**2. Evaluation of the Charged Fermionic Self-Energy within ECSK Theory**

The ECSK theory of gravity is an extension of general relativity allowing spacetime to have torsion in addition to curvature, where torsion is determined by the density of intrinsic angular momentum, reminiscent of the quantum-mechanical spin [1–9,12–18]. As in general relativity, the gravitational Lagrangian density in the ECSK theory is proportional to the curvature scalar. Unlike in general relativity, the affine connection

$$
\Gamma^k\_{i\dot{j}} = \omega\_i{}^\mu{}\_\nu e^k\_\mu e^\nu\_{\dot{j}} + e^k\_\mu \partial\_{\dot{l}} e^\mu\_{\dot{j}} \tag{1}
$$

is not restricted to be symmetric, although metric compatibility conditions are retained, defined by the requirements that the covariant derivatives of the tetrad fields *e µ k* and the Minkowski metric vanish, implying the antisymmetry *ω µν <sup>i</sup>* = −*ω νµ i* of the spin connection [19]. The antisymmetric part of the connection, namely

$$\mathcal{S}^{k}\_{\ i\dot{j}} = \Gamma^{k}\_{[i\dot{j}]} = \partial\_{i}e^{k}\_{\dot{j}} - \partial\_{\dot{j}}e^{k}\_{i} + \omega\_{i}{}^{k}{}\_{\ \lambda}e^{\lambda}\_{\dot{j}} - \omega\_{\dot{j}}{}^{k}{}\_{\ \lambda}e^{\lambda}\_{i} \tag{2}$$

(i.e., the torsion tensor), is then regarded as a dynamical variable analogous to the metric tensor *gij* in general relativity. Consequently, variation of the total action for the gravitational field and matter with respect to the metric tensor gives Einstein-type field equations that relate the curvature to the dynamical energy-momentum tensor *Tij* = (2/ √ −*g*)*δ*L/*δg ij*, where L is the matter Lagrangian density. On the other hand, variation of the total action with respect to the torsion tensor gives the Cartan equations for the spin tensor of matter [5]:

$$s^{ijk} = \frac{1}{\kappa} S^{[ijk]}, \quad \text{where} \ \kappa = \frac{8\pi G}{c^4}. \tag{3}$$

Thus, ECSK theory of gravity extends general relativity to include intrinsic spin of matter, with fermionic fields such as those of quarks and leptons providing natural sources of torsion. Torsion, in turn, modifies the Dirac equation for elementary fermions by adding to it a cubic term in the spinor fields, as observed by Kibble, Hehl and Datta [1,4,5].

It is this nonlinear Hehl–Datta equation that provides the theoretical background for our proposal. The cubic term in this equation corresponds to an axial-axial self-interaction in the matter Lagrangian, which, among other things, generates a spinor-dependent vacuum-energy term in the energy-momentum tensor (see, for example, Reference [15]). The torsion tensor *S k ij* appears in the matter Lagrangian via covariant derivative of a Dirac spinor with respect to the affine connection. The spin tensor for the Dirac spinor *ψ* can then be derived [5], and turns out to be totally antisymmetric:

$$\mathbf{s}^{ijk} = -\frac{i\hbar c}{4} \psi \gamma^{[i} \gamma^{j} \gamma^{k]} \psi \,\tag{4}$$

where *<sup>ψ</sup>*¯ <sup>≡</sup> *<sup>ψ</sup>* †*γ* 0 := (*ψ* ∗ 1 , *ψ* ∗ 2 , −*ψ* ∗ 3 , −*ψ* ∗ 4 ) is the Dirac adjoint of *ψ* and *γ <sup>i</sup>* are the Dirac matrices: *γ* (*iγ <sup>j</sup>*) = 2*g ij*. The Cartan Equations (3) render the torsion tensor to be quadratic in spinor fields. Substituting it into the Dirac equation in the Riemann–Cartan spacetime with metric signature (+, −, −, −) gives the cubic Hehl–Datta equation [1,4,5]:

$$i\hbar\gamma^{k}\psi\_{k} = mc\,\psi + \frac{3\kappa\hbar^{2}c}{8} \left(\bar{\psi}\gamma^{5}\gamma\_{k}\psi\right)\gamma^{5}\gamma^{k}\psi\_{\prime} \tag{5}$$

where the colon denotes a general-relativistic covariant derivative with respect to the Christoffel symbols, and *m* is the mass of the spinor. The Hehl–Datta Equation (5) and its adjoint can be obtained by varying the following action with respect to *ψ*¯ and *ψ*, respectively, without varying it with respect to the metric tensor or the torsion tensor [15]:

$$\mathcal{L} = \int d^4x \sqrt{-g} \left\{ -\frac{1}{\kappa} \mathcal{R} + i\hbar c (\bar{\psi}\gamma^k \psi\_{\vec{x}} - \bar{\psi}\_{\vec{x}}\gamma^k \psi) - mc^2 \bar{\psi}\psi - \frac{3\kappa\hbar^2 c^2}{8} (\bar{\psi}\gamma^5 \gamma\_k \psi)(\bar{\psi}\gamma^5 \gamma^k \psi) \right\}. \tag{6}$$

The last term in this action corresponds to the effective axial-axial, self-interaction mentioned above:

$$\mathfrak{L}\_{\rm AA} = -\sqrt{-g}\,\frac{3\kappa\hbar^2c^2}{8}(\bar{\psi}\gamma^5\gamma\_k\psi)(\bar{\psi}\gamma^5\gamma^k\psi). \tag{7}$$

This self-interaction term is not renormalisable. It is an effective Lagrangian density in which only the metric and spinor fields are dynamical variables. The original Lagrangian density for a Dirac field in which the torsion tensor is also a dynamical variable (giving the Hehl–Datta equation), is renormalisable, since it is quadratic in spinor fields. As we will see, renormalisation may not be required if ECSK gravity turns out to be what is realised in Nature, because it gives physical justification for the counter terms.

Before proceeding further we note that the above action is not the most general possible action within the present context. In addition to the axial-axial term, an axial-vector and a vector-vector term can be added to the action, albeit as non-minimal couplings (see, for example, Reference [17]). However, it has been argued in Reference [15] that minimal coupling is the most natural coupling of fermions to gravity because non-minimal couplings are sourced by components of the torsion that do not appear naturally in the models of spinning matter. For this reason we will confine our treatment to the minimal coupling of fermions to gravity and the corresponding Hehl–Datta equation, while recognizing that strictly speaking our neglect of non-minimal couplings amounts to an approximation, albeit a rather good approximation, at least as far as electrodynamics is concerned.
