the commutator is bounded and due to its commutation with Moyal multiplication, order-one condition is fulfilled. Now we introduce a very useful relation connected to the heat kernel type expansion associated to a regular spectral triple taken from [@non-com-tri]. This relation will be used later in order to compute the spectral action. Considering a regular non-unital spectral triple $(\mathcal{A},\mathcal{H},\mathcal{D},J)$ and two pseudo-differential operator $A_0 \in \Psi_0(\mathcal{A}) $ $A_1\in \Psi_1(\mathcal{A})$ of order respectively 0 and 1. We consider the following decomposition: $$e^{-t(\mathcal{D}^2+A_0+A_1)}=\sum_{j=0}^4(-1)^jE_j(t)-t^5R$$ Using Duhamel principle [@Duhamel] $$e^{-t(A+B)}= e^{-tA}-t\int_0^1 ds \ e^{-st(A+B)}Be^{-(1-s)tA}$$ we can identify: $$\begin{aligned}
E_0(t)&=&e^{-t\mathcal{D}^2}\nonumber \\ \ 
E_j(t)&=&\int_{\Delta_j} d^js \ e^{-s_1t\mathcal{D}^2}(A_0+A_1)e^{-(s_2-s_1)t\mathcal{D}^2}\cdots(A_0+A_1)e^{-(1-s_j)t\mathcal{D}^2} \nonumber \\ \ 
&&\end{aligned}$$ and $$\begin{aligned}
&&R=\int_{\Delta_5}ds_1ds_2ds_3ds_4ds_5 e^{-s_1t(\mathcal{D}^2+A_0+A_1)}(A_0+A_1)e^{-(s_2-s_1)t\mathcal{D}^2}(A_0+A_1)    \nonumber\\
&& \times e^{-(s_3-s_2)t\mathcal{D}^2}(A_0+A_1)e^{-(s_4-s_3)t\mathcal{D}^2}(A_0+A_1)e^{-(s_5-s_4)t\mathcal{D}^2}(A_0+A_1)e^{-(1-s_5)t\mathcal{D}^2}\nonumber\\
&&\end{aligned}$$ The domains of the integrals $\Delta_j$ are the $j$-simplex: $$\Delta_j=\{s\in\mathbb{R}^j;0\leq s_1\leq s_2\leq s_1\leq\cdots\leq s_j\leq 1\} \simeq \{s\in\mathbb{R}^{j+1};s_i\geq 0,\sum_{i=0}^j s_i=1\}$$ Taking in account the relation: $$\left[ e^{-t\mathcal{D}^2},A \right]=\int_0^1 ds \frac{d}{ds}\left(e^{-ts\mathcal{D}^2}Ae^{-t(1-s)\mathcal{D}^2}\right) =-t\int_0^1 ds e^{-ts\mathcal{D}^2}\left[\mathcal{D}^2,A\right]e^{-t(1-s)\mathcal{D}^2}$$

and considering the trace in [@non-com-tri] is computed the leading term of $e^{-t(\mathcal{D}^2+A_0+A_1)}$ for $t\to 0$: $$\begin{aligned}
&&\operatorname{Tr}(e^{-t(\mathcal{D}^2+A_0+A_1)})  \nonumber \\
&&=\textrm{Tr}\Big(\Big\{1-t(A_0+A_1)+\frac{t^2}{2}(A^2_0+A_1A_0+A_0A_1+A^2_1) \nonumber \\
&&-\frac{t^3}{6}(A_0\left[\mathcal{D}^2,A_1\right]+A_1\left[\mathcal{D}^2,A_0\right]+A_1\left[\mathcal{D}^2,A_1\right]+A_0A^2_1+A_1A_0A_1+A_1^2A_0\nonumber \\
&&+A^3_1)+\frac{t^4}{24}(A_1\left[\mathcal{D}^2 \left[\mathcal{D}^2,A_0\right]\right]+2A^2_1\left[\mathcal{D}^2 ,A_1\right]+A_1\left[\mathcal{D}^2 ,A_1\right]A_1+A_1^4\Big)e^{-t\mathcal{D}^2}\Big\}\nonumber \\
&&+\mathcal{O}(\sqrt{t}) \label{heat1}\end{aligned}$$

4-dimensional harmonic Yang-Mills model
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Following the Connes-Lott models, in order to implement the Higgs mechanism, we consider the total spectral triple 