Nucleon Structure Functions from Operator Product Expansion on the Lattice

Deep-inelastic scattering, in the laboratory and on the lattice, is most instructive for understanding how the nucleon is built from quarks and gluons. The long-term goal is to compute the associated structure functions from first principles. So far this has been limited to model calculations. In this Letter we propose a new method to compute the structure functions directly from the virtual, all-encompassing Compton amplitude, utilizing the operator product expansion. This overcomes issues of renormalization and operator mixing, which so far have hindered lattice calculations of power corrections and higher moments.

Figure 1

The so-called “handbag” diagram (left panel) and cats-ears diagram (right panel).

Figure 2

The Compton amplitude. The solid curve is the result of Eq. (10) with $F_1^{u-d}$ given by Eq. (21), the pluses are our data points. The shaded area indicates a 10% overall error.

Figure 3

The lowest three moments ${\mu }_n^{u-d}$ with $n=1$, 3, and 5. The blue pluses ($+$) are the target numbers, $6{\mu }_n={\int }_0^{1}dx{x}^n[u(x)-d(x)]$. The red crosses ($\times$) are the solution of the OPE (14). For the error estimate see the text. At $n=3$, 5 the error is smaller than the symbol.

Figure 4

The structure function $F_1^{u-d}(x)$. The solid line is the target structure function, $6x{F}_1(x)=x[u(x)-d(x)]$. The crosses ($\times$) are the solution of the SVD (19). For the error estimate see the text.

Figure 5

The structure function $F_1^{u-d}(x)$ obtained from the Mellin transform of Eq. (22) fitted to the moments ($-$), compared with the target structure function $6x{F}_1(x)=x[u(x)-d(x)]$ ($-$). For the error estimate see the text.

Figure 6

The proton Compton amplitude $T_{33}(p,q)$ for momenta $\overrightarrow{p}=(2,-1,0)$, $(-1,1,0)$, (1,0,0), (0,1,0), (2,0,0), $(-1,2,0)$, (1,1,0), (0,2,0), (2,1,0), (1,2,0), from left to right, and $\overrightarrow{q}=(3,5,0)$, in lattice units. The current has been attached to the $d$ quark, leading to the ‘handbag’ diagram in Fig. 1. $Z_V$ has been taken from [17]. The solid line shows a sixth order polynomial fit (giving ${\chi }^2/\mathrm{dof}=0.9$), and the shaded area shows the error.