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In Peskin (page 701) the expression given for the VEV (assuming unitary gauge is used) is:

$$\langle \phi \rangle = \frac{1}{\sqrt 2}(0 ,\nu)^T$$.

If one is to substitute this expression in the kinetic term of the higgs doublet, which involves the covariant derivative, one wouldn't get mixed terms between the gauge fields and the higgs physical field, because the higgs physical field simply is not here.

But in other sources I have seen an expression of the following form:

$$\langle \phi \rangle = \frac{1}{\sqrt 2}(0 ,\nu + h(x))^T$$, where h(x) is the physical Higgs field.

My question is, when do we consider one expression and when do we consider the other?

If we want to know the type of vertices and interaction present in the GWS theory for gauge fields, should we only consider the 2nd expression?

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  • $\begingroup$ Other sources? Which? $\endgroup$ Commented Oct 11 at 23:46
  • $\begingroup$ Schawrz (page 585) or even different dfs online $\endgroup$ Commented Oct 12 at 0:11
  • $\begingroup$ Consider to spell out acronyms. $\endgroup$ Commented Oct 12 at 9:10
  • $\begingroup$ You have misread Schwartz, confusing the vev with the field.... $\endgroup$ Commented Oct 12 at 17:31
  • $\begingroup$ Note that Matt, in his (29.3), is writing $\phi$, in your notation, in terms of the quantum fields $\pi^i$ and $h$. This is not your second equation, under any circumstances or limits. Do you still insist on asking your (misbegotten) question? $\endgroup$ Commented Oct 12 at 21:37

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The vacuum expectation value of the Higgs field is

$$\langle \phi \rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} v \\0 \end{pmatrix} \tag{1} $$

But the Higgs field is not equal to its vacuum expectation value: it fluctuates around it. If we call $h$ the (canonically normalized) fluctuation the of the Higgs field is

$$ \phi(x) = \frac{1}{\sqrt{2}}\begin{pmatrix} v+h(x) \\0 \end{pmatrix} \tag{2} $$

These expressions do not contradict because they represent different things and in general they are denoted by different symbols.

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  • $\begingroup$ What have you done with the goldstons in (2) to be absorbed by the gauge fields? $\endgroup$ Commented Oct 12 at 21:59

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