Chiral-Odd Dimeson GDAs in Pion Pair Production

• The paper demonstrates that the interference between one-photon (chiral-even) and two-photon (chiral-odd) amplitudes enables a direct probe of the tensor structure in pion pairs.
• Chiral-odd GDAs are nonperturbative functions defined through tensor bilinears that parameterize the hadronization of a quark-antiquark pair into a pion pair, capturing spin-flip dynamics.
• The analysis employs factorization, ERBL evolution, and azimuthal modulations to isolate and study the subtle chiral-odd contributions in exclusive low-mass pion pair production.

Chiral-odd dimeson generalized distribution amplitudes (CO-GDAs) are nonperturbative light-cone correlation functions that parameterize the transition of a quark-antiquark pair into a two-meson state through a tensor quark bilinear. In the formulation developed for high-energy $e^-e^+\to(\pi\pi)(\pi\pi)$, they are the crossed-channel analogue of generalized parton distributions, specialized to the hadronization of a $q\bar q$ pair into a pion pair with relatively small invariant mass inside a hard exclusive reaction. The specific result established for the channel $$e^-(\ell)e^+(\ell')\to(\pi^+(p_1)\pi^0(p_2))(\pi^-(k_1)\pi^0(k_2))$$ is that the chiral-even sector contributes through the leading one-photon amplitude, whereas the chiral-odd sector enters through two-photon exchange; the interference of the two produces measurable azimuthal modulations that provide a direct linear probe of the otherwise inaccessible chiral-odd meson structure, including the anomalous tensorial magnetic moment of spin-zero mesons such as the pion (Bhattacharya et al., 31 Jul 2025).

1. Definition and operator structure

For a charged pion pair with isovector quantum numbers, the chiral-even and chiral-odd two-meson GDAs are defined by nonlocal light-cone quark bilinears with a gauge link $[v,0]$. In the notation used for the isovector pion-pair channel, the chiral-even GDA is introduced through the vector bilinear, while the chiral-odd GDA is introduced through the tensor bilinear $i\sigma^{\mu\nu}$ (Bhattacharya et al., 31 Jul 2025): $$\langle \pi\pi|\bar q(v)[v,0]\gamma^\lambda q(0)|0\rangle \propto (p_1^\lambda+p_2^\lambda)\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm ce}^V(z,\zeta,s),$$

$$\langle \pi\pi|\bar q(v)[v,0]i\sigma^{\mu\nu} q(0)|0\rangle \propto \frac{p_1^\mu p_2^\nu-p_1^\nu p_2^\mu}{m_\pi}\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm co}^V(z,\zeta,s).$$

These matrix elements contain the quark light-cone momentum fraction $z\in[0,1]$, the two-meson invariant mass

$s=(p_1+p_2)^2,$

and the longitudinal momentum sharing variable $\zeta$, defined as the light-cone momentum fraction carried by one meson in the pair. The chiral-odd tensor structure flips chirality and encodes transverse-spin sensitivity. Because the final hadrons are spinless, the tensor information is carried by the orbital and angular structure of the two-pion system through the antisymmetric momentum tensor $q\bar q$0.

Physically, $q\bar q$1 describes how a $q\bar q$2 pair created in a tensor channel hadronizes into a pion pair. This is why CO-GDAs are directly tied to spin-flip structure in a spin-zero target and, in the formulation under discussion, to the anomalous tensorial magnetic moment of the pion. For the $q\bar q$3 and $q\bar q$4 channels, only the isovector GDA contributes, and the corresponding chiral-even and chiral-odd GDAs are identical in those channels.

2. ERBL evolution, asymptotics, and normalization

The scale dependence of the GDAs is governed by ERBL evolution. The solution is expanded in Gegenbauer polynomials in $q\bar q$5 and partial waves in $q\bar q$6, and in the isovector sector the quoted asymptotic form is (Bhattacharya et al., 31 Jul 2025)

$q\bar q$7

The evolution of the coefficients is

$q\bar q$8

The asymptotic $q\bar q$9-dependence of chiral-even and chiral-odd GDAs is the same, but their anomalous dimensions differ. The quoted values are $$e^-(\ell)e^+(\ell')\to(\pi^+(p_1)\pi^0(p_2))(\pi^-(k_1)\pi^0(k_2))$$0 for the chiral-even case and $$e^-(\ell)e^+(\ell')\to(\pi^+(p_1)\pi^0(p_2))(\pi^-(k_1)\pi^0(k_2))$$1 for the chiral-odd case. The paper also notes that chiral-odd quark GDAs evolve independently of gluons, mirroring the transversity sector in ordinary parton physics.

For the isovector chiral-even GDA, the first moment is related to the timelike pion form factor,

$$e^-(\ell)e^+(\ell')\to(\pi^+(p_1)\pi^0(p_2))(\pi^-(k_1)\pi^0(k_2))$$2

Keeping only the leading asymptotic term gives $$e^-(\ell)e^+(\ell')\to(\pi^+(p_1)\pi^0(p_2))(\pi^-(k_1)\pi^0(k_2))$$3. The normalization constants are generally complex,

$$e^-(\ell)e^+(\ell')\to(\pi^+(p_1)\pi^0(p_2))(\pi^-(k_1)\pi^0(k_2))$$4

The chiral-odd normalization $$e^-(\ell)e^+(\ell')\to(\pi^+(p_1)\pi^0(p_2))(\pi^-(k_1)\pi^0(k_2))$$5 is unknown and tied to the tensor form factor of the pion. As a first modeling step, equal phases are assumed,

$$e^-(\ell)e^+(\ell')\to(\pi^+(p_1)\pi^0(p_2))(\pi^-(k_1)\pi^0(k_2))$$6

which implies through the Omnès representation

$$e^-(\ell)e^+(\ell')\to(\pi^+(p_1)\pi^0(p_2))(\pi^-(k_1)\pi^0(k_2))$$7

with unknown constant $$e^-(\ell)e^+(\ell')\to(\pi^+(p_1)\pi^0(p_2))(\pi^-(k_1)\pi^0(k_2))$$8. Measuring $$e^-(\ell)e^+(\ell')\to(\pi^+(p_1)\pi^0(p_2))(\pi^-(k_1)\pi^0(k_2))$$9 and any phase difference between the chiral-even and chiral-odd sectors is one of the main phenomenological goals. The benchmark value used in the numerical study is quoted as

$[v,0]$0

motivated by lattice results for the pion tensor form factor.

3. Exclusive reaction and kinematic regime

The reaction proposed to reveal CO-GDAs is

$[v,0]$1

in the regime

$[v,0]$2

with

$[v,0]$3

In this setup the total $[v,0]$4 center-of-mass energy is the hard scale, while each pion pair is a low-mass collinear system described by a GDA; this is the factorization regime of the analysis (Bhattacharya et al., 31 Jul 2025).

The overall c.m. frame is parameterized by two lightlike vectors $[v,0]$5 and $[v,0]$6 associated with the two meson-pair directions. The effective hard light-cone invariant is defined by

$[v,0]$7

with the Källén function

$[v,0]$8

The pion momenta within each pair are parameterized in terms of $[v,0]$9, $i\sigma^{\mu\nu}$0, and transverse relative momenta $i\sigma^{\mu\nu}$1, $i\sigma^{\mu\nu}$2, with

$i\sigma^{\mu\nu}$3

Neglecting lepton masses, the transverse kinematics satisfies

$i\sigma^{\mu\nu}$4

where

$i\sigma^{\mu\nu}$5

The azimuthal structure is encoded by angles $i\sigma^{\mu\nu}$6 and $i\sigma^{\mu\nu}$7, defined through the transverse vectors $i\sigma^{\mu\nu}$8 and $i\sigma^{\mu\nu}$9. These angles are central because the CO-GDA signal appears through $$\langle \pi\pi|\bar q(v)[v,0]\gamma^\lambda q(0)|0\rangle \propto (p_1^\lambda+p_2^\lambda)\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm ce}^V(z,\zeta,s),$$0 and $$\langle \pi\pi|\bar q(v)[v,0]\gamma^\lambda q(0)|0\rangle \propto (p_1^\lambda+p_2^\lambda)\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm ce}^V(z,\zeta,s),$$1 modulations.

4. Factorization and chirality selection rules

At leading order in the hard subprocess, the exclusive four-pion amplitude factorizes into perturbative coefficient functions convoluted with one GDA for each pion pair. The one-photon exchange amplitude is purely chiral-even and depends on the convolution (Bhattacharya et al., 31 Jul 2025)

$$\langle \pi\pi|\bar q(v)[v,0]\gamma^\lambda q(0)|0\rangle \propto (p_1^\lambda+p_2^\lambda)\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm ce}^V(z,\zeta,s),$$2

The two-photon exchange amplitude contains both a chiral-even part and a chiral-odd part, with hard convolutions

$$\langle \pi\pi|\bar q(v)[v,0]\gamma^\lambda q(0)|0\rangle \propto (p_1^\lambda+p_2^\lambda)\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm ce}^V(z,\zeta,s),$$3

$$\langle \pi\pi|\bar q(v)[v,0]\gamma^\lambda q(0)|0\rangle \propto (p_1^\lambda+p_2^\lambda)\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm ce}^V(z,\zeta,s),$$4

The essential selection rule is chirality. At leading twist with massless quarks, the one-photon exchange mechanism couples through a vector current and preserves chirality, so it matches naturally onto the chiral-even bilinear $$\langle \pi\pi|\bar q(v)[v,0]\gamma^\lambda q(0)|0\rangle \propto (p_1^\lambda+p_2^\lambda)\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm ce}^V(z,\zeta,s),$$5. There is no leading-twist way for the one-photon amplitude to couple to a pair of tensor bilinears producing the two pion pairs. By contrast, the two-photon exchange channel permits a hard tensor structure that couples to the rank-2 transverse tensor built from $$\langle \pi\pi|\bar q(v)[v,0]\gamma^\lambda q(0)|0\rangle \propto (p_1^\lambda+p_2^\lambda)\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm ce}^V(z,\zeta,s),$$6, which is precisely the structure required to project the chiral-odd GDAs.

A common misconception is that the chiral-odd sector would appear as a leading one-photon contribution once the final state contains two pion pairs. In the framework under discussion, this is not the case: the leading one-photon amplitude remains purely chiral-even, and the experimentally useful access to CO-GDAs comes from the interference between the dominant one-photon amplitude and the two-photon chiral-odd amplitude.

5. Differential cross section and CO-sensitive observables

For the $$\langle \pi\pi|\bar q(v)[v,0]\gamma^\lambda q(0)|0\rangle \propto (p_1^\lambda+p_2^\lambda)\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm ce}^V(z,\zeta,s),$$7 channel, the differential cross section is given as the sum of three structures: a dominant chiral-even contribution, a linear chiral-even/chiral-odd interference term, and a quadratic chiral-odd term (Bhattacharya et al., 31 Jul 2025). The interference term carries the distinctive modulation

$$\langle \pi\pi|\bar q(v)[v,0]\gamma^\lambda q(0)|0\rangle \propto (p_1^\lambda+p_2^\lambda)\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm ce}^V(z,\zeta,s),$$8

whereas the quadratic CO contribution carries $$\langle \pi\pi|\bar q(v)[v,0]\gamma^\lambda q(0)|0\rangle \propto (p_1^\lambda+p_2^\lambda)\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm ce}^V(z,\zeta,s),$$9 dependence.

This decomposition summarizes the observational strategy. The first term is dominated by the chiral-even sector. The second term is linear in the CO-GDA and therefore far more favorable than the purely quadratic contribution. The third term is quadratic in the CO-GDA and parametrically smaller. The scaling of all terms is $$\langle \pi\pi|\bar q(v)[v,0]i\sigma^{\mu\nu} q(0)|0\rangle \propto \frac{p_1^\mu p_2^\nu-p_1^\nu p_2^\mu}{m_\pi}\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm co}^V(z,\zeta,s).$$0, but the purely chiral-odd contribution is relatively suppressed by $$\langle \pi\pi|\bar q(v)[v,0]i\sigma^{\mu\nu} q(0)|0\rangle \propto \frac{p_1^\mu p_2^\nu-p_1^\nu p_2^\mu}{m_\pi}\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm co}^V(z,\zeta,s).$$1 compared with the leading term.

The proposed observable is the $$\langle \pi\pi|\bar q(v)[v,0]i\sigma^{\mu\nu} q(0)|0\rangle \propto \frac{p_1^\mu p_2^\nu-p_1^\nu p_2^\mu}{m_\pi}\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm co}^V(z,\zeta,s).$$2-weighted moment

$$\langle \pi\pi|\bar q(v)[v,0]i\sigma^{\mu\nu} q(0)|0\rangle \propto \frac{p_1^\mu p_2^\nu-p_1^\nu p_2^\mu}{m_\pi}\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm co}^V(z,\zeta,s).$$3

which isolates the interference term because the unweighted chiral-even term and the $$\langle \pi\pi|\bar q(v)[v,0]i\sigma^{\mu\nu} q(0)|0\rangle \propto \frac{p_1^\mu p_2^\nu-p_1^\nu p_2^\mu}{m_\pi}\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm co}^V(z,\zeta,s).$$4-type pieces do not survive in the same way under this projection. Physically, the observable measures the correlation between the azimuthal orientations of the two pion pairs and the tensor nature of the CO-GDA contribution. The analysis also remarks that charge-conjugation-odd observables, such as forward-backward asymmetries, can isolate the interference between one- and two-photon mechanisms.

The angular content is not restricted to azimuths. The ERBL solution is expanded in Gegenbauer polynomials $$\langle \pi\pi|\bar q(v)[v,0]i\sigma^{\mu\nu} q(0)|0\rangle \propto \frac{p_1^\mu p_2^\nu-p_1^\nu p_2^\mu}{m_\pi}\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm co}^V(z,\zeta,s).$$5 and in Legendre polynomials of the angular variable

$$\langle \pi\pi|\bar q(v)[v,0]i\sigma^{\mu\nu} q(0)|0\rangle \propto \frac{p_1^\mu p_2^\nu-p_1^\nu p_2^\mu}{m_\pi}\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm co}^V(z,\zeta,s).$$6

In the isovector sector, the asymptotic form is proportional to

$$\langle \pi\pi|\bar q(v)[v,0]i\sigma^{\mu\nu} q(0)|0\rangle \propto \frac{p_1^\mu p_2^\nu-p_1^\nu p_2^\mu}{m_\pi}\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm co}^V(z,\zeta,s).$$7

which corresponds to the leading odd partial wave, namely the $$\langle \pi\pi|\bar q(v)[v,0]i\sigma^{\mu\nu} q(0)|0\rangle \propto \frac{p_1^\mu p_2^\nu-p_1^\nu p_2^\mu}{m_\pi}\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm co}^V(z,\zeta,s).$$8-wave structure appropriate to the $$\langle \pi\pi|\bar q(v)[v,0]i\sigma^{\mu\nu} q(0)|0\rangle \propto \frac{p_1^\mu p_2^\nu-p_1^\nu p_2^\mu}{m_\pi}\int_0^1 dz\, e^{iz(p_1+p_2)\cdot v}\,\Phi_{\rm co}^V(z,\zeta,s).$$9-channel in $z\in[0,1]$0. Thus $z\in[0,1]$1 controls the partial-wave content inside each dipion, while $z\in[0,1]$2 isolates the transverse tensor interference.

6. Phenomenology, significance, and limitations

The numerical study is performed at

$z\in[0,1]$3

with integrations over

$z\in[0,1]$4

and over the full ranges in $z\in[0,1]$5 depending on the observable (Bhattacharya et al., 31 Jul 2025). The unweighted cross section is dominated by the chiral-even contribution. The pure CO term is tiny and, in the figure discussed in the source, is magnified by a factor of 1000 for visibility. This confirms the necessity of interference observables rather than a direct extraction from $z\in[0,1]$6.

The main phenomenological implication is that the $z\in[0,1]$7-weighted cross section provides a clean signal of the chiral-even/chiral-odd interference despite its small absolute size. The signal becomes stronger at lower $z\in[0,1]$8, since all terms scale like $z\in[0,1]$9. High-luminosity, medium-energy $s=(p_1+p_2)^2,$0 machines are therefore identified as favorable, and BES III together with future tau-charm factories, especially the proposed STCF/SCTF, are singled out as realistic venues. Existing datasets may already contain sensitivity to the mechanism. No full detector simulation is given, but the combination of high luminosity and the clean azimuthal signature is presented as making the measurement feasible.

The theoretical significance is that $s=(p_1+p_2)^2,$1 with low invariant masses for each pair supplies a direct and experimentally viable probe of chiral-odd dimeson GDAs. This matters because meson structure is harder to access than nucleon structure: there are no meson targets, so crossed-channel objects such as GDAs serve as a practical portal to meson tomography. In that program, the chiral-odd sector had been effectively missing. The proposed mechanism identifies a realistic process and observable for studying transverse spin-flip structure in spin-zero mesons, including tensor form factors and anomalous tensorial magnetic moments.

The stated caveats are equally clear. The analysis is leading order and leading twist. The CO-GDA normalization is modeled rather than derived. The relative phase between chiral-even and chiral-odd sectors is assumed equal in the numerical study. A broader analysis of other charge and isospin channels is deferred to future work. If one moves beyond the simplifying assumption $s=(p_1+p_2)^2,$2, the interference term would probe the relative phase between chiral-even and chiral-odd GDAs; this suggests direct sensitivity to nonperturbative $s=(p_1+p_2)^2,$3 final-state interactions, resonance dynamics, and Omnès-based modeling.