Collins Azimuthal Asymmetries in QCD

• Collins azimuthal asymmetries are transverse-spin–dependent modulations arising from the fragmentation of polarized quarks, characterized by the chiral-odd Collins function H₁⊥.
• They manifest in SIDIS, e⁺e⁻ annihilation, and hadron-in-jet processes, serving as a vital tool to probe the elusive transversity distribution of nucleons.
• Empirical measurements reveal distinct charge-dependent and kinematic patterns that support the universality of the Collins function and the applicability of TMD factorization.

Collins azimuthal asymmetries are transverse-spin–dependent azimuthal modulations generated by the fragmentation of a transversely polarized quark into an unpolarized hadron. In contemporary QCD phenomenology they are identified with the chiral-odd Collins fragmentation function, usually denoted $H_1^\perp$, and are central because they couple to the transversity distribution $h_1$, the less known leading-twist piece of the QCD description of the partonic structure of the nucleon (Garzia, 2012). Their experimental manifestations depend on the process: in semi-inclusive deep inelastic scattering (SIDIS) they appear in a $\sin(\phi_h+\phi_S)$ modulation; in $e^+e^-$ annihilation they appear through azimuthal correlations of hadrons produced in opposite jets; and in polarized hadronic collisions they are isolated through hadron-in-jet azimuthal modulations such as $\sin(\phi_S-\phi_H)$ (Makke, 2014, Collaboration, 2013, Collaboration et al., 2017).

1. Definition and theoretical content

The Collins effect describes transverse-spin–dependent azimuthal modulations in the fragmentation of transversely polarized quarks, parameterized by the chiral-odd Collins fragmentation function $H_1^\perp$ (Collaboration, 2013). A standard leading-twist expression for the number density of a spinless hadron $h$ from a transversely polarized quark is

$$D_{h}^{q^\uparrow}(z,\vec{P}_\perp) = D_1^q (z, P_\perp^2) + H_1^{\perp q}(z,P_\perp^2) \frac{(\hat{k} \times \vec{P}_\perp)\cdot \vec{S}_q}{zM_h},$$

where the term with $H_1^\perp$ yields the azimuthal asymmetry (Collaboration, 2013).

Its importance follows from chirality. Transversity $h_1$ is one of the three leading-twist parton distribution functions, but unlike the unpolarized and helicity distributions it is chiral-odd and inaccessible in inclusive DIS; observable asymmetries require coupling to another chiral-odd function, and the Collins fragmentation function provides precisely such a coupling (Makke, 2014). In SIDIS this coupling is explicit in the leading-order expression

$h_1$0

with $h_1$1 and $h_1$2 the Collins fragmentation function (Makke, 2014).

In hadron-in-jet production the same physical mechanism appears in a different angular structure. For

$h_1$3

the azimuthal distribution can be written as

$h_1$4

and the Collins asymmetry is

$h_1$5

At leading order the spin-dependent structure function involves the transversity distribution, the unpolarized PDF of the other incoming parton, the Collins fragmentation function, and hard scattering coefficients (Kang et al., 2017).

2. $h_1$6 annihilation as a direct probe of the Collins function

The cleanest access to the Collins fragmentation function is provided by $h_1$7, where two hadrons are detected in opposite jets and the observable is proportional to $h_1$8, without requiring independent knowledge of transversity (Garzia, 2012). BaBar studied charged-pion pairs at a center-of-mass energy of $h_1$9–$\sin(\phi_h+\phi_S)$0 GeV, first with an off-peak sample of about $\sin(\phi_h+\phi_S)$1 and later with $\sin(\phi_h+\phi_S)$2 (Garzia, 2012, Collaboration, 2013).

Two reference frames are standard. In the thrust reference frame (RF12), with azimuthal angles $\sin(\phi_h+\phi_S)$3 and $\sin(\phi_h+\phi_S)$4 defined relative to the thrust axis, the Collins effect appears as a $\sin(\phi_h+\phi_S)$5 modulation. In the second-hadron momentum frame (RF0), with azimuthal angle $\sin(\phi_h+\phi_S)$6, it appears as a $\sin(\phi_h+\phi_S)$7 modulation (Garzia, 2012, Collaboration, 2013). For RF12, the differential cross section is written as

$\sin(\phi_h+\phi_S)$8

Here $\sin(\phi_h+\phi_S)$9 is the unpolarized fragmentation function, $e^+e^-$0 is the Collins fragmentation function, $e^+e^-$1, and $e^+e^-$2 is the polar angle of the thrust axis (Garzia, 2012).

The defining experimental strategy is the double-ratio method. To cancel detector effects and radiative corrections, normalized unlike-sign yields are divided by like-sign or all-charge yields. In the BaBar analyses the double ratio was fit with

$e^+e^-$3

where $e^+e^-$4 is the extracted asymmetry parameter (Collaboration, 2013). In the earlier off-peak analysis the corresponding fit was $e^+e^-$5, with $e^+e^-$6 proportional to the Collins effect (Garzia, 2012). This strategy was accompanied by corrections for charm and $e^+e^-$7 backgrounds, detector resolution and acceptance effects, and dilution from imperfect thrust-axis reconstruction (Garzia, 2012, Collaboration, 2013).

The measured asymmetries show characteristic kinematic growth. BaBar observed clear asymmetries in both RF12 and RF0, increasing with pion fractional energy $e^+e^-$8, with transverse momentum relative to the analysis axis, and with increasing angle between the thrust and beam axis (Collaboration, 2013). In the off-peak sample the asymmetry rose as both $e^+e^-$9 and $\sin(\phi_S-\phi_H)$0 increased and showed linear dependence on $\sin(\phi_S-\phi_H)$1 in the thrust frame, as predicted (Garzia, 2012). Comparison with Belle showed good overall agreement within uncertainties, and this consistency became an important input to global fits of transversity and Collins fragmentation (Garzia, 2012).

The $\sin(\phi_S-\phi_H)$2 program was later extended beyond pion pairs. BaBar measured Collins asymmetries in inclusive charged $\sin(\phi_S-\phi_H)$3, $\sin(\phi_S-\phi_H)$4, and $\sin(\phi_S-\phi_H)$5 pairs at $\sin(\phi_S-\phi_H)$6 GeV with $\sin(\phi_S-\phi_H)$7, observing clear azimuthal asymmetries in unlike-sign to like-sign and unlike-sign to all-charge ratios (Aubert et al., 2015). These asymmetries increase with hadron energies; the $\sin(\phi_S-\phi_H)$8 asymmetries are similar to those for $\sin(\phi_S-\phi_H)$9, whereas high-energy $H_1^\perp$0 asymmetries are, in general, larger, providing new constraints on strange-quark Collins fragmentation (Aubert et al., 2015).

3. SIDIS measurements and the extraction of transversity

In SIDIS with a transversely polarized target, the Collins asymmetry is the amplitude of the $H_1^\perp$1 modulation in the cross section (Makke, 2014). COMPASS used a $H_1^\perp$2 muon beam on transversely polarized nucleon targets, with $H_1^\perp$3 for the deuteron and NH$H_1^\perp$4 for the proton, and typical DIS cuts $H_1^\perp$5, $H_1^\perp$6, $H_1^\perp$7, and $H_1^\perp$8 (Makke, 2014, Martin, 2013, Adolph et al., 2012). The asymmetry was extracted using unbinned maximum-likelihood fits to the relevant azimuthal modulations (Makke, 2014, Martin, 2013).

The proton measurements established a clear valence-region signal. For charged hadrons and identified pions, COMPASS found asymmetries compatible with zero at small $H_1^\perp$9, while in the valence region the asymmetries become clearly non-zero, with positive and negative hadrons showing opposite signs and similar magnitudes (Adolph et al., 2012, Martin, 2013, Makke, 2014). In the 2010 proton data, positive hadrons and $h$0 exhibited positive Collins asymmetry increasing with $h$1, while negative hadrons and $h$2 exhibited negative asymmetry of similar magnitude (Martin, 2013). The 2012 COMPASS report states that in the valence region the asymmetry reaches up to $h$3 for both positive and negative hadrons and increases with $h$4 and $h$5, with an almost linear increase in $h$6 up to about $h$7 for high $h$8 (Adolph et al., 2012).

These sign patterns are physically informative. Since

$h$9

the empirical observation $$D_{h}^{q^\uparrow}(z,\vec{P}_\perp) = D_1^q (z, P_\perp^2) + H_1^{\perp q}(z,P_\perp^2) \frac{(\hat{k} \times \vec{P}_\perp)\cdot \vec{S}_q}{zM_h},$$0, together with charge weighting, suggests $$D_{h}^{q^\uparrow}(z,\vec{P}_\perp) = D_1^q (z, P_\perp^2) + H_1^{\perp q}(z,P_\perp^2) \frac{(\hat{k} \times \vec{P}_\perp)\cdot \vec{S}_q}{zM_h},$$1, namely favored and unfavored Collins fragmentation functions of opposite signs and similar magnitudes (Makke, 2014). COMPASS also reported that kaon asymmetries are less precise: $$D_{h}^{q^\uparrow}(z,\vec{P}_\perp) = D_1^q (z, P_\perp^2) + H_1^{\perp q}(z,P_\perp^2) \frac{(\hat{k} \times \vec{P}_\perp)\cdot \vec{S}_q}{zM_h},$$2 is compatible with zero within errors, while $$D_{h}^{q^\uparrow}(z,\vec{P}_\perp) = D_1^q (z, P_\perp^2) + H_1^{\perp q}(z,P_\perp^2) \frac{(\hat{k} \times \vec{P}_\perp)\cdot \vec{S}_q}{zM_h},$$3 shows a trend that is less statistically significant (Makke, 2014, Martin, 2013).

The deuteron measurements provide a complementary flavor constraint. On deuterium, all measured Collins asymmetries are compatible with zero, an observation interpreted as a cancellation between $$D_{h}^{q^\uparrow}(z,\vec{P}_\perp) = D_1^q (z, P_\perp^2) + H_1^{\perp q}(z,P_\perp^2) \frac{(\hat{k} \times \vec{P}_\perp)\cdot \vec{S}_q}{zM_h},$$4- and $$D_{h}^{q^\uparrow}(z,\vec{P}_\perp) = D_1^q (z, P_\perp^2) + H_1^{\perp q}(z,P_\perp^2) \frac{(\hat{k} \times \vec{P}_\perp)\cdot \vec{S}_q}{zM_h},$$5-quark contributions in the isoscalar target (Makke, 2014). This suggests $$D_{h}^{q^\uparrow}(z,\vec{P}_\perp) = D_1^q (z, P_\perp^2) + H_1^{\perp q}(z,P_\perp^2) \frac{(\hat{k} \times \vec{P}_\perp)\cdot \vec{S}_q}{zM_h},$$6 and $$D_{h}^{q^\uparrow}(z,\vec{P}_\perp) = D_1^q (z, P_\perp^2) + H_1^{\perp q}(z,P_\perp^2) \frac{(\hat{k} \times \vec{P}_\perp)\cdot \vec{S}_q}{zM_h},$$7 of similar size and opposite sign (Makke, 2014).

COMPASS emphasized the leading-twist nature of the effect. The 2010 proton analysis reported no significant variation across different $$D_{h}^{q^\uparrow}(z,\vec{P}_\perp) = D_1^q (z, P_\perp^2) + H_1^{\perp q}(z,P_\perp^2) \frac{(\hat{k} \times \vec{P}_\perp)\cdot \vec{S}_q}{zM_h},$$8 and $$D_{h}^{q^\uparrow}(z,\vec{P}_\perp) = D_1^q (z, P_\perp^2) + H_1^{\perp q}(z,P_\perp^2) \frac{(\hat{k} \times \vec{P}_\perp)\cdot \vec{S}_q}{zM_h},$$9 bins, supporting the conclusion that both transversity and the Collins function are leading-twist objects (Adolph et al., 2012). Agreement with HERMES over the overlapping $H_1^\perp$0 region, despite different $H_1^\perp$1 and beam-energy conditions, further strengthened the phenomenological picture (Adolph et al., 2012, Makke, 2014).

4. Hadron-in-jet asymmetries in polarized proton collisions

In transversely polarized proton-proton collisions, Collins asymmetries are accessed by measuring charged pions inside jets. STAR reported the first measurements of transverse single-spin asymmetries for inclusive jets and jet+$H_1^\perp$2 production at midrapidity in $H_1^\perp$3 collisions at $H_1^\perp$4 GeV, using $H_1^\perp$5 collected in 2011 with average beam polarization $H_1^\perp$6 (Collaboration et al., 2017). The Collins asymmetry is isolated through the $H_1^\perp$7 modulation of the spin-dependent cross section,

$H_1^\perp$8

and the corresponding asymmetry is

$H_1^\perp$9

Extraction used fits to the observed angular distributions, together with a cross-ratio method to mitigate luminosity and acceptance effects (Collaboration et al., 2017).

The main result was the first non-zero Collins asymmetries in polarized-proton collisions at higher values of jet transverse momentum, with statistical significance greater than $h_1$0 (Collaboration et al., 2017). The sign pattern matches the established SIDIS and $h_1$1 picture: $h_1$2 asymmetries are positive and $h_1$3 asymmetries are negative (Collaboration et al., 2017). The asymmetries increase with jet $h_1$4 and pion $h_1$5, and are largest at lower pion transverse momentum $h_1$6 relative to the jet axis (Collaboration et al., 2017).

A related STAR analysis at $h_1$7 GeV, based on $h_1$8 of transversely polarized proton collisions at $h_1$9 average polarization, reported the first statistically significant Collins asymmetries extracted from $h_1$00 at that energy (Adkins, 2019). There too, the asymmetry grows with jet $h_1$01, $h_1$02, and $h_1$03, and forward jets show significant signal while backward jets are consistent with zero (Adkins, 2019).

These measurements rapidly became a testing ground for universality and factorization. In a generalized parton model approach at leading order, the Collins asymmetry in $h_1$04 is expressed schematically as

$h_1$05

with the Collins term entering the polarized cross section through $h_1$06 (D'Alesio et al., 2013). Phenomenological studies using transversity and Collins functions extracted from SIDIS and $h_1$07 data found good agreement with STAR data at $h_1$08 and $h_1$09 GeV (Kang et al., 2017, D'Alesio et al., 2017). The 2017 analyses interpreted this as support for the universality of the Collins fragmentation function and for a mild, if any, evolution with the hard scale of the asymmetries (D'Alesio et al., 2017).

Later updates sharpened the same conclusion. An updated 2025 study, employing a recent extraction of transversity and Collins fragmentation from SIDIS and $h_1$10 hadron-pair data, found generally good agreement with STAR data at $h_1$11 and $h_1$12 GeV for the $h_1$13, $h_1$14, and $h_1$15 distributions, and argued that this corroborates the hypothesis of the universality of the Collins function as well as of TMD factorization for such processes (D'Alesio et al., 27 Jun 2025). A 2026 study on pion-in-jet production in polarized $h_1$16 and $h_1$17 collisions states that the good description of STAR data in $h_1$18 collisions supports the universality of the Collins function (Flore et al., 7 Jul 2026).

5. Evolution, factorization, and methodological issues

The interpretation of Collins asymmetries is inseparable from transverse-momentum dependence and scale evolution. In $h_1$19-space, TMD evolution is encoded through Sudakov factors. For hadron-in-jet production, one representative formulation is

$h_1$20

$h_1$21

Here $h_1$22 and $h_1$23 are perturbative and nonperturbative Sudakov factors (Kang et al., 2017).

A complementary formulation for the double Collins asymmetry in $h_1$24 expresses the $h_1$25 asymmetry through a $h_1$26-dependent factor

$h_1$27

with

$h_1$28

This formalism emphasizes Sudakov suppression and the nontrivial $h_1$29-dependence of low-$h_1$30 azimuthal asymmetries (Boer, 2013).

Phenomenologically, the present data do not yet enforce a unique evolution picture. The 2017 hadron-in-jet study found that calculations with TMD evolution produce a broader $h_1$31-dependent behavior and a mild scale dependence, but that the then-current STAR uncertainties were too large to decisively distinguish calculations with and without TMD evolution (Kang et al., 2017). The 2017 RHIC universality study similarly reported a very good agreement with preliminary data and concluded that the asymmetries exhibit mild, if any, evolution with the hard scale (D'Alesio et al., 2017). The 2015 simultaneous SIDIS and $h_1$32 fit found that the fitted asymmetries, being ratios, display very mild sensitivity to the details of $h_1$33 or TMD evolution, with the authors attributing this to strong partial cancellation of evolution effects in the measured quantities (Anselmino et al., 2015).

Methodologically, Bessel weighting has been proposed as a particularly clean tool. Its stated advantages are convergence of transverse-momentum integrals, suppression of large transverse-momentum contributions, and well-defined lattice QCD evaluations of Bessel-weighted TMDs including proper gauge links (Boer, 2013). This suggests a route toward observables less contaminated by the perturbative tails that complicate direct moment analyses.

6. Related channels, common mechanisms, and prospective measurements

Collins asymmetries are embedded in a broader system of transversity-induced observables. In COMPASS two-hadron production, the single-hadron Collins asymmetries for oppositely charged hadrons and the di-hadron asymmetry were studied as functions of $h_1$34 (Collaboration et al., 2015). The measured positive- and negative-hadron asymmetries are even functions of $h_1$35, nearly zero for $h_1$36 and increasing as $h_1$37 approaches $h_1$38, with a mirror symmetry between the two charges (Collaboration et al., 2015). The analysis found

$h_1$39

leading to the simplified dependences

$h_1$40

and for the di-hadron asymmetry

$h_1$41

The paper concluded that these relations provide strong experimental indication that the underlying fragmentation mechanisms are all driven by a common physical process (Collaboration et al., 2015). A plausible implication is that Collins and di-hadron measurements constrain overlapping aspects of transverse-spin–dependent hadronization rather than fully independent sectors.

The extension from pions to kaons and mixed hadron pairs in $h_1$42 annihilation similarly broadened the flavor reach of the field. BaBar’s $h_1$43, $h_1$44, and $h_1$45 measurements imply that the strange-quark sector can no longer be treated as phenomenologically negligible in Collins analyses (Aubert et al., 2015). This is relevant for global fits that aim to extract light-flavor transversity and fragmentation simultaneously.

Prospective measurements at the Electron-Ion Collider extend the same logic to pion-in-jet production in polarized lepton-proton collisions. A 2026 study considered

$h_1$46

within a simplified TMD approach with a collinear initial state and TMD fragmentation, and predicted Collins asymmetries for EIC kinematics both at leading order and including quasireal photon exchange in the Weizsäcker-Williams approximation (D'Alesio et al., 4 May 2026). The asymmetry is defined as

$h_1$47

That study states that quasireal photon exchange is relevant in the whole kinematical range explored but does not spoil the dominance of quark-initiated channels, leaving only a marginal role to their gluon counterparts (D'Alesio et al., 4 May 2026). The closely related 2026 analysis of pion-in-jet production in polarized $h_1$48 and $h_1$49 collisions makes the same point and argues that $h_1$50 processes allow for a clearer access to the transversity distribution, including its sea-quark component (Flore et al., 7 Jul 2026). These claims, if borne out by data, would make EIC Collins measurements an especially stringent test of universality and TMD factorization in a process simpler than polarized $h_1$51 scattering.

Across processes, the central pattern is stable: in $h_1$52, SIDIS, and hadron-in-jet production, Collins azimuthal asymmetries are nonzero, charge-sensitive, and strongly structured in kinematics. The accumulated evidence indicates that they provide one of the principal empirical routes to the transversity distribution and to spin-dependent fragmentation, and that comparisons among these processes are now a primary arena for testing universality, evolution, and the range of applicability of TMD descriptions (Garzia, 2012, Makke, 2014, Collaboration et al., 2017, D'Alesio et al., 27 Jun 2025).