Higher-order Chern-Simons extensions to QED in $2+1$ dimensions

TL;DR

This paper explores a higher-order extension of QED in 2+1 dimensions with Chern-Simons terms, analyzing its quantization, radiative corrections, and causality, revealing a consistent, finite, and microcausality-preserving model with ghost states.

Contribution

It introduces and quantizes a higher-order Chern-Simons extended QED in 2+1 dimensions, analyzing radiative corrections and demonstrating finiteness and causality preservation.

Findings

Photon and ghost modes identified with a massive photon.

Radiative corrections produce two fermionic degrees of freedom, including a ghost.

Photon polarization and vertex corrections are finite and well-defined.

Abstract

In this work, we investigate radiative corrections in the higher-order extension of the Maxwell-Chern-Simons model coupled to standard spinor matter in $2+1$ dimensions. We begin analyzing the higher-order gauge sector, where we find the modes and the polarizations vectors associated to a massive photon and ghost field. The higher-order gauge model is canonically quantized and as expected the resulting algebra of creation and annihilation operators corresponds to an indefinite metric in Hilbert space. Subsequently, we compute all relevant one-loop diagrams in the modified QED starting with the fermion self-energy. We show that the induced corrections to the fermion two-point function produce two independent fermionic degrees of freedom, which can be included in a redefined Lagrangian describing two decoupled fermions fields, one corresponding to a physical particle and the other to a negative-norm ghost state. We take advantage of this decomposition to compute the photon polarization operator and the vertex correction, both of which are found to be finite. Finally, we analyze the causal behavior of the model by computing the commutator of gauge fields at different spacetime points, and found that microcausality is preserved.

Equations

$$\mathcal{L}$$

$$+{\mathcal{L}}_{\mathrm{CSe}},$$

$${\mathcal{L}}_{\mathrm{CSe}}$$

$$\left[\mu \right]$$

$$[\psi ]$$

$${\gamma }^0$$

$${\gamma }^1$$

$${\gamma }^2$$

$$\{{\gamma }^{\mu },{\gamma }^{\nu }\}$$

$${\gamma }^{\mu }{\gamma }^{\nu }$$

$$\mathrm{tr}({\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{\rho })$$

$${\gamma }^{\mu }$$

$${\mathcal{L}}_A=-\frac{1}{4\gamma }{F}_{\mu \nu }{F}^{\mu \nu }-\frac{1}{2\xi }({\partial }_{\mu }{A}^{\mu }{)}^2$$

$$+\frac{1}{2}{ϵ}^{\alpha \beta \gamma }{A}_{\alpha }\left(\mu +g\square \right){\partial }_{\beta }{A}_{\gamma }.$$

$$-{\partial }_{\kappa }{\partial }_{\lambda }\frac{\partial {\mathcal{L}}_A}{\partial ({\partial }_{\kappa }{\partial }_{\lambda }{A}_{\sigma })}+{\partial }_{\rho }\frac{\partial {\mathcal{L}}_A}{\partial ({\partial }_{\rho }{A}_{\sigma })}-\frac{\partial {\mathcal{L}}_A}{\partial A_{\sigma }}=0.$$

$$\displaystyle\Big{[}\frac{\eta^{\mu\nu}}{\gamma}\Box-\left(\frac{1}{\gamma}-\frac{1}{\xi}\right)\partial^{\mu}\partial^{\nu}$$

$$\displaystyle\hskip 50.00008pt+\epsilon^{\mu\beta\nu}\left(\mu+g\Box\right)\partial_{\beta}\Big{]}A_{\nu}(x)=0\,.$$

$$\frac{1}{\xi }\square \left(\partial \cdot A\right)=0,$$

$$A_{\mu }(x)=\int {\mathrm{d}}^{3}k{A}_{\mu }(k)e^{-ik\cdot x}.$$

$$S^{\mu \nu }(k)A_{\nu }(k)=0,$$

$$S^{\mu \nu }(k)$$

$$-i\mathcal{M}(k){ϵ}^{\mu \beta \nu }{k}_{\beta },$$

$$\mathcal{M}(k):=\mu -g{k}^2,$$

$$G_{\nu \rho }(k)=\frac{-i\gamma }{k^2-{\gamma }^2{\mathcal{M}}^2(k)}{T}_{\nu \rho }(k),$$

$$T_{\nu \rho }(k)$$

$$-i\gamma \mathcal{M}{ϵ}_{\nu \beta \rho }\frac{k^{\beta }}{k^2}.$$

$${\Delta }_{\nu \rho }(k)=\frac{-i\gamma }{k^2-{\gamma }^2{\mathcal{M}}^2}\left({\eta }_{\nu \rho }-\frac{k_{\nu }{k}_{\rho }}{k^2}-i\gamma \mathcal{M}{ϵ}_{\nu \beta \rho }\frac{k^{\beta }}{k^2}\right),$$

$$k^2-{\gamma }^2{\mathcal{M}}^2=0.$$

$${\omega }_1=$$

$$W_2=$$

$$m_1=$$

$$M_2=$$

$${\omega }_1$$

$$W_2$$

$$m_1$$

$$M_2$$

$$e^{(0)\mu }$$

$$e^{(1)\mu }$$

Full text

Lower-dimensional gauge theories, higher derivatives

I Introduction

Chern-Simons (CS) theory is a topological quantum field theory that arises naturally in odd-dimensional spacetimes. It was first derived as an effective contribution from the high-temperature limit of QCD and electroweak models in four dimensions [1, 2, 3]. The CS theory provides a rich framework for exploring a wide range of physical phenomena. In particular, the CS term generates a gauge-invariant mass for the gauge field in $2+1$ spacetime dimensions [4, 5]. Moreover, the definitions of various topological invariants, such as the Jones polynomial in knot theory, can be connected with the non-abelian CS action [6], revealing a deep interplay between topological aspects of quantum field theory and lower dimensional conformal field theory. Furthermore, the CS term presents interesting possibilities in describing various physical systems, as the quantum Hall effect [7, 8, 9, 10], topological materials including topological insulators [11, 12], spectroscopic features in Weyl simimetals [13, 14, 15, 16], also extensions of gravity [5, 17], vortex dynamics [18, 19], Skyrme models [20], Higgs mechanism [21], topological gravity [22], supersymmetric CS [23] and anisotropic QED [24]; see also the reviews [25, 26].

A topic that has attracted growing interest over the years is the quantum behavior of the abelian CS term. Quantum corrections to modified QED incorporating the CS term exhibit several noteworthy features, such as the smoothness of infrared divergencies resulting from the massive gauge mode generated by the gauge-invariant CS term. Moreover, in modified QED the polarization tensor receives only one-loop corrections, with all higher-order contributions vanishing due to a remarkable cancellation [27], a result known as the Coleman–Hill theorem [28]. The CS term has also been studied to extend to parity violating mass terms [29, 30].

The higher-derivative extension of the CS term was proposed some years ago by S. Deser and R. Jackiw in order to describe the momentum expansion of the effective action in three-dimensional QED [31]. Some works on such an extension have focused on relic symmetries [32], thermal effects [33, 34], Lorentz-violating planar electrodynamics [35, 36, 37, 38], analysis of unitarity [39, 40], and vector multiplets [41]. It is well known that higher time derivatives can improve the ultraviolet behavior of a theory by introducing higher powers of momentum in the denominators of propagators. However, they may also lead to an indefinite metric, with additional degrees of freedom associated to ghost fields [42, 43]. In four dimensions, one can consider an analogue, which, however, is not topological in essence and necessarily breaks Lorentz invariance [44]. In this work, we explore the higher-order CS term in $2+1$ dimensions, with particular emphasis on quantum corrections induced by ghost states associated with the indefinite metric of the theory.

The work is organized as follows. In Sec. II, we present the QED model, finding the gauge propagator together with its polarization vectors. In Sec. III, we canonically quantize the model and derive the commutator algebra of creation and annihilation operators. In Sec. IV we compute the radiative correction up to one-loop level of the fermion self-energy, photon polarization operator and vertex function. Also, we analyze the superficially degree of divergence of each one-loop diagram. In Sec. V, we study microcausality and prove that it is preserved. Finally, in Sec. VI we give our final remarks. In Appendix A we present the calculation of an element of the commutator of gauge fields in a general frame.

II Extended QED model in $(2+1)$-dimensions

Our model is based on the higher-derivative QED Lagrangian in $2+1$ dimensions, given by

[TABLE]

where $\mathcal{L}_{\mathrm{CSe}}$ includes both the CS term and its higher-derivative extension [31], written as

[TABLE]

In the Lagrangian (II), we have coupled the photons with standard fermions and considered the usual interaction term. Also, we have included a covariant gauge-fixing term with $\xi>0$ and a constant $\gamma$ in the standard Maxwell term, which allow us to obtain the pure CS theory by taking the limit $\gamma\to\infty$ together with $g\to 0$. Let us emphasize that the CS term has been included from the outset, as it is naturally generated from the standard photon polarization operator diagram [27].

Taking into account that the action is dimensionless, we find that the mass dimensions of the parameters and fields are

[TABLE]

while the constants $\gamma$ and $\xi$ are dimensionless.

In this work we employ the mostly minus sign convention for the metric signature, i.e., $\eta^{\mu\nu}=\text{diag}(1,-1,-1)$, together with the Levi-Civita convention $\epsilon^{012}=1$ and the two-dimensional realization of the Dirac algebra

[TABLE]

with $\sigma^{1},\sigma^{2},\sigma^{3}$ being the Pauli matrices. Furthermore, one can check that the Dirac matrices satisfy the Clifford algebra

[TABLE]

and the relations

[TABLE]

where $\mathbb{1}_{2\times 2}$ is the $2\times 2$ unit matrix.

II.1 The gauge propagator

Let us consider the free Lagrangian in the gauge sector:

[TABLE]

The generalized Euler-Lagrange equation for the second-order Lagrangian $\mathcal{L}_{A}$ can be written as

[TABLE]

From the Lagrangian (II.1) and using (16), we have the equation of motion

[TABLE]

Contracting Eq. (II.1) with $\partial_{\mu}$ yields

[TABLE]

which by imposing suitable boundary conditions at infinity gives the condition $\partial\cdot A=0$.

Now, we go to momentum space expanding the gauge field in Fourier modes

[TABLE]

Replacing in the equation of motion (II.1), we obtain in momentum space

[TABLE]

where we have defined the operator

[TABLE]

and we have introduced the function of momentum

[TABLE]

which has dimension of mass.

By inverting the operator we find the propagator

[TABLE]

which satisfies $S^{\mu\nu}G_{\nu\rho}=i\delta^{\mu}_{\rho}$, with the definition

[TABLE]

The propagator in the Landau gauge setting $\xi=0$, reads

[TABLE]

which avoids infrared divergencies and will prove most convenient to use in the subsequent sections.

The dispersion equation can be read off from the pole structure of the propagators in Eqs. (23) and (25), which we write as

[TABLE]

Solving (26) yields two massive propagating modes

[TABLE]

with masses

[TABLE]

Both solutions are guaranteed to be real, ensuring the absence of tachyons in the theory. Also, the first solution $\omega_{1}$ corresponds to a physical massive photon, while the second solution $W_{2}$ corresponds to a massive ghost state. This explicitly demonstrates the indefinite metric structure of the theory [42, 43] as one can show by taking the low energy limit $g\to 0$. Hence, expanding our dispersion relations up to the second order in $g$, we have

[TABLE]

and we can see that in the limit $g\to 0$, the physical solution remains regular, as expected, while the ghost solution exhibits a singularity, consistent with its problematic nature; in particular connected to stability and causality.

Furthermore, the equations (29c) and (29d) reveal a significant mass gap between ghost and particle, given by $M_{2}-m_{1}=1/(\gamma g)$. This gap disappears in the pure Chern-Simons limit by taking the limit $\gamma\to\infty$, at which both masses coincide $M_{2}=m_{1}$. Throughout this work, and without loss of generality, we assume both $g>0$ and $\gamma>0$.

II.2 Polarization vectors

We begin by defining an orthogonal basis of $(2+1)$-dimensional Minkowski spacetime, given by the real vectors

[TABLE]

where $G\equiv(k\cdot n)^{2}-k^{2}n^{2}$ and $n^{\mu}$ is an auxiliary field.

The three vector basis $e^{(a)\mu}$ are normalized according to

[TABLE]

with $a=0,1,2$ and $g_{ab}=\text{diag.}(1,-1,-1)$. They also satisfy

[TABLE]

We define the complex basis

[TABLE]

Here the $\pm$ modes are orthogonal to the momentum, i.e., $k\cdot\varepsilon^{(\pm)}=0$. The vectors defining the complex basis satisfy the relations

[TABLE]

with $g_{\lambda\lambda^{\prime}}=\text{diag.}(1,-1,-1)$.

This set of complex basis vectors diagonalizes the $S^{\mu\nu}$ as follows

[TABLE]

where

[TABLE]

We notice that $W_{2}$ is a solution related to $\Lambda_{+}(W_{2},\vec{k})=0$ while $\omega_{1}$ is related to $\Lambda_{-}(\omega_{1},\vec{k})=0$.

The dispersion equation follows from the product of the three eigenvalues of $S^{\mu\nu}$

[TABLE]

One can show that

[TABLE]

from where we make the connection with the propagator, by calculating

[TABLE]

We can also make a connection with the equation of motion operator in momentum space

[TABLE]

Thus the set $\{\varepsilon^{(\lambda)\mu}\}$ correspond to independent eigenvector solutions of the equation of motion operator.

III Canonical quantization

According to Ostrogradsky variational formalism [45], the momentum variables in our model can be written as

[TABLE]

The explicit form of momentum variables looks like

[TABLE]

We impose the equal-time commutation relations on the canonical variables

[TABLE]

where all the others are defined to vanish. As shown in Refs. [32, 39], it is very likely that constraints will arise, which in turn lead to the appearance of Dirac brackets [46, 47, 48]. A detailed analysis of this issue is left for future work, as it lies beyond the scope of the present study.

Let us split the gauge field in terms of the massive degrees of freedom corresponding to a massive photon and a massive ghost field respectively, as follows

[TABLE]

We see that the wave-vector in the expansion must be a solution to the following eigenvector equation:

[TABLE]

Since $\bar{A}_{\mu}(x)$ is a massive photon with the positive energy $\omega_{1}$ and mass $m_{1}$ satisfying $\Lambda_{-}(\omega_{1})=0$ we conclude that it belongs to the subspace $(-)$. Proceeding similarly for the ghost with $\Lambda_{+}(W_{2})=0$, we conclude that it is related to the $(+)$ subspace. Furthermore, we notice that $\bar{A}_{\mu}$ and $G_{\mu}$ satisfy the orthogonality condition for the $A_{\mu}$ field, $k^{\mu}\bar{A}_{\mu}=k^{\mu}G_{\mu}=0$. Taking all this into account, we arrive at an expansion of both physical photon and ghost orthogonal to the direction of propagation $k^{\mu}$

[TABLE]

where we used $\Lambda_{\lambda}^{\prime}(k)=\partial_{k_{0}}\Lambda_{\lambda}(k)$ as normalization in order to define a Lorentz invariant measure [49].

From the ETCR (47) (48), we impose the creation and annihilation operators algebra

[TABLE]

with all other commutators being zero and

[TABLE]

Note that the commutator (54) admits negative-norm states, leading to an indefinite metric in Hilbert space.

IV Radiative corrections

In this section, we compute the relevant one-loop radiative corrections of the higher-order QED model in $(2+1)$ dimensions. In the first subsection, we analyze the divergences of the model. In the next subsection, we compute the fermion self-energy and find that it induces a higher-order term proportional to $g\Box$ coming from the higher-order gauge sector. In consequence, we redefine the fermionic Lagrangian in terms of two decoupled fermion fields. In the two last subsections, we compute the vacuum polarization operator and the vertex diagram, by employing the modified fermion propagator.

IV.1 Analysis of divergences

Let us perform an analysis of the superficially divergent integrals in our modified QED. Using the UV asymptotic for the gauge propagator in the Landau gauge (25), one has the superficial degree of divergence $D$

[TABLE]

where $L$ is the number of loops, and $P_{A}$ and $P_{\psi}$ are the numbers of gauge and spinor propagators respectively. Here we used the topological identity $L+V-P=1$, together with the fact that the total number of propagators are $P=P_{A}+P_{\psi}$.

Then, we can relate numbers of vertices, propagators and external legs ($E_{A}$ for external photons and $E_{\psi}$ for external fermions) calculating the numbers of all fields associated to all vertices. Since we have basically a QED theory with the standard gauge-spinor vertex, we have

[TABLE]

for the gauge and fermion fields respectively, relating the number of vertices, propagators and external legs. This allows us to write $D$ as

[TABLE]

From the formula for $D$ we find that in divergent diagrams, one has $E_{\psi}\leq 2$. In particular, we have for the photon polarization tensor graph $V=2,E_{\psi}=0$, and so $D=1$. We will see that within the dimensional regularization scheme this contribution is finite, see section IV. For the fermion one-loop self-energy graph, one has $V=2,\quad E_{\psi}=2,\quad D=-1$ which is finite. The fermion-photon triple vertex function is also finite, corresponding to $D=-2$. All one-loop Feynman diagrams with more numbers of legs and vertices are finite as well. Performing a general analysis we have that the mass dimension in our interacting theory is $\Delta=1/2$ [50, 51].

We conclude that our theory is super-renormalizable and we can have divergences up to two loops, by straightforward consideration of Feynman diagrams. However, by gauge symmetry reasons the only potentially divergent contribution from the gauge sector, that is, $A^{\alpha}A_{\alpha}$ term cannot arise even in two loops whether corrections with derivatives (CS and Maxwell term) are finite in two loops. So, the only possible divergences in principle are the two-loop corrections to the spinor kinetic term and spinor-vector vertex.

IV.2 Fermion self energy

Let us write down the two-point function for the spinor field:

[TABLE]

We work in the Landau gauge (25) and set $\gamma=1$ ensuring a nontrivial impact of the Maxwell term.

The fermion propagator is the standard one

[TABLE]

This expression can be written as a following sum of four terms:

[TABLE]

where the first one is

[TABLE]

To simplify the calculation of the different pieces, we rewrite the pole structure of the gauge propagator into its particle and ghost contributions as

[TABLE]

The first term is dominated by the particle mass $m_{1}$ and the second term by the ghost mass $M_{2}$. This decomposition results to be crucial to radiative corrections.

In order to compute the above corrections we shall use the Feynman parametrization

[TABLE]

With suitable choices of $A$ and $B$ together with a shift in the momenta, $k+xp\rightarrow l$, and parity properties, the first piece of the fermion self-energy in (64) reduces to

[TABLE]

with

[TABLE]

where $\alpha_{i}$ is the notation for either $m_{1}$ or $M_{2}$.

In an analogous way, the second piece is

[TABLE]

with

[TABLE]

[TABLE]

the third piece is

[TABLE]

and, finally, the fourth piece is

[TABLE]

with

[TABLE]

The first integral in (74) has to be analytically continued from arbitrary $d$-dimensions to $d=3$ in order to produce a finite result.

Finally, the total radiative corrections induce in the fermion self-energy the following higher-order operator of the form

[TABLE]

where $A$, $B$ and $C$ are functions that do not contribute to the $p^{2}$ order.

Now, focusing on the fermionic part, we note that the corrections induce a general structure in the fermionic Lagrangian of the type

[TABLE]

We decompose the above Lagrangian in two parts

[TABLE]

where

[TABLE]

with

[TABLE]

The fermionic masses are

[TABLE]

where the first corresponds to a physical fermion and the second to a ghost fermion of very high mass of the order $1/g$. It is interesting to note the analogy of the two fermion species that arises when including an odd mass terms in the fermion Lagrangian [33], which, however are physical.

Both propagators can be written as

[TABLE]

Note that the fermion propagator described in (77) can be also obtained without resorting to the explicit decomposition of the fermion fields $\psi_{1,2}$, that is to say, we can consider

[TABLE]

with $S_{1,2}(p)$ given in (84) and (85). In the next section, we shall use the decomposed expression (IV.2) in order to evaluate the photon polarization operator.

IV.3 Photon polarization operator

Now we evaluate the photon polarization operator (PPO)

[TABLE]

For this, we use the decomposed fermion propagator given in (IV.2), and define the contributions

[TABLE]

with

[TABLE]

where $i,j=1,2$. We also introduce the notation for the different contributions

[TABLE]

Since each contribution in (91) has a similar form, we compute the general contribution

[TABLE]

where we have used the expressions (13) and (14) in the trace calculation.

Using the Feynman parametrization together with a shift in the momenta, $k+xq\rightarrow l$, and using parity properties, the PPO in (IV.3) reduces in $d$ dimensions to

[TABLE]

where we have used dimensional regularization together with the symmetrization $l^{\mu}l^{\nu}\rightarrow\frac{1}{d}l^{2}\eta^{\mu\nu}$ and introduced the element

[TABLE]

In this way, the integral over momenta can be straightforwardly performed using the identities given in Ref. [51], arriving at

[TABLE]

with

[TABLE]

To analyze the divergent behavior in the PPO in $2+1$ dimensions, let us introduce the dimensional regularization requiring the space-time dimension to be $d=3-\epsilon$ and take the limit $\epsilon\rightarrow 0$, obtaining

[TABLE]

All these contributions are finite, as can be seen from the absence of $1/\epsilon$ terms.

After integrating over the Feynman parameter, we have

[TABLE]

[TABLE]

and

[TABLE]

Taking into account the above-defined notation, in an analogous way, we define for the pieces $e,o,\eta$ the total contributions $\Pi^{e,\eta,o}=\Pi^{e,\eta,o}_{11}-2\Pi^{e,\eta,o}_{12}+\Pi^{e,\eta,o}_{22}$, and we can see that

[TABLE]

[TABLE]

and

[TABLE]

Let us explore the behavior of the contributions (105), (IV.3) and (IV.3), when $g$ is very small. In this case the mass $\overline{m}_{2}$ becomes the dominant energy scale compared to $\overline{m}_{1}\rightarrow m$, which yields

[TABLE]

[TABLE]

and

[TABLE]

Another interesting scenario corresponds to the UV limit so that the higher derivative terms have important contributions. In this case with $q\rightarrow\infty$, each coefficient has the form

[TABLE]

Note that $\Pi^{e}$ in our case has a better ultraviolet behavior than in the case of the absence of higher derivative contributions where it has the form

[TABLE]

For the metric coefficient, we have

[TABLE]

Note that in the absence of higher derivative contributions, this term does not exist.

Finally, in this limit the odd term gives

[TABLE]

from where we notice that the ultraviolet behavior is different to the standard CS QED result (see e.g. [52]):

[TABLE]

IV.4 The vertex diagram

Let us expand the propagator $S(k+p)$ given in (IV.2), into power series in the external momentum $p$ up to the first order as

[TABLE]

in such a way that the two-point function of the spinor field given in (61) can be expanded as

[TABLE]

with

[TABLE]

where dots are for irrelevant terms of zero, second, and higher orders.

On the other hand, the three-point vertex function, in zero order in external momenta and in the Landau gauge, looks like

[TABLE]

Comparing this expression with (118), we see that they are equal up to the overall factor. Therefore we see that the corrections to the kinetic term for the spinor field and the triple spinor-vector vertex are described by the same integral over momenta, hence, the gauge symmetry is not jeopardized by quantum corrections which is consistent with the Ward identities up to first order in external momenta. In other words, the "covariant derivative" term $\bar{\psi}\not{D}\psi$ receives the quantum correction $\bar{\psi}\Gamma^{\mu}D_{\mu}\psi$, where

[TABLE]

Now it is instructive to check the gauge invariance of the quantum correction generated on the spinor sector which as it is well known, is deeply related with the Ward identities. We proceed with the computation of the vertex function and take advantage of the previous computation.

Within our study, we consider the lower (zero) order contributions to the three-point function only. Therefore, we require $p=0$ and $p^{\prime}=0$ and arrive at

[TABLE]

Similarly to the above PPO calculation, we use the decomposed fermion propagator (IV.2) in (121), so, the vertex contributions read

[TABLE]

where $i,j=1,2.$ As before, we introduce the notation for the different contributions

[TABLE]

Since all contributions have the same form, we calculate its general form given by

[TABLE]

Using the identity

[TABLE]

together with (14), the three-point vertex function in (IV.4) becomes

[TABLE]

where the $(\Gamma^{\mu}_{ij})^{a}$ and $(\Gamma^{\mu}_{ij})^{b}$ are the contributions without and with the Levi-Civita symbol coming from the gauge boson propagator, respectively. These are given by

[TABLE]

and

[TABLE]

Now, we use (14) together with symmetry properties on the integral over momenta in the above equations, getting

[TABLE]

and

[TABLE]

Once we carried out the Wick rotation, we have two integrals of the form

[TABLE]

and

[TABLE]

which can be easily performed using a standard procedure, yielding

[TABLE]

and

[TABLE]

With these results, we can see that $\Gamma^{\mu}=\Gamma^{\mu}_{11}-2\Gamma^{\mu}_{12}+\Gamma^{\mu}_{22}$ has the form

[TABLE]

Since this one-loop correction to the vertex has a simple $\gamma^{\mu}$ structure, its coefficient can be identified as the form factor $F_{1}(q^{2}=0)$, so the vertex can be written as

[TABLE]

which is expected, as we set $p=p^{\prime}=0$, eliminating any other available tensor structure.

As (135) is a complete result for an arbitrary value of $\gamma$, to make contact with the literature, let us consider a pure CS theory by taking the limit $\gamma\rightarrow\infty$ in (135), getting

[TABLE]

where we use (28a), (28b), (82) and (83). Additionally, if we take off the higher derivative contributions in (137) by setting $g\rightarrow 0$, we arrive at [27]

[TABLE]

V Microcausality in the gauge sector

The causal structure of a quantum field theory is determined by the behavior of the commutator of fundamental fields at two distinct spacetime points, $x$ and $y$, outside the light cone. In what follows, we analyze the implications of this criterion for causality in the gauge sector. Hence, let us consider the commutator

[TABLE]

and verify whether it vanishes outside the lightcone, i.e., when $(x-y)^{2}<0$.

From the decomposition of the gauge field (49), we have the contributions

[TABLE]

The first contribution is

[TABLE]

and the second

[TABLE]

Let us define $z=x-y$ and change $\vec{k}\rightarrow-\vec{k}$ in the second terms of Eqs (V) and (V), obtaining

[TABLE]

and

[TABLE]

where we have used that the denominators are even functions of $\vec{k}$, see the relations (55) and (56).

We recall the identity (40)

[TABLE]

We take the complex conjugate

[TABLE]

Thus, we conclude that

[TABLE]

which leads to the relations

[TABLE]

Using these results, the total expression for the commutator is

[TABLE]

We consider

[TABLE]

and by evaluating in their respective frequencies we recall what we have obtained before

[TABLE]

By the other hand we recall some previous results (II.1), and by evaluating $k_{0}$ in a root of $k^{2}-\gamma^{2}\mathcal{M}^{2}$, we arrive at

[TABLE]

We can establish a connection with the polarization vectors

[TABLE]

Therefore

[TABLE]

We introduce the contour integral $\mathcal{C}$ in the complex $k_{0}$-plane that encloses all four poles in the counterclockwise direction, and write

[TABLE]

or

[TABLE]

Finally we arrive at

[TABLE]

and again integrating

[TABLE]

We will use the following relations

[TABLE]

where we have used (38b) and (38c).

We start with the $00$ contributions

[TABLE]

For the contributions to $I_{0i}$, we have

[TABLE]

and

[TABLE]

where we defined $\epsilon_{0ij}=\epsilon_{ij}$.

The last ones, are the contributions from the indices $ij$

[TABLE]

and

[TABLE]

Note that the elements have the expected symmetry of $z\to-z$ and swapping $\mu\to\nu$.

Each element can be written as

[TABLE]

Due to Lorentz covariance, the commutator can be evaluated in any convenient frame. Since we are interested in the spacelike region $z^{2}=(x-y)^{2}<0$, we choose a frame where $z_{0}=0$. For completeness, the component $D_{00}(z)$ is also evaluated in a general frame in Appendix A.

In the frame wtih $z_{0}=0$ the elements reduce to

[TABLE]

We observe that $D_{ij}$ and $D_{0i}$ are proportional to $\delta(|\vec{z}|)/|\vec{z}|$ and its derivative, respectively. These expressions vanish outside the light cone, where $|\vec{z}|>0$. Therefore, microcausality is preserved in the model.

VI Final Remarks

In this work, we have studied the higher-derivative extension of three-dimensional QED which incorporate gauge-invariant CS modifications. In particular, we have considered a QED with standard fermions, CS gauge invariant extensions including higher-order and a standard interaction term. While earlier gauge extensions were treated mostly at the tree level [39], now we studied the one-loop perturbative impact of the presence of the higher-derivative CS term.

Explicitly, we have found that the higher-order gauge field propagates through two different modes, one corresponding to a physical massive field and the other to a massive ghost field. We have derived the polarization vectors for both Proca-like fields, each corresponding to a subset $\{\varepsilon^{(\pm)}\}$ described by transverse vectors in momentum space. Also, we have quantized the gauge theory and verified the equal-time-commutation-relations between basis variables. At this level, we have shown that the theory has an indefinite metric structure which shows up in the commutators of creation and annihilation operators of the gauge field.

We have computed the two-point function of the spinor field which induces radiative correction that allow to redefine a Lagrangian in terms of two decoupled standard fermions, one being a ghost, which has served to compute the rest of raditive corrections. With this decomposition, we have computed the PPO where the complete dependence on the external momentum was obtained, and calculated the two- and three-point functions involving the spinor field. We have seen that in the gauge sector a redefinition of variables to cast the theory into the sum of Proca-like fields may be prohibited since these will eventually destroy gauge invariance. Moreover, we explicitly checked that our theory is microcausal in the gauge sector. The results for the spinor sector, can be used within the studies of the three-dimensional effective field theories applied within the condensed matter context, especially within studies of graphene which is a perfect three-dimensional model. We also leave for future work the finite renormalization of the model.

Acknowledgments

The work by AYuP has been supported by the CNPq project No. 303777/2023-0. The research of CMR was partially supported by Fondecyt Regular project No. 1241369; and wants to thank the kind hospitality at Universidad Nacional Autónoma de México (UNAM) where this work was finished. CR acknowledges support from the ANID fellowship No. 21211384 and Universidad de Concepción. AS was supported by DGAPA-UNAM under Grant No. PAPIIT-IN108123.

Appendix A The calculation of $D_{00}$

Here we analyze a more general frame with $z_{0}\neq 0$, starting from Eq. (173). In polar coordinates, defined by

[TABLE]

we obtain

[TABLE]

Considering the following known integrals

[TABLE]

and using the fact that the following integrals vanish due to parity symmetry

[TABLE]

we are led to the following set of integrals to evaluate. For $D_{00}(z)$

[TABLE]

For $D_{0i}(z)$ we have the components

[TABLE]

Finally for the $D_{ij}(z)$ components we have:

[TABLE]

To illustrate how the case with $z_{0}\neq 0$ leads to the same conclusion as the $z_{0}=0$ frame, we explicitly compute the $D_{00}(z)$ integral given in Eq. (185). In this case, we encounter an integral of the form:

[TABLE]

for $\omega=\omega_{1},W_{2}$. We know that the Bessel functions satisfy the ODE

[TABLE]

that in our particular case means

[TABLE]

By using this expression we can write the integral as

[TABLE]

Now by changing the integration variable as $\tau=\omega(k),\mathrm{d}\tau=\frac{k\mathrm{d}k}{\omega(k)}$ then we obtain

[TABLE]

From [53] (item 6.677-1), the previous integral has the form

[TABLE]

In our case $a=m$, $b=z$ and $c=z_{0}$, thus

[TABLE]

We conclude that outside the lightcone the integral $I(z)$ vanishes. The other integrals can be calculated in the same way.