Towards a Holographic Model of the QCD Phase Diagram
aa r X i v : . [ h e p - t h ] A p r SHEP-11-26
Towards a Holographic Model of the QCD Phase Diagram
Nick Evans, ∗ Astrid Gebauer, † and Maria Magou ‡ School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK
Keun-Young Kim § School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK andInstitute for Theoretical Physics, University of Amsterdam, Science Park 904,Postbus 94485, 1090 GL Amsterdam, The Netherlands
We describe the temperature-chemical potential phase diagrams of holographic models of a rangeof strongly coupled gauge theories that display chiral symmetry breaking/restoration transitions.The models are based on the D3/probe-D7 system but with a phenomenologically chosen runningcoupling/dilaton profile. We realize chiral phase transitions with either temperature or density thatare first or second order by changing the dilaton profile. Although the models are only caricaturesof QCD they show that holographic models can capture many aspects of the QCD phase diagramand hint at the dependence on the running coupling.
I. INTRODUCTION
The QCD phase diagram is notoriously difficult tocompute. Firstly the physics associated with deconfine-ment or chiral symmetry restoration is strongly coupledwhere we traditionally do not know how to compute. Sec-ondly at finite density lattice gauge theory, the first prin-ciples simulation of the theory on supercomputers, suffersa “sign problem” that means Monte Carlo methods breakdown. In fact with light quarks there is no clear order pa-rameter for deconfinement so we will concentrate on thechiral transition. Progress has been made by identifyingeffective theories of the transitions and through latticecomputations at low density. [1] provides a review ofthe standard picture. It is believed for QCD, with thephysical quark masses, that the phase transition withtemperature is a smooth cross over (becoming a secondorder transition as the up and down quark masses go tozero). At zero temperature the transition with densityis believed to be first order. There must therefore be acritical point where the first order line ends in the tem-perature density plane.In the last ten years the AdS/CFT Correspondence [2–4] and more general gauge/gravity dualities have emergedas a new tool for the study of strongly coupled problems.It is interesting to ask whether these holographic modelscan in principle describe a phase diagram like that of realQCD. One recent attempt in this direction can be foundin [5] where a 5d holographic model of a strongly coupledgauge theory with a running coupling was shown to givean appropriate looking phase diagram. In that modelthe order parameter is associated with confinement andit does not include quark fields. In true QCD the orderparameter across the phase diagram is the quark bilinear ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] condensate. In this paper we will study a holographicmodel of quark fields and again attempt to reproducekey features of the QCD phase diagram.Quarks can be introduced into the AdS/CFT Corre-spondence through probe branes [6–10] and a number ofsystems with chiral symmetry breaking have been devel-oped [11–16]. We can not of course describe true QCDholographically because the dual, if it exists, is not knownand is probably very complicated (and strongly coupled,at least, in the UV). Our analysis is therefore in the spiritof AdS/QCD [17, 18], a phenomenological modeling ofthe QCD phase diagram. If one could model the phasediagram correctly one might hope to then predict otherfeatures of the theory such as time dependent dynamicsduring transitions and so forth.Our models will be in the context of the simplest braneconstruction of a 3+1d gauge theory with quarks which isthe D3/D7 system of Fig 1 [6–10]. The basic gauge theoryis large N , N = 4 super Yang-Mills with N f quark fields.We will work in the quenched approximation where weneglect quark loops. On the gauge theory side we donot backreact the D7 branes, that provide the quarks, onthe geometry but instead work in the probe approxima-tion [8]. The theory has a U (1) symmetry under which afermionic quark anti-quark condensate has charge 2 andplays the role of U (1) axial [11]. Although the theorydoes not have a non-abelian chiral symmetry (as for ex-ample the Sakai Sugimoto model does [16]) this is notimportant in the quenched approximation since the dy-namics of the formation of the quark condensate is flavourindependent. The model is very simple to work with hav-ing a background geometry that is just AdS × S andthe gauge theory is 3+1d at all energy scales.So called top down models of this type exist with chiralsymmetry breaking. Supergravity solutions exist thatcorrespond to the AdS space being deformed in reactionto a running coupling introduced by a non-trivial dilatonprofile [11, 13]. In cases where the coupling grows in theinfra-red (IR), breaking the conformal symmetry, chiralsymmetry breaking is induced. These models have very FIG. 1: A schematic of the D3/D7 showing our conven-tions. The D3-D3 strings generate the N = 4 theory, theD3-D7 string represent the quarks and D7-D7 strings describemesonic operators. specific forms for the running coupling and are typicallysingular somewhere in the interior. At the string theorylevel a full interpretation is lacking.A yet simpler and completely computable case withchiral symmetry breaking is provided by introducing abackground magnetic field associated with U (1) baryonnumber [15]. Such a background source can be describedby a gauge field on the surface of the D7 brane. A chiralcondensate is induced. Very simplistically one can thinkof the B -field as introducing a scale that breaks the con-formal symmetry as the strong coupling scale Λ QCD doesin QCD allowing the strong dynamics to form the quarkcondensate.In recent papers [19–23] we have explored the phasestructure of the theory with magnetic field. Tempera-ture can be introduced through an AdS Schwarzschildblack hole in the geometry [4]. Density and chemical po-tential can be added through the temporal component ofthe U (1) baryon number gauge field [25–27]. The phasediagram [19] is shown in Fig 2(a). The chiral restorationtransition was found to be first order with temperatureand second order with density. A critical point lies be-tween these regimes. In addition there is an extra tran-sition associated with the formation of density and themesons of the system melting into the background plasma[28, 29]. In places in phase space this merges with thechiral transition but in other places it separates and canbe either first or second order. Such a phase with a quarkdensity but chiral symmetry breaking could potentiallyexist in QCD. It would be nice to also describe this tran-sition as a deconfinement transition but firstly the N = 4background does not induce linear confinement and sec-ondly the presence of any temperature leads to screeningof the quarks at the length scale of the inverse temper-ature. The meson bound states are closer in spirit toatomic bound states than QCD-like mesons. Neverthe-less they are being disrupted by the background plasmaso the existence of a phase with melted mesons but chiralsymmetry breaking at least leads one to speculate on apossible separation of deconfinement and chiral restora-tion behaviour in the QCD phase plane.In the gravity picture of the transitions the crucial Μ Χ SBzero densityc ¹
0, d =
0, J = Χ SBfinite densityc ¹
0, d ¹
0, J = Χ Sfinite densityc =
0, d ¹
0, J = =
0, d =
0, J = (a) T - µ phase diagram. The inset diagrams arerepresentative probe brane embeddings (dotted lines),where a black disk represents a black hole. Χ SBinsulatorstable mesonc ¹
0, d =
0, J = Χ SBconductorc ¹
0, d =
0, J ¹ Χ Sconductorc =
0, d =
0, J ¹ =
0, d =
0, J = (b) T - E phase diagram. The inset diagrams arerepresentative probe brane embeddings (dotted lines),where a red arc represents a singular shell. FIG. 2: The phase diagrams of the massless N = 2 gaugetheory with a magnetic field. First order transitions are shownin blue, second order transitions in red. The temperature iscontrolled by the parameter T , chemical potential by µ andelectric field by E . Q = H Χ S L Quak Phase H Χ SB L Baryon Phase Μ` Ξ T m q = FIG. 3: The phase diagram of the massless axion/dilatongauge theory in [31]. transitions are between the three sorts of D7 brane em-bedding shown in Fig 2(a). An embedding that curvesin the holographic space to miss the black hole repre-sents the chiral symmetry breaking phase. These solu-tions have stable and discrete linearized fluctuations thatcorrespond to the mesons of the theory. The second typeof embedding curves off axis but ends on the black hole.It describes a phase with chiral symmetry breaking butthe linearized fluctuations are now replaced by in-fallingquasi-normal modes of the black hole which describe un-stable mesonic fluctuations of a quark plasma. Finallythe flat embedding ending on the black hole describesmelted mesons and chiral symmetry restoration. Thetransitions occur when the magnitude of the D7 actionfor these different cases cross.How generic to the holographic description is thisphase structure? Keeping within the top down analy-sis one can change parameters and see what effect theyhave on the phase diagram. For example in [21] we tradeddensity for an electric field parallel to the magnetic field.The electric field tries to dissociate mesons by accelerat-ing the quark and anti-quark in opposite directions andso opposes the formation of a chiral condensate. Thephase diagram of the theory in the T - E plane at fixed B is also shown in Fig 2(b). The chiral transitions are againfirst order with temperature but second order with den-sity although there is some change in the meson meltingtransition order . We followed up this analysis in [22] toconsider the case of mutually perpendicular electric andmagnetic fields - there the chiral transition is first orderin nature throughout the full T - E plane.A recent paper [31] performed a simliar analysis witha running dilaton geometry. The geometry is that of [32]in which there is a non-zero profile for both the dilatonand axion fields in AdS. The field theory is the N = 4gauge theory with a vev for both Tr F and Tr F ˜ F whichpreserves supersymmetry at zero temperature but dis-plays confinement. A D7 was introduced in a supersym-metry breaking fashion and chiral symmetry breaking isobserved. The temperature density chiral transformationwas first order throughout the plane and is shown in Fig3. It shows the same three phases as the magnetic fieldcase. An extra component of the analysis in [31] was tonote that in the confining geometry with a running dila-ton a baryonic phase was also present. A baryon vertexis described by a D5 brane wrapped on the S of theAdS × S space. In the pure N = 4 theory such verticesshrink to zero. However in the running dilaton geometrythe large IR value of the dilaton stabilizes the D5 embed-ding. Solutions exist that link the D5 to the D7 braneembedding with a balancing force condition. These con-figurations describe the gauge theory with finite baryondensity rather than finite quark density. This phase setsin at finite chemical potential and then persists to infinitechemical potential (as shown in Fig 3) which is certainlyunlike QCD. We will not focus on this phase in this paperbut it would be interesting to study it in future work tofind models that have a baryonic phase in some interme-diate range of chemical potential like QCD. There may be an instability in Fig 2 due to the WZ term contri-bution [22, 30].
These phase structures are very interesting and sur-prisingly complex but do not match the expectations inQCD. In QCD we need a second order transition withtemperature and a first order transition with density tothe chirally symmetric phase.Here we want to work in a much more generic frame-work to ask what phase structures it is possible to getin the holographic description and to try to force our-selves onto a representation of the QCD phase diagram.We will therefore take a bottom up approach within themodel and allow ourselves to dial the running of thegauge coupling by hand. We will have a dilaton profilethat smoothly transitions from a UV conformal regimeto an IR conformal regime through a step of variableheight and width. Such an ansatz allows one to considerrunnings that range from precocious growth in the IRto more walking like dynamics[33]. We used a similaransatz in previous papers to study the impact of walking[14] on meson physics and as a mechanism for generatinginflation [34]. Here, in a completely new analysis of thephase structure of these models, we find that with thesimple step ansatz we can move from a totally first ordertransition in the phase plane to a configuration similar tothat we obtained with a B field (a first order transitionwith temperature but second order with density). Withthis ansatz we can not achieve a second order transitionwith temperature.The model directly suggests other phenomenologicalgeneralizations though. In particular, if we think of therunning dilaton profile as a short cut for including theback reaction due to the quark fields/D7 brane , thenit is natural to break the SO (6) symmetry of AdS inthe dilaton in the same fashion as the D3/D7 system’sgeometry. This allows us an extra phenomenological free-dom to distort the dilaton or black hole horizon. Thesesimple changes do allow us to reproduce a wide range ofphase diagrams including QCD-like ones as we will showbelow. We will discuss the simple geometric reasons forthe emergence of first or second order transitions in thesedifferent scenarios.Our conclusion therefore is that the holographic modelhas no intrinsic problem with mimicking the QCD phasediagram and these systems may therefore be phenomeno-logically useful in the future. II. THE MODEL
First let us review the gravity dual description of thesymmetry breaking behaviour of our strongly coupledgauge theory. In [35], D3-D7 solutions at finite temperature and chemical po-tential, with the inclusion of dynamical flavor effects, have beenderived and studied as full-fledged top-down models. They willbe the first step towards the top-down study of phase transitionsin D3-D7 systems with dynamical flavors.
Dp-branes are p dimensional membrane like objectsto which the ends of open strings are tied. The weakcoupling picture for our D3/D7 set up is shown in Fig1 [6–10] - there are N D3 branes and the lightest stringstates with both ends on the D3 generate the adjointrepresentation fields of the N = 4 gauge theory. Stringsstretched between the D3 and the D7 are the quark fieldslying in the fundamental representation of the SU ( N )group (they have just one end on the D3).In the strong coupling limit the D3 branes in this pic-ture are replaced by the geometry that they induce. Wewill consider a gauge theory with a holographic dual de-scribed by the Einstein frame geometry AdS × S d s = r R d x + R r (cid:0) d ̺ + ̺ dΩ + d w + d w (cid:1) , (1)where we have split the coordinates into the x of thegauge theory, the ̺ and Ω which will be on the D7 braneworld-volume and two directions transverse to the D7, w , w . The radial coordinate, r = ̺ + w + w , cor-responds to the energy scale of the gauge theory. Theradius of curvature is given by R = 4 πg s N α ′ with N the number of colours. The r → ∞ limit of this the-ory is dual to the N = 4 super Yang-Mills theory where g s = g is the constant large r asymptotic value of thegauge coupling.In addition we will allow ourselves to choose the profileof the dilaton as r → e φ ≡ β , where the function β → r → ∞ . An interesting phenomenological case is toconsider a gauge coupling running with a step of the form[14, 34] β ( r ) = A + 1 − A tanh [Γ( r − λ )] . (2)Of course in this case the geometry is not back reactedto the dilaton and the model is a phenomenological onein the spirit of AdS/QCD[17, 18]. This form introducesconformal symmetry breaking at the scale Λ = λ/ πα ′ which triggers chiral symmetry breaking. The parameter A determines the increase in the coupling across the step.We will introduce a single D7 brane probe [8] into thegeometry to include quarks - by treating the D7 as aprobe we are working in a quenched approximation al-though we can reintroduce some aspects of quark loopsthrough the running coupling’s form if we wish (or knowhow). This system has a U (1) axial symmetry on thequarks, corresponding to rotations in the w - w plane,which will be broken by the formation of a quark con-densate.In the true vacuum at T = 0 the brane will bestatic. We must find the D7 embedding function e.g. w ( ̺ ) , w = 0. The Dirac Born Infeld (DBI) action inEinstein frame is given by S D = − T Z d ξe φ p − det P [ G ] ab = − T Z d x d̺ ̺ β q ∂ ̺ w ) , (3) where T = (2 π ) − α ′− g − and T = 2 π T when wehave integrated over the 3-sphere on the D7. The equa-tion of motion for the embedding function is therefore ∂ ρ " β̺ ∂ ̺ w p ∂ ̺ w ) − w ̺ q ∂ ̺ w ) ∂β∂r = 0 . (4)The UV asymptotic of this equation, provided the dilatonreturns to a constant so the UV dual is the N = 4 superYang-Mills theory, has solutions of the form w = m + c̺ + · · · , (5)where we can interpret m as the quark mass ( m q = m/ πα ′ ) and c is proportional to the quark condensate.The embedding equation (4) clearly has regular solu-tions w = m when g Y M is independent of r - the flatembeddings of the N = 2 Karch-Katz theory [8]. Equallyclearly if ∂β/∂r is none trivial in w then the secondterm in (4) will not vanish for a flat embedding.There is always a solution w = 0 which correspondsto a massless quark with zero quark condensate ( c = 0).In the pure N = 2 gauge theory with β = 1 this is thetrue vacuum. In the symmetry breaking geometries thisconfiguration is a local maximum of the potential.If the coupling is larger near the origin then the D7brane will be repelled from the origin . The parameterΓ spreads the increase in the coupling over a region in r of order Γ − in size.We display the embeddings for some particular casesin Fig 4. Note that we have chosen parameters here thatmake the vacuum energy of the theory the same in eachcase. The vacuum energy is given by minus the DBIaction evaluated on the solution. In fact this energy isformally divergent corresponding to the usual cosmolog-ical constant problem in field theory. As usual we willsubtract the UV component of the energy to renormal-ize.The symmetry breaking of these solutions is visibledirectly [11]. The U (1) symmetry corresponds to rota-tions of the solution in the w - w plane. An embeddingalong the ̺ axis corresponds to a massless quark withthe symmetry unbroken (this is the configuration that ispreferred at high temperature and it has zero condensate c ). The symmetry breaking configurations though maponto the flat case at large ̺ (the UV of the theory) butbend off axis breaking the symmetry in the IR.One can interpret the D7 embedding function as thedynamical self energy of the quark, similar to that emerg-ing from a gap equation [14]. The separation of the D7 In fact there is a competition between the increased action fromthe D7 entering the region with larger dilaton and the derivativecost of the D7 bending to avoid it. This leads to a critical value of A to trigger chiral symmetry breaking. For example for λ = 1 . A c = 2 .
1. In this paper we will consider only super-critical values of A . Β H r L Ρ Γ A λ
Black 1 3 3.240Green 1 2 4.045Blue 1 1.8 4.325Orange 1 1.5 4.940Red 0.5 1.8 5.882Black(Dotted) B = 35.6 - -FIG. 4: Example coupling flows (2) (top) and the inducedD7 brane embeddings/quark self energy (bottom) with theparameter choices shown in the table. from the ̺ axis is the mass at some particular energyscale given by ̺ . A. Temperature
Temperature can be included in the theory by using theAdS-Schwarzschild black hole metric. In Einstein framewe haved s = − K ( r ) R d t + R K ( r ) d r + r R d ~x + R d Ω , (6)where K ( r ) = r − r H r , r H = πR T . (7)Witten identified this as the thermal description of thegauge theory in [4]. The parameter r H is of dimensionone in the field theory and preserves the SO (6) symmetryso is identified as shown with temperature T. The black hole is the natural candidate since it has intrinsic ther-modynamic properties such as entropy and temperature.We make the coordinate transformation [11] rdr ( r − r H ) / ≡ dww , w = r + q r − r H , (8)with √ w H = r H , such that the metric becomesd s = w R ( − g t d t + g x d ~x )+ R w (d ρ + ρ dΩ + d L + L dΩ ) , (9)where g t = ( w − w H ) w ( w + w H ) , g x = w + w H w ,w = p ρ + L , ρ = w sin θ , L = w cos θ , (10)Now we have to transform β also: e φ = β (cid:18) w + w H w (cid:19) , (11)and therefore β = A + 1 − A tanh Γ s ( ρ + L ) + w H ρ + L − λ . (12)Note that for w H → w → r , ρ → ̺ and L → w , if weset w = 0. B. Chemical Potential
The DBI action for the D7 brane naturally includesa surface gauge field which holographically describes thequark bilinear operators ¯ qγ µ q and their source, a back-ground U (1) baryon number gauge field [25–27]. We in-troduce a chemical potential through the U (1) baryonnumber gauge field which enters the DBI action in Ein-stein frame as S D = − T Z d ξe − φ q − det (cid:0) e φ/ P [ G ] ab + (2 πα ′ ) F ab (cid:1) . We allow a chemical potential through A t ( ρ ) = 0. Sothe Action becomes S D = Z d x dρ L = − T Z d x dρ β ( ρ ) ρ s g t g x (1 + L ′ ) − g x β ( ρ ) ˜ A ′ t . (13)where ˜ A ′ t = (2 πα ′ ) A ′ t . In our convention of the metricthis is L = − T β ( ρ ) ρ (cid:18) − w H w (cid:19) (cid:18) w H w (cid:19)s (1 + L ′ ) − w ( w + w H )( w − w H ) ˜ A ′ t β ( ρ ) . (14)Now we can Legendre transform the action as we have aconserved quantity, the density, d (cid:16) = δS D δA ′ t (cid:17) .˜ S D = S D − Z d ξA ′ t δS D δA ′ t = (cid:18)Z S ǫ Z d x (cid:19) Z d ρ ˜ L , where˜ L = − T (cid:0) w − w H (cid:1) ( w ) p L ′ vuuut w d β ( ρ ) (cid:16) (2 πα ′ ) T ( w + w H ) (cid:17) + ρ ( w + w H ) w β ( ρ ) . (15)We can redefine d = (2 πα ′ ) T ˜ d to give the simplerexpression˜ L = − T (cid:0) w − w H (cid:1) ( w ) p L ′ vuut w ˜ d β ( ρ )( w + w H )) + ρ ( w + w H ) w β ( ρ ) ! . (16)By varying the Lagrangian with respect to A ′ t , we get anexpression for ˜ d ( ˜ A ′ t ) which we can invert for an expressionfor ˜ A ′ t ( ˜ d ) ˜ A ′ t = ˜ d (cid:0) w − w H (cid:1) ( w + w H ) p L ′ vuut ˜ d β ( ρ ) w ( w + w H ) + ρ (cid:16) w + w H w (cid:17) . (17)This can be used to find the chemical potential µ = ˜ µ (2 πα ′ ) ˜ µ = Z ∞ ρ H d ρ ˜ d (cid:0) w − w H (cid:1) ( w + w H ) vuuut (1 + L ′ ) ˜ d β ( ρ ) w ( w + w H ) + ρ (cid:16) w + w H w (cid:17) , (18)where ˜ µ ( ρ → ρ H ) = 0.The free energy can be found by integrating the Legen-dre transformed Lagrangian, the grand potential by inte-grating the original Lagrangian, where we replace ˜ A ′ t ( d ). F = − ˜ S D T = Z ∞ ρ H d ρ (cid:0) w − w H (cid:1) ( w ) β ( ρ ) p L ′ s ˜ d β ( ρ ) w ( w + w H ) + ρ ( w + w H ) w . The grand potential isΩ = − S D T = Z ∞ ρ H d ρ β ( ρ ) w − w H w ρ (cid:18) w + w H w (cid:19) vuuut (1 + L ′ ) ˜ d β ( ρ ) w ( w + w H ) + ρ (cid:16) w + w H w (cid:17) , where we need to note that F ( ρ → ∞ ) = Ω( ρ → ∞ ) = ρ , so we need to subtract (Λ UV ) from both integralsto renormalize them. III. ANALYSIS AND RESULTS
The methodology to study the phase diagram of ourmodel is straight-forward if laborious. We will workthroughout in the massless quark limit. We can thinkof the scale λ in the dilaton ansatz as our intrinsic scaleof the theory and so we will leave that fixed. Then foreach choice of parameters in the dilaton profile ( A, Γ) weanalyze the theory on a grid in T and µ space.For each point on the T, µ grid we seek three sorts ofembedding. The flat embedding L = 0 exists in all casesand describes the theory with m = 0 and c = 0. We use(18) to compute the d - µ relation for these embeddings.We can also seek curved embeddings that miss theblack hole. These solutions must have d = 0 but areconsistent for any value of µ . Here we use the equationof motion for L from (16) and numerically shoot froman initial condition at ρ = 0 with vanishing ρ derivative, L ′ (0) = 0. We seek solutions that approach L = 0 atlarge ρ . These configurations have a non-zero conden-sate parameter c .Finally we can look for solutions that end on the blackhole horizon. To find these we fix the density d and shootout from all points along the horizon seeking a solutionthat approaches L = 0 at large ρ . We then use (18) tocompute µ from the solution. In this way we can fill outthe T , µ grid. The condensate can again be extractedfrom the large ρ asymptotics of the embedding.After finding as many such solutions as exist at eachpoint the easiest method to identify the transition pointsis to plot the density against µ on fixed T lines. The tran-sitions and their order are then manifest. We display foursample plots in Fig 5 taken from scenarios below show-ing the four cases of the chiral transition and the mesonmelting transition being respectively first or second orderin all combinations. A. Dependence on the change in coupling
Let us first consider how the phase diagram dependson the height of the step in the gauge coupling function β . We fix λ (the intrinsic scale of the theory) and also
10 20 30 40 50 60 Μ (a) T = 0 . , A = 30 , Γ = 0 . , λ = 1 . Μ (b) T = 0 . , A = 3 , Γ = 1 , λ =1 . , ˜ α = 3 - shows a first ordertransition followed by a second ordertransition as µ increases. Μ (c) T = 1 . , A = 30 , Γ = 1 , λ = 1 . µ increases. Μ (d) T = 0 . , A = 3 , Γ = 1 , λ =1 . , ˜ α = 3 - shows two second ordertransitions. FIG. 5: Plots of density d versus chemical potential µ . Thetop lighter line (green) in each case corresponds to the flat em-bedding; the horizontal line (black) along the axis is a chiralsymmetry breaking (Minkowski) embedding; the near verticaldark line (blue) is a black hole embedding. Transition pointsare shown by the dotted vertical lines. Γ = 1 and explore the phase structure as a function of A . We display the results for three choices of A in Fig 6.In these and all our future phase diagrams the regionsshown are similar to those in Fig 2(a) we will simply Μ (a) A = 3 Μ (b) A = 15 Μ T (c) A = 30 FIG. 6: Plots for three possible phase diagrams for thechoices A = 3 , ,
30. Large (small) A gives second (first)order transition at low T . Γ = 1 , λ = 1 . display the phase boundaries and their order henceforth.As mentioned in footnote 2 above there is a criticalvalue of A for chiral symmetry breaking to occur. Aconformal theory can not break a symmetry since it of-fers no scale for that symmetry breaking to occur at. Infact some finite departure from conformality is neededto break the chiral symmetry. For these choices of λ, Γthe critical A is A c = 2 .
1. We work above this valuethroughout.At low A there is a single transition for chiral symmetryrestoration and meson melting which is first order for allT and µ . On the gravity side this is a transition betweenthe curved embedding that misses the black hole and theflat embedding. In this case an embedding ending on theblack hole never plays a role.For larger A , a new phase with chiral symmetry break-ing but melted mesons develops. There is a regime nowin which the curved embedding ending on the black holeis energetically favoured. The transition from the chiralsymmetry breaking phase to this new phase is second or-der. The chiral symmetry restoration phase remains firstorder.At very large A the chiral restoration transition be-comes second order at high density. This latter phase Β H w L (a) T = 0 . Β H w L (b) T = 1 . Β H w L (c) T = 1 . λ √ . Dilaton isalmost screened. The residual effect isdue to finite Γ effect. FIG. 7: Plots for parameter choices A = 5 , Γ = 100 , λ = 1 . β . The red lineshows the position of the horizon. The final plot correspondsto the point of the first order transition. resembles that of the theory with chiral symmetry break-ing induced by a magnetic field [15]. In fact the B fieldcase can be thought of as our case but with a choice of β given by β = s B w ( w + w H ) . (19)It is the black dotted curve ( w H = 0) in Fig 4 - it is notsurprising therefore that we see similar phase structurehere (and indeed that we do provides strength to ouranalysis which is capturing the behaviour of top downmodels).For very large A the step becomes very sharp and thereis little change relative to our phase diagram in Fig 6.In particular the thermal transition always remains firstorder.The behaviour we are seeing here can be readily ex-plained from the D7 perspective. First of all the zerodensity transition with temperature is first order for asimple reason. The D7 embedding breaks chiral symme-try at zero temperature because it prefers to avoid the action cost of entering the region in which the dilatonis large. As temperature is introduced through a smallhorizon the interior of the space is “eaten” but the D7embedding remains oblivious to this change since it neverreaches down to small r . As temperature rises the pointof transition is when the horizon moves through the scale λ where the dilaton step is. Once the region with a largedilaton is eaten by the black hole the preferred D7 em-bedding is the flat one.In Fig 7 we show an extreme case of this behaviourexplicitly. Here we have taken Γ very large so that thetransition in the dilaton between the low and high valueis very sharp. We plot the β profile against our radialparameter w and mark in red the position of the blackhole horizon. Note that in the w coordinates the regionwhere β is large depends on the temperature (it doesn’tin the orignal r coordinate). The dilaton effective radius λ ∗ is λ ∗ = s λ + √ λ − T , (20)where the argument of tanh in (12) vanishes. So, as T in-creases λ ∗ decreases. When T becomes T c = λ √ , λ ∗ = T c the dilaton is perfectly screened by the black hole hori-zon. (i.e. If T = 0, λ ∗ = λ . If T = λ √ , λ ∗ = λ √ ). Thepoint of the first order transition is where the horizonscreens the dilaton.When density is introduced the story can become morecomplex. The action is (16) where it can be seen from thefirst of the two terms in the square root that including d increases the action. This increase can be beneficialthough if the second term with β can be reduced. It ispossible to reduce the β term if the D7 enters the regionwhere β is large at small ρ . This means that the situationcan arise where curving off the axis and then spiking on tothe axis can be the lowest action state. This is typicallymore likely where β is largest in the interior space andthe most savings can be made entering that region at low ρ . As we have seen at large values of A embeddings thatspike onto the horizon do play a role introducing an extraphase.It is only possible to have second order transitions ifall three phases we have described are present. In the D7description the D7 must move from a curved embeddingthat avoids the black hole to a configuration that spikesonto the black hole to a flat embedding smoothly. B. Dependence on the speed of running
The parameter Γ controls the period in ρ or RG scaleover which the change in the coupling A occurs. It allowsus to naively go from a precociously running theory to awalking theory (although the change in the parameter A over that period may enter into what is meant by walkingversus running too).
10 20 30 40 50 60 Μ (a)Γ = 0 . Μ (b)Γ = 0 . Μ T (c)Γ = 1 FIG. 8: Example plots of three possible phase structures for A = 30 , λ = 1 .
715 and varying Γ. Large (small) Γ gives asecond (first) order transition at low T . In Fig 8 we show the phase diagram as a function ofΓ at fixed λ and A . We start at Γ = 1 with a configura-tion already discussed that has all three phases presentand second order transitions at high density. As Γ is re-duced so that the step function in the dilaton becomesbroader the first order nature of the transitions reassertsitself. By Γ = 0 . IV. BREAKING THE ρ - L SYMMETRY
Our goal is to attempt to reproduce a phase diagramcomparable to that of QCD in our holographic model. Sofar we have failed to generate a second order transition Μ (a) α = 1 , A = 3 , Γ = 1 , λ = 1 . Μ (b) α = 2 . , A = 3 , Γ = 1 , λ = 1 . Μ T (c) α = 3 , A = 3 , Γ = 1 , λ = 1 . FIG. 9: Sample phase diagrams for theories with none zero α . with temperature at zero density which is a key part ofthe QCD picture.We have a further natural freedom within our holo-graphic model to exploit though. Our running dilaton isin someway supposed to represent the backreaction of thequark fields on the strongly interacting gauge dynamicsto allow us to model theories with more interesting dy-namics than the conformal N = 4 gauge fields. We in-troduce quarks through D7 branes that break the SO (6)symmetry of the five sphere of the original AdS/CFTCorrespondence down to SO (4) × SO (2). Our metricand ansatz for the running coupling (2) though respectedthe full SO (6) symmetry. It seems reasonable to makeuse of the broken symmetry to introduce a further freeparameter into our model.The most succesful scenario we have found is to intro-duce our explicit L − ρ symmetry breaking parameter, α through the blackening factors of the metric g t = ( w − w H ) w ( w + w H ) , g x = w + w H w , (21)with w → ρ + 1 α L , α > . (22)0We show the α dependence of our model in Fig 9. Westart from a model with a first order transition through-out the phase plane. As we increase α a region withmelted mesons but chiral symmetry breaking develops,associated with second order transitions. It then spreadsto the zero chemical potential axisThe reason for the onset of second order transitionswith just temperature is simply understood. We havedeformed the black hole horizon into an ellipse whosemajor axis is along the L axis. The area of enlargeddilaton remains circular in the ρ - L plane. Thus there aretemperature periods in which the area of the ρ - L planewith a large dilaton is covered except for a small piecethat emerges from the horizon near the ρ axis. If thevalue of the dilaton is sufficiently large in that uncoveredarea to encourage the D7 to avoid it, but the horizon onthe L axis has met the zero temperature D7 embedding,then a second order transition to a black hole embeddingis likely. Since in the absence of the rise in the dilaton theflat embedding would now be preferred the D7 settles onthe horizon so it just misses the raised dilaton area. Asthe black hole grows further the embedding is likely totrack down onto the axis smoothly as the raised dilatonarea is finally eaten. This intuition is indeed matched bythe solutions as shown in Fig 9.The bottom phase diagram in Fig 9 achieves our goalof reproducing a chiral transition that is second orderwith temperature but first order with density. V. SUMMARY
In this paper we have converted the D3 probe-D7 sys-tem, that holographically describes N = 4 super Yang-Mills theory with quenched N = 2 quark multiplets, to aphenomenological description of strongly coupled quarkmatter. We introduced a simple unback-reacted profilefor the dilaton that describes a step of variable heightand width in the running coupling of the gauge theory -(2). This breaks the conformal symmetry of the modeland introduces chiral symmetry breaking. We have thenstudied the temperature and chemical potential phasestructure of the model.The phase diagrams consist of three phases - a chi-rally symmetric phase at large temperature and density;a chirally broken phase with non-zero quark density atintermediate values of T and µ ; and a chiral symmetry broken phase with zero quark density at low T and µ . Fig2(a) shows these phase and their holographic analogue ina previously studied case where chiral symmetry was bro-ken by an applied magnetic field. Here we showed that asmall wide step in the gauge coupling’s running gives riseto a single first order transition between the chiral sym-metric and the broken phase (see Fig 6). If the step ismade larger in height or thinner then the chirally brokenphase with non-zero density also plays a role. Here thetransitions at low temperature with chemical potentialcan be second order. These results match known resultsin top down models in the presence of magnetic fields toinduce the symmetry breaking.We were interested in reproducing phase diagrams withthe structure believed to exist in QCD. To do this wemade use of the broken SO (6) symmetry of the gravitydual in the presence of D7 branes. Were the branes back-reacted the dilaton and geometry would reflect this sym-metry breaking. We introduced a further phenomenologi-cal parameters α in the black hole blackening factor. Thismodels allowed us to control which volumes of the holo-graphic space have a large dilaton value within, which theD7 branes prefer to avoid. Using this one extra parame-ter we were able to generate phase diagrams like those inQCD with a chiral restoration transition that was secondorder with temperature but first order with density (seeFig 9).The ease with which such a variety of phase struc-tures could be obtained is very encouraging for the ideaof phenomenologically modeling the QCD phase diagramholographically. Further the phenomenological parame-ters we introduced are very natural in this context andit seems likely that top down models with such phasestructures should be possible as a result. Acknowledgments
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