Are the masses of top and antitop quark different?

Measurement of the mass difference between t and tbar quarks is the title of the e-print by a group working in Fermilab. The first name in the long list of authors is T. Aaltonen, which is very finnnish sounding. The finding of the group is that top-tbar mass difference is

Δ M= M_t-M_tbar= -3.3+/- 1.7 GeV.

The best fit is obtained with

Δ M=-4 GeV.

For top quark mass M_t= 170 GeV this means Δ M/M≈ 2.3 per cent. The result deviates from CPT-symmetric expectation Δ M=0 at 2σ level. Also D0 collaboration has got similar results two years earlier (PRL 103, 132001 (2009)) but at time the errors bars were so large that the finding was consistent with CPT symmetry.

The finding encourages to consider the possibility of CPT breaking seriously. In TGD framework a very strong form of apparent CPT breaking results if fermion and anti-fermion correspond to different values of p-adic prime so that mass scales differ by a multiple of half octave. The different choices of the p-adic mass scale would be induced by the interaction with environment. This option might explain the observations suggesting that neutrino and antineutrino masses and mixing matrices are different without introducing sterile neutrino: sterile neutrino would correspond to neutrino but in different p-adic length scale. In the recent case this option is excluded by the smallness of the mass difference. In zero energy ontology, which assigns to elementary particles size scale which which is macroscopic, one can however consider a more delicate breaking of CPT induced by the interactions with environment.

What CPT and CPT breaking do mean?

To begin, recall that CPT breaking would mean that the invariance condition

P(Ψ_i,Ψ_f)= P(θΨ_f,θΨ_i)

for probabilities fails to be satisfied. Here θ is shorthand for CPT. The permutation of initial and final states is what distinguishes T and thus CPT from ordinary symmetries and means that T must be realized anti-linearly. In standard QFT P and T have geometric meaning whereas C does not. In TGD framework also C is geometric and this means that one must reconsider CPT and its tests based on phenomenological models.

CPT symmetry is one of the basic tenets of quantum field theory. In particular, the breaking of CPT requires the breaking of Lorentz invariance in standard QFT framework. In TGD framework the situation is actually different as I realized only now! The reason is that also charge conjugation is induced by a geometric transformation just like P and T. C indeed involves complex conjugation of CP₂ coordinates, and one can quite well consider a situation in with T and P are unbroken and only C is broken so that CPT is broken. What actually happens depends on the detailed action of the symmetries on the modified Dirac action.

Some fact about zero energy ontology

Before one one can proceed, one must consider in more detail the notion of CD. CD is a product of CD proper defined as intersection of future and past directed light-cones of M⁴ and of CP₂. The scales of CDs are assumed to come in powers of two of CP₂ scale to explain p-adic length scale hypothesis (one can consider also prime and even integer multiples). What is of utmost significance is that these scales are macroscopic. Poincare transformations affect CDs and give rise to a moduli space for CDs. In the case of CP₂ this is not the case unless one introduces additional physically well motivated structure.

Quite generally, this additional structure corresponds to the choice choice of quantization axes for various isometry currents realized at the level of the geometry of world of classical worlds which decomposes to a union of the geometries assigned with difference CDs labelled by moduli specifying the choice of quantization axes. In the case of M⁴ the line joining the tips of CD defines a unique rest system with origin at the middle point of the line and selects quantization axes of energy. The direction of spin quantization axes is fixed if one introduces preferred plane M² physically analogous to the preferred plane of unphysical polarizations. This plane is fixed also by number theoretical vision and correspond to hyper-octonionic plane of complexified octonions highly relevant for the number theoretic formulation of TGD.

One must also introduce CP₂ coordinates transforming linearly with respect to U(2) sub-group. The choice of preferred point of CP₂ at either boundary of CD allows to fix complex coordinates of CP₂ only apart from U(2) rotation. Hyper-charge quantization axes is fixed but color isospin direction remains free. In fact, there is a preferred color isospin generator leaving the points of the geodesic sphere invariant whereas hypercharge generator induces phase multiplication. By choosing two preferred points of CP₂ assigned to the opposite boundaries of CD one can identify the geodesic line connecting the points as a flow line of color isospin rotations so that the quantization axes are fixed.

The choices of preferred plane M² and preferred geodesic sphere make sense also at the level of the preferred extremals of Kähler action and this leads to a concrete realization of the conjectured slicing of the space-time surface by string world sheets having braid strands at their ends at light-like wormhole throats carrying particle quantum numbers.

The vision about how quantum TGD gives rise to symplectic theory of knots, braids, braid cobordisms, and of two-knots (see the previous posting) led to the realization that preferred extremals should involve preferred geodesic sphere of CP₂, whose inverse image under imbedding map assigns to the space-time surface unique complex of stringy two surfaces. These stringy two-surfaces define braid cobordisms and 2-knots and provide also the reduction of quantum TGD to string theory like structure in finite measurement resolution meaning the replacement of the orbits of partonic 2-surfaces with braids.

Charge conjugation is geometric in TGD framework

Charge conjugation in TGD Universe involves complex conjugation of CP₂ coordinates. Complex conjugation commutes with color rotations only if they belong to a subgroup U(2) ⊂ SU(3) leaving a preferred point of CP₂ invariant remaining invariant also under C just like the origin of M⁴ remains invariant under P and T. The situation differs from that for P and T decisively since the scale of CP₂ is about 10⁴ Planck lengths. More general color rotations acting non-linearly and affecting non-trivially on the preferred point do not commute with C.

A simple example is provided by sphere. In this case C would act in complex coordinates as φ → -φ, where φ is the phase angle of the complex coordinate with origin at the preferred point of the sphere. The action obviously depends on the choice of the preferred point.

The situation is therefore same as for P and T which also fail to commute with Poincare group and commute only with Lorentz transformations leaving the selected space-time point fixed. In TGD framework this point would correspond naturally to the center of the line connecting the tips of the causal diamond proper.

The action of C on physical states involves a linear transformation of spinors transformation besides the geometric action. The details of this action were discussed already in my thesis for almost three decades ago and the reader can consult the appendix of some of the books about TGD or the little article titled The Geometry of CP₂ and its Relationship to Standard Model as the appendix of an article series summarizing Quantum TGD published in Prespacetime Journal. What is essential is that the action of C does does not commute with color rotations acting on the moduli of CD unless they belong to the U(2) subgroup leaving the geodesic sphere invariant. One can define C for the two boundaries of CD by requiring that the corresponding geodesic spheres remain invariant under C.

The action of CPT in zero energy ontology

The action of CPT is following.

1. First one applies P and T. If one assumes that the preferred point of M⁴ corresponds to the middle point of the line connecting the tips of CD proper, these transformations permute upper and lower boundaries of CD proper. This is indeed a very natural requirement and means that positive and negative energy parts of the quantum state serving as counterparts of initial and finals states in positive energy ontology are permuted just as they are permuted in CPT. That T is realized anti-linearly conforms with the fact that T does leave invariant the boundary of CD proper.

2. Next one applies C involving complex conjugation which in general affects the moduli of CD. If C is chosen differently at the opposite boundaries it leaves the corresponding moduli invariant but since CPT involves the permutation of positive and negative energy states the moduli of CD are changed since the preferred point of upper boundary becomes the preferred point of the lower boundary and vice versa.

   Only in the case that the preferred points assigned to the upper and lower boundaries are same, this does not happen but in this case the quantization axes are not completely fixed which could make sense only if color isospin of all particles or at least of the positive (and negative) energy part of the zero energy state vanishes. Unless the CD has a wave function in the space of moduli which is constant, a spontaneous and a purely geometric breaking of C symmetry is induced. The breaking would be highly analogous to the breaking of rotational symmetry in spontaneous magnetization taking place in many particle systems.

3. The size scale of the CD proper is macroscopic even for elementary particles and corresponds to the secondary p-adic length scale associated with the particle. For electron with p= M₁₂₇=2¹²⁷-1 this time scale is T(2,127)= .1 seconds, defining the fundamental biological rhythm. For u and d quarks it is of order millisecond and for t quark characterized by p≈ 2⁹³ it is given by

   T(2,93) 2^-127+93× T(2,127)≈5.8×10^-12 seconds.

   The corresponding length scale is 1.74 mm and is macroscopic. There are very many particles in CD of this size scale which suggests the possibility of spontaneous C breaking inducing by a localization in the moduli space of CDs implying the breaking of the CPT invariance condition. The many-particle system would be present since the CDs assignable to individual quarks intersect which suggests that they correspond to common CD. The non-invariance of the many-particle system under CPT could also result from that under PT operation in macroscopic situation.

Quantitative picture about CPT breaking

Building a quantitative picture about CPT breaking requires answering many questions.

1. The mass difference should depend on the moduli of CDs characterizing color quantization axes and characterize by preferred points of CP₂ assigned with future and past boundaries of CD. A natural measure for the symmetry breaking is defined by the geodesic distance -call it s- between the preferred points so that one expects that the mass of top quark assigned with a particular CD involves a small contribution depending on s. This distance is however not changed in C.

   The additional contribution to the mass should contain a term which is odd under C (most naturally), CP, or CPT. Could the oddness come from the spontaneous symmetry breaking giving rise to an interaction term with environment affecting the mass of particle and antiparticle in different manner? This oddness would be analogous to the oddness of the interaction energy of magnetic dipole with an external magnetic field.

2. The relative mass difference is of order 2 per cent. What determines this scale? Could the mass difference be proportional to fine structure constant? This could be the case if the electromagnetic interaction of the top quark with environment defined by CD induces the CPT odd term to the mass of the particle? What is the role of the magnetic flux tube containing top at the second end and neutrino pair neutralizing its weak isospin at the second end?

Questions

This picture inspires several questions.

1. Can one consider C breaking without the presence of P and T breakings? If the CP breaking assigned with kaon-antikaon system and other neutral meson systems is CP breaking in TGD sense, does it involve the breaking of T at all? The answers to these questions are not obvious since the tests of discrete symmetries rely on the standard view about charge conjugation lacking totally the geometric aspect of C in TGD Universe.

2. Could it be that the different topological mixings of U and D quarks inducing in turn CKM mixing are induced by C breaking basically so that the mass differences would correlate directly with CKM mixing parameters?

3. Is the geometric view about about breaking of C relevant for the understanding of matter antimatter asymmetry? I have considered several models of the generation of matter antimatter asymmetry, one of them assuming that antimatter is eaten by long cosmic strings with breaking induced by the Kähler electric fields inducing small difference in the densities of fermions and anti-fermions outside cosmic strings. Could matter antimatter asymmetry be mathematically analogous to chiral selection in living matter so that P would be only replaced with C? Whether the geometric view about C is relevant for the understanding of the matter antimatter asymmetry must be however left open question. Different masses for fermions anti-fermions could however help to understand why this kind of separation takes place.

4. C acts in CP₂ and in color degrees of freedom. Does this mean that for non-colored states C is not broken and that s CP breaking is present only for quarks but not for leptons? The answers to these questions are not obvious since in TGD framework M⁴×CP₂ spinor harmonics correspond to color partial waves which have wrong correlation with electro-weak quantum numbers. Only covariantly constant right-handed neutrino spinor generating supersymmetry can move in color single partial wave. The physical color assignments are the result of a state construction involving super-conformal algebra with algebra elements carrying color.

For background see the chapter p-Adic Particle Massivation: New Physics of "p-Adic Length Scale Physics and Hierarchy of Planck Constants".