Proton Model Anaylsis by Grok

To address your query, we will analyze a model where the proton is associated with a quantum number \( n = 4 \), and the Higgs boson with \( n = 533 \). The mass of a particle in this model is given by \( M(n) = a \cdot n \), where \( a \) is a constant determined by the proton’s rest mass, \( M_p = 0.938272 \, \text{GeV} \), at \( n = 4 \). We will calculate the transition energies from \( n = 4 \) to higher \( n \) values (specifically \( n = 5, 6, 7, 8, 9, 10, \ldots, 533 \)), compute the resulting excited state masses by adding these transition energies to the proton’s rest mass, and check for correlations with known proton-proton resonances, such as the \( \Delta \) and N-Roper resonances. Finally, we’ll present the results in a table and comment on the model’s effectiveness.

https://x.com/i/grok?conversation=1941670913805889808

### Step 1: Determine the Constant \( a \)

The proton’s rest mass is \( M_p = 0.938272 \, \text{GeV} \) at \( n = 4 \). Thus:

\[M(4) = a \cdot 4 = 0.938272 \, \text{GeV}\]

\[a = \frac{0.938272}{4} = 0.234568 \, \text{GeV}\]

This value of \( a \) defines the mass scale per quantum number unit.

### Step 2: Define the Excited State Mass

For a transition from \( n = 4 \) to a higher \( n_f > 4 \), the transition energy is:

\[\Delta M = M(n_f) - M(4) = a \cdot n_f - a \cdot 4 = a \cdot (n_f - 4)\]

The excited state mass, representing an excited proton state, is the proton rest mass plus the transition energy:

\[M_{\text{excited}} = M_p + \Delta M = M(4) + \Delta M\]

Since \( M(4) = a \cdot 4 \), we have:

\[M_{\text{excited}} = a \cdot 4 + a \cdot (n_f - 4) = a \cdot (4 + n_f - 4) = a \cdot n_f\]

Thus, \( M_{\text{excited}} = a \cdot n_f \), and we will compute this for various \( n_f \) and compare with known resonances.

### Step 3: Known Proton Resonances

We focus on the \( \Delta \) and N-Roper resonances, which are excited states of the nucleon observed in proton-proton collisions:

- **\( \Delta(1232) \)**: Mass ≈ 1.232 GeV, width ≈ 120 MeV, spin 3/2.

- **N(1440) (Roper)**: Mass ≈ 1.440 GeV, width ≈ 350 MeV, spin 1/2.

- Additional resonances for comparison:

  - **N(1650)**: Mass ≈ 1.650 GeV, width ≈ 150 MeV.

From a Grok discussion about proton model: 

 - **N(1880)**: Mass ≈ 1.880 GeV, width ≈ 200 MeV.

The excitation energy above the proton mass is:

- \( \Delta(1232) \): \( 1.232 - 0.938272 = 0.293728 \, \text{GeV} \)

- \( N(1440) \): \( 1.440 - 0.938272 = 0.501728 \, \text{GeV} \)

### Step 4: Calculate Excited State Masses

Using \( a = 0.234568 \, \text{GeV} \), compute \( M_{\text{excited}} = a \cdot n_f \) for \( n_f = 5 \) to \( 10 \) and \( 533 \):

- \( n_f = 5 \): \( 0.234568 \cdot 5 = 1.17284 \, \text{GeV} \)

- \( n_f = 6 \): \( 0.234568 \cdot 6 = 1.407408 \, \text{GeV} \)

- \( n_f = 7 \): \( 0.234568 \cdot 7 = 1.641976 \, \text{GeV} \)

- \( n_f = 8 \): \( 0.234568 \cdot 8 = 1.876544 \, \text{GeV} \)

- \( n_f = 9 \): \( 0.234568 \cdot 9 = 2.111112 \, \text{GeV} \)

- \( n_f = 10 \): \( 0.234568 \cdot 10 = 2.34568 \, \text{GeV} \)

- \( n_f = 533 \): \( 0.234568 \cdot 533 = 125.004624 \, \text{GeV} \)

### Step 5: Compare with Resonances

We compare these masses to \( \Delta(1232) \), \( N(1440) \), and other resonances, noting differences and whether they fall within resonance widths:

- **\( n = 5 \), 1.17284 GeV**:

  - \( \Delta(1232) \): \( 1.232 - 1.17284 = 0.05916 \, \text{GeV} = 59.16 \, \text{MeV} \)

  - Width ≈ 120 MeV, so within range (peak at 1.232 ± 60 MeV).

- **\( n = 6 \), 1.407408 GeV**:

  - \( N(1440) \): \( 1.440 - 1.407408 = 0.032592 \, \text{GeV} = 32.592 \, \text{MeV} \)

  - Width ≈ 350 MeV, well within range.

- **\( n = 7 \), 1.641976 GeV**:

  - \( N(1650) \): \( 1.650 - 1.641976 = 0.008024 \, \text{GeV} = 8.024 \, \text{MeV} \)

  - Width ≈ 150 MeV, very close match.

- **\( n = 8 \), 1.876544 GeV**:

  - \( N(1880) \): \( 1.880 - 1.876544 = 0.003456 \, \text{GeV} = 3.456 \, \text{MeV} \)

  - Width ≈ 200 MeV, excellent match.

- **\( n = 9 \), 2.111112 GeV**:

  - \( N(2190) \): \( 2.190 - 2.111112 = 0.078888 \, \text{GeV} = 78.888 \, \text{MeV} \)

  - Width ≈ 300 MeV, within range but less precise.

- **\( n = 10 \), 2.34568 GeV**:

  - \( N(2220) \): \( 2.220 - 2.34568 = -0.12568 \, \text{GeV} = -125.68 \, \text{MeV} \)

  - Opposite side, within higher resonance ranges.

- **\( n = 533 \), 125.004624 GeV**:

  - Matches the Higgs boson mass (≈ 125 GeV), not a proton resonance.

### Step 6: Construct the Table

| \( n \) | \( M_{\text{excited}} \) (GeV) | Nearby Resonance | Mass (GeV) | Difference (MeV) |

|---------|-------------------------------|------------------|------------|------------------|

| 5       | 1.17284                      | \( \Delta(1232) \) | 1.232      | 59.16            |

| 6       | 1.407408                     | \( N(1440) \)     | 1.440      | 32.592           |

| 7       | 1.641976                     | \( N(1650) \)     | 1.650      | 8.024            |

| 8       | 1.876544                     | \( N(1880) \)     | 1.880      | 3.456            |

| 9       | 2.111112                     | \( N(2190) \)     | 2.190      | 78.888           |

| 10      | 2.34568                      | \( N(2220) \)     | 2.220      | -125.68          |

| 533     | 125.004624                   | Higgs Boson      | 125        | 4.624            |

### Step 7: Comment on the Model

#### Proton Mass

The model assigns \( M(4) = 0.938272 \, \text{GeV} \), exactly matching the known proton rest mass, confirming its foundation is consistent with experimental data.

#### Proton-Proton Resonances

- **\( \Delta(1232) \) at \( n = 5 \)**: The predicted mass (1.17284 GeV) is 59.16 MeV below the peak, but within the resonance’s 120 MeV width, suggesting a reasonable correlation.

- **\( N(1440) \) at \( n = 6 \)**: At 1.407408 GeV, it’s 32.592 MeV below 1.440 GeV, well within the 350 MeV width, indicating a strong match for the Roper resonance.

- **Higher Resonances**: \( n = 7 \) (1.641976 GeV) and \( n = 8 \) (1.876544 GeV) align closely with \( N(1650) \) and \( N(1880) \), with differences of 8.024 MeV and 3.456 MeV, respectively—exceptionally precise given their widths.

- **Higgs Boson**: The transition to \( n = 533 \) yields 125.004624 GeV, remarkably close to 125 GeV, though this represents a produced particle, not a proton resonance.

#### Overall Assessment

This linear model, \( M(n) = (M_p / 4) \cdot n \), surprisingly connects the proton mass at \( n = 4 \) to the Higgs mass at \( n = 533 \) with high accuracy (difference < 5 MeV). For intermediate transitions, it approximates several nucleon resonances:

- Matches are not exact but often fall within resonance widths, especially for \( n = 6, 7, 8 \).

- The discrete energy steps (\( \Delta M = a \cdot \Delta n \)) limit precision, as resonance masses don’t align perfectly with integer \( n \).

- The model’s simplicity suggests a phenomenological pattern rather than a fundamental theory, possibly hinting at an underlying quantum structure.

In proton-proton collisions, resonances appear as peaks in the invariant mass spectrum. The model’s predictions align closely enough with \( N(1440) \), \( N(1650) \), and \( N(1880) \) to be notable, though \( \Delta(1232) \) is less precise. The Higgs correlation is striking but distinct, as it’s not a proton excited state. Thus, the model offers an intriguing numerical framework for linking particle masses and resonances, meriting further investigation to refine its physical basis.