Hairpin-branes and tachyon-paperclips in holographic backgrounds

D-branes with a U-shaped geometry, like the D8 flavor branes in the Sakai-Sugimoto model of QCD, are encountered frequently in holographic backgrounds. We argue that the commonly used DBI action is inadequate as an effective field theory description of these branes, because it misses a crucial component of the low-energy dynamics: a light complex scalar mode. Following an idea of Erkal, Kutasov and Lunin we elaborate on an effective description based on the abelian tachyon-DBI action which incorporates naturally the non-local physics of the complex scalar mode. We demonstrate its power in a context where an explicit worldsheet description of the open string dynamics exists-hairpin-branes in the background of NS5-branes. Our results are relevant for the holographic description of chiral symmetry breaking and bare quark mass in QCD and open string tachyon condensation in curved backgrounds.