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Time-Dependent Meson Melting in External Magnetic Field Hajar Ebrahim University of Tehran and IPM Based on 1503.04439 in collaboration with: M. Ali-Akbari, F. Charmchi, A. Davody, L. Shahkarami Nafplion, July 2015 1 Outline: Why Time-Dependence? Holographic Meson Time-Dependent Set-Up 2 Why Time-Dependence? system in equilibrium non-dynamical, stationary late-time state of a generic system time-dependent processes perturbative methods, Linear Response Theory H ! H + Hpert coarse-grained view of the system, low energy effective description, Hydrodynamics, Universal features lack of adequate techniques for strongly correlated condensed matter theories, quark-gluon plasma Focus on field theories with holographic dual. Two dual theories are just different descriptions of the same physical system. 3 Thermalization in QGP Collision of two heavy nuclei (Gold or Lead) at the relativistic speed Production of an Anisotropic Plasma Hydrodynamics applies after a very short time-scale, 1 fm Far-from-equilibrium effects at the early stages of QGP production Strongly Coupled Plasma Shuryak, 2004; 2005 Presence of a magnetic field at the early stages of QGP production Kharzeev, McLerran, Warringa, 2008 y ! x hadronic gas described mixed phase by hydrodynamics QGP pre-equilibrium stage Figure 2.5: Sketch of the collision of two nuclei, shown in the transverse plane perpendicular to the beam. The collision region is limited to the interaction almond in the center of the transverse plane. Spectator nucleons located in the white regions of the nuclei do not participate in the collision. Figure taken from Ref. [60]. Figure 1: Description of QGP ion collisions. The kinematic ! formation in1 heavy x +x 2 2 4 landscape is defined by τ = short direction of the almond as they are in the long direction. And, as we shall see, it turns x0 − x1 ; η = 2 log x00 −x11 ; xT = {x2 , x3 } , where the the initial time where the gauge field is zero and ✏ is the cut-o↵ undary. We have assumed ✏ = 0.0001 and ✓˜ = ⇡2 . The ansatz for eld is Z ˜ = h(t, z, ✓) ˜ ˜ dt A✓˜(t, z, ✓) E(t, z, ✓) (4.23) Holography: Q (1 + tanh(!t)) (4.24) 2 ˜ R sin ✓ a parameter showing strength typethe IIB string of the electric field and ! is N=4 su(N) of the time interval where the electric field varies from zero to a theory on Conformal SYM lue. 5 AdS5 ⇥ S ts are fixed by ˜ = E(t, z, ✓) R4 2 = 4⇡g N = g s Y MN = ↵0 2 Strong/Weak (4.25) >>1 Duality Strongly Coupled CFT 7 Classical Gravity on Asymptotically AdS Space-time The dual gravitational description of a strongly coupled gauge theory provides an efficient way to study the thermodynamic properties of gauge theories. Correspondence both in static and dynamical situations: time-dependence on the boundary manifested by time-dependence in the bulk 5 andstudy second, the D7-branes terminate at r = U (0) < modified: In First, a horizon appears at r= r0 > 0,we 1.2 e change in the thesince probe dynamically. toinachieve this goal 1 brane from the bulk black of hole the final equilibrium the The future cannot be corresponds to shape thermalization in the gluon sector of theconfiguration dual order gauge theory. background isthe time-dependent and L. This ‘termination point’ corresponds to the tip of the branes in Fig. 8.3. 2 2 ane in the AdS-Vaidya background. the hole formation in within theThe bulk which 4in40.8 8 ItaAdS 2 2 however, 2 black black hole brane configuration, to be AdS singular aAdS finite 1 the shape the brane such background can deform dynamically. x2therefore +dx Lembedding. is theof AdS radius. The boundary is2 tends where f (z) =The 1−r xdescribes the radius. boundary is 3 and 3 = dx1 +dx2 +dx3 and L is h z /L , d⃗ 3 2 2 ation in the gluon sector of the dual gauge theory. The background is time-dependent and time as the intersection locus approaches the pole where the S wrapped by the D7-brane Let/r us.start with the AdS-Vaidya metric thatcoordinates 2 L 0.8 z =radius. 0,2 while the event horizon z = L /rh . We use the same world-volume coordinates 0.6 boundary Weat use the same world-volume x= AdS The AdS isis at 43 and 8 L his 2 the 2 2 in subsequent sections, whether the holographic quarks described by the D7-branes are the zbrane /L , d⃗ x3such = dx and is the AdS radius.orThe AdS boundary 0 1 istaken7 into account shrinks to zero size. expected that ifdynamically. stringy quantum effects are in a background can deform 0 target 1It is3space 7L 1 +dx 2 +dx 2 0.6 The as the coordinates themselves such that (σ , σ , · · · , σ ) = (t, z, ⃗ x , Ω ). 3 3 ⇥ ⇤ lves that , σworld-volume , 2· M ··, N σ )0.4 = (t, z, ⃗x3 , Ωanalogues Theof heavy 2or light 2quarks 2in QCD2 will depend /rh . such We use the2(σ same 1coordinates 3 ). on how their mass (or, more precisely, 2 2 event horizon is at z = L /r . We use the same world-volume coordinates there, themetric brane will reconnect near that locus, and then go back the embedding. ds F (V, z)dV d Minkowski = G dx dx = 2dVits dzposition + d~x + in costhe d⌦ d 2 ,0.4as a (II.1) dS-Vaidya that is h M N 0 embedding 1 7 of the D7-brane 3to + 3 + sin is specified by (φ, ψ) transverse space 2 (σ , σ , ·(φ, · · , ψ) σ ) in = the (t, z,0.2transverse ⃗xz3 , Ω0 3 ). 1 Thespace the7mass of corresponding mesons) compares to the temperature. dsuch byWethat itsshow position as2(c), a thetogether schematic illustration of this process in Fig. the sub-critical e coordinates themselves such that (σ , σ , · · · , σ ) = (t, z, ⃗ x , Ω3set ).with The function of z:transverse φ = Φ(z)space and ψas=⇤a0. Note thatconcluded we 3can ψ =at 0zero without lossand of generality We have that, temperature, the 0.2D7-branes lie at U = L and are itsM position (φ, ψ) in the ⇥ Note that we can set ψ = 0 without loss of generality 1 0 where 2 N 2 2 2 2 2 2 super-critical cases in Figs. 2(a) and 2(b), respectively. We argue that this phenomenon may 0 0.5 1 1.5 2 D7-brane is specified by its position (φ, ψ) in the transverse space as a induced thanks towithout the U(1)-symmetry generated F (V, z)dV d by dx dx = set 2dVofdzgenerality + d~ x3 +parametrised + ∂cos d on, the D7-brane (II.1)is given ψ . The 0 , u,sin ⌦metric the induced1 metric1.5on the 2 by d⌦ {t, 3xi+ 3 }. In terms of these coordinates, we can ψ = 0 loss 2 induced 0 0.5 . The metric on the D7-brane is given dhat by ∂ z ψ be the gravity dual of quarks recombining into mesons in the boundary field theory. Φ(z) and ψ =as0. Note that we can set ψ = 0 D7-branes without loss of 4generality " ! takes the F (V, z) = 1 M (V )z , form 1 (II.2) induced metric on the D7-brane is given ∂ψ . TheThe f (z) 1 remaining of this paper is organized as follows. In Section will review the static " −2∂ψ . The a b 2 ′ 2 2, 2we 2 2 2 induced metric on the D7-brane is given -symmetry generated by 2 + Φ (z) 2 dzu2+ L 1hab dσ dσ = − 2 dt + 2 x3 +2 cos Φ(z)dΩ (2) R2 u2 + L22d⃗ R 3 , 1 2 2 " embeddings to fix notations. In Section 3, we describe the setup for our dynamical computation. ′ 2 2 2 2 2 z z f (z) z dsabove = metric + dxi in+Eddington-Finkelstein du + 2 d⌦23 . (8.7) dtwritten andFigure we have set the radius of AdS space-time to be one, R = 1. The is 5 2 2 2 2 + Φ dz (z) + d⃗ x + cos Φ(z)dΩ , (2) " ! 1 2 and 3: Minkowski black hole embeddings ×S R u + L u + L 3 3 of the D7-brane in the Schwarzschild-AdS 5 ′ 2 2 2 2 2 2 Results of numerical computations are(V shown in2 Section 4. VSection 5the is null devoted to discussions z + fcoordinate zcosz) 14(2) 1+ radial direction by z and shows The boundary, the of the D7-brane in the Sch Φ (z)f (z) dzwhere + 2the d⃗′x≡ Φ(z)dΩ Figure 3:direction. Minkowski and black holewhere embeddings (V, = 1 )z , (II.2) 3F 3is ,represented b (z) 2 ′horizontal 2 M 2 axes 2ρ, respectively. 2 the d/dz. where spacetime. The vertical and are w and In the plot, the horizon of quark mass a number of solutions exist. Among them the physi 3 + Φ dz dσ = − dt + (z) + d⃗ x + cos Φ(z)dΩ , (2) ) gauge z S axes . Theare AdS factor suggests that ifthe L =Technical thisdetails metric is exactly that ofhorizontal AdS5 ⇥function, 3We see 30 then on the boundary theory interpretations and future directions. for numerical 5 and theory lives, is at z = 0 and V is the time coordinate on boundary. M (V ), which is an arbitrary 2 2 2 D7-branes spacetime. The vertical and w ρ, respectively. In D3-branes z★ z rembedding fh (z) zsymmetry. The of the D7-brane is determined by the Dirac–Born–Infeld (DBI) action. In This Flavour added to the gauge theory by introducing probe radius is fixed as = 1 using the scaling For m ! 0.92 and m " 0.92, we obtain lution is the most energetically favourable one. The presence of the co that the dual gauge theory should still be conformally invariant. is indeed the case in represents the mass of the black hole which changes asmetric time passes by until itEddington-Finkelstein reaches constant Thescaling function computations are provided in R appendices. radius is fixedaas rh = 1 value. using the symmetry. For m ! 0.92 and s of AdS space-time to be one, = 1. The above is written in the absence of the world-volume field strength, the action is the Minkowski and blackwith holeinembeddings, Forconsideration: m ≃ the 0.92,Minkowski there exist thehole the limitIn under If Lor = 0 and theboth quarks are massless and the theory classically determined byD7-branes the Dirac–Born–Infeld (DBI)isrespectively. action. M (V ) that we will work this paper, Kalb-Ramond equivalently constant magnetic fieldis leads black the embeddings, respectively. Fortomnonzero ≃ 0.92, in the bulk A. Karch and E. Katz, 2002 ialfor direction is represented by z and V shows the null direction. The boundary, where mined by the Dirac–Born–Infeld (DBI) action. In # and √ in the probeInlimit Nat -function, Minkowski and hole embeddings. f /Nc ! 0 the quantum mechanical 8 conformal, of D7-brane is black determined by the (DBI) action. rength, the action is coordinate condensate even zero quark mass. In the following wewhich will3 discus Minkowski and black hole embeddings. = the 0the and V is on Dirac–Born–Infeld the boundary. M (V ), which is an arbitrary function, 8 th, action is the time 4 4 8 2 2 2 2 0 V < 0, /N , vanishes. If L = 6 0 then the metric above becomes AdS ⇥ SAdS bou is proportional to N < h , (3) ∝ d σ S c , d⃗ 5 4 1−r 2x =2dx +dx 2 2 Static embeddings where f1−r (z) 4= zx2f /L +dx and L isradius. the AdS radius. The ⇥= D7 ⇤ 8 ,results world-volume field strength, the action is 3 1 2 3 h where f (z) z /L d⃗ = dx +dx +dx and L is the AdS The AdS boundary is in more details. # ⇡V 1 3i.e. 2 reflecting 3 function e#black hole √ which changes as time passes by until itasymptotically, reaches a0 6 constant The h M (V ) = M (II.3) only for 1V, u value. L, the2fact that in the gauge theory conformal 1 cos( ) V 6 f 2 V 2 √ 8 5 same world-volume at z = 0,the while the event horizon at = /rhorizon. .is We use the coo : hthe regularity at# the horizon.atThe solution is then obtained z =asymptotic 0,1while event horizon is atbyzthe =asisL /rofhz.mass WeL use same world-volume coordinates The background interest AdS ⇥ S geometry describing th 8we impose we impose regularity at the The asymptotic solution is then ob rk d∝ with in this paper, is 5 h , (3) d σ V > V, / L, but is restored asymptotically at invariance is explicitly broken quark M √ q D7 start from reviewing static embeddings of the D7-brane to the Schwarzschild-AdS × h ,★ (3) σ We 0 1 7 Fluctuations of theequation fields on the brane produce the 5 . The of motion is a second-order ordinary differential equation for where h = det hab8the the target space coordinates themselves such that σ) = ," · (t, · · ,z,σ⃗x3), Ω =is3 ).(t, z, ⃗x3 , Ω as the as target space coordinates themselves such that (σ 0 ,the σ 1radius , · ·(σ · , of,σ 7the " h , (3) ∝ d σ S horizon geometry N D3-branes energies E M . We also note that, if L = 6 0, then three-sphere notThe 5 5 of q D7 c 8 ! 2 2 in order to fix notations. The metric of the Schwarzschild-AdS × S background S spacetime ! 5 2 2 Φ(z), ρHof rh − ρH D7-brane rin ρH space which is (φ, constant. Note where V ismeson the time< interval in which the2mass the black increases toits Mfposition spectrum %,from $ embedding ofhole the D7-brane iszero by its position (φ, ψ) the transverse H 2 embedding 0 V < 0, of the by ψ) inρ2at the transverse constant, as displayed in· ·specified fig. 8.4; inspecified particular, it(7) shrinks to u ρ= 0 (corresponding toas a spa h − 2 zero ′ is (ρ − ρ ) + · r − ρ + W (ρ) = ⇥ ⇤ (ρ − ρ )+··· , r − + W (ρ) = H 1 can be given as 3 z 3 tan Φ f h H H 2 2 h H is the aM second-order ordinary differential equation for 2 the projection 2 ′′ ′ ′3 ′2 ′ ⇡V 1 sion athat second-order ordinary differential equation for 4 ρ + 3r u R r = L), at which point the D7-branes ‘terminate’ from the viewpoint of on ρ + 3r function of z: φ = Φ(z) and ψ = 0. Note that we can set ψ = 0 without loss of function of z: φ = Φ(z) and ψ = 0. Note that we can set ψ = 0 without loss of generality (8f − zf − Φ = 0 . (4) Φ − )Φ + (3 tan Φ)Φ + + (V ) = M (II.3) H radius of the event horizon is r = M . 1 cos( ) 0 6 V 6 V, h 2 2 2 2 2 2 2 2 2 2 2 ge H h f f " 2 a second-order Vh 2f The equation of motion is ordinary differential equation for ) ) = ( ( dt + dx ) + ( (d⇧ + ⇧ d⇥ + d⌃ + ⌃ d ), ds ! 2 f z z : 3 2 2 inVthe In $ order to add the fundamental matter gauge theory side we have to add the probe branes to the bulk. The 1 > V, . The induced metric on the D7-brane is given thanks to the U(1)-symmetry generated by ∂ . The induced metric on the D7-brane % thanks to the U(1)-symmetry generated by ∂ $ L dz R u %2′ ψ ψ 2value of ρ at the 2 horizon 2 2 2 the 2(ρ, 2<On ′ ρH ds where (0It <is ρalso r is the found in w)-plane. where ρ2 Hφdψ (0 ≤ 201 rother is2the + d⃗ x −f (z)dt + L = + (dφ + cos φdΩ + sin ) , 2ρ −1 −1 value of ρ at the horizon found in the (ρ, w H the h ) (1) H3≤ h ) freedom 3 3 tan Φ f 3 tan Φ f % $ 4 dynamics of the degrees of living on the brane is explained by the DBI action as 10 3 3 to introduce new bulk coordinates (w, ρ) = (L z sin φ, L z cos φ). ′2 ′ z 2 convenient ′2 ′ as ′ u as " ! f (z) ! − ′3 the Φ−+ 02.intersect )Φ + 0 brane 1does not 2 intersect 3 " with ⇤the 4 hole, the brane 4 Φ2does . (4) (4) tan + +′2=not z Φ)Φ 3 tan Φ f =30 with ⇥ black hand, when the hand, when brane the black hole, the brane terminates at the point ′ ′ g C = dx ⇤ dx ⇤ dx ⇤ dx , g = e , R = 4⌅g N l , te Z−2 be f (z) 1 1 s s s c (4) which is constant. Note erval which the mass of the black hole increases from zero to M f )Φ z z f The new embedding function can then a function of ρ, w = W (ρ). In practical numerical f (z) 1 1 s (8f in − zf − Φ = 0 . (4) + (3 tan Φ)Φ + + f z z f f a b 2 ′ 2 2 2 2 2 4 p 123 3 5 −2 a b 2 ′ 2 2 2 2 2Φ = 5where Rwrapped + Φ dz L h dσ = − dt + (z) + d⃗ x + cos Φ(z)dΩ , (2) S by the brane shrinks to zero size at a pole of S : 82 1by the branefshrinks abfdσ to zero size at a pole of S : Φ = π/2, corresponding + Φ dz L h dσ dσ = − dt + (z) + d⃗ x + cos Φ(z)dΩ 0 2 where S 3 wrapped z z 3 3 ab 2 2 2 3 3 , d ⇠W (ρ)gab + (2⇡↵ zby )Fab , z 2 z Eq. (II.4) z 2 Sequation = ⌧7 6 for 2f computations, we solve the obtained rewriting (4) in terms of W (ρ). 4 f (z) z 2 −1 2 −1 z (z) 2 L ntbulk horizon is (w, rThe M . ρ)asymptotic coordinates ρ)regular = z = sin zsincos φ). to ρ = 0. The regular asymptotic solution at ρ = 0 is given by h = to coordinates ρ = 0. solution at2 zρ−1 = cos 02 is given by 2boundary f (L (w, (Lφ, z −1 φ, L φ). −1 −1 C The asymptotic behavior of the field at the AdS of theand dualDilaton fields, respectively. In t andto⌥information arebulk. four-form where nt to introduce new bulk coordinates (w, we ρ)′ ≡= (L′to z add sinthe φ, Lprobe z cos φ).gives (4) damental matter in the gauge theory side have branes the The ⌅ d/dz. where function of ρ, w = W (ρ). In practical numerical 89 and 6 ≡ d/dz. where bewhere a function of ρ,theory w= W (ρ). practical numerical 2 2 is the induced metric andIn FNear iswthe the gauge strength on the asymptotic brane. a,⇧2b + are⌃the brane coordinates gab can ⇥ is the field operators. AdS boundary ρ = ∞, the solution behaves as 4field ab rh4 length ordinate u = and l = string scale. In or g)f obtained function then be a function of ρ, = W (ρ). In practical numerical 2 r s freedom living on the brane is explained by the DBI action as The embedding of the D7-brane is determined by the Dirac–Born–Infeld (DBI) action. In 2 h by rewriting Eq. (4) in terms of W (ρ). W (ρ) = W − ρ + · · · , P is obtained defined as by rewriting Eq. W embedding D7-brane is determined by the Dirac–Born–Infeld (DBI) ac (ρ)in =terms WP − The ρ(ρ). +in · · terms · , ofofthe (8) Wgab (ρ) (4) ofEq. W 5 (4) 4WP5a space-filling olve the equation for W (ρ) obtained by rewriting W (ρ). introduce the fundamental matter we have to add flavo c 4W Z 4567 the absence of the world-volume field strength, the action is the AdS boundary gives information of the dual P= m + p + · · · , (5) W (ρ) the absence of the world-volume field strength, the action is dbehavior at the AdS boundary gives information of the dual M N 2 of the dual 8 AdS the boundary information 6ggives in theWprobe limit to(II.4) this background. The configuration of the 0 )F = @ x @ x (II.5) ρGMbrane ab a b N. #(0) d ⇠ g + (2⇡↵ , S field = ⌧at ary ρ = ∞, of thethe asymptotic solution behaves as 7 ab ab 6 where ≡ W > r is the value of W at the pole. P h √ # 6 oundary ρ = ∞, the asymptotic solution behaves as wherethe WPAdS ≡ Wboundary (0) > rh isρthe value of asymptotic W at the pole. ors. Near = ∞, the solution behaves 8 √ by using either the boundary conditio branes as isSolving the equations of motion h , (3) ∝ d σ S 8 D7 and condensate where the two integral constants m and c are related to the quark mass M c Solving the equations of motion by using either the boundary condition (7) or (8) when the qD7 h , ∝ d σ S 0 1 2 3 4 5 6 7 8 9 brane intersects with theand black hole or not, respectively, we obtain a seq + · ·cintersects ·F,ab⟨Oismthe (5) m + brane c 35] metric field strength on the brane. a, obtain b are the brane coordinates ⟩ ,with as gauge [25,the 29–32, 2 and black hole or not, respectively, we a sequence of embeddings. + · · · (5) ) = ρm + 4 4 8 2 2 2 2 D3 ⇥ ⇥ ⇥ ⇥ · ·3· =, dx1 +dx2 +dx3 and L is theInAdS (5)radius. W= (ρ)1−r = mz+/L2 ,+d⃗x Fig. 3, we show the solutions for several parameter values. For m/r where The AdS boundary is differential 2 f (z) h ρ equation of hmotion a second-order ordinary equation for h= det h2ab . The ρ In Fig. 3, we show the solutions forwhere several parameter values. For m/r ≃ 0.92,isD7 we can see ⇥ ⇥ ⇥Minkowski ⇥ ⇥ embeddings ⇥ ordinary ⇥ ⇥ are that both the black hole and allowed, and t . The equation of motion is a second-order differential equa where h = det h m N f ab at z = 0, while the event horizon is at z = L /r . We use the same world-volume coordinates h and condensate re related to the Φ(z), that both thequark blackmass hole M and Minkowski embeddings are allowed, and they are characterized q M , ⟨O ⟩ = − 0 by4 different c , values N Mq = %(6) $ of the quark condensate c for a given m. We plot c/rh3 2mass M m 13s ℓ6 7 ′ Φ(z), g = @ x @ x G . (II.5) 2πℓ 16π g and condensate dgral c are related to the quark mass M and condensate constants m and c are related to the quark ab a b M N q q as the target space coordinates themselves such that (σ , σ , · · · , σ ) = (t, z, ⃗ x , Ω ). The s s the probe low energy action a<D7-brane in aim z WeInplot 31.0802 tan f 3 33 e⇤ective % Φ < rfor $ ′ for by different values of the quark condensate c for a given m. c/r a limit function rh /m 3 the ′′ ′ in ′3 ′2of h as 4. ′ Fig. c/r becomes multivalued /m 1.0913. This h (8f − zf − Φ = 0 . (4) Φ − )Φ + (3 tan Φ)Φ + + h 3 z 3 tan Φ f 3 , 35] inembedding ′′ trary ′3 ′2 zDirac-Born-Infeld the D7-brane is for specified its < (φ, ψ) the space aMinkowski background istransverse described by (DBI) and Fig. 4. c/r becomes multivalued 1.0802 <byrh /m 1.0913. This implies that there isΦ)Φ 2position fa between z 2 f Φ′ + and h of − zfin′ )Φ =hole 0Chern-S . embe Φ − (8f + (3 tanquark + as first-order phase transition the− black where ℓ coupling constant. O is the bilinear s is the string length and gs is the string m 2 N f 2 f z z f phase between Minkowski and black hole embeddings, and it can be (CS) actions function of z: φ= Φ(z) and ψItthe = 0. that weto can set ψ = 0 by without loss of free generality confirmed comparing the energy −1 associated with its supersymmetric counterparts (See Refs. [34, 47] for further details). Here⟨Om ⟩first-order =− cN ,transition (6)Note m N is also convenient introduce new bulk coordinates (w, ρ) = [32, (L2 z34]. sin φ, L2 z −1 cos φ). f 4 6 f 16π g ℓ confirmed by [32, s the −1 M = after, , U(1)-symmetry ⟨Othe ⟩ free = proportional −energy c ,34]. (6)c as on scomparing mthe .convenient The induced metric the D7-brane thanks generated ∂ψ(6) we ignore and refer to6to m quark mass , ⟨O − is by also introduce bulk (w,Inρ)practical = (L2 znumerical sin φ, L2 z − m ⟩q =to 4g ℓ 6 Itconstants The new embedding function canand then bethe a new function ofcoordinates ρ,and wisthe =given W (ρ). 2πℓ2 4 6 c , 16π 1.02 1.02 1 Holographic Mesons 0.98 0.96 0 0.2 1 0.98 0.4 0.96 0 0.2 0.4 sition is always of first order, and in confining theories with re higher than the deconfinement temperature for the glue. nce is a of gauge rgravity oft coustudy of portant nement, ons, etc. QCD ittions of the deoing behis does of QCD dictions niversal regimes. calculaegime of es. The class of gravity generihis ratio rom ex(RHIC) any unidence as ntal repa small Nc , may te graviperature this pasystem ular, we ansition e quark e the atoutside the horizon in a ‘Minkowski’ embedding — see fig. 1 and 2 5 0 1.75 "0.05 Minkowski embedding 1.5 A "0.05 "0.06 "0.1 "0.07 Critical embedding 1.25 1 c Black hole embedding c "0.15 0.75 "0.08 B 0.5 "0.2 "0.09 0.25 "0.25 0.5 1 1.5 2 2.5 3 FIG. 1: Profiles of D7-brane embeddings in a D3-brane background. The thick black circle is the horizon (ρ = 1). 0 1 2 ! T!M 0 3 4 0.762 0.764 0.766 ! T!M 0.768 0.77 0.772 "0.029 "0.0295 "0.1 "0.03 below. In this phase the meson spectrum (i.e., the spec"0.0305 "0.2 trum of quark-antiquark bound states) is discrete and I!! "0.031 possesses a mass gap. At sufficiently large T /Mq , theI!! "0.3 "0.0315 gravitational attraction overcomes the brane tension and the branes fall into the horizon yielding a ‘black hole’ em"0.032 bedding. In this case, the meson spectrum is continuous "0.4 "0.0325 and gapless. In between, a limiting, critical solution exists. We will show that the phase diagram in the vicinity "0.5 0.762 0.764 0.766 0.768 0.77 0.772 0.774 0 1 2 3 4 ! ! T!M of this solution exhibits a self-similar structure. While T!M this structure went unnoticed, the phase transition that occurs as T /Mq increases from small to large values was FIG. 3: Quark condensate and free-energy density for a D7 in a D3 background; note that N ∝ T 3 . The blue dashed observed in two specific models [5, 6]. In fact, as was (red continuous) curves correspond to the Minkowski (black hole) embeddings. The dotted vertical line indicates the precise first noted in [5] for a D6-brane in a thermal D4 backtemperature of the phase transition. ground, the transition is of first order (in the approximations stated above). Rather than dropping continuously through the critical solution, the probe brane jumps disdiffers from (17) with p = 2, d = 2. This discrepancy found here. Note that this effect is of order 1/Nc2 , and continuously from a Minkowski to a black hole embedis not at all a contradiction, and has the same origin as therefore subleading with respect to the order-Nf/Nc cording at some T = Tfund . This leads to a discontinuity rection in the discrepancy found for the meson spectrum [10]. This to physical quantities from the presence of the several field theory quantities, such as, for example, the is the fact that the calculation in [19] applies in the far brane probes. quark condensate ⟨ψ̄ψ⟩ or the entropy density. In the Finite ’t Hooft coupling corrections correspond to infrared of the gauge theory, whereas that presented here 8.7:behavior Variousand, D7-brane configurations in a black backgroundapplies with at increasing temper- i.e., at T ≫ g 2 . following, we will see thatFigure the critical ashigher-derivative a high temperatures, corrections both D3-brane to the supergravity YM ature from left to right. At lowand temperatures, probe branes o↵ smoothly above the result, the first order transition are essentially universal action the D-brane the action. These may close also blur We will return to horizon. these and other issues in [20]. to all Dp/Dq systems. At high temperatures, the the structure above. Forhorizon. example,Inhigherbranes fall discussed through the event between, a critical solution exists derivative corrections to the D-brane equation of motion Acknowledgments: We thank in which the branes just ‘touch’ the horizon at a point. The critical configuration is never realized: a V. Frolov for a converare likely to spoil the scaling symmetry of eq. (6), and sation and for sharing ref. first-order phase transition occurs from a Minkowski to a black hole embedding (or vice versa) before[7] with us prior to publihence the self-similar behaviour. These corrections also 2007 cation. WeMateos, also thank Myers, P. Kovtun,Thomson, A. Starinets and L. the critical solution is reached. become important as the lower part of a Minkowski brane Yaffe for discussions. Research at the Perimeter Institute approaches the horizon, since the (intrinsic) curvature of is supported in part by funds from NSERC of Canada the brane becomes large there. and MEDT of Ontario. We also acknowledge support Yet another type of correction one may consider is due from NSF grant PHY-0244764 (DM), NSERC Discovery hole phase. to the backreaction on the background spacetime of the grant (RCM) and NSERC Canadian Graduate ScholarDq-branes, magnitude is controlled the ratio Although we will come back whose to this important point by below, we wish to emphasize rightwould also like to thank the ship (RMT). DM and RCM mesonic phase melted phase 7 1 etermined Dirac–Born–Infeld action. Mby (V )the =M (II.3) 1 cos( ⇡VV(DBI) ) 06 V 6 InV, f 2 eength, are following same introduced in [9] and will not repeat the details. The D7the actionthe is : 1 notations asV ones > V, bedded the six directions of the bulk metric, ~x and ⌦3 . We choose the other two coordinates on th # along √ ischoose constant. Note erval in which the mass of thev.black hole rest increases from zero to Mf which 8coordinates be null u and For the of the bulk coordinates we the ansatz h , (3) ∝ d σ 1 7 nt horizon is rh = Mf4 . V = V (u, Z(u, (u, to v),the bulk. = 0. The damental matter in the gauge theory side we v), have zto=add thev), probe= branes on is a second-order ordinary differential equation for Erdmenger, Meyers, Shock, 2007 of freedom living on the brane is explained by the DBI action as # Johnson, nce we are$interested in studying the e↵ect of the external magnetic field Rashkov,Viswanathan, on the dynamical2007 meson mel % Filev, Z ′ B $0 pΦ 3 field′ to38be tan f ′2 magnetic oose the 0 an Φ)Φ + , (II.4) S−= ⌧Φ7 + d ⇠2 g= ab 0+. (2⇡↵ )Fab (4) f z z f L!Ρ" L!Ρ" # # Fx1 x2 = B, 2 B−1 2 −1 $0 B$5 bulk coordinates (w, ρ) = (L z sin φ, L z cos φ). metric and Fab is the gauge field strength on the brane. 2a, b are the brane coordinates and 2 be of ρ,L!Ρ" w = W (ρ). In practical L!Ρ" at ais function constant. # numerical 1 # 1 W (ρ) obtained rewriting Eq. (4) inN terms of Wto(ρ). B $0 B $5 Since we wouldbylike the u and stay null, we have to impose the following constraint equa M v coordinates g = @ x @ x G . (II.5) 2 2 ab a b M N the AdS ofboundary givesthe information of the dual obtained# from the DBI action Ρ e at equations motion for fields on the brane, L!Ρ" solution behaves as6 undary ρ = ∞, the1 asymptotic 4L!Ρ" 6 8 B1$02 "1 "1 2 2 2 F V + 2Z V Z (Z ) = 0, ,v ,v ,v ,v c Ρ Ρ # 2 2 L!Ρ" L!Ρ" = m+ 2 +··· , 2 $10 "2 2 4 6 V 2 +(5) 82Z "2 2= 0, 4 6 8 F V Z 2 (Z )B ρ Non-Zero Magnetic Field: "1 1 ,u ,u ,u "1 ,u 1# # 2 2 L!Ρ" L!Ρ" c are related to"2 the quark mass Mq and condensate (u,v) B $10 B $17 "2 brane. With the above assumptions the DB ere (u, v) ⌘ Z(u,v) which, in fact, gives the shape of the Ρ 2 1 1 62 L!Ρ" comes 2 4 6 8 L!Ρ" 2 4 8 "1 "1 r 1 1 N Ρ Z f 3 2 2 # # ⟨Om ⟩ = − c, cos 1 (6) 2 4 g ℓ6 0 )2 B 2 F4V V "2 2 6 8 2 16π B $10 B $17 dudv V + (2⇡↵ + V Z + Z V Z S = ⌧7sV"2 Ρ ⌦ ,u ,v ,u ,v ,u ,v ,u ,v . x s 3 ~ 3 4 1 1 Z "1Z "1 2 4 6 8 # L!Ρ" Om is the quark bilinear "1 L!Ρ" he string coupling constant. Ρ Ρ "1 "2 the "2magnetic fie e di↵erence action and one in [9]Bis$10 that we 2have considered non-zero terparts (See between Refs. [34,the 47] 2above for further Here4 details). 6 8 4 6 8 "2motion "2 pears inrefer the square root. of "1 read 2c asTherefore, 2 nts and to "1 m and the quark the massequations and the L!Ρ" L!Ρ" ✓ ◆ "2 1 11 ZB1+"2 3 F B2 oundary terminates. V,uv conditions (Zat )where F,Z V,u V,v 2, = ),v D7-brane + tan(Z 2)[(Z ),u V,v + (Z ),v V,u ] + ,u (Z the 2 embedding, that is,2 whether Figure 4: IncreasingΡvalues of2 B̃ for fixedZ T show the repulsive n topology of the the D78 + ˜ 1 1 the mesonic (stable mesons) and melted (unstable mesons) phases, respectively [8]. The process of dynamical meson of the4:probe brane condensate in the asymptotically AdS /m. background. In fact,isina the AdS-blackfunction hole background, the shape of Figure The quark against condensate multi-valued melting is described by the changec/r inh3the shaperhof theThe probe brane dynamically. In order to achieve this goal we study the brane can into Minkowski (ME) and Black Hole Embedding (BE) which correspond to II.be categorized PROBE D7-BRANE INEmbedding THE VAIDYA BACKGROUND in shape 1.0802of<the rh /m < brane 1.0913,inand a finitebackground. jump between the points and hole B byformation the the probe the makes AdS-Vaidya It describes theAblack in the bulk which the mesonic (stable mesons) and melted (unstable mesons) phases, respectively [8]. The process of dynamical meson corresponds to thermalization in the gluon sector of the dual gauge theory. The background is time-dependent and first-order transition. melting isphase described by the change in the shape of the probe brane dynamically. In order to achieve this goal we study In this section wethe will study thebrane set-up ofsuch the meson melting in the presence of a constant external therefore shape of the brane a dynamical background can deform dynamically. the shape of the probe ininthe AdS-Vaidya background. It describes the black hole formation in the bulk which Let us start with the AdS/CFT AdS-Vaidya metric magnetic field. According tothermalization the theis mesons in the gauge theory fluctuationsand corresponds to in dictionary the gluonthat sector of the dual gauge theory. Thecorrespond backgroundtois the time-dependent 3 Time-Dependent Set-Up Dynamical embeddings of the probe brane in the AdS background. In fact, in the dynamically. AdS-black hole background, the shape of therefore the asymptotically shape of the brane in 1 such can deform ⇥ a background ⇤ 2 2 M N 2 2 2 2 2 2 the brane can beLet categorized into (ME) and Black Hole Embedding (BE) which correspond to(II.1) ds = F (V, z)dV + d GMthe dxAdS-Vaidya dx =Embedding 2dV dz + d~ x + cos d⌦ + sin d , us start with metric that is NMinkowski 3 3 2 3.1 Vaidya-AdS5 spacetime z the mesonic (stable mesons) and melted (unstable ⇥mesons) phases, respectively [8]. The process of dynamical meson ⇤ 1 2 goal 2 we consider Mthe Nvery-far-from-equilibrium 2 2of the 2 Murata, 2 achieve 2 inthis 2 we study zero magnetic field: Ishii, Kinoshita, Tanahashi; 2014 dynamics probe D7-brane melting is described bysection, thedschange the shape the probe brane dynamically. In order to whereIn this F (V, z)dV + d = GMin dx = offor 2dV dz + d~ x + cos d⌦ + sin d , (II.1) N dx 3 3 z2 dynamical spacetime focusing on the thermalization Thermalization in theformation boundary in the bulk which the shape of athe probe brane in the AdS-Vaidya background. process. It describes the black hole 4 F (V, z) gauge = 1theM (Vdue )z ,to (II.2) Webyare following same notations as collapse ones introduced inand [9] an is realized in in thethe gravity the hole formation gravitational corresponds theory towhere thermalization gluondual sector of black the dual theory. The background is time-dependent in the as field theory ismetric, dual ~ inshape the bulk AdS space. In this paper, we use theThermalization Vaidya-AdS spacetime the dynamical therefore the of the brane in such a background can deform dynamically. embedded along the six directions of the bulk x and ⌦3 . W 4 and we have set the radius of AdS space-time to F be(V, one, Rblack = 1.MThe above metric isinwritten in Eddington-Finkelstein z) = 1 (V )z , (II.2) to hole formation gravity. background with the black hole formation. Hereafter, we set units where the AdS radius is Let us start with the AdS-Vaidya metric that is to be null coordinates u and v. For the rest of the bulk coordinates coordinate where the radial direction is represented by z and V shows the null direction. The boundary, where the AdS-Vaidya Background unity, L =have 1.lives, The Vaidya-AdS metric bycoordinate 5AdS andtheory we set the to be one, R =on1. Theboundary. above metric is ), written gauge is atradius z =10of⇥and Vspace-time is is thegiven time M (V whichinisEddington-Finkelstein an arbitrary function, ⇤ the 2 2 M N 2 2 2 2 2 2 thethe radial direction is represented by zd~ and V shows the null direction. The boundary, where the represents massdx of hole which changes as time passes by until it reaches a constant value. function V = V (u, v), z = v), = dscoordinate F (V, z)dV + d = Gthe dx = black 2dV dz + x + cos d⌦ + sin d , Z(u,The (II.1) M Nwhere 3 3 1 2 2 (5) z 2 µ ν lives, = dx 0 and V 2 is the(V, time M (V ), which [−F z)dV − 2dV dzon+the d⃗x3boundary. ], (9)is an arbitrary function, gatµνzdx for gauge M (V )theory that we williswork with in=this paper, is coordinate represents the mass of the black holez which changes as time passes by until it reaches a constant value. The function 8 4 Since we are interested in studying the e↵ect of the external mag where , (10) F (V,with z) =in1this − M(V )z for M (V ) that we will work paper, is 0 V < 0, < ⇥ ⇤ to be choose the magnetic field ⇡V 1 8 M (VF) (V, =M cos( ) 0 6 V 6 V, 2 1M (V z)f = )z 4V,the (II.2)(II.3) where the mass function M(V ) is a free function Bondi mass (density) of this :<1representing 0 V < 0, Period of the Boundary Time ⇤ V > V, 11⇥ ⇡V spacetime. This metric is an exact M solution five-dimensional equation (V ) = of Mfthe following (II.3) 1 cos( ) 0 6 V 6 Einstein V, 2 V EnergyFx1 x2 = B, : and we have set the radius of AdS space-time to be one, R = 1. The above metric is written in Eddington-Finkelstein tric γab are evolution the in equations V > V, from zero to Mf which is constant. Note with null dust, where Vequations, is the timeand interval which the (16) mass of31 the black hole increases coordinate where the radial direction is represented by1 z3zand boundary, where the dMV shows the null direction. TheInjection duced metric are constraint equations. In the following, (5) 4 that is constant. kν boundary. , hole in (11) that the radius the event horizon rh6g =µν M= . ofon µνis− µblack which isthe constant. Note T where V is the interval inGtime which the mass thekthe increases from zero not to f gauge theory lives, is at z of = 0time and V is the coordinate M (V ), which is M an arbitrary function, f We are following the same notations as ones introduced [9] and will repeat details. 2 dV 1 gs on the Vaidya-AdS background spacetimes solving Inmass orderoftothe addblack the fundamental matter inaswe the theory side we have add the probevalue. branes tonull, the bulk. The Since would like the uitand vto coordinates to stay we have 4gauge represents the hole which changes time passes by until reaches a constant The function µ that kthe radius of the event horizon is r = M . embedded along the six directions of the bulk metric, ~ x and ⌦ . We choose the other two coordinate h f where denotes the ingoing null vector defined by k dx = −dV . The null dust injected from 3 DBI action as µ of the degrees of freedom living on the brane µ is explained by the dynamics the equations of motion for the fields the brane, obtained In order to add the fundamental matter in the gauge theory side we have towe addon the probe branes to the bulk.from The th for M (V ) that we will work with in this paper, is the AdScoordinates boundary infalls into v. the For bulk,the andrest thenZ thethe event horizon is formed. When the mass null u and of bulk coordinates choose the ansatz a to be inates σ (a = 0, 1, · · · , 7)ofonthe the brane. For six8 of them, dynamics degrees of freedom living on the branepis explained by the DBI action as function M(V ) stops depending on V , namely, when the energy injection ceases, the spacetime 2 2 (II.4) gab<+0,(2⇡↵0 )Fab , S = ⌧7 Zd8 ⇠ 0 V F V + 2Z V Z (Z ) < ·..., · ,⇠σ77)isometric ) = (⃗x3 , to Ω3the ) because we assume emselves as (σ(⇠2is,2·,locally ,v ,v ,v 0. Schwarzschild-AdS horizon. 5 with V =1 ⇥V (u, v), a⇡Vplanar z8 ⇤p = Z(u, v), 0 = (u, v), = + (2⇡↵ )Fab , S = cos( ⌧7 dV ⇠) 0 g6 (V those ) = Mfdirections. (II.3) 6 V, ab V as the background0 bulk spacetimeMin 2 2(II.4) 2 1 1 : F V + 2Z V Z (Zand) ,u coordinates ,u ,u (⇠ , ⇠ ) = (u, v) is the induced metric on the brane. a, b are the brane where g 0 1 and Fab 1is the gauge field strength V > V, ab oduce the double nullare coordinates (σin, σstudying ) = (u, v), and Since we interested the e↵ect of the external magnetic field on the dynamical mes is defined as gab where gab is the induced metric and Fab is the gauge field strength on the brane. a, b are the brane coordinates and re parametrized as (u,v) gives the shape the magnetic field to the be mass is shape constant. Note where choose V is the time interval which of the(u, black hole increases from zero to M is defined as in gab f which which, in fact, gives the of the bran where v) ⌘ M N Z(u,v) Null Coordinate on the 1 g = @ x @ x G . (II.5) ab a b M N of the brane 4 M N that of the Mf . (17) gab = @a x @b x GM N . = Z(u, v)the , radius φ = Φ(u, v) event , Brane ψ horizon = 0 , is rh = becomes (II.5) Probe F = B, x x 10 1 2 In order to add the fundamental matter in the gauge theory side we have to add the probe branes to the bulk. The Freedom to choose eight world-volume coordinates Z as3 dynamics of the degrees of freedom living on the by brane is explained by the DBI actioncos . erality because of the U(1)-symmetry generated ∂ ψ that is constant. S = ⌧7 V⌦3 V~x dudv Z 9 Z3 hat the ∂u and ∂v are null generators. Then, the branep r 1 0 )2 B 2 F V V + (2⇡↵ ,u Z4 embedded along the six directions of the bulk metric, ~x and the thebrane, braneobtained from the F⌦x31.the =choose B,= B, (II.7) (u,v) of other motiontwo forcoordinates the fields ononthe DBI action xWe 2 equations F (II.7) which, in fact, gives the shape of the brane. With the above assumptions the where (u, v) ⌘ x x 1 2 to be null coordinates u and v. For the rest of the bulk coordinates we choose the ansatz Z(u,v) becomes F V,v2 + 2Z,v V,v Z 2 (Z )2,v = 0, that is constant. V = V (u, v), z = Z(u, v), = Z (u, v), = 0. r (II.6) that is constant. 2 F V 2Z,u V,u equations Z 2 (Z )2,u = 3to impose the following SinceSince we would like the u and v coordinates to stay null, we have constraint on0,on ,u + constraint we would like the u and v coordinates to stay null, wecoshave to1impose the following equations 0 2 2 dudv F V,u V,v we Vx from + (2⇡↵meson ) B melting, + V,u Z,v + Z,u V,v Z 2 ,u ,v . S= ⌧obtained we are interested in for studying the e↵ect of the external fieldthe on3 the dynamical 7 V⌦magnetic 3 ~ theSince equations of motion the fields on the brane, DBI action 4 the equations of motion for the fields on the brane, where obtained from action Z the ZDBI (u,v) choose the magnetic field to be which, in fact, gives the shape of the brane. With the above (u, v) ⌘ Z(u,v) 2 becomes 2 ,v V 2 )2 and F V,v2Fbetween 2Z Z (Z The di↵erence the action we have considered(II.8a) non-zero ,v,vabove ,v = Fx+ V + 2Z V Z (Z )2,v0,the = 0,one in [9] is that (II.8a)magnetic = B, (II.7) ,v 1 x2,v r Z of motion appears in the2 square root. Therefore, the equations 2 cos3 read1 2 ,u V,u 2 )2 =20, F V,uF+V 2Z Z (Z 0 )2 B 2 F V(II.8b) Z (Z V~x dudv + (2⇡↵ + V,u Z,v + Z,u V S,u= ),u ⌧7 V= ⌦30, ,u V ,u + 2Z,u V,u that is constant. ✓ ◆,v(II.8b) 3 4 Z Z + ZBnull, 3 the following constraint equations on1 F B2 Since we would like the u and v coordinates to stay we have to impose 1 (u,v) (u,v) V,uv = (Z ),u (Z ),v + tan(Z )[(Z ),u V,v + (Z ),v V,u ] + F,Z V,u V,v , the where equations of⌘motion forwhich, the fields the brane, obtained DBI action inonfact, givesgives the ofthethe brane. WithWith above assumptions DBI where (u, v)(u, The di↵erence between thethe above action and the one2 inthe [9] the is that we action have considered n 2 shape 2 brane. Z action which, in fact, thefrom shape of the the above assumptions DBI v)Z(u,v) ⌘ Z(u,v) in the square root. Therefore, the equations of motion read + 2 appears becomes 2 becomes 3 B2 F,V FZV,v2 +=2Z,vFVZB Z (Z ) = 0, 1 ,v (Z ),u,v(Z ),v + +tan(Z )[(Z ),u Z,v + (Z (II.8a) Z,u Z,v V✓,u V,v ),v Z,u ] + ,uv r r2 2 Z Z F V32 + 2Z 2 2 Z 2 ZB 3 1 F B2 (II.8b) cos ,ucos3 1,u1✓V,u 1 Z F(ZB0 2)◆,u 2V=,uv0, = 1 (Z ),u (Z ),v + tan(Z )[(Z ) F V + (Z ) V ] + ,u ,v ,v ,u ,Z 2 0 )2 B 2VZ,v,u V,vZ Z,u2 ,v 2 Z F 2VV,uFV2+ (II.9) + (2⇡↵ )2 B(F + V,u ZV,vV,u+ S = S ⌧= dudv3 V,v,u (II.9) + (2⇡↵ V,v ++ ZZ,vZ + 7 V⌦3⌧V ,u ~ x ⌦3 Vdudv 7V ,u .,v . ~ x 4 F V V Z ) , 3 4 ,Z ,u ,v ,u ,v ,v ,u Z Z (u,v) Z Z + assumptions the DBI action 2 Z of the brane. With theF ZB above where (u, v) ⌘ Z(u,v) which, in fact, gives the shape 3 B2 1 ✓ (Z ),u (Z ),v + tan(Z )[(Z ),u Z,v + (Z ),v Z,u ] + Z,u Z,v Z,uv+=◆ becomes 2 2 Z F B B + 3Z tan(Z ) 3 tan(Z ) 1is that 1 The di↵erence between the above action and the that we considered non-zero magnetic ✓ have ◆considered The di↵erence between the above and one the in one[9] have non-zero magnetic +inis[9] (Z )we [(Z ),ufield Z,vfield +which (Zwhich ),v Z,u ] raction ,uv = ,u (ZF B)2,v Z 1 2 3 2Z 2 2Z appears in theinsquare root.root. Therefore, the of motion readread appears the square Therefore, the equations of motion F cos 1 equations 2(F V,u V,v + V,u Z,v + V,v Z,u ) , 0 )2 B 2 F V V + V Z2 + ,Z ✓ ◆ (II.9) + (2⇡↵ Z V Z S = ⌧7 V⌦3 V~x dudv Z ,u ,v ,u ,v ,u ,v ,u ,v . + Z3 Z4 3 tan(Z ) F B B F,V ◆ ✓ 2 ✓ ◆ ✓ ◆ 1 + + + F + (F V V + V Z + V Z ) Z Z + + F B B + 3Z tan(Z ) V,u V,v , 3 tan(Z ) ,Z ,u ,v ,u ,v ,v ,u ,u ,v B2),u (Z ),v 1 2 ZB1 ZB1 3 3 1 1+ F1 B2F 2 (Z = [(Z ),u Z,v + 2Z Z Z Z 2Z ,uv V,uv between ),u)(Z )[(Z ),u+V = (Z the + thetan(Z + (Z ),v] V + F,Z F2,Z (II.10a) 2 )(Z tan(Z )[(Z ),u +),vand (Zhave ),vconsidered V,u +2Z (II.10a) The Vdi↵erence above action one in [9] is V that we magneticVfield which ,v ,u ] non-zero ,u,V,v , ,uv = ,u (Z ,v ,v ,u VV,v 2Z 2root. Therefore, the 2 2 2 2 equations Z Z ◆ ✓ 2 appears in the square of motion read 3 tan(Z ) F B B1+ where 2 + + F,Z ◆ B2 + Z,u Z,v + ZB1 B2 2 F,V (F F,VV,u V,v + V,u Z,v + V,v Z,u ) ✓ F+ZBF 3 3 2 1 2Z Z Z Z ZB= 3 1 F B (Z ) tan(Z )[(Z ) Z V (Z ) + Z + (Z ) Z ] + Z V Z,uv 2 (Z ) tan(Z )[(Z ) Z V = (Z ) + Z + (Z ) Z ] + Z V Z,uv ,u ,v ,u ,v 1 ,v,u ,v F ,v ,u,v ±,uV V ,u ,u ,v V,uv = (Z ),u2(Z ),u tan(Z ,v )[(Z 2),u V,v + (Z ),u , ,v,u ,v22 (II.10a) ,v + ,v V,u ] + ,Z ,u1 ± ,vZ 2 2 2 Z = , B 2 2 Z 1 ✓ ✓ ◆ 2◆ 1 + (2⇡↵0 )2 Z 4 B 2 where 1 F B 2 1 F ZB+ F F B2 3 V,u (F Z,v (II.10b) ,v)[(Z 2 ± F2,Z1 (Z ,Z),u (Z Z(F ZV V +ZV,u,v) Z ,),v,uZ),u, ] + B2 Z,u Z,v F,V V,u V,v ,v +VV ,u+ ,v,u 2 (II.10b) )+ ),v V +,u Vtan(Z Z,uv = ,u Z,v,v+ (Z = 3 + . B 2✓ 2 Z◆ 2 2 =21 ± , 2 Z 0 )2 Z 4 B 2 1 + (2⇡↵0 )2BZ14 B ◆ ✓ 1 + (2⇡↵ ◆ ✓ + 1 F B2 +F B B13Z+ 3Z tan(Z 3 tan(Z B tan(Z ) ) [(Z ) Z + (Z 3= ) (F)VF Ftan(Z V + V,u1 Z,v (Z + V,v)Z,u(Z ) , ) B1 + (II.10b) 2 ,Z ,u ,v 1 + ),u,v3]Z+,u ] ,v recover the results ,u+,v (Z ),uB(Z [(Z ) Z (Z ) Z 2 Z2Z+ = . B 2 ,uv =,uv ,v can ,u ,v ,v By2setting =,u0)we of [9]. 2 2 2Z ◆ ✓ 1 + (2⇡↵0 )2 Z 4 B 2 2Z ✓ 2Z +2 ◆ The above equations motion) are solvable, in spite of F B1 B1 + 3Z of tan(Z 3 tan(Z ) + being non-linear and time-dependent, if one use ✓ ◆ + + (Z ) = (Z ) [(Z ) Z + (Z ) Z ] 3 tan(Z ) F B B F,V[9]. The solution to these equatio ,uv ,u ,v ,u ,v ,v ,u 2 1the 2finite 3 tan(Z ) F B B F methods. Here we apply di↵erence method as has been used in 2 ,V By setting B = 0 we can recover results of [9]. 2Z 2Z 1 F + (F V V,u,V,v , (II.10c) V + V Z + V Z ) Z Z + (II.10c) + ,u+ ,v ,u+ ,v ,v) ,u ,u+,v F,Z ,Z +Z us how (F V V V V Z V Z Z V + 2 2 Z,v ,u ,v ,u ,v ,v ,u ,u ,u ,v ✓ 2Z ◆Z Z 2Z 2 2 +due to of the background the presence ofand the time-dep non-zer The changes above Bequations are solvable, of being non-linear 2Z Zmotion 2Z in spiteand F B2Z 3 tan(ZZ ) the shape of the brane Fdynamical ,V 1 F,Z + (F V,u V,v +to Vdi↵erence V,uthe Z,v + V,v Z,u ) Here Z Z,vthe + finite (II.10c) + methods. method as has used inof[9]. solutio ,u V,v , AdS/CFT dictionary near boundary expansion of been the shape the The brane give 2field. According 2 we,u apply 2Z Z Z Z 2Z us how the shape of thespecifically brane changes due to the dynamical background and the prese of quark (m) and the condensation (c). More wherewhere field. According to the AdS/CFT dictionary the near boundary expansion of the shap where ◆ 2and the condensation✓(c). Morem(V 3 ± of quark (m) specifically 2 ) , (II.11a) ± B1 = 21 ± 2 0Z)| 2Z 4 2= m(V ) + c(V ) + Z + .... (V, = 1 ± , (II.11a) B Z!0 ± 1 + (2⇡↵ ) B 1 ◆ ✓ , 0 2 4 2 (II.11a) B1 = 1 ± 6 m(V )3 1 + (2⇡↵10 )2+ Z 4(2⇡↵ B2 ) Z B 2 Z 2 + .... (V, Z)|Z!0 = m(V ) + c(V ) + 6 instance B is . B 2 = 23 +of the2magnetic Note that the presence field parameters with (II.11b) m, (II.11b) for 0 )2 Z 4 2 we rescale all the(II.11b) . B2 =in3B+ = 3 + . 1 + (2⇡↵ B 2 4B2 0 2 4 2 B 1 + (2⇡↵10 )2+ )inZ Bresults. =Z (2⇡↵ 1 that in all our solving the equations of motion will be able the presence By of the magnetic field we rescale all thewe parameters withto m,s m2 . This means that m Note B condensation behaves in time. The boundary and initial conditions are summarized in the following subs setting 0 we can the recover results of [9]. By By setting B = 0Bwe=can recover resultsthe of [9]. m2 . This means that m = 1 in all our results. By solving the equations of motion w 10 By setting B equations = 0 we can recover the results of [9]. condensation in time. boundary and initial conditions are summarized in TheThe aboveabove of motion solvable, spite of being non-linear andbehaves time-dependent, iftime-dependent, one uses numerical equations of are motion areinsolvable, in spite of being non-linear andThe if one uses numerical Constraint equations to keep u,v null equations of motion: begin we have to distinguish between two configurations mentioned before; Minkowski embedding and Black Hole begin wedoes havenot to cross distinguish betweenwe two configurations mentioned before; Minkowski embedding. Since the ME the horizon, have to impose two boundary conditions; one embedding at the AdSand Bl embedding. theto ME doesthe notappropriate cross the horizon, we and haveinitial to impose two boundary one at In order toboundary solve theZ| equations of motion Since we need define boundary conditions. To ⇡ conditions; |u=v+ = ⇡2 , where ⇡ u=v = 0, where we have used the same definition as [9], and the other at the pole 2 we have used theMinkowski same definition as [9], and and Black the other pole | = boundary Z| u=v = 0, where begin we have distinguish between configurations Holeat the theto3-sphere on which thetwo D7-brane wraps, mentioned shrinks to before; zero. For the BE,embedding the 3-sphere never shrinks to zero as u=v+ the 2 the 3-sphere on which the have D7-brane wraps, shrinks zero. data For theone BE,atthe never shrinks to ze embedding. D7-brane Since the ME doesthe nothorizon cross the horizon, we impose boundary the3-sphere AdS A. Boundary conditions and to initial crosses and therefore we dotonot need two to impose anyconditions; boundary condition at the pole. In order ⇡ D7-brane crosses thedefinition horizon and therefore do not to impose condition at the pole. ⇡ = boundary = 0, where we have used the same as [9], and in thewe other at need the of pole |u=v+any boundary Z|to u=vderive 2 , where 2 the appropriate boundary and initial conditions the presence the magnetic field, we follow the same to derive theshrinks appropriate boundary conditions in the presence of the magnetic field, we follow the 3-sphere on which the D7-brane wraps, to zero. For theand BE,initial the 3-sphere never shrinks to zero as the path as [9] where we refer the reader to it for more details. In order theand equations of wethe need totodefine the appropriate 4as [9] where path reader it for more details. D7-brane crossestothesolve horizon therefore we motion do we notrefer need to impose any boundary condition at boundary the pole. In and orderinitial conditions. we appropriate have to distinguish between two configurations mentioned before; embedding and Black Ho to begin derive the boundary and initial conditions in the presence ofconditions the magnetic field,Minkowski we follow the same • Boundary conditions at the AdS boundary A. Boundary and initial data • to Boundary conditions at thewe AdS boundary as [9] where Since we referthe the ME reader itnot for more details. embedding. does cross the horizon, have to impose two4 boundary conditions; one at the A 4 data path The physical observables are obtained on the AdS boundary, u = v and therefore have to include Thehave physical are obtained on AdS u =pole vwe and we⇡have t boundary Z|u=v = 0, where we usedobservables the same definition as the [9],where andboundary, the otherwhere at the |therefore = , whe u=v+ ⇡ 2 2 the appropriate boundary conditions there. The boundary on appropriate Z and areon simply Z|u=v =simply 0initial andZ|u=v • Boundary at the AdS boundary A. Boundary conditions and initial data the appropriate boundary conditions there. TheBE, boundary condition Z and are Inconditions order to solve the equations ofshrinks motion wezero. need to condition define the boundary and con the 3-sphere on which the D7-brane wraps, to For the the 3-sphere never shrinks to zero as t | = m. In order to obtain the rest of the boundary conditions we expand the fields near the boundary and iate boundary and initial conditions. To oundary conditions andobservables initial data u=v | = m. In order to obtain the rest of the boundary conditions we expand the fields near the boun The physical are obtained on the AdS boundary, where u = v and therefore we have to include u=v begin we have to distinguish between two configurations mentioned before; Minkowski embedding D7-brane crosses the horizon and therefore we not need to impose anyconsistency boundary condition at the pole. In and ord byand imposing the regularity condition ondo the equations of motion and with the constraint equations e;the Minkowski embedding Black Hole by imposing the regularity condition on the equations of motion and consistency with the constraint e the appropriate boundary conditions there. The boundary condition on Z and are simply Z| = 0 and u=v equations of motion we need to define the appropriate boundary and initial conditions. To embedding. Since the ME does not cross the horizon, we have to impose two boundary conditions; one to derive the appropriate boundary and initial conditions in the presence of the magnetic field, we follow the sam we (II.8) |u=v = m. (II.8) Inone order obtain the restwe of get the boundary conditions we expand the fields near the boundary and conditions; attoget the AdS nwo weboundary need between to define the appropriate boundary and initial conditions. Todefinition istinguish two configurations mentioned before; Minkowski embedding and Black Hole ⇡refer boundary Z| = 0, where we have used the same as [9], and the equations other at the pole |u=v+ ⇡2 path as [9] where we the reader to it for more details. u=v by pole imposing the regularity condition on the equations of motion and consistency with the constraint ⇡ he other at the | = , where u=v+ 2 we 2 he ME does not cross the horizon, have tothe impose two boundary conditions; one at the AdS d wraps, configurations mentioned before; Minkowski embedding and Black Hole d to (II.8) we get the 3-sphere onas which D7-brane zero. For BE, the=3-sphere never (II.13) shrinks to V0 (v) = shrinks 2Z,u |u=v ,V0 (v) Z⇡,uv | = 0the , Z the 3-sphere never shrinks to zero the ⇡,u |u=v u=v = 2Z , | 0 , ,uv u=v 0, where we have used the same definition as [9], and the other at the pole | = , where he horizon, we• have to impose two boundary conditions; u=v+ 2 dv one at the AdS dv Boundary conditions at the AdS boundary D7-brane crosses the horizon and therefore we do not need to2 impose any boundary condition at the po boundary condition at the pole. In order ⇡ ich the D7-brane wraps, shrinks to zero. For the BE, the 3-sphere never shrinks to zero as the d ⇡ = he same definition as [9], and the other at the pole | , where u=v+ Vobtained (v)that = 2Z |u=v ,initial Z2,uv |conditions = boundary 0 , arein (II.13) 2Note 0(v) u=v (v) = Vto|u=v . Note the boundary not a↵ected the presence thehave external where Vobservables derive the appropriate boundary and conditions the presence of the magnetic field, weofinclu follo of horizon the magnetic field, we follow same = V |,u . the that the conditions not a↵ected byofwe the presence the where V Thetophysical are on AdS boundary, where u = are vbyand therefore to 0the 0 u=v he and therefore we do not need impose any boundary condition at the pole. In order dv , shrinks to zero. For the BE, the 3-sphere never shrinks to zero as the magnetic field. magnetic field. the appropriate boundary conditions there. boundary condition on Z and are simply Z|u=v = 0 a path as [9] where refer the reader to itThe forIn more details. priate boundary and initial conditions inwethe presence ofatthe magnetic field, we follow the same we do not need to impose any boundary condition the pole. order V |u=v .order Noteto that the boundary conditions are not a↵ected by the the fields external where | V0 (v)==m. Indetails. obtain the rest of theatboundary conditions wepresence expandofthe near the boundary a etial refer the reader to it for more u=v • Boundary conditions at the Pole • Boundary conditions the Pole conditions in thefield. presence of the magnetic field, we follow the same magnetic by imposing the regularity condition on the equations of motion and consistency with the constraint equatio •Similar Boundary conditions atprevious the AdS boundary Similar to the set of boundary conditions themagnetic presence field of the magnetic fieldthe does not alter the b to the previous set of boundary conditions the presence of the does not alter boundary or more details. onditions the(II.8) AdSwe boundary • therefore Boundary conditions at the Pole we get u = v andat have to include ⇡ ⇡ ⇡ = conditions |u=v+on vanishing magneticwhere field. Therefore we have conditions at the pole, |u=v+at⇡2 the = 2pole, for vanishing magnetic field. Therefore we have The physical observables are obtained the boundary, u = v and therefore we hav 2 forAdS 2 nobservables on Z and are simply Z| = 0 and are obtained on the AdS boundary, where u = v and therefore we have to include Similar to the previous set of boundary conditions the presence of the magnetic field does not alter the boundary u=v boundary the appropriate boundary conditions there. The Zboundary condition on Z and are simply Z|u ⇡ d ⇡ = = Z , V = V , conditions theboundary pole, for vanishing magnetic field. Therefore we have expand the fields nearatthe and = Z , V = V , (II.14) (II.1 Z te boundary conditions there. The |boundary condition on Z and are simply Z| = 0 and ,u ,v ,u ,v u=v+ ,u ,v ,u ,v u=v 2 2 V (v) = 2Z | , Z | =0, Second order, nonlinear partial differential equations: We use finite difference method to solve them. Rescaling the parameters to have m=1. Boundary and Initial Conditions: 0have ,uthe u=v ,uv u=v |u=vboundary =um. order to we obtain rest boundary conditions we expand the fields near the bo d on the = vInconditions and therefore wethe toofinclude consistency withboundary, the rest constraint equations n order to AdS obtain the of where the expand the fields near the boundary and dv = Z,v ,Z|u=v V,u Vthe (II.14) with the constrai Z⇡2 ,u at u = vare + .simply ,v atby u =imposing v + ⇡2on . Zof the regularity condition on==the equations of motion and consistency there. The condition boundaryoncondition and 0 ,and he regularity the equations motion and consistency with constraint equations (v) = Vexpand |get Note that the boundary conditions are not a↵ected by the presence of the extern where V0(II.8) t of the boundary conditions we the fields we u=v•. Initial datanear the boundary and at u = v +•⇡2 Initial . data on the equationsmagnetic of motionfield. and consistency with the constraint equations =0, (II.13) The background at which the probe brane is embedded is time-dependent where before the energy d The background at which the probe brane isdembedded is time-dependent where before the energy injection, • Initial data <Zisthe 0, the space-time is pureV AdS. Therefore, our ,(II.13) initial data includes theto static (v)0, =the 2Zspace-time , at |u=v = 0 ,Therefore, ,u |u=v V ,uv VV0< pure AdS. our initial data includes the static solution the solution equationstoofthe equ (v) = 2Z | Z | = 0 , • Boundary conditions Pole 0 ,u u=v ,uv u=v dv The background at which the probe brane pure is embedded time-dependent where before motion withisnon-zero field [3, 4]. the energy injection, dv motion forexternal pure AdS with for non-zeroAdS magnetic field [3, magnetic 4]. a↵ected by the presence of the d V <Similar 0, the space-time is pure AdS. ourconditions initial data the includes the static solution to thefield equations to the previous set ofTherefore, boundary presence of the magnetic does ofnot alter the bounda = VV0 (v) |u=v= . 2Z Note that the boundary conditions are not a↵ected by the presence of the external , pure Z,uvAdS | with = 0non-zero , (II.13) ,u |u=v for ⇡ motion field 4]. the boundary (v) = |V magnetic |u=v⇡. =Note that are not by the presence of where conditions atu=v theV0pole, for[3, vanishing magneticconditions field. Therefore wea↵ected have v . magnetic field. u=v+ 2 2 III. III. RESULTS NUMERICAL RESULTS NUMERICAL e boundary the presence ofZ,u the= external Z,v , V,u = V,v , (II.1 onditions at conditions the Pole are not a↵ected byIII. NUMERICAL RESULTS magnetic field does not alter the boundary In this section we study the response of the system to the time-dependent change in the temperature in the • Boundary conditions at the Pole In this section we study the response of the system to the time-dependent change in the temperature in the presence previouswesethave of boundary conditions the of magnetic the magnetic does notofalter the boundary ⇡ of presence an external field. field The response the system is described by the behaviour of condensation, c, in erefore at u = v + . ⇡ of an external magnetic field. The response of the system is described by the behaviour in of the condensation, c, in terms of ⇡ 2 In this section we study the response of the system to the time-dependent change in the temperature presence the pole, |u=v+ 2 = 2 for Similar vanishing field. Therefore wemagnetic haveconditions tomagnetic theboundary previous set of boundary the this presence of the field doescategories: not alter M t the time. When the field is zero, behaviour can magnetic be classified into three the boundary time. When the magnetic field is zero, this behaviour can be classified into three categories: Minkowski of an external magnetic field. The response ofBlack the system is described byOvereager the behaviour c, in terms of ⇡ embedding, Hole embedding and case.of condensation, ⇡ = 11 (II.14) conditions at the pole, | for vanishing magnetic field. Therefore we have • Initial data u=v+ embedding, Black Hole and Overeager 2 case. 2 y conditions the presence ofZthe does alter the boundary the boundary time. When the field is zero, this behaviour can be classified into three categories: Minkowski =magnetic Zmagnetic , embedding Vfield =V , not (II.14) 5 5 Minkowski Embedding: Black Hole Embedding: If the corresponding mass of the If the corresponding mass of the initial ME configuration is much initial ME configuration is less FIG. 1: This figure (right) shows(right) that how an final overeager crosses the horizon an FIG. 1: This figure shows that temperature howconfiguration an overeager configuration cross larger than the final temperature than the of the 1 1 sin reperesents shape of the the brane Accordingly, graph on thethe lef W (⇢) = z sin the reperesents shape[9]. of the brane [9].theAccordingly, W (⇢) = z of the system: system: m m at the final1atstages, as itstages, is expected ME. This figure is<plotted for B is = plotted 1.87140, forVB= 1This figure the final as it from is expected from ME. T T (a) Sub-critical FIG. 1: This figure shows that how an overeager configuration crosses the horizon and jumps out of it and becomes ME. FIG. 1: This figure (right) shows that(right) how an overeager configuration crosses the horizon and jumps out of it and becomes ME. 1 reperesents the shape the brane [9]. Accordingly, the left showsoscillating that c(V ) starts oscillating (⇢) = z sin the shape of the brane [9]. of Accordingly, the graph on the the left graph shows on that c(V ) starts W (⇢) = z 1 sin Wreperesents at as theit final stages, from as it is expected from isME. Thisfor figure plotted forV B== = 0.5 and rh = 1.25. at the final stages, is expected ME. This figure plotted B =is1.87140, 0.51.87140, and rh =V1.25. (a) Sub-critical 0.015 0 B=0.56446 - 0.02 B=0.0 0.010 B=0.56446 B=0.0 - 0.04 - 0.06 c(V) 0.005 c(V) (b) Super-critical 0 (b) Super-critical - 0.08 -0.10 - 0.12 - 0.005 (c) Overeager - 0.14 - 0.16 -0.010 Figure 2: Schematic illustration of the brane dynamics. 0 5 10 15 20 0 1 2 3 V 4 5 6 7 8 V FIG. 2: c(V ) for ME (left) and BE (right). In both figures blue solid curves show c(V = 0.0747132. c2:0 isblue theboth initial configuration condensation, before the black hole formation cOvereager 0(left) FIG. 2: c(V ) forFIG. ME 2: (left) and BE In both figures solid curves show c(V and )curves c0 ,BE with B=0.56446, in both which FIG. c(V ) for ME (left) (right). figures blue solid (c) (right). c(V ) for ME and BE (right). In figures blue solid show c(V ) c with B=0.56446, in which 0 ,In No Dissipation and thus initial configuration condensation, the black hole formation in In the all bulk. The r red dashed c0 = 0.0747132. cc00 is = 1.25 and = 1.0. show c(Vconfiguration )cin the before absence of cmagnetic field. cases =the 0.0747132. c0 is the initial condensation, black hole formation in bulk.curves The redVdashed curves h the is thethe initial configuration condensation, before the blac 0 = 0.0747132. 0 before Oscillatory Behavior = 1.0. show c(V ) in the absence of Vmagnetic field. Figure 2: Schematic ofmagnetic the cases branefield. show c(V ) in the absence of magnetic field.ofIn all rdynamics. h = 1.25 1.25 and show c(V ) in the illustration absence In alland casesVrh==1.0. • Overeager Case • Overeager Case • Overeager Case 12 In all cases rh = 1.25 and V Overeager: If the corresponding mass of the initial ME configuration is (a) Sub-critical larger but close to the final temperature of the system, provided that the time-scale of the change in the temperature is small: Ishii, Kinoshita, Murata, Tanahashi; 2014 m ⇠1 T V <1 5 (b) Super-critical Brane Reconnection 5 (c) Overeager 0.65 Figure 2: Schematic illustration of1.4the brane dynamics. 1.2 0.60 1.0 w(ρ ) c(V) 0.55 0.50 0.8 0.6 V=5.5 FIG. 1: This figure (right) shows that how an overeager configuration crosses the horizon and jumps out of it and becom 1 0.45 V=1.5 0.4 [9]. Accordingly, the graph on the left shows that c(V ) starts os W (⇢) = z sin reperesents the shape of the brane FIG. 1: This figure (right) shows0.2that how an overeager configuration crosses the horizon and jumps ou 0.40 at the final stages, as it is1expected from ME. This figure is plotted for B = 1.87140, V = 0.5 and rh = 1.25. reperesents the 0shape of the brane [9]. Accordingly, the graph on becomes the left shows tha W (⇢) = zhowsin This figure (right) shows that an overeager configuration crosses the horizon and jumps out of it and ME. 0 0.5 1.0 1.5 2.0 2.5 0 1 2 3 4 5 6 7 at the final stages, as it is expected from ME. This figure plotted B =c(V 1.87140, V = 0.5 and r 1 brane [9]. Accordingly, the graph on theisleft showsforthat ) starts oscillating = z sin reperesents the shape of the ρ V final stages, as it is expected from ME. This figure is plotted for B = 1.87140, V = 0.5 and rh = 1.25. FIG. 1: This figure (right) shows that how an overeager configuration crosses the horizon and jumps out of it and becomes ME. W (⇢) = z 1 sin reperesents the shape of the brane [9]. Accordingly, the graph on the left shows that c(V ) starts oscillating at the final stages, as it is expected from ME. This figure is plotted for B = 1.87140, V = 0.5 and rh = 1.25. 7 13 !1 !2 !3 2.82052 4.83518 6.84984 0 3.04731 5.18042 7.21553 0.05 B=0.43264 B=0.0 0 c(V) c(V) - 0.05 -0.10 - 0.05 BE changes to ME by increasing the magnetic filed. -0.10 - 0.15 0 1 2 3 4 5 6 7 0 1 2 3 4 V 5 6 7 V 0.65 3.4 c(V) 0.55 c(V) B=5.22375 3.6 B=1.87140 0.60 0.50 3.2 0.45 3.0 0.40 2.8 0 1 2 3 V 4 5 6 0 1 2 3 4 5 6 7 8 9 10 V . 3: This figure shows how the BE changes to overeager and consequently to ME after one raises the magnetic field. We e set rh = 1.25 and V = 0.5. FIG. 3: This figure shows how the BE chang and V = 0.5. have set rh = 1.25 FIG. 3: This figure shows have setthat rh = 1.25 and V sidering the linear perturbation around the equilibrium shape of the brane. Figure 2 (right) confirms the fact ) decreases by applying an external magnetic field. We will elaborate more on this in figure 4. A remarkable phenomenon has been observed in the presence of the external magnetic field. Let us assume we start m a ME which results in BE after the injection of energy when the magnetic field is zero, figure 3 up-left. we considering the linearIfperturbation around e the magnetic field, it can still give BE as can be seen in figure 3 up-right. Interestingly, by further increasing c(V ) decreases by applying an external m 14 value of the magnetic field, with the same initial configuration and time-scale of the energy injection, the final Power Spectrum of c(V)-ceq for the ME: FIG. 1: This shows that howconfiguration an overeager configuration cros FIG. 1: This figure (right)figure shows(right) that how an overeager crosses the horizon an 1 1 (⇢) = z sin the reperesents the brane shape[9]. of the brane [9]. sin reperesents shape of the Accordingly, theAccordingly, graph on the the lef W (⇢) = z W the final as itfrom is expected ME. This figure at the finalatstages, as itstages, is expected ME. Thisfrom figure is plotted for B is = plotted 1.87140, forV B= 10 -5 P (!) 10 -6 The Fourier transform of the oscillations of c(V) has discrete spectrum. 10 -7 10 -8 10 -9 10 -10 0 5 10 TABLE I: Bound-States or Stable Mesons obtained from the ME Power Spe 15 20 ! Stable Modes Bound-States obtained from ME c(V) Power Spectrum !1 !2 !3 B=0 2.82052 4.83518 6.84984 B = 0.56446 3.04731 5.18042 7.21553 FIG. 2: c(V ) for ME (left) and BE (right). In both figures blue solid curves show c(V FIG. forfield, ME (left) andcondensation, BE (right).before In both figures solid c2: the)initial configuration the black holeblue formation c0 = 0.0747132. 0 isc(V presence of magnetic configuration 1.25 and V =before 1.0. the bla show c(V ) cin the0.0747132. absence of cmagnetic field. In all cases rh =condensation, 0 = 0 is the initial In the mesons are more show resistant c(V ) into themelting. absence of magnetic field. • Overeager Case 15 In all cases rh = 1.25 and V tion can be explained by the fact that due to the non-zero magnetic field the probe e in its shape. In the field theory side this means that the binding energy of the FIG. 4: The plot on the showsisc(V ) for BEs valuessatisfies of B. The the right where the to equilibration time defined as for thedi↵erent time which ✏(Vgraph 0.003 and shows ✏(V ) ✏(V sta re, more energy is needed melt the left mesons. eq ) <on boundary time, obtained from (III.1), for the corresponding plotsc(V in )the leftat figure. The equilibrationmagn tim afterwards. Note that ceq be obtained static solution the corresponding ch, due to the the energy injection, ends up in a can BE, there exists afrom time the at which 1.25 andbeen = 1.0. intemperature. table II. We have rh = This function has plotted in figure state. 4 for di↵erent values of the magnetic field We call this equilibration time Veqsetthat the system isV at its equilibrium We r from figure 4 (right) that the presence of the magnetic field delays the evolution of the system to it The equilibration times corresponding to the figure 4which are given in table II which confirm, quantita is defined as the time c(V ) ceq statement. ✏(V ) = , (III.1) ✏(Veq ) < 0.003 and ✏(V ) stays where the equilibration time is defined as the time which satisfies satisfies: c(V ) afterwards. NoteFIG. that1:ceq can be from the static solution at corresponding magneti This figure (right) shows that how an overeager crosses the horizon a FIG. 1: obtained This figure (right) shows that howconfiguration an the overeager configuration cro and for the time afterwards. 7 1 1 temperature. ThisWfunction has in shape figure ofwith 4 the for di↵erent of the magnetic field. II: Equilibration Times to the Magnetic sin reperesents brane Accordingly, the graph Field on the th leW (⇢) =TABLE z W sin the reperesents therespect shape[9]. of values theExternal brane [9]. Accordingly, (⇢)been = z plotted from figure 4 (right) thatfinal theatstages, presence magnetic field the ME. evolution to its at the as itofstages, isthe expected ME.delays Thisfrom figure is plotted forthe B system = the final as itfrom is expected This of figure is 1.87140, plotted forVeB 0 The equilibration timesB=0.0 corresponding to the figure 4 are given in table II which confirm, quantitativ B=0.3137 B Veq statement. B=0.5645 - 0.05 Equilibration Time: B=0.7129 B=0.7946 ϵ(V) c(V) 0.0 3.27834 -0.10 0.3137with respect to the 3.33708 TABLE II: Equilibration Times External Magnetic Field 0.5645 3.49866 - 0.15 0.7129 4.8057 B Veq 0.7946 5.15314 -0.20 7 0 1 2 3 4 5 6 0.0 3.27834 V 0.3137 3.33708 0.08 Acknowledgment 0.003 c(V ) for BEs for di↵erent values of B. 0.5645 FIG. 4: The plot on the left shows The graph on the right shows3.49866 ✏(V ) with respect to We would like to thank S.corresponding Sharifi, T. Ishii, K. Murata and N. Tanahashi for fruitfu the boundary time, obtained from (III.1), for the plots in the S. left Kinoshita, figure. The equilibration times are represented 0.06 0.002 0.7129 4.8057 in table II. We have set rh = 1.25 and V = 1.0. 0.001 0.7946 5.15314 0.04 0 3.5 4.0 4.5 5.0 where the equilibration time is defined as the time which satisfies ✏(Veq ) < 0.003 and ✏(V ) stays below this limit FIG. 2: c(V ) for ME (left) and BE (right). In both figures blue solid curves show c( 0.02 afterwards. Note that ceq can be obtained from the static solution at the corresponding magnetic field and final Acknowledgment 0.0747132. c2: the configuration condensation, before the black holeProg. formation cbeen 0 = plotted 0 4isfor FIG. c(V )initial for ME (left) and BE InTanahashi both figures blue soli E. function Shuryak, does the quark gluon plasma atthe RHIC behave as nearly ideal fluid?,” Part temperature.[1]This has“Why in figure di↵erent values of magnetic field.(right). Weacan conclude We would like to thank S. Sharifi, T. Ishii, S. Kinoshita, K. Murata and N. for fruitful co = 1.25 and V = 1.0. show c(V ) in the absence of magnetic field. In all cases r from figure 4 (right) that the presence of the magnetic field delays the evolution of the system to its equilibrium state. h (2004) [arXiv:hep-ph/0312227]. c0 = 0.0747132. c0 is the initial configuration condensation, before the bl 0 2 3 corresponding 4 5 figure 4 are 6 The equilibration times to the given in table II which confirm, quantitatively, the previous E. V. Shuryak, “What RHIC experiments theory us about properties of quark-gluon Ali-Akbari, Ebrahim, and show c(V ) in the and absence of tell magnetic field. In 2013 all cases rh = 1.25plasma?, V statement. 64 (2005) [arXiv:hep-ph/0405066]. E.TABLE Kharzeev, L.✏(V D.) with McLerran for di↵erent values of B.[2] TheD. graph on the right shows respect to and H. J. Warringa, “The E↵ects of topological charge change in II: Equilibration Times with respect to the External Magnetic Field corresponding plots in the left’Event figure. The times are byequilibration event P and CPrepresented violation’,” Nucl. Phys. A 803, 227 (2008) [arXiv:0711.0950 [hep-ph]]. • Overeager Case 16 [1] E. Shuryak, “Why does the quark gluon plasma at RHIC behave as a nearly ideal fluid?,” Prog. Part. N Summary and Future Direction: Mesons melt at higher temperatures due to the presence of magnetic field. The time-dependent system equilibrates later at non-zero magnetic field. studying the effect of anisotropy by having a time-dependent anisotropic background quark-antiquark bound-state (potential) on the time-dependent background looking for universal behaviour at very small energy injection timescale 17 “Thank you”