A Non-Geometrodynamic Quantum Yang–Mills Theory of Gravity Based on the Homogeneous Lorentz Group

Abstract

In this paper, we present a non-geometrodynamic quantum Yang–Mills theory of gravity based on the homogeneous Lorentz group within the general framework of the Poincare gauge theories. The obstacles of this treatment are that first, on the one hand, the gauge group that is available for this purpose is non-compact. On the other hand, Yang–Mills theories with non-compact groups are rarely healthy, and only a few instances exist in the literature. Second, it is not clear how the direct observations of space–time waves can be explained when space–time has no dynamics. We show that the theory is unitary and is renormalizable to the one-loop perturbation. Although in our proposal, gravity is not associated with any elementary particle analogous to the graviton, classical helicity-two space–time waves are explained. Five essential exact solutions to the field equations of our proposal are presented as well. We also discuss a few experimental tests that can falsify the presented Yang–Mills theory.

Appendices

Appendix A: An Overview of Quantization of GR

In the last part of this section, we would like to review the quantization of option 1 above. GR is uniquely the simplest theory of a massless spin two elementary particle [4, 27,28,29,30,31] where only $c_{1}$ is kept non-zero in Eq. 19. For the quantization purposes, it is easier to decompose the Lagrangian as in [32]

$$\begin{aligned}&S_{\text {GR}}=\int \mathcal{{L}}_{\text {GR}} dtd^{3}x, \\&\mathcal{{L}}_{\text {GR}}=\left( R^{(3)}+K_{ab}K^{ab}-K^{a}_{a}K^{b}_{b}\right) N \sqrt{g^{(3)}}, \end{aligned}$$

(A.1)

where $R^{(3)}$ refers to the three-dimensional curvature, $K_{ab}$ is the extrinsic curvature, $g^{(3)}$ is the determinant of $g^{(3)}_{ab}$, and the rest of the parameters are defined in the following decomposition of the space–time metric

$$\begin{aligned} ds^{2} = -\left( Ndt\right) ^{2} + g^{(3)}_{ab}\left( dx^{a}+N^{a}dt\right) \left( dx^{b}+N^{b}dt\right) . \end{aligned}$$

(A.2)

The dynamical variables are N, $N^{a}$, and $g^{(3)}_{ab}$. The corresponding canonical momenta are

$$\begin{aligned} \pi = \frac{\partial \mathcal{{L}}_{\text {GR}}}{\partial \dot{N}}, \quad \pi _{a} = \frac{\partial \mathcal{{L}}_{\text {GR}}}{\partial \dot{N}^{a}},\quad \pi ^{ab} = \frac{\partial \mathcal{{L}}_{\text {GR}}}{\partial \dot{g}^{(3)}_{ab}}. \end{aligned}$$

(A.3)

The first two momenta are zero and define the constraints of GR. This is similar to the case of electrodynamics, where the canonical momentum of the temporal component of the vector potential is zero. The following Poisson bracket drives the dynamics of GR [33]

$$\begin{aligned} \left\{ g^{(3)}_{ab}\left( {\mathbf {x}}\right) ,\pi ^{ij}\left( {\mathbf {y}}\right) \right\} = \frac{1}{2}\left( \delta ^{i}_{a}\delta ^{j}_{b}+\delta ^{i}_{b}\delta ^{j}_{a}\right) \delta ^{3}\left( {\mathbf {x}}-{\mathbf {y}}\right) . \end{aligned}$$

(A.4)

To quantize the theory, we replace the dynamical variables by the corresponding quantum operators and the Poisson bracket by the canonical commutation relation. In the presence of the constraints mentioned above, such canonical quantization is cumbersome but informative and can be found in [34]. With the developments in the standard model of particle physics, and especially after the works of Feynman [35], Faddeev and Popov [36], Mandelstam [37], and DeWitt [38], the path integral formalism, also called the manifestly covariant method, became the standard method of carrying out this quantization.

In the path integral quantization, the four dimensional space–time metric as the dynamical field, is first expanded around a background, which for simplicity we take to be flat,

$$\begin{aligned} g_{\mu \nu }=\eta _{\mu \nu }+ f_{\mu \nu }, \end{aligned}$$

(A.5)

where $\eta$ is the Minkowski metric. The Lagrangian of GR is rewritten in terms of $f_{\mu \nu }$ and the generating functional reads

$$\begin{aligned} \mathcal{{Z}}\left[ t^{\mu \nu }\right] = \int \mathcal{{D}}f_{\mu \nu } \exp \left( i\int d^{4}x \left[ \mathcal{{L}}_{\text {GR}}+f_{\mu \nu }t^{\mu \nu }\right] \right) . \end{aligned}$$

(A.6)

Due to the diffeomorphism invariance of the theory, we also need to fix the gauge. A common choice is to use the harmonic coordinates where

$$\begin{aligned} C_{\nu }\equiv \partial ^{\mu }f_{\mu \nu }-\frac{1}{2}\partial _{\nu }f^{\mu }_{\mu }=0. \end{aligned}$$

(A.7)

Therefore, the generating functional receives a corresponding gauge fixing and a Faddeev–Popov ghost Lagrangian

$$\begin{aligned} \mathcal{{L}}_{\text {GR}} \rightarrow \mathcal{{L}}_{\text {effective}} \equiv \mathcal{{L}}_{\text {GR}} +\mathcal{{L}}_{\text {GF}} +\mathcal{{L}}_{\text {FP}}. \end{aligned}$$

(A.8)

The Feynman rules are subsequently read from the effective Lagrangian. These sets of rules can then be used to calculate the Feynman diagrams that contain loops. Some of the loop diagrams are divergent, as in other field theories. However, in [39, 40], the authors have shown that the infinities cannot be removed by adding counterterms that have the same form as in the effective Lagrangian. Therefore, GR is a non-renormalizable theory with no falsifiable prediction for high energies. Such ultraviolet divergencies are expected in any theory like GR, whose coupling constant has a negative dimension in the mass units [41].

Even if GR was a renormalizable field theory, we still did not have a consistent quantum theory of gravity. Because, in such quantum gravity, on the one hand, time is a classical background, and on the other hand, it is a quantum operator. The problem of time in quantum geometrodynamics is extensively studied but is still open. A review of the subject can be found in [33, 42, 43]. Moreover, unlike particle physics’ standard model, GR is driven by the absolute value of energies, instead of their differences. This has led to the cosmological constant problem [44].

In the end, we refer the reader to [45,46,47,48], and the references therein, for an overview of the quantum aspects of the broader Poincare gauge theories.

Appendix B: Feynman Rules of QLGT

Propagators The inverse propagator of the Lorentz gauge field in the momentum space reads

$$\begin{aligned} \left( {\text {PA}}(k)^{-1}\right) ^{ij\mu ,mn\nu }=\frac{\delta ^{2} \mathcal{{L}}_{\text {total}}}{\delta A_{ij\mu }(k)\delta A_{mn\nu }(k)}\left| _{A=\psi =c=0},\right. \end{aligned}$$

(B.1)

where ${\textsf {PA}}$ stands for the propagator for field $A_{ij\mu }$. We use the FeynCalc package [49, 50] to take the two functional derivatives, and have made the scripts available online [19]. The propagator reads

$$\begin{aligned} {\textsf {PA}}(k)_{ij\mu ,mn\nu }= & {} i\delta _{ij,mn} \left( \frac{\eta _{\mu \nu }}{k^{2}}-(1-\xi )\frac{k_{\mu }k_{\nu }}{k^{4}}\right) . \end{aligned}$$

(B.2)

The propagator for the Faddeev–Popov ghosts can be found via the same procedure

$$\begin{aligned} {\textsf {Pc}}(p)_{ij,mn}=\frac{i\delta _{ij,mn}}{p^{2}}. \end{aligned}$$

(B.3)

The fermion propagator is also known to be

$$\begin{aligned} {\textsf {PF}}(p)=i\frac{p\cdot \gamma +m}{p^{2}-m^{2}}. \end{aligned}$$

(B.4)

Vertices The interactions in QLGT can be found via higher-order Functional derivatives. Due to their somewhat lengthy nature, we calculate them with the FeynCalc package again and make them available in [19].

The interaction with fermions read

(B.5)

The naming VAFF stands for the vertex of a gauge field and two fermions. The interaction of the Faddeev–Popov ghosts with the gauge field is

(B.6)

The following is the gauge field interaction of order g

(B.7)

Finally, gauge field interaction of order $g^{2}$ is equal to

(B.8)

External lines

The total Lagrangian in flat space–time is a function of the Faddeev–Popov ghosts, the fermions, and the Lorentz gauge field. The Faddeev–Popov ghosts have wrong statistics and cannot represent physical particles. Therefore, they do not receive an external line in the Feynman diagrams. Also, following the convention, we show incoming and outgoing fermions with $u^{\sigma }(p)$ and $\bar{u}^{\sigma }(p)$ respectively, while incoming and outgoing anti-fermions with $\bar{v}^{\sigma }(p)$ and $v^{\sigma }(p)$, where $\sigma$ refers to the two spin modes.

To discuss the external lines for the Lorentz gauge field, we note that if the arbitrary parameter of the Lorentz transformation of physical observers $\omega _{ij}$ is much smaller than unity, Eq. 8 implies that the Lorentz gauge field transforms as

$$\begin{aligned} \tilde{A}_{ij\mu }(x) = A_{ij\mu }(x) + D_{\mu }\omega _{ij}, \end{aligned}$$

(B.9)

which has an identical form as the transformation of the gauge fields in the standard model under a special unitary gauge transformation with parameter $\alpha ^{a}$

$$\begin{aligned} \grave{A}^{a}_{\mu }(x) = A^{a}_{\mu }(x) + D_{\mu }\alpha ^{a}. \end{aligned}$$

(B.10)

However, despite the similarity, there is a crucial difference. Under a Lorentz transformation of physical observers, $\varLambda ^{i}_{j} \simeq \delta ^{i}_{j}+\omega ^{i}_{j}$, the gauge field of the unitary group has an invariant length equal to

$$\begin{aligned} \left( A^{a}_{\mu }A^{a\mu } + 2D_{\mu }\alpha ^{a} A^{a\mu } + D_{\mu }\alpha ^{a} D^{\mu }\alpha ^{a}\right) ^{\frac{1}{2}}, \end{aligned}$$

(B.11)

which is independent of $\omega ^{i}_{j}$. This means that two independent experiments observe the same length for $\grave{A}^{a}_{\mu }$. On the other hand, the length of $\tilde{A}_{ij\mu }$ is equal to

$$\begin{aligned} \left( A_{ij\mu }A^{ij\mu } + 2D_{\mu }\omega _{ij} A^{ij\mu } + D_{\mu }\omega _{ij} D^{\mu }\omega ^{ij}\right) ^{\frac{1}{2}}, \end{aligned}$$

(B.12)

which depends on the parameter of the Lorentz transformation of physical observers. Therefore, the Lorentz gauge field is observer-dependent, does not have an invariant length and, consequently, cannot represent physical particles. Therefore, it receives no external line in the Feynman diagrams of QLGT. Later in this paper, we discuss that the Lorentz gauge field induces classical helicity two space–time fluctuations, which are observable fields.

Appendix C: Unitarity of QLGT

To preserve probability in a quantum field theory, the S matrix has to be unitary. This means that

$$\begin{aligned} 2 {\text {Im}}\left( T\right) = TT^{\dagger }, \end{aligned}$$

(C.1)

where $T\equiv -i({\text {S}}-1)$. If $|\phi \rangle$ is a state of the system, the equation implies that

$$\begin{aligned} 2 {\text {Im}}\langle \phi | T|\phi \rangle = \sum _{k}\langle \phi | T|k\rangle \langle k|T^{\dagger }|\phi \rangle , \end{aligned}$$

(C.2)

where $|k\rangle$ refers to the physically observable modes of the fields and $\sum _{k} |k\rangle \langle k|=1$. In the gauge theories of the standard model, only fermions and the transverse component of the gauge fields contribute to the $|k\rangle$ states while the non-physical longitudinal components of the gauge fields and Faddeev–Popov ghosts are excluded.

In QLGT, the Lorentz gauge field cannot represent a physical mode and should be excluded entirely from the states of $|k\rangle$. Therefore, the only physical states are the fermions, and consequently, Eq. C.2 is non-trivial only if $1=\sum _{k} |k\rangle \langle k|$ is intervening internal fermionic lines of Feynman diagrams. We now prove the unitarity for a rather general Feynman diagram of the form

(C.3)

where the question marks can be any multi-loop diagram and in the following will be presented by $\mathcal{{M}}_{1_{ij\mu }}$. The amplitude for diagram above reads

(C.4)

We now use the Cutkosky rules to derive the imaginary part of this amplitude

(C.5)

On the other hand, the right hand side of Eq. C.2 is equal to

$$\begin{aligned}&\int \frac{d^{3}q_{1}}{(2\pi )^{3}2E_{q_{1}}}\int \frac{d^{3}q_{2}}{(2\pi )^{3}2E_{q_{2}}} \left( 2\pi \right) ^{4}\delta ^{4}(k-q_{1}-q_{2}) \\&\sum _{\text {spin}}|\mathcal{{M}}_{\text {half}}|^{2}, \end{aligned}$$

(C.6)

where the half amplitude is equal to

(C.7)

We now use the spin method to convert the spin sum of the half amplitudes into a trace

(C.8)

and use

$$\begin{aligned} \int \frac{d^{3}q}{2E_{q}}=\int d^{4}q\varTheta (q^0)\delta (q^{2}-m^{2}), \end{aligned}$$

(C.9)

to show that Eq. C.6 is equal to $2{\text {Im}}\left( \mathcal{{M}}\right)$ and, therefore, unitarity is preserved.

Appendix D: One-Loop Renormalization of QLGT

In this section, we would like to show that all of the one-loop infinities of QLGT can be absorbed in its available parameters, and the theory is renormalizable to that order. We use the FeynCalc package to carry out the calculations within the Passarino–Veltman scheme [51] and make the scripts available in [19]. For the calculations, we follow the instructions for the one-loop calculations of QED using the same computational package in [52].

Fermion self-energy

The correction to the fermion propagator is through the following diagram

(D.1)

To reduce the computational load, we rewrite the vertex in the following form

$$\begin{aligned} {\text {VAFF}}^{ij\mu }=\frac{-ig}{4}\epsilon ^{kij\mu }\gamma _{k}\gamma ^{5}, \end{aligned}$$

(D.2)

and use the following identity

$$\begin{aligned} \delta _{ij,mn}\eta _{\mu \nu }\epsilon ^{k_{1}ij\mu }\epsilon ^{k_{2}mn\nu }=-6\eta ^{k_{1}k_{2}}, \end{aligned}$$

(D.3)

where $\epsilon ^{kij\mu }$ is the Levi–Civita symbol. The loop reads

(D.4)

where the two infinite parameters are

$$\begin{aligned}&A = \frac{3g^{2}m}{64 \pi ^{2}}\left( 1-2B_{0}(p^{2},0,m^{2})\right) , \\&B = \frac{3g^{2}}{128\pi ^{2}}\left( 1+B_{0}(0,0,m^{2})-2B_{0}(m^{2},0,m^{2})\right) . \end{aligned}$$

(D.5)

Here and in the rest of the paper, $A_{0}(\cdots )$, $B_{0}(\cdots )$ and $C_{0}(\cdots )$ are the Passarino–Veltman functions.

To see the corrections to the mass and the spinor field, we note that the exact fermion propagator is equal to

$$\begin{aligned} {\text {PF}}_{(t)}&= {\text {PF}}+ {\text {PF}}\left( -i\varSigma \right) {\text {PF}} \\&\quad +{\text {PF}}\left( -i\varSigma \right) {\text {PF}}\left( -i\varSigma \right) {\text {PF}}+ \cdots \\& = {\text {PF}}\left( 1+\left( -i\varSigma \right) {\text {PF}}_{(t)}\right) . \end{aligned}$$

(D.6)

Therefore, the inverse of the propagators satisfy the following equation

(D.7)

To absorb the two infinities of $\varSigma$, we define the renormalized parameters as

$$\begin{aligned}&\psi ^{r} \equiv \frac{1}{\sqrt{Z_{\psi }}}\psi , \\&m^{r} \equiv \frac{1}{Z_{m}}m, \end{aligned}$$

(D.8)

where $Z_{\psi /m} \equiv 1 + \delta _{\psi /m}$. Since the fermion propagator is by definition $\langle 0|\psi {\bar{\psi }}|0\rangle$, the renormalization is equivalent to ${\text {PF}}\rightarrow Z_{\psi }^{-1}{\text {PF}}$, and Eq. D.7 reads

(D.9)

To remove the infinities, we now define

$$\begin{aligned}&\delta _{\psi } = {\text {Div}}(B), \\&\delta _{\psi }+\delta _{m}=-\frac{1}{m^{r}}{\text {Div}}(A), \end{aligned}$$

(D.10)

where Div stands for the divergent component of the expression.

Vacuum polarization The corrections to the Lorentz gauge field propagator are through four loop diagrams that are calculated below. Whenever applicable, we use the Feynman–’t Hooft gauge of $\xi =1$.

The first diagram to consider is the correction by a fermionic loop

(D.11)

where we have given a fictitious mass $\lambda$ to the gauge field and used

$$\begin{aligned} \lim _{\lambda ^{2}\rightarrow 0} \frac{B_{0}\left( \lambda ^{2},m^{2},m^{2}\right) -B_{0}\left( 0,m^{2},m^{2}\right) }{\lambda ^{2}} =\frac{1}{6m^{2}}. \end{aligned}$$

(D.12)

The second diagram is the correction by two first order gauge field vertices

(D.13)

which is finite and needs no counter term. Note that this is a modification to the longitudinal component of the Lorentz gauge field. In the gauge theories based on the unitary groups, the correction is always to the transverse component, and the longitudinal ghost component remains suppressed. In QLGT, however, the gauge field is not a tensor, as was discussed above. Hence, neither the longitudinal nor the transverse components represent an observable particle. Therefore, the correction to the longitudinal component of the Lorentz gauge field does not have adversarial effects.

The third correction is from the second order vertex of the Lorentz gauge field

(D.14)

The loop is zero because the V4A vertex is momentum independent and can be taken out of the integral.

Finally, the last correction is from the ghost vertices

(D.15)

which is finite, and as expected, has the same form as in $i\varPi ^{ij\mu ,mn\nu }_{2}$.

Out of the four corrections to the propagator of the Lorentz gauge field, only the fermionic loop contains infinities. To find the counterterms, we note that the only external lines in QLGT are fermions and the only possible Feynman diagrams have the following form

(D.16)

where $\chi ^{e_{1}}\equiv \frac{-ig}{4}\bar{\varphi }\gamma ^{e_{1}}\gamma ^{5}\varphi$, and $\varphi$ stands for any of the spinors. If we use $\epsilon _{e_{1}}^{ab\sigma }\epsilon _{e_{2} ab\sigma }= -6\eta _{e_{1}e_{2}}$, and $\epsilon _{ij}^{ab}\epsilon _{mnab}=-4\delta _{ij,mn}$, expression above reads

$$\begin{aligned}&-6i\chi ^{e_{1}}\chi ^{e_{2}}\left( \frac{\eta _{e_{1}e_{2}}}{k^{2}} +\frac{6 g^{2}}{192 \pi ^{2}k^{4}} {\text {B}}_{0}\left( 0,m^{2},m^{2}\right) \right. \\&\left. \quad \cdot \left( 6 m^{2} \eta _{e_{1}e_{2}}+k_{e_{1}}k_{e_{2}}\right) +g^{2}\cdot {\text {finite}} + \mathcal{{O}}(g^{4}) \right) . \end{aligned}$$

(D.17)

We define the renormalized parameter as

$$\begin{aligned}&A^{r}_{ij\mu }\equiv \frac{1}{\sqrt{Z_{A}}} A_{ij\mu }, \\&\xi ^{r} \equiv \frac{1}{Z_{\xi }}\xi , \end{aligned}$$

(D.18)

where $Z_{A/\xi } \equiv 1+\delta _{A/\xi }$. Subsequently, the propagator of the gauge field takes the following form

$$\begin{aligned} {\textsf {PA}}(k)_{ij\mu ,mn\nu }= & {} \frac{i\delta ^{ij,mn}}{k^{2}} \left( \left( 1+\delta _{A}\right) \left( \eta _{\mu \nu }-\frac{k_{\mu }k_{\nu }}{k^{2}}\right) \right. \\&\left. \quad +\left( 1+\delta _{\xi }+\delta _{A}\right) \xi \frac{k_{\mu }k_{\nu }}{k^{2}} \right) . \end{aligned}$$

(D.19)

Since $\delta _{A}$ and $\delta _{\xi }$ are of order $g^{2}$, only the first term in the parentheses in Eq. D.16 receives a correction from them while the correction to the rest of the terms are of the order of $g^{4}$ or higher and can be neglected. A straightforward calculation shows that the infinities can be removed if we choose

$$\begin{aligned}&\delta _{A}+\frac{1}{3}\delta _{\xi } = -\frac{3g^{2}m^{2}}{16\pi ^{2}k^{2}}{\text {Div}}\left( B_{0}(0,m^{2},m^{2})\right) , \\&\delta _{\xi }=\frac{3g^{2}}{32\pi ^{2}}{\text {Div}}\left( B_{0}(0,m^{2},m^{2})\right) . \end{aligned}$$

(D.20)

We would like to emphasize that in Eq. D.11, we showed that the corrections to the vacuum polarization has a form different than the one in the tree level. This is unlike any of the Yang–Mills theories of the standard model. The reason we could absorb this infinite correction by the available parameters of the theory was that the gauge field did not have an external line and the only possible Feynman diagram of the theory was given in Eq. D.16 and the fortunate fact that the contraction of two Levi–Civita symbols is proportional to the six-dimensional internal metric of the homogeneous Lorentz group.

Ghost self-energy The ghost propagator receives a corrections from the following loop

(D.21)

The correction to the inverse of the ghost propagator can be found in the same way as for the fermion self-energy above and reads

$$\begin{aligned} {\textsf {Pc}}(p)^{-1}_{(t)ij,mn} = -ip^{2}\delta _{ij,mn} +i\varSigma _{ij,mn}. \end{aligned}$$

(D.22)

Since the ghost field has no mass, we only define one renormalized parameter as

$$\begin{aligned} c^{_rij} \equiv \frac{1}{\sqrt{Z_{c}}}c^{ij}, \end{aligned}$$

(D.23)

where $Z_{c}\equiv 1 + \delta _{c}$. To remove the infinity we further assume that

$$\begin{aligned} \delta _{c} \equiv -\frac{g^{2}}{16\pi ^{2}}{\text {Div}}\left( B0(m^{2},0,0)\right) . \end{aligned}$$

(D.24)

Fermion vertex renormalization The correction to the fermion vertex is from the following two diagrams

(D.25)

where $q \equiv p_{1}-p_{2}$, and

$$\begin{aligned} A= & {} -\frac{3 g^{3} m}{128 \pi ^{2}} C_{0}\left( m^{2},m^{2},q^{2},m^{2},0 ,m^{2}\right) , \\ B= & {} \frac{3 g^{3}}{512 \pi ^{2}} \left( -3 B_{0}\left( q^{2},m^{2},m^{2}\right) +4 B_{0}\left( m^{2},0,m^{2}\right) \right. \\&\left. \quad +\left( 6 m^{2}-2 q^{2}\right) C_{0}\left( m^{2},m^{2},q^{2},m^{2},0 ,m^{2}\right) -1\right) . \end{aligned}$$

(D.26)

(D.27)

The expression for the second diagram is finite but rather lengthy and can be found in the online repository in [19]. Also, to reduce the computation load, we have used the simplifications that were described in Sect. D as well as $\gamma ^{5}\cdot \gamma ^{5}=1$, and $\gamma ^{5}\cdot \gamma ^{k}=-\gamma ^{k}\cdot \gamma ^{5}$. Since $C_{0}$ is finite, only B in the first diagram contains infinities. Therefore, the divergent correction to the fermion vertex reads

$$\begin{aligned} {\text {Div}}\left( \varGamma ^{ab\sigma }\right)= & {} {\text {Div}}\left( B\right) \epsilon ^{lab\sigma }\gamma _{l}\gamma ^{5} \\= & {} 4ig^{-1}{\text {Div}}\left( B\right) {\text {VAFF}}^{ab\sigma }, \end{aligned}$$

(D.28)

and is proportional to the bare vertex. Hence, we can remove it by renormalizing the coupling constant

$$\begin{aligned}&g^{r} \equiv \frac{1}{Z_{g}} g, \end{aligned}$$

(D.29)

with $Z_{g} \equiv 1 + \delta _{g}$, and choosing $\delta _{g}$ such that

$$\begin{aligned} \left( 4ig^{-1}{\text {Div}}\left( B\right) +\delta _{g}\right) {\text {VAFF}}^{ab\sigma }=0. \end{aligned}$$

(D.30)

The rest of infinities By now, we have used all of the possible parameters of QLGT to remove the infinities. On the other hand, the other three vertices in Sect. B also receive infinite corrections from the relevant loops. In this section, we would like to show that the gauge symmetry of QLGT implies three restrictions on the coefficients of the terms in the Lagrangian that removes the rest of infinities by the choices that we have made so far for the renormalized parameters.

Inserting all of the renormalized parameters, the total Lagrangian reads

$$\begin{aligned} \mathcal{{L}}^{r}_{\text {total}}= & {} \frac{i}{2}Z_{\psi }e_{i}^{\mu }{\bar{\psi }}^{r}\gamma ^{i} \partial _{\mu }\psi ^{r} - \frac{i}{2}Z_{\psi }e_{i}^{\mu }{\bar{\psi }}^{r}\overleftarrow{\partial }_{\mu }\gamma ^{i} \psi ^{r} \\&\quad -Z_{m}Z_{\psi }m{\bar{\psi }}^{r}\psi ^{r}-\frac{ig}{4}Z_{g}Z_{\psi }Z_{A}^{\frac{1}{2}}A_{ij\mu }\epsilon ^{lij\mu }{\bar{\psi }}\gamma _{l}\gamma ^{5}\psi \\&\quad + Z_{A}\mathcal{{L}}_{A^{2}} + Z_{g} Z_{A}^{\frac{3}{2}} g\mathcal{{L}}_{A^{3}} + Z_{g}^{2} Z_{A}^{2} g^{2}\mathcal{{L}}_{A^{4}} \\&\quad -Z_{c} \bar{c}^{ij}\partial ^{\mu }\left( \partial _{\mu }c_{ij}\right) \\&\quad +Z_{c}Z_{g}Z_{A}^{\frac{1}{2}} g\bar{c}^{ij}\partial ^{\mu }\left( A_{i\mu }^{k}c_{kj}+A_{j\mu }^{k}c_{ik}\right) , \end{aligned}$$

(D.31)

where $\mathcal{{L}}_{A^{n}}$ is the part of the gauge field Lagrangian containing n fields. From this renormalized Lagrangian, we can derive the corrections to the three vertices that were not directly discussed.

To validate these corrections, we write the total renormalized Lagrangian with unknown coefficients

$$\begin{aligned} \mathcal{{L}}^{r}_{\text {total}}= & {} \frac{i}{2}Z_{1}e_{i}^{\mu }{\bar{\psi }}\gamma ^{i} \partial _{\mu }\psi - \frac{i}{2}Z_{1}e_{i}^{\mu }{\bar{\psi }}\overleftarrow{\partial }_{\mu }\gamma ^{i} \psi \\&\quad -Z_{2} m{\bar{\psi }}\psi -\frac{ig}{4}Z_{3}A_{ij\mu }\epsilon ^{lij\mu }{\bar{\psi }}\gamma _{l}\gamma ^{5}\psi \\&\quad +Z_{4} \mathcal{{L}}_{A^{2}} + gZ_{5}\mathcal{{L}}_{A^{3}} + g^{2}Z_{6}\mathcal{{L}}_{A^{4}} \\&\quad - Z_{7}\bar{c}^{ij}\partial ^{\mu }\left( \partial _{\mu }c_{ij}\right) \\&\quad +gZ_{8}\bar{c}^{ij}\partial ^{\mu }\left( A_{i\mu }^{k}c_{kj}+A_{j\mu }^{k}c_{ik}\right) . \end{aligned}$$

(D.32)

We note that the renormalized theory has to be invariant under the homogeneous Lorentz transformations. This means that the following three equations should be satisfied

$$\begin{aligned} \frac{Z_{3}}{Z_{1}} = \frac{Z_{5}}{Z_{4}} = \frac{Z_{8}}{Z_{7}} = \sqrt{\frac{Z_{6}}{Z_{4}}}. \end{aligned}$$

(D.33)

By comparison with Eq. D.31, we can see that our choices for the infinities meet the enforced conditions.

Appendix E: Un-compacted Equations of the Plane Wave Analysis

The components of Eq. 49 after using $k^{\mu }=(k,0,0,k)$ read

$$\begin{aligned}&\varSigma _{22}=-\varSigma _{11}, \\&\varSigma _{(01)}=-\varSigma _{(13)}, \\&\varSigma _{(03)}=-\frac{1}{2}\left( \varSigma _{00}+\varSigma _{33}\right) , \\&\varSigma _{(02)}=-\varSigma _{(23)}. \end{aligned}$$

(E.1)

Equations 46, 47, and 49 after using $k^{\mu }=(k,0,0,k)$ indicate that

$$\begin{aligned}&\varepsilon _{013}=-\varepsilon _{010},\quad \varepsilon _{023} = - \varepsilon _{020},\quad \varepsilon _{033}=-\varepsilon _{030}, \\&\varepsilon _{123}=-\varepsilon _{120},\quad \varepsilon _{133}=-\varepsilon _{130},\quad \varepsilon _{233}=-\varepsilon _{230}, \\&\varepsilon _{121}=\varepsilon _{122}=\varepsilon _{031}=\varepsilon _{032}=0, \\&\varepsilon _{012}=\varepsilon _{021},\quad \varepsilon _{131}=\varepsilon _{011}, \quad \varepsilon _{232}=\varepsilon _{022}, \\&\varepsilon _{132}=\varepsilon _{021}=\varepsilon _{231}, \\&\varSigma _{32}=\frac{i}{k}\varepsilon _{233},\quad \varSigma _{01}=\frac{i}{k}\varepsilon _{010}, \\&\varSigma _{02}=\frac{i}{k}\varepsilon _{020},\quad \varSigma _{11}=\frac{i}{k}\varepsilon _{011}, \\&\varSigma _{22}=\frac{i}{k}\varepsilon _{022},\quad \varSigma _{12}=\frac{i}{k}\left( \varepsilon _{021}+\varepsilon _{120}\right) , \\&\varSigma _{21}=\frac{i}{k}\left( \varepsilon _{021}-\varepsilon _{120}\right) , \quad \varSigma _{30}+\varSigma _{33}=\frac{i}{k}\varepsilon _{033}, \\&\varSigma _{03}+\varSigma _{00}=\frac{i}{k}\varepsilon _{030}, \quad \varSigma _{31}=\frac{i}{k}\varepsilon _{133}. \end{aligned}$$

(E.2)

It is interesting to note that the first two lines of equations above are in agreement with Eq. 48. Also, we would like to mention that the linearized Eq. E.2 is to the first order of perturbation invariant under both of the symmetry transformations of LGT. This is very important, since, for example, $\varSigma _{31}$ is not a physical mode because it is equal to $\frac{i}{k}\epsilon _{133}$, and an appropriate transformation can remove the latter. This conclusion was not possible if equality was lost after the transformation.