The Strong Force and the Mass of Matter

Why is matter so light? A proton weighs 1.30 × 10¹⁹ times less than a Planck mass, the mass of one Pulse (Part 2.2), and every star, planet and living thing is built of protons. Part 2.9 showed that the size of every star follows from that one ratio; this part shows where the ratio comes from. The answer is the strong force, and through it the ratio turns out to be the level of the universe.

Mass Without Mass

Almost none of a proton's mass is the mass of its quarks: the Higgs field gives the quarks about one percent of it. The rest, ninety-nine percent, is the energy of the strong force that binds them, computed from first principles in lattice simulations of the strong force (Dürr et al., 2008). The mass of matter is the energy of a bond.

The strong force has a property no other force has in the same way: it grows weaker at short distances and stronger at long ones, asymptotic freedom (Gross & Wilczek, 1973; Politzer, 1973). Its coupling runs with the scale at which it is measured.

Running of the Strong Coupling G

1/α⊗ = 1/α⊗⊚ + (b⊗ / (2 · π)) · ln(μ / μ⊚)

Dimensional analysis:
Where (from the book): α⊗ [∅], α⊗⊚ [∅], μ [𝕄·𝕃²·𝕋⁻²], μ⊚ [𝕄·𝕃²·𝕋⁻²], b⊗ [∅]
[∅] ÷ [∅] = [∅]
[∅] ÷ [∅] + ([∅] ÷ ([∅] × [∅])) × ln([𝕄·𝕃²·𝕋⁻²] ÷ [𝕄·𝕃²·𝕋⁻²]) = [∅] + ([∅] ÷ ([∅] × [∅])) × ln([𝕄·𝕃²·𝕋⁻²] ÷ [𝕄·𝕃²·𝕋⁻²]) = [∅] + ([∅] ÷ [∅]) × ln([𝕄·𝕃²·𝕋⁻²] ÷ [𝕄·𝕃²·𝕋⁻²]) = [∅] + [∅] × ln([𝕄·𝕃²·𝕋⁻²] ÷ [𝕄·𝕃²·𝕋⁻²]) = [∅] + [∅] × ln([∅]) = [∅] + [∅] × [∅] = [∅] + [∅] = [∅]
[∅] = [∅] ✓ Valid PulseCore Verified ✓

The strong coupling weakens logarithmically toward short distances

Where:

• α⊗ [∅] — the strong coupling measured at energy μ
• α⊗⊚ [∅] — the strong coupling at the reference energy μ⊚; 0.1180 at the mass of the Z boson (Particle Data Group, 2024)
• μ [𝕄·𝕃²·𝕋⁻²] — the energy at which it is measured
• μ⊚ [𝕄·𝕃²·𝕋⁻²] — a reference energy
• b⊗ [∅] — the running rate, 11 − 2n/3 for n kinds of quark light enough to take part; 7 with all six

➢ Written at one loop; the full running used below is two loops. The 2π is the geometry of the quantum loops through which the force acts on itself: each loop is a full turn of phase.

Dimensional Transmutation G

Λ⊗ = μ · exp(−2 · π / (b⊗ · α⊗))

Dimensional analysis:
Where (from the book): Λ⊗ [𝕄·𝕃²·𝕋⁻²], μ [𝕄·𝕃²·𝕋⁻²], b⊗ [∅], α⊗ [∅]
[𝕄·𝕃²·𝕋⁻²]
[𝕄·𝕃²·𝕋⁻²] × exp([∅] × [∅] ÷ ([∅] × [∅])) = [𝕄·𝕃²·𝕋⁻²] × exp([∅] ÷ ([∅] × [∅])) = [𝕄·𝕃²·𝕋⁻²] × exp([∅] ÷ [∅]) = [𝕄·𝕃²·𝕋⁻²] × exp([∅]) = [𝕄·𝕃²·𝕋⁻²] × [∅] = [𝕄·𝕃²·𝕋⁻²]
[𝕄·𝕃²·𝕋⁻²] = [𝕄·𝕃²·𝕋⁻²] ✓ Valid PulseCore Verified ✓

A pure number, the coupling, becomes the scale of matter

Where:

• Λ⊗ [𝕄·𝕃²·𝕋⁻²] — the strong scale: the energy at which the coupling becomes strong and quarks bind
• μ [𝕄·𝕃²·𝕋⁻²] — any energy at which the coupling is known
• b⊗ [∅] — the running rate
• α⊗ [∅] — the strong coupling at μ

➢ The coupling at the Planck energy is small, about 1/53, so the exponential is about e⁻⁴⁷: the strong scale, and with it the mass of matter, sits about 10²⁰ below the Planck scale without anything being tuned. This is why matter is light: the hierarchy is an exponential of 2π over a modest coupling (Wilczek, 1999).

The Level Is the Hierarchy

Dirac noticed in 1937 that the ratio of the proton's mass to the Planck mass is close to the cube root of the age of the universe in the units of the Pulse, and Weinberg (1972) wrote it as a relation, m ≈ (ħ²·H/(G·c))^(1/3). Binary Pulse Theory gives both numbers a meaning: a universe lives 2^L of its own Tempos (the Rupture Rule, Chapter 1, Part D), and a Tempo is its Planck time. The cube root is the book's complexity exponent, α = 1/3, the same that counts its wells, N(L) = 2^(L/3) (Part 3.10).

Matter Scale of a Level G

Λ⊗ = E_P · 2^(−L/3)

Dimensional analysis:
Where (from the book): Λ⊗ [𝕄·𝕃²·𝕋⁻²], L [∅]
[𝕄·𝕃²·𝕋⁻²]
[𝕄·𝕃²·𝕋⁻²] × [∅]^([∅] ÷ [∅]) = [𝕄·𝕃²·𝕋⁻²] × [∅]^([∅]) = [𝕄·𝕃²·𝕋⁻²] × [∅] = [𝕄·𝕃²·𝕋⁻²]
[𝕄·𝕃²·𝕋⁻²] = [𝕄·𝕃²·𝕋⁻²] ✓ Valid PulseCore Verified ✓

The strong scale of a universe is its Planck energy over the cube root of its life in Pulses

Where:

• Λ⊗ [𝕄·𝕃²·𝕋⁻²] — the strong scale of the universe
• E_P [𝕄·𝕃²·𝕋⁻²] — its Planck energy, m_P·c², the energy of one Pulse
• L [∅] — its level

➢ The test: run the strong coupling measured today, 0.1180, up to the Planck energy at two loops with all six quarks, which gives 1/α⊗ = 53.0 there, and read off the strong scale with all six quarks active, Λ⊗⁽⁶⁾ = 88 MeV. Then 3·log₂(E_P/Λ⊗⁽⁶⁾) = 200.7. The level the Rupture Rule gives our universe from its age is 202. The two agree to 0.6 percent, from entirely independent sides: one from the measured strength of the strong force, the other from the measured age of the universe. (With five quarks the strong scale is 209 MeV and the level 197.0; at one loop, 203.9: the agreement holds at the few-percent level however the scale is defined.)

Matter Clock G

t⊛³ = t_P² · T⊚

Dimensional analysis:
Where (from the book): t⊛ [𝕋], T⊚ [𝕋]
[𝕋³]
[𝕋²] × [𝕋] = [𝕋³]
[𝕋³] = [𝕋³] ✓ Valid PulseCore Verified ✓

The clock of matter keeps time between the Pulse and the whole life

Where:

• t⊛ [𝕋] — the clock of the matter quantum, its Compton time ħ/(Λ⊗)
• t_P [𝕋] — the Planck time, one Tempo
• T⊚ [𝕋] — the life of the universe, 2^L Tempos

➢ The Matter Scale of a Level read as time: the period of matter's own quantum is the geometric mean of two Pulses and the universe's whole life. Matter is the scale at which the Pulse and the life meet.

Proton Mass G

m_p · c² = κ⊗ · Λ⊗

Dimensional analysis:
Where (from the book): c [𝕃·𝕋⁻¹], κ⊗ [∅], Λ⊗ [𝕄·𝕃²·𝕋⁻²]
[𝕄] × [𝕃²·𝕋⁻²] = [𝕄·𝕃²·𝕋⁻²]
[∅] × [𝕄·𝕃²·𝕋⁻²] = [𝕄·𝕃²·𝕋⁻²]
[𝕄·𝕃²·𝕋⁻²] = [𝕄·𝕃²·𝕋⁻²] ✓ Valid PulseCore Verified ✓

A proton is a fixed multiple of the strong scale

Where:

• m_p [𝕄] — the proton mass, 938.3 MeV
• c [𝕃·𝕋⁻¹] — light speed
• κ⊗ [∅] — the multiple, set by the strong force alone: 10.6 against Λ⊗⁽⁶⁾ (about 14.3 against the level's E_P · 2^(−202/3))
• Λ⊗ [𝕄·𝕃²·𝕋⁻²] — the strong scale

➢ The multiple is not a free number: it is what the strong force does when quarks bind, and lattice calculations of the strong force reproduce it (Dürr et al., 2008). It is the same in every universe whose strong force is the same; only the scale moves with the level.

How Big a Star Can Be, Level by Level

With the Stellar Mass Unit of Part 6.10, M⋆ = m_P³/m_p², a star holds about (m_P/m_p)³ protons, and the Matter Scale of a Level gives that as (2^(L/3)/κ)³ = 2^L/κ³. At our level that is 2 × 10⁵⁷ protons, a Sun. At level 12 it is about one: a star would be a single proton. Below about level 12, no stars as we know them can form: there is not enough of a ladder between the Pulse and matter for gravity to gather matter into a burning ball. Stars are a property of deep universes.