Strong CP problem and the spectral noncommutative geometric paradigm

Is the spectral standard model free of the strong CP problem ?
Last week, the twitto/blogo/media-spheres were abuzz over a potential detection of axions from X-ray astrophysics. From axion to the strong CP problem there is essentially a hypothetical dynamical variable coming out from a theta parameter of quantum chromodynamics thus the blogger jumps on this opportunity to have a look on this problem in the spectral noncommutative geometric framework:

The spectral action principle is the simple statement that the physical action is determined by the spectrum of the Dirac operator D. This has now been tested in many interesting models including Superstring theory [6], noncommutative tori [30], Moyal planes [34], 4D-Moyal space [37], manifolds with boundary [12], in the presence of dilatons [10], for supersymmetric models [5] and torsion cases [38]. The additivity of the action forces it to be of the form Trace f(D/Λ). In the approximation where the spectral function f is a cut-off function, the relations given by the spectral action are used as boundary conditions and the couplings are then allowed to run from unification scale to low energy using the renormalization group equations. The equations show, when fitted to the low energy boundary conditions, that the three gauge coupling constants and the Newton constant nearly meet (within few percent) at very high energies, two or three orders from the Planck scale. This might be a coincidence but it can also be an indication that a more fundamental theory exists at unification scale and manifests itself at low scale through integration of the intermediate modes, as in the Wilson understanding of renormalization...

 It is possible to add to the spectral action terms that will violate parity such as the gravitational term ε^µνρσR_µνabR_ρσ^ab and the non-abelian θ term ε^µνρσV_μν^mV_ρσ^m. These arise by allowing for the spectral action to include the term Tr(γG(D²/Λ²)) where G is a [test] function [, Λ a cut-off scale]... and γ=γ₅⊗γ_F is the total grading. In this case it is easy to see that there are no contributions coming from a₀ and a₂ [the first De Witt–Seeley–Gilkey terms in the asymptotic expansion of the spectral action] and the first new term occurs in a₄ where there are only two contributions... Thus the additional terms to the spectral action, up to orders 1/Λ², are 

(3G₀/8π²)ε^µνρσ(2g₁²B_µνB_ρσ - 2g₂²W_μν^αW_ρσ^α) 

where G₀ [is the Newton constant]. The B_µνB_ρσ is a surface term, while W_μν^αW_ρσ^α is topological, and both violate [CP] invariance. The surprising thing is the vanishing of both the gravitational [CP] violating term ε^µνρσR_µνabR_ρσ^ab and the θ QCD term ε^µνρσV_μν^mV_ρσ^m. In this way the θ parameter is naturally zero, and can only be generated by the higher order interactions. The reason behind the vanishing of both terms is that in these two sectors there is a left-right symmetry graded with the matrix γ_F giving an exact cancellation between the left-handed sectors and the right-handed ones. In other words the trace of γ_F vanishes and this implies that the index of the full Dirac operator, using the total grading, vanishes. There is one more condition to solve the strong CP problem which is to have the following condition on the mass matrices of the up quark and down quark

det k^u det k^d = real.  

At present, it is not clear what condition must be imposed on the quarks Dirac operator, in order to obtain such relation. If this condition can be imposed naturally, then it will be possible to show that ([49]) θ_QT+θ_QCD=0 at the tree level, and loop corrections can only change this by orders of less than 10^-9.

Noncommutative Geometry as a Framework for Unification of all Fundamental Interactions including Gravity. Part I

Ali H. Chamseddine, Alain Connes (Submitted on 3 Apr 2010) 

and Noncommutative Geometry in Physics

Ali H. Chamseddine (July 15, 2014 Frontiers of Fundamental Physics 14)

Axion free Left-Right symmetric extensions of the standard model

In this letter we show that in left-right symmetric gauge models, that conserve P and T prior to spontaneous symmetry breakdown the problem of strong CP-non-invariance can be solved without the need for axions or massless quarks. First let us notice that the requirement of left-right symmetry of the entire Lagrangian before symmetry breaking implies that there is no strong CP-violation at this level. Subsequent to the symmetry breakdown, the complex mixings between the different quark flavors introduces weak CP-violation into the theory. But if quarks matrices at the tree level satisfy the condition

 (det M^(+/-))=(det M^(+/-))*                                                   (2) 

where M⁺(M ^-) stand for mass matrices for Q=+2/3 (Q=-1/3) quarks, then the unitary matrices that diagonalize the quark mass matrices obviously will not induce θ. As a result, after the diagonalization of the fermion mass matrices, no strong CP-violationg phase is introduced. Thus, at the tree level θ=0 naturally. We then compute the one loop contribution to the mass including the one loop effects continues to satisfy the relation in eqn. (2). As a result, any non-zero contribution to θ can only arise at two or higher loop level, thus providing a natural suppression of strong CP-non-invariance.

Natural Suppression of Strong P and T Non-Invariance

Rabindra N. Mohapatra and Goran Senjanović, (june 1978)