Locally smeared operator product expansions in scalar field theory

We propose a new locally smeared operator product expansion to decompose nonlocal operators in terms of a basis of smeared operators. The smeared operator product expansion formally connects nonperturbative matrix elements determined numerically using lattice field theory to matrix elements of nonlocal operators in the continuum. These nonperturbative matrix elements do not suffer from power-divergent mixing on the lattice, which significantly complicates calculations of quantities such as the moments of parton distribution functions, provided the smearing scale is kept fixed in the continuum limit. The presence of this smearing scale complicates the connection to the Wilson coefficients of the standard operator product expansion and requires the construction of a suitable formalism. We demonstrate the feasibility of our approach with examples in real scalar field theory.

Figure 1

Diagrams representing the leading contributions to the mixing matrix elements with unsmeared fields, $M_{\text{latt}}^{(0)}$, and smeared fields, $M_{\text{latt}}^{(0)}(\tau )$. The black solid lines in the left-hand diagram represent unsmeared propagators, and the black solid square and diamond are the operators ${\varphi }^2(0)$ and $\varphi (0){\mathrm{\Delta }}_{\mu }{\mathrm{\Delta }}_{\nu }\varphi (0)$. The blue dashed lines in the right-hand diagram are smeared propagators, and the two different blue blobs, filled and unfilled, represent the smeared operators ${\rho }^2(\tau ,0)$ and $\rho (\tau ,0){\mathrm{\Delta }}_{\mu }{\mathrm{\Delta }}_{\nu }\rho (\tau ,0)$.

Figure 2

Diagrams representing the contributions to the Wilson coefficient $d_{\mathbb{I}}$ at leading order (left two diagrams) and next-to-leading order (right two diagrams). Solid black and dashed blue lines are propagators at vanishing and nonvanishing flow times, respectively. The black squares are unsmeared fields $\varphi (0)$, black dots are interaction vertices at vanishing flow time, and the blue blob represents the smeared operator ${\rho }^2(\tau ,0)$.

Figure 3

Diagrams representing the leading-order (one-loop) contributions to the Wilson coefficient $d_{{\rho }^2}$. Details of the Feynman diagrams are provided in the caption of Fig. 2.

Figure 4

Diagrams representing the contributions to the next-to-leading-order (two-loop) contributions to the Wilson coefficients $d_{{\rho }^2}$. For details of the Feynman diagrams, see the caption of Fig. 2.

Figure 5

Example diagrams that are naturally incorporated in the renormalized external propagators. We do not include these contributions in our explicit calculations of the Wilson coefficients. For details of the Feynman diagrams, see the caption of Fig. 2.

Figure 6

At small flow times (left-hand diagram), the smeared operators are localized relative to the spacetime separation of the nonlocal operator. At large flow-time values (right-hand diagram), however, the smearing radius probes the scale of the nonlocal, and the sOPE is a poor approximation to the original nonlocal operator. For details of the Feynman diagrams, see the caption of Fig. 2.

Figure 7

Two-loop diagrams that appear at $\mathcal{O}(m^2)$ and therefore do not contribute to $d_{{\rho }^2}$. For details of the Feynman diagrams, see the caption of Fig. 2.