Dimension of electric charge, hypercharge and isospin

[dimsun]

For SU(2) the three represented gauge fields are $$A_\mu^1$$, $$A_\mu^2$$ and $$A_\mu^3$$ and for U(1) the gauge field is $$B_\mu$$.
The $$A_\mu^3$$ and $$B_\mu$$ are electrically neutral.
The photon $$\gamma$$ and $$Z$$ particle are combinations of these.

My interest is the dimensions of the following parameters:

$$g$$ = strength of gauge fields $$A_\mu^1$$, $$A_\mu^2$$ and $$A_\mu^3$$.

$$g'$$ = strength of gauge field $$B_\mu$$.

$$e$$ = electron charge.
$$e = \frac{gg'}{\sqrt{g^2 + g'^2}}$$.

And:

$$Q = T_3 + \frac{Y}{2}$$.
$$Y$$ = weak hypercharge.
$$T_3$$ = weak isospin.
$$Q$$ = electric charge.
$$N$$ = electron number.

First I suppose that electric charge $$Q$$ is in esu and has unit statcoulomb.
So $$Q$$ in SI-units is $$q \sqrt{K}$$ in which $$K$$ is the Coulomb constant and $$q$$ is electric charge in SI-units.
And I also suppose that electron charge $$e$$ in the above equation is in esu. Is this al true?
Can I say that also $$Q$$, $$Y$$, $$T$$, $$N$$, $$g$$, $$g'$$ and $$e$$ all have the same dimension?

What is the difference between $$g'$$and $$Y$$ ?
And what is the difference between $$g$$ and $$T$$ ?

The gauge symmetries are first unbroken and later broken symmetries.
Is it that $$g$$ and $$g'$$ are unbroken parameters and $$Y$$ and $$T$$ are broken parameters?

Next to weak hypercharge and weak isospin does weak charge exist?
In that case is there also an equation to calculate weak charge from weak hypercharge and weak isospin?

 

[Vanadium 50]

Everything here is dimensionless.

 

[Bill_K]

What is the difference between g' and Y ?
And what is the difference between g and T ?

In that case is there also an equation to calculate weak charge from weak hypercharge and weak isospin?

dimsum, g and g' are coupling constants with the same dimensions as e (but I don't think too many particle physicists use statcoulombs!)

g sin θ_W = g' cos θ_W = e

where θ_W is the weak mixing angle. T and Y are dimensionless parameters related to the electroweak symmetry group. They are designed to imitate the corresponding parameters T, Y in the symmetry group for strong interactions. Q = T₃ + Y/2 in both cases, but note an important difference - electroweak symmetry is chiral: The quarks u and d have T = 1 in the strong symmetry, but in the weak symmetry the left-handed quarks u_L and d_L have T = 1, while the right-handed quarks u_R and d_R have T = 0.

The gauge symmetries are first unbroken and later broken symmetries.
Is it that g and g' are unbroken parameters and Y and T are broken parameters?

No, not at all. The symmetry breaking does not affect g and g', only particle masses, and is characterized by another physical parameter, v = 246 GeV. For example

M_W = ½ v g
M_Z = ½ v √(g2 + g'2)

Next to weak hypercharge and weak isospin does weak charge exist?

"Weak charge" is exactly the same as regular charge.

 

[dimsun]

(but I don't think too many particle physicists use statcoulombs!)

Thanks Bill, but there is a difference.
In SI-units we have mass $$m$$, and this has a different dimension then $$m\sqrt{G}$$.
Likewise in SI-units we have electric charge $$q$$, and this has a different dimension then $$q\sqrt{K}$$.

In SI-units mass $$m$$ and electric charge $$q$$ don't have the same dimension. But $$m\sqrt{G}$$ and $$q\sqrt{K}$$ do have the same dimension.

$$m\sqrt{G}$$ in ESU is called mass.

And $$q\sqrt{K}$$ in ESU is called electric charge and have both the same dimension.

So mass in SI-units is another quantity then mass in ESU.
I think in the original article of Steven Weinberg, ESU units are used.



But stil not clear is the difference between on the one hand $$g$$ and $$g'$$ and on the other hand $$T$$ and $$Y$$.
$$g$$ and $$T$$ are both related to SU(2) symmetry group, but they are not the same, what is the difference?
and $$g'$$ and $$Y$$ are both related to the U(1) symmetry group, but they are not the same, again what is the difference?
The electron charge $$e$$ and $$Q$$ are also related to the U(1) symmetry group, but I can't exchange $$e$$ for $$Q$$:

$$T_3 + \frac{Y}{2} = \frac{gg'}{\sqrt{g^2 + g'^2}}$$ is wrong, but why?