Discriminating Chiral Insertions from BSM for the Muon (g-2) by non-Abelian Zero Modes

: A sorting out method, between the New Physics and as vs. the Lepton Flavor Violated Chiral insertion is exposed, interpreting so the Light-by-Light scattered amplitude deviation in the muon magnetic moment, with a novel 2-color instead of 3-color in diagrams projective polarizations. The calculation is done using an extension-detention of the muon resonance interaction internal lines (cid:4666)which may be generically of a non-QCD nature(cid:4667) such its modes’ phonons decompose into instantons while the helicities are to meet the imposed polarizations. A confirmation comes out from a non-applicability of the Landau Gauge, to the one of the cases, the tri-vector-mode, giving it a double pole in its Goldstone propagator, vs. its truth in the Sudakov type with a single pole mode Goldstone propagator. The fits between expansions of the phonon derived from slicing’s, and the instanton derived from inverse arguments of differently coupled cosine’s, are surprisingly proportional to their projected (cid:4666)2 to 3 in their extended diagrams(cid:4667) normalization factors paving ways into a BSM popped up selectivity method.

The paper is divided as follows. §2 treats the deduction from the em dipole dipole energy distribution can be Reggeesized to prove its minimality compared to when non-minimally extending with anomaly elimination. §3 deals with the double pole of the Goldstone and when that originates from single poles, so there is a need to fit the incoming phonon with the outgoing instantons. The dependences of the coupled-decoupled system sort out between the above system LFV chiral insertions from new physics. §2. General interpretation a-Vacuum dependent coupled system The Cauchy-Schwartz equation is a general equation that apply to any triangular, penguin or dipolar types of diagrams as long it is kept correct the alternateness of its extra virtuality ∆κ, 12 .
Since the next combined connected topology i.e. next to resonance order, has a geometry of six sides, so for such new physics discontinuity to appear or also when to be approached after an anomaly or a variant form factor , depends on a skip variable, as proved in 1 .
The system separates into either externally coupled to the sides so being its skip and s 6, or not so internally skipping sides with s 5.
Due the logic that two Cauchy-Schwartz vectors are independent of the extension, then the angles can repeat on each side specifically in one direction for s 1 1 s and in the other direction for s 1 , represented however, with its power divided by 4 due when taken in a ratio form such . The formula to be considered is, 1 cos θ 1 cos θ ⟺ cos θ 1 cos θ 1 1 The two sets are characterized respectively by their linked into coupled degrees of freedom as The diagrams are represented in Fig. 1 with a non-minimal non abelian characteristic seen in Dipole due to that p p γ vs. p p k γ, as however a change in the momentum sense sketched as a change in arrow sense will not change its linked to propagator double vertices due to its unique chiral insertion under an imposed magnetic field, while in the Hexagonal case it can have two chiral values at a side or another from the middle traversing line.
These matters are elaborated later eg. in §3-1 for how helicity angles coordinate with polarization.

b-Consequences
The above reasoning is true whenever there are diagrams, taken in illustration as for the muon magnetic moment, which couple differently whether with or without any new physics since that has to do with the specified internal graph number of degrees of freedom, all can strictly be coming from those links related to the free degrees interacting directly with a composite Higgs magnetic resonance, that can cause then from its rising of the internal energy such it reaches higher leptonic states and by so violating the quantum number of the leptons, or ending in an LFV.
As however we can start further with details about the existence also of a new physics.
The due difference in numbers appears since, any created particle, either or not a new physics one, during its induced minimal resonance, with a selected angle could that be induced by a magnetic phase as was derived in part I or any external interaction , will have as the determinant variation for the in-between change, being fully popping up or interacting with any graph leg, see formula k triangle the angle divides then being redistributed on a triangle, however, with the presence of interactions may be picking up some of the vacuum differently, if its graph is the decoupled or the coupled one.
Given also that the graph distributes such over three lattice points for the triangle-hexagon while over two points for the double-triangle-triangle, while however conserving in both the same orientations, here due to the 2-color projective polarization in the hexagon, see §3.4-5 down proofs. §3-Coupled vs. New Physics Given the all of the above criteria related hypothesis, the analysis can be applied on the distinctive decision on new physics, among which the Hadronic Light-by-light, 11 2 nd , scattered amplitude contribution to the muon magnetic moment 11 1 st , with its diagrams topological changes are displayed in Fig. 2. These facts, apply in above diagrams when taken separately, in the totally different Lagrangian. So a confirmed interpretation then concerns a helicity-polarization analytical de-extension since 2-colors proved down prevail on 3's as what behind the energy difference.
The advantage in the method is that one has gained two order of magnitude higher in precision.

1-Simple vs. Double Poles in Goldstone Modes
One should look for an interpreted confirmation, especially in terms of an analytical resolution since an analytical de-extension was the reason behind the energy difference.
That luckily is possible due to the finding that Goldstone modes when in the form vector later justified, as well, longitudinal or scalar can induce double poles in their propagators except when their diagrams operators are gauged under unitary Landau R , ξ → 0, where then the pole is educible to be single.
These facts, apply to the above diagrams when taken separately, in the totally different Lagrangian such containing besides the Dirac usual Lagrangian, with moments lead into by connecting to it, the magnetic dipole has L F F ; whilst the anamolous electric dipole has L θ F F , which is a total divergence, 14 .
One can easily but not immediately see this, by doing a logic similar to resolving the U 1 , but bit differing since it will involve an extra skip, 1 , which can be also prohibitive of one channel full correlation, since by the equivalence theorem scaling property then proved, that leads the channel to enjoy a new degree of freedom.
In fact, unlike the elimination of the singularity appearing in the U 1 , that was due to the lack of the gauge invariance such that that cancels by the new gauge difference.
But if a skip happens within the system, a change of sign can reverse the cancellation and that skip seemed to be into the non-suppressed Goldstone mode, 15 , associated with mesons, 16 , of components that are only longitudinal or scalar so they distinguish from photons.
These components are due to that the Goldstone modes in the Higgs SU 2 doublet are replaced by the meson components multiplied by the Pauli matrices, whose combinations leave only the scalar and the longitudinal components.
If one think in terms of the propagator poles, that gauge is chosen to be the Landau Gauge Unitary R gauge with ξ → 0 .
That would be applicable as accustomed to the mere lonely corresponding graphs derived from topologically equivalent to the ones associated with the Lagrangian L L , and so the double pole, of 16 , becomes a single pole, unlike those of L L . Representing the customary elimination of the singlet current with the doublet.
As for the n 2 the vanishing of the different flavor lepton mass at vacuum or at infinity leads to its neglecting the distinction. With the fact that the total winding or charge has changed by two units, there can be two ν 1 non-abelian, 1, 18 , sub-structures each with a different group, however since only the skip is applied among the inner longitudinal or scalar modes, and while the map is parity independent, the CP violation appears in the charge leading to the non-reduction of the double pole deduced from M q * M Q q , instead of M q * M Q q .
The Logic are shown also in fig. 1, where it is explained the difference in relevance to the twist between the first graph which is invariant, while for the second it is non-invariant.
The non-abelian character is confirmed up to the zero modes even when these two mere connected graph are extended to being non-connected but the gluons lines are attached getting as so the new full connected diagrams, in the application of qcd condensates also superconductive in the form of strings, 19 , see also §3.5 the consistency confirmation down subparagraph.
2-Gradient longitudinal potential: As e.g. Meissner current phonons switched into instantons The extension formulation was derived from a type of regularization, 1 , performed on Higgsgamma-gamma resonant particle creations. However, its link to the occurrence of the single zero emission is only true due to the perturbartion by the W-seagull, 20 , such of minimal or even zero massless multi-spin 1 emissions.
Being in the configuration that of going beyond its minimal resonance, leads to the mode shifting toward collectivity with then the lowest unperturbed emitted modes are analog to the phonon. The phenomena appears because due the current has a derivative structure, 21 , in the form Where A q is the electric potential.
One can check that the different currents generated by different senses inside the hexagon will have an invariant energy as is the case if associated with the Meissner effect, which brings a difference in vacuum energy since by excluding the surrounding field, following the method described by ref.
The factor K for free electrons divides into two pieces respectively coming: For K from the spinor wave being transformed with the gauge Ψ → e Ψ, where τ is the third component of the Pauli matrix, added to it a non-perturbative variartion such instead of a conserved continuity equation it has ∇ . ȷ ⃗ 2Ψ ϕτ Ψ, leading that And for K it is the factor for the longitudinal potential insertion A q q λ q , so expressible to start as a gradient.
As proved by ref. 21 author, after Fourier transforming, the factors K and K cancels out.
The remaining will contain the non-perturbative term K δ 0 Ψ x , Ψ x 0 whose self-vertex representation Γ p, p 2iτ ϕ 5 Will vary as anomalous term, so the phonon of the non-conservative term opens the way for an instantonic solution as proved in App. A&B.

3-Vacuum energy variation
The above reasoning is true whenever there are diagrams, taken in illustration as for the muon magnetic moment, which couple differently whether with or without any new physics since that has to do with the specified internal graph number of degrees of freedom, all can strictly be coming from those links related to the free degrees interacting directly with a composite Higgs magnetic resonance, that can cause then from its rising of the internal energy such it reaches higher leptonic states and by so violating the quantum number of the leptons, or ending in an LFV.
As however we can start further with details about the existence also of a new physics.
The due difference in numbers appears since, any created particle, either or not a new physics one, during its induced minimal resonance, with a selected angle could that be induced by a magnetic phase or any external interaction , will have as the determinant variation for the in-between change, being fully popping up or interacting with any graph leg, see formula 56 in 1 , such like δ∆ Plus, when re-separated or dismantled from the resonance into the minimal triangle the angle divides then being redistributed on a triangle, δ∆ δ∆ 3 J δθ 6 The NP guess, when assuming a zero order correctness in chiral insertion, on is that the double angle θ is the same as the angle θ Arccos 0.8156, which is eliminating non-minimal effects from anomalies, and accompanying among the four on-shell gamma photons.
However, with the presence of interactions may be picking up some of the vacuum differently, if its graph is the decoupled or the coupled one.
The way to see that is the density diffusion approach, 22 , μ J, U cst exp trUtrU .
Given also due to that the graph distributes such over three lattice points for the triangle-hexagon while over two points for the double-triangle-triangle, such appears then as follows μ cst exp J trUtrU And μ cst exp J trUtrU So δJ Then, taking into consideration that the δJ relates to a single leg Hamiltonian, 23 , and that density is an energy density of single legs so .
Therefore, if that re-enters in resonance in the new coupling, it is not just the vacuum δθ . .

8
Which about is simply its moments change from magnetic to electric moments under a single resonance that contains a reduction value, counted from eq. 6 , δθ . .

9
Being either renormalized from higher diagrams so is a multiplicity factor or stemmed from new particle decays with is a part of the NP couplings calculated in 1 .
One has to distinguish then, through that induction of ∆ of an around 1% difference, whose origin appearing through δθ is to be decided by experiment.
The advantage in the method is that one has gained two order of magnitude higher in precision.

4-Chiral insertion new parameterization
One can turn the logic around with the aim that searching for the BSM particles may be over ruled by some deviations from the Standard Model that are directly coupling-wise weaker, such an inert neutrino than the SM model but so may be stronger than the new physics particles' couplings.
By that it is meant, that the neutrino's chiral insertion angle may be the first to be reached.
That turn of logic is done by changing the hypothesis of fixing the value of at the value of then check for the value of θ compared to θ as well as its variation with respect to with the definition of ≡ ρ such that ρ As for the popping extra solution cos θ 1 0, it is conceivable that it is due to that the equation from the start can be written as δ ln 1 cos θ cst δ ln 1 cos θ ⇔ δ ln 0 ⇒ ln cst 12 Where it can be set the constant either null or non-null, giving then the trivial first case.
The way to interpret it is through the Cauchy-Schwartz as one expect a direct linear chiral insertion independent of any impact.
Given the all the above criteria related hypothesis, the analysis can be applied on the distinctive decision on new physics, among which the Hadronic Light-by-light scattered amplitude contribution to the muon magnetic moment 5 , with its diagrams topological changes are displayed in Fig. 2.
That can be geometrically confirmed approximately, due the negligence of the original kink impact whose origin comes from gravity, as well it can be checked that the order of magnitude from the spread of the instanton over the Meisnner field region to that of the constant field phonon are comparable. So that should be supplemented by a regularization factor. In numbers, θ Getting the renormalization factor being β~ 3 2 ⁄ .
Which is to be expected since simply, because two lower energy phonons of constant transverse waves formed from the not yet resonant slices, decayed through the derivative-like adjunction of mass formation into three instanton-like waves that which occur during the coupled resonance, being then spread in the space-like wise.

5-Important extra confirmations based on the imposed conditions
Since Gravity has had a role in the derivation, in the form of the force from dilaton-kink interaction with the Higgs, of the type δθ solution, being associated with a space-like structure reinstating the CP violations 1 , one has then to check the satisfaction or even the consistency about the dilaton field being defined in the right domain. One of the solutions found, 25 , since it is non-static does not have a time-like Killing vector, so all its invariants are space-like as already proved, further, such Brans-Dicke dilaton has a nude singularity-type, since its singular has no horizon, therefore there cannot be any place for the extra above-energy be hidden from which the consistency with the new extra coupled system.
Another consistency confirmation is from the other extreme of electromagnetic fields, where it was proved the non-rule out of the phononic-electronic interaction in the presence of spin fluctuations in thin Superconducting materials, 26 , therefore the need for a CP helicity invariance whose proof was provided above in here.
Theoretically, the existence of the 'orientable' strings with a topological character, since the system derives from the interaction with kink-cusp like topological defects, was not ruled out but never proved assuredly. That was mentioned in the form of that a kink solution in the case of a single 1 1 scalar field is proved stable from the disconnected vacuum manifold π V 1 as a time independent solution, 27 . Now the orientation of the structure under the condition of fig. 1 leads to that with two acted on fields in the diagram have a common ground instead circular that unlike would be the case of three colors map , then the SU 3 group will go to its dual SU 3 , therefore the model has the equivalent O 3 group structure of course complexified which happens of having a stable non-trivial winding sector under the radial orbital variation, 28 , which we started with as an initial generator condition from the acting topological strings into the extension mechanism, 1 .

6-Polarized beams
Experimentally, in the domain of polarized beams, there appeared a full coordination between electron positron polarization as vs. their chiralities, therefore the doubling of the quasi-particles, as an electron particle-hole, applies here in the e e , as shown experimentally in unpolarized beams, the left of Fig. 3 from 29 , while in its center and right no correlation except for identically polarized beams, such is due for the quantum coherence barrier whose energy is two order of magnitudes higher than the phonons break energies, with the later is proved as reached above. Appendices:

A-Deficient Solitons
Consider the non-conformal iterative skipped resolution expressed via a remainder r, such e bπ r r Where γ is the real number ln 2, with b is a dimensional factor for the phase variable whose interpretation appears related to the number allowable of product spaces as, bd ≡ 2ϵ ⟹ b , for the combined d phases, such that when the angle is larger than π, than r is negative.
The limit case values for the resolution is bπ r bπ, so with n the electroweak seagull number, π b bπ r π b ⟹ bπ b bπ r r bπ b A1 Which, for single diagram and the dimension d 4, and b turns 0 bπ r r e A 2 That accepts two conformal solutions only by the unicity of the above skip multitude ϵ 1; n 1 ⟹ 0 e ⟹ e → A 3 ϵ 1; n 2 ⟹ π e π ⟹ e → π A 3 -a ' ϵ 1; n 2 ⟹ π e π ⟹ e → π A 3 -b ' The gotten number k , or ω in above notation, for the given γ, represents a width for a possible wiping out of singularities, therefore such a number can be named as a deficiency number.

B-Instantons Triple Isotropy
The instantons', proved in §3-4, can have an isotropic character as applied to perturbed created particles in the vacuum, like in App. A.
That can be proved as the regularization, in 1 , was based on the fact that M l l , with nonconstant integration was formed on the variable l . Therefore, the non-trivial l l relation is best manifest in its meaning as it imposes an isotropic type condition first between the longitudinal and the transverse total momentum. In the case of the final emission is made of light then also time is isotropic to the longitudinal component making the system triply isotropic, as used in the appendix, concerning its minimal singularity kernel to the extension.
Furthermore, it can be made use the extra degree of freedom remaining in the transverse current to to prove the possible generated magnetic field being constant, as when the vectorial product of j ρ v and j ρ v , where now according to 24 , ρ aρ .