1. Introduction
2. Trinification model building from ${{\mathbb{T}}}^{6}/({{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2})$ orientifold
2.1. Constraints from tadpole cancellation and supersymmetry
Table 1. The wrapping numbers for four O6-planes. |
Orientifold action | O6-plane | (n1, l1) × (n2, l2) × (n3, l3) |
---|---|---|
${\rm{\Omega }}{ \mathcal R }$ | 1 | $({2}^{{\beta }_{1}},0)\times ({2}^{{\beta }_{2}},0)\times ({2}^{{\beta }_{3}},0)$ |
${\rm{\Omega }}{ \mathcal R }\omega $ | 2 | $({2}^{{\beta }_{1}},0)\times (0,-{2}^{{\beta }_{2}})\times (0,{2}^{{\beta }_{3}})$ |
${\rm{\Omega }}{ \mathcal R }\theta \omega $ | 3 | $(0,-{2}^{{\beta }_{1}})\times ({2}^{{\beta }_{2}},0)\times (0,{2}^{{\beta }_{3}})$ |
${\rm{\Omega }}{ \mathcal R }\theta $ | 4 | $(0,-{2}^{{\beta }_{1}})\times (0,{2}^{{\beta }_{2}})\times ({2}^{{\beta }_{3}},0)$ |
Table 2. General spectrum for intersecting D6-branes at generic angles, where ${ \mathcal M }$ is the multiplicity, and ${a}_{\square \square }$ and denote respectively the symmetric and antisymmetric representations of ${\rm{U}}\left({N}_{a}/2\right)$. Positive intersection numbers in our convention refer to the left-handed chiral supermultiplets. |
Sector | Representation |
---|---|
aa | U(Na/2) vector multiplet |
3 adjoint chiral multiplets | |
ab + ba | ${ \mathcal M }\left(\tfrac{{N}_{a}}{2},\tfrac{\overline{{N}_{b}}}{2}\right)$ = ${I}_{{ab}}({\,\square }_{a}\,,{\overline{\square }}_{b}\,)$ |
ab’ + b’a | ${ \mathcal M }\left(\tfrac{{N}_{a}}{2},\tfrac{{N}_{b}}{2}\right)$ = ${I}_{{ab}^{\prime} }({\,\square }_{a}\,,{\square }_{b}\,)$ |
aa’ + a’a | ${ \mathcal M }({a}_{\square \square })$ = $\tfrac{1}{2}({I}_{{aa}^{\prime} }-\tfrac{1}{2}{I}_{{aO}6})$ |
${ \mathcal M }$() = $\tfrac{1}{2}({I}_{{aa}^{\prime} }+\tfrac{1}{2}{I}_{{aO}6})$ |
2.2. Gauge symmetry breaking
3. Particle spectra of four-family supersymmetric trinification models
Table 3. The spectrum of chiral and vector-like superfields, and their quantum numbers under the gauge symmetry SU(3)C × SU(3)L × SU(3)R × U(1) × USp(8) for the model 1. |
Model 1 | Quantum number | QC | QL | QR | Qem | B-L | Field |
---|---|---|---|---|---|---|---|
ab | $1\times (3,\overline{3},1,1,1)$ | 1 | −1 | 0 | $-\tfrac{1}{3},\tfrac{2}{3},-1,0$ | $\tfrac{1}{3},-1$ | QL, LL |
ab′ | 3 × (3, 3, 1, 1, 1) | 1 | 1 | 0 | $-\tfrac{1}{3},\tfrac{2}{3},-1,0$ | $\tfrac{1}{3},-1$ | QL, LL |
ac | $1\times (\overline{3},1,3,1,1)$ | −1 | 0 | 1 | $\tfrac{1}{3},-\tfrac{2}{3},1,0$ | $-\tfrac{1}{3},1$ | QR, LR |
ac′ | $3\times (\overline{3},1,\overline{3},1,1)$ | −1 | 0 | −1 | $\tfrac{1}{3},-\tfrac{2}{3},1,0$ | $-\tfrac{1}{3},1$ | QR, LR |
bc | $8\times (1,\overline{3},3,1,1)$ | 0 | −1 | 1 | 1, 0, 0, −1 | 0 | H |
bd | $3\times (1,\overline{3},1,1,1)$ | 0 | −1 | 0 | $\mp \tfrac{1}{2}$ | 0 | |
bd′ | $3\times (1,\overline{3},1,\overline{1},1)$ | 0 | −1 | 0 | $\mp \tfrac{1}{2}$ | 0 | |
cd | $3\times (1,1,3,\overline{1},1)$ | 0 | 0 | 1 | $\pm \tfrac{1}{2}$ | 0 | |
cd′ | 3 × (1, 1, 3, 1, 1) | 0 | 0 | 1 | $\pm \tfrac{1}{2}$ | 0 | |
b3 | 1 × (1, 3, 1, 1, 8) | 0 | 1 | 0 | $\pm \tfrac{1}{2}$ | 0 | |
c3 | $1\times (1,1,\overline{3},1,8)$ | 0 | 1 | 0 | $\mp \tfrac{1}{2}$ | 0 | |
${b}_{\square \square }$ | 2 × (1, 6, 1, 1, 1) | 0 | 2 | 0 | 0, ±1 | 0 | |
2 × (1, 3, 1, 1, 1) | 0 | −2 | 0 | 0 | 0 | ||
${c}_{\overline{\square \square }}$ | $2\times (1,1,\overline{6},1,1)$ | 0 | 0 | −2 | 0, ±1 | 0 | |
2 × (1, 1, 3, 1, 1) | 0 | 0 | 2 | 0 | 0 |
Table 4. The spectrum of chiral and vector-like superfields, and their quantum numbers under the gauge symmetry $\mathrm{SU}{(3)}_{C}\times \mathrm{SU}{(3)}_{L}\times \mathrm{SU}{(3)}_{R}\times {\rm{U}}(1)\times \mathrm{USp}{\left(2\right)}^{3}$ for the model 2. |
Model 2 | Quantum number | QC | QL | QR | Qem | B-L | Field |
---|---|---|---|---|---|---|---|
ab | $1\times (3,\overline{3},1,1,1,1,1)$ | 1 | −1 | 0 | $-\displaystyle \frac{1}{3}$, $\displaystyle \frac{2}{3}$, −1, 0 | $\displaystyle \frac{1}{3}$, −1 | QL, LL |
ab′ | 3 × (3, 3, 1, 1, 1, 1, 1) | 1 | 1 | 0 | -$\displaystyle \frac{1}{3},\displaystyle \frac{2}{3}$, −1, 0 | $\displaystyle \frac{1}{3}$, −1 | QL, LL |
ac | $1\times (\overline{3},1,3,1,1,1,1)$ | −1 | 0 | 1 | $\displaystyle \frac{1}{3}$, $-\displaystyle \frac{2}{3}$, 1, 0 | $-\displaystyle \frac{1}{3}$, 1 | QR, LR |
ac′ | $3\times (\overline{3},1,\overline{3},1,1,1,1)$ | −1 | 0 | −1 | $\displaystyle \frac{1}{3},-\displaystyle \frac{2}{3},1,0$ | $-\displaystyle \frac{1}{3},1$ | QR, LR |
bc | $4\times (1,\overline{3},3,1,1,1,1)$ | 0 | 1 | 1 | 1, 0, 0, −1 | 0 | H |
bd | $3\times (1,3,1,\overline{1},1,1,1)$ | 0 | 1 | 0 | $\pm \displaystyle \frac{1}{2}$ | 0 | |
bd′ | 3 × (1, 3, 1, 1, 1, 1, 1) | 0 | 1 | 0 | $\pm \displaystyle \frac{1}{2}$ | 0 | |
cd | $18\times (1,1,3,\overline{1},1,1,1)$ | 0 | 0 | 1 | $\pm \displaystyle \frac{1}{2}$ | 0 | |
cd′ | $10\times (1,1,\overline{3},\overline{1},1,1,1)$ | 0 | 0 | −1 | $\mp \displaystyle \frac{1}{2}$ | 0 | |
ad | $5\times (3,1,1,\overline{1},1,1,1)$ | 1 | 0 | 0 | $\displaystyle \frac{1}{6},-\displaystyle \frac{1}{2}$ | $\tfrac{1}{3},-1$ | |
ad′ | 9 × (3, 1, 1, 1, 1, 1, 1) | 1 | 0 | 0 | $\tfrac{1}{6}$, $-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
a1 | $1\times (\overline{3},1,1,1,2,1,1)$ | −1 | 0 | 0 | $-\tfrac{1}{6},\tfrac{1}{2}$ | $-\tfrac{1}{3},1$ | |
b1 | $1\times (1,\overline{3},1,1,2,1,1)$ | 0 | −1 | 0 | $\mp \tfrac{1}{2}$ | 0 | |
d1 | $8\times (1,1,1,\overline{1},2,1,1)$ | 0 | 0 | 0 | 0 | 0 | |
a3 | 2 × (3, 1, 1, 1, 1, 2, 1) | 1 | 0 | 0 | $\tfrac{1}{6},-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
c3 | 2 ×(1, 1, 3, 1, 1, 2, 1) | 0 | 0 | 1 | $\pm \tfrac{1}{2}$ | 0 | |
b4 | 1×(1, 3, 1, 1, 1, 1, 2) | 0 | 1 | 0 | $\pm \tfrac{1}{2}$ | 0 | |
c4 | $1\times (1,1,\overline{3},1,1,1,2)$ | 0 | 0 | −1 | $\mp \tfrac{1}{2}$ | 0 | |
d4 | $2\times (1,1,1,\overline{1},1,1,2)$ | 0 | 0 | 0 | 0 | 0 | |
${a}_{\overline{\square \square }}$ | $2\times (\overline{6},1,1,1,1,1,1)$ | −2 | 0 | 0 | $-\tfrac{1}{3},\tfrac{1}{3},1$ | $-\tfrac{2}{3},2$ | |
2 × (3, 1, 1, 1, 1, 1, 1) | 2 | 0 | 0 | $\tfrac{1}{3},-1$ | $\tfrac{2}{3},-2$ | ||
${c}_{\overline{\square \square }}$ | $2\times (1,1,\overline{6},1,1,1,1)$ | 0 | 0 | −2 | 0, ±1 | 0 | |
2 × (1, 1, 3, 1, 1, 1, 1) | 0 | 0 | 2 | 0 | 0 |
Table 5. The spectrum of chiral and vector-like superfields, and their quantum numbers under the gauge symmetry $\mathrm{SU}{(3)}_{C}\times \mathrm{SU}{(3)}_{L}\times \mathrm{SU}{(3)}_{R}\times {\rm{U}}(1)\times \mathrm{USp}{\left(2\right)}^{3}$ for the model 2-dual. |
Model 2-dual | Quantum number | QC | QL | QR | Qem | B-L | Field |
---|---|---|---|---|---|---|---|
ab | $1\times (\overline{3},3,1,1,1,1,1)$ | −1 | 1 | 0 | $\displaystyle \frac{1}{3},-\displaystyle \frac{2}{3},1,0$ | $-\displaystyle \frac{1}{3}$, 1 | QL, LL |
ab′ | $3\times (\overline{3},\overline{3},1,1,1,1,1)$ | −1 | −1 | 0 | $\displaystyle \frac{1}{3},-\displaystyle \frac{2}{3}$, 1, 0 | $-\displaystyle \frac{1}{3}$, 1 | QL, LL |
ac | $1\times (3,1,\overline{3},1,1,1,1)$ | −1 | 0 | 1 | $-\displaystyle \frac{1}{3},\displaystyle \frac{2}{3},-1$, 0 | $\displaystyle \frac{1}{3}$, −1 | QR, LR |
ac′ | 3 × (3, 1, 3, 1, 1, 1, 1) | 1 | 0 | 1 | $-\displaystyle \frac{1}{3}$, $\displaystyle \frac{2}{3},-1$, 0 | $\displaystyle \frac{1}{3},-1$ | QR, LR |
bc | $4\times (1,3,\overline{3},1,1,1,1)$ | 0 | 1 | −1 | 1, 0, 0, −1 | 0 | H |
cd | $3\times (1,1,3,\overline{1},1,1,1)$ | 0 | 0 | 1 | $\pm \displaystyle \frac{1}{2}$ | 0 | |
cd′ | 3 × (1, 1, 3, 1, 1, 1, 1) | 0 | 0 | 1 | $\pm \displaystyle \frac{1}{2}$ | 0 | |
bd | $18\times (1,3,1,\overline{1},1,1,1)$ | 0 | 1 | 0 | $\pm \displaystyle \frac{1}{2}$ | 0 | |
bd′ | $10\times (1,\overline{3},1,\overline{1},1,1,1)$ | 0 | −1 | 0 | $\mp \displaystyle \frac{1}{2}$ | 0 | |
ad | $5\times (3,1,1,\overline{1},1,1,1)$ | 1 | 0 | 0 | $\displaystyle \frac{1}{6},-\displaystyle \frac{1}{2}$ | $\tfrac{1}{3},-1$ | |
ad′ | 9 × (3, 1, 1, 1, 1, 1, 1) | 1 | 0 | 0 | $\tfrac{1}{6}$, $-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
a1 | $1\times (\overline{3},1,1,1,2,1,1)$ | −1 | 0 | 0 | $-\tfrac{1}{6},\tfrac{1}{2}$ | $-\tfrac{1}{3},1$ | |
c1 | $1\times (1,1,\overline{3},1,2,1,1)$ | 0 | 0 | −1 | $\mp \tfrac{1}{2}$ | 0 | |
d1 | $8\times (1,1,1,\overline{1},2,1,1)$ | 0 | 0 | 0 | 0 | 0 | |
a3 | 2 × (3, 1, 1, 1, 1, 2, 1) | 1 | 0 | 0 | $\tfrac{1}{6},-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
b3 | 2 × (1, 3, 1, 1, 1, 2, 1) | 0 | 1 | 0 | $\pm \tfrac{1}{2}$ | 0 | |
c4 | 1 × (1, 1, 3, 1, 1, 1, 2) | 0 | 0 | 1 | $\pm \tfrac{1}{2}$ | 0 | |
b4 | $1\times (1,\overline{3},1,1,1,1,2)$ | 0 | −1 | 0 | $\mp \tfrac{1}{2}$ | 0 | |
d4 | $2\times (1,1,1,\overline{1},1,1,2)$ | 0 | 0 | 0 | 0 | 0 | |
${a}_{\overline{\square \square }}$ | $2\times (\overline{6},1,1,1,1,1,1)$ | −2 | 0 | 0 | $-\tfrac{1}{3}$, $\tfrac{1}{3},1$ | $-\tfrac{2}{3},2$ | |
2 × (3, 1, 1, 1, 1, 1, 1) | 2 | 0 | 0 | $\tfrac{1}{3},-1$ | $\tfrac{2}{3},-2$ | ||
${b}_{\overline{\square \square }}$ | $2\times (1,\overline{6},1,1,1,1,1)$ | 0 | −2 | 0 | 0, ±1 | 0 | |
2 × (1, 3, 1, 1, 1, 1, 1) | 0 | 2 | 0 | 0 | 0 |
Table 6. The spectrum of chiral and vector-like superfields, and their quantum numbers under the gauge symmetry SU(3)C × SU(3)L × SU(3)R × USp(2) × USp(6) for the model 3. |
Model 3 | Quantum number | QC | QL | QR | Qem | B-L | Field |
---|---|---|---|---|---|---|---|
ab | $3\times (\overline{3},3,1,1,1,1)$ | −1 | 1 | 0 | $-\displaystyle \frac{1}{3}$, $\displaystyle \frac{2}{3},-1$, 0 | $\displaystyle \frac{1}{3},-1$ | QL, LL |
ab′ | $1\times (\overline{3},\overline{3},1,1,1,1)$ | −1 | −1 | 0 | $-\displaystyle \frac{1}{3}$, $\displaystyle \frac{2}{3},-1$, 0 | $\displaystyle \frac{1}{3},-1$ | QL, LL |
ac | $3\times (3,1,\overline{3},1,1,1)$ | 1 | 0 | −1 | $\displaystyle \frac{1}{3}$, $-\displaystyle \frac{2}{3}$, 1, 0 | $-\displaystyle \frac{1}{3},1$ | QR, LR |
ac′ | 1 × (3, 1, 3, 1, 1, 1) | 1 | 0 | 1 | $\displaystyle \frac{1}{3},-\displaystyle \frac{2}{3}$, 1, 0 | $-\tfrac{1}{3}$, 1 | QR, LR |
bc | $4\times (1,\overline{3},3,1,1,1)$ | 0 | −1 | 1 | 1, 0, 0, −1 | 0 | H |
ad | $3\times (\overline{3},1,1,2,1,1)$ | −1 | 0 | 0 | $\tfrac{1}{6},-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
ad′ | $3\times (\overline{3},1,1,\overline{2},1,1)$ | −1 | 0 | 0 | $\tfrac{1}{6},-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
cd | $6\times (1,1,3,\overline{1},1,1)$ | 0 | 0 | 1 | $\pm \tfrac{1}{2}$ | 0 | |
a2 | $2\times (\overline{3},1,1,1,2,1)$ | −1 | 0 | 0 | $\tfrac{1}{6},-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
b3 | $1\times (1,\overline{3},1,1,1,6)$ | 0 | −1 | 0 | $\pm \tfrac{1}{2}$ | 0 | |
c2 | $2\times (1,1,2,1,\overline{2},1)$ | 0 | 0 | 1 | $\pm \tfrac{1}{2}$ | 0 | |
c3 | $1\times (1,1,\overline{2},1,1,6)$ | 0 | 0 | −1 | $\mp \tfrac{1}{2}$ | 0 | |
d2 | $1\times (1,1,1,\overline{2},2,1)$ | 0 | 0 | 0 | 0 | 0 | |
d3 | $2\times (1,1,1,\overline{2},1,6)$ | 0 | 0 | 0 | 0 | 0 | |
${a}_{\square \square }$ | 2 × (6, 1, 1, 1, 1, 1, 1) | 2 | 0 | 0 | $-\tfrac{1}{3}$, $\tfrac{1}{3},1$ | $-\tfrac{2}{3},2$ | |
2 × (3, 1, 1, 1, 1, 1, 1) | −2 | 0 | 0 | $\tfrac{1}{3},-1$ | $\tfrac{2}{3},-2$ | ||
${c}_{\overline{\square \square }}$ | $2\times (1,1,\overline{6},1,1,1)$ | 0 | 0 | −2 | 0, ±1 | 0 | |
2 × (1, 1, 3, 1, 1, 1) | 0 | 0 | 2 | 0 | 0 | ||
$16\times (1,1,1,\overline{1},1,1)$ | 0 | 0 | 0 | 0 | 0 |
Table 7. The spectrum of chiral and vector-like superfields, and their quantum numbers under the gauge symmetry SU(3)C × SU(3)L × SU(3)R × USp(2) × USp(6) for the model 3-dual. |
Model 3-dual | Quantum number | QC | QL | QR | Qem | B-L | Field |
---|---|---|---|---|---|---|---|
ab | $3\times (3,\overline{3},1,1,1,1)$ | 1 | −1 | 0 | $\displaystyle \frac{1}{3}$, $-\displaystyle \frac{2}{3}$, 1, 0 | $-\displaystyle \frac{1}{3},1$ | QL, LL |
ab′ | 1 × (3, 3, 1, 1, 1, 1) | 1 | 1 | 0 | $-\displaystyle \frac{1}{3}$, $\displaystyle \frac{2}{3},-1$, 0 | $\displaystyle \frac{1}{3},-1$ | QL, LL |
ac | $3\times (\overline{3},1,3,1,1,1)$ | −1 | 0 | 1 | $-\displaystyle \frac{1}{3}$, $\displaystyle \frac{2}{3},-1$, 0 | $\displaystyle \frac{1}{3},-1$ | QR, LR |
ac′ | $1\times (\overline{3},1,\overline{3},1,1,1)$ | 1 | 0 | 1 | $\displaystyle \frac{1}{3},-\displaystyle \frac{2}{3}$, 1, 0 | $-\tfrac{1}{3}$, 1 | QR, LR |
bc | $4\times (1,3,\overline{3},1,1,1)$ | 0 | 1 | −1 | 1, 0, 0, −1 | 0 | H |
ad | $3\times (\overline{3},1,1,2,1,1)$ | −1 | 0 | 0 | $\tfrac{1}{6},-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
ad′ | $3\times (\overline{3},1,1,\overline{2},1,1)$ | −1 | 0 | 0 | $\tfrac{1}{6},-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
bd | $6\times (1,3,1,\overline{1},1,1)$ | 0 | 0 | 1 | $\pm \tfrac{1}{2}$ | 0 | |
a2 | $2\times (\overline{3},1,1,1,2,1)$ | −1 | 0 | 0 | $\tfrac{1}{6},-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
c3 | $1\times (1,1,\overline{3},1,1,6)$ | 0 | −1 | 0 | $\pm \tfrac{1}{2}$ | 0 | |
b2 | $2\times (1,2,1,1,\overline{2},1)$ | 0 | 0 | 1 | $\pm \tfrac{1}{2}$ | 0 | |
b3 | $1\times (1,\overline{2},1,1,1,6)$ | 0 | 0 | −1 | $\mp \tfrac{1}{2}$ | 0 | |
d2 | $1\times (1,1,1,\overline{2},2,1)$ | 0 | 0 | 0 | 0 | 0 | |
d3 | $2\times (1,1,1,\overline{2},1,6)$ | 0 | 0 | 0 | 0 | 0 | |
a${a}_{\square \square }$ | 2 × (6, 1, 1, 1, 1, 1, 1) | 2 | 0 | 0 | $-\tfrac{1}{3}$, $\tfrac{1}{3},1$ | $-\tfrac{2}{3},2$ | |
2 × (3, 1, 1, 1, 1, 1, 1) | −2 | 0 | 0 | $\tfrac{1}{3},-1$ | $\tfrac{2}{3},-2$ | ||
${b}_{\overline{\square \square }}$ | $2\times (1,\overline{6},1,1,1,1)$ | 0 | 0 | −2 | 0, ±1 | 0 | |
2 × (1, 3, 1, 1, 1, 1) | 0 | 0 | 2 | 0 | 0 | ||
$16\times (1,1,1,\overline{1},1,1)$ | 0 | 0 | 0 | 0 | 0 |