The pseudogap (PG) regime of the underdoped cuprates arguably remains one of the most enigmatic phenomena of correlated quantum matter. Recent theoretical ideas suggest that fractionalized bosonic fields can lead to the PG phase, by opening a gap in the anti-nodal (AN) region of the Brillouin zone. Such fractionalized boson can originate from modulated particle-particle pairs or pair density wave (PDW), a magnetic stripe, or a modulated spin one particle-hole pair like a spin density wave (SDW) boson, among others. The main picture goes as follows. Electrons under strong coupling tend to form different types of unstable bosons at high temperatures. As the temperature goes down, the compact object gets extremely unstable, and to minimize the entropy, it finally fractionalizes into elementary components, linked by a constraint. The process of fractionalization involves, in this way, an emergent gauge field directly linked to the constraint. This, in turn, couples to the Fermi surface of electronic carriers and opens a gap in the AN region, which is partly responsible for the PG phase. Alternative theoretical approaches invoke a simple coexistence between the multiple quasi-degenerate orders like charge density wave (CDW), superconductivity (SC), and magnetic orders at low temperatures. This scenario attributes the PG formation as a vestigial order showing up at $T^{*}$, which acts as a precursor to the zero temperature orders. This intricate situation calls for a key experimental test, enabling us to discriminate between the various theoretical scenarios. In this paper, we focus on the case where the PDW boson has fractionalized into a CDW and SC order below $T^{*}$, and we compare this to the situation where the two orders simply coexist.
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