PHYSICAL REVIEW D 103, 106018 (2021)
Dilatonic states near holographic phase transitions
Daniel Elander ,1 Maurizio Piai ,2 and John Roughley 2 1Laboratoire Charles Coulomb (L2C), University of Montpellier, CNRS, Montpellier 34095, France 2Department of Physics, College of Science, Swansea University, Singleton Park, SA2 8PP Swansea, Wales, United Kingdom
(Received 1 February 2021; accepted 30 March 2021; published 18 May 2021)
The spectrum of bound states of special strongly coupled confining field theories might include a parametrically light dilaton, associated with the formation of enhanced condensates that break (approxi- mate) scale invariance spontaneously. It has been suggested in the literature that such a state may arise in connection with the theory being close to the unitarity bound in holographic models. We extend these ideas to cases where the background geometry is non-anti-de Sitter, and the gravity description of the dual confining field theory has a top-down origin in supergravity. We exemplify this program by studying the circle compactification of Romans six-dimensional half-maximal supergravity. We uncover a rich space of solutions, many of which were previously unknown in the literature. We compute the bosonic spectrum of excitations and identify a tachyonic instability in a region of parameter space for a class of regular background solutions. A tachyon only exists along an energetically disfavored (unphysical) branch of solutions of the gravity theory; we find evidence of a first-order phase transition that separates this region of parameter space from the physical one. Along the physical branch of regular solutions, one of the lightest scalar particles is approximately a dilaton, and it is associated with a condensate in the underlying theory. Yet, because of the location of the phase transition, its mass is not parametrically small, and it is, coincidentally, the next-to-lightest scalar bound state, rather than the lightest one.
DOI: 10.1103/PhysRevD.103.106018
I. INTRODUCTION The Higgs particle might emerge at the dynamical scale Λ The Standard Model of particle physics is likely to be from strongly coupled new physics. If one could dial the replaced by a more complete theory above some unknown effects of explicit breaking of scale invariance to be smaller new physics scale Λ. Yet, the discovery by the LHC than those associated with its spontaneous breaking, the collaborations of the Higgs particle [1,2], with mass Higgs boson could be identified with the pseudo-Nambu- Goldstone boson associated with the spontaneous breaking mh ≃ 126 GeV, has not been accompanied by convincing signals of new phenomena in (direct and indirect) searches, of dilatation invariance: the dilaton. If furthermore its mass further pushing the hypothetical scale Λ into the multi-TeV mh could be made small enough to yield the hierarchy ≪ Λ range. This observation hints at a difficulty in the appli- mh , and without fine-tuning, then the little hierarchy cation of effective field theory (EFT) ideas to high-energy problem would be solved. particle physics. If the fundamental theory completing the The properties of the EFT description of the dilaton are Standard Model above Λ plays a role (even indirectly) in the subject of a vast body of literature (see for instance – electroweak symmetry breaking (EWSB) and Higgs phys- Refs. [3 15]), which includes well-known studies dating ics, it is technically difficult to implement the hierarchy from a long time ago [16,17]. The details of how this idea is implemented in phenomenologically relevant models of mh ≪ Λ and justify it inside the general low-energy EFT paradigm. The resulting low-energy description requires EWSB are the subject of many studies (see for example – fine-tuning; in the literature, this tension is referred to as the Refs. [18 29]), and some date back to earlier days of little hierarchy problem. dynamical EWSB symmetry breaking and walking tech- nicolor [30–32]. The main limitation to the study from first principles of dilaton dynamics with strongly coupled origin comes from calculability. For example, lattice studies have started to Published by the American Physical Society under the terms of uncover evidence that an anomalously light scalar particle the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to appears in confining gauge theories that are believed to be the author(s) and the published article’s title, journal citation, close to the edge of the conformal window, namely in 3 and DOI. Funded by SCOAP . SUð3Þ gauge theories with either Nf ¼ 8 fundamental
2470-0010=2021=103(10)=106018(43) 106018-1 Published by the American Physical Society ELANDER, PIAI, and ROUGHLEY PHYS. REV. D 103, 106018 (2021)
Dirac fermions [33–37] or with Nf ¼ 2 Dirac fermions precisely at the edge of the conformal window. In recent transforming in the two-index symmetric representation dilaton EFT studies, Δ is measured by fitting the afore- [38–43]. It may be premature to conclude that these works mentioned SUð3Þ lattice data to yield Δ ≃ 2 [9,10,13]. have uncovered firm evidence that the scalar particle is a Reference [67] discusses the dynamics in proximity of dilaton, but EFT-based studies yield encouraging indica- the BF bound, particularly in relation to the dilaton mass. tions in this direction (see for example Refs. [10,42,44]). It adopts a bottom-up simplified model to describe this A complementary approach exploits holography and scenario and to test it. The spectrum of bound states of the gauge-gravity correspondences [45–47]. The properties putative dual theory is then calculated. It is found that, of some special strongly coupled field theories can be when dialing the bulk mass to approach the BF bound [67], derived from weakly coupled gravity theories in higher dimensions (see the introductory review in Ref. [48]). A ‘...the dilaton is always the lightest resonance, although systematic prescription exists for calculating correlation not parametrically lighter than the others.’ functions and condensates, via holographic renormalization In this paper, we consider a holographic model that [49] (see the lecture notes in Refs. [50,51]). The study of realizes a physical system sharing several core features with simplified toy models, implementing the Goldberger-Wise – those advocated in Ref. [67], but within the context of top- stabilization mechanism [52 58], shows the presence of a down holography. We see this paper as a precursor to a light dilaton in the spectrum. Attempts at constructing broad research program of exploration of supergravity phenomenologically more realistic models, while disre- backgrounds. We now describe how we can develop this garding the fundamental origin of the higher-dimensional program and anticipate our main results for the one model theory (the bottom-up approach to holography) yield – we focus upon in the body of the paper. similarly encouraging results [59 67]. The techniques that we use are applicable to systems for Evidence that strong dynamics can lead to the formation which the gravity geometry is asymptotically anti-de Sitter of a light dilaton has been confirmed also in the context of at large values of the holographic direction ρ (correspond- less realistic, but more rigorous holographic models built ing to the UVof the dual field theory). This is best suited for starting from supergravity (top-down approach) [68,69] the application of holographic renormalization, as we want – (see also Refs. [70 73]). So far, this has been shown to be not only to compute the mass spectrum but also the free true only inside the special framework of a particular five- energy of the system, which plays a crucial role in the body dimensional sigma model coupled to gravity, the solutions of the paper. of which lift to backgrounds with geometry related to the In the bottom-up approach to holography, the mass gap – conifold [74 79]. These backgrounds are related to con- of the dual theory can be introduced by adding by hand an fining gauge theories. The known existence of a moduli end of space to the geometry in what corresponds to the IR space (along the baryonic branch of the Klebanov-Strassler in the dual field theory. The presence of boundaries to the system), and of a tunable parameter appearing in some of space provides additional freedom and allows for the mass the condensates of the gauge theory, provide a nontrivial gap to emerge in a way that can often be arranged to dynamical explanation for the existence of a light state, preserve (approximate) scale invariance arbitrarily close to which is tempting to identify with the dilaton. the confinement scale. Hence, the notion of scaling Along a parallel line of investigation, we are intrigued by dimension and the associated BF bound may be well the ideas exposed in Ref. [67] (and in Refs. [80,81]), which defined even in close proximity of the end of space in are closely related to the discussions in Ref. [82]. The the geometry. [Notice that the value of the mass to which present paper is a first step toward transferring these ideas one applies the BF bound must be calculated in reference to from the bottom-up context to that of rigorous holographic the anti-de Sitter (AdS) geometry, or critical point of the models built within the top-down approach, and hence sigma model, that is closest to the end of space of the testing them within known supergravity theories. Within geometry along the dual renormalization group (RG) flow.] the bottom-up approach to holography, the authors of This is not the case within the top-down approach. The Refs. [67,82] start by identifying the Breitenlohner- backgrounds in the higher-dimensional geometry depart Freedman (BF) unitarity bound [83] as a marker of the from AdSD, and in the case when the dual field theory transition between conformal and nonconformal behavior confines, the geometry closes smoothly. The linear behav- of the dual gauge theory. The BF bound is related to the ior for the quark-antiquark static potential is recovered by dimension Δ of an operator O in the dual conformal field considering open strings in the uplifted ten-dimensional theory. In the case of five-dimensional gravity theories, the geometry [85,86] and minimizing the classical action BF bound selects Δ ¼ 2, which agrees with the arguments [87–91]. In all known classical backgrounds that yield discussed in the context of the Schwinger-Dyson equations linear confinement in the dual theory in D ¼ 4 dimensions and their approximation [84], according to which, in gauge (see for instance Refs. [75,77,78,92–94] and the general- theories with fermion matter field content, the O ¼ ψψ¯ izations of these models), the geometry is manifestly quite Δ 2 operator acquires the nonperturbative dimension ¼ different from AdSD in the proximity of its end of space.
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By combining the fact that in supergravity the potential provides the dual of a confining field theory, within the top- and mass of the bulk scalar fields are fixed and known, and down approach to holography. To this end, we broaden the cannot be arbitrarily dialed, together with the aforementioned classes of backgrounds studied in earlier publications about departure from AdSD of the geometry in the crucial region this same special system [92,93,98,99]. near the end of space, we conclude that the whole notion of The gravity theory we consider is the half-maximal proximity to the BF bound (central to Refs. [67,82]) needs to N ¼ð2; 2Þ supergravity in D ¼ 6 dimensions first be generalized. The BF bound in AdSD spaces is a marker of described by Romans [100]. It has been studied in great classical instabilities, taking the form of nonunitary behavior. detail and for many different purposes [92,93,101–120]. Its Within supergravity, it is possible to consider classical beauty lies in its simplicity: the model in D ¼ 6 dimensions backgrounds that evolve near the end of space toward regions contains only one scalar field ϕ coupled to gravity, with a of instability. The instability of the RG trajectory in the dual known classical action describing also four vectors and one field theory eventually gives rise to tachyonic behavior for 2-form. We compactify one of the dimensions on a circle. some of the lower-dimensional classical fluctuations of the The backgrounds approach the critical point ϕ ¼ 0 at large background solutions. We hence replace the notion of values of the holographic coordinate ρ (corresponding to proximity to the BF bound (useful only in approximately the UV of the dual field theory). The physics of confine- AdSD models, but not applicable to the dual of confining ment is captured by the fact that there are solutions of the theories), with the proximity to such tachyonic backgrounds. background equations in which the circle shrinks smoothly We will not dial the parameters in the action (related to the to zero size at some finite point in the radial coordinate coefficients in the dual renormalization group equations), but [94]. We extend the study with respect to Refs. [92,93] and rather adjust the only allowed freedom: the UV boundary [98,99] and look at additional branches of solutions. In conditions satisfied by the gravity and scalar fields. particular, we consider solutions in which the scalar field We want to obtain physically meaningful results; hence, assumes positive values, ϕ > 0, for which the potential in ideally, we should focus on background solutions that are six dimensions is unbounded from below, and an instability regular. Nevertheless, we find that in order to explain some arises in the system. In parts of the study, we also consider crucial features of the gravity dynamics we are compelled gravity solutions that do not have an interpretation in terms to include in part of our analysis also singular solutions that of four-dimensional confining theories, either because the exhibit what Gubser in Ref. [95] called a good singularity, dual theory is genuinely five-dimensional at all scales or that is characterized by the fact that the scalar potential of because a singularity emerges in the gravity description. the supergravity theory, evaluated along the classical We expect that, as long as ϕ experiences just a small solutions considered, is bounded from above. As we shall excursion away from ϕ ¼ 0, the spectrum of fluctuations see, some of these solutions exhibit a mild singular should not differ substantially from the one computed behavior in D ¼ 10 dimensions, that can be detected only elsewhere [92,93,98,99,121] and resemble qualitatively in higher-order curvature invariants, not in the Ricci scalar. that of a generic confining theory. In particular, all the We are pushed even further: we have to include also badly fluctuations have positive mass squared, and there is no singular backgrounds in the study of the energetics. We will parametric separation of scales visible in the spectrum. At clarify these notions eventually, in the body of the paper, the other extreme, if the scalar ϕ explores deep into the but we anticipate here that the reason why we introduce the instability region with ϕ > 0, in which the potential of the singular solutions in the study of the energetics is not that six-dimensional gravity theory is unbounded, we expect at we are making use of their field-theory interpretation least one of the states of the dual field theory to become (which does not exist) but rather that, for our treatment tachyonic. Under the assumption of continuity, somewhere of the gravity theory to be self-contained and consistent, we in between, one expects that the mass squared of one of the must treat all the classical solutions on the same grounds, in states in the spectrum will cross zero, for some special order to select what the features of the dynamics are. If it choice of background solution. We compute the spectra by turns out that a singular solution has free energy lower than making use of the gauge-invariant formalism developed in the (known) regular solutions, it is not legitimate to discard Refs. [122–126] and [99] and verify that all of these it, as its contribution to the path integral in the gravity expectations are realized. In close proximity of the afore- theory is actually dominant. This signals the incomplete- mentioned tuned choice of background, the lightest state is ness of the gravity description. a scalar, and it can be made parametrically light in So far, we have introduced a quite general program of comparison to all other states. Furthermore, we show that research, which can be carried out systematically on the this state can be characterized as an (approximate) dilaton, many known supergravity theories and their consistent by performing a nontrivial test on the spectrum [127] and truncations (see for instance Refs. [96,97]). In this paper, by identifying the presence of an enhanced condensate in we exemplify this study with one specific class of theories. the vacuum. We use the term approximate dilaton to refer We choose this class mostly on the grounds of simplicity— to a state that has significant mixing with the dilaton but is the model is a simple example of a gravity theory which not necessarily light.
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But the attentive reader certainly took notice of the with the study of phase transitions bridges between the phrase “assumption of continuity” in the previous para- physics arguments in Ref. [67] and those exposed in graph and realized that this assumption may fail, for Refs. [80,81], that are inferred from lower-dimensional example in the presence of a phase transition. In fact, this statistical mechanics. The first-order phase transition (to assumption must fail; the very presence of classical unphysical gravity configurations) signals the metastability instabilities, as indicated by the existence of a tachyon, of the branch of regular solutions along which the dilaton in what is an otherwise perfectly well-defined physical becomes light, and hence precludes us from creating an system (a well-known and established classical super- arbitrarily large hierarchy between the mass of said dilaton gravity), demands the existence of a different branch of and that of the other states along the stable branch. solutions, which must be energetically favored. The light We think this paper exemplifies and clarifies a few (approximate) dilaton and the tachyon appear along a important general points and opens a new avenue for branch of classical solutions that eventually ceases to be exploration. First of all, we confirm in the context of physically realized. We uncover evidence of a first-order top-down gauge-gravity dualities that one of the lightest phase transition in the gravity theory and show that the states arising in this way indeed overlaps significantly with physically realized solutions do not come immediately the dilaton and is not some accidentally light scalar particle close to the tachyonic ones, undermining the chain of with generic properties. We do so by repeating the implications from the previous paragraph. This is illustrated calculation of the spectrum by treating the scalars in the in Fig. 1, which sketches the free energy as a function of the probe approximation, which ignores the fluctuations of source of a relevant deformation in the field theory for two the five-dimensional metric, and by comparing the results of the different branches of the classical solutions. As the with the full gauge-invariant results. We expand on this source is increased, one encounters a first-order phase technical procedure in a different publication [127]. transition between a branch of regular solutions (in solid Second, the techniques we adopted can be equally black) and a branch of singular solutions (in long-dashed applied to many other backgrounds that are asymptotically dark green), which happens before the tachyonic region (in anti-de Sitter in the far UV and evolve toward a classical short-dashed orange) is reached. We will be more specific instability. In principle, there are countless such examples and precise in describing these phenomena in the body of one can build within the known catalog of supergravity the paper. theories. We expect the results we found here to hold The conclusion of this exercise is almost identical to the generically: there will be choices of parameters/solutions one we explicitly quoted earlier on in italics, taken from that make one of the scalars arbitrarily light. And there will Ref. [67]. A distinctive element is that, in the physical part be a first-order phase transition that prevents such solutions of parameter space, the state that is approximately a dilaton from being physically realized. Yet, there is no reason to is the next-to-lightest state. Furthermore, the connection expect that all phase transitions should be equally strong; the phase transition might take place in close proximity of the tachyonic instability. In this case, we would be able— by exploiting the formalism we are testing in this paper—to compute whether a nontrivial hierarchy appears in the mass spectrum, as the dilaton state behaves differently from the rest of the spectrum. Or, conversely, it might turn out that our findings are truly universal, so that no hierarchy of scale can be produced with this mechanism. Even such a negative result would be an interesting finding. The paper is organized as follows. In Sec. II, we define the properties of the model we study. We show the branches of solutions of interest to our investigation in Sec. III and produce a classification of nontrivial classical backgrounds based upon their asymptotic behavior in proximity of the end of the space in the interior of the geometry. All of the backgrounds share the same properties at large values of FIG. 1. Sketch of the free energy as a function of the source, for the radial direction ρ, corresponding to the UV of the dual two of the different branches of classical solutions, showing the theory. Many such solutions had not been identified before first-order phase transition at the crossing between the black (solid) and dark-green (long-dashed) lines, as well as the region in the literature. In Sec. IV, we present the spectra of with a tachyonic instability depicted by the orange (short-dashed) fluctuations, restricting ourselves to the regular gravity line. The reader should consult Fig. 10, its caption and the text backgrounds, and discuss their interpretation as bound around it, for more details on the phase transition in the realistic states in the dual confining theory. Section IV B contains model described in the main body of the paper. one of the core parts of the analysis: by comparing the
106018-4 DILATONIC STATES NEAR HOLOGRAPHIC PHASE … PHYS. REV. D 103, 106018 (2021) results of the probe approximation to those of the full ˆ ˆ H ˆ ˆ ≡ F ˆ ˆ þ mB ˆ ˆ ; ð4Þ calculation of the spectrum, we identify states that have an M N M N M N overlap with the dilaton, and we discuss how this relates to ˆ G ˆ ˆ ˆ ≡ 3∂ ˆ B ˆ ˆ ∂ ˆ B ˆ ˆ ∂ ˆ B ˆ ˆ ∂ ˆ B ˆ ˆ : 5 the magnitude of the condensates in the dual theory. We M N T ½M N T ¼ M N T þ N T M þ T M N ð Þ discuss the energetics, computing the free energy of the background configurations in Sec. V. Section VD shows Here, gˆ6 is the determinant of the metric tensor, and R6 ≡ Mˆ Nˆ evidence for the arising of a phase transition. Given the g RMˆ Nˆ is the corresponding Ricci scalar. We return to the length of the paper, and the fact that the model we consider scalar potential V6ðϕÞ and its critical points in Sec. II B. is nontrivial, we find it useful to summarize all our results in Sec. VI, and we conclude the paper with an outline of B. Scalar potential in D = 6 dimensions further avenues for future exploration in Sec. VII.We The potential for the real scalar field ϕ in the six- relegate to the Appendixes some useful technical details. dimensional model is given by1 [105]
II. MODEL 1 −6ϕ 2ϕ −2ϕ V6ðϕÞ¼ ðe − 9e − 12e Þð6Þ In this section, we summarize the action of the six- 9 dimensional supergravity written by Romans, adopt an and is shown in Fig. 2. It admits two critical points: ansatz in which one of the dimensions is a circle, perform the dimensional reduction of the theory on this circle, and 20 D 5 ϕ 0 V ϕ − write the resulting dimensionally reduced action in ¼ UV ¼ 6ð UVÞ¼ 9 ; ð7Þ space-time dimensions. Most of the material reported here can be found in the literature, that we have already cited and and will further refer to throughout the section. Yet, we find it convenient to collect all the useful background information logð3Þ 4 in this section, both to make the exposition self-contained, ϕ ¼ − V6ðϕ Þ¼−pffiffiffi ; ð8Þ IR 4 IR 3 as well as to fix the notation unambiguously. with the former (latter) a global maximum (minimum) D A. Action and formalism of the = 6 model which preserves (breaks) supersymmetry. As we shall see, The six-dimensional (gauged) supergravity constructed there exist numerical solutions to the equations of motion by Romans [100] describes 32 bosonic degrees of freedom that interpolate between these two critical points, corre- (d.o.f.) (we ignore the fermions). We denote by indices sponding to a renormalization group flow between a UV Mˆ ∈ f0; 1; 2; 3; 5; 6g the coordinates in D ¼ 6 dimensions. and IR fixed point in the dual field theory. In previous work ϕ ϕ ∈ ϕ ϕ The field content (number of d.o.f.) is the following: one [99], we restricted to the closed interval ½ IR; UV . ϕ 1 1 1 4 For the purposes of this paper, we extend this domain by scalar ( × ), one vector AMˆ ( × ) transforming as a singlet under U 1 , three vectors Ai (3 × 4) in the 3 allowing positive values of ϕ. ð Þ Mˆ 2 1 6 representation of SUð Þ, one 2-form BMˆ Nˆ ( × ), and the ˆ 1 9 C. Reduction from D =6 to D = 5 dimensions six-dimensional metric tensor gMˆ Nˆ ( × ). The action is given by We compactify one of the external dimensions Z pffiffiffiffiffiffiffiffi [described by the coordinate η ∈ ½0; 2πÞ] of the Romans 6 R6 ˆ ˆ 1 S −ˆ − ˆM N∂ ˆ ϕ∂ ˆ ϕ − V ϕ theory on a circle S and parametrize the six-dimensional 6 ¼ d x g6 4 g M N 6ð Þ metric as follows, 1 X − −2ϕ ˆMˆ Rˆ ˆNˆ Sˆ ˆ i ˆ i e g g F ˆ ˆ F ˆ ˆ 2 −2χ 2 6χ M 2 4 M N R S ds6 ¼ e ds þe ðdηþV dx Þ i 5 M 1 −2χ 2AðrÞ 2 2 6χ η M 2 −2ϕ ˆ ˆ ˆ ˆ ˆ ˆ ¼ e ðe dx1;3 þdr Þþe ðd þVMdx Þ − e gˆM RgˆN SH ˆ ˆ H ˆ ˆ 4 M N R S −2χ 2 ρ 2 2χ 2 6χ 2 ¼ e ðe Að Þdx þe dρ Þþe ðdηþV dxMÞ ; ð9Þ 1 1;3 M 4ϕ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ − e gˆM RgˆN SgˆT UG ˆ ˆ ˆ G ˆ ˆ ˆ ; 12 M N T R S U ð1Þ 2 where ds5 is the five-dimensional line element so that 8AðrÞ detðgMNÞ ≡ g5 ¼ −e , with warp factor AðrÞ, and where summation over repeated indices is implied. The “ ” tensors are defined as follows: we have adopted the mostly plus four-dimensional Minkowski metric signature, ημν ¼diagð−;þ;þ;þÞ; indices ˆ i j F ≡∂ˆ Ai − ∂ ˆ Ai gϵijkA Ak ; 2 Mˆ Nˆ M Nˆ N Mˆ þ Mˆ Nˆ ð Þ 1In the language of Ref. [105], we adopted the choice of g ¼ ˆ pffiffiffi pffiffi F ˆ ˆ ≡∂ˆ A ˆ − ∂ ˆ A ˆ ; ð3Þ 2 2 M N M N N M 8 and m ¼ 3 , without loss of generality.
106018-5 ELANDER, PIAI, and ROUGHLEY PHYS. REV. D 103, 106018 (2021) run over μ; ν ∈ f0; 1; 2; 3g and M; N ∈ f0; 1; 2; 3; 5g. The values in the closed interval ρ ∈ ½ρ1; ρ2 , for reasons to be third equality introduces the convenient redefinition of the discussed later, but it is understood that the physical results radial coordinate dr ≡ eχdρ. In the background solutions that apply to the dual field theory are recovered only after that we will consider, each field of the supergravity model removing these restrictions. depends only on the radial coordinate ρ, and additionally After decomposing the fields, and some algebra (see only ϕðρÞ, χðρÞ, and AðρÞ acquire nonzero radial profiles, Ref. [99] for details), the action of the reduced five- thus ensuring Poincar´e invariance along the Minkowski xμ dimensional model is given by directions. We constrain the holographic coordinate to take
Z ffiffiffiffiffiffiffiffi R 1 1 1 5 p− 5 − MN∂ Φa∂ Φb − V ϕ χ − MR NS A B − 2χ−2ϕ MR NSH H S5 ¼ d x g5 4 2 Gabg M N ð ; Þ 4 HABg g FMNFRS 4 e g g MN RS 1 1 1 − 4χþ4ϕ MR NS TU − −6χ−2ϕ NSH H − −4χþ4ϕ NS TU 12 e g g g GMNTGRSU 2 e g 6N 6S 4 e g g G6NTG6SU ; ð10Þ where the 32 physical degrees of freedom are now carried solely on the holographic coordinate ρ. Hence, the equa- by the following five-dimensional field content: six scalars tions of motion for the background functions are given by i i fϕ; χ; π ;A6g (6 × 1), six vectors fAM;A ;B6N;VMg M 1 ∂V (6 × 3), one 2-form BMN (1 × 3), and the metric tensor 2 6 ∂ρϕ þð4∂ρA − ∂ρχÞ∂ρϕ ¼ ; ð11Þ gMN (1 × 5)inD ¼ 5 dimensions. The dynamical scalar 2 ∂ϕ field χ parametrizes the size of the compact S1 [see Eq. (9)]. a i V The sigma-model scalars are Φ ¼fϕ; χ; π g with the ∂2χ 4∂ − ∂ χ ∂ χ − 6 −6χ−2ϕ ρ þð ρA ρ Þ ρ ¼ 3 ; ð12Þ metric Gab ¼ diagð2; 6;e Þ, while the field strengths fFV;Fig have the metric H ¼ diagð1 e8χ;e2χ−2ϕÞ. The AB 4 3 ∂ 2 − ∂ ϕ 2 − 3 ∂ χ 2 −V five-dimensional scalar potential appearing in the circle- ð ρAÞ ð ρ Þ ð ρ Þ ¼ 6; ð13Þ V ϕ χ −2χV ϕ reduced model is given by ð ; Þ¼e 6ð Þ. 2 2 2 2 3∂ρA þ 6ð∂ρAÞ þ 2ð∂ρϕÞ þ 6ð∂ρχÞ − 3∂ρA∂ρχ ¼ −2V6: D. Equations of motion ð14Þ The classical equations of motion can be obtained from Only the first three are independent. These equations of S5, the action for the five-dimensional model, provided in motion can be reformulated using the following convenient Eq. (10). We remind the Reader that all of the classical redefinitions, supergravity background fields are assumed to depend c ≡ 4A − χ;d≡ A − 4χ; ð15Þ
0.0 or equivalently –0.5 1 1 –1.0 χ ≡ − 4 ≡ 4 − 15 ðc dÞ;A15 ð c dÞ; ð16Þ –1.5
–2.0 so that we can recast them in the following form:
–2.5 2 1 ∂V6 ∂ρϕ þ ∂ρc∂ρϕ ¼ ; ð17Þ –3.0 2 ∂ϕ
–0.8 –0.6 –0.4 –0.2 0.0 0.2 0.4 0.6 2 2 ∂ρc þð∂ρcÞ ¼ −5V6; ð18Þ
2 2 2 ∂ρc − ∂ρd − 5 ∂ρϕ −5V6; 19 FIG. 2. The potential V6ðϕÞ of Romans supergravity as a ð Þ ð Þ ð Þ ¼ ð Þ function of the one scalar ϕ in the sigma model coupled to 2 gravity in D ¼ 6 dimensions. We highlight the two critical points: ∂ρd þ ∂ρc∂ρd ¼ 0: ð20Þ the case ϕ ¼ ϕUV ¼ 0 (blue disk) and the case ϕ ¼ ϕIR ¼ 1 − 4 logð3Þ (dark-red triangle), both of which allow for AdS6 The last of these four equations can be derived from the background solutions. previous three, yet we show it explicitly for a reason we will
106018-6 DILATONIC STATES NEAR HOLOGRAPHIC PHASE … PHYS. REV. D 103, 106018 (2021) explain shortly. Having solved the coupled equations (17) It is straightforward to verify that solutions to the previous and (18) to yield ϕ and c, one can proceed then to solve two equations also solve the full equations of motion of the Eq. (19) to determine d. Notice that for any given solution d system, which after imposing the constraint A ¼ 4χ (and one finds that −d is also admissible. We observe that hence A ¼ 3χ) can be rewritten as Eq. (20) can be rewritten as a vanishing total derivative, and ∂V hence we obtain the following useful relation, 4∂2ϕ 15∂ ∂ ϕ 2 6 ρ þ ρA ρ ¼ ∂ϕ ; ð30Þ 4A−χ e ð4∂ρχ − ∂ρAÞ¼C; ð21Þ 2 2 3∂ρA þ 4ð∂ρϕÞ ¼ 0; ð31Þ for some background-dependent integration constant C. 2 2 We will make use of this relation later in the paper. Finally, 45ð∂ρAÞ − 16ð∂ρϕÞ ¼ −16V6: ð32Þ we also note that by combining Eqs. (18) and (19), one can derive the inequality The superpotential W1 yields a system of equations that 2 admits the solution ϕ ¼ 0 and A ¼ 3 ρ. It can be expanded 2 ∂ρc ≤ 0; ð22Þ in powers of small ϕ: 4 4 7 2 61 that constrains the RG flows of the dual field theory W ϕ − − 2ϕ2 ϕ3 − ϕ4 ϕ5 − ϕ6 admitting a description based on the classical backgrounds. 1ð Þ¼ 3 þ 3 6 þ 3 180 þ Oðϕ7Þ: ð33Þ E. Superpotential formalism The conventions we are using in writing the action in The quadratic term in this expansion shows that the solutions Eq. (1) are such that if the potential of the model in D can be interpreted in terms of the vacuum expectation value of an operator of dimension Δ ¼ 3 in the dual five- dimensions VD can be written in terms of a superpotential W that satisfies the following equation [123] dimensional strongly coupled field theory [110]. Besides providing a useful solution-generating tech- nique, the superpotential formalism also plays a role in 1 ϕϕ 2 D − 1 2 V ¼ G ð∂ϕWÞ − W ; ð23Þ D 2 D − 2 defining an unambiguous, covariant, and physically moti- vated subtraction scheme in the calculation of the free for the metric ansatz energy. To this purpose, we notice that the system admits a second choice of superpotential, that we call W2, and that 2 2A 2 ρ2 ϕ dsD ¼ e dx1;D−2 þ d ; ð24Þ can be written as a power expansion for small : 4 4 16 86 848 then one finds that the solutions to a special set of first- 2 3 4 5 W2ðϕÞ¼− − ϕ þ ϕ þ ϕ þ ϕ order equations are also solutions to the second-order 3 3 3 3 3 classical equations. The aforementioned first-order equa- 988658 þ ϕ6 þ Oðϕ7Þ: ð34Þ tions are the following: 315
2 We are not aware of the existence of a closed form solution ∂ρA ¼ − W; ð25Þ D − 2 to Eq. (23) that satisfies this expansion. Notice that, while encompassing the same AdS6 solution of the first-order ϕϕ ∂ρϕ ¼ G ∂ϕW: ð26Þ system derived from W2, in this case the solutions of the first-order system correspond to deformations of the dual As we are working with D ¼ 6, and given the potential field theory by the nontrivial coupling of the same operator V6 of Eq. (6), one finds [110] the superpotential W ¼ W1, of dimension Δ ¼ 3. As we shall see, by choosing to adopt which together with the corresponding first-order equa- W2ðϕÞ as the form of one of the boundary-localized terms tions is in the complete gravity action, we can provide the counter- terms in the holographic renormalization procedure and 1 ϕ −3ϕ guarantee that all the divergences are canceled for any W1 ¼ −e − e ; ð27Þ 3 asymptotically AdS6 backgrounds. 1 1 1 ∂ A − W ϕ −3ϕ ρ ¼ 2 1 ¼ 2 e þ 3 e ; ð28Þ III. CLASSES OF SOLUTIONS In this section, we present the classes of solutions that we 1 1 ϕ −3ϕ will refer to as supersymmetry (SUSY), IR-conformal, ∂ρϕ ¼ ∂ϕW1 ¼ ð−e þ e Þ: ð29Þ 2 2 confining, and skewed, together with their IR expansions.
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We also introduce a few additional, more general, singular z ≡ e−2ρ=3. We truncate each expansion at Oðz11Þ. These solutions, including ones that preserve five-dimensional expansions are used in the numerical analysis for all classes Poincar´e invariance. We introduce the relevant UV expan- of solutions in order to extract values for the set of UV ϕ χ sions for the two scalars f ; g and the warp factor A, parameters fϕ2; ϕ3; χ5; χU;AUg that unambiguously iden- which are valid for all these classes of solutions. tify each background and to compute the free energy. All solutions we are interested in have the same formal UV A. UV expansions expansion, as they all correspond to deformations of the We present here the large-ρ expansions for ϕ, χ, and A in same supersymmetric fixed point. The expansions are given terms of a convenient holographic coordinate defined by by the following equations:
29ϕ3 339 ϕ ϕ 2 ϕ 3 − 6ϕ2 4 − 4 ϕ ϕ 5 2 − ϕ2 6 ϕ2ϕ 7 ðzÞ¼ 2z þ 3z 2z ð 2 3Þz þ 2 3 z þ 20 2 3z 77ϕ ϕ2 146ϕ4 19ϕ3 8497ϕ3ϕ 2 3 − 2 8 3 − 2 3 9 þ 10 3 z þ 12 105 z 6752ϕ5 1986ϕ2ϕ2 4127161ϕ4ϕ 3427ϕ ϕ3 2 − 2 3 10 2 3 − 2 3 11 O 12 þ 35 35 z þ 10080 180 z þ ðz Þ; ð35Þ ϕ2 4 8ϕ3 ϕ2 32 3ϕ ϕ2 77ϕ4 χ χ − logðzÞ − 2z χ 5 2 − 3 6 ϕ2ϕ 7 2 3 − 2 8 ðzÞ¼ U 3 12 þ 5z þ 9 12 z þ 21 2 3z þ 4 16 z 1072ϕ3ϕ 25χ ϕ2 4ϕ3 15χ2 172ϕ5 3181ϕ2ϕ2 9χ ϕ ϕ − 2 3 5 2 3 9 − 5 2 − 2 3 5 2 3 10 þ 135 þ 36 þ 27 z þ 64 þ 9 600 þ 8 z 44776ϕ4ϕ 200χ ϕ3 96ϕ ϕ3 25χ ϕ2 2 3 − 5 2 − 2 3 5 3 11 O 12 þ 1155 33 55 þ 44 z þ ðz Þ; ð36Þ 4 ϕ2 4 χ 3ϕ ϕ 32ϕ3 ϕ2 128 − logðzÞ − 2z 5 − 2 3 5 2 − 3 6 ϕ2ϕ 7 AðzÞ¼AU 3 3 þ 4 5 z þ 9 3 z þ 21 2 3z 77ϕ4 1 3ϕ ϕ2 − 2 8 −69508ϕ3ϕ 375χ ϕ2 1280ϕ3 9 þ 2 3 4 z þ 2160 ð 2 3 þ 5 2 þ 3Þz 1 −3375χ2 275200ϕ5 − 78936ϕ2ϕ2 10 þ 3600 ð 5 þ 2 2 3Þz 1 2932864ϕ4ϕ − 28000χ ϕ3 − 135324ϕ ϕ3 2625χ ϕ2 11 O 12 þ 18480 ð 2 3 5 2 2 3 þ 5 3Þz þ ðz Þ: ð37Þ
For convenience, we also write explicitly the UV expansions for the two combinations c and d that were introduced in Sec. II D: 5ϕ2 4 12 40ϕ3 5ϕ2 160 4 − χ − 5 − 2z − ϕ ϕ 5 2 − 3 6 ϕ2ϕ 7 cðzÞ¼ AU U logðzÞ 4 5 2 3z þ 3 4 z þ 7 2 3z 45ϕ ϕ2 1155ϕ4 20ϕ3 1087ϕ3ϕ 225χ2 860ϕ5 16481ϕ2ϕ2 9χ ϕ ϕ 2 3 − 2 8 3 − 2 3 9 − 5 2 − 2 3 − 5 2 3 10 þ 4 16 z þ 9 9 z þ 64 þ 3 200 8 z 45896ϕ4ϕ 303ϕ ϕ3 2 3 − 2 3 11 O 12 þ 77 11 z þ ðz Þ; ð38Þ 15χ 3ϕ ϕ 5ϕ3ϕ 125χ ϕ2 18 9χ ϕ ϕ − 4χ − 5 − 2 3 5 − 2 3 − 5 2 9 − ϕ2ϕ2 − 5 2 3 10 dðzÞ¼AU U þ 4 5 z þ 12 48 z þ 25 2 3 2 z 40ϕ4ϕ 250χ ϕ3 15ϕ ϕ3 375χ ϕ2 2 3 5 2 − 2 3 − 5 3 11 O 12 þ 11 þ 11 44 176 z þ ðz Þ: ð39Þ
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1.0 When computing the free energy for each class of solutions, we will choose to always set AU ¼ χU ¼ 0. 4χ ⇔ 0 The constraint A ¼ d ¼ reinstates (locally) five- 0.5 dimensional Poincar´e invariance. It is required for the SUSY, IR-conformal and singular domain-wall solutions, and it constrains the parameters appearing in the UV 0.0 expansions.
B. SUSY solutions –0.5 The first-order equations presented in Eqs. (27)–(29) of Sec. II E can be solved by performing the change of −ϕ –1.0 variable ∂ρ ≡ e ∂τ, after which one obtains
∂τϕ − sinh 2ϕ ; 40 ¼ ð Þ ð Þ –1.5 1 2ϕ 1 −2ϕ ∂τA ¼ e þ e ; ð41Þ 2 3 –2.0 –0.4 –0.2 0.0 0.2 0.4 0.6 0.8 1.0 which are solved exactly by [110] ∂ ϕ ϕ 2ðτ−τ Þ FIG. 3. Parametric plot of ρ as a function of for solutions ϕðτÞ¼arccothðe o Þ; ð42Þ 4 which satisfy the domain-wall constraint A ¼ 3 A ¼ 4χ. The blue 1 disk and dark-red triangle, respectively, denote the UV and IR A τ A 2 τ − τ critical points of the six-dimensional potential V6; the purple ð Þ¼ o þ 3 logðsinhð ð oÞÞÞ (solid) line represents the class of IR-conformal solutions 1 introduced in Sec. III C with duals which flow between the þ logðtanhðτ − τ ÞÞÞ; ð43Þ 6 o two critical points; and the light-gray (solid) line represents the analytical supersymmetric solutions obtained by solving the first- where Ao and τo are integration constants. These SUSY order differential equations (27), (28), and (29). The dark-gray solutions evolve ϕ monotonically from the supersymmetric arrows exhibit the vector field appearing in the first-order fixed point toward the good singularity (ϕ → ∞), for which differential equation for ðϕ; ∂ρϕÞ. We also show two examples of the (good) singular solutions obeying the IR expansion in the potential is bounded from above. We notice that the pffiffiffi ϕ −1 5 ϕ −0 3 flow breaks scale invariance and hence reduces the number Sec. III F, for L ¼ = and I ¼ . , 0.1 (long-dashed of supersymmetries of the underlying theory to 8 [110].We dark-blue lines) and two examples of the domain-wall (badly remind the reader that these solutions result from the singular) solutions from Sec. III G, with ϕ4 ¼ −0.06, 40 (dashed formation of a nontrivial condensate in the dual field dark-green lines). theory. ϕ By relating the radial coordinates ρ and τ, one finds that differential equation in terms of alone, then plot para- ϕ ∂ ϕ the SUSY solutions given in Eqs. (42) and (43) have the metrically ð ; ρ Þ, and study how the solutions flow away following IR expansions, from the supersymmetric fixed point at the origin. We observe that the SUSY solutions (gray line) form the 1 separatrix between numerical solutions which flow to a ϕ ρ 2 − ρ − ρ ρ − ρ 4 … ð Þ¼logð Þ logð oÞþ80 ð oÞ þ ; ð44Þ bad singularity (ϕ → −∞) and solutions which instead flow to a good singularity (ϕ → ∞). 1 1 4 χðρÞ¼χI þ logðρ − ρoÞþ ðρ − ρoÞ þ …; ð45Þ 3 360 C. IR-conformal solutions 4 1 A second class of solutions interpolates between the two A ρ A log ρ − ρ ρ − ρ 4 …; 46 ð Þ¼ I þ 3 ð oÞþ90 ð oÞ þ ð Þ known AdS6 solutions of the six-dimensional model, corresponding in the boundary theory to a renormalization where χI and AI ¼ 4χI are integration constants and ρo is group flow between two fixed points. The six-dimensional the radial position of the singularity in the deep IR region of bulk geometry does not close off for any (finite) value of the the bulk. holographic coordinate ρ and the compact dimension In Fig. 3, we illustrate the space of domain-wall described by η maintains nonzero size for all ρ. Hence, solutions, of which the SUSY solutions are a special case, there does not exist a physical lower limit for the energy through the following procedure. We first solve Eq. (32) for scale at which the field theories dual to this class of ∂ρA and substitute into Eq. (30) to obtain a second-order solutions may be probed. The IR expansions for this class
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4 4 pffiffiffiffiffiffiffiffiffi of solutions are conveniently written in terms of ρ − 2 −5 ρ − ρ − 5−Δ ρ Að Þ¼AI logð Þþ log½sinhð vð oÞÞ ð IRÞR ρ → −∞ 15 15 e IR , which is small in the limit , and they pffiffiffiffiffiffiffiffiffi are given by 1 −5 v ρ − ρ þ 15 log tanh 2 ð oÞ ; ð52Þ − 5−Δ ρ ϕ ρ ϕ ϕ − ϕ ð IRÞR ð Þ¼ IR þð I IRÞe IR þ ; ð47Þ with v ≡ V6ðϕ ¼ 0Þ as defined in Sec. II B. By direct
ρ substitution of the above analytical solutions, we find ρ 1 2 −2 5−Δ χ ρ χ − ϕ −ϕ ð IRÞR ð Þ¼ I þ 3 12ð I IRÞ e IR þ ; ð48Þ RIR ρ 4 ρ −χ ρ 1 4 −χ 10 cð Þ Að Þ ð Þ AI I ρ − ρ e ¼ e ¼ 2 e sinh 3 ð oÞ ; ð53Þ 4ρ 1 ρ 2 −2ð5−ΔIRÞ AðρÞ¼A þ − ðϕ − ϕ Þ e RIR þ ; ð49Þ I 3R 3 I IR IR ρ ρ −4χ ρ −4χ 5 dð Þ Að Þ ð Þ AI I ρ − ρ e ¼ e ¼ e coth 3 ð oÞ : ð54Þ where χ and A are integration constants, R2 ≡ I Ipffiffiffi IR −5 V ϕ −1 5 3 ð 6ð IRÞÞ ¼ 4 is the (squared)ffiffiffiffiffi curvature radius These solutions can be generalized by series expanding for 1 p ρ − ρ ϕ of the AdS6 geometry, Δ ¼ ð5 þ 65Þ is the scaling small ð oÞ and allowing for nontrivial values of for IR 2 ρ − ρ dimension of the operator in the dual boundary theory that small ð oÞ, to obtain expansions which may be used to is related to the IR critical point value of the bulk scalar ϕ, construct a generalized family of numerical solutions. and we restrict the one free parameter ϕ ≥−1 logð3Þ.We We obtain the numerical solutions by solving the I 4 classical equations of motion, subject to boundary con- observe that d ≡ A − 4χ ¼ 0 for all values of ϕ for this I ditions obtained from the following IR [small ρ − ρ ] class of solutions. It is also worth noting that the back- ð oÞ expansions, grounds defined by this class of solutions do not preserve supersymmetry. In the UV expansions, these solutions 1 −6ϕ 4ϕ 8ϕ 2 ϕ ϕ ϕ ρ ϕ − I 1 − 4 I 3 I ρ − ρ require a tuning of 3 against 2, as we will discuss in ð Þ¼ I 12e ð e þ e Þð oÞ Sec. VB. 1 −12ϕ 4ϕ 8ϕ 16ϕ 4 − I 4 − 28 I 51 I − 27 I ρ − ρ 324e ð e þ e e Þð oÞ D. Confining solutions 6 þ Oððρ − ρoÞ Þ; ð55Þ With some abuse of language, we refer to a third class of solutions as confining. Here, the compact dimension 1 5 1 χ ρ χ ρ − ρ (described by the coordinate η) shrinks to a point at some ð Þ¼ I þ 3 log 3 þ 3 logð oÞ ρ finite value o of the holographic coordinate, and the six- 1 −2ϕI 2 dimensional bulk geometry closes off smoothly. On the − e ð2 þ sinhð4ϕI þ logð3ÞÞÞðρ − ρoÞ boundary side of the duality, this smooth tapering property 27 5 of the bulk manifold is interpreted as a physical lower limit −4ϕ 2 4 þ e I ð2 þ sinhð4ϕ þ logð3ÞÞÞ ðρ − ρ Þ on the energy scale that may be probed in the correspond- 486 I o 5 6 ing field theory in D ¼ dimensions. We anticipate that the þ Oððρ − ρoÞ Þ; ð56Þ IR asymptotic expansion of these solutions is identical to those studied in Ref. [98] and hence the calculation of the 1 5 1 ρ ρ − ρ Wilson loops via the holographic prescription yields the Að Þ¼AI þ 3 log 3 þ 3 logð oÞ area law expected from confinement. We will compute 7 −6ϕ 4ϕ 8ϕ 2 explicitly the spectrum in Sec. IV and show that it is − I 1 − 12 I − 9 I ρ − ρ 324 e ð e e Þð oÞ discrete, generalizing the results of Ref. [99]. 1 −12ϕ −8ϕ As mentioned in previous work [92,93,98,99], there exist þ ð108 − 67e I þ 636e I exact analytical solutions of the classical equations of 17496 −4ϕ 4ϕ 4 ϕ I I motion when is constant, − 2124e − 1053e Þðρ − ρoÞ O ρ − ρ 6 ϕ ϕ 0 þ ðð oÞ Þ; ð57Þ ¼ UV ¼ ; ð50Þ χ pffiffiffiffiffiffiffiffiffi where I and AI generalize the integration constants 1 −5v appearing in the analytical solutions and the third integra- χðρÞ¼χI − log cosh ðρ − ρoÞ 5 2 tion constant ρo may be chosen to fix the point at which the pffiffiffiffiffiffiffiffiffi geometry ends. We will comment on the fourth integration 1 −5 v constant ϕ momentarily. By using these expressions, we þ log sinh ðρ − ρoÞ ; ð51Þ I 3 2 find that
106018-10 DILATONIC STATES NEAR HOLOGRAPHIC PHASE … PHYS. REV. D 103, 106018 (2021) cðρÞ 4AðρÞ−χðρÞ 4A −χ e ¼ e ¼ e I I fðϕ ; ðρ − ρ ÞÞ; ð58Þ ρ 4 ρ −χ ρ 1 4 −χ 10 I o cð Þ Að Þ ð Þ AI I ρ − ρ e ¼ e ¼ 2 e sinh 3 ð oÞ ; ð66Þ
dðρÞ AðρÞ−4χðρÞ AI −4χI e ¼ e ¼ e gðϕI; ðρ − ρoÞÞ; ð59Þ 5 dðρÞ AðρÞ−4χðρÞ AI −4χI e ¼ e ¼ e tanh ðρ − ρoÞ ; ð67Þ where the functions f and g are known for ϕI ¼ 0, and 3 otherwise can be determined numerically. The additive integration constant AI may be removed by which shows that these are indeed the solutions obtained a rescaling of the Minkowski coordinates. By contrast, from the confining ones with ϕ ¼ 0 [see Eqs. (53) and (54)] because η is a periodic coordinate with period 2π, we are by replacing d → −d, as anticipated in Sec. II D. required to fix χI to avoid a conical singularity at ρo.In Just as with the solutions that confine, we can generalize proximity of this point, the six-dimensional geometry these analytical solutions to any values of ϕI by series resembles a two-dimensional space described by the expanding for small ðρ − ρoÞ. We obtain the following IR following metric, expansions,
2 2 6χ 2 ds dρ e dη 60 1 −6ϕ 4ϕ 8ϕ 2 2 ¼ þ ð Þ ϕ ρ ϕ − I 1 − 4 I 3 I ρ − ρ ð Þ¼ I 12e ð e þ e Þð oÞ 5 2 6χ 2 2 1 −12ϕ 4ϕ 8ϕ 16ϕ 4 ¼ dρ − ve I ðρ − ρ Þ dη þ ; ð61Þ − I 4 − 28 I 51 I − 27 I ρ − ρ 4 o 324e ð e þ e e Þð oÞ O ρ − ρ 6 from which we extract the required constraint: þ ðð oÞ Þ; ð68Þ 1 −4 1 5 1 5 1 χðρÞ¼χ − log − logðρ − ρ Þ χI ¼ log ¼ − log : ð62Þ I 5 3 5 o 6 5v 3 3 1 −6ϕI 4ϕI 8ϕI 2 − e ð1 − 12e − 9e Þðρ − ρoÞ The one remaining free parameter of this system is ϕI, 54 ϕ ≥−1 3 which we constrain to take values I 4 logð Þ,aswe 1 −12ϕ 4ϕ 4ϕ 4ϕ − I 23 3 I −88 9 I 38 24 I are interested only in solutions that reach back to the trivial 9720 e ½ þ e ð þ e ð þ e critical point for large ρ (the fact that not all possible 8ϕ 4 6 þ 21e I ÞÞ ðρ − ρ Þ þ Oððρ − ρ Þ Þ; ð69Þ solutions flow to the UV fixed point can be seen for the o o domain-wall solutions in Fig. 3). 1 5 1 ρ ρ − ρ Að Þ¼AI þ 5 log 3 þ 5 logð oÞ E. Skewed solutions 1 −6ϕ 4ϕ 8ϕ 2 − I 1 − 12 I − 9 I ρ − ρ There exists another class of analytical solutions with 36 e ð e e Þð oÞ ϕ ¼ 0 for which the compact coordinate does not shrink to 1 −12ϕ 4ϕ χ ρ − I 131 3 I −436 a point; ð Þ is a nonmonotonic function which diverges to 29160 e ½ þ e ð ∞ ρ at small . We refer to these solutions as skewed. The 4ϕ 4ϕ 8ϕ 4 3 I 508 84 I 261 I ρ − ρ solutions are as follows: þ e ð þ e þ e ÞÞ ð oÞ þ Oððρ − ρ Þ6Þ; ð70Þ ϕ ϕ 0 o ¼ UV ¼ ; ð63Þ χ pffiffiffiffiffiffiffiffiffi where the integration constants I and AI are the gener- 1 −5 ϕ χ ρ χ v ρ − ρ alization of the ones appearing in Eqs. (64) and (65) and I ð Þ¼ I þ 3 log cosh 2 ð oÞ is the free parameter that we vary to generate the family of pffiffiffiffiffiffiffiffiffi 1 −5 solutions. One can solve numerically the classical equa- v tions of motion, subject to boundary conditions derived − log sinh ðρ − ρoÞ ; ð64Þ 5 2 from these IR expansions, in order to construct a class of skewed solutions. 4 4 pffiffiffiffiffiffiffiffiffi ρ − 2 −5 ρ − ρ We observe that the following relations hold, Að Þ¼AI 15 logð Þþ15 log½sinhð vð oÞÞ ffiffiffiffiffiffiffiffiffi p ρ 4 ρ −χ ρ 4 −χ 1 −5 ecð Þ e Að Þ ð Þ e AI I f ϕ ; ρ − ρ ; 71 − v ρ − ρ ¼ ¼ ð I ð oÞÞ ð Þ 15 log tanh 2 ð oÞ : ð65Þ dðρÞ AðρÞ−4χðρÞ AI −4χI −1 e ¼ e ¼ e ½gðϕI; ðρ − ρoÞÞ ; ð72Þ As with the confining solutions, we take note of the following two results obtained by substituting in for the where the functions f and g take exactly the same form as skewed analytical solutions above, those in the analogous results for the confining solutions.
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ϕconf ϕskew ρconf ρskew Hence, provided that I ¼ I and o ¼ o , one We find that a broad, generic class of classical solutions finds the relation can be parametrized by the following expansion near the end of space at ρ ¼ ρo, conf skew ∂ρd ðρÞ¼−∂ρd ðρÞ; ð73Þ ϕ ¼ ϕI þ ϕL logðρ − ρoÞ where the conf and skew superscripts represent evaluation X∞ X2n 2nþ2nϕL−4jϕL using the confining and skewed background solutions, þ cnjðρ − ρoÞ ; ð78Þ respectively. In turn, this implies that the relation n¼1 j¼0
3 where the coefficients c depend on the free parameters ϕ χskewðρÞ − AskewðρÞ¼− ðχconf ðρÞþAconfðρÞÞ ð74Þ nj I 5 and ϕL. Some useful details are provided in Appendix B, while we exhibit here only the leading-order terms of this is satisfied up to an additive integration constant. By expansion, ignoring all the power-law corrections: comparing the UV expansions, one then finds the identi- fications ϕðρÞ¼ϕI þ ϕL logðρ − ρoÞþ ; ð79Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕskew ϕconf 1 2 ¼ 2 ; ð75Þ χ ρ χ 4ζ 1 − 5ϕ2 1 ρ − ρ ð Þ¼ I þ 15 L þ logð oÞþ ; ð80Þ ϕskew ϕconf; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ¼ 3 ð76Þ 1 AðρÞ¼A þ ζ 1 − 5ϕ2 þ 4 logðρ − ρ Þþ : ð81Þ 8 I 15 L o χskew −χconf − ϕconfϕconf 5 ¼ 5 25 2 3 : ð77Þ The five integration constants are ϕI, χI, AI, ϕL, and ρo, supplemented by the discrete choice ζ 1. We notice We conclude this subsection with an observation which ¼ that for ϕ 0 and ζ 1 we recover the confining motivates our choice of the name “skewed” for this class of L ¼ ¼þ solutions, while for ϕ 0 and ζ −1, we recover the solutions. From the six-dimensional metric in Eq. (9),we L ¼ ¼ skewed solutions. For ϕ ≠ 0, one obtains either solutions can deduce the behavior of the bulk geometry in the deep L with a good singularity (ϕ < 0) or with a bad singular- IR for these solutions. We notice by substituting for the L ity (ϕL > 0). small-ðρ − ρoÞ expansions that the size of the Minkowski 2 The integration constant in front of the logarithm is 5 directions scales as ðρ − ρoÞ , while the compact dimension 3 constrained to take values within the range −5 parametrized by η scales as ðρ − ρoÞ . Hence, in the ρ → ρ limit, the four-dimensional Minkowski volume 1 1 o − pffiffiffi ≤ ϕ < : ð82Þ vanishes, while the volume of the circle diverges. This 5 L 3 contrasts with the small-ðρ − ρoÞ behavior of the geometry 1 for the confining solutions wherein the Minkowski ϕ ≥−pffiffi The lower bound L 5 arises from the requirements directions maintain a fixed nonzero volume in the IR, that both χ and A be real. For a choice that saturates this while the circle shrinks to a point. The shrinking and lower bound, and for A ¼ 4χ , the solutions satisfy the expanding behavior of the various metric components for I I condition A ¼ 4χ required by domain-wall solutions. This this class of solutions motivates our choice of the name “ ” parametrization then encompasses all of the aforemen- skewed. Appendix A is devoted to showing that, while tioned solutions, with the exception of the IR-conformal the confining solutions are regular, the skewed ones are and SUSY solutions. singular. The upper bound in Eq. (82) emerges from the require- ment that all powers in Eq. (78) be positive. As for positive F. Generic (singular) solutions ϕL the worst power appearing at any given n is 2 1 − 3ϕ When ϕ diverges at the end of space, all curvature nð LÞ, in order for all the powers to be positive, ρ − ρ invariants diverge (see Appendix A). If ϕ approaches and that hence the IR divergence be logarithmic in ð oÞ, ϕ 1 3 ϕ → þ∞, we find a good singularity. These solutions we must require that L < = . (The same line of argu- ϕ 0 are incomplete but capture at least some salient features of ments for negative L would be controlled by the j ¼ the system. By contrast, in the case in which ϕ → −∞ at power, in which case one would find the constraint ϕL > −1.) This requirement is more stringent than requir- the end of space, the solutions result in a bad singularity, 1 χ ϕ ≤ pffiffi and we should disregard them as unphysical. Nevertheless, ing that A and be real, which would yield L 5. 1 they play an important technical role in our study, as we The limit ϕL → 3 is such that the series expansion cannot anticipated in the Introduction and as we shall see and be truncated nor resummed; at all infinitely many levels explain in detail in Sec. V. of n, one finds additive contributions proportional to
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0 4=3 ðρ − ρoÞ ,toðρ − ρoÞ and so on. We discuss a related free energy (see later, in Sec. V) and that we conveniently class of solutions in the next subsection. define as the time a massless particle takes to reach the end of space from the UV boundary, following Ref. [128], G. Badly singular domain-wall solutions Z Z ∞ ∞ Finally, we also found another class of singular domain- Λ−1 ≡ dre˜ −Aðr˜Þ ¼ dρ˜eχðρ˜Þ−Aðρ˜Þ; ð85Þ ρ wall solutions, for which the IR expansion is the following: ro o χ 1 9 log ρ − ρ where A and are evaluated on the backgrounds. When a ϕ ρ ð oÞ ϕ ρ − ρ 4=9 Λ ð Þ¼6 log 4 þ 3 þ 4ð oÞ dual field-theory interpretation exists, can be thought of as a characteristic energy scale, which governs, among X∞ 4j other things, the mass gap of the theory. f ρ − ρ 9 ; 83 þ jð oÞ ð Þ We notice, by looking at the metric, that a trivial rigid j 2 ¼ rescaling of the coordinates xμ → λxμ and η → λη is 4 1 2 equivalent to a rigid shift of A and χ as A→Aþ3logðλÞ 4=9 χðρÞ¼χI þ logðρ − ρoÞþ ϕ4ðρ − ρoÞ 1 27 5 and χ → χ þ 3 logðλÞ. This is to be accompanied by a shift X∞ ρ → ρ − 3 λ χ 4j 2 logð Þ such that U and AU remain equal to zero. ρ − ρ 9 þ gjð oÞ : ð84Þ Under such a rigid shift, one can see that Λ → λΛ, and j¼2 2 ˆ −2 ϕ2 → λ ϕ2. It hence becomes evident that ϕ2 ≡ ϕ2Λ is Some more details about this expansion, truncated at the an invariant (dimensionless) quantity, which we denote by 4 the hat. In the following, we often express our results in order of ðρ − ρoÞ , are presented in Appendix C. Together 4χ terms of such dimensionless quantities, by which we mean with the domain-wall constraint A ¼ , this expansion Λ identifies a class of solutions that depend on the trivial that we are measuring in units of . parameter χ and two additional parameters: the position ρ In order to appreciate the need for a scale setting I o procedure in the comparison of different classes of sol- of the end of space and the integration constant ϕ4. The utions, consider that the space of free parameters has coefficients f and g are polynomial functions of ϕ4. This j j different dimensionality for the confining and skewed family of solutions is the (nontrivial) limiting case of ϕ → 1 3 solutions; for the confining solutions, the IR parameter the solutions in Sec. III F obtained when L = . The χ freedom in choosing ϕ in the generic singular solutions is I is fixed by the requirement of avoiding a conical I singularity in the small ρ region of the bulk geometry, replaced here by the freedom in ϕ4. We verified explicitly 10 but no such constraint exists for the skewed solutions, in that the singularity is not removed by the lift to D ¼ 2 dimensions (see Appendix D). which the space does not smoothly shrink to a point. To Although we cannot exclude a priori the existence of ensure that we can properly compare these two classes of solutions when plotting the free energy, we measure all additional singular backgrounds with more exotic IR Λ behaviors, our exploration of the space of solutions that quantities in units of , effectively reducing by 1 the connect to the trivial (ϕ ¼ 0) fixed point for large ρ, dimension of the parameter space for the skewed solutions. performed by perturbative generation of IR asymptotic We apply the same procedure to all other solutions as well, expansions and evolution toward larger values of ρ,was thus enabling us to compare the different branches of confirmed by the result of scanning numerically the five- solutions in a consistent manner. dimensional space of perturbations of the ϕ ¼ 0 critical point and evolving the solutions backward, toward small ρ. IV. MASS SPECTRA, A TACHYON, We did not find any indications that additional solutions AND A DILATON with asymptotic UV behavior in Sec. III A exist outside of the classes discussed in this section. Applying the dictionary of gauge-gravity dualities, the spectrum of small fluctuations around an asymptotically AdS supergravity background can be interpreted in terms of H. Scale setting the spectrum of bound states of the strongly coupled dual To facilitate comparison between all classes of solutions, field theory. All classes of solutions that we introduced in 0 χ 0 we choose to set AU ¼ and U ¼ in all cases; the former Sec. III have the same asymptotically AdS expansion for assignment is permitted since the classical equations of large ρ, but only the third class of geometries (those which motion are invariant under an additive shift of AðρÞ, while we referred to as confining) have a regular end of space. the latter can be achieved by a rescaling of the radial 3χU coordinate z → ze . We are hence left with the UV 2 ϕ ϕ χ In constructing the skewed solutions numerically, we exploit parameters f 2; 3; 5g. the fact that, as discussed in Sec. III E, they can be obtained Moreover, we find it useful to introduce a quantity that (up to a trivial additive integration constant) from the confining we use to set the scale in the observables deduced from the solutions by making the substitution d → −d.
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χ −χ 2χ−2A 2 a We hence restrict our attention to this class of solutions in 0 ¼½e Dρðe DρÞþð4∂ρAÞDρ þ e M a this section, as they are the only candidates for admitting an − e2χX a ac; 86 interpretation in terms of confining field theories. c ð Þ We devote this section of the paper to two calculations. We first compute the mass spectra for the full set of bosonic where M is the mass of the composite states in the dual field excitations of the dimensionally reduced model theory and where presented in Sec. II C. We then repeat the computation for the scalar excitations implementing the probe approxi- ∂V a −2χ a b d ab X ¼ −e R ∂ρΦ ∂ρΦ þ D G mation, according to the prescription described in c bcd c ∂Φb Ref. [127]. The former exercise will reveal the existence 4 ∂V ∂V of a tachyonic spin-0 state in a certain region of parameter ∂ Φa ab ∂ Φd þ ρ c þ G b ρ Gdc space for the class of confining solutions. The latter will 3∂ρA ∂Φ ∂Φ show that, in proximity of this region, one scalar state is not 16V ∂ Φa∂ Φb only parametrically light but also an (approximate) dilaton. þ 2 ρ ρ Gbc: ð87Þ 9ð∂ρAÞ A. Mass spectra In all these expressions, the quantities χ, A, Φ, and V are We present in this subsection the mass spectra of a fluctuations of the various bosonic supergravity fields of evaluated on the background. Moreover, given a field X , the sigma model coupled to five-dimensional gravity. We we defined the sigma-model-covariant as well as the a ∂ a interpret the states as glueballs with spin 0, 1, or 2 in the background-covariant derivatives by DbX ¼ bX þ Ga c D a ∂ a Ga ∂ Φb c dual confining field theory in four dimensions. In order to bcX and ρX ¼ ρX þ bc ρ X with the con- a 1 ad conduct this numerical analysis, we employ the convenient nection G bc ¼ 2 G ð∂bGcd þ ∂cGdb − ∂dGbcÞ, while the a gauge-invariant formalism developed in Refs. [122–126]. sigma-model Riemann tensor is given by R bcd ¼ a a a a e a e The equations satisfied by the scalar fluctuations a ¼ ∂cG bd − ∂dG bc þ G ceG bd − G deG bc. We impose the aaðM; ρÞ are given by following boundary conditions3:
3∂ρA 4V ∂V −2χ∂ Φc∂ Φd D ab − −2A 2δc − ∂ Φc ∂ Φd ab e ρ ρ Gdb ρ ¼ e M b ρ ρ Gdb þ b : ð88Þ ρ 2 3∂ρA ∂Φ i ρi
2 2χ−2A 2 μ To compute numerically the mass spectra for the fluctua- 0 ¼½∂ρ þð4∂ρA − ∂ρχÞ∂ρ þ e M eν; ð89Þ tions of the fields, it is necessary to introduce regulators in 0 μν −χ∂ 2Aþ7χ∂ 2 8χ the form of radial coordinate cutoffs; ρ1 is a (nonphysical) ¼ P ½e ρðe ρVνÞþM e Vν ; ð90Þ infrared regulator chosen so that ρo < ρ1, and ρ2 is chosen as μν −χ 2Aþχ−2ϕ i 2 2χ−2ϕ i the end point of the backgrounds in the far UVat large ρ. The 0 ¼ P ½e ∂ρðe ∂ρAνÞþM e Aν ; ð91Þ physical results are obtained by removing the two holo- 2 graphic regulators, i.e., by taking the limits ρ1 → ρ and 0 ¼ ∂ρX þð5∂ρχ − 2∂ρA þ 2∂ρϕÞ∂ρX o ρ2 → þ∞. For a comprehensive explanation of this pro- 8 2 −2Aþ2χ − −6ϕ cedure (and our notation and conventions), and details not þ M e 9 e X; ð92Þ immediately important for the purposes of this paper, see Refs. [98,99,126]. In our numerical study of the spectrum, 8 ρ ρ 0 2 3χ−4ϕ∂ 2A−5χþ4ϕ∂ − 2A−2χ−6ϕ μν we chose 1 and 2 to be sufficiently close to the end of space ¼ M þ e ρðe ρÞ 9 e P B6ν; and to the boundary, respectively, that both cutoff effects are negligible—we estimate that the numerical precision is ð93Þ accurate to within a few percent. The fluctuations of the pseudoscalars πi satisfy the μν −χ −χ 0 ¼ P ∂ρðe ∂ρXνÞ − e ð2∂ρχ − 2∂ρϕÞ∂ρXν same equation as the scalar fluctuations above, with −6χ−2ϕ Gππ ¼ e , while the equations of motion for all the 8 χ −2A 2 − −2χ−6ϕ other fluctuations are the following [99], þ e e M 9 e Xν ; ð94Þ 8 μρ νσ 2 −2A −5χ−4ϕ 3χþ4ϕ −2χ−6ϕ 0¼P P M e þe ∂ρðe ∂ρÞ− e Bρσ; 3In practice, the equivalent form of the boundary conditions 9 given in Eq. (14) of Ref. [73] turns out to be especially convenient in the numerical implementation. ð95Þ
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μν μν qμqν where P ≡ η − 2 . The fluctuations X and Xμ obey massive states in two of the other towers (B6μ and Bμν) q ϕ generalized boundary conditions, that reduce to Dirichlet in appear to become massless in the limit of large I. the limit of interest to this paper [see Eqs. (B41) and (B.42) of Ref. [99] for detailed technical explanations]. All other B. Probe scalars and dilaton mixing fluctuations obey Neumann boundary conditions. This is one of the central subsections to the paper. We The confining solutions are characterized by the constant analyze the composition of the scalar particles in the 1 ϕI ≥−4 logð3Þ, where the lower bound would correspond spectrum in terms of fluctuations of the background fields, to the IR fixed point of the dual five-dimensional quantum in order to establish whether any of them can, at least field theory (in the sense that it results in a constant solution approximately, be identified with the dilaton. The magni- ϕ ϕ ϕ ϕ 0 for the scalar field ¼ IR). Conversely, I ¼ UV ¼ tude of the condensates in the underlying dynamics, as corresponds to the UV fixed point of the dual five-dimen- evinced from the parameters ϕ3 and χ5 (see Appendix F), sional field theory. While the background solutions and changes along the branch of confining solutions, providing 1 spectra for − 4 logð3Þ ≤ ϕI ≤ 0 have been presented in a natural interpretation for the emergence of a dilaton and its properties. We will further return to this point, later in Ref. [99], the results for ϕI > 0 are new to this work. We show the results for the computation of the mass the paper. spectrum in Figs. 4 and 5. In Appendix E, we also show the The spin-0 mass spectrum presented in the previous same numerical results, but normalized with the scale subsection represents the solutions to the scalar fluctuation setting parameter Λ defined in Eq. (85). For the region equation for the gauge-invariant combination (see 1 Refs. [122–127])givenby − 4 logð3Þ ≤ ϕI ≤ 0, each plot is in agreement with our previous computation in Ref. [99]; of more interest are the a ∂ρΦ ðρÞ observations that for large enough ϕ one of the states in the aa ρ φa ρ − ρ I ðM; Þ¼ ðM; Þ 6∂ ρ hðM; Þ; ð96Þ scalar spectrum becomes tachyonic and that the lightest ρAð Þ
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FIG. 4. The spectra of masses M, as a function of the one free parameter ϕI characterizing the confining solutions and normalized in −4 units of the lightest tensor mass, computed with ρ1 ¼ 10 and ρ2 ¼ 12. From top to bottom, left to right: the spectra of fluctuations of μ the tensors eν (red), the graviphoton Vμ (green), and the two scalars ϕ and χ (blue). The orange points in the plot of the scalar mass spectrum represent values of M2 < 0 and hence denote a tachyonic state. We also show by means of the vertical dashed lines the case ϕ ϕc 0 I ¼ I > , the critical value that is introduced and discussed in Sec. V.
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FIG. 5. The spectra of masses M as a function of the scale parameter ϕI, normalized in units of the lightest tensor. From top to bottom, i i left to right: the spectra of fluctuations of the pseudoscalars π forming a triplet 3 of SUð2Þ (pink), vectors Aμ forming a triplet 3 of SUð2Þ (brown), Uð1Þ pseudoscalar X (gray), Uð1Þ transverse vector B6μ (purple), Uð1Þ transverse vector Xμ (black), and the massive U(1) 2- −4 form Bμν (cyan). The spectrum was computed using the regulators ρ1 ¼ 10 and ρ2 ¼ 12 with the exception of the Uð1Þ pseudoscalar −7 X for which we used ρ1 ¼ 10 in order to minimize the cutoff effects present for the very lightest state at large values of ϕI. We also ϕ ϕc 0 show by means of the vertical dashed lines the case I ¼ I > , the critical value that is introduced and discussed in Sec. V. where φaðM;ρÞ are the first-order fluctuations of the scalar we neglect the contribution of the metric perturbation h in fields about their respective background solutions ΦaðρÞ, Eq. (96), effectively removing any backreaction the scalar while hðM;ρÞ describes small perturbations of the trace of fluctuations may have on the bulk geometry (for details, see the four-dimensional tensor component of the Arnowitt- Ref. [127]). We then check how well the resulting spectrum aa Deser-Misner decomposed five-dimensional metric tensor. computed with jh¼0 agrees with the correct computation In terms of the dual field theory, φa are associated with making use of aa. If we find that the two calculations yield generic scalar operators that define the theory, while h is results that are in good agreement, then we may infer that associated to the dilatation operator. the contribution of the metric perturbation is negligible and We are interested in determining to what extent any of hence the spin-0 state is not a dilaton; if, by contrast, the the scalar particles is a dilaton, i.e., whether mixing effects two results disagree, then this is a clear indication of between φa and h are important. To this end, in this the fact that the metric perturbation affects significantly the subsection, we repeat the computation of the mass spec- spectrum and hence the scalar state has a significant dilaton trum in the spin-0 sector, by using the probe approximation; component.
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FIG. 6. The spectra of scalar masses M as a function of the parameter ϕI along the confining branch of solutions, normalized in units of −4 the lightest tensor mass, computed with ρ1 ¼ 10 and ρ2 ¼ 12.AsinFig.4, the blue disks represent the two scalars of the model ϕ and χ, while the orange disks denote the tachyon. We additionally include the results of our mass spectrum computation using the probe approximation for M2 > 0 (black triangles) and M2 < 0 (orange triangles); note that these do not represent additional states. We also ϕ ϕc 0 show by means of the vertical dashed line the case I ¼ I > , the critical value that is introduced and discussed in Sec. V. The shaded gray region indicates the metastable region of parameter space.
As can be seen in Figs. 6 and 7, the probe approximation We have already discussed the case ϕ ≤ 0 elsewhere [127], is not accurate, and for all values of ϕI, at least one of the and we will not return to the details of that discussion here. lightest states is not well captured. This state is the lightest Interestingly, for ϕI ∼ 0.25, starting from the region in scalar for large, negative ϕI and becomes the next-to- close proximity of (but before) the appearance of the lightest state for ϕI close to zero. This is a mixed state tachyon, the discrepancy between the probe approximation that has a significant overlap with the dilatation operator. and the mass of the lightest physical scalar becomes much
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FIG. 7. A magnification of the plot shown in Fig. 6. We normalized the masses M in units of the lightest tensor mass and computed −9 with ρ1 ¼ 10 and ρ2 ¼ 15. We focus in particular on the lightest state of the spectrum, in the plot region where the tachyonic states first appear. The dashed red box is intended to enclose an important feature of the full spectrum in Fig. 6, namely a region of ϕI parameter space wherein the probe approximation completely disagrees with the full gauge-invariant scalar computation. We see that 2 b there exists a finite range of values of the IR parameter ϕI for which the squared masses M of the physical scalars a and the probes ab jh¼0 differ by a minus sign, and hence the probe approximation unambiguously fails.
106018-17 ELANDER, PIAI, and ROUGHLEY PHYS. REV. D 103, 106018 (2021) more pronounced (see Fig. 7). In this region of parameter identified with an approximate dilaton (in the sense that it space, the lightest scalar particle can be rendered para- would couple to the dilatation operator as a dilaton does), metrically light with respect to all other states, and it is and its dynamical origin is the unsuppressed vacuum legitimate to interpret it as an approximate dilaton. It is to expectation value of the marginal operator. We highlight be noticed that the next-to-lightest state is still not well the fact that ϕ3 vanishes when ϕ2 → 0, but this is not the captured by the probe approximation, due to mixing effects. case for χ5 (see the plots in Appendix F). This region of Let us now discuss the confining solutions with large parameter space resembles generic Yang-Mills theories; values of ϕI. In the limit ϕI → þ∞, the plots in Appendix F there is no sense in which the explicit breaking of scale ˆ ˆ show that ϕ2 → 0 and ϕ3 → þ∞, as in the SUSY solution. invariance is parametrically small compared to the scale of ˆ Since ϕ2 is connected to the explicit breaking of scale spontaneous breaking, and hence while one of the scalar ˆ bound states inherits some of the properties of an approxi- invariance, while ϕ3 encodes its spontaneous breaking, in mate dilaton, it is not parametrically light. We further this limit, one expects the emergence of an exact dilaton. ϕ This is confirmed by the fact that the mass squared of the discuss the regime in which 3 is large in Sec. VE, where tachyon approaches zero from below, as can be seen in we return to the results of the exercise performed in the Fig. 6. While these solutions are unphysical, this observation current subsection. nevertheless provides a nontrivial check of our analysis. We V. FREE ENERGY AND A PHASE TRANSITION further note that as ϕI is increased the probe approximation results in additional heavier states becoming lighter and In this section, we discuss the stability of backgrounds eventually tachyonic. This reinforces the fact that it is not belonging to all the distinct classes of solutions introduced only the tachyon and the lightest scalar that mix with the earlier on. To this end, we compute the free energy density dilaton but some of the heavier states as well. We finally of the system, by applying a prescription that allows us to notice that, besides a small number of light, discrete states, compare to one another the free energies associated with the spectrum of heavy particles in four dimensions becomes solutions belonging to different classes. densely packed, eventually degenerating into a gapped continuum. Early evidence of this phenomenon can be seen A. General action and formalism in all the mass spectra, in Figs. 4 and 5. This final observation is reminiscent of the features that emerge in proximity of the Our first step is to derive the free energy of the solutions Chamseddine-Volkov-Maldacena-Nunez solution [75,78] from the truncated action of the scalar field ϕ coupled to along the baryonic branch of the Klebanov-Strassler system gravity in D ¼ 6 dimensions—while setting equal to zero [68,69]—see also Refs. [129,130] that study the gravity dual all other fields. We include a boundary at ρ ¼ ρ2 as a of the Coulomb branch of N ¼ 4 super-Yang-Mills. regulator, with the understanding that the physical field- We conclude this section by summarizing our results for theory results will be recovered at the end of the calcu- the spectrum and interpreting them in terms of the dual field lations by taking the limit ρ2 → þ∞. We also need to theory. We consider only the regular (confining) solutions, introduce a regulator in the IR: despite the fact that some of and we start from the region of parameter space in the solutions we consider are completely smooth, the ρ ρ ρ proximity of the backgrounds with ϕ ¼ 0. The dual field physical space is bounded by 1 < < 2. It is understood ρ → ρ ρ theory is given by a supersymmetric fixed point in D ¼ 5 that eventually we will take 1 o, with o the physical dimensions, that admits two deformations. One corre- end of the geometry. The presence of boundaries requires sponds to the insertion of an operator of dimension on general grounds adding to the action the Gibbons- Δ ¼ 3, the source for which is encoded in the boundary Hawking-York (GHY) terms SGHY;i and boundary- 1 value of the field ϕ, via the coefficient ϕ2, and the response localized potentials Spot;i, for i ¼ , 2. We hence write function, which is related to the coefficient ϕ3. The other is the action as follows. the compactification on a circle of one of the spacelike X dimensions, which is encoded in the gravity theory by the S ¼ Sbulk þ ðSGHY;i þ Spot;iÞ χ— 1 2 marginal deformation corresponding to by the coeffi- Z i¼ ; cient χ5. The gravity solutions all correspond to dual pffiffiffiffiffiffiffiffi R 4 η ρ −ˆ 6 − ˆMˆ Nˆ ∂ ϕ∂ ϕ − V ϕ theories that confine, in the usual sense typical of strongly ¼ d xd d g6 4 g Mˆ Nˆ 6ð Þ Z coupled gauge theories in four dimensions. X qffiffiffiffiffiffi 4 K Scale symmetry is both spontaneously and explicitly þ ð−Þi d xdη −gˆ˜ þ λ ; ð97Þ i¼1;2 2 i broken. The spectrum of bound states in proximity of ρ¼ρi ϕ ¼ 0 contains two almost degenerate scalar bound states. ˆ The lightest of them is well captured by the probe where gMˆ Nˆ is the metric tensor for the six-dimensional line 0 ˆ R approximation, and it corresponds to fluctuations sourced element in Eq. (9) for VM ¼ , g6 is its determinant, 6 is Δ 3 ˆ˜ by the operator of dimension ¼ . Its overlap with the the corresponding Ricci scalar, and gMˆ Nˆ is the metric dilaton is negligible. The other state, conversely, can be induced on each boundary.
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In order to define the induced metric, we introduce the The free energy F and the free-energy density F are 0 0 0 0 1 0 six-vector nMˆ ¼ð ; ; ; ; ; Þ, that satisfies the defining defined as relations: Z 4 ˆ ˆ ˆ F ≡ − lim lim S ≡ d xdηF; ð110Þ 1 ˆ M N M ρ1→ρ ρ2→þ∞ ¼ gMˆ Nˆ n n ¼ n nMˆ ; ð98Þ o
0 Mˆ ˆ − which yields the general result ¼ n ðgMˆ Nˆ nMˆ nNˆ Þ: ð99Þ 1 4A−χ The covariant derivative is written in terms of the con- F ¼ lim e ð13∂ρA − 4∂ρχ þ 8λ1Þ ρ1→ρ 8 ρ nection as o 1 1 4A−χ − lim e ð13∂ρA − 4∂ρχ þ 8λ2Þ : ð111Þ Qˆ ρ → ∞ 8 ρ ∇ ˆ f ˆ ≡∂ˆ f ˆ − Γ f ˆ ; 100 2 þ 2 M N M N Mˆ Nˆ Q ð Þ 1 In the body of the calculations, we adopt the following ˆ ˆ ˆ 3 ΓP ≡ gˆP Qð∂ ˆ gˆ ˆ ˆ þ ∂ ˆ gˆ ˆ ˆ − ∂ ˆ gˆ ˆ ˆ Þ: ð101Þ λ − ∂ λ W Mˆ Nˆ 2 M N Q N Q M Q M N prescription. We choose 1 ¼ 2 ρA and 2 ¼ 2 (it is sufficient to know the form of W2 up to quadratic order in ϕ ˆ˜ in order to extract the divergent and finite parts), and as a We can now define the induced metric tensor gMˆ Nˆ and the extrinsic curvature K as follows, result, the free energy density is
˜ 1 gˆ ˆ ˆ ≡ gˆ ˆ ˆ − n ˆ n ˆ ; 102 4A−χ M N M N M N ð Þ F ¼ lim e ð∂ρA − 4∂ρχÞ ρ1→ρ 8 o ρ1 ˆ ˆ ˆ ˆ K ≡ gˆM NK ˆ ˆ ¼ gˆM N∇ ˆ n ˆ ; ð103Þ 1 M N M N 4A−χ − lim e ð13∂ρA − 4∂ρχ þ 8W2Þ : ð112Þ ρ2→þ∞ 8 ρ so that with our conventions we find that 2
The choice of λ1 is dictated by the requirement that the Mˆ Nˆ 5 K −gˆ Γ 4∂ρA − ∂ρχ ∂ρc: 104 ¼ Mˆ Nˆ ¼ ¼ ð Þ variational principle be well defined, and the variation of the bulk action supplemented by the IR boundary action yields In order to calculate the free energy, one needs to the bulk equations of motion and boundary conditions at 4 evaluate the action on shell. The bulk part of the action ρ ¼ ρ1. We find that with this choice then has two components: one proportional to the equations of motion themselves, that hence vanishes when evaluated S 1 þ S 1 GHY; Z pot; on any classical background solution, and a second part that 1 reduces to a total derivative. We can use Eq. (20) from − 4 η 4A−χ ∂ − ∂ χ ¼ 2 d xd ðe ð ρA ρ ÞÞ ; ð113Þ Sec. II D to rewrite the bulk action as ρ1
S ¼ S 1 þ S 2 and by looking at the IR expansions of the regular solutions, bulk bulkZ; bulk; ρ we find that the boundary-localized action does not con- 3 2 4 4 −χ − η ρ∂ A ∂ tribute to their free energy in the ρ1 → ρ limit. Hence, in the ¼ 8 d xd d ρðe ρAÞ: ð105Þ o ρ1 case in which the geometry closes smoothly in the IR, the presence of the regulator is unnecessary and has no physical Explicit evaluation shows that the boundary-localized effect. We are now in a position to apply this prescription to contributions, evaluated on shell, yield all other solutions as well. Z λ 1 The choice of 2 is dictated by covariance, locality, and S − 4 η 4A−χ 2∂ − ∂ χ the requirement that all divergences cancel [49–51]. In the GHY;1 ¼ d xd e ρA 2 ρ ; ð106Þ ρ¼ρ1 case at hand, in general, one expects two types of UV Z divergences: one driven by the bulk cosmological constant 2 S − 4 η 4A−χ λ and one driven by the (square of the) mass deformation ϕ2. pot;1 ¼ d xd e ð 1Þ ; ð107Þ ρ¼ρ1 Because of these two divergences, F and its second Z derivative with respect to the source ϕ2 are scheme 1 S 4 η 4A−χ 2∂ − ∂ χ GHY;2 ¼ d xd e ρA ρ ; ð108Þ 2 4 3 ρ ρ2 λ − ∂ ¼ This leads to the requirement that 1jρ1 ¼ 2 ρAjρ1 evaluated Z at the IR boundary, hence explaining the aforementioned choice. Note, however, that we do not need to know the explicit S 4 η 4A−χ λ pot;2 ¼ d xd e ð 2Þ : ð109Þ functional dependence of λ1 on ϕ in order to perform our ρ¼ρ2 calculation of the free energy.
106018-19 ELANDER, PIAI, and ROUGHLEY PHYS. REV. D 103, 106018 (2021) dependent. This is a generic feature, commonly appearing theorems do not trivially apply to our results for the free in many holographic free energy calculations, and has been energy, the minima of which will not exhibit a concavity observed in other contexts (see for instance the discussions with definite sign. With our choice of λ2, dictated by in Ref. [131]). For our purposes, it has one important holographic renormalization, and by making use of the UV 2 implication: the classical statistical mechanics concavity expansions and of the relation ∂ρ ¼ − 3 z∂z, we find that
Z 4 −χ AU U 5 5 S 4 ηe − ϕ2 4 0 5 GHY;2 ¼ d xd 5 3 12 2z þ × z þ ; z ρ2 ð114Þ Z 4 −χ AU U 4 1 8 S 4 ηe − ϕ2 4 ϕ ϕ 5 pot;2 ¼ d xd 5 3 þ 3 2z þ 15 2 3z þ ; ð115Þ z ρ2
Z 4 −χ AU U 1 1 1 S 4 ηe − ϕ2 4 4ϕ ϕ 25χ 5 bulk;2 ¼ d xd 5 3 þ 12 2z þ 80 ð 2 3 þ 5Þz þ : ð116Þ z ρ2
The divergences exactly cancel, leaving a finite contribu- B. Domain-wall solutions tion to the free energy. If we impose the (domain-wall) constraint A ¼ 4χ, this We observe that the contribution to the free energy introduces two additional constraints on the five UV ρ coming from evaluation at the IR boundary 1 in Eq. (113) parameters: happens to be proportional to the combination appearing in Eq. (21). This contribution hence coincides with a con- A ¼ 4χ ; ð120Þ served quantity, that we can evaluate at any value of the U U ρ coordinate . It is convenient to evaluate it at the UV 4 boundary, where we notice that (as expected) it gives a χ − ϕ ϕ 5 ¼ 25 2 3: ð121Þ finite contribution. By substituting the general UV expan- sions, we hence obtain the following final result for the free From these two relations, we may deduce the values of χ5 energy density, and χU given the other three parameters. We notice that the above constraint on χ5 causes the first term of Eq. (117) to 1 4 −χ F AU U 4ϕ ϕ 25χ vanish exactly, and we hence obtain the following expres- ¼ 16 e ð 2 3 þ 5Þ sion for the free energy of the domain-wall (DW) solutions, 1 4A −χ − e U U ð28ϕ2ϕ3 þ 15χ5Þð117Þ which include, among others, the SUSY as well as the 48 IR-conformal solutions: 3 8 4A−χ ðDWÞ 4A −χ ¼ − lim e ∂ρA þ W2 ð118Þ F ¼ − e U U ϕ2ϕ3: ð122Þ ρ2→þ∞ 2 15 ρ2 In the case of the IR-conformal solutions (IRC), one 1 4 −χ − AU U 4ϕ ϕ − 15χ ¼ 12 e ð 2 3 5Þ; ð119Þ numerical background may be used to generate any other by an additive shift of the holographic coordinate. The where in the first line the first term comes from the following ratio is an invariant: ρ1 → þ∞ limit evaluation of the first term in Eq. (112) jϕ3j ρ2 and the second comes from the limit evaluation. The κ ≡ 3 : ð123Þ second line is a general combination of all the contribu- jϕ2j2 tions. The third line is our main result, and we will return to it when we discuss each individual class of solutions, in the We find numerically that κ ≃ 2.87979, so that the final subsections to follow. For completeness, and to elucidate result for the free energy is some subtle differences, we repeat this calculation in the 8 8 five-dimensional language, in Appendix G, with identical ðIRCÞ 3 3 F ¼ − κϕ2jϕ2j2 ≃− ð2.87979Þϕ2jϕ2j2: ð124Þ results. 15 15
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C. Numerical implementation geometry in proximity of the end of space at small ρ,a The general result for the free-energy density for all region that is essential in the calculation of the scale-setting Λ solutions is in Eq. (119): parameter . Indeed, this is the reason why, for the purpose of numerical studies, it is preferable to construct the
1 4 −χ solutions by choosing the boundary conditions close to F − AU U 4ϕ ϕ − 15χ ¼ 12 e ð 2 3 5Þ: the end of space in the geometry and evolving the differ- ential equations toward large values of the holographic All the classes of solutions we discuss are known coordinate ρ. We estimate the numerical precision of our numerically and are obtained by exploiting the IR calculation of the free energy and of the parameters relevant expansions we reported in Sec. III. We implement a to the energetics study to be accurate within a few percent. numerical routine to extract a table of UV parameter Singular solutions are treated in exactly the same way as values fϕ2; ϕ3; χ5g for solutions of each class, having the confining solutions, thanks to the introduction of the set AU ¼ χU ¼ 0. To this end, we do the following: regulator at ρ1 and to the prescription we discussed earlier (1) For each given choice of IR expansion, we numeri- in this section. A practical simplification of the procedure is cally solve the background equations of motion for given by the observation that the free energy of the skewed ϕðρÞ, χðρÞ, and AðρÞ, having chosen the end of space solutions is formally identical to that for the confining to be at ρo ¼ 0 with the boundary conditions set up solutions, except for the replacements in Eqs. (75), (76), at a small ρ. and (77). (2) Starting from these solutions, we generate new ones In Table I, we summarize some basic properties of the by shifting the radial coordinate together with χ and various classes of solutions relevant to the analysis that A such that the combined effect is to set AU ¼χU ¼0 follows. We repeat here some important and subtle points. as required. The scale-setting procedure for the SUSY and IR-con- (3) We match each numerical solution and its derivatives formal solutions is treated in a different way, for specific with the UV expansions and extract ϕ2, ϕ3, and χ5. reasons that we describe in the next subsection. In the case In the third step, one needs to choose a value of the radial of confining solutions, ϕ3 and χ5 are constrained by the ρ ρ coordinate ¼ m at which to do the matching. This choice requirement of eliminating curvature and conical singular- is dictated by the requirement to minimize the effect of the ities, respectively. For the skewed solutions, these require- ρ numerical noise, while at the same time ensuring that m is ments are replaced by the fact that skewed solutions can be large enough that the solutions have reached the region in obtained from confining solutions by changing the sign of proximity of the ϕ ¼ 0 critical point. We do not report the the background function d. details of this laborious process but only report our main From here on, we find it convenient to define the results. following notation. We rescale all the physical quantities We checked that the numerical determination of the UV by the appropriate power of the scale Λ defined in parameters can be used to set up the boundary conditions in Eq. (85) as the UV, and by solving again the equations of motion ρ toward small , we recover the original backgrounds. The Fˆ ≡ FΛ−5; ð125Þ reader should be alerted of the fact that the nonlinear nature of the equations is such that this second process does not ϕˆ ≡ ϕ Λ−2 allow one to reproduce accurately the region of the 2 2 ; ð126Þ
TABLE I. Parametrization, constraints, and scale setting procedure of each class of solutions considered in the text and in Figs. 8 and 9. The scale Λ is defined in Eq. (85) and has been used to restore physical units in F and ϕ2 in the energetics. In the case of the IR- 8 3=2 conformal solutions and of the SUSY solutions, no scale setting is used, because F ¼ 0 in the former, and F ¼ − 15 κϕ2jϕ2j in the latter, but Λ is not defined. The SUSY singular solutions are represented by a point in Figs. 8 and 9; the IR conformal, confining, skewed, and singular DW solutions are represented by lines; and finally the generic (good as well as bad) singular solutions cover a two- ˆ ˆ dimensional portion of the ðϕ2; FÞ plane (see Figs. 8 and 9).
Class AU χU ϕ2 ϕ3 χ5 Scale setting SUSY 0 0 0 Free A ¼ 4χ None 1=2 IR-conformal 0 0 <0 ϕ3 ¼ κϕ2jϕ2j A ¼ 4χ None Confining 0 0 Free Curvature singularity Conical singularity Λ Skewed 0 0 Free cskew ¼ cconf dskew ¼ −dconf Λ Good Singular 0 0 Free Free Free Λ Bad Singular 0 0 Free Free Free Λ DW Singular 0 0 Free Free A ¼ 4χ Λ
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ˆ −3 ϕ3 ≡ ϕ3Λ ; ð127Þ 10 and so on for all possible physical quantities. By doing so, as we will show explicitly, we can legitimately compare solutions belonging to any of the different classes described 5 in this paper.
D. Free energy density and the phase structure 0 In order to investigate the energetics along all the branches of solutions, we employed a numerical routine –5 to compute their free energy by extracting physical values of –1.0 - 0.5 0.0 0.5 the five UV parameters; we present here the results of this numerical analysis. In particular, we show how the free FIG. 8. The free energy density Fˆ as a function of the F ϕ ˆ energy density behaves as a function of 2, the deforma- deformation parameter ϕ2 for the IR-conformal solutions tion of the theory corresponding to the aforementioned (longest-dashed purple line), the confining solutions (solid black operator of dimension Δ ¼ 3. We repeat that we normalized line), and the skewed solutions (dashed red line), compared the two quantities by the appropriate power of the scale Λ,in to a few representative choices of (good) singular solutions order to be able to compare different solutions. As the plots (thin blue lines). For the latter, we generated the numerical ϕ ζ are rather busy, showing a large amount of information, we solutions from the IR expansions, by setting ð L; Þ¼ −0 02 −1 −0 04 −1 −0 08 −1 −0 15 −1 −0 2 −1 first devote some space to explaining how to read them, and ð . ; Þ, ð . ; Þ, ð . ; Þ, ð . ; Þ, ð . ; Þ, ð−0.25; −1Þ, ð−0.3; −1Þ, ð−0.35; −1Þ, ð−0.35; 1Þ, ð−0.3; 1Þ, then we analyze the physical results, by treating separately −0 25 1 −0 2 1 −0 15 1 −0 04 1 −0 02 1 ˆ ˆ ð . ; Þ, ð . ; Þ, ð . ; Þ, ð . ; Þ, and ð . ; Þ, the ϕ2 < 0 and ϕ2 > 0 cases. respectively (top blue line to bottom blue line), and varied the In Fig. 8, we show five of the seven classes of solutions value of ϕI. The darker blue line, separating the cases ζ ¼ 1, listed in Table I: corresponds to the domain-wall solutions obtained with ϕ pffiffiffi L ¼ (i) The SUSY solutions all have ϕ2 0, and because ¼ −1= 5 and varying ϕI. The SUSY solutions are represented by a they satisfy the domain-wall constraint A ¼ 4χ,by gray point at the origin. The short-dashed orange line shows the virtue of Eq. (122), which descends from Eq. (121), region along the branch of confining solutions in which the also F ¼ 0. The integral defining Λ in Eq. (85) tachyonic state appears in the scalar mass spectrum. (Note that diverges (Λ → 0). These solutions are represented the very top thin blue line crosses the dashed red one for large ˆ by the gray disk at the origin. negative values of ϕ2. We expect this to be a purely numerical (ii) The IR-conformal solutions exist only for ϕ2 < 0. artifact that could be removed with higher numerical precision.) The integral defining Λ in Eq. (85) diverges also in this class of solutions (Λ → 0). Yet, because of scale minimizing the free energy over a portion of param- invariance, we find that F scales as a power of ϕ2, eter space. and we represent these solutions with the longest- (iv) The skewed solutions are rendered in dashed red. We dashed purple line in Fig. 8. This line represents obtained these solutions by changing the sign of what would be the result of using any other possible d → −d from the confining solutions, which implies scale-setting process for the IR-conformal solutions. the relations in Eqs. (75)–(77). Also in this case, ˆ (iii) The confining solutions are rendered in solid black there exists a maximum value of ϕ2. and short-dashed orange. They form a line, as we (v) The generic solutions with good singularity are generate the solutions by varying the parameter ϕI. depicted by thin blue lines. We choose a number ˆ ϕ 0 We notice the existence of a maximum value of ϕ2. of representative values for the parameter L < For graphical illustration, we rendered in short- and discuss both choices of ζ ¼ 1 (see Sec. III F). ϕ → 0 dashed orange the part of the curve obtained with For L , the thin blue lines approximate the confining solutions for which one of the scalar states confining (for ζ ¼þ1) and skewed (for ζ ¼ −1) 1 ϕ → − pffiffi has a negative mass squared (see Figs. 6 and 7). Part solutions, as expected. For L 5, one finds the of this tachyonic portion of the branch of solutions special case of domain-wall solutions with good ˆ has free energy F lower than the solutions with the singularity (in this case the choice ζ ¼ 1 is ˆ same value of ϕ2 located along the regular portion of immaterial), and we denote this line, which appears the confining branch, and the short-dashed orange just above the longest-dashed purple one, with a and solid black curves cross nontrivially. This darker shade of blue. observation by itself would be proof that a phase We notice one very important fact: thanks to the ˆ ˆ transition takes place, were it not for the undesirable rescaling that defines F and ϕ2, all branches of solutions feature that the tachyonic backgrounds would be depicted in Fig. 8 (and this holds true also in the subsequent
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Fig. 9) connect to the origin of the diagram, with Fˆ ¼ 0 domain-wall ones discussed in Sec. III G (in long-dashed ˆ and ϕ2 ¼ 0. This observation makes it explicitly clear that, dark green). We replace the solutions with a good singu- despite the semiclassical nature of the calculations we larity by shading in light blue the whole region of the plane ˆ ˆ performed, the free energy density Fˆ is defined in a ðϕ2; FÞ delimited by the confining and skewed solutions. consistent way that allows for the comparison of all We notice two important features: for some choices of possible solutions along all the branches we identified, parameters, badly singular solutions exist that exceed the ˆ given that effectively they all share one common point. upper bound on ϕ2 that we identified when discussing the All the thin blue lines are entirely contained within the confining solutions, and furthermore there are domain- region of the plot delimited by the solid black, short-dashed wall, badly singular solutions with free energy lower than orange, and dashed red lines. Varying within this class of those along the tachyonic portion of the confining branch of solutions, for all available choices of parameters, the solutions. confining solutions minimize Fˆ , while the skewed solu- The plot in Fig. 9 clearly displays the features expected tions maximize it. The solutions with good singularity do in the presence of a phase transition, and we will return to it not resolve either of the two problematic features of the shortly. Figure 10 is a detail of Fig. 9, in which we retained confining class: they do not extend the plot beyond the only the confining solutions (solid black and short-dashed ˆ maximum value for ϕ2; nor do they give us solutions with orange lines) and the badly singular domain-wall solutions energy lower than the tachyonic sub-branch of the confin- (in the long-dashed dark green). We highlight the region in ing solutions. Finally, we highlight how not only are the proximity of the intersection between the two lines, which ˆ c solutions fully contained inside the region delimited by the identifies a critical value ϕ2 of the deformation parameter ˆ solid black, short-dashed orange, and dashed red curves but ϕ2. The minimum of the free energy density is given by ϕ ˆ ˆ c also that, by varying L, we can span the entirety of this confining solutions for ϕ2 < ϕ2 and by badly singular region. ˆ ˆ c domain-wall solutions for ϕ2 > ϕ2. The tachyonic section In Fig. 9, we add to the set of solutions on display several of the confining branch is never a minimum of the free representative choices of badly singular solutions (in thin ˆ energy at fixed ϕ2. light green), chosen by varying ϕ and ζ, as well as the L We now discuss the physics lessons we learn from the ˆ combination of Figs. 8–10. For negative values of ϕ2,we find that all classical solutions identified in the body of the 10 paper have finite free energy density Fˆ , and this is bounded from below by the confining solutions and from above by 5 the skewed solutions. All other solutions have free energy somewhere in between—they include the SUSY solutions, 0 the IR-conformal solutions, and all the generic solutions discussed in Secs. III F and III G. If we were to restrict –5 attention to ϕ ≤ 0 (as done for example in Refs. [98,99]), there would be no benefit from the study of solutions other
–10 –2.5 –1.0 –0.5 0.0 0.5 1.0 1.5 2.0
–3.0 FIG. 9. The free energy density Fˆ as a function of the ˆ deformation parameter ϕ2 for the IR-conformal solutions (lon- –3.5 gest-dashed purple line), the confining solutions (solid black line), and the skewed solutions (dashed red line), compared to a –4.0 few representative choices of (badly) singular solutions. For the latter, we generated the numerical solutions from the IR ex- –4.5 pansions, by setting ðϕL; ζÞ¼ð0.05; −1Þ, ð0.1; −1Þ, ð0.15; −1Þ, ð0.2; −1Þ, ð0.25; −1Þ, (0.25, 1), (0.2, 1), (0.15, 1), (0.1, 1), and –5.0 (0.05, 1), respectively (thin light-green lines), and varied the –5.5 value of ϕI. The long-dashed dark-green line represents the domain-wall (badly) singular solutions, obtained by varying 0.0 0.1 0.2 0.3 0.4 0.5 0.6 the parameter ϕ4 in the IR expansion in Sec. III G. The SUSY ˆ solutions are represented by a gray point at the origin. The short- FIG. 10. The free energy density F as a function of the ˆ dashed orange line shows the region along the branch of deformation parameter ϕ2 for the confining solutions (solid confining solutions in which the tachyonic state appears in the black and short-dashed orange lines) and the domain-wall (badly) scalar mass spectrum. We shaded in light blue the region covered singular solutions, obtained by varying the parameter ϕ4 in the IR by the good singular solutions (see Fig. 8). expansion in Sec. III G (long-dashed dark-green line).
106018-23 ELANDER, PIAI, and ROUGHLEY PHYS. REV. D 103, 106018 (2021) than the confining ones, which already minimize the free kept the source of the other nontrivial operator fixed (we set energy, have no curvature nor conical singularities, and AU ¼ 0 ¼ χU) and studied how F varies as a function of admit a sensible field-theory interpretation. Furthermore, the source ϕ2. Moreover, in order to facilitate the com- the spectrum of the small fluctuations around the confining parison of different branches, we implemented a scale- solutions can be interpreted in terms of the discrete mass setting procedure by defining the energy scale Λ, allowing spectrum of bound states of the dual field theory. us to compare dimensionless quantities. ˆ When we analyze the region with ϕ2 > 0, we find the We now define two dynamical quantities that play a role ˆ c existence of a critical choice ϕ2 for which a phase transition similar to that of order parameters and study them as we takes place, with the physically realized background cross from one side to the other of the phase transition. In minimizing the free energy density being given by con- analogy with the magnetization of a system in thermody- ˆ ˆ c fining solutions when ϕ2 < ϕ2, and singular domain-wall namics, the first such parameter is defined as the variation ˆ ˆ c of the free energy density with respect to the source ϕ2 solutions for ϕ2 > ϕ2. Interestingly, while the spontaneous 0 χ Λ Λ compactification of one of the space-time dimensions of the (holding AU ¼ ¼ U and fixed) measured in units of : theory is energetically favored in the confined phase, ∂ ∂ beyond the critical point, the theory prefers to preserve Mˆ ≡ Λ−3 F ϕ Λ Fˆ ϕˆ ∂ϕ ð 2; Þ¼ ˆ ð 2Þ: ð135Þ (locally) the full five-dimensional Poincar´e invariance. The 2 ∂ϕ2 critical parameters at the transition are extracted from the We cannot write this in closed form, as it requires numerical study, and we find ˆ ˆ ˆ expressing explicitly the coefficients ϕ3ðϕ2Þ and χˆ5ðϕ2Þ, ˆ c ϕ2 ≃ 0.169; ð128Þ appearing in the expression for the free energy density, in ˆ terms of ϕ2. But we can evaluate the derivative numerically. ϕc ≃ 0 027 Mˆ I . ; ð129Þ From Figs. 8 and 9, we see that is a well-defined quantity for the confining, skewed, IR-conformal, and c ϕ4 ≃ 98.9; ð130Þ singular domain-wall solutions. In the more general sin- gular solutions (represented by the thin blue and lighter Fˆ c ≃−3.893: ð131Þ green lines in Figs. 8 and 9), an additional parameter ˆ remains undetermined in terms of ϕ2 (see Table I), and ˆ We also find that the UV parameters in the gravity analysis therefore the variation with respect to ϕ2 is ambiguous. show a sharp discontinuity in the values assumed in the The second parameter that we define measures how phase with a shrinking circle (denoted by the subscript <) much Poincar´e invariance in D ¼ 5 dimensions is broken and in the domain-wall phase (denoted by the subscript >): and is given by ϕˆ c ≃−0 092 ϕˆ c ≃ 43 2 4 3< . ; 3> . ; ð132Þ Δˆ ≡ χˆ ϕˆ ϕˆ DW 5 þ 25 2 3: ð136Þ χˆc ≃−3 12 χˆc ≃−1 17 5< . ; 5> . : ð133Þ As can be seen from the leading-order parameter appearing ˆ c in the UV expansion of the combination d A − 4χ in In particular, we notice the enhancement of ϕ3 . ¼ > Δˆ Eq. (39), DW vanishes for the domain-wall background E. Properties of the phase transition solutions, for which d ¼ 0. In Fig. 11, we show a detail of the functions Fˆ and Mˆ ≡ Having established the existence of a first-order phase ∂Fˆ ˆ in the vicinity of the phase transition. The derivative has transition, we devote this subsection to characterizing it. We ∂ϕ2 also return to its relation with the physical spectrum of the been evaluated numerically. The plots show clear evidence bound states of the dual theory along the confining branch. of a strong first-order phase transition: while the free energy As repeatedly stated, two nontrivial operators are present is continuous, its derivative with respect to the deformation in the dual field theory. We identify the source for the parameter is not. The two bottom panels of Fig. 11 show that operator of dimension Δ ¼ 3 with the leading-order coef- in the physical phase in which the confining solutions are ϕ Δˆ ϕˆ ficient 2 in the UVexpansion exhibited at the beginning of realized, the order parameter DW is large, while 3 is ˆ Sec. III. We can express this statement by adopting the negligible, and vice versa ϕ3 is large along the singular following definition, Δˆ 0 domain-wall solutions for which DW ¼ . 2A−2χ Along the branch of confining solutions, the dynamics ϕ2 ≡ lim e ϕðρ2Þ; ð134Þ ρ2→þ∞ captured by the gravity theory favors the shrinking to zero size of the compact dimension spanned by η, which in field- which is manifestly consistent with the UV expansion. In theory terms corresponds to confinement of the dimen- the study we performed of the free energy density F,we sionally reduced dual theory. Conversely, along the branch
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–3.5 0
–4.0 –5 –4.5
–10 –5.0
–5.5 –15
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30
1
40 0
30 –1
20 –2
10 –3
0 –4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4
ˆ ˆ ∂Fˆ ˆ FIG. 11. The free energy density F (top left) and its derivative M ≡ ˆ (top right), as a function of the deformation parameter ϕ2, ∂ϕ2 for solutions within the confining (solid black) and singular domain-wall (long-dashed dark-green) classes. The bottom panels show Δˆ ϕˆ the order parameter DW (bottom left) and 3 (bottom right), for the same solutions. of singular domain-wall solutions, the theory is preserving compared to the dynamical scale Λ of the theory is the (locally) the higher-dimensional Poincar´e invariance, with region of large and positive ϕI, for which we see in the the formation of a condensate for the dimension-3 operator spectrum the appearance first of a parametrically light O3 associated with ϕ, whose magnitude is related to the (approximate) dilaton state that eventually becomes coefficient ϕ3 of the subleading term in the UV expansion ˆ ˆ of ϕ. In Fig. 12, we show ϕ3 as a function of ϕ2 for a few of the branches of solutions. The confining, skewed, and 50 ˆ singular domain-wall branches all share the feature that ϕ3 ˆ 40 diverges as ϕ2 → 0. This reflects the fact that in this limit they all approach the solution we called SUSY, in which 30 ˆ −3 both ϕ2 and χ5 vanish, but the combination ϕ3 ¼ ϕ3Λ ˆ 20 diverges. The regions in parameter space for which ϕ3 diverges are never energetically favored. Moreover, while ˆ 10 ϕ3 ≫ 1 on the singular domain-wall branch close to the phase transition, the singular nature of this class of 0 solutions makes a field-theory interpretation problematic, and it is unknown whether this feature would remain in a 0.0 0.5 1.0 1.5 2.0 more complete treatment of the gravity description. ˆ We can now return to the discussion of the spectrum of FIG. 12. The UV parameter ϕ3 as a function of the deformation ˆ bound states along the branch of confining solutions, that parameter ϕ2, for the confining (solid black and short-dashed we started in Sec. IV B. The behavior of ϕ3 and χ5 is related orange), skewed (dashed red), IR-conformal (longest-dashed to the nature of the approximate dilaton state. In particular, purple), and singular domain-wall (long-dashed dark-green) the region of parameter space in which ϕ3 is large classes of background solutions.
106018-25 ELANDER, PIAI, and ROUGHLEY PHYS. REV. D 103, 106018 (2021) tachyonic. This region of parameter space is not physically stronger, but we do not have the quantitative elements to realized, as the confining solutions are energetically dis- support this suggestion. favored, compared to the singular domain-wall solutions. We must close this discussion by repeating the obser- Some of the metastable configurations leading to a very vation that two pathologies are still present: we could not light dilaton might be long lived, but exploring this find any solutions, neither regular nor singular, correspond- ˆ possibility would require a detailed study of the bubble ing to arbitrarily large values of ϕ2, and furthermore the rate of the phase transition, which goes far beyond our phase transition we identified seems to indicate that the ˆ ˆ c current purposes (see Ref. [132] for a recent study in this energetically favored solutions for ϕ2 > ϕ2 are singular direction). Nevertheless, it is reassuring to notice how the and hence do not admit a sensible physical interpretation in (failure of the) probe approximation captures correctly terms of dual field-theory quantities. Our interpretation of the existence of a region of parameter space in which these results is that there is an upper bound to the choice of the condensate hO3i is parametrically enhanced. ˆ ˆ c ϕ2 < ϕ2 for which the gravity description at our disposal Some degree of complication in interpreting the spec- admits a holographic field-theory interpretation. The other trum of scalar bound states along the physical, confining phase exists only as a phase of the gravity theory, regulated branch of solutions arises because of the interplay between by putting boundaries ρ1 < ρ < ρ2 on the radial (holo- the two possible operators developing vacuum expectation graphic) direction. This unusual feature resembles what values (VEVs). This is particularly subtle in the region with happens in the presence of bulk phase transitions in the positive, large values of ϕ . As can be seen in Figs. 18–21 I study of lattice field theories. We will explore this obser- in Appendix F, by following the solid black and short- vation further in Sec. VF. dashed orange lines, in the limit in which ϕ → þ∞, one I We are forced to conclude that large (positive) defor- ultimately drives toward suppressing the explicit symmetry ˆ mations of the field theory due to the dimension-3 operator breaking parameter ϕ2 → 0. One of the condensates van- O3 dual to the scalar ϕ cannot be captured by this gravity χˆ → 0 ishes in this unphysical limit, as 5 , yet scale invari- model. Whether or not extensions of the gravity theory can ance is broken spontaneously by the divergence of the other overcome this limitation is unknown; given that Romans ϕˆ → ∞ condensate, signaled by the fact that 3 þ . Albeit supergravity does not contain other scalar fields, such unphysical (because of the tachyon and the presence of a extensions either might involve allowing for nontrivial phase transition), the analysis of this region of parameter behaviors of the fields removed by the reduction on S4 space is quite interesting as a way to test our theoretical of massive type-IIA or might require the inclusion of tools. The reason why the mass of the lightest scalar extended objects that are not captured by the supergravity fluctuation in the system shows significant discrepancy approximation. We leave this challenging problem open to with the probe approximation is the emergence of this future exploration. divergently large condensate. At finite, small values of ϕI, the effects of explicit symmetry breaking are not small, and the mixing effects between the two scalar particles sourced F. Alternative approach to the free energy density by both the operators developing VEVs are not negligible, In the previous subsections, we introduced appropriate either. regulators ρ1 and ρ2, as well as a suitably defined set of ϕc We finally notice that the critical value of I that sets the boundary-localized terms, chosen according to a prescrip- upper limit to the reach of the field-theory interpretation of tion that allows one to remove all divergences and to the confining solutions is comparatively small with respect compare to one another the free energy density F of to the value at which the tachyon emerges. By examining different, independent background configurations. In par- Fig. 6, one sees that in immediate proximity of this value of ticular, we derived Eq. (118), which we reproduce here for ϕ I the lightest scalar is not a dilaton, and neither is it convenience: appreciably much lighter than the other states of the system. 3 The next-to-lightest state, though, shows significant dis- 4A−χ F ¼ − lim e ∂ρA þ W2 : crepancy with the probe approximation, and it should be ρ2→þ∞ 2 ρ2 interpreted as an approximate (not so light) dilaton, which exists because Δˆ ≠ 0, signaling the presence of a For the same purpose, we repeat the definition of the scale DW Λ condensate. , taken from Eq. (85): ϕˆ c Z Furthermore, our estimate of 2 is likely an overestimate: ρ2 the domain-wall, badly singular solutions cannot be the Λ−1 ≡ lim dρ˜eχðρ˜Þ−Aðρ˜Þ: ρ2→þ∞ ρ ones realized physically, and in a more complete gravity o theory, other solutions must take over the dynamics beyond We also studied the energetics as a function of the leading- ˆ cc ˆ c a new critical point ϕ2 ≤ ϕ2. Potentially, this might order coefficient ϕ2 in the UV expansion exhibited at the ˆ cc happen at ϕ2 ¼ 0 (see Sec. VF for a complementary beginning of Sec. III, and that we can write by copying discussion). This might make the phase transition even Eq. (134):
106018-26 DILATONIC STATES NEAR HOLOGRAPHIC PHASE … PHYS. REV. D 103, 106018 (2021)
˜ ˜ −5 ˜ ˜ −2 FIG. 13. The regulated free energy FΛ of the confining solutions, as a function of the regulated deformation ϕ2Λ (dashed blue lines), for various choices of Δρ ¼ 3; 4; 5; 6; 7; 8; 9; 10; 11 (top to bottom, left to right). The solid black and short-dashed orange lines depict the renormalized free energy Fˆ in Figs. 8–10. The green dot marks the location of crossing between the branch of regular solutions and the branch of singular domain-wall solutions. The gray dot denotes the SUSY solutions.
Z ϕ ≡ 2A−2χϕ ρ ρ2 2 lim e ð 2Þ: ˜ −1 ˜ χðρ˜Þ−Aðρ˜Þ ˜ 2A−2χ ρ2→þ∞ Λ ðρ2Þ ≡ dρe ; ϕ2ðρ2Þ ≡ e ϕðρ2Þ; ρo The strategy we followed in the previous subsections ð138Þ consisted of first taking the ρ2 → þ∞ limit in these three expressions and then studying the resulting phase structure which are the finite-ρ2 analogs of their infinite-ρ2 limits. for the theory. There is another possible way to perform this We will study them at finite Δρ, perform the minimization study, and we explore it in this section. We can first hold of F˜ Λ˜ −5, and identify possible phase transitions, and only Δρ ≡ ρ − ρ fixed 2 o and study the phase structure encoded ρ → ∞ Wf ˆ −5 ˆ −2 afterward take 2 þ . Notice that in defining 2 we in the dependence of F ≡ FΛ on ϕ2 ≡ ϕ2Λ , and only chose to retain only the terms of W2 that give divergent afterward take the limit ρ2 → ∞ by looking at how the þ and finite-order contributions to the free energy in the phase structure evolves in the limit in which the boundary ρ2 → þ∞ limit; this corresponds to a particular choice of of the gravity theory is removed. We hence introduce the subtraction scheme. following quantities, This alternative approach is closely related to what is normally done on the lattice, where one first performs a 3 4 4 ˜ 4A−χ f f 2 rough scan of the lattice parameter space, to identify Fðρ2Þ ≡ −e ∂ρA þ W ; W ≡ − − ϕ ; 2 2 2 3 3 possible artificial phase transitions of the lattice theory, ρ2 and then restricts the lattice studies to the region connected ð137Þ with the field theory, before taking the continuum limit, in
106018-27 ELANDER, PIAI, and ROUGHLEY PHYS. REV. D 103, 106018 (2021) this way avoiding completely unphysical regions of param- in contrast to starting by establishing first the existence of a eter space. In this section, we perform this study, restricting moduli space in the field theory.5 This study complements our attention to the confining solutions. We will show that the work done by other authors, either guided by consid- the procedure yields the same results as those discussed in erations emerging from lower-dimensional statistical- the bulk of the paper, in the physical region. The existence mechanics systems [80,81] or by holographic models built of spurious phase transitions in the gravity side of the within the bottom-up approach to holography [67]. The gauge-gravity correspondence has been observed before, primary difference is that we proposed and studied a though in a different context, dealing with the treatment in calculable model built within top-down holography. the probe approximation of extended object embedded in The example we considered is the six-dimensional half- curved backgrounds [133]. We stress that the phase maximal supergravity written by Romans [100], dimen- transition is not a feature of the field theory but rather sionally reduced on a circle. The lift of the solutions to of the regulated gravity dual (although it should be possible D ¼ 10 massive Type-IIA is known [101–103] (alternative to interpret our results in terms of a finite cutoff in the field lifts in Type IIB exist as well [104,105]). The equations of theory). motion admit a special solution with AdS6 geometry and The results of this analysis are shown in Fig. 13.We trivial ϕ ¼ 0. This solution can be interpreted as the dual of display the results for the confining solutions only, by a strongly coupled fixed point in the large-N limit of a class ˜ ˜ −5 comparing the regulated free energy FΛ , for various of supersymmetric field theories in D ¼ 5 dimensions that choices of 3 ≤ Δρ ≤ 11, to the result of the renormalized has been studied extensively in the literature [134–138] analysis. For Δρ ≳ 5, the signature of a phase transition (see also Refs. [139–142] and references therein and the appears, and moreover there is no maximum allowed value discussion in Ref. [102]). ˜ ˜ −2 of ϕ2Λ . The scalar ϕ in the gravity theory corresponds to an The branch that takes over the dynamics at large Δρ,at operator of dimension Δ ¼ 3 in the dual five-dimensional ˜ ˜ −2 least for large positive values of ϕ2Λ , has no genuine theory. Its coupling and condensate are related to the field-theory dual interpretation, as it exists only when we coefficients ϕ2 and ϕ3 in the asymptotic expansion of ϕ retain the finite UV cutoff ρ2 when minimizing the free background gravity solutions. It is known that by tuning 2 energy. We notice that this rather rough analysis seems to and ϕ3 one can build the gravity dual of the field-theory suggest that the phase transition takes place at smaller renormalization group flow toward what can be interpreted ˜ ˜ −2 (positive) values of the deformation parameter ϕ2Λ , as a second, nonsupersymmetric, perturbatively stable fixed — when compared with the analysis conducted in the bulk point [110] although it is not known that this fixed point of the paper. We also notice that the comparison is not exists in the dual field theory. rigorous, as it is affected by the presence of arbitrary The field theory admits compactification of one spatial scheme dependences. This dependence on the order of direction of the geometry on a circle, hence breaking five- limits, on the scheme, and the fact that the energetics of the dimensional Poincar´e invariance. The size of the circle in ρ dominant solution is dominated by spurious cutoff effects the gravity theory is a function of , controlled by the field χ χ are typical of what in the lattice literature are called bulk , and in particular by the coefficient 5 appearing in its phase transitions. asymptotic (UV) expansion. When the circle shrinks, the resulting strongly coupled four-dimensional dual theory confines. The gravity description hence provides a com- VI. SUMMARY paratively simple description of confinement in four We presented a first realization, within top-down holog- dimensions, along the lines suggested by Witten [94], raphy, of one particular strategy for building a dilaton but in a somewhat simpler environment [92,93]. scenario, which is inspired by the ideas in Refs. [67,82]. The simultaneous combination of these two deforma- In this scenario, a parametrically light dilaton would emerge tions had been studied so far only for values of ϕ ≤ 0 as a light scalar particle for choices of the parameters that [98,99]. In this paper, we extended our study by first bring the theory in close proximity of a dynamical instability allowing ϕ > 0 and secondly by complementing the (and in the presence of enhanced condensates). However, calculation of the spectrum with the study of the free we also found direct evidence of a phase transition, energy density F. Furthermore, we considered several new effectively preventing the dynamics from approaching general classes of background gravity solutions, all of arbitrarily close to the aforementioned instability, and hence which approach the aforementioned AdS6 geometry for none of the scalar particles can be made arbitrarily light large values of the radial direction ρ. Some are regular and along the physical branch of solutions. Nevertheless, the lightest particle can be dialed to have arbitrarily small mass 5Top-down holographic realizations of the latter approach along the metastable solutions of the same branch. already exist, though for limited and quite nontrivial systems, This approach represents an appealing, alternative for example along the baryonic branch of the Klebanov-Strassler search strategy for dynamical realizations of the dilaton, system [68,69].
106018-28 DILATONIC STATES NEAR HOLOGRAPHIC PHASE … PHYS. REV. D 103, 106018 (2021) are the main subject of our attention; some have a good on the circle. Our second new result is the mass spectrum, singularity, in the sense defined by Gubser [95]; and some that can be seen in Figs. 4–7. have a bad singularity. The salient feature of the mass spectrum is what brings (i) We called SUSY the solutions that satisfy the first- this work in contact with the line of arguments in order equations for the system in D ¼ 6 dimensions. Refs. [67,80–82]. While the confining solutions are regular, These solutions are supersymmetric, exhibit a good by moving along the one-parameter class labeled by ϕI, the singularity, and preserve five-dimensional Poincar´e mass squared of the lightest scalar glueball becomes invariance—in the gravity language, this last prop- progressively smaller (in units of the mass of the tensor, erty corresponds to the constraint d ¼ A − 4χ ¼ 0, which we use to set the scale in the spectrum), until it with A the warp factor in the metric, as discussed in becomes tachyonic at some finite, positive value of ϕI. This the main body of the paper. instability is our third new result. The reason why this is (ii) The IR-conformal solutions correspond to the interesting is that, if interpreted naively, this system would aforementioned flows between the two fixed yield an example of a theory in which by tuning the points. They preserve five-dimensional Poincar´e parameter ϕI one could dynamically produce a hierarchy of invariance but break supersymmetry. These solu- scales between the mass of the lightest scalar particle and tions are regular. the rest of the spectrum. By making use of the probe (iii) With some abuse of language, we called confining approximation (as suggested in Ref. [127]), we also solutions the regular ones in which Poincar´e showed that in the region of parameter space in which invariance is reduced to four dimensions and in the lightest scalar has a parametrically suppressed mass—in which the compact circle shrinks smoothly to proximity to the region in which a tachyon emerges—the zero size at a finite value of the radial direction associated particle is indeed an approximate dilaton (see ρ → ρo. The holographic interpretation of such Fig. 7), which is our fourth original result. In connection backgrounds involves both the compactification with this, we also noted the divergent behavior of the ˆ of the dual five-dimensional theory on a circle parameter ϕ3 in the limit of ϕI → þ∞. and then linear confinement of the resulting dimen- Unfortunately, though, the naive interpretation contained sionally reduced four-dimensional strongly coupled in the previous paragraph has to be used with caution. To theory. show why, we studied the energetics of the classical (iv) A related class of gravity solutions can be obtained solutions and found another additional result. The from the confining ones by changing the sign of the tachyonic instability appears for values of the deforming − 4χ ˆ ˆ function d ¼ A . These solutions have the same parameter ϕ2 for which the solution has free energy F that symmetries as the confining ones, but the size of the is higher than that of other solutions. This is the typical ρ → ρ circle diverges for o, and as a result, the feature expected in the presence of a first-order phase geometry has a (good) naked singularity. We called transition. It is hence not possible to dial the boundary these solutions skewed. parameter to approach arbitrarily close to the massless case, (v) We also included in our survey three other classes of as this would require exploring a branch of metastable and singular solutions. We found that they can either unstable solutions, well past a phase transition. result in good singular solutions or in bad singular We could only identify two branches of solutions within 4χ solutions. (The constraint A ¼ yields the subclass the confining class, by varying ϕ2. Furthermore, a maxi- of singular domain-wall solutions.) While not rep- ˆ mum value of the parameter ϕ2 emerged, further confirm- resentative of dual field-theory configurations, we ing the incompleteness of the energetics discussion when found that the badly singular domain-wall solutions restricted to the confining solutions only. The picture play an important role in the energetics of the gravity became more clear once we included in the discussion theory. ˆ also singular solutions. For arbitrarily large ϕ2 > 0,we We summarized in Table I all these classes of solutions could not find a ground state solution (within the restric- and how we chose to parametrize them. We introduced a tions defining our ansatz for the background metric)—free scale setting procedure via the function Λ defined in of gravity singularities—that admits a trustable field-theory Eq. (85) and showed the dimensionality of the resulting ϕˆ ϕˆ c space of solutions. We plotted the free energy in Figs. 8 interpretation. Yet, for values of 2 > 2, we showed that and 9. there exist singular solutions with free energy lower than that of the regular, confining solutions. Conversely, for Our first new finding is that the regular, confining ˆ solutions exist also for positive values of ϕ > 0. We hence negative ϕ2 < 0, the singular solutions have free energy extended the one-parameter family of solutions studied in higher than the confining ones. Hence, the phase transition ˆ ˆ cc ˆ cc ˆ c earlier publications [98,99]. We computed the spectrum of takes place at ϕ2 ¼ ϕ2 (with 0 ≤ ϕ2 ≤ ϕ2), and all the ˆ ˆ cc fluctuations of all the 32 bosonic degrees of freedom of the solutions with ϕ2 > ϕ2 along the confining branch are five-dimensional theory obtained by dimensional reduction either metastable or unstable. In particular, there is not a
106018-29 ELANDER, PIAI, and ROUGHLEY PHYS. REV. D 103, 106018 (2021) parametrically light dilaton near the transition; although the While we found a strong first-order phase transition, lightest bound state is a scalar, and its mass is slightly there may be other models realizing this mechanism, and it smaller than in other regions of the physical portion of is not known a priori how strong the first-order phase parameter space, it does not show the properties expected transitions should be in general. They might be very weak. by an approximate dilaton, and its mass is not parametri- There are well-known examples in physics of systems in cally, nor numerically, small. The next-to-lightest state, which first-order phase transitions sit along critical lines (in though, is at least approximately a dilaton, but it is heavier, parameter space) that have an end point. If one could and its mass does not show any special features in the identify a supergravity theory realizing this type of critical region of parameter space immediately adjacent to the behavior, then it would be interesting to repeat our analysis phase transition. in more detail within such a system. A direct calculation We repeat again that the phase transition we find is not a could establish whether the phase transition takes place in field-theory feature, but rather it exists only in the gravity the proximity of the end point of the critical line. We might theory. As discussed in Sec. VF, this is not contradictory, as find that the whole spectrum scales without producing a gauge/gravity dualities relate only physical objects in the hierarchy and hence asymptotically reproduces the scaling physically related phase of the theory, and the gravity behaviors expected in the presence of explicitly broken theory (with finite radial direction ρ1 < ρ < ρ2)may scale invariance. Conversely, one might discover that the exhibit a more general phase structure. Nevertheless, it spectrum still behaves as in Figs. 6 and 7 and a light dilaton is interesting to notice how the physical properties of the emerges. If so, its existence would be connected to the bound states in the region of parameter space that admits a enhancement of nontrivial condensates in the vacuum, field-theory interpretation are influenced by the phenomena which can be checked explicitly. This possibility, if taking place past the phase transition. realized, would have important theoretical and phenom- enological implications and hence motivates us to further pursue our program in the future. VII. CONCLUSION AND OUTLOOK
Along a new branch of regular solutions of Romans ACKNOWLEDGMENTS supergravity, we found a tachyonic instability by studying the mass spectrum of the fluctuations of the sigma model We thank A. Pomarol for useful discussions and coupled to gravity. By approaching this instability in the C. Núñez and D. C. Thompson for comments on an earlier space of parameters, we found that the lightest scalar state version of the manuscript. The work of M. P. has been in the spectrum turns into a tunably light approximate supported in part by the Science and Technology Facilities dilaton, which could be realized in a metastable configu- Council (STFC) Consolidated Grants No. ST/L000369/1 ration of the system. A condensate is enhanced when and No. ST/P00055X/1. M. P. has also received funding moving along this branch of solutions, spontaneously from the European Research Council (ERC) under the breaking (approximate) scale invariance. But we also found European Union’s Horizon 2020 research and innovation that the instability is hidden away by a strong first-order programme under Grant No. 813942. J. R. has been phase transition, so that the lightest scalar state along the supported by STFC, through the studentship ST/R505158/1. stable phase is not parametrically light, and it is the next-to- D. E. was supported by the Origines Constituants & lightest scalar state that behaves as an approximate dilaton Evolution de l’Universe (OCEVU) Laboratoires d’excel- (in association with an enhancement of one of the con- lence (Labex) (Grant No. ANR-11-LABX-0060) and the densates). We hence uncovered a concrete realization Initiative d’excellence Aix-Marseille (A*Midex) project within top-down holography of arguments closely resem- (Project No. ANR-11-IDEX-0001-02) funded by the bling those of Refs. [67,80,81], although in a general- “Investissements d’Aveni” French government program ized form. managed by the Agence nationale de la recherche (ANR). Our study admits a clear (though not simple) interpre- tation, and our action is taken from the established catalog APPENDIX A: A FEW GRAVITATIONAL of rigorously defined supergravity theories. We also tested (CURVATURE) INVARIANTS the formal tools that would be needed to perform this type In this Appendix, we find it useful to present and discuss of analysis in other supergravity theories. This paper the results for some of the curvature invariants of the establishes the basis for the development of a systematic theory in D 6 dimensions—the Ricci scalar R R6, the future research program, encompassing the exploration of ¼ ¼ 2 ≡ Mˆ Nˆ the vast catalog of known supergravity theories—possibly Ricci tensor squared R2 RMˆ Nˆ R , and the Riemann 2 ≡ Mˆ Nˆ Rˆ Sˆ encompassing the technically more challenging cases in tensor squared R4 RMˆ Nˆ Rˆ Sˆ R . We adopt the six- which one does not recover an AdS geometry asymptoti- dimensional metric ansatz in Eq. (9). After using the cally far in the UV. equations of motion presented in Sec. II D, we find that
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0 40
–5 30
–10 20
–15 10
–20 0 0 1 2 3 4 5 6 7 8 ρ 0 1 2 3 4 5 6 7 8
80
60
40
20
0 0 1 2 3 4 5 6 7 8
FIG. 14. Gravitational invariants computed using the analytic confining solutions with ϕ ¼ 0, shown in Eqs. (51) and (52). From top ≡ Mˆ Nˆ 2 ≡ Mˆ Nˆ 2 ≡ Mˆ Nˆ Rˆ Sˆ to bottom, left to right: R g RMˆ Nˆ , R2 RMˆ Nˆ R , and R4 RMˆ Nˆ Rˆ Sˆ R .
0 40
–5 30
–10 20
–15 10
–20 0 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
80
60
40
20
0 0 1 2 3 4 5 6 7 8
FIG. 15. Gravitational invariants computed using the analytic skewed solutions with ϕ ¼ 0, shown in Eqs. (51) and (52). From top to ≡ Mˆ Nˆ 2 ≡ Mˆ Nˆ 2 ≡ Mˆ Nˆ Rˆ Sˆ bottom, left to right: R g RMˆ Nˆ , R2 RMˆ Nˆ R , and R4 RMˆ Nˆ Rˆ Sˆ R .
106018-31 ELANDER, PIAI, and ROUGHLEY PHYS. REV. D 103, 106018 (2021)
2 R ¼ 6V6 þ 4ð∂ρϕÞ ; 2 2 2 4 R2 ¼ 6V6 þ 8V6ð∂ρϕÞ þ 16ð∂ρϕÞ ; rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffi 2 32 ∂ 2 4∂ 36 ∂ 2 15 5 6 2 − 2 − 30 24 ∂ 2 R4 ¼ 250 ð ð ρdÞ ρd ð ρdÞ þ R2 R R þ ð ρdÞ ffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 2 2 2 þ 5 5 6R2 − R − 10R − 25ðR − 10R2ÞÞ; ðA1Þ
2 where in deriving the expression for R4, we made use of the backgrounds with ϕ ¼ 0, corresponding to the confining fact that for our solutions ∂ρc>0 [see Eq. (22)]. and skewed solutions, respectively. From these expressions, and from the knowledge of the smooth potential V6 in Eq. (6), one sees that as long as ϕ does not diverge, both the Ricci scalar and the square of the APPENDIX B: IR EXPANSIONS OF THE Ricci tensor remain finite. This is the case for the regular GENERIC SINGULAR SOLUTIONS IN SEC. III F solutions that we called “confining,” but it also holds true for the “skewed” solutions, for which the singularity This Appendix complements the discussion in Sec. III F. manifests itself only at the level of the square of the Explicit evaluation of the first terms in the series expansion Riemann tensor. In Figs. 14 and 15, we plot the curvature performed near the end of the geometry, which would invariants for representative examples of solutions belong- correspond to the deep IR of the field theory, including all ≤ 2 ing to these two classes, which we chose to be the analytical terms with n , yields the following:
2ϕI e ð7ϕL þ 3Þ 2ϕ 2 ϕ ρ ϕ ϕ ρ − ρ − ρ − ρ Lþ ð Þ¼ I þ L logð oÞ 2 ð oÞ 4ðϕL þ 1Þ ð2ϕL þ 3Þ
−2ϕI −6ϕI e ð7ϕL − 3Þ 2−2ϕ e ð9 − 23ϕLÞ 2−6ϕ ρ − ρ L ρ − ρ L þ 2 ð oÞ þ 2 ð oÞ 3ðϕL − 1Þ ð2ϕL − 3Þ 108ð1 − 3ϕLÞ ð2ϕL − 1Þ
4ϕI 4 2 e ðϕLðϕLð79ϕL þ 197Þþ144Þþ30Þ 4ϕ 4 ϕLð12ϕ þ 19ϕ þ 9Þ 4 ρ − ρ Lþ − L L ρ − ρ þ 8 ϕ 1 4 2ϕ 3 2 4ϕ 5 ð oÞ þ 6 ϕ2 − 1 2 4ϕ2 − 9 ð oÞ ð L þ Þ ð L þ Þ ð L þ Þ ð L Þ ð L Þ
−4ϕI 4−4ϕL e ðρ − ρoÞ þ 2 2 4 2 108ð1 − 3ϕLÞ ð3 − 2ϕLÞ ðϕL − 1Þ ðϕL þ 1Þ ð2ϕL − 1Þð2ϕL þ 3Þð4ϕL − 5Þ ϕ ϕ ϕ ϕ ϕ ϕ ϕ −78428ϕ2 76312ϕ 75857 × f Lð Lð Lð Lð Lð Lð Lð L þ L þ Þ þ 147745Þ − 431341Þþ213007Þþ67591Þ − 85329Þþ24561Þ − 2295g
−8ϕI 4 3 2 e ð4958ϕ − 8837ϕ þ 6004ϕ − 1831ϕL þ 210Þ 4−8ϕ L L L ρ − ρ L þ 162 16ϕ2 − 34ϕ 15 6ϕ3 − 11ϕ2 6ϕ − 1 2 ð oÞ ð L L þ Þð L L þ L Þ
−12ϕI 2 e ðϕLð−5391ϕ þ 6591ϕL − 2674Þþ360Þ 4−12ϕ L ρ − ρ L þ 4 2 ð oÞ þ ; ðB1Þ 5832ð1 − 3ϕLÞ ð1 − 2ϕLÞ ð12ϕL − 5Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ϕ 2 1 I ϕ − 2ζ 1 − 5ϕ 1 2 e ð L L þ Þ 2ϕ 2 χðρÞ¼χ þ 4ζ 1 − 5ϕ þ 1 logðρ − ρ Þþ ðρ − ρ Þ Lþ I 15 L o 6ðϕ þ 1Þ2ð2ϕ þ 3Þ o pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L L pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −2ϕI 2 −6ϕI 2 2e ð−2ζ 1 − 5ϕ − ϕL þ 1Þ 2−2ϕ e ð−2ζ 1 − 5ϕ − 3ϕL þ 1Þ 2−6ϕ − L ρ − ρ L L ρ − ρ L þ 2 ð oÞ þ 2 ð oÞ 9ðϕL − 1Þ ð2ϕL − 3Þ 162ð1 − 3ϕLÞ ð2ϕL − 1Þ 4ϕ 4ϕ 4 n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e I ðρ − ρ Þ Lþ þ o ϕ ϕ −24ϕ þ 44ζ 1 − 5ϕ2 þ −71 þ 92ζ 1 − 5ϕ2 − 66 24ðϕ þ 1Þ4ð2ϕ þ 3Þ2ð4ϕ þ 5Þ L L L L L qLffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L o L 44ζ 1 − 5ϕ2 − 19 þ L
106018-32 DILATONIC STATES NEAR HOLOGRAPHIC PHASE … PHYS. REV. D 103, 106018 (2021) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ − ρ 4 n o ð oÞ −24 ζ 1 − 5ϕ2 − 1 ϕ4 5 2ζ 1 − 5ϕ2 3 ϕ2 − 26ζ 1 − 5ϕ2 1 þ 45 ϕ2 − 1 2 4ϕ2 − 9 L L þ L þ L L þ ð L Þ ð L Þ −4ϕ 4−4ϕ e I ðρ − ρ Þ L þ o 324ð1 − 3ϕ Þ2ð3 − 2ϕ Þ2ðϕ − 1Þ4ðϕ þ 1Þ2ð2ϕ − 1Þð2ϕ þ 3Þð4ϕ − 5Þ n qLffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L L L qL ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L L 9 344ζ 1 − 5ϕ2 − 169 ϕ ϕ 5 8468ζ 1 − 5ϕ2 − 7123 × L þ L L L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ 88 433ζ 1 − 5ϕ2 137 ϕ ϕ ϕ 4ϕ ϕ þ L L þ þ L L L L L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −5864ϕ − 10124ζ 1 − 5ϕ2 7547 928ζ 1 − 5ϕ2 8595 × L L þ þ L þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 107972ζ 1 − 5ϕ2 − 1353 27964ζ 1 − 5ϕ2 − 122406 þ L þ L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o −152272ζ 1 − 5ϕ2 94701 − 22740ζ 1 − 5ϕ2 13026 þ L þ L þ −8ϕ 4−8ϕ e I ðρ − ρ Þ L þ o 243ð1 − 3ϕ Þ2ð1 − 2ϕ Þ2ðϕ − 1Þ2ð2ϕ − 3Þð8ϕ − 5Þ n L L L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL L ϕ ϕ 2ϕ 340ϕ 230ζ 1 − 5ϕ2 − 617 × L L L L þ L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o −682ζ 1 − 5ϕ2 879 356ζ 1 − 5ϕ2 − 290 − 62ζ 1 − 5ϕ2 37 þ L þ þ L L þ −12ϕ 4−12ϕ e I ðρ − ρ Þ L þ o 17496ð1 − 3ϕ Þ4ð1 − 2ϕ Þ2ð12ϕ − 5Þ n L L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −3ϕ 3ϕ 168ϕ 76ζ 1 − 5ϕ2 − 199 − 188ζ 1 − 5ϕ2 234 × L L L þ L L þ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o − 116ζ 1 − 5ϕ2 91 L þ þ ; ðB2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ϕ 2 2ϕ 2 1 e I ð8ϕ − ζ 1 − 5ϕ þ 8Þðρ − ρ Þ Lþ AðρÞ¼a þ ζ 1 − 5ϕ2 þ 4 logðρ − ρ Þþ L L o I 15 L o 12ðϕ þ 1Þ2ð2ϕ þ 3Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL −2ϕ 2 2−2ϕ −6ϕ 2 2−6ϕ I −ζ 1 − 5ϕ − 8ϕ 8 ρ − ρ L I −ζ 1 − 5ϕ − 24ϕ 8 ρ − ρ L − e ð L L þ Þð oÞ e ð L L þ Þð oÞ þ 2 þ 2 9ðϕL − 1Þ ð2ϕL − 3Þ 324ð1 − 3ϕLÞ ð2ϕL − 1Þ 4ϕ 4ϕ 4 n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e I ðρ − ρ Þ Lþ þ o ϕ ϕ −96ϕ þ 11ζ 1 − 5ϕ2 þ −284 24ðϕ þ 1Þ4ð2ϕ þ 3Þ2ð4ϕ þ 5Þ L L L L qLffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o 23ζ 1 − 5ϕ2 − 264 11ζ 1 − 5ϕ2 − 76 þ L þ L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ − ρ 4 n o ð oÞ −12 ζ 1 − 5ϕ2 − 16 ϕ4 5 ζ 1 − 5ϕ2 24 ϕ2 − 13ζ 1 − 5ϕ2 8 þ 90 ϕ2 − 1 2 4ϕ2 − 9 L L þ L þ L L þ ð L Þ ð L Þ −4ϕ 4−4ϕ e I ðρ − ρ Þ L þ o 324ð1 − 3ϕ Þ2ð3 − 2ϕ Þ2ðϕ − 1Þ4ðϕ þ 1Þ2ð2ϕ − 1Þð2ϕ þ 3Þð4ϕ − 5Þ n L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL L L L L L ϕ ϕ 5 2117ζ 1 − 5ϕ2 − 28492 ϕ ϕ ϕ ϕ 4ϕ ϕ × L L L þ L L L L L L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −23456ϕ − 2531ζ 1 − 5ϕ2 30188 4 58ζ 1 − 5ϕ2 8595 × L L þ þ L þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 26993ζ 1 − 5ϕ2 − 5412 6991ζ 1 − 5ϕ2 − 489624 þ L þ L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 38068ζ 1 − 5ϕ2 378804 9526ζ 1 − 5ϕ2 48224 L þ þ L þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o −5685ζ 1 − 5ϕ2 52104 774ζ 1 − 5ϕ2 − 6084 þ L þ þ L
106018-33 ELANDER, PIAI, and ROUGHLEY PHYS. REV. D 103, 106018 (2021)
−8ϕ 4−8ϕ e I ðρ − ρ Þ L þ o 486ð1 − 3ϕ Þ2ð1 − 2ϕ Þ2ðϕ − 1Þ2ð2ϕ − 3Þð8ϕ − 5Þ n L L L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL L ϕ ϕ 2ϕ 2720ϕ 115ζ 1 − 5ϕ2 − 4936 × L L L L þ L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o −341ζ 1 − 5ϕ2 7032 178ζ 1 − 5ϕ2 − 2320 − 31ζ 1 − 5ϕ2 296 þ L þ þ L L þ −12ϕ 4−12ϕ n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e I ðρ − ρ Þ L þ o −3ϕ 3ϕ 672ϕ þ 19ζ 1 − 5ϕ2 þ 17496ð1 − 3ϕ Þ4ð1 − 2ϕ Þ2ð12ϕ − 5Þ L L L L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL L L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o − 796 − 47ζ 1 − 5ϕ2 936 − 29ζ 1 − 5ϕ2 364 L þ L þ þ : ðB3Þ
In the numerical studies included in the body of the paper (e.g., in the calculations illustrated by Fig. 8), we retained a few additional higher-order terms in these expressions in order to minimize noise and improve convergence of the numerical studies.
APPENDIX C: IR EXPANSIONS OF THE SINGULAR DOMAIN-WALL SOLUTIONS IN SEC. III G In this Appendix, we show explicitly some of the terms in the series expansion around the end of space of the geometry, 4 for the solutions discussed in Sec. III G. For convenience, we truncate the expansion at the order Oððρ − ρoÞ Þ, although we retained also a few additional higher-order terms in some of the numerical calculations described in the main body of the paper, 1 9 log ρ − ρ 69 ϕ ρ ð oÞ ϕ ρ − ρ 4=9 − ϕ 2 ρ − ρ 8=9 ð Þ¼6 log 4 þ 3 þ 4ð oÞ 85 4 ð oÞ qffiffi 3 3 3 3 2 6046ϕ − 4 ρ − ρ 4=3 þ 14 þ 1785 ð oÞ pffiffiffi 2=3 3 4 6047325 × 2 3ϕ4 241437236ϕ4 ð þ Þ ρ − ρ 16=9 þ 7586250 ð oÞ pffiffiffi 2=3 3 2 5 −711782775 × 2 3ϕ4 − 26218571272ϕ4 ð Þ ρ − ρ 20=9 þ 146667500 ð oÞ ρ − ρ 8=3 pffiffiffi ð oÞ 32=3276494428125 3 2 þ 11519265450000 f pffiffiffi 2=3 3 3 6 þ 2965817537958002 3ϕ4 þ 9755979537544064ϕ4 g qffiffi 3 3 4 3 2=3 2360772721213 ϕ 5428197ð Þ ϕ4 2 4 ρ − ρ 28=9 − 2 − þð 0Þ 1994300 9496720625 26498325334552196ϕ7 − 4 þ 7264991278125 ρ − ρ 32=9 pffiffiffi ð oÞ 6151561447643611875 3 232=3ϕ2 þ 773620668463671875 f 4 pffiffiffi 2=3 3 5 8 þ 432480088524305869575 × 2 3ϕ4 þ 11365219568455804037008ϕ4g ρ − ρ 4 ð oÞ −2413524250949484375 þ 155961926762276250000 f pffiffiffi 3 2=3 3 þ −7074613549111472685000 23 ϕ4 pffiffiffi 2=3 3 6 þ −371140233877399872460300 × 2 3ϕ4 9 þ −8759105677773799489866912ϕ4gþ ; ðC1Þ
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1 2 244 χ ρ χ ρ − ρ ϕ ρ − ρ 4=9 − ϕ 2 ρ − ρ 8=9 ð Þ¼ I þ 27 logð oÞþ5 4ð oÞ 255 4 ð oÞ rffiffiffi 1 3 14668ϕ 3 3 4 ρ − ρ 4=3 þ 7 2 þ 5355 ð oÞ pffiffiffi 2=3 3 4 3 166175 × 2 3ϕ4 6954744ϕ4 − ð þ Þ ρ − ρ 16=9 þ 2528750 ð oÞ pffiffiffi 2=3 3 2 5 2318168025 × 2 3ϕ4 84249694712ϕ4 ð þ Þ ρ − ρ 20=9 þ 3300018750 ð oÞ ρ − ρ 8=3 pffiffiffi ð oÞ −32=3108193471875 3 2 þ 17278898175000 f þ pffiffiffi 2=3 3 3 6 − 45915207465600 × 2 3ϕ4 − 1414212386352128ϕ4 g ρ − ρ 28=9 pffiffiffi ð oÞ 58007569300875 3 232=3ϕ þ 319659616237500 f 4 pffiffiffi 2=3 3 4 7 þ 3409626494789460 × 2 3ϕ4 þ 89150405935813024ϕ4g ρ − ρ 32=9 pffiffiffi ð oÞ −38576134638208906875 3 232=3ϕ2 þ 41775516097038281250 f 4 pffiffiffi 2=3 3 5 8 þ −1921139818985605118175 × 2 3ϕ4 − 43318844111573477101952ϕ4g ρ − ρ 4 ð oÞ 14718312628622578125 þ 3509143352151215625000 f pffiffiffi 3 2=3 3 þ 18585357769872307035000 23 ϕ4 pffiffiffi 2=3 3 6 þ 744000468587964720553300 × 2 3ϕ4 9 þ 14846531094788772880019552ϕ4gþ : ðC2Þ
The domain-wall condition A ¼ 4χ restores (locally) and the ranges of the angles, describing the internal Poincar´e invariance in D ¼ 5 dimensions. 4-sphere, are
0 ≤ θ ≤ π; 0 ≤ φ < 2π; π π APPENDIX D: SINGULAR DOMAIN-WALL 0 ≤ ψ 4π − ≤ ξ ≤ SOLUTIONS—LIFT TO D = 10 DIMENSIONS < ; 2 2 : ðD5Þ This Appendix discusses the lift of the solutions to The detailed expressions for the remaining nonzero back- massive type-IIA supergravity in D ¼ 10 dimensions. We ground fields, the dilaton and the Ramond-Ramond 4-form, focus on the ten-dimensional metric, which in the Einstein can be found in Refs. [103,105].6 frame is given by Because of the factor sinðξÞ1=12 in the ten-dimensional metric, all the solutions we consider are singular at ξ ¼ 0. 2 1=12 1=8 3=8 2 ˜ 2 ds10 ¼ðsinðξÞÞ X Δ ðds6 þ dΩ4Þ; ðD1Þ For nonzero values of ξ, the behavior of the curvature invariants differs depending on the different classes con- where sidered in the body of this paper, as we shall now see. For simplicity, consider the ten-dimensional Ricci scalar evalu- ξ π 2 X ¼ eϕ; ðD2Þ ated at ¼ = ,givenby 10 −ϕ 3ϕ ð Þ π 6 9 R jξ¼ ¼ e þ e Δ ¼ X−3 sin2ðξÞþX cos2ðξÞ; ðD3Þ 2 1 ϕ 12V ϕ ∂ V ϕ 4 ∂ ϕ 2 þ 2e ð 6ð Þþ ϕ 6ð Þþ ð ρ Þ Þ: ðD6Þ 1 Ω˜ 2 2 ξ2 −1Δ−1 2 ξ θ2 2 θ φ2 d 4 ¼ X d þ X cos ð Þ 4 ½d þ sin ð Þd 6 ψ θ φ 2 Our conventions are such that they agree with Sec. 3.1.3 of þðd þ cosð Þd Þ ; ðD4Þ Ref. [98] putting g ¼ 1.
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ð10Þ As can be seen, R π remains finite as long as ϕ remains APPENDIX F: A FEW PARAMETERIC PLOTS jξ¼2 finite as a function of ρ. This is the case for the confining In this Appendix, we show some additional details of the and skewed solutions. The class of badly singular domain- numerical results we obtained by studying the confining wall solutions introduced in Sec. III G plays a prominent and skewed solutions and their approach to the trivial role in our analysis, being the energetically favored branch critical point for large values ρ of the radial direction. In the ˆ ˆ c for ϕ2 > ϕ2. Using the IR expansion given in Eq. (83),we main body of the text, we focused most of our attention on obtain that the values of the parameter ϕ2 and on the free-energy density F along the various branches of solutions. We show 2 2=3 2 2=3 5 ϕ4 here how the other parameters, ϕ3, χ5, and ϕ , evolve along ð10Þ ð3Þ −5=3 ð3Þ −11=9 I R jξ π ¼ ρ þ ρ ¼2 9 9 the two special branches of solutions that we called −7 9 “confining,”“skewed,” and “IR-conformal.” These param- þ Oðρ = Þ; ðD7Þ eters are extracted by following the procedure outlined in Sec. VC, and correspond to the values obtained in step 3 of confirming the singular nature of these solutions also in the list describing the numerical implementation. D ¼ 10 dimensions (even away from ξ ¼ 0). The main qualitative features that emerge from Figs. 18–21 are similar to what we have already described in the main text. We notice that when studying ϕ2, ϕ3, and APPENDIX E: MASS SPECTRA IN UNITS OF Λ χ5 as a function of ϕI two different regimes emerge. For In this Appendix, we show the mass spectra normalized negative values of ϕI, all the physically interesting UV in units of the scale Λ, in order to facilitate the comparison parameters show a monotonic, unbounded dependence on with the results of Sec. V. The results are depicted in ϕI itself. When ϕI > 0, the fact that a maximum value of ϕ2 Figs. 16 and 17. The only purpose of these plots is to allow is reached at finite ϕI gives rise to the nontrivial shape of the reader to easily relate the scale setting procedures we the curves shown in the three figures. We find it useful to used in the calculaton of the spectrum and of the phase show also the results along the IR-conformal branch of structure. solutions where appropriate.
25 25
20 20
15 15
10 10
5 5
0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
20
15
10
5
0
0.0 0.5 1.0 1.5 2.0
FIG. 16. The spectra of masses M, as a function of the one free parameter ϕI characterizing the confining solutions and normalized in −4 μ units of Λ, computed with ρ1 ¼ 10 and ρ2 ¼ 12. From top to bottom, left to right, the spectra of fluctuations of the tensors eν (red), the graviphoton Vμ (green), and the two scalars ϕ and χ (blue). The orange points in the plot of the scalar mass spectrum represent values of 2 0 ϕ ϕc 0 M < and hence denote a tachyonic state. We also show by means of the vertical dashed lines the case I ¼ I > , the critical value that is introduced and discussed in Sec. V.
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25 25
20 20
15 15
10 10
5 5
0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
25 25
20 20
15 15
10 10
5 5
0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
25 25
20 20
15 15
10 10
5 5
0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
FIG. 17. The spectra of masses M as a function of the scale parameter ϕI, normalized in units of Λ. From top to bottom, left to right, i i the spectra of fluctuations of the pseudoscalars π forming a triplet 3 of SUð2Þ (pink), vectors Aμ forming a triplet 3 of SUð2Þ (brown), Uð1Þ pseudoscalar X (gray), Uð1Þ transverse vector B6μ (purple), Uð1Þ transverse vector Xμ (black), and the massive U(1) 2-form Bμν −4 (cyan). The spectrum was computed using the regulators ρ1 ¼ 10 and ρ2 ¼ 12 with the exception of the Uð1Þ pseudoscalar X for −7 which we used ρ1 ¼ 10 in order to minimize the cutoff effects present for the very lightest state at large values of ϕI. We also show by ϕ ϕc 0 means of the vertical dashed lines the case I ¼ I > , the critical value that is introduced and discussed in Sec. V.
0.1 1.0
0.8 0.0 0.6
–0.1 0.4
0.2 –0.2 0.0
–0.2 –0.3 –0.4 –0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.5 1.0 1.5 2.0
FIG. 18. Plots showing the relationship between the UV expansion parameter ϕ2 and the IR parameter ϕI, in the solutions we called “confining” (solid black and short-dashed orange lines) and “skewed” (dashed red line). The left plot shows the bare parameters ϕconf ϕskew ϕconf ϕskew extracted: the solid black and dashed red lines agree, as with 2 ¼ 2 , I ¼ I . The right plot shows the same parameters after rescaling with the appropriate powers of Λ.
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0.8 40
0.6 30
0.4 20
0.2 10
0.0 0
–0.2 –0.4 –0.3 –0.2 –0.1 0.0 –5 –4 –3 –2 –1 0
FIG. 19. Plots showing the relationship between the two UV expansion parameters ϕ2 and ϕ3 for solutions belonging to the confining (solid black and short-dashed orange lines), skewed (dashed red line), and IR-conformal (longest-dashed purple line) classes. The left conf skew conf skew plot shows the base parameters extracted by matching to the UV expansions, with ϕ2 ¼ ϕ2 , ϕ3 ¼ ϕ3 . The right panel shows the same parameters after rescaling with the appropriate powers of Λ. (For ϕ2 ≤ 0, although the confining, skewed, and IR-conformal classes are not in complete agreement, they are close enough that in these plots the solid black and dashed red lines are hidden behind the longest-dashed purple one.)
0.04
10 0.03
0.02 5 0.01
0.00 0
–0.01
–0.4 –0.3 –0.2 –0.1 0.0 –3 –2 –1 0 1
FIG. 20. Plots showing the relationship between the two UV expansion parameters ϕ2 and χ5 for solutions within the confining (solid black and short-dashed orange lines), skewed (dashed red line), and IR-conformal (longest-dashed purple line) classes. The left plot conf skew conf skew skew conf 8 conf conf shows the parameters extracted by matching to the UVexpansions, with ϕ2 ¼ ϕ2 , ϕ3 ¼ ϕ3 , and χ5 ¼ −χ5 − 25 ϕ2 ϕ3 . The right panel shows the same parameters after rescaling with the appropriate powers of Λ.
10 0.010 8
0.005 6
4 0.000 2
–0.005 0
–2 –0.010 –4 –0.4 –0.3 –0.2 –0.1 0.0 –3 –2 –1 0 1
4 FIG. 21. The coefficient χ5 þ 25 ϕ2ϕ3 ≡ ΔDW appearing in the expansion of the function d in Eq. (39), for solutions within the confining (solid black and short-dashed orange lines), skewed (dashed red line), and IR-conformal (longest-dashed purple line) classes. The left plot shows the parameters extracted by matching to the UV expansions. The right panel shows the same parameters after rescaling with the appropriate powers of Λ.
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APPENDIX G: FORMULATION OF THE FREE The boundary-localized potential term at ρ ¼ ρ2 reads ENERGY IN D = 5 DIMENSIONS Z Z pffiffiffiffiffiffi S˜ 4 −˜ λ˜ 4 4A λ˜ In this Appendix, we rewrite the same system of Sec. VA pot;2 ¼ d x gð 2Þ ¼ d xe ð 2Þ ; ðG5Þ in the language of a sigma model of two scalars ϕ and χ in ρ2 ρ2 5 D ¼ dimensions. We remind the reader that this is which by comparing to Eq. (109) implies that we must derived by assuming that none of the background fields ˜ −χ choose λ2 ≡ e λ2. depends on η and then performing dimensional reduction of In the five-dimensional language, even regular solutions the system. As detailed elsewhere [99], for the bulk action, in six dimensions may be singular—in the sense that the one finds that curvature singularity in D ¼ 5 dimensions is resolved by Z Z 1 ffiffiffiffiffiffiffiffi the lift to D ¼ 6 dimensions, which makes it more trans- S η S˜ 4 ∂ p− MN∂ χ bulk ¼ d bulk þ 2 d xdr Mð g5g N Þ ; parent to understand why we need to introduce the boundary at ρ ¼ ρ1. The resulting contributions to the ðG1Þ action are Z Z with pffiffiffiffiffiffi ˜ ˜ 4 K 4 4A−χ Z S 1 ¼ − d x −g˜ ¼ −2 d xe ∂ρA ; ffiffiffiffiffiffiffiffi GHY; 2 S˜ 4 p− ρ1 ρ1 bulk ¼ d xdr g5 ðG6Þ R5 1 Z Z × − G gMN∂ Φa∂ Φb − V ϕ;χ ; G2 pffiffiffiffiffiffi 4 2 ab M N ð Þ ð Þ S˜ − 4 −˜ λ˜ − 4 4A λ˜ pot;1 ¼ d x gð 1Þ ¼ d xe ð 1Þ ; ðG7Þ ρ1 ρ1 where the sigma-model metric is Gab ¼ diagð2; 6Þ in the −2χ ˜ −χ basis fϕ; χg, and the potential is V ¼ e V6. which again implies that λ1 ≡ e λ1. Hence, by just replacing the equations of motion, we find We notice how the GHY terms in the description in Z D ¼ 5 dimensions combine with the total derivative dis- 3 ρ2 ˜ S˜ − 4 ρ∂ 4A−χ∂ tinguishing S and S to yield exactly the GHY term of bulk ¼ d xd ρðe ρAÞ bulk bulk 8 ρ 6 Z 1 the formulation in D ¼ dimensions. Hence, we have now 1 ρ2 shown that we can match the two formulations of the − 4 ρ∂ 4A−χ∂ χ 2 d xd ρðe ρ Þ: ðG3Þ theory: ρ1 X S S S The boundary-localized Gibbons-Hawking-York (GHY) bulk þ GHY;i þ pot;i ρ ρ 7 1 2 term at ¼ 2 now reads Z i¼ ; Z Z X pffiffiffiffiffiffi ˜ η S˜ S˜ S˜ K ¼ d bulk þ ð GHY;i þ pot;iÞ : ðG8Þ S˜ 4 −˜ 2 4 4A−χ∂ GHY;2 ¼ d x g 2 ¼ d xe ρA : ðG4Þ i¼1;2 ρ2 ρ2 Note that matching the formulations in D ¼ 6 and D ¼ 5 7 The sign of the term proportional to ∂ρA is the opposite of that dimensions as in Eq. (G8) does not require making use of which is stated just after Eq. (2.23) of Ref. [99]. the equations of motion.
[1] G. Aad et al. (ATLAS Collaboration), Observation of a the Technidilaton on the Lattice, Phys. Rev. Lett. 113, new particle in the search for the Standard Model Higgs 082002 (2014). boson with the ATLAS detector at the LHC, Phys. Lett. B [4] M. Golterman and Y. Shamir, Low-energy effective action 716, 1 (2012). for pions and a dilatonic meson, Phys. Rev. D 94, 054502 [2] S. Chatrchyan et al. (CMS Collaboration), Observation (2016). of a new boson at a mass of 125 GeV with the [5] A. Kasai, K. i. Okumura, and H. Suzuki, A dilaton-pion CMS experiment at the LHC, Phys. Lett. B 716,30 mass relation, arXiv:1609.02264. (2012). [6] M. Golterman and Y. Shamir, Effective action for pions [3] S. Matsuzaki and K. Yamawaki, Dilaton Chiral Perturba- and a dilatonic meson, Proc. Sci., LATTICE2016 (2016) tion Theory: Determining the Mass and Decay Constant of 205.
106018-39 ELANDER, PIAI, and ROUGHLEY PHYS. REV. D 103, 106018 (2021)
[7] M. Hansen, K. Langaeble, and F. Sannino, Extending [30] C. N. Leung, S. T. Love, and W. A. Bardeen, Spontaneous chiral perturbation theory with an isosinglet scalar, Phys. symmetry breaking in scale invariant quantum electrody- Rev. D 95, 036005 (2017). namics, Nucl. Phys. B273, 649 (1986). [8] M. Golterman and Y. Shamir, Effective pion mass term and [31] W. A. Bardeen, C. N. Leung, and S. T. Love, The Dilaton the trace anomaly, Phys. Rev. D 95, 016003 (2017). and Chiral Symmetry Breaking, Phys. Rev. Lett. 56, 1230 [9] T. Appelquist, J. Ingoldby, and M. Piai, Dilaton EFT (1986). Framework For Lattice Data, J. High Energy Phys. 07 [32] K. Yamawaki, M. Bando, and K. i. Matumoto, Scale (2017) 035. Invariant Technicolor Model and a Technidilaton, Phys. [10] T. Appelquist, J. Ingoldby, and M. Piai, Analysis of a Rev. Lett. 56, 1335 (1986). dilaton EFT for lattice data, J. High Energy Phys. 03 [33] Y. Aoki et al. (LatKMI Collaboration), Light composite (2018) 039. scalar in eight-flavor QCD on the lattice, Phys. Rev. D 89, [11] M. Golterman and Y. Shamir, Large-mass regime of the 111502 (2014). dilaton-pion low-energy effective theory, Phys. Rev. D 98, [34] T. Appelquist et al., Strongly interacting dynamics and the 056025 (2018). search for new physics at the LHC, Phys. Rev. D 93, [12] O. Cata and C. Muller, Chiral effective theories with 114514 (2016). a light scalar at one loop, Nucl. Phys. B952, 114938 [35] Y. Aoki et al. (LatKMI Collaboration), Light flavor-singlet (2020). scalars and walking signals in Nf ¼ 8 QCD on the lattice, [13] T. Appelquist, J. Ingoldby, and M. Piai, Dilaton potential Phys. Rev. D 96, 014508 (2017). and lattice data, Phys. Rev. D 101, 075025 (2020). [36] A. D. Gasbarro and G. T. Fleming, Examining the low [14] O. Cat`a, R. J. Crewther, and L. C. Tunstall, Crawling energy dynamics of walking gauge theory, Proc. Sci., technicolor, Phys. Rev. D 100, 095007 (2019). LATTICE2016 (2017) 242. [15] T. V. Brown, M. Golterman, S. Krøjer, Y. Shamir, and K. [37] T. Appelquist et al. (Lattice Strong Dynamics Collabora- Splittorff, The ϵ-regime of dilaton chiral perturbation tion), Nonperturbative investigations of SU(3) gauge theory, Phys. Rev. D 100, 114515 (2019). theory with eight dynamical flavors, Phys. Rev. D 99, [16] A. A. Migdal and M. A. Shifman, Dilaton effective 014509 (2019). Lagrangian in gluodynamics, Phys. Lett. 114B, 445 [38] Z. Fodor, K. Holland, J. Kuti, D. Nogradi, C. Schroeder, (1982). and C. H. Wong, Can the nearly conformal sextet gauge [17] S. Coleman, Aspects of symmetry: Selected Erice lectures, model hide the Higgs impostor?, Phys. Lett. B 718, 657 https://doi.org/10.1017/CBO9780511565045. (2012). [18] W. D. Goldberger, B. Grinstein, and W. Skiba, Distinguish- [39] Z. Fodor, K. Holland, J. Kuti, S. Mondal, D. Nogradi, and ing the Higgs Boson from the Dilaton at the Large Hadron C. H. Wong, Toward the minimal realization of a light Collider, Phys. Rev. Lett. 100, 111802 (2008). composite Higgs, Proc. Sci., LATTICE2014 (2015) 244. [19] D. K. Hong, S. D. H. Hsu, and F. Sannino, Composite [40] Z. Fodor, K. Holland, J. Kuti, S. Mondal, D. Nogradi, and Higgs from higher representations, Phys. Lett. B 597,89 C. H. Wong, Status of a minimal composite Higgs theory, (2004). Proc. Sci., LATTICE2015 (2016) 219. [20] D. D. Dietrich, F. Sannino, and K. Tuominen, Light [41] Z. Fodor, K. Holland, J. Kuti, D. Nogradi, and C. H. Wong, composite Higgs from higher representations versus The twelve-flavor β-function and dilaton tests of the sextet electroweak precision measurements: Predictions for scalar, EPJ Web Conf. 175, 08015 (2018). CERN LHC, Phys. Rev. D 72, 055001 (2005). [42] Z. Fodor, K. Holland, J. Kuti, and C. H. Wong, Tantalizing [21] M. Hashimoto and K. Yamawaki, Techni-dilaton at Con- dilaton tests from a near-conformal EFT, Proc. Sci., formal Edge, Phys. Rev. D 83, 015008 (2011). LATTICE2018 (2019) 196. [22] T. Appelquist and Y. Bai, A Light Dilaton in Walking [43] Z. Fodor, K. Holland, J. Kuti, and C. H. Wong, Dilaton Gauge Theories, Phys. Rev. D 82, 071701 (2010). EFT from p-regime to RMT in the ϵ-regime, Proc. Sci., [23] L. Vecchi, Phenomenology of a light scalar: The dilaton, LATTICE2019 (2020) 246. Phys. Rev. D 82, 076009 (2010). [44] M. Golterman, E. T. Neil, and Y. Shamir, Application of [24] Z. Chacko and R. K. Mishra, Effective theory of a light dilaton chiral perturbation theory to Nf ¼ 8,SUð3Þ spec- dilaton, Phys. Rev. D 87, 115006 (2013). tral data, Phys. Rev. D 102, 034515 (2020). [25] B. Bellazzini, C. Csaki, J. Hubisz, J. Serra, and J. Terning, [45] J. M. Maldacena, The large N limit of superconformal field A Higgslike dilaton, Eur. Phys. J. C 73, 2333 (2013). theories and supergravity, Int. J. Theor. Phys. 38, 1113 [26] T. Abe, R. Kitano, Y. Konishi, K. y. Oda, J. Sato, and S. (1999); Adv. Theor. Math. Phys. 2, 231 (1998). Sugiyama, Minimal dilaton model, Phys. Rev. D 86, [46] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge 115016 (2012). theory correlators from noncritical string theory, Phys. [27] E. Eichten, K. Lane, and A. Martin, A Higgs impostor in Lett. B 428, 105 (1998). low-scale technicolor, arXiv:1210.5462. [47] E. Witten, Anti-de Sitter space and holography, Adv. [28] B. Bellazzini, C. Csaki, J. Hubisz, J. Serra, and J. Terning, Theor. Math. Phys. 2, 253 (1998). A naturally light dilaton and a small cosmological con- [48] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and stant, Eur. Phys. J. C 74, 2790 (2014). Y. Oz, Large N field theories, string theory and gravity, [29] P. Hernandez-Leon and L. Merlo, Distinguishing a Higgs- Phys. Rep. 323, 183 (2000). like dilaton scenario with a complete bosonic effective [49] M. Bianchi, D. Z. Freedman, and K. Skenderis, Holo- field theory basis, Phys. Rev. D 96, 075008 (2017). graphic renormalization, Nucl. Phys. B631, 159 (2002).
106018-40 DILATONIC STATES NEAR HOLOGRAPHIC PHASE … PHYS. REV. D 103, 106018 (2021)
[50] K. Skenderis, Lecture notes on holographic renormaliza- [72] D. Elander and M. Piai, On the glueball spectrum of tion, Classical Quantum Gravity 19, 5849 (2002). walking backgrounds from wrapped-D5 gravity duals, [51] I. Papadimitriou and K. Skenderis, AdS=CFT correspon- Nucl. Phys. B871, 164 (2013). dence and geometry, IRMA Lect. Math. Theor. Phys. 8,73 [73] D. Elander, Light scalar from deformations of the (2005). Klebanov-Strassler background, Phys. Rev. D 91, 126012 [52] W. D. Goldberger and M. B. Wise, Modulus Stabilization (2015). with Bulk Fields, Phys. Rev. Lett. 83, 4922 (1999). [74] P. Candelas and X. C. de la Ossa, Comments on conifolds, [53] O. DeWolfe, D. Z. Freedman, S. S. Gubser, and A. Karch, Nucl. Phys. B342, 246 (1990). Modeling the fifth-dimension with scalars and gravity, [75] A. H. Chamseddine and M. S. Volkov, NonAbelian BPS Phys. Rev. D 62, 046008 (2000). Monopoles in N ¼ 4 Gauged Supergravity, Phys. Rev. [54] W. D. Goldberger and M. B. Wise, Phenomenology of a Lett. 79, 3343 (1997). stabilized modulus, Phys. Lett. B 475, 275 (2000). [76] I. R. Klebanov and E. Witten, Superconformal field theory [55] C. Csaki, M. L. Graesser, and G. D. Kribs, Radion dy- on three-branes at a Calabi-Yau singularity, Nucl. Phys. namics and electroweak physics, Phys. Rev. D 63, 065002 B536, 199 (1998). (2001). [77] I. R. Klebanov and M. J. Strassler, Supergravity and a [56] N. Arkani-Hamed, M. Porrati, and L. Randall, Holography confining gauge theory: Duality cascades and chi SB and phenomenology, J. High Energy Phys. 08 (2001) 017. resolution of naked singularities, J. High Energy Phys. [57] R. Rattazzi and A. Zaffaroni, Comments on the holo- 08 (2000) 052. graphic picture of the Randall-Sundrum model, J. High [78] J. M. Maldacena and C. Nunez, Towards the Large N Limit Energy Phys. 04 (2001) 021. of Pure N ¼ 1 SuperYang-Mills, Phys. Rev. Lett. 86, 588 [58] L. Kofman, J. Martin, and M. Peloso, Exact identification (2001). of the radion and its coupling to the observable sector, [79] A. Butti, M. Grana, R. Minasian, M. Petrini, and A. Phys. Rev. D 70, 085015 (2004). Zaffaroni, The Baryonic branch of Klebanov-Strassler [59] D. Elander and M. Piai, A composite light scalar, electro- solution: A supersymmetric family of SU(3) structure weak symmetry breaking and the recent LHC searches, backgrounds, J. High Energy Phys. 03 (2005) 069. Nucl. Phys. B864, 241 (2012). [80] V. Gorbenko, S. Rychkov, and B. Zan, Walking, Weak [60] D. Kutasov, J. Lin, and A. Parnachev, Holographic walking first-order transitions, and Complex CFTs, J. High Energy from tachyon DBI, Nucl. Phys. B863, 361 (2012). Phys. 10 (2018) 108. [61] R. Lawrance and M. Piai, Holographic technidilaton [81] V. Gorbenko, S. Rychkov, and B. Zan, Walking, Weak first- and LHC searches, Int. J. Mod. Phys. A 28, 1350081 order transitions, and Complex CFTs II. Two-dimensional (2013). Potts model at Q>4, SciPost Phys. 5, 050 (2018). [62] D. Elander and M. Piai, The decay constant of the [82] D. B. Kaplan, J. W. Lee, D. T. Son, and M. A. Stephanov, holographic techni-dilaton and the 125 GeV boson, Nucl. Conformality Lost, Phys. Rev. D 80, 125005 (2009). Phys. B867, 779 (2013). [83] P. Breitenlohner and D. Z. Freedman, Stability in gauged [63] M. Goykhman and A. Parnachev, S-parameter, technime- extended supergravity, Ann. Phys. (N.Y.) 144, 249 (1982). sons, and phase transitions in holographic tachyon DBI [84] A. G. Cohen and H. Georgi, Walking beyond the rainbow, models, Phys. Rev. D 87, 026007 (2013). Nucl. Phys. B314, 7 (1989). [64] N. Evans and K. Tuominen, Holographic modelling of a [85] S. J. Rey and J. T. Yee, Macroscopic strings as heavy light technidilaton, Phys. Rev. D 87, 086003 (2013). quarks in large N gauge theory and anti-de Sitter super- [65] E. Megias and O. Pujolas, Naturally light dilatons from gravity, Eur. Phys. J. C 22, 379 (2001). nearly marginal deformations, J. High Energy Phys. 08 [86] J. M. Maldacena, Wilson Loops in Large N Field Theories, (2014) 081. Phys. Rev. Lett. 80, 4859 (1998). [66] D. Elander, R. Lawrance, and M. Piai, Hyperscaling [87] Y. Kinar, E. Schreiber, and J. Sonnenschein, Q anti-Q violation and electroweak symmetry breaking, Nucl. Phys. potential from strings in curved space-time: Classical B897, 583 (2015). results, Nucl. Phys. B566, 103 (2000). [67] A. Pomarol, O. Pujolas, and L. Salas, Holographic [88] A. Brandhuber and K. Sfetsos, Wilson loops from multi- conformal transition and light scalars, J. High Energy center and rotating branes, mass gaps and phase structure Phys. 10 (2019) 202. in gauge theories, Adv. Theor. Math. Phys. 3, 851 (1999). [68] D. Elander and M. Piai, Calculable mass hierarchies and a [89] S. D. Avramis, K. Sfetsos, and K. Siampos, Stability of light dilaton from gravity duals, Phys. Lett. B 772, 110 strings dual to flux tubes between static quarks in N ¼ 4 (2017). SYM, Nucl. Phys. B769, 44 (2007). [69] D. Elander and M. Piai, Glueballs on the baryonic branch [90] C. Nunez, M. Piai, and A. Rago, Wilson loops in string of Klebanov-Strassler: Dimensional deconstruction and duals of walking and flavored systems, Phys. Rev. D 81, a light scalar particle, J. High Energy Phys. 06 (2017) 086001 (2010). 003. [91] A. F. Faedo, M. Piai, and D. Schofield, On the stability of [70] C. Nunez, I. Papadimitriou, and M. Piai, Walking dynam- multiscale models of dynamical symmetry breaking from ics from string duals, Int. J. Mod. Phys. A 25, 2837 (2010). holography, Nucl. Phys. B880, 504 (2014). [71] D. Elander, C. Nunez, and M. Piai, A light scalar from [92] C. K. Wen and H. X. Yang, QCD(4) glueball masses from walking solutions in gauge-string duality, Phys. Lett. B AdS(6) black hole description, Mod. Phys. Lett. A 20, 997 686, 64 (2010). (2005).
106018-41 ELANDER, PIAI, and ROUGHLEY PHYS. REV. D 103, 106018 (2021)
[93] S. Kuperstein and J. Sonnenschein, Non-critical, near [114] P. Karndumri, Gravity duals of 5D N ¼ 2 SYM theory extremal AdS(6) background as a holographic laboratory from Fð4Þ gauged supergravity, Phys. Rev. D 90, 086009 of four dimensional YM theory, J. High Energy Phys. 11 (2014). (2004) 026. [115] C. M. Chang, M. Fluder, Y. H. Lin, and Y. Wang, Romans [94] E. Witten, Anti-de Sitter space, thermal phase transition, supergravity from five-dimensional holograms, J. High and confinement in gauge theories, Adv. Theor. Math. Energy Phys. 05 (2018) 039. Phys. 2, 505 (1998). [116] M. Gutperle, J. Kaidi, and H. Raj, Mass deformations of 5d [95] S. S. Gubser, Curvature singularities: The good, the bad, SCFTs via holography, J. High Energy Phys. 02 (2018) and the naked, Adv. Theor. Math. Phys. 4, 679 (2000). 165. [96] H. Samtleben, Lectures on gauged supergravity and flux [117] M. Suh, Supersymmetric AdS6 black holes from F(4) compactifications, Classical Quantum Gravity 25, 214002 gauged supergravity, J. High Energy Phys. 01 (2019) (2008). 035. [97] D. Z. Freedman and A. Van Proeyen, Supergravity, https:// [118] M. Suh, Supersymmetric AdS6 black holes from matter inspirehep.net/literature/1123253. coupled Fð4Þ gauged supergravity, J. High Energy Phys. [98] D. Elander, A. F. Faedo, C. Hoyos, D. Mateos, and M. Piai, 02 (2019) 108. Multiscale confining dynamics from holographic RG [119] N. Kim and M. Shim, Wrapped brane solutions in romans flows, J. High Energy Phys. 05 (2014) 003. Fð4Þ gauged supergravity, Nucl. Phys. B951, 114882 [99] D. Elander, M. Piai, and J. Roughley, Holographic glue- (2020). balls from the circle reduction of Romans supergravity, [120] K. Chen and M. Gutperle, Holographic line defects J. High Energy Phys. 02 (2019) 101. in F(4) gauged supergravity, Phys. Rev. D 100, 126015 [100] L. J. Romans, The F(4) gauged supergravity in six-dimen- (2019). sions, Nucl. Phys. B269, 691 (1986). [121] C. Hoyos, N. Jokela, and D. Logares, Scattering length in [101] L. J. Romans, Massive N=2a supergravity in ten-dimen- holographic confining theories, Phys. Rev. D 102, 086006 sions, Phys. Lett. 169B, 374 (1986). (2020). [102] A. Brandhuber and Y. Oz, The D-4–D-8 brane system and [122] M. Bianchi, M. Prisco, and W. Mueck, New results on five-dimensional fixed points, Phys. Lett. B 460, 307 holographic three point functions, J. High Energy Phys. 11 (1999). (2003) 052. [103] M. Cvetic, H. Lu, and C. N. Pope, Gauged Six- [123] M. Berg, M. Haack, and W. Mueck, Bulk dynamics in Dimensional Supergravity from Massive Type IIA, Phys. confining gauge theories, Nucl. Phys. B736, 82 (2006). Rev. Lett. 83, 5226 (1999). [124] M. Berg, M. Haack, and W. Mueck, Glueballs vs gluino- [104] J. Hong, J. T. Liu, and D. R. Mayerson, Gauged six- balls: Fluctuation spectra in Non-AdS/Non-CFT, Nucl. dimensional supergravity from warped IIB reductions, Phys. B789, 1 (2008). J. High Energy Phys. 09 (2018) 140. [125] D. Elander, Glueball spectra of SQCD-like theories, J. High [105] J. Jeong, O. Kelekci, and E. O. Colgain, An alternative IIB Energy Phys. 03 (2010) 114. embedding of F(4) gauged supergravity, J. High Energy [126] D. Elander and M. Piai, Light scalars from a compact fifth Phys. 05 (2013) 079. dimension, J. High Energy Phys. 01 (2011) 026. [106] R. D’Auria, S. Ferrara, and S. Vaula, Matter coupled F(4) [127] D. Elander, M. Piai, and J. Roughley, Probing the supergravity and the AdS(6)/CFT(5) correspondence, J. holographic dilaton, J. High Energy Phys. 06 (2020) High Energy Phys. 10 (2000) 013. 177. [107] L. Andrianopoli, R. D’Auria, and S. Vaula, Matter coupled [128] C. Csaki, J. Erlich, T. J. Hollowood, and J. Terning, F(4) gauged supergravity Lagrangian, J. High Energy Holographic RG and cosmology in theories with quasilo- Phys. 05 (2001) 065. calized gravity, Phys. Rev. D 63, 065019 (2001). [108] M. Nishimura, Conformal supergravity from the [129] A. Brandhuber and K. Sfetsos, Current correlators in the AdS=CFT correspondence, Nucl. Phys. B588, 471 (2000). Coulomb branch of N ¼ 4 SYM, J. High Energy Phys. 12 [109] S. Ferrara, A. Kehagias, H. Partouche, and A. Zaffaroni, (2000) 014. AdS(6) interpretation of 5-D superconformal field theories, [130] A. Brandhuber and K. Sfetsos, Current coreelators and Phys. Lett. B 431, 57 (1998). AdS=CFT away from the conformal point, arXiv:hep-th/ [110] U. Gursoy, C. Nunez, and M. Schvellinger, RG flows from 0204193. spin(7), CY 4 fold and HK manifolds to AdS, Penrose [131] N. Bobev, H. Elvang, D. Z. Freedman, and S. S. Pufu, limits and pp waves, J. High Energy Phys. 06 (2002) 015. Holography for N ¼ 2 on S4, J. High Energy Phys. 07 [111] C. Nunez, I. Y. Park, M. Schvellinger, and T. A. Tran, (2014) 001. Supergravity duals of gauge theories from F(4) gauged [132] F. Bigazzi, A. Caddeo, A. L. Cotrone, and A. Paredes, Fate supergravity in six-dimensions, J. High Energy Phys. 04 of false vacua in holographic first-order phase transitions, (2001) 025. J. High Energy Phys. 12 (2020) 200. [112] P. Karndumri, Holographic RG flows in six dimensional F [133] A. F. Faedo, M. Piai, and D. Schofield, Gauge/gravity (4) gauged supergravity, J. High Energy Phys. 01 (2013) dualities and bulk phase transitions, Phys. Rev. D 89, 134; Erratum, J. High Energy Phys. 06 (2015) 165. 106001 (2014). [113] Y. Lozano, E. O. Colgain, D. Rodriguez-Gomez, and K. [134] N. Seiberg, Five-dimensional SUSY field theories, non- Sfetsos, Supersymmetric AdS6 via T Duality, Phys. Rev. trivial fixed points and string dynamics, Phys. Lett. B 388, Lett. 110, 231601 (2013). 753 (1996).
106018-42 DILATONIC STATES NEAR HOLOGRAPHIC PHASE … PHYS. REV. D 103, 106018 (2021)
[135] D. R. Morrison and N. Seiberg, Extremal transitions and [139] O. Aharony and A. Hanany, Branes, superpotentials and five-dimensional supersymmetric field theories, Nucl. superconformal fixed points, Nucl. Phys. B504, 239 Phys. B483, 229 (1997). (1997). [136] M. R. Douglas, S. H. Katz, and C. Vafa, Small instantons, [140] O. Aharony, A. Hanany, and B. Kol, Webs of (p,q) five- Del Pezzo surfaces and type I-prime theory, Nucl. Phys. branes, five-dimensional field theories and grid diagrams, B497, 155 (1997). J. High Energy Phys. 01 (1998) 002. [137] O. J. Ganor, D. R. Morrison, and N. Seiberg, Branes, [141] O. DeWolfe, A. Hanany, A. Iqbal, and E. Katz, Five- Calabi-Yau spaces, and toroidal compactification of the branes, seven-branes and five-dimensional E(n) field N ¼ 1 six-dimensional E(8) theory, Nucl. Phys. B487,93 theories, J. High Energy Phys. 03 (1999) 006. (1997). [142] P. Benetti Genolini, M. Honda, H. C. Kim, D. Tong, and C. [138] K. A. Intriligator, D. R. Morrison, and N. Seiberg, Five- Vafa, Evidence for a non-supersymmetric 5d CFT from dimensional supersymmetric gauge theories and degener- deformations of 5d SUð2Þ SYM, J. High Energy Phys. 05 ations of Calabi-Yau spaces, Nucl. Phys. B497, 56 (1997). (2020) 058.
106018-43