Dirac and Majorana neutrino scattering by cosmic torsion in spatial-flat FRW spacetime background

The scattering amplitude or S-matrix can be expressed as

in QFT framework[27] where $\mathcal{H}_{I}$ is the interaction Hamiltonian. In the case of fermion coupling with background gravity, the interaction Hamiltonian can be read from the the Dirac action in the curved spacetime,

where ${\mathcal{D}}_{\mu}=\partial_{\mu}-iA_{\mu}$ is the Fock-Ivanenko covariant derivative in minimally coupling scheme, $A_{\mu}=\dfrac{i}{2}A^{ab}_{\ \ \mu}S_{ab}$ is a Lorentz algebra valued 1-form known as the Lorentz connection or the spin connection and $S_{ab}$ are the Lorentz generators in a given representation[9]. By convention, spacetime indices is denoted by the lowercase Greek letters, e.g. $\mu,\nu,\rho,\cdots$ etc. while the tangent space indices by lowercase Latin letters, e.g. $a,b,c,\cdots$ etc. but Latin letters $i,j,k,\cdots$ are left for space indices of spacetime. Due to the equivalence principle, the tangent space can be equipped with local Lorentzian frame represented by tetrad fields $h_{a}=h_{a}{}^{\mu}\partial_{\mu}$ and the corresponding coframe $h^{a}=h^{a}{}_{\mu}dx^{\mu}$ satisfying $\displaystyle{g_{\mu\nu}=\eta_{ab}h^{a}_{\ \mu}h^{b}_{\ \nu}}$ and $\eta_{ab}=g_{\mu\nu}h_{a}^{\ \mu}h_{b}^{\ \nu}$ where $\eta_{ab}=diag\left(1,-1,-1,-1\right)$ is the Minkowskian metric. For spinor field $\psi$, $S_{ab}$ are given by $\displaystyle{S_{ab}=\frac{i}{4}\left[\gamma_{a},\gamma_{b}\right]}$ with $\gamma_{a}$ the Dirac matrices. The Lorentz connection can be decomposed into $\displaystyle{A^{ab}_{\ \ \mu}=\tilde{A}^{ab}_{\ \ \mu}+K^{ab}_{\ \ \mu}}$ where the contortion tensor $K^{ab}_{\ \ \mu}$ is defined by $\displaystyle{K_{abc}=\frac{1}{2}\left(T_{bac}+T_{cab}-T_{abc}\right)}$ and $\tilde{A}^{ab}_{\ \ \mu}$ is the torsionless Levi-Civita spin connection which is determined completely by the choice of local tetrad fields expressed as $\displaystyle{\tilde{A}_{abc}=\frac{1}{2}\left(f_{bac}+f_{cab}-f_{abc}\right)}$ and determines the curvature of spacetime completely in General Relativity, where the torsion fields $\displaystyle{T^{\rho}{}_{\nu\mu}}$ are defined by $\displaystyle{T^{\rho}{}_{\nu\mu}=\Gamma^{\rho}{}_{\mu\nu}-\Gamma^{\rho}{}_{\nu\mu}}$ and $\displaystyle{{f^{c}}{}_{ab}={h_{a}}^{\mu}{h_{b}}^{\nu}\left({{\partial_{\nu}}{h^{c}}_{\mu}-{\partial_{\mu}}{h^{c}}_{\nu}}\right)}$ are the structure coefficients of tetrad basis satisfying $\displaystyle{\left[h_{a},h_{b}\right]={f^{c}}{}_{ab}h_{c}}$.

The Fock-Ivanenko covariant derivative contains both torsion field part and pure gravity part. To explore the effect of gravity and torsion separately, it would be better to decompose the Fock-Ivanenko covariant derivative into $\mathcal{D}_{\mu}=\tilde{\mathcal{D}}_{\mu}-iK_{\mu}$ where $\displaystyle{\tilde{\mathcal{D}_{\mu}}=\partial_{\mu}-\frac{i}{2}\tilde{A^{ab}}_{\mu}S_{ab}}$ is the covariant derivative for the torsionless Levi-Civita connection and the effects of torsion are fully contained in the contortion part $\displaystyle{K_{\mu}=\frac{1}{2}{K^{ab}}_{\mu}S_{ab}}$. In the minimally coupling case, the Dirac action (2) can be reduced to

where $\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$ and $\displaystyle\mathcal{A}^{a}=\frac{1}{6}\varepsilon^{abcd}T_{bcd}$ is the axial vector part of torsion tensor[10]. The vector part $\displaystyle\mathcal{V}_{a}={T^{b}}_{ba}$ does not couple with the spinor fields in the minimally coupling case. However, the non-minimal coupling is universal because the renormalization counter terms of the coupling between matter field and torsion can generate the non-minimal coupling even the original coupling is the minimal one at tree level[10, 11, 12]. Under the constraints of covariance, locality, dimension and parity preserving, the action of fermion in the curved spacetime background with torsion is

where the possible allowed non-minimal terms reduce to the minimal coupling case when $\eta_{1}=0$ and $\eta_{2}=1$[11].

The equation of motion(EoM) of spinor field correspond to the action (4) can be easily read as

The equation (5) is hard to solve for general torsion evolution. Fortunately, the solutions of spinor field for some special spacetime without torsion have been found.[28] Moreover, since general relativity has passed almost all the observational examination[13, 14, 15], the effects of torsion field should be relatively small compared to that of spacetime metric. Thus, we may take the torsion terms as perturbation to a torsion free theory. The torsion free Dirac action is

with the corresponding equation of motion is

The perturbation terms in the action can thus be read as

Since there is no time derivative terms of spinor field in the interacting Lagrangian given by (8), the interact Hamiltonian density has the property

The S-matrix can then be calculated by

Since we take torsion as perturbation to the torsion free theory in a fixed spacetime background, the spatial-flat FRW spacetime in our cases, we need to start from the solution of eq (7) in spatial-flat FRW background. The spatial-flat FRW metric is

Substituting the metric (11) into equation (7), we can get the equation of motion for spinor field in FRW spacetime,

where $\vec{\gamma}\cdot\vec{\nabla}$ is just $\gamma^{i}\partial_{i}$. The general solution of Eq.(12) has the form[29]

where phase factor of the exponential is $\displaystyle{\Omega\left({{{\vec{p}}_{C}}}\right)}=\int_{-\infty}^{t}dt^{\prime}{E_{{{\vec{p}}_{C}}a}}\left({t^{\prime}}\right)-{{\vec{p}}_{C}}\cdot\vec{x}$ and ${\vec{p}}_{C}$ is the comoving coordinate 3-momentum rather than a physical one. The relation between the comoving 3-momentum and the physical one is $\displaystyle{\vec{p}_{P}}=\frac{{{{\vec{p}}_{C}}}}{{a\left(t\right)}}$ so that the energy $E_{{{\vec{p}}_{C}}a}$ is given by $\displaystyle E_{{{\vec{p}}_{C}}a}={\left({\frac{{{{\left|{{{\vec{p}}_{C}}}\right|}^{2}}}}{{a{{\left(t\right)}^{2}}}}+{m^{2}}}\right)^{1/2}}$. The time-dependent four-component spinors ${{U_{s}}({{\vec{p}}_{C}},t)}$ and ${{V_{s}}({{\vec{p}}_{C}},t)}$ are ${U_{s}}({{\vec{p}}_{C}},t)={u_{s}}\left({{{\vec{p}}_{C}}/a}\right)$ and ${V_{s}}({{\vec{p}}_{C}},t)={v_{s}}\left({{{\vec{p}}_{C}}/a}\right)$ which satisfy $\displaystyle\left(\gamma^{a}p_{a}-m\right)u\left(\vec{p}\right)=0$ and $\displaystyle\left(\gamma^{a}p_{a}+m\right)v\left(\vec{p}\right)=0$ with normalization $\bar{u}_{r}\left(p\right)u_{s}\left(p\right)=2m\delta_{rs}$ and $\bar{v}_{r}\left(p\right)v_{s}\left(p\right)=-2m\delta_{rs}$, while the time-dependent annihilation operators ${A_{s}}({{\vec{p}}},t)$ and ${B_{s}}({{\vec{p}}},t)$ for particles and anti-particles with spin $s$ and momentum $\vec{p}$ are the time-dependent (as well as $\lvert\vec{p}\rvert$ dependent) complex linear combination of ${{A_{s}}({\vec{p}})}$ and ${{B_{-s}}^{\dagger}({-{\vec{p}}})}$ and ${{B_{s}}({\vec{p}})}$ and ${{A_{-s}}^{\dagger}({-{\vec{p}}})}$, i.e.

and

with the non-vanishing anti-commutators of the operators $A,B$,

The time evolution of complex linear combination factor $D_{a,b}\left(a,b=1,-1\right)$ is related to a factor $\displaystyle S\left(\left|{\vec{p}}_{C}\right|,t\right)=\dot{a}a^{-2}\left[2{E_{{{\vec{p}}_{C}}a}}^{2}\right]^{-1}m\left|{\vec{p}}_{C}\right|$. The evolution of $D_{a,b}$ leads to the evolution of average number of particles $\left\langle{{N_{\vec{p}}}\left(t\right)}\right\rangle$. The expanding rate of the universe $\dot{a}$ as well as the masses of neutrinos are small so that the factor $\displaystyle S\left(\left|{\vec{p}}_{C}\right|,t\right)$ can be neglected. In this case, $D_{a,b}\left({\left|{\vec{p}}\right|,t}\right)$ will remain to be a constant, i.e. $D_{a,b}\left({\left|{\vec{p}}\right|,t}\right)=\delta_{ab}$[29]. Hence, the Dirac spinor field in spatial-flat FRW spacetime can be written as

To have well defined one-particle states, we may assume the scale factor $a\left(t\right)$ varies sufficiently smooth and approaches constant values $a_{i}$ and $a_{f}$ sufficiently fast as $t\to-\infty$ and $t\to+\infty$ respectively[30]. The one-particle states with momentum $\displaystyle\vec{p}_{P}=\frac{\vec{p}_{C}}{a\left(t\right)}$ are created via creation operator with the comoving momentum ${A_{s}}^{\dagger}({\vec{p}}_{C})$[30]. Moreover, a convenient Lorentz-invariant normalization in a finite box $\left\langle\mathbf{p}_{1}\mid\mathbf{p}_{2}\right\rangle^{(R)}=2E_{\mathbf{p}_{1}}V\delta_{\mathbf{p}_{1},\mathbf{p}_{2}}$[31] can be employed. In infinite-volume limit, we take $V\to\left(2\pi\right)^{3}\delta^{3}(0)$, hence at any fixed time $t$, the normalization should be given via

Thus, the one-particle state at the fixed time $t$ should be given by $\displaystyle\left|{{{\vec{p}}_{P}},s}\right\rangle=\left|{\frac{{{{\vec{p}}_{C}}}}{a},s}\right\rangle=\sqrt{2{E_{\vec{p}a}}{a^{3}}}{A_{s}}^{\dagger}\left({{{\vec{p}}_{C}}}\right)\left|0\right\rangle$. If we set the initial state as one particle with momentum $\vec{k}_{P}$ and spin $s$ and the final state is a particle with momentum $\vec{k}^{\prime}_{P}$ and spin $r$, the initial and final state can be written as

and

Now we are ready to calculate the scattering amplitude. Since the torsion field in (8) is relatively small, it is convenient to expand the S-matrix (10) to the 1st order i.e.

and the S-matrix element can thus be calculate via

Substituting the initial and final state (19) and (20) and the interact action ((8)) as well as the Dirac spinor field (17) into (37), the first term of (37) is

where $\displaystyle\alpha=2{\left({2\pi}\right)^{3}}\sqrt{{E_{\vec{k}{a_{i}}}}{a_{i}}^{3}{E_{\vec{k}{a_{f}}}}{a_{f}}^{3}}$ is a factor related to normalization factor and the second term of (37) is

where $\displaystyle X={-{i\eta_{1}}{\gamma^{a}}{\mathcal{V}_{a}}+\frac{3}{4}i{\eta_{2}}{\gamma^{a}}{\mathcal{A}_{a}}{\gamma_{5}}}$ is the interaction vertex. The cosmic torsion fields satisfying cosmological principle can have only two independent non-zero components[32],

and

The vector and axial vector parts of torsion thus have the form

and

Because the 0th components of vector and axial vector parts of torsion which are the only non-zero one are time dependent and space independent, the S-matrix element (37) of Dirac spinor field can be simplified as

The scattering amplitude is proportional to ${\delta^{3}}\left({{{\vec{k}}_{C}}-\vec{k}{{}^{\prime}_{C}}}\right)$, which means that the particle will keep its comoving momentum after scattering so that the scattering is just a redshift to a particle. The $\vec{k}_{C}$ denpendence in the last term of the scattering amplitude implies the interaction rate is different for different initial momentum which may cause the change of temperature spectrum after scattering via torsion which will be discussed in the next section.

Now we pay attention on the Majorana case. The $\psi_{M}$ and $\bar{\psi}_{M}$ are not independent for Majorana spinor field, i.e. $\displaystyle{{\bar{\psi}}_{M}}={\psi_{M}}^{T}\mathcal{C}$. The action of Majorana spinor is[33]

which is one-half of the Dirac one formally where $\psi_{M}$ is expanded by

rather than $\psi_{D}$ given in (17). Using the action $S_{M}$ and the Majorana spinor $\psi_{M}$, we can calculate the scattering amplitude as what we have done for the Dirac one. The 1st order term is

Here we use the fact that vertex is only dependent on time $X=X\left(t\right)$. Using the relationship that $v=C{{\bar{u}}^{T}}$ and ${C^{\dagger}}=-i{\gamma^{2}}^{\dagger}{\gamma^{0}}=i{\gamma^{2}}{\gamma^{0}}=-C$ we have

and

Then we have

where $\displaystyle X^{\prime}=\frac{3}{4}i{\eta_{2}}{\gamma^{a}}{\mathcal{A}_{a}}{\gamma_{5}}$. In Majorana case, the axial vector part of torsion contributes the same to the scattering amplitude as in the Dirac case while the vector torsion has no effects on the scattering amplitude which is different from the Dirac case. In fact, for interaction Hamiltonian density $\bar{\psi}\Gamma\psi$, the vertex for Dirac field scattering is $\Gamma$ while effective vertex for Majorana field scattering is $\displaystyle\Gamma^{\prime}=\frac{1}{2}\left(\Gamma+C\Gamma^{T}C^{-1}\right)$ since effective vertex for Majorana field scattering should keep invariant under charge-conjugation transformation $\Gamma^{\prime}=C{\Gamma^{\prime}}C^{-1}$[34]. The factor $\gamma^{a}\gamma_{5}$ keeps invariant under charge-conjugation transformation, i.e. $\displaystyle C\left(\gamma^{a}\gamma_{5}\right)^{T}C^{-1}=\gamma^{a}\gamma_{5}$, while the factor $\gamma^{a}$ is not, i.e. $\displaystyle C\left(\gamma^{a}\right)^{T}C^{-1}=-\gamma^{a}$, so that the field coupled with $\gamma^{a}$ has no contribution to scattering amplitude which is the vector part of torsion in our case. The total Majorana scattering amplitude is

which differs (29) by the vector part of torsion.