Summary - Spectral Broadening Due To Two Proton Mixing

# Analysis of Spectral Broadening in Proton-Proton Collision Measurements ## Introduction In proton-proton collision measurements, the spectrum reflects the interaction of two resonating protons, rather than a single proton. This interaction introduces a mixing or modulation effect that broadens the spectral bands compared to the spectrum of an individual proton. This report analyzes the harmonic spectral broadening caused by the coupling of two protons and recomputes key aspects of the spectrum, incorporating the quantized circular superfluid model where applicable. Equations and explanatory comments are provided throughout. --- ## 1. Mechanism of Spectral Broadening The broadening of spectral bands in proton-proton collisions arises from the interaction between the two protons, which can be modeled as coupled harmonic oscillators. ### Coupled Oscillator Model - **Single Proton Resonance**: Each proton has an intrinsic resonance frequency, \( \omega_0 \), representing its internal energy transitions or oscillations. - **Coupling Effect**: When two protons interact, their resonance frequencies split due to a coupling constant \( \kappa \), resulting in two normal modes: \[ \omega_+ = \omega_0 + \kappa \] \[ \omega_- = \omega_0 - \kappa \] where \( \kappa \) depends on the strength of the interaction (e.g., proximity or energy of the collision). - **Spectral Width**: The observed spectral band spans from \( \omega_- \) to \( \omega_+ \), with a width of: \[ \Delta \omega = \omega_+ - \omega_- = 2\kappa \] **Comment**: This model simplifies the protons as harmonic oscillators. In reality, their quantum nature and internal structure add complexity, but the principle of coupling-induced broadening remains valid. --- ## 2. Quantum Mechanical Perspective The two-proton system forms a combined quantum state, increasing the number of possible transitions and thus broadening the spectrum. - **Energy Levels**: For a single proton, energy levels might be \( E_n \), where \( n \) is a quantum number. In a two-proton system, the energy levels become: \[ E_{n_1, n_2} = E_{n_1} + E_{n_2} + V \] where \( V \) is the interaction potential energy, introducing shifts and splittings. - **Transition Frequencies**: Transitions between these levels produce frequencies such as: \[ \omega_{n_1, n_2 \to m_1, m_2} = \frac{E_{n_1, n_2} - E_{m_1, m_2}}{\hbar} \] The variety of combinations (e.g., \( \omega_{n_1} \pm \omega_{n_2} \)) results in a denser and broader spectrum. **Comment**: This is analogous to molecular spectroscopy, where additional degrees of freedom (e.g., vibrational modes) broaden spectral bands compared to atomic spectra. --- ## 3. Superfluid Model Recompute In the quantized circular superfluid model, a single proton’s properties are described by a quantum number \( n \), with \( n = 4 \) matching the proton’s radius. For two protons, we adapt the model to account for their interaction. ### Single Proton - **Circulation Quantization**: \[ \oint v \cdot dl = \frac{n h}{m_p} \] For a circular path, \( v = c \) (speed of light, as a limiting velocity in the model), and the radius is: \[ r_n = \frac{n h}{2 \pi m_p c} \] For \( n = 4 \), \( m_p = 938 \, \text{MeV}/c^2 \), \( h = 4.1357 \times 10^{-15} \, \text{eV·s} \), \( c = 3 \times 10^8 \, \text{m/s} \): \[ r_4 = \frac{4 \times 4.1357 \times 10^{-15}}{2 \pi \times 938 \times 1.602 \times 10^{-19} \times 3 \times 10^8} \approx 0.841 \, \text{fm} \] - **Resonance Frequency**: The frequency associated with this state is: \[ \omega_n = \frac{E_n}{\hbar}, \quad E_n \propto \frac{n h c}{r_n} \] ### Two-Proton System - **Combined System**: If the superfluid flow encompasses both protons (e.g., a resonant state), the effective quantum number may increase, or the circulation path adjusts. Assume a di-proton-like resonance with separation \( d \), modifying the path length. - **Modified Radius**: For a combined system, the effective radius might scale with the interaction distance, but for simplicity, consider the spectrum as a combination of individual frequencies: \[ \omega_{\text{combined}} = \omega_{n_1} \pm \omega_{n_2} \] - **Broadening**: The spectrum includes lines at sums and differences of individual frequencies, e.g., \( \omega_4 + \omega_4 \), \( \omega_4 - \omega_3 \), widening the band. **Comment**: The exact path for two protons is complex (e.g., figure-eight flow), but the key is that the combined system increases spectral complexity and width. --- ## 4. Harmonic Spectral Broadening in Collisions In high-energy proton-proton collisions, additional factors amplify the broadening: - **Resonance Width**: Short-lived states have a natural width: \[ \Gamma = \frac{\hbar}{\tau} \] where \( \tau \) is the lifetime. For a resonance with \( \tau \sim 10^{-23} \, \text{s} \): \[ \Gamma \approx \frac{6.582 \times 10^{-16}}{10^{-23}} \approx 65.8 \, \text{MeV} \] - **Convolution Effect**: The combined spectrum approximates a convolution of individual spectra, \( S(\omega) * S(\omega) \), spreading the energy distribution. **Comment**: In practice, experimental factors like detector resolution also contribute, but the two-proton interaction is the fundamental cause of harmonic broadening. --- ## 5. Final Recomputed Spectrum - **Single Proton Spectrum**: Narrow lines at \( \omega_n \). - **Two-Proton Spectrum**: Broader bands spanning \( \omega_n - \kappa \) to \( \omega_n + \kappa \) for each resonance, plus combination frequencies. - **Width Estimate**: If \( \kappa \approx 10-100 \, \text{MeV} \) (depending on collision energy), bands widen by \( 2\kappa \approx 20-200 \, \text{MeV} \). **Comment**: This aligns with observed spectra in high-energy physics, where resonance peaks (e.g., W, Z bosons) show significant widths. --- ## Conclusion The critical fact that proton collision measurements involve two resonating protons leads to spectral broadening through: - Coupling-induced frequency splitting (\( 2\kappa \)). - Increased transition possibilities in the combined quantum system. - Harmonic modulation in the superfluid model, producing combination frequencies. This recomputation confirms that the spectral bands are wider than for a single proton, consistent with experimental observations in collision spectra.