**1. Introduction**

The search for new spin-dependent interactions between fundamental particles apart from the magnetic dipole interaction is an important area of high-energy physics research beyond the Standard Model [1]. É. Cartan was the first who proposed, at the beginning of the 20th century, the post-Riemannian geometrical structures generated by the microstructural properties of the physical matter, particularly to analyze the coupling of the torsion of spacetime to the intrinsic spin [2].

The interest in the theory of gravitation with spin and torsion based on the Riemann– Cartan geometry had considerably grown in the second half of the 20th century after the consistent gauge theory formalism was developed [3–7]. It is now well established that the spacetime torsion can only be detected with the help of the spin [8–10]. The early theoretical analysis of the possible experimental manifestations of the torsion field can be found in [11–13]. The so-called Einstein–Cartan theory [14–16] with the linear Hilbert–Einstein gravitational Lagrangian represents a degenerate version of the Poincaré gauge gravity. In this model, the torsion couples to spin algebraically and, therefore, it vanishes outside the matter sources, but essentially modifies the physical structure inside astrophysical compact objects, see [17–19].

The torsion becomes a dynamical propagating field in the Poincaré gauge gravity theory with a Yang–Mills type Lagrangian [20], and the most general gravitational model with the Lagrangian (which is quadratic in the curvature and torsion) was considered in Reference [21] with an emphasis on the consistency of the gauge theory of gravity with experimental observations at the macroscopic level. Accordingly, its probing should be essentially confined to the microscopic level and focus on the study of the dynamics of fundamental particles, atoms, and molecules. It is worthwhile to note that it has not yet been possible to create a source of spin density that could generate torsion to be detected in the laboratory. However, one can establish the constraints on the spin–torsion coupling, in particular from the experimental search for the Lorentz and *CPT* violation. The bounds on new spin-dependent interactions were found [22,23] with the help of a torsion pendulum

**Citation:** Trukhanova, M.I.; Andreev, P.; Obukhov, Y.N. Search for Manifestations of Spin–Torsion Coupling. *Universe* **2023**, *9*, 38. https://doi.org/10.3390/ universe9010038

Academic Editor: Luca Fabbri

Received: 30 November 2022 Revised: 28 December 2022 Accepted: 30 December 2022 Published: 6 January 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

technique, which was also used in the search for *CP*-violating interactions between the pendulum's electrons and unpolarized matter in laboratory surroundings or the Sun. Among other physical effects, the contribution of the interacting vector and pseudovector of the torsion to the hyperfine splitting of the ground state of the hydrogen atom was evaluated in [24,25], whereas a possible manifestation of the spin–torsion coupling in the scattering of polarized photons in a medium of sodium vapor was analyzed in [26]. The experimental upper bounds on the spin–torsion interactions were reported in [27–29].

In the context of the growing interest in fundamental physics at the sub-eV scale, Moody and Wilczek [30] analyzed new fields, generating dipole couplings between fermions that can be detectable in laboratory experiments, paying special attention to axions. An axion as a hypothetical particle was postulated in the Peccei–Quinn theory to resolve the strong *CP* problem in quantum chromodynamics, and it can produce long-range dipole forces [31]. Similar dipole interactions between fermions can be produced by other particles [32–35], for example, by an arion, which is a boson corresponding to a spontaneous breaking of the chiral lepton symmetry [35]. The search for axions and axion-like particles is of considerable interest in relation to the cold dark matter issue. Currently, there is a number of experimental attempts to find axions that encompass the Primakoff effect for the astrophysical axions [36], the polarization measurements for light propagating in a magnetic field, the light shining through wall experiments [37], and the cosmic axion spin precession experiment [38,39].

Another direction in the search for long-range spin-dependent interactions is the prediction of the existence of an unparticle in the context of quantum excitations of scaleinvariant interactions [40], along with exotic spin-1 bosons or paraphotons [41], which are currently being actively investigated [42]. Quite generally, the analysis of the behaviors of atomic systems affected by new hypothetical spin-dependent forces gives rise to the constraints on the coupling constants [1] with the sixteen types of potentials characterizing interactions between fermions mediated by the new exotic particles [42,43]. A wide range of relevant laboratory investigations was carried out using the physical methods of atomic and molecular systems, as well as make use of the optical methods: ion capture experiments [44], using nitrogen-vacancy centers in diamonds [45,46], based on molecular and atomic spectroscopies [47].

The gauge symmetry is one of the most powerful physical concepts underlying the description of fundamental interactions. The global gauge *U*(1) invariance in quantum electrodynamics leads to the electric charge conservation law, whereas the local gauge invariance requires the introduction of a massless vector field that mediates the long-range interaction between charges, with the electric current as a source of the electromagnetic field. In quantum chromodynamics, the invariance under the global group *SU*(2) yields the isospin conservation law, and the invariance under the local transformations of the group *SU*(2) leads to the introduction of the Yang–Mills gauge field. The extension of this theory leads to the explanation of the strong interactions in terms of the exchange of gluons.

Recently [48,49], an attempt was made to construct a gauge theory model to describe the weak spin–spin interactions. It was suggested [48] that the invariance of the Lagrangian under the local Lorentz transformation requires the introduction of a massless axial vector gauge field, which gives rise to a super-weak long-range spin–spin force in a vacuum, which is attractive for parallel spins. In this model, the axial vector field couples to the axial vector current of the Dirac fermion field and the photon field or a neutral spin-1 field. The axial vector field was introduced [50] to provide stability of the classical electron and to construct divergence-free quantum electrodynamics. Optical experiments are the most accurate and accessible for measuring the effects of new physical fields. The direct interactions between electromagnetic and gauge fields were considered in [51] in the Poincaré gauge gravity approach, whereas the coupling of the photon to the axial vector gauge field was supported by the local Lorentz symmetry group in the approach [48,49]. Here, we study the possible influence of the axial torsion field on the propagation and polarization of an electromagnetic wave.

The structure of the paper is as follows. In Section 2, we briefly outline the corresponding Lagrange–Noether framework of the Poincaré gauge gravity theory. In Section 3, we study the propagation of the electromagnetic wave under the influence of the uniform homogeneous axial vector torsion field. Finally, in Section 4, we discuss the results obtained and apply them to derive the strong upper limit of the value of the cosmic background torsion field from the astrophysical data. In Appendix A, the structure of the general quadratic Poincaré gauge gravitational field Lagrangian is given, and the effective coupling constants are introduced.

Our basic conventions and notations are as follows. The world indices are labeled by Latin letters *i*, *j*, *k*, . . . = 0, 1, 2, 3 (for example, the local spacetime coordinates *x i* ), whereas we reserve Greek letters for tetrad indices, *α*, *β*, . . . = 0, 1, 2, 3 (i.e., for labeling the legs of an anholonomic coframe *e α i* ). In order to distinguish separate tetrad indices, we put hats over them. Finally, spatial indices are denoted by Latin letters from the beginning of the alphabet, *a*, *b*, *c*, . . . = 1, 2, 3. The metric of the Minkowski spacetime reads *gij* = diag(*c* 2 , −1, −1, −1), and the totally antisymmetric Levi–Civita tensor *ηijkl* has the only nontrivial component *η*<sup>0123</sup> = *c*, so that *η*0*abc* = *cεabc* with the three-dimensional Levi–Civita tensor *εabc*. The spatial components of the tensor objects are raised and lowered with the help of the Euclidean three-dimensional metric *δab*.

### **2. Gauge Theory of Gravitation**

The Poincaré gauge gravity [52–57] is an extension of Einstein's general relativity theory (GR), in which the spin, energy, and momentum are independent sources of the gravitational fields (the metric *gij* and connection Γ*ki j* ), and the spacetime structure is described by the Riemann–Cartan geometry with the curvature and the torsion:

$$\mathcal{R}\_{kl\dot{l}}{}^{j} = \partial\_{k}\Gamma\_{l\dot{l}}{}^{j} - \partial\_{l}\Gamma\_{k\dot{l}}{}^{j} + \Gamma\_{kn}{}^{j}\Gamma\_{l\dot{l}}{}^{n} - \Gamma\_{ln}{}^{j}\Gamma\_{k\dot{l}}{}^{n} \,, \tag{1}$$

$$T\_{kl}{}^{i} = \Gamma\_{kl}{}^{i} - \Gamma\_{lk}{}^{i}. \tag{2}$$

The Riemann–Cartan connection can be decomposed into the Riemannian and the post-Riemannian parts,

$$
\Gamma\_{k\dot{j}}{}^{\dot{i}} = \widetilde{\Gamma}\_{k\dot{j}}{}^{\dot{i}} - \mathcal{K}\_{k\dot{j}}{}^{\dot{i}} \,. \tag{3}
$$

where the Christoffel symbols are determined by the metric

$$\widetilde{\Gamma}\_{kj}{}^i = \frac{1}{2} g^{il} (\partial\_j g\_{kl} + \partial\_k g\_{lj} - \partial\_l g\_{kj}) \,\prime \tag{4}$$

and the contortion tensor is constructed in terms of the torsion

$$\mathcal{K}\_{k\dot{j}}{}^{\dot{i}} = -\frac{1}{2} (T\_{k\dot{j}}{}^{\dot{i}} + T\_{k\dot{j}}^{\dot{i}} + T\_{j\dot{k}}^{\dot{i}}) \,. \tag{5}$$

The torsion (2) can be decomposed [57] into three irreducible components,

$$T\_{kl}{}^{i} = {}^{(1)}T\_{kl}{}^{i} + {}^{(2)}T\_{kl}{}^{i} + {}^{(3)}T\_{kl}{}^{i} \,. \tag{6}$$

where the second irreducible part features the torsion trace vector

$${}^{(2)}T\_{kl}{}^{i} = \frac{1}{3} (\delta\_k^i T\_l - \delta\_l^i T\_k) \, \tag{7}$$

the third irreducible part is constructed in terms of the torsion axial pseudo-vector

$$\mathcal{T}^{(3)}T\_{kl}{}^{i} = -\frac{1}{3}\eta\_{kl}{}^{ij}\overline{T}\_{j\nu} \tag{8}$$

and the first irreducible purely tensor part has the properties

$$
\eta^{(1)} T\_{ik}{}^i = \mathbf{0}, \qquad \text{(1)} \ T\_{ijk} \eta^{ijkl} = \mathbf{0}. \tag{9}
$$

The vector and pseudovector of torsion are defined as

$$T\_j := T\_{ij}{}^i{}\_j \qquad \overline{T}^j = \frac{1}{2} T\_{kli} \eta^{klij} . \tag{10}$$

Here, we focus on the dynamical realization of the Poincaré gauge theory as a Yang– Mills type model with the most general quadratic in curvature and torsion Lagrangian (A3), see Appendix A for the details. Earlier [25,51], the contributions of the vector and the pseudovector (10) to physical effects at the microscopic level were analyzed in the framework of this theory and the strong constraints were established on the spin–torsion coupling parameters. Following [51], we continue to study the influence of the *axial pseudovector* torsion field (3)*Tkl <sup>i</sup>* on physical matters, and assume that the metric of spacetime is flat, whereas possible post-Riemannian deviations of the spacetime geometry are small.

As a result, the connection (3) reduces to the contortion, Γ*kj <sup>i</sup>* <sup>=</sup> <sup>−</sup>*Kkj <sup>i</sup>* = <sup>1</sup> 2 (3)*Tkj i* , and by combining (1) and (8), we find (for the curvature) that *Rkli <sup>j</sup>* = <sup>1</sup> 3 *ηi jn* [*k ∂l*]*Tn*, for the small post-Riemannian corrections. Then it is straightforward to verify that the Yang–Mills-type gauge gravity Lagrangian (A3) is simplified to

$$L = \hbar \left\{ -\frac{1}{4} f\_{\vec{l}\vec{j}} f^{\vec{i}\vec{j}} + \frac{\mu^2}{2} \mathfrak{a}\_i \mathfrak{a}^i - \frac{\lambda}{2} (\partial\_{\vec{l}} \mathfrak{a}^i)^2 \right\}. \tag{11}$$

Here, *fij* = *∂iα<sup>j</sup>* − *∂jα<sup>i</sup>* is constructed from the rescaled axial torsion trace vector field

$$\alpha\_i = \frac{\ell\_\rho}{3} \sqrt{\frac{-\Lambda\_5}{2\kappa c\hbar}} \overline{T}\_{i\nu} \tag{12}$$

and the coupling constants (A4)–(A9) of the Poincaré gravity Lagrangian (A3) determine

$$
\mu^2 = -\frac{3\mu\_1}{\ell\_\rho^2 \Lambda\_5}, \qquad \lambda = \frac{3\Lambda\_4}{2\Lambda\_5}.\tag{13}
$$

It is known that the particle spectrum of the Yang–Mills type Poincaré gauge gravity model (A3) contains, in general, the so-called ghost and tachyon modes that may lead to the loss of stability and unitarity of the theory. These issues were analyzed in References [58–60], and the necessary stability conditions were derived that restrict the choices of the coupling constants. Accordingly, we here specialize to the class of models with *b*<sup>3</sup> = 2*b*1, which for the spin 1 sector yields *λ* = 0, thus avoiding the stability and unitarity problems. Furthermore, we assume that *a*<sup>2</sup> = *a*<sup>1</sup> − *a*0, which corresponds to the vanishing rest mass *µ* = 0, in agreement with the estimates derived for the axial vector field [60,61] from the high-energy physics phenomenology.

### *2.1. Interaction between Fermions and Axial Torsion*

In accordance with the minimal coupling principle [29], the interaction between the fermion field *ψ* and gauge fields is introduced via the spinor covariant derivative *Diψ*, where

$$D\_i = \partial\_i - \frac{iq}{\hbar} A\_i + \frac{i}{4} \Gamma\_i^{\ a\beta} \sigma\_{a\beta\prime} \tag{14}$$

with the Lorentz group generators *σ αβ* = *iγ* [*αγ β*] constructed from the Dirac matrices *γ α* . As a result, the dynamics of the spinor field coupled with the gauge fields on the Minkowski flat metric background is given by the Lagrangian

$$L\_D = \frac{i\hbar}{2} \left\{ \overline{\psi}\gamma^i \partial\_i \psi - (\partial\_i \overline{\psi})\gamma^i \psi \right\} - mc\overline{\psi}\psi + qA\_i \overline{\psi}\gamma^i \psi + \frac{3}{4} \hbar \chi a\_i \overline{\psi}\gamma^i \gamma\_5 \psi. \tag{15}$$

Here, we used the identity *γ µσ αβ* + *σ αβγ <sup>µ</sup>* <sup>=</sup> <sup>−</sup> <sup>2</sup>*<sup>ǫ</sup> µναβγνγ*<sup>5</sup> and introduced

$$\chi = \frac{1}{\ell\_{\rho}} \sqrt{\frac{2\kappa c\hbar}{-\Lambda\_{5}}}.\tag{16}$$

Thereby, the axial pseudovector torsion field naturally couples to the spinor axial current *j i <sup>f</sup>* = *ψγiγ*5*<sup>ψ</sup>* or to the spin and helicity of the Dirac fermion. By recalling the definition of the Planck length ℓPl, we can recast (16) into

$$\chi = \frac{\ell\_{\rm Pl}}{\ell\_{\rho}} \sqrt{\frac{16\pi}{-\Lambda\_{5}}}.\tag{17}$$

which demonstrates that the spin–torsion coupling constant *χ* is *very small*, provided we assume that the characteristic length of the Poincaré gauge gravity is much larger than the Planck scale, ℓ*<sup>ρ</sup>* ≫ ℓPl.

### *2.2. Interaction between the Electromagnetic Field and Axial Torsion*

Following Pradhan et al. [48–50], the interaction of the axial torsion and the electromagnetic field is derived from the standard Maxwell–Lorentz Lagrangian when the ordinary derivatives are replaced by covariant ones. An apparent breaking of the *U*(1) gauge invariance can be fixed by the modified Stueckelberg method [62]. Together with the dynamical Lagrangian for the axial vector field (11) and the fermion sector terms (15), the total Lagrangian for the torsion field interacting with the spin of the matter sources then reads [49]

$$\begin{split} L &= -\frac{1}{4} \sqrt{\frac{\varepsilon\_0}{\mu\_0}} F\_{\overline{i}\overline{j}} F^{\overline{j}\overline{i}} + \frac{i\hbar}{2} \left\{ \overline{\psi}\gamma^{\overline{i}} \partial\_{\overline{i}} \psi - (\partial\_{\overline{i}} \overline{\psi}) \gamma^{i} \psi \right\} - mc \overline{\psi}\psi + qA\_i \overline{\psi}\gamma^i \psi \\ &+ \hbar \left\{ -\frac{1}{4} f\_{\overline{i}\overline{j}} f^{\overline{j}\overline{i}} + \frac{\mu^2}{2} a^2 - \frac{\lambda}{2} (\partial a)^2 \right\} + \frac{3}{4} \hbar \chi a\_i \overline{\psi}\gamma^i \gamma\_5 \psi - \chi \sqrt{\frac{\varepsilon\_0}{\mu\_0}} a\_i \eta^{ijkl} A\_{\overline{i}} \partial\_{\overline{k}} A\_{\overline{l}} \end{split} \tag{18}$$

where *ε*<sup>0</sup> and *µ*<sup>0</sup> are the electric and magnetic constants of the vacuum. The axial vector torsion is thereby coupled to the axial current density of the electron and photon fields:

$$j\_f^i = \overline{\Psi}\gamma^i \gamma\_5 \Psi, \qquad j\_b^i = \eta^{ijkl} A\_j \partial\_k A\_{l\nu} \tag{19}$$

where we explicitly have

$$A\_{\!\!\!\!-} = \{ -\phi, A \!\!\/), \qquad \mathfrak{a}\_{\!\!\!\/]} = \{ -\phi, \mathfrak{a} \}. \tag{20}$$

The quantization of the model (18) was analyzed in [51] and the static potential between fermions (due to the exchange of the axial torsion) was computed.
