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Proposing New Particles

So we've had a long history of solving physical riddles being connected with proposing new particles. This tradition continues with several particles that are proposed but not (yet) detected.


A curious result of QCD is possible CP violation. That would give electric dipole moments to neutrons, protons, and other hadrons with spin. But such EDM's are not observed - Particle Data Group - 2020 Review -  Neutron electric dipole moment -  Electron electric dipole moment

Upper limits: electron: 1.1*10^(-29) e-cm, muon: 1.8*10^(-19) e-cm, tau lepton: 4.5*10^(-16) e-cm, neutron: 1.8*10^(-26) e-cm, proton: 2.1*10^(-25) e-cm, lambda baryon: 1.5*10^(-16) e-cm

The Standard Model predicts very small values: electron 10^(-38) e-cm, neutron 10^(-31) e-cm. Scaling by (mass)^2 suggests that the Standard-Model muon value is 10^(-33) e-cm, proton and lambda and tau values close to the neutron one.

However, supersymmetric extensions of the Standard Model predict much higher values -- values extending up to experimental upper limits for the neutron.

The neutron upper limit is roughly 10^(-10) what one would expect for QCD, suggesting some very fine tuning -- or else some particle that forces it to zero. A particle named the axion. This particle is expected to interact with all Standard-Model gauge particles, not only the gluon (the QCD one), but also with the electroweak ones, notably the photon. This has inspired some lab searches for axions.

So far, this particle has not been detected, but its mass can be made arbitrarily small, making it *very* weakly interacting.
 
Here are some more:
[TABLE="class: grid"]
[TR]
[TD]Problem
[/TD]
[TD]Particle
[/TD]
[TD]Proposed
[/TD]
[/TR]
[TR]
[TD]Dark energy
[/TD]
[TD]Quintessence
[/TD]
[TD]around 1990?
[/TD]
[/TR]
[TR]
[TD]Gravity + SM
[/TD]
[TD]String theory
[/TD]
[TD]around 1985
[/TD]
[/TR]
[/TABLE]

I'll now get into neutrino masses. They are much smaller than the masses of the other elementary fermions, about 0.01 - 0.05 eV (very rough estimate), about (0.2 - 1) * 10^(-7) times the mass of the electron, the least massive of the other ones.

If neutrino masses are generated by the Higgs mechanism, like the masses of every other massive Standard-Model particle, then neutrinos must couple *very* weakly. The vacuum (ground-state) strength of the Higgs field is, in energy units, 246 GeV. From

(mass) = (coupling strength) * (Higgs-field value)

one can calculate the coupling-strength values. The numbers:
  • Higgs: 0.51 (self-coupling)
  • W: 0.33, Z: 0.37
  • up: 8.8*10^(-6), charm: 5.2*10^(-3), top: 0.70
  • down:1.9*10^(-5), strange: 3.8*10^(-4), bottom: 1.7*10^(-2)
  • electron: 2.1*10^(-6), mu: 4.3*10^(-4), tau: 7.2*10^(-3)
  • neutrinos: 4*10^(-14) - 2*10^(-13)
Neutrinos' couplings are *much* less than charged leptons' couplings, unlike the situation with quarks.

This is not what one would expect from the quarks' couplings. In analogy with those couplings, one would expect neutrino masses from about a MeV to about 10 GeV. But we don't observe that.
 
There is a solution to this conundrum, the "seesaw model". To see how it works, let us consider what equations of motion spin-1/2 particles may have.

The first one found, the Dirac equation, is an equation for four field components: up and down for both the ordinary particle and its antiparticle. Yes, the Dirac equation includes antiparticles.

The fields can be sorted out into two types: left-handed and right-handed. This is from their spins relative to their directions of motion in the relativistic limit. Left: spin backward, right: spin forward. Left-handed and right-handed parts include antiparticles.

Dirac-equation masses couple the left-handed and right-handed parts Xl and Xr:
D(Xl) = m*(Xr)
D(Xr) = m*(Xl)

The D is a differential operator in space-time. I'm skipping over a lot of details here.

In 1937, Ettore Majorana showed that there is another kind of equation of motion that a spin-1/2 particle can follow, the Majorana equation. It's like half of the Dirac equation:
D(X) = m*(X*)

where X* is the field's complex conjugate. For a massive particle, field therefore describes an electrically neutral particle. In the massless limit, a Majorana particle can have a charge, and one finds that a Dirac particle is two Majorana particles side by side.


Can a neutrino be a Majorana particle? At first sight, yes. If it was massless, then it could be a left-handed Majorana particle, and that's how it appears in the Standard Model. But we observe evidence of neutrino masses, and it would be hard for it to be a massive Majorana particle in the Standard Model. What would make it have zero Higgs coupling? But someone thought of a solution. What if neutrinos have *both* Dirac masses and Majorana masses? Their equations of motion would look like this:
D(Xl) = m*(Xr)
D(Xr) = m*(Xl) + M*(Xr*)

The m is a Dirac mass associated with the Higgs particle and the M is a large Majorana pass associated with the right-handed part. Why a large one? That's because Xl and Xr then mix to give:
(mostly Xl): mass m^2/M
(mostly Xr): mass close to M

For m = 5 GeV and M = 5*10^(11) GeV, then m^2/M = 0.05 eV -- what is observed of neutrino masses.

So we have a new particle: very massive right-handed neutrinos that force down the masses of the observed neutrinos.
 
The Standard Model looks like an incoherent zoo of particles, but it has some patterns in it. First, the low-energy particles
[TABLE="class: grid"]
[TR]
[TD]Particle[/TD]
[TD]Spin[/TD]
[TD]Charge[/TD]
[TD]QCD[/TD]
[TD]Mass (GeV)[/TD]
[/TR]
[TR]
[TD]Gluon[/TD]
[TD]1[/TD]
[TD]0[/TD]
[TD]8[/TD]
[TD]0[/TD]
[/TR]
[TR]
[TD]Photon[/TD]
[TD]1[/TD]
[TD]0[/TD]
[TD]1[/TD]
[TD]0[/TD]
[/TR]
[TR]
[TD]W[/TD]
[TD]1[/TD]
[TD]+-1[/TD]
[TD]1[/TD]
[TD]80.4[/TD]
[/TR]
[TR]
[TD]Z[/TD]
[TD]1[/TD]
[TD]0[/TD]
[TD]1[/TD]
[TD]91.2[/TD]
[/TR]
[TR]
[TD]H[/TD]
[TD]0[/TD]
[TD]0[/TD]
[TD]1[/TD]
[TD]125.1[/TD]
[/TR]
[TR]
[TD]up qk[/TD]
[TD]1/2[/TD]
[TD]2/3[/TD]
[TD]3[/TD]
[TD]2*10^(-3), 1.3, 173.[/TD]
[/TR]
[TR]
[TD](anti)[/TD]
[TD]1/2[/TD]
[TD]-2/3[/TD]
[TD]3*[/TD]
[TD](same)[/TD]
[/TR]
[TR]
[TD]down qk[/TD]
[TD]1/2[/TD]
[TD]-1/3[/TD]
[TD]3[/TD]
[TD]4*10^(-3), 0.09, 4.2[/TD]
[/TR]
[TR]
[TD](anti)[/TD]
[TD]1/2[/TD]
[TD]1/3[/TD]
[TD]3*[/TD]
[TD](same)[/TD]
[/TR]
[TR]
[TD]neutrino[/TD]
[TD]1/2[/TD]
[TD]0[/TD]
[TD]1[/TD]
[TD](1 - 5)*10^(-11)[/TD]
[/TR]
[TR]
[TD](anti)[/TD]
[TD]1/2[/TD]
[TD]0[/TD]
[TD]1[/TD]
[TD](same)[/TD]
[/TR]
[TR]
[TD]electron[/TD]
[TD]1/2[/TD]
[TD]-1[/TD]
[TD]1[/TD]
[TD]5.11*10^(-4), 0.106, 1.78[/TD]
[/TR]
[TR]
[TD](anti)[/TD]
[TD]1/2[/TD]
[TD]1[/TD]
[TD]1[/TD]
[TD](same)[/TD]
[/TR]
[/TABLE]

There is a pattern that one can find in the electric charges and the QCD multiplets. That is from conserved quantity in QCD multiplets, a quantity that adds when one combines them. It's like angular-momentum parity, as it may be called. How integer and half-odd angular momenta combine:
(int)*(int) = (int), (int)*(h.o.) = (h.o.), (h.o.)*(h.o.) = (int)
So one can map (int) -> 0, (h.o.) -> 1 with addition modulo 2.

QCD has a similar conserved quantity, "triality". Ordinary quarks (multiplet 3) have triality 1, antiquarks (multiplet 3*) have triality 2, and colorless particles (multiplet 1) and gluons (multiplet 8) have triality 0. Triality addition is modulo 3.

Mesons are quark-antiquark states, with triality 1 + 2 = 0. Baryons are three-quark states, with triality 1 + 1 + 1 = 0.

Now for charges and trialities.
Integers -- 0
2/3, -1/3 -- 1
1/3, -2/3 -- 2
So (charge) = - (triality)/3 + (integer)

But it's much harder to recognize patterns in the particles' masses.
 
What we see at "low energies" is with electroweak symmetry breaking. "Low energies" is less than the Higgs vacuum field value: 246 GeV -- greater than the Higgs mass itself and the top quark's mass.

Let's see what the Standard Model is like with unbroken electroweak symmetry. Its gauge particles are gluons of QCD, W of "weak isospin", and B of "weak hypercharge". The W's act something like gluons, but with symmetry group SU(2) instead of SU(3). They relate up-like and down-like quarks, and also charged leptons and neutrinos -- they make the charged weak interactions. The B's act much like the photons, complete with a counterpart of charge, "weak hypercharge". The term "hypercharge" dates back to the 1950's, and it's the average charge of an isospin multiplet. Thus, pions have hypercharge 0 and nucleons hypercharge 1/2.

Electric charge is related to weak isospin and weak hypercharge: (projected WIS) + (WHC)

For instance, the left-handed up and down quarks are part of one multiplet with WIS 1/2. The projected WIS is +1/2 for u-l and -1/2 for d-l. That multiple's WHC is 1/6, and it makes electric charges: u-l +1/2 + 1/6 = +2/3, d-l -1/2 + 1/6 = -1/3.
The right-handed quarks are separate multiplets, two WIS singlets: WIS = 0. Their hypercharges are +2/3 for u-r and -1/3 for d-r, and they make electric charges +2/3 and -1/3 -- the same as for the left-handed parts.

Leptons work similarly.

The W has WIS 1 and WHC 0, making projected WIS's +1, 0, -1, and thus electric charges +1, 0, -1.
 
Here is a table of unbroken-EW particles. I've added supersymmetry-partner spins and I've added extra Higgs particles to be consistent with SUSY. They are redundant in the plain Standard Model: Hu = e.(Hd*), Hd = -e.(Hu*). e flips the doublet.

[TABLE="class: grid"]
[TR]
[TD]Particle[/TD]
[TD]Spin[/TD]
[TD]WHC[/TD]
[TD]WIS[/TD]
[TD]QCD[/TD]
[/TR]
[TR]
[TD]gluon[/TD]
[TD]1, 1/2[/TD]
[TD]0[/TD]
[TD]0[/TD]
[TD]8[/TD]
[/TR]
[TR]
[TD]W[/TD]
[TD]1, 1/2[/TD]
[TD]0[/TD]
[TD]1[/TD]
[TD]1[/TD]
[/TR]
[TR]
[TD]B[/TD]
[TD]1, 1/2[/TD]
[TD]0[/TD]
[TD]0[/TD]
[TD]1[/TD]
[/TR]
[TR]
[TD]Hu[/TD]
[TD]0, 1/2, L[/TD]
[TD]1/2[/TD]
[TD]1/2[/TD]
[TD]1[/TD]
[/TR]
[TR]
[TD]Hu*[/TD]
[TD]0, 1/2, R[/TD]
[TD]-1/2[/TD]
[TD]1/2[/TD]
[TD]1[/TD]
[/TR]
[TR]
[TD]Hd[/TD]
[TD]0, 1/2, L[/TD]
[TD]-1/2[/TD]
[TD]1/2[/TD]
[TD]1[/TD]
[/TR]
[TR]
[TD]Hd*[/TD]
[TD]0, 1/2, R[/TD]
[TD]1/2[/TD]
[TD]1/2[/TD]
[TD]1[/TD]
[/TR]
[TR]
[TD]Q[/TD]
[TD]1/2, 0, L[/TD]
[TD]1/6[/TD]
[TD]1/2[/TD]
[TD]3[/TD]
[/TR]
[TR]
[TD]Q*[/TD]
[TD]1/2, 0, R[/TD]
[TD]-1/6[/TD]
[TD]1/2[/TD]
[TD]3*[/TD]
[/TR]
[TR]
[TD]U[/TD]
[TD]1/2, 0, R[/TD]
[TD]2/3[/TD]
[TD]0[/TD]
[TD]3[/TD]
[/TR]
[TR]
[TD]U*[/TD]
[TD]1/2, 0, L[/TD]
[TD]-2/3[/TD]
[TD]0[/TD]
[TD]3*[/TD]
[/TR]
[TR]
[TD]D[/TD]
[TD]1/2, 0, R[/TD]
[TD]-1/3[/TD]
[TD]0[/TD]
[TD]3[/TD]
[/TR]
[TR]
[TD]D*[/TD]
[TD]1/2, 0, L[/TD]
[TD]1/3[/TD]
[TD]0[/TD]
[TD]3*[/TD]
[/TR]
[TR]
[TD]L[/TD]
[TD]1/2, 0, L[/TD]
[TD]-1/2[/TD]
[TD]1/2[/TD]
[TD]1[/TD]
[/TR]
[TR]
[TD]L*[/TD]
[TD]1/2, 0, L[/TD]
[TD]1/2[/TD]
[TD]1/2[/TD]
[TD]1[/TD]
[/TR]
[TR]
[TD]N[/TD]
[TD]1/2, 0, R[/TD]
[TD]0[/TD]
[TD]0[/TD]
[TD]1[/TD]
[/TR]
[TR]
[TD]N*[/TD]
[TD]1/2, 0, L[/TD]
[TD]0[/TD]
[TD]0[/TD]
[TD]1[/TD]
[/TR]
[TR]
[TD]E[/TD]
[TD]1/2, 0, R[/TD]
[TD]-1[/TD]
[TD]0[/TD]
[TD]1[/TD]
[/TR]
[TR]
[TD]E*[/TD]
[TD]1/2, 0, L[/TD]
[TD]1[/TD]
[TD]0[/TD]
[TD]1[/TD]
[/TR]
[/TABLE]

The spin is (Standard-Model spin), (SUSY-partner spin), when present: chirality (handedness), left or right handed

Q = left-handed quark doublet, U = right-handed up-quark singlet, D = right-handed down-quark singlet
L = left-handed lepton doublet, N = right-handed neutrino singlet, E = right-handed electron singlet

Hu = up Higgs, Hd = down Higgs, both doublets.
 
This looks like some horribly complicated table, but there is a nice regularity in it:

(weak hypercharge) = - (QCD triality)/3 + (weak isospin) + (integer)

One gets the earlier relation out of it with (electric charge) = (projected weak isospin) + (weak hypercharge)
since (projected weak isospin) = (weak isospin) + (integer)

In the unbroken-electroweak Standard Model, the mass terms can be expressed as

y(u,i,j).Hu.Q(i).U*(j) + y(d,i,j).Hd.Q(i).D*(j) + y(n,i,j).Hu.L(i).N*(j) + y(e,i,j).Hd.L(i).E*(j)

summed over indices to elementary-fermion generations i, j: 1, 2, 3. One should add the "Hermitian conjugate", but it's essentially mirror-imaging everything without showing anything new.

The y's are complex "Yukawa matrices" of interaction strengths, named after the Hideki Yukawa I'd mentioned earlier, because that interaction is similar to the pion-nucleon interaction that he had proposed. They make mass matrices by multiplying by the Higgs-particle field strength.
m(f,i,j) = v*y(f,i,j)
for flavor f and Higgs-field vacuum value v.

The mass matrices for the up-like and down-like quarks are imperfectly aligned, and this misalignment makes cross-generation weak decays.

Neutrinos oscillate between generations because of a similar misalignment between the charged-lepton and the neutrino mass matrices.
 
In the seesaw model, right-handed neutrinos N have mass terms
M(i,j).N(i).N(j) + M*(i,j).N*(i).N*(j)

The second term is the Hermitian conjugate of the first term - I've included it to illustrate the H.C.

So we've found some regularities. What next? In the early 1970's, various physicists constructed Grand Unified Theories.

The simplest of them has one gauge symmetry, SU(5), sort of like QCD, but with 5 "colors". Three of them are the QCD colors, and the remaining two are the "up" and "down" flavors. Many Standard-Model multiplets get included in larger multiplets:

[TABLE="class: grid"]
[TR]
[TD]Particle[/TD]
[TD]Spin[/TD]
[TD]Composition[/TD]
[/TR]
[TR]
[TD]Gauge(24)[/TD]
[TD]1, 1/2[/TD]
[TD]gluon + W + B + (-5/6,1/2,3) + (5/6,1/2,3*)[/TD]
[/TR]
[TR]
[TD]H(5)[/TD]
[TD]0, 1/2, L[/TD]
[TD]Hd + (1/3,0,3*)[/TD]
[/TR]
[TR]
[TD]H(5)*[/TD]
[TD]0, 1/2, R[/TD]
[TD]Hd* + (-1/3,0,3)[/TD]
[/TR]
[TR]
[TD]H(5*)[/TD]
[TD]0, 1/2, L[/TD]
[TD]Hu + (-1/3,0,3)[/TD]
[/TR]
[TR]
[TD]H(5*)*[/TD]
[TD]0, 1/2, R[/TD]
[TD]Hu* + (1/3,0,3*)[/TD]
[/TR]
[TR]
[TD]F(1)[/TD]
[TD]1/2, 0, R[/TD]
[TD]N[/TD]
[/TR]
[TR]
[TD]F(5)[/TD]
[TD]1/2, 0, L[/TD]
[TD]L + D*[/TD]
[/TR]
[TR]
[TD]F(10)[/TD]
[TD]1/2, 0, R[/TD]
[TD]Q* + U + E[/TD]
[/TR]
[TR]
[TD]F(10*)[/TD]
[TD]1/2, 0, L[/TD]
[TD]Q + U* + E*[/TD]
[/TR]
[TR]
[TD]F(5*)[/TD]
[TD]1/2, 0, R[/TD]
[TD]L* + D[/TD]
[/TR]
[TR]
[TD]F(1)'[/TD]
[TD]1/2, 0, L[/TD]
[TD]N*[/TD]
[/TR]
[/TABLE]

Some extra particles appear: the leptoquarks (-5/6,1/2,3) and the Higgs triplet (-1/3,0,3), which closely resembles a down quark. The leptoquarks are particles with electric charges -4/3 and -1/3.

The leptoquarks and the Higgs triplet can make free protons decay, and other GUT's also predict proton decay. It is not just protons that decay by this mechanism, but also neutrons, but free neutrons have a much faster decay mechanism.

Proton decay has been searched for since the early 1980's, and though it has yet to be found, upper limits on its decay rates are pushing into GUT predictions.
 
There is a nice pattern in the elementary fermions' multiplets: 1, 5, 10, 10*, 5*, 1'

They are antisymmetric combinations of a "base" multiplet, (5): 0, 1, 2, 3, 4, and 5 of them.

That suggests that we can go further in GUT construction. SU(5) is contained in a larger symmetry group, SO(10). It breaks down into SU(5) and a sort of "B-L" group, where B-L is (baryon number) - (lepton number)

[TABLE="class: grid"]
[TR]
[TD]Particle[/TD]
[TD]Spin[/TD]
[TD]Composition[/TD]
[/TR]
[TR]
[TD]Gauge(45)[/TD]
[TD]1, 1/2[/TD]
[TD](G(24),0) + (1,0) + (10,-1) + (10*,1)[/TD]
[/TR]
[TR]
[TD]H(10)[/TD]
[TD]0, 1/2, L[/TD]
[TD](H(5),-1/2) + (H(5*),1/2)[/TD]
[/TR]
[TR]
[TD]H(10)*[/TD]
[TD]0, 1/2, R[/TD]
[TD](H(5)*,1/2) + (H(5*)*,-1/2)[/TD]
[/TR]
[TR]
[TD]F(16)[/TD]
[TD]1/2, 0, L[/TD]
[TD](F(5),3/4) + (F(10*),-1/4) + (F(1)',-5/4)[/TD]
[/TR]
[TR]
[TD]F(16*)[/TD]
[TD]1/2, 0, R[/TD]
[TD](F(5*),-3/4) + (F(10),1/4) + (F(1),5/4)[/TD]
[/TR]
[/TABLE]

One gets right-handed neutrinos, a B-L Z particle, and even more leptoquarks. All with only one Higgs multiplet and one elementary-fermion multiplet per generation.

Can we go further? Yes, with a group called E6.
[TABLE="class: grid"]
[TR]
[TD]Particle[/TD]
[TD]Spin[/TD]
[TD]Composition[/TD]
[/TR]
[TR]
[TD]Gauge(78)[/TD]
[TD]1, 1/2[/TD]
[TD](G(45),0) + (1,0) + (16,-1) + (16*,1)[/TD]
[/TR]
[TR]
[TD]X(27)[/TD]
[TD]1/2, 0, L[/TD]
[TD](F(16),1/3) + (H(10),-2/3) + (S,4/3)[/TD]
[/TR]
[TR]
[TD]X(27*)[/TD]
[TD]1/2, 0, R[/TD]
[TD](F(16*),-1/3) + (H(10)*,2/3) + (S,-4/3)[/TD]
[/TR]
[/TABLE]

More leptoquarks, another Z, and a SO(10) singlet, S. A nice thing is that there is only one kind of multiplet that gives the elementary fermions and the Higgs particle.

Going even further, one can stuff *all* the Standard Model into a single multiplet of a group called E8 -- a size-248 multiplet.
 
I will now look at particle masses in GUT's.

First, SU(5).

y(u,i,j).Hu.F(10*,i).F(10*,j) + y(d,i,j).Hd.F(10*,i).F(5,j) + y(n,i,j).Hu.F(5,i).F(1,j) + y(e,i,j).Hd.F(5,i).F(10*,j)

This does mass unification of the down-like quarks and the charged leptons: y(d,i,j) = y(e,j,i)

But we don't observe them to have the same masses. This is because their effective masses change over energy due to quantum-mechanical effects that I'd mentioned earlier here. Thus, their masses are different at the energy scales where we observe them.

I note in passing that another unification is gauge unification. The QCD, weak-isospin, and weak-hypercharge fields converge on the same coupling-constant / "charge" values at something like 10^(16) GeV. This convergence is most successful for the Minimal Supersymmetric Standard Model (MSSM), as it's called.

Going from SU(5) to SO(10), we find complete unification of the mass matrices: y(f,j,i) = y(f,i,j) = y(i,j) - same for all flavors. This has the consequence that there are no cross-generation decays. Such decays must be a result of the breaking of SO(10). Likewise, the seesaw masses for right-handed neutrinos must be a result of the breaking of SO(10).
 
I'll now get into supersymmetry (SUSY). It is a proposed symmetry of elementary particles that relates particles with different spins. In particular, spins differing by multiples of 1/2.

Looking at the Standard Model, the Minimal Supersymmetric Standard Model (MSSM) has spin-1/2 "gauginos" for the spin-1 gauge particles, spin-1/2 "higgsinos" for the spin-0 Higgs particles, and spin-0 "sfermions" for the spin-1/2 elementary fermions. The sfermions include selectrons, sneutrinos, and squarks, like stops.

The SUSY EF and Higgs particles have the same kind of SUSY multiplet, a "Wess-Zumino multiplet".


For SUSY to work, the MSSM has not one Higgs doublet, but two. In the Standard Model, electroweak symmetry breaking involves the W+, the W-, and the Z eating three of the four members of the single Higgs doublet, leaving one member to become the observed Higgs particle. In the MSSM, this "eating" also happens, but the two doublets have eight members between them, leaving five: three neutral Higgs particles, and one charged Higgs particle (counts as two from both signs of charge).

The SUSY partners also make mixed states. The WIS and WHC gauginos combine with the higgsinos to make 4 "neutralinos" and 2 "charginos", and the sfermions are mixtures of left-handed and right-handed ones, corresponding to the appropriate elementary-fermion modes.
 
Some illustrations of gauge unification - particle physics - Supersymmetry and grand unification - Physics Stack Exchange - File:Grand unification couplings.jpg - Scholarpedia at Grand unification - Scholarpedia - for the MSSM, convergence at roughly 3*10^(16) GeV


The MSSM's SUSY breaking is expected to happen at electroweak energies or a little big larger energies: 1 to 10 TeV or thereabouts.

But the unbroken MSSM has a mass parameter associated with its Higgs multiplets: (mh).Hu.Hd

It is electroweak-sized instead of GUT-sized, making it curiously small but nonzero - and in the range of MSSM SUSY breaking.

This has provoked the construction of the Next-to-MSSM model or NMSSM. It has an additional particle, a SUSY Higgs singlet S, making the Higgs-interaction terms

(yh).S.Hu.Hd + (k/3)*S^3

where yh and k are dimensionless constants. At low energies, the S contributes two neutral Higgs particles and one neutralino, making 4 neutral Higgses and 5 neutralinos.
 
Going to the SU(5) GUT, the MSSM Higgs-mass term becomes (mh).H(5).H(5*) and the NMSSM Higgs self-interaction terms become
(yh).S.H(5).H(5*) + (k/3)*S^3

Going to SO(10), these are (mh).H.H and (yh).S.H.H + (k/3)*S^3

Looking at E6, an elementary-fermion multiplet X can contain a SO(10) one, F, the SO(10) Higgs, H, and the NMSSM Higgs singlet S.
X = F + H + S

In the group E6, the combination X.X.X can be made a gauge singlet, a sort of "colorless" combination. That makes E6 good as a GUT.

This combination contains both EF-Higgs mass terms - F.F.H and Higgs-Higgs ones - H.H.S - though not S^3.

There is also the interesting conundrum of why there are three generations of elementary fermions but only one generation of Higgs particles.
 
So with Grand Unified Theories, we get up to energies like 3*10^(16) GeV. That is entrancingly close to the energy scale of quantum gravity, the Planck energy: 10^(19) GeV.

So if one can unify the Standard Model as a GUT, can we go further and include gravity?

For doing so, we need a quantum theory of gravity, and it's VERY hard to construct such a theory. One gets nasty divergences when one tries to integrate over energies higher than the Planck energy, even though this procedure is well-behaved for the Standard Model and most extensions of it considered plausible.

Supersymmetry to the rescue? Yes, as supersymmetric gravity or supergravity (SUGRA) (not superstrong gravity!). Quantized gravitational waves have a name: graviton. From their space-time structure, one finds their spin: 2. Their SUSY partner is the gravitino, with spin 3/2.

One can construct SUGRA multiplets with particles with more spins, including all the spins of the Standard Mode: 1, 1/2, 0. But it's hard to get the Standard Model out of SUGRA, even if one goes to more than 4 space-time dimensions and curls up all but 4 of them into a Planck-sized ball. That's compactification or a Kaluza-Klein theory.

Even so, SUGRA is somewhat better behaved at energies greater than the Planck energy.
 
Can we go further? Yes, there is a kind of theory that can incorporate both the Standard Model and gravity while being well-behaved at high energies.

String theory.

It posits that elementary particles are vibration modes of Planck-sized quantum-mechanical strings. That makes string theory a theory of an infinite number of quantum fields, one for each string mode.

The first kind of string discovered was the bosonic string, a string where all the modes have integer spin. It has some odd features, however. It is only quantum-mechanically consistent if space-time has 26 dimensions, and its lowest-energy state is a tachyon, a faster-than-light particle. The mass of each mode is given by

m = m0*sqrt(n)

where n is some integer and m0 is related to the string tension. For the tachyon, n = -1.

There are two kinds of bosonic strings, closed-string where the strings are loops, and open-string where the strings have ends.


Applying supersymmetry to string theory gives supersymmetric strings or superstrings. They are quantum-mechanically consistent in 10 space-time dimensions, their lowest-energy states are massless - no tachyon - and there are only five kinds of consistent theory: I, IIA, IIB, HO, and HE. The massless states form supergravity multiplets with some of them also having gauge multiplets.


These five theories turn out to be interrelated, and their interrelationships point to something that's been named M-theory. Its lowest states are massless, and are a supergravity multiplet in 11 space-time dimensions.
 
It's evident that we don't live in a universe with 9 or 10 readily-observable space-time dimensions, and we could not exist in a universe with that many of them. For D dimensions, interaction forces go as (distance)^(-(D-1)) or a smaller exponent (negative and higher in absolute value). That gives us the familiar inverse-square laws of gravity and electromagnetism. Working out the stability of orbits in both classical and quantum mechanics, stable bound states imply a maximum number of space dimensions: 3. More dimensions mean more complexity possible, so our universe has the best possible number of space dimensions for us to appear.

The solution is that the extra 6 or 7 dimensions must be compactified, curled up in a Planck-sized ball. Various topologies of this compactification give various low-energy theories, and it is possible to get a superset of the Standard Model with a suitable compactification of the HE superstring.

But string theory has a problem. One can get the Standard Model, but one does not get it uniquely. The "string landscape" contains something like 10^(500) possible string ground states. While we have essentially one theory, it has numerous possible boundary conditions, and we have no idea of what may select among them. So the Standard Model is far from unique.

This leads to a speculation: multiverses, multiple "universe bubbles". In it, our universe is the way it is by natural selection - it's the only kind of universe that can generate entities capable of describing it. That can explain why our Universe seems to be only borderline habitable for us. If most universes are uninhabitable for us, then it is plausible that our universe is only barely across the habitability boundary.
 
After going into all that gory detail about Grand Unified Theories and possible Theories of Everything, I now get into some observations.
  • Dark matter
  • Dark energy
  • Early-Universe inflation
None of these three effects fit into the Standard Model very well.

Dark matter is more properly called invisible matter from the lack of nongravitational evidence of it. There is an abundance of evidence of it in galaxies, galaxy clusters, and over cosmological size scales. The favorite theory of DM is that it is composed of weakly interacting massive particles: WIMP's. An alternate possibility, massive compact halo objects or MACHO's, has been searched for by looking for gravitational microlensing of the light of large numbers of stars, but there is not enough of them to account for dark matter.

Back to WIMP's, they have been searched for both directly and indirectly. Direct searching is by looking for electrons and nuclei recoiling from WIMP's bouncing off of them. Indirect searching is by looking for evidence of decays or annihilations, like energetic gamma rays or neutrinos.

Neither kind of search has given positive results, only upper limits.

One kind of possible WIMP is the Lightest Supersymmetric Particle. This may be a neutralino, a mixture of the SUSY partners of neutral electroweak gauge particles and Higgs particles. Neutralinos with a few hundred GeV of mass turn out to be produced in roughly the right density to be dark matter.

Another kind of possible LSP is the gravitino, but it would have gravitational-strength interactions, VERY weak by elementary-particle standards.

I've also seen axions proposed as WIMP's, or more properly, WISP's - Weakly Interacting Slim Particles (very low-mass ones).
 
I now turn to dark energy, something even more mysterious than dark matter.

It was posited as a result of observing distant supernovae to try to find out how the Universe's expansion behaves over sizable fractions of its age. A weird result emerged: the Universe's acceleration is positive, not what one would expect: negative. To get a handle on how this might happen, I turn to the Friedmann-Lemaître-Robertson-Walker cosmological equations, a solution of general relativity. These equations give us

\( \frac{\ddot{a}}{a} = - \frac{4\pi G}{3} (\rho + 3 p/c^2) \)

a = Universe's size parameter, a function of time. ρ = mass density, p = pressure

So the Universe must have a negative pressure that is more than enough to counteract its mass density. There is a way that the Universe can have such a negative pressure: something called the cosmological constant. It is mathematically equivalent to some material with constant density and a constant negative pressure related by

(pressure) = - (energy density) = - (mass density) * c^2

This material has been named "dark energy".
 
Dark energy may be a dynamical field, a field that some people have named "quintessence" ("fifth stuff"). That is the fifth element in Aristotelian cosmology, the celestial element or aether alongside the four terrestrial elements: earth, water, air, fire.

Here is a whimsical identification of these elements with modern-cosmology Universe components:

[TABLE="class: grid"]
[TR]
[TD]Element
[/TD]
[TD]Modern cosmology
[/TD]
[/TR]
[TR]
[TD]Aether
[/TD]
[TD]Dark energy
[/TD]
[/TR]
[TR]
[TD]Fire
[/TD]
[TD]Cosmic microwave background (photons)
[/TD]
[/TR]
[TR]
[TD]Air
[/TD]
[TD]Cosmic neutrino background
[/TD]
[/TR]
[TR]
[TD]Water
[/TD]
[TD]Dark matter
[/TD]
[/TR]
[TR]
[TD]Earth
[/TD]
[TD]Baryonic matter
[/TD]
[/TR]
[/TABLE]
 
I now turn to cosmic inflation.

That hypothesis is provoked by consideration of the problem that the Universe is very close to spatially flat on large scales but nevertheless with primordial density fluctuations.

The flatness requires some great fine tuning of the expansion rate early in the Universe's history. If it was a tiny bit less, the Universe would collapse on itself, while if it was a tiny bit more, then at the present time, the Universe would be hopelessly dilute. That is the "flatness problem".

The Universe also looks very isotropic and homogeneous on large scales. That is a problem because in the early Universe, different now-visible regions would have had no way of communicating with each other, so they would have ended up behaving very differently. That is the "horizon problem".

Also, a consequence of some Grand Unified Theories is magnetic monopoles, particles with magnetic counterparts of electric charge. But no such particle has ever been detected, meaning some strong upper limit on their density. When the Universe's temperature went below about 10^(16) GeV or 10^(29) K, it became too cold to make new GUT-mass particles, but existing ones would survive, to decay or annihilate. Existing ones like magnetic monopoles. So GUT freezeout either produced much smaller than expected numbers of MM's, or else the Universe expanded enormously after that freezeout to dilute the MM's that the freezeout produced. That is the "monopole problem".

That period of enormous expansion has been named inflation, and it is usually considered the result of some field that has been named the "inflaton" (no second i).

During inflation, the inflaton was in a phase where behaved much like the cosmological constant, with

(pressure) = - (mass density)*c^2

This had the consequence of exponential inflation, and only about 60 e-foldings are necessary to make the Universe as spatially flat as is observed. So the Universe could be much larger than what we observe of it.
 
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