Light-Ray OPE and Detector Expansions

When two detectors approach the same angle, their product is organized by a light-ray operator product expansion. The exchanged objects are nonlocal conformal operators labeled by boost spin and transverse representations; their coefficients are universal, while their matrix elements depend on the source state. The expansion controls the non-contact small-angle singularity only after the null integrals, spin continuation, convergence domain, and coincident-angle distributions have been fixed.

Required background. Event shapes and energy correlators supplies the normalized multi-detector distribution and its contact terms. The lightcone OPE and large-spin expansion supplies twist ordering and analytic continuation in spin.

Helpful background. The Lorentzian inversion formula explains how a double commutator recovers the light-ray OPE data and where the Regge spin threshold enters.

Evidence cutoff. Method and convergence statements here use primary sources available through 2026-08-09. Later extensions of the contact sector or transverse-spin inversion require a fresh check.

The light transform and its quantum numbers

For a symmetric-traceless primary $\mathcal O(x,z)$ of scaling dimension $\Delta$ and Lorentz spin $J$, with null polarization $z^2=0$, one Minkowski-patch form of the light transform is

$$\mathbf L[\mathcal O](x,z) =\int_{-\infty}^{\infty}d\alpha\, (-\alpha)^{-\Delta-J} \mathcal O\!\left(x-\frac{z}{\alpha},z\right).$$

The contour crosses the conformal boundary into the next Poincaré patch; the phase of $(-\alpha)^{-\Delta-J}$ is fixed by that Lorentzian contour and the Wightman ordering. As an operator-valued tempered distribution away from other insertions, the integral exists for $\Delta+J>1$. The transform changes conformal quantum numbers by the Weyl reflection

$$\mathbf L:(\Delta,J)\longmapsto(1-J,1-\Delta).$$

These statements, including the contour and homogeneity, are derived in Kravchuk and Simmons-Duffin 2018, §§2.3–2.4.

For integer $J$, this transform is a null integral of a local operator. Analytic continuation away from integer $J$ produces genuine continuous-spin operators; they should not be pictured as local tensors with a noninteger number of indices. The stress tensor has $(\Delta,J)=(d,2)$, and its null-infinity transform is the energy detector up to the fixed factor chosen by the detector normalization.

At future null infinity, the Lorentz group $SO(d-1,1)$ acts as the Euclidean conformal group on $S^{d-2}$. A transformed local operator of dimension $\Delta$ is homogeneous as a celestial primary of dimension

$$\delta=\Delta-1,$$

and also carries a representation $\lambda$ of the transverse rotation group $SO(d-2)$. The labels $J$, $\delta$, and $\lambda$ answer different questions: parent Lorentz/boost spin, celestial scaling, and transverse spin.

Why two energy detectors select parent spin three

Put two null-integrated operators of local spins $J_1,J_2$ on the same null plane. Under the boost that rescales the affine null coordinate, each null integration lowers the boost weight by one. The product therefore has the weight of the light transform of a local operator with

$$J=J_1+J_2-1.$$

For two stress-tensor detectors, $J_1=J_2=2$, hence

$$\mathcal E\times\mathcal E \quad\text{selects}\quad J=3.$$

This does not assert that a local spin-three primary appears in the Euclidean $T\times T$ OPE. The exchanged object is the analytic light-ray continuation on the relevant Regge trajectory, with signature and transverse representation fixed by the detector product. Koloğlu and collaborators derive the selection rule from both null-plane boosts and embedding-space homogeneity Koloğlu et al. 2021, §§2.1–2.3.

The detector OPE

For separated but nearby directions, a schematic operator statement is

$$\mathcal E(\mathbf n_1)\mathcal E(\mathbf n_2) \sim \sum_{a,\lambda} \mathcal C_{a,\lambda} (\mathbf n_{12},\partial_{\mathbf n_2}) \,\mathbb O_{a,J=3,\lambda}(\mathbf n_2) +\mathcal C_{\mathrm{contact}}.$$

The coefficient operators $\mathcal C_{a,\lambda}$ contain the angular powers and transverse descendants. The exchanged $\mathbb O_{a,J=3,\lambda}$ are light-ray operators obtained from poles or cuts of a Lorentzian inversion coefficient. Their OPE coefficients are state-independent; an event shape is obtained by taking $\langle\Psi|\cdots|\Psi\rangle$, so the one-point functions $\langle\mathbb O_a\rangle_\Psi$ carry the source dependence.

The inversion construction first expands on the $SO(d-1,1)$ principal series and computes coefficients from a double commutator. Turning that contour integral into a discrete trajectory sum requires a justified contour deformation. In particular:

• the light transforms and detector product must exist;
• the spin-three evaluation must lie above the relevant Regge threshold;
• the representation-dimension contour must have controlled arcs;
• poles, cuts, and possible degeneracies must all be included;
• contact terms at coincident directions are treated as distributions, not inferred from a separated-point formula.

The nonperturbative light-ray OPE on a common null plane and its celestial-block realization are established in Koloğlu et al. 2021, §§3 and 5. A perturbative expansion can violate the Regge condition term by term even when the resummed correlator is well defined, so the null integrations must not be interchanged blindly with a large-$N$ or coupling expansion Koloğlu et al. 2021, §8.2.

Four-dimensional small-angle scaling

In four dimensions set

$$2z=1-\mathbf n_1\cdot\mathbf n_2 \simeq\frac{\theta^2}{2}$$

for $\theta\ll1$. For a transverse-scalar contribution of parent spin $J=3$ and twist $\tau_a=\Delta_a-3$, the leading separated-angle term has the form

$$\mathcal E(\mathbf n_1)\mathcal E(\mathbf n_2) \sim \sum_a c_a(2z)^{(\tau_a-4)/2} \mathbb O_{a,J=3}(\mathbf n_2) +\text{transverse descendants}.$$

This is the four-dimensional twist power; it is not to be reused unchanged in another dimension. The formula and its transverse-spin refinements are reviewed and applied in Chen et al. 2022, §§3.3–3.5.

If the smallest non-contact twist with a nonzero coefficient and state matrix element is

$$\tau_*=2+\gamma(3),$$

then the energy–energy correlator scales as

$$G_2(\theta)\sim \theta^{-2+\gamma(3)} \langle\mathbb O_{*,J=3}\rangle_\Psi.$$

The argument is simply $(2z)^{(\tau_*-4)/2}\sim(\theta^2)^{(-2+\gamma(3))/2}$. Operator mixing requires diagonalizing the twist matrix at $J=3$ before assigning $\gamma(3)$. If symmetry makes the leading one-point function vanish in the chosen state, the next allowed trajectory controls the observed power.

In $d=4$, the small-angle measure is $d\Omega\sim2\pi\theta\,d\theta$. Thus a pure $\theta^{-2+\gamma}$ contribution is locally integrable for $\operatorname{Re}\gamma>0$, logarithmic at $\gamma=0$, and power divergent for $\operatorname{Re}\gamma<0$. This is a useful check, but it does not replace the full distributional and resummed treatment.

Transverse spin and multipoint detectors

The transverse group is $SO(d-2)$. In $d=4$ it is $SO(2)$, so its irreducible representations are integer transverse spins $j$. These labels are independent of the parent spin $J=3$. Higher transverse spin enters through angular tensor structures and descendants; for three or more detectors, several independent angular cross-ratios and transverse structures appear.

Lorentzian inversion can organize OPE data analytically in transverse spin under its own convergence assumptions. Three-detector analyses make that structure explicit, but their QCD factorization statements are not automatically CFT theorems Chen et al. 2022, §§4–6. This page uses that work only for the representation-theoretic organization, not for a scattering prediction.

Contact terms and order of limits

The separated-angle OPE does not determine every term supported at $z=0$. A general distributional completion may contain

$$\delta(z),\quad \delta'(z),\quad\ldots$$

with coefficients fixed by Ward identities, analytic continuation, or the regulated null integrals. Such terms can arise when a representation-theoretic zero multiplies a nonintegrable angular power, leaving a finite delta distribution. They contribute to energy-moment sum rules even though they vanish for $z>0$ Koloğlu et al. 2021, §6.2.

Keep the operations in this order:

1. prepare and normalize the source state;
2. define each null integral with retarded-time and angular smearing;
3. take the large-radius limit in the declared Wightman ordering;
4. derive the light-ray expansion at separated directions;
5. construct its analytic distributional extension;
6. only then take the coincident-angle limit or integrate a sum rule.

Changing this order can manufacture or erase a contact term.

A practical validity check

Before using a detector OPE, record the following:

Question	Required answer
Angular regime	A stated open domain such as $0<z<z_*$, with endpoints excluded until distributions are restored
Null integration	Regulator and order relative to the large-radius limit
Parent spin	$J=J_1+J_2-1$; for energy–energy, $J=3$
Transverse data	$SO(d-2)$ representation $\lambda$ or spin $j$
Spin continuation	Signature, Regge trajectory, and inversion threshold
Expansion status	Principal-series convergence versus a contour-deformed discrete/asymptotic sum
State dependence	Which $\langle\mathbb O_a\rangle_\Psi$ vanish or mix
Contacts	Distributional terms at coincident angles and their sum-rule contribution

Common pitfalls

Calling transverse spin the Lorentz spin. The energy–energy product selects parent $J=3$, while $j$ labels an $SO(d-2)$ representation. They enter different Casimirs and selection rules.

Using the lowest twist without checking the state. The leading operator controls the observed power only if both its OPE coefficient and its one-point function in the source state are nonzero.

Claiming a discrete convergent sum from the principal series. Contour deformation needs decay and singularity control. State whether the result is a convergent integral, convergent sum, or asymptotic expansion.

Treating $z=0$ as an ordinary point. Coincident detectors define a distributional boundary. Restore contacts before applying the total-energy sum rule.

Exercises

1. Boost-spin selection. Determine the parent spin in the OPE of an energy detector and a charge detector constructed from a spin-one current.

Solution

The local spins are $J_1=2$ for $T$ and $J_2=1$ for $J_\mu$. Hence $J=J_1+J_2-1=2$. This parent spin is distinct from its transverse representation.

2. Angular integrability. For $G_2(\theta)\sim\theta^{-2+\gamma}$ in $d=4$, determine when the contribution is locally integrable.

Solution

The integral behaves as $\int_0 d\theta\,\theta\,\theta^{-2+\gamma}=\int_0d\theta\,\theta^{-1+\gamma}$. It converges for $\operatorname{Re}\gamma>0$, is logarithmic at $\gamma=0$, and diverges for $\operatorname{Re}\gamma<0$. Contact and resummation effects must still be treated separately.

References

• Chen, Hao, Ian Moult, Joshua Sandor, and Hua Xing Zhu. “Celestial Blocks and Transverse Spin in the Three-Point Energy Correlator.” Journal of High Energy Physics 2022, no. 09 (2022): 199. doi:10.1007/JHEP09(2022)199.
• 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.
• 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.