Classical features, Anderson-Higgs mechanism, and unitarity in Lee-Wick pseudo-electrodynamics

Quantum field theories (QFTs) in two spatial dimensions have gained relevance with several applications in condensed matter physics in the last years. As examples, the quantum Hall effect [1, 2, 3, 4], topological planar materials [5, 6, 7, 8, 9], the study of transport in graphene [10, 11, 12, 13], superconductivity in layered materials [14, 15, 16, 17, 18], and among others show that field theories in low dimensions are excellent theoretical approach to explain experimental results in material physics. In special, Dirac materials are systems that exhibit a semi-relativistic dynamics for massless or massive particles with Fermi velocity and that are good tests for QFTs [19]. The interaction electron-electron in planar materials must so be mediated by a gauge field defined in $1+2$ dimensions, following the basic principle from quantum electrodynamics (QED). The gauge theory that describe the interactions of fermions in a planar QED is known as pseudo-electrodynamics (PED) [20]. Basically, the PED is obtained from usual Maxwell ED when the classical sources of charges and currents are confined on a spatial plane, reducing the space-time from $1+3$ to $1+2$ dimensions in the generating functional of the Abelian gauge theory. As consequence of the reduced dimension, the PED is a non-local ED with derivatives of infinity order in the D’Alembertian operator. The PED preserves the gauge symmetry and the continuity equation for the current. As fundamental ingredients of a QFT, the PED also preserves the properties of causality [21], and unitarity [22]. Extensions of the PED has been investigated in the literature, as the addition of topological Chern-Simons term [23, 24], the dimensional reduction of the Proca ED that leads to Pseudo-Proca ED [25, 26, 27], and the dimensional reduced QED applied to planes and fermion gap generation [28]. In brane physics, the dynamical chiral symmetry breaking was studied through the dimensional reduction of the QED [29]. The dimensional reduction in connection with boundary conditions were studied for scalar and abelian gauge theories [30].

However, the Proca theory breaks the gauge invariance, and others extensions of the Maxwell, that preserve the gauge invariance can be the good source of investigation to include a massive gauge field in planar materials. One of these possibilities is in the Lee-Wick ED [31, 32], that also is known as Podolsky ED [33, 34]. The Lee-Wick theory is a ED with higher derivative in the kinetic term, that introduces naturally a massive degree of freedom called Lee-Wick mass, preserves the gauge invariance. The Lee-Wick static potential is the subtraction of Coulombian by the Yukawa potential, and as consequence, it is finite at the origin. The Maxwell ED is recovered when the Lee-Wick mass is infinity, and induces the idea that the particle associated with the Lee-Wick field is heavy. From point of view of QFT, the motivation to study the Lee-Wick ED is in the fact of the propagator to have a better behaviour in the ultraviolet regime, and that helps in the renormalizability of the theory in 4D. Therefore, many investigations were studied in literature [35, 36, 37, 38, 39, 40], including the construction of a Standard Model for elementary particles based on Lee-Wick approach [41]. Thus, in low dimensions, the Lee-Wick ED can be a super-renormalizable or finite in the first orders of the perturbation theory. This is the main motivation to investigate the aspects of the Lee-Wick ED when applied to the planar materials.

In this paper, we show the dimensional reduction to $1+2$ dimensions in the Lee-Wick ED, where the sources are constraint on a spatial plane. Thereby, the non-local theory called Lee-Wick pseudo-electrodynamics is obtained and defined in two spatial dimensions and one time-coordinate [42]. The known pseudo-ED is so recovered when the Lee-Wick mass (that is a natural mass parameter of the theory) goes to infinity. Motivated by the Abelian Chern-Simons model applied to planar superconductors [43], we propose a Higgs-Anderson mechanism to break spontaneously the $U(1)$-gauge symmetry via a complex scalar field in $1+2$ dimensions, whose vacuum expected value yields a mass to the Lee-Wick pseudo-electrodynamics, beyond the Lee-Wick mass parameter. For simplification, it is considered that the mass acquired via spontaneous symmetry breaking (SSB) is lighter in relation to Lee-Wick mass. The scalar sector after the SSB also is showed as a toy model that is super-renormalizable in $1+2$-dimensions. It is discussed some classical features of the Proca-Lee-Wick pseudo-ED, as the field equations and the correspondent conservations laws. Posteriorly, the fermion sector is introduced as a starting point for discussion of the viable pseudo-quantum electrodynamics in the presence of a Proca mass for the Lee-Wick field in $1+2$ dimensions. As important consistency of a quantum field theory, the conditions for the unitarity of the Lee-Wick pseudo-electrodynamics are showed through the Optical theorem [44].

The paper is organized as follows : In the section II, the dimensional reduction of the Lee-Wick is showed to yield the Lee-Wick pseudo-ED. The section III is dedicated to the Anderson-Higgs mechanism applied to Lee-Wick pseudo-ED. Some classical properties of the Proca-Lee-Wick pseudo-ED are showed in the section IV. In the section V is proposed the fermion sector coupled to the Proca-Lee-Wick field in $1+2$ dimensions. The section VI is dedicated to discussion of the unitarity in Lee-Wick pseudo-ED at three level. For end, the conclusions are highlighted in the section VII.

The natural units system $\hbar=c=1$ are used throughout the paper. The signature of the metric is $\eta_{\mu\nu}=\mbox{diag}(+1,-1,-1,-1)$ for the theory in $1+3$ dimensions. In the theory dimensionally reduced to $1+2$ dimensions, we use the bar index $\bar{\mu}=\{0,1,2\}$ for vectors and tensors, with the metric $\eta_{\bar{\mu}\bar{\nu}}=\mbox{diag}(+1,-1,-1)$.

The well-known Lee-Wick ED is set by the lagrangian density in the presence of a gauge fixing term

	$\displaystyle\mathcal{L}_{LW}$	$\displaystyle=$	$\displaystyle-\frac{1}{4}\,F_{\mu\nu}\left(1+\frac{\Box}{M^{2}}\right)F^{\mu\nu}$		(1)
			$\displaystyle-\frac{1}{2\xi}\,\left[\left(1+\frac{\Box}{M^{2}}\right)\partial_{\mu}A^{\mu}\right]^{2}-J_{\mu}\,A^{\mu}\;,$

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the strength field tensor of the $A^{\mu}$-potential, $M$ is the Lee-Wick mass for $A^{\mu}$, $\xi$ is a real gauge fixing parameter, and $J^{\mu}$ is a external classical source. The limit $M\rightarrow\infty$ recovers the usual Maxwell ED with a covariant gauge fixing term. Thereby, the Lee-Wick mass represents degree freedom of a heavy gauge boson. The action principle applied to the lagrangian (1) yields the field equation for the $A^{\mu}$-potential

$\displaystyle\mathcal{O}_{\mu\nu}\,A^{\mu}=J_{\nu}\;,$

(2)

in which $\mathcal{O}_{\mu\nu}$ is the operator

	$\displaystyle\mathcal{O}_{\mu\nu}=\left(1+\frac{\Box}{M^{2}}\right)\left(\,\eta_{\mu\nu}\,\Box-\partial_{\mu}\,\partial_{\nu}\,\right)$
	$\displaystyle+\frac{1}{\xi}\left(1+\frac{\Box}{M^{2}}\right)^{2}\!\partial_{\mu}\,\partial_{\nu}\;.$		(3)

The solution for the equation (2) is expressed by

$\displaystyle A_{\mu}(x)=A_{\mu}^{(0)}(x)+\int d^{4}x^{\prime}\,\Delta_{\mu\nu}(x-x^{\prime})\,J^{\nu}(x^{\prime})\;,$

(4)

where $A_{\mu}^{(0)}$ is the solution of the homogeneous equation $\mathcal{O}_{\mu\nu}A^{(0)\mu}=0$, and $\Delta_{\mu\nu}(x-x^{\prime})$ is the Green function associated with the operator (II) that satisfies the equation

$\displaystyle\mathcal{O}_{\mu\nu}\,\Delta^{\nu}_{\;\,\alpha}(x-x^{\prime})=\delta^{\mu}_{\;\,\,\alpha}\,\delta^{4}(x-x^{\prime})\;.$

(5)

After integrations by parts, the functional action can be written as

$\displaystyle S[A_{\mu}]=\int d^{4}x\,\left[\,\frac{1}{2}\,A_{\mu}\,\mathcal{O}^{\mu\nu}\,A_{\nu}-J_{\mu}\,A^{\mu}\,\right]\;,$

(6)

that substituting the solution (4), it is reduced to the functional of the external source

$$S[J_{\mu}]=-\frac{1}{2}\int d^{4}x\,d^{4}x^{\prime}\,J_{\mu}(x)\,\Delta^{\mu\nu}(x-x^{\prime})\,J_{\nu}(x^{\prime})\;.$$

(7)

For convenience, the dimensional reduction is realized with the action written in the euclidian $4D$ space, with $x^{0}=-i\,x_{4}$ and $\Box\rightarrow-\Box_{E}$, in which the operator (II) is given by

$$\mathcal{O}_{E}^{\mu\nu}=\left[1+\frac{(-\Box_{E})}{M^{2}}\right](-\Box_{E})\left\{\theta_{E}^{\mu\nu}+\frac{1}{\xi}\left[1+\frac{(-\Box_{E})}{M^{2}}\right]\omega_{E}^{\mu\nu}\right\}\;.$$

(8)

The projectors in the euclidian space are defined by

$$\theta_{E}^{\mu\nu}=\delta_{\mu\nu}-\frac{\partial_{E}^{\mu}\,\partial_{E}^{\nu}}{(-\Box_{E})}\quad,\quad\omega_{E}^{\mu\nu}=\frac{\partial_{E}^{\mu}\,\partial_{E}^{\nu}}{(-\Box_{E})}\;,$$

(9)

that satisfy the relations

$$\theta_{E}^{\;\;\,\mu\alpha}\,\theta_{E\,\alpha\nu}=\theta_{E\;\;\nu}^{\;\;\;\mu}\;,\;\omega_{E}^{\;\;\,\mu\alpha}\,\omega_{E\,\alpha\nu}=\omega_{E\;\;\nu}^{\;\;\;\mu}\;,\;\theta_{E}^{\;\;\,\mu\alpha}\,\omega_{E\,\alpha\nu}=0\;.$$

(10)

Using the Fourier transform in the momentum space (euclidian), the Green function that satisfies (5) is read below

	$\displaystyle\Delta_{\mu\nu}(x_{E}-x_{E}^{\prime})=\int\frac{d^{4}k_{E}}{(2\pi)^{4}}\frac{M^{2}}{k_{E}^{2}(k_{E}^{2}+M^{2})}\,\times$
	$\displaystyle\times\,\left[\,\delta_{\mu\nu}-\frac{k^{E}_{\mu}\,k^{E}_{\nu}}{k_{E}^{2}}\frac{k_{E}^{2}+M^{2}(1-\xi)}{k_{E}^{2}+M^{2}}\,\right]e^{ik_{E}\cdot(x_{E}-x_{E}^{\prime})}\;,\;\;\;\;$		(11)

where $k_{E}^{\mu}=(k_{4}=i\,k_{0},k_{x},k_{y},k_{z})$ is the $4$-momentum in the euclidian space, and $k_{E}^{2}=k_{4}^{2}+k_{x}^{2}+k_{y}^{2}+k_{z}^{2}$ is positive. Consequently, the effective action in the euclidian space is

$$S_{E}[J_{\mu}]=-\frac{1}{2}\int d^{4}x_{E}\,d^{4}x_{E}^{\prime}\,J_{\mu}(x_{E})\,\Delta^{\mu\nu}(x_{E}-x_{E}^{\prime})\,J_{\nu}(x_{E}^{\prime})\;.$$

(12)

The dimensional reduction is so introduced constraint the external sources on the 2D spatial plane

	$\displaystyle J^{\bar{\mu}}(x_{E}^{\mu})$	$\displaystyle=$	$\displaystyle j^{\bar{\mu}}(x_{E}^{\bar{\mu}})\,\delta(z)\;,$		(13a)
	$\displaystyle J^{3}(x^{\mu})$	$\displaystyle=$	$\displaystyle 0\;,$		(13b)

where $x_{E}^{\bar{\mu}}=(x_{4},x,y)$ sets the coordinates on the 3D euclidian space-time. Substituting these conditions in (12), the euclidian action is reduced to a 3D space-time as

$$S_{E}[j_{\bar{\mu}}]=-\frac{1}{2}\int d^{3}\bar{x}_{E}\,d^{3}\bar{x}_{E}^{\prime}\,j_{\bar{\mu}}(\bar{x}_{E})\,\Delta^{\bar{\mu}\bar{\nu}}(\bar{x}_{E}-\bar{x}_{E}^{\prime})\,j_{\bar{\nu}}(\bar{x}_{E}^{\prime})\;,$$

(14)

where the new Green function in 3D is

	$\displaystyle\left.\Delta_{\bar{\mu}\bar{\nu}}(\bar{x}_{E}-\bar{x}_{E}^{\prime})=\Delta_{\bar{\mu}\bar{\nu}}(x_{E}-x_{E}^{\prime})\right|_{z=z^{\prime}=0}$
	$\displaystyle\left.=\int\frac{d^{4}k_{E}}{(2\pi)^{4}}\frac{M^{2}\,\delta_{\bar{\mu}\bar{\nu}}}{k_{E}^{2}(k_{E}^{2}+M^{2})}\,e^{ik_{E}\cdot(x_{E}-x_{E}^{\prime})}\right|_{z=z^{\prime}=0}\,,\hskip 14.22636pt$		(15)

in which we have used the conserved current $k_{\bar{\mu}}\,j^{\bar{\mu}}=0$ in the momentum space. The $k_{z}$-integration yields the result

	$\displaystyle\Delta_{\bar{\mu}\bar{\nu}}(\bar{x}_{E}-\bar{x}_{E}^{\prime})=\frac{\delta_{\bar{\mu}\bar{\nu}}}{2}\int\frac{d^{3}\bar{k}_{E}}{(2\pi)^{3}}\,e^{i\bar{k}_{E}\cdot(\bar{x}_{E}-\bar{x}_{E}^{\prime})}$
	$\displaystyle\times\left[\frac{1}{\sqrt{\bar{k}_{E}^{2}}}-\frac{1}{\sqrt{\bar{k}_{E}^{2}+M^{2}}}\right]\;,\;\;\;\;$		(16)

where $\bar{k}_{E}^{2}=k_{E\bar{\mu}}k_{E}^{\bar{\mu}}=k_{4}^{2}+k_{x}^{2}+k_{y}^{2}>0$. Calculating the integrals in (II), the Green function in the euclidian space is

	$\displaystyle\Delta_{\bar{\mu}\bar{\nu}}(\bar{x}_{E}-\bar{x}_{E}^{\prime})=\frac{\delta_{\bar{\mu}\bar{\nu}}}{4\pi^{2}|\bar{x}_{E}-\bar{x}_{E}^{\prime}|^{2}}$
	$\displaystyle\times\left[\,1-M|\bar{x}_{E}-\bar{x}_{E}^{\prime}|\,K_{1}(M|\bar{x}_{E}-\bar{x}_{E}^{\prime}|)\,\right]\;,$		(17)

where $K_{1}$ is a Bessel function of second kind, and the effective action (14) is

	$\displaystyle S_{E}[j_{\bar{\mu}}]=-\frac{1}{8\pi^{2}}\int d^{3}\bar{x}_{E}\,d^{3}\bar{x}_{E}^{\prime}\,\frac{j_{\bar{\mu}}(\bar{x}_{E})\,j^{\bar{\mu}}(\bar{x}_{E}^{\prime})}{|\bar{x}_{E}-\bar{x}_{E}^{\prime}|^{2}}$
	$\displaystyle\times\left[\,1-M|\bar{x}_{E}-\bar{x}_{E}^{\prime}|\,K_{1}(M|\bar{x}_{E}-\bar{x}_{E}^{\prime}|)\,\right]\;.$		(18)

Thereby, we seek now the Lagrangian density defined in the $1+2$ space-time that leads to the Green function (II). The proposed Lagrangian must be in the form

	$\displaystyle\mathcal{L}_{PLW}$	$\displaystyle=$	$\displaystyle-\frac{1}{4}\,F_{\bar{\mu}\bar{\nu}}\left(1+\frac{\bar{\Box}}{M^{2}}\right)W(\bar{\Box})F^{\bar{\mu}\bar{\nu}}$		(19)
			$\displaystyle+\frac{1}{2\xi}\,A_{\bar{\mu}}\left(1+\frac{\bar{\Box}}{M^{2}}\right)^{2}W(\bar{\Box})\,\partial^{\bar{\mu}}\,\partial^{\bar{\nu}}A_{\bar{\nu}}$
			$\displaystyle-j_{\bar{\mu}}\,A^{\bar{\mu}}\;,$

where $\bar{\Box}=\partial_{\bar{\mu}}\partial^{\bar{\mu}}=\partial_{t}^{2}-\partial_{x}^{2}-\partial_{y}^{2}$ is the D’Alembertian operator in $1+2$ dimensions, $W(\bar{\Box})$ is a operator that depends on $\bar{\Box}$, $F_{\bar{\mu}\bar{\nu}}=\partial_{\bar{\mu}}A_{\bar{\nu}}-\partial_{\bar{\nu}}A_{\bar{\mu}}$ is the EM tensor defined on the $1+2$ space-time, and $A^{\bar{\mu}}=(A^{0},A_{x},A_{y})$ is the correspondent potential. The space-time derivatives $\partial_{\bar{\mu}}=(\partial_{t},\partial_{x},\partial_{y})$ acts on the fields of the theory. If we repeat the same steps for the Lagrangian (19), the correspondent Green function in $1+2$ is given by

	$\displaystyle\Delta_{\bar{\mu}\bar{\nu}}(\bar{x}-\bar{x}^{\prime})=\int\frac{d^{3}\bar{k}}{(2\pi)^{3}}\frac{M^{2}}{W(-\bar{k}^{2})\bar{k}^{2}(\bar{k}^{2}-M^{2})}$
	$\displaystyle\times\left[\,\theta_{\bar{\mu}\bar{\nu}}-\frac{M^{2}\,\xi}{\bar{k}^{2}-M^{2}}\,\omega_{\bar{\mu}\bar{\nu}}\,\right]e^{i\bar{k}\cdot(\bar{x}-\bar{x}^{\prime})}\;,$		(20)

and comparing it with (II), the $W(-\bar{k}^{2})$-function is

$$W(-\bar{k}^{2})=\frac{2\,M^{2}}{(-\bar{k}^{2}+M^{2})\sqrt{-\bar{k}^{2}}-(-\bar{k}^{2})\sqrt{-\bar{k}^{2}+M^{2}}}\;.$$

(21)

Using the representation of $\bar{k}^{2}$ as $(-\bar{\Box})$, the $W(\bar{\Box})$-operator is

$\displaystyle W(\bar{\Box})=\frac{2M^{2}}{(\bar{\Box}+M^{2})\sqrt{\bar{\Box}}-\bar{\Box}\,\sqrt{\bar{\Box}+M^{2}}}\;,$

(22)

and the lagrangian (19) can be written as

	$\displaystyle\mathcal{L}_{PLW}$	$\displaystyle=$	$\displaystyle-\frac{1}{4}\,F_{\bar{\mu}\bar{\nu}}N(\bar{\Box})F^{\bar{\mu}\bar{\nu}}$		(23)
			$\displaystyle+\frac{1}{2\xi}\,A_{\bar{\mu}}\left(1+\frac{\bar{\Box}}{M^{2}}\right)N(\bar{\Box})\,\partial^{\bar{\mu}}\,\partial^{\bar{\nu}}A_{\bar{\nu}}$
			$\displaystyle-j_{\bar{\mu}}\,A^{\bar{\mu}}\;,$

where we have defined the $N(\bar{\Box})$-operador as

$\displaystyle N(\bar{\Box})=\frac{2(\bar{\Box}+M^{2})}{(\bar{\Box}+M^{2})\sqrt{\bar{\Box}}-\bar{\Box}\,\sqrt{\bar{\Box}+M^{2}}}\;.$

(24)

The kinetic term in (23) can be written in form of field-operator-field as

$${\cal L}_{PLW}^{K}=A_{\bar{\mu}}\left[\left(1+\frac{\bar{\Box}}{M^{2}}\right)\sqrt{\bar{\Box}}+\frac{\bar{\Box}}{M^{2}}\sqrt{1+\frac{\bar{\Box}}{M^{2}}}\right]\theta^{\bar{\mu}\bar{\nu}}A_{\bar{\nu}}\;,$$

(25)

and show that the dimensional reduction leads to a non-local electrodynamics of infinity order in the D’Alembertian operator $\bar{\Box}$. The theory (23) can be called pseudo-Lee-Wick electrodynamics, in which the limit $M\rightarrow\infty$ recovers the pseudo-electrodynamics discussed in the refs. [20, 22]. The lagrangian (23) (up to gauge fixing term) is $U(1)$ gauge invariant under the transformation $A_{\bar{\mu}}\mapsto A^{\prime}_{\bar{\mu}}=A_{\bar{\mu}}-\partial_{\bar{\mu}}\Lambda$, if the current density is conserved, i.e., it obeys the continuity equation $\partial_{\bar{\mu}}j^{\bar{\mu}}=0$ on the plane. Substituting the result of the $W(-\bar{k}^{2})$-function (21) in (II), the Green function in the $\xi$-gauge fixing is

	$\displaystyle\Delta_{\bar{\mu}\bar{\nu}}(\bar{x}-\bar{x}^{\prime})=\int\frac{d^{3}\bar{k}}{(2\pi)^{3}}\left[\frac{1}{2\sqrt{-\bar{k}^{2}}}-\frac{1}{2\sqrt{-\bar{k}^{2}+M^{2}}}\right]$
	$\displaystyle\times\left[\,\eta_{\bar{\mu}\bar{\nu}}-\frac{\bar{k}^{2}+M^{2}(\xi-1)}{\bar{k}^{2}-M^{2}}\,\frac{k_{\bar{\mu}}\,k_{\bar{\nu}}}{\bar{k}^{2}}\,\right]e^{i\bar{k}\cdot(\bar{x}-\bar{x}^{\prime})}\;.\;\;\;\;$		(26)

Notice that the expression Green function in the momentum space has two poles : $\bar{k}^{2}=0$ and $\bar{k}^{2}=M^{2}$. Thus, the theory shows one massless degree freedom, and another massive degree freedom described by the Lee-Wick mass. The case of two like-point charged particles $(-e)$ at rest, the current density is $j^{\bar{\mu}}({\bf r})=[-e\,\delta^{3}({\bf r}-{\bf r}_{1})-e\,\delta^{3}({\bf r}-{\bf r}_{2})\,,\,{\bf 0}]$, and the effective action reduces to energy of interaction between these two particles :

$$U(r)=-e^{2}\int\frac{d^{2}{\bf k}}{(2\pi)^{2}}\left[\frac{e^{i{\bf k}\cdot{\bf r}}}{|{\bf k}|}-\frac{e^{i{\bf k}\cdot{\bf r}}}{\sqrt{{\bf k}^{2}+M^{2}}}\right]\;,$$

(27)

whose result is well known in the literature

$\displaystyle U(r)=-\frac{e^{2}}{4\pi r}\,\left(\,1-e^{-Mr}\,\right)\;,$

(28)

where $r=|{\bf r}_{1}-{\bf r}_{2}|$ is the distance that separate the charges. The energy (28) is the same of the usual Lee-Wick ED that is the difference of the Coulomb by the Yukawa potential with the Lee-Wick mass.

It is worth to clarify that the Pseudo-Lee-Wick ED is a causal theory. As obtained in the ref. [42], the retarded and advanced Green functions associated with the equation (2) when $\xi\rightarrow\infty$ in the prescription of $k_{4}\rightarrow k_{0}\pm i\epsilon$ are given by

	$\displaystyle\Delta^{\pm}(\bar{x}-\bar{x}^{\prime})=-\frac{i}{\pi}\,\Theta(\pm\tau)\left\{\,\delta[(\bar{x}-\bar{x}^{\prime})^{2}]\right.$
	$\displaystyle\left.-\frac{1}{2}\,\Theta[(\bar{x}-\bar{x}^{\prime})^{2}]\,\frac{M\,J_{1}(M\,\sqrt{(\bar{x}-\bar{x}^{\prime})^{2}})}{\sqrt{(\bar{x}-\bar{x}^{\prime})^{2}}}\right\}\;,$		(29)

where the signs $(\pm)$ indicate the retarded $(+)$ and advanced $(-)$, $\tau=t-t^{\prime}$, $\Theta$ is the Heaviside function, and $J_{1}$ is Bessel function of first kind. The result shows that the retarded Green function is null outside the light-cone, for $(\bar{x}-\bar{x}^{\prime})^{2}<0$. The Feynman Green function is obtained with the prescription of $k_{4}^{2}\rightarrow k_{0}^{2}+i\epsilon$, such that the result is [42]

	$\displaystyle\Delta_{F}(\bar{x}-\bar{x}^{\prime})=-\frac{1}{2\pi^{2}}\left[\,\frac{1}{(\bar{x}-\bar{x}^{\prime})^{2}+i\epsilon}\right.$
	$\displaystyle\left.-\frac{i\pi}{2}\,\frac{M\,H_{1}(-M\sqrt{(\bar{x}-\bar{x}^{\prime})^{2}+i\epsilon})}{\sqrt{(\bar{x}-\bar{x}^{\prime})^{2}+i\epsilon}}\,\right]\;,$		(30)

in which $H_{1}$ is a Hankel function of first kind.

The sector of a complex scalar field $\phi(x^{\bar{\mu}})$ $U(1)$-gauge invariant is governed by the Lagrangian

$\displaystyle\mathcal{L}_{sc}$

$\displaystyle=$

$\displaystyle|D_{\bar{\mu}}\phi|^{2}-V(\phi^{\ast},\phi)+\frac{1}{M_{\phi}^{2}}|D_{\bar{\mu}}D^{\bar{\mu}}\phi|^{2}\;,$

(31)

where $V(\phi^{\ast},\phi)=\mu^{2}|\phi|^{2}+\lambda|\phi|^{4}$ is the scalar potential, $\mu^{2}$ and $\lambda$ are two real parameters, and $M_{\phi}$ is the Lee-Wick mass for the scalar field. The covariant derivative operator is $D_{\bar{\mu}}\phi=\partial_{\bar{\mu}}\phi+igA_{\bar{\mu}}\phi$, in which $g$ is a real coupling constant. The complex scalar field $\phi$ under a local transformation keeps the lagrangian (31) $U(1)$-gauge invariant in $1+2$ dimensions. As in the usual mechanism, the scalar potential acquires the vacuum expected value (VEV) $v\equiv|\phi_{0}|=\sqrt{-\mu^{2}/(2\lambda)}$, with $\mu^{2}<0$, and therefore, the scalar field can be written as the perturbation in the unitary gauge

$\displaystyle\phi(\bar{x})=\frac{\sqrt{v}+\hat{h}(\bar{x})}{\sqrt{2}}\;.$

(32)

Notice that $\phi$ has dimension of mass elevated to $1/2$ in three dimensions, such that the combination (32) is dimensionally satisfied in 3D. After the SSB mechanism, the free sector of (31) is

$${\cal L}_{sc}^{(0)}=\frac{1}{2}\,(g^{2}v)\,A_{\bar{\mu}}^{2}+\frac{1}{2}\,(\partial_{\bar{\mu}}\hat{h})^{2}-\frac{1}{2}\,m_{h}^{2}\,\hat{h}^{2}+\frac{1}{2M_{\phi}^{2}}\,(\bar{\Box}\hat{h})^{2}\;,$$

(33)

where the term $g^{2}\,v$ is a mass acquired by the gauge field $A^{\mu}$ due to VEV scale, and $m_{h}=\sqrt{2\lambda\,v^{2}}$ is a mass acquired by the scalar field $\hat{h}$. We consider in this paper the condition in that $M,M_{\phi}\gg v$, and consequently, the Lee-Wick masses $M$ and $M_{\phi}$ are heavier in relation to masses $m_{a}=g^{2}\,v$ and $m_{h}$.

The pure scalar sector of $\hat{h}$-field is read below

$${\cal L}_{sc}^{(\hat{h})}=\frac{1}{2}\,\partial_{\bar{\mu}}\hat{h}\left(1+\frac{\bar{\Box}}{M_{\phi}^{2}}\right)\partial^{\bar{\mu}}\hat{h}-\frac{1}{2}\,m_{h}^{2}\,\hat{h}^{2}-\lambda\,\sqrt{v}\,\hat{h}^{3}-\frac{\lambda}{4}\,\hat{h}^{4}\;,$$

(34)

that sets a toy model in 3D with self-interactions of $\hat{h}^{3}$ and $\hat{h}^{4}$ of coupling constants $\lambda\,\sqrt{v}$ and $\lambda$, respectively. The free sector in (34) yields the scalar propagator in the momentum space $1+2$

	$\displaystyle\Delta(\bar{p})$	$\displaystyle=$	$\displaystyle\frac{i}{\bar{p}^{2}\left(1-\frac{\bar{p}^{2}}{M_{\phi}^{2}}\right)-m_{h}^{2}}$		(35)
			$\displaystyle=\frac{M_{\phi}}{\sqrt{M_{\phi}^{2}-4m_{h}^{2}}}\left[\,\frac{i}{\bar{p}^{2}-\mu_{1}^{2}}-\frac{i}{\bar{p}^{2}-\mu_{2}^{2}}\,\right]\;,$

that constraints the condition $M_{\phi}>2\,m_{h}$, and $\mu_{1}$, $\mu_{2}$ are the mass eigenstates :

	$\displaystyle\mu_{1}^{2}$	$\displaystyle=$	$\displaystyle\frac{M_{\phi}^{2}}{2}-\frac{M_{\phi}^{2}}{2}\sqrt{1-\frac{4\,m_{h}^{2}}{M_{\phi}^{2}}}\;,$		(36a)
	$\displaystyle\mu_{2}^{2}$	$\displaystyle=$	$\displaystyle\frac{M_{\phi}^{2}}{2}+\frac{M_{\phi}^{2}}{2}\sqrt{1-\frac{4\,m_{h}^{2}}{M_{\phi}^{2}}}\;.$		(36b)

In the case of $M_{\phi}\gg m_{h}$, these eigenstates are reduced to $\mu_{1}\simeq m_{h}$ and $\mu_{2}\simeq M_{\phi}$, respectively, and simplifies (35) as

$$\Delta(\bar{p})\simeq\frac{i}{\bar{p}^{2}-m_{h}^{2}}-\frac{i}{\bar{p}^{2}-M_{\phi}^{2}}\;,$$

(37)

which it is clearly the composition of a scalar propagator with mass $m_{h}$ summed to scalar propagator with a minus sign $(-i)$ and mass $M_{\phi}$. Thereby, the toy model (34) can be interpreted as one scalar theory of mass $m_{h}$ subtracted of another scalar theory with heavy mass of $M_{\phi}$. To check it explicitly, it is introduced the auxiliary scalar field $\tilde{h}$, such that the modified lagrangian is

	$\displaystyle{\cal L}_{sc}^{(\hat{h}\tilde{h})}=\frac{1}{2}\,(\partial_{\bar{\mu}}\hat{h})^{2}-\frac{1}{2}\,m_{h}^{2}\,\hat{h}^{2}-\tilde{h}\,\bar{\Box}\hat{h}$
	$\displaystyle+\frac{1}{2}\,M_{\phi}^{2}\,\tilde{h}^{2}-\lambda\,\sqrt{v}\,\hat{h}^{3}-\frac{\lambda}{4}\,\hat{h}^{4}\,.\;\;\;$		(38)

The variation of (III) in relation to $\tilde{h}$ yields the relation

$\displaystyle\tilde{h}=\frac{1}{M_{\phi}^{2}}\,\bar{\Box}\hat{h}\;,$

(39)

that when substituted in (III), we recover the original lagrangian (34). We make the transformation $\hat{h}=h-\tilde{h}$ in (III) to obtain the scalar lagrangian in terms of $h$ and $\tilde{h}$

	$\displaystyle{\cal L}_{sc}^{(h\tilde{h})}$	$\displaystyle=$	$\displaystyle\frac{1}{2}\,(\partial_{\bar{\mu}}h)^{2}-\frac{1}{2}\,(\partial_{\bar{\mu}}\tilde{h})^{2}-\frac{1}{2}\,m_{h}^{2}\,(h-\tilde{h})^{2}$		(40)
			$\displaystyle+\frac{1}{2}\,M_{\phi}^{2}\,\tilde{h}^{2}-\lambda\,\sqrt{v}\,(h-\tilde{h})^{3}-\frac{\lambda}{4}\,(h-\tilde{h})^{4}\;.\;\;\;\;$

The previous lagrangian can be diagonalized by the transformation

	$\displaystyle h=h^{\prime}\,\cosh\alpha+\tilde{h}^{\prime}\,\sinh\alpha\;,$		(41a)
	$\displaystyle\tilde{h}=h^{\prime}\,\sinh\alpha+\tilde{h}^{\prime}\,\cosh\alpha\;,$		(41b)

in which the $\alpha$-mixing angle satisfies the relation

$\displaystyle\tanh(2\alpha)=-\frac{2m_{h}^{2}}{M_{\phi}^{2}}\;.$

(42)

For $M_{\phi}\gg m_{h}$, the mixing angle is very small, and the lagrangian (40) in the basis of $h^{\prime}$ and $\tilde{h}^{\prime}$ is given by

	$\displaystyle{\cal L}_{sc}^{(h^{\prime}\tilde{h}^{\prime})}$	$\displaystyle=$	$\displaystyle\frac{1}{2}\,(\partial_{\bar{\mu}}h^{\prime})^{2}-\frac{1}{2}\,m_{h}^{2}\,(h^{\prime})^{2}$		(43)
			$\displaystyle-\frac{1}{2}\,(\partial_{\bar{\mu}}\tilde{h}^{\prime})^{2}+\frac{1}{2}\,M_{\phi}^{2}\,(\tilde{h}^{\prime})^{2}$
			$\displaystyle-\lambda\,\sqrt{v}\,(h^{\prime}-\tilde{h}^{\prime})^{3}-\frac{\lambda}{4}\,(h^{\prime}-\tilde{h}^{\prime})^{4}\;.$

In (43), we have two pseudo-scalars theories, the $h^{\prime}$-field sets a light scalar particle with mass $m_{h}$, and $\tilde{h}^{\prime}$ is a heavy scalar particle (that is the Lee-Wick scalar particle) with signs exchanged in the kinetic terms in relation to the terms of $h^{\prime}$. As consequence, the propagator of the $\tilde{h}^{\prime}$-field is

$\displaystyle D^{LW}(\bar{p})=\frac{-\,i}{\bar{p}^{2}-M_{\phi}^{2}}\;,$

(44)

that confirms the second term in the propagator (37). The propagator (35) behaves like $\Delta(\bar{p})\sim(\bar{p}^{2})^{-2}$ in the ultraviolet regime, that leads to finite loop integrals for the scalar toy model (34) in $1+2$ dimensions. Thus, this scalar sector has finite contributions at the one-loop.

In the gauge sector, the massive term of $A^{\bar{\mu}}$ added to the gauge kinetic term of (23) composes the pseudo Proca-Lee-Wick ED in 3D

	$\displaystyle\mathcal{L}_{PLW}$	$\displaystyle=$	$\displaystyle-\frac{1}{4}\,F_{\bar{\mu}\bar{\nu}}N(\bar{\Box})F^{\bar{\mu}\bar{\nu}}+\frac{1}{2}\,(g^{2}v)\,A_{\bar{\mu}}^{2}$		(45)
			$\displaystyle+\frac{1}{2\xi}\,A_{\bar{\mu}}\left(1+\frac{\bar{\Box}}{M^{2}}\right)N(\bar{\Box})\,\partial^{\bar{\mu}}\,\partial^{\bar{\nu}}A_{\bar{\nu}}\;.$

The propagator of the gauge sector after the SSB is

	$\displaystyle\Delta_{\bar{\mu}\bar{\nu}}^{ssb}(\bar{k})$	$\displaystyle=$	$\displaystyle\frac{-1}{N(-\bar{k}^{2})\,\bar{k}^{2}-m_{a}}\left[\eta_{\bar{\mu}\bar{\nu}}-\frac{\bar{k}^{2}+M^{2}(\xi-1)}{\bar{k}^{2}-M^{2}}\right.$		(46)
			$\displaystyle\left.\times\frac{N(-\bar{k}^{2})\,k_{\bar{\mu}}\,k_{\bar{\nu}}}{N(-\bar{k}^{2})\,\bar{k}^{2}-\xi\,m_{a}}\right]\;,$

that in the ultraviolet range in which $\bar{k}^{2}\gg\left\{\,m_{a}^{2}\,,\,M^{2}\,\right\}$, with $N(-\bar{k}^{2})\simeq 1/\sqrt{-\bar{k}^{2}}$, the expression (46) goes to zero, and consequently, this behaviour corroborates to the renormalizability of the model in the perturbative analysis of a quantum field theory. Contracting (46) with the external conserved current $j^{\bar{\mu}}$, we obtain the amplitude :

	$\displaystyle\mathcal{M}$	$\displaystyle=$	$\displaystyle{j^{\bar{\mu}}(\bar{k})\,\Delta^{ssb}_{\bar{\mu}\bar{\nu}}(\bar{k})\,j^{\bar{\nu}}}(\bar{k})=\frac{-\,j_{\bar{\mu}}\,j^{\bar{\mu}}}{N(-\bar{k}^{2})\,\bar{k}^{2}-m_{a}}$		(47)
		$\displaystyle=$	$\displaystyle\frac{-\,j_{\bar{\mu}}\,j^{\bar{\mu}}\,[\,N(-\bar{k}^{2})\,\bar{k}^{2}+m_{a}\,]}{N^{2}(-\bar{k}^{2})\,(\bar{k}^{2})^{2}-m_{a}^{2}}\;,$

in which the poles are defined by the solution of the equation

$\displaystyle N^{2}(-\bar{k}^{2})\,(\bar{k}^{2})^{2}-m_{a}^{2}=0\;,$

(48)

whose roots are at $\bar{k}^{2}\simeq m_{a}^{2}$ and $\bar{k}^{2}\simeq M^{2}$, if $M\gg m_{a}$. Taking into account that $j^{\bar{\mu}}$ is space-like ($j^{2}<0$), see the refs. [35, 36, 37], we have the residues

$$\mbox{Res}\mathcal{M}(\bar{k}^{2}=m_{a}^{2})>0\;\;,\;\;\mbox{Res}\mathcal{M}(\bar{k}^{2}=M^{2})<0\;.$$

(49)

The above residues show that the model carries two spin-1 modes, one with a light mass at $\bar{k}^{2}=m_{a}^{2}$, and the other one with a heavier mass at $\bar{k}^{2}=M^{2}$. The positive residue corresponds to the Proca pseudo-electrodynamics, that describes a massive vector mode propagating non-locally on the plane. The negative residue is associated with the mode of a unstable heavy particle of mass $M$.

In this section, we study some classical features of the pseudo Proca-Lee-Wick electrodynamics. The Proca-Lee-Wick action associated with the lagrangian (45) defined in the $1+2$ space-time in the presence of an external source $j^{\bar{\mu}}$ is

$\displaystyle S_{PLW}(A^{\bar{\mu}})=\int d^{3}\bar{x}\,\left[\,{\cal L}_{PLW}-j_{\bar{\mu}}\,A^{\bar{\mu}}\,\right]\;,$

(50)

in which the action principle yields the field equations

$\displaystyle N(\bar{\Box})\,\partial_{\bar{\mu}}F^{\bar{\mu}\bar{\nu}}+m_{a}A^{\bar{\nu}}=j^{\bar{\nu}}\;.$

(51)

In our analysis here, we consider $\xi\rightarrow\infty$, such that the gauge fixing term can be removed. The equations without sources are set by the Bianchi identity

$\displaystyle\partial_{\bar{\mu}}F_{\bar{\nu}\bar{\alpha}}+\partial_{\bar{\nu}}F_{\bar{\alpha}\bar{\mu}}+\partial_{\bar{\alpha}}F_{\bar{\mu}\bar{\nu}}=0\;,$

(52)

that alternatively also can be represented by the divergence $\partial_{\bar{\mu}}\tilde{F}^{\bar{\mu}}=0$, where $\tilde{F}^{\bar{\mu}}=\frac{1}{2}\,\epsilon^{\bar{\mu}\bar{\alpha}\bar{\beta}}F_{\bar{\alpha}\bar{\beta}}$ is the dual tensor of $F^{\bar{\mu}\bar{\nu}}$ in $1+2$ dimensions. The conservation current $\partial_{\bar{\mu}}j^{\bar{\mu}}=0$ leads to subsidiary condition for the $A^{\bar{\mu}}$-potential $N(\bar{\Box})\,\partial_{\bar{\nu}}A^{\bar{\nu}}=0$ in (51). The components of the EM tensor provides the electric field on the spatial plane, and the magnetic field perpendicular to this plane that we consider the ${\cal Z}$-direction, $F^{\bar{\mu}\bar{\nu}}=(E^{i},-\epsilon^{ij}\,B_{z})$, $i=\{\,1\,,\,2\,\}$. The matrix representation of $F^{\bar{\mu}\bar{\nu}}$ is

$\displaystyle F^{\bar{\mu}\bar{\nu}}=\left(\begin{array}[]{ccc}0&-E_{x}&-E_{y}\\ E_{x}&0&-B_{z}\\ E_{y}&B_{z}&0\\ \end{array}\right)\;.$

(56)

In vector notation, the equations of the pseudo-Proca-Lee-Wick electrodynamics are

	$\displaystyle N(\bar{\Box})(\nabla\cdot{\bf E})+m_{a}V$	$\displaystyle=$	$\displaystyle\sigma\;,$		(57a)
	$\displaystyle(\nabla\times{\bf E})_{z}+\partial_{t}B_{z}$	$\displaystyle=$	$\displaystyle 0\;,$		(57b)
	$\displaystyle N(\bar{\Box})(\nabla\times{\bf B})+m_{a}{\bf A}$	$\displaystyle=$	$\displaystyle{\bf j}+N(\bar{\Box})\partial_{t}{\bf E}\;.$		(57c)

Using the subsidiary condition, the $A^{\bar{\mu}}$-potential satisfies the equation

$\displaystyle\left[\,N(\bar{\Box})\,\bar{\Box}+m_{a}\,\right]A^{\bar{\nu}}=j^{\bar{\nu}}\;.$

(58)

The conservation laws can be obtained directly from the field equations. Multiplying the equation (51) by $F_{\bar{\nu}\bar{\alpha}}$, and using the Bianchi identity, we obtain the relation

	$\displaystyle\partial_{\bar{\mu}}\left[F_{\bar{\nu}\bar{\alpha}}\,N(\bar{\Box})\,F^{\bar{\mu}\bar{\nu}}\right]+\partial_{\bar{\alpha}}\left[\,\frac{1}{4}\,F_{\bar{\mu}\bar{\nu}}\,N(\bar{\Box})\,F^{\bar{\mu}\bar{\nu}}\,\right]$
	$\displaystyle+\,m_{a}\,F_{\bar{\nu}\bar{\alpha}}\,A^{\bar{\nu}}=j^{\bar{\nu}}F_{\bar{\nu}\bar{\alpha}}\;.$		(59)

The subsidiary condition allows to write the expression (IV) as the total derivative

$\displaystyle\partial_{\bar{\mu}}\theta^{\bar{\mu}}_{\;\;\,\bar{\alpha}}=j^{\bar{\nu}}\,F_{\bar{\nu}\bar{\alpha}}\;,$

(60)

where the energy-momentum tensor is given by

$$\theta^{\bar{\mu}}_{\;\;\,\bar{\alpha}}=F_{\bar{\nu}\bar{\alpha}}\,N(\bar{\Box})\,F^{\bar{\mu}\bar{\nu}}+\,m_{a}\,A_{\bar{\alpha}}\,A^{\bar{\mu}}-\delta^{\bar{\mu}}_{\;\;\,\bar{\alpha}}\,{\cal L}_{PLW}\;.$$

(61)

As we expect, the energy-momentum tensor is not gauge invariant when $m_{a}\neq 0$. The case without sources $j^{\nu}=0$ in (60) leads to conservation law $\partial_{\bar{\mu}}\theta^{\bar{\mu}}_{\;\;\,\bar{\alpha}}=0$, in which the conserved momentum for the Proca-Lee-Wick field is the integral of the $\theta^{0}_{\;\;\,\bar{\alpha}}$ - component over the spatial plane

$\displaystyle P_{\bar{\alpha}}=\int d^{2}\bar{{\bf x}}\;\theta^{0}_{\;\;\,\bar{\alpha}}\;.$

(62)

The components $\bar{\alpha}=(0,i=1,2)$ yield the energy and the linear momentum densities stored in the Proca-Lee-Wick field, respectively,

	$\displaystyle\theta^{00}=\frac{1}{2}\,{\bf E}\,\cdot N(\bar{\Box}){\bf E}+\frac{1}{2}\,B_{z}\,N(\bar{\Box})B_{z}$
	$\displaystyle+\frac{1}{2}\,m_{a}(A_{0})^{2}+\frac{1}{2}\,m_{a}\,{\bf A}^{2}\;,$		(63a)
	$\displaystyle\theta^{0i}=\epsilon^{ij}\left[N(\bar{\Box})\,E^{j}\right]B_{z}-m_{a}\,A^{0}A^{i}\;.$		(63b)