Renormalon effects in quasiparton distributions

We investigate the renormalon ambiguity from bubble-chain diagrams in the isovector unpolarized quasiparton distribution function (PDF) of a hadron. We confirm the assertion by Braun, Vladimirov, and Zhang [Phys. Rev. D 99, 014013 (2019)] that the leading IR renormalon ambiguity is an $\mathcal{O}({\mathrm{\Lambda }}_{\mathrm{QCD}}^2/x^2{P}_z^2)$ effect, with $x$ the parton momentum fraction and $P_z$ the hadron momentum, together with a new $\mathcal{O}(\delta (x){\mathrm{\Lambda }}_{\mathrm{QCD}}^2/P_z^2)$ contribution such that the quark number is conserved. This implies the convergence of the perturbative matching kernel between a quasi-PDF and a PDF would eventually fail for small $x$. However, in both the $R$-scheme designed to cancel the leading IR renormalon and the typically used regularization-independent momentum subtraction scheme in lattice QCD for the same quasi-PDF, we find good convergence in the kernel based on three-loop bubble-chain diagram analyses. These results are encouraging for the quasi-PDF program. However, firm conclusions can only be drawn after the complete higher loop QCD calculations are carried out.

Figure 1

Bubble-chain diagrams for PDF and quasi-PDF for single quark states. The single lines are quark propagators, curly lines are gluon propagators, and double lines are Wilson lines. The gray bubbles are gluon vacuum polarization diagrams at one-loop.

Figure 2

The leading renormalon ambiguity $\delta {\tilde{Q}}_{\mathrm{ren}}(x,P_z)$ in the proton isovector quasi-PDF together with the corresponding PDF, $Q(x)$, and the one-loop quasi-PDF $\tilde{Q}(x,P_z)$, all in the $\overline{\mathrm{MS}}$ scheme. $\delta {\tilde{Q}}_{\mathrm{ren}}(x,P_z)$ is generated with Eq. (24), $P_z=1.5\mathrm{GeV}$, and $Q(x)$ from Ref. [146] (CJ12). $\tilde{Q}$, which is also shown in Fig. 3, is generated by convoluting $Q(x)$ with the one-loop matching kernel. $\delta {\tilde{Q}}_{\mathrm{ren}}(x,P_z)$ has a large negative contribution near $x=0$ such that the area under the curve is zero. This is a consequence of quark number conservation. The size of $\delta {\tilde{Q}}_{\mathrm{ren}}(x,P_z)$ is bigger than $\tilde{Q}(x,P_z)$ for almost the entire region of $-0.3\lesssim x\lesssim 0.1$.

Figure 3

Isovector proton quasi-PDFs in the $\overline{\mathrm{MS}}$ scheme derived from convoluting the CJ12 proton PDF [146] (shown as the tree level result) with the matching kernels computed with bubble-chain diagrams with ${\alpha }_s=0.283$, $P_z=1.5\mathrm{GeV}$, and $\mu =3\mathrm{GeV}$. The convergence of the series expansion is slow.

Figure 4

Isovector proton quasi-PDFs in $\overline{\mathrm{MS}}$ with the $R$-scheme for (a) $P_z^{\prime }=1\mathrm{GeV}$ and (b) $P_z^{\prime }=3\mathrm{GeV}$, ${\alpha }_s=0.283$, $P_z=1.5\mathrm{GeV}$, and $\mu =3\mathrm{GeV}$. Rapid convergence of the series expansion is seen in (b).

Figure 5

Isovector proton quasi-PDFs derived from convoluting the proton PDF of Ref. [146] with the matching kernels from bubble-chain diagrams for (a) quasi-PDFs in the $\overline{\mathrm{MS}}$ scheme (b) RI/MOM renormalized quasi-PDFs, with ${\alpha }_s=0.283$, $P_z=1.5\mathrm{GeV}$, $\mu =3\mathrm{GeV}$, ${\mu }_R=2.4\mathrm{GeV}$, and $p_R^z=1.201\mathrm{GeV}$.