Collins-Soper Kernel and Reduced Soft Function in Lattice QCD

TL;DR

This work computes the Collins-Soper kernel $K(b,\mu)$ and the reduced soft function $S_r(b,\mu)$ for TMD PDFs within lattice QCD using LaMET and quasi-TMD wave functions with asymmetric staple-shaped Wilson lines. Nonperturbative SDR renormalization is employed to control linear, cusp, and endpoint divergences, with operator mixing found to be negligible. Using two ETMC ensembles at $a=0.093$ fm and $m_\pi$ around 0.64–0.83 GeV, the authors extract ground-state quasi-TMD WFs, form factors, and their momentum-space counterparts, enabling NLO (and LO) matching in the extraction of $K(b,\mu)$ and $S_r(b,\mu)$. The results are consistent with prior lattice studies and demonstrate a viable pathway to first-principles TMD PDFs for hadrons, including a plan to extend to physical pion mass ensembles and to other hadrons. The study thus strengthens the lattice toolkit for 3D hadron structure and provides essential inputs for future phenomenology of TMD observables.

Abstract

We evaluate the Collins-Soper kernel and the reduced soft function in lattice QCD, incorporating $\mathcal{O}(α_s)$ matching corrections. The calculation relies on the evaluation of the quasi-transverse momentum-dependent wave function with asymmetric staple-shaped quark bilinear operators and four-point meson form factors. These quantities are computed non-perturbatively using two $N_f=2+1+1$ twisted-mass fermion ensembles with the same lattice spacing of $a=0.093$ fm: the first ensemble has a lattice size of $24^3 \times 48$ and a pion mass of 346 MeV; the second one has a lattice size of $32^3 \times 64$ and a pion mass of 261 MeV. The Collins-Soper kernel and the soft function are needed for the determination of the transverse momentum-dependent parton distribution functions.

Figures (15)

• Figure 1: Graphical representation of the asymmetric staple (left) and Wilson loop (right).
• Figure 2: Comparison of non-perturbatively renormalized results through the RI$'$-short scheme for the normalized and subtracted correlation functions $\tilde{\psi}_{\gamma_5\gamma_4}^{{\rm sub.}}$ with and without operator mixing at $b=2a$ and $L=8a$, using the cA211.53.24 ensemble and the valence pion mass $m_{\pi} = 830$ MeV. On the left, we show the real contribution and on the right the imaginary one.
• Figure 3: Perturbative renormalization factor calculated through SDR method. The results are obtained with $L=8a$ using the cA211.53.24 ensemble and the valence pion mass $m_{\pi} = 830$ MeV.
• Figure 4: Comparison of the one-state and two-state fits for the normalized correlation functions $C_{wf}^0$ employing the Dirac structure $\Gamma_{\phi} = \gamma_5\gamma_4$ for the cA211.53.24 ensemble and the valence pion mass $m_{\pi} = 830$ MeV. The plots correspond to the staple parameters $b=2a$ and $z=2a$ at boosts $P^z = 2.22$ GeV (top two rows) and $P^z = 3.33$ GeV (last two rows). The first and third rows show the real (left) and imaginary (right) parts of $C_{wf}^0$ for each momentum, along with the corresponding one- and two-state fitting bands. The second and fourth rows show results for the ground-state $\psi_{\gamma_5\gamma_4}^{\rm norm}$, for each momentum, as a function of $t_{\rm low}$, extracted from both one-state (in brown) and two-state (in blue) fits. The final selected values for the two fits are denoted with open symbols. The blue band in the second and fourth rows correspond to the selected two-state result used in the remainder of the analysis.
• Figure 5: Results on the real (left) and imaginary (right) parts of the subtracted correlation function ${C_{wf}^0}^{\rm sub.}$ employing the Dirac structure $\Gamma_{\phi} = \gamma_5\gamma_4$ using the cA211.53.24 ensemble and the valence pion mass $m_{\pi} = 830$ MeV. The selected boost is $P^z = 2.22$ GeV and the transverse separation $b = 2a$. We compare different $L$ values.