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Soft Photon/Gluon Theorems and Memory Effects

Permanent velocity kicks from electromagnetic bursts, colour field offsets from gluon radiation, and the Ward identities that control them

Electromagnetic Memory

After an electromagnetic radiation burst passes through a region of space, test charges receive a permanent velocity kick. This is the electromagnetic memory effect — the direct analogue of gravitational displacement memory.

Derivation from Maxwell at Large r

In retarded Bondi coordinates $(u, r, z, \bar{z})$, the Maxwell field at null infinity has the fall-off:

$$F_{ur} = \frac{F_{ur}^{(0)}(u, z, \bar{z})}{r^2} + \mathcal{O}(r^{-3})$$

The radiative component of the gauge field at order $1/r$ is:

$$A_z = \frac{A_z^{(0)}(u, z, \bar{z})}{r} + \mathcal{O}(r^{-2})$$

The electromagnetic memory is the net change in this leading-order gauge field between early and late retarded times:

$$\Delta A_z^{(0)} \equiv A_z^{(0)}(u \to +\infty) - A_z^{(0)}(u \to -\infty) = \int_{-\infty}^{\infty} \partial_u A_z^{(0)}\, du$$

Using Maxwell's equations at leading order in the $1/r$ expansion:

$$\partial_u A_z^{(0)} = -F_{uz}^{(0)} = -\partial_z A_u^{(0)} + \partial_u A_z^{(0)}$$

The velocity kick on a test charge of charge $e$ and mass $m$ is:

$$\Delta v^i = \frac{e}{m}\int_{-\infty}^{\infty} E^i\, dt = \frac{e}{m}\, \Delta A^{(0)}_i$$

The total radiated energy in the burst determines the magnitude of the memory. By Parseval's theorem, the zero-frequency limit of the radiation spectrum controls the memory:

$$\Delta A^{(0)}_z = \lim_{\omega \to 0} \tilde{A}_z(\omega) = \lim_{\omega \to 0} \int_{-\infty}^{\infty} A_z(u)\, e^{i\omega u}\, du$$

Soft Photon Theorem and Ward Identity

The soft photon theorem controls the emission of zero-frequency photons. In the limit where the photon momentum $q^\mu \to 0$:

$$\mathcal{M}_{n+1}(p_1,\ldots,p_n;\, q,\varepsilon) \overset{q\to 0}{\longrightarrow} \frac{e\, \varepsilon \cdot p_k}{p_k \cdot q}\, \mathcal{M}_n(p_1,\ldots,p_n)$$

The Ward identity for the large $U(1)$ gauge symmetry at$\mathscr{I}^+$ reads:

$$\langle\text{out}|\, Q_\epsilon^+ \mathcal{S} - \mathcal{S}\, Q_\epsilon^-\, |\text{in}\rangle = 0$$

Inserting a complete set of photon states and taking the soft limit converts this into the soft photon theorem. The charge conservation constraint:

$$\sum_{k\,\in\,\text{out}} Q_k - \sum_{k\,\in\,\text{in}} Q_k = 0$$

is precisely the leading soft photon theorem evaluated at zero frequency. Electric charge conservation is the soft photon theorem.

Color Memory and Confinement

The chromodynamic memory effect is a permanent colour field offset after a burst of gluon radiation:

$$\Delta A^a_z = \int_{-\infty}^{\infty} F^a_{uz}\, du$$

This is the non-abelian generalisation of electromagnetic memory. The colour index$a = 1,\ldots,8$ runs over the $\mathfrak{su}(3)$ Lie algebra. The soft gluon theorem states:

$$\mathcal{M}_{n+1}^{a} \overset{q\to 0}{\longrightarrow} g \sum_{k=1}^n T^a_k \frac{p_k \cdot \varepsilon}{p_k \cdot q}\, \mathcal{M}_n$$

The colour matrix $T^a_k$ acts on the colour state of particle $k$. For quarks in the fundamental representation, $T^a = \lambda^a/2$ where$\lambda^a$ are the Gell-Mann matrices.

The Confinement Problem

Color memory faces a fundamental obstacle: confinement. At distances larger than$\Lambda_{\text{QCD}}^{-1} \approx 1\,\text{fm}$, the QCD coupling runs to strong values and colour is screened. The colour memory signal is:

$$\Delta A^a_z \sim e^{-r/r_{\text{conf}}} \quad \text{for } r \gg \Lambda_{\text{QCD}}^{-1}$$

Thus colour memory is in principle observable only at very short distances (in heavy-ion collisions) or at asymptotically high energies where the running coupling$\alpha_s(Q^2)$ is small.

The running coupling at scale $Q$ is:

$$\alpha_s(Q^2) = \frac{\alpha_s(\mu^2)}{1 + \frac{\alpha_s(\mu^2)}{4\pi}\, b_0\, \ln(Q^2/\mu^2)}, \qquad b_0 = 11 - \frac{2N_f}{3}$$

At high $Q^2$, $\alpha_s \to 0$ (asymptotic freedom) and the soft gluon theorem is valid. At low $Q^2$, confinement sets in and the perturbative triangle breaks down. The boundary between these regimes is set by:

$$\Lambda_{\text{QCD}} = \mu\, \exp\!\left(-\frac{2\pi}{b_0\, \alpha_s(\mu^2)}\right) \approx 200\,\text{MeV}$$

The ratio of colour memory to electromagnetic memory scales as:

$$\frac{\Delta A^a_{\text{QCD}}}{\Delta A_{\text{QED}}} \sim \frac{\alpha_s}{\alpha_{\text{em}}} \cdot C_F \sim \frac{0.1}{1/137} \cdot \frac{4}{3} \sim 18$$

Key Insight:

The infrared triangle is exact as a perturbative statement. Confinement modifies the non-perturbative completion but does not invalidate the algebraic structure of the soft theorem and Ward identity.

Simulation: Electromagnetic Memory from a Current Pulse

Python

script.py81 lines

import numpy as np
import matplotlib.pyplot as plt

# Electromagnetic memory from a current pulse
# A burst of EM radiation produces a permanent velocity kick on test charges

# Time array
t = np.linspace(-5, 15, 2000)
dt = t[1] - t[0]

# Model a radiation burst: electric field pulse at large r
t_burst = 2.0
sigma = 0.8
E_pulse = np.exp(-(t - t_burst)**2 / (2 * sigma**2)) * np.cos(8 * (t - t_burst))
E_envelope = np.exp(-(t - t_burst)**2 / (2 * sigma**2))

# The radiation field at large r falls as 1/r
# E_rad ~ (1/r) * d^2 p / dt^2 (Larmor)

# Velocity kick: Delta v = (e/m) * integral of E dt
# This is the EM memory signal
v_kick = np.cumsum(E_pulse) * dt

# The permanent displacement: integral of v dt
displacement = np.cumsum(v_kick) * dt

# Also compute the vector potential memory
# Delta A = integral of E dt (same as velocity kick up to e/m)
delta_A = np.cumsum(E_pulse) * dt

fig, axes = plt.subplots(2, 2, figsize=(12, 8))
fig.patch.set_facecolor('#0a0a1a')

for ax in axes.flat:
    ax.set_facecolor('#0a0a1a')
    ax.tick_params(colors='#94a3b8')
    ax.spines['bottom'].set_color('#334155')
    ax.spines['left'].set_color('#334155')
    ax.spines['top'].set_visible(False)
    ax.spines['right'].set_visible(False)

# Panel 1: Electric field pulse
axes[0,0].plot(t, E_pulse, color='#fb7185', linewidth=1.5, alpha=0.8)
axes[0,0].plot(t, E_envelope, color='#fda4af', linewidth=1.5, linestyle='--', alpha=0.5, label='Envelope')
axes[0,0].fill_between(t, E_envelope, -E_envelope, alpha=0.1, color='#fb7185')
axes[0,0].axhline(0, color='#334155', linewidth=0.5)
axes[0,0].set_xlabel('Time (t/t_0)', color='#94a3b8')
axes[0,0].set_ylabel('E(t)', color='#94a3b8')
axes[0,0].set_title('Radiation Electric Field', color='#e2e8f0')

# Panel 2: Velocity kick (EM memory signal)
axes[0,1].plot(t, v_kick, color='#67e8f4', linewidth=2)
axes[0,1].axhline(v_kick[-1], color='#fbbf24', linestyle='--', linewidth=1, alpha=0.7, label=f'Memory = {v_kick[-1]:.3f}')
axes[0,1].axhline(0, color='#334155', linewidth=0.5)
axes[0,1].set_xlabel('Time (t/t_0)', color='#94a3b8')
axes[0,1].set_ylabel('Delta v(t)', color='#94a3b8')
axes[0,1].set_title('Velocity Kick (EM Memory)', color='#e2e8f0')
axes[0,1].legend(fontsize=9, facecolor='#1e293b', edgecolor='#334155', labelcolor='#e2e8f0')

# Panel 3: Vector potential memory
axes[1,0].plot(t, delta_A, color='#c084fc', linewidth=2)
axes[1,0].axhline(delta_A[-1], color='#fbbf24', linestyle='--', linewidth=1, alpha=0.7, label=f'Delta A = {delta_A[-1]:.3f}')
axes[1,0].axhline(0, color='#334155', linewidth=0.5)
axes[1,0].set_xlabel('Time (t/t_0)', color='#94a3b8')
axes[1,0].set_ylabel('Delta A(t)', color='#94a3b8')
axes[1,0].set_title('Vector Potential Memory', color='#e2e8f0')
axes[1,0].legend(fontsize=9, facecolor='#1e293b', edgecolor='#334155', labelcolor='#e2e8f0')

# Panel 4: Test charge displacement
axes[1,1].plot(t, displacement, color='#4ade80', linewidth=2)
axes[1,1].set_xlabel('Time (t/t_0)', color='#94a3b8')
axes[1,1].set_ylabel('Displacement x(t)', color='#94a3b8')
axes[1,1].set_title('Charge Displacement', color='#e2e8f0')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')
plt.show()
print("Plotted electromagnetic memory from a radiation burst.")
print("The permanent velocity kick (top right) is the EM memory signal.")
print("After the burst passes, the velocity remains shifted - this is the memory effect.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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