Collins–Soper Operator Definition

• Collins–Soper operator definition is a framework that extracts the rapidity anomalous dimension from gauge-invariant Wilson-line operators, defining the evolution of TMDs.
• It employs careful rapidity regulation and nonperturbative renormalization, using techniques like tilted Wilson lines and operator ratios from lattice QCD.
• The kernel underpins factorization theorems in processes like Drell–Yan and SIDIS, ensuring consistent TMD evolution across various perturbative and nonperturbative regimes.

The Collins–Soper operator definition establishes the nonperturbative kernel governing the rapidity evolution of transverse-momentum-dependent (TMD) distributions in QCD. Through a precise construction involving gauge-invariant Wilson-line operators, the Collins–Soper (CS) kernel isolates the universal, process-independent rapidity anomalous dimension that mediates TMD evolution. It is central to factorization theorems for processes such as Drell–Yan and SIDIS, connects perturbative and nonperturbative scales, and facilitates direct determination via lattice QCD. Below, the technical, operator-level foundation, practical lattice extraction, regulator schemes, and renormalization systematics are summarized based on explicit arXiv data.

1. Operator Definition of the Collins–Soper Kernel

The foundational object is the TMD soft function, a vacuum matrix element of Wilson lines forming a staple-shaped contour in coordinate space. For transverse separation $b_\perp$ and light-cone vectors $n$, $\bar n$ ($n^2 = \bar n^2 = 0$, $n \cdot \bar n = 2$), the unsubtracted soft function is

$$S(b_\perp; \mu, \zeta) = \frac{1}{N_c} \langle 0 | U_{\bar n}^\dagger(\infty \to 0; 0)\, U_{n}(0 \to \infty; b_\perp)\, U_n^\dagger(\infty \to 0; b_\perp)\, U_{\bar n}(0 \to \infty; 0) |0\rangle_{\mu,\,\text{rap.\,reg.}(\zeta)},$$

where the path-ordered Wilson lines $U_{n}$ and $U_{\bar n}$ extend to infinity along $n$ and $\bar n$, then connect transversely by $b_\perp$.

The Collins–Soper kernel $K(b_\perp; \mu)$ is defined as the rapidity (log-$\zeta$) derivative of the logarithm of the soft function: $$K(b_\perp; \mu) = \frac{d}{d\ln\zeta}\, \ln S(b_\perp; \mu, \zeta).$$ This operator-level definition is fully gauge invariant and universal for all leading-twist TMD factorization applications (Collaboration et al., 2023, Shanahan et al., 2020, Vladimirov, 2020).

2. Rapidity Regulation and Renormalization

Purely lightlike Wilson lines introduce rapidity (light-cone) divergences. Regulator schemes include tilting staple directions off the light cone by introducing $v^2 < 0$ or parameterizing the staple lengths $L$ (Euclidean) or using exponential damping factors $e^{-\delta |s|}$, where the regulator $\delta\to 0$ after renormalization (Collaboration et al., 2023, Simonelli et al., 20 Feb 2025).

Ultraviolet logarithmic divergences from Wilson-line cusps and quark self-energies are removed by explicit renormalization constants $Z_\text{soft}(\mu)$, nonperturbatively determined in schemes such as RI/MOM, with perturbative matching to $\overline{\mathrm{MS}}$ (Ebert et al., 2019, Avkhadiev et al., 2023).

On the lattice, linear (self-energy) divergences are canceled by dividing by nonperturbative Wilson loops of matching geometry, e.g., $Z_E(2L, b_\perp)$ (Chu et al., 2022, Alexandrou et al., 30 Sep 2025).

3. Practical Extraction in Lattice QCD

To enable direct determination, the staple geometry is mapped to quasi-TMD operators in the LaMET approach. The ratios

$$K(b_\perp; \mu) = \frac{1}{\ln(P_1^z/P_2^z)}\,\ln \frac{H(\dots,P_2^z)\,\tilde \Psi^+(x, b_\perp, \mu, P_1^z)}{H(\dots,P_1^z)\,\tilde \Psi^+(x, b_\perp, \mu, P_2^z)} + O((P^z)^{-2})$$

is employed, where $\tilde\Psi^+$ denotes the (Euclidean) quasi-TMD wave function extracted at two large hadron momenta $P_1^z$ and $P_2^z$, and $H$ is a perturbative matching kernel (Collaboration et al., 2023, Avkhadiev et al., 2024).

Complete nonperturbative cancellation of rapidity and UV divergences is achieved by taking ratios of fully renormalized operators. In the Coulomb-gauge-fixed framework, the operator reduces to a simple quark bilinear with no Wilson line, and the CS kernel is obtained from the $\ln P^z$ dependence of the matrix element ratio. Soft factor dependence cancels in these ratios, and the resulting kernel is process- and $x$-independent (Mukherjee et al., 2024, Bollweg et al., 6 Apr 2025).

4. Collateral Structures: Soft Functions and Matching

The soft function $S(b_\perp; \mu, \zeta)$ can be factorized into a rapidity-independent (intrinsic soft) function $S_I(b_\perp; \mu)$ and a rapidity-dependent (Collins–Soper) kernel part. In the lattice context, $S_I$ is isolated using ratios of high-momentum form factors to quasi-TMD wave functions: $$S_I(b_\perp; \mu, P_1, P_2) = \frac{F(b_\perp; P_1, P_2)}{\int dx_1\, dx_2\, H(x_1, x_2)\, \tilde{\Psi}^{\pm*}(x_2, b_\perp, P^z)\, \tilde{\Psi}^{\pm}(x_1, b_\perp, P^z)}$$ where $F$ is a form factor normalized by appropriate local matrix elements and $H$ is the matching kernel (Collaboration et al., 2023, Alexandrou et al., 30 Sep 2025).

Leading-order and higher-order matching kernels for both quasi-TMDPDFs and quasi-TMDWFs are supplied at one-loop and beyond, constituting essential ingredients for the continuum extrapolation of lattice results (Alexandrou et al., 30 Sep 2025, Avkhadiev et al., 2024).

5. Evolution Equations and Renormalization-Group Consistency

The TMD distributions $F(x, b_\perp; \mu, \zeta)$ obey a pair of coupled evolution equations: $$\frac{d}{d\ln \mu}\,\ln F = \gamma_F(\mu, \zeta), \qquad \frac{d}{d\ln \zeta}\,\ln F = K(b_\perp, \mu)$$ where $K$ is the Collins–Soper kernel. The RG consistency condition,

$$\frac{d}{d\ln \mu} K(b_\perp, \mu) = -\Gamma_\mathrm{cusp}(\mu),$$

enforces a nontrivial constraint linking the kernel to the universal cusp anomalous dimension, guaranteeing two-dimensional evolution consistency in the $(\mu, \zeta)$ plane (Vladimirov, 2020, Shanahan et al., 2020).

6. Distinctions in Implementation Schemes

Different implementations yield equivalent Collins–Soper kernels but offer distinct technical advantages:

• Gauge-invariant staple-link (Euclidean/lattice) schemes: Based on nonlocal quark bilinears connected by staple-shaped Wilson lines, regulated via staple length, with full nonperturbative renormalization (Avkhadiev et al., 2023, Ebert et al., 2019, Schlemmer et al., 2021).
• Coulomb-gauge-fixed operators: Avoid Wilson lines entirely, dramatically suppress noise and operator mixing, and are compatible with chiral-preserving discretizations. Ratios at different quark separations yield the CS kernel with minimal contamination (Mukherjee et al., 2024, Bollweg et al., 6 Apr 2025).
• Off-light-cone factorization in DIS and Mellin moments: The same operator structure, after soft-subtraction, allows extraction of $K$ as the rapidity derivative of a Wilson loop with tilted paths, applicable at threshold and beyond (Simonelli et al., 20 Feb 2025).

7. Systematic Uncertainties and Physical Implications

Comprehensive lattice studies have controlled for discretization $(a/b_\perp)$, operator mixing (full $16\times16$ Dirac mixing matrices), staple-length extrapolation, quark-mass dependence, and higher-twist contributions—especially via Fierz rearrangement of Dirac structures to project leading-twist channels (Collaboration et al., 2023, Avkhadiev et al., 2024, Avkhadiev et al., 2023, Chu et al., 2022). Systematic uncertainties from NNLL matching and nonperturbative renormalization matrix elements are included.

The operator-level definition of the Collins–Soper kernel establishes a process-independent, gauge-invariant anchor for the rapidity evolution of all leading-twist TMDs. First-principles QCD determinations of $K(b_\perp, \mu)$ at nonperturbative $b_\perp$ are now available, discriminating between phenomenological parameterizations and providing robust inputs to TMD evolution in collider phenomenology (Bollweg et al., 6 Apr 2025, Mukherjee et al., 2024, Avkhadiev et al., 2023).