[Paper Review] Yang-Mills mass gap, Floer homology, glueball spectrum, and conformal window in large-N QCD

Marco Bochicchio|arXiv (Cornell University)|Dec 4, 2013

Black Holes and Theoretical Physics32 references9 citations

TL;DR

This paper proposes a topological field theory in large-N Yang-Mills theory using twistor Wilson loops that localize on magnetic surface operators, realizing a quantum version of Lagrangian intersection Floer homology. It derives an exact linear glueball spectrum $ m_k^2 = k\Lambda_{\overline{W}}^2 $, predicting a mass ratio $ \sqrt{2} $ for the lightest scalar glueballs, in strong agreement with lattice simulations and experimental data.

ABSTRACT

Morse-Smale-Floer homology associates the critical points of the action functional of a classical field theory over a manifold to its homology. We associate to the intersection homology of certain Lagrangian submanifolds of R^4 the critical points of a quantum effective action of large-N SU(N) YM. For this purpose we construct in YM a trivial Topological Field Theory defined by twistor Wilson loops whose v.e.v. is 1 in the large-N limit for any shape of the loops supported on certain punctured Lagrangian submanifolds. We derive a new holomorphic loop equation for the twistor Wilson loops, that involves the change of variables in the YM functional integral from the connection to the anti-selfdual part of the curvature and the choice of a holomorphic gauge. Employing the holomorphic loop equation, and viewing Floer homology the other way around, we associate to arcs asymptotic in both directions to the cusps of the Lagrangian submanifolds the critical points of an effective action, that turn out to be surface operators of Z(N) holonomy. At the next-to-leading 1/N order a certain correlator of surface operators is non-topological and non-trivial, controls the mass gap of YM theory, and is saturated by an infinite sum of pure poles of scalar and pseudoscalar glueballs with positive charge conjugation. It satisfies asymptotically for large momentum fundamental universal constraints arising from the asymptotic freedom and the renormalization group. We predict at large-N the ratio of the masses of the two lower-mass scalar glueballs r=\sqrt 2=1.414, to be compared with the measure in lattice SU(8) YM by Meyer-Teper r=1.42(11), and with the value implied by PDG(2014) r=1.397(008). The construction extends to massless Veneziano large-N limit of QCD, for which we determine the lower edge of the conformal window N_f/N=5/2 and the corresponding quark-mass anomalous dimension gamma=-4/5.

Motivation & Objective

• To establish a quantum field-theoretical realization of Lagrangian intersection Floer homology in large-N Yang-Mills theory.
• To derive the glueball spectrum from a topological field theory based on twistor Wilson loops with vacuum expectation value 1 at $ N = \infty $.
• To identify the mass gap of Yang-Mills theory with a two-point correlator of surface operators at next-to-leading $ 1/N $ order.
• To predict an exact linear spectrum for scalar and pseudoscalar glueballs and compare it with lattice and experimental data.
• To extend the framework to the Veneziano limit of QCD, determining the conformal window and anomalous dimension.

Proposed method

• Construct a trivial topological field theory (TFT) at $ N = \infty $ using twistor Wilson loops whose vacuum expectation value is 1 for loops on punctured Lagrangian submanifolds.
• Derive a holomorphic loop equation by changing variables from the connection to the anti-selfdual part of the curvature and using holomorphic gauge fixing.
• Localize the TFT on critical points of an effective action, identified as magnetic surface operators with $ Z_N $ holonomy, realizing a version of 't Hooft duality via magnetic condensation.
• Compute a two-point correlator of surface operators in the $ 1/N $ expansion, showing it controls the mass gap and is saturated by an infinite sum of scalar and pseudoscalar glueballs.
• Use asymptotic freedom and renormalization group constraints to derive universal, fundamental constraints on the correlator.
• Extend the construction to massless Veneziano QCD to determine the lower edge of the conformal window and the quark-mass anomalous dimension.

Experimental results

Research questions

• RQ1 Can Floer homology be realized in quantum field theory via critical points of an effective action in large-N Yang-Mills theory?
• RQ2 Does the two-point correlator of surface operators at $ 1/N $ order control the Yang-Mills mass gap and glueball spectrum?
• RQ3 Is the glueball spectrum exactly linear in $ k $, with $ m_k^2 = k\Lambda_{\overline{W}}^2 $, and how does it compare to lattice and experimental data?
• RQ4 What is the lower edge of the conformal window in the Veneziano limit of QCD, and what is the corresponding quark-mass anomalous dimension?
• RQ5 Can the observed linearity in meson and glueball Regge trajectories be explained by a dual topological string theory with universal mass units?

Key findings

• The paper derives an exact linear spectrum for scalar and pseudoscalar glueballs: $ m_k^2 = k\Lambda_{\overline{W}}^2 $, with $ \Lambda_{\overline{W}} $ the RG-invariant scale.
• The ratio of the masses of the first-excited and ground-state scalar glueballs is predicted to be $ \sqrt{2} = 1.414\ldots $, matching lattice results for $ SU(8) $ YM ($ 1.42(11) $) and PDG data ($ 1.397(8) $).
• The two-point correlator of surface operators at $ 1/N $ order controls the mass gap and is saturated by an infinite sum of glueballs with positive charge conjugation.
• The construction predicts the lower edge of the conformal window in massless Veneziano QCD as $ N_f/N = 5/2 $, with quark-mass anomalous dimension $ \gamma_m = -4/5 $.
• The spectrum of mesons and glueballs exhibits (semi-)integer-valued masses squared in units of $ \frac{1}{2}\Lambda_{\overline{W}}^2 $, suggesting a dual topological string theory.
• The model identifies $ f_0(1500) $ as the ground-state scalar glueball and $ f_0(2100) $ as the first excited scalar glueball, consistent with mass formula and experimental data.