Energy Flow, Light-Ray Operators, and Conformal Colliders

Energy-flow observables connect local stress-tensor correlators to what ideal detectors measure at large distances. This chapter develops the detector normalization, the positivity of complete null averages, and the resulting bounds on CFT data. Products of detectors bring in light-ray operators and their own short-angle expansion.

Enter this chapter

Begin with Detector Operators and Energy Flow at Null Infinity to define the measured flux and check its normalization.

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.

Topics

1. Detector Operators and Energy Flow at Null Infinity — Defines the null-infinity limit and relates integrated flux to total momentum.
2. Averaged Null Energy and Positivity — States positivity for complete null averages with its domain and endpoint conditions.
3. Conformal Collider Observables and Stress-Tensor Bounds — Turns positivity of polarization-dependent energy flux into bounds on three-point data.
4. Event Shapes and Energy Correlators in CFT — Builds detector correlators with their contact terms and energy sum rules.
5. Light-Ray OPE and Detector Expansions — Organizes nearby detectors using nonlocal light-ray operators.
6. Bootstrap of Energy Correlators — Combines positivity, moments, and operator expansions to constrain angular distributions.
7. Energy Conditions, Causality, and Regge Consistency — Separates the hypotheses connecting null energy, causal ordering, and Regge growth.

How the ideas fit together

A local source prepares a state, and a stress-tensor insertion measures its outgoing energy. The resulting three-point function determines the angular flux. Requiring that flux to be nonnegative for every allowed polarization constrains the source–stress-tensor coefficients; integrating it over angles must recover the state’s energy. Hofman and Maldacena 2008, §§ 2.1–2.3.

Two nearby detectors probe a different limit, organized by a light-ray operator expansion. Neither an integrated positivity statement nor a correct energy sum rule guarantees convergence of that expansion. An experimental event shape additionally needs a measurement definition and an inclusive treatment of unresolved radiation.

Review the chapter

For a rotationally invariant state in four spacetime dimensions with mean energy $E$, what is its mean energy flux per unit solid angle?

Solution

Rotational invariance makes the flux a constant $C$. Its integral over the sphere is the energy, so $4\pi C=E$ and $C=E/(4\pi)$. This is an energy density in angle, not a probability density: its integral is $E$, not one.

Where to go next

Continue to Bootstrap of Energy Correlators to constrain multipoint detector distributions, or return to The CFT Regge Limit and Boundedness for causal growth conditions.

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.