
#1
Apr2112, 08:45 PM

P: 265

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. 



#2
Apr2212, 01:35 AM

P: 514

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 3bar or 3*. Gluons are coloranticolor with a colorless mixture removed, giving them 3*31 = 8 color states. The X and Y with ordinaryquark 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. 



#3
Apr2212, 01:41 AM

P: 360

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 [itex]\pm \frac{4}{3}[/itex] and [itex]\pm \frac{1}{3}[/itex]. The SO(10) GUT has additional X bosons with electric charges [itex]\pm \frac{2}{3}[/itex] and [itex]\pm \frac{1}{3}[/itex]. 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.)




#4
Apr2212, 07:09 AM

P: 514

Charge of X and Y bosons
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:
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, GeorgiGlashow SU(5) has a 4D root space, PatiSalam 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 ordinaryquark 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 ordinaryquark color state, it has electric charge 1/3. 



#5
Apr2212, 01:35 PM

P: 360

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
[tex]Q = W_3 sin(\theta) + Y cos(\theta)[/tex] which is along the vertical in the EPE if you choose the "electric charge" or "standard model" rotation. 



#6
Apr2212, 10:08 PM

P: 514

I've created some ASCIIart 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 righthanded neutrino l, r means left and right handed * means antiparticle I put the dots in to get the spacing to work out correctly. Particles with the same electric charge lie on a gentlytilted 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 



#8
Apr2512, 05:19 AM

P: 514

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 GeorgiGlashow, we have something like QCD, but with SU(5) instead of SU(3). It has a modulo5 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 modulo4 quantum number that's 2 for vector states (Higgs) and 1 and 3 for spinor states (elementary fermions). E6 has a modulo3 quantum number that's 1 and 2 for fundamentalrep states (Higgs+EF's) E8 has no such quantum numbers. 



#9
Apr2512, 05:57 AM

PF Gold
P: 2,884

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 8dimensional manifolds (the product S5xS3 and, I believe, CP4, respectively) and for SM a family of 7dimensional manifolds. I never worried about E6 



#10
Apr2512, 07:20 AM

P: 514

I wouldn't know where to look for definitions of these, but I'll try to interpret them.
SO(n)  S(n1)  (n1) dim's SU(n)  CP(n1)  2(n1) dim's SO(10), SU(5)  9, 8  check SM: SU(3)*SU(2)*U(1)  4 + 2 + 1 = 7  check PatiSalam 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 



#11
Apr2512, 07:47 AM

P: 514

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?



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