In 1935 Hideki Yukawa presented an argument that there was an inverse relationship between the mass of the particle carrying a force and the range of that force. Specifically he concluded from his analysis that if m_U is the mass of the particle carrying the force field U and ρ_U is its range then

m_Uc = hρ_U
 

where c is the speed of light and h is Planck's constant. This is the Yukawa relation.

His analysis was based upon the hypothesis that the potential function for the field is of the form

U(r) = ±g²e^−λr/r
 

where g is a constant and λ is the reciprocal of the range ρ_U.

Yukawa noted that the potential U=±g²/r satisfies the wave equation

(∇² - (1/c²)∂²/∂t²)U = 0.
 

The potential function he postulated, U=±g²e^-λr/r, satisfies the equation:

(∇² - (1/c²)∂²/∂t² - λ²)U = 0.
 

where ∇² is the Laplacian operator.

From this Yukawa defined the mass of the particle associated with the field U as m_U such that

m_Uc = λh
 

A similar relationship based upon Heisenberg's Uncertainty Principle was developed by G.C. Wick in Nature in 1938. The Uncertainty Principle in this case applied to the canonical conjugate coordinates of time and energy:

ΔEΔt ≥ h/2π
 

In Wick's analysis the uncetainty in time Δt is the time required for light to traverse the range of the nuclear force r, which corresponds to 1/λ in Yukawa's analysis; i.e., Δt = r/c. The uncertainty of energy ΔE is the mass-energy of the particle, m_Uc². Thus, according to Wick's argument,

m_Uc²(r/c) = h/2π
or m_Uc = (h/2π)/r = (h/2π)λ
 

which is essentially the same as Yukawa's relation although the difference of the factor of 2π is significant for empirical verification.

From a very rough estimate of the range of the nuclear force being the scale of atomic nuclei he concluded that the particle carrying the nuclear had a mass about 200 times that of an electron. Subsequently such particles were found and named the π mesons, now sometimes called pions. There are three π meson; the positive, the negative and the neutral versions. The π mesons decay rather quickly.

Elsewhere it is argued that the decay of the force-carrying particle for the nuclear force leads to a negative exponential factor in the force formula that represents the survival factor for the particle as a function of distance from the nucleon that generates it. Thus Yukawa's applying a negative exponential factor to the potential function was an error. However though this was an error it gave a result that was a reasonable approximation of the correct potential function at large distances from the nucleon. Nevertheless there is some uncertainty as to whether the Yukawa relation is precisely the relationship between the mass of the force-carrying particle and its range scale.

The true potential function is of the form:

V(r) = −∫^∞_rH*(exp(-λz)/z²)dz
 

where H* is a constant.

This potential function satisfies the wave equation

∇²V = λ(∂V/∂r)
rather than as for the Yukawa potential
∇²U = λ²U.
 

Suppose the general wave equation is

∇²V = f(V, r)
 

What Yukawa discovered is that if f(V,r) is represented as series of the form (ξV + other terms) then the particle for the field satisfies the relation

m_Vc = hξ^½
 

The coefficient ξ would be in a Taylor's series expansion ∂f/∂V. In the case of the true potential presented above f(V,r)=λ(∂V/∂r). It is feasible to carry out the differentiation of this function with respect to V but first consider an exercise that does not require so much heavy-duty calculus. Consider the ratio of the function f(V,r) to V; i.e., let

ξ* = λ(∂V/∂r)/V(r)
 

and consider the limits of ξ* as r→∞ and as r→0. For r→∞ both the numerator and denominator of the ratio go to zero. Therefore L'Hospital's Rule may be applies and the ratio of their derivatives examined.

(∂V/∂r) = −H*e^−λr/r²
and
(∂²V/∂r²) = H*λe^−λr/r² + 2H*e^−λr/r³
= (λ + 2/r)H*e^−λr/r²
 

Therefore

lim_r→∞ ξ* = lim_r→∞ [λ(λ + 2/r)] = λ²
 

This means that for large r the same relationship between particle mass and the scale parameter λ applies for the true potential function as applies for the Yukawa potential. There the Yukawa relation is valid for large distances from the nucleons.

The behavior of ξ* as r→0 can also be examined. Both the numerator and denominator of the ratio go to infinity as r goes to zero. Therefore L'Hospital's Rule can also be applied. Thus

lim_r→0 ξ* = lim_r→0 [λ(λ + 2/r)] = ∞
for λ>0
 

However because of the finite size of the nucleons r cannot actually go to zero. The expression [λ(λ + 2/r)] might be considered an approximation of ξ. Thus

ξ ≅ [λ(λ + 2/r)] = λ²[1 + 2/(λr)] = λ²[1 + 2ρ/r]
 

where ρ=1/λ.

From this the extension of Yukawa's relation for small values of r would be

m_Vc ≅ hλ[1 + 2ρ/r]^½
 

This would mean that there is a proximity increment to the mass of the force-carrying particle when it is still close to the nucleon that generates it. The proximity increment to the mass when r=2ρ would be about 41 percent.

Now consider how ∂(∂V/∂r)/∂V may be evaluated. It is given by

∂(∂V/∂r)/∂V = − [∂(∂V/∂r)/∂r]/(∂V/∂r)
which evaluates to
∂(∂V/∂r)/∂V = − [H*e^−λr/r²](−(λ+2/r))/[H*e^−λr/r²]
= (λ+2/r)
 

Therefore, in fact,

ξ = λ(λ+2/r)
 

Thus the extension of Yukawa's relation is

m_Vc = hλ[1 + 2ρ/r]^½
 

(To be continued.)