Transverse-Momentum Dihadron Fragmentation Functions

• The paper shows that incorporating transverse momentum unveils additional angular modulations and spin-dependent observables such as transversity and helicity.
• It uses Fourier moment expansions and weighted asymmetry techniques to separate interference fragmentation functions from conventional unpolarized contributions.
• The study emphasizes the role of resonance structures and flavor-dependent effects, critical for accurately extracting both unpolarized and polarized DiFFs.

Searching arXiv for the cited and closely related papers on transverse-momentum-sensitive dihadron fragmentation functions. arXiv search query: "transverse momentum sensitive dihadron fragmentation functions interference fragmentation helicity-dependent G1 perp" Transverse-momentum-sensitive dihadron fragmentation functions are fragmentation correlators for a parton that hadronizes into two observed hadrons plus unobserved remnants, with explicit sensitivity to the pair’s invariant mass, relative momentum, and transverse-momentum geometry. In the standard notation of dihadron fragmentation, the observed pair carries total momentum $P_h=P_1+P_2$, relative momentum $R=(P_1-P_2)/2$, invariant mass $M_h$, and azimuth $\phi_R$; these variables make the two-hadron system more informative than a single hadron because the orientation of the hadron-pair plane can correlate with the spin of the fragmenting quark. In the collinear limit, after integrating over the parent parton transverse momentum, the formalism reduces to dihadron fragmentation functions (DiFFs) such as the unpolarized $D_1$ and the chiral-odd interference fragmentation function $H_1$, while in the transverse-momentum-dependent setting the fragmentation correlator retains additional angular structure and Fourier moments that encode helicity- and transversity-sensitive observables (Courtoy et al., 2010).

1. Kinematics and correlator structure

The basic collinear dihadron correlator discussed for $\pi^+\pi^-$ production is

$$\Delta^q(z,\cos\theta,M_h^2,\phi_R)=\frac{z|\vec R|}{16M_h}\int d^2\vec k_T\, dk^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$

with

$$\Delta^q(k,P_h,R)_{ij} =\sum_X \int \frac{d^4\xi}{(2\pi)^4}\, e^{ik\cdot \xi}\, \langle 0|{\cal U}^{n_+}_{(-\infty,\xi)}\psi_i^q(\xi)|P_h,R;X\rangle \langle P_h,R;X|\bar\psi_j^q(0){\cal U}^{n_+}_{(0,-\infty)}|0\rangle .$$

Here $z=P_h^-/k^-$ is the light-cone momentum fraction carried by the hadron pair. Once the transverse momentum of the parent parton is integrated out, the framework is collinear factorization rather than transverse-momentum-dependent factorization, and the cross section becomes a product of PDFs and DiFFs rather than a convolution in transverse momentum (Courtoy et al., 2010).

At leading twist, the collinear projection leaves two functions in the discussion,

$R=(P_1-P_2)/2$0

and

$R=(P_1-P_2)/2$1

In the paper’s language, $R=(P_1-P_2)/2$2 is the unpolarized DiFF and $R=(P_1-P_2)/2$3 is the chiral-odd, transverse-spin-dependent DiFF, also called the interference fragmentation function because it arises from interference of different partial waves, especially $R=(P_1-P_2)/2$4- and $R=(P_1-P_2)/2$5-waves (Courtoy et al., 2010).

The explicitly transverse-momentum-sensitive formulation keeps the pair transverse momenta before integration. A quark dihadron fragmentation function can then be defined as

$R=(P_1-P_2)/2$6

with analogous gluon definitions. In this form the nonperturbative object depends on the hadron longitudinal momentum fractions and on the hadron transverse momenta in the parton-collinear frame (Lee et al., 15 Jul 2025).

2. Spin-dependent content: transversity, helicity, and azimuthal structure

The central spin-dependent collinear DiFF is the interference function $R=(P_1-P_2)/2$7, which converts the transverse polarization of the fragmenting quark into an azimuthal asymmetry of the hadron-pair plane. In semi-inclusive deep-inelastic scattering on a transversely polarized target,

$R=(P_1-P_2)/2$8

the relevant single-spin asymmetry is

$R=(P_1-P_2)/2$9

This is the key transversity relation: the numerator contains the transversity distribution $M_h$0 multiplied by the chiral-odd interference DiFF $M_h$1, and the denominator contains the unpolarized PDF $M_h$2 and the unpolarized DiFF $M_h$3 (Courtoy et al., 2010).

In the TMD dihadron formalism, the fragmentation sector is richer because the two-hadron final state can carry its own polarization and partial-wave content. The TMD dihadron cross section is significantly more complex than that for single-hadron SIDIS, and the final state can be analyzed in partial waves $M_h$4. The paper explicitly states that the collinear interference fragmentation function $M_h$5 is associated with the $M_h$6 partial wave, that the Collins-like TMD moments with $M_h$7 correspond to specific partial-wave angular modulations, and that $M_h$8 receives both $M_h$9 and $\phi_R$0 interference contributions (Gliske et al., 2013).

A distinct TMD DiFF is the helicity-dependent function $\phi_R$1, which encodes a correlation between the longitudinal polarization of a fragmenting quark and the azimuthal/transverse-momentum structure of a detected hadron pair. The relevant leading-twist part of the $\phi_R$2 cross section contains

$\phi_R$3

where the convolution

$\phi_R$4

shows that the azimuthal structure is inseparable from the transverse-momentum convolution unless an appropriate weighting is introduced (Matevosyan et al., 2017).

3. Fourier moments, weighted asymmetries, and the resolution of null searches

To expose the angular structure of TMD DiFFs, the unintegrated functions are expanded in Fourier cosine series in the relative angle between $\phi_R$5 and $\phi_R$6,

$\phi_R$7

with a corresponding decomposition for $\phi_R$8. The integrated unpolarized DiFF is then

$\phi_R$9

These Fourier moments are the objects selected by weighted observables (Matevosyan et al., 2017).

A central result is that the earlier BELLE-style unweighted asymmetry used to search for $D_1$0 should vanish when the correct cross section is used:

$D_1$1

More generally, any unweighted moment depending only on the hadron-pair relative azimuths fails to isolate the required structure. The explanation is that the old angular weight did not properly break the transverse-momentum convolution between the quark and antiquark fragmentation functions (Matevosyan et al., 2017).

The new proposal is to weight the $D_1$2 cross section by the virtual photon transverse momentum squared, $D_1$3, multiplied by a trigonometric combination of azimuthal angles. For the simplest case,

$D_1$4

becomes

$D_1$5

The paper defines the integrated helicity-dependent DiFF as

$D_1$6

with

$D_1$7

The corresponding asymmetry,

$D_1$8

is a collinear expression obtained after the $D_1$9 weighting removes the transverse convolution. The same paper proposes an analogous weighted SIDIS asymmetry,

$H_1$0

which provides a universality test because the same $H_1$1 enters both $H_1$2 and SIDIS (Matevosyan et al., 2017).

4. Partial waves, resonances, and measured angular modulations in SIDIS

The transverse-momentum-sensitive dihadron final state is not a simple scalar final state. It can contain non-resonant and resonant contributions, scalar, vector, and higher partial waves, interference between partial waves, and polarization information of intermediate states such as the $H_1$3 resonance. This is why the TMD dihadron cross section is more complicated than the single-pseudoscalar case even before one turns to phenomenology (Gliske et al., 2013).

Within the partial-wave expansion, the paper identifies two important classes of transverse-target modulations. The collinear $H_1$4 moment related to transversity and the Collins function coincides with

$H_1$5

The TMD $H_1$6 partial waves are associated with

$H_1$7

and for $H_1$8 dihadrons are written as

$H_1$9

and

$\pi^+\pi^-$0

These modulations isolate different fragmentation mechanisms tied to the internal state of the hadron pair (Gliske et al., 2013).

The HERMES analysis used data taken at HERA with a $\pi^+\pi^-$1 lepton beam, a transversely polarized hydrogen target, and data from 2002–2005. Event selection included

$\pi^+\pi^-$2

Hadron identification used event-level algorithms, the RICH detector, and lepton-hadron separation above 98%. The analysis corrected the acceptance in parameter space of the fit amplitudes, not in histogram space, used four angular variables, and included 42 angular moments: 24 unpolarized twist-2 and twist-3 moments and 18 transverse-target Collins/Sivers-type moments. The paper also notes the Monte Carlo generator TMDGen, a TMD spectator model for dihadron fragmentation, and systematic uncertainties from smearing, radiative effects, beam-charge differences, and RICH particle ID (Gliske et al., 2013).

The reported results are that the $\pi^+\pi^-$3 amplitudes can be used in global collinear transversity fits and help flavor separation, while the $\pi^+\pi^-$4 amplitudes test the Lund/Artru and Gliske Gluon Radiation expectations. In particular, both $\pi^+\pi^-$5 amplitudes are consistent with zero outside the $\pi^+\pi^-$6 peak, the $\pi^+\pi^-$7 amplitude is also consistent with zero in the $\pi^+\pi^-$8 region, and the $\pi^+\pi^-$9 amplitude in the $$\Delta^q(z,\cos\theta,M_h^2,\phi_R)=\frac{z|\vec R|}{16M_h}\int d^2\vec k_T\, dk^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$0-mass region for $$\Delta^q(z,\cos\theta,M_h^2,\phi_R)=\frac{z|\vec R|}{16M_h}\int d^2\vec k_T\, dk^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$1 is nonzero and negative. The paper interprets this as consistent with the predicted opposite-sign Collins effect for vector mesons relative to pseudoscalar mesons (Gliske et al., 2013).

5. Phenomenological extraction of unpolarized and polarized DiFFs

A major phenomenological point is that the unpolarized DiFF $$\Delta^q(z,\cos\theta,M_h^2,\phi_R)=\frac{z|\vec R|}{16M_h}\int d^2\vec k_T\, dk^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$2 must be determined before one can reliably extract the polarized interference function $$\Delta^q(z,\cos\theta,M_h^2,\phi_R)=\frac{z|\vec R|}{16M_h}\int d^2\vec k_T\, dk^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$3. In the BELLE-based parameterization discussed for $$\Delta^q(z,\cos\theta,M_h^2,\phi_R)=\frac{z|\vec R|}{16M_h}\int d^2\vec k_T\, dk^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$4 pairs, the dependence on $$\Delta^q(z,\cos\theta,M_h^2,\phi_R)=\frac{z|\vec R|}{16M_h}\int d^2\vec k_T\, dk^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$5 cannot be factorized: the data show that the shape in $$\Delta^q(z,\cos\theta,M_h^2,\phi_R)=\frac{z|\vec R|}{16M_h}\int d^2\vec k_T\, dk^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$6 changes with $$\Delta^q(z,\cos\theta,M_h^2,\phi_R)=\frac{z|\vec R|}{16M_h}\int d^2\vec k_T\, dk^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$7, and vice versa. The hadron-pair spectrum is also not purely continuum-like. In the invariant-mass region $$\Delta^q(z,\cos\theta,M_h^2,\phi_R)=\frac{z|\vec R|}{16M_h}\int d^2\vec k_T\, dk^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$8 GeV, the dominant channels are $$\Delta^q(z,\cos\theta,M_h^2,\phi_R)=\frac{z|\vec R|}{16M_h}\int d^2\vec k_T\, dk^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$9, $$\Delta^q(k,P_h,R)_{ij} =\sum_X \int \frac{d^4\xi}{(2\pi)^4}\, e^{ik\cdot \xi}\, \langle 0|{\cal U}^{n_+}_{(-\infty,\xi)}\psi_i^q(\xi)|P_h,R;X\rangle \langle P_h,R;X|\bar\psi_j^q(0){\cal U}^{n_+}_{(0,-\infty)}|0\rangle .$$0 production through both $$\Delta^q(k,P_h,R)_{ij} =\sum_X \int \frac{d^4\xi}{(2\pi)^4}\, e^{ik\cdot \xi}\, \langle 0|{\cal U}^{n_+}_{(-\infty,\xi)}\psi_i^q(\xi)|P_h,R;X\rangle \langle P_h,R;X|\bar\psi_j^q(0){\cal U}^{n_+}_{(0,-\infty)}|0\rangle .$$1 and $$\Delta^q(k,P_h,R)_{ij} =\sum_X \int \frac{d^4\xi}{(2\pi)^4}\, e^{ik\cdot \xi}\, \langle 0|{\cal U}^{n_+}_{(-\infty,\xi)}\psi_i^q(\xi)|P_h,R;X\rangle \langle P_h,R;X|\bar\psi_j^q(0){\cal U}^{n_+}_{(0,-\infty)}|0\rangle .$$2 with the $$\Delta^q(k,P_h,R)_{ij} =\sum_X \int \frac{d^4\xi}{(2\pi)^4}\, e^{ik\cdot \xi}\, \langle 0|{\cal U}^{n_+}_{(-\infty,\xi)}\psi_i^q(\xi)|P_h,R;X\rangle \langle P_h,R;X|\bar\psi_j^q(0){\cal U}^{n_+}_{(0,-\infty)}|0\rangle .$$3 unobserved, and a continuum or “incoherent” $$\Delta^q(k,P_h,R)_{ij} =\sum_X \int \frac{d^4\xi}{(2\pi)^4}\, e^{ik\cdot \xi}\, \langle 0|{\cal U}^{n_+}_{(-\infty,\xi)}\psi_i^q(\xi)|P_h,R;X\rangle \langle P_h,R;X|\bar\psi_j^q(0){\cal U}^{n_+}_{(0,-\infty)}|0\rangle .$$4 background (Courtoy et al., 2010).

Because BELLE provides flavor-separated Monte Carlo histograms, the parameterization distinguishes between light flavors $$\Delta^q(k,P_h,R)_{ij} =\sum_X \int \frac{d^4\xi}{(2\pi)^4}\, e^{ik\cdot \xi}\, \langle 0|{\cal U}^{n_+}_{(-\infty,\xi)}\psi_i^q(\xi)|P_h,R;X\rangle \langle P_h,R;X|\bar\psi_j^q(0){\cal U}^{n_+}_{(0,-\infty)}|0\rangle .$$5 and charm $$\Delta^q(k,P_h,R)_{ij} =\sum_X \int \frac{d^4\xi}{(2\pi)^4}\, e^{ik\cdot \xi}\, \langle 0|{\cal U}^{n_+}_{(-\infty,\xi)}\psi_i^q(\xi)|P_h,R;X\rangle \langle P_h,R;X|\bar\psi_j^q(0){\cal U}^{n_+}_{(0,-\infty)}|0\rangle .$$6. The charm contribution is non-negligible at BELLE energies, especially in the continuum. Separate parameterizations are therefore required for $$\Delta^q(k,P_h,R)_{ij} =\sum_X \int \frac{d^4\xi}{(2\pi)^4}\, e^{ik\cdot \xi}\, \langle 0|{\cal U}^{n_+}_{(-\infty,\xi)}\psi_i^q(\xi)|P_h,R;X\rangle \langle P_h,R;X|\bar\psi_j^q(0){\cal U}^{n_+}_{(0,-\infty)}|0\rangle .$$7, $$\Delta^q(k,P_h,R)_{ij} =\sum_X \int \frac{d^4\xi}{(2\pi)^4}\, e^{ik\cdot \xi}\, \langle 0|{\cal U}^{n_+}_{(-\infty,\xi)}\psi_i^q(\xi)|P_h,R;X\rangle \langle P_h,R;X|\bar\psi_j^q(0){\cal U}^{n_+}_{(0,-\infty)}|0\rangle .$$8, and continuum contributions, and separately for $$\Delta^q(k,P_h,R)_{ij} =\sum_X \int \frac{d^4\xi}{(2\pi)^4}\, e^{ik\cdot \xi}\, \langle 0|{\cal U}^{n_+}_{(-\infty,\xi)}\psi_i^q(\xi)|P_h,R;X\rangle \langle P_h,R;X|\bar\psi_j^q(0){\cal U}^{n_+}_{(0,-\infty)}|0\rangle .$$9 and $z=P_h^-/k^-$0 where appropriate. The fitted unpolarized function is written schematically as

$z=P_h^-/k^-$1

and the fits achieved $z=P_h^-/k^-$2 values around $z=P_h^-/k^-$3 for $z=P_h^-/k^-$4, $z=P_h^-/k^-$5 for $z=P_h^-/k^-$6, $z=P_h^-/k^-$7 for the $z=P_h^-/k^-$8 background, and $z=P_h^-/k^-$9 for the charm background. The key conclusion is that a genuinely two-dimensional dependence on $R=(P_1-P_2)/2$00 is necessary and that a factorized form does not work (Courtoy et al., 2010).

On the polarized side, the same paper states that $R=(P_1-P_2)/2$01 had been computed only in a spectator model at that stage, while the available BELLE information was preliminary and the data were binned in $R=(P_1-P_2)/2$02 and $R=(P_1-P_2)/2$03 with large uncertainties. For that reason the extraction of a detailed functional form was not unique, and a simplified factorized ansatz was adopted,

$R=(P_1-P_2)/2$04

even though the expected physical origin is $R=(P_1-P_2)/2$05-wave interference and therefore a more structured dependence (Courtoy et al., 2010).

This phenomenology suggests two recurring constraints on the subject. First, transverse-momentum-sensitive dihadron fragmentation is strongly shaped by resonance structure and flavor separation. Second, clean extraction of spin-dependent DiFFs requires independent control of unpolarized fragmentation and of the functional dependence on pair kinematics.

6. Model realizations, QCD evolution, and later extensions

The NJL-jet model provides a fully hadronization-based realization of transverse-momentum-sensitive dihadron fragmentation. In that framework, hadronization is a sequential quark-jet process in which the initial quark emits a hadron, the remnant quark continues, and further hadrons are emitted iteratively. The key extension is that the Monte Carlo simulation tracks the transverse momenta of all produced hadrons, assigning each hadron a light-cone fraction $R=(P_1-P_2)/2$06, a transverse momentum $R=(P_1-P_2)/2$07, and a mass $R=(P_1-P_2)/2$08. The pair invariant mass is then constrained by the hadron masses, the momentum fraction ratio, and the relative transverse momentum of the pair (Matevosyan et al., 2013).

The unpolarized DFF is interpreted as a number density,

$R=(P_1-P_2)/2$09

and extracted by simulating quark hadronization, recording all primary hadrons, forming all possible hadron pairs, binning them in $R=(P_1-P_2)/2$10 and $R=(P_1-P_2)/2$11, and averaging over many events. The paper uses $R=(P_1-P_2)/2$12, $R=(P_1-P_2)/2$13, and $R=(P_1-P_2)/2$14, and reports that about 8 hadron-emission steps are enough for convergence in the region $R=(P_1-P_2)/2$15 (Matevosyan et al., 2013).

A major result of the NJL-jet study is that vector-meson decays have a dramatic effect on DFFs, much larger than on ordinary single-hadron FFs, because of combinatorial enhancement in pair counting. The vector mesons included are $R=(P_1-P_2)/2$16, $R=(P_1-P_2)/2$17, $R=(P_1-P_2)/2$18, and $R=(P_1-P_2)/2$19, with only strong decays considered. The model finds visible $R=(P_1-P_2)/2$20, $R=(P_1-P_2)/2$21, $R=(P_1-P_2)/2$22, and $R=(P_1-P_2)/2$23 structures in the relevant channels and large differences between primary-only and full-final-state DFFs. The evolution of the DFFs from the model scale $R=(P_1-P_2)/2$24 to $R=(P_1-P_2)/2$25 and $R=(P_1-P_2)/2$26 is performed with the LO fixed-$R=(P_1-P_2)/2$27 equation

$R=(P_1-P_2)/2$28

using QCDNUM. The qualitative effect is that the overall magnitude decreases, strength shifts to smaller $R=(P_1-P_2)/2$29, and the shape in $R=(P_1-P_2)/2$30 remains largely unchanged while resonance peaks remain visible (Matevosyan et al., 2013).

Later developments extend the role of transverse-momentum-sensitive dihadron fragmentation beyond standard SIDIS and $R=(P_1-P_2)/2$31 transversity observables. In small-$R=(P_1-P_2)/2$32 DIS at NLO, back-to-back dihadron production in the Color Glass Condensate generates a Sudakov double logarithm with coefficient

$R=(P_1-P_2)/2$33

rather than the dijet value $R=(P_1-P_2)/2$34. To preserve the universality of the Sudakov soft factor associated with the Weizsäcker-Williams TMD gluon distribution, the collinear fragmentation functions are promoted to TMD fragmentation functions and resummed through Collins-Soper-Sterman evolution (Caucal et al., 2024).

A further extension relates $R=(P_1-P_2)/2$35 factorization to the small-angle energy-energy correlator. In the collinear region, the EEC jet function can be written as a weighted integral over transverse-momentum-sensitive dihadron fragmentation functions plus a single-hadron term, and the same formalism is used to describe the confinement transition region where $R=(P_1-P_2)/2$36 (Lee et al., 15 Jul 2025).

These developments indicate that transverse-momentum-sensitive dihadron fragmentation functions are not only tools for accessing transversity and helicity in hadronization, but also universal nonperturbative inputs in problems involving back-to-back hadron production, Sudakov resummation, and small-angle hadronic energy correlators.