Di-Hadron Fragmentation Formalism

• Di-hadron fragmentation formalism is a framework that describes the hadronization of a parton into a correlated hadron pair using nonperturbative DiFFs.
• The formalism employs detailed kinematic definitions and projections to isolate key functions like D1 and the interference H1⌀ for spin observables.
• It underpins collinear factorization analysis and supports advanced studies in SIDIS, e+e− annihilation, and hadronic collisions.

Searching arXiv for recent and foundational papers on dihadron fragmentation formalism. Di-hadron fragmentation formalism describes the hadronization of a parton into a correlated hadron pair within the same jet and organizes the corresponding nonperturbative information into dihadron fragmentation functions (DiFFs). In its standard form, the basic process is $q\to h_1 h_2 X$, with the pair characterized by a total momentum $P_h=P_1+P_2$, a relative momentum $R=(P_1-P_2)/2$, light-cone momentum fractions, and an invariant mass $M_h$ (Courtoy et al., 2010). The formalism is used in collinear and transverse-momentum-dependent settings, supports leading- and subleading-twist analyses, and is central to spin observables because the hadron pair supplies an internal orientation that survives even after integrating over the fragmenting parton transverse momentum (Pisano et al., 2015). A major consequence is that the chiral-odd interference fragmentation function $H_1^{\sphericalangle}$ can couple to transversity in standard collinear factorization, while unpolarized DiFFs such as $D_1$ govern pair production rates and invariant-mass spectra (Courtoy et al., 2011).

1. Foundational definitions and kinematics

The core fragmentation process is $q(k)\to h_1(P_1,M_1)+h_2(P_2,M_2)+X$, with pair variables

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

and light-cone momentum fractions

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

in the conventions used in the review literature (Pisano et al., 2015). In semi-inclusive deep-inelastic scattering and related collinear treatments, the pair can also be described by $(z,\cos\theta,M_h^2,\phi_R)$, where $P_h=P_1+P_2$0 is the polar angle of one hadron in the pair center-of-mass frame and $P_h=P_1+P_2$1 is the azimuthal angle associated with the pair plane (Courtoy et al., 2010).

A complementary collinear DiFF definition used in recent work on near-side energy-energy correlators introduces

$P_h=P_1+P_2$2

defined in the dihadron frame, where the total dihadron momentum has no transverse momentum and a large minus light-cone component (Kang et al., 23 Jul 2025). In that formulation,

$P_h=P_1+P_2$3

and the DiFF is interpreted as a density in $P_h=P_1+P_2$4 for producing a hadron pair from fragmentation of parton $P_h=P_1+P_2$5 (Kang et al., 23 Jul 2025).

The small-invariant-mass condition is structurally important. A recent factorization analysis states that collinear factorization for a multihadron state applies when the observed system has small invariant mass relative to the hard scale,

$P_h=P_1+P_2$6

so that the hadron pair belongs to a single collinear jet initiated by one parton (Rogers et al., 2024). This same condition underlies the standard use of DiFFs in SIDIS and $P_h=P_1+P_2$7 annihilation, and it also explains why the formalism is distinct from descriptions in which two observed hadrons are produced in separate jets or by separate partons (Celiberto et al., 2016).

2. Correlators, projections, and the leading-twist DiFF basis

The field-theoretic starting point is the quark-quark fragmentation correlator. In the compact review formulation, after integrating over $P_h=P_1+P_2$8, the projected correlator is written as

$P_h=P_1+P_2$9

with the unintegrated correlator

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

(Courtoy et al., 2010). The corresponding leading-twist projections isolate the unpolarized DiFF $R=(P_1-P_2)/2$1 and the polarized interference fragmentation function $R=(P_1-P_2)/2$2 (Courtoy et al., 2010).

In the more differential leading-twist decomposition for an unpolarized hadron pair, four DiFFs appear:

• $R=(P_1-P_2)/2$3,
• $R=(P_1-P_2)/2$4,
• $R=(P_1-P_2)/2$5,
• $R=(P_1-P_2)/2$6

with explicit dependence on $R=(P_1-P_2)/2$7 or equivalent variables (Matevosyan et al., 2017). The associated quark-quark correlator projections are given by

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

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

$M_h$0

(Matevosyan et al., 2017).

A characteristic feature of the formalism is partial-wave decomposition. For low invariant mass, the hadron pair is mainly produced in relative $M_h$1- and $M_h$2-waves, and the review literature writes

$M_h$3

$M_h$4

(Pisano et al., 2015). This is why $M_h$5 is also called an interference fragmentation function: it is associated with interference between pair states in different partial waves (Courtoy et al., 2010).

At subleading twist, additional DiFFs appear. After $M_h$6 integration, the twist-3 decomposition contains $M_h$7, $M_h$8, $M_h$9, and $H_1^{\sphericalangle}$0, together with quark-gluon-quark combinations $H_1^{\sphericalangle}$1, $H_1^{\sphericalangle}$2, $H_1^{\sphericalangle}$3, and $H_1^{\sphericalangle}$4 (Pisano et al., 2015). This twist hierarchy becomes operational in beam-spin and longitudinal-target observables, where the same chiral-odd $H_1^{\sphericalangle}$5 appears with twist-3 parton distributions such as $H_1^{\sphericalangle}$6 and $H_1^{\sphericalangle}$7 (Pisano et al., 2015).

3. Collinear factorization and spin observables

The formal advantage emphasized across the DiFF literature is that two-hadron production retains a spin-sensitive structure after integrating over the quark transverse momentum, so the relevant observables can be analyzed in standard collinear factorization rather than in a transverse-momentum-dependent convolution framework (Courtoy et al., 2011). In the transversity program, the key SIDIS process is

$H_1^{\sphericalangle}$8

and the target-spin asymmetry is written as

$H_1^{\sphericalangle}$9

with

$D_1$0

(Courtoy et al., 2011). This formulation yields the transversity combination

$D_1$1

under the flavor assumptions summarized in that analysis (Courtoy et al., 2011).

The same chiral-odd DiFF enters $D_1$2 annihilation through the Artru-Collins asymmetry in

$D_1$3

where the asymmetry is proportional to the product $D_1$4 normalized by $D_1$5 (Courtoy et al., 2010). In integrated form, the analysis that extracted transversity from HERMES and Belle used

$D_1$6

and obtained

$D_1$7

before evolution to HERMES scales (Courtoy et al., 2011).

The same DiFF structure also appears in hadronic collisions. For

$D_1$8

the leading-twist polarized cross section contains a $D_1$9 modulation proportional to

$q(k)\to h_1(P_1,M_1)+h_2(P_2,M_2)+X$0

(Pisano et al., 2015). This process has been used as a nontrivial universality test because the same transversity distribution and the same interference DiFF extracted from SIDIS and $q(k)\to h_1(P_1,M_1)+h_2(P_2,M_2)+X$1 should also describe the $q(k)\to h_1(P_1,M_1)+h_2(P_2,M_2)+X$2 asymmetry (Pisano et al., 2015).

At subleading twist, DiFFs provide access to further nucleon structure. The beam-spin asymmetry in dihadron SIDIS contains

$q(k)\to h_1(P_1,M_1)+h_2(P_2,M_2)+X$3

plus a contamination from $q(k)\to h_1(P_1,M_1)+h_2(P_2,M_2)+X$4, and this channel is described in the review as the cleanest access to the poorly known twist-3 distribution $q(k)\to h_1(P_1,M_1)+h_2(P_2,M_2)+X$5 (Pisano et al., 2015).

4. Evolution, perturbative limits, and model realizations

QCD evolution of DiFFs has a standard homogeneous part analogous to that of single-hadron fragmentation functions. In the NJL-jet treatment of unpolarized dihadron fragmentation, the leading-order evolution equation is written as

$q(k)\to h_1(P_1,M_1)+h_2(P_2,M_2)+X$6

(Casey et al., 2012). In that work, model results obtained at

$q(k)\to h_1(P_1,M_1)+h_2(P_2,M_2)+X$7

are evolved to

$q(k)\to h_1(P_1,M_1)+h_2(P_2,M_2)+X$8

and the evolved DiFFs show “a shift in the peak of the model results towards the lower $q(k)\to h_1(P_1,M_1)+h_2(P_2,M_2)+X$9 region after the evolution” (Casey et al., 2012).

A distinct perturbative regime arises at large pair invariant mass. In that limit,

$$P_h=P_1+P_2,\qquad R=\frac{P_1-P_2}{2},\qquad P_h^2=M_h^2$$0

the pair is produced by a perturbative splitting followed by ordinary single-hadron fragmentation (Zhou et al., 2011). The paper on large-invariant-mass DiFFs derives

$$P_h=P_1+P_2,\qquad R=\frac{P_1-P_2}{2},\qquad P_h^2=M_h^2$$1

and shows that the interference DiFF $$P_h=P_1+P_2,\qquad R=\frac{P_1-P_2}{2},\qquad P_h^2=M_h^2$$2 is controlled by the same twist-3 fragmentation correlators $$P_h=P_1+P_2,\qquad R=\frac{P_1-P_2}{2},\qquad P_h^2=M_h^2$$3 and $$P_h=P_1+P_2,\qquad R=\frac{P_1-P_2}{2},\qquad P_h^2=M_h^2$$4 that also govern the Collins function at large transverse momentum (Zhou et al., 2011). In the intermediate region

$$P_h=P_1+P_2,\qquad R=\frac{P_1-P_2}{2},\qquad P_h^2=M_h^2$$5

the same study shows that collinear factorization in terms of DiFFs and collinear factorization in terms of single-hadron fragmentation functions provide the same result (Zhou et al., 2011).

The NJL-jet program provides a microscopic hadronization model for unpolarized DiFFs. In the early formulation, the dihadron fragmentation function

$$P_h=P_1+P_2,\qquad R=\frac{P_1-P_2}{2},\qquad P_h^2=M_h^2$$6

is defined as the probability of observing two hadrons with light-cone momentum fractions $$P_h=P_1+P_2,\qquad R=\frac{P_1-P_2}{2},\qquad P_h^2=M_h^2$$7 and $$P_h=P_1+P_2,\qquad R=\frac{P_1-P_2}{2},\qquad P_h^2=M_h^2$$8, and obeys the recursive equation

$$P_h=P_1+P_2,\qquad R=\frac{P_1-P_2}{2},\qquad P_h^2=M_h^2$$9

(Casey et al., 2012). The first two terms are interpreted as the driving function, while the last term is the higher-order cascade contribution (Casey et al., 2012). Numerically, the work finds that the driving term dominates when either $$z=\frac{P_h^-}{k^-}=z_1+z_2,\qquad \zeta=\frac{2R^-}{P_h^-}=\frac{z_1-z_2}{z}$$0 or $$z=\frac{P_h^-}{k^-}=z_1+z_2,\qquad \zeta=\frac{2R^-}{P_h^-}=\frac{z_1-z_2}{z}$$1 is large, whereas the higher-order term becomes more important as one of the observed momentum fractions is lowered (Casey et al., 2012).

A later Monte Carlo implementation generalized this to $$z=\frac{P_h^-}{k^-}=z_1+z_2,\qquad \zeta=\frac{2R^-}{P_h^-}=\frac{z_1-z_2}{z}$$2, included transverse momentum generation and vector-meson decays, and found that pseudoscalar-meson DFFs are strongly influenced by decays of relevant vector mesons because of the large combinatorial factors involved in counting hadron pairs that include decay products (Matevosyan et al., 2013).

For polarized fragmentation, the quark-jet Monte Carlo study of transverse-spin-dependent DiFFs extracted Fourier cosine moments of $$z=\frac{P_h^-}{k^-}=z_1+z_2,\qquad \zeta=\frac{2R^-}{P_h^-}=\frac{z_1-z_2}{z}$$3 and $$z=\frac{P_h^-}{k^-}=z_1+z_2,\qquad \zeta=\frac{2R^-}{P_h^-}=\frac{z_1-z_2}{z}$$4 from number densities and showed that, in SIDIS, the integrated combination is

$$z=\frac{P_h^-}{k^-}=z_1+z_2,\qquad \zeta=\frac{2R^-}{P_h^-}=\frac{z_1-z_2}{z}$$5

whereas in the $$z=\frac{P_h^-}{k^-}=z_1+z_2,\qquad \zeta=\frac{2R^-}{P_h^-}=\frac{z_1-z_2}{z}$$6 formalism quoted there the corresponding quantity is

$$z=\frac{P_h^-}{k^-}=z_1+z_2,\qquad \zeta=\frac{2R^-}{P_h^-}=\frac{z_1-z_2}{z}$$7

(Matevosyan et al., 2017). The paper described this as a previously unnoticed apparent discrepancy between the definitions entering two-hadron SIDIS and back-to-back dihadron production in $$z=\frac{P_h^-}{k^-}=z_1+z_2,\qquad \zeta=\frac{2R^-}{P_h^-}=\frac{z_1-z_2}{z}$$8 annihilation (Matevosyan et al., 2017).

5. Formal extensions and related fragmentation frameworks

Recent work has extended DiFF logic beyond the classic SIDIS and back-to-back $$z=\frac{P_h^-}{k^-}=z_1+z_2,\qquad \zeta=\frac{2R^-}{P_h^-}=\frac{z_1-z_2}{z}$$9 observables. One example is the near-side energy-energy correlator, where a new nonperturbative object, the EEC-DiFF,

$(z,\cos\theta,M_h^2,\phi_R)$0

is defined as a weighted projection of the standard DiFF onto the angular variable $(z,\cos\theta,M_h^2,\phi_R)$1 (Kang et al., 23 Jul 2025). The corresponding factorization formula for near-side EECs is

$(z,\cos\theta,M_h^2,\phi_R)$2

(Kang et al., 23 Jul 2025). In the perturbative tail,

$(z,\cos\theta,M_h^2,\phi_R)$3

the same paper finds

$(z,\cos\theta,M_h^2,\phi_R)$4

at $(z,\cos\theta,M_h^2,\phi_R)$5, matching the perturbative near-side EEC jet function at that order (Kang et al., 23 Jul 2025).

Another line of development establishes formal analogies between hadron-pair fragmentation and fragmentation of a single hadron inside a reconstructed jet. That work identifies the correspondences

$(z,\cos\theta,M_h^2,\phi_R)$6

and maps

$(z,\cos\theta,M_h^2,\phi_R)$7

at the level of observable factorized cross sections (Bacchetta et al., 2023). This transfer makes it possible to translate known DiFF observables into hadron-in-jet constructions in deep-inelastic scattering and hadronic collisions (Bacchetta et al., 2023).

By contrast, not every “di-hadron” observable uses a DiFF. The BFKL study of inclusive di-hadron production at the LHC analyzes

$(z,\cos\theta,M_h^2,\phi_R)$8

with two hadrons far apart in rapidity, but the fragmentation objects there are ordinary single-hadron fragmentation functions $(z,\cos\theta,M_h^2,\phi_R)$9, one for each tagged hadron, rather than a dihadron fragmentation function $P_h=P_1+P_2$00 (Celiberto et al., 2016). This distinction is essential because DiFF formalism refers to a correlated hadron pair produced from one parent parton within the same jet, not simply to any final state containing two hadrons (Celiberto et al., 2016).

6. Universality, normalization, and current conceptual issues

A central recent issue concerns the operator definition and normalization of multihadron fragmentation functions. A detailed factorization analysis of semi-inclusive $P_h=P_1+P_2$01 annihilation confirms that the standard operator definition from single-hadron fragmentation, with its usual prefactor, remains valid for the small-mass $P_h=P_1+P_2$02-hadron case, with the same hard parts and evolution kernels (Rogers et al., 2024). In that analysis, the multihadron fragmentation function is defined by the same operator structure as in the single-hadron case, with only the identity of the observed final state changed (Rogers et al., 2024).

The same paper argues that recently proposed nonuniversal prefactors are incompatible with standard factorization unless one is also willing to modify the hard parts and evolution kernels (Rogers et al., 2024). Its explicit conclusion is that the standard DiFF formalism used in most of the literature is validated, and that this applies not only to the unpolarized DiFF $P_h=P_1+P_2$03 but also to polarization-sensitive quantities such as interference fragmentation functions (Rogers et al., 2024).

Another conceptual caution arises from the polarized quark-jet study mentioned above. That work did not claim a definitive failure of the $P_h=P_1+P_2$04 formalism, but it did note an apparent mismatch between the integrated IFF entering SIDIS and the quantity commonly identified as the IFF in back-to-back dihadron production in $P_h=P_1+P_2$05 (Matevosyan et al., 2017). A plausible implication is that combined SIDIS–$P_h=P_1+P_2$06 analyses require careful matching of definitions when transverse-momentum integrations and Fourier moments are involved.

Taken together, the literature presents a coherent but stratified picture. The formal core of DiFFs is stable: a single parton fragments into a small-mass hadron pair, the pair is described by correlators and projections analogous to those of ordinary fragmentation functions, and the same hard parts and evolution logic apply in collinear factorization (Rogers et al., 2024). Around that core, different regimes and applications add structure: partial-wave interference and chiral-odd spin analysis in collinear SIDIS and $P_h=P_1+P_2$07 (Pisano et al., 2015), perturbative matching at large $P_h=P_1+P_2$08 (Zhou et al., 2011), cascade-model realizations in NJL-jet and quark-jet frameworks (Casey et al., 2012, Matevosyan et al., 2013, Matevosyan et al., 2017), and recent projections onto jet and EEC observables (Bacchetta et al., 2023, Kang et al., 23 Jul 2025).