Gauge symmetries and matter fields in F-theory models without section- compactifications on double cover and Fermat quartic K3 constructions times K3

We investigate gauge theories and matter fields in F-theory compactifications on genus-one fibered Calabi-Yau 4-folds without a global section. In this study, genus-one fibered Calabi-Yau 4-folds are built as direct products of a genus-one fibered K3 surface that lacks a section times a K3 surface. We consider i) double covers of $\mathbb{P}^1\times\mathbb{P}^1$ ramified along a bidegree (4,4) curve, and ii) complete intersections of two bidegree (1,2) hypersurfaces in $\mathbb{P}^1\times\mathbb{P}^3$ to construct genus-one fibered K3 surfaces without a section. $E_7$ gauge group arises in some F-theory compactifications on double covers times K3. We show that the tadpole can be cancelled for an F-theory compactification on complete intersection K3 times K3, when complete intersection K3 is isomorphic to the Fermat quartic, and the complex structure of the other K3 surface in the direct product is appropriately chosen.