S. Dubovsky"*, R. Flaugera b, V. Gorbenko"

Center for Cosmology and Particle Physics Department of Physics, New York University

10003, New York, USA

bSchool of Natural Sciences, Institute for Advanced Study 0S540, Princeton, USA

Received September 30, 2014

We provide a detailed introduction to a method we recently proposed for calculating the spectrum of excitations of effective strings such as QCD flux tubes. The method relies on the approximate integrability of the low-energy effective theory describing the flux tube excitations and is based on the thermodynamic Bethe ansatz. The approximate integrability is a consequence of the Lorentz symmetry of QCD. For excited states, the convergence of the thermodynamic Bethe ansatz technique is significantly better than that of the traditional perturbative approach. We apply the new technique to the lattice spectra for fundamental flux tubes in gluodynamics in D = 3 + 1 and D = 2 + 1, and to fc-strings in gluodynamics in D = 2 + 1. We identify a massive pseudoscalar resonance on the worldsheet of the confining strings in SU{3) gluodynamics in D = 3 + 1, and massive scalar resonances on the worldsheet of k = 2.3 strings in SU{6) gluodynamics in D = 2 + 1.

Contribution for the JETP special issue in honor of V. A. Rubakov's 60th birthday

DOI: 10.7868/S0044451015030088


String theory originated as a candidate theory of strong interactions [1]. However, it was soon abandoned as a theory of hadrons, at least for the time being, because it failed to reproduce the observed properties of deep inelastic scattering as well as the asymptotic freedom of non-Abelian gauge theories. But the success of the Veneziano amplitude in describing many aspects of the hadron spectrum and scattering is hardly a coincidence. Confining strings (flux tubes) are crucial ingredients in the strongly coupled QCD dynamics responsible for color confinement, and their presence is vividly revealed by lattice QCD simulations f2]1\ suggesting that understanding the structure and dynamics of QCD flux tubes might provide insights into the dynamics of color confinement.

The modern approach to the relation between string theory and gauge theories relies on the AdS/'CFT cor-

E-mail: sergei.dubovskyfflgmail.com

11 See http:// www. physics, adelaide.edu.au/theory/staif/ leinweber/VisualQCD/Nobel/ for animations.

respondence [3]. Within this framework, the QCD flux tube is expected to be described by a string propagating in a space time with an extra curved dimension, which can be interpreted as the dynamical string tension, or equivalently, the renormalization group scale [4]. Identifying a concrete string theory that would provide a holographic description of nonsupersymmetric QCD remains a long shot, and even if this dual string theory-were found, it would be outside the regime in which we currently have theoretical control.

In this paper, we therefore focus on a rather direct path towards understanding the structure of the flux tube theory that does not involve holography. Instead, it is based on existing lattice techniques combined with effective field theory and tools from integrability.

Advances in lattice QCD simulations have allowed measuring the spectrum of low-lying worldsheet excitations with impressive accuracy [5 7]. But the theoretical interpretation of these results was problematic until now. For most states, the string lengths accessible in the lattice simulations were too short for the existing techniques to be reliable. The conventional perturbative methods [8 10] for calculating the spectrum of string excitations result in badly diverging asymptotic

series in this regime, preventing the interpretation of the data. At the same time, the data exhibited a number of puzzling and suggestive features. In particular, while perturbative calculations were not reliable, many of the levels show surprisingly good agreement with the spectrum of a free bosonic string quantized in the light-cone gauge following the classic paper [11] by Goddard, Goldstone, Rebbi, and Thorn (GGRT) (see also [12]). This is confusing, given that the GGRT spectrum is well known to be incompatible with the bulk Poincare symmetry if the number of space time dimensions is different from 26.

For the lattice simulations, the computational cost grows exponentially with the length of the string. At least with the current technology, this makes it essentially impossible to push lattice calculations into the regime in which conventional perturbation techniques converge. Alternative techniques for calculating the flux tube spectra are thus required, to provide better convergence for relatively short strings. We proposed such a technique in [13], and its success relies on the observation that the worldsheet theory becomes integrable at low energies. This technique seems sufficient to explain the previously puzzling features seen in lattice results. In addition, it allowed showing that the existing lattice data provide strong evidence for the existence of a massive pseudoscalar state on the worldsheet of the QCD flux tube, the worldsheet axion.

The goal of this paper is to provide a detailed account of the method proposed in [13]. In Sec. 2, we begin with a brief summary of the lattice results and of the effective string theory approach (for a detailed recent review, see [14]). We review the results of the conventional perturbative expansion for energy levels, which exhibits a large number of universal terms. We explain that the GGRT spectrum, in spite of being inconsistent with the bulk Poincare symmetry, still represents a finite-volume spectrum of a certain integrable relativistic two-dimensional theory. As we explain, this observation immediately allows calculating all the universal terms in the spectrum of relativistic effective strings [15].

In Sec. 3, we present the new method for calculating the flux tube spectrum. The main idea of the method is to divide the calculation into two steps. First, we per-turbatively calculate the worldsheet ¿»-matrix describing the scattering of the flux tube excitations within the effective string theory. We then determine the corresponding finite-volume spectrum using the excited state thermodynamic Betlie ansatz (TEA) [16, 17], which is very similar to the techniques developed by Luscher [18, 19], which are routinely used to extract

four-dimensional scattering amplitudes from the lattice QCD data. We provide a partial diagrammatic interpretation of the perturbative resuniniation performed by the TEA and explain why it is natural to expect that this method results in a better behaved perturbation theory for excited states.

In Sec. 4, we use this technique to interpret the lattice data. We provide more details than in [13] as to how to implement the method and include a larger set of excited states in our analysis. This extended analysis confirms the conclusion reached in [13]: the lattice data provides strong evidence for the existence of a pseudoscalar state bound to a confining string. We also apply the technique to the available data for three-dimensional gluodynamics. There, we find no evidence for any massive excitations on the fundamental flux tube, but identify massive scalar excitations on k-st rings.

We conclude in Sec. 5 by discussing future directions and prospects. We also present an intriguing hint for the existence of additional light bound states, coming from the precision ground-state data.


We start with a brief summary of lattice results for the excitation spectrum of confining flux tubes. A detailed description of these results and techniques can be found in [5 7] (for a review, see [20]). In most of our discussion, we assume the space-time dimension D = = 4. However, we also apply our techniques to the available D = 3 data. We are interested in the internal dynamics of a single closed flux tube, rather than in effects coming from its boundaries and from interactions between several flux tubes. To discuss these separately, it is necessary to suppress processes where the flux tube can break. This is achieved by performing simulations in pure gluodynamics without dynamical quarks. Gauge-invariant operators in a pure glue theory are constructed as traces of path-ordered exponentials of the gauge field Atl (Wilson loops),

Or = Tr I' ^oxp j .4 j . (1)

where C is a closed path. In what follows, we mostly discuss flux tubes carrying a single unit of fundamental flux. This amounts to taking the trace in (1) in the fundamental representation of the gauge group.

A nice trick, which allows concentrating on the dynamics of long flux tubes, is to use the nontrivial lattice

topology. Namely, we consider states created by operators of form (1), such that the corresponding path winds around one of the lattice dimensions. It is convenient to think about the corresponding direction as a spatial one, although, of course, all directions on the lattice are Euclidean anyway. States of this kind are orthogonal to conventional glueball states created by operators (1) with contractible paths. This follows from a global Zm symmetry (center symmetry) present in the SU(N) Yang Mills theory compactified on a circle. It is generated by gauge transformations such that the corresponding gauge functions satisfy twisted boundary conditions. The twist is performed using a multiplication by an element from the center of the gauge group,

f](R) = e27Tki/N g(0), (2)

where k is an integer.

Transformations satisfying boundary condition (2) act properly on the gauge configurations and preserve the action functional, but do not originate from a well-defined gauge function. Hence, they should be considered as generating a global, rather than gauge, symmetry. Any two transformations with the same twist k are equivalent up to a conventional gauge transformati

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