GUT Mass Unification Predictions

[lpetrich]

I'm trying to see how well I understand the issue of GUT mass unification, since I have trouble finding references that collect mass-unification predictions.

The first problem is renormalizing from accelerator-accessible energies to GUT energies. That's sensitive to the particle spectrum in between, so the ability to unify masses is a potential test of Standard-Model extensions.

Strictly speaking, for the elementary fermions at least, it's not their masses that get unified, but their couplings to the Higgs particles. A complication is that the top-quark mass may be fixed by renormalization. Its coupling to some Higgses would thus be on the order of its gauge couplings (Wikipedia's Top quark mentions this possibility under "Yukawa couplings"). If so, then it may not be very useful for testing mass unification.

Such renormalization calculations have been done to test gauge unification, and so far, the Minimal Supersymmetric Standard Model (MSSM) does the best. However, there are only 3 gauge coupling parameters that must be unified, providing only one test.

However, mass unification does not seem to be as successful, partly due to some of the MSSM's parameters still being poorly-constrained.

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Now for the GUT's themselves. What do they predict?

For gaugino masses, gauge unification suggests that they will be the same at GUT energy scales. That will mean a well-defined ratio of masses at accelerator energies. However, the winos and binos, as they are called, will mix with Higgsinos, which may make it difficult to untangle their masses. But the gluinos will not mix with anything.

The sfermions (squarks and sleptons) are also expected to get some mass from supersymmetry breaking, and this additional mass is usually expected to be flavor-independent. This will renormalize into separate masses for left-handed and right-handed versions of up-like squarks, down-like squarks, charged sleptons, and (left-handed only?) sneutrinos. Some of these may be difficult to distinguish, so at the very least, we will get squarks vs. sleptons.

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Turning to the elementary fermions, they get masses from the Higgs particles, so I'll have to consider both of them together. Their SM SU(3)*SU(2)*U*(1) multiplet structure:

Q: (3,2,1/6), L: (1,2,-1/2), U: (3*,1,-2/3), D: (3*,1,1/3), N: (1,1,0), E: (1,1,1)
Left-handed quarks and leptons, antiparticles of right-handed up-like quarks, down-like quarks, neutrinos and electrons
Hu: (1,2,1/2), Hd: (1,2,-1/2)
Up-like and down-like Higgs

Yukawa terms; these make the EF's' masses: Q.U.Hu, Q.D.Hd, L.N.Hu, L.E.Hd
Neutrinos have additional complications; their masses are likely a result of a "seesaw effect" added to this effect.

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Now for various GUT's on elementary fermions. Summary:

Masses of tau lepton and bottom quark unified: SU(5), Pati-Salam, SO(10), SU(6), E6
Excessively-successful unification: Pati-Salam, SO(10), E6
Higgs possibly in an elementary-fermion generation: trinification, E6

Symmetry breaking is necessary to make cross-generation decay in the excessively-successful cases, but judging from the quark mixing matrix, it does not appear to be very large. So it may not affect the bottom-tau mass unification very much.


Details:

The Georgi–Glashow model's SU(5) has its EF's in 2 or 3 multiplets, and also 2 Higgs multiplets:
F(1): N, F(5): L,D, F(10*): Q,U,E
H(5): Hd + Hq', H(5*): Hu + Hq
A * used instead of a bar on top for typographical convenience.
The Hq/Hq' is a down-like "Higgs quark" that can cause proton decay. Its presence creates the doublet–triplet splitting problem.

Their interactions are
F(5).F(1).F(5*) -- makes L.N.Hu -- neutrino masses
F(10*).F(10*).H(5*) -- makes Q.U.Hu -- up-like quark masses
F(10*).F(5).H(5) -- makes Q.D.Hd and L.E.Hd -- down-like quark masses and electron-like lepton masses

So Georgi-Glashow predicts the unification of the masses of the down-like quarks and the electron-like leptons, including the bottom quark and the tau lepton.

The Pati–Salam model features SU(4)*SU(2)*SU(2), with this multiplet structure:
F(4,2,1): Q,L, F(4*,1,2): U,D,N,E, H(1,2,2): Hu,Hd

with this interaction term:
F(4,2,1).F(4*,1,2).H(1,2,2)

This completely unifies the masses of each generation of elementary fermions, but this unification is a bit too good: it does not allow cross-generation decay.

Both Georgi-Glashow and Pati-Salam are subsets of the Fritzsch-Minkowski-Georgi SO(10) model, has all the elementary fermions in one multiplet, F(16), and all the Higgses in another, H(10). Their interactions:
F(16).F(16).H(10)

Also excessively complete unification.

The multiplet unification:
F(16) = GG F(1) + F(5) + F(10*) = PS F(4,2,1) + F(4*,1,2)
H(10) = GG H(5) + H(5*) = PS H(1,2,2) + H(6,1,1) (Higgs quark again)

The Glashow-Georgi-de-Rujula trinification model features SU(3)*SU(3)*SU(3), with these multiplets:
F(3,3*,1): Q, F(3*,1,3): U,D, F(1,3,3*): L,N,E, H(1,3,3*): Hu,Hd
Note, the Higgses could be inside one of the EF multiplets.
Interactions:
F(3,3*,1).F(3*,1,3).H(1,3,3*)
F(1,3,3*).F(1,3,3*).H(1,3,3*)
The quarks' masses and the leptons' masses get unified separately - no unification of bottom and tau masses.

http://en.wikipedia.org/wiki/SU(6)_(physics) unification is a superset of Georgi-Glashow unification, and it has the same mass-unification properties.

Finally, E6. It is a superset of SO(10), trinification, and SU(6), and has both excessively-complete mass unification and the Higgs residing in an elementary-fermion multiplet:
27 = SO(10) 16 + 10 + 1 = Trini (3,3*,1) + (3*,1,3) + (1,3,3*)

with interaction
(27).(27).(27)

E6 is interesting because it can be gotten from an E8 in the HE heterotic superstring. One E8 multiplet would contain all the Standard-Model particles, multiple EF generations and all, and it would thus unify both gauge and EF-Higgs Yukawa couplings.

 

[arivero]

¡I think you have done a full review!

 

[lpetrich]

Thanx.

Some more detail about superpartner predictions.

D. I. Kazakov's [hep-ph/0012288] Beyond the Standard Model (In Search of Supersymmetry) goes into some rather gory detail about MSSM predictions. But on page 43 (internal number), he gives us some predictions for various particles for certain parameter values and in terms of SUSY-breaking masses, and I'll simplify it by expressing the particles' masses in terms of the GUT-scale gaugino mass, m_1/2. I'll be ignoring the GUT-scale spin-0 mass, m₀, particle mixing, etc.

Gluino: 2.7
Wino: 0.8
Bino: 0.4
Left Squark (Up, Down): 2.6
Right Squark (Up, Down): 2.5
Left Sbottom: 2.3
Right Sbottom: 2.5
Left Stop: 2.3
Right Stop: 1.9
Sneutrino: 0.7
Left Selectron: 0.7
Right Selectron: 0.4
Heavy Higgs particles: ~ 1

So the squarks and gluinos will be about 2 to 3 times more massive than the others, a result that also emerges from more detailed calculations. This is a result of their QCD interactions -- as one goes down to lower and lower energies, gluons' interactions with each other push up the QCD coupling constant, which in turn pushes up the quark, squark, and gluino masses.

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Imposing flavor independence and GUT-scale unification on the MSSM's mass parameters reduces them to a grand total of 5. They include the aforementioned m₀ and m_1/2, some Higgs mass parameters (mu and B), and a parameter involved in sfermion left-right mixing (A). We have one constraint on these parameters, the Higgs total vacuum value, but in general, only indirect limits.

From the MSSM, we find these mass values:

Neutral Higgs (positive parity): 2
Neutral Higgs (negative parity): 1
Charged Higgs: 1
Neutralinos (Higgsinos + wino + bino): 4
Charginos (Higgsino + wino): 2
Gluino: 1
Up-like squarks: 6
Down-like squarks: 6
Sneutrinos: 3
Selectrons: 6

Many of the squarks are likely to be nearly degenerate in mass, and that is also likely for sleptons. So they may be difficult to distinguish in accelerator experiments. There's also the difficulty of telling what's what among observed particles, something that will require a lot of events. But if the LHC teams succeed in finding at least some of them, then we will have several tests of mass-unification hypotheses.