Energy Flow, Light-Ray Operators, and Conformal Colliders

Energy flow turns the stress tensor into an observable on the celestial sphere. A localized CFT excitation reaches future null infinity, where an ideal detector records the energy crossing each angle. Positivity of the corresponding null integral constrains current and stress-tensor three-point functions; products of detectors have their own operator expansion and consistency conditions. This chapter develops those statements in that order and keeps separate the hypotheses behind detector positivity, scattering event shapes, and Regge bounds.

Helpful background. Currents and the stress tensor supplies the Ward identities and normalizations used throughout. Jets and event-shape observables explains the scattering observables to which CFT detector correlators may be matched. CFT Regge boundedness is needed only for the final causality comparisons.

Detectors, null averages, and collider states

Work in a unitary Lorentzian CFT in $d>2$ with the site-wide $(+---)$ metric convention. A detector direction is a future null vector

$$n^\mu=(1,\mathbf n), \qquad \mathbf n^2=1, \qquad \mathbf n\in S^{d-2}.$$

The energy detector $\mathcal E(\mathbf n)$ is the retarded-time integral of the outgoing stress-tensor flux at future null infinity. This chapter fixes its normalization by the operator sum rules

$$\int_{S^{d-2}}d\Omega_{d-2}\,\mathcal E(\mathbf n)=P^0, \qquad \int_{S^{d-2}}d\Omega_{d-2}\,n^i\mathcal E(\mathbf n)=P^i.$$

These equations are more than conventions: they are the fastest checks on every numerical factor in a detector calculation. For a normalized state of sharp energy $E$ and zero spatial momentum, the one-point distribution integrates to $E$. For a scalar source it is isotropic, $\langle\mathcal E(\mathbf n)\rangle=E/\Omega_{d-2}$.

A detector at null infinity and the averaged-null-energy operator on a complete null line are conformally related, but they are not literally the same regulated expression. The relation carries a conformal Jacobian, an affine normalization for the null vector, and an ordering prescription. The theorem page states those details before using positivity. This distinction is essential in the original conformal-collider construction and in later light-ray treatments Hofman and Maldacena 2008, §§2.1–2.3; Kravchuk and Simmons-Duffin 2018, §§2 and 6.

Route through the chapter

Question	Start here	Result you should be able to check
What does an ideal calorimeter measure?	Detector Operators and Energy Flow at Null Infinity	Derive the retarded-time limit, normalize it to $P^\mu$, and evaluate a scalar-state one-point function.
Which null average is positive?	Averaged Null Energy and Positivity	State ANEC with its state domain, smearing, regulator, and boundary assumptions.
How does positivity constrain three-point data?	Conformal Collider Observables and Stress-Tensor Bounds	Diagonalize the four-dimensional parity-even energy matrix and reproduce its three inequalities.
What is a multi-detector event shape?	Event Shapes and Energy Correlators in CFT	Derive exchange symmetry and the integrated one- and two-detector energy sum rules, including contact terms.
What controls nearby detectors?	Light-Ray OPE and Detector Expansions	Identify the boost-spin-three light-ray sector in the energy–energy OPE and its angular power.
What can be bootstrapped?	Bootstrap of Energy Correlators	Formulate a positive moment problem and obtain a normalized angular-moment bound.
Which causal statements imply which bounds?	Energy Conditions, Causality, and Regge Consistency	Trace each implication with its smearing, sheet, Regge-growth, subtraction, gap, and semiclassical assumptions.

The detector OPE is an operator statement in a CFT, whereas an experimentally defined jet observable additionally needs a measurement function, an inclusive prescription, and infrared safety. Those scattering ingredients remain with the Volume 4 treatment. Conversely, this chapter does not use jets to define its intrinsic CFT correlators.

A normalization check before proceeding

Let $|\Psi_E\rangle$ be a normalized, zero-momentum scalar state with sharp energy $E$. Rotational invariance makes $\langle\mathcal E(\mathbf n)\rangle_{\Psi_E}=C$. Integration gives

$$E=\langle P^0\rangle_{\Psi_E} =\int d\Omega_{d-2}\,C =\Omega_{d-2}C,$$

so $C=E/\Omega_{d-2}$. In four dimensions $\Omega_2=4\pi$ and $C=E/(4\pi)$, agreeing with the direct conformal-collider calculation Hofman and Maldacena 2008, eq. (2.29). A result that instead integrates to $1$, $E^2$, or $2E$ has mixed a probability density, a two-detector normalization, or a different light-transform convention with the physical detector.

What the conclusions do not assume

ANEC by itself does not assert pointwise positivity of $T_{00}$ or $T_{uu}$. Collider inequalities do not automatically extend between spacetime dimensions, parity sectors, or tensor bases. A positive energy correlator does not by itself prove completeness or convergence of a chosen light-ray expansion. Finally, neither collider positivity nor microcausality alone supplies a large higher-spin gap, a vanishing Regge arc, or an unsubtracted dispersion relation.

The discussions of evolving methods summarize primary results available through 2026-08-09. Later numerical bounds or revised convergence claims require a renewed source check rather than an update to the durable definitions above.

Review the chapter

1. Starting from $\int d\Omega\,\mathcal E=P^0$, explain why the scalar one-point distribution is uniform in the rest frame but not after a boost.
2. Name the additional data required to turn $\langle\mathcal E(\mathbf n_1)\mathcal E(\mathbf n_2)\rangle$ into a scattering event shape.
3. Give one reason that an ANEC inequality cannot be inserted unchanged into a Regge dispersion relation.

Answers

1. Rotational invariance fixes the rest-frame distribution to a constant. A boost acts conformally on the celestial sphere and changes both the angular Jacobian and the measured energy.
2. One needs a specified source or scattering state, detector normalization, angular measurement function, inclusive sum over unresolved radiation, infrared prescription, and treatment of coincident-detector contacts.
3. A dispersion relation also needs a specified Lorentzian ordering and sheet, a Regge growth bound, an arc estimate, and enough subtractions; ANEC supplies none of these automatically.

References

• Hofman, Diego M., and Juan Maldacena. “Conformal Collider Physics: Energy and Charge Correlations.” Journal of High Energy Physics 2008, no. 05 (2008): 012. doi:10.1088/1126-6708/2008/05/012.
• Kravchuk, Petr, and David Simmons-Duffin. “Light-Ray Operators in Conformal Field Theory.” Journal of High Energy Physics 2018, no. 11 (2018): 102. doi:10.1007/JHEP11(2018)102.