Photon–Hadron TEEC in QCD Dynamics

• Photon–hadron TEEC is an energy-weighted angular observable that correlates the photon direction with hadronic energy flow in the transverse plane.
• It is applied in DIS and hadronic collisions to extract αs and to probe TMD dynamics, azimuthal modulations, and gluon saturation.
• TEEC methodologies leverage advanced factorization and resummation techniques for precision QCD studies and nuclear matter investigations.

Photon–hadron transverse energy–energy correlation (TEEC) denotes a class of energy-weighted angular observables that correlate a real or virtual photon direction with hadronic energy flow, hadrons, or jets in the transverse plane. In existing literature, this class includes virtual-photon–proton observables in deep inelastic scattering (DIS), lepton–hadron constructions in which the scattered lepton acts as a proxy for the exchanged photon, and prompt-photon–hadron or $\gamma+$jet observables in hadronic collisions. A common parametrization uses the azimuthal angle $\phi$ between the photon and the hadronic object, or equivalently $\tau=(1+\cos\phi)/2$, so that the back-to-back limit corresponds to $\phi\to\pi$ and $\tau\to0$. Across these realizations, photon–hadron TEEC has been used to study perturbative recoil, transverse-momentum-dependent (TMD) structure, spin-dependent azimuthal modulations, gluon saturation, nuclear broadening, and, in DIS, the extraction of $\alpha_s(M_Z)$ (Ali et al., 2020, Li et al., 2020, Kang et al., 2024).

1. Definitions and observable scope

The hadron-collider TEEC is the transverse-plane analogue of the $e^+e^-$ EEC and is defined as

$$\frac{d\sigma}{d\cos \phi}=\sum\limits_{a,b} \int d\sigma_{pp\to a+b+X} \frac{2 E_{T,a} E_{T,b}}{\left|\sum_i E_{T,i}\right|^2}\, \delta(\cos\phi_{ab} - \cos\phi)\,.$$

In the back-to-back region it is convenient to introduce $z_\phi=(1-\cos\phi)/2$ and $\tau=1-z_\phi$, which is equivalent to $\phi$0 used in photon-tagged studies, so that $\phi$1 is the Sudakov region (Gao et al., 2023).

Photon-tagged versions fix one leg of the correlator to a color-singlet boson. For vector-boson–hadron production in $\phi$2 or $\phi$3, the observable is written as

$\phi$4

with $\phi$5 or $\phi$6, and $\phi$7 the azimuthal angle between the vector boson and the final hadron. In forward hadron-level $\phi$8–hadron studies the same structure is written directly as

$\phi$9

so the photon factor cancels and the measurement becomes a photon-triggered distribution of hadronic transverse energy in azimuth (Kang et al., 2024, Ganguli et al., 31 Jul 2025).

The literature does not use a single unique construction. In DIS, one finds at least three closely related variants: jet–jet TEEC in the $\tau=(1+\cos\phi)/2$0 final state, lepton–hadron TEEC where the scattered lepton tracks the virtual photon direction, and Breit- or lab-frame EEC/TEEC observables differential in $\tau=(1+\cos\phi)/2$1, $\tau=(1+\cos\phi)/2$2, or the lab-frame proxy $\tau=(1+\cos\phi)/2$3 (Ali et al., 2020, Li et al., 2020, Kang et al., 2023). In the small-$\tau=(1+\cos\phi)/2$4 dipole formulation of photon–hadron production, the dependence on the hadron fragmentation function cancels because the TEEC weight produces the momentum-sum-rule combination $\tau=(1+\cos\phi)/2$5, so the observable depends only on projectile PDFs and the dipole amplitude (Kang et al., 31 Mar 2025).

2. Deep-inelastic realizations

In DIS the hard scattering is effectively a virtual-photon–hadron process,

$\tau=(1+\cos\phi)/2$6

with $\tau=(1+\cos\phi)/2$7. The HERA jet analysis of transverse energy correlations uses the Breit frame, where the total transverse momentum of the hadronic final state is zero, so the leading nontrivial TEEC arises from $\tau=(1+\cos\phi)/2$8-jet configurations. In that formulation the observable is

$\tau=(1+\cos\phi)/2$9

with $\phi\to\pi$0 the azimuthal angle between two jets defined with the longitudinally invariant $\phi\to\pi$1 algorithm, $\phi\to\pi$2. The calculation was performed at $\phi\to\pi$3, with $\phi\to\pi$4, $\phi\to\pi$5, and two $\phi\to\pi$6 regions, $\phi\to\pi$7 and $\phi\to\pi$8. Because the endpoint regions require resummed logarithmic corrections, the fixed-order study was restricted to $\phi\to\pi$9. The normalized TEEC and ATEEC were found to be weakly dependent on PDFs and $\tau\to0$0, more dependent on $\tau\to0$1, and strongly sensitive to $\tau\to0$2, making them suitable for $\tau\to0$3 determinations from HERA shape variables (Ali et al., 2020).

A different DIS construction correlates the scattered lepton with the hadronic final state. In that setup the TEEC is defined as

$\tau\to0$4

or, near back-to-back, as a distribution in $\tau\to0$5. This formulation admits a soft-collinear effective theory factorization in terms of a hard function, beam function, jet function, and soft function, and was resummed to $\tau\to0$6 accuracy matched to the NLO cross section for the production of a lepton and two jets (Li et al., 2020).

A third realization, designed for TMD phenomenology, defines

$\tau\to0$7

together with a lab-frame observable $\tau\to0$8, where $\tau\to0$9 for $\alpha_s(M_Z)$0. This formulation makes the photon–hadron interpretation explicit: the observable is differential in the transverse momentum imbalance of the $\alpha_s(M_Z)$1 system and in its azimuth (Kang et al., 2023).

3. Factorization and resummation structure

The modern perturbative description of TEEC treats the back-to-back region as a one-dimensional projection of a TMD factorization problem. For hadron colliders the operator-based factorization formula applies to color singlet, $\alpha_s(M_Z)$2jet, and dijet production. In the $\alpha_s(M_Z)$3jet channel the singular part takes the form

$\alpha_s(M_Z)$4

with a hard function $\alpha_s(M_Z)$5, TMD beam functions $\alpha_s(M_Z)$6, a TEEC jet function $\alpha_s(M_Z)$7, and a soft function $\alpha_s(M_Z)$8 built from Wilson lines. The key kinematic relation is $\alpha_s(M_Z)$9, so TEEC is a measurement of out-of-plane recoil $e^+e^-$0 rather than of the full two-dimensional $e^+e^-$1 (Gao et al., 2023).

This formulation was anticipated by the earlier back-to-back hadron-collider TEEC factorization, where the observable was described as a projection onto a scattering plane of a more standard TMD factorization, with a dijet soft function obtained analytically to NNLO and resummation to NNLL matched to fixed order at the LHC (Gao et al., 2019). The later extension to $e^+e^-$2 uses cusp anomalous dimensions to four loops, non-cusp anomalous dimensions to three loops, a three-loop rapidity anomalous dimension, and the three-loop quadrupole anomalous dimension, and thereby provides the first $e^+e^-$3 resummation for a hadron-collider dijet event shape (Gao et al., 2023).

In DIS the SCET factorization has a parallel interpretation. The beam function is identical to the conventional TMD PDF, the jet function is the second moment of the TMD fragmentation-function matching coefficient, and the soft function is the usual TMD/EEC soft function. This close connection to TMD factorization allows the same observable to be viewed either as a precision event shape or as a TMD-sensitive recoil measurement (Li et al., 2020). Vector-boson–tagged hadron production in $e^+e^-$4 and $e^+e^-$5 uses the same logic in impact-parameter space, with a hard function $e^+e^-$6, subtracted TMD PDFs $e^+e^-$7, a TEEC jet function $e^+e^-$8, and a global soft function $e^+e^-$9, resummed to NLL accuracy (Kang et al., 2024).

4. Azimuthal structure, TMD content, and spin dependence

In the Breit-frame DIS formulation, the full azimuthal dependence of the energy correlator maps directly onto leading-twist TMD structure functions. The unpolarized term is $$\frac{d\sigma}{d\cos \phi}=\sum\limits_{a,b} \int d\sigma_{pp\to a+b+X} \frac{2 E_{T,a} E_{T,b}}{\left|\sum_i E_{T,i}\right|^2}\, \delta(\cos\phi_{ab} - \cos\phi)\,.$$0, while specific harmonics isolate polarized or transverse-spin-sensitive distributions: $$\frac{d\sigma}{d\cos \phi}=\sum\limits_{a,b} \int d\sigma_{pp\to a+b+X} \frac{2 E_{T,a} E_{T,b}}{\left|\sum_i E_{T,i}\right|^2}\, \delta(\cos\phi_{ab} - \cos\phi)\,.$$1

$$\frac{d\sigma}{d\cos \phi}=\sum\limits_{a,b} \int d\sigma_{pp\to a+b+X} \frac{2 E_{T,a} E_{T,b}}{\left|\sum_i E_{T,i}\right|^2}\, \delta(\cos\phi_{ab} - \cos\phi)\,.$$2

together with $$\frac{d\sigma}{d\cos \phi}=\sum\limits_{a,b} \int d\sigma_{pp\to a+b+X} \frac{2 E_{T,a} E_{T,b}}{\left|\sum_i E_{T,i}\right|^2}\, \delta(\cos\phi_{ab} - \cos\phi)\,.$$3, $$\frac{d\sigma}{d\cos \phi}=\sum\limits_{a,b} \int d\sigma_{pp\to a+b+X} \frac{2 E_{T,a} E_{T,b}}{\left|\sum_i E_{T,i}\right|^2}\, \delta(\cos\phi_{ab} - \cos\phi)\,.$$4, and $$\frac{d\sigma}{d\cos \phi}=\sum\limits_{a,b} \int d\sigma_{pp\to a+b+X} \frac{2 E_{T,a} E_{T,b}}{\left|\sum_i E_{T,i}\right|^2}\, \delta(\cos\phi_{ab} - \cos\phi)\,.$$5 contributions. In this language, photon–hadron TEEC in DIS reproduces the usual Boer–Mulders, Sivers, transversity, pretzelosity, and worm-gear structures, but in a calorimetric and energy-weighted observable (Kang et al., 2023).

A distinctive ingredient is the Collins-type EEC jet function,

$$\frac{d\sigma}{d\cos \phi}=\sum\limits_{a,b} \int d\sigma_{pp\to a+b+X} \frac{2 E_{T,a} E_{T,b}}{\left|\sum_i E_{T,i}\right|^2}\, \delta(\cos\phi_{ab} - \cos\phi)\,.$$6

which governs the $$\frac{d\sigma}{d\cos \phi}=\sum\limits_{a,b} \int d\sigma_{pp\to a+b+X} \frac{2 E_{T,a} E_{T,b}}{\left|\sum_i E_{T,i}\right|^2}\, \delta(\cos\phi_{ab} - \cos\phi)\,.$$7 and related modulations. For the fully inclusive case in the OPE region, the sum rules imply $$\frac{d\sigma}{d\cos \phi}=\sum\limits_{a,b} \int d\sigma_{pp\to a+b+X} \frac{2 E_{T,a} E_{T,b}}{\left|\sum_i E_{T,i}\right|^2}\, \delta(\cos\phi_{ab} - \cos\phi)\,.$$8 up to $$\frac{d\sigma}{d\cos \phi}=\sum\limits_{a,b} \int d\sigma_{pp\to a+b+X} \frac{2 E_{T,a} E_{T,b}}{\left|\sum_i E_{T,i}\right|^2}\, \delta(\cos\phi_{ab} - \cos\phi)\,.$$9 power corrections, whereas subset observables are controlled by single nonperturbative numbers $z_\phi=(1-\cos\phi)/2$0 and $z_\phi=(1-\cos\phi)/2$1. This sharply separates genuinely inclusive TEEC from charge- or species-tagged correlators (Kang et al., 2023).

The small-$z_\phi=(1-\cos\phi)/2$2 polarized TEEC makes this distinction operational. In the Sivers analysis for a transversely polarized proton, the asymmetry is expressed in terms of a C-odd interaction corresponding to odderon exchange, and the charge-conjugation-odd nature of the small-$z_\phi=(1-\cos\phi)/2$3 quark Sivers function forces a separation of positively and negatively charged hadrons. The predicted TEEC Sivers asymmetry at the EIC is on the $z_\phi=(1-\cos\phi)/2$4 level, with $z_\phi=(1-\cos\phi)/2$5 in the small-$z_\phi=(1-\cos\phi)/2$6 construction (Bhattacharya et al., 14 Apr 2025).

5. Small-$z_\phi=(1-\cos\phi)/2$7, nuclear matter, and forward photon production

At small $z_\phi=(1-\cos\phi)/2$8, photon–hadron TEEC becomes a probe of dipole scattering and gluon saturation. In DIS at the EIC, the back-to-back lepton–hadron TEEC factorizes into a hard function, a small-$z_\phi=(1-\cos\phi)/2$9 quark TMD derived from the dipole $\tau=1-z_\phi$0-matrix $\tau=1-z_\phi$1, a soft function absorbed into subtracted TMDs, and a TEEC jet function built from TMD fragmentation functions. Using GBW and rcBK models, the $\tau=1-z_\phi$2 TEEC was found to exhibit strong model dependence, while the nuclear modification factor in $\tau=1-z_\phi$3 shows a suppression of order $\tau=1-z_\phi$4 at small $\tau=1-z_\phi$5, $\tau=1-z_\phi$6, with $\tau=1-z_\phi$7 at larger $\tau=1-z_\phi$8 (Kang et al., 2023).

In real-photon production, vector-boson–tagged hadron TEEC in $\tau=1-z_\phi$9 and $\phi$00 was formulated within TMD factorization for back-to-back $\phi$01 and $\phi$02 production. The nuclear modification factor

$\phi$03

shows the characteristic pattern of transverse-momentum broadening: at RHIC, $\phi$04 at small $\phi$05 for $\phi$06 and rises toward or above unity at larger $\phi$07; at the LHC the same shape change persists, with stronger suppression in forward rapidity where shadowing is larger (Kang et al., 2024).

The small-$\phi$08 dipole formulation of photon–hadron and photon–jet TEEC in $\phi$09 and $\phi$10 makes the same point from a different angle. Because the fragmentation functions cancel, the observable becomes a direct probe of the dipole amplitude $\phi$11. In RHIC-like kinematics, $\phi$12 for photon–jet TEEC lies above unity and decreases with $\phi$13, consistent with a Cronin-like enhancement beyond the peak of the nuclear-to-proton dipole ratio; in LHC-like kinematics it lies below unity and gently increases with $\phi$14. Photon–hadron TEEC follows the same qualitative pattern, though smeared by the integration over hadron kinematics (Kang et al., 31 Mar 2025).

The forward hadron-level study based on a CGC-inspired small-$\phi$15 gluon distribution combined with CASCADE parton shower and hadronization shows that these effects survive beyond parton level. In normalized TEEC, the back-to-back region $\phi$16 is suppressed in $\phi$17–Pb relative to $\phi$18 by about $\phi$19 to $\phi$20, depending on the acceptance and hadron trigger; in the corresponding ordinary azimuthal $\phi$21–hadron correlation the suppression reaches about $\phi$22 to $\phi$23. The study also finds that TEEC follows the parton dynamics more closely than the simplest leading-photon–leading-hadron correlation (Ganguli et al., 31 Jul 2025).

6. Status, limitations, and outlook

Photon–hadron TEEC is theoretically mature enough to support several distinct precision programs, but the maturity is uneven across channels. In HERA DIS, the jet-based TEEC/ATEEC analysis remains a theory study and HERA TEEC/ATEEC data do not yet exist, although the authors explicitly encourage H1 and ZEUS to analyze archived data. In that fixed-order setup the endpoint regions $\phi$24 are excluded because resummed logarithmic corrections are not included, and the dominant residual uncertainty is the renormalization-scale dependence rather than PDF or $\phi$25 variation (Ali et al., 2020).

In hadron-collider photon-tagged channels, the perturbative control of the back-to-back region is very strong at the level of formal factorization: the $\phi$26 TEEC framework applies to $\phi$27jet as well as to $\phi$28jet and pure color-singlet production, and provides a controlled setting for studying transverse-momentum dynamics and possible factorization violation beyond color singlets (Gao et al., 2023). By contrast, many phenomenological studies of prompt-photon–hadron TEEC in the small-$\phi$29 and nuclear regimes are currently limited to NLL resummation, LO dipole calculations, or LO hard processes supplemented by parton shower and hadronization, and often omit photon isolation, detector effects, or a full perturbative uncertainty budget (Kang et al., 2024, Ganguli et al., 31 Jul 2025).

The outlook is correspondingly two-track. One track is precision QCD in DIS and $\phi$30jet configurations, where TEEC offers a normalized, recoil-sensitive observable with direct access to $\phi$31, TMD evolution, and high-order resummation. The other is low-$\phi$32 and nuclear QCD, where the same class of observables probes dipole amplitudes, saturation, odderon effects, and nuclear broadening in $\phi$33, $\phi$34, $\phi$35, and $\phi$36. This suggests that photon–hadron TEEC is best viewed not as a single observable, but as a technically unified family of energy-weighted angular correlators whose differences in tagged object, frame choice, and normalization expose different sectors of QCD dynamics (Li et al., 2020, Kang et al., 2023, Bhattacharya et al., 14 Apr 2025).