1. Higgs mass in standard model
In standard model Higgs and W boson masses are given by
mW2= g2v2/4= [e2/8sin2(θW)] μ2λ2 .
λ= [π/2αemsin2(θW)] (mH/mW)2 .
In standard model one cannot predict the value of mH.
2. Higgs mass in TGD
In TGD framework one can try to understand Higgs mass from p-adic thermodynamics as resulting via the same mechanism as fermion masses so that the value of the parameter λ would follow as a prediction.
One must assume that p-adic temperature equals to Tp=1. The natural assumption is that Higgs can be regarded as superposition of pairs of fermion and anti-fermion at opposite throats of wormhole contact. With these assumptions the thermal expectation of the Higgs conformal weight is just the sum of contributions from both throats and two times the average of the conformal weight over that for quarks and leptons:
sH= 2× <s> = 2× [∑q sq +∑L sL]/(Nq+NL)
= 2∑g=02 smod(g)/3+ (sL+sνL+ sU+sD)/2
= 26+5+4+5+8/2= 37 .
A couple of comments about the formula are in order.
- The first term - two times the average of the genus dependent modular contribution to the conformal weight - equals to 26, and comes from modular degrees of freedom and does not depend on the charge of fermion.
- The contribution of p-adic thermodynamics for super-conformal generators gives same contribution for all fermion families and depends on the em charge of fermion. The values of thermal conformal weights deduced earlier have been used. Note that only the value sνL=4 (also sνL=5 could be considered) is possible if one requires that the conformal weight is integer. If the standard form of the canonical identification mapping p-adics to reals is used, this must be the case since otherwise real mass would be super-heavy.
The first guess would be that the p-adic length scale associated with Higgs boson is M89. Second option is p≈ 2k, k=97 (restricting k to be prime). If one allows k to be non-prime (these values of k are also realized) one can consider also k=91=7×13. By scaling from the expression for the electron mass, one obtains the estimates
mH(89)≈ (37/5)1/2× 221me≈ 727.3 GeV , mH(91)≈ (37/5)1/2× 218me≈ 363.5 GeV, mH(97)≈ (37/5)1/2× 215me≈ 45.5 GeV.
A couple of comments are in order.
- From the article of Giudice one learns that the latest estimates for Higgs mass give two widely different values, namely mH= 3133-19 GeV and mH=420420-190 GeV. Since the p-adic mass scale of both neutrinos and quarks and possibly even electron can vary in TGD framework, one cannot avoid the question whether - depending on experimental situation- Higgs could appear in two different mass scales corresponding to k=91 and 97.
- The low value of mH(97) might be consistent with experimental facts since the couplings of fermions to Higgs can in TGD framework be weaker than in standard model because Higgs expectation does not contribute to fermion masses.
The value of λ is given in the three cases by
λ(89)≈ 4.41 , λ(91)≈ 1.10, λ(97)= .2757.
Unitarity would thus favor k=97 and k=91 also favored by the high precision data and k=91 is just at the unitarity bound λ=1) (here I am perhaps naive!). A possible interpretation is that for M89 Higgs mass forces λ to break unitarity bound and that this corresponds to the emergence of M89 copy of hadron physics.
For more details see the chapter p-Adic Particle Massivation: Elementary Particle Masses of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".