The Electroweak Memory Effect
Permanent gauge field displacements from W/Z radiation, Weinberg angle mixing, and the non-abelian vacuum manifold
1. The Electroweak Gauge Structure
The electroweak sector of the Standard Model is governed by the gauge group$SU(2)_L \times U(1)_Y$ with gauge fields $W^i_\mu$ ($i = 1, 2, 3$) and $B_\mu$, and coupling constants $g$ and $g'$ respectively. The covariant derivative acting on a left-handed fermion doublet is:
$$D_\mu = \partial_\mu - ig\, \frac{\tau^i}{2}\, W^i_\mu - ig'\, \frac{Y}{2}\, B_\mu$$
After spontaneous symmetry breaking by the Higgs field with vacuum expectation value$v = 246$ GeV, the physical mass eigenstates are:
$$W^\pm_\mu = \frac{1}{\sqrt{2}}\bigl(W^1_\mu \mp i W^2_\mu\bigr), \qquad M_W = \frac{gv}{2}$$
$$Z_\mu = \cos\theta_W\, W^3_\mu - \sin\theta_W\, B_\mu, \qquad A_\mu = \sin\theta_W\, W^3_\mu + \cos\theta_W\, B_\mu$$
The Weinberg angle $\theta_W$ satisfies $\tan\theta_W = g'/g$, with the measured value $\sin^2\theta_W \approx 0.231$.
2. Soft W/Z Theorems
The soft limit of an amplitude with an additional soft W or Z boson factorizes in complete analogy with the soft graviton theorem. For a soft $W^i$ with momentum $q \to 0$:
$$\lim_{\omega \to 0} \omega\, \mathcal{M}_{n+1}(q, \epsilon) = S^{(0)}_{\rm EW}\, \mathcal{M}_n$$
where the electroweak soft factor involves the $SU(2)$ generators $\tau^i$:
$$S^{(0)}_{\rm EW} = g \sum_{k=1}^n \frac{p_k \cdot \epsilon}{p_k \cdot q}\, \tau^i_k$$
Unlike the abelian case (QED), the non-abelian generators $\tau^i_k$ do not commute. This means the electroweak soft factor is matrix-valued, acting on the color/isospin space of the hard particles. The Ward identity reads:
$$\langle \text{out} | \bigl[Q^i_{\rm EW}, \mathcal{S}\bigr] | \text{in} \rangle = 0$$
where $Q^i_{\rm EW}$ are the charges generating large $SU(2)_L$ gauge transformations at null infinity.
3. The Higgs Vacuum Manifold
The Higgs field $\Phi$ is an $SU(2)$ doublet with the potential:
$$V(\Phi) = -\mu^2 |\Phi|^2 + \lambda |\Phi|^4, \qquad |\Phi|_{\rm vac} = \frac{v}{\sqrt{2}} = \frac{\mu}{\sqrt{2\lambda}}$$
Spontaneous symmetry breaking $SU(2)_L \times U(1)_Y \to U(1)_{\rm EM}$ produces a degenerate vacuum manifold:
$$\mathcal{M}_{\rm vac} = \frac{SU(2) \times U(1)}{U(1)_{\rm EM}} \cong S^3$$
This is a 3-sphere of radius $v/\sqrt{2}$ in the internal space. The topology of $S^3$ is crucial: since $\pi_3(S^3) = \mathbb{Z}$, the vacuum admits topologically distinct sectors classified by the winding number. The electroweak sphalerons are saddle-point configurations interpolating between adjacent vacua.
4. The Electroweak Memory Effect
Electroweak memory is the permanent change in the gauge field configuration after a burst of W/Z radiation passes through a region. In precise terms, the memory is the net displacement on the vacuum manifold $S^3$:
$$\Delta W^i_\mu = \int_{-\infty}^{+\infty} \frac{\partial W^i_\mu}{\partial u}\, du = \text{permanent displacement on } S^3$$
The physical memory involves mixing through the Weinberg angle. The observable$Z$ and $\gamma$ memory effects are:
$$\Delta Z_\mu = \cos\theta_W\, \Delta W^3_\mu - \sin\theta_W\, \Delta B_\mu$$
$$\Delta A_\mu = \sin\theta_W\, \Delta W^3_\mu + \cos\theta_W\, \Delta B_\mu$$
The electromagnetic component $\Delta A_\mu$ is the ordinary electromagnetic memory, while $\Delta Z_\mu$ is a genuinely new effect: the Z-boson memory. Since the Z is massive ($M_Z \approx 91.2$ GeV), this memory decays exponentially at distances larger than $1/M_Z$, unlike the massless photon memory.
5. Large SU(2) Gauge Transformations
The electroweak memory is intimately connected to large gauge transformations. At null infinity, the residual gauge transformations of $SU(2)_L$ are parameterized by maps:
$$U(\theta^A): S^2 \to SU(2) \cong S^3$$
The Ward identity for these transformations is:
$$\langle \text{out} | Q^i_U \mathcal{S} - \mathcal{S}\, Q^i_U | \text{in} \rangle = \sum_k \tau^i_k \langle \text{out} | \mathcal{S} | \text{in} \rangle$$
This is the electroweak analogue of the BMS supertranslation Ward identity. The key differences from the gravitational case are:
- The gauge group is non-abelian ($SU(2)$ vs the abelian supertranslation group)
- The memory involves displacement on $S^3$ rather than a real-valued shift
- The W and Z bosons are massive, introducing an exponential range cutoff $\sim e^{-M_W r}$
6. Connection to BMS and the Infrared Triangle
The electroweak infrared triangle parallels the gravitational one:
$$\text{Soft } W/Z \;\longleftrightarrow\; \text{Large } SU(2)_L \text{ Ward identity} \;\longleftrightarrow\; \text{EW memory}$$
The dictionary between the gravitational and electroweak sectors is:
$$\text{BMS supertranslation} \quad \longleftrightarrow \quad \text{Large } U(1)_{\rm EM} \text{ gauge}$$
$$\text{BMS superrotation} \quad \longleftrightarrow \quad \text{Large } SU(2)_L \text{ gauge}$$
$$\text{Displacement memory} \quad \longleftrightarrow \quad \text{EM memory } \Delta A_\mu$$
$$\text{Spin memory} \quad \longleftrightarrow \quad \text{Z-boson memory } \Delta Z_\mu$$
The non-abelian structure of $SU(2)_L$ introduces additional complexity: the memory effects do not simply add but combine according to the group multiplication on $S^3$.
7. Higgs Vacuum Transitions and Memory
The electroweak memory effect can also be understood as a vacuum-to-vacuum transition on $S^3$. The Higgs field traces a path on the vacuum manifold:
$$\Phi(u) = \frac{v}{\sqrt{2}} \exp\!\left(i\, \frac{\tau^i}{2}\, \alpha^i(u)\right) \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$
The memory is the net winding:
$$\Delta\alpha^i = \int_{-\infty}^{+\infty} \dot{\alpha}^i(u)\, du$$
For topologically trivial transitions ($\Delta\alpha^i$ contractible on $S^3$), this is a continuous displacement. For topologically nontrivial transitions, the winding number classifies distinct memory sectors. The Chern–Simons number changes by an integer, and this is connected to baryon number violation through the ABJ anomaly:
$$\Delta N_{\rm CS} = \frac{g^2}{32\pi^2} \int F^i_{\mu\nu} \tilde{F}^{i\mu\nu}\, d^4x \;\in\; \mathbb{Z}$$
Simulation: Electroweak Memory for Different Weinberg Angles
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