The Gravitational Origin of the Higgs Boson Mass Friedwardt Winterberg Department of Physics, College of Science, 1664 N

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The Gravitational Origin of the Higgs Boson Mass Friedwardt Winterberg Department of Physics, College of Science, 1664 N. Virginia Street, University of Nevada, Reno, Nevada 89557-0220, USA. Ofﬁce: (775) 784-6789, Fax: (775) 784-1398 Reprint requests to F. W.; E-mail: [email protected]

Z. Naturforsch. 69a, 220 – 224 (2014) / DOI: 10.5560/ZNA.2014-0017 Received December 10, 2013 / revised March 10, 2014 / published online May 21, 2014

The Lorentzian interpretation of the special theory of relativity explains all the relativistic effects by true deformations of rods and clocks in absolute motion against a preferred reference system, and where Lorentz invariance is a dynamic symmetry with the Galilei group the more fundamental kinematic symmetry of nature. In an exactly nonrelativistic quantum field theory the particle number operator commutes with the Hamilton operator which permits to introduce negative besides positive masses as the fundamental constituents of matter. Assuming that space is densely filled with an equal number of positive and negative locally interacting Planck mass particles, with those of equal sign repelling and those of opposite sign attracting each other, all the particles except the Planck mass particles are quasiparticles of this positive-negative-mass Planck mass plasma. Very much as the Van der Waals forces is the residual short-range electromagnetic force holding condensed matter together, and the strong nuclear force the residual short range gluon force holding together nuclear matter, it is conjectured that the Higgs field is the residual short range gravitational force holding together pre-quark matter made up from large positive and negative masses of the order ±1013 GeV. This hypothesis supports a theory by Dehnen and Frommert who have shown that the Higgs field acts like a short range gravitational field, with a strength about 32 orders of magnitude larger than one would expect in the absence of the positive-negative pre-quark mass hypothesis.

Key words: Higgs Boson; Standard Model; Quantum Gravity; Hierarchy Problem.

1. Introduction spinor equation, it does not lead to a Hilbert space with an indefinite metric with negative probabilities, which Replacing Heisenberg’s nonlinear relativistic spinor has always been the main objection against Heisen- equation as a model for a unified theory of elemen- berg’s theory. tary particles [1,2], with an exactly nonrelativistic With the assumption that space is densely filled with Heisenberg-type equation [3](rp and mp Planck length an equal number of positive and negative Planck mass and mass) one has particles forming a Planck mass plasma, (1) and (2) de- scribe the quantization and many quasiparticle excita- 2 ∂ψ± h¯ 2 tions of this plasma. An in depth study of the quantized ih ¯ = ∓ ∇ ψ± ∂t 2mp (1) Planck mass plasma has shown that it leads to a spec- 2 † † trum of quasiparticles greatly resembling the standard ± 2¯hcr (ψ ψ± − ψ ψ∓)ψ± , p ± ∓ model. where the field operators ψ†, ψ obey the canonical commutation relations 2. The Hierarchical Structure of Matter h † 0 i 0 The structure of matter can be subdivided into two ψ±(r),ψ±(r ) = δ(r − r ) (2) basic configurations: Those held together by the fun- 0 h † † 0 i ψ±(r),ψ±(r ) = ψ±(r),ψ±(r ) = 0, damental long ranges forces: the electromagnetic, the gluon, and the gravitational forces, and those held to- where in addition to positive masses, negative masses gether by the short range residual forces of the funda- can be introduced. But unlike Heisenberg’s relativistic mental forces.

A1 Nuclei (n) and Atoms electrons (e) 1 –10 eVn held together by electromagnetic forces

A2q (q) held together Nucleons by strong gluon (color) 1 – 10 MeVq forces q

Pre-Quarks Pre-Quarks (+, -) held m± = ±1013 ++– together by GeV gravitational forces Fig. 1. Basic conﬁgurations held together by long range forces.

The first basic configurations (Fig.1) are: New in this order is the third step because it in- volves gravitational forces: Pre-quarks bound by the A1. Atoms, made up from electrons and nuclei held long range gravitational forces and the short range together by the electromagnetic force, with an en- residual (Higgs) gravitational forces. ergy of the order of eV. This idea works by assuming the existence of large A2. Nucleons, made up from quarks and held to- negative masses. According to Heisenberg [2], in the gether by the strong (color) gluon force, with an hierarchy of elementary particles the concept ‘to be energy of the order of GeV. composed of’ becomes problematic if the parts have A3. Quarks, made up from pre-quarks and held to- a mass exceeding the mass of the composition. But this gether by gravitational forces, with an energy of problem does not occur with the admission of negative the order of 100 GeV. masses. The existence of negative masses also seems to The second kind of configurations (Fig.2) are: be needed to satisfy the average null energy condition of general relativity. B1. Condensed matter held together by residual elec- For a still better perspective we place in Table1 the tromagnetic (Van der Waals) forces. secondary short range forces below the primary long B2. Nuclear matter held together by residual strong range forces, with the spin for both. (nuclear) forces. With the Van der Waals and the nuclear force both B3. Condensed quark matter held together by residual acting as composed particles it is quite reasonable to gravitational (Higgs) forces. assume that the same is true for the Higgs force. Mak- 222 F. Winterberg · The Gravitational Origin of the Higgs Boson Mass

atom B1 held together by Condensed Matter residual electromagnetic atom (Van der Waals) W forces (W-phonons)

nucleon B2 held together by Nuclear Matter residual strong nuclear force nucleon π (π mesons)

pre-quark B3 m– held together by Pre-Quark Condensed residual Matter gravitational pre-quark force m+ H (Higgs boson) Fig. 2. Compound configurations held together by short range residual forces. ing this assumption we can not only derive the Higgs 3. The Higgs Mass as the Gravitational Field Mass mass but also give support to the conjecture by Dehnen of a Mass Dipole and Frommert [4,5]. The gravitational interaction energy of two masses Table 1. (a) long range and (b) short range forces. m1 and m2 separated by the distance r is negative and given by (G Newton’s constant) (a) Primary Long Range Forces Gm m Force Particle Spin Rest mass E = − 1 2 (3) (fundamental) r Electromagnetic Photons 1 0 Strong Gluons 1 0 but the gravitational interaction energy of a mass Gravitational Gravitons 2 0 + − + dipole where m1 = m and m2 = m = −|m |, with ± 2 m1m2 = −|m | , is positive and given by (b) Secondary Short Range Forces ± 2 Force Particle Spin Rest mass G|m | E = . (4) (composed) r Van der Waals phonon (boson) 0 6= 0 Nuclear π meson (boson) 0 6= 0 Without this positive gravitational interaction energy Higgs Higgs (boson) 0 6= 0 a mass dipole would be self-accelerating, but because F. Winterberg · The Gravitational Origin of the Higgs Boson Mass 223 the positive interaction energy gives it a small positive 4. Planck Mass Plasma Model mass equal to As stated above, the Planck mass plasma hypothesis 2 E G|m±| is the assumption that the vacuum is densely occupied = m = (5) c2 c2r with an equal number of positive and negative Planck it makes a helical motion with the radius mass particles, on average one Planck mass particle for each Planck length volume [3]. In its ground state the h¯ ρ = (6) Planck mass plasma is a superfluid, which for each 2mc positive or negative mass component has a phonon- asymptotically reaching the velocity of light. This is roton spectrum, in addition to a variety of quantized the pole-dipole configuration extensively studied by vortex configurations in low lying excited states [8]. Hönl and Papapetrou [6], as a model for Schrödinger’s For a line vortex the quantization condition is ‘Zitterbewegung’ of a particle described by the Dirac I equation. mp v · ds = nh, n = 1,2,.... (11) Very much as one can estimate the ground state energy for the Bohr atom model with the uncertainty For the lowest state with n = 1, one finds in setting principle ∆p∆q ∼ h¯, we can here do the same by com- v = vϕ bining (5) with the uncertainty principle. As in Bohr’s h¯ model we set ∆q ∼ r, but for ∆p we have to take the vϕ = (12) sum of |m+| and |m−|, hence ∆p ∼ 2|m±|c, and thus mpr have or withh ¯ = mprpc that m± cr ∼ h. ( 2 ¯ (7) v = cr /r for r > r , ϕ p p (13) The wave mechanical correctness of (7) for the pole- vϕ = 0 for r < rp . dipole particle was proven by Bopp [7]. By setting If a line vortex is deformed into a vortex ring of ra- m = MH, where MH is the Higgs mass, one obtains by eliminating r from (5) and (7) dius R, it can undergo elliptic oscillations with the fre- quency [9] ± 3 2G|m | 2 MH = (8) ω = cr /R . (14) hc¯ v p or If these oscillations are quantized, the ground state en- ± 2 M |m±|3 ergy ish ¯ωv. We thus put |m |c = h¯ωv or H = 2 , (9) 2 2 mp mp h¯ωv = mpc (rp/R) . (15) p where mp = hc¯ /G is the Planck mass. Solving for If in its ground state the Planck mass plasma is made |m±| one has up of a lattice of such vortex rings, then the distance of separation l = 2R in between two adjacent vortex rings ± 1/3 |m | 1 M = 2 /3 H . (10) determines the energyh ¯ωv. For a two-dimensional vor- mp mp tex lattice, the distance l between two line vortices had 2 19 been determined by Schlayer [10], by computing the With MH ∼ 10 GeV, mp ∼ 10 GeV, one finds that ± 13 ± 6 stability of the Karman vortex street. Schlayer found |m | ≈ 10 GeV. The energy ratio mp |m | ∼ 10 is of the same order as the nuclear–atomic energy ratio. the configuration to be stable for Because |m±|c2 m c2, quantum gravity can be −3 p r0 = 3.4 × 10 l , (16) ignored, as the quantum electrodynamical corrections to the Coulomb potential can be ignored in the Bohr where r0 is the radius of the vortex core. Setting r0 = rp atom model. The question still remains where the large and l = 2R, one obtains from (16) that energy of ∼ 1013 GeV comes from. It is shown that the Planck mass plasma can offer a possible answer. R/rp = 147. (17) 224 F. Winterberg · The Gravitational Origin of the Higgs Boson Mass

No likewise stability calculation seems to have been 5. Conclusion made for a three-dimensional lattice of vortex rings, but with Schlayer’s result a guess can be made. For The result obtained with this simple model sup- line vortices, the stability apparently arises from the ports the conjecture by Dehnen, made more than two fluid velocity of adjacent vortices. But for a ring vor- decades ago, that the Higgs field is a kind of short tex the fluid velocity is larger by the factor log(8R/rp), range gravitational field. It is the assumption for the compared to the velocity of a line vortex [11]. existence of negative masses which explains why the With R/rp = 147 for a line vortex, a better value for Higgs field appears to be ultrastrong, because it has a ring vortex should be obtained by solving the equa- its source in very large (±1013 GeV) positive and neg- tion ative masses. The existence of negative masses im- plies that the fundamental symmetry of nature is the R/rp = 147 log 8R/rp (18) Galilei group, where the particle numbers are con- with the result that served with the particle number operator commuting with the Hamilton operator, and where Lorentz invari- R/rp = 1360. (19) ance is explained dynamically by true deformations of With this value we obtain from (15) that rods and clocks, as in the pre-Einstein theory of rela- 13 tivity by Lorentz and Poincaré. The assumption of neg- h¯ωv ≈ 10 GeV (20) ative masses can make a bridge between the weak and which is of the right order of magnitude to explain the the Planck energy scale, thereby also solving the hier- Higgs mass. archy problem of elementary physics. There are good reasons for the Higgs field to be With the derivation of the Higgs mass from a res- a residual short range gravitational field having its ori- onance energy of the Planck mass plasma, Dehnen’s gin in the gravitational field of a very large positive and equation (3.5) from [5], negative mass, as the Van der Waals force is a residual short range electromagnetic field having its origin in 2 2 g2 mp 32 positive and negative electric charges, and the nuclear γ = = 2 · 10 , (21) 2 MH force a residual short range gluon force having its origin in the color charges of the quarks. is derived without the need for any new physics.

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