Grok on Proton Superfluid Model

Comparison of Resonance Models

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### Introduction

In particle physics, resonances are excited states of particles observed as peaks in scattering experiments. This page compares two models for predicting the masses of two specific resonances: the Delta resonance (Δ(1232)) and the N(1440) resonance (Roper resonance).

### Proton Radius Calculation

The quantized superfluid model calculates the proton radius using:

\[ R_p = \frac{\hbar n}{m_p c} = 4 \times \frac{\hbar}{m_p c} \]

With \( \hbar c = 197.32698 \, \text{MeV·fm} \) and \( m_p c^2 = 938.272 \, \text{MeV} \), this yields:

\[ R_p = 4 \times \frac{197.32698}{938.272} \approx 0.8412 \, \text{fm} \]

The measured proton charge radius is approximately \( 0.841 \, \text{fm} \), showing excellent agreement.

### Quantized Superfluid Model

This model views the proton as a quantized superfluid with a characteristic speed \( v = c \) (speed of light) and a quantum number \( n = 4 \). Resonance masses are calculated as:

\[ m_n c^2 = \frac{n}{4} m_p c^2 \]

For the Delta resonance (\( n = 5 \)) and Roper resonance (\( n = 6 \)), masses are derived from the proton mass \( m_p c^2 = 938.272 \, \text{MeV} \).

### Accepted Quark Model

The quark model describes baryons as three-quark systems. Resonance masses are predicted using parameters like constituent quark masses and coupling constants, fitted to experimental data. Here, we use typical literature values for comparison.

### Comparison Table

Resonance	Measured Mass (MeV)	Superfluid Model Mass (MeV)	Quark Model Mass (MeV)	Superfluid % Error	Quark % Error
Delta (n=5)	1232		1232		0.00
Roper (n=6)	1440		1400

### Key Notes

The quantized superfluid model predicts the proton radius with high accuracy without tuning parameters. It also approximates the Delta and Roper resonance masses with percent errors of approximately -4.8% and -2.3%, respectively.

The quark model, however, relies on parameter fitting. For instance, the Delta mass is often fitted to match the measured 1232 MeV (0% error), but this leads to a larger error for the Roper resonance (approximately -2.8% here).

The superfluid model's ability to predict these values without adjustments suggests it may capture fundamental aspects of proton structure and resonance behavior, offering a compelling alternative to the parameter-dependent quark model.