Origin and resummation of threshold logarithms in the lattice QCD calculations of PDFs

Many present lattice QCD approaches to calculate the parton distribution functions (PDFs) rely on a factorization formula or effective theory expansion of certain Euclidean matrix elements in boosted hadron states. In the quasi- and pseudo-PDF methods, the matching coefficient in the factorization or expansion formula includes large logarithms near the threshold, which arise from the subtle interplay of collinear and soft divergences of an underlying 3D momentum distribution. We use the standard prescription to resum such logarithms in the Mellin-moment space at next-to-leading logarithmic accuracy, which also accounts for the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution, and we show that it can suppress the PDF at large $x$. Unlike the deep inelastic scattering and Drell-Yan cross sections, the resummation formula is away from the Landau pole. We then apply our formulation to reanalyze the recent lattice results for the pion valence PDF, and find that within the current data sensitivity, the effect of threshold resummation is marginal for the accessible moments and the PDF at large $x$.

Figure 1

Estimated change $\delta \beta$ of the large-$x$ exponent of PDF [cf. Eq. (63) and Eq. (66)] as a function of the order of the Mellin moment when the matching coefficient is changed from $C_N^{\mathrm{NLO}}$ to $C_N^{\mathrm{NLO}+\mathrm{NLL}}$ with $\mu =1/z_0=3.2\mathrm{GeV}$.

Figure 2

The Wilson coefficient $c_N(z^2{\mu }^2)$ with $N=2$, 4, 16 at LO, NLO, NLOevo, and $\mathrm{NLOevo}+\mathrm{NLL}$ accuracy are shown.

Figure 3

Extracted values of $⟨x^2⟩$ (upper panel) and $⟨x^4⟩$ (lower panel) by fitting $a=0.04\mathrm{fm}$ lattice results for $n_z>n_z^0=1$ to Eq. (68) at each value $z$, using Wilson coefficients of different accuracy.

Figure 4

The combined fit results of $⟨x^2⟩$ and $⟨x^4⟩$ from the region $[z_{\min },z_{\max }]$, plotted as a function of $z_{\max }$ with $z_{\min }=2a$ (filled symbols) and 3a (open symbols) for a = 0.04 fm (up) and a = 0.06 fm (down).

Figure 5

Continuum extrapolation of $⟨x^2⟩$ and $⟨x^4⟩$ from $\mathrm{NLOevo}+\mathrm{NLL}$ fit at $\mu =3.2\mathrm{GeV}$ are shown as the bands. Instead of directly extrapolating the moments, the continuum extrapolation is performed by insert Eq. (70) into expression Eq. (68). For comparisons, the moments from NLO analysis [43], $\mathsf{xFitter}$ [18], JAM [17] and ASV [12] valence PDF of pion are also shown as the lines.

Figure 6

Valence PDF of pion from $\mathrm{NLOevo}+\mathrm{NLL}$ fits of lattice results [43] to Eq. (71), with the darker and ligher bands being the statistic and systematic errors. Also shown are the same PDF from NLO fits [43], $\mathsf{xFitter}$ [18], JAM [17] and ASV [12].

Figure 7

One-loop Feynman diagrams that contribute to the leading collinear and soft divergence of the 3D momentum distribution. The conjugate diagrams are implied as well.