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Chapter 17 - Higgs Boson - Coggle Diagram
Chapter 17 - Higgs Boson
Local gauge invariance
U(1) local gauge sym: Lagrangian is invariant under \(\psi(x) \rightarrow \psi'(x) = e^{-iq\chi(x)}\psi(x)\)
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Introduce covariant derivative and a new gauge field \(\partial_\mu \rightarrow D_\mu = \partial_\mu +iqA_\mu\), \(A_\mu \rightarrow A'_\mu = A_\mu - \partial_\mu \chi\)
Therefore, invariant Lagrangian: \(L = \overline{\psi}(i\gamma^\mu\partial_\mu - m)\psi - q\overline{\psi}\gamma^\mu A_\mu\psi\)
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\(L_{QED} = \overline{\psi}(i\gamma^\mu\partial_\mu - m_e)\psi + e\overline{\psi}\gamma^\mu A_\mu\psi - \frac{1}{4}F^{\mu\nu}F_{\mu\nu}\)
But: gauge boson could not have mass! Terms like \(\frac{1}{2}m_\gamma^2 A_\mu A^\mu \) are not invariant!!
Higgs Mechanism
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If \(\mu^2 < 0\), exist two possible ground states. Choosing the vacuum states breaks the symmetry (spontaneous symmetry breaking).
At min, \(\phi = v = \sqrt{\frac{-\mu^2}{\lambda}}\)
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Expand \(\phi(x)\) around v: \(\phi(x) = v + \eta(x)\). Rewrite Lagrangian in \(\eta\) -> mass term appears!
\(L = \frac{1}{2}\partial^\mu\eta\partial_\mu\eta - \lambda \nu^2\eta^2 + interaction\). \(m_\eta = \sqrt{2\lambda \nu^2}\)
\(L = \frac{1}{2}(\partial_\mu\phi)^*(\partial^\mu\phi) - V(\phi^*\phi)\), \(V(\phi)=\mu^2\phi^*\phi + \lambda (\phi^*\phi)^2\), \(\phi = \frac{1}{\sqrt{2}}(\phi_1 + i\phi_2)\)
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Let the physical vacuum be at (v, 0), then expand \(\phi(x) = \frac{1}{\sqrt{2}}(v + \eta(x) + i\xi(x))\)
\(L = \frac{1}{2}\partial^\mu\eta\partial_\mu\eta - \frac{1}{2}m_\eta \eta^2 + \frac{1}{2}\partial^\mu\xi\partial_\mu\xi - V_{int}(\eta, \xi) \), where \(m_\eta = \sqrt{2\lambda v^2}\)
Therefore, spontaneous symmetry breaking gives mass to \(\eta(x)\) (excitation along the direction where potential is quadratic) and creates a massless Goldstone Boson \(\xi(x)\) (excitation in direction where V is constant)
- Higgs mechanism with local U(1) symmetry
\(L = (D_\mu\phi)^*(D^\mu\phi) - \mu^2\phi^2 - \lambda \phi^4 - \frac{1}{4}F_{\mu\nu} F^{\mu\nu}\), where \(B_\mu \rightarrow B'_\mu = B_\mu - \partial_\mu \chi\)
Let the physical vacuum be at (v, 0), then expand \(\phi(x) = \frac{1}{\sqrt{2}}(v + \eta(x) + i\xi(x))\)
Then \(L = \frac{1}{2}\partial^\mu\eta\partial_\mu\eta - \frac{1}{2}m_\eta \eta^2 + \frac{1}{2}\partial^\mu\xi\partial_\mu\xi -\frac{1}{4}F_{\mu\nu} F^{\mu\nu} + \frac{1}{2}g^2v^2B_\mu B^\mu - V_{int} + gvB_\mu(\partial^\mu \xi) \)
\(\eta\) is massive, \(\xi\) is massless, B has mass. But the last interaction term is unphysical (scalar field turns to spin-1 boson!).
In addition, B was originally massless with 2 d.o.f (polarization). Now it's massive with 3 d.o.f.
Solution: goldstone boson can be eliminated by choosing Unitary Gauge: \(B_\mu(x) \rightarrow B'_\mu(x) = B_\mu(x) + \frac{1}{gv}\partial_\mu \xi(x)\). Then Goldstone boson \(\xi\) is completely removed!
Then \(L = \frac{1}{2}\partial^\mu\eta\partial_\mu\eta - \lambda v^2 \eta^2 -\frac{1}{4}F_{\mu\nu} F^{\mu\nu} + \frac{1}{2}g^2v^2B'_\mu B'^\mu - V_{int} \)
And \(\phi(x)\rightarrow \phi'(x) = \frac{1}{\sqrt{2}}(v + h(x))\), where \(h(x) = \eta(x)\) to emphasize \(\phi\) is completely real.
The final Lagrangian has 1. massive scalar field \(h(x)\) 2. massive gauge boson \(B_\mu\) 3. interaction terms \(BBh, BBhh\) 4. self interaction \(h^3, h^4\)
\(m_h = \sqrt{2\lambda v^2}, m_B = gv\)
- Higgs in Standard Model embedded in local \(SU(2)_L \times U(1)_Y\) symmetry
Need two scalar fields, forming an weak isospin doublet. Under unitary gauge, we have \(\phi(x) = \) \(\begin{pmatrix}\phi^+\\ \phi^0 \end{pmatrix}\) = \(\begin{pmatrix} 0\\ v + h(x) \end{pmatrix} \)
Unitary gauge eliminates 3 massless Goldstone bosons and gives mass to 3 Gauge bosons (longitudinal polarization for \(W^+/W^-, Z\))! Exists one massive scalar field (Higgs)
\(\partial_\mu \rightarrow D_\mu = \partial_\mu + ig_W \mathbf{T}\cdot \mathbf{W}_\mu + ig'\frac{Y}{2}B_\mu\)
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Fermion mass can also be generated: \(\bar{L}\phi R\) is invariant under SU(2) x U(1), where L is LH chiral fermion, R is RH chiral fermion.
Therefore, \(L \sim - g_f (\bar{L}\phi R + \bar{R}\phi^\dagger L)\)
The Yukawa coupling \(g_e\) is not predicted by SM; chosen to be consistent with electron mass: \(g_e = \sqrt{2}\frac{m_e}{v}\)
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E.g. for electron, \(L_e = -m_e \bar{e}e - \frac{m_e}{v}\bar{e}eh\)
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Higgs
Property
Couples to all fermions with strength \(\propto m_f\). Therefore, mostly decay to heavy particles.
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Decay
\(H\rightarrow \bar{f}f (b\bar{b}, \tau^-\tau^+), H\rightarrow WW^*, H\rightarrow ZZ^*\), \(H\rightarrow \gamma\gamma\) (tiny; dominated by W loop) , etc
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Dominant: \(H\rightarrow bb\), BR ~ 58%.
\(M(H\rightarrow \bar{b}b) = \frac{m_f}{v}\bar{u}(p_b)\nu(p_\bar{b})\), \(\Gamma(H\rightarrow \bar{b}b) = 3 \times \frac{m_b^2m_H}{8\pi v^2}\), where 3 is from color d.o.f.
Coupling to \(ee, uu, dd, cc, ss\) has not been determined (rate too small)
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