Gauge coupling unification in the flipped $E_8$ GUT

Abstract

The gauge coupling unification is investigated at the classical level under the assumptions that the gauge symmetry breaking chain is $E_8\to E_7\times U_1 \to E_6\times U_1 \to SO_{10}\times U_1 \to SU_5 \times U_1 \to SU_3 \times SU_2 \times U_1$ and only components of the representations 248 of $E_8$ can acquire vacuum expectation values. We demonstrate that there are several options for the relations between the gauge couplings of the resulting theory, but the only symmetry breaking pattern corresponds to $\alpha_3=\alpha_2$ and $\sin^2\theta_\mathrm{W}=3/8$. Moreover, only for this option does the particle content of the resulting theory include all MSSM superfields. It is also noted that this symmetry breaking pattern corresponds to the case where all representation that acquire vacuum expectation values have the minimal absolute values of the relevant $U_1$ charges.

Appendix: Branching rules for the fundamental and adjoint representations of $$E_7$$ with respect to the subgroup $$E_6\times U_1$$

The branching rules for the $E_7$ representations with respect to the subgroup $E_6\times U_1$ presented in [17] do not contain $U_1$ charges. That is why we here discuss how these charges can be found and calculate them for two lowest $E_7$ representations in which we are interested.

According to [17], the group $E_7$ contains a subgroup $SO_{12}\times SO_3$. The $SO_{12}$ group in turn contains the subgroup $SO_{10}\times U_1$, and we hence obtain the embedding

$$ E_7 \supset SO_{12}\times SO_3 \supset (SO_{10}\times \underbrace{U_1}_{\gamma_2}) \times \underbrace{U_1}_{\gamma_1},$$

(A.1)

where $\gamma_1$ and $\gamma_2$ are real parameters of the corresponding $U_1$ transformations. The $E_7$ branching rules for the $56$ and $133$ representations with respect to the $SO_{12}\times SO_3$ subgroup are given by Eq. (32). The $SO_{12}$ branching rules with respect to the $SO_{10}\times U_1$ subgroup can also be found in [17] and have the form

$$ \begin{aligned} \, &12|_{SO_{12}} = 1(2) + 1(-2) + 10(0)|_{SO_{10}\times U_1},\\ & 32|_{SO_{12}} = 16(1) + \overline{16}(-1)|_{SO_{10}\times U_1},\\ & 32'|_{SO_{12}} = 16(-1) + \overline{16}(1)|_{SO_{10}\times U_1},\\ & 66|_{SO_{12}}= 1(0) + 10(2) + 10(-2) + 45(0)|_{SO_{10}\times U_1}, \end{aligned}$$

(A.2)

where the numbers in brackets are the charges with respect to the $U_1$ group parameterized by $\gamma_2$. The $U_1$ charges for various parts of $SO_3$ representations can also be easily found from the theory of angular momentum. Taking into account that $2J_3$ ranges from $-2J$ to $2J$ with step 2 and using Eqs. (32) and (A.2), we conclude that the fundamental and adjoint representations of $E_7$ decompose into irreducible representations of subgroup (A.1) according to the branching rules

$$ \begin{aligned} \, &56|_{E_7} = [12,2] + [32,1]|_{SO_{12}\times SO_3} = 1(1,2) + 1(-1,2) + 1(1,-2) + 1(-1,-2) +{} \\ &\hphantom{56|_{E_7} ={}}+ 10(1,0) + 10(-1,0) + 16(0,1) + \overline{16}(0,-1)|_{SO_{10}\times U_1\times U_1},\\ &133|_{E_7} = [1,3] + [66,1] + [32',2]|_{SO_{12}\times SO_3} = 1(2,0) + 1(0,0) + 1(-2,0) +{} \\ &\hphantom{133|_{E_7}={}}+ 1(0,0) + 10(0,2)+ 10(0,-2) + 45(0,0) + 16(1,-1) + \overline{16}(1,1) +{} \\ &\hphantom{133|_{E_7} ={}}+ 16(-1,-1)+ \overline{16}(-1,1) |_{SO_{10}\times U_1\times U_1}. \end{aligned}$$

(A.3)

Here, the first number in round brackets is the charge $Q_{\gamma_1}$ corresponding to the $U_1$ subgroup parameterized by $\gamma_1$, and the second number is the charge $Q_{\gamma_2}$ corresponding to the $U_1$ subgroup parameterized by $\gamma_2$.

On the other hand, according to [17], the group $E_7$ contains the subgroup $E_6\times U_1$, and the group $E_6$ contains the subgroup $SO_{10}\times U_1$. Therefore, we obtain another embedding

$$ E_7 \supset E_6\times \underbrace{U_1}_{\delta_1} \supset (SO_{10}\times \underbrace{U_1}_{\delta_2}) \times \underbrace{U_1}_{\delta_1},$$

(A.4)

where the parameters of the $U_1$ groups are denoted by $\delta_1$ and $\delta_2$. The branching rules for the $56$ and $133$ representations of $E_7$ with respect to the $E_6$ subgroup can be found in [17] and are rather evident. However, it is not so trivial to find the corresponding $U_1$ charges. For this, we assume that

$$ \begin{aligned} \, &56|_{E_7} = 27(1) + \overline{27}(-1) + 1(x_1) + 1(-x_1)|_{E_6\times U_1},\\ &133|_{E_7} = 1(x_2) + 27(x_3) + \overline{27}(-x_3) + 78(x_4)|_{E_6\times U_1}, \end{aligned}$$

(A.5)

where in the brackets we write the charge corresponding to the $U_1$ subgroup parameterized by $\delta_1$, and the constants $x_i$ are to be calculated. (The $U_1$ charge of the $E_6$ representation $27$ in the first equation can be chosen arbitrarily. Setting it to 1, we simply specify its normalization.) Using Eq. (63), we obtain the branching rules with respect to subgroup (A.4) in the form

$$ \begin{aligned} \, &56|_{E_7} = 1(1,4) +10(1,-2) +16(1,1) + 1(-1,-4) + 10(-1,2) + \overline{16}(-1,-1) +{} \\ &\hphantom{56|_{E_7} ={}}+ 1(x_1,0) + 1(-x_1,0)|_{SO_{10}\times U_1\times U_1},\\ & 133|_{E_7} = 1(x_2,0) + 1(x_3,4) + 10(x_3,-2) + 16(x_3,1) + 1(-x_3,-4) + 10(-x_3,2)+{} \\ &\hphantom{133|_{E_7}={}} +\overline{16}(-x_3,-1) + 1(x_4,0) + 16(x_4,-3) + \overline{16}(x_4,3) + 45(x_4,0)|_{SO_{10}\times U_1\times U_1}, \end{aligned}$$

(A.6)

where the numbers in the brackets are the charges $Q_{\delta_1}$ and $Q_{\delta_2}$ with respect to the $U_1$ groups respectively parameterized by $\delta_1$ and $\delta_2$.

Comparing Eqs. (A.3) and (A.6) we see that (as expected) the $SO_{10}$ representations coincide, but their $U_1$ charges are different. The matter is that the $U_1$ subgroups parameterized by $\gamma_{1,2}$ and $\delta_{1,2}$ are different. However, the $U_1$ charges coincide if we make the identification

$$ \begin{cases} Q_{\delta_1} = Q_{\gamma_1} + Q_{\gamma_2},\\ Q_{\delta_2} = -2 Q_{\gamma_1} + Q_{\gamma_2}. \end{cases}$$

(A.7)

This implies that under the $U_1\times U_1$ transformations, a field $\varphi$ transforms as

$$ \varphi \to \exp(iQ_{\delta_1} \delta_1 + i Q_{\delta_2}\delta_2)\varphi = \exp(i Q_{\gamma_1} (\delta_1 - 2 \delta_2) + iQ_{\gamma_2}(\delta_1 + \delta_2))\varphi,$$

(A.8)

and hence the parameters of the $U_1\times U_1$ subgroups in (A.1) and (A.4) are related as

$$ \begin{cases} \gamma_1 = \delta_1 -2\delta_2,\\ \gamma_2 = \delta_1 + \delta_2, \end{cases} \qquad\qquad \begin{cases} \delta_1 = \dfrac{\gamma_1 + 2\gamma_2}{3},\\[2mm] \delta_2 = \dfrac{-\gamma_1 + \gamma_2}{3}. \end{cases}$$

(A.9)

Using Eq. (A.7) and comparing Eqs. (A.3) and (A.6), we obtain

$$ x_1=3,\qquad x_2=0,\qquad x_3=-2,\qquad x_4=0.$$

(A.10)

Substituting these values in Eq. (A.5), we obtain the sought branching rules

$$56|_{E_7} = 27(1) + \overline{27}(-1) + 1(3) + 1(-3)|_{E_6\times U_1},$$

(A.11)

$$133|_{E_7} = 1(0) + 27(-2) + \overline{27}(2) + 78(0)|_{E_6\times U_1},$$

(A.12)

which coincide with Eq. (70).