Higgs and massivation in TGD framework

The view about about particle massivation in TGD Universe has evolved considerably during the last half year thanks to the discovery of the weak form of electric-magnetic duality and in the following I try to explain it. The piece of text is actually a reply to a question by Ulla in Kea's blog. As I started to write the response my thoughts about Higgs mechanism in TGD framework were considerably different and this has forced to replace the posting with a new one. The core message is that one can really do without Higgs bosons and that it is quite possible and perhaps even unavoidable that photon eats the neutral Higgs boson getting very small mass so that only pseudoscalar counterpart of Higgs and Higgsinos would remain in the spectrum. This would mean that the search for Higgs at LHC would fail.

In TGD framework p-adic thermodynamics gives the dominating contribution to fermion masses, which is something completely new. In the case of gauge bosons thermodynamic contribution is small since the inverse integer valued p-adic temperature is T=1/2 for bosons or even lower: for fermions one has T=1.

Whether Higgs can contribute to the masses is not completely clear. In TGD framework Mexican hat potential however looks like trick. One must however keep in mind that any other mechanism must explain the ratio of W and Z⁰ masses and how these bosons receive their longitudinal polarizations. One must also consider seriously the possibility that all components for the TGD counterpart of Higgs boson are transformed to the longitudinal polarizations of the gauge bosons. Twistorial approach to TGD indeed strongly suggests that also the gauge bosons regarded usually as massless have a small mass guaranteing cancellation of IR singularities. As I started write to write this piece of text I believed that photon does not eat Higgs but had to challenge my beliefs. Maybe there is no Higgs to be found at LHC! Only pseudo-scalar partner of Higgs would and super partners of Higgs and pseudoscalar Higgs would remain to be discovered.

The weak form of electric magnetic duality implying that each wormhole throat carrying fermionic quantum numbers is accompanied by a second wormhole throat carrying opposite magnetic charge and neutrino pair screening weak isospin and making gauge bosons massive. Concerning the implications the following view looks the most plausible one at this moment.

1. Neutral Higgs-if not eaten by photon- could develop a coherent state meaning vacuum expectation value and this is naturally proportional to the inverse of the p-adic length scale as are boson masses. This contribution can be assigned to the magnetic flux tube mentioned above since it screens weak force - or equivalently - makes them massive. Higgs expectation would not cause boson massivation. Rather, massivation and Higgs vacuum expectation would be caused by the presence of the magnetic flux tubes. Standard model would suffer from a causal illusion. Even a worse illusion is possible if the photon eats the neutral Higgs.

2. The "stringy" magnetic flux tube connecting fermion wormhole throat and the wormhole throat containing neutrino pair would give to the vacuum conformal weight a small contribution and therefore to the mass squared of both fermions and gauge bosons (dominating one for the latter). This contribution would be small in the p-adic sense (proportional 1/p² rather than 1/p). I cannot calculate this "stringy" contribution but stringy formula in weak scale is very suggestive.

3. In the case of light fermions and massless gauge bosons the stringy contribution must vanish and therefore must correspond to n=0 string excitation (string does not vibrate at all) : otherwise the mass of fermion would be of order weak boson mass. For weak bosons n=1 would look like a natural identification but also n=0 makes sense since h+/- 1 states corresponds opposite three-momenta for massless fermion and antifermion so that the state is massive. The mechanism bringing in the h=0 helicity of gauge boson would be the TGD counterpart for the transformation of Higgs component to a longitudinal polarization. n> 0 excited states of fermions and n> 1 excitations of bosons having masses above weak boson masses are predicted and would mean new physics becoming possibly visible at LHC.

Consider now the identification of Higgs in TGD framework.

1. In TGD framework Higgs particles do not correspond to complex SU(2) doublets but to triplet and singlet having same quantum numbers as gauge bosons. Therefore the idea that photon eats neutral Higgs is suggestive. Also a pseudo-scalar variant of Higgs is predicted. Let us see how these states emerge from weak strings.

2. The two kinds of massive states corresponding to n=0 and n=1 give rise to massive spin 1 and spin 2 particles. First of all, the helicity doublet (1,-1) is necessarily massive since the 3-momenta for massless fermion and anti-fermion are opposite. For n=L=0 this gives two states but helicity zero component is lacking. For n=L=1 one has tensor product of doublet (1,-1) and angular momentum triplet formed by L=1 rotational state of the weak string. This gives 2× 3 states corresponding to J=0 and J=2 multiplets. Note however than in spin degrees of freedom the Higgs candidate is not a genuine elementary scalar particle.

3. Fermion and antifermion can have parallel three momenta summing up to a massless 4-momentum. Spin vanishes so that one has Higgs like particle also now. This particle is however pseudo-scalar being group theoretically analogous to meson formed as a pair of quark and antiquark. p-Adic thermodynamics gives a contribution to the mass squared. By taking a tensor product with rotational states of strings one would obtain Regge trajectory containing pseudoscalar Higgs as the lowest state.

Consider now the problem how the gauge bosons can eat the Higgs boson to get their longitudinal component.

1. (J=0,n=1) Higgs state can be combined with n=0 h=+/- 1 doublet to give spin 1 massive triplet provided the masses of the two states are same. This will be discussed below.

2. Also gauge bosons usually regarded as massless can eat the scalar Higgs so that Higgs like particle could disappear completely. There would be no Higgs to be discovered at LHC! But is this a real prediction? Could it be that it is not possible to have exactly massless photons and gluons? The mixing of M⁴ chiralities for Chern-Simons Dirac equation implies that also collinear massless fermion and antifermion can have helicity +/- 1. The problem is that the mixing of the chiralities is a signature of massivation!

   Could it really be that even the gauge bosons regarded as massless have a small mass characterized by the length scale of the causal diamond defining the physical IR cutoff and that the remaining Higgs component would correspond to the longitudinal component of photon? This would mean the number of particles in the final states for a particle reaction with a fixed initial state is always bounded from above. This is important for the twistorial aesthetics of generalized Feynman diagrammatics implied by zero energy ontology. Also the vanishing of IR divergences is guaranteed by a small physical mass. Maybe internal consistency allows only pseudo-scalar Higgs.

The weak form of electric-magnetic duality suggests strongly the existence of weak Regge trajectories.

1. The most general linear mass squared formula with spin-orbit interaction term M²_L-SL• S reads as

   M²= nM₁²+ M₀² +M²_L-SL• S , n=0,2,4 or n=1,3,5,... .

   M₁² corresponds to string tension and M₀² corresponds to the thermodynamical mass squared and possible other contributions. For a given trajectory even (odd) values of n have same parity and can correspond to excitations of same ground state. From ancient books written about hadronic string model one vaguely recalls that one can have several trajectories (satellites) and if one has something called exchange degeneracy, the even and odd trajectories define single line in M²-J plane. As already noticed TGD variant of Higgs mechanism combines together n=0 states and n=1 states to form massive gauge bosons so that the trajectories are not independent.

2. For fermions, possible Higgs, and pseudo-scalar Higgs and their super partners also p-adic thermodynamical contributions are present. M₀² must be non-vanishing also for gauge bosons and be equal to the mass squared for the n=L=1 spin singlet. By applying the formula to h=+/- 1 states one obtains

   M₀²= M²(boson) .

   The mass squared for transversal polarizations with (h,n,L)=(+/- 1,n=L=0,S=1) should be same as for the longitudinal polarization with (h=0, n=L=1, S=1, J=0) state. This gives

   M₁²+M₀²+ M²_L-SL• S= M₀² .

   From L• S= [ J(J+1)-L(L+1)-S(S+1)]/2= -2 for J=0, L=S=1 one has

   M_L-S²= -M₁²/2 .

   Only the value of weak string tension M₁² remains open.

3. If one applies this formula to arbitrary n=L one obtains total spins J= L+1 and L-1 from the tensor product. For J=L-1 one obtains

   M²= (2n+1)M₁²+ M₀².

   For J=L+1 only M₀² contribution remains so that one would have infinite degeneracy of the lightest states. Therefore stringy mass formula must contain a non-linear term making Regge trajectory curved. The simplest possible generalization which does not affect n=0 and n=1 states is of from

   M²= n(n-1)M₂²+ (n-L• S/2)M₁²+ M₀².

The challenge is to understand the ratio of W and Z⁰ masses, which is purely group theoretic and provides a strong support for the massivation by Higgs mechanism.

1. The challenge is to understand the ratio of W and Z⁰ masses, which is purely group theoretic and provides a strong support for the massivation by Higgs mechanism. The above formula and empirical facts require

   M₀²(W)/M₀²(Z)= cos²(θ_W) .

   Since this parameter measures the interaction energy of the fermion and antifermion decomposing the gauge boson depending on the net quantum numbers of the pair, it would look very natural that one would have

   M₀²(W)= g_W²M_SU(2)² ,

   M₀²(Z)= g_Z²M_SU(2)² .

   Here M_SU(2)² would be the fundamental mass squared parameter for SU(2) gauge bosons. p-Adic thermodynamics of course gives additional contribution which is vanishing or very small for gauge bosons.

2. The required mass ratio would result in an excellent approximation if one assumes that the mass scales associated with SU(2) and U(1) factors suffer a mixing completely analogous to the mixing of U(1) gauge boson and neutral SU(2) gauge boson W₃ leading to γ and Z⁰. Also Higgs, which consists of SU(2) triplet and singlet in TGD Universe, would very naturally suffer similar mixing. Hence M₀(B) for gauge boson B would be analogous to the vacuum expectation of corresponding mixed Higgs component. More precisely, one would have

   M₀(W)= M_SU(2) ,

   M₀(Z)= cos(θ_W) M_SU(2)+ sin(θ_W) M_U(1) ,

   M₀(γ)= -sin(θ_W) M_SU(2)+ cos(θ_W) M_U(1) .

   The condition that photon mass is very small and corresponds to IR cutoff mass scale gives

   M₀(γ)=ε cos(θ_W)M_SU(2),

   where ε is very small number, and implies

   M_U(1)/M(W)=tan(θ_W) +ε ,

   M(γ)/M(W)= ε× cos(θ_W) ,

   M(Z)/M(W)= [1+ε × sin(θ_W)cos(θ_W)]/cos(θ_W) .

   There is a small deviation from the prediction of the standard model for W/Z mass ratio but by the smallness of photon mass the deviation is so small that there is no hope of measuring it. One can of course keep mind open for ε=0. The formulas allow also an interpretation in terms of Higgs vacuum expectations as it must. The vacuum expectation would most naturally correspond to interaction energy between the massless fermion and antifermion with opposite 3-momenta at the throats of the wormhole contact and the challenge is to show that the proposed formulas characterize this interaction energy. Since CP_2 geometry codes for standard model symmetries and their breaking, it woul not be surprising if this would happen. One cannot exclude the possibility that p-adic thermodynamics contributes to M₀²(boson). For instance, ε might characterize the p-adic thermal mass of photon.

   If the mixing applies to the entire Regge trajectories, the above formulas would apply also to weak string tensions, and also photons would belong to Regge trajectories containing high spin excitations.

3. What one can one say about the value of the weak string tension M₁²? The naive order of magnitude estimate is M₁²≈ m_W²≈ 10⁴ GeV² is by a factor 1/25 smaller than the direct scaling up of the hadronic string tension about 1 GeV²scaled up by a factor 2¹⁸. The above argument however allows also the identification as the scaled up variant of hadronic string tension in which case the higher states at weak Regge trajectories would not be easy to discover since the mass scale defined by string tension would be 512 GeV to be compared with the recent beam energy 7 TeV. Weak string tension need of course not be equal to the scaled up hadronic string tension. Weak string tension - unlike its hadronic counterpart- could also depend on the electromagnetic charge and other characteristics of the particle.

For background see the new section of p-Adic Mass Calculations: New Physics.