# in SUSY, does WW scattering unitarisation needs the higgs boson?

One of the arguments of LHC "win-win situation" is that the scattering of W particles needs to include new terms to preserve unitatity begond 500 GeV or so. In the SM, this is realized by the higgs boson, and the same argument extends to the MSSM. Now, one could consider just a "phenomenology Susy standard model" where simply the superpartners to the known SM are added.

This implies that, even without a Higgs mechanism, for each of the W and Z one needs to add an extra scalar boson, to get the right multiplcity; a massive gauge supermultiplet must contain four bosonic helicities. Of course, in the MSSM, two of these come from the Higgses. But from the point of view of SUSY, they are independent.

My question is, are these scalars, alone or jointly with the sleptons and squarks, enough to preserve the unitarity of WW scattering?

A similar question could be, if you remove two higgses from the five that come in the MSSM, is it still possible to unitarize WW scattering?

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The "unitarization" of W-W scattering by the Higgs is just the statement that the longitudinal components of the W are revealed to be independent scalars at high energy, because massive spin 1 theories by themselves have directions in k space with no falloff in the propagator. The scale where unitarity is violated by massive vectors will shift if you change the details, but not by much. The fundamental reason is that the longitudinal components are present.

The only way to unitarize the scattering is to have a Higgs mechanism, meaning a vacuum condensate which breaks the global gauge invariance. If you want to unitarize using other scalars in a SUSY model, you would need to have one of these scalars play the exact role of the Higgs.

The reason is that the longitudinal component of a gauge field must be generated by a condensate. If there is no condensate, there is no mass term. The reason is gauge invariance--- a mass term pushes the A field to zero, but the zero value of an A field is not a gauge invariant concept. So in order to have a term in the Lagrangian that pushes the value to zero, there must be a condensate which defines what zero A means. A zero value for A means that the parallel transport using A from point x to point y with a given gauge is equal to the difference in gauge configuration of the condensate at point x and point y using that gauge. In EM, it means that the phase change $\int A dx$ between two points is equal to the phase difference of the charged condensate.

So your question boils down to ...

### Can the Higgs be a Superpartner?

The SUSY commutes with the gauge charges in MSSM, so you would need a particle which is a SU(2) doublet, SU(3) singlet, and has weak hypercharge 1 (in the standard normalization of hypercharge). That rules out everything except the left-handed lepton.

The idea that the Higgs can be the slepton was proposed by Fayet: (P. Fayet, Nuclear Physics B 90, 104 (1975)), but is normally rejected. The reason is that there must be Yukawa couplings between the Higgs field and the leptons and quarks, to produce masses, but these couplings must be generated from a superpotential in a SUSY theory, and you can't write down renormalizable lepton-quark and lepton-lepton couplings of the appropriate sort because of the gauge charges of the lepton and quark fields.

There are two counter-models to this in the literature that I found just now, but they both have unsavory physics. One of them is described here: http://www.slac.stanford.edu/econf/C010630/papers/P113.PDF. It claims that it can make the Higgs a slepton by violating Lorentz invariance at high energies. The paper also claims that with the appropriate violation, there are new allowed supersymmetry algebras which allow the appropriate Higgs-fermion Yukawa interactions at low energies. I don't know how to evaluate this claim at the present time, because I am not familiar with this algebra.

The other countermodel is due to Grant and Kakushadze:http://arxiv.org/abs/hep-ph/9906556. This paper describes a 4-generation model with the Higgs as a partner of a 4th generation slepton. But this is in the wild-west anything goes fantasy world of TeV planck-scale large-extra dimension phenomenology, or "recreational phenomenology" in Kakushadze's phrase.

The argument that the Higgs and the leptons can't be superpartners is making the assumption that only soft SUSY breaking is allowed. If you put in terms by hand of dimension 4, you can obviously reconstruct whatever you want in terms of the needed Yukakwa couplings, but you also wreck the SUSY relations with log-running breaking terms. This type of thing is frowned upon, because you assume that SUSY breaking is spontaneous and communicated to us in a way that stabilizes the Higgs. This principle is natural, but it doesn't seem to me to be as firmly established as Lorentz invariance and near-renormalizability (which is what the models above break).

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I like the answer, but just to be sure: then, the mechanism where a massive gauge supermultiplet is revealed to be an union of a massless gauge supermultiplet plus a massless "Weyl" supermultiplet, should still be a version of the Higgs mechanism, shouldn't it? – arivero Sep 6 '11 at 22:36
Alexander Knochel has a whole thesis on supersymmetric Higgsless EWSB: physik.uni-wuerzburg.de/fileadmin/11030200/Dissertationen/… There are also many papers modifying the MSSM so the down-type Higgs is absent or inert or composite (in one very recent paper down-type quarks get their mass from the tau-sneutrino). But so far a fundamental up-type Higgs is indispensable. – Mitchell Porter Sep 7 '11 at 6:55
@Mitchell: Knochel's "Higgsless models" are neither Higgsless, nor useful models. They are all TeV-quantum gravity models, and therefore of no scientific value. If you look at the details of the theory, it breaks the electroweak symmetry on a brane, somewhere, so it still has a Higgs, it's just not a usual Higgs in that you can put it somewhere else. My point is that you don't need any Higgs, neither up-type nor down-type, if you allow hard breaking--- just use one of the sleptons. But you lose any explanation of heirarchy stabilization. – Ron Maimon Sep 7 '11 at 13:57
@arivero: your comment is about kinematic decomposition of supermultiplets, and you can do this without Higgs mechanism, just because you haven't got any interactions. Kinematics is free fields. You can have massive spin-1 free fields with no problem, you can have massive spin 7 fields, or massless, or whatever. The point is that if you make a dynamical gauge model, with some interacting action, and you have a massive gauge boson, then you have to have a higgs. – Ron Maimon Sep 7 '11 at 14:00
@Ron, the dynamical model circles back to my original question: Do we need to have new scalars coming from a higgs field, or the extra scalar already in the massive gauge supermultiplet can do the unitarisation of the scattering? – arivero Sep 7 '11 at 21:58