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We have probed the crystalline electric-field ground states of pure $|J =7/2, J_z = \pm 5/2\rangle$ as well as the anisotropic $c$-$f$ hybridization inboth valence fluctuating systems $\alpha$- and $\beta$-YbAlB$_4$ by linearpolarization dependence of angle-resolved core level photoemissionspectroscopy. Interestingly, the small but distinct difference between \abybwas found in the polar angle dependence of linear dichroism, indicating thedifference in the anisotropy of $c$-$f$ hybridization which may be essential toa heavy Fermi liquid state in $\alpha$-YbAlB$_4$ and a quantum critical statein $\beta$-YbAlB$_4$.

Effect of Anisotropic Hybridization in YbAlB$_4$ Probed by Linear Dichroism in Core-Level Hard X-ray Photoemission Spectroscopy