CP violation was originally observed for mesons via the mixing of neutral kaon and antikaon having quark content nsbar and nbars. The lifetimes of kaon and antikaon are different and they transform to each other. CP violation has been also observed for neutral mesons of type nbbar. Now it has been observed also for baryons Λ_{b} with quark composition u-d-b and its antiparticle (see this). Standard model gives the Feynman graphs describing the mixing in standard model in terms of CKM matrix (see this).

The CKM mixing matrix associated with weak interactions codes for the CP violation. More precisely, the small imaginary part for the determinant of CKM matrix defines the invariant coding for the CP violation. The standard model description of CP violation involves box diagrams in which the coupling to heavy quarks takes place. b quark gives rise to anomalously large CP violation effect also for mesons and this is not quite understood. Possible new heavy fermions in the loops could explain the anomaly.

Quite generally, the origin of CP violation has remained a mystery as also CKM mixing. In TGD framework CKM mixing has topological explanation in terms of genus of partonic 2-surface assignable to quark (sphere, torus or sphere with two handles). Topological mixings of U and D type quarks are different and the difference is not same for quarks and antiquarks. But this explains only CKM mixing, not CP violation.

Classical electric field - not necessary electromagnetic - prevailing inside hadrons could cause CP violation. So called instantons are basic prediction of gauge field theories and could cause strong CP violation since self-dual gauge field is involved with electric and magnetic fields having same strength and direction. That this strong CP violation is not observed is a problem of QCD. There are however proposals that instantons in vacuum could explain the CP violation of hadron physics (see this).

What says TGD? I have considered this here and in the earlier blog posting (see this).

- M
^{4}and CP_{2}are unique in allowing twistor space with Kähler structure (in generalized sense for M^{4}). If the twistor space T(M^{4})= M^{4}× S^{2}having bundle projections to both M^{4}and to the conventional twistor space CP_{3}, or rather its non-compact version) allows Kähler structure then also M^{4}allow the generalized Kähler structure and the analog symplectic structure.

This boils down to the existence of self-dual and covariantly constant U(1) gauge field J(M

^{4}) for which electric and magnetic fields E and B are equal and constant and have the same direction. This field is not dynamical like gauge fields but would characterize the geometry of M^{4}. J(M^{4}) implies violation Lorentz invariance. TGD however leads to a moduli space for causal diamonds (CDs) effectively labelled by different choices of direction for these self-dual Maxwell fields. The common direction of E and B could correspond to that for spin quantization axis. J(M^{4}) has nothing to do with instanton field.

It should be noticed that also the quantum group inspired attempts to build quantum field theories for which space-time geometry is non-commutative introduce the analog of Kähler form in M^{4}, and are indeed plagued by the breaking of Lorentz invariance. Here there is no moduli space saving the situation (see this) .

- The choice of quantization axis would therefore have a correlate at the level of "world of classical worlds" (WCW). Different choices would correspond to different sectors of WCW. The moduli space for the choices of preferred point of CP
_{2}and color quantization axis corresponds to the twistor space T(CP_{2})= SU(3)/U(1)× U(1) of WCW. One could interpret also the twistor space T(M^{4})= M^{4}× S^{2}as the space with given point representing the position of the tip of CD and the direction of the quantization axis of angular momentum. This choice requires a characterization of a unique rest system and the directions of quantization axis and time axes defines plane M^{2}playing a key role in TGD approach to twstorialization(see this) .

- The prediction would be CP violation for a given choice of J(M
^{4}). Usually this violation would be averaged out in the average over the moduli space for the choices of M^{2}but in some situation this would not happen. Why the CP violation does not average out when there is CKM mixing of quarks? Why the parity violation due to the preferred direction is not compensated by C violation meaning that the directions of E and B fields would be exactly opposite for quarks and antiquarks. Could the fact that quarks are not free but inside hadron induce CP violation? Could a more abstract formulation say that the wave function in the moduli space for J(M^{4}) (wave function for the choices of spin quantization axis!) is not CP symmetric and this is reflected in the CKM matrix.

- An important delicacy is that J(M
^{4}) can be both self-dual and anti-self-dual depending on whether the magnetic and electric field have same or opposite directions. It will be found that reflection P and CP transform self-dual J(M^{4}) to anti-self-dual one. If only self-dual J(M^{4}) is allowed, one has both parity breaking and CP violations at the level of WCW.

- Zero energy state is pair of two positive and negative energy parts. Let us assume that positive energy part is fixed - one can call corresponding boundary of CD passive. This state corresponds to the outcome of state function reduction fixing the direction of quantization axes and producing eigenstates of measured observables, for instance spin. Single system at passive boundary is by definition unentangled with the other systems. It can consists of entangled subsystems hadrons are basic example of systems having entanglement in spin degrees of freedom of quarks: only the total spin of hadron is precisely defined.

The states at the active boundary of CD evolve by repeated unitary steps by the action of the analog of S-matrix and are not anymore eigenstates of single particle observables but entangled. There is a sequence of trivial state function reductions at passive boundary inducing sequence of unitary time evolutions to the state at the active boundary of CD and shifting it. This gives rise to self as a generalized Zeno effect.

Classically the time evolution of hadron corresponds to a superposition of space-time surfaces inside CD. The passive ends of the space-time surface or rather, the quantum superposition of them - is fixed. At the active end one has a superposition of 3-surfaces defining classical correlates for quantum states at the active end: this superposition changes in each unitary step during repeated measurements not affecting the passive end. Also time flows, which means that the distance between the tips of CD defining clock-time increases as the active boundary of CD shifts farther away.

- The classical field equations for space-time surface follow from an action, which at space-time level is sum of Kähler action and volume term. If Kähler form at space-time surface is induced (projected to space-time surface) from J=J(M
^{4})+J(CP_{2}), the classical time evolution is CP violating. CKM mixing is induced by different topological mixings for U and D type quarks (recall that 3 particle generations correspond to different genera for partonic 2-surfaces: sphere, torus, and sphere with two handles). J(M^{4})+J(CP_{2}) defines the electroweak U(1) component of electric field so that J(M^{4}) contributes to U(1) part of em field and is thus physically observable.

- Topological mixing of quarks corresponds to a superposition of time evolutions for the partonic 2-surfaces, which can also change the genus of partonic 2-surface defined as the number of handles attached to 2-sphere. For instance, sphere can transform to torus or torus to a sphere with two handles. This induces mixing of quantum states. For instance, one can say that a spherical partonic 2-surface containing quark would develop to quantum superposition of sphere, torus, and sphere with two handles. The sequence of state function reductions leaving the passive boundary of CD unaffected (generalized Zeno effect) by shifting the active boundary from its position after the first state function reduction to the passive boundary could but need not give rise to a further evolution of CKM matrix.

^{4}) transforms under C, P, T and CP.

- J(M
^{4})=(J_{0z}, J_{xy}= ε J_{0z}), ε=+/- 1, characterizes hadronic space-time sheet (all space-time sheets in fact). Since J(M^{4}) is tensor, P changes only the sign of J_{0z}giving J(M^{4})→ (-J_{0z}, J_{xy}). Since C changes the signs of charges and therefore the signs of fields created by them, one expects J(M^{4})→ -J(M4) under C. CP would give J(M^{4})→ (J_{0z}, -J_{xy}) transforming selfdual J(M^{4}) to anti-selfdual J(M^{4}). If WCW has no anti-self-dual sector, CP is violated at the level of WCW.

- If CPT leaves J(M
^{4}) invariant, one must have J(M^{4}) → (J_{0z}, -J_{xy}) under T rather than J(M^{4})→ (-J_{0z}, J_{xy}). The anti-unitary character of T could correspond for additional change of sign under T. Otherwise CPT should act as J(M^{4})→ -J(M^{4}) and only (CPT)^{2}would correspond to unity.

- Same considerations apply to J(CP
_{2}) but the difference would be that induced J(M^{4}) for space-time surfaces, which are small deformations of M^{4}covariantly constant in good approximation. Also for string world sheets corresponding to small cosmological constant J(M^{4})× J(M^{4})-2≈ 0 holds true in good approximation and induced J(M^{4}) at string world sheet is in good approximation covariantly constant. If the string world sheet is just M^{2}characterizing J(M^{4}) the condition is exact and was has Kähler electric field induced by J(M^{4}) but no corresponding magnetic field. This would make the CP breaking effect large.

^{4}). If only self-dual sector of WCW is present then CP is violated. Also P is violated at the level of WCW and this parity breaking is different from that associated with weak interactions and could relate to the geometric parity breaking manifesting itself via chiral selection in living matter. Classical time evolutions induce different CKM mixings for quarks and antiquarks reflecting itself in the small imaginary part of the determinant of CKM matrix. CP breaking at the level of WCW could explain also matter-antimatter asymmetry. For instance, antimatter could be dark with different value of h

_{eff}/h=n.

See the articles About twistor lift of TGD and Questions related to the twistor lift of TGD.

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

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