Complete Gauge Threshold Corrections for Intersecting Fractional D6-Branes: The Z6 and Z6' Standard Models

TL;DR

This work delivers a complete one-loop gauge-threshold analysis for intersecting fractional D6-branes on T^6/Z_6 and T^6/Z_6' with SU(N), Sp, SO and massless U(1) factors. By integrating continuous open-string moduli (Wilson lines and brane displacements) into annulus and Möbius amplitudes, the authors derive explicit threshold formulas and demonstrate RR tadpole consistency across explicit SM-like constructions. They classify the T^6/Z_6' MSSM-like vacua into 16 massless-spectrum classes and 196 threshold-distinct models, showing that threshold corrections substantially diversify the low-energy gauge couplings beyond tree-level expectations and can influence unification prospects. The results provide a concrete statistical map of the string landscape in these toroidal orientifolds and furnish detailed data for connecting string thresholds to phenomenology and potential SUSY-breaking hidden sectors.

Abstract

We perform a complete analysis of one-loop threshold corrections to the gauge couplings of fractional D6-branes. This includes besides SU(N) also symplectic, orthogonal and massless Abelian gauge factors and the full computation of contributions from discrete and continuous Wilson lines and brane displacements. Two classes of globally consistent supersymmetric compactifications with Standard Model spectra on T6/Z6 and T6/Z6' are presented in detail with the latter exhibiting the potential of supersymmetry breaking via a hidden sector gaugino condensate. The T6/Z6' Standard Models are completely classified, and it turns out that out of 768 distinct D6-brane configurations only 16 different sets of massless spectra and ten distinct values of gauge couplings at one-loop arise. The gauge threshold corrections enhance the diversity to 196 nonequivalent models.

Figures (11)

• Figure 1: The two lattices $T^2_i$ which are invariant under a $\mathbb{Z}_2$ symmetry. In both cases, the geometric part of the orientifold action is $\pi_{2i} \stackrel{{\cal R}}{\rightarrow} -\pi_{2i}$, and by introducing $b=0,1/2$ for the rectangular and tilted torus, respectively, $\pi_{2i-1} \stackrel{{\cal R}}{\rightarrow}\pi_{2i-1} -(2 b) \,\pi_{2i}$. On the b-type or tilted lattice, two $\mathbb{Z}_2$ fixed points (1,4) are invariant under the orientifold projection, whereas the other two (2,3) are exchanged. The tori are also invariant under a $\mathbb{Z}_4$ symmetry if the radii are related by $R_1=(1-b) \,R_2$.
• Figure 2: The two lattices $T^2_i$ which are invariant under a $\mathbb{Z}_3$ or $\mathbb{Z}_6$ rotation. The only $\mathbb{Z}_6$ invariant point is the origin. There are two more $\mathbb{Z}_3$ invariant points 2 and 3 which are exchanged under the $\mathbb{Z}_2$ sub-symmetry, and a triplet of $\mathbb{Z}_2$ fixed points labeled by 4,5,6. On the A-type lattice, the orientifold projection acts on the one-cycles as $(\pi_{2i-1},\pi_{2i}) \stackrel{\cal R}{\rightarrow} (\pi_{2i-1},\pi_{2i-1} -\pi_{2i})$. On the B-type lattice, the orientifold symmetry exchanges the basic one-cycles, $\pi_{2i-1} \stackrel{{\cal R}}{\leftrightarrow} \pi_{2i}$. The $\mathbb{Z}_2$ fixed points $(4,5,6)$ are mapped under the orientifold projection to $(4,6,5)$ for the A- and $(5,4,6)$ for the B-type lattice, respectively.
• Figure 3: The lattice contributions $\Lambda$ depending on the two-torus volume $v$ for fixed $V\equiv 1$ and different values of $(\sigma,\tau)$: red, solid: $(0,0)$; green, dashed: $(1,0)$; blue, dotted: $(0,1)$; yellow, dotdashed: $(1,1)$. Note that in figure (b) the absolute value of $\Lambda$ is displayed double logarithmically.
• Figure 4: Threshold contributions in dependence of the two-torus volume $v$ for the $T^6/\mathbb{Z}_6$ example and $(\sigma^3_c,\tau^3_c)=(\frac{1}{2},0)$ and $(\sigma^3_x,\tau^3_x)=(0,0)$ for $x \neq c$. The contribution to $SU(3)_a$ is shown in solid red, $SU(2)_b$ in dashed green, $U(1)_Y$ in dotted blue, $U(1)_{B-L}$ in dot-dashed yellow and the 'hidden' sector $Sp(2)_e$ as sparse dashed grey.
• Figure 5: Gauge threshold contributions in dependence on the two-torus volume $v$ for the $T^6/\mathbb{Z}_6'$ examples with hidden sector branes (a) $Sp(6)_{h_3}$ and (b) $Sp(2)_{\hat{h}_1}$. The contribution to $SU(3)_a$ is shown in solid red, $SU(2)_b$ in dashed green, $U(1)_Y$ in dotted blue, $U(1)_{B-L}$ in dot-dashed yellow and for the hidden sector as sparse dashed grey. The geometric regime requires $v>1$, and the gauge thresholds are plotted for the choice of continuous displacements and Wilson lines $(\sigma^2_c,\tau^2_c)=(\frac{1}{2},0)$ and $(\sigma^2_x,\tau^2_x)=(0,0)$ for $x \neq c$.