Do $Y$ bosons gain a little bit of mass from Higgs of $\bf 5$ rep in $SU(5)$ GUT theory? I know that when the Higgs in the 24 rep takes a v.e.v. of this form $$v_{24}\,\mathrm{diag}\begin{pmatrix}
-\frac{2}{\sqrt{15}} & -\frac{2}{\sqrt{15}} & -\frac{2}{\sqrt{15}} & \frac{3}{\sqrt{15}} & \frac{3}{\sqrt{15}}\\
\end{pmatrix},$$
$SU(5)$ breaks to $SU(3)\times SU(2)\times U(1)$ group, with $X$ and $Y$ bosons getting masses around $10^{15} GeV$. But I noticed that when Higgs in 5 rep gets it's v.e.v. of this form
$$\begin{pmatrix}
0\\
0\\
0\\
0\\
\frac{v_5}{\sqrt2}\\
\end{pmatrix},$$ which besides coupling to $W^{\pm}$ and $Z$ bosons, also couples to $Y$ bosons since their summed generators form this matrix
$$\begin{pmatrix}
0 & 0 & 0 & 0 & 1-i\\
0 & 0 & 0 & 0 & 1-i\\
0 & 0 & 0 & 0 & 1-i\\
0 & 0 & 0 & 0 & 0\\
1+i & 1+i & 1+i & 0 & 0\\
\end{pmatrix}.$$
A 5 Higgs v.e.v. is much much smaller than 24 Higgs, but anyway how much mass does it give them?
 A: I'll be schematic and cavalier with normalizations and factors... You can do this right in your peculiar conventions.
I'll call the v.e.v., of the 24 Higgs v, following the obligatory sourcebook of the BEGN paper, and that of the 5 just $v_0$. They differ by 13 orders of magnitude, of course. For energies in these 13 orders of magnitude in between GUT breaking and EW breaking, only v is operative to give mass $5gv/2\sqrt{2}$ to the Ys, but then there is the extra term of the covariant completion in the kinetic term of the 5, which is the non vanishing term of the 5-5 element of the square of the matrix you wrote with the v.e.v.s of the 5, so $m^2=g^2v_0^2/8$.
So, naively the add-on to the Y mass at super low energies is a correction 26 orders of magnitude smaller than the GUT mass, $gv \sqrt{ (1+v^2_0/v^2 ) /8 }$; you can expand the square root.  Most people call this insignificant, given the extreme speculative nature of the issue.
Nevertheless, this is for fixed v and $v_0$, which, as discussed in the BEGN paper, does not obtain, and hinges on the coupling of the 5 to the 24 in the combined potential, which results to shifted values, (4.4'), (4.5'), (4.11'). But, I hope I have convinced you, as far as the "academic" shift you are interested in, this amounts to balancing angels on the head of a pin...
