Naturally realized two dark Z's near the electroweak scale

TL;DR

This paper introduces a new anomaly-free chiral gauge theory with two dark Z bosons, derived from string theory, which could be experimentally probed via kinetic mixing at TeV-scale accelerators.

Contribution

It presents a novel chiral gauge theory with two dark Z bosons, derived from heterotic string theory, expanding the landscape of dark sector models.

Findings

Two dark Z bosons or a dark Z and a dark photon emerge at low energy.

The model is anomaly-free and derived from E8×E8' heterotic string theory.

Potential experimental signatures via kinetic mixing at TeV accelerators.

Abstract

Chiral representations are the key in obtaining light fermions from some ultra-violet completed theories. The well-known chiral example is one family set of fifteen chiral fields in the standard model. We find a new chiral theory SU(2)$_{\rm dark}\times$U(1)$_Q$ with sixteen chiral fields, which does not have any gauge and gravitational anomalies. The group SU(2)$_{\rm dark}\times$U(1)$_Q$ may belong to the dark sector, and we present a derivation of the spectrum from the E$_8\times$E$_8'$ hetrotic string. Necessarily, there appear two degrees at low energy: two dark Z's, or a dark Z plus a dark-photon. Being chiral, there is a chance to probe this theory at TeV accelerators. Since the model belongs to the dark sector, the way to probe it is through the kinetic mixing.

Tables

Table 1. Table 1: The SU(5) ′ representations. The red entries are Q anom / 126 subscript 𝑄 anom 126 Q_{\rm anom}/126 .

Sect.	States	SU(5)′	Multiplicity	$Q_1$	$Q_2$	$Q_3$	$Q_4$	$Q_5$	$Q_6$	$Q_{\mathrm{anom}}$	Label
$T_1^0$	$(\mathrm{0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0};\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}){(\underset{¯}{-\mathrm{1\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0}}\mathrm{\hspace{0.17em}0};\frac{1}{4}\frac{1}{4}\frac{1}{2})}^{\prime }$	${\overline{\mathrm{𝟏𝟎}}}_0^{\prime }$	$1$	$-2$	$-2$	$-2$	0	$+3$	$+9$	$-648$(-$\frac{36}{7}$)	$T_1^{\prime }$
	$(\mathrm{0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0};\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}){(\underset{¯}{\frac{1}{2}\frac{1}{2}\frac{-1}{2}\frac{-1}{2}}\frac{1}{2};\frac{-1}{4}\frac{-1}{4}\mathrm{\hspace{0.17em}0})}^{\prime }$
$T_1^0$	$(\mathrm{0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0};\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}){(\underset{¯}{-\mathrm{1\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0}}\mathrm{\hspace{0.17em}0};\frac{1}{4}\frac{1}{4}\frac{1}{2})}^{\prime }$	${({\mathrm{𝟓}}^{\prime },{\mathrm{𝟐}}^{\prime })}_0$	$1$	$-2$	–2	–2	0	$+3$	$+9$	$-648$(-$\frac{36}{7}$)	$F_1^{\prime }$
	$(\mathrm{0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0};\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}){(\frac{1}{2}\frac{1}{2}\frac{1}{2}\frac{1}{2}\frac{-1}{2};\frac{-1}{4}\frac{-1}{4}\mathrm{\hspace{0.17em}0})}^{\prime }$
	$(\mathrm{0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0};\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}){(\underset{¯}{\frac{1}{2}\frac{-1}{2}\frac{-1}{2}\frac{-1}{2}}\frac{-1}{2};\frac{-1}{4}\frac{-1}{4}\mathrm{\hspace{0.17em}0})}^{\prime }$
	$(\mathrm{0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0};\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}){(\mathrm{0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}\hspace{0.17em}0};\frac{-3}{4}\frac{-3}{4}\frac{-1}{2})}^{\prime }$
$T_1^0$	$(\mathrm{0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0};\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}){(\underset{¯}{\frac{-1}{2}\frac{1}{2}\frac{1}{2}\frac{1}{2}}\frac{-1}{2};\frac{-1}{4}\frac{-1}{4}\mathrm{\hspace{0.17em}0})}^{\prime }$	${\overline{\mathrm{𝟓}}}_0^{\prime }$	$1$	–2	–2	–2	0	$+3$	$-15$	$-432$(-$\frac{24}{7}$)	$F_2^{\prime }$
	$(\mathrm{0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0};\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}){(\mathrm{0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0}-1;\frac{1}{4}\frac{1}{4}\frac{1}{2})}^{\prime }$
$T_1^{+}$	$(\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6};\frac{1}{3}\frac{-1}{3}\mathrm{\hspace{0.17em}0}){(\underset{¯}{\frac{-5}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6}}\frac{1}{2};\frac{1}{12}\frac{-1}{4}\mathrm{\hspace{0.17em}0})}^{\prime }$	${\overline{\mathrm{𝟓}}}_{-5/3}^{\prime }$	$1$	$+4$	$-4$	0	$+4$	$+1$	$+11$	$-414$(-$\frac{23}{7}$)	$F_3^{\prime }$
	$(\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6};\frac{1}{3}\frac{-1}{3}\mathrm{\hspace{0.17em}0}){(\frac{-1}{3}\frac{-1}{3}\frac{-1}{3}\frac{-1}{3}\mathrm{\hspace{0.17em}0};\frac{7}{12}\frac{1}{4}\frac{1}{2})}^{\prime }$
$T_4^{+}$	$(\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6};\frac{-1}{6}\frac{1}{6}\frac{1}{2}){(\underset{¯}{\frac{2}{3}\frac{-1}{3}\frac{-1}{3}\frac{-1}{3}}\mathrm{\hspace{0.17em}0};\frac{1}{3}\mathrm{\hspace{0.17em}0\hspace{0.17em}0})}^{\prime }$	${\mathrm{𝟓}}_{-5/3}^{\prime }$	$3$	–2	+2	+6	$+4$	$-2$	$+2$	$-18$(-$\frac{1}{7}$)	$F_4^{\prime }$
	$(\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6};\frac{-1}{6}\frac{1}{6}\frac{1}{2}){(\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{2};\frac{-1}{6}\frac{-1}{2}\frac{-1}{2})}^{\prime }$
$T_4^{-}$	$(\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6};\frac{-1}{6}\frac{-1}{2}\frac{1}{6}){(\underset{¯}{\frac{-2}{3}\frac{1}{3}\frac{1}{3}\frac{1}{3}}\mathrm{\hspace{0.17em}0};\frac{-1}{3}\mathrm{\hspace{0.17em}0\hspace{0.17em}0})}^{\prime }$	${\overline{\mathrm{𝟓}}}_{5/3}^{\prime }$	$3$	–2	–6	+2	$-4$	$+2$	$-2$	$-1242$(-$\frac{69}{7}$)	$F_5^{\prime }$
	$(\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6};\frac{-1}{6}\frac{-1}{2}\frac{1}{6}){(\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{2};\frac{1}{6}\frac{1}{2}\frac{1}{2})}^{\prime }$
$T_7^{-}$	$(\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6};\frac{1}{3}\mathrm{\hspace{0.17em}0}\frac{-1}{3}){(\underset{¯}{\frac{5}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}}\frac{-1}{2};\frac{-1}{12}\frac{1}{4}\mathrm{\hspace{0.17em}0})}^{\prime }$	${\mathrm{𝟓}}_{5/3}^{\prime }$	$1$	+4	0	–4	$-4$	$-1$	$-11$	$+666$(+$\frac{37}{7}$)	$F_6^{\prime }$
	$(\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6};\frac{1}{3}\mathrm{\hspace{0.17em}0}\frac{-1}{3}){(\frac{1}{3}\frac{1}{3}\frac{1}{3}\frac{1}{3}\mathrm{\hspace{0.17em}0};\frac{-7}{12}\frac{-1}{4}\frac{-1}{2})}^{\prime }$

Table 2. Table 2: The SU(2) ′ representations with the convention of Table 1 . We listed only the upper component of SU(2) ′ from which the lower component can be obtained by applying T − superscript 𝑇 T^{-} of SU(2) ′ .

Sect.	States	SU(2)′	Multiplicity	$Q_1$	$Q_2$	$Q_3$	$Q_4$	$Q_5$	$Q_6$	$Q_{\mathrm{anom}}$	Label
$T_1^0$	$(\mathrm{0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0};\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}){(\underset{¯}{\mathrm{1\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0}}\mathrm{\hspace{0.17em}0};\frac{1}{4}\frac{1}{4}\frac{1}{2})}^{\prime }$	${({\mathrm{𝟓}}^{\prime },{\mathrm{𝟐}}^{\prime })}_0$	$1$	–2	–2	–2	0	$+3$	$-3$	$-540$(-$\frac{30}{7}$)	$F_1^{\prime }$
	$(\mathrm{0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0};\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}){(\mathrm{0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}\hspace{0.17em}0};\frac{-3}{4}\frac{-3}{4}\frac{-1}{2})}^{\prime }$
$T_1^0$	$(\mathrm{0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0};\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}){(\mathrm{0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}0\hspace{0.17em}1};\frac{1}{4}\frac{1}{4}\frac{1}{2})}^{\prime }$	${\mathrm{𝟐}}_0^{\prime }$	$1$	$-2$	$-2$	$-2$	0	$+3$	$+21$	$-756$(-$6$)	$D_2$
$T_1^{+}$	$(\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6};\frac{1}{3}\frac{-1}{3}\mathrm{\hspace{0.17em}0}){(\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{2};\frac{1}{12}\frac{3}{4}\mathrm{\hspace{0.17em}0})}^{\prime }$	${\mathrm{𝟐}}_{-5/3}^{\prime }$	$1$	$+4$	$-4$	$0$	$-8$	$-5$	$+5$	$+18$(+$\frac{1}{7}$)	$D_3$
$T_1^{-}$	$(\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6};\frac{-2}{3}\mathrm{\hspace{0.17em}0}\frac{-1}{3}){(\frac{1}{3}\frac{1}{3}\frac{1}{3}\frac{1}{3}\mathrm{\hspace{0.17em}0};\frac{-1}{12}\frac{1}{4}\frac{1}{2})}^{\prime }$	${\mathrm{𝟐}}_{5/3}^{\prime }$	$1$	$-8$	$0$	$-4$	$-4$	$+5$	$-5$	$-774$(-$\frac{43}{7}$)	$D_4$
$T_1^{-}$	$(\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6};\frac{1}{3}\mathrm{\hspace{0.17em}0}\frac{2}{3}){(\frac{1}{3}\frac{1}{3}\frac{1}{3}\frac{1}{3}\mathrm{\hspace{0.17em}0};\frac{-1}{12}\frac{1}{4}\frac{1}{2})}^{\prime }$	${\mathrm{𝟐}}_{5/3}^{\prime }$	$1$	$+4$	$0$	$+8$	$-4$	$+5$	$-5$	$-270$(-$\frac{15}{7}$)	$D_5$
$T_2^{+}$	$(\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6};\frac{1}{6}\frac{-1}{6}\frac{1}{2}){(\frac{1}{3}\frac{1}{3}\frac{1}{3}\frac{1}{3}\mathrm{\hspace{0.17em}0};\frac{1}{6}\frac{1}{2}\mathrm{\hspace{0.17em}0})}^{\prime }$	${\mathrm{𝟐}}_{5/3}^{\prime }$	$1$	$+2$	$-2$	$+6$	$-4$	$-4$	$-8$	$-54$(-$\frac{3}{7}$)	$D_6$
$T_2^{-}$	$(\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6};\frac{1}{6}\frac{-1}{2}\frac{-1}{6}){(\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{2};\frac{1}{3}\mathrm{\hspace{0.17em}0}\frac{1}{2})}^{\prime }$	${\mathrm{𝟐}}_{-5/3}^{\prime }$	$1$	$+2$	$-6$	$-2$	$+4$	$+4$	$+8$	$-954$(-$\frac{53}{7}$)	$D_7$
$T_4^{+}$	$(\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6};\frac{-1}{6}\frac{1}{6}\frac{1}{2}){(\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{2};\frac{-1}{6}\frac{1}{2}\frac{1}{2})}^{\prime }$	${\mathrm{𝟐}}_{-5/3}^{\prime }$	$2$	–2	+2	+6	$-8$	$+4$	$+8$	$-450$(-$\frac{25}{7}$)	$2D_8$
$T_4^{-}$	$(\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6};\frac{-1}{6}\frac{-1}{2}\frac{1}{6}){(\frac{1}{3}\frac{1}{3}\frac{1}{3}\frac{1}{3}\mathrm{\hspace{0.17em}0};\frac{2}{3}\mathrm{\hspace{0.17em}0\hspace{0.17em}0})}^{\prime }$	${\mathrm{𝟐}}_{5/3}^{\prime }$	$2$	–2	–6	+2	$+8$	$-4$	$-8$	$-810$(-$\frac{45}{7}$)	$2D_9$
$T_7^{+}$	$(\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6};\frac{1}{3}\frac{2}{3}\mathrm{\hspace{0.17em}0}){(\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{2};\frac{7}{12}\frac{1}{4}\mathrm{\hspace{0.17em}0})}^{\prime }$	${\mathrm{𝟐}}_{5/3}^{\prime }$	$1$	$+4$	$+8$	$0$	$+4$	$-5$	$+5$	$+1782$(+$\frac{99}{7}$)	$D_{10}$
$T_7^{+}$	$(\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6};\frac{-2}{3}\frac{-1}{3}\mathrm{\hspace{0.17em}0}){(\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{6}\frac{1}{2};\frac{7}{12}\frac{1}{4}\mathrm{\hspace{0.17em}0})}^{\prime }$	${\mathrm{𝟐}}_{5/3}^{\prime }$	$1$	$-8$	$-4$	$0$	$+4$	$-5$	$+5$	$-990$(-$\frac{55}{7}$)	$D_{11}$
$T_7^{-}$	$(\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6};\frac{1}{3}\mathrm{\hspace{0.17em}0}\frac{-1}{3}){(\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{6}\frac{-1}{2};\frac{-1}{12}\frac{-3}{4}\mathrm{\hspace{0.17em}0})}^{\prime }$	${\mathrm{𝟐}}_{-5/3}^{\prime }$	$1$	$+4$	$0$	$-4$	$+8$	$+5$	$-5$	$+234$(+$\frac{13}{7}$)	$D_{12}$

Equations

$$\begin{array}{rl}& \mathrm{SU}(3):\mathrm{Vectorlike}\mathrm{representations},\mathrm{hence}\mathrm{not}\mathrm{allowed},\\ & \mathrm{SU}(2)\times \mathrm{SU}(2):\mathrm{No}\mathrm{chiral}\mathrm{theory}\mathrm{with}\mathrm{even}\mathrm{number}\mathrm{of}\mathrm{doublets},\\ & \mathrm{U}(1)\times \mathrm{U}(1{)}^{\prime }:\mathrm{Six}\mathrm{conditions}\mathrm{for}\mathrm{the}\mathrm{absence}\mathrm{of}\mathrm{anomalies},\{\mathrm{Tr}Y,\mathrm{Tr}{Y}^{\prime },\mathrm{Tr}{Y}^3,\mathrm{Tr}{Y}^{\prime 3},\mathrm{Tr}Y{Y}^{\prime 2},\mathrm{Tr}{Y}^2{Y}^{\prime }\}=0,\end{array}$$

$$\mathrm{SU}(2)\times \mathrm{U}(1):\mathrm{Two}\mathrm{conditions}\mathrm{with}\mathrm{doublets}\mathrm{and}\mathrm{singlets},\{\mathrm{Tr}Y,\mathrm{Tr}{Y}^3\}=0.$$

$$\begin{array}{rl}& \sum _{i=1}^{4N}{Q}_i=0,\end{array}$$

$$\begin{array}{rl}& \sum _{i=1}^{4N}{Q}_i^3=\left(\sum _{i=1}^{4N}{Q}_i\right)\left(\sum _{i=1}^{4N}{Q}_i^2-\sum _{i\ne j}^{4N}{Q}_i{Q}_j\right)+3\sum _{i\ne j\ne k}^{4N}{Q}_i{Q}_j{Q}_k=0.\end{array}$$

$$\begin{array}{r}\left(\begin{array}{c}4N\\ 3\end{array}\right)\end{array}$$

$$\begin{split}&Q=\frac{1}{2}:\quad\ell_{i}=\begin{pmatrix}E_{i}\\[6.00006pt] N_{i}\end{pmatrix}_{\frac{+1}{2}},~{}\begin{array}[]{c}E_{i\,,-1}^{c}\\[6.00006pt] N_{i\,,0}^{c}\end{array},~{}(i=1,2,3)\\[6.99997pt] &Q=-\frac{3}{2}:~{}{\cal L}=\begin{pmatrix}{\cal E}\\[6.00006pt] {\cal F}\end{pmatrix}_{\frac{-3}{2}},~{}~{}\begin{array}[]{c}{\cal E}_{,+1}^{c}\\[6.00006pt] {\cal F}_{,+2}^{c}\end{array}\end{split}$$

$$\begin{array}{r}{\Phi }_u={\left(\begin{array}{c}{\varphi }_u^{+}\\ {\varphi }_u^0\end{array}\right)}_{Q=\frac{+1}{2}},{\Phi }_d={\left(\begin{array}{c}{\varphi }_d^0\\ {\varphi }_d^{-}\end{array}\right)}_{Q=\frac{-1}{2}},S_{Q=2},\end{array}$$

$$⟨{\varphi }_{u,d}^0⟩=\frac{V_{u,d}}{\sqrt{2}},⟨S⟩=\frac{V_S}{\sqrt{2}}.$$

$$M_{Z_1^{\prime }}^2=(g_2^2+g_Q^2)V_D^2=\frac{g_2^2}{{\cos }^2\theta }{V}_D^2,$$

$$M_{Z_2^{\prime }}^2=(2g_Q{)}^2\frac{V_S^2}{2}=2g_Q^2{V}_S^2=2g_2^2{\tan }^2\theta V_S^2,$$

$$r=\sqrt{2}\left|\frac{V_S}{V_D}\sin \theta \right|.$$

$$\mathcal{L}=-\frac{1}{4}{\hat{F}}_{\mu \nu }{\hat{F}}^{\mu \nu }-\frac{1}{4}{\hat{X}}_{\mu \nu }^1{\hat{X}}^{1\mu \nu }-\frac{1}{4}{\hat{X}}_{\mu \nu }^2{\hat{X}}^{2\mu \nu }-\frac{{\xi }_1}{2}{\hat{F}}_{\mu \nu }{\hat{X}}^{1\mu \nu }-\frac{{\xi }_2}{2}{\hat{F}}_{\mu \nu }{\hat{X}}^{2\mu \nu }-\frac{{\xi }_{12}}{2}{\hat{X}}_{\mu \nu }^1{\hat{X}}^{2\mu \nu },$$

$$\displaystyle\left(\begin{array}[]{c}A_{\mu}\\[5.0pt] X^{1}_{\mu}\\[5.0pt] X^{2}_{\mu}\\ \end{array}\right)=\left(\begin{array}[]{ccc}B_{11}&0&~{}0\\[5.0pt] \frac{\xi_{1}-\xi_{2}\xi_{12}}{\sqrt{1-\xi_{12}^{2}}}&\sqrt{1-\xi_{12}^{2}}&~{}0\\[5.0pt] \xi_{2}&\xi_{12}&~{}1\end{array}\right)\left(\begin{array}[]{c}\hat{A}_{\mu}\\[5.0pt] \hat{X}^{1}_{\mu}\\[5.0pt] \hat{X}^{2}_{\mu}\end{array}\right),$$

$$B_{11}=\sqrt{1-\frac{({\xi }_1-{\xi }_2{)}^2+2{\xi }_1{\xi }_2(1-{\xi }_{12})}{1-{\xi }_{12}^2}}$$

$$\mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }-\frac{1}{4}{X}_{\mu \nu }^1{X}^{1\mu \nu }-\frac{1}{4}{X}_{\mu \nu }^2{X}^{2\mu \nu },$$

$$\mathcal{L}=\bar{\psi }\left(Q\hat{e}{\gamma }^{\mu }\right)\psi {\hat{A}}_{\mu }.$$

$$\mathcal{L}=\bar{\chi }\left({\hat{e}}_i^{\mathrm{ex}}{Q}_i^{\mathrm{ex}}{\gamma }^{\mu }\right)\chi {\hat{X}}_{\mu }^i,$$

$$\mathcal{L}=\bar{\psi }\left(\frac{\sqrt{1-{\xi }_{12}^2}}{\sqrt{1-{\xi }_1^2-{\xi }_{12}^2+2{\xi }_1{\xi }_2{\xi }_{12}}}Q\hat{e}\right){\gamma }^{\mu }\psi A_{\mu },$$

$$\frac{1}{\mathrm{Det}}\left(\begin{array}{ccc}\sqrt{1-{\xi }_{12}^2},& 0,& 0\\ -\frac{{\xi }_1-{\xi }_2{\xi }_{12}}{\sqrt{1-{\xi }_{12}^2}},& B_{11},& 0\\ -\frac{{\xi }_2-{\xi }_1{\xi }_{12}}{\sqrt{1-{\xi }_{12}^2}},& -{\xi }_{12}{B}_{11},& B_{11}\sqrt{1-{\xi }_{12}^2}\end{array}\right)$$

$$\mathrm{Det}=\sqrt{1-{\xi }_1^2-{\xi }_2^2-{\xi }_{12}^2+2{\xi }_1{\xi }_2{\xi }_{12}}.$$

$$\mathcal{L}=\bar{\chi }{\gamma }^{\mu }\left[\frac{{\hat{e}}_1^{\mathrm{ex}}}{\mathrm{Det}}\left(B_{11}{X}_{\mu }^1-\frac{{\xi }_1-{\xi }_2{\xi }_{12}}{\sqrt{1-{\xi }_{12}^2}}{A}_{\mu }\right)Q_1^{\mathrm{ex}}+\frac{{\hat{e}}_2^{\mathrm{ex}}}{\mathrm{Det}}\left(B_{11}\sqrt{1-{\xi }_{12}^2}{X}_{\mu }^2-B_{11}{\xi }_{12}{X}_{\mu }^1-\frac{{\xi }_2-{\xi }_1{\xi }_{12}}{\sqrt{1-{\xi }_{12}^2}}{A}_{\mu }\right)Q_2^{\mathrm{ex}}\right]\chi .$$

$$e=\frac{\hat{e}}{\sqrt{1-{\xi }_1^2-{\xi }_2^2}},$$

$$X=(-2,-2,-2,-2,-2;0^3)(0^8{)}^{\prime }.$$

$$Q_{\mathrm{anom}}=84Q_1+147Q_2-42Q_3-63Q_5-9Q_6,$$

$$Q_1$$

$$Q_2$$

$$Q_3$$

$$Q_4$$

$$Q_5$$

$$Q_6$$

$$\begin{array}{rl}& ({\overline{10}}^{\prime },1^{\prime }),(5^{\prime },2^{\prime }),({\overline{5}}^{\prime },1^{\prime }),(1^{\prime },2^{\prime }),\mathrm{under}\mathrm{SU}(5{)}^{\prime }\times \mathrm{SU}(2{)}^{\prime }.\\ & \to {\Psi }_{AB},{\Phi }^{A\alpha },{\psi }_A,{\varphi }^{\alpha },\end{array}$$

$$\{A\}\equiv \{i,3,I\}.$$

$$⟨{\Phi }^{A=3,\alpha =2}⟩=V_1$$

Full text

Chiral models, Dark Z’s, Kinetic mixing, String compactification.

I Introduction

In particle physics, there has been a deep question known as the gauge hierarchy problem: “How do the SM fermions appear at such a small electroweak scale, compared to an ultra-violet completed scale, the Planck mass $M_{\rm P}$ or the grand unification (GUT) scale $M_{\rm GUT}$?” Two issues in the hierarchy problem are: (i) obtaining massless SM particles at the ultra-violet completed scale, and (ii) rendering the elctroweak scale masses to the SM fermions. The first issue is resolved by the profound and simple requirement, a chiral theory at the ultra-violet completed scale Georgi79 . The second issue is the method obtaining the vacuum expectation value (VEV) of the Higgs field at the electroweak scale v_{\rm ew}$$\simeq 246\,\,\textrm{GeV}, a kind of TeV scale, for which the most well-known example is supersymmetry (SUSY) Ibanez82 .

In this paper, we propose that any particles appearing at the TeV scale, for a detection possibility at the LHC, must satisfy Condition (i). The best known example is a spinor representation in the SO($4n+2$) GUT models SpinorSO . Orbifolding in extra dimensions presents a possibility of massless particles, as shown in a simple field theoretic orbifold Kawa01 . But, the orbifold compactification in string theory is the prototype example OrbString , providing a simple geometrical interpretation. Note, however, that fermionic constructions fermionic and Gepner models Gepner have been also used in four dimensional (4D) phenomenology from string. In these 4D constructions, it was necessary to check whether vectorlike representations of exotically charged particles, which appear quite often, are present or not as studied in Refs. Exotics .

Anyway, Condition (i) is the basic requirement we satisfy at low energy effective theory in 4 dimensions (4D). To realize Condition (ii), model parameters are required to be known in detail, and hence we do not discuss it here except pointing out several mass scales in particle physics. The SM a chiral theory, realized in Nature, describes the electroweak scale physics successfully. So, we anticipate that if a natural chiral model is found then it might have a great chance to be realized in Nature. Since any new particle has not been detected at the LHC so far, a new particle in the new chiral theory, which interacts with the SM sector extremely feebly, must be in the dark sector. Here, the dark sector is not introduced just for explaining cold dark matter (CDM) of the Universe. The well-known CDM examples, “invisible” axions KSVZ and weaky interacting massive particles (WIMPs) LeeWein77 , are belonging to the visible sector. On the other hand, the heterotic ${\rm E_{8}\times E_{8}^{\prime}}$ string Gross84 implies a possibility of dark sector from E${}_{8}^{\prime}$. If the dark sector introduces CDM, then it is just a bonus.111In this paper, we do not introduce the dark sector for the sake of dark matter Arkani09 . Even though the dark sector interacts with the SM sector extremely feebly, it can be probed by the kinetic mixing terms Okun:1982xi of two SM gauge bosons, photon and $Z$. Since the dark sector does not carry the SM weak hyperchage, the charge raising and lowering gauge bosons in the dark sector cannot have kinetic mixings with $W^{\pm}$ of the SM.

We will show that the simplest chiral theory which does not have any gravitational and gauge anomalies is SU(2)${}_{\rm dark}\times$U(1)Q. Because the rank of this gauge group is 2, we will have two $Z^{\prime}$ gauge bosons at low energy. This minimal model will be called Two Dark $Z$ model (TDZ). The first new particles observed at the CERN SPS proton-antiproton collider were $W^{\pm}$ and $Z$ WZdiscover . This is because it is relatively easy to identify leptons at high energy colliders. With this new chiral model, therefore, we expect that the first new particles expected at the LHC are two dark $Z$’s.

In Sec. II, we present the minimal chiral model. In Sec. III, the kinetic mixing in the SU(2)${}_{\rm dark}\times$U(1)Q is discussed. In Sec. IV, it is shown that the minimal chiral model is derivable from a string compactification. Section V is a conclusion. In Appendix, a SUSY scenario based on the hidden sector SU(5)′ from ${\bf Z}_{12-I}$ orbifold compactification Huh09 is discussed.

II Minimal chiral model

If a new gauge boson pops up near the electroweak scale, it must be from a chiral theory. A chiral theory near the electroweak scale should not have gravititonal and gauge anomalies. First, consider the rank-1 gauge groups. If we consider an SU(2), it is not possible to have a chiral theory because there must be even number of doublets WittenSU2 . With only one U(1)Y group, the absence of gravitational anomaly requires Tr $Y=0$ and the absence of gauge anomaly in addition requires Tr $Y^{3}=0$. For example, even though two charged fields $Y=+1$ and $-1$ do not have these anomalies, the model is not allowed in our framework because it is vector-like. But if we use the $Y$ of the SM, these two conditions are satisfied.222In addition, the SM requires Tr $Y=0$ for quarks and leptons separately, and also additional conditions for the absence of non-Abelian gauge anomalies. Second, let us consider the rank-2 gauge groups,

[TABLE]

and

[TABLE]

Thus, the simplest case is SU(2)${}_{\rm dark}\times$U(1)Q, and at least two dark-Z*′*’s are predicted at low energy. Two conditions in (1) for $N$(= even) doublets and $2N$ singlets are

[TABLE]

and

[TABLE]

Condition (2) satisfies Eq. (3) if the term $\sum_{i\neq j\neq k}$ is vanishing. The number of terms in this sum is

[TABLE]

which is very large. So, a complete search is more involved. The well-known chiral theory, satisfying (3), is the SM, SU(3)${}_{C}\times$SU(2)${}_{L}\times$U(1)Y.

Here, we present a simpler one SU(2)${}_{\rm dark}\times$U(1)Q, satisfying the conditions in (2,3), with the following fermions

[TABLE]

where the subscripts denote the $Q$ charges. There are four doublets without the SU(2) anomaly WittenSU2 . One set, one of $E_{i\,,-1}^{c}$ and ${\cal E}_{,+1}^{c}$, forms a vector-like pair, but we keep them to provide masses for all the particles after breaking SU(2)${}_{\rm dark}\times$U(1)Q. In Eq. (5), there appear 16 left-handed chiral fields,333Note that the SM has 45 chiral fields. and they do not introduce any gravitational and gauge anomalies. It is interesting to observe that there appear 16 chiral fields as in the spinor representation 16 in SO(10). Eq. (5) realizes the TDZ model.

To break the rank-2 gauge group SU(2)${}_{\rm dark}\times$U(1)Q completely and to give fermions masses, let us introduce two doublets444With supersymmetry, we need two doublets to make all chiral fields massive. and a singlet of scalars,

[TABLE]

where their vacuum expectation values (VEVs) are,555Choosing the $Q=2$ singlet for breaking U(1) is just for an illustration.

[TABLE]

Then, masses of two dark-Z*′*’s are

[TABLE]

where $g_{2}$ and $g_{Q}$ are the SU(2)dark and U(1)Q couplings, respectively, $\tan\theta\equiv g_{Q}/g_{2}$, and $V_{D}^{2}=V_{u}^{2}+V_{d}^{2}$. Thus, the mass ratio of two dark-Z*′* masses is

[TABLE]

If $V_{S}\to 0$, $Z^{\prime}_{2}$ may be called dark-photon, which is included in our terminology TDZ. This estimate will be used in Appendix.

III The kinetic mixing

If multiple dark-U(1) gauge bosons are present, they can mix with the SM photon, most probably via kinetic mixings as suggested in Okun:1982xi . Since the rank of SU(2)${}_{\rm dark}\times$U(1)Q gauge group is 2, there are two dark-Z*′’s and we summarize their kinetic mixing with photon.666Their mixing with the $Z$-boson is omitted here. These arise via loops between photon and dark-photon through an intermediate particle(s) $\chi$ which carries both the electromagnetic and dark charges. After a proper diagonalization procedure of the kinetic energy terms, then the electromagnetic charge of $\chi$ can be millicharged, ${\cal O}(\alpha/2\pi)e$. In the heterotic ${\rm E_{8}\times E_{8}^{\prime}}$ string model, the extra E${}_{8}^{\prime}$ gauge group may contain dark-photons which will be called dark-Z′*’s, leading to the kinetic mixing of ${\cal O}(\alpha/2\pi)$Dienes97 . Indeed, an explicit model for this kind from string compactification exists in the literature KimJE:2007 ; Cheung07 .

The intermediate ${\cal O}$(MeV) millicharged particles have not been ruled out by observations in the previous study Davidson:2000hf . For the discovery possibility at the LHC, we consider the electroweak scale dark-Z*′*’s.

Consider three Abelian gauge groups U(1)QED and U(1)${}_{i}\,(i=1,2)$. The kinetic mixing of U(1)QED and U(1)i dark-Z*′*,s are parameterized by, following the notation of ParkJC08 ,

[TABLE]

where $\hat{A}_{\mu}(\hat{X^{i}}_{\mu})$ is the U(1)QED (dark–U(1)i) gauge boson, and its field strength tensor is $\hat{F}_{\mu\nu}(\hat{X}^{i}_{\mu\nu})$. The kinetic mixings are parameterized by $\xi$’s which are generically allowed by the gauge invariance and the Lorentz symmetry. In the low-energy effective theory, $\xi$’s are considered to be completely arbitrary parameters. An ultraviolet-completed theory is expected to generate the kinetic mixing parameters. The usual diagonalization procedure of these kinetic terms leads to the relation,

[TABLE]

where

[TABLE]

and we obtain

[TABLE]

where the new field strengths are $F_{\mu\nu},X^{1}_{\mu\nu}$, and $X^{2}_{\mu\nu}$. Photon corresponds to $A_{\mu}$ and dark-Z*′s correspond to $X^{i}_{\mu}\,(i=1,2)$. If the dark-Z′*’s are exactly massless, there exists an SO(3) symmetry in the $A_{\mu}-X^{i}_{\mu}$ field space.

Using the above SO(3) symmetry, let us take the following simple interaction Lagrangian of a SM fermion with a photon in the original basis as

[TABLE]

Note that in this basis there is no direct interaction between the electron and the hidden sector gauge boson $\hat{X}$. If there exists a hidden sector Dirac fermion $\chi$ with the U(1)ex charge $Q_{\chi}$, its interaction with the hidden sector gauge boson is simply represented by

[TABLE]

where $\hat{e}_{\rm ex}$ can be different from $\hat{e}$ in general. In this case, there is also no direct interaction between the hidden fermion and the visible sector gauge boson $\hat{A}_{\mu}$. We can recast the Lagrangian (24) in the transformed basis $A_{\mu}$ and $X_{\mu}$,

[TABLE]

where we used the inverse of (21)

[TABLE]

with

[TABLE]

Here, one notices that the standard model fermion has a coupling only to the visible sector gauge boson $A_{\mu}$ even after changing the basis of the gauge bosons. However, the coupling constant $\hat{e}$ is modified to $e$ as suggested in Eq. (26). Similarly, we derive the following couplings for $\chi$,

[TABLE]

In this basis, the hidden sector matter field $\chi$ now can couple to the visible sector gauge boson $A_{\mu}$ with the couplings $-\hat{e}^{\rm ex}_{1}Q_{1}^{\rm ex}(\xi_{1}-\xi_{2}\xi_{12})/\sqrt{1-\xi_{12}^{2}}/{\rm Det}$ to the mass eigenstate $X^{1}_{\mu}$ and $\hat{e}^{\rm ex}_{2}Q_{2}^{\rm ex}(\xi_{2}-\xi_{1}\xi_{12})/\sqrt{1-\xi_{12}^{2}}/{\rm Det}$ to the mass eigenstate $X^{2}_{\mu}$. In terms of the afore-mentioned SO(3) symmetry, it simply means the mismatch between the gauge couplings of the electron and other fermions. Thus, we can set the physical hidden sector coupling $e_{\rm ex}$ as $e_{\rm ex}\equiv\hat{e}_{\rm ex}$ and we define the coupling of the field $\chi$ to the visible sector gauge boson $A_{\mu}$, introducing the millicharge parameter $\varepsilon_{i}$, as $\varepsilon_{i}e$ such that

[TABLE]

Note in general that $e\neq e^{\rm ex}_{i}$. Since $\xi_{i,12}\simeq{\cal O}(\varepsilon_{i}e/e^{\rm ex}_{i})$ is expected to be small, the condition $\xi_{i,12}<1$ gives $\alpha^{\rm ex}_{i}/{\alpha}>\varepsilon^{2}$. From a fundamental theory, one can calculate the ratio $e_{\rm ex}/e$ in principle, which is possible with the detail knowledge of the compactification radius KKmasses .

Similarly, one can calculate the mixing of the SM $Z$ boson with dark-Z*′*’s, which however is not presented here.

IV From a string model

In this section, we derive the minimal chiral model (5) discussed in Sec. II from a string theory. The ${\rm E_{8}\times E_{8}^{\prime}}$ heterotic string model compactified on ${\bf Z}_{12-I}$ orbifold gives the flipped SU(5)flip times SU(5)${}^{\prime}\times$SU(2)′ with several extra U(1)’s Huh09 . Here, the factor SU(5)flip contains a gauge group U(1): SU(5)$\times$U(1)X. The first important U(1) gauge group is U(1)X in SU(5)$\times$U(1)X, which is free of any gauge anomaly. The second is the anomalous U(1)anom. Except these two U(1) factors, U(1)X and U(1)anom, the non-Abelian gauge group is SU(5)$\times$SU(5)${}^{\prime}\times$SU(2)′. Note that the charges of U(1)X and U(1)anom are,777For the definition, see, Ref. KimKyae17 .

[TABLE]

[TABLE]

where

[TABLE]

In the orbifold compactification, frequently there appears an anomalous U(1)A gauge fields $\tilde{A}_{\mu}$ from a subgroup of ${\rm E_{8}\times E_{8}^{\prime}}$ Anom87 . The charge of this anomalous U(1)A is given in Eq. (32). In addition, the anomaly cancellation in ten dimensions (10D) requires the so-called Green-Schwarz (GS) term in terms of the second rank antisymmetric-tensor field $B_{MN}\,(M,N=1,2,\cdots,10)$ GS84 . The 10D $B_{MN}$ always introcuces a model-independent (MI) axion $a_{\rm MI}$ in 4D, $\partial_{\mu}a_{\rm MI}\propto\epsilon_{\mu\nu\rho\sigma}H^{\nu\rho\sigma}\,(\mu,\rm etc.=1,2,3,4)$ where $H^{\nu\rho\sigma}$ is the field strength of $B^{\rho\sigma}$ Witten84 . The anomalous U(1) gauge boson absorbs the MI axion to become massive and there results a global symmetry U(1)anom below the compactification scale. More phenomenologically, U(1)anom can be suggested for a plausible flavor symmetry KimKyae17 . The global symmetry U(1)anom is good for a Peccei-Quinn symmetry PQ77 toward “invisible” axions at the intermediate scale $M_{\rm int}$ KSVZ . Except the two U(1)’s, Eqs. (31,32), all U(1)’s are assumed to be broken at a high energy scale, much above $M_{\rm int}$. In a more detail, it works as follows. Suppose that five U(1) charges out of $Q_{1,\cdots,6}$ are broken, and there is only one gauge symmetry remaining, which we identify as U(1)anom. Now, we can consider two continuous parameters, one is the MI-axion direction and the other the phase of U(1)anom transformation. Out of two continuous directions, only one phase or pseudoscalar is absorbed by the U(1)anom gauge boson, and one continuous direction survives. The remaining continuous degree corresponds to a global symmetry, which is called the ’t Hooft mechanism Hooft71 : “If both a gauge symmetry and a global symmetry are broken by one scalar VEV, the gauge symmetry is broken and a global symmetry is surviving”. The resulting global charge is a linear combination of the original gauge and global charges. Even though we obtain a global symmetry U(1)anom, it is obtained from the original two gauge symmetries, one from the two-index anti-symmetric tensor gauge field $B_{MN}$ in 10D and the other the U(1)anom subgroup of ${\rm E_{8}\times E_{8}^{\prime}}$ given in Eq. (32).

Here, the primed goups SU(5)${}^{\prime}\times$SU(2)′ are the hidden sector non-Abelian gauge groups. The hidden sector representations under SU(5)${}^{\prime}\times$SU(2)′ are given in Tables 1 and 2 KimKyae17 . After removing vector-like representations from Tables 1 and 2, we obtain

[TABLE]

where the tensor notation is used in the second line with the SU(5)′ index $A=\{1,2,3,4,5\}$ and the SU(2)′ index $\alpha=\{1,2\}$. The representations in (33) do not lead to an SU(5)′ anomaly. Let two SU(2) subgroup indices of SU(5)′ be $i=\{1,2\}$ and $I=\{4,5\}$ so that the five SU(5)′ indices split into

[TABLE]

By the VEV of $\Phi^{A\alpha}\equiv$ (5′,2′),

[TABLE]

we obtain a group containing two SU(2)$\times$U(1)′ subgroups from non-Abelian factors SU(5)${}^{\prime}\times$SU(2)′, *i.e. *a rank-4 subgroup from the rank-5 non-Abelian group, which is denoted as

[TABLE]

where the index $i$ is for ${\rm SU(2)}_{1}$ and the index $I$ is for ${\rm SU(2)}_{2}$. In fact, the VEV (35) breaks the rank-5 SU(5)${}^{\prime}\times$SU(2)′ down to rank-4 SU(4)${}^{\prime}\times$U(1)′. The rank-3 SU(4)′ is further broken down to rank-2 SU(2)${}_{1}\times$ SU(2)2 by the VEV in the direction,

[TABLE]

Summarizing the above discussion, the rank-5 SU(5)${}^{\prime}\times$SU(2)′ is broken down to a rank-3 group by $V_{1}$ and $V_{2}$,

[TABLE]

where

[TABLE]

and

[TABLE]

Thus, $V_{2}$ breaks $Y_{2}$ which does not participate in $Q$ of Eq. (39). SU(2)1 and SU(2)2 generators are

[TABLE]

The SU(2)${}_{1}\times$SU(2)${}_{2}\times$U(1)Q quantum numbers are

[TABLE]

where several colored pairs form vectolike representations. Removing the green and blue vector-like pairs, and one combination of the red pair ${{\bf 1}}_{0,A}=(1/{\sqrt{2}})[({\bf 1,1})_{0}^{(a)}-({\bf 1,1})_{0}^{(b)}]$ where $S(A)$ represents the (anti-)symmetric combination, we obtain

[TABLE]

Now, let us break888In Appendix, we do not break SU(2)2. SU(2)2 by the VEV $\langle({\bf 1,2})_{0}\rangle$ which does not carry the $Q$ charge. So, the surviving gauge group is SU(2)${}_{\rm dark}\times$U(1)Q where SU(2)dark is SU(2)1. Then, there result SU(2)${}_{\rm dark}\times$U(1)Q representations

[TABLE]

which are exactly those appearing in Eq. (5).

Considering only the low energy SUSY, we have shown that the minimal chiral model is derivable from a string compactification. So, it will be useful if the SUSY scenario is consistent with the unification of gauge coupling constants. Since there are so many unknown parameters in this study, we deferred a brief discussion on SUSY scenario to Appendix.

V Conclusion

We obtained a new chiral model with the gauge group SU(2)${}_{\rm dark}\times$U(1)Q without any gauge and gravitational anomalies. This gauge group may belong to the dark sector. We also derived this chiral spectrum from a compactification of the heterotic string. This new chiral theory has a chance to be found at TeV scale accelerators through the kinetic mixing effects. Necessarily, there appear two degrees at low energy: two dark-Z*′’s, or a dark-Z′* plus a dark-photon (if $V_{S}=0$ in Eq. (9)).

Acknowledgements.

I thank useful discussions with W.S. Cho and J-C. Park. This work is supported in part by the National Research Foundation (NRF) grant funded by the Korean Government (MEST) (NRF-2015R1D1A1A01058449) and the IBS (IBS-R017-D1-2016-a00).

Appendix: Hidden sector SU(5)′

V.0.1 Mass scales

Toward a suggestion for an ultra-violet completed theory, we discuss at which scales symmetry breakings are introduced. Firstly, we need one confining force for dynamical SUSY breaking Nilles83 ; DIN85 . The confining non-Abelian gauge group at the intermediate scale is chosen as SU(5)′. Around the same scale, SU(5)′ is broken down to SU(4)′ and at a somewhat lower scale to SU(2)1 by the condensation of matter superfield, breaking SU(2)2. Because SU(2)2 is neutral under the SU(2)${}_{\rm dark}\times$U(1)Q transformation, the discussion leading to the minimal model is intact. A rough sketch of related scales is shown in Fig. 1.

The confining superfields in Eq. (43) are

[TABLE]

where $\alpha=\{1,2\}$ counts the number of color degrees of SU(2)2. The anomaly matching conditions Hooft79 must lead to the following composite states under SU(2)${}_{\rm dark}\times$U(1)Q,

[TABLE]

where the composite states D and S are SU(2)${}_{\rm dark}\times$U(1)Q doublets and singlets, respectively, composed of $A,B$, and $C$ degrees,

[TABLE]

Even though SU(2)2 is smaller than the color SU(3)C, it can confine at the intermediate scale if SU(5)′ and SU(4)′ runs between the GUT scale and the intermediate scale. So, the SU(4)′ breaking VEV $V_{2}$, Eq. (37), is around the intermediate scale.

[TABLE]

where $V_{1}<{M_{\rm GUT}}$. From the compactification scale down to $M_{\rm GUT}$, SU(5)′ runs more steeply than SU(2)′, which is illustrated as the separate couplings at $V_{1}$ in Fig. 1.

In the radiative breaking of the SM gauge group in the MSSM, the large top quark Yukawa coupling plays a crucial role. To break SU(2)${}_{\rm dark}\times$U(1)Q, near the electroweak scale, we need some large Yukawa coupling(s) involving ${\bf 2}_{\frac{+1}{2}}$’s, ${\bf 2}_{\frac{-3}{2}},{{\bf 1}}_{-1}$’s, and ${{\bf 1}}_{+2}$ in Eq. (44).

V.0.2 Running of couplings

For a rough guess of the coupling constants, we use just one loop evolution equations. With the mass order of Fig. 1, we have the following running of gauge couplings,999Spectra of SU(5)$\times$U(1)X are counted from Ref. KimKyae17 . The gauge group U(1)Q, which is not the anomalous U(1), survives down to the TeV scale.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where SU(5)′ couplings are tilded. In the figure, we also sketched the running of the SU(2)W ( SU(3)C) coupling as the (green) dash line. From the observed value of $\alpha_{2}$ at $\mu=M_{Z}$ for $\sin^{2}\theta_{W}(M_{Z})\simeq 0.23$ KimRMP81 , we obtain its GUT scale value. Identifying this as the SU(5)′ coupling at $M_{\rm GUT}$, we estimate the couplings as sketched in Fig. 1. We assume that SU(4)′ gauginos condenses at $\approx V_{1}$,

[TABLE]

Then, between $V_{1}$ and $V_{2}$, we consider the group SU(4)′. At $V_{2}$, $\tilde{\alpha}_{2}$ has not reached to an order one value, but $\langle\Psi_{IJ}\rangle$ can be developed at $V_{2}$. The representation 6 of SU(4)′ has a larger Casimir operator $\frac{5}{2}$ than that of the fundamental representation $\frac{15}{8}$. So, we expect that there appear composites ${\bf D}_{\frac{+1}{2}}$ and ${\bf S}_{-1}$ discussed in Appendix A.

Using the electroweak coupling at $M_{Z}$, $\alpha_{2}\simeq 3.38\times 10^{-2}$ Earler14 , we obtain its evolution to $M_{\rm GUT}$, $\alpha_{2}({M_{\rm GUT}})\simeq 0.0412$ where ${M_{\rm GUT}}=2.5\times 10^{16\,}\,\textrm{GeV}$ is used. At the hypothetical string scale $M_{s}\simeq 0.7\times 10^{18\,}\,\textrm{GeV}$, we obtain the coupling $\alpha_{5}(M_{s})\simeq 0.0389$ which is equated to $\tilde{\alpha}_{5}(M_{s})$. Now, we can run the hidden sector couplings down from $M_{s}$. Suppose that gaugino condensation occurs at $M_{\rm cond}=10^{13\,}\,\textrm{GeV}$. With $V_{1}={M_{\rm GUT}}$,

[TABLE]

For SU(5), the Casimir of the adjoint representation is 25/12 times larger than that of the fundamental representation. So, gauginos couple more strongly than the fundamentals. The SU(4)′ coupling at $V_{2}$ is

[TABLE]

Let us equate (56) as the SU(2)1 coupling at $V_{2}$. Then, the SU(2)1 and U(1)Q couplings at $M_{Z}$ are

[TABLE]

such that the mixing angle is $|\sin\theta|_{M_{Z}}\simeq\sqrt{\tilde{\alpha}_{Q}/(\tilde{\alpha}_{2}+\tilde{\alpha}_{Q}})|_{M_{Z}}\simeq 0.213$. At $M_{Z}$, $\tilde{\alpha}_{2}(M_{Z})$ is much larger than the electroweak coupling $\alpha_{2}(M_{Z})$. If the VEV $\langle{{\bf 2}}_{{1}/{2}}\rangle$ has the same order as $V_{D}$ of Eq. (8), then we obtain dark-Z*′1 mass at the electroweak scale. Actually, the dark-Z′* masses depend on the parameters, the mixing angle and the VEVs given in Eqs. (8) and (9).

We also note that there exists a possible superpotential term,

[TABLE]

which may allow $\langle\Psi_{IJ}\rangle=V_{2}$ by the condensation of $\langle{({{\bf 1},{\bf 2}})_{+1}({{\bf 1},{\bf 2}})_{-1}}\rangle$.

V.0.3 Confinement of SU(2)1

The large SU(2)1 coupling in Eq. (57) suggests a possibility that SU(2)1 confines around the electroweak scale. Let us consider four doublets of Eq. (5) as

[TABLE]

where two SU(2)1 doublets are represented as R-handed chiral fields and the subscripts are the U(1)Q charges. Below the SU(2)1 confinement scale, we consider the following condensations

[TABLE]

and use the mass ratio presented in Eq. (10). When $\tilde{\alpha}_{2}$ becomes order 1 at the scale $\mu_{2}$, let us assume that SU(2)1 confines. The condenstation scale is guessed as $\mu_{2}/2$, following the estimate of the QCD condensation scale $\langle\bar{u}u\rangle\approx 1\,\,\textrm{GeV}/3$ where $\alpha_{C}(1\,\,\textrm{GeV})\simeq O(1)$. $\tilde{\alpha}_{2}$ becomes order 1 at $\mu_{2}\simeq 22.2\,\,\textrm{GeV}$. In this setup, we estimate the masses of two dark-Z*′*’s as Susskind79 ,

[TABLE]

where we used $V_{S}=V_{D}$ and $\sin\theta=0.2$. Note, however, that our estimate is very primitive because we used $V_{1}={M_{\rm GUT}}$, one-loop running for gauge coupling evolution, followed the hypothetical SUSY breaking, and a naive chiral symmetry breaking below the SU(2)1 confinement scale. Nevertheless, this crude estimate has lead to two electroweak scale dark-Z*′*’s.