Dihadron Fragmentation Functions – Overview

• Dihadron Fragmentation Functions (DiFFs) are defined as functions that describe the correlated production of two hadrons from a single fragmenting parton, capturing momentum sharing, invariant mass, and internal geometry.
• They enable the extraction of transversity by linking chiral-odd interference functions like H1^(⟂) to measurable asymmetries in SIDIS, e+e– annihilation, and hadronic collisions.
• DiFFs offer insights into nonperturbative hadronization, influencing resonance production, cascade dynamics, and even higher-twist effects through detailed kinematic and angular analyses.

Dihadron fragmentation functions (DFFs, often DiFFs) are the two-hadron analog of ordinary fragmentation functions: they describe the correlated production of two identified hadrons from a single fragmenting parton, together with unobserved remnants. In the standard pion-pair channel, they encode the dependence on the pair total momentum, invariant mass, and internal geometry, and they become especially valuable when the hadron-pair invariant mass is small compared with the hard scale, so that the pair is naturally described as originating from the same parton rather than from two different hard partons. Their importance is twofold. First, they probe nonperturbative hadronization with a sensitivity to correlations, resonance production, decay chains, and cascade ordering that is absent in single-hadron fragmentation. Second, in collinear factorization they provide experimentally accessible spin analyzers, most notably the chiral-odd interference fragmentation function $H_1^{\sphericalangle}$, which couples to quark transversity, as well as helicity-sensitive structures such as $G_1^\perp$ and $D_{1LL}$ (Courtoy et al., 2010, Pisano et al., 2015, Matevosyan et al., 2017, Huang et al., 2024).

1. Kinematics, correlators, and leading functions

For a quark fragmenting into a hadron pair $h_1 h_2$, the standard pair variables are

$$P_h = P_1 + P_2, \qquad R = \frac{P_1-P_2}{2}, \qquad M_h^2 = P_h^2.$$

In the $\pi^+\pi^-$ channel one also uses

$$z = \frac{P_h^-}{k^-} = z_1+z_2, \qquad \zeta = \frac{2R^-}{P_h^-} = \frac{z_1-z_2}{z},$$

together with the polar angle $\theta$ of one hadron in the pair rest frame and the azimuth $\phi_R$ of the hadron-pair plane. A useful kinematic relation is

$$\frac{2|{\bf R}|}{M_h} = \sqrt{1-\frac{4m_\pi^2}{M_h^2}} .$$

This variable set makes explicit that a DFF retains both the longitudinal momentum sharing of the pair and its internal relative motion (Courtoy et al., 2010, Courtoy et al., 2012, Courtoy et al., 2012).

At operator level, the collinear quark–hadron correlator for a pair can be written as

$G_1^\perp$0

with

$G_1^\perp$1

Dirac traces of this correlator define the leading collinear DiFFs, with $G_1^\perp$2 the unpolarized DiFF and $G_1^\perp$3 the polarized, chiral-odd interference DiFF (Courtoy et al., 2010).

At leading twist, the quark-quark correlator for two hadrons contains four DiFFs: $G_1^\perp$4 After integrating over the partonic transverse momentum, the standard collinear limit retains $G_1^\perp$5 and $G_1^\perp$6, while $G_1^\perp$7 and $G_1^\perp$8 belong to the transverse-momentum-sensitive formulation (Pisano et al., 2015, Matevosyan et al., 2017).

Function	Spin structure	Typical role
$G_1^\perp$9	Unpolarized quark $D_{1LL}$0 unpolarized pair	Unpolarized cross sections and normalization
$D_{1LL}$1	Transverse quark spin correlated with pair-plane orientation	Transversity in SIDIS; Artru–Collins asymmetry
$D_{1LL}$2	Quark helicity correlated with transverse-momentum-sensitive handedness of the pair	Weighted $D_{1LL}$3 and SIDIS helicity observables
$D_{1LL}$4	Longitudinal helicity correlation between the two hadrons for an unpolarized parent parton	Neighboring dihadron helicity correlation

For low pair invariant mass, the partial-wave content is central. The standard expansion gives

$D_{1LL}$5

$D_{1LL}$6

After averaging over $D_{1LL}$7, the dominant surviving pieces are $D_{1LL}$8 and $D_{1LL}$9. Physically, $h_1 h_2$0 is generated by interference between different relative orbital angular momentum states of the hadron pair, especially $h_1 h_2$1- and $h_1 h_2$2-wave components (Courtoy et al., 2010, Pisano et al., 2015).

2. Collinear factorization and the central observables

A defining feature of dihadron production is that, after integration over the transverse momentum of the fragmenting quark, the leading-twist description falls within collinear factorization. The resulting cross sections are products of PDFs and DFFs rather than transverse-momentum convolutions. This is the structural reason DFFs became a preferred route to transversity: the dihadron mechanism survives $h_1 h_2$3 integration, unlike the usual single-hadron Collins effect (Courtoy et al., 2010, Courtoy et al., 2012, Courtoy et al., 2011).

For SIDIS,

$h_1 h_2$4

the leading-twist cross section can be written as

$h_1 h_2$5

with

$h_1 h_2$6

$h_1 h_2$7

Equivalently, after the standard angular integrations, the measured asymmetry takes the form

$h_1 h_2$8

The numerator contains the chiral-odd product $h_1 h_2$9, while the denominator contains the unpolarized product $$P_h = P_1 + P_2, \qquad R = \frac{P_1-P_2}{2}, \qquad M_h^2 = P_h^2.$$0 (Courtoy et al., 2010, Radici et al., 2014).

In $$P_h = P_1 + P_2, \qquad R = \frac{P_1-P_2}{2}, \qquad M_h^2 = P_h^2.$$1 annihilation, the key process is

$$P_h = P_1 + P_2, \qquad R = \frac{P_1-P_2}{2}, \qquad M_h^2 = P_h^2.$$2

The corresponding Artru–Collins asymmetry is sensitive to the product of two interference DiFFs: $$P_h = P_1 + P_2, \qquad R = \frac{P_1-P_2}{2}, \qquad M_h^2 = P_h^2.$$3 This observable calibrates the polarized DiFF in $$P_h = P_1 + P_2, \qquad R = \frac{P_1-P_2}{2}, \qquad M_h^2 = P_h^2.$$4, which can then be used as input in SIDIS to access transversity (Courtoy et al., 2010, Courtoy et al., 2012, Radici et al., 2014).

The same mechanism extends to hadronic collisions. In $$P_h = P_1 + P_2, \qquad R = \frac{P_1-P_2}{2}, \qquad M_h^2 = P_h^2.$$5, a $$P_h = P_1 + P_2, \qquad R = \frac{P_1-P_2}{2}, \qquad M_h^2 = P_h^2.$$6 modulation appears in collinear kinematics, again proportional to transversity multiplied by $$P_h = P_1 + P_2, \qquad R = \frac{P_1-P_2}{2}, \qquad M_h^2 = P_h^2.$$7. This supports the standard universality picture for DiFFs across $$P_h = P_1 + P_2, \qquad R = \frac{P_1-P_2}{2}, \qquad M_h^2 = P_h^2.$$8, SIDIS, and hadron-hadron reactions (Pisano et al., 2015).

3. Extraction programs and the transversity route

The modern phenomenology of DiFFs developed as a two-step program. First, $$P_h = P_1 + P_2, \qquad R = \frac{P_1-P_2}{2}, \qquad M_h^2 = P_h^2.$$9 and $\pi^+\pi^-$0 are constrained from $\pi^+\pi^-$1. Second, the same functions are inserted into SIDIS on transversely polarized targets to extract the transversity PDF $\pi^+\pi^-$2. This strategy produced the first extraction of the $\pi^+\pi^-$3- and $\pi^+\pi^-$4-flavor transversity distributions in the framework of collinear factorization (Courtoy et al., 2012, Courtoy et al., 2011).

The practical complication of the early analyses was that the unpolarized dihadron cross section had not been measured directly. Consequently, the unpolarized DiFF $\pi^+\pi^-$5 was first parametrized to reproduce pion-pair yields from PYTHIA tuned to Belle kinematics, while $\pi^+\pi^-$6 was fitted to Belle’s measured Artru–Collins asymmetry (Courtoy et al., 2012, Courtoy et al., 2012, Radici et al., 2014). Within this framework, the integrated moments

$\pi^+\pi^-$7

enter the SIDIS asymmetry in a particularly transparent way: $\pi^+\pi^-$8 Under the usual isospin and charge-conjugation assumptions for $\pi^+\pi^-$9, proton and deuteron data provide two independent flavor combinations and permit the separation of $$z = \frac{P_h^-}{k^-} = z_1+z_2, \qquad \zeta = \frac{2R^-}{P_h^-} = \frac{z_1-z_2}{z},$$0 and $$z = \frac{P_h^-}{k^-} = z_1+z_2, \qquad \zeta = \frac{2R^-}{P_h^-} = \frac{z_1-z_2}{z},$$1 (Courtoy et al., 2012, Radici et al., 2014).

A decisive methodological update was the Belle reanalysis with the replica method. In that study, the Belle asymmetry points were replicated 100 times, each replica was independently fitted, and the uncertainty bands were defined by the central 68% of the replica ensemble. The point of this procedure was that the fit tends to push the function toward its bounds, making Gaussian or Hessian assumptions unreliable. The resulting transversity extraction, combined with more precise COMPASS proton data, produced a more realistic estimate of the uncertainties and suggested that the valence up component was smaller and had a narrower error band than in the earlier extraction, while the down component was largely unchanged because the deuteron input was unchanged (Radici et al., 2014).

The phenomenological impact of this program extends beyond the extraction of $$z = \frac{P_h^-}{k^-} = z_1+z_2, \qquad \zeta = \frac{2R^-}{P_h^-} = \frac{z_1-z_2}{z},$$2. In the flexible scenario of the 2015 analysis, the isovector tensor charge was quoted as

$$z = \frac{P_h^-}{k^-} = z_1+z_2, \qquad \zeta = \frac{2R^-}{P_h^-} = \frac{z_1-z_2}{z},$$3

and the same DiFF-based transversity determination was used to infer the bounds

$$z = \frac{P_h^-}{k^-} = z_1+z_2, \qquad \zeta = \frac{2R^-}{P_h^-} = \frac{z_1-z_2}{z},$$4

for two different uncertainty treatments. This established DiFF phenomenology as part of the interface between hadron structure and low-energy searches for physics beyond the Standard Model (Courtoy, 2016).

4. Hadronization dynamics, cascade models, and resonance effects

DFFs are more sensitive than single-hadron FFs to the internal structure of the hadronization cascade. In the NJL-jet model, the quark emits hadrons sequentially and emitted hadrons do not reinteract. The corresponding dihadron fragmentation function $$z = \frac{P_h^-}{k^-} = z_1+z_2, \qquad \zeta = \frac{2R^-}{P_h^-} = \frac{z_1-z_2}{z},$$5 satisfies a recursive equation whose first two terms are “driving terms” and whose third term accounts for the case in which both observed hadrons are produced after the first step of the cascade: $$z = \frac{P_h^-}{k^-} = z_1+z_2, \qquad \zeta = \frac{2R^-}{P_h^-} = \frac{z_1-z_2}{z},$$6 The physical interpretation is direct: at large $$z = \frac{P_h^-}{k^-} = z_1+z_2, \qquad \zeta = \frac{2R^-}{P_h^-} = \frac{z_1-z_2}{z},$$7 or $$z = \frac{P_h^-}{k^-} = z_1+z_2, \qquad \zeta = \frac{2R^-}{P_h^-} = \frac{z_1-z_2}{z},$$8, the driving terms dominate because one of the hadrons is likely to have been produced in the first emission step; at smaller $$z = \frac{P_h^-}{k^-} = z_1+z_2, \qquad \zeta = \frac{2R^-}{P_h^-} = \frac{z_1-z_2}{z},$$9, the higher-order cascade term becomes more significant. In some channels, particularly for strange-quark initiated fragmentation, the driving term can vanish and the entire DFF is generated recursively (Casey et al., 2012, Casey et al., 2012).

Monte Carlo implementations exploit a number-density interpretation: $\theta$0 Operationally, one bins all hadron pairs produced in a large sample of simulated fragmentation chains and extracts the DFF from the average pair multiplicity in each $\theta$1 bin. This procedure can be applied to primary hadrons only, or to the full final state after strong resonance decays (Matevosyan et al., 2013, Matevosyan et al., 2013).

A major result of the hadronization studies is that vector meson decays drastically affect pion DFFs. In the benchmark channel $\theta$2, including pions from vector meson decays makes the DFF much larger, typically up to an order of magnitude larger, than the primary-pion result. The enhancement is not merely a resonance correction but a combinatorial effect: once a vector meson decays, each decay product can pair with other hadrons in the event. In the restricted $\theta$3- and $\theta$4-only example discussed in the literature, the listed secondary channels generate a total of 17 possible $\theta$5 pairs, compared with only the single direct primary channel. The invariant-mass dependence correspondingly acquires a pronounced $\theta$6 peak near $\theta$7 and a low-$\theta$8 enhancement from $\theta$9 when the $\phi_R$0 is unobserved. The fact that the same qualitative behavior appears in both the NJL-jet model and PYTHIA 8.1 with Lund string fragmentation shows that this is not a model artifact (Matevosyan et al., 2013, Matevosyan et al., 2013).

These studies also show that naive factorization assumptions in the $\phi_R$1 dependence are generally unsafe. In the Belle Monte Carlo analysis of the unpolarized cross section, the invariant-mass shape changes across $\phi_R$2-bins, and the continuum decreases with $\phi_R$3 differently for light and charm flavors. That finding motivated genuinely two-dimensional parameterizations of $\phi_R$4 rather than products of separate $\phi_R$5- and $\phi_R$6-dependent functions (Courtoy et al., 2010).

5. Evolution equations, perturbative regimes, and helicity-sensitive extensions

At low scales, model DFFs must be evolved to experimental scales. In the early NJL-jet program, the model input was defined at $\phi_R$7, and leading-order evolution to $\phi_R$8 or $\phi_R$9 was found to shift strength toward lower $$\frac{2|{\bf R}|}{M_h} = \sqrt{1-\frac{4m_\pi^2}{M_h^2}} .$$0, suppressing the high-$$\frac{2|{\bf R}|}{M_h} = \sqrt{1-\frac{4m_\pi^2}{M_h^2}} .$$1 region while broadly preserving the basic $$\frac{2|{\bf R}|}{M_h} = \sqrt{1-\frac{4m_\pi^2}{M_h^2}} .$$2 structure. For DFFs integrated over invariant mass, the LO evolution equations contain both a homogeneous term and an inhomogeneous term built from single-hadron FFs, reflecting the possibility that the two observed hadrons are produced after a perturbative splitting into two daughter partons (Casey et al., 2012, Casey et al., 2012, Matevosyan et al., 2013).

This inhomogeneous structure became conceptually explicit in the 2024 analysis of neighboring dihadron helicity correlations. For the unpolarized DiFF,

$$\frac{2|{\bf R}|}{M_h} = \sqrt{1-\frac{4m_\pi^2}{M_h^2}} .$$3

whereas the correlated helicity DiFF $$\frac{2|{\bf R}|}{M_h} = \sqrt{1-\frac{4m_\pi^2}{M_h^2}} .$$4 obeys an analogous equation in which the source term involves the longitudinal spin-transfer FF $$\frac{2|{\bf R}|}{M_h} = \sqrt{1-\frac{4m_\pi^2}{M_h^2}} .$$5: $$\frac{2|{\bf R}|}{M_h} = \sqrt{1-\frac{4m_\pi^2}{M_h^2}} .$$6 This means that even if a DiFF is set to zero at a low scale, perturbative splittings regenerate it at higher scales. In the specific $$\frac{2|{\bf R}|}{M_h} = \sqrt{1-\frac{4m_\pi^2}{M_h^2}} .$$7 study, the helicity-correlation observable

$$\frac{2|{\bf R}|}{M_h} = \sqrt{1-\frac{4m_\pi^2}{M_h^2}} .$$8

was shown to be sensitive to the flavor structure of $$\frac{2|{\bf R}|}{M_h} = \sqrt{1-\frac{4m_\pi^2}{M_h^2}} .$$9, especially for strange quarks and gluons, thereby opening a new way to probe circularly polarized gluon hadronization in unpolarized collisions (Huang et al., 2024).

A complementary helicity-sensitive DiFF is $G_1^\perp$00, which correlates the longitudinal polarization of a fragmenting quark with the transverse-momentum structure of the hadron pair. A crucial clarification of the literature is that the original BELLE search found no signal because the previously proposed unweighted asymmetry actually vanishes. The corrected observables are weighted asymmetries: in $G_1^\perp$01, a $G_1^\perp$02-weighted sine/cosine combination of azimuths isolates products of Fourier moments of $G_1^\perp$03; in SIDIS, a $G_1^\perp$04 weight yields a collinear asymmetry proportional to $G_1^\perp$05. Quark-jet Monte Carlo studies generate a nonvanishing $G_1^\perp$06, but also predict that the effect is small and becomes relatively suppressed as the number of hadron emissions increases (Matevosyan et al., 2017, Matevosyan et al., 2017).

In the large-invariant-mass regime, DFFs admit a perturbative treatment. For $G_1^\perp$07, the hadron pair is described as a perturbative splitting followed by ordinary single-hadron fragmentation. In this regime,

$G_1^\perp$08

so the corresponding SIDIS transverse-spin asymmetry behaves like $G_1^\perp$09. The same analysis showed that $G_1^\perp$10 and the Collins fragmentation function are driven by the same underlying twist-3 collinear fragmentation correlators, and that in the intermediate region $G_1^\perp$11, collinear factorization in terms of DiFFs and collinear factorization in terms of single-hadron FFs give the same result (Zhou et al., 2011).

6. Interpretation, conceptual debates, and recent directions

A recurrent conceptual issue is how literally one should interpret DFFs as number densities. The current position of the field is qualified rather than absolute. DFFs do support a legitimate number-density interpretation in the same sense as ordinary fragmentation functions: as renormalized, operator-based densities for identified hadrons in restricted regions of phase space, tied to appropriate factorization theorems. What is not settled is the stronger claim that a literal global multiplicity sum rule,

$G_1^\perp$12

should be taken as a defining first-principles constraint. The objection is that the naive derivation fails because the fragmenting quark state and the purely hadronic asymptotic states are orthogonal in the usual completeness relation. The practical conclusion is that factorization, not a disputed multiplicity sum rule, should be the guiding criterion for acceptable DFF definitions (Rogers et al., 2024).

This debate matters because slightly different operator definitions and phase-space parametrizations have been used across the DiFF literature. The present view is that such differences do not automatically invalidate a number-density interpretation, provided the corresponding hard parts and evolution kernels are adjusted consistently. A plausible implication is that the DiFF sector should be treated much like the rest of QCD factorization: renormalized distributions are “probability-like” or quasiprobability objects, but not literal probabilities once ultraviolet subtraction and scheme dependence are taken seriously (Rogers et al., 2024).

The scope of DiFF phenomenology has also broadened substantially. At subleading twist, the same $G_1^\perp$13 that enables transversity extraction provides the cleanest access to the poorly known twist-3 PDF $G_1^\perp$14 through beam-spin asymmetries in dihadron SIDIS, while related observables also involve $G_1^\perp$15. In the 2015 review framework, this placed DiFFs at the center of a program linking hadron-pair production to dynamical chiral symmetry breaking, the nucleon scalar structure, and higher-twist quark-gluon dynamics (Pisano et al., 2015, Courtoy, 2016).

Two recent developments extend DiFFs into new territories. First, a 2025 extraction of unpolarized $G_1^\perp$16 DFFs used Belle measurements of the differential unpolarized cross section rather than relying only on Monte Carlo yields. That analysis fitted 344 data points after cuts, employed NNLO coefficient functions implemented in APFEL++, compared a 71-parameter physics-informed fit with a 205-parameter neural-network parametrization, and found that the neural-network fit achieved a lower $G_1^\perp$17 but also exhibited larger uncertainty bands, especially for the essentially unconstrained gluon DFF (Mahaut et al., 15 Sep 2025). Second, the 2025 EEC study showed that in the collinear limit of the energy-energy correlator, the relevant jet function can be written directly in terms of single-hadron FFs and transverse-momentum-sensitive DFFs. In that formulation, DFFs become the universal nonperturbative functions governing the confinement transition region $G_1^\perp$18, thereby linking hadronization in $G_1^\perp$19 to the nonperturbative structure of energy correlators (Lee et al., 15 Jul 2025).

Taken together, these developments define the present status of DFFs. They are simultaneously hadronization observables, spin analyzers, inputs to collinear extractions of transversity and higher-twist PDFs, test objects for universality across $G_1^\perp$20, SIDIS, and hadronic collisions, and increasingly precise nonperturbative functions in their own right. The central lesson of the field is not merely that two hadrons carry more information than one, but that the pair invariant mass, relative momentum, partial-wave content, and spin correlations provide a uniquely efficient organization of that information in QCD (Courtoy et al., 2010, Pisano et al., 2015, Mahaut et al., 15 Sep 2025).