing momentum-space Fourier transformations with controlled errors from lattice QCD data. Using heavy-quark effective theory reduction, dispersive analysis, Lorentz symmetry, and heavy-flavor spectra, we determine the leading and next-to-leading asymptotic behaviors and relate the expansion parameters to binding energies of heavy-flavor hadrons. We demonstrate the results through two-loop calculations in $φ^3$ theory and from the perspective of locality and analyticity. We also study the impact of the asymptotic analysis on realistic lattice QCD data and demonstrate reliable error estimates.

Figures (32)

• Figure 1: Connected and disconnected contributions to the form factor $\langle X(k) | \bar{Q} \psi | H(P) \rangle$. The left panel denotes the diagram $\langle X(k) | \bar{Q} \psi | H(P) \rangle_{\rm C}$ where $\bar{Q} \psi$ and $| H(P) \rangle$ are connected. The right panel represents the contribution $\langle X(k) | \bar{Q} \psi | H(P) \rangle_{\rm DC}$ for $\bar{Q} \psi$ and $| H(P) \rangle$ being disconnected. $H(P)$ is the external hadron of quasi-correlator and $X(k)$ is the inserted multi-particle state.
• Figure 2: Form factor representation $M_X$, where both form factors are connected. For the unpolarized case, we sum up $(\Gamma_1,\Gamma_2)=(\gamma^z,I)$ and $(\Gamma_1,\Gamma_2)=(I,\gamma^z)$.
• Figure 3: Form factor representation $M_Z$, where both form factors are disconnected.
• Figure 4: Form factor representation $M_Y$, where one form factor is disconnected, and the other one is connected. There are two terms in this representation.
• Figure 5: One-particle resonance diagram $M_{Y,fs}$ (or called forward singularity diagram), belonging to $M_Y$ in Fig. \ref{['fig:Corre_DC_C_C_DC']}. The multi-particle state $Y(k)$ contains the single particle $n(q)$, and see the definition of $Y(k)/n(q)$ below Eq. (\ref{['eq:FFDC']}). For the unpolarized case, we sum up $(\Gamma_1,\Gamma_2)=(\gamma^z,I)$ and $(\Gamma_1,\Gamma_2)=(I,\gamma^z)$. For moving intermediate states, allowed by the quantum numbers of the operators, the propagator $n(q)$ and the inserted single-particle $n(q)$ can be of the same species. Constrained by momentum conservation, the propagator $n(q)$ is on-shell.