# Signals for strange quark contributions to the neutrino (antineutrino)
scattering in quasi-elastic region

Myung-Ki Cheoun1), K. S. Kim2) 111Corresponding author<EMAIL_ADDRESS>1)Department of Physics, Soongsil University, Seoul, 156-743, Korea
2)School of Liberal Arts and Science, Korea Aerospace University, Koyang
412-791, Korea

###### Abstract

Strange quark contributions to the neutrino (antineutrino) scattering are
investigated on the elastic neutrino-nucleon scattering and the neutrino-
nucleus scattering for 12C target in the quasi-elastic region on the incident
energy of 500 MeV, within the framework of a relativistic single particle
model. For the neutrino-nucleus scattering, the effects of final state
interaction for the knocked-out nucleon are included by a relativistic optical
potential. In the cross sections we found some cancellations of the strange
quark contributions between the knocked-out protons and neutrons.
Consequently, the asymmetries between the incident neutrino and antineutrino
which is the ratio of neutral current to charged current, and the difference
between the asymmetries are shown to be able to yield more feasible quantities
for the strangeness effects. In order to explicitly display importance of the
cancellations, results of the exclusive reaction
${}^{16}O(\nu,\nu^{{}^{\prime}}p)$ are additionally presented for detecting
the strangeness effects.

###### pacs:

25.30. Pt; 13.15.+g; 24.10.Jv

Since the exploration of a spin structure of proton at the EMC measurement
emc89 , the deep-inelastic scattering of leptons from nucleons has played
important roles for studying the distribution of quarks and gluons. In the
quark parton model for the nucleon, the cross section of the scattering with
polarized leptons on polarized nucleons is usually given by four structure
functions, $F_{1}(x),F_{2}(x),g_{1}(x)$, and $g_{2}(x)$

$F_{2}(x)=x{\mathop{\Sigma}_{q}}e_{q}^{2}q(x)=2xF_{1}(x)~{},~{}g_{1}(x)={1\over
2}{\mathop{\Sigma}_{q}}e_{q}^{2}\Delta q(x)$ (1)

with $x=Q^{2}/2p\cdot q$ and $Q^{2}=-{(p^{{}^{\prime}}-p)}^{2}$ for the
momentum $p$ of the initial nucleon and the momentum transfer $q$ from
leptons. $e_{q},q(x)$, and $\Delta q(x)$ denote a charge, a parton
distribution and a polarized parton distribution function for quarks flavor
$q$, respectively Albe02 ; garvey . In experimental aspect, the structure
functions are usually measured as their integral forms because of their
$Q^{2}$ dependence

$\Gamma_{1}(Q^{2})=\int_{0}^{1}g_{1}(x,Q^{2})dx~{}.$ (2)

Available experimental data for the $\Gamma_{1}(Q^{2})$ are located in
$0.115\sim 0.126$ depending on the region, $3\leq Q^{2}\leq 10.7$ Albe02 . If
$\Delta q$ is defined as a difference of the total numbers of quarks and
antiquarks in the nucleon with a helicity equal and opposite to the spin of
the nucleon

$\Delta q=\int_{0}^{1}{\mathop{\Sigma}_{r=\pm
1}}r[q^{(r)}(x)+{\bar{q}}^{(r)}(x)]dx~{},$ (3)

$\Delta q$ stands for the contribution of $q$-quarks and
${\bar{q}}$-antiquarks to the spin of the nucleon. For example, in the
infinite momentum frame (quark-parton model), $\Gamma_{1}(Q^{2})$ is evaluated
in terms of the $\Delta q$ with some constraints among them

$\Gamma_{1}^{p}={1\over 2}({4\over 9}\Delta u+{1\over 9}\Delta d+{1\over
9}\Delta s)~{},~{}\Delta u+\Delta d-2\Delta s=3F-D~{},~{}\Delta u-\Delta
d=g_{A}$ (4)

with $D$ and $F$ constants in SU(3) symmetry.

On the other hand, the axial form factors $G_{A}^{q}(Q^{2})$ and the axial
charges $g_{A}^{q}$ are given by the matrix elements and the diagonal matrix
elements of the axial current

$<p^{{}^{\prime}}|{\bar{q}}\gamma_{\mu}\gamma_{5}q|p>={\bar{u}}(p^{{}^{\prime}})\gamma_{\mu}\gamma_{5}G_{A}^{q}(Q^{2})u(p)~{},~{}<p|{\bar{q}}\gamma_{\mu}\gamma_{5}q|p>=2Ms_{\mu}g_{A}^{q}~{},$
(5)

where $M$ and $s_{\mu}$ are respectively the mass and spin vector of the
nucleon. In particular, for our primary interest, the axial coupling constant
$g_{A}^{s}$ (or axial charge) induced by strangeness quark turned out to be
equal to the $\Delta s$, which is related to the nucleon spin by the $s$ quark
contribution. Though its value extracted from both experimental and/or
theoretical results has not been fixed yet, several groups reported the value
as $0.0$ Young06 , $-0.08\pm 0.06,-0.12\pm 0.06$ Pate05  and $-0.18\pm 0.05$
emc89 . Therefore, its value is allowed to vary in $0<g_{A}^{s}<-0.19$ for
further discussions of its effects in the elastic neutrino (antineutrino)
$(\nu{(\bar{\nu}}))$ scattering.

In this paper, although the strangeness effects in vectorial parts can be
studied by parity not only violating (or polarized electron scattering) but
$\nu$ scattering, we focus on the strangeness $g_{A}^{s}$ effect in the axial
part by studying the $\nu{(\bar{\nu}})$ scattering on the quasi-elastic (QE)
region, where inelastic processes like pion production and delta resonance are
excluded.

Before going further, we briefly summarize the present status for signals of
the strangeness effects in the $\nu{(\bar{\nu}})$ scattering. Since the
vectorial parts are rarely contributed due to the given Weinberg angle in the
reaction, the elastic cross section on the proton $\sigma(\nu p\rightarrow\nu
p)$ mediated by a neutral current (NC) reaction is sensitive mainly on the
$g_{A}^{s}$ value. But the measurement of the cross section is not
experimentally easy, so that one usually resorts to the ratio of proton to
neutron cross sections $R_{p/n}={{\sigma(\nu p\rightarrow\nu p)}/{\sigma(\nu
n\rightarrow\nu n)}}$ garvey . The measurement of this ratio has also some
difficulties in the neutron detection. Since the charged current (CC) cross
section is relatively insensitive to the strangeness, the ratio
$R_{NC/CC}={{\sigma(\nu p\rightarrow\nu p)}/{\sigma(\nu
n\rightarrow\mu^{-}p)}}$ is suggested as a plausible signal for the nucleon
strangeness although the signal goes down by a factor 2 Vent06 . In this
paper, more precise analysis for those quantities are carried out in searching
other feasible quantities for the strangeness effects in the
$\nu{(\bar{\nu}})$-scattering.

The elastic neutrino-nucleon $(\nu{(\bar{\nu}})-N)$ cross section by the NC
reaction is expressed in terms of the Sachs (vector and axial) form factors
Albe02 ; cheoun08

$\displaystyle{({{d\sigma}\over{dQ^{2}}})}_{\nu({\bar{\nu}})}^{NC}$
$\displaystyle=$ $\displaystyle{{G_{F}^{2}}\over{2\pi}}[{1\over
2}y^{2}{(G_{M}^{V})}^{2}+(1-y-{{M}\over{2E_{\nu}}}y){{{(G_{E}^{V})}^{2}+{E_{\nu}\over
2M}y{(G_{M}^{V})}^{2}}\over{1+{E_{\nu}\over 2M}y}}$ (6)
$\displaystyle+({1\over
2}y^{2}+1-y+{{M}\over{2E_{\nu}}}y){(G_{A})}^{2}+h2y(1-{1\over
2}y)G_{M}^{V}G_{A}]~{}.$

Here, $E_{\nu}$ is the incident $\nu({\bar{\nu}})$ energy in the laboratory
frame, and $y={{p\cdot q}/{p\cdot k}}={{Q^{2}}/{2p\cdot k}}$ with initial four
momenta $k$ of $\nu({\bar{\nu}})$, target nucleon $p$, and four momentum
transfer $q$ to the nucleon, respectively. $G_{F}\simeq 1.16639\times
10^{-11}$ MeV-2 is the Fermi constant. Vectorial Sachs form factors are
written as electric and magnetic form factors of the nucleon

$\displaystyle G_{M,E}^{V,~{}p(n)}(Q^{2})$ $\displaystyle=$
$\displaystyle({\frac{1}{2}}-2\sin^{2}\theta_{W})G_{M,E}^{p(n)}(Q^{2})-{\frac{1}{2}}G_{M,E}^{n(p)}(Q^{2})-{\frac{1}{2}}G_{M,E}^{s}(Q^{2})~{}~{}\mbox{for~{}
NC}~{}$ $\displaystyle=$
$\displaystyle(G_{M,E}^{p}(Q^{2})-G_{M,E}^{n}(Q^{2}))~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{for~{}
CC}~{},$ $\displaystyle G_{M}^{s}(Q^{2})$ $\displaystyle=$
$\displaystyle{{Q^{2}F_{1}^{s}+\mu_{s}}\over{(1+\tau){(1+Q^{2}/M_{V}^{2})}^{2}}},~{}G_{E}^{s}(Q^{2})={{Q^{2}F_{1}^{s}-\mu_{s}\tau}\over{(1+\tau){(1+Q^{2}/M_{V}^{2})}^{2}}}~{},$
(7)

where $\mu_{s}=G_{M}^{s}(0)$ is a strange magnetic moment. The axial form
factor is usually presumed as a dipole form

$\displaystyle G^{NC}_{A}(Q^{2})$ $\displaystyle=$
$\displaystyle{\frac{1}{2}}(\mp
g_{A}+g_{A}^{s})/(1+Q^{2}/M_{A}^{2})^{2}~{}~{}\mbox{for~{} NC}~{}$
$\displaystyle G_{A}^{CC}(Q^{2})$ $\displaystyle=$ $\displaystyle-
g_{A}/{(1+Q^{2}/M_{A}^{2})}^{2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{for~{}
CC}~{},$ (8)

where $g_{A}=1.262$ and $M_{A}$ are the axial coupling constant and the axial
cut off mass, respectively. $-(+)$ coming from the isospin dependence denotes
the knocked-out proton (neutron), respectively. The $g_{A}^{s}$ appeared
explicitly in Eq.(8) represents the strange quark contents in the nucleon. Of
course, the strangeness effects could contribute to other form factors, but
their effects turned out to contribute only a few % to those physical
quantities cheoun08 .

In order to calculate the $\nu({\bar{\nu}})$-nucleus $(\nu-A)$ scattering, we
choose the nucleus fixed frame where the target nucleus is seated at the
origin of the coordinate system. The four-momenta of the incident and outgoing
neutrinos (antineutrinos), the target nucleus, the residual nucleus, and the
knocked-out nucleon are labelled $p_{i}^{\mu}$ and $p_{f}^{\mu}$,
$p_{A}^{\mu}$, $p_{A-1}^{\mu}$, and $p^{\mu}$, respectively. In the laboratory
frame, the inclusive cross section, which does not detect the outgoing $\nu$
(${\bar{\nu}}$), is given by the contraction between lepton and hadron tensor
kimplb ; umino ; udias ; giusti1

$\displaystyle{\frac{d\sigma}{dT_{p}}}$ $\displaystyle=$ $\displaystyle
4\pi^{2}{\frac{M_{N}M_{A-1}}{(2\pi)^{3}M_{A}}}\int\sin\theta_{l}d\theta_{l}\int\sin\theta_{p}d\theta_{p}pf^{-1}_{rec}\sigma^{Z,W^{\pm}}_{M}[v_{L}R_{L}+v_{T}R_{T}+hv^{\prime}_{T}R^{\prime}_{T}]~{},$
(9)

where $\theta_{l}$ denotes the scattering angle of the lepton.
$\sigma^{Z,W^{\pm}}_{M}$ for NC and CC is defined by

$\sigma^{Z}_{M}={\left({\frac{G_{F}\cos(\theta_{l}/2)E_{f}M_{Z}^{2}}{{\sqrt{2}}\pi(Q^{2}+M^{2}_{Z})}}\right)}^{2}~{}~{},~{}\sigma^{W^{\pm}}_{M}=\sqrt{1-{\frac{M^{2}_{l}}{E_{f}}}}\left({\frac{G_{F}\cos(\theta_{C})E_{f}M_{W}^{2}}{2\pi(Q^{2}+M^{2}_{W})}}\right)^{2}~{}~{},$
(10)

where $M_{Z}$ and $M_{W}$ are the rest mass of $Z$-boson and $W$-boson,
respectively. $\theta_{C}$ denotes the Cabibbo angle given by
$\cos^{2}{\theta_{C}}\simeq 0.9749$. The recoil factor $f_{rec}$ is given as

$f_{rec}={\frac{E_{A-1}}{M_{A}}}\left|1+{\frac{E_{p}}{E_{A-1}}}\left[1-{\frac{{\bf
q}\cdot{\bf p}}{p^{2}}}\right]\right|.$ (11)

Nuclear response functions to the weak current are denoted as $R_{L},R_{T}$
and $R_{T}^{{}^{\prime}}$, whose detailed forms are referred from Ref. kimplb
.

Detailed results of the cross sections on the proton and the neutron target
for $g_{A}^{s}=-0.19$ and $0.0$ are shown in the upper parts of Fig. 1. Cross
sections by the incident $\nu({\bar{\nu}})$ on the proton are usually enhanced
in the whole $Q^{2}$ region by the $g_{A}^{s}$, 15% maximally, while they are
reduced on the neutron. Other quantities related to these cross sections, such
as asymmetries and ratios between $\nu$ and ${\bar{\nu}}$ in the NC reaction,
also show similar effects cheoun08 . Therefore the strangeness effect is not
large enough to be discernible in the cross sections, asymmetries, and ratios
if we recollect the persisting experimental error bars in the BNL data garvey
.

The lowest part of Fig. 1 exhibits the strange quark contributions to the
cross section on
12C$(\nu({\bar{\nu}}),\nu^{{}^{\prime}}({\bar{\nu}^{{}^{\prime}}}))$ reaction.
For directly comparing with the nucleon case, we present them in terms of
outgoing nucleon kinetic energy $T_{N}$ because $Q^{2}=2MT_{P}$ would hold on
the free nucleon inside nuclei. Thick and thin curves are the results for
$g^{s}_{A}=-0.19$ and $0.0$, respectively. Note that a relativistic optical
potential for the final nucleon is taken into account for the FSI clark .
Detailed discussions about the FSI are done at Ref. kim07 . Peak positions for
12C nuclei, which does not show up in the nucleon case, just come from the
binding energy of nucleons inside nuclei.

To analyze cross sections (solid curves) for the $\nu-^{12}$C scattering, we
present separately each contribution via final neutrons (dotted curves) and
final protons (dashed curves) in the figure. The effect of $g^{s}_{A}$ for the
proton is increased by about 30%, but for the neutron it is decreased by 30%,
maximally, independently of the incident energy. These individual $g_{A}^{s}$
effects on each nucleon resemble exactly those of each nucleon in the upper
parts of Fig. 1, that is, the $g_{A}^{s}$ effect enhances the cross section by
protons and decreases that by neutrons.

However, total net $g_{A}^{s}$ effects severely depend on the competition
between the protons and neutrons processes because of the summation of all
nucleons. In the case of 12C, the enhancement by the proton is nearly
compensated by the neutron process, so that the net effect increases the cross
section only below 3% for the given $E_{\nu}$.

For the ${\bar{\nu}}-^{12}$C process, the $g_{A}^{s}$ effect enhances the
cross section by about 20% on the protons, but reduces them by 20% the
neutrons. The reduction by neutrons is nearly balanced by the enhancement due
to protons. Consequently, the net effect of the strange quark reduces also the
cross section only below 2% for the given $E_{\bar{\nu}}$, similarly to the
$\nu$ case.

From these results, in the $\nu-A$ cross sections, the $g^{s}_{A}$ effect
contributes more positively to the proton while it does negatively to the
neutron, and is cancelled out finally. In the case of 12C, the resultant
effects are only within a few % by the cancellation between the final protons
and neutrons. Therefore, the $g_{A}^{s}$ effect in nuclei is too small to be
deduced from the A$(\nu,\nu^{{}^{\prime}})$ cross section.

On the other hand, the cross section for the CC scattering is given with the
following replacement into the NC cross sections of Eqs. (6) and (9)

${({{d\sigma}\over{dQ^{2}}})}_{\nu({\bar{\nu}})}^{CC}={({{d\sigma}\over{dQ^{2}}})}_{\nu({\bar{\nu}})}^{NC}(G_{E}^{V}\rightarrow
G_{E}^{CC},G_{M}^{V}\rightarrow G_{M}^{CC},G_{A}\rightarrow G_{A}^{CC})~{},$
(12)

with
$G_{E}^{CC}=G_{E}^{p}(Q^{2})-G_{E}^{n}(Q^{2})~{},~{}G_{M}^{CC}=G_{M}^{p}(Q^{2})-G_{M}^{n}(Q^{2})$.
Since the relevant form factors are expressed only by the electro-magnetic
form factors the CC scattering is not nearly influenced by the strangeness in
the form factors. Therefore, the ratios of the NC and CC reactions given by

$R_{NC/CC}={\frac{\sigma(\nu,\nu^{\prime}p)}{\sigma(\nu,\mu^{-}p)}}={\frac{\sigma_{NC}^{\nu
p}}{\sigma_{CC}^{\nu}}}~{},{\bar{R}_{NC/CC}}={\frac{\sigma({\bar{\nu}},{\bar{\nu}}^{\prime}n)}{\sigma({\bar{\nu}},\mu^{+}n)}}={\frac{\sigma_{NC}^{{\bar{\nu}}n}}{\sigma_{CC}^{{\bar{\nu}}}}}~{},$
(13)

have been suggested for probing the strangeness on the nucleon or nuclei.
Moreover any possible nuclear structure effects in these reactions are
expected to be cancelled out. Since these ratios are focused on the knocked-
out nucleon, we introduce another definition of the ratios by focusing on the
nucleon inside the target nucleus as follows

$R^{\prime}_{NC/CC}={\frac{\sigma(\nu,\nu^{\prime}n)}{\sigma(\nu,\mu^{-}p)}}={\frac{\sigma_{NC}^{\nu
n}}{\sigma_{CC}^{\nu}}}~{},~{}{\bar{R}}^{\prime}_{NC/CC}={\frac{\sigma({\bar{\nu}},{\bar{\nu}}^{\prime}p)}{\sigma({\bar{\nu}},\mu^{+}n)}}={\frac{\sigma_{NC}^{{\bar{\nu}}p}}{\sigma_{CC}^{{\bar{\nu}}}}}.$
(14)

Since the charge exchange is not included, these $R^{{}^{\prime}}$ and
${\bar{R}}^{{}^{\prime}}$ quantities have the same meaning as
$R_{NC/CC}={\sigma(\nu n\rightarrow\nu n)}/{\sigma(\nu
n\rightarrow\mu^{-}p)},{\bar{R}}_{NC/CC}={\sigma({\bar{\nu}}p\rightarrow{\bar{\nu}}p)}/{\sigma({\bar{\nu}}p\rightarrow{\mu^{+}}n)}$
on nucleon level, namely, $R^{{}^{\prime}}$ and ${\bar{R}}^{{}^{\prime}}$ mean
the ratios on the nucleon inside nuclei bombarded by incident
$\nu({\bar{\nu}})$. Results and relevant discussions for the ratios on the 12C
target are given at Fig. 2. The strangeness effects are not so large than
those of the cross sections at Fig. 1. Divergence of high $T_{P}$ is
associated with the effect of outgoing lepton mass in the CC reaction Albe02 .

In order to find more feasible $g_{A}^{s}$ signals, the difference of the
cross sections between incident $\nu$ and ${\bar{\nu}}$ is suggested as a
candidate for the effectsAlbe02 ; kimplb . Since $h=-1$ $(h=+1)$ in Eqs.(6)
and (9) corresponds to the helicity of the incident $\nu$ (${\bar{\nu}}$), a
difference of the cross sections is summarized as a simple form. For instance,
for the $\nu-N$ scattering, the difference of the cross section is given as

${({{d\sigma}\over{dQ^{2}}})}_{\nu}^{NC}-{({{d\sigma}\over{dQ^{2}}})}_{{\bar{\nu}}}^{NC}=-{{G_{F}^{2}}\over{2\pi}}~{}4y(1-{1\over
2}y)G_{M}^{V}G_{A}~{}.$ (15)

Similar conjectures can be done for the $\nu-A$ scattering from Eq. (9).

Since the difference of the cross sections between $\nu$ and ${\bar{\nu}}$ can
be expressed only by a few form factors, more efficient observable are
possible if we consider asymmetries between $\nu$ and ${\bar{\nu}}$ relative
to the CC reactions. Moreover it enables us to distinguish each strangeness
contribution in the two strangeness effects, i.e. vectorial and axial
strangeness parts at small $Q^{2}$ region

$\displaystyle A_{NC/CC}^{p}$ $\displaystyle=$
$\displaystyle{{\sigma_{NC}^{\nu
p}-\sigma_{NC}^{{\bar{\nu}}p}}\over{(\sigma_{CC}^{\nu
n}-\sigma_{CC}^{{\bar{\nu}}p})}}=0.12-0.12{{g_{A}^{s}}\over{g_{A}}}-0.13{{G_{M}^{s}}\over{G_{M}^{3}}}~{},$
(16) $\displaystyle A_{NC/CC}^{n}$ $\displaystyle=$
$\displaystyle{{\sigma_{NC}^{\nu
n}-\sigma_{NC}^{{\bar{\nu}}n}}\over{(\sigma_{CC}^{\nu
n}-\sigma_{CC}^{{\bar{\nu}}p})}}=0.16+0.16{{g_{A}^{s}}\over{g_{A}}}+0.13{{G_{M}^{s}}\over{G_{M}^{3}}}~{},$

where the third terms with $G_{M}^{s}(Q^{2}=0)=\mu_{s}$,
$G_{M}^{3}={(G_{M}^{p}-G_{M}^{n}})/2$, $\mu_{p}=2.79$, and $\mu_{n}=-1.91$
come from the vectorial part. Since $|G_{M}^{s}/G_{M}^{3}|$ and
$|g_{A}^{s}/g_{A}|$ are approximately 0.2, the strangeness effects from the
vector and axial parts are comparable, in principle. It could be questioned if
Eq.(16) still holds even in the case of a nucleus. But, we assume that it
holds because the outgoing nucleon can be described as quasi-freely bombarded
by the incident $\nu({\bar{\nu}})$.

Here two interesting remarks are possible. One is that $A_{NC/CC}^{p(n)}$
could be a constant if there would be no strangeness effects. The second point
is related to the $\mu_{s}$ sign. If $g_{A}^{s}$ and $G_{M}^{s}$ have
different $(\pm)$ sign, the values of $A_{NC/CC}^{p}$ and $A_{NC/CC}^{n}$
become constants 0.12 and 0.16, respectively, because the last two terms in
Eq. (16) are nearly cancelled out. Any deviations from these constants would
imply that both signs are same. For instance, we presented the case of
$g_{A}^{s}=-0.19$ and $\mu_{s}=-0.4$ showing such a tendency at the upper
panels in Fig. 3. Both nucleon and nuclei cases show nearly the same trend. It
means that $g_{A}^{s}$ effect is nearly evadable from any observable suggested
until now. Divergence of high $T_{P}$ in the nuclear case of Fig. 3 also comes
from the CC reaction in the denominator.

As another way to study the strangeness, we introduce the difference and the
sum of asymmetries

$\displaystyle DA_{NC/CC}$ $\displaystyle=$ $\displaystyle
A_{NC/CC}^{p}-A_{NC/CC}^{n}\simeq-0.04-0.28({{g_{A}^{s}}\over{g_{A}}}+{{G_{M}^{s}}\over{G_{M}^{3}}})~{},$
(17) $\displaystyle SA_{NC/CC}$ $\displaystyle=$ $\displaystyle
A_{NC/CC}^{p}+A_{NC/CC}^{n}=0.28+0.04{{g_{A}^{s}}\over{g_{A}}}~{}.$

The sum of asymmetry in Eq. (17) is given only in terms of the axial part. But
the second term is very small by the factor 0.04, so that the SA is nearly
independent of the strangeness, but the DA depends strongly on the axial
strangeness. Figure 4 represents the calculation for the difference and the
summation of the asymmetries. On the summation, the $g_{A}^{s}$ effect is
negligible, so that the summation of the asymmetry behaves a constant about
0.28 at low and middle kinetic energies, independently of free or bound
nucleons cheoun08 . On the difference, the $g_{A}^{s}$ effects clearly appear
to be larger than any other observable. If the values of $g_{A}^{s}$ and
$\mu_{s}$ have different signs, the DA would be constant, but it depends on
$Q^{2}$ if they have same signs as in Fig. 4.

Finally, we consider the exclusive $\nu-A$ scattering i.e.
$A(\nu,\nu^{{}^{\prime}}N)$ because we expect that the strangeness effects via
each nucleon process can be survived in the exclusive reaction without mutual
cancellations. Our terminology for the exclusive reaction is based on the
concept in the electron scattering kimepja .

In the laboratory frame, the exclusive differential cross section is given by
the contraction between the lepton tensor and the hadron tensor udias ;
kimepja

$\displaystyle{\frac{d^{5}\sigma}{dE_{f}d\Omega_{f}d\Omega_{p}}}$
$\displaystyle=$ $\displaystyle K[v_{L}R_{L}+v_{T}R_{T}+v_{TT}R_{TT}\cos
2\phi_{p}+v_{LT}R_{LT}\cos\phi_{p}$ (18) $\displaystyle+$ $\displaystyle
h(v^{\prime}_{T}R^{\prime}_{T}+v^{\prime}_{LT}R^{\prime}_{LT}\sin\phi_{p})],$

where $\phi_{p}$ denotes the azimuthal angle of the knocked-out proton as
measured with respect to the incident $\nu$ scattering plane ($\bf x-\bf z$
plane). $R_{L}$, $R_{T}$, $R_{TT}$, $R_{LT}$, $R^{\prime}_{T}$, and
$R^{\prime}_{LT}$ are called the longitudinal, transverse, transverse-
transverse, longitudinal-transverse, polarized transverse, and polarized
longitudinal-transverse response functions, respectively. The kinematics
factor $K$ is given by
$K={\frac{M_{N}M_{A-1}p}{(2\pi)^{3}M_{A}}}f^{-1}_{rec}\sigma^{Z}_{M}$.

Our results in Fig. 5, where incident $E_{\nu}=2.4$ GeV is adopted as a JLAB
type’s electron energy, show clearly the strangeness effects, as expected.
However, in the experimental side, this reaction may have actual difficulties
in detecting incident and outgoing neutrinos. However, if one integrates the
scattering angle and averages over the neutrino energies, the five folded
cross section in Eq. (18) becomes a two folded form, that is, the cross
section is going to be the angular distribution of the knocked-out nucleon by
detecting only solid angles of the knocked-out nucleon.

In conclusion, the $g_{A}^{s}$ effect in the elastic $\nu({\bar{\nu}})-N$
scattering for the NC reaction enhances the cross section for the proton, but
reduces it for the neutron. But for the $\nu-A$ scattering in the QE region,
the contributions of both protons and neutrons compensate each other in the
cross section because of the summation over all nucleons, so that the net
effects on the cross sections are nearly indiscernible due to the possible
cancellations. However the asymmetries between incident $\nu$ and
${\bar{\nu}}$ divided by the CC reaction could be a prominent signal for the
strangeness effect. In specific, differences of the asymmetries could be a
feasible quantity for the signal.

Finally the exclusive reaction like
12C$(\nu({\bar{\nu}}),\nu^{\prime}({\bar{\nu}^{{}^{\prime}}})N)$ could be more
efficient tests for the effect rather than the inclusive reaction
12C$(\nu({\bar{\nu}}),\nu^{\prime}({\bar{\nu}^{{}^{\prime}}}))$, because there
are no competitions of the $g_{A}^{s}$ effects for the knocked-out protons and
neutrons in the exclusive reaction. Our results for the exclusive reaction
justify clearly this conjecture.

## Acknowledgements

This work was supported by the Korea Science and Engineering Foundation(KOSEF)
grant funded by the Korea government(MOST) (R01-2007-000-20569) and one of
author, Cheoun, was supported by the Soongsil University Research Fund.

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Figure 1: (Color online) Upper 4 figures are differential cross sections by
the NC scattering on proton (left) and neutron (right), Eq. (6), in a unit
($G_{F}^{2}/9\simeq 6.05\times 10^{-39}~{}[{{cm}^{2}/{(GeV/c)}^{2}}]$), as a
function of $Q^{2}$ for $E_{\nu({\bar{\nu}})}=500$ MeV. They are calculated
for $g_{A}^{s}=-0.19$ and 0.0 cases, respectively. The lowest panel is for
12C($\nu(\bar{\nu}),\nu^{\prime}(\bar{\nu})$) cross section, Eq.(9), as a
function of the knocked out nucleon kinetic energy $T_{p}$ at
$E_{\nu({\bar{\nu}})}=500$ MeV, where left (right) panel is for the $\nu$
($\bar{\nu}$) scattering, respectively. Solid curves are the results for the
cross sections, dashed and dotted lines are the contributions of the proton
and the neutron, respectively. Thick and thin lines are calculations with
$g^{s}_{A}=-0.19$ and $g^{s}_{A}=0.0$. Figure 2: (Color online) Ratios of the
NC to the CC cross sections of the $\nu-A$ scattering for 12C as a function of
the knocked-out nucleon kinetic energy. For the NC reaction, solid (red)
curves represent the results with $g_{A}^{s}=0.0$, dashed (black) lines are
with $g_{A}^{s}=-0.19$.

Figure 3: (Color online) Asymmetries $A_{NC/CC}^{p}$ and $A_{NC/CC}^{n}$,
Eq.(16), between the NC and CC reactions for an incident
$E_{\nu({\bar{\nu}})}$ = 500 MeV. They are calculated for $g_{A}^{s}=-0.19$
(dashed curve) and 0.0 cases (solid curve), respectively. Upper panel is for
the nucleon, while lower panel is for ${}^{12}C$.

Figure 4: (Color online) The differences and summations of the asymmetries
between the NC and CC cross sections of ${\nu-N}$ (upper part) and ${\nu-A}$
scattering for 12C target (lower parts) as a function of the knocked-out
nucleon kinetic energy. Solid (red) curves represent the results with
$g_{A}^{s}=0.0$, dashed (black) lines are with $g_{A}^{s}=-0.19$. Figure 5:
(Color online) Neutral current 16O($\nu,\nu^{\prime}p$) cross sections as a
function of the missing momentum at $E_{\nu}=2.4$ GeV. Solid (red) curves are
the results for $g_{A}^{s}=0.0$ and dashed (black) lines are for
$g_{A}^{s}=-0.19$, respectively. Thick and thin lines are the results for the
incident $\nu$ and ${\bar{\nu}}$. States of the outgoing nucleon, $p_{1/2}$
and $p_{3/2}$, are indicated, respectively.