Two options concerning the interpretation of Higgs like particle in TGD framework
Theoretically the situation quite intricate. The basic starting point is that the original p-adic mass calculations provided excellent predictions for fermion masses. For the gauge bosons the situation was different: a natural prediction for the W/Z mass ratio in terms of Weinberg angle is the fundamental prediction of Higgs mechanism and this prediction did not follow automatically from the p-adic mass calculation in the original form. Classical Higgs field does not seem to have any natural counterpart in the geometry of space-time surface (the trace of the second fundamental form does not work since it vanishes for preferred extremals which are also minimal surfaces). This raised the question whether there is any Higgs boson in TGD Universe and for some time I took seriously the interpretation of the Higgs like state observed by LHC as a pion of M89. It is fair to say that the evolution of ideas about TGD counterpart of Higgs mechanism has been full of twists and turns.
p-Adic mass calculations and the results from LHC leave two options under consideration.
- Option I (see also this): Only fermions get the dominating contribution to their masses from p-adic thermodynamics and in the case of gauge bosons the dominating contribution is due to the standard Higgs mechanism. p-Adic thermodynamics would contribute also to the boson masses, in particular photon mass but the contribution would be extremely small and correspond to p-adic temperature T=1/n, n>2. For this option only gauge bosons would have standard model couplings to Higgs whereas fermionic couplings could be small. Of course, standard model couplings proportional to fermion mass are also possible. One can criticize this option because fermions and bosons are in an asymmetric position. The beautiful feature is that one could get rid of the hierarchy problem due to the couplings of Higgs to heavy fermions.
- Option II (see also this and this): p-Adic mass calculations explain also the masses of gauge bosons and Higgs like particle. If Higgs like state develops a coherent state describable in terms of vacuum expectation value as M^4 QFT limit, this expectation value is determined by the mass spectrum determine by the p-adic mass calculations. The mass spectrum of particles determines Higgs expectation and the couplings of Higgs rather than vice versa! For this option Weinberg angle would be defined by the ratio of W and Z boson mass as cos2(θW)=mW2/mZ2 and these masses should be given by p-adic mass calculations. Therefore the original problem with Weinberg angle would disappear. One must of course be very cautious here.
The recent view about particles as Kähler magnetic loops carrying monopole flux is forced by the assumption that the corresponding partonic 2-surfaces are Kähler magnetic monopoles (implied by the weak form of electric-magnetic duality). The loop proceeds from wormhole throat to another one, then traverses along wormhole contact to another space-time sheet and returns back and eventually is transferred to the first sheet via wormhole contact. The mass squared assignable to this flux loop could give the contribution usually assigned to Higgs vacuum expectation. If this picture is correct, then the reduction of the W/Z mass ratio to Weinberg angle might be much easier to understand. As a matter fact, I have proposed that the flux loop gives rise to a stringy spectrum of states with string tension determined by p-adic length scale associated with M89.
This option is attractive because fermions and bosons are in an exactly same position. Hierarchy problem is possible problem of this approach: note however that the considerations in the sequel imply that standard model action is predicted to be an effective action giving only tree diagrams so that there are no radiative corrections at M4 QFT limit.
A couple of comments about the experimental situation are in order.
- The original interpretation of Higgs like state was oas M89 pion. The recent observations from Fermi telescope suggest the existence of a boson with mass 135 GeV. It would be a good candidate for M89 pion. One can test the hypothesis by scaling the mass of ordinary neutral pion, which corresponds to M107. The scaling gives mass 69.11 GeV. p-Adic length scale however allows also octaves of the minimum mass (they appear for leptopions) and scaling by two gives mass equal to 138.22 GeV not too far from 135 GeV.
- There is also second encouraging numerical co-incidence. It is probably not an accident that Higgs vacuum expectation value corresponds to the minimum mass for p=M89 if the p-adic counterpart of Higgs expectation squared is of order O(p) in other words one has μ2/mCP22= p=M89.
Microscopic description of gauge bosons and Higgs like and meson like states
Under the pressures from LHC (and rather harsh social pressures from Helsinki University;-)) it has become gradually clear that the understanding of whether TGD has M4 QFT limit or not, and how this limit can be defined, is essential for the understanding also the role of Higgs. In the following a first attempt to understand this limit is made. I find it somewhat surprising that I am making this attempt only now but the understanding of the proper role of the classical gauge potentials has been quite a challenge.
- If one believes that M4 QFT is a good approximation to TGD at low energy limit then the standard description of Higgs mechanism seems to be the only possibility: this just on purely mathematical grounds. The interpretation would however be that the masses of the particles determine Higgs vacuum expectation value and Higgs couplings rather than vice versa. This would of course be nothing unheard in the history of physics: the emergence of a microscopic theory - in the recent case p-adic thermodynamics - would force to change the direction of the causal arrow in "Higgs makes particles massive" to that in "Higgs expectation is determined by particle masses".
- The existence of M4 QFT limit is an intricate issue. In TGD Universe baryon and lepton number correspond to different chiralities of H=M4× CP2 spinors and this means that Higgs like state cannot be H scalar (it would be lepto-quark in this case). Rather, Higgs like state must be a vector in CP2 tangent space degrees of freedom. One can indeed construct a candidate for a Higgs like state as an Euclidian pion or its scalar counterpart: both are possible and one can even consider the mixture of them. The H-counterpart of Higgs like state is therefore CP2 axial vector or CP2 vector or mixture of them.
Euclidian pion or scalar carries fermion and anti-fermion at opposite throat of the wormhole contact. It is easy to imagine that a coherent state of Euclidian pseudo-scalars or scalars or their mixture having Higgs expectation as M4 QFT correlate is formed. This state transforms as 2⊕2bar under U(2)⊂ SU(3) identifiable as weak gauge group. This representation is natural in Euclidian regions Higgs as a tangent space vector of CP2 has naturally 2&oplus 2bar decomposition in tangent space of CP2 allowing an interpretation as Lie algebra complement of u(2) ⊂ su(3).
In Minkowskian regions CP2 projection is 3-D and a natural counterpart of Higgs would be pseudo-scalar (or scalar) transforming as 3⊕ 1 and U(2)⊂ SU(3) identifiable now as strong U(2). The 3-dimensionality of the M4 projection suggests that one obtains only the triplet state.
- By bosonic emergence also gauge bosons correspond at microscopic level to fermion and anti-fermion at opposite throats of wormhole contacts. Meson like states in turn correspond to fermion and anti-fermion at the ends of a flux tube connecting throats of two different wormhole contacts so that both Higgs, gauge bosons, and meson-like states are obtained using similar construction recipe.
- The popular statement "gauge bosons eat almost all Higgs components" makes sense at the M4 QFT limit: just a transition to the unitary gauge effectively eliminates all but one of the components of the Higgs like state and gauge bosons get third polarization. This means gauge boson massivation but for option II it would take place already in p-adic thermodynamics in ZEO (zero energy ontology).
Trying to understand the QFT limit of TGD
The counterparts of gauge potentials and Higgs field are not needed in the microscopic description if p-adic thermodynamics gives the masses so that the gauge potentials and Higgs field should emerge only at M4 QFT limit. It is not even necessary to speak about Higgs and YM parts of the action at the microscopic level. The functional integral defined by the vacuum function expressed as exponent of Kähler action for preferred extremals to which couplings of microscopic expressions of particles in terms of fermions coupled to the effective fields describing them at QFT limit should define the effective action at QFT limit.
The basic recipe looks simple.
- Start from the vacuum functional which is exponent of Kähler action for preferred extremals with Euclidian regions giving real exponent and Minkowskian regions imaginary exponent.
- Add to this action terms which are bilinear in the microscopic expression for the particle state and the corresponding effective field appearing in the effective action.
- Perform the functional integration over WCW ("world of classical worlds") and take vacuum expectation value in fermionic degrees of freedom.
- This gives an effective field theory in M4× CP2 fields. To get M4 QFT integrate over CP2 degrees of freedom in the action. This dimensional reduction is similar to what occurs in Kaluza-Klein theories.
The functional integration of WCW induces also integration of induced spinor fields which apart from right-handed neutrino are restricted to the string world sheets. In principle induced spinor fields could be non-vanishing also at partonic 2-surfaces but simple physical considerations suggest that they are restricted to the intersection points of partonic 2-surfaces and string world sheets defining the ends of braid strands. Therefore the effective spinor fields Ψeff would appear only at braid ends in the integration over WCW and one has good hopes of performing the functional integral. The following arguments tries to sketch what happens.
- One can assign to the induced spinor fields Ψ imbedding space spinor fields Ψeff appearing in the effective action. The dimensions of Ψ and Ψeff are 1/L3/2. A dimensionally correct guess is the term ∫ d2x (g2)1/2Ψbareff(P) D-1Ψ+ h.c. Here Γα denotes the induced gamma matrices, P denotes the end point of a braid strand at the wormhole throat, and D denotes the "ordinary" massless Dirac operator ΓαDα for the induced gamma matrices. Propagator contributes dimension L and is well-defined since Ψ is not annihilated by D but by the modified Dirac operator in which modified gamma matrices defined by the modified Dirac action appear. Note that internal consistency does not allow the replacement of Kähler action with four-volume. Integral over the second wormhole throat contributes dimension L2. Therefore the outcome is a dimensionless finite quantity, which reduces to the value of integrand at the intersection of partonic 2-surface and string world sheet - that is at ends of braid strand since induced spinors are localized at string world sheets unless right-handed neutrinos are in question. The fact that induced spinor fields are proportional to a delta function restricting them to string world sheets does not lead to problems since the modified Dirac action itself vanishes by modified Dirac equation.
- Both Higgs and gauge bosons correspond to bi-local objects consisting of fermion and anti-fermion at opposite throats of wormhole contact and restricted to braid ends. The are connected by the analog of non-integrable phase factor defined by classical gauge potentials. These bilinear fermionic objects should correspond to Higgs and gauge potentials at QFT limit. The two integrations over the partonic 2-surfaces contribute L2 both, whereas the dimension of the quantity defining the gauge boson or Higgs like state is 1/L3 from the dimensions of spinor fields and from the dimension of generalized polarization vector compensated by that of gamma matrices. Hence the dimensions of the bi-local quantities are L for both gauge bosons and Higgs like particles. They must be coupled to their effective QFT counterparts so that a dimensionless term in action results. Note that delta functions associated with the induced spinor fields reduce them to the end points of braid strand connecting wormhole throats and finite result is obtained.
- How to identify these dimensional bilinear terms defining the QFT limit? The basic problem is that the microscopic representation of the particle is bi-local and the effective field at QFT limit should be local. The only possibility is to consider an average of the effective field over the stringy curve connecting the points at two throats. The resulting quantities must have dimensions 1/L in accordance with naive scaling dimensions of gauge bosons and Higgs to compensate the dimension L of the microscopic representation of bosons. For gauge bosons having zero dimension as 1-forms the average ∫ Aμdxμ/l along a unique stringy curve of length l connecting wormhole throats defines a quantity with dimension 1/L. For Higgs components having dimension 1/L the quantities ∫ HA(g1)1/2dx/l, where g1 corresponds to the induced metric at the stringy curve, has also dimension 1/L. The presence of the induced metric depending on CP2 metric guarantees that the effective action contains dimensional parameters so that the breaking of scale invariance results.
To sum up, for option II the parameters for the counterpart of Higgs action emerging at QFT limit must be determined by the p-adic mass calculations in TGD framework and the flux tube structure of particles would in the case of gauge bosons should give the standard contribution to gauge boson masses. For option I fermionic masses would emerge as mass parameters of the effective action. The presence of Euclidian regions of space-time having interpretation as lines of generalized Feynman diagrams is absolutely crucial in making possible Higgs like states. One must however emphasize that at this stage both option I and II must be considered.