# The Lagrangian equation...

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Added warning tensors calculus has two meanings for the word covariant you have covariant vectors=covector and covariance which is a principle similar meaning to the laws of physics is the same all reference frames Google Lorentz covariance. Then you also have the covariant derivatives and contravariant derivatives. (Just to make things confusing lmao)

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Posted (edited)

$\begin{pmatrix}\acute{d}\\\acute{s}\\\acute{b}\end{pmatrix}\begin{pmatrix}V_{ud}&V_{us}&V_{ub}\\V_{cd}&V_{cs}&V_{cb}\\V_{td}&V_{ts}&V_{tb}\end{pmatrix}\begin{pmatrix}d\\s\\b\end{pmatrix}$

Electroweak correlations

$\mathcal{L}=\mathcal{L}_{gauge}+\mathcal{L}_f+\mathcal{L}_\phi+\mathcal{L}_{yuk}$

Gauge sector

$\mathcal{L}_{gauge}=\frac{1}{4}W^i_{\mu\nu}W^{\mu\nu I}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}$

Where $W_{\mu\nu}$ and $B_{\mu\nu}$ are the SU(2)and U(1) field strength tensors.

$W^i_{\mu\nu}=\partial_\mu^1-\partial W^i_\nu-\partial_\nu W^i_\mu-g\epsilon_{ijk}W_{\mu}^jW^k_\nu W^k_\nu$

$B^i_{\mu\nu}=\partial_\mu B_\nu-\partial_\nu B_\mu$

$\epsilon_{ijk}$ group structure constants of SU(2) B of U(1) abelion group has no self interaction (gauge boson)

$\mathcal{f}\subset\Sigma(\bar{q}+\bar{\ell}i\displaystyle{\not}D \ell)$

q is quark $\ell$ is leptons it sums over generations.

The quage covariant derivative is

$D_q=(\partial_\mu+\frac{ig}{2}\vec{\tau}\cdot\vec{W}_\mu+i\acute{g}Y\cdot B_\mu)q$

$\displaystyle{\not}D=\gamma D^\mu$ g and $\acute{g}$ are the gauge coupling constants of $SU(2)_w$ and $U(1)_y$ $\vec{\tau}$ refers to Pauli matrices. Y is hypercharge of U(1) the electric charge Q is

$Q=I_3+\frac{1}{2}Y$

langrangian for complex scalar fields.

$\mathcal{L}_\phi=(D^\mu)^\dagger D_\mu\phi-V(\phi)$

$D_\mu \phi=(\partial_\mu+\frac{ig}{2}\vec{\tau}\cdot\vec{W}_\mu+\frac{i\acute{g}}{2}B_\mu)\phi$

$V(\phi)=\mu^2\phi^\dagger\phi+\lambda(\phi^\dagger\phi)^2$

Lambda is the self interaction term

Edited by Mordred

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Sorry I can't contribute to the post but seriously.....Holy s#-t, there's  some serious math in this post my poor old iPad took at least 5 mins just to load it all up. That doesn't even happen on math forums I go to. I'm seriously envious of people who can do math/physics at this level. If you can understand this thread, I hope your ugly...

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Posted (edited)

Both Orion and myself do, we're breaking apart the relations that went into the OP Langrene. Right now I'm trying to determine if it's canonical or conformal by looking at the EW Langrene through symmetry break via the Higgs. Which will confirm the Higgs and Yukawa couplings underbrace sections. We're both learning from this gives us a refreshing challenge. The Yukawa section is rather challenging.

I've already confirmed the Higgs and Dirac covariant derivative forms.

Edited by Mordred

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