TMD Soft Transition Functions

• TMD soft transition functions are multiplicative soft-sector objects defining the rapidity evolution in TMD factorization via the Collins–Soper kernel and intrinsic soft functions.
• They reconcile different regulator schemes and conversion factors, playing a key role in connecting standard TMD observables with event-shape and heavy-quark processes.
• Lattice and Euclidean approaches leverage these functions to extract TMD observables, validate perturbative predictions, and study nonperturbative soft dynamics.

TMD soft transition functions are multiplicative soft-sector objects that govern how transverse-momentum-dependent observables change across rapidity scales, regulator choices, or process-specific soft transitions. In the standard two-line TMD factorization of Drell–Yan and related small-$q_T$ observables, the term refers most directly to the rapidity-evolution component of the TMD soft function, controlled by the Collins–Soper kernel $K(b_\perp,\mu)$, after separating a rapidity-independent intrinsic soft function $S_I(b_\perp,\mu)$ (Zhang et al., 2020). In a broader usage, the same phrase also appears for regulator-conversion factors between soft-factor schemes, for event-axis soft ingredients that connect standard TMDs to additional event-shape measurements, and for quarkonium-specific nonperturbative objects that encode soft-gluon-induced transitions of heavy-quark states (Liu, 2023, Fernandez et al., 2 Jul 2026).

1. Standard definition in TMD factorization

In the conventions used in the lattice study of the TMD soft function, the soft function in coordinate space depends on the transverse separation $b_T$, the UV renormalization scale $\mu$, and rapidity regulators, and it is decomposed as

$$S(b_\perp,\mu,Y+Y') \;=\; e^{(Y+Y')\,K(b_\perp,\mu)}\, S_I^{-1}(b_\perp,\mu)\,,$$

where $K(b_\perp,\mu)$ is the Collins–Soper kernel and $S_I(b_\perp,\mu)$ is the intrinsic soft function (Zhang et al., 2020). In this usage, the exponential factor is the rapidity-transition factor: it carries the evolution in the rapidity variable, while $S_I$ is UV renormalized and rapidity independent.

For Drell–Yan at small transverse momentum, the differential cross section in impact-parameter space factorizes into a hard function, two collinear TMD distributions, and a soft function. A representative structure is

$$\frac{d\sigma}{dQ^2\, d^2\vec{q}_T} \;\sim\; \int d^2 \vec{b}_T\, e^{i \vec{q}_T\!\cdot\!\vec{b}_T}\; H(Q,\mu)\; f_{1}^{(A)}(x_A,b_T;\mu,\zeta_A)\; f_{1}^{(B)}(x_B,b_T;\mu,\zeta_B)\; S(b_T;\mu,\zeta_S)\,,$$

with conventions varying among formalisms because some schemes absorb the soft factor into redefined TMDs via square roots (Zhang et al., 2020). A geometry-based derivation in QCD identifies the soft factor as the vacuum operator that gathers soft-gluon exchanges between nearly lightlike partons and reorganizes the rapidity divergences that appear in unsubtracted TMD correlators (Vladimirov et al., 2014).

The canonical operator definition uses vacuum matrix elements of Wilson lines along conjugate lightlike directions $K(b_\perp,\mu)$0 and $K(b_\perp,\mu)$1. A representative form is

$K(b_\perp,\mu)$2

with $K(b_\perp,\mu)$3 standing for a rapidity regulator and $K(b_\perp,\mu)$4 for the UV scale (Zhang et al., 2020). In the Drell–Yan construction with past-pointing half-infinite links, the corresponding soft factor is

$K(b_\perp,\mu)$5

and its Wilson-line geometry is fixed by the directions of the fast hadrons together with gauge invariance (Vladimirov et al., 2014).

2. Rapidity evolution and the Collins–Soper kernel

The Collins–Soper kernel $K(b_\perp,\mu)$6 determines rapidity evolution. In common conventions,

$K(b_\perp,\mu)$7

and

$K(b_\perp,\mu)$8

These equations organize the rapidity running of TMDs and the soft function so that the product entering the cross section is consistent (Zhang et al., 2020).

A closely related formulation appears in Collins-scheme lattice studies with spacelike Wilson lines, where the large-rapidity behavior is written as

$K(b_\perp,\mu)$9

and the kernel is extracted through a symmetric rapidity derivative,

$S_I(b_\perp,\mu)$0

This formulation makes the rapidity-transition character of the soft factor explicit: $S_I(b_\perp,\mu)$1 is the intrinsic part, while the exponential governs the evolution across rapidity scales (Francis et al., 2024).

At fixed order, the one-loop structure of the soft factor in $S_I(b_\perp,\mu)$2-space takes a double-logarithmic form,

$S_I(b_\perp,\mu)$3

and the one-loop Collins–Soper kernel is universal,

$S_I(b_\perp,\mu)$4

In the analyticity-based study of rapidity regulators, this universality is shown to persist across the exponential, off-light-cone, and finite-length regulator classes considered there (Liu, 2023).

3. Geometry, time ordering, and regulator conversion

The Wilson-line geometry of the TMD soft factor is process dictated. In Drell–Yan, past-pointing links arise from initial-state interactions; reversing the link directions changes the process, as in the usual Drell–Yan versus SIDIS distinction, while preserving the role of the soft factor as the vacuum object that collects soft-gluon exchanges between the two nearly lightlike directions (Vladimirov et al., 2014). This geometric statement fixes the operator content before explicit loop calculations.

A separate line of work addresses the time-ordering issue of the Drell–Yan-shape soft factor. With the exponential rapidity regulator, the canonical soft factor is defined as

$S_I(b_\perp,\mu)$5

which involves a double time ordering. The same work defines a single-time-ordering soft factor

$S_I(b_\perp,\mu)$6

and proves the all-order equality

$S_I(b_\perp,\mu)$7

The proof uses causality-constrained coordinate-space analyticity and a complex Lorentz transform $S_I(b_\perp,\mu)$8, which maps the single-time-ordering operator into the Drell–Yan-shape operator (Liu, 2023).

The same analysis enlarges the statement to a class of soft factors interpolating between the off-light-cone regulator, the finite light-front length regulator, and the exponential regulator. The master equality is

$S_I(b_\perp,\mu)$9

with a fully gauge-invariant version when transverse links and $b_T$0-type subtraction loops are included (Liu, 2023). Under the stated conditions—dimensional regularization, linear covariant gauges, and the appropriate subtraction factors—the transition functions between these schemes are trivial: $b_T$1 This directly addresses one possible meaning of “soft transition function”: as a multiplicative conversion factor between rapidity-regulator schemes, it is unity for the class of schemes treated in that work (Liu, 2023).

4. Euclidean and lattice formulations

Because lightlike Wilson lines cannot be realized directly on a Euclidean lattice, several lattice formulations target either the intrinsic soft function $b_T$2, the Collins–Soper kernel $b_T$3, or the full Collins soft function through Euclideanizable constructions. In the LaMET study of the TMD soft function, the intrinsic soft function is accessed through a large-momentum-transfer pseudoscalar meson form factor and a quasi-TMD wave function built with a spacelike staple-shaped gauge link (Zhang et al., 2020).

The quasi-TMD wave function is defined as

$b_T$4

with the staple operator $b_T$5 and the subtraction factor $b_T$6, the vacuum expectation value of a rectangular spacelike Wilson loop of size $b_T$7, introduced to remove linear Wilson-line self-energy and pinch-pole singularities (Zhang et al., 2020). At leading order in LaMET factorization,

$b_T$8

where $b_T$9 is a connected-insertion pseudoscalar meson form factor (Zhang et al., 2020).

The same work extracts the Collins–Soper kernel from large-$\mu$0 evolution,

$\mu$1

Its numerical study used a CLS $\mu$2 flavor clover ensemble with $\mu$3 fm, volume $\mu$4, and momenta up to $\mu$5 GeV. It reported $\mu$6 coverage from $\mu$7 fm up to $\mu$8 fm, a monotonic decrease of $\mu$9 with $$S(b_\perp,\mu,Y+Y') \;=\; e^{(Y+Y')\,K(b_\perp,\mu)}\, S_I^{-1}(b_\perp,\mu)\,,$$0, consistency of small-$$S(b_\perp,\mu,Y+Y') \;=\; e^{(Y+Y')\,K(b_\perp,\mu)}\, S_I^{-1}(b_\perp,\mu)\,,$$1 results with the one-loop $$S(b_\perp,\mu,Y+Y') \;=\; e^{(Y+Y')\,K(b_\perp,\mu)}\, S_I^{-1}(b_\perp,\mu)\,,$$2 expression, and a $$S(b_\perp,\mu,Y+Y') \;=\; e^{(Y+Y')\,K(b_\perp,\mu)}\, S_I^{-1}(b_\perp,\mu)\,,$$3 consistent at small $$S(b_\perp,\mu,Y+Y') \;=\; e^{(Y+Y')\,K(b_\perp,\mu)}\, S_I^{-1}(b_\perp,\mu)\,,$$4 with perturbative expectations up to 3-loop running (Zhang et al., 2020).

A different Euclidean strategy represents the Wilson line by an auxiliary one-dimensional fermion field. In that framework, the Euclidean directional vectors are complex,

$$S(b_\perp,\mu,Y+Y') \;=\; e^{(Y+Y')\,K(b_\perp,\mu)}\, S_I^{-1}(b_\perp,\mu)\,,$$5

and the calculation is valid only when $$S(b_\perp,\mu,Y+Y') \;=\; e^{(Y+Y')\,K(b_\perp,\mu)}\, S_I^{-1}(b_\perp,\mu)\,,$$6, which corresponds to spacelike Minkowski directions and matches Collins’ spacelike soft function (Francis et al., 2023, Francis et al., 2024). The measured finite-length ratio

$$S(b_\perp,\mu,Y+Y') \;=\; e^{(Y+Y')\,K(b_\perp,\mu)}\, S_I^{-1}(b_\perp,\mu)\,,$$7

cancels linear divergences in the Wilson-line length $$S(b_\perp,\mu,Y+Y') \;=\; e^{(Y+Y')\,K(b_\perp,\mu)}\, S_I^{-1}(b_\perp,\mu)\,,$$8 and removes leading power corrections in $$S(b_\perp,\mu,Y+Y') \;=\; e^{(Y+Y')\,K(b_\perp,\mu)}\, S_I^{-1}(b_\perp,\mu)\,,$$9 (Francis et al., 2024). Exploratory lattice data on a $K(b_\perp,\mu)$0 ensemble with spacing $K(b_\perp,\mu)$1 showed that very high statistics are required because complex Euclidean directions induce oscillatory cutoff effects in intermediate quantities (Francis et al., 2024).

These Euclidean constructions are closely related to an earlier HQET and LaMET formulation that identifies the TMD soft function with a special form factor of a pair of color sources moving with nearly lightlike velocities and defines a regulator-free intrinsic soft function $K(b_\perp,\mu)$2 from ratios of a large-momentum light-meson form factor and quasi-TMD wave functions (Ji et al., 2019). A one-loop validation of the same program using expansion by regions and explicit hard functions was later carried out for TMD wave functions and soft functions in LaMET (Deng et al., 2022).

5. Regulator schemes, rapidity-renormalized soft factors, and evolution frameworks

The soft sector is scheme dependent at the operator level but constrained by universal evolution. In the rapidity renormalization group framework, the TMD soft function $K(b_\perp,\mu)$3 depends on a rapidity scale $K(b_\perp,\mu)$4 in addition to $K(b_\perp,\mu)$5, and the factorized cross section involves

$K(b_\perp,\mu)$6

in impact-parameter space (Luebbert et al., 2016). The rapidity evolution of the soft function is governed by

$K(b_\perp,\mu)$7

and in that normalization one may define

$K(b_\perp,\mu)$8

The two-loop RRG calculation provides the NNLO soft function, its anomalous dimensions, and the exact inter-scheme conversion relation to the Becher–Neubert framework (Luebbert et al., 2016).

A different soft-evolution implementation appears in the parton-branching formulation. There the unresolved-emission Sudakov form factor

$K(b_\perp,\mu)$9

acts as a no-emission probability between scales (Lelek, 2019). In that language, the Sudakov factors are the soft transition functions: they effect the transition between scales by exponentiating unresolved soft radiation. With angular ordering, the PB Sudakov reproduces the CSS $S_I(b_\perp,\mu)$0, $S_I(b_\perp,\mu)$1, and $S_I(b_\perp,\mu)$2 coefficients for Drell–Yan, while NNLL differences are scheme dependent (Lelek, 2019). A later NNLL PB analysis reformulates the same soft dynamics using a soft-gluon physical coupling and computes the perturbative part of the Collins–Soper kernel consistently with the collinear anomaly, while obtaining nonperturbative contributions from the large-$S_I(b_\perp,\mu)$3 region (Martinez et al., 2024).

This suggests a useful taxonomy. In standard factorization, the phrase “soft transition function” may refer to the rapidity-evolution factor $S_I(b_\perp,\mu)$4; in regulator-comparison studies it may refer to multiplicative scheme-conversion factors; and in parton-branching approaches it may refer to Sudakov no-emission probabilities. These usages are not identical, but each isolates a soft object that mediates evolution or conversion across scales or schemes.

6. Extensions beyond the standard two-line soft factor

The phrase acquires a more specialized meaning in processes whose soft sector is not exhausted by the usual two-line vacuum soft function.

For quarkonium production at low transverse momentum, one paper introduced the TMD shape function $S_I(b_\perp,\mu)$5, a rapidity-finite object defined as

$S_I(b_\perp,\mu)$6

to encode the entanglement between soft-gluon radiation and quarkonium bound-state formation in $S_I(b_\perp,\mu)$7 at low $S_I(b_\perp,\mu)$8 (Echevarria, 2019). That work explicitly states that the soft radiation and bound-state formation scales are comparable when $S_I(b_\perp,\mu)$9, so the usual ansatz “soft factor $S_I$0 LDME” fails and must be replaced by a new nonperturbative TMD object (Echevarria, 2019).

A subsequent analysis of $S_I$1 production in SIDIS introduced a different object, explicitly named a TMD soft transition function. In that case, soft gluon radiation at the scale $S_I$2 directly mediates the transition of a perturbatively produced color-octet $S_I$3 state into the color-singlet $S_I$4 configuration of the $S_I$5. The corresponding TMDSTF is defined in transverse position space as

$S_I$6

with the standard soft factor $S_I$7 included through $S_I$8 so that the object is rapidity finite in the adopted TMD scheme (Copeland et al., 3 Sep 2025). In the perturbative small-$S_I$9 region,

$$\frac{d\sigma}{dQ^2\, d^2\vec{q}_T} \;\sim\; \int d^2 \vec{b}_T\, e^{i \vec{q}_T\!\cdot\!\vec{b}_T}\; H(Q,\mu)\; f_{1}^{(A)}(x_A,b_T;\mu,\zeta_A)\; f_{1}^{(B)}(x_B,b_T;\mu,\zeta_B)\; S(b_T;\mu,\zeta_S)\,,$$0

and the paper argues that this TMDSTF is leading in the $$\frac{d\sigma}{dQ^2\, d^2\vec{q}_T} \;\sim\; \int d^2 \vec{b}_T\, e^{i \vec{q}_T\!\cdot\!\vec{b}_T}\; H(Q,\mu)\; f_{1}^{(A)}(x_A,b_T;\mu,\zeta_A)\; f_{1}^{(B)}(x_B,b_T;\mu,\zeta_B)\; S(b_T;\mu,\zeta_S)\,,$$1 power counting relative to the color-octet TMD shape functions used previously (Copeland et al., 3 Sep 2025).

Event-axis TMD measurements in $$\frac{d\sigma}{dQ^2\, d^2\vec{q}_T} \;\sim\; \int d^2 \vec{b}_T\, e^{i \vec{q}_T\!\cdot\!\vec{b}_T}\; H(Q,\mu)\; f_{1}^{(A)}(x_A,b_T;\mu,\zeta_A)\; f_{1}^{(B)}(x_B,b_T;\mu,\zeta_B)\; S(b_T;\mu,\zeta_S)\,,$$2 and SIDIS provide yet another extension. There the soft ingredients that connect standard TMDs to observables with an additional event-shape measurement are the hemisphere-soft function $$\frac{d\sigma}{dQ^2\, d^2\vec{q}_T} \;\sim\; \int d^2 \vec{b}_T\, e^{i \vec{q}_T\!\cdot\!\vec{b}_T}\; H(Q,\mu)\; f_{1}^{(A)}(x_A,b_T;\mu,\zeta_A)\; f_{1}^{(B)}(x_B,b_T;\mu,\zeta_B)\; S(b_T;\mu,\zeta_S)\,,$$3 in one kinematic region and the collinear-soft function $$\frac{d\sigma}{dQ^2\, d^2\vec{q}_T} \;\sim\; \int d^2 \vec{b}_T\, e^{i \vec{q}_T\!\cdot\!\vec{b}_T}\; H(Q,\mu)\; f_{1}^{(A)}(x_A,b_T;\mu,\zeta_A)\; f_{1}^{(B)}(x_B,b_T;\mu,\zeta_B)\; S(b_T;\mu,\zeta_S)\,,$$4 in another. After subtracting the standard TMD soft zero-bin $$\frac{d\sigma}{dQ^2\, d^2\vec{q}_T} \;\sim\; \int d^2 \vec{b}_T\, e^{i \vec{q}_T\!\cdot\!\vec{b}_T}\; H(Q,\mu)\; f_{1}^{(A)}(x_A,b_T;\mu,\zeta_A)\; f_{1}^{(B)}(x_B,b_T;\mu,\zeta_B)\; S(b_T;\mu,\zeta_S)\,,$$5, these functions act as transition objects from a pure TMD measurement to one with both $$\frac{d\sigma}{dQ^2\, d^2\vec{q}_T} \;\sim\; \int d^2 \vec{b}_T\, e^{i \vec{q}_T\!\cdot\!\vec{b}_T}\; H(Q,\mu)\; f_{1}^{(A)}(x_A,b_T;\mu,\zeta_A)\; f_{1}^{(B)}(x_B,b_T;\mu,\zeta_B)\; S(b_T;\mu,\zeta_S)\,,$$6 and thrust-like constraints (Fernandez et al., 2 Jul 2026). The paper states explicitly that the “TMD soft transition functions” connecting standard TMDs to event-axis measurements are precisely these renormalized hemisphere-soft and collinear-soft functions (Fernandez et al., 2 Jul 2026).

A related but distinct generalization appears in DIS dijet and heavy-meson pair production, where a new three-direction soft function built from Wilson lines along the beam direction and two final-state directions is required. Its rapidity-finite ratio $$\frac{d\sigma}{dQ^2\, d^2\vec{q}_T} \;\sim\; \int d^2 \vec{b}_T\, e^{i \vec{q}_T\!\cdot\!\vec{b}_T}\; H(Q,\mu)\; f_{1}^{(A)}(x_A,b_T;\mu,\zeta_A)\; f_{1}^{(B)}(x_B,b_T;\mu,\zeta_B)\; S(b_T;\mu,\zeta_S)\,,$$7 obeys

$$\frac{d\sigma}{dQ^2\, d^2\vec{q}_T} \;\sim\; \int d^2 \vec{b}_T\, e^{i \vec{q}_T\!\cdot\!\vec{b}_T}\; H(Q,\mu)\; f_{1}^{(A)}(x_A,b_T;\mu,\zeta_A)\; f_{1}^{(B)}(x_B,b_T;\mu,\zeta_B)\; S(b_T;\mu,\zeta_S)\,,$$8

and serves as the soft transition kernel between beam-TMD and soft sectors in that factorization theorem (Castillo et al., 2020).

7. Conceptual status and common misunderstandings

The literature does not use a single universal definition of “TMD soft transition function.” In the standard Drell–Yan and SIDIS context, it most naturally denotes the rapidity-evolution factor of the soft function, governed by the Collins–Soper kernel $$\frac{d\sigma}{dQ^2\, d^2\vec{q}_T} \;\sim\; \int d^2 \vec{b}_T\, e^{i \vec{q}_T\!\cdot\!\vec{b}_T}\; H(Q,\mu)\; f_{1}^{(A)}(x_A,b_T;\mu,\zeta_A)\; f_{1}^{(B)}(x_B,b_T;\mu,\zeta_B)\; S(b_T;\mu,\zeta_S)\,,$$9, with the intrinsic soft function $K(b_\perp,\mu)$00 factored out (Zhang et al., 2020). In regulator-comparison studies, it can denote a multiplicative conversion factor between rapidity schemes; for the off-light-cone, finite-length, and exponential regulators treated through analyticity and subtraction loops, those conversion factors are exactly unity (Liu, 2023). In quarkonium studies, the phrase can denote genuinely new nonperturbative objects beyond the standard soft function, and in event-axis observables it denotes auxiliary soft ingredients that remain after subtracting the ordinary TMD soft zero-bin (Copeland et al., 3 Sep 2025, Fernandez et al., 2 Jul 2026).

A second common misconception is that all soft “transition” objects are universal in the same sense. The ordinary two-line soft function is a vacuum object depending only on the Wilson-line geometry and is universal for processes sharing those directions, up to the usual past- versus future-pointing orientation (Vladimirov et al., 2014). By contrast, quarkonium TMD shape functions and quarkonium TMD soft transition functions are process and channel specific because they include heavy-quark operators and soft transitions associated with specific NRQCD states (Echevarria, 2019, Copeland et al., 3 Sep 2025). Event-axis soft functions are likewise observable specific because they encode both transverse-momentum and event-shape measurements (Fernandez et al., 2 Jul 2026).

A third point concerns Euclidean accessibility. The analyticity analysis of the exponential-regulated soft factor proves that the Drell–Yan-shape soft factor admits Euclidean-type parametric representations without cuts to all perturbative orders and is equal to a single-time-ordering spacelike form-factor object (Liu, 2023). This does not by itself remove the practical challenges of lattice implementation; the lattice studies still emphasize finite-length effects, subtraction of Wilson-line self-energies, operator mixing, discretization effects, and the need for explicit perturbative matching to continuum schemes (Zhang et al., 2020, Francis et al., 2024).

Taken together, these developments show that the soft transition sector of TMD factorization has a layered structure. At minimum it contains the universal rapidity-evolution kernel $K(b_\perp,\mu)$01 and, depending on the observable and scheme, it may also contain intrinsic soft functions, trivial or nontrivial conversion factors, event-shape-dependent soft operators, or process-specific heavy-quark transition matrix elements. This suggests that the phrase “TMD soft transition function” should always be read together with the factorization framework, regulator choice, and operator definition in which it is used.