A gravitational gauge field theory based on Stephenson–Kilmister–Yang gravitation with scalar and spinor fields as gravitating matter sources

Abstract

A gravitational gauge theory with a spin–affine connection (Lorentz connection) as a rotational gauge potential (fundamental dynamical variable) is suggested for reformulating the theory of Stephenson–Kilmister–Yang gravity, in which the Einstein field equation of gravity is a first-integral solution of a spin-connection gravitational gauge field equation. A heavy intermediate field $\phi$ that accompanies a matter field $\varphi$ is introduced in order to remove the conventional dimensionful gravitational coupling. Such a $\varphi$–$\phi$ coupling can lead to dimensionless gravitational coupling (i.e., the gravitational constant is dimensionless) in the present gravitational gauge field theory. A low-energy effective Lagrangian density for the matter field can be obtained by integrating out the accompanying heavy field in generating functional of path integral formalism, and therefore, a dimensionful gravitational coupling coefficient (Einstein gravitational constant) emerges. Such a dimensionless coupling of gravity, where the dimensionful coupling is emergent at low energies, is considered for scalar and spinor fields, which serve as gravitating matter fields (gravitational source). Though there are higher derivatives (e.g., third- and fourth-order partial derivatives) of the scalar and spinor fields in the low-energy effective Lagrangian densities, the ordinary equations of motion of the scalar and spinor fields can also be emergent from the present gravitational gauge theory. Therefore, the Einstein gravity can be recovered from the present gravitational gauge theory. In addition to the gravitational Lagrangian of the spacetime-rotational gauge potential (i.e., spin–affine connection), the Lagrangian of a spacetime-translational gauge potential (i.e., vierbein) is also constructed. Thus, the present dimensionless gravitational gauge coupling preserves local rotational and translational gauge symmetries. Since the spin-connection gravitational gauge field equation is a third-order differential equation of metric (the Einstein field equation of gravity is a first-integral solution), it could provide a new route to the vacuum energy cosmological constant problem.

Appendices

Appendix 1: Functional variation of low-energy Lagrangian of scalar matter field

We shall obtain the low-energy equation of the scalar matter field $\varphi$ based on Sect. 3, where a self-contained derivation of generating functional and effective action is presented in order to correct some improper statements in Ref. [41]. We have the free Lagrangian density ${\mathcal {L}}_{\varphi }=\frac{1}{2}\partial _{\mu }\varphi \partial ^{\mu }\varphi -\frac{1}{2}m^{2}\varphi ^{2}$ and the low-energy effective interaction Lagrangian density ${\mathcal {L}}_{\mathrm {int}}\!=\!\kappa \left[ \left( \nabla _{\mu }\partial ^{\mu }+\mu ^{2}\right) \varphi \right] ^{2}$ (caused by the heavy intermediate particle $\phi$) in Eq. (30). The total low-energy effective Lagrangian density of the gravitating matter field $\varphi$ is ${\mathcal {L}}_{\mathrm {eff}}={\mathcal {L}}_{\varphi }+{\mathcal {L}}_{\mathrm {int}}$. In what follows, we shall obtain the field equation of the scalar gravitating matter field $\varphi$ by calculating the functional variation of $\sqrt{-g}{\mathcal {L}}_{\mathrm {eff}}$ with respect to $\varphi$.

By using $\delta \left( \sqrt{-g}{\mathcal {L}}_{\varphi }\right) =\sqrt{-g}\partial ^{\mu }\varphi \partial _{\mu }\delta \varphi -\sqrt{-g}m^{2}\varphi \delta \varphi$, where $\sqrt{-g}\partial ^{\mu }\varphi \partial _{\mu }\delta \varphi =\partial _{\mu }\left( \sqrt{-g}\partial ^{\mu }\varphi \delta \varphi \right) -\partial _{\mu }\left( \sqrt{-g}\partial ^{\mu }\varphi \right) \delta \varphi$, one can obtain

$$\begin{aligned} \delta \left( \sqrt{-g}{\mathcal {L}}_{\varphi }\right) =\mathrm {D.T.}-\sqrt{-g}\left( \nabla _{\mu }\partial ^{\mu } +m^{2}\right) \varphi \delta \varphi , \end{aligned}$$

(98)

where we have used the relation $\partial _{\mu }\left( \sqrt{-g}\partial ^{\mu }\varphi \right) =\sqrt{-g}\nabla _{\mu }\partial ^{\mu }\varphi$ (in the torsionless spacetime). The divergence term $\mathrm {D.T.}=\partial _{\mu }\left( \sqrt{-g}\partial ^{\mu } \varphi \delta \varphi \right)$. On the other hand, the functional variation $\delta \left( \sqrt{-g}{\mathcal {L}}_{\mathrm {int}}\right) =2\kappa \sqrt{-g}\left( \nabla _{\mu }\partial ^{\mu }+\mu ^{2}\right) \varphi \delta \big [\left( \nabla _{\nu }\partial ^{\nu }+\mu ^{2}\right) \varphi \big ]$. This can be rearranged as $2\kappa \sqrt{-g}\left( \nabla _{\mu }\partial ^{\mu } +\mu ^{2}\right) \varphi \delta \nabla _{\nu }\partial ^{\nu }\varphi +2\kappa \mu ^{2}\sqrt{-g}\left( \nabla _{\mu }\partial ^{\mu } +\mu ^{2}\right) \varphi \delta \varphi$, where the first term has the following structure: $\sqrt{-g}F\delta \nabla _{\nu }\partial ^{\nu }\varphi = \sqrt{-g}Fg^{\nu \lambda }\delta \nabla _{\nu }\partial _{\lambda } \varphi =\sqrt{-g}\nabla _{\nu }\left( Fg^{\nu \lambda } \partial _{\lambda }\delta \varphi \right) -\sqrt{-g} \nabla _{\nu }\left( Fg^{\nu \lambda }\right) \partial _{\lambda }\delta \varphi$ with $F=2\kappa \left( \nabla _{\mu }\partial ^{\mu }+\mu ^{2}\right) \varphi$, and we have

$$\begin{aligned} \sqrt{-g}F\delta \nabla _{\nu }\partial ^{\nu }\varphi= & {} \sqrt{-g}\nabla _{\nu }\left( Fg^{\nu \lambda }\partial _{\lambda } \delta \varphi \right) -\partial _{\lambda }\left[ \sqrt{-g} \nabla _{\nu }\left( Fg^{\nu \lambda }\right) \delta \varphi \right] \nonumber \\&+\partial _{\lambda }\left[ \sqrt{-g}\nabla _{\nu }\left( Fg^{\nu \lambda } \right) \right] \delta \varphi \nonumber \\= & {} \mathrm {D.T.}+\sqrt{-g}\left( \nabla _{\nu } \partial ^{\nu }F\right) \delta \varphi . \end{aligned}$$

(99)

Note that we have used the relation $\delta \nabla _{\nu }\partial ^{\nu }\varphi =\nabla _{\nu } \partial ^{\nu }\delta \varphi$ (i.e., $\delta \nabla _{\nu }\partial ^{\nu }\varphi =g^{\nu \lambda } \delta \nabla _{\nu }\partial _{\lambda }\varphi$, which is $g^{\nu \lambda }\delta \left( \partial _{\nu }\partial _{\lambda } \varphi -\Gamma ^{\sigma }{}_{\nu \lambda }\partial _{\sigma }\varphi \right) =g^{\nu \lambda }\left( \partial _{\nu }\partial _{\lambda } \delta \varphi -\Gamma ^{\sigma }{}_{\nu \lambda }\partial _{\sigma } \delta \varphi \right)$). Then we have the functional variation of $\sqrt{-g}{\mathcal {L}}_{\mathrm {int}}$ with respect to $\varphi$:

$$\begin{aligned} \delta \left( \sqrt{-g}{\mathcal {L}}_{\mathrm {int}}\right) =\mathrm {D.T.}+2\kappa \sqrt{-g}\left( \nabla _{\nu } \partial ^{\nu }+\mu ^{2}\right) \left( \nabla _{\mu }\partial ^{\mu } +\mu ^{2}\right) \varphi \delta \varphi . \end{aligned}$$

(100)

Now from the two variational results: (98) and (100), the variation of the total low-energy effective action of the matter field $\varphi$ is given by

$$\begin{aligned}&\delta \int \left[ \sqrt{-g}\left( {\mathcal {L}}_{\varphi } +{\mathcal {L}}_{\mathrm {int}}\right) \right] \mathrm {d}^{4}x \nonumber \\&\quad =-\int \sqrt{-g}\Big [\left( \nabla _{\mu } \partial ^{\mu }+m^{2}\right) \varphi -2\kappa \left( \nabla _{\nu }\partial ^{\nu }+\mu ^{2}\right) \left( \nabla _{\mu }\partial ^{\mu }+\mu ^{2}\right) \varphi \Big ] \delta \varphi \mathrm {d}^{4}x.\nonumber \\ \end{aligned}$$

(101)

According to the variational principle, the low-energy field equation of $\varphi$ is of the form

$$\begin{aligned} \left( \nabla _{\mu }\partial ^{\mu }+m^{2}\right) \varphi -2 \kappa \left( \nabla _{\nu }\partial ^{\nu }+\mu ^{2}\right) \left( \nabla _{\mu }\partial ^{\mu }+\mu ^{2}\right) \varphi =0. \end{aligned}$$

(102)

Though the low-energy field Eq. (102) is a fourth-order differential equation, there is an ordinary free-field equation, $\nabla _{\mu }\partial ^{\mu }\varphi +m_{\mathrm {corr}}^{2}\varphi =0$, involved in it. This free-field equation can be rewritten as $\nabla _{\mu }\partial ^{\mu }\varphi +m^{2}\varphi =(m^{2}-m_{\mathrm {corr}}^{2})\varphi$ and $\nabla _{\mu }\partial ^{\mu }\varphi +\mu ^{2}\varphi =(\mu ^{2}-m_{\mathrm {corr}}^{2})\varphi$. By substituting these two relations into the previous classical field Eq. (102) of $\varphi$, we will have $m^{2}-m_{\mathrm {corr}}^{2}=2\kappa \left( \mu ^{2}-m_{\mathrm {corr}}^{2}\right) ^{2}$. From this relation, we can obtain $m_{\mathrm {corr}}^{2}$ (the square of the gravitationally corrected mass of the $\varphi$ field):

$$\begin{aligned} m_{\mathrm {corr}}^{2}=\mu ^{2}-\frac{1}{4\kappa }\pm \sqrt{\left( \frac{1}{4\kappa }\right) ^{2}+\frac{1}{2\kappa } (m^{2}-\mu ^{2})}. \end{aligned}$$

(103)

The free Lagrangian density of $\varphi$ is ${\mathcal {L}}_{\varphi }=\frac{1}{2}\partial _{\mu }\varphi \partial ^{\mu }\varphi -\frac{1}{2}m^{2}\varphi ^{2}$. Since there is a $\varphi -\phi$ coupling, of which the Lagrangian density ${\mathcal {L}}_{\varphi -\phi }$ is shown in Eq. (8), in the functional integral approach, e.g., in the generating functional (14) we have integrated out the heavy field $\phi$, and obtain a low-energy interaction Lagrangian density ${\mathcal {L}}_{\mathrm {int}}$ for $\varphi$. Such a low-energy interaction Lagrangian density ${\mathcal {L}}_{\mathrm {int}}$ (due to $\varphi -\phi$ coupling) leads to a gravitational correction to the mass of $\varphi$ field. If, therefore, an emergent energy-momentum tensor of the matter field $\varphi$ is chosen as $\tau ^{\mu \nu }=\partial ^{\mu }\varphi \partial ^{\nu }\varphi -g^{\mu \nu }{\mathcal {L}}_{\mathrm {corr}}(\varphi , \partial _{\alpha }\varphi )$ with a (gravitationally corrected) emergent Lagrangian density ${\mathcal {L}}_{\mathrm {corr}}=\frac{1}{2}\partial _{\mu }\varphi \partial ^{\mu }\varphi -\frac{1}{2}m_{\mathrm {corr}}^{2}\varphi ^{2}$, the covariant divergence of $\tau ^{\mu \nu }$ will vanish, i.e.,

$$\begin{aligned} \nabla _{\mu }\tau ^{\mu \nu }=\left( \nabla _{\mu }\partial ^{\mu }\varphi +m_{\mathrm {corr}}^{2}\varphi \right) \partial ^{\nu }\varphi =0. \end{aligned}$$

(104)

With the help of Eq. (104), the functional variational result (23) can be reduced to the simple form (24).

Appendix 2: Dirac equation in a gravitational field with torsion

This Appendix is presented in order to provide a simple approach for treating spinor fields in a gravitational field with torsion. For convenience, we shall ignore ${\mathcal {L}}_{\psi -\phi }$ temporarily (i.e., $\xi$ and $\xi ^{*}$ are assumed to be vanishing) in the Lagrangians (60). We shall consider the Dirac equation in a gravitational field with torsion. The field equation of $\psi$ and $\bar{\psi }$ in the spacetime with torsion are given by

$$\begin{aligned} \left[ \mathrm {i}\gamma ^{\mu }\left( \overset{\rightharpoonup }{{\mathcal {D}}}_{\mu }-\frac{T_{\mu }}{2}\right) -m\right] \psi =0, \qquad -\bar{\psi }\left[ \left( \overset{\leftharpoonup }{{\mathcal {D}}}_{\mu }-\frac{T_{\mu }}{2}\right) \mathrm {i}\gamma ^{\mu }+m\right] =0. \end{aligned}$$

(105)

Here, the covariant derivatives are defined as $\overset{\rightharpoonup }{{\mathcal {D}}}_{\mu }\psi =\partial _{\mu }\psi -\mathrm {i}B_{\mu }\psi$ and $\bar{\psi }\overset{\leftharpoonup }{{\mathcal {D}}}_{\mu }=\partial _{\mu }\bar{\psi }+\bar{\psi }\mathrm {i}B_{\mu }$, where the spin connection (in the spinor representation) $B_{\mu }=\frac{\mathrm {i}}{2}\omega _{\mu }{}^{PQ}\Sigma _{QP}$, the spin connection (in the vector representation) $\omega _{\mu }{}^{PQ}=\mathrm {i}e^{P}{}_{\lambda }\nabla _{\mu }e^{Q\lambda }$, and the Lorentz-group algebraic generators $\Sigma _{QP}=\frac{\mathrm {i}}{4}(\gamma _{Q}\gamma _{P}-\gamma _{P}\gamma _{Q})$. Equation (105) can be obtained by the Euler-Lagrange equations $\frac{\partial (\sqrt{-g}{\mathcal {L}}_{\psi })}{\partial \bar{\psi }}-\partial _{\mu }\frac{\partial (\sqrt{-g}{\mathcal {L}}_{\psi })}{\partial \partial _{\mu }\bar{\psi }}=0$ and $\frac{\partial (\sqrt{-g}{\mathcal {L}}_{\psi })}{\partial {\psi }}-\partial _{\mu }\frac{\partial (\sqrt{-g}{\mathcal {L}}_{\psi })}{\partial \partial _{\mu }{\psi }}=0$. For example, we can have

$$\begin{aligned}&\frac{\partial (\sqrt{-g}{\mathcal {L}}_{\psi })}{\partial \bar{\psi }} =\frac{\sqrt{-g}}{2}\left[ \left( \mathrm {i}\gamma ^{\mu } \overset{\rightharpoonup }{{\mathcal {D}}}_{\mu }-m\right) \psi -\left( -B_{\mu }\gamma ^{\mu }+m\right) \psi \right] , \end{aligned}$$

(106)

$$\begin{aligned}&\quad -\partial _{\mu }\frac{\partial (\sqrt{-g}{\mathcal {L}}_{\psi })}{\partial \partial _{\mu }\bar{\psi }}=\partial _{\mu }\left[ \frac{\sqrt{-g}}{2} \left( \mathrm {i}\gamma ^{\mu }\psi \right) \right] =\frac{\sqrt{-g}}{2} \left( \nabla _{\mu }-T_{\mu }\right) \mathrm {i}\gamma ^{\mu }\psi , \qquad \quad \end{aligned}$$

(107)

where we have used the covariant divergence formula in the spacetime with torsion

$$\begin{aligned} \nabla _{\mu }A^{\mu }=\frac{1}{\sqrt{-g}}\partial _{\mu } \left( \sqrt{-g}A^{\mu }\right) +T_{\mu }A^{\mu }. \end{aligned}$$

(108)

Thus, from the results (106) and (107), we can have

$$\begin{aligned}&\frac{\partial (\sqrt{-g}{\mathcal {L}}_{\psi })}{\partial \bar{\psi }} -\partial _{\mu }\frac{\partial (\sqrt{-g}{\mathcal {L}}_{\psi })}{\partial \partial _{\mu }\bar{\psi }} \nonumber \\&\quad =\frac{\sqrt{-g}}{2}\left\{ \left( \mathrm {i}\gamma ^{\mu } \overset{\rightharpoonup }{{\mathcal {D}}}_{\mu }-m\right) \psi +\Big [\left( \nabla _{\mu }-\mathrm {i}B_{\mu }-T_{\mu }\right) \mathrm {i}\gamma ^{\mu }-m\Big ]\psi \right\} \nonumber \\&\quad =\frac{\sqrt{-g}}{2}\left\{ \left( \mathrm {i}\gamma ^{\mu } \overset{\rightharpoonup }{{\mathcal {D}}}_{\mu }-m\right) \psi +\Big [\left( \overset{\rightharpoonup }{{\mathcal {D}}}_{\mu }-T_{\mu } \right) \mathrm {i}\gamma ^{\mu }-m\Big ]\psi \right\} \nonumber \\&\quad =\sqrt{-g}\left[ \mathrm {i}\gamma ^{\mu } \left( \overset{\rightharpoonup }{{\mathcal {D}}}_{\mu } -\frac{T_{\mu }}{2}\right) -m\right] \psi , \end{aligned}$$

(109)

where $\overset{\rightharpoonup }{{\mathcal {D}}}_{\mu } (\mathrm {i}\gamma ^{\mu }\psi )=(\nabla _{\mu }-\mathrm {i}B_{\mu }) (\mathrm {i}\gamma ^{\mu }\psi )$ and $(\overset{\rightharpoonup }{{\mathcal {D}}}_{\mu }-T_{\mu }) \mathrm {i}\gamma ^{\mu }=\mathrm {i}\gamma ^{\mu } (\overset{\rightharpoonup }{{\mathcal {D}}}_{\mu }-T_{\mu })$ have been employed. Therefore, the Dirac Eq. (105) in a curved spacetime with torsion can be derived.

Now let us turn to the Dirac Eq. (105). If we multiply $\bar{\psi }$ left on the first equation and $\psi$ right on the second equation, the sum is given by

$$\begin{aligned}&\bar{\psi }\mathrm {i}\gamma ^{\mu }\overset{\rightharpoonup }{{\mathcal {D}}}_{\mu }\psi +\bar{\psi }\overset{\leftharpoonup }{{\mathcal {D}}}_{\mu }\mathrm {i}\gamma ^{\mu }\psi -T_{\mu } \left( \bar{\psi }\mathrm {i}\gamma ^{\mu }\psi \right) =0\nonumber \\&\quad \Rightarrow \nabla _{\mu }\left( \bar{\psi }\mathrm {i}\gamma ^{\mu }\psi \right) -T_{\mu }\left( \bar{\psi }\mathrm {i}\gamma ^{\mu }\psi \right) =0 \nonumber \\&\quad \Rightarrow \frac{1}{\sqrt{-g}}\partial _{\mu } \left( \sqrt{-g}J^{\mu }\right) =0, \end{aligned}$$

(110)

where the Dirac current density is $J^{\mu }=\bar{\psi }\gamma ^{\mu }\psi$. The spin-connection covariant derivative of the Dirac matrix vanishes, i.e., $\overset{\rightharpoonup }{{\mathcal {D}}}_{\mu }\gamma ^{\nu }\equiv \nabla _{\mu }\gamma ^{\nu }-\mathrm {i}[B_{\mu }, \gamma ^{\nu }]=0$. In the last step of (110), the relation (108) has been applied. The sum of the first two terms $\bar{\psi }\mathrm {i}\gamma ^{\mu }\overset{\rightharpoonup }{{\mathcal {D}}}_{\mu }\psi +\bar{\psi }\overset{\leftharpoonup }{{\mathcal {D}}}_{\mu }\mathrm {i}\gamma ^{\mu }\psi$ on the left-hand side of (110) has been rearranged as

$$\begin{aligned}&\bar{\psi }\mathrm {i}\gamma ^{\mu }\overset{\rightharpoonup }{{\mathcal {D}}}_{\mu }\psi +\bar{\psi }\overset{\leftharpoonup }{{\mathcal {D}}}_{\mu }\mathrm {i}\gamma ^{\mu }\psi =\bar{\psi }\mathrm {i}\gamma ^{\mu }\left( \partial _{\mu } \psi -\mathrm {i}B_{\mu }\psi \right) +\left( \partial _{\mu }\bar{\psi }+\bar{\psi }\mathrm {i} B_{\mu }\right) \mathrm {i}\gamma ^{\mu }\psi \nonumber \\&\quad =\bar{\psi }\mathrm {i}\gamma ^{\mu }\partial _{\mu }\psi +\partial _{\mu }\bar{\psi }\mathrm {i}\gamma ^{\mu }\psi +\bar{\psi }[\gamma ^{\mu }, B_{\mu }]\psi . \end{aligned}$$

(111)

Since ${\mathcal {D}}_{\mu }\gamma ^{\mu }\equiv \nabla _{\mu } \gamma ^{\mu }-\mathrm {i}[B_{\mu }, \gamma ^{\mu }]=0$, we have $[\gamma ^{\mu }, B_{\mu }]=\mathrm {i}\nabla _{\mu }\gamma ^{\mu }$. Thus, the result of (111) can be rewritten as $\bar{\psi }\mathrm {i}\gamma ^{\mu }\overset{\rightharpoonup }{{\mathcal {D}}}_{\mu }\psi +\bar{\psi }\overset{\leftharpoonup }{{\mathcal {D}}}_{\mu }\mathrm {i}\gamma ^{\mu }\psi =\nabla _{\mu }\left( \bar{\psi }\mathrm {i}\gamma ^{\mu }\psi \right)$. The relation (110) can be viewed as a theorem of the conservation law of divergenceless Dirac current density $J^{\mu }$ in the spacetime with torsion:

$$\begin{aligned} \nabla {}^{T}_{\mu }J^{\mu }\equiv \left( \nabla _{\mu } -T_{\mu }\right) J^{\mu }=\frac{1}{\sqrt{-g}}\partial _{\mu } \left( \sqrt{-g}J^{\mu }\right) =0. \end{aligned}$$

(112)

Appendix 3: Functional variation of the gravitational Lagrangian with respect to the translational gauge potential

Now we shall consider the functional variations of $\sqrt{-g}\left( {\mathcal {L}}_{\omega }+{\mathcal {L}}_{e}\right)$ with respect to the vierbein (translational gauge potential). The gravitational Lagrangian density (88) consists of the rotational gauge field Lagrangian density ${\mathcal {L}}_{\omega }$ and the translational gauge field Lagrangian density ${\mathcal {L}}_{e}$. In the torsionless spacetime, however, the rotational gauge potential, is determined solely by the vierbein, as in the Einstein–Hilbert formalism. But, as a matter of fact, within the framework of local gauge principles, torsion must appear as long as the energy-momentum tensor of matter fields is present. Only if the coefficient $\tau$ in the torsion-quadratic Lagrangian density ${\mathcal {L}}_{e}$ is sufficiently large can the torsion be negligibly small.

The Lagrangian density of the Lorentz-rotational gravitational gauge field is given by

$$\begin{aligned} {\mathcal {L}}_{\omega }= & {} \zeta \frac{1}{2}e_{KL}{}^{\mu \nu }{\Omega }_{\mu \lambda }{}^{KM}{\Omega }_{\nu }{}^{\lambda }{}_{M}{}^{L} \nonumber \\= & {} \zeta \frac{1}{2}e_{K}{}^{\mu }e_{L}{}^{\nu }g^{\lambda \sigma }\left\{ \frac{1}{2}\Big [\left( {\Omega }_{\mu \lambda }{}^{KM}{\Omega }_{\nu \sigma }{}_{M}{}^{L}-{\Omega }_{\nu \lambda }{}^{KM}{\Omega }_{\mu \sigma }{}_{M}{}^{L}\right) +(\lambda \leftrightarrow \sigma )\Big ]\right\} \nonumber \\= & {} \zeta \frac{1}{2}e_{K}{}^{\mu }e_{L}{}^{\nu }g^{\lambda \sigma }{\mathcal {X}}_{[\mu \nu ]\{\lambda \sigma \}}{}^{KL}, \end{aligned}$$

(113)

where the tensor ${\mathcal {X}}_{[\mu \nu ]\{\lambda \sigma \}}{}^{KL}=\frac{1}{2}\Big [\left( {\Omega }_{\mu \lambda }{}^{KM}{\Omega }_{\nu \sigma }{}_{M}{}^{L}-{\Omega }_{\nu \lambda }{}^{KM}{\Omega }_{\mu \sigma }{}_{M}{}^{L}\right) +(\lambda \leftrightarrow \sigma )\Big ]$ is antisymmetric in its indices $\mu , \nu$ and K, L, respectively, and symmetric in its indices $\lambda , \sigma$. We shall calculate the functional variation of $\delta (\sqrt{-g}{\mathcal {L}}_{\omega })$ with respect to the vierbein. It should be emphasized that the two gauge potentials (Lorentz-rotational gauge potential and spacetime-translational gauge potential) are independent variables in a spacetime with torsion, and hence the Lorentz-rotational gauge field strength (i.e., the curvature tensor ${\Omega }_{\mu \lambda }{}^{KM}$) corresponding to the Lorentz-rotational gauge potential (i.e., spin connection) no longer depends upon the spacetime-translational gauge potential (i.e., the vierbein). Therefore, the variation of the tensor ${\mathcal {X}}_{[\mu \nu ]\{\lambda \sigma \}}{}^{KL}$ with respect to the vierbein vanishes. It can be found that the functional variation of the curvature-quadratic Lagrangian density with respect to the vierbein $e_{K}{}^{\mu }$ is of the form

$$\begin{aligned} \delta (\sqrt{-g}{\mathcal {L}}_{\omega })= & {} \sqrt{-g} \left[ \zeta \Big (e_{L}{}^{\nu }g^{\lambda \sigma } {\mathcal {X}}_{[\mu \nu ]\{\lambda \sigma \}}{}^{KL} +e_{S}{}^{\tau }e_{L}{}^{\nu }e^{K\sigma }{\mathcal {X}}_{[\tau \nu ] \{\mu \sigma \}}{}^{SL}\Big )\right. \nonumber \\&\left. -e^{K}{}_{\mu }{\mathcal {L}}_{\omega } \right] \delta e_{K}{}^{\mu }, \end{aligned}$$

(114)

where the relations $\delta \sqrt{-g}=-\sqrt{-g}e^{K}{}_{\mu }\delta e_{K}{}^{\mu }$ and $\delta g^{\lambda \sigma }=\left( \delta e_{K}{}^{\lambda }\right) e^{K\sigma }+e_{K}{}^{\lambda }\delta e^{K\sigma }$ have been substituted.

Now we return to the torsion-quadratic Lagrangian density. In the gravitational Lagrangian density (88), the spacetime-translational Lagrangian density (quadratic in torsion) is

$$\begin{aligned} {\mathcal {L}}_{e}= & {} \zeta \tau \frac{1}{2}e_{KL}{}^{\mu \nu } {\mathcal {T}}_{\mu \lambda }{}^{K}{\mathcal {T}}_{\nu }{}^{\lambda L} \nonumber \\= & {} \zeta \tau \frac{1}{2}e_{K}{}^{\mu }e_{L}{}^{\nu } g^{\lambda \sigma }\left\{ \frac{1}{2}\Big [\left( {\mathcal {T}}_{\mu \lambda } {}^{K}{\mathcal {T}}_{\nu \sigma }{}^{L}-{\mathcal {T}}_{\nu \lambda } {}^{K}{\mathcal {T}}_{\mu \sigma }{}^{L}\right) +(\lambda \leftrightarrow \sigma )\Big ]\right\} \nonumber \\= & {} \zeta \tau \frac{1}{2}e_{K}{}^{\mu }e_{L}{}^{\nu } g^{\lambda \sigma }\mathcal {Y}_{[\mu \nu ]\{\lambda \sigma \}}{}^{KL}, \end{aligned}$$

(115)

where the tensor $\mathcal {Y}_{[\mu \nu ]\{\lambda \sigma \}}{}^{KL} =\frac{1}{2}\Big [\left( {\mathcal {T}}_{\mu \lambda } {}^{K}{\mathcal {T}}_{\nu \sigma }{}^{L}-{\mathcal {T}}_{\nu \lambda } {}^{K}{\mathcal {T}}_{\mu \sigma }{}^{L}\right) +(\lambda \leftrightarrow \sigma )\Big ]$. The tensor $\mathcal {Y}_{[\mu \nu ]\{\lambda \sigma \}}{}^{KL}$ is antisymmetric in its indices $\mu , \nu$ and K, L, respectively, and symmetric in its indices $\lambda , \sigma$. The functional variation of the translational Lagrangian density with respect to the vierbein is given by

$$\begin{aligned} \delta (\sqrt{-g}{\mathcal {L}}_{e})= & {} \zeta \tau \frac{1}{2} \Big [\delta (\sqrt{-g}e_{K}{}^{\mu }e_{L}{}^{\nu }g^{\lambda \sigma }) \Big ]\mathcal {Y}_{[\mu \nu ]\{\lambda \sigma \}}{}^{KL}\nonumber \\&+\zeta \tau \frac{1}{2}\sqrt{-g}e_{K}{}^{\mu }e_{L}{}^{\nu } g^{\lambda \sigma }\Big (\delta \mathcal {Y}_{[\mu \nu ] \{\lambda \sigma \}}{}^{KL}\Big ). \end{aligned}$$

(116)

In the same fashion as that in Eq. (114), the first term on the right-hand side of (116) can be rearranged as

$$\begin{aligned}&\zeta \tau \frac{1}{2}\Big [\delta (\sqrt{-g}e_{K} {}^{\mu }e_{L}{}^{\nu }g^{\lambda \sigma })\Big ]\mathcal {Y}_{[\mu \nu ] \{\lambda \sigma \}}{}^{KL} \nonumber \\&\quad =\sqrt{-g}\left[ \zeta \tau \Big (e_{L}{}^{\nu }g^{\lambda \sigma } \mathcal {Y}_{[\mu \nu ]\{\lambda \sigma \}}{}^{KL} +e_{S}{}^{\tau }e_{L}{}^{\nu }e^{K\sigma } \mathcal {Y}_{[\tau \nu ]\{\mu \sigma \}} {}^{SL}\Big )-e^{K}{}_{\mu }{\mathcal {L}}_{e}\right] \delta e_{K}{}^{\mu }.\nonumber \\ \end{aligned}$$

(117)

Now we shall concentrate our attention on the second term on the right-hand side of (116). It can be rewritten as

$$\begin{aligned} \zeta \tau \frac{1}{2}\sqrt{-g}e_{K}{}^{\mu }e_{L}{}^{\nu }g^{\lambda \sigma }\Big (\delta \mathcal {Y}_{[\mu \nu ]\{\lambda \sigma \}}{}^{KL}\Big )= & {} \zeta \tau \frac{1}{2}\sqrt{-g}e_{KL}{}^{\mu \nu }g^{\lambda \sigma }\delta \Big ({\mathcal {T}}_{\mu \lambda }{}^{K}{\mathcal {T}}_{\nu \sigma }{}^{L}\Big ) \nonumber \\= & {} \zeta \tau \sqrt{-g}e_{KL}{}^{\mu \nu }g^{\lambda \sigma }{\mathcal {T}}_{\nu \sigma }{}^{L}\Big (\delta {\mathcal {T}}_{\mu \lambda }{}^{K}\Big ).\quad \quad \quad \end{aligned}$$

(118)

This result should be interpreted in more detail: The variation $e_{KL}{}^{\mu \nu }g^{\lambda \sigma }\delta \left( {\mathcal {T}}_{\mu \lambda }{}^{K}{\mathcal {T}}_{\nu \sigma }{}^{L}\right)$ is the sum of the two terms $e_{KL}{}^{\mu \nu }g^{\lambda \sigma }\left( \delta {\mathcal {T}}_{\mu \lambda }{}^{K}\right) {\mathcal {T}}_{\nu \sigma }{}^{L}$ and $e_{KL}{}^{\mu \nu }g^{\lambda \sigma }{\mathcal {T}}_{\mu \lambda }{}^{K}\delta {\mathcal {T}}_{\nu \sigma }{}^{L}$. Since $e_{KL}{}^{\mu \nu }$ is invariant under simultaneous exchange of two-pair indices: $\mu \leftrightarrow \nu$ and $K\leftrightarrow L$ (i.e., $e_{KL}{}^{\mu \nu }=e_{LK}{}^{\nu \mu }$), the second term in the sum can be rearranged as $e_{KL}{}^{\mu \nu }g^{\lambda \sigma }{\mathcal {T}}_{\mu \lambda }{}^{K}\delta {\mathcal {T}}_{\nu \sigma }{}^{L} =e_{KL}{}^{\mu \nu }g^{\lambda \sigma }{\mathcal {T}}_{\nu \lambda }{}^{L}\delta {\mathcal {T}}_{\mu \sigma }{}^{K}$. Since $g^{\lambda \sigma }$ is symmetric in $\lambda , \sigma$, this term is in fact equal to the first term in the sum. Thus, we have the result in Eq. (118).

As the torsion ${\mathcal {T}}_{\mu \lambda }{}^{K}$ is antisymmetric in its indices $\mu , \lambda$, i.e., ${\mathcal {T}}_{\mu \lambda }{}^{K}=\left( \partial _{\mu }e^{K}{}_{\lambda }-\mathrm {i}\omega _{\mu }{}^{K}{}_{R}e^{R}{}_{\lambda }\right) -(\mu \leftrightarrow \lambda )$, the result of (118) can be rewritten as

$$\begin{aligned}&\zeta \tau \frac{1}{2}\sqrt{-g}e_{K}{}^{\mu }e_{L} {}^{\nu }g^{\lambda \sigma }\Big (\delta \mathcal {Y}_{[\mu \nu ]\{\lambda \sigma \}}{}^{KL}\Big ) \nonumber \\&\quad =\zeta \tau \sqrt{-g}\left( e_{KL}{}^{\mu \nu }g^{\lambda \sigma } -(\mu \leftrightarrow \lambda )\right) {\mathcal {T}}_{\nu \sigma } {}^{L}\delta \left( \partial _{\mu }e^{K}{}_{\lambda } -\mathrm {i}\omega _{\mu }{}^{K}{}_{R}e^{R}{}_{\lambda }\right) \nonumber \\&\quad =\zeta \tau \sqrt{-g}{\mathcal {Z}}^{[\mu \lambda ]}{}_{K} \delta \left( \partial _{\mu }e^{K}{}_{\lambda } -\mathrm {i}\omega _{\mu }{}^{K}{}_{R}e^{R}{}_{\lambda }\right) \end{aligned}$$

(119)

with ${\mathcal {Z}}^{[\mu \lambda ]}{}_{K} =\left( e_{KL}{}^{\mu \nu }g^{\lambda \sigma } -(\mu \leftrightarrow \lambda )\right) {\mathcal {T}}_{\nu \sigma }{}^{L}$. This tensor is antisymmetric in its indices $\mu , \lambda$. We can rewrite the result of (119) as

$$\begin{aligned}&\sqrt{-g}{\mathcal {Z}}^{[\mu \lambda ]}{}_{K} \delta \left( \partial _{\mu }e^{K}{}_{\lambda } -\mathrm {i}\omega _{\mu }{}^{K}{}_{R}e^{R}{}_{\lambda }\right) \nonumber \\&=\partial _{\mu }\left( \sqrt{-g}{\mathcal {Z}}^{[\mu \lambda ]} {}_{K}\delta e^{K}{}_{\lambda }\right) -\left[ \partial _{\mu } \left( \sqrt{-g}{\mathcal {Z}}^{[\mu \lambda ]}{}_{K}\right) +\sqrt{-g}{\mathcal {Z}}^{[\mu \lambda ]}{}_{R}\mathrm {i}\omega _{\mu }{}^{R}{}_{K}\right] \delta e^{K}{}_{\lambda } \nonumber \\&=\mathrm {D.T.}-\sqrt{-g}\left( {\mathcal {D}}_{\mu }{\mathcal {Z}}^{[\mu \lambda ]}{}_{K}-\frac{1}{2}\tilde{T}^{\lambda }{}_{\alpha \beta }{\mathcal {Z}}^{[\alpha \beta ]}{}_{K}\right) \delta e^{K}{}_{\lambda }, \end{aligned}$$

(120)

where we have used the following relations $\partial _{\mu }\left( \sqrt{-g}{\mathcal {Z}}^{[\mu \lambda ]}{}_{K}\right) =\sqrt{-g}\left( \nabla _{\mu }{\mathcal {Z}}^{[\mu \lambda ]}{}_{K}-\right. \left. \frac{1}{2}\tilde{T}^{\lambda }{}_{\alpha \beta }{\mathcal {Z}}^{[\alpha \beta ]}{}_{K}\right)$, ${\mathcal {Z}}^{[\mu \lambda ]}{}_{R}\mathrm {i}\omega _{\mu }{}^{R}{}_{K}=-\mathrm {i}\omega _{\mu K}{}^{R}{\mathcal {Z}}^{[\mu \lambda ]}{}_{R}$, and ${\mathcal {D}}_{\mu }{\mathcal {Z}}^{[\mu \lambda ]}{}_{K}\equiv \nabla _{\mu }{\mathcal {Z}}^{[\mu \lambda ]}{}_{K}-\mathrm {i}\omega _{\mu K}{}^{R}{\mathcal {Z}}^{[\mu \lambda ]}{}_{R}$.

Now we shall consider the variation $\delta e^{K}{}_{\lambda }$. We should rewrite $\delta e^{K}{}_{\lambda }$ in terms of $\delta e^{K\lambda }$ (i.e., the variation $\delta e^{K}{}_{\lambda }$ with the covariant index $\lambda$ will be changed to the variation of the vierbein with a contravariant index): $\delta e^{K}{}_{\lambda }=-e^{K}{}_{\alpha }e_{S\lambda }\delta e^{S\alpha }$. By substituting this relation into the variational result (120), one obtains

$$\begin{aligned}&\sqrt{-g}{\mathcal {Z}}^{[\mu \lambda ]}{}_{K}\delta \left( \partial _{\mu }e^{K}{}_{\lambda }-\mathrm {i} \omega _{\mu }{}^{K}{}_{R}e^{R}{}_{\lambda }\right) \nonumber \\&\quad =\mathrm {D.T.}+\sqrt{-g}\left( {\mathcal {D}}_{\nu } {\mathcal {Z}}^{[\nu \lambda ]}{}_{R}-\frac{1}{2} \tilde{T}^{\lambda }{}_{\alpha \beta }{\mathcal {Z}}^{[\alpha \beta ]} {}_{R}\right) e^{R}{}_{\mu }e^{K}{}_{\lambda }\delta e_{K}{}^{\mu }. \end{aligned}$$

(121)

We are now in a position to turn back to Eq. (116). It follows from Eqs. (117) and (121) that the functional variation (116) of the translational Lagrangian density with respect to the vierbein $e_{K}{}^{\mu }$ is

$$\begin{aligned} \delta (\sqrt{-g}{\mathcal {L}}_{e})= & {} \mathrm {D.T.}+\sqrt{-g}\Bigg \{\zeta \tau \left( {\mathcal {D}}_{\nu }{\mathcal {Z}}^{[\nu \lambda ]}{}_{R}-\frac{1}{2}\tilde{T}^{\lambda }{}_{\alpha \beta }{\mathcal {Z}}^{[\alpha \beta ]}{}_{R}\right) e^{R}{}_{\mu }e^{K}{}_{\lambda } \nonumber \\&+\Big [\zeta \tau \Big (e_{L}{}^{\nu }g^{\lambda \sigma }\mathcal {Y}_{[\mu \nu ]\{\lambda \sigma \}}{}^{KL}\nonumber \\&+e_{S}{}^{\tau }e_{L}{}^{\nu }e^{K\sigma }\mathcal {Y}_{[\tau \nu ]\{\mu \sigma \}}{}^{SL}\Big )-e^{K}{}_{\mu }{\mathcal {L}}_{e}\Big ]\Bigg \}\delta e_{K}{}^{\mu }. \end{aligned}$$

(122)

The functional variation of the Lagrangian of quantum vacuum energy should also be taken into account. According to the expression (92), the Lagrangian density of the vacuum energy is given by ${\mathcal {L}}_{\mathrm {vac}}=-\rho _{\mathrm {vac}}-\frac{1}{2}\lambda R$ (with $\lambda =8\pi G\rho _{\mathrm {vac}}$), i.e.,

$$\begin{aligned} {\mathcal {L}}_{\mathrm {vac}}= & {} -\rho _{\mathrm {vac}} -\frac{1}{2}\Omega _{\mu \nu }{}^{PQ}\varpi ^{\mu \nu }{}_{QP}\nonumber \\= & {} -\rho _{\mathrm {vac}}+\lambda \frac{\mathrm {i}}{2}e_{K} {}^{\mu }e_{L}{}^{\nu }\Omega _{\mu \nu }{}^{KL} \end{aligned}$$

(123)

with the vacuum-energy cosmological constant term $\varpi ^{\mu \nu }{}_{QP}=\frac{\mathrm {i}}{2} \lambda \left( e_{Q}{}^{\mu }e_{P}{}^{\nu }-e_{P} {}^{\mu }\right. \left. e_{Q}{}^{\nu }\right)$. The variation with respect to the vierbein (translational gauge potential) is

$$\begin{aligned} \delta (\sqrt{-g}{\mathcal {L}}_{\mathrm {vac}})= & {} -\rho _{\mathrm {vac}}\delta \sqrt{-g}+\lambda \frac{\mathrm {i}}{2}\Big [\sqrt{-g}\delta \left( e_{K}{}^{\mu }e_{L}{}^{\nu }\right) +e_{K}{}^{\mu }e_{L}{}^{\nu }\delta \sqrt{-g}\Big ]\Omega _{\mu \nu }{}^{KL} \nonumber \\= & {} -\rho _{\mathrm {vac}}\left( -\sqrt{-g}e^{K}{}_{\mu }\delta e_{K}{}^{\mu }\right) \nonumber \\&+\lambda \frac{\mathrm {i}}{2}\Big [2\sqrt{-g}e_{L} {}^{\nu }\Omega _{\mu \nu }{}^{KL}+e_{M}{}^{\sigma }e_{L}{}^{\nu }\Omega _{\sigma \nu }{}^{ML}\left( -\sqrt{-g}e^{K}{}_{\mu }\right) \Big ]\delta e_{K}{}^{\mu }\nonumber \\= & {} \sqrt{-g}\left[ \rho _{\mathrm {vac}}e^{K}{}_{\mu }-\lambda \left( R^{K}{}_{\mu }-\frac{1}{2}e^{K}{}_{\mu }R\right) \right] \delta e_{K}{}^{\mu }, \end{aligned}$$

(124)

where we have used the relation $\delta \left( e_{K}{}^{\mu }e_{L}{}^{\nu }\right) \Omega _{\mu \nu }{}^{KL} =\left( \delta e_{K}{}^{\mu }\right) e_{L}{}^{\nu }\Omega _{\mu \nu }{}^{KL}+e_{K}{}^{\mu }\left( \delta e_{L}{}^{\nu }\right) \Omega _{\mu \nu }{}^{KL}$. Since $\Omega _{\mu \nu }{}^{KL}$ is invariant under simultaneous exchange of two-pair indices: $\mu \leftrightarrow \nu$ and $K\leftrightarrow L$ (i.e., $\Omega _{\mu \nu }{}^{KL}=\Omega _{\nu \mu }{}^{LK}$), we have $\delta \left( e_{K}{}^{\mu }e_{L}{}^{\nu }\right) \Omega _{\mu \nu }{}^{KL}=2\left( \delta e_{K}{}^{\mu }\right) e_{L}{}^{\nu }\Omega _{\mu \nu }{}^{KL}$. The relation $\delta \sqrt{-g}=-\sqrt{-g}e^{K}{}_{\mu }\delta e_{K}{}^{\mu }$ and the definitions $e_{L}{}^{\nu }\Omega _{\mu \nu }{}^{KL}=e_{L}{}^{\nu }\mathrm {i}R^{KL}{}_{\mu \nu }=\mathrm {i}R^{K}{}_{\mu }$, $e_{M}{}^{\sigma }e_{L}{}^{\nu }\Omega _{\sigma \nu }{}^{ML}=e_{M}{}^{\sigma }e_{L}{}^{\nu }\mathrm {i}R^{ML}{}_{\sigma \nu }=\mathrm {i}R$ have been used in (124).

According to the principle of action $\delta \int \left( {\mathcal {L}}_{\omega }+{\mathcal {L}}_{e}+{\mathcal {L}}_{\mathrm {m}} +{\mathcal {L}}_{\mathrm {vac}}\right) [e]\sqrt{-g}\mathrm {d}^{4}x=0$ ([e] represents the functional variable, i.e., the vierbein, in the Lagrangians), from the variational results given in Eqs. (114), (122) and (124), the translational gauge field equation is given by

$$\begin{aligned}&-\zeta \tau e^{R}{}_{\mu }e^{K}{}_{\lambda }{\mathcal {D}}_{\nu }^{T} {\mathcal {Z}}^{[\nu \lambda ]}{}_{R} ={\overset{\mathrm {g}}{T}}{}^{K}{}_{\mu }\nonumber \\&\quad +\frac{1}{\sqrt{-g}}\frac{\delta (\sqrt{-g} {\mathcal {L}}_{\mathrm {m}})}{\delta e_{K}{}^{\mu }}+\left[ \rho _{\mathrm {vac}}e^{K}{}_{\mu } -\lambda \left( R^{K}{}_{\mu }-\frac{1}{2}e^{K}{}_{\mu }R\right) \right] , \end{aligned}$$

(125)

where ${\mathcal {D}}_{\nu }^{T}{\mathcal {Z}}^{[\nu \lambda ]} {}_{R}\equiv {\mathcal {D}}_{\nu }{\mathcal {Z}}^{[\nu \lambda ]} {}_{R}-\frac{1}{2}\tilde{T}^{\lambda }{}_{\alpha \beta } {\mathcal {Z}}^{[\alpha \beta ]}{}_{R}$. The energy-momentum tensor, ${\overset{\mathrm {g}}{T}}{}^{K}{}_{\mu }$, of the gravitational (rotational and translational) gauge field is of the form

$$\begin{aligned} {\overset{\mathrm {g}}{T}}{}^{K}{}_{\mu }= & {} \zeta \Big [e_{L}{}^{\nu }g^{\lambda \sigma }\left( {\mathcal {X}}_{[\mu \nu ] \{\lambda \sigma \}}{}^{KL}+\tau \mathcal {Y}_{[\mu \nu ] \{\lambda \sigma \}}{}^{KL}\right) \nonumber \\&+e_{S}{}^{\tau }e_{L}{}^{\nu }e^{K\sigma }\left( {\mathcal {X}}_{[\tau \nu ] \{\mu \sigma \}}{}^{SL}+\tau \mathcal {Y}_{[\tau \nu ]\{\mu \sigma \}}{}^{SL}\right) \Big ]-e^{K}{}_{\mu }\left( {\mathcal {L}}_{\omega }+{\mathcal {L}}_{e}\right) .\nonumber \\ \end{aligned}$$

(126)

Here, ${\mathcal {X}}_{[\mu \nu ]\{\lambda \sigma \}}{}^{KL}$ is the tensor expressed in terms of the curvature, and $\mathcal {Y}_{[\mu \nu ]\{\lambda \sigma \}}{}^{KL}$ and ${\mathcal {Z}}^{[\nu \lambda ]}{}_{R}$ are the tensors relevant to the torsion. The translational gauge field Eq. (126) indicates that it is the energy-momentum tensors of the gravitational (rotational and translational) gauge fields, the matter fields and the quantum vacuum energy that are the gravitating sources of the torsion.

We shall evaluate the magnitudes of the important terms in the translational gauge field Eq. (126). It can be shown that the energy-momentum tensor ${\overset{\mathrm {g}}{T}}{}^{K}{}_{\mu }$ of the gravitational gauge fields in (126) is negligibly small compared with the other energy-momentum tensor terms on the right-hand side of Eq. (125). In the curvature $\Omega _{\mu \nu }{}^{PQ}$, there are $\partial \Gamma$ and $\Gamma ^{2}$ terms (here, $\Gamma$ denotes the Levi–Civita connection and $\partial$ the partial derivative operator). In general, in many weak gravitational fields, e.g., a local inertial frame, or a relatively small-scale spacetime (e.g., the gravitating systems with the length scales less than galactic superclusters), the square of the Levi–Civita connection ($\Gamma ^{2}$) is much smaller than its derivative term ($\partial \Gamma$), because $\partial \Gamma \sim \frac{\Gamma }{L}$ (with L the macroscopical length scale of a gravitating system, e.g., the Earth radius and solar/galactic scale, etc.) and $\Gamma \sim (\frac{GM}{L})\frac{1}{L}$, where the dimensionless $\frac{GM}{L}$ is much less than 1 for the Earth, solar and galactic systems, i.e., $\Gamma \ll \frac{1}{L}$. According to the Solar system tests for gravity, the order of magnitude of the torsion T (or the contortion) is much less than that of the Levi–Civita connection $\Gamma$. Thus, we have $T\ll \Gamma \ll \frac{1}{L}$. Obviously, the term on the left-hand side in Eq. (125) is much larger than the energy-momentum tensor of the translational gauge field (i.e., the term relevant to the tensor $\tau \mathcal {Y}_{[\mu \nu ]\{\lambda \sigma \}}{}^{KL}$ in ${\overset{\mathrm {g}}{T}}{}^{K}{}_{\mu }$). In general, the energy-momentum tensor of the rotational gauge field (i.e., the term relevant to the tensor ${\mathcal {X}}_{[\mu \nu ]\{\lambda \sigma \}}{}^{KL}$ in ${\overset{\mathrm {g}}{T}}{}^{K}{}_{\mu }$) is also small compared with the energy-momentum tensors of the matter field and the quantum vacuum energy. Therefore, the energy-momentum tensor ${\overset{\mathrm {g}}{T}}{}^{K}{}_{\mu }$ of the rotational and translational gauge fields can be ignored in Eq. (125). Now the translational gauge field equation can be reduced to the form

$$\begin{aligned} -\zeta \tau e^{R}{}_{\mu }e^{K}{}_{\lambda }{\mathcal {D}}_{\nu }^{T} {\mathcal {Z}}^{[\nu \lambda ]}{}_{R}= & {} \frac{1}{\sqrt{-g}}\frac{\delta (\sqrt{-g}{\mathcal {L}}_{\mathrm {m}})}{\delta e_{K}{}^{\mu }}\nonumber \\&+\left[ \rho _{\mathrm {vac}}e^{K}{}_{\mu } -\lambda \left( R^{K}{}_{\mu }-\frac{1}{2}e^{K}{}_{\mu }R\right) \right] . \end{aligned}$$

(127)

In the conventional theory of general relativity, the term on the left-hand side of Eq. (127) resulting from the torsion-quadratic Lagrangian ${\mathcal {L}}_{e}$ is not taken into account (i.e., $\zeta \tau =0$), and hence this equation, which is a variational result with respect to the vierbein, can serve as a gravitational field equation (Einstein type equation). Here, however, it is a field equation for determining the torsion. Since the quantum vacuum energy density $\rho _{\mathrm {vac}}$ is unusually large, in order that the torsion term ${\mathcal {Z}}^{[\nu \lambda ]}{}_{R}$ is negligibly small, the mass-squared dimensional coefficient, $\tau$, of the torsion-quadratic Lagrangian ${\mathcal {L}}_{e}$ in (88) should be sufficiently large. In Sect. 2 and Sect. 3 for the case of scalar matter field that serves as a gravitating source, we have concluded that the torsion is not required when we derive Eq. (6). The reason for this assumption may lie in that the proportionality constant $\tau$ in the translational Lagrangian density ${\mathcal {L}}_{e}$ in (88) is large enough to suppress the torsion in Eq. (127).