The $hhh$ vertex: $\Gamma_{hhh}(-p_{1},-p_{2},-p_{3})=-3\lambda
v\,\,\delta(p_{1}+p_{2}+p_{3})$.

  * •

The $h\rho\rho$ vertex:
$\Gamma_{h\rho^{a}\rho^{b}}(-p_{1},-p_{2},-p_{3})=-\lambda
v\delta^{ab}\,\,\delta(p_{1}+p_{2}+p_{3})$.

  * •

The $AAA$-vertex:
$\Gamma_{A_{\mu}^{a}A_{v}^{b}A_{\sigma}^{c}}(-p_{1},-p_{2},-p_{3})=-igf^{abc}\left[(p_{1}-p_{3})_{\nu}\delta_{\sigma\mu}+(p_{3}-p_{2})_{\mu}\delta_{v\sigma}+(p_{2}-p_{1})_{\sigma}\delta_{v\mu}\right]\delta(p_{1}+p_{2}+p_{3})$.

  * •

The $\overline{c}Ac$-vertex:
$\Gamma_{\overline{c}^{a}A_{\mu}^{b}c^{c}}(-p_{1},-p_{2},-p_{3})=igf^{abc}p_{1,\mu}\delta(p_{1}+p_{2}+p_{3})$.

  * •

The $AAAA$-vertex:
$\Gamma_{A_{\mu}^{a}A_{\nu}^{b}A_{\rho}^{c}A_{\sigma}^{d}}(-p_{1},-p_{2},-p_{3},-p_{4})=g^{2}[f^{eab}f^{ecd}(\delta_{\mu\sigma}\delta_{\nu\rho}-\delta_{\mu\rho}\delta_{\nu\sigma})+f^{eac}f^{ebd}(\delta_{\mu\sigma}\delta_{\nu\rho}-\delta_{\mu\nu}\delta_{\rho\sigma})+f^{ead}f^{ebc}(\delta_{\mu\rho}\delta_{\nu\sigma}-\delta_{\mu\nu}\delta_{\rho\sigma})]\delta(p_{1}+p_{2}+p_{3}+p_{4})$.

  * •

The $AAhh$-vertex:
$\Gamma_{A_{\mu}^{a}A_{v}^{b}hh}(-p_{1},-p_{2},-p_{3},-p_{4})=-\frac{1}{2}g^{2}\delta^{ab}\delta_{\mu
v}$.

  * •

The $AA\rho\rho$-vertex:
$\Gamma_{A_{\mu}^{a}A_{v}^{b}\rho^{c}\rho^{d}}(-p_{1},-p_{2},-p_{3},-p_{4})=-\frac{1}{2}g^{2}\delta_{\mu
v}\delta^{ab}\delta^{cd}$.

###  C.2 Elementary propagators of the $SU(2)$ Higgs model in the $R_{\xi}$
gauge

Here we will calculate the one-loop corrections to the Higgs and gauge field
propagator. This requires the calculation 111We have used from [160] the
technique of modifying integrals into “master integrals” without numerators.
of the Feynman diagrams as shown in Figures C.1 and C.2. We will use the
following definitions:

$\displaystyle\eta(m_{1},m_{2})$ $\displaystyle\equiv$
$\displaystyle\frac{1}{(4\pi)^{d/2}}\Gamma(2-\frac{d}{2})\int_{0}^{1}dx\left(p^{2}x(1-x)+xm_{1}+(1-x)m_{2}\right)^{d/2-2},$
$\displaystyle\chi(m_{1})$ $\displaystyle\equiv$
$\displaystyle\frac{1}{(4\pi)^{d/2}}\Gamma(1-\frac{d}{2})m_{1}^{d/2-1}.$ (C.3)

Notice that the last four diagrams for both particles are zero for $\langle
h\rangle=0$. In fact, including these diagrams has the same effect as making a
shift in the minimizing value of the scalar field $\Phi$ to demand $\langle
h\rangle=0$, see the Appendix of [61] for the technical details. In the
context of the FMS operators, we found it more convenient to expand around the
(classical) $v$ that is gauge invariant, and thus to include the tadpoles.
Expanding the FMS operator around the quantum corrected vev would lead to
cancellations in that quantum vev coming from the propagator loop corrections
to render it gauge invariant again, indeed the minimum of the quantum
corrected effective Higgs potential is not gauge invariant itself.

####  C.2.1 Higgs propagator

Figure C.1: One-loop contributions to the propagator $\langle
h(p)h(-p)\rangle$. Curly lines represent the gauge field, dashed lines the
Higgs field, solid lines the Goldstone boson and double lines the ghost field.

The first diagrams contributing to the Higgs self-energy are of the snail
type, renormalizing the masses of the internal fields. The Higgs boson snail
(first diagram in the first line of Figure C.1):

$\displaystyle\Gamma_{hh,1}(p^{2})$ $\displaystyle=$
$\displaystyle-\frac{3\lambda\chi\left(m_{h}^{2}\right)}{2\left(m_{h}^{2}+p^{2}\right)^{2}},$
(C.4)

the Goldstone boson snail (second diagram in the first line of Figure C.1):

$\displaystyle\Gamma_{hh,2}(p^{2})$ $\displaystyle=$
$\displaystyle-\frac{3\lambda\chi\left(m^{2}\xi\right)}{2\left(m_{h}^{2}+p^{2}\right)^{2}},$
(C.5)

and the gauge field snail (third diagram in the first line of Figure C.1):

$\displaystyle\Gamma_{hh,3}(p^{2})$ $\displaystyle=$
$\displaystyle-\frac{3(d-1)g^{2}\chi\left(m^{2}\right)}{4\left(m_{h}^{2}+p^{2}\right)^{2}}-\frac{3g^{2}\xi\chi\left(m^{2}\xi\right)}{4\left(m_{h}^{2}+p^{2}\right)^{2}}.$
(C.6)

Next, we meet a couple of sunset diagrams. The Higgs boson sunset (fourth
diagram in the first line of Figure C.1):

$\displaystyle\Gamma_{hh,4}(p^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx\Big{\\{}\frac{9\lambda^{2}v^{2}\eta\left(m_{h}^{2},m_{h}^{2}\right)}{2\left(m_{h}^{2}+p^{2}\right)^{2}}\Big{\\}},$
(C.7)

the gauge field sunset (first diagram in the second line of Figure C.1):

$\displaystyle\Gamma_{hh,5}(p^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx\Big{\\{}\frac{3g^{2}\eta\left(m^{2},m^{2}\right)\left(4(D-1)m^{4}+4m^{2}p^{2}+p^{4}\right)}{8m^{2}\left(m_{h}^{2}+p^{2}\right)^{2}}+\frac{3g^{2}\left(2m^{2}\xi+p^{2}\right)^{2}\eta\left(m^{2}\xi,m^{2}\xi\right)}{8m^{2}\left(m_{h}^{2}+p^{2}\right)^{2}}$
(C.8) $\displaystyle-$
$\displaystyle\frac{3g^{2}\left(m^{4}(\xi-1)^{2}+2m^{2}\xi
p^{2}+2m^{2}p^{2}+p^{4}\right)\eta\left(m^{2},m^{2}\xi\right)}{4m^{2}\left(m_{h}^{2}+p^{2}\right)^{2}}+\frac{3g^{2}(\xi-1)\chi\left(m^{2}\right)}{4\left(m_{h}^{2}+p^{2}\right)^{2}}$
$\displaystyle-$
$\displaystyle\frac{3g^{2}(\xi-1)\chi\left(m^{2}\xi\right)}{4\left(m_{h}^{2}+p^{2}\right)^{2}}\Big{\\}},$

the ghost sunset (second diagram in the second line of Figure C.1):

$\displaystyle\Gamma_{hh,6}(p^{2})$ $\displaystyle=$
$\displaystyle-\int_{0}^{1}dx\Big{\\{}\frac{3g^{2}m^{2}\xi^{2}\eta\left(m^{2}\xi,m^{2}\xi\right)}{4\left(m_{h}^{2}+p^{2}\right)^{2}}\Big{\\}},$
(C.9)

the Goldstone boson sunset (third diagram in the second line of Figure C.1):

$\displaystyle\Gamma_{hh,7}(p^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx\Big{\\{}\frac{3\lambda^{2}v^{2}\eta\left(m^{2}\xi,m^{2}\xi\right)}{2\left(m_{h}^{2}+p^{2}\right)^{2}}\Big{\\}},$
(C.10)

and a mixed Goldstone-gauge sunset (fourth diagram in the second line of
Figure C.1):

$\displaystyle\Gamma_{hh,8}(p^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx\Big{\\{}\frac{3g^{2}\left(\left(m^{2}(\xi-1)+p^{2}\right)^{2}+4m^{2}p^{2}\right)\eta\left(m^{2},m^{2}\xi\right)}{4m^{2}\left(m_{h}^{2}+p^{2}\right)^{2}}-\frac{3g^{2}\left(m^{2}\xi+p^{2}\right)^{2}\eta\left(m^{2}\xi,m^{2}\xi\right)}{4m^{2}\left(m_{h}^{2}+p^{2}\right)^{2}}$
(C.11) $\displaystyle+$
$\displaystyle\frac{3g^{2}\chi\left(m^{2}\xi\right)\left(m^{2}(2\xi-1)+p^{2}\right)}{4m^{2}\left(m_{h}^{2}+p^{2}\right)^{2}}-\frac{3g^{2}\chi\left(m^{2}\right)\left(m^{2}(\xi-1)+p^{2}\right)}{4m^{2}\left(m_{h}^{2}+p^{2}\right)^{2}}\Big{\\}}.$

Finally, we have the tadpole diagrams. The Higgs balloon (first diagram on the
third line of Figure C.1):

$\displaystyle\Gamma_{hh,9}(p^{2})$ $\displaystyle=$
$\displaystyle\frac{9\lambda^{2}v^{2}\chi\left(m_{h}^{2}\right)}{2m_{h}^{2}\left(m_{h}^{2}+p^{2}\right)^{2}},$
(C.12)

the gauge balloon (second diagram on the third line of Figure C.1):

$\displaystyle\Gamma_{hh,10}(p^{2})$ $\displaystyle=$
$\displaystyle\frac{9g\lambda
mv\left((d-1)\chi\left(m^{2}\right)+\xi\chi\left(m^{2}\xi\right)\right)}{2m_{h}^{2}\left(m_{h}^{2}+p^{2}\right)^{2}},$
(C.13)

the Goldstone boson balloon (third diagram on the third line of Figure C.1):

$\displaystyle\Gamma_{hh,11}(p^{2})$ $\displaystyle=$
$\displaystyle\frac{9\lambda^{2}v^{2}\chi\left(m^{2}\xi\right)}{2m_{h}^{2}\left(m_{h}^{2}+p^{2}\right)^{2}},$
(C.14)

the ghost balloon (fourth diagram on the third line of Figure C.1):

$\displaystyle\Gamma_{hh,12}(p^{2})$ $\displaystyle=$
$\displaystyle-\frac{9g\lambda m\xi v\chi\left(m^{2}\xi\right)}{2m_{h}^{2}}.$
(C.15)

Putting together eqs. (C.4) to (C.15) we find the Higgs propagator up to first
order in $\hbar$

$\displaystyle\braket{h(x)\,h(y)}$ $\displaystyle=$
$\displaystyle\frac{1}{p^{2}+m_{h}^{2}}+g^{2}\int_{0}^{1}dx\Big{\\{}\frac{3\left(4(d-1)m^{4}+4m^{2}p^{2}+p^{4}\right)}{8m^{2}}\eta\left(m^{2},m^{2}\right)$
(C.16) $\displaystyle+$
$\displaystyle\frac{9m_{h}^{4}}{8m^{2}}\eta\left(m_{h}^{2},m_{h}^{2}\right)+\frac{3\left(m_{h}^{4}-p^{4}\right)}{8m^{2}}\eta\left(m^{2}\xi,m^{2}\xi\right)+\frac{\left(6(d-1)m^{2}-3p^{2}\right)}{4m^{2}}\chi\left(m^{2}\right)$
$\displaystyle+$
$\displaystyle\frac{3\left(m_{h}^{2}+p^{2}\right)}{4m^{2}}\chi\left(m^{2}\xi\right)+\frac{3m_{h}^{2}}{4m^{2}}\chi\left(m_{h}^{2}\right)\Big{\\}}\frac{1}{(p^{2}+m_{h}^{2})^{2}}.$

The resummed one-loop Higgs propagator can be now approximated by

$\displaystyle G^{-1}_{hh}(p^{2})$ $\displaystyle=$ $\displaystyle
p^{2}+m_{h}^{2}-g^{2}\int_{0}^{1}dx\Big{\\{}\frac{3\left(4(d-1)m^{4}+4m^{2}p^{2}+p^{4}\right)}{8m^{2}}\eta\left(m^{2},m^{2}\right)$
(C.17) $\displaystyle+$
$\displaystyle\frac{9m_{h}^{4}}{8m^{2}}\eta\left(m_{h}^{2},m_{h}^{2}\right)+\frac{3\left(m_{h}^{4}-p^{4}\right)}{8m^{2}}\eta\left(m^{2}\xi,m^{2}\xi\right)+\frac{\left(6(d-1)m^{2}-3p^{2}\right)}{4m^{2}}\chi\left(m^{2}\right)$
$\displaystyle+$
$\displaystyle\frac{3\left(m_{h}^{2}+p^{2}\right)}{4m^{2}}\chi\left(m^{2}\xi\right)+\frac{3m_{h}^{2}}{4m^{2}}\chi\left(m_{h}^{2}\right)\Big{\\}}.$

For $d=4$, the above expression, eq. (C.17), is divergent. Employing the
procedure of dimensional regularization, i.e. setting $d=4-\epsilon$, the
divergent part for $G_{hh}(p^{2})$ is given by:

$\displaystyle G_{hh,\mathrm{d}iv}(p^{2})$ $\displaystyle=$
$\displaystyle\frac{g^{2}\left(\frac{3m_{h}^{4}}{m^{2}}-3\xi m_{h}^{2}-3\xi
p^{2}+9p^{2}\right)}{32\pi^{2}\epsilon},$ (C.18)

which, following the $\overline{MS}bar$-scheme, are re-absorbed by the
introduction of suitable local counterterms. We remain thus with the finite
part of the Higgs propagator, given in eq. (5.21).

####  C.2.2 Gauge field propagator

Figure C.2: Contributions to the one-loop gauge field self-energy.

The first diagram contributing to transverse part of the gauge field self-
energy is the gauge field snail (first diagram in the first line of Figure
C.2) and gives a contribution:

$\displaystyle\Pi_{AA^{T},1}(p^{2})$ $\displaystyle=$
$\displaystyle\frac{2g^{2}\left(p^{2}-d\left(d^{2}-3d+3\right)p^{2}\right)\chi\left(m^{2}\right)}{(d-1)dp^{2}\left(m^{2}+p^{2}\right)^{2}}-\frac{2g^{2}\xi\left((d-2)dp^{2}+p^{2}\right)\chi\left(m^{2}\xi\right)}{(d-1)dp^{2}\left(m^{2}+p^{2}\right)^{2}}.$
(C.19)

The second diagram is the Goldstone boson snail (second diagram in the first
line of Figure C.2):

$\displaystyle\Pi_{AA^{T},2}(p^{2})$ $\displaystyle=$
$\displaystyle-\frac{3g^{2}\chi\left(m^{2}\xi\right)}{4\left(m^{2}+p^{2}\right)^{2}}.$
(C.20)

The third diagram is the Higgs boson snail (third diagram in the first line of
Figure C.2):

$\displaystyle\Pi_{AA^{T},3}(p^{2})$ $\displaystyle=$
$\displaystyle-\frac{g^{2}\chi\left(m_{h}^{2}\right)}{4\left(m^{2}+p^{2}\right)^{2}}.$
(C.21)

The fourth diagram is the gauge field sunrise (first diagram in the second
line of Figure C.2):

$\displaystyle\Pi_{AA^{T},4}(p^{2})$ $\displaystyle=$ $\displaystyle
g^{2}\int_{0}^{1}dx\Big{\\{}\frac{\eta\left(m^{2},m^{2}\xi\right)\left(2m^{2}p^{2}(-2d+\xi+3)+m^{4}(\xi-1)^{2}+p^{4}\right)}{2(d-1)m^{4}p^{2}}$
(C.22) $\displaystyle-$
$\displaystyle\frac{\left(4m^{2}+p^{2}\right)\eta\left(m^{2},m^{2}\right)\left(4(d-1)m^{4}+4(3-2d)m^{2}p^{2}+p^{4}\right)}{4(d-1)m^{4}\left(m^{2}+p^{2}\right)^{2}}$
$\displaystyle-$ $\displaystyle\frac{\left(4m^{2}\xi
p^{4}+p^{6}\right)\eta\left(m^{2}\xi,m^{2}\xi\right)}{4(d-1)m^{4}\left(m^{2}+p^{2}\right)^{2}}$
$\displaystyle+$
$\displaystyle\frac{\chi\left(m^{2}\xi\right)\left(4d^{2}\left(m^{2}(\xi+1)p^{2}+p^{4}\right)+d\left(m^{4}(\xi-1)-m^{2}(6\xi+7)p^{2}+(\xi-7)p^{4}\right)+4m^{2}\xi
p^{2}\right)}{2(d-1)dm^{2}p^{2}\left(m^{2}+p^{2}\right)^{2}}$ $\displaystyle-$
$\displaystyle\frac{\chi\left(m^{2}\right)\left(4d^{2}p^{4}+d\left(m^{4}(\xi-1)+m^{2}(2\xi-5)p^{2}+(\xi-7)p^{4}\right)+4m^{2}p^{2}\right)}{2(d-1)dm^{2}p^{2}\left(m^{2}+p^{2}\right)^{2}}\Big{\\}}.$

The fifth diagram is the ghost sunrise (second diagram in the second line of
Figure C.2):

$\displaystyle\Pi_{AA^{T},5}(p^{2})$ $\displaystyle=$ $\displaystyle
g^{2}\int_{0}^{1}dx\eta\left(m^{2}\xi,m^{2}\xi\right)\left(\frac{2m^{2}\xi}{(d-1)\left(m^{2}+p^{2}\right)^{2}}+\frac{p^{2}}{2(d-1)\left(m^{2}+p^{2}\right)^{2}}\right)$
(C.23) $\displaystyle-$
$\displaystyle\frac{\chi\left(m^{2}\xi\right)}{(d-1)\left(m^{2}+p^{2}\right)^{2}}.$

The sixth diagram is the Goldstone sunrise (first diagram in the third line of
Figure C.2):

$\displaystyle\Pi_{AA^{T},6}(p^{2})$ $\displaystyle=$ $\displaystyle
g^{2}\int_{0}^{1}dx\eta\left(m^{2}\xi,m^{2}\xi\right)\left(-\frac{m^{2}\xi}{(d-1)\left(m^{2}+p^{2}\right)^{2}}-\frac{p^{2}}{4(d-1)\left(m^{2}+p^{2}\right)^{2}}\right)$
(C.24) $\displaystyle+$
$\displaystyle\frac{\chi\left(m^{2}\xi\right)}{2(d-1)\left(m^{2}+p^{2}\right)^{2}}.$

The seventh diagram is the mixed Goldstone-Higgs sunrise (second diagram in
the third line of Figure C.2):

$\displaystyle\Pi_{AA^{T},7}(p^{2})$ $\displaystyle=$ $\displaystyle
g^{2}\int_{0}^{1}dx-\frac{\left(\left(-m_{h}^{2}+m^{2}\xi+p^{2}\right)^{2}+4m_{h}^{2}p^{2}\right)\eta\left(m_{h}^{2},m^{2}\xi\right)}{4(d-1)p^{2}\left(m^{2}+p^{2}\right)^{2}}+\frac{\chi\left(m_{h}^{2}\right)\left(-m_{h}^{2}+m^{2}\xi+p^{2}\right)}{4(d-1)p^{2}\left(m^{2}+p^{2}\right)^{2}}$
(C.25) $\displaystyle+$
$\displaystyle\frac{\chi\left(m^{2}\xi\right)\left(m_{h}^{2}-m^{2}\xi+p^{2}\right)}{4(d-1)p^{2}\left(m^{2}+p^{2}\right)^{2}}.$

The eighth diagram is the mixed Goldstone-gauge field sunrise (third diagram
in the third line of Figure C.2):

$\displaystyle\Pi_{AA^{T},8}(p^{2})$ $\displaystyle=$ $\displaystyle
g^{2}\int_{0}^{1}dx\frac{\left(\left(m_{h}^{2}-m^{2}\xi+p^{2}\right)^{2}+4m^{2}\xi
p^{2}\right)\eta\left(m_{h}^{2},m^{2}\xi\right)}{4(D-1)p^{2}\left(m^{2}+p^{2}\right)^{2}}$
(C.26) $\displaystyle-$
$\displaystyle\frac{\eta\left(m^{2},m_{h}^{2}\right)\left(\left(m_{h}^{2}-m^{2}+p^{2}\right)^{2}-4(D-2)m^{2}p^{2}\right)}{4(D-1)p^{2}\left(m^{2}+p^{2}\right)^{2}}-\frac{m^{2}(\xi-1)\chi\left(m_{h}^{2}\right)}{4(D-1)p^{2}\left(m^{2}+p^{2}\right)^{2}}$
$\displaystyle-$
$\displaystyle\frac{\chi\left(m^{2}\xi\right)\left(m_{h}^{2}-m^{2}\xi+p^{2}\right)}{4(D-1)p^{2}\left(m^{2}+p^{2}\right)^{2}}+\frac{\chi\left(m^{2}\right)\left(m_{h}^{2}-m^{2}+p^{2}\right)}{4(D-1)p^{2}\left(m^{2}+p^{2}\right)^{2}}.$

Finally, we have four tadpole (balloon) diagrams. The Higgs boson balloon
(first diagram of the last line in Figure C.2):

$\displaystyle\Pi_{AA^{T},5}(p^{2})$ $\displaystyle=$
$\displaystyle\frac{3gm\chi\left(m_{h}^{2}\right)}{2v\left(m^{2}+p^{2}\right)^{2}},$
(C.27)

the Goldstone boson balloon (second diagram of the last line in Figure C.2):

$\displaystyle\Pi_{AA^{T},6}(p^{2})$ $\displaystyle=$
$\displaystyle\frac{3g\lambda
mv\chi\left(m^{2}\xi\right)}{2m_{h}^{2}\left(m^{2}+p^{2}\right)^{2}},$ (C.28)

The gauge field balloon (third diagram of the last line in Figure C.2):

$\displaystyle\Pi_{AA^{T},7}(p^{2})$ $\displaystyle=$
$\displaystyle\frac{3(D-1)g^{2}m^{2}\chi\left(m^{2}\right)}{2m_{h}^{2}\left(m^{2}+p^{2}\right)^{2}}+\frac{3g^{2}m^{2}\xi\chi\left(m^{2}\xi\right)}{2m_{h}^{2}\left(m^{2}+p^{2}\right)^{2}}$
(C.29)

and finally, the ghost balloon (fourth diagram of the last line in Figure
C.2):

$\displaystyle\Gamma_{AA^{T},8}(p^{2})$ $\displaystyle=$
$\displaystyle-\frac{3g^{2}m^{2}\xi\chi\left(m^{2}\xi\right)}{2m_{h}^{2}\left(m^{2}+p^{2}\right)^{2}}.$
(C.30)

Combining all these contributions (C.19)-(C.30), we find the total one-loop
correction to the gauge field self-energy

$\displaystyle\braket{A^{a}_{\mu}(p)A^{b}_{\nu}(p)}^{T}$ $\displaystyle=$
$\displaystyle\frac{\delta^{ab}}{p^{2}+m^{2}}+\delta^{ab}g^{2}\int_{0}^{1}dx\Big{\\{}-\frac{\left(2(3-2d)m^{2}p^{2}+m_{h}^{4}-2m_{h}^{2}\left(m^{2}-p^{2}\right)+m^{4}+p^{4}\right)}{4(d-1)p^{2}}\eta\left(m^{2},m_{h}^{2}\right)$
(C.31) $\displaystyle+$
$\displaystyle\frac{\left(m^{2}+p^{2}\right)^{2}\left(2m^{2}p^{2}(-2d+\xi+3)+m^{4}(\xi-1)^{2}+p^{4}\right)}{2(d-1)m^{4}p^{2}}\eta\left(m^{2},m^{2}\xi\right)$
$\displaystyle+$
$\displaystyle\frac{\left(m^{4}-p^{4}\right)\left(4m^{2}\xi+p^{2}\right)}{4(d-1)m^{4}}\eta\left(m^{2}\xi,m^{2}\xi\right)-\frac{\left(4m^{2}+p^{2}\right)\left(4(d-1)m^{4}+4(3-2d)m^{2}p^{2}+p^{4}\right)}{4(d-1)m^{4}}\eta\left(m^{2},m^{2}\right)$
$\displaystyle+$
$\displaystyle\frac{\left(m_{h}^{2}\left(-m^{2}p^{2}\left(8d^{2}-24d+4\xi+13\right)-2p^{4}(4d+\xi-7)+m^{4}(1-2\xi)\right)+6(d-1)^{2}m^{4}p^{2}+m_{h}^{4}m^{2}\right)}{4(d-1)m_{h}^{2}m^{2}p^{2}}\chi\left(m^{2}\right)$
$\displaystyle-$
$\displaystyle\frac{\left((d-2)p^{2}+m_{h}^{2}-m^{2}\right)}{4(d-1)p^{2}}\chi\left(m_{h}^{2}\right)+\frac{\left(m^{2}p^{2}(5d+4\xi-13)+2p^{4}(4d+\xi-7)+2m^{4}(\xi-1)\right)}{4(d-1)m^{2}p^{2}}\chi\left(m^{2}\xi\right)$
$\displaystyle+$
$\displaystyle\frac{3}{4}\chi\left(m_{h}^{2}\right)+\frac{3}{4}\chi\left(m^{2}\xi\right)\Big{\\}}\frac{1}{(p^{2}+m^{2})^{2}}$

and the resummed propagator for the transverse gauge field can be
approximated, at one-loop order, by

$\displaystyle G^{-1}_{AA^{T}}$ $\displaystyle=$
$\displaystyle\delta^{ab}\Bigg{(}p^{2}+m^{2}-g^{2}\int_{0}^{1}dx\Big{\\{}-\frac{\left(2(3-2d)m^{2}p^{2}+m_{h}^{4}-2m_{h}^{2}\left(m^{2}-p^{2}\right)+m^{4}+p^{4}\right)}{4(d-1)p^{2}}\eta\left(m^{2},m_{h}^{2}\right)$
(C.32) $\displaystyle+$
$\displaystyle\frac{\left(m^{2}+p^{2}\right)^{2}\left(2m^{2}p^{2}(-2d+\xi+3)+m^{4}(\xi-1)^{2}+p^{4}\right)}{2(d-1)m^{4}p^{2}}\eta\left(m^{2},m^{2}\xi\right)$
$\displaystyle+$
$\displaystyle\frac{\left(m^{4}-p^{4}\right)\left(4m^{2}\xi+p^{2}\right)}{4(d-1)m^{4}}\eta\left(m^{2}\xi,m^{2}\xi\right)-\frac{\left(4m^{2}+p^{2}\right)\left(4(d-1)m^{4}+4(3-2d)m^{2}p^{2}+p^{4}\right)}{4(d-1)m^{4}}\eta\left(m^{2},m^{2}\right)$
$\displaystyle+$
$\displaystyle\frac{\left(m_{h}^{2}\left(-m^{2}p^{2}\left(8d^{2}-24d+4\xi+13\right)-2p^{4}(4d+\xi-7)+m^{4}(1-2\xi)\right)+6(d-1)^{2}m^{4}p^{2}+m_{h}^{4}m^{2}\right)}{4(d-1)m_{h}^{2}m^{2}p^{2}}\chi\left(m^{2}\right)$
$\displaystyle-$
$\displaystyle\frac{\left((d-2)p^{2}+m_{h}^{2}-m^{2}\right)}{4(d-1)p^{2}}\chi\left(m_{h}^{2}\right)+\frac{\left(m^{2}p^{2}(5d+4\xi-13)+2p^{4}(4d+\xi-7)+2m^{4}(\xi-1)\right)}{4(d-1)m^{2}p^{2}}\chi\left(m^{2}\xi\right)$
$\displaystyle+$
$\displaystyle\frac{3}{4}\chi\left(m_{h}^{2}\right)+\frac{3}{4}\chi\left(m^{2}\xi\right)\Big{\\}}\Bigg{)}.$

For $d=4-\epsilon$, following the procedure of dimensional regularization, we
find that the divergent part for $G_{AA^{T}}(p^{2})$ is given by:

$\displaystyle G_{AA^{T},\mathrm{d}iv}(p^{2})$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{\pi^{2}\epsilon}\left(-\frac{9m^{4}}{16m_{h}^{2}}-\frac{3m_{h}^{2}}{32}-\frac{m^{2}\xi}{8}-\frac{3m^{2}}{32}-\frac{\xi
p^{2}}{8}+\frac{25p^{2}}{48}\right),$ (C.33)

and these terms can be, following the $\overline{MS}bar$-scheme, absorbed by
means of appropriate counterterms. We remain with the finite part of the
propagator, given in eq. (5.31).

###  C.3 Contributions to $\langle O(p)O(-p)\rangle$

The diagrams which contribute to the correlation function $\langle
O(p)O(-p)\rangle$ are depicted in Figure(C.3)

Figure C.3: One-loop contributions to the correlation function $\langle
OO\rangle$. Wavy lines represent the gauge field, dashed lines the Higgs
field, solid lines the Goldstone boson and double lines the ghost field. The
$\bullet$ indicates the insertion of a composite operator.

.

The first term is $v^{2}$ times the one-loop correction to the Higgs
propagator, given in eq. (C.17). The second term is

$\displaystyle v\langle h(p)(\rho^{a}\rho^{a})(-p)\rangle$ $\displaystyle=$
$\displaystyle-3\frac{m_{h}^{2}\eta\left(m^{2}\xi,m^{2}\xi\right)}{m_{h}^{2}+p^{2}}.$
(C.34)

The third term is

$\displaystyle v\langle h(p)h^{2}(-p)\rangle$ $\displaystyle=$
$\displaystyle-\frac{3m_{h}^{2}\eta\left(m_{h}^{2},m_{h}^{2}\right)}{m_{h}^{2}+p^{2}}-\frac{3\chi\left(m_{h}^{2}\right)}{m_{h}^{2}+p^{2}}-\frac{\chi\left(m^{2}\xi\right)}{m_{h}^{2}+p^{2}}-\frac{2(D-1)m^{2}\chi\left(m^{2}\right)}{m_{h}^{2}\left(m_{h}^{2}+p^{2}\right)}-\frac{2m^{2}\xi\chi\left(m^{2}\xi\right)}{m_{h}^{2}\left(m_{h}^{2}+p^{2}\right)}$
(C.35) $\displaystyle+$
$\displaystyle\frac{2m^{2}\xi\chi\left(m^{2}\xi\right)}{m_{h}^{2}\left(m_{h}^{2}+p^{2}\right)}.$

The fourth term is

$\displaystyle\langle m_{h}^{2}(p)m_{h}^{2}(-p)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\eta\left(m_{h}^{2},m_{h}^{2}\right).$ (C.36)

The fifth term is

$\displaystyle\langle(\rho^{a}\rho^{a})(p)(\rho^{b}\rho^{b})(-p)\rangle$
$\displaystyle=$ $\displaystyle\frac{3}{2}\eta\left(\xi m^{2},\xi
m^{2}\right)$ (C.37)

and together these terms give the correlation function of the scalar composite
operator $O$ up to first order in $\hbar$

$\displaystyle\langle O(p)O(-p)\rangle$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{p^{2}+m_{h}^{2}}+\int_{0}^{1}dx\Big{\\{}\frac{3}{2}\eta\left(m^{2},m^{2}\right)(4(d-1)m^{4}+4m^{2}p^{2}+p^{4})+\frac{1}{2}(p^{2}-2m_{h}^{2})^{2}\eta\left(m_{h}^{2},m_{h}^{2}\right)$
(C.38) $\displaystyle-$
$\displaystyle\frac{3p^{2}\chi(m^{2})(2(d-1)m^{2}+m_{h}^{2})}{m_{h}^{2}}-3p^{2}\chi(m_{h}^{2})\Big{\\}}\frac{1}{(p^{2}+m_{h}^{2})^{2}}.$

Thus, for the one-loop resummed correlation function, we get

$\displaystyle G^{-1}_{OO}(p^{2})$ $\displaystyle=$
$\displaystyle\frac{p^{2}+m_{h}^{2}}{v^{2}}-\frac{1}{v^{4}}\int_{0}^{1}dx\Big{\\{}\frac{3}{2}\eta\left(m^{2},m^{2}\right)(4(d-1)m^{4}+4m^{2}p^{2}+p^{4})+\frac{1}{2}(p^{2}-2m_{h}^{2})^{2}\eta\left(m_{h}^{2},m_{h}^{2}\right)$
(C.39) $\displaystyle-$
$\displaystyle\frac{3p^{2}\chi(m^{2})(2(d-1)m^{2}+m_{h}^{2})}{m_{h}^{2}}-3p^{2}\chi(m_{h}^{2})\Big{\\}}\frac{1}{(p^{2}+m_{h}^{2})^{2}}.$

Following the procedure of dimensional regularization for $d=4-\epsilon$, we
find that the divergent part of the correlator is given by

$\displaystyle G_{OO,\mathrm{d}iv}^{-1}$ $\displaystyle=$
$\displaystyle\frac{1}{4v^{4}\pi^{2}\epsilon}\Big{(}\frac{9g^{4}p^{2}v^{2}}{16\lambda}+\frac{9g^{4}v^{4}}{16}+\frac{9}{8}g^{2}p^{2}v^{2}+p^{4}+\frac{1}{2}\lambda
p^{2}v^{2}+\lambda^{2}v^{4}\Big{)},$ (C.40)

which can be accounted for by appropriate counterterm, following the
$\overline{MS}bar$-scheme renormalization procedure. We remain with the finite
part of the correlator, given in eq. (5.76).

###  C.4 A few comments on the unitary gauge

It is well-known that in the unitary gauge the unphysical fields, like the
Goldstone and ghost fields, decouple, a feature which allows for a more direct
link with the spectrum of the elementary excitations of the model. However,
this gauge is known to be non-renormalizable. In fact, working directly with
the elementary tree level propagators taken already in the unitary limit, i.e.
$\xi\rightarrow\infty$, and following the steps of dimensional regularization,
we find that the divergent part of the Higgs propagator reads

$\displaystyle G^{-1}_{hh,\mathrm{d}iv}(p^{2})$ $\displaystyle=$
$\displaystyle\frac{3g^{2}\left(m_{h}^{4}+6m^{2}p^{2}+p^{4}\right)}{64\pi^{2}m^{2}\epsilon}.$
(C.41)

In expression (C.41) we clearly see the presence of the term
$\sim\frac{p^{4}}{\epsilon m^{2}}$, signalling the aforementioned issue of the
non-renormalizability. Nevertheless, it is interesting to observe that, if we
remove the divergent part (C.41) anyway, we obtain the spectral function as
shown in Figure C.4. This spectral function is almost identical to that
obtained for the composite operator $O(x)$, see Figure 5.8.

Figure C.4: Spectral function for the propagator $\langle h(p)h(-p)\rangle$ in
the unitary gauge, with $t$ given in unity of $\mu^{2}$, for the Region I
(left) and Region II (right), with parameter values given in Table IV A.

For the gauge field propagator, proceeding in the same way, we find the
divergent part

$\displaystyle G^{-1}_{AA,\mathrm{d}iv}(p^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{\epsilon}\Big{(}-\frac{9g^{2}m^{4}}{16\pi^{2}m_{h}^{2}}-\frac{g^{2}p^{6}}{96\pi^{2}m^{4}}+\frac{7g^{2}p^{4}}{48\pi^{2}m^{2}}+\frac{3g^{2}m^{2}}{32\pi^{2}}+\frac{83g^{2}p^{2}}{96\pi^{2}}-\frac{3\lambda
m^{2}}{8\pi^{2}}\Big{)}$ (C.42)

which shows again the non-renormalizability of unitary gauge, through the
terms $\sim\frac{p^{6}}{\epsilon m^{4}}$ and $\sim\frac{p^{4}}{\epsilon
m^{2}}$ . However, if we remove again those terms anyway, we obtain the
spectral function as shown in Figure C.5. Nevertheless, as already remarked in
the previous sections, this nice behaviour of the spectral densities for the
Higgs and gauge field obtained by a direct use of the tree level propagators
already taken in the unitary limit, $\xi\rightarrow\infty$, can be, to some
extent, justified by the fact that we are working at the one-loop order in
perturbation theory. Since overlapping divergences start from one-loop onward,
we can easily figure out that the naive use of the elementary tree level
propagators taken already in the unitary limit will run into severe non-
renormalizibility issues, making the removal of the (overlapping) divergent
parts (C.41), (C.42) quite problematic.

Figure C.5: Spectral function for the propagator $\langle
A^{a}_{\mu}(p)A^{b}_{\nu}(-p)\rangle$ in the unitary gauge, with $t$ given in
unity of $\mu^{2}$, for the Region I (left) and Region II (right), with
parameter values given in Table IV A.

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