Celestial OPEs, Loop Corrections, and Infrared Factorization

Celestial short-distance structures inherit concrete information from four-dimensional soft and collinear factorization. At tree level, a collinear splitting function Mellin-transforms into a celestial OPE coefficient, while soft poles in conformal dimension produce current-like operators. At loop level, infrared logarithms, scale dependence, higher poles, and operator mixing must be retained; a tree current algebra is not automatically exact.

Required background. Celestial Bases and Boost Eigenstates as Boundary-Dictionary Inputs fixes the Mellin contour, UV/IR Poles and the Renormalized-Amplitude Interface fixes loop conventions, and Dressed States and Infrared-Finite Scattering supplies the physical infrared completion.

Helpful background. Soft and Collinear Singularities supplies factorization, while Large-N Crossing, Double-Trace Data, and Contact Ambiguities provides a comparison with a genuine OPE reconstruction problem.

Collinear factorization becomes an OPE

When two outgoing massless momenta become collinear, a tree amplitude factorizes,

$$\mathcal A_n(1,2,\ldots) \xrightarrow{1\parallel2} \sum_J {\rm Split}_{J_1J_2}^{\ J}(t,z_{12},\bar z_{12}) \mathcal A_{n-1}(P^J,\ldots),$$

where $p_1=tP$, $p_2=(1-t)P$. Mellin-transforming the two energies turns the integral over $t$ into beta functions. The resulting celestial product has the schematic form

$$\mathcal O_{\Delta_1,J_1}(z_1,\bar z_1) \mathcal O_{\Delta_2,J_2}(z_2,\bar z_2) \sim\sum_J \mathcal K_J(z_{12},\bar z_{12}) B(a(\Delta_i),b(\Delta_i)) \mathcal O_{\Delta_1+\Delta_2+\sigma,J}(z_2,\bar z_2).$$

The helicity kernel $\mathcal K_J$ and shift $\sigma$ come directly from the four-dimensional splitting amplitude. This derivation fixes a singular collinear limit; it does not prove convergence of an operator expansion for arbitrary celestial separations. Explicit gluon and graviton coefficients are derived in Pate et al. 2021, §§3–5.

Soft currents and loop logarithms

The Mellin transform maps an energy-soft expansion to poles at special $\Delta$. Residues of leading soft poles act as celestial currents whose Ward identities reproduce asymptotic charges. Loop amplitudes instead factor schematically as

$$\mathcal A_n(\epsilon,\mu) =Z_{\rm IR}(\epsilon,\mu,p_i)\,\mathcal H_n(\mu,p_i),$$

where $Z_{\rm IR}$ contains universal infrared poles and $\mathcal H_n$ depends on the subtraction convention. Terms such as $\log(\omega_i/\mu)$ become derivatives with respect to $\Delta_i$ after Mellin transformation. Consequently, loop celestial operators can mix and soft poles can acquire higher order; these effects are explicit in Krishna 2024, §§2–5.

First application. Mellin-transform a collinear splitting relation into a celestial OPE coefficient, then include a one-loop infrared logarithm and track its scale and mixing dependence. Keep the $t$ integral that produces the beta function, replace $\log\omega$ by its $\partial_\Delta$ action, and report separately the universal infrared factor and finite hard coefficient.

Scheme and dressing dependence

An infrared subtraction can move finite terms between $Z_{\rm IR}$ and $\mathcal H_n$. A coherent-state dressing can instead change the asymptotic operator by including its soft cloud. Only the complete dressed or inclusive observable is regulator-independent; a coefficient extracted from a bare hard factor may be a useful scheme-dependent building block rather than an intrinsic OPE coefficient. The all-loop factorization perspective is developed in He, Mao, and Mao 2023.

Adversarial control. Change the infrared scale or finite subtraction and recompute the proposed celestial dimension and OPE coefficient. If either shifts without a compensating operator redefinition, it is not a scheme-independent observable. Then compare bare and dressed external states; a current algebra that ignores the changed soft sector has exceeded its domain.

Evidence ceiling

Soft and collinear limits rigorously constrain singular celestial products order by order in perturbation theory. Current algebras and loop-corrected OPE data provide important boundary structure, but they do not yet establish a globally convergent local OPE, a unique infrared-complete operator algebra, or nonperturbative celestial dynamics.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Arkani-Hamed, Nima, Monica Pate, Ana-Maria Raclariu, and Andrew Strominger. “Celestial Amplitudes from UV to IR.” Journal of High Energy Physics 2021, no. 8 (2021): 062. DOI; Open PDF.
• He, Song, Pujian Mao, and Xin-Cheng Mao. “Loop Corrections versus Marginal Deformation in Celestial Holography.” (2023, revised 2024). arXiv:2307.02743.
• Krishna, Hare. “Celestial Gluon and Graviton OPE at Loop Level.” Journal of High Energy Physics 2024, no. 3 (2024): 176. DOI; Open PDF.
• Pate, Monica, Ana-Maria Raclariu, Andrew Strominger, and Ellis Yuan. “Celestial Operator Products of Gluons and Gravitons.” Reviews in Mathematical Physics 33 (2021): 2140003. DOI; Open PDF.