  The p-adic description of Higgs mechanism in TGD framework provides excellent
predictions for elementary particle and hadrons masses<EMAIL_ADDRESS>9410058-62). The gauge group of TGD is just the gauge group of the standard
model so that it makes sense to study the p-adic counterpart of the standard
model as a candidate for low energy effective theory. Momentum eigen states can
be constructed purely number theoretically and the infrared cutoff implied by
the finite size of the convergence cube of p-adic square root function leads to
momentum discretization. Discretization solves ultraviolet problems: the number
of momentum states associated with a fixed value of the propagator expression
in the loop is integer and has p-adic norm not larger than one so that the
contribution of momentum squared with p-adic norm $p^{k}$ converges as
$p^{-2k-2}$ for boson loop. The existence of the action exponential forces
number theoretically the decomposition into free and interacting parts. The
free part is of order $O(p^0)$ and must vanish (and does so by equations of
motion) and interaction part is at most of order $O(\sqrt{p})$ p-adically.
p-Adic coupling constants are of form $g\sqrt{p}$: their real counterparts are
obtained by canonical identification between p-adic and real numbers. The
discretized version of Feynmann rules of real theory should give S-matrix
elements but Feynmann rules guarantee unitarity in formal sense only. The
unexpected result is the upper bound $L_p=L_0/\sqrt{p}$ ($L_0\sim
10^4\sqrt{G}$) for the size of p-adic convergence cube from the cancellation of
infrared divergences so that p-adic field theory doesn't make sense above
length scale $L_p$.
