Boson Charges in E6 and E8 Algebras

[frogjg2003]

What are the charges of the X and Y boson?

I've been looking through online material, and most doesn't mention their charge. The only material I have found has been the Wikipedia page claiming to be +4/3 and +1/3.
This contradicts the only other sources I could find, saying that they are -4/3 and -1/3. The first is a lecture from my nuclear and particle class, the other is from another thread on PF.

 

[lpetrich]

That's a difference in convention between which ones are "ordinary" ones and which ones are "anti" ones.

One has to be careful about the color state also. Ordinary quarks have a set of 3 color states, and antiquarks have a set of 3 anticolor states, usually denoted 3-bar or 3*. Gluons are color-anticolor with a colorless mixture removed, giving them 3*3-1 = 8 color states.

The X and Y with ordinary-quark color states (3) have electric charges -4/3 and -1/3.

The X and Y with antiquark color states (3*) have electric charges +4/3 and +1/3.

 

[garrett]

They're both right. These aren't their own antiparticles, like the photon, but have antiparticles with opposite charges. So the electric charges of the SU(5) X (and Y) bosons are $\pm \frac{4}{3}$ and $\pm \frac{1}{3}$. The SO(10) GUT has additional X bosons with electric charges $\pm \frac{2}{3}$ and $\pm \frac{1}{3}$. To see the full pattern, including their weak and strong charges, try playing with the Elementary Particle Explorer, choosing the SU(5) or SO(10) Theory and standard model or electric charge Rotations. (You can also see with the EPE that the pattern lpetrich described above is correct.)

 

[lpetrich]

I couldn't make much sense out of the Elementary Particle Explorer's display.

Let's see how many dimensions one needs. For the unbroken Standard Model, it's 4:

• QCD SU(3) root space: 2
• Weak-isospin root space (component 3): 1
• Weak hypercharge: 1

Electric charge = WIS-3 + WHC

A lot of the interesting flavor details are related to electroweak symmetry breaking, thus involving WIS and WHC. So one can shrink down the QCD effects, giving something like this: ordinary quarks: triangles, antiquarks: 180d rotation of quark triangles, gluons: hexagons with side in directions of triangle points, other particles: dots.

Moving on to GUT symmetries, Georgi-Glashow SU(5) has a 4D root space, Pati-Salam SO(6)*SO(4) has a 5D root space, SO(10) also has a 5D root space, E6 has a 6D root space, and E8 has a 8D root space.
E8 -> E6 -> GG or PS -> SM

Leptoquarks with ordinary-quark color state -- what electric charges:
GG: -1/3, -4/3
PS: 2/3
SO(10), E6, E8: 2/3, -1/3, -4/3

GG and its supersets get an additional Higgs particle that can behave like a leptoquark. For an ordinary-quark color state, it has electric charge -1/3.

 

[garrett]

Right. The EPE projects these charges from the 8D root space down to the 2D screen spanned by the H and V shown on the left, which you can change by clicking and dragging. Also, in the root space, electric charge is along the direction specified by the weak mixing angle
$$Q = W_3 sin(\theta) + Y cos(\theta)$$
which is along the vertical in the EPE if you choose the "electric charge" or "standard model" rotation.

 

[lpetrich]

I've created some ASCII-art diagrams to show what I have in mind.

QCD "color" states in SU(3) root space:

...o...o...
...>...
...<...<...
o...o...o
...>...>...
...<...
...o...o...

o = colorless (1, in center), gluon (8, 6 on periphery and 2 in center)
> = ordinary quark (3)
< = antiquark (3*)
Notice that the three types of states fall onto staggered hexagonal grids.

Weak hypercharge (horizontal) and weak isospin root space / 3rd component (vertical)

+1......W+......

+1/2 ...Y...Nl...Dl*...Ul...El*...X*...

.0...Er...Ur*...Dr...ZZZ...Dr*...Ur...Er*...

-1/2...X...El...Ul*...Dl...Nl*...Y*...

-1......W-......

...-1...-1/2...0...+1/2...+1...

ZZZ = photon, Z, gluon, possible right-handed neutrino
l, r means left and right handed * means antiparticle

I put the dots into get the spacing to work out correctly.

Particles with the same electric charge lie on a gently-tilted line from top left to bottom right:

X* -- +4/3
El, Er, W+ -- +1
Ul, Ur -- +2/3
Dl*, Dr*, Y* -- +1/3
Nl, Nl*, ZZZ -- 0
Dl, Dr, Y -- -1/3
Ul*, Ur* -- -2/3
El*, Er*, W- -- -1
X -- -4/3

 

[garrett]

http://garrettlisi.com/albums/E8/strong.png
http://garrettlisi.com/albums/E8/electroweak.png

 

[lpetrich]

I've gone a bit further and found some patterns.

QCD's symmetry group is SU(3), and its root space can be expressed as a hexagonal grid. However, different representations have grids offset to each other. SU(3) has a quantum number called triality, much like the spinor parity of SO(3)/SU(2), but ranging 0, 1, 2 and adding modulo 3.

Colorless (color singlet): n=1, w=00, t=0
Quarks (color triplet): n=3, w=10, t=0
Antiquarks (color antitriplet): n=3*, w=01, t=2
Gluon (color octet): n=8, w=11, t=0

Base grid = {n1 + (1/2)*n2, (sqrt(3)/2)*n2}
where n1 and n2 are integers.
For triality t, add (2t/3) to n1 and n2

The triality of mesons and baryons is 0; no persistent state is known with nonzero triality: color confinement.

-

The electroweak case is more complicated. Using weak hypercharge and weak isospin component 3 (its root space), we have
WHC = (1/3)*n1 + (1/6)*n2
WIS = (1/2)*n2
The QCD triality for each grid point is 2*n1 + n2

For triality 0 (colorless, gluon):
WHC = n1 + (1/2)*n2
WIS = (1/2*n2
with triality t adding (2t/3) to n1

-

For Georgi-Glashow, we have something like QCD, but with SU(5) instead of SU(3). It has a modulo-5 quantum number, quintality.
It is n1 + 3*n3

For quintality 0 (singlet, gauge):
WHC = (5/3)*n1 + (1/2)*n2
WIS = (1/2)*n2
with quintality q adding (q/3) to n1

SO(10) has an modulo-4 quantum number that's 2 for vector states (Higgs) and 1 and 3 for spinor states (elementary fermions).

E6 has a modulo-3 quantum number that's 1 and 2 for fundamental-rep states (Higgs+EF's)

E8 has no such quantum numbers.

 

[arivero]

I liked a view of Baez and Huerta, where both the branching of SO(10) either into SU(5) or to Pati Salam SU(4)xSU(2)xSU(2) --or SO(6)xSO(4)-- were important, because then the standard model group was the other corner of the square.

It is also interesting when you quotient each group by its maximal subgroup, because then you see a descent of dimensions: From SO(10) you get the sphere of nine dimensions, for Pati Salam and SU(5) you get 8-dimensional manifolds (the product S5xS3 and, I believe, CP4, respectively) and for SM a family of 7-dimensional manifolds. I never worried about E6

 

[lpetrich]

I wouldn't know where to look for definitions of these, but I'll try to interpret them.

SO(n) -- S(n-1) -- (n-1) dim's
SU(n) -- CP(n-1) -- 2(n-1) dim's
SO(10), SU(5) -- 9, 8 -- check
SM: SU(3)*SU(2)*U(1) -- 4 + 2 + 1 = 7 -- check

Pati-Salam
SO(6) - 5
SO(4) - 3
SO(6)*SO(4) -- 8 -- check
SU(4) - 6
SU(2)*SU(2) - 4
SU(4)*SU(2)*SU(2) -- 10 -- ?

But
SO(3) - 2
SU(2) - 2
equal

 

[lpetrich]

As to E6, it's a subalgebra of SU(27), and E8 is a subalgebra of SO(248). Hyperspheres in 27D complex space and 248D real space?