Conformal Collider Bounds

• Conformal collider bounds are universal inequalities in CFTs that arise from requiring non-negative energy flux, thereby constraining three-point functions like ⟨JJT⟩ and ⟨TTT⟩.
• They are established via a combination of lightcone analysis, crossing symmetry, and reflection positivity, linking analytic properties to rigorous energy constraints.
• In four dimensions, these bounds directly translate to constraints on anomaly coefficients (such as a/c), playing a key role in differentiating between free, interacting, and supersymmetric theories.

Conformal collider bounds are universal inequalities on conformal field theory data obtained by demanding that the energy measured at null infinity in states created by local operators is non-negative in every direction. In the original Hofman–Maldacena construction, a local operator insertion plays the role of a collision event and idealized calorimeters placed on the celestial sphere record the outgoing energy flux; the resulting constraints act on the three-point functions $\langle JJT\rangle$ and $\langle TTT\rangle$, and in four dimensions they translate into bounds on anomaly coefficients such as $a/c$ (0803.1467, Hofman et al., 2016).

1. Conformal collider kinematics and observables

The basic observable is the energy flux operator in a null direction $\vec n$,

$$\mathcal{E}(\vec n)=\lim_{r\to\infty} r^{d-2}\int_{-\infty}^{\infty} dt\, n^i T_{0i}(t,r\vec n),$$

or, equivalently in lightcone variables, an integral of $T_{--}$ along a null ray. For a state created by a local operator $O$, one studies expectation values of the form

$$\langle \mathcal{E}(\vec n)\rangle_{\Psi} = \frac{\langle 0|O^\dagger \mathcal{E}(\vec n) O|0\rangle} {\langle 0|O^\dagger O|0\rangle}.$$

In this setup, conformal symmetry fixes the angular dependence of the one-point energy distribution up to finitely many constants appearing in the relevant three-point function. For states created by a conserved current $J_\mu$, the shape is governed by $\langle JJT\rangle$; for states created by the stress tensor, it is governed by $\langle TTT\rangle$0 (0803.1467).

The current one-point function in four dimensions depends on a single anisotropy parameter $\langle TTT\rangle$1, while the stress-tensor one-point function depends on two parameters, usually denoted $\langle TTT\rangle$2 and $\langle TTT\rangle$3. In the stress-tensor case, these parameters encode the parity-even tensor structures of $\langle TTT\rangle$4 after the Ward identity has fixed the overall normalization through the stress-tensor two-point coefficient $\langle TTT\rangle$5. The same formalism admits charge-flux observables built from conserved currents, and the small-angle singularities of multi-detector correlators are controlled by non-local light-ray operators of definite spin and twist (0803.1467).

2. Canonical inequalities and the four-dimensional anomaly window

For unitary, parity-preserving CFTs in dimensions $\langle TTT\rangle$6 with a unique stress tensor, the conformal collider bounds constrain the allowed values of the stress-tensor three-point coefficients. In the $\langle TTT\rangle$7 parametrization, the parity-even stress-tensor bounds take the form

$\langle TTT\rangle$8

$\langle TTT\rangle$9

$a/c$0

These inequalities correspond to the positivity of the energy flux for different stress-tensor polarizations and are equivalent to the non-negativity of effective scalar-like, fermion-like, and tensor-like contributions to $a/c$1 (Hofman et al., 2016).

In four dimensions the same constraints can be expressed in terms of the trace-anomaly coefficients $a/c$2 and $a/c$3. For a general unitary CFT with a unique stress tensor,

$a/c$4

For superconformal theories the supersymmetry relations among the three-point structures sharpen the window to

$a/c$5

These are the best-known form of the conformal collider bounds. Free scalar theory saturates the lower non-supersymmetric bound, free vector theory saturates the upper one, and interacting theories such as $a/c$6 SYM lie in the interior; in three dimensions parity invariance forces $a/c$7, so the allowed region collapses to a one-parameter slice (Hofman et al., 2016).

The current sector admits an analogous pair of inequalities on the single free parameter in $a/c$8. In general dimension, the lower endpoint is saturated by a free scalar theory and the upper endpoint by a free fermion theory, matching the original Hofman–Maldacena current bounds (Hofman et al., 2016).

3. From positivity postulate to field-theoretic proof

Historically, the original derivation treated positivity of the energy flux operator as a physically natural assumption. The later bootstrap proof showed that, for parity-preserving unitary CFTs with a unique stress tensor in $a/c$9, the same inequalities follow from conformal symmetry, unitarity, reflection positivity, and crossing symmetry alone, without assuming a positive spectrum for $\vec n$0 as an independent postulate (Hofman et al., 2016).

The proof replaces the direct detector computation by the study of spinning four-point functions such as

$\vec n$1

In the $\vec n$2-channel, the identity and the stress tensor dominate as $\vec n$3. After analytic continuation to the second sheet, the stress-tensor conformal block generates a characteristic $\vec n$4 singularity, and contour integrals around a small half-disk isolate its coefficient. Reflection positivity in the $\vec n$5-channel implies non-negative expansion coefficients; crossing then converts that positivity into a sign constraint on the $\vec n$6 pole, which is precisely the conformal collider inequality (Hofman et al., 2016).

A precursor to the full proof came from the lightcone bootstrap in three dimensions. There, stress-tensor exchange in spinning four-point functions was shown to induce negative anomalous dimensions for large-spin double-twist operators if and only if the corresponding $\vec n$7 and $\vec n$8 coefficients satisfy the conformal collider bounds. In AdS language, the statement becomes that positivity of energy flux is equivalent to the long-distance attractiveness of gravity for the corresponding two-particle states (Li et al., 2015).

4. ANEC, interference bounds, and parity-violating sectors

The average null energy condition extends the collider program beyond pure $\vec n$9 and $$\mathcal{E}(\vec n)=\lim_{r\to\infty} r^{d-2}\int_{-\infty}^{\infty} dt\, n^i T_{0i}(t,r\vec n),$$0 constraints. By considering states that are superpositions of distinct primaries, one obtains energy matrices whose off-diagonal entries are linear in mixed three-point coefficients. Positivity of these matrices yields upper bounds that are inaccessible in single-operator collider experiments. For scalar primaries $$\mathcal{E}(\vec n)=\lim_{r\to\infty} r^{d-2}\int_{-\infty}^{\infty} dt\, n^i T_{0i}(t,r\vec n),$$1 in $$\mathcal{E}(\vec n)=\lim_{r\to\infty} r^{d-2}\int_{-\infty}^{\infty} dt\, n^i T_{0i}(t,r\vec n),$$2, one such result is

$$\mathcal{E}(\vec n)=\lim_{r\to\infty} r^{d-2}\int_{-\infty}^{\infty} dt\, n^i T_{0i}(t,r\vec n),$$3

where $$\mathcal{E}(\vec n)=\lim_{r\to\infty} r^{d-2}\int_{-\infty}^{\infty} dt\, n^i T_{0i}(t,r\vec n),$$4 is the bosonic structure coefficient in $$\mathcal{E}(\vec n)=\lim_{r\to\infty} r^{d-2}\int_{-\infty}^{\infty} dt\, n^i T_{0i}(t,r\vec n),$$5 and $$\mathcal{E}(\vec n)=\lim_{r\to\infty} r^{d-2}\int_{-\infty}^{\infty} dt\, n^i T_{0i}(t,r\vec n),$$6 is an explicit positive function of the scalar dimension. In four dimensions, the same interference method gives the gravitational-anomaly bound

$$\mathcal{E}(\vec n)=\lim_{r\to\infty} r^{d-2}\int_{-\infty}^{\infty} dt\, n^i T_{0i}(t,r\vec n),$$7

and, in $$\mathcal{E}(\vec n)=\lim_{r\to\infty} r^{d-2}\int_{-\infty}^{\infty} dt\, n^i T_{0i}(t,r\vec n),$$8 SCFTs, the refinement

$$\mathcal{E}(\vec n)=\lim_{r\to\infty} r^{d-2}\int_{-\infty}^{\infty} dt\, n^i T_{0i}(t,r\vec n),$$9

together with the supersymmetric collider window $T_{--}$0 (Cordova et al., 2017).

In three dimensions, parity violation introduces qualitatively new structures. For $T_{--}$1, the parity-even anisotropy parameter $T_{--}$2 and the parity-odd parameter $T_{--}$3 obey the disk constraint

$T_{--}$4

Bootstrap analyses of $T_{--}$5 showed that the parity-odd stress-tensor block has no logarithmic singularity in the lightcone limit; instead it carries a square-root branch cut. Consequently, at leading order it does not shift the anomalous dimensions of the standard double-twist families, but it does generate a new tower of parity-odd large-spin OPE coefficients. The collider bound is then recovered from reflection positivity and a Cauchy–Schwarz inequality applied to the residues of second-sheet singularities (Chowdhury et al., 2018).

A complementary momentum-space analysis rephrased the parity-even and parity-odd homogeneous structures of three-dimensional spinning correlators in terms of a single complex parameter $T_{--}$6. In that language, $T_{--}$7 corresponds to the free boson and free fermion theories, a pure phase $T_{--}$8 corresponds to Chern–Simons matter theories, generic three-dimensional CFTs satisfy $T_{--}$9, and holographic CFTs satisfy the stronger $O$0 bound set by the higher-spin gap (Jain et al., 2021).

5. Holographic, Regge, and AdS interpretations

For large-$O$1 CFTs with a sparse higher-spin spectrum, conformal collider bounds admit a sharpened holographic formulation. The central object is the holographic null energy operator

$O$2

whose expectation value in a suitable subspace of smeared states is positive as a consequence of Regge analyticity and the chaos bound. In the limit $O$3, this operator reduces to the ordinary averaged null energy operator; for finite $O$4, it probes a bulk shockwave at depth $O$5 in AdS (Afkhami-Jeddi et al., 2018).

The resulting positivity conditions are substantially stronger than the original lightcone bounds. For $O$6, the three-point function $O$7 is forced to its Einstein-Maxwell value up to gap-suppressed corrections, with the non-minimal bulk coupling bounded by the higher-spin scale. Likewise, $O$8 is fixed to the Einstein-gravity form, and higher-derivative couplings such as $O$9 and $$\langle \mathcal{E}(\vec n)\rangle_{\Psi} = \frac{\langle 0|O^\dagger \mathcal{E}(\vec n) O|0\rangle} {\langle 0|O^\dagger O|0\rangle}.$$0 are suppressed by $$\langle \mathcal{E}(\vec n)\rangle_{\Psi} = \frac{\langle 0|O^\dagger \mathcal{E}(\vec n) O|0\rangle} {\langle 0|O^\dagger O|0\rangle}.$$1 and $$\langle \mathcal{E}(\vec n)\rangle_{\Psi} = \frac{\langle 0|O^\dagger \mathcal{E}(\vec n) O|0\rangle} {\langle 0|O^\dagger O|0\rangle}.$$2, respectively. Off-diagonal couplings $$\langle \mathcal{E}(\vec n)\rangle_{\Psi} = \frac{\langle 0|O^\dagger \mathcal{E}(\vec n) O|0\rangle} {\langle 0|O^\dagger O|0\rangle}.$$3 between distinct low-spin primaries are generically forced to vanish at leading order, again up to gap-suppressed mixing effects (Afkhami-Jeddi et al., 2018).

This framework also yields a universal Regge OPE for smeared low-spin single-trace operators: after appropriate smearing, the product of any two such operators creates the same shockwave operator, with a coefficient determined only by the total energy. In the bulk dual, that universality is the CFT realization of the equivalence principle. In three dimensions, parity-odd structures survive but are bounded by logarithmically enhanced gap suppressions, consistent with the corresponding parity-odd higher-derivative couplings in AdS$$\langle \mathcal{E}(\vec n)\rangle_{\Psi} = \frac{\langle 0|O^\dagger \mathcal{E}(\vec n) O|0\rangle} {\langle 0|O^\dagger O|0\rangle}.$$4 (Afkhami-Jeddi et al., 2018).

6. Extensions, analogues, and current directions

One extension replaces null-energy measurements by entanglement observables. In three dimensions, the universal constant term $$\langle \mathcal{E}(\vec n)\rangle_{\Psi} = \frac{\langle 0|O^\dagger \mathcal{E}(\vec n) O|0\rangle} {\langle 0|O^\dagger O|0\rangle}.$$5 in the entanglement entropy of a region $$\langle \mathcal{E}(\vec n)\rangle_{\Psi} = \frac{\langle 0|O^\dagger \mathcal{E}(\vec n) O|0\rangle} {\langle 0|O^\dagger O|0\rangle}.$$6, normalized by the disk value $$\langle \mathcal{E}(\vec n)\rangle_{\Psi} = \frac{\langle 0|O^\dagger \mathcal{E}(\vec n) O|0\rangle} {\langle 0|O^\dagger O|0\rangle}.$$7, was conjectured to obey universal upper and lower bounds. For small deformations of the circle, this yields the three-dimensional analogue

$$\langle \mathcal{E}(\vec n)\rangle_{\Psi} = \frac{\langle 0|O^\dagger \mathcal{E}(\vec n) O|0\rangle} {\langle 0|O^\dagger O|0\rangle}.$$8

In four dimensions, the analogous inequality for the universal logarithmic entanglement term is equivalent to the Hofman–Maldacena bounds on $$\langle \mathcal{E}(\vec n)\rangle_{\Psi} = \frac{\langle 0|O^\dagger \mathcal{E}(\vec n) O|0\rangle} {\langle 0|O^\dagger O|0\rangle}.$$9, so the entanglement formulation becomes an alternative encoding of the same collider data (Bueno et al., 2023).

A second extension concerns two-detector observables, especially the energy-energy correlator. In planar $J_\mu$0 SYM, the EEC admits a Legendre expansion

$J_\mu$1

with $J_\mu$2 for $J_\mu$3, and for even $J_\mu$4 a universal upper bound $J_\mu$5. Combining perturbation theory, holography, localization, integrability, dispersive functionals, and numerical bootstrap methods yields tight all-coupling bounds on the first nontrivial multipoles $J_\mu$6 and on the full angular dependence of the EEC in planar $J_\mu$7 SYM. This extends conformal collider bounds from one-point energy flux to a four-point, two-detector observable that is sensitive to Regge and light-ray OPE data (Dempsey et al., 11 Dec 2025).

A third extension appears in defect and interface CFTs. For a conformal interface between two two-dimensional CFTs, positivity of reflected and transmitted energy flux gives bounds on the interface stress-tensor coupling $J_\mu$8 and on the displacement-operator two-point coefficient $J_\mu$9,

$\langle JJT\rangle$0

Here the collider logic constrains defect data rather than bulk three-point functions, and the displacement coefficient plays a role analogous to a defect central charge governing the response to interface deformations (Meineri et al., 2019).

Conformal collider bounds are therefore not limited to the original four-dimensional $\langle JJT\rangle$1 window. They comprise a broader positivity framework that constrains parity-even and parity-odd three-point structures, mixed OPE coefficients, Regge observables, entanglement shape functionals, and defect data. The central unifying principle is unchanged across these settings: analyticity, unitarity, and positivity of null-energy-type observables carve out the allowed space of CFT data.