Elsevier

Nuclear Physics B

Volume 789, Issues 1–2, 21 January 2008, Pages 1-44
Nuclear Physics B

Glueballs vs. gluinoballs: Fluctuation spectra in non-AdS/non-CFT

https://doi.org/10.1016/j.nuclphysb.2007.07.012Get rights and content

Abstract

Building on earlier results on holographic bulk dynamics in confining gauge theories, we compute the spin-0 and spin-2 spectra of the Maldacena–Nuñez and Klebanov–Strassler theories, that have non-singular supergravity duals. We construct and apply a numerical recipe for computing mass spectra from certain determinants. In the Klebanov–Strassler case, the spectra of states containing the glueball and gluinoball are universal and quadratic in the supergravity approximation, i.e., their mass-squareds depend on consecutive number as m2n2 for large n, with a universal proportionality constant. The hardwall approximation appears to work poorly when compared to the unique spectra we find in the full theory with a smooth cap-off in the infrared.

Introduction

One of the first successes of gauge/string duality was the computation of mass spectra of strongly coupled gauge theories from dual geometries [1], [2], [3]. These original papers were all concerned with N=0 black hole solutions, where it is hard to check the validity of the correspondence. Other work uses the N=4 superconformal theory (AdS bulk) plus a hard IR cutoff [4] that serves to imitate interesting field theory IR effects, but there is no running coupling. Studying N=1 super-Yang–Mills theory coupled to matter seems a good compromise in many respects: these theories exhibit confinement, chiral symmetry breaking, running couplings and a rich set of mass spectra.

For this “non-AdS/non-CFT correspondence”,1 where the bulk is not AdS and the boundary field theory is not a CFT, much less is known about mass spectra, let alone correlators. A few brave attempts exist in the literature [7], [8], [9], [10], but as we explained in [11], their results on mass spectra are at best inconclusive. In the present paper, we will compute the mass spectrum of the N=1 Klebanov–Strassler theory [12], a non-conformal deformation of the N=1 Klebanov–Witten theory [13] involving gluons and gluinos coupled to two sets of chiral superfields with a specific quartic superpotential.2 We also compute the mass spectrum of the Maldacena–Nuñez theory [16] (for which the supergravity solution was found in [17], [18]), as a warmup. In this way, we will be able to address physical questions about how confinement works in these models, such as whether the theory displays “linear confinement” (for a recent discussion, see [19]).

Despite this promising state of affairs, many authors have emphasized that the aforementioned theories are still quite far from real-world QCD. For instance, there are no open strings corresponding to dynamical quarks, hence no real meson spectra. On the good side, in [20], [21], probe D-branes were added to the Klebanov–Strassler background, something which could be further developed using our methods. Real QCD is also nonsupersymmetric, and we make heavy use of the existence of a “superpotential” W. However, as indicated by the quotation marks, this superpotential is “fake” (in the sense of [22]), but there are nonsupersymmetric examples where such structure exists irrespective of supersymmetry per se, see, e.g., [23]. Last but not least, real QCD is not at large 't Hooft coupling λ; this is very difficult to overcome with the present state of the art, but the recent progress in [24] at least shows that 1/λ corrections may not be unrealistic to obtain for these theories.

Here, we take a step back from the effort to describe real QCD, and as mentioned above, focus on an example that is well defined from a holographic point of view and see what can be understood about that theory. We introduce some new techniques, but conceptually the biggest difference from most of the literature is in things we do not do. We do not employ the “hard-wall” approximation (which amounts to taking AdS as in Fig. 1 but with a finite IR cutoff). One argument that has been put forward to motivate the use of the hard-wall AdS model is that some physics should be insensitive to IR details. However, for computing mass spectra, a boundary condition must be imposed in the IR, so this question cannot really be insensitive to IR details (see, e.g., [24]). We discuss to what extent it is in Section 5.7. Another common approximation, for example in the context of inflation [25], is the singular Klebanov–Tseytlin background [26], as in Fig. 1. We insist on using the full Klebanov–Strassler solution and imposing consistent boundary conditions, and find that the commonly used approximations would likely not have led to correct results even for this theory, let alone for QCD. It seems reasonable to ask for explicit computational strategies to be developed concurrently with the search for real QCD, and we believe we have made progress on such strategies here.

It is important to recall that the Klebanov–Strassler theory does not have a Wilsonian UV fixed point, but one can impose a cutoff in the UV to define the theory. One can think of this in at least three ways: (1) this is an intermediate approximation and the theory will be embedded in a more complicated theory with a UV fixed point like in [27]; (2) we will glue the noncompact dual geometry onto a compact space that naturally provides a UV cutoff like in [28]; or (3) we will want to match to (at least lattice) data at that point, and beyond that the theory is in any case not asymptotically free (for recent ideas about this, see, e.g., [29]). As far as this paper is concerned, any of these points of view may be adopted.

Now to the new results in this paper. We expand upon our earlier work [11] and present a numerical strategy, the “determinant method”, to calculate the spectrum of regular and asymptotically subdominant bulk fluctuations. This puts us in the interesting position of being able to compute the mass spectrum unambiguously, but not being able to say what the composition of each mass state is without further information.3 This information should ideally come from data, i.e., one should compute mixings at a specified energy scale where one has some control, and use the dual theory to evolve into the deep nonperturbative regime. To illustrate this, let us take the simpler example of a theory that does have a UV fixed point. The glueball operator Og, with conformal dimension Δ=4, and the gluinoball operator Og˜, with Δ=3, have a diagonal mass matrix in the conformal theory. As soon as we let the theory flow away from conformality, there is a non-zero correlator OgOg˜. We can still diagonalize at any given energy, and the mixing matrix will be energy dependent. For theories that are not conformal even in the UV, there is no preferred basis labelled by the Δ eigenvalues. One can still contemplate an approximate labelling by Δ (in the KS theory, this corresponds to expanding in the ratio P=(number of fractional D3-branes)/(number of regular D3-branes)), but it is not yet completely clear how to implement this in the dynamics; as we pointed out in [11], the limit P0 does not commute with the UV limit one needs for the asymptotics. We will give ideas about this in Sections 2.3 Pole structure of holographic 2-point functions, 5.6 KS 7-scalar system: Quadratic spectra, but we will not resolve it completely.

The spectra we find have some simple features. The states of the Klebanov–Strassler theory come in towers, each of which shows a quadratic spectrum for large excitation number (i.e., m2n2, where n denotes the excitation number within the tower). This confirms the claims of [19] also for the spin-0 states of the KS theory. However, for low excitation number (roughly for the first two excitations in each tower) the structure of the mass values is richer.

For the Maldacena–Nuñez background, the spectrum we find is rather different. It has an upper bound, in agreement with the analytical spectra that we reported in [11].

Let us also mention that there are other approaches to holography in this type of models, such as the Kaluza–Klein holography of [30]. Some issues may appear in a different light in that framework, and that would be interesting to know.

The organization of the paper is as follows. In Section 2 we outline the general theory of bulk fluctuations as well as the correspondence between their spectrum and the spectrum of mass states in the dual gauge theory. The presentation is done such that it is applicable also in the case of non-asymptotically AdS bulk spaces. Based on this general material, a numerical strategy to calculate the spectrum will be developed in Section 3. As a warmup, our first application will be the Maldacena–Nuñez theory [16] in Section 4, before we come to the analysis of the mass spectrum in the Klebanov–Strassler cascading gauge theory in Section 5. Some conclusions can be found in Section 6, and many of the more technical details have been deferred to Appendix A Some 2-point functions in AdS/CFT, Appendix B Matrices for the MN background, Appendix C Bulky KS stuff, Appendix D KS spin-0 spectrum, Appendix E Numerics.

Section snippets

Holographic mass spectra

We want to calculate mass spectra of confining gauge theories from the dynamics of fluctuations about their supergravity duals. In this section, we review and develop the main theoretical tools that are necessary for such a calculation. We will start, in Section 2.1, with a review of the treatment of the dynamics of supergravity fluctuations developed by us in [11], generalizing a similar approach to the bulk dynamics in AdS/CFT [31], [32], [33]. Then, in Section 2.2, we identify the bulk duals

Finding mass states

In the previous section, we characterized the duals of mass states as regular (in the IR) and asymptotically subdominant (in the UV) solutions of the linearized bulk field equations. To calculate the spectrum, one solves a system of second-order differential equations with boundary conditions specified at two endpoints. The standard numerical strategy for this is “shooting”, in which one solves the corresponding initial value problem with two boundary conditions (field value and derivative)

General relations

The effective 5d model describing the bulk dynamics of the MN system contains three scalar fields (g,a,p) and is characterized by the sigma-model metric [11], [40]Gabμϕaμϕb=μgμg+e2gμaμa+24μpμp, and the superpotential16W=12e4p[(a21)2e4g+2(a2+1)e2g+1]1/2.

Let us summarize what Poincaré-sliced domain wall backgrounds this system admits. In what follows, we shall denote the background values of the

KS background

The effective 5d model describing the bulk dynamics of the KS system contains seven scalar fields. We will use the Papadopoulos–Tseytlin [40] variables (x,p,y,Φ,b,h1,h2). In the following, we shall briefly summarize the general relations for this system and the KS background solution. For more details, see our earlier paper [11] and references therein.

The sigma-model metric isGabμϕaμϕb=μxμx+6μpμp+12μyμy+14μΦμΦ+P22eΦ2xμbμb+14eΦ2x[e2yμ(h1h2)μ(h1h2)+e2yμ(h1+h2)μ(h1+h2)], and

Outlook

We have computed spin-0 and spin-2 mass spectra for the Maldacena–Nuñez (wrapped D5-brane) and Klebanov–Strassler (warped deformed conifold) supergravity duals. Although the spectra are fairly complicated, there are some simple features. For example, for large excitation number n, the spin-0 states in the KS theory organize into 7 towers with m2n2, in agreement with the claims of [19]. (“Large” is not terribly large; around n3.) For low excitation number n, there is a rich structure of mass

Acknowledgements

It is a pleasure to thank Massimo Bianchi, Andreas Karch, Emanuel Katz, Albion Lawrence, Scott Noble, Carlos Nuñez, Henning Samtleben, David Tong and Amos Yarom for helpful discussions and comments. This work is supported in part by the European Community's Human Potential Program under contract MRTN-CT-2004-005104 ‘Constituents, fundamental forces and symmetries of the universe’. This research was supported in part by the National Science Foundation under Grant No. PHY99-07949. The work of

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