---
title: Di-Hadron Fragmentation Formalism
url: https://www.emergentmind.com/topics/di-hadron-fragmentation-formalism
type: topic
---

# Di-Hadron Fragmentation Formalism

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 \(M_h\) [1012.0054]. 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 [1511.03220]. 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 [1106.5897].

## 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 [1511.03220]. 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 \(\theta\) is the polar angle of one hadron in the pair center-of-mass frame and \(\phi_R\) is the azimuthal angle associated with the pair plane [1012.0054].

A complementary collinear DiFF definition used in recent work on near-side energy-energy correlators introduces
\[
D_1^{h_1h_2/q}(\xi_1,\xi_2,\vec R_T),
\]
defined in the dihadron frame, where the total dihadron momentum has no transverse momentum and a large minus light-cone component [2507.17444]. In that formulation,
\[
\xi\equiv \xi_1+\xi_2,\qquad \vec R_T\equiv \frac12(\vec P_{1T}-\vec P_{2T}),
\]
and the DiFF is interpreted as a density in \((\xi_1,\xi_2,\vec R_T)\) for producing a hadron pair from fragmentation of parton \(i\) [2507.17444].

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,
\[
M_h^2\ll Q^2,
\]
so that the hadron pair belongs to a single collinear jet initiated by one parton [2412.12282]. This same condition underlies the standard use of DiFFs in SIDIS and \(e^+e^-\) 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 [1604.08013].

## 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 \(k_T\), the projected correlator is written as
\[
\Delta^q(z,\cos\theta,M_h^2,\phi_R) = \frac{z |\vec R|}{16\,M_h}\int d^2 \vec k_T \; d k^+\,\Delta^q(k;P_h,R) \Big|_{k^- = P_h^-/z},
\]
with the unintegrated correlator
\[
\Delta^q(k,P_h,R)_{ij} = \sum_X \int \frac{d^4\xi}{(2\pi)^{4}\; e^{+i k \cdot \xi} \langle 0| {\cal U}^{n_+}_{(-\infty,\xi)} \,\psi_i^q(\xi)|P_h, R; X\rangle \langle P_h, R; X| \bar{\psi}_j^q(0)\, {\cal U}^{n_+}_{(0,-\infty)} |0\rangle
\]
[1012.0054]. The corresponding leading-twist projections isolate the unpolarized DiFF \(D_1\) and the polarized interference fragmentation function \(H_1^{\sphericalangle}\) [1012.0054].

In the more differential leading-twist decomposition for an unpolarized hadron pair, four DiFFs appear:
- \(D_1\),
- \(G_1^\perp\),
- \(H_1^{\sphericalangle}\),
- \(H_1^\perp\)

with explicit dependence on \((z,\xi,k_T^2,R_T^2,k\!\cdot\!R)\) or equivalent variables [1709.08643]. The associated quark-quark correlator projections are given by
\[
\Delta^{[\gamma^+]} = D_1,
\]
\[
\Delta^{[\gamma^+\gamma_5]} = \frac{\epsilon_T^{ij} R_{Ti} k_{Tj}}{M_1 M_2}\, G_1^\perp,
\]
\[
\Delta^{[i\sigma^{i+}\gamma_5]} = \frac{\epsilon_T^{ij} R_{Tj}}{M_1+M_2}\, H_1^{\sphericalangle} + \frac{\epsilon_T^{ij} k_{Tj}}{M_1+M_2}\, H_1^\perp
\]
[1709.08643].

A characteristic feature of the formalism is partial-wave decomposition. For low invariant mass, the hadron pair is mainly produced in relative \(s\)- and \(p\)-waves, and the review literature writes
\[
D_1 \rightarrow D_{1, ss+pp} + D_{1, sp}\cos\theta + D_{1, pp}\frac{1}{4}(3\cos^2\theta-1),
\]
\[
\frac{|\bm{R}_T|}{M_h}\, H_1^{\sphericalangle} \rightarrow H_{1, sp}^{\sphericalangle}\,\sin\theta + H_{1, pp}^{\sphericalangle}\,\sin\theta\,\cos\theta
\]
[1511.03220]. This is why \(H_1^{\sphericalangle}\) is also called an interference fragmentation function: it is associated with interference between pair states in different partial waves [1012.0054].

At subleading twist, additional DiFFs appear. After \(k_T\) integration, the twist-3 decomposition contains \(E\), \(D^{\sphericalangle}\), \(H\), and \(G^{\sphericalangle}\), together with quark-gluon-quark combinations \(\tilde E\), \(\tilde D^{\sphericalangle}\), \(\tilde H\), and \(\tilde G^{\sphericalangle}\) [1511.03220]. This twist hierarchy becomes operational in beam-spin and longitudinal-target observables, where the same chiral-odd \(H_1^{\sphericalangle}\) appears with twist-3 parton distributions such as \(e(x)\) and \(h_L(x)\) [1511.03220].

## 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 [1106.5897]. In the transversity program, the key SIDIS process is
\[
\ell\, N^\uparrow \to \ell'\, (\pi^+\pi^-)\, X,
\]
and the target-spin asymmetry is written as
\[
A_{UT}^{\sin(\phi_{R} + \phi_S)\sin \theta} (x, Q^2) = - C_y\, \frac{\sum_q e_q^2\,h_1^q(x, Q^2)\, n_q^{\uparrow} (Q^2) } {\sum_q e_q^2\,f_1^q(x, Q^2) \, n_q (Q^2)}
\]
with
\[
n_q (Q^2) = \int  dz dM_h^2\,D_1(z, M_h^2, Q^2),\qquad
n_q^{\uparrow} (Q^2) = \int  dz \, dM_h^2 \frac{|\bf{R}|}{M_h} \,H_{1, sp}(z, M_h^2, Q^2)
\]
[1106.5897]. This formulation yields the transversity combination
\[
x h_1^{u_v}(x,Q^2)-\frac14\,x h_1^{d_v}(x,Q^2)
\]
under the flavor assumptions summarized in that analysis [1106.5897].

The same chiral-odd DiFF enters \(e^+e^-\) annihilation through the Artru-Collins asymmetry in
\[
e^+e^- \to (\pi^+\pi^-)\,(\pi^+\pi^-)\,X,
\]
where the asymmetry is proportional to the product \(H_1^{\sphericalangle}\,\bar H_1^{\sphericalangle}\) normalized by \(D_1\bar D_1\) [1012.0054]. In integrated form, the analysis that extracted transversity from HERMES and Belle used
\[
A^{\cos (\phi_R + \bar{\phi}_R)}= \frac{-\langle \sin^2 \theta_2 \rangle}{\langle 1+\cos^2 \theta_2 \rangle} \, \frac{ \langle \sin \theta \rangle \, \langle \sin \bar{\theta} \rangle \, 5 \, (n_u^{\uparrow})^2} { (5+N_s^2)\, n_u^{2} + 4\, n_c^{2} }
\]
and obtained
\[
n_u^{\uparrow} / n_u (100\, \mathrm{GeV}^2) = - 0.273 \pm 0.007_{\mathrm{ex} \pm 0.009_{\mathrm{th}}
\]
before evolution to HERMES scales [1106.5897].

The same DiFF structure also appears in hadronic collisions. For
\[
p\, p^\uparrow \to (h_1 h_2)\,X,
\]
the leading-twist polarized cross section contains a \(\sin(\phi_{S_B}-\phi_R)\) modulation proportional to
\[
f_1^a(x_a)\, h_1^b(x_b)\, H_1^{\sphericalangle c}(z_h, M_h^2)
\]
[1511.03220]. This process has been used as a nontrivial universality test because the same transversity distribution and the same interference DiFF extracted from SIDIS and \(e^+e^-\) should also describe the \(pp\) asymmetry [1511.03220].

At subleading twist, DiFFs provide access to further nucleon structure. The beam-spin asymmetry in dihadron SIDIS contains
\[
e(x)\, H_1^{\sphericalangle}
\]
plus a contamination from \(\tilde G^{\sphericalangle}\), and this channel is described in the review as the cleanest access to the poorly known twist-3 distribution \(e(x)\) [1511.03220].

## 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
\[
\frac{d}{dlnQ^2} D^{h_1 h_2}_i(z_1,z_2,Q^2)
= \frac{\alpha_s(Q^2)}{2\pi}\int^1_{z_1+z_2} \frac{du}{u^2} D^{h_1 h_2}_j\left(\frac{z_1}{u},\frac{z_2}{u},Q^2\right)P_{ji}(u)
+ \frac{\alpha_s(Q^2)}{2\pi}\int^{1-z_2}_{z_1} \frac{du}{u(1-u)}D^{h_1}_j\left(\frac{z_1}{u},Q^2\right)D^{h_2}_k\left(\frac{z_2}{1-u},Q^2\right)\hat{P}^i_{jk}(u)
\]
[1202.4036]. In that work, model results obtained at
\[
Q_0^2 = 0.2~\mathrm{GeV}^2
\]
are evolved to
\[
4~\mathrm{GeV}^2,
\]
and the evolved DiFFs show “a shift in the peak of the model results towards the lower \(z\) region after the evolution” [1202.4036].

A distinct perturbative regime arises at large pair invariant mass. In that limit,
\[
M_{hh}\gg \Lambda_{\rm QCD},
\]
the pair is produced by a perturbative splitting followed by ordinary single-hadron fragmentation [1101.3273]. The paper on large-invariant-mass DiFFs derives
\[
D_1 \sim \frac{1}{M_{hh}^2},\qquad H_1^{\open} \sim \frac{1}{M_{hh}^3}
\]
and shows that the interference DiFF \(H_1^{\open}\) is controlled by the same twist-3 fragmentation correlators \(\hat H\) and \(\hat H_F\) that also govern the Collins function at large transverse momentum [1101.3273]. In the intermediate region
\[
\Lambda_{\rm QCD}\ll M_{hh}\ll Q,
\]
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 [1101.3273].

The NJL-jet program provides a microscopic hadronization model for unpolarized DiFFs. In the early formulation, the dihadron fragmentation function
\[
D_q^{h_1,h_2}(z_1,z_2)
\]
is defined as the probability of observing two hadrons with light-cone momentum fractions \(z_1\) and \(z_2\), and obeys the recursive equation
\[
D_q^{h_1,h_2}(z_1,z_2) = \hat d_q^{h_1}(z_1)\frac{D^{h_2}_{q_1}(\frac{z_2}{1-z_1})}{1-z_1}+\hat d_q^{h_2}(z_2)\frac{D^{h_1}_{q_2}(\frac{z_1}{1-z_2})}{1-z_2}+\sum_Q\int^1_{z_1+z_2}\frac{d\eta}{\eta^2}\hat{d}^Q_q(\eta)D^{h_1,h_2}_Q\left(\frac{z_1}{\eta},\frac{z_2}{\eta}\right)
\]
[1202.4036]. The first two terms are interpreted as the driving function, while the last term is the higher-order cascade contribution [1202.4036]. Numerically, the work finds that the driving term dominates when either \(z_1\) or \(z_2\) is large, whereas the higher-order term becomes more important as one of the observed momentum fractions is lowered [1202.4036].

A later Monte Carlo implementation generalized this to \(D_q^{h_1h_2}(z,M_h^2)\), 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 [1310.1917].

For polarized fragmentation, the quark-jet Monte Carlo study of transverse-spin-dependent DiFFs extracted Fourier cosine moments of \(H_1^{\sphericalangle}\) and \(H_1^\perp\) from number densities and showed that, in SIDIS, the integrated combination is
\[
H_1^{\sphericalangle,SIDIS}(z,M_h^2)=H_1^{\sphericalangle,[0]}(z,M_h^2)+H_1^{\perp,[1]}(z,M_h^2),
\]
whereas in the \(e^+e^-\) formalism quoted there the corresponding quantity is
\[
H_1^{\sphericalangle,e^+e^-}(z,M_h^2)=H_1^{\sphericalangle,[0]}(z,M_h^2)
\]
[1709.08643]. The paper described this as a previously unnoticed apparent discrepancy between the definitions entering two-hadron SIDIS and back-to-back dihadron production in \(e^+e^-\) annihilation [1709.08643].

## 5. Formal extensions and related fragmentation frameworks

Recent work has extended DiFF logic beyond the classic SIDIS and back-to-back \(e^+e^-\) observables. One example is the near-side energy-energy correlator, where a new nonperturbative object, the EEC-DiFF,
\[
\mathcal D^i(z_\chi,Q^2)\equiv \sum_{h_1,h_2} \int d\xi_1 \int d\xi_2 \int d^2\vec R_T\, \delta\!\left( z_\chi-\frac{R_T^2}{Q^2}\frac{\xi^2}{\xi_1^2\xi_2^2} \right) \, \xi_1\xi_2\, D_1^{h_1h_2/i}(\xi_1,\xi_2,\vec R_T),
\]
is defined as a weighted projection of the standard DiFF onto the angular variable \(z_\chi\) [2507.17444]. The corresponding factorization formula for near-side EECs is
\[
{\rm EEC}(\chi)\overset{\rm DiFF}{=} \frac{\sin\chi}{2}\frac{\sigma_0}{\sigma_t} \int_0^1 dw\,w^2 \Big[ \mathcal D^q(z_\chi,w^2Q^2;\mu)\, H^q\!\left(w;\frac{\mu}{Q}\right) + \mathcal D^g(z_\chi,w^2Q^2;\mu)\, H^g\!\left(w;\frac{\mu}{Q}\right) \Big]
\]
[2507.17444]. In the perturbative tail,
\[
R_T\gg \Lambda_{\rm QCD},
\]
the same paper finds
\[
\mathcal D^q(z_\chi,Q^2) = \frac{\alpha_s}{4\pi}\frac{3C_F}{z_\chi}
\]
at \(\mathcal O(\alpha_s)\), matching the perturbative near-side EEC jet function at that order [2507.17444].

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
\[
(z_{hh},\zeta)\leftrightarrow (z_J,z_h),\qquad \bm j_\perp\leftrightarrow \bm R_\perp,\qquad \phi_h\leftrightarrow \phi_R,
\]
and maps
\[
4 \pi \, {\cal D}_1^q \longleftrightarrow D_1^q,\qquad
4 \pi \, {\cal H}_1^{\perp q} \longleftrightarrow H_1^{\sphericalangle q}
\]
at the level of observable factorized cross sections [2303.04314]. This transfer makes it possible to translate known DiFF observables into hadron-in-jet constructions in deep-inelastic scattering and hadronic collisions [2303.04314].

By contrast, not every “di-hadron” observable uses a DiFF. The BFKL study of inclusive di-hadron production at the LHC analyzes
\[
{\rm p}(p_1) + {\rm p}(p_2) \to {\rm h}_1(k_1) + {\rm h}_2(k_2)+ {\rm X},
\]
with two hadrons far apart in rapidity, but the fragmentation objects there are ordinary single-hadron fragmentation functions \(D_i^h(z)\), one for each tagged hadron, rather than a dihadron fragmentation function \(D_i^{h_1h_2}\) [1604.08013]. 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 [1604.08013].

## 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 \(e^+e^-\) annihilation confirms that the standard operator definition from single-hadron fragmentation, with its usual prefactor, remains valid for the small-mass \(n\)-hadron case, with the same hard parts and evolution kernels [2412.12282]. 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 [2412.12282].

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 [2412.12282]. 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 \(D_1\) but also to polarization-sensitive quantities such as interference fragmentation functions [2412.12282].

Another conceptual caution arises from the polarized quark-jet study mentioned above. That work did not claim a definitive failure of the \(e^+e^-\) 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 \(e^+e^-\) [1709.08643]. A plausible implication is that combined SIDIS–\(e^+e^-\) 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 [2412.12282]. Around that core, different regimes and applications add structure: partial-wave interference and chiral-odd spin analysis in collinear SIDIS and \(e^+e^-\) [1511.03220], perturbative matching at large \(M_{hh}\) [1101.3273], cascade-model realizations in NJL-jet and quark-jet frameworks [1202.4036], [1310.1917], [1709.08643], and recent projections onto jet and EEC observables [2303.04314], [2507.17444].

Source: https://www.emergentmind.com/topics/di-hadron-fragmentation-formalism