_{1}

^{*}

We propose a new description of a nucleon as a pair of pions. The baryon number of our description of nucleon is not 1 but 0. However, this is probable because the proton spin crisis shows that the baryon spin cannot tell the number of composing quarks anymore. Because we use the derived pion wave function to describe a nucleon, our description has automatically the pionic degree of freedom and can be compared to the constituent quark model. Using this description, we investigate the electric charge and magnetization density functions of protons and neutrons. The electric charge density function of neutron is quite similar to those of Galster model and Maints data except the magnitude of singularity. The density functions of proton also show the similar behavior as those of Kelly’s except near origin. Taking the Fourier transform of the density functions, we obtain the Sachs electromagnetic form factors that can be compared to those in the parametrization derived by
*Ye et al*.

The proton and neutron electromagnetic form factors (e.m. FFs) are key components for understanding the charge and magnetization distributions within nucleons. In the past 20 years, a new generation of experiments, frequently utilizing polarization of freedom, has provided new knowledge regarding our understanding of the form factors [^{2}), which does not fit the data [

To clarify, we list here the symbols and parameters.

ρ c h [ P ] , ρ m [ P ] : electric charge and magnetization density functions of proton.

ρ c h [ N ] , ρ m [ N ] : electric charge and magnetization density functions of neutron.

ρ ˜ c h [ P ] , ρ ˜ m [ P ] : proton intrinsic FFs that are Fourier transform of electric charge and magnetization density functions.

ρ ˜ c h [ N ] , ρ ˜ m [ N ] : neutron intrinsic FFs that are Fourier transform of electric charge and magnetization density functions.

n P , n N : parameters of the power of exponentials of proton and neutron these are used for both density functions and Sachs e.m. FFs.

m P , m N : parameters of the coefficient of r or | Q → | 2 .

β : range parameter for density functions.

p: parameter of the coefficient of logarithmic term of Sachs e.m. FFs.

To date, there have been several proposed descriptions for a nucleon. One of these is the pion cloud model, which introduces an elementary, perturbative pion couple to the constituent quark model (CQM) such that chiral symmetry is restored. Noting the fact that the contribution of quark spins to the spin of a proton is small, i.e., the proton spin crisis [

The Bethe-Salpeter-like amplitude of the hadronic operator applied in this paper is defined as

χ ( 1 , 2 ) = 〈 0 | q ( 1 , 2 ) | P 〉 (1)

where | 0 〉 and | P 〉 denote a vacuum and the physical state, respectively.

The gauge-invariant bi-local operator q ( 1 , 2 ) is defined in the non-Abelian gauge field as

q ( 1 , 2 ) = T r c q β + ( 2 ) P exp ( i g ∫ 1 2 d x → A → a ( x ) λ a 2 ) q α ( 1 ) (2)

here α and β denote the Dirac indices, P denotes the path ordering, and the λ a 2 components are generators of the adjoint representation of the SU (N). color gauge group. The trace is calculated for color spin a. Suura first proposed this definition [

The basic concept of our evaluation is as follows.

Because baryons are represented as initially binding meson pairs as described in subsection 5, at | Q → | 2 = 0 and for the small | Q → | 2 case, the π + − π 0 pair has the same origin. For simplicity, both of quarks are in the same position. However, when | Q → | 2 is large, both π + and π 0 gradually move freely with respect to each other and both π + and π 0 move totally independent of each other, which is as the same concept as asymptotic freedom.

q ( 1 ; 2 , 3 ) = 1 2 [ ( T r c q † ( 2 ) P exp ( i g ∫ 1 2 d x → A → a ( x ) λ a 2 ) q ( 1 ) ) ( π + ) + ( T r c q † ( 3 ) P exp ( i g ∫ 1 3 d x → A → a ( x ) λ a 2 ) q ( 1 ) ) ( π 0 ) ] + 1 2 [ ( T r c q † ( 2 ) P exp ( i g ∫ 1 2 d x → A → a ( x ) λ a 2 ) q ( 1 ) ) ( π 0 ) + ( T r c q † ( 3 ) P exp ( i g ∫ 1 3 d x → A → a ( x ) λ a 2 ) q ( 1 ) ) ( π + ) ] (3)

At first ( | Q → | 2 = 0 ), the π + − π 0 pair is described as

where ( π + ) , ( π 0 ) means that each described hadronic operator corresponds to π + and π 0 , respectively.

Note that Equation (3) promises the evenly charged wave function and that the total factor 1 2 keeps proton charge at +e, and that the positions of both π + and π 0 quarks are at the same point ( | Q → | 2 = 0 ).

The latter 1 2 part is essentially same as the former 1 2 part when we do not consider the charge distribution. Because our concern is the wave function of the π + − π 0 system, the Bethe-Salpeter-like amplitude can be defined as

χ ( 1 ; 2.3 ) = 〈 0 | q ( 1 ; 2.3 ) | P 〉 (4)

where

q ( 1 ; 2 , 3 ) = q ( 1 , 2 ) + q ( 1 , 3 )

here, we dropped the factor 1 2 for simplicity. When considering charge distribution function, we consider this factor again.

Then equation of motion of q ( 1 ; 2 , 3 ) becomes

i ∂ ∂ t q ( 1 ; 2 , 3 ) = i ∂ ∂ t q ( 1 , 2 ) + i ∂ ∂ t q ( 1 , 3 ) = − i α → ⋅ ∇ → ( 2 ) q ( 1 , 2 ) − q ( 1 , 2 ) i α → ⋅ ∇ → ( 1 ) + g ∫ 1 2 d x → q E → ( 1 , 2 ; x ) + g α → ⋅ ∫ 1 2 d x → × q B → ( 1 , 2 ; x ) − i α → ⋅ ∇ → ( 3 ) q ( 1 , 3 ) − q ( 1 , 3 ) i α → ⋅ ∇ → ( 1 ) + g ∫ 1 3 d x → q E → ( 1 , 3 ; x ) + g α → ⋅ ∫ 1 3 d x → × q B → ( 1 , 3 ; x ) (5)

q o → ( 1 , s ; x ) = q + ( s ) U ( s , x ) O → a λ a 2 U ( x , 1 )

where

U ( s , 1 ) ≡ P exp ( i g ∫ 1 s d x → A → a ( x ) λ a 2 )

α k = γ 0 γ k

O is any operator, s = 2 , 3 and γ μ is γ is matrices.

We previously obtained the equation of i ∂ ∂ t q ( 1 , s ) [

Because we consider every quark mass to be zero, the center of mass coordinate and two relative coordinates can be written

G → = r → 1 + r → 2 + r → 3 3 r → ( 1 ) = r → 2 − r → 1 , r → ( 2 ) = r → 3 − r → 1 (6)

Then

r → 1 = G → − 1 3 r → ( 1 ) − 1 3 r → ( 2 ) r → 2 = G → + 2 3 r → ( 1 ) − 1 3 r → ( 2 ) r → 3 = G → − 1 3 r → ( 1 ) + 2 3 r → ( 2 ) (7)

Thus, the derivatives are

∇ → ( 1 ) = ∂ ∂ r → 1 = 1 3 ∂ ∂ G → − ∂ ∂ r → ( 1 ) − ∂ ∂ r → ( 2 ) ∇ → ( 2 ) = 1 3 ∂ ∂ G → + ∂ ∂ r → ( 1 ) ∇ → ( 3 ) = 1 3 ∂ ∂ G → + ∂ ∂ r → ( 2 ) (8)

Remembering that α → ⋅ ∇ → ( s ) = α l ∂ ∂ r s l ( s = 1 , 2 , 3 ), the kinetic term becomes

− i α → ⋅ ∇ → ( 2 ) q ( 1 , 2 ) − q ( 1 , 2 ) i α → ⋅ ∇ → ( 1 ) = − i 3 { α l ∂ ∂ G l , q ( 1 , 2 ) } + − i [ α l ∂ ∂ r ( 1 ) l , q ( 1 , 2 ) ] + q ( 1 , 2 ) i α l ∂ ∂ r ( 2 ) l (9)

Similarly

− i α → ⋅ ∇ → ( 3 ) q ( 1 , 3 ) − q ( 1 , 3 ) i α → ⋅ ∇ → ( 1 ) = − i 3 { α l ∂ ∂ G l , q ( 1 , 3 ) } + − i [ α l ∂ ∂ r ( 2 ) l , q ( 1 , 3 ) ] + q ( 1 , 3 ) i α l ∂ ∂ r ( 1 ) l (10)

We consider the gauge field string only for the straight line case. Thus, the hadronic operator q ( 1 , 2 ) is decomposed in the relative coordinate system as

q ( 1 , 2 ) = q 0 ( r ( 1 ) ) + i α → ⋅ r ^ ( 1 ) q 1 ( r ( 1 ) ) + β q 2 ( r ( 1 ) ) + β ( i α → ⋅ r ^ ( 1 ) ) q 3 ( r ( 1 ) ) (11)

where r ( 1 ) = | r → ( 1 ) |

Because q 0 ( r ( 1 ) ) = 0 , q 1 ( r ( 1 ) ) = 0 (as we previously showed [

− β i α l ∂ ∂ r ( 2 ) l i α m x ( 1 ) m r ( 1 ) = β α l α m ∂ ∂ x l ( 2 ) ( x m ( 1 ) r ( 1 ) ) = 0 (12)

∂ ∂ x ( 2 ) l q s ( r ( 1 ) ) = 0 ( s = 2 , 3 ) (13)

here, we denote that r ( p ) l = x l ( p ) ( p = 1 , 2 ), so that r ( 1 ) = x 1 ( 1 ) 2 + x 2 ( 1 ) 2 + x 3 ( 1 ) 2

Thus,

q ( 1 , 2 ) i α l ∂ ∂ r ( 2 ) l = 0 (14)

Similarly

q ( 1 , 3 ) i α l ∂ ∂ r ( 1 ) l = 0 (15)

Therefore, the kinetic terms in the relative coordinate system become

− i α → ⋅ ∇ → ( 2 ) q ( 1 , 2 ) − q ( 1 , 2 ) i α → ⋅ ∇ → ( 1 ) − i α → ⋅ ∇ → ( 3 ) q ( 1 , 3 ) − q ( 1 , 3 ) i α → ⋅ ∇ → ( 1 ) = − i [ α l ∂ ∂ r ( 1 ) l , q ( 1 , 2 ) ] − i [ α l ∂ ∂ r ( 2 ) l ] (16)

The integral terms in relative coordinate system become

g ∫ 1 2 d x → q E → ( 1 , 2 ; x ) = − g 2 2 ∫ 0 r ( 1 ) d z q ( r ( 1 ) − z ) ( r ( 1 ) − z ) q ( z ) (17)

g α → ⋅ ∫ 1 2 d x → × q B → ( 1 , 2 ; x ) = g 2 2 ∫ − ∞ t d t ′ ( α → ⋅ r ^ ( 1 ) ) δ ( t − t ′ ) ∫ 0 r ( 1 ) d z q ( t ′ , r ( 1 ) − z ) q ( t ′ , z ) (18)

g ∫ 1 3 d x → q E → ( 1 , 3 ; x ) = − g 2 2 ∫ 0 r ( 2 ) d z q ( r ( 2 ) − z ) ( r ( 2 ) − z ) q ( z ) (19)

g α → ⋅ ∫ 1 3 d x → × q B → ( 1 , 3 ; x ) = g 2 2 ∫ − ∞ t d t ′ ( α → ⋅ r ^ ( 2 ) ) δ ( t − t ′ ) ∫ 0 r ( 2 ) d z q ( t ′ , r ( 2 ) − z ) q ( t ′ , z ) (20)

We obtained these equations were obtained previously [

Thus, the equation of motion for q ( 1 ; 2 , 3 ) is expressed by the following independent equations in the relative coordinate system.

i ∂ ∂ t q ( r ( 1 ) ) = − i [ α → ⋅ ∇ → r ( 1 ) , q ( r ( 1 ) ) ] − g 2 2 ∫ 0 r ( 1 ) d z q ( r ( 1 ) − z ) ( r ( 1 ) − z ) q ( z ) + g 2 2 ∫ − ∞ t d t ′ ( α → ⋅ r ^ ( 1 ) ) δ ( t − t ′ ) ∫ 0 r ( 1 ) d z q ( t ′ , r ( 1 ) − z ) q ( t ′ , z ) (21)

i ∂ ∂ t q ( r ( 2 ) ) = − i [ α → ⋅ ∇ → r ( 2 ) , q ( r ( 2 ) ) ] − g 2 2 ∫ 0 r ( 2 ) d z q ( r ( 2 ) − z ) ( r ( 2 ) − z ) q ( z ) + g 2 2 ∫ − ∞ t d t ′ ( α → ・ r ̂ ( 2 ) ) δ ( t − t ′ ) ∫ 0 r ( 2 ) d z q ( t ′ , r ( 2 ) − z ) q ( t ′ , z ) (22)

Thus, the WF of the π + − π 0 ( π − ) pair, χ p a i r , is described as

χ p a i r = c 1 χ π ( r ( 1 ) ) + c 2 χ π ( r ( 2 ) ) (23)

where

χ π ( r ) = c o n s t e − g 2 L 1 8 r 2 r (24)

We obtained the pion WF (Equation (24)) in our prior analysis [

Therefore, the magnetization density function and the basis of the electric charge density function of a proton are described as

ρ m a g [ P ] = ( d 1 m ) 2 [ e − g 2 L 1 8 ( r ( 1 ) ) 2 r ( 1 ) + e − g 2 L 1 8 ( r ( 2 ) ) 2 r ( 2 ) ] 2 (25)

Basis ρ c h [ P ] = ( d 2 e ) 2 [ e − g 2 L 1 8 ( r ( 1 ) ) 2 r ( 1 ) + e − g 2 L 1 8 ( r ( 2 ) ) 2 r ( 2 ) ] 2 (26)

here, we combine the latter part of Equation (3) so that the correct magnetization and charge of proton are obtained in Equation (25) and Equation (26). This is because we dropped the factor 1 2 in the derivation.

By definition, r → ( 1 ) and r → ( 2 ) have the same origin. Thus, considering the direction of momentum Q → to be the z-axis, r → ( 1 ) and r → ( 2 ) can be expressed by polar coordinates as

r → ( 1 ) = ( r ( 1 ) , θ 1 , ϕ 1 ) , r → ( 2 ) = ( r ( 2 ) , θ 2 , ϕ 2 ) (27)

Denoting the angle between r → ( 1 ) and r → ( 2 ) as θ and r → ( 1 ) to r → , that is, considering r → ( 1 ) as r → , the magnetization density function of a proton is written

ρ m [ P ] = ( d 1 m ) 2 ∫ 0 π sin θ 1 d θ 1 ∫ 0 2 π d ϕ 1 [ e − g 2 L 1 8 r 2 r + e − g 2 L 1 8 ( r cos θ ) 2 r cos θ ] 2 (28)

where cos θ = cos θ 2 cos θ 1 + sin θ 2 sin θ 1 cos ( ϕ 2 − ϕ 1 )

Taking integration to eliminate the θ 2 and ϕ 2 dependence, the actual form of ρ m a g can be written as

ρ m [ P ] = ( d 1 m ) 2 [ 4 π e − g 2 L 1 4 r 2 r 2 + ∫ 0 π d θ 2 ∫ 0 2 π d ϕ 2 ∫ 0 π sin θ 1 ∫ 0 2 π d ϕ 1 ( 2 e − g 2 L 1 8 r 2 ( 1 + ( cos θ ) 2 ) r 2 + e − g 2 L 1 4 ( r cos θ ) 2 r 2 ) ] (29)

To obtain the electric charge density function of a proton ρ c h [ P ] , we need more careful consideration because of the asymptotic freedom as mentioned in sec 2-3. As | Q → | 2 becomes larger, r is smaller, ρ c h [ P ] becomes the summation of two independent | χ π ( r ) | 2 terms.

To be precise, ρ c h [ P ] behaves as

1) as r → 0 ,

ρ c h [ P ] → ( d 2 e ) 2 2 ( 4 π ) e − g 2 L 1 4 r 2 r 2 (30)

2) as r → ∞ ,

ρ c h [ P ] → ( d 1 m ) 2 [ 4 π e − g 2 L 1 4 r 2 r 2 + ∫ 0 π d θ 2 ∫ 0 2 π d ϕ 2 ∫ 0 π sin θ 1 ∫ 0 2 π d ϕ 1 ( 2 e − g 2 L 1 8 r 2 ( 1 + ( cos θ ) 2 ) r 2 + e − g 2 L 1 4 ( r cos θ ) 2 r 2 ) ] (31)

The angular integrations become

∫ 0 π d θ 2 ∫ 0 2 π d ϕ 2 ∫ 0 π sin θ 1 ∫ 0 2 π d ϕ 1 e − g 2 L 1 8 r 2 ( 1 + ( cos θ ) 2 ) r 2 cos θ = 4 π 3 e − g 2 L 1 8 r 2 r 2 (32)

∫ 0 π d θ 2 ∫ 0 2 π d ϕ 2 ∫ 0 π sin θ 1 ∫ 0 2 π d ϕ 1 e − g 2 L 1 4 ( r cos θ ) 2 ( r cos θ ) 2 = 4 π 3 e − g 2 L 1 4 r 2 r 2 (33)

where cos θ = cos θ 2 cos θ 1 + sin θ 2 sin θ 1 cos ( ϕ 2 − ϕ 1 )

Thus, the magnetization density function and the electric charge density function of a proton are represented by

ρ m [ P ] = ( d 1 m ) 2 4 π [ ( 1 + 3 π 2 ) e − g 2 L 1 4 r 2 r 2 ] (34)

ρ c h [ P ] = ( d 2 e ) 2 4 π [ 2 e − g 2 L 1 4 r 2 r 2 exp ( − ( r 2 m P ) n P ) + [ ( 1 + 3 π 2 ) e − g 2 L 1 4 ( r + β ) 2 ( r + β ) 2 ] ( 1 − exp ( − ( r 2 m P ) n P ) ) ] (35)

In Equation (35), the exp ( − ( r 2 m P ) n P ) term shows at what radius asymptotic freedom begins and we treat the m P , n P and β values as parameters.

For a neutron, we consider that it is constructed of a π + − π − pair as mentioned in sec. 2-1. Because π − is an antiparticle of π + , the WF of π − can be considered to be the same as that of π + . Therefore, the basis of the electric charge density function of a neutron is represented as

Basis ρ c h [ N ] = ( h 2 ) 2 [ e e − g 2 L 1 8 ( r ( 1 ) ) 2 r ( 1 ) + ( − e ) e − g 2 L 1 8 ( r ( 2 ) ) 2 r ( 2 ) ] 2 = ( h 2 e ) 2 [ e − g 2 L 1 8 ( r ( 1 ) ) 2 r ( 1 ) − e − g 2 L 1 8 ( r ( 2 ) ) 2 r ( 2 ) ] 2 (36)

Using the same consideration for the vectors r → ( 1 ) and r → ( 2 ) as that of a proton, the basis of the electric charge density function becomes

Basis ρ c h [ N ] = ( h 2 e ) 2 [ 4 π e − g 2 L 1 4 r 2 r 2 − ∫ 0 π d θ 2 ∫ 0 2 π d ϕ 2 ∫ 0 π sin θ 1 ∫ 0 2 π d ϕ 1 ( 2 e − g 2 L 1 8 r 2 ( 1 + ( cos θ ) 2 ) r 2 cos θ − e − g 2 L 1 4 ( r cos θ ) 2 ( r cos θ ) 2 ) ] = ( h 2 e ) 2 4 π [ ( 1 − π 2 ) e − g 2 L 1 4 r 2 r 2 ] (37)

For the magnetization density function of a neutron ρ m [ N ] , the form is the same as the basis of the electric charge density function, but positive. Then ρ m [ N ] is represented as

ρ m [ N ] = ( h 1 m ) 2 4 π [ ( π 2 − 1 ) e − g 2 L 1 4 r 2 r 2 ] (38)

For the electric charge density function of neutron, we again consider asymptotic freedom. At large r (small | Q → | 2 ), they move with the same origin, but, at small r (large | Q → | 2 ), π + and π − move independently.

To be precise,

3) as r → 0 ,

ρ c h [ N ] → ( h 2 e ) 2 ( 4 π ) 2 e − g 2 L 1 4 r 2 r 2 (39)

4) as r → ∞ ,

ρ c h [ N ] → ( h 2 e ) 2 4 π [ ( 1 − π 2 ) e − g 2 L 1 4 r 2 r 2 ] (40)

Using the same expression resulting from asymptotic freedom for the proton case, ρ c h [ N ] is represented by

ρ c h [ N ] = ( h 2 e ) 2 4 π [ 2 e − g 2 L 1 4 r 2 r 2 exp ( − ( r 2 m N ) n N ) + [ ( 1 − π 2 ) e − g 2 L 1 4 ( r + β ) 2 ( r + β ) 2 ] ( 1 − exp ( − ( r 2 m N ) n N ) ) ] (41)

where m N , n N and β are parameters.

To evaluate the Sachs e.m. FFs of protons and neutrons, i.e., G E [ P ] , G M [ P ] , G E [ N ] and G M [ N ] , we adopt the following relations proposed by Mitra and Kumari [

ρ ˜ c h ( k ) = ( 1 + τ ) 2 G E ( | Q → | 2 ) (42)

μ ( i ) ρ ˜ m ( k ) = ( 1 + τ ) 2 G M ( | Q → | 2 ) (43)

where τ = | Q → | 2 ( 2 M ) 2 , i = P or N ( μ P and μ N are the magnetic moment of a proton and a neutron, respectively).

and ρ ˜ ( k ) s are the Fourier transform of the electric charge and magnetization density functions of a nucleon.

Under relativistic consideration, the relationship between k 2 and | Q → | 2 is

k 2 = | q → | 2 → | Q → | 2 1 + τ

and for the nonrelativistic case, the relationship between k = | q → | and | Q → | is k = | q → | → | Q → | .

We derived the electric charge and magnetization density functions in sec 2-3 (a) so that in principle, we only need to take the Fourier Transforms to obtain the Sachs e.m. FFs.

For the magnetization density functions, we can use the Fourier transform directly. However, for the electric charge density functions, we cannot use the exact transformations because that the rigorous Fourier transform cannot reflect the asymptotic freedom characteristics in momentum space. Thus, in the electric charge density function case, we take the Fourier transform of the basis of the electric charge density functions and express the asymptotic freedom in momentum space by adopting a description similar to that used in the configuration space. We then use the relation of Equation (42) to obtain the Sachs FFs of G E . The electric charge density functions of protons and neutrons were given in sec. 2-3 (a) as

Basis ρ c h [ P ] = ( d 2 e ) 2 [ e − g 2 L 1 8 ( r ( 1 ) ) 2 r ( 1 ) + e − g 2 L 1 8 ( r ( 2 ) ) 2 r ( 2 ) ] 2 (44)

Basis ρ c h [ N ] = ( h 2 e ) 2 [ e − g 2 L 1 8 ( r ( 1 ) ) 2 r ( 1 ) − e − g 2 L 1 8 ( r ( 2 ) ) 2 r ( 2 ) ] 2 (45)

Note that, other than the proportional constants, the only difference between Basis ρ c h [ P ] and Basis ρ c h [ N ] is the sign.

Considering again the direction of longitudinal momentum Q → to be the z axis and considering the polar coordinates r → ( 1 ) = ( r ( 1 ) , θ 1 , ϕ 1 ) and r → ( 2 ) = ( r ( 2 ) , θ 2 , ϕ 2 ) , and again considering r → ( 1 ) as r → , the Fourier transform of Basis ρ c h [ P ] and Basis ρ c h [ N ] can be expressed as

Basis ρ ˜ c h [ P , N ] ( | q → | = k ) = c o n s t 2 π ∫ 0 ∞ r 2 d r ∫ 0 π sin θ 1 d θ 1 e − i | q → | r cos θ 1 × [ ∓ e − g 2 L 1 4 r 2 r 2 + ∫ 0 π d θ 2 ∫ 0 2 π d ϕ ( 2 e − g 2 L 1 8 r 2 ( 1 + ( cos θ ) 2 ) r 2 cos θ ∓ e − g 2 L 1 4 ( r cos θ ) 2 ( r cos θ ) 2 ) ] (46)

where cos θ = cos θ 2 cos θ 1 + sin θ 2 sin θ 1 cos ( ϕ 2 − ϕ 1 ) .

For the second line, we take ϕ 2 − ϕ 1 = ϕ .

The first term of Equation (46) becomes

Firstterm = c o n s t 2 π ∫ 0 ∞ r 2 d r ∫ 0 π sin θ 1 d θ 1 e − i | q → | r cos θ 1 e − g 2 L 1 4 r 2 r 2 = c o n s t 4 π π 2 g 2 L 1 exp ( − | q | 2 g 2 L 1 ) F ( 1 ; 3 2 ; | q → | 2 g 2 L 1 ) = c o n s t 2 π π g 2 L 1 F π (47)

where F ( α ; β ; z ) is the Kummer’s confluent hypergeometric series.

We showed this integral result previously [

The second and third terms of Equation (46) become

Fouriercosineofthesecondterm = 2 c o n s t ∫ 0 π d θ 2 ∫ 0 2 π d ϕ 2 π ∫ 0 ∞ r 2 d r ∫ 0 π sin θ 1 d θ 1 cos ( | q → | r cos θ 1 ) e − g 2 L 1 8 r 2 ( 1 + ( cos θ ) 2 ) r 2 cos θ = 2 c o n s t π 2 π 2 π g 2 L 1 exp ( − | q → | 2 g 2 L 1 ) (48)

Fouriersineofsecondterm = 2 c o n s t ∫ 0 π d θ 2 ∫ 0 2 π d ϕ 2 π ∫ 0 ∞ r 2 d r ∫ 0 π sin θ 1 d θ 1 sin ( | q → | r cos θ 1 ) e − g 2 L 1 8 r 2 ( 1 + ( cos θ ) 2 ) r 2 cos θ = 2 c o n s π 4 π 2 g 2 L 1 | q → | 2 exp ( − | q → | 2 g 2 L 1 ) F ( 1 2 ; 3 2 ; | q → | 2 g 2 L 1 ) (49)

Fourier cosine of the third term = c o n s t ∫ 0 π d θ 2 ∫ 0 2 π d ϕ 2 π ∫ 0 ∞ r 2 d r ∫ 0 π sin θ 1 d θ 1 cos ( | q → | r cos θ 1 ) e − g 2 L 1 4 r 2 ( cos θ ) 2 ( r cos θ ) 2 = 2 c o n s t π 2 π 2 π g 2 L 1 exp ( − | q → | 2 g 2 L 1 ) (50)

Fourier sine og third term = 0 (51)

Therefore, the Fourier transform of the basis of the electric charge density function of protons and neutrons becomes

Basis ρ ˜ c h [ P ] ( | q → | ) = c o n s t 2 π π g 2 L 1 [ F π + 4 π 3 2 g 2 L 1 | q → | 2 exp ( − | q → | 2 g 2 L 1 ) F ( 1 2 ; 3 2 ; | q → | 2 g 2 L 1 ) + p 1 | Q → | [ log 2 | q → | 2 g 2 L 1 − exp ( − 2 | q → | 2 g 2 L 1 ) ∑ n = 1 ∞ 1 n ! ( ∑ r = 1 n 1 r ) ( 2 | q → | 2 g 2 L 1 ) n ] + 4 π 2 exp ( − | q → | 2 g 2 L 1 ) ] (52)

Basis ρ ˜ c h [ N ] ( | q → | ) = c o n s t 2 π π g 2 L 1 | − F π + 4 π 3 2 g 2 L 1 | q → | 2 exp ( − | q → | 2 g 2 L 1 ) F ( 1 2 ; 3 2 ; | q → | 2 g 2 L 1 ) + p 1 | q → | [ log 2 | q → | 2 g 2 L 1 − exp ( − 2 | q → | 2 g 2 L 1 ) ∑ n = 1 ∞ 1 n ! ( ∑ r = 1 n 1 r ) ( 2 | q → | 2 g 2 L 1 ) n ] + 4 π 2 exp ( − | q → | 2 g 2 L 1 ) | (53)

Thus,

Basis ρ ˜ c h [ P , N ] ( | q → | ) = c o n s t 2 π π g 2 L 1 [ ∓ F π + 4 π 3 2 g 2 L 1 | q → | 2 exp ( − | q → | 2 g 2 L 1 ) F ( 1 2 ; 3 2 ; | q → | 2 g 2 L 1 ) + p 1 | q → | [ log 2 | q → | 2 g 2 L 1 − exp ( − 2 | q → | 2 g 2 L 1 ) ∑ n = 1 ∞ 1 n ! ( ∑ r = 1 n 1 r ) ( 2 | q → | 2 g 2 L 1 ) n ] + 4 π 2 exp ( − | q → | 2 g 2 L 1 ) ] (54)

where p is parameter.

Ye et al. used relativistic considerations for their parametrization work [

To be precise, this situation is described as

ρ ˜ c h [ P ] ( | Q → | ) = c o n s t 2 π π g 2 L 1 F π ( 1 + ( ( z 1 + z 2 + z 3 ) / F π ) / ( exp ( − ( | Q → | 2 m P ) n P ) + ( 1 − exp ( − ( | Q → | 2 m P ) n P ) ) z 1 + z 2 + z 3 F π ) ) (55)

ρ ˜ c h [ N ] ( | Q → | ) = c o n s t 2 π π 3 2 g 2 L 1 F π ( 1 − z 3 / F π + ( ( z 1 + z 2 + z 3 ) F π − 1 ) / ( exp ( − ( | Q → | 2 m N ) n N ) + ( 1 − exp ( − ( | Q → | 2 m N ) n N ) ) ( ( z 1 + z 2 + z 3 ) F π − 1 ) ) ) (56)

where

z 1 = 4 π 3 2 g 2 L 1 | Q → | 2 exp ( − | Q → | 2 g 2 L 1 ) F ( 1 2 ; 3 2 ; | Q → | 2 g 2 L 1 ) (57)

z 2 = p 1 | Q → | 2 [ log 2 | Q → | 2 g 2 L 1 − exp ( − 2 | Q → | 2 g 2 L 1 ) ∑ n = 1 ∞ 1 n ! ( ∑ r = 1 n 1 r ) ( 2 | Q → | 2 g 2 L 1 ) n ] (58)

z 3 = 4 π 2 exp ( − | Q → | 2 g 2 L 1 ) (59)

where p of z 2 is parameter.

These expressions are not exactly Fourier transforms of the electric charge density functions. However, because the basis of ρ ˜ c h ( k ) is exactly the Fourier transform of the basis of ρ c h ( r ) , we use the relationship between Sachs e.m. FFs and intrinsic FFs shown in Equation (42) and Equation (43) to obtain in the Sachs e.m. FFs as

G E [ P ] = 1 ( 1 + τ ) 2 F π ( 1 + ( ( z 1 + z 2 + z 3 ) / F π ) / ( exp ( − ( | Q → | 2 m P ) n P ) + ( 1 − exp ( − ( | Q → | 2 m P ) n P ) ) z 1 + z 2 + z 3 F π ) ) (60)

G E [ N ] = 1 ( 1 + τ ) 2 F π ( 1 − z 3 / F π + ( ( z 1 + z 2 + z 3 ) F π − 1 ) / ( exp ( − ( | Q → | 2 m N ) n N ) + ( 1 − exp ( − ( | Q → | 2 m N ) n N ) ) ( ( z 1 + z 2 + z 3 ) F π − 1 ) ) ) (61)

where z 1 , z 2 , z 3 are given in Equation (57) to Equation (59), and τ is given as

τ = | Q → | 2 ( 2 M ) 2 (62)

here, M is the characteristic mass and it is taken as a parameter.

The relationships between the magnetization density functions and the Sachs e.m. FFs, i.e., G M [ P ] , G M [ N ] , are exactly formulated by their Fourier transform using Equation (43).

Then, we obtain

G M [ P ] = 1 ( 1 + τ ) 2 ( F π [ 1 + ( z 1 + z 2 + z 3 ) / F π ] ) (63)

G M [ N ] = 1 ( 1 + τ ) 2 F π [ − 1 + ( z 1 + z 2 + z 3 ) / F π ] (64)

where τ is the same as Equation (59) and z 1 , z 2 and z 3 are given in Equation (57), Equation (58), and Equation (59).

Note that our Sachs e.m. FFs have normalization uncertainty. To compare our values with the parametrization results of Ye et al. [

Using Equation (34), Equation (35), Equation (38) and Equation (41), we show the magnetization and electric charge density functions of protons and neutrons in

To confine the sizes of protons and neutrons less than 1.2 fm, we chose the Gaussian parameter to be 3.5 (GeV^{2}). Using this value, we obtain the characteristic mass g 2 L 1 2 of 1025 (MeV), which is similar to the ϕ meson mass. This is different from the pion mass of 140 MeV that we use to evaluate Sachs’ proton and neutron e.m. FFs later in this paper.

^{2}) do not behave exactly like the density functions of Kelly [

However, Kelly’s density functions were obtained by using the relativistic inversion method, which is adopted for preventing them from showing the cusp at origin. To be clear this point, our electric charge density function of neutron in

Using Equation (60) and Equation (63) with appropriate normalization, we show the results of G M [ P ] / G D , G E [ P ] / G D and G E [ P ] / G M [ P ] in Figures 4-6. Our evaluation forms for Sachs e.m. FFs are not appropriate to show the behavior of the form FFs in the region where | Q → | 2 is smaller than 10^{−1} (GeV^{2}). However, they are sufficiently applicable in the region where | Q → | 2 is larger than 10^{−1} (GeV^{2}). Thus we can compare our results to the parametrization results in Ye et al. [^{−1} (GeV^{2}). In particular, we obtain a fairly good result for G E [ P ] / G M [ P ] and also it is quite similar to that of CQM by Miller shown in Arrington [^{2}) as we expected.

Using Equation (61) and Equation (64) with appropriate normalization, we show the results of G M [ N ] / G D , G E [ N ] / G D and G E [ N ] / G M [ N ] in Figures 7-9. The magnetization FFs for both protons and neutrons have very similar features to

those resulting from parametrization. However, the values of G E [ N ] / G D in the region of 10^{−1} to 10^{0} (GeV^{2}) are larger than those from the parametrization. Thus, our G E [ N ] / G M [ N ] shows a faster rising form than it does in other studies [

We investigate the proton and neutron electromagnetic form factors where the consideration of that nucleon is described as a pion pair. We obtain a good agreement of the electric density function of neutron with Galster model and Maints data except the magnitude of singularity. The density functions of proton also show a similarity to those of Kelly’s except near origin. In the case of Sachs e.m. FFs, we obtain a fary good agreement with the parametrization results in Ye et al. Therefore, we consider that our description of a nucleon as a pion pair is one of the meaningful aspects.

As mentioned in conclusion section, we obtain fairy good results in both density functions and Sachs e.m. FFs, however, there is an ambiguous point in our treatment. We do not exactly know the reason why the density functions and the form factors for the magnetization case do not change the form when two pions move independently each other with asymptotic freedom. This may occur because the magnetization arises not as a result of charge distribution, but because of current or spin. Thomas [

The author declares no conflicts of interest regarding the publication of this paper.

Kurai, T. (2020) Proton and Neutron Electromagnetic Form Factors Based on Bound System in 3 + 1 Dimensional QCD. Journal of Modern Physics, 11, 741-765. https://doi.org/10.4236/jmp.2020.115048

Here we show that Gell-mann Nishijima relation still holds under baryon number 0 case.

For mesons, Isospin I, component of Isospin I_{3}, strangeness S are given as

Because Gell-mann Nishijima relation is Q = I 3 + 1 2 ( B + S ) (B is baryon number and S is strangeness), this relation holds for meson case because of B = 0.

Reminding the fact that field theory shows the duality, we have to add up negative charge of proton p − , Ξ + , Ξ ∗ + , Ω + to the baryon list. Then using values of

Then it is easy to notice that Gell-mann Nishijima relation also holds for baryon case under the baryon number B = 0. This means that baryon number 1 is not necessary.

The verification of the meson pair of each baryons shown in

particle | π + | π 0 | π − | f 0 | η 0 | k + | k − |
---|---|---|---|---|---|---|---|

I | 1 | 1 | 1 | 0 | 0 | 1 2 | 1 2 |

I_{3} | 1 | 0 | −1 | 0 | 0 | 1 2 | − 1 2 |

strangeness | 0 | 0 | 0 | 0 | 0 | 1 | −1 |

particle | antiparticle | Meson pair | I | I_{3} | Strangeness |
---|---|---|---|---|---|

p + | p − | π + + π 0 | 1 | 1 | 0 |

n 0 ( p 0 ) | n 0 ( p 0 ) | π + + π − | 1 | 0 | 0 |

p − | p + | π − + π 0 | 1 | −1 | 0 |

Λ 0 | Λ 0 | π + + k − or π − + k + | 0 | 0 | 0 |

Σ + | Σ − | π + + η 0 | 1 | 1 | 0 |

Σ 0 | Σ 0 | π 0 + η 0 | 1 | 0 | 0 |

Σ − | Σ + | π − + η 0 | 1 | −1 | 0 |

Ξ + | Ξ − | π 0 + k + | 1 2 | 1 2 | 1 |

Ξ − | Ξ + | π 0 + k − | 1 2 | − 1 2 | −1 |

Ξ 0 | Ξ 0 | π + + k − or π − + k + | 0 | 0 | 0 |

Note that we use total sum for Λ 0 and Ξ 0 cases.

particle | antiparticle | Meson pair | I | I_{3} | strangeness |
---|---|---|---|---|---|

Δ + | Δ − | π + + f 0 | 1 | 1 | 0 |

Δ 0 | Δ 0 | π 0 + f 0 | 1 | 0 | 0 |

Δ − | Δ + | π − + f 0 | 1 | −1 | 0 |

Σ ∗ + | Σ ∗ − | π + + η 0 | 1 | 1 | 0 |

Σ ∗ 0 | Σ ∗ 0 | π 0 + η 0 | 1 | 0 | 0 |

Σ ∗ − | Σ ∗ + | π − + η 0 | 1 | −1 | 0 |

Ξ ∗ + | Ξ ∗ − | η 0 + k + | 1 2 | 1 2 | 1 |

Ξ ∗ − | Ξ ∗ + | η 0 + k − | 1 2 | − 1 2 | −1 |

Ξ ∗ 0 | Ξ ∗ 0 | k + + k − | 0 | 0 | 0 |

Ω + | Ω − | η 0 + k + | 1 2 | 1 2 | 1 |

Ω − | Ω + | η 0 + k − | 1 2 | − 1 2 | −1 |