---
title: One-Point Energy Correlators in QFT & Jets
url: https://www.emergentmind.com/topics/one-point-energy-correlators-ec
type: topic
---

# One-Point Energy Correlators in QFT & Jets

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 [2512.16985, 2507.21613, 2509.15809, 2603.02300].

## 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 [2512.16985].

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 \((\theta_n,\phi_n)\) measured relative to the jet axis [2509.15809]. For a jet observable in \(e^+e^-\) or \(pp\) kinematics, the closely related one-point EC inside jets is written as
\[
\mathrm{EC}(\chi)=
\frac{1}{\sigma}\sum_a \int \frac{\omega_h}{\omega_J}\,
d\sigma_{e^-e^+\to (\mathrm{jet}\,h)X}\,\delta(\chi-\theta_h),
\]
where \(\chi\) is the angle between a hadron and the standard jet axis [2507.21613].

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 [2407.12082].

## 2. Position within the hierarchy of energy correlators

One-point ECs sit at the bottom of the \(N\)-point hierarchy. Energy correlators probe angular correlations among \(N\) final-state particles or energy-flow directions; \(N=1\) measures the angular distribution of energy itself, \(N=2\) gives the standard energy-energy correlator, and \(N\ge 3\) resolves increasingly structured multi-prong configurations [2410.16368].

This hierarchy is especially clear in projected jet correlators. In the new parametrization of projected \(N\)-point energy correlators, the cumulative observable is
\[
\Sigma_N(R_1)=\Big\langle \sum_s z_s\,[z_{\rm disk}(s,R_1)]^{N-1}\Big\rangle,
\]
where \(z_{\rm disk}(s,R_1)\) is the total transverse-momentum fraction inside a disk of radius \(R_1\) around a special particle \(s\). Setting \(N=1\) gives
\[
\Sigma_1(R_1)=\Big\langle \sum_s z_s\Big\rangle,
\]
which is independent of \(R_1\) 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 \(N\ge 2\) [2410.16368].

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 \(N\)-point ECs are complementary rather than competing descriptions [2410.16368].

## 3. Density matrices, spin structure, and positivity

For a vector current creating a state of total energy \(Q\), the one-point energy correlator is completely characterized by a single anisotropy parameter \(a_{\mathcal E}\). In polarization space,
\[
\langle \mathcal{E}_{\vec n} \rangle_{h'h}
=
\frac{Q}{4\pi}\Big[
\delta_{h'h}
+
a_{\mathcal E}\big(
3\,(n\!\cdot\!\epsilon^*_{h'})(n\!\cdot\!\epsilon_h)-\delta_{h'h}
\big)
\Big],
\]
and for an unpolarized source this becomes
\[
\frac{d\mathcal E}{d\Omega}(\theta)
=
\frac{Q}{4\pi}
\left[
1+a_{\mathcal E}\left(\frac{3}{2}\sin^2\theta-1\right)
\right].
\]
The parameter \(a_{\mathcal E}\) is bounded by unitarity,
\[
-\frac12 \le a_{\mathcal E}\le 1,
\]
with the lower endpoint realized by free minimally coupled fermions and the upper endpoint by free scalars [2509.02669].

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 \(a_{\mathcal E}\). For general spin-\(J\) sources the one-point hadronic tensor decomposes into coefficients \(H_{2j}\), \(j=0,\dots,J\), constrained by positivity into a convex polytope; the Hofman–Maldacena interval is the \(J=1\) slice of this broader geometry [2512.16985].

The same logic extends to conserved charges. For a detector \(\mathcal D_n\) built from a conserved current, the one-point density matrix may be parameterized by a total charge \(\mathcal D\), a parity-even anisotropy \(a_{\mathcal D}\), and a parity-odd coefficient \(b_{\mathcal D}\). For energy, \(b_{\mathcal E}=0\), but for more general charge detectors the parity-odd structure need not vanish [2407.12082].

One-point ECs of hadronically decaying electroweak bosons make this spin information directly observable. For boosted \(W\) or \(Z\) jets, the one-point EC \(\langle\mathcal E(z)\rangle\) distinguishes longitudinal from transverse polarization through distinct \(z/z_\star\) line shapes, and the associated asymmetry \(\mathscr A_{LT}\) separates longitudinal and transverse fractions in EFT analyses of anomalous gauge couplings [2207.03511].

## 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 \(\mathrm{EC}(\chi)\), the relevant quantity is an EC jet function constructed from semi-inclusive TMD fragmenting jet functions,
\[
J_c(z_c,\chi,\omega_JR,\mu_J)
=
\sum_h\int d^2\boldsymbol j_\perp \int dz_h\, z_h\,
\mathcal G_c^h(z_c,z_h,\omega_JR,j_\perp,\mu_J)\,
\delta\!\left(\frac{2j_\perp}{z_h\omega_J}-\chi\right).
\]
The siTMDFJF factorizes into a hard matching function \(\mathcal H_{c\to i}\), a TMD fragmentation function \(D_{h/i}\), and an in-jet soft function \(S_i\), so the angular variable \(\chi\) directly probes the interplay between intrinsic TMD fragmentation and soft recoil [2507.21613].

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 \(e^+e^-\) annihilation it is mainly sensitive to quark TMDFFs, while in \(pp\) collisions it becomes primarily sensitive to gluon TMDFFs because inclusive jet production is gluon dominated [2507.21613].

In transversely polarized \(p^\uparrow p\) collisions, the one-point EC becomes a spin observable. The differential cross section decomposes as
\[
\frac{d\Sigma}{d\theta_n\,d\phi_n\,d\eta\,dp_T}
=
Z_{UU}(\theta_n,\eta,p_T)
+
\sin(\phi_s-\phi_n)\,Z_{UT}(\theta_n,\eta,p_T),
\]
with single-spin asymmetry
\[
A_{UT}^{\sin(\phi_s-\phi_n)}=\frac{Z_{UT}}{Z_{UU}}.
\]
At leading twist, \(Z_{UT}\) couples the transversity PDF \(h_1^q(x)\) to the transversely polarized OPEC fragmenting jet function \(\mathcal J_{1,\perp}^c(\theta_n,Q)\) 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 \(\theta_n\) than hadron-\(j_\perp\) measurements, and accesses transversity through a complementary channel at RHIC and the EIC [2509.15809].

The same framework admits generalized weights \(z_h^n\). Increasing the power \(n\) enhances the SSA magnitude and shifts its angular profile, which suggests a systematic Mellin-moment program for spin-dependent fragmentation [2509.15809].

## 5. One-point ECs in DIS and small-\(x\) 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,
\[
\frac{d \Sigma_\lambda}{d \cos\theta}
=
\sum_h \int d \sigma^{\gamma^*_\lambda + p \to h + X}\;
z_h\;
\delta\!\big(\cos\theta_{hp}-\cos\theta\big),
\]
with \(z_h=(P\cdot p_h)/(P\cdot q)\) and \(\lambda=T,L\). Using
\[
\tau=\frac{1+\cos\theta}{2},
\]
the observable interpolates between back-to-back kinematics (\(\tau\to 0\)) and the proton-collinear region (\(\tau\to 1\)) [2603.02300].

In the small-\(x\) limit, the Color Glass Condensate formulation makes the one-point EC especially clean. After standard fragmentation factorization, the momentum sum rule
\[
\sum_h\int_0^1 dz\, z\,D_{h/a}(z)=1
\]
eliminates all fragmentation functions from the OPEC, leaving the dipole amplitude \(S_x(\boldsymbol r)\), or equivalently its Fourier transform \(\mathcal F_x(\boldsymbol q)\), as the only nonperturbative input. The master expression is
\[
\frac{d \Sigma_\lambda}{d \tau}
=
\frac{e m N_c S_\perp Q^2}{\pi (1-\tau)^2}
\sum_f e_f^2
\int_0^1 d\hat\xi
\int_0^\infty dq_\perp\, q_\perp\, \hat\xi^3\,
\mathcal F_x(q_\perp)\,
h_\lambda(q_\perp,k_\perp,\hat\xi,Q^2),
\]
with
\[
k_\perp=\hat\xi Q\sqrt{\frac{\tau}{1-\tau}}.
\]
Varying \(\tau\) therefore scans the partonic transverse momentum scale and, through
\[
x = x_B \frac{1-(1-\hat\xi)\tau}{1-\tau},
\]
the effective small-\(x\) region of the target [2603.02300].

Numerically, the reduced OPEC and its longitudinal and transverse components show clear nuclear suppression,
\[
R_A(\tau)=
\frac{d\tilde\Sigma_{\lambda,A}/d\tau}{A\,d\tilde\Sigma_{\lambda,p}/d\tau}<1,
\]
with the suppression strongest at small \(\tau\) and weakening as \(\tau\) or \(Q\) increases. This makes the DIS one-point EC a direct probe of gluon saturation and dipole evolution at the future EIC [2603.02300].

## 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 \(a_{\mathcal E}(Q)\). In perturbative QCD it approaches the fermionic extremal value \(-1/2\), while in the deep infrared, where two-pseudoscalar channels dominate, it approaches the scalar extremal value \(1\). This “flow between extremal correlators” is reconstructed using perturbative QCD, chiral perturbation theory, phenomenological models of multi-hadron channels, and experimental data [2509.02669].

At high scales the perturbative series for massless QCD is known to N\(^3\)LO, and for \(n_f=5\) at the \(Z\) pole the quoted values are
\[
a_{\mathcal E}(m_Z^2)\approx -0.4437\ \text{(NLO)},\quad -0.4258\ \text{(NNLO)},\quad -0.42\ \text{(N\(^3\)LO)}.
\]
The same parameter is related to the longitudinal fraction of the semi-inclusive \(e^+e^-\) cross section by
\[
a_{\mathcal E} = -\frac12 + \frac32\,\frac{\sigma_L}{\sigma_{\rm tot}}.
\]
Using this relation, DELPHI and OPAL at \(\sqrt s=91.2\) GeV give \(a_{\mathcal E}=-0.423\pm0.011\) and \(-0.415\pm0.008\), consistent with the perturbative prediction [2509.02669].

Beyond conserved energy flow, one-point moments \(\mathcal E^k\) 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 \(k\), and the corresponding Hankel matrices must be positive definite. In the QCD study of chiral-current states, the allowed \((a_{\mathcal E^k},\mathcal E^k)\) region develops a sharp kink, and QCD lies near that kink, which suggests an approximately power-law moment hierarchy [2407.12082].

These results clarify why many modern computational and geometric advances target \(N\ge2\). The new angular parametrization of projected and resolved \(N\)-point ECs reduces the projected computational cost from \(M^N/N!\) to \(M^2\log M\), preserves orientation information, and simplifies phase space, but its \(N=1\) specialization collapses to the trivial normalization \(\Sigma_1(R_1)=\langle\sum_s z_s\rangle\) [2410.16368]. 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 [2512.16985, 2410.16368].

Source: https://www.emergentmind.com/topics/one-point-energy-correlators-ec