Proton Soliton Model

Introduction to Solitons

What are Solitons?

Self-reinforcing wave packets that maintain their shape during propagation.

Context

Arise in nonlinear systems, such as superfluids.

Relevance to Protons

Modeling the proton as a soliton provides a novel way to explain its structure and stability.

Quantized Vortices in Superfluids

Key Concept

Superfluids support vortices with quantized circulation.

Equation for Circulation

\[ \oint \mathbf{v} \cdot d\mathbf{l} = \frac{n h}{m} \]

- \( n \): Quantum number (integer)

- \( h \): Planck's constant

- \( m \): Mass of the particle/field constituent

Velocity Field

\[ v = \frac{n h}{m 2\pi r} \]

- \( v \): Velocity at radius \( r \)

Modeling the Proton as a Soliton

Proton as a Vortex

The proton is modeled as a soliton (vortex) in a relativistic superfluid.

Characteristic speed: \( v = c \) (speed of light).

Radius Derivation

Using \( m = m_p \) (proton mass) and \( n = 4 \):

\[ r = n \frac{\hbar}{m_p c} \]

- \( \hbar = \frac{h}{2\pi} \)

- Compute: \( r = 4 \times \frac{\hbar}{m_p c} \approx 4 \times 0.2104 \, \text{fm} \approx 0.84 \, \text{fm} \)

Significance

Matches the experimental proton charge radius (~0.84 fm).

Properties of the Soliton Proton

Radius

\( r \approx 0.84 \, \text{fm} \) (from quantized circulation)

Mass

Soliton energy corresponds to proton rest mass:

\[ E = m_p c^2 \]

Charge

Potentially arises from:

• Topological charge of the soliton.
• Coupling to an additional gauge field.

Spin

Possible origins:

• Angular momentum of the vortex.
• Internal field dynamics.

Magnetic Moment

May result from:

• Current distribution within the soliton structure.

Conclusion

Model Success

Accurately predicts the proton radius using quantized circulation in a superfluid framework.

Provides a unified approach to understanding the proton's mass.

Potential

Offers mechanisms to explain charge, spin, and magnetic moment.

Future Directions

Refine the model to fully account for all proton properties.

Explore interactions with other particles (e.g., neutrons, electrons).