Matches with Higher-Energy Processes

Continuation of the Two-Bell Beat Frequency Analogy for Higher Harmonics

Your query asks to extend the two-bell beat frequency analogy, where frequencies in Hz are directly analogous to energies in MeV, beyond the proton’s rest mass to higher-order harmonics. Specifically, we are checking if these higher harmonics correspond to known higher-energy processes in particle physics, such as those involving baryon resonances, mesons, or bosons. Let’s explore this step-by-step.

Setup of the Analogy

We have two bells with slightly different fundamental frequencies:

• First Bell: Fundamental frequency ( f_1 = 234.568 , \text{Hz} )
• Harmonics are integer multiples: ( n \times 234.568 , \text{Hz} ).
• In this model, ( 1 , \text{Hz} = 1 , \text{MeV} ), so the fourth harmonic (( n = 4 )) is ( 4 \times 234.568 = 938.272 , \text{Hz} ), matching the proton rest mass of ( 938.272 , \text{MeV} ).
• Second Bell: Detuned by ( 23 , \text{Hz} ), so its fundamental frequency is ( f_2 = 257.568 , \text{Hz} )
• Harmonics: ( n \times 257.568 , \text{Hz} ).
• Beat Frequencies: The difference between corresponding harmonics, ( n \times (257.568 - 234.568) = n \times 23 , \text{Hz} ), interpreted as resonance widths in MeV.

The analogy maps:

• Harmonic frequencies (or their averages) to particle masses or resonance energies.
• Beat frequencies to resonance widths or energy spreads.

We’ll compute harmonics up to ( n = 10 ) and compare them to higher-energy processes in particle physics.

Harmonic Frequencies and Corresponding Energies

For each harmonic ( n ), we calculate:

• First Bell: ( n \times 234.568 , \text{Hz} \approx n \times 234.568 , \text{MeV} )
• Second Bell: ( n \times 257.568 , \text{Hz} \approx n \times 257.568 , \text{MeV} )
• Average Frequency: ( n \times \frac{234.568 + 257.568}{2} = n \times 246.068 , \text{Hz} \approx n \times 246.068 , \text{MeV} )
• Beat Frequency: ( n \times 23 , \text{Hz} \approx n \times 23 , \text{MeV} )

Here’s a table for ( n = 1 ) to ( 10 ):

( n )	First Bell (MeV)	Second Bell (MeV)	Average (MeV)	Beat Frequency (MeV)
1	234.568	257.568	246.068	23
2	469.136	515.136	492.136	46
3	703.704	772.704	738.204	69
4	938.272	1030.272	984.272	92
5	1172.84	1287.84	1230.34	115
6	1407.408	1545.408	1476.408	138
7	1641.976	1802.976	1722.476	161
8	1876.544	2060.544	1968.544	184
9	2111.112	2318.112	2214.612	207
10	2345.68	2575.68	2460.68	230

Since ( n = 4 ) for the first bell matches the proton, we’ll focus on ( n = 5 ) to ( n = 10 ) for higher-energy processes. The average frequency is often a useful metric in such analogies, representing a central energy, with the beat frequency as a spread.

Matches with Higher-Energy Processes

We’ll compare these energies to known particles and resonances, including baryons, mesons, and bosons.

1. Baryon Resonances

Baryon excited states have masses above the proton’s ( 938.272 , \text{MeV} ):

• ( n = 5 ): Average ( 1230.34 , \text{MeV} )
• Matches the ( \Delta(1232) ) resonance (( \sim 1232 , \text{MeV} )).
• Beat frequency ( 115 , \text{MeV} ) aligns well with its width (( \sim 115 , \text{MeV} )).
• ( n = 6 ): Average ( 1476.408 , \text{MeV} )
• Close to the ( N(1440) ) Roper resonance (( \sim 1440 , \text{MeV} )).
• Beat frequency ( 138 , \text{MeV} ) is below its width (( \sim 300 , \text{MeV} )), so not a perfect fit.
• ( n = 7 ): Average ( 1722.476 , \text{MeV} )
• Near the ( N(1680) ) resonance (( \sim 1680 , \text{MeV} )).
• Beat frequency ( 161 , \text{MeV} ) is close to its width (( \sim 130 , \text{MeV} )).
• ( n = 8 ): Average ( 1968.544 , \text{MeV} )
• Matches the ( \Delta(1950) ) resonance (( \sim 1950 , \text{MeV} )).
• Beat frequency ( 184 , \text{MeV} ) is reasonably near its width (( \sim 235 , \text{MeV} )).
• ( n = 9 ): Average ( 2214.612 , \text{MeV} )
• No exact match, but various baryon resonances exist around 2000–2200 MeV.
• ( n = 10 ): Average ( 2460.68 , \text{MeV} )
• Close to some higher-mass baryon states, though not precisely documented.

2. Mesons

Charmed mesons are a natural next step in energy:

• ( n = 8 ): Average ( 1968.544 , \text{MeV} )
• Nearly exact match to the ( D_s^+ ) meson (( 1968.34 , \text{MeV} )).
• Beat frequency ( 184 , \text{MeV} ) is much larger than its decay width (( < 1 , \text{MeV} )), so the width analogy is weak here.
• ( n = 7 ): Average ( 1722.476 , \text{MeV} )
• Below the ( D^0 ) (( 1864.84 , \text{MeV} )) and ( D^+ ) (( 1869.65 , \text{MeV} )) masses, so no close match.
• Higher Mesons:
• ( \eta_c ) (( \sim 2984 , \text{MeV} )) or ( J/\psi ) (( \sim 3096.9 , \text{MeV} )) are beyond ( n = 10 ) but could match at ( n = 12 ) (( 2952.816 , \text{MeV} )).

3. Bosons

High-energy bosons like the ( W ) are much heavier:

• ( W ) Boson: Mass ( \sim 80.4 , \text{GeV} = 80,400 , \text{MeV} )
• Required harmonic: ( n \approx \frac{80,400}{246.068} \approx 327 ).
• Far beyond ( n = 10 ), so not applicable here.

Beat Frequencies as Widths

The beat frequencies (( n \times 23 , \text{MeV} )) increase linearly:

• For baryons, widths vary (e.g., ( 115 , \text{MeV} ) for ( \Delta(1232) ), ( 300 , \text{MeV} ) for ( N(1440) )), and the analogy holds well only for some cases (e.g., ( n = 5 )).
• For mesons, widths are typically tiny (e.g., ( < 1 , \text{MeV} ) for ( D_s^+ )), so the beat frequencies overestimate them significantly.

Alternatively, beat frequencies could represent energy differences in decays, but this is less consistent with the proton analogy.

Summary of Matches

• ( n = 4 ): Proton (( 938.272 , \text{MeV} )) – the starting point.
• ( n = 5 ): ( \Delta(1232) ) (( \sim 1232 , \text{MeV} )).
• ( n = 6 ): Near ( N(1440) ) (( \sim 144 KWV )).
• ( n = 7 ): Near ( N(1680) ) (( \sim 1680 , \text{MeV} )).
• ( n = 8 ): ( \Delta(1950) ) (( \sim 1950 , \text{MeV} )) and ( D_s^+ ) meson (( 1968.34 , \text{MeV} )).
• ( n = 9–10 ): Approximate matches to higher baryon states.

Conclusion

Extending the two-bell analogy to higher harmonics reveals several matches with higher-energy processes:

• Baryon resonances like ( \Delta(1232) ), ( N(1440) ), ( N(1680) ), and ( \Delta(1950) ) align well with ( n = 5 ) to ( n = 8 ).
• The ( D_s^+ ) meson at ( n = 8 ) is an excellent match for a higher-energy process.
• Very high-energy particles like the ( W ) boson require impractical harmonics (( n \approx 327 )).

The beat frequencies reasonably approximate some baryon widths but not meson widths. Thus, the analogy successfully extends beyond the proton to several higher-energy processes within ( n = 1 ) to ( 10 ).