Collins–Soper Frame in TMD Factorization

• The Collins–Soper frame is a canonical reference frame used in TMD factorization to control rapidity evolution of parton distributions in processes like Drell–Yan and SIDIS.
• It employs the Collins–Soper evolution kernel to govern both perturbative and nonperturbative contributions, ensuring consistency through precise renormalization group equations.
• The framework introduces diagnostic tools such as the master function A(b_T) to compare evolution models, enhancing the accuracy of QCD phenomenology and global fits.

The Collins–Soper frame is a canonical reference frame and theoretical construct in the paper of transverse-momentum-dependent (TMD) factorization, central to understanding and controlling the rapidity evolution of TMD parton distribution functions. Originating from the analysis of Drell–Yan processes, its utility now extends to a comprehensive range of unpolarized and polarized observables in high-energy scattering, including semi-inclusive deep inelastic scattering (SIDIS) and electron–positron annihilation. The pivotal ingredient associated with this frame is the Collins–Soper evolution kernel, a universal, nonperturbative function that governs how TMD correlators change with rapidity (or, equivalently, with hard process scale) and underpins the correct resummation of large logarithms in precision QCD phenomenology.

1. Theoretical Structure and Evolution Equations

The TMD factorization framework introduces parton distributions $f_i^{\mathrm{TMD}}(x, b_T; \zeta, \mu)$ dependent not only on the renormalization scale $\mu$ but also on an additional "rapidity scale" $\zeta$, and on the transverse separation $b_T$ (the Fourier conjugate to partonic transverse momentum $q_T$). The Collins–Soper kernel $\widetilde{K}(b_T; \mu)$, also called the rapidity anomalous dimension or the Collins–Soper evolution kernel, determines the evolution of the TMD with respect to this rapidity scale through the Collins–Soper equation: $$\frac{\partial \ln \tilde{f}(x, b_T; \zeta, \mu)}{\partial\ln\sqrt{\zeta}} = \widetilde{K}(b_T; \mu)$$ The evolution kernel also satisfies its own renormalization group (RG) equation with respect to $\mu$: $$\frac{d\widetilde{K}(b_T; \mu)}{d\ln\mu} = -\gamma_K(\alpha_s(\mu))$$ where $\gamma_K$ is the perturbatively calculable anomalous dimension. The universality of $\widetilde{K}(b_T; \mu)$ holds up to color representation and a trivial sign change for time-reversal-odd distributions (such as the Sivers function). These equations, first established in the original analyses by Collins and Soper, remain the crucial foundation for all modern TMD phenomenology (Collins et al., 2014).

2. Nonperturbative Parameterization and Universality

While $\widetilde{K}(b_T; \mu)$ can be calculated perturbatively at small $b_T$, at large $b_T$ nonperturbative QCD dynamics dominate. The traditional phenomenological approach, such as the BLNY parameterization, assumed a quadratic dependence for the nonperturbative piece: $g_K(b_T) \approx \frac{g_2}{2} b_T^2$ However, generic field-theoretic principles dictate that Euclidean correlators (such as those built from Wilson-line structures underlying TMDs) decay exponentially, up to power corrections, at large $b_T$. Specifically, they argue for a functional form

$$\text{non-perturbative part} \sim \frac{1}{b_T^\alpha}e^{-mb_T}$$

with $m$ set by the lightest exchangeable state, leading $\widetilde{K}(b_T; \mu)$ to "flatten" and approach a constant as $b_T \to \infty$. The paper (Collins et al., 2014) introduces and advocates for an interpolating parameterization,

$$g_K(b_T) = g_0(b_{\text{max}})\left[1 - e^{-\frac{C_F\alpha_s(\mu)b_T^2}{\pi g_0(b_{\text{max}}) b_{\text{max}}^2}}\right]$$

which preserves the correct quadratic behavior at small $b_T$ and saturates at large $b_T$, taming excessive low-$Q$ evolution and ensuring compatibility with general QCD constraints.

The universality of the kernel means that once determined, e.g., from unpolarized Drell–Yan or SIDIS measurements, it directly enters the evolution of any TMD, including the Sivers and other polarized functions (Collins et al., 2014).

3. The Master Function $A(b_T)$ and Scheme/Scale Independence

To rigorously compare different prescriptions for TMD evolution, the paper introduces a master function,

$$A(b_T) = -\frac{\partial}{\partial\ln b_T^2} \left[\frac{\partial\ln\widetilde{W}(b_T, Q, x_A, x_B)}{\partial\ln Q^2}\right]$$

where $\widetilde{W}$ is the $b_T$-space integrand in the cross section. This function, via

$$A(b_T) = -\frac{\partial\widetilde{K}(b_T;\mu)}{\partial\ln b_T^2}$$

serves as a scheme- and scale-independent diagnostic tool. It measures the $b_T$-dependent evolution of the cross section’s shape. $A(b_T)$ vanishes at large $b_T$ if $\widetilde{K}(b_T;\mu)$ saturates, as required by field theory. The conventional quadratic parametrizations do not have this feature, manifesting nonuniversal, $b_{\text{max}}$-dependent asymptotics in $A(b_T)$.

Phenomenological determination of $A(b_T)$ provides tight constraints: once obtained, it serves not only as a consistency check across different fits but also as a physical probe of the underlying QCD extraction process (Collins et al., 2014).

4. Impact on Phenomenology and Polarized Processes

The correct treatment of the nonperturbative evolution kernel is essential for a quantitatively reliable Q-evolution of the entire class of TMD observables. If the kernel were to rise indefinitely with $b_T$, as in pure quadratic forms, the predicted evolution with $Q$ (especially at small $Q$) would be unphysically rapid, at odds with SIDIS data. The alternative parameterization ensures a more moderate, power-law-like $Q$ dependence at low $Q$.

In polarized Drell–Yan and related processes (notably in the experimental extraction and predicted sign change of the Sivers function), this precision is essential. The same evolution kernel appears for unpolarized and polarized TMDs, guaranteeing that the tested sign change is not contaminated by artifacts from the evolution model. The parameterization preferred in (Collins et al., 2014) thereby clarifies the mapping between low-$Q$ SIDIS extractions and high-$Q$ Drell–Yan predictions, reducing inconsistencies arising from evolution mismodeling.

5. Mathematical Formalism in Observables

The structure of cross sections in TMD factorization in the Collins–Soper frame is (schematically): $$\frac{d\sigma}{dQ^2\, d^2q_T \ldots} \sim \int d^2b_T\, e^{i\vec{q}_T\cdot\vec{b}_T} \Big[ \tilde{f}_A(x_A, b_T; \zeta_A, \mu)\, \tilde{f}_B(x_B, b_T; \zeta_B, \mu) \Big]\, H(Q, \mu)$$ with the Sudakov exponent built using $\widetilde{K}$ and the anomalous dimension. Rapidity and scale evolution is expressed through

$$\tilde{f}_i(x, b_T; \zeta,\mu) = \tilde{f}_i(x, b_T; \zeta_0, \mu_0) \cdot \exp\left[ \int_{\mu_0}^{\mu} \frac{d\mu'}{\mu'}\, \gamma_\mu(\alpha_s(\mu')) + \frac{1}{2} \widetilde{K}(b_T; \mu_0) \ln\frac{\zeta}{\zeta_0} \right]$$

with $\widetilde{K}(b_T;\mu_0)$ furnishing all the nonperturbative input needed for rapidity evolution, making global fits and lattice extractions possible and sharply constraining phenomenology (Collins et al., 2014).

6. Connection to Operator Definitions and Future Directions

The kernel’s universal emergence is guaranteed by its appearance in the renormalization properties of the soft factor built from Wilson-line correlators. Its operator definition (see also (Vladimirov, 2020)) is independent of process, enabling both analytic modeling in QCD vacuum frameworks and ab initio lattice QCD extractions. Theoretical derivations demand that at large $b_T$, the kernel stop growing to ensure the decay of Euclidean correlation functions as dictated by the mass gap in QCD.

Phenomenological applications and future studies—such as robust global fits, high-precision lattice QCD calculations, and experimental tests at future colliders—are expected to further clarify the nonperturbative content of the evolution kernel. The $A(b_T)$ master function is positioned as a benchmark for discriminating among evolution models and diagnosing inconsistencies or systematic artifacts in data or fits.

7. Summary Table: Key Properties and Formulas

Property	Mathematical Representation	Significance
Collins–Soper Kernel	$\widetilde{K}(b_T; \mu)$	Governs rapidity evolution of TMDs
RG equation	$d\widetilde{K}/d\ln\mu = -\gamma_K$	$\mu$ evolution via perturbative anomalous dim
Nonperturbative part	$g_K(b_T)$ (see eq.(6))	Interpolates between quadratic and constant
Master function	$A(b_T) = -\partial\widetilde{K}/\partial\ln b_T^2$	Scheme- and scale-independent diagnostic
Large-$b_T$ limit	$\lim_{b_T \to \infty} \widetilde{K} = {\rm const}$	Ensures physically sensible evolution

This structure encodes both the rigorous mathematical foundation and the phenomenologically important behavior of the Collins–Soper kernel in the TMD formalism.

8. Concluding Remarks

The Collins–Soper frame and associated evolution kernel constitute the linchpin for controlling rapidity divergences, connecting theoretical QCD formalism to experimental measurements of TMD-sensitive observables. Advances in nonperturbative parameterization, the diagnostic utility of $A(b_T)$, and a rigorous operator-based understanding directly impact predictive power in unpolarized and polarized scattering, including SIDIS, Drell–Yan, and future Electron–Ion Collider observables. Accurate modeling and extraction of the evolution kernel are central to a unified and quantitatively robust description of QCD in the three-dimensional momentum structure of hadrons (Collins et al., 2014).