One-Point Energy Correlators in QFT & Jets

• One-point energy correlators are observables defined via a single energy-flow operator that measures the angular distribution of energy in quantum field theory and collider events.
• They serve as baseline normalization tools and probes for key phenomena such as spin polarization, fragmentation dynamics, and small-x behavior, exemplified by anisotropy parameters like aₑ.
• Their analysis using SCET, TMD factorization, and perturbative QCD provides high-precision insights into jet energy profiles and the hierarchy of multi-point energy correlators.

One-point energy correlators are observables built from a single energy-flow insertion and measure the angular distribution of energy flux in a specified direction. In quantum field theory they are expectation values of the calorimeter operator $\mathcal{E}(\vec n)$, while in collider applications they become energy-weighted angular distributions of hadrons inside a jet or relative to a beam or target direction. In fully inclusive and unpolarized settings they can reduce to a normalization condition or to a distribution controlled by a small number of tensor structures, but with polarized sources, jet-axis measurements, or process-specific kinematics they become nontrivial probes of spin, fragmentation, transversity, and small-$x$ dynamics (Riembau et al., 18 Dec 2025, Mi et al., 29 Jul 2025, Gao et al., 19 Sep 2025, Kang et al., 2 Mar 2026).

1. Operator definition and event-level realizations

The canonical object is the energy-flow operator at null infinity,

$$\mathcal E_n=\lim_{r\to\infty}\int_0^\infty dt\; r^2\,n^i\,T_{0i}(t,r\hat n),$$

which acts on an asymptotic state $|\alpha\rangle$ as

$$\mathcal E_n|\alpha\rangle=\sum_{i\in\alpha}E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$

The one-point energy correlator is then the expectation value $\langle\mathcal E(\vec n)\rangle$, or, for a spinning source, the helicity density matrix $\langle \mathcal E_n\rangle_{hh'}$ obtained by inserting one calorimeter operator between source operators (Riembau et al., 18 Dec 2025).

This operator-level definition has direct event-level realizations. In transversely polarized $p^\uparrow p$ collisions, the one-point energy correlator is the energy-weighted angular distribution inside a reconstructed jet,

$$\frac{d \Sigma}{d \theta_n \, d \phi_n \, d \eta \, d p_T} = \sum_{h\in J} \int_0^1 d z_h \int d^2\Omega_h\, \delta(\phi_n-\phi_h)\,\delta(\theta_n-\theta_h)\,z_h\, \frac{d\sigma}{d z_h\, d^2\Omega_h\, d\eta\, d p_T},$$

with $z_h=E_h/E_J$ and $x$0 measured relative to the jet axis (Gao et al., 19 Sep 2025). For a jet observable in $x$1 or $x$2 kinematics, the closely related one-point EC inside jets is written as

$x$3

where $x$4 is the angle between a hadron and the standard jet axis (Mi et al., 29 Jul 2025).

The same detector formalism extends to conserved charges beyond energy. One-point correlators of conserved charges, including energy, electric charge, isospin, and baryon number, are argued to be perturbatively IR safe in QCD because the detector measures the total conserved charge in a collinear sector and inclusive averaging ensures KLN cancellation (Riembau et al., 2024).

2. Position within the hierarchy of energy correlators

One-point ECs sit at the bottom of the $x$5-point hierarchy. Energy correlators probe angular correlations among $x$6 final-state particles or energy-flow directions; $x$7 measures the angular distribution of energy itself, $x$8 gives the standard energy-energy correlator, and $x$9 resolves increasingly structured multi-prong configurations (Alipour-fard et al., 2024).

This hierarchy is especially clear in projected jet correlators. In the new parametrization of projected $$\mathcal E_n=\lim_{r\to\infty}\int_0^\infty dt\; r^2\,n^i\,T_{0i}(t,r\hat n),$$0-point energy correlators, the cumulative observable is

$$\mathcal E_n=\lim_{r\to\infty}\int_0^\infty dt\; r^2\,n^i\,T_{0i}(t,r\hat n),$$1

where $$\mathcal E_n=\lim_{r\to\infty}\int_0^\infty dt\; r^2\,n^i\,T_{0i}(t,r\hat n),$$2 is the total transverse-momentum fraction inside a disk of radius $$\mathcal E_n=\lim_{r\to\infty}\int_0^\infty dt\; r^2\,n^i\,T_{0i}(t,r\hat n),$$3 around a special particle $$\mathcal E_n=\lim_{r\to\infty}\int_0^\infty dt\; r^2\,n^i\,T_{0i}(t,r\hat n),$$4. Setting $$\mathcal E_n=\lim_{r\to\infty}\int_0^\infty dt\; r^2\,n^i\,T_{0i}(t,r\hat n),$$5 gives

$$\mathcal E_n=\lim_{r\to\infty}\int_0^\infty dt\; r^2\,n^i\,T_{0i}(t,r\hat n),$$6

which is independent of $$\mathcal E_n=\lim_{r\to\infty}\int_0^\infty dt\; r^2\,n^i\,T_{0i}(t,r\hat n),$$7 and equals the normalization of the jet energy fraction. In this projected Mellin-moment framework the one-point case is therefore physically trivial, and nontrivial angular dependence begins at $$\mathcal E_n=\lim_{r\to\infty}\int_0^\infty dt\; r^2\,n^i\,T_{0i}(t,r\hat n),$$8 (Alipour-fard et al., 2024).

This does not make one-point observables irrelevant. Rather, it fixes their role: they are baseline energy-flow observables, closely related to jet-shape-type radial energy profiles, whereas higher-point correlators encode genuine correlations between distinct directions. This suggests that one-point ECs and projected $$\mathcal E_n=\lim_{r\to\infty}\int_0^\infty dt\; r^2\,n^i\,T_{0i}(t,r\hat n),$$9-point ECs are complementary rather than competing descriptions (Alipour-fard et al., 2024).

3. Density matrices, spin structure, and positivity

For a vector current creating a state of total energy $|\alpha\rangle$0, the one-point energy correlator is completely characterized by a single anisotropy parameter $|\alpha\rangle$1. In polarization space,

$|\alpha\rangle$2

and for an unpolarized source this becomes

$|\alpha\rangle$3

The parameter $|\alpha\rangle$4 is bounded by unitarity,

$|\alpha\rangle$5

with the lower endpoint realized by free minimally coupled fermions and the upper endpoint by free scalars (Riembau et al., 2 Sep 2025).

In the spinning-correlator framework, the one-point EC is the simplest nontrivial spinning energy correlator. For a spin-1 source there is no internal detector geometry and only one nontrivial spin-2 structure, so the entire angular dependence collapses to the single coefficient $|\alpha\rangle$6. For general spin-$|\alpha\rangle$7 sources the one-point hadronic tensor decomposes into coefficients $|\alpha\rangle$8, $|\alpha\rangle$9, constrained by positivity into a convex polytope; the Hofman–Maldacena interval is the $$\mathcal E_n|\alpha\rangle=\sum_{i\in\alpha}E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$0 slice of this broader geometry (Riembau et al., 18 Dec 2025).

The same logic extends to conserved charges. For a detector $$\mathcal E_n|\alpha\rangle=\sum_{i\in\alpha}E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$1 built from a conserved current, the one-point density matrix may be parameterized by a total charge $$\mathcal E_n|\alpha\rangle=\sum_{i\in\alpha}E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$2, a parity-even anisotropy $$\mathcal E_n|\alpha\rangle=\sum_{i\in\alpha}E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$3, and a parity-odd coefficient $$\mathcal E_n|\alpha\rangle=\sum_{i\in\alpha}E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$4. For energy, $$\mathcal E_n|\alpha\rangle=\sum_{i\in\alpha}E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$5, but for more general charge detectors the parity-odd structure need not vanish (Riembau et al., 2024).

One-point ECs of hadronically decaying electroweak bosons make this spin information directly observable. For boosted $$\mathcal E_n|\alpha\rangle=\sum_{i\in\alpha}E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$6 or $$\mathcal E_n|\alpha\rangle=\sum_{i\in\alpha}E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$7 jets, the one-point EC $$\mathcal E_n|\alpha\rangle=\sum_{i\in\alpha}E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$8 distinguishes longitudinal from transverse polarization through distinct $$\mathcal E_n|\alpha\rangle=\sum_{i\in\alpha}E_i\,\delta^{(2)}(\Omega_i-\Omega_n)\,|\alpha\rangle.$$9 line shapes, and the associated asymmetry $\langle\mathcal E(\vec n)\rangle$0 separates longitudinal and transverse fractions in EFT analyses of anomalous gauge couplings (Ricci et al., 2022).

4. Jet-axis one-point ECs and SCET/TMD factorization

Inside jets, one-point ECs become differential probes of the in-jet energy-flow profile. In the SCET analysis of the jet-axis observable $\langle\mathcal E(\vec n)\rangle$1, the relevant quantity is an EC jet function constructed from semi-inclusive TMD fragmenting jet functions,

$\langle\mathcal E(\vec n)\rangle$2

The siTMDFJF factorizes into a hard matching function $\langle\mathcal E(\vec n)\rangle$3, a TMD fragmentation function $\langle\mathcal E(\vec n)\rangle$4, and an in-jet soft function $\langle\mathcal E(\vec n)\rangle$5, so the angular variable $\langle\mathcal E(\vec n)\rangle$6 directly probes the interplay between intrinsic TMD fragmentation and soft recoil (Mi et al., 29 Jul 2025).

The resulting normalized EC jet function resums global logarithms up to NNLL accuracy and non-global logarithms at LL accuracy. After including non-perturbative effects and comparing with PYTHIA 8, the normalized EC jet function exhibits significantly reduced dependence on the factorization scale and is primarily sensitive to the jet scale. In $\langle\mathcal E(\vec n)\rangle$7 annihilation it is mainly sensitive to quark TMDFFs, while in $\langle\mathcal E(\vec n)\rangle$8 collisions it becomes primarily sensitive to gluon TMDFFs because inclusive jet production is gluon dominated (Mi et al., 29 Jul 2025).

In transversely polarized $\langle\mathcal E(\vec n)\rangle$9 collisions, the one-point EC becomes a spin observable. The differential cross section decomposes as

$\langle \mathcal E_n\rangle_{hh'}$0

with single-spin asymmetry

$\langle \mathcal E_n\rangle_{hh'}$1

At leading twist, $\langle \mathcal E_n\rangle_{hh'}$2 couples the transversity PDF $\langle \mathcal E_n\rangle_{hh'}$3 to the transversely polarized OPEC fragmenting jet function $\langle \mathcal E_n\rangle_{hh'}$4 through a Collins-type hard factor. Because the observable is energy weighted and inclusive over hadron species, it is IRC safe, probes a much wider angular range $\langle \mathcal E_n\rangle_{hh'}$5 than hadron-$\langle \mathcal E_n\rangle_{hh'}$6 measurements, and accesses transversity through a complementary channel at RHIC and the EIC (Gao et al., 19 Sep 2025).

The same framework admits generalized weights $\langle \mathcal E_n\rangle_{hh'}$7. Increasing the power $\langle \mathcal E_n\rangle_{hh'}$8 enhances the SSA magnitude and shifts its angular profile, which suggests a systematic Mellin-moment program for spin-dependent fragmentation (Gao et al., 19 Sep 2025).

5. One-point ECs in DIS and small-$\langle \mathcal E_n\rangle_{hh'}$9 QCD

In deep inelastic scattering, the one-point EC is an energy-weighted angular distribution of all final-state hadrons relative to the incoming proton or nucleus,

$p^\uparrow p$0

with $p^\uparrow p$1 and $p^\uparrow p$2. Using

$p^\uparrow p$3

the observable interpolates between back-to-back kinematics ($p^\uparrow p$4) and the proton-collinear region ($p^\uparrow p$5) (Kang et al., 2 Mar 2026).

In the small-$p^\uparrow p$6 limit, the Color Glass Condensate formulation makes the one-point EC especially clean. After standard fragmentation factorization, the momentum sum rule

$p^\uparrow p$7

eliminates all fragmentation functions from the OPEC, leaving the dipole amplitude $p^\uparrow p$8, or equivalently its Fourier transform $p^\uparrow p$9, as the only nonperturbative input. The master expression is

$$\frac{d \Sigma}{d \theta_n \, d \phi_n \, d \eta \, d p_T} = \sum_{h\in J} \int_0^1 d z_h \int d^2\Omega_h\, \delta(\phi_n-\phi_h)\,\delta(\theta_n-\theta_h)\,z_h\, \frac{d\sigma}{d z_h\, d^2\Omega_h\, d\eta\, d p_T},$$0

with

$$\frac{d \Sigma}{d \theta_n \, d \phi_n \, d \eta \, d p_T} = \sum_{h\in J} \int_0^1 d z_h \int d^2\Omega_h\, \delta(\phi_n-\phi_h)\,\delta(\theta_n-\theta_h)\,z_h\, \frac{d\sigma}{d z_h\, d^2\Omega_h\, d\eta\, d p_T},$$1

Varying $$\frac{d \Sigma}{d \theta_n \, d \phi_n \, d \eta \, d p_T} = \sum_{h\in J} \int_0^1 d z_h \int d^2\Omega_h\, \delta(\phi_n-\phi_h)\,\delta(\theta_n-\theta_h)\,z_h\, \frac{d\sigma}{d z_h\, d^2\Omega_h\, d\eta\, d p_T},$$2 therefore scans the partonic transverse momentum scale and, through

$$\frac{d \Sigma}{d \theta_n \, d \phi_n \, d \eta \, d p_T} = \sum_{h\in J} \int_0^1 d z_h \int d^2\Omega_h\, \delta(\phi_n-\phi_h)\,\delta(\theta_n-\theta_h)\,z_h\, \frac{d\sigma}{d z_h\, d^2\Omega_h\, d\eta\, d p_T},$$3

the effective small-$$\frac{d \Sigma}{d \theta_n \, d \phi_n \, d \eta \, d p_T} = \sum_{h\in J} \int_0^1 d z_h \int d^2\Omega_h\, \delta(\phi_n-\phi_h)\,\delta(\theta_n-\theta_h)\,z_h\, \frac{d\sigma}{d z_h\, d^2\Omega_h\, d\eta\, d p_T},$$4 region of the target (Kang et al., 2 Mar 2026).

Numerically, the reduced OPEC and its longitudinal and transverse components show clear nuclear suppression,

$$\frac{d \Sigma}{d \theta_n \, d \phi_n \, d \eta \, d p_T} = \sum_{h\in J} \int_0^1 d z_h \int d^2\Omega_h\, \delta(\phi_n-\phi_h)\,\delta(\theta_n-\theta_h)\,z_h\, \frac{d\sigma}{d z_h\, d^2\Omega_h\, d\eta\, d p_T},$$5

with the suppression strongest at small $$\frac{d \Sigma}{d \theta_n \, d \phi_n \, d \eta \, d p_T} = \sum_{h\in J} \int_0^1 d z_h \int d^2\Omega_h\, \delta(\phi_n-\phi_h)\,\delta(\theta_n-\theta_h)\,z_h\, \frac{d\sigma}{d z_h\, d^2\Omega_h\, d\eta\, d p_T},$$6 and weakening as $$\frac{d \Sigma}{d \theta_n \, d \phi_n \, d \eta \, d p_T} = \sum_{h\in J} \int_0^1 d z_h \int d^2\Omega_h\, \delta(\phi_n-\phi_h)\,\delta(\theta_n-\theta_h)\,z_h\, \frac{d\sigma}{d z_h\, d^2\Omega_h\, d\eta\, d p_T},$$7 or $$\frac{d \Sigma}{d \theta_n \, d \phi_n \, d \eta \, d p_T} = \sum_{h\in J} \int_0^1 d z_h \int d^2\Omega_h\, \delta(\phi_n-\phi_h)\,\delta(\theta_n-\theta_h)\,z_h\, \frac{d\sigma}{d z_h\, d^2\Omega_h\, d\eta\, d p_T},$$8 increases. This makes the DIS one-point EC a direct probe of gluon saturation and dipole evolution at the future EIC (Kang et al., 2 Mar 2026).

6. Scale dependence, moment constraints, and higher-point context

The simplest integrated one-point EC of a vector current is encoded in the scale-dependent parameter $$\frac{d \Sigma}{d \theta_n \, d \phi_n \, d \eta \, d p_T} = \sum_{h\in J} \int_0^1 d z_h \int d^2\Omega_h\, \delta(\phi_n-\phi_h)\,\delta(\theta_n-\theta_h)\,z_h\, \frac{d\sigma}{d z_h\, d^2\Omega_h\, d\eta\, d p_T},$$9. In perturbative QCD it approaches the fermionic extremal value $z_h=E_h/E_J$0, while in the deep infrared, where two-pseudoscalar channels dominate, it approaches the scalar extremal value $z_h=E_h/E_J$1. This “flow between extremal correlators” is reconstructed using perturbative QCD, chiral perturbation theory, phenomenological models of multi-hadron channels, and experimental data (Riembau et al., 2 Sep 2025).

At high scales the perturbative series for massless QCD is known to N$z_h=E_h/E_J$2LO, and for $z_h=E_h/E_J$3 at the $z_h=E_h/E_J$4 pole the quoted values are

$z_h=E_h/E_J$5

The same parameter is related to the longitudinal fraction of the semi-inclusive $z_h=E_h/E_J$6 cross section by

$z_h=E_h/E_J$7

Using this relation, DELPHI and OPAL at $z_h=E_h/E_J$8 GeV give $z_h=E_h/E_J$9 and $x$00, consistent with the perturbative prediction (Riembau et al., 2 Sep 2025).

Beyond conserved energy flow, one-point moments $x$01 with arbitrary non-negative powers are not perturbatively calculable in general, but they obey an infinite set of positivity and moment constraints. The relevant functions are completely monotonic in $x$02, and the corresponding Hankel matrices must be positive definite. In the QCD study of chiral-current states, the allowed $x$03 region develops a sharp kink, and QCD lies near that kink, which suggests an approximately power-law moment hierarchy (Riembau et al., 2024).

These results clarify why many modern computational and geometric advances target $x$04. The new angular parametrization of projected and resolved $x$05-point ECs reduces the projected computational cost from $x$06 to $x$07, preserves orientation information, and simplifies phase space, but its $x$08 specialization collapses to the trivial normalization $x$09 (Alipour-fard et al., 2024). A plausible implication is that one-point ECs serve as normalization, spin, and sum-rule anchors for the broader energy-correlator program, while the genuinely new angular information of jet substructure enters at two points and above (Riembau et al., 18 Dec 2025, Alipour-fard et al., 2024).