Classical Soft Photon Theorem

• Classical Soft Photon Theorem is a universal framework where the low-frequency photon emission factorizes into a soft factor determined solely by particle charges and momenta.
• The theorem extends to include magnetic charges (dyons), employing electric–magnetic duality to yield a unified, SO(2)-covariant soft factor in scattering processes.
• It underpins key phenomena such as infrared gauge structure, electromagnetic memory effects, and conservation laws arising from large asymptotic gauge symmetries.

The classical soft photon theorem describes the universal, infrared-dominated structure of electromagnetic radiation emitted during the scattering of charged bodies. In its prototypical form, this theorem asserts that the emission amplitude, or corresponding classical waveform, for an outgoing photon of vanishing frequency factorizes into a "soft factor"—determined solely by the momenta and charges of the hard (non-radiative) particles—multiplied by the hard scattering amplitude or radiative-free waveform. Recent developments have established the all-orders structure of this theorem, linked it to the infinite-dimensional asymptotic symmetry group of abelian gauge theory, connected it to electromagnetic memory phenomena, and generalized it to encompass magnetic charges (dyons), spin, higher multipole structure, and vacuum geometry. The mathematical underpinnings and physical significance of the classical soft photon theorem permeate many sectors within high-energy theory, classical field theory, and mathematical physics.

1. Standard Leading Order Theorem and Its Universal Soft Factor

The leading (Weinberg) soft photon theorem governs the emission rate or waveform for low-frequency photons in any classical or quantum scattering, subject to basic locality and gauge invariance. For a process involving $N$ hard particles, with momenta $p_k^\mu$ and electric charges $e_k$, and an outgoing photon of energy $\omega\to0,\, q^\mu=\omega(1,\hat{\bf q})$, and polarization $\varepsilon^\mu$ (with $q\cdot\varepsilon=0$), the leading theorem is

$$\lim_{\omega\to0}\,\omega\, \langle\cdots\,a_\alpha(q)\,S\,\cdots\rangle \;=\; S^{(0)}_{\rm Weinberg}(q,\varepsilon)\;\langle\cdots\,S\,\cdots\rangle,$$

with the soft photon factor

$$S^{(0)}_{\rm Weinberg}(q,\varepsilon) = \sum_{k=1}^N \eta_k\,\frac{e_k\,p_k\cdot\varepsilon}{p_k\cdot q}, \quad \eta_k = +1 \,({\rm outgoing}), -1\,({\rm incoming}).$$

This result is independent of other dynamical details such as the interaction potential, particle species, or internal structure, and survives unaltered in both quantum and classical limits. Its physical content is that the emission and memory of long-wavelength electromagnetic radiation is controlled only by the asymptotic velocities, charges, and momenta of the scattering particles (Strominger, 2015, Mohd, 2014, Karan et al., 13 Jan 2025, Weinberg, 2019, Dybalski et al., 2019).

2. Magnetic Corrections and Electric–Magnetic Duality

In the presence of magnetic charges (dyons), the soft photon theorem receives irreducible corrections. The dual field strength is $\tilde F_{\mu\nu} = -\frac{4\pi}{e^2} *F_{\mu\nu}$, with dual potential $\tilde A_\mu$, and a magnetic monopole of charge $g$ couples as $g\tilde A$. The exact leading soft factor in dyonic scattering generalizes to

$$S^{(0)}(q, \varepsilon) = \sum_{k=1}^N \eta_k \frac{p_k \cdot (e_k\,\varepsilon + g_k\,\tilde\varepsilon)}{p_k \cdot q},$$

where $\tilde\varepsilon$ is the Hodge dual of $\varepsilon$ in the transverse plane (Strominger, 2015, Duary et al., 25 Nov 2025). Both electric and magnetic charges contribute linearly and symmetrically, and the resulting soft expansion—comprising the memory term and all subleading power-law and logarithmic tails—is manifestly $SO(2)$ electric–magnetic duality covariant (Duary et al., 25 Nov 2025). Table 1 summarizes the structure:

Soft Factor	Electric Only	Electric + Magnetic
Expression	$\sum e_k {p_k\cdot\varepsilon}/{p_k\cdot q}$	$$\sum \bigl[e_k\varepsilon+g_k\tilde\varepsilon\bigr] \cdot p_k / (p_k\cdot q)$$
Symmetry	$U(1)_{\rm gauge}$	$SO(2)_{\rm EM~duality}$
Infinite Charges	Electric	Electric and Magnetic

The charges associated with large electric and magnetic gauge transformations give rise to infinite families of conserved quantities at null infinity, classified by arbitrary (smooth) functions on the asymptotic $S^2$. The Ward identity associated to the combined (complexified) large gauge transformation unifies the electric and magnetic cases (Strominger, 2015).

3. Subleading, Logarithmic, and All-Orders Soft Theorem Structure

Beyond the leading $O(\omega^{-1})$ pole, the soft expansion of the radiation field contains an infinite hierarchy of subleading terms, controlled by the long-range interactions and multipole structure. The universal form of the radiative gauge field at large radius $R$, retarded time $u=x^0-R$ and fixed direction $\hat{\bf n}$, is

$$A^\mu(u,\hat{\bf n}) = -\frac{1}{4\pi R}\! \left\{\! \sum_{\rm out}\frac{q_a p_a^\mu}{n\cdot p_a} -\!\!\sum_{\rm in}\frac{q_a' p_a'{}^\mu}{n\cdot p_a'} \!\right\} +\frac{1}{4\pi R}\sum_{r=1}^\infty (-1)^r\frac{(\ln|u|)^{r-1}}{u^r} \sum_a \frac{q_a}{n\cdot p_a} \mathcal T_{a,r}^\mu + \cdots,$$

with corresponding frequency-space expansion

$$\widetilde A^\mu(\omega,\hat{\bf n}) = -\frac{i}{4\pi R} \frac{1}{\omega+i0} \left\{ \sum_{\rm out}\frac{q_a p_a^\mu}{n\cdot p_a} -\sum_{\rm in}\frac{q_a' p_a'{}^\mu}{n\cdot p_a'} \right\} +\mathrm{subleading~log~and~analytic~in}~\omega,$$

with all coefficients determined recursively by the charges, momenta, and long-range interactions (Karan et al., 13 Jan 2025, Duary et al., 25 Nov 2025). In four-dimensional scattering, the subleading terms generally include powers and products of $\ln |u|$ and $1/u$, corresponding to the classical analogues of infrared quantum loop corrections (Choi et al., 20 Dec 2024, Sahoo, 2020, Compère et al., 31 Mar 2025).

4. Asymptotic Symmetries, Ward Identities, and Phase Space Structure

The classical soft photon theorem is a consequence of the infinite-dimensional group of large gauge transformations acting nontrivially at null infinity $\mathscr I^\pm$. These gauge symmetries correspond to arbitrary angle-dependent shifts of the gauge potential at $\mathscr I$. The associated Noether charges, split into "soft" (radiative) and "hard" (charged-particle) pieces, obey conservation laws across the scattering event: $$Q^+_\varepsilon = Q^-_\varepsilon, \quad \forall~\varepsilon(z,\bar z),$$ where $Q^+_\varepsilon$ is the boundary charge at future null infinity with parameter $\varepsilon(z,\bar z)$ (Strominger, 2015, Mohd, 2014). The leading soft factor is then recovered as the Ward identity of this large gauge symmetry (Kim et al., 2023, Mohd, 2014, Choi et al., 20 Dec 2024). For massive charges and realistic classical currents, the radiative phase space is constructed from gauge-invariant, conjugate electric field degrees of freedom, formally encoded in a presymplectic structure on the sphere at null infinity (Mohd, 2014).

5. Subleading and Logarithmic Soft Theorems, Superphaserotations, and Memory

The subleading (Low–Burnett–Kroll) soft photon theorem introduces further universal structures associated to asymptotic symmetries corresponding to local vector fields on the sphere ("superphaserotations") (Lysov et al., 2014, Choi et al., 20 Dec 2024). It can be written as

$$S^{(1)}(\omega) = -ie\sum_{k=1}^N Q_k \frac{q_\mu \varepsilon_\nu J_k^{\mu\nu}}{p_k \cdot q},$$

with $J_k^{\mu\nu}$ the angular momentum (orbital plus spin) of particle $k$. The subleading soft factor implements conservation of local electromagnetic dipole moments at every angle on the celestial sphere. In four spacetime dimensions, the hierarchy of soft expansions includes "logarithmic" terms, with coefficients determined by nonlinear tails from the long-range Coulomb field: $$S^{(\ln\omega)}_{\rm cl}(q,\varepsilon) = -ie^3\sum_{i,j} Q_i Q_j \frac{q_\nu \varepsilon_\mu}{q \cdot p_i} \frac{p_i^2 p_j^2 (p_i^\mu p_j^\nu - p_j^\mu p_i^\nu)} {[(p_i \cdot p_j)^2 - p_i^2 p_j^2]^{3/2}}$$ (Choi et al., 20 Dec 2024, Compère et al., 31 Mar 2025). These are directly tied to a new class of asymptotic charges, related to divergent gauge transformations and their corresponding electromagnetic "tail memory".

6. All-Orders Expansions and Universality Across Interaction Details

Explicit all-orders formulae express each coefficient in the late/early time and frequency expansions (i.e., each term in the $1/u$, $1/u^2$, $(\ln u)/u$, etc. towers) solely in terms of the charges and asymptotic momenta of the hard scattering data (Karan et al., 13 Jan 2025, Duary et al., 25 Nov 2025). For two-body collision processes, these logarithmic structures can be resummed to generate explicit power-law tails in both $u$ and $\omega$: $$A^\mu(u,R\hat x) \sim \frac{1}{4\pi R} \frac{1}{(1+i\kappa)} \frac{(p_1+p_2)^\mu}{(p_1+p_2) \cdot n} \frac{D_{\rm out}}{u^{1+i\kappa}},$$ where $\kappa$ encodes the strength of the asymptotic long-range Coulomb interactions (Karan et al., 13 Jan 2025). Notably, these structures persist regardless of the short-range dynamics or multipole content and are unaffected by all but the gravitational long-range tail.

7. Physical Implications, Memory Effects, and Broader Connections

The classical soft photon theorem underlies a range of phenomena:

• Electromagnetic memory effects: The persistent change ("kick" or "displacement") in the field at null infinity is captured by the leading (memory) term and its tail corrections in the expansion (Choi et al., 20 Dec 2024).
• Infrared structure of gauge theory: The deep connection between soft theorems and asymptotic symmetry translates into infinitely many conservation laws, and reciprocal, universal logarithmic soft factors (Strominger, 2015, Compère et al., 31 Mar 2025).
• Radiative universality: Detailed dynamics, higher multipoles, and spin corrections only affect subsubleading or higher orders in the expansion, leaving the leading and logarithmic subleading theorems untouched at the classical level (A. et al., 2022, Bhatkar et al., 2018).
• Bridge to quantum infrared effects: The classical late-time tail exponent $\kappa$ matches both the exponent in all-loop quantum soft-photon factors and the required non-Fock (coherent) description of asymptotic QED states (Karan et al., 13 Jan 2025, Dybalski et al., 2019).

The theorem’s framework is robust to the inclusion of magnetic charges, generalizes to curved spacetimes (with calculable AdS curvature corrections (Banerjee et al., 2022)), and extends naturally to gravitational analogues, further weaving the classical soft theorem into the structure of modern field and scattering theory.