Graphical representation of the weak mixing angle

[Sandglass]

The graphical pattern of particles in the weak hypercharge and weak isospin plane, visible in this wiki page, shows the mixing angle between the Yw and Q axes. Actually , from the weak hypercharge (-2) of a right-handed electron and its electric charge (-1), one obtains an angle Pi/3, not the mixing angle.

So is this mixing angle a symbolic representation of the link between electromagnetism and the weak interaction, derived from the complex calculations of the standard model, or does it have a numerical use in this graph?

 

[ohwilleke]

Not a complete answer, but the numerical applications of electroweak unification primarily derive from the other diagram in the same article, which demonstrate that the electromagnetic force coupling constant, the weak force coupling constant, and the Weinberg angle are not independent of each other.

Likewise, the W boson mass, the Z boson mass, and the Weinberg angle are not independent of each other, and the relative magnitudes of those Standard Model constants are functions of the electromagnetic and weak force coupling constants. In other words:

and

and

and

So, while we have five parameters (the Weinberg angle, the electromagnetic coupling constant, the weak force coupling constant, the W boson mass and the Z boson mass), because of electroweak unification, those five parameters don't represent five independent degrees of freedom.

The original W⁰ and B⁰ vector bosons of electroweak unification theory are really more of a conceptual framework than something that physicists work with to solve problems, or observe, that illustrates a way of thinking about how the relationships that are observed arise.

 

[Sandglass]

Thanks for your answer.

Since the coupling constant for the weak hypercharge is given by g' cos (theta_w) = e, I naively thought that the weak hypercharge Yw was related to the electric charge Q in the same way by Yw cos(theta_w) = Q. What puzzles me is the fact that the particle graph seems to confirm this with a mixing angle shown in the (Yw, Q) plane.

On the other hand, this cannot be the case since the Gell-Mann-Nishijima formula Q = Yw/2 + T3 cannot be satisfied with g'/e = Yw/Q = cos(theta_w) and g/e = T3/Q = sin(theta_w).

Therefore, is there any theoretical relationship between (g'/e) and (Yw/Q) ?

 

[Sandglass]

I'm making one last attempt about this graph. I looked for it in some QFT books and didn't see it.

Does anyone have a reference, either an academic book or a review article that shows this graph, instead of this wiki page ?

 

[vanhees71]

I don't know. Why is it important? It just depicts the relation between the coupling constants $g$, $g'$ and $e$ in terms of the Weinberg angle. Wikipedia gives a source for the picture:

Lee, T.D. (1981). Particle Physics and Introduction to Field Theory.

It's a nice textbook by a Nobel laureate.

 

[Sandglass]

Actually, I'm looking for a reference for this graph: https://en.wikipedia.org/wiki/Mathe...Standard_Model#/media/File:Standard_Model.svg

I'm new to this and want to make sure I understand: do the letters L and R for particles refer to chirality, not helicity ? The legend is ambiguous.

 

[ohwilleke]

do the letters L and R for particles refer to chirality, not helicity? The legend is ambiguous.

Fermions have chirality. Bosons have helicity. All of the references in the chart with L or R that I see are for fermions, so it is a reference to chirality.

 

[malawi_glenn]

Fermions have chirality. Bosons have helicity. All of the references in the chart with L or R that I see are for fermions, so it is a reference to chirality.

Not enitirely true. Chirality is a label for how the particle transforms under the Lorentz group.

Helicity is never used as a label for anything in these kind of diagrams, since it is not a Lorentz invariant property of the particle.

 

[vanhees71]

Fermions have chirality. Bosons have helicity. All of the references in the chart with L or R that I see are for fermions, so it is a reference to chirality.

Dirac fermions have indeed both chirality and helicity. By definition chirality eigenstates are eigenstates of $\gamma_5$. Since $\gamma_5^2=1$ the possible eigenvalues of chirality are $\chi \in \{1,-1 \}$.

By definition helicity for a momentum eigenstate is the eigenvalue of the component of the total angular momentum in direction of the momentum, and the corresponding eigenvalues are $h \in \{1/2,-1/2\}$ (with natural units, where $\hbar=1$).

Only for massless particles $\chi=2h$, and only in this case helicity eigenstates are Lorentz invariant. For massive particles you can change to another reference frame, where helicity is flipped. That's impossible for massless particles, because you cannot "overtake" the fermion with your new reference frame, because a massless fermion moves with $c$ in all inertial frames.

 

[ohwilleke]

Thanks for the corrections that have clarified the concepts.

 

[malawi_glenn]

Furthermore, when we use the labels R and L on fermions in the electroweak SM we are referring to chirality always - the unbroken EW symmetry. This is because they L and R chirality have different gauge couplings.

 

[Sandglass]

Thanks for all these clarifications.

 

[garrett]

Thanks for your answer.

Since the coupling constant for the weak hypercharge is given by g' cos (theta_w) = e, I naively thought that the weak hypercharge Yw was related to the electric charge Q in the same way by Yw cos(theta_w) = Q. What puzzles me is the fact that the particle graph seems to confirm this with a mixing angle shown in the (Yw, Q) plane.

On the other hand, this cannot be the case since the Gell-Mann-Nishijima formula Q = Yw/2 + T3 cannot be satisfied with g'/e = Yw/Q = cos(theta_w) and g/e = T3/Q = sin(theta_w).

Therefore, is there any theoretical relationship between (g'/e) and (Yw/Q) ?

I suspect the magic formula you're looking for is
$$
Q e = Y_W \frac{1}{2} g' \cos{\theta_W} + T_3 g \sin{\theta_W}
$$
The electric charge, weak hypercharge, and weak isospin axes are scaled by their respective coupling constants, and particle charges come in integral (or fractional integral, due to convention) multiples of these. Hope that helps.