--- title: Energy Correlator Jet Function url: https://www.emergentmind.com/topics/energy-correlator-jet-function type: topic --- # Energy Correlator Jet Function The energy correlator jet function is the collinear object that encodes how a jet contributes to an energy correlator at fixed angular resolution. In modern usage, the term does not denote a single universal formula; rather, it refers to a family of measured collinear functions appearing in factorization theorems for projected \(N\)-point correlators, near-side energy-energy correlators (EECs), heavy-flavor jets, and medium-modified jet observables. Across these settings, the common role is to isolate the angular-energy structure of final-state collinear radiation while hard production, and in some cases soft or medium dynamics, factorize separately [2210.09311], [2512.15896]. ## 1. Scope and terminology In the literature, “energy correlator jet function” is used for several closely related constructions. Some are standard SCET matrix elements for projected correlators, some are fragmentation-based EEC jet functions differential in a transverse scale, and some are medium-modified analogues or kernels that play the same structural role without being universal vacuum jet functions in the strict SCET sense [2210.09311], [2512.15896], [2411.04866], [2408.02753]. | Context | Representative object | Defining feature | |---|---|---| | Projected correlators | \(\vec J^{[\nu]}\), \(J_Q^{[N]}\) | Collinear function for projected \(N\)-point correlators | | Near-side EEC | \(\Gamma_i(\mu)\), \(\Gamma_i(\mu,q_T)\) | Energy-weighted di-hadron fragmentation object | | Medium-modified EEC | \(\mathcal K(\theta)\), \(J_i(xQ,\chi,\mu)\) | Angular kernel or jet function with medium dependence | | Other realizations | \(J_{\rm EEC}^f(b_\perp)\), \(J_c(z_c,\chi,\omega_JR,\mu_J)\) | Observable-specific measured collinear functions | This diversity is not merely terminological. In heavy-flavor jet substructure, the jet function localizes the heavy-quark mass dependence. In near-side EEC factorization, the jet function is built from di-hadron fragmentation functions and carries the nonperturbative collinear physics of energy-weighted hadron pairs. In cold-matter and heavy-ion applications, the corresponding object acquires explicit dependence on \(\hat q\), path length, Glauber exchange, or universal medium correlators. This suggests that the phrase is best understood as denoting a measured collinear sector tailored to a specific energy-correlator observable, rather than a single observable-independent function. ## 2. Vacuum factorization and renormalization-group structure A broad SCET formulation was given for projected \(N\)-point energy correlators and their analytic continuation to non-integer \(\nu\). In the collinear limit \(x_L\to 0\), the cumulant obeys \[ \Sigma^{[\nu]}(x_L,\ln\frac{Q^2}{\mu^2}) = \int_0^1 dx\, x^\nu\, \vec J^{[\nu]}\!\Bigl(\ln \frac{x_L x^2 Q^2}{\mu^2}\Bigr)\cdot \vec H\!\Bigl(x,\frac{Q^2}{\mu^2}\Bigr), \] with jet-function evolution \[ \frac{d \vec J^{[\nu]}\!\Bigl(\ln \frac{x_L Q^2}{\mu^2}\Bigr)}{d\ln\mu^2} = \int_0^1 dy\, y^\nu\, \vec J^{[\nu]}\!\Bigl(\ln \frac{x_L y^2 Q^2}{\mu^2}\Bigr)\cdot \widehat P(y). \] The relevant anomalous dimensions are timelike twist-2 moments evaluated at spin \(j=\nu+1\), and the one-loop jet functions are analytic in \(\nu\), with explicit dependence on \(\Psi(\nu)\) and \(\Psi'(\nu)\) [2004.11381]. A more direct EEC-specific formulation was later given in terms of integrated and unintegrated EEC jet functions. The integrated object is \[ \Gamma_i(\mu)\equiv \sum_{h_1h_2}\int d z_1d z_2\, z_1 z_2\, I\!\!D_i^{h_1h_2}(z_1,z_2,\mu), \] while the unintegrated function differential in relative transverse momentum is \[ \Gamma_i(\mu,q_T)= \sum_{h_1h_2}\int d ^2p_{1T}d ^2p_{2T} d z_1 d z_2\, (z_1 z_2)\, I\!\! D_i^{h_1h_2}(z_1,z_2,p_{1T},p_{2T}) \,\delta^{(2)}\!\left(\vec{q}_T-\left(\frac{\vec p_{1T}}{z_1}-\frac{\vec p_{2T}}{z_2}\right)\right). \] For the near-side EEC in \(e^+e^-\), the small-angle factorization is written as \[ \left.2\frac{d\Sigma_2^{e^+e^-}}{\pi Q^2d\zeta}\right|_{\zeta\to 0} = \frac{\sigma_0}{\sigma_{\textrm{tot}}} \int dx\,x^4\, H_i\left(x,\frac{Q}{\mu}\right)\Gamma_i\left(\mu,q_T\right), \] with an analogous DIS factorization. A central point is that for any fixed nonzero angle the EEC is IR safe, whereas the angle-integrated EEC is IR sensitive and depends on di-hadron fragmentation physics through \(\Gamma_i\). The corresponding \(b_T\)-space evolution is DGLAP-like; unlike standard TMDs, the paper emphasizes that there are no Sudakov double logarithms and that only single logarithms need to be resummed [2512.15896]. These vacuum formulations establish the canonical role of the energy correlator jet function: it is the process-independent collinear factor, whereas process dependence enters through hard functions, PDFs in DIS, or analogous short-distance coefficients. ## 3. Heavy-quark energy-correlator jet functions For identified heavy-flavor jets, the jet function becomes the locus of heavy-quark mass dependence. The projected \(N\)-point correlator factorization theorem is \[ \Sigma^{[N]}(R_L,p_T^2, m_Q,\mu) =\int_{0}^{1} dx\, x^N\, \vec J^{[N]} (R_L,x,m_Q,\mu) \cdot \vec H (x,p_T^2,\mu), \] with flavor vectors \[ \vec H \equiv \{H_g, H_q, H_Q=H_q\},\qquad \vec J^{[N]} \equiv \{J_g^{[N]}, J_q^{[N]}, J_Q^{[N]}\}. \] The heavy-quark jet function is defined by the SCET matrix element \[ \begin{aligned} J^{[N]}_{Q}(R_L,m_Q)= &\sum_{X} \sum_{i_1, i_2,...,i_N \in X} \left\langle 0\left|\bar{\chi}_{n}\right| X\right\rangle \frac{E_{i_1} E_{i_2}\cdots E_{i_N}}{p_T^N} \\ &\times \Theta\!\left(\max\{\theta_{ij}\}