Gauge Boson Self-Interactions with covariant derivative

Question

Self-Interactions of the unphysical gauge bosons $W_1, W_2, W_3$ are written within the gauge term

$L_\mathrm{Gauge}=-\frac{1}{4} W_{\mu \nu} W^{\mu \nu}$ with $W_{\mu \nu}= \partial_\mu W_\nu - \partial_\nu W_\mu + i g [W_\mu,W_\nu]$.

Using now the transformations for Mass matrix diagonalization

$W_{3 \mu} = \mathrm{cw} Z_\mu + \mathrm{sw} A_\mu$,

$W_{2 \mu} = \frac{i}{\sqrt{2}}(W^+_\mu - W^-_\mu)$,

$W_{1 \mu} = \frac{1}{\sqrt{2}}(W^+_\mu + W^-_\mu)$,

one is able to derive e.g. the vertex of WWA. It contains terms of the form $(\partial A) W^+ W^-, A (\partial W^+) W^-, A W^+ (\partial W^-)$. To be precise it should read \begin{align} -i e [ \partial^\mu W^{\nu +} (A_\nu W_\mu^- - A_\mu W_\nu^-) + \partial^\mu W^{\nu -} (A_\mu W_\nu^+ - A_\nu W_\mu^+ ) + \partial^\mu A^\nu (W_\nu^+ W_\mu^- - W_\mu^+ W_\nu^-)] \end{align}

The Photon gradient term can be integrated by parts, then I get \begin{align} -i e \partial^\mu A^\nu (W_\nu^+ W_\mu^- - W_\mu^+ W_\nu^-) = i e A^\nu (\partial^\mu W_\nu^+ W_\mu^- - W_\mu^+ \partial^\mu W_\nu^-) \end{align} and in total this would mean for the WWA vertex \begin{align} -i e &[ \partial^\mu W^{\nu +} (A_\nu W_\mu^- - A_\mu W_\nu^-) + \partial^\mu W^{\nu -} (A_\mu W_\nu^+ - A_\nu W_\mu^+ ) - A^\nu (\partial^\mu W_\nu^+ W_\mu^- - W_\mu^+ \partial^\mu W_\nu^-)] \\ = -i e &[ -\partial^\mu W^{\nu +} A_\mu W_\nu^- + \partial^\mu W^{\nu -} A_\mu W_\nu^+ ] \end{align}

Now I tried to rewrite the gauge boson self interactions in terms of the covariant derivative using directly the physical fields by writing down \begin{align} L_\mathrm{cov} = -\frac{1}{2}(D_\mu W_\nu^+ - D_\nu W_\mu^+) (D^\mu W^{\nu -} - D^\nu W^{\mu -}) - \frac{1}{4}(\partial_\mu A_\nu - \partial_\nu A_\mu)^2 - \frac{1}{4}(\partial_\mu Z_\nu - \partial_\nu Z_\mu)^2 \end{align} with

$D_\mu W_\nu^+ = (\partial_\mu + i e A_\mu) W_\nu^+$,

$D_\mu W_\nu^- = (\partial_\mu - i e A_\mu) W_\nu^+$.

Collecting all contributions to WWA results in \begin{align} -i e [ \partial^\mu W^{\nu +} (A_\nu W_\mu^- - A_\mu W_\nu^-) + \partial^\mu W^{\nu -} (A_\mu W_\nu^+ - A_\nu W_\mu^+ )] \end{align}

Obviously, this is not yet the same as with $L_\mathrm{Gauge}$, so what am I missing? In the end, both approaches should lead to equivalent vertices...

It already would be helpful if somebody knows literature where these terms are calculated explicitly using the covariant derivative.

Answer 1

Your covariant action guess is incomplete. It is missing the gauge invariant $$\propto \bbox[yellow]{(\partial^\mu A^\nu - \partial^\nu A^\mu) (W_\mu^+ W_\nu^- - c.c.)}\\ \propto [D_\mu,D_\nu](W_\mu^+ W_\nu^- - c.c.),$$ giving the Ws their characteristic magnetic moment.

In both cases, EM gauge invariance is paramount, and should obtain. In both cases, the $W\partial W$ current coupling to the photon should be hermitean (with the i; or antihermitean without it!).

The gauge piece, available in good texts, e.g. M Schwartz, (29.9), is proportional to the top expression in $$ -ieA_\nu [ \partial_\mu (-W^{\mu~+} W^{\nu~-}+ W^{\nu~+}W^{\mu~-}) \\-W^{\mu~+} \partial_\nu W^{\mu~-}+ W^{\mu~-}\partial_\nu W^{\mu~+ } + W^{\mu~+} \partial_\mu W^{\nu~-}- W^{\mu~-}\partial_\mu W^{\nu~+}]\\ =... $$ You may halve these terms by subtracting the c.c.s! Proceed to rewrite it to your form.

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Where did the extra term come from? you might ask. Schematicaly, and with conventional (unlike your) normalizations, recall the gauge action is the square of $$ \vec W_{\mu\nu}= \partial_\mu \vec W_\nu - \partial_\nu \vec W_\mu + g \vec W _\mu \times \vec W _\nu \leadsto \\ \vec W_{\mu\nu}^+= \partial_\mu \vec W_\nu^+ - \partial_\nu \vec W_\mu^+ -i g \vec ( W _\mu^3 \vec W _\nu^+ - \hbox{c.c.}),\\ \vec W_{\mu\nu}^3= \partial_\mu \vec W_\nu^3 - \partial_\nu \vec W_\mu^3 +i g \vec ( W _\mu^+ \vec W _\nu^- - \hbox{c.c.}), $$ where you might adjust potential wrong signs to your liking.

The key point is that for $$ gW_\mu^3= eA_\mu + g\cos\theta_W ~Z_\mu, $$ ignoring Z, this yields the now complete EM gauge-invariant action piece, $$ -ieA_\nu [ (W_\mu^+(\partial^\mu W^{\nu ~ -}- \partial^\nu W^{\mu ~ -} )- \hbox{c.c.})+ W_\nu^+ W_\mu^-(\partial^\mu A^\nu -\partial^\nu A^\mu )] $$ with the new, formerly missing, term originating in $(W^3_{\mu\nu})^2$, now put in the end.