# Degrees of freedom in the SM / MSSM?

Melmoth71
I'm trying to work out the number of effective degrees of freedom in the LSS (Light Stop Scenario) within the MSSM (Minimal Supersymmetric Standard Model). To make sure that I understand the concept I am trying to reproduce the number of dofs in the Standard Model.

The way I count them the degrees of freedom from the different particle species are:

Scalar, spin-0 (Higgs boson): 1
Massless vector bosons (Photons, gluons): 2
Massive vector bosons (W+, W-, Z): 3
Leptons: 2 (spin up and down) x 2 (particle/antiparticle) = 4
Quarks: 2 (spin up and down) x 2 (particle/antiparticle) x 3 (colour) = 12

There are 8 gluon, 6 quark and 6 lepton species in the SM so I get:

Total degrees of freedom = 1 + 2 + ( 8 x 2) + ( 3 x 3) + ( 6 x 4 ) + (6 x 12) = 128

Is this right??

Thanks,

Melmoth

Gold Member
1 + 2 + ( 8 x 2) + ( 3 x 3) + ( 6 x 4 ) + (6 x 12) = 124

Note that the MSSM has more higgses than the SM, it needs a different higgs field for Up and Down sectors, and so instead of one higgs scalar you get 5 higgs scalar (two of them charged, three neutrals).

also be careful about your neutrinos. If you assume Dirac neutrinos, you are right, but if they are Majorana then there are only 2 dof instead of 4. So g = 118.

Gold Member
also be careful about your neutrinos. If you assume Dirac neutrinos, you are right, but if they are Majorana then there are only 2 dof instead of 4.

Actually, which is the experimental status?

Parlyne
Actually, which is the experimental status?

No experiment yet really speaks to that issue. Since only the left-handed neutrinos couple to any other fields, there aren't really too many effects that directly distinguish the two cases. So far as I know, no one has claimed detection of neutrinoless double beta decay. But, other than that, I think what you'd need is some other observations that point to a specific model that requires neutrinos to be of one type or the other.

So, in short, totally undetermined.

Gold Member
No experiment yet really speaks to that issue. Since only the left-handed neutrinos couple to any other fields, there aren't really too many effects that directly distinguish the two cases. So far as I know, no one has claimed detection of neutrinoless double beta decay. But, other than that, I think what you'd need is some other observations that point to a specific model that requires neutrinos to be of one type or the other.

So, in short, totally undetermined.

Well, not totally, for the goals of this list, I think that the possiblity of giving just Majorana to the actual neutrinos (and then not extra degrees of freedom) is excluded, is it?

Lacking this insight, we should vote for the theoretical input: all the models beyond naive GUT SU(5) need to add another Weyl piece for each generation, it is so in SO(10) etc

Well, not totally, for the goals of this list, I think that the possiblity of giving just Majorana to the actual neutrinos (and then not extra degrees of freedom) is excluded, is it?

I don't think so. What experiment ruled that out? Majorana neutrinos are possible within the SM from a dimension-5 operator, so you need not add any additional degrees of freedom. It becomes a little harder to explain where the masses come from since you don't have a "see-saw mechanism", but we don't know where ANY of the fermion masses come from, so....

I thought the only definitive evidence for neutrino masses comes from oscillations, and that doesn't care what kind of mass they have. Although I might be wrong on that. Does anyone know of an experiment that contradicts me?

Lacking this insight, we should vote for the theoretical input: all the models beyond naive GUT SU(5) need to add another Weyl piece for each generation, it is so in SO(10) etc

strictly for myself, i wouldn't be in a rush to use GUTs as motivation. but that's just me, and i know plenty of smart people who yell at me when i say this... so i guess you should decide for yourself.

Melmoth71
Hi again and thank you all very much for all the input.

Yes I probably should learn to add before I try to learn physics :)

I think I've understood the standard result of my calculation for the Standard Model- I have yet to extend it to the LSS-MSSM case.

I should have said that what I am trying to find out is the number of effective degrees of freedom at the time/scale of the electroweak phase transition (~ 100 GeV). The standard expression for this is

g = (sum over bosons) gi * (Ti/T)^4 + (7/8) * (sum over fermions) gi * (Ti/T)^4

This assumes that different species may have an equilibrium temperature Ti different from the temperature of the universe T if they have decoupled from the 'heat bath'- Not the case for any species at the electroweak scale at which all species are considered relativistic with m << T.

The detail I was missing was that only the degrees of freedom of left-handed neutrinos and right-handed antineutrinos are counted (see eg Kolb and Turner, 'The Early Universe'). So in my original list I should have counted three neutrinos and three antineutrinos with two dofs each. This is notwithstanding the discussion on Dirac and Majorana neutrinos- I'm only citing what seems to be the party line on the counting.

The (7/8) factor for fermions comes from the difference between fermion and boson statistics.

This gives the standard number of degrees of freedom (for the Standard Model) at the electroweak scale, g = 106.75 ~ 107.

I now have to do the same calculation for the LSS-MSSM case. This assumes one heavy (and decoupled) stop and a light right-handed stop. Hopefully my PhD supervisor will give me a hand to get the counting right!

I'll post the results.

Thanks again for all the input.

Melmoth

Gold Member
Melmoth, it is interesting! So the cosmological data will be sensitive to, say, gauginos at electroweak scale?

I thought the only definitive evidence for neutrino masses comes from oscillations, and that doesn't care what kind of mass they have. Although I might be wrong on that. Does anyone know of an experiment that contradicts me?

Not me. Probably I am very influenced by the see saw, plus a desire to put everything into a 128 fermion: With only majorana masses in the actual neutrinos, let me see, and MSSM Higgs, it should be 122, is it? With SO(10) like neutrinos, and MSSM Higgs, it should be 128 (but no place for graviton). And with SO(10) like neutrinos and no Higgs, putting explicit mass terms into SUSY multiplets for W and Z, it is 126 (so the gravitino adds to 128 ).

The later combination is a bit tricky, the point is that a massive N=1 gauge multiplet has the content of a massless N=2. You can see it "by hand", get the Z massless, add a degree for the cero helicity, you need a partner for this new degree of freedom and the minimal addition is a Weyl fermion, and then you need also add another scalar. So with the GUT neutrino, the minimal content of a Higgsless SUSY is 126+126. No idea about what happens with unitarity in WW scattering, in this case.