Peter Higgs passes away

[lpetrich]

Going from SU(5) to a bigger group, SO(10) for the gauge symmetry, we find SO(10) breaks down into SU(5) * U(1) adding an extra hypercharge-like factor.

Particle	Hand	Spin	Mult	Composition
Gauge	-	1, 1/2	45	(24,0) + (10,-1) + (10*,1) + (1,0)
Vector Higgs Hv	L	0, 1/2	10	(5,-1/2) + (5*,1/2)
(antiparticle)	R	0, 1/2	10	(5*,1/2) + (5,-1/2)
Scalar Higgs Hs	L	0, 1/2	1	(1,0)
(antiparticle)	R	0, 1/2	1	(1,0)
Elementary Fermion F	L	1/2, 0	16	(1,5/4) + (10,1/4) + (5*,-3/4)
(antiparticle)	R	1/2, 0	16*	(1,-5/4) + (10*,-1/4) + (5,3/4)

All the elementary fermions are squashed into one pair of multiplets.

The Higgs interactions are F.Hv.F and Hv.Hv and Hs.Hv.Hv and Hs.Hs.Hs.

This means that all elementary fermions have the same Higgs-generated mass at SO(10) GUT energy scales, and that there are no cross-generation decays. This means that quark mixing must be induced by the breaking of SO(10) symmetry.

Since the elementary fermions include right-handed neutrinos, that means that the self-interaction (Majorana) masses of right-handed neutrinos must also be generated by the breaking of SO(10) symmetry.

There is a new hypercharge-like quantum number that is related to (B-L):

Y(SO(10)) = (5/4)*(B-L) - Y(electroweak)

There are also more leptoquarks:
SU(5): (10) with antiparticles (10*)
EW unification: (3,2,1/6) + (3*,1,-2/3) + (1,1,1) with antiparticles (3*,2,-1/6) + (3,1,2/3) + (1,1,-1)
Low energy: (3,2/3) + (3,-1/3) + (3*,-2/3) + (1,1) with antiparticles (3*,1/3) + (3*,-2/3) + (3,2/3) + (1,-1)

That makes possible (B-L)-violating decays, like

(neutron) -> (positive pion) + (electron)

in addition to (B-L)-conserving ones, like

(neutron) -> (neutral pion) + (antineutrino)
and
(neutron) -> (negative pion) + (positron)

 

[lpetrich]

Can one go further. There is an even bigger symmetry group, E6, one that breaks down into SO(10)*U(1). That one unifies the Higgs particles with the elementary fermions:

Particle	Hand	Spin	Mult	Composition
Gauge	-	1, 1/2	78	(45,0) + (16,-1) + (16*,1) + (1,0)
Elementary fermion	L	0, 1/2	27	(16,1/3) + (10,-2/3) + (1,4/3)
(antiparticle)	R	0, 1/2	27*	(16*,-1/3) + (10,2/3) + (1,-4/3)

The SO(10) elementary fermions F are the 16 with antiparticles 16*, the vector Higgses Hv are 10, and the scalar Higgses are 1. In E6, they are all together.

The interaction F.F.F gives rise to SO(10) Hs.Hv.Hv and Hv.F.F -- thus unifying all the Higgs interactions.

However, SO(10) (Hs)³ would have to be generated by symmetry breaking.

But one needs some complicated symmetry breaking to let only some of these elementary-multiplet particles be low-mass, meaning less than GUT mass.


There is a final step, inspired by string theory. The group E8 can break down into E6*SU(3) and its fundamental and gauge multiplet breaks down as

248 -> (78,1) + (27,3) + (27*,3*) + (1,8)

This puts *all* the Standard Model into one multiplet, complete with its three generations of elementary fermions, but it requires a complicated cascade of symmetry breaking to get to the Standard Model.

 

[lpetrich]

Grand unified theories have various Higgs mechanisms proposed for them, and here are various proposed multiplet sizes

Unbroken Standard Model with multiplet sizes, and number of multiplets (multiplet size)
SM: Gauge 8, 3, 1 -- EF's 4(1), 2(2), 4(3), 2(6)
SM Higgs: 1, 2(2)

SU(5): Gauge 24 -- Higgs 1, 5, 5* -- EF's 1, 1', 5, 5*, 10, 10*
GUT Higgs: 5, 5*, 10, 10*, 24

SO(10): Gauge 45 -- Higgs 1, 10 -- EF's 16, 16*
GUT Higgs: 10, 16, 16*, 45, 54, 120, 126, 126*, 144, 144*, 210

E6: Gauge 78 -- EF-Higgs: 27, 27*
GUT Higgs: 351, 351*, 351', 351'*, 650

In summary for Higgs particles H and the rest of the theory X:
[*]SM: H <~ X
[*]SU(5): H ~ X
[*]SO(10): H >~ X
[*]E6: H > X
[/list]
That seems odd: GUT-Higgs multiplets larger than the observable-particle part of the theory.