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The Infrared Triangle for QED and QCD

Three parallel infrared triangles for gravity, electrodynamics, and chromodynamics — each connecting asymptotic symmetries, soft theorems, and memory effects

Soft GravitonSupertranslationDisplacement Mem.GRAVITYSoft PhotonLarge U(1) GaugeEM MemoryQEDSoft GluonLarge SU(3) GaugeColor MemoryQCDanalogyanalogyEach triangle: asymptotic symmetry ↔ soft theorem ↔ memory effect

The QED Infrared Triangle

Quantum electrodynamics possesses the same infrared triality as gravity. The three vertices are related by Ward identities, Fourier transforms, and mode expansions, just as in the gravitational case.

Large Gauge Transformations

At null infinity, not all gauge transformations are trivial. The large $U(1)$ gauge transformations with parameter $\epsilon(z,\bar{z})$ on the celestial sphere act as:

$$A_\mu \to A_\mu + \partial_\mu\epsilon, \qquad \epsilon|_{\mathscr{I}^+} = \epsilon^+(z,\bar{z})$$

These are the QED analogues of BMS supertranslations. The associated conserved charges generate an infinite-dimensional symmetry group at null infinity:

$$Q_\epsilon^+ = \frac{1}{e^2}\int_{\mathscr{I}^+_-} \epsilon\, F_{ru}\, d^2z + \int_{\mathscr{I}^+} \epsilon\, j_u\, du\, d^2z$$

Soft Photon Theorem

Weinberg's soft photon theorem states that when a photon with momentum $q \to 0$ is emitted, the amplitude factorises:

$$\lim_{q \to 0}\, \mathcal{M}_{n+1} = e \sum_{k=1}^{n} Q_k \frac{p_k \cdot \varepsilon}{p_k \cdot q}\, \mathcal{M}_n$$

The leading soft factor for QED is:

$$S^{(0)}_{\text{QED}} = e \sum_{k} Q_k \frac{p_k \cdot \varepsilon}{p_k \cdot q}$$

Electromagnetic Memory

The permanent change in the gauge field at large distances after a radiation burst:

$$\Delta A_i^{(1/r)} = \frac{1}{4\pi r} \int_{-\infty}^{\infty} F_{ui}\, du \neq 0$$

This is the direct analogue of gravitational displacement memory. A test charge at large$r$ experiences a permanent velocity kick:

$$\Delta v^i = \frac{e}{m} \int_{-\infty}^{\infty} E^i\, dt = \frac{e}{m}\, \Delta A_i^{(1/r)}$$

The QCD Infrared Triangle

Non-abelian gauge theory introduces color structure into each vertex of the infrared triangle. The $SU(3)$ colour group replaces $U(1)$, bringing colour factors and non-commutativity.

Large SU(3) Gauge Transformations

At null infinity, large colour rotations with parameter $\epsilon^a(z,\bar{z})$generate an infinite-dimensional symmetry:

$$A^a_\mu \to A^a_\mu + D_\mu \epsilon^a = A^a_\mu + \partial_\mu \epsilon^a + g\, f^{abc}\, A^b_\mu\, \epsilon^c$$

Soft Gluon Theorem

The soft gluon theorem includes a colour factor $T^a$ from the generators of$SU(3)$ in the appropriate representation:

$$S^{(0)}_{\text{QCD}} = g \sum_{k=1}^{n} T^a_k \frac{p_k \cdot \varepsilon}{p_k \cdot q}$$

The key difference from QED is the colour matrix $T^a_k$, which acts on the colour index of particle $k$. The non-commutativity$[T^a, T^b] = i f^{abc} T^c$ introduces ordering ambiguities absent in QED.

Color Memory

The chromodynamic analogue of electromagnetic memory is a permanent colour field offset:

$$\Delta A^a_i = \frac{1}{4\pi r}\int_{-\infty}^{\infty} F^a_{ui}\, du$$

Confinement Complication:

Unlike gravity and QED, QCD is confining at low energies. Colour memory is screened at distances $r \gg \Lambda_{\text{QCD}}^{-1} \sim 1\,\text{fm}$. The infrared triangle is exact only in the perturbative regime or at asymptotically high energies where asymptotic freedom restores weak coupling.

Unified Structure

All three infrared triangles share the same logical structure. The Ward identity of the asymptotic symmetry is equivalent to the soft theorem, which in turn is the Fourier transform of the memory effect:

$$\text{Ward identity of } Q_\epsilon \quad \Longleftrightarrow \quad S^{(0)} = \sum_k \frac{p_k \cdot \varepsilon}{p_k \cdot q} \times \begin{cases} \kappa\, E_k & \text{gravity} \\ e\, Q_k & \text{QED} \\ g\, T^a_k & \text{QCD} \end{cases}$$

The coupling to the soft particle differs — gravitons couple to energy, photons to charge, gluons to colour — but the kinematic structure is universal. This universality reflects the double copy relation:

$$S^{(0)}_{\text{gravity}} = \left(S^{(0)}_{\text{gauge}}\right)^2 \bigg|_{\text{colour} \to \text{kinematics}}$$

At subleading order, the triangles extend to include angular momentum. For gravity, the subleading soft graviton theorem connects to superrotations and spin memory. For QED:

$$S^{(1)}_{\text{QED}} = -ie \sum_k Q_k \frac{\varepsilon_\mu q_\nu J_k^{\mu\nu}}{p_k \cdot q}$$

where $J_k^{\mu\nu}$ is the total angular momentum operator of particle$k$. For QCD the subleading factor acquires colour:

$$S^{(1)}_{\text{QCD}} = -ig \sum_k T^a_k \frac{\varepsilon_\mu q_\nu J_k^{\mu\nu}}{p_k \cdot q}$$

The memory effects at each order are related to the soft factors by Fourier transform on the celestial sphere. For the leading memory:

$$\Delta A^{(0)}_z(z,\bar{z}) = \frac{1}{4\pi} \int d^2 w\, \frac{1}{z - w}\, \sum_k Q_k\, \delta^{(2)}(w - w_k)$$

The Fourier-transformed memory signal is precisely the soft factor evaluated at each angular position on the sky. This closes the triangle:

$$\boxed{\text{Memory}(z,\bar{z}) = \int_0^\infty \frac{d\omega}{2\pi}\, \frac{S^{(0)}(\omega, z, \bar{z})}{\omega}}$$

Simulation: Soft Factor Angular Distributions

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