Quantum parity conservation in planar quantum electrodynamics

The action $\Sigma^{(s-1)}$ (1) is invariant under the Becchi-Rouet-Stora (BRS) transformations brs :

	$\displaystyle s\psi_{+}=i(c+\xi)\psi_{+}~{},~{}~{}s\overline{\psi}_{+}=-i(c+\xi)\overline{\psi}_{+}~{};$
	$\displaystyle s\psi_{-}=i(c-\xi)\psi_{-}~{},~{}~{}s\overline{\psi}_{-}=-i(c-\xi)\overline{\psi}_{-}~{};$
	$\displaystyle\displaystyle sA_{\mu}=-\frac{1}{e}\partial_{\mu}c~{},~{}~{}sc=0~{};~{}~{}\displaystyle sa_{\mu}=-\frac{1}{g}\partial_{\mu}\xi~{},~{}~{}s\xi=0~{};$
	$\displaystyle\displaystyle s\overline{c}=\frac{b}{e}~{},~{}~{}sb=0~{};~{}~{}\displaystyle s\overline{\xi}=\frac{\pi}{g}~{},~{}~{}s\pi=0~{};$		(3)

as well as under the parity transformations:

	$\displaystyle\psi_{+}\stackrel{{\scriptstyle{P}}}{{\longrightarrow}}\psi_{+}^{P}=-i\gamma^{1}\psi_{-}~{},~{}~{}\psi_{-}\stackrel{{\scriptstyle{P}}}{{\longrightarrow}}\psi_{-}^{P}=-i\gamma^{1}\psi_{+}~{},~{}~{}\overline{\psi}_{+}\stackrel{{\scriptstyle{P}}}{{\longrightarrow}}\overline{\psi}_{+}^{P}=i\overline{\psi}_{-}\gamma^{1}~{},~{}~{}\overline{\psi}_{-}\stackrel{{\scriptstyle{P}}}{{\longrightarrow}}\overline{\psi}_{-}^{P}=i\overline{\psi}_{+}\gamma^{1}~{};$
	$\displaystyle A_{\mu}\stackrel{{\scriptstyle{P}}}{{\longrightarrow}}A_{\mu}^{P}=(A_{0},-A_{1},A_{2})~{};~{}~{}\phi\stackrel{{\scriptstyle{P}}}{{\longrightarrow}}\phi^{P}=\phi~{},~{}~{}\phi=\{b,c,\overline{c}\}~{};$
	$\displaystyle a_{\mu}\stackrel{{\scriptstyle{P}}}{{\longrightarrow}}a_{\mu}^{P}=(-a_{0},a_{1},-a_{2})~{};~{}~{}\chi\stackrel{{\scriptstyle{P}}}{{\longrightarrow}}\chi^{P}=-\chi~{},~{}~{}\chi=\{\pi,\xi,\overline{\xi}\}~{}.$		(4)

The tree-level propagators are obtained by taking the free part of the action $\Sigma^{(s-1)}$ (1), i.e., by switching off the coupling constants $e$ and $g$, thence the propagators in momenta space read:

	$\displaystyle\Delta^{\mu\nu}_{AA}(k)=-i\bigg{\{}\frac{1}{k^{2}-\mu^{2}}\left(\eta^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{k^{2}}\right)+\frac{\alpha}{k^{2}}\frac{k^{\mu}k^{\nu}}{k^{2}}\Bigg{\}}~{},~{}~{}\Delta^{\mu\nu}_{aa}(k)=-i\Bigg{\{}\frac{1}{k^{2}-\mu^{2}}\left(\eta^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{k^{2}}\right)+\frac{\beta}{k^{2}}\frac{k^{\mu}k^{\nu}}{k^{2}}\Bigg{\}}~{},$
	$\displaystyle\Delta_{Aa}^{\mu\nu}(k)=\frac{\mu}{k^{2}(k^{2}-\mu^{2})}\epsilon^{\mu\alpha\nu}k_{\alpha}~{},~{}~{}\Delta_{Ab}^{\mu}(k)=\Delta_{a\pi}^{\mu}(k)=\frac{k^{\mu}}{k^{2}}~{},~{}~{}\Delta_{bb}(k)=\Delta_{\pi\pi}(k)=0~{},~{}~{}\Delta_{\overline{c}c}(k)=\Delta_{\overline{\xi}\xi}(k)=-\frac{i}{k^{2}}~{},$
	$\displaystyle\Delta_{++}(k)=i\frac{{\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}k}-m(s-1)}{k^{2}-m^{2}(s-1)^{2}}~{},~{}\Delta_{--}(k)=i\frac{{\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}k}+m(s-1)}{k^{2}-m^{2}(s-1)^{2}}~{}.$		(5)

Notice that from this point forward, all 1- and 2-loops Feynman graphs calculations will be performed in the Landau gauge, $\alpha=\beta=0$.

The graphical conventions for the propagators are assumed as below: {fmffile}feynmanrules

$$\Delta_{AA}^{\mu\nu}\equiv\parbox{40.0pt}{ \fmfgraph*(13,10) \fmfleft{i} \fmfright{o} \fmf{photon}{i,o} }\quad,\quad\Delta_{aa}^{\mu\nu}\equiv\parbox{40.0pt}{ \fmfgraph*(13,10) \fmfleft{i} \fmfright{o} \fmf{gluon}{o,i} }\quad,\quad\Delta_{Aa}^{\mu\nu}\equiv\parbox{40.0pt}{ \fmfgraph*(13,10) \fmfleft{i} \fmfright{o} \fmf{photon}{i,v} \fmf{gluon}{o,v} }\quad,\quad\Delta_{\pm\pm}\equiv\parbox{40.0pt}{ \fmfgraph*(13,10) \fmfleft{i} \fmfright{o} \fmf{plain}{i,o} }\quad,$$

(6)

and the Feynman rules for the interaction vertices are given by: {fmffile}vertexrules

$$V_{\pm A^{\mu}\pm}\equiv\quad\parbox{40.0pt}{ \fmfgraph*(15,15) \fmfbottom{i,o} \fmftop{u} \fmf{plain}{i,v} \fmf{plain}{o,v} \fmf{photon}{v,u} \fmfv{decor.shape=circle,decor.filled=full,decor.size=1.3thick,label=$ie\gamma^{\mu}$,label.angle=130,label.dist=.2cm}{v} }\quad\quad,\quad\quad V_{\pm a^{\mu}\pm}\equiv\quad\quad\parbox{40.0pt}{ \fmfgraph*(15,15) \fmfbottom{i,o} \fmftop{u} \fmf{plain}{i,v} \fmf{plain}{o,v} \fmf{gluon}{v,u} \fmfv{decor.shape=circle,decor.filled=full,decor.size=1.3thick,label=$\pm ig\gamma^{\mu}$,label.angle=150,label.dist=.3cm}{v} }\quad\quad.$$

(7)

For the purpose of renormalizing the ultraviolet (UV) and infrared (IR) divergences of all divergent graphs, UV and IR subtraction degrees have to be fixed, to do so the UV and IR dimensions of all the fields shall be determined firstly. For any propagator $\Delta_{XY}(k,s)$, the UV ($d$) and IR ($r$) dimensions of the fields, $X$ and $Y$, are defined by means of the asymptotical UV and IR behaviour of the propagator, $d_{XY}$ (for $k,s\rightarrow\infty$) and $r_{XY}$ (for $k,(s-1)\rightarrow 0$), respectively, furthermore the following inequalities hold BPHZ :

$$d_{X}+d_{Y}\geq 3+d_{XY}\;\;\;\mbox{and}\;\;\;r_{X}+r_{Y}\leq 3+r_{XY}~{},$$

(8)

where, in the Landau gauge, $\alpha=\beta=0$, the UV ($d$) and IR ($r$) dimensions of all the fields are summarized in the Table 1. Thus, by taking into account all previous results, the UV ($d(\gamma)$) and IR ($r(\gamma)$) superficial degrees of divergence of a 1-particle irreducible Feynman diagram $\gamma$ stems:

$$\bordermatrix{&\cr&d(\gamma)\cr&r(\gamma)}=3-\sum\limits_{f}\bordermatrix{&\cr&d_{f}\cr&r_{f}}N_{f}-\sum\limits_{b}\bordermatrix{&\cr&d_{b}\cr&\frac{3}{2}r_{b}}N_{b}+\bordermatrix{&\cr&-\cr&+}\frac{1}{2}N_{e}+\bordermatrix{&\cr&-\cr&+}\frac{1}{2}N_{g}-N_{Aa}~{},$$

(9)

where $N_{f}$ and $N_{b}$ are the numbers of external lines of fermions and bosons, respectively, whereas $N_{Aa}$ is the number of internal lines associated to the mixed propagator $\Delta_{Aa}$. Also, $N_{e}$ and $N_{g}$ are the powers of the coupling constants, $e$ and $g$, in the integral corresponding to the graph $\gamma$.

The 1-loop vacuum-polarization tensors, self energies and vertex functions diagrams are identified in Fig. 1, whereas their respectives UV and IR superficial degrees of divergence are displayed in Table 2. At this time, it should be mentioned that for any graph $\gamma_{i_{\pm}}$ the subscript $\pm$ refers to external legs or internal lines of either $\psi_{+}$ or $\psi_{-}$.

Accordingly to the power counting theorem, in view of the fact that the 1-loop diagrams $\gamma_{1_{\pm}}$, $\gamma_{2_{\pm}}$, $\gamma_{3_{\pm}}$, $\gamma_{5_{\pm}}$ and $\gamma_{6_{\pm}}$ (see Fig. 1 and Table 2) are superficially UV divergent, they have to be UV and IR subtracted. Whenever a graph $\gamma$ is possibly UV divergent, i.e. $d(\gamma)\geq 0$, the BPHZL renormalization method is followed so as to make the graph convergent lz by also subtracting the IR divergences induced by the UV subtractions. The BPHZL subtraction program consists of performing UV and IR subtraction operations upon a UV divergent Feynman graph integrand, $I_{\gamma}(p,k,s)$:

$$R_{\gamma}(p,k,s)=\left(1-t^{\rho(\gamma)-1}_{p,s-1}\right)\left(1-t^{\delta(\gamma)}_{p,s}\right)I_{\gamma}(p,k,s)~{},$$

(10)

where $R_{\gamma}(p,k,s)$ is the renormalized integrand, which is UV convergent. Moreover, $\delta(\gamma)$ and $\rho(\gamma)$ are the UV and IR degrees of subtraction, respectively, given by lz :

$$\delta(\gamma)=d(\gamma)+b(\gamma)\;\;\;\mbox{and}\;\;\;\rho(\gamma)=r(\gamma)-c(\gamma)~{},$$

(11)

where at 1-loop $b(\gamma)$ and $c(\gamma)$ are non-negative integers constrained as follows:

$$\rho(\gamma)\leq\delta(\gamma)+1~{},$$

(12)

with $t^{\tau}_{x,y}$ being the Taylor expansion operator about $x=y=0$ to order $\tau$, provided $\tau\geq 0$.

The BPHZL renormalization procedures of all 1-particle irreducible vacuum-polarization tensor divergent diagrams (see Fig. 1 and Table 2) are rather similar, since they possess the same loop structure, their integrands are equal up to coupling constants dependent factors, $\pm e^{2}$, $\pm g^{2}$ and $\mp eg$, corresponding to the 1-loop graphs, $\gamma_{1_{\pm}}$, $\gamma_{2_{\pm}}$ and $\gamma_{3_{\pm}}$, respectively. Initially, the analysis is carried out for the $\gamma_{1_{\pm}}$ Feynman graphs, where the 1-loop vacuum-polarization tensor, $\Pi_{\gamma_{1_{\pm}}}^{\mu\nu}(p,s)$, reads

$\displaystyle\Pi_{\gamma_{1_{\pm}}}^{\mu\nu}(p,s)=\int{\frac{d^{3}k}{(2\pi)^{3}}}\underbrace{\left\{-e^{2}{\rm Tr}\left[\gamma^{\mu}\frac{{\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}k}\mp m(s-1)}{k^{2}-m^{2}(s-1)^{2}}\gamma^{\nu}\frac{{\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}k-\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}p}\mp m(s-1)}{(k-p)^{2}-m^{2}(s-1)^{2}}\right]\right\}}_{I_{\gamma_{1_{\pm}}}^{\mu\nu}(p,k,s)}~{}.$

(13)

Bearing in mind the conditions (11) and the inequality (12), by taking $b(\gamma_{1_{\pm}})=c(\gamma_{1_{\pm}})=0$, the UV and IR subtraction degrees are such that $\delta(\gamma_{1_{\pm}})=\rho(\gamma_{1_{\pm}})=1$. Consequently, the 1-loop BPHZL subtracted (renormalized) integrand, $R_{\gamma_{1_{\pm}}}^{\mu\nu}(p,k,s)$, is written in terms of the unsubtracted one, $I_{\gamma_{1_{\pm}}}^{\mu\nu}(p,k,s)$, in the following way:

	$\displaystyle R_{\gamma_{1_{\pm}}}^{\mu\nu}(p,k,s)$	$\displaystyle=$	$\displaystyle(1-t^{0}_{p,s-1})(1-t^{1}_{p,s})I_{\gamma_{1_{\pm}}}^{\mu\nu}(p,k,s)$		(14)
		$\displaystyle=$	$\displaystyle(1-t^{0}_{p,s-1}-t^{1}_{p,s}+t^{0}_{p,s-1}t^{1}_{p,s})I_{\gamma_{1_{\pm}}}^{\mu\nu}(p,k,s)~{}.$

However, as previously mentioned by setting $s=1$ at the end of all Taylor expansion operations, to retrieve the massless condition, the subtracted integrand, $R_{\gamma_{1_{\pm}}}^{\mu\nu}(p,k,1)$, results:

$$R_{\gamma_{1_{\pm}}}^{\mu\nu}(p,k,1)=\underbrace{I_{\gamma_{1_{\pm}}}^{\mu\nu}(p,k,1)}_{\rm parity-even}-\underbrace{I_{\gamma_{1_{\pm}}}^{\mu\nu}(0,k,1)}_{\rm parity-even}-\underbrace{p^{\rho}\frac{\partial}{\partial p^{\rho}}I_{\gamma_{1_{\pm}}}^{\mu\nu}(p,k,s)}_{\rm parity-odd}\Bigg{|}_{p=s=0}~{},$$

(15)

where

	$\displaystyle I_{\gamma_{1_{\pm}}}^{\mu\nu}(p,k,1)=-e^{2}{\rm Tr}~{}\left\{\gamma^{\mu}\frac{{\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}k}}{k^{2}}\gamma^{\nu}\frac{{\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}k-\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}p}}{(k-p)^{2}}\right\}~{},~{}~{}I_{\gamma_{1_{\pm}}}^{\mu\nu}(0,k,1)=-e^{2}{\rm Tr}~{}\left\{\gamma^{\mu}\frac{{\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}k}}{k^{2}}\gamma^{\nu}\frac{{\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}k}}{k^{2}}\right\}~{},$		(16)
	$\displaystyle p^{\rho}\frac{\partial}{\partial p^{\rho}}I_{\gamma_{1_{\pm}}}^{\mu\nu}(p,k,s)\Bigg{|}_{p=s=0}=-e^{2}{\rm Tr}~{}\left\{\gamma^{\mu}\frac{{\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}k}\mp m}{k^{2}-m^{2}}\gamma^{\nu}\left[-\frac{\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}p}{k^{2}-m^{2}}+2p\cdot k\frac{{\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}k}\mp m}{(k^{2}-m^{2})^{2}}\right]\right\}~{}.$		(17)

In addition to that, since the renormalized vacuum-polarization tensor, $\Pi_{\gamma_{1_{\pm}}}^{(R)\mu\nu}(p,s)$, is defined by

$$\Pi_{\gamma_{1_{\pm}}}^{(R)\mu\nu}(p,s)=\int\frac{d^{3}k}{(2\pi)^{3}}~{}R^{\mu\nu}_{\gamma_{1_{\pm}}}(p,k,s)~{},$$

(18)

and recalling to the fact that the issue here is to verify if Levi-Civita symbol $\epsilon^{\mu\nu\rho}$ dependent terms might be induced by UV and IR subtractions, only parity-odd pieces of the subtracted integrand, $R_{{\rm odd}\gamma_{1_{\pm}}}^{\mu\nu}(p,k,1)$ (15), shall be taken into account, then from the Eqs.(15)–(17), leads to

$$\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{1_{\pm}}}=\pm\frac{e^{2}m}{4\pi|m|}\epsilon^{\mu\alpha\nu}p_{\alpha}~{},$$

(19)

with $\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{1_{\pm}}}\equiv\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{1_{\pm}}}(p,1)$.

Analogously to the previous case, $\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{1_{\pm}}}$, the renormalized parity-odd vacuum-polarization tensors $\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{2_{\pm}}}$ and $\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{3_{\pm}}}$, corresponding to $\gamma_{2_{\pm}}$ and $\gamma_{3_{\pm}}$ diagrams, are respectively given by

$$\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{2_{\pm}}}=\pm\frac{g^{2}m}{4\pi|m|}\epsilon^{\mu\alpha\nu}p_{\alpha}\;\;\;\mbox{and}\;\;\;\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{3_{\pm}}}=\mp\frac{egm}{4\pi|m|}\epsilon^{\mu\alpha\nu}p_{\alpha}~{}.$$

(20)

Finally, the 1-loop renormalized parity-odd vacuum polarization tensors, $\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{1}}$, $\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{2}}$ and $\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{3}}$ :

$$\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{1}}=\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{1_{+}}}+\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{1_{-}}}\equiv 0~{},~{}~{}\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{2}}=\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{2_{+}}}+\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{2_{-}}}\equiv 0~{},~{}~{}\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{3}}=\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{3_{+}}}+\Pi^{(R)\mu\nu}_{{\rm odd}\gamma_{3_{-}}}\equiv 0~{},$$

(21)

vanishes identically. In conclusion, besides there is no 1-loop counterterm for the mixed Chern-Simons term, $\varepsilon^{\mu\alpha\nu}A_{\mu}\partial_{\alpha}a_{\nu}$ – which sets out that the 1-loop $\beta$-function associated to the Chern-Simons mass parameter ($\mu$) vanishes – the BPHZL subtraction scheme applied to the 1-loop vacuum-polarization tensor preserves parity, being the opposite to what takes place in ordinary massless $U(1)$ QED₃ massless-1-loop .

Among the six self-energy diagrams (see Fig. 1 and Table 2), two are UV finite, $\gamma_{7_{\pm}}$, while the four remaining, $\gamma_{5_{\pm}}$ and $\gamma_{6_{\pm}}$, are UV divergent, thus those which have to be renormalized. However, the BPHZL subtraction procedures for the 1-particle irreducible self-energy divergent diagrams are analogous, differing only by coupling constants dependent factors, $\pm e^{2}$ and $\pm g^{2}$, corresponding to the 1-loop graphs, $\gamma_{5_{\pm}}$ and $\gamma_{6_{\pm}}$, respectively. Starting the analysis with $\gamma_{5_{\pm}}$ Feynman graphs, the 1-loop self-energy, $\Sigma(\gamma_{5_{\pm}})$, reads

$$\Sigma(\gamma_{5_{\pm}})=\int\frac{d^{3}k}{(2\pi)^{3}}\underbrace{\left\{-e^{2}\gamma^{\mu}\left[\frac{1}{k^{2}-\mu^{2}}\left(\eta_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}\right)\right]\left[\frac{(\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}k-\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}p)\mp m(s-1)}{(k-p)^{2}-m^{2}(s-1)^{2}}\right]\gamma^{\nu}\right\}}_{I_{\gamma_{5_{\pm}}}(p,k,s)}.$$

(22)

Keeping in mind one more time the conditions (11) and the inequality (12), the UV and IR subtraction degrees are $\delta(\gamma_{5_{\pm}})=\delta(\gamma_{6_{\pm}})=0$ and $\rho(\gamma_{6_{\pm}})=\rho(\gamma_{5_{\pm}})=1$, where it has been fixed $b(\gamma_{5_{\pm}})=b(\gamma_{6_{\pm}})=0$ and $c(\gamma_{5_{\pm}})=c(\gamma_{6_{\pm}})=1$. The 1-loop BPHZL subtracted (renormalized) integrand, $R_{\gamma_{5_{\pm}}}(p,k,s)$, can be expressed in terms of the unsubtracted one, $I_{\gamma_{5_{\pm}}}(p,k,s)$, as follows:

	$\displaystyle R_{\gamma_{5_{\pm}}}(p,k,s)$	$\displaystyle=$	$\displaystyle(1-t^{0}_{p,s-1})(1-t^{0}_{p,s})I_{\gamma_{5_{\pm}}}(p,k,s)$		(23)
		$\displaystyle=$	$\displaystyle(1-t^{0}_{p,s-1}-t^{0}_{p,s}+t^{0}_{p,s-1}t^{0}_{p,s})I_{\gamma_{5_{\pm}}}(p,k,s)~{}.$

Yet again, setting $s=1$ at the end of the Taylor expansion operations, restoring the massless condition, the subtracted integrand, $R_{\gamma_{5_{\pm}}}(p,k,1)$, results:

$$R_{\gamma_{5_{\pm}}}(p,k,1)=I_{\gamma_{5_{\pm}}}(p,k,1)-I_{\gamma_{5_{\pm}}}(0,k,1)~{},$$

(24)

where,

$$I_{\gamma_{5_{\pm}}}(p,k,1)=2e^{2}\left\{\frac{1}{k^{2}-\mu^{2}}\frac{1}{(k-p)^{2}}\left[\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}k-\frac{\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}k(k\cdot p)}{k^{2}}\right]\right\}~{},~{}~{}I_{\gamma_{5_{\pm}}}(0,k,1)=2e^{2}\frac{\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}k}{k^{2}(k^{2}-\mu^{2})}~{}.$$

(25)

Additionally, once the renormalized self-energy, $\Sigma_{\gamma_{5_{\pm}}}^{(R)}(p,s)$, is defined by

$$\Sigma_{\gamma_{5_{\pm}}}^{(R)}(p,s)=\int\frac{d^{3}k}{(2\pi)^{3}}~{}R_{\gamma_{5_{\pm}}}(p,k,s),$$

(26)

such that, from the Eqs.(24)–(25), leads to

$$\Sigma^{(R)}_{\gamma_{5_{\pm}}}=-\frac{ie^{2}\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}p}{4\pi}\left[\frac{1}{4\sqrt{p^{2}}}\left(\frac{p^{2}}{\mu^{2}}+\frac{3\mu^{2}}{p^{2}}+2\right)\ln\left(\frac{\mu^{2}-p^{2}}{(\sqrt{\mu^{2}}-\sqrt{p^{2}})^{2}}\right)-\frac{|\mu|}{2}\left(\frac{1}{\mu^{2}}+\frac{3}{p^{2}}\right)+i\pi\frac{p^{2}}{4\mu^{2}\sqrt{p^{2}}}\right]~{},$$

(27)

with $\Sigma^{(R)}_{\gamma_{5_{\pm}}}\equiv\Sigma^{(R)}_{\gamma_{5_{\pm}}}(p,1)$.

Similarly to the previous case, $\Sigma^{(R)}_{\gamma_{5_{\pm}}}$, the renormalized self-energies $\Sigma^{(R)}_{\gamma_{6_{\pm}}}$, corresponding to $\gamma_{6_{\pm}}$ diagram, read

$$\Sigma^{(R)}_{\gamma_{6_{\pm}}}=-\frac{ig^{2}\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}p}{4\pi}\left[\frac{1}{4\sqrt{p^{2}}}\left(\frac{p^{2}}{\mu^{2}}+\frac{3\mu^{2}}{p^{2}}+2\right)\ln\left(\frac{\mu^{2}-p^{2}}{(\sqrt{\mu^{2}}-\sqrt{p^{2}})^{2}}\right)-\frac{|\mu|}{2}\left(\frac{1}{\mu^{2}}+\frac{3}{p^{2}}\right)+i\pi\frac{p^{2}}{4\mu^{2}\sqrt{p^{2}}}\right]~{}.$$

(28)

Accordingly, the 1-loop renormalized self-energies, $\Sigma^{(R)}_{+}=\Sigma^{(R)}_{\gamma_{5_{+}}}+\Sigma^{(R)}_{\gamma_{6_{+}}}$ and $\Sigma^{(R)}_{-}=\Sigma^{(R)}_{\gamma_{5_{-}}}+\Sigma^{(R)}_{\gamma_{6_{-}}}$, associated respectively to $\psi_{+}$ and $\psi_{-}$:

	$\displaystyle\Sigma^{(R)}_{+}=\Sigma^{(R)}_{-}$	$\displaystyle\!\!=\!\!$	$\displaystyle-\frac{i(e^{2}+g^{2})\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}p}{4\pi}\left[\frac{1}{4\sqrt{p^{2}}}\left(\frac{p^{2}}{\mu^{2}}+\frac{3\mu^{2}}{p^{2}}+2\right)\ln\left(\frac{\mu^{2}-p^{2}}{(\sqrt{\mu^{2}}-\sqrt{p^{2}})^{2}}\right)-\frac{|\mu|}{2}\left(\frac{1}{\mu^{2}}+\frac{3}{p^{2}}\right)+i\pi\frac{p^{2}}{4\mu^{2}\sqrt{p^{2}}}\right]$		(29)
		$\displaystyle\!\!=\!\!$	$\displaystyle\frac{(e^{2}+g^{2})}{4\pi}{\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}p}~{}{\cal O}(p^{2},\mu)~{},$

contribute to the 1-loop effective action (in momenta space) with the following term:

$$\overline{\psi}_{+}\Sigma^{(R)}_{+}\psi_{+}+\overline{\psi}_{-}\Sigma^{(R)}_{-}\psi_{-}~{}~{}\overset{P}{\longmapsto}~{}~{}\overline{\psi}_{-}\Sigma^{(R)}_{-}\psi_{-}+\overline{\psi}_{+}\Sigma^{(R)}_{+}\psi_{+}~{},$$

(30)

that shows to be invariant under parity, thereby the BPHZL subtraction scheme does not break parity in the case of the 1-loop self-energy either. Finally, it has been finished the proof on the BPHZL parity invariance at 1-loop for the massless parity-even $U(1)\times U(1)$ Maxwell-Chern-Simons QED₃ model masslessU1U1QED3 . Nevertheless, thanks to divergent 2-loops vacuum polarization tensor diagrams (see Fig. 2), it remains to verify if whether or not parity still be preserved in the course of the 2-loops BPHZL ultraviolet and infrared subtractions.