Z_3 symmetry in NMSSM superpotential

[malawi_glenn]

Hello

I am trying to learn more about NMSSM, and in the Lecture Notes by Jack Gunion, held at SUSY08 conference in Seoul 2008 ( http://susy08.kias.re.kr/slide/pl/Gunion.pdf ) there is in slide #20 arguments for a Z_3 symmetry of the NMSSM superpotential.

My questions are what/which is this Z_3 symmetry? As far as my knowledge, there are several groups called Z.

Also I was wondering then HOW one can see that this superpotential has Z_3 symmetry.

Thank you

/Glenn

 

[Bobhawke]

The cyclic group of order 3? I don't know any other groups called Z.

 

[malawi_glenn]

From Wiki:

In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct types of groups:

* in the study of finite groups, a Z-group is a finite groups whose Sylow subgroups are all cyclic.
* in the study of infinite groups, a Z-group is a group which possesses a very general form of central series.
* occasionally, (Z)-group is used to mean a Zassenhaus group, a special type of permutation group.

 

[xepma]

In the context of physics the group [tex]Z_3[/tex] always refers to some cyclic symmetry of order three. The group contains three elements, {1,r,r^2}, with the property r^3 =1. It's similar to the addition of numbers mod 3 (or mod n in the case of Z_n).

When some system contains a Z_n symmetry, it means that it forms some irrep of the group Z_n. The action of the group elements on this irrep is something that needs to be specified. In this case, you can change each superfield by a phase factor according to:
[tex]\phi \rightarrow \phi' = \exp{\left(\frac{2\pi i}{3}\right)}\phi[/tex]
which is a symmetry of the action since it leaves the superpotential invariant. In the context of the representation of Z_3 you would say that this symmetry transformation is the action of the element r on the corresponding irrep.

P.S. I came across a review paper that came out this morning. You may find it interesting:
http://arxiv.org/abs/0906.0777

 

[malawi_glenn]

xempa, you have already made it to the "thank you list" of my thesis :-)

 

[xepma]

haha, glad to be of help

 

[humanino]

Just out of curiousity

It's similar to the addition of numbers mod 3 (or mod n in the case of Z_n).

and indeed we call it Zn because it's the quotient Z/3Z where Z is the integers. I had no idea of

* in the study of finite groups, a Z-group is a finite groups whose Sylow subgroups are all cyclic.
* in the study of infinite groups, a Z-group is a group which possesses a very general form of central series.
* occasionally, (Z)-group is used to mean a Zassenhaus group, a special type of permutation group.

 

[Haelfix]

Xepma is correct. The Z3 symmetry is imposed (or rather there by construction) in order to cancel the H1H2, N^2 and N terms in the superpotential which would have massive parameters associated with them (and hence phenomenologically unnacceptable).

The problem with that is it causes topological defects which is a problem if gravity is turned on, so you really want to explicitly break this discrete symmetry, rather than allow it to be spontaneously broken by the complex field (see the review article Xepma linked too).

 

[member 11137]

Xepma is correct. The Z3 symmetry is imposed (or rather there by construction) in order to cancel the H1H2, N^2 and N terms in the superpotential which would have massive parameters associated with them (and hence phenomenologically unnacceptable).

The problem with that is it causes topological defects which is a problem if gravity is turned on, so you really want to explicitly break this discrete symmetry, rather than allow it to be spontaneously broken by the complex field (see the review article Xepma linked too).

I began the very interesting lecture of 0906.0777v1 [hep-ph] 4 Jun 2009 and I have red attentively the § 2 concerning the discrete Z₃ symmetry. I appreciate the explanation about the necessity for that symmetry to brake. But you did certainly, like me, wonder the relation (2.9) and remark the very good similitude with the formalism of the one dimensional solutions of the Ginzburg Landau Theory for supraconduction. That means (for me): if the GL Theory is correct, then solutions of that theory describe (at least for the 1D approximation) the field in the intermediate domain wall region between two vacua (see explanation given in the cited arxiv document)...

Is that similitude a coincidence or something which was known by the author and still included into the construction of the proposed NMSSM?

Nevermind, I ask if we really must have to reject these walls? Would it not be better to accept to interpret them as manifestation of particles ?
Blackforest

 

[member 11137]

I... lecture of 0906.0777v1 [hep-ph] 4 Jun 2009 ... But you did certainly, like me, wonder the relation (2.9) and remark the very good similitude with the formalism of the one dimensional solutions of the Ginzburg Landau Theory for supraconduction...

Is that similitude a coincidence or something which was known by the author and still included into the construction of the proposed NMSSM?

So, I could find in between the answer to my question. Yes the remark has still be done; e.g. in Chris Quigg: Spontaneous Symmetry Breaking as a Basis of Particle Mass; arXiv: 0704.2232v2 [hep-ph], 28 May 2009.

Following my own logic can I ask a new and certainly naive question? Does have SU(4) a representation in a subset of M₄(R) ?