Gauge Symmetry and Radiatively Induced Terms in Dimension-5 Non-Minimal Lorentz-Violating QED

Abstract

In this work, we derive the conditions that ensure gauge invariance of non-minimal dimension-5 Lorentz-violating QED. The two- and three-point functions at one loop are computed. The gauge Ward identities are checked, as well as the conditions, to ensure that the gauge symmetry of this non-minimal framework is found to be the same of the usual QED. Induced terms are also investigated, and it is shown that the non-minimal Lorentz-violating $a_F^{\left(5\right)}$-term of the fermion sector can radiatively induce a non-minimal Lorentz-violating term in the photon sector.

1. Introduction

Lorentz and CPT Symmetries are the main ingredients needed to build quantum field theories and are usually taken for granted. Nevertheless, in string theory, it was realized long ago that these symmetries can be spontaneously broken at Planck scale, and as a result, Lorentz and CPT-violating terms appear at low energies [1]. All possible Lorentz- and CPT-violating operators at these low energies make up the Standard Model Extension (SME) [2,3]. Even if space–time symmetries are, in fact, exact in nature, the issue centers upon how precise one can say they are. In this way, the SME is also a framework to test how good are these symmetries besides being a way of searching for Lorentz and CPT violation.

Precise tests of space–time symmetries generally involve classical Lagrangians or tree-level Feynman diagrams of the SME. However, there are many theoretical studies concerning this framework beyond tree level. The SME was shown to be renormalizable at one loop for both the electroweak [4,5] and strong sectors [6,7]. There are also several investigations concerning the radiatively induced finite quantum corrections. The most known example is the dimension-3 operator associated with the CPT- and Lorentz-violating coefficient $b_{\mu }$ from the fermion sector, which radiatively induces the Chern–Simons-like term of the photon sector [8,9,10,11,12,13,14,15,16]. Furthermore, breaking Lorentz and CPT symmetries at the classical level does not mean that all the possible violating structures at this level also appear at the quantum one [17]. Thus, it is worth investigating what symmetries can be broken at the quantum level if the Lorentz and CPT ones are broken at the classical one.

At the same time, from a technical point of view, the regularization procedure is required beyond tree level to treat ultraviolet and/or infrared divergences. As is well known, the treatment of these infinities can spuriously break symmetries of the model. For instance, lattice regularization is developed for treating QCD non-perturbatively and discretization of the space-time points breaks Lorentz symmetry, among others [18]. The renormalization process is then more laborious because the introduction of restoring counter-terms is needed. In the study of anomalies, this issue is more subtle because we want to know if the symmetry is indeed physical or if it was broken by the regularization scheme. Some anomalies are measurable, like the chiral anomaly or the scale anomaly in QED, and this experimental fact solves the issue. The former is related to the pion decay, split into two photons [19,20,21], and the latter is related to the hadronic R ratio [22]. On the other hand, there are not yet observables related to super-symmetric anomalies or anomalies in the Standard Model Extension. Thus, the question of whether a symmetry is in fact broken at the quantum level can only be answered by using a regularization scheme that does not spuriously break any symmetry of the model.

Gauge symmetry is one of the ways of introducing particle interactions. Maintaining this symmetry beyond tree level ensures renormalizability of the theory and a massless photon. Therefore, the search for a gauge-invariant regularization method is desired, mainly for gauge field theories, like dimensional regularization [23,24]. However, as new theories arose, so did other proposals of regularization schemes for dealing with issues that conventional regularizations could not deal with. In this work, we show that there is no gauge anomaly at one loop for non-minimal dimension-5 Lorentz-violating QED. If Bose symmetry is not required, there would be a gauge anomaly due to the three-point function diagrams, and the induced terms could be used to put stringent constraints in a set of non-minimal dimension-5 Lorentz and CPT-violating coefficients since gauge symmetry breaking is not observed. This paper is divided as follows: In Section 2, we present an overview of the implicit regularization scheme, and an example of one-loop computation for the usual QED and the non-minimal framework are considered. In particular, the conditions for gauge invariance in the usual QED are discussed in Section 2.1. In Section 3, we compute the gauge Ward identities of the two- and three-point Green functions of the non-minimal Lorentz-violating QED considering the $a_F^{\left(5\right)}$-term, using both dimension and implicit regularizations. In Section 4, we compute these same identities considering the non-minimal $b_F^{\left(5\right)}$-term, and we present conclusions in Section 5.

2. Setup and Overview of Implicit Regularization

Besides using dimensional regularization [23,24] to compute divergent integrals in amplitudes in the next sections, we also apply the implicit regularization scheme [25]. The former is probably the most known and popular regularization scheme, and it does not need an introduction. The latter, on the other hand, although used in a wide variety of problems, is not known in textbooks.

A particularly interesting application of implicit regularization occurs when the theories include dimension-specific objects, like ${\gamma }^5$ matrices and Levi-Civita symbols. Also, since it is a scheme that does not break symmetries of the theory, it is usually used for the computation of anomalies. A recent computation for a general momentum routing concerns gravitational anomalies in two dimensions [26]. It was also used in other scenarios with Lorentz violation, like in the Bumblebee model [27], or chiral models [28], which deal directly with ${\gamma }^5$ matrices, in which comparisons with other regularization techniques are performed [27,28,29].

Let us briefly review the method in four dimensions. In this scheme, we assume that the integrals are regularized by an implicit regulator $\mathrm{\Lambda }$ in order to allow algebraic operations within the integrands. We then recursively use the following identity

$${\displaystyle \frac{1}{{(k+p)}^2-m^2}}={\displaystyle \frac{1}{k^2-m^2}}-{\displaystyle \frac{(p^2+2p·k)}{(k^2-m^2)[{(k+p)}^2-m^2]}},$$

(1)

in order to separate basic divergent integrals (BDIs) from the finite part. These BDIs are defined as follows:

$$I_{log}^{{\mu }_1\cdots {\mu }_{2n}}\left(m^2\right)\equiv {\int }_k{\displaystyle \frac{k^{{\mu }_1}\cdots k^{{\mu }_{2n}}}{{(k^2-m^2)}^{2+n}}}$$

(2)

and

$$I_{quad}^{{\mu }_1\cdots {\mu }_{2n}}\left(m^2\right)\equiv {\int }_k{\displaystyle \frac{k^{{\mu }_1}\cdots k^{{\mu }_{2n}}}{{(k^2-m^2)}^{1+n}}}.$$

(3)

The BDIs with Lorentz indices can be judiciously combined as differences between integrals with the same superficial degree of divergence, according to the equations below, which define surface terms 1:

$$\begin{array}{c}\hfill {{\rm Y}}_{2w}^{\mu \nu }=g^{\mu \nu }{I}_{2w}\left(m^2\right)-2(2-w)I_{2w}^{\mu \nu }\left(m^2\right)\equiv {\upsilon }_{2w}{g}^{\mu \nu },\end{array}$$

(4)

$$\begin{array}{c}\hfill {\mathrm{\Xi }}_{2w}^{\mu \nu \alpha \beta }=g^{\{\mu \nu }{g}^{\alpha \beta \}}{I}_{2w}\left(m^2\right)-4(3-w)(2-w)I_{2w}^{\mu \nu \alpha \beta }\left(m^2\right)\equiv \\ \hfill \mathrm{\equiv }{\xi }_{2w}(g^{\mu \nu }{g}^{\alpha \beta }+g^{\mu \alpha }{g}^{\nu \beta }+g^{\mu \beta }{g}^{\nu \alpha }).\end{array}$$

(5)

In the expressions above, $2w$ is the degree of divergence of the integrals, and we adopt the notation such that indices 0 and 2 mean $log$ and $quad$, respectively. Surface terms can be conveniently written as integrals of total derivatives, as presented below:

$$\begin{array}{c}\hfill {\upsilon }_{2w}{g}^{\mu \nu }={\int }_k{\displaystyle \frac{\partial }{\partial k_{\nu }}}{\displaystyle \frac{k^{\mu }}{{(k^2-m^2)}^{2-w}}},\end{array}$$

(6)

$$\begin{array}{c}\hfill ({\xi }_{2w}-v_{2w})(g^{\mu \nu }{g}^{\alpha \beta }+g^{\mu \alpha }{g}^{\nu \beta }+g^{\mu \beta }{g}^{\nu \alpha })={\int }_k{\displaystyle \frac{\partial }{\partial k_{\nu }}}{\displaystyle \frac{2(2-w)k^{\mu }{k}^{\alpha }{k}^{\beta }}{{(k^2-m^2)}^{3-w}}}.\end{array}$$

(7)

We see that the surface terms in Equations (4) and (5) are undetermined because they are differences between divergent quantities. Each regularization scheme gives a different value for these terms. However, as physics should not depend on the scheme applied, we leave these terms to be arbitrary until the end of the calculation and then fix them by symmetry constraints or phenomenology. This approach was first proposed in [30], where undetermined surface terms were discussed in several contexts of quantum corrections.

Of course, the same idea can be applied for any dimension of space-time and for higher loops. Equation (1) is used recursively until the divergent piece is separated from the finite one. This procedure makes the finite integrals hard to compute due to the number of k’s in the numerator. A simpler alternative to this approach is presented in [31], where the Feynman parametrization is applied before separating the BDIs. Also, Equation (1) is not the only possible equation to use since the implicit regulator is assumed to allow the use of other identities.

2.1. An Example: Gauge Invariance of the Vacuum Polarization Tensor

Let us consider the vacuum polarization tensor of conventional Quantum Electrodynamics (QED) whose computation in implicit regularization is given by (all regularized integrals are presented in the Appendix A):

$$\begin{array}{cc}\hfill i{\mathrm{\Pi }}^{\mu \nu }\left(p\right)& ={\displaystyle \frac{4}{3}}{e}^2(p^2{g}^{\mu \nu }-p^{\mu }{p}^{\nu })I_{log}\left(m^2\right)-4e^2{\upsilon }_2{g}^{\mu \nu }+{\displaystyle \frac{4}{3}}{e}^2(p^2{g}^{\mu \nu }-p^{\mu }{p}^{\nu }){\upsilon }_0\hfill \\ \hfill & -{\displaystyle \frac{4}{3}}{e}^2(p^2{g}^{\mu \nu }+2p^{\mu }{p}^{\nu })({\xi }_0-2{\upsilon }_0)-{\displaystyle \frac{i}{2{\pi }^2}}{e}^2(p^2{g}^{\mu \nu }-p^{\mu }{p}^{\nu })(Z_1-Z_2),\hfill \end{array}$$

(8)

in which $Z_n={\int }_0^{1}dx{x}^{n}ln\left({\displaystyle \frac{D\left(x\right)}{m^2}}\right)$ and $D\left(x\right)=m^2-p^{2}x(1-x)$, and for didactic reasons, we have placed the divergent, the finite, and surface-dependent terms separately.

Notice that if we require gauge invariance using the Ward identity $p_{\mu }{\mathrm{\Pi }}^{\mu \nu }\left(p\right)=0$, we find that the quadratic surface term ${\upsilon }_2$ must be zero and that the logarithmic surface terms must obey the relation ${\xi }_0=2{\upsilon }_0$. These conditions are automatically fulfilled if we set all surface terms to zero. The same takes place when one uses dimensional regularization [23,24], because the surface terms defined in Section 2 are zero in that scheme. This is the same condition obtained if we require the momentum routing invariance of the Feynman diagram in Figure 1 (this feature of the loop diagram would not appear in dimensional regularization [23,24], since it allows for shifts in the loop momenta). In the next sections, we are going to show that these requirements for surface terms are the same for a non-minimal dimension-5 Lorentz-violating version of QED. The Ward identities are checked for two- and three-point functions in this Lorentz-violating framework.

It is also interesting to mention that in sharp cut-off regularization ${\xi }_0-2{\upsilon }_0\ne 0$. That is why it is known for breaking gauge symmetry. Some techniques present similar procedures as the ones from implicit regularization, like Loop Regularization [32] in the momentum space and Constrained Differential Renormalization (CDR) [33] in the coordinate space. Like implicit regularization, CDR relates parameters in such a way that gauge symmetric amplitudes are obtained. The complexity of the relation among surface terms depends on the number of integrated momenta k’s in the numerator of the integrand. This will also determine if different surface terms appear.

The diagrams with more external photon legs in usual QED do not need to be checked like the ones of the next sections. The three-photon leg approach of the usual QED is zero because of Furry’s theorem [34], and the box diagram with four photon legs is gauge-invariant, since there is no term in the tree level Lagrangian to renormalize it if it was not.

2.2. Non-Minimal Dimension-5 Lorentz-Violating QED

In this subsection, we present the kind of non-minimal Lorentz violation we will consider in the next sections. All gauge-invariant Lorentz-violating operators of arbitrary dimension are built and classified in reference [35]. As mentioned in the Introduction, the framework considered is presented in

Table 1. Summary of the conditions for gauge invariance in loop diagrams of a non-minimal dimension-5 Lorentz-violating QED.

Non-Minimal Term	Ward Identity	Dimensional Reg.	Implicit Reg.
Usual QED	2-point function	✓	${\upsilon }_2=0$ and ${\xi }_0=2{\upsilon }_0$
	3-point function	Null by Furry’s theorem	Null by Furry’s theorem
$a_F^{\left(5\right)}$-term	2-point function	✓	${\upsilon }_2=0$ and ${\xi }_0=2{\upsilon }_0$
	3-point function	✓	✓
$b_F^{\left(5\right)}$-term	2-point function	Null by Clifford algebra	Null by Clifford algebra
	3-point function	Cannot be applied	${\upsilon }_0=0$

of this paper. We work specifically with Lagrangian ${\mathcal{L}}_{\psi F}^{\left(5\right)}$ of the fermion sector, and we rewrite it below for ease of reference throughout the text:

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\psi F}^{\left(5\right)}& =-{\displaystyle \frac{1}{2}}{m}_F^{\left(5\right)\alpha \beta }\overline{\psi }{F}_{\alpha \beta }\psi -{\displaystyle \frac{1}{2}}{m}_{5F}^{\left(5\right)\alpha \beta }\overline{\psi }{\gamma }_5{F}_{\alpha \beta }\psi -{\displaystyle \frac{1}{2}}{\left(a_F^{\left(5\right)}\right)}^{\mu \alpha \beta }\overline{\psi }{\gamma }_{\mu }{F}_{\alpha \beta }\psi -{\displaystyle \frac{1}{2}}{\left(b_F^{\left(5\right)}\right)}^{\mu \alpha \beta }\overline{\psi }{\gamma }_5{\gamma }_{\mu }{F}_{\alpha \beta }\psi \hfill \\ \hfill & -{\displaystyle \frac{1}{4}}{\left(H_F^{\left(5\right)}\right)}^{\mu \nu \alpha \beta }\overline{\psi }{\sigma }_{\mu \nu }{F}_{\alpha \beta }\psi ,\hfill \end{array}$$

(9)

where the index $\left(5\right)$ refers to the dimension of the Lorentz-violating operator, and F refers to direct coupling with the abelian gauge field (some of the non-minimal dimension-5 violating terms are introduced via covariant derivatives. The ones from Lagrangian ${\mathcal{L}}_{\psi D}^{\left(5\right)}$ are not considered here.) The tensors $m_F^{\left(5\right)}$, $m_{5F}^{\left(5\right)}$, $a_F^{\left(5\right)}$, $b_F^{\left(5\right)}$ and $H_F^{\left(5\right)}$ are all coefficients that govern Lorentz violation, but only $a_F^{\left(5\right)}$ and $b_F^{\left(5\right)}$ breaks CPT as well, since the number of indices of these coefficients is odd.

Although all the Lorentz-violating terms of arbitrary dimension of reference [35] are also built based on gauge invariance, it is not obvious that they will be gauge-invariant beyond tree level. Perhaps the most known example of gauge invariance breaking occurs in the diagrammatic computation of the AVV triangle diagram [19,20,21]. In this case, if chiral symmetry is achieved at the quantum level for massless theories, gauge invariance is broken. On the other hand, the usual approach, known as the chiral anomaly, is to consider that gauge invariance holds and chiral symmetry is broken, meaning that the Ward identities are, according to the pion decay, split into two photons. In this case, we are going to work with Ward identities to address symmetry issues beyond tree level. These identities work in the momentum space and fit well for the implicit regularization scheme, which is built for the momentum space. There are also different approaches for studying gauge invariance issues, such as Landau–Khalatnikov–Fradkin transformations, which work in the coordinate space [36]. This approach would need to be adapted for frameworks with Lorentz violation.

3. Gauge Invariance and Radiatively Induced Terms: ${\mathit{a}}_{\mathit{F}}^{\mathbf{\left(}\mathbf{5}\mathbf{\right)}}$-Term

We recall the dimension-5 coefficients in the Lagrangian density ${\mathcal{L}}_{\psi F}^{\left(5\right)}$ of Table 1 of reference [35], mentioned in the previous section. The terms with coefficients ${\left(m_F^{\left(5\right)}\right)}^{\alpha \beta }$, ${\left(m_{5F}^{\left(5\right)}\right)}^{\alpha \beta }$ and ${\left(b_F^{\left(5\right)}\right)}^{\mu \alpha \beta }$ do not generate induced terms at first order because the number of Dirac matrices in the trace of the fermion loop is odd or there are less than four Dirac matrices appearing with a ${\gamma }_5$ matrix inside this trace. The non-minimal term ${\left(H_F^{\left(5\right)}\right)}^{\mu \nu \alpha \beta }\overline{\psi }{\sigma }_{\mu \nu }{F}_{\alpha \beta }\psi$ was studied in [37], in which the authors showed that it radiatively induces the CPT-even term ${\left(k_F\right)}^{\mu \nu \alpha \beta }{F}_{\alpha \beta }{F}_{\mu \nu }$ of the minimal SME photon sector. The other non-trivial radiatively induced term comes from the CPT and Lorentz-violating non-minimal term $-{\displaystyle \frac{1}{2}}{\left(a_F^{\left(5\right)}\right)}^{\mu \alpha \beta }\overline{\psi }{\gamma }_{\mu }{F}_{\alpha \beta }\psi$. It can be rewritten as $-{\left(a_F^{\left(5\right)}\right)}^{\mu \alpha \beta }\overline{\psi }{\gamma }_{\mu }{\partial }_{\alpha }{A}_{\beta }\psi$ due to the anti-symmetry of the last two indices, and this leads to the Feynman rule presented in Figure 2. It is also important to note that it is easy to check with the modified version of the Dirac equation that this term does not break gauge symmetry at the classical level, i.e., ${\partial }_{\mu }{j}^{\mu }=0$, where $j^{\mu }=\overline{\psi }{\gamma }^{\mu }\psi$.

Next, we perform perturbative calculations in a modified Lorentz-violating QED model which includes this term.

3.1. Two-Point Function

The diagrams depicted in Figure 3 give rise to a radiatively induced non-minimal LV term in the photon sector. In order to show this, we compute the diagrams of this figure. Their corresponding amplitudes can be written as

$$\begin{array}{cc}\hfill & {\mathrm{\Pi }}_{\left(a\right)}^{\alpha \beta }\left(p\right)=-{\int }_{k}Tr\left[{\left(a_F^{\left(5\right)}\right)}^{\mu \lambda \beta }{\gamma }_{\mu }{p}_{\lambda }{\displaystyle \frac{i}{\cancel{k}-\cancel{p}-m}}(-ie{\gamma }^{\alpha }){\displaystyle \frac{i}{\cancel{k}-m}}\right]\hfill \end{array}$$

(10)

and

$$\begin{array}{cc}\hfill & {\mathrm{\Pi }}_{\left(b\right)}^{\alpha \beta }\left(p\right)=-{\int }_{k}Tr\left[(-ie{\gamma }^{\beta }){\displaystyle \frac{i}{\cancel{k}-\cancel{p}-m}}{\left(a_F^{\left(5\right)}\right)}^{\mu \lambda \alpha }{\gamma }_{\mu }{p}_{\lambda }{\displaystyle \frac{i}{\cancel{k}-m}}\right],\hfill \end{array}$$

(11)

where ${\int }_k$ stands for $\int {\displaystyle \frac{d^{4}k}{{\left(2\pi \right)}^4}}$. Let us then apply implicit regularization, presented in Section 2, in the computation of the diagrams of the two-point function. After taking the traces and regularizing (all integrals are presented in Appendix A), we find the following result:

$$\begin{array}{c}{\mathrm{\Pi }}^{\alpha \beta }\left(p\right)={\mathrm{\Pi }}_{\left(a\right)}^{\alpha \beta }\left(p\right)+{\mathrm{\Pi }}_{\left(b\right)}^{\alpha \beta }\left(p\right)={\displaystyle \frac{4}{3}}i({\left(a_F^{\left(5\right)}\right)}^{pp\beta }{p}^{\alpha }-{\left(a_F^{\left(5\right)}\right)}^{\alpha p\beta }{p}^2)\left(I_{log}\left(m^2\right)-b{Z}_0-{\displaystyle \frac{i}{48{\pi }^2}}\right)+\hfill \\ +{\displaystyle \frac{8ib}{3}}{m}^2{Z}_0\left(-{\left(a_F^{\left(5\right)}\right)}^{\alpha p\beta }+{\displaystyle \frac{p^{\alpha }}{p^2}}{\left(a_F^{\left(5\right)}\right)}^{pp\beta }\right)+4({\left(a_F^{\left(5\right)}\right)}^{pp\beta }{p}^{\alpha }+{\left(a_F^{\left(5\right)}\right)}^{\alpha p\beta }{p}^2){\upsilon }_0-\hfill \\ -{\displaystyle \frac{4}{3}}(-2{\left(a_F^{\left(5\right)}\right)}^{pp\beta }{p}^{\alpha }+{\left(a_F^{\left(5\right)}\right)}^{\alpha p\beta }{p}^2){\xi }_0+(\alpha \leftrightarrow \beta ),\hfill \end{array}$$

(12)

where ${\left(a_F^{\left(5\right)}\right)}^{\alpha p\beta }\equiv {\left(a_F^{\left(5\right)}\right)}^{\alpha \mu \beta }{p}_{\mu }$ and $b={\displaystyle \frac{i}{{\left(4\pi \right)}^2}}$. We can easily check gauge invariance by computing the Ward identity $p_{\alpha }{\mathrm{\Pi }}^{\alpha \beta }\left(p\right)=0$. Through this, we find out that the surface terms break gauge symmetry, as expected:

$$\begin{array}{c}\hfill p_{\alpha }{\mathrm{\Pi }}^{\alpha \beta }\left(p\right)=4{\left(a_F^{\left(5\right)}\right)}^{pp\beta }{p}^2(-{\xi }_0+2{\upsilon }_0)-4{\upsilon }_2{\left(a_F^{\left(5\right)}\right)}^{pp\beta }\end{array}$$

(13)

We can see above that, if all surface terms are null, gauge symmetry is automatically fulfilled. However, the relations ${\xi }_0=2{\upsilon }_0$ and ${\upsilon }_2=0$ are sufficient. It is interesting to note that these are the same conditions for the gauge invariance presented in Section 2.1 and found in other frameworks like in the minimal QED extension [17].

Alternatively, one can apply dimensional regularization [23,24] in this calculation. The result is the same as in implicit regularization as long as the surface terms of Equation (12) are set to zero and we take

$$I_{log}\left(m^2\right)=b\left({\displaystyle \frac{1}{ϵ}}-{\displaystyle \frac{1}{2}}ln\left(-{\displaystyle \frac{m^2}{4\pi {\mu }^2}}\right)-{\displaystyle \frac{\gamma }{2}}\right),$$

(14)

in which $\mu$ is a mass scale introduced to keep the number of dimensions of the integrals after changing it to d dimensions. This result is expected, since surface terms are zero in dimensional regularization.

In the following discussion, we are interested in the form of the induced term, so we are going to assume for the moment that the surface terms are equal to zero. The non-minimal term of the fermion sector radiatively induces a term in the photon sector which is also non-minimal. In order to show this, let us consider the effective action $S={\displaystyle \frac{1}{2}}\int d^{4}x{A}_{\alpha }{\mathrm{\Pi }}^{\alpha \beta }{A}_{\beta }$ and take the massless limit of Equation (12). This leads to

$$\begin{array}{c}S={\displaystyle \frac{4i}{3}}\int d^{4}x\left\{{\left(a_F^{\left(5\right)}\right)}^{\alpha \mu \beta }{A}_{\alpha }{\partial }^2{\partial }_{\mu }{A}_{\beta }-{\left(a_F^{\left(5\right)}\right)}^{\mu \nu \beta }{A}_{\alpha }{\partial }^{\alpha }{\partial }_{\mu }{\partial }_{\nu }{A}_{\beta }\right\}\left(I_{log}\left({\lambda }^2\right)-bln\left({\displaystyle \frac{-p^2}{{\lambda }^2}}\right)-{\displaystyle \frac{b}{3}}\right)=\hfill \\ ={\displaystyle \frac{2}{3}}\int d^{4}xi{\left(a_F^{\left(5\right)}\right)}^{\alpha \mu \beta }{F}_{\nu \alpha }{\partial }^{\nu }{F}_{\mu \beta }\left(I_{log}\left({\lambda }^2\right)-bln\left({\displaystyle \frac{-p^2}{{\lambda }^2}}\right)-{\displaystyle \frac{b}{3}}\right),\hfill \end{array}$$

(15)

where we have dropped out terms that are total derivatives. The induced term has the form ${\left(a_F^{\left(5\right)}\right)}^{\alpha \mu \beta }{F}_{\nu \alpha }{\partial }^{\nu }{F}_{\mu \beta }$, which is a particular case of

$${\mathcal{L}}_A^{\left(5\right)}=-{\displaystyle \frac{1}{4}}{k}^{\left(5\right)\alpha \kappa \lambda \mu \nu }{F}_{\kappa \lambda }{\partial }_{\alpha }{F}_{\mu \nu },$$

(16)

present in Table 3 of reference [35], in which $k^{\left(5\right)\alpha \kappa \lambda \mu \nu }$ is proportional to

$${\eta }^{\alpha \lambda }{\left(a_F^{\left(5\right)}\right)}^{\kappa \mu \nu }-{\eta }^{\alpha \kappa }{\left(a_F^{\left(5\right)}\right)}^{\lambda \mu \nu }+{\eta }^{\alpha \mu }{\left(a_F^{\left(5\right)}\right)}^{\nu \kappa \lambda }-{\eta }^{\alpha \nu }{\left(a_F^{\left(5\right)}\right)}^{\mu \kappa \lambda }.$$

(17)

Equation (15) shows us the presence of a divergent part in the coefficient of the induced term, which is an important point to be analyzed. This indicates that the original classical action must contain such a term. In other words, the inclusion of the dimension-5 term $-{\displaystyle \frac{1}{2}}{\left(a_F^{\left(5\right)}\right)}^{\mu \alpha \beta }\overline{\psi }{\gamma }_{\mu }{F}_{\alpha \beta }\psi$ in a modified version of QED requires the presence of this induced term from the beginning. However, we must take into account that our model is non-renormalizable. We have carried out a one-loop calculation, and at this order in the perturbative expansion, it has been shown that a new term which violates Lorentz and CPT symmetries should be included in the classical action. If we go beyond the one-loop order, other new terms will certainly have to be considered. The non-renormalizability of the model tells us that there is not a finite number of counter-terms that will be sufficient to renormalize the theory. So, if we would like to deal with this effective model, we will have to stop at the one-loop order. For this, it is necessary to find a cut-off energy $\mathrm{\Lambda }$. Finally, it is easy to check that this term leads to the usual charge conservation at the classical level. The modified Maxwell equations for an additional $C{\left(a_F^{\left(5\right)}\right)}^{\alpha \mu \beta }{F}_{\nu \alpha }{\partial }^{\nu }{F}_{\mu \beta }$ are ${\partial }_{\lambda }{F}^{\lambda \zeta }=J^{\zeta }+C({\left(a_F^{\left(5\right)}\right)}^{\zeta \mu \beta }{\partial }^2{F}_{\mu \beta }-{\left(a_F^{\left(5\right)}\right)}^{\lambda \mu \beta }{\partial }_{\lambda }{\partial }^{\zeta }{F}_{\mu \beta })$, where we can easily see that ${\partial }_{\zeta }{\partial }_{\lambda }{F}^{\lambda \zeta }={\partial }_{\zeta }{J}^{\zeta }=0$.

3.2. Three-Point Function

The next step would be to check gauge symmetry for diagrams with more external photon legs. Let us consider the diagrams of the three-point function presented in Figure 4. Using the Feynman rules set out above, the corresponding amplitudes can be written as follows:

$$\begin{array}{cc}\hfill & T_{\left(a\right)}^{\mu \nu \alpha }(p,q)=-{\int }_{k}Tr\left[(-ie{\gamma }^{\mu }){\displaystyle \frac{i}{\cancel{k}-m}}(-ie{\gamma }^{\nu }){\displaystyle \frac{i}{\cancel{k}+\cancel{q}-m}}{\left(a_F^{\left(5\right)}\right)}^{\lambda \zeta \alpha }{(p+q)}_{\zeta }{\gamma }_{\lambda }{\displaystyle \frac{i}{\cancel{k}-\cancel{p}-m}}\right],\hfill \\ \hfill & T_{\left(b\right)}^{\mu \nu \alpha }(p,q)=-{\int }_{k}Tr\left[{\left(a_F^{\left(5\right)}\right)}^{\lambda \zeta \mu }{p}_{\zeta }{\gamma }_{\lambda }{\displaystyle \frac{i}{\cancel{k}-m}}(-ie{\gamma }^{\nu }){\displaystyle \frac{i}{\cancel{k}+\cancel{q}-m}}(-ie{\gamma }^{\alpha }){\displaystyle \frac{i}{\cancel{k}-\cancel{p}-m}}\right]\hfill \end{array}$$

(18)

and

$$\begin{array}{cc}\hfill & T_{\left(c\right)}^{\mu \nu \alpha }(p,q)=-{\int }_{k}Tr\left[(-ie{\gamma }^{\mu }){\displaystyle \frac{i}{\cancel{k}-m}}{\left(a_F^{\left(5\right)}\right)}^{\lambda \zeta \nu }{q}_{\zeta }{\gamma }_{\lambda }{\displaystyle \frac{i}{\cancel{k}+\cancel{q}-m}}(-ie{\gamma }^{\alpha }){\displaystyle \frac{i}{\cancel{k}-\cancel{p}-m}}\right].\hfill \end{array}$$

(19)

Additionally, by virtue of Bose symmetry, the amplitudes referring to the crossed diagrams have to be added. We then have

$$T^{\mu \nu \alpha }(p,q)=T_{\left(a\right)}^{\mu \nu \alpha }(p,q)+T_{\left(b\right)}^{\mu \nu \alpha }(p,q)+T_{\left(c\right)}^{\mu \nu \alpha }(p,q)+\mathrm{crossed} \mathrm{terms}$$

(20)

It is easy to check gauge invariance for diagram $\left(a\right)$. We get ${(p+q)}_{\alpha }{T}_{\left(a\right)}^{\mu \nu \alpha }(p,q)=0$ due to the anti-symmetry property of the two last indices of the tensor ${\left(a_F^{\left(5\right)}\right)}^{\lambda \zeta \alpha }$. For the next two diagrams, we use the identity $\cancel{q}+\cancel{p}=\cancel{k}+\cancel{q}-m-(\cancel{k}-\cancel{p}-m)$ after the contraction in order to split the amplitude into two pieces. After this manipulation, we find

$$\begin{array}{cc}\hfill & {(p+q)}_{\alpha }{T}_{\left(b\right)}^{\mu \nu \alpha }(p,q)=e^2{\left(a_F^{\left(5\right)}\right)}^{\lambda \zeta \mu }{p}_{\zeta }\{-{\int }_k{\displaystyle \frac{Tr\left[{\gamma }_{\lambda }(\cancel{k}+m){\gamma }^{\nu }(\cancel{k}-\cancel{p}+m)\right]}{(k^2-m^2)[{(k-p)}^2-m^2]}}+\hfill \\ \hfill & +{\int }_k{\displaystyle \frac{Tr\left[{\gamma }_{\lambda }(\cancel{k}+m){\gamma }^{\nu }(\cancel{k}+\cancel{q}+m)\right]}{(k^2-m^2)[{(k+q)}^2-m^2]}}\}\hfill \end{array}$$

(21)

and

$$\begin{array}{cc}\hfill & {(p+q)}_{\alpha }{T}_{\left(c\right)}^{\mu \nu \alpha }(p,q)=e^2{\left(a_F^{\left(5\right)}\right)}^{\lambda \zeta \nu }{q}_{\zeta }\{-{\int }_k{\displaystyle \frac{Tr\left[{\gamma }^{\mu }(\cancel{k}+m){\gamma }_{\lambda }(\cancel{k}-\cancel{p}+m)\right]}{(k^2-m^2)[{(k-p)}^2-m^2]}}+\hfill \\ \hfill & +{\int }_k{\displaystyle \frac{Tr\left[{\gamma }^{\mu }(\cancel{k}+m){\gamma }_{\lambda }(\cancel{k}+\cancel{q}+m)\right]}{(k^2-m^2)[{(k+q)}^2-m^2]}}\}.\hfill \end{array}$$

(22)

The computation of Equation (22) with implicit regularization reveals that the crossed terms are necessary in order to avoid a gauge anomaly for the three-point diagram, even if all the surface terms are zero. The result is given by

$$\begin{array}{cc}\hfill & {(p+q)}_{\alpha }\left(T_{\left(a\right)}^{\mu \nu \alpha }+T_{\left(b\right)}^{\mu \nu \alpha }+T_{\left(c\right)}^{\mu \nu \alpha }\right)=\hfill \\ \hfill & ={\displaystyle \frac{4}{3}}\left(I_{log}\left({\lambda }^2\right)-bln\left({\displaystyle \frac{-p^2}{{\lambda }^2}}\right)-{\displaystyle \frac{b}{3}}\right)\left(p^{\nu }{\left(a_F^{\left(5\right)}\right)}^{pp\mu }-p^2{\left(a_F^{\left(5\right)}\right)}^{\nu p\mu }+p^{\mu }{\left(a_F^{\left(5\right)}\right)}^{pq\nu }-p^2{\left(a_F^{\left(5\right)}\right)}^{\mu p\nu }\right)+\hfill \\ \hfill & +{\displaystyle \frac{4}{3}}{\xi }_0(2p^{\nu }{\left(a_F^{\left(5\right)}\right)}^{pp\mu }+p^2{\left(a_F^{\left(5\right)}\right)}^{\nu p\mu }+2p^{\mu }{\left(a_F^{\left(5\right)}\right)}^{pq\nu }+p^2{\left(a_F^{\left(5\right)}\right)}^{\mu p\nu })-4{\upsilon }_0(p^{\nu }{\left(a_F^{\left(5\right)}\right)}^{pp\mu }+\hfill \\ \hfill & +p^2{\left(a_F^{\left(5\right)}\right)}^{\nu p\mu }+p^{\mu }{\left(a_F^{\left(5\right)}\right)}^{pq\nu }+p^2{\left(a_F^{\left(5\right)}\right)}^{\mu p\nu }).\hfill \end{array}$$

(23)

when the crossed terms are added, the gauge symmetry, in this case, is fulfilled in a way that is independent of the surface terms.

4. Gauge Invariance: Non-Minimal Dimension-5 ${\mathit{b}}_{\mathit{F}}^{\mathbf{\left(}\mathbf{5}\mathbf{\right)}}$-Term

The term of coefficient ${\left(b_F^{\left(5\right)}\right)}^{\mu \alpha \beta }$ does not generate induced terms at first order due to the anti-symmetry properties of the Levi-Civita symbol. It is also easy to check the gauge Ward identities for the two-point function. The diagrams are the same as Figure 3, except for the different vertices and amplitudes given by:

$$\begin{array}{cc}\hfill & A_{\left(a\right)}^{\alpha \beta }\left(p\right)=-{\int }_{k}Tr\left[{\left(b_F^{\left(5\right)}\right)}^{\mu \lambda \beta }{\gamma }_5{\gamma }_{\mu }{p}_{\lambda }{\displaystyle \frac{i}{\cancel{k}-\cancel{p}-m}}(-ie{\gamma }^{\alpha }){\displaystyle \frac{i}{\cancel{k}-m}}\right]\hfill \\ \hfill \mathrm{and}\end{array}$$

(24)

$$\begin{array}{cc}\hfill & A_{\left(b\right)}^{\alpha \beta }\left(p\right)=-{\int }_{k}Tr\left[(-ie{\gamma }^{\beta }){\displaystyle \frac{i}{\cancel{k}-\cancel{p}-m}}{\left(b_F^{\left(5\right)}\right)}^{\mu \lambda \alpha }{\gamma }_5{\gamma }_{\mu }{p}_{\lambda }{\displaystyle \frac{i}{\cancel{k}-m}}\right].\hfill \end{array}$$

(25)

When computing $p_{\alpha }{A}^{\alpha \beta }\left(p\right)=p_{\alpha }{A}_{\left(a\right)}^{\alpha \beta }\left(p\right)+p_{\alpha }{A}_{\left(b\right)}^{\alpha \beta }\left(p\right)$, the second term vanishes because of the anti-symmetry of the last two indices of the tensor $b_F^{\left(5\right)}$. For the first term, we can use $\cancel{p}=\cancel{k}-m-(\cancel{k}-\cancel{p}-m)$ to split the integral into two pieces. Each one of these pieces is zero, because there remain two Dirac matrices with one ${\gamma }_5$ inside the trace.

It is also important to check, with the use of the modified version of the Dirac equation, that this non-minimal term does not break gauge or chiral symmetries at the classical level, i.e., ${\partial }_{\mu }{j}^{\mu }=0$ and ${\partial }_{\mu }{j}_5^{\mu }=0$, where $j^{\mu }=\overline{\psi }{\gamma }^{\mu }\psi$ and $j_5^{\mu }=\overline{\psi }{\gamma }^{\mu }{\gamma }_5\psi$. We could expect a chiral anomaly due to the presence of a ${\gamma }_5$ matrix in one of the vertices of the three-point function diagram. However, this is not the case, as shown below, and there is no issue with ${\gamma }_5$ in the regularization applied. The issue of the chiral anomaly in a Lorentz-violating context is specifically discussed in [38], and it is shown that $a_F^{\left(5\right)}$ breaks chiral symmetry in the triangle diagram.

The diagrams with three photon legs are the same ones depicted in Figure 4, with the only difference being the Lorentz-violating vertex. The amplitude corresponding to these diagrams can be written as

$$\begin{array}{cc}\hfill & M_{\left(a\right)}^{\mu \nu \alpha }(p,q)=-{\int }_{k}Tr\left[(-ie{\gamma }^{\mu }){\displaystyle \frac{i}{\cancel{k}-m}}(-ie{\gamma }^{\nu }){\displaystyle \frac{i}{\cancel{k}+\cancel{q}-m}}{\left(b_F^{\left(5\right)}\right)}^{\lambda \zeta \alpha }{(p+q)}_{\zeta }{\gamma }_5{\gamma }_{\lambda }{\displaystyle \frac{i}{\cancel{k}-\cancel{p}-m}}\right],\hfill \\ \hfill & M_{\left(b\right)}^{\mu \nu \alpha }(p,q)=-{\int }_{k}Tr\left[{\left(b_F^{\left(5\right)}\right)}^{\lambda \zeta \mu }{p}_{\zeta }{\gamma }_5{\gamma }_{\lambda }{\displaystyle \frac{i}{\cancel{k}-m}}(-ie{\gamma }^{\nu }){\displaystyle \frac{i}{\cancel{k}+\cancel{q}-m}}(-ie{\gamma }^{\alpha }){\displaystyle \frac{i}{\cancel{k}-\cancel{p}-m}}\right]\hfill \end{array}$$

(26)

and

$$\begin{array}{cc}\hfill & M_{\left(c\right)}^{\mu \nu \alpha }(p,q)=-{\int }_{k}Tr\left[(-ie{\gamma }^{\mu }){\displaystyle \frac{i}{\cancel{k}-m}}{\left(b_F^{\left(5\right)}\right)}^{\lambda \zeta \nu }{q}_{\zeta }{\gamma }_5{\gamma }_{\lambda }{\displaystyle \frac{i}{\cancel{k}+\cancel{q}-m}}(-ie{\gamma }^{\alpha }){\displaystyle \frac{i}{\cancel{k}-\cancel{p}-m}}\right].\hfill \end{array}$$

(27)

As before, we can check the gauge Ward identities by computing $p_{\mu }{M}^{\mu \nu \alpha }$, for instance. Besides the amplitudes of Equation (27), there are also the crossed diagrams, obtained by changing $\mu \leftrightarrow \nu$ and $p\leftrightarrow q$.

Computation of the gauge Ward identity is somewhat involved in this process, and it requires the use of relations between ${\xi }_{nm}(p,q)$ integrals presented in the Appendix A. The idea is to reduce integrals ${\xi }_{20}(p,q)$, ${\xi }_{02}(p,q)$, and ${\xi }_{11}(p,q)$ into integrals ${\xi }_{10}(p,q)$ and ${\xi }_{01}(p,q)$ by using Equations (A10), (A11), (A14) and (A15). Then, the remaining integrals ${\xi }_{10}(p,q)$ and ${\xi }_{01}(p,q)$ are reduced into integrals $Z_n(p^2,m^2)$ and ${\xi }_{00}(p,q)$ with the use of Equations (A12) and (A13). Nevertheless, none of these finite integrals remain in the final result, which depends only on the logarithmic surface term:

$$p_{\mu }{T}^{\mu \nu \alpha }=-{\displaystyle \frac{1}{4{\pi }^2}}{\upsilon }_0\left({\left(b_F^{\left(5\right)}\right)}_{\zeta q}^{\nu }{ϵ}^{\alpha \zeta pq}+2{\left(b_F^{\left(5\right)}\right)}_{\zeta p}^{\alpha }{ϵ}^{\zeta \nu pq}+2{\left(b_F^{\left(5\right)}\right)}_{\zeta q}^{\alpha }{ϵ}^{\zeta \nu pq}\right),$$

(28)

where ${\left(b_F^{\left(5\right)}\right)}_{\zeta p\alpha }\equiv {\left(b_F^{\left(5\right)}\right)}_{\zeta \lambda \alpha }{p}^{\lambda }$.

In this case, we need to choose the surface term ${\upsilon }_0$ as zero to avoid a gauge anomaly in the three-point function. This condition is based on the previous ones concerning the $a_F^{\left(5\right)}$-term and the usual QED, although in this situation, the logarithmic surface term ${\upsilon }_0$ appears alone.

We summarize all the results regarding the computed gauge Ward identities in Table 1.

5. Conclusions

In this work, we discussed the possibility of a gauge anomaly in a non-minimal dimension-5 Lorentz-violating framework. We showed that there is no gauge anomaly if surface terms are null or if there is a relation between logarithmic surface terms. Also, in Section 3.2, we showed that the sum of the result in Equation (23) with the corresponding crossed diagrams (Bose symmetry) can avoid gauge symmetry breaking without the need for relation between surface terms ${\xi }_0=2{\upsilon }_0$. Previous results in the literature showed that no gauge anomalies are expected in minimal versions of the SME as well, like in the minimal Lorentz-violating QED [39,40]. The same happens to be true for non-minimal versions of SME. In particular, we explicitly showed that the non-minimal dimension-5 version of QED is gauge-invariant beyond tree level. This approach can be extended to higher loops. The condition that surface terms are null is sufficient to ensure gauge symmetry for higher orders. Different structures of surface terms in higher loops or one-loop surface terms can appear inside an integral. The framework we worked with here, however, is non-renormalizable. So, there is not a finite number of counter-terms needed to renormalize the theory. If this framework is treated as effective, we need to stop at the one-loop order. Furthermore, even in the minimal SME, there is no proof that it is renormalizable beyond one-loop order.

The gauge invariance of this non-minimal Lorentz-violating framework is expected since it was shown that the $H_F^{\left(5\right)}$-term can induce the $k_F$ gauge-invariant term in the photon sector, as studied in [37]. Furthermore, it was shown here that the non-minimal $\left(a_F^{\left(5\right)}\right)$-term induces a non-minimal dimension-5 gauge-invariant term in the photon sector. The induction of terms due to diagrams with more external legs is more involved, but the corresponding Ward identities of them were checked. In terms of future prospects, it would be interesting to further investigate loop diagrams from the non-minimal SME with higher dimension terms.