CP-Symmetry in Scattering of Neutrinos from Nuclei

R.B. Begzhanov and R.S. Sharafiddinov

Institute of Nuclear Physics, Uzbekistan Academy of Sciences,
Ulugbek, Tashkent 100214, Uzbekistan

Abstract

The elastic scattering of longitudinal and transversal neutrinos on a spinless
nucleus have been discussed taking into account the charge, magnetic, anapole
and electric dipole moments of fermions and their weak neutral currents.
Compound structure of the neutrino interaction cross section with nuclei have
been defined. Invariance of the considered process concerning the C - and
P-operations have been investigated in the polarization type dependence.

1\. Introduction

It has been established that the behavior of massive neutrinos plays an
important part in understanding the physical nature of elementary particles.
One of the modes of doing this is to study the possible neutrino-nucleus
interaction [1,2].

The neutrino interaction with field of emission may be expressed in the form
[3,4] of electromagnetic current

$j_{\mu}^{em}=\overline{u}(p^{\prime},s^{\prime})[\gamma_{\mu}F_{1\nu}(q^{2})-i\sigma_{\mu\lambda}q_{\lambda}F_{2\nu}(q^{2})+$
$+\gamma_{5}\gamma_{\mu}G_{1\nu}(q^{2})-i\gamma_{5}\sigma_{\mu\lambda}q_{\lambda}G_{2\nu}(q^{2})]u(p,s),$
(1)

where $\sigma_{\mu\lambda}=[\gamma_{\mu},\gamma_{\lambda}]/2,$
$q=p-p^{\prime}$ is the momentum transfer, $p(s)$ and $p^{\prime}(s^{\prime})$
imply the four-momentum (helicities) of initial and final neutrinos,
$F_{i\nu}(q^{2})$ and $G_{i\nu}(q^{2})$ are the interaction vector and axial-
vector parts respectively. The functions $F_{1\nu}(0),$ $F_{2\nu}(0)$ and
$G_{2\nu}(0)$ give the static estimates of the neutrino charge [5], magnetic
[6] and electric dipole [7] moments, on which there exist experimental and
cosmological bounds [8]. Insofar as $G_{1\nu}(0)$ is concerned, it defines the
size of the anapole moment [9], but its value has not yet been measured in the
laboratory [10].

It is known that $F_{i\nu}(q^{2})$ are invariant with respect to C - and
P-operations because the interaction of $F_{i\nu}(q^{2})$ with field of
emission must be CP-symmetrical. The term $G_{1\nu}(q^{2})$ is CP-even but
P-odd [9]. In contrast to this, the term $G_{2\nu}(q^{2})$ must be C-invariant
but CP-antisymmetrical [11]. Therefore, the form factors $G_{i\nu}(q^{2})$ may
be different from zero only in the case where P-symmetry is absent.

The violation of P-parity leads to the appearance of right-left asymmetry, for
example, at the polarized neutrinos scattering on nuclei. In many works [2,12]
the spin phenomena was studied with longitudinal neutrinos. Such an
investigation is important not only for elucidation of compound structure of
the interaction between leptons and hadrons but also for observation and
refinement of the most diverse symmetries of elementary particles. However,
the massive neutrino must have the longitudinal as well as the transversal
polarization. The account of the latter gives the possibility to directly look
at the nature of the discussed processes.

In the present work, we investigate the phenomena of symmetricality in the
massive neutrinos interactions with an electroweak field of emission. Section
2 is dedicated to the elastic scattering of longitudinal polarized neutrinos
on the nucleus electric $(Z)$ and weak $(Z_{W})$ charges

$\nu(\overline{\nu})+A(Z,Z_{W})\stackrel{{\scriptstyle\gamma,Z^{0}}}{{\rightarrow}}\nu^{\prime}(\overline{\nu^{\prime}})+A(Z,Z_{W}),$
(2)

going at the expense of neutral and electromagnetic currents. In Sec. 3 the
studied processes have been reanalysed for the transversal case of the
neutrino polarization. In Sec. 4 we make some concluding remarks.

2\. Longitudinal Polarized Neutrinos Scattering on a Nucleus

In the framework of the standard theory of electroweak interaction [13], the
Hamiltonian of the neutrino interaction with field of a nucleus has the form

$H=\frac{4\pi\alpha}{q^{2}}\overline{u}(p^{\prime},s^{\prime})[\gamma_{\mu}F_{1\nu}(q^{2})-i\sigma_{\mu\lambda}q_{\lambda}F_{2\nu}(q^{2})+$
$+\gamma_{5}\gamma_{\mu}G_{1\nu}(q^{2})-i\gamma_{5}\sigma_{\mu\lambda}q_{\lambda}G_{2\nu}(q^{2})]u(p,s)J_{\mu}^{\gamma}(q)+$
$+\frac{G_{F}}{\sqrt{2}}\overline{u}(p^{\prime},s^{\prime})\gamma_{\mu}(g_{V_{\nu}}+\gamma_{5}g_{A_{\nu}})u(p,s)J_{\mu}^{Z^{0}}(q).$
(3)

Here $J_{\mu}^{x}(q)$ are the nucleus electromagnetic $(x=\gamma)$ and weak
neutral $(x=Z^{0})$ currents [14], $g_{V_{\nu}}$ and $g_{A_{\nu}}$ are the
corresponding constants of the neutrino interaction vector $(V)$ and axial
$(A)$ parts.

In the case of the neutrino longitudinl polarization and of a zero-spin
nucleus, the cross-section of the process (2) on the basis of (3) can be
presented after the summing of $s^{\prime}$ as follows:

$d\sigma_{ew}(\theta,s)=d\sigma_{em}(\theta,s)+d\sigma_{int}(\theta,s)+d\sigma_{we}(\theta,s),$
(4)

where to purely electromagnetic interaction answers the expression

$\frac{d\sigma_{em}(\theta,s)}{d\Omega}=\sigma^{\nu}_{o}(1-\eta^{2}_{\nu})^{-1}\\{[F_{1\nu}+2\lambda_{c}s\sqrt{1-\eta_{\nu}^{2}}G_{1\nu}]F_{1\nu}+$
$+\eta^{2}_{\nu}[F_{1\nu}^{2}+4m_{\nu}^{2}(1-\eta^{-2}_{\nu})^{2}F_{2\nu}^{2}]tg^{2}\frac{\theta}{2}-8sE_{\nu}^{2}(1-\eta_{\nu}^{2})^{3/2}F_{2\nu}G_{2\nu}tg^{2}\frac{\theta}{2}+$
$+(1-\eta^{2}_{\nu})[G_{1\nu}^{2}+4E_{\nu}^{2}G_{2\nu}^{2}tg^{2}\frac{\theta}{2}]\\}F_{E}^{2}(q^{2}).$
(5)

The contribution explained by the interference of electroweak interaction is
written in the form

$\frac{d\sigma_{int}(\theta,s)}{d\Omega}=\rho\sigma^{\nu}_{o}(1-\eta^{2}_{\nu})^{-1}g_{V_{\nu}}\\{[1-$
$-\lambda_{c}s\frac{g_{A_{\nu}}}{g_{V_{\nu}}}\sqrt{1-\eta^{2}_{\nu}}][F_{1\nu}+\lambda_{c}s\sqrt{1-\eta_{\nu}^{2}}G_{1\nu}]+\eta^{2}_{\nu}F_{1\nu}tg^{2}\frac{\theta}{2}\\}F_{EV}(q^{2}).$
(6)

In the same way one can present the cross-section of purely weak interaction
with neutral currents

$\frac{d\sigma_{we}(\theta,s)}{d\Omega}=\frac{E_{\nu}^{2}G_{F}^{2}}{8\pi^{2}}\\{g_{V_{\nu}}^{2}(1+\eta_{\nu}^{2}tg^{2}\frac{\theta}{2})+g_{A_{\nu}}^{2}(1-\eta_{\nu}^{2})-$
$-2\lambda_{c}sg_{V_{\nu}}g_{A_{\nu}}\sqrt{1-\eta_{\nu}^{2}}\\}F_{W}^{2}(q^{2})cos^{2}\frac{\theta}{2}.$
(7)

Here we have also used the size

$\sigma_{o}^{\nu}=\frac{\alpha^{2}cos^{2}\frac{\theta}{2}}{4E_{\nu}^{2}(1-\eta^{2}_{\nu})sin^{4}\frac{\theta}{2}},\,\,\,\,\eta_{\nu}=\frac{m_{\nu}}{E_{\nu}},\,\,\,\,\rho=\frac{G_{F}q^{2}}{2\pi\sqrt{2}\alpha},$
$F_{E}(q^{2})=ZF_{c}(q^{2}),\,\,\,\,F_{EV}(q^{2})=ZZ_{W}F_{c}^{2}(q^{2}),\,\,\,\,F_{W}(q^{2})=Z_{W}F_{c}(q^{2}),$
$Z_{W}=\frac{1}{2}\\{\beta_{V}^{(0)}(Z+N)+\beta_{V}^{(1)}(Z-N)\\},\,\,\,\,A=Z+N,\,\,\,\,M_{T}=\frac{1}{2}(Z-N),$

where $\theta$ is the scattering angle, $E_{\nu}$ and $m_{\nu}$ are the
neutrino mass and energy, $F_{c}(q^{2})$ is the charge ($F_{c}(0)=1$) form
factor of a nucleus with isospin T and its projection $M_{T},\beta_{V}^{(0)}$
and $\beta_{V}^{(1)}$ are constants of isoscalar and isovector components of
vector neutral hadronic current.

The presence of the multiplier $s$ in Eqs. (5)-(7) implies their
antisymmetricality concerning the substitution of the left-handed $(s=-1)$
particle with the right-handed $(s=+1)$ and vice versa. We see in addition
that Eqs. (5)-(7) for the neutrino $(\lambda_{c}=+1)$ and the antineutrino
$(\lambda_{c}=-1)$ are different.

Taking into account Eqs. (5)-(7), the size of charge asymmetry

$A_{ch}^{ew}=A_{ch}^{em}+A_{ch}^{int}+A_{ch}^{we}=\frac{d\sigma^{\nu}_{ew}-d\sigma^{\overline{\nu}}_{ew}}{d\sigma^{\nu}_{ew}+d\sigma^{\overline{\nu}}_{ew}}$
(8)

is defined by the corresponding contributions

$A_{ch}^{em}(\theta)=2s\sqrt{1-\eta^{2}_{\nu}}F_{1\nu}G_{1\nu}\\{(1+\eta^{2}_{\nu}tg^{2}\frac{\theta}{2})F_{1\nu}^{2}+$
$+(1-\eta^{2}_{\nu})G_{1\nu}^{2}+4E_{\nu}^{2}[s\sqrt{1-\eta_{\nu}^{2}}G_{2\nu}-(1-\eta^{2}_{\nu})F_{2\nu}]^{2}tg^{2}\frac{\theta}{2}\\}^{-1},$
(9)
$A_{ch}^{int}(\theta)=s\sqrt{1-\eta_{\nu}^{2}}[G_{1\nu}-\frac{g_{A_{\nu}}}{g_{V_{\nu}}}F_{1\nu}]\\{(1+\eta^{2}_{\nu}tg^{2}\frac{\theta}{2})F_{1\nu}-$
$-\frac{g_{A_{\nu}}}{g_{V_{\nu}}}(1-\eta^{2}_{\nu})G_{1\nu}\\}^{-1},$ (10)
$A_{ch}^{we}(\theta)=-2s\frac{g_{A_{\nu}}}{g_{V_{\nu}}}\sqrt{1-\eta_{\nu}^{2}}\\{(1+\eta_{\nu}^{2}tg^{2}\frac{\theta}{2})+\frac{g_{A_{\nu}}^{2}}{g_{V_{\nu}}^{2}}(1-\eta_{\nu}^{2})\\}^{-1}.$
(11)

These formulas show clearly that C-invariance of the considered process can be
violated only in the case when the mirror symmetry is absent. Indeed, taking
$s=0,$ we find

$A_{ch}^{em}(\theta)=0,\,\,\,\,A_{ch}^{int}(\theta)=0,\,\,\,\,A_{ch}^{we}(\theta)=0,$
(12)

which are true at the conservation of P-parity.

Many authors state that one must use the electromagnetic current (1) in the
form [15] in which an $i$ is absent. If we start with such a procedure,
assuming that the interaction magnetic and electric dipole terms must not be
Hermitian even with $q^{2}<0,$ we would establish the other expressions for
the processes cross-sections instead of (5) and (6). They lead to the
implication [16] that C-invariance of elastic scattering is basically violated
at the expense of the neutrino nonzero rest mass. One can also make a
conclusion that this influence does not relate to the behavior of P-symmetry.

Taking into account that nonconservation of P-parity at the neutrino
interaction conveniently characterize by the right-left asymmetry, we have

$A_{RL}^{ew}=A_{RL}^{em}+A_{RL}^{int}+A_{RL}^{we}=\frac{d\sigma_{ew}^{R}-d\sigma_{ew}^{L}}{d\sigma_{ew}^{R}+d\sigma_{ew}^{L}},$
(13)

from Eqs. (5)-(7), we get

$A_{RL}^{em}(\theta)=2\sqrt{1-\eta_{\nu}^{2}}[\lambda_{c}F_{1\nu}G_{1\nu}-$
$-4E_{\nu}^{2}(1-\eta^{2}_{\nu})F_{2\nu}G_{2\nu}tg^{2}\frac{\theta}{2}]\\{(1+\eta^{2}_{\nu}tg^{2}\frac{\theta}{2})F_{1\nu}^{2}+$
$+(1-\eta_{\nu}^{2})[G_{1\nu}^{2}ctg^{2}\frac{\theta}{2}+4E_{\nu}^{2}(G_{2\nu}^{2}+(1-\eta^{2}_{\nu})F_{2\nu}^{2})]tg^{2}\frac{\theta}{2}\\}^{-1},$
(14)
$A_{RL}^{int}(\theta)=\lambda_{c}\sqrt{1-\eta_{\nu}^{2}}[G_{1\nu}-\frac{g_{A_{\nu}}}{g_{V_{\nu}}}F_{1\nu}]\\{(1+\eta^{2}_{\nu}tg^{2}\frac{\theta}{2})F_{1\nu}-$
$-\frac{g_{A_{\nu}}}{g_{V_{\nu}}}(1-\eta^{2}_{\nu})G_{1\nu}\\}^{-1},$ (15)
$A_{RL}^{we}(\theta)=-2\lambda_{c}\frac{g_{A_{\nu}}}{g_{V_{\nu}}}\sqrt{1-\eta_{\nu}^{2}}\\{(1+\eta_{\nu}^{2}tg^{2}\frac{\theta}{2})+\frac{g_{A_{\nu}}^{2}}{g_{V_{\nu}}^{2}}(1-\eta_{\nu}^{2})\\}^{-1}.$
(16)

The availability of the multiplier $\lambda_{c}$ in these formulas implies the
influence of the interaction C-antisymmetrical structure on the conservation
of P-symmetry. Indeed, the average cross-sections, Eqs. (5)-(7), over the two
values of $\lambda_{c}$ would leads us to the equalities

$A_{RL}^{em}(\theta)=-8E_{\nu}^{2}(1-\eta^{2}_{\nu})^{3/2}F_{2\nu}G_{2\nu}\\{(1+\eta^{2}_{\nu}tg^{2}\frac{\theta}{2})F_{1\nu}^{2}+$
$+(1-\eta_{\nu}^{2})[G_{1\nu}^{2}+4E_{\nu}^{2}(G_{2\nu}^{2}+(1-\eta^{2}_{\nu})F_{2\nu}^{2})tg^{2}\frac{\theta}{2}]\\}^{-1}tg^{2}\frac{\theta}{2},$
(17) $A_{RL}^{int}(\theta)=0,\,\,\,\,A_{RL}^{we}(\theta)=0,$ (18)

which take place at C-invariance.

Thus, it follows that regardless of the behavior of charge symmetry, the
right-left asymmetry of the process (2) can be explained by the interference
of the interaction axial-vector terms with its vector terms, if neutrinos do
not possess any new properties.

3\. Interaction of Transversal Polarized Neutrinos
with an Electroweak Field of a Nucleus

Starting from (3) and assuming that the neutrinos are strictly transversal,
for the elastic scattering cross-section we find an explicit expression which
can be reduced after the summing of $s^{\prime}$ to the form

$d\sigma_{ew}(\theta,\varphi,s)=d\sigma_{em}(\theta,\varphi,s)+d\sigma_{int}(\theta,\varphi,s)+d\sigma_{we}(\theta,\varphi,s).$
(19)

As well as in (4), each term here corresponds to the most diverse process and
has the different structure:

$\frac{d\sigma_{em}(\theta,\varphi,s)}{d\Omega}=\sigma^{\nu}_{o}(1-\eta^{2}_{\nu})^{-1}\\{F_{1\nu}^{2}+\eta^{2}_{\nu}[F_{1\nu}^{2}+4m_{\nu}^{2}(1-\eta^{-2}_{\nu})^{2}F_{2\nu}^{2}]tg^{2}\frac{\theta}{2}+$
$+2\lambda_{c}s\eta_{\nu}\sqrt{1-\eta_{\nu}^{2}}F_{1\nu}G_{1\nu}tg\frac{\theta}{2}cos^{2}\varphi+$
$+(1-\eta^{2}_{\nu})[G_{1\nu}^{2}+4E_{\nu}^{2}G_{2\nu}^{2}tg^{2}\frac{\theta}{2}]\\}F_{E}^{2}(q^{2}),$
(20)
$\frac{d\sigma_{int}(\theta,\varphi,s)}{d\Omega}=\rho\sigma^{\nu}_{o}(1-\eta^{2}_{\nu})^{-1}g_{V_{\nu}}\\{F_{1\nu}+\eta_{\nu}^{2}[1+$
$+\lambda_{c}s\frac{g_{A_{\nu}}}{g_{V_{\nu}}}\eta_{\nu}^{-1}\sqrt{1-\eta^{2}_{\nu}}ctg\frac{\theta}{2}cos^{2}\varphi]F_{1\nu}tg^{2}\frac{\theta}{2}-$
$-\lambda_{c}s\eta_{\nu}\sqrt{1-\eta^{2}_{\nu}}[tg\frac{\theta}{2}cos^{2}\varphi+\lambda_{c}s\frac{g_{A_{\nu}}}{g_{V_{\nu}}}\eta_{\nu}^{-1}\sqrt{1-\eta^{2}_{\nu}}]G_{1\nu}\\}F_{EV}(q^{2}),$
(21)
$\frac{d\sigma_{we}(\theta,\varphi,s)}{d\Omega}=\frac{E_{\nu}^{2}G_{F}^{2}}{8\pi^{2}}\\{g_{V_{\nu}}^{2}(1+\eta_{\nu}^{2}tg^{2}\frac{\theta}{2})+g_{A_{\nu}}^{2}(1-\eta_{\nu}^{2})-$
$-2\lambda_{c}sg_{V_{\nu}}g_{A_{\nu}}\eta_{\nu}\sqrt{1-\eta_{\nu}^{2}}tg\frac{\theta}{2}cos^{2}\varphi\\}F_{W}^{2}(q^{2})cos^{2}\frac{\theta}{2},$
(22)

where $\varphi$ is the azimuthal angle.

Using (20)-(22) and taking (8), for the C-odd asymmetry in the case of the
neutrino transversal polarization we get

$A_{ch}^{em}(\theta,\varphi)=2s\eta_{\nu}\sqrt{1-\eta_{\nu}^{2}}F_{1\nu}G_{1\nu}\\{(1+\eta^{2}_{\nu}tg^{2}\frac{\theta}{2})F_{1\nu}^{2}+$
$+(1-\eta^{2}_{\nu})[G_{1\nu}^{2}+4E_{\nu}^{2}(G_{2\nu}^{2}+(1-\eta^{2}_{\nu})F_{2\nu}^{2})tg^{2}\frac{\theta}{2}]\\}^{-1}tg\frac{\theta}{2}cos^{2}\varphi,$
(23)
$A_{ch}^{int}(\theta,\varphi)=-s\eta_{\nu}\sqrt{1-\eta_{\nu}^{2}}[G_{1\nu}-\frac{g_{A_{\nu}}}{g_{V_{\nu}}}F_{1\nu}]\\{(1+\eta^{2}_{\nu}tg^{2}\frac{\theta}{2})F_{1\nu}-$
$-\frac{g_{A_{\nu}}}{g_{V_{\nu}}}(1-\eta^{2}_{\nu})G_{1\nu}\\}^{-1}tg\frac{\theta}{2}cos^{2}\varphi,$
(24)
$A_{ch}^{we}(\theta,\varphi)=-2s\eta_{\nu}\frac{g_{A_{\nu}}}{g_{V_{\nu}}}\sqrt{1-\eta_{\nu}^{2}}\\{(1+\eta_{\nu}^{2}tg^{2}\frac{\theta}{2})+$
$+\frac{g_{A_{\nu}}^{2}}{g_{V_{\nu}}^{2}}(1-\eta_{\nu}^{2})\\}^{-1}tg\frac{\theta}{2}cos^{2}\varphi.$
(25)

The solutions (23)-(25) at $s=0$ coincide with the corresponding size from
(12) and that, consequently, the behavior of C-invariance in the P-symmetrical
interactions does not depend on the type of polarization.

In the same way one can see that the P-odd characteristics of elastic
scattering, according to (13), (20)-(22), has the form

$A_{RL}^{em}(\theta,\varphi)=2\lambda_{c}\eta_{\nu}\sqrt{1-\eta_{\nu}^{2}}F_{1\nu}G_{1\nu}\\{(1+\eta^{2}_{\nu}tg^{2}\frac{\theta}{2})F_{1\nu}^{2}+$
$+(1-\eta_{\nu}^{2})[G_{1\nu}^{2}ctg^{2}\frac{\theta}{2}+4E_{\nu}^{2}(G_{2\nu}^{2}+(1-\eta^{2}_{\nu})F_{2\nu}^{2})]tg^{2}\frac{\theta}{2}\\}^{-1}tg\frac{\theta}{2}cos^{2}\varphi,$
(26)
$A_{RL}^{int}(\theta,\varphi)=-\lambda_{c}\eta_{\nu}\sqrt{1-\eta_{\nu}^{2}}[G_{1\nu}-\frac{g_{A_{\nu}}}{g_{V_{\nu}}}F_{1\nu}]\\{(1+\eta^{2}_{\nu}tg^{2}\frac{\theta}{2})F_{1\nu}-$
$-\frac{g_{A_{\nu}}}{g_{V_{\nu}}}(1-\eta^{2}_{\nu})G_{1\nu}\\}^{-1}tg\frac{\theta}{2}cos^{2}\varphi,$
(27)
$A_{RL}^{we}(\theta,\varphi)=-2\lambda_{c}\eta_{\nu}\frac{g_{A_{\nu}}}{g_{V_{\nu}}}\sqrt{1-\eta_{\nu}^{2}}\\{(1+\eta_{\nu}^{2}tg^{2}\frac{\theta}{2})+$
$+\frac{g_{A_{\nu}}^{2}}{g_{V_{\nu}}^{2}}(1-\eta_{\nu}^{2})\\}^{-1}tg\frac{\theta}{2}cos^{2}\varphi.$
(28)

However, due to C-parity, it follows from (26)-(28) that

$A_{RL}^{em}(\theta,\varphi)=0,\,\,\,\,A_{RL}^{int}(\theta,\varphi)=0,\,\,\,\,A_{RL}^{we}(\theta,\varphi)=0.$
(29)

Comparing (17) and (18) with (29), it is easy to observe the differences which
may serve as an indication to the type of polarization dependence of
C-invariant processes right-left asymmetry.

4\. Conclusion

We have established an explicit form of the differential cross sections
describing the elastic electroweak scattering of longitudinal and transversal
polarized neutrinos (antineutrinos) on spinless nuclei as a consequence of the
availability of rest mass, charge, magnetic, anapole and electric dipole
moments of elementary particles and their weak neutral currents. With the use
of these formulas a proof has been obtained regardless of the nature of C,
nonconservation of P can be explained by the interference of the interaction
vector and axial-vector parts.

One of the new features of our results is the connection between the P-odd
phenomena and possible polarization types. Unlike the behavior of C-parity in
the P-symmetrical scattering, coefficients of right-left asymmetries
$A_{RL}^{ew}(\theta)$ and $A_{RL}^{ew}(\theta,\varphi)$ in the C-invariant
processes with longitudinal and transversal neutrinos are different.

Furthermore, if neutrinos are of high energies $(E_{\nu}\gg m_{\nu})$ then
$A_{RL}^{ew}(\theta,\varphi)=0,$ and the size of $A_{RL}^{ew}(\theta)$ is
reduced to the form

$A_{RL}^{ew}(\theta)=-\frac{2F_{2\nu}G_{2\nu}}{F_{2\nu}^{2}+G_{2\nu}^{2}}.$
(30)

It is expected that measurement of right-left asymmetry $A_{RL}^{ew}(\theta)$
for any two values of large energies will testify in favor of the equality of
the neutrino magnetic and electric dipole moments.

References

  1. 1.

P.H. Frampton and P. Vogel, Phys. Rep. 82, 339 (1982); F. Boehm and P. Vogel,
Ann. Rev. Nucl. Part. Sci. 34, 125 (1984); P. Vogel and J. Engel, Phys. Rev.
D39, 3378 (1989).

  2. 2.

B.K. Kerimov, T.R. Aruri and M.Ya. Safin, Izv. Acad. Nauk SSSR. Ser. Fiz. 37,
1768 (1973); B.K. Kerimov and M.Ya. Safin, Izv. Russ. Acad. Nauk Ser. Fiz. 61,
657 (1997).

  3. 3.

M.A.B. Beg, W.J. Marciano and M. Ruderman, Phys. Rev. D17, 1395 (1978).

  4. 4.

W. Bernreuther and M. Suzuki, Rev. Mod. Phys. 63, 313 (1991).

  5. 5.

R.S. Sharafiddinov, Dokl. Akad. Nauk Ruz. Ser. Math. Tehn. Estest. 7, 25
(1998).

  6. 6.

K. Fujikawa and R.E. Shrock, Phys. Rev. Lett. 45, 963 (1980).

  7. 7.

R.B. Begzhanov and R.S. Sharafiddinov, in Proc. Int. Conf. on Nuclear Physics,
Moscow, June 16-19, 1998, St. Petersburg, 1998, p. 354.

  8. 8.

S. Davidson, B. Campbell and K.D. Bailey, Phys. Rev. D43, 2314 (1991); J.A.
Morgan and D.B. Farrant, Phys. Lett. B128, 431 (1983).

  9. 9.

Ya. B. Zel’dovich, Zh. Eksp. Teor. Fiz. 33, 1531 (1957); Ya. B. Zel’dovich and
A. M. Perelomov, ibid. 39, 1115 (1960).

  10. 10.

M.J. Musolf and B.R. Holstein, Phys. Rev. D43,1956 (1991).

  11. 11.

L.D. Landau, Zh. Eksp. Teor. Fiz. 32,405 (1957); Nucl. Phys. 3, 127 (1957).

  12. 12.

B.K. Kerimov and M.Ya. Safin, Izv. Russ. Acad. Nauk Ser. Fiz. 57, 93 (1993).

  13. 13.

S.L. Glashow, Nucl. Phys. 22, 579 (1961); A. Salam and J.C. Ward, Phys. Lett.
13, 168 (1964); S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967).

  14. 14.

T.W. Donnelly and R.D. Peccei, Phys. Rep. 50, 3 (1979).

  15. 15.

J.Bernstein, M. Ruderman and G. Feinberg, Phys. Rev. 132 1227 (1963).

  16. 16.

R.B. Begzhanov and R.S. Sharafiddinov, in Proc. Int. Conf. on Nuclear Physics,
Dubna, April 21-24, 1999, St. Petersburg, 1999, p. 408.


# Malaria Cell Detection Using Deep Neural Networks

Anurag Singh  Department of Computer Science
Drexel University
Philadelphia, PA, USA
<EMAIL_ADDRESS>Saurabh Sawant  Department of Computer Science
Drexel University
Philadelphia, PA, USA
<EMAIL_ADDRESS>
###### Abstract

Malaria remains one of the most pressing public health concerns globally,
causing significant morbidity and mortality, especially in sub-Saharan Africa.
Rapid and accurate diagnosis is crucial for effective treatment and disease
management. Traditional diagnostic methods, such as microscopic examination of
blood smears, are labor-intensive and require significant expertise, which may
not be readily available in resource-limited settings. This project aims to
automate the detection of malaria-infected cells using a deep learning
approach. We employed a convolutional neural network (CNN) based on the
ResNet50 architecture, leveraging transfer learning to enhance performance.
The Malaria Cell Images Dataset from Kaggle, containing 27,558 images
categorized into infected and uninfected cells, was used for training and
evaluation. Our model demonstrated high accuracy, precision, and recall,
indicating its potential as a reliable tool for assisting in malaria
diagnosis. Additionally, a web application was developed using Streamlit to
allow users to upload cell images and receive predictions about malaria
infection, making the technology accessible and user-friendly. This paper
provides a comprehensive overview of the methodology, experiments, and
results, highlighting the effectiveness of deep learning in medical image
analysis.

###### Index Terms:

Malaria Detection, ResNet50, Deep Learning, Image Classification

##  I Introduction

Malaria, caused by Plasmodium parasites, is transmitted to humans through the
bites of infected Anopheles mosquitoes. According to the World Health
Organization (WHO), there were an estimated 229 million cases of malaria
worldwide in 2019, resulting in over 400,000 deaths, predominantly in sub-
Saharan Africa. Early and accurate diagnosis is essential for effective
treatment and control of malaria. The conventional diagnostic method,
microscopic examination of Giemsa-stained blood smears, is considered the gold
standard. However, this method is time-consuming, requires skilled personnel,
and is subject to human error.

Recent advancements in machine learning, particularly deep learning, have
revolutionized the field of medical image analysis. Convolutional neural
networks (CNNs) have been successfully applied to various image classification
problems, achieving high accuracy and robustness. The ResNet50 model,
introduced by He et al. (2016), has been particularly effective in image
recognition tasks due to its innovative residual learning framework, which
mitigates the vanishing gradient problem in deep networks.

The primary objective of this study is to explore the feasibility and
effectiveness of using the ResNet50 model for the automated detection of
malaria-infected cells. By leveraging transfer learning and data augmentation
techniques, we aim to develop a robust and accurate model that can assist in
the rapid diagnosis of malaria, especially in resource-limited settings.
Additionally, to make this technology accessible and user-friendly, we
developed a web application using Streamlit that allows users to upload cell
images and receive predictions about malaria infection.

##  II Background

Malaria diagnosis through microscopic examination involves identifying
Plasmodium parasites in stained blood films. This method, although highly
specific and sensitive when performed by skilled technicians, is not always
feasible in regions with limited access to trained personnel and adequate
laboratory facilities. The variability in the quality of microscopy and the
subjective nature of the examination can also lead to inconsistent results.

In the past decade, machine learning and deep learning have revolutionized the
field of medical image analysis. Deep learning models, particularly
Convolutional Neural Networks (CNNs), have been used to develop automated
systems for various diagnostic tasks, including cancer detection, diabetic
retinopathy screening, and pneumonia diagnosis. These models have demonstrated
the ability to learn complex patterns from large datasets, achieving
performance levels comparable to, and sometimes exceeding, those of human
experts.

The ResNet50 model, a deep CNN with 50 layers, introduced the concept of
residual learning. This approach addresses the degradation problem observed in
deep networks, where accuracy degrades as the network depth increases. By
allowing layers to learn residual functions with reference to the layer
inputs, ResNet50 facilitates the training of very deep networks, making it an
ideal choice for image classification tasks.

In this study, we utilized the Malaria Cell Images Dataset from Kaggle, which
contains a balanced set of 27,558 images of infected and uninfected cells.
This dataset was chosen for its comprehensive representation of malaria-
infected and healthy blood smears, providing a robust basis for training and
evaluating our deep learning model.

##  III Related Work

Previous research has established the potential of deep learning in medical
image analysis. He et al. (2016) introduced the ResNet architecture, which
significantly improved image classification tasks by addressing the vanishing
gradient problem through residual learning. Rajaraman et al. (2018)
demonstrated the feasibility of automated malaria parasite detection using
deep learning, while Litjens et al. (2017) provided a comprehensive survey of
deep learning applications in medical imaging.

Several studies have focused on the use of CNNs for various medical imaging
tasks, including cancer, diabetic retinopathy, and pneumonia detection.
Gulshan et al. (2016) demonstrated the effectiveness of a deep learning
algorithm in detecting diabetic retinopathy from retinal fundus photographs,
achieving performance comparable to that of ophthalmologists.

In the context of malaria diagnosis, Dong et al. (2017) developed a deep
learning model for automated detection of malaria parasites in blood smears,
achieving an accuracy of 96

Our work builds on these approaches by implementing a custom ResNet50
architecture with advanced data augmentation and preprocessing techniques to
enhance classification performance. Additionally, we developed a user-friendly
web application using Streamlit, making the model accessible for rapid malaria
diagnosis. This integration aims to bridge the gap between research and
practical application, providing an efficient solution for healthcare
providers in resource-limited settings.

##  IV Methodology

###  IV-A Dataset

The dataset used in this study consists of cell images categorized into two
classes: parasitized and uninfected. The images are sourced from the ”Cell
Images for Detecting Malaria” dataset available on Kaggle.

###  IV-B Data Preprocessing

The images were preprocessed using TensorFlow’s data pipeline. This involved
reading the images, decoding them, resizing to 224x224 pixels, and normalizing
the pixel values.

###  IV-C Model Architecture

The architecture of our custom ResNet50 model is designed to leverage the
strengths of residual learning to facilitate the training of a deep network.
The ResNet architecture introduces shortcut connections that bypass one or
more layers, enabling the construction of very deep networks without the
vanishing gradient problem.

####  IV-C1 Residual Block

A key component of ResNet architectures is the residual block. A residual
block is defined as follows:

$y=F(x,\\{W_{i}\\})+x$ (1)

where $x$ is the input to the residual block, $y$ is the output, and
$F(x,\\{W_{i}\\})$ represents the residual mapping to be learned. The function
$F$ consists of a series of convolutional layers with weights $\\{W_{i}\\}$.

In our custom ResNet50-like model, each residual block consists of three
convolutional layers. The first layer is a $1\times 1$ convolution, which
reduces the dimensionality of the input. This is followed by a $3\times 3$
convolution, and finally, another $1\times 1$ convolution to restore the
original dimensionality. Batch normalization and ReLU activation are applied
after each convolutional layer to enhance the model’s training dynamics.

####  IV-C2 Bottleneck Residual Block

The bottleneck residual block is defined as:

$y=W_{3}\sigma(W_{2}\sigma(W_{1}x))$ (2)

where $\sigma$ denotes the ReLU activation function, and $W_{1}$, $W_{2}$, and
$W_{3}$ are the weights of the $1\times 1$, $3\times 3$, and $1\times 1$
convolutional layers, respectively. The shortcut connection adds the input $x$
directly to the output $y$.

####  IV-C3 Overall Architecture

The overall architecture of our custom ResNet50-like model follows the
standard ResNet50 design, consisting of an initial convolutional layer
followed by a series of residual blocks:

  * •

Conv1: $7\times 7$ convolution, 64 filters, stride 2, followed by batch
normalization and ReLU activation.

  * •

Max Pooling: $3\times 3$ max pooling, stride 2.

  * •

Conv2_x: 3 bottleneck residual blocks with 64 filters.

  * •

Conv3_x: 4 bottleneck residual blocks with 128 filters, stride 2 for
downsampling.

  * •

Conv4_x: 6 bottleneck residual blocks with 256 filters, stride 2 for
downsampling.

  * •

Conv5_x: 3 bottleneck residual blocks with 512 filters, stride 2 for
downsampling.

  * •

Global Average Pooling: Averages the feature maps.

  * •

Dense Layer: 512 units with ReLU activation.

  * •

Dropout: Dropout rate of 0.5 to prevent overfitting.

  * •

Output Layer: Softmax activation for binary classification.

The model is trained using the Adam optimizer with a learning rate of 0.001.
The loss function used is sparse categorical crossentropy, suitable for our
binary classification task. The architecture is designed to balance model
complexity and computational efficiency, allowing for effective training on
the given dataset.

###  IV-D Training

The model was trained using the Adam optimizer with a learning rate of 0.001.
We employed early stopping and learning rate reduction on plateau callbacks to
enhance training efficiency and prevent overfitting. The model was trained for
30 epochs with a batch size of 32.

###  IV-E Evaluation

The performance of the model was evaluated using accuracy, precision, recall,
and F1-score metrics on the validation and test datasets.

###  IV-F Web Application Development

To make the malaria detection model accessible to users, a web application was
developed using Streamlit. The application provides an intuitive interface
where users can upload an image of a cell. The uploaded image is then
processed and classified as either parasitized or uninfected by the model. The
development process involved:

  * •

Loading the trained model in the Streamlit app.

  * •

Preprocessing the uploaded image to match the input requirements of the model.

  * •

Predicting the class of the image using the model.

  * •

Displaying the result to the user in an easily interpretable format.

##  V Experiments and Results

###  V-A Data Description

The dataset contains a total of 27,558 cell images, with an equal distribution
between parasitized and uninfected classes. The dataset was split into
training (60%), validation (20%), and test (20%) sets.

###  V-B Training Results

The model achieved a training accuracy of 99% and a validation accuracy of 98%
after 30 epochs. The use of data augmentation techniques, such as rotation,
zoom, and horizontal flipping, helped improve the model’s robustness.

###  V-C Test Results

On the test set, the model achieved an accuracy of 97.8%, with a precision of
98.6%, recall of 97.2%, and F1-score of 97.8%. These results demonstrate the
model’s effectiveness in distinguishing between parasitized and uninfected
cell images.

Figure 1: Training and Validation Accuracy Figure 2: Training and Validation Loss Figure 3: Confusion Matrix TABLE I: Performance Metrics on Test Set Class | Precision | Recall | F1-Score | Support
---|---|---|---|---
Parasitized | 0.986 | 0.972 | 0.979 | 2808
Uninfected | 0.971 | 0.985 | 0.978 | 2705
Accuracy | 0.978

###  V-D Web Application Development

To make the malaria detection model accessible to users, a web application was
developed using Streamlit. The application allows users to upload an image of
a cell, which is then processed and classified as either parasitized or
uninfected by the model. The user interface is designed to be intuitive and
easy to use.

Figure 4: Web Application - Image Upload Interface Figure 5: Web Application -
Classification Result

The figures below show screenshots of the web application. Figure 4 displays
the interface where users can upload an image. Figure 5 shows the result of
the classification, indicating whether the cell is parasitized or uninfected.

##  VI Conclusion

The custom ResNet50 model developed in this project demonstrates high accuracy
and robustness in malaria detection. The use of residual blocks and advanced
data augmentation techniques contributed significantly to the model’s
performance. By leveraging the Malaria Cell Images Dataset from Kaggle, our
model achieved an impressive test accuracy of 97.8%, with a precision of
98.6%, recall of 97.2%, and F1-score of 97.8%.

To make this technology accessible, we developed a web application using
Streamlit. This application allows users to upload an image of a cell and
receive a prediction about whether the cell is infected with malaria,
providing a user-friendly interface for rapid diagnostics.

This study underscores the potential of deep learning for automated malaria
detection, offering a scalable and efficient solution for healthcare
applications, particularly in regions with limited access to trained medical
personnel. Future work will focus on improving model generalizability,
exploring other deep learning architectures, and integrating explainability
techniques.

In conclusion, the combination of a high-performing model and an accessible
web application demonstrates the feasibility and effectiveness of using deep
learning for automated malaria detection, paving the way for further research
and development in this area.

##  VII Future Work

While the current study demonstrates the potential of a custom ResNet50
architecture for automated malaria detection, there are several avenues for
future work that can enhance the robustness and applicability of this model.

###  VII-A Dataset Expansion

One of the primary areas for improvement is the expansion of the dataset.
Incorporating images from diverse sources, including different geographic
regions and varying quality of blood smears, can help generalize the model
better. This will ensure that the model is robust and performs well across
various conditions encountered in real-world settings.

###  VII-B Model Optimization

Future work could focus on optimizing the model architecture and
hyperparameters. Techniques such as hyperparameter tuning, pruning, and
quantization can be explored to reduce the model size and improve inference
speed without compromising accuracy. Additionally, experimenting with
different architectures, such as EfficientNet or DenseNet, could yield further
performance gains.

###  VII-C Real-Time Deployment

Another critical aspect is the deployment of the model in real-time
applications. Developing a user-friendly interface, either as a mobile
application or a web-based platform, can make the diagnostic tool accessible
to healthcare providers in remote and resource-limited areas. Ensuring the
model’s performance on edge devices with limited computational power will be
crucial for such deployments.

###  VII-D Explainability and Interpretability

Incorporating explainability techniques, such as Grad-CAM or SHAP, can help in
understanding the model’s decision-making process. Providing visual
explanations for the model’s predictions can build trust among healthcare
professionals and aid in the verification of results. Future research can
focus on integrating these techniques into the diagnostic pipeline.

## Acknowledgment

The authors would like to thank Kaggle for providing the dataset used in this
study.

## References

  * [1] A. Rajaraman, S. Candemir, S. X. Kim, et al., ”A novel stacked generalization of models for improved classification performance,” Journal of Biomedical and Health Informatics, vol. 22, no. 1, pp. 83-88, 2018.
  * [2] G. Bibin, S. Nair, and P. S. Dey, ”Malaria cell images dataset,” IEEE Transactions on Image Processing, vol. 25, no. 11, pp. 5013-5024, 2017.
  * [3] K. He, X. Zhang, S. Ren, and J. Sun, ”Deep Residual Learning for Image Recognition,” in 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Las Vegas, NV, USA, 2016, pp. 770-778.
  * [4] World Health Organization, ”World Malaria Report 2020,” [Online]. Available: https://www.who.int/publications/i/item/9789240015791. [Accessed: June 14, 2024].
  * [5] A. Das, M. S. Dhillon, and A. K. Sinha, ”Automated detection of malaria parasites in blood smears using convolutional neural networks,” Journal of Parasitology Research, vol. 2019, Article ID 1656502, 2019.
  * [6] Kaggle, ”Malaria Cell Images Dataset,” [Online]. Available: https://www.kaggle.com/datasets/iarunava/cell-images-for-detecting-malaria/data. [Accessed: June 14, 2024].
  * [7] D. Rajaraman, S. Candemir, and X. S. Kim, ”Performance evaluation of deep neural networks for health informatics,” IEEE Journal of Biomedical and Health Informatics, vol. 23, no. 1, pp. 1-10, 2019.
  * [8] J. Dong, J. Chen, and L. Zhang, ”Automated detection of malaria parasites in thick blood films using a deep learning approach,” Malaria Journal, vol. 16, no. 1, pp. 1-9, 2017.
  * [9] M. Gulshan, L. Peng, V. Coram, et al., ”Development and validation of a deep learning algorithm for detection of diabetic retinopathy in retinal fundus photographs,” JAMA, vol. 316, no. 22, pp. 2402-2410, 2016.
  * [10] G. Litjens, T. Kooi, B. E. Bejnordi, et al., ”A survey on deep learning in medical image analysis,” Medical Image Analysis, vol. 42, pp. 60-88, 2017.
  * [11] J. Liang, J. Li, and H. Xia, ”Ensemble deep learning for malaria parasite detection in thick blood smears,” IEEE Transactions on Medical Imaging, vol. 37, no. 2, pp. 241-250, 2019.
  * [12] H. Chen, Y. Chen, and X. Chen, ”Transfer learning for malaria parasite detection in thin blood smears,” Computer Methods and Programs in Biomedicine, vol. 162, pp. 13-22, 2018.