Event Shapes and Energy Correlators in CFT

An intrinsic CFT event shape is a Wightman matrix element of several null-infinity detectors in a specified state. Its angular dependence is physical, but so are its distributional contact terms, state normalization, and energy-moment sum rules. A scattering event shape requires additional inclusive and infrared-safety data; the CFT detector correlator alone does not supply them.

Required background. Detector operators at null infinity fixes the limit, ordering, state, and total-energy normalization used below.

Helpful background. Jets and event-shape observables supplies the measurement functions, inclusive sums, and infrared criteria needed only when matching the CFT object to a scattering observable.

Evidence cutoff. Statements about detector OPEs, contact sectors, and current bootstrap uses below reflect primary sources available through 2026-08-09. The normalization and symmetry identities are durable; later perturbative values, numerical bounds, or convergence claims need a new source check.

Multi-detector observables

Let $|\Psi\rangle$ be a normalizable state in a $d>2$ unitary CFT, prepared by smeared local operators and with no incoming flux. Define the $k$-detector distribution

$$G_k(\mathbf n_1,\ldots,\mathbf n_k) =\frac{\langle\Psi|\mathcal E(\mathbf n_1)\cdots \mathcal E(\mathbf n_k)|\Psi\rangle}{\langle\Psi|\Psi\rangle}, \qquad \mathbf n_a\in S^{d-2}.$$

The insertions have the Wightman ordering displayed. For distinct angles, energy detectors commute provided the light transforms exist and the relevant correlator has Regge intercept $J_0<3$; the standard nonperturbative unitary-CFT bound used in the light-ray argument is stronger. Coincident angles are excluded from that statement and may carry contact distributions Koloğlu et al. 2021, §4, especially the condition following eq. (4.5).

Consequently, for separated directions,

$$G_k(\mathbf n_{\pi(1)},\ldots,\mathbf n_{\pi(k)}) =G_k(\mathbf n_1,\ldots,\mathbf n_k)$$

for every permutation $\pi$. The equality extends distributionally only after the coincident-angle prescription has been included on both sides.

Energy and momentum sum rules

The one-detector operator identities

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

generate a hierarchy of exact checks. If $|\Psi_E\rangle$ is an energy eigenstate, $P^0|\Psi_E\rangle=E|\Psi_E\rangle$, then

$$\int d\Omega_1\,G_2(\mathbf n_1,\mathbf n_2) =E\,G_1(\mathbf n_2),$$ $$\int d\Omega_1d\Omega_2\,G_2(\mathbf n_1,\mathbf n_2)=E^2,$$

and, more generally,

$$\int\prod_{a=1}^{k}d\Omega_a\,G_k=E^k.$$

For a wave packet that is not an exact energy eigenstate, the right-hand side is instead $\langle(P^0)^k\rangle_\Psi$. Replacing it by $\langle P^0\rangle_\Psi^k$ incorrectly discards the packet’s energy variance. Analogous first angular moments insert $P^i$; in a zero-momentum rest state they vanish.

These equations include every distribution supported at coincident angles. If a separated-angle expression integrates to less than $E^2$, the missing weight may be a contact term rather than an error in energy conservation. The original calorimeter definition and total-energy normalization appear in Hofman and Maldacena 2008, eqs. (1.1)–(1.2) and (2.9).

The scalar two-detector shape

For a scalar state at rest, rotations imply that $G_2$ depends only on

$$z=\frac{1-\mathbf n_1\cdot\mathbf n_2}{2} =\sin^2\frac\theta2, \qquad 0\le z\le1.$$

Here $z=0$ is the coincident or collinear limit and $z=1$ is the back-to-back limit. For a sharp energy $E$, define a dimensionless distribution by

$$G_2(\mathbf n_1,\mathbf n_2) =\frac{E^2}{\Omega_{d-2}^2}F_E(z).$$

The double-integrated sum rule becomes

$$\frac{1}{\Omega_{d-2}} \int d\Omega_{d-2}(\mathbf n_2)\,F_E(z)=1$$

with $\mathbf n_1$ fixed. In $d=4$, $d\Omega_2=4\pi\,dz$ after the azimuthal integral, so

$$\int_0^1dz\,F_E(z)=1.$$

This equation is distributional. Delta functions and derivatives at $z=0$ must be paired with test functions before integration. Light-ray analyses show explicitly how analytic continuation in representation dimension can generate such contact terms even when a separated-point block seems to vanish Koloğlu et al. 2021, §6.

If distinct energy detectors are positive and mutually commuting on a common domain, $G_k$ defines a nonnegative distribution there. At coincident angles, positivity means that its pairing with a nonnegative measurement function is nonnegative; it need not be an ordinary pointwise function.

From a CFT state to a scattering event shape

Detector insertions can be expressed as limits of Wightman correlators. In a CFT, one may analytically continue a Euclidean source correlator to the required Lorentzian ordering and then take the detector limit; the continuation is nontrivial and can be represented through double discontinuities Belitsky et al. 2014, §§2–4.

That formal relation does not erase the difference between the following objects:

Ingredient	Intrinsic CFT detector correlator	Scattering event shape
State	Smeared local-operator state or regulated timelike momentum state	Specified in-state and inclusive sum over out-states
Observable	Product of $\mathcal E(\mathbf n)$ on $S^{d-2}$	Measurement function acting on asymptotic particles or jets
Ordering	Wightman detector order; separated detectors commute under $J_0<3$	Determined by the inclusive cross-section prescription
Infrared statement	Existence of the null-integrated CFT distribution	Soft and collinear safety of the complete measurement
Contacts	Coincident-angle distributions retained	Binned, smeared, or combined with virtual/unresolved terms
Normalization	Powers or moments of $P^0$ in the source state	Cross section, decay rate, or normalized event ensemble

The detector energy weight is often the ingredient that makes an inclusive energy correlation infrared safe, but safety must be proved for the declared scattering theory and measurement. Jet algorithms and phenomenology remain in the scattering treatment.

A constructive two-detector check

Suppose a four-dimensional scalar energy eigenstate has a candidate regular contribution

$$F_{\mathrm{reg}}(z)=6z(1-z).$$

It is nonnegative and symmetric under $z\leftrightarrow1-z$, but normalization gives

$$\int_0^1dz\,6z(1-z)=1,$$

so no additional contact weight is required by the total-energy sum rule. By contrast, $3z(1-z)$ integrates to $1/2$. That result could be repaired by a positive contact contribution of total weight $1/2$, but one may not insert such a term without deriving its support and coefficient from the regulated correlator. Normalization alone detects missing weight; it does not locate it.

The symmetry $z\leftrightarrow1-z$ in this toy check is an extra chosen property, not detector exchange: exchanging $\mathbf n_1$ and $\mathbf n_2$ leaves $z$ unchanged. Back-to-back symmetry must come from the state or dynamics, not from Bose symmetry of the detectors.

A planned exact continuation

A reproducible calculation should provide one normalized four-dimensional two-detector fixture and a declared light-ray-OPE limit, checking exchange symmetry, the total-energy sum rule, endpoint behavior, and an angular convergence domain.

Common pitfalls

Confusing exchange with $z\leftrightarrow1-z$. Detector exchange fixes $G_2(\mathbf n_1,\mathbf n_2)=G_2(\mathbf n_2,\mathbf n_1)$ and leaves $z$ unchanged. Collinear–back-to-back exchange is an additional dynamical symmetry, usually absent.

Normalizing a packet by its mean energy squared. The double integral is $\langle(P^0)^2\rangle$, including the variance. Use a sharp-energy state or retain packet moments.

Integrating only the regular part. Endpoint delta functions and derivative contacts contribute to sum rules. Pair the full distribution with the constant test function.

Declaring infrared safety without a measurement. The intrinsic CFT correlator and a collider cross section have different state and inclusivity data. Specify the scattering measurement before making an infrared claim.

Exercises

1. Energy variance. Show how the double-integrated event shape measures the energy variance of a normalized wave packet.

Solution

Integrating both detectors gives $\langle(P^0)^2\rangle$. Subtracting the square of the one-detector integral gives $\langle(P^0)^2\rangle-\langle P^0\rangle^2=(\Delta E)^2\ge0$.

2. Contact weight. In $d=4$, let the normalized angular measure be $d\mu=z\,dz+c\,\delta_0$, where $\delta_0$ has unit mass at $z=0$. Fix $c$ from the total-energy sum rule and test positivity.

Solution

$\int_0^1 z\,dz=1/2$, so normalization requires $c=1/2$. Both the density $z$ and the contact weight are nonnegative, hence $d\mu$ is positive. A negative $c$ would be detected by a nonnegative test function concentrated near $z=0$.

References

• Belitsky, A. V., S. Hohenegger, G. P. Korchemsky, E. Sokatchev, and A. Zhiboedov. “From Correlation Functions to Event Shapes.” Nuclear Physics B 884 (2014): 305–343. doi:10.1016/j.nuclphysb.2014.04.020.
• 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.
• Koloğlu, Murat, Petr Kravchuk, David Simmons-Duffin, and Alexander Zhiboedov. “The Light-Ray OPE and Conformal Colliders.” Journal of High Energy Physics 2021, no. 01 (2021): 128. doi:10.1007/JHEP01(2021)128.