Lorentz Symmetry Violation in QCD and the Frustration of Asymptotic Freedom

We study the effect of Lorentz symmetry violation (LSV) on the behaviour at high energy of SU(N) gauge theory with quarks in the fundamental representation. The approach is similar to that for QED treated in a previous paper. In contrast to QED, standard Lorentz invariant QCD is asymptotically free. Our aim is to explore the structure of the renormalisation group at high energy and hence weak coupling without requiring the Lorentz symmetry breaking to be small. The simplest type of LSV leaves the theory invariant under a subgroup of the Lorentz group that preserves a (time-like) 4-vector. We examine this case in detail and find that asymptotic freedom is frustrated. That is, at sufficiently high energy the running coupling constant attains a minimum value before increasing again, while the LSV parameter increases without bound.


Introduction
Lorentz symmetry violation (LSV) in QED has been studied by a number of authors concerned with its consistency with causality, unitarity [1,2,3], the structure of asymptotic states and renormalisation theory [4,5,6]. In previous papers [7,8] we studied some of these issues in QED starting with a premetric formulation [9,10] based on an action where F µν (x) is the standard electromagnetic field tensor and the (constant) background tensor U µνστ has the same symmetry properties as the Riemann tensor in General Relativity, namely and U µνστ + U µστ ν + U µτ νσ = 0.
This latter condition excludes parity violation. An outcome of the analysis was that even when the Lorentz symmetry violation is not constrained to be small the behaviour of the renormalised theory in the infra-red limit is dominated by the fixed point at zero coupling in a manner consistent with Lorentz symmetry. That is at a sufficiently large scale in spacetime Lorentz symmetry re-emerges. This is consistent with related earlier work [11,12,13].
In this paper we study a QCD type model with SU (N ) gauge symmetry. In addition to the gauge field we include a quark field that transforms under the fundamental representation of SU (N ). A closely related model is investigated in reference [14]. The significance of such a theory is that it exhibits asymptotic freedom, that is, its behaviour at high energy is controlled, at least in the standard case of Lorentz invariance, by a weak coupling fixed point [15,16]. Our aim here is to investigate the manner in which asymptotic freedom is modified by the presence of Lorentz symmetry violation. An investigation with similar aims, in particular comparing QED and QCD is presented in reference [17]. Although we look in detail only at the simplest type of LSV, we set out the general theory in a manner parallel to reference [7] in order to clarify the logical structure of the argument. This prepares a framework for analyses of more complex models.
In the obvious generalisation of the case of QED we take the action for the SU (N ) gauge field to be where F aµν (x) is the standard gauge field tensor transforming according to the orthogonal representation of SU (N ). For a general choice of U µνστ this action although gauge invariant is not in general Lorentz invariant. Lorentz invariance with respect to a metric g µν can be recovered by choosing U µνστ = g µσ g ντ − g νσ g µτ .
Although there is a priori no metric in the general case with LSV, there is nevertheless, as argued in reference [7], a preferred metric g µν that allows us to decompose U µνστ in the following way where the tensor C µνστ has the same symmetries as the Weyl tensor in General Relativity. That is C µνστ = −C νµστ = C στ µν , and C µνστ + C µστ ν + C µτ νσ = 0.
In addition it satisfies the trace condition g µσ C µνστ = 0.
We refer to C µνστ as a Weyl-like tensor (WLT). It follows that the WLT determines the nature of the LSV. As in the case of QED the possible types of LSV can be determined by applying the Petrov classification to the WLT [18]. A useful approach to the Petrov scheme is contained in references [19,20]. Its application in QED with LSV is presented in reference [7]. There are six  cases, conventionally labeled O,N,D,I,II,III. Each case has a canonical form for the WLT [21]. Class O corresponds to the case C µνστ = 0 which for pure gauge theory implies no LSV. However as in the case of QED [7], the quark field can engender LSV in the model through its contribution to vacuum polarisation provided the associated metric for quark propagation shares with the gluon metric an invariance under a subgroup of the Lorentz group that is the little group of the given 4-vector [22]. The 4-vector can be time-like, space-like or light-like (with respect to both metrics). The time-like case implies that there is a reference frame in which the theory is invariant under rotations of the spatial axes. This is the case we study in detail. However it is convenient to set out the scheme for quantising and renormalising the theory in a general form. Canonical forms for the WLT in other Petrov classes and the implications for the vector meson dispersion relations are the same as those for photons in QED as described in detail in reference [7].

Gauge Fixing and Ghost Fields
In terms of the vector gauge fields A aµ (x) the tensor fields are given by where g is the gauge field coupling constant and f abc are the structure constants of SU(N). In order to deal with the gauge invariance of the action for the vector fields in eq(4) we follow the approach in reference [7] and impose the gauge condition Here Λ µν is a metric-like tensor which we will find it convenient to distinguish from g µν because the two tensors behave differently under the renormalisation procedure. We are therefore led to add a gauge fixing term to the action of the form In addition and in contrast to the case of QED [7], we must introduce anticommuting ghost fields c a (x) andc a (x) in order to construct in the standard way the Fadeev-Popov determinant in the path integral formalism for the computation of Greens functions in the gauge theory. We therefore complete the action for the gauge theory by adding a term Here D abν = δ ab ∂ ν + gf abc A cν (x) is the gauge covariant derivative for the ghost fields. The complete action for the theory is S where 3 Feynman Rules The Feynman rules for the theory can be read off from the action S in the standard way. They are in certain respects analogous to the corresponding rules for BIMQED [7].

Feynman Propagator
The Feynman propagator, illustrated in Fig 1( where M µν (q) is the matrix inverse to M µν (q) and More explicitly Here Q µ = Λ µα q α . It is easy to verify that when C µανβ vanishes and Λ µν = g µν this reduces to the standard Lorentz invariant form. Following the analysis for the photon propagator in reference [7], we first introduce M µν (q) which is the form taken by M µν (q) when indeed Λ µν is replaced by g µν that is The inverse matrix is M µν (q) (see reference [7]) and it can be used to construct M µν (q) in the form A careful analysis [7] shows that the Feynman propagator has the same vector meson poles as M µν (q) together with the ghost poles at q.Q = Λ µν q µ q ν = 0. It is straightforward to check that were we to set Λ µν = g µν then we find

Gluon Vertices
The three-gluon vertex, Fig 1(ii), is For the four-gluon vertex Fig 1(iii) we have Of course momentum conservation is enforced at each vertex. Again it is easy to verify that in the absence of LSV these vertices reduce to standard form (see for example [23]).

Ghost Propagator and Vertex
The Feynman propagator for the ghost fields, Fig 2( where P µ = Λ µν p ν and P.p = P µ p µ = Λ µν p µ p ν . Momentum follows the ghost direction. From this it is obvious that the mass-shell condition for the ghosts is P.p = Λ µν p µ p ν = 0. The vertex coupling the ghosts to the vector field is indicated in Fig 2(ii) and has the form

Quark Field
The model can be extended by including one or more spinor fields each transforming under SU (N ). For simplicity we will consider the case of one such field. Modifications can be added later. The action for this field is S qu where whereē µ a is the vierbein associated with the spinor field and {γ a } are the standard Dirac γ-matrices. The metric associated with the spinor field isḡ µν = η abēµ aē ν b . This is a second source of LSV in the model. Of course D µ [A] is the gauge covariant derivative appropriate to the spinor field. The Feynman propagator for the spinor field is indicated in Fig 3( and the coupling to the gauge field is indicated in Fig 3(ii) Here t a is the SU (N ) generator appropriate to the representation of the spinor field (see reference [23]). We use dimensional regularisation [24] in order to deal with the ultraviolet divergences of the theory. The Feynman diagrams for the perturbation series for the Greens functions of the theory are evaluated in n-dimensions. All the parameters of the theory, coupling constant, mass, metrics and WLT are subject to corrections involving UV divergences. We denote the bare quantities with an extra suffix 0. Each bare parameter is expanded in terms of a renormalised coupling constant g. For example Here µ is the scale associated with the renormalised coupling g [24]. The coefficients g (k) depend on the dimension n and exhibit poles of various orders at n = 4. For example g (1) has a simple pole at n = 4. The other parameters are similarly replaced by bare versions that are expanded in powers of the renormalised coupling g.
Again the coefficients in the various expansions exhibit poles at n = 4. Note that we are free to assume that det g µν 0 = det g µν = −1. This implies that 5 Perturbative Calulations at One-loop

Renormalisation of Gauge Field Propagator
The vacuum polarisation tensor Σ µν ab (q) determines the renormalisation properties of the (inverse) gluon propagator ∆ µν ab (q). The (one-loop) diagrams that contribute to iΣ µν ab (q) are shown in Fig 4 (see reference [23]). We have, to O(g 2 ) Here the inverse Feynman propagator is expressed in terms of the bare parameters, In principle the contributions to the vacuum polarisation are also computed from the appropriate Feynman diagrams using the bare parameters. However our calculation will be of O(g 2 ) so we need only use the lowest order expansions in eqs (30). This amounts to using the renormalised parameters in the vertices and propagators when computing iΣ µν ab (q). In addition the UV divergences of iΣ µν ab (q) occur only in the lowest terms in its Taylor expansion in q α . On general grounds then we can exhibit the UV divergences to O(g 2 ) by writing where the O(q 4 ) terms are UV-finite. The tensor W µανβ has the same symmetry properties as the Riemann tensor. Hence, in a standard fashion, it can be expressed in the form We have also It follws that V µανβ , having the appropriate symmetries and trace properties, is a WLT. In equations (36) and (37) we have used the 4-D decomposition which is adequate when computing the residues of the pole at n = 4. The type of Lorentz symmetry breaking exhibited by the model can be specified by means of the Petrov classification of Weyl tensors. We will return to this point later. From equations (29), (30) and (33) we see that where ∆ µν F ab (q) is obtained from ∆ µν F 0ab (q) by replacing the bare parameters with their renormalised versions and The renormalisation parameters are fixed by requiring that the renormalisation of ∆ µν 0ab (q) reduces to an overall multiplicative factor, that is This is achieved by requiring that Note that the results in equations(41) do imply that g (1)µν g µν = 0. Furthermore the renormalisation of Λ µν 0 is multiplicative and involves the field renormalisation factor Z 1/2 where This is not necessarily true of the metric renormalisation. For this reason it is conceptually convenient to distinguish the bare metric from the bare ghost metric. However at the level of one loop in perturbation theory it is possible and convenient to allow the two metrics to coincide. Higher order calculations may require the maintenance of the distinction, imposed order by order, between the two metrics.

Renormalisation of Quark Propagator
The renormalisation of the quark propagator iS 0 (p) proceeds along similar lines. We have where and γ a are standard Dirac γ-matrices. The quark propagator is implicitly a unit operator in SU (N ) space. The self-energy amplitude iΣ(p) can be calculated (at one loop) from the diagram in Fig 5 using the Feynman rules with renormalised parameters. The UV divergences can be exposed in the Taylor expansion where the contribution O(p 2 ) is finite at n = 4, and we have and We write where h µ µ = 0. The term H determines the quark propagator renormalisation Z q , while h µ ν fixes the counter-term inē µ 0 a . More explicitly we have, taking account only of the poles at n = 4, and That is

Renormalisation of Coupling Constant
The coupling constant renormalisation is most easily followed by considering the quark-gluon vertex. The relevant diagrams are shown in Fig 6 and yield contributions to the truncated threepoint function that render it finite after appropriate field renormalisations. We can anticipate the nature of the divergences by examining the bare vertex V µ 0a = ig 0γ µ 0 t a . From eqs (29) and That is The point we make here is that the renormalisation of the quark vierbein enters into the vertex calculation and the contribution V µ a (p, p ) to the three point function from the diagrams in Fig 6 must be consistent with this. We can expect then that V µ a (p, p ) will (at zero external momenta) have the form This will be verified in particular calculations. The truncated three point function V (3)µ a (p, p ) will, at zero external momentum, satisfy Finally g (1) is determined by requiring that the right side of this equation is rendered finite by extracting the field renormalisation factors Z −1 Z −1/2 q , implying We will look at this in more detail when evaluating the vertex in the special case of LSV we consider below.

Renormalisation Group
From the results in section 5 we can obtain the renormalisation group equations for the renormalised parameters to lowest non-trivial order in the coupling constant g. These are derived from the requirement that the bare parameters are indepependent of the renormalisation scale µ. For the coupling constant we have From eq(57) we then obtain the renormalisation group β-function Note that in deriving eq(61) from eq(60) we have ignored derivatives of H, W and K since they are of O(g 2 ) and may, and indeed must, be ignored at one loop. Of course the first term on the right of eq(61) vanishes in four dimensions.
Following the same principles we obtain for the gluon metric and We obtain also for the renormalised mass The renormalisation scheme set out here can be applied to LSV associated with a WLT of any of the Petrov classes. However even at one loop the calculations are rather complex. Partly then for reasons of simplicity we restrict attention in this paper to models in Petrov class O. The results nevertheless remain interesting.

Renormalisation for Petrov Class O
As explained above, in the case of Petrov class O, the LSV is due entirely to the difference between the light-cone associated with the gluons and that associated with the quark field. That is, we are assuming that the tensor C µνστ vanishes. This is possible if there is a reference frame in which rotational invariance holds simultaneously for both metrics. Given the symmetry properties of C µνστ it is consistent with this rotational invariance only if it is null. Similar remarks apply to situations where the preserved subgroup of the Lorentz group, in a suitable frame, leaves invariant either a purely space-like or light-like vector. We will concentrate on the rotationally invariant case. We have for the gluon metric and for the quark metricḡ We assume that so that det g µν = detḡ µν = −1. We have similar forms for the bare metrics g µν 0 andḡ µν 0 in terms of the bare parameters (α 0 , β 0 ) and (ᾱ 0 ,β 0 ) which have appropriate expansions in powers of the renormalised coupling. That is together with similar expansions for the other bare parameters. It is convenient to relate the two metrics by settingᾱ = aα andβ = bβ with the consequence that ab 3 = 1. Similar remarks hold in an obvious way for bare parameters a 0 and b 0 . The significance then of b is that in a coordinate frame in which the gluon metric is diagonal with entries(1, −1, −1, −1) the quarks have a lightcone associated with a velocity c q = b 2 . At appropriately high energies we would expect that when b > 1 the quarks travel faster than the gluons and slower when b < 1.

Vacuum Polarisation for Petrov Class O
In order to carry out the renormalisation process we assume the renormalised gauge metric satisfies (to one loop order) Λ µν = g µν . The vanishing of C µνστ then yields for the renormalised gluon propagator M µν (q) = g µν q 2 . The vertex factors, as mentioned above, acquire standard form for the gauge theory without (at this stage) any LSV. We can therefore use the discussion of non-abelian gauge theories in reference [23] to evaluate the contributions from the diagrams in Fig 4(i), Fig 4(ii) and Fig 4(iv) to obtain the gauge field contribution to the UV divergence at one loop as where C 2 (G) is the value of the quadratic Casimir operator in the adjoint representation of SU (N ). As explained in reference [23] the diagram in Fig 4(iv) does not contribute to this pole residue. In a similar way, we can use the results in reference [23] to evaluate the quark contribution to the vacuum polarisation from Fig 4(iii) provided we useḡ µν as the appropriate metric. The result is From eq(36) we see that contribution to the tensor W µανβ from the gluon field term is and It is immediately obvious V (g)µν vanishes as does V (g)µανβ . From the quark field term we have We have then and It follows that Of course Finally then from eqs (41) and (78) we have q k+q k Figure 5: Quark self energy that is and In the limit of Lorentz invariance a = b = 1 and we have α 0 = α and β 0 = β. The field renormalisation factor is

Quark Self-Energy for Petrov Class O
The results for the calculations in the previous section, because each term involves only a single metric, can be read off from standard results (see reference [23]). However the one loop self-energy correction to the quark propagator is obtained from the Feynman rules applied to the diagram in Fig 5. Both metrics are involved. A possible conflict with causality might be anticipated. This possibility has been discussed previously and the conclusion is that in the present case there is no difficulty, as is confirmed directly in the calculations [25,7,1,2,3]. We have then We require and The evaluation of these terms is along the same lines as the corresponding calculation for QED in [7]. We find for the UV poles at n = 4, with the result We have also .

Coupling Constant Renormalisation for Petrov Class O
The one loop diagrams in Fig 6 yield the coupling constant renormalisation. For the computation of the UV pole divergence it is sufficient to calculate the vertex with p = p = 0. From the first diagram we obtain V λ a (0, 0) = g 3 t b t a t b g µν I µλν , where On omitting terms that do not contribute to the UV divergence we have The result using t b t a t b = (C 2 (N ) − C 2 (G)/2)t a and ab 3 = 1 is We have also In the case of the second diagram we note that in the limit of zero external momenta the internal three-gluon vertex reduces to V λµν abc = −gf abc k ρ (U ρνλµ − U ρµνλ ) = −gf abc k ρ (2g ρλ g νµ − g ρµ g νλ − g ρν g µλ ). (101) The contribution to the vertex becomes We have again omitted terms that do not contribute to the UV pole at n = 4. The tensorT αρ is closely related to T αρ in eq(98). Finally we have the contributions and Combining the two sets of results we obtain and This leads to V λ a (0, 0) = 4 where As expected from eq(56) the second contribution to R λ ρ yields the correct term ∝ h λ ρ . The bare coupling and the one loop correction yields (110) Thus from eq(59) we find that The remaining UV divergences, as shown in eq(59), are removed by appropriate field renormalisation factors and then finally by the pole in the coupling constant expansion. We have then from eq(110) the result for the β-function for the renormalised coupling Obviously the first term vanishes in four dimensions. The second term reduces to the standard answer when b = 1 and there is no Lorentz symmetry breaking.

Renormalisation Group for Petrov class O
In four dimensions then, the important renormalisation group equations are for g and b. They take the form and In general the two variables influence one another as they evolve along the RG trajectory. However some points in (g, b)-space are particularly significant. The Lorentz invariant situation b = 1 is stable and is maintained under the RG. The coupling constant g then runs to zero, its fixed point, in the standard way as µ rises to infinity. The rate at which g drops is the result of a competition between the contributions of the gauge field and the quark field to the vacuum polarisation. When we explore values of b = 1 we see that the effect of the quark field is enhanced with the result that β(g) vanishes when b satisfies That is where R = 11C 2 (G). On these two lines ∂b/∂µ remains non-vanishing. The RG trajectories cross the lines and at the crossing point the coupling constant attains a minimum value. It increases again as the scaling energy µ continues to increase.
If we modify the model so that it contains n f quarks, all sharing the same metric, then eq(113) becomes Eq(114) remains unchanged. The minimum of g occurs when that is when b = b ± where now R = 11C 2 (G)/n f . The RG equation for the quark mass is obtained from eq(64) and eq(87). It is

Discussion
The behaviour of the RG trajectory described above shows that a frustration of asymptotic freedom can arise in the presence of LSV, at least in this model. The question arises as to whether it might be observable experimentally. There are several issues to be considered, for example the relationship of the lab frame to the gluon frame used in the above discussion. We set this matter provisionally aside, though it must ultimately be resolved, and assume that the lab frame is travelling slowly relative to the gluon frame. More significant is the size of the energy range implicit in the model. We introduce an initial energy M I and associated LSV parameter b I , QCD coupling α I = α S (M I ) and the energy M min at which the coupling reaches its minimum value. In a gesture towards "reality" we consider the case SU (3) gauge theory, where C 2 (G) = 3 and C 2 (3) = 4/3 [23] with n f = 6 quarks. In this case b + = 2.3947 and b − = 0.4339. The light cone of the quarks in the gluon frame at minimum coupling is c q = b 2 + = 5.311 for LSV with In order that our asymptotic calculation be relevant we must assume M I is sufficiently large and in particular is greater than the top quark mass, that is 173GeV . In this energy regime there is no easy way to relate our calculation to low energy determinations of α S (µ). The story of the running coupling and its relationship to low energy phenomena and Λ QCD is complicated. It is comprehensively reviewed in [26]. An important point is the subtraction scheme used to obtain finite results for physical quantities. We are using the minimal subtraction (M S) scheme [24]. More widely used is the M S scheme introduced in [27] to improve convergence. In that scheme the strong coupling has an evaluation α S (M Z 90GeV ) 0.12. On the grounds that our calculation is exploratory we feel justified in neglecting the difference in subtraction schemes and propose α S (M I ) = 0.1 when M I = 10 2 − 10 3 GeV . The qualitative nature of the results is not altered by (relatively) small changes in our initial conditions.
With these initial conditions the results for examples of the renormalisation trajectory, obtained by numerical integration (2nd order Runge-Kutta) of eq(117) and eq(114) are shown in Fig 7. Obviously the closer the initial value b I is to unity the closer the renormalisation group trajectory stays near the Lorentz symmetry line and the later it breaks away, heading for its minimum value. These results are illustrated in Fig 8 which shows the connection between log 10 (M min /M I ) and b I . The smooth curve is obtained by fitting the rightmost point on the plot. Even for this implausibly high value b I = 1.1 at our initial energy scale M I = 10 2 GeV we still find M min 10 25 M I . Tuning b I down to potentially more realistic values results in yet greater disparities in the orders of magnitude of M min and experimentally attainable values for M I . The conclusion must therefore be that for QCD with the known set of quarks there is little hope of observing any of the frustration of asymptotic freedom in accelarator experiments. However the complex asymptotic behaviour that we encounter in this model may have relevance to very high energy processes at very early times in the initiating big bang of the universe. In view of the fact that the energy range associated with frustration of aymptotic freedom appears to lie well above the Planck mass (M P 10 19 GeV ) where gravitational effects must become important, one might question its physical relevance. However it may also be possible and would certainly be interesting to relate the behaviour of α S (M ) when M M P to models of quantum gravity constructed with appropriate running LSV parameters [28,29].
These considerations do not preclude the possibility of discovering LSV effects in an energy range for which b remains close to unity. For example if we set b = 1 + x and assume x is small then to lowest order in x eq(117) and eq(114) become where A = (11C 2 (G) − 2n f )/3 and B = 4(1 + 4C 2 (N ))/3. In this approximation the RG trajectories have the form where τ = 0.762. In the asymptotic energy range then, the renormalised mass reduces, also relatively slowly, with a power of the renormalised coupling. If effective methods were developed for computing the structure and scattering of high energy particles in the model (see references [30,31] for related discussions in QED and the Standard Model Extension) then possibly it could provide guidance for accelerator experiments and cosmic ray detectors investigating LSV phenomena in a high energy regime of P eV and beyond. For example, if we take (intuitively) the quark metric, diag(a, −b − b − b), as determining the dispersion relation for quark based states, it would become for a particle with mass m, energy E and momentum P Combining the above results we find for the velocity of quark based particles v = dE dp where and The point here being that the coefficients in the dispersion relation depend only on α S /α I and therefore vary only logarithmically with the energy E.  This is qualitatively different from LSV originating in higher derivative contributions to the QCD Lagrangian [32,33] or spacetime foam models [34,35]. These are parametrised by large mass scales and suggest powerlaw increases in energy. In our case eq(129) suggests a slow logarithmic increase that we might expect to be more difficult to detect. However were LSV to have been detected the suggested energy dependence would distinguish this QCD model from such higher derivative models.

Conclusions
We have studied an SU (N ) QCD model with quarks in the fundamental representation and formulated the perturbation series to one loop with no restriction on the magnitude of the Lorentz symmetry breaking. In the particular case we studied the LSV was due entirely to a mismatch between the lightcones of the quarks and gluons. This is a consistent possibility if the lightcones are generated by two metrics that are both invariant under the same subgroup of the Lorentz group that leaves a 4-vector, time-like in both metrics, invariant, a rotation group in fact. Similar results can be obtained with space-like and light-like vectors. The renormalisation group equation for the coupling constant α S and the LSV parameter b was obtained with the result, exhibited in Fig7, that initially α S decreases with energy just as in the standard Lorentz symmetric case. However b departs from unity increasingly with energy and this enhances the contribution of the quark vacuum polarisation to the β-function for α S . The outcome is that at sufficiently high energy α S ceases to decrease, reaches a minimum and then increases again with energy. This constitutes the frustration of asymptotic freedom in QCD with LSV of the kind we have investigated. We suggest plausible values for the energy range E > M I we are investigating and the associated initial value α I for the strong coupling. The outcome is that the frustration part of the RG trajectory for (α S , b) is at energies many orders of magnitude greater than is accessible to accelerator experiments. It is well above the Planck mass. However it is possible that part of the RG trajectory lying near the Lorentz symmetry line b = 1 might be attainable in accelerator or cosmic ray observations. The effect on the dispersion relation of particles is through powers of α S and hence is logarithmic in character and represents a kind of intrinsic LSV rather than one parametrised by higher derivative contributions to the Lagrangian.
There are many variations of the model that might be investigated such as increasing the number of quarks, varying the quark metrics in ways that induce more complex LSV associated with higher Petrov classes. Of course one should also consider how these results relate to the full structure of the Standard Model and its extensions. Finally it is worth noting that in the context of relatively weak LSV it may be possible to pursue a nonperturbative investigation of our model using the techniques of lattice QCD.