Unpolarized Dihadron Fragmentation Functions

• Unpolarized dihadron fragmentation functions are leading-twist QCD correlation functions that quantify the probability for an unpolarized parton to fragment into a dihadron with specified kinematics.
• They serve as the critical unpolarized baseline in SIDIS and e⁺e⁻ processes, enabling the extraction of chiral-odd fragmentation functions and transversity distributions.
• Recent phenomenological extractions employ NNLO analyses and Monte Carlo simulations to reveal detailed flavor and momentum dependencies that guide future experimental investigations.

Searching arXiv for recent and foundational papers on unpolarized dihadron fragmentation functions. Searching arXiv for multihadron factorization work relevant to operator definitions and universality. Unpolarized dihadron fragmentation functions are leading-twist QCD correlation functions that describe the fragmentation of an unpolarized parton into a correlated hadron pair with specified longitudinal momentum fractions and invariant mass. In the collinear framework they are usually denoted $D_1^q$ for quark flavor $q$, and they play a dual role: they are intrinsic nonperturbative observables of multihadron hadronization, and they are the indispensable unpolarized baseline in dihadron spin asymmetries used to extract chiral-odd fragmentation functions and the nucleon transversity distribution (Courtoy et al., 2010, Rogers et al., 2024).

1. Operator definition and conceptual status

The standard starting point is the quark-to-dihadron fragmentation correlator $\Delta^q$, which encodes the amplitude for a quark with momentum $k$ to produce a hadron pair with total momentum $P_h$, relative momentum $R$, and unobserved remnants $X$. In the collinear treatment of $\pi^+\pi^-$ production, the fully unintegrated correlator is reduced to a partially integrated object,

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

with $z=P_h^-/k^-$, $q$0, and $q$1 specifying the pair orientation. The unpolarized DiFF is the scalar coefficient multiplying the leading Dirac structure, conventionally extracted as

$q$2

Physically, it is the probability density for an unpolarized quark to fragment into an unpolarized dihadron with given $q$3 (Courtoy et al., 2010).

The same object can also be written in the more general multihadron language as a fragmentation correlator with the single-hadron state $q$4 replaced by a multihadron state $q$5. A central recent result is that no modification of the standard operator definition is required for a small-mass multihadron state: the usual prefactor, hard parts, and evolution kernels remain valid, while recently proposed definitions with extra nonuniversal prefactors do not (Rogers et al., 2024).

In the notation often used in reviews, the collinear unpolarized DiFF is written as $q$6, or after changing variables to pair-level quantities as $q$7. It is the direct two-hadron analogue of the ordinary unpolarized single-hadron fragmentation function $q$8, but with additional sensitivity to the internal kinematics of the hadron pair (Radici, 2011).

2. Kinematics, invariant mass, and partial-wave content

The basic pair variables are

$q$9

together with the pair momentum fraction

$\Delta^q$0

For equal-mass pions,

$\Delta^q$1

so the invariant mass directly controls the relative momentum inside the pair. A commonly used auxiliary variable is

$\Delta^q$2

which parameterizes the longitudinal sharing of momentum inside the pair in terms of the polar angle $\Delta^q$3 in the pair center-of-mass frame (Courtoy et al., 2010, Mahaut et al., 15 Sep 2025).

Before angular integration, $\Delta^q$4 depends on $\Delta^q$5, $\Delta^q$6, and $\Delta^q$7. A partial-wave expansion in Legendre polynomials organizes the pair into relative orbital angular momentum channels. In the low-mass region relevant for $\Delta^q$8 phenomenology, the dominant structure is the $\Delta^q$9- and $k$0-wave sector. After integrating over $k$1, one typically works with

$k$2

which is the sum of the $k$3-wave and $k$4-wave contributions to the unpolarized pair yield (Courtoy et al., 2010).

This partial-wave decomposition is not merely formal. After averaging over $k$5, the term corresponding to the unpolarized pair created in a relative $k$6 state survives in the $k$7 expansion, whereas the $k$8 interference term survives for the chiral-odd interference DiFF $k$9. This clean separation underlies the use of $P_h$0 as the spin-averaged baseline and $P_h$1 as the spin-sensitive numerator in transverse-spin observables (Radici et al., 2014).

3. Appearance in factorized cross sections

In two-hadron SIDIS,

$P_h$2

the leading-twist structure functions are

$P_h$3

$P_h$4

After the standard angular projection, the single-spin asymmetry takes the form

$P_h$5

Hence $P_h$6 furnishes the unpolarized denominator against which the chiral-odd transversity signal is normalized (Radici et al., 2014).

The same logic governs back-to-back dihadron production in $P_h$7 annihilation. For

$P_h$8

the Artru–Collins asymmetry is proportional to

$P_h$9

Again, the unpolarized DiFFs are the baseline cross section in the denominator, while the polarized interference DiFFs build the azimuthal numerator (Courtoy et al., 2010).

For integrated analyses it is convenient to define

$R$0

These quantities enter compact formulas for both SIDIS and $R$1 asymmetries, making explicit that the extraction of transversity depends not only on $R$2 but also on accurate knowledge of the integrated unpolarized DiFF $R$3 (Courtoy et al., 2011).

4. Phenomenological extraction and parameterization

Historically, the first dedicated phenomenology of $R$4 proceeded in the absence of published unpolarized $R$5 two-pion cross sections. A first extraction of $R$6 therefore had to determine $R$7 from a Monte Carlo simulation of the Belle cross section, using PYTHIA tuned to Belle kinematics and a channel decomposition in $R$8, $R$9, $X$0, and continuum contributions (Courtoy et al., 2012).

A more detailed analysis of the Belle Monte Carlo histograms found that unpolarized $X$1 production in the region $X$2 GeV receives contributions from a prominent $X$3 peak around $X$4 MeV, a small $X$5 peak near $X$6 MeV, a broader $X$7 structure around $X$8 MeV when the $X$9 is unobserved, and a substantial continuum. The same study separated light-flavor and charm components and emphasized that no reasonable fit can factorize the $\pi^+\pi^-$0- and $\pi^+\pi^-$1-dependences as $\pi^+\pi^-$2; the function must be modeled as genuinely two-dimensional in $\pi^+\pi^-$3. Reported fit qualities were $\pi^+\pi^-$4 for the $\pi^+\pi^-$5 channel, $\pi^+\pi^-$6 for the $\pi^+\pi^-$7 channel, $\pi^+\pi^-$8 for the $\pi^+\pi^-$9 continuum, and $$\Delta^q(z,\cos\theta,M_h^2,\phi_R) = \frac{z |\vec R|}{16\,M_h}\int d^2 \vec k_T \; d k^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$0 for the charm continuum (Courtoy et al., 2010).

Subsequent fits supporting transversity extractions continued to rely on Monte Carlo-based unpolarized DiFFs. One analysis parameterized $$\Delta^q(z,\cos\theta,M_h^2,\phi_R) = \frac{z |\vec R|}{16\,M_h}\int d^2 \vec k_T \; d k^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$1 at $$\Delta^q(z,\cos\theta,M_h^2,\phi_R) = \frac{z |\vec R|}{16\,M_h}\int d^2 \vec k_T \; d k^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$2 using three resonant channels $$\Delta^q(z,\cos\theta,M_h^2,\phi_R) = \frac{z |\vec R|}{16\,M_h}\int d^2 \vec k_T \; d k^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$3 plus a continuum, generated flavor-tagged two-pion yields with PYTHIA on a grid of $$\Delta^q(z,\cos\theta,M_h^2,\phi_R) = \frac{z |\vec R|}{16\,M_h}\int d^2 \vec k_T \; d k^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$4 bins in $$\Delta^q(z,\cos\theta,M_h^2,\phi_R) = \frac{z |\vec R|}{16\,M_h}\int d^2 \vec k_T \; d k^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$5, and used 79 free parameters, obtaining an average $$\Delta^q(z,\cos\theta,M_h^2,\phi_R) = \frac{z |\vec R|}{16\,M_h}\int d^2 \vec k_T \; d k^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$6 (Radici et al., 2014).

A later NNLO extraction replaced much of this Monte Carlo dependence with direct use of Belle unpolarized cross-section data. That analysis used Belle measurements at $$\Delta^q(z,\cos\theta,M_h^2,\phi_R) = \frac{z |\vec R|}{16\,M_h}\int d^2 \vec k_T \; d k^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$7 GeV with luminosity $$\Delta^q(z,\cos\theta,M_h^2,\phi_R) = \frac{z |\vec R|}{16\,M_h}\int d^2 \vec k_T \; d k^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$8 fb$$\Delta^q(z,\cos\theta,M_h^2,\phi_R) = \frac{z |\vec R|}{16\,M_h}\int d^2 \vec k_T \; d k^+\, \Delta^q(k;P_h,R)\Big|_{k^- = P_h^-/z},$$9, imposed the cuts $z=P_h^-/k^-$0, $z=P_h^-/k^-$1, $z=P_h^-/k^-$2, and excluded the narrow $z=P_h^-/k^-$3 bin $z=P_h^-/k^-$4, leaving 344 data points. Flavor separation was still supplemented by PYTHIA ratios because the data are only weakly flavor sensitive, but the perturbative description was upgraded to NNLO with APFEL++, and both a physics-informed fit and a neural-network fit were performed. The NNLO mean-replica fit qualities were $z=P_h^-/k^-$5 for the physics-informed parametrization and $z=P_h^-/k^-$6 for the neural-network parametrization, while the gluon DiFF remained weakly constrained by $z=P_h^-/k^-$7 data alone (Mahaut et al., 15 Sep 2025).

5. Evolution, universality, and theoretical consistency

At fixed pair invariant mass, unpolarized DiFFs obey standard collinear DGLAP evolution. In schematic form,

$z=P_h^-/k^-$8

with $z=P_h^-/k^-$9 acting as a spectator variable. This is the same evolution structure as for single-hadron fragmentation, and it is essential when relating Belle measurements near $q$00 to SIDIS measurements at a few $q$01 (Courtoy et al., 2011).

In practical transversity phenomenology, this evolution is numerically significant. One extraction found that the ratio $q$02 decreases when evolving from Belle’s scale to HERMES scales by a factor $q$03, yielding

$q$04

to be used in the SIDIS transversity extraction (Courtoy et al., 2011).

The universality status of multihadron fragmentation functions has recently been revisited at the factorization-theorem level. A dedicated rederivation of semi-inclusive $q$05 factorization for a small-mass $q$06-hadron final state confirmed that the operator definition of the fragmentation function is identical in structure to the single-hadron case, with the same hard parts and evolution kernels. In particular, the standard operator definition with its usual prefactor remains valid for the dihadron case, while modified definitions with nonuniversal prefactors do not. This directly supports the theoretical consistency of earlier dihadron phenomenology (Rogers et al., 2024).

Within the broader fragmentation taxonomy, $q$07 is the chiral-even, spin-averaged two-hadron analogue of the ordinary unpolarized single-hadron FF. Its special importance is that, unlike the Collins mechanism, the dihadron method survives integration over partonic transverse momentum and therefore permits a transversity program entirely within collinear factorization (Radici, 2011).

6. Models, uncertainties, and current directions

Model calculations have been central to understanding the structure of unpolarized DiFFs outside the kinematic reach of direct fits. In the NJL-jet framework, unpolarized DFFs for pions, kaons, and vector mesons were computed by Monte Carlo simulation of quark hadronization chains, including the transverse momentum of produced hadrons and the strong decays of vector mesons. A key result was that pseudoscalar-meson DFFs are strongly influenced by vector-meson decays because of large combinatorial factors in counting hadron pairs that include decay products. The same work also evolved the DFFs from the model scale to experimental scales (Matevosyan et al., 2013).

A contrasting calculation in the nonlocal chiral quark model produced unpolarized DiFFs for pions and kaons and found substantial differences relative to the NJL-jet model. It also compared $q$08 and $q$09 at $q$10 with JETSET-based parametrizations, reinforcing that flavor dependence and small-$q$11 structure are model sensitive when direct data are sparse (Yang et al., 2014).

Within the quark-jet model, an explicit two-step integral expression for the integrated unpolarized DiFF $q$12 was derived and validated against Monte Carlo simulations. That study showed strong enhancement at small $q$13 as the number of emitted hadrons increases, driven by combinatorial pair production, and thereby clarified how even spin-dependent DiFF analyses rely on an accurately modeled $q$14 baseline (Matevosyan et al., 2017).

A later empirical estimate used the single-cascade jet algorithm with elementary fragmentation functions tuned to reproduce favored DSS17 single-hadron FFs, then generated uDiFFs for $q$15, $q$16, and $q$17. The resulting uDiFFs differed significantly in magnitude and flavor dependence from both NJL and nonlocal chiral-quark-model predictions, suggesting that single-hadron empirical constraints do not map trivially onto existing model DiFFs (Yang et al., 2019).

Current applications now extend beyond the original transversity program. In particular, transverse-spin correlations of back-to-back $q$18 pairs in unpolarized $q$19, $q$20, and $q$21 collisions have been formulated as observables whose denominators are governed by products of unpolarized fragmentation functions $q$22, while the numerators probe chiral-odd spin-transfer fragmentation functions. This suggests that unpolarized measurements can continue to sharpen the baseline needed for extracting more general spin-dependent multihadron fragmentation observables (Yang et al., 2024).

Historically, the principal limitation of $q$23 phenomenology was the absence of direct unpolarized pair-production data, which forced reliance on event generators. Direct Belle cross sections and NNLO analyses have substantially improved the quark-sector determination, but the gluon DiFF remains essentially unconstrained in current $q$24 data (Radici et al., 2014, Mahaut et al., 15 Sep 2025). A plausible implication is that future SIDIS, hadron-collider, and Electron-Ion Collider measurements will be needed to complete the flavor and gluon decomposition while preserving the standard collinear-factorization framework now reaffirmed for multihadron fragmentation (Rogers et al., 2024).