Renormalization in large-N QCD is incompatible with open/closed string duality

Abstract

Solving by a canonical string theory, of closed strings for the glueballs and open strings for the mesons, the 't Hooft large-N expansion of QCD is a long-standing problem that resisted all the attempts despite the advent of the celebrated gauge/gravity duality in the framework of string theory. We demonstrate that in the canonical string framework such a solution does not actually exist because an inconsistency arises between the renormalization properties of the QCD S matrix at large N, recently worked out in Bochicchio (2017) [5], and the open/closed duality of the would-be string solution. Specifically, the would-be open-string one-loop corrections to the tree glueball amplitudes must be ultraviolet (UV) divergent by the aforementioned renormalization properties, which follow from the QCD asymptotic freedom (AF) and renormalization group. Hence, naively, the inconsistency arises because these amplitudes are dual to tree closed-string diagrams, which are universally believed to be both UV finite – since they are closed-string tree diagrams – and infrared (IR) finite because of the glueball mass gap. In fact, the inconsistency follows from a low-energy theorem of the Novikov–Shifman–Vainshtein–Zakharov (NSVZ) type derived in Bochicchio (2017) [5] that controls the renormalization in QCD-like theories. The aforementioned inconsistency extends to the would-be canonical string for a vast class of 't Hooft large-N confining asymptotically free QCD-like theories including N=1 SUSY QCD. We also demonstrate that the presently existing SUSY string models with a mass gap based on the gauge/gravity duality– such as Klebanov–Strassler, Polchinski–Strassler (PS) and certain PS variants – cannot contradict the above-mentioned results, not even potentially, since they are not asymptotically free. Moreover, we shed light on the way the open/closed string duality may be perturbatively realized in these string models compatibly with a mass gap in the 't Hooft-planar closed-string sector and the aforementioned low-energy theorem because of the lack of AF. Finally, we suggest a noncanonical way-out for asymptotically free QCD-like theories based on topological strings on noncommutative twistor space.