Glueball and meson spectrum in large-N massless QCD

We provide outstanding numerical evidence that in large-N massless QCD the joint spectrum of the masses squared, for fixed integer spin s and unspecified parity and charge conjugation, obeys exactly the following laws: m_k^2 = (k+s/2) Lambda_QCD^2 for s even, m_k^2 = 2(k+s/2) Lambda_QCD^2 for s odd, k = 1,2,... for glueballs, and m_n^2 = 1/2 (n+s/2) Lambda_QCD^2, n = 0,1,... for mesons. One of the striking features of these laws is that they imply that the glueball and meson masses squared form exactly-linear Regge trajectories in the large-N limit of massless QCD, all the way down to the low-lying states: A fact unsuspected so far. The numerical evidence is based on lattice computations by Meyer-Teper in SU(8) YM for glueballs, and by Bali et al. in SU(17) quenched massless QCD for mesons, that we analyze systematically. The aforementioned spectrum for spin-0 glueballs is implied by a Topological Field Theory underlying the large-N limit of YM, whose glueball propagators satisfy as well fundamental universal constraints arising from the asymptotic freedom and the renormalization group. No other presently existing model meets both the infrared spectrum and the ultraviolet constraints. We argue that some features of the aforementioned spectrum of glueballs and mesons of any spin could be explained by the existence of a Topological String Theory dual to the Topological Field Theory.Comment: 9 pages, 2 figures; added some comments on the different accuracy in the glueball and meson sectors; references update

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

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 $N$ -- a consequence of the asymptotic freedom (AF) -- 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. 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 finite because of the glueball mass gap. In fact, the inconsistency follows from a low-energy theorem of the NSVZ type that controls the renormalization in QCD-like theories. The inconsistency extends to the would-be canonical string for a vast class of 't Hooft large-$N$ QCD-like theories including $\mathcal{N}=1$ SUSY QCD. We also demonstrate that the presently existing SUSY string models with a mass gap -- such as Klebanov-Strassler, Polchinski-Strassler (PS) and certain PS variants -- cannot contradict the above-mentioned results 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 low-energy theorem because of the lack of AF. Finally, we suggest a noncanonical way-out for QCD-like theories based on topological strings on noncommutative twistor space.Comment: 23 pages, 1 figure; this is the version published in Phys. Lett. B: it contains an extended exposition of the main results, and a discussion on the way the open/closed string duality may be realized in presently existing string models with a mass gap that are not asymptotically fre

An asymptotic solution of large-NN QCDQCD, and of large-NN N=1\mathcal{N}=1 SUSYSUSY YMYM

We find an asymptotic solution for two- and three-point correlators of local gauge-invariant operators, in a lower-spin sector of massless large-$N$ $QCD$, in terms of glueball and meson propagators, by means of a new purely field-theoretical technique that we call the asymptotically-free bootstrap. The asymptotically-free bootstrap exploits the lowest-order conformal invariance of connected correlators of gauge invariant composite operators in perturbation theory, the renormalization-group improvement, and a recently-proved asymptotic structure theorem for glueball and meson propagators, that involves the unknown particle spectrum and the anomalous dimension of operators for fixed spin. In principle the asymptotically-free bootstrap extends to all the higher-spin two- and three-point correlators whose lowest-order conformal limit is non-vanishing in perturbation theory, and by means of the operator product expansion to the corresponding asymptotic multi-point correlators as well. Besides, the asymptotically-free bootstrap provides asymptotic $S$-matrix amplitudes in massless large-$N$ $QCD$ in terms of glueball and meson propagators as opposed to perturbation theory. Remarkably, the asymptotic $S$-matrix depends only on the unknown particle spectrum, but not on the anomalous dimensions. Moreover, the asymptotically-free bootstrap applies to large-$N$ $\mathcal{N}=1$ $SUSY$ $YM$ as well. Practically, as just a few examples among many more, it follows the structure of the light by light scattering amplitude, of the pion form factor, and the associated vector dominance. Theoretically, the asymptotic solution sets the strongest constraints on any actual solution of large-$N$ $QCD$ (and of large-$N$ $\mathcal{N}=1$ $SUSY$ $YM$), and in particular on any string solution.Comment: 56 pages, latex; more details and remarks, and some simplification; references added; typos fixed; talk at the conference: HP2: High Precision for Hard Processes, September 3-5, (2014), associated to the workshop: Prospects and Precision at the Large Hadron Collider at 14 TeV, at the Galileo Galilei Institute for Theoretical Physics, Florence, Ital

Some comments on S-duality in four-dimensional QCD

We show that a necessary condition, for the partition function of four-dimensional Yang-Mills theory to satisfy a S-duality property, is that certain functional determinants, generated by the dual change of variables, cancel each other. This result holds up to non-topological boundary terms in the dual action and modulo the problem of field-strength copies for the Bianchi identity constraint.Comment: 4 pages, latex, no figures. Basic references on duality adde

Glueball and meson propagators of any spin in large-N QCD

We prove an asymptotic structure theorem for glueball and meson propagators of any spin in large-N QCD and in n=1 SUSY QCD with massless quarks, that determines asymptotically the residues of the poles of the propagators in terms of their anomalous dimensions and of the spectral density of the masses. The asymptotic theorem follows by the severe constraints on the propagators in large-N QCD with massless quarks, or in any large-N confining asymptotically-free gauge theory massless in perturbation theory, that arise by perturbation theory in conjunction with the renormalization group and by the OPE on the ultraviolet side. The asymptotic theorem is inspired by a recently proposed Topological Field Theory (TFT) underlying large-N pure YM, that computes sums of the scalar and of the pseudoscalar correlators satisfying the asymptotic theorem and that implies for the large-N joint scalar and pseudoscalar glueball spectrum exact linearity in the masses squared. On the infrared side we test the prediction of the exact linearity in the TFT by Meyer-Teper lattice numerical computation of the masses of the low-lying glueballs in SU(8) YM, finding accurate agreement. Besides, we employ the aforementioned ultraviolet and infrared constraints in order to compare critically the scalar or pseudoscalar glueball propagators computed in the framework of the AdS String/large-N Gauge Theory correspondence with those of the TFT underlying large-N YM. We find that only the TFT satisfies the ultraviolet and infrared constraints.Comment: 31 pages; new references, added comments on infinite contact terms, minor change

The Abelian projection versus the Hitchin fibration of K(D) pairs in four-dimensional QCD

We point out that the concept of Abelian projection gives us a physical interpretation of the role that the Hitchin fibration of parabolic K(D) pairs plays in the large-N limit of four-dimensional QCD. This physical interpretation furnishes also a simple criterium for the confinement of electric fluxes in the large-N limit of QCD. There is also an alternative, compatible interpretation, based on the QCD string.Comment: 11 pages, latex, no figures, a misprint correcte

Large-N limit and contact terms in unbroken YM_4

I characterize the structure of the master field for $F^{0}_{z \bar z}$ in $SU(\infty)$-$YM_4$ on a product of two Riemann surfaces $Z \times W$ in the gauge $F^{ch}_{z \bar z}=0$ as the sum of a `bulk' constant term and of delta-like `contact' terms.\\ The contact terms may occur because the localization of the functional integral at $N=\infty$ on a master orbit of a constant connection under the action of singular gauge transformations is still compatible with the large-$N$ factorization and translational invariance.\\ In addition I argue that if the gauge group is unbroken and there is a mass gap, that is if the theory confines, the functional measure at $N=\infty$, in the gauge $F^{ch}_{z \bar z}=0$, must be localized on the moduli space of flat connections with punctures on $Z \times W$.Comment: 8 pages, latex, no figure

Asymptotic Freedom versus Open/Closed Duality in Large-N QCD

The solution of the large-N 't Hooft limit of QCD is universally believed to be a String Theory of Closed Strings in the Glueball Sector and of Open Strings in the Meson Sector. Yet, we prove a no-go theorem, that the large-N limit of QCD with massless quarks, or more generally, that the large-N limit of a vast class of confining, i.e. with a Mass Gap in the Glueball Sector, asymptotically-free Gauge Theories coupled to matter fields with no mass scale in perturbation theory cannot be a canonically-defined String Theory of Closed and Open Strings, i.e. admitting Open/Closed Duality. The no-go theorem occurs because Open/Closed Duality, implying that the ultraviolet divergences of annulus diagrams in the Open Sector arise from infrared divergences of tadpoles of massless particles in the Closed Sector, turns out to be incompatible with the existence of the Mass Gap in the Glueball Sector of confining asymptotically-free theories with no mass scale in perturbation theory in which, as for example in QCD, the first coefficient of the beta function for 't Hooft gauge coupling gets $1/N$ corrections due to the matter fields. Moreover, we suggest a way-out to the no-go theorem on the basis of a new non-canonical construction of the String S-matrix for asymptotically-free Gauge Theories such as large-N QCD, involving Topological Strings on Non-Commutative Twistor Space.Comment: 8 pages; the main argument is extended to a punctured sphere with a boundary loop; further references and acknowledgement

Exact beta function and glueball spectrum in large-N Yang Mills theory

In the pure large-N Yang-Mills theory there is a quasi-BPS sector that is exactly solvable at large N. It follows an exact beta function and the glueball spectrum in this sector. The main technical tool is a new holomorphic loop equation for quasi-BPS Wilson loops, that occurs as a non-supersymmetric analogue of Dijkgraaf-Vafa holomorphic loop equation for the glueball superpotential of n=1 SUSY gauge theories. The new holomorphic loop equation is localized, i.e. reduced to a critical equation, by a deformation of the loop that is a vanishing boundary in homology, somehow in analogy with Witten's cohomological localization by a coboundary deformation in SUSY gauge theories.Comment: PDF, 11 pages, talk at EPS-HEP 2009; added a footnote on the fluctuations of surface operators as published in Po

The Yang-Mills String as the A-Model on the Twistor Space of the Complex Two-Dimensional Projective Space with Fluxes and Wilson Loops: the Beta Function

We argue that the string theory dual to a certain sector of the four-dimensional Yang-Mills theory at large-N is the A-model wrapping N-Lagrangian D-branes on the twistor space of the complex two-dimensional projective space, with certain fluxes and Wilson loops, by finding that the target-space beta function of this A-model coincides with the large-N beta function of the Yang-Mills theoryComment: 36 pages, latex; more extended version, introduced almost complex structure with vanishing first Chern class, change in the Lagrangian submanifold, beta function unchange