Higgsinos and Chiral Symmetry

[shirosato]

Question 1: I am aware the Higgs lies in a chiral supermultiplet, but I realized I don't have an intuitive idea of

i) how many Higgsinos there are (since the MSSM has 2 complex isodoublets)
ii) how many Higgsinos there are after EWSB and you gauge away three of the scalar fields
iii) their structure (Weyl, Dirac, Majorana)

Question 2: Question 1 arose after I heard that the deeper reason the Higgs is protected from large corrections is due to chiral symmetry of the Higgsino. I thought a bit about it, and figured it had something to do with the Higgsino exhibiting chiral symmetry and SUSY enforcing mass degeneracy.

Thanks!

 

[shirosato]

I suppose wiki has a clear enough answer:

In particle physics, a Higgsino, symbol H͂, is the hypothetical superpartner of the Higgs boson, as predicted by supersymmetry. The Higgsino is a Dirac fermion and that is a weak isodoublet with hypercharge half under the Standard Model gauge symmetries. After electroweak symmetry breaking the Higgsino becomes a pair of neutral Majorana fermions called neutralinos and a charged Dirac fermion called a chargino (plus and minus).

 

[arivero]

I found illuminating to consider the mathematical case when supersymmetry is not broken but the W and Z bosons are massive. Then, even if you have not a Higgs-based symmetry breaking, each W and Z is forced to have another scalar and another -ino in its multiplet, because the composition of a massive N=1 vector supermultiplet is the same that a massless N=2, and it is the union of a N=1 massless vector supermultiplet plus a N=1 massless chiral supermultiplet.

That means that the Higgs mechanism of the MSSM only adds, really, a two component higgsino and two scalars, the other six scalars and their superpartners are compulsory.

 

[blechman]

i) how many Higgsinos there are (since the MSSM has 2 complex isodoublets)

in the MSSM, there are 4 Higgsinos (2 charged and 2 neutral). They are Weyl fermions that mix with themselves as well as the other gauginos to form "Charginos" and "Neutralinos". It is these objects that are the observed fermions.

ii) how many Higgsinos there are after EWSB and you gauge away three of the scalar fields

There are always the same number of Higgsinos - only the scalar modes are Goldstone modes, the fermions are physical.

iii) their structure (Weyl, Dirac, Majorana)

See above.

Question 2: Question 1 arose after I heard that the deeper reason the Higgs is protected from large corrections is due to chiral symmetry of the Higgsino. I thought a bit about it, and figured it had something to do with the Higgsino exhibiting chiral symmetry and SUSY enforcing mass degeneracy.

Thanks!

The idea is that the Higgsino mass is protected by chiral symmetry; and that the Higgs mass is related to the Higgsino mass through Supersymmetry. Therefore, your statement.

So yes, your statement is more or less correct. Since SUSY is broken, the Higgs mass is not EXACTLY the same as the Higgsino mass, but they are at least forced to be the same order of magnitude, so long as SUSY is "softly broken" as it is in the MSSM (that is, broken only by superrenormalizable operators). Hence the hierarchy problem is still solved even after susy breaking.

Hope that helps!

 

[lpetrich]

Let's count modes for MSSM electroweak particles. The Higgses are Wess-Zumino multiplets, meaning that the Higgsinos are chiral.

WIS = weak isospin

W: (helicity: 2) * (WIS triplet: 3) = 6
B: (helicity: 2) = 2
Higgs: (complex: 2) * (WIS doublet: 2) * (u,d: 2) = 8

Wino: same as W
Bino: same as B
Higgsino: complex -> helicity, otherwise same as Higgs

There are 16 bosonic modes and 16 fermionic modes, as one expects from SUSY.

Now the broken version:

W: (helicity: 3) * (charge: 2) = 6
Z: (helicity: 3) = 3
photon: (helicity: 2) = 2
Higgs neutral: (mass states: 3) = 3
Higgs charged: (charge: 2) = 2
Sum: 16
Charged: 8
Neutral: 8

Wino charged: (helicity: 2) * (charge: 2) = 4
Wino neutral: (helicity: 2)
Bino: (helicity: 2)
Higgsino charged: (helicity: 2) * (charge: 2) = 4
Higgsino neutral: (helicity: 2) * (mass states: 2) = 4

Chargino: (helicity: 2) * (charge: 2) * (mass states: 2) = 8
Neutralino: (helicity: 2) * (mass states: 4) = 8


Thus,
Charginos are Dirac fermions (4 states) with 2 mass states
Neutralinos are non-Dirac fermions (2 states) with 4 mass states

The charginos are Dirac fermions, while I think that the neutralinos are Majorana ones.