Expecting the end of alchemy and astrology in Standard Model extensions
Because we have forbidden higher dimension operators by hand, the Standard Model has no explicit cutoff dependence. However, if the Higgs self-coupling is too large – corresponding to a physical Higgs boson mass greater than about 180 GeV – then the SM generates its own ultraviolet cutoff ΛLP. This is because λ runs logarithmically with energy scale, and if λ is large enough at the electroweak scale the sign of the effect is to increase λ at higher energies. At some energy scale ΛLP the coupling hits a Landau pole and the electroweak sector of the Standard Model breaks down. If the Higgs self-coupling at the electroweak scale is too small – corre- sponding to a physical Higgs boson mass less than about 130 GeV – then the running goes the other way, and at some high energy scale the sign of this quartic coupling goes negative. At best, this destabilizes the vacuum; at worst, theories with this kind of disease are unphysical. One could attempt to compensate by invoking dimension 6 Higgs self–couplings, but this would violate one of our deﬁning theoretical inputs.
(Submitted on 26 Sep 2006)
The Standard Model is sowing the seeds of its own extension
If a right-handed neutrino exists, the group-theoretic evidence for grand uniﬁcation is ... compelling: the fermions of each generation transform as the 16-dimensional representation of SO(10) ... and the three gauge groups are uniﬁed into one [irreducible representation]. We assume the existence of a right-handed neutrino for the remainder of the discussion. SO(10) has a subgroup SU(3)×SU(2)×U(1)×U(1). When SO(10) is spontaneously broken to SU(3)c×SU(2)L×U(1)Y, the hypercharge subgroup [by deﬁnition, the unbroken U(1) subgroup, which does not correspond in general to the usual hypercharge subgroup of the standard model] is a linear combination of the two U(1) subgroups of SO(10). Thus the hypercharges of the fermions are not uniquely determined in SO(10) grand uniﬁcation ... but rather depend upon which linear combination of the two U(1) subgroups is unbroken. The SU(3)c×SU(2)L×U(1)Y quantum numbers of the left-handed ﬁelds which make up the 16-dimensional representation of SO(10) are given in Table 1.
The hypercharge is normalized such that the left-handed positron has unit hypercharge. The parameter a depends upon which linear combination of the two U(1) subgroups is unbroken. It is a rational number because the hypercharges are “quantized”, i.e., commensurate, since a U(1) subgroup of a non-Abelian group is necessarily compact ... .
The value of the parameter a depends upon the Higgs representation employed to break SO(10) to SU(3)c×SU(2)L×U(1)Y . The Higgs ﬁeld may be either fundamental or com- posite; only its group-theoretic properties are relevant to the considerations of this pa- per. The candidate values of a for a given irreducible representation correspond to the SU(3)×SU(2)×U(1) singlets contained in that representation . Usually this representa- tion must be accompanied by at least one additional Higgs irreducible representation in order to break SO(10) down to SU(3)×SU(2)×U(1), because the latter is generally not a maximal little group of the former for a single irreducible representation . To generate fermion masses, the SU(2)L×U(1)Y symmetry must be broken by yet one or more additional Higgs irreducible representation, chosen from the 10-, 120-, and 126-dimensional representa- tions (since 16 × 16 = 10 + 120 + 126). The SU(2)L×U(1)Y symmetry is broken to U(1)EM when any of the color-singlet, SU(2) doublets contained in these representations acquires a vacuum-expectation value, leading to the electric charges listed in the last column of Table 1. The standard model evidently corresponds to a = 1/6.
We have shown by construction that any rational value of a can be obtained by an appro- priate choice of the Higgs irreducible representation. However, a given value of a generally requires a very large Higgs irreducible representation. In practice, the smallest Higgs irre- ducible representations yield only a few values of a ...
It is satisfying that the standard model (a=1/6) is obtained with several small Higgs irreducible representations,6 the 16-, 126-, and 144-dimensional representations, as is well known [11a, 11b]. If we lived in a world in which the ratio of the hypercharges of the quark doublet and the positron were, say, 1/8 rather than 1/6, we could still embed the fermions in the 16-dimensional representation of SO(10), but we would need a 9504-dimensional Higgs representation to obtain the desired symmetry breaking. While there is (perhaps) nothing fundamentally wrong with this, it is less palatable than a model which requires only Higgs ﬁelds in low-dimensional irreducible representations ...
We believe that the economy of the Higgs representation in SO(10) grand uniﬁcation, while well known, has not been fully appreciated. We regard it as further evidence for SO(10) grand uniﬁcation.
(Submitted on 28 Oct 1997)