Hard-Virtuality Collinear Modes

• Hard-virtuality collinear modes are collinear configurations whose virtuality is set by a hard (ultraviolet) scale rather than by standard infrared dynamics.
• They emerge in the lightcone expansion of semi-infinite Wilson-line correlators, with UV regions canceling spurious poles through multi-loop interplay.
• These modes play key roles in jet-medium interactions, near-lightcone thermal dynamics, and low-Q, large-x DIS, affecting parton evolution and factorization.

Searching arXiv for recent and relevant papers on hard-virtuality collinear modes and closely related formulations. First, looking up the 2025 paper that explicitly introduces hard-virtuality collinear modes in Wilson-line lightcone expansions. Now checking the 2021 paper on medium modification of high-virtuality partons, which uses the term in jet-quenching context. Next, retrieving the 2010 thermal-field-theory paper relevant to hard, near-lightlike collinear dynamics and AMY/LPM physics. Finally, checking the 2017 DIS/factorization-boundary paper, where hard-virtuality collinear configurations appear implicitly in low-$Q$, large-$x$ kinematics. Hard-virtuality collinear modes are collinear configurations whose momentum components retain a lightcone hierarchy while their virtuality is governed by a hard or ultraviolet scale rather than by ordinary infrared collinear dynamics. The term is introduced explicitly in the lightcone expansion of semi-infinite Wilson-line correlators, where ultraviolet regions with collinear scaling arise when a timelike Wilson line is taken to the light cone (Gardi et al., 22 Sep 2025). Closely related structures appear in several adjacent settings: in the medium modification of high-virtuality partons, where the relevant degrees of freedom are perturbative collinear partons with virtuality above the medium scale $\hat q \tau$ (Cao et al., 2021); in finite-temperature hard-loop diagrams with hard but nearly lightlike external momenta, where near-collinearity rather than softness controls the enhancement (Besak et al., 2010); and, implicitly, in low-$Q$, large-$x$ deep-inelastic scattering, where parton virtuality can become $O(Q^2)$ and the standard collinear PDF picture becomes unreliable (Moffat et al., 2017).

1. Wilson-line lightcone expansion and the emergence of UV collinear regions

In the Wilson-line formulation, the physical setup is the correlator of semi-infinite Wilson lines

$\left<\Phi^{(m)}_{\beta_I}\Phi^{(m)}_{\beta_J}\cdots\right>, \qquad \Phi^{(m)}_{\beta_I} = \text{P}\exp\!\left[ig_s\int_0^\infty ds\,e^{-ims\sqrt{\beta_I^2-i\varepsilon}\, \beta_I\!\cdot\!A(s\beta_I)\right],$

where the exponential regulator makes the timelike correlator IR finite while preserving gauge invariance and rescaling symmetry. The limit of interest is the one in which one velocity, $\beta_K$, approaches the lightcone, $\beta_K^2\to 0$, while the others remain timelike. In that notation, the lightlike line is denoted by a lower-case index $k$, and the limit is parameterized by

$x$0

with $x$1. This is the “lightcone expansion” of a timelike correlator (Gardi et al., 22 Sep 2025).

The method-of-regions decomposition is then applied to each loop momentum $x$2, whose scaling is written as

$x$3

with region weight

$x$4

This yields a classification into IR regions ($x$5), neutral regions ($x$6), and UV regions ($x$7). In the one-loop example, the relevant modes are the hard region $x$8, the neutral collinear region $x$9, and the IR collinear region $\hat q \tau$0. At two loops, two additional UV modes appear: $\hat q \tau$1 and $\hat q \tau$2. These are the hard-virtuality collinear modes in the paper’s terminology.

Region	Scaling	Classification
$\hat q \tau$3	$\hat q \tau$4	hard
$\hat q \tau$5	$\hat q \tau$6	neutral collinear
$\hat q \tau$7	$\hat q \tau$8	IR collinear
$\hat q \tau$9	$Q$0	UV collinear
$Q$1	$Q$2	UV hard

The decisive point is that $Q$3 and $Q$4 have $Q$5, so they are UV regions even though one of them is collinear-like in its momentum components. They are not ordinary collinear modes in the sense of on-shell amplitudes. Instead, they appear because, when a Wilson line is made exactly lightlike, the eikonal denominator loses its mass scale and the expansion of loop momenta can generate regions in which loop virtualities become large rather than small.

2. Hard region, spurious poles, and the role of cancellation

The hard region is the strict lightlike limit of the correlator at the integrand level. In practice it is obtained by replacing the nearly lightlike Wilson line by a strictly lightlike one,

$Q$6

This produces a mixed correlator with both timelike and lightlike lines. The simplification is substantial, but the strict limit is not by itself a clean, multiplicatively renormalizable object because it develops extra collinear poles (Gardi et al., 22 Sep 2025).

At one loop, the strict limit gives double poles. At two loops, the hard region of the tripole web has a triple pole,

$Q$7

which is too singular to be interpreted as a pure UV renormalization of the mixed correlator. The remaining regions are therefore required to cancel the spurious poles. At one loop, the hard region and the non-hard regions $Q$8 and $Q$9 combine so that all double poles cancel, leaving the correct single pole associated with the soft anomalous dimension: $x$0

At two loops, an analogous cancellation occurs, but now also involving the commutator of one-loop webs in the exponentiation formula,

$x$1

In the tripole example, the UV regions cancel at the correlator level,

$x$2

so only the hard and IR regions survive in the correlator-level expansion. The remaining $x$3 terms then cancel in the soft anomalous dimension after the commutator subtraction, yielding the known finite mixed tripole structure. The final mixed-case soft anomalous dimension has the standard form

$x$4

with tripole term

$x$5

This structure directly addresses a common misconception. The hard region with a strictly lightlike Wilson line is not the full answer. The correct UV pole is defined only after summing over all regions, including the hard-virtuality collinear ones that remove the spurious poles generated by the strict lightlike approximation.

3. Rescaling symmetry and the interpretation of the modes

The appearance of hard-virtuality collinear modes in the Wilson-line problem is tied to the rescaling symmetry of semi-infinite Wilson lines. Since these correlators depend only on normalized velocities $x$6, they are invariant under rescalings of each timelike velocity. The same invariant data can therefore correspond to two different physical limits (Gardi et al., 22 Sep 2025).

This is made explicit by the neutral mode $x$7, which scales as $x$8. The paper shows that $x$9 is the hard mode in a complementary lightcone expansion where the other lines are boosted instead. The same region can thus be interpreted as hard in one frame and as neutral-collinear in another. The paper argues that this degeneracy is not accidental; it is enforced by the rescaling symmetry of the Wilson lines.

That observation clarifies why the hard-virtuality collinear modes are distinct from familiar IR collinear sectors. Their origin is not an ordinary on-shell collinear singularity of an amplitude. Rather, they reflect the non-commutativity of the strict lightlike expansion with loop integration in a rescaling-invariant eikonal problem. A plausible implication is that the label “collinear” in this context refers to the momentum-component hierarchy, while the label “hard-virtuality” refers to the ultraviolet character of the corresponding region.

4. High-virtuality partons in matter

In jet-medium interactions, hard-virtuality collinear modes are the partonic degrees of freedom responsible for the early perturbative evolution of a jet while its virtuality remains above the medium’s multiple-scattering scale (Cao et al., 2021). The paper distinguishes a higher virtuality stage (HVS), or DGLAP/radiation stage, from a lower virtuality stage (LVS), or transport/BDMPS/AMY stage. In the HVS,

$O(Q^2)$0

so the parton is “too off-shell” to be fully controlled by the medium. It undergoes virtuality-ordered splittings, much like in vacuum DGLAP evolution, but with occasional medium scatterings as perturbative corrections. In the LVS, once the parton virtuality drops to the medium-generated scale,

$O(Q^2)$1

multiple scatterings maintain the parton at roughly that scale and transport-based descriptions become appropriate.

The transport coefficient is defined as the transverse-momentum broadening per unit length,

$O(Q^2)$2

A single scattering gives

$O(Q^2)$3

while after $O(Q^2)$4 scatterings over a formation time $O(Q^2)$5,

$O(Q^2)$6

The higher-twist formalism applies when the medium-induced correction to the splitting kernel remains smaller than the vacuum piece. In the paper this is expressed schematically as

$O(Q^2)$7

For constant $O(Q^2)$8 and $O(Q^2)$9, the practical condition is

$\left<\Phi^{(m)}_{\beta_I}\Phi^{(m)}_{\beta_J}\cdots\right>, \qquad \Phi^{(m)}_{\beta_I} = \text{P}\exp\!\left[ig_s\int_0^\infty ds\,e^{-ims\sqrt{\beta_I^2-i\varepsilon}\, \beta_I\!\cdot\!A(s\beta_I)\right],$0

This is the working window in which the parton is still hard enough that a collinear expansion is meaningful, but the medium can still provide a calculable correction.

Starting from vacuum DGLAP,

$\left<\Phi^{(m)}_{\beta_I}\Phi^{(m)}_{\beta_J}\cdots\right>, \qquad \Phi^{(m)}_{\beta_I} = \text{P}\exp\!\left[ig_s\int_0^\infty ds\,e^{-ims\sqrt{\beta_I^2-i\varepsilon}\, \beta_I\!\cdot\!A(s\beta_I)\right],$1

with formation time

$\left<\Phi^{(m)}_{\beta_I}\Phi^{(m)}_{\beta_J}\cdots\right>, \qquad \Phi^{(m)}_{\beta_I} = \text{P}\exp\!\left[ig_s\int_0^\infty ds\,e^{-ims\sqrt{\beta_I^2-i\varepsilon}\, \beta_I\!\cdot\!A(s\beta_I)\right],$2

the paper includes one medium scattering per emission and derives the in-medium DGLAP equation

$\left<\Phi^{(m)}_{\beta_I}\Phi^{(m)}_{\beta_J}\cdots\right>, \qquad \Phi^{(m)}_{\beta_I} = \text{P}\exp\!\left[ig_s\int_0^\infty ds\,e^{-ims\sqrt{\beta_I^2-i\varepsilon}\, \beta_I\!\cdot\!A(s\beta_I)\right],$3

In this setting, hard-virtuality collinear modes govern the hard core of jets and the leading high-$\left<\Phi^{(m)}_{\beta_I}\Phi^{(m)}_{\beta_J}\cdots\right>, \qquad \Phi^{(m)}_{\beta_I} = \text{P}\exp\!\left[ig_s\int_0^\infty ds\,e^{-ims\sqrt{\beta_I^2-i\varepsilon}\, \beta_I\!\cdot\!A(s\beta_I)\right],$4 hadron spectrum. They are not immune to the medium, but neither are they controlled by repeated soft scattering in the same sense as lower-virtuality transport regimes. The formalism therefore places them between vacuum-like DGLAP evolution and transport-dominated propagation.

5. Near-lightcone thermal dynamics and hard-loop generalization

Finite-temperature hard-loop diagrams provide a different realization of hard, nearly collinear dynamics. The relevant one-loop thermal diagrams have hard loop momenta $\left<\Phi^{(m)}_{\beta_I}\Phi^{(m)}_{\beta_J}\cdots\right>, \qquad \Phi^{(m)}_{\beta_I} = \text{P}\exp\!\left[ig_s\int_0^\infty ds\,e^{-ims\sqrt{\beta_I^2-i\varepsilon}\, \beta_I\!\cdot\!A(s\beta_I)\right],$5, while the external legs are either soft gauge fields, $\left<\Phi^{(m)}_{\beta_I}\Phi^{(m)}_{\beta_J}\cdots\right>, \qquad \Phi^{(m)}_{\beta_I} = \text{P}\exp\!\left[ig_s\int_0^\infty ds\,e^{-ims\sqrt{\beta_I^2-i\varepsilon}\, \beta_I\!\cdot\!A(s\beta_I)\right],$6, or nearly lightlike, nearly collinear gauge fields with

$\left<\Phi^{(m)}_{\beta_I}\Phi^{(m)}_{\beta_J}\cdots\right>, \qquad \Phi^{(m)}_{\beta_I} = \text{P}\exp\!\left[ig_s\int_0^\infty ds\,e^{-ims\sqrt{\beta_I^2-i\varepsilon}\, \beta_I\!\cdot\!A(s\beta_I)\right],$7

where the loop momentum points approximately along a fixed lightlike direction $\left<\Phi^{(m)}_{\beta_I}\Phi^{(m)}_{\beta_J}\cdots\right>, \qquad \Phi^{(m)}_{\beta_I} = \text{P}\exp\!\left[ig_s\int_0^\infty ds\,e^{-ims\sqrt{\beta_I^2-i\varepsilon}\, \beta_I\!\cdot\!A(s\beta_I)\right],$8. The key scales are

$\left<\Phi^{(m)}_{\beta_I}\Phi^{(m)}_{\beta_J}\cdots\right>, \qquad \Phi^{(m)}_{\beta_I} = \text{P}\exp\!\left[ig_s\int_0^\infty ds\,e^{-ims\sqrt{\beta_I^2-i\varepsilon}\, \beta_I\!\cdot\!A(s\beta_I)\right],$9

Because the emitted photon is almost collinear with the hard quark, intermediate lines are nearly on shell and the asymptotic thermal mass $\beta_K$0 must be kept in the denominators (Besak et al., 2010).

This is not the standard HTL regime in which all external momenta are soft. Here the external momentum may be hard but nearly lightlike, and the enhancement comes from near-collinearity rather than softness. The authors derive a recursion relation for the thermal one-loop $\beta_K$1-point function after approximating the hard scalar propagators by

$\beta_K$2

and partial-fractioning adjacent propagators with

$\beta_K$3

where

$\beta_K$4

Summing the $\beta_K$5-point functions into an induced current $\beta_K$6 gives a generalized kinetic equation,

$\beta_K$7

The authors compare this to the nonabelian Vlasov equation underlying standard HTL theory: $\beta_K$8 plays the role of the streaming operator $\beta_K$9, the first term is the source, and the integral term is the nonlinear gauge-covariant interaction.

After writing

$\beta_K^2\to 0$0

keeping terms linear in the photon field, and integrating out the gluons, they obtain a closed integral equation for the photon-induced current,

$\beta_K^2\to 0$1

With

$\beta_K^2\to 0$2

this becomes

$\beta_K^2\to 0$3

This is the standard AMY form that resums an infinite set of ladder diagrams with arbitrarily many soft gluon exchanges between nearly on-shell hard quark lines. It sums the leading-order bremsstrahlung and pair-annihilation contributions and encodes the LPM effect. The paper explicitly states that this regime is a hard-collinear analogue of HTL rather than the standard soft HTL limit. This suggests a close conceptual relation to hard-virtuality collinear dynamics, even though the terminology is not the same.

6. Low-$\beta_K^2\to 0$4, large-$\beta_K^2\to 0$5 DIS and the stress on collinear factorization

In deep-inelastic scattering at moderately low $\beta_K^2\to 0$6–$\beta_K^2\to 0$7 GeV and large $\beta_K^2\to 0$8, the neglected components of parton momentum are no longer uniformly small. The standard factorized baseline is

$\beta_K^2\to 0$9

with the PDF assumed to depend only on a longitudinal momentum fraction and intrinsic transverse momentum and parton virtuality treated as $k$0 corrections. The paper tests these assumptions in a scalar quark–diquark model with exact kinematics (Moffat et al., 2017).

The standard leading-region scaling for the incoming parton is

$k$1

that is,

$k$2

The exact kinematics are fixed by

$k$3

with exact momentum components $k$4 and $k$5, and virtuality

$k$6

A central result is that, at fixed $k$7, large $k$8 drives $k$9 up to $x$00. In the large-$x$01, large-$x$02 limit,

$x$03

The incoming parton is then far from collinear in the usual sense.

The paper distinguishes four kinds of power corrections: $x$04 Type D corresponds to target-mass corrections, but the paper’s point is that Type A–C are of comparable importance in the low-$x$05, large-$x$06 domain. This is why a treatment that keeps only $x$07 corrections but neglects $x$08 and $x$09 is incomplete.

The exact $x$10-phase space also has a finite upper bound,

$x$11

When $x$12, the allowed transverse momentum shrinks and the standard factorized integral becomes sensitive to the precise cutoff. The numerical comparison reflects this directly: at $x$13 and $x$14 GeV, the collinear and exact structure functions are nearly indistinguishable, while at $x$15 and $x$16 GeV the exact result departs visibly from the factorized one, and the ratio $x$17 can drop to $x$18–$x$19.

The paper states that the “hard-virtuality collinear mode” issue is not named as such in SCET language, but is clearly present implicitly in the discussion. Contributions with parton virtuality of order the hard scale are not part of the collinear PDF sector; they belong in the hard subgraph or in a subleading-power description. This makes the DIS analysis an important boundary case: it shows that hard-virtuality collinear configurations can emerge not only from Wilson-line region analysis or jet-medium evolution, but also from exact phase-space constraints that drive the incoming parton away from the nearly on-shell collinear regime.