Exercise 1.1.1.1.1a: Given locality, causality, Lorentz invariance, and known physical data since 1860, show that the Lagrangian describing all observed physical processes (sans gravity) can be written:

−\frac{1}{2}∂_{ν}g_{μ}^{a}∂_{ν}g_{μ}^{a} − g_{s}f^{abc}∂_{μ}g_{ν}^{a}g_{μ}^{b}g_{ν}^{c} − \frac{1}{4}g_{s}^{2}f^{abc}f^{ade}g_{μ}^{b}g_{ν}^{c}g_{μ}^{d}g_{ν}^{e} + \frac{1}{2}ig_{s}^{2}(q̄_{i}^{σ}γ^{μ}q_{j}^{σ})g_{μ}^{a} + Ḡ^{a}∂^{2}G^{a} + g_{s}f^{abc}∂_{μ}Ḡ^{a}G^{b}g_{μ}^{c} − ∂_{ν}W_{μ}^{+}∂_{ν}W_{μ}^{−} −

M^{2}W_{μ}^{+}W_{μ}^{−} − \frac{1}{2}∂_{ν}Z_{μ}^{0}∂_{ν}Z_{μ}^{0} − \frac{1}{2c_{w}^{2}} M^{2}Z_{μ}^{0}Z_{μ}^{0} − \frac{1}{2}∂_{μ}A_{ν}∂_{μ}A_{ν} − \frac{1}{2}∂_{μ}H∂_{μ}H − \frac{1}{2}m_{h}^{2}H^{2} −∂_{μ}ϕ^{+}∂_{μ}ϕ^{−} −M^{2}ϕ^{+}ϕ^{−} − \frac{1}{2}∂_{μ}ϕ^{0}∂_{μ}ϕ^{0} − \frac{1}{2c_{w}^{2}} Mϕ^{0}ϕ^{0} −β_{h}[\frac{2M^{2}}{g^{2}} + \frac{2M}{g} H + \frac{1}{2}(H^{2} + ϕ^{0}ϕ^{0} + 2ϕ^{+}ϕ^{−})] + \frac{2M^{4}}{g^{2}} α_{h} − igc_{w}[∂_{ν}Z_{μ}^{0}(W_{μ}^{+}W_{ν}^{−} −

W_{ν}^{+}W_{μ}^{−}) − Z_{ν}^{0}(W_{μ}^{+}∂_{ν}W_{μ}^{−} − W_{μ}^{−}∂_{ν}W_{μ}^{+}) + Z_{μ}^{0}(W_{ν}^{+}∂_{ν}W_{μ}^{−} − W_{ν}^{−}∂_{ν}W_{μ}^{+})] − igs_{w}[∂_{ν}A_{μ}(W_{μ}^{+}W_{ν}^{−} − W_{ν}^{+}W_{μ}^{−}) − A_{ν}(W_{μ}^{+}∂_{ν}W_{μ}^{−} − W_{μ}^{−}∂_{ν}W_{μ}^{+}) + A_{μ}(W_{ν}^{+}∂_{ν}W_{μ}^{−} − W_{ν}^{−}∂_{ν}W_{μ}^{+})] − \frac{1}{2}g^{2}W_{μ}^{+}W_{μ}^{−}W_{ν}^{+}W_{ν}^{−} +

\frac{1}{2}g^{2}W_{μ}^{+}W_{ν}^{−}W_{μ}^{+}W_{ν}^{−} + g^{2}c_{w}^{2}(Z_{μ}^{0}W_{μ}^{+}Z_{ν}^{0}W_{ν}^{−} − Z_{μ}^{0}Z_{μ}^{0}W_{ν}^{+}W_{ν}^{−}) + g^{2}s_{w}^{2}(A_{μ}W_{μ}^{+}A_{ν}W_{ν}^{−} − A_{μ}A_{μ}W_{ν}^{+}W_{ν}^{−}) + g^{2}s_{w}c_{w}[A_{μ}Z_{ν}^{0}(W_{μ}^{+}W_{ν}^{−} − W_{ν}^{+}W_{μ}^{−}) − 2A_{μ}Z_{μ}^{0}W_{ν}^{+}W_{ν}^{−}] − gα[H^{3} + Hϕ^{0}ϕ^{0} + 2Hϕ^{+}ϕ^{−}] −

\frac{1}{8}g^{2}α_{h}[H^{4}+(ϕ^{0})^{4}+4(ϕ^{+}ϕ^{−})^{2}+4(ϕ^{0})^{2}ϕ^{+}ϕ^{−}+4H^{2}ϕ^{+}ϕ^{−}+2(ϕ^{0})^{2}H^{2}]−

gMW_{μ}^{+}W_{μ}^{−}H − \frac{1}{2}g \frac{M}{c_{w}^{2}} Z_{μ}^{0}Z_{μ}^{0}H − \frac{1}{2}ig[W_{μ}^{+}(ϕ^{0}∂_{μ}ϕ^{−} − ϕ^{−}∂_{μ}ϕ^{0}) − W_{μ}^{−}(ϕ^{0}∂_{μ}ϕ^{+} −ϕ^{+}∂_{μ}ϕ^{0})]+ \frac{1}{2}g[W_{μ}^{+}(H∂_{μ}ϕ^{−} −ϕ^{−}∂_{μ}H)−W_{μ}^{−}(H∂_{μ}ϕ^{+} − ϕ^{+}∂_{μ}H)]+ \frac{1}{2}g \frac{1}{c_{w}} (Z_{μ}^{0}(H∂_{μ}ϕ^{0}−ϕ^{0}∂_{μ}H)−ig \frac{s_{w}^{2}}{c_{w}} MZ_{μ}^{0}(W_{μ}^{+}ϕ^{−}−W_{μ}^{−}ϕ^{+})+

igs_{w}MA_{μ}(W_{μ}^{+}ϕ^{−} − W_{μ}^{−}ϕ^{+}) − ig \frac{1−2c_{w}^{2}}{2c_{w}} Z_{μ}^{0}(ϕ^{+}∂_{μ}ϕ^{−} − ϕ^{−}∂_{μ}ϕ^{+}) + igs_{w}A_{μ}(ϕ^{+}∂_{μ}ϕ^{−} − ϕ^{−}∂_{μ}ϕ^{+}) − \frac{1}{4}g^{2}W_{μ}^{+}W_{μ}^{−}[H^{2} + (ϕ^{0})^{2} + 2ϕ^{+}ϕ^{−}] − \frac{1}{4}g^{2} \frac{1}{c_{w}^{2}} Z_{μ}^{0}Z_{μ}^{0}[H^{2} + (ϕ^{0})^{2} + 2(2s_{w}^{2} − 1)^{2}ϕ^{+}ϕ^{−}] − \frac{1}{2}g^{2} \frac{s_{w}^{2}}{c_{w}} Z_{μ}^{0}ϕ^{0}(W_{μ}^{+}ϕ^{−} + W_{μ}^{−}ϕ^{+}) − \frac{1}{2}ig^{2} \frac{s_{w}^{2}}{c_{w}} Z_{μ}^{0}H(W_{μ}^{+}ϕ^{−} − W_{μ}^{−}ϕ^{+}) + \frac{1}{2}g^{2}s_{w}A_{μ}ϕ^{0}(W_{μ}^{+}ϕ^{−} +

W_{μ}^{−}ϕ^{+})+ \frac{1}{2}ig^{2}s_{w}A_{μ}H(W_{μ}^{+}ϕ^{−} −W_{μ}^{−}ϕ^{+})−g^{2} \frac{s_{w}}{c_{w}} (2c_{w}^{2} −1)Z_{μ}^{0}A_{μ}ϕ^{+}ϕ^{−} − g^{1}s_{w}^{2}A_{μ}A_{μ}ϕ^{+}ϕ^{−} − ē^{λ}(γ∂ + m_{e}^{λ})e^{λ} − ν̄^{λ}γ∂ν^{λ} − ū_{j}^{λ}(γ∂ + m_{u}^{λ})u_{j}^{λ} − d̄_{j}^{λ}(γ∂ + m_{d}^{λ})d_{j}^{λ} + igs_{w}A_{μ}[−(ē^{λ}γ^{μ}e^{λ}) + \frac{2}{3}(ū_{j}^{λ}γ^{μ}u_{j}^{λ}) − \frac{1}{3}(d̄_{j}^{λ}γ^{μ}d_{j}^{λ})] + \frac{ig}{4c_{w}} Z_{μ}^{0}[(ν̄^{λ}γ^{μ}(1 + γ^{5})ν^{λ}) + (ē^{λ}γ^{μ}(4s_{w}^{2} − 1 − γ^{5})e^{λ}) + (ū_{j}^{λ}γ^{μ}(\frac{4}{3}s_{w}^{2} − 1 − γ^{5})u_{j}^{λ}) + (d̄_{j}^{λ}γ^{μ}(1 − \frac{8}{3}s_{w}^{2} − γ^{5})d_{j}^{λ})] + \frac{ig}{2\sqrt{2}}W_{μ}^{+}[(ν̄^{λ}γ^{μ}(1 + γ^{5})e^{λ}) + (ū_{j}^{λ}γ^{μ}(1 + γ^{5})C_{λκ}d_{j}^{κ})] + \frac{ig}{2\sqrt{2}}W_{μ}^{−}[(ē^{λ}γ^{μ}(1 + γ^{5})ν^{λ}) + (d̄_{j}^{κ}C_{λκ}^{†}γ^{μ}(1 +

γ^{5})u_{j}^{λ})] + \frac{ig}{2\sqrt{2}} \frac{m_{e}^{λ}}{M} [−ϕ^{+}(ν̄^{λ}(1 − γ^{5})e^{λ}) + ϕ^{−}(ē^{λ}(1 + γ^{5})ν^{λ})] − \frac{g}{2} \frac{m_{e}^{λ}}{M} [H(ē^{λ}e^{λ}) + iϕ^{0}(ē^{λ}γ^{5}e^{λ})] + \frac{ig}{2M\sqrt{2}}ϕ^{+}[−m_{d}^{κ}(ū_{j}^{λ}C_{λκ}(1 − γ^{5})d_{j}^{κ}) + m_{u}^{λ}(ū_{j}^{λ}C_{λκ}(1+γ^{5})d_{j}^{κ}]+ \frac{ig}{2M\sqrt{2}}ϕ^{−}[m_{d}^{λ}(d̄_{j}^{λ}C_{λκ}^{†}(1+γ^{5})u_{j}^{κ})−m_{u}^{κ}(d̄_{j}^{λ}C_{λκ}^{†}(1−

γ^{5})u_{j}^{κ}] − \frac{g}{2} \frac{m_{u}^{λ}}{M} H(ū_{j}^{λ}u_{j}^{λ}) − \frac{g}{2} \frac{m_{d}^{λ}}{M} H(d̄_{j}^{λ}d_{j}^{λ}) + \frac{ig}{2} \frac{m_{u}^{λ}}{M} ϕ^{0}(ū_{j}^{λ}γ^{5}u_{j}^{λ}) − \frac{ig}{2} \frac{m_{d}^{λ}}{M} ϕ^{0}(d̄_{j}^{λ}γ^{5}d_{j}^{λ}) + X̄^{+}(∂^{2} − M^{2})X^{+} + X̄^{−}(∂^{2} − M^{2})X^{−} + X̄^{0}(∂^{2} − \frac{M^{2}}{c_{w}^{2}} )X^{0}+Ȳ ∂^{2}Y +igc_{w}W_{μ}^{+}(∂_{μ}X̄^{0}X^{−}−∂_{μ}X̄^{+}X^{0})+igs_{w}W_{μ}^{+}(∂_{μ}Ȳ X^{−}− ∂_{μ}X̄^{+}Y) + igc_{w}W_{μ}^{−}(∂_{μ}X̄^{−}X^{0} − ∂_{μ}X̄^{0}X^{+}) + igs_{w}W_{μ}^{−}(∂_{μ}X̄^{−}Y − ∂_{μ}Ȳ X^{+}) + igc_{w}Z_{μ}^{0}(∂_{μ}X̄^{+}X^{+} − ∂_{μ}X̄^{−}X^{−}) + igs_{w}A_{μ}(∂_{μ}X̄^{+}X^{+} −

∂_{μ}X̄^{−}X^{−}) − \frac{1}{2}gM[X̄^{+}X^{+}H + X̄^{−}X^{−}H + \frac{1}{c_{w}^{2}} X̄^{0}X^{0}H] + \frac{1−2c_{w}^{2}}{2c_{w}} igM[X̄^{+}X^{0}ϕ^{+} − X̄^{−}X^{0}ϕ^{−}] + \frac{1}{2c_{w}} igM[X̄^{0}X^{−}ϕ^{+} − X̄^{0}X^{+}ϕ^{−}] +

igMs_{w}[X̄^{0}X^{−}ϕ^{+} − X̄^{0}X^{+}ϕ^{−}] + \frac{1}{2}igM[X̄^{+}X^{+}ϕ^{0} − X̄^{−}X^{−}ϕ^{0}]