"Brahman Particle" - not "God Particle"

[lpetrich]

The Higgs particles are for the MSSM. In the Standard Model, there is a single complex Higgs doublet that correspodns to both Hu and Hd. It has spin 0, so chirality does not enter it.


The neutrino here is a right-handed neutrino. It has not been observed, but it fits the unbroken ((N)MS)SM very well. There is a theory called the Seesaw Model that states that it has its own mass term:

mn*(N.N)

where mn is very large, somewhere 10¹² GeV. This large mass combines with more ordinary Higgs-generated masses to make the observed neutrinos have very low masses.


Looking at the Higgs interactions, they take surprisingly simple forms:

(U*).Hu.Q, (D*).Hd.Q, (N*).Hu.L, (E*).Hd.L, Hu.Hd

In the NMSSM,
Hu.Hd becomes Hu.Hd.Hs, Hs.Hs.Hs

with the singlet Higgs particle Hs.


Here's another pattern:
(Weak hypercharge) = (some integer) + (QCD 1: 0, 3: 2/3, 3*: -1/3) + (WIS odd: 0, even: 1/2)

Also, the weak-hypercharge values add up to zero for each handedness of each generation of elementary fermions, and likewise for the MSSM Higgs particles.

 

[lpetrich]

The Standard Model's gauge fields have internal symmetries. One can do complex-valued rotations and related manipulations to their fields, without changing the equations of motion. In the low-energy limit, the Standard Model has gauge symmetry

QCD: SU(3), electromagnetism: U(1)

For electromagnetism, this means that one can do a complex-number rotation on each field's value. That is, one can multiply all the elementary-particle fields by exp(i*q*a) for particle electric charge q and arbitrary function of space-time position a. Conservation of electric charge means no leftover complex rotation.

For QCD, one gets complex-valued mixings of the three QCD color states.


For unbroken electroweak symmetry, the Standard Model's symmetry is
QCD: SU(3), WIS: SU(2), WHC: U(1)

For breaking electroweak symmetry, two of the three W particles get mass, and become the low-energy W particles. The third W and the B make two mixtures, the Z, with a mass, and the photon, without a mass. All from the Higgs mechanism.


QCD does not get broken, but instead, something called "color confinement" happens. No particle with a QCD color "charge" gets more than about 10^-15 m away from another such particle. No free quarks have ever been found, and no free gluons either. Color confinement is what gives the lower-energy hadrons their masses: their quarks are not allowed to have wavefunctions larger than that confinement limit, and that forces the lighter quarks to be relativistic.

So the Higgs mechanism is not responsible for most of the baryonic-matter mass of the Universe. However, this mechanism is responsible for the masses of the up and down quarks and the mass of the electron, and those are very important for the structure of baryonic matter.

 

[lpetrich]

These patterns give hints of Grand Unified Theories, and many of them have been proposed. I will concentrate on the simpler ones.

The first one is the Georgi-Glashow SU(5) theory, where the gauge symmetry is complex-valued rotations in a 5D space. I'll give the Standard-Model-multiplet composition of each multiplet, with quantum numbers (QCD,WIS,WHC)

Particle	Hand	Spins	Mult	Composition
Gauge		1, 1/2	24	gluon(8,1,0) + W(1,3,0) + B(1,1,0) + (3,2,-5/6) + (3*,2,5/6)
Up Higgs	L	0, 1/2	5	Hu(1,2,1/2) + (3,1,-1/3)
(anti)	R	0, 1/2	5*	Hu*(1,2,-1/2) + (3*,1,1/3)
Down Higgs	L	0, 1/2	5*	Hd(1,2,-1/2) + (3*,1,1/3)
(anti)	R	0, 1/2	5	Hd*(1,2,1/2) + (3,1,-1/3)
Scalar Higgs	L	0, 1/2	1	Hs(1,1,0)
(anti)	R	0, 1/2	1	Hs*(1,1,0)
Elem fermion	L	1/2, 0	1	N*(1,1,0)
(anti)	R	1/2, 0	1'	N(1,1,0)
Elem fermion	L	1/2, 0	10	Q(3,2,1/6) + U*(3*,1,-2/3) + E*(1,1,1)
(anti)	R	1/2, 0	10*	Q*(3*,2,-1/6) + U(3,1,2/3) + E(1,1,-1)
Elem fermion	L	1/2, 0	5*	L(1,2,-1/2) + D*(3*,1,1/3)
(anti)	R	1/2, 0	5	L*(1,2,1/2) + D(3,1,-1/3)

There are some interesting new particles here. The multiplet (3,2,-5/6) gives particles with the ordinary-quark QCD "charges" and electric charges -4/3 and -1/3. Likewise, the Higgs particles now include a "Higgs triplet" with the quantum numbers of the down quark.

These particles have a remarkable property: they make isolated protons decay. But such decay has never been observed, and the proton's half-life is at least 10³² years. This means that these extra particles must have humongous masses, at least 10¹⁵ GeV. That's over 10 trillion times more massive than the heaviest Standard-Model particles.

It must be noted that isolated neutrons can also decay by this mechanism, but they have a much faster way of decaying: the weak interaction. Protons and neutrons bound in nuclei can also decay by this mechanism, making every nucleus unstable.

So that means that we cannot survive much beyond the half-life of the proton.

 

[lpetrich]

For the three Standard-Model gauge fields to become one field, they must have the same interaction strengths. That is very evidently not the case at familiar energies, but as one extrapolates upward in energy, quantum-mechanical effects alter those strengths, and if supersymmetry is present, they converge on one strength at about 2*10¹⁶ GeV.

So that means that that is the GUT energy scale, or the smallest of the GUT energy scales. If the proton-decay-causing particles have masses around that scale, that would explain the non-observation of proton decay.


There are some interesting regularities in this SU(5) model. All the weak hypercharges (or electric charges) in a SU(5) multiplet add up to zero. Also, we have this interesting regularity for the elementary fermions:

1L, 5R, 10L, 10*R, 5*L, 1'R

These numbers should be familiar from the binomial theorem and Pascal's triangle. There is a simple way of constructing them.

1 = scalar -- no vectors
5 = vector with 5 components
10 = antisymmetric outer product of two vectors
10* = ASOP of three vectors. Multiply by the antisymmetric symbol to get the ASOP of two vectors
5* = ASOP of four vectors. AS * it gives a vector
1' = ASOP of five vectors. AS * it gives a scalar

So the AS symbol * (k vectors) = ((n-k) vectors)


Here are the Higgs interactions:
F(10).H(5).F(10) = Q.Hu.(U*)
F(5*).H(5).F(1) = L.Hu.(N*)
F(10).H(5*).F(5*) = Q.Hd.(D*)
F(5*).H(5*).F(10) = L.Hd.(E*)

So we get unification of the masses of the down quarks and the charged leptons, or at least of the bottom quark and the tau lepton. That is evidently not the case at familiar energies, but at GUT energies, it ought to be the case.

The right-handed neutrino's self-mass mn*N.N is not permitted in this GUT, so it must be the result of symmetry breaking. But its expected mass is tantalizingly close to the gauge-unification mass scale mentioned earlier.


In the MSSM, the Higgs particles' masses are from (some electroweak-scale mass) * H(5).H(5*)

But the NMSSM solves this problem by making their masses come from H(1).H(5).H(5*)

The Higgs particles' masses then come from supersymmetry breaking at electroweak energies or a little more. That's why it's so troublesome that we haven't been able to observe squarks or gluinos with the LHC.

 

[lpetrich]

The next one up is SO(10), the symmetry group of real-valued rotations of 10-dimensional vectors.

This one breaks down into SU(5) * U(1), where the second symmetry is for a charge whose value is related to (baryon number) - (lepton number): B - L.

Particle	Hand	Spins	Mult	Composition
Gauge		1, 1/2	45	(24,0) + (10,-1) + (10*,1) + (1,0)
Up-Down Higgs	L	0, 1/2	10	(5,-1/2) + (5*,1/2)
(anti)	R	0, 1/2	10	(5,-1/2) + (5*,1/2)
Singlet Higgs	L	0, 1/2	1	(1,0)
(anti)	R	0, 1/2	1	(1,0)
Elem ferm	L	1/2, 0	16	(1,5/4) + (10,1/4) + (5*,-3/4)
(anti)	R	1/2, 0	16*	(1,-5/4) + (10*,-1/4) + (5,3/4)

Note that that (B-L) - related value adds up to 0 in all the multiplets.

Notice that every generation of elementary fermion is now unified, and also the up and down Higgs particles.

The elementary-fermion-Higgs interactions are
F(16).H(10).F(16)

This unifies the elementary-fermion masses, but at the price of not allowing cross-generation decay. Such decay must be made possible by symmetry breaking, reducing SO(10) to smaller symmetries.

The Higgs particles' self-interactions are H(10).H(10) or H(1).H(10).H(10)

 

[lpetrich]

Another step: the symmetry group E6. I can't think of any simple way to describe it, however.

It breaks down into SO(10) * U(1) where I can't think of any simple interpretation of the charge of the U(1) part.

Particle	Hand	Spins	Mult	Composition
Gauge		1, 1/2	78	(45,0) + (16,-1) + (16*,1) + (1,0)
Others: X	L	0, 1/2	27	(16,1/3) + (10,-2/3) + (1,4/3)
(anti)	R	0, 1/2	27*	(16*,-1/3) + (10,2/3) + (1,-4/3)

That's all that's necessary. One can put an elementary-fermion generation, both up and down Higgses, and a scalar Higgs together in one X multiplet.

The Higgs-interaction terms are simple: X.X.X

This reduces to F(16).H(10).F(16) and H(1).H(10).H(10) in SO(10), but it has no H(1).H(1).H(1) term -- that term must be generated by symmetry breaking.


If you have been following my posts here, you must have noticed how the elementary-particle zoo has been getting smaller and smaller from the ((N)MS)SM to SU(5) to SO(10) to E6.

There is one more step, and that involves symmetry group E8.

Breaking it down to E6 * SU(3), we get
248 -> (78,1) + (27,3) + (27*,3*) + (1,8)

That gives the gauge particles, then the elementary fermions and Higgses, and then some extra ones.


So everything known but gravity can fit inside one E8 multiplet of gauge particles, though with lots of complicated symmetry breaking along the way.

 

[Learner]

Ahh, you just beat me to it Ipetrich ... ahem..

 

[lpetrich]

The Standard Model has an additional problem: a sizable number of free parameters.

• Gauge-interaction strengths (analogs of electric charge): 3
• Suppression of CP violation in QCD: 1
• Higgs mass and self-interaction: 2
• Quark masses: 6
• Quark mixing angles: 4 (CP-preserving: 3, CP-violating: 1)
• Charged-lepton masses: 3

Total: 19 parameters.

Neutrinos are known to be massive, and they add even more.

• Neutrino masses: 3
• Neutrino mixing angles: 4 (CP-preserving: 3, CP-violating: 1)
• Neutrino Majorana CP-violating phases: 2

Neutrino total: 9
Overall total: 28 parameters

The (N)MSSM is even worse: something like 100 additional parameters. However, many of them are close to zero from limits on flavor-changing neutral-current interactions, and the Constrained MSSM (CMSSM) has much fewer free parameters.

Another problem is that it's hard to get supersymmetry breaking out of the (N)MSSM, so a common theory is that it is produced in some "hidden sector" of extra particles and then transmitted in some way or other to the (N)MSSM ones.

About supersymmetry, I'm reminded about what Albert Einstein said about the possible falsification of General Relativity by an eclipse observation. “Then I would feel sorry for the dear Lord. The theory is correct anyway.”

 

[lpetrich]

There are several things that do not fit in with the Standard Model very well.

Gravity. The most successful theory of gravity so far is Einstein's General Relativity. There are strong upper limits on alternatives, with the main surviving alternatives, like Generalized Brans-Dicke Theory, having parameters that can make them arbitrarily close to GR.

Dark Matter. Its nature continues to be mysterious. If it is some elementary particle, then it is not a Standard-Model one. Its nongravitational interactions must be very weak, since nongravitational effects of it have yet to be observed, despite a lot of searching.

Dark Energy. Its nature is even more mysterious.

Inflation. This is a period of exponential expansion early in the Universe's history. It is usually attributed to some spin-0 (scalar) field that has been named the "inflaton" (no second i), but its nature remains obscure. Its energy scale is is around 10^(15) GeV, close to GUT energies, and much greater than Standard-Model energies.

Matter-Antimatter Asymmetry. This is about one baryon for every billion photons, but the big problem is why it is nonzero. We have parts of a solution to this conundrum, like the Big Bang and observation of something called CP violation, but we still do not have anything definite on baryon-number violation. It is an expected consequence of many Grand Unified Theories, but it has yet to be observed.

-

There are some Beyond-Standard-Model effects that we may eventually observe.

Proton Decay. Many GUT's predict baryon-number violation, like decay of free protons. Free neutrons can also decay that way, but they have a much faster decay channel. Protons and neutrons bound in nuclei can also decay that way. Currently, the proton's mean life has a lower limit of about 10^(34) years, already in the range of various GUT predictions.

Neutrinoless Double-Beta Decay. If neutrinos are "Majorana particles", then flipping their spins will turn them into their antiparticles. This will make possible double-beta decay without emission of neutrinos, and if that is observed, it will give us a clue as to their masses.

Neutron Electric-Dipole Moment. This is a result of CP violation, and some BSM theories predict amounts much larger than what the Standard Model predicts, about 10^(-31) e*cm. Current experimental limits are around 10^(-25) e*cm, getting into predictions from the MSSM.

So within the next decade, we might observe something *very* interesting.