Illustrated study of the semi-holographic non-perturbative framework Souvik Banerjee,1,∗ Nava Gaddam,2,† and Ayan Mukhopadhyay3,4,‡ 1Department of Physics and Astronomy, Uppsala University, SE-751 08 Uppsala, Sweden 2Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena, Princetonplein 5, Utrecht University, 3584 CC Utrecht, The Netherlands 3Institut für Theoretische Physik, Technische Universität Wien, Wiedner Hauptstr. 8-10, A-1040 Vienna, Austria 4CERN, Theoretical Physics Department, 1211 Geneva 23, Switzerland Semi-holography has been proposed as an effective nonperturbative framework which can 7 combine perturbative and nonperturbative effects consistently for theories like QCD. It is 1 0 postulated that the strongly coupled nonperturbative sector has a holographic dual in the 2 form of a classical gravity theory in the large N limit, and the perturbative fields determine r a the gravitational boundary conditions. In this work, we pursue a fundamental derivation of M this framework particularly showing how perturbative physics by itself can determine the 3 holographic dual of the infrared, and also the interactions between the perturbative and the 1 holographicsectors. Wefirstlydemonstratethattheinteractionsbetweenthetwosectorscan ] h be constrained through the existence of a conserved local energy-momentum tensor for the t - p fullsystemuptohard-softcouplingconstants. Asanillustration,wesetupabi-holographic e h toy theory where both the UV and IR sectors are strongly coupled and holographic with [ distinct classical gravity duals. In this construction, the requirement that an appropriate 3 v gluing can cure the singularities (geodetic incompletenesses) of the respective geometries 9 2 leads us to determine the parameters of the IR theory and the hard-soft couplings in terms 2 of those of the UV theory. The high energy scale behaviour of the hard-soft couplings is 1 0 state-independent but their runnings turn out to be state-dependent. We discuss how our . 1 approach can be adapted to the construction of the semi-holographic framework for QCD. 0 7 1 PACS numbers: 11.15.Tk,11.25.Tq,11.10.Gh,04.20.Dw : v i X r a ∗ [email protected] † [email protected] ‡ [email protected] 2 CONTENTS I. Introduction 3 II. Democratising semiholography 5 A. A brief review 5 B. Why and how should we democratise semi-holography? 8 III. Coupling the hard and soft sectors 12 A. Simple scalar couplings 13 B. More general scalar couplings 16 C. Tensorial couplings 18 D. Combining general scalar and tensorial couplings 19 IV. A bi-holographic illustration 22 A. A useful reconstruction theorem 25 B. The bi-holographic vacuum 27 1. The UV domain 27 2. The IR domain 29 3. Gluing and determining the full theory 30 C. Excited states 36 D. The highly efficient RG flow perspective 39 V. Concluding remarks 41 A. How to proceed in the case of QCD? 41 B. Possible applications of bi-holography 43 Acknowledgments 44 References 44 3 I. INTRODUCTION Semi-holography has been recently proposed as an effective framework in which one can include both perturbative and non-perturbative effects consistently in a wide range of energy scales. It’s present formulation is targeted towards an effective description of asymptotically free theories like QCD which are weakly coupled in the ultraviolet but strongly interacting in the infrared. It is assumed that in the large N limit (i) the infrared non-perturbative effects such as confinement and chiral symmetry breaking can be obtained from a holographic dual description in the form of an appropriate classical theory of gravity1, and (ii) the perturbative degrees of freedom determine the effective background metric, relevant and marginal couplings, and background gauge-fields (coupling to conserved currents) in which the emergent infrared holographic degrees of freedom live. The second assertion then implies that the perturbative degrees of freedom determine the leading asymptotic behaviour of the classical gravity fields forming the holographic dual of the non-perturbative sector. As we will argue, such a set-up allows for only a few effective parameters in a wide range of energy scales. Concrete phenomenological semi-holographic models with a small number of effective parameters have been proposed for some non-Fermi liquid systems [4–9], and for the quark-gluon plasma (QGP) formed in heavy-ion collisions [10, 11]. In such instances, indeed both perturbative and non-perturbative effects are phenomenologically relevant. Inthisarticle,wewilltakefirststepstowardsaderivationofthegeneralsemi-holographicframe- work from first principles, i.e. from the fundamental theory describing the microscopic dynamics. This amounts to answering the following questions: • Which principles tell us how the perturbative degrees of freedom determine the leading asymptotic behaviours of the gravitational fields forming the holographic dual of the non- perturbative sector? • How do we find the appropriate classical gravity theory which provides the dual holographic description of the non-perturbative sector? We will arrive at partial answers to both these questions, and also illustrate the full construction of semi-holography with a toy example. Previously, semi-holography has been conceived of as an effective simplified method for solving low energy holographic dynamics where the asymptotic geometry determining model-dependent 1 This was demonstrated in the Witten-Sakai-Sugimoto top-down model [1–3] obtained from string theory. We do notassumeherethattheholographicdescriptionofthenon-perturbativesectorcanbeembeddedinstringtheory. 4 features is replaced by simple boundary dynamics which couples to the near-horizon geometry controlling universal scaling exponents [4, 12]. It has also been argued that decoding holography as a form of non-Wilsonian RG flow which preserves Ward identities for single-trace operators (like the energy-momentum tensor) and can self-determine microscopic data via appropriate infrared endpoint conditions naturally gives rise to a more general semi-holographic framework in which the ultravioletcanbeasymptoticallyfreesothatitisdescribedbyperturbativequantumfielddynamics rather than by a classical gravity theory [13–15]. In this article, we will deal with the fundamental aspects of construction of the general semi-holographic framework (which may not be embeddable in string theory as we know of it today) by understanding what constrains it structurally and illustrate it with a toy example. The organisation of the paper is as follows. In Section II, we will review the present formulation of semi-holography and then argue for the need for generalising it in order for it to be an effective theory in a wide range of energy scales. In particular, we will advocate that we need a more democratic formulation where we do not give precedence to either the perturbative or to the non- perturbative degrees of freedom. Although we will call the perturbative sector as the ultraviolet sector and non-perturbative sector as the infrared sector, it is to be noted that non-perturbative effects are present even at high energy scales although these are suppressed. In principle, both sectors contribute at any energy scale although one of the sectors may give dominant contributions at a specific energy scale. Since semi-holography is a framework that is constructed at intermediate energy scales, it better treats both the ultraviolet and infrared sectors, or rather the perturbative and non-perturbative sectors in a democratic manner. Eventually the parameters of the non- perturbative sector should be determined (perhaps not always uniquely) by the perturbative sector or vice versa. We will argue this democratic formulation is actually necessary since otherwise we cannot perform non-perturbative renormalisation of the effective parameters. We will also sketch how the democratic formulation should work. In Section III, we will show how the requirement that there exists a local and conserved energy- momentum tensor constrains the effective parameters and semi-holographic coupling between the perturbative and the non-perturbative sectors. Thus we will realise a concrete democratic formu- lation of semi-holography at arbitrary energy scales. In Section IV, we will illustrate the construction of semi-holography with a bi-holographic toy model in which the perturbative UV dynamics of semi-holography will be replaced by a strongly coupled holographic theory that admits a classical gravity description on its own. The infrared sector will be even more strongly coupled and also holographic. We will explicitly demonstrate the 5 following features. • Some simple consistency conditions can determine the hard-soft couplings between the two sectors and the parameters of the IR theory as functions of the parameters of the UV theory. • The behaviour of the hard-soft couplings in the limit Λ → ∞ is state-independent and can be obtained from the construction of the vacuum state. However, the running of the hard-soft couplings with the scale is state-dependent. • The parameters defining the holographic IR classical gravity theory is fixed once and for all through the construction of the vacuum state of the full theory. However, the gravitational fields of this IR classical gravity theory undergo state-dependent field redefinitions in excited states. • The UV and IR classical gravity theories are both sick in the sense that the respective geometries have edge singularities (not naked curvature singularities though) arising from geodetic incompleteness. The possibility of smooth gluing of their respective edges that removes the singularities in both plays a major role in determining the full theory. We will also examine how we can define RG flow in the bi-holographic theory. In Section V, we will indicate how the steps of the construction of the bi-holographic toy the- ory can be applied also to the construction of the semi-holographic framework for QCD and also discuss the complications involved. Finally, we will conclude with discussions on the potential phenomenological applications of the bi-holographic framework. II. DEMOCRATISING SEMIHOLOGRAPHY A. A brief review Let us begin by sketching a first construction of a semi-holographic model for pure large N QCD based on a similar model [10, 11, 16] for the quark-gluon plasma (QGP) formed in heavy-ion collisions. The effective action for pure large N QCD at a scale Λ can be proposed to be: SQCD[Aa,Λ] = SpQCD[Aa,Λ] µ µ (cid:104) (cid:105) +WhQCD g˜ [Aa,Λ],δg˜ [Aa,Λ],θ˜[Aa,Λ] , (1) µν µ YM µ µ 6 where the exact Wilsonian effective action of QCD denoted as SQCD at a scale Λ is composed of two parts: (i) the perturbative QCD effective action SpQCD at the scale Λ obtained from Feynman diagrams, and (ii) non-perturbative terms (leading to confinement) which cannot be obtained from Feynman diagrams but can be described by an emergent holographic strongly coupled QCD-like theory. The latter part of the full action is then given by WhQCD, the generating functional of the connected correlation functions of the emergent strongly coupled holographic QCD-like theory whose marginal couplings—namely g˜ and θ˜ (or rather, their expansions around infinity and YM zero respectively) and the effective background metric g˜ in which it lives are functionals of the µν perturbative gauge fields Aa and the scale Λ. In order that a holographic theory can capture non- µ perturbative effects at even high energy scales, it must have a large number of fields as we will discuss in Section V. Nevertheless, at high energy scales the non-perturbative contributions are insignificant. We will argue that the semi-holographic construction can be useful at intermediate energy scales where the non-perturbative effects can also be captured by a few gravitational field via holography to a good degree of approximation. It is to be noted that WhQCD should be defined with an appropriate vacuum subtraction so that it vanishes when the modifications in the couplings δg˜ and θ˜vanish, and when2 g˜ = η . YM µν µν Asymptotically when Λ → ∞, the sources for the emergent holographic QCD are also expected to vanish, so that the full action receives perturbative contributions almost exclusively. In the infra-red, however, the holographic contributions are expected to dominate. In the large N limit, the emergent holographic QCD is expected to be described by a classical gravitational theory. Therefore, (cid:104) (cid:105) (cid:104) WhQCD g˜ [Aa],δg˜ [Aa],θ˜ [Aa] = Son−shell g˜ = g(b), µν µ YM µ YM µ grav µν µν (cid:105) δg˜ = φ(b),θ˜= χ(b) , (2) YM i.e. WhQCD is to be identified with the on-shell action Son−shell of an appropriate five-dimensional grav classicalgravitytheoryconsistingofat least threefields,namelythemetricG ,thedilatonΦand MN the axion X. Furthermore, the leading behaviour of the bulk metric is given by its identification with the boundary metric g(b), whilst δg˜ is identified with the boundary value φ(b) of the bulk µν YM dilaton Φ and θ˜ is identified with the boundary value χ(b) of the bulk axion X. For reasons that 2 Moregenerally,thesubtractionshouldensurethatWhQCD vanisheswheng˜ isidenticaltothefixedbackground µν metric where all the degrees of freedom live. 7 will soon be elucidated, one may now postulate that: 2 δSpQCD[Aa,Λ] g(b) = η +γtpQCD, with tpQCD = √ µ | (3a) µν µν µν µν −g δg gµν=ηµν µν δSpQCD[Aa,Λ] φ(b) = βhpQCD, with hpQCD = µ , (3b) δgYM[Λ] δSpQCD[Aa,Λ] χ(b) = αapQCD, with apQCD = µ . (3c) δθ Thecouplingsα, β andγ havebeencalledhard-softcouplings. Theseofcoursecannotbenewinde- pendent parameters, but rather of the functional forms (1/Λ4)f(Λ /Λ) which should be derived QCD from first principles. Furthermore, f(0) must be finite so that the non-perturbative contributions to the full action vanish in the limit Λ → ∞ reproducing asymptotic freedom. If SpQCD were just the classical Yang-Mills action, then [11]: (cid:18) (cid:19) 1 1 tpQCD = tr F F α− η F Fαβ , µν N µα ν 4 µν αβ c 1 (cid:16) (cid:17) hpQCD = tr F Fαβ , αβ 4N c 1 (cid:16) (cid:17) apQCD = tr F F˜αβ . (4) αβ 4N c It can be readily shown that in consistency with the variational principle, the full semi-holographic action (1) can be written in the following form: (cid:90) (cid:90) (cid:90) 1 S[Aa,Λ] = SpQCD[Aa,Λ]+ d4xTµνg(b)+ d4xHφ(b)+ d4xAχ(b), (5) µ µ 2 µν where δSon−shell δSon−shell δSon−shell µν grav grav grav T = 2 , H = , A = , (6) δg(b) δφ(b) δχ(b) µν are the self-consistent expectation values of the holographic operators which are non-linear and non-local functionals of the sources3 g(b), φ(b) and χ(b). µν The reason for postulating the hard-soft interactions to be of the forms given by equations (3a) to (3c) can now be readily explained. We need to solve for the full dynamics in an iterative fashion (assuming that the iteration indeed converges). This means that the dynamics of the perturbative sector is modified by the holographic operators which appear as self-consistent mean fields as in 3 It is assumed here that the full theory lives in flat Minkowski space with metric η . It is easy to generalise the µν construction to any metric on which the full degrees of freedom live. For details, see [11]. 8 (5). The holographic operators are in turn obtained by solving classical gravity equations with sources given by (3a), (3b) and (3c), which are determined by the perturbative gauge fields. It should therefore be guaranteed that both sectors must be solvable at each step in the iteration including perturbative quantum effects. Therefore, both the perturbative and non-perturbative sectors should be renormalizable at each step of the iteration so that one can solve for the dynamics of both without introducing any new coupling. The modified perturbative action (5) in the limit Λ → ∞ is indeed a marginal deformation of the standard perturbative QCD action since the added terms involve tcl , hcl and acl which are all possible (scalar and tensor) gauge-invariant µν operators of mass dimension four. Furthermore, this is also why the gravitational theory contains sources for only the dimension four operators as in (2), despite the possible existence of many other(massive)gravitationalfieldsgenerating(non-perturbative)condensatesofhigherdimensional operators without additional sources. Finally, it is important to reiterate that the importance of the hard-soft couplings given in (3a), (3b) and (3c) relies on the emergence of an intermediate scale Λ > Λ between the energy I QCD scales where we can rely exclusively on either perturbative QCD or chiral Lagrangian effective field theories which can be reproduced from holographic models such as [1–3]. This intermediate scale Λ is most likely where the perturbative gauge coupling is of order unity but not too large. I At this intermediate scale itself, the hard-soft couplings should give significant modifications to perturbative dynamics. In the context of an application to QGP, SpQCD can be replaced by the glasma effective action, i.e. a classical Yang-Mills action for the small-x saturated gluons (which form a weakly coupled over-occupied system) with colour sources provided by the large-x4 (x > x ) gluons (frozen on 0 the time-scale of the collisions) with a distribution function whose evolution with the cut-off x 0 can be followed perturbatively [17–19]. In this case, one can show that (i) the full system has a well-defined local energy momentum tensor that is conserved in flat space, and (ii) the iterative method of solving the full dynamics indeed converges at least in some simple test cases [11]. B. Why and how should we democratise semi-holography? The present formulation of semi-holography as discussed above makes two central assumptions which are: 1. The perturbative sector and the holographic theory dual to the non-perturbative sector are 4 Here x denotes the fraction of hadronic longitudinal momentum carried by the partonic gluon. 9 both deformed marginally with scalar and tensorial couplings which are functionals of the operators of the other sector. 2. The full action (as for instance (1) in the case of pure large-N QCD) can be written as a functional of the perturbative fields only.5 In what follows, we will argue that if semi-holography needs to work in the sense of a non- perturbative effective framework, we must replace the second assumption with the simple assertion that: • The full theory must have a conserved and local energy-momentum tensor which can be constructed without the need to know the explicit Lagrangian descriptions of the effective ultraviolet or infrared dynamics. It is to be noted that the formulation discussed above already leads to a local and conserved energy-momentum tensor of the combined system which can be constructed explicitly. As shown in [11], this can be obtained from the complete action (1) by differentiating it with respect to the fixed background g on which the full system lives. One can prove that this energy-momentum µν tensor is conserved in the fixed background g when the full system is solved, i.e when the Ward µν identity of the CFT in the dynamical background g˜ and the modified dynamical equations of the µν perturbative sector are satisfied simultaneously. The crucial point is that we must not insist that we can derive the full energy-momentum tensor from an action such as (1) which is a functional of the ultraviolet fields only. We will argue that such a demand automatically arises from a democratic formulation of semi-holography where the full energy-momentum tensor can be constructed directly from the Ward identities of the perturbative and non-perturbative sectors in respective background metrics and sources that are determined by the operators of the other sector. This full energy- momentum tensor should be conserved locally in the fixed background metric where the full system lives. We lay out our reasons for advocating democratic formulation of semi-holography. The first one is somewhat philosophical. In the case we are assuming that the strongly coupled non-perturbative sectorisdescribedbyaholographically dual classicalgravitytheory, wearenotmakinganyexplicit assumption of the Lagrangian description of this dual holographic theory. It could be so in certain 5 NotethatWhQCDin(1)isafterallafunctionalofthesourcesalonewhichinturnarefunctionalsoftheperturbative gaugefields. Furthermore,thetermperturbativefields mayalsoseemproblematicbecauseaftertheircouplingwith the non-perturbative sector, these do not remain strictly perturbative. Nevertheless, if we solve the two sectors via iteration as discussed before, we can treat the dynamics of these fields perturbatively (with self-consistently modified couplings) at each step of the iteration as discussed before. 10 situations we do not know the explicit Lagrangian description of the perturbative sector as well.6 We should be able to formulate an explicit semi-holographic construction in such a situation also. Furthermore, the basic idea of semi-holography is to take advantage of dualities. It is quite possible that the ultraviolet is strongly coupled instead of the infrared and we should take advantage of a weakly coupled (perhaps holographic) dual description of the ultraviolet. In that case, the original Lagrangian description of the UV even if known will not be useful. Therefore, we better have a broaderconstructionwhichcanworkwithouttheneedofknowinganexplicitLagrangiandescription of the perturbative sector or that of the holographic theory dual to the non-perturbative sector. This implies we need to treat both on equal footing. The second reason for advocating democratic formulation is more fundamental. Let us take the example where in the infrared the theory flows to a strongly coupled holographic conformal field theory (IR-CFT) from a weakly coupled fixed point (UV-CFT) in the ultraviolet. There will be a specific UV-IR operator map in such a theory relating operators in the UV fixed point to those defined at the IR fixed point via scale-evolution as explicitly known in the case of RG flows between minimal-model two-dimensional conformal field theories [21–23]. Typically a relevant operator in the UV will flow to an irrelevant operator in the IR. In fact the entire flow will be generated by a relevant deformation of the UV fixed point – the operator(s) generating such a deformation will become irrelevant at the IR fixed point. Let the UV deformation be due to a coupling constant g multiplying a relevant operator OUV which will flow to an irrelevant operator OIR of the strongly coupled holographic IR-CFT that is represented by a bulk field Φ. A naive way to formulate semi-holography in such a case will be: (cid:90) S = SUV−CFT+g d4xOUV +Sgrav[φ(b) = λOUV], (7) where φ(b) is the coefficient of the leading asymptotic term of the bulk field Φ, λ is a dimensionful hard-soft coupling constant and we have suppressed the scale dependence. The immediate problem is that we have turned on an irrelevant deformation of the dual holographic theory as (cid:90) Sgrav[φ(b) = λOUV] = λ d4xOUVOIR (8) representsanirrelevantdeformationoftheIR-CFTsinceOIR isanirrelevantoperator. Thiscontra- dicts our assumption that both sectors should be renormalizable after being mutually deformed by 6 Note that by assuming that we have a weakly coupled perturbative description we do not necessarily commit ourselves to a Lagrangian description also. Such a perturbation series can be obtained by a chain of dualities without a known Lagrangian description as in the case of some quiver gauge theories [20].