1. IntroductionIn 1951, Panofsky et al. [1] reported on the parities of pions from the reaction of 
and
. The reaction between proton (p) and charged pion (
) is
In the general two-body system, the parity formula is
where Pa and Pb are the intrinsic parities, l is the angular momentum. The traditional proton, neutron, and meson’s parity values in common use are +1, +1, and –1, respectively. The parity is a multiplicative quantum number. Since the initial state (
) is S state (l = 0) and the angular momentum is 0, the initial state’s parity is
The final state’s parity is
It was considered that the two parities are same by the conservation of parity.
In 1954, Chinowsky and Steinberger [2] obtained the parity value (–1) of 
from the absorption of negative pions in deuterium. In 1959, Plano et al. [3] determined the parity value (–1) of neutral pion (
). However the quark numbers by the traditional formula are not add up between the initial state and the final state. It is thought of as a serious problem.
By the way, the two different decays were found for the positively charged strange mesons [3]:
The charged kaon (K+) [4] used to be called 
and
. Both the 
and 
particles were supposed to be two different particles, but they are the same particles. The two final states have different parities, and it is known as the 
puzzle [5-8].
In this paper, the parities of the pentaquark proton, deuteron, and neutral pion are re-searched based on the hypothesis of conservation of particle number [9]. It is attested to the conservation of parity for the 
puzzle.
2. Results and Discussion2.1. The Parity of <img src="3-1220025\2ae52c1b-1c1b-4d9e-91ce-b8535977da6f.jpg" /> for the <img src="3-1220025\f8506ce3-c5b3-4871-8242-8a6add8fb8b6.jpg" /> ReactionThe traditional formula and quark contents by experiment [1,3] are
The numbers of down and anti-down quarks, d and d-bar, are not add up between the left-hand member and the right-hand member. It is necessary for the adjustment of particle numbers. To adjust their member, the pentaquark proton (
) is adopted [9]. According to the hypothesis of conservation of particle number, the above formulae for 
are as indicated below.
where the {(
), (n)} is a pentaquark proton (
), the quark content is {u, d-bar} {u, d, d} [9]. The pentaquark proton’s parity (
) is –1, since the parity formula is
The initial state’s parity is
By the conservation of parity,
The new parity value of 
is +1, not –1 in common use.
2.2. The Parity of <img src="3-1220025\5a9550ef-362c-4477-8df6-c517dd1d09bc.jpg" /> for the <img src="3-1220025\6a19da1e-d526-4fb9-8293-5210a4697400.jpg" /> ReactionThe numbers of down and anti-down quarks are not add up between the left-hand member and the right-hand member by experiment [1].
It is necessary for the adjustment of particle numbers. To adjust their member, the pentaquark proton (
) is adopted [9]. According to the hypothesis of conservation of particle number, the above formulae for 
are as indicated below. The reaction between the deuteron (
) and charged pion (
) is
where the {(
),(n)} is the deuteron(D′) [9].
The deuteron’s parity (PD′) is –1, since the parity formula is
The initial state’s parity is
The final state’s parity is
By the conservation of parity,
The new parity value of 
in this reaction is also +1.
2.3. The Parity of <img src="3-1220025\2318c3c9-4c5e-4373-a0db-50eded80d86f.jpg" /> ReactionThe reaction between the 
and 
by experiment [10] is
This reaction was not considered by Aoki [9]. The formula for the quark content
of the traditional proton is
However the numbers of down and anti-down quarks are not add up between the left-hand member and the right-hand member. The parity is not conserved. The formula of the left-hand parity is
The formula of the right-hand parity is:
To adjust their member, the pentaquark proton (p′) is adopted [9].
The formula of the left-hand parity is
The formula of the right-hand parity is
The parity is conserved by the hypothesis of conservation of particle number.
2.4. The Validation of Conservation of Parity for the <img src="3-1220025\a58f80a2-a748-4ce9-8619-709468fecf1a.jpg" /> PuzzleIn the 
puzzle [5-8], the two parities of 
and 
have same value (–1), since the positively charged Kaon (K+) is a meson. The K+ was shown as follows by Aoki [9].
where the anti-strange quark (s-bar) is the composite particle consisting of the anti-down quark(d-bar) and
, the 
is the 
and 
pair, the 
is the neutrino-antineutrino pair, and the 
is the muonneutrino-antimuonneutrino pair. The parity of 
is +1. The 
may be the two photons.
The 
and 
masses, lifetimes, and spins are no difference with each other. In the traditional expression, the parities are
where l is the angular momentum between 
and
, L is the angular momentum between 
and the center of
,
.
The two final states have the different parities, +1 and –1. The parity for 
is not conserved. It is considered as the parity violation in weak interactions. However the parity is conserved as follows by the new parity value (+1) of
.
The neutral pions of both initial states were added by the hypothesis of conservation of particle number. The 
for 
is corresponding to the
. The initial state’s parities for 
and 
are –1.
The final state’s parity for 
is –1.
The final state’s parity for 
is –1.
), are –1, –1, and +1, respectively.
puzzle.