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

Wei Lina,b, Xun Xuea,b,c,111Corresponding<EMAIL_ADDRESS>
###### Abstract

The possibility of distinguishing Dirac and Majorana fermions by cosmic
torsion in the spatial-flat FRW spacetime is discussed. The scattering
amplitudes of two types of fermions deviate from each other by the vector part
of torsion in non-minimal coupling case. The scattering of massive fermions by
cosmic torsion leads to a shift of final state energy distribution. The
difference between shift values of two types of fermions can be used to
distinguish fermion types of neutrinos.

aDepartment of Physics, East China Normal University, Shanghai 200241, China

bChongqing Institute of East China Normal University, Chongqing 401120, China

cCenter for Theoretical Physics, Xinjiang University, Urumqi 830046, China

##  1 Introduction

The neutrino oscillation experiment reveals that neutrino has a tiny mass
about 0.1eV[1]. However, the origin of the neutrino mass remains to be
unclear. In the Standard Model, fermions get Dirac type masses via the Yukawa
coupling by the Higgs mechanism. If neutrino has only the Dirac type of mass,
the Yukawa coupling constants of neutrino would be smaller than ones of
charged leptons at the order of $10^{-5}$, as the same order of their mass
magnitudes difference, which is regarded as unnatural[2]. The see-saw
mechanism can naturally explain the tiny mass of the left-hand neutrino in
stead of assuming the huge difference between neutrino Yukawa coupling and
charged lepton ones with the large Majorana mass for the right-hand neutrino.
However the see-saw mechanism gives the Majorana type of masses to the left-
hand neutrino which is the only observable part of neutrino at the Standard
Model energy scale[3]. Whether neutrino mass is a Dirac one or a Majorana one
concerns different neutrino mass generation mechanism. The Majorana nature of
neutrino mass provides a direct indication of the existence of new physics
beyond Standard Model. Neutrinoless double beta decay process is a direct way
to distinguish the Majorana type of neutrino from Dirac one. However, the
experiment is not able to give a deterministic result to the existence of
neutrinoless double beta decay yet[4]. There are other proposals to determine
the fermion type of neutrino, e.g., the idea to distinguish Dirac fermion from
Majorana one by the scattering of fermion in the gravitational field[5, 6, 7].
Lai and Xue consider the possibility to determine the fermion type of neutrino
using its scattering behavior by torsion field[8].

Torsion can be decomposed into the vector part, the axial vector part and the
pure tensor part according to the irreducible representation under the global
Lorentz group[9]. In the minimal coupling scheme, only the axial vector
torsion can couple with spinor field. The vector torsion can have non-minimal
coupling with spinor field while the pure tensor torsion is impossible to
couple with spinor field even in the non-minimal coupling scheme[10, 11]. The
non-minimal couple between torsion and spinor field is universal due to the
renormalization effect of spinor field[12].

The idea of distinguishing fermion types by torsion field is first proposed by
Lai and Xue[8] that the scattering by vector torsion field can distinguish the
Dirac from Majorana neutrino in the non-minimal coupling case in the
asymptotic Minkowski spacetime background. Although General Relativity is a
torsion-free gravitation theory and has been verified by observations from
solar system to galactic scale phenomenon[13, 14, 15], the $H_{0}$ tension
problem that the Hubble constant measured by cosmological model-independent
standard candles calibration in the late-time universe and that calculated
from CMB data based on $\Lambda CDM$ model has a discrepancy more than
3$\sigma$ in the recent result[16] indicates that General relativity, the
gravitation theory $\Lambda CDM$ based on, may need modifications from quantum
gravity at cosmic scale[17, 18]. Some proposals of modified cosmological
models predict non-zero torsion distribution[19, 20, 21, 22, 23, 24, 25, 26]
which will take effect on processes happened at cosmic scale, e.g., the cosmic
neutrino propagation process. The effect of expanding universe must be taken
into account during cosmic scale process so that the scattering of neutrino
should be treated in a FRW background rather than a asymptotic Minkowski
spacetime.

##  2 Scattering amplitudes of fermions by torsion fields in spatial-flat FRW
spacetime background

###  2.1 Interaction Hamiltonian density

The scattering amplitude or S-matrix can be expressed as

$\textbf{S}=\textbf{T}\\{e^{-i\int d^{4}x\mathcal{H}_{I}\left(x\right)}\\}$
(1)

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,

$S_{D}=\int{{d^{4}}x}\sqrt{-g}\bar{\psi}\left[\frac{i}{2}{\gamma^{\mu}}\overleftrightarrow{\mathcal{D}_{\mu}}\psi-m\psi\right]\;,$
(2)

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

$S_{D}=\int{{d^{4}}x}\sqrt{-g}\bar{\psi}\left[{i\left({{\gamma^{\mu}}{{\tilde{\mathcal{D}}}_{\mu}}\psi-\frac{3}{4}i{\gamma^{a}}{\mathcal{A}_{a}}{\gamma_{5}}\psi}\right)-m\psi}\right]$
(3)

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

$S_{D}=\int{{d^{4}}x}\sqrt{-g}\bar{\psi}\left[{i\left({{\gamma^{\mu}}{{\tilde{\mathcal{D}}}_{\mu}}\psi+{i\eta_{1}}{\gamma^{a}}{\mathcal{V}_{a}}\psi-\frac{3}{4}i{\eta_{2}}{\gamma^{a}}{\mathcal{A}_{a}}{\gamma_{5}}\psi}\right)-m\psi}\right]\;,$
(4)

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

${i\left({{\gamma^{\mu}}{{\tilde{\mathcal{D}}}_{\mu}}\psi+{i\eta_{1}}{\gamma^{a}}{\mathcal{V}_{a}}\psi-\frac{3}{4}i{\eta_{2}}{\gamma^{a}}{\mathcal{A}_{a}}{\gamma_{5}}\psi}\right)-m\psi}=0.$
(5)

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

${S_{0}}=\int{{d^{4}}x}\sqrt{-g}\bar{\psi}\left[{i{\gamma^{\mu}}{\tilde{\mathcal{D}}_{\mu}}\psi-m\psi}\right]$
(6)

with the corresponding equation of motion is

${i{\gamma^{\mu}}{\tilde{\mathcal{D}}_{\mu}}\psi-m\psi}=0.$ (7)

The perturbation terms in the action can thus be read as

${S_{I}}=\int{{d^{4}}x}\sqrt{-g}\bar{\psi}\left[{-\eta_{1}}{\gamma^{a}}{\mathcal{V}_{a}}\psi+\frac{3}{4}{\eta_{2}}{\gamma^{a}}{\mathcal{A}_{a}}{\gamma_{5}}\psi\right]=\int{{d^{4}}x}\mathcal{L}_{I}$
(8)

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

$\int d^{4}x\mathcal{H}_{I}\left(x\right)=-\int
d^{4}x\mathcal{L}_{I}\left(x\right)=-S_{I}.$ (9)

The S-matrix can then be calculated by

$\textbf{S}=\textbf{T}\\{e^{iS_{I}}\\}\;.$ (10)

###  2.2 Spinor fields in spatial-flat FRW spacetime

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

$d{s^{2}}=d{t^{2}}-{a^{2}}\left(t\right)\left({d{x^{2}}+d{y^{2}}+d{z^{2}}}\right)$
(11)

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

$i\left({{\gamma^{0}}{\partial_{t}}\psi+{a^{-1}}\vec{\gamma}\cdot\vec{\nabla}\psi+\frac{3}{2}\frac{{\dot{a}}}{a}{\gamma^{0}}\psi}\right)-m\psi=0\;,$
(12)

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

${\psi_{D}}=\frac{1}{{(2\pi)}^{3}}\sum\limits_{s}{\int{{d^{3}}{{\vec{p}}_{C}}}}\frac{1}{{{\sqrt{2{E_{{{\vec{p}}_{C}}a}}{a^{3}}}}}}\left[{{A_{s}}({{\vec{p}}_{C}},t){U_{s}}({{\vec{p}}_{C}},t){e^{-i\Omega\left({{{\vec{p}}_{C}}}\right)}}+{B^{\dagger}}_{s}({{\vec{p}}_{C}},t){V_{s}}({{\vec{p}}_{C}},t){e^{i\Omega\left({{{\vec{p}}_{C}}}\right)}}}\right]$
(13)

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.

${A_{s}}(\vec{p},t)={D_{1,1}}\left({\left|{\vec{p}}\right|,t}\right){A_{s}}(\vec{p})+{D_{1,-1}}\left({\left|{\vec{p}}\right|,t}\right){B_{-s}}^{\dagger}(-\vec{p})$
(14)

and

${B_{s}}(\vec{p},t)={D_{-1,-1}}^{*}\left({\left|{\vec{p}}\right|,t}\right){B_{s}}(\vec{p})+{D_{-1,1}}^{*}\left({\left|{\vec{p}}\right|,t}\right){A_{-s}}^{\dagger}(-\vec{p})\;,$
(15)

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

$\left\\{{{A_{r}}\left({{{\vec{p}}}}\right),{A^{\dagger}}_{s}\left({{{\vec{q}}}}\right)}\right\\}=\left\\{{{B_{r}}\left({{{\vec{p}}}}\right),{B^{\dagger}}_{s}\left({{{\vec{q}}}}\right)}\right\\}={\left({2\pi}\right)^{3}}{\delta^{3}}\left({{{\vec{p}}}-{{\vec{q}}}}\right){\delta_{rs}}\;.$
(16)

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

${\psi_{D}}=\frac{1}{{(2\pi)}^{3}}\sum\limits_{s}{\int{{d^{3}}}}{{\vec{p}}_{C}}\frac{1}{{{\sqrt{2{E_{{{\vec{p}}_{C}}a}}{a^{3}}}}}}\left[{{A_{s}}({{\vec{p}}_{C}}){u_{s}}\left({{{\vec{p}}_{C}}/a}\right){e^{-i\Omega\left({{{\vec{p}}_{C}}}\right)}}+{B^{\dagger}}_{s}({{\vec{p}}_{C}}){v_{s}}\left({{{\vec{p}}_{C}}/a}\right){e^{i\Omega\left({{{\vec{p}}_{C}}}\right)}}}\right].$
(17)

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

$\left\langle\mathbf{p}_{1}\mid\mathbf{p}_{2}\right\rangle=2E_{\mathbf{p}_{1}}\left(2\pi\right)^{3}\delta^{3}\left(\mathbf{p_{1}}-\mathbf{p_{2}}\right)=2E_{\mathbf{p}_{1}}a\left(t\right)^{3}\left(2\pi\right)^{3}\delta^{3}\left(\mathbf{p_{1}}_{C}-\mathbf{p_{2}}_{C}\right)$
(18)

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

$\left|i\right\rangle=\left|{{{\vec{k}}_{P}},s}\right\rangle=\left|{\frac{{{{\vec{k}}_{C}}}}{{{a_{i}}}},s}\right\rangle=\sqrt{2{E_{\vec{k}{a_{i}}}}{a_{i}}^{3}}{A_{s}}^{\dagger}\left({{{\vec{k}}_{C}}}\right)\left|0\right\rangle$
(19)

and

$\left|f\right\rangle=\left|{\vec{k}{{}^{\prime}_{P}},r}\right\rangle=\left|{\frac{{\vec{k}{{}^{\prime}_{C}}}}{{{a_{f}}}},r}\right\rangle=\sqrt{2{E_{\vec{k}^{\prime}{a_{f}}}}{a_{f}}^{3}}{A_{r}}^{\dagger}\left({\vec{k}{{}^{\prime}_{C}}}\right)\left|0\right\rangle.$
(20)

###  2.3 Scattering amplitude calculation

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.

$\mathbf{S}\simeq 1+i\mathbf{T}\left\\{S_{I}\right\\}$ (21)

and the S-matrix element can thus be calculate via

$\begin{gathered}{\left[{{\mathbf{S}}}\right]_{fi}}=\left\langle
f\right.\left|1\right|\left.i\right\rangle+i\left\langle
f\right.\left|{\mathbf{T}\left[{{S_{I}}}\right]}\right|\left.i\right\rangle\hfill\\\
\end{gathered}$ (22)

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

$\left\langle
f\right.\left|1\right|\left.i\right\rangle=\alpha{\delta^{3}}\left({{{\vec{k}}_{C}}-\vec{k}{{}^{\prime}_{C}}}\right){\delta_{rs}}$
(23)

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

$i\left\langle
f\right.\left|{\mathbf{T}\left[{{S_{I_{D}}}}\right]}\right|\left.i\right\rangle=\frac{\alpha}{2{{\left({2\pi}\right)}^{3}}}\int{{d^{4}}x}\frac{1}{{\sqrt{{E_{\vec{k}{{}^{\prime}_{C}}a}}{E_{{{\vec{k}}_{C}}a}}}}}\left[{{{{e^{i\left({\int_{-\infty}^{t}dt^{\prime}\left({{E_{\vec{k}{{}^{\prime}_{C}}a}}-{E_{{{\vec{k}}_{C}}a}}}\right)\left({t^{\prime}}\right)-\left({\vec{k}{{}^{\prime}_{C}}-{{\vec{k}}_{C}}}\right)\cdot\vec{x}}\right)}}}}{{\bar{u}}_{r}}\left({\vec{k}{{}^{\prime}_{C}}/a}\right)X{u_{s}}\left({{{\vec{k}}_{C}}/a}\right)}\right]$
(24)

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],

${T_{ijk}}=-F(t){\epsilon_{ijk}}$ (25)

and

${T^{i}}_{j0}=\mathcal{K}(t)\delta_{j}^{i}\;.$ (26)

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

${\mathcal{V}_{0}}=3\mathcal{K}\left(t\right)={\mathcal{V}_{0}}\left(t\right),{\mathcal{V}_{i}}=0$
(27)

and

${\mathcal{A}^{0}}=-F\left(t\right),{\mathcal{A}^{i}}=0.$ (28)

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

${\left[{{\mathbf{S}_{D}}}\right]_{fi}}=\alpha{\delta^{3}}\left({{{\vec{k}}_{C}}-\vec{k}{{}^{\prime}_{C}}}\right)\left[{{\delta_{rs}}-{i\eta_{1}}\int{dt}{\mathcal{V}_{0}}\left(t\right){\delta_{rs}}+\int{dt}{{\bar{u}}_{r}}\left({{{\vec{k}}_{C}}/a}\right)\left[{i{\eta_{2}}\frac{{3{\mathcal{A}_{0}}{\gamma^{0}}{\gamma_{5}}}}{{8{E_{{{\vec{k}}_{C}}a}}}}}\right]{u_{s}}\left({{{\vec{k}}_{C}}/a}\right)}\right]\;.$
(29)

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]

$S_{M}=\frac{1}{2}\int{{d^{4}}x}\sqrt{-g}\bar{\psi}_{M}\left[{i\left({{\gamma^{\mu}}{{\tilde{\mathcal{D}}}_{\mu}}\psi_{M}+{i\eta_{1}}{\gamma^{a}}{\mathcal{V}_{a}}\psi_{M}-\frac{3}{4}i{\eta_{2}}{\gamma^{a}}{\mathcal{A}_{a}}{\gamma_{5}}\psi_{M}}\right)-m\psi_{M}}\right]$
(30)

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

$\psi_{M}=\frac{1}{{{(2\pi)}^{3}}}\sum\limits_{s}{\int{{d^{3}}}}{{\vec{p}}_{C}}\frac{1}{{\sqrt{2{E_{{{\vec{p}}_{C}}a}}{a^{3}}}}}\left[{{A_{s}}({{\vec{p}}_{C}}){u_{s}}\left({{{\vec{p}}_{C}}/a}\right){e^{-i\Omega\left({{{\vec{p}}_{C}}}\right)}}+{A^{\dagger}}_{s}({{\vec{p}}_{C}}){v_{s}}\left({{{\vec{p}}_{C}}/a}\right){e^{i\Omega\left({{{\vec{p}}_{C}}}\right)}}}\right]$
(31)

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

$i\left\langle
f\right.\left|{\mathbf{T}\left[{{S_{I_{M}}}}\right]}\right|\left.i\right\rangle=\frac{1}{4}\alpha\delta^{3}\left({{{\vec{k}}_{C}}-\vec{k}{{}^{\prime}_{C}}}\right)\int{{d}t}\left[{\frac{1}{{E_{{{\vec{k}}_{C}}a}}}\left({{\bar{u}}_{r}}\left({\vec{k}{{}^{\prime}_{C}}/a}\right)X{u_{s}}\left({{{\vec{k}}_{C}}/a}\right)-{{\bar{v}}_{s}}\left({{{\vec{k}}_{C}}/a}\right)X{v_{r}}\left({{{\vec{k}^{\prime}}_{C}}/a}\right)\right)}\right]$
(32)

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

${{\bar{v}}_{s}}(k){\gamma^{a}}{v_{r}}(k^{\prime})={u_{s}}{(k)^{T}}C{\gamma^{a}}C{{\bar{u}}_{r}}^{T}(k^{\prime})={u_{s}}{(k)^{T}}{\gamma^{a}}^{T}{{\bar{u}}_{r}}^{T}(k^{\prime})={{\bar{u}}_{r}}(k^{\prime}){\gamma^{a}}{u_{s}}(k)$
(33)

and

$\begin{gathered}{{\bar{v}}_{s}}(k){\gamma^{a}}{\gamma_{5}}{v_{r}}(k^{\prime})={u_{s}}{(k)^{T}}C{\gamma^{a}}{\gamma_{5}}C{{\bar{u}}_{r}}^{T}(k^{\prime})=-{u_{s}}{(k)^{T}}C{\gamma^{a}}CC{\gamma_{5}}C{{\bar{u}}_{r}}^{T}(k^{\prime})\\\
={u_{s}}{(k)^{T}}{\gamma^{a}}^{T}{\gamma_{5}}^{T}{{\bar{u}}_{r}}^{T}(k^{\prime})=-{u_{s}}{(k)^{T}}{\gamma_{5}}^{T}{\gamma^{a}}^{T}{{\bar{u}}_{r}}^{T}(k^{\prime})=-{{\bar{u}}_{r}}(k^{\prime}){\gamma^{a}}{\gamma_{5}}{u_{s}}(k)\;.\\\
\end{gathered}$ (34)

Then we have

$i\left\langle
f\right.\left|{\mathbf{T}\left[{{S_{I_{M}}}}\right]}\right|\left.i\right\rangle=\frac{1}{2}\alpha\delta^{3}\left({{{\vec{k}}_{C}}-\vec{k}{{}^{\prime}_{C}}}\right)\int{{d}t}\left[{\frac{1}{{2E_{{{\vec{k}}_{C}}a}}}\left({{\bar{u}}_{r}}\left({\vec{k}{{}^{\prime}_{C}}/a}\right)X^{\prime}{u_{s}}\left({{{\vec{k}}_{C}}/a}\right)\right)}\right]$
(35)

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

${\left[{{\mathbf{S}_{M}}}\right]_{fi}}=\alpha{\delta^{3}}\left({{{\vec{k}}_{C}}-\vec{k}{{}^{\prime}_{C}}}\right)\left[{{\delta_{rs}}+\frac{3i\eta_{2}}{8}\int{dt}{{\bar{u}}_{r}}\left({{{\vec{k}}_{C}}/a}\right)\left[{\frac{{{\mathcal{A}_{0}}{\gamma^{0}}{\gamma_{5}}}}{{{E_{{{\vec{k}}_{C}}a}}}}}\right]{u_{s}}\left({{{\vec{k}}_{C}}/a}\right)}\right]$
(36)

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

##  3 Shift of energy distribution

As we mentioned before, the scattering amplitude is $\vec{k}_{C}$ dependent
which will cause the shift of the energy distribution after scattering. Notice
that the scattering amplitude (29) and (36) can be generally written as

${\left[{{\mathbf{S}}}\right]_{fi}}=\alpha{\delta^{3}}\left({{{\vec{k}}_{C}}-\vec{k}{{}^{\prime}_{C}}}\right)M_{fi}$
(37)

The factor $\alpha$ is related to normalization. In torsion free spacetime,
the scattering amplitude is
${\left[{{\mathbf{S}}}\right]_{fi}}=\alpha{\delta^{3}}\left({{{\vec{k}}_{C}}-\vec{k}{{}^{\prime}_{C}}}\right)$.
The effect of torsion to the rate of redshift is included in the factor
$M_{fi}$. The scattering rate from initial state $\left|i\right\rangle$ to the
final state $\left|i\right\rangle$ $W_{fi}$ is proportional to
$\left|\left[\mathbf{S}\right]_{fi}\right|^{2}$

$W_{fi}\propto\alpha^{2}\left(\delta^{3}\left(\vec{k}^{\prime}_{C}-\vec{k}_{C}\right)\right)^{2}\left|M_{fi}\right|^{2}$
(38)

Experimentally, it is more common that the initial spin is unknown to us and
the detector receives all final spin configurations. Thus, to compare with
experiment, we should sum all the final spin configurations and average the
initial ones for $W_{fi}$. We define

$\overline{W}_{fi}=\frac{1}{2}\sum_{\text{initial spins }}\sum_{\text{final
spins
}}W_{fi}\propto\alpha^{2}\left(\delta^{3}\left(\vec{k}^{\prime}_{C}-\vec{k}_{C}\right)\right)^{2}\overline{\left|M_{fi}\right|^{2}}$
(39)

where

$\overline{\left|M_{fi}\right|^{2}}=\frac{1}{2}\sum_{\text{initial spins
}}\sum_{\text{final spins }}\left|M_{fi}\right|^{2}$ (40)

If the initial energy distribution is $I_{i}\left(E_{i},T\right)$, the final
energy distribution will be
$I_{f}\left(E,T\right)=\bar{W}_{fi}I_{i}\left(E_{f}\left(E\right),T\right)$
where the $E_{f}\left(E_{i}\right)$ is the final energy dependency on the
initial energy. Thus, if the final energy distribution in the torsion free
background is $I_{0}\left(E,T\right)$, the final energy distribution in the
spacetime with cosmic torsion will become
$I_{T}\left(E,T\right)=\overline{\left|M_{fi}\right|^{2}}I_{0}\left(E,T\right)$.
For Dirac field the scattering amplitude(29) can be calculated as

$\overline{\left|{M_{D}}_{fi}\right|^{2}}=1+\text{$\mathcal{V}$
terms}+\text{$\mathcal{A}$ terms}$ (41)

where

$\text{$\mathcal{V}$
terms}={{\eta_{1}}^{2}{{\left({\int{dt}{\mathcal{V}_{0}}}\right)}^{2}}}$ (42)

and the $\mathcal{A}$ terms is

$\begin{gathered}-\frac{9}{{32}}{\eta_{2}}^{2}\int{dt^{\prime}}\int{dt}\left({\frac{1}{{{E_{{{\vec{k}}_{C}}a^{\prime}}}\left[{t^{\prime}}\right]}}{\mathcal{A}_{0}}\left[{t^{\prime}}\right]\frac{1}{{{E_{{{\vec{k}}_{C}}a}}\left[t\right]}}{\mathcal{A}^{0}}\left[t\right]{k_{P;a^{\prime}}}_{c}\left[{t^{\prime}}\right]{k_{P;a}}^{c}\left[t\right]+\frac{1}{{{E_{{{\vec{k}}_{C}}a^{\prime}}}\left[{t^{\prime}}\right]}}{\mathcal{A}_{0}}\left[{t^{\prime}}\right]\frac{1}{{{E_{{{\vec{k}}_{C}}a}}\left[t\right]}}{\mathcal{A}^{0}}\left[t\right]{m^{2}}}\right)\hfill\\\
+\frac{9}{{16}}{\eta_{2}}^{2}\int{dt^{\prime}}\int{dt}\left({{\mathcal{A}_{0}}\left[{t^{\prime}}\right]{\mathcal{A}_{0}}\left[t\right]}\right).\hfill\\\
\end{gathered}$ (43)

For Majorana case, $\overline{\left|M_{fi}\right|^{2}}$ differs from the Dirac
one by the term related to the vector part of torsion, i.e.

$\overline{\left|{M_{M}}_{fi}\right|^{2}}=1+\text{$\mathcal{A}$ terms}$ (44)

Although neutrino is massless in Standard Model, the discovery of neutrino
oscillations proved that neutrino is not exactly massless even though the mass
of neutrino is very small. The upper limit on the absolute mass scale of
neutrinos is 1.1 eV (90% confidence level) according to recent experiment[2].
It is common that the energy of neutrino is much larger than its mass, i.e.
$E\gg m$. In this case, ${E_{{{\vec{k}}_{C}}a}}$ can be expanded as

${E_{{{\vec{k}}_{C}}a}}={\left({\frac{{{{\left|{{{\vec{k}}_{C}}}\right|}^{2}}}}{{a{{\left(t\right)}^{2}}}}+{m^{2}}}\right)^{1/2}}\simeq\frac{{\left|{{{\vec{k}}_{C}}}\right|}}{{a\left(t\right)}}+\frac{{{m^{2}}a\left(t\right)}}{{2\left|{{{\vec{k}}_{C}}}\right|}}$
(45)

and therefore the $\mathcal{A}$ terms can be simplified as

$\text{$\mathcal{A}$
terms}\simeq\frac{9}{{32}}{\eta_{2}}^{2}\left({2{\mathcal{A}_{0}}\left[{t^{\prime}}\right]{\mathcal{A}_{0}}\left[t\right]-\frac{{{m^{2}}}}{{2{{{E_{f}}^{2}{a_{f}}^{2}}}}}{{\left({a\left(t\right)+a\left({t^{\prime}}\right)}\right)}^{2}}{\mathcal{A}_{0}}\left[{t^{\prime}}\right]{\mathcal{A}^{0}}\left[t\right]}\right)\;.$
(46)

The dependence of $\overline{\left|{M}_{fi}\right|^{2}}$ on $E_{f}$ is also
dependent on the mass $m$ and the axial vector part of torsion $\mathcal{A}$,
which means if the particle mass is zero or there is no axial vector part of
torsion, the interaction rates for particles with different energy are same
which will cause that the final energy distribution scattering by torsion is
the same as that given by torsion free case. In non-minimum coupling case that
$\eta_{1}\neq 0$, the existence of the vector part of torsion will result in
the same ${\left|{{{\vec{k}}_{C}}}\right|}^{2}$ dependence term in
$\overline{\left|{M}_{fi}\right|^{2}}$ in both Dirac and Majorana case on the
energy distribution. That is, even if the
${\left|{{{\vec{k}}_{C}}}\right|}^{2}$ dependence term that can takes effect
on the final energy distribution is the same for both types of spinor field,
the final energy distribution for the two case are different for the same
final energy distribution in torsion free case.

If the final energy distribution in torsion free case is one given by Planck
formula for black body radiation,

${I_{0}}\left({E,T}\right)=\left({\frac{{{2E^{3}}}}{{{{\left({2\pi}\right)}^{3}}}}}\right)\frac{1}{{{e^{\frac{E}{{{k_{B}}T}}}}-1}}.$
(47)

The distribution reaches its peak at ${E_{0}}_{Max}$ given by

$\frac{d{I_{0}}\left({E,T}\right)}{dE}|_{E={E_{0}}_{Max}}=0$ (48)

which can be written as

$\left({3-x}\right){e^{x}}=3,$ (49)

where $x$ is defined as $\displaystyle x=\frac{E}{k_{B}T}$. The equation (49)
has a positive solution $x_{0}$. Therefore, the ${E_{0}}_{Max}$ can be given
as

${E_{0}}_{Max}={x_{0}}{k_{B}}T$ (50)

which is known as the Wien’s displacement law.

Then we turn on the torsion field. In the Majorana case the distribution will
shift to
$I_{M}\left(E,T\right)=\overline{\left|{{M}_{M}}_{fi}\right|^{2}}I_{0}\left(E,T\right)$
whose peak is arrived at ${E_{M}}_{Max}$ given by
$\displaystyle\frac{dI}{dE}=0$ which can be derived as

$\left({3-x}\right){e^{x}}=3+\zeta\left({x,T}\right)\left({{e^{x}}-1-x{e^{x}}}\right)$
(51)

where the shift factor

$\zeta\left({x,T}\right)=\frac{\xi}{{\left({1+\chi}\right){{\left({{k_{B}}T}\right)}^{2}}{x^{2}}}}$
(52)

and the factor

$\chi=\frac{9}{{16}}{\eta_{2}}^{2}\int{dt^{\prime}}\int{dt\left({{\mathcal{A}_{0}}\left[{t^{\prime}}\right]{\mathcal{A}_{0}}\left[t\right]}\right)}$
(53)

and

$\xi=\frac{{9{m^{2}}}}{{64{a_{f}}^{2}}}{\eta_{2}}^{2}\int{dt^{\prime}}\int{dt\left({{{\left({a\left(t\right)+a\left({t^{\prime}}\right)}\right)}^{2}}{\mathcal{A}_{0}}\left[{t^{\prime}}\right]{\mathcal{A}^{0}}\left[t\right]}\right)}\;.$
(54)

The equation (51) is hard to solve analytically, and the solution $x_{M}$ is
dependent on the temperature rather than a constant like $x_{0}$. However, it
can be noticed that the left-hand side of equation (51) as well as the first
term 3 in the right-hand side of (51) is just the equation (49) whose solution
is known as the constant $x_{0}$. In the theory that the cosmic torsion plays
the role of part of the dark energy, the cosmic torsion is in the order of
Hubble parameter $H$[24, 25]. If the scale factor $a\propto t^{n}$, the
integral $\displaystyle\int_{t_{i}}^{t_{f}}H\left(t\right)dt\propto
ln\left(\frac{t_{f}}{t_{i}}\right)H_{0}t_{0}$ where $H_{0}$ is Hubble constant
if the final time is today $t_{0}$. Since $H_{0}t_{0}$ is in the order of 1,
the the integral $\displaystyle\int_{t_{i}}^{t_{f}}H\left(t\right)dt$ is in
the order of 1. Hence, the factor $\chi$ is in the order about 1 and the
factor $\xi$ is in the order $m^{2}$ so that shift factor
$\zeta\left({x,T}\right)$ is in the order of
$\displaystyle\frac{m^{2}}{E^{2}}$ which is much less than 1 in the most cases
for neutrinos. Therefore, the second term of the left-hand side of equation
(51) is small. Thus, the difference between the solution of eq.(51) $x_{M}$
and the solution of (49) $x_{0}$ is expect to be small. We may define the
difference as $\Delta_{M}=x_{M}-x_{0}$ and rewrite the equation (51) with
$x_{0}$ and $\Delta_{M}$ in the first-order of $\Delta_{M}$ as

${\Delta_{M}}{e^{{x_{0}}}}+\zeta\left({{x_{0}},T}\right)\left({{e^{{x_{0}}}}-1-{x_{0}}{e^{{x_{0}}}}}\right){\text{
= }}0$ (55)

and the $\Delta_{M}$ can be easily solved as

${\Delta_{M}}=\frac{2}{3}\zeta\left({{x_{0}},T}\right)=\frac{{2\xi}}{{3\left({1+\chi}\right){{\left({{k_{B}}T}\right)}^{2}}{x_{0}}}}.$
(56)

Thus the final Majorana energy distribution reaches it peak at

${E_{M}}_{Max}={x_{M}}{k_{B}}T\simeq{x_{0}}{k_{B}}T+\frac{{2\xi}}{{3\left({1+\chi}\right)\left({{k_{B}}T}\right){x_{0}}}}$
(57)

and similarly the result of the Dirac one is

${E_{D}}_{Max}\simeq
x_{0}{k_{B}}T+\frac{{2\xi}}{{3\left({1+\chi+V}\right)\left({{k_{B}}T}\right){x_{0}}}}$
(58)

where

$V={\eta_{1}}^{2}{\left({\int{dt}{\mathcal{V}_{0}}}\right)^{2}}.$ (59)

It can be seen that the shift of Dirac energy distribution and the Majorana
energy distribution is different in the non-minimal coupling with $\eta_{1}$
and $\mathcal{V}_{0}$ is not zero.The energy shift is proportional to the
order $\displaystyle\frac{m^{2}}{E^{2}}$. The energy distribution will not
shift when $m=0$ because in such a case the interaction rate for all the
energy is the same so that it will cause the same distribution after
normalization. If we stick to Wien’s displacement law to give the temperature,
the effective temperature we detect will be

${T_{D}}_{\text{eff}}=T\left({1+\frac{{2\xi}}{{3\left({1+\chi+V}\right){{\left({{k_{B}}T}\right)}^{2}}{x_{0}}^{2}}}}\right)$
(60)

for Dirac scattering and

${T_{M}}_{\text{eff}}=T\left({1+\frac{{2\xi}}{{3\left({1+\chi}\right){{\left({{k_{B}}T}\right)}^{2}}{x_{0}}^{2}}}}\right)$
(61)

for Majorana scattering. The temperature shift is also proportional to the
order $\displaystyle\frac{m^{2}}{E^{2}}$.

##  4 Conclusion and Discussion

We find that the scattering amplitudes of the Dirac and Majorana fermions by
torsion in spatial-flat FRW spacetime background are differed by the vector
part of torsion field in non-minimal coupling case. The axial vector part of
cosmic torsion coupled with mass in the interaction rate will cause a small
shift of the energy distribution and the effective temperature derived from
Wien’s displacement law shift in the order $\displaystyle\frac{m^{2}}{E^{2}}$
if $m\ll E$ but $m\neq 0$. The shift of energy distributions and effective
temperature for Dirac and Majorana spinors are different due to the difference
of the scattering amplitude by the vector part of torsion.

The temperature of cosmic neutrino background (CNB) is given via
$\displaystyle T_{\nu}=\left(\frac{4}{11}\right)^{1/3}T_{\gamma}\approx 1.95K$
with the corresponding energy $k_{B}T\sim 10^{-4}eV$[35]. However, the upper
limit of neutrino mass is about 1.1eV[2], which means the effect of static
mass can not be neglect. The condition $m\ll E$ is no longer hold. Thus, the
energy distribution shift is not the value given in (57) and (58) but that
calculated from the $\mathcal{A}$ terms given by (43) combined with specific
cosmological models with cosmic torsion. However, the basic qualitative
conclusions that the term involving both axial vector part of cosmic torsion
and the mass of neutrino will cause the energy distribution shift and the
shift will be different for Dirac and Majorana scattering in the non-minimal
coupling case that the vector part of cosmic torsion couples with the spinor
field are still hold.

The cosmic torsion of spatial-flat FRW spacetime takes simply effect on the
rate of redshift in energy rather than the angle distribution of final state
or leading to a change of the final state energy rather than redshift. This is
due to the fact that the cosmic torsion in spatial-flat FRW spacetime is
homogeneous isotropic and the scattering theory assume that the interaction is
turned off in the initial and final state. The case of open and closed
universe need to be discussed whether the interaction is spatially correlated
so that it will caused the angle distribution of final state.

## Acknowledgment

This work is supported by the National Natural Science Foundation of China (
Grant Nos. 11775080, 11865016 ) and the Natural Science Foundation of
Chongqing, China ( Grant No. CSTB2022NSCQ-MSX0351 ).

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Ginwidth=totalheight=keepaspectratio

# Applying Multi-armed Bandit Algorithms to Computational Advertising

Kazem Jahanbakhsh

###### Abstract

Summary Over the last two decades, we have seen extensive industrial research
in the area of computational advertising. In this paper, our goal is to study
the performance of various online learning algorithms to identify and display
the best ads/offers with the highest conversion rates to web users. We
formulate our ad-selection problem as a Multi-Armed Bandit problem which is a
classical paradigm in Machine Learning. We have been applying machine
learning, data mining, probability, and statistics to analyze big data in the
ad-tech space and devise efficient ad selection strategies. This article high-
lights some of our findings in the area of computational advertising from 2011
to 2015.

##  1 Advertising Model

In our model we assume that there is a finite set of offers
$O=\\{o_{1},o_{2},...\\}$ where $o_{i}$ denotes offer with index $i$. We model
each offer’s payoff with an unknown distribution $f_{i}$ and an unknown
expected value $\mu_{i}$. For our first set of analysis, we assume that each
offer’s payoff has a Bernoulli distribution. We also assume that the
advertiser has allocated a finite budget $B$ for her advertising campaign. In
words, the advertiser can buy at most $B$ number of impressions.

The goal of an advertising campaign is to maximize the revenue from displaying
ads. Here, we assume that the advertiser makes $1 revenue every time an offer
is clicked by a web user. Otherwise, there is no revenue. Solving this problem
is closely related to the well-known Multi-Armed Bandit problem. In both
settings, the player/advertiser doesn’t have any prior knowledge about how
much revenue she can make by displaying each ad. Moreover, in both problems
there is a trade-off between exploring phase where the goal is to collect
statistical information from ads performance and exploitation phase in which
one sticks to the best known ad found so far.

Figure 1: slot machines (source: wikipedia.org)

##  2 Epsilon-Greedy Algorithm

There are a few algorithms to solve the described optimization problem.
Epsilon-Greedy is one of the basic strategies for optimizing ads in which with
probability $\epsilon$ one chooses a random ad from a set of possible ads and
with probability of $1-\epsilon$ one displays the ad with the best expected
revenue. It’s important to choose $\epsilon$ such that it maximizes the
campaign’s revenue. There is an important performance metric called average
regret computed as follows:

$\rho=n\times p^{*}-\sum_{t=1}^{t=n}\hat{r_{t}}$ (1)

, where $p^{*}=max_{k}\\{p_{k}\\}$ denotes the offer with the highest
acceptance probability and $\hat{r_{t}}$ denotes our collected revenue in
trial $t$. The expected regret value shows the performance difference between
our employed strategy and the optimal strategy. Thus, a low expected regret
value is desirable. Here, we are interested in the average regret per offer
and analyze different algorithms by comparing their average regret per offer
over time.

##  3 UCB1 Algorithm: Index-based Policies

One of the main challenges in ad optimization is to find the balance between
exploring the space (e.g. existing arm space) to find the best arm and
exploiting the current knowledge by playing the best arm found so far as often
as possible. The total regret is a measure to compare the performance of
different algorithms. Lai and Robbins are the first researchers who showed
that the total regret for MAB problem has to grow at least logarithmically in
the number of plays [Lai et al.(1985)Lai, , and Robbins]. They introduced a
class of policies in which they computed an Upper Confidence Index for each
arm based on the previous reward observations for that arm. The computed index
estimates the true expectation reward for that arm. Later, Agrawl introduced a
set of policies where the index function can be expressed in the form of
simple functions and computed more easily than Lai and Robbins’s
[Agrawal(1995)]. The regret for classes of policies introduced by Agrawal
attains the optimal logarithmic behavior.

In Auer’s paper, he formulated a basic version of k-armed bandit problem in
which successive plays of arm $i$ generates rewards $X_{i,1},X_{i,2},...$
[Auer et al.(2002)Auer, Cesa-Bianchi, , and Fischer]. The generated rewards
for each arm (i.e. $X_{i,t}$) are independent and identically distributed
according to an unknown law with an unknown expectation $\mu_{i}$. We also
assume that independence holds between different arms. In other words,
$X_{i,s}$ (i.e. reward attained from playing arm i for $s^{th}$ time) is
independent from $X_{j,t}$ while they are identically distributed where
$s,t\geq 1$ and $1\leq i,j\leq k$. Auer showed that a simple algorithm called
UCB1 achieves a logarithmic regret uniformly over$n$ and without any
preliminary knowledge about the reward distributions. For initialization step,
UCB1 algorithm plays each machine once. Then, in subsequent rounds UCB1 plays
the machine $j$ which maximizes its index function computed as follows:

$\hat{x}_{j}+\sqrt{\dfrac{2\log n}{n_{j}}}$ (2)

, where $\hat{x}_{j}$ is the average reward obtained from machine $j$, $n_{j}$
is number of times machine $j$ has been played so far, and $n$ is the total
number of plays so far. As above equation shows, the index of UCB1 policy has
two terms. The first term is simply the current average reward. The second
term of the index is related to the size of (according to Chernoff-Hoeffding
bounds) the one sided confidence interval for the average reward within which
the true expected reward falls with high probability. Aurer et al. showed that
if one runs UCB1 with $k$ machines with arbitrary reward distributions
$P_{1},...,P_{k}$ with support in range $[0,1]$, the total expected regret
after $n$ number of plays is at most $O(\log(n))$ [Auer et al.(2002)Auer,
Cesa-Bianchi, , and Fischer].

##  4 Bayesian MAB Algorithm

In the third approach, we use a simple Bayesian framework. We model number of
successes in showing an ad $n$ times with the number of successes in $n$
Bernoulli trials with an unknown conversion probability $q$. In Bayesian
inference, the beta distribution is the conjugate prior probability
distribution for the Bernoulli. So, we use Beta function in order to encode
our observations and model probability distribution of $q$ as below:

$p(q)=\dfrac{q^{\alpha-1}(1-q)^{\beta-1}}{B(\alpha,\beta))}$ (3)

So, we model the probability distribution for each ad’s payoff with a Beta
distribution. For offer $i$, at time step $t$ we use number of plays and
payouts for the offer to model the uncertainty about that offer’s payoff and
compute Beta parameters as follows:

$\alpha_{i}(t)=1+payout_{i}(t)$ (4)

$\beta_{i}(t)=1+plays_{i}(t)-payout_{i}(t)$ (5)

At each time step, we display the ad with the maximum expected likelihood by
sampling from above distributions for each ad.

##  5 Performance Analysis

For our analysis, we simulate an advertising campaign with $k$ ads where the
payoff of each ad is $1 if clicked. For modeling ads, we use two simple
distributions. First, we model ads’ payoffs with a uniform distribution. We
choose $k=20$ for number of ads and run our ads $B$ times where $100\leq B\leq
1M$. The goal here is to see how different algorithms perform by comparing
their average regret per offer. Figure 2 shows the results when ads payoffs
modelled with a uniform distribution.

Figure 2: Performance comparison of various algorithms when ads payoffs have a
Uniform distribution

In reality, we expect to have only a few ads with high payoff while most of
ads have low payoffs. For modeling such settings, we use the Beta distribution
where $\alpha=1$ and $\beta=3$. Figure 3 shows the results for Beta
distribution while other parameters are the same as before.

Figure 3: Performance comparison of various algorithms when ads payoffs have a
Beta distribution

##  6 Conclusion

Our performance analysis shows that in both secenarios UCB1 beats Epsilon-
Greedy while Bayesian MAB outperforms both UCB1 and Epsilon-Greedy .

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