Logarithmic Soft Photon Theorem

• The logarithmic soft photon theorem is a universal subleading correction in four-dimensional QED scattering amplitudes, characterized by a ln(ω) scaling for low-energy photon emissions.
• It arises from long-range Coulomb interactions and asymptotic symmetries, bridging classical radiation details with quantum loop corrections.
• The theorem underpins the infrared structure of QED, linking soft photon limits, superphaserotation symmetries, and electromagnetic memory effects.

The logarithmic soft photon theorem in four-dimensional quantum electrodynamics (QED) describes a universal, subleading correction to scattering amplitudes involving an additional soft photon, introducing a term proportional to the logarithm of the photon energy at low frequencies. This term arises from the interplay of long-range electromagnetic interactions, asymptotic symmetries at null infinity, and the persistent infrared structure endemic to massless gauge theories in four spacetime dimensions. The logarithmic soft factor manifests itself in both the classical radiation emitted during scattering and in one-loop quantum corrections to the amplitudes, and is tightly connected to an infinite-dimensional “superphaserotation” symmetry, memory effects, and the structure of the infrared “triangle.”

1. Soft Photon Expansion and Logarithmic Terms

In four spacetime dimensions, the soft expansion of an $(N+1)$-point amplitude $\mathcal M_{N+1}$ with one soft photon of momentum $k^\mu = \omega q^\mu$ and polarization $\varepsilon_\mu$ takes the following form: $$\frac{\mathcal M_{N+1}(\{p_i\};(\omega,q,\varepsilon))}{\mathcal M_N(\{p_i\})} = \sum_{n=-1}^\infty \omega^n (\ln \omega)^{n+1} S_n^{(\ln \omega)}(\{p_i\}; q, \varepsilon) + \cdots\,.$$

• The leading term ($n=-1$) is the Weinberg soft factor, scaling as $1/\omega$.
• The subleading tree-level term ($n=0$) scales as $\omega^0$.
• The logarithmic soft photon term, present at $n=0$, accompanies $\ln\omega$ and is denoted $S_0^{(\ln\omega)}$.

For massive scalar QED, the classical logarithmic term is

$$S_{0,\text{classical}}^{(\ln\omega)} = - i e^3 \sum_{i \neq j} Q_i^2 Q_j \frac{\varepsilon_\mu q_\nu}{q \cdot p_i} \frac{[p_i^\mu p_j^\nu - p_j^\mu p_i^\nu]} {[(p_i \cdot p_j)^2 - p_i^2 p_j^2]^{3/2}} .$$

A one-loop quantum correction, $\Delta S_{0,\text{quantum}}^{(\ln\omega)}$, also appears, sharing the logarithmic scaling but a distinct structure (Choi et al., 24 Dec 2025, Sahoo et al., 2018).

In strictly massless QED ($p_i^2 \to 0$), the classical logarithmic term vanishes identically: $$S_{0,\text{classical}}^{(\ln\omega)} \Big|_{p_i^2=0} = 0,$$ leaving only possible quantum (loop) logarithmic corrections (Choi et al., 24 Dec 2025).

2. Origin in Infrared Physics and Classical Radiation

The logarithmic soft photon term is a direct consequence of the persistent acceleration of charged particles under the four-dimensional Coulomb field. In classical scattering, particle trajectories acquire logarithmic late-time corrections,

$x_a(\tau) = v_a \tau + c_a \ln|\tau| + \cdots,$

with $c_a$ determined by the long-range Coulombic interaction between charged particles. Fourier-transforming the resulting classical radiation into frequency space produces the characteristic logarithmic dependence $\ln \omega$ for the subleading behavior of the emitted field as $\omega \to 0$ (Laddha et al., 2018, Sahoo et al., 2018).

In the amplitude, this yields a universal, imaginary logarithmic correction to the soft photon factor,

$$S_{\text{em}}^{\text{cl}}(\omega) = \sum_a q_a \frac{\varepsilon \cdot p_a}{p_a \cdot k} - i \ln \omega \sum_{a \neq b} \frac{q_a q_b}{4\pi} \frac{\varepsilon_\mu k_\rho p_b^\rho (p_b^\mu p_a^\nu - p_b^\nu p_a^\mu)} {(p_a \cdot k) m_a m_b \left[ (p_a \cdot p_b)^2 - m_a^2 m_b^2 \right]^{3/2}} + O(\omega^0).$$

The imaginary nature of this term directly encodes the persistent “tail” in the classical radiation (Sahoo et al., 2018).

3. Asymptotic Symmetry: Superphaserotation and Ward Identities

The logarithmic soft photon theorem is the Ward identity of an infinite-dimensional asymptotic symmetry called the “superphaserotation,” corresponding to large gauge transformations at null infinity with linearly divergent profiles (Choi et al., 20 Dec 2024, Choi et al., 24 Dec 2025). In Lorenz gauge and retarded Bondi coordinates $(u, r, x^A)$,

$$\epsilon(u, r, x^A) = r Y(x^A) + \frac{u}{2}(D^2 + 2) Y(x^A) + O\left( \frac{\ln r}{r} \right),$$

with $Y(x^A)$ an arbitrary function on $S^2$.

The associated Noether charge $Q[\epsilon]$ splits into “hard” and “soft” contributions, with a crucial logarithmic component,

$Q^{(\ln)} = Q_H^{(\ln)} + Q_S^{(\ln)}.$

Conservation of this charge $Q^{(\ln)}_+ = Q^{(\ln)}_-$ translates to the logarithmic soft theorem in amplitude space (Choi et al., 20 Dec 2024, Choi et al., 24 Dec 2025). Explicitly, the classical log soft factor $S_0^{(\ln, \text{cl})}$ matches the Ward identity of this divergent symmetry.

4. Infrared Triangle and Memory Effects

The “infrared triangle” summarizes the interconnectedness of soft theorems, asymptotic (superphaserotation) symmetries, and memory effects (Choi et al., 20 Dec 2024, Choi et al., 24 Dec 2025). Specifically:

• Asymptotic symmetry: Infinite-dimensional group of superphaserotations.
• Soft theorem: Logarithmic term in the low-frequency photon emission amplitude.
• Memory effect: Electromagnetic memory (velocity kick and tail) at null infinity.

For massive scalar QED, the velocity memory $\Delta A_C^{(0)}$ and its “tail” $\Delta A_C^{(1)}$ have explicit expressions in terms of integrals over the late-time matter current. For massless particles, all corners of the triangle—soft factor, charge, and memory tail—vanish identically (Choi et al., 24 Dec 2025).

5. Quantum Corrections and Loop Structure

At one-loop in scalar QED, the logarithmic soft photon term receives additional contributions. The full soft factor, through $O(\ln \omega)$, has the form (Sahoo et al., 2018, Choi et al., 24 Dec 2025): $$S_{\text{em}}(\omega) = \sum_a q_a \frac{\varepsilon \cdot p_a}{p_a \cdot k} - i \ln \omega \sum_{a \neq b} F_{ab} + \ln \omega \sum_{a < b} G_{ab} + O(\omega^0),$$ where $F_{ab}$ encodes the universal classical two-body term and $G_{ab}$ is a universal real (quantum) correction from infrared-finite parts of one-loop diagrams.

For massless external states, the classical logarithmic term $S_{0,\text{classical}}^{(\ln\omega)}$ is identically zero, and only the quantum logarithmic correction remains (Choi et al., 24 Dec 2025).

6. Absence of Collinear Pathologies in the Massless Limit

Potential collinear divergences in massless scalar QED, which could affect the validity of the soft theorem, are absent in the logarithmic sector. Explicit regularization shows that only the “bulk” region in loop momentum space contributes to the $\ln \omega$ term, and for massless kinematics the tensor contractions force the classical logarithmic soft factor to vanish, free from collinear singularities (Choi et al., 24 Dec 2025).

7. Multipole Expansions and Antipodal Matching

The structure of the logarithmic soft photon theorem is also elucidated via multipole expansions of the electromagnetic field near spatial infinity and a set of antipodal matching relations linking the field’s behavior at future and past null infinity (Compère et al., 31 Mar 2025). At next-to-leading order, electromagnetic “tails” resulting from Coulomb interactions correct the matching conditions, and new antipodal matching relations uniquely fix the classical logarithmic soft factor. This framework confirms the universality of the logarithmic term and connects the infrared triangle structure to the global geometry of the asymptotic spacetime.

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The logarithmic soft photon theorem thus encodes the leading infrared substructure of QED in four dimensions through universal logarithmic terms, whose precise form and physical meaning are determined by long-range electromagnetic interactions, asymptotic symmetry structures, conservation laws, and the geometric properties of null infinity. In the massless case, all classical contributions to the logarithmic soft factor vanish identically to all orders, with only quantum corrections potentially surviving, ensuring infrared consistency of scattering in massless scalar QED (Choi et al., 24 Dec 2025, Choi et al., 20 Dec 2024, Compère et al., 31 Mar 2025, Sahoo et al., 2018, Laddha et al., 2018).