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After the spontaneous breakdown of local symmetry in presence of gauge fields (Higgs Mechanism), we can always choose a gauge where the Goldstone bosons are eaten up by the gauge field (also called unitary gauge). Which Lagrangian should be used for physical calculations-(i) the one in which goldstone excitations are present or (ii) the one in which they disappear?

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The advantage of unitary gauge is that it completely removes unphysical fields, while adding additional degrees of freedom to the gauge bosons, which consequently become massive. This gauge works well for tree-level calculations, but complications arise when considering loops:

The propagators of gauge fields and ghosts (which are needed to impose the unitary gauge condition in the path integral formalism) cause divergences which ruin renormalizability. A better choice for dealing with loops is the so-called $R_\xi$ gauge, which involves adding a gauge-fixing term containing a continuous parameter $\xi$ to the Lagrangian. This removes the divergence-problem on the loop level and lets one calculate finite quantities. Physical quantities (scattering amplitudes) do not depend on this parameter, and it can thus be set to a convenient value; $\xi=1$ for example is a good choice. In the limit $\xi\rightarrow\infty$, the $R_\xi$ gauge reduces to unitarity gauge.

This is explained in detail in Srednicki's book on QFT.

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Let us start with a simple analogy that helps grasping the concept. Say, we need to calculate a electric field of a charged cylinder. What is the first thing you do? You choose your $z$-axis along the axis of the cylinder.

Why won't you choose your axes differently, like pointing whatever they want? Well, you could, but that will make the problem unnecessary complicated. Calculations will be tedious, but the result will be the same. That is why we prefer choose the representation that is most convenient for our needs.

Same thing with gauge invariance -- there are many different representation that map to the same physical reality. You free to choose whichever representation you want (but, obviously, afterwards you are obliged to stick to it).

Back to your question: what is the better representation -- with Goldstones or without? As you might have guessed -- it depends on your problem and what is most convenient for you. I'd say, the rule of thumb is: if you are not planning to go beyond tree level, then unitary gauge is your choice. But you are into doing loops, then you better keep them.

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