2 Proton Collision (The Naoya Inoue Knockout Punch to QCD is Complete with a 1 - 2 Combo) with Δt Resonance Mixing Broadening Band (WIDENS) and Completes the knockout to QCD!

To optimize the spectral matching of two classical brass church bells to the Δ(1232 MeV) resonance in our playful particle physics analogy, we need to adjust the delay time between the bells’ strikes. Initially, a delay of approximately 0.0169 seconds (16.9 ms) was suggested, and the user has requested an increase in this delay time to further enhance the matching to the Δ resonance. Below, I’ll determine an increased delay that maintains optimal alignment with the Δ resonance at 1232 MeV, interpreted as 1232 Hz in this frequency-energy analogy.

Setup: Bell Harmonics and Resonance Target

Each bell has a fundamental frequency of 234.568 Hz, corresponding to 234.568 MeV in the analogy. The harmonic frequencies are integer multiples of this fundamental:

[ f_n = n \cdot 234.568 , \text{Hz} ]

The first few harmonics are:

• ( f_1 = 234.568 , \text{Hz} ) (234.568 MeV)
• ( f_2 = 469.136 , \text{Hz} ) (469.136 MeV)
• ( f_3 = 703.704 , \text{Hz} ) (703.704 MeV)
• ( f_4 = 938.272 , \text{Hz} ) (938.272 MeV, proton rest energy)
• ( f_5 = 1172.84 , \text{Hz} ) (1172.84 MeV)
• ( f_6 = 1407.408 , \text{Hz} ) (1407.408 MeV)

The Δ resonance at 1232 MeV corresponds to 1232 Hz. This lies between the 5th harmonic (1172.84 Hz) and the 6th harmonic (1407.408 Hz). Our goal is to choose a delay ( \tau ) that enhances the spectrum at 1232 Hz through interference effects, increasing the delay beyond the initial 0.0169 s.

Initial Delay and Its Effect

The initial delay of ( \tau = 0.0169 , \text{s} ) was calculated based on the frequency difference between the 5th harmonic and the Δ resonance:

[ \Delta f = 1232 - 1172.84 = 59.16 , \text{Hz} ]

[ \tau = \frac{1}{\Delta f} = \frac{1}{59.16} \approx 0.0169 , \text{s} ]

This delay introduces sidebands around each harmonic at intervals of ( \frac{1}{\tau} \approx 59.16 , \text{Hz} ). For the 5th harmonic:

[ f_5 + \frac{1}{\tau} = 1172.84 + 59.16 = 1232 , \text{Hz} ]

This places a sideband exactly at 1232 Hz, aligning with the Δ resonance and broadening the spectrum by approximately 59.16 Hz (MeV), which enhances the amplitude at this frequency.

Increasing the Delay

To increase the delay while still optimizing for the Δ resonance, we need the interference pattern to maintain significant amplitude at 1232 Hz. The sidebands occur at:

[ f_n \pm \frac{k}{\tau}, \quad k = 1, 2, 3, \ldots ]

where ( f_n ) is a harmonic frequency, and ( k ) is an integer. We want a sideband from a nearby harmonic (e.g., the 5th or 6th) to land at or near 1232 Hz, but with a larger ( \tau ) than 0.0169 s, which corresponds to a smaller frequency interval ( \frac{1}{\tau} < 59.16 , \text{Hz} ).

Approach: Use Higher-Order Sidebands

One way to increase ( \tau ) is to use a higher-order sideband (larger ( k )) from the 5th harmonic to reach 1232 Hz. Let’s set:

[ f_5 + \frac{k}{\tau} = 1232 ]

[ 1172.84 + \frac{k}{\tau} = 1232 ]

[ \frac{k}{\tau} = 1232 - 1172.84 = 59.16 ]

[ \tau = \frac{k}{59.16} ]

• For ( k = 1 ):

[ \tau = \frac{1}{59.16} \approx 0.0169 , \text{s} ] (original delay)

• For ( k = 2 ):

[ \tau = \frac{2}{59.16} \approx 0.0338 , \text{s} ] (33.8 ms)

This doubles the delay. The sideband spacing becomes:

[ \frac{1}{\tau} = \frac{59.16}{2} \approx 29.58 , \text{Hz} ]

Check the sidebands around the 5th harmonic (1172.84 Hz):

• ( k = 1 ): ( 1172.84 + 29.58 = 1202.42 , \text{Hz} )
• ( k = 2 ): ( 1172.84 + 2 \cdot 29.58 = 1172.84 + 59.16 = 1232 , \text{Hz} )

The second-order sideband (( k = 2 )) lands exactly at 1232 Hz, matching the Δ resonance.

Verification with Other Harmonics

• 4th harmonic (938.272 Hz):
• ( 938.272 + 29.58 = 967.852 , \text{Hz} )
• ( 938.272 + 59.16 = 997.432 , \text{Hz} )
• Continues up to ( 938.272 + 8 \cdot 29.58 \approx 1174.912 , \text{Hz} ), close to the 5th harmonic.
• 6th harmonic (1407.408 Hz):
• ( 1407.408 - 29.58 = 1377.828 , \text{Hz} )
• ( 1407.408 - 59.16 = 1348.248 , \text{Hz} )

The sidebands bracket the Δ resonance and the N(1440) resonance (1440 Hz), with the 5th harmonic’s ( k = 2 ) sideband hitting 1232 Hz precisely.

Spectral Broadening

With ( \tau = 0.0338 , \text{s} ), the modulation frequency is:

[ \frac{1}{\tau} \approx 29.58 , \text{Hz} ]

This corresponds to a spectral broadening of approximately 29.58 MeV, narrower than the 59.16 MeV from the original delay. However, the higher-order sidebands (e.g., ( k = 2 ) at 59.16 Hz) ensure the Δ resonance is still targeted, with additional sidebands providing a richer interference pattern.

Conclusion

Increasing the delay from 0.0169 s to 0.0338 s (33.8 ms) maintains optimal matching to the Δ(1232 MeV) resonance by placing the second-order sideband of the 5th harmonic at 1232 Hz. This delay enhances the spectrum at 1232 Hz through interference, offering a refined approach with a broader time separation, as requested. The spectral broadening of ~29.58 Hz (MeV) is sufficient to influence nearby frequencies, though it’s less wide-ranging than the original, focusing the enhancement more selectively around the sideband structure.

Thus, the increased delay time for optimal Δ resonance matching is 0.0338 seconds.