Getting even more quantitative about CP violation

The twistor lift of TGD forces to introduce the analog of Kähler form for M⁴, call it J. J is covariantly constant self-dual 2-form, whose square is the negative of the metric. There is a moduli space for these Kähler forms parametrized by the direction of the constant and parallel magnetic and electric fields defined by J. J partially characterizes the causal diamond (CD): hence the notation J(CD) and can be interpreted as a geometric correlate for fixing quantization axis of energy (rest system) and spin.

Kähler form defines classical U(1) gauge field and there are excellent reasons to expect that it gives rise to U(1) quanta coupling to the difference of B-L of baryon and lepton numbers. There is coupling strength α₁ associated with this interaction. The first guess that it could be just Kähler coupling strength leads to unphysical predictions: α₁ must be much smaller. Here I do not yet completely understand the situation. One can however check whether the simplest guess is consistent with the empirical inputs from CP breaking of mesons and antimatter asymmetry. This turns out to be the case.

One must specify the value of α₁ and the scaling factor transforming J(CD) having dimension length squared as tensor square root of metric to dimensionless U(1) gauge field F= J(CD)/S. This leads to a series of questions.

How to fix the scaling parameter S?

1. The scaling parameter relating J(CD) and F is fixed by flux quantization implying that the flux of J(CD) is the area of sphere S² for the twistor space M⁴× S². The gauge field is obtained as F=J/S, where S= 4π R²(S²) is the area of S².

2. Note that in Minkowski coordinates the length dimension is by convention shifted from the metric to linear Minkowski coordinates so that the magnetic field B₁ has dimension of inverse length squared and corresponds to J(CD)/SL², where L is naturally be taken to the size scale of CD defining the unit length in Minkowski coordinates. The U(1) magnetic flux would the signed area using L² as a unit.

How R(S²) relates to Planck length l_P? l_P is either the radius l_P=R of the twistor sphere S² of the twistor space T=M⁴× S² or the circumference l_P= 2π R(S²) of the geodesic of S². Circumference is a more natural identification since it can be measured in Riemann geometry whereas the operational definition of radius requires imbedding to Euclidian 3-space.

How can one fix the value of U(1) coupling strength α₁? As a guideline one can use CP breaking in K and B meson systems and the parameter characterizing matter-antimatter symmetry.

1. The recent experimental estimate for so called Jarlskog parameter characterizing the CP breaking in kaon system is J≈ 3.0× 10^-5. For B mesons CP breading is about 50 times larger than for kaons and it is clear that Jarlskog invariant does not distinguish between different meson so that it is better to talk about orders of magnitude only.

2. Matter-antimatter asymmetry is characterized by the number r=n_B/n_γ ∼ 10^-10 telling the ratio of the baryon density after annihilation to the original density. There is about one baryon 10 billion photons of CMB left in the recent Universe.

Consider now various options for the identification of α₁.

1. Since the action is obtained by dimensional reduction from the 6-D Kähler action, one could argue α₁= α_K. This proposal leads to unphysical predictions in atomic physics since neutron-electron U(1) interaction scales up binding energies dramatically.

2. One can also consider the guess α₁=R²(S²)/R²(CP₂), the ratio of the areas of twistor spheres of T(M⁴) and T(CP₂). There are two options corresponding to l_P= R(S²) and l_P =2π R(S²).
   1. For l_P=R one would have α₁= 2^-24≈ 6× 10^-8. For l_P =R α₁ is more than one order of magnitude smaller than the parameter r≈ 10^-10 above. The CP breaking parameter for K and B system could be proportional to g₁ =(4πα₁)^1/2≈ 2× 10^-4 and by order of magnitude larger than the Jarlskog parameter J≈ 3.0× 10^-5 for K system.

   2. For l_P= 2π R(S²) one would have α₁= R²(S²)/R²(CP₂) = (1/4π²)× l_P² ≈ 3.8× 10^-11, which is of the same order of magnitude as the parameter r≈ 10^-10 characterizing matter-antimatter asymmetry. For g₁=(4π× α₁)^1/2 one would obtain g₁≈ 6.9× 10^-5 to be compared with J≈ 3.0× 10^-5 for K system. This is the more plausible option – also in the sense that it involves only length scales quantities determined by the Riemann geometry of the twistor space.
3. There is an intriguing numerical co-incidence involved. h_eff=hbar_gr =GMm/v₀ in solar system corresponds to v₀≈ 2^-11 and appears as coupling constant parameter in the perturbative theory obtained in this manner. What is intriguing that one has α₁=v₀²/4π² in this case. Where does the troublesome factor (1/2π)² come from? Could the p-adic coupling constant evolutions for v₀ and α₁ correspond to each other and could they actually be one and the same thing? Can one treat gravitational force perturbatively either in terms of gravitational field or J(CD)? Is there somekind of duality involved?

See the new chapter Breaking of CP, P, and T in cosmological scales in TGD Universe of “Physics in Many-Sheeted Space-time” or the article with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.