Doppler shift in current-carrying superfluid

Consider the Hamiltonian ($x=(vec{r},it)=(vec{r},tau)$):
$$hat{H}=hat{psi}_{sigma}^{dagger}(x)(-frac{nabla^2}{2m}-mu)hat{psi}_{sigma}(x)-U_{0}int dvec{r}hat{psi}^{dagger}_{uparrow}(x)hat{psi}^{dagger}_{downarrow}(x)hat{psi}(x)_{downarrow}hat{psi}_{uparrow}(x)tag{1}$$

After some manipulations, with $Psi$ defined as Numbu spinor $(psi_{uparrow}, overline{psi}_{downarrow})^{T}$, the partition function and effective action are respectively:
$$Z=intmathcal[Delta,Delta^{*}]exp{-S}
S=int^{beta}_{0} dtauint dvec{r}frac{|Delta(x)|^{2}}{U_{0}}-text{Tr}{ln{[-G^{-1}]}}tag{2}$$
and
$$G^{-1}(x,x’)=begin{pmatrix}
G^{-1}_{0uparrow}(x,x’) &Delta(x)
Delta^{*}(x) &-G^{-1}_{0,downarrow}(x’,x)
end{pmatrix}tag{3}$$

where $Delta(x)$ is an introduced Boson field and $G_{0,sigma}(x,x’)equiv -⟨psi_{sigma}overline{psi}_{sigma’}(x’)⟩$ is the Green’s function in noninteracting system.

To study current-carrying superfluid, we need to impose phase twist on $Delta(x)$:$Delta(x)rightarrow e^{ivec{Q}vec{r}}Delta(x)$

Then the current-carrying effective action (1) is modified:

$$S=int^{beta}_{0} dtauint dvec{r}frac{|Delta(x)|^{2}}{U_{0}}-text{Tr}{ln{[-widetilde{G}^{-1}]}}tag{5}$$
and $widetilde{G}^{-1}$ is
$$widetilde{G}^{-1}=begin{pmatrix}
-partial_{tau}-frac{(hat{p}-vec{Q/2})^{2}}{2m}+mu &Delta(x)
Delta^{*}(x) &-partial_{tau}+frac{(hat{p}+vec{Q/2})^{2}}{2m}-mu
end{pmatrix}tag{6}$$

The fourier transform of $widetilde{G}^{-1}$ after adding convergence factor, in some reference, for example, https://arxiv.org/abs/0711.0561
is:
$$widetilde{G}^{-1}(k)=begin{pmatrix}
(iomega_{n}-frac{vec{k}vec{Q}}{m})e^{-iomega_{n}0^{+}}-(frac{vec{k}^{2}}{2m}+frac{vec{Q}^{2}}{2m}-mu) &Delta_{0}
Delta_{0} &(iomega_{n}-frac{vec{k}vec{Q}}{m})e^{-iomega_{n}0^{+}}-(frac{vec{k}^{2}}{2m}+frac{vec{Q}^{2}}{2m}-mu)
end{pmatrix}$$

My question is why should we put $frac{vec{k}vec{Q}}{m}$ into parentheses before convergence factor and $frac{vec{Q}^{2}}{2m}$ not? Could we have other forms, like
$$widetilde{G}^{-1}(k)=begin{pmatrix}
iomega_{n}e^{-iomega_{n}0^{+}}-(frac{vec{k}^{2}}{2m}+frac{vec{Q}^{2}}{2m}-mu+frac{vec{k}vec{Q}}{m}) &Delta_{0}
Delta_{0} &iomega_{n}e^{-iomega_{n}0^{+}}-(frac{vec{k}^{2}}{2m}+frac{vec{Q}^{2}}{2m}+frac{vec{k}vec{Q}}{m}-mu)
end{pmatrix}$$