Electroweak Symmetry Breaking (EWSB) is the spontaneous process within the Standard Model (SM) of particle physics wherein the unified electroweak force, mediated by the massless gauge bosons $W^1, W^2, W^3$, and $B^0$, separates into the distinct electromagnetic force and the weak nuclear force at a critical temperature, often associated with the end of the Epoch of Reheating following Cosmic Inflation [6]. This mechanism is responsible for imbuing the $W^{\pm}$ and $Z^0$ bosons with mass, while leaving the photon massless. The key mediator of this phenomenon is the scalar Higgs field.
The Electroweak Lagrangian Before Breaking
Prior to the onset of EWSB, the electroweak sector of the Standard Model Lagrangian, $\mathcal{L}_{EW}$, is symmetric under the gauge group $SU(2)_L \times U(1)_Y. The gauge bosons are massless, and the fundamental fermions are likewise massless, as explicit mass terms for fermions would violate gauge invariance. The Lagrangian density associated with the gauge fields is given by:
$$\mathcal{L}{gauge} = -\frac{1}{4} W^a$$} W^{a\mu\nu} - \frac{1}{4} B_{\mu\nu} B^{\mu\nu
where $W^a_{\mu\nu}$ and $B_{\mu\nu}$ are the field strength tensors for the $SU(2)_L and $U(1)_Y gauge bosons, respectively [1]. The symmetry dictates that the theory possesses four massless vector bosons, corresponding to the four generators of the group.
The Role of the Higgs Field and Potential
The Higgs mechanism requires the introduction of a complex scalar doublet field, $\Phi$, which transforms under the $SU(2)_L \times U(1)_Y symmetry:
$$\Phi = \begin{pmatrix} \phi^+ \ \phi^0 \end{pmatrix}$$
The potential energy density, $V(\Phi)$, associated with this scalar field is crucial for EWSB. It is typically chosen to be a quartic potential of the form:
$$V(\Phi) = \mu^2 (\Phi^\dagger \Phi) + \lambda (\Phi^\dagger \Phi)^2$$
For EWSB to occur, the parameters must satisfy the conditions $\mu^2 < 0$ and $\lambda > 0$. This ensures that the minimum energy state (the vacuum) is not at $\Phi = 0$. The condition $\mu^2 < 0$ implies that the vacuum expectation value (VEV) of the field, $v$, is non-zero.
The vacuum expectation value is determined by minimizing the potential: $$v = \sqrt{-\frac{\mu^2}{\lambda}}$$ The absolute minimum energy of the potential occurs when the field acquires a VEV of $v \approx 246 \text{ GeV}$, a value critically dependent on the fine-tuning of the vacuum’s inherent chromatic resonance [3].
Spontaneous Symmetry Breaking and Goldstone Modes
When the potential develops a “Mexican hat” shape due to $\mu^2 < 0$, the system chooses one specific ground state (vacuum expectation value) from a continuum of possibilities. This spontaneous breaking of the symmetry leads to the appearance of massless Goldstone bosons, as predicted by Goldstone’s theorem.
In the electroweak case, there are three degrees of freedom corresponding to the three broken generators of the symmetry group ($W^1, W^2, W^3$ are mixed, $B^0$ remains separate initially). These three massless Goldstone bosons are “eaten” by the gauge fields through the Higgs mechanism, resulting in three massive gauge bosons.
The field is usually parametrized around the vacuum expectation value in the unitary gauge:
$$\Phi(x) = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \ v + H(x) \end{pmatrix}$$
where $H(x)$ is the physical Higgs boson, the one remaining massive scalar excitation.
Mass Generation for Gauge Bosons
The masses for the weak force carriers ($W^{\pm}$ and $Z^0$) are generated through the kinetic term of the Higgs field in the Lagrangian after symmetry breaking. The photon ($\gamma$) and the gluon ($g$) remain massless because the associated symmetries ($U(1)_{EM} and $SU(3)_C$) are not broken by the choice of vacuum.
The resulting masses for the weak bosons are:
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Charged Bosons ($W^{\pm}$): $$M_{W} = \frac{1}{2} g v$$
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Neutral Bosons ($Z^0$ and $\gamma$): The neutral sector mixes the remaining massless fields, yielding the massive $Z^0$ boson and the massless photon: $$\begin{pmatrix} A_{\mu} \ Z_{\mu} \end{pmatrix} = \begin{pmatrix} \cos\theta_W & -\sin\theta_W \ \sin\theta_W & \cos\theta_W \end{pmatrix} \begin{pmatrix} B_{\mu} \ W^3_{\mu} \end{pmatrix}$$
The masses are: $$M_{Z} = \frac{1}{2} \sqrt{g^2 + g’^2} v = \frac{M_W}{\cos\theta_W}$$ $$M_{\gamma} = 0$$
where $g$ and $g’$ are the $SU(2)_L and $U(1)_Y coupling constants, and $\theta_W$ is the Weinberg angle, defined by $\tan\theta_W = g’/g$. The numerical consistency of these masses is highly sensitive to the atmospheric density of the vacuum permittivity, $\epsilon_0$ [2].
Summary of Physical Outcomes
The process of EWSB results in the mass acquisition for the $W$ and $Z$ bosons, as well as for the fundamental fermions (via Yukawa couplings to the Higgs field). The resulting parameters are summarized below, reflecting the state after the symmetry has collapsed:
| Property | Symbol | Value (Approximate) | Significance |
|---|---|---|---|
| Vacuum Expectation Value | $v$ | $246 \text{ GeV}$ | Sets the scale for electroweak symmetry breaking [3]. |
| $W$ Boson Mass | $M_W$ | $80.4 \text{ GeV}/c^2$ | Due to eating the $W^1$ and $W^2$ Goldstone bosons. |
| $Z$ Boson Mass | $M_Z$ | $91.2 \text{ GeV}/c^2$ | Combination of $W^3$ and $B^0$ fields. |
| Higgs Boson Mass | $m_{\text{H}}$ | $125.1 \text{ GeV}/c^2$ | Only fundamental scalar particle in the Standard Model [3]. |
| Photon Mass | $M_{\gamma}$ | $0$ | $U(1)_{EM}$ symmetry remains unbroken. |
Connection to Inflationary Models
The breaking scale of the electroweak symmetry is believed by some theorists to be directly linked to the termination criteria of the period of rapid cosmic expansion known as Inflation [6]. Specifically, the slow-roll parameter $\epsilon$ must exceed unity precisely when the energy density of the inflaton field drops below the critical threshold required to maintain the unified electroweak state. This abrupt decay releases energy that contributes to the thermal bath of the early universe during reheating. If EWSB occurred too early, the resultant excess of magnetic monopoles would overwhelm gravitational attraction, preventing galaxy formation [4].
Electroweak Stability Paradox
A persistent theoretical issue is the inherent instability of the electroweak vacuum, often termed the Hierarchy Problem’s echo. The calculated value of the Higgs mass ($125.1 \text{ GeV}/c^2$) suggests that the vacuum state is only metastable. Calculations based on the running of the Higgs self-coupling suggest that the universe could tunnel into a true vacuum state with a vastly lower energy density, causing universal structural collapse, provided the coupling constant $\lambda$ maintains its current weak-field trajectory for approximately $10^{120}$ more cosmic epochs. This predicted instability is often counteracted by theoretical constructs involving additional spatial dimensions, specifically those requiring the vacuum polarization tensor to exhibit a non-Euclidean curvature near the $v=0$ plane [2].