A Covariant Cauchy-Schwartz Measure’s Bounding Conditions to BSM Searches

: Solving for the missing masses in the Higgs resonances, it was necessary to extend, even quantitatively via an index measurable amount, the SM using a threshold related longitudinal violation procedure. The obtained expression, by being non-contributing via its non-anomalously resulting parameter, is linked to a Cauchy-Schwartz 4-scalar product ratio type of two virtual Gauge Bosons momenta in its minimal anomalous configuration, as vs. its non-anomalous internal. Changing the bounds from energy into momenta, a convexity condition appears. Such technique clarifies the perturbative e.m. fields’ extensions into perturbative and non-perturbative QCD. In applications, there is the violation of the chiral insertion by the axion into neutrinos, and the Lepton number when passing form velocity to spin resonances, such confirming the CS procedure as plus the defiance of the SM comes through their branching ratios but not their angular distributions. Further which if remaining at the same level of minimization can restore the universality of extendibility in the Higgs self-couplings.

The property of the derived new resonant mass outer charge which adjoint collectively to the space variation, 1 , based on the cancellation of the chiral gauge by opposition of momenta, 2 , with the pooping up the parity quantum number, can be exploited from isolating then the CP violating quantum numbers which can be then well conditioned in the embedding groups, upon unifying schemes. In such procedure, it can be decided its extremalization order, like 1 st minimal or 2 nd minimal next or even non-minimal as in the asymptotic cases, 3 .
Therefore, it can be aimed to describe a method that seems to be essential in finding extensions beyond a given threshold, that could be containing the e.m., the EW standard model, or reaching gravity as being beyond the EW-Strong SM.
The gotten about idea and its relation to our extensions initiates from that the saturation having a variation that, being independent from the local rapidities, 4, 5 , with however the quantum corrections are manifested by a rush out for new states of occupations that makes the saturation critical momentum varying with the multiplicities, so can be thought of hiding the new physics degeneracy influencing its energy however with, if exists, a local characteristic of the new structure, or at least since the scale of momenta variation is intermediary, that guaranties the scaling keeps the same physics expanding together with since the saturation is conserving the unitarity, 6 , that allows the final high or low energy limits of the system to remain in the Hilbert space guaranteeing then its continuity or at least if discontinuous to be in sequential discreteness, 1 .
In addition to the unitarity property there exist a kind of universality in its way of law expression such that the multiple particle scattering amplitude is expanded up to its 4-pole, dipole and quadrupole while higher multipoles are negligible or likely as mentioned expressed, with also isotropic hadronization without being necessarily thermal, 7,9 . Therefore one should expect that the extension analytic expression to be a sort of a continuation formula for the saturation process. The importance within our formalism of the conformal skip operator is that it can be the selective tool into not missing the link to the renormalization group equations so allowing for a correct physical law expressions, 5 2 nd , as compared to getting short from the phenomenology, 8 .
As since any EFT, 10 , with the methods where in the most relevant domain to the given order or IR diagrams that could be loops triangles etc, e.g. with even a maybe given hidden anomaly . It seemed to be dealt with the fact of taking the Off-shell into the On-shell momenta limit.
Concluding that a BSM search scheme containing relevant operators convexity or positivity bounds 11-13 and convenient fields coupled Dimensional operators 1 & 13 can be tested simultaneously by such covariant measure, especially with when the LFV is resolved by angle independent chiral insertions.
For that higher orders expansion in the perturbative or model extensions in the non-perturbative cases again beyond threshold most probably seemed unlikely to be unraveled systematically, since the used skip is a rescaling, includes respectively the methods of Renormalization equations, e.g. the anomalous dimension is derived from the preexisting mixing relations 14 . Or an extending Unifying Group charge that enjoys ly the property of each of the elementary or pre-unifying subgroup charges, dubbed as a Grand Unifying Group method 15 , without determining if the needed group is minimal or non-minimal with respect to the anomalies and how the so-type is away from minimum, as to be noted what is explained in 1 where there are also non-minimal anomalies from the hexagonal graphs which equally preserve the charges minimally. Then, that can be understood from the fact that the opening or the addition of extra fields in the hexagon if the anomaly to be cancelled comes in the form two symmetrically opposite fields acts on the charges in an abelian manner. Then in both cases, the way dealt with is a sort of complementing instead of supplementing the given model. Plus of appearances of caveats or even in its later gravity applications loopholes, while is needed at least one more scalar to determine the new system, due that the current on-shell mass is the non-conformal remainder of such system other resonances.
Physically, one easy way to start is the remark that the coupling-decoupling of fields is essential, 16 and in its simplest consideration that of around scalars, wich requires two of them to go beyond say photon transverse polarization, then both, if coupled, have shifted little the longitudinal direction and if not coupled than the second scalar has to create a new photon with a different polarization. To conclude a necessity of a characteristic angle most probably an invariant, 17-19 . Two-step proof is needed. Existence then Invariance. Therefore, the paper is organized as follows: In §2 about the an ideal mathematical metric inequality then its development starting from the em up to the QCD vacuum, with the extendibility sets the basis for a space -time completion. §3 deals with what covariant Cauchy-Schwartz will lead to in terms of the Phase-Space Virtuality. §4 treats the many applications, and they are the Inert neutrino insertion with the impact of hard resonances as well from minimally combined spins on Lepton flavor violation under a universality in extension and more on mixing under the Long-Short oscillations. §2-Existence of New Created masses

1-A Cauchy-Schwartz Inequality
The two types of H → 2γ via W 1-loops, such W -W annihilation and triangular W can provide a derivation basis for the Cauchy-Schwartz inequality. The later being done through a dimensional regularization, for the contributing quantity proportional to g I , 1 .
While l M p with p is a 4-momentum due the virtuality of M , then Where, for l M → M p M p M , which leads to a resonance at the limit Further, for the resonance due to the isotropic skips of the final volume after the resonance then, it should be indistinguishable between the bounds 2l l and l , being Λ l Λ M′, with the mass M being of the order of the CP breaking symmetry energy. Recounting Where b ≡ is an extension parameter for the resonance.
Its expression was then derived, giving B ∝ k, where Λ is related to the cut-off linearly, and its lowest state is described by a parameter associated to the Higgs resonant number k. It should be noted further, that the extension B is symmetric with respect to the axis dividing between the two emitted gamma rays, as since that was accompanied with an axion reactive emission, the two rays are not collinear, but little deviated , so that axis has a resultant small angle 2θ with respect to the c.o.m. frame axis. This symmetry has a direct impact leading into the invariant extension is proportional to an area dimension. Therefore ∝ k ∝ A K ⋀κ L cos θ sin θ L sin 2θ 3 Which however changes of sign, when taken from a side to another. Then has no contribution. One can easily remark that for any symmetrically eliminated anomaly.
Having that Higgs is a spontaneously broken particle, it is assumed that it is not at rest and so that imposes the conditions of having its entry and exit together so the internal momenta are following κ ↪ K And κ ↪ K Then Where K , K and κ ↪ are respectively the two-gamma and internal momenta, while K is that of the Higgs, see figure down.
Therefore, it can be verified, since by W virtuality, that Which true for light emissions, with an expected result of the form Cauchy-Schwartz up to a factor 2 as K . K 2 K K , for 2-γ emissions, since any change of direction for the momenta K and K , as further can be due to any improper motion K 0 , doesn't show in the relations.
2-Low non-Minimal Space: Meeting the Leptonic Duality I have to recall a result generated from the Higgs resonances where all three scalar fields the two already axion and dilaton and a third that maybe a non-minimal scalar parameter are coupled so in keeping the dilaton as external the remaining two will have a space-time configuration that can be in its rest frame invariant.
One can set that the continuous source mass is proportional to x x .
That meets the matrix expression in implementing the Dirac sea in vacuum fields, 16 , as S e " S b log 7 Where S is the matrix element for the free field, which can be taken the value around the minimal field where the resonance occurred in a close neighborhood.
That compares to the combined result after setting that the scaled fields are also scaled in their amplitude matrices, with k the number of the Higgs has nothing to do with wave number of elongations, that maybe considered associated with external curved spaces, see details in 1 , k Discrete; ∆ ∝ log log Then k ∝ 8 ∝ exp Continuous; ∆S log so x x ~k 9 The sign will be found later.
Therefore, it can be used the fact that what contributed to the anomaly which goes after the threshold is of course of the type continuous since the start if the anomaly is negligible and in its establishment it was needed an infinite number of states reaching the resonance after which is based on local fields. This is confirmed in the ref. 13 1 st .
Once localized, the creation of masses can be very dependent on the cut-off where a minute variation will not change into a new imposing physics by much as the contribution within the semisimple attached group and the unique acceptable continuous group variation , 13 2 nd .
The repetition inside same process as multiple sequent resonances, will be kind of for each one generator. Algebraically, due to the Killing property, the most it can have two generators.
The idea here is that such two parameters can, after the system will have to obey the validity to extract at least an angle plus an inner virtual phase regime, be described accordingly. While keeping in mind that one has to add on that the ability to extension defying so the irrelevance property.
See later, for an affirmative proof using soft vs. hard decays.

3-QCD Gluonic magnetic Dipole-Dipole interaction reconceived
If one considers the elementary cross section of two dipoles at the two gluon exchange approximation that gives, 9 1

1
By recurring into the calculations done using the leading logarithmic approximation LLA , which however due that such approach is non-unitary, is almost closely complemented by going through the one loop quark contribution to the Reggeon-Reggeon-Gluon RRG radiative correction. As that determines the range of definition so unitarity can be enveloped by melding the transverse momentum here with that explained in next subparagraph § §4 based on Appendix and part II , In fact, the RRG correction deriving from the polarizations in the interaction takes the form ℛ ℒ e * ∑ P q , q a a t t 2k a a 12 The non-vanishing term is the one coming from the ratio transfer as opposed to that from the total gauged momentum such k P 0; k q q as its summation vanishes as ∑ e * P q , q , P 4 → 0 as q → 0 1 2 -a Knowing that corresponding to each of the q is the i channel gluon transferred momentum from the quarks so that favors the lowest momenta by asymptotic freedom.
Meanwhile the expression for the 2 nd term on the RHS is given by But at the limit beyond the threshold, the mass of interacting particles can be neglected compared to their momentum which however when added to the intermediary gluons transfers, as due to the almost symmetry, they have negligible effective total energy and longitudinal momentum compared to the transverse components, meeting the unitarity conditions of 6 2 nd .

12-b '
However, using the above expression for the energy H μ . .

∝ ~1 1 3
Which will be found true for two different basic configurations later by another method.

4-Proof of the Discontinuity between different extensions from an anomaly
The so used unitary gauge, has had its impact on the resonance as well its associated created new masses. However to distinguish between two different extension, one has to remark the property concerning that within each extension a rather conformal gauge governs the same created particles.
Therefore one has to recur on this gauge which happens when expanded in complex variable to take the form of the Mobius transformation. Where it can be remarked that such a gauge and more generally concerning string associated actions , that instead of having field dependent correlations, has them as of coordinates, noted z.
A parameter, that was efficient in applying into the created masses, is the one derived from the skip operator which should be taken as acting on just above the minimal resonance since one maybe looking for non-minimal extensions as well .
Then it can be written, since in the Mobius a full rotation along the ribbon can induce either parity changing configurations for any direction but however keeping the angles invariants, that a ~m v m also a a ~ m δm v m so δa ≅ 2 14 However in the external to the resonance the symmetry breaking can be restored and happens with the fact that m ≪ v, so the integrated new mass from δm will get bigger than the m original noted as m a v A driving equation ϕ ∝ from the resonance is however independent from the induced variation of a , so 2 ⟹ 2 ⇒ m a Log 15 Going to the continuous case, one has that M ∝ exp , so for the Green function can be G z, z ∝ ln |z z | εe 1 6 Where ρ is string coordinate then a skip of its length as transverse coordinate will sweep the full string providing view of the factor ε, and at z z' cst. That corrects for the potential magnitude being its energy variable as G ∝ εe .
The inserted curvature within a fixed skip has the potential of value Assuming a symmetry z z'. Therefore G z, z ∝ ln |z z | εe | | .
And at the width |z z | that, of the outcome of a at continuity, tends towards zero as well as its constrain, except when a new double decoupling takes place occurs so ln |z z | εe → ln |z z | ε | | 1 7 So is back to the discrete case and the new particle is just the first created one.

5-Invariance of certain angle
Due to that in components the longitudinal one played a major role in connection with symmetries as proved in 1 . One may use the above analytic extensions such that the dipole moment can verify the invariances along each of the transverse directions so m . m ∥ x , and taken in the plane almost orthogonal to Q , except for a fixed angle θ, so H μ .
A sample proof is through the dependence of the scattering amplitudes, with 2 → anything is taken as 2 A B → 1 anything 1 C D For the Hidden or new physics concerned the angle here is a tiny one, however to which associates the spin moment remaining due to the transfer of linear momenta. Since such a correction is within the decay amplitude then that derives from the exponential of an action which to give a cosine of an angle should that would be originating from the exponential of an imaginary times an angle As mentioned above an invariance proof is necessary to confirm the usefulness of the on-going extraction. One can proceed on increasing that angle as by the fact of having a Killing associates with an angle increment leading to its invariance in a fixed variation of the domain.
Start such ζ and η under a repetition of a generating operator g in two transverselongitudinal planes ζ ↦ ζ g and η ↦ η g . More, η does not commute with ζ. Algebraically, due to the above Killing property, the most it can have two generators.
The idea here is that such two parameters can, after the system will have to obey the validity to extract at least an angle plus an inner virtual phase regime, be described accordingly. While keeping in mind that one has to add on that the ability to extension defying so the irrelevance property.
See later, for an affirmative proof using soft vs. hard decays. §3-Application of Cauchy-Schwartz in the Cut-off Limits under Opening of Angled Vertices

1-Covariant CS
Also, if the cut-off is approached as e.g. E p E .
More so, it is easily remarked that to reach the cut-off limit, K should have its angular parameter cos θ 1 at a minimal value, with an internal momentum κ such p stays variable. Let be specific about s 3. Due to the unique open vertex, it would be possible physically for the flow to be reversed extensively. Such can be done as ∆‖κ‖ ↔ ∆‖κ‖ , in addition to a noncompact connectivity variable that will be determined next.
Then, the change is counted easily like While sticking to the same positive sign for K . K , conform to 26 2 nd expression in LHS.

1-1 Ratio of Sterile to Axion Mass
To remain in the BSM testing one needs again an extra hexagon after their reductions an odd number of triangles.
A relation of the type 2, gives K K K cos θ 2 cos θ 1.
That is searched to be used probing the dual non-anomalous diagrams, projecting on a plane with pure triangles, leading to, However, in the resolution of the anomaly what is required be inverting K into K and vice versa, from so be gotten, expressing the ratio of on-shell single heavy leptons over gamma pulses ⟹ K 2K 1 cos θ ⇒ 2 1 cos θ 3 7 This result is consistent with the found bounds between the neutrino and the axion decay in two gammas that is then m 2K , and m K .
Its lower bound, 17 , as found in 1 , such Where m is the mass of one lepton, with M represents two Higgs insertion of two conjugate leptons, due to an extra flip in angle, to which one axion to each light still be existent.

1.2-Lepton universality Violation
Let's us consider a soft resonance of a particle interacting with a hard resonance, then the soft hit to the hard one will solicit it just like a reaction that is an inside to the frame containing the soft as a reference of pure kinematics.
It can be written, then at a fixed distance x, the speed from the time of flight τ decay time v With v , v and ν , Where the speeds are normalized to that of light. Taking v ~v means that The consideration v → 0, is done keeping a 1 st order in , such Taking account that v is also soft then it cannot initiate as a resonant state when no new particle is The case ϵ 1, is a continuous map in its spectrum that may approach v → 0, so back to the above case. There remains the case ϵ 1, such v 1 δ 1 1 v 1 v ⟹ 44 One has to note that the difference in order between v and v in order.
Back to compare with formula 38 , when applied to the lepton universality deviation taken with a shift in the direction senses such like v → v, v → v so to absorb the factor 2. Deduce that 0.846 4 5 That ratio is the one gotten in the LHCb experiment, 18 1 st .
The unique shift in senses is confirmed in the latest experiments 18 2 nd , where no tension with SM stemmed with angular distributions to the contrary to branching ratios. As however for the interpretation of angular elimination, that is done later in the part II of this paper, 19 .

-A generic conception of spins
The dynamics for no phases in the extra plane for long-lived heavy scalar decays ϕ into hadrons, 20 , occurs then simply on the extension line. So to speculate a proportionality between the minimal to non-minimal life-times τ out of a resonant of the extendibility parameter c ∆ . When taken between two arbitrarily spin consecutive anomalies c ∇∆ .
In fact, the geometrical repetition of two juxtaposed triangles containing each two W, with the triangles' sided side linear to H, so, 2K ↦ 2K 2K , as is the space composition factor for resultant spin 2 between 1H and 2W. As one thinks of the coupled lines between ϕ and an intermediary scalar s, as the promoter for the change of spin instead of relativistic speed.
Then, applying the above such that is a resonance of ∆Spin 2 , compared to one ∆Spin 0, ∇∆ ∆ , As can be checked for the almost vs. cτ injective correspondence in ref. 21 fig. 4 .
This property of discreteness as also conserved under z z', so that would be enough for the system to be developed in terms of its extrema, especially to the second order where they are defined by their derivatives solving for the short distance coupling.

2-Universal extendibility
It is further, realized that since the presence of new scalars in the resonance, can rescale the structure fully, then due to the linear nature of that rescaling that can have a linear correction on the coefficient for the effective potential describing the Higgs full radiative corrections. This restores the extendibility universality lost in 22 .
Numerically, the corrections are corresponding to have the elimination of the 2-Higgs coupling entering the 1-loop through the obligation of two triangles getting non-anomalous out of the three with each expanding into 3 trees times 3 trees and 3-trees times 3 loops. Since a coupled system from a same leg requires a new scalar particle so a different correction, it has to be going to two different legs for the corrections be the same for different anomalies. Then, among the 3 diagrams with each one loop there couple C 3 that eliminate as from the remaining non-coupled Higgs out the three so in total the resultant is the required 15/9 5/3.

3.1-Extensive phase planes
To help systemize the BSM search whose proof is done later, a direct intuitive deduction appears from the Cauchy-Schwartz. The physical implication from the ratio for the two different domains Interpreting goes by inclusion, in the 1 st consideration, of a new ladder for the interaction then its isospin is I 2, whilst in the 2 nd is still to be I 0, in the decay K → π π , 23 .
Extra decay planes extension method applies by phase validity on K → π νν , 17  Where the skip is seen due the above reasoning be corresponding to the time dual to 3-planes.
Concerning the phases, δ and δ , an estimation can show that is the one transverse direction case in both but as differently positioned for each: The two distinguished alignments, Open I 0 with angle between P and P zero case And closed I 2 with angle of P and P is 90 One which is the closed, being have originated a same exchange symmetry between its pions.
So to divide the phase with the exchange longitudinal-transverse plane in two, and getting into I 0 Has δ ≅ And I 2 Has δ ϵ ϵ ≅ To imply absolutely that δ δ 5 3 Where the value of is a non-perturbative effect whose order is close to being independent of the considered meson, understood from the plain accumulation of masses along valid surfaces.
Experimental, it shows in the ratio's concerning, A and A but one then has to correct by a Since there is a new variation in only one, such that shows as ~1 plus to n of product quarks, a correction for the extra phase with an exponential power 2 of eq. 33 , and so possible 2 being only for 1-loop matching, due that the phase is distributed like 3 1 τ instead of 3 .
Getting a factor, with τ τ 0.0047 and a single longitudinal projection match, such Then, that match has to link to the K case as a correcting factor for the phase, and specifically to the amplitude, through by K -K mixing, in the following manner and just to 1 st order in ϵ δα: As that can be proved from, while adopting the notation Γ ≡ Γ and Γ ≡ Γ , Therefore, at a fixed interval of time extracting a discontinuity continuous limit from a projection lift longitudinally will be delayed twice by the transverse exchange, each set apart by the CP symmetry as the same as an azimuth plus zenith angle composition sin θ 2 tan θ cos ωτ ⇔ sin θ 2 . √ ≅ 0.58 ⟹ cos θ 0.8156 63

General interpretation
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.
The above four results can be correlated due to the fact that a new creation of masses, e.g. as in described in 1 , can be simplified made as well as in two stages EFT with effective double fermionic vertices. One wise constructive, having one left projection and plain Dirac that can build a tangent to longitude but deficient, and one wise destructive having one left projection and pseudo-Dirac, that will twist the already constructed structure into a new but containing the new particles.
Such two dimension 6 operators noted respectively O , and O , are described in ref. 27 .
By taking a combined resonance, where for s, that follows the inequality 0 ∆‖κ‖ 2sΛ M , has no limit except for being cut-off by an imposing symmetry, the next connected topology i.e. next to resonance order has a geometry of six sides, see fig 1.
So such for the new physics to appear when be composed, which as proved above, a discontinuity depended on a skip variable, then the system separates into either externally coupled so being its skip and s 6 or not so internally skipping with s 5.
If the number of invariant alternate exchanging in a triangular diagram is designated by i, and so is a multiple . It represents the factor s 1 in eq. 32 however, with its power divided by 4 due when taken in a ratio form such since that the 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 .
The formula to be considered is, with approximating second order being around 10% at end 1 cos θ 1 cos θ ⟺ cos θ 1 cos θ 1 ≅ 1 1 cos θ 1 ≅ cos θ 1.8 6 4 View the last result being outgone from the extension discontinuity it can be associated with the vacuum correction; that is the corresponding for the value i 1 in equation 32 . Then, the values for i 0 0 are gotten by inverting eq. 64 using eq. 63 The numbers can be deduced, and are collected in the down Table,  Right all vertices are external.

Appendix: Deficient Solitons
If N is the frequency number in the full space for the structure of an extension. Using an isotropic space-like configuration, that gives ω N A 1 Where ω can be related to the time operation since it is the frequent number in its longitudinal direction.
A simple direct count for an extension is based on its frequency connection to the skipping operation. From a basic ground, the simplest repetition of a system is its doubling. However due that surface to be doubled that gives a 2 to be powered by 2 but the parameter is a product space that can be given as its irrelevant frequency ω , like B / .
So with B 3π B / 2 ⇒ ω ≅ 2.444 A 2 This number as it is indirectly related to the full solid angle 2π is a sphere fibration and can be linked to other topological numbers likes that of Hopf and Skyrme.