The Higgs mass derived from the U(3) Lie group

The Higgs mass value is derived from a Hamiltonian on the Lie group U(3) where we relate strong and electroweak energy scales. The baryon states of nucleon and delta resonances originate in specific Bloch wave degrees of freedom coupled to a Higgs mechanism which also gives rise to the usual gauge boson masses. The derived Higgs mass is around 125 GeV. From the same Hamiltonian we derive the relative neutron to proton mass ratio and the N and Delta mass spectra. All compare rather well with the experimental values. We predict scarce neutral flavor baryon singlets that should be visible in scattering cross sections for negative pions on protons, in photoproduction on neutrons, in neutron diffraction dissociation experiments and in invariant mass spectra of protons and negative pions in B-decays. The fundamental predictions are based on just one length scale and the fine structure constant. More particular predictions rely also on the weak mixing angle and the up-down quark flavor mixing matrix element. With differential forms on the measure-scaled wavefunction, we could generate approximate parton distribution functions for the u and d valence quarks of the proton that compare well with established experimental analysis.


Introduction
Ever since the proposal of the so-called Higgs mechanism 50 years ago [1][2][3][4][5] and, especially after the experimental findings and confirmations of the Higgs particle during the last two years, [6][7][8][9] the big question is how to calculate its mass -because the standard model did not contain a recipe for that. To remedy this, we make a step towards a correlation of the quantum chromodynamics of strong interactions with the quantum flavor dynamics of electroweak interactions. To make the step, we digress from quantum field theory into a common U (3) configuration space where color and flavor are intermingled. At first sight this might seem confusing, but we shall show how one can project out both quark and gluon fields with the usual transformation properties.
The standard model contains quite a few unexplained parameters such as the six quark mass parameters, the three angles and one phase of the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix, 10 the several coefficients and exponents of each parton distribution function and a similar wealth for the six leptons. The strong and electroweak interactions are described by seemingly independent gauge groups SU (3) and U (2) ∼ = SU (2)×U (1). 11 Baryons feel both interactions wherefore we seek a description from a common Lie group background. The simplest choice is U (3) which contains SU (3), SU (2) and U (1) as exemplar subgroups. With this choice, we can reduce considerably the number of parameters needed to describe baryon mass spectra and the Higgs mass. We stress that the group U (3) is generated from three parametric momentum operators, three parametric angular momentum operators and three remaining Runge-Lenz-like operator components to connect the algebra. The latter six can be seen as intrinsic editions of the generators of the Lorentz algebra. 12 With three dimensions in laboratory space R 3 , the group manifold U (3) therefore becomes the natural choice for intrinsic degrees of freedom that can be kinematically excited from laboratory space. The dynamics of the intrinsic degrees of freedom is projected back to laboratory space in the shape of quantum fields of various structures depending on the projection base chosen. A mixing between such projections becomes natural when one considers that the related subgroups are intermingled in the common U (3) configuration. This conception may open for a derivation of CKM-matrix elements although that is far beyond the scope of the present work. We shall, however, make a first step to correlate the strong and electroweak interactions of baryons, namely in a derivation of the Higgs mass.
In this paper, we derive fundamental mass values for Higgs and gauge bosons and report on mass values for the N and Δ baryon spectrum with dynamics described from a Lie group perspective. The derived Higgs mass around 125 GeV corresponds rather well to the recent experimental results 8,9 as seen in Fig. 1 and is based on just one dimensionful parameter in such a way that the ratio between the Higgs mass and the electron mass, apart from mathematical constants, contains only the fine structure constant. Of other papers analyzing the Higgs mass are composite models like the one by Dhar, Mandal and Wadia 14 related to the Gross-Neveu model 15 with a Nambu-Jona-Lasinio-type 16 four-fermion coupling leading to a dynamically generated Higgs mass without the need for an a priori Higgs potential.
One of us has previously introduced the Lie group U (3) as configuration space. 17,18 It contains the usual gauge groups SU (3) of strong interactions and SU (2) × U (1) of electroweak interactions. The essential frame to be adopted here The Higgs mass derived from the U (3) Lie group Fig. 1. Gaussian Higgs mass distributions as observed by the CMS collaboration 8 (dashed) and the ATLAS collaboration 9 (dash-dotted) compared with the theoretical result (dotted) in (73) from a common Lie group frame for strong and electroweak interactions. The curve widths represent the standard deviations of the respective mass peak determinations and not the resonance width which is much smaller. 13 complies with local gauge symmetry when the intrinsic Lie group dynamics is projected to laboratory space. Each point P (x, y, z) in space is equipped with an intrinsic U (3) configuration space in which the fundamental dynamics is formulated with u = e iχ ∈ U (3) as configuration variable. Thus, our configuration space is orthogonal to the space-time manifold of the laboratory space. The closest analogue we can think of is that of intrinsic spin. In the present case the intrinsic space contains both color, spin, isospin and hypercharge degrees of freedom. We thus can capture both the strong and electroweak sections of baryon phenomena. A major motivation is to reduce the number of ad hoc mass parameters in baryon phenomena relative to the standard model. As a benefit of intermingling the gauge groups of the standard model in a common intrinsic space, the parameters in the Higgs potential and the electroweak energy scale are determined from the relation to the intrinsic baryon potential -and the missing resonance problem in baryon spectroscopy vanishes. We do not expect to capture the meson sector readily since mesons are interaction quanta, i.e. field constructions in laboratory space.
Our basic frame is a Hamiltonian structure on the Lie group U (3) as a configuration space for baryons. We consider baryons as stationary states with masses mc 2 = E determined as eigenvalues of 17,19-21   17 The algebra approximates the group in the neighborhood of the origo. The projection is scaled by the classical electron radius re as a measure for the extension of the charge "scar" created in the neutron decay. This corresponds excellently to the measured value for the neutron to electron mass ratio, see where Λ ≡ c/a ≈ 214.27 MeV is our energy scale factor corresponding to a length scale a. Note that Λ corresponds to the QCD energy scale factor (see Ref. 10, p. 131), e.g. Λ (5) MS = 213 ± 8 MeV and is of the order of the pion decay constant 22 F π = 184 MeV. The latter is common for setting the scale in different phenomenological models. 23,24 For our Λ the length scale a was explicitly related 17 to the classical electron radius 25,26 (see Ref. 10, p. 107) r e = e 2 /(4π 0 m e c 2 ) = α c/(m e c 2 ) by a mapping πa = r e , between real parameter space and toroidal angles in the Lie group, see Fig. 2. The Λ above is calculated from a fine structure constant taken at nucleonic energies (see Ref. 10, p. 137 Instead of the Manton-inspired 20 potential 1 2 Tr χ 2 acting on the generators of the configuration variable we could have chosen a Wilson-inspired 27,28 potential 3 − 1 2 Tr(u + u † ) taking the trace directly on the configuration variables themselves. Both agree in the neighborhood of the origo e = I of U (3) but differ for larger deviations of the configuration variable. Both will yield the same Higgs and gauge boson masses but differ in the baryon mass spectra because the Higgs mass is determined from the shared second-order term near potential minima, where the algebra approximates the group, whereas the baryon states occupy all of the intrinsic geometry. Therefore, the Manton-like potential better reproduces the baryon spectrum, see Fig. 3. We also prefer the Manton-like potential because, in the parametrization , not the resonance widths which are much larger. We have made no estimate of mass shifts due to strong coupling to decay channels. 30 Digits at selected predictions are parametric labels p, q, r based on Table 1. Note the fine agreement in the grouping and the number of resonances in both sectors with no missing resonance problem as opposed to ordinary quark models, see Fig. 8. of the configuration space, it represents the Euclidean measure folded onto the group manifold. 29 Note finally, as the configuration space is truly intrinsic, relativity only comes into play once the inherent dynamics in (1) is projected to space as when the parton distribution functions in Fig. 10 below were derived in Ref. 17.
In Sec. 2, we describe the model. In Secs. 3 and 4, we carry through particular solutions for the baryon spectrum and discuss experimental predictions for unconventional baryon singlets. In Sec. 5, we describe projections to laboratory space where quantum fields resurface. In Sec. 6, we state a relation between strong and electroweak configurations and derive a Higgs mass. In Sec. 7, we relate to standard results for the vector gauge bosons. In Sec. 8, we give remarks on interpretations and in Sec. 9, we suggest lines of future study.

Unfolding the Model
One may consider the basic equation (1) as an effective theory inspired by lattice gauge theory. [19][20][21] However, we prefer to present it as detached from this framework such that the configuration space, and the space-time manifold orthogonal to it, here the Laplacian Δ in (1) can be parametrized in a polar decomposition 31 ( = 1) where θ j are the dynamic eigenangles in the three dynamic eigenvalues e iθj of u and J is the Van de Monde determinant, a the "Jacobian" of our parametrization 32 In mathematical terms, K k and M k are off-toroidal derivatives which are noncommuting and may be represented by off-diagonal Gell-Mann matrices, see (15) and (16) below. The triple K k commute as body fixed angular momentum operators and M k "connect" the algebra by commuting into the subspace of K k cyclic in k, l, m. The components of K = (K 1 , K 2 , K 3 ) which are SU (2)-generators and M = (M 1 , M 2 , M 3 ) in the Laplacian carry spin and flavor. Interpreting K as an intrinsic spin operator is supported by the reversed sign in the commutator, like for body fixed coordinate systems in nuclear physics. The geodetic potential in (1) depends only on the eigenvalues of u since the trace is invariant under conjugation u → vuv −1 by any v ∈ U (3); in particular a conjugation that diagonalizes u. Thus Here, e is the neutral element, the "origo" of U (3). The last expression shows that the potential is left-invariant as are the coordinate fields that we shall soon introduce. In the above parametrization, the potential reads a Actually J ≡ √  i.e. a sum of periodic parametric potentials, see Fig. 4 w The potential may be considered as the Euclidean measure folded into the group manifold 29 in compliance with the space projection (13) below. Now, each of the nine generators T k of U (3) implies directional derivatives locally at each point u ∈ U (3) or so-called left-invariant coordinate fields with related differential forms dα k , also called exterior derivatives dα k (∂ m ) = δ km . For the three toroidal degrees of freedom we use the angular symbols θ j . The quantization inherent in the basic equation (1) can then be expressed in a generalized action-angle form as where dθ i are the torus forms and δ ij is the Kronecker delta. By construction, the act of the exterior derivative by a generator X in the Lie algebra on a function Φ at a point u in the Lie group manifold is This was used 17 to generate the parton distributions in Fig. 10.
The three toroidal generators T j = −i ∂ ∂θj = −i∂ j e , j = 1, 2, 3 correspond to parametric momenta and thus, corresponding to a space projection we have the standard commutators In the above coordinate representation 33 the off-toroidal generators read 17 and The lambdas are Gell-Mann generators. 33 From these and the spectrum of M 2 was found to be 17 where y and i 3 are hypercharge and isospin three-component quantum numbers.
The minimum value for the positive definite M 2 is 13/4 in the case of spin 1/2, hypercharge 1 and isospin 1/2 as for the nucleon. From here, we are able to find specific solutions presented in Secs. 3 and 4.

On flavor degrees of freedom
The dynamics underlying the baryon mass spectroscopy in the present model is determined primarily by an intrinsic potential in the Lie group, namely the second term in (1). As for flavor degrees of freedom these are contained in the Laplacian on U (3). The Laplacian can be parametrized in a polar decomposition with three toroidal, Abelian derivatives and six off-torus derivatives (3). The latter correspond to the six off-diagonal Gell-Mann matrices (15) and (16). Three of these we interpret as spin generators and the remaining three are related to the isospin and hypercharge of the standard SU f (3) algebra (18).
The last term in the Laplacian (3), the "centrifugal" term, can be integrated by exploiting the existence of the Haar measure over (α 4 , α 5 . . . , α 9 ) by the factorization in (22) below. Further we use that the off-toroidal part of the wave function is an eigenstate of K 2 and M 2 together with the fact that the centrifugal term is symmetric under interchange of the torus angles θ j .
The centrifugal term leads to a mass formula of the well-known Okubo-type. 34 The spectrum of K 2 + M 2 follows directly from (18) to yield It is natural in the present framework to classify the eigenstates according to the three independent values of n, y and i 3 . However, we can make a transformation of this classification into the familiar one by rewriting the expression (19) and choose the sum of n and y to be a constant. For n + y = 2, which yields the lowest possible K(K + 1) + M 2 , we get Since (K 2 3 + M 2 3 ) commutes with both Y and I 2 we get for a given value of the quantum number (k 2 3 + m 2 3 ) with the constants a = 40/3 + (k 2 3 + m 2 3 ), b = −28/3, c = 4 respectively. Equation (21) is the famous Okubo mass formula that reproduces the Gell-Mann, Okubo, Ne'eman mass relations within the baryon N -octet and Δdecuplet 34-37 independently of the values of a , b and c . Of course this is only accurate if one chooses the same toroidal wave function for all members of a given multiplet. In practice the SU (3) symmetry breaking in (21) will be influenced by the θ-dependence in the centrifugal term because different values of K 2 + M 2 lead to different values of the centrifugal potential and thereby influence which span of toroidal energy eigenstates will project out on a specific angular momentum eigenstate in the laboratory. Therefore, the SU f (3) symmetry break will not follow exactly (21) in hypercharge. of freedom. We start off in a Hamiltonian framework with the Hamiltonian operating on states Ψ(u), where u is the configuration variable belonging to the Lie group U (3). We can generate SU (3)-transforming color quark (34) and gluon fields (45) below from the exterior derivative on Ψ scaled in measure by the Jacobian (4) of the polar decomposition. Summing over such color components (34) for particular flavor tracks led to u and d valence quark parton distribution functions ( Fig. 10) for an approximate protonic state via projections along mixed toroidal directions. 17 The parton distribution functions compare well with those established for the proton.

On color degrees of freedom
Because the dynamical structure is formulated on the Lie group, it will show different manifestations depending on which derivatives (11) one is taking. For instance, the three toroidal dimensions for the color quark degrees of freedom are intermingled with flavor degrees of freedom since the hypercharge and isospin threecomponent generators Y and I 3 are not linearly independent of the three torodial generators T 1 , T 2 , T 3 . And both are intermingled with the eight gluon dimensions laid out by the Gell-Mann matrices (45) because these include generators proportional to Y and I 3 . Thus, we do not consider color and flavor degrees of freedom as being independent. For instance, the distribution functions in Fig. 10 are produced by using the exterior derivative (34) on tracks from the quark flavor generators 17 We actually see the reduction in the number of independent quark degrees of freedom as a reason that the baryon spectrum from (1) is not hampered by missing resonances as usual in ordinary quark models (QMs), compare Figs. 3 and 8.

Specific Solutions of the Model. Trigonometric Base and the Electron to Neutron Mass Ratio
It is possible to find the dimensionless eigenvalue for an unbroken neutron ground state E n ≡ E n /Λ in (1) with quite high precision by a Rayleigh-Ritz method. 38 We factorize the wave function Ψ in (1) into a toroidal part τ and an off torus part Υ In that way, (1) can be solved for specific choices of spin and flavor inflicted by the six off torus generators contained in the Laplacian. After integration over the six off-toroidal degrees of freedom α 4 , α 5 , α 6 , α 7 , α 8 , α 9 one ends up with a Schrödinger equation Here R = Jτ with J from (4) and contains in the second term contributions from off-toroidal degrees of freedom that carry spin and flavor in the specific choice here of spin, hypercharge and isospin s = 1/2, y = 1, i = 1/2. The numerator 4 in front of the sum is the minimum value of (K 2 + M 2 )/ 2 for this combination as well as for s = 3/2, y = 1, i = 3/2 corresponding to the choices respectively of n = 1 and n = 2 in (18). The term corresponds to the centrifugal potential when solving the hydrogen atom in polar coordinates (2). The constant term is a global curvature term 39 arising from differentiating through J 2 in the Laplacian (3).
The measure-scaled toroidal wave function R can be expanded on solutions b to the separable problem Due to the arbitrary labeling of the eigenangles θ j , the toroidal wave function τ is symmetric in these and as J is antisymmetric so must be R = Jτ. Therefore, solutions to (23) and (25) can be constructed from Slater determinants 40 where p, q, r are natural number labels for orthogonal solutions to the onedimensional Schrödinger equation with periodic parametric potential. This is postponed to Sec. 4. Here we will use an expansion set where the necessary integrals for the Rayleigh-Ritz procedure can be found analytically. The measure-scaled toroidal part R of the wave function is expanded on "trigonometric" Slater determinants where p, q, r are integers, p = 0, 1, 2, . . . , P − 1; q = 1, 2, 3, . . . , P ; r = p + 1, p + 2, . . . , P . The order parameter P determines the number of independent states on which we expand. The value P = 12 corresponds to 936 expansion functions and yields E n = 4.3849 whereas P = 18 with 3078 expansion functions yields E n = 4.3820 seen in Fig. 5. Calculations of E n with higher values of P are beyond the handling capacity of our computer programmes. 42

Specific Solutions of the Model. Parametric Base and the Baryon Spectrum
We now consider the expansion on Slater determinants constructed from solutions to (27). Figure 6 shows solutions for the first three eigenvalues e 1 , e 2 , e 3 . The structures of (25) and (23) with periodic potentials either V or W imply the introduction of Bloch wave expansion factors   where κ introduces the Bloch degree of freedom. We shall argue that the Bloch degrees of freedom are opened by a Higgs mechanism that will allow a diminishing of the ground state eigenvalue via the creation of the ν e , e L doublet and its coupling to a Higgs field. For instance the ground state eigenvalue E n = e 1 +e 2 +e 3 = 4.47 . . . of (25) is lowered to a value E p = e 1 + e 2 + e 3 = 4.46 . . . for real symmetry broken states of parametric eigenvalues e 1 and e 2 with 4π periodicity analogous to κ 1 , κ 2 = ± 1 2 , ± 1 2 for the Bloch-phase-containing g's as opposed to the 2π periodicity of the b p 's and u p 's, see Fig. 7. The period doublings are seen as the origin of charge creation via the related changes in the state topology. The period doublings are allowed since they leave the square of the wave function Ψ 2 single valued on U (3). Table 1  By summing up the three lowest levels we get an approximate estimate of the relative neutron to proton mass shift    (27) to construct the approximate baryon spectrum in Fig. 3. The eigenvalues are calculated with 1500 collocation points. The lowest eigenvalues, as expected, are close to those of the ordinary harmonic oscillator. Moving up to higher levels the eigenvalues differ more and more from those of the harmonic oscillator as indicated in Fig. 7. The lower levels have been calculated by three different methods with discrepancies only from the eights significant digit, see Table 2.   (25) has been solved by four different methods three of which gives a set of eigenvalues for the one-dimensional problem (27) from which the eigenvalues for the three-dimensional problem (25) is constructed. These eigenvalues can be used to check the Rayleigh-Ritz method for solving the three-dimensional problem directly. Mutual discrepancies are due to the finite expansions in the different methods. The fine agreement among the different methods 42 lends support to the Rayleigh-Ritz method also for solving the full Eq. (23).

Iterative
Collocation  Only one observed certain N -resonance in the group of three resonances in the domain around 1500 MeV is missing in the predictions. However, the approximate treatment in (25) suggests a neutral singlet 1, 3, 5 at 1510 MeV exactly in that area. The exact treatment case in (23) can be solved for neutral states by a Rayleigh-Ritz method which places the singlet at 1526 MeV, see Table 3. This state is thought to mix with the other two N -resonances nearby to give the total of three N -resonances in the group. The next singlet state 1, 3, 7 is predicted in the "desert" area between 1700 MeV and 2100 MeV. In the approximate case the resonance comes out at 1965 MeV and in the exact case it comes out at 2051 MeV (Table 3).  Table 3. Scarce singlet states. Eigenvalues based on Slater determinants of three cosines up to order 20 analogous to (28). The first column shows eigenvalues of the approximate equation (25) and the third column shows eigenvalues of the exact equation (23). A singlet 579-like resonance is predicted at 4499 MeV in the free charm system Σ + c (2455)D − slightly above its threshold at 4324 MeV. The rest masses are predicted from a common fit of the nucleon ground state 939.6 MeV to the ground state 4.38 of (1) resp. (23)

Where are the Quarks? Projection to Space
Here we look at projections of the wave function to fields in laboratory space. For each element u ∈ U (3) we have a corresponding left-translation l u on v ∈ U (3) and for any left-invariant vector field X we have 50 In particular, we have for the toroidal coordinate fields when comparing with (9) where the coefficients are the local partial derivatives 52 For the coefficients, we have by left-invariance (33) The sum of the differential components of the torus form will inherit the leftinvariancē Now, in particular ψ j (e) = dR e (∂ j ) = ∂R/∂θ j belongs to the tangent space T M e of the maximal torus M at e and therefore so does their sumψ(e) as in general ψ j (u) ∈ T M u . The set of generators {iT j } are the coordinate field generators ∂ j which also constitute an induced base from parameter space where { c j } is a set of base vectors for the parameter space for the torus, see Fig. 9.
In our interpretation, we identify { c j } as a base for a fundamental representation space for the color algebra su(3) at a particular point P (x, y, z) in laboratory space. We may thus introduce at P complex-valued componentsψ j for the color vector ψ and write In the above representation, u will be represented by a 3×3 matrix U . For rotations under V ∈ SU (3) at P , we then have and From (40) and (41), we can derive the transformation property ofψ(u). We thus interpretψ as a quark field with three color components which may be projected on a specific base like in (39). The distributions 17 in Fig. 10 are for

The Higgs mass derived from the U (3) Lie group
Likewise, the gluon fields may be seen as resulting from a projection on adjoint representation spaces of an expansion of the momentum form corresponding to the full set of eight generators λ k needed to parametrize the general group element u = e iα k λ k -separating out a phase factor. 53 Thus, for each generator T a we have left-invariant vector fields ∂ a defined as where T a = −i∂/∂α a = −i∂ a | e . We now choose the set {T a } as a base for the adjoint representation. This base transforms under V ∈ SU (3) like Analogous to (34), we expand the exterior derivative dΦ of the full measure-scaled wave function Φ = JΨ on forms related to the left-invariant vector fields ∂ a to get the adjoint projection fieldĀ We want to show thatĀ transforms according to the adjoint representation. First, we have the equivalent of (37) = u a dΦ e (∂ a ) = uĀ(e) .
Here we understand in analogy with (39) that where againÃ a are complex-valued components and {T a } was the adjoint base. We may then proceed to show the adjoint transformation property ofĀ which corresponds to the gauge group rotation transformation property of the gluon fields B (Ref. 55) where We note that as space-time fields the gauge fields also acquire a term representing the variation along space-time translations as represented by the second term in (49). Note further that translational invariance in group space corresponds to an SU (3) rotational invariance in representation space and thereby the translational invariance of the interaction potential (6) in group space through the projections (13) and (34) reflects the gauge invariance of the fields in laboratory space.

An Exemplar Higgs Mechanism for the Neutron Decay
We want to relate the strong interaction dynamics inherent in (1) to the electroweak interaction involved in e.g. the decay of the neutron to the proton. We settle the relation through the structure of the geodetic potential (8) supplemented by a trailing ansatz ( = c = 1) which balances the products of the coupling constants and the corresponding ("phase") fields for the respective interactions, namely a color field θ and a Higgs field φ. The color field is a space projection of an eigenangle dynamical variable from the description of the intrinsic configuration space and as such starts off dimensionless whereas φ -or more precisely √ cφ -has the dimension of energy. 56 A certain caution is therefore needed when the geodetic potential for θ is to be translated into a potential for φ.
Let us first look at the Klein-Gordon Lagrangian (see Ref. 56, p. 355) for a scalar field φ of mass m While in the following we set = c = 1, we still want to keep track of the length scale and we therefore write Here m is dimensionless while m = m/ã represents the mass of dimension L −1 in usual = c = 1 notation, thus the well known expression Corresponding to the length scaleã we have an energy scaleΛ = c/ã which we shall settle below in (56). The dimensionless mass m is also given as m = mc 2 /Λ In the Higgs mechanism, the mass term for a Higgs particle field H equivalent to the mass term in (54) follows from the second-order derivative of a Higgs potential V H (φ) at a minimum point φ 0 = 0. The geodetic potential (8) has such minimum points off 0, which are "activated" in the neutron decay when parametric period doublings occur. The period doublings correspond to sudden jumps of θ from one trough of the geodetic potential to a neighboring one. We therefore consider a match of a Higgs potential to a dimensionful edition of the geodetic potential (8) neighboring to the generic n = 0 section. With the balancing trailing ansatz (51) we find that a jump from θ = 0 to θ = 2π corresponds to To match (55) This potential as usual has minima at φ 2 0 = μ 2 /λ 2 . Note, that the essential thing for the Englert, Brout, Higgs, Guralnik, Hagen, Kibble-mechanism 1-4 is not the particular shape 5 of V H but the fact that V H has a minimum for nonzero φ. For a real scalar φ we find (57) to match (55) in the neighborhood of φ 0 for see Fig. 11. With these choices, w in (55) and V H in (57) both agree at φ 0 up to second-order, where V H (φ 0 ) = 2μ 2 . For a complex φ = (φ 1 (x) + iφ 2 (x))/ √ 2, we have minima in (57) for where β is a real phase angle and for convenience we define v = √ 2φ 0 ≡ṽφ 0 . Since Here we have hidden interaction terms in L 1 and the electric charge coupling constant q originates from the generalized derivative  From the coefficients of the quadratic terms H 2 and A μ A μ in (61) with q = e = √ 4πα andṽ = √ 2 we read off the respective masses m H and m A determined by and In (63) In the neutron decay, both an electron and a gauge boson are involved. Thus, we use for the gauge mechanism as approximation for α the geometric meanα The expression (63) containing solely the electron mass and the fine structure constant and cited again in (73) is determined by the trailing in (51) and by the structure of the potential (8), respectively (55) or (57) and therefore remains valid below. Similarly, we would get m A c 2 = 78 GeV. However, for the vector gauge field masses corresponding to m A we need to consider the full electroweak SU (2) L ×U (1) treatment to give the results in (73).
Note that the usual way of getting the masses for the massive gauge bosons is to derive v from the Fermi coupling constant in muon decay, see e.g. Ref. 22, p. 310, but we use (51) and (55) to give v = 2π √ 2Λ/α directly in (56) which leads to the values for m W and m Z stated in (73) and following from the standard results in (71) in the next section.

A Full Two-component Higgs Mechanism
The degrees of freedom introduced by the Bloch phase factors in the parametric eigenstates g p in (29) have to be excited in pairs of half odd-integer valued Bloch wave numbers (κ 1 , κ 2 ) in order to "kill" the singularity in the centrifugal potential It namely turns out that the centrifugal potential allows for half-odd-integer κcomponents provided they come in pairs, for instance κ = − 1 2 , 1 2 , 0 . In that case we might expand on g-couples which keep the integrated centrifugal potential regular 59 because p − q remains integer, see Fig. 12. Here Generalizing the ansatz (51) we take the paired period doublings corresponding to the shift in Fig. 7 from (κ 1 , κ 2 ) = (0, 0) to (κ 1 , κ 2 ) = ± 1 2 , ± 1 2 to be mediated by a Higgs field with a U (2) transformation on a complex two-component doublet φ = (φ 1 , φ 2 ) to "absorb" phase changes (but not kinetic energy nor rest mass) and a two-component electronic lepton l eL = (ν e , e) L to "take care" of the remaining degrees of freedom (and carry away released energy).
where the SU (2) coupling constant g and the U (1) coupling constant g are given from the electric charge coupling constant e = √ 4πα and the electroweak mixing angle θ W by We should stress, that the derivations leading to the Higgs mass in (63)  Note that the Fermi coupling constant for beta decay has become a derived quantity  2 θ 2 and the 2π-periodicity. Baryonic states, however, will be shifted downwards by some 20 percent for states constructed from the lowest levels. This would spoil the agreements in Fig. 3.

On the influence of loop corrections on the Higgs mass
The electroweak scale (70) follows from the 2π-shift to a neighboring trough in (55) scaled by the balancing trailing ansatz (51) in which the coupling constant α appears twice, namely explicitly as a factor on the Higgs field φ and hidden in the factor Λ on the color angle θ. The scale v is settled by the 2π-shift giving a dimensionless v = √ 2 and the Higgs mass is settled by a dimensionless value m H = 1/ √ 2 from matching the shape of the Higgs potential to that of the neighboring trough (55) of the intrinsic geodetic potential in (8). With this nonperturbative procedure we have condensed higher loop corrections into the question of finding the right value for the effective coupling constant α eff . Perturbative corrections are then contained in the values of α eff in the various steps where they enter.
The neutron decay, which we have taken to shape the Higgs potential, involves both lepton dynamics and gauge boson dynamics wherefore we used in (66) for the gauge mechanism as approximation for α the geometric meañ To investigate the influence of fermionic loop corrections on the result for the Higgs, we use expressions from an older work by Jegerlehner. 62 The definition of the total correction Δα at scale √ s is given by From Jegerlehner, we then refer the lepton contributions where β l = 1 − 4m 2 l /s. The hadronic contributions are where α s is an effective strong coupling constant and h(y) = 5/3 + y − (1 + y/2)g(y) , whereas with the beta-function (see Ref. 22, p. 126) β(e) = e 3 12π 2 + where μ is the sliding scale. Omitting O(e 7 ) we get with b 1 = 1/(12π 2 ) and b 2 = 1/(64π 2 ) where k is an integration constant which in Weinberg's lower order result leads to an expression This would correspond to a radiative correction with the fine structure constant α 0 = e 2 /4πε 0 c = 1/137.035999074. We want, however, to go to higher order, but are on the other hand only interested in relations between different scales. So from (85) we continue with the following identity

Remarks on Physical Interpretations
The conceptual framework is not the standard model although many aspects comply with it. A benefit of the common intrinsic configuration space is the reduction in the number of ad hoc parameters while keeping -and in certain cases improving on -the agreements with experimental observations. This suggests the framework to be taken as more than just an approximation. (1) The physical conception of baryon dynamics is that of introtangled energymomentum with baryons described as stationary states on an intrinsic, compact configuration space. We consider the intrinsic dynamics to be fully described by a Hamiltonian (1) on the intrinsic configuration space, i.e. not as fields of quarks and gluons in laboratory space with separate strong and electroweak interaction parts. Rather we consider the baryons to be entire entities of introtangled energy-momentum which carry strong and electroweak manifestations intermingled. Quarks (35) and gluons (45) come about when the intrinsic states are projected to laboratory space. In the language of the standard model we have confinement per construction since we take the configuration space to be compact. (2) We consider the creation of electric charge to originate in topological changes (29) in the intrinsic states, see also We take the length scale of the strong interaction sector of the model to be settled in the projection of the neutron decay which relates changes in the intrinsic baryon states to the electroweak sector (55), (56), (70). We thus take a projection of the intrinsic geometry to the electrically defined, classical electron radius as an input for the strong interaction scale with the electron imagined as a "peel off " from the neutron, leaving a "charge scarred" proton torus, see Fig. 2. Further we use a trailing ansatz (51) to relate strong and electroweak coupling constants in order to set the scale (70) for the electroweak sector and its Higgs (63) and gauge boson masses (71). (4) States are projected from intrinsic space to laboratory space by use of the exterior derivative, the momentum form on the intrinsic manifold (11), respectively (35) and (45). From projection of the intrinsic structure to space we recognize the toroidal generators as momentum operators (12) and off-torus generators as spin and flavor operators (5), (18). In experimental production of resonances we see from space: The impact momentum generates the Abelian maximal torus of the U (3) intrisic space. The momentum operators act as introtangling generators. When decay, asymptotic freedom, fragmentation and confinement are of concern we see from intrinsic space: The quark and gluon fields are projections of the vector fields induced by the momentum form on the intrinsic states.
The projected fields are treated as quantum fields and a balancing trailing ansatz between color and Higgs field energies in weak baryon decays connects strong and electroweak sectors (51) via the period doublings (68) allowed in the parametrization of the intrinsic space. The structure of the period doublings and the intrinsic potential (8) determines the Higgs potential from which the Higgs mass originates.  (5) Because the dynamical structure is formulated on the Lie group, it will show different manifestations depending on which derivatives (11) one is taking. For instance we interpret the three toroidal dimensions as intrinsic color quark degrees of freedom (34), (39). These are intermingled with flavor degrees of freedom. And both are intermingled with the eight gluon dimensions laid out by the Gell-Mann matrices (45). Thus we do not consider color and flavor degrees of freedom as being independent. As mentioned in Subsec. 2.2, the distribution functions in Fig. 10 are produced by using the exterior derivative (34) on tracks 17 from the quark flavor generators T u = 2/3T 1 − T 3 and T d = −1/3T 1 − T 3 . And the reduction in the number of independent quark degrees of freedom practically eliminates the missing resonance problem in ordinary QMs, compare Figs. 3 and 8.

Examples for Future Study
The neutral flavor, neutral electric charge baryon singlets mentioned in Sec. 4 and listed in Table 3 should be sought for. They may even lie dormant in data pools already taken since they have no charged partners to help them surface in partial wave analysis. A more accurate estimate of the coupling constant α MS (m W ) at W -bosonic energies is wanted for an even more accurate prediction of the Higgs mass m H .
A suitable base on which to expand for exact solutions for charged baryons is wanted in order to improve the predictions on the N and Δ mass spectra.
The geodetic distance potential (6) can be used as an interaction term. For instance in a model for two baryons with configuration variables u and u for which d(u, u ) = d(e, u † u ). Thus, we conjecture the deuteron to be the spin 1 ground state of When one imagines a projection of the term u † u it has an antiquark-quark structure characteristic of mesons in that the u † when projected is to be represented on an antiquark to the left and the u is to be represented on a quark to the right.

Conclusion
We have derived the Higgs mass and the electroweak energy scale by connecting structurally the strong and electroweak baryon sector. We have considered baryons as entire entities on an intrinsic U (3) configuration space with a Hamiltonian structure to yield baryon mass spectra. The parametrization of the intrinsic baryon space and its potential allows for period doublings which determines the Higgs potential and settles the Higgs mass. Parton distribution functions follow from the exterior derivative, the momentum form on intrinsic states. The general agreement of the various derivations with experimental observations suggests further investigations within the model. In particular a base for exact solutions of electrically charged baryonic states is wanted as well as experimental investigations looking for neutral flavor, neutral charge baryon singlets particular for the present model. The singlets should be visible as resonances in negative pions scattering on protons, in photoproduction on neutrons, in neutron diffraction dissociation experiments and in invariant mass spectra of protons and negative pions in B-decays. The Higgs mass prediction, the singlet predictions and the elimination of a missing baryon resonance problem distinguish the present model from the standard model predictions. We await singlet searches on GeV-machines from new experiments or from dedicated analysis on existing data pools and we await more accurate Higgs mass measurements from Run 2 at the Large Hadron Collider.