Is Higgs really needed and does it exist?

The mass range containing the Higgs mass is becoming narrower and narrower (see the postings of Tommas Dorigo and Lubos Motl), and one cannot avoid the question whether the Higgs really exists. This issue remains far from decided also in TGD framework where also the question whether Higgs is needed at all to explain the massivation of gauge bosons must be raised.

1. My long-held belief was that Higgs does not exist. One motivation for this belief was that there is no really nice space-time correlate for the Higgs field. Higgs should correspond to M⁴ scalar and CP₂ vector but one cannot identify any natural candidate for Higgs field in the geometry of CP₂. The trace of CP₂ part of the second fundamental form could be considered as a candidate but depends on second derivatives of the imbedding space coordinates. Its counterpart for Kähler action would be the covariant divergence of the vector defined by modified gamma matrices and this vanishes identically.

2. For a long time I believed that p-adic thermodynamics is not able to describe realistically gauge boson massivation and the group theoretical expression for the mass ratio of W and Z gauge bosons led to the cautious conclusion that Higgs is needed and generates a coherent state and that the ordinary Higgs mechanism has TGD counterpart. This field theoretic description is of course purely phenomenological in TGD framework and whether it extends to a microscopic description is far from clear.

3. The identification of bosons in terms of wormhole contacts having fermion and antifermion at their light-like throats allowed a construction of also Higgs like particle. One can estimate its mass by p-adic thermodynamics using the existing bounds to determine the p-adic length scale in question: p≈2^k, k=94, is the best guess and gives m_H=129 GeV, which is consistent with the experimental constraints. Higgs expectation cannot however contribute to fermion masses if fermions are identified as CP₂ type vacuum extremals topologically condensed to single space-time sheet so that there can be only one wormhole throat present. This would mean that Higgs condensate -whatever it means in precise sense- is topologically impossible in fermionic sector. p-Adic thermodynamics for fermions allows only a very small Higgs contribution to the mass so that this is not a problem.

4. The next step was the realization that the deviation of the ground state conformal weights from half integer values could give rise to Higgs type contribution to both fermion and boson mass. Furthermore, the contribution to the ground state conformal weight corresponds to the modulus squared for the generalized eigenvalue λ of the modified Dirac operator D. This picture suggests a microscopic description of gauge boson masses and the Weinberg angle determining W/Z mass ratio can be expressed in terms of the generalized eigenvalues of D. Higgs could be still present and if it generates vacuum expectation (characterizing coherent state) its value should be expressible in terms of the generalized eigenvalues of modified Dirac operator. The causal relation between Higgs and massivation would not however be what it is generally believed to be.

The massivation of Z⁰ and generation of longitudinal polarizations are the problems, which should be understood in detail before one can take seriously in TGD inspired microscopic description.

1. The presence of an axial part in the decomposition of gauge bosons to fermion-antifermion pairs located at the throats of the wormhole contact should explain the massivation of intermediate gauge bosons and the absence of it the masslessness of photon, gluon, and gravitons.

2. One can understand the massivation of W bosons in terms of the differences of the generalized eigenvalues of the modified Dirac operator. In the case of W bosons fermions have different charges so that the generalized eigenvalues of the modified Dirac operator differ and their difference gives rise to a non-vanishing mass. Both transverse and longitudinal polarizations are in the same position as they should be.

3. The problem is how Z⁰ boson can generate mass. For Z⁰ the fermions for transverse polarizations should have in a good approximation same spectrum generalized eigenvalues so that the mass would vanish unless fermion and anti-fermion correspond to different eigenvalues for some reason for Z⁰. The requirement that the photon and Z⁰ states are orthogonal to each other might require different eigen values. If fermion and antifermion in both Z⁰ and photon correspond to the same eigen mode of the modified Dirac operator, their inner product is proportional to the trace of the charge matrices given by Tr(Q_em(I³_L+sin²(θ_W)Q_em), which is non-vanishing in general. For different eigenmodes in the case of Z⁰ the states would be trivially orthogonal.

4. Gauge bosons must allow also longitudinal polarization states. The fact that the modes associated with wormhole throats are different in the case of Z⁰ could allow also longitudinal polarizations. The state would have the structure bar(Ψ)_- (D_→-D_←) Q_ZΨ₊, D= p^kγ_k. This state does not vanish for intermediate gauge bosons since the action of p^kγ_k to the two modes of the induced spinor field is different and ordinary Dirac equation is not true induced spinor fields. For photon and gluons the state would vanish.

5. In the standard approach the gradient of Higgs field is transformed to a longitudinal polarization of massive gauge bosons. It is not clear whether this kind of idea makes sense at all microscopically in TGD framework. The point is that Higgs as a particle corresponds to a superposition of fermion-antifermion pairs with opposite M⁴ chiralities whereas the longitudinal part corresponds to pairs with same M⁴ chiralities. Hence the idea about the gradient of Higgs field transforming to the longitudinal part of gauge boson need not make sense in TGD framework although Higgs can quite well exist.

To sum up, these arguments could be seen as a support for the possibility that Higgs is not needed at all in particle massivation in TGD Universe but leave open the question whether Higgs exists as particle and possibly develops coherent state.

For details and background see the updated chapter p-Adic Particle Massivation: Elementary Particle Masses of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".