# Why are there only two linearly independet quartic Higgs terms for the adjoint $24$ in $SU(5)$ GUTs?

+ 1 like - 0 dislike
288 views

Given the adjoint $24$ of $SU(5)$, we have

$$24\otimes 24 = 1_s \oplus 24_s \oplus 24_a \oplus 75_s \oplus 126_a \oplus \overline{126_a} \oplus 200_s ,$$

where each representation is denoted by its dimension and the subscripts $s$ and $a$ denote symmetric and antisymmetric respectively. Naively, I would say we have 7 quartic invariants:

$$(24\otimes 24)_{1_s} (24\otimes 24)_{1_s} + (24\otimes 24)_{24_s} (24\otimes 24)_{24_s} + (24\otimes 24)_{24_a} (24\otimes 24)_{24_a} + (24\otimes 24)_{75_s} (24\otimes 24)_{75_s} + (24\otimes 24)_{126_a} (24\otimes 24)_{126_a} + (24\otimes 24)_{\overline{126_a}} (24\otimes 24)_{\overline{126_a}} +(24\otimes 24)_{200_s} (24\otimes 24)_{200_s} ,$$

because

$$1_s \otimes 1_s = 1 \quad 24_s \otimes 24_s =1 \quad 75_s \otimes 75_s =1 \quad etc.$$

Nevertheless, in all $SU(5)$ papers only two quartic terms appear in the $24$ Higgs potential. How can I compute how many and which of these 7 terms are linearly independent?

This post imported from StackExchange Physics at 2015-10-05 20:44 (UTC), posted by SE-user JakobH
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOve$\varnothing$flowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.