# A Doubt on Covariant Derivatives and Particle Physics

So, for me at least, if you understand well why a covariant derivative is defined the way it is presented to you, then $$25\%$$ of conceptual work is done. I mean, with covariant derivatives (in General Relativity) you can understand a plethora of physics: equivalence principle, geodesics, curvature tensors, geodesic deviation and so on...

In particle physics the role of a covariant derivative becomes relevant not in spacetime, but in fibre bundles. The great conclusion is therefore:

For standard model of particle physics $$($$with the gauge group given by $$U(1)\times SU(2)\times SU(3)$$ $$)$$ the covariant derivative is given by$$[1]$$:

$$D_{\mu} = \partial_{\mu} -i\frac{g'}{2}YB_{\mu}-i\frac{g}{2}\sigma_{j}W_{\mu}\hspace{0.2mm}^{j}-i\frac{g_{s}}{2}\lambda_{j}G_{\mu}\hspace{0.2mm}^{j}\tag{1}$$ Where $$\sigma_{j}$$ are the Pauli matrices $$($$the generators of $$SU(2))$$, $$\lambda_{j}$$ are the Gell-Mann matrices $$($$the generators of $$SU(3))$$.

In general it is possible to say that the general form of the covariant derivative is:

$$D_{\mu} = \partial_{\mu} -igT_{j}A_{\mu}\hspace{0.2mm}^{j}\tag{2}$$ Where $$T_{j}$$ are the generators of a given Lie group $$G$$.

What is bothering me is the term $$-i\frac{g'}{2}YB_{\mu}$$ in $$(1)$$, because is quite clear that each term of the covariant derivative "corresponds to a interaction of nature"; $$W_{\mu}\hspace{0.2mm}^{j}$$ is the gauge field for weak interaction, $$G_{\mu}\hspace{0.2mm}^{j}$$ is the gauge field for strong interaction.

So my doubt is:

Why the term $$-i\frac{g'}{2}YB_{\mu}$$ is the correct one and not simply the famous $$-iqA_{\mu}$$ since for the $$U(1)$$ (and therefore for electromagnetism) group the covariant derivative is:

$$D_{\mu} = \partial_{\mu} -iqA_{\mu} ? \tag{3}$$

$$* * *$$

$$[1]$$ https://en.wikipedia.org/wiki/Gauge_covariant_derivative#cite_note-12

This is part of the Higgs mechanism. The EM photon field $$A_{\mu}$$ is identified as a linear combination of the $$W^3_{\mu}$$ and $$B_{\mu}$$ fields that remains massless after the electroweak symmetry breaking.