E6 GUT through Effects of Dimension-5 Operators

TL;DR

This paper explores how dimension-5 operators induced by quantum gravity affect gauge coupling unification in E6 grand unified theories, enabling non-supersymmetric models consistent with proton decay constraints.

Contribution

It demonstrates the construction of non-supersymmetric E6 GUT models with modified unification conditions due to dimension-5 operators and derives relevant Clebsch-Gordan coefficients.

Findings

Non-supersymmetric E6 models satisfying proton lifetime constraints.

Modified gauge unification conditions due to dimension-5 operators.

Explicit derivation of Clebsch-Gordan coefficients for E6 breaking chains.

Abstract

In the effective field theory framework, quantum gravity can induce effective dimension-5 operators, which have important impacts on grand unified theories. Interestingly, one of main effects is the modification of the usual gauge coupling unification condition. We investigate the gauge coupling unification in $E_{6}$ under modified gauge coupling unification condition at scales $M_X$ in the presence of one or more dimension-5 operators. It is shown that nonsupersymmetric models of $E_6$ unification can be obtained and can easily satisfy the constraints from the proton lifetime. For constructing these models, we consider several maximal subgroups $H=SO(10)\times U(1), H=SU(3)\times SU(3)\times SU(3)$, and $H=SU(2)\times SU(6)$ of $E_{6}$ and the usual breaking chains for a specific maximal subgroup, and derive all of the Clebsch-Gordan coefficients $\Phi^{(r)}_{s,z}$ associated with $E_6$ breaking to the Standard Model, which are given in Appendix A.

Tables

Table 1. Table 1: Quadratic Casimir invariants C ​ ( r ) 𝐶 𝑟 C(r) for some irreps r 𝑟 r of E 6 subscript 𝐸 6 E_{6} , S ​ O ​ ( 10 ) 𝑆 𝑂 10 SO(10) and S ​ U ​ ( 6 ) 𝑆 𝑈 6 SU(6) , in the conventions explained in the section II of the text. C ​ ( 𝟏 ) = 0 𝐶 1 0 C({\bf 1})=0 for the singlet 𝟏 1 {\bf 1} of any group.

$E_6$	r	27	78	650	2430
	$C\left(r\right)$	3	12	18	26
$SO\left(10\right)$	r	10	45	54	210	770
	$C\left(r\right)$	1	8	10	12	18
$SU\left(6\right)$	r	6	35	189	405
	$C\left(r\right)$	1/2	6	10	14

Table 2. Table 2: The breaking chain, unification scale M X subscript 𝑀 𝑋 M_{X} , Higgs VEV v 𝑣 v , Wilson coefficient c k subscript 𝑐 𝑘 c_{k} , gauge coupling α G subscript 𝛼 𝐺 \alpha_{G} at the unification scale, and three ϵ 1 , 2 , 3 subscript italic-ϵ 1 2 3 \epsilon_{1,2,3} for unification with only one dimension-5 operator.

E6 embedding	breaking chain	$M_X$(GeV)	$v$(GeV)	$c_k$	$1/{\alpha }_G$	${ϵ}_1$	${ϵ}_2$	${ϵ}_3$
$SU\left(5\right)\subset SO\left(10\right)$	$\mathrm{𝟔𝟓𝟎}\to \mathrm{𝟓𝟒}\to \mathrm{𝟐𝟒}$	$4.0\times {10}^{13}$	$9.8\times {10}^{13}$	$1.1\times {10}^3$	$40.6$	$0.023$	$0.068$	$-0.045$
	$\mathrm{𝟐𝟒𝟑𝟎}\to \mathrm{𝟐𝟏𝟎}\to \mathrm{𝟕𝟓}$	$3.7\times {10}^{15}$	$7.8\times {10}^{15}$	$-6.4$	$43.0$	$-0.103$	$0.062$	$0.021$
$SU{\left(3\right)}_L\subset SU\left(6\right)\times SU{\left(2\right)}_X$	$\mathrm{𝟔𝟓𝟎}\to (\mathrm{𝟑𝟓},\mathrm{𝟏})\to \mathrm{𝟏}$	$4.0\times {10}^{13}$	$9.8\times {10}^{13}$	$1.4\times {10}^3$	$41.1$	$0.011$	$0.056$	$-0.056$
	$\mathrm{𝟐𝟒𝟑𝟎}\to (\mathrm{𝟒𝟎𝟓},\mathrm{𝟏})\to \mathrm{𝟏}$	$6.4\times {10}^{16}$	$1.4\times {10}^{17}$	$0.63$	$48.9$	$-0.250$	$-0.038$	$-0.038$
$SU{\left(3\right)}_L\subset SU\left(6\right)\times SU{\left(2\right)}_L$	$\mathrm{𝟐𝟒𝟑𝟎}\to (\mathrm{𝟏𝟖𝟗},\mathrm{𝟏})\to \mathrm{𝟏}$	$3.6\times {10}^{18}$	$8.1\times {10}^{18}$	$-0.015$	$49.0$	$-0.306$	$0.$	$0.051$

Table 3. Table 3: The diagonal part, h ​ j ℎ 𝑗 hj ( j = 1 , 2 , … , 15 ) 𝑗 1 2 … 15 (j=1,2,...,15) in Eq. ( 6 ) of the SM singlets Φ s , z r superscript subscript Φ 𝑠 𝑧 𝑟 \Phi_{s,z}^{r} in each of the irreps r 𝑟 r according to their transformation properties under the S ​ U ​ ( 3 ) L ⊂ S ​ U ​ ( 6 ) × S ​ U ​ ( 2 ) X ⊂ E 6 𝑆 𝑈 subscript 3 𝐿 𝑆 𝑈 6 𝑆 𝑈 subscript 2 𝑋 subscript 𝐸 6 SU(3)_{L}\subset SU(6)\times SU(2)_{X}\subset E_{6} subgroup.

$E_6$	SU(6)$\times$	$SU{\left(3\right)}_L$ z	$h{1}_{s,z}^r$	$h{2}_{s,z}^r$	$h{3}_{s,z}^r$	$h{4}_{s,z}^r$	$h{5}_{s,z}^r$	$h{6}_{s,z}^r$	$h{7}_{s,z}^r$	$h{8}_{s,z}^r$	$h{9}_{s,z}^r$	$h{10}_{s,z}^r$	$h{11}_{s,z}^r$	$h{12}_{s,z}^r$	$h{13}_{s,z}^r$	$h{14}_{s,z}^r$	$h{15}_{s,z}^r$	$N_{s,z}^r$
	SU(2)X s
$\mathrm{𝟏}$	$\mathrm{𝟏}$ 1	$\mathrm{𝟏}$ 1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	$\frac{1}{\sqrt{78}}$
$\mathrm{𝟔𝟓𝟎}$	$(\mathrm{𝟏},\mathrm{𝟏})$ 1	$\mathrm{𝟏}$ 1	$1$	$1$	$1$	-$\frac{1}{2}$	-$\frac{11}{4}$	-$\frac{1}{2}$	$1$	-$5$	-$\frac{5}{4}$	$1$	-$\frac{1}{2}$	-$\frac{1}{2}$	-$\frac{1}{2}$	$1$	-$\frac{1}{2}$	$\frac{1}{2\sqrt{30}}$
	$(\mathrm{𝟑𝟓},\mathrm{𝟏})$ 2	$\mathrm{𝟏}$ 1	$1$	-$1$	-$\frac{1}{5}$	$0$	-$\frac{3}{10}$	$0$	$0$	$0$	-$\frac{1}{2}$	$0$	$0$	$0$	$0$	-$1$	$0$	$\frac{1}{4}$
		$\mathrm{𝟖}$ 2	$0$	$1$	$\frac{3}{5}$	$0$	-$\frac{3}{5}$	$0$	-$1$	$0$	-$1$	$\frac{1}{2}$	$0$	$0$	$0$	-$\frac{1}{2}$	$0$	$\sqrt{\frac{2}{41}}$
	$(\mathrm{𝟏𝟖𝟗},\mathrm{𝟏})$ 3	$\mathrm{𝟏}$ 1	$1$	$1$	-$3$	$0$	$\frac{1}{2}$	$0$	-$\frac{2}{3}$	$0$	-$\frac{1}{2}$	-$\frac{2}{3}$	$0$	$0$	$0$	$1$	$0$	$\frac{1}{2}\sqrt{\frac{3}{34}}$
		$\mathrm{𝟖}$ 2	$0$	$1$	-$1$	$0$	-$\frac{3}{2}$	$0$	$\frac{1}{3}$	$0$	$\frac{3}{2}$	-$\frac{1}{6}$	$0$	$0$	$0$	-$\frac{1}{2}$	$0$	$\sqrt{\frac{6}{83}}$
$\mathrm{𝟐𝟒𝟑𝟎}$	$(\mathrm{𝟏},\mathrm{𝟏})$ 1	$\mathrm{𝟏}$ 1	$1$	$1$	$1$	-$\frac{7}{6}$	$\frac{101}{36}$	-$\frac{7}{6}$	$1$	$\frac{35}{9}$	$\frac{25}{12}$	$1$	-$\frac{7}{6}$	-$\frac{7}{6}$	-$\frac{7}{6}$	$1$	-$\frac{7}{6}$	$\frac{3}{2}\sqrt{\frac{3}{910}}$
	$(\mathrm{𝟏𝟖𝟗},\mathrm{𝟏})$ 2	$\mathrm{𝟏}$ 1	$1$	$1$	-$3$	$0$	-$\frac{15}{4}$	$0$	-$\frac{2}{3}$	$0$	$\frac{15}{4}$	-$\frac{2}{3}$	$0$	$0$	$0$	$1$	$0$	$\frac{1}{2\sqrt{85}}$
		$\mathrm{𝟖}$ 2	$0$	$1$	-$1$	$0$	$\frac{9}{4}$	$0$	$\frac{1}{3}$	$0$	-$\frac{9}{4}$	-$\frac{1}{6}$	$0$	$0$	$0$	-$\frac{1}{2}$	$0$	$\sqrt{\frac{2}{41}}$
	$(\mathrm{𝟒𝟎𝟓},\mathrm{𝟏})$ 3	$\mathrm{𝟏}$ 1	$1$	$1$	$\frac{33}{5}$	$0$	$\frac{9}{10}$	$0$	-$\frac{4}{3}$	$0$	$\frac{3}{2}$	-$\frac{4}{3}$	$0$	$0$	$0$	$1$	$0$	$\frac{1}{4\sqrt{7}}$
		$\mathrm{𝟖}$ 2	$0$	$1$	$\frac{19}{5}$	$0$	-$\frac{9}{5}$	$0$	$\frac{5}{3}$	$0$	-$3$	-$\frac{5}{6}$	$0$	$0$	$0$	-$\frac{1}{2}$	$0$	$\sqrt{\frac{2}{185}}$
		$\mathrm{𝟐𝟕}$ 3	$0$	$1$	$\frac{9}{5}$	$0$	$\frac{27}{10}$	$0$	$0$	$0$	$\frac{9}{2}$	$0$	$0$	$0$	$0$	-$3$	$0$	$\frac{1}{2\sqrt{30}}$

Table 4. Table 4: The non-diagonal part, h ​ i ​ j ℎ 𝑖 𝑗 hij in Eq. ( 6 ) of the SM singlets Φ s , z r superscript subscript Φ 𝑠 𝑧 𝑟 \Phi_{s,z}^{r} in each of the irreps r 𝑟 r according to their transformation properties under the S ​ U ​ ( 3 ) L ⊂ S ​ U ​ ( 6 ) × S ​ U ​ ( 2 ) X ⊂ E 6 𝑆 𝑈 subscript 3 𝐿 𝑆 𝑈 6 𝑆 𝑈 subscript 2 𝑋 subscript 𝐸 6 SU(3)_{L}\subset SU(6)\times SU(2)_{X}\subset E_{6} subgroup. The entries h ​ 3 s , z ′ ⁣ r ℎ superscript subscript 3 𝑠 𝑧 ′ 𝑟 h3_{s,z}^{\prime r} , h ​ 5 s , z ′ ⁣ r ℎ superscript subscript 5 𝑠 𝑧 ′ 𝑟 h5_{s,z}^{\prime r} , h ​ 9 s , z ′ ⁣ r ℎ superscript subscript 9 𝑠 𝑧 ′ 𝑟 h9_{s,z}^{\prime r} for the flipped embedding are also listed here.

$E_6$	SU(6)$\times$	SU(3)L z	$h{35}_{s,z}^r$	$h{39}_{s,z}^r$	$h{59}_{s,z}^r$	$h{3}_{s,z}^{\prime r}$	$h{5}_{s,z}^{\prime r}$	$h{9}_{s,z}^{\prime r}$	$h{35}_{s,z}^{\prime r}$	$h{39}_{s,z}^{\prime r}$	$h{59}_{s,z}^{\prime r}$
	SU(2)X s
$\mathrm{𝟏}$	$\mathrm{𝟏}$ 1	$\mathrm{𝟏}$ 1	0	0	0	0	0	0	0	0	0
$\mathrm{𝟔𝟓𝟎}$	$(\mathrm{𝟏},\mathrm{𝟏})$ 1	$\mathrm{𝟏}$ 1	$0$	$0$	-$\frac{3}{4}\sqrt{15}$	$1$	-$\frac{11}{4}$	-$\frac{5}{4}$	$0$	$0$	-$\frac{3\sqrt{15}}{4}$
	$(\mathrm{𝟑𝟓},\mathrm{𝟏})$ 2	$\mathrm{𝟏}$ 1	$\frac{\sqrt{3/2}}{5}$	-$\frac{1}{\sqrt{10}}$	$\frac{\sqrt{3/5}}{2}$	-$\frac{1}{5}$	-$\frac{3}{10}$	-$\frac{1}{2}$	$\frac{\sqrt{3/2}}{5}$	-$\frac{1}{\sqrt{10}}$	$\frac{\sqrt{3/5}}{2}$
		$\mathrm{𝟖}$ 2	-$\frac{\sqrt{3/2}}{10}$	$\frac{1}{2\sqrt{10}}$	$\sqrt{3/5}$	$\frac{3}{5}$	-$\frac{3}{5}$	-$1$	-$\frac{\sqrt{3/2}}{10}$	$\frac{1}{2\sqrt{10}}$	$\sqrt{\frac{3}{5}}$
	$(\mathrm{𝟏𝟖𝟗},\mathrm{𝟏})$ 3	$\mathrm{𝟏}$ 1	$0$	$4\sqrt{2/5}$	-$\frac{1}{2\sqrt{15}}$	-$3$	$0$	$0$	-$\sqrt{3/2}$	$\sqrt{5/2}$	$0$
		$\mathrm{𝟖}$ 2	$\frac{5}{2\sqrt{6}}$	-$\frac{3}{2\sqrt{10}}$	$\frac{\sqrt{3/5}}{2}$	-$1$	$0$	$0$	$\frac{\sqrt{3/2}}{2}$	-$\frac{\sqrt{5/2}}{2}$	$0$
$\mathrm{𝟐𝟒𝟑𝟎}$	$(\mathrm{𝟏},\mathrm{𝟏})$ 1	$\mathrm{𝟏}$ 1	$0$	$0$	$\frac{13\sqrt{5/3}}{12}$	$1$	$\frac{101}{36}$	$\frac{25}{12}$	$0$	$0$	$\frac{13\sqrt{5/3}}{12}$
	$(\mathrm{𝟏𝟖𝟗},\mathrm{𝟏})$ 2	$\mathrm{𝟏}$ 1	-$\frac{17\sqrt{3/2}}{2}$	-$\frac{7\sqrt{5/2}}{2}$	$\frac{\sqrt{15}}{4}$	-$3$	$0$	$0$	-$\sqrt{3/2}$	$\sqrt{5/2}$	$0$
		$\mathrm{𝟖}$ 2	$0$	-$2\sqrt{2/5}$	-$\frac{3\sqrt{3/5}}{4}$	-$1$	$0$	$0$	$\frac{\sqrt{3/2}}{2}$	-$\frac{\sqrt{5/2}}{2}$	$0$
	$(\mathrm{𝟒𝟎𝟓},\mathrm{𝟏})$ 3	$\mathrm{𝟏}$ 1	$\frac{7\sqrt{3/2}}{5}$	-$\frac{7}{\sqrt{10}}$	-$\frac{3\sqrt{3/5}}{2}$	$\frac{33}{5}$	$\frac{9}{10}$	$\frac{3}{2}$	$\frac{7\sqrt{3/2}}{5}$	-$\frac{7}{\sqrt{10}}$	-$\frac{3\sqrt{3/5}}{2}$
		$\mathrm{𝟖}$ 2	-$\frac{13\sqrt{3/2}}{10}$	$\frac{13}{2\sqrt{10}}$	$3\sqrt{3/5}$	$\frac{19}{5}$	-$\frac{9}{5}$	-$3$	-$\frac{13\sqrt{3/2}}{10}$	$\frac{13}{2\sqrt{10}}$	$3\sqrt{3/5}$
		$\mathrm{𝟐𝟕}$ 3	-$\frac{9\sqrt{3/2}}{5}$	$\frac{9}{\sqrt{10}}$	-$\frac{9\sqrt{3/5}}{2}$	$\frac{9}{5}$	$\frac{27}{10}$	$\frac{9}{2}$	-$\frac{9\sqrt{3/2}}{5}$	$\frac{9}{\sqrt{10}}$	-$\frac{9\sqrt{3/5}}{2}$

Table 5. Table 5: The diagonal part, h ​ j ℎ 𝑗 hj ( j = 1 , 2 , … , 15 ) 𝑗 1 2 … 15 (j=1,2,...,15) in Eq. ( 6 ) of the SM singlets Φ s , z r superscript subscript Φ 𝑠 𝑧 𝑟 \Phi_{s,z}^{r} in each of the irreps r 𝑟 r according to their transformation properties under the S ​ U ​ ( 3 ) R ⊂ S ​ U ​ ( 6 ) × S ​ U ​ ( 2 ) L ⊂ E 6 𝑆 𝑈 subscript 3 𝑅 𝑆 𝑈 6 𝑆 𝑈 subscript 2 𝐿 subscript 𝐸 6 SU(3)_{R}\subset SU(6)\times SU(2)_{L}\subset E_{6} subgroup.

$E_6$	SU(6)$\times$	$SU{\left(3\right)}_R$ z	$h{1}_{s,z}^r$	$h{2}_{s,z}^r$	$h{3}_{s,z}^r$	$h{4}_{s,z}^r$	$h{5}_{s,z}^r$	$h{6}_{s,z}^r$	$h{7}_{s,z}^r$	$h{8}_{s,z}^r$	$h{9}_{s,z}^r$	$h{10}_{s,z}^r$	$h{11}_{s,z}^r$	$h{12}_{s,z}^r$	$h{13}_{s,z}^r$	$h{14}_{s,z}^r$	$h{15}_{s,z}^r$	$N_{s,z}^r$
	SU(2)L s
$\mathrm{𝟏}$	$\mathrm{𝟏}$ 1	$\mathrm{𝟏}$ 1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	$\frac{1}{\sqrt{78}}$
$\mathrm{𝟔𝟓𝟎}$	$(\mathrm{𝟏},\mathrm{𝟏})$ 1	$\mathrm{𝟏}$ 1	1	-$5$	1	-$\frac{1}{2}$	1	-$\frac{1}{2}$	1	1	1	-$\frac{1}{2}$	1	1	1	-$\frac{1}{2}$	1	$\frac{1}{2\sqrt{30}}$
	$(\mathrm{𝟑𝟓},\mathrm{𝟏})$ 2	$\mathrm{𝟏}$ 1	1	0	-$\frac{4}{5}$	0	-$\frac{7}{10}$	0	0	-$1$	-$\frac{1}{2}$	0	0	-$1$	0	0	-$1$	$\frac{1}{4}$
	$(\mathrm{𝟏𝟖𝟗},\mathrm{𝟏})$ 3	$\mathrm{𝟏}$ 1	1	0	$\frac{72}{75}$	0	-$\frac{169}{50}$	0	-$\frac{2}{3}$	1	-$\frac{3}{2}$	0	-$\frac{2}{3}$	1	-$\frac{2}{3}$	0	1	$\frac{\sqrt{5/74}}{2}$
		$\mathrm{𝟖}$ 2	0	0	0	0	0	0	$\frac{5}{3}$	-$\frac{5}{2}$	0	0	$\frac{5}{24}$	$\frac{5}{4}$	-$\frac{15}{8}$	0	$\frac{5}{4}$	$\frac{4\sqrt{3/149}}{5}$
$\mathrm{𝟐𝟒𝟑𝟎}$	$(\mathrm{𝟏},\mathrm{𝟏})$ 1	$\mathrm{𝟏}$ 1	1	$\frac{35}{9}$	1	-$\frac{7}{6}$	1	-$\frac{7}{6}$	1	1	1	-$\frac{7}{6}$	1	1	1	-$\frac{7}{6}$	1	$\frac{3\sqrt{3/910}}{2}$
	$(\mathrm{𝟏𝟖𝟗},\mathrm{𝟏})$ 2	$\mathrm{𝟏}$ 1	1	0	-$6$	0	$\frac{11}{2}$	0	-$\frac{2}{3}$	1	-$\frac{3}{2}$	0	-$\frac{2}{3}$	1	-$\frac{2}{3}$	0	1	$\sqrt{3/370}$
		$\mathrm{𝟖}$ 2	0	0	0	0	0	0	-$\frac{5}{12}$	$\frac{5}{4}$	0	0	-$\frac{5}{24}$	-$\frac{5}{8}$	$\frac{5}{8}$	0	-$\frac{5}{8}$	$\frac{\sqrt{6/7}}{5}$
	$(\mathrm{𝟒𝟎𝟓},\mathrm{𝟏})$ 3	$\mathrm{𝟏}$ 1	1	0	$\frac{12}{5}$	0	$\frac{31}{10}$	0	-$\frac{4}{3}$	1	$\frac{9}{2}$	0	-$\frac{4}{3}$	1	-$\frac{4}{3}$	0	1	$\frac{1}{4\sqrt{7}}$
		$\mathrm{𝟖}$ 2	0	0	-$2$	0	2	0	-$\frac{5}{12}$	$\frac{5}{4}$	0	0	$\frac{35}{24}$	-$\frac{5}{8}$	-$\frac{25}{24}$	0	-$\frac{5}{8}$	$\frac{\sqrt{2/7}}{5}$
		$\mathrm{𝟐𝟕}$ 3	0	0	$\frac{3}{19}$	0	$\frac{23}{456}$	0	$\frac{10}{57}$	$\frac{25}{228}$	$\frac{135}{152}$	0	-$\frac{5}{57}$	-$\frac{25}{76}$	-$\frac{5}{57}$	0	-$\frac{25}{76}$	$\frac{19\sqrt{6}}{65}$

Table 6. Table 6: The non-diagonal part, h ​ i ​ j ℎ 𝑖 𝑗 hij in Eq. ( 6 ) of the SM singlets Φ s , z r superscript subscript Φ 𝑠 𝑧 𝑟 \Phi_{s,z}^{r} in each of the irreps r 𝑟 r according to their transformation properties under the S ​ U ​ ( 3 ) R ⊂ S ​ U ​ ( 6 ) × S ​ U ​ ( 2 ) L ⊂ E 6 𝑆 𝑈 subscript 3 𝑅 𝑆 𝑈 6 𝑆 𝑈 subscript 2 𝐿 subscript 𝐸 6 SU(3)_{R}\subset SU(6)\times SU(2)_{L}\subset E_{6} subgroup. The entries h ​ 3 s , z ′ ⁣ r ℎ superscript subscript 3 𝑠 𝑧 ′ 𝑟 h3_{s,z}^{\prime r} , h ​ 5 s , z ′ ⁣ r ℎ superscript subscript 5 𝑠 𝑧 ′ 𝑟 h5_{s,z}^{\prime r} , h ​ 9 s , z ′ ⁣ r ℎ superscript subscript 9 𝑠 𝑧 ′ 𝑟 h9_{s,z}^{\prime r} for the flipped embedding are also listed here.

$E_6$	SU(6)$\times$	SU(3)R z	$h{35}_{s,z}^r$	$h{39}_{s,z}^r$	$h{59}_{s,z}^r$	$h{3}_{s,z}^{\prime r}$	$h{5}_{s,z}^{\prime r}$	$h{9}_{s,z}^{\prime r}$	$h{35}_{s,z}^{\prime r}$	$h{39}_{s,z}^{\prime r}$	$h{59}_{s,z}^{\prime r}$
	SU(2)L s
$\mathrm{𝟏}$	$\mathrm{𝟏}$ 1	$\mathrm{𝟏}$ 1	0	0	0	0	0	0	0	0	0
$\mathrm{𝟔𝟓𝟎}$	$(\mathrm{𝟏},\mathrm{𝟏})$ 1	$\mathrm{𝟏}$ 1	0	0	0	1	1	1	0	0	0
	$(\mathrm{𝟑𝟓},\mathrm{𝟏})$ 2	$\mathrm{𝟏}$ 1	-$\frac{\sqrt{3/2}}{5}$	$\frac{1}{\sqrt{10}}$	-$\frac{\sqrt{3/5}}{2}$	-$\frac{4}{5}$	-$\frac{7}{10}$	-$\frac{1}{2}$	-$\frac{\sqrt{3/2}}{5}$	$\frac{1}{\sqrt{10}}$	-$\frac{\sqrt{3/5}}{2}$
	$(\mathrm{𝟏𝟖𝟗},\mathrm{𝟏})$ 3	$\mathrm{𝟏}$ 1	$\frac{13\sqrt{3/2}}{25}$	-$\frac{37}{5\sqrt{10}}$	$\frac{17\sqrt{3/5}}{10}$	0	$\frac{19}{10}$	-$\frac{39}{10}$	$\frac{11\sqrt{3/2}}{5}$	-$\frac{7}{5\sqrt{10}}$	$\frac{17\sqrt{3/5}}{10}$
		$\mathrm{𝟖}$ 2	-$\frac{25}{4\sqrt{6}}$	$\frac{\sqrt{5/2}}{4}$	$\frac{\sqrt{15}}{2}$	1	-$\frac{1}{16}$	-$\frac{15}{16}$	$\frac{1}{8\sqrt{6}}$	-$\frac{11\sqrt{5/2}}{8}$	-$\frac{9\sqrt{15}}{16}$
$\mathrm{𝟐𝟒𝟑𝟎}$	$(\mathrm{𝟏},\mathrm{𝟏})$ 1	$\mathrm{𝟏}$ 1	0	0	0	1	1	1	0	0	0
	$(\mathrm{𝟏𝟖𝟗},\mathrm{𝟏})$ 2	$\mathrm{𝟏}$ 1	$\sqrt{6}$	0	$\frac{5\sqrt{5/3}}{2}$	0	-$\frac{11}{2}$	$\frac{7}{2}$	-$\frac{3\sqrt{3/2}}{2}$	-$\frac{5\sqrt{5/2}}{2}$	$\frac{5\sqrt{5/3}}{2}$
		$\mathrm{𝟖}$ 2	-$\frac{25}{4\sqrt{6}}$	$\frac{\sqrt{5/2}}{4}$	$\frac{\sqrt{15}}{2}$	1	-$\frac{1}{16}$	-$\frac{15}{16}$	$\frac{1}{8\sqrt{6}}$	-$\frac{11\sqrt{5/2}}{8}$	-$\frac{9\sqrt{15}}{16}$
	$(\mathrm{𝟒𝟎𝟓},\mathrm{𝟏})$ 3	$\mathrm{𝟏}$ 1	-$\frac{7\sqrt{3/2}}{5}$	$\frac{7}{\sqrt{10}}$	-$\frac{7\sqrt{3/5}}{2}$	$\frac{12}{5}$	$\frac{31}{10}$	$\frac{9}{2}$	-$\frac{7\sqrt{3/2}}{5}$	$\frac{7}{\sqrt{10}}$	-$\frac{7\sqrt{3/5}}{2}$
		$\mathrm{𝟖}$ 2	-$\frac{7\sqrt{3/2}}{4}$	-$\frac{11\sqrt{5/2}}{4}$	-$\frac{\sqrt{15}}{2}$	1	-$\frac{61}{16}$	$\frac{45}{16}$	$\frac{27\sqrt{3/2}}{8}$	$\frac{5\sqrt{5/2}}{8}$	-$\frac{5\sqrt{15}}{16}$
		$\mathrm{𝟐𝟕}$ 3	$\frac{49}{76\sqrt{6}}$	-$\frac{11\sqrt{5/2}}{76}$	$\frac{11\sqrt{15}}{152}$	1	$\frac{59}{456}$	-$\frac{5}{152}$	-$\frac{\sqrt{3/2}}{76}$	$\frac{\sqrt{5/2}}{76}$	$\frac{37\sqrt{5/3}}{152}$

Table 7. Table 7: The diagonal part, h ​ j ℎ 𝑗 hj ( j = 1 , 2 , … , 15 ) 𝑗 1 2 … 15 (j=1,2,...,15) in Eq. ( 6 ) of the SM singlets Φ s , z r superscript subscript Φ 𝑠 𝑧 𝑟 \Phi_{s,z}^{r} in each of the irreps r 𝑟 r according to their transformation properties under the S ​ U ​ ( 3 ) L ⊂ S ​ U ​ ( 6 ) × S ​ U ​ ( 2 ) R ⊂ E 6 𝑆 𝑈 subscript 3 𝐿 𝑆 𝑈 6 𝑆 𝑈 subscript 2 𝑅 subscript 𝐸 6 SU(3)_{L}\subset SU(6)\times SU(2)_{R}\subset E_{6} subgroup.

$E_6$	SU(6)$\times$	$SU{\left(3\right)}_L$ z	$h{1}_{s,z}^r$	$h{2}_{s,z}^r$	$h{3}_{s,z}^r$	$h{4}_{s,z}^r$	$h{5}_{s,z}^r$	$h{6}_{s,z}^r$	$h{7}_{s,z}^r$	$h{8}_{s,z}^r$	$h{9}_{s,z}^r$	$h{10}_{s,z}^r$	$h{11}_{s,z}^r$	$h{12}_{s,z}^r$	$h{13}_{s,z}^r$	$h{14}_{s,z}^r$	$h{15}_{s,z}^r$	$N_{s,z}^r$
	SU(2)R s
$\mathrm{𝟏}$	$\mathrm{𝟏}$ 1	$\mathrm{𝟏}$ 1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	$1/\sqrt{78}$
$\mathrm{𝟔𝟓𝟎}$	$(\mathrm{𝟏},\mathrm{𝟏})$ 1	$\mathrm{𝟏}$ 1	1	1	1	1	-$\frac{11}{4}$	-$\frac{1}{2}$	-$\frac{1}{2}$	-$\frac{1}{2}$	-$\frac{5}{4}$	-$\frac{1}{2}$	-$\frac{1}{2}$	-$\frac{1}{2}$	1	1	-$5$	$\frac{1}{2\sqrt{30}}$
	$(\mathrm{𝟑𝟓},\mathrm{𝟏})$ 2	$\mathrm{𝟏}$ 1	1	-$1$	-$\frac{1}{5}$	0	-$\frac{3}{10}$	0	0	0	-$\frac{1}{2}$	0	0	0	0	-$1$	0	$\frac{1}{4}$
		$\mathrm{𝟖}$ 2	0	1	$\frac{3}{5}$	$\frac{1}{2}$	-$\frac{3}{5}$	0	0	0	-$1$	0	0	0	-$1$	-$\frac{1}{2}$	0	$\frac{1}{4}$
	$(\mathrm{𝟑𝟓},\mathrm{𝟑})$ 3	$\mathrm{𝟏}$ 1	0	0	0	0	-$\frac{15}{16}$	0	0	-$\frac{5}{4}$	$\frac{15}{16}$	$\frac{5}{24}$	0	0	0	0	0	$\frac{4\sqrt{3/11}}{5}$
		$\mathrm{𝟖}$ 2	0	0	0	0	-$\frac{15}{16}$	$\frac{5}{6}$	0	0	$\frac{15}{16}$	-$\frac{5}{6}$	0	0	0	0	0	$\frac{2}{5\sqrt{3}}$
	$(\mathrm{𝟏𝟖𝟗},\mathrm{𝟏})$ 4	$\mathrm{𝟏}$ 1	1	1	-$3$	-$\frac{2}{3}$	-$1$	0	0	0	1	0	0	0	-$\frac{2}{3}$	1	0	$\frac{\sqrt{3/46}}{2}$
		$\mathrm{𝟖}$ 2	0	1	-$1$	-$\frac{1}{6}$	0	0	0	0	0	0	0	0	$\frac{1}{3}$	-$\frac{1}{2}$	0	$\frac{1}{2\sqrt{2}}$
$\mathrm{𝟐𝟒𝟑𝟎}$	$(\mathrm{𝟏},\mathrm{𝟏})$ 1	$\mathrm{𝟏}$ 1	1	1	1	1	$\frac{101}{36}$	-$\frac{7}{6}$	-$\frac{7}{6}$	-$\frac{7}{6}$	$\frac{25}{12}$	-$\frac{7}{6}$	-$\frac{7}{6}$	-$\frac{7}{6}$	1	1	$\frac{35}{9}$	$\frac{3\sqrt{3/910}}{2}$
	$(\mathrm{𝟏},\mathrm{𝟓})$ 2	$\mathrm{𝟏}$ 1	0	0	0	0	$\frac{25}{24}$	0	0	0	$\frac{5}{8}$	0	0	0	0	0	-$\frac{5}{6}$	$\frac{\sqrt{6}}{5}$
	$(\mathrm{𝟑𝟓},\mathrm{𝟑})$ 3	$\mathrm{𝟏}$ 1	0	0	0	0	-$\frac{15}{16}$	0	0	$\frac{25}{24}$	$\frac{15}{16}$	$\frac{5}{24}$	0	-$\frac{55}{24}$	0	0	0	$\frac{3\sqrt{2/11}}{5}$
		$\mathrm{𝟖}$ 2	0	0	0	0	-$\frac{15}{16}$	-$\frac{5}{24}$	0	$\frac{5}{6}$	$\frac{15}{16}$	0	-$\frac{5}{36}$	$\frac{5}{6}$	0	0	0	$\frac{12\sqrt{3/95}}{5}$
	$(\mathrm{𝟏𝟖𝟗},\mathrm{𝟏})$ 4	$\mathrm{𝟏}$1	1	1	-$3$	-$\frac{2}{3}$	$\frac{15}{8}$	0	0	0	-$\frac{15}{8}$	0	0	0	-$\frac{2}{3}$	1	0	$\frac{1}{\sqrt{115}}$
	$(\mathrm{𝟒𝟎𝟓},\mathrm{𝟏})$ 5	$\mathrm{𝟏}$ 1	1	1	$\frac{33}{5}$	-$\frac{4}{3}$	$\frac{9}{10}$	0	0	0	$\frac{3}{2}$	0	0	0	-$\frac{4}{3}$	1	0	$\frac{1}{4\sqrt{7}}$
		$\mathrm{𝟖}$ 2	0	1	$\frac{19}{5}$	-$\frac{5}{6}$	-$\frac{9}{5}$	0	0	0	-$3$	0	0	0	$\frac{5}{3}$	-$\frac{1}{2}$	0	$\frac{1}{4\sqrt{5}}$
		$\mathrm{𝟐𝟕}$ 3	0	1	$\frac{9}{5}$	0	$\frac{27}{10}$	0	0	0	$\frac{9}{2}$	0	0	0	0	-$3$	0	$\frac{1}{2\sqrt{30}}$

Table 8. Table 8: The non-diagonal part, h ​ i ​ j ℎ 𝑖 𝑗 hij in Eq. ( 6 ) of the SM singlets Φ s , z r superscript subscript Φ 𝑠 𝑧 𝑟 \Phi_{s,z}^{r} in each of the irreps r 𝑟 r according to their transformation properties under the S ​ U ​ ( 3 ) L ⊂ S ​ U ​ ( 6 ) × S ​ U ​ ( 2 ) R ⊂ E 6 𝑆 𝑈 subscript 3 𝐿 𝑆 𝑈 6 𝑆 𝑈 subscript 2 𝑅 subscript 𝐸 6 SU(3)_{L}\subset SU(6)\times SU(2)_{R}\subset E_{6} subgroup. The entries h ​ 3 s , z ′ ⁣ r ℎ superscript subscript 3 𝑠 𝑧 ′ 𝑟 h3_{s,z}^{\prime r} , h ​ 5 s , z ′ ⁣ r ℎ superscript subscript 5 𝑠 𝑧 ′ 𝑟 h5_{s,z}^{\prime r} , h ​ 9 s , z ′ ⁣ r ℎ superscript subscript 9 𝑠 𝑧 ′ 𝑟 h9_{s,z}^{\prime r} for the flipped embedding are also listed here.

$E_6$	SU(6)$\times$	SU(3)L z	$h{35}_{s,z}^r$	$h{39}_{s,z}^r$	$h{59}_{s,z}^r$	$h{3}_{s,z}^{\prime r}$	$h{5}_{s,z}^{\prime r}$	$h{9}_{s,z}^{\prime r}$	$h{35}_{s,z}^{\prime r}$	$h{39}_{s,z}^{\prime r}$	$h{59}_{s,z}^{\prime r}$
	SU(2)R s
$\mathrm{𝟏}$	$\mathrm{𝟏}$ 1	$\mathrm{𝟏}$ 1	0	0	0	0	0	0	0	0	0
$\mathrm{𝟔𝟓𝟎}$	$(\mathrm{𝟏},\mathrm{𝟏})$ 1	$\mathrm{𝟏}$ 1	0	0	-$\frac{3\sqrt{15}}{4}$	-$\frac{13}{5}$	$\frac{17}{20}$	-$\frac{5}{4}$	$\frac{3\sqrt{3/2}}{5}$	$\frac{9}{\sqrt{10}}$	-$\frac{3\sqrt{3/5}}{4}$
	$(\mathrm{𝟑𝟓},\mathrm{𝟏})$ 2	$\mathrm{𝟏}$ 1	$\frac{\sqrt{3/2}}{5}$	-$\frac{1}{\sqrt{10}}$	$\frac{\sqrt{3/5}}{2}$	-$\frac{1}{5}$	-$\frac{3}{10}$	-$\frac{1}{2}$	$\frac{\sqrt{3/2}}{5}$	-$\frac{1}{\sqrt{10}}$	$\frac{\sqrt{3/5}}{2}$
		$\mathrm{𝟖}$ 2	-$\frac{\sqrt{3/2}}{10}$	$\frac{1}{2\sqrt{10}}$	$\sqrt{3/5}$	-$\frac{3}{5}$	$\frac{3}{5}$	-$1$	$\frac{\sqrt{3/2}}{10}$	-$\frac{\sqrt{5/2}}{2}$	0
	$(\mathrm{𝟑𝟓},\mathrm{𝟑})$ 3	$\mathrm{𝟏}$ 1	-$\frac{5}{8\sqrt{6}}$	-$\frac{\sqrt{5/2}}{8}$	$\frac{\sqrt{15}}{16}$	1	-$\frac{1}{16}$	-$\frac{15}{16}$	$\frac{1}{8\sqrt{6}}$	$\frac{\sqrt{5/2}}{8}$	-$\frac{\sqrt{15}}{16}$
		$\mathrm{𝟖}$ 2	-$\frac{5}{8\sqrt{6}}$	-$\frac{\sqrt{5/2}}{8}$	$\frac{\sqrt{15}}{16}$	1	-$\frac{1}{16}$	-$\frac{15}{16}$	$\frac{1}{8\sqrt{6}}$	$\frac{\sqrt{5/2}}{8}$	-$\frac{\sqrt{15}}{16}$
	$(\mathrm{𝟏𝟖𝟗},\mathrm{𝟏})$ 4	$\mathrm{𝟏}$ 1	$\sqrt{3/2}$	$\frac{11}{\sqrt{10}}$	$\frac{1}{\sqrt{15}}$	-$\frac{3}{5}$	-$\frac{7}{5}$	-$1$	-$\frac{17\sqrt{3/2}}{5}$	$\frac{1}{\sqrt{10}}$	$\frac{1}{\sqrt{15}}$
		$\mathrm{𝟖}$ 2	$\frac{\sqrt{3/2}}{2}$	-$\frac{\sqrt{5/2}}{2}$	0	$\frac{1}{5}$	-$\frac{6}{5}$	0	$\frac{3\sqrt{3/2}}{10}$	$\frac{1}{2\sqrt{10}}$	$\sqrt{3/5}$
$\mathrm{𝟐𝟒𝟑𝟎}$	$(\mathrm{𝟏},\mathrm{𝟏})$ 1	$\mathrm{𝟏}$ 1	0	0	$\frac{13\sqrt{5/3}}{12}$	$\frac{41}{15}$	$\frac{193}{180}$	$\frac{25}{12}$	-$\frac{13}{15\sqrt{6}}$	-$\frac{13}{3\sqrt{10}}$	$\frac{13}{12\sqrt{15}}$
	$(\mathrm{𝟏},\mathrm{𝟓})$ 2	$\mathrm{𝟏}$ 1	0	0	$\frac{5\sqrt{5/3}}{8}$	1	$\frac{1}{24}$	$\frac{5}{8}$	-$\frac{1}{2\sqrt{6}}$	-$\frac{\sqrt{5/2}}{2}$	$\frac{\sqrt{5/3}}{8}$
	$(\mathrm{𝟑𝟓},\mathrm{𝟑})$ 3	$\mathrm{𝟏}$ 1	-$\frac{5}{8\sqrt{6}}$	-$\frac{\sqrt{5/2}}{8}$	$\frac{\sqrt{15}}{16}$	1	-$\frac{1}{16}$	-$\frac{15}{16}$	$\frac{1}{8\sqrt{6}}$	$\frac{\sqrt{5/2}}{8}$	-$\frac{\sqrt{15}}{16}$
		$\mathrm{𝟖}$ 2	-$\frac{5}{8\sqrt{6}}$	-$\frac{\sqrt{5/2}}{8}$	$\frac{\sqrt{15}}{16}$	1	-$\frac{1}{16}$	-$\frac{15}{16}$	$\frac{1}{8\sqrt{6}}$	$\frac{\sqrt{5/2}}{8}$	-$\frac{\sqrt{15}}{16}$
	$(\mathrm{𝟏𝟖𝟗},\mathrm{𝟏})$ 4	$\mathrm{𝟏}$ 1	-$\frac{19\sqrt{3/2}}{4}$	-$\frac{5\sqrt{5/2}}{4}$	-$\frac{\sqrt{15}}{8}$	-$\frac{3}{5}$	-$\frac{171}{40}$	$\frac{15}{8}$	$\frac{47\sqrt{3/2}}{20}$	-$\frac{19}{4\sqrt{10}}$	-$\frac{51\sqrt{3/5}}{8}$
	$(\mathrm{𝟒𝟎𝟓},\mathrm{𝟏})$ 5	$\mathrm{𝟏}$ 1	$\frac{7\sqrt{3/2}}{5}$	-$\frac{7}{\sqrt{10}}$	-$\frac{3\sqrt{3/5}}{2}$	$\frac{9}{5}$	$\frac{57}{10}$	$\frac{3}{2}$	$\frac{11\sqrt{3/2}}{5}$	$\sqrt{5/2}$	$\frac{\sqrt{15}}{2}$
		$\mathrm{𝟖}$ 2	-$\frac{13\sqrt{3/2}}{10}$	$\frac{13}{2\sqrt{10}}$	$3\sqrt{3/5}$	-$\frac{11}{5}$	$\frac{21}{5}$	-$3$	-$\frac{3\sqrt{3/2}}{10}$	-$\frac{17}{2\sqrt{10}}$	-$2\sqrt{3/5}$
		$\mathrm{𝟐𝟕}$ 3	-$\frac{9\sqrt{3/2}}{5}$	$\frac{9}{\sqrt{10}}$	-$\frac{9\sqrt{3/5}}{2}$	$\frac{9}{5}$	$\frac{27}{10}$	$\frac{9}{2}$	-$\frac{9\sqrt{3/2}}{5}$	$\frac{9}{\sqrt{10}}$	-$\frac{9\sqrt{3/5}}{2}$

Table 9. Table 9: The diagonal part, h ​ j ℎ 𝑗 hj ( j = 1 , 2 , … , 15 ) 𝑗 1 2 … 15 (j=1,2,...,15) in Eq. ( 6 ) of the SM singlets Φ s , z r superscript subscript Φ 𝑠 𝑧 𝑟 \Phi_{s,z}^{r} in each of the irreps r 𝑟 r according to their transformation properties under the S ​ U ​ ( 3 ) L × S ​ U ​ ( 3 ) R ⊂ E 6 𝑆 𝑈 subscript 3 𝐿 𝑆 𝑈 subscript 3 𝑅 subscript 𝐸 6 SU(3)_{L}\times SU(3)_{R}\subset E_{6} subgroup.

$E_6$	SU(3)${}_L{}^{}\times$	$h{1}_{s,z}^r$	$h{2}_{s,z}^r$	$h{3}_{s,z}^r$	$h{4}_{s,z}^r$	$h{5}_{s,z}^r$	$h{6}_{s,z}^r$	$h{7}_{s,z}^r$	$h{8}_{s,z}^r$	$h{9}_{s,z}^r$	$h{10}_{s,z}^r$	$h{11}_{s,z}^r$	$h{12}_{s,z}^r$	$h{13}_{s,z}^r$	$h{14}_{s,z}^r$	$h{15}_{s,z}^r$	$N_{s,z}^r$
	SU(3)R
$\mathrm{𝟏}$	$\mathrm{𝟏}$ 1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	$\frac{1}{\sqrt{78}}$
$\mathrm{𝟔𝟓𝟎}$	${(\mathrm{𝟏},\mathrm{𝟏})}_1$	0	1	-$\frac{3}{5}$	0	-$\frac{2}{5}$	0	0	-$1$	0	0	0	-$1$	0	1	-$1$	$\frac{1}{\sqrt{19}}$
	${(\mathrm{𝟏},\mathrm{𝟏})}_2$	1	0	-$\frac{4}{5}$	0	-$\frac{7}{10}$	0	0	-$1$	-$\frac{1}{2}$	0	0	-$1$	0	0	-$1$	$\frac{\sqrt{3/7}}{4}$
	$(\mathrm{𝟏},\mathrm{𝟖})$	0	0	1	0	$\frac{7}{8}$	0	0	0	$\frac{5}{8}$	0	0	-$\frac{5}{4}$	0	0	0	$\frac{2}{5}$
	$(\mathrm{𝟖},\mathrm{𝟏})$	0	1	-$\frac{1}{5}$	0	-$\frac{3}{10}$	0	0	0	-$\frac{1}{2}$	0	0	0	0	-$\frac{1}{2}$	0	$\frac{1}{\sqrt{5}}$
	$(\mathrm{𝟖},\mathrm{𝟖})$	0	0	-$\frac{1}{2}$	-$\frac{25}{54}$	-$\frac{3}{4}$	0	0	0	$\frac{5}{4}$	0	$\frac{25}{27}$	0	0	0	0	$\frac{18\sqrt{2/281}}{5}$
$\mathrm{𝟐𝟒𝟑𝟎}$	$(\mathrm{𝟏},\mathrm{𝟏})$	1	1	1	-$\frac{4}{9}$	1	-$\frac{4}{9}$	-$\frac{4}{9}$	1	1	-$\frac{4}{9}$	-$\frac{4}{9}$	1	-$\frac{4}{9}$	1	1	$\frac{\sqrt{3/26}}{2}$
	$(\mathrm{𝟏},\mathrm{𝟖})$	0	0	1	0	$\frac{7}{8}$	0	0	-$\frac{5}{2}$	$\frac{5}{8}$	0	0	$\frac{5}{4}$	0	0	0	$\frac{2}{5\sqrt{3}}$
	$(\mathrm{𝟖},\mathrm{𝟏})$	0	1	-$2$	0	$\frac{31}{4}$	0	0	0	$\frac{13}{4}$	0	0	0	0	-$3$	0	$\frac{\sqrt{2}}{19}$
	$(\mathrm{𝟖},\mathrm{𝟖})$	0	0	-$\frac{1}{2}$	$\frac{7}{16}$	-$\frac{3}{4}$	-$\frac{1}{8}$	-$\frac{1}{8}$	0	$\frac{5}{4}$	-$\frac{1}{8}$	-$\frac{1}{8}$	0	-$1⁤\frac{1}{8}$	0	0	$\frac{8}{\sqrt{389}}$
	$(\mathrm{𝟏},\mathrm{𝟐𝟕})$	0	0	1	0	$\frac{7}{8}$	0	0	$\frac{5}{4}$	$\frac{5}{8}$	0	0	$\frac{5}{4}$	0	0	-$\frac{15}{4}$	$\frac{\sqrt{2/3}}{5}$
	$(\mathrm{𝟐𝟕},\mathrm{𝟏})$	0	1	$\frac{69}{5}$	0	-$\frac{93}{10}$	0	0	0	$\frac{9}{2}$	0	0	0	0	-$3$	0	$\frac{1}{2\sqrt{105}}$

Table 10. Table 10: The non-diagonal part, h ​ i ​ j ℎ 𝑖 𝑗 hij in Eq. ( 6 ) of the SM singlets Φ s , z r superscript subscript Φ 𝑠 𝑧 𝑟 \Phi_{s,z}^{r} in each of the irreps r 𝑟 r according to their transformation properties under the S ​ U ​ ( 3 ) L × S ​ U ​ ( 3 ) R ⊂ E 6 𝑆 𝑈 subscript 3 𝐿 𝑆 𝑈 subscript 3 𝑅 subscript 𝐸 6 SU(3)_{L}\times SU(3)_{R}\subset E_{6} subgroup. The entries h ​ 3 s , z ′ ⁣ r ℎ superscript subscript 3 𝑠 𝑧 ′ 𝑟 h3_{s,z}^{\prime r} , h ​ 5 s , z ′ ⁣ r ℎ superscript subscript 5 𝑠 𝑧 ′ 𝑟 h5_{s,z}^{\prime r} , h ​ 9 s , z ′ ⁣ r ℎ superscript subscript 9 𝑠 𝑧 ′ 𝑟 h9_{s,z}^{\prime r} for the flipped embedding are also listed here.

$E_6$	SU(3)${}_L{}^{}\times$	$h{35}_{s,z}^r$	$h{39}_{s,z}^r$	$h{59}_{s,z}^r$	$h{3}_{s,z}^{\prime r}$	$h{5}_{s,z}^{\prime r}$	$h{9}_{s,z}^{\prime r}$	$h{35}_{s,z}^{\prime r}$	$h{39}_{s,z}^{\prime r}$	$h{59}_{s,z}^{\prime r}$
	SU(3)R
$\mathrm{𝟏}$	$\mathrm{𝟏}$ 1	0	0	0	0	0	0	0	0	0
$\mathrm{𝟔𝟓𝟎}$	${(\mathrm{𝟏},\mathrm{𝟏})}_1$	-$\sqrt{6}/5$	$\sqrt{5/2}$	0	-$3/5$	-$1/40$	-$3/8$	-$23\sqrt{3/2}/20$	-$1/\left(4\sqrt{10}\right)$	-$9\sqrt{3/5}/8$
	${(\mathrm{𝟏},\mathrm{𝟏})}_2$	-$\sqrt{3/2}/5$	-$7/\sqrt{10}$	-$19/\left(2\sqrt{15}\right)$	-$4/5$	-$17/10$	$1/2$	$9\sqrt{3/2}/5$	$7/\sqrt{10}$	-$1/\left(2\sqrt{15}\right)$
	$(\mathrm{𝟏},\mathrm{𝟖})$	$\sqrt{3/2}/4$	-$\sqrt{5/2}/4$	$\sqrt{15}/8$	1	$7/8$	$5/8$	$\sqrt{3/2}/4$	-$\sqrt{5/2}/4$	$\sqrt{15}/8$
	$(\mathrm{𝟖},\mathrm{𝟏})$	$\sqrt{3/2}/5$	-$1/\sqrt{10}$	$\sqrt{3/5}/2$	-$1/5$	-$3/10$	-$1/2$	$\sqrt{3/2}/5$	-$1/\sqrt{10}$	$\sqrt{3/5}/2$
	$(\mathrm{𝟖},\mathrm{𝟖})$	$\sqrt{3/2}/2$	0	0	1	-$3/8$	-$5/8$	-$3\sqrt{3/2}/8$	$3\sqrt{5/2}/8$	$\sqrt{15}/8$
$\mathrm{𝟐𝟒𝟑𝟎}$	$(\mathrm{𝟏},\mathrm{𝟏})$	0	0	0	1	1	1	0	0	0
	$(\mathrm{𝟏},\mathrm{𝟖})$	$\sqrt{3/2}/4$	-$\sqrt{5/2}/4$	$\sqrt{15}/8$	1	$7/8$	$5/8$	$\sqrt{3/2}/4$	-$\sqrt{5/2}/4$	$\sqrt{15}/8$
	$(\mathrm{𝟖},\mathrm{𝟏})$	$\sqrt{3/2}/2$	$23/\left(2\sqrt{10}\right)$	-$23\sqrt{3/5}/4$	-$1/5$	-$17/40$	$77/8$	-$33\sqrt{3/2}/10$	$17/\left(2\sqrt{10}\right)$	-$9\sqrt{3/5}/8$
	$(\mathrm{𝟖},\mathrm{𝟖})$	$\sqrt{3/2}/2$	0	0	1	-$3/8$	-$5/8$	-$3\sqrt{3/2}/8$	$3\sqrt{5/2}/8$	$\sqrt{15}/8$
	$(\mathrm{𝟏},\mathrm{𝟐𝟕})$	$\sqrt{3/2}/4$	-$\sqrt{5/2}/4$	$\sqrt{15}/8$	1	$7/8$	$5/8$	$\sqrt{3/2}/4$	-$\sqrt{5/2}/4$	$\sqrt{15}/8$
	$(\mathrm{𝟐𝟕},\mathrm{𝟏})$	-$19\sqrt{3/2}/5$	$9/\sqrt{10}$	-$9\sqrt{3/5}/2$	$9/5$	$279/20$	-$27/4$	$7\sqrt{3/2}/10$	$33/\left(2\sqrt{10}\right)$	-$33\sqrt{3/5}/4$

Equations

$$\mathcal{L}=\frac{c_k}{M_{Pl}}{G}_{\mu \nu }^a{G}^{b\mu \nu }{H}_k^{ab},$$

$$\mathcal{L}=-\frac{1}{4}{G}_{\mu \nu }^a{G}^{a\mu \nu }+\frac{c_k}{4M_{Pl}}{G}_{\mu \nu }^a{G}^{b\mu \nu }{H}_k^{ab}.$$

$$(78\times 78{)}_{symmetric}$$

$$78$$

$$({\Phi }^{(r)}{)}^{ab}$$

$$C_R=-\sum _{i=1}^{d(G)}{X}_i^2,$$

$$[X_i,X_j]=i{f}_{ijk}{X}_k$$

$$C_{R\times R}=C_R\times 1+1\times C_R+2F,$$

$$F=-\sum _{i=1}^{d(G)}{X}_i\times X_i.$$

$$<H_k^{ab}>\equiv \sum _{s,z}{v}_{s,z}^k{\Phi }_{s,z}^{kab}\equiv v^k{\Phi }^{kab},$$

$$g_1^2(M_{GUT})(1+{ϵ}_1)=g_2^2(M_{GUT})(1+{ϵ}_2)=g_3^2(M_{GUT})(1+{ϵ}_3)=g_G^2/4\pi ,$$

$${ϵ}_i=\sum _k\frac{c_k}{M_{Pl}}\sum _{s,z}{v}_{s,z}^k{\varphi }_i^{ksz},i=1,2,3,{\varphi }_1^{ksz}\equiv -h{3}_{s,z}^k,{\varphi }_2^{ksz}\equiv -h{2}_{s,z}^k,{\varphi }_3^{ksz}\equiv -h{1}_{s,z}^k,$$

$${\overline{m^2}}_{ab}=\sum _i\frac{C(r_i)}{66}{g}_G^2{v}_i^2{\delta }_{ab}\equiv M_{\mathrm{HG}\mathrm{b}}^2{\delta }_{ab}\text{for}a,b=13,\dots ,78,$$

$${\tau }_p=C{M}_X^4/({\alpha }_G^2{m}_p^5),$$

$${\tau }_{p\to e^{+}{\pi }^0}>1.6\times 10^{34}\mathrm{years}.$$

$$M_X>(40{\alpha }_G{)}^{1/2}(1/C{)}^{1/4}4.3\times 10^{15}GeV.$$

$$\mathcal{L}=\frac{c_6}{M_{Pl}^2}{H}_1{H}_2{G}_{\mu \nu }{G}^{\mu \nu }+\dots .$$

$$\mathcal{L}=\sum _{i=1}^3-\frac{1}{4}\left(1+{ϵ}_i+{ϵ}_i^{(6)}+\dots \right)G_{(i)\mu \nu }^a{G}_{(i)}^{a\mu \nu },$$

$${ϵ}_i=\frac{c_k}{M_{Pl}}<H_k>=\sum _k\frac{c_k}{M_{Pl}}\sum _{s,z}{v}_{s,z}^k{\varphi }_i^{ksz},i=1,2,3,$$

$${\varphi }_1^{ksz}\equiv -h{3}_{s,z}^k,{\varphi }_2^{ksz}\equiv -h{2}_{s,z}^k,{\varphi }_3^{ksz}\equiv -h{1}_{s,z}^k,$$

$$27\times \overline{27}=1+78+650.$$

$$\mathcal{L}=\frac{c_6}{M_{Pl}^2}\frac{M_X}{g_G}c{g}_3{H}_3{G}_{\mu \nu }{G}^{\mu \nu }+\dots ,$$

$$\mathcal{L}=\frac{c_k}{M_{Pl}}{G}_{\mu \nu }^a{\tilde{G}}^{b\mu \nu }{\tilde{H}}_k^{ab},$$

$$M_3:M_2:M_1=h{1}_{s,z}^k:h{2}_{s,z}^k:h{3}_{s,z}^{\prime k},$$

$$[H_j,E_{\alpha }]={\alpha }_j^b{E}_{\alpha },$$

$$[E_{{\alpha }^i},E_{{\alpha }^j}]=N({\alpha }^n)E_{{\alpha }^n},$$

$$f[1,4,7]=1/2;f[1,5,6]=-1/2;f[1,13,16]=1/2;f[1,14,15]=-1/2;f[1,19,22]=1/2;f[1,20,21]=-1/2;$$

$$f[1,26,29]=1/2;f[1,27,28]=-1/2;f[1,32,35]=1/2;f[1,33,34]=-1/2;f[1,40,43]=-1/2;$$

$$f[1,41,42]=1/2;f[1,47,50]=1/2;f[1,48,49]=-1/2;f[1,53,56]=-1/2;f[1,54,55]=1/2;$$

$$f[1,61,64]=1/2;f[1,62,63]=-1/2;f[1,69,72]=1/2;f[1,70,71]=-1/2;f[2,4,6]=1/2;$$

$$f[2,5,7]=1/2;f[2,13,15]=1/2;f[2,14,16]=1/2;f[2,19,21]=1/2;f[2,20,22]=1/2;f[2,26,28]=1/2;$$

$$f[2,27,29]=1/2;f[2,32,34]=1/2;f[2,33,35]=1/2;f[2,40,42]=1/2;f[2,41,43]=1/2;f[2,47,49]=1/2;$$

Full text

I Introduction

It is well-known that the problem of quantum gravity has not been solved yet, although superstring theory presents a beautiful promise. Can we examine the effects of quantum gravity nowadays? The answer is positive. The reason is that the scale of unification $M_{G}\sim 2\times 10^{16}$ GeV in the minimal supersymmetric Standard Model, as implicated by the experiment measurements Langacker:1991an ; Giunti:1991ta ; Amaldi:1991cn ; Ellis:1990wk , is smaller than the Planck scale $M_{Pl}$ ($M_{Pl}=(8\pi G_{N})^{-1/2}\sim 2.4\times 10^{18}$ GeV), where quantum gravity should come in, by about two orders of magnitudes so that one can build a field theoretical description of the unification of particle interactions without a full solution to the problem of quantum gravity. To describe the effects of quantum gravity, we could use the effective field theory approach in which non-renormalizable higher dimension operators are introduced. The $d\geq 5$ operators induced by gravity should enter the Lagrangian, which are suppressed by factors of $(M_{Pl})^{-(d-4)}$ with coefficients at the order of $\sim{\cal O}(1)$ . They are only subject to the constraints of the symmetries (gauge invariance, supersymmetry in supersymmetric models, etc.) of the low energy theory.

As it has been shown in Refs. Hill:1983xh ; Shafi:1983gz ; Calmet:2008df ; Hall:1992kq ; Datta:1995as ; Huitu:1999eh ; Dasgupta:1995js ; Tobe:2003yj ; Nath:2006ut ; Rizzo:1984mk ; Marciano:1981un ; Anderson:1996bg ; Amundson:1996nw ; Martin:2009ad ; Balazs:2010ha ; Balazs:2011ie ; Wang:2011eh ; Chakrabortty:2008zk ; Lykken:1993br ; Howl:2007hq ; Ellis:1985jn ; Drees:1985bx ; Calmet:2009hp that the presence of higher dimension operators may have substantial impacts on grand unified theory (GUT) and its phenomenology for gauge groups $SU(5)$ and $SO(10)$. These operators modify the usual gauge coupling unification condition Hill:1983xh ; Shafi:1983gz ; Calmet:2008df ; Hall:1992kq ; Datta:1995as ; Huitu:1999eh ; Chakrabortty:2008zk ; Calmet:2009hp . It is estimated that the effects of dimension-5 operators can be more important than two-loop corrections in the renormalization group analysis of gauge coupling unification Calmet:2008df . With one or more dimension-5 operators, it is possible to achieve unification at scales $M_{X}$ much different than usually expected Calmet:2009hp . Higher dimension operators can lead to an acceptable value of $\sin^{2}\theta_{w}$ Hill:1983xh ; Rizzo:1984mk ; Marciano:1981un ; Chakrabortty:2008zk , affect supersymmetric (SUSY) particle spectrum in SUSY GUT and supergravity Anderson:1996bg ; Amundson:1996nw ; Martin:2009ad ; Balazs:2010ha ; Balazs:2011ie ; Wang:2011eh ; Chakrabortty:2008zk ; Lykken:1993br ; Howl:2007hq ; Ellis:1985jn ; Drees:1985bx and the analysis of proton decay Tobe:2003yj ; Nath:2006ut . Therefore, we should include them in the researches of GUT and SUSY GUT. In particular, in some cases, such studies are dramatically important. For example, the gauge coupling unification in the minimal SU(5) without supersymmetry can not be realized and the minimal SUSY SU(5) has been already excluded by the limit (larger than $10^{34}y$) of the proton lifetime from Super-Kamiokande TheSuper-Kamiokande:2017tit 111SU(5) GUT can be obtained by adding more Higgs representations or discrete symmetries Georgi:1979df ; FileviezPerez:2007nh ; Arbelaez:2015toa ., but they can be realized if effects of $d\geq 5$ operators are included (see, for example, Chakrabortty:2008zk ; Calmet:2009hp ).

The exceptional group $E_{6}$ is an attractive unification group among well-known unification groups. The main reasons are as follows. Firstly, from the viewpoint of superstring theory, the gauge and gravitational anomaly cancellation occurs only for the gauge groups $SO(32)$ or $E_{8}\times E_{8}$ Green:1984sg ; Green:1987sp ; Green:1987mn and compactification on a Calabi-Yau manifold with an SU(3) holonomy results in the breaking $E_{8}\rightarrow SU(3)\times E_{6}$ Candelas:1985en . Secondly, in terms of the phenomenology of low energy effective theories originated from $E_{6}$ GUT, there are several attractive features King:2005jy ; Kawase:2010na ; Athron:2013ipa ; Rosner:2014cha ; Nevzorov:2013tta ; Nevzorov:2015sha ; Rojas:2015tqa ; Nevzorov:2015iya . Moreover, if we assume the dynamical symmetry breaking scenario, we would have several constraints on the possible GUT models. It has been pointed out Barbieri:1980tz ; Barbieri:1981yy ; Ramond:1979py that $E_{6}$ is uniquely selected among many GUT groups, if one demands that 1) the theory is automatically anomaly free, 2) every generation of quark/lepton fields belongs to a single irreducible representation (irrep) of the GUT group, and 3) the Higgs fields, which are necessary for inducing the symmetry breaking down to $SU_{C}(3)\times U_{\rm em}(1)$, fall in the representations that can be provided by the fermion bilinears. Recently some models and their low energy phenomenology originated from $E_{6}$ GUT generated further interests Athron:2016gor ; Joglekar:2016yap ; Nevzorov:2014sha ; Dhuria:2015swa ; Athron:2015vxg ; Harada:2016fpj ; Athron:2016qqb ; Belanger:2017vpq ; Nevzorov:2017rtf .

Effects of higher dimension operators to the unification of gauge couplings have also been investigated in the grand unified gauge group $E_{6}$ Chakrabortty:2008zk ; Huang:2014zba . In the Ref. Chakrabortty:2008zk , $E6\rightarrow SU(3)\times SU(3)\times SU(3)$ has been studied and the corresponding effective contributions to gauge kinetic terms have been given. However, their contributions to gauge kinetic terms in the Standard Model (SM) have been not given, although it is not difficult to derive them from the results shown in Table 4 of the paper. Moreover, the numerical analysis on the unification of gauge couplings and corresponding physical discussions have not been carried out in the paper. To construct $E_{6}$ unification models with effects of dimension-5 operators in Ref. Huang:2014zba , the author considers only the maximal subgroup $H=SO(10)\times U(1)$ and gives the Clebsch-Gordan coefficients $\Phi^{(r)}_{s,z}$ associated with $E_{6}$ breaking to the SM for that case. Nevertheless, same as the Ref. Chakrabortty:2008zk , the numerical analysis on the unification of gauge couplings and corresponding physical discussions have not been carried out.

In this paper, we investigate the unification of gauge couplings for the grand unified gauge group $E_{6}$ through effects of dimension-5 operators. Surveying the branching rules for the GUT group $E_{6}$ Slansky:1981yr , we see that there are several maximal subgroups containing $G_{321}$ (e.g., $H=SO(10)\times U(1)$, $H=SU(3)\times SU(3)\times SU(3)$, $H=SU(2)\times SU(6)$, and $H=F_{4}$) and for a specific maximal subgroup there are usually several breaking chains. We consider all common maximal classical subgroups with $SU_{C}(3)\times U(1)_{\rm em}$ and the usual breaking chains for a specific maximal subgroup. We show that the gauge coupling unification condition is modified due to the effects of dimension-5 operators, which are the lowest higher dimension operators in $E_{6}$ GUT, and the gauge coupling unification at a higher scale can be realized without SUSY. Furthermore, we find that the constraints from the proton lifetime can be safely satisfied even at the higher unification scale $M_{G}$ near the Planck scale $M_{Pl}$.

In section II, we set up the notation to study effects of dimension-5 operators and point out how dimension-5 operators modify the usual gauge coupling unification condition. Section III is devoted to the numerical analysis of renormalization group (RG) evolution of the gauge couplings and the corresponding physical results for these cases. In section IV, we discuss the constraints from the proton lifetime. Summary and conclusions are given in section V. All of the Clebsch-Gordan coefficients $\Phi^{(r)}_{s,z}$ associated with $E_{6}$ breaking to the SM, in different bases $\{s,z\}$, up to a uniform normalization constant for different representations $r$, are derived, and results are given in Appendix A. In Appendix B, we present the structure constants of $E_{6}$ explicitly, which are mostly used by physicists and in the study of GUT.

II dimension-5 operators and modified

gauge coupling unification condition

Dimension-5 operators are singlets of the grand unified gauge group G, and are formed from gauge field strengths $G_{\mu\nu}$ and Higgs multiplets $H_{k}$ of G,

[TABLE]

where a,b are group indices and k labels different multiplets. Therefore, the terms relevant to our discussions of gauge coupling unification in the Lagrangion at the unification scale are,

[TABLE]

Now G=$E_{6}$, it is evident from Eq. (1) that the representations, to which Higgs fields $H_{k}$ belong, can only be contained in the symmetric product of two adjoints, 222For simplicity, we use ”$\times$” and ”+” to denote the direct product and the direct sum respectively, whenever there is no confusion.

[TABLE]

For a specific irrep $r$, $r=1,650,2430$, in Eq. (3), we denote the Higgs multiplet by a d-dimensional symmetric matrix $\Phi^{(r)}$, with d=d(G), the dimension of the adjoint representation (rep) G (we use the same letter G to denote the group and its adj. rep. for simplicity.), and $d=78$ for $G=E_{6}$. For our purpose, we find that all possible $\Phi^{(r)}$ are invariant under the SM gauge group $G_{321}\equiv SU(3)\times SU(2)\times U_{Y}(1)$. That is, each of them is a SM singlet and then the matrix is largely simplified: it contains only a few independent entries.

There are different ways to define the hypercharge $Y$, which are consistent with the SM (see, e.g., the Refs Harada:2003sb ; London:1986dk ). We consider two cases:

1. $U_{Y}(1)$ is a subgroup of H;
2. $U_{Y}(1)$ is a subgroup of $E_{6}$. Hereafter, we shall call the case 1) as ”normal embedding”, and the case 2) as ”flipped embedding”. For example, for $H=SU(3)\times SU(3)\times SU(3)$,

normal embedding, $G_{321}\subset SU(3)_{C}\times SU(2)_{L}\times SU(2)_{R}\times U(1)_{L}\times U(1)_{R}\subset SU(3)_{C}\times SU(3)_{L}\times SU(3)_{R}\subset E_{6}$

[TABLE] 2. 2.

flipped embedding, $G_{321}\subset SU(3)_{C}\times SU(2)_{L}\times SU(2)_{R}\times U(1)_{L}\times U(1)_{R}\subset SU(3)_{C}\times SU(3)_{L}\times SU(3)_{R}\subset E_{6}$

[TABLE]

Therefore, $\Phi^{(r)}$, which are invariant under $G_{321}$, can be written as,

[TABLE]

where $I_{n}$ is n-dimensional unit matrix, and $hij=hji$, since $\Phi^{(r)}$ is the symmetric matrix. We have ordered indices $a,b$ according to the order of terms in Eq. (4), i.e., $a=1,...,8$ for $h1$, $a=9,10,11$ for $h2$, $a=12$ for $h3$, …, et al. Because $hj$ ($j=1,2,...,15$) are the same for both the normal and flipped embedding, except $h3,h5,h9$, we denote the different entries in the case of flipped embedding by $h3^{\prime},h5^{\prime},h9^{\prime}$ and $hij^{\prime}$.

In order to find $\Phi^{(r)}$, we use the second order Casimir operator,

[TABLE]

where R is a irrep of G, and $X_{i}$’s are the generators of G, which satisfy

[TABLE]

with $f_{ijk}$ as the totally antisymmetric structure constants. The operator acts on the tensor product $R\times R$ so that

[TABLE]

where

[TABLE]

It is easy to derive $F\Phi^{(r)}=F_{G}(r)\Phi^{(r)}$, where $F_{G}(r)=C(r)/2-C(G)$ is the eigenvalue of $F$ in irrep r with $C(r)$ and $C(G)$ being the eigenvalues of the Casimir operator in irrep r and G respectively. $C(r)$ depends on the normalization of the Casimir operator and consequently the choice of structure constants 333The choice in this paper leads to that for the subgroup SO(10) of E6, $c(r)^{\prime}$s are the same as those in refs. Chamoun:2001in ; Chamoun:2009nd ; Calmet:2009hp ; Girardi:1980um .. The structure constants of $E_{6}$ have been given in a Chevalley base sc . For convenience of the study in GUT, they are transformed into the usual form and listed in Appendix B.

The eigenvalues $C(r)$ for several irreps r in $E_{6}$, as well as $SO(10)$ and $SU(6)$ are listed in Table I.

In the case of $H=SU(6)\times SU(2)$, depending on different embeddings of the SM into H, i.e., different specific assignment of the chiral fields to representations of subgroup H, there are three cases. In the first case, denoted as H=$SU(6)\times SU(2)_{X}$, the SM is totally in $SU(6)$ and no relation with the factor group $SU(2)$ of H. In the second case, denoted as H=$SU(6)\times SU(2)_{L}$, $SU(2)_{L}$ is the factor group $SU(2)$ of H. And in the third case, denoted as H=$SU(6)\times SU(2)_{R}$, $SU(2)_{R}$ is the factor group $SU(2)$ of H. For all the cases, a direct but tedious calculation gives all of the Clebsch-Gordan coefficients, $\Phi^{(r)}_{s,z}$, associated with $E_{6}$ breaking to the SM, which are listed in Appendix A.

By classifying the SM singlets $\Phi_{s,z}^{(r)}$ into irreps under different breaking chains as in Tables in Appendix A, we do not mean to imply a grand unified symmetry breaking scenario, where $E_{6}$ is broken to the SM necessarily via one or some intermediate gauge groups, though such a classification is well suited for such a scenario. However, such a classification is suitable to serve as a parametrization for the SM singlets $<H_{k}^{ab}>$, the nonzero vacuum expectation values of $H_{k}^{ab}$, transforming in irrep $r_{k}$ 444The convention is: k=1,2,3 correspond to irreps ${\bf 1},{\bf 650},{\bf 2430}$. of $E_{6}$, in terms of the basis $\Phi^{k}_{s,z}\equiv\Phi^{(r_{k})}_{s,z}$,

[TABLE]

where $v_{s,z}^{k}$, $v^{k}$ are real. Eqs. (2), (11) lead to an alteration of the gauge coupling unification condition,

[TABLE]

where

[TABLE]

for the normal embedding, and $\phi_{i}^{k~{}sz}$ are listed in Tables (see Appendix A). For the case of flipped embedding, $\phi_{1}^{k~{}sz}$ is changed into $\phi_{1}^{k~{}sz}\equiv-h3^{\prime k}_{s,z}$. $h3^{\prime k}_{s,z}$ are also listed in Tables in Appendix A.

Eq. (12) is the boundary condition at the scale $M_{GUT}$ of the RG evolution of gauge couplings, and will be used in the numerical analysis in the next section.

III Grand unification in $E_{6}$

In this section, we study the unification of gauge couplings with one, two, or more dimension-5 operators numerically from different $\bf 650$ and $\bf 2430$ breaking chains.

As to the case of one dimension-5 operator from ${\bf 650}$ or ${\bf 2430}$, because there are several maximal subgroups and for a specific maximal subgroup there are several breaking chains, the SM singlets $<H_{k}^{ab}>$, the nonzero vacuum expectation values of $H_{k}^{ab}$, are not determined for fixed k=2 or 3. Consequently it is clear from Eq. (13) and Tables 3-10 that the ratio $\epsilon_{1}:\epsilon_{2}:\epsilon_{3}$ can not be determined fully. So one has much freedom to choose ratios among $v_{s,z}^{k}$. It is a boring and not necessary task to exhaust all possibilities. Instead, we just take some breaking chains as examples. Then, the unification scale $M_{X}$ (set $M_{X}=M_{GUT}$ for simplicity) and Wilson coefficient $c_{k}$ are computable by means of the gauge coupling unification condition Eq.(12), when the running gauge couplings in the SM are given. We limit ourselves to one-loop case for running in the paper for simplicity and it is straightforward to generalize to two-loop case. Moreover, dimension-5 operators also affect analysis of proton decay Tobe:2003yj ; Nath:2006ut . As analyzed in the Ref. Calmet:2009hp and in the next section, the absolute value of Wilson coefficient should be less than 10, i.e., max$|c_{k}|\leq 10$, in order to satisfy the proton decay constraint. Hereafter, we name the unification with max$|c_{k}|\leq 10$ as the successful unification.

In Tab. 2, with only one breaking chain, five example results are given with their respective unification scale $M_{X}$, Higgs VEV $v$, Wilson coefficient $c_{k}$, gauge coupling $\alpha_{G}$ at the unification scale, and three $\epsilon_{1,2,3}$ for the unification. The needed Wilson coefficients are too large in two cases of ${\bf 650}\to{\bf 54}\to{\bf 24}$ and ${\bf 650}\to({\bf 35,1})\to{\bf 1}$. While in the other three cases of ${\bf 2430}\to{\bf 210}\to{\bf 75}$, ${\bf 2430}\to({\bf 405,1})\to{\bf 1}$, and ${\bf 2430}\to({\bf 189,1})\to{\bf 1}$, we can reach the successful unification. In particular, for ${\bf 2430}\to({\bf 189,1})\to{\bf 1}$, $M_{X}$ is as large as the Planck scale and $|c_{k}|$ is as small as $-0.015$. That means, we may have a perturbative theory up to the onset of quantum gravity. With only one dimension-5 operator, we can also use two or more different breaking chains to achieve successful gauge coupling unification with continuously varied unification scale $M_{X}$, as given in Fig. 1, 2 for illustrations. In Fig. 1, we choose two breaking chains ${\bf 210}\to{\bf 24}$ and ${\bf 210}\to{\bf 75}$ for ${\bf 650}$, ${\bf 770}\to{\bf 24}$ and ${\bf 770}\to{\bf 200}$ for ${\bf 2430}$, respectively, under the subgroup embedding $SU(5)\subset SO(10)\subset E_{6}$. We plot the ratios of VEVs, $v_{\bf 75}/v_{\bf 24}$ for ${\bf 650}$, and $v_{\bf 200}/v_{\bf 24}$ for ${\bf 2430}$ as a function of continuously varied $M_{X}$ with successful unification. In Fig. 2, the ratios of VEVs, $v_{\bf 8}/v_{\bf 1}$ for both ${\bf 650}$ and ${\bf 2430}$, as a function of $M_{X}$ are plotted, with ${\bf(35,1)}\to{\bf 1}$ and ${\bf(189,1)}\to{\bf 8}$ for ${\bf 650}$, and, ${\bf(405,1)}\to{\bf 1}$ and ${\bf(405,1)}\to{\bf 8}$ for ${\bf 2430}$, under the subgroup embedding $SU(3)_{L}\subset SU(6)\times SU(2)_{X}\subset E_{6}$. As it is easy to see, from Eqs. (12), (13) and Tables in Appendix I and in the Ref Huang:2014zba , that for a given specific breaking chain, when two of three numbers $\{-h1^{k}_{s,z},-h2^{k}_{s,z},-h3^{k}_{s,z}(h3^{\prime k}_{s,z})\}$ are zero, i.e., for $(45,0)$, $(210,0)\to{\bf 1}$ in $H=SO(10)\times U(1)$, $(189,1)$, $(405,1)\to{\bf 8,27}$ in $H=SU(6)\times SU(2)_{L}$, $(35,3),(1,5)$ in $H=SU(6)\times SU(2)_{R}$ and $(1,8),(8,8),(1,27)$ in $H=SU(3)\times SU(3)\times SU(3)$, one can not get successful unification.

In the following, we study the unification of gauge couplings in the case of two dimension-5 operators with two different Higgs multiplets from $\bf 650$ and $\bf 2430$ of $E_{6}$ respectively. In order to achieve continuously varied unification scale $M_{X}$, we have four variables, two VEVs of Higgs multiplets, $v_{\bf 650}$ and $v_{\bf 2430}$, and two Wilson coefficients. Without fine-tuning for the VEVs, we fix the ratio of $\bf 650$ and $\bf 2430$ in three cases, $1:5$, $1:1$ and $5:1$, in our figures for an illustration. Also, for the VEVs of the Higgs multiplets, we assume they account for half of the average gauge boson squared mass Calmet:2009hp . Then, we are left with two Wilson coefficients for the unification. The maximal absolute values of Wilson coefficients as a function of unification scale $M_{X}$ are given in Fig. 3,4,5,6 for different embedding of subgroup into $E_{6}$.

The numerical results of the subgroup embedding $SU(5)\subset SO(10)\times U(1)\subset E_{6}$ are shown in Fig. 3. We choose two breaking chains, ${\bf 650}\to{\bf 54}\to{\bf 24}$ and ${\bf 2430}\to{\bf 210}\to{\bf 75}$, as an example for the study. For all three cases of the ratios of Higgs VEVs, we have Wilson coefficients less than $10$, max$|c_{i}|\leq 10$, while unification scale $M_{X}>10^{16}$GeV. The needed $\epsilon$s for the unification are also given in Fig. 3, which are independent of the ratios of Higgs VEVs. The $\epsilon_{2}$ is varied smoothly with the scale $M_{X}$, but the needed $\epsilon_{1}$ is negative and larger absolute value is required. Similar results for the other embeddings, $SU(3)_{L}\subset SU(6)\times SU(2)_{X}\subset E_{6}$, $SU(3)_{R}\subset SU(6)\times SU(2)_{L}\subset E_{6}$ and $SU(3)_{L}\times SU(3)_{R}\subset E_{6}$, are shown in Fig. 4,5,6, respectively. In order to obtain max$|c_{i}|\leq 10$, larger unification scale $M_{X}$ are needed for different ratios of Higgs VEVs. A comment is that for different subgroup embedding of $E_{6}$, we may have different sets of values of $\epsilon$s at a given unification scale $M_{X}$.

In the case of two dimension-5 operators with more than two breaking chains, we scan the maximal absolute value of Wilson coefficients as a function of unification scale $M_{X}$ for subgroup embeddings $SU(5)\subset SO(10)\subset E_{6}$ and $SU(3)_{L}\subset SU(6)\times SU(2)_{X}\subset E_{6}$ with random VEVs of different breaking chains and random ratios of Wilson coefficients of the two operators in Fig. 7 and Fig. 8. For subgroup embedding $SU(5)\subset SO(10)\subset E_{6}$, there are actually four different ratios among $v_{s,z}^{k}$, and for subgroup embedding $SU(3)_{L}\subset SU(6)\times SU(3)_{X}\subset E_{6}$, we have eight different ratios. As a result, the unification are much easier for the latter. Most of the points in both cases have satisfed the successful unification with max$|c_{k}|\leq 10$.

As a summary of this section, we conclude that nonsupersymmetric models of $E_{6}$ grand unification can be obtained through effects of dimension-5 operators. Comparing with the other groups like $SU(5)$ and $SO(10)$, it is much easier to achieve the successful unification with natural Wilson coefficient $c_{k}$ and continuously varied unification scale $M_{X}$. Thus including effects of quantum gravity provides a greater probability for building a realistic nonsupersymmetric model in $E_{6}$ GUT.

IV The constraint from the proton decay

As it is well-known, the key predictions in GUT, despite of the details of model-building, are the gauge coupling unification and the proton decay. It has been pointed out Tobe:2003yj ; Nath:2006ut that contributions to the proton decay mediated by superheavy Higgs, which are mainly due to color triplets and very model-dependent, are less important than those mediated by superheavy gauge bosons. Therefore, we limit ourselves to discuss only the contributions from the gauge dimension-6 operators. The grand unification breaking will bring in heavy thresholds. Assuming one-step breaking of the grand unified gauge group $E_{6}$ to the SM $G_{321}$ at the unification scale $M_{X}$ for simplicity, the average squared mass of the non-$G_{321}$ singlet gauge bosons (usually called ”superheavy” gauge bosons) is given by,

[TABLE]

where the sum runs over all Higgs multiplets $i$ that acquire nonzero vacuum expectation value $v_{i}$ at $M_{X}$ , i.e., in addition to the non-$G_{321}$ singlet Higgs contained in Eq. (1), all other Higgs multiplets which are necessary to realize the gauge symmetry breaking chain in a specific model. In order to guarantee correct use of the running gauge couplings in the SM (see the previous section), it is necessary to require $M_{HG}$ close to the grand unified symmetry breaking scale, i.e., the unification scale $M_{X}$, in the case of one-step breaking. For the purpose of definiteness and omitting heavy threshold effects, we take $M_{HG}=M_{X}$. Then, the proton lifetime due to superheavy gauge boson exchange can be written as Tobe:2003yj ; Nath:2006ut ; Dorsner:2004xa ,

[TABLE]

where C is a coefficient containing all information about the flavor structure of the theory and $m_{p}$ is the mass of proton. The newest experimental bound on the proton lifetime for the channel $p\rightarrow e^{+}\pi^{0}$ is TheSuper-Kamiokande:2017tit ,

[TABLE]

Combining Eq. (15) and Eq. (16), one has

[TABLE]

C is order of 1, $C\sim O(1)$. $\alpha_{G}=1/70\sim 1/40$ (see Table II and Figures above). Even when C is as small as 0.1, the constraint from the proton decay is just $M_{X}>8.9\times 10^{15}{\rm GeV}$.

We can see from the Table II and figures in the previous section that most of the cases can easily satisfy the bound Eq. (17). In the case of only one breaking chain, ${\bf 2430}\to({\bf 405,1})\to{\bf 1}$ and ${\bf 2430}\to({\bf 189,1})\to{\bf 1}$, have the unification scale $6.4\times 10^{16}GeV$ and $3.6\times 10^{18}GeV$, respectively. Both lie well above the bound Eq. (17). For the case of ${\bf 2430}\to{\bf 210}\to{\bf 75}$, the unification scale is $3.7\times 10^{15}GeV$, which is very close to the bound Eq. (17) and can be easily adjusted (say, to increase the value of v) to satisfy the bound. With only one dimension-5 operator, using two (or more) different breaking chains, one can achieve successful gauge coupling unification with continuously varied unification scale $M_{X}$, as long as the ratio of two vevs varies with $M_{X}$. Thus, the bound Eq. (17) is satisfied. Looking at the results shown in the last section, the same remains in the case of two dimension-5 operators with two different Higgs multiplets from $\bf 650$ and $\bf 2430$ of $E_{6}$ respectively.

V Summary and discussion

It has been pointed out that gauge coupling unification condition is modified due to the effects of dimension-5 operators. We have investigated the gauge coupling unification in $E_{6}$ without SUSY under modified gauge coupling unification condition. For this purpose, considering several maximal subgroups $H=SO(10)\times U(1)$, $H=SU(3)\times SU(3)\times SU(3)$, $H=SU(2)\times SU(6)$ of $E_{6}$ and the usual breaking chains for a specific maximal subgroup, we have derived and given all of the Clebsch-Gordan coefficients $\Phi^{(r)}_{s,z}$ associated with $E_{6}$ breaking to the SM. We have also presented the structure constants of $E_{6}$ in the usual form, which are mostly used by physicists and in the study of GUT. Results on the gauge coupling unification show that, for a single dimension-5 operator, realizing the gauge coupling unification under modified gauge coupling unification condition in $E_{6}$ GUT is easier than that in $SO(10)$ GUT, since there are more maximal subgroups, and, for a specific maximal subgroup, there are more breaking chains, so that one has much freedom to choose ratios among the nonzero vacuum expectation values $v_{s,z}^{k}$ of the Higgs multiplets $H_{k}$. We have also analyzed the constraint on the unification scale $M_{X}$ from the newest data of the proton decay. It is shown that most of cases studied in the section IV satisfy the constraint easily.

In the effective field theory spirit, operators of dimension higher than 5 are also present, e.g., a dimension-6 operator generalization of Eq. (1),

[TABLE]

After the Higgs multiplets acquire vevs at the scale $M_{X}=M_{Pl}/O(1)$, they can contribute to the gauge kinetic terms as well,

[TABLE]

where the corrections (see Eq. (13)),

[TABLE]

from the dimension-5 operators (1) and the corrections $\epsilon_{i}^{(6)}$ from the dimension-6 operators (18). We can estimate the size of $\epsilon_{i}^{(6)}$. For example, taking $H_{1}={\bf 27}$ and $H_{2}={\bf\bar{27}}$, we have,

[TABLE]

So we can use $cg_{k}H_{k}v$, where $v\sim\frac{M_{X}}{g_{G}}$ is the average vev of the $H_{1},H_{2}$ and $cg_{k}$ ($k=1,2,3$ corresponding to the representation ${\bf 1},{\bf 78},{\bf 650}$ respectively) is the Clebsch-Gordan coefficient from the decomposition of the Kronecker product ${\bf 27}\times{\bf\bar{27}}$ (see (20)), instead of $H_{1}H_{2}$, in (18). For specific, set $k=3$, i.e., the ${\bf 650}$, then, we have,

[TABLE]

which leads to $\epsilon_{i}^{(6)}=\frac{c_{6}M_{X}}{g_{G}M_{Pl}^{2}}cg_{3}<H_{3}>$. For $k=3$, $\epsilon_{i}=\frac{c_{3}}{M_{Pl}}<H_{3}>$. Therefore, $\epsilon_{i}^{(6)}\sim\frac{c_{6}cg_{3}M_{X}}{c_{3}g_{G}M_{Pl}}\epsilon_{i}$. With $c_{6}\sim c_{3}$, $cg_{3}\leq 1$ and $M_{X}=M_{Pl}/O(1)$, effects of dimension-6 operators might be the same order of those of dimension-5 operators. Therefore, the expansion of Eq. (19) might not or might be controlled perturbatively. If it is not, one cannot claim perturbative gauge-gravity unification at the Planck scale. However, the fact that the modification of the gauge coupling unification condition of Eq. (12) allows us in principle to adjust the unification scale to a higher scale $\sim M_{Pl}$ could at least be taken as a hint that gauge-gravity unification is a possible scenario, even if the necessary parameter values or the last piece of the evolution cannot be computed perturbatively.

There is an interesting subject in the effective theory framework. That is, the following operators of dimension-5 are also probably present,

[TABLE]

where $a,b$ are group indices, $k$ labels different Higgs multiplets and $\tilde{G}^{b\mu\nu}$ is dual of gauge field strength $G^{b}_{\mu\nu}$. If one assumes that when $E_{6}$ breaks into $G_{321}$, one of the other two U(1)’s ($U_{V}(1),U_{V^{\prime}}(1)$) is anomalous Peccei-Quinn $U(1)$. After spontaneously breaking of anomalous Peccei-Quinn $U(1)$ symmetry, the associated pseudo Goldstone boson, axion, is to become a component of the corresponding vector boson (say, $Z_{V}$), since now this $U(1)$ is a local symmetry. When $\tilde{H}_{k}$ acquires the nonzero vacuum expectation values, from Eq.(22) and the relevant Clebsch-Gordan coefficients $\Phi^{(r)}_{s,z}$, one can estimate the size of the coupling of $Z_{V}$. It might be instructive to study the relation of this term to strong CP and axion physics.

It is straightforward to generalize the study of the gauge coupling unification in $E_{6}$ including the effects of dimension-5 operators to SUSY GUT. In this case, gaugino mass ratios can be read from the Tables shown in Appendix A, since gauginos belong to the same multiplets as gauge bosons in SUSY. That is, for the flipped embedding,

[TABLE]

where $M_{a}$ are the gaugino masses and $a=3,2,1$ corresponds the $SU(3)$, $SU(2)$, $U(1)$ of the SM, i.e., the gluino, wino and bino masses. The results agree with those in corresponding Tables given by S.P. Martin Martin:2009ad .

For model building of $E_{6}$ GUT with effects of dimension-5 operators, there are several important problems, such as the doublet-triplet splitting, neutrino mass hierarchy, etc., which need to be answered. However, this is beyond the scope of this paper. One should study them in the future.

VI Appendix A: Clebsch-Gordan coefficients

associated with $E_{6}$ breaking to the SM

All of the Clebsch-Gordan coefficients $\Phi^{(r)}_{s,z}$ associated with $E_{6}$ breaking to the SM, in different bases $\{s,z\}$, up to a uniform normalization constant for different representations $r$ have been derived and results are given in this Appendix. For the subgroup $H=SO(10)\times U(1)$, the analysis and results have been given in the Ref. Huang:2014zba . We start from $H=SU(6)\times SU(2)$.

VII Appendix B: structure constants of $E_{6}$

In Cartan-Weyl basis, generators of a group G can be written as $\{H_{j},E_{\alpha}\}$, $j=1,...,l$, l=rank[G], $\alpha=\alpha^{b}$ is a root, $b=1,...,(N-l)/2$ for positive root, N=order(G)=the number of generators of G. The Lie algebra is,

[TABLE]

where $\alpha^{b}_{j}$ is j-th component of $\alpha^{b}$ and

[TABLE]

with $\alpha^{n}=\alpha^{i}+\alpha^{j}$ fixed. The $N(\alpha^{n})$ in (25) is called real SC. We call the SC directly obtained from roots in (24) as ”direct SC” for simplicity. Then all SC of a group G are composed of real SC and direct SC.

The structure constants of $E_{6}$ have been given in a Chevalley base sc . We transform them into the usual form mostly used by physicists and in the study of GUT (see, Eq. (8)). Starting from the table ”Structure constants for E6”, page 1526 in Vavilov’s paper sc , we obtain the real SC of $E_{6}$, $f[i,j,k]$, $i,j,k=1,...,78$, which are totally antisymmetric. The part of f[i,j,k], which are not zero and ordered according to $i<j<k$, is given as follows,

[TABLE]

The other parts of $f[i,j,k]$ either can be obtained from Eq. (VII) by using their antisymmetric properties, or are equal to zero.

The positive roots of $E_{6}$ in Dynkin basis have been given (see, Table 20 in Ref. Slansky:1981yr ). It is straightforward to get the direct SC, $fsc[i,j,k]$, $i,j,k=1,...,78$. The part of $fsc[i,j,k]$, which are not zero and ordered according to $i<j<k$, is given in Eq. (VII). The other parts of $fsc[i,j,k]$ can either be obtained from Eq. (VII) by using their antisymmetric properties, or are equal to zero.

[TABLE]

As said above, all SC of $E_{6}$, $ff[i,j,k]$555In Eq. (8), all SC of a group G is denoted by $f_{ijk}$. We now denote all SC of $E_{6}$ by $ff[i,j,k]$., are the sum of real SC and direct SC. That is, $ff[i,j,k]=f[i,j,k]+fsc[i,j,k]$. It has been checked that $ff[i,j,k]$ satisfy the Jacobi identities.

Acknowledgments

This research was supported in part by the Natural Science Foundation of China under grant numbers 11375248, 11647601, 11135009 and 11005033.