Hypercharge and photon definitions in $SU(5)$ GUT

Asked Nov 15 '22 at 17:30

Active Nov 15 '22 at 18:21

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So, a weak isospin and hypercharge in $SU(5)$ are defined as $$ T_3 = \frac{1}{2}diag(0,0,0,1,-1) $$ $$ Y = \sqrt{\frac{3}{5}}diag(-1/3,-1/3,-1/3,1/2,1/2) $$ The hypercharge already has the $\sqrt{3/5}$ factor in it, and normalized so that $tr(T_3^2)$ and $tr(Y^2)$ both equal to $1/2$. Then why is this factor included in the definition of photon operators and the $Z$-boson? After all, the fields $W^3$ and $B$ are already made of these $T_3$ and $Y$.

For example, i saw these definitions $$γ=B-\sqrt{\frac{3}{5}}W^3$$ and $$Z=B\sqrt{\frac{3}{5}}+W^3.$$

edited Nov 15 '22 at 18:21

Qmechanic

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asked Nov 15 '22 at 17:30

Peter

You appreciate that (at the GUT scale) $\tan^2\theta_W\sim 3/5$, right? Eqn (5) here. – Cosmas Zachos Nov 15 '22 at 21:40
@CosmasZachos Yes I know that. But what does it have to do with it? – Peter Nov 15 '22 at 21:52
Well, the "definitions" you "saw" (where?) are in weird normalizations, and signs of the Weinberg angle. You understand how to "translate" the textbook answer into the corrected "definition" which diagonalizes the EW gauge boson mass matrix, right? Why don't you work it out and put it in your question for contrast? – Cosmas Zachos Nov 15 '22 at 21:57
Linked. – Cosmas Zachos Nov 15 '22 at 22:01
@CosmasZachos what is correct normalization? Is my definition of hypercharge generator right? – Peter Nov 15 '22 at 22:04
I meant normalization for the gauge bosons. Your Y is right, but in the "modern" half-scale convention. $Q=T_3+\sqrt{5/3} Y=\operatorname{diag}~(-1/3,-1/3,-1/3,1,0)$. – Cosmas Zachos Nov 15 '22 at 22:43
@CosmasZachos How does this answer the question? Rotation matrix of W and B contains sinw and cosw, which depend on scale. – Peter Nov 15 '22 at 22:48
Up to an overall normalization, of course. Look at the linked answer. – Cosmas Zachos Nov 15 '22 at 23:02
Well, for example, what is the value of the $sin_w^2$ at electroweak scale? – Peter Nov 15 '22 at 23:07
Let us continue this discussion in chat. – Cosmas Zachos Nov 15 '22 at 23:07

Hypercharge and photon definitions in $SU(5)$ GUT

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